fundamentals of satellite and inertial navigation

fundamentals of satellite and inertial navigation
GLOBAL
POSITIONING SYSTEMS,
INERTIAL NAVIGATION,
AND INTEGRATION
SECOND EDITION
MOHINDER S. GREWAL
LAWRENCE R. WEILL
ANGUS P. ANDREWS
WILEY-INTERSCIENCE
A John Wiley & Sons, Inc., Publication
GLOBAL
POSITIONING SYSTEMS,
INERTIAL NAVIGATION,
AND INTEGRATION
GLOBAL
POSITIONING SYSTEMS,
INERTIAL NAVIGATION,
AND INTEGRATION
SECOND EDITION
MOHINDER S. GREWAL
LAWRENCE R. WEILL
ANGUS P. ANDREWS
WILEY-INTERSCIENCE
A John Wiley & Sons, Inc., Publication
Copyright © 2007 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data is available.
ISBN-13 978-0-470-04190-1
ISBN-10 0-470-04190-0
Printed in the United States of America.
10 9 8 7 6 5 4 3 2 1
M. S. G. dedicates this book to the memory of his parents, Livlin Kaur and
Sardar Sahib Sardar Karam Singh Grewal.
L. R. W. dedicates his work to his late mother, Christine R. Weill, for her love
and encouragement in pursuing his chosen profession.
A. P. A. dedicates his work to his wife Jeri, without whom it could not have
been done.
CONTENTS
Preface to the Second Edition
xvii
Acknowledgments
xix
Acronyms
xxi
1 Introduction
1
1.1 GNSS/INS Integration Overview, 1
1.2 GNSS Overview, 2
1.2.1 GPS, 2
1.2.2 GLONASS, 4
1.2.3 Galileo, 5
1.3 Differential and Augmented GPS, 7
1.3.1 Differential GPS (DGPS), 7
1.3.2 Local-Area Differential GPS, 7
1.3.3 Wide-Area Differential GPS, 8
1.3.4 Wide-Area Augmentation System, 8
1.4 Space-Based Augmentation Systems (SBASs), 8
1.4.1 Historical Background, 8
1.4.2 Wide-Area Augmentation System (WAAS), 9
1.4.3 European Geostationary Navigation Overlay System (EGNOS),
10
vii
viii
CONTENTS
1.4.4 Japan’s MTSAT Satellite-Based Augmentation System
(MSAS), 11
1.4.5 Canadian Wide-Area Augmentation System (CWAAS), 12
1.4.6 China’s Satellite Navigation Augmentation System (SNAS), 12
1.4.7 Indian GPS and GEO Augmented Navigation System
(GAGAN), 12
1.4.8 Ground-Based Augmentation Systems (GBASs), 12
1.4.9 Inmarsat Civil Navigation, 14
1.4.10 Satellite Overlay, 15
1.4.11 Future Satellite Systems, 15
1.5 Applications, 15
1.5.1 Aviation, 16
1.5.2 Spacecraft Guidance, 16
1.5.3 Maritime, 16
1.5.4 Land, 16
1.5.5 Geographic Information Systems (GISs), Mapping, and
Agriculture, 16
Problems, 17
2 Fundamentals of Satellite and Inertial Navigation
2.1 Navigation Systems Considered, 18
2.1.1 Systems Other than GNSS, 18
2.1.2 Comparison Criteria, 19
2.2 Fundamentals of Inertial Navigation, 19
2.2.1 Basic Concepts, 19
2.2.2 Inertial Navigation Systems, 21
2.2.3 Sensor Signal Processing, 28
2.2.4 Standalone INS Performance, 32
2.3 Satellite Navigation, 34
2.3.1 Satellite Orbits, 34
2.3.2 Navigation Solution (Two-Dimensional Example), 34
2.3.3 Satellite Selection and Dilution of Precision, 39
2.3.4 Example Calculation of DOPs, 42
2.4 Time and GPS, 44
2.4.1 Coordinated Universal Time Generation, 44
2.4.2 GPS System Time, 44
2.4.3 Receiver Computation of UTC, 45
2.5 Example GPS Calculations with no Errors, 46
2.5.1 User Position Calculations, 46
2.5.2 User Velocity Calculations, 48
Problems, 49
18
ix
CONTENTS
3 Signal Characteristics and Information Extraction
53
3.1 Mathematical Signal Waveform Models, 53
3.2 GPS Signal Components, Purposes, and Properties, 54
3.2.1 50-bps (bits per second) Data Stream, 54
3.2.2 GPS Satellite Position Calculations, 59
3.2.3 C/A-Code and Its Properties, 65
3.2.4 P-Code and Its Properties, 70
3.2.5 L1 and L2 Carriers, 71
3.3 Signal Power Levels, 72
3.3.1 Transmitted Power Levels, 72
3.3.2 Free-Space Loss Factor, 72
3.3.3 Atmospheric Loss Factor, 72
3.3.4 Antenna Gain and Minimum Received Signal Power, 73
3.4 Signal Acquisition and Tracking, 73
3.4.1 Determination of Visible Satellites, 73
3.4.2 Signal Doppler Estimation, 74
3.4.3 Search for Signal in Frequency and C/A-Code Phase, 74
3.4.4 Signal Detection and Confirmation, 78
3.4.5 Code Tracking Loop, 81
3.4.6 Carrier Phase Tracking Loops, 84
3.4.7 Bit Synchronization, 87
3.4.8 Data Bit Demodulation, 88
3.5 Extraction of Information for Navigation Solution, 88
3.5.1 Signal Transmission Time Information, 89
3.5.2 Ephemeris Data, 89
3.5.3 Pseudorange Measurements Using C/A-Code, 89
3.5.4 Pseudorange Measurements Using Carrier Phase, 91
3.5.5 Carrier Doppler Measurement, 92
3.5.6 Integrated Doppler Measurements, 93
3.6 Theoretical Considerations in Pseudorange and Frequency Estimation,
95
3.6.1 Theoretical versus Realizable Code-Based Pseudoranging
Performance, 95
3.6.2 Theoretical Error Bounds for Carrier-Based Pseudoranging, 97
3.6.3 Theoretical Error Bounds for Frequency Measurement, 98
3.7 Modernization of GPS, 98
3.7.1 Deficiencies of the Current System, 99
3.7.2 Elements of the Modernized GPS, 100
3.7.3 Families of GPS Satellites, 103
3.7.4 Accuracy Improvements from Modernization, 104
3.7.5 Structure of the Modernized Signals, 104
Problems, 107
x
CONTENTS
4 Receiver and Antenna Design
111
4.1 Receiver Architecture, 111
4.1.1 Radiofrequency Stages (Front End), 111
4.1.2 Frequency Downconversion and IF Amplification, 112
4.1.3 Digitization, 114
4.1.4 Baseband Signal Processing, 114
4.2 Receiver Design Choices, 116
4.2.1 Number of Channels and Sequencing Rate, 116
4.2.2 L2 Capability, 118
4.2.3 Code Selections: C/A, P, or Codeless, 119
4.2.4 Access to SA Signals, 120
4.2.5 Differential Capability, 121
4.2.6 Pseudosatellite Compatibility, 123
4.2.7 Immunity to Pseudolite Signals, 128
4.2.8 Aiding Inputs, 128
4.3 High-Sensitivity-Assisted GPS Systems (Indoor Positioning), 129
4.3.1 How Assisting Data Improves Receiver Performance, 130
4.3.2 Factors Affecting High-Sensitivity Receivers, 134
4.4 Antenna Design, 135
4.4.1 Physical Form Factors, 136
4.4.2 Circular Polarization of GPS Signals, 137
4.4.3 Principles of Phased-Array Antennas, 139
4.4.4 The Antenna Phase Center, 141
Problems, 142
5 Global Navigation Satellite System Data Errors
5.1 Selective Availability Errors, 144
5.1.1 Time-Domain Description, 147
5.1.2 Collection of SA Data, 150
5.2 Ionospheric Propagation Errors, 151
5.2.1 Ionospheric Delay Model, 153
5.2.2 GNSS Ionospheric Algorithms, 155
5.3 Tropospheric Propagation Errors, 163
5.4 The Multipath Problem, 164
5.5 How Multipath Causes Ranging Errors, 165
5.6 Methods of Multipath Mitigation, 167
5.6.1 Spatial Processing Techniques, 167
5.6.2 Time-Domain Processing, 169
5.6.3 MMT Technology, 172
5.6.4 Performance of Time-Domain Methods, 182
5.7 Theoretical Limits for Multipath Mitigation, 184
5.7.1 Estimation-Theoretic Methods, 184
5.7.2 MMSE Estimator, 184
5.7.3 Multipath Modeling Errors, 184
144
xi
CONTENTS
5.8
5.9
5.10
5.11
5.12
Ephemeris Data Errors, 185
Onboard Clock Errors, 185
Receiver Clock Errors, 186
Error Budgets, 188
Differential GNSS, 188
5.12.1 PN Code Differential Measurements, 190
5.12.2 Carrier Phase Differential Measurements, 191
5.12.3 Positioning Using Double-Difference Measurements, 193
5.13 GPS Precise Point Positioning Services and Products, 194
Problems, 196
6 Differential GNSS
199
6.1 Introduction, 199
6.2 Descriptions of LADGPS, WADGPS, and SBAS, 199
6.2.1 Local-Area Differential GPS (LADGPS), 199
6.2.2 Wide-Area Differential GPS (WADGPS), 200
6.2.3 Space-Based Augmentation Systems (SBAS), 200
6.3 Ground-Based Augmentation System (GBAS), 205
6.3.1 Local-Area Augmentation System (LAAS), 205
6.3.2 Joint Precision Approach Landing System (JPALS), 205
6.3.3 LORAN-C, 206
6.4 GEO Uplink Subsystem (GUS), 206
6.4.1 Description of the GUS Algorithm, 207
6.4.2 In-Orbit Tests, 208
6.4.3 Ionospheric Delay Estimation, 209
6.4.4 Code–Carrier Frequency Coherence, 211
6.4.5 Carrier Frequency Stability, 212
6.5 GUS Clock Steering Algorithms, 213
6.5.1 Primary GUS Clock Steering Algorithm, 214
6.5.2 Backup GUS Clock Steering Algorithm, 215
6.5.3 Clock Steering Test Results Description, 216
6.6 GEO with L1 /L5 Signals, 217
6.6.1 GEO Uplink Subsystem Type 1 (GUST) Control Loop
Overview, 220
6.7 New GUS Clock Steering Algorithm, 223
6.7.1 Receiver Clock Error Determination, 226
6.7.2 Clock Steering Control Law , 227
6.8 GEO Orbit Determination, 228
6.8.1 Orbit Determination Covariance Analysis, 230
Problems, 235
7 GNSS and GEO Signal Integrity
7.1 Receiver Autonomous Integrity Monitoring (RAIM), 236
7.1.1 Range Comparison Method of Lee [121], 237
236
xii
CONTENTS
7.2
7.3
7.4
7.5
7.1.2 Least-Squares Method [151], 237
7.1.3 Parity Method [182, 183], 238
SBAS and GBAS Integrity Design, 238
7.2.1 SBAS Error Sources and Integrity Threats, 240
7.2.2 GNSS-Associated Errors, 240
7.2.3 GEO-Associated Errors, 243
7.2.4 Receiver and Measurement Processing Errors, 243
7.2.5 Estimation Errors , 245
7.2.6 Integrity-Bound Associated Errors, 245
7.2.7 GEO Uplink Errors, 246
7.2.8 Mitigation of Integrity Threats, 247
SBAS example, 253
Conclusions, 254
GPS Integrity Channel (GIC), 254
8 Kalman Filtering
255
8.1 Introduction, 255
8.1.1 What Is a Kalman Filter?, 255
8.1.2 How It Works, 256
8.2 Kalman Gain, 257
8.2.1 Approaches to Deriving the Kalman Gain, 258
8.2.2 Gaussian Probability Density Functions, 259
8.2.3 Properties of Likelihood Functions, 260
8.2.4 Solving for Combined Information Matrix, 262
8.2.5 Solving for Combined Argmax, 263
8.2.6 Noisy Measurement Likelihoods, 263
8.2.7 Gaussian Maximum-Likelihood Estimate, 265
8.2.8 Kalman Gain Matrix for Maximum-Likelihood Estimation, 267
8.2.9 Estimate Correction Using Kalman Gain, 267
8.2.10 Covariance Correction for Measurements, 267
8.3 Prediction, 268
8.3.1 Stochastic Systems in Continuous Time, 268
8.3.2 Stochastic Systems in Discrete Time, 273
8.3.3 State Space Models for Discrete Time, 274
8.3.4 Dynamic Disturbance Noise Distribution Matrices, 275
8.3.5 Predictor Equations, 276
8.4 Summary of Kalman Filter Equations, 277
8.4.1 Essential Equations, 277
8.4.2 Common Terminology, 277
8.4.3 Data Flow Diagrams, 278
8.5 Accommodating Time-Correlated Noise, 279
8.5.1 Correlated Noise Models, 279
8.5.2 Empirical Sensor Noise Modeling, 282
8.5.3 State Vector Augmentation, 283
xiii
CONTENTS
8.6 Nonlinear and Adaptive Implementations, 285
8.6.1 Nonlinear Dynamics, 285
8.6.2 Nonlinear Sensors, 286
8.6.3 Linearized Kalman Filter, 286
8.6.4 Extended Kalman Filtering, 287
8.6.5 Adaptive Kalman Filtering, 288
8.7 Kalman–Bucy Filter, 290
8.7.1 Implementation Equations, 290
8.7.2 Kalman–Bucy Filter Parameters, 291
8.8 GPS Receiver Examples, 291
8.8.1 Satellite Models, 291
8.8.2 Measurement Model, 292
8.8.3 Coordinates, 293
8.8.4 Measurement Sensitivity Matrix, 293
8.8.5 Implementation Results, 294
8.9 Other Kalman Filter Improvements, 302
8.9.1 Schmidt–Kalman Suboptimal Filtering, 302
8.9.2 Serial Measurement Processing, 305
8.9.3 Improving Numerical Stability, 305
8.9.4 Kalman Filter Monitoring, 309
Problems, 313
9 Inertial Navigation Systems
9.1 Inertial Sensor Technologies, 316
9.1.1 Early Gyroscopes, 316
9.1.2 Early Accelerometers, 320
9.1.3 Feedback Control Technology, 323
9.1.4 Rotating Coriolis Multisensors, 326
9.1.5 Laser Technology and Lightwave Gyroscopes, 328
9.1.6 Vibratory Coriolis Gyroscopes (VCGs), 329
9.1.7 MEMS Technology, 331
9.2 Inertial Systems Technologies, 332
9.2.1 Early Requirements, 332
9.2.2 Computer Technology, 332
9.2.3 Early Strapdown Systems, 333
9.2.4 INS and GNSS, 334
9.3 Inertial Sensor Models, 335
9.3.1 Zero-Mean Random Errors, 336
9.3.2 Systematic Errors, 337
9.3.3 Other Calibration Parameters, 340
9.3.4 Calibration Parameter Instability, 341
9.3.5 Auxilliary Sensors, 342
9.4 System Implementation Models, 343
9.4.1 One-Dimensional Example, 343
9.4.2 Initialization and Alignment, 344
316
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CONTENTS
9.4.3 Earth Models, 347
9.4.4 Gimbal Attitude Implementations, 355
9.4.5 Strapdown Attitude Implementations, 357
9.4.6 Navigation Computer and Software Requirements, 363
9.5 System-Level Error Models, 364
9.5.1 Error Sources, 365
9.5.2 Navigation Error Propagation, 367
9.5.3 Sensor Error Propagation, 373
9.5.4 Examples, 377
Problems, 381
10 GNSS/INS Integration
382
10.1 Background, 382
10.1.1 Sensor Integration, 382
10.1.2 The Influence of Host Vehicle Trajectories on Performance,
383
10.1.3 Loosely and Tightly Coupled Integration, 384
10.1.4 Antenna/ISA Offset Correction, 385
10.2 Effects of Host Vehicle Dynamics, 387
10.2.1 Vehicle Tracking Filters, 388
10.2.2 Specialized Host Vehicle Tracking Filters, 390
10.2.3 Vehicle Tracking Filter Comparison, 402
10.3 Loosely Coupled Integration, 404
10.3.1 Overall Approach, 404
10.3.2 GNSS Error Models, 404
10.3.3 Receiver Position Error Model, 407
10.3.4 INS Error Models, 408
10.4 Tightly Coupled Integration, 413
10.4.1 Using GNSS for INS Vertical Channel Stabilization, 413
10.4.2 Using INS Accelerations to Aid GNSS Signal Tracking
, 414
10.4.3 Using GNSS Pseudoranges, 414
10.4.4 Real-Time INS Recalibration, 415
10.5 Future Developments, 423
Appendix A Software
425
A.1 Software Sources, 425
A.2 Software for Chapter 3, 426
A.2.1 Satellite Position Determination Using Ephemeris Data•, 426
A.2.2 Satellite Position Determination Using Almanac Data for All
Satellites, 426
A.3 Software for Chapter 5, 426
A.3.1 Ionospheric Delays, 426
A.4 Software for Chapter 8, 426
xv
CONTENTS
A.5 Software for Chapter 9, 427
A.6 Software for Chapter 10, 428
Appendix B Vectors and Matrices
B.1 Scalars, 429
B.2 Vectors, 430
B.2.1 Vector Notation, 430
B.2.2 Unit Vectors, 430
B.2.3 Subvectors, 430
B.2.4 Transpose of a Vector, 431
B.2.5 Vector Inner Product, 431
B.2.6 Orthogonal Vectors, 431
B.2.7 Magnitude of a Vector, 431
B.2.8 Unit Vectors and Orthonormal Vectors, 431
B.2.9 Vector Norms, 432
B.2.10 Vector Cross-Product, 432
B.2.11 Right-Handed Coordinate Systems, 433
B.2.12 Vector Outer Product, 433
B.3 Matrices, 433
B.3.1 Matrix Notation, 433
B.3.2 Special Matrix Forms, 434
B.4 Matrix Operations, 436
B.4.1 Matrix Transposition, 436
B.4.2 Subscripted Matrix Expressions, 437
B.4.3 Multiplication of Matrices by Scalars, 437
B.4.4 Addition and Multiplication of Matrices, 437
B.4.5 Powers of Square Matrices, 438
B.4.6 Matrix Inversion, 438
B.4.7 Generalized Matrix Inversion, 438
B.4.8 Orthogonal Matrices, 439
B.5 Block Matrix Formulas, 439
B.5.1 Submatrices, Partitioned Matrices, and Blocks, 439
B.5.2 Rank and Linear Dependence, 440
B.5.3 Conformable Block Operations, 441
B.5.4 Block Matrix Inversion Formula, 441
B.5.5 Inversion Formulas for Matrix Expressions, 441
B.6 Functions of Square Matrices, 442
B.6.1 Determinants and Characteristic Values, 442
B.6.2 The Matrix Trace, 444
B.6.3 Algebraic Functions of Matrices, 444
B.6.4 Analytic Functions of Matrices, 444
B.6.5 Similarity Transformations and Analytic Functions, 446
B.7 Norms, 447
B.7.1 Normed Linear Spaces, 447
B.7.2 Matrix Norms, 447
429
xvi
CONTENTS
B.8 Factorizations and Decompositions, 449
B.8.1 Cholesky Decomposition, 449
B.8.2 QR Decomposition (Triangularization), 451
B.8.3 Singular-Value Decomposition, 451
B.8.4 Eigenvalue–Eigenvector Decompositions of Symmetric
Matrices, 452
B.9 Quadratic Forms, 452
B.9.1 Symmetric Decomposition of Quadratic Forms, 453
B.10 Derivatives of Matrices, 453
B.10.1 Derivatives of Matrix-Valued Functions, 453
B.10.2 Gradients of Quadratic Forms, 455
Appendix C Coordinate Transformations
456
C.1 Notation, 456
C.2 Inertial Reference Directions, 458
C.2.1 Vernal Equinox, 458
C.2.2 Polar Axis of Earth, 459
C.3 Coordinate Systems, 460
C.3.1 Cartesian and Polar Coordinates, 460
C.3.2 Celestial Coordinates, 461
C.3.3 Satellite Orbit Coordinates, 461
C.3.4 ECI Coordinates, 463
C.3.5 ECEF Coordinates, 463
C.3.6 LTP Coordinates, 470
C.3.7 RPY Coordinates, 473
C.3.8 Vehicle Attitude Euler Angles, 473
C.3.9 GPS Coordinates, 475
C.4 Coordinate Transformation Models, 477
C.4.1 Euler Angles, 477
C.4.2 Rotation Vectors, 478
C.4.3 Direction Cosines Matrix, 493
C.4.4 Quaternions, 497
References
502
Index
517
PREFACE TO THE SECOND EDITION
This book is intended for people who need to combine global navigation satellite
systems (GNSSs), inertial navigation systems (INSs), and Kalman filters. Our
objective is to give our readers a working familiarity with both the theoretical
and practical aspects of these subjects. For that purpose we have included “realworld” problems from practice as illustrative examples. We also cover the more
practical aspects of implementation: how to represent problems in a mathematical model, analyze performance as a function of model parameters, implement
the mechanization equations in numerically stable algorithms, assess its computational requirements, test the validity of results, and monitor performance
in operation with sensor data from GNSS and INS. These important attributes,
often overlooked in theoretical treatments, are essential for effective application
of theory to real-world problems.
The accompanying CD-ROM contains MATLAB m-files to demonstrate the
workings of the Kalman filter algorithms with GNSS and INS data sets, so that
the reader can better discover how the Kalman filter works by observing it in
action with GNSS and INS. The implementation of GNSS, INS, and Kalman
filtering on computers also illuminates some of the practical considerations of
finite-wordlength arithmetic and the need for alternative algorithms to preserve
the accuracy of the results. Students who wish to apply what they learn, must
experience all the workings and failings of Kalman Filtering—and learn to recognize the differences.
The book is organized for use as a text for an introductory course in GNSS
technology at the senior level or as a first-year graduate-level course in GNSS,
INS, and Kalman filtering theory and application. It could also be used for selfinstruction or review by practicing engineers and scientists in these fields.
This second edition includes some significant changes in GNSS/INS technology since 2001, and we have taken advantage of this opportunity to incorporate
xvii
xviii
PREFACE TO THE SECOND EDITION
many of the improvements suggested by reviewers and readers. Changes in this
second edition include the following:
1. New signal structures for GPS, GLONASS, and Galileo
2. New developments in augmentation systems for satellite navigation, including
(a) Wide-area differential GPS (WADGPS)
(b) Local-area differential GPS (LADGPS)
(c) Space-based augmentation systems (SBASs)
(d) Ground-based augmentation systems (GBASs)
3. Recent improvements in multipath mitigation techniques, and new clock
steering algorithms
4. A new chapter on satellite system integrity monitoring
5. More thorough coverage of INS technology, including development of error
models and simulations in MATLAB for demonstrating system performance
6. A new chapter on GNSS/INS integration, including MATLAB simulations
of different levels of tight/loose coupling
The CD-ROM enclosed with the second edition has given us the opportunity to
incorporate more background material as files. The chapters have been reorganized to incorporate the new material.
Chapter 1 informally introduces the general subject matter through its history
of development and application. Chapters 2–7 cover the basic theory of GNSS
and present material for a senior-level class in geomatics, electrical engineering,
systems engineering, and computer science.
Chapters 8–10 cover GNSS and INS integration using Kalman filtering. These
chapters could be covered in a graduate-level course in electrical, computer, and
systems engineering. Chapter 8 gives the basics of Kalman filtering: linear optimal filters, predictors, nonlinear estimation by “extended” Kalman filters, and
algorithms for MATLAB implementation. Applications of these techniques to the
identification of unknown parameters of systems are given as examples. Chapter
9 is a presentation of the mathematical models necessary for INS implementation and error analysis. Chapter 10 deals with GNSS/INS integration methods,
including MATLAB implementations of simulated trajectories to demonstrate
performance.
Mohinder S. Grewal, Ph.D., P.E.
California State University at Fullerton
Lawrence R. Weill, Ph.D.
California State University at Fullerton
Angus P. Andrews, Ph.D.
Rockwell Science Center (retired) Thousand Oaks, California
ACKNOWLEDGMENTS
M. S. G. acknowledges the assistance of Mrs. Laura Cheung, graduate student at
California State University at Fullerton, for her expert assistance with the MATLAB programs, and Dr. Jya-Syin Wu of the Boeing Company for her assistance
in reviewing the earlier manuscript.
L. R. W. is indebted to the people of Magellan Navigation who so willingly
shared their knowledge of the Global Positioning System during the development
of the first handheld receiver for the consumer market.
A. P. A. thanks Captains James Black and Irwin Wenzel of American Airlines
for their help in designing the simulated takeoff and landing trajectories for
commercial jets, and Randall Corey from Northrop Grumman and Michael Ash
from C. S. Draper Laboratory for access to the developing Draft IEEE Standard
for Inertial Sensor Technology. He also thanks Dr. Michael Braasch at GPSoft,
Inc. for providing evaluation copies of the GPSoft INS and GPS MATLAB
Toolboxes.
xix
ACRONYMS AND ABBREVIATIONS
A/D
ADC
ADR
ADS
AGC
AHRS
AIC
AIRS
ALF
ALS
altBOC
AODE
AOR-E
AOR-W
AR
ARMA
ASD
ASIC
ASQF
A-S
ATC
BOC
BPSK
C/A
C&V
CDM
Analog-to-digital (conversion)
Analog-to-digital converter
Accumulated delta range
Automatic dependent surveillance
Automatic gain control
Attitude and heading reference system
Akaike information-theoretic criterion
Advanced inertial reference sphere
Atmospheric loss factor
Autonomous landing system
Alternate binary offset carrier
Age of data word, ephemeris
Atlantic Ocean Region East (WAAS)
Atlantic Ocean Region West (WAAS)
Autoregressive
Autoregressive moving average
Amplitude spectral density
Application-specific integrated circuit
Application-Specific Qualification Facility (EGNOS)
Antispoofing
Air traffic control
Binary offset carrier
Binary phase-shift keying
Coarse acquisition (channel or code)
Correction and verification (WAAS)
Code-division multiplexing
xxi
xxii
CDMA
CEP
CNMP
CONUS
CORS
COSPAS
CPS
CRC
CWAAS
DGNSS
DGPS
DME
DOD
DOP
ECEF
ECI
EGNOS
EIRP
EMA
EMA
ENU
ESA
ESG
ESGN
EU
EWAN
FAA
FEC
FLL
FM
FOG
FPE
FSLF
FT
GAGAN
GBAS
GCCS
GDOP
ACRONYMS AND ABBREVIATIONS
Code-division multiple access
Circle error probable
Code noise and multipath
Conterminous United States, also Continental United
States
Continuously operating reference station
Acronym from transliterated Russian title
“Cosmicheskaya Sistyema Poiska Avariynich
Sudov,” meaning “Space System for the Search of
Vessels in Distress”
Chips per second
Cyclic redundancy check
Canadian WAAS
Differential GNSS
Differential GPS
Distance measurement equipment
Department of Defense (USA)
Dilution of precision
Earth-centered, earth-fixed (coordinates)
Earth-centered inertial (coordinates)
European (also Geostationary) Navigation Overlay
System
Effective isotropic radiated power
Electromagnetic accelerator
Electromagnetic accelerometer
East–north–up (coordinates)
European Space Agency
Electrostatic gyroscope
Electrically Supported Gyro Navigation (System;
USA)
European Union
EGNOS Wide-Area (communication) Network
(EGNOS)
Federal Aviation Administration (USA)
Forward error correction
Frequency-lock loop
Frequency modulation
Fiberoptic gyroscope
Final prediction error (Akaike’s)
Free-space loss factor
Feet
GPS & GEO Augmented Navigation (India)
Ground-based augmentation system
GEO communication and control segment
Geometric dilution of precision
ACRONYMS AND ABBREVIATIONS
GEO
GES
GIC
GIPSY
GIS
GIVE
GLONASS
GNSS
GOA
GPS
GUS
GUST
HDOP
HMI
HOW
HRG
ICAO
ICC
IDV
IF
IFOG
IGP
IGS
ILS
IMU
Inmarsat
INS
IODC
IODE
IONO
IOT
IRU
ISA
ITRF
JPALS
JTIDS
LAAS
LADGPS
LD
LEM
LHCP
LORAN
LOS
LPV
xxiii
Geostationary earth orbit
GPS Earth Station COMSAT
GPS Integrity Channel
GPS Infrared Positioning System
Geographic information system(s)
Grid ionosphere vertical error
Global Orbiting Navigation Satellite System
Global navigation satellite system
GIPSY/OASIS analysis
Global Positioning System
GEO uplink subsystem
GEO uplink subsystem type 1
Horizontal dilution of precision
Hazardously misleading information
Handover word
Hemispheric resonator gyroscope
International Civil Aviation Organization
Ionospheric correction computation
Independent Data Verification (of WAAS)
Intermediate frequency
Integrating or interferometric Fiberoptic gyroscope
Ionospheric grid point (for WAAS)
International GNSS Service
Instrument landing system
Inertial measurement unit
International Mobile (originally “Maritime”) Satellite
Organization
Inertial navigation system
Issue of data, clock
Issue of data, ephemeris
Ionosphere, Ionospheric
In-orbit test
Inertial reference unit
Inertial sensor assembly
International Terrestrial Reference Frame
Joint precision approach and landing system
Joint Tactical Information Distribution System
Local-Area Augmentation System
Local-area differential GPS
Location determination
Lunar Excursion module
Left-hand circularly polarized
Long-range navigation
Line of sight
Lateral positioning with vertical guidance
xxiv
LSB
LTP
M
MBOC
MCC
MCPS
MEDLL
MEMS
ML
MLE
MMSE
MMT
MOPS
MSAS
MTSAT
MVUE
MWG
NAS
NAVSTAR
NCO
NED
NGS
NLES
NPA
NSRS
NSTB
OASIS
OBAD
OD
OPUS
OS
PA
PACF
P-code
pdf
PDOP
PI
PID
PIGA
PLL
PLRS
PN
POR
ACRONYMS AND ABBREVIATIONS
Least significant bit
Local tangent plane
Meter
Modified BOC
Mission/Master Control Center (EGNOS)
Million Chips Per Second
Multipath-estimating delay-lock loop
Microelectromechanical system(s)
Maximum likelihood
Maximum-likelihood estimate (or estimator)
Minimum mean-squared error (estimator)
Multipath mitigation technology
Minimum Operational Performance Standards
MTSAT Satellite-based Augmentation System (Japan)
Multifunctional Transport Satellite (Japan)
Minimum-variance unbiased estimator
Momentum wheel gyroscope
National Airspace System
Navigation system with time and ranging
Numerically controlled oscillator
North–east–down (coordinates)
National Geodetic Survey (USA)
Navigation Land Earth Station(s) (EGNOS)
Nonprecision approach
National Spatial Reference System
National Satellite Test Bed
Orbit analysis simulation software
Old but active data
Orbit determination
Online Positioning User Service (of NGS)
Open service (of Galileo)
Precision approach
Performance Assessment and Checkout Facility
(EGNOS)
Precision code
portable document format
Position dilution of precision
Proportional and integral (controller)
Process Input Data (of WAAS); Proportional, integral,
and differential (control)
Pulse integrating gyroscopic accelerometer
Phase-lock loop
Position Location and Reporting System (U.S. Army)
Pseudorandom noise
Pacific Ocean Region
ACRONYMS AND ABBREVIATIONS
PPS
PPS
PR
PRN
PRS
PSD
RAG
RAIM
RF
RHCP
RIMS
RINEX
RLG
RMA
RMS
RPY
RTCA
RTCM
RTOS
RVCG
s
SA
SAR
SARP
SARSAT
SAW
SBAS
SBIRLEO
SCOUT
SCP
SF
SIS
SM
SNAS
SNR
SOL
SPS
STF
SV
SVN
TCS
TCXO
TDOA
TDOP
xxv
Precise Positioning Service
Pulse(s) per second
Pseudorange
Pseudorandom noise or pseudorandom number (=SVN
for GPS)
Public Regulated service (of Galileo)
Power spectral density
Receiver antenna gain (relative to isotropic)
Receiver autonomous integrity monitoring
Radiofrequency
Right-hand circularly polarized
Ranging and Integrity Monitoring Station(s) (EGNOS)
Receiver independent exchange format (for GPS data)
Ring laser gyroscope
Reliability, maintainability, availability
Root-mean-squared; reference monitoring station
Roll–pitch–yaw (coordinates)
Radio Technical Commission for Aeronautics
Radio Technical Commission for Maritime Service
Real-time operating system
Rotational vibratory coriolis gyroscope
second
Selective availability (also abbreviated “S/A”)
Search and Rescue (service; of Galileo)
Standards and Recommended Practices (Japan)
Search and rescue satellite–aided tracking
Surface acoustic wave
Space-based augmentation system
Space-based infrared low earth orbit
Scripps coordinate update tool
Satellite Correction Processing (of WAAS)
Scale Factor
Signal in space
Solar magnetic
Satellite Navigation Augmentation System (China)
Signal-to-noise ratio
Safety of Life Service (of Galileo)
Standard Positioning Service
Signal Task Force (of Galileo)
Space vehicle
Space vehicle number (= PRN for GPS)
Terrestrial communications subsystem (for WAAS)
Temperature-compensated Xtal (crystal) oscillator
Time difference of arrival
Time dilution of precision
xxvi
TEC
TECU
TLM
TOA
TOW
TTA
TTFF
UDRE
UERE
URE
USAF
USN
UTC
UTM
VAL
VCG
VDOP
VHF
VOR
VRW
WAAS
WADGPS
WGS
WMS
WN
WNT
WRE
WRS
ZLG
ACRONYMS AND ABBREVIATIONS
Total electron content
Total electron content units
Telemetry word
Time of arrival
Time of week
Time to alarm
Time to first fix
User differential range error
User-equivalent range error
User range error
United States Air Force
United States Navy
Universal Time, Coordinated (or Coordinated
Universal Time)
Universal Transverse Mercator
Vertical alert limit
Vibratory coriolis gyroscope
Vertical dilution of precision
Very high frequency (30–300 MHz)
VHF Omnirange (radionavigation aid)
Velocity Random Walk
Wide-Area Augmentation System (U.S.)
Wide-area differential GPS
World Geodetic System
Wide-area Master Station
Week number
WAAS network time
Wide-area reference equipment
Wide-area Reference Station
Zero-Lock Gyroscope (“Zero Lock Gyro” and “ZLG”
are trademarks of Northrop Grumman Corp.)
1
INTRODUCTION
There are five basic forms of navigation:
1. Pilotage, which essentially relies on recognizing landmarks to know where
you are and how you are oriented. It is older than humankind.
2. Dead reckoning, which relies on knowing where you started from, plus
some form of heading information and some estimate of speed.
3. Celestial navigation, using time and the angles between local vertical and
known celestial objects (e.g., sun, moon, planets, stars) to estimate orientation, latitude, and longitude [186].
4. Radio navigation, which relies on radiofrequency sources with known locations (including global navigation satellite systems satellites).
5. Inertial navigation, which relies on knowing your initial position, velocity,
and attitude and thereafter measuring your attitude rates and accelerations.
It is the only form of navigation that does not rely on external references.
These forms of navigation can be used in combination as well [18, 26, 214].
The subject of this book is a combination of the fourth and fifth forms of navigation using Kalman filtering.
1.1 GNSS/INS INTEGRATION OVERVIEW
Kalman filtering exploits a powerful synergism between the global navigation
satellite systems (GNSSs) and an inertial navigation system (INS). This synergism is possible, in part, because the INS and GNSS have very complementary
Global Positioning Systems, Inertial Navigation, and Integration, Second Edition, by M. S. Grewal, L. R. Weill, and A. P. Andrews
Copyright © 2007 John Wiley & Sons, Inc.
1
2
INTRODUCTION
error characteristics. Short-term position errors from the INS are relatively small,
but they degrade without bound over time. GNSS position errors, on the other
hand, are not as good over the short term, but they do not degrade with time.
The Kalman filter is able to take advantage of these characteristics to provide a
common, integrated navigation implementation with performance superior to that
of either subsystem (GNSS or INS). By using statistical information about the
errors in both systems, it is able to combine a system with tens of meters position
uncertainty (GNSS) with another system whose position uncertainty degrades at
kilometers per hour (INS) and achieve bounded position uncertainties in the order
of centimeters [with differential GNSS (DGNSS)] to meters.
A key function performed by the Kalman filter is the statistical combination of
GNSS and INS information to track drifting parameters of the sensors in the INS.
As a result, the INS can provide enhanced inertial navigation accuracy during
periods when GNSS signals may be lost, and the improved position and velocity
estimates from the INS can then be used to cause GNSS signal reacquisition to
occur much sooner when the GNSS signal becomes available again.
This level of integration necessarily penetrates deeply into each of these subsystems, in that it makes use of partial results that are not ordinarily accessible to
users. To take full advantage of the offered integration potential, we must delve
into technical details of the designs of both types of systems.
1.2 GNSS OVERVIEW
There are currently three global navigation satellite systems (GNSSs) operating
or being developed.
1.2.1 GPS
The Global Positioning System (GPS) is part of a satellite-based navigation system developed by the U.S. Department of Defense under its NAVSTAR satellite
program [82, 84, 89–94, 151–153].
1.2.1.1 GPS Orbits The fully operational GPS includes 24 or more (28 in
March 2006) active satellites approximately uniformly dispersed around six circular orbits with four or more satellites each. The orbits are inclined at an angle of
55◦ relative to the equator and are separated from each other by multiples of 60◦
right ascension. The orbits are nongeostationary and approximately circular, with
radii of 26,560 km and orbital periods of one-half sidereal day (≈11.967 h). Theoretically, three or more GPS satellites will always be visible from most points
on the earth’s surface, and four or more GPS satellites can be used to determine
an observer’s position anywhere on the earth’s surface 24 h per day.
1.2.1.2 GPS Signals Each GPS satellite carries a cesium and/or rubidium atomic
clock to provide timing information for the signals transmitted by the satellites.
Internal clock correction is provided for each satellite clock. Each GPS satellite
transmits two spread spectrum, L-band carrier signals—an L1 signal with carrier
frequency f1 = 1575.42 MHz and an L2 signal with carrier frequency f2 = 1227.6
GNSS OVERVIEW
3
MHz. These two frequencies are integral multiples f1 = 1540f0 and f2 = 1200f0
of a base frequency f0 = 1.023 MHz. The L1 signal from each satellite uses binary
phase-shift keying (BPSK), modulated by two pseudorandom noise (PRN) codes
in phase quadrature, designated as the C/A-code and P-code. The L2 signal from
each satellite is BPSK modulated by only the P-code. A brief description of the
nature of these PRN codes follows, with greater detail given in Chapter 3.
Compensating for Propagation Delays This is one motivation for use of two
different carrier signals, L1 and L2 . Because delay varies approximately as the
inverse square of signal frequency f (delay ∝ f −2 ), the measurable differential
delay between the two carrier frequencies can be used to compensate for the
delay in each carrier (see Ref. 128 for details).
Code-Division Multiplexing Knowledge of the PRN codes allows users independent access to multiple GPS satellite signals on the same carrier frequency.
The signal transmitted by a particular GPS signal can be selected by generating
and matching, or correlating, the PRN code for that particular satellite. All PRN
codes are known and are generated or stored in GPS satellite signal receivers
carried by ground observers. A first PRN code for each GPS satellite, sometimes
referred to as a precision code or P-code, is a relatively long, fine-grained code
having an associated clock or chip rate of 10f0 = 10.23 MHz. A second PRN
code for each GPS satellite, sometimes referred to as a clear or coarse acquisition code or C/A-code, is intended to facilitate rapid satellite signal acquisition
and handover to the P-code. It is a relatively short, coarser-grained code having
an associated clock or chip rate f0 = 1.023 MHz. The C/A-code for any GPS
satellite has a length of 1023 chips or time increments before it repeats. The full
P-code has a length of 259 days, during which each satellite transmits a unique
portion of the full P-code. The portion of P-code used for a given GPS satellite
has a length of precisely one week (7.000 days) before this code portion repeats.
Accepted methods for generating the C/A-code and P-code were established by
the satellite developer1 in 1991 [61, 97].
Navigation Signal The GPS satellite bit stream includes navigational information on the ephemeris of the transmitting GPS satellite and an almanac for all GPS
satellites, with parameters providing approximate corrections for ionospheric signal propagation delays suitable for single-frequency receivers and for an offset
time between satellite clock time and true GPS time. The navigational information is transmitted at a rate of 50 baud. Further discussion of the GPS and
techniques for obtaining position information from satellite signals can be found
in Chapter 3 (below) and in Ref. 125, pp. 1–90.
1.2.1.3 Selective Availability Selective availability (SA) is a combination of
methods available to the U.S. Department of Defense to deliberately derating
the accuracy of GPS for “nonauthorized” (i.e., non-U.S. military) users during
1
Satellite Systems Division of Rockwell International Corporation, now part of the Boeing Company.
4
INTRODUCTION
periods of perceived threat. Measures may include pseudorandom time dithering
and truncation of the transmitted ephemerides. The initial satellite configuration
used SA with pseudorandom dithering of the onboard time reference [212] only,
but this was discontinued on May 1, 2000.
Precise Positioning Service Formal, proprietary service Precise Positioning Service (PPS) is the full-accuracy, single-receiver GPS positioning service provided
to the United States and its allied military organizations and other selected agencies. This service includes access to the unencrypted P-code and the removal of
any SA effects.
Standard Positioning Service without SA Standard Positioning Service (SPS)
provides GPS single-receiver (standalone) positioning service to any user on
a continuous, worldwide basis. SPS is intended to provide access only to the
C/A-code and the L1 carrier.
Standard Positioning Service with SA The horizontal-position accuracy, as
degraded by SA, currently is advertised as 100 m, the vertical-position accuracy
as 156 m, and time accuracy as 334 ns—all at the 95% probability level. SPS
also guarantees the user-specified levels of coverage, availability, and reliability.
1.2.2 GLONASS
A second configuration for global positioning is the Global Orbiting Navigation
Satellite System (GLONASS), placed in orbit by the former Soviet Union, and
now maintained by the Russian Republic [108, 123].
1.2.2.1 GLONASS Orbits GLONASS also uses 24 satellites, but these are
distributed approximately uniformly in three orbital planes (as opposed to six for
GPS) of eight satellites each (four for GPS). Each orbital plane has a nominal
inclination of 64.8◦ relative to the equator, and the three orbital planes are separated from each other by multiples of 120◦ right ascension. GLONASS orbits
have smaller radii than GPS orbits, about 25,510 km, and a satellite period of
8
revolution of approximately 17
of a sidereal day. A GLONASS satellite and a
GPS satellite will complete 17 and 16 revolutions, respectively, around the earth
every 8 days.
1.2.2.2 GLONASS Signals The GLONASS system uses frequency-division
multiplexing of independent satellite signals. Its two carrier signals corresponding
to L1 and L2 have frequencies f1 = (1.602 + 9k/16) GHz and f2 = (1.246 +
7k/16) GHz, where k = 0, 1, 2, . . . , 23 is the satellite number. These frequencies
lie in two bands at 1.597–1.617 GHz (L1 ) and 1240–1260 GHz (L2 ). The L1
code is modulated by a C/A-code (chip rate = 0.511 MHz) and by a P-code
(chip rate = 5.11 MHz). The L2 code is presently modulated only by the P-code.
The GLONASS satellites also transmit navigational data at a rate of 50 baud.
Because the satellite frequencies are distinguishable from each other, the P-code
and the C/A-code are the same for each satellite. The methods for receiving and
GNSS OVERVIEW
5
analyzing GLONASS signals are similar to the methods used for GPS signals.
Further details can be found in the patent by Janky [97].
GLONASS does not use any form of SA.
1.2.3 Galileo
The Galileo system is the third satellite-based navigation system currently under
development. Its frequency structure and signal design is being developed by the
European Commission’s Galileo Signal Task Force (STF), which was established
by the European Commission (EC) in March 2001. The STF consists of experts
nominated by the European Union (EU) member states, official representatives of
the national frequency authorities, and experts from the European Space Agency
(ESA).
1.2.3.1 Galileo Navigation Services The EU intends the Galileo system to
provide the following four navigation services plus one search and rescue (SAR)
service.
Open Service (OS) The OS provides signals for positioning and timing, free of
direct user charge, and is accessible to any user equipped with a suitable receiver,
with no authorization required. In this respect it is similar to the current GPS L1
C/A-code signal. However, the OS will be of higher quality, consisting of six
different navigation signals on three carrier frequencies. OS performance will be
at least equal to that of the modernized Block IIF GPS satellites, which began
launching in 2005, and the future GPS III system architecture currently being
investigated. OS applications will include the use of a combination of Galileo
and GPS signals, thereby improving performance in severe environments such
as urban canyons and heavy vegetation.
Safety of Life Service (SOL) The SOL service is intended to increase public
safety by providing certified positioning performance, including the use of certified navigation receivers. Typical users of SOL will be airlines and transoceanic
maritime companies. The EGNOS regional European enhancement of the GPS
system will be optimally integrated with the Galileo SOL service to have independent and complementary integrity information (with no common mode of
failure) on the GPS and GLONASS constellations. To benefit from the required
level of protection, SOL operates in the L1 and E5 frequency bands reserved for
the Aeronautical Radionavigation Services.
Commercial Service (CS) The CS service is intended for applications requiring
performance higher than that offered by the OS. Users of this service pay a fee
for the added value. CS is implemented by adding two additional signals to the
OS signal suite. The additional signals are protected by commercial encryption
and access protection keys are used in the receiver to decrypt the signals. Typical
value-added services include service guarantees, precise timing, ionospheric delay
models, local differential correction signals for very high-accuracy positioning
applications, and other specialized requirements. These services will be developed
by service providers, which will buy the right to use the two commercial signals
from the Galileo operator.
6
INTRODUCTION
Public Regulated Service (PRS) The PRS is an access-controlled service for
government-authorized applications. It will be used by groups such as police,
coast guards, and customs. The signals will be encrypted, and access by region
or user group will follow the security policy rules applicable in Europe. The
PRS will be operational at all times and in all circumstances, including periods
of crisis. A major feature of PRS is the robustness of its signal, which protects
it against jamming and spoofing.
Search and Rescue (SAR) The SAR service is Europe’s contribution to the international cooperative effort on humanitarian search and rescue. It will feature near
real-time reception of distress messages from anywhere on Earth, precise location
of alerts (within a few meters), multiple satellite detection to overcome terrain
blockage, and augmentation by the four low earth orbit (LEO) satellites and the
three geostationary satellites in the current COSPAS-SARSAT system.
1.2.3.2 Galileo Signal Characteristics Galileo will provide 10 right-hand circularly polarized navigation signals in three frequency bands. The various signals
fall into four categories: F/Nav, I/Nav, C/Nav, and G/Nav. The F/Nav and I/Nav
signals are used by the Open Service (OS), Commercial Service (CS) and Safety
of Life (SOL) service. The I/Nav signals contain integrity information, while the
F/Nav signals do not. The C/Nav signals are used by the Commercial Service
(CS), and the G/Nav signals are used by the Public Regulated Service (PRS). At
the time of this writing not all of the signal characteristics described below have
been finalized.
E5a – E5b Band This band, which spans the frequency range from 1164 to 1214
MHz, contains two signals, denoted E5a and E5b , which are respectively centered
at 1176.45 and 1207.140 MHz. Each signal has an in-phase component and a
quadrature component. Both components use spreading codes with chipping rate
of 10 Mcps (million chips per second). However, the in-phase components are
modulated by navigation data, while the quadrature components, called pilot signals, are data-free. The data-free pilot signals permit arbitrarily long coherent
processing, thereby greatly improving detection and tracking sensitivity. A major
feature of the E5a and E5b signals is that they can be treated as either separate
signals or a single wide-band signal. Low-cost receivers can use either signal,
but the E5a signal might be preferred, since it is centered at the same frequency
as the modernized GPS L5 signal and would enable the simultaneous reception
of E5a and L5 signals by a relatively simple receiver without the need for reception on two separate frequencies. Receivers with sufficient bandwidth to receive
the combined E5a and E5b signals would have the advantage of greater ranging
accuracy and better multipath performance.
Even though the E5a and E5b signals can be received separately, they actually
are two spectral components produced by a single modulation called alternate
binary offset carrier (altBOC) modulation. This form of modulation retains the
simplicity of standard BOC modulation (used in the modernized GPS M-code
DIFFERENTIAL AND AUGMENTED GPS
7
military signals) and has a constant envelope while permitting receivers to differentiate the two spectral lobes. The current modulation choice is altBOC(15,10),
but this may be subject to change.
The in-phase component of the E5a signal is modulated with 50 symbols
per second (sps) navigation data without integrity information, and the in-phase
component of the E5b signal is modulated with 250 sps (symbols per second)
data with integrity information. Both the E5a and E5b signals are available to the
Open Service (OS), CS, and SOL services.
E6 Band This band spans the frequency range from 1260 to 1300 MHz and
contains a C/Nav signal and a G/Nav signal, each centered at 1278.75 MHz. The
C/Nav signal is used by the CS service and has both an in-phase and quadrature
pilot component using a BPSK spreading code modulation of 5 Mcps. The inphase component contains 1000 sps data modulation, and the pilot component is
data-free. The G/Nav signal is used by the PRS service and has only an in-phase
component modulated by a BOC(10,5) spreading code and data modulation with
a symbol rate that is to be determined.
E2 –L1 –E1 Band The E2 –L1 –E1 band (sometimes denoted as L1 for convenience) spans the frequency range from 1559 to 1591 MHz and contains a
G/Nav signal used by the PRS service and an I/Nav signal used by the OS,
CS, and SOL services. The G/Nav signal has only an in-phase component with
a BOC spreading code and data modulation; the characteristics of both are still
being decided. The I/Nav signal has an in-phase and quadrature component.
The in-phase component will contain 250 sps data modulation and will likely
use BOC(1,1) spreading code, but this has not been finalized. The quadrature
component is data-free.
1.3 DIFFERENTIAL AND AUGMENTED GPS
1.3.1 Differential GPS (DGPS)
Differential GPS (DGPS) is a technique for reducing the error in GPS-derived
positions by using additional data from a reference GPS receiver at a known
position. The most common form of DGPS involves determining the combined
effects of navigation message ephemeris, conospheric and satellite clock errors
(including the effects of SA) at a reference station and transmitting pseudorange
corrections, in real time, to a user’s receiver, which applies the corrections in the
process of determining its position [94, 151, 153].
1.3.2 Local-Area Differential GPS
Local-area differential GPS (LAGPS) is a form of DGPS in which the user’s
GPS receiver also receives real-time pseudorange and, possibly, carrier phase
corrections from a local reference receiver generally located within the line of
sight. The corrections account for the combined effects of navigation message
8
INTRODUCTION
ephemeris and satellite clock errors (including the effects of SA) and, usually,
propagation delay errors at the reference station. With the assumption that these
errors are also common to the measurements made by the user’s receiver, the
application of the corrections will result in more accurate coordinates.
1.3.3 Wide-Area Differential GPS
Wide-area DGPS (WADGPS) is a form of DGPS in which the user’s GPS receiver
receives corrections determined from a network of reference stations distributed
over a wide geographic area. Separate corrections are usually determined for
specific error sources—such as satellite clock, ionospheric propagation delay, and
ephemeris. The corrections are applied in the user’s receiver or attached computer
in computing the receiver’s coordinates. The corrections are typically supplied
in real time by way of a geostationary communications satellite or through a
network of ground-based transmitters. Corrections may also be provided at a
later date for postprocessing collected data [94].
1.3.4 Wide-Area Augmentation System
The WAAS enhances the GPS SPS over a wide geographic area. The U.S. Federal
Aviation Administration (FAA), in cooperation with other agencies, is developing
WAAS to provide WADGPS corrections, additional ranging signals from geostationary earth orbit (GEO) satellites, and integrity data on the GPS and GEO
satellites.
1.4 SPACE-BASED AUGMENTATION SYSTEMS (SBASS)
Four space-based augmentation systems (SBASs) were under development at
the beginning of the third millennium. These are the Wide-Area Augmentation System (WAAS), European Geostationary Navigation Overlay System
(EGNOS), Multifunctional Transport Satellite (MTSAT)–based Augmentation
System (MSAS), and GPS & GEO Augmented Navigation (GAGAN) by India.
1.4.1 Historical Background
Although GPS is inherently a very accurate system for positioning and time
transfer, some applications require accuracies unobtainable without some form
of performance augmentation, such as differential GPS (DGPS), in which position relative to a base (or reference) station can be established very accurately
(in some cases within millimeters). A typical DGPS system employs an additional GPS receiver at the base station to measure the GPS signals. Because the
coordinates of the base station are precisely known, errors in the received GPS
signals can be calculated. These errors, which include satellite clock and position
error, as well as tropospheric and ionospheric error, are very nearly the same for
users at a sufficiently small distance from the base station. In DGPS the error
values determined by the base station are transmitted to the user and applied as
corrections to the user’s measurements.
SPACE-BASED AUGMENTATION SYSTEMS (SBASS)
9
However, DGPS has a fundamental limitation in that the broadcast corrections
are good only for users in a limited area surrounding the base station. Outside this
area the errors tend to be decorrelated, rendering the corrections less accurate.
An obvious technical solution to this problem would be to use a network of
base stations, each with its own communication link to serve its geographic area.
However, this would require a huge number of base stations and their associated
communication links.
Early on it was recognized that a better solution would be to use a spacebased augmentation system (SBAS) in which a few satellites can broadcast the
correction data over a very large area. Such a system can also perform sophisticated computations to optimally interpolate the errors observed from relatively
few ground stations so that they can be applied at greater distances from each
station.
A major motivation for SBAS has been the need for precision aircraft landing
approaches without requiring separate systems, such as the existing instrument
landing systems (ILSs) at each airport. An increasing number of countries are
currently developing their own versions of SBAS, including the United States
(WAAS), Europe (EGNOS), Japan (NSAS), Canada (CWAAS), China (SNAS),
and India (GAGAN).
1.4.2 Wide-Area Augmentation System (WAAS)
In 1995 the United States began development of the Wide Area Augmentation
System (WAAS) under the auspices of the Federal Aviation Administration (FAA)
and the Department of Transportation (DOT), to provide precision approach capability for aircraft. Without WAAS, ionospheric disturbances, satellite clock drift,
and satellite orbit errors cause too much error in the GPS signal for aircraft to
perform a precision landing approach. Additionally, signal integrity information
as broadcast by the satellites is insufficient for the demanding needs of public
safety in aviation. WAAS provides additional integrity messages to aircraft to
meet these needs.
WAAS includes a core of approximately 25 wide-area ground reference stations (WRSs) positioned throughout the United States that have precisely surveyed coordinates. These stations compare the GPS signal measurements with
the measurements that should be obtained at the known coordinates. The WRS
send their findings to a WAAS master station (WMS) using a land-based communications network, and the WMS calculates correction algorithms and assesses
the integrity of the system. The WMS then sends correction messages via a
ground uplink system (GUS) to geostationary (GEO) WAAS satellites covering
the United States. The satellites in turn broadcast the corrections on a per-GPS
satellite basis at the same L1 1575.42 MHz frequency as GPS. WAAS-enabled
GPS receivers receive the corrections and use them to derive corrected GPS
signals, which enable highly accurate positioning.
On July 10, 2003, Phase 1 of the WAAS system was activated for general
aviation, covering 95% of the conterminous United States and portions of Alaska.
10
INTRODUCTION
In September 2003, improvements enabled WAAS-enabled aircraft to approach
runways to within 250 ft altitude before requiring visual control.
Currently there are two Inmarsat III GEO satellites serving the WAAS area:
the Pacific Ocean Region (POR) satellite and the West Atlantic Ocean Region
(AOR-W) satellite.
In March 2005 two additional WAAS GEO satellites were launched (PanAmSat Galaxy XV and Telesat Anik F1R), and are now operational. These satellites
plus the two existing satellites will improve coverage of North America and all
except the northwest part of Alaska. The four GEO satellites will be positioned
at 54◦ , 107◦ , and 133◦ west longitude, and at 178◦ east longitude.
WAAS is currently available over 99% of the time, and its coverage will
include the full continental United States and most of Alaska. Although primarily intended for aviation applications, WAAS will be useful for improving
the accuracy of any WAAS-enabled GPS receiver. Such receivers are already
available in low-cost handheld versions for consumer use.
Positioning accuracy using WAAS is currently quoted at less than 2 m of
lateral error and less than 3 m of vertical error, which meets the aviation Category
I precision approach requirement of 16 m lateral error and 4 m vertical error.
Further details of the WAAS system can be found in Chapter 6.
1.4.3 European Geostationary Navigation Overlay System (EGNOS)
The European Geostationary Navigation Overlay System (EGNOS) is Europe’s
first venture into satellite navigation. It is a joint project of the European Space
Agency (ESA), the European Commission (EC), and Eurocontrol, the European
organization for the safety of air navigation. Inasmuch as Europe does not yet
have its own standalone satellite navigation system, initially EGNOS is intended
to augment both the United States GPS and the Russian GLONASS systems,
providing differential accuracy and integrity monitoring for safety-critical applications such as aircraft landing approaches and ship navigation through narrow
channels.
EGNOS has functional similarity to WAAS, and consists of four segments:
space, ground, user, and support facilities segments.
1.4.3.1 Space Segment The space segment consists of three geostationary
(GEO) satellites, the Inmarsat-3 AOR-E, Inmarsat-3 AOR-W, and the ESA
Artemis, which transmit wide-area differential corrections and integrity information throughout Europe. Unlike the GPS and GLONASS satellites, these satellites
will not have signal generators aboard, but will be transponders relaying uplinked
signals generated on the ground.
1.4.3.2 Ground Segment The EGNOS ground segment includes 34 Ranging
and Integrity Monitoring Stations (RIMSs), four Mission/Master Control Centers
SPACE-BASED AUGMENTATION SYSTEMS (SBASS)
11
(MCCs), six Navigation Land Earth Stations (NLESs), and an EGNOS Wide-Area
Network (EWAN).
The RIMS stations monitor the GPS and GLONASS signals. Each station
contains a GPS/GLONASS/EGNOS receiver, an atomic clock, and network communications equipment. The RIMS tasks are to perform pseudorange measurements, demodulate navigation data, mitigate multipath and interference, verify
signal integrity, and to packetize and transmit data to the MCC centers.
The MCC centers monitor and control the three EGNOS GEO satellites, as
well as perform real-time software processing. The MCC tasks include integrity
determination, calculation of pseudorange corrections for each satellite, determination of ionospheric delay, and generation of EGNOS satellite ephemeris data.
The MCC then sends all the data to the NLES stations. Every MCC has a backup
station that can take over in the event of failure.
The NLES stations receive the data from the MCC centers and generate the
signals to be sent to the GEO satellites. These include a GPS-like signal, an
integrity channel, and a wide-area differential (WAD) signal. The NLES send
this data on an uplink to the GEO satellites.
The EWAN links all EGNOS ground-based components.
1.4.3.3 User Segment This segment consists of the user receivers. Although
EGNOS has been designed primarily for aviation applications, it can also be used
with land or marine EGNOS-compatible receivers, including low-cost handheld
units.
1.4.3.4 Support Facilities Segment Support for development, operations, and
verifications is provided by this segment.
The EGNOS system is currently operational. Positioning accuracy obtainable
from use of EGNOS is approximately 5 m, as compared to 10–20 m with unaided
GPS. There is the possibility that this can be improved with further technical
development.
1.4.4 Japan’s MTSAT Satellite-Based Augmentation System (MSAS)
The Japanese MSAS system, currently under development by Japan Space
Agency and the Japan Civil Aviation Bureau, will improve the accuracy, integrity,
continuity, and availability of GPS satellite signals throughout the Japanese Flight
Information Region (FIR) by relaying augmentation information to user aircraft
via Japan’s Multifunctional Transport Satellite (MTSAT) geostationary satellites.
The system consists of a network of Ground Monitoring Stations (GMS) in Japan,
Monitoring and Ranging Stations (MRSs) outside of Japan, Master Control Stations (MCSs) in Japan with satellite uplinks, and two MTSAT geostationary
satellites.
MSAS will serve the Asia–Pacific region with capabilities similar to the United
States WAAS system. MSAS and WAAS will be interoperable and are compliant with the International Civil Aviation Organization (ICAO) Standards and
Recommended Practices (SARP) for SBAS systems.
12
INTRODUCTION
1.4.5 Canadian Wide-Area Augmentation System (CWAAS)
The Canadian CWAAS system is basically a plan to extend the U.S. WAAS
coverage into Canada. Although the WAAS GEO satellites can be received in
much of Canada, additional ground reference station sites are needed to achieve
valid correctional data outside the United States. At least 11 such sites, spread
over Canada, have been evaluated. The Canadian reference stations are to be
linked to the U.S. WAAS system.
1.4.6 China’s Satellite Navigation Augmentation System (SNAS)
China is moving forward with its own version of a SBAS. Although information
on their system is incomplete, at least 11 reference sites have been installed in and
around Beijing in Phase I of the program, and further expansion is anticipated.
Receivers manufactured by Novatel, Inc. of Canada have been delivered for
Phase II.
1.4.7 Indian GPS and GEO Augmented Navigation System (GAGAN)
In August 2001 the Airports Authority of India and the Indian Space Research
Organization signed a memorandum of understanding for jointly establishing the
GAGAN system. The system is not yet fully operational, but by 2007 a GSAT-4
satellite should be in orbit, carrying a transponder for broadcasting correction
signals. On the ground, eight reference stations are planned for receiving signals
from GPS and GLONASS satellites. A Mission Control Center, as well as an
uplink station, will be located in Bangalore.
Once GAGAN is operational, it should materially improve air safety over
India. There are 449 airports and airstrips in the country, but only 34 have
instrument landing systems (ILSs) installed. With GAGAN, aircraft will be able
to make precision approaches to any airport in the coverage area. There will
undoubtedly be other uses for GAGAN, such as tracking of trains so that warnings
can be issued if two trains appear likely to collide.
1.4.8 Ground-Based Augmentation Systems (GBASs)
Ground-based augmentation systems (GBASs) differ from the SBAS in that
backup, aiding, and/or correction information is broadcast from ground stations
instead of from satellites. Three major GBAS are LAAS, JPALS, and LORAN-C.
1.4.8.1 Local-Area Augmentation System (LAAS) LAAS is an augmentation
to GPS that services airport areas approximately 20–30 mi in radius, and has been
developed under the auspices of the Federal Aviation Administration (FAA). It
broadcasts GPS correction data via a very high-frequency (VHF) radio data link
from a ground-based transmitter, yielding extremely high accuracy, availability,
and integrity deemed necessary for aviation Categories I, II, and III precision
landing approaches. LAAS also provides the ability for flexible, curved aircraft
SPACE-BASED AUGMENTATION SYSTEMS (SBASS)
13
approach trajectories. Its demonstrated accuracy is less than 1 m in both the
horizontal and vertical directions.
A typical LAAS system, which is designed to support an aircraft’s transition
from en route airspace into and throughout terminal area airspace, consists of
ground equipment and avionics. The ground equipment consists of four GPS
reference receivers, a LAAS ground facility, and a VHF radio data transmitter.
The avionics equipment includes a GPS receiver, a VHF radio data receiver, and
computer hardware and software.
The GPS reference receivers and the LAAS ground facility work together
to measure errors in GPS position that are common to the reference receiver
and aircraft locations. The LAAS ground facility then produces a LAAS correction message based on the difference between the actual and GPS-calculated
positions of the reference receivers. The correction message includes integrity
parameters and approach-path information. The LAAS correction message is
sent to a VHF data broadcast transmitter, which broadcasts a signal containing
the correction/integrity data throughout the local LAAS coverage area, where it
is received by incoming aircraft.
The LAAS equipment in the aircraft uses the corrections for position, velocity,
and time to generate instrument landing system (ILS) lookalike guidance as low
as 200 ft above touchdown. It is anticipated that further technical improvements
will eventually result in vertical accuracy below 1 m, enabling ILS guidance
all the way down to the runway surface, even in zero visibility (Category III
landings).
A major advantage of LAAS is that a single installation at a major airport
can be used for multiple precision approaches within its local service area. For
example, if an airport has 12 runway ends, each with a separate ILS, all 12
ILS facilities can be replaced with a single LAAS installation. Furthermore, it
is generally agreed that the Category III level of accuracy anticipated for LAAS
cannot be supported by WAAS.
1.4.8.2 Joint Precision Approach and Landing System (JPALS) JPALS is
basically a military version of LAAS that supports fixed-base, tactical, specialmission, and shipboard landing environments. It will allow the military to overcome problems of age and obsolescence of ILS equipment, and also will afford
greater interoperability, both among systems used by the various services and
between military and civilian systems.
The main distinction between LAAS and JPALS is that the latter can be quickly
deployed almost anywhere and makes full use of military GPS functionality,
which includes the use of the encrypted M-codes not available for civilian use.
The requirement for deployment in a variety of locations not optimized for good
GPS reception places great demands on the ability of JPALS equipment to handle
poor signal environments and multipath. Such problems are not as severe for
LAAS installations, where there is more freedom in site selection for best GPS
performance of the reference receivers. Additionally, JPALS GPS receivers must
be designed to foil frequent attempts by the enemy to jam the received GPS
signals.
14
INTRODUCTION
1.4.8.3 Long-Range Navigation (LORAN-C) LORAN-C is a low-frequency
ground-based radionavigation and time reference system that uses stable 100
kHz transmissions to provide an accurate regional positioning service. Unlike
LAAS and JPALS, LORAN-C is an independent, standalone system that does
not provide corrections to GPS signals, but instead uses time difference of arrival
(TDOA) to establish position.
LORAN-C transmitters are organized into chains of 3–5 stations. Within a
chain one station is designated as the master (M) and the other secondary stations (slaves) are identified by the letters W, X, Y, and Z. The sequence of signal
transmissions consists of a pulse group from the master station followed at precise time intervals by pulse groups from the secondary stations. All LORAN-C
stations operate on the same frequency of 100 kHz, and all stations within a
given chain use the same group repetition interval (GRI) to uniquely identify the
chain. Within a chain, each of the slave stations transmits its pulse group with
a different delay relative to the master station in such a way that the sequence
of the pulse groups from the slaves is always received in the same order, independent of the location of the user. This permits identification of the individual
slave station transmissions.
The basic measurements made by LORAN-C receivers are TDOAs between
the master station signal pulses and the signal pulses from each of the secondary
stations in a chain. Each time delay is measured to a precision of about 0.1 μs
or better. LORAN-C stations maintain integrity by constantly monitoring their
transmissions to detect signal abnormalities that would render the system unusable
for navigation. If a signal abnormality is detected, the transmitted pulse groups
“blink” on and off to notify the user that the transmitted signal does not comply
with the system specifications.
LORAN-C, with an accuracy approaching approximately 30 m in regions
with good geometry, is not as precise as GPS. However, it has good repeatability, and positioning errors tend to be stable over time. A major advantage of
using LORAN-C as an augmentation to GPS is that it provides a backup system
completely independent of GPS. A failure of GPS that would render LAAS or
JPALS inoperable does not affect positioning using LORAN-C. On the other
hand, LORAN-C is only a regional and not a truly global navigation system,
covering significant portions, but not all, of North America, Canada, and Europe,
as well as some other areas.
1.4.9 Inmarsat Civil Navigation
The Inmarsat overlay is an implementation of a wide-area differential service.
Inmarsat is the International Mobile Satellite Organization (IMSO), an 80-nation
international consortium, originally created in 1979 to provide maritime2 mobile
services on a global basis but now offering a much wider range of mobile satellite
services. Inmarsat launched four geostationary satellites that provide complete
2
The “mar” in the name originally stood for “maritime.”
APPLICATIONS
15
coverage of the globe from ±70◦ latitude. The data broadcast by the satellites
are applicable to users in regions having a corresponding ground station network.
The U.S. region is the continental U.S. (CONUS) and uses Atlantic Ocean Region
West (AOR-W) and Pacific Ocean Region (POR) geostationary satellites. This
is called the WAAS and is being developed by the FAA. The ground station
network is operated by the service provider, that is, the FAA, whereas Inmarsat
is responsible for operation of the space segment. Inmarsat affiliates operate the
uplink Earth stations (e.g., COMSAT in the United States). WAAS is discussed
further in Chapter 6.
1.4.10 Satellite Overlay
The Inmarsat Civil Navigation Geostationary Satellite Overlay extends and complements the GPS and GLONASS satellite systems. The overlay navigation
signals are generated at ground-based facilities. For example, for WAAS, two
signals are generated from Santa Paula, California—one for AOR-W and one
for POR. The backup signal for POR is generated from Brewster, Washington.
The backup signal for AOR-W is generated from Clarksburg, Maryland. Signals
are uplinked to Inmarsat-3 satellites such as AOR-W and POR. These satellites
contain special satellite repeater channels for rebroadcasting the navigation signals to users. The use of satellite repeater channels differs from the navigation
signal broadcast techniques employed by GLONASS and GPS. GLONASS and
GPS satellites carry their own navigation payloads that generate their respective
navigation signals.
1.4.11 Future Satellite Systems
In Europe, activities supported by the European Tripartite Group [European Space
Agency (ESA), European Commission (EC), EUROCONTROL] are underway
to specify, install, and operate a future civil global navigation satellite system
(GNSS) (GNSS-2 or Galileo).
Based on the expectation that GNSS-2 will be developed through an evolutionary process as well as long-term augmentations [e.g., EGNOS], short to
midterm augmentation systems (e.g., differential systems) are being targeted.
The first steps toward GNSS-2 will be made by the Tripartite Group. The
augmentations will be designed such that the individual elements will be suitable
for inclusion in GNSS-2 at a later date. This design process will provide the user
with maximum continuity in the upcoming transitions.
In Japan, the Japanese Commercial Aviation Board (JCAB) is currently developing the MSAS.
1.5 APPLICATIONS
Both GPS and GLONASS have evolved from dedicated military systems into
true dual-use systems. Satellite navigation technology is utilized in numerous
16
INTRODUCTION
civil and military applications, ranging from golf and leisure hiking to spacecraft
navigation. Further discussion on applications can be found in Chapters 6 and 7.
1.5.1 Aviation
The aviation community has propelled the use of GNSS and various augmentations (e.g., WAAS, EGNOS, GAGAN, MSAS). These systems provide guidance
for en route through precision approach phases of flight. Incorporation of a data
link with a GNSS receiver enables the transmission of aircraft location to other
aircraft and/or to air traffic control (ATC). This function is called automatic
dependent surveillance (ADS) and is in use in the POR. Key benefits are ATC
monitoring for collision avoidance and optimized routing to reduce travel time
and fuel consumption [153].
1.5.2 Spacecraft Guidance
The Space Shuttle utilizes GPS for guidance in all phases of its operation (e.g.,
ground launch, on-orbit and reentry, and landing). NASA’s small satellite programs use and plan to use GPS, as does the military on SBIRLEO (space-based
infrared low earth orbit) and GBI (ground-based interceptor) kill vehicles.
1.5.3 Maritime
GNSS has been used by both commercial and recreational maritime communities.
Navigation is enhanced on all bodies of water, from oceanic travel to riverways,
especially in inclement weather.
1.5.4 Land
The surveying community depends heavily on DGPS to achieve measurement
accuracies in the millimeter range. Similar techniques are used in farming, surface mining, and grading for real-time control of vehicles and in the railroad
community to obtain train locations with respect to adjacent tracks. GNSS is a
key component in intelligent transport systems (ITSs). In vehicle applications,
GNSS is used for route guidance, tracking, and fleet management. Combining
a cellular phone or data link function with this system enables vehicle tracing
and/or emergency messaging.
1.5.5 Geographic Information Systems (GISs), Mapping, and Agriculture
Applications include utility and asset mapping and automated airborne mapping,
with remote sensing and photogrammetry. Recently, GIS, GPS, and remote sensing have matured enough to be used in agriculture. GIS companies such as the
Environmental System Research Institute (Redlands, California) have developed
software applications that enable growers to assess field conditions and their
relationship to yield. Real time kinematic and differential GNSS applications for
precision farming are being developed. This includes soil sampling, yield monitoring, chemical, and fertilizer applications. Some GPS analysts are predicting
precision site-specific farming to become “the wave of the future.”
APPLICATIONS
17
PROBLEMS
1.1 How many satellites and orbit planes exist for GPS, GLONASS, and Galileo?
What are the respective orbit plane inclinations?
1.2 List the differences in signal characteristics between GPS, GLONASS, and
Galileo.
2
FUNDAMENTALS OF SATELLITE
AND INERTIAL NAVIGATION
2.1 NAVIGATION SYSTEMS CONSIDERED
This book is about GNSS and INS and their integration. An inertial navigation
system can be used anywhere on the globe, but it must be updated within hours
of use by independent navigation sources such as GNSS or celestial navigation.
Thousands of self-contained INS units are in continuous use on military vehicles,
and an increasing number are being used in civilian applications.
2.1.1 Systems Other than GNSS
GNSS signals may be replaced by LORAN-C signals produced by three or more
long-range navigation (LORAN) signal sources positioned at fixed, known locations for outside-the-building location determination. A LORAN-C system relies
on a plurality of ground-based signal towers, preferably spaced 100–300 km
apart, that transmit distinguishable electromagnetic signals that are received and
processed by a LORAN signal antenna and LORAN signal receiver/
processor that are analogous to the Satellite Positioning System signal antenna
and receiver/processor. A representative LORAN-C system is discussed in the
U.S. DOT LORAN-C User Handbook [127]. LORAN-C signals use carrier frequencies of the order of 100 kHz and have maximum reception distances of
hundreds of kilometers. The combined use of FM signals for location determination inside a building or similar structure can also provide a satisfactory location
determination (LD) system in most urban and suburban communities.
Global Positioning Systems, Inertial Navigation, and Integration, Second Edition, by M. S. Grewal, L. R. Weill, and A. P. Andrews
Copyright © 2007 John Wiley & Sons, Inc.
18
FUNDAMENTALS OF INERTIAL NAVIGATION
19
There are other ground-based radiowave signal systems suitable for use as
part of an LD system. These include Omega, Decca, Tacan, JTIDS Relnav (U.S.
Air Force Joint Tactical Information Distribution System Relative Navigation),
and PLRS (U.S. Army Position Location and Reporting System) (see summaries
in Ref. 125 pp. 6–7 and 35–60).
2.1.2 Comparison Criteria
The following criteria may be used in selecting navigation systems appropriate
for a given application system:
1.
2.
3.
4.
5.
6.
7.
8.
Navigation method(s) used
Coordinates provided
Navigational accuracy
Region(s) of coverage
Required transmission frequencies
Navigation fix update rate
User set cost
Status of system development and readiness
2.2 FUNDAMENTALS OF INERTIAL NAVIGATION
The fundamental idea for inertial navigation (also called Newtonian navigation)
comes from high-school physics:
The second integral of acceleration is position.
Given sensors that can measure the three components of acceleration over
time, and initial values for position and velocity, the approach would appear to
be relatively straightforward. As is often the case, however, “the devil is in the
details.”
This introductory section presents a descriptive overview of the fundamental
concepts that have evolved in reducing this idea to practice. A more mathematical
treatment is presented in Chapter 9, and additional application-specific details can
be found in the literature [27, 36, 63, 88, 108, 120, 124, 139, 168, 169, 181, 189].
2.2.1 Basic Concepts
Inertia is the propensity of bodies to maintain constant translational and rotational
velocity, unless disturbed by forces or torques, respectively (Newton’s first law
or motion).
An inertial reference frame is a coordinate frame in which Newton’s laws
of motion are valid. Inertial reference frames are neither rotating nor accelerating. They are not necessarily the same as the navigation coordinates, which are
20
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
typically dictated by the navigation problem at hand. For example, “locally level”
coordinates used for navigation near the surface of the earth are rotating (with
the earth) and accelerating (to counter gravity). Such rotations and accelerations
must be taken into account in the practical implementation of inertial navigation.
Inertial sensors measure inertial accelerations and rotations, both of which are
vector-valued variables.
•
•
•
Accelerometers are sensors for measuring inertial acceleration, also called
specific force to distinguish it from what we call “gravitational acceleration.” Accelerometers do not measure gravitational acceleration, which is
perhaps more accurately modeled as a warping of the spacetime continuum
in a gravitational field. An accelerometer in free fall (e.g., in orbit) in a
gravitational field has no detectable input. What accelerometers measure is
modeled by Newton’s second law as a = F /m, where F is the physically
applied force (not including gravity), m is the mass it is applied to, and
specific force is the ratio F /m.
Gyroscopes (usually shortened to gyros) are sensors for measuring rotation. Rate gyros measure rotation rate, and displacement gyros (also called
whole-angle gyros) measure accumulated rotation angle. Inertial navigation
depends on gyros for maintaining knowledge of how the accelerometers are
oriented in inertial and navigational coordinates.
The input axis of an inertial sensor defines which vector component of
acceleration or rotation rate it measures. Multiaxis sensors measure more
than one component.
An inertial sensor assembly (ISA) is an ensemble of inertial sensors rigidly
mounted to a common base to maintain the same relative orientations, as illustrated in Fig. 2.1. Inertial sensor assemblies used in inertial navigation usually
contain three accelerometers and three gyroscopes, as shown in the figure, or an
equivalent configuration using multiaxis sensors. However, ISAs used for some
other purposes (e.g., dynamic control applications such as autopilots or automotive steering augmentation) may not need as many sensors, and some designs
provide more than three input axis directions for the accelerometers and gyroscopes. The term inertial reference unit (IRU) usually refers to an inertial sensor
system for attitude information only (i.e., using only gyroscopes). Other terms
used for the ISA are instrument cluster and (for gimbaled systems) stable element
or stable platform.
An inertial measurement unit (IMU) includes an ISA and its associated support electronics for calibration and control of the ISA. The support electronics
may also include thermal control or compensation, signal conditioning, and
input/output control. The IMU may also include an IMU processor, and—for
gimbaled systems—the gimbal control electronics.
An inertial navigation system (INS) consists of an IMU plus the following:
•
Navigation computers (one or more) to calculate the gravitational acceleration (not measured by accelerometers) and process the outputs of the
21
FUNDAMENTALS OF INERTIAL NAVIGATION
3 ACCELEROMETERS
INPUT
AXES
COMMON
MOUNTING
BASE
3 GYROSCOPES
(a)
Fig. 2.1
view.
•
•
(b)
Inertial sensor assembly (ISA) components: (a) top-Front view; (b) bottom-back
accelerometers and gyroscopes from the IMU to maintain an estimate of
the position of the IMU. Intermediate results of the implementation method
usually include estimates of velocity, attitude, and attitude rates of the IMU.
User interfaces, such as display consoles for human operators and analog
and/or digital data interfaces for vehicle guidance1 and control functions.
Power supplies and/or raw power conditioning for the complete INS.
2.2.1.1 Host Vehicles The term host vehicle is used to refer to the platform
on or in which an INS is mounted. This could be a spacecraft, aircraft, surface
ship, submarine, land vehicle, or pack animal (including humans).
2.2.1.2 What an INS Measures An INS estimates the position of its ISA, just
as a GNSS receiver estimates the position of its antenna. The relative locations
of the ISA and GNSS antenna on the host vehicle must be taken into account in
GNSS/INS integration.
2.2.2 Inertial Navigation Systems
The first known inertial navigation systems are of the type you carry around in
your head. There are two of them, and they are part of the vestibular system in
1
Guidance generally includes the generation of command signals for controlling the motion and
attitude of a vehicle to follow a specified trajectory or to arrive at a specified destination.
22
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
GIMBAL-STABILIZED
INERTIAL SENSOR
ASSEMBLY (ISA)
INNER RING
INNER BEARING
MIDDLE BEARING
MIDDLE RING
OUTER BEARING
OUTER GIMBAL RING
QUASI-HARDMOUNTED
TO HOST VEHICLE FRAME
Fig. 2.2
Gimbaled IMU.
your inner ears. Each includes three rotational acceleration sensors (semicircular
canals) and two dual-axis accelerometers (otolith organs). They are not accurate
enough for long-distance navigation, but they do enable you to balance and walk
in total darkness.
Engineers have designed many different inertial navigation systems with better
long-distance performance characteristics. They generally fall into two categories:
•
•
Gimbaled or floated systems, in which the inertial sensor assembly (ISA) is
isolated from rotations of the host vehicle, as illustrated in Figs. 2.2 and 2.3.
This rotation-isolated ISA is also called an inertial platform, stable platform,
or stable element. In this case, the IMU includes the ISA, the gimbal/float
structure and all associated electronics (e.g., gimbal wiring, rotary slip rings,
gimbal bearing angle encoders, signal conditioning, gimbal bearing torque
motors, and thermal control).
Strapdown systems are illustrated in Fig. 2.4. In this case, the ISA is not
isolated from rotations, but is “quasirigidly” mounted to the frame structure
of the host vehicle.
2.2.2.1 Shock and Vibration Isolation We use the term quasirigid for IMU
mountings that can provide some isolation of the IMU from shock and vibration
transmitted through the host vehicle frame. Many host vehicles produce severe
mechanical noise within their propulsion systems, or through vehicle contact with
the environment. Both strapdown and gimbaled systems may require shock and
23
FUNDAMENTALS OF INERTIAL NAVIGATION
SPHERICAL ENCLOSURE
(IMMERSED IN LIQUID)
INTERNAL
INERTIAL
SENSOR
ASSEMBLY
THRUSTERS
FLOATED SYSTEM
IS BALANCED AND
NEUTRALLY BUOYANT
Fig. 2.3 Floated IMU.
INERIAL SENSORS
MOUNTED ON
COMMON
BASE
(COMMON
BASE ILLUSTRATED
AS TRANSLUCENT BOX)
Fig. 2.4
COMMON
BASE MOUNTED
ON HOST VEHICLE FRAME
Strapdown ISA.
24
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
NO ROTATIONAL
ISOLATION ABOUT
THIS AXIS
90 deg PITCH-UP
Fig. 2.5 Gimbal lock.
vibration isolators to dampen the vibrational torques and forces transmitted to
the inertial sensors. These isolators are commonly made from “lossy” elastomers
that provide some amount of damping, as well.
2.2.2.2 Gimbaled Systems Two-axis gimbals were in use in China (for carrying liquids with less sloshing) around the beginning of the Current Era. Gimbaled
inertial navigation systems using feedback control technology were first developed around the middle of the twentieth century, when computers were too slow
for strapdown calculations and too heavy for inflight applications.
Gimbals are nested ringlike structures with orthogonal rotation bearings (also
called gimbals) that allow isolation of the inside from rotations of the outside. As
illustrated in Fig. 2.2, three sets of gimbal bearings are sufficient for complete
rotational isolation in applications with limited attitude mobility (e.g., surface
ships), but applications in fully maneuverable hosts require an additional gimbal
bearing to avoid the condition shown in Fig. 2.5, known as gimbal lock , in
which the gimbal configuration no longer provides isolation from outside rotations
about all three axes. The example shown in Fig. 2.5 cannot isolate the INS from
rotations about the axis illustrated by the rotation vector.
Gyroscopes inside the gimbals can be used to detect any rotation of that frame
due to torques from bearing friction or load imbalance, and torquing motors in
the gimbal bearings can then be used to servo the rotation rates inside the gimbals
to zero. For navigation with respect to the rotating earth, the gimbals can also be
servoed to maintain the sensor axes fixed in locally level coordinates.
25
FUNDAMENTALS OF INERTIAL NAVIGATION
HEADING
ROLL
PITCH
HOST
VEHICLE FRAME
Fig. 2.6
Gimbals reading attitude Euler angles.
Design of gimbal torquing servos is complicated by the motions of the gimbals
during operation, which determines how the torquing to correct for sensed rotation
must be applied to the different gimbal bearings. This requires a bearing angle
sensor for each gimbal axis.
The gimbal arrangement shown in Fig. 2.2 and 2.6, with the outer gimbal axis
aligned to the roll (longitudinal) axis of the host vehicle and the inner gimbal
axis maintained in the vertical direction, is a popular one. If the ISA is kept
aligned with locally level east–north–up directions, the gimbal bearing angles
shown in Fig. 2.6 will equal the heading, pitch and roll Euler angles defining the
host vehicle attitude relative to east, north, and upward directions. These are the
same Euler angles used to drive attitude and heading reference systems (AHRSs)
(e.g., compass card and artificial horizon displays) in aircraft cockpits.
Advantages The principal advantage of both gimbaled and floated systems is
the isolation of the inertial sensors from high angular rates, which eliminates
many rate-dependent sensor errors (including gyro-scale factor sensitivity) and
generally allows for higher accuracy sensors. Also, gimbaled systems can be
self-calibrated by orienting the ISA with respect to gravity (for calibrating the
accelerometers) and with respect to the earth rotation axis (for calibrating the
gyros), and by using external optical autocollimators with mirrors on the ISA to
independently measure its orientation with respect to its environment.
The most demanding INS applications for “cruise” applications (i.e., at ≈ 1 g)
are probably for nuclear missile-carrying submarines, which must navigate submerged for about 3 months. The gimbaled Electrically Supported Gyro Navigation (ESGN, DOD designation AN/WSN-3 [34]) system developed in the 1970s
at the Autonetics Division of Rockwell International for USN Trident-class submarines was probably the most accurate INS of that era [129].
26
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
Disadvantages The principal drawbacks of gimbals are cost, weight, volume,
and gimbal flexure (in high-g environments). In traditional designs, electrical
pathways are required through the gimbal structure to provide power to the
IMU, and to carry power and the encoder, torquer, and sensor signals. These
require slip rings (which can introduce noise) or cable wraps at the gimbal bearings. However, more recent designs have used wireless methods for signal and
power transmission. The gimbal structure can interfere with air circulation used
to maintain uniform temperatures within the IMU, and it can hamper access to
the sensors during test and operation (e.g., for autocollimation off the ISA).
2.2.2.3 Floated Systems Gimbals and gimbal lock can be eliminated by floating the inertial sensor assembly in a liquid and operating it like a robotic
submersible, using liquid thrusters to maintain its orientation and keep itself centered within the flotation cavity, as illustrated in Fig. 2.3. The floated assembly
must also be neutrally buoyant and balanced to eliminate acceleration-dependent
disturbances.
Advantages Floated systems have the advantage over gimbaled systems in that
there are no gimbal structures to flex under heavy acceleration loading. Floated
systems also have the same advantages as gimbaled systems over strapdown
systems; isolation of the inertial sensors from high angular rates eliminates many
rate-dependent error effects and generally allows for higher accuracy sensors. The
ability to orient the sphere allows self-calibration capability, but the problem of
autocollimation off the stable element is more difficult than for gimbaled systems.
The floated advanced inertial reference sphere (AIRS) designed at the C. S.
Draper Laboratory for MX/Peacekeeper and Minuteman III missiles is probably
the most accurate (and most expensive) high-g INS ever developed [129].
Disadvantages A major disadvantage of floated systems is the difficulty of
accessing the inertial sensor assembly for diagnostic testing, maintenance, or
repair. The flotation system must be disassembled and the fluid drained for
access, and then reassembled for operation. Floated systems also require means
for determining the attitude of the floated assembly relative to the host vehicle,
and wireless methods for providing power to the floated assembly and passing
commands and sensor signals through the fluid.
2.2.2.4 Carouseling and Indexing
Carouseling A carousel is an amusement ride using continuous rotation of a
circular platform about a vertical axis. The term “carouseling” has been applied
to an implementation for gimbaled or floated systems in which the inertial sensor
assembly revolves slowly around the local vertical axis—at rates in the order
of a revolution per minute. The 3-gimbal configuration shown in Fig. 2.2 can
implement carouseling using only the inner (vertical) gimbal axis. Carouseling
significantly reduces long-term navigation errors due to some types of sensor
errors (uncompensated biases of nominally level accelerometers and gyroscopes,
in particular).
FUNDAMENTALS OF INERTIAL NAVIGATION
27
Indexing Alternative implementations called “indexing” or “gimbal flipping”
use discrete rotations (usually by multiples of 90◦ degrees) to the same effect.
2.2.2.5 Strapdown Systems Strapdown systems use an inertial measurement
unit that is not isolated from rotations of its host vehicle—except possibly by
shock and vibration isolators. The gimbals are effectively replaced by software
that uses the gyroscope outputs to calculate the equivalent accelerometer outputs
in an attitude-stabilized coordinate frame, and integrates them to provide updates
of velocity and position. This requires more computation (which is cheap) than
does the gimbaled implementation, but it eliminates the gimbal system (which
may not be cheap). It also exposes the accelerometers and gyroscopes to relatively
high rotation rates, which can cause attitude-rate-dependent sensor errors.
Advantages The principal advantage of strapdown systems over gimbaled or
floated systems is cost. The cost or replicating software is vanishingly small, compared to the cost of replicating a gimbal system for each IMU. For applications
requiring attitude control of the host vehicle, strapdown gyroscopes generally provide more accurate rotation rate data than do the attitude readouts of gimbaled
or floated systems.
Disadvantages Strapdown sensors must operate at much higher rotation rates,
which can increase sensor cost. The dynamic ranges of the inputs to strapdown
gyroscopes may be orders of magnitude greater than those for gyroscopes in
gimbaled systems. To achieve comparable navigation performance, this generally requires orders of magnitude better scale factor stability for the strapdown
gyroscopes. Strapdown systems generally require much shorter integration intervals—especially for integrating gyroscope outputs—and this increases the computer costs relative to gimbaled systems. Another disadvantage for strapdown is
the cost of gyroscope calibration and testing, which requires a precision rate table.
(Rate tables are not required for whole-angle gyroscopes—including electrostatic
gyroscopes—or for the gyroscopes used in gimbaled systems.)
2.2.2.6 Strapdown Carouseling and Indexing For host vehicles that are nominally upright during operation (e.g., ships), a strapdown system can be rotated
about the host vehicle yaw axis. So long as the vehicle yaw axis remains close
to the local vehicle, slow rotation (carouseling) or indexing about this axis can
significantly reduce the effects of uncompensated biases of the nominally level
accelerometers and gyroscopes. The rotation is normally oscillatory, with reversal of direction after a full rotation, so that the connectors can be wrapped to
avoid using slip rings (an option not generally available for gimbaled systems).
Disadvantages Carouseling or indexing of strapdown systems requires the addition of a rotation bearing and associated motor drive, wiring, and control electronics. The benefits are not without cost. For gimbaled and floated systems, the
additional costs are relatively insignificant.
28
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
Fig. 2.7 Essential navigation signal processing for gimbaled INS.
2.2.3 Sensor Signal Processing
2.2.3.1 Gimbaled Implementations Figure 2.7 shows the essential navigation
signal processing functions for a gimbaled INS with inertial sensor axes aligned
to locally level coordinates, where
fspecific is the specific force (i.e., the sensible acceleration, exclusive of gravitational acceleration) applied to the host vehicle.
inertial is the instantaneous inertial rotation rate vector of the host vehicle.
A denotes a specific force sensor (accelerometer).
E denotes the ensemble of gimbal angle encoders, one for each gimbal angle
(there are several possible formats for the gimbal angles, including digitized
angles, 3-wire synchro signals, or sin / cos pairs).
G denotes an inertial rotation rate sensor (gyroscope).
position is the estimated position of the host vehicle in navigation coordinates
(e.g., longitude, latitude, and altitude relative to sea level).
velocity is the estimated velocity of the host vehicle in navigation coordinates
(e.g., east, north, and vertical).
FUNDAMENTALS OF INERTIAL NAVIGATION
29
attitude is the estimated attitude of the host vehicle relative to locally level
coordinates. For some 3-gimbal systems, the gimbal angles are the Euler
angles representing vehicle heading (with respect to north), pitch, and roll.
Output attitude may also be used to drive cockpit displays such as compass
cards or artificial horizon indicators.
accel. error comp. and gyro error comp. denote the calibrated corrections
for sensor errors. These generally include corrections for scale factor variations, output biases and input axis misalignments for both types of sensors,
and acceleration-dependent errors for gyroscopes.
gravity denotes the gravity model used to compute the acceleration due to
gravity as a function of position.
coriolis denotes the acceleration correction for coriolis effect in rotating coordinates.
leveling denotes the rotation rate correction in locally level coordinates moving over the surface of the earth.
earthrate denotes the model used to calculate the earth rotation rate in
locally level INS coordinates.
torquing denotes the servo loop gain computations used in stabilizing the
INS in locally level coordinates.
Not shown in the figure is the input altitude reference (e.g., barometric altimeter or GPS) required for vertical channel (altitude) stabilization.2
Initializing INS Alignment This signal processing schematic in Fig. 2.7 is for
operation in the navigation mode. It does not show the implementation used for
initial alignment of the sensor axes, which is done while the INS is essentially
stationary. During initial alignment, the outputs of the east and north accelerometers (denoted by AE and AN ) are used for leveling the INS, and the output of
the east gyroscope (denoted by GE ) is used for aligning the INS in heading.
When the INS is aligned, the east- and north-pointing accelerometer outputs will
always be zero, and the east-pointing gyroscope output will also be zero.
A magnetic compass may be used to get an initial rough estimate of alignment,
which can speed up the alignment process. Also—if the host vehicle has not been
moved—the alignment information at system shutdown can be saved to initialize
alignment at the next system turnon.
Initializing INS Position and Velocity The integrals shown in Fig. 2.7 require
initial values for velocity and position. The INS normally remains stationary
(i.e., with zero velocity) during INS alignment initialization, which solves the
velocity initialization problem. The angle between the sensed acceleration vector
and the sensed earthrate vector can be used to estimate latitude, but INS position
2
Vertical channel instability of inertial navigation is caused by the decrease in modeled gravity with
increasing altitude.
30
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
(including longitude and altitude) must ordinarily be initialized from external
sources (such as GPS). If the vehicle has not been moved too far during shutdown,
the position from the last shutdown can be used to initialize position at turnon.
2.2.3.2 Strapdown Implementations The basic signal processing functions for
a strapdown INS navigation are diagrammed in Fig. 2.8, where the common
symbols used in Fig. 2.7 have the same meaning as before, and
G is the estimated gravitational acceleration, computed as a function of estimated position.
POSNAV is the estimated position of the host vehicle in navigation coordinates.
VELNAV is the estimated velocity of the host vehicle in navigation coordinates.
ACCNAV is the estimated acceleration of the host vehicle in navigation coordinates, which may be used for trajectory control (i.e., vehicle guidance).
ACCSENSOR is the estimated acceleration of the host vehicle in sensor-fixed
coordinates, which may be used for steering stabilization and control.
SENSOR
CNAV
is the 3 × 3 coordinate transformation matrix from sensor-fixed coordinates to navigation coordinates, representing the attitude of the sensors
in navigation coordinates.
SENSOR is the estimated angular velocity of the host vehicle in sensor-fixed
coordinates, which may be used for vehicle attitude stabilization and control.
NAV is the estimated angular velocity of the host vehicle in navigation coordinates, which may be used in a vehicle pointing and attitude control loop.
The essential processing functions include double integration (represented by
boxes containing integration symbols) of acceleration to obtain position, and
computation of (unsensed) gravitational acceleration as a function of position.
The sensed angular rates also need to be integrated to maintain the knowledge
of sensor attitudes. The initial values of all the integrals (i.e., position, velocity,
and attitude) must also be known before integration can begin.
The position vector POSNAV is the essential navigation solution. The other
outputs shown are not needed for all applications, but most of them (except NAV )
are intermediate results that are available “for free” (i.e., without requiring further
processing). The velocity vector VELNAV , for example, characterizes speed and
heading, which are also useful for correcting the course of the host vehicle to
bring it to a desired location. Most of the other outputs shown would be required
for implementing control of an unmanned or autonomous host vehicle to follow
a desired trajectory and/or to bring the host vehicle to a desired final position.
Navigation functions that are not shown in Fig. 2.8 include
1. How initialization of the integrals for position, velocity, and attitude is
implemented. Initial position and velocity can be input from other sources
(e.g., GNSS), and attitude can be inferred from some form of trajectory
matching (using GNSS, e.g.) or by gyrocompassing (described below).
FUNDAMENTALS OF INERTIAL NAVIGATION
31
Fig. 2.8 Essential navigation signal processing for strapdown INS.
2. How attitude rates are integrated to obtain attitude. Because rotation operations are not commutative, attitude rate integration is not as straightforward
as the integration of acceleration to obtain velocity and position. Special
techniques required for attitude rate integration are described in Chapter 9.
3. For the case that navigation coordinates are earth-fixed, the computation of
navigational coordinate rotation due to earthrate as a function of position,
and its summation with sensed rates before integration.
4. For the case that navigation coordinates are locally level, the computation
of the rotation rate of navigation coordinates due to vehicle horizontal
velocity, and its summation with sensed rates before integration.
5. Calibration of the sensors for error compensation. If the errors are sufficiently stable, it needs to be done only once. Otherwise, it can be implemented using GNSS/INS integration techniques.
2.2.3.3 Gyrocompass Alignment
Alignment is the term used for a procedure to determine the orientation of the
ISA relative to navigation coordinates.
32
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
Measured
variables
w
→
Estimated
variables
North
→
a
Up
East
Rotating
earth
Fig. 2.9 Gyrocompassing determines sensor orientations with respect to east, north, and
up.
A gyrocompass is an instrument that levels itself relative to true vertical,
detects the rotation of the vertical due to the rotation of the earth, and uses
the direction of rotation to align itself to true north.
Gyrocompassing of an INS is a procedure for self-contained alignment of its
ISA.
Gyrocompassing must be performed when the host vehicle is essentially stationary. Under stationary conditions, the INS can use its accelerometers to determine
the direction of the local vertical relative to the sensors, and its gyroscopes to
determine the direction of the earth rotation axis relative to the sensors. The
cross-product of local vertical vector and the earth rotation axis vector in sensorfixed coordinates will point east, as illustrated in Fig. 2.9. The cross-product of
local vertical (i.e., “Up”) and “East,” will be “North,” as shown in the figure.
Performance Degrades Near the Poles Near the north or south pole of the earth,
the direction of the earth rotation axis comes close to the local vertical direction,
and their cross-product is not well determined. For that reason, gyrocompassing
is not accurate near the poles.
2.2.4 Standalone INS Performance
2.2.4.1 Free Inertial Operation Operation of an INS without external aiding of any sort is called “free inertial” or “pure inertial.” Because free inertial
FUNDAMENTALS OF INERTIAL NAVIGATION
33
navigation in the near-earth gravitational environment is unstable3 in the vertical direction, aiding by other sensors (e.g., barometric altimeters for aircraft or
surface vehicles, radar altimeters for aircraft over water, or hydrostatic pressure
for submersibles) is required to avoid vertical error instability. For that reason,
performance of free-inertial navigation systems is usually specified for horizontal
position errors only.
2.2.4.2 INS Performance Metrics
Simplified Error Model INS position is initialized by knowing where you are
starting from at initial time t0 . The position error may start out very small,
but it tends to increase with time due to the influence of sensor errors. Double
integration of accelerometer output errors is a major source of this growth over
time. Experience has shown that the variance and standard deviation of horizontal
position error
∝
2
σposition
(t) ≈ (t − t0 )2 ,
(2.1)
σposition (t) ≈ C × |t − t0 | ,
(2.2)
with unknown positive constant C. This constant C would then characterize
performance of an INS in terms of how fast its RMS position error grows.
A problem with this model is that actual horizontal INS position errors are
two-dimensional, and we would need a 2 × 2 covariance matrix in place of C.
That would not be very useful in practice. As an alternative, we replace C with
something more intuitive and practical.
CEP The radius of a horizontal circle centered at the estimated position, and of
sufficient radius such that it is equally probable that the true horizontal position
is inside or outside the circle is called circular error probable (CEP). CEP is also
used as an acronym for “circle of equal probability” (of being inside or outside).
CEP rate The time rate of change of circular error probable is CEP rate. Traditional units of CEP rate are nautical miles per hour or kilometers per hour. The
nautical mile was originally intended to designate a surface distance equivalent
to one arc minute of latitude change at sea level, but that depends on latitude.
The SI-derived nautical mile is 1.852 km.
2.2.4.3 Performance Levels In the 1970s, before GPS became a reality, the
U.S. Air Force had established the following levels of performance for INS:
High-accuracy systems have free inertial CEP rates in the order of 0.1 nautical miles per hour (nmi/h) (≈ 185 m/h) or better. This is the order of
3
This is due to the falloff of gravity with increasing altitude. The issue is covered in Section 9.5.2.2.
34
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
magnitude in accuracy required for intercontinental ballistic missiles and
missile-carrying submarines, for example.
Medium-accuracy systems have free inertial CEP rates in the order of 1 nmi/h
(nautical miles per hour) (≈ 1.85 km/h). This is the level of accuracy
deemed sufficient for most military and commercial aircraft [34].
Low-accuracy systems have free inertial CEP rates in the order of 10 nmi/h
(nautical miles per hour) (≈ 18.5 km/h) or worse. This range covered
the requirements for many short-range standoff weapons such as guided
artillery or tactical rockets.
However, after GPS became available, GPS/INS integration could make a
low-accuracy INS behave more like a high-accuracy INS.
2.3 SATELLITE NAVIGATION
The GPS is widely used in navigation. Its augmentation with other space-based
satellites is the future of near-earth navigation.
2.3.1 Satellite Orbits
GPS satellites occupy six orbital planes inclined 55◦ from the equatorial plane,
as illustrated in Figs. 2.10 and 2.11. Each of the six orbit planes in Fig. 2.11
contains four or more satellites.
2.3.2 Navigation Solution (Two-Dimensional Example)
Antenna location in two dimensions can be calculated by using range measurements [65].
2.3.2.1 Symmetric Solution Using Two Transmitters on Land In this case,
the receiver and two transmitters are located in the same plane, as shown in Fig.
2.12, with known positions x1 ,y1 and x2 ,y2 . Ranges R1 and R2 of two transmitters
from the user position are calculated as
R1 = c T1 ,
(2.3)
R2 = c T2 ,
(2.4)
where
c = speed of light (0.299792458 m/ns)
T1 = time taken for the radiowave to travel from
transmitter 1 to the user
T2 = time taken for the radiowave to travel from
transmitter 2 to the user
(X, Y ) = user position
35
SATELLITE NAVIGATION
Fig. 2.10
Parameters defining satellite orbit geometry.
Fig. 2.11
GPS orbit planes.
36
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
Fig. 2.12 Two transmitters with known 2D positions.
The range to each transmitter can be written as
R1 = [(X − x1 )2 + (Y − y1 )2 ]1/2 ,
(2.5)
R2 = [(X − x2 )2 + (Y − y2 )2 ]1/2 .
(2.6)
Expanding R1 and R2 in Taylor series expansion with small perturbation in
X by x and Y by y yields
R1 =
∂R1
∂R1
x +
y + u1 ,
∂X
∂Y
(2.7)
R2 =
∂R2
∂R2
x +
y + u2 ,
∂X
∂Y
(2.8)
where u1 and u2 are higher order terms. The derivatives of Eqs. 2.5 and 2.6 with
respect to X, Y are substituted into Eqs. 2.7 and 2.8, respectively.
37
SATELLITE NAVIGATION
Thus, for the symmetric case, we obtain
R1 =
X − x1
x
[(X − x1 + (Y − y1 )2 ]1/2
(2.9)
)2
+
Y − y1
y + u1 ,
[(X − x1 )2 + (Y − y1 )2 ]1/2
= sin θ x + cos θ y + u1 ,
(2.10)
R2 = − sin θ x + cos θ y + u2 .
(2.11)
To obtain the least-squares estimate of (X, Y ), we need to minimize the quantity
J = u21 + u22 ,
(2.12)
which is
⎛
⎞2
⎛
⎞2
J = ⎝R1 − sin θ x − cos θ y ⎠ + ⎝R2 + sin θ x − cos θ y ⎠ .
u1
u2
(2.13)
The solution for the minimum can be found by setting ∂J/∂x = 0 = ∂J/∂y,
then solving for x and y:
∂J
∂x
= 2(R1 − sin θ x − cos θ y) (− sin θ ) + 2(R2
0=
+ sin θ x − cos θ y) (sin θ )
= R2 − R1 + 2 sin θ x,
(2.14)
(2.15)
(2.16)
(2.17)
with solution
x =
R1 − R2
.
2 sin θ
(2.18)
The solution for y may be found in similar fashion as
y =
R1 + R2
.
2 cos θ
(2.19)
38
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
Navigation Solution Procedure Transmitter positions x1 ,y1 ,x2 ,y2 are given. Signal travel times T1 , T2 are given. Estimated user position X̂u ,Ŷu are assumed.
Set position coordinates X, Y equal to their initial estimates:
X = X̂u ,
Y = Ŷu ,
Compute the range errors:
Geometric ranges
pseudoranges
Measured
2
2 1/2
R1 = [(X̂u − x1 ) + (Ŷu − y1 ) ] −
CT1 ,
(2.20)
R2 = [(X̂u − x2 )2 + (Ŷu − y2 )2 ]1/2 −
(2.21)
CT2 .
Compute the theta angle:
θ = tan−1
X̂u − x1
(2.22)
Ŷu − y1
= sin−1 X̂u − x1
2 2 .
X̂u − x1 + Ŷu − y1
(2.23)
Compute user position corrections:
1
(R1 − R2 ),
2 sin θ
1
(R1 + R2 ).
y =
2 cos θ
x =
(2.24)
(2.25)
Compute a new estimate of position:
X = X̂u + x, Y = Ŷu + y.
(2.26)
Continue to compute θ , R1 and R2 from these equations with new values of
x and y.
Iterate Eqs. 2.20–2.26:
Correction equations
1
(R1 − R2 ),
Xbest = 2 sin
θ
1
Ybest = 2 cos θ (R1 + R2 ),
Iteration equations
Xnew = Xold + Xbest ,
Ynew = Yold + Ybest .
39
SATELLITE NAVIGATION
2.3.3 Satellite Selection and Dilution of Precision
Just as in a land-based system, better accuracy is obtained by using reference
points well separated in space. For example, the range measurements made to
four reference points clustered together will yield nearly equal values. Position
calculations involve range differences, and where the ranges are nearly equal,
small relative errors are greatly magnified in the difference. This effect, brought
about as a result of satellite geometry, is known as dilution of precision (DOP).
This means that range errors that occur from other causes such as clock errors
are also magnified by the geometric effect.
To find the best locations of the satellites to be used in the calculations of the
user position and velocity, DOP calculations are needed.
The observation equations in three dimensions for each satellite with known
coordinates (xi ,yi ,zi ) and unknown user coordinates (X, Y , Z) are given by
Zρ i = ρ i =
(xi − X)2 + (yi − Y )2 + (zi − Z)2 + Cb .
(2.27)
These are nonlinear equations that can be linearized using Taylor series (see, e.g.,
Chapter 5 of Ref. 66).
Let the vector of ranges be Zρ = h(x), a nonlinear function h(x) of the fourdimensional vector x representing user position and receiver clock bias, and
expand the left-hand side of this equation in a Taylor series about some nominal
solution xnom for the unknown vector
x = [X, Y, Z, Cb ]T
(2.28)
of variables
def
X = east component of the user’s antenna location
def
Y = north component of the user’s antenna location
def
Z = upward vertical component of the user’s antenna location
def
Cb = receiver clock bias
for which
Zρ
= h(x) = h(xnom ) + ∂h(x)
+ H.O.T
∂x δx
x=xnom
δx = x − xnom ,
δZρ
,
= h(x) − h(xnom ),
where H.O.T stands for “higher-order terms.”
(2.29)
40
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
These equations become
∂h(x) δx = H [1] δx,
δZρ =
∂x x=x nom
(2.30)
δx = X − Xnom , δy = Y − Ynom , δz = Z − Znom ,
where H [1] is the first-order term in the Taylor series expansion
δZρ = ρ(X, Y , Z) − ρr (Xnom , Ynom , Znom )
∂ρr δx + vρ
≈
∂X Xnom , Ynom , Znom
(2.31)
(2.32)
H [1]
for vρ = noise in receiver measurements. This vector equation can be written in
scalar form where i = satellite number as
⎫
∂ρri
−(x
−
X)
⎪
i
⎪
=
⎪
∂X
⎪
(xi − X)2 + (yi − Y )2 + (zi − Z)2 X=Xnom , Ynom , Znom ⎪
⎪
⎪
⎪
−(x
−
X
)
⎪
i
nom
⎪
= ⎬
2
2
2
(xi − Xnom ) + (yi − Ynom ) + (zi − Znom )
(2.33)
i
∂ρr
−(yi − Ynom )
⎪
⎪
=
⎪
⎪
∂Y
⎪
(xi − Xnom )2 + (yi − Ynom )2 + (zi − Znom )2
⎪
⎪
⎪
⎪
∂ρri
−(z
−
Z
)
⎪
i
nom
⎭
=
∂Z
2
2
2
(xi − Xnom ) + (yi − Ynom ) + (zi − Znom )
for
i = 1, 2, 3, 4 (i.e., four satellites)
We can combine Eqs. 2.32 and 2.33 into
⎡ 1
∂ρr
∂ρr1
∂ρr1
⎡ 1 ⎤
∂x
∂y
∂z
δzρ
⎢ ∂ρ 2 ∂ρ 2 ∂ρ
2
r
r
r
⎢ δzρ2 ⎥ ⎢
⎢
∂x
∂y
∂z
⎢ 3 ⎥=⎢ 3
⎣ δzρ ⎦ ⎢ ∂ρr ∂ρr3 ∂ρr3
∂y
∂z
⎣ ∂x
δzρ4
∂ρr4
∂ρr4
∂ρr4
∂x
∂y
∂z
4×1
the matrix equation
⎤
⎡
⎤ ⎡
1
δx
⎥
⎥
⎢
1 ⎥ ⎢
δy ⎥
⎥+⎢
⎥ ⎢
⎣
⎣
⎦
δz
1 ⎥
⎦
Cb
1
4×1
4×4
which we can write in symbolic form as
4×4
4×1
4×1
4×1
[1]
δx + vk
δZρ = H
(see Table 5.3 in Ref. 66).
(2.34)
vρ1
vρ2
vρ3
vρ4
4×1
⎤
⎥
⎥,
⎦
41
SATELLITE NAVIGATION
To calculate H [1] , one needs satellite positions and the nominal value of the
user’s position.
To calculate the geometric dilution of precision (GDOP) (approximately), we
obtain
4×1
4×1
4×1
[1]
δx .
δZρ = H
(2.35)
Known are δZρ and H [1] from the pseudorange, satellite position, and nominal
value of the user’s position. The correction δx is the unknown vector.
If we premultiply both sides of Eq. 2.35 by H [1]T , the result will be
4×4
H
[1]T
4×4
[1]T
[1]
δZρ = H
H δx.
(2.36)
4×4
−1
Then we premultiply Eq. 2.36 by H [1]T H [1] :
−1 [1]T
H
δZρ .
δx = H [1]T H [1]
(2.37)
If δx and δZρ are assumed random with zero mean, the error covariance
E(δx) (δx)T T
−1 [1]T
−1 [1]T
H
δZρ H [1]T H [1]
H
δZρ = E H [1]T H [1]
−1 [1]T
−1
= H [1]T H [1]
H
EδZρ δZρT H [1]T H [1]
.
(2.38)
(2.39)
The pseudorange measurement covariance is assumed uncorrelated satelliteto-satellite with variance σ 2 :
EδZρ δZρT = σ 2 I4 ,
(2.40)
a 4 × 4 matrix.
Substituting Eq. 2.40 into Eq. 2.39 gives
Eδx(δx)T = σ 2 (H [1]T H [1] )−1 (H [1]T H [1] )(H [1]T H [1] )−1
(2.41)
I
= σ (H
2
[1]T
H
[1] −1
) ,
(2.42)
42
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
for
⎤
E
⎢ N ⎥
⎥
δx = ⎢
⎣ U ⎦ ,
Cb
⎡
4×1
and
⎞
locally
⎟
⎜
level
⎟
⎜
⎝ coordinate ⎠ ,
frame
⎛
E = east error
N = north error
U = up error
and the covariance matrix becomes
⎤
⎡
E EU E ECb
E E 2
E EN
⎢ E N E
E N Cb ⎥
E N 2
E N
⎥.
U
E δx(δx)T = ⎢
2
⎦
⎣
E U E E U N E U
E U
Cb
2
4×4
E CbE E CbN E CbU E Cb
(2.43)
We are principally interested in the diagonal elements of
⎡
(H [1] T H [1] )−1
A11
⎢ A21
⎢
=⎣
A31
A41
A12
A22
A32
A42
A13
A23
A33
A43
⎤
A14
A24 ⎥
⎥,
A34 ⎦
A44
(2.44)
with σ 2 = 1 m2 in the following combinations (see Fig. 2.13):
GDOP
PDOP
HDOP
VDOP
TDOP
=
=
=
=
=
√
√A11 + A22 + A33 + A44
√A11 + A22 + A33
√A11 + A22
√A33
A44
(geometric DOP) ,
(position DOP) ,
(horizontal DOP) ,
(vertical DOP) ,
(time DOP) .
Hereafter, all DOPs represent the sensitivities to pseudorange errors.
2.3.4 Example Calculation of DOPs
2.3.4.1 Four Satellites The best accuracy is found with three satellites equally
spaced on the horizon, at minimum elevation angle, with the fourth satellite
directly overhead, as listed in Table 2.1.
43
SATELLITE NAVIGATION
GDOP
PDOP
TDOP
HDOP
VDOP
Fig. 2.13
DOP hierarchy.
The diagonal of the unscaled covariance matrix H [1] T H [1] then has the terms
⎡
⎢
⎢
⎣
⎤
(east DOP)2
⎥
⎥ ,
⎦
(north DOP)2
(vertical DOP)2
(time DOP)2
where
GDOP =
!
∂ρ trace(H [1] T H [1] )−1 , H [1] =
∂x Xnom
.
Ynom , Znom
TABLE 2.1. Example with Four Satellites
Satellite location
Elevation (deg)
Azimuth (deg)
1
2
3
4
5
0
5
120
5
240
90
0
44
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
Typical example values of H [1] for this geometry are
⎤
0.000
0.996 0.087 1.000
⎢ 0.863 −0.498 0.087 1.000 ⎥
⎥
=⎢
⎣ −0.863 −0.498 0.087 1.000 ⎦ .
0.000
0.000 1.000 1.000
⎡
H [1]
The GDOP calculations for this example are
⎡
(H [1]T H [1] )−1
0.672
⎢ 0.000
⎢
= ⎣
0.000
0.000
⎤
0.000
0.000
0.000
0.672
0.000
0.000 ⎥
⎥ ,
0.000
1.600 −0.505 ⎦
0.000 −0.505
0.409
√
GDOP =
0.672 + 0.672 + 1.6 + 0.409
= 1.83,
HDOP
VDOP
PDOP
TDOP
=
=
=
=
1.16,
1.26,
1.72,
0.64.
2.4 TIME AND GPS
2.4.1 Coordinated Universal Time Generation
Coordinated Universal Time (UTC) is the timescale based on the atomic second, but occasionally corrected by the insertion of leap seconds, so as to keep
it approximately synchronized with the earth’s rotation. The leap second adjustments keep UTC within 0.9 s of UT1, which is a timescale based on the earth’s
axial spin. UT1 is a measure of the true angular orientation of the earth in space.
Because the earth does not spin at exactly a constant rate, UT1 is not a uniform
timescale [3].
2.4.2 GPS System Time
The timescale to which GPS signals are referenced is referred to as GPS time. GPS
time is derived from a composite or “paper” clock that consists of all operational
monitor station and satellite atomic clocks. Over the long run, it is steered to
keep it within about 1 μs of UTC, as maintained by the master clock at the U.S.
Naval Observatory, ignoring the UTC leap seconds. At the integer second level,
GPS time equaled UTC in 1980. However, due to the leap seconds that have
been inserted into UTC, GPS time was ahead of UTC by 14 s in February 2006.
45
TIME AND GPS
2.4.3 Receiver Computation of UTC
The parameters needed to calculate UTC from GPS time are found in subframe 4
of the navigation data message. These data include a notice to the user regarding
the scheduled future or recent past (relative to the navigation message upload)
value of the delta time due to leap seconds tLSF , together with the week number
WNLSF and the day number DN at the end of which the leap second becomes
effective. The latter two quantities are known as the effectivity time of the leap
second. “Day 1” is defined as the first day relative to the end/start of a week and
the WNLSF value consists of the eight least significant bits (LSBs) of the full
week number.
Three different UTC/GPS time relationships exist, depending on the relationship of the effectivity time to the user’s current GPS time:
1. First Case. Whenever the effectivity time indicated by the WNLSF and
WN values is not in the past relative to the user’s present GPS time, and the
user’s present time does not fall in the timespan starting at DN + 34 and ending
at DN + 54 , the UTC time is calculated as:
tUTC = (tE − tUTC )
(modulo 86400)s,
where tUTC is in seconds, 86400 is the number of seconds per day, and
tUTC = tLS + A0 + A1 [tE − t0t + 604800(WN − WNt )]s,
where 604800 is the number of seconds per week, and
tE = user GPS time from start of week (s)
tLS = delta time due to leap seconds
A0 = a constant polynomial term from the ephemeris message
A1 = a first-order polynomial term from the ephemeris message
t0t = reference time for UTC data
WN = current week number derived from subframe 1
WNt = UTC reference week number
The user GPS time tE is in seconds relative to the end/start of the week, and
the reference time t0t for UTC data is referenced to the start of that week, whose
number WNt is given in word 8 of page 18 in subframe 4. The WNt value
consists of the eight LSBs of the full week number. Thus, the user must account
for the truncated nature of this parameter as well as truncation of WN, WNt , and
WNLSF due to rollover of the full week number. These parameters are managed
by the GPS control segment so that the absolute value of the difference between
the untruncated WN and WNt values does not exceed 127.
46
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
2. Second Case. Whenever the user’s current GPS time falls within the timespan from DN + 34 to DN + 54 , proper accommodation of the leap second event
with a possible week number transition is provided by the following expression
for UTC:
tUTC = W [modulo(86400 + tLSF − tLS )] seconds,
where
W = (tE − tUTC − 43200) (modulo 86400) + 43200 seconds,
and the definition of tUTC given previously applies throughout the transition
period.
3. Third Case. Whenever the effectivity time of the leap second event, as
indicated by the WNLSF and DN values, is in the past relative to the user’s
current GPS time, the expression given for tUTC in the first case above is valid
except that the value of tLSF is used instead of tLS . The GPS control segment
coordinates the update of UTC parameters at a future upload in order to maintain
a proper continuity of the tUTC timescale.
2.5 EXAMPLE: USER POSITION CALCULATIONS WITH NO
ERRORS
2.5.1 User Position Calculations
This section demonstrates how to go about calculating the user position, given
ranges (pseudoranges) to satellites, the known positions of the satellites, and
ignoring the effects of clock errors, receiver errors, propagation errors, and so
on.
Then, the pseudoranges will be used to calculate the user’s antenna location.
2.5.1.1 Position Calculations Neglecting clock errors, let us first determine
position calculation with no errors:
ρr
x, y, z
X, Y, Z
=
=
=
pseudorange (known),
satellite position coordinates (known),
user position coordinates (unknown),
where x, y, z, X, Y, Z are in the earth-centered, earth-fixed (ECEF) coordinate
system.
Position calculation with no errors is
ρr = (x − X)2 + (y − Y )2 + (z − Z)2 .
(2.45)
47
EXAMPLE: USER POSITION CALCULATIONS WITH NO ERRORS
Squaring both sides yields
ρr2 = (x − X)2 + (y − Y )2 + (z − Z)2
(2.46)
2
2
2
2
2
2
=X
+ Y + Z +x + y + z
r 2 +Crr
−2Xx − 2Y y − 2Zz,
ρr2
(2.47)
− (x + y + z ) − r = Crr − 2Xx − 2Y y − 2Zz,
2
2
2
2
(2.48)
where r equals the radius of earth and Crr is the clock bias correction. The
four unknowns are (X, Y, Z, Crr ). Satellite position (x, y, z) is calculated from
ephemeris data. For four satellites, Eq. 2.48 becomes
ρr21
ρr22
ρr23
ρr24
− (x12 + y12 + z12 ) − r 2
− (x22 + y22 + z22 ) − r 2
− (x32 + y32 + z32 ) − r 2
− (x42 + y42 + z42 ) − r 2
= Crr − 2Xx1 − 2Y y1 − 2Zz1 ,
= Crr − 2Xx2 − 2Y y2 − 2Zz2 ,
= Crr − 2Xx3 − 2Y y3 − 2Zz3 ,
= Crr − 2Xx4 − 2Y y4 − 2Zz4 ,
(2.49)
with unknown 4 × 1 state vector
⎡
⎤
X
⎢ Y ⎥
⎢
⎥
⎣ Z ⎦ .
Crr
We can rewrite the four equations in matrix form as
⎡
ρr21
⎢ ρr2
⎢ 22
⎣ ρr
3
ρr24
⎤ ⎡
− (x12 + y12 + z12 ) − r 2
−2x1 − 2y1 − 2z1
⎢ −2x2 − 2y2 − 2z2
− (x22 + y22 + z22 ) − r 2 ⎥
⎥=⎢
− (x32 + y32 + z32 ) − r 2 ⎦ ⎣ −2x3 − 2y3 − 2z3
−2x4 − 2y4 − 2z4
− (x42 + y42 + z42 ) − r 2
⎤ ⎡
X
1
⎢ Y
1 ⎥
⎥ ⎢
1 ⎦ ⎣ Z
Crr
1
⎤
⎥
⎥
⎦
or
4×1
4×4
4×1
Y = M χρ ,
where
Y = vector (known),
M = matrix (known),
χρ = vector (unknown).
(2.50)
48
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
Then we premultiply both sides of Eq. 2.50 by M −1 :
M −1 Y
= M −1 Mχρ
= χ
⎡ρ
⎤
X
⎢ Y ⎥
⎥
= ⎢
⎣ Z ⎦ .
Crr
If the rank of M (defined in Section B.5.2), the number of linearly independent
columns of the matrix M, is less than 4, then M will not be invertible. In that
case, its determinant (defined in Section B.6.1) is given as
det M = |M| = 0.
2.5.2 User Velocity Calculations
The governing equation in this case is
ρ̇r =
(x − X)(ẋ − Ẋ) + (y − Y )(ẏ − Ẏ ) + (z − Z)(ż − Ż)
,
ρr
(2.51)
where
ρ̇r
ρr
(x, y, z)
(ẋ, ẏ, ż)
(X, Y, Z)
(Ẋ, Ẏ , Ż)
=
=
=
=
=
=
range rate (known)
range (known)
satellite positions (known)
satellite rates (known)
user position (known from position calculations)
user velocity (unknown)
and from Eq. 2.51,
−ρ̇r +
− X) + ẏ(y − Y ) + ż(z − Z)]
y−Y
z−Z
= x−X
Ẋ
+
Ẏ
+
Ż
.
ρr
ρr
ρr
1
ρr [ẋ(x
(2.52)
For three satellites, Eq. 2.52 becomes
⎤
⎡
−ρ̇r1 + ρ1r [ẋ1 (x1 − X) + ẏ1 (y1 − Y ) + ż1 (z1 − Z)]
⎢ −ρ̇ + 11 [ẋ (x − X) + ẏ (y − Y ) + ż (z − Z)] ⎥
r2
2 2
2 2
⎦
⎣
ρr2 2 2
1
−ρ̇r3 + ρr [ẋ3 (x3 − X) + ẏ3 (y3 − Y ) + ż3 (z3 − Z)]
3
⎡
⎢
=⎢
⎣
(x1 −X) (y1 −Y ) (z1 −Z)
ρr1
ρr1
ρr1
(x2 −X) (y2 −Y ) (z2 −Z)
ρr2
ρr2
ρr2
(x3 −X) (y3 −Y ) (z3 −Z)
ρr2
ρr3
ρr3
⎤
⎡
⎤
Ẋ
⎥
⎥ ⎣ Ẏ ⎦ .
⎦
Ż
(2.53)
EXAMPLE: USER POSITION CALCULATIONS WITH NO ERRORS
3×1
49
3×3 3×1
Dξ = N U∨ ,
(2.54)
3×1
U∨ = N −1 Dξ .
(2.55)
However, if the rank of N (defined in Section B.5) is <3, N will not be invertible.
PROBLEMS
Refer to Appendix C for coordinate system definitions, and to Eqs. C.103 and
C.104 for satellite orbit equations.
2.1 Which of the following coordinate systems is not rotating?
(a) North–east–down (NED)
(b) East–north–up (ENU)
(c) Earth-centered, earth-fixed (ECEF)
(d) Earth-centered inertial (ECI)
(e) Moon-centered, moon-fixed
2.2 What is the minimum number of two-axis gyroscopes (i.e., gyroscopes with
two, independent, orthogonal input axes) required for inertial navigation?
(a) 1
(b) 2
(c) 3
(d) Not determined
2.3 What is the minimum number of gimbal axes required for gimbaled inertial
navigators in fully maneuverable host vehicles? Explain your answer.
(a) 1
(b) 2
(c) 3
(d) 4
2.4 Define specific force.
2.5 An inertial sensor assembly (ISA) operating at a fixed location on the surface of the earth would measure
50
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
(a) No acceleration
(b) 1 g acceleration downward
(c) 1 g acceleration upward
2.6 Explain why an inertial navigation system is not a good altimeter.
2.7 The inertial rotation rate of the earth is
(a) 1 revolution per day
(b) 15 degrees per hour
(c) 15 arc-seconds per second
(d) ≈ 15.0411 arc-seconds per second
2.8 Define CEP and CEP rate for an INS.
2.9 The CEP rate for a medium accuracy INS is in the order of
(a) 2 meters per second (m/s)
(b) 200 meters per hour (m/h)
(c) 2000 m/h
(d) 20 km/h
2.10 For the following GPS satellites, find the satellite position in ECEF coordinates at t = 3 s. (Hint: see Appendix C.) 0 and θ0 are given below at
time t0 = 0:
(a)
(b)
0 (deg)
326
26
θ0 (deg)
68
34
2.11 Using the results of the previous problem, find the satellite positions in
the local reference frame. Reference should be to the COMSAT facility in
Santa Paula, California, located at 32.4◦ latitude, −119.2◦ longitude. Use
coordinate shift matrix S = 0. (Refer to Appendix C, Section C.3.9.)
2.12 Given the following GPS satellite coordinates and pseudoranges:
Satellite
Satellite
Satellite
Satellite
1
2
3
4
0 (deg)
326
26
146
86
θ0 (deg)
68
340
198
271
ρ (m)
2.324 × 107
2.0755 × 107
2.1103 × 107
2.3491 × 107
(a) Find the user’s antenna position in ECEF coordinates.
EXAMPLE: USER POSITION CALCULATIONS WITH NO ERRORS
51
(b) Find the user’s antenna position in locally level coordinates referenced
to 0◦ latitude, 0◦ longitude. Coordinate shift matrix S = 0.
(c) Find the various DOPs.
2.13 Given two satellites in north and east coordinates
x(1) = 6.1464 × 106 , y(1) = 2.0172 × 107 in meters,
x(2) = 6.2579 × 106 , y(2) = −7.4412 × 106 in meters,
with pseudoranges
c t (1) = ρr (1) = 2.324 × 107 in meters,
c t (2) = ρr (2) = 2.0755 × 107 in meters,
and starting with an initial guess of (xest , yest ), find the user’s antenna
position.
2.14 Rank VDOP, HDOP and PDOP from smallest (best) to largest (worst) under
normal conditions:
(a) VDOP ≤ HDOP≤PDOP
(b) VDOP≤PDOP≤HDOP
(c) HDOP≤VDOP≤PDOP
(d) HDOP≤PDOP≤VDOP
(e) PDOP≤HDOP≤VDOP
(f) PDOP≤VDOP≤HDOP
2.15 UTC time and the GPS time are offset by an integer number of seconds
(e.g., 14 s as of January 1, 2006) as well as a fraction of a second. The
fractional part is approximately:
(a) 0.1–0.5 s
(b) 1–2 ms
(c) 100–200 ns
(d) 10–20 ns
ECEF
ENU
ECEF
× CECEF
= I , the 3 × 3 identity matrix. (Hint: CENU
=
2.16 Show that CENU
" ENU #T
CECEF .)
2.17 A satellite position at time t = 0 is specified by its orbital parameters as
0 = 92.847◦ , θ0 = 135.226◦ , α = 55◦ , R = 26, 560, 000 m.
(a) Find the satellite position at t = 1 s, in ECEF coordinates.
52
FUNDAMENTALS OF SATELLITE AND INERTIAL NAVIGATION
(b) Convert the satellite position from (a) with user at
⎡
⎤
⎤
⎡
−2.430601
Xu
⎣ Yu ⎦
= ⎣ −4.702442 ⎦ × 106 meter
3.546587
Zu ECEF
to WGS84 east–north–up (ENU) coordinates with origin at
◦
θ = local reference longitude = 32.4
◦
φ = local reference latitude = −119.2
3
SIGNAL CHARACTERISTICS AND
INFORMATION EXTRACTION
Why is the GPS signal so complex? GPS was designed to be readily accessible
to millions of military and civilian users. Therefore, it is a receive-only passive
system for a user, and the number of users that can simultaneously use the
system is unlimited. Because there are many functions that must be performed,
the GPS signal has a rather complex structure. As a consequence, there is a
correspondingly complex sequence of operations that a GPS receiver must carry
out in order to extract desired information from the signal. In this chapter we
characterize the signal mathematically, describe the purposes and properties of
the important signal components, and discuss generic methods for extracting
information from these components.
3.1 MATHEMATICAL SIGNAL WAVEFORM MODELS
Each GPS satellite simultaneously transmits on two L-band frequencies denoted
by L1 and L2 , which are 1575.42 and 1227.60 MHz, respectively. The carrier of
the L1 signal consists of an in-phase and a quadrature-phase component. The inphase component is biphase modulated by a 50-bps (bits per second) data stream
and a pseudorandom code called the C/A-code consisting of a 1023-chip sequence
that has a period of 1 ms and a chipping rate of 1.023 MHz. The quadraturephase component is also biphase modulated by the same 50-bps (bits per second)
data stream but with a different pseudorandom code called the P-code, which
Global Positioning Systems, Inertial Navigation, and Integration, Second Edition, by M. S. Grewal, L. R. Weill, and A. P. Andrews
Copyright © 2007 John Wiley & Sons, Inc.
53
54
SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
has a 10.23-MHz chipping rate and a one-week period. The mathematical model
of the L1 waveform is
s(t) =
2PI d(t)c(t) cos(ωt + θ ) + 2PQ d(t)p(t) sin(ωt + θ ),
(3.1)
where PI and PQ are the respective carrier powers for the in-phase and quadraturephase carrier components, d(t) is the 50-bps (bits per second) data modulation,
c(t) and p(t) are the respective C/A and P pseudorandom code waveforms, ω is
the L1 carrier frequency in radians per second, and θ is a common phase shift in
radians. The quadrature carrier power PQ is approximately 3 dB less than PI .
In contrast to the L1 signal, the L2 signal is modulated with only the 50bps (bits per second) data and the P-code, although there is the option of not
transmitting the 50-bps (bits per second) data stream. The mathematical model
of the L2 waveform is
s(t) =
2PQ d(t)p(t) sin(ωt + θ ).
(3.2)
Figures 3.1 and 3.2 show the structure of the in-phase and quadrature-phase
components, respectively, of the L1 signal. The 50-bps (bits per second) data
bit boundaries always occur at an epoch of the C/A-code. The C/A-code epochs
mark the beginning of each period of the C/A-code, and there are precisely 20
code epochs per data bit, or 20,460 C/A-code chips. Within each C/A-code chip
there are precisely 1540 L1 carrier cycles. In the quadrature-phase component
of the L1 signal there are precisely 204,600 P-code chips within each 50-bps
(bits per second) data bit, and the data bit boundaries always coincide with the
beginning of a P-code chip [61, 84].
3.2 GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES
3.2.1 50-bps (bits per second) Data Stream
The 50-bps (bits per second) data stream conveys the navigation message, which
includes, but is not limited to, the following information:
1. Satellite Almanac Data. Each satellite transmits orbital data called the
almanac, which enables the user to calculate the approximate location of
every satellite in the GPS constellation at any given time. Almanac data
are not accurate enough for determining position but can be stored in a
receiver where they remain valid for many months. They are used primarily to determine which satellites are visible at a given location so that the
receiver can search for those satellites when it is first turned on. They can
also be used to determine the approximate expected signal Doppler shift to
aid in rapid acquisition of the satellite signals.
GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES
55
Fig. 3.1 Structure of in-phase component of the L1 signal.
2. Satellite Ephemeris Data. Ephemeris data are similar to almanac data but
enable a much more accurate determination of satellite position needed to
convert signal propagation delay into an estimate of user position. In contrast to almanac data, ephemeris data for a particular satellite are broadcast
only by that satellite, and the data are valid for only several hours.
3. Signal Timing Data. The 50-bps (bits per second) data stream includes
time tagging, which is used to establish the transmission time of specific
points on the GPS signal. This information is needed to determine the
satellite-to-user propagation delay used for ranging.
4. Ionospheric Delay Data. Ranging errors due to ionospheric effects can be
partially canceled by using estimates of ionospheric delay that are broadcast
in the data stream.
5. Satellite Health Message. The data stream also contains information regarding the current health of the satellite, so that the receiver can ignore that
satellite if it is not operating properly.
3.2.1.1 Structure of the Navigation Message The information in the navigation message has the basic frame structure shown in Fig. 3.3. A complete message
56
SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
Fig. 3.2
Structure of quadrature-phase component of the L1 signal.
consists of 25 frames, each containing 1500 bits. Each frame is subdivided into
five 300-bit subframes, and each subframe consists of 10 words of 30 bits each,
with the most significant bit (MSB) of the word transmitted first. Thus, at the
50-bps (bits per second) rate it takes 6 s to transmit a subframe and 30 s to
complete one frame. Transmission of the complete 25-frame navigation message
requires 750 s, or 12.5 min. Except for occasional updating, subframes 1, 2, and
3 are constant (i.e., repeat) with each frame at the 30-s frame repetition rate. On
the other hand, subframes 4 and 5 are each subcommutated 25 times. The 25
versions of subframes 4 and 5 are referred to as pages 1–25. Hence, except for
occasional updating, each of these pages repeats every 750 s, or 12.5 min.
A detailed description of all information contained in the navigation message
is beyond the scope of this text. Therefore, we give only an overview of the
fundamental elements. Each subframe begins with a telemetry word (TLM). The
first 8 bits of the TLM is a preamble that enables the receiver to determine when a
subframe begins. The remainder of the TLM contains parity bits and a telemetry
message that is available only to authorized users and is not a fundamental item.
The second word of each subframe is called the handover word (HOW).
3.2.1.2 Z-Count Information contained in the HOW is derived from a 29-bit
quantity called the Z-count. The Z-count is not transmitted as a single word, but
part of it is transmitted within the HOW. The Z-count counts epochs generated
by the X1 register of the P-code generator in the satellite, which occur every 1.5 s.
GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES
Fig. 3.3
57
Navigation message frame structure.
The 19 LSBs of the Z-count, called the time-of-week (TOW) count, indicate the
number of X1 epochs that have occurred since the start of the current week. The
start of the current week occurs at the X1 epoch, which occurs at approximately
midnight of Saturday night/Sunday morning. The TOW count increases from zero
at the start of the week to 403199 and then rolls over to zero again at the start
of the following week. A TOW count of zero always occurs at the beginning of
subframe 1 of the first frame (the frame containing page 1 of subcommutated
subframes 4 and 5). A truncated version of the TOW count, containing its 17
MSBs, constitutes the first 17 bits of the HOW. Multiplication of this truncated
count by 4 gives the TOW count at the start of the following subframe. Since
the receiver can use the TLM preamble to determine precisely the time at which
each subframe begins, a method for determining the time of transmission of any
part of the GPS signal is thereby established. The relationship between the HOW
counts and TOW counts is shown in Fig. 3.4.
3.2.1.3 GPS Week Number The 10 MSBs of the Z-count contain the GPS
week number (WN), which is a modulo-1024 week count. The zero state is
defined to be that week that started with the X1 epoch occurring at approximately
midnight on the night of January 5, 1980/morning of January 6, 1980. Because
WN is a modulo-1024 count, an event called the week rollover occurs every 1024
58
SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
Fig. 3.4
Relationship between HOW counts and TOW counts.
weeks (a few months short of 20 years), and GPS receivers must be designed to
accommodate it.1 . The WN is not part of the HOW but instead appears as the
first 10 bits of the third word in subframe 1.
Frame and Subframe Identification Three bits of the HOW are used to identify
which of the five subframes is being transmitted. The frame being transmitted
(corresponding to a page number from 1 to 25) can readily be identified from
the TOW count computed from the HOW of subframe 5. This TOW count is the
TOW at the start of the next frame. Since there are 20 TOW counts per frame,
the frame number of that frame is simply (TOW/20) (mod 25).
3.2.1.4 Information by Subframe In addition to the TLM and HOW, which
occur in every subframe, the following information is contained within the
remaining eight words of subframes 1–5 (only fundamental information is
described):
1. Subframe 1. The WN portion of the Z-count is part of word 3 in this
subframe. Subframe 1 also contains GPS clock correction data for the
satellite in the form of polynomial coefficients defining how the correction
varies with time. Time defined by the clocks in the satellite is commonly
called SV time (space vehicle time); the time after corrections have been
applied is called GPS time. Thus, even though individual satellites may not
have perfectly synchronized SV times, they do share a common GPS time.
1
The most recent rollover occurred at GPS time zero on August 22, 1999, with little difficulty
GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES
59
Additional information in subframe 1 includes the quantities t0c , TGD , and
IODC. The clock reference time t0c is used as a time origin to calculate
satellite clock error, the ionospheric group delay TGD is used to correct
for ionospheric propagation delay errors, and IODC (issue of date, clock)
indicates the issue number of the clock data set to alert users to changes
in clock parameters.
2. Subframes 2 and 3. These subframes contain the ephemeris data, which
are used to determine the precise satellite position and velocity required
by the navigation solution. Unlike the almanac data, these data are very
precise, are valid over a relatively short period of time (several hours), and
apply only to the satellite transmitting it. The components of the ephemeris
data are listed in Table 3.1, and the algorithm that should be used to compute satellite position in WGS84 coordinates is given in Table 3.2. The
satellite position computation using these data is implemented in the MATLAB m-file ephemeris.m on the accompanying CD. The IODE (issue of
date, ephemeris) informs users when changes in ephemeris parameters have
occurred. Each time new parameters are uploaded from the GPS control
segment, the IODE number changes.
3. Subframe 4. The 25 pages of this subframe contain the almanac for satellites with PRN (pseudorandom code) numbers 25 and higher, as well as
special messages, ionospheric correction terms, and coefficients to convert
GPS time to UTC time. There are also spare words for possible future
applications. The components of an almanac are very similar to those of
the ephemeris, and the calculation of satellite position is performed in
essentially the same way.
4. Subframe 5. The 25 pages of this subframe includes the almanac for satellites with PRN numbers from 1 to 24.
It should be noted that since each satellite transmits all 25 pages, almanac data
for all satellites are transmitted by every satellite. Unlike ephemeris data, almanac
data remain valid for long periods (months) but are much less precise. Additional
data contained in the navigation message are user range error (URE), which
estimate the range error due to errors in satellite ephemeris, timing errors, and
selective availability (SA) and flags to indicate the health status of the satellites.
3.2.2 GPS Satellite Position Calculations
3.2.2.1 Transmission of Satellite Ephemerides The interface between the GPS
space and user segments consists of two radiofrequency (RF) links, L1 and L2 .
The carriers of the L-band links are modulated by up to two bit trains, each of
which normally is a composite generated by the modulo-2 addition of a PRN
ranging code and the downlink system data. Utilizing these links, the space
vehicles of the GPS space segment should provide continuous earth coverage
for signals that provide to the user segment the ranging codes and system data
needed to accomplish the GPS navigation mission. These signals are available
60
SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
to a suitably equipped user with RF visibility to a space vehicle. Therefore, the
GPS users continuously receive navigation information from the space vehicles
in the form of modulated data bits.
The received information is computed and controlled by the control segment
and includes the satellite’s time, its clock correction and ephemeris parameters,
almanacs and health for all GPS space vehicles, and text messages. The precise
position and clock offset of the space vehicle antenna phase center in the ECEF
coordinates can be computed by receiving this information.
The ephemeris parameters describe the orbit during the interval of time (at
least 1 h) for which the parameters are transmitted. This representation model
is characterized by a set of parameters that is an extension (including drag)
to the Keplerian orbital parameters. They also describe the ephemeris for an
additional interval of time (at least 0.5 h) to allow time for the user to receive
the parameters for the new interval of time. The definitions of the parameters are
given in Table 3.1.
The age of data word (AODE) provides a confidence level in the ephemeris
representation parameters. The AODE represents the time difference (age) between
TABLE 3.1. Components of Ephemeris Data
Term
M0
n
E
√
a
0
i0
ω
˙
IDOT
Cuc
Cus
Crc
Crs
Cic
Cis
t0e
IODE
a
Unitsa
Description
Mean anomaly at reference time
Mean motion difference from computed value
Eccentricity
Square root of semimajor axis
Longitude of ascending node of orbit plane at
weekly epoch
Inclination angle at reference time
Argument of perigee
Rate of right ascension
Rate of inclination angle
Amplitude of cosine harmonic correction term
to the argument of latitude
Amplitude of sine harmonic correction term to
the argument of latitude
Amplitude of cosine harmonic correction term
to the orbit radius
Amplitude of sine harmonic correction term to
the orbit radius
Amplitude of cosine harmonic correction term
to the angle of inclination
Amplitude of sine harmonic correction term to
the angle of inclination
Ephemeris reference time
Issue of data, ephemeris
Units used in MATLAB m-file ephemeris.m are different
Semicircle
Semicircle/s
Dimensionless
m1/2
Semicircle
Semicircle
Semicircle
Semicircle/s
Semicircle/s
rad
rad
m
m
rad
rad
s
Dimensionless
GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES
61
TABLE 3.2. Algorithm for Computing Satellite Position
μ = 3.986005 × 1014 m3 /s2
˙ e = 7.292115167 × 10−5 rad/s
√
a = (!a)2
$
n0 = μ a 3
a
tk = t − t0e
n = n0 + n
Mk = M0 + ntk
Mk = Ek −e sin Ek cos Ek −e
fk = cos−1 1−e
%√ cos Ek &
1−e2 sin Ek
fk = sin−1
1−e cos Ek
e+cos fk
Ek = cos−1 1+e
cos fk
φk = fk + ω
δμk = Cμc cos 2φk + Cμs sin 2φk
δgk = Crc cos 2φk + Crs sin 2φk
δgk = Cic cos 2φk + Cis sin 2φk
μk = φk + δμk
rk = a(1 − e cos Ek ) + δrk
ik = i0 + δik + (I DOT )tk
xk
= rk cos μk
yk
= rk sin μk
˙ e )tk − ˙ e t0e
k = 0 + ( − xk = xk
cos k − yk
cos ik sin k
yk = xk
sin k + yk
cos ik cos k
zk = yk
sin ik
WGS84 value of earth’s universal gravitational
parameter
WGS84 value of earth’s rotation rate
Semimajor axis
Computed mean motion, rad/s
Time from ephemeris reference epoch
Corrected mean motion
Mean anomaly
Kepler’s equation for eccentric anomaly
True anomaly from cosine
True anomaly from sine
Eccentric anomaly from cosine
Argument of latitude
Second-harmonic correction to argument of latitude
Second-harmonic correction to radius
Second-harmonic correction to inclination
Corrected argument of latitude
Corrected radius
Corrected inclination
X coordinate in orbit plane
Y coordinate in orbit plane
Corrected longitude of ascending node
ECEF X coordinate
ECEF Y coordinate
ECEF Z coordinate
a
t is in GPS system time at time of transmission, i.e., GPS time corrected for transit time (range/speed
of light). Furthermore, tk shall be the actual total time difference between the time t and the time
epoch t0e and must account for beginning or end of week crossovers. Thus, if tk is greater than
302,400 s, subtract 604800 s from tk ; if tk is less than −302400 s, add 604,800 s to tk
the reference time (t0e ) and the time of the last measurement update (tL ) used to
estimate the representation parameters.
The ECEF coordinates for the phase center of the satellite’s antennas can be
calculated using a variation of the equations shown in Table 3.2. In this table, time
t is the GPS system time at the time of transmission, that is, GPS time corrected
for transit time (range/speed of light). Further, tk is the actual total time difference
between time t and epoch time t0e and must account for beginning- or end-ofweek crossovers. Thus, if tk is greater than 302400 s, subtract 604800 s from tk ;
if tk is less than −302400 ss, add 604800 s to tk .
3.2.2.2 Ephemeris Data Transmitted The ephemeris parameters and algorithms used for computing satellite positions are given in Tables 3.1 and 3.2,
62
SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
Fig. 3.5
Geometric relationship between true anomaly f and eccentric anomaly E.
respectively. The problem of determining satellite position from these data and
equations is called the Kepler problem.
3.2.2.3 True, Eccentric, and Mean Anomaly Orbit phase variables used for
determining the position of a satellite in its orbit are illustrated in Fig. 3.5. The
variable f in the figure is called true anomaly in orbit mechanics. The hardest part
of the Kepler problem, is the problem of determining true anomaly as a function
of time. This problem was eventually solved by introducing two intermediate
“anomaly” variables:
E, the eccentric anomaly, defined as a geometric function of true anomaly,
as shown in Fig. 3.5. Eccentric anomaly E is defined by projecting the
satellite position on the elliptical orbit out perpendicular to the semimajor
axis a and onto the circumscribed circle. Eccentric anomaly is then defined
as the central angle to this projection point on the circle, as shown in the
figure. The shaded area represents the area swept out by the radius from
the earth to the satellite as the satellite moves from its perigee. Kepler
had determined that this area grows linearly with time, and he used this
relationship to derive his equation for mean anomaly.
M, the mean anomaly, defined as a linear function of time:
2π t − tperigee
M(t) =
(in radians),
(3.3)
Tperiod
GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES
63
where t is the time in seconds at which true anomaly is to be determined;
tperigee is the time at which the satellite was at its perigee, closest to the
earth; and Tperiod is the orbit period in seconds.
From this equation, calculating mean anomaly as a function of time is relatively easy, but the solution for true anomaly as a function of eccentric
anomaly is much more difficult.
For the low eccentricities of GPS orbits, the numerical values of the true,
eccentric, and mean anomalies are quite close together. However, the precision
required in calculating true anomaly will require that they be treated as separate
variables.
3.2.2.4 Kepler’s Equation The equation
Mk = Ek − e sin Ek ,
(3.4)
in Table 3.2 is called Kepler’s equation. It relates the eccentric anomaly Ek of the
kth satellite to its mean anomaly Mk and the orbit eccentricity e. This equation
is the most difficult of all the equations in Table 3.2 to solve for Ek as a function
of Mk .
3.2.2.5 Solution of Kepler’s Equation Kepler’s equation (Eq. 3.4) includes a
transcendental function of eccentric anomaly Ek . It is impractical to solve for
Ek in any way except by approximation. Standard practice is to solve the true
anomaly equation iteratively for Ek , using the second-order Newton–Raphson
method to solve
def
εk = Mk + Ek − e sin Ek
= 0,
(3.5)
(3.6)
and then use the resulting value of Ek to calculate true anomaly. It starts by
assigning an initial guess Ek[0] for Ek (the mean anomaly will do), and then forming successively better estimates Ek[n+1] by the second-order Newton–Raphson
formula
⎧
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
(n)
⎪
⎪
⎨
⎬
εk Ek
[n+1]
[n]
Ek
= Ek −
(3.7)
⎡ ∂2ε ⎤ .
(n)
⎪
⎪
k
ε
[E
]
⎪
⎪
[n]
k
k
2
⎪ ∂εk
⎪
⎪
⎣ ∂Ek ∂εEk=Ek
⎦⎪
⎪
⎪
⎪
⎪
⎩ ∂Ek Ek= Ek[n] −
⎭
2 k
[n]
∂Ek E E
k= k
The iteration of Eq. 3.7 can stop when the difference in the estimated Ek is
sufficiently small, say
[n+1]
− Ek[n] < 10−6 .
Ek
64
SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
3.2.2.6 Other Calculation Considerations The satellite’s antenna phase center position is very sensitive to small perturbations
in most ephemeris parameters.
√
The sensitivity of position to the parameters a, Crc , and Crs is about 1 m/m.
This sensitivity to angular parameters is on the order of 108 m/semicircle and to
the angular rate parameters on the order of 1012 m/semicircle/s. Because of this
extreme sensitivity to angular perturbations, the required value of π (a mathematical constant, the ratio of a circle’s circumference to its diameter) used in the
curve fit is given as
π = 3.1415926535898.
The user must correct the time received from the space vehicle in seconds with
the equation
t = tsv − tsv ,
(3.8)
where t is GPS system time in seconds, tsv is the effective SV PRN code phase
time at message transmission time in seconds, and tsv is the SV PRN code
phase time offset in seconds. The SV PRN code phase offset is given by
tsv = af 0 + af 1 (t − t0c ) + af 2 (t − t0c )2 + tr ,
(3.9)
where af 0 , af 1 , af 2 are polynomial coefficients given in the ephemeris data
file; t0c is the clock data reference time in seconds; and tr is the relativistic
correction term in seconds, given by
√
tr = F e a sin Ek .
(3.10)
In Eq. 3.10, F is a constant whose value is given as
√
−2 μ
F =
c2
√
= −4.442807633 × 10−10 [s/ m],
(3.11)
(3.12)
where the speed of light c = 2.99792458 × 108 m/s. Note that Eqs. 3.8 and 3.9
are coupled. While the coefficients af 0 , af 1 , and af 2 are generated by using GPS
time as indicated in Eq. 3.9, sensitivity of tsv to t is negligible. This negligible
sensitivity will allow the user to approximate t by tsv in Eq. 3.9. The value of
t must account for beginning- or end-of-week crossovers. Thus, if the quantity
t − t0c is greater than 302,400 s, subtract 604,800 s from t; if the quantity t − t0c
is less than −302, 400 s, add 604,800 s to t.
By using the value of the ephemeris parameters for satellite PRN 2 in the
set of equations in Table 3.1 and Eqs. 3.8–3.12, we can calculate the space
vehicle time offset and the ECEF coordinates of the satellite position [46]. The
MATLAB m-file (ephemeris.m) on the accompanying CD calculates satellite
position for one set of ephemeris data and one time. Other programs calculate
satellite positions for a range of time. (See Appendix A.)
GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES
65
3.2.3 C/A-Code and Its Properties
The C/A-code has the following functions:
1. To enable accurate range measurements and resistance to errors caused by
multipath. To establish the position of a user to within 10–100 m, accurate
user-to-satellite range estimates are needed. The estimates are made from
measurements of signal propagation delay from the satellite to the user. To
achieve the required accuracy in measuring signal delay, the GPS carrier
must be modulated by a waveform having a relatively large bandwidth.
The needed bandwidth is provided by the C/A-code modulation, which
also permits the receiver to use correlation processing to effectively combat
measurement errors due to thermal noise. Because the C/A-code causes the
bandwidth of the signal to be much greater than that needed to convey the
50-bps (bits per second) data stream, the resulting signal is called a spreadspectrum signal. Using the C/A-code to increase the signal bandwidth also
reduces errors in measuring signal delay caused by multipath (the arrival
of the signal via multiple paths such as reflections from objects near the
receiver antenna) since the ability to separate the direct path signal from
the reflected signal improves as the signal bandwidth is made larger.
2. To permit simultaneous range measurement from several satellites. The use
of a distinct C/A-code for each satellite permits all satellites to use the same
L1 and L2 frequencies without interfering with each other. This is possible
because the signal from an individual satellite can be isolated by correlating
it with a replica of its C/A-code in the receiver. This causes the C/A-code
modulation from that satellite to be removed so that the signal contains only
the 50-bps (bits per second) data and is therefore narrowband. This process
is called despreading of the signal. However, the correlation process does
not cause the signals from other satellites to become narrowband, because
the codes from different satellites are orthogonal. Therefore the interfering
signals can be rejected by passing the desired despread signal through a
narrowband filter, a bandwidth-sharing process called code-division multiplexing (CDM) or code-division multiple access (CDMA).
3. To provide protection from jamming. The C/A-code also provides a measure
of protection from intentional or unintentional jamming of the received
signal by another man-made signal. The correlation process that despreads
the desired signal has the property of spreading any other signal. Therefore,
the signal power of any interfering signal, even if it is narrowband, will be
spread over a large frequency band, and only that portion of the power lying
in the narrowband filter will compete with the desired signal. The C/A-code
provides about 20–30 dB of improvement in resistance to jamming from
narrowband signals.
We next detail important properties of the C/A-code.
66
SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
3.2.3.1 Temporal Structure Each satellite has a unique C/A-code, but all the
codes consist of a repeating sequence of 1023 chips occurring at a rate of
1.023 MHz with a period of 1 ms, as previously shown in Fig. 3.1. The leading edge of a specific chip in the sequence, called the C/A-code epoch, defines
the beginning of a new period. Each chip is either positive or negative with the
same magnitude. The polarities of the 1023 chips appear to be randomly distributed but are in fact generated by a deterministic algorithm implemented by
shift registers. The algorithm produces maximal-length Gold codes, which have
the property of low cross-correlation between different codes (orthogonality) as
well as reasonably small autocorrelation sidelobes.
3.2.3.2 Autocorrelation Function The autocovariance function of the C/Acode is
*
1 T
y (t) =
c (t) c (t − t) dt,
(3.13)
T 0
where c(t) is the idealized C/A-code waveform (with chip values of ±1), τ is
the relative delay measured in seconds, and T is the code period (1 ms). The
autocorrelation function is periodic in τ with a period of 1 ms. A single period is
plotted in Fig. 3.6, which is basically a triangle two chips wide at its base with a
peak located at τ = 0 [in reality ψ(τ ) contains small-sidelobe structures outside
the triangular region, but these are of little consequence].
The C/A-code autocorrelation function plays a substantial role in GPS receivers, inasmuch as it forms the basis for code tracking and accurate user-to-satellite
range measurement. In fact, the receiver continually computes values of this
function in which c(t) in the integral in Eq. 3.13 is the signal code waveform
and c(t − τ ) is an identical reference waveform (except for the relative delay τ )
Fig. 3.6 Autocorrelation functions of C/A- and P(Y)-codes.
GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES
67
generated in the receiver. Special hardware and software enable the receiver to
adjust the reference waveform delay so that the value of τ is zero, thus enabling
determination of the time of arrival of the received signal.
3.2.3.3 Power Spectrum The power spectrum (f ) of the C/A-code describes
how the power in the code is distributed in the frequency domain. It can be
defined in terms of either a Fourier series expansion of the code waveform or,
equivalently, the code autocorrelation function. Using the latter, we have
1
T →∞ 2T
(f ) = lim
*
T
−T
ψ (τ ) e−j 2πf τ dτ.
(3.14)
A plot of (f ) is shown as a smooth curve in Fig. 3.7; however, in reality (f )
consists of spectral lines with 1-kHz spacing due to the 1-ms periodic structure
of ψ(τ ). The power spectrum (f ) has a characteristic sin2 (x)/x 2 shape with
first nulls located 1.023 MHz from the central peak. Approximately 90% of the
signal power is located between these two nulls, but the smaller portion lying
outside the nulls is very important for accurate ranging. Also shown in the figure
for comparative purposes is a typical noise power spectral density found in a
GPS receiver after frequency conversion of the signal to baseband (i.e., with
carrier removed). It can be seen that the presence of the C/A-code causes the
Fig. 3.7
Power spectra of C/A- and P(Y)-codes.
68
SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
entire signal to lie well below the noise level, because the signal power has been
spread over a wide frequency range (approximately ±1 MHz).
3.2.3.4 Despreading of the Signal Spectrum The mathematical model of the
signal modulated by the C/A-code is
s(t) =
2PI d(t)c(t) cos(ωt + θ ),
(3.15)
where PI is the carrier power, d(t) is the 50-bps (bits per second) data modulation, c(t) is the C/A-code waveform, ω is the L1 carrier frequency in radians per
second, and θ is the carrier phase shift in radians. When this signal is frequencyshifted to baseband and tracked with a phase-lock loop, the carrier is removed
and only the data modulation and the C/A-code modulation remain. The resulting
signal, which in normalized form is
s(t) = d(t)c(t),
(3.16)
has a power spectrum similar to that of the C/A-code in Fig. 3.7. As previously
mentioned, the signal in this form has a power spectrum lying below the receiver
noise level, making it inaccessible. However, if the signal is multiplied by a
replica of c(t) in exact alignment with it, the result is
s(t)c(t) = d(t)c(t)c(t) = d(t)c2 (t) = d(t),
(3.17)
where the last equality arises from the fact that the values of the ideal C/A-code
waveform are ±1 (in reality the received C/A-code waveform is not ideal, due
to bandlimiting in the receiver; however, the effects are usually minor). This
procedure, called code despreading, removes the C/A-code modulation from the
signal. The resulting signal has a two-sided spectral width of approximately
100 Hz due to the 50-bps (bits per second) data modulation. From the above
equation it can be seen that the total signal power has not been changed in
this process, but it now is contained in a much narrower bandwidth. Thus the
magnitude of the power spectrum is greatly increased, as indicated in Fig. 3.8.
In fact, it now exceeds that of the noise, and the signal can be recovered by
passing it through a small-bandwidth filter (signal recovery filter) to remove the
wideband noise, as shown in the figure.
3.2.3.5 Role of Despreading in Interference Suppression At the same time
that the spectrum of the desired GPS signal is narrowed by the despreading
process, any interfering signal that is not modulated by the C/A-code will instead
have its spectrum spread to a width of at least 2 MHz, so that only a small
portion of the interfering power can pass through the signal recovery filter. The
amount of interference suppression gained by using the C/A-code depends on
the bandwidth of the recovery filter, the bandwidth of the interfering signal, and
the bandwidth of the C/A-code. For narrowband interferors whose signal can be
GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES
69
Fig. 3.8 Despreading of the C/A-code.
modeled by a nearly sinusoidal waveform and a signal recovery filter bandwidth
of 1000 Hz or more, the amount of interference suppression in decibels is given
approximately by
&
%
Wc
dB,
(3.18)
η = 10 log
Wf
where Wc and Wf are respectively the bandwidths of the C/A-code (2.046 MHz)
and the signal recovery filter. If Wf = 2000 Hz, about 30 dB of suppression
can be obtained for narrowband interferors. When the signal recovery filter has
a bandwidth smaller than 1000 Hz, the situation is more complicated, since the
despread interfering sinusoid will have discrete spectral components with a 1000Hz spacing. As the bandwidth of the interfering signal increases, the C/A-code
despreading process provides a decreasing amount of interference suppression.
For interferors having a bandwidth greater than that of the signal recovery filter,
the amount of suppression in decibels provided by the C/A-code is approximately
%
&
WI + W c
η = 10 log
dB,
(3.19)
WI
where WI is the bandwidth of the interferor. When WI >> Wc , the C/A-code
provides essentially no interference suppression at all compared to the use of an
unspread carrier.
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
3.2.3.6 Code-Division Multiplexing Property The C/A-codes from different
satellites are orthogonal, which means that for any two codes c1 (t) and c2 (t)
from different satellites, the cross-covariance
1
T
*
T
0
c1 (t) c2 (t − τ ) dt ∼
= 0 for all τ.
(3.20)
Thus, when a selected satellite signal is despread using a replica of its code, the
signals from other satellites look like wideband interferors that are below the noise
level. This permits a GPS receiver to extract a multiplicity of individual satellite
signals and process them individually, even though all signals are transmitted at
the same frequency. This process is called code-division multiplexing (CDM).
3.2.4 P-Code and Its Properties
The P-code, which is used primarily for military applications, has the following
functions:
1. Increased Jamming Protection. Because the bandwidth of the P-code is
10 times greater than that of the C/A-code, it offers approximately 10 dB
more protection from narrowband interference. In military applications the
interference is likely to be a deliberate attempt to jam (render useless) the
received GPS signal.
2. Provision for Antispoofing. In addition to jamming, another military tactic
that an enemy can employ is to radiate a signal that appears to be a GPS
signal (spoofing), but in reality is designed to confuse the GPS receiver.
This is prevented by encrypting the P-code. The would-be spoofer cannot
know the encryption process and cannot make the contending signal look
like a properly encrypted signal. Thus the receiver can reject the false signal
and decrypt the desired one.
3. Denial of P-Code Use. The structure of the P-code is published in the open
literature, so than anyone may generate it as a reference code for despreading the signal and making range measurements. However, encryption of
the P-code by the military will deny its use by unauthorized parties.
4. Increased Code Range Measurement Accuracy. All other parameters being
equal, accuracy in range measurement improves as the signal bandwidth
increases. Thus, the P-code provides improved range measurement accuracy
as compared to the C/A-code. Simultaneous range measurements using both
codes is even better. Because of its increased bandwidth, the P-code is also
more resistant to range errors caused by multipath.
3.2.4.1 P-Code Characteristics Unlike the C/A-code, the P-code modulates
both the L1 and L2 carriers. Its chipping rate is 10.23 MHz, which is precisely 10
times the C/A rate, and it has a period of one week. It is transmitted synchronously
with the C/A-code in the sense that each chip transition of the C/A-code always
GPS SIGNAL COMPONENTS, PURPOSES, AND PROPERTIES
71
corresponds to a chip transition in the P-code. Like the C/A-code, the P-code
autocorrelation function has a triangular central peak centered at τ = 0, but with
one-tenth the base width, as shown in Fig. 3.6. The power spectrum also has a
sin2 (x)/x 2 characteristic, but with 10 times the bandwidth, as indicated in Fig. 3.6.
Because the period of the P-code is so long, the power spectrum may be regarded
as continuous for practical purposes. Each satellite broadcasts a unique P-code.
The technique used to generate it is similar to that of the C/A-code, but somewhat
more complicated, and will not be covered in this book.
3.2.4.2 Y-Code The encrypted form of the P-code used for antispoofing and
denial of the P-code to unauthorized users is called the Y-code. The Y-code
is formed by multiplying the P-code by an encrypting code called the W-code.
The W-code is a random-looking sequence of chips that occur at a 511.5-kHz
rate. Thus there are 20 P-code chips for every W-code chip. Since both the
P-code and the W-code have chip values of ±1, the resulting Y-code has the
same appearance as the P-code; that is, it also has a 10.23-MHz chipping rate.
However, the Y-code cannot be despread by a receiver replica P-code unless it is
decrypted. Decryption consists of multiplying the Y-code by a receiver-generated
replica of the W-code that is made available only to authorized users. Since the
encrypting W-code is also not known by the creators of spoofing signals, it is
easy to verify that such signals are not legitimate.
3.2.5 L1 and L2 Carriers
The L1 (or L2 ) carrier is used for the following purposes:
1. To provide very accurate range measurements for precision applications
using carrier phase.
2. To provide accurate Doppler measurements. The phase rate of the received
carrier can be used for accurate determination of user velocity. The integrated Doppler, which can be obtained by counting the cycles of the
received carrier, is often used as a precise delta range observable that
can materially aid the performance of code tracking loops. The integrated
Doppler history is also used as part of the carrier phase ambiguity resolution
process.
3.2.5.1 Dual-Frequency Operation The use of both the L1 and L2 frequencies
provides the following benefits:
1. Provides accurate measurement of ionospheric signal delay. A major source
of ranging error is caused by changes in both the phase velocity and group
velocity of the signal as it passes through the ionosphere. Range errors of
10–20 m are commonplace and sometimes much larger. Because the delay
induced by the ionosphere is known to be inversely proportional to the
square of frequency, ionospheric range error can be estimated accurately
72
SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
by comparing the times of arrival of the L1 and L2 signals. Details on the
calculations appear in Chapter 5.
2. Facilitates carrier phase ambiguity resolution. In high-accuracy GPS differential positioning, the range estimates using carrier phase measurements
are precise but highly ambiguous due to the periodic structure of the carrier. The ambiguity is more easily resolved (by various methods) as the
carrier frequency decreases. By using L1 and L2 carrier frequencies, the
ambiguity resolution can be based on their frequency difference (1575.42–
1227.6 MHz), which is smaller than either carrier frequency alone, and
hence will result in better ambiguity resolution performance.
3. Provides system redundancy (primarily for the military user).
3.3 SIGNAL POWER LEVELS
3.3.1 Transmitted Power Levels
The L1 C/A-code signal is transmitted at a minimum level of 478.63 W
(26.8 dBW) effective isotropic radiated power (EIRP), which means that the
minimum received power is the same as that that would be obtained if the satellite radiated 478.63 W from an isotropic antenna. This effective power level is
reached by radiating a smaller total power in a beam approximately 30◦ wide
toward the earth. The radiated power level was chosen to provide a signal-tonoise ratio sufficient for tracking of the signal by a receiver on the earth with
an unobstructed view of the satellite. However, the chosen power has been criticized as being inadequate in light of the need to operate GPS receivers under
less desirable conditions, such as in heavy vegetation or in urban canyons where
considerable signal attenuation often occurs. For this reason, future satellites may
have higher transmitted power.
3.3.2 Free-Space Loss Factor
As the signal propagates toward the earth, it loses power density due to spherical
spreading. The loss is accounted for by a quantity called the free-space loss factor
(FSLF), given by
%
&
λ 2
FSLF =
.
(3.21)
4πR
The FSLF is the fractional power density at a distance R meters from the transmitting antenna compared to a value normalized to unity at the distance λ/4π
meters from the antenna phase center. Using R = 2 × 107 and λ = 0.19 m at the
L1 frequency, the FSLF is about 5.7 × 10−19 , or −182.4 dB.
3.3.3 Atmospheric Loss Factor
An additional atmospheric loss factor (ALF) of about 2.0 dB occurs as the signal
becomes attenuated by the atmosphere. If the receiving antenna is assumed to be
73
SIGNAL ACQUISITION AND TRACKING
TABLE 3.3. Calculation of Minimum Received Signal Power
Minimum transmitted signal
power (EIRP)
Free-space loss factor (FSLF)
Atmospheric loss factor (ALF)
Receiver antenna gain relative
to isotropic (RAG)
Minimum received signal power
(EIRP − FSLF − ALF + RAG)
a
26.8a
dBW
−182.4
−2.0
3.0
dB
dB
dB
−154.6
dBW
Including antenna gain
isotropic, the received signal power is EIRP − FSLF − ALF = 26.8 − 182.4 −
2.0 = −157.6dBW.
3.3.4 Antenna Gain and Minimum Received Signal Power
Since a typical GPS antenna with right-hand circular polarization and a hemispherical pattern has about 3.0 dB of gain relative to an isotropic antenna, the
minimum received signal power for such an antenna is about 3.0 dB larger. These
results are summarized in Table 3.3.
3.4 SIGNAL ACQUISITION AND TRACKING
When a GPS receiver is turned on, a sequence of operations must ensue before
information in a GPS signal can be accessed and used to provide a navigation
solution. In the order of execution, these operations are as follows:
1.
2.
3.
4.
5.
6.
7.
8.
Determine which satellites are visible to the antenna.
Determine the approximate Doppler of each visible satellite.
Search for the signal both in frequency and C/A-code phase.
Detect the presence of a signal and confirm detection.
Lock onto and track the C/A-code.
Lock onto and track the carrier.
Perform data bit synchronization.
Demodulate the 50-bps (bits per second) navigation data.
3.4.1 Determination of Visible Satellites
In many GPS receiver applications it is desirable to minimize the time from
receiver turnon until the first navigation solution is obtained. This time interval
is commonly called time to first fix (TTFF). Depending on receiver characteristics,
the TTFF might range from 30 s to several minutes. An important consideration
in minimizing the TTFF is to avoid a fruitless search for those satellite signals
that are blocked by the earth, that is, below the horizon. A receiver can restrict
74
SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
its search to only those satellites that are visible if it knows its approximate location (within several hundred miles) and approximate time (within approximately
10 min) and has satellite almanac data obtained within the last several months.
The approximate location can be manually entered by the user or it can be the
position obtained by GPS when the receiver was last in operation. The approximate time can also be entered manually, but most receivers have a sufficiently
accurate real-time clock that operates continuously, even when the receiver is
off.
Using the approximate time, approximate position, and almanac data, the
receiver calculates the elevation angle of each satellite and identifies the visible satellites as those whose elevation angle is greater than a specified value,
called the mask angle, which has typical values of 5˚ to 15˚. At elevation angles
below the mask angle, tropospheric attenuation and delays tend to make the
signals unreliable.
Most receivers automatically update the almanac data when in use, but if the
receiver is just “out of the box” or has not been used for many months, it will
need to search “blind” for a satellite signal to collect the needed almanac. In
this case the receiver will not know which satellites are visible, so it simply
must work its way down a predetermined list of satellites until a signal is found.
Although such a “blind” search may take an appreciable length of time, it is
infrequently needed.
3.4.2 Signal Doppler Estimation
The TTFF can be further reduced if the approximate Doppler shifts of the visible
satellite signals are known. This permits the receiver to establish a frequency
search pattern in which the most likely frequencies of reception are searched
first. The expected Doppler shifts can be calculated from knowledge of approximate position, approximate time, and valid almanac data. The greatest benefit is
obtained if the receiver has a reasonably accurate clock reference oscillator.
However, once the first satellite signal is found, a fairly good estimate of
receiver clock frequency error can be determined by comparing the predicted
Doppler shift with the measured Doppler shift. This error can then be subtracted
out while searching in frequency for the remaining satellites, thus significantly
reducing the range of frequencies that need to be searched.
3.4.3 Search for Signal in Frequency and C/A-Code Phase
Why is a signal search necessary? Since GPS signals are radio signals, one might
assume that they could be received simply by setting a dial to a particular frequency, as is done with AM and FM broadcast band receivers. Unfortunately,
this is not the case.
1. GPS signals are spread-spectrum signals in which the C/A- or P-codes
spread the total signal power over a wide bandwidth. The signals are therefore virtually undetectable unless they are despread with a replica code in the
SIGNAL ACQUISITION AND TRACKING
75
receiver that is precisely aligned with the received code. Since the signal cannot be detected until alignment has been achieved, a search over the possible
alignment positions (code search) is required.
2. A relatively narrow postdespreading bandwidth (perhaps 100–1000 Hz) is
required to raise the signal-to-noise ratio to detectable and/or usable levels. However, because of the high carrier frequencies and large satellite velocities used by
GPS, the received signals can have large Doppler shifts (as much as ±5 kHz),
which may vary rapidly (by as much as 1 Hz/s). The observed Doppler shift also
varies with location on earth, so that the received frequency will generally be
unknown a priori. Furthermore, the frequency error in typical receiver reference
oscillators will typically cause several kilohertz or more of frequency uncertainty
at L-band. Thus, in addition to the code search, there is also the need for a search
in frequency.
Therefore, a GPS receiver must conduct a two-dimensional search in order to
find each satellite signal, where the dimensions are C/A-code delay and carrier
frequency. A search must be conducted across the full delay range of the C/Acode for each frequency searched. A generic method for conducting the search is
illustrated in Fig. 3.9, in which the received waveform is multiplied by delayed
replicas of the C/A-code, translated by various frequencies, and then passed
through a baseband correlator containing a lowpass filter which has a relatively
small bandwidth (perhaps 100–1000 Hz). The output energy of the detection
filter serves as a signal detection statistic and will be significant only if both the
selected code delay and frequency translation match that of the signal. When the
Fig. 3.9
Signal search method.
76
SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
energy exceeds a predetermined threshold β, a tentative decision is made that
a signal is being received, subject to later confirmation. The value chosen for
the threshold β is a compromise between the conflicting goals of maximizing
the probability PD of detecting the signal when it is actually present at a given
Doppler and code delay and minimizing the probability PFA of false alarm when
it is not.
3.4.3.1 Searching in Code Delay For each frequency searched, the receiver
generates the same PRN code as that of the satellite and moves the delay of this
code in discrete steps (typically 0.5 chip) until approximate alignment with the
received code (and also a match in Doppler) is indicated when the correlator
output energy exceeds threshold β. A step size of 0.5 code chip, which is used
by many GPS receivers, is an acceptable compromise between the conflicting
requirements of search speed (enhanced by a larger step size) and guaranteeing
a code delay that will be located near the peak value of the code correlation
function (enhanced by a smaller step size). For a search conducted in 1-chip
increments, the best situation occurs when one of the delay positions is at the
correlation function peak, and the worst one occurs when there are two delay
positions straddling the peak, as indicated in Fig. 3.10. In the latter case, the
effective SNR is reduced by as much as 6 dB. However, the effect is ameliorated
because, instead of only one delay position with substantial correlation, there are
two that can be tested for the presence of signal.
An important parameter in the code search is the dwell time used for each
code delay position, since it influences both the search speed and the detection/falsealarm performance. The dwell time should be an integral multiple of
Fig. 3.10 Effect of 1-chip step size in code search.
SIGNAL ACQUISITION AND TRACKING
77
1 ms to assure that the correct correlation function, using the full range of 1023
code states, is obtained. Satisfactory performance is obtained with dwell times
from 1 to 4 ms in most GPS receivers, but longer dwell times are sometimes
used to increase detection capability in weak-signal environments. However, if
the dwell time for the search is a substantial fraction of 20 ms (the duration of
one data bit), it becomes increasingly probable that a bit transition of the 50-Hz
data modulation will destroy the coherent processing of the correlator during the
search and lead to a missed detection. This imposes a practical limit for a search
using coherent detection.
The simplest type of code search uses a fixed dwell time, a single detection
threshold value β, and a simple yes/no binary decision as to the presence of
a signal. Many receivers achieve considerable improvement in search speed by
using a sequential detection technique in which the overall dwell time is conditioned on a ternary decision involving an upper and a lower detection threshold.
Details on this approach can be found in the treatise by Wald [199].
3.4.3.2 Searching in Frequency The range of frequency uncertainty that must
be searched is a function of the accuracy of the receiver reference oscillator, how
well the approximate user position is known, and the accuracy of the receiver’s
built-in real-time clock. The first step in the search is to use stored almanac data to
obtain an estimate of the Doppler shift of the satellite signal. An interval [f1 ,f2 ]
of frequencies to be searched is then established. The center of the interval is
located at fc + fd , where fc is the L1 (or L2 ) carrier frequency and fd is the
estimated carrier Doppler shift. The width of the search interval is made large
enough to account for worst-case errors in the receiver reference oscillator, in
the estimate of user position, and in the real-time clock. A typical range for the
frequency search interval is fc + fd ± 5 kHz.
The frequency search is conducted in N discrete frequency steps that cover
the entire search interval. The value of N is (f2 − f1 )/f , where f is the
spacing between adjacent frequencies (bin width). The bin width is determined
by the effective bandwidth of the correlator. For the coherent processing used in
many GPS receivers, the frequency bin width is approximately the reciprocal of
the search dwell time. Thus, typical values of f are 250–1000 Hz. Assuming
a ±5-kHz frequency search range, the number N of frequency steps to cover the
entire search interval would typically be 10–40.
3.4.3.3 Frequency Search Strategy Because the received signal frequency
is more likely to be near to—rather than far from—the Doppler estimate, the
expected time to detect the signal can be minimized by starting the search at
the estimated frequency and expanding in an outward direction by alternately
selecting frequencies above and below the estimate, as indicated in Fig. 3.11. On
the other hand, the unknown code delay of the signal can be considered to be
uniformly distributed over its range so that each delay value is equally likely.
Thus, the delays used in the code search can simply sequence from 0 to 1023.5
chips in 0.5-chip increments.
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
Fig. 3.11 Frequency search strategy.
3.4.3.4 Sequential versus Parallel Search Methods Almost all current GPS
receivers are multichannel units in which each channel is assigned a satellite
and processing in the channels is carried out simultaneously. Thus, simultaneous
searches can be made for all usable satellites when the receiver is turned on.
Because the search in each channel consists of sequencing through all possible
frequency and code delay steps, it is called a sequential search. In this case, the
expected time required to acquire as many as eight satellites is typically 30–100 s,
depending on the specific search parameters used.
Certain applications (mostly military) demand that the satellites be acquired
much more rapidly (perhaps within a few seconds). This can be accomplished
by using a parallel search technique in which extra hardware permits many
frequencies and code delays to be searched at the same time. However, this
approach is seldom used in commercial receivers because of its high cost.
3.4.4 Signal Detection and Confirmation
As previously mentioned, there is a tradeoff between the probability of detection
PD and false alarm PFA . As the detection threshold β is decreased, PD increases
but PFA also increases, as illustrated in Fig. 3.12. Thus, the challenge in receiver
design is to achieve a sufficiently large PD so that a signal will not be missed
but at the same time keep PFA small enough to avoid difficulties with false
detections. When a false detection occurs, the receiver will try to lock onto and
track a nonexistent signal. By the time the failure to track becomes evident, the
receiver will have to initiate a completely new search for the signal. On the other
SIGNAL ACQUISITION AND TRACKING
Fig. 3.12
79
Illustration of tradeoff between PD and PFA .
hand, when a detection failure occurs, the receiver will waste time continuing to
search remaining search cells that contain no signal, after which a new search
must be initiated.
3.4.4.1 Detection Confirmation One way to achieve both a large PD and a
small PFA is to increase the dwell time so that the relative noise component of the
detection statistic is reduced. However, to reliably acquire weak GPS signals, the
required dwell time may result in unacceptably slow search speed. An effective
way around this problem is to use some form of detection confirmation.
To illustrate the detection confirmation concept, suppose that to obtain the
detection probability PD = 0.95 with a typical medium-strength GPS signal, we
obtain the false-alarm probability PFA = 10−3 . (These are typical values for a
fixed search dwell time of 3 ms.) This means that on the average, there will be
one false detection in every 1000 frequency/code cells searched. A typical twodimensional GPS search region might contain as many as 40 frequency bins and
2046 code delay positions, for a total of 40 × 2046 = 81, 840 such cells. Thus
we could expect about 82 false detections in the full search region. Given the
implications of a false detection discussed previously, this is clearly unacceptable.
However, suppose that we change the rules for what happens when a detection
(false or otherwise) occurs by performing a confirmation of detection before turning the signal over to the tracking loops. Because a false detection takes place
only once in 1000 search cells, it is possible to use a much longer dwell (or
a sequence of repeated dwells) for purposes of confirmation without markedly
80
SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
Fig. 3.13 Global and confirmation search regions.
increasing the overall search speed, yet the confirmation process will have an
extremely high probability of being correct. In the event that confirmation indicates no signal, the search can continue without interruption by the large time
delay inherent in detecting the failure to track. In addition to using longer dwell
times, the confirmation process can also perform a local search in which the
frequency/code cell size is smaller than that of the main, or global, search, thus
providing a more accurate estimate of signal frequency and code phase when a
detection is confirmed. Figure 3.13 depicts this scheme. The global search uses
a detection threshold β that provides a high PD and a moderate value of PFA .
Whenever the detection statistic exceeds β at a frequency/delay cell, a confirmation search is performed in a local region surrounding that cell. The local
region is subdivided into smaller cells to obtain better frequency delay resolution,
and a longer dwell time is used in forming the detection statistic . The longer
dwell time makes it possible to use a value of β that provides both a high PD
and a low PFA .
3.4.4.2 Adaptive Signal Searches Some GPS receivers use a simple adaptive
search in which shorter dwell times are first used to permit rapid acquisition
of moderate to strong signals. Whenever a search for a particular satellite is
unsuccessful, it is likely that the signal from that satellite is relatively weak, so
the receiver increases the dwell time and starts a new search that is slower but
has better performance in acquiring weak signals.
SIGNAL ACQUISITION AND TRACKING
81
3.4.4.3 Coordination of Frequency Tuning and Code Chipping Rate As the
receiver is tuned in frequency during search, it is advantageous to precess the
chipping rate of the receiver generated code so that it is in accordance with
the Doppler shift under consideration. The relationship between Doppler shift
and the precession rate of the C/A-code is given by p(t) = fd /1540, where
p(t) is the code precession rate in chips per second, fd is the Doppler shift in
hertz, and a positive precession rate is interpreted as an increase in the chipping
rate. Precession is not required while searching because the dwell times are so
short. However, when detection of the signal occurs, it is important to match the
incoming and reference code rates during the longer time required for detection
confirmation and/or initiation of code tracking to take place.
3.4.5 Code Tracking Loop
At the time of detection confirmation the receiver-generated reference C/A-code
will be in approximate alignment with that of the signal (usually within 0.5 chip),
and the reference code chipping rate will be approximately that of the signal.
Additionally, the frequency of the signal will be known to within the frequency
bin width f . However, unless further measures are taken, the residual Doppler
on the signal will eventually cause the received and reference codes to drift out
of alignment and the signal frequency to drift outside the frequency bit at which
detection occurred. If the code alignment error exceeds one chip in magnitude,
the incoming signal will no longer despread and will disappear below the noise
level. The signal will also disappear if it drifts outside the detection frequency bin.
Thus there is the need to continually adjust the timing of the reference code so
that it maintains accurate alignment with the received code, a process called code
tracking. The process of maintaining accurate tuning to the signal carrier, called
carrier tracking, is also necessary and will be discussed in following sections.
Code tracking is initiated as soon as signal detection is confirmed, and the goal
is to make the receiver-generated code line up with incoming code as precisely
as possible. There are two objectives in maintaining alignment:
1. Signal Despreading. The first objective is to fully despread the signal so
that it is no longer below the noise and so that information contained in the
carrier and the 50-bps (bits per second) data modulation can be recovered.
2. Range Measurements. The second objective is to enable precise measurement of the time of arrival (TOA) of received code for purposes of measuring range. Such measurements cannot be made directly from the received
signal, since it is below the noise level. Therefore, a code tracking loop,
which has a large processing gain, is employed to generate a reference
code precisely aligned with that of the received signal. This enables range
measurements to be made using the reference code instead of the much
noisier received signal code waveform.
Figure 3.14 illustrates the concept of a code tracking loop. It is assumed that
a numerically controlled oscillator (NCO) has translated the signal to complex
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
Fig. 3.14 Code tracking loop concept.
baseband form (i.e., zero frequency). Each component (I and Q) of the baseband
signal is multiplied by three replicas of the C/A-code that are formed by delaying
the output of a single code generator by three delay values called early, punctual,
and late. In typical GPS receivers the early and late codes respectively lead and
lag the punctual code by 0.05 to 0.5 code chips and always maintain these relative
positions. Following each multiplier is a lowpass filter (LPF) or integrator that,
together with its associated multiplier, forms a correlator. The output magnitude
of each correlator is proportional to the cross-correlation of its received and
reference codes, where the cross-correlation function has the triangular shape
previously shown in Fig. 3.6. In normal operation the punctual code is aligned
with the code of the incoming signal so that the squared magnitude IP2 + Q2P
of the punctual correlator output is at the peak of the cross-correlation function,
and the output magnitudes of the early and late correlators have smaller but
equal values on each side of the peak. To maintain this condition, a loop error
signal
ec (τ ) = IL2 + Q2L − IE2 + Q2E ,
(3.22)
is formed, which is the difference between the squared magnitudes of the late
and early correlators. The loop error signal as a function of received code delay
is shown in Fig. 3.15. Near the tracking point the error is positive if the received
code is delayed relative to the punctual code and negative if it is advanced.
SIGNAL ACQUISITION AND TRACKING
Fig. 3.15
83
Code tracking loop error signal.
Alignment of the punctual code with the received code is maintained by using the
error signal to delay the reference code generator when the error signal is positive
and to advance it when the error signal is negative. Since ec (τ ) is generally quite
noisy, it is sent through a lowpass loop filter before it controls the timing of the
reference code generator, as indicated in Fig. 3.14. The bandwidth of this filter is
usually quite small, resulting in a closed-loop bandwidth typically less than 1 Hz.
This is the source of the large processing gain that can be realized in extracting
the C/A-code from the signal.
When the code tracking loop is first turned on, the integration time T for
the correlators is usually no more than a few milliseconds, in order to minimize
corruption of the correlation process by data bit transitions of the 50-bps (bits per
second) data stream whose locations in time are not yet known. However, after
bit synchronization has located the data bit boundaries, the integration interval
can span a full data bit (20 ms) in order to achieve a maximum contribution to
processing gain.
3.4.5.1 Coherent versus Noncoherent Code Tracking If the error signal is
formed from only the squared magnitudes of the (complex) early and late correlator outputs as described above, the loop is called a noncoherent code tracking
loop. A distinguishing feature of such a loop is its insensitivity to the phase of the
received signal. Insensitivity to phase is desirable when the loop is first turned
on, since at that time the signal phase is random and not yet under any control.
On the other hand, once the phase of the signal is being tracked, a coherent code tracker can be employed, in which the outputs of the early and late
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
correlators are purely real. In this situation the loop error signal can be formed
directly from the difference of the early and late squared magnitudes from only
the I correlator. By avoiding the noise in the Q correlator outputs, a 3-dB SNR
advantage is thereby gained in tracking the code. However, a price is paid in that
the code loop error signal becomes sensitive to phase error in tracking the carrier.
If phase tracking is ever lost, complete failure of the code tracking loop could
occur. This is a major disadvantage, especially in mobile applications where the
signal can vary rapidly in magnitude and phase. Since noncoherent operation
is much more robust in this regard and is still needed when code tracking is
initiated, most GPS receivers use only noncoherent code tracking.
3.4.5.2 Factors Affecting Code Tracking Performance The bandwidth of the
code tracking loop is determined primarily by the loop filter and needs to be
narrow for best ranging accuracy but wide enough to avoid loss of lock if the
receiver is subject to large accelerations that can suddenly change the apparent
chipping rate of the received code. Excessive accelerations cause loss of lock by
moving the received and reference codes too far out of alignment before the loop
can adequately respond. Once the alignment error exceeds approximately 1 code
chip, the loop loses lock because it no longer has the ability to form the proper
error signal.
In low-dynamics applications with lower-cost receivers, code tracking loop
bandwidths on the order of 1 Hz permit acceptable performance in handheld
units and in receivers with moderate dynamics (e.g., in automobiles). For highdynamics applications, such as missile platforms, loop bandwidths might be on
the order of 10 Hz or larger. In surveying applications, which have no appreciable
dynamics, loop bandwidths can be as small as 0.01 Hz to obtain the required
ranging accuracy. Both tracking accuracy and the ability to handle dynamics are
greatly enhanced by means of carrier aiding from the receiver’s carrier phase
tracking loop, which will be be discussed subsequently.
3.4.6 Carrier Phase Tracking Loops
The purposes of tracking carrier phase are to
1. Obtain a phase reference for coherent detection of the GPS biphase modulated data
2. Provide precise velocity measurements (via phase rate)
3. Obtain integrated Doppler for rate aiding of the code tracking loop
4. Obtain precise carrier phase pseudorange measurements in high-accuracy
receivers
Tracking of carrier phase is usually accomplished by a phase-lock loop (PLL).
A Costas-type PLL or its equivalent must be used to prevent loss of phase coherence induced by the biphase data modulation on the GPS carrier. The origin
of the Costas PLL is described in [40]. A typical Costas loop is shown in
SIGNAL ACQUISITION AND TRACKING
85
Fig. 3.16 Costas PLL.
Fig. 3.16. In this design the output of the receiver last intermediate-frequency
(IF) amplifier is converted to a complex baseband signal by multiplying the
signal by both the in-phase and quadrature-phase outputs of an NCO and integrating each product over each 20-ms data bit interval to form a sequence of
phasors. The phase angle of each phasor is the phase difference between the
signal carrier and the NCO output during the 20-ms integration. A loop phase
error signal is formed by multiplying together the I and Q components of each
phasor. This error signal is unaffected by the biphase data modulation because
the modulation appears on both I and Q and is removed in forming the I × Q
product. After passing through a lowpass loop filter the error signal controls
the NCO phase to drive the loop error signal I × Q to zero (the phase-locked
condition). In some receivers the error signal is generated by forming twice the
four-quadrant arctangent of the I and Q phasor components, as indicated in the
figure.
Because the Costas loop is unaffected by the data modulation, it will achieve
phase lock at two stable points where the NCO output phase differs from that
of the signal carrier by either 0˚ or 180˚, respectively. This can be seen by
considering I = A cos θ and Q = A sin θ , where A is the phasor amplitude and
θ is its phase. Then
I × Q = A2 cos θ sin θ = 12 A2 sin 2θ.
(3.23)
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
There are four values of θ in [0, 2π) where the error signal I × Q = 0. Two
of these are the stable points, namely, θ = 0 and θ = 180◦ , toward which the
loop tends to return if perturbed. Since sin 2θ is unchanged by 180◦ changes in θ
caused by the data bits, the data modulation will have no effect. At either of the
two stable points the Q integrator output is nominally zero and the I integrator
output contains the demodulated data stream, but with a polarity ambiguity that
can be removed by observing frame preamble data. Thus the Costas loop has the
additional feature of serving as a data demodulator.
In the Costas loop design shown the phase of the signal is measured by
comparing the phase of the NCO output with a reference signal. Normally the
reference signal frequency is a rational multiple of the same crystal-controlled
oscillator that is used in frequency shifting the GPS signal down to the last IF.
When the NCO is locked to the phase of the incoming signal, the measured
phase rate will typically be in the range of ±5 kHz due to signal Doppler shift.
Two types of phase measurement are usually performed on a periodic basis
(the period might be every 20 ms). The first is an accurate measurement of
the phase modulo 2π, which is used in precision carrier phase ranging. The
second is the number of cycles (including the fractional part) of phase change
that have occurred from a defined point in time up to the present time. The
latter measurement is often called integrated Doppler and is used for aiding the
code tracking loop. By subtracting consecutive integrated Doppler measurements,
extremely accurate average frequency measurements can be made, which can be
used by the navigation filter to accurately determine user velocity.
Although the Costas loop is not disturbed by the presence of data modulation,
at low SNR its performance degrades considerably from that of a loop designed
for a pure carrier. The degradation is due to the noise × noise component of the
I × Q error signal. Furthermore, the 20-ms duration of the I and Q integrations
represents a limit to the amount of coherent processing that can be achieved. If
it is assumed that the maximum acceptable bit error rate for the 50-bps (bits per
second) data demodulation is 10−5 , GPS signals become unusable when C/N0
falls below about 25 dB-Hz.
The design bandwidth of the PLL is determined by the SNR, desired tracking
accuracy, signal dynamics, and ability to “pull in” when acquiring the signal or
when lock is momentarily lost.
3.4.6.1 PLL Capture Range An important characteristic of the PLL is the
ability to “pull in” to the frequency of a received signal. When the PLL is
first turned on following code acquisition, the difference between the signal
carrier frequency and the NCO frequency must be sufficiently small, or the
PLL will not lock. In typical GPS applications, the PLL must have a relatively small bandwidth (1–10 Hz) to prevent loss of lock due to noise. However,
this results in a small pullin (or capture) range (perhaps only 3–30 Hz), which
would require small (hence many) frequency bins in the signal acquisition search
algorithm.
SIGNAL ACQUISITION AND TRACKING
87
3.4.6.2 Use of Frequency-Lock Loops for Carrier Capture Some receivers
avoid the conflicting demands of the need for a small bandwidth and a large
capture range in the PLL by using a frequency-lock loop (FLL). The capture
range of a FLL is typically much larger than that of a PLL, but the FLL cannot
lock to phase. Therefore, a FLL is often used to pull the NCO frequency into
the capture range of the PLL, at which time the FLL is turned off and the PLL
is turned on. A typical FLL design is shown in Fig. 3.17. The FLL generates
a loop error signal eFLL that is approximately proportional to the rotation rate
of the baseband signal phasor and is derived from the vector cross-product of
successive baseband phasors [I (t − τ ), Q(t − τ )] and [I (t), Q(t)], where τ is a
fixed delay, typically 1–5 ms. More precisely
eFLL = Q(t)I (t − τ ) − I (t)Q(t − τ ).
(3.24)
3.4.6.3 PLL Order The order of a PLL refers to its capability to track different
types of signal dynamics. Most GPS receivers use second- or third-order PLLs.
A second-order loop can track a constant rate of phase change (i.e., constant
frequency) with zero average phase error and a constant rate of frequency change
with a nonzero but constant phase error. A third-order loop can track a constant
rate of frequency change with zero average phase error and a constant acceleration
of frequency with nonzero but constant phase error. Low-cost receivers typically
use a second-order PLL with fairly low bandwidth because the user dynamics are
minimal and the rate of change of the signal frequency due to satellite motion is
sufficiently low (<1 Hz/s) that phase tracking error is negligible. On the other
hand, receivers designed for high dynamics (i.e., missiles) will sometimes use
third-order or even higher-order PLLs to avoid loss of lock due to the large
accelerations encountered.
The price paid for using higher-order PLLs is a somewhat lower robust performance in the presence of noise. If independent measurements of platform
dynamics are available (such as accelerometer or INS outputs), they can be used
to aid the PLL by reducing stress on the loop. This can be advantageous because
it often renders the use of higher-order loops unnecessary.
3.4.7 Bit Synchronization
Before bit synchronization can occur, the PLL must be locked to the GPS signal.
This is accomplished by running the Costas loop in a 1-ms integration mode
where each interval of integration is over one period of the C/A-code, starting
and ending at the code epoch. Since the 50-Hz biphase data bit transitions can
occur only at code epochs, there can be no bit transitions while integration is
taking place. When the PLL achieves lock, the output of the I integrator will
be a sequence of values occurring once per millisecond or 20 times per data bit.
With nominal signal levels the processing gain of the integrator is sufficient to
guarantee with high probability that the polarity of the 20 integrator outputs will
remain constant during each data bit interval and will change polarity when a
data bit transition occurs.
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
Fig. 3.17
Frequency-lock loop.
A simple method of bit synchronization is to clock a modulo-20 counter
with the epochs of the receiver-generated reference C/A-code and record the
count each time the polarity of the I integrator output changes. A histogram
of the frequency of each count is constructed, and the count having the highest
frequency identifies the epochs that mark the data bit boundaries.
3.4.8 Data Bit Demodulation
Once bit synchronization has been achieved, demodulation of the data bits can
occur. As previously described, many GPS receivers demodulate the data by
integrating the in-phase (I ) component of the baseband phasor generated by a
Costas loop, which tracks the carrier phase. Each data bit is generated by integrating the I component over a 20-ms interval from one data bit boundary to
the next. The Costas loop causes a polarity ambiguity of the data bits that can
be resolved by observation of the subframe preamble in the navigation message
data.
3.5 EXTRACTION OF INFORMATION FOR NAVIGATION
SOLUTION
After data demodulation has been performed, the essential information in the
signal needed for the navigation solution is at hand. This information can be
EXTRACTION OF INFORMATION FOR NAVIGATION SOLUTION
89
classified into the following three categories:
1. The information needed to determine signal transmission time
2. The information needed to establish the position and velocity of each satellite
3. The various pseudorange and Doppler measurements made by the receiver
3.5.1 Signal Transmission Time Information
In our previous discussion of the Z-count, we saw that the receiver can establish
the time of transmission of the beginning of each subframe of the signal and of
the corresponding C/A-code epoch that coincides with it. Since the epochs are
transmitted precisely 1 ms apart, the receiver labels subsequent C/A-code epochs
merely by counting them. This enables the determination of the transmission time
of any part of the signal by a process to be described later.
3.5.2 Ephemeris Data
The ephemeris data permit the position and velocity of each satellite to be computed at the signal transmission time. The calculations were outlined in Table
3.2.
3.5.3 Pseudorange Measurements Using C/A-Code
In its basic form, finding the three-dimensional position of a user would consist
of determining the range, that is, the distance of the user from each of three or
more satellites having known positions in space, and mathematically solving for
a point in space where that set of ranges would occur. The range to each satellite
can be determined by measuring how long it takes for the signal to propagate
from the satellite to the receiver and multiplying the propagation time by the
speed of light.
Unfortunately, however, this method of computing range would require very
accurate synchronization of the satellite and receiver clocks used for the time
measurements. GPS satellites use very accurate and stable atomic clocks, but
it is economically infeasible to provide a comparable clock in a receiver. The
problem of clock synchronization is circumvented in GPS by treating the receiver
clock error as an additional unknown in the navigation equations and using measurements from an additional satellite to provide enough equations for a solution
for time as well as for position. Thus the receiver can use an inexpensive clock for
measuring time. Such an approach leads to perhaps the most fundamental measurement made by a GPS receiver: the pseudorange measurement, computed as
ρ = c(trcve − txmit ),
(3.25)
where trcve is the time at which a specific, identifiable portion of the signal is
received, txmit is the time at which that same portion of the signal is transmitted,
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
Fig. 3.18 Pseudorange measurement concept.
and c is the speed of light (2.99792458 × 108 m/s). It is important to note that
trcve is measured according to the receiver clock, which may have a large time
error, but txmit is in terms of GPS time, which in turn is SV (spacecraft vehicle)
time plus a time correction transmitted by the satellite. If the receiver clock were
synchronized to GPS time, then the pseudorange measurement would in fact be
the range to the satellite.
Figure 3.18 shows the pseudorange measurement concept with four satellites,
which is the minimum number needed for a three-dimensional position solution
without synchronized clocks. The raw measurements are simultaneous snapshots
at time trcve of the states of the received C/A-codes from all satellites. This is
accomplished indirectly by observation of the receiver-generated code state from
each code tracking loop. For purposes of simplicity we define the state of the
C/A-code to be the number of chips (including the fractional part) that have
occurred since the last code epoch. Thus the state is a real number in the interval
[0, 1023).
As discussed earlier, the receiver has been able to tag each code epoch with
its GPS transmission time. Thus, it is a relatively simple matter to compute the
time of transmission of the code state that is received at time trcve . For a given
EXTRACTION OF INFORMATION FOR NAVIGATION SOLUTION
91
satellite let te denote the GPS transmission time of the last code epoch received
prior to trcve , let X denote the code state observed at trcve , and let cr denote the
C/A-code chipping rate (1.023 × 106 chips/s). Then the transmission time of that
code state is
X
txmit = te + .
cr
(3.26)
3.5.3.1 Basic Positioning Equations If pseudorange measurements can be
made from at least four satellites, enough information exists to solve for the
unknown position (X, Y , Z) of the GPS user and for the receiver clock error Cb
(often called the clock bias). The equations are set up by equating the measured
pseudorange to each satellite with the corresponding unknown user-to-satellite
distance plus the distance error due to receiver clock bias:
ρ1 = (x1 − X)2 + (y1 − Y )2 + (z1 − Z)2 + Cb ,
ρ2 = (x2 − X)2 + (y2 − Y )2 + (z2 − Z)2 + Cb ,
..
. ρn = (xn − X)2 + (yn − Y )2 + (zn − Z)2 + Cb ,
(3.27)
where ρi denotes the measured pseudorange of the ith satellite whose position
in ECEF coordinates at txmit is (xi , yi , zi ) and n ≥ 4 is the number of satellites observed. The unknowns in this nonlinear system of equations are the user
position (X, Y , Z) in ECEF coordinates and the receiver clock bias Cb .
3.5.4 Pseudorange Measurements Using Carrier Phase
Although pseudorange measurements using the C/A-code are the most commonly
employed, a much higher level of measurement precision can be obtained by
measuring the received phase of the GPS L1 or L2 carrier. Because the carrier waveform has a very short period (6.35 × 10−10 s at the L1 frequency),
the noise-induced error in measuring signal delay by means of phase measurements is typically 10–100 times smaller than that encountered in code delay
measurements.
However, carrier phase measurements are highly ambiguous because phase
measurements are simply modulo 2π numbers. Without further information such
measurements determine only the fractional part of the pseudorange when measured in carrier wavelengths. Additional measurements are required to effect
ambiguity resolution, in which the integer number of wavelengths in the pseudorange measurement can be determined. The relation between the measured signal
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
phases φi and the unambiguous pseudoranges ρi can be expressed as
φ1
2π
φ2
λ 2π
ρ1 = λ
+ k1 ,
+ k2 ,
ρ2 =
..
.
φn
+ kn ,
ρn = λ 2π
(3.28)
where n is the number of satellites observed, λ is the carrier wavelength, and ki is
the unknown integral number of wavelengths contained in the pseudorange. The
additional measurements required for determination of the ki may include C/Aand/or P(Y)-code pseudorange measurements from the same satellites used for the
phase measurements. Since the code measurements are unambiguous, they significantly narrow the range of admissible integer values for the ki . Additionally,
phase measurements made on both the L1 and L2 signals can be used to obtain
a virtual carrier frequency equal to the difference of the two carrier frequencies
(1575.42 − 1227.60 = 347.82 MHz). The 86.3-cm wavelength of this virtual carrier thins out the density of pseudorange ambiguities by a factor of about 4.5,
making the ambiguity resolution process much easier. Redundant code and phase
measurements from extra satellites can also be used to aid the process; the extra
code measurements further narrow the range of admissible integer values for the
ki , and the extra phase measurements thin out the phase ambiguity density by
virtue of satellite geometry.
Because of unpredictable variations in propagation delay of the code and
carrier due to the ionosphere and other error sources, it is virtually impossible to
obtain ambiguity resolution with single-receiver positioning. Therefore, carrier
phase measurements are almost always relegated to high-accuracy applications
in which such errors are canceled out by differential operation with an additional
receiver (base station).
In GPS receivers, carrier phase is usually measured by sampling the phase
of the reference oscillator of the carrier tracking loop. In most receivers this
oscillator is an NCO that tracks the phase of the incoming signal at a relatively
low intermediate frequency. The signal phase is preserved when the incoming
signal is frequency-downconverted. The NCO is designed to provide a digital
output of its instantaneous phase in response to a sampling signal. Phase-based
pseudorange measurements are made by simultaneously sampling at time trcve the
phases of the NCOs tracking the various satellites. As with all receiver measurements, the reference for the phase measurements is the receiver’s clock reference
oscillator.
3.5.5 Carrier Doppler Measurement
Measurement of the received carrier frequency provides information that can be
used to determine the velocity vector of the user. Although this could be done
EXTRACTION OF INFORMATION FOR NAVIGATION SOLUTION
93
by forming differences of code-based position estimates, frequency measurement
is inherently much more accurate and has faster response time in the presence of
user dynamics. The equations relating the measurements of Doppler shift to the
user velocity are
fd1 = λ1 (v · u1 − v 1 · u1 ) + fb ,
fd2 = λ1 (v · u2 − v 2 · u2 ) + fb ,
..
.
(3.29)
fdn = λ1 (v · un − v n · un ) + fb ,
where the unknowns are the user velocity vector v = (vx , vy , vz ) and the receiver
reference clock frequency error fb in hertz and the known quantities are the carrier wavelength λ and the measured Doppler shifts fdi in hertz, satellite velocity
vectors vi , and unit satellite direction vectors ui (pointing from the receiver
antenna toward the satellite antenna) for each satellite index i. The unit vectors
ui are determined by computing the user-to-ith satellite displacement vectors ρ i
and normalizing them to unit length:
ρi
ui
= [(xi − X), (yi − Y ), (zi − Z)]T ,
ρi
.
=
|ρ i |
(3.30)
In these expressions the ith satellite position (xi , yi , zi ) at time txmit is computed
from the ephemeris data and the user position (X, Y , Z) can be determined from
solution of the basic positioning equations using the C/A- or P(Y)-codes.
In GPS receivers, the Doppler measurements fdi are usually derived by sampling the frequency setting of the NCO (Fig. 3.16) that tracks the phase of the
incoming signal. An alternate method is to count the output cycles of the NCO
over a relatively short time period, perhaps 1 s or less. However, in either case,
the measured Doppler shift is not the raw measurement itself, but the deviation
from what the raw NCO measurement would be without any signal Doppler shift,
assuming that the receiver reference clock oscillator had no error.
3.5.6 Integrated Doppler Measurements
Integrated Doppler can be defined as the number of carrier cycles of Doppler
shift that have occurred in a given interval [t0 , t]. For the ith satellite the relation
between integrated Doppler Fdi and Doppler shift fdi is given by
*
Fdi (t) =
t
fdi (t) dt.
(3.31)
t0
However, accurate calculation of integrated Doppler according to this relation
would require that the Doppler measurement be a continuous function of time.
Instead, GPS receivers take advantage of the fact that by simply observing the
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
output of the NCO in the carrier tracking loop (Fig. 3.16), the number of cycles
that have occurred since initial time t0 can be counted directly.
Integrated Doppler measurements have several uses:
1. Accurate Measurement of Receiver Displacement over Time. Motion of the
receiver causes a change in the Doppler shift of the incoming signal. Thus,
by counting carrier cycles to obtain integrated Doppler, precise estimates of
the change in position (delta position) of the user over a given time interval
can be obtained. The error in these estimates is much smaller than the error in
establishing the absolute position using the C/A- or P(Y)-codes. The capability
of accurately measuring changes in position is used extensively in real-time
kinematic surveying with differential GPS. In such applications the user needs to
determine the locations of many points in a given land area with great accuracy
(perhaps to within a few centimeters). When the receiver is first turned on, it may
take a relatively long time to acquire the satellites, to make both code and phase
pseudorange measurements, and to resolve phase ambiguities so that the location
of the first surveyed point can be determined. However, once this is done, the
relative displacements of the remaining points can be found very rapidly and
accurately by transporting the receiver from point to point while it continues to
make integrated Doppler measurements.
2. Positioning Based on Received Signal Phase Trajectories. In another form
of differential GPS, a fixed receiver is used to measure the integrated Doppler
function, or phase trajectory curve, from each satellite over relatively long periods
of time (perhaps 5–20 min). The position of the receiver can be determined
by solving a system of equations relating the shape of the trajectories to the
receiver location. The accuracy of this positioning technique, typically within a
few decimeters, is not as good as that obtained with carrier phase pseudoranging
but has the advantage that there is no phase ambiguity. Some handheld GPS
receivers employ this technique to obtain relatively good positioning accuracy at
low cost.
3. Carrier Rate Aiding for the Code Tracking Loop. In the code tracking
loop, proper code alignment is achieved by using observations of the loop error
signal to determine whether to advance or retard the state of the otherwise freerunning receiver-generated code replica. Because the error signal is relatively
noisy, a narrow loop bandwidth is desirable to maintain good pseudoranging
accuracy. However, this degrades the ability of the loop to maintain accurate
tracking in applications where the receiver is subject to substantial accelerations.
The difficulty can be substantially mitigated with carrier rate aiding, in which
the primary code advance/retard commands are not derived from the code discriminator (early–late correlator) error signal but instead are derived from the
Doppler-induced accumulation of carrier cycles in the integrated Doppler function. Since there are 1540 carrier cycles per C/A-code chip, the code will therefore
be advanced by precisely one chip for every 1540 cycles of accumulated count
of integrated Doppler. The advantage of this approach is that, even in the presence of dynamics, the integrated Doppler can track the received code rate very
95
THEORETICAL CONSIDERATIONS IN PSEUDORANGE
accurately. As a consequence, the error signal from the code discriminator is
“decoupled” from the dynamics and can be used for very small and infrequent
adjustments to the code generator.
3.6 THEORETICAL CONSIDERATIONS IN PSEUDORANGE AND
FREQUENCY ESTIMATION
In a well-designed GPS receiver the major source of measurement error within
the receiver is thermal noise, and it is useful to know the best performance that
is theoretically possible in its presence. Theoretical bounds on errors in estimating code-based and carrier-based pseudorange, as well as in Doppler frequency
estimates, have been developed within an interesting branch of mathematical
statistics called estimation theory. There it is seen that a powerful estimation
approach called the method of maximum likelihood (ML) can often approach
theoretically optimum performance (see Section 7.2.4). ML estimates of pseudorange (using either the code or the carrier) and frequency are unbiased, which
means that the expected value of the error due to random noise is zero.
An important lower bound on the error variance of any unbiased estimator is provided by the Cramer–Rao bound, and any estimator that reaches this
lower limit is called a minimum-variance unbiased estimator (MVUE). It can
be shown that at the typical SNRs encountered in GPS, ML estimates of code
pseudorange, carrier pseudorange, and carrier frequency are all MVUEs. Thus,
these estimators are optimal in the sense that no unbiased estimator has a smaller
error variance [197].
3.6.1 Theoretical versus Realizable Code-Based Pseudoranging
Performance
It can be shown that a ML estimate τML of signal delay based on code measurements is obtained by maximizing the cross-correlation of the received code cr (t)
with a reference code cref (t) that is an identical replica (including bandlimiting)
of the received code:
* T
tML = max
(3.32)
cr (t) cref (t − τ ) dt,
τ
0
where [0, T ] is the signal observation interval. Here we assume coherent processing for purposes of simplicity. This estimator is a MVUE, and it can be shown
that the error variance of τML (which equals the Cramer–Rao bound) is
στ2ML =
2
+T "
0
N0
#2 .
cr
(t) dt
(3.33)
This is a fundamental relation that in temporal terms states that the error variance
is proportional to the power spectral density N0 of the noise and inversely
proportional to the integrated square of the derivative of the received code
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
waveform. It is generally more convenient to use an expression for the standard deviation, rather than the variance, of delay error, in terms of the bandwidth
of the C/A-code. The following formula is derived in Ref. 201:
3.444 × 10−4
στML = √
.
(C/N0 )W T
(3.34)
In this expression it is assumed that the received code waveform has been bandlimited by an ideal lowpass filter with one-sided bandwidth W . The signal
observation time is still denoted by T , and C/N0 is the ratio of power in the
code waveform to the one-sided power spectral density of the noise. A similar
expression is√obtained for the error variance using the P(Y)-code, except the
numerator is 10 times smaller.
Figure 3.19 shows the theoretically achievable pseudoranging error using the
C/A-code as a function of signal observation time for various values of C/N0 .
The error is surprisingly small if the code bandwidth is sufficiently large. As
an example, for a moderately strong signal with C/N0 = 31, 623 (45 dB-Hz),
a bandwidth W = 10 MHz, and a signal observation time of 1 s, the standard
deviation of the ML delay estimate obtained from Eq. 3.34 is about 6.2 × 10−10
s, corresponding to 18.6 cm after multiplying by the speed of light.
Fig. 3.19
Theoretically achievable C/A-code pseudoranging error.
THEORETICAL CONSIDERATIONS IN PSEUDORANGE
97
3.6.1.1 Code Pseudoranging Performance of Typical Receivers Most GPS
receivers approximate the ML estimator by correlating the incoming signal with
an ideal code waveform that does not include bandlimiting effects and use early
and late correlators in the code tracking loop that straddle the location of the
correlation function peak rather than find its actual location. As a result, the
code tracking error can be significantly larger than the theoretical minimum
discussed above. One-chip early–late spacing of the tracking correlators was common practice for the several decades preceding the early 1990s. It is somewhat
surprising that the substantial amount of performance degradation resulting from
this approach went unnoticed for so long. Not until 1992 was it widely known
that significant error reduction could be obtained by narrowing the spacing down
to 0.1–0.2 C/A-code chips in combination with a large precorrelation bandwidth.
Details of this approach, dubbed narrow-correlator technology, can be found in
Ref. 195. With narrow early–late spacing the random noises on the early and late
correlator outputs become highly correlated and therefore tend to cancel when
the difference error signal is formed. A large precorrelation bandwidth sharpens
the peak of the correlation function so that the closely spaced early and late
correlators can still operate on the high-slope portion of the correlation function,
thus preserving SNR in the loop.
It can be shown that the variance of the code tracking error continues to decrease
as the early–late spacing approaches zero but approaches a limiting value. Some
researches are aware that forming a difference signal with early and late correlators
is mathematically equivalent to a single correlation with the difference of the early
and late codes, which in the limit (as the early–late spacing goes to zero) becomes
equivalent to polarity-modulated sampling of the received code at the punctual
reference code transitions and summing the sample values to produce the loop
error signal. Some GPS receivers already put this principle into practice.
Figure 3.20 [201] compares the performance of several correlator schemes,
including the narrow correlator, with theoretical limits. It is seen that the narrow
correlator approaches the theoretical performance limit given by the Cramer–Rao
bound as the early–late spacing 2e approaches zero.
3.6.2 Theoretical Error Bounds for Carrier-Based Pseudoranging
At typical GPS signal-to-noise ratios the ML estimate τML of signal delay using
carrier phase is a MVUE, and it can be shown that the error standard deviation is
στML =
2πfc
√
1
,
2(C/N0 )T
(3.35)
where fc is the GPS carrier frequency and C/N0 and T have the same meaning
as in Eq. 3.34. This result is also reasonably accurate for a carrier tracking loop
if T is set equal to the reciprocal of the loop bandwidth. As an example of
the much greater accuracy of carrier phase pseudoranging compared with code
pseudoranging, a signal at C/N0 = 45 dB-Hz observed for 1 s can theoretically
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
Fig. 3.20 Performance of various correlators. (Reproduced from Ref. 76, with permission.)
yield an error standard deviation of 4 × 10−13 s, which corresponds to only
0.12 mm. However, typical errors of 1–3 mm are experienced in most receivers
as a result of random phase jitter in the reference oscillator.
3.6.3 Theoretical Error Bounds for Frequency Measurement
The ML estimate fML of the carrier frequency is also a MVUE, and the expression
for its error standard deviation is
,
3
.
(3.36)
σfML =
2π 2 (C/N0 )T 3
A 1-s observation of a despread GPS carrier with C/N0 = 45 dB-Hz yields
a theoretical error standard deviation of about 0.002 Hz, which could also be
obtained with a phase tracking loop having a bandwidth of 1 Hz. As in the case
of phase estimation, however, phase jitter in the receiver reference oscillator
yields frequency error standard deviations from 0.05 to 0.1 Hz.
3.7 MODERNIZATION OF GPS
Since it was declared fully operational in April 1995, the GPS has been operating
continuously with 24 or more operational satellites, and user equipment has
MODERNIZATION OF GPS
99
evolved rapidly, especially in the civil sector. As a result, radically improved
levels of performance have been reached in positioning, navigation, and time
transfer. However, the availability of GPS has also spawned new and demanding
applications that reveal certain shortcomings of the present system. Therefore,
since the mid-1990s numerous governmental and civilian committees have investigated the needs and deficiencies of the existing system in order to conceive a
plan for GPS modernization.
The modernization of GPS is a difficult and complex task that requires tradeoffs in many areas. Major issues include spectrum needs and availability, military
and civil performance, signal integrity and availability, financing and cost containment, and potential competition from Europe’s Galileo system. However, after
many years of hard work it now appears that critical issues have been resolved.
Major decisions have been made for the incorporation of new civil frequencies,
new civil and military signals, and higher transmitted power levels.
3.7.1 Deficiencies of the Current System
The changes that are planned for GPS address the following needs:
1. Civil users need two-frequency ionospheric correction capability in
autonomous operation. Since only the encrypted P-code appears at the L2 frequency, civil users are denied the benefit of dual-frequency operation to remove
ionospheric range error in autonomous (i.e., nondifferential) operation. Although
special techniques such as signal squaring can be used to recover the L2 carrier,
the P-code waveform is lost and the SNR is dramatically reduced. Consequently,
such techniques are of little value to the civil user in reducing ionospheric range
error.
2. Signal blockage and attenuation are often encountered. In some applications heavy foliage in wooded areas can attenuate the signal to an unusable level.
In certain locations, such as in urban canyons, a satellite signal can be completely
blocked by buildings or other features of the terrain. In such situations there will
not be enough visible satellites to obtain a navigation solution. New applications,
such as emergency 911 position location by GPS receivers embedded in cellular
telephone handsets, will require reliable operation of GPS receivers inside buildings, despite heavy signal attenuation due to roof, floors, and walls. Weak signals
are difficult to acquire and track.
3. Ability to resolve ambiguities in phase measurements needs improvement.
High-accuracy differential positioning at the centimeter level by civil users
requires rapid and reliable resolution of ambiguities in phase measurements.
Ambiguity resolution with single-frequency (L1 ) receivers generally requires a
sufficient length of time for the satellite geometry to change significantly. Performance is improved with dual-frequency receivers. However, the effective SNR
of the L2 signal is dramatically reduced because the encrypted P-code cannot be
despread by the civil user.
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
4. Selective Availability is detrimental to performance in civil applications. SA
has been suspended as of 8 p.m. EDT on May 1, 2000. The degradation in autonomous positioning performance by SA (about 50 m RMS error) is of concern in
many civil applications requiring the full accuracy of which GPS is capable. A
prime example is vehicle tracking systems in which an accuracy of 5–10 m RMS is
needed to establish the correct city street on which a vehicle is located. Moreover,
many civil and military committees have found that a military adversary can easily
mitigate errors due to SA by using differential positioning. In the civil sector, a
large and costly infrastructure has developed to overcome its effects.
5. Improvements in system integrity and robustness are needed. In applications involving public safety the integrity of the current system is judged to be
marginal. This is particularly true in aviation landing systems that demand the
presence of an adequate number of healthy satellite signals and functional crosschecks during precision approaches. Additional satellites and higher transmitted
power levels are desirable in this context.
6. Improvement is needed in multipath mitigation capability. Multipath
remains a dominant source of GPS positioning error and cannot be removed
by differential techniques. Although certain mitigation techniques, such as multipath mitigation technology (MMT), approach theoretical performance limits for
in-receiver processing, the required processing adds to receiver costs. In contrast,
effective multipath rejection could be made available to all receivers by using
new GPS signal designs.
7. The military needs improved acquisition capability and jamming immunity. Because the P(Y)-code has an extremely long period (seven days), it is
difficult to acquire unless some knowledge of the code timing is known. In the
current system P(Y) timing information is supplied by the HOW. However, to
read the HOW, the C/A-code must first be acquired to gain access to the navigation message. Unfortunately, the C/A-code is relatively susceptible to jamming,
which would seriously impair the ability of a military receiver to acquire the
P(Y) code. It would be far better if direct acquisition of a high-performance code
were possible.
3.7.2 Elements of the Modernized GPS
3.7.2.1 Civil Spectrum Modernization The upper part of Fig. 3.21 outlines the
current civil GPS signal spectrum and the additional codes and signal frequencies
in the plans for modernization. The major elements are as follows:
1. A New L2 Civil Signal (L2C ) Modulated with a New Code Structure. The
L2C signal, described in detail below, will offer civilian users the following
improvements:
(a) Two-frequency ionospheric error correction becomes possible. The 1/f 2
dispersive delay characteristic of the ionosphere can be used to accurately
estimate errors in propagation delay.
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MODERNIZATION OF GPS
Civil Signals
Aeronautical Radio navigation
Services Band (ARNS)
960–1215
Current
L1
Radio navigation
RNSS Bands
Satellite Services
Band (RNSS) 1215–1240 1560–1610
Frequency in MHz
L5
1575.42
L1
L2
Modernized
C/A-codes
C/A-codes +
Proposed L1 Civil
in-phase codes
L2 Civil
Codes
1164.45
1188.45
1215
Proposed L1 Civil
quadrature codes
1227.6
1575.42
1176.45
Military Signals
Current
RNSS Bands
1215–1240 1560–1610
L2
Frequency in MHz
P/Y codes
P/Y codes
1227.6
M-codes
Modernized
Note :
Civil F-codes (Fine codes) at L5 and
Military M-codes are in definition.
Band occupancy L5 is TBD
C/A-codes at L1 or Civil and Military Uses
Fig. 3.21
C/A-codes
1215.6
M-codes
C/A-codes
M-codes
L2
P/Y codes
1227.6
1575.42
M-codes
P/Y codes
1239.6
1563.42
1575.42
1587.42
Existing and modernized GPS signal spectrum.
(b) Carrier phase ambiguity resolution will be significantly improved, The
accessibility of the L1 and L2 carriers provides “wide lane” phase measurements having ambiguities that are much easier to resolve.
(c) The additional L2 signal will improve robustness in acquisition and tracking and improve C/A code positioning accuracy.
The existing C/A code at the L1 frequency will be retained for legacy
purposes.
2. A New L5 Signal Modulated by a New Code Structure. Although the use
of the L1 and L2 frequencies can satisfy most civil users, there are concerns that
the L2 frequency band may be subject to unacceptable levels of interference for
applications involving public safety, such as aviation. The potential for interference arises because the International Telecommunications Union (ITU) has
authorized this band on a coprimary basis with radiolocation services, such as
high-power radars. As a result of FAA requests, the Department of Transportation and Department of Defense have determined a new civil GPS frequency,
called L5 , in the Aeronautical Radio Navigation System band at 1176.45 MHz.
To gain maximum performance, the L5 spread-spectrum codes will have a higher
chipping rate and longer period than do the C/A-codes. Proposed codes have
a 10.23-megachip/s chipping rate and a period of 10,230 chips. Additionally,
the plan is to transmit two signals in phase quadrature, one of which will
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
not carry data modulation. The L5 signal will provide the following system
improvements:
(a) Ranging accuracy will improve. Pseudorange errors due to random noise
will be reduced below levels obtainable with the C/A-codes, due to the
larger bandwidth of the proposed codes. As a consequence, both codebased positioning accuracy and phase ambiguity resolution performance
will improve.
(b) Errors due to multipath will be reduced. The larger bandwidth of the new
codes will sharpen the peak of the code autocorrelation function, thereby
reducing the shift in the peak due to multipath signal components.
(c) Carrier phase tracking will improve. Weak-signal phase tracking performance of GPS receivers is severely limited by the necessity of using
a Costas (or equivalent-type) PLL to remove carrier phase reversals of
the data modulation. Such loops rapidly degrade below a certain threshold (about 25–30 dB-Hz) because truly coherent integration of the carrier
phase is limited to the 20-ms data bit length. In contrast, the “data-free”
quadrature component of the L5 signal will permit coherent integration
of the carrier for arbitrarily long periods, which will permit better phase
tracking accuracy and lower tracking thresholds.
(d) Weak-signal code acquisition and tracking will be enhanced. The “datafree” component of the L5 signal will also permit new levels of positioning
capability with very weak signals. Acquisition will be improved because
fully coherent integration times longer than 20 ms will be possible. Code
tracking will also improve by virtue of better carrier phase tracking for
the purpose of code rate aiding.
(e) The L5 signal will further support rapid and reliable carrier phase ambiguity resolution. Because the difference between the L5 and L2 frequencies
is only 51.15 MHz as opposed to the 347.82-MHz difference between the
L1 and L2 frequencies, carrier phase ambiguity will be possible using an
extrawide lane width of about 5.9 m instead of 0.86 m. The inevitable
result will be virtually instantaneous ambiguity resolution, a critical issue
in high-performance real-time kinematic modes of GPS positioning.
(f) The codes will be better isolated from each other. The longer length of
the L5 codes will reduce the size of cross-correlation between codes from
different satellites, thus minimizing the probability of locking onto the
wrong code during acquisition, even at the increased power levels of the
modernized signals.
3. Higher Transmitted Power Levels. For safety, cost, and performance, many
in the GPS community are advocating a general increase of 3–6 dB in the signal
power at at all three civil frequencies.
4. A Proposed New L1 Civil Signal (L1C ) Using Binary Offset Carrier (BOC )
Modulation. Although a decision to use this signal has not yet been made, if
MODERNIZATION OF GPS
103
implemented it will be the first civilian signal to use BOC modulation. The
characteristics of the L1C signal are described in more detail below.
3.7.2.2 Military Spectrum Modernization The lower part of Fig. 3.21 shows
the current and modernized spectrum used by the military community. The current
signals consist of C/A-codes and P/Y-codes transmitted in quadrature in the L1
band and only P/Y-codes in the L2 band. The primary elements of the modernized
spectrum are as follows:
1. All existing signals will be retained for legacy purposes.
2. New M-codes will also be transmitted in both the L1 and L2 bands. These
codes are BOC(10,5) codes in which a 5.115 Mcps chipping sequence modulates
a 10.23 MHz square wave subcarrier. The resulting spectrum has two lobes, one
on each side of the band center, and for this reason the M-codes are sometimes
called “split-spectrum codes”. They will be transmitted in the same quadrature
channel as the C/A-codes, that is, in phase quadrature with the P(Y)-codes. Civil
use of these codes will be denied by as yet unannounced encryption techniques.
The M-codes will provide the following advantages to military users:
(a) Direct acquisition of the M-codes will be possible. The design of these
codes will eliminate the need to first acquire the L1 C/A-code with its
relatively high vulnerability to jamming.
(b) Better ranging accuracy will result. As can be seen in Fig. 3.21, the Mcodes have significantly more energy near the edges of the bands, with a
relatively small amount of energy near band center. Since most of the C/Acode power is near band center, potential interference between the codes
is mitigated. The effective bandwidth of the M-codes is much larger than
that of the P(Y)-codes, which concentrate most of their power near the
L1 or L2 carrier. Because of the modulated subcarrier, the autocorrelation
function of the M-codes has, not just one peak, but several peaks spaced
one subcarrier period apart, with the largest at the center. The modulated subcarrier will cause the central peak to be significantly sharpened,
significantly reducing pseudorange measurement error.
(c) Error due to multipath will be reduced. The sharp central peak of the Mcode autocorrelation function is less susceptible to shifting in the presence
of multipath correlation function components.
3.7.3 Families of GPS Satellites
The families of satellites prior to modernization have been Block I (1978–1985),
Block II (1989–1990), and Block IIA (1990–1997). These satellites all carry the
standard L1 C/A-, P-, and L2 P-codes.
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
In 1997 a new family, the Block IIR satellites, began to replace the older Block
II/IIA family. The Block IIR satellites have several improvements, including
reprogrammable processors enabling problem fixes and upgrades in flight. Eight
Block IIR satellites are being modernized to include the new military M-code
signals on both the L1 and L2 frequencies, as well as the new L2C signal on L2 .
The first modernized Block IIR was launched in September 2005.
The Block IIF family is the next generation of GPS satellites, retaining all the
capabilities of the previous blocks, but with many improvements, including an
extended design life of 12 years, faster processors with more memory, and the
inclusion of the new L5 signal on a third L5 frequency (1176.45 MHz). The first
Block IIF satellite is scheduled to launch in 2007.
The GPS system of the future is the Block III family, still under development.
The first satellite is scheduled to launch in 2010. Military improvements may
include two high-power spot beams for the L1 and L2 military M-code signals,
giving 20 dB higher received power over the earlier M-code signals. It is also
likely that there will be two additional channels providing navigation signals for
civilian use in local, regional and national safety-of-life applications for improved
positioning, navigation, and timing, perhaps with higher power as well. Perhaps
the proposed L1C signal will be considered for this purpose. The entire Block III
constellation is expected to remain operational through at least 2030.
3.7.4 Accuracy Improvements from Modernization
The progress in standalone positioning accuracy prior to and after the steps of
modernization can be summarized as follows:
•
20–100 m with C/A code and with selective availability
(SA) on (prior to May 2002)
• 10–20 m using C/A code with SA off (typical at the
present time)
• 5–10 m by 2009 using L1 C/A-code and L2C code
together for dual-frequency ionospheric correction
• 1–5 m by 2013 using L1 C/A code, L2C code, and L5
code
3.7.5 Structure of the Modernized Signals
The bandwidths of all modernized GPS signals will be at least 24 MHz. Assuming equal received power and filtered bandwidth, the ranging performance (with
or without multipath) on a GPS signal is determined by its spectral shape (or
equivalently, the shape of the autocorrelation function), where fine structure is
ignored. In this sense, the L1 C/A-coded and L2 civil signals are equivalent, as are
the P/Y and L5 civil signals. The military M-coded signal and the proposed civil
L1 signal use BOC modulation, but are not equivalent in performance because
they use different subcarrier frequencies and chipping rates. However, they both
place more of the signal power near the band edges, resulting in a multilobed
autocorrelation function.
MODERNIZATION OF GPS
105
3.7.5.1 L1 C/A-Coded and L2 Civil Signals
The C/A-Coded Signal The GPS modernization program retains the C/A-code at
the L1 carrier frequency (1575.42 MHz) for legacy purposes, mostly for civilian
users. These codes are maximal-length direct-sequence Gold codes, each consisting of a 1023-chip sequence transmitted at 1.023 × 106 chips/s, which repeats
every 1 ms.
The L2 Civil Signal Originally the modernization plan also called for the C/Acode at the L2 carrier frequency (1227.60 MHz) to provide the civilian community with ionospheric correction capability as well as additional flexibility and
robustness. However, late in the planning process it was realized that additional
advantages could be obtained by replacing the planned L2 C/A signal with a
new L2 civil signal (L2C ). The decision was made to use this new signal, and
its structure was made public early in 2001. Both the L2C and the new military
M-code signal (to be described) will appear on the L2 in-phase (I) channel, with
the P/Y coded signal on the quadrature (Q) channel.
Like the C/A code, the L2C code appears to be a 1.023 × 106 chip/s. sequence.
However, it is generated by 2:1 time-division multiplexing of two independent
subcodes, each having half the chipping rate, namely 511.5 × 103 chips/s. Each
of these subcodes is made available to the receiver by demultiplexing. The first
subcode (CM) has a moderate length of 10,230 chips, a 20-ms period, and is
modulated with either a 25-s or a 50-bps (bits per second) navigation message.
The moderate length of this code permits relatively easy acquisition of the signal
although the 2:1 multiplexing results in a 3 dB acquisition and data demodulation
loss. The second subcode (CL) has a length of 707,250 chips, a 1.5-s period, and
is data-free. With no data there is no limit on coherent processing time, thereby
permitting better code and carrier tracking performance, especially at low SNR.
Full-cycle carrier tracking is possible, with a 6 dB improvement in the tracking
threshold compared to that using only the CM code, where squaring loss is
incurred in removing data phase changes. The relatively long CL code length
also generates smaller correlation sidelobes as compared to the CM (or C/A)
code.
Details on the L2 civil signal are given by Fontana et al. [56].
3.7.5.2 P/Y-Coded and L5 Civil Signals
The P/Y-Coded Signal For legacy purposes, GPS modernization will retain the
P/Y-code on both the L1 and L2 frequencies. This code will be in phase quadrature with the C/A-code and the military M-code at the L1 frequency, and at
L2 will be in quadrature with the new L2 civil signal and the M-code. The
P/Y-code is transmitted at 10.23 × 106 chips/s in either unencrypted (P-code) or
encrypted (Y-code) form. The P-code sequence is publicly known and has a very
long period of 1 week. The Y-code is formed by modulating the P-code with a
slower sequence of encrypting chips, called a W-code, generated at 511.5 × 103
chips/sec.
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
The L5 Civil Signal GPS modernization calls for a completely new civil signal
at a carrier frequency of 1176.45 MHz with the total received signal power
divided equally between in-phase (I ) and quadrature (Q) components. Each
component is modulated with a different but synchronized 10,230-chip direct
sequence L5 code transmitted at 10.23 × 106 chips/s, the same rate as the P/Ycode, but with a 1-ms period (the same as the C/A-code period). The I channel
is modulated with a 100-symbol-per-second (sps) data stream, which is obtained
by applying 1/2-rate, constraint length 7, forward error correction (FEC) convolutional coding to a 50-bps (bits per second) navigation data message that
contains a 24-bit cyclic redundancy check (CRC). The Q-channel is unmodulated by data. However, both channels are further modulated by Neuman–
Hoffman (NH) synchronization codes, which provide additional spectral spreading of narrowband interference, improve bit and symbol synchronization, and
also improve cross-correlation properties between signals from different GPS
satellites.
Compared to the C/A code, the 10-times larger chip count of the I - and Qchannel civil L5 -codes provides lower autocorrelation sidelobes, and the 10-times
higher chipping rate substantially improves ranging accuracy, provides better
interference protection, and substantially reduces multipath errors at longer path
separations (far multipath). Additionally, these codes were selected to reduce as
much as possible the cross-correlation between satellite signals. The absence of
data modulation on the Q-channel permits longer coherent processing intervals
in code and carrier tracking loops, with full-cycle carrier tracking in the latter. As
a result, tracking capability and phase ambiguity resolution become more robust.
Further details on the civil L5 signal structure can be found in Ref. 45.
3.7.5.3 The Proposed L1 Civil (L1C ) Signal Although the current C/A code
will remain on the L1 frequency (1575.42 MHz), a new L1 civil signal has
recently been proposed. Like the L5 civil signal, it will have a data-free quadrature component. However, it is unique among the civil signals in that it will
use binary offset carrier (BOC) code modulation. The modulation candidates
under consideration are BOC(1,1) and several versions which time-multiplex
BOC(1,1) waveforms. These are collectively known as MBOC. The BOC(1,1)
modulation consists of a 1.023-MHz square-wave subcarrier modulated by a
1.023-megachip/second (Mcps) spreading sequence. Each spreading chip subtends exactly one cycle of the subcarrier, with the rising edge of the first subcarrier
half-cycle coincident with initiation of the spreading chip. The MBOC codes
provide a larger RMS bandwidth compared to pure BOC(1,1).
Some GPS receiver manufacturers prefer the pure BOC(1,1) modulation for
the following reasons:
1. The BOC(1,1) RMS bandwidth is smaller than that of MBOC modulation,
permitting a lower digital sampling rate in the receiver, which is desirable
to keep the receiver cost and power consumption as small as possible.
MODERNIZATION OF GPS
107
2. Although a low-cost, narrow bandwidth receiver can use any of the MBOC
signal candidates, the higher frequency components of an MBOC signal will
be lost, resulting in some loss of received signal power. This disadvantage
is most serious in very high-sensitivity receivers designed for indoor use.
On the other hand, MBOC modulation provides better ranging accuracy and is
inherently more robust against multipath, since its rms bandwidth is considerably
greater than that of BOC(1,1).
Details of the proposed L1C signal can be found in T. Stansell, “BOC or
MBOC,” Inside GNSS, Jul/Aug 2006, pub. by Gibbons Media & Research,
Eugene, Oregon.
3.7.5.4 The M-Coded Signal New military M-coded signals will be transmitted on the L1 and L2 carriers, with the capability of using different codes on the
two frequencies. The nominal received power level will be -158 dBW over the
entire portion of the earth viewed by the satellite. The received L1 M-code will
appear in the I channel additively superimposed on the C/A-code, and the L2 Mcode will appear in the I channel superimposed on the civil L2 code. However,
in the fully modernized Block III satellites, the M-coded signal component can
be radiated as a physically distinct signal from a separate antenna on the same
satellite. This is done in order to enable optional transmission of a spot beam
for greater antijam resistance within a selected local region on the earth. Spot
beam nominal received power will be 20 dB greater than that of normal earth
coverage.
The M-code, denoted a BOC(10,5) code, consists of a 10.23-MHz squarewave subcarrier modulated by a 5.115 × 106 chip/second spreading sequence.
Each spreading chip subtends exactly two cycles of the subcarrier, with the
rising edge of the first subcarrier cycle coincident with initiation of the spreading
chip. The spectrum of the BOC(10,5) code has considerably more relative power
near the edges of the signal bandwidth than any of the C/A-coded, L2 civil,
L5 civil, or P/Y-coded signals. As a consequence, the M-coded signal not only
offers the best pseudoranging accuracy and resistance to multipath but also has
minimal spectral overlap with the other GPS transmitted signals, which permits
transmission at higher power levels without mutual interference.
Details on the BOC(10,5) code can be found in a paper by Parker et al. [149].
PROBLEMS
3.1 An important signal parameter is the maximum Doppler shift due to satellite
motion, which must be accommodated by a receiver. Find its approximate
value by assuming that a GPS satellite has a circular orbit with a radius of
27,000 km, an inclination angle of 55˚, and a 12-h period. Is the rotation
rate of the earth significant? At what latitude(s) would one expect to see
the largest possible Doppler shift?
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
3.2 Another important parameter is the maximum rate of Doppler shift in hertz
per second that a phase-lock loop must be able to track. Using the orbital
parameters of the previous problem, calculate the maximum rate of Doppler
shift of a GPS signal one would expect, assuming that the receiver is stationary with respect to the earth.
3.3 Find the power spectrum of the 50-bps (bits per second) data stream containing the navigation message. Assume that the bit values are −1 and 1
with equal probability of occurrence, that the bits are uncorrelated random
variables, and that the location of the bit boundary closest to t = 0 is a
uniformly distributed random variable on the interval [−0.01 s, 0.01 s].
[Hint: First find the auto-correlation function R(τ ) of the bit stream and
then take its Fourier transform.]
3.4 In two-dimensional positioning, the user’s altitude is known, so only three
satellites are needed. Thus, there are three pseudorange equations containing
two position coordinates (e.g., latitude and longitude) and the receiver clock
bias term B. Since the equations are nonlinear, there will generally be more
than one position solution, and all solutions will be at the same altitude.
Determine a procedure that isolates the correct solution.
3.5 Some civil receivers attempt to extract the L2 carrier by squaring the
received waveform after it has been frequency-shifted to a lower IF. Show
that the squaring process removes the P(Y)-code and the data modulation,
leaving a sinusoidal signal component at twice the frequency of the original IF carrier. If the SNR in a 20-MHz IF bandwidth is −30 dB before
squaring, find the SNR of the double-frequency component after squaring
if it is passed through a 20-MHz bandpass filter. How narrow would the
bandpass filter have to be to increase the SNR to 0 dB?
3.6 The relativistic effect in a GPS satellite clock which is compensated by a
deliberate clock offset is about
(a) 4.5 parts per million
(b) 4.5 parts per 100 million
(c) 4.5 parts per 10 billion
(d) 4.5 parts per trillion
3.7 The following component of the ephemeris error contributes the most to
the range error:
(a) Along-track error
(b) Cross-track error
(c) Both along-track and cross-track error
(d) Radial error
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MODERNIZATION OF GPS
3.8 The differences between pseudorange and carrier phase observations are
(a) Integer ambiguity, multipath errors, and receiver noise
(b) Satellite clock, integer ambiguity, multipath errors, and receiver noise
(c) Integer ambiguity, ionospheric errors, multipath errors, and receiver
noise
(d) Satellite clock, integer ambiguity, ionospheric errors, multipath errors,
and receiver noise
3.9 GPS week number started incrementing from zero at
(a) Midnight of Jan. 5–6, 1980
(b) Midnight of. Jan. 5–6, 1995
(c) Midnight of Dec. 31–Jan. 1, 1994–1995
(d) Midnight of Dec. 31–Jan. 1, 1999–2000
3.10 The complete set of satellite ephemeris data comes once in every
(a) 6 s
(b) 30 s
(c) 12.5 s
(d) 12 s
3.11 Describe how the time of travel (from satellite to receiver) of the GPS
signal is determined.
3.12 Describe how the receiver locks on via code correlation. (Use sketches if
it helps.)
1
Satellite
-1
1 1
-1 -1 1 1
User Set
-1 -1
1
-1
1
-1
1 1
-1 -1
1 1 1
-1 -1
-1
1 1
-1 -1
3.13 For high accuracy of the carrier phase measurements, the most suitable
carrier tracking loop will be
(a) PLL with low loop bandwidth
(b) FLL with low loop bandwidth
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SIGNAL CHARACTERISTICS AND INFORMATION EXTRACTION
(c) PLL with high loop bandwidth
(d) FLL with high loop bandwidth
3.14 Which of the following actions does not reduce the receiver noise (code)?
(a) Reducing the loop bandwidth
(b) Decreasing the predetection integration time
(c) Spacing the early–late correlators closer
(d) Increasing the signal strength
4
RECEIVER AND ANTENNA DESIGN
4.1 RECEIVER ARCHITECTURE
Although there are many variations in GPS receiver design, all receivers must
perform certain basic functions. We will now discuss these functions in detail,
each of which appears as a block in the diagram of the generic receiver shown
in Fig. 4.1.
4.1.1 Radiofrequency Stages (Front End)
The purpose of the receiver front end is to filter and amplify the incoming GPS
signal. As was pointed out earlier, the GPS signal power available at the receiver
antenna output terminals is extremely small and can easily be masked by interference from more powerful signals adjacent to the GPS passband. To make
the signal usable for digital processing at a later stage, RF amplification in the
receiver front end provides as much as 35–55 dB of gain. Usually the front
end will also contain passband filters to reduce out-of-band interference without
degradation of the GPS signal waveform. The nominal bandwidth of both the L1
and L2 GPS signals is 20 MHz (±10 MHz on each side of the carrier), and sharpcutoff bandpass filters are required for out-of-band signal rejection. However, the
small ratio of passband width to carrier frequency makes the design of such filters infeasible. Consequently, filters with wider skirts are commonly used as a
first stage of filtering, which also helps prevent front-end overloading by strong
interference, and the sharp-cutoff filters are used later after downconversion to
intermediate frequencies (IFs).
Global Positioning Systems, Inertial Navigation, and Integration, Second Edition, by M. S. Grewal, L. R. Weill, and A. P. Andrews
Copyright © 2007 John Wiley & Sons, Inc.
111
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Fig. 4.1 Generic GPS receiver.
4.1.2 Frequency Downconversion and IF Amplification
After amplification in the receiver front end, the GPS signal is converted to a
lower frequency called an intermediate frequency for further amplification and
filtering. Downconversion accomplishes several objectives:
1. The total amount of signal amplification needed by the receiver exceeds
the amount that can be performed in the receiver front end at the GPS
carrier frequency. Excessive amplification can result in parasitic feedback
oscillation, which is difficult to control. In addition, since sharp-cutoff filters
with a GPS signal bandwidth are not feasible at the L-band, excessive
front-end gain makes the end-stage amplifiers vulnerable to overloading by
strong nearby out-of-band signals. By providing additional amplification at
an IF different from the received signal frequency, a large amount of gain
can be realized without the tendency toward oscillation.
2. By converting the signal to a lower frequency, the signal bandwidth is
unaffected, and the increased ratio of bandwidth to center frequency permits
the design of sharp-cutoff bandpass filters. These filters can be placed ahead
of the IF amplifiers to prevent saturation by strong out-of-band signals. The
filtering is often by means of surface acoustic wave (SAW) devices.
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RECEIVER ARCHITECTURE
3. Conversion of the signal to a lower frequency makes the sampling of the
signal required for digital processing much more feasible.
Downconversion is accomplished by multiplying the GPS signal by a sinusoid
called the local-oscillator signal in a device called a mixer. The local-oscillator
frequency is either larger or smaller than the GPS carrier frequency by an amount
equal to the IF. In either case the IF signal is the difference between the signal and
local-oscillator frequencies. Sum frequency components are also produced, but
these are eliminated by a simple bandpass filter following the mixer. An incoming
signal either above or below the local-oscillator frequency by an amount equal
to the IF will produce an IF signal, but only one of the two signals is desired.
The other signal, called the image, can be eliminated by bandpass filtering of the
desired signal prior to downconversion. However, since the frequency separation
of the desired and image signals is twice the IF, the filtering becomes difficult if a
single downconversion to a low IF is attempted. For this reason downconversion
is often accomplished in more than one stage, with a relatively high first IF
(30–100 MHz) to permit image rejection.
Whether it is single-stage or multistage, downconversion typically provides
a final IF that is low enough to be digitally sampled at feasible sampling rates
without frequency aliasing. In low-cost receivers typical final IFs range from 4
to 20 MHz with bandwidths that have been filtered down to several MHz. This
permits a relatively low digital sampling rate and at the same time keeps the
lower edge of the signal spectrum well above 0 Hz to prevent spectral foldover.
However, for adequate image rejection either multistage downconversion or a
special single-stage image rejection mixer is required. In more advanced receivers
there is a trend toward single conversion to a signal at a relatively high IF
(30–100 MHz), because advances in technology permit sampling and digitizing
even at these high frequencies.
4.1.2.1 Signal-to-Noise Ratio An important aspect of receiver design is the
calculation of signal quality as measured by the signal-to-noise ratio (SNR) in
the receiver IF bandwidth. Typical IF bandwidths range from about 2 MHz in
low-cost receivers to the full GPS signal bandwidth of 20 MHz in high-end units,
and the dominant type of noise is the thermal noise in the first RF amplifier stage
of the receiver front end (or the antenna preamplifier if it is used). The noise
power in this bandwidth is given by
N = kTe B
(4.1)
where k = 1.3806 × 10−23 J/K, B is the bandwidth in Hz, and Te is the effective
noise temperature in degrees Kelvin. The effective noise temperature is a function
of sky noise, antenna noise temperature, line losses, receiver noise temperature,
and ambient temperature. A typical effective noise temperature for a GPS receiver
is 513 K, resulting in a noise power of about −138.5 dBW in a 2-MHz bandwidth
and −128.5 dBW in a 20-MHz bandwidth. The SNR is defined as the ratio of
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signal power to noise power in the IF bandwidth, or the difference of these powers
when expressed in decibels. Using −154.6 dBW for the received signal power
obtained in Section 3.3, the SNR in a 20-MHz bandwidth is seen to be −154.6
−(−128.5) = −26.1 dB. Although the GPS signal has a 20-MHz bandwidth,
about 90% of the C/A-code power lies in a 2-MHz bandwidth, so there is only
about 0.5 dB loss in signal power. Consequently the SNR in a 2-MHz bandwidth
is (−154.6 − 0.5) − (−138.5) = −16.6 dB. In either case it is evident that the
signal is completely masked by noise. Further processing to elevate the signal
above the noise will be discussed subsequently.
4.1.3 Digitization
In modern GPS receivers digital signal processing is used to track the GPS signal,
make pseudorange and Doppler measurements, and demodulate the 50-bps (bits
per second) data stream. For this purpose the signal is sampled and digitized
by an analog-to-digital converter (ADC). In most receivers the final IF signal is
sampled, but in some the final IF signal is converted down to an analog baseband
signal prior to sampling. The sampling rate must be chosen so that there is no
spectral aliasing of the sampled signal; this generally will be several times the
final IF bandwidth (2–20 MHz).
Most low-cost receivers use 1-bit quantization of the digitized samples, which
not only is a very-low cost method of analog-to-digital conversion, but has the
additional advantage that its performance is insensitive to changes in voltage
levels. Thus, the receiver needs no automatic gain control (AGC). At first glance
it would appear that 1-bit quantization would introduce severe signal distortion.
However, the noise, which is Gaussian and typically much greater than the signal
at this stage, introduces a dithering effect that, when statistically averaged, results
in an essentially linear signal component. One-bit quantization does introduce
some loss in SNR, typically about 2 dB, but in low-cost receivers this is an
acceptable tradeoff. A major disadvantage of 1-bit quantization is that it exhibits
a capture effect in the presence of strong interfering signals and is therefore quite
susceptible to jamming.
Typical high-end receivers use anywhere from 1.5-bit (three-level) to 3-bit
(eight-level) sample quantization. Three-bit quantization essentially eliminates
the SNR degradation found in 1-bit quantization and materially improves performance in the presence of jamming signals. However, to gain the advantages of
multibit quantization, the ADC input signal level must exactly match the ADC
dynamic range. Thus the receiver must have AGC to keep the ADC input level
constant. Some military receivers use even more than 3-bit quantization to extend
the dynamic range so that jamming signals are less likely to saturate the ADC.
4.1.4 Baseband Signal Processing
Baseband signal processing refers to a collection of high-speed real-time algorithms implemented in dedicated hardware and controlled by software that acquire
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RECEIVER ARCHITECTURE
and track the GPS signal, extract the 50-bps (bits per second) navigation data,
and provide measurements of code and carrier pseudoranges and Doppler.
4.1.4.1 Carrier Tracking Tracking of the carrier phase and frequency is
accomplished by using feedback control of a numerically controlled oscillator
(NCO) to frequency shift the signal to precisely zero frequency and phase.
Because the shift to zero frequency results in spectral foldover of the signal
sidebands, both in-phase (I ) and a quadrature (Q) baseband signal components
are formed in order to prevent signal information loss. The I component is generated by multiplying the digitized IF by the NCO output, and the Q component
is formed by first introducing a 90◦ phase lag in the NCO output before multiplication. Feedback is accomplished by using the measured baseband phase to
control the NCO so that this phase is driven toward zero. When this occurs,
signal power is entirely in the I component, and the Q component contains only
noise. However, both components are necessary in order to measure the phase
error for feedback and to provide full signal information during acquisition when
phase lock has not yet been achieved. The baseband phase θbaseband is defined by
θbaseband = atan2(I, Q)
(4.2)
where atan2 is the four-quadrant arctangent function. The phase needed for feedback is recovered from I and Q after despreading of the signal. When phase lock
has been achieved, the output of the NCO will match the incoming IF signal in
both frequency and phase but will generally have much less noise due to lowpass filtering used in the feedback loop. Comparing the NCO phase to a reference
derived from the receiver reference oscillator provides the phase measurements
needed for carrier phase pseudoranging. Additionally, the cycles of the NCO
output can be accumulated to provide the raw data for Doppler, delta-range, and
integrated Doppler measurements.
4.1.4.2 Code Tracking and Signal Spectral Despreading The digitized IF
signal, which has a wide bandwidth due to the C/A- (or P-) code modulation, is completely obscured by noise. The signal power is raised above the
noise power by despreading, in which the digitized IF signal is multiplied by a
receiver-generated replica of the code precisely time-aligned with the code on
the received signal. Typically the individual baseband I and Q signals from the
controlled NCO mixer are despread in parallel, as previously shown in Fig. 3.13.
The despreading process removes the code from the signal, thus concentrating
the full signal power into the approximately 50-Hz baseband bandwidth of the
data modulation. Subsequent filtering (usually in the form of integration) can
now be employed to dramatically raise the SNR to values permitting observation
and measurement of the signal. As an example, recall that in a GPS receiver
a typical SNR in a 2-MHz IF bandwidth is −16.6 dB. After despreading and
50-Hz lowpass filtering the total signal power is still about the same, but the
bandwidth of the noise has been reduced from 2 MHz to about 50 Hz, which
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RECEIVER AND ANTENNA DESIGN
increases the SNR by the ratio 2 × 106 /50, or 46 dB. The resulting SNR is
therefore −16.6 + 46.0 = 29.4 dB.
4.2 RECEIVER DESIGN CHOICES
4.2.1 Number of Channels and Sequencing Rate
GPS receivers must observe the signal from at least four satellites to obtain threedimensional position and velocity estimates. If the user altitude is known, three
satellites will suffice. There are several choices as to how the signal observations
from a multiplicity of satellites can be implemented. In early designs, reduction
of hardware cost and complexity required that the number of processing channels
be kept at a minimum, often smaller than the number of satellites observed. In
this case, each channel must sequentially observe more than one satellite. As a
result of improved lower-cost technology, most modern GPS receivers have a
sufficient number of channels to permit one satellite to be continuously observed
on each channel.
4.2.1.1 Receivers with Channel Time Sharing
Single-Channel Receivers In a single-channel receiver, all processing, such as
acquisition, data demodulation, and code and carrier tracking, is performed by a
single channel in which the signals from all observed satellites are time-shared.
Although this reduces hardware complexity, the software required to manage the
time-sharing process can be quite complex, and there are also severe performance
penalties. The process of acquiring satellites can be very slow and requires a
juggling act to track already-acquired satellites while trying to acquire others.
The process is quite tricky when receiving ephemeris data from a satellite, since
about 30 s of continuous reception is required. During this time the signals from
other satellites are eclipsed, and resumption of reliable tracking can be difficult.
After all satellites have been acquired and their ephemeris data stored, two
basic techniques can be used to track the satellite signals in a single-channel
receiver. In slow-sequencing designs the signal from each satellite is observed
for a duration (dwell time) on the order of 1 s. Since a minimum of four satellites must typically be observed, the signal from each satellite is eclipsed for an
appreciable length of time. For this reason, extra time must be allowed for signal
reacquisition at the beginning of each dwell interval. Continually having to reacquire the signal generally results in less reliable operation, since the probability
of losing a signal is considerably greater as compared to the case of continuous
tracking. This is especially critical in the presence of dynamics, in which unpredictable user platform motion can take place during signal eclipse. Generally the
positioning and velocity accuracy is also degraded in the presence of dynamics.
If a single-channel receiver does not have to accurately measure velocity,
tracking can be accomplished with only a frequency-lock loop (FLL) for carrier
tracking. Since a FLL typically has a wider pull-in range and a shorter pull-in
RECEIVER DESIGN CHOICES
117
time than does a phase-lock loop (PLL), reacquisition of the signal is relatively
fast and the sequencing dwell time can be as small as 0.25 s per satellite. Because
loss of phase lock is not an issue, this type of receiver is also more robust in the
presence of dynamics. On the other hand, if accurate velocity determination is
required, a PLL must be used and the extra time required for phase lock during
signal reacquisition pushes the dwell time up to about 1–1.5 s per satellite, with
an increased probability of reacquisition failure due to dynamics.
A single-channel receiver requires relatively complex software for managing
the satellite time-sharing process. A typical design employs only one pseudonoise
(PN) code generator and one PPL in hardware. Typical tasks that the software
must perform during the dwell period for a specific satellite are as follows:
1. Select the PN code corresponding to the satellite observed.
2. Compute the current state of the code at the start of the dwell based on the
state at the end of the last dwell, the signal Doppler, and the eclipse time
since the last dwell.
3. Load the code state into the code generator hardware.
4. Compute the initial Doppler frequency of the FLL/PLL reference.
5. Load the Doppler frequency into the FLL/PLL hardware.
6. Initiate the reacquisition process by turning on the code and carrier tracking
loops.
7. Determine when reacquisition (code/frequency/phase lock) has occurred.
8. Measure pseudorange/carrier phase/carrier phase rate during the remainder
of the dwell.
In addition to these tasks, the software must be capable of ignoring measurements
from a satellite if the signal is momentarily lost and must permanently remove
the satellite from the sequencing cycle when its signal becomes unusable, such as
when the satellite elevation angle is below the mask angle. The software must also
have the capability of acquiring new satellites and obtaining their ephemeris data
as their signals become available while at the same time not losing the satellites
already being tracked. A satellite whose ephemeris data are being recorded must
have a dwell time (about 30 s) much longer than those of other satellites that
are only being tracked, which causes a much longer eclipse time for the latter.
The software must therefore modify the calculations listed above to take this into
account.
Because current technology makes the hardware costs of a multichannel receiver almost as small as that for a single channel, the single-channel approach has
been almost entirely abandoned in modern designs.
Another method of time sharing that can be used in single-channel receivers
is multiplexing, in which the dwell time is much shorter, typically 5–10 ms
per satellite. Because the eclipse time is so short, the satellites do not need
to be reacquired at each dwell. However, a price is paid in that the effective
SNR is significantly reduced in proportion to the number of satellites being
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RECEIVER AND ANTENNA DESIGN
tracked. Resistance to jamming is also degraded by values of 7 dB or more.
Additionally, the process of acquiring new satellites without disruption is made
more demanding because the acquisition search must be broken into numerous
short time intervals. Because of the rapidity with which satellites are sequenced,
a common practice with a two-channel receiver is to use a full complement of
PN code generators that run all the time, so that high-speed multiplexing of a
single code generator can be avoided.
Two-Channel Receivers The use of two channels permits the second channel to
be a “roving” channel, in which new satellites can be acquired and ephemeris data
collected while on the first channel satellites can be tracked without slowdown in
position/velocity updates. However, the satellites must still be time-shared on the
first channel. Thus the software must still perform the functions listed above and
in addition must be capable of inserting/deleting satellites from the sequencing
cycle. As with single-channel designs, either slow sequencing or multiplexing
may be used.
Receivers with Three to Five Channels In either slow-sequencing or multiplexed
receivers, additional channels will generally permit better accuracy and jamming
immunity as well as more robust performance in the presence of dynamics. A
major breakthrough in receiver performance occurs with five or more channels,
because four satellites can be simultaneously tracked without the need for time
sharing. The fifth channel can be used to acquire a new satellite and collect its
ephemeris data before using it to replace one of the satellites being tracked on
the other four channels.
Multichannel All-in-View Receivers The universal trend in receiver design is
to use enough channels to receive all satellites that are visible. In most cases
eight or fewer useful satellites are visible at any given time; for this reason
modern receivers typically have no more than 10–12 channels, with perhaps
several channels being used for acquisition of new satellites and the remainder
for tracking. Position/velocity accuracy is materially improved because satellites
do not have to be continually reacquired as is the case with slow sequencing,
there is no reduction in effective SNR found in multiplexing designs, and the
use of more than the minimum number of satellites results in an overdetermined
solution. In addition, software design is much simpler because each channel has
its own tracking hardware that tracks only one satellite and does not have to be
time shared.
4.2.2 L2 Capability
GPS receivers that can utilize the L2 frequency (1227.60 MHz) gain several
advantages over L1 -only receivers. Currently the L2 carrier is modulated only
with a military-encrypted P-code, called the Y-code, and the 50-bps (bits per
second) data stream. Because of the encryption, civilians are denied the use of
the P-code. However, it is still possible to recover the L2 carrier, which can
provide significant performance gains in certain applications.
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119
4.2.2.1 Dual-Frequency Ionospheric Correction Because the pseudorange
error caused by the ionosphere is inversely proportional to the square of frequency, it can be calculated in military receivers by comparing the P-code
pseudorange measurements obtained on the L1 and L2 frequencies. After subtraction of the calculated error from the pseudorange measurements, the residual
error due to the ionosphere is typically no more than a few meters as compared
to an uncorrected error of 5–30 m. Although civilians do not have access to
the P-code, in differential positioning applications the L2 carrier phase can be
extracted without decryption, and the ionospheric error can then be estimated by
comparing the L1 and L2 phase measurements.
4.2.2.2 Improved Carrier Phase Ambiguity Resolution in High-Accuracy Differential Positioning High-precision receivers, such as those used in surveying,
use carrier phase measurements to obtain very precise pseudoranges. However,
the periodic nature of the carrier makes the measurements highly ambiguous.
Therefore, solution of the positioning equations yields a grid of possible positions separated by distances on the order of one to four carrier wavelengths,
depending on geometry. Removal of the ambiguity is accomplished by using
additional information in the form of code pseudorange measurements, changes
in satellite geometry, or the use of more satellites than is necessary. In general, ambiguity resolution becomes less difficult as the frequency of the carrier
decreases. By using both the L1 and L2 carriers, a virtual carrier frequency
of L1 − L2 = 1575.42 − 1227.60 = 347.82 MHz can be obtained, which has a
wavelength of about 86 cm as compared to the 19 cm wavelength of the L1
carrier. Ambiguity resolution can therefore be made faster and more reliable by
using the difference frequency.
4.2.3 Code Selections: C/A, P, or Codeless
All GPS receivers are designed to use the C/A-code, since it is the only code
accessible to civilians and is used by the military for initial signal acquisition.
Most military receivers also have P-code capability to take advantage of the
improved performance it offers. On the other hand, commercial receivers seldom have P-code capability because the government does not make the needed
decryption equipment available to the civil sector. Some receivers, notably those
used for precision differential positioning application, also incorporate a codeless mode that permits recovery of the L2 carrier without knowledge of the code
waveform.
4.2.3.1 The C/A-Code The C/A-code, with its 1.023-MHz chipping rate and
1-ms period, has a bandwidth that permits a reasonably small pseudorange error
due to thermal noise. The code is easily generated by a few relatively small shift
registers. Because the C/A-code has only 1023 chips per period, it is relatively
easy to acquire. In military receivers direct acquisition of the P-code would
be extremely difficult and time-consuming. For this reason these receivers first
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RECEIVER AND ANTENNA DESIGN
acquire the C/A-code on the L1 frequency, allowing the 50-bps (bits per second)
data stream to be recovered. The data contains a hand-over word that tells the
military receiver a range in which to search for the P-code.
4.2.3.2 The P-Code The unencrypted P-code has a 10.23-MHz chipping rate
and is known to both civilian and military users. It has a very long period of
one week. The Y-code is produced by biphase modulation of the P-code by
an encrypting code known as the W-code. The W-code has a slower chipping
rate than does the P-code; there are precisely 20 P-code chips per W-code chip.
Normally the W-code is known only to military users who can use decryption to
recover the P-code, so that the civilian community is denied the full use of the
L2 signal. However, as will be indicated shortly, useful information can still be
extracted from the L2 signal in civilian receivers without the need for decryption.
Advantages of the P-code include the following:
Improved Navigation Accuracy Because the P-code has 10 times the chipping
rate of the C/A-code, its spectrum occupies a larger portion of the full 20-MHz
GPS signal bandwidth. Consequently, military receivers can typically obtain 3
times better pseudoranging accuracy compared to that obtained with the C/Acode.
Improved Jamming Immunity The wider bandwidth of the P-code gives about
40 dB suppression of narrowband jamming signals as compared to about 30 dB
for the C/A-code, which is of obvious importance in military applications.
Better Multipath Rejection In the absence of special multipath mitigation techniques, the P-code provides significantly smaller pseudorange errors in the presence of multipath as compared to the C/A-code. Because the P-code correlation
function is approximately one-tenth as wide as that of the C/A-code, there is less
opportunity for a delayed-path component of the receiver-generated signal correlation function to cause range error by overlap with the direct-path component.
4.2.3.3 Codeless Techniques Commercial receivers can recover the L2 carrier
without knowledge of the code modulation simply by squaring the received signal
waveform or by taking its absolute value. Because the a priori SNR is so small,
the SNR of the recovered carrier will be reduced by as much as 33 dB because
the squaring of the signal greatly increases the noise power relative to that of the
signal. However, the squared signal has extremely small bandwidth (limited only
by Doppler variations), so that narrowband filtering can make up the difference.
4.2.4 Access to SA Signals
Selective Availability (SA) refers to errors that may be intentionally introduced
into the satellite signals by the military to prevent full-accuracy capability by
the civilian community. SA was suspended on May 1, 2000 but can be turned
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121
on again at the discretion of the DoD. The errors appear to be random, have a
zero long-term average value, and typically have a standard deviation of 30 m.
Instantaneous position errors of 50–100 m occur fairly often and are magnified
by large position dilution of precision (PDOP) values. Part of the SA error is
in the ephemeris data transmitted by the satellite, and the rest is accomplished
by dithering of the satellite clock that controls the timing of the carrier and
code waveforms. Civil users with a single receiver generally have no way to
eliminate errors due to SA, but authorized users (mostly military) have the key
to remove them completely. On the other hand, civilians can remove SA errors
by employing differential operation, and a large network of differential reference
stations has been spawned by this need.
4.2.5 Differential Capability
Differential GPS (DGPS) is a powerful technique for improving the performance
of GPS positioning. This concept involves the use of not only the user’s receiver
(sometimes called the remote or roving unit) but also a reference receiver at an
accurately known location within perhaps 200 km of the user. Because the location of the reference receiver is known, pseudorange errors common to the user
and reference receivers can be measured and removed in the user’s positioning
calculations.
4.2.5.1 Errors Common to Both Receivers The major sources of error common to the reference and remote receivers, which can be removed (or mostly
removed) by differential operation, are the following:
1. Selective Availability Error. As mentioned previously, these are typically
about 30 m, 1σ .
2. Ionospheric Delays. Ionospheric signal propagation group delay, which is
discussed further in Chapter 5, can be as much as 20–30 m during the
day to 3–6 m at night. Receivers that can utilize both the L1 and L2
frequencies can largely remove these errors by applying the inverse squarelaw dependence of delay on frequency.
3. Tropospheric Delays. These delays, which occur in the lower atmosphere,
are usually smaller and more predictable than ionospheric errors, and typically are in the 1–3 m range but can be significantly larger at low satellite
elevation angles.
4. Ephemeris Errors. Ephemeris errors, which are the difference between the
actual satellite location and the location predicted by satellite orbital data,
are typically less than 3 m and will undoubtedly become smaller as satellite
tracking technology improves.
5. Satellite Clock Errors. These are the difference between the actual satellite
clock time and that predicted by the satellite data.
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Differential operation can almost completely remove satellite clock errors,
errors due to SA, and ephemeris errors. For these quantities the quality of
correction has little dependence on the separation of the reference and roving
receivers. However, because SA errors vary quite rapidly, care must be taken in
time-synchronizing the corrections to the pseudorange measurements of the roving receiver. The degree of correction that can be achieved for ionospheric and
tropospheric delays is excellent when the two receivers are in close proximity,
say, up to 20 km. At larger separations the ionospheric/tropospheric propagation
delays to the receivers become less correlated, and residual errors after correction are correspondingly larger. Nonetheless, substantial corrections can often be
made with receiver separations as large as 100–200 km.
Differential operation is ineffective against errors due to multipath, because
these errors are strictly local to each of the two receivers.
4.2.5.2 Corrections in the Measurement Domain versus the Solution Domain
In the broadest sense there are two ways that differential corrections can be
made. In the measurement domain, corrections are determined for pseudorange
measurements to each satellite in view of the reference receiver, and the user
simply applies the corrections corresponding to the satellites the roving receiver
is observing. On the other hand, in the solution-domain approach, the reference
station computes the position error that results from pseudorange measurements
to a set of satellites, and this is applied as a correction to the user’s computed
position. A significant drawback to the solution-domain approach is that the user
and reference station must use exactly the same set of satellites if the position
correction is to be valid. In most cases the reference station does not know which
satellites can be received by the roving receiver (e.g., some might be blocked
by obstacles) and therefore would have to transmit the position corrections for
many possible sets of satellites. The impracticality of doing this strongly favors
the use of the measurement-domain method.
4.2.5.3 Real-Time versus Postprocessed Corrections In some applications,
such as surveying, it is not necessary to obtain differentially corrected position solutions in real time. In these applications it is common practice to obtain
corrected positions at a later time by bringing together recorded data from
both receivers. No reference-to-user data link is necessary if the recorded data
from both receivers can be physically transported to a common computer for
processing.
However, in the vast majority of cases it is imperative that corrections be
applied as soon as the user has enough pseudorange measurements to obtain a
position solution. When the user needs to know his or her corrected position in
real time, current pseudorange corrections can be transmitted from the reference
receiver to the user via a radio or telephone link, and the user can use them in
the positioning calculations. This capability requires a user receiver input port
for receiving and using differential correction messages. A standardized format
of these messages has been recommended by Special Committee 104 (SC-104),
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123
established by the Radio Technical Commission for Maritime Service (RTCM)
in 1983. Details on this format appear in Ref. 103.
4.2.6 Pseudosatellite Compatibility
Although differential GPS can improve the reliability, integrity, and accuracy
of GPS navigation, it cannot overcome inherent limitations that are critical to
successful operation in specific applications. A major limitation is poor satellite
geometry, which can be caused by signal failure of one or more satellites, signal
blockage by local objects and/or terrain, and occasional periods of high PDOP,
which can occur even with a full constellation of satellites. Vertical positioning
error is usually more sensitive to this effect, which is bad news for aviation
applications. In some cases a navigation solution may not exist because not
enough satellite signals can be received.
The use of pseudolites can solve these problems within a local area. A pseudolite is simply a ground-based transmitter that acts as an additional GPS satellite
by transmitting a GPS-like signal. This signal can be utilized by a receiver for
pseudoranging and can also convey messages to the receiver to improve reliability and signal integrity. The RTCM SC-104 was formed in 1983 to study
pseudolite system and receiver design issues. The recommendations of SC-104
can be found in Ref. 180. The major improvements offered by pseudolites are
the following:
1. Improvement in Geometry. Pseudolites, acting as additional satellites, can
provide major improvements in geometry, hence in positioning accuracy,
within their region of coverage. Vertical (VDOP) as well as horizontal
(HDOP) dilution of precision can be dramatically reduced, which is of
major importance to aviation. Experiments have shown that PDOP of about
3 over a region having a radius of 20–40 km can be obtained by using
several pseudolites even when there are fewer than the minimum of four
satellites that would otherwise be needed for a navigation solution.
2. Improvement in Signal Availability. Navigation solutions with fewer than
the minimum required number of GPS satellites are made possible by using
the additional signals provided by pseudolites.
3. Inherent Transmission of Differential Corrections. The GPS-like signals
transmitted by a pseudolite include messaging capability that can be received
directly by the GPS receiver, thus allowing the user to receive differential
corrections without the need for a separate communications link.
4. Self-Contained Failure Notification. The additional signals provided by
pseudolites permit users to perform their own failure assessments. For
example, if pseudorange measurements from four satellites and one pseudolite are available, a problem can be detected by examining the consistency
of the measurements. If two pseudolites are available, not only can the failure of a single signal be detected, but the offending signal can be identified
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as well. These advantages are especially important in aviation, where pilot
notification of signal failures must occur very rapidly (within 1–10 s).
5. Solution of Signal Blockage Problems. The additional signals from pseudolites can virtually eliminate problems due to blockage of the satellite
signals by objects, terrain, or the receiving platform itself.
4.2.6.1 Pseudolite Signal Structure Ideally the pseudolite signal structure
would permit reception by a standard GPS receiver with little or no modification
of the receiver design. Thus it would seem that the pseudolite signal should have
a unique C/A-code with the same characteristics as the C/A-codes used by the
satellites. However, with this scheme it would be difficult to prevent a pseudolite
signal from interfering with the reception of the satellite signals, even if its C/Acode were orthogonal to the satellite codes. The fundamental difficulty, which
is called the near–far problem, occurs because of the inverse square-law dependence of received signal power with range. The near–far problem does not occur
with the GPS satellite signals because variation in the user-to-satellite range is
relatively small compared to its average value. However, with pseudolites this is
not the case. The problem is illustrated by considering that the received signal
strength of a pseudolite must be at least approximately that of a satellite. If the
pseudolite signal equals that of a satellite when the user is, say, 50 km from the
pseudolite, then that same signal will be 60 dB stronger when the user is 50 m
from the pseudolite. At this close range the pseudolite signal would be so strong
that it would jam the weaker GPS satellite signals.
Several solutions to the near–far problem involving both pseudolite signal
design and receiver design have been proposed [180] for the 60-dB received
signal dynamic range discussed above.
4.2.6.2 Pseudolite Signal Design Approaches
1. Use of High-Performance Pseudorandom Codes. The 60 dB of jamming
protection would require the pseudolite to transmit a code much longer
than a C/A-code and clocked at a much higher rate. This has been judged
to be an impractical solution because it would reduce compatibility with
the GPS signal structure and significantly increase receiver costs.
2. Pseudolite Frequency Offset. By moving the frequency of the pseudolite
signal sufficiently far away from the 1575.42-MHz L1 frequency, filters in
the receiver could prevent the pseudolite signals from interfering with the
satellite signals. Again, however, this approach would significantly increase
receiver costs and reduce compatibility with the GPS signal structure.
3. Low-Duty-Cycle Time-Division Multiplexing. A preferred approach is for
the pseudolite to transmit at the L1 frequency using short, low-duty-cycle
pulses that interfere with the satellite signals only a small fraction of the
time. The impact on receiver design is minimal because modifications are
primarily digital and low in cost. This approach retains compatibility with
the GPS signal structure by using a new set of 51 pseudolite Gold codes
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125
with the same chipping rate, period, and number of chips per period as the
satellite C/A-codes and a 50-bps (bits per second) data stream. Although
the codes run continuously in both the pseudolite and the user receiver, the
pseudolite signal is gated on only during eleven 90.91-μs intervals in each
10-ms (half-data-bit) interval. Each of the 11 gate intervals transmits 93 new
chips of the code, so that all 1023 chips get transmitted in 10 ms. However,
the timing of the gate intervals is randomized in order to randomize the
signal spectrum. Further details of the signal structure can be found in
Ref. 180.
4.2.6.3 Pseudolite Characteristics
1. Pseudolite Identification. Identification of a pseudolite is accomplished by
both its unique Gold code and its physical location, which appears in its
50-bps (bits per second) message. Since pseudolite signals are low power
and thus can be received only within a relatively small coverage area, it is
possible for pseudolites spaced sufficiently far apart to use the same Gold
code. In this case correct identification is effected by noting the location
transmitted by the pseudolite.
2. Pseudolite Clock Offset. Since the pseudolite can monitor GPS signals
over extended time periods, it can determine GPS time. This permits the
transmitted epochs of the pseudolite signal to be correct in GPS time and
avoids the necessity of transmitting pseudolite clock corrections. The time
reference for the differential pseudorange corrections transmitted by the
pseudolite is also GPS time.
3. Transmitted Signal Power. The primary use of pseudolite signals is for aircraft in terminal areas, so that a typical maximum reception range is 50 km.
At this range a half-hemisphere omnidirectional transmitting antenna fed
with approximately 30 mW of signal power will provide a signal level
comparable to that typical of a GPS satellite (−116 dBm). At a range of
50 m the signal level will be 60 dB larger (−56 dBm).
4. Pseudolite Message Structure. Although the pseudolite data stream is 50
bps (bits per second) to ensure compatibility with GPS receivers, its structure must be modified to transmit information that differs somewhat from
that transmitted by the GPS satellites. A proposed structure can be found
in Ref. 180.
5. Minimum Physical Spacing of Pseudolites. Placement of pseudolites involves considerations that depend on whether the pseudolites use the same
or different Gold codes.
4.2.6.4 Separation of Pseudolites Using the Same Code One approach when
two pseudolites use the same code is to synchronize the timing of the gated signals
of the pseudolites and separate the pseudolites by a distance that guarantees that
received transmissions from different pseudolites will not overlap. This requires
that the pseudolites be separated by at least 130 km, which guarantees that a
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user 50 km from the desired pseudolite will be at least 80 km from the undesired
pseudolite. The pulses from the latter will then travel at least 30 km further than
those from the desired pseudolite, thus arriving at least 100 μs later. Since the
width of pulses is 90.91 μs, pulses from two pseudolites will not overlap and
interference is thereby avoided.
However, a more conservative approach is to separate two pseudolites by a
distance that is sufficient to guarantee that when the user is at the maximum
usable range from one pseudolite, the signal from the other is too weak to interfere. Suppose that each pseudolite is set to achieve a received signal level of
−126 dBm at a maximum service radius of 50 km and that an undesired pseudolite signal must be at least 14 dB below the desired signal to avoid interference.
A simple calculation involving the inverse square power law shows that this can
be achieved with a minimum spacing of 300 km between the two pseudolites, so
that the minimum distance to the undesired pseudolite will be 250 km when the
user is 50 km from the desired pseudolite.
4.2.6.5 Separation of Pseudolites Using Different Codes When the user must
receive several pseudolites simultaneously, separation of the signals from different pseudolites might be possible by using different timing offsets of the
transmitted pulses. However, this would substantially complicate system design.
A preferred approach is to use synchronous transmissions but space the pseudolites so that when the received pulses do overlap, they can still be recovered by
using a suitable low-cost receiver design. The situation is clarified by considering
the two pseudolites shown in Fig. 4.2, which are separated by at least 27.25 km,
the distance traveled by a signal in the time required to transmit a single pulse.
With synchronous pulse transmissions from the pseudolites there exists a central
Fig. 4.2 Minimum spacing of pseudolites.
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127
region bounded on the left and right by two hyperbolic curves 27.25 km apart
along the baseline connecting the pseudolites. This distance is independent of the
separation of the pseudolites, but the curvature of the hyperbolas decreases as
the pseudolite separation increases. Outside the central region the received pulses
will not overlap and can easily be recovered by the receiver. The difficulty of
separating the overlapping pulses within the central region is a function of the
pseudolite separation. Separation is most difficult when the receiver is located at
the intersection of a hyperbola and the baseline where the stronger of the two
signals has its largest value, thus having the potential to overpower the weaker
signal. This problem can be avoided by adequate separation of the pseudolites,
but the separation required is a function of receiver design.
It will be seen later that a typical receiver designed for pseudolite operation
might clip the incoming signal at ±2σ of the precorrelation noise power in order
to limit the received power of strong pseudolite signals. Under this assumption
and an assumed ±1-MHz precorrelation bandwidth, the clipping threshold in a
receiver with a 4-dB noise figure would be −104 dBm. Assuming that the pseudolites are designed to produce a −116-dBm power level at 50 km, a receiver
receiving overlapping pulses would need to be at least 12.5 km from both pseudolites to avoid the capture effect in the clipping process. Thus, the two pseudolites
in Fig. 4.2 should each be moved 12.5 km from the boundaries of the central
region, resulting in a minimum distance of 52.25 km between them.
4.2.6.6 Receiver Design for Pseudosatellite Compatibility Major design issues
for a GPS receiver that receives pseudosatellite signals (often called a participating
receiver) are as follows:
1. Continuous Reception. Because the receiver must continuously recover the
pseudolite data message, a channel must be dedicated to this task. For
this reason a single-channel slow-sequencing receiver could not be used.
This is really not a problem, since almost all modern receivers use parallel
channels.
2. Ability to Track Pseudolite Gold Codes. The receiver must be capable of
generating and tracking each of the 51 special C/A-codes specified for the
pseudolite signals. These codes and their method of generation can be found
in Ref. 61. Although the codes can be tracked with standard GPS tracking
loops, optimum performance demands that the noise between pseudolite
pulses be blanked to obtain a 10-dB improvement in SNR.
3. Reduction of Pseudosatellite Interference to GPS Signal Channels. In a
GPS satellite channel a pseudolite signal appears as pulsed interference that
can be 60 dB greater above the satellite signal level. The resulting degradation of the GPS satellite signal can be reduced to acceptable levels by
properly designed wideband precorrelation signal clipping in the receiver.
This approach, which generally improves with increasing precorrelation
bandwidth and decreasing clipping level, typically results in a reduction in
the GPS SNR of 1–2 dB. A somewhat more effective approach is to blank
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the GPS signal ahead of the correlator during the reception of a pseudolite
pulse, which results in a GPS SNR reduction of about 0.5 dB.
4. Ability to Receive Overlapping Pseudolite Pulses. A group of pseudolites
designed to be utilized simultaneously must be located relatively close
together, inevitably causing received pulse overlap in certain portions of the
coverage area. Consequently, receiver design parameters must be chosen
carefully to ensure that overlapping pulses from different pseudolites can
be separated. The signal level from a nearby pseudolite often can be strong
enough to overcome the approximately 24 dB of interference suppression
provided by the cross-correlation properties of distinct Gold codes and
also can obliterate a second overlapping signal by saturating the receiver
amplifiers. Both of these problems can be solved by properly designed
wideband precorrelation signal clipping, in which there are two conflicting
requirements. Deep (severe) clipping significantly reduces the amount of
interfering power from a strong signal but gives the stronger signal more
ability to blank out the weaker one (capture effect). On the other hand, more
modest clipping levels reduce the capture effect at the expense of passing
more power from the stronger signal into the correlators. As a result, more
stress is put on the Gold codes to separate the weaker pulses from the
stronger ones in the correlation process. An acceptable compromise for
most purposes is to clip the received signal at about ±2 standard deviations
of the precorrelation noise power.
4.2.7 Immunity to Pseudolite Signals
A receiver that is not designed to receive pseudolite signals (a so-called nonparticipating receiver) must be designed so that a pseudolite signal, which might
be 60 dB stronger than a satellite signal, will not interfere with the latter. The
importance of this requirement cannot be overstated, since it is expected that use
of pseudolites will grow dramatically, especially near airports. Therefore, purchasers of nonparticipating receivers would be well advised to obtain assurances
of immunity to jamming by pseudolites.
Pseudolite immunity in a nonparticipating receiver can be effected by designing the front-end amplifier circuits for quick recovery from overload in combination with precorrelation hard limiting of the signal. This approach is suitable
for low-cost receivers such as handheld units. More sophisticated receivers using
more than 1 bit of digital quantization to avoid quantization loss may still be
designed to operate well if the clipping level is the same as that used in participating receivers. The design issues for obtaining immunity to pseudolite interference
have been analyzed by RTCM SC 104 and can be found in Ref. 180.
4.2.8 Aiding Inputs
Although GPS can operate as a standalone system, navigation accuracy and coverage can be materially improved if additional information supplements the received
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GPS signals. Basic sources include the following:
1. INS Aiding. Although GPS navigation is potentially very accurate, periods
of poor signal availability, jamming, and high-dynamics platform environments often limit its capability. INSs are relatively immune to these
situations and thus offer powerful leverage in performance under these
conditions. On the other hand, the fundamental limitation of INS long-term
drift is overcome by the inherent calibration capability provided by GPS.
Incorporation of INS measurements is readily achieved through Kalman
filtering.
2. Aiding with Additional Navigation Inputs. Kalman filtering can also use
additional measurement data from navigation systems such as LORAN-C,
vehicular wheel sensors, and magnetic compasses, to improve navigation
accuracy and reliability.
3. Altimeter Aiding. A fundamental property of GPS satellite geometry causes
the greatest error in GPS positioning to be in the vertical direction. Vertical
error can be significantly reduced by inputs from barometric, radar, or laser
altimeter data. Coupling within the system of positioning equations tends
to reduce the horizontal error as well.
4. Clock Aiding. An external clock with high stability and accuracy can materially improve navigation performance. It can be continuously calibrated
when enough satellite signals are available to obtain precise GPS time.
During periods of poor satellite visibility it can be used to reduce the
number of satellites needed for positioning and velocity determination.
4.3 HIGH-SENSITIVITY-ASSISTED GPS SYSTEMS (INDOOR
POSITIONING)
The last decade (since 1996) has seen increasing interest in the development of
very high-sensitivity GPS receivers for use in poor signal environments. More
generally, such receivers can be designed for use with any global navigation
satellite system (GNSS), such as the Russian GLONASS and European Galileo
systems. A major application is incorporation of such receivers in cell phones,
thus enabling a user to automatically transmit his location to rescue authorities in
emergencies such as a 911 call. Such a receiver must be able to reliably operate
deep within buildings or heavy vegetation, which severely attenuates the GPS
signals.
In order to achieve the requisite reliable and rapid positioning for such applications, assisting data from a base station receiver (the server) at a location having
good signal reception is sent to the user’s receiver (the client). The assisting
data can include base station location, satellite ephemeris data, the demodulated
navigation data bit stream, frequency calibration data, and timing information. In
addition the base station can provide pseudorange and/or carrier phase measurements that enable differential operation. The assisting data can be transmitted via
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a cell phone or other radiolink. In some case the assisting information can be
transmitted over the Internet and relayed to a cell phone via a local-area wireless
link. An example of an assisted GPS system can be found in Ref. 207.
Assisting data can not only increase the sensitivity of the client receiver, but
also can significantly reduce the time required to obtain a position. A typical
standalone GPS receiver can acquire signals down to about −145 dBm, and
might require a minute or more to obtain a position from a cold start On the
other hand, high-sensitivity assisted GPS receivers are currently being produced
by a number of major manufacturers who are claiming sensitivities in the −155
to −165 dBm range and a cold start time to first fix (TTFF) under 10 seconds. To
gain the required sensitivity and processing speed, assisted GPS receivers usually
capture several seconds of received signal in a memory that can be accessed at
high speed to facilitate the signal processing operations.
4.3.1 How Assisting Data Improves Receiver Performance
4.3.1.1 Reduction of Frequency Uncertainty To achieve rapid positioning,
the range of frequency uncertainty in acquiring the satellites at the client receiver
must be reduced as much as practicable in order to reduce the search time.
Reducing the number of searched frequency bins also increases receiver sensitivity because the acquisition false-alarm rate is reduced. Two ways that frequency
uncertainty can be reduced are as follows:
1. Transmission of Doppler Information. The server can accurately calculate
signal Doppler shifts at its location and transmit them to the user. For best
results, the user must either be reasonably close to the server’s receiver or
must know his or her approximate position to avoid excessive uncompensated differential Doppler shift between server and client.
2. Transmission of a Frequency Reference. If only Doppler information is
transmitted to the user, the frequency uncertainty of the client receiver
local oscillator still remains an obstacle to rapid acquisition. Today’s technology can produce oscillators that have a frequency uncertainty on the
order of 1 part per million at a cost low enough to permit incorporation
into a consumer product such as a cell phone. Even so, 1 part per million
translates into about ±1575 Hz of frequency uncertainty at the GPS L1 frequency. Assuming that the coherent integration time during satellite search
is 20 ms (the length of a navigation message data bit), the frequency bins
in the search would have a 50-Hz spacing, and a total of 2 × 1575/50 = 63
frequency bins might have to be searched to find the first satellite. Once
the first satellite is acquired, the local-oscillator offset can be determined,
and the frequency uncertainty in searching for the remaining satellites can
thereby be reduced to a small value. To remedy the problem of acquiring
the first satellite in a sufficiently short time, some assisted GPS systems use
an accurate frequency reference transmitted from the server to the client,
in addition to satellite Doppler measurements. However, this requirement
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131
significantly complicates the design of the server-to-client communication
system, and is certainly undesirable when trying to use an existing communication system for assisting purposes. If the communication system is
a cell phone network, every cell tower would need to transmit a precise
frequency reference.
4.3.1.2 Determination of Accurate Time In order to obtain accurate pseudoranges, a conventional GPS receiver obtains time information from the navigation
data message that permits the precise GPS time of transmission of any part of
the received signal to be tracked at the receiver. When a group of pseudorange
measurements is made, the time of transmission from each satellite is used for
two purposes: (1) to obtain an accurate position of each satellite at the time
of transmission and (2) to compute pseudorange by computing the difference
between signal reception time (according to the receiver clock) and transmission
time.
In order to obtain time information from the received GPS signal, a conventional receiver must go through the steps of acquiring the satellite signal, tracking
it with a phase-lock loop to form a coherent reference for data demodulation,
achieve bit synchronization, demodulate the data, achieve frame synchronization,
locate the portion of the navigation message that contains the time information,
and finally, continue to keep track of time (usually by counting C/A-code epochs
as they are received).
However, it is desirable to avoid these numerous and time-consuming steps in
a positioning system that must reliably obtain a position within several seconds
of startup in a weak-signal environment. Because the navigation data message
contains time information only once per 6-s subframe, the receiver may have to
wait a minimum of 6 s to obtain it (additionally, more time is needed to phaselock to the signal and achieve bit and frame synchronization). Furthermore, if
the signals are below about −154 dBm, demodulation of the navigation data
message has an error rate that precludes the reading of time from the signal.
If the approximate position of the client is known with sufficient accuracy
(perhaps within 100 km), it is possible to resolve the difference in times of
transmission. This is possible because the times of transmission of the C/Acode epochs are known to be integer multiples of 1 ms according to SV time
(which can be corrected to GPS time using slowly changing time correction data
sent from the server). This integer ambiguity in differences of time transmission
is resolved by using approximate ranges to the satellites, which are calculated
from the approximate position of the client and insertion of approximate time
into satellite ephemeris data sent by the server to the client. For this purpose
the accuracy of the approximated time needs to be sufficiently small to avoid
excessive uncertainty in the satellite positions. Generally a time accuracy of
better than 10 s will suffice.
Once the ambiguity of the differences in transmission times has been resolved,
accurate positioning is possible if the positions of the satellites at transmission
time are known with an accuracy comparable to the positioning accuracy desired.
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However, since the satellites are moving at a tangential orbital velocity of approximately 3800 m/s, the accuracy in knowledge of signal transmission time for the
purpose of locating the satellites must be significantly more accurate than that
required for the ambiguity resolution previously described.
Most weak-signal assisted GPS rapid positioning systems obtain the necessary
time accuracy for locating the satellites by using time information transmitted
from the server. It is important to recognize that such time information must be
in “real time”; that is, it must have a sufficiently small uncertainty in latency
as it arrives at the client receiver. For example, a latency uncertainty of 0.1 s
could result in a satellite position error of 380 m along its orbital path, causing
a positioning error of the same order of magnitude.
Transmission of time from the server with small latency uncertainty has a
major impact on the design of the server-to-client communication system, and is
a major disadvantage in getting the providers of existing communication systems,
such as cellular networks, to become involved in providing indoor assisted GPS
positioning service.
4.3.1.3 Transmission of Satellite Ephemeris Data Due to the structure of the
GPS navigation message, up to 30 s is required for a standalone GPS receiver to
obtain the ephemeris data necessary to determine the position of a satellite. This
delay is undesirable in emergency applications. Furthermore, in indoor operation the signal is likely to be too weak to demodulate the ephemeris data. The
problem is solved if the server transmits the data to the client via a high-speed
communication link. The Internet can even be used for this purpose if the client
receiver has access to a high-speed internet connection.
4.3.1.4 Provision of Approximate Client Location Some servers (e.g., a cell
phone network) can transmit the approximate position of the client receiver to
the user. As mentioned previously, this information can be used to resolve the
ambiguity in times of signal transmission from the satellites.
4.3.1.5 Transmission of the Demodulated Navigation Bit Stream The ultimate achievable receiver sensitivity is affected by the length of the signal capture
interval and the presence of navigation data modulation on the GPS signal.
Fully Coherent Processing If the GPS signal were modulated only by the C/Acode and contained no navigation data modulation, maximum theoretically possible acquisition sensitivity would result from fully coherent delay and Doppler
processing. In this form of processing the baseband signal in the receiver is
frequency-shifted and precession-compensated in steps (Doppler bins), and for
each step the signal is cross-correlated with a replica of the C/A-code spanning
the entire signal observation interval. Alternatively, the 1-msec periods of the
C/A-code could be synchronously summed prior to cross-correlation.
However, the presence of the 50-bps (bits per second) navigation data modulation precludes the use of fully coherent processing over signal capture intervals
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exceeding 20 ms unless some means is available to reliably strip the modulation
from the signal. If the server can send the demodulated data bit stream to the
client, the data modulation can be stripped from the user’s received signal, thus
enabling fully coherent processing. However, the timing of the bit stream must
be known with reasonable accuracy (within approximately 1 ms); otherwise a
search for time alignment must be made.
Partially Coherent Processing In the absence of a demodulated data bit stream
from the server, a common method of dealing with the presence of data modulation is to coherently process the signal within each data bit interval, followed by
noncoherent summation of the results. Assuming that the timing of the data bit
boundaries is available from the server, the usual implementation of this technique
is to first coherently sum the 20 periods of the complex baseband C/A coded signal within each data bit. For each data bit a waveform is produced that contains
one 1-ms period of the C/A-code, with a processing gain of 10 log(20) = 13
dB. Each waveform is then cross-correlated with a replica of the C/A-code
to produce a complex-valued cross-correlation function. The squared magnitudes of the cross-correlation functions are computed and summed to produce
a single function spanning 1 ms, and the location of the peak value of the function is the signal delay estimate. We shall call this form of processing partially
coherent.
When 1 second of signal is observed by the user, fully coherent processing gives a sensitivity approximately 3–4 dB over partially coherent processing.
It is important to note that fully coherent processing has a major drawback–many more delay/Doppler bins must be processed, which either dramatically
slows down processing speed or requires a large amount of parallel processing
to maintain that speed.
Data Detection and Removal by the Client Receiver An alternate method of
achieving fully coherent processing is to have the client receiver detect the
data bits and use them to homogenize the polarity of the signal, thus permitting coherent processing over the full signal capture interval. In order for this
method to be effective, the signal must be strong enough to ensure reliable
data bit detection. Furthermore, a phase reference is needed, and it should be
estimated using the entire signal observation. A practical technique for estimating phase, which approaches theoretically optimum results, is the method
of maximum likelihood (ML). We shall call this methodology coherent processing with data stripping, or simply data stripping for short. At low signal
power levels (less than about −160 dBm) its performance approaches that of partially coherent processing, and at high signal levels its performance approaches
that of fully coherent processing. At first glance, it seems that data stripping
might give a worthwhile advantage over partially coherent processing. However, it shares a common disadvantage with fully coherent processing in that a
larger number of delay/Doppler bins must be processed, and the cost is often
prohibitive.
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4.3.2 Factors Affecting High-Sensitivity Receivers
In a good signal environment, a certain amount of signal-to-noise ratio (SNR)
implementation loss is tolerable. Typical standalone GPS receivers for outdoor
use may have total losses as great as 3–6 dB. However in a high-sensitivity
receiver the maintenance of every decibel of signal-to-noise ratio is important,
thus requiring attention to minimizing losses that would otherwise not be of
concern. The following are some of the more important issues that arise in highsensitivity receiver design.
4.3.2.1 Antenna and Low-Noise RF Design A good antenna and a low-noise
receiver front end are mandatory elements of a high-sensitivity receiver.
4.3.2.2 Degradation Due to Signal Phase Variations With fully coherent processing over long time intervals, performance is adversely affected by signal
phase variations from sources including Doppler curvature due to satellite motion,
receiver oscillator phase stability, and motion of the receiver. Doppler curvature
can be partially predicted from assisting almanac or ephemeris data, but its accuracy depends on knowledge of the approximate position of the user. On the other
hand, oscillator phase noise is random and unpredictable, and hence resistant to
compensation (for this reason research efforts are currently underway to produce
a new generation of low-cost atomic and optical frequency sources). Especially
pernicious is motion of a GPS receiver in the user’s hand. Because of the short
wavelengths of GPS signals, such motion can cause phase variations of more
that a full cycle during the time that the receiver is searching for a signal, thus
seriously impairing acquisition performance.
4.3.2.3 Signal Processing Losses There are various forms of processing loss
that must be minimized in a high-sensitivity GPS receiver.
Digitization Losses due to quantization of the analog-to-digital converter (ADC)
digital output must be minimized. The 1-bit quantization often used in low-cost
receivers causes almost 2 dB of SNR loss. Hence it is desirable to use an ADC
with at least 2 bits in high-sensitivity applications.
Sampling Considerations The bandwidth of the receiver should be large enough
to avoid SNR loss. However, this generally requires higher sampling rates with
an attendant increase in power consumption and processing loads, a factor that
is detrimental to low-cost, low-power consumer applications.
Correlation Losses Rapid signal acquisition drives the need for coarser quantization of correlator reference code phase during signal search. However, this
causes correlation loss, and an acceptable tradeoff must be made. Correlation
loss is further exacerbated if the receiver bandwidth is made small to reduce the
required sampling rate.
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135
Doppler Compensation Losses One source of these losses is “scalloping loss,”
caused by the discretization of the Doppler frequencies used in searching for the
satellites. Scalloping loss can be as large as 2 dB in some receivers. Another
source is phase quantization of the Doppler compensation, which can introduce
a degradation of as much as 1 dB in the simplest designs.
4.3.2.4 Multipath Fading It is common in poor signal environments, especially indoors, for the signal to have large and/or numerous multipath components.
In addition to causing pseudorange biases, multipath can significantly reduce
receiver sensitivity when phase cancellation of the signal occurs.
4.3.2.5 Susceptibility to Interference and Strong Signals As receiver sensitivity is increased, so does the susceptibility to various forms of interference.
Although this is seldone a problem with receivers of normal sensitivity, in a
high-sensitivity receiver steps must be taken to prevent erroneous acquisition
of lower-level PN code correlation sidelobes from both desired and undesired
satellite signals.
4.3.2.6 The Problem of Time Synchronization In assisted GPS systems designed for rapid positioning (within a few seconds) using weak signals, the user’s
receiver does not have time to read unambiguous time from the received signal
itself. The need for the base station to transmit to the user low-latency time
accurate enough to establish the position of the satellites is a major disadvantage
in getting the providers of existing communication networks, such as cellular
networks, to provide this capability.
4.3.2.7 Difficulties in Reliable Sensitivity Assessment Realistic assessment
of receiver sensitivity is a challenging task. At the extremely low signal levels
for which a high-sensitivity receiver has been designed, laboratory signal generators often have signal leakage, which causes the signal levels to be higher
than indicated by the generator. For this and other reasons, published sensitivity
specifications should at least be regarded with healthy skepticism. A meaningful
comparison of competing specifications can be a daunting task if there is not an
adequate description of the conditions under which the sensitivity measurements
are made.
4.4 ANTENNA DESIGN
Although there is a wide variety of GPS antennas, most are normally right-hand
circularly polarized to match the incoming signal and the spatial reception pattern
is nominally a hemisphere. Such a pattern permits reception of satellites in any
azimuthal direction from zenith down to the horizon. The short wavelengths at
the L1 and L2 frequencies permit very compact designs. In low-cost handheld
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receivers the antenna is often integrated with the receiver electronics in a rugged
case. In more sophisticated applications it is often desirable that the antenna
be separate from the receiver in order to site it more advantageously. In these
situations the signal is fed from the antenna to the receiver via a low-loss coaxial
cable. At L-band frequencies the cable losses are still quite large and can reach
1 dB for every 10 ft of cable. Thus, it is often necessary to use a low-noise
preamplifier at the antenna (active antenna). Preamplifier gain is usually in the
range of 20–40 dB, and DC power is commonly fed to the preamplifier via
the coaxial cable itself, with appropriate decoupling filters to isolate the signal
from the DC power voltage. The preamplifier sets the noise figure for the entire
receiver system and typically has a noise figure of 1.5–3 dB.
4.4.1 Physical Form Factors
Figure 4.3 shows several common physical forms of GPS antennas.
Patch Antennas The patch antenna, the most common antenna type, is often
used in low-cost handheld receivers. In typical designs the antenna elements are
formed by etching the copper foil on a printed-circuit board, which forms a
very rugged low-profile unit. This is advantageous in some aviation applications,
because it is relatively easy to integrate the antenna into the skin of the aircraft.
Dome Antennas These antennas are housed in a bubblelike housing.
Blade Antennas The blade antenna, also commonly used in aviation applications, resembles a small airfoil protruding from its base.
Helical (Volute) Antennas Helical antennas contain elements that spiral along an
axis that typically points toward the zenith. In some designs the helical elements
Fig. 4.3
Types of GPS antennas.
ANTENNA DESIGN
137
are etched from a cylindrical copper-clad laminate to reduce cost. Helical antennas are generally more complex and costly to manufacture than patch antennas
but tend to be somewhat more efficient. Some handheld receivers use this type
of antenna as an articulated unit that can be adjusted to point skyward while the
case of the receiver can be oriented for comfortable viewing by the user. A popular design is the quadrifilar helix, which consists of four helixes symmetrically
wound around a circular insulating core.
Choke-Ring Designs In precision applications, such as surveying, choke-ring
antennas are used to reduce the effects of multipath signal components reflected
from the ground. These antennas are usually of the patch or helical type with a
groundplane containing a series of concentric circular troughs one-quarter wavelength deep that act as transmission lines shortcircuited at the bottom ends so
that their top ends exhibit a very high impedance at the GPS carrier frequency.
Low-elevation angle signals, including ground-reflected components, are nulled
by the troughs, reducing the antenna gain in these directions. The size, weight,
and cost of a choke-ring antenna are significantly greater than that of simpler
designs.
Phased-Array Antennas Although most applications of GPS require a nominally
hemispherical antenna pattern, certain applications (especially military) require
that the antenna be capable of forming beams in specified directions to obtain
better spatial gain or to form nulls in the direction of intentional jamming signals
to reduce their effect on the desired GNSS signals. Principles of operation of
phased-array antennas are outlined in Section 4.4.3.
Needless to say, phased-array antennas are much more costly than simpler
designs and historically have only been used by the military. However, civilian
applications have recently begun to emerge, primarily for the purpose of improving positioning performance in the presence of multipath. An introduction to
multipath-mitigation antennas and a design example can be found in [41].
4.4.2 Circular Polarization of GPS Signals
An important design goal of GPS antennas is to obtain good performance with
the right-hand circularly polarized (RHCP) electromagnetic field characteristic of
these signals. It is also desirable for the antenna to have little or no response
to multipath signal components in which the sense of polarization is typically
changed to left-hand circular polarization (LHCP) by reflection from objects in
the vicinity of the antenna.
Figure 4.4 is a somewhat oversimplified illustration of the response of an
antenna designed for RHCP signals. The antenna consists of two orthogonal
dipoles, which can be assumed to lie in a horizontal plane, with the incoming
signal coming directly from above. The centers of the dipoles lie on the origin
138
RECEIVER AND ANTENNA DESIGN
y
2
w
e(t)
1
1
x
2
cos wt
(dipole 1)
2 cos wt
sin wt
(dipole 2)
π/2
Fig. 4.4
Antenna responsive to RHCP signals.
of an x –y coordinate system, with dipole 1 on the x axis and dipole 2 on the
y axis. The positive directions of the x and y axes are respectively indicated by
unit vectors i and j.
The normalized vector electrostatic field e(t) of an arriving RHCP signal in
the x-y plane can be represented by
e (t) = i cos (ωt) − j sin (ωt) ,
(4.3)
which is a unit vector rotating clockwise at angular rate ω, where ω is the
frequency of the GPS carrier in radians per second (rad/s). Dipole 1 responds
only to the x component of the arriving signal and dipole 2, only to the y
component. The polarities of the dipole outputs are such that their respective
normalized output voltages are
s1 (t) = cos ωt,
s2 (t) = sin ωt.
(4.4)
The output of dipole 2 is phase-shifted by π/2 radians and summed with the
output of dipole 1 to obtain the signal
s (t) = cos ωt + sin (ωt + π/2)
= 2 cos ωt.
(4.5)
139
ANTENNA DESIGN
On the other hand, if the incoming signal is changed to LHCP, then
e (t)
s1 (t)
s2 (t)
s (t)
=
=
=
=
=
i cos (ωt) + j sin (ωt)
cos ωt
− sin ωt
cos ωt − sin (ωt + π/2)
0,
⎫
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎭
(4.6)
and there is no response to the signal.
4.4.3 Principles of Phased-Array Antennas
Two general types of phased-array antennas are used with GNSS receivers: the
single output adaptive nulling antenna and the multiple output beam-steering
antenna, as illustrated in Fig. 4.5. Both types employ N antenna elements whose
outputs are amplitude- and/or phase-weighted to produce the desired spatial reception pattern.
4.4.3.1 The Single-Output Adaptive Nulling Antenna This type of antenna
is used to sense the presence of an interfering signal and adaptively place nulls
in the direction of jamming signals. The objective of the adaptive algorithm is
to adjust the weighting of the antenna elements so that the power in the sum of
Fig. 4.5
Phased-array antenna operation.
140
RECEIVER AND ANTENNA DESIGN
the weighted signal outputs is minimized, subject to a constraint that prevents
the minimized power from being zero. Typically the constraint is provided by
fixing the weight of one of the antenna elements and allowing the other weights
to be adjusted. In mathematical terms the weights are adjusted to minimize the
-2 of the sum
average power |r|
r (t) =
N
.
wj sj (t),
(4.7)
j =1
where the unweighted output signal from the j th antenna element is sj (t) and
wj is the complex weight (characterized by amplitude and phase) applied to the
j th element. Without loss of generality it can be assumed that the constraint
is applied by fixing the weight w1 at unity. Since there are N − 1 degrees of
freedom in adjusting the weights, as many as N − 1 nulls in the antenna spatial
pattern can be generated. Because the direction of arriving GNSS signals is not
taken into account, there is the possibility that the antenna gain might be low in
some signal directions.
Since the single output adaptive nulling antenna provides only one output
signal r (t), it can be used with a standard receiver plus the hardware and software
for performing the adaptive nulling.
4.4.3.2 The Multiple-Output Beamforming Antenna Instead of forming nulls
to reject interfering signals, this type of antenna uses beamforming to produce
M independent beams. Thus, there are M corresponding output signals that the
receiver must be able to process independently. Each beam can be pointed toward
a satellite to achieve a first-order spatial gain of 10 log10 N dB relative to an
isotropic antenna. A separate set of weights is required for each beam. Instead
of being altered adaptively, the weights are typically computed on the basis of
antenna attitude information and satellite ephemeris data, so that each beam points
toward a satellite. The beamformer signal outputs are
rk (t) =
N
.
wkj sj (t) k = 1, 2, . . . , M
(4.8)
j =1
where wkj is the complex weight for the kth beam and the j th antenna element.
For each k the weights wkj , j = 1, 2, . . . , N are computed to generate a narrow
beam in the direction of a specified unit vector uk .
If jamming signals arrive from directions sufficiently different from the directions of the satellites, the multiple-output beamsteering antenna can also provide
reasonably good suppression of jammers.
ANTENNA DESIGN
141
4.4.3.3 Space–Frequency Processing The interference reduction capability
of both types of phased-array antennas is best when the interfering signals are
narrowband. Broadband jammers present a problem, because for a given set
of weights the antenna spatial gain pattern varies with frequency. Thus, a set
of weights optimal at one frequency will not be optimal at another frequency.
Space–frequency processing solves this problem by dividing the frequency band
into multiple narrow subbands, typically by using a fast Fourier transform. For
each subband a set of optimal weights is used to obtain a corresponding desired
antenna spatial pattern for that subband. By combining the nullformed or beamformed signals from all the subbands, an optimal wideband antenna spatial pattern
is obtained. Similar results can be obtained by using a bank of narrowband filters
in the time domain.
4.4.3.4 Relative Merits of Adaptive Nulling and Beamforming Antennas
•
•
•
•
•
Adaptive nulling is much simpler and cheaper than beamforming, since only
one output emerges from the process, enabling use with a standard GNSS
receiver. A beamforming antenna produces one output for each beam, so
a considerably more complex receiver is required to process each output
independently.
Beamforming can produce significant spatial gain in the direction of the
GNSS satellites, while adaptive nulling makes no attempt to maximize gain
in the desired directions.
Jamming reduction is usually greater with an adaptive nulling antenna,
because it can place deep nulls in the direction of jamming signals. Beamforming makes no attempt to do this, so its jamming performance is determined by the ratio of gain in the direction of a satellite to the gain in the
direction of a jammer.
Beamformers tend to have high spatial gain in the direction of desired
signals and lower gain in other directions. Multipath arriving from a lowgain direction is therefore attenuated relative to the desired signal. Adaptive
nulling is less effective against multipath because it does not emphasize
signals arriving from a desired direction.
Because the large physical extent of the antenna array, beamformers and
nullers have the common problem of causing biases in signal delay caused
by movement of the antenna phase center as a function of the weight values.
In some cases biases of 100◦ in carrier phase and one meter in code phase
can occur. For high-precision systems, these errors can be significant.
4.4.4 The Antenna Phase Center
GNSS positioning locates the antenna position, not the receiver position. However, since the antenna has physical extent, it is not simply a point in space. This
is not significant in most applications in which positioning errors on the order of
142
RECEIVER AND ANTENNA DESIGN
decimeters or more are usually encountered. However, in high-precision differential applications where accuracy at the centimeter level or below is needed, the
definition of antenna location becomes important. The phase center of an antenna
can be defined as the point in space where the electrostatic field of the signal
exactly matches the signal emerging from the antenna terminals (or equivalently,
matches the emerging signal except for a known delay).
The phase center can vary with the arrival direction of the signal, usually
within the range of 1 cm or less. There are two basic methods of dealing with this
problem. One method is to calibrate the phase center as a function of signal arrival
direction. The calibration uses a physical point on the antenna as a reference
point. Calibration is usually performed in an anechoic chamber with a very precise
signal source that can be moved to different positions around the antenna. Another
method is to use identical antennas oriented in the same way at the two receivers
in a differential GNSS system.
PROBLEMS
4.1 An ultimate limit on the usability of weak GPS signals occurs when the bit
error rate (BER) in demodulating the 50-bps (bits per second) navigation
message becomes unacceptably large. Find the signal level in dBm at the
output of the receiver antenna that will give a BER of 10−5 . Assume an
effective receiver noise temperature of 513◦ K, and that all signal power
has been translated to the baseband I channel with optimal demodulation
(integration over the 20-ms bit duration followed by polarity detection).
4.2 Support the claim that a 1-bit analog-to-digital converter (ADC) provides an
essentially linear response to a signal deeply buried in Gaussian noise by
solving the following problem. Suppose that the input signal sin to the ADC
is a DC voltage embedded in zero-mean additive Gaussian noise n(t) with
standard deviation σin and that the power spectral density of n(t) is flat in
the frequency interval [−W, W ] and zero outside the interval. Assume that
the 1-bit ADC is modeled as a hard limiter that outputs a value vout = 1 if
the polarity of the signal plus noise is positive and vout = −1 if the polarity
is negative. Define the output signal sout by
sout = E[vout ],
(4.9)
where E denotes expectation, and let σout be the standard deviation of
the ADC output. The ADC input signal-to-noise ratio SNRin can then be
defined by
s
SNRin = in
σin
(4.10)
143
ANTENNA DESIGN
and the ADC output signal-to-noise ratio SNRout by
s
SNRout = out ,
σout
(4.11)
where sout and σout are respectively the expected value and the standard
deviation of the ADC output. Show that if sin << σin , then sout = Ksin ,
where K is a constant, and
SNRout
2
= .
SNRin
π
(4.12)
Thus, the signal component of the ADC output is linearly related to the input
signal component, and the output SNR is about 2 dB less than that of the
input.
4.3 Some GPS receivers directly sample the signal at an IF instead of using
mixers for the final frequency shift to baseband. Suppose that you wish to
sample a GPS signal with a bandwidth of 1 MHz centered at an IF of 3.5805
MHz. What sampling rates will not result in frequency aliasing? Assuming
that a sampling rate of 2.046 MHz were used, show how a digitally sampled
baseband signal could be obtained from the samples.
4.4 Instead of forming a baseband signal with I and Q components, a singlecomponent baseband signal can be created simply by multiplying the incoming L1 (or L2 ) carrier by a sinusoid of the same nominal frequency, followed
by lowpass filtering. Discuss the problems inherent in this approach. (Hint:
Form the product of a sinusoidal carrier with a sinusoidal local-oscillator signal, use trigonometric identities to reveal the sum and difference frequency
components, and consider what happens to the difference frequency as the
phase of the incoming signal assumes various values.)
4.5 Write a computer program using C or another high-level language that produces the 1023-chip C/A-code used by satellite SV1. The code for this
satellite is generated by two 10-stage shift registers called the G1 and G2
registers, each of which is initialized with all 1s. The input to the first stage
of the G1 register is the exclusive OR of its 3rd and 10th stages. The input
to the first stage of the G2 register is the exclusive OR of its 2nd, 3rd, 6th,
8th, 9th, and 10th stages. The C/A-code is the exclusive OR of stage 10 of
G1, stage 2 of G2, and stage 6 of G2.
5
GLOBAL NAVIGATION SATELLITE
SYSTEM DATA ERRORS
5.1 SELECTIVE AVAILABILITY ERRORS
Prior to May 1, 2000, Selective Availability (SA) was a mechanism adopted by the
Department of Defense (DOD) to control the achievable navigation accuracy by
nonmilitary GPS receivers. In the GPS SPS mode, the SA errors were specified
to degrade navigation solution accuracy to 100 m (2D RMS) horizontally and
156 m (RMS) vertically.
In a press release on May 1, 2000, the President of the United States announced
the decision to discontinue this intentional degradation of GPS signals available to
the public. The decision to discontinue SA was coupled with continuing efforts
to upgrade the military utility of systems using GPS and supported by threat
assessments that concluded that setting SA to zero would have minimal impact
on United States national security. The decision was part of an ongoing effort to
make GPS more responsive to civil and commercial users worldwide.
The transition as seen from Colorado Springs, Colorado (USA) at the GPS
Support Center is shown in Fig. 5.1. The figure shows the horizontal and vertical
errors with SA, and after SA was suspended, midnight GMT (8 p.m. EDT),
May 1, 2000. Figure 5.2 shows mean errors with and without SA, with satellite
PRN numbers.
Aviation applications will probably be the most visible user group to benefit
from the discontinuance of SA. However, precision approach will still require
some form of augmentation to ensure that integrity requirements are met. Even
though setting SA to zero reduces measurement errors, it does not reduce the
need for and design of WAAS and LAAS ground systems and avionics.
Global Positioning Systems, Inertial Navigation, and Integration, Second Edition, by M. S. Grewal, L. R. Weill, and A. P. Andrews
Copyright © 2007 John Wiley & Sons, Inc.
144
SELECTIVE AVAILABILITY ERRORS
Fig. 5.1
145
Change in errors when SA is turned off.
Time and frequency users may see greater effects in the long term via communication systems that can realize significant future increases in effective bandwidth use due to tighter synchronization tolerances. The effect on vehicle tracking
applications will vary. Tracking in the trucking industry requires accuracy only
good enough to locate in which city the truck is, whereas public safety applications can require the precise location of the vehicle. Maritime applications
have the potential for significant benefits. The personal navigation consumer will
benefit from the availability of simpler and less expensive products, resulting in
more extensive use of GPS worldwide.
Because SA could be resumed at any time, for example, in time of military
alert, one needs to be aware of how to minimize these errors.
There are at least two mechanisms to implement SA. Mechanisms involve
the manipulation of GPS ephemeris data and dithering the satellite clock (carrier
frequency). The first is referred to as epsilon-SA (ε-SA), and the second as clockdither SA. The clock-dither SA may be implemented by physically dithering
the frequency of the GPS signal carrier or by manipulating the satellite clock
correction data or both.
Although the mechanisms for implementation of SA and the true SA waveform
are classified, a variety of SA models exist in the literature [e.g., [4, 24, 35, 212]].
These references show various models. One proposed by Braasch [24] appears
146
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
Fig. 5.2 Location errors with and without SA.
147
SELECTIVE AVAILABILITY ERRORS
to be the most promising and suitable. Another used with some success for
predicting SA is a Levinson predictor [10].
The Braasch model assumes that all SA waveforms are driven by normal white
noise through linear system [autoregressive moving average (ARMA)] models
(see Chapter 3 of Ref. 66). Using the standard techniques developed in system
and parameter identification theory, it is then possible to determine the structure
and parameters of the optimal linear system that best describes the statistical
characteristics of SA. The problem of modeling SA is estimating the model of a
random process (SA waveform) based on the input/output data.
The technique used to find an SA model involves three basic elements:
The observed SA
A model structure
A criterion for determination of the best model from the set of candidate
models
There are three choices of model structures:
1. An ARMA model of order (p,q), which is represented as ARMA(p,q)
2. An ARMA model of order (p,0) known as the moving-average MA(p)
model
3. An ARMA model of order (q,0), the auto regression AR(q) model
Selection from these three models is performed with physical laws and past
experience.
5.1.1 Time-Domain Description
Given observed SA data, the identification process repeatedly selects a model
structure and then calculates its parameters. The process is terminated when a
satisfactory model, according to a certain criterion, is found.
We start with the general ARMA model. Both the AR and MA models can
be viewed as special cases of an ARMA model. An ARMA(p, q) model is
mathematically described by
a1 yk + a2 yk−1 + · · · + aq yk−q+1 = b1 xk + b2 xk−1 + · · · + bp xk−p+1 + ek ,
(5.1)
or in a concise form by
q
.
i=1
ai yk−i+1 =
p
.
bj xk−j +1 + ek ,
(5.2)
j =1
where ai , i = 1, 2, . . ., q and bj , j = 1, 2, . . ., p are the sets of parameters that
describe the model structure, xk and yk are the input and output to the model at
any time k for k = 1, 2, . . ., and ek is the noise value at time k. Without loss of
generality, it is always assumed that al = 1.
148
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
Once the model parameters ai and bj are known, calculation of yk for an
arbitrary k can be accomplished by
yk = −
q
.
ai yk−i+1 +
p
.
bj xk−j +1 + ek .
(5.3)
j =1
i=2
It is noted that when all the ai in Eq. 5.3 take the value of 0, the model is reduced
to the MA (p,0) model or simply MA(p). When all of the bj take the value of
0, the model is reduced to the AR(0,q) model or AR(q). In the latter case, yk is
calculated by
yk = −
q
.
ai yk−i+1 + ek .
(5.4)
i=2
5.1.1.1 Model Structure Selection Criteria Two techniques, known as
Akaike’s final prediction error (FPE) criterion and the closely related Akaike
information-theoretic criterion (AIC), may be used to aid in the selection of
model structure. According to Akaike’s theory, in the set of candidate models,
the one with the smallest values of FPE or AIC should be chosen. The FPE is
calculated as
FPE =
1 + n/N
V,
1 − n/N
(5.5)
where n is the total number of parameters of the model to be estimated, N is
the length of the data record, and V is the loss function for the model under
consideration. Here, V is defined as
V =
n
.
ei2 ,
(5.6)
i=1
where e is as defined in Eq. 5.2. The AIC is calculated as
AIC = log [(1 + 2n/N ) V ] .
(5.7)
In the following, an AR(12) model was chosen to characterize SA. This selection
was based primarily on Braasch’s recommendation [24]. As such, the resulting
model should be used with caution before the validity of this model structure
assumption is further studied using the above criteria.
5.1.1.2 Frequency-Domain Description The ARMA models can be equivalently described in the frequency domain, which provides further insight into
model behavior. Introducing a one-step delay operator Z −1 , Eq. 5.2 can be
rewritten as
A Z −1 yk = B Z −l xk + ek ,
(5.8)
149
SELECTIVE AVAILABILITY ERRORS
where
A(Z −1 ) =
q
.
ai Z −i+l ,
(5.9)
bi Z i+1 ,
(5.10)
i=1
B(Z
−1
)=
p
.
i=1
and
Z −1 yk = yk−1 .
(5.11)
It is noted that A(Z −1 ) and B(Z −1 ) are polynomials of the timeshift operator Z −1
and normal arithmetic operations may be carried out under certain conditions.
Defining a new function H (Z −1 ) as B(Z −1 ) divided by A(Z −1 ) and expanding
the resulting H (Z −1 ) in terms of operator Z −1 , we have
∞
.
−1 B Z −1
H Z
hi Z i+1 .
= −1 =
A Z
i=1
(5.12)
The numbers of {hi } are the impulse responses of the model. It can be shown
that hi is the output of the ARMA model at time i = 1, 2, . . . when the model
input xi takes the value of zero at all times except for i = 1. The function
H (Z −1 ) is called the frequency function of the system. By evaluating its value
for Z −1 = ej ω , the frequency response of the model can be calculated directly.
Note that this process is a direct application of the definition of the discrete
Fourier transform (DFT) of hi .
5.1.1.3 AR Model Parameter Estimation The parameters of an AR model
with structure
A Z −1 yk = ek ,
(5.13)
may be estimated using the least-squares (LS) method. If we rewrite Eq. 5.13 in
matrix format for k = q, q + 1, . . ., n, we get
⎡
⎤⎡
⎤ ⎡
⎤
yn
a1
en
yn−l · · · yn−q+1
⎢ yn−l yn−2 · · · yn−q ⎥ ⎢ a2 ⎥ ⎢ en−1 ⎥
⎢
⎥⎢
⎥ ⎢
⎥
(5.14)
⎢ ..
⎥ ⎢ .. ⎥ = ⎢ .. ⎥ ,
..
..
..
⎣ .
⎦⎣ . ⎦ ⎣ . ⎦
.
.
.
yq
yq−1
···
y1
aq
eq
or
H · A = E,
(5.15)
150
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
where
⎡
⎢
⎢
H =⎢
⎣
yn
yn−l
..
.
yn−l
yn−2
..
.
yq
yq−1
· · · yn−q+1
· · · yn−q
..
..
.
.
···
y1
⎤
⎥
⎥
⎥,
⎦
(5.16)
"
#T
A = a1 a2 a3 · · · aq ,
(5.17)
E = [en en−1 en−2 · · · eq ]T .
(5.18)
and
The LS estimation of the parameter matrix A can then be obtained by
−1 T
H E.
A = HT H
(5.19)
5.1.2 Collection of SA Data
To build effective SA models, samples of true SA data must be available. This
requirement cannot be met directly as the mechanism of SA generation and
the actual SA waveform are classified. The approach we take is to extract SA
from flight test data. National Satellite Test Bed (NSTB) flight tests recorded
the pseudorange measurements at all 10 RMS (reference monitoring station)
locations. These pseudorange measurements contain various clock, propagation,
and receiver measurement errors, and they can, in general, be described as
PRM = ρ + Tsat + Trcvr + Tiono + Ttrop + Tmultipath + SA + tnoise ,
(5.20)
where ρ is the true distance between the GPS satellite and the RMS receiver;
Tsat and Trcvr are the satellite and receiver clock errors; Tiono and Ttrop are
the ionosphere and troposphere propagation delays, Tmultipath is the multipath
error; SA is the SA error; and tnoise is the receiver measurement noise.
To best extract SA from PRM , values of the other terms were estimated. The
true distance ρ is calculated by knowing the RMS receiver location and the precise orbit data available from the National Geodetic Survey (NGS) bulletin board.
GIPSY1 /OASIS analysis (GOA) was used for this calculation, which recreated
the precise orbit and converted all relevant data into the same coordinate system. Models for propagation and satellite clock errors have been built into GOA,
and these were used to estimate Tsat , Tiono , and Ttrop . The receiver clock
errors were estimated by the NSTB algorithm using data generated from GOA
1
GPS Positioning System.
151
IONOSPHERIC PROPAGATION ERRORS
for the given flight test conditions. From these, a simulated pseudorange PRsim
was formed
PRsim = ρsim + Tsatsim + Trcvrsim + Tionosim + Ttrop
sim
,
(5.21)
where Tsatsim , Trcvrsim , Tionosim , and Ttropsim are, respectively, the estimated
values of Tsat , Trcvr , Tiono , and Ttrop in the simulation.
From Eqs. 5.20 and 5.21, pseudorange residuals are calculated
PR = PRM − PRsim = SA + Tmultipath + tnoise + Tmodels ,
(5.22)
where Tmodels stands for the total modeling error, given by
Tmodels = (ρ − ρsim ) + Tsat − Tsatsim + Trcvr − Trcvrsim
+ Tiono − Tionosim + Ttrop − Ttropsim .
(5.23)
It is noted that the terms Tmultipath and tnoise should be significantly smaller
than SA, although it is not possible to estimate their values precisely. The term
Tmodels should also be negligible compared to SA. It is, therefore, reasonable
to use PR as an approximation to the actual SA term to estimate SA models.
Examination of all available data show that their values vary between ±80 m.
These are consistent with previous reports on observed SA and with the DoD’s
specification of SPS accuracy.
5.2 IONOSPHERIC PROPAGATION ERRORS
The ionosphere, which extends from approximately 50 to 1000 km above the
surface of the earth, consists of gases that have been ionized by solar radiation.
The ionization produces clouds of free electrons that act as a dispersive medium
for GPS signals in which propagation velocity is a function of frequency. A particular location within the ionosphere is alternately illuminated by the sun and
shadowed from the sun by the earth in a daily cycle; consequently the characteristics of the ionosphere exhibit a diurnal variation in which the ionization is
usually maximum late in midafternoon and minimum a few hours after midnight.
Additional variations result from changes in solar activity.
The primary effect of the ionosphere on GPS signals is to change the signal propagation speed as compared to that of free space. A curious fact is
that the signal modulation (the code and data stream) is delayed, while the
carrier phase is advanced by the same amount. Thus the measured pseudorange using the code is larger than the correct value, while that using the
carrier phase is equally smaller. The magnitude of either error is directly proportional to the total electron content (TEC) in a tube of 1 m2 cross section along
the propagation path. The TEC varies spatially, due to spatial nonhomogeneity of the ionosphere. Temporal variations are caused not only by ionospheric
152
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
dynamics but also by rapid changes in the propagation path due to satellite
motion. The path delay for a satellite at zenith typically varies from about
1 m at night to 5–15 m during late afternoon. At low elevation angles the
propagation path through the ionosphere is much longer, so the corresponding delays can increase to several meters at night and as much as 50 m during
the day.
Since ionospheric error is usually greater at low elevation angles, the impact of
these errors could be reduced by not using measurements from satellites below a
certain elevation mask angle. However, in difficult signal environments, including
blockage of some satellites by obstacles, the user may be forced to use lowelevation satellites. Mask angles of 5◦ –7.5◦ offer a good compromise between
the loss of measurements and the likelihood of large ionospheric errors.
The L1 -only receivers in nondifferential operation can reduce ionospheric
pseudorange error by using a model of the ionosphere broadcast by the satellites, which reduces the uncompensated ionospheric delay by about 50% on the
average. During the day errors as large as 10 m at midlatitudes can still exist
after compensation with this model and can be much worse with increased solar
activity. Other recently developed models offer somewhat better performance.
However, they still do not handle adequately the daily variability of the TEC,
which can depart from the modeled value by 25% or more.
The L1 /L2 receivers in nondifferential operation can take advantage of the
dependence of delay on frequency to remove most of the ionospheric error. A
relatively simple analysis shows that the group delay varies inversely as the
square of the carrier frequency. This can be seen from the following model of
the code pseudorange measurements at the L1 and L2 frequencies:
ρi = ρ ±
k
,
fi2
(5.24)
where ρ is the error-free pseudorange, ρi is the measured pseudorange, and k is
a constant that depends on the TEC along the propagation path. The subscript
i = 1, 2 identifies the measurement at the L1 or L2 frequencies, respectively,
and the plus or minus sign is identified with respective code and carrier phase
pseudorange measurements. The two equations can be solved for both ρ and k.
The solution for ρ for code pseudorange measurements is
ρ=
f 21
f 22
ρ
−
ρ2 ,
1
f 21 − f 22
f12 − f 22
(5.25)
where f1 and f2 are the L1 and L2 carrier frequencies, respectively, and ρ1 and
ρ2 are the corresponding pseudorange measurements.
An equation similar to Eq. 5.25 can be obtained for carrier phase pseudorange
measurements. However, in nondifferential operation the residual carrier phase
pseudorange error can be greater than either an L1 or L2 carrier wavelength,
making ambiguity resolution difficult.
153
IONOSPHERIC PROPAGATION ERRORS
With differential operation ionospheric errors can be nearly eliminated in many
applications, because ionospheric errors tend to be highly correlated when the
base and roving stations are in sufficiently close proximity. With two L1 -only
receivers separated by 25 km, the unmodeled differential ionospheric error is
typically at the 10–20-cm level. At 100 km separation this can increase to as
much as a meter. Additional error reduction using an ionospheric model can
further reduce these errors by 25–50%.
5.2.1 Ionospheric Delay Model
J. A. Klobuchar’s model [54, 111] for ionospheric delay in seconds is given by
0
/
x4
x2
Tg = DC + A 1 −
+
2
24
for |x| ≤
π
,
2
(5.26)
where
x =
2π(t − Tp )
rad
P
DC = 5 ns (constant offset)
Tp = phase
= 50,400 s
A = amplitude
P = period
t = local time of the earth subpoint of the signal
intersection with mean ionospheric height (s)
The algorithm assumes this latter height to be 350 km. The DC and phasing Tp
are held constant at 5 ns and 14 h (50,400 s) local time.
Amplitude (A) and period (P ) are modeled as third-order polynomials:
A=
3
.
n
αn φm
(s),
n
βn φm
(s),
n=0
P =
3
.
n=0
where φm is the geomagnetic latitude of the ionospheric subpoint and αn ,βn are
coefficients selected (from 370 such sets of constants) by the GPS master control
station and placed in the satellite navigation upload message for downlink to the
user.
For Southbury, Connecticut, we obtain
"
#
αn = 0.8382 × 10−8 , −0.745 × 10−8 , − 0.596 × 10−7 , 0.596 × 10−7 ,
#
"
βn = 0.8806 × 105 , −0.3277 × 105 , − 0.1966 × 106 , 0.1966 × 106 .
154
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
The parameter φm is calculated as follows:
1. Subtended earth angle (EA) between user and satellite is given by the
approximation
%
&−4
445
EA ≈
(deg)
el + 20
where el is the elevation of the satellite and with respect to the user equals
15.5◦ .
2. Geodetic latitude (lat) and longitude (long) of the ionospheric subpoint are
found using the approximations
Iono lat φI = φuser + EA cos AZ
(deg),
EA cos AZ
Iono long λI = λuser +
cos φI
(deg),
where φuser is geodetic latitude = 41◦ , λuser is geodetic longitude = −73◦ ,
and AZ is azimuth of the satellite with respect to the user = 112.5◦ .
3. The geodetic latitude is converted to a geomagnetic coordinate system using
the approximation
◦
◦
φm ≈ φI + 11.6 cos λI − 291
(deg)
4. The final step in the algorithm is to account for elevation angle effect by
scaling with an obliquity scale factor (SF):
/
SF = 1 + 2
96◦ − el
90◦
03
(unitless) .
With scaling, time delay due to ionospheric becomes
&
%
⎧
2
4
π
⎪
x
x
⎪
|x| < ,
+
⎨ SF(DC) + A 1 − 2
24
2
Tg =
⎪
π
⎪
⎩
|x| ≥ ,
SF(DC),
2
TG = CTg
C = speed of light
t=
λI
+ UTC,
15
where Tg is in seconds and TG is in meters.
The MATLAB programs Klobuchar fix.m and Klobuchar(PRN) for computing ionospheric delay (for PRN = satellite number) are described in Appendix A.
155
IONOSPHERIC PROPAGATION ERRORS
5.2.2 GPS Ionospheric Algorithms
The ionospheric correction computation algorithms (ICC) enable the computation of the ionospheric delays applicable to a signal on L1 and to the GPS and
WRS (Wide-Area Reference Station) L1 and L2 interfrequency biases. These
algorithms also calculate GIVEs (grid Ionospheric vertical errors), empirically
derived error bounds for the broadcast ionospheric corrections. The ionospheric
delays are employed by the SBAS user to correct the L1 measurements, as well
as internally to correct the WRSs’ L1 GEO measurement for orbit determination
if dual-frequency corrections are not available from GEOs. The interfrequency
biases are needed internally to convert the dual-frequency-derived SBAS corrections to single-frequency corrections for the SBAS users. The vertical ionospheric delay and GIVE information is broadcast to the SBAS user via Message
Types 18 and 26. See MOPs for details on the content and usage of the SBAS
messages [167].
The algorithms used to compute ionospheric delays and interfrequency biases
are based on those originated at the Jet Propulsion Laboratory [130]. The ICC
models assume that ionospheric electron density is concentrated on a thin shell
of height 350 km above the mean earth surface. The estimates of interfrequency
biases and ionospheric delays are derived using a pair of Kalman filters, herein
referred to as the L1 L2 and iono filters. The purpose of the L1 L2 filter is to
estimate the interfrequency biases, while the purpose of the IONO filter is to estimate the ionospheric delays. The inputs to both filters are leveled WRS receiver
slant delay measurements (L2 minus L1 differential delay), which are output
from the data. Both filters perform their calculations in total electron count units
(TECU) (1m of L1 ranging delay = 6.16 TECU, and 1 m of L1 − L2 differential
delay = 9.52 TECU). Conceptually, the measurement equation is (neglecting the
noise term):
τTECU = 9.52 × τm
= 9.52 × tL2 , m − tL1 , m
r
s
+ bm
+ TECTECU
= 9.52 × bm
r
s
+ bTECU
+ TECTECU ,
= bTECU
(5.27)
(5.28)
(5.29)
(5.30)
where τ is differential delay, br and bs are the interfrequency biases of the respective receiver and satellite, and TEC is the ionospheric delay. The subscripts m
(meters) and TECU denote the corresponding units of each term. The ionospheric
delay in meters for a signal on the L1 frequency is
τmL1 = 1.5457 ×
=
1
TECTECU
9.52
1
TECTECU .
6.16
(5.31)
(5.32)
156
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
Both Kalman filters contain the vertical delays at the vertices of a triangular
spherical grid of height 350 km fixed in the solar-magnetic coordinate frame as
states. The L1 L2 filter also contains interfrequency biases as states. In contrast,
the IONO filter does not estimate the interfrequency biases, but instead they
are periodically forwarded to the IONO filter, along with the variances of the
estimates, from the L1 L2 filter. Each slant measurement is modeled as a linear
combination of the vertical delays at the three vertices surrounding the corresponding measurement pierce point (the intersection of the line of sight and the
spherical grid), plus the sum of the receiver and satellite biases, plus noise. The
ionospheric delays computed in the IONO filter are eventually transformed to
a latitude–longitude grid that is sent to the SBAS users via Message Type 26.
Because SBAS does not have any calibrated ground receivers, the interfrequency
bias estimates are all relative to a single receiver designated as a reference, whose
L1 L2 interfrequency bias filter covariance is initialized to a small value, and to
which no process noise is applied.
The major algorithms making up the ICC discussed here are
•
•
•
•
Initialization—the L1 L2 and IONO filters are initialized using either the
Klobuchar model or using previously recorded data.
Estimation—the actual computation of the interfrequency biases and ionospheric delays involves both the L1 L2 and IONO filters.
Thread switch—the measurements from a WRS may come from an alternate
WRS receiver. In this case, the ICC must compensate for the switch by
altering the value of the respective receiver’s interfrequency bias state in the
L1 L2 filter. In the nominal case, an estimate of the L1 L2 bias difference
is available.
Anomaly processing—the L1 L2 filter contains a capability to internally
detect when a bias estimate is erroneous. Both thread switch and anomaly
processing algorithms may also result in the change of the reference
receiver.
5.2.2.1 L1 L2 Receiver and Satellite Bias and Ionospheric Delay Estimations
System Model The ionospheric delay estimation Kalman filter uses a randomwalk system model. A state of the Kalman filter at time tk is modeled to be equal
to that state at the previous time tk−1 , plus a random process noise representing
the uncertainty in the transition from time tk−1 to time tk ; that is
xk = xk−1 + wk ,
where xk is the state vector of the Kalman filter at time tk and wk is a white
process noise vector with known covariance Q. The state vector xk consists of
three subgroups of states: the ionospheric vertical delays at triangular tile vertices,
157
IONOSPHERIC PROPAGATION ERRORS
the satellite L1 L2 biases, and the receiver L1 /L2 biases; that is
⎡
x1,k
..
.
⎢
⎢
⎢
xN V ,k
⎢
⎢
xN V +1,k
⎢
⎢
..
xk = ⎢
.
⎢
⎢
x
⎢
N V +NS,k
⎢
⎢ xN V +NS+1,k
⎢
..
⎣
.
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎥
⎥
⎦
xN V +NS+NR,k
where N V is the number of triangular tile vertices, N S is the number of GPS
satellites, and N R is the number of WRSs. The values of N V , N S, and N R
must be adjusted to fit the desired configuration. In simulations, one can use
24 GPS satellites in the real orbits generated by GIPSY using ephemeris data
downloaded from the GPS bulletin board. The number of WRSs is 25 and these
WRSs are placed at locations planned for SBAS operations.
Observation Model The observation model or measurement equation establishes the relationship between a measurement and the Kalman filter state vector.
For any GPS satellite in view, there is an ionospheric slant delay measurement
corresponding to each WRS–satellite pair. Ionospheric slant delay measurement
is converted to the vertical delay at its corresponding pierce point through an
obliquity factor. At any time tk , there are approximately 80–200 pierce points
and hence the same number of ionospheric vertical delay measurements that can
be used to update the Kalman filter state vector.
Denote the ionospheric vertical delay measurement at tk for the ith satellite
and j th WRS as zij k . Thus
zij k = iij k +
bsj
bsi
+
+ vij k
qij k
qij k
where iij k is the vertical ionospheric delay at the piece point corresponding to
satellite i and WRS j , bsi and bsj are the L1 /L2 interfrequency biases for satellite
i and WRS j , respectively, qij k is the obliquity factor, and vij k is the receiver
measurement noise, white with covariance R.
To establish an observation model, we need to relate iij k , bsi , and bsj to
the state vector of the ionospheric delay estimation Kalman filter. Note that bsi
and bsj are the elements of the state vector labeled N V + i and N V + N S + j ,
respectively. The relationship between iij k and the state vector is established
below. The value iij k is modeled as a linear combination of the vertical delay
values at the three vertices of the triangular tile in which the piece point is
located, as shown in Fig. 5.3.
158
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
Fig. 5.3
Bilinear interpolation.
In Fig. 5.3, assume a pierce point P is located arbitrarily in the triangular
tile ABC. The ionospheric delay at pierce point P is obtained from the vertical
delay values at vertices A, B, and C using a bilinear interpolation as follows.
Draw a line from point A to point P and find the intersection point D between
this line and the line BC. The bilinear interpolation involves two simple linear
interpolations—the first yields the vertical delay value at point D from points B
and C; the second yields the vertical delay value at point P from points D and
A. The result can be summarized as
IP = wA IA + wB IB + wC IC ,
where IP , IA , IB , and IC are the ionospheric vertical delay values at points
P , A, B, and C, respectively, and wA , wB , and wC are the bilinear weighting
coefficients from points A, B, and C, respectively, to point P . The values of
wA , wB , and wC can be readily calculated from the geometry involved. It is
recognized that IA , IB , and IC are three elements of the Kalman filter state
vector. In summary, the measurement equation can be written as
zij k = hij k xk + vij k ,
where hij k is the measurement matrix and vij k is the measurement noise, respectively, for the pierce point measurement for the satellite with index i and WRS
with index j at time tk . Here, hij k is an (N V + N S + N R) dimension row
vector with all elements equal to zeros except five elements. The first three of
these five nonzero elements correspond to the vertices of the tile that contains the
pierce point under consideration, and the other two correspond to the ith satellite
and j th WRS, which yields the ionospheric slant delay measurement zij k .
U DU T Kalman Filter (See Chapter 8) As noted previously, there are approximately 180–200 pierce points at any time tk . Each pierce point corresponds to
one of the possible combinations of a satellite and a WRS, which further corresponds to an ionospheric vertical delay measurement at that pierce point. The
IONOSPHERIC PROPAGATION ERRORS
159
ionospheric estimation Kalman filter is designed so that its state vector is updated
upon the reception of each ionospheric vertical delay measurement.
SM (Solar Magnetic)-to-ECEF Transformation At the end of each 5-min interval (Kalman filter cycle), the ionospheric vertical delays at the vertices of all tiles
are converted from the SM coordinates to the ECEF coordinates. This conversion
is completed by first transforming the SBAS IGPs from the ECEF coordinates to
the SM coordinates. For each IGP converted to SM coordinates, the triangular
tile which contains this IGP is found. A bilinear interpolation identical to the
one described in Fig. 5.3 is then used to calculate the ionospheric vertical delay
values at this IGP. (Transformations are given in Appendix C.)
In new GEOs (3rd, PRN 135, at 133◦ longitude; 4th, PRN 138, at 107◦ longitude) will have L1 /L5 frequencies (see Chapter 6). Ionospheric delays can be
calculated at the WRSs directly instead of using ionospheric delay provided by
ionospheric grids from SBAS broadcast messages.
5.2.2.2 Kalman Filter In estimating the ionospheric vertical delays in the SM
coordinate system by the Kalman filter, there are three types of estimation errors:
1. Estimation error due to ionospheric slant delay measurement noise error
2. Estimation error due to the temporal variation of the ionosphere
3. Estimation error due to nonlinear spatial variation of the ionosphere
Each of the three sources of error can be individually minimized by adjusting
the values of the covariances Q and R. However, the requirements to minimize
the errors due to noise and temporal variations are often in conflict.
Intuitively, to minimize the measurement noise implies that we want the Q and
R values to result in a Kalman gain that averages out the measurement noise. That
is, we want the Kalman gain to take values so that for each new measurement,
the value of innovation is small, such that a relatively large noise component
of the measurement results in a relatively small estimation error. On the other
hand, if we want to minimize the estimation error due to temporal variations,
then we want to have a Kalman gain that can produce a large innovation, so
that the component in the measurement that represents the actual ionospheric
delay variation with time can be quickly reflected in the new state estimate. This
suggests that we usually need to compromise in selecting the values of Q and R
when a conventional nonadaptive Kalman filter is used.
Although the Kalman filter estimation error is the dominant source of error, it
is not the only source. The nonlinear spatial variation introduces additional error
when converting the ionospheric vertical delay estimated by the Kalman filter
in the SM coordinate system to the SBAS IGP in the ECEF coordinate system.
This is because bilinear interpolation is used and there is an implicit assumption
that interpolation is a strictly valid procedure. However, if the actual value of the
vertical delay was measured at some location, it would not be equal to the value
found by interpolation. Violation of this assumption results in interpolation error
160
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
during the transformation. It can be shown by simulations that, under certain
conditions, this conversion error can be significant and non-negligible.
In order to isolate the sources of errors and understand how the algorithm
responds to various conditions, consider seven scenarios, with each testing one
aspect of the possible estimation error, and all their possible combinations.
Scenario 1: Measurement Noise. In this scenario, the ionospheric vertical delay
is assumed to be a time-invariant constant anywhere over the earth’s surface. The
Kalman filter estimation errors due to temporal and spatial variations are zero. For
each of the ionospheric slant delay measurements, a zero mean white Gaussian
noise is added. The magnitude of the noise is characterized by its variance. The
measurement noise is added to the slant delay rather than the vertical delay
because this is where the actual measurement noise is introduced by a GPS
receiver.
Scenario 2: Temporal Variation. In this scenario, the ionosphere is assumed to
be uniformly distributed spatially, but its TEC values change with time; that is,
the ionospheric vertical delays vary with time, but these variations are identical
everywhere. Various time variation functions, such as a sinusoidal function, a
linear ramp, a step function, or an impulse function, can be used to study this
scenario. In a simulation using a sinusoidal time variation function, the sinusoidal
function is characterized by two parameters—its amplitude and frequency. The
values of these two parameters are chosen to produce a time variation that is
similar in magnitude to the ionospheric delay variation data published in the
literature. The measurement noise is zero. Kalman filter estimation errors due
to both the measurement noise and spatial variation are fixed at zero (for this
scenario).
Scenario 3: Spatial variation. In this scenario, the ionosphere is assumed to
be a constant at any fixed location when observed in the SM coordinate system. The ionospheric delays at different locations in the SM coordinate system,
however, are different. Various spatial variation functions can be used to study
this scenario. Here, we use a three-dimensional surface constructed from two
orthogonal sinusoidal functions of varying amplitude and frequency to model the
values of ionospheric vertical delays over the earth. The values of the parameters
of the two sinusoidal functions are chosen to produce gradients in TEC similar
in magnitude to the ionospheric delay variation data published in the literature.
The measurement noise is zero. Kalman filter estimation errors due to both the
measurement noise and temporal variations are fixed at zero for this scenario.
Scenario 4: Noise + Temporal. Scenarios 1 and 2 are combined, and the Kalman
filter estimation error due to spatial variation is zero.
Scenario 5: Noise + Spatial. Scenarios 1 and 3 are combined. In this scenario,
the Kalman filter estimation error due to temporal variation is zero.
IONOSPHERIC PROPAGATION ERRORS
161
Scenario 6: Temporal + Spatial. Here, the Kalman filter estimation error due
to measurement noise is zero. The combined values of temporal and spatial
variations define the “truth ionosphere” in the simulation.
Scenario 7: Noise + Temporal + Spatial. In this scenario, the parameters that
define the “true ionosphere” and “measurement noise” can be configured to mimic
any ionospheric conditions.
In the simulations, the GPS satellite orbits used are the precise orbits generated
by GIPSY using GPS satellite ephemeris data downloaded from the GPS bulletin
board. The WRS locations used are those currently recommended by the FAA.
These locations may be adjusted to evaluate the impact of other WRS locations
or additions WRSs.
5.2.2.3 Selection of Q and R Theoretically, a Kalman filter yields optimal
estimation of the states of a system, given a knowledge of the system dynamics
and measurement equations, when both the system process noise and measurement noise are zero-mean Gaussian at each epoch and white in time and
their variances are known. However, in practice, the system dynamics are often
unknown and system modeling errors are introduced when the actual system
dynamics differ from the assumptions. In addition, the system process noise and
the measurement noise are often non-Gaussian and their variances are not known
precisely. To ensure a stable solution, a relatively large value of Q is often used,
sacrificing estimation accuracy. Careful selection of Q and R values impacts the
performance of the Kalman filter in practical applications, including the SBAS
ionospheric estimation filter.
In each phase of the validation, many parameters are tuned. The procedures
and rationale involved in selecting the final values of these parameters include
an effort to distinguish those parameters for which the performance is particularly sensitive. For many parameters, performance is not particularly sensitive.
Table 5.1 shows typical values of the parameters used in two Kalman filters. The
L1 L2 filter can be eliminated and thereby use the IONO filter, including the
satellite and receiver biases, may be sufficient to estimate the biases and IONO
delays. This reduces the computational load and simplifies the process.
The algorithms must be validated to ensure that the estimation accuracy is
good enough to ultimately support downstream precision-approach requirements.
Convergence properties of the estimation algorithms must be examined, and the
logic associated with restarting the estimation using recorded data must be analyzed. The capabilities to perform thread switches and detect anomalies must
be examined, and the special cases necessitating a change of reference receiver.
In each phase of validation, the critical test is whether there is any significant
degradation in accuracy as compared with nominal performance, and whether the
nominal performance itself is adequate.
5.2.2.4 Calculation of Ionospheric Delay Using Pseudoranges The problem
of calculating ionospheric propagation delay from P-code and C/A-code can be
162
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
TABLE 5.1. Representative Kalman Filter Parameter Values
Parameter Term
Value
L1 L2 filter bias process noise update interval
L1 L2 filter TEC process noise
L1 L2 filter TEC process noise update interval
Iono filter process noise
Iono filter process noise update interval
Iono meas floor
Iono meas scale
L1 L2 filter bias process noise
L1 L2 next bias distribution time interval
L1 L2 cold start bias distribution time interval
L1 L2 cold start time interval
Iono a priori covariance matrix
L1 L2 bias a priori covariance matrix
(ref receiver)
Maximum initial TEC
Nominal initial TEC
Units
300
0.05
300
0.05
300
9
0
4.25 × 10−4
300
300
86,400
400 = 202
10, 000 = 1002
10−10
1000
25
s
TECU/s1/2
s
TECU/s1/2
s
TECU2
TECU/s1/2
s
s
s
TECU2
TECU2
TECU2
TECU
TECU
formulated in terms of the following measurement equalities:
PRL1 = ρ + L1iono + cτRX1 + cτGD ,
PRL2 = ρ + L1iono
τGD
$ 2 + cτRX 2 + c $ 2 ,
fL2 fL1
fL2 fL1
where
PRL1
PRL2
ρ
fL1
fL2
τRX1
τRX2
τGD
c
(5.33)
= L1 pseudorange
= L2 pseudorange
= geometric distance between GPS satellite
transmitter and GPS receiver, including
nondispersive contributions such as
tropospheric refraction and clock drift
= L1 frequency
= 1575.42 MHz
= L2 frequency
= 1227.6 MHz
= receiver noise as manifested in code
(receiver and calibration biases) at L1 (ns)
= receiver noise as manifested in code
(receiver and calibration biases) at L2 (ns)
= satellite group delay (interfrequency bias)
= speed of light
= 0.299792458 m/ns
(5.34)
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
.
(5.35)
TROPOSPHERIC PROPAGATION ERRORS
163
Subtracting Eq. 5.34 from Eq. 5.33, we get
L1 iono =
PRL1 − PRL2
c (τRX1 − τRX2 )
$ 2 −
$ 2 − cτGD .
1 − fL1 fL2
1 − fL1 fL2
(5.36)
What is actually measured in the ionospheric delay is the sum of receiver bias and
interfrequency bias. The biases are determined and taken out from the ionospheric
delay calculation. These biases may be up to 10 ns (3 m) [50, 142].
However, the presence of ambiguities N1 and N2 in carrier phase measurements of L1 and L2 preclude the possibility of using these in the daytime by
themselves. At night, these ambiguities can be calculated from the pseudoranges
and carrier phase measurements may be used for ionospheric calculations.
The MATLAB program Iono delay(PRN#) (described in Appendix A) uses
pseudorange and carrier phase data from L1 and L2 signals.
5.3 TROPOSPHERIC PROPAGATION ERRORS
The lower part of the earth’s atmosphere is composed of dry gases and water
vapor, which lengthen the propagation path due to refraction. The magnitude of
the resulting signal delay depends on the refractive index of the air along the
propagation path and typically varies from about 2.5 m in the zenith direction
to 10–15 m at low satellite elevation angles. The troposphere is nondispersive
at the GPS frequencies, so that delay is not frequency dependent. In contrast to
the ionosphere, tropospheric path delay is consequently the same for code and
carrier signal components. Therefore, this delay cannot be measured by utilizing
both L1 and L2 pseudorange measurements, and either models and/or differential
positioning must be used to reduce the error.
The refractive index of the troposphere consists of that due to the dry-gas
component and the water vapor component, which respectively contribute about
90% and 10% of the total. Knowledge of the temperature, pressure, and humidity
along the propagation path can determine the refractivity profile, but such measurements are seldom available to the user. However, using standard atmospheric
models for dry delay permits determination of the zenith delay to within about
0.5 m and with an error at other elevation angles that approximately equals the
zenith error times the cosecant of the elevation angle. These standard atmospheric
models are based on the laws of ideal gases and assume spherical layers of constant refractivity with no temporal variation and an effective atmospheric height
of about 40 km. Estimation of dry delay can be improved considerably if surface
pressure and temperature measurements are available, bringing the residual error
down to within 2–5% of the total.
The component of tropospheric delay due to water vapor (at altitudes up to
about 12 km) is much more difficult to model, because there is considerable
spatial and temporal variation of water vapor in the atmosphere. Fortunately, the
wet delay is only about 10% of the total, with values of 5–30 cm in continental
164
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
midlatitudes. Despite its variability, an exponential vertical profile model can
reduce it to within about 2–5 cm.
In practice, a model of the standard atmosphere at the antenna location would
be used to estimate the combined zenith delay due to both wet and dry components. Such models use inputs such as the day of the year and the latitude
and altitude of the user. The delay is modeled as the zenith delay multiplied by
a factor that is a function of the satellite elevation angle. At zenith, this factor
is unity, and it increases with decreasing elevation angle as the length of the
propagation path through the troposphere increases. Typical values of the multiplication factor are 2 at 30◦ elevation angle, 4 at 15◦ , 6 at 10◦ , and 10 at 5◦ . The
accuracy of the model decreases at low elevation angles, with decimeter level
errors at zenith and about 1 m at 10◦ elevation.
Much research has gone into the development and testing of various tropospheric models. Excellent summaries of these appear in the literature [84, 96,
177].
Although a GPS receiver cannot measure pseudorange error due to the troposphere, differential operation can usually reduce the error to small values by
taking advantage of the high spatial correlation of tropospheric errors at two
points within 100–200 km on the earth’s surface. However, exceptions often
occur when storm fronts pass between the receivers, causing large gradients in
temperature, pressure, and humidity.
5.4 THE MULTIPATH PROBLEM
Multipath propagation of the GPS signal is a dominant source of error in differential positioning. Objects in the vicinity of a receiver antenna (notably the ground)
can easily reflect GPS signals, resulting in one or more secondary propagation paths. These secondary-path signals, which are superimposed on the desired
direct-path signal, always have a longer propagation time and can significantly
distort the amplitude and phase of the direct-path signal.
Errors due to multipath cannot be reduced by the use of differential GPS,
since they depend on local reflection geometry near each receiver antenna. In a
receiver without multipath protection, C/A-code ranging errors of 10 m or more
can be experienced. Multipath can not only cause large code ranging errors but
also severely degrade the ambiguity resolution process required for carrier phase
ranging such as that used in precision surveying applications.
Multipath propagation can be divided into two classes: static and dynamic. For
a stationary receiver, the propagation geometry changes slowly as the satellites
move across the sky, making the multipath parameters essentially constant for
perhaps several minutes. However, in mobile applications there can be rapid
fluctuations in fractions of a second. Therefore, different multipath mitigation
techniques are generally employed for these two types of multipath environments.
Most current research has been focused on static applications, such as surveying,
where greater demand for high accuracy exists. For this reason, we will confine
our attention to the static case.
HOW MULTIPATH CAUSES RANGING ERRORS
165
5.5 HOW MULTIPATH CAUSES RANGING ERRORS
To facilitate an understanding of how multipath causes ranging errors, several
simplifications can be made that in no way obscure the fundamentals involved. We
will assume that the receiver processes only the C/A-code and that the received
signal has been converted to complex (i.e., analytic) form at baseband (nominally zero frequency), where all Doppler shift has been removed by a carrier
tracking phase-lock loop. It is also assumed that the 50-bps (bits per second)
GPS data modulation has been removed from the signal, which can be achieved
by standard techniques. When no multipath is present, the received waveform is
represented by
r(t) = aej φ c (t − τ ) + n(t),
(5.37)
where c(t) is the normalized, undelayed C/A-code waveform as transmitted,
τ is the signal propagation delay, a is the signal amplitude, φ is the carrier
phase, and n(t) is Gaussian receiver thermal noise having flat power spectral
density. Pseudoranging consists of estimating the delay parameter τ . As we have
previously seen, an optimal estimate (i.e., a minimum-variance unbiased estimate)
of τ can be obtained by forming the cross-correlation function
*
R(τ ) =
T2
r(t)cr (t − τ ) dt,
(5.38)
T1
of r(t) with a replica cr (t) of the transmitted C/A-code and choosing as the delay
estimate that value of τ that maximizes this function. Except for an error due
to receiver thermal noise, this occurs when the received and replica waveforms
are in time alignment. A typical cross-correlation function without multipath for
C/A-code receivers having a 2-MHz precorrelation bandwidth is shown by the
solid lines Fig. 5.4 (these plots ignore the effect of noise, which would add small
random variations to the curves).
If multipath is present with a single secondary path, the waveform of Eq. 5.37
changes to
r(t) = aej φ1 c (t − τ1 ) + bej φ2 c (t − τ2 ) + n(t),
(5.39)
where the direct and secondary paths have respective propagation delays τ1 and
τ2 , amplitudes a and b, and carrier phases φ1 and φ2 . In a receiver not designed
expressly to handle multipath, the resulting cross-correlation function will now
have two superimposed components, one from the direct path and one from the
secondary path. The result is a function with a distortion depending on the relative
amplitude, delay, and phase of the secondary-path signal, as illustrated at the top
of Fig. 5.4 for an in-phase secondary path and at the bottom of the figure for an
out-of-phase secondary path. Most importantly, the location of the peak of the
function has been displaced from its correct position, resulting in a pseudorange
error.
166
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
Fig. 5.4
Effect of multipath on C/A-code cross-correlation function.
In vintage receivers employing standard code tracking techniques (early and
late codes separated by one C/A-code chip), the magnitude of pseudorange error
caused by multipath can be quite large, reaching 70–80 m for a secondary-path
signal one-half as large as the direct-path signal and having a relative delay of
approximately 250 m. Further details can be found in Ref. 78.
METHODS OF MULTIPATH MITIGATION
167
5.6 METHODS OF MULTIPATH MITIGATION
Processing against slowly changing multipath can be broadly separated into two
classes: spatial processing and time-domain processing. Spatial processing uses
antenna design in combination with known or partially known characteristics of
signal propagation geometry to isolate the direct-path received signal. In contrast, time domain processing achieves the same result by operating only on the
multipath-corrupted signal within the receiver.
5.6.1 Spatial Processing Techniques
5.6.1.1 Antenna Location Strategy Perhaps the simplest form of spatial processing is to locate the antenna where it is less likely to receive reflected signals.
For example, to obtain the position of a point near reflective objects, one can
first use GPS to determine the position of a nearby point “in the clear” and then
calculate the relative position of the desired point by simple distance and/or angle
measurement techniques. Another technique that minimizes ever-present ground
signal reflections is to place the receiver antenna directly at ground level. This
causes the point of ground reflection to be essentially coincident with the antenna
location so that the secondary path has very nearly the same delay as the direct
path. Clearly such antenna location strategies may not always be possible but
can be very effective when feasible.
5.6.1.2 Groundplane Antennas The most common form of spatial processing
is an antenna designed to attenuate signals reflected from the ground. A simple
design uses a metallic groundplane disk centered at the base of the antenna to
shield the antenna from below. A deficiency of this design is that when the signal wavefronts arrive at the disk edge from below, they induce surface waves
on the top of the disk that then travel to the antenna. The surface waves can be
eliminated by replacing the groundplane with a choke ring, which is essentially a
groundplane containing a series of concentric circular troughs one-quarter wavelength deep. These troughs act as transmission lines shorted at the bottom ends
so that their top ends exhibit a very high impedance at the GPS carrier frequency.
Therefore, induced surface waves cannot form, and signals that arrive from below
the horizontal plane are significantly attenuated. However, the size, weight, and
cost of a choke-ring antenna is significantly greater than that of simpler designs.
Most importantly, the choke ring cannot effectively attenuate secondary-path signals arriving from above the horizontal, such as those reflecting from buildings
or other structures. Nevertheless, such antennas have proven to be effective when
signal ground bounce is the dominant source of multipath, particularly in GPS
surveying applications.
5.6.1.3 Directive Antenna Arrays A more advanced form of spatial processing
uses antenna arrays to form a highly directive spatial response pattern with high
gain in the direction of the direct-path signal and attenuation in directions from
168
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
which secondary-path signals arrive. However, inasmuch as signals from different
satellites have different directions of arrival and different multipath geometries,
many directivity patterns must be simultaneously operative, and each must be
capable of adapting to changing geometry as the satellites move across the sky.
For these reasons, highly directive arrays seldom are practical or affordable for
most applications.
5.6.1.4 Long-Term Signal Observation If a GNSS signal is observed for sizable fractions of an hour to several hours, one can take advantage of changes in
multipath geometry caused by satellite motion. This motion causes the relative
delays between the direct and secondary paths to change, resulting in measurable variations in the received signal. For example, a periodic change in signal
level caused by alternate phase reinforcement and cancellation by the reflected
signals is often observable. Although a variety of algorithms have been proposed for extracting the direct-path signal component from measurements of
the received signal, the need for long observation times rules out this technique for most applications. However, it can be an effective method of multipath
mitigation at a fixed site, such as at a differential GNSS base station. In this
case, it is even possible to observe the same satellites from one day to the
next, looking for patterns of pseudorange or phase measurements that repeat
daily.
Multipath Calculation from Long-Term Observations Delays can be computed
as follows by using pseudoranges and carrier phases over long signal observations (one day to next). This technique may be ruled out for most applications.
Ambiguities and cycle slips have been eliminated or mitigated.
Let
⎫
λ1 = 19.03 cm, wavelength of L1 ⎪
⎪
⎪
⎪
λ2 = 24.42 cm, wavelength of L2 ⎪
⎪
⎪
⎪
⎪
⎪
φL1 = carrier phase for L1
⎪
⎪
⎪
⎪
⎪
φL2 = carrier phase for L2
⎪
⎪
⎪
fL1 = L1 frequency = 1575.42 MHz ⎪
⎪
⎪
⎪
⎪
⎪
fL2 = L2 frequency = 1227.6 MHz ⎪
⎬
ρ = geometrical pseudorange
.
(5.40)
⎪
⎪
⎪
PRL1 = pseudorange L1
⎪
⎪
⎪
⎪
⎪
PRL2 = pseudorange L2
⎪
⎪
⎪
⎪
⎪
I = ionospheric delay
⎪
⎪
⎪
⎪
⎪
IL1 = Ionospheric delay in L1
⎪
⎪
⎪
⎪
⎪
MPL1 = multipath in L1
⎪
⎪
⎭
MPL2 = multipath in L2
169
METHODS OF MULTIPATH MITIGATION
For dual-frequency GNSS receivers, one obtains
λ1 φL1 = ρ −
λ2 φL2 = ρ −
I
(fL1 )2
I
(fL2 )2
,
(5.41)
.
(5.42)
Subtracting Eq. 5.42 from Eq. 5.41, one can obtain
λ1 φL1 − λ2 φL2 =
IL1 =
K=
I (fL1 )2 − I (fL2 )2
(fL1 )2 (fL2 )2
,
(λ1 φL1 − λ2 φL2 ) (fL2 )2
,
(fL1 )2 − (fL2 )2
(fL2 )2
(fL1 )2 − (fL2 )2
,
IL1 = K (λ1 φL1 − λ2 φL2 ) ,
I
.
PRL1 = ρ +
(fL1 )2
(5.43)
(5.44)
Subtracting Eq. 5.41 from Eq. 5.44, one obtains the multipath as
MPL1 = PRL1 − λ1 φL1 − 2IL1 ,
where
IL1 =
I
(fL1 )2
(5.45)
.
Substitute Eq. 5.43 into Eq. 5.45 to obtain
MPL1 = PRL1 − λ1 φL1 − 2K (λ1 φL1 − λ2 φL2 )
= PRL1 − [(1 + 2K) λ1 φL1 + 2K λ2 φL2 ] ,
(5.46)
the multipath solution for L1 .
5.6.2 Time-Domain Processing
Although time-domain processing against GPS multipath errors has been the subject of active research for at least two decades, there is still much to be learned,
both at theoretical and practical levels. Most of the practical approaches have
been developed by receiver manufacturers, who are often reluctant to explicitly
reveal their methods. Nevertheless, enough information about multipath processing exists to gain insight into its recent evolution.
170
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
5.6.2.1 Narrow-Correlator Technology (1990–1993) The first significant
means to reduce GPS multipath effects by receiver processing made its debut in
the early 1990s. Until that time, most receivers had been designed with a 2-MHz
precorrelation bandwidth that encompassed most, but not all, of the GPS spreadspectrum signal power. These receivers also used one-chip spacing between the
early and late reference C/A-codes in the code tracking loops. However, the 1992
paper [195] makes it clear that using a significantly larger bandwidth combined
with much closer spacing of the early and late reference codes would dramatically
improve the ranging accuracy both with and without multipath. It is somewhat
surprising that these facts were not recognized earlier by the GPS community,
given that they had been well known in radar circles for many decades.
A 2-MHz precorrelation bandwidth causes the peak of the direct-path crosscorrelation function to be severely rounded, as illustrated in Fig. 5.4. Consequently, the sloping sides of a secondary-path component of the correlation
function can significantly shift the location of the peak, as indicated in the figure.
The result of using an 8-MHz bandwidth is shown in Fig. 5.5, where it can be
noted that the sharper peak of the direct-path cross-correlation function is less
easily shifted by the secondary-path component. It can also be shown that at
larger bandwidths the sharper peak is more resistant to disturbance by receiver
thermal noise, even though the precorrelation signal-to-noise ratio is increased.
Another advantage of a larger precorrelation bandwidth is that the spacing
between the early and late reference codes in a code tracking loop can be made
smaller without significantly reducing the gain of the loop; hence the term narrow
correlator. It can be shown that this causes the noises on the early and late
Fig. 5.5 Reduced multipath error with larger precorrelation bandwidth.
METHODS OF MULTIPATH MITIGATION
171
correlator outputs to become more highly correlated, resulting in less noise on
the loop error signal. An additional benefit is that the code tracking loop will be
affected only by the multipath-induced distortions near the peak of the correlation
function.
5.6.2.2 Leading-Edge Techniques Because the direct-path signal always precedes secondary-path signals, the leading (left-hand) portion of the correlation
function is uncontaminated by multipath, as is illustrated in Fig. 5.5. Therefore, if
one could measure the location of just the leading part, it appears that the directpath delay could be determined with no error due to multipath. Unfortunately, this
seemingly happy state of affairs is illusory. With a small direct-/secondary-path
separation, the uncontaminated portion of the correlation function is a minuscule
piece at the extreme left, where the curve just begins to rise. In this region, not
only is the signal-to-noise ratio relatively poor, but the slope of the curve is also
relatively small, which severely degrades the accuracy of delay estimation.
For these reasons, the leading-edge approach best suits situations with a moderate to large direct-/secondary-path separation. However, even in these cases
there is the problem of making the delay measurement insensitive to the slope
of the correlation function leading edge, which can vary with signal strength.
Such a problem does not occur when measuring the location of the correlation
function peak.
5.6.2.3 Correlation Function Shape-Based Methods Some GPS receiver designers have attempted to determine the parameters of the multipath model from
the shape of the correlation function. The idea has merit, but for best results
many correlations with different values of reference code delay are required
to obtain a sampled version of the function shape. Another practical difficulty
arises in attempting to map each measured shape into a corresponding directpath delay estimate. Even in the simple two-path model (Eq. 5.39) there are six
signal parameters, so that a very large number of correlation function shapes
must be handled. An example of a heuristically developed shape-based approach
called the early—late slope (ELS) method can be found in Ref. 190, while a
method based on maximum-likelihood estimation called the multipath-estimating
delay-lock loop (MEDLL) is described in Ref. 191.
5.6.2.4 Modified Correlator Reference Waveforms A relatively new approach
to multipath mitigation alters the waveform of the correlator reference PRN code
to provide a cross-correlation function with inherent resistance to errors caused
by multipath. Examples include the strobe correlator [58], the use of special code
reference waveforms to narrow the correlation function developed in Refs. 202
and 203, and the gated correlator developed in Ref. 138. These techniques take
advantage of the fact that the range information in the received signal resides
primarily in the chip transitions of the C/A-code. By using a correlator reference
waveform that is not responsive to the flat portions of the C/A-code, the resulting correlation function can be narrowed down to the width of a chip transition,
172
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
Fig. 5.6 Multipath-mitigating reference code waveform.
thereby being almost immune to multipath having a primary/secondary-path separation greater than 30–40 m. An example of such a reference waveform and the
corresponding correlation function are shown in Fig. 5.6.
5.6.3 MMT Technology
The latest approach to time-domain multipath mitigation is called multipath
mitigation technology (MMT), and is incorporated a number of GPS receivers
manufactured by NovAtel Corporation of Canada. The MMT technique not only
reaches theoretical performance limits described in Section 5.7 for both code and
carrier phase ranging but also, compared to existing approaches, has the advantage that its performance improves as the signal observation time is lengthened.
A description of MMT follows in Section 5.6.3 and also appears in a patent
[208].
5.6.3.1 Description MMT is based on maximum-likelihood (ML) estimation.
Although the theory of ML estimation is well-developed, its application to GPS
multipath mitigation has not been feasible until now, due to the large amount of
computation required. However, recent mathematical breakthroughs have solved
METHODS OF MULTIPATH MITIGATION
173
this problem. Before introducing the MMT algorithm we first briefly describe the
process of ML estimation in the context of the multipath problem.
5.6.3.2 Maximum-Likelihood Multipath Estimation Maximum likelihood
estimation (MLE) is described in detail in Chapter 8. Its application to multipath
mitigation is described below.
5.6.3.3 The Two-Path ML Estimator (MLE) The simplest ML estimator designed for multipath is based on a two-path model (one direct path and one
secondary delayed path). For simplicity in describing MMT we consider only
this model, although generalization to additional paths is straightforward, and
the MMT algorithm can be implemented for such cases. It is assumed that the
received signal has been frequency-shifted to baseband, and the navigation data
have been stripped off. The two-path signal model is
r(t) = A1 ej φ1 m(t − τ1 ) + A2 ej φ2 m(t − τ2 ) + n(t).
(5.47)
In this model the parameters A1 , φ1 , τ1 are respectively the direct path signal
amplitude, phase, and delay, and the parameters A2 , φ2 , and τ2 are the corresponding parameters for the secondary path. The code modulation is denoted by
m(t), and the noise function n(t) is an additive zero-mean complex Gaussian
noise process with a flat power spectral density. It will be convenient to group
the multipath parameters into the vector
θ = [A1 , φ1 , τ1 , A2 , φ2 , τ2 ].
(5.48)
Observation of the received signal r(t) is accomplished by sampling it on the
time interval [0,T ] to produce a complex observed vector r.
ˆ
The ML estimate of the multipath parameters is
the vector θ of parameter
values that maximizes the likelihood function p(r θ ) , which is the probability
density of the received signal vector conditioned on the values of the multipath
parameters. In this maximization the vector r is held fixed at its observed value.
Within the vector θˆ the estimates τ̂1 and φ̂1 of direct-path delay and carrier
phase are normally the only ones of interest. However,
the ML estimate of these
parameters requires that the likelihood function p(r θ ) be maximized over the
six-dimensional (6D) space of all multipath parameters (components of θ ). For
this reason the unwanted parameters are called nuisance parameters.
Since
the natural logarithm is a strictly increasing
function,
the
maximization
of p(r θ ) is equivalent to maximization of L r; θ = ln p r θ , which is called
the log-likelihood function. The log-likelihood function is often simpler than the
likelihood function itself, especially when the noise in the observations is additive
and Gaussian. In our application
this is the case.
Maximization of L r; θ by standard techniques is a daunting task. A bruteforce approach is to find the maximum by a search over the 6D multipath
174
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
parameter space, but it takes too long to be of practical value. Reliable gradientbased or hill climbing methods are too slow to be useful. Finding the maximum
using differential calculus is difficult, because of the nonlinearity of the resulting
equations and the possibility of local maxima that are not global maxima. Iterative solution techniques are often difficult to analyze and may not converge to
the correct solution in a timely manner, if they converge at all. As we shall see,
the MMT algorithm solves these problems by reducing the dimensionality of the
search space.
5.6.3.4 Asymptotic Properties of ML Estimators ML estimation is used by
MMT not only because it can be made computationally simple enough to be
practical but also because ML estimators have desirable asymptotic2 properties:
1. The ML estimate of a parameter asymptotically converges in probability
to the true parameter value.
2. The ML estimate is asymptotically efficient, that is, the ratio of the variance
of the estimation error to the Cramer–Rao bound approaches unity.
3. The ML estimate is asymptotically Gaussian.
5.6.3.5 The MMT Multipath Mitigation Algorithm The MMT algorithm uses
several mathematical techniques to solve what would otherwise be intractable
computational problems. The first of these is a nonlinear transformation on
the multipath parameter space to permit rapid computation of a log-likelihood
function that has been partially maximized with respect to all of the multipath
parameters except for the path delays. Thus, final maximization requires a search
in only two dimensions for the two-path case, aided by acceleration techniques.
A new method of signal compression, described in Section 5.6.3.10, is used
to transform the received signal into a very small vector on which MMT can
operate very rapidly.
A major advantage of the MMT algorithm is that its performance improves
with increasing E/N0 , the ratio of signal energy E to noise power spectral density.
This is not true for most GNSS multipath mitigation methods, because their
estimation error is in the form of an irreducible bias. Additionally, the MMT
algorithm provides ML estimates of all parameters in the multipath model, and
can utilize known bounds on the magnitudes of the secondary paths, if available,
to improve performance.
5.6.3.6 The MMT Baseband Signal Model In the complex baseband signal r(t) given by Eq. (5.47) it is assumed that the signal has been Dopplercompensated and stripped of the 50-bps (bits per second) navigation data modulation. In developing the MMT algorithm it is useful to separate r(t) into its real
2
Asymptotic refers to the behavior of an estimator when the error becomes small. In GNSS this
occurs when E/N0 is sufficiently large.
175
METHODS OF MULTIPATH MITIGATION
component x(t) and imaginary component y(t):
x(t) = A1 cos φ1 m(t − τ1 )
+A2 cos φ2 m(t − τ2 ) + nx (t) ,
y(t) = A1 sin φ1 m(t − τ1 )
+A2 sin φ2 m(t − τ2 ) + ny (t) ,
(5.49)
where nx (t) and ny (t) are independent, real-valued, zero-mean Gaussian noise
processes with flat power spectral density.
5.6.3.7 Baseband Signal Vectors The real and imaginary signal components
are synchronously sampled on [0,T ] at the Nyquist rate 2W , corresponding to
the lowpass baseband bandwidth W , to produce the vectors
x = (x1 , x2 , . . . , xM ) ,
y = (y1 , y2 , . . . , yM ) ,
(5.50)
in which the noise components of distinct samples are essentially uncorrelated
(hence independent, since the noise is Gaussian).
5.6.3.8 The Log-Likelihood Function The ML estimates of the six parameters
in the vector θ given by (5.48) are obtained by maximizing the log-likelihood
function with respect to these parameters. For MMT the log-likelihood function is
#
" L x, y θ = ln p x, y θ
= ln C1
02
M /
.
xk − A1 cos θ1 mk (τ1 )
− C2
−A2 cos θ2 mk (τ2 )
k=1
02
M /
.
yk − A1 sin θ1 mk (τ1 )
− C2
,
−A2 sin θ2 mk (τ2 )
(5.51)
k=1
where
%
C1 =
√
1
&M
2πσ
1
C2 =
2σ 2
σ 2 = noise variance of x (t) and y(t)
mk (τ1 ) = kth sample of m (t − τ1 )
mk (τ2 ) = kth sample of m (t − τ2 ) .
(5.52)
Replacing the summations in (5.51) by integrals and utilizing the fact that C1
and −C2 are negative constants that do not depend on the multipath parameters,
176
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
maximization of Eq.5.51 is equivalent to minimization of
⎡
*T
⎣
=
0
*T
+
x(t) − A1 cos φ1 m (t − τ1 )
− A2 cos φ2 m (t − τ2 )
⎡
⎣
⎤2
⎦ dt
⎤2
y(t) − A1 sin φ1 m (t − τ1 )
(5.53)
⎦ dt.
− A2 sin φ2 m (t − τ2 )
0
with respect to the six multipath parameters. This is a highly coupled, nonlinear
minimization problem on the 6D space spanned by the parameters A1 , φ1 , τ1 ,
A2 , φ2 , and τ2 . Standard minimization techniques such as a gradient search on
this space or ad hoc iterative approaches are either unreliable or too slow to be
useful.
However, a major breakthrough results by using the invertible transformation
a = A1 cos φ1
b = A2 cos φ2
c = A1 sin φ1
d = A2 sin φ2
1
.
(5.54)
When this transformation is applied and the integrands in (5.53) are expanded,
the problem becomes one of minimizing
*T
=
#
x 2 (t) + y 2 (t) dt
"
0
+ a 2 + b2 + c2 + d 2 Rmm (0)
(5.55)
− 2aRxm (τ1 ) − 2bRxm (τ2 ) + 2abRmm (τ1 − τ2 )
− 2cRym (τ1 ) − 2dRym (τ2 ) + 2cdRmm (τ1 − τ2 ) .
Note that in Eq. 5.55 is quadratic in a, b, c, and d, and uses the correlation
functions
*T
Rxm (τ ) =
x (t) m (t − τ ) dt
0
*T
Rym (τ ) =
y (t) m (t − τ ) dt
0
*T
Rmm (τ ) =
m (t) m (t − τ ) dt.
0
(5.56)
METHODS OF MULTIPATH MITIGATION
177
Thus, minimization of Eq. 5.55 with respect to a, b, c, and d can be accomplished
by taking partial derivatives with respect to these parameters, resulting in the
linear system
∂
∂a
∂
0=
∂b
∂
0=
∂c
∂
0=
∂d
0=
= 2aRmm (0) − 2Rxm (τ1 ) + 2bRmm (τ1 − τ2 )
= 2bRmm (0) − 2Rxm (τ2 ) + 2aRmm (τ1 − τ2 )
(5.57)
= 2cRmm (0) − 2Rym (τ1 ) + 2dRmm (τ1 − τ2 )
= 2dRmm (0) − 2Rym (τ2 ) + 2cRmm (τ1 − τ2 ) .
For each pair of values of τ1 and τ2 this linear system can be explicitly solved
for the minimizing values of a, b, c, and d. Thus the space to be searched for a
minimum of (5.55) (i.e., Eq. 5.55) is now 2D instead of 6D. The minimization
procedure is as follows. Search the (τ1 , τ2 ) domain. At each point (τ1 , τ2 ) compute the values of the correlation functions in system (5.57) and then solve the
system to find the
values of a, b, c, and d that minimize at that point. Identify
the point τ̂1 , τ̂2 ML where the smallest of all such minima is obtained, as well
as the associated minimizing values of a, b, c, and d. Transform these values
of a, b, c, and d back to the estimates Â1ML , Â2ML , φ̂1ML , φ̂2ML by using the
inverse of transformation (5.54), which is
1
√
√
A1 = a 2 + c2
A2 = b2 + d 2
.
φ1 = arctan 2 (a, c) φ2 = arctan 2 (b, d)
(5.58)
5.6.3.9 Secondary-Path Amplitude Constraint In the majority of multipath
scenarios, the amplitudes of secondary-path signals are smaller than that of the
direct path. The multipath mitigation performance of MMT can be significantly
improved by minimizing in (5.55) subject to the constraint
A2
≤ α,
A1
(5.59)
where α is a positive constant (a typical value is 0.7). The constraint in terms of
the transformed parameters a, b, c, and d is
b2 + d 2 ≤ α 2 a 2 + c2 .
(5.60)
The constrained minimization of (5.55) uses the method of Lagrange multipliers.
5.6.3.10 Signal Compression In the MMT algorithm the correlation functions Rxm (τ ), Rym (τ ), and Rmm (τ ) defined by (5.56) and appearing in (5.57)
178
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
are computed very rapidly by first using a process called signal compression, in
which the large number of signal samples (on the order of 108 –109 ) that would
normally be involved is reduced to only a few tens of samples (the exact number
depends on which type of GPS signal is being processed). This processing is
easily done in real time.
The correlation functions appearing in (5.56) have the form
*T
R (τ ) =
r (t) m (t − τ ) dt,
(5.61)
0
where r (t) is a given function and m (t) is a replica of the code modulation, which
includes the effects of filtering in the satellite and receiver. The calculation of
R (τ ) in a conventional receiver is ordinarily not computationally difficult because
in such receivers m (t) can be an ideal chipping sequence with only the values
±1, and the multiplications of samples of the integrand of (5.61) then become
trivial. Furthermore, conventional receivers track only the peak of the correlation
function so that R (τ ) needs to be computed for only a few values of τ (usually
for early, punctual, and late correlations). However, the MMT algorithm cannot
employ these simplifications. The function m (t) used by MMT must include
the aforementioned effects of filtering, thus requiring multibit multiplications
(typically numbering in the millions) in the calculation of R (τ ). Furthermore,
R (τ ) must be calculated for many values of τ to obtain high resolution for
accurate estimation of direct-path delay in the presence of multipath.
These difficulties are circumvented by using signal compression. To simplify
its description, we assume that the correlation function R (τ ) in (5.61) is a cyclic
correlation over one period T of the replica code m (t) in which m (t − τ ) is a
rotation by τ (right for positive τ and left for negative τ ). However, compression
can be accomplished over an arbitrary interval of observation of the function
r (t) in which many periods of a received PN code occur, and furthermore the
correlation function need not be cyclic.
A single period of replica code can be written as
m (t) =
N
−1
.
εk c (t − kTc ),
(5.62)
k=0
where Tc is the duration of each chip, εk is the chip polarity (either +1 or −1),
and N is the number of chips in one period of the code. The function c (t) is the
response of the combined satellite and receiver filtering to a single ideal chip of
the code. This ideal chip has a constant value of 1 on the interval 0 ≤ t ≤ Tc .
Because the filtering is linear and time-invariant, it follows that m (t) is the filter
response to the entire code sequence. The index k identifies the individual chips
of the code, where k = 0 identifies the epoch chip, defined as the first chip of
the chipping sequence.
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METHODS OF MULTIPATH MITIGATION
The compressed signal r̃ (t) is defined by
r̃ (t) =
N
−1
.
εk r (t + kTc ).
(5.63)
k=0
In this expression εk r (t + kTc ) is r (t) weighted by εk and left-rotated by kTc . In
GPS applications the compressed signal has the very nice property that essentially
all of its energy (excluding noise) is concentrated into a pulse of one filtered chip
in duration. This is made evident by noting that the received signal r (t) without
multipath can be expressed as
⎡
⎤
N
−1
.
r (t) = am (t − τ0 ) + n (t) = a ⎣
(5.64)
εj c (t − τ0 − j Tc )⎦ + n (t) ,
j =0
where a is the signal amplitude, τ0 is the signal delay, n (t) is noise, and all
timeshifts are rotations (i.e., cyclic over one code period). Substitution of this
expression into (5.63) gives
r̃ (t) =
N−1
.
k=0
=
=
=
N−1
.
εk r (t + kTc )
⎧⎡
⎫
⎤
−1
⎨ N.
⎬
εk ⎣
εj c (t − τ0 + kTc − j Tc )⎦ + n (t + kTc )
⎩
⎭
k=0
N−1
−1
. N.
k=0 j =0
N−1
−1
. N.
j =0
−1
"
# N.
εk εj c t − τ0 + (k − j ) Tc +
εk n (t + kTc )
(5.65)
k=0
"
#
εk εj c t − τ0 + (k − j ) Tc + ñ (t) ,
k=0 j =0
where the double summation is the compressed signal component, and the single summation is the compressed noise function ñ (t). The terms in the double
summation can be grouped into N groups such that each group contains N terms
having the same value of k − j modulo N . Thus, r̃ (t) will be the summation of
N group sums plus ñ(t).
" The group# sum corresponding to particular value p of
k − j modulo N is c t − τ0 + pTc weighted by the sum of terms εj εk , which
satisfy k − j = p modulo N . Since Tc is the duration of c(t) before filtering, it
can be seen that r̃ (t) consists of a concatenation of N weighted and translated
copies of c(t) which do not overlap, except for a trailing transient from each
copy due to filtering.
5.6.3.11 Properties of the Compressed Signal If the number of chips N is
sufficiently large (on the order of 103 or more), the autocorrelation function of the
GPS chipping sequence has the property that the group sums in which k − j = 0
180
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
modulo N are negligible compared to the group sum in which k − j = 0 modulo
N . Furthermore, the sum of all of these small group sums is also negligible
because the translations of the weighted copies of c(t) prevent the small group
sums from accumulating to large values. Thus, to a very good approximation,
the double summation in (5.65) is just the sum of the terms where k − j =
0 moduloN :
2N −1
3
.
r̃ (t) ∼
(5.66)
ε2 c (t − τ0 ) + ñ (t) = N c (t − τ0 ) + ñ (t) .
=
k
k=0
This is a very significant result, because it tells us that the compressed received
signal is essentially just the single weighted filtered chip N c (t − τ0 ) plus noise,
with small “sidelobe” chips to either side. Furthermore, the compression process
provides a processing gain of 10 log N dB. Since a receiver can measure the
delay τ0 , a window can be constructed that need be long enough only to contain
N c (t − τ0 ), and the sidelobe chips as well as all noise outside this window
can be rejected. The required length of the window is Tc + δ, where δ is large
enough to accommodate the measurement uncertainty of τ0 , the trailing transient
due to filtering, and any multipath components with delays larger than τ0 (almost
certainly the only multipath components having significant amplitude are found
within 1 chip of the direct path delay) . Thus the window length is somewhat
larger than one chip duration of the code, a quantity much smaller than the length
T of the observed signal r (t), which must include all N chips of the code. It is
because of this result that r̃ (t) can justifiably be called a compressed signal. An
illustration of the compressed signal is shown in Fig. 5.7.
Normalized baseband
code waveform r(t)
(noise omitted)
1
0
−1
N = 1023 for compression
N
of 1023 C/A chips
Sidelobe region
of compressed
signal
0
Epoch
chip
Compressed waveform
Sidelobe region
of compressed
signal
t0 t0 + Tc
Compression window
Fig. 5.7 Compression of the received signal.
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METHODS OF MULTIPATH MITIGATION
If N is sufficiently large, the processing gain is great enough to make the
compressed signal within the window visible with very little noise, so that small
subtleties in the chip waveshape due to multipath or other causes can easily be
seen. This property is very beneficial for signal integrity monitoring. It has been
put to practical use in GPS receivers sold by the NovAtel Corporation, which
calls its implementation the Vision Correlator.
The compressed signal also enjoys a linearity property: If r (t) = a1 r1 (t) +
a2 r2 (t), then r̃ (t) = a1 r̃1 (t) + a2 r̃2 (t). The linearity property is essential for the
MMT to properly process a multipath corrupted signal.
5.6.3.12 The Compression Theorem Most importantly, the compressed signal
can be used to drastically reduce the amount of computation of the correlation
function R (τ ) in (5.61). The basis for this assertion is the following theorem:
The correlation function
*T
R (τ ) =
r (t) m (t − τ ) du,
(5.67)
0
can be computed by the alternate method
*T
r̃ (t) c (t − τ ) du.
R (τ ) =
(5.68)
0
Proof :
R (τ ) =
+T
r (t) m (t − τ ) dt
0
=
+T
r (t)
0
=
N−1
4 +T
k=0 0
/N −1
4
0
εk c (t − kTc − τ ) dt
k=0
εk r (t) c (t − kTc − τ ) dt
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
=
εk r (u + kTc ) c (u − τ ) du (using u = t − kTc ) ⎪
⎪
⎪
⎪
k=0 0
⎪
⎪
⎪
/
0
⎪
⎪
T
−1
+ N4
⎪
⎪
⎪
=
εk r (u + kTc ) c (u − τ ) du
⎪
⎪
⎪
k=0
0
⎪
⎪
⎪
⎪
⎪
+T
⎪
⎪
⎪
=
r̃ (u) c (u − τ ) du
⎭
N−1
4 +T
0
. (5.69)
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GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
This theorem shows that R (τ ) can be computed by cross-correlating the compressed signal r̃ (t) with the very short function c (t). Furthermore, since we have
already noted that the significant portion of r̃ (t) also spans a short time interval,
the region surrounding the correlation peak of R (τ ) can be obtained with far
less computation than the original correlation (5.67). The bottom line is that the
cross-correlations in (5.56) used by MMT can be calculated very efficiently by
using the compressed versions of the signals x (t), y (t), and m (t).
5.6.4 Performance of Time-Domain Methods
5.6.4.1 Ranging with the C/A-Code Typical C/A-code ranging performance
curves for several multipath mitigation approaches are shown in Fig. 5.8 for the
case of an in-phase secondary path with amplitude one-half that of the direct
path. Even with the best available methods (other than MMT), peak range errors
of 3–6 m are not uncommon. It can be observed that the error tends to be
largest for “close-in” multipath, where the separation of the two paths is less
that 20–30 m. Indeed, this region poses the greatest challenge in multipath mitigation research because the extraction of direct-path delay from a signal with
small direct/secondary-path separation is an ill-conditioned parameter estimation
problem.
A serious limitation of most existing multipath mitigation algorithms is that
the residual error is mostly in the form of a bias that cannot be removed by further
filtering or averaging. On the other hand, the above mentioned MMT algorithm
overcomes this limitation and also appears to have significantly better performance than other published algorithms, as is indicated by curve F of Fig. 5.8.
Fig. 5.8 Performance of various multipath mitigation approaches.
METHODS OF MULTIPATH MITIGATION
183
Fig. 5.9 Residual multipath phase error using MMT algorithm.
5.6.4.2 Carrier Phase Ranging The presence of multipath also causes errors
in estimating carrier phase, which limits the performance in surveying and other
precision applications, particularly with regard to carrier phase ambiguity resolution. Not all current multipath mitigation algorithms are capable of reducing
multipath-induced phase error. The most difficult situation occurs at small separations between the direct and secondary paths (less than a few meters). It can
be shown that under such conditions essentially no mitigation is theoretically
possible. Typical phase error curves for the MMT algorithm, which appears to
have the best performance of published methods, is shown in Fig. 5.9.
5.6.4.3 Testing Receiver Multipath Performance Conducting meaningful tests
of receiver multipath mitigation performance on either an absolute or a comparative basis is no easy matter. There are often two conflicting goals. On the one
hand, the testing should be under strictly controlled conditions, so that the signal
levels and true multipath parameters are precisely known; otherwise the measured
performance cannot be linked to the multipath conditions that actually exist. Generally this will require precision signal simulators and other ancillary equipment
to generate accurately characterized multipath signals.
On the other hand, receiver end users place more credence on how well a
receiver performs in the field. However, meaningful field measurements pose a
daunting challenge. It is extremely difficult to know the amount and character of
the multipath, and great difficulty can be experienced in isolating errors caused
by multipath from those of other sources. To add to these difficulties, it is not
clear that either the receiver manufacturers or the users have a good feel for
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GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
the range of multipath parameter values that represent typical operation in the
field.
5.7 THEORETICAL LIMITS FOR MULTIPATH MITIGATION
5.7.1 Estimation-Theoretic Methods
Relatively little has been published on multipath mitigation from the fundamental
viewpoint of statistical estimation theory, despite the power of its methods and
its ability to reach theoretical performance limits in many cases. Knowledge
of such limits provides a valuable benchmark in receiver design by permitting
an accurate assessment of the potential payoff in developing techniques that are
better than those in current use. Of equal importance is the revelation of the signal
processing operations that can reach performance bounds. Although it may not
be feasible to implement the processing directly, its revelation often leads to a
practical method that achieves nearly the same performance.
5.7.1.1 Optimality Criteria In discussing theoretical performance limits, it is
important to define the criterion of optimality. In GPS the optimal range estimator is traditionally considered to be the minimum-variance unbiased estimator
(MVUE), which can be realized by properly designed receivers. However, in
Ref. 204 it is shown that the standard deviation of a MVUE designed for multipath becomes infinite as the primary-to-secondary-path separation approaches
zero. For this reason it seems that a better criterion of optimality would be the
minimum RMS error, which can include both random and bias components.
Unfortunately, it can be shown that no estimator exists having minimum RMS
error for every combination of true multipath parameters.
5.7.2 MMSE Estimator
There is an estimator that can be claimed optimal in a weaker sense. The
minimum-mean-square-error (MMSE) estimator has the property that no other
estimator has a uniformly smaller RMS error. In other words, if some other estimator has smaller RMS error than the MMSE estimator for some set of true
multipath parameter values, then that estimator must have a larger RMS error
than the MMSE estimator for some other set of values.
The MMSE estimator also has an important advantage not possessed by most
current multipath mitigation methods in that the RMS error decreases as the
length of the signal observation interval is increased.
5.7.3 Multipath Modeling Errors
Although a properly designed estimation-theoretic approach such as the MMSE
estimator will generally outperform other methods, the design of such estimators
requires a mathematical model of the multipath-contaminated signal containing
ONBOARD CLOCK ERRORS
185
parameters to be estimated. If the actual signal departs from the assumed model,
performance degradation can occur. For example, if the model contains only two
signal propagation paths but in reality the signal is arriving via three or more
paths, large bias errors in range estimation can result. On the other hand, poorer
performance (usually in the form of random error cause by noise) can also occur
if the model has too many degrees of freedom. Striking the right balance in the
number of parameters in the model can be difficult if little information exists
about the multipath reflection geometry.
5.8 EPHEMERIS DATA ERRORS
Small errors in the ephemeris data transmitted by each satellite cause corresponding errors in the computed position of the satellite (here we exclude the ephemeris
error component of SA, which is regarded as a separate error source). Satellite
ephemerides are determined by the master control station of the GPS ground
segment based on monitoring of individual signals by four monitoring stations.
Because the locations of these stations are known precisely, an “inverted” positioning process can calculate the orbital parameters of the satellites as if they
were users. This process is aided by precision clocks at the monitoring stations
and by tracking over long periods of time with optimal filter processing. Based on
the orbital parameter estimates thus obtained, the master control station uploads
the ephemeris data to each satellite, which then transmits the data to users via
the navigation data message. Errors in satellite position when calculated from the
ephemeris data typically result in range errors less than 1 m. Improvements in
satellite tracking will undoubtedly reduce this error further.
5.9 ONBOARD CLOCK ERRORS
Timing of the signal transmission from each satellite is directly controlled by
its own atomic clock without any corrections applied. This time frame is called
space vehicle (SV) time. A schematic of a rubidium atomic clock is shown in
Fig. 5.10. Although the atomic clocks in the satellites are highly accurate, errors
can be large enough to require correction. Correction is needed partly because it
would be difficult to directly synchronize the clocks closely in all the satellites.
Instead, the clocks are allowed some degree of relative drift that is estimated by
ground station observations and is used to generate clock correction data in the
GPS navigation message. When SV time is corrected using this data, the result
is called GPS time. The time of transmission used in calculating pseudoranges
must be in GPS time, which is common to all satellites.
The onboard clock error is typically less than 1 ms and varies slowly. This
permits the correction to be specified by a quadratic polynomial in time whose
186
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
Fig. 5.10 Schematic of a rubidium atomic clock.
coefficients are transmitted in the navigation message. The correction has the
form
tsv = af 0 + af 1 (tsv − t0c ) + af 2 (tsv − t0c )2 + tR ,
(5.70)
tGPS = tsv − tsv ,
(5.71)
with
where af0 , af1 , af2 are the correction coefficients, tsv is SV time, and tR is a
small relativistic clock correction caused by the orbital eccentricity. The clock
data reference time t0c in seconds is broadcast in the navigation data message.
The stability of the atomic clocks permits the polynomial correction given by
Eq. 5.70 to be valid over a time interval of 4–6 h. After the correction has been
applied, the residual error in GPS time is typically less than a few nanoseconds,
or about 1 m in range. Complete calculations of GPS time are given as exercises
in Problems 3.6–3.10 (in Chapter 3).
5.10 RECEIVER CLOCK ERRORS
Because the navigation solution includes a solution for receiver clock error, the
requirements for accuracy of receiver clocks is far less stringent than for the
187
RECEIVER CLOCK ERRORS
GPS satellite clocks. In fact, for receiver clocks short-term stability over the
pseudorange measurement period is usually more important than absolute frequency accuracy. In almost all cases such clocks are quartz crystal oscillators
with absolute accuracies in the 1–10 ppm range over typical operating temperature ranges. When properly designed, such oscillators typically have stabilities
of 0.01–0.05 ppm over a period of a few seconds.
Receivers that incorporate receiver clock error in the Kalman filter state vector
need a suitable mathematical model of the crystal clock error. A typical model
in the continuous-time domain is shown in Fig. 5.11, which is easily changed to
a discrete version for the Kalman filter. In this model the clock error consists of
a bias (frequency) component and a drift (time) component. The frequency error
component is modeled as a random walk produced by integrated white noise.
The time error component is modeled as the integral of the frequency error after
additional white noise (statistically independent from that causing the frequency
error) has been added to the latter. In the model the key parameters that need
to be specified are the power spectral densities of the two noise sources, which
depend on characteristics of the specific crystal oscillator used.
The continuous time model has the form
ẋ1 = w1 ,
(5.72)
ẋ2 = x1 + w2 ,
(5.73)
where w1 (t) and w2 (t) are independent zero-mean white-noise processes with
known variances.
The equivalent discrete-time model has the state vector
/
0
x1
x=
,
(5.74)
x2
and the stochastic process model
0
/
0
/
1 0
w1,k−1
,
xk−1 +
xk =
w2,k−1
t 1
Fig. 5.11
Crystal clock error model.
(5.75)
188
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
where t is the discrete-time step and {w1,k−1 }, {w2,k−1 } are independent zeromean white-noise sequences with known variances.
5.11 ERROR BUDGETS
For purposes of analyzing the effects of the errors discussed above, it is convenient to convert each error into an equivalent range error experienced by a user,
which is called the user-equivalent range error (UERE). In general, the errors
from different sources will have different statistical properties. For example, satellite clock and ephemeris errors tend to vary slowly with time and appear as biases
over moderately long time intervals, perhaps hours. On the other hand, errors due
to receiver noise and quantization effects may vary much more rapidly, perhaps
within seconds. Nonetheless, if sufficiently long time durations over many navigation scenarios are considered, all errors can be considered as zero-mean random
processes that can be combined to form a single UERE. This is accomplished by
forming the root sum square of the UERE errors from all sources:
5
6 n
6.
UERE =7
(UERE)2 .
i
(5.76)
i=1
Figure 5.12 depicts the various GPS UERE errors and their combined effect for
both C/A-code and P(Y)-code navigation at the 1-σ level.
When SA is on, the UERE for the C/A-code user is about 36 m and reduces
to about 19 m when it is off. Aside from SA, it can be seen that for such a user
the dominant error sources in nondifferential operations are multipath, receiver
noise/resolution, and ionospheric delay (however, recent advances in receiver
technology have in some cases significantly reduced receiver noise/resolution
errors). On the other hand, the P(Y)-code user has a significantly smaller UERE
of about 6 m, for the following reasons:
1. Errors due to SA can be removed, if present. The authorized user can
employ a key to eliminate them.
2. The full use of the L1 and L2 signals permits significant reduction of
ionospheric error.
3. The wider bandwidth of the P(Y)-codes greatly reduces errors due to multipath and receiver noise.
5.12 DIFFERENTIAL GNSS
Differential GNSS (DGNSS) is a technique for improving the performance
of GNSS positioning. The basic idea of DGNSS is to compute the spatial
189
DIFFERENTIAL GNSS
Fig. 5.12 GPS UERE budget.
displacement vector of the user’s receiver (sometimes called the roving or remote
receiver) relative to another receiver (usually called the reference receiver or base
station). In most DGNSS applications the coordinates of the reference receiver
are precisely known from highly accurate survey information; thus, the accurate location of the roving receiver can be determined by vector addition of the
reference receiver coordinates and the reference-to-rover displacement vector.
The positioning accuracy of DGNSS depends on the error in estimating the
reference-to-rover displacement vector. This error can be made considerably
smaller than the positioning error of a standalone receiver, because major components of pseudorange measurement errors are common to the roving and reference
receivers and can be canceled out by using the difference between the reference
and rover measurements to compute the displacement vector.
There are basically two ways that errors common to the roving and reference
receiver can be canceled. The first method is called the measurement or solutiondomain technique, in which both receivers individually compute their positions
and the reference-to-rover displacement vector is simply the difference of these
positions. However, the two receivers must use exactly the same set of satellites
for this method to be effective. Since this requirement is often impossible to fulfill
(e.g., due to blockage of signals at the roving receiver), this method is seldom
190
GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
used. A far better method, which offers more flexibility, is to use only difference
the measurements from the set of satellites which are viewed in common by both
receivers. Therefore, only this method will be described.
The two primary types of differential measurements are code measurements
and carrier phase measurements.
5.12.1 Code Differential Measurements
To obtain code differential measurements, the roving and reference receivers each
make a pseudorange measurement of the following form for each satellite
ρM = ρ + c dt − c dT + dION + dTROP + dEPHEM + dρ ,
(5.77)
where ρM is the measured pseudorange in meters, ρ is the true receiver-tosatellite geometric range in meters, c is the speed of light in meters per second,
dt is satellite clock error in seconds, dT is receiver clock error in seconds,
dION is ionospheric delay error in meters, dTROP is tropospheric delay error in
meters, dEPHEM is delay error in meters due to satellite ephemeris error, and dρ
represents other pseudorange errors in meters, such as multipath, interchannel
receiver biases, thermal noise, and selective availability (when turned on). The
pseudorange measurements made by both receivers must occur at a common GPS
time, or if not, corrections must be applied to extrapolate the measurements to a
common time.
5.12.1.1 First Difference Observations A code first difference observation
is determined by subtracting an equation of the form (5.77) for the reference
receiver from a similar equation for the roving receiver, where both equations
relate to the same satellite. The result is
ρM = ρ − cdT + dION + dTROP + dEPHEM + dρ ,
(5.78)
where the symbol denotes the difference between the corresponding terms in
the two equations of the form (5.77). Note that the term cdt representing the
satellite clock error has disappeared, since the satellite clock error is the same
for the pseudorange measurements made by each receiver. Furthermore, if the
distance between the roving and reference receivers is sufficiently small (say, <
20 km), the terms dION , dTROP , and dEPHEM will be nearly canceled out,
since errors due to the ionosphere, troposphere, and ephemeredes vary slowly
with position.
5.12.1.2 Second Difference Observations A code second difference measurement is formed by subtraction of the first difference observation of the form
(5.78) for one satellite from a similar first difference observation for another
191
DIFFERENTIAL GNSS
satellite. Thus, if there are N first difference observations corresponding to N
satellites, there will be N − 1 independent second difference observations that
can be formed. The second difference observations have the form
∇ρM = ∇ρ + ∇dION + ∇dTROP + ∇dEPHEM + ∇dρ ,
(5.79)
where the symbol ∇ denotes difference between the corresponding difference
terms in the two equations of the form (5.78). Note that the double difference error
term c∇dT involving receiver clock error has been cancelled out, since receiver
clock error is constant across all satellite measurements in both the reference and
roving receivers. Furthermore, for a sufficiently small distance between the rover
and reference receivers, the first difference errors dION , dTROP , and dEPHEM
are so small that the corresponding double difference errors ∇dION , ∇dTROP ,
and ∇dEPHEM can be neglected. In this case the second difference observations
become
∇ρM ∼
= ∇ρ + ∇dρ ,
(5.80)
which can result positioning accuracies that are often in the submeter range.
Although DGPS is effective in removing satellite and receiver clock errors,
ionospheric and tropospheric errors, ephemeris errors, and selective availability
errors, it cannot remove errors due to multipath, receiver interchannel biases,
and thermal noise, since these errors are not common to the roving and reference
receivers.
5.12.2 Carrier Phase Differential Measurements
Because carrier phase pseudorange measurements have a significantly smaller
error than do those using the code, positioning accuracy is potentially much
more accurate. However, since only the fractional and not the integer part of a
carrier cycle can be observed, some method of finding the integer part must be
employed. This is the classic ambiguity resolution problem.
First and second difference observations can be obtained from carrier phase
pseudorange measurements having the form
λφM = ρ + cdt − cdT + λN − dION + dTROP + dEPHEM + dφ ,
(5.81)
where the new variables are the carrier wavelength λ (0.1903 m for L1 and
0.2442 m for L2 ), the measured carrier phase φM in cycles, the carrier phase
ambiguity N in cycles, and other errors dφ in meters. Because the ionospheric
group delay for the carrier is opposite that of the code, the ionospheric error is
reversed in sign in (5.81).
In some cases triple differences of carrier phase measurements are used, as
will be described subsequently.
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GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
5.12.2.1 First Difference Observations Each carrier phase first difference
observation is determined in the same manner as for PN code by subtracting
an equation of the form (5.81) for the reference receiver from a similar equation
for the roving receiver, where both equations relate to the same satellite. The
result is
λφM = ρ − cdT + λN − dION + dTROP + dEPHEM + dφ ,
(5.82)
where, as before, the satellite clock error term has disappeared and the terms
dION , dTROP , and dEPHEM are small for small distances between the rover
and reference receivers.
5.12.2.2 Second Difference Observations A second difference carrier phase
measurement is formed by subtraction of the first difference observation of the
form (5.82) for one satellite from a similar first difference observation for another
satellite. The second difference observations have the form
∇λφM = ∇ρ + λ∇N − ∇dION + ∇dTROP + ∇dEPHEM + ∇dφ ,
(5.83)
Again, the receiver clock error term has disappeared, and the terms ∇dION ,
∇dTROP , and ∇dEPHEM are usually small. However, the value of N in the
phase ambiguity term λ∇N must be determined by some method of ambiguity
resolution.
5.12.2.3 Third Difference Observations Triple difference carrier observations
are sometimes used in DGPS, primarily to detect and correct cycle slips during
carrier tracking. These observations have the form
δ∇λφM = δ∇ρ − δ∇dION + δ∇dTROP + δ∇dEPHEM + δ∇dφ ,
(5.84)
where δ is the time difference between two successive double difference observations. Cycle slips can be detected by observing the deviation of successive triple
difference observations from their predicted values as the carrier is tracked.
5.12.2.4 Combinations of L1 and L2 Carrier Phase Observations Summing
the L1 and L2 second difference carrier observations results in higher-resolution
phase measurements than can be obtained at either frequency alone. Such narrowlane measurements result in more precision, but place greater demands on phase
ambiguity resolution. On the other hand, it is easier to resolve the phase ambiguity
of wide-lane measurements obtained by differencing the L1 and L2 observations
at the expense of reduced resolution.
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DIFFERENTIAL GNSS
5.12.3 Positioning Using Double-Difference Measurements
5.12.3.1 Code-Based Positioning The linearized matrix equation for positioning using code double-difference measurements from four satellites has the form
3×3
3×1
3×1
3×1
[1]
δZ∇ρ = H∇ρ δx + νρ ,
(5.85)
which is the same form as shown in Section 2.3.3. However, because the doubledifference measurements have eliminated receiver clock error as an unknown,
the unknowns are simply the x, y, and z coordinates of the roving receiver, constituting the components of the 3×1 vector δx. Thus, the measurement matrix
H [1] is 3×3 and the partial derivatives in it are partial derivatives of the double differences ∇ρM with respect to user position coordinates x,y,z instead of
partial derivatives of the pseudorange measurements.. Accordingly, the measurement vector δZρ and the measurement noise vector νρ are 3×1. As indicated in
Section 2.3.3, a solution for position can be found by computing the measurement
vector associated with an assumed initial position x, finding the difference δZρ
between the computed and actual measurement vectors, solving (5.85) (omitting
the measurement noise vector) for the position correction δx, and obtaining the
new value x + δx for x. Iteration of this process is used to produce a sequence
of positions which converges to the position solution.
5.12.3.2 Carrier Phase–Based Positioning For positioning using carrier phase
double difference measurements the linearized matrix equation from four satellites used for iterative position solution has the form
3×3
3×1
3×1
3×1
[1]
δZ∇ρ = H∇ρ δx + νρ ,
(5.86)
where the measurement matrix H [1] contains the partial derivatives of the double
differences ∇λφM with respect to user position coordinates x,y,z. As compared
to code-based positioning, the measurement noise term νρ is much smaller, often
in the centimeter range. However, the major difference is that the ambiguity in the
phase measurements can cause convergence to any one of many possible positions
in a spatial grid of points. Only one of these points is the correct position. Various
techniques for resolving the ambiguity have been developed. A simple method
is to use the position solution from the code second difference measurements
as the initial position x in the carrier phase iterative position solution. If this
initial position is sufficiently accurate, convergence to the correct solution will
be obtained.
5.12.3.3 Real-Time Processing versus Postprocessing Since double differencing combines measurements made in the roving and reference receivers, these
measurements must be brought together for processing. Often the processing site
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GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
is at the roving receiver, although in other applications it can be at the reference
station or at another off-site location. In real-time processing measurements are
transmitted to the processing site using wireless communication or a telephone
link. In post processing the data can be physically carried to the processing site
in a storage medium such as a floppy disk or CD-ROM. Another post processing
option is to transmit the data via the Internet.
5.13 GPS PRECISE POINT POSITIONING SERVICES AND
PRODUCTS
The cost and inconvenience of setting up one’s own DGPS system can be eliminated because there are numerous services and software packages available to
the user, some of which are free. There are too many to describe completely, so
only a few of them are described in this section.
The International GNSS Service (IGS) Many of the DGPS services are subsumed under the IGS, which is a voluntary federation of more than 200 worldwide
agencies that pool resources and permanent GPS and GLONASS station data to
generate precise DGPS positioning services. The IGS is committed to providing the highest quality data and products as the standard for global navigation
satellite systems (GNSSs) in support of earth science research, multidisciplinary
applications, and education. The IGS also intends to incorporate future GNSS
systems, such as Galileo, as they become operational.
Continuously Operating Reference Stations (CORSs) The National Geodetic
Survey (NGS), an office of NOAA’s National Ocean Service, manages two
networks of CORS: the National CORS network and the Cooperative CORS
network. These networks consist of numerous base stations containing DGPS
reference receivers that operate continuously to generate pseudorange and other
DGPS data for postprocessing. The data is disseminated to a wide variety of
users. Surveyors, GIS/LIS professionals, engineers, scientists, and others can
apply CORS data to their own GPS measurements to obtain positioning accuracies approaching a few centimeters relative to the National Spatial Reference
System, both horizontally and vertically. The CORS program is a multipurpose
cooperative endeavor involving more than 130 government, academic, and private organizations, each of which operates at least one CORS site. In particular,
it includes all existing National Differential GPS (NDGPS) sites and all existing
FAA Wide-Area Augmentation System (WAAS) sites. New sites are continually
being evaluated according to established criteria.
Typical uses of CORS include land management, coastal monitoring, civil
engineering, boundary determination, mapping and geographic information systems (GISs), geophysical and infrastructure modeling, as well as future improvements to weather prediction and climate modeling.
GPS PRECISE POINT POSITIONING SERVICES AND PRODUCTS
195
All national CORS data are available from NGS at their original sampling rate
for 30 days, after which the data are decimated to a 30-s sampling rate. Cooperative CORS data are available from a large number of participating organizations
that operate individual sites. Most of the CORS data are available on the internet.
GPS-Inferred Positioning System (GIPSY) and Orbit Analysis Simulation Software (OASIS) The GIPSY-OASIS II (GOA II) package consists of extremely
versatile software that can be used for GPS positioning and satellite orbit analysis. Developed by the Caltech Jet Propulsion Laboratory (JPL), it can provide
centimeter-level DGPS positioning accuracy over short to intercontinental baselines. It is capable of unattended, automated, low-cost operation in near real
time for precise positioning and time transfer in ground, sea, air, and space
applications.
GOA II also includes many force models useful for orbit determination, such
as earth/sun/moon/planet (and tidal) gravity perturbations, solar pressure, thermal radiation, and drag, which make it useful in non-GPS satellite positioning
applications. To augment its potential accuracy, models are included for earth
characteristics, such as tides, ocean/atmospheric loading, and crustal plate motion.
Parameter estimation for positioning and time transfer is state-of-the-art. A
general estimator can be used for GPS and non-GPS data. Matrix factorization
is used to maintain robustness of solutions, and the estimator can intelligently
identify, correct, or exclude questionable data. A general and flexible noise model
is included.
Australia’s Online GPS Processing System (AUPOS) AUPOS provides users
with the facility to submit via the Internet dual-frequency geodetic quality GPS
RINEX data observed in a “static” mode and receive rapid-turnaround precise
position coordinates. The service is free and provides both International Terrestrial Reference Frame (ITRF) and Geocentric Datum of Australia (GDA94)
coordinates. This Internet service takes advantage of both IGS products and the
IGS GPS network and can handle GPS data collected anywhere on earth.
Scripps Coordinate Update Tool (SCOUT) SCOUT, managed by the Scripps
Institute of Oceanography, is also a system which provides precise positioning
for users who submit GPS RINEX data from their receiver via the Internet.
The reference stations are by default the three nearest sites for which data have
been collected and are available for the specific day the user’s data are taken.
However, the user can specify the reference stations if desired. Station maps are
provided to assist the user in specifying nearby reference sites. When SCOUT has
finished determining a DGPS position solution, it sends a report of the results
to the user via the Internet. The report contains both Cartesian and geodetic
coordinates, standard deviations, and the locations of the reference sites that
were used. The reported Cartesian coordinates are referenced to the International
Terrestrial Reference Frame 2000 (ITRF2000), and the geodetic coordinates are
referenced to both ITRF2000 and the World Geodetic System 1984 (WGS84)
ellipsoid.
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GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
The Online Positioning User Service (OPUS) The National Geodetic Survey
(NGS) operates OPUS as a means to provide GPS users easier access to the
National Spatial Reference System (NSRS). OPUS users submit their GPS data
files to the NGS Internet site. The NGS computers and software determine a
position by using reference receivers from three CORS sites. The position is
reported back to the user by email in both ITRF and NAD83 (North American
Datum 1983) coordinates, as well as Universal Transverse Mercator (UTM), and
State Plain Coordinates (SPC) northing and easting. Results are typically obtained
within a few minutes. OPUS is intended for use in the coterminous United States
and most U.S. territories. It is NGS policy not to publish geodetic coordinates
outside the United States without the agreement of the affected countries.
PROBLEMS
5.1 Using the values provided for Klobuchar’s model in Section 5.2, for Southbury, Connecticut, calculate the ionospheric delay and plot the results.
5.2 Assume that a direct-path GPS L1 C/A-code signal arrives with a phase
such that all of the signal power lies in the baseband I channel, so that
the baseband signal is purely real. Further assume an infinite signal bandwidth so that the cross-correlation of the baseband signal with an ideal C/A
reference code waveform will be an isosceles triangle 600 m wide at the
base.
(a) Suppose that in addition to the direct-path signal there is a secondarypath signal arriving with a relative time delay of precisely 250 L1 carrier
cycles (so that it is in phase with the direct-path signal) and with an
amplitude one-half that of the direct path. Calculate the pseudorange
error that would result, including its sign, under noiseless conditions.
Assume that pseudorange is measured with a delay-lock loop using
0.1-chip spacing between the early and late reference codes. (Hint: The
resulting cross-correlation function is the superposition of the crosscorrelation functions of the direct- and secondary-path signals.)
(b) Repeat the calculations of part (a) but with a secondary-path relative
time delay of precisely 250 12 carrier cycles. Note that in this case the
secondary-path phase is 180◦ out of phase with the direct-path signal,
but still lies entirely in the baseband I channel.
5.3 (a) Using the discrete matrix version of the receiver clock model given by
Eq. 5.75, find the standard deviation σw1 of the white-noise sequence
w1,k needed in the model to produce a frequency standard deviation σx1
of 1Hz after 10 min of continuous oscillator operation. Assume that the
initial frequency error at t = 0 is zero and that the discrete-time step
t is 1 s.
GPS PRECISE POINT POSITIONING SERVICES AND PRODUCTS
197
(b) Using the assumptions and the value of σw1 found in part (a), find
the standard deviation σx2 of the bias error after 10 min. Assume that
σw2 = 0.
(c) Show that σx1 and σx2 approach infinity as the time t approaches infinity.
Will this cause any problems in the development of a Kalman filter that
includes estimates of the clock frequency and bias error?
5.4 The peak electron density in the ionosphere occurs in a height range of
(a) 50–100 km
(b) 250–400 km
(c) 500–700 km
(d) 800–1000 km
5.5 The refractive index of the gaseous mass in the troposphere is
(a) Slightly higher than unity
(b) Slightly lower than unity
(c) Unity
(d) Zero
5.6 If the range measurements for two simultaneously tracking satellites in a
receiver are differenced, then the differenced measurement will be free of
(a) Receiver clock error only
(b) Satellite clock error and orbital error only
(c) Ionospheric delay error and tropospheric delay error only
(d) Ionospheric delay error, tropospheric delay error, satellite clock error,
and orbital error only.
5.7 Zero baseline test (code) can be performed to estimate
(a) Receiver noise and multipath
(b) Receiver noise
(c) Receiver noise, multipath, and atmospheric delay errors
(d) None of the above
5.8 What are the purposes of selective availability (SA) and antispoofing (AS)?
5.9 What are GNSS
(a) Single difference?
(b) Double difference?
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GLOBAL NAVIGATION SATELLITE SYSTEM DATA ERRORS
(c) Triple difference?
(d) Wide lane?
(e) Narrow lane?
5.10 What is a CORS site?
5.11 Derive the multipath formula equivalent to Eq. 5.46 for L2 , using the same
notation as in Eq. 5.40.
5.12 Calculate the ionospheric delay using dual-frequency carrier phases.
6
DIFFERENTIAL GNSS
6.1 INTRODUCTION
Differential global navigation satellite system (differential GNSS; abbreviated
DGNSS) is a technique for reducing the error in GPS-derived positions by using
additional data from a reference GNSS receiver at a known position. The most
common form of DGNSS involves determining the combined effects of navigation message ephemeris and satellite clock errors [including the effects of
propagation] at a reference station and transmitting delays corrections, in real
time, to a user’s receiver. The receiver applies the corrections in the process of
determining its position [94]. They include:
•
•
•
Corrections for completed: (1) selective availability (if present) (2) satellite
ephemeris and clock errors.
(3) ionospheric delay error and (4) tropospheric delay error.
Still other error sources cannot be corrected with DGNSS: (1) multipath
errors and (2) user receiver errors.
6.2 DESCRIPTIONS OF LADGPS, WADGPS, AND SBAS
6.2.1 Local-Area Differential GPS (LADGPS)
LADGPS is a form of DGPS in which the user’s GPS receiver receives real-time
pseudorange and, possibly, carrier phase corrections from a reference receiver
Global Positioning Systems, Inertial Navigation, and Integration, Second Edition, by M. S. Grewal, L. R. Weill, and A. P. Andrews
Copyright © 2007 John Wiley & Sons, Inc.
199
200
DIFFERENTIAL GNSS
generally located within the line of sight. The corrections account for the combined effects of navigation message ephemeris and satellite clock errors (including the effects of SA) and, usually, atmospheric propagation delay errors at the
reference station. With the assumption that these errors are also common to the
measurements made by the user’s receiver, the application of the corrections will
result in more accurate coordinates [117].
6.2.2 Wide-Area Differential GPS (WADGPS)
WADGPS is a form of DGPS in which the user’s GPS receiver receives corrections determined from a network of reference stations distributed over a wide
geographic area. Separate corrections are usually determined for specific error
sources, such as satellite clock, ionospheric propagation delay, and ephemeris.
The corrections are applied in the user’s receiver or attached computer in computing the receiver’s coordinates. The corrections are typically supplied in real
time by way of a geostationary communications satellite or through a network of
ground-based transmitters. Corrections may also be provided at a later date for
postprocessing collected data [117].
6.2.3 Space-Based Augmentation Systems (SBAS)
6.2.3.1 Wide-Area Augmentation System (WAAS) WAAS enhances the GPS
SPS and is available over a wide geographic area. The WAAS being developed
by the Federal Aviation Administration, together with other agencies, will provide WADGPS corrections, additional ranging signals from geostationary (GEO)
satellites, and integrity data on the GPS and GEO satellites [117].
Each GEO uplink subsystem includes a closed-loop control algorithm and
special signal generator hardware. These ensure that the downlink signal to the
users is controlled adequately to be used as a ranging source to supplement the
GPS satellites in view.
The primary mission of WAAS is to provide a means for air navigation for all
phases of flight in the National Airspace System (NAS) from departure, en route,
arrival, and through approach. GPS augmented by WAAS offers the capability
for both nonprecision approach (NPA) and precision approach (PA) within a
specific service volume. A secondary mission of the WAAS is to provide a WAAS
network time (WNT) offset between the WNT and Coordinated Universal Time
(UTC) for nonnavigation users.
WAAS provides improved en route navigation and PA capability to WAAScertified avionics. The safety critical WAAS system consists of the equipment and
software necessary to augment the Department of Defense (DOD)-provided GPS
SPS. WAAS provides a signal in space (SIS) to WAAS-certified aircraft avionics
using the WAAS for any FAA-approved phase of flight. The SIS provides two
services: (1) data on GPS and GEO satellites and (2) a ranging capability.
The GPS satellite data is received and processed at widely dispersed wide-area
reference stations (WRSs), which are strategically located to provide coverage
DESCRIPTIONS OF LADGPS, WADGPS, AND SBAS
Fig. 6.1
201
WAAS top-level view.
over the required WAAS service volume. Data are forwarded to wide-area master
stations (WMSs), which process the data from multiple WRSs to determine the
integrity, differential corrections, and residual errors for each monitored satellite
and for each predetermined ionospheric grid point (IGP). Multiple WMSs are
provided to eliminate single-point failures within the WAAS network. Information
from all WMSs is sent to each GEO uplink subsystem (GUS) and uplinked along
with the GEO navigation message to GEO satellites. The GEO satellites downlink
these data to the users via the GPS SPS L-band ranging signal (L1 ) frequency
with GPS-type modulation. Each ground-based station/subsystem communicates
via a terrestrial communications subsystem (TCS). (See Fig. 6.1).
In addition to providing augmented GPS data to the users, WAAS verifies its
own integrity and takes any necessary action to ensure that the system meets
the WAAS performance requirements. WAAS also has a system operation and
maintenance function that provides status and related maintenance information
to FAA airway facilities (AFs) NAS personnel.
WAAS has a functional verification system (FVS) that is used for early development test and evaluation (DT&E), refinement of contractor site installation
procedures, system-level testing, WAAS operational testing, and long-term support for WAAS.
Correction and Verification (C&V) processes data from all WRSs to determine
integrity, differential corrections, satellite orbits, and residual error bounds for
each monitored satellite. It also determines ionospheric vertical delays and their
residual error bounds at each of the IGPs. C&V schedules and formats WAAS
messages and forwards them to the GUSs for broadcast to the GEO satellites.
C&V’s capabilities are as follows:
202
DIFFERENTIAL GNSS
1. Control C&V Operations and Maintenance (COM) supports the transfer of files, performs remotely initiated software configuration checks,
and accepts requests to start and stop execution of the C&V application
software.
2. Control C&V Modes manage mode transitions in the C&V subsystem
while the application software is running.
3. Monitor C&V (MCV) reports line replaceable unit (LRU) faults and configuration status. In addition, it monitors software processes and provides
performance data for the local C&V subsystems.
4. Process Input Data (PID) selects and monitors data from the wide-area reference equipment (WREs). Data that passes PID screening is repackaged
for other C&V capabilities. PID performs clock and L1 GPS Precision
Positioning Service L-band ranging signal (L2 ) receiver bias calculations,
cycle slip detection, outlier detection, data smoothing, and data monitoring. In addition, PID calculates and applies the windup correction to
the carrier phase, accumulates data to estimate the pseudorange to carrier phase bias, and computes the ionosphere corrected carrier phase and
measured slant delay.
5. Satellite Orbit Determination (SOD) determines the GPS and GEO satellite orbits and clock offsets, WRE receiver clock offsets, and troposphere
delay.
6. Ionosphere Correction Computation (ICC) determines the L1 IGP vertical
delays, grid ionosphere vertical error (GIVE) for all defined IGPs, and
L1 –L2 interfrequency bias for each satellite transmitter and each WRS
receiver.
7. Satellite Correction Processing (SCP) determines the fast and long-term
satellite corrections, including the user differential range error (UDRE).
It determines the WNT and the GEO and WNT clock steering commands [154].
8. Independent Data Verification (IDV) compares satellite corrections, GEO
navigation data, and ionospheric corrections from two independent computational sources, and if the comparisons are within limits, one source
is selected from which to build the WAAS messages. If the comparisons
are not within limits, various responses may occur, depending on the data
being compared, all the way from alarms being generated to the C&V
being faulted.
9. Message Output Processing (MOP) transmits messages containing independently verified results of C&V calculations to the GUS processing
(GP) for broadcast.
10. C&V Playback (PLB) processes the playback data that has been recorded
by the other C&V capabilities.
11. Integrity Data Monitoring (IDM) checks both the broadcast and the to-bebroadcast UDREs and GIVEs to ensure that they are properly bounding
their errors. In addition, it monitors and validates that the broadcast
DESCRIPTIONS OF LADGPS, WADGPS, AND SBAS
203
messages are sent correctly. It also performs the WAAS time-to-alarm
validation [1, 154].
WRS Algorithms Each WRS collects raw pseudorange (PR) and accumulated
delta range (ADR) measurements from GPS and GEO satellites selected for
tracking. Each WRS performs smoothing on the measurements and corrects for
atmospheric effects, that is, ionospheric and tropospheric delays. These smoothed
and atmospherically corrected measurements are provided to the WMS.
WMS Foreground (Fast) Algorithms The WMS foreground algorithms are applicable to real-time processing functions, specifically the computation of fast correction, determination of satellite integrity status and WAAS message formatting.
This processing is done at a 1-Hz rate.
WMS Background (Slow) Algorithms The WMS background processing consists
of algorithms that estimate slowly varying parameters. These algorithms consist
of WRS clock error estimation, grid ionospecific delay computation, broadcast
ephemeris computation, satellite orbit determination, satellite ephemeris error
computation, and satellite visibility computation.
Independent Data Verification and Validation Algorithms This includes a set of
WRS and at least one WMS, which enable monitoring the integrity status of
GPS and the determination of wide-area DGPS correction data. Each WRS has
three dual-frequency GPS receivers to provide parallel sets of measurement data.
The presence of parallel data streams enables independent data verification and
validation (IDV&V) to be employed to ensure the integrity of GPS data and
their corrections in the WAAS messages broadcast via one or more GEOs. With
IDV&V active, the WMS applies the corrections computed from one stream to
the data from the other stream to provide verification of the corrections prior to
transmission. The primary data stream is also used for the validation phase to
check the active (already broadcast) correction and to monitor their SIS performance. These algorithms are continually being improved. The latest versions can
be found in references the literature [68, 151, 152, 154, 217; 153, pp. 397–425].
6.2.3.2 European Global Navigation Overlay System (EGNOS) EGNOS is
a joint project of the European Space Agency, the European Commission, and
the European Organization for the Safety of Air Navigation (Eurocontrol). Its
primary service area is the ECAC (European Civil Aviation Conference) region.
However, several extensions of its service area to adjacent and more remote areas
are under study. An overview of the EGNOS system architecture is presented in
Fig. 6.2, where
[RIMS] are the Ranging and Integrity Monitoring Stations
[MCC] is the Mission and Control Center
[NLES] are the Navigation Land Earth Stations
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DIFFERENTIAL GNSS
GPS
GLONASS
GEO
GEO
GEO
USERS
PACF/
RIMS
MCC
MCC
MCC
NLES
NLES
NLES
DVP/
ASQF
EWAN
Fig. 6.2
European Global Navigation Overlay System architecture.
Fig. 6.3
Current and planned SBAS service areas.
[PACF] is the Performance Assessment and Checkout Facility
[DVP] is the Development and Verification Platform
[ASQF] is the Application Specific Qualification Facility
[EWAN] is the EGNOS Wide Area (communication) Network
6.2.3.3 Other SBAS Service areas of current and future SBAS systems are
mapped in Fig, 6.3, and the acronyms are listed in Table 6.1.
GROUND-BASED AUGMENTATION SYSTEM (GBAS)
205
TABLE 6.1. Worldwide SBAS System Coverages
Country
Acronym
Title
USA
Europe
WAAS
EGNOS
Japan
MSAS
Canada
CWAAS
China
SNAS
India
GAGAN
Wide-Area Augmentation System
European Geostationary Navigation
Overlay System
MTSAT Satellite-based Augmentation
System
Canadian Wide Area Augmentation
System
Satellite Navigation Augmentation
System
GPS & GEO Augmented Navigation
6.3 GROUND-BASED AUGMENTATION SYSTEM (GBAS)
6.3.1 Local-Area Augmentation System (LAAS)
The Local-Area Augmentation System (near airports) will be designed to provide differential GPS corrections in support of navigation and landing systems.
The system provides monitoring functions via LAAS Ground Facility (LGF)
and includes individual measurements, ranging sources, reference receivers, navigation data, data broadcast, environment sensors, and equipment failures. Each
identified monitor has a corresponding system response including alarms, alerts
and service alerts. (See Fig. 6.4.)
6.3.2 Joint Precision Approach Landing System (JPALS)
The Joint Precision Approach Landing System (JPALS) is in the “concept and
technology development” phase by the Department of Defense. Concept exploration led to the determination that the Local-Area Differential Global Positioning
System (LADGPS) was the best precision approach and landing solution.
The objective of upcoming “component advanced development” (CAD) work
effort is to provide sufficient evidence that key technical risks for LADGPS have
been reduced and to help define the JPALS technical architecture. The CADevelopment effort will operate under the belief that a fully developed LADGPS will
give both conventional manned aircraft and possible unmanned aerial vehicles
(UAVs) fully coupled (automatic) landing capability in any weather, and in any
mission environment.
LADGPS-complementing technologies are also being explored. These include:
GPS-based local- and wide-area augmentation systems (LAAS and WAAS) that
will enhance civil interoperability, and autonomous landing capability (ALC),
which will greatly improve a pilot’s visibility by filtering out meteorological
conditions like snow and fog.
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DIFFERENTIAL GNSS
Reference Station
Processor/VDB transmitter
Pseudolite
VDB = Very High Frequency Data Broadcast
Fig. 6.4
Local-Area Augmentation System (LAAS).
6.3.3 LORAN-C
Long-range navigation (LORAN) uses signal phase information from three or
more long-range navigation signal sources positioned at fixed, known locations.
The LORAN-C system relies upon a plurality of ground-based signal towers,
spaced 100-300 km apart. Antenna towers transmit distinguishable electromagnetic signals that are received and processed by a LORAN signal antenna and
LORAN signal receiver/processor that are analogous to GPS antenna and receiver/
processor.
6.4 GEO UPLINK SUBSYSTEM (GUS)
Corrections from the WMS are sent to the ground uplink subsystem (GUS) for
uplink to the GEO. The GUS receives integrity and correction data and WAAS
specific messages from the WMS, adds forward error correction (FEC) encoding,
and transmits the messages via a C-band uplink to the GEO satellites for broadcast
to the WAAS user. The GUS signal uses the GPS standard positioning service
waveform (C/A-code, BPSK modulation); however, the data rate is higher (250
(bits per second)). The 250 (bits per second) of data are encoded with a onehalf rate convolutional code, resulting in a transmission rate of 500 symbols per
second (sps).
GEO UPLINK SUBSYSTEM (GUS)
207
Each symbol is modulated by the C/A-code, a 1.023 × 106 -chips/s pseudo
random sequence to provide a spread-spectrum signal. This signal is then BPSKmodulated by the GUS onto an IF carrier, upconverted to a C-band frequency,
and uplinked to the GEO. It is the C/A-code modulation that provides the ranging
capability if its phase is properly controlled.
Control of the carrier frequency and phase is also required to eliminate uplink
Doppler and to maintain coherence between code and carrier. The GUS monitors
the C-band and L1 downlinks from the GEO to provide closed-loop control of
the PRN code and L1 carrier coherency. WAAS short-and long-term code–carrier
coherence requirements are met.
6.4.1 Description of the GUS Algorithm
The GUS control loop algorithm “precorrects” the code phase, carrier phase,
and carrier frequency of the GEO uplink signal to maintain GEO broadcast
code–carrier coherence. The uplink effects such as ionospheric code–carrier
divergence, uplink Doppler, equipment delays, and frequency offsets must be
corrected in the GUS control loop algorithm.
Figure 6.5 provides an overview of the functional elements of the GUS control
loop. The control loop contains algorithm elements (shaded boxes) and hardware
elements that either provide inputs to the algorithm or are controlled or affected
by outputs from the algorithm. The hardware elements include a WAAS GPS
receiver, GEO satellite, and GUS signal generator.
Downlink ionospheric delay is estimated in the ionospheric delay and rate
estimator using pseudorange measurements form the WAAS GPS receiver on L1
and L2 (downconverted from the GEO C-band downlink at the GUS). This is a
two-state Kalman filter that estimates the ionospheric delay and delay rate.
At each measurement interval, a range measurement is taken and fed into the
range, rate, and acceleration estimator. This measurement is the average between
the reference pseudorange from the GUS signal generator PR sign and the received
pseudorange from the L1 downlink as measured by the WAAS GPS receiver
(PR geo ) and adjusted for estimated ionospheric delay (PR iono ). The equation for
the range measurement is then
z=
1
[(P Rgeo − P Riono ) + P Rsign ] − TCup − TL1dwnS ,
2
where TCup is C-band uplink delay (m) and TL1dwnS is L1 receiver delay of the
GUS (m).
The GUS signal generator is initialized with a pseudorange value from satellite
ephemeris data. This is the initial reference from which corrections are made.
The range, rate and acceleration estimator is a three-state Kalman filter that
drives the frequency and code control loops.
The code control loop is a second-order control system. The error signal for
this control system is the difference between the WAAS pseudorange (Prsign ) and
208
DIFFERENTIAL GNSS
Fig. 6.5 GUS control loop block diagram.
the estimated pseudorange from the Kalman filter. The loop output is the code
rate adjustments to the GUS signal generator.
The frequency control loop has two modes. First, it adjusts the signal generator
frequency to compensate for uplink Doppler effects. This is accomplished using
a first-order control system. The error signal input is the difference between the
L1 Doppler frequency from the WAAS GPS receiver and the estimated range
rate (converted to a Doppler frequency) from the Kalman filter.
Once the frequency error is below a threshold value, the carrier phase is
controlled. This is accomplished using a second-order control system. The error
signal input to this system is the difference between the L1 carrier phase and a
carrier phase estimate based on the Kalman filter output. This estimated range
is converted to carrier cycles using the range estimate at the time carrier phase
control starts as a reference. Fine-tuning adjustments are made to the signal
generator carrier frequency to maintain phase coherence [52, 67, 68, 69, 145].
6.4.2 In-Orbit Tests (IOT)
Two separate series of in-orbit tests (IOTs) were conducted, one at the COMSAT
GPS Earth Station (GES) in Santa Paula, California with Pacific Ocean Region
(POR) and Atlantic Ocean Region West (AOR-W) Inmarsat-3 (I-3) satellites and
GEO UPLINK SUBSYSTEM (GUS)
209
Fig. 6.6 IOT test GUS setup.
the other at the COMSAT GES in Clarksburg, Maryland, using AOR-W. The IOTs
were conducted to validate a prototype version of the GUS control loop algorithm. Data were collected to verify the ionospheric estimation and code–carrier
coherence performance capability of the control loop and the short–term carrier
frequency stability of the Inmarsat-3 satellites with a prototype ground station.
The test results were also used to validate the GUS control loop simulation.
Figure 6.6 illustrates the IOT setup at a high level. Prototype ground station
hardware and software were used to assess algorithm performance at two different
ground stations with two different Inmarsat-3 satellites.
6.4.3 Ionospheric Delay Estimation
The GUS control loop estimates the ionospheric delay contribution of the GEO
C-band uplink to maintain code–carrier coherence of the broadcast SIS. Figure
6.7 shows the delay estimates for POR using the Santa Paula AOR-W. The plot
shows the estimated ionospheric delay (output of the two-state Kalman filter)
versus the calculated delay using the L1 and C pseudorange data from a WAAS
GPS receiver. Calculated delay is noisier and varying about 1 m/s, whereas the
estimated delay by the Kalman filter is right in middle of the measured delay, as
shown in Fig. 6.7. Delay measurements were calculated using the equation
"
#2
PRL1 − PRC − τ L1 + τ C1 − L1 freq
Ionospheric delay =
$"
#2
1 − [L1 freq]2 C freq
210
DIFFERENTIAL GNSS
Fig. 6.7
Measured and estimated ionospheric delay, AOR-W, Santa Paula.
where
PRL1 = L1 pseudorange (m)
PRC = C pseudorange (m)
τ L1 = L1 downlink delay (m)
τ C = C downlink delay (m)
L1 freq = L1 frequency = 1575.42 MHz
Cfreq = C frequency = 3630.42 MHz
The ionosphere during the IOTs was fairly benign with no high levels of solar
activity observed. Table 6.2 provides the ionospheric delay statistics (in meters)
between the output of the ionospheric Kalman filter in the control loop, and the
calculated delay from the WAAS GPS receiver’s L1 and L2 pseudoranges. The
statistics show that the loop’s ionospheric delay estimation is very close (low
TABLE 6.2. Observed RMS WAAS Ionospheric Correction
Errors
In-Orbit Test
Santa Paula GES, Oct. 10, 1997, POR
Santa Paula GES, Dec. 1, 1997, AOR-W
Clarksburg GES, March 20, 1998, AOR-W
RMS Error (m)
0.20
0.45
0.34
211
GEO UPLINK SUBSYSTEM (GUS)
RMS) to the ionospheric delay calculated using the measured pseudorange from
the WAAS GPS receiver.
6.4.4 Code–Carrier Frequency Coherence
The GEO’s broadcast code–carrier frequency coherence requirement is specified
in the WAAS System Specification and Appendix A of Ref. 165. It states:
The lack of coherence between the broadcast carrier phase and the code phase shall
be limited. The short term fractional frequency difference between the code phase
rate and the carrier frequency will be less than 5 × 10−11 . That is,
fcode
fcarrier
−11
1.023 MHz − 1575.42 MHz < 5 × 10
Over the long term, the difference between the broadcast code phase (1/1540) and
the broadcast carrier phase will be within one carrier cycle, one sigma. This does
not include code–carrier divergence due to ionospheric refraction in the downlink
propagation path.
For the WAAS program, short term is defined as less than 10 s and long term,
less than 100 s. Pseudorange minus the ionospheric estimates averaged over τ
seconds is expressed as
FPR =
P RL1 (t) − ionoestimate (t)
τ
m/s.
Carrier phase minus the ionospheric estimate average over τ seconds is expressed
as
FPH =
−φL1 (t) + (ionoestimate (t)/λL1 )
τ
cycles/s.
For long-term code–carrier coherence calculations, a τ of 60 s was chosen to
mitigate receiver bias errors in the pseudorange and carrier phase measurements
of the WAAS GPS receiver. For short-term code–carrier coherence a shorter
30-s averaging time was selected. The code–carrier coherence requirement is
specified at the output of the GEO and not the receiver, so data averaging has
to be employed to back out receiver effects such as multipath and noise. Each
averaging time was based on analyzing GPS satellite code–carrier coherence data
and selecting the minimum averaging time required for GPS to meet the WAAS
code–carrier coherence requirements.
For long-term code–carrier coherence calculations, the difference between the
pseudorange and the phase measurements is given by
PR−PH =
FPR
− FPH
λL1
cycles/s,
212
DIFFERENTIAL GNSS
TABLE 6.3. Code–Carrier Coherence Test Results
Carrier Coherence Requirements
Test Location
Santa Paula
prototyping
Clarksburg
prototyping
a
b
Short-Terma (10 s) Long-Termb (100 s)
<5 × 10−11
<1 cycle
Test Date
POR, Oct. 10, 1997
1.89 × 10−11
0.326
AOR-W, Dec. 1, 1997
AOR-W, March 20, 1998
1.78 × 10−11
1.92 × 10−11
0.392
0.434
Data averaging 30 s for short term.
Data averaging 60 s for long term.
where λL1 is the wavelength of the L1 carrier frequency and long-term coherence
equals |PR−PH (t + 100) –PR−PH (t)| cycles.
For short-term code–carrier coherence calculations, the difference between
the pseudorange and the phase measurements is given by
δPR−PH =
FPR − FPH
,
10 × c (speed of light)
and short-term coherence is |δPR−PH (t + 10) − δPR−PH (t)|.
The IOT long- and short-term code–carrier results from Santa Paula and
Clarksburg are shown in Table 6.3. The results indicate that the control loop
algorithm performance meets the long- and short-term code–carrier requirements
of WAAS with the Inmarsat-3 satellites.
6.4.5 Carrier Frequency Stability
Carrier frequency stability is a function of both the uplink frequency standard,
GUS signal generator, and Inmarsat-3 transponder. The GEO’s short-term carrier
frequency stability requirement is specified in the WAAS System Specification
and Appendix A of Ref. 165. It states: “The short term stability of the carrier
frequency (square root of the Allan variance) at the input of the user’s receiver
antenna shall be better than 5 × 10−11 over 1 to 10 seconds, excluding the effects
of the ionosphere and Doppler.”
The Allan variance [2] is calculated on the second difference of L1 phase data
divided by the center frequency over 1–10 s. Effects of smoothed ionosphere and
Doppler are compensated for in the data prior to this calculation. Test results in
Table 6.4 show that the POR and AOR-W Inmarsat-3 GEOs, in conjunction with
WAAS ground station equipment, meet the short-term carrier frequency stability
requirement of WAAS.
213
GUS CLOCK STEERING ALGORITHMS
TABLE 6.4. Carrier Frequency Stability Requirements Satisfied
<5 × 10−11 (1 s)
<5 × 10−11 (10 s)
Oct. 10, 1997, POR
4.52 × 10−11
5.32 × 10−12
Dec. 1, 1997, AOR-W
March 20, 1998
3.93 × 10−11
4.92 × 10−11
4.5 × 10−12
4.73 × 10−12
Requirements for L1 :
Santa Paula
prototyping
Clarksburg
prototyping
6.5 GUS CLOCK STEERING ALGORITHMS
The local oscillator (cesium frequency standard) at the GUS is not perfectly stable
with respect to WAAS network time (WNT). Even though the cesium frequency
standard is very stable, it has inherent drift. Over a long period of operation, as
in the WAAS scenario, this slow drift will accumulate and result in an offset so
large that the value will not fit in the associated data fields in the WAAS Type 9
message. This is why a clock steering algorithm is necessary at the GUS. This
drifting effect will cause GUS time and WNT to slowly diverge. The GUS can
compensate for this drift by periodically re-synchronizing the receiver time with
the WNT using the estimated receiver clock offset [a0 (tk )]. This clock offset is
provided by the WMS in WAAS Type 9 messages. (See Fig. 6.8.)
GUS steering algorithms for the primary and backup GEO uplink subsystems
[66, 158] are discussed in the next section.
Fig. 6.8
WMS-to-GUS clock steering.
214
DIFFERENTIAL GNSS
The primary GUS clock steering is closed loop via the signal generator, GEO,
WRS, WMS, to the GUS processor. The backup GUS clock steering is an openloop system, because the backup does not uplink to the GEO. The clock offset
is calculated using the estimated range and the range calculated from the C&V
provided GEO positions.
The GUS also contains the WAAS clock steering algorithm. This algorithm
uses the WAAS Type 9 messages from the WMS to align the GEO’s epoch
with the GPS epoch. The WAAS Type 9 message contains a term referred to
as a0 , or clock offset. This offset represents a correction, or time difference,
between the GEOs epoch and WNT. WNT is the internal time reference scale
of WAAS and is required to track the GPS timescale, while at the same time
providing the users with the translation to UTC. Since GPS master time is not
directly obtainable, the WAAS architecture requires that WNT be computed at
multiple WMSs using potentially differing sets of measurements from potentially
differing sets of receivers and clocks (WAAS reference stations). WNT is required
to agree with GPS to within 50 ns. At the same time, the WNT-to-UTC offset
must be provided to the user, with the offset being accurate to 20 ns. The GUS
calculates local clock adjustments. In accordance with these clock adjustments,
the frequency standard can be made to speed up or slow the GUS clock. This will
keep the total GEO clock offset within the range allowed by the WAAS Type 9
message so that users can make the proper clock corrections in their algorithms.
6.5.1 Primary GUS Clock Steering Algorithm
The GUS clock steering algorithm calculates the fractional frequency control
adjustment required to slowly steer the GUS’s cesium frequency standard to
align the GEO’s epoch. These frequency control signals are very small so normal
operation of the code and frequency control loops of any user receiver is not
disturbed. Figure 6.8 shows the primary GUS’s closed-loop control system block
diagram. The primary GUS is the active uplink dedicated to either the AOR-W
or POR GEO satellite. If this primary GUS fails, then the hot “backup GUS” is
switched to primary.
The clock steering algorithm is designed using a proportional and integral (PI)
controller. This algorithm allows one to optimize by adjusting the parameters a,
b, and T . Values of a and b are optimized to 0.707 damping ratio.
The value a 0 (tk ) is the range residual for the primary GUS:
N
1 .
a 0 (tk ) =
a0 (tk−n ).
N
n=1
The value fc (tk ) is the frequency control signal to be applied at time tk to the
GUS cesium frequency standard:
/
0
* tk
α
β
fc (tk ) = −
a 0 (tk ) + 2
a 0 (t) dt ,
T
T 0
215
GUS CLOCK STEERING ALGORITHMS
Fig. 6.9
Clock steering block diagram.
where
T = large time constant
α, β = control parameters
N = number of data points within period t
t = time of averaging period
tk = time when the frequency control signal is applied to
the cesium frequency standard
a0 (tk ) = time offset for GEO at timetk provided by WMS
for primary GUS
S = Laplace transform variable(see Fig. 6.9)
6.5.2 Backup GUS Clock Steering Algorithm
The backup GUS must employ a different algorithm for calculating the range
residual. Since the backup GUS is not transmitting to the satellite, the WMS
cannot model the clock drift at the backup GUS, and therefore an a0 term is not
present in the WAAS Type 9 message. In lieu of the a0 term provided by the
WMS, the backup GUS calculates an equivalent a0 parameter.
The range residual a0 (tk ) for the backup GUS is calculated as follows [64]:
a0 (tk ) =
BRE − RWMS
,
c − S(tk )
216
DIFFERENTIAL GNSS
Fig. 6.10 Primary GP clock steering parameters, AOR-W, Clarksburg.
where BRE is range estimate in the backup range estimator, RWMS is range estimate calculated from the GEO position supplied by WMs Type 9 message, c is
the speed of light, and S(tk ) is Sagnac effect correction in an inertial frame.
The backup GUS uses the same algorithm fc (tk ) as the primary GUS.
6.5.3 Clock Steering Test Results Description
6.5.3.1 AOR-W Primary (Clarksburg, MD) Figure 6.10 shows the test results
for the first 9 days. The first two to three days had cold-start transients and WMS
switch overs (LA to DC and DC to LA). From the third to the sixth day, the
clock stayed within ±250 ns. At the end of the sixth day, a maneuver took place
and caused a small transient and the clock offset went to –750 ns. On the eighth
day, the primary GUS was switched to Santa Paula, and another transient was
observed. Clock steering command limits are ±138.89 × 10−13 . Limits on the
clock offset from the WAAS Type 9 messages are ±953.7 ns.
6.5.3.2 POR Backup (Santa Paula, CA) Figure 6.11 shows cold-start transients. After initial transients, the backup GUS stayed within ±550 ns for 9
days.
GEO WITH L1 /L5 SIGNALS
Fig. 6.11
217
Backup GPS clock steering parameters for POR (Santa Paula, CA).
The clock offsets in all four cases are less than ±953.7 ns (limit on WAAS
Type 9 Message) for 9 days.
6.6 GEO WITH L1 /L5 SIGNALS
The space-based augmentation system (SBAS) uses geostationary earth orbit
(GEO) satellites to relay correction and integrity information to users. A secondary use of the GEO signal is to provide users with a GPS-like ranging source.
The ranging signal is generated on the ground and provided via C-band uplink
to the GEO, where the navigation payload translates the uplinked signal to an
L1 downlink frequency. The GEO incorporates an additional C-band downlink
to provide ionospheric delay observations to the GEO uplink ground station. The
GEO Communication and control segment (GCCS) will add new L1 L5 GEOs
and ground stations to SBAS.
A key feature of GCCS is the addition of a second independently generated
and controlled uplink signal. In contrast to SBAS, which uplinks and controls
a single C-band signal, GCCS uplinks two independent C-band signals, which
218
DIFFERENTIAL GNSS
are translated to L1 and L5 downlink signals. Closed-loop control of the GEO’s
L1 and L5 broadcast signals in space is necessary to ensure that the algorithms
compensate for various sources of uplink divergence between the code and carrier, including uplink ionospheric delay, uplink Doppler, and divergence due to
carrier frequency translation errors induced by the GEO’s transponder. Use of
two independent broadcast signals creates unique challenges in estimating biases
and maintaining coherency between the two signals.
Raytheon Company is developing a subsystem for GCCS uplink signal generation under a subcontract to Lockheed Martin. GCCS will add new GEOs to the
space-based augmentation system (SBAS), which Raytheon is developing under
contract with the Federal Aviation Administration (FAA). SBAS is a GPS-based
navigation system that is intended to become the primary navigational aid for
aviation during all phases of flight.
The SBAS makes use of a network of wide-area reference stations (WRSs)
distributed throughout the United States. These reference stations collect pseudorange measurements and send them to the SBAS wide-area master stations
(WMSs). The master stations process the data to provide correction and integrity
information for each geostationary earth orbit (GEO) and GPS satellite in view.
The corrections information includes satellite ephemeris errors, clock bias, and
ionospheric estimation data. The corrections from the WMS are sent to the GEO
uplink subsystem (GUS) for uplink to the GEO.
The GUS receives SBAS messages from the WMS, adds forward error correction (FEC) encoding, and transmits the messages via a single C-band uplink
to the GEO satellite for broadcast to SBAS users. The GUS uplink signal uses
the GPS standard positioning service waveform (C/A code, BPSK modulation);
however, the data rate is higher (250 (bits per second)). The 250 bits of data are
encoded with a one-half rate convolutional code, resulting in a 500 symbols per
second transmission rate.
New GEO satellites will be added to SBAS under the SBAS GEO Communication and Control Segment (GCCS) contract. Raytheon is currently developing
control algorithms for GCCS under a subcontract to Lockheed Martin for the
FAA. A key feature of GCCS is that satellite broadcasts will be available at
both the GPS L1 and L5 frequencies. Unlike current SBAS broadcasts, which
utilize a single uplink signal frequency translated into two downlinks, the future
GEOs will uplink two independent C-band signals, which the transponder will
frequency translate and broadcast as independent L1 and L5 downlink signals.
Figure 6.12 provides a top-level view of the GCCS architecture.
For L1 loop, each symbol is modulated by the C/A code, a 1.023 × 106 chipsper-second (cps) pseudorandom sequence to provide a spread spectrum signal.
This signal is then BPSK modulated by the GUS onto an intermediate frequency
(IF) carrier, upconverted to a C-band frequency, and uplinked to the GEO. The
satellite’s navigation transponder translates the signal in frequency to both Lband (GPS L1 ) and C-band downlink frequencies. The GUS monitors the L1 and
C-band downlink signals from the GEO to provide closed-loop control of the
219
GEO WITH L1 /L5 SIGNALS
code and L1 carrier. When properly controlled, the SBAS GEO provides ranging
signals, as well as GPS corrections and integrity data, to end users.
The L5 spread-spectrum signal will be generated by modulating each message symbol with a 10.23 × 106-cps pseudorandom code, which is an order of
magnitude longer than that of the L1 C/A-code. As with L1 , the L5 signal will
then be BPSK modulated onto an IF carrier, upconverted to a C-band frequency,
and uplinked to the GEO. The GEO transponder will independently translate
the two uplink signals to L band for broadcast to SBAS end users. Use of two
independent broadcast signals creates unique challenges in estimating biases and
maintaining coherency between the two signals.
An important aspect of the downlink signals is coherence between the code
and carrier frequency. To ensure code carrier coherency, closed-loop control algorithms, implemented in the safety computer’s SBAS message processors (WMPs),
are used to maintain the code chipping rate and carrier frequency of the received
L1 signal at a constant ratio of 1:1540. The C-band downlink is used by the control algorithms to estimate and correct for ionospheric delay on the uplink signal.
Control algorithms also correct for other uplink effects such as Doppler, equipment delays, and transponder offsets in order to maintain the correct Doppler and
ionospheric divergence as observed by the user.
Closed loop control of each signal is required to maintain coherence between
its code and carrier frequency, as described above. With two independent signal
paths, it is also required that coherence between the two carriers be maintained
for correct ionospheric delay estimation. As before, the control-loop algorithms
“pre-correct” the code phase, carrier phase, and carrier frequency of the L1 and
L5 signals to remove uplink effects such as ionospheric delays, uplink Doppler,
equipment delays, and frequency offsets. In addition, differential biases between
the L1 and L5 signals must be estimated and corrected.
PanAmSat
Galaxy XV
GEO L1
GPS
L1, L2
L5 & L1
GPS L1, L2
C1 & C5
Clock Steer to
GPS
L5 & L1
KPAs
GEO L1
C5 U/C
C1 U/C
RF
Load
L5 & L1 IF Out
TLT
GUS T-1
Receiver
Primary
GUS-Type 1
Safety Computer
WMPs
L1/L5 Signal
Generator
GUS Processor
Fig. 6.12
WMS
LINCS
TCN
GCCS Top Level View.
Backup GUSType 1
220
DIFFERENTIAL GNSS
L1 up
L1 down
Geo Satellite
L5 up
WAAS GUS TYPE1
Signal Generator
L1 code control
L1 Frequency
control
L5 down
Code
control loop
Frequency
control
loop
L1 Control loop
Range
kalman
filter
Iono
kalman
filter
XXX
Code
control loop
L5 Frequency control
Frequency
control
loop
Fig. 6.13
L5 Control loop
Range
kalman
filter
XXX
XXX
WAAS GUS TYPE1
Receiver
XXX
L1/L5 Bias
estimator
L5 code control
X
WAAS Iono
Input
Iono
kalman
filter
XXX
XXX
Primary GUST control loop functional block diagram.
Each control algorithm contains two Kalman filters and two control loops. One
Kalman filter estimates the ionospheric delay and its rate of change from L1 and
L5 pseudorange measurements. The second Kalman filter estimates range, range
rate, range acceleration, and acceleration rate from raw pseudorange measurements. Range estimates are adjusted for ionospheric delay, as estimated by the
first Kalman filter. Each code control loop generates a code chip rate command
and chip acceleration command to compensate for uplink ionospheric delay and
for the uplink Doppler effect. Each frequency control loop generates a carrier
frequency command and a frequency rate command. A final estimator is used to
calculate bias between the L1 and L5 signals.
Results of laboratory tests utilizing live L1 /L5 hardware elements and simulated satellite effects follow.
6.6.1 GEO Uplink Subsystem Type 1 (GUST) Control Loop Overview
The primary GUST control loop functional block diagram is shown in Figure
6.13. The backup GUST control loop is similar to the primary GUST control
loop except that the uplink signal is radiated into a dummy load. The operation
of the backup GUST control loop is different from the primary GUST because
of the latter.
Each of the L1 and L5 control loops in the primary GUST consists of an iono
Kalman filter, a range Kalman filter, a code control function, and a frequency
control function. In addition, there is an L1 /L5 bias estimation function. These
control loop functions reside inside the safety computer. The external inputs
to the control loop algorithm are the pseudorange, carrier phase, Doppler, and
carrier-to-noise ratio from the receiver.
221
GEO WITH L1 /L5 SIGNALS
6.6.1.1 Ionospheric Kalman Filters The L1 and L5 ionospheric (iono) Kalman
filters are two-state filters.
/
0
iono delay
x=
.
iono delay rate
During every 1-s timeframe in the safety computer, the ionospheric Kalman
filter states and the covariance are propagated. The equations for Kalman filter
propagation are given in Table 8.1 (in Chapter 8).
The L1 filter measurement is formulated as follows:
z=
(P RL1 − dL1 ) − (P RL5 − dL5 )
$
,
(1 − L1 freq)2 (L5 freq)2
where P RL1 is the L1 pseudorange, P RL5 is the L5 pseudorange, dL1 and dL5
are the predetermined L1 and L5 downlink path hardware delays, L5 freq is the
L1 nominal frequency of 1575.42 MHz, and L5 freq is the L5 nominal frequency
of 1176.45 MHz.
The L5 ionospheric Kalman filter design is similar to that for L1 , with the
filter measurement as follows:
z=
(P RL1 − dL1 ) − (P RL5 − dL5 )
.
$
(L5 freq)2 (L1 freq)2 − 1
(6.1)
6.6.1.2 Range Kalman Filter The L1 and L5 range Kalman filters use four
state variables:
⎤
⎡
range
⎥
def ⎢ range rate
⎥.
(6.2)
x=⎢
⎦
⎣ acceleration
acceleration rate
During every 1-s timeframe in the safety computer, the range Kalman filter states
and their covariance of uncertainty are propagated (predicted) in the Kalman filter.
After the filter propagation, if L1 pseudorange is valid, the L1 range estimate and covariance are updated in the Kalman filter using the L1 pseudorange
measurement correction. Likewise, if L5 pseudorange is valid, the L5 range estimate and covariance are updated in the Kalman filter using the L5 pseudorange
measurement correction.
The L1 range Kalman filter measurement is
Z1 = L1 pseudorange
−predetermined L1 downlink path hardware delay
−L1 iono delay estimate.
222
DIFFERENTIAL GNSS
Likewise, the L5 range Kalman filter measurement is
Z5 = L5 pseudorange
−predetermined L5 downlink path hardware delay
−L5 iono delay estimate.
The required Kalman filter equations are given in Table 8.1.
6.6.1.3 Code Control Function The L1 and L5 code control functions compute
the corresponding code chip rate commands, and the chip acceleration commands
to be sent to the signal generator. The signal generator adjusts its L1 and L5 chip
rates according to these commands. The purpose of code control is to compensate
for any initial GEO range estimation error, the iono delay on the uplink C-band
signal, and the Doppler effects due to the GEO movement on the uplink signal
code chip rate. This compensation will ensure that the GEO signal code phase
deviation is within the required limit.
The receiver and signal generator timing [1-pulse-per-second (pps)] errors
also affect the GEO signal code phase deviation. These errors are compensated
separately by the clock steering algorithm [72].
Measurement errors in the predetermined hardware delays of the two signal
paths (both uplink and downlink) will result in additional code phase deviation
for the GEO signal due to the closed-loop control. This additional code phase
deviation will be interpreted as GEO satellite clock error by the master station’s
GEO orbit determination. Since the clock steering algorithm will use the SBAS
broadcast Type 9 message GEO clock offset as part of the input to the clock
steering controller [72], the additional code phase deviation due to common
measurement errors will be compensated for by the clock steering function.
There are several inputs to the code control function: the uplink range, the
projected range of the GEO for the next one-second timeframe, the estimated iono
delay, and so on. The uplink range is the integration of the commanded chip rate,
and this integration is performed in the safety computer. The commanded chip
acceleration is computed on the basis of the estimated acceleration from the
Kalman filter (see Table 8.1).
6.6.1.4 Frequency Control Function The L1 and L5 frequency control functions compute the corresponding carrier frequency commands and the frequency
change rate (acceleration) commands to be sent to the signal generator. The signal generator adjusts the L1 and L5 IF outputs according to these commands. The
purpose of frequency control is to compensate for the Doppler effects due to the
GEO movement on the carrier of the uplink signal, the effect of iono rate on the
uplink carrier, and the frequency offset of the GEO transponders. This function
also continuously estimates the GEO transponder offset, which could drift during
the lifetime of the GEO satellite.
NEW GUS CLOCK STEERING ALGORITHM
223
6.6.1.5 L1 /L5 Bias Estimation Function This function estimates the bias
between the L1 and L5 that is due to differential measurement errors in the
predetermined hardware delays of the two signal paths. If not estimated and compensated, the bias between L1 and L5 will be indistinguishable from iono delay,
as shown in the equations below. L1 and L5 pseudorange can be expressed as
PRL1 = R + IL1 + true dL1 + clock error + tropo delay,
(6.3)
PRL5 = R + IL5 + true dL5 + clock error + tropo delay,
(6.4)
where R is the true range, IL1 is the true L1 iono delay, IL5 is the true L5 iono
delay, true dL1 is the true L1 downlink path hardware delay, and true dL5 is the
true L5 downlink path hardware delay.
This becomes
PRL1 − dL1 = R + IL1 + true dL1 + clock error + tropo delay − dL1 , (6.5)
PRL5 − dL5 = R + IL5 + true dL5 + clock error + tropo delay − dL5 , (6.6)
where dL1 is the predetermined (measured) L1 downlink path hardware delay and
dL5 is the predetermined (measured) L5 downlink path hardware delay.
Let dL1 = true dL1 − dL1 and dL5 = true dL5 − dL5 . The measurement for
the L1 iono Kalman filter becomes
z=
(P RL1 − dL1 ) − (P RL5 − dL5 )
,
$
(1 − L1 freq)2 (L5 freq)2
(6.7)
(dL1 − dL5 )
$
.
(1 − L1 freq)2 (L5 freq)2
(6.8)
= IL1 +
The term (dL1 − dL5 ) /(1- L1 freq)2 / L5 freq)2 is the differential L1 /L5 bias
term, and it becomes an error in the L1 iono delay estimation. The L5 iono
Kalman filter is similarly affected by the L1 /L5 bias term.
6.7 NEW GUS CLOCK STEERING ALGORITHM
Presently, the SBAS wide-area master station (WMS) calculates SBAS network
time (WNT) and estimates clock parameters (offset and drift) for each satellite.
The GEO uplink system (GUS) clock is an independent free running clock. However, the GUS clock must track WNT (GPS time) to enable accurate ranging from
the GEO signal in space (SIS). Therefore, a clock steering algorithm is necessary. The GUS clock steering algorithms reside in the SBAS Message Processor
(WMP). The SBAS Type 9 message (GEO navigation message) is used as input
to the GUS WMP, provided by the WMS.
In the new algorithm, the GUS clock is steered to the GPS time epoch (see also
Fig. 6.14). The GUS receiver clock error is the deviation of its one-second pulse
224
DIFFERENTIAL GNSS
GPS
1 PPS
Phase
Noise
Enhancer
10 MHz
Frequency
Dist Amp
GU ST Receiver
Engineering Model
10 MHz
Dist Amp
1176.46
MHz RF
1575.42
MHz RF
Pulse
5 MHz
10 MHz
1 PPS
10 MHz
Atomic Clock
L1
Upconverter
L5
Upconverter
Receiver
Logs
10 MHz
10 MHz
Safety Computer
(Computer Algorithm)
Status/Control
70 MHz
70 MHz
IF
IF
L1/L5 Signal
Generator
Reciever Logs
AIX Computer
Modified
Receiver Logs
Signal Generator
Test Tool (SGTT)
Fig. 6.14 Control loop test setup.
from the GPS epoch. The clock error is computed in the GUS processor by calculating the user position error by combining (in the least-squares sense, weighted
with expected error statistics) multiple satellite data (pseudorange residuals called
MOPS 1 residuals) into a position error estimate with respect to surveyed GUS
position. The clock steering algorithm is initialized with the SBAS Type 9 message (GEO navigation message). This design keeps the GUS receiver clock 1
pulse per second synchronized with the GPS time epoch. Since the 10-MHz frequency standard is the frequency reference for the receiver, its frequency output
needs to be controlled so that the 1 pps is adjusted. A proportional, integral and
differential (PID) controller has been designed to synchronize to the GPS time
at GUS locations. Two sets of prototype clock steering results are shown. Clock
adjustment commands as applied to the frequency standard to null the MOPS
clock offset are given.
This new algorithm also decouples the GUS clock from orbit errors and
increases the observability of orbit errors in the orbit determination filter in the
correction processor of the WMS. It also synchronizes GUS clocks at all GUS
locations to GPS time. This section shows the algorithm prototype results of
GUS clock steering to GPS time. It also shows the improvements in the GEO
(AOR-W) orbits when the GEO clock state is known, thereby making the GEO
a valid ranging source. The GEO range errors with the known clock solution are
found to be up to a factor of ∼5 better than those for the fielded SBAS with real
1
Reference 165 specifies Minimum Operational Performance Standards (MOPS) for GPS/WAAS
airborne equipment, including pseudorange residuals.
NEW GUS CLOCK STEERING ALGORITHM
225
data using correction verification simulation. This will increase the availability
and continuity of SBAS services since the signal is already processed by the user.
Raytheon Company is currently proposing to upgrade the SBAS to achieve
full operational capability (FOC) for the Federal Aviation Administration (FAA).
One of the new features will be to steer the GUS clock to GPS time. SBAS is a
GPS-based navigation system that is intended to become the primary navigational
aid for aviation during all phases of flight, from en route through lateral and
vertical navigation (LNAV/VNAV) approach. SBAS makes use of a network of
wide-area reference stations (WRSs) distributed throughout the U.S. National
Airspace System. Figure 6.1 provides a top-level view of the SBAS architecture.
In the L1 path, the GUS receives integrity and correction data and SBAS
specific messages from the WMS, adds forward error correction (FEC) encoding,
and transmits the messages via a C-band uplink to the GEO satellites for broadcast
to the SBAS user. The GUS uplink signal uses the GPS standard positioning
service waveform (C/A-code, BPSK modulation); however, the data rate is higher
(250 bits per second). The 250 bits of data are encoded with a one-half rate
convolutional code, resulting in a 500-(symbols per second) transmission rate.
Each symbol is modulated by the C/A code, a 1.023 × 106 -cps pseudorandom
sequence, to provide a spread-spectrum signal. This signal is then binary phaseshift keying (BPSK) modulated by the GUS onto an intermediate-frequency (IF)
carrier, upconverted to a C-band frequency, and uplinked to the GEO.
The existing GUS in SBAS contains a clock steering algorithm. This algorithm
uses SBAS Type 9 messages from the WMS to align the GEO’s epoch with the
GPS epoch. The SBAS Type 9 message contains a term referred to as aGf 0
or clock offset. This offset represents a correction, or time difference, between
the GEO’s epoch and SBAS network time (WNT). WNT is the internal time
reference scale of SBAS and is required to track the GPS timescale, while at the
same time providing users with the translation to Universal Time, Coordinated
(UTC). Since GPS master time is not directly obtainable, the SBAS architecture
requires that WNT be computed at multiple WMSs using potentially differing sets
of measurements from potentially differing sets of receivers and clocks (SBAS
reference stations). WNT is required to agree with GPS to within 50 ns. At the
same time, the WNT to UTC offset must be provided to the user, with the offset
being accurate to 20 ns. The GUS calculates local clock adjustments. On the
basis of these clock adjustments, the frequency standard can be made to speed
up, or slow the GUS clock. This will keep the total GEO clock offset within the
range allowed by the SBAS Type 9 message so that users can make the proper
clock corrections in their algorithms [33, 192].
The new algorithm in the GUS will use the clock steering method described
above during the initial 24 h after it becomes primary. Once the GUS clock is
synchronized with WNT, a second steering method of clock steering is used. The
algorithm now uses the composite of the MOPS [165] solution for the receiver
clock error, and the average of the aGf0 , and the average of the MOPS solution
as the input to the clock steering controller.
226
DIFFERENTIAL GNSS
6.7.1 Receiver Clock Error Determination
Determination of receiver clock error is based on the user position solution algorithm described in the SBAS MOPS. The clock bias (Cb ) is a resultant of the
MOPS weighted least-squares solution.
Components of the weighted least-squares solution are the observation matrix
(H), the measurement weighting matrix (W) and the MOPS residual column
vector (ρ). The weighted gain matrix (K) is calculated using H and W (see
Eq. 2.37):
K = (HT WH)−1 HT W.
(6.9)
From this, the column vector for the user position error and the clock bias solution is:
X = Kρ
(6.10)
= (HT WH)−1 HT Wρ,
where
⎤
X(U )
⎢ X(E) ⎥
⎥
X = ⎢
⎣ X(N ) ⎦
Cb
(6.11)
⎡
(6.12)
and X(U ) is the up error, X(E) is the east error, X(N ) is the north error,
and Cb is the clock bias or receiver clock error.
The n × 4 observation matrix (H) is computed in up–east–north (UEN) reference frame using the line-of-sight (LOS) azimuth (Azi ) and LOS elevation (Eli )
from the GUS omni antenna to the space vehicle (SV). The value n is the number
of satellites in view. The formula for calculating the observation matrix is
⎡
⎤
cos(El1 ) cos(Az1 ) cos(E11 ) sin(Az1 ) sin(E11 ) 1
⎢ cos(El2 ) cos(Az2 ) cos(E12 ) sin(Az2 ) sin(E12 ) 1 ⎥
⎢
⎥
(6.13)
H=⎢
..
..
..
.. ⎥ .
⎣
.
.
.
. ⎦
cos(Eln ) cos(Azn ) cos(E1n ) sin(Azn ) sin(E1n ) 1
The n × n weighting matrix (W) is a function of the total variance (σi2 ) of the
individual satellites in view. The inverse of the weighting matrix is
⎡
⎤
..
2
σ
0
0
.
0
⎢ 1
⎥
⎢
⎥
..
⎢ 0 σ2 0
.
0 ⎥
2
⎢
⎥
⎥.
..
(6.14)
W−1 = ⎢
2
⎢ 0
.
0 ⎥
0 σ3
⎢
⎥
⎢ ..
..
.. . .
. ⎥
⎣ .
. .. ⎦
.
.
0
0
0
0
σn2
227
NEW GUS CLOCK STEERING ALGORITHM
The equation to calculate the total variance (σi2 ) is
%
σi2 =
UDREi
3.29
&2
%
+
Fppi × GIVEi
3.29
&2
2
+ σL1,
nmp, i +
2
σtropo,
i
sin2 (Eli )
.
(6.15)
The algorithms for calculating user differential range error (UDREi ), user
grid ionospheric vertical error (GIVEi ), LOS obliquity factor (Fppi ), standard
deviation of uncertainty for the vertical troposphere delay model (σtropo,i ) and the
standard deviation of noise and multipath on the L1 omni pseudorange (σL1,nmp,i )
are found in the SBAS MOPS [165].
The MOPS residuals (ρ) are the difference between the smoothed MOPS
measured pseudorange (P RM,i ) and the expected pseudorange (P Rcorr, i ):
⎡
⎤
P RM, 1 − P Rcorr, 1
⎢ P RM, 2 − P Rcorr, 2 ⎥
⎢
⎥
(6.16)
ρ = ⎢ P RM, 3 − P Rcorr, 3 ⎥ .
⎣
⎦
..
.
The MOPS measured pseudorange (P RM,i ) in earth-centered earth-fixed (ECEF)
reference is corrected for earth rotation, for SBAS clock corrections, for ionospheric effects and for tropospheric effects. The equation to calculate PRM,i is
P RM,i = P RL,i + P RCC,i + P RFC,i + P RER,i − P RT ,i − P RI,i .
(6.17)
The algorithms used to calculate smoothed L1 omni pseudorange (P RL,i ), pseudorange clock correction (P RCC,i ), pseudorange fast correction (P RFC,i ),
pseudorange earth rotation correction (P RER,i ), pseudorange troposphere correction (P RT ,i ), and pseudorange ionosphere correction (P RI,i ) are found in
the SBAS MOPS [165].
Expected pseudorange (P Rcorr,i ), ECEF, at the time of GPS transmission is
computed from broadcast ephemeris corrected for fast and long term corrections.
The calculation is
!
P Rcorr,i = (Xcorr,i − XGUS )2 + (Ycorr,i − YGUS )2 + (Zcorr,i − ZGUS )2 . (6.18)
The fixed-position parameters of the WRE (XGUS , YGUS , ZGUS ) are site-specific.
6.7.2 Clock Steering Control Law
In the primary GUS, the clock steering algorithm is initialized with SBAS Type 9
message (GEO navigation message). After the initialization, composite of MOPS
solution and Type 9 message for the receiver clock error is used as the input to
the control law (see Fig. 6.15). For the backup GUS, the MOPS solution for the
receiver clock error is used as the input to the control law (see Fig. 6.16).
228
DIFFERENTIAL GNSS
Fig. 6.15
Primary GUS clock steering.
For both the primary and backup clock steering algorithm, the control law is a
proportional, integral, and differential (PID) controller. The output of the control
law will be the frequency adjustment command. This command is sent to the
frequency standard to adjust the atomic clock frequency. The output frequency
to the receiver causes the 1 pps to approach the GPS epoch. Thus, a closed loop
control of the frequency standard is established.
6.8 GEO ORBIT DETERMINATION
The purpose of WAAS is to provide pseudorange and ionospheric corrections for
GPS satellites to improve the accuracy for the GPS navigation user and to protect
the user with “integrity.” Integrity is the ability to provide timely warnings to the
user whenever any navigation parameters estimated using the system are outside
tolerance limits. WAAS may also augment the GPS constellation by providing
additional ranging sources using GEO satellites that are being used to broadcast
the WAAS signal.
229
GEO ORBIT DETERMINATION
Fig. 6.16
Backup GUS clock steering.
The two parameters having the most influence on the integrity bounds for
the broadcast data are user differential ranging error (UDRE) for the pseudorange corrections and grid ionospheric vertical error (GIVE) for the ionospheric
corrections. With these, the onboard navigation system estimates the horizontal
protection limit (HPL) and the vertical protection limit (VPL), which are then
compared to the horizontal alert limit (HAL) and the vertical alert limit (VAL)
requirements for the particular phase of flight involved, that is, oceanic/remote,
en route, terminal, nonprecision approach, and precision approach. If the estimated protection limits are greater than the alert limits, the navigation system
is declared unavailable. Therefore, the UDRE and GIVE values obtained by the
WAAS (in concert with the GPS and GEO constellation geometry and reliability) essentially determine the degree of availability of the WADGPS navigation
service to the user.
The WAAS algorithms calculate the broadcast corrections and the corresponding UDREs and GIVEs by processing the satellite signals received by the network
of ground stations. Therefore, the expected values for UDREs and GIVEs are
230
DIFFERENTIAL GNSS
dependent on satellite and station geometries, satellite signal and clock performance, receiver performance, environmental conditions (such as multipath,
ionospheric storms, etc.) and algorithm design [71, 147].
6.8.1 Orbit Determination Covariance Analysis
A full WAAS algorithm contains three Kalman filters—an orbit determination
filter, an ionospheric corrections filter, and a fast corrections filter. The fast corrections filter is a Kalman filter that estimates the GEO, GPS, and ground station
clock states every second. In this section, we derive an estimated lower bound of
the GEO UDRE for a WAAS algorithm that contains only the orbit determination
Kalman filter, called the UDRE (OD), where OD refers to orbit determination.
A method is proposed to approximate the UDRE obtained for a WAAS
including both the orbit determination filter and the fast corrections filter from
UDRE(OD). From case studies of the geometries studied in the previous section,
we obtain the essential dependence of UDRE on ground station geometry.
A covariance analysis on the orbit determination is performed using a simplified version of the orbit determination algorithms. The performance of the
ionospheric corrections filter is treated as perfect, and therefore, the ionospheric
filter model is ignored. The station clocks are treated as if perfectly synchronized using the GPS satellite measurements. Therefore, the station clock states
are ignored. This allows the decoupling of the orbit determinations for all the
satellites from each other, simplifying the orbit determination problem to that for
one satellite with its corresponding ground station geometry and synchronized
station clocks. Both of these assumptions are liberal; therefore, the UDRE(OD)
obtained here is a lower bound for the actual UDRE(OD). Finally, we consider
only users within the service volume covered by the stations and, therefore, ignore
any degradation factors depending on user location.
To simulate the Kalman filter for the covariance matrix P , the following four
matrices are necessary (Table 8.1):
= state transition matrix,
H = measurement sensitivity matrix,
Q = process noise covariance matrix,
R = measurement noise covariance matrix.
The methods used to determine these matrices are described below.
The state vector for the satellite is
⎡
⎤
r
x = ⎣ ṙ ⎦ ,
Cb
231
GEO ORBIT DETERMINATION
where
r ≡ [x y z]T
is the satellite position in the ECI frame,
ṙ ≡ [ẋ, ẏ, ż]T
is the satellite velocity in the ECI frame, and Cb is the satellite clock offset relative
to the synchronized station clocks. Newton’s second and third (gravitational) laws
provide the equations of motion for the satellite:
r̈ ≡
μE r
d 2r
= − 3 + M,
dt 2
|r|
where r̈ is the acceleration in the ECI frame, μE is the gravitational constant
for the earth, and M is the total perturbation vector in the ECI frame containing
all the perturbing accelerations. For this analysis, only the perturbation due to
the oblateness of the earth is included. The effect of this perturbation on the
behavior of the covariance is negligible, and therefore higher-order perturbations
are ignored. (Note that although the theoretical model is simplified, the process
noise covariance matrix Q is chosen to be consistent with a far more sophisticated
orbital model.)
Therefore
3 μE a 2
M = − J2 3 E2 [I3 × 3 + 2ẑẑT ]r,
2 |r| |r|
where aE is the semimajor axis of the earth-shape model, J2 is the second zonal
harmonic coefficient of the earth-shape model, and ẑ ≡ [0, 0, 1]T [11].
The second-order differential equation of motion can be rewritten as a pair of
first-order differential equations
ṙ1 = r2 ,
ṙ2 =
μE r1
+ M,
|r|3
(6.19)
where r 1 and r 2 are vectors, which therefore gives a system of six first-order
equations.
The variational equations are differential equations describing the rates of
change of the satellite position and velocity vectors as functions of variations in
the components of the estimation state vector. These lead to the state transition
matrix used in the Kalman filter. The variational equations are
Ÿ (t) = A(t)Y (t) + B(t)Ẏ (t),
(6.20)
232
DIFFERENTIAL GNSS
where
2%
Y (tk )3 × 6 ≡
2%
Ẏ (tk )3 × 6 ≡
A(t)3 ×3 ≡
=
B(t)3 × 3 ≡
∂r(tk )
∂r(tk−1)
∂ ṙ(tk )
∂r(tk−1)
&
%
3×3
&
%
3×3
∂r(tk)
∂ ṙ(tk−1)
3
&
3×3
∂r(tk)
∂ ṙ(tK−1)
&
,
(6.21)
,
(6.22)
3
3×3
∂ r̈
∂r
3 μE a 2
−μE
[I3 ×3 − 3r̂r̂T ] − J2 3 E2 ,
3
|r|
2 |r| |r|
"
T
× I3 × 3 + 2ẑẑ − 10(r̂T ẑT )(ẑr̂T + r̂ẑT )
#
+(10(r̂T ẑ)2 − 5)(r̂r̂T ) ,
(6.23)
∂ r̈
= 03 ×3 ,
∂ ṙ
(6.24)
$
where r̂ = r |r| .
Equations 6.21–6.24 are substituted into Eq. 6.20 and Eq. 6.19, and the differential equations are solved using the fourth-order Runge–Kutta method. The
time step used is a 5-min interval. The initial conditions for the GEO are specified
for the particular case given and propagated forward for each time step, whereas
the initial conditions for the Y terms are
Y (tk−1 )3 ×6 = [I3 × 3
Ẏ (tk )3 × 6 = [03 × 3
03 × 3 ] ,
I3 ×3 ]
and reset for each timestep. This is due to the divergence of the solution of the
differential equation used in this method to calculate the state transition matrix
for the Kepler problem.
This gives the state xT = rT1 rT2 and the state transition matrix
⎡
k,k−17 × 7
Y (tk )3 × 6
= ⎣ Ẏ (tk )3 × 6
01 × 6
⎤
03 × 1
03 × 1 ⎦
I1 × 1
(6.25)
for the Kalman filter.
The measurement sensitivity matrix is given by
HN × 7 ≡
∂ρ
=
∂x
/%
∂ρ
∂r
&
N ×3
%
∂ρ
∂ ṙ
%
&
N ×3
∂ρ
∂(ct)
0
&
,
N ×1
233
GEO ORBIT DETERMINATION
where ρ is the pseudorange for a station and N is the number of stations in view
of the satellite. Note that this is essentially the same H as in the previous section.
Ignoring relativistic corrections and denoting the station position by the vector
rS ≡ [xS yS zS ]T , the matrices above are given by
∂ρ
[r − rS ]T ∂r(tk )
=
,
∂r
|r − rS | ∂r(tk−1 )
[r − rS ]T ∂r(tk )
∂ρ
=
,
∂ ṙ
|r − rS | ∂ ṙ(tk−1 )
and
∂ρ
= 1.
∂(ct)
(6.26)
The station position is calculated with the WGS84 model for the earth and converted to the ECI frame using the J2000 epoch. (See Appendix C.)
These are then combined with the measurement noise covariance matrix, R
and the process noise covariance matrix Q to obtain the Kalman filter equations
for the covariance matrix P , as shown in Table 8.1.
The initial condition, P0 (+), and Q are chosen to be consistent with the WAAS
algorithms. The value of R is chosen by matching the output of the GEO covariance for AOR-W with R = σ 2 I and is used as the input R for all other satellites
and station geometries (note that this therefore gives approximate results). This
corresponds to carrier phase ranging for the stations. The results corresponding
to the value of R for code ranging are also presented.
From this covariance, the lower bound on the UDRE is obtained by
UDRE ≥ EMRBE+KSS tr(P ),
where EMRBE is the estimated maximum range and bias error. To obtain the
.999 level of bounding for the UDRE with EMRBE = 0, KSS = 3.29. Finally,
since the message is broadcast every second, t = 1, so the trace can be used
for the velocity components as well.
Figure 6.17 shows the relationship between UDRE and GDOP for various
GEO satellites and WRS locations. Table 6.5 describes the various cases considered in this analysis.
The numerical values used for the filter are as follows [all units are Système
International (SI)]:
•
Earth parameters:
μE
aE
= 3.98600441 × 1014 ,
= 6, 378, 137.0,
J2
bE
= 1082.63 × 10−6 ,
= 6, 356, 752.3142.
234
DIFFERENTIAL GNSS
Fig. 6.17 Relationship between UDRE and GDOP.
TABLE 6.5. Cases Used in Geometry-per-Station Analysis
Case
UDRE
GDOP
Satellite
Geometry
1
2
3
4
5
6
17.9
45.8
135
4.5
5.8
4.0
905
2,516
56,536
254
212
154
AOR-W
AOR-W
AOR-W
AOR-W
AOR-W
AOR-W
7
7.5
439
AOR-W
8
8.6
337
AOR-W
9
6.6
271
AOR-W
10
11
12
47.7
21.5
16.4
2,799
1,405
1,334
AOR-W
AOR-W
AOR-W
13
14
15
16
17
18
19
20
28.5
45.4
31.1
55.0
6.7
8.3
6.7
21.0
1,686
3,196
1,898
4,204
257
338
257
1,124
POR
POR
POR
POR
POR
POR
POR
MTSAT
WAAS stations (25),21 in view
4 WAAS stations (CONUS)a
4 WAAS stations (NE)b
WAAS stations + Santiago
WAAS stations + London
WAAS stations + Santiago + London
4 WAAS stations (CONUS) + Santiago
4 WAAS stations (CONUS) + London
4 WAAS stations (CONUS) + Santiago + London
4 WAAS stations (NE) + Santiago
4 WAAS stations (NE) + London
4 WAAS stations (NE) + Santiago
+ London
WAAS stations (25), 8 in view
WAAS stations, Hawaii
WAAS stations, Cold Bay
WAAS stations, Hawaii, Cold Bay
WAAS stations + Sydney
WAAS stations + Tokyo
WAAS stations + Sydney + Tokyo
MSAS stations, 8 in view
235
GEO ORBIT DETERMINATION
TABLE 6.5. (continued )
Case
UDRE
GDOP
Satellite
Geometry
21
22
23
24
25
26
22.0
24.9
54.6
22.0
29.0
54.8
1,191
1,407
4,149
1,198
1,731
4,164
MTSAT
MTSAT
MTSAT
MTSAT
MTSAT
MTSAT
27
A
B
C
D
E
F
13.2
609
139
422
3,343
13,211
67
64
MTSAT
TEST
TEST
TEST
TEST
TEST
TEST
MSAS stations–Hawaii
MSAS stations–Australia
MSAS stations –Hawaii, Australia
MSAS stations –Ibaraki
MSAS stations –Ibaraki, Australia
MSAS stations –Ibaraki, Australia,
Hawaii
MSAS stations + Cold Bay
θ = 75◦
θ = 30◦
θ = 10◦
θ = 5◦
41 stations
41 + 4 stations
a
4 WAAS stations (CONUS) are Boston, Miami, Seattle, and Los Angeles.
WAAS stations (NE) are Boston, New York, Washington DC, and Cleveland.
b4
•
Filter parameters:
Qpos
P0,pos
σR
•
= 0,
= 144.9,
= 0.013.
Qvel
P0,vel
= 0.75 × 106 ,
= 1 × 10−4 ,
Qct
P0,ct
= 60,
= 100.9,
Curve fit parameters:
σQ,fit
= 6.12,
εφ,fit
= 0.0107.
PROBLEMS
6.1 Determine the code–carrier coherence at the GUS location using L1 code
and carrier.
6.2 Determine the frequency stability of the AOR and POR transponder using
Allan variance for the L1 using 1–10-s intervals.
7
GNSS AND GEO SIGNAL INTEGRITY
7.1 RECEIVER AUTONOMOUS INTEGRITY MONITORING (RAIM)
Navigation system integrity refers to the ability of the system to provide timely
warnings to users when the system should not be used for navigation. The basic
GPS (as described in Chapter 3) provides integrity information to the user via
navigation message, but this may not be timely enough for some applications,
such as civil aviation. Therefore, additional methods of providing integrity are
necessary.
Two different methods will be discussed—GPS-only receiver (TSO-C129compliant) autonomous integrity monitoring (RAIM) and use of ground monitoring stations to monitor the health of the satellites, as is done via SBAS and GBAS
(TSO-C145-compliant receivers). Three RAIM methods have been proposed in
recent papers on GPS integrity [150, 151, 152, 153]:
1. Range comparison method
2. Least-squares residual method
3. Parity method
We will briefly discuss the RAIM methods, then discuss SBAS and GBAS
integrity design.
Global Positioning Systems, Inertial Navigation, and Integration, Second Edition, by M. S. Grewal, L. R. Weill, and A. P. Andrews
Copyright © 2007 John Wiley & Sons, Inc.
236
RECEIVER AUTONOMOUS INTEGRITY MONITORING (RAIM)
237
7.1.1 Range Comparison Method of Lee [121]
For the GNSS navigation problem described in Chapter 2, Section 2.5, there
are four unknowns (three position coordinates [x, y, z] and clock bias Cb) and
more than four satellites in view (e.g., six satellites). One can solve the position
and time equations for the first four satellites, ignoring noise, and find the user
position. This solution can then be used to predict the remaining two pseudorange
measurements, and the predicted values could be compared with actual measured
values. If the two differences (residuals) are small, we have near consistency in
the measurements and the detection algorithm can declare “no failure.” It only
remains to quantify what we mean by “small” or “large,” and then assess the
decision rule performance on actual data.
7.1.2 Least-Squares Method [151]
The basic measurement equation with noise (Eq. 2.32 from Chapter 2) is
δZρ = H δx + vρ ,
(7.1)
where the additive white noise vρ ∈ N 0, σ 2 .
Let us suppose six satellites are in view and four unknowns, as in Section
7.1.1, and solve for the four unknowns by the least-squares method.
The least-squares solution is given by Eq. 2.37:
8 = HT H −1 HT δZρ .
δx
(7.2)
The least-squares solution can be used to predict the six measurements, in accordance with
9 ρ (predicted) = H δx.
8
δZ
(7.3)
We can get a formula for the sum-squared residual error S by substituting δx
from Eq. 7.3 into Eq. 7.2:
9 ρ (residual error)
Zρ = δZρ − δZ
−1 T = I − H HT H
H δZρ
S = ZρT Zρ , the sum-squared error.
(7.4)
(7.5)
(7.6)
This sum of squared error has three properties that are important in the decision
rule:
1. S is a nonnegative scalar quantity. Choose a threshold value τ of S such
that S < τ will be considered safe and S ≥ τ will be declared a failure.
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GNSS AND GEO SIGNAL INTEGRITY
2. If the vρ have the same independent zero-mean Gaussian distribution, then
the statistical distribution of S is completely independent of the satellite
geometry for any number of satellites (n). Thresholds are precalculated,
that results in the desired alarm rate for the various anticipated values of
n. Then the real-time algorithm sets the threshold appropriately for the
number of satellites in view at the moment.
3. With the vρ , from above, S has an unnormalized chi-square (χ 2 ) distribution
√
with (n − 4) degrees of freedom. Parkinson and Axelrad [150] use S/n −
4 as the test statistic. Calculating the test statistic involves the same matrix
manipulation, but these are no worse than calculating DOP.
7.1.3 Parity Method [182, 183]
The parity RAIM method is somewhat similar to the range comparison method,
except that the way in which the test statistic is formed is different. In the
parity method, perform a linear transformation on the measurement vector δZρ
as follows:
/
0 / −1 T 0
δx
HT H
H
δZρ .
=
(7.7)
p
P
The lower portion of Eq. 7.7, which yields p, is the result of operating on δZρ
with the special (n − 4)×n matrix P, whose rows are mutually orthogonal, unity
magnitude and orthogonal to the columns of H.
Under the same assumptions about the noise vρ as above, the following statements can be made:
1
Ep
= 0
,
(7.8)
Epp T = σ 2 I (covariance of p)
where σ 2 is the variance associated with vρ . Use p as the test statistic in this
method. For detection, obtain all the information needed about p from its magnitude or magnitude squared. Thus, in the parity method, the test statistic for
detection reduces to a scalar, as in the least squares method.
7.2 SBAS AND GBAS INTEGRITY DESIGN
The objectives of the space-based augmentation system (SBAS) and the groundbased augmentation system (GBAS) are to provide integrity, accuracy, availability and continuity for GPS, GLONASS, and Galileo Standard Positioning Service
(SPS). Integrity is defined as the ability of the system to provide timely warnings
to the user when individual corrections or certain satellites should not be used
for navigation, specifically, the prevention of hazardously misleading information
(HMI) data transmission to the user. The system should not be used for navigation when hardware, software or environmental errors directly pose a threat to
SBAS AND GBAS INTEGRITY DESIGN
239
the user or indirectly pose a threat by obscuring HMI from the integrity monitors.
SBAS integrity is based on the premise that errors not detected or corrected in
the operational environment can become threats to integrity and, if not mitigated,
can become hazards to the user.
An SBAS design should mitigate the majority of these data errors with corrections that are proved to bound the integrity hazard to an acceptable level.
The leftover data errors (referred to as residual errors) are mitigated by the
transmission of residual error bounding information. The threat of potential underbounding of integrity information is mitigated by integrity monitors. This chapter
examines both the faulted and unfaulted cases and mitigation strategies for these
cases. These SBAS corrections improve the accuracy of satellite signals. The
integrity data ensure that the residual errors are bounded. The SBAS integrity
monitors help ensure that the integrity data have not been corrupted by SBAS
failures.
The chapter addresses the data errors, error detection and correction pitfalls,
and how such threats can become HMI to the user, as well as fault conditions,
failure conditions, threats, and mitigation, and how safety integrity requirements
are satisfied. Safety integrity assurance rules will be evaluated. Results from real
SIS (signal in space) data, a high-level overview of the required SBAS safety
architecture, and a data processing path protection approach are included.
This chapter provides information that defines how a safety-of-life-critical
SBAS system should be designed and implemented in order to ensure mitigation of the entire International Civil Aviation Organization (ICAO) threat space
to the required level less than 10−7 . It provides as an example, the rationale,
background, and references to show that the SBAS can be used as a trusted
navigational aid to augment the Global Positioning System (GPS) for lateral
positioning with vertical guidance (LPV).
Rail integrity is one of the most stringent operational requirements, as evidenced by ERTMS (European Rail Traffic Management Systems) required integrity levels, which are in the order of 10−11 . Train detection will require an
equally high level of positive integrity.
The chapter addresses the hazardous/severe–major integrity failure condition
using LPV as an illustrative example. The ICAO integrity requirement is based on
the premise that errors not detected or corrected in the operational environment
system can become threats to the integrity and, if not mitigated, can become
hazards to the user. These errors in the operational environment (referred to as
data errors) can affect both the user and the SBAS system. Integrity in this
context is defined as the ability of the system to provide timely warning to users
when individual corrections or satellites should not be used for navigation, that
is, the prevention of hazardously misleading information (HMI) data transmission
to the user. The system should not be used for navigation when data errors in
the environment, such as the ionosphere, and data processing, such as multipath,
render the integrity data erroneous. The user must be protected from residual
errors that can become threats to the integrity data that could result in HMI
being transmitted to the user [171, 200].
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GNSS AND GEO SIGNAL INTEGRITY
An SBAS design mitigates the majority of these data errors with “corrections.”
The leftover data errors (referred to as residual errors) are mitigated by the
transmission of residual error bounding information. The threat of potential underbounding of the integrity information is mitigated by integrity monitors and point
design features that protect the integrity of the information within the SBAS
system. Additionally, analytic safety analyses are required to provide evidence
and proof that the residual errors are acceptable (i.e., that the probability of HMI
transmission to the user is sufficiently low).
Table 7.1 list the SBAS error sources. Mitigation of these errors when they
become integrity threats are presented in Section 7.2.8. Section 7.3 gives an
application of these techniques to SBAS for threat mitigation. Section 7.11 shows
the conclusions. GPS Integrity Channel (GIC) is discussed in Section 7.5.
TABLE 7.1. List of SBAS Error Sources
GPS satellite
GEO satellites
Reference receiver
Estimation
Integrity bound associated
Message uplink
Environment (ionosphere and troposphere)
7.2.1 SBAS Error Sources and Integrity Threats
The SBAS operational environment contains data errors. The SBAS ensures that
these data errors do not become threats to the integrity data, so that HMI is not
broadcasted to the user with a PHMI greater than 10−7 .
The data used by an SBAS to calculate the correction and/or integrity data
are assumed to contain errors, such as, GPS satellite clock offset, which must
be sufficiently mitigated. The errors discussed are inherent in any SBAS design
that utilizes GPS, Galileo, GLONASS, or GEO satellites; reference receivers;
corrections; and integrity bounds. Depending on the system architecture, other
error sources may also exist. Table 7.1 summarizes the error sources that every
SBAS system must address [65].
The integrity threats associated with each of these error sources generally have
two cases, shown in Fig. 7.1. The fault-free case addresses the nominal errors
associated with each error source and the faulted case represents the errors when
one or more of the system’s components cause errors. The defining quality of
an SBAS system that meets the ICAO standards is the mitigation of the faulted
case and the fault-free case.
7.2.2 GPS-Associated Errors
GPS error sources are mitigated in an SBAS system by using corrections and
integrity bounds. Generally, the SBAS system corrects the errors as well as possible, and then bounds the residual errors with integrity bounds that are broadcast
241
SBAS AND GBAS INTEGRITY DESIGN
Miltigated By
Safety Design
*Data Errors
Error
Detection
&
Correction
Residual Errors
Fault Free Case
Correction
Bounded
by initial
UDRE/GIVE
* Source of Errors:
-
Environment
Satellites (GPS,GEO)
Estimation Errors
Software Code
Hardware
Residual Errors
Faulted Case
Faults(s)
HW & SW
Fault
Conditions
Residual Error not big
Faults(s)
Residual Errors
Failure enough to be a threat
No Threat
Conditions
Residual Error Shown to be
Residual
Acceptable
by Safety Analyses
Errors
Threats
Failure(s)
Residual Error
Loss of Function
Failure(s)
Integrity Threat
Correction
Underbounded
(Navigation System Error)
Integrity
Integrity Threat
Threat
Mitigated by Integrity
Monitors if with in the
TTA requirement
HMI
to the user
Fig. 7.1 Integrity mitigation within an SBAS.
to the user. The nominal GPS satellite errors are well understood. The literature
includes many techniques for mitigating these errors. GPS failure modes are not
as well understood and often require careful study to define the threat, which
must be accounted for in threat models.
7.2.2.1 GPS Clock Error Each GPS satellite broadcasts a navigation data
message containing an estimate of its clock offset (relative to GPS time) and
drift rate. The GPS satellite clock value is utilized to correct the satellite’s pseudorange, the measurement used to calculate the distance (range) from the satellite
to the receiver (either the user’s receiver or the reference receiver).
Under fault-free conditions the SBAS can accurately compute these corrections and mitigate this error source. Simple statistical techniques can be used to
characterize these errors. The SBAS must also address satellite failures that cause
the clock to rapidly accelerate, rendering the corrections suddenly invalid. As a
result, the error bounds may not be bounding the residual error in the corrections.
These types of failures have been observed many times in the history of GPS.
7.2.2.2 GPS Ephemeris Error Each GPS also broadcasts a navigation data
message containing a prediction of its orbital parameters—Keplerian orbital
parameters. The satellite’s ephemeris data enable determination of the satellite
position and velocity. Any difference between the satellite’s calculated position
and velocity and the true position is a potential source of error.
Under fault-free conditions the SBAS provides corrections relative to the GPS
broadcast ephemeris data. The SBAS can accurately compute these corrections
and mitigate this error source using standard statistical techniques. The satellite
may experience an unexpected maneuver, rendering the corrections suddenly
invalid. This threat includes geometric constraints that may be insufficient for
the SBAS to adequately detect the orbit error.
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GNSS AND GEO SIGNAL INTEGRITY
7.2.2.3 GPS Code and Carrier Incoherence The GPS signal consists of a
radiofrequency carrier encoded with a pseudorandom spread-spectrum code. The
user’s receiver performs smoothing of its pseudorange measurements using the
carrier phase measurements. If the code and carrier are not coherent, there will be
an error in this pseudorange smoothing process. This error is caused by a satellite
failure. Incoherence between the code and carrier phase can increase the range
error, ultimately resulting in the user incorrectly determining the code/carrier
ambiguity.
7.2.2.4 GPS Signal Distortion A satellite may fail in a manner that distorts
the pseudorange portion (PRN encoding) of the GPS transmission. This causes an
error in the user’s pseudorange measurements that may be different from the error
that the SBAS receiver experiences. In 1993, the SV-19 GPS satellite experienced
a failure that fits into this category. This error is caused by a satellite failure. If a
satellite experiences this type of failure, the SBAS may not be able to estimate the
satellite clock corrections that are aligned with the user’s measurements, which
could result in HMI.
7.2.2.5 GPS L1 L2 Bias The GPS L1 and L2 signals are utilized together to
compute the ionospheric delay so the delay can be removed from the range
calculations. The satellite has separate signal paths for these two frequencies;
therefore, the signals can have different delays. The difference in the delays
must be modeled accurately to be able to properly calibrate and use L1 and L2
signals together.
Under nominal conditions, the SBAS estimation process is very accurate, and
this error is easily modeled with standard statistical techniques. If a satellite
experiences a fault, the L1 L2 bias can suddenly change, resulting in a large
estimation error. A large estimation error can lead to excessive errors in correction
processing.
7.2.2.6 Environment Errors: Ionosphere As the GPS L1 and L2 signals propagate through the ionosphere, the signals are delayed by charged particles. The
density of the charged particles, and therefore the delay, varies with location,
time of day, angle of transmission through the ionosphere, and solar activity.
This delay will cause an error in range measurements and must be corrected and
properly accounted for in the SBAS measurement error models. As discussed
earlier in Chapter 5, during calm ionospheric conditions, modeling errors are
well understood and can be handled using standard statistical techniques. Ionospheric storms pose a multitude of threats for SBAS users. The model used in
the error estimation may become invalid. The user may experience errors that
are not observable to the SBAS, due to the geometry of the reference station
pierce points. The error in the corrections may increase over time due to rapid
fluctuations in the ionosphere.
SBAS AND GBAS INTEGRITY DESIGN
243
7.2.2.7 Environment Errors: Troposphere As the GPS L1 and L2 signals
propagate through the troposphere, the signals are delayed. This delay is dependent on temperature, humidity, angle of transmission through the atmosphere, and
atmospheric pressure. This delay will cause an error in range measurements and
must be corrected and properly accounted for in the measurement error models.
Tropospheric modeling errors manifest themselves in the algorithms that generate
the corrections. The user utilizes a separate tropospheric model that may have
errors due to tropospheric modeling.
7.2.3 GEO-Associated Errors
7.2.3.1 GEO Code and Carrier Incoherence The GEO signal consists of a
radiofrequency carrier encoded with a pseudorandom spread-spectrum code. The
user’s receiver performs smoothing of its GEO pseudorange measurements using
the carrier phase measurements. If the code and carrier are not coherent, there will
be an error in this pseudorange smoothing process. Under fault free conditions,
some incoherence is possible (due to environmental effects). This will be a very
small error that is easily modeled by the ground system. Under faulted conditions,
severe divergence and potentially large errors are theoretically possible if the
GEO uplink subsystem fails.
7.2.3.2 Environment Errors: Ionosphere As the GEO L1 signal propagates
through the ionosphere, the signal is delayed by charged particles. The density of
the charged particles, and therefore the delay, varies with location, time of day,
angle of transmission through the ionosphere, and solar activity. GEO satellites
that are available today broadcast single-frequency (L1 ) signals that do not allow
a precise determination of the ionospheric delay at a reference station. Without
dual-frequency measurements, uncertainty in the calculated ionospheric delay
estimates bleed into the corrections. New GEO satellites [PRN 135,138] have
two frequencies L1 , L5 . Ionospheric delay is calculated using those frequencies.
(see Chapter 5.)
7.2.3.3 Environment Errors: Troposphere Like GPS satellite signals, the
GEO L1 signal is delayed as it propagates through the troposphere. This delay
will cause an error in range measurements and must be corrected and properly
accounted for in the measurement error models.
7.2.4 Receiver and Measurement Processing Errors
Measurement errors affect an SBAS system in two ways. They can corrupt or
degrade the accuracy of the corrections. They can also mask other system errors
and result in HMI slipping through to the user. The errors given below must all
be mitigated and residual errors bounded.
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GNSS AND GEO SIGNAL INTEGRITY
7.2.4.1 Receiver Measurement Error The receiver outputs pseudorange and
carrier phase measurements for all satellites that are in view. The receiver and
antenna characteristics limit the measurement accuracy. Under fault-free conditions these errors can be addressed using well-documented processes. A receiver
could fault and output measurement data that are in error for any or all of the
satellites in view. A latent common-mode failure in the receiver firmware could
cause all measurements in the system to simultaneously fail. Erroneous measurements pose two threats. They cannot only result in correction errors; they can
also fool the integrity monitors and let HMI slip through to the user.
7.2.4.2 Intercard Bias For receiver designs that include multiple correlators,
the internal delays in the subreceivers are different. This creates a different apparent clock for each subreceiver, called an intercard bias. Under nominal conditions,
the intercard bias estimate is extremely accurate and the intercard bias error is
easily accounted for. Any failure condition in the receiver or the algorithm computing the bias will result in an increase in the measurement data error.
7.2.4.3 Multipath Under nominal conditions, the dominant source of noise is
multipath. Multipath is caused by reflected signals arriving at the receiver delayed
relative to the direct signal. The amount of error is dependent on the delay time
and the receiver correlator type. (See discussion of multipath mitigation methods
in Chapter 5.)
7.2.4.4 L1 L2 Bias The GPS L1 and L2 signals are utilized together to compute the ionospheric delay so that the delay can be removed from the range
calculations. The receivers and antenna will experience different delays in the
electronics when monitoring these two frequencies. The difference in the delays
must be accurately modeled to be able to remove the bias and use the L1 and L2
signals together. If a receiver fails, the L1 /L2 bias can suddenly change, resulting
in large estimation errors. Under nominal conditions, the estimation process is
very accurate and the error is not significant.
7.2.4.5 Receiver Clock Error A high-quality receiver generally utilizes a
(cesium) frequency standard that provides a long-term stable time reference
(clock). This clock does drift. If a receiver fails, the clock bias can suddenly
change, resulting in a large estimation error. Under nominal conditions, the SBAS
is able to accurately account for receiver clock bias and drift. If a receiver fails,
the clock may accelerate, introducing errors into the corrections and the integrity
monitoring algorithms.
7.2.4.6 Measurement Processing Unpack/Pack Corruption The measurement
processing software that interfaces with the receiver needs to unpack and repack
the GPS ephemeris. A software failure or network transmission failure could corrupt the GPS ephemeris data and result in the SBAS using an incorrect ephemeris.
SBAS AND GBAS INTEGRITY DESIGN
245
7.2.5 Estimation Errors
The SBAS system provides corrections to improve the accuracy of the GPS
measurements and mitigate the GPS/GEO error sources. Estimation of parameters
and corrections described in Sections 7.2.5.1–7.2.5.4 cause these errors, which
must be accounted for.
7.2.5.1 Reference Time Offset Estimation Error The difference between the
SBAS and GPS reference time must be less than 50 nano-seconds. If the user
is en route and mixing SBAS corrected satellite data with non-SBAS-corrected
satellite data, then the offset (error) between the SBAS reference time and the
GPS reference time could affect the user’s receiver position solution. Under faultfree conditions, this error varies slowly. If one or more GPS satellites fail, the
offset between GPS time and the SBAS reference time could vary rapidly.
7.2.5.2 Clock Estimation Error The SBAS system must compute estimates
of the reference receiver clocks and GPS/GEO satellite clock errors. An error in
this estimation results in errors in the user’s position solution. The error in the
estimation process must be accounted for in the integrity bounds.
7.2.5.3 Ephemeris Correction Error The SBAS computes estimates of each
satellite’s orbit (ephemeris) and then uses these estimates to compute corrections.
Error in the orbit (ephemeris) estimation process will result in erroneous corrections. Sources of error include measurement noise, troposphere modeling error,
and orbital parameter modeling error. The error in the estimation process must
be accounted for in the integrity bounds.
7.2.5.4 L1 /L2 WRE and GPS Satellite Bias Estimation Error The L1 /L2 bias
of the satellites and the receivers is used to generate the SBAS corrections. SBAS
users utilize single-frequency corrections while corrections are generated using
dual-frequency measurements that are unaffected by ionospheric delay errors. An
error in the estimation process will result in erroneous corrections. Sources of
estimation error include measurement error, time in view, ionospheric storms,
and receiver/satellite malfunctions. The error in the estimation process must be
accounted for in the integrity bounds.
7.2.6 Integrity-Bound Associated Errors
The integrity monitoring functionality in an SBAS system ensures that the system
meets the allocated integrity requirement. This processing includes functionality
that must be performed on a “trusted” platform with software developed to the
proper RTCA/ DO178-B safety level.
The ICAO HMI hazard has been evaluated to be a “Hazardous/severe–major”
failure condition. This requires all software that be responsible for preventing
HMI to be developed using a process that meets all the RTCA/DO-178B Level
B objectives.
246
GNSS AND GEO SIGNAL INTEGRITY
A critical aspect of mitigating an integrity threat is the determination of the
threat model. Threats originating in the RTCA/DO-178B Level B software can
be characterized using observed performance, provided all the inputs originate
from Level B software and the algorithms have been designed in an analytic
methodology.
7.2.6.1 Ionospheric Modeling Errors The SBAS system uses an underlying
characterization to transmit ionospheric corrections to the user. During periods
of high solar activity the ionospheric decorrelation can be quite rapid and large
and the true delay variation around the grid point may not match the underlying
characterization. In this case, the SBAS-estimated delay measurement and the
associated error bound may not be accurate or the SBAS may not sample a
particular ionospheric event that is affecting a user.
7.2.6.2 Fringe Area Ephemeris Error Errors may be present in the SBAS
GPS position estimates that are not observable from the reference receivers.
These errors could cause position errors in a user’s position solution that are not
observable to the reference receivers.
7.2.6.3 Small-Sigma Errors It is possible that any quantity of satellites could
contain small or medium-sized errors that combine in such a manner that creates
an overall position error that is unbounded to a user.
7.2.6.4 Missed Message—Old But Active Data (OBAD) The user could have
missed one or more messages and is allowed to use old corrections and integrity
data. The use of these old data could result in an increased error compared to
users that have not missed messages.
7.2.6.5 TTA Exceeded If there is an underbound condition, the SBAS is
required to correct that condition within a specified period of time. This is called
the time to alarm (TTA). This alarm is a series of messages that contain the
new information, such as an increased error bound or new corrections that are
needed to correct the situation and prevent HMI. Different types of failure, such
as hardware, software, or network transmission delay, could occur and cause the
alarm messages to be delayed in excess of the required time.
7.2.7 GEO Uplink Errors
Errors caused by the uplink system can also be a source of HMI to the user.
7.2.7.1 GEO Uplink System Fails to Receive SBAS Message Any hardware
or software along the path to the satellite could fault, causing the message to be
delayed or not broadcast at all.
SBAS AND GBAS INTEGRITY DESIGN
247
7.2.8 Mitigation of Integrity Threats
This section describes some approaches that may be used to eliminate and
minimize data errors, mitigate integrity threats, and satisfy the safety integrity
requirements.
Safety design and safety analyses are utilized to protect the data transmission
path into the integrity monitors and out to the user through the Geostationary
satellite.
Such integrity monitors written to DO-178B Level B standards to provide
adjustments to the integrity bounds, must test the associated integrity data, user
differential range error (UDRE) or grid ionospheric vertical error (GIVE) in
an analytically tractable manner. The test prevents HMI by either passing the
integrity data with no changes, increasing the integrity data to bound the residual
error in the corrections, or setting the integrity data to “not monitored” or “don’t
use”. Each integrity monitor must carefully account for the uncertainty in each
component of a calculation. Noisy measurements or poor quality corrections will
result in large integrity bounds.
The examples given in this section are for a system that utilizes either a
“calculate then monitor” or “monitor then calculate” design. Both techniques are
used in the examples to fully illustrate the types of mitigation needed to meet
the general SBAS integrity requirements. Under the “calculate then monitor”
design, corrections and error bounds are computed assuming that the inputs to the
system follow some observed or otherwise predetermined model. A monitoring
system then verifies the validity of these corrections and error bounds against
the integrity threats. With the “monitor then calculate design” the measurements
inputs to the monitor are carefully screened and forced to meet strict integrity
requirements. The corrections and the error bounds are then computed in an
analytically tractable manner and no further testing is required. Both designs
must address all of the errors associated with an SBAS system in an analytically
tractable manner.
7.2.8.1 Mitigation of GPS Associated Errors
GPS Clock Error
Fault-free case—the clock corrections are computed in a Kalman filter. The
broadcast UDRE should be constructed using standard statistical techniques
to ensure that the nominal errors in the fast corrections and long-term clock
corrections are bounded.
Faulted case—a monitor is designed to ensure that the probability of a large
fast correction error and/or long-term clock correction error is less than the
allocation on the fault tree. The monitor must use measurements that are
independent of the measurements used to compute the corrections. Error
models for each input into the monitor must be determined and validated.
The monitor either passes the UDRE or increases the UDRE or sets it to
“not monitored” or “don’t use” depending on the size of the GPS clock
error.
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GNSS AND GEO SIGNAL INTEGRITY
GPS Ephemeris Error
Fault-free case—the orbit corrections are computed in a Kalman filter. The
broadcast UDRE would be constructed using standard statistical techniques
to ensure that the nominal errors in the long-term position corrections are
bounded.
Faulted case—clock errors are easily observed by a differential GPS system.
The ability of an SBAS to observe orbit errors is dependent on the location
of the system’s reference stations. The SBAS can generate a covariance
matrix and package it in SBAS Message Type 28. This message provides a
location-specific multiplier for the broadcast UDRE. The covariance matrix
must take into account the quality of the measurements from the reference
stations and the quality of the ephemeris corrections broadcast from the
SBAS. When the GPS ephemeris is grossly in error, the SBAS must either
detect and correct the problem or increase the uncertainty in the UDRE.
Under faulted conditions, the SBAS must account for the situation where
clock error cancels with the ephemeris error at one or more of the reference
stations.
GPS Code and Carrier
Fault-free case—GPS code–carrier divergence results from a failure on the GPS
satellite and errors do not need to be mitigated in the fault free case.
Faulted case—a monitor must be developed to detect and alarm if the GPS
code and carrier phase become incoherent. The monitor must account for
differences in the SBAS measurement smoothing algorithm and the user’s
measurement smoothing algorithm. The most difficult threat to detect and
mitigate is one where the code–carrier divergence occurs shortly (within
seconds) after the user acquires the satellite. In this case, the error has an
immediate effect on the user and a gradual effect on the SBAS.
GPS Signal Distortion
Fault-free case—GPS signal distortion results from a failure on the GPS satellite
and errors do not need to be mitigated in the fault free case.
Faulted case—a monitor can be developed to mitigate the errors from GPS
signal distortion. The measurement error incurred from signal distortion is
receiver-dependent. The monitor must mitigate the errors regardless of the
type of equipment the user is employing.
GPS L1 L2 Bias
Fault-free case—L1 L2 bias errors can be computed with a Kalman filter. These
corrections are not sent to the user, but used in the other monitors. Nominal
error bounds are computed with standard statistical techniques.
SBAS AND GBAS INTEGRITY DESIGN
249
Faulted case—if the SBAS design utilizes the L1 L2 bias corrections in the
integrity monitors, then they must account for the faulted case. The L1 L2
bias can suddenly change due to an equipment failure on board the GPS
satellite. The SBAS must be designed so that this type of failure does
not “blind” the monitors. One approach to this design is to form a singlefrequency integrity monitor that tests the corrections without using the L1 L2
bias corrections.
Environment (Ionosphere) Errors
Fault-free case—under calm ionospheric conditions, the GIVE is computed in
a fashion that accounts for measurement uncertainty, L1 L2 bias errors, and
nominal fluctuations in the ionosphere.
Faulted case—the integrity monitors must ensure that an ionospheric storm cannot cause HMI. One approach to this problem is to create an ionospheric
storm detector that is sensitive to spatial and/or temporal changes in the
ionospheric delay. Proving such a detector mitigates HMI is a difficult
endeavor since the ionosphere is unpredictable during ionospheric storms.
It is possible for ionospheric storms to exist in regions where the SBAS
does not sample the event. An additional factor can be added to the GIVE
to account for unobservable ionospheric storms. In some cases (when a
reference receiver is out or the grid point on the edge of the service volume) this term can be quite large. The GIVE must also account for rapid
fluctuations in the ionosphere between ionospheric correction updates. One
way to mitigate such errors is to run the monitor frequently and send alarm
messages if such an event occurs.
Environment (Troposphere) Errors
Both cases—tropospheric delay errors are built into many of the SBAS corrections. The SBAS must determine error bounds on the tropospheric delay
error and build them into the UDRE.
7.2.8.2 Mitigation of GEO-Associated Errors
GEO Code and Carrier and Environment Errors For GEO code-associated
errors, fault-free and faulted, see Section 7.2.8.1, subsection “GPS Code and
Carrier.”
Fault-free case—since GEO measurements are single-frequency, the dual-frequency techniques utilized for GPS integrity monitoring have to be modified.
One approach to working with single-frequency measurements is to compensate for the iono delay using the broadcast ionospheric grid delays. The
uncertainty of the iono corrections (GIVE) needs to be accounted for in
the integrity monitors.
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GNSS AND GEO SIGNAL INTEGRITY
Faulted case—during ionospheric storms, the GIVE is likely to be substantially
inflated. The inflated values will “blind” the other integrity monitors from
detecting small GEO clock and ephemeris errors, resulting in a large GEO
UDRE.
For both faulted and fault-free cases, of environment (troposphere) errors, see
Section 7.2.8.1, subsection “Environment (Troposphere) Errors,” Both cases.
7.2.8.3 Mitigation of Receiver and Measurement Processing Errors
Receiver Measurement Error
Fault-free case—The integrity monitors must account for the noise in the reference station measurements. A bound on the noise can be computed and
utilized in the integrity monitors. In the “calculate then monitor” approach,
integrity monitors must use measurements that are uncorrelated with the
measurements used to compute the corrections. Otherwise, error cancellation may occur.
Faulted case—in the faulted case, one or more receivers may be sending out
erroneous measurements. An integrity monitor must be built to detect such
events and ensure that erroneous measurements do not blind the integrity
monitors.
Intercard Bias Both cases—Intercard bias errors appear to be measurement
errors and are mitigated by the methods discussed in Section 7.2.4.1.
Code Noise and Multipath (CNMP)
Fault-free case—Small multipath errors are accounted for in the receiver measurement error discussed in Section 7.2.8.3.
Faulted case—Large multipath errors must be detected and screened from the
integrity monitors or accounted for in the measurement noise error bounds.
WRE L1 /L2 Bias
Fault-free case—The WRE L1 L2 bias can be computed in a manner similar to
that for the GPS L1 L2 bias. The nominal errors in this computation must
be bounded and accounted for in the integrity monitors.
Faulted case—A receiver can malfunction causing the L1 /L2 bias to suddenly
change. The L1 L2 bias is used in the correction and integrity monitoring
functions and such a change must be detected and corrected to prevent
HMI. As discussed in Section 4.1.5, a single-frequency monitor can be
created that tests the corrections without using L1 /L2 bias as an input.
SBAS AND GBAS INTEGRITY DESIGN
251
WRE Clock Error
Fault-free case—The receiver clock error can be computed using a Kalman
filter. Standard statistical techniques can be used to determine the error in
the WRE clock estimates. This error bound can be utilized by the integrity
monitors.
Faulted case—if bad data is received in the Kalman filter, erroneous WRE
clock corrections could result. An integrity monitor can be built that does
not utilize the WRE clock estimates from the Kalman filter to test the
corrections when the WRE clock estimates are bad.
7.2.8.4 Mitigation of Estimation Errors
Reference Time Offset Estimation Error
Fault-free case—in the fault-free case, the difference between the GPS reference
time and the SBAS reference time are accounted for by the user, provided
the difference is less than 50 ns.
Faulted case—in the faulted case, due to some system fault or GPS anomaly, the
difference in the SBAS reference time and the GPS reference time exceeds
50 ns. A simple monitor can be constructed to measure the difference
between the two references. The monitor would respond to a large offset by
setting all satellites not monitored, stopping the user from mixing corrected
and uncorrected satellites.
Clock Estimation Error, Ephemeris Correction Error, L1 /L2 WRE, and GPS Satellite Bias Estimation Error See Section 7.2.8.1, “GPS Clock Error,” “GPS
Ephemeris Error,” and “GPS L1 /L2 Bias,” and Section 7.2.8.3 “WRE L1 L2 Bias.”
7.2.8.5 Mitigation of Integrity-Bound-Associated Errors
Ionospheric Modeling Error
Fault-free case—extensive testing of the models used in the SBAS will provide assurance that the iono model error is properly bounding under quiet
ionospheric conditions.
Faulted case—during an ionospheric storm, the validity of the model is in
question. A monitor can be constructed to test the validity of the model
and increase the GIVE when the model is in question.
Fringe Area Ephemeris Error
Fault-free case—this error is mitigated by Message Type 28 as discussed in
Section 7.2.8.1, “GPS Ephemeris Error.”
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GNSS AND GEO SIGNAL INTEGRITY
Faulted case—special considerations must be taken to ensure that the integrity
monitors are sensitive to satellite ephemeris errors on the fringe of coverage. Errors in the satellite ephemeris are not well viewed by the SBAS on
the edge of the service region. A specific proof of the monitors’ sensitivity to errors of this nature is required. Additional inflation factors may be
needed to adjust the UDRE for this error.
Small-Sigma Errors
Fault-free case—tests can easily be performed on individual corrections; the
user, however, must be protected from the combination of all error sources.
An analysis can be performed to demonstrate that any combination of errors
observed in the fault-free case is bounded by the broadcast integrity bounds.
An example of this analysis is discussed in Ref. 10.
Faulted case—under faulted conditions, small biases may occur which can
“add” in the user position solution to cause HMI. This threat can be
mitigated by monitoring the accuracy of the user position solution at the
reference stations.
Missed Message—OBAD
Fault-free case—the old but active data deprivation factors broadcast by the
SBAS account for aging data.
Faulted case—the integrity monitors must ensure that every combination of
active SBAS messages meets the integrity requirements. Two methods are
suggested for this threat. First, the integrity monitors can run on every
active set of broadcast messages to check their validity after broadcast. If a
large error is detected, an alarm will be sent. A second, preferable, approach
is to test the messages against every active data set before broadcast and
adjust the corrections/integrity bounds accordingly.
TTA Exceeded
Fault-free case—the system is designed to meet the time-to-alarm requirement
by continually monitoring the satellite signals and responding to integrity
faults with alarms.
Faulted case—A monitor can be designed to test the “loop back” time in the
system and continually ensure that the time to alarm requirement is met.
The monitor sends a test message every minute and measures the time it
takes for message to loop back through the system.
SBAS EXAMPLE
253
7.3 SBAS EXAMPLE
The process for identifying, characterizing, and mitigating a failure condition is
illustrated by the following SBAS example.
SBAS broadcasts corrections to compensate for range errors incurred as the
signal passes through the ionosphere. The uncertainty in these corrections is
computed and sent to the user along with the corrections. HMI would result if
the SBAS broadcasts erroneous integrity data (error bounds) and does not alert
the user to the erroneous integrity data within a specified time limit. This time
limit is referred to as the time to alarm (TTA).
1. Identify error conditions that can cause HMI. Error conditions can be caused
by internal or external hardware or software failures or fluctuations in environmental conditions. The onset of an ionospheric storm represents a failure
condition that could result in large errors in the ionospheric corrections,
ultimately resulting in an increased probability of HMI.
2. Precisely characterize the threat. On days with nominal ionospheric behavior, the ionospheric threats are well understood and reasonably easy to
quantify. Scientists are not yet able to characterize the ionosphere during
storm conditions. For these reasons, SBAS has generated specific threat
models for the ionosphere based on real data collected during the worst
ionospheric activity from the solar maximum period (an 11-year solar
cycle). An important aspect of this model is the ionospheric irregularity
detector, which assures the validity of the model and inflates the error
bounds if the validity of the model is in question.
3. Identify error detection mechanisms. In the SBAS, errors in ionospheric
corrections are mitigated by a monitor located in a “safety processor” and
a special detector called the “ionospheric irregularity detector.”
4. Analytically determine that the threat is mitigated. It’s tempting to take
an RMA (reliability, maintainability, availability) approach to dealing with
ionospheric storms.
(a) Ionospheric storms are “infrequent events.”
(b) “We haven’t seen them cause HMI yet . . . .”
(c) “They don’t last very long.”
(d) “The system has other margins . . . ”
The a priori probability of a storm is not the mitigation of the threat. SBAS
must meet its 10−7 integrity allocation during ionospheric storms. The analysis
must account for worst-case events, like storms that are not well sampled by
the ground system. Furthermore, it is not necessarily the storms with the highest
magnitude that are the hardest to detect or most likely to cause HMI. Extensive
analysis is needed to characterize the threat.
In general, every requirement in a system’s specification is tested by some
type of formal demonstration. Most of the SBAS system-level requirements fall
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GNSS AND GEO SIGNAL INTEGRITY
into this category; however ‘the SBAS integrity requirement does not. Testing
fault-tree allocations of 10−7 and smaller requires on the order of 100,000,000
independent points (1 sample every 5 min for 950 years). Integrity can only be
demonstrated where reference stations exist. Integrity must be proved for every
satellite/user geometry. Every user at every point in space must be protected at
all times. Demonstrations cannot be conducted where data are not available. In
addition, every satellite geometry (subset) must be tested. Since GPS orbits repeat,
then, if at a specific airport a satellite/user geometry exists with an increased
probability of HMI, the situation will repeat every day at the same time until the
constellation changes. It is because of these considerations that analytic proofs
are required to satisfy integrity requirements.
The identification, characterization, and mitigation of a threat to the SBAS
user should be carefully scrutinized by a panel of experts in the SBAS field. The
analysis supporting claims is formally documented, scrutinized, and approved by
this panel. This four-step process should be completed for every error identified
in the system [200].
7.4 CONCLUSIONS
The data used by an SBAS to calculate the corrections and integrity data are
assumed to contain errors which have been sufficiently mitigated. The errors
discussed are inherent in any SBAS design that utilizes GPS satellites. An SBAS
design mitigates the majority of these errors with “corrections,” thereby making
it a trusted navigation aid. The leftover errors, referred to as residual errors,
are mitigated by the transmission of residual error bounding information. The
threat of potential underbounding of integrity information is mitigated by integrity
monitors. Both faulted and unfaulted cases are examined and mitigation strategies
are discussed in this chapter. These SBAS corrections improve the accuracy of
satellite signals. The integrity data ensure that the residual errors are bounded. The
SBAS integrity monitors ensure that the integrity data have not been corrupted
by SBAS failures. Following the integrity design guidelines given in this chapter
is an important factor in obtaining certification and approval for use of the SBAS
system.
SBAS integrity concepts may be applied to GBAS. In GBAS, the integrity
will be broadcast from the ground.
7.5 GPS INTEGRITY CHANNEL (GIC)
This GPS data integrity channel will be provided in the next generation of GPS
satellites such as GPS IIF and GPS III.
8
KALMAN FILTERING
8.1 INTRODUCTION
Kalman’s paper introducing his now-famous filter was first published in 1960
[104], and its first practical implementation was for integrating an inertial navigator with airborne radar aboard the C5A military aircraft [137].
The application of interest here is quite similar. We want to integrate an
onboard inertial navigator with a different electromagnetic ranging system (GPS).
There are many ways to do this [18], but nearly all involve Kalman filtering.
The purpose of this chapter is to familiarize you with theoretical and practical
aspects of Kalman filtering that are important for GPS/INS integration, and the
presentation is primarily slanted toward this application. We have also included
a brief derivation of the Kalman gain matrix, based on the maximum-likelihood
estimation (MLE) model. Broader treatments of the Kalman filter are presented
in Refs. 6, 30, 59, and 101; more basic introductions can be found in Refs. 48
and 218, more mathematically rigorous derivations can be found in Ref. 99; and
more extensive coverage of the practical aspects of Kalman filtering can be found
in Refs. 29 and 66.
8.1.1 What Is a Kalman Filter?
The Kalman filter is an extremely effective and versatile procedure for combining
noisy sensor outputs to estimate the state of a system with uncertain dynamics,
where
Global Positioning Systems, Inertial Navigation, and Integration, Second Edition, by M. S. Grewal, L. R. Weill, and A. P. Andrews
Copyright © 2007 John Wiley & Sons, Inc.
255
256
KALMAN FILTERING
The noisy sensors could be just GPS receivers and inertial navigation systems,
but may also include subsystem-level sensors (e.g., GPS clocks or INS
accelerometers and gyroscopes) or auxiliary sensors such as speed sensors (e.g., wheel speed sensors for land vehicles, water speed sensors for
ships, air speed sensors for aircraft, or Doppler radar), magnetic compasses,
altimeters (barometric or radar), or radionavigation aids (e.g., DME, VOR,
LORAN).
The system state in question may include the position, velocity, acceleration,
attitude, and attitude rate of a vehicle on land, at sea, in the air, or in
space. The system state may also include ancillary “nuisance variables”
for modeling time-correlated noise sources such as ionospheric propagation
delays of GPS signals, and time-varying parameters of the sensors, GPS
receiver clock frequency and phase, or scale factors and output biases of
accelerometers or gyroscopes.
Uncertain dynamics includes unpredictable disturbances of the host vehicle,
whether caused by a human operator or by the medium (e.g., winds, surface
currents, turns in the road, or terrain changes), but it may also include
unpredictable changes in the sensor parameters.
8.1.2 How it Works
8.1.2.1 Estimates and Uncertainties The Kalman filter maintains two types
of variables:
1. An estimate x̂ of the state vector x. The components of the estimated state
vector include the following:
(a) The variables of interest (i.e., what we want or need to know, such as
position and velocity).
(b) “Nuisance variables” that are of no intrinsic interest, but may be necessary to the estimation process. These nuisance variables may include,
for example, the effective propagation delay errors in GPS signals. We
generally do not wish to know their values but may be obliged to
calculate them to improve the receiver estimate of position.
The Kalman filter state variables for a specific application ordinarily include
all those system dynamic variables that are measurable by the sensors used
in that application. For example, A Kalman filter for an INS containing
accelerometers and rate gyroscopes might include accelerations and rotation
rates to which these instruments respond. However, simplified INS models
might ignore the accelerometers and angular rate sensors and model the
INS itself as a position-only sensor, or as a position and velocity sensor.
In similar fashion, the Kalman filter state variables for GPS-only navigation might include the pseudoranges between the satellites and the receiver
KALMAN GAIN
257
antenna, or they might contain the position coordinates of the receiver
antenna. Position could be represented by geodetic latitude, longitude, and
altitude with respect to a reference ellipsoid, or geocentric latitude, longitude, and altitude with respect to a reference sphere, or ECEF Cartesian
coordinates, or ECI coordinates, or any equivalent coordinates.
2. An estimate of estimation uncertainty. Uncertainty is modeled by a covariance matrix
:
T ;
def
P = E x̂ − x x̂ − x
(8.1)
of estimation error x̂ − x , where x̂ is the estimated state vector, x is the
actual state vector and E is the expectancy operator. The equations used to
propagate the covariance matrix (collectively called the Riccati equation)
model and manage uncertainty, taking into account how sensor noise and
dynamic uncertainty contribute to uncertainty about the estimated system
state.
By maintaining an estimate of its own estimation uncertainty and the relative
uncertainty in the various sensor outputs, the Kalman filter is able to combine all
sensor information “optimally,” in the sense that the resulting estimate minimizes
any linear quadratic loss function of estimation error, including the mean-squared
value of any linear combination of state estimation errors. The Kalman gain is
the optimal weighting matrix for combining new sensor data with a prior estimate
to obtain a new estimate. The Kalman gain is usually obtained as a partial result
in the solution of the Riccati equation.
8.1.2.2 Prediction Updates and Correction Updates The Kalman filter is a
two-step process, the steps of which we call “prediction” and “correction.” The
filter can start with either step.
The correction step updates the estimate and estimation uncertainty, based
on new information obtained from sensor measurements. It is also called the
observational update or measurement update, and the Latin prepositional phrase
a posteriori is often used for the corrected estimate and its associated uncertainty.
The prediction step updates the estimate and estimation uncertainty, taking
into account the effects of uncertain system dynamics over the times between
measurements. It is also called the temporal update, and the Latin phrase a priori
is often used for the predicted estimate and its associated uncertainty.
8.2 KALMAN GAIN
The Kalman gain matrix K is the crown jewel of Kalman filtering. All the effort of
solving the matrix Riccati equation is for the purpose of computing the “optimal”
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KALMAN FILTERING
Fig. 8.1 Estimate correction using kalman gain.
value of the gain matrix K used as shown in Fig. 8.1 for correcting an estimate
x̂, based on a measurement
z = Hx + noise
(8.2)
that is a linear function of the vector variable x to be estimated, plus additive
noise with known statistical properties.
8.2.1 Approaches to Deriving the Kalman Gain
Kalman’s original derivation of his gain matrix made no assumptions about the
underlying probability distributions, but this requires a level of mathematical
rigor that is a bit beyond standard engineering mathematics. As an alternative, it
has become common practice to derive the formula for the Kalman gain matrix K
based on an analogous filter called the Gaussian maximum-likelihood estimator.
It uses the analogies shown in Fig. 8.2 between concepts in Kalman filtering,
Gaussian probability distributions, and maximum-likelihood estimation.
Fig. 8.2
Analogous concepts in three different contexts.
259
KALMAN GAIN
This derivation is given in the following subsections. It begins with some
background information on the properties of Gaussian probability distributions
and Gaussian likelihood functions, then development of models for noisy sensor
outputs and a derivation of the associated maximum-likelihood estimate (MLE)
for combining prior estimates with noisy sensor measurements.
8.2.2 Gaussian Probability Density Functions
Probability density functions (PDFs) are nonnegative integrable functions whose
integral equals unity (i.e., 1). The density functions of Gaussian probability distributions all have the form
p(x) = √
1
exp( − 12 [x − μ]T P−1 [x − μ]) ,
(2π)n det P
(8.3)
where n is the dimension of P (i.e., P is an n × n matrix) and the parameters
def
μ = Ex∈N (μ,P ) x
*
*
def
=
dx1 · · ·
dxn p(x) x
x1
(8.5)
xn
def
P = Ex∈N (μ,P ) (x − μ)(x − μ)T *
*
def
=
dx1 · · ·
dxn p(x)(x − μ)(x − μ)T .
x1
(8.4)
(8.6)
(8.7)
xn
The parameter μ is the mean of the distribution. It will be a column vector with
the same dimensions as the variate x.
The parameter P is the covariance matrix of the distribution. By its definition,
it will always be an n × n, symmetric, nonnegative definite matrix. However,
because its determinant appears in the denominator of the square root and its
inverse appears in the exponential function argument, it must be positive definite
as well; that is, its eigenvalues must be real and positive for the definition to
work.
The constant factor 1/ (2π)n det P in Eq. 8.3 is there to make the integral
of the probability density function equal to unity, a necessary condition for all
probability density functions.
The operator E · is the expectancy operator, also called the expected-value
operator.
The notation x ∈ N (μ, P) denotes that the variate (i.e., random variable) x is
drawn from the Gaussian distribution with mean μ and covariance P. Gaussian
distributions are also called normal (the source of the “N ” notation) or Laplace
distributions.
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KALMAN FILTERING
Fig. 8.3
Maximum-likelihood estimate.
8.2.3 Properties of Likelihood Functions
Likelihood functions are similar to probability density functions, except that their
integrals are not constrained to equal unity, or even required to be finite. They
are useful for comparing relative likelihoods and for finding a value
m ∈ argmax [L(x)]
(8.8)
of the unknown independent variable x at which the likelihood function L
achieves its maximum,1 as illustrated in Fig. 8.3.
8.2.3.1 Gaussian Likelihood Functions Gaussian likelihood functions have
the form
<
1
1
T
L(x, m, Y) = c exp − [x − m] Y[x − m] ,
(8.9)
2
where c > 0 is an arbitrary scaling variable and m (defined in Eq. 8.8) is a value
of x at which L achieves its maximum value.
Information Matrices The parameter Y in Eq. 8.9 is called the information
matrix of the likelihood function. It replaces P−1 in the Gaussian probability
density function. If the information matrix Y is nonsingular, then its inverse
Y−1 = P, a covariance matrix. However, an information matrix is not required to
be nonsingular. This property of information matrices is important for representing the information from a set of measurements (sensor outputs) with incomplete
1
It is possible that a likelihood function will achieve its maximum value at more than one value of
x, but that will not matter in the derivation.
261
KALMAN GAIN
Fig. 8.4
Likelihood without unique maximum.
information for determining the unknown vector x. Thus, the measurements may
provide no information about some linear combinations of the components of x,
as illustrated in Fig. 8.4.
8.2.3.2 Scaling of Likelihood Functions Maximum-likelihood estimation is
based on the argmax of the likelihood function, but for any positive scalar c > 0,
argmax(cL) = argmax(L).
(8.10)
Thus, positive scaling of likelihood functions will have no effect on the
maximum-likelihood estimate. As a consequence, likelihood functions can have
arbitrary positive scaling.
8.2.3.3 Independent Likelihood Functions The joint probability P (A&B) of
independent events A and B is the product P (A&B) = P (A) × P (B). The analogous effect on independent likelihood functions LA and LB is the pointwise
product; that is, at each “point” x
LA&B (x) = LA (x) × LB (x).
(8.11)
8.2.3.4 Pointwise Products of Likelihood Functions One of the remarkable
attributes of Gaussian likelihood functions is that their pointwise products are
also Gaussian likelihood functions, as illustrated in Fig. 8.5.
A Lemma Given two Gaussian likelihood functions with parameter sets {mA , YA }
and {mB , YB }, their pointwise product is a scaled Gaussian likelihood function with
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KALMAN FILTERING
Fig. 8.5 Pointwise products of Gaussian likelihood functions.
parameters {mA&B , YA&B } such that, for all x, one obtains
&
1
T
exp − [x − mA&B ] YA&B [x − mA&B ]
2
&
%
1
T
= c × exp − [x − mA ] YA [x − mA ]
2
&
%
1
T
× exp − [x − mB ] YB [x − mB ]
2
%
(8.12)
for some constant c > 0.
This is the fundamental lemma about Gaussian likelihood functions, and proving it by construction will give us the Kalman gain matrix as a corollary.
8.2.4 Solving for Combined Information Matrix
One can solve Eq. 8.12 for the parameters mA&B and YA&B as functions of the
parameters mA , YA , mB , YB .
Taking logarithms of both sides of Eq. 8.12 will produce the equation
1
1
− [x − mA&B ]T YA&B [x − mA&B ] = log(c) − [x − mA ]T YA [x − mA ]
2
2
1
− [x − mB ]T YB [x − mB ] . (8.13)
2
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KALMAN GAIN
Next, taking the first and second derivatives with respect to the independent
variable x will produce the equations
YA&B (x − mA&B ) = YA (x − mA ) + YB (x − mB ),
YA&B = YA + YB ,
(8.14)
(8.15)
respectively.
8.2.4.1 Information is Additive The information matrix of the combined likelihood function (YA&B in Eq. 8.15) equals the sum of the individual information
matrices of the component likelihood functions (YA and YB in Eq. 8.15).
8.2.5 Solving for Combined Argmax
Equation 8.14 evaluated at x = 0 is
YA&B mA&B = YA mA + YB μB ,
(8.16)
which can be solved for
mA&B = Y†A&B (YA mA + YB mB )
= (YA + YB )† (YA mA + YB μB ),
(8.17)
(8.18)
where † denotes the Moore–Penrose inverse of a matrix (defined in Section
B.1.4.7).
8.2.5.1 Combined Maximum-Likelihood Estimate is a Weighted Average Equations 8.15 and 8.18 are key results for deriving the formula for Kalman gain.
All that remains is to define likelihood function parameters for noisy sensors.
8.2.6 Noisy Measurement Likelihoods
The term measurements refers to outputs of sensors that are to be used in estimating the argument vector x of a likelihood function. Measurement models
represent how these measurements are related to x, including those errors called
measurement errors or sensor noise. These models can be expressed in terms of
likelihood functions with x as the argument.
8.2.6.1 Measurement Vector The collective output values from a multitude of sensors can be represented as the components of a vector
⎡
⎤
z1
⎢ z2 ⎥
⎢
⎥
def ⎢ z ⎥
(8.19)
z = ⎢ 3 ⎥,
⎢ .. ⎥
⎣ . ⎦
z
called the measurement vector, a column vector with rows.
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KALMAN FILTERING
8.2.6.2 Measurement Sensitivity Matrix We suppose that the measured values
zi are linearly2 related to the unknown vector x we wish to estimate, namely
z = Hx,
(8.20)
where H is the measurement sensitivity matrix.
8.2.6.3 Measurement Noise Measurement noise is the unpredictable error at
the output of the sensors. It is assumed to be additive:
z = Hx + v
(8.21)
z = Hx + Jv,
(8.22)
or
where the measurement noise vector v is assumed to be zero-mean Gaussian
noise with known covariance R:
def
E v = 0,
def
R = EvvT .
(8.23)
(8.24)
8.2.6.4 Sensor Noise Distribution Matrix The matrix J in Eq. 8.22 is called
a “sensor noise distribution matrix.” It models “common mode” sensor noise,
in which a lower-dimensional noise source (e.g., power supply noise, electromagnetic interference, or temperature variations) corrupts multiple sensor outputs.
8.2.6.5 Measurement Likelihood A measurement vector z and its associated
covariance matrix of measurement noise R define a likelihood function for the
“true” value of the measurement (i.e., without noise). This likelihood function
will have its argmax at
mz = z
(8.25)
Yz = R−1 ,
(8.26)
and information matrix
assuming that R is nonsingular.
8.2.6.6 Unknown Vector Likelihoods The same parameters defining measurement likelihoods also define an inferred likelihood function for the true value of
2
The Kalman filter is defined in terms of the measurement sensitivity matrix H, but the extended
Kalman filter (described in Section 8.6.4) can be defined in terms of a suitably differentiable vectorvalued function h(x).
265
KALMAN GAIN
the unknown vector, with argmax
mx = H† mz
=H z
†
(8.27)
(8.28)
and information matrix
Yx = HT Yz H
= HT R−1 H,
(8.29)
(8.30)
where the n× matrix H† is defined as the Moore–Penrose generalized inverse
(defined in Appendix B) of the × n matrix H. This information matrix will be
singular if < n (i.e., if there are fewer sensor outputs than unknown variables),
which is not unusual for GPS/INS integration.
8.2.7 Gaussian Maximum-Likelihood Estimate
8.2.7.1 Variables Gaussian MLE uses the following variables:
x̂, the maximum-likelihood estimate of x. It will always equal the argmax of an
associated Gaussian likelihood function, but it can have different values:
x̂(−), the predicted value, representing the likelihood function prior to
using the measurement results.
x̂(+), the corrected value, representing the likelihood function after using
the measurement results.
P, the covariance matrix of estimation uncertainty. It will always equal the
inverse of the information matrix Y of the associated likelihood function.
It also can have two values:
P(−), representing the likelihood function prior to using the measurements.
P(+), representing the likelihood function after using the measurements.
z, the vector of measurements.
H, the measurement sensitivity matrix.
R, the covariance matrix of sensor noise.
8.2.7.2 Maximum-Likelihood Correction Equations The MLE formula for
correcting the variables x̂ and P to reflect the effect of measurements can be
derived from Eqs. 8.15 and 8.18 with initial likelihood parameters
mA = x̂(−),
(8.31)
the MLE before measurements, and
YA = P(−)−1 ,
(8.32)
266
KALMAN FILTERING
the inverse of the covariance matrix of MLE uncertainty before measurements.
The likelihood function of x inferred from the measurements alone (i.e., without
taking into account the prior estimate) is represented by the likelihood function
parameters
YB = HT R−1 H,
(8.33)
the information matrix of the measurements, and
mB = H† z,
(8.34)
where z is the measurement vector and † represents the Moore–Penrose generalized matrix inverse.
8.2.7.3 Covariance Update The Gaussian likelihood function with parameters
mA&B , YA&B of Eqs. 8.15 and 8.18 then represents the state of knowledge about
the unknown vector x combining both sources (i.e., the prior likelihood and the
effect of the measurements). That is, the covariance of MLE uncertainty after
using the measurements will be
P(+) = Y−1
A&B ,
(8.35)
and the MLE of x after using the measurements will then be
x̂(+) = mA&B .
(8.36)
Equation 8.15 can be simplified by applying the following general matrix formula:3
(A−1 + BC−1 D)−1 = A − AB(C + DAB)−1 DA,
where
A−1
A
B
C
D
=
=
=
=
=
(8.37)
⎫
⎪
YA , the prior information matrix for x̂
⎪
⎪
⎪
P(−), the prior covariance matrix for x̂
⎬
T
H , the transpose of the measurement sensitivity matrix ,
⎪
⎪
R,
⎪
⎪
⎭
H, the measurement sensitivity matrix,,
(8.38)
so that Eq. 8.35 becomes
P(+) =
=
=
=
⎫
Y−1
⎪
A&B
⎪
⎬
(YA + HT R−1 H)−1
(Eq. 8.15)
, (8.39)
−1 T
−1 T
−1
−1
Y−1
(Eq. 8.37) ⎪
⎪
A − YA H (HYA H + R) HYA
⎭
P(−) − P(−)HT (HP(−)HT + R)−1 HP(−),
a form better conditioned for computation.
3
A formula with many discoverers. Henderson and Searle [81] list some earlier ones.
267
KALMAN GAIN
8.2.7.4 Estimate Correction Equation 8.18 with substitutions from Eqs. 8.31–
8.34 will have the form shown in Fig. 8.1
⎫
x̂(+) = mA&B
(Eq. 8.36) ⎪
⎪
⎪
= (YA + YB )† (YA mA + YB mB )
(Eq. 8.18) ⎪
⎪
⎪
⎪
T −1
†
⎪
= P(+)[P(−)−1 x̂(-) + H
R
H
H
z
]
⎪
⎪
⎪
⎪
⎪
(8.33)
8.34
⎪
(8.35)
(8.32) (8.31)
⎪
⎪
T
T
−1
⎪
= [P(−) − P(−)H (HP (−)H + R) HP (−)]
⎪
⎪
⎪
−1
T −1
†
× [P(−) x̂(−) + H R HH z]
(Eq. 8.39) ⎪
⎪
⎪
⎪
⎪
= [I − P(−)HT (HP(−)HT + R)−1 H]
⎪
⎪
⎪
T −1
†
⎪
× [x̂(−) + P(−)H R HH z]
⎬
T
T
−1
= x̂(−) + P(−)H (HP(−)H + R)
,
⎪
⎪
× {[(HP(−)HT + R)R−1
⎪
⎪
⎪
⎪
− HP(−)HT R−1 ]z − Hx̂(−)}
⎪
⎪
⎪
T
T
−1
⎪
= x̂(−) + P(−)H (HP(−)H + R)
⎪
⎪
⎪
T −1
T −1
⎪
× {[HP(−)H R + I − HP(−)H R ]z
⎪
⎪
⎪
⎪
− Hx̂(−)}
⎪
⎪
⎪
T
T
−1
⎪
= x̂(−) + P(−)H (HP(−)H + R)
⎪
⎪
⎪
⎪
⎪
⎪
K
⎪
⎭
× {z − Hx̂(−)},
(8.40)
where the matrix K has a special meaning in Kalman filtering.
8.2.8 Kalman Gain Matrix for Maximum-Likelihood Estimation
The last line in Eq. 8.73 has the form of the equation in Fig. 8.1 with Kalman
gain matrix
K = P(−)HT [HP(−)HT + R]−1 ,
(8.41)
which completes the derivation of the Kalman gain matrix based on Gaussian
MLE.
8.2.9 Estimate Correction Using Kalman Gain
The Kalman gain expression from Eq. 8.41 can be substituted into Eq. 8.73 to
yield
#
"
x̂(−) = x̂(−) + K z − Hx̂(−) ,
(8.42)
the estimate correction equation to account for the effects of measurements.
8.2.10 Covariance Correction for Measurements
The act of making a measurement and correcting the estimate on the basis of the
information obtained reduces the uncertainty about the estimate. The effect this
268
KALMAN FILTERING
has on the covariance of estimation uncertainty P can be found by substituting
Eq. 8.41 into Eq. 8.39. The result is a simplified equation for the covariance
matrix update to correct for the effects of using the measurements:
P(+) = P(−) − KHP(−).
(8.43)
8.3 PREDICTION
The rest of the Kalman filter is the prediction step, in which the estimate x̂
and its associated covariance matrix of estimation uncertainty P are propagated
from one time epoch to another. This is the part where the dynamics of the
underlying physical processes come into play. The “state” of a dynamic process
is a vector of variables that completely specify enough of the initial boundary
value conditions for propagating the trajectory of the dynamic process forward
in time, and the procedure for propagating that solution forward in time is called
“state prediction.” The model for propagating the covariance matrix of estimation uncertainty is derived from the model used for propagating the state
vector.
8.3.1 Stochastic Systems in Continuous Time
The word stochastic derives from the Greek expression for aiming at a target,
indicating some degree of uncertainty in the dynamics of the projectile between
launch and impact. That idea has been formalized mathematically as stochastic
systems theory, in which a stochastic process is a model for the evolution over
time of a probability distribution.
8.3.1.1 White-Noise Processes A white noise process in continuous time is
a function whose value at any time is a sample from a zero-mean Gaussian
distribution, statistically independent of the values sampled at other times. White
noise processes are not integrable functions in the ordinary (Riemann) calculus.
A special calculus is required to render them integrable. It is called the stochastic
calculus. See Ref. 99 for more details on this.
8.3.1.2 Stochastic Differential Equations Ever since the differential calculus
was introduced (more or less simultaneously) by Sir Isaac Newton (1643–1727)
and Gottfried Wilhelm Leibnitz (1646–1716), we have been using ordinary differential equations as models for the dynamical behavior of systems of all sorts.
In 1827, botanist Robert Brown (1773–1858) described the apparently random motions of small particles in fluids, and the phenomenon came to be called
Brownian motion. In 1908, French physicist Paul Langevin4 (1872–1946) published a mathematical model for Brownian motion as a differential equation. It
included a random function of time that was eventually characterized as a white4
Langevin also invented and developed sonar.
269
PREDICTION
noise process. When the dependent variables in a differential equation include
white-noise processes w(t), it is called a stochastic differential equation.
Uncertain dynamical systems are modeled by linear stochastic differential
equations of the sort
d
x(t) = F(t)x(t) + w(t)
dt
(8.44)
d
x(t) = F(t) x(t) + G(t) w(t),
dt
(8.45)
or
where x(t) is the system state vector, a column vector with n rows; F(t) is the
dynamic coefficient matrix, an n × n matrix; G(t) is a dynamic noise distribution
matrix, which can be an identity matrix; and w(t) is a zero-mean white-noise
vector representing dynamic disturbance noise, also called process noise.
Example 8.1: Stochastic Differential Equation Model for Harmonic Resonator. Dynamical behavior of the one-dimensional damped mass–spring system shown schematically in Fig. 8.6 is modeled by the equations
m
d 2ξ
dξ
= ma = F = −Cdamping
− Cspring ξ +
dt 2
dt
damping force
spring force
w(t)
disturbance
or
Cspring
Cdamping dξ
w(t)
d 2ξ
+
ξ=
,
+
2
dt
m
dt
m
m
Fig. 8.6 Schematic model for dynamic system of Example 8.1.
(8.46)
270
KALMAN FILTERING
where m is the mass attached to spring and damper, ξ is the upward displacement
of the mass from its rest position, Cspring is the spring constant, Cdamping is the
damping coefficient of the dashpot, and w(t) is the random disturbing force acting
on the mass.
8.3.1.3 Systems of First-Order Linear Differential Equations The so-called
state space models for dynamic systems replace higher-order differential equations
with systems of first-order differential equations. This can be done by defining
the first n − 1 derivatives of an nth-order differential equation as state variables.
Example 8.2: State Space Model for Harmonic Resonator. Equation 8.46 is
a linear second-order (n = 2) differential equation. It can be transformed into a
system of two linear first-order differential equations with state variables
def
x1 = ξ (mass displacement),
def
x2 =
dξ
(mass velocity),
dt
for which
dx1
= x2
dt
−Cspring
−Cdamping
w(t)
dx2
=
x1 +
x2 +
.
dt
m
m
m
(8.47)
(8.48)
8.3.1.4 Representation in Terms of Vectors and Matrices State space models
using systems of linear first-order differential equations can be represented more
compactly in terms of a state vector, dynamic coefficient matrix, and dynamic
disturbance vector.
Systems of linear first-order differential equations represented in “longhand”
form as
dx1
dt
dx2
dt
dx3
dt
..
.
= f11 x1
= f21 x1
= f31 x1
..
.
+ f12 x2
+ f22 x2
+ f32 x2
..
.
+ f13 x3
+ f23 x3
+ f33 x3
..
.
+
+
+
···
···
···
..
.
+ f1n xn
+ f2n xn
+ f3n xn
..
.
+ w1 ,
+ w2 ,
+ w3 ,
..
.
dxn
dt
= fn1 x1
+ fn2 x2
+ fn3 x3
+
···
+ fnn xn
+
wn
can be represented more compactly in matrix form as
d
x = Fx + w,
dt
(8.49)
271
PREDICTION
where the state vector x, dynamic coefficient matrix F, and dynamic disturbance
vector w are given as
⎡
⎢
⎢
⎢
x=⎢
⎢
⎣
x1
x2
x3
..
.
⎤
⎥
⎥
⎥
⎥,
⎥
⎦
⎡
⎢
⎢
⎢
F=⎢
⎢
⎣
xn
f11
f21
f13
..
.
f12
f22
f23
..
.
f13
f23
f33
..
.
fn1
fn2
fn3
· · · f1n
· · · f2n
· · · f3n
..
..
.
.
· · · fnn
⎡
⎤
⎢
⎢
⎢
w=⎢
⎢
⎣
⎥
⎥
⎥
⎥,
⎥
⎦
w1
w2
w3
..
.
⎤
⎥
⎥
⎥
⎥,
⎥
⎦
wn
respectively.
Example 8.3: Harmonic Resonator Model in Matrix Form. For the system
of linear differential equations 8.47 and 8.48, we obtain
/
x=
/
F=
/
w(t) =
x1
x2
0
,
0
1
0
−Cspring /m −Cdamping /m
0
0
.
w(t)/m
8.3.1.5 Eigenvalues of Dynamic Coefficient Matrices The coefficient matrix
F of a system of linear differential equations ẋ = Fx + w has effective units of
reciprocal time, or frequency (in units of radians per second). It is perhaps then not
surprising that the characteristic values (eigenvalues) of F are the characteristic
frequencies of the dynamic system represented by the differential equations.
The eigenvalues of an n × n matrix F are the roots {λi } of its characteristic
polynomial
det(λI − F) =
n
.
an λn = 0.
(8.50)
k=0
The eigenvalues of F have the same interpretation as the poles of the related
system transfer function, in that the dynamic system ẋ = Fx + w is stable if and
only if the solutions of the characteristic equation det(λI − F) = 0 lie in the left
half-plane.
Example 8.4: Damping and Resonant Frequency for Underdamped Harmonic Resonator. For the dynamic coefficient matrix
/
0
0
1
F=
(8.51)
−Cspring /m −Cdamping /m
272
KALMAN FILTERING
in Example 8.3, the eigenvalues of F are the roots of its characteristic polynomial
0
/
λ
−1
det(λI − F) = det
Cspring /m λ + Cdamping /m
= λ2 +
Cspring
Cdamping
λ+
,
m
m
which are
λ=−
Cdamping
1 ! 2
±
Cdamping − 4mCspring .
2m
2m
If the discriminant
2
Cdamping
− 4mCspring < 0,
then the mass–spring system is called underdamped, and its eigenvalues are a
complex conjugate pair
λ=−
1
± ωresonant i
τdamping
with real part
−
1
τdamping
=−
Cdamping
2m
and imaginary part
ωresonant =
1 !
2
4mCspring − Cdamping
.
2m
(8.52)
The alternative parameter
τdamping =
2m
Cdamping
is called the damping time constant of the system, and the other parameter
ωresonant is the resonant frequency in units of radians per second.
The dynamic coefficient matrix for the damped harmonic resonator model
can also be expressed in terms of the resonant frequency and damping time
constant as
0
/
0
1
Fharmonic resonator =
.
(8.53)
−ω2 − 1/τ 2 −2/τ
As long as the damping coefficient Cdamping > 0, the eigenvalues of this system
will lie in the left half-plane. In that case, the damped mass–spring system is
guaranteed to be stable.
273
PREDICTION
8.3.1.6 Matrix Exponential Function The matrix exponential function is defined in Section B.6.4 of Appendix B (on the CD-ROM) as
def
exp (M) =
∞
.
1 k
M
k!
(8.54)
k=0
for square matrices M. The result is a square matrix of the same dimension as M.
This function has some useful properties:
1. The matrix N = exp (M) is always invertible and N−1 = exp (−M).
2. If M is antisymmetric (i.e., its matrix transpose MT = −M), then N =
exp (M) is an orthogonal matrix (i.e., its matrix transpose NT = N−1 ).
3. The eigenvalues of N = exp (M) are the (scalar) exponential functions of
the eigenvalues of M.
4. If M(s) is an integrable function of a scalar s, then the derivative
%* t
&
%* t
&
d
M(s) ds = M(t)
M(s) ds .
(8.55)
dt
8.3.1.7 Forward Solution The forward solution of a differential equation is
a solution in terms of initial conditions. The property of the matrix exponential function shown in Eq. 8.55 can be used to define the forward solution of
Eq. 8.49 as
%* t
& /
% * s
&
0
* t
x(t) = exp
F(s) ds
x(t0 ) +
exp −
F(r) dr w(s) ds ,
t0
t0
t0
(8.56)
where x(t0 ) is the initial value of the state vector x for t ≥ t0 .
8.3.1.8 Time-Invariant Systems If the dynamic coefficient matrix F of
Eq. 8.49 does not depend on t (time), then the problem is called time invariant.
In that case
* T
(8.57)
Fds = (t − t0 ) F,
t0
and the forward solution
x(t) = exp [(t − t0 ) F]
<
*
x (t0 ) +
t
1
exp [− (s − t0 ) F] w(s) ds .
(8.58)
t0
8.3.2 Stochastic Systems in Discrete Time
8.3.2.1 Zero-Mean White Gaussian Noise Sequences A zero-mean white
Gaussian noise process in discrete time is a sequence of independent samples
274
KALMAN FILTERING
. . . , wk−1 , wk , wk+1 , . . . from a normal probability distribution N (0, Qk )
with zero-mean and known finite covariances Qk . In Kalman filtering, it is not
necessary (but not unusual) that the covariance of all samples be the same.
Sampling is called independent if the expected values of outer products
E wi wTj =
<
0, i = j,
Qi , i = j,
(8.59)
for all integer indices i and j of the random process.
Zero-mean white Gaussian noise sequences are the fundamental random processes used in Kalman filtering. However, it is not necessary that all noise
sources in the modeled sensors and dynamic systems be zero-mean white Gaussian noise sequences. It is only necessary that they can be modeled in terms of
such processes.
8.3.2.2 Gaussian Linear Stochastic Processes in Discrete Time A linear stochastic processes model in discrete time has the form
xk = k−1 xk−1 + wk−1 ,
(8.60)
where wk is a zero-mean white Gaussian noise process with known covariances
Qk and the vector x represents the state of a dynamic system.
This model for “marginally random” dynamics is quite useful for representing
physical systems (e.g., land vehicles, seacraft, aircraft) with zero-mean random
disturbances (e.g., winds or currents). The state transition matrix k represents the
known dynamic behavior of the system, and the covariance matrices Qk represent
the unknown random disturbances. Together, they model the propagation of the
necessary statistical properties of the state variable x.
Example 8.5: Harmonic Resonator with White Acceleration Disturbance
Noise. If the disturbance acting on the harmonic resonator of Examples 8.1–
2
8.6 were zero-mean white acceleration noise with variance σdisturbance
, then its
disturbance noise covariance matrix would have the form
/
0
0
0
Q=
.
(8.61)
2
0 σdisturbance
8.3.3 State Space Models for Discrete Time
Measurements are the outputs of sensors sampled at discrete times · · · < tk−1 <
tk < tk+1 < · · ·. The Kalman filter uses these values to estimate the state of the
associated dynamic systems at those discrete times.
If we let . . . , xk−1 , xk , xk+1 , . . . be the corresponding state vector values of
a linear dynamic system at those discrete times, then each of these values can be
275
PREDICTION
determined from the previous value by using Eq. 8.58 in the form
xk = k−1 xk−1 + wk−1 ,
=*
>
tk
def
k−1 = exp
F(s) ds ,
(8.62)
(8.63)
tk−1
def
wk−1 = k
*
tk
= *
exp −
tk−1
>
tk
F(s) ds
w(t) dt.
(8.64)
tk−1
Equation 8.62 is the discrete-time dynamic system model corresponding to the
continuous-time dynamic system model of Eq. 8.49.
The matrix k−1 (defined in Eq. 8.63) in the discrete-time model (Eq. 8.62)
is called a state transition matrix for the dynamic system defined by F. Note that
depends only on F, and not on the dynamic disturbance function w(t).
The noise vectors wk are the discrete-time analog of the dynamic disturbance
function w(t). They depend on their continuous-time counterparts F and w.
Example 8.6: State Transition Matrix for Harmonic Resonator Model. The
underdamped harmonic resonator model of Example 8.4 has no time-dependent
terms in its coefficient matrix (Eq. 8.51), making it a time-invariant model with
state transition matrix
= exp(t F)
2
= e−t/τ
cos (ω t) + sin(ω t)/ωτ
"
#"
#
− sin(ω t)/ωτ 2 1 + ω2 τ 2
sin(ω t)/ω
(8.65)
3
cos(ω t) − sin(ω t)/ωτ
,
(8.66)
where ω = ωresonant , the resonant frequency; τ = τdamping , the damping time constant; and t is the discrete timestep.
The eigenvalues of F were shown to be −1/τdamping ± iωresonant , so the eigenvalues of F t will be -tτdamping ± i t ωresonant and the eigenvalues of will be
%
exp −
t
τdamping
&
± i ωresonant t
= e−t/τ [cos (ω t) ± i sin (ω t)] .
A discrete-time dynamic system will be stable only if the eigenvalues of lie
inside the unit circle (i.e., |λ | < 1).
8.3.4 Dynamic Disturbance Noise Distribution Matrices
A common noise source can disturb more than one independent component of the
state vector representing a dynamic system. Forces applied to a rigid body, for
example, can affect rotational dynamics as well as translational dynamics. This
276
KALMAN FILTERING
sort of coupling of common disturbance noise sources into different components
of the state dynamics can be represented by using a noise distribution matrix G
in the form
d
x = Fx + Gw (t) ,
dt
(8.67)
where the components of w(t) are the common disturbance noise sources and
the matrix G represents how these disturbances are distributed among the state
vector components.
The covariance of state vector disturbance noise will then have the form
GQw GT , where Qw is the covariance matrix for the white-noise process w(t).
The analogous model in discrete time has the form
xk = k−1 xk−1 + Gk−1 wk−1 ,
(8.68)
where {wk } is a zero-mean white-noise process in discrete time.
In either case (i.e., continuous or discrete time), it is possible to use the noise
distribution matrix for noise scaling, as well, so that the components of wk can be
independent, uncorrelated unit normal variates and the noise covariance matrix
Qw = I, the identity matrix.
8.3.5 Predictor Equations
The linear stochastic process model parameters and Q can be used to calculate
how the discrete-time process variables μ (mean) and P (covariance) evolve over
time.
Using Eq. 8.60 and taking expected values, we obtain
x̂k = μk
def
= E xk = E k−1 xk−1 + wk−1 = k−1 Exk−1 + Ewk−1 = k−1 x̂k−1 + 0
= k−1 x̂k−1 ,
(8.69)
def
Pk = E(x̂k − xk )(x̂k − xk )T = E(k−1 x̂k−1 − k−1 xk−1 − wk−1 )(k−1 x̂k−1 − k−1 xk−1 − wk−1 )T = k−1 E(xk−1 − xk−1 )(xk−1 − xk−1 )T Tk−1 + Ewk−1 wTk−1 Pk−1
Qk−1
+ terms with zero expected value
= k−1 Pk−1 Tk−1 + Qk−1 .
(8.70)
277
SUMMARY OF KALMAN FILTER EQUATIONS
Equations 8.69 and 8.70 are the essential predictor equations for Kalman
filtering.
8.4 SUMMARY OF KALMAN FILTER EQUATIONS
8.4.1 Essential Equations
The complete equations for the Kalman filter are summarized in Table 8.1.
8.4.2 Common Terminology
The symbols used in Table 8.1 for the variables and parameters of the Kalman
filter are essentially those used in the original paper by Kalman [104], and this
notation is fairly common in the literature.
The following are some terms commonly used for the symbols in Table 8.1:
H is the measurement sensitivity matrix or observation matrix.
Hx̂k (−) is the predicted measurement.
z − H x̂k (−), the difference between the measurement vector and the predicted
measurement, is the innovations vector.
K is the Kalman gain.
Pk (−) is the predicted or a priori value of estimation covariance.
Pk (+) is the corrected or a posteriori value of estimation covariance.
Qk is the covariance of dynamic disturbance noise.
R is the covariance of sensor noise or measurement uncertainty.
x̂k (−) is the predicted or a priori value of the estimated state vector.
TABLE 8.1. Essential Kalman Filter Equations
Predictor (Time or Temporal Updates)
Predicted state vector:
x̂k (−)
=
k x̂k−1 (+)
Predicted covariance matrix:
=
k Pk−1 (+)Tk + Qk−1
Pk (−)
Corrector (Measurement or Observational Updates)
Kalman gain:
Kk
=
Pk (−)HTk (Hk Pk (−)HTk + Rk )−1
Corrected state estimate:
x̂k (+)
=
x̂k (−) + Kk (zk − Hk x̂k (−))
Corrected covariance matrix:
Pk (+)
=
Pk (−) − Kk Hk Pk (−)
(Eq. 8.69)
(Eq. 8.70)
(Eq. 8.41)
(Eq. 8.42)
(Eq. 8.43)
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KALMAN FILTERING
x̂k (+) is the corrected or a posteriori value of the estimated state vector.
z is the measurement vector or observation vector.
8.4.3 Data Flow Diagrams
The matrix-level data flow of the Kalman filter implementation for a time-varying
problem is diagrammed in Fig. 8.7, with the inputs shown on the left, the outputs
Fig. 8.7 Kalman filter data array flows for time-varying system.
ACCOMMODATING TIME-CORRELATED NOISE
279
(corrected estimates) on the right, and the symbol z−1 representing the unit delay
operator.
The dashed lines in the figure enclose two computation loops. The top loop
is the estimation loop, with the feedback gain (Kalman gain) coming from the
bottom loop. The bottom loop implements the Riccati equation solution used to
calculate the Kalman gain. This bottom loop runs “open loop,” in that there is no
feedback mechanism to stabilize it in the presence of roundoff errors. Numerical
instability problems with the Riccati equation propagation loop were discovered
soon after the introduction of the Kalman filter.
8.5 ACCOMMODATING TIME-CORRELATED NOISE
The fundamental noise processes in the basic Kalman filter model are zero-mean
white Gaussian noise processes {wk }, called dynamic disturbance, plant noise, or
process noise and {vk }, called sensor noise, measurement noise, or observation
noise.
GPS signal propagation errors and INS position errors are not white noise processes, but are correlated over time. Fortunately, time-correlated noise processes
can easily be accommodated in Kalman filtering by adding state variables to the
Kalman filter model. A correlated noise process ξ k can be modeled by a linear
stochastic system model of the sort
ξ k = k−1 ξ k−1 + wk−1 ,
(8.71)
where {wk } is a zero-mean white Gaussian noise process, and then augment the
state vector by appending the new variable ξ k
0
/
xoriginal
(8.72)
xaugmented =
ξ
and modify the parameter matrices , Q, and H accordingly.
8.5.1 Correlated Noise Models
8.5.1.1 Autocovariance Functions Correlation of a random sequence {ξ } is
characterized by its discrete-time autocovariance function Pξ [k], a function of
the delay index k defined as
def
Pξ [k] = Ek (ξ k − μξ )(ξ k+k − μξ )T ,
? @
where μξ is the mean value of the random sequence ξ k .
For white-noise processes
<
0, k = 0,
P [k] =
Q, k = 0,
where Q is the covariance of the white-noise process.
(8.73)
(8.74)
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KALMAN FILTERING
8.5.1.2 Random Walks Random walks, also called Wiener processes, are
cumulative sums of white-noise processes {wk }
ξk = ξk−1 + wk−1 ,
(8.75)
a stochastic process model with state transition matrix = I, an identity matrix.
Random walks are notoriously unstable, in the sense that the covariance of
the variate ξ k grows linearly with k and without bound as k → ∞. In general,
if any of the eigenvalues of a state transition matrix fall on or outside the unit
circle in the complex plane (as they all do for identity matrices), the variate of
the stochastic process can fail to have a finite steady-state covariance matrix.
However, as was demonstrated by R. E. Kalman in 1960, the covariance of
uncertainty in the estimated system state vector can still converge to a finite
steady-state value, even if the process itself is unstable.
8.5.1.3 Exponentially Correlated Noise Exponentially correlated random processes have finite, constant steady-state covariances. A scalar exponentially random process {ξk } has a model of the sort
ξk = e−t/τ ξk−1 + wk−1 ,
(8.76)
where t is the time period between samples and τ is the exponential decay
time constant of the process. The steady-state variance σ 2 of such a process is
the solution to its steady-state variance equation
2
2
σ∞
= e−2 t/τ σ∞
+Q
=
Q
,
1 − e−2t/τ
(8.77)
(8.78)
where Q is the variance of the scalar zero-mean white-noise process {wk }.
The autocovariance sequence of an exponentially correlated random process
in discrete time has the general form
2
P [k] = σ∞
exp
−k
,
Nc
(8.79)
2
which falls off exponentially on either side of its peak value σ∞
(the process
variance) at k = 0. The parameter Nc is called the correlation number of the
process, where Nc = τ/t for correlation time τ and sample interval t.
8.5.1.4 Harmonic Noise Harmonic noise includes identifiable frequency components, such as those from AC power or from mechanical or electrical resonances. A stochastic process model for such sources has already been developed
in the examples of this chapter.
8.5.1.5 SA Noise Autocorrelation A pseudorandom clock dithering algorithm
is described in U.S. Patent 4,646,032 [212] including a parametric model of
ACCOMMODATING TIME-CORRELATED NOISE
281
the autocorrelation function (autocovariance function divided by variance) of the
resulting timing errors. Knowledge of the dithering algorithm does not necessarily give the user any advantage, but there is at least a suspicion that this may be
the algorithm used for SA dithering of the individual GPS satellite time references. Its theoretical autocorrelation function is plotted in Fig. 8.8 along with an
exponential correlation curve. The two are scaled to coincide at the autocorrelation coefficient value of 1/e ≈ 0.36787944. . ., the argument at which correlation
time is defined. Unlike exponentially correlated noise, this source has greater
short-term correlation and less long-term correlation.
The correlation time of SA errors determined from GPS signal analysis is on
the order of 102 – 103 s. It is possible that the actual correlation time is variable,
which might explain the range of values reported in the literature.
Although this is not an exponential autocorrelation function, it could perhaps
be modeled as such.
8.5.1.6 Slow Variables SA timing errors (if present) are only one of a number
of slowly varying error sources in GPS/INS integration. Slow variables may also
include many of the calibration parameters of the inertial sensors, which can
be responding to temperature variations or other unknown but slowly changing
influences. Like SA errors, these other slow variations of these variables can often
be tracked and compensated by combining the INS navigation estimates with the
Fig. 8.8 Autocorrelation function for pseudonoise algorithm.
282
KALMAN FILTERING
GPS-derived estimates. What is different about the calibration parameters is that
they are involved nonlinearly in the INS system model.
8.5.2 Empirical Sensor Noise Modeling
Noise models used in Kalman filtering should be reasonably faithful representations of the true noise sources. Sensor noise can often be measured directly
and used in the design of an appropriate noise model. Dynamic process noise
is not always so accessible, and its models must often be inferred from indirect
measurements.
8.5.2.1 Spectral Characterization Spectrum analyzers and spectrum analysis
software make it relatively easy to calculate the power spectral density of sampled
noise data, and the results are useful for characterizing the type of noise and
identifying likely noise models.
The resulting noise models can then be simulated using pseudorandom sequences, and the power spectral densities of the simulated noise can be compared
to that of the sampled noise to verify the model.
The power spectral density of white noise is constant across the spectrum,
and each successive integral changes its slope by -20 dB/decade of frequency,
as illustrated in Fig. 8.9.
8.5.2.2 Shaping Filters The spectrum of white noise is flat, and the amplitude
spectrum of the output of a filter with white-noise input will have the shape of the
Fig. 8.9 Spectral properties of some common noise types.
283
ACCOMMODATING TIME-CORRELATED NOISE
Fig. 8.10 Putting white noise through shaping filters: (a) white-noise source; (b) linear
shaping filter; (c) shaped noise source.
amplitude transfer function of the filter, as illustrated in Fig. 8.10. Therefore, any
noise spectrum can be approximated by white noise passed through a shaping
filter to yield the desired shape. All correlated noise models for Kalman filters
can be implemented by shaping filters.
8.5.3 State Vector Augmentation
8.5.3.1 Correlated Dynamic Disturbance Noise A model for a linear stochastic process model in discrete time with uncorrelated and correlated disturbance
noise has the form
xk = x, k−1 xk−1 + Gwx ,k−1 wk−1 + Dξ,k−1 ξ k−1 ,
(8.80)
where wk−1 is zero-mean white (i.e., uncorrelated) disturbance noise, Gwx ,k−1 is
white-noise distribution matrix, ξ k−1 is zero-mean correlated disturbance noise,
and Dξ x, k−1 is correlated noise distribution matrix.
If the correlated dynamic disturbance noise can be modeled as yet another
linear stochastic process
ξ k = ξ,k−1 ξ k−1 + Gwξ , k−1 wξ, k−1
(8.81)
?
@
with only zero-mean white-noise inputs wu,k , then the augmented state vector
def
xaug,k =
/
xk
ξk
0
(8.82)
has a stochastic process model
0
x, k−1 Dξ x, k−1
xaug, k−1
0
ξ,k−1
0/
0
/
Gwx , k−1
0
wx, k−1
+
0
Gwξ , k−1
wξ, k−1
/
xaug, k =
(8.83)
284
KALMAN FILTERING
having only uncorrelated disturbance noise with covariance
0
/
0
Qwx,k−1
.
Qaug, k−1 =
0
Qwξ,k−1
(8.84)
The new measurement sensitivity matrix for this augmented state vector will have
the block form
Haug, k =
"
Hk
0
#
.
(8.85)
The augmenting block is zero in this case because the uncorrelated noise source
is dynamic disturbance noise, not sensor noise.
8.5.3.2 Correlated Sensor Noise
? @The same sort of state augmentation can be
done for correlated sensor noise ξ k ,
zk = Hk xk + Ak vk + Bk ξk ,
(8.86)
with the same type of model for the correlated noise (Eq. 8.81) and using the
same augmented state vector (Eq. 8.82), but now with a different augmented state
transition matrix
0
/
0
x,k−1
(8.87)
aug, k−1 =
0
ξ,k−1
and augmented measurement sensitivity matrix
Haug,k = [Hk Bk ] .
(8.88)
8.5.3.3 Correlated Noise in Continuous Time There is an analogous procedure for state augmentation using continuous-time models. If ξ (t) is a correlated
noise source defined by a model of the sort
d
ξ = Fξ ξ + wξ
dt
(8.89)
for wξ (t) a white-noise source, then any stochastic process model of the sort
d
x = Fx x + wx (t) + ξ (t)
dt
(8.90)
with this correlated noise source can also be modeled by the augmented state
vector
/ 0
x
def
xaug =
(8.91)
ξ
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NONLINEAR AND ADAPTIVE IMPLEMENTATIONS
as
d
xaug =
dt
/
Fx
0
I
Fξ
0
/
xaug +
wx
wξ
0
(8.92)
with only uncorrelated disturbance noise.
8.6 NONLINEAR AND ADAPTIVE IMPLEMENTATIONS
Although the Kalman filter is defined for linear dynamic systems with linear
sensors, it has been applied more often than not to real-world applications without
truly linear dynamics or sensors—and usually with remarkably great success.
The following subsections show how this is done.
8.6.1 Nonlinear Dynamics
State dynamics for nonlinear systems can be expressed in the functional form
d
x(t) = f (x, t) + w (t) .
dt
(8.93)
For this to be linearized, the function f must be differentiable, with Jacobian
matrix
∂f def ∂f F(x, t) =
or
,
(8.94)
∂x x̂(t)
∂x xnominal (t)
extended
linearized
where the extended Kalman filter uses the estimated trajectory for evaluating the
Jacobian, and linearized Kalman filtering uses a nominal trajectory xnominal (t),
which may come from a simulation.
8.6.1.1 Nonlinear Dynamics with Control In applications with control variables u(t), Eq. 8.93 can also be expressed in the form
d
x = f [x, u (t) , t] + w (t) ,
dt
(8.95)
in which case the control vector u may also appear in the Jacobian matrix F.
8.6.1.2 Propagating Estimates The estimate x̂ is propagated by solving the
differential equation
d
x̂ = f x̂, t ,
dt
(8.96)
using whatever means necessary (e.g., Runge–Kutta integration). The solution is
called the trajectory of the estimate.
286
KALMAN FILTERING
8.6.1.3 Propagating Covariances The covariance matrix for nonlinear systems is also propagated over time as the solution to the matrix differential
equation
d
P (t) = F (x (t) , t) P (t) + P (t) FT (x (t) , t) + Q (t) ,
dt
(8.97)
where the values of F(x, t) from Eq. 8.63 must be calculated along a trajectory
x(t). This trajectory can be the solution for the estimated value x̂ calculated
using the Kalman filter and Eq. 8.94 (for the extended Kalman filter) or along
any “nominal” trajectory (for the “linearized” Kalman filter).
8.6.2 Nonlinear Sensors
Nonlinear Kalman filtering can accommodate sensors that are not truly linear but
can at least be represented in the functional form
zk = hk (xk ) + vk ,
(8.98)
where h is a smoothly differentiable function of x. For example, even linear sensors with nonzero biases (offsets) bsensor will have sensor models of
the sort
h(x) = Hx + bsensor ,
(8.99)
in which case the Jacobian matrix
∂h
= H.
∂x
(8.100)
8.6.2.1 Predicted Sensor Outputs The predicted value of nonlinear sensor
outputs uses the full nonlinear function applied to the estimated state vector:
ẑk = hk (x̂k ).
(8.101)
8.6.2.2 Calculating Kalman Gains The value of the measurement sensitivity
matrix H used in calculating Kalman gain is evaluated as a Jacobian matrix
∂h ∂h Hk =
,
(8.102)
or
∂x ∂x x=xnominal
x=x̂
extended
linearized
where the first value (used for extended Kalman filtering) uses the estimated
trajectory for evaluation of partial derivatives and the second value uses a nominal
trajectory (used for the linearized Kalman filtering).
8.6.3 Linearized Kalman Filter
Perhaps the simplest approach to Kalman filtering for nonlinear systems uses
linearization of the system model about a nominal trajectory. This approach is
287
NONLINEAR AND ADAPTIVE IMPLEMENTATIONS
necessary for preliminary analysis of systems during the system design phase,
when there may be several potential trajectories defined by different mission
scenarios. The essential implementation equations for this case are summarized
in Table 8.2.
8.6.4 Extended Kalman Filtering
This approach is due to Stanley F. Schmidt, and it has been used successfully in
an enormous number of nonlinear applications. It is a form of nonlinear Kalman
filtering with all Jacobian matrices (i.e., H and/or F) evaluated at x̂, the estimated state. The essential extended Kalman filter equations are summarized in
Table 8.3; the major differences from the conventional Kalman filter equations
of Table 8.1 are
1. Integration of the nonlinear integrand ẋ = f(x) to predict x̂k (−)
2. Use of the nonlinear function hk (x̂k (−)) in measurement prediction
3. Use of the Jacobian matrix of the dynamic model function f as the dynamic
coefficient matrix F in the propagation of the covariance matrix
4. Use of the Jacobian matrix of the measurement function h as the measurement sensitivity matrix H in the covariance correction and Kalman gain
equations
TABLE 8.2. Linearized Kalman Filter Equations
Predictor (Time Updates)
Predicted state vector: +
tk
x̂k (−) = x̂k−1 (+) + tk−1
f(x̂, t) dt
Predicted covariance matrix:
Ṗ = FP+ PFT + Q(t)
∂f F = ∂x
x=xnom (t)
or
Pk (−) = k Pk−1 (+)Tk + Qk−1
(Eq. 8.93)
(Eq. 8.97)
(Eq. 8.94)
(Eq. 8.70)
Corrector (Measurement Updates)
Kalman gain:
T
T
−1
Kk = Pk (−)H
(Eq. 8.41)
k [Hk Pk (−)Hk + Rk ]
∂f (Eq. 8.102)
Hk = ∂x x=x
nom
Corrected state estimate:
x̂k (+) = x̂k (−) + Kk [zk − hk (x̂k (−))]
(Eqs. 8.42, 8.101)
Corrected covariance matrix:
Pk (+) = Pk (−) − Kk Hk Pk (−)
(Eq. 8.43)
288
KALMAN FILTERING
TABLE 8.3. Extended Kalman Filter Equations
Predictor (Time Updates)
Predicted state vector: +
tk
x̂k (−) = x̂k−1 (+) + tk−1
f(x̂, t) dt
(Eq. 8.93)
Predicted covariance matrix:
Ṗ = FP+ PFT + Q(t)
(Eq. 8.97)
∂f (Eq.
8.94)
F = ∂x
x=x̂( t)
or
(Eq. 8.70)
Pk (−) = k Pk−1 (+)Tk + Qk−1
Corrector (Measurement Updates)
Kalman gain:
T
T
−1
Kk = Pk (−)H
(Eq. 8.41)
k [Hk Pk (−)Hk + Rk ]
∂f (Eq. 8.102)
Hk = ∂x x=x̂
Corrected state estimate:
x̂k (+) = x̂k (−) + Kk [zk − hk (x̂k (−))]
(Eqs. 8.42, 8.101)
Corrected covariance matrix:
(Eq. 8.43)
Pk (+) = Pk (−) − Kk Hk Pk (−)
8.6.5 Adaptive Kalman Filtering
In adaptive Kalman filtering, nonlinearities in the model arise from making
parameters of the model into functions of state variables. For example, the time
constant τ of a scalar exponentially correlated process
xk = exp
−t
xk−1 + wk
τ
may be unknown or slowly time-varying, in which case it can be made part of
the augmented state vector
/
x̂aug =
x̂
τ̂
0
with state transition matrix
/
=
0
exp −t/τ̂
t exp(−t/τ̂ )x̂/τ̂ 2
,
0
exp(−t/τ ∗ )
where τ ∗ >> τ is the correlation time constant of the variations in τ̂ .
Example 8.7: Tracking Time-Varying Frequency and Damping. Consider
the problem of tracking the phase components of a damped harmonic oscillator
with slowly time-varying resonant frequency and damping time constant. The
289
NONLINEAR AND ADAPTIVE IMPLEMENTATIONS
state variables for this nonlinear dynamic system are x1 , the in-phase component of the oscillator output signal (i.e, the only observable component); x2 , the
quadrature-phase component of the signal; x3 , the damping time constant of the
oscillator (nominally 5 s); and x4 , the frequency of oscillator (nominally 2 π
rad/s, or 1 Hz).
The dynamic coefficient matrix will be
⎡
−1/x3
⎢ −x4
F=⎢
⎣ 0
0
x4
−1/x3
0
0
⎤
x2
−x1 ⎥
⎥,
⎦
0
−1/τω
$
x1 $x32
x2 x32
−1/ττ
0
where ττ is the correlation time for the time-varying oscillator damping time
constant and τω is the correlation time for the time-varying resonant frequency
of the oscillator.
If only the in-phase component or the oscillator output can be sensed, then
the measurement sensitivity matrix will have the form
H=
"
1 0 0 0
#
Figure 8.11 is a sample output of the MATLAB m-file osc ekf.m on the accompanying CD-ROM, which implements this extended Kalman filter. Note that it
tracks the phase, amplitude, frequency, and damping of the oscillator.
Fig. 8.11 Extended Kalman filter tracking simulated time-varying oscillator.
290
KALMAN FILTERING
The unknown or time-varying parameters can also be in the measurement
model. For example, a sensor output with time-varying scale factor S and bias
b can be modeled by the nonlinear equation z = Sx + b and linearized using
augmented state vector
⎡
⎤
x
xaug = ⎣ S ⎦
b
and measurement sensitivity matrix
H=
"
Ŝ
x̂
1
#
.
8.7 KALMAN–BUCY FILTER
The discrete-time form of the Kalman filter is well suited for computer implementation, but is not particularly natural for engineers, who find it more natural
think about dynamic systems in terms of differential equations.
The analog of the Kalman filter in continuous time is the Kalman–Bucy filter,
developed jointly by Richard Bucy5 and Rudolf Kalman [105].
8.7.1 Implementation Equations
The fundamental equations of the Kalman–Bucy filter are shown in Table 8.4.
People already familiar with differential equations may find the Kalman–Bucy
filter more intuitive and easier to work with than the Kalman filter—despite
complications of the stochastic calculus. To its credit, the Kalman–Bucy filter
requires only one equation each for propagation of the estimate and its covariance,
whereas the Kalman filter requires two (for prediction and correction).
However, if the result must eventually be implemented in a digital processor, then it will have to be put into discrete-time form. Formulas for this
transformation are given below. Those who prefer to “think in continuous time”
TABLE 8.4. Kalman–Bucy Filter Equations
State equation (unified predictor/corrector):
d
dt x̂(t)
= F(t) x̂(t) + P(t)HT (t)R−1 (t)[z(t) − H(t)x(t)]
KKB (t)
Covariance equation (unified predictor/corrector):
T
Ṗ(t) = F(t) P(t) + P(t) FT (t) + Q(t) − KKB (t)R(t)KKB
5
Bucy recognized the covariance equation as a form of the nonlinear differential equation studied
by Jacopo Francesco Riccati [162] (1676–1754), and that the equation was equivalent to spectral
factorization in the Wiener filter.
291
GPS RECEIVER EXAMPLES
can develop the problem solution first in continuous time as a Kalman–Bucy
filter, then transform the result to Kalman filter form for implementation.
8.7.2 Kalman–Bucy Filter Parameters
Formulas for the Kalman filter parameters Qk and Rk as functions of the Kalman–
Bucy filter parameters Q(t) and R(t) can be derived from the process models.
8.7.2.1 Q(t) and Qk The relationship between these two distinct matrix parameters depends on the coefficient matrix F(t) in the stochastic system model:
*
Qk =
%*
tk
%*
t
&T
tk
F(s) ds Q(t) exp
exp
tk−1
&
tk
F(s) ds
dt.
t
8.7.2.2 R(t) and Rk This relationship will depend on how the sensor outputs
in continuous time are filtered before sampling for the Kalman filter. If the sensor
outputs were simply sampled without filtering, then
Rk = R(tk ).
(8.103)
However, it is common practice to use antialias filtering of the sensor outputs
before sampling for Kalman filtering. Filtering of this sort can also alter the
parameter H between the two implementations. For an integrate-and-hold filter
(an effective antialiasing filter), this relationship has the form
* tk
Rk =
R(t) dt,
(8.104)
tk−1
in which case the measurement sensitivity matrix for the Kalman filter will be
HK = tHKB , where HKB is the measurement sensitivity matrix for the Kalman–
Bucy filter.
8.8 GPS RECEIVER EXAMPLES
The following is a simplified example of the expected performance of a GPS
receiver using (1) DOP calculations and (2) covariance analysis using the Riccati
equations of a Kalman filter6 for given sets of GPS satellites. These examples
are implemented in the MATLAB m-file GPS perf.m on the accompanying CD.
8.8.1 Satellite Models
This example demonstrates how the Kalman filter converges to its minimum
error bound and how well the GPS system performs as a function of the different
phasings of the four available satellites. In the simulations, the available satellites
6
There are more Kalman filter models for GNSS in Section 10.2.
292
KALMAN FILTERING
and their respective initial phasings include the following:
Satellite No.
1
2
3
4
5
0 (deg)
326
26
146
86
206
θ0 (deg)
68
340
198
271
90
The simulation runs two cases to demonstrate the criticality of picking the
correctly phased satellites. Case 1 chooses satellites 1, 2, 3, and 4 as an example
of an optimum set of satellites. Case 2 utilizes satellites 1, 2, 3, and 5 as an
example of a nonoptimal set of satellites that will result in the dreaded “GDOP
chimney” measure of performance.
Here, the GPS satellites are assumed to be in a circular orbital trajectory at
a 55◦ inclination angle. The angle 0 is the right ascension of the satellite and
θ0 is the angular location of the satellite in its circular orbit. It is assumed that
the satellites orbit the earth at a constant rate θ̇ with a period of approximately
43,082 s or slightly less than half of a day. The equations of motion that describe
the angular phasing of the satellites are given, as in the simulation
˙
(t) = 0 − t,
θ (t) = θ0 + θ̇ t,
where the angular rates are given as
π
,
86164
π
,
θ̇ = 2
43082
˙ =2
where t is in seconds. The projects simulate the GPS system from t = 0 s to
3600 s as an example of the available satellite visibility window.
8.8.2 Measurement Model
In both the GDOP and Kalman filter models, the common observation matrix
equations for discrete points is
zk = Hk xk + vk ,
where z, H , and v are the vectors and matrices for the kth observation point in
time k. This equation is usually linearized when calculating the pseudorange by
defining z = ρ - ρ 0 = H [1] x + v.
Measurement noise v is usually assumed to be N (0, R) (normally distributed
with zero mean and variance R). The covariance of receiver error R is usually
assumed to be the same error for all measurements as long as all the same
conditions exist for all time intervals (0–3600 s) of interest. By defining the
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GPS RECEIVER EXAMPLES
measurement Z as the difference in position, the measurement sensitivity matrix
H can be linearized and approximated as H [1] (i.e., first-order linear approximation) by defining
H [1] =
∂ρri
,
∂xi
where i refers to the n different states of the Kalman filter and ρ r is the reference
pseudorange.
8.8.3 Coordinates
The orbital frame coordinates used in this simulation simplify the mathematics
by using a linear transformation between the ECEF coordinate system to a locally
level coordinate frame as the observer’s local reference frame. Then, the satellite
positions become
x = y, y = z, z
= x − Re ,
where (x , y , z
) are the locally level coordinates of the satellites and (x, y, z)
are the original ECEF coordinates. Here, Re is the earth’s radius. This assumes
a user position at (0,0,0) in locally level coordinates, which makes the math
simpler because now the pseudorange can be written as
ρ1 (t) = (x1 (t) − 0)2 + (y1 (t) − 0)2 + (z1 (t) − 0)2 ,
h[1]
x (t) = −(x1 (t) − 0)/ρ1 (t),
where h[1] represents the partial of the pseudorange with respect to x (component
of the H [1] matrix). Therefore, the default earth and orbit constants are defined as
Rsat = 26560000.0,
Re = 6380000.0,
◦
α = 55 .
8.8.4 Measurement Sensitivity Matrix
The definition of the different elements of the H [1] matrix are
x1 (t) = Rsat {cos[θ (t)] sin[(t)] + sin[θ (t)] cos[(t)] cos α},
y1 (t) = Rsat {sin[θ (t)]} sin α,
z1 (t) = Rsat {cos[θ (t)] cos[(t)] − sin[θ (t)] sin[(t)] cos α} − Re
ρ1 (t) = [x1 (t)]2 + [y1 (t)]2 + [z1 (t)]2 ,
hx (t) = −x1 (t)/ρ1 (t),
hy (t) = −y1 (t)/ρ1 (t),
hz (t) = −z1 (t)/ρ1 (t),
and likewise for each of the other four satellites.
294
KALMAN FILTERING
The complete H [1] matrix can then be defined as
⎡
h1x (t) h1y (f )
⎢ h2x (t) h2y (t)
H (t) = ⎢
⎣ h3x (t) h3y (t)
h4x (t) h4y (t)
h1z (t)
h2z (t)
h3z (t)
h4z (t)
1
1
1
1
⎤
0
0 ⎥
⎥ ,
0 ⎦
0
where the last two columns refer to the clock bias and clock drift.
In calculating GDOP, only the clock bias is used in the equations, so the H
matrix becomes
⎡
h1x (t)
⎢ h2x (t)
H (t) = ⎢
⎣ h3x (t)
h4x (t)
h1y (t)
h2y (t)
h3y (t)
h4y (t)
h1z (t)
h2z (t)
h3z (t)
h4z (t)
⎤
1
1 ⎥
⎥.
1 ⎦
1
The calculation of the GDOP and various other DOPs are then defined in terms
of this H (t) matrix as a function of time t:
A(t) = [H (t)T H (t)]−1 ,
GDOP(t) = tr[A(t)],
PDOP(t) = A(t)1,1 + A(t)2,2 + A(t)3,3 ,
HDOP(t) = A(t)1,1 + A(t)2,2, ,
VDOP(t) = A(t)3,3 ,
TDOP(t) = A(t)4,4 .
8.8.5 Implementation Results
8.8.5.1 DOP Calculations In the MATLAB implementation, the GDOP,
PDOP, HDOP, VDOP, and TDOP, are defined and plotted for the two different
cases of satellite phasings:
Case 1: Good Geometry. The results from case 1 (satellites 1, 2, 3, and 4)
show an excellent GDOP ranging to less 3.2 as a function of time. Figure
8.12 shows the variation of GDOP in meters as a function of time. This
is a reasonable GDOP. Figure 8.13 shows all of the DOPs in meters as a
function of time.
Case 2: Bad Geometry. Case 2 satellite phasing results in the infamous GDOP
“chimney peak” during that time when satellite geometry fails to provide
observability of user position. Figure 8.14 shows the resulting GDOP plots.
295
GPS RECEIVER EXAMPLES
Fig. 8.12 Case 1 GDOP.
Fig. 8.13
Case 1 DOPs.
Fig. 8.14 Case 2 GDOP.
296
KALMAN FILTERING
Fig. 8.15
Case 2 DOPs.
It shows that two satellites out of four are close to each other and thereby do
not provide linearly independent equations. This combination of satellites
cannot be used to find the user position, clock drift, and biases. Figure 8.15
is a multiplot of all the DOPs.
8.8.5.2 Kalman Filter Implementation For the second part of the example, a
covariance analysis of the GPS/Kalman filter system is used to evaluate the performance of the system, given initial position estimates and estimates of receiver
R and system dynamic Q noise. This type of analysis is done if actual measurement data is not available and can serve as a predictor of how well the system
will converge to a residual error estimate in the position and time. The masking
error in the Q matrix is
⎡
⎢
⎢
Q=⎢
⎣
0.333
0
0
0
0
0
0.333
0
0
0
0
0
0.333
0
0
0
0
0
0.0833
0
0
0
0
0
0.142
⎤
⎥
⎥
⎥ ,
⎦
(8.105)
where dimensions are in meters squared and meters squared per second
squared.
The receiver noise R matrix in meters squared is
⎤
225 0
0
0
⎢ 0 225 0
0 ⎥
⎥ .
R=⎢
⎣ 0
0 225 0 ⎦
0
0
0 225
⎡
297
GPS RECEIVER EXAMPLES
The initial transformation matrix between the first and next measurement is the
matrix
⎤
⎡
1 0 0 0 0
⎢ 0 1 0 0 0 ⎥
⎥
⎢
=⎢ 0 0 1 0 0 ⎥ .
⎣ 0 0 0 1 1 ⎦
0 0 0 0 1
The assumed initial error estimate was 100 m and is represented by the P0 (+)
matrix and is an estimate of how far off the initial measurements are from the
actual points:
⎤
⎡
10, 000
0
0
0
0
0
10, 000
0
0
0 ⎥
⎢
⎥
⎢
0
0
10, 000
0
0 ⎥ .
(8.106)
P0 (+) = ⎢
⎣
0
0
0
90, 000 0 ⎦
0
0
0
0
900
These assume a clock bias error of 300 m and a drift of 30 m/s. The discrete extended Kalman filtering equations, as listed in Table 8.3, are the a priori
covariance matrix
Pk (−) = Pk−1 (+)T + Qk−1 ,
the Kalman gain equation
[1]T
Kk = Pk (−)H[1]T
[H[1]
+ Rk ]−1 ,
k
k Pk (−)Hk
and the a posteriori covariance matrix
Pk (+) = {I − Kk H[1]
k }Pk (−).
The diagonal elements of the covariance matrices Pk (−) (predicted) and Pk (+)
(corrected) are plotted as an estimate of how well the individual x, y, z and clock
drift errors converge as a function of time for t = 1 s to t = 150 s.
In a real system, the Q, R, and matrices and Kalman gain estimates are
under control of the designer and need to be varied individually to obtain an
acceptable residual covariance error. This example only analyzes the covariance estimates for the given Q, R, and matrices, which turned out to be a
satisfactory set of inputs.
Simulation Procedure Start simulation for t = 0, . . . , 3600. Case 1: Satellites
1, 2, 3, and 4:
π
,
180
π
0 3 = 146
,
180
0 1 = 326
π
,
180
π
0 4 = 86
,
180
0 2 = 26
298
KALMAN FILTERING
π
π
, θ0 2 = 340
,
180
180
π
π
θ0 3 = 198
, θ0 4 = 271
.
180
180
θ0 1 = 68
Define rate variables:
˙r =2
π
π
, θ̇r = 2
.
86, 164
43, 082
The angular rate equations are
1(t) = 0 1 − r t,
θ 1(t) = θ0 1 + θr t,
2(t) = 0 2 − r t,
θ 2(t) = θ0 2 + θr t,
3(t) = 0 3 − r t,
θ 3(t) = θ0 3 + θr t,
4(t) = 0 4 − r t,
θ 4(t) = θ0 4 + θr t
The default earth constants are
Rsat = 26560000.0,
Re = 6380000.0,
◦
cos α = cos 55 ,
◦
sin α = sin 55 ,
For satellite 1:
x1 (t) = Rsat {cos[θ1 (t)] sin[1 (t)] + sin[θ1 (t)] cos[1 (t)] cos[α]}
y1 (t) = Rsat {sin[θ1 (t)] sin[α]
z1 (t) = Rsat {cos[θ1 (t)] cos[1 (t)] − sin[θ (t)] sin[1 (t)] cos[α]}] − Re ,
ρ1 (t) = [(x1 (t)]2 + [y1 (t)]2 + [z1 (t)]2
and the H matrix elements are
h1x (t) =
−x1 (t)
,
ρ1 (t)
h1y (t) =
−y1 (t)
,
ρ1 (t)
h1z (t) =
−z1 (t)
.
ρ1 (t)
For satellite 2
x2 (t) = Rsat {cos[θ2 (t)] sin[2 (t)] + sin[θ2 (t)] cos[2 (t)] cos[α]}
y2 (t) = Rsat sin[θ2 (t)] sin[α]
z2 (t) = Rsat {cos[θ2 (t)] cos[2 (t)] − sin[θ2 (t)] sin[2 (t)] cos[α]} − Re
ρ2 (t) = [x2 (t)]2 + [y2 (t)]2 + [z2 (t)]2 ,
and the H matrix elements are
h2x (t) =
−x2 (t)
,
ρ2 (t)
h2y (t) =
−y2 (t)
,
ρ2 (t)
h2z (t) =
−z2 (t)
ρ2 (t)
299
GPS RECEIVER EXAMPLES
For satellite 3
x3 (t) = Rsat {cos[θ3 (t)] sin[3 (t)] + sin[θ3 (t)] cos[3 (t)] cos[α]}
y3 (t) = Rsat sin[θ3 (t)] sin[α]
z3 (t) = Rsat {cos[θ3 (t)] cos[3 (t)] − sin[θ3 (t)] sin[3 (t)] cos[α]} − Re
ρ3 (t) = [x3 (t)]2 + [y3 (t)]2 + z3 (t)]2 ,
and the H matrix elements are
h3x (t) =
−x3 (t)
,
ρ3 (t)
h3y (t) =
−y3 (t)
,
ρ3 (t)
h3z (t) =
−z3 (t)
.
ρ3 (t)
For satellite 4
x4 (t) = Rsat {cos[θ4 (t)] sin[4 (t)] + sin[θ4 (t)] cos[4 (t)] cos[α]}
y4 (t) = Rsat sin[θ4 (t)] sin[α]
z4 (t) = Rsat {cos[θ4 (t)] cos[4 (t)] − sin[θ4 (t)] sin[4 (t)] cos[α]} − Re
ρ4 (t) = [x4 (t)]2 + [y4 (t)]2 + [z4 (t)]2 ,
and the H matrix elements are
h4x (t) =
−x4 (t)
,
ρ4 (t)
h4y (t) =
−y4 (t)
,
ρ4 (t)
h4z (t) =
−z4 (t)
.
ρ4 (t)
Complete H[1] matrix:
⎡
h1x (t)
⎢ h2x (t)
[1]
H (t) = ⎢
⎣ h3x (t)
h4x (t)
h1y (t)
h2y (t)
h3y (t)
h4y (t)
h1z (t)
h2z (t)
h3z (t)
h4z (t)
The H matrix used in the GDOP calculation is
⎡
h1x (t) h1y (t)
⎢
h2
x (t) h2y (t)
H [1] (t) = ⎢
⎣ h3x (t) h3y (t)
h4x (t) h4y (t)
h1z (t)
h2z (t)
h3z (t)
h4z (t)
1
1
1
1
⎤
0
0 ⎥
⎥ .
0 ⎦
0
⎤
1
1 ⎥
⎥ .
1 ⎦
1
The noise matrix is
⎡
⎢
⎢
Q=⎢
⎣
0.333
0
0
0
0
0
0.333
0
0
0
0
0
0.333
0
0
0
0
0
0.0833
0
0
0
0
0
0.142
⎤
⎥
⎥
⎥ .
⎦
300
KALMAN FILTERING
The initial guess of the P0 (+) matrix is
⎡
⎢
⎢
P0 (+) = ⎢
⎣
10, 000
0
0
0
0
0
10, 000
0
0
0
0
0
10, 000
0
0
0
0
0
90, 000 0
0
0
0
0
900
⎤
⎥
⎥
⎥
⎦
and the R matrix is
⎤
225 0
0
0
⎢ 0 225 0
0 ⎥
⎥ ,
R=⎢
⎣ 0
0 225 0 ⎦
0
0
0 225
⎡
A(t) = [H [1]T (t)H [1] (t)]−1 ,
GDOP(t) = tr[A(t)].
Kalman Filter Simulation Results Figure 8.16 shows the square roots of the
covariance terms P11 (RMS east position uncertainty), both predicted (dashed
line) and corrected (solid line). After a few iterations, the RMS error in the
x position is less than 5 m. Figure 8.17 shows the corresponding RMS north
position uncertainty in meters, and Figure 8.18 shows the corresponding RMS
vertical position uncertainty in meters.
Figures 8.19 and 8.20 show the square roots of the error covariances in clock
bias and clock drift rate in meters.
Fig. 8.16
RMS east position error.
301
GPS RECEIVER EXAMPLES
Fig. 8.17 RMS north position error.
Fig. 8.18 RMS vertical position error.
Fig. 8.19 RMS clock error.
302
KALMAN FILTERING
Fig. 8.20 RMS drift error.
8.9 OTHER KALMAN FILTER IMPROVEMENTS
There have been many “improvements” in the Kalman filter since 1960. Some are
changes in the methods of computation, some use the Kalman filter model to solve
related nonfiltering problems, and some make use of Kalman filtering variables
for addressing other applications-related problems. We present here some that
have been found useful in GPS/INS integration. More extensive coverage of the
underlying issues and solution methods is provided in Ref. 66.
8.9.1 Schmidt–Kalman Suboptimal Filtering
This is a method proposed by Stanley F. Schmidt [173] for reducing the processing and memory requirements for Kalman filtering, with predictable performance
degradation. It has been used in GPS navigation as a means of eliminating
additional variables (one per GPS satellite) required for Kalman filtering with
time-correlated pseudorange errors (originally for SA errors, also useful for
uncompensated ionospheric propagation delays).
8.9.1.1 State Vector Partitioning Schmidt–Kalman filtering partitions the state
vector into “essential” variables (designated by the subscript e) and “unessential”
variables (designated by the subscript u)
/
x=
xe
xu
0
,
(8.107)
where xe is the ne × 1 subvector of essential variables to be estimated, xu is the
nu × 1 subvector that will not be estimated, and
ne + nu = n, the total number of state variables.
(8.108)
OTHER KALMAN FILTER IMPROVEMENTS
303
Even though the subvector xu of nuisance variables is not estimated, the effects
of not doing so must be reflected in the covariance matrix Pee of uncertainty in
the estimated variables. For that purpose, the Schmidt–Kalman filter calculates
the covariance matrix Puu of uncertainty in the unestimated state variables and
the cross-covariance matrix Pue between the two types. These other covariance
matrices are used in the calculation of the Schmidt–Kalman gain.
8.9.1.2 Implementation Equations The essential implementation equations
for the Schmidt–Kalman (SK) filter are listed in Table 8.5. These equations have
been arranged for reusing intermediate results to reduce computational requirements.
8.9.1.3 Simulated Performance in GPS Position Estimation Figure 8.21 is
the output of the MATLAB m-file SchmidtKalmanTest.m on the accompanying
CD. This is a simulation using the vehicle dynamic model Damp2, described in
Section 10.2.2.5, with 29 GPS satellites (almanac of March 8, 2006), 9–11 of
which were in view (15◦ above the horizon) at any one time, and
ne = 9, the number of essential state variables (3 each of position, velocity,
acceleration)
nu = 29, the number of unessential state variables (propagation delays)
TABLE 8.5. Summary Implementation of Schmidt–Kalman
Filter
Corrector (Observational Update)
"
#
T
T
C
=
He, k Pee,
" k (−)He,k T+ Peu,k (−)Hu,k T #
P
(−)H + Puu, k (−) Hu, k
+H
" u, k ue,T k
#
KSK, k
=
Pee, k (−)He, k + Peu, k (−) HTu, k C
#
"
x̂e,k (+)
=
x̂e,k (−) + KSK, k zk − Hek xe,k (−)
A
=
Ine − KSK, k He, k
T
B
=
APeu,k (−) HTu,k KSK, k
T
Pee,k (+)
=
APee,k − A − B − BT
T
+ KSK, k (Hu, k Puu, k − HTu, k + Rk )KSK, k
Peu, k (+)
=
APeu, k (−) − KSK, k Hu, k Puu, k (−)
=
Peu, k (+)T
Pue, k (+)
Puu, k (+)
=
Puu, k (−)
Predictor (Time Update)
x̂e, k+1 (−)
Pee, k+1−
Peu, k+1 (−)
Pue, k+1 (−)
Puu, k+1−
=
=
=
=
=
e, k x̂e, k (+)
e, k Pee, k (+) Te, k + Qee
e, k Peu, k (+)Tu, k
Peu, k+1 (−)T
u, k Puu, k+ Tu, k + Quu
304
KALMAN FILTERING
Fig. 8.21
Simulation comparing Schmidt–Kalman and Kalman filters.
t = 1 s, time interval between filter updates
σpos (0) = 20 m, initial position uncertainty, RMS/axis
σvel = 200 m/s, RMS of random vehicle speed(∼ 447 mi/h)
σacc = 0.5 g, RMS of random vehicle acceleration
σprop = 10 m, RMS propagation delay uncertainty (steady-state)
σρ = 10 m, RMS pseudorange measurement noise (white, zero-mean)
τprop = 150 s, correlation time constant of propagation delay
τacc = 120 s, correlation time constant of random vehicle acceleration
The variables plotted in the top graph are the “un-RSS” differences
σdifference =
!
2
σSK
− σK2
between the mean-squared position uncertainties of the Schmidt–Kalman filter
2
(σSK
) and the Kalman filter (σK2 ≈ 102 m2 ). The peak errors introduced by the
Schmidt–Kalman filter are a few meters to several meters, and transient. The
error spikes generally coincide with the changes in the number of satellites used
(plotted in the bottom graph). This would indicated that, for this GPS application
anyway, the Schmidt–Kalman filter performance comes very close to that of the
Kalman filter, except when a new satellite with unknown propagation delay is
first used. Even then, the errors introduced by using a new satellite generally die
down after a few correlation time constants of the propagation delay errors.
OTHER KALMAN FILTER IMPROVEMENTS
305
8.9.2 Serial Measurement Processing
It has shown been [106] that it is more efficient to process the components of a
measurement vector serially, one component at a time, than to process them as
a vector. This may seem counterintuitive, but it is true even if its implementation requires a transformation of measurement variables to make the associated
measurement noise covariance R a diagonal matrix (i.e., with noise uncorrelated
from one component to another).
8.9.2.1 Measurement Decorrelation If the covariance matrix R of measurement noise is not a diagonal matrix, then it can be made so by UDUT decomposition (Eq. B.22) and changing the measurement variables
Rcorrelated = UR DR UTR ,
def
Rdecorrelated = DR (a diagonal matrix),
def
zdecorrelated = UR \zcorrelated ,
def
Hdecorrelated = UR \Hcorrelated ,
(8.109)
(8.110)
(8.111)
(8.112)
where Rcorrelated is the nondiagonal (i.e., correlated component-to-component)
measurement noise covariance matrix and the new decorrelated measurement
vector zdecorrelated has a diagonal measurement noise covariance matrix Rdecorrelated
and measurement sensitivity matrix Hdecorrelated .
8.9.2.2 Serial Processing of Decorrelated Measurements The components of
zdecorrelated can now be processed one component at a time using the corresponding
row of Hdecorrelated as its measurement sensitivity matrix and the corresponding
diagonal element of Rdecorrelated as its measurement noise variance.
A “pidgin-MATLAB” implementation for this procedure is listed in Table 8.6,
where the final line is a “symmetrizing” procedure designed to improve
robustness.
8.9.3 Improving Numerical Stability
8.9.3.1 Effects of Finite Precision Computer roundoff limits the precision
of numerical representation in the implementation of Kalman filters. It has been
known to cause severe degradation of filter performance in many applications, and
alternative implementations of the Kalman filter equations (the Riccati equations,
in particular) have been shown to improve robustness against roundoff errors.
Computer roundoff for floating-point arithmetic is often characterized by a
single parameter εroundoff , which is the smallest number such that
1 + εroundoff > 1 in machine precision.
(8.113)
306
KALMAN FILTERING
TABLE 8.6. Implementation
Equations for Serial Measurement
Update
x = x̂k (−);
P = Pk (−);
for j = 1:,
z = zk (j);
H = Hk (j, :);
R = Rdecorrelated (j, j);
K = PH
/(HPH
+ R);
x̂ = K(z − Hx);
P = P − KHP;
end;
x̂k (+) = x;
Pk (+) = (P + P
)/2; (symmetrize)
It is the value assigned to the parameter eps in MATLAB. In 64-bit ANSI/IEEE
Standard floating-point arithmetic (MATLAB precision on PCs) eps = 2−52 .
The following example, due to Dyer and McReynolds [49] , shows how a
problem that is well conditioned, as posed, can be made ill-conditioned by the
filter implementation.
Example 8.8: Ill-Conditioned Measurement Sensitivity. Consider the filtering problem with measurement sensitivity matrix
/
0
1 1
1
H=
1 1 1+δ
and covariance matrices
P0 = I3 , and R = δ 2 I2 ,
where In denotes the n × n identity matrix and the parameter δ satisfies the
constraints
δ 2 < εroundoff but δ > εroundoff .
In this case, although H clearly has rank 2 in machine precision, the product
HP0 HT with roundoff will equal
/
0
3
3+δ
,
3 + δ 3 + 2δ
which is singular. The result is unchanged when R is added to HP0 HT . In this
case, then, the filter observational update fails because the matrix HP0 HT + R
is not invertible.
307
OTHER KALMAN FILTER IMPROVEMENTS
8.9.3.2 Alternative Implementations The covariance correction process
(observational update) in the solution of the Riccati equation was found to be the
dominant source of numerical instability in the Kalman filter implementation, and
the more common symptoms of failure were asymmetry of the covariance matrix
(easily fixed) or (worse by far) negative terms on its diagonal. These implementation problems could be avoided for some problems by using more precision,
but they were eventually solved for most applications by using alternatives to
the covariance matrix P as the dependent variable in the covariance correction
equation. However, each of these methods required a compatible method for
covariance prediction. Table 8.7 lists several of these compatible implementation
methods for improving the numerical stability of Kalman filters.
Figure 8.22 illustrates how these methods perform on the ill-conditioned problem of Example 8.8 as the conditioning parameter δ → 0. For this particular test
case, using 64-bit floating-point precision (52-bit mantissa), the accuracy of the
Carlson [31] and Bierman [17] implementations degrade more gracefully than
do the others as δ → ε, the machine precision limit. The Carlson and√Bierman
solutions still maintain about nine digits (≈30 bits) of accuracy at δ ≈ ε, when
the other methods have essentially no bits of accuracy in the computed solution.
These results, by themselves, do not prove the general superiority of the Carlson and Bierman solutions for the Riccati equation. Relative performance of
alternative implementation methods may depend on details of the specific application, and for many applications, the standard Kalman filter implementation will
suffice. For many other applications, it has been found sufficient to constrain the
covariance matrix to remain symmetric.
The MATLAB m-file shootout.m on the accompanying CD generates Fig.
8.22, using m-files with the same names as those of the solution methods in
Fig. 8.22. For detailed derivations of these methods, see Ref. 66.
Conditioning and Scaling Considerations The data formatting differences between Cholesky factors (Carlson implementation) and modified Cholesky factors
(Bierman–Thornton implementation) are not always insignificant, as illustrated
by the following example.
TABLE 8.7. Compatible Methods for Solving the Riccati Equation
Covariance Matrix Format
Riccati Equation
Implementation Methods
Corrector
Symmetric nonnegative definite
Square Cholesky factor C
Triangular Cholesky factor C
Triangular Cholesky factor C
Modified Cholesky factors U, D
a
From unpublished sources.
Predictor
Kalman [104]
Kalman [104]
Joseph [30]
Kalman [104]
Potter [12, 155]
Ck+1 (−) = k Ck (+)
Carlson [31]
Kailath–Schmidta
Morf–Kailath combined [143]
Bierman [17]
Thornton [187]
308
KALMAN FILTERING
Fig. 8.22
Degradation of numerical solutions with problem conditioning.
Example 8.9: Cholesky Factor Scaling and Conditioning. The n × n covariance matrix
⎤
⎡ 0
10
0
0
···
0
⎥
⎢ 0 10−2
0
···
0
⎥
⎢
−4
⎥
⎢ 0
0
10
···
0
(8.114)
P=⎢
⎥
⎥
⎢ ..
..
..
.
.
.
.
⎦
⎣ .
.
.
.
.
0
0
0
· · · 102−2n
has condition number 102n−2 . Its Cholesky factor
⎡ 0
10
0
0
···
⎢ 0 10−1
0
···
⎢
−2 · · ·
⎢ 0
0
10
C=⎢
⎢ ..
..
..
..
⎣ .
.
.
.
0
0
0
0
0
0
..
.
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(8.115)
· · · 101−n
has condition number 10n−1 . However, its modified Cholesky factors are U = In
(condition number = 1) and D = P (condition number = 102n−2 ).
The condition numbers of the different factors are plotted versus matrix dimension n in Fig. 8.23. As a rule, one would like matrix condition numbers to be
1/, where is the machine precision limit (the smallest number such that
309
OTHER KALMAN FILTER IMPROVEMENTS
Fig. 8.23 Example 8.9: Conditioning of Cholesky factors.
1 + > 1 in machine precision, equal to 2−52 in the IEEE 64-bit precision used
by MATLAB on most PCs). This threshold is labeled 1/ on the plot.
If the implementation is to be done in fixed-point arithmetic, scaling also
becomes important. For this example, the nonzero elements of D and C will
have the same relative dynamic ranges as the condition numbers.
8.9.4 Kalman Filter Monitoring
8.9.4.1 Rejecting Anomalous Sensor Data Anomalous sensor data can result
from sensor failures or from corruption of the signals from sensors, and it is
important to detect these events before the anomalous data corrupt the estimate.
The filter is not designed to accept errors due to sensor failures or signal corruption, and they can seriously degrade the accuracy of estimates. The Kalman
filter has infinite impulse response, so errors of this sort can persist for some
time.
Detecting Anomalous Sensor Data Fortunately, the Kalman filter implementation includes parameters that can be used to detect anomalous data. The Kalman
gain matrix
Kk = Pk (−)HTk (Hk Pk (−)HTk + Rk )−1
(8.116)
Yvk
includes the factor
Yvk = (Hk Pk (−)HTk + Rk )−1 ,
(8.117)
310
KALMAN FILTERING
the information matrix of innovations. The innovations are the measurement
residuals
def
v k = zk − Hk x̂k (−),
(8.118)
the differences between the apparent sensor outputs and the predicted sensor
outputs. The associated likelihood function for innovations is
L(v k ) = exp(− 12 v Tk Yvk v k ),
(8.119)
log[L(v k )] = −v Tk Yvk v k ,
(8.120)
and the log-likelihood is
which can easily be calculated. The equivalent statistic
χ2 =
v Tk Yvk v k
(8.121)
(i.e., without the sign change and division by 2, but divided by the dimension
of v k ) is nonnegative with a minimum value of zero. If the Kalman filter were
perfectly modeled and all white-noise sources were Gaussian, this would be
a chi-squared statistic with distribution as plotted in Fig. 8.24. An upper limit
threshold value on χ 2 can be used to detect anomalous sensor data, but a practical
value of that threshold should be determined by the operational values of χ 2 ,
not the theoretical values. Thus, first its range of values should be determined
2
by monitoring the system in operation, then a threshold value χmax
chosen such
2
2
that the fraction of good data rejected when χ > χmax will be acceptable.
Fig. 8.24
Chi-squared distribution.
311
OTHER KALMAN FILTER IMPROVEMENTS
Exception Handling for Anomalous Sensor Data The log-likelihood test can be
used to detect and reject anomalous data, but it can also be important to use the
measurement innovations in other ways:
1. As a minimum, to raise an alarm whenever something anomalous has been
detected
2. To tally the relative frequency of sensor data anomalies, so that trending
or incipient failure may be detectable
3. To aid in identifying the source, such as which sensor or system may have
failed.
8.9.4.2 Monitoring Filter Health Filter health monitoring methods are useful
for detecting disparities between the physical system and the model of the system
used in Kalman filtering (useful in filter development), for detecting numerical
instabilities in the solution of the Riccati equation, and for detecting the onset of
poor observability conditions. We have discussed in Section 8.9.4.1 the monitoring methods for detecting when sensors fail, or for detecting gradual degradation
of sensors.
Covariance Analysis Covariance analysis in this context means monitoring
selected diagonal terms of the covariance matrix P of estimation uncertainty.
These are the variances of state estimation uncertainty. System requirements are
often specified in terms of the variance or RMS uncertainties of key state variables, and this is a way of checking whether these requirements are being met.
It is not always possible to cover all operational trajectories in the design of the
sensor system. It is possible that situations can occur when these requirements
are not being met in operation, and it can be useful to know that.
Checking Covariance Symmetry The so-called “square root” filtering methods
presented in Section 8.9.3 are designed to ensure that the covariance matrix of
estimation uncertainty (the dependent variable of the matrix Riccati equation)
remains symmetric and positive definite. Otherwise, the fidelity of the solution
of the Riccati equation can degrade to the point that it corrupts the Kalman gain,
and that can corrupt the estimate. If you should choose not to use square-root
filtering, then you may need some assurance that the decision was justified.
Verhaegen and Van Dooren [198] have shown that asymmetry of P is one
of the factors contributing to numerical instability of the Riccati equation. If
square-root filtering is not used, then the covariance matrix can be “symmetrized”
occasionally by adding it to its transpose and rescaling:
P := 12 (P + PT ).
(8.122)
This trick has been used for many years to head off numerical instabilities.
Checking Innovations Means and Autocorrelations Innovations are the differences between what comes out of the sensors and what was expected, based on
312
KALMAN FILTERING
the estimated system state. If the system were perfectly modeled in the Kalman
filter, the innovations would be a zero-mean white-noise process and its autocorrelation function would be zero except at zero delay. The departure of the
empirical autocorrelation of innovations from this model is a useful tool for
analysis of mismodeling in real-world applications.
Calculation of Autocovariance and Autocorrelation Functions The mean of the
innovations should be zero. If not, the mean must be subtracted from the innovations before calculating the autocovariance and autocorrelation functions of the
innovations.
For vector-valued innovations, the autocovariance function is a matrix-valued
function, defined as
def
Acovar,k = Ei zv,i zTv,i+k ,
(8.123)
def
zv,i = zi − Hi x̂i (−) (innovations),
(8.124)
and the autocorrelation function is defined by
−1
Acorrel,k = D−1
σ Acovar,k Dσ ,
⎡
σ1 0 0 · · ·
⎢ 0 σ2 0 · · ·
⎢
def ⎢
Dσ = ⎢ 0 0 σ 3 · · ·
⎢ ..
..
.. . .
⎣ .
.
.
.
def
0
0
0
..
.
0 0 0 · · · σ
!?
@
def
σj =
Acovar, 0 jj ,
⎤
⎥
⎥
⎥
⎥,
⎥
⎦
(8.125)
(8.126)
(8.127)
@
?
where the j th diagonal element Acovar, 0 jj of Acovar,0 is the variance of the
j th component of the innovations vector.
Calculation of Spectra and Cross-Spectra The Fourier transforms of the diagonal elements of the autocovariance function Acovar, k (i.e., as functions of k)
are the power spectral densities (spectra) of the corresponding components of
the innovations, and the Fourier transforms of the off-diagonal elements are the
cross-spectra between the respective components.
Simple patterns to look for in interpretation of the results include the following:
1. Nonzero means of innovations may indicate the presence of uncompensated
sensor output biases, or mismodeled output biases. The modeled variance
of the bias may be seriously underestimated, for example.
2. Short-term means increasing or varying with time may indicate output noise
that is a random walk or an exponentially correlated process.
313
OTHER KALMAN FILTER IMPROVEMENTS
3. Exponential decay of the autocorrelation functions is a reasonable indication of unmodeled (or mismodeled) random walk or exponentially correlated noise.
4. Spectral peaks may indicate unmodeled harmonic noise, but it could also
indicate that there is an unmodeled harmonic term in the state dynamic
model.
5. The autocovariance function at zero delay, Acovar,0 , should equal HPHT +
R for time-invariant or very slowly time-varying systems. If Acovar,0 is
much bigger than HPHT + R, it could indicate that R is too small or that
the process noise Q is too small, either of which may cause P to be too
small. If Acovar,0 is much smaller than HPHT + R, than R and/or Q may
be too large.
6. If the off-diagonal elements of Acorrel,0 are much bigger than those of
D−1
HPHT + R D−1
σ
σ , then there may be unmodeled correlations between
sensor outputs. These correlations could be caused by mechanical vibration
or power supply noise, for example.
PROBLEMS
8.1 Given the scalar plant and observation equations
xk = xk−1 ,
zk = xk + vk ∼ N (0, σv2 )
and white noise with σv2 = 1
E(x0 ) = 1,
E(x02 ) = P0 ,
find the estimate of xk and the steady-state covariance.
8.2 Given the vector plant and scalar observation equations
/
xk =
1
0
1
1
0
xk−1 + wk−1 (normal and white),
zk = [1 0]xk + vk , (normal and white),
/
0
0 0
Ewk = 0, Qk =
,
0 1
Evk = 0,
Rk = 1 + (−1)k ,
/
find the covariances and Kalman gains for k = 10,
P0 =
10 0
0 10
0
.
314
KALMAN FILTERING
8.3 Given
/
xk =
1
0
1
1
0
/
xk−1 +
0
1
(−g),
1
2
zk = [1 0]xk + vk ∼ normal and white,
where g is gravity, find x̂1 , Pk (+) for k = 6:
/
0
/
90
10
x̂0 =
, P0 =
1
0
E(vk ) = 0,
0
2
0
,
E(vk2 ) = 2
Z1 = 1
8.4 Given
xk = −2xk−1 + wk−1 ,
zk = xk + vk ∼ normal and white,
E(vk ) = 0,
E(vk2 ) = 1,
E(wk ) = 0,
E(wk wj ) = e−|k−j | ,
find the covariances and Kalman gains for k = 3, P0 = 10.
8.5 Given
E[(wk − 1)(wj − 1)] = e−|k−j | ,
find the discrete time equation model for the sequence {wk }.
8.6 Given
E[w(t1 ) − 1][w(t2 ) − 1] = e−|t1 −t2 | ,
find the stochastic differential equation model.
8.7 Based on the 24-satellite GPS constellation, five satellite trajectories are
selected, and their parameters tabulated accordingly:
α = 55◦
Satellite ID 0 (deg)
6
272.847
7
332.847
8
32.847
9
92.847
10
152.847
0 (deg)
268.126
80.956
111.876
135.226
197.046
OTHER KALMAN FILTER IMPROVEMENTS
315
(a) Choose correctly phased satellites of four.
(b) Calculate DOPs to show their selection by plots for 5 satellites.
(c) Use Kalman filter equations for Pk (−), K k , and Pk (+) to show the errors.
Draw the plots. This should be done with good GDOP for 4 and 5
satellites.
Choose user positions at (0, 0, 0) for simplicity. (see section 8.8.)
9
INERTIAL NAVIGATION SYSTEMS
A descriptive overview of the fundamental concepts of inertial navigation is
presented in Chapter 2. The focus here is on the necessary details for INS implementation and GNSS/INS integration.
We begin with an overview of some of the technologies used for INS, and
how these devices and systems are modeled mathematically. Next, we cover the
mathematical modeling necessary for system implementation. Finally, we will
delve into mathematical models characterizing how INS errors propagate over
time. These last models will be essential for GNSS/INS integration.
9.1 INERTIAL SENSOR TECHNOLOGIES
A sampling of inertial sensor technologies used in inertial navigation is presented in Table 9.1. There are many more, but these will serve to illustrate the
great diversity of technologies applied to inertial navigation. How these devices
function will be explained briefly. A more thorough treatment of inertial sensor
designs is given in [190].
The histories by Draper [47], Mackenzie [129], Mueller [144], and Wrigley
[215] contain much more information on the history of inertial systems and sensor
technologies and the individuals involved. See also Refs. 8, 63, 110, 168 and the
cited references below for more historical and technical details.
9.1.1 Early Gyroscopes
9.1.1.1 Momentum Wheel Gyroscopes (MWGs) The earth itself is a giant
momentum wheel gyroscope, the spin axis of which remains pointing at the pole
Global Positioning Systems, Inertial Navigation, and Integration, Second Edition, by M. S. Grewal, L. R. Weill, and A. P. Andrews
Copyright © 2007 John Wiley & Sons, Inc.
316
317
INERTIAL SENSOR TECHNOLOGIES
TABLE 9.1. Some Basic Inertial Sensor Technologies
Sensor
Physical effect used
Sensor
implementation
methods
Gyroscope
Conservation of
angular
momentum
Angular
displacement
Torque rebalance
Accelerometer
Coriolis
effect
Sagnac
effect
Gyroscopic
precession
Electromagnetic
force
Strain under
load
Vibration
Ring laser
Angular
displacement
Drag cup
Piezoelectric
Rotation
Fiberoptic
Torque rebalance
Electromagnetic
Piezoresistive
star Polaris. The toy top is essentially a MWG that is probably older than recorded
history. These are devices with stored angular momentum, a vector quantity that
tends to remain constant and pointing in a fixed inertial direction unless disturbed
by torques.
Jean Bernard Léon Foucault (1819–1868) had used a momentum wheel gyroscope to measure the rotation of the earth in 1852, and it was he who coined the
term “gyroscope” from the Greek words for “turn” (γυ
ρoς) and “view” (σ κoπ óς).
9.1.1.2 Gyrocompass Technology Gyroscope technology advanced significantly in the early 1900s, when gyrocompasses were developed to replace magnetic compasses, which would not work on iron ships (see Section 2.2.3.3). The
first patent for a gyrocompass was issued to M. G. van den Bos in 1885, but
the first known practical device was designed by Hermann Anschütz-Kaempfe1
in 1903. It performed well in the laboratory but did not do well in sea trials—especially when the host ship was heading northeast-southwest or northwest–southeast in heavy seas. Anschütz-Kaempfe’s cousin Maximilian Schuler
analyzed its dynamics and determined that lateral accelerations due to rolling of
the ship were the cause of the observed errors. Schuler also found that the gyrocompass suspension could be tuned to eliminate this error sensitivity [174]. This
has come to be called Schuler tuning. It essentially tunes the pendulum period of
the suspended mass to mimic a gravity pendulum with an effective arm length
equal to the radius of curvature of the earth (called a Schuler pendulum). The
period of this pendulum (called the Schuler period) equals the orbital period of
a satellite at the same altitude (about 84.4 min at sea level).
Inertial navigators also experience oscillatory errors at the Schuler period, as
described in Section 9.5.2.1.
9.1.1.3 Bearing Technologies The earliest gyroscopes used bearing technology from wheeled vehicles, industrial rotating machinery and clocks—including
sleeve bearings, thrust bearings, jewel bearings and (later on) ball bearings. Bearing technologies developed explicitly for gyroscopes include the following:
1
When the American inventor Elmer Sperry attempted to patent his gyrocompass in Europe in 1914,
he was sued by Anschütz-Kaempfe. Sperry hired the former Swiss patent examiner Albert Einstein
as an expert witness, but Einstein’s testimony tended to support the claims of Anschütz-Kaempfe,
which prevailed.
318
INERTIAL NAVIGATION SYSTEMS
Dry-tuned gyroscopes use flexible coupling between the momentum wheel and
its shaft, as illustrated in Fig. 9.1, with spring constants “tuned” so that
the momentum wheel is effectively decoupled from bearing torques at the
wheel rotation rate. This is not so much a bearing technology as a momentum wheel isolation technology. The very popular AN/ASN-141 military
INS (USAF Standard Navigator) used dry-tuned gyroscopes.
Gas bearing gyroscopes support the momentum wheel on a spherical bearing,
as illustrated in Fig. 9.2, with a thin layer of gas between the moving
parts. Once operating, this type of bearing has essentially no wear. It was
used in the gyroscopes of the USAF Minuteman I–III ICBMs, which were
operated for years without being turned off.
Electrostatic gyroscopes (ESGs) have spherical beryllium rotors suspended by
electrostatic forces inside a spherical cavity lined with suspension electrodes, as illustrated in Fig. 9.3. The electrostatic gyroscope in the USAF
B-52 INS in the 1980s used optical readouts from a pattern etched on the
hollow rotor to determine the direction of the rotor spin axis. The Electrically Supported Gyro Navigation (ESGN) system in USN Trident-class
submarines in the late twentieth century used a solid rotor with deliberate
radial mass unbalance, and determined the direction of the rotor spin axis
from the effect this has on the suspension servo signals. Neither of these
Fig. 9.1
Dry-tuned gyroscope.
INERTIAL SENSOR TECHNOLOGIES
Fig. 9.2 Gas bearing gyroscope, with momentum wheel split to show bearing.
Fig. 9.3 Electrostatic gyroscope.
319
320
INERTIAL NAVIGATION SYSTEMS
TABLE 9.2. Performance Grades for Gyroscopes
Performance
Parameter
Performance
Units
Maximum input
deg/h
deg/s
part/part
deg/h
deg/s
√
deg/√h
deg/ s
Scale factor
Bias stability
Bias drift
Performance Grades
Inertial
Intermediate
Moderate
102 –106
10−2 –102
10−6 –10−4
10−4 –10−2
10−8 –10−6
10−4 –10−3
10−6 –10−5
102 –106
10−2 –102
10−4 –10−3
10−2 –10
10−6 –10−3
10−2 –10−1
10−5 –10−4
102 –106
10−2 –102
10−3 –10−2
10–102
10−3 –10−2
1–10
10−4 –10−3
gyroscopes was torqued, except for induction torquing during spinup and
spindown.
9.1.1.4 Gyroscope Performance Grades Gyroscopes used in inertial navigation are called “inertial grade,” which generally refers to a range of sensor performance, depending on INS performance requirements. Table 9.2 lists
some generally accepted performance grades used for gyroscopes, based on their
intended applications but not necessarily including integrated GNSS/INS applications.
These are only rough order-of-magnitude ranges for the different error characteristics. Sensor requirements are determined largely by the application. For
example, gyroscopes for gimbaled systems can generally use much smaller input
ranges than can those for strapdown applications.
9.1.2 Early Accelerometers
Pendulum clocks were used in the eighteen century for measuring the acceleration
due to gravity, but these devices were not usable on moving platforms.
9.1.2.1 Drag Cup Accelerometer An early “integrating” accelerometer design
is illustrated in Fig. 9.4. It has two independent moving parts, both able to rotate
on a common shaft axis:
1. A bar magnetic, the rotation rate of which is controlled by a dc motor.
2. A nonferrous metal “drag cup” that will be dragged along by the currents
induced in the metal by the moving magnetic field, producing a torque on
the drag cup that is proportional to the magnet rotation rate. The drag cup
also has a proof mass attached to one point, so that acceleration along the
“input axis” direction shown in the illustration will also create a torque on
the drag cup.
INERTIAL SENSOR TECHNOLOGIES
321
Fig. 9.4 Drag cup accelerometer.
The DC current to the motor is servoed to keep the drag cup from rotating, so that
the magnet rotation rate will be proportional to acceleration and each rotation of
the magnet will be proportional to the resulting velocity increment over that time
period. At very low input accelerations (e.g., during gimbaled IMU leveling),
inhomogeneities in the drag cup material can introduce harmonic noise in the
output.
This same sort of drag cup, without the proof mass and with a torsion spring
restraining the drag cup, has been used for decades for automobile speedometers.
A flexible shaft from the drive wheels drove the magnet, so that the angular
deflection of the drag cup would be proportional to speed.
9.1.2.2 Vibrating-Wire Accelerometer This is another early digital accelerometer design, with the output a frequency difference proportional to input acceleration.
The resonant frequencies of vibrating wires (or strings) depend upon the
length, density, and elastic modulus of the wire and on the square of the tension in
the wire. The motions of the wires must be sensed (e.g., by capacitance pickoffs)
and forced (e.g., electrostatically or electromagnetically) to be kept in resonance.
The wires can then be used as digitizing force sensors, as illustrated in Fig. 9.5.
The configuration shown is for a single-axis accelerometer, but the concept can
be expanded to a three-axis accelerometer by attaching pairs of opposing wires
in three orthogonal directions.
In the “push–pull” configuration shown, any lateral acceleration of the proof
mass will cause one wire frequency to increase and the other to decrease. Furthermore, if the preload tensions in the wires are servoed to keep the sum of their
322
INERTIAL NAVIGATION SYSTEMS
Fig. 9.5
Single-axis vibrating-wire accelerometer.
frequencies constant, then the difference frequency
2
2
− ωright
∝ ma,
ωleft
ωleft − ωright ∝
ma
,
ωleft + ωright
∝ a.
(9.1)
(9.2)
(9.3)
Both the difference frequency ωleft − ωright and the sum frequency ωleft + ωright
(used for preload tension control) can be obtained by mixing and filtering the
two wire position signals from the resonance forcing servo loop. Each cycle of
the difference frequency then corresponds to a constant delta velocity, making
the sensor inherently digital.
9.1.2.3 Gyroscopic Accelerometers Some of the earlier designs for accelerometers for inertial navigation used the acceleration-sensitive precession of momentum wheel gyroscopes, as illustrated in Fig. 9.6.
Fig. 9.6
Acceleration-sensitive precession of momentum wheel gyroscope.
323
INERTIAL SENSOR TECHNOLOGIES
This has the center of support offset from the center of mass of the momentum
wheel, a condition known as “mass unbalance.” For a mass-unbalanced design
like the one shown in the figure, precession rate will be proportional to acceleration. If the angular momentum and mass offset of the gyro can be kept constant,
this relationship will extremely linear over several orders of magnitude.
Gyroscopic accelerometers are integrating accelerometers. Angular precession
rate is proportional to acceleration, so the change in precession angle will be
proportional to velocity change along the input axis direction.
Accelerometer designs based on gyroscopic precession are still used in the
most accurate floated system [129].
9.1.2.4 Accelerometer Performance Ranges Table 9.3 lists accelerometer and
gyroscope performance ranges compatible with the INS performance ranges listed
in Chapter 2, Section 2.2.4.3.
9.1.3 Feedback Control Technology
9.1.3.1 Feedback Control Practical inertial navigation began to evolve in
the 1950s, using technologies that had evolved in the early twentieth century,
or were coevolving throughout the mid-twentieth century. These technologies
include classical control theory and feedback2 control technology. Its mathematical underpinnings included analytic function theory, Laplace transforms and
Fourier transforms, and its physical implementations were dominated by analog
electronics and electromagnetic transducers.
Feedback Control Servomechanisms Servomechanisms are electronic and/or
electromechanical devices for implementing feedback control. The term “servo”
is often used as a noun (short for servomechanism), adjective (e.g., “servo control”) and verb (meaning to control with a servomechanism).
TABLE 9.3. INS and Inertial Sensor Performance Ranges
Performance Ranges
System or Sensor
INS
Gyroscopes
Accelerometers
a Nautical
High
−1
≤ 10
≤ 10−3
≤ 10−7
Medium
Low
Units
≈1
≈ 10−2
≈ 10−6
≥ 10
≥ 10−1
≥ 10−5
nmi/ha
deg/h
g (9.8 m/s2 )
miles per hour CEP rate (defined in Section 2.2.4.2).
2
A concept discovered by Harold S. Black (1898–1983) in 1927 and applied to the operational
amplifier [20]. According to his own account, the idea of negative feedback occurred to Black when
he was commuting to his job at the West Street Laboratories of Western Electric in New York (later
part of Bell Labs) on the Hudson River Ferry. He wrote it down on the only paper available to
him (a copy of the New York Times), dated it and signed it, and had it witnessed and signed by
a colleague when he arrived at work. His initial patent application was refused by the U.S. Patent
Office on the grounds that it was a “perpetual motion” device.
324
INERTIAL NAVIGATION SYSTEMS
Transducers These are devices which convert measurable physical quantities
to electrical signals, and vice versa. Early servo transducers for INS included
analog shaft angle encoders (angle to signal) and torquers (signal to torque) for
gimbal bearings, and torquers for controlling the direction of angular momentum
in momentum wheel gyroscopes.
9.1.3.2 Gimbal Control In the mid-1930s, Robert H. Goddard used momentum wheel gyroscopes for feedback attitude control of rockets, and gyros and
accelerometers were used for missile guidance in Germany during World War II
[8]. These technologies (along with many of their developers) were transferred
to the United States and the Soviet Union immediately after the war [47, 215].
All feedback control loops are used to null something, usually the difference
between some reference signal and a measured signal. Servos are used in gimbaled systems for controlling the gimbals to keep the gyro outputs at specified
values (e.g., earthrate), which keeps the ISA in a specified orientation relative to
navigation coordinates, independent of host vehicle dynamics.
9.1.3.3 Torque Feedback Gyroscopes These use a servo loop to apply just
enough torque on the momentum wheel to keep the spin axis from moving
relative to its enclosure, and use the applied torque (or the motor current required
to generate it) as a measure of rotation rate. If the feedback torque is delivered
in precisely repeatable pulses, then each pulse represents a fixed angular rotation
δθ , and the pulse count in a fixed time interval t will be proportional to the
net angle change θ over that time period (plus quantization error). The result
is a digital integrating gyroscope.
Pulse Quantization Quantization pulse size determines quantization error, and
smaller quantization levels are preferred. The feedback pulse quantization size
also has an effect on outer control loops, such as those used for nulling the east
gyro output to align a gimbaled IMU in heading. When the east gyro output is
close to being nulled and its pulse rate approaches zero, quantization pulse size
will determine how long one has to wait for an LSB of the gyro output.
9.1.3.4 Torque Feedback Gyroscopic Accelerometers These use a torque
feedback loop to keep the momentum wheel rotation axis in a gyroscopic
accelerometer from moving relative to the instrument housing. In the pulse integrating gyroscopic accelerometer (PIGA), the feedback torque is delivered in
repeatable pulses. Each pulse then represents a fixed velocity change δv along
the input acceleration axis, and the net velocity change during a fixed time interval
t will be proportional to the pulse count in that period.
Nulling the outputs of the north and east accelerometers of a gimbaled IMU
during leveling is affected by pulse quantization the same way that nulling the
east gyro outputs is influenced by quantization (see “Pulse Quantization” above).
325
INERTIAL SENSOR TECHNOLOGIES
(a)
Fig. 9.7
(b)
Pendulous (a) and beam (b) accelerometers.
9.1.3.5 Force Feedback Accelerometers Except for gyroscopic accelerometers, all other practical accelerometers measure (in various ways) the specific
force required to make a proof mass follow the motions of the host vehicle.3
Pendulous Accelerometers One of the design challenges for accelerometers is
how to support a proof mass rigidly in two dimensions and allow it to be completely free in the third dimension. Pendulous accelerometers use a hinge to
support the proof mass in two dimensions, as illustrated in Fig. 9.7a, so that it
is free to move only in the input axis direction, normal to the “paddle” surface.
This design requires an external supporting force to keep the proof mass from
moving in that direction, and the force required to do it will be proportional to
the acceleration that would otherwise be disturbing the proof mass.
Electromagnetic Accelerometer (EMA) Electromagnetic accelerometers (EMAs)
are pendulous accelerometers using electromagnetic force to keep the paddle from
moving. A common design uses a voice coil attached to the paddle, as illustrated
in Fig. 9.8. Current through the voice coil provides the force on the proof mass to
keep the paddle centered in the instrument enclosure. This is similar to the speaker
cone drive in permanent magnet speakers, with the magnetic flux through the coils
provided by permanent magnets. The coil current is controlled through a feedback
servo loop including a paddle position sensor such as a capacitance pickoff. The
current in this feedback loop through the voice coil will be proportional to the
disturbing acceleration.
Integrating Accelerometers For pulse-integrating accelerometers, the feedback
current is supplied in discrete pulses with very repeatable shapes, so that each
pulse is proportional to a fixed change in velocity. An up/down counter keeps
3
This approach was turned inside-out around 1960, when satellites designed to measure low levels
of atmospheric drag at the outer edges of the atmosphere used a free-floating proof mass inside the
satellite, protected from drag forces, and measured the thrust required to make the satellite follow
the drag-free proof mass.
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INERTIAL NAVIGATION SYSTEMS
Fig. 9.8
Electromagnetic accelerometer (EMA).
track of the net pulse count between samples of the digitized accelerometer output. The pulse feedback electromagnetic accelerometer is an integrating accelerometer, in that each pulse output corresponds to a constant increment in velocity
δv. The electromagnetic accelerometer (EMA) illustrated in Fig. 9.8 is another
type of integrating accelerometer, similar to the PIGA, as is the beam accelerometer of Fig. 9.7(b) with SAW strain sensor.
9.1.4 Rotating Coriolis Multisensors
9.1.4.1 Coriolis Effect Gustav Gaspard de Coriolis (1792–1843) published a
report in 1835 [39] describing the effects of coordinate rotation on Newton’s laws
of motion. Bodies with no applied acceleration maintain constant velocity in nonrotating coordinates, but appear to experience additional apparent accelerations
in rotating frames. The “Coriolis effect” is an apparent acceleration of the form
acoriolis = −2ω ⊗ vrotating ,
(9.4)
where ω is the coordinate rotation rate vector, ⊗ represents the vector crossproduct, and vrotating is the velocity of the body measured in rotating coordinates.
9.1.4.2 Rotating Coriolis Gyroscope These are gyroscopes that measure the
coriolis acceleration on a rotating wheel. An example of such a two-axis gyroscope is illustrated in Fig. 9.9. For sensing rotation, it uses an accelerometer
mounted off axis on the rotating member, with its acceleration input axis parallel
to the rotation axis of the base. When the entire assembly is rotated about any
axis normal to its own rotation axis, the accelerometer mounted on the rotating
base senses a sinusoidal coriolis acceleration.
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INERTIAL SENSOR TECHNOLOGIES
(a)
(b)
Fig. 9.9 Rotating coriolis gyroscope: (a) Function; (b) sensing.
The position x and velocity v of the rotated accelerometer with respect to
inertial coordinates will be
⎡
⎤
cos(drive t)
x(t) = ρ ⎣ sin(drive t) ⎦ ,
(9.5)
0
v(t) =
d
x(t),
dt
⎡
(9.6)
⎤
− sin(drive t)
= ρdrive ⎣ cos(drive t) ⎦ ,
0
(9.7)
where drive is the drive rotation rate and ρ is the offset distance of the accelerometer from the base rotation axis.
The input axis of the accelerometer is parallel to the rotation axis of the base,
so it is insensitive to rotations about the base rotation axis (z axis). However,
if this apparatus is rotated with components x, input and y, input orthogonal to
the z axis, then the coriolis acceleration of the accelerometer will be the vector
cross-product
⎡
⎤
x, input
acoriolis (t) = − ⎣ y, input ⎦ ⊗ v(t)
(9.8)
0
⎤
⎡
⎤ ⎡
x, input
− sin(drive t)
= −ρdrive ⎣ y, input ⎦ ⊗ ⎣ cos(drive t) ⎦
(9.9)
0
0
⎡
⎤
0
⎦ (9.10)
0
= ρ drive ⎣
−x, input cos(drive t) + y, input sin(drive t)
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INERTIAL NAVIGATION SYSTEMS
The rotating z-axis accelerometer will then sense the z-component of coriolis
acceleration,
az, input (t) = ρdrive [x, input cos(drive t) − y, input sin(drive t)],
(9.11)
which can be demodulated to recover the phase components ρ drive x (in phase)
and ρ drive y, input (in quadrature), each of which is proportional to a component
of the input rotation rate. Demodulation of the accelerometer output removes the
DC bias, so this implementation is insensitive to accelerometer bias errors.
9.1.4.3 Rotating Multisensor Another accelerometer can be mounted on the
moving base of the rotating coriolis gyroscope, but with its input axis tangential
to its direction of motion. Its outputs can be demodulated in similar fashion to
implement a two-axis accelerometer with zero effective bias error. The resulting
multisensor is a two-axis gyroscope and two-axis accelerometer.
9.1.5 Laser Technology and Lightwave Gyroscopes
Lasers are phase-coherent light sources. Phase-coherent light traveling around a
closed planar path will experience a slight phase shift each lap that is proportional to the inertial rotation rate of its planar path (the Sagnac effect). Lightwave
gyroscopes compare the phases of two phase-coherent beams traveling in opposite directions around the same path. All require mechanical stability to optical
tolerances, and all exhibit some level of angle random walk.
The two common types of laser gyroscopes are illustrated in Fig. 9.10 and
described below.
9.1.5.1 Ring Laser Gyroscopes (RLGs) Ring laser gyroscopes use a lasing
segment within a closed polygonal light path with mirrors at the corners. These
are effectively digital rate integrating gyroscopes, with the phase rate between
the counterrotating beams proportional to inertial rotation rate.
The first ring laser gyroscopes were developed in the 1960s, soon after the first
practical lasers had been developed. It would take about a decade to make them
(a)
(b)
Fig. 9.10 Laser gyroscopes: (a) ring laser gyro (RLG); (b) fiberoptic gyro.
INERTIAL SENSOR TECHNOLOGIES
329
practical, however. The problem was scattering of the counterrotating beams off
the RLG mirrors that causes the frequencies of the two beams to “lock in,”
creating a serious dead-zone near zero input rate. Most practical RLG designs
employ some sort of dithering to avoid the dead zone. A Zero Lock Gyroscope 4
(ZLG) uses a combination of laser frequency splitting and out-of-plane path
segments to eliminate lockin.
9.1.5.2 Fiberoptic Gyroscopes (FOGs) Fiberoptic gyroscopes use thousands
of turns of optical fiber to increase the phase sensitivity, multiplying the number
of turns by the phase shift per turn. A common external laser source can be used
for both beams. There are two basic strategies for sensing rotation rates:
Open-Loop FOGs Open-loop designs compare the phases of the two counterrotating beams. They are effectively rate gyroscopes, with the relative phase change
between the counterrotating light beams proportional to the inertial rotation rate
normal the plane of the lightpath. Those used in inertial navigation typically have
dynamic ranges in the order of 103 , sensitivities (i.e., minimum detectable inputs)
≥ 10−2 degrees per hour and bias stabilities in the order of 1 degree per hour
(or more) [196].
Closed-Loop Integrating FOGs (IFOGs) Closed-loop designs use feedback of
the output phase to a light modulator in the loop to null the output. These are
effectively rate integrating gyroscopes. They can have dynamic ranges in the
order of 106 , nonlinearity errors ≤ 10−5 , and bias stability in the order of 10−2
degrees per hour (or better) [196].
9.1.6 Vibratory Coriolis Gyroscopes (VCGs)
9.1.6.1 VCG Principles The first functional vibrational coriolis gyroscope was
probably the 1851 Foucault pendulum, in which the coriolis effect from the rotation of the earth causes the plane of pendulum mass oscillation to rotate. Modern
designs have replaced the gravity pendulum (which does not travel well) with
relatively high-frequency mechanical resonances, but the principle of operation
remains essentially the same. A mass particle resonating with velocity v0 cos (t)
fixed to a body rotating at rate ωinput would experience a time-varying coriolis
acceleration
"
#
acoriolis (t) = −2ωinput ⊗ v0 cos (t) ,
(9.12)
which is at the same frequency as the driving acceleration, but at right angles to
the particle velocity.
Equation 9.12 only describes the mass particle acceleration due to rotation.
There are dozens of electromechanical gyro designs using this mass particle
dynamic model to provide an output signal proportional to rotation rate [88].
4
“Zero Lock Gyro” and “ZLG” are trademarks of Northrop Grumman Corp.
330
INERTIAL NAVIGATION SYSTEMS
(a)
(b)
Fig. 9.11 Vibration modes of tuning fork gyroscope: (a) input mode; (b) output mode.
Transmission of vibrational energy to the supporting structure is a significant
error mechanism for most vibratory sensors. Two designs which do provide good
vibration isolation are the tuning fork gyro and the hemispherical resonator gyro.
9.1.6.2 Tuning Fork Gyroscope A tuning fork gyro illustrated in Fig. 9.11 is
driven in a balanced vibration mode with its tines coming together and apart in
unison (Fig. 9.11a), creating no vibrational stress in the handle. Its sensitive axis
is parallel to the handle. Rotation about this axis is orthogonal to the direction
of tine velocity, and the resulting coriolis acceleration will be in the direction
of ω ⊗ v, which excites the output vibration mode shown in Fig. 9.11b. This
unbalanced “twisting” mode will create a torque couple through the handle, and
some designs use a double-ended fork to transfer this mode to a second set of
output tines.
9.1.6.3 Hemispherical Resonator Gyroscope (HRG) In 1890, physicist G. H.
Bryan observed that the nodes of the resonant modes of wine glasses (produced
by rubbing a wet finger around the rim) precessed when the wine glass was
rotated. This became the basis for the hemispherical resonator gyroscope (HRG)
(also called the “wine glass” gyroscope), which uses resonant modes of bowlshaped structures on stems (similar to a wine glass) with vibratory displacements
normal to the edges of the bowl. When the device is rotated about its stem axis
(its input axis), the nodes of the vibration modes rotate around the stem at a
rate proportional to input rotation rate. Like many gyroscopes, they exhibit angle
random walk (< 10−3 degree per root hour in some HRG designs). They are
very rugged. They can be made to operate through the radiation bursts from
nuclear events, because mechanical resonances will persist and the phase change
will continue to accumulate during periods of high neutron fluence when the
drive and sensing electronics are turned off, then recover the accumulated angular
displacements after turnon. (Most momentum wheel gyroscopes can do this, too.)
331
INERTIAL SENSOR TECHNOLOGIES
9.1.7 MEMS Technology
Microelectromechanical systems (MEMS) evolved from silicon semiconductor
manufacturing in the late 1970s as an inexpensive mass manufacturing technology for sensors at sub-millimeter scales. At these scales, the ratio of surface area
to volume becomes enormous, and electrostatic forces are significant. Vibration
frequencies also scale up as size shrinks, and this makes vibratory coriolis gyroscopes very effective at MEMS scales. Electrostatic or piezoelectric forcing is
used in most MEMS vibratory coriolis gyroscopes.
9.1.7.1 Open-Loop MEMS Accelerometers Many MEMS accelerometers are
“open loop,” in the sense that no force-feedback control loop is used on the proof
mass. The cantilever beam accelerometer design illustrated in Fig. 9.7b senses
the strain at the root of the beam resulting from support of the proof mass under
acceleration load. The surface strain near the root of the beam will be proportional
to the applied acceleration. This type of accelerometer can be manufactured
relatively inexpensively using MEMS technologies, with a surface strain sensor
(e.g., piezoelectric capacitor or ion implanted piezoresistor) to measure surface
strain.
9.1.7.2 Rotational Vibratory Coriolis Gyroscope (RVCG) Many vibratory coriolis gyroscopes are MEMS devices. The rotational vibratory coriolis gyroscope
is a MEMS device first developed at C. S. Draper Laboratory in the 1980s, then
jointly with industry. It uses a momentum wheel coupled to a torsion spring and
driven by a rotational electrostatic “comb drive” at resonance to create sinusoidal
angular momentum in the wheel. If the device is turned about any axis in the
plane of the wheel, the coriolis effect will introduce sinusoidal tilting about
the orthogonal axis in the plane of the wheel, as illustrated in Fig. 9.12a. This
sinusoidal tilting is sensed by four capacitor sensors in close proximity to the
wheel underside, as illustrated in Fig. 9.12b. All the supporting electronics for
controlling the momentum wheel and extracting the angular rate measurements
fits on an application-specific integrate circuit (ASIC) that is only slightly bigger
than the active device.
(a)
(b)
Fig. 9.12 Rotational vibratory coriolis gyroscope: (a) Function; (b) sensing.
332
INERTIAL NAVIGATION SYSTEMS
9.2 INERTIAL SYSTEMS TECHNOLOGIES
9.2.1 Early Requirements
The development of inertial navigation in the United States started around 1950,
during the Cold War with the Soviet Union. Cold War weapons projects in the
United States that would need inertial navigation included the following:
1. Long-range bombers could not rely on radionavigation technologies of
World War II for missions into the Soviet Union, because they could easily
be jammed. Efforts started around 1950 and led by Charles Stark Draper in
the Servomechanisms Laboratory at MIT (now the C. S. Draper Laboratory)
were focused on developing airborne inertial navigation systems. The first
successful flights with INS5 across the United States had terminal errors in
the order of a few kilometers.
2. Hymnan Rickover began studying and promoting nuclear propulsion for the
U.S. Navy immediately after World War II, leading to the development of
the first nuclear submarine, Nautilus, launched in 1954. Nuclear submarines
would be able to remain submerged for months, and needed an accurate
navigation method that did not require exposing the submarine to airborne
radar detection.
3. The Navaho Project started in the early 1950s to develop a long-range airbreathing supersonic cruise missile to carry a 15,000-lb payload (the atomic
bomb of that period) 5500 miles with a terminal navigation accuracy of
about one nautical mile (1.85 km). The prime contractor, North American
Aviation, developed an INS for this system. The project was canceled in
1957, when nuclear weaponry and rocketry had improved to the point that
thermonuclear devices could be carried on rockets. However, a derivative
of the Navaho INS survived, and the nuclear submarine Nautilus used it
to successfully navigate under the arctic ice cap in 1958.
4. The intercontinental ballistic missiles (ICBMs) that replaced Navaho would
not have been practical without an INS to guide them. The INS made each
missile self-contained and able to control itself from liftoff without any
external aiding. Their accuracy requirements were not far from those for
Navaho.
The accuracy requirements of many of these systems was determined by the
radius of destruction of nuclear weapons.
9.2.2 Computer Technology
9.2.2.1 Early Computers Computers in 1950 used vacuum-tube technology,
and they tended to be expensive, large, heavy, and power-hungry. Computers
5
One of these systems did not use gimbal rings, but had the INS stabilized platform mounted on a
spherical bearing [129].
INERTIAL SYSTEMS TECHNOLOGIES
333
of that era occupied large rooms or whole buildings [160], they were extremely
slow by today’s standards, and the cost of each bit of memory was in the order
of a U.S. dollar in today’s money.
9.2.2.2 The Silicon Revolution Silicon transistors and integrated circuits began to revolutionize computer technology in the 1950s. Some early digital inertial navigation systems of that era used specialized digital differential analyzer
circuitry to doubly integrate acceleration. Later in the 1950s, INS computers
used magnetic core or magnetic drum memories. The Apollo moon missions
(1969–1972) used onboard computers with magnetic core memories, and magnetic memory technology would dominate until the 1970s. The cost per bit would
drop as low as a few cents before semiconductor memories came along. Memory
prices fell by several orders of magnitude in the next few decades.
9.2.2.3 Impact on INS Technology Faster, cheaper computers enabled the
development of strapdown inertial technology. Many vehicles (e.g., torpedos)
had been using strapdown gyroscopes for steering control since the early twentieth century, but now they could be integrated with accelerometers to make
a strapdown INS. This eliminated the need for expensive gimbals, but it also
required considerable progress in attitude estimation algorithms [23]. Computers
also enabled “modern” estimation and control, based on state space models. This
would have a profound effect on sensor integration capabilities for INS.
9.2.3 Early Strapdown Systems
A gimbaled INS was carried on each of nine Apollo command modules from
the earth to the moon and back between December 1968 and December 1972,
but a strapdown INS was carried on each of the six6 Lunar Excursion Modules
(LEMs) that shuttled two astronauts from lunar orbit to the lunar surface and
back.
By the mid-1970s, strapdown systems could demonstrate “medium” accuracy (1 nmi/h CEP). A strapdown contender for the U.S. Air Force Standard
Navigator contract in 1979 was the Autonetics7 N73 navigator, using electrostatic gyroscopes, electromagnetic accelerometers and a navigation computer with
microprogrammed instructions and nonvolatile“magnetic wire” memory. In that
same time period, the first INSs with ring laser gyros appeared in commercial
aircraft.
A few years later, GPS appeared.
6
Two additional LEMs were carried to the moon but did not land there. The Apollo 13 LEM did
not make its intended lunar landing, but played a far more vital role in crew survival.
7
Then a division of Rockwell International, now part of the Boeing Company.
334
INERTIAL NAVIGATION SYSTEMS
9.2.4 INS and GNSS
9.2.4.1 Advantages of INS The main advantages of inertial navigation over
other forms of navigation are as follows:
1. It is autonomous and does not rely on any external aids or on visibility
conditions. It can operate in tunnels or underwater as well as anywhere
else.
2. It is inherently well suited for integrated navigation, guidance, and control
of the host vehicle. Its IMU measures the derivatives of the variables to be
controlled (e.g., position, velocity, attitude).
3. It is immune to jamming and inherently stealthy. It neither receives nor
emits detectable radiation and requires no external antenna that might be
detectable by radar.
9.2.4.2 Disadvantages of INS These include the following:
1. Mean-squared navigation errors increase with time.
2. Cost, including
(a) Acquisition cost, which can be an order of magnitude (or more) higher
than that of GNSS receivers.
(b) Operations cost, including the crew actions and time required for initializing position and attitude. Time required for initializing INS attitude
by gyrocompass alignment is measured in minutes. Time to first fix for
GNSS receivers is measured in seconds.
(c) Maintenance cost. Electromechanical avionics systems (e.g., INS) tend
to have higher failure rates and repair costs than do purely electronic
avionics systems (e.g., GPS).
3. Size and weight, which have been shrinking:
(a) Earlier INS systems weighed tens to hundreds of kilograms.
(b) Later “mesoscale” INSs for integration with GPS weighed 1–10 kgms.
(c) Developing MEMS sensors are targeted for gram-size systems.
INS weight has a multiplying effect on vehicle system design, because it
requires increased structure and propulsion weight as well.
4. Power requirements, which have been shrinking along with size and weight
but are still higher than those for GPS receivers.
5. Temperature control and heat dissipation, which is proportional to (and
shrinking with) power consumption.
9.2.4.3 Competition from GPS Since the 1970s, U.S. commercial air carriers
have been required by FAA regulations to carry two INS systems on all flights
over water. The cost of these two systems is on the order of 105 U.S. dollars. The
relatively high cost of INS was one of the factors leading to the development of
INERTIAL SENSOR MODELS
335
GPS. After deployment of GPS in the 1980s, the few remaining applications for
“standalone” (i.e., unaided) INS include submarines, which cannot receive GPS
signals while submerged, and intercontinental ballistic missiles, which cannot
rely on GPS availability in time of war.
9.2.4.4 Synergism with GNSS GNSS integration has not only made inertial navigation perform better, it has made it cost less. Sensor errors that were
unacceptable for stand-alone INS operation became acceptable for integrated
operation, and the manufacturing and calibration costs for removing these errors
could be eliminated. Also, new low-cost MEMS manufacturing methods could
be applied to meet the less stringent sensor requirements for integrated
operation.
The use of integrated GNSS/INS for mapping the gravitational field near the
earth’s surface has also enhanced INS performance by providing more detailed
and accurate gravitational models.
Inertial navigation also benefits GNSS performance by carrying the navigation
solution during loss of GNSS signals and allowing rapid reacquisition when
signals become available.
Integrated systems have found applications that neither GNSS nor INS could
perform alone. These include low-cost systems for precise autonomous control of
vehicles operating at the surface of the earth, including automatic landing systems
for aircraft and autonomous control of surface mining equipment, surface grading
equipment, and farm equipment.
9.3 INERTIAL SENSOR MODELS
Mathematical models for how inertial sensors perform are used throughout the
INS development cycle. They include the following:
1. Models used in designing the sensors to meet specified performance metrics.
2. Models used to calibrate and compensate for fixed errors, such as scale factor and bias variations. The extreme performance requirements for inertial
sensors cannot be met within manufacturing tolerances. Fortunately, the
last few orders of magnitude improvement in performance can be achieved
through calibration. These models are generally of two types:
(a) Models based on engineering data and the principles of physics, such
as the models carried over from the design trade offs. These models
generally have a known cause for each observed effect.
(b) Abstract, general-purpose mathematical models such as polynomials,
used to fit observed error data in such a way that the sensor output
errors can be effectively corrected.
336
INERTIAL NAVIGATION SYSTEMS
3. Error models used in GNSS/INS integration for determining the optimal weighting (Kalman gain) in combining GNSS and INS navigation
data.
4. Sensor models used in GNSS/INSS integration for recalibrating the INS
continuously while GNSS data are available. This approach allows the INS
to operate more accurately during periods of GNSS signal outage.
9.3.1 Zero-Mean Random Errors
These are the standard types of error models used in Kalman filtering, described
in the previous chapter.
9.3.1.1 White Sensor Noise This is usually lumped together under “electronic
noise,” which may come from power supplies, intrinsic noise in semiconductor
devices, or from quantization errors in digitization.
9.3.1.2 Exponentially Correlated Noise Temperature sensitivity of sensor bias
will often look like a time-varying additive noise source, driven by external
ambient temperature variations or by internal heat distribution variations.
9.3.1.3 Random-Walk Sensor Errors Random-walk errors are characterized
by variances that grow linearly with time and power spectral densities that fall
off as 1/frequency2 (i.e., 20 dB per decade; see Section 8.5.1.2).
There are specifications for random walk noise in inertial sensors, but mostly
for the integrals of their outputs, and not in the outputs themselves. For example,
the “angle random walk” from a rate gyroscope is equivalent to white noise
in the angular rate outputs. In similar fashion, the integral of white noise in
accelerometer outputs would be equivalent to “velocity random walk.”
The random-walk error model has the form
k = k−1 + wk−1
def σk2 = E k2
2
2
+ E wk−1
= σk−1
= σ02 + k Qw for time-invariant systems,
def
Qw = Ek wk2 .
The value of Qw will be in units of squared-error per discrete time step t.
Random-walk error sources are usually specified in terms of standard deviations,
that is, error units per square-root of time unit.√ Gyroscope angle random walk
errors, for example, might be specified in deg/ h. Most navigation-grade gyroscopes (including
√ RLG, HRG, IFOG) have angle random-walk errors in the order
of 10−3 deg/ h or less.
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INERTIAL SENSOR MODELS
9.3.1.4 Harmonic Noise Temperature control schemes (including building
HVAC systems) often introduce cyclical errors due to thermal transport lags, and
these can cause harmonic errors in sensor outputs, with harmonic periods that
scale with device dimensions. Also, suspension and structural resonances of host
vehicles introduce harmonic accelerations, which can excite acceleration-sensitive
error sources in sensors.
9.3.1.5 “1/f ” Noise This noise is characterized by power spectral densities
that fall off as 1/f , where f is frequency. It is present in most electronic devices,
its causes are not well understood, and it is usually modeled as some combination
of white noise and random walk.
9.3.2 Systematic Errors
These are errors that can be calibrated and compensated.
9.3.2.1 Sensor-Level Models These are sensor output errors in addition to
additive zero-mean white noise and time-correlated noise considered above. The
same models apply to accelerometers and gyroscopes. Some of the more common
types of sensor errors are illustrated in Fig. 9.13:
(a) Bias, which is any nonzero sensor output when the input is zero
(b) Scale factor error, often resulting from aging or manufacturing tolerances
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 9.13 Common input/output error types: (a) bias; (b) scale Factor; (c) nonlinearity;
(d) ± asymmetry; (e) dead zone; (f ) quantization.
338
INERTIAL NAVIGATION SYSTEMS
(c) Nonlinearity, which is present in most sensors to some degree
(d) Scale factor sign asymmetry (often from mismatched push–pull amplifiers)
(e) A dead zone, usually due to mechanical stiction or lockin [for a ring laser
gyroscope (RLG)]
(f ) Quantization error, inherent in all digitized systems. It may not be zeromean when the input is held constant, as it could be under calibration
conditions.
We can recover the sensor input from the sensor output so long as the input/output
relationship is known and invertible. Dead-zone errors and quantization errors are
the only ones shown with this problem. The cumulative effects of both types (dead
zone and quantization) often benefit from zero-mean input noise or dithering.
Also, not all digitization methods have equal cumulative effects. Cumulative
quantization errors for sensors with frequency outputs are bounded by ±0.5 least
significant bit (LSB) of the digitized output, but the variance of cumulative errors
from independent sample-to-sample A/D conversion errors can grow linearly with
time.
9.3.2.2 ISA-Level Models For a cluster of N ≥ 3 gyroscopes or accelerometers, the effects of individual biases, scale factors, and input axis misalignments
can be modeled by an equation of the form
zinput = Mscale factor & misalignment zoutput + bz ,
3×1
3×N
N ×1
(9.13)
3×1
where the components of the vector bz are three aggregate biases, the components
of the zinput and zoutput vectors are the sensed values (accelerations or angular
rates) and output values from the sensors, respectively, and the elements mij of
the “scale factor and misalignment matrix” M represent the individual scale factor
deviations and input axis misalignments as illustrated in Fig. 9.14 for N = 3
orthogonal input axes. The larger arrows in the figure represent the nominal
input axis directions (labeled #1, #2, and #3) and the smaller arrows (labeled
mij ) represent the directions of scale factor deviations (i = j ) and input axis
misalignments (i = j ).
Equation 9.13 is in “compensation form”; that is, it represents the inputs as
functions of the outputs. The corresponding “error form” is
zoutput = M† zinput − bz
(9.14)
where † represents the Moore–Penrose matrix inverse.
The compensation model of Eq. 9.13 is the one used in system implementation
for compensating sensor outputs using a single constant matrix M and vector bz .
339
INERTIAL SENSOR MODELS
Fig. 9.14
Directions of modeled sensor cluster errors.
9.3.2.3 Calibrating Sensor Biases, Scale Factors, and Misalignments Calibration is the process of taking sensor data to characterize sensor inputs as
functions of sensor outputs.
" It amounts to# estimating the values of M and bz ,
given input-output pairs zinput, k , zoutput, k , where zinput, k is known from controlled calibration conditions (e.g., for accelerometers, from the direction and
magnitude of gravity, or from conditions on a shake table or centrifuge) and
zoutput, k is measured under these conditions.
Calibration Data Processing The full set of data under K sets of calibration
conditions yields a system of 3K linear equations
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
z1, input, 1
z2, input, 1
z3, input, 1
..
.
z3, input, K
3K knowns
⎤
⎡
⎥ ⎢
⎥ ⎢
⎥ ⎢
⎥=⎢
⎥ ⎢
⎥ ⎢
⎦ ⎣
z1, output, 1
0
0
..
.
0
z2, output, 1 z3, output, 1
0
0
0
0
..
..
.
.
0
0
Z, a 3K×(3N+3) matrix of knowns
⎤ ⎡
m1, 1
··· 0
⎢ m1, 2
··· 0 ⎥
⎥ ⎢
⎥ ⎢
· · · 1 ⎥ ⎢ m1, 3
⎥ ⎢
. ⎥ ⎢ .
..
. .. ⎦ ⎣ ..
b3, z
··· 1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
3N+3 unknowns
(9.15)
in the 3N unknown parameters mi, j (the elements of the matrix M) and 3
unknown parameters bi, z (rows of the 3-vector bz ), which will be overdetermined
for K > N + 1. In that case, the system of linear equations may be solvable for
340
INERTIAL NAVIGATION SYSTEMS
the 3(N + 1) calibration parameters by using the method
⎡
⎤
⎡
m1, 1
z1, input, 1
⎢ m1, 2 ⎥
⎢ z2, input, 1
⎢
⎥
⎢
⎢ m1, 3 ⎥ " T #−1 T ⎢ z3, input, 1
Z ⎢
⎢
⎥= Z Z
⎢ .. ⎥
⎢
..
⎣ . ⎦
⎣
.
b3, z
of least-squares
⎤
⎥
⎥
⎥
⎥,
⎥
⎦
(9.16)
z3, input, K
provided that the matrix ZT Z is nonsingular.
Calibration Parameters The values of M and bz determined in this way are
called calibration parameters.
Estimation of the calibration parameters can also be done using Kalman filtering, a by product of which would be the covariance matrix of calibration
parameter uncertainty. This matrix is also useful in modeling system-level performance.
9.3.3 Other Calibration Parameters
9.3.3.1 Nonlinearities Sensor input/output nonlinearities are generally modeled by polynomials:
zinput =
N
.
i
ai zoutput
,
(9.17)
i=0
where the first two parameters a0 = bias and a1 = scale factor. The polynomial
input output model of Eq. 9.17 is linear in the calibration parameters, so they
can still be calibrated using a system of linear equations—as was used for scale
factor and bias.
The generalization of Eq. 9.17 to vector-valued inputs and outputs includes
all the cross-power terms between different sensors, but it also includes multidimensional data structures in place of the scalar parameters ai . Such a model
would, for example, include the acceleration sensitivities of gyroscopes and the
rotation rate sensitivities of accelerometers.
9.3.3.2 Sensitivities to Other Measurable Conditions Most inertial sensors
are also thermometers, and part of the art of sensor design is to minimize their
temperature sensitivities. Other bothersome sensitivities include acceleration sensitivity of gyroscopes and rotation rate sensitivities of accelerometers (already
mentioned above).
Compensating for temperature sensitivity requires adding one or more thermometers to the sensors and taking calibration data over the expected operational
temperature range, but the other sensitivities can be “cross-compensated” by using
the outputs of the other inertial sensors. The accelerometer outputs can be used
in compensating for acceleration sensitivities of gyroscopes, and the gyro outputs
can be used in compensating for angular rate sensitivities of accelerometers.
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INERTIAL SENSOR MODELS
9.3.3.3 Other Accelerometer Models
Centrifugal Acceleration Effects Accelerometers have input axes defining the
component(s) of acceleration that they measure. There is a not-uncommon superstition that these axes must intersect at a point to avoid some unspecified error
source. That is seldom the case, but there can be some differential sensitivity
to centrifugal accelerations due to high rotation rates and relative displacements
between accelerometers. The effect is rather weak, but not always negligible. It
is modeled by the equation
ai, centrifugal = ω2 ri ,
(9.18)
where ω is the rotation rate and ri is the displacement component along the input
axis from the axis of rotation to the effective center of the accelerometer. Even
manned vehicles can rotate at ω ≈ 3 rad/s, which creates centrifugal accelerations
of about 1 g at ri = 1 m and 0.001 g at 1 mm. The problem is less significant,
if not insignificant, for MEMS-scale accelerometers that can be mounted within
millimeters of one another.
Center of Percussion Because ω can be measured, sensed centrifugal accelerations can be compensated, if necessary. This requires designating some reference
point within the instrument cluster and measuring the radial distances and directions to the accelerometers from that reference point. The point within the
accelerometer required for this calculation is sometimes called its “center of
percussion.” It is effectively the point such that rotations about all axes through
the point produce no sensible centrifugal accelerations, and that point can be
located by testing the accelerometer at differential reference locations on a rate
table.
Angular Acceleration Sensitivities Pendulous accelerometers are sensitive to
angular acceleration about their hinge lines, with errors equal to ω̇hinge , where
ω̇ is the angular acceleration in radians per second squared and hinge is the displacement of the accelerometer proof mass (at its center of mass) from the hinge
line. This effect can reach the 1-g level for hinge ≈ 1 cm and ω̇ ≈ 103 rad/s2 ,
but these extreme conditions are rarely persistent enough to matter in most applications.
9.3.4 Calibration Parameter Instability
INS calibration parameters are not always exactly constant. Their values can
change over the operational life of the INS. Specifications for calibration stability
generally divide these calibration parameter variations into two categories: (1)
changes from one system turnon to the next and (2) slow “parameter drift” during
operating periods.
9.3.4.1 Calibration Parameter Changes Between Turn-ons These are changes that occur between a system shutdown and the next startup. They may be
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INERTIAL NAVIGATION SYSTEMS
caused by temperature transients during shutdowns and turnons, or by what is
termed “aging.” They are generally considered to be independent from turn-on
to turnon, so the model for the covariance of calibration errors for the kth turnon
would be of the form
Pcalib, k = Pcalib, k−1 + P calib ,
(9.19)
where P calib is the covariance of turnon-to-turnon parameter changes. The initial value Pcalib, 0 at the end of calibration is usually determinable from error
covariance analysis of the calibration process. Note that this is the covariance
model for a random walk, the covariance of which grows without bound.
9.3.4.2 Calibration Parameter Drift This term applies to changes that occur in
the operational periods between startups and shutdowns. The calibration parameter uncertainty covariance equation has the same form as Eq. 9.19, but with
P calib. now representing the calibration parameter drift in the time interval
t = tk − tk−1 between successive discrete times within an operational period.
Detecting Error Trends Incipient sensor failures can sometimes be predicted by
observing the variations over time of the sensor calibration parameters. One of the
advantages of tightly coupled GNSS/INS integration is that INS sensors can be
continuously calibrated all the time that GNSS data are available. System health
monitoring can then include tests for the trends of sensor calibration parameters,
setting threshold conditions for failing the INS system, and isolating a likely set
of causes for the observed trends.
9.3.5 Auxilliary Sensors
9.3.5.1 Attitude Sensors Nongyroscopic attitude sensors can also be used as
aids in inertial navigation. These include the following:
Magnetic sensors are used primarily for coarse heading initialization, to speed
up INS alignment.
Star trackers are used primarily for space-based or near-space applications. The
U-2 spy plane, for example, used an inertial-platform-mounted star tracker
to maintain INS alignment on long flights.
Optical alignment systems used on some space launch systems. Some use Porro
prisms mounted on the inertial platform to maintain optical line-of-sight
reference through ground-based theodolites to reference directions at the
launch complex.
GNSS receiver systems using antenna arrays and carrier phase interferometry
have been used for heading initialization in artillery fire control systems,
for example, but the same technology could be used for INS aiding. The
systems generally have baselines in the order of several meters, which
could limit their utility for some host vehicles.
SYSTEM IMPLEMENTATION MODELS
343
9.3.5.2 Altitude Sensors These include barometric altimeters and radar altimeters. Without GNSS inputs, some sort of altitude sensor is required to stabilize
INS vertical channel errors.
9.4 SYSTEM IMPLEMENTATION MODELS
9.4.1 One-Dimensional Example
This example is intended as an introduction to INS technology for the uninitiated.
It illustrates some of the key properties of inertial sensors and inertial system
implementations.
If we all lived in one-dimensional “line land,” then there could be no rotation and no need for gyroscopes. In that case, an INS would need only one
accelerometer and navigation computer (all one-dimensional, of course), and its
implementation would be as illustrated in Fig. 9.15 (in two dimensions), where
the dependent variable x denotes position in one dimension and the independent
variable t is time.
This implementation for one dimension has many features common to implementations for three dimensions:
1. Accelerometers cannot measure gravitational acceleration. An accelerometer effectively measures the force acting on its proof mass to make it
follow its mounting base, which includes only nongravitational accelerations applied through physical forces acting on the INS through its host
vehicle. Satellites, which are effectively in free fall, experience no sensible
accelerations.
2. Accelerometers have scale factors, which are the ratios of input acceleration units to output signal magnitude units (e.g., meters per second squared
per volt). The signal must be rescaled in the navigation computer by multiplying by this scale factor.
Fig. 9.15 INS functional implementation for a one-dimensional world.
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INERTIAL NAVIGATION SYSTEMS
3. Accelerometers have output errors, including
(a) Unknown constant offsets, also called biases.
(b) Unknown constant scale factor errors.
(c) Unknown sensor input axis misalignments.
(d) Unknown nonconstant variations in bias and scale factor.
(e) Unknown zero-mean additive noise on the sensor outputs, including
quantization noise and electronic noise. The noise itself is not predictable, but its statistical properties may be used in Kalman filtering
to estimate drifting scale factor and biases.
4. Gravitational accelerations must be modeled and calculated in the navigational computer, then added to the sensed acceleration (after error and
scale compensation) to obtain the net acceleration ẍ of the INS.
5. The navigation computer must integrate acceleration to obtain velocity.
This is a definite integral and it requires an initial value, ẋ(t0 ); that is, the
INS implementation in the navigation computer must start with a known
initial velocity.
6. The navigation computer must also integrate velocity (ẋ) to obtain position
(x). This is also a definite integral and it also requires an initial value,
x(t0 ). The INS implementation in the navigation computer must start with
a known initial location, too.
Inertial navigation in three dimensions requires more sensors and more signal
processing than in one dimension, and it also introduces more possibilities for
implementation (e.g., gimbaled or strapdown)
9.4.2 Initialization and Alignment
9.4.2.1 Navigation Initialization INS initialization is the process of determining initial values for system position, velocity, and attitude in navigation
coordinates. INS position initialization ordinarily relies on external sources such
as GNSS or manual entry by crew members. INS velocity initialization can be
accomplished by starting when it is zero (i.e., the host vehicle is not moving) or
(for vehicles carried in or on other vehicles) by reference to the carrier velocity.
(See alignment method 3 below.) INS attitude initialization is called alignment.
9.4.2.2 Sensor Alignment INS alignment is the process of aligning the stable
platform axes parallel to navigation coordinates (for gimbaled systems) or that
of determining the initial values of the coordinate transformation from sensor
coordinates to navigation coordinates (for strapdown systems).
Alignment Methods Four basic methods for INS alignment are as follows:
1. Optical alignment, using either of the following:
(a) Optical line-of-sight reference to a ground-based direction (e.g., using
a ground-based theodolite and a mirror on the platform). Some space
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SYSTEM IMPLEMENTATION MODELS
boosters have used this type of optical alignment, which is much faster
and more accurate than gyrocompass alignment. Because it requires a
stable platform for mounting the mirror, it is applicable only to gimbaled systems.
(b) An onboard star tracker, used primarily for alignment of gimbaled or
strapdown systems in space or near-space (e.g., above all clouds).
2. Gyrocompass alignment of stationary vehicles, using the sensed direction of
acceleration to determine the local vertical and the sensed direction of rotation to determine north. Latitude can be determined by the angle between
the earth rotation vector and the horizontal, but longitude must be determined by other means and entered manually or electronically. This method
is inexpensive but the most time-consuming (several minutes, typically).
3. Transfer alignment in a moving host vehicle, using velocity matching with
an aligned and operating INS. This method is typically several times faster
than gyrocompass alignment, but it requires another INS on the host vehicle and it may require special maneuvering of the host vehicle to attain
observability of the alignment variables. It is commonly used for in-air INS
alignment for missiles launched from aircraft and for on-deck INS alignment for aircraft launched from carriers. Alignment of carrier-launched
aircraft may also use the direction of the velocity impulse imparted by the
steam catapult.
4. GNSS-aided alignment using position matching with GNSS to estimate the
alignment variables. It is an integral part of integrated GNSS/INS implementations. It does not require the host vehicle to remain stationary during
alignment, but there will be some period of time after turnon (a few minutes,
typically) before system navigation errors settle to acceptable levels.
Gyrocompass alignment is the only one of these that requires no external aiding. Gyrocompass alignment is not necessary for integrated GNSS/INS, although
many INSs may already be configured for it.
INS Gyrocompass Alignment Accuracy A rough rule of thumb for gyrocompass
alignment accuracy is
2
2
σgyrocompass
> σacc
+
2
σgyro
152 cos2 (φgeodetic )
,
(9.20)
where σgyrocompass is the minimum achievable RMS alignment error in radians,
σacc is the RMS accelerometer accuracy in g values, σgyro is the RMS gyroscope
accuracy in degrees per hour, 15◦ per hour is the rotation rate of the earth, and
φgeodetic is the latitude at which gyrocompassing is performed.
Alignment accuracy is also a function of the time allotted for it, and the
time required to achieve a specified accuracy is generally a function of sensor
error magnitudes (including noise) and the degree to which the vehicle remains
stationary.
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INERTIAL NAVIGATION SYSTEMS
Gimbaled INS Gyrocompass Alignment Gyrocompass alignment for gimbaled
systems is a process for aligning the inertial platform axes with the navigation
coordinates using only the sensor outputs while the host vehicle is essentially
stationary. For systems using ENU navigation coordinates, for example, the platform can be tilted until two of its accelerometer inputs are zero, at which time
both input axes will be horizontal. In this locally leveled orientation, the sensed
rotation axis will be in the north–up plane, and the platform can be slewed about
the vertical axis to null the input of one of its horizontal gyroscopes, at which
time that gyroscope input axis will point east–west. That is the basic concept
used for gyrocompass alignment, but practical implementation requires filtering8 to reduce the effects of sensor noise and unpredictable zero-mean vehicle
disturbances due to loading activities and/or wind gusts.
Strapdown INS Gyrocompass Alignment Gyrocompass alignment for strapdown
systems (see Fig. 9.16) is a process for “virtual alignment” by determining the
sensor cluster attitude with respect to navigation coordinates using only the sensor
outputs while the system is essentially stationary.
If the sensor cluster could be firmly affixed to the earth and there were no
sensor errors, then the sensed acceleration vector aoutput in sensor coordinates
would be in the direction of the local vertical, the sensed rotation vector ωoutput
would be in the direction of the earth rotation axis, and the unit column vectors
1U =
aoutput
,
|aoutput |
(9.21)
1N =
woutput − (1TU woutput )1U
,
|woutput − (1TU woutput )1U |
(9.22)
1E = 1N ⊗ 1U
(9.23)
Fig. 9.16 Outputs (in angular brackets) of simple strapdown INS.
8
The vehicle dynamic model used for gyrocompass alignment filtering can be “tuned” to include the
major resonance modes of the vehicle suspension.
SYSTEM IMPLEMENTATION MODELS
347
would define the initial value of the coordinate transformation matrix from sensorfixed coordinates to ENU coordinates:
T
Csensor
ENU = [1E |1N |1U ] .
(9.24)
In practice, the sensor cluster is usually mounted in a vehicle that is not moving over the surface of the earth but may be buffeted by wind gusts and/or
disturbed by fueling and payload handling. Gyrocompassing then requires some
amount of filtering to reduce the effects of vehicle buffeting and sensor noise.
The gyrocompass filtering period is typically on the order of several minutes for
a medium-accuracy INS but may continue for hours or days for high-accuracy
systems.
9.4.3 Earth Models
Inertial navigation and satellite navigation require models for the shape, gravity,
and rotation of the earth.
9.4.3.1 Navigation Coordinates Descriptions of the major coordinates used
in inertial navigation and GNSS/INS integration are described in Appendix C
(on the CD). These include coordinate systems used for representing the trajectories of GNSS satellites and user vehicles in the near-earth environment and for
representing the attitudes of host vehicles relative to locally level coordinates,
including the following:
1. Inertial coordinates:
(a) Earth-centered inertial (ECI), with origin at the center of mass of the
earth and principal axes in the directions of the vernal equinox (defined
in Section C.2.1) and the rotation axis of the earth.
(b) Satellite orbital coordinates, as illustrated in Fig. C.4 and used in GPS
ephemerides.
2. Earth-fixed coordinates:
(a) Earth-centered, earth-fixed (ECEF), with origin at the center of mass
of the earth and principal axes in the directions of the prime meridian
(defined in Section C.3.5) at the equator and the rotation axis of the
earth.
(b) Geodetic coordinates, based on an ellipsoid model for the shape of
the earth. Longitude in geodetic coordinates is the same as in ECEF
coordinates, and geodetic latitude as defined as the angle between the
equatorial plane and the normal to the reference ellipsoid surface.
Geodetic latitude can differ from geocentric latitude by as much as
12 arc minutes, equivalent to about 20 km of northing distance.
(c) Local tangent plane (LTP) coordinates, also called “locally level coordinates,” essentially representing the earth as being locally flat. These
coordinates are particularly useful from a human factors standpoint for
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INERTIAL NAVIGATION SYSTEMS
representing the attitude of the host vehicle and for representing local
directions. They include
i. East–north–up (ENU), shown in Fig. C.7
ii. North–east–down (NED), which can be simpler to relate to vehicle
coordinates and
iii. Alpha wander, rotated from ENU coordinates through an angle α
about the local vertical (see Fig. C.8)
3. Vehicle-fixed coordinates:
(a) Roll–pitch–yaw (RPY) (axes shown in Fig. C.9).
Transformations between these different coordinate systems are important for
representing vehicle attitudes, for resolving inertial sensor outputs into inertial
navigation coordinates, and for GNSS/INS integration. Methods used for representing and implementing coordinate transformations are also presented in
Appendix C, Section C.4.
9.4.3.2 Earth Rotation Our earth is the mother of all clocks. It has given us
the time units of days, hours, minutes, and seconds we use to manage our lives.
Not until the discovery of atomic clocks based on hyperfine transitions were we
able to observe the imperfections in our earth clock. Despite these, we continue
to use earth rotation as our primary time reference, adding or subtracting leap
seconds to atomic clocks to keep them synchronized to the rotation of the earth.
These time variations are significant for GNSS navigation, but not for inertial
navigation.
WGS84 Earthrate Model The value of earthrate in the World Geodetic System
1984 (WGS84) earth model used by GPS is 7, 292, 115, 167 × 10−14 radians per
second, or about 15.04109◦ /h. This is its sidereal rotation rate with respect to
distant stars. Its mean rotation rate with respect to the nearest star (our sun), as
viewed from the rotating earth, is 15◦ /h, averaged over one year.
9.4.3.3 GPS Gravity Models Accurate gravity modeling is important for maintaining ephemerides for GPS satellites, and models developed for GPS have been
a boon to inertial navigation as well. However, spatial resolution of the earth gravitational field required for GPS operation may be a bit coarse compared to that for
precision inertial navigation, because the GPS satellites are not near the surface
and the mass concentration anomalies that create surface gravity anomalies. GPS
orbits have very little sensitivity to surface-level undulations of the gravitational
field on the order of 100 km or less, but these can be important for high-precision
inertial systems.
9.4.3.4 INS Gravity Models Because an INS operates in a world with gravitational accelerations it is unable to sense and unable to ignore, it must use a
reasonably faithful model of gravity.
SYSTEM IMPLEMENTATION MODELS
349
Gravity models for the earth include centrifugal acceleration due to rotation of
the earth as well as true gravitational accelerations due to the mass distribution
of the earth, but they do not generally include oscillatory effects such as tidal
variations.
Gravitational Potential Gravitational potential is defined to be zero at a point
infinitely distant from all massive bodies and to decrease toward massive bodies
such as the earth. That is, a point at infinity is the reference point for gravitational
potential.
In effect, the gravitational potential at a point in or near the earth is defined
by the potential energy lost by a unit of mass falling to that point from infinite
altitude. In falling from infinity, potential energy is converted to kinetic energy,
2
mvescape
/2, where vescape is the escape velocity. Escape velocity at the surface of
the earth is about 11 km/s.
Gravitational Acceleration Gravitational acceleration is the negative gradient of
gravitational potential. Potential is a scalar function, and its gradient is a vector.
Because gravitational potential increases with altitude, its gradient points upward
and the negative gradient points downward.
Equipotential Surfaces An equipotential surface is a surface of constant gravitational potential. If the ocean and atmosphere were not moving, then the surface of
the ocean at static equilibrium would be an equipotential surface. Mean sea level
is a theoretical equipotential surface obtained by time averaging the dynamic
effects.
Ellipsoid Models for Earth Geodesy is the process of determining the shape of
the earth, often using ellipsoids as approximations of an equipotential surface
(e.g., mean sea level), as illustrated in Fig. 9.17. The most common ones are
ellipsoids of revolution, but there are many reference ellipsoids based on different
survey data. Some are global approximations and some are local approximations.
The global approximations deviate from a spherical surface by about ±10 km,
and locations on the earth referenced to different ellipsoidal approximations can
differ from one another by 102 − 103 m.
Geodetic latitude on a reference ellipsoid is measured in terms of the angle
between the equator and the normal to the ellipsoid surface, as illustrated in
Fig. 9.17.
Orthometric height is measured along the (curved) plumbline.
WGS84 Ellipsoid The World Geodetic System (WGS) is an international standard for navigation coordinates. WGS84 is a reference earth model released in
1984. It approximates mean sea level by an ellipsoid of revolution with its rotation axis coincident with the rotation axis of the earth, its center at the center of
mass of the earth, and its prime meridian through Greenwich. Its semimajor axis
(equatorial radius) is defined to be 6,378,137 m, and its semiminor axis (polar
radius) is defined to be 6,356,752.3142 m.
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INERTIAL NAVIGATION SYSTEMS
Fig. 9.17 Equipotential surface models for earth.
Geoid Models Geoids are approximations of mean sea-level orthometric height
with respect to a reference ellipsoid. Geoids are defined by additional higher
order shapes, such as spherical harmonics of height deviations from an ellipsoid,
as illustrated in Fig. 9.17. There are many geoid models based on different data,
but the more recent, most accurate models depend heavily on GPS data. Geoid
heights deviate from reference ellipsoids by tens of meters, typically.
The WGS84 geoid heights vary about ±100 m from the reference ellipsoid.
As a rule, oceans tend to have lower geoid heights and continents tend to have
higher geoid heights. Coarse 20-m contour intervals are plotted versus longitude
and latitude in Fig. 9.18, with geoid regions above the ellipsoid shaded gray.
9.4.3.5 Longitude and Latitude Rates The second integral of acceleration in
locally level coordinates should result in the estimated vehicle position. This
integral is somewhat less than straightforward when longitude and latitude are
the preferred horizontal location variables.
The rate of change of vehicle altitude equals its vertical velocity, which is
the first integral of net (i.e., including gravity) vertical acceleration. The rates of
change of vehicle longitude and latitude depend on the horizontal components of
vehicle velocity, but in a less direct manner. The relationship between longitude
and latitude rates and east and north velocities is further complicated by the
oblate shape of the earth.
The rates at which these angular coordinates change as the vehicle moves
tangent to the surface will depend on the radius of curvature of the reference
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SYSTEM IMPLEMENTATION MODELS
Fig. 9.18
WGS84 geoid heights.
surface model, which is an ellipsoid of revolution for the WGS84 model. Radius
of curvature can depend on the direction of travel, and for an ellipsoidal model
there is one radius of curvature for north–south motion and another radius of
curvature for east–west motion.
Meridional Radius of Curvature The radius of curvature for north–south motion
is called the “meridional” radius of curvature, because north–south travel is along
a meridian (i.e., line of constant longitude). For an ellipsoid of revolution (the
WGS84 model), all meridians have the same shape, which is that of the ellipse
that was rotated to produce the ellipsoidal surface model. The tangent circle
with the same radius of curvature as the ellipse is called the “osculating” circle (osculating means “kissing”). As illustrated in Fig. 9.19 for an oblate earth
model, the radius of the meridional osculating circle is smallest where the geocentric radius is largest (at the equator), and the radius of the osculating circle
is largest where the geocentric radius is smallest (at the poles). The osculating circle lies inside or on the ellipsoid at the equator and outside or on the
ellipsoid at the poles and passes through the ellipsoid surface for latitudes in
between.
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INERTIAL NAVIGATION SYSTEMS
Fig. 9.19
Ellipse and osculating circles.
The formula for meridional radius of curvature as a function of geodetic
latitude (φ geodetic ) is
rM =
b2
a[1 − e2 sin2 (φgeodetic )]3/2
="
a(1 − e2 )
1 − e2 sin2 (φgeodetic )
#3/2 ,
(9.25)
(9.26)
where a is the semimajor axis of the ellipse, b is the semiminor axis, and e2 =
(a 2 –b2 )/a 2 is the eccentricity squared.
Geodetic Latitude Rate The rate of change of geodetic latitude as a function of
north velocity is then
dφgeodetic
vN
=
,
dt
rM + h
(9.27)
and geodetic latitude can be maintained as the integral
φgeodetic (tnow ) = φgeodetic (tstart )
* tnow
vN(t) dt
, (9.28)
+
2
2
2
3/2 + h(t)]
tstart a(1 − e )/{1 − e sin [φgeodetic (t)]}
SYSTEM IMPLEMENTATION MODELS
353
Fig. 9.20 Transverse osculating circle.
where h(t) is height above (+) or below (–) the ellipsoid surface and φ geodetic (t)
will be in radians if vN (t) is in meters per second and rM (t) and h(t) are in
meters.
Transverse Radius of Curvature The radius of curvature of the reference ellipsoid surface in the east–west direction (i.e., orthogonal to the direction in which
the meridional radius of curvature is measured) is called the transverse radius of
curvature. It is the radius of the osculating circle in the local east–up plane, as
illustrated in Fig. 9.20, where the arrows at the point of tangency of the transverse osculating circle are in the local ENU coordinate directions. As this figure
illustrates, on an oblate earth, the plane of a transverse osculating circle does not
pass through the center of the earth, except when the point of osculation is at
the equator. (All osculating circles at the poles are in meridional planes.) Also,
unlike meridional osculating circles, transverse osculating circles generally lie
outside the ellipsoidal surface, except at the point of tangency and at the equator,
where the transverse osculating circle is the equator.
The formula for the transverse radius of curvature on an ellipsoid of
revolution is
a
rT = !
,
1 − e2 sin2 (φgeodetic )
(9.29)
where a is the semimajor axis of the generating ellipse and e is its eccentricity.
Longitude Rate The rate of change of longitude as a function of east velocity
is then
vE
dθ
=
dt
cos(φgeodetic ) (rT + h)
(9.30)
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INERTIAL NAVIGATION SYSTEMS
and longitude can be maintained by the integral
θ (tnow ) = θ (tstart )
* tnow
+
tstart
vE (t) dt
% !
&,
2
2
cos[φgeodetic (t)] a/ 1 − e sin (φgeodetic (t)) + h(t)
(9.31)
where h(t) is height above (+) or below (−) the ellipsoid surface and θ will
be in radians if vE (t) is in meters per second and rT (t) and h(t) are in meters.
Note that this formula has a singularity at the poles, where cos(φ geodetic ) = 0, a
consequence of using latitude and longitude as location variables.
WGS84 Reference Surface Curvatures The apparent variations in meridional
radius of curvature in Fig. 9.19 are rather large because the ellipse used in generating Fig. 9.19 has an eccentricity of about 0.75. The WGS84 ellipse has an
eccentricity of about 0.08, with geocentric, meridional, and transverse radius of
curvature as plotted in Fig. 9.21 versus geodetic latitude. For the WGS84 model:
•
•
•
Mean geocentric radius is about 6371 km, from which it varies by –14.3
km (–0.22%) to +7.1 km (+0.11%).
Mean meridional radius of curvature is about 6357 km, from which it varies
by –21.3 km (–0.33%) to 42.8 km (+0.67%).
Mean transverse radius of curvature is about 6385 km, from which it varies
by –7.1 km (–0.11%) to +14.3 km (+0.22%).
Fig. 9.21
Radii of WGS84 reference ellipsoid.
SYSTEM IMPLEMENTATION MODELS
355
Because these vary by several parts per thousand, one must take radius of
curvature into account when integrating horizontal velocity increments to obtain
longitude and latitude.
9.4.4 Gimbal Attitude Implementations
The primary function of gimbals is to isolate the ISA from vehicle rotations, but
they are also used for other INS functions.
9.4.4.1 Accelerometer Recalibration Navigation accuracy is very sensitive
to accelerometer biases, which can shift as a result of thermal transients in
turnon/turnoff cycles, and can also drift randomly over time. Fortunately, the
gimbals can be used to calibrate accelerometer biases in a stationary 1-g environment. In fact, both bias and scale factor can be determined by using the
gimbals to point the accelerometer input axis straight up and straight down,
and recording
the respective
accelerometer outputs
aup and adown . Then the bias
abias = aup + adown /2 and scale factor s = aup − adown /2 glocal , where glocal
is the local gravitational acceleration.
9.4.4.2 Gyrocompass Alignment This is the process of determining the orientation of the ISA with respect to locally level coordinates (e.g., NED or ENU).
Gyrocompassing allows the ISA to be oriented with its sensor axes aligned parallel to the north, east, and vertical directions. It is accomplished using three
servo loops. The two “leveling” loops slew the ISA until the outputs of the nominally “north” and “east” accelerometer outputs are zeroed, and the “heading”
loop slews the ISA around the third orthogonal axis (i.e., the vertical one) until
the output of the nominally “east-pointing” gyro is zeroed.
9.4.4.3 Vehicle Attitude Determination The gimbal angles determine the vehicle attitude with respect to the ISA, which has a controlled orientation with respect
to locally level coordinates. Each gimbal angle encoder output determines the relative rotation of the structure outside gimbal axis relative to the structure inside
the gimbal axis, the effect of each rotation can be represented by a 3 × 3 rotation matrix, and the coordinate transformation matrix representing the attitude of
vehicle with respect to the ISA will be the ordered product of these matrices.
For example, in the gimbal structure shown in Fig. 2.6, each gimbal angle represents an Euler angle for vehicle rotations about the vehicle roll, pitch and yaw
axes. Then the transformation matrix from vehicle roll–pitch–yaw coordinates
to locally level east–north–up coordinates will be
⎡
RPY
CENU
SY CP
= ⎣ CY CP
SP
CR CY + SR SY SP
−CR SY + SR CY SP
−SR CP
⎤
−SR CY + CR SY SP
SR SY + CR CY SP ⎦ , (9.32)
−CR CP
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INERTIAL NAVIGATION SYSTEMS
where
SR = sin (roll angle) ,
CR = cos (roll angle) ,
SP = sin (pitch angle) ,
CP = cos (pitch angle) ,
SY = sin (yaw angle) ,
CY = cos (yaw angle) .
9.4.4.4 ISA Attitude Control The primary purpose of gimbals is to stabilize
the ISA in its intended orientation. This is a 3-degree-of-freedom problem, and
the solution is unique for three gimbals. That is, there are three attitude-control
loops with (at least) three sensors (the gyroscopes) and three torquers. Each
control loop can use a PID controller, with the commanded torque distributed
to the three torquers according to the direction of the torquer/gimbal axis with
respect to the gyro input axis, somewhat as illustrated in Fig. 9.22, where
disturbances includes the sum of all torque disturbances on the individual gimbals and the ISA, including those due to mass unbalance and acceleration,
air currents, torque motor errors, etc.
gimbal dynamics is actually quite a bit more complicated than the rigid-body
torque equation
τ = Minertia ω̇,
Fig. 9.22 Simplified control flow diagram for three gimbals.
SYSTEM IMPLEMENTATION MODELS
357
which is the torque analog of F = ma, where Minertia is the moment of
inertia matrix. The IMU is not a rigid body, and the gimbal torque motors
apply torques between the gimbal elements (i.e., ISA, gimbal rings and
host vehicle).
desired rates refers to the rates required to keep the ISA aligned to a moving
coordinate frame (e.g., locally level).
resolve to gimbals is where the required torques are apportioned among the
individual torquer motors on the gimbal axes.
The actual control loop is more complicated than that shown in the figure, but it
does illustrate in general terms how the sensors and actuators are used.
For systems using four gimbals to avoid gimbal lock, the added gimbal adds
another degree of freedom to be controlled. In this case, the control law usually
adds a fourth constraint (e.g., maximize the minimum angle between gimble axes)
to avoid gimbal lock.
9.4.5 Strapdown Attitude Implementations
9.4.5.1 Strapdown Attitude Problem Early on, strapdown systems technology
had an “attitude problem,” which was the problem of representing attitude rate in
a format amenable to accurate computer integration. The eventual solution was
to represent attitude in different mathematical formats as it is processed from raw
gyro outputs to the matrices used for transforming sensed acceleration to inertial
coordinates for integration.
Figure 9.23 illustrates the resulting major gyro signal processing operations,
and the formats of the data used for representing attitude information. The processing starts with gyro outputs and ends with a coordinate transformation matrix
from sensor coordinates to the coordinates used for integrating the sensed accelerations.
Fig. 9.23
Strapdown attitude representations.
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INERTIAL NAVIGATION SYSTEMS
Fig. 9.24 Attitude representation formats and MATLAB transformations.
9.4.5.2 MATLAB Implementations The diagram in Fig. 9.24 shows four different representations used for relative attitudes, and the names of the MATLAB
script m-files (i.e., with the added ending .m) on the accompanying CD-ROM
for transforming from one representation to another.
9.4.5.3 Coning Motion This type of motion is a problem for attitude integration when the frequency of motion is near or above the sampling frequency. It
is usually a consequence of host vehicle frame vibration modes where the INS
is mounted, and INS shock and vibration isolation is often designed to eliminate
or substantially reduce this type of rotational vibration.
Coning motion is an example of an attitude trajectory (i.e., attitude as a function of time) for which the integral of attitude rates does not equal the attitude
change. An example trajectory would be
⎡
⎤
cos coning t
⎥
⎢
ρ(t) = θcone ⎣ sin coning t ⎦ ,
(9.33)
0
⎡
⎤
− sin coning t
⎥
⎢
(9.34)
ρ̇(t) = θcone coning ⎣ cos coning t ⎦ ,
0
where θcone is the cone angle of the motion and coning is the coning frequency
of the motion, as illustrated in Fig. 9.25.
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SYSTEM IMPLEMENTATION MODELS
Fig. 9.25
Coning motion.
The coordinate transformation matrix from body coordinates to inertial coordinates (Eq. C.112) will be
body
Cinertial (ρ) = cos θ I + (1 − cos θ )
⎤
⎡
2
cos coning t
sin coning t cos coning t 0
2
⎥
⎢ × ⎣sin coning t cos coning t
sin coning t
0 ⎦
0
0
0
⎡
⎤
0
0
sin coning t + sin θ ⎣
0
0
−
cos
coning t ⎦ ,
0
− sin coning t cos coning t
(9.35)
and the measured inertial rotation rates in body coordinates will be
ωbody = Cinertial
body ρ̇ inertial
⎡
− sin coningt
T
body
= θcone coning Cinertial ⎣ cos coning t
0
⎡
−θcone coning sin coning t cos (θcone )
= ⎣ θcone coning cos coning t cos (θcone )
− sin (θcone ) θcone coning
⎤
⎦,
(9.36)
(9.37)
⎤
⎦.
(9.38)
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INERTIAL NAVIGATION SYSTEMS
Fig. 9.26
Coning error for 1 deg cone angle, 1kHz coning rate
The integral of ωbody
*
"
# ⎤
−θcone cos (θcone ) 1 − cos
coning
t
⎦ , (9.39)
ωbody (s)ds = ⎣
θcone cos (θcone ) sin coning t
s=0
− sin (θcone ) θcone coning t
⎡
t
which is what a rate integrating gyroscope would measure.
The solutions for θcone = 0.1◦ and coning = 1 k Hz are plotted over one cycle
(1 ms) in Fig. 9.26. The first two components are cyclical, but the third component accumulates linearly over time at about −1.9 × 10−5 radians in 10− 3
second, which is a bit more than −1◦ /s. This is why coning error compensation
is important.
9.4.5.4 Rotation Vector Implementation This implementation is primarily
used at a faster sampling rate than the nominal sampling rate (i.e., that required
for resolving measured accelerations into navigation coordinates). It is used to
remove the nonlinear effects of coning and skulling motion that would otherwise
corrupt the accumulated angle rates over the nominal intersample period. This
implementation is also called a “coning correction.”
Bortz Model for Attitude Dynamics This exact model for attitude integration
based on measured rotation rates and rotation vectors was developed by John
Bortz [23]. It represents ISA attitude with respect to the reference inertial coordinate frame in terms of the rotation vector ρ required to rotate the reference
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SYSTEM IMPLEMENTATION MODELS
Fig. 9.27 Rotation vector representing coordinate transformation.
inertial coordinate frame into coincidence with the sensor-fixed coordinate frame,
as illustrated in Fig. 9.27.
The Bortz dynamic model for attitude then has the form
ρ̇ = ω + fBortz (ω, ρ) ,
(9.40)
where ω is the vector of measured rotation rates. The Bortz “noncommutative
rate vector”
B
A
1
ρ sin (ρ)
1
# ρ ⊗ (ρ ⊗ ω) , (9.41)
fBortz (ω, ρ) = ρ ⊗ ω +
1− "
2
ρ2
2 1 − cos (ρ)
|ρ| <
π
.
2
(9.42)
Equation 9.40 represents the rate of change of attitude as a nonlinear differential equation that is linear in the measured instantaneous body rates ω. Therefore,
by integrating this equation over the nominal intersample period [0, t] with
initial value ρ(0) = 0, an exact solution of the body attitude change over that
period can be obtained in terms of the net rotation vector
* t
ρ (t) =
ρ̇ (ρ (s) , ω (s)) ds
(9.43)
0
that avoids all the noncommutativity errors, and satisfies the constraint of Eq. 9.42
as long as the body cannot turn 180◦ in one sample interval t. In practice, the
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INERTIAL NAVIGATION SYSTEMS
integral is done numerically with the gyro outputs ω1 , ω2 , ω3 sampled at intervals
δt t. The choice of δt is usually made by analyzing the gyro outputs under
operating conditions (including vibration isolation), and selecting a sampling frequency 1/δt above the Nyquist frequency for the observed attitude rate spectrum.
The frequency response of the gyros also enters into this design analysis.
The MATLAB function fBrotz.m on the CD-ROM calculates fBortz (ω)
defined by Eq. 9.41.
9.4.5.5 Quaternion Implementation The quaternion representation of vehicle
attitude is the most reliable, and it is used as the “holy point” of attitude representation. Its value is maintained using the incremental rotations ρ from the
rotation vector representation, and the resulting values are used to generate the
coordinate transformation matrix for accumulating velocity changes in inertial
coordinates.
Converting Incremental Rotations to Incremental Quaternions An incremental
rotation vector ρ from the Bortz coning correction implementation of Eq. 9.43
can be converted to an equivalent incremental quaternion q by the operations
θ = |ρ| (rotation angle in radians),
1
ρ
θ
⎡
⎤
u1
= ⎣ u2 ⎦ (unit vector),
u3
⎤
⎡
cos θ2
⎢
⎥
⎢ u1 sin θ ⎥
⎢
2 ⎥
q = ⎢
⎥,
⎢ u2 sin θ ⎥
2 ⎦
⎣
θ u3 sin 2
⎡
⎤
q0
⎢ q1 ⎥
⎥
=⎢
⎣ q2 ⎦ (unit quaternion).
q3
u=
(9.44)
(9.45)
(9.46)
(9.47)
(9.48)
Quaternion Implementation of Attitude Integration If qk−1 is the quaternion
representing the prior value of attitude, q is the quaternion representing the
change in attitude, and qk is the quaternion representing the updated value of
attitude, then the update equation for quaternion representation of attitude is
qk = q × qk−1 × q! ,
where the postsuperscript
!
represents the conjugate of a quaternion.
(9.49)
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SYSTEM IMPLEMENTATION MODELS
9.4.5.6 Direction Cosines Implementation The coordinate transformation
body
matrix Cinertial from body-fixed coordinates to inertial coordinates is needed for
transforming discretized velocity changes measured by accelerometers into inertial coordinates for integration. The quaternion representation of attitude is used
body
for computing Cinertial .
body
Quaternions to Direction Cosines Matrices The direction cosines matrix Cinertial
from body-fixed coordinates to inertial coordinates can be computed from its
equivalent unit quaternion representation
⎡
⎤
q0
⎢ q1 ⎥
body
⎥
(9.50)
qinertial = ⎢
⎣ q2 ⎦
q3
as
body
Cinertial
⎡
⎤ ⎡
⎤T
⎡
⎤
q1
q1
q1
= 2 q02 − 1 I3 + 2 ⎣ q2 ⎦ × ⎣ q2 ⎦ − 2 q0 ⎣ q2 ⎦ ⊗
q3
q3
q3
⎡ 2
2 q0 − 1 + 2 q12 (2 q1 q2 + 2 q0 q3 ) (2 q1 q3 − 2 q0 q2 )
2
⎢
= ⎣ (2 q1 q2 − 2 q0 q3 ) 2 q02 − 1 + 2 q22
2 q2 + 2 q0 q1
2
2
2 q2 − 2 q0 q1
2 q0 − 1 + 2 q32
(2 q1 q3 + 2 q0 q2 )
(9.51)
⎤
⎥
⎦.
(9.52)
9.4.6 Navigation Computer and Software Requirements
Inertial navigation systems operate under conditions of acceleration, shock, and
vibration that may preclude the use of hard disks or standard mounting and
interconnect methods. As a consequence, INS computers tend to be somewhat
specialized. The following sections list some of the requirements placed on navigation computers and software that tend to set them apart.
9.4.6.1 Physical and Operational Requirements These include
1. Size, weight, form factor, and available input power.
2. Environmental conditions such as shock/vibration, temperature, and electromagnetic interference (EMI).
3. Memory (how much and how fast), throughput (operations per second),
wordlength/precision.
4. Time required between power-on and full operation, and minimum time
between turnoff and turnon. (Some vehicles shut down all power during
fueling, for example.)
5. Reliability, shelf-life, and storage requirements.
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INERTIAL NAVIGATION SYSTEMS
6. Operating life. Some applications (e.g., missiles) have operating lifetimes
of minutes or seconds, others (e.g., military and commercial aircraft) may
operate nearly continuously for decades.
7. Additional application-specific requirements such as radiation hardening.
9.4.6.2 Operating Systems Inertial navigation is a real-time process. The tasks
of sampling the sensor outputs, and of integrating attitude rates, accelerations
and velocities must be scheduled at precise time intervals, and the results must
be available after limited delay times. The top-level operating system which
prioritizes and schedules these and other tasks must be a real time operating
system (RTOS). It may also be required to communicate with other computers
in various ways.
9.4.6.3 Interface Requirements These not only include the operational interfaces to sensors and displays but may also include communications interfaces
and specialized computer interfaces to support navigation software development
and verification.
9.4.6.4 Software Development Because INS failures could put host vehicle
crews and passengers at risk, it is very important during system development to
demonstrate high reliability of the software. INS software is usually developed
offline on a general-purpose computer interfaced to the navigation computer.
Software development environments for INS typically include code editors, crosscompilers, navigation computer emulators, hardware simulators, hardware-in-theloop interfaces, and specialized source–code–online interfaces to the navigation
computer for monitoring, debugging and verifying the navigation software on the
navigation computer. Software developed for manned missions must be acceptably reliable, which requires metrics for demonstrating reliability. In addition,
real-time programmers for INS do tend to be a different breed of cat from generalpurpose programmers, and software development cost can be a significant fraction
of overall system development cost [28].
9.5 SYSTEM-LEVEL ERROR MODELS
The system-level implementation models discussed in the previous section are
for the internal implementation of the inertial navigation system, itself. These
are models for the peculiarities of the sensors and software that contribute to
navigation errors. They are used in INS design analysis for predicting performance as a function of component characteristics. They are also used within the
implementation for compensating the corrupting influence of sensor and software
error tolerances on the measured and inferred vehicle dynamic variables.
The system-level error models in this section are for implementing GNSS/INS
integration. These models represent how the resulting navigation errors will propagate over time, as functions of the error parameters of the inertial sensors. They
are used in two ways:
SYSTEM-LEVEL ERROR MODELS
365
1. In so-called “loosely coupled” approaches for keeping track of the uncertainty in the INS navigation solution to use in a Kalman gain for combining
the GNSS and INS navigation solutions to maintain an integrated navigation solution. When GNSS signals are not available, the model can still
be used to propagate the estimated INS errors and subtract them from the
uncompensated INS navigation solution. The resulting compensated INS
navigation solution can then be used to speed up detection and reacquisition
of GNSS signals—if and when they become available again.
2. In more “tightly coupled” approaches using GNSS measurements to estimate and compensate for random variations in the calibration parameters of
individual sensors in the INS. These approaches continually re-calibrate the
INS when GNSS signals are available. They are functionally similar to the
loosely coupled approaches, in that they still carry forward the calibrationcompensated INS navigation solution when GNSS signals are unavailable.
Unlike the loosely coupled approaches, however, they are feedback-based
and not as sensitive to modeling errors.
Model Diversity There is no universal INS error model for GNSS/INS integration, because there is no standard design for an INS. There may be differences
between different GNSS systems, and between generations of GPS satellites, but
GNSS error models are all quite similar. Differences between error models for
INSs, on the other hand, can be anything but minor. There are some broad INS
design types (e.g., gimbaled vs. strapdown), but there are literally thousands of
different inertial sensor designs that can be used for each INS type.
Methodology We present here a variety of inertial system error models, which
will be sufficient for many of the sensors in common use, but not for every
conceivable inertial sensor. For applications with sensor characteristics different
from those used here, the use of these error models in GNSS/INS integration will
serve to illustrate the general integration methodology, so that users can apply
the same methodology to GNSS/INS integration with other sensor error models,
as well.
9.5.1 Error Sources
9.5.1.1 Initialization Errors Inertial navigators can only integrate sensed accelerations to propagate initial estimates of position and velocity. Systems without
GNSS aiding require other sources for their initial estimates of position and
velocity. Initialization errors are the errors in these initial values.
9.5.1.2 Alignment Errors Most standalone INS implementations include an
initial period for alignment of the gimbals (for gimbaled systems) or attitude
direction cosines (for strapdown systems) with respect to the navigation axes.
Errors remaining at the end of this period are the alignment errors. These include
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INERTIAL NAVIGATION SYSTEMS
tilts (rotations about horizontal axes) and heading errors (rotations about the
vertical axis).
Tilt errors introduce acceleration errors through the miscalculation of gravitational acceleration, and these propagate primarily as Schuler oscillations (see
Section 9.5.2.1) plus a non-zero-mean position error approximately equal to the
tilt error in radians times the radius from the earth center. Initial azimuth errors
primarily rotate the system trajectory about the starting point, and there are secondary effects due to coriolis accelerations and excitation of Schuler oscillations.
9.5.1.3 Sensor Compensation Errors Sensor calibration is a procedure for
estimating the parameters of models used in sensor error compensation. It is
not uncommon for these modeled parameters to change over time and between
turnons, and designing sensors to make the parameters sufficiently constant
can also make the sensors relatively expensive. Costs resulting from stringent
requirements for parameter stability can be reduced significantly for sensors that
will be used in integrated GNSS/INS applications, because Kalman filter–based
GNSS/INS integration can use the differences between INS-derived position and
GNSS-derived position to make corrections to the calibration parameters.
These nonconstant sensor compensation parameters are not true parameters
(i.e., constants), but “slow variables,” which change slowly relative to the other
dynamic variables. Other slow variables in the integrated system model include
the satellite clock offsets for selective availability (SA).
The GNSS/INS integration filter implementation requires models for how variations in the compensation parameters propagate into navigation errors. These
models are derived in Section 9.3 for the more common types of sensors and
their compensation parameters.
9.5.1.4 Gravity Model Errors The influence of unknown gravity modeling
errors on vehicle dynamics is usually modeled as a zero-mean exponentially
correlated acceleration process (see Section 8.5)
δak = e−t/τcorrelation δak−1 + wk ,
(9.53)
where t is the filter period, the correlation time
τcorrelation ≈
dcorrelation
,
|vhorizontal |
(9.54)
vhorizontal is horizontal velocity, dcorrelation is the horizontal correlation distance
of gravity anomalies (usually on the order of 104 –105 m), wk is a zero-mean
white-noise process with covariance matrix
def Qgravity model = E wk wTk
2
(1 − e−2t/τ ) I,
≈ aRMS
(9.55)
(9.56)
2
aRMS
is the variance of acceleration error, and I is an identity matrix. The correlation distance dcorrelation and RMS acceleration disturbance aRMS will generally
367
SYSTEM-LEVEL ERROR MODELS
depend upon the local terrain. Here, dcorrelation tends to be larger and aRMS smaller
as terrain becomes more gentle or (for aircraft) as altitude increases.
The effects of gravity modeling errors in the vertical direction will be mediated
by vertical channel stabilization.
9.5.2 Navigation Error Propagation
The dynamics of INS error propagation are strongly influenced by the fact that
gravitational accelerations point toward the center of the earth and decrease in
magnitude with altitude and is somewhat less influenced by the fact that the earth
rotates.
9.5.2.1 Schuler Oscillations of INS Errors The dominant effect of alignment errors on free-inertial navigation is from tilts, also called leveling errors.
These are errors in the estimated direction of the local vertical in sensor-fixed
coordinates. The way in which tilts translate into navigation errors is through
a process called Schuler oscillation. These are oscillations at the same period
that Schuler had identified for gyrocompassing errors (see Section 9.1.1.) The
analogy between these Schuler oscillations of INS errors and those of a simple gravity pendulum is illustrated in Figure 9.28. The physical force acting
on the mass of a gravity pendulum is the vector sum of gravitational acceleration (mg) and the tension T in the support are, as shown in Figure 9.28a.
The analogous acceleration driving inertial navigation errors is the difference
between the modeled gravitational acceleration (which changes direction with
estimated location) and the actual gravitational acceleration, as shown in Figure
9.28b. In the case of the gravity pendulum, the physical mass of the pendulum
is oscillating. In the case of INS errors, only the estimated position, velocity,
and acceleration errors oscillate. The gravity pendulum is a physical device,
but the Schuler pendulum is a theoretical model to illustrate how INS errors
behave.
In either case, the restoring acceleration is approximately related to displacement δ from the equilibrium position by the harmonic equation
δ̈ ≈ −
g
δ,
L or R
(9.57)
= −ω2 δ
g
,
ω=
L or R
(9.58)
δ(t) = |δ|max cos (ω t + φ) ,
(9.60)
(9.59)
the solution for which is
where g is the acceleration due to gravity at the surface of the earth, L is the
length of the support arm of the gravity pendulum, R is the radius to the center
368
INERTIAL NAVIGATION SYSTEMS
Fig. 9.28 Pendulum model for Schuler oscillations of INS errors: (a) gravity pendulum
model; (b) Schuler oscillation model.
369
SYSTEM-LEVEL ERROR MODELS
of the earth, |δ|max is the peak displacement, ω is the oscillation frequency (in
radians per second), and φ is an arbitrary oscillation phase angle.
In the case of the gravity pendulum, the period of oscillation
2π
,
ω
√
2π L
= √ ,
g
Tgravity pendulum =
(9.61)
(9.62)
and for the Schuler pendulum
TSchuler =
√
2π R
√ ,
g
(9.63)
√
2 × 3.1416 × 6378170 (m),
≈
9.8 (m/s2 )
(9.64)
≈ 5069 (s)
(9.65)
≈ 84.4 (min)
(9.66)
at the surface of the earth.
This ≈ 84.4-min period is called the Schuler period. It is also the orbital period
of a satellite at that radius (neglecting atmospheric drag), and the exponential
time constant of altitude error instability in pure inertial navigation (see Section
9.5.2.2).
The corresponding angular frequency (Schuler frequency)
2π
TSchuler
≈ 0.00124 (rad/s)
(9.68)
≈ 0.0002 (Hz).
(9.69)
Schuler =
(9.67)
Dependence on Position and Direction. A spherical earth model was used to
illustrate the Schuler pendulum. The period of Schuler oscillation actually depends
on the radius of curvature of the equipotential surface of the gravity model, which
can be different in different directions and vary with longitude, latitude, and altitude. However, the variations in Schuler period due to these effects are generally
in the order of parts per thousand, and are usually ignored.
Impact on INS Performance Schuler oscillations include variations in the INS
errors in position, velocity, and acceleration (or tilt), which are all related harmonically. Thus, if the peak position displacement from the median location is
370
INERTIAL NAVIGATION SYSTEMS
|δ|max , then
δ(t) = |δ|max cos (Schuler t + φ) (position error),
(9.70)
δ̇(t) = − |δ|max Schuler sin (Schuler t + φ) (velocity error),
(9.71)
δ̈(t) =
δ̈(t) =
− |δ|max 2Schuler
− |δ|max 2Schuler
g
cos (Schuler t + φ) (acceleration error),
(9.72)
cos (Schuler t + φ) (acceleration error in g values),
(9.73)
= τ (t) (tilt error in radians).
(9.74)
Note, however, that when the initial INS error is a pure tilt error (i.e., no position
min from the starting
error), the peak position error will be 2 |δ|max after
≈ 42.2
√ location, and the RMS position error will be 1 + 1/ 2 |δ|max . If the initial
INS error is a pure tilt error, then the true INS position would at one end of
the Schuler pendulum swing—not the middle—and the peak and RMS position
errors
|δ|peak =
|δ|RMS =
2 τinitial g
,
2Schuler
1 + √12 τinitial g
2Schuler
(9.75)
,
(9.76)
as plotted in Fig. 9.29. This shows why alignment tilt errors are so important in
free inertial navigation. Tilts as small as one milliradian can cause peak position
excursions as big as 10 km after 42 min.
Figure 9.30 is a plot generated using the MATLAB INS Toolbox from GPSoft,
showing how an initial northward velocity error of 0.1 ms excites Schuler oscillations in the INS navigation errors, and how coriolis accelerations rotate the
direction of oscillation—just like a Foucault pendulum with Schuler period. The
total simulated time is about/14 h, enough time for 10 Schuler oscillation periods.
For a maximum velocity error |δ̇|max = 0.1 m/s, the maximum expected position error would be
|δ|max =
≈
|δ̇|max
Schuler
0.1 m/s
0.00124 rad/s
≈ 80 meter,
which is just about the maximum excursion seen in Fig. 9.30.
SYSTEM-LEVEL ERROR MODELS
Fig. 9.29 Effect of tilt errors on Schuler oscillations.
Fig. 9.30 GPSoft INS Toolkit simulation of Schuler oscillations.
371
372
INERTIAL NAVIGATION SYSTEMS
9.5.2.2 Vertical Channel Instability The modeled vertical acceleration due to
gravity in the downward direction, as a function of height h above the reference
geoid, is dominated by the first term
agravity =
GM
(R + h)2
+ less significant terms,
(9.77)
where R is the reference radius and GM is the model gravitational constant,
≈ 398,600 km3 /s2 for the WGS84 gravity model. Consequently, the vertical
gradient of downward gravity in the downward direction will be
−
∂agravity
agravity
≈
∂h
(R + h)
agravity
,
≈
R
(9.78)
(9.79)
which is positive. Therefore, if δh is the altitude estimation error, it satisfies a
differential equation of the form
δ̈h ≈
agravity
δh ,
R
(9.80)
which is exponentially unstable with solution
/
δh (t) ≈ δh (t0 ) exp
0
agravity
(t − t0 ) .
R
(9.81)
Not surprisingly, the exponential time constant of this vertical channel
instability
,
Tvertical channel =
R
agravity
≈ 84.4 min,
(9.82)
(9.83)
the Schuler period (see Section 9.5.2.1).
9.5.2.3 Coriolis Coupling The coriolis effect couples position error rates (velocities) into into their second derivative (accelerations) as
⎡
⎤
⎡
δv
0
E
d ⎣
δvN ⎦ = −2 earth ⎣ sin φ
dt
− cos φ
δvU
− sin φ
0
0
earth ≈ 7.3 × 10−5 (rad/s) (earthrate) ,
⎤⎡
⎤
cos φ
δvE
0 ⎦ ⎣ δvN ⎦ , (9.84)
0
δvU
(9.85)
373
SYSTEM-LEVEL ERROR MODELS
where φ is geodetic latitude. Adding Schuler oscillations yields the model
d
δvE = −2Schuler δE + 2earth sin φ δvN − 2earth cos φ δvU ,
dt
d
δvN = −2Schuler δN − 2earth sin φ δvE ,
dt
d
δvU = 2Earth cos φ δvE ,
dt
(9.86)
(9.87)
(9.88)
where δE is east position error and δN is north position error. This effect can
be seen in Fig. 9.30, in which the coriolis acceleration causes the error trajectory
to swerve to the right. The sign of sin φ in Eq. 9.86 is negative in the southern
hemisphere, so it would swerve to the left there.
Note that, through coriolis coupling, vertical channel errors couple into east
channel errors.
9.5.3 Sensor Error Propagation
Errors made in compensating for inertial sensor errors will cause navigation
errors. Here, we derive some approximating formulas for how errors in individual
compensation parameters propagate into velocity and position errors.
9.5.3.1 Scale Factors, Biases, and Misalignments The models for bias, scale
factor and input axis misalignment compensation are the same for gyroscopes
and accelerometers. The compensated sensor output
zcomp , = Mzoutput + bz
(9.89)
for z = ω (for gyroscopes) or z = a (for accelerometers). The sensitivity of the
compensated output to bias is then
∂zcomp
= I3 ,
∂bz
(9.90)
the 3×3 identity matrix, and the sensitivity to the elements mj, k of M are
∂zcomp, i
=
∂mj,k
<
0,
zoutput, k
i = j
i=j
(9.91)
If we put these 12 calibration parameters in vector form as
pcomp =
"
bz, 1
bz, 2
m2, 3
m3, 1
bz, 3
m3, 2
m1, 1
m1, 2
#T
m3, 3
,
m1, 3
m2, 1
m2, 2 . . .
(9.92)
374
INERTIAL NAVIGATION SYSTEMS
then the matrix of partial derivatives
⎡
∂zcomp, i
∂pcomp
1
0
0
⎢
⎢
⎢
⎢
⎢ zoutput, 1
⎢
⎢ z
⎢ output, 2
⎢
⎢ zoutput, 3
=⎢
⎢
0
⎢
⎢
0
⎢
⎢
0
⎢
⎢
⎢
0
⎢
⎣
0
0
0
1
0
0
0
0
zoutput, 1
zoutput, 2
zoutput, 3
0
0
0
0
0
1
0
0
0
0
0
zoutput, 1
zoutput, 2
zoutput, 3
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥.
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(9.93)
For analytical purposes, this matrix of partial derivatives would be evaluated
under “nominal” conditions, which could be for M = I and bz = 0. In that case,
zoutput = zinput , the bias sensitivities will be unitless (e.g., g/g), the scale factor
sensitivities will be in units of parts-per-part and the misalignment sensitivities
will be in output units per radian.
This matrix can be augmented with additional calibration parameters, such
as acceleration-sensitivities for gyroscopes or temperature sensitivities. It only
requires taking the necessary partial derivatives.
9.5.3.2 Accelerometer Compensation Error Propagation Acceleration errors
due to accelerometer compensation errors in sensor-fixed coordinates and navigation coordinates will then be
δasensor ≈
∂acomp
δpacc.comp ,
∂pacc.comp
δanav = Csensor
nav δasensor
≈ Csensor
nav
∂acomp
δpacc.comp ,
∂pacc.comp
(9.94)
(9.95)
(9.96)
where δpacc.comp is the vector of accelerometer compensation parameter errors,
the partial derivative matrix is the one in Eq. 9.93 with z = a, and Csensor
is
nav
the coordinate transformation matrix from sensor-fixed coordinates to navigation
coordinates. For example
•
For gimbaled systems aligned to locally level coordinates with sensor axes
pointing north and east
Csensor
= I, the identity matrix.
nav
(9.97)
375
SYSTEM-LEVEL ERROR MODELS
•
For a carouseled gimbal system aligned to locally level coordinates
⎡
Csensor
nav
•
cos α
= ⎣ − sin α
0
sin α
cos α
0
⎤
0
0 ⎦,
1
where α is the carousel angle.
For a strapdown system aligned to body coordinates of the host vehicle
Csensor
= Cbody
nav
nav .
•
(9.98)
(9.99)
For a carouseled strapdown system rotating about the body-fixed yaw axis
⎡
Csensor
nav
cos α
⎣
−
sin α
= Cbody
nav
0
sin α
cos α
0
⎤
0
0 ⎦.
1
(9.100)
Velocity and Position Errors. The velocity error sensitivities to each of the
compensation parameters will be the integral over time of the acceleration sensitivities, and the position error sensitivities to each of the compensation parameters
will be the integral over time of the velocity sensitivities. However, the accelerations must be transformed into navigation coordinates before integration:
*
t
δvnav (t) = δvnav (t0 ) +
*
= δvnav (t0 ) +
*
≈ δvnav (t0 ) +
δanav (s) ds
(9.101)
Csensor
nav (s) δasensor (s) ds
(9.102)
0
t
t0
t
t0
Csensor
nav (s)
∂acomp
(s) δpacc.comp ,
∂pacc.comp
δxnav (t) ≈ δxnav (t0 ) + (t − t0 ) δvnav (t0 )
* * t
∂acomp
+
Csensor
(s) δpacc.comp ds,
nav (s)
∂pacc.comp
t0
(9.103)
(9.104)
(9.105)
where δxnav is the navigation position error due to compensation parameter errors.
The GNSS navigation solution will not include δxnav , and it is the difference
between the INS and GNSS solutions that can be used to estimate the compensation parameter errors.
9.5.3.3 Gyroscope Compensation Error Propagation The principal effect of
gyroscope compensation errors on inertial navigation position errors is from the
376
INERTIAL NAVIGATION SYSTEMS
Fig. 9.31 Acceleration errors due to tilts.
miscalculation of gravitational acceleration due to the resulting tilt errors, as
illustrated in Fig. 9.31, where
δaE ≈ −gδρN ,
(9.106)
δaN ≈ gδρE ,
(9.107)
and δρN is a small rotation about the north-pointing coordinate axis and δρE is
the corresponding east tilt component.
Tilt Errors Small tilt errors due to calibration errors can be approximated as the
horizontal north and east components of a rotation vector δρ:
d
δρ (t) ≈ δωnav (t)
dt nav
= Csensor
nav (t)δ ωsensor (t)
≈ Csensor
nav (t)
∂ωcomp
(t) δpgyro.comp
∂pgyro.comp
(9.108)
(9.109)
(9.110)
The east and north tilt components can then be substituted into Eqs. 9.106 and
9.107 to obtain the equations for position error due to tilts. Schuler oscillations
are excited when these position errors, in turn, cause tilts.
Velocity Errors For the tilt error angles δρE , δρN in radians and g ≈ 9.8 m/s2 ,
the corresponding velocity errors will propagate as
d
δvE (t) ≈ gδρN (t),
dt
d
δvN (t) ≈ −gδρE (t).
dt
(9.111)
(9.112)
377
SYSTEM-LEVEL ERROR MODELS
Effect of Heading Errors Navigation error sensitivity to rotational error ρU about
the local vertical (i.e., heading error) is usually smaller than the sensitivities
to tilt-related errors. The time rate-of-change of position errors due to heading
error are
d
δxE (t) ≈ −δρU vN ,
dt
(9.113)
d
δxN (t) ≈ δρU vE ,
dt
(9.114)
where δxE and δxN are the navigation error components due to heading error δρU
(measured counterclockwise in radians) and vE and vN are the vehicle velocity
components in the east and north directions, respectively.
9.5.4 Examples
9.5.4.1 Damping Vertical Channel Errors The propagation of altitude error
δh over time is governed by Eq. 9.80:
agravity
d2
δh
δh ≈
dt 2
R+h
δh
,
≈ 2
TSchuler
(9.115)
where the Schuler period TSchuler ≈ 5046 s at sea level. The INS initialization
procedure can make the initial value δh(t0 ) very small (e.g., in the order of
a meter). It can also make the vertical component of the initial gravity model
error very small (e.g., by making it match the sensed vertical acceleration during
alignment). Thereafter, the vertical channel navigation error due to zero-mean
white accelerometer noise w(t) will propagate according to the model
3/
0 2
0 /
0
/
0
1
d
δh(t)
0
δh(t)
1
=
+
, (9.116)
0
δvU (t)
w(t)
δvU (t)
2
dt
TSchuler
Fvert.chan
where δvU (t) = dtd δh(t) is vertical velocity error.
The equivalent state transition matrix in discrete time with timestep t is
vert.chan = exp (F t)
/
0
t
1
1
TSchuler
= e TSchuler
−1
TSchuler
1
2
/
0
1 − T t
1
−TSchuler
,
+ e Schuler
−1
−TSchuler
1
2
(9.117)
378
INERTIAL NAVIGATION SYSTEMS
and the corresponding Riccati equation for propagation of the covariance matrix
Pvert.chan of vertical channel navigation errors has the form
0
/
0
0
, (9.118)
Pvert.chan, k = vert.chan Pvert.chan, k−1 Tvert.chan +
0 qaccelerometer
where qaccelerometer is the incremental variance of velocity uncertainty per timestep
t due to vertical altimeter noise. For example, for velocity random walk errors
specified as having VRW meter per second per root hour, we obtain
qaccelerometer =
VRW2 t
.
3600
(9.119)
Figure 9.32 is a plot of altitude uncertainty versus time after INS initialization
√
for a range of accelerometer white noise levels, from 10−2 to 102 m/s/ hr. All
the solid-line plots will increase over time without bound.
Barometric Altimeter for Vertical Channel Damping. If a barometric altimeter is
to be used for vertical channel stabilization, then the altimeter error δhaltimeter will
not be zero-mean white noise, but something more like a zero-mean exponentially
correlated error. This sort of error has a discrete-time model of the form
δhaltimeter, k = exp
t
τaltimeter
δhaltimeter, k + wk ,
Fig. 9.32 INS vertical instability, with and without altimeter aiding.
(9.120)
379
SYSTEM-LEVEL ERROR MODELS
where τaltimeter is the correlation time and the white-noise sequence {wk } has
variance
&
%
−2t
2
σaltitude
,
(9.121)
qaltimeter = 1 − exp
τaltimeter
2
for steady-state altitude variance σaltitude
.
In this case, the augmented vertical channel state vector
⎤
δh(t)
⎥
⎢
= ⎣ δvU (t) ⎦ ,
δhaltimeter
⎡
xaug.vert.chan
(9.122)
and the resulting Riccati equations for state uncertainty will be
Paug.vert.chan, k(−) = aug.vert.chan Paug.vert.chan, k−1(+) Taug.vert.chan
⎡
0
0
⎢
+ ⎣ 0 qaccelerometer
0
0
/
aug.vert.chan =
vert.chan
0
0
0
⎤
⎥
⎦ (a priori),
(9.123)
qaltimeter
0
exp (−t / τaltimeter )
0
(state transition),
(9.124)
Paug.vert.chan, k(+) = Paug.vert.chan, k(−) − Kk HPaug.vert.chan, k(−) (a posteriori),
(9.125)
T
Paug.vert.chan, k(−) H
(Kalman gain), (9.126)
HPaug.vert.chan, k(−) HT + Raltimeter
"
#
H = 1 0 1 (measurement sensitivity),
(9.127)
Kk =
where Raltimeter is mean-squared altimeter noise, exclusive of the correlated component δhaltimeter .
The dotted lines in Fig. 9.32 are plots of altitude uncertainty with vertical
channel damping, using a barometric altimeter. The assumed atmospheric and
altimeter model parameters are written on the plot. These show much better
performance than does the undamped case, over the same range of accelerometer
noise levels. In all cases, the damped results do not continue to grow without
bound.
Figure 9.32 was generated by the m-file VertChanErr.m on the CD-ROM.
380
INERTIAL NAVIGATION SYSTEMS
9.5.4.2 Carouseling Accelerometer bias errors δbacc couple into horizontal
navigation errors, as modeled in Eqs. 9.86–9.87 and 9.96:
⎤
⎡
⎤ ⎡
0
0
1
0
δE
⎥
⎥ ⎢
d ⎢
0
0
0
1
⎥
⎢ δN ⎥ = ⎢
2
⎣
⎦
⎣
⎦
−
δv
0
0
2
sin
(φ)
dt
E
Schuler
earth
δvN
0
−Schuler 2 −2 earth sin (φ)
0
⎡
⎤ ⎡
⎤
02×1
δE
⎢ δN ⎥ ⎢
⎥
⎥+⎢ /
⎥,
0
×⎢
⎣ δvE ⎦ ⎣
⎦
1 0 0
sensor
Cnav δbacc
0 1 0
δvN
(9.128)
where δbacc is the vector of accelerometer biases and Csensor
can have any of the
nav
values given in Eqs. 9.97–9.100.
Figure 9.33 is a plot of fourteen hours of simulated INS position errors resulting from 10-μg north accelerometer bias on a gimbaled INS, with and without
carouseling. The simulated carousel rotation period is 5 min, and the resulting
navigation errors are reduced by more than an order of magnitude. This shows
why carouseling (and indexing) is a popular implementation scheme.
The plot in Figure 9.33 was generated by the MATLAB m-file AccBiasCarousel.m on the CD-ROM. Note that it exhibits the same coriolis-coupled
Fig. 9.33 Fourteen hours of simulated gimbaled INS navigation errors, with and without
carouseling.
SYSTEM-LEVEL ERROR MODELS
381
Schuler oscillations as in Fig. 9.30, which was the result of an initial north velocity error. In this case, it is a north acceleration error, the result of which is that
the Schuler oscillation center is offset from the starting point.
PROBLEMS
9.1 In the one-dimensional “line land” world of Section 9.4.1, “an INS requires
no gyroscopes. How many gyroscopes would be required for two-dimensional
navigation in “flat land?”
9.2 Derive the equivalent formulas in terms of Y (yaw angle), P (pitch angle),
and R (roll angle) for unit vectors 1R , 1P , 1Y in NED coordinates and 1N , 1E ,
1D in RPY coordinates, corresponding to Eqs. C.86–C.91 of Appendix C.
9.3 Explain why accelerometers cannot sense gravitational accelerations.
body
9.4 Show that the matrix Cinertial defined in Eq. 9.52 is orthogonal by showing
body
body T
that Cinertial × Cinertial = I, the identity matrix. (Hint: Use q02 + q12 + q22 +
q32 = 1.)
9.5 Calculate the numbers of computer multiplies and adds required for
(a) Gyroscope scale factor/misalignment/bias compensation (Eq. 9.13 with
N = 3)
(b) Accelerometer scale factor/misalignment/bias compensation (Eq. 9.13
with N = 3)
(c) Transformation of accelerations to navigation coordinates (Fig. 9.16)
using quaternion rotations (Eq. C.243) requiring two quaternion products
(Eq. C.234).
If the INS performs these 100 times per second, how many operations per
second will be required?
9.6 Calculate the maximum tilt error for the Schuler oscillations shown in
Fig. 9.30. Does this agree with the sensitivities plotted in Fig. 9.29?
10
GNSS/INS INTEGRATION
10.1 BACKGROUND
10.1.1 Sensor Integration
GNSS/INS integration is a form of sensor integration or sensor fusion, which
involves combining the outputs of different sensor systems to obtain a better
estimate of what they are sensing.
A GNSS receiver is a position sensor. It may use velocity estimates to reduce
filter lags, but its primary output is the position of its antenna relative to an earthcentered coordinate system. GNSS position errors will depend on the availability
and geometric distribution of GNSS satellites it can track, and other error sources
described in Chapter 5. The resulting RMS position errors will be bounded, except
for those times when there are not enough satellite signals available for a position
solution.
An INS uses acceleration and attitude (or attitude rate) sensors, but its primary
output as a sensor system is also position—the position of its ISA relative to
an earth-centered coordinate system. INS position errors depend on the quality
of its inertial sensors and earth models, described in Chapter 9. Although their
short-term position errors are very smooth, RMS position errors are not bounded.
They do tend to grow over time, and without bound.
This chapter is about practical methods for combining GNSS and INS outputs
to improve overall system performance metrics, including
Global Positioning Systems, Inertial Navigation, and Integration, Second Edition, by M. S. Grewal, L. R. Weill, and A. P. Andrews
Copyright © 2007 John Wiley & Sons, Inc.
382
BACKGROUND
•
•
•
•
•
383
RMS position estimation error under nominal GNSS conditions, when the
receiver can track enough satellites to obtain a good position estimate.
RMS position estimation error when the GNSS receiver cannot track enough
satellites to obtain a good position estimate. This can happen as a result of
• Background noise or signal jamming,
• Blocking by the leaf canopy, buildings, tunnels, etc.
• GNSS system failures
• Vehicle attitudes pointing the GNSS antenna pattern downward, or
• Violent vehicle dynamics causing loss of signal phase lock.
RMS velocity estimation error, which is important for
• Aircraft autonomous landing systems (ALS),
• Attitude transfer alignment to auxiliary INS
• Guided weapon delivery
• Rendezvous and docking maneuvers
RMS attitude estimation error, which is important for
• Sensor pointing
• Host vehicle guidance and control
Maintaining GNSS satellite signal lock, which can be difficult under severe
dynamic conditions. In effect, INS accelerometers measure the derivative of
signal Doppler shift, which can be used to improve GNSS receiver phaselock control margins.
Evaluating these metrics for a proposed system application requires statistical
models for how the component error sources contribute to overall system performance under conditions of the intended mission trajectories.
The purpose of this chapter is to present these models, and the techniques for
applying them to evaluate the expected integrated performance for a specific INS
design, GNSS configuration and ensemble of mission trajectories.
10.1.2 The Influence of Host Vehicle Trajectories on Performance
Host vehicle dynamics have a strong influence on GNSS navigation performance and INS performance. We will demonstrate in Section 10.2 how host
vehicle dynamic uncertainty affects the achievable navigation accuracy of GNSS
receivers. Trajectory profiles of acceleration and attitude rate also affect INS performance, because the errors due to inertial sensor scale factors tend to grow
linearly with the input accelerations and attitude rates.
The combined influence of host vehicle dynamics on integrated GNSS/INS
performance can usually be characterized by a representative set of host vehicle trajectories, from which we can estimate the expected performance of the
integrated GNSS/INS system over an ensemble of mission applications.
For example, a representative set of trajectories for a regional passenger jet
might include several trajectories from gate to gate, including taxiing, takeoff,
384
GNSS/INS INTEGRATION
climbout, cruise, midcourse heading and altitude changes, approach, landing, and
taxiing. Trajectories with different total distances and headings should represent
the expected range of applications. These can even be weighted according to the
expected frequency of use. With such a set of trajectories, one can assess expected
performances with different INS error characteristics and different satellite and
pseudolite geometries.
Similarly, for a standoff air-to-ground weapon, an ensemble of trajectories with
different approach and launch geometries and different target impact constraints
can be used to evaluate RMS miss distances with GNSS jamming at different
ranges from the target.
We will here demonstrate integrated GNSS/INS performance using some simple trajectory simulators, just to show the benefits of GNSS/INS integration. In
order to quantify the expected performance for a specific application, however,
you must use your own representative set of trajectories.
10.1.3 Loosely and Tightly Coupled Integration
The design process for GNSS/INS integration includes the tradeoff between performance and cost, and cost can be strongly influenced by the extent to which
GNSS/INS integration requires some modification of the inner workings of the
GNSS receiver or INS. The terms “loosely coupled” and “tightly coupled” are
used to describe this attribute of the problem.
10.1.3.1 Loosely Coupled Implementations The most loosely coupled implementations use only the standard outputs of the GNSS receiver and INS as inputs
to a sensor system integrating filter (often a Kalman filter), the outputs of which
are the estimates of “navigation variables” (including position and velocity) based
on both subsystem outputs. Although each subsystem (GNSS or INS) may already
include its own Kalman filter, the integration architecture does not modify it in
any way.
10.1.3.2 More Tightly Coupled Implementations The more tightly coupled
implementations use less standard subsystem outputs, such as pseudoranges from
GNSS receivers or raw accelerations from inertial navigators. These outputs generally require software changes within “standalone” GNSS receivers or INSs, and
may even require hardware changes. The filter model used for system integration
may also include variables such as GNSS signal propagation delays or accelerometer scale factor errors, and the estimated values of these variables may used in
the internal implementations of the GNSS receiver or INS.
Tightly coupled implementations may impact the internal inputs within the
GNSS receiver or INS, as well. The acceleration outputs from the INS can be
used to tighten the GNSS Doppler tracking loops, and the position estimates
from the INS can be used for faster reacquisition after GNSS outages. Also, the
INS accelerometers and gyroscopes can be recalibrated in real time to improve
free-inertial performance during GNSS signal outages.
385
BACKGROUND
10.1.3.3 Examples There are many possible levels of coupling between the
extremes. Figure 10.1 shows some of these intermediate approaches.
10.1.3.4 Incomplete Ordering The loose/tight ordering not “complete,” in the
sense that it is not always possible to decide whether one implementation is looser
or tighter than another.
A loosely coupled implementation designed for a given GNSS receiver, INS
and surface ship, for example, may differ significantly from one designed for a
highly maneuverable aircraft using the same GNSS receiver and INS. Both may
be equally “loose,” in the sense that they require no modifications of the GNSS
receiver or INS, but the details of the integrating filter will almost certainly differ.
The possibilities for equally loose but different implementations only multiplies
when we consider different GNSS receivers and INS hardware.
Similarly, there is no unique and well-defined “ultimately tightly coupled”
implementation, because there is no unique GNSS receiver or INS. As the technology advances, the possibilities for tighter coupling continue to grow, and
the available hardware and software for implementing it will almost certainly
continue to improve. Something even better may always come along.
10.1.4 Antenna/ISA Offset Correction
The “holy point” for a GNSS navigation receiver is its antenna. It is where
the relative phases of all received signals are determined, and it is the location
determined by the navigation solution.
The INS holy point is its inertial sensor assembly (ISA), which is where the
accelerations and attitude rates of the host vehicle are measured and integrated.
The distance between the two navigation solutions can be large enough1 to
be accounted for when combining the two navigation solutions. In that case, the
displacement of the antenna from the ISA can be specified as a parameter vector
⎡
⎤
δant, R
δ ant,RPY = ⎣ δant, P ⎦
(10.1)
δant, Y
in body-fixed roll–pitch–yaw (RPY) coordinates. Then the displacement in
north–east–down (NED) coordinates will be
δ ant, NED = CRPY
NED δ ant, RPY
⎡
CY PP −SY CR + CY SP SR
CY CR + SY SP SR
= ⎣ SY CP
−SP
CP SR
⎤
(10.2)
SY SR + CY SP CR
−CY SR + SY SP CR ⎦ δ ant, RPY ,
CP CR
(10.3)
1
It can be tens of meters for ships, where the INS may located well below deck and the antenna is
mounted high on a mast.
Fig. 10.1 Loosely and tightly coupled implementations.
386
GNSS/INS INTEGRATION
EFFECTS OF HOST VEHICLE DYNAMICS
387
where CRPY
NED is the coordinate transformation from RPY to NED coordinates,
defined in terms of the vehicle attitude Euler angles by
def
SR = sin (roll) ,
def
CR = cos (roll) ,
def
SP = sin (pitch) ,
def
CP = cos (pitch) ,
def
SY = sin (yaw) ,
def
CY = cos (yaw) ,
def
roll = vehicle roll angle,
def
pitch = vehicle pitch angle,
def
yaw = vehicle yaw/heading angle.
Usually, the matrix CRPY
NED and/or the roll, pitch, and yaw angles are variables in
INS implementation software.
Once δ ant., NED is computed, it can be used to relate the two navigation positions in NED coordinates:
xant, NED = xISA, NED + δ ant, NED ,
(10.4)
xISA, NED = xant, NED − δ ant, NED ,
(10.5)
eliminating the potential source of error.
This correction is generally included in integrated GNSS/INS systems. It
requires a procedure for requesting and entering the components of δ ant, NED
during system installation.
10.2 EFFECTS OF HOST VEHICLE DYNAMICS
Host vehicle dynamics impact GNSS/INS performance in a number of ways,
including the following:
1. A GNSS receiver is primarily a position sensor. To counter unpredictable
dynamics, it must use a vehicle dynamic model with higher derivatives
of position. This adds state variables that dilute the available (pseudorange) information, the net effect of which is reduction of position accuracy
with increased host vehicle dynamic activity. The filtering done within
GNSS receivers influences loosely coupled GNSS/INS integration through
the impact it has on GNSS position errors. The examples in Section 8.8
388
GNSS/INS INTEGRATION
introduced Kalman filter models for estimating GPS receiver antenna position and clock bias. sections 10.2.1–10.2.3 expand the modeling to include
filters optimized for specific classes of host vehicle trajectories.
2. Host vehicle dynamics translate into phase dynamics of the received satellite signals. This impacts phase-lock capability in three ways:
(a) It increases RMS phase error, which increases pseudorange error. This
contributes to position solution error. It also degrades to clock error
correction capability.
(b) It can cause the phase-lock loop to fail, causing momentary loss of
signal information.
(c) At best, it increases signal acquisition time. At worst, it may cause
signal acquisition to fail.
3. High dynamic rates increase the sensitivity of INS performance to inertial sensor errors, especially scale factor errors. For strapdown systems, it
greatly increases the influence of gyroscope scale factor errors on navigation performance.
The effects are reduced by GNSS/INS integration. To understand how, we will
need to examine how these effects work and how integration can mitigate their
effects. Impacts on GNSS host vehicle dynamic filtering and mitigation methods are discussed in this section. Mitigation methods for the other effects are
discussed in the next two sections.
10.2.1 Vehicle Tracking Filters
Starting around 1950, radar systems were integrated with computers to detect
and track Soviet aircraft that might invade the continental United States [160].
The computer software included filters to identify and track individual aircraft
within a formation. These “tracking filters” generated estimates of position and
velocity for each aircraft, and they could be tuned to follow the unpredictable
maneuvering capabilities of Soviet bombers of that era.
The same sorts of tracking filters are used in GNSS receivers to estimate
the position and velocity of GNSS antennas on host vehicles with unpredictable
dynamics. Important issues in the design and implementation of these filters
include the following:
1. In what ways does vehicle motion affect GNSS navigation performance?
2. Which characteristics of vehicle motions influence the choice of tracking
filter models?
3. How do we determine these characteristics for a specified vehicle type?
These issues are addressed in the following subsections.
EFFECTS OF HOST VEHICLE DYNAMICS
389
10.2.1.1 Dynamic Dilution of Information In addition to the “dilution of
precision” related to satellite geometry, there is a GNSS receiver “dilution of
information” problem related to vehicle dynamics. In essence, if more information
(in the measurements) is required to make up for the uncertainty of vehicle
movement, then less information is left over for determining the instantaneous
antenna position and clock bias.
For example, the location of a receiver antenna at a fixed position on the earth
can be specified by three unknown constants (i.e., position coordinates in three
dimensions). Over time, as more and more measurements are used, the accuracy
of the estimated position should improve. If the vehicle is moving, however, only
the more recent measurements relate to the current antenna position.
10.2.1.2 Effect on Position Uncertainty Figure 10.2 is a plot of the contribution vehicle dynamic characteristics make to GPS position estimation uncertainty
for a range of host vehicle dynamic capabilities. In order to indicate the contributions that vehicle dynamics make to position uncertainty, this demonstration
assumes that other contributory error sources are either negligible or nominal, e.g.,
•
•
•
No receiver clock bias. (That will come later.)
10 m RMS time-correlated pseudorange error due to iono spheric delay,
receiver bias, interfrequency biases, etc.
60 s pseudorange error correlation time.
Fig. 10.2 DAMP2 tracker performance versus σacc and τacc .
390
•
•
•
•
•
•
•
GNSS/INS INTEGRATION
Pseudoranges of each available satellite sampled every second.
10 m RMS pseudorange uncorrelated measurement noise.
29-satellite GPS configuration of March 8, 2006.
Only those satellites more than 15◦ above the horizon were used.
200 m/s RMS host vehicle velocity, representing a high-performance aircraft
or missile.
Host vehicle at 40◦ north latitude.
Results averaged over 1 h of simulated operation.
Figure 10.2 is output from the MATLAB m-file Damp2eval.m on the accompanying CD-ROM. It performs a set of GPS tracking simulations using the “DAMP2”
tracking filter described in Table 10.1 and Section 10.2.2.5. This filter allows the
designer to specify the RMS velocity, RMS acceleration, and acceleration correlation time of the host vehicle, and the plot shows how these two dynamic
characteristics influence position estimation accuracy.
These results would indicate that navigation performance is more sensitive
to vehicle acceleration magnitude than to its correlation time. Five orders of
magnitude variation in correlation time do not cause one order of magnitude
variation in RMS position estimation accuracy. At short correlation times, five
orders of magnitude variation in RMS acceleration2 cause around three orders
of magnitude variation in RMS position estimation accuracy. These simulations
were run at 40◦ latitude. Changing simulation conditions may change the results
somewhat.
The main conclusion is that unpredictable vehicle motion does, indeed, compromise navigation accuracy.
10.2.2 Specialized Host Vehicle Tracking Filters
In Kalman filtering, dynamic models are completely specified by two matrix
parameters:
1. The dynamic coefficient matrix F (or equivalent state transition matrix )
2. The dynamic disturbance covariance matrix Q
The values of these matrix parameters for six different vehicle dynamic models are listed in Table 10.1. They are all time-invariant (i.e., constant). As a
consequence, the corresponding state transition matrices
= exp (Ft)
are also constant, and can be computed using the matrix exponential function
(expm in MATLAB).
2
The RMS acceleration used here does not include the acceleration required to counter gravity.
EFFECTS OF HOST VEHICLE DYNAMICS
391
Table 10.1 also lists the independent and dependent parameters of the models.
The independent parameters can be specified by the filter designer. Because the
system model is time-invariant, the finite dependent variables are determinable
from the steady-state matrix Riccati differential equation,
0 = FP∞ + P∞ FT + Q,
(10.6)
the solution of which exists only if the eigenvalues of F lie in the left-half complex
plane. However, even in those cases where the full matrix Riccati differential
equation has no finite solution, a reduced equation with a submatrix of P∞
and corresponding submatrix of F may still have a finite steady-state solution.
For those with “closed form” solutions that can be expressed as formulas, the
solutions are listed below with the model descriptions.
The TYPE2 filter, for example, does not have a steady-state solution for its
Riccati equation without measurements. As a consequence, we cannot use meansquared velocity as a TYPE2 filter parameter for modeling vehicle maneuverability. However, we can still solve the Riccati equation with GNSS measurements
(which is not time-invariant) to characterize position uncertainty as a function of mean-squared vehicle acceleration (modeled as a zero-mean white noise
process).
10.2.2.1 Unknown Constant Tracking Model This model was used in
Section 8.8. There are no parameters for vehicle dynamics, because there are
no vehicle dynamics. The Kalman filter state variables are three components of
position, shown below as NED coordinates. The only model parameters are three
2 (0) for three direction components. These represent the initial posivalues of σpos
2 (0) can be different
tion uncertainties before measurements start. The value of σpos
in different directions. The necessary Kalman filter parameters for a stationary
antenna are then
⎡ 2
⎤
σnorth
0
0
2
σeast
P0 = ⎣ 0
0 ⎦ , the initial mean-squared position uncertainty,
2
0
0
σdown
= I, the 3 × 3 identity matrix,
Q = 0, the 3 × 3 zero matrix.
2
(0), may also influence GNSS
The initial position uncertainty, as modeled by σpos
signal acquisition search time. The other necessary Kalman filter parameters (H
and R) come from the pseudorange measurement model, which was addressed
in Section 8.8.
10.2.2.2 Damped Harmonic Resonator GNSS antennas can experience harmonic displacements in the order of several centimeters from host vehicle resonant modes, which are typically at frequencies in the order of ≈ 1 Hz (the
DAMP3
DAMP2
DAMP1
TYPE2
Harmonic resonator
(Example 8.3)
0
1
1
−2
−ω − 2
/ τ 0τ
0 1
0 0
0
/
0
1
0 −1/τvel
⎤
⎡
0
1
0
−1
⎥
⎢
⎢ 0
1 ⎥
⎥
⎢
τvel
⎣
−1 ⎦
0
0
τacc
⎡ −1
⎤
1
0
⎢ τpos
⎥
⎢
⎥
−1
⎢
⎥
⎢ 0
1 ⎥
⎢
⎥
τvel
⎣
−1 ⎦
0
0
τacc
2
3
0
⎣ 0
0
⎡
0
⎣ 0
0
⎡
0
0
0
/ 0
0
0
/
/
0
0
0
0
0
0
0
0
⎦
0
2
σjerk
t 2
⎤
⎤
0
⎦
0
2
σjerk
t 2
0
0
2 t 2
σacc
0
0
2
σacc
t 2
0
2
σacc
t 2
0
0
Unknown constant
2
Model Parameters (Each Axis)
F
Q
Model Name
TABLE 10.1. Vehicle Dynamic Models for GNSS Receivers
2
σpos
2
σvel
2
σacc
τacc
2
σvel
2
σacc
τacc
2
σvel
τvel
2
σacc
2
σpos
(0)
2
σpos
ω
τ
Independent Variables
→
→
→
∞
∞
∞
τpos
τvel
ρpos,vel
ρpos,acc
ρvel,acc
2
σjerk
2
σpos
→ ∞
τvel
ρvel,acc
2
σjerk
2
σpos
2
σvel
2
σpos
2
σacc
2
σacc
None
Dependendent Variables
392
GNSS/INS INTEGRATION
393
EFFECTS OF HOST VEHICLE DYNAMICS
suspension resonance of most passenger cars) to several Hz—but the effect is
small compared to other error sources.
However, a model of this sort (developed in Examples 8.1–8.7) is needed for
INS gyrocompass alignment, which is addressed in Chapter 9.
10.2.2.3 TYPE2 Tracking Model The TYPE2 tracker is older than Kalman
filtering. Given sufficient measurements, it can estimate position and velocity in
three dimensions. (Type 1 trackers do not estimate velocity.) The tracker uses a
host vehicle dynamic model with zero-mean white-noise acceleration, unbounded
steady-state mean-squared velocity (not particularly reasonable), and unbounded
steady-state mean-squared position variation (quite reasonable). When GNSS signals are lost, the velocity uncertainty variance will grow without bound unless
something is done about it—such as limiting velocity variance to some maximum value. Trackers based on this model can do an adequate job when GNSS
signals are present.
The model parameters shown in Table 10.1 are for a single-direction component, and do not include position. The full tracking model will include three
position components and three velocity components. The necessary Kalman filter
parameters for a 3D TYPE2 tracking filter include
⎡
⎢
⎢
⎢
P0 = ⎢
⎢
⎢
⎣
⎡
⎢
⎢
⎢
=⎢
⎢
⎣
⎡
⎢
⎢
⎢
Q=⎢
⎢
⎢
⎣
2
σnorth
0
0
0
0
0
2
σeast
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0 t
0 0
1 0
0 1
0 0
0 0
0
t
0
0
1
0
2
σv,north
0
0
⎤
0
0 ⎥
⎥
t ⎥
⎥,
0 ⎥
0 ⎦
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
0 σacc
t 2
0
0
0
0
0
0
0
0
2
σacc
t 2
0
0
0
0
2
σdown
0
0
0
0
0
0
0
0
0
0
2
σv,east
0
0
0
0
0
0
2
σacc
t 2
0
0
0
0
0
⎤
⎥
⎥
⎥
⎥,
⎥
⎥
⎦
2
σv,down
⎤
⎥
⎥
⎥
⎥,
⎥
⎥
⎦
2
where σacc
is the only adjustable parameter value. Adjusting it for a particular
application may take some experimenting.
394
GNSS/INS INTEGRATION
10.2.2.4 DAMP1 Tracking Model This type of tracking filter is based on the
Langevin equation3
d
1
v(t) = −
v(t) + w(t),
dt
τvel
(10.7)
F
where v(t) is a velocity component and w(t) is a zero-mean white-noise process
in continuous time.
It differs from the TYPE2 tracker in that it includes a velocity damping time
constant τvel , which is enough to put an eigenvalue of F in the left-half complex
plane and allow a steady-state variance for velocity. This is more realistic as a
model for a vehicle with finite speed capabilities. Also, the parameter τvel is a
measure of persistence of velocity, which would be useful for distinguishing the
dynamics of an oil tanker, say, from those of a jet ski.
The values of P0 and Q will be the same as for the TYPE2 tracker, and the
state transition matrix
⎡
⎢
⎢
⎢
=⎢
⎢
⎣
1
0
0
0
0
0
0
1
0
0
0
0
0 t
0 0
1 0
0 ε
0 0
0 0
0
t
0
0
ε
0
0
0
t
0
0
ε
⎤
⎥
⎥
⎥
⎥
⎥
⎦
ε = exp (−t/τvel ) .
The steady-state solution of the Riccati equation can be used to solve for
#
"
2
2
σacc
= σvel
1 − exp (−2t/τvel ) /t 2 .
(10.8)
Thus, one can specify the vehicle maneuver capability in terms of its mean2
square velocity σvel
and velocity correlation time τvel , and use Eq. 10.8 to specify
compatible values for the modeled Q matrix.
10.2.2.5 DAMP2 Tracking Model This is an even more realistic model for a
vehicle with finite speed and acceleration capabilities. It also includes an acceleration time correlation constant τacc , which is useful for distinguishing the more
lively vehicle types from the more sluggish ones. The corresponding steady-state
Riccati equation used for making the model a function of RMS velocity and
acceleration is not as easy to solve in closed form, however.
3
The first known stochastic differential equation, was published by Paul Langevin (1872–1946) in
1908 [116]. Langevin was a prolific scientist with pioneering work in many areas, including paraand diamagnetism, and sonar.
395
EFFECTS OF HOST VEHICLE DYNAMICS
The 2 × 2 submatrix of the
acceleration along a single axis
/
φ1, 1
vel, acc =
φ2, 1
state transition matrix relating velocity and
has the form
0
φ1, 2
(10.9)
φ2, 2
φ1, 1 = exp (−t/τvel )
#
"
τvel τacc exp (−t/τvel ) − exp (−t/τacc )
φ1, 2 =
τvel − τacc
φ2, 1 = 0
(10.11)
φ2, 2 = exp (−t/τacc ) ,
(10.13)
(10.10)
(10.12)
and the corresponding 2 × 2 submatrix of Q will be
/
Qvel, acc =
0
0
2
t 2
0 σjerk
0
.
(10.14)
The corresponding steady-state Riccati equation 10.6 can be solved for
#
"
2
2
σjerk
= σacc
1 − exp (−2t/τacc ) /t 2 ,
(10.15)
the analog of Eq. 10.8.
The steady-state Riccati equation can also be solved for the correlation coefficient
ρvel, acc = −τvel τacc σacc exp(−t/τacc )[exp(−t/τvel ) − exp(−t/τacc )]
σvel [1− exp(−t/τacc ) exp(−t/τvel )](τvel − τacc )
(10.16)
between the velocity and acceleration components. But solving the remaining
element p∞, 1, 1 = 0 of the Riccati equation 10.6 for τvel as variable dependent on
the independent variables requires solving a transcendental equation. It is solved
numerically in the MATLAB function Damp2Params.m on the accompanying
CD-ROM.
Figure 10.2 was generated by the MATLAB m-file Damp2eval.m, and
Fig. 8.21 was generated by the m-file SchmidtKalmanTest.m on the accompanying CD-ROM. Both include solutions of the Riccati equation for a DAMP2
GNSS position filter.
10.2.2.6 DAMP3 Tracking Model This type of filter is designed for vehicles
with limited but nonzero position variation, such as the altitudes of some surface
watercraft (e.g., riverboats) and land vehicles. Ships that remain a sea level are
at zero altitude, by definition. They need no vertical navigation, unless they are
trying to estimated tides. Flatwater boats and land vehicles in very flat areas can
probably do without vertical navigation, as well.
396
GNSS/INS INTEGRATION
Continuous-Time Solutions It is generally easier to solve the steady-state covariance equation in continuous time
0 = F3 P3 + P3 FT3 + Q3
(10.17)
for the parameters in the steady-state solution
⎡
⎤
p1,1 p1,2 p1,3
P3 = ⎣ p1,2 p2,2 p2,3 ⎦ ,
p1,3 p2,3 p3,3
where the other model parameters are
⎡
−τpos −1
1
⎢
0
−τvel −1
F3 = ⎢
⎣
0
0
⎡
⎤
0 0
0
⎢
⎥
⎥.
0
0
0
Q3 = ⎢
⎣
⎦
0 0 qc,3,3
⎤
0
1
−τacc
(10.18)
−1
⎥
⎥,
⎦
(10.19)
(10.20)
The six scalar equations equivalent to the symmetric 3 × 3 matrix equation
(10.17) are
⎫
0 = −p1,1 + p1,2 τpos
⎪
Eq1,1 ⎪
⎪
⎪
0 = −p1,2 τvel + p2,2 τpos τvel − p1,2 τpos + p1,3 τpos τvel
⎪
Eq1,2 ⎪
⎬
0 = −p1,3 τacc + p2,3 τpos τacc − p1,3 τpos
Eq1,3 .
0 = −p2,2 + p2,3 τvel
⎪
Eq2,2 ⎪
⎪
⎪
0 = −p2,3 τacc + p3,3 τvel τacc − p2,3 τvel
⎪
Eq2,3 ⎪
⎭
0 = −2 p3,3 + qc,3,3 τacc
Eq3,3
(10.21)
We wish to solve for the steady-state covariance matrix P, where the following
independent variables are to be specified
τacc , the acceleration correlation time constant
p1,1 , the mean-squared position excursion
p2,2 , the mean-squared velocity variation
p3,3 , the mean-squared acceleration variation
are to be specified, and the dependent variables are to be determined from
Eq. 10.21:
397
EFFECTS OF HOST VEHICLE DYNAMICS
τvel , the velocity correlation time constant
τpos , the position correlation time constant
p1,2 , the cross-covariance of position and velocity
p1,3 , the cross-covariance of position and acceleration
p2,3 , the cross-covariance of velocity and acceleration
qc,3,3 , the continuous-time disturbance noise variance
From the last of these (labeled Eq3,3 ), we obtain
qc,3,3 = 2
p3,3
.
τacc
(10.22)
From Eq2,2 and Eq2,3 , we have
p2,2
,
τvel
p3,3 τvel τacc
,
=
τacc + τvel
p2,3 =
(10.23)
(10.24)
and from equating the two solutions, we obtain
τvel =
p2,2 +
p2,2 2 + 4 p3,3 τacc 2 p2,2
.
2 p3,3 τacc
(10.25)
Similarly, from Eq1,1 and Eq1,2 , we have
p1,2 =
p1,1
,
τpos
p1,3 = −
(10.26)
p2,2 τpos 2 τvel − p1,1 τvel − p1,1 τpos
,
τpos 2 τvel
(10.27)
and from Eq1,3 , we obtain
p1,3 =
p2,3 τpos τacc
.
τacc + τpos
(10.28)
By equating the two independent solutions for p1,3 , we obtain a cubic polynomial
in τpos :
2
3
+ c3 τpos
0 = c0 + c1 τpos + c2 τpos
(10.29)
398
GNSS/INS INTEGRATION
with coefficients
c0 = p1,1 τvel τacc ,
(10.30)
c1 = p1,1 (τacc + τvel ) ,
(10.31)
c2 = −p2,2 τvel τacc + p1,1 ,
c3 = −τvel p2,3 τacc + p2,2 ,
(10.32)
(10.33)
which can be solved numerically using the MATLAB function roots.
The MATLAB solution sequence is then
0
1
2
3
4
5
Given: p1,1 , p2,2 , p3,3 and τacc .
Solve for τvel using Eq. 10.25.
Solve for p2,3 using Eq. 10.23.
Solve for τpos using Eq. 10.29 and the MATLAB function roots.
Solve for p1,2 using Eq, 10.26.
Solve for p1,3 using Eq. 10.28.
This solution is implemented in the MATLAB m-file DAMP3Params.m on the
CD-ROM.
This leaves the problem of solving for the discrete-time process noise covariance matrix Q3, discrete , which is not the same as the analogous matrix Q3 in
continuous time (solved using Eq. 10.22). There is a solution formula
/*
t
Q3, discrete = exp (t F)
0
T
exp (−s F) Q3 exp −s F ds exp t FT ,
0
(10.34)
but—given and P∞ —it is easier to use the steady-state formula
Q3, discrete = P3 − P 3 T ,
(10.35)
which is how the solution is implemented in DAMP3Params.m.
10.2.2.7 Tracking Models for Highly Constrained Trajectories Racecars in
some televised races have begun using integrated GPS/INS on each vehicle to
determine their positions on the track. The estimated positions are telemetered
to the television control system, where they are used in generating television
graphics (e.g., an arrow or dagger icon) to designate on the video images where
each car is on the track at all times. The integrating filters constrain the cars to be
on the 2D track surface, which improves the estimation accuracy considerably.
399
EFFECTS OF HOST VEHICLE DYNAMICS
FIG8 Tracking Model As a simple example of how this works, we will use a
one-dimensional “figure-8” track model, with the position on the track completely
specified by the down-track distance from a reference point.
The trajectory of a vehicle on the track is specified in terms of a formula,
⎡
δ pos
⎤
Northing
⎦
= ⎣ Easting
−Altitude
⎡
3 S sin (ω t + φ)
⎢
=⎢
⎣ 2 S sin (ω t + φ) cos (ω t + φ)
−1/2 h cos (ω t + φ)
(10.36)
⎤
⎥
⎥
⎦
(10.37)
where S is a track scaling parameter, ≈ [track length (m)]/14.94375529901562,
h is half the vertical separation where the track crosses over itself, ω = 2 π ×
[average speed (m/s)]/[track length (m)], and φ is an arbitrary phase angle (rad).
The phase rate φ̇ can be modeled as a random walk or exponentially correlated
process, to simulate speed variations. This model is implemented in the MATLAB function Fig8Mod1D, which also calculates vehicle velocity, acceleration,
attitude, and attitude rates. This m-file is on the accompanying CD-ROM. The
resulting trajectory is illustrated in Fig. 10.3.
Fig. 10.3 Figure-8 trajectory of length 1500 m.
400
GNSS/INS INTEGRATION
The resulting Kalman filter is implemented in the MATLAB m-file GPSTrackingDemo.m on the CD-ROM. This particular implementation is for a 1.5-km
track with vehicle speeds of 90 km/h ±10% RMS random variation. The Kalman
filter model in GPSTrackingDemo.m uses only two vehicle states: (1) the phase
angle φ and (2) its derivative φ̇, which is modeled as an exponentially correlated random process with a correlation time constant of 10 s and RMS value
equivalent to ±10% variation in speed.
10.2.2.8 Filters for Spacecraft Unpowered vehicles in space do not have sufficiently random dynamics to justify a tracking filter. They may have unknown,
quasiconstant orbit parameters, but their trajectories over the short term are essentially defined by a finite set of parameters. GNSS vehicle tracking then becomes
an orbit determination problem. The orbital parameters may change during brief
orbit changes, but the problem remains an orbit determination problem with
increased uncertainty in initial conditions (velocity, in particular).
10.2.2.9 Other Specialized Vehicle Filter Models The list of models in Table
10.1 is by no means exhaustive. It does not include the FIG8 filter described
above. Other specialized filters have been designed for vehicles confined to narrow corridors within a limited area, such as race cars on a 2D track or motor
vehicles on streets and highways. Specialized filters are also required for trains,
which need to know where they are on a 1D track, and possibly which set of
parallel rails they are on.
Still, the models listed in Table 10.1 and described above should cover the
majority of GNSS applications.
10.2.2.10 Filters for Different Host Vehicle Types Table 10.2 lists some
generic host vehicle types, along with names of models in Table 10.1 that might
be used for GNSS position tracking on such vehicles.
TABLE 10.2. Filter Models for Unknown Vehicle Dynamics
Host Vehicles
Filter Models
Horizontal Directions
None (fixed to earth)
Parked
Ships
Land vehicles
Aircraft and missiles
Spacecraft
Unknown constant
Damping harmonic resonator
Vertical Direction
Unknown constant
Damping harmonic
resonator
DAMP1, DAMP2
Unknown constant
DAMP1, DAMP2
DAMP3
DAMP1, DAMP2
DAMP2, DAMP3
In free fall, use orbit estimation models
After maneuvers, increment velocity uncertainty
EFFECTS OF HOST VEHICLE DYNAMICS
401
TABLE 10.3. Statistical Parameters of Host Vehicle
Dynamics
Symbol
Definition
2
σpos
2
σvel
2
σacc
2
σjerk
ρi,j
Mean-squared position excursionsa
Mean-squared vehicle velocity (E |v|2 )
Mean-squared vehicle acceleration (E |a|2 )
Mean-squared jerk (E |ȧ|2 )
Correlation coefficients between position,
velocity, and acceleration variations
Position correlation time
Velocity correlation time
Acceleration correlation time
Suspension resonant frequency
Suspension damping time constant
τpos
τvel
τacc
ωresonant
τdamping
a
Mean-squared position excursions generally grow without
bound, except for altitudes of ships (and possibly land vehicles).
10.2.2.11 Parameters for Vehicle Dynamics Table 10.3 contains descriptions
of the tracking filter parameters shown in Table 10.1. These are statistical parameters for characterizing random dynamics of the host vehicle.
10.2.2.12 Empirical Modeling of Vehicle Dynamics The most reliable vehicle
dynamic models are those based on data from representative vehicle dynamics.
Empirical modeling of the uncertain dynamics of host vehicles requires data (i.e.,
position and attitude, and their derivatives) recorded under conditions representing the intended mission applications.
The ideal sensor for this purpose is an INS, or at least an inertial sensor
assembly (ISA) capable of measuring and recording 3D accelerations and attitude
rates (or attitudes) during maneuvers of the host vehicle.
The resulting data are the sum of three types of motion:
1. Internal motions due to vibrating modes of the vehicle, excited by propulsion noise and flow noise in the surrounding medium. The oscillation
periods for this noise generally scale with the size of the host vehicle,
but are generally in the order of a second or less.
2. Short-term perturbations of the host vehicle that are corrected by steering,
such as turbulence acting on aircraft or potholes acting on wheeled vehicles.
These also excite the vibrational modes of the vehicle.
3. The intended rigid-body motions of the whole vehicle to follow the planned
trajectory. The frequency range of these motions is generally 10 Hz and
often < 1 Hz.
Only the last of these is of interest in tracking. It can often be separated from
the high-frequency noise by lowpass filtering, ignoring the high-frequency end of
402
GNSS/INS INTEGRATION
the power spectral densities and cross-spectral densities of the data. The inverse
Fourier transforms of the low-end power spectral data will yield autocovariance
functions that are useful for modeling purposes. The statistics of interest in these
autocovariance functions are the variances σ 2 (values at zero correlation time)
and the approximate exponential decay times τ of the autocovariances.
10.2.3 Vehicle Tracking Filter Comparison
The alternative GNSS receiver tracking filters of the previous section were evaluated using the figure-8 track model described in Section 10.2.2.7. This is a
trajectory confined in all three dimensions, and more in some dimensions than
others.
10.2.3.1 Simulated Trajectory The simulated trajectory is that of an automobile on a banked figure-8 track, as illustrated in Fig. 10.3. The MATLAB m-file
Fig8TrackDemo.m on the CD-ROM generates a series of plots and statistics of
the simulated trajectory. It calls the MATLAB function Fig8Mod1D, which generates the simulated dynamic conditions on the track, and it outputs the following
statistics of nominal dynamic conditions:
RMS N-S Position Excursion=
RMS E-W Position Excursion=
RMS Vert. Position Excursion=
RMS N-S Velocity=
RMS E-W Velocity=
RMS Vert. Velocity=
RMS N-S Acceleration=
RMS E-W Acceleration=
RMS Vert. Acceleration=
RMS Delta Velocity North=
=
RMS Delta Velocity East=
=
RMS Delta Velocity Down=
=
N. Position Correlation
E. Position Correlation
Vertical Position Corr.
N. Velocity Correlation
E. Velocity Correlation
Vertical Velocity Corr.
N. Acceler. Correlation
E. Acceler. Correlation
Vertical Acceler. Corr.
Time=
Time=
Time=
Time=
Time=
Time=
Time=
Time=
Time=
212.9304 meter
70.9768 meter
3.5361 meter
22.3017 m/s
14.8678 m/s
0.37024 m/s
2.335 m/s/s
3.1134 m/s/s
0.038778 m/s/s
0.02335 m/s at t =0.01 sec.
2.334 m/s at t =1 sec.
0.031134 m/s at t =0.01 sec.
3.1077 m/s at t =1 sec.
0.00038771 m/s at t =0.01 sec.
0.038754 m/s at t =1 sec.
13.4097 sec.
7.6696 sec.
9.6786 sec.
9.6786 sec.
21.4921 sec.
13.4097 sec.
13.4097 sec.
7.6696 sec.
9.6786 sec.
403
EFFECTS OF HOST VEHICLE DYNAMICS
TABLE 10.4. Comparison of Alternative GNSS
Filters on 1.5-km Figure-8 Track Simulation
GNSS
Filter
RMS Position Estimated Errorsa (m)
North
East
Down
TYPE2
DAMP2
DAMP3
FIG8
42.09
22.98
7.34
0.53
a
40.71
25.00
10.52
0.31
4.84
3.51
3.31
0.01
Clock errors not included.
These statistics are used for “tuning” the filter parameters for each of the alternative vehicle tracking filters—within the capabilities of the tracking filter.
10.2.3.2 Results The MATLAB m-file GPSTrackingDemo.m on the accompanying CD-ROM simulates the GPS satellites, the vehicle, and all four types
of filters on a common set of pseudorange measurements over a period of 2 h.
The position estimation results are summarized in Table 10.4 for one particular
simulation.
The m-file GPSTrackingDemo.m generates many more plots to demonstrate
how the different filters are working, including plots of the simulated and estimated pseudorange errors for each of the 29 satellites, some of which are not
in view. Because the simulation uses a pseudo-random-number generator, the
results can change from run to run.
10.2.3.3 Model Dimension versus Model Constraints These results indicate
that dilution of information is not just a matter of state vector dimension. One
might expect that the more variables there are to estimate, the less information
will be available for each variable. In Table 10.4, the DAMP3 model has three
more state variables than do the TYPE2 or DAMP2 models, yet it produces better
results. The other issue at work here is the degree to which the model constrains
the solution, and this factor better explains the ordering of estimation accuracy.
The degree to which the model constrains the solution increases downward in
the table, and simulated performance improves monotonically with the degree of
constraint.
10.2.3.4 Role of Model Fidelity These results strongly suggest some performance advantage to be gained by tuning the vehicle tracking filter structure and
parameters to the problem at hand. For the simulated trajectory, the accelerations,
velocities and position excursions are all constrained, and the model that takes
greatest advantage of that is FIG8, the track-specific model.
404
GNSS/INS INTEGRATION
10.3 LOOSELY COUPLED INTEGRATION
10.3.1 Overall Approach
At an abstract level, loosely coupled implementations represent the two sensor
systems in a mathematical model of the sort
zGNSS = hGNSS (xhostveh ) + δ GNSS
d
δ GNSS ≈ fGNSS (δ GNSS , xhostveh ) + wGNSS (t)
dt
zINS = hINS (xhostveh ) + δ INS
d
δ INS ≈ fINS (δ INS , xhostveh ) + wINS (t)
dt
(10.38)
(10.39)
(10.40)
(10.41)
where zGNSS represents the GNSS output; zINS represents the INS output; xhostveh
represents either the “true” navigation state of the host vehicle (the one both
GNSS receiver and INS are mounted on), or the best available estimate of the
navigation state; δ GNSS represents GNSS output error; fGNSS (δ GNSS , xhostveh ) and
the white-noise process wGNSS (t) represent the dynamic model for δ GNSS assumed
in the GNSS/INS integration scheme; δ INS represents INS output error; and
fINS (δ INS , xhostveh ) and the white-noise process wINS (t) represent the dynamic
model for δ INS assumed in the GNSS/INS integration scheme.
Different approaches to loosely coupled integration are free to assume quite
different mathematical forms for the functions fGNSS and fINS , yet each approach
would still be considered loosely coupled.
10.3.2 GNSS Error Models
10.3.2.1 Empirical Modeling The trajectory of the antenna has some influence
on receiver positioning errors. However, getting data on the actual GNSS position
errors is not that easy unless the receiver antenna is stationary. Otherwise, in
order to measure the actual receiver position errors, one needs an independent,
more accurate means of measuring antenna position while it is moving—or use
simulation of receiver errors during simulation.
We can use the results of the figure-8 track simulation to illustrate how the
trajectory influences positioning errors, and how receiver error data for a stationary antenna can be used to develop a model for loosely coupled GNSS/INS
integration.
Figure-8 Track Simulation Results Figures 10.4–10.6 show plots of the power
spectral densities of position errors from using TYPE2, DAMP2 and DAMP3
filters, respectively, in the simulations summarized in Table 10.4 and generated
by the MATLAB m-file GPSTrackingDemo.m.
Harmonic Errors These simulations were for one-minute trajectories around a
figure-8 track, and they exhibit harmonics at one cycle per minute (≈ 0.01667 Hz)
LOOSELY COUPLED INTEGRATION
Fig. 10.4
Power spectral densities of TYPE2 GPS filter errors.
Fig. 10.5
Power spectral densities of DAMP2 GPS filter errors.
405
406
GNSS/INS INTEGRATION
Fig. 10.6
Power spectral densities of DAMP3 GPS filter errors.
and two cycles per minute (≈ 0.0333 Hz), standing 10–30 dB above background
noise. The slight broadening of these spectral peaks is due to simulated random
vehicle speed variations of ±10% RMS from the mean speed.
These spectral peaks are most likely due to filter lags.4 The track altitude
model has only one harmonic at one cycle per minute, and that is the only
harmonic peak evident in the plots of the vertical components of position errors.
PSD for Stationary GPS Antenna Figure 10.7 is a plot of the PSDs analogous to
Figs. 10.4–10.6, but with a stationary antenna. The spectra are almost identical
to those in Figs. 10.4–10.6, but without the harmonic peaks. This would indicate
that the harmonic peaks in Fig. 10.6 are the dominant effects of the figure-8
trajectory, and the rest of the spectrum is due to the other GPS tracking error
sources (random ranging errors).
Exponentially Correlated Errors The background noise in the power spectral
densities in Fig. 10.7 looks much like the exponentially correlated noise in
Fig. 8.9, which flattens out at the low-frequency end but falls off at about −20
dB/decade at frequencies 1/τ , where τ is the correlation time. These correlated errors are most likely due to the correlated errors in the simulated GPS
pseudoranges (i.e., signal delay errors exponentially correlated with one minute
correlation time).
4
A diligent engineer would always take the PSD of receiver error data to help in understanding its
statistical properties. Seeing these pronounced harmonic peaks, she or he would probably be lead to
using an alternative tracking filter such as the FIG8 model.
407
LOOSELY COUPLED INTEGRATION
Fig. 10.7 Error power spectral densities without motion.
Indeed, the empirically calculated autocorrelations functions plotted in
Fig. 10.8 do appear to be exponential functions. Exponential functions would
look like straight lines on these semi-log plots, and that is about what the plots
show.
The estimated exponential decay time-constants will equal the lag times at
which these straight lines cross the 1/e threshold, and these are the values shown
in Table 10.5. These vary a bit, depending on the receiver filter used, but the
correlation times of the horizontal position components are relatively consistent
for a given filter.
10.3.3 Receiver Position Error Model
We will use the DAMP3 receiver filter model results to show how they can be
used in deriving an appropriate receiver position error model for loosely-coupled
GPS/INS integration. This will be an exponentially damped position error model
of the sort
⎡
⎤ ⎡
⎤⎡
⎤
−1/τhor
δpGNSS N
0
0
δpGNSS N
d
⎣ δpGNSS E ⎦ = ⎣
⎦ ⎣ δpGNSS E ⎦
0
−1/τhor
0
dt
0
0
−1/τ
δp
δp
GNSS D
⎡
⎤
whor (t)
+ ⎣ whor (t) ⎦ ,
wvert (t)
vert
GNSS D
(10.42)
408
GNSS/INS INTEGRATION
Fig. 10.8 Error autocorrelation functions without motion.
TABLE 10.5. Receiver Position Error
Correlation Times
Correlation Times (s)
Receiver
Filter
North
East
Down
TYPE2
DAMP2
DAMP3
49
42
37
49
30
38
111
23
32
with independent noise on all three channels.
10.3.4 INS Error Models
Error models for GNSS receivers will depend on uncertainty about the internal
clock and host vehicle dynamics, and the internal filtering in the receiver for
coping with it. The distributions of “upstream” errors in the signal information
at the receiver antenna are essentially fixed by the GNSS system error budget,
which is constant.
Models for INS errors, on the other hand, depend to some lesser degree on
vehicle dynamics, but are dominated by the INS error budget, which depends
very much on system design and the quality of sensors used. We will first use a
very simple INS model to demonstrate how loosely coupled GNSS/INS integration performance depends on INS performance. This simplified model splits the
409
LOOSELY COUPLED INTEGRATION
loosely coupled GNSS/INS integration problem into two problems:
1. An altitude problem, solved by a two-state Kalman filter using GNSS altitude outputs to stabilize the otherwise unstable INS vertical channel.
2. The remaining horizontal navigation problem, which is solved using an
independent eight-state Kalman filter.
This splitting ignores the coriolis coupling between the vertical and horizontal
channels, but the coriolis coupling is not a dominating error mechanism anyway.
10.3.4.1 Using GNSS Altitude for INS Vertical Channel Stabilization This is
quite similar to the barometric altimeter implementation used in Section 9.5.4.1,
with a GNSS altitude error model replacing the altimeter error model.
10.3.4.2 Random-Walk Tilt Model This is a generic INS system-level error
model designed to demonstrate how INS CEP rate influences integrated GPS/INS
performance. It is not a particularly faithful error model for any INS, but it does
serve to demonstrate in very general terms how the gross error characteristics of
inertial navigators are mitigated by GNSS/INS integration.
Without carouseling, INS errors tend to be dominated by tilt errors, which can
result from accelerometer bias errors or gyro errors. A relatively simple model
for horizontal error dynamics uses the six state variables:
δpN
δpE
δvN
δvE
ρN
ρE
=
=
=
=
=
=
north component of position error.
east component of position error.
north component of velocity error.
east component of velocity error.
tilt error rotation about north axis.
tilt error rotation about east axis.
The corresponding dynamic model for this state vector will be
⎡
⎤ ⎡
0
0
1
0
0
δpN
0
0
0
1
0
⎢ δpE ⎥ ⎢
⎥ ⎢
2
d ⎢
−
0
0
−2s
−g
⎢ δvN ⎥ ⎢
Sch.
⎢
⎥=⎢
0
−2Sch. 2s
0
0
dt ⎢ δvE ⎥ ⎢
⎢
⎣ ρ ⎦ ⎣
0
0
0
0
0
N
ρE
0
0
0
0
0
⎡
⎢
⎢
⎢
+⎢
⎢
⎣
0
0
0
0
wρ (t)
wρ (t)
⎤
⎥
⎥
⎥
⎥,
⎥
⎦
0
0
0
g
0
0
⎤⎡
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎢
⎥⎣
⎦
δpN
δpE
δvN
δvE
ρN
ρE
⎤
⎥
⎥
⎥
⎥
⎥
⎦
Fhor
(10.43)
410
GNSS/INS INTEGRATION
s = sin (Latitude) ,
def
Sch. = Schuler frequency
≈ 0.00124 (rad/s),
def
= earth ratation rate
≈ 7.3 × 10−5 (rad/s),
def
g = gravitational acceleration
≈ 9.8 m
s−2 .
(10.44)
(10.45)
(10.46)
(10.47)
(10.48)
(10.49)
(10.50)
The CEP rate of the modeled INS will depend only on the single parameter
qcontinuous = E wρ2 (t)
t
or its equivalent discrete-time counterpart
qdiscrete = E wk .
k
The resulting relationship between CEP rate and the discrete-time parameter
to qdiscrete is computed by the MATLAB m-file HorizINSperfModel.m on the
accompanying CD-ROM, the result of which is plotted in Fig. 10.9. The values
of q for some specific CEP rates are listed in Table 10.6.
Fig. 10.9 CEP rate versus model parameter q.
411
LOOSELY COUPLED INTEGRATION
TABLE 10.6. q-Parameter Values
versus CEP Rates
CEP Rate (nmi/h)a
0.01
0.1
1
10
100
1000
a
q (rad2 /s)
4.5579 × 10−16
4.5579 × 10−14
4.5579 × 10−12
4.5579 × 10−10
4.5579 × 10−8
4.5579 × 10−6
Nautical miles per hour.
10.3.4.3 Performance Analysis Performance advantages for GNSS/INS integration include improvements in velocity, acceleration, attitude, and attitude
rate estimates, as well as improving position estimates. Nevertheless, the results
shown here are for position uncertainty, only.
The full Kalman filtering model used for loosely coupled integration includes
20 state variables:
•
•
•
Three state variables for time-correlated GPS output position errors.
The nine state variables of the DAMP3 vehicle dynamics model (three
components each of acceleration, velocity, and position).
Eight state variables for INS output errors (three components each of horizontal velocity and position errors, plus two tilt errors). This part of the
model includes one parameter characterizing the CEP rate of the INS navigator, and one parameter (RMS accelerometer noise) characterizing relative
vertical channel instability.
The six measurements include three position outputs from the GPS receiver and
three position outputs from the INS.
Figures 10.10 and 10.11 were generated by the MATLAB m-file GPSINSwGPSpos.m to show the theoretical relative performance of a GPS receiver, an INS,
and a loosely coupled integrated GPS/INS system. Different plots show performances for several levels of INS performance, as characterized by CEP rate (for
horizontal accuracy) and accelerometer noise (for vertical channel accuracy).
Figure 10.10 shows predicted RMS horizontal position uncertainty as a function of time over 2 h, and INS performances ranging from 0.1 nmi/h (nautical
miles per hour) (“high accuracy”) to 100 nmi/h (very low accuracy). The plot
illustrates how INS-only performance degrades over time, GPS-only performance
remains essentially constant over time, and loosely coupled integrated GPS/INS
performance is better than either.
Figure 10.11 shows predicted RMS altitude uncertainty as a function of time
over 2 h, and accelerometer noise ranging from 0.001 m/s2 per root hour to
1 m/s2 per root hour. The plot illustrates how INS-only altitude diverges over
412
GNSS/INS INTEGRATION
Fig. 10.10 Theoretical horizontal performance of loosely coupled GPS/INS integration.
Fig. 10.11 Theoretical vertical performance of loosely coupled GPS/INS integration.
TIGHTLY COUPLED INTEGRATION
413
time, GPS-only performance remains essentially constant over time, and loosely
coupled integrated GPS/INS performance is better than either. In this example,
GPS altitude uncertainty is about 1.5 m RMS, because the DAMP3 filter used
for vehicle tracking constrains vehicle altitude to a small dynamic range (based
on figure-8 track dynamics).
There are many other ways to implement loosely coupled GNSS/INS integration, but this particular simple example5 demonstrates how weakly loosely
coupled GPS/INS performance depends on standalone INS performance. This
can have a profound effect on overall system cost, because INS cost tends to
vary inversely as CEP rate. This, in turn, has spurred greater interest in developing lower-cost, lower-performance inertial sensors for integrated systems. The
major advantages of GNSS/INS integration come from improvements in the accuracy of velocity, attitude, and other parameters, not just position alone, and these
other performance improvements are also driving lower-cost sensor development
interests.
10.4 TIGHTLY COUPLED INTEGRATION
There are many GNSS/INS integration approaches requiring changes in the internal software and hardware implementations of GNSS receivers and/or inertial
navigation systems. The following subsections describe some that are in use or
under development.
10.4.1 Using GNSS for INS Vertical Channel Stabilization
Inertial navigators designed for submarines have used depth (water pressure)
sensors for vertical channel stabilization,6 and early inertial systems designed for
aircraft or surface vehicles7 used a barometric altimeter interface for the same
reason. GNSS/INS integration has limited applicability for submarines, whose
commanders are reluctant to pop up an antenna while submerged, for fear of being
detected. For aircraft and surface vehicles, however, GNSS is more accurate than
a barometric altimeter.
This would be considered a less-than-loosely coupled integration, because it
requires some INS software changes. However, because vertical channel stabilization has always been part of INS implementations, the changeover to using
GNSS may require only relatively minor changes to INS software.
5
This example would seem to imply that one can effectively stabilize the INS vertical channel
without modifying the internal INS software, but this cannot be done in practice. The coriolis effect
would couple large vertical velocity errors within the INS implementation into horizontal acceleration
errors. INS vertical channel stabilization is best implemented within the INS software.
6
Submarines can also use electromagnetic waterspeed sensors (EM logs) for velocity error damping.
7
Except those operating at the sea surface, which can stabilize the vertical channel by fixing altitude
to a constant.
414
GNSS/INS INTEGRATION
10.4.2 Using INS Accelerations to Aid GNSS Signal Tracking
In addition to the effects due to dilution of information, vehicle dynamics have
a serious impact on the ability of a GNSS receiver to maintain phase lock on
the satellite signals, and to reacquire lock after it is lost. These effects can be
mitigated significantly in GNSS/INS integration, by using the acceleration measurements of the INS to predict the Doppler shift changes before they accumulate
as phase tracking errors.
If aLOS is the sensed acceleration component along the line of sight toward a
particular GNSS satellite, the rate of change of a carrier frequency fcarrier from
that satellite, due to vehicle acceleration, will be
aLOS fcarrier
d
fcarrier =
,
dt
c
(10.51)
where c = 299792458 m/s is the speed of light. This computed value of frequency
shift rate can be applied to the carrier phase-lock loop to head off an otherwise
rapidly accumulating carrier phase tracking error.
The phase-lock loops used for maintaining GNSS signal phase lock are primarily PI controllers with control margins designed for relatively benign vehicle
dynamics. Their control margins can be extended to more highly maneuverable vehicles by using accelerations in a PID controller, because acceleration is
related to the derivative of frequency. This level of GNSS/INS integration generally requires hardware and software changes to the receiver signal interfaces
and phase lock control, and may require INS output signal changes, as well.
10.4.3 Using GNSS Pseudoranges
All GNSS navigation receivers use pseudoranges as their fundamental range
measurements. The filtering they do to estimate antenna position increases data
latency at the receiver outputs. If these outputs are filtered again for GNSS/INS
integration, the lags only increase. Time tagging and output of the pseudoranges
helps to control and reduce data latency in GNSS/INS integration.
GNSS/INS integration software using pseudorange data may include receiver
clock bias and drift estimation, even though this function is already implemented
in the receiver software. The reason for wanting to include clock error estimation in the GNSS/INS integration software is that the integration filter uses
pseudoranges which are corrupted by residual clock errors. The integrating filter has access to INS measurements, which should allow it to make even better
estimates of clock errors than those within the receiver.
Using these improved clock corrections within the receiver may not be necessary, especially if the improvements are rather modest. It simplifies the receiver
interface if the clock control loop can be maintained within the receiver itself.
10.4.3.1 Example This is essentially the same model used in Section 10.3.4.3,
but with pseudoranges used in place of GPS receiver position estimates. It is
TIGHTLY COUPLED INTEGRATION
415
implemented in the MATLAB m-file GPSINSwPRs.m on the accompanying CDROM. As before, the INS system-level error model is a generic model designed
to demonstrate how INS CEP rate influences integrated GPS/INS performance.
In this case, however, simulated GPS satellite geometry variations will introduce
some irregularity into the plots.
The full Kalman filter sensor integration model for this example includes 46
state variables:
•
•
•
The nine state variables of the DAMP3 vehicle dynamics model (three
components each of acceleration, velocity, and position).
Twenty-nine state variables for time-correlated pseudorange errors. (This
had been three state variables for GPS receiver position errors in the loosely
coupled example.)
Eight state variables for INS output errors (three components each of horizontal velocity and position errors, plus two tilt errors). This part of the
model includes one parameter characterizing the CEP rate of the INS navigator, and one parameter (RMS accelerometer noise) characterizing relative
vertical channel instability.
The measurements include three position outputs from the INS and the pseudoranges from all acquired satellite signals.
Simulation results are plotted in Figures 10.12–10.16. Some of this is “comparing apples and oranges,” because the vertical disturbance model for INS errors
is driven by vertical accelerometer noise, which is not present in the GPS-only
model.
10.4.4 Real-Time INS Recalibration
There are some potential GNSS/INS integration applications for which the host
vehicle must navigate accurately through GNSS signal outages lasting a minute
or more. These applications include manned or unmanned vehicles operating
under leaf cover, through tunnels or in steep terrain (including urban canyons).
They also include “standoff” precision weapons for attacking high-value targets
protected by GNSS jamming. If the INS can use GNSS to continuously calibrate
its sensors while GNSS is available, then the INS can continue to navigate over
the short term without GNSS, starting with freshly recalibrated sensors. If and
when GNSS signals become available again, the improved position and velocity
information from the INS can speed up signal reacquisition.
10.4.4.1 Example This example integration architecture is tightly coupled in
two ways:
•
•
It uses pseudoranges directly for a GNSS receiver.
It feeds back the estimated calibration parameters to the INS software.
416
GNSS/INS INTEGRATION
Fig. 10.12 Theoretical RMS horizontal position of tightly coupled GPS/INS
integration.
Fig. 10.13 Theoretical RMS altitude of tightly coupled GPS/INS integration.
TIGHTLY COUPLED INTEGRATION
417
Fig. 10.14 Theoretical RMS horizontal velocity of tightly coupled GPS/INS
integration.
Fig. 10.15
Theoretical RMS vertical velocity of tightly coupled GPS/INS integration.
418
GNSS/INS INTEGRATION
Fig. 10.16 Theoretical RMS tilt uncertainty of tightly coupled GPS/INS integration.
Performance statistics of interest for this example problem would include RMS
position, velocity, and attitude uncertainty after periods of a minute or more
without GNSS signals.
The software for GNSS/INS integration can run in a relatively slow loop (1 s)
compared to the INS dynamic integration loop (1–10 ms, typically), because it
is primarily tracking the slowly changing parameters of the inertial sensors. This
reduces the added computational requirements considerably.
The example GNSS/INS integration model shown here uses 58 state variables:
15 state variables related to INS navigation errors (position, velocity, acceleration, attitude, and attitude rate) due to sensor calibration errors. The entire
INS system-level error model used before had only 8 state variables, but
these are insufficient for modeling the effects of calibration errors (below).
12 state variables related to calibration parameter errors (2 state variables to
model time-varying calibration parameters of each of the six inertial sensors). This increases the INS error model state vector size to 27 variables,
where there had been only 8 before.
2 state variables for the receiver clock (bias and drift). These are included
because residual clock errors will corrupt the pseudorange measurements
output from the receiver.
29 state variables for pseudorange biases (one for each of the 29 GPS satellites
for the March 2006 ephemerides used in simulations)
419
TIGHTLY COUPLED INTEGRATION
The last 31 of these are essentially nuisance variables that must be included
in the state vector because they corrupt the measurements (pseudoranges, plus
INS errors in position, velocity, acceleration, attitude, and attitude rate). Some
reductions are possible. The number of state variables devoted to pseudorange
biases can be reduced to just those usable at any time, or it can be eliminated
altogether by using Schmidt–Kalman filtering. The number of variables used for
sensor calibration can be reduced to just those deemed sufficiently unstable to
warrant recalibration.
The number of variables used for sensor calibration can also be increased to
include drift rates of the calibration parameters.
Integration Filter Models Mathematically, the Kalman filtering model for
GNSS/INS integration is specified by
1. The model state vector x and two matrix parameters of the state dynamic
model:
(a) The state transition matrix k or dynamic coefficient matrix F(t),
related by
=*
>
tk
k = exp
F(s) ds .
tk−1
(b) The dynamic disturbance noise covariance matrix Q.
2. The model measurement vector z and two matrix parameters of the measurement model:
(a) The measurement sensitivity matrix H.
(b) The measurement noise covariance matrix R.
Potential State Variables The components of a potential state vector x for this
example problem include the following:
1
2
3
4
5
6
7
8
9
10
δlat
δlon
δalt
δvN
δvE
δvD
δaN
δaE
δaD
δρN
=
=
=
=
=
=
=
=
=
=
11
12
13
14
δρE
δρD
δωN
δωE
=
=
=
=
error in INS latitude (rad),
error in INS longitude (rad),
error in INS altitude (m),
error in north velocity of INS (m/s),
error in east velocity of INS (m/s),
error in downward velocity of INS (m/s),
error in north acceleration of INS(m s1 s−1 ),
error in east acceleration of INS(m s1 s−1 ),
error in downward acceleration of ISA(m s1 s−1 ),
north component of INS attitude error rotation
vector, representing INS tilt (rad),
east component of INS attitude error (rad),
downward component of INS attitude error (rad),
north component of vehicle rotation rate error (rad/s),
east component of vehicle rotation rate error (rad/s),
420
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
GNSS/INS INTEGRATION
δωD
δab1
δab2
δab3
δaS1
δaS2
δaS3
δgb1
δgb2
δgb3
δgS1
δgS2
δgS3
δclockb
δclockd
δPRN1
δPRN2
58 δPRN29
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
..
.
downward component of vehicle rotation rate error (rad/s),
first accelerometer bias error(m s1 s−1 ),
second accelerometer bias error(m s1 s−1 ),
third accelerometer bias error(m s1 s−1 ),
relative error in first accelerometer scale factor (unitless),
relative error in second accelerometer scale factor (unitless),
relative error in third accelerometer scale factor (unitless),
first gyroscope bias error (rad/s),
second gyroscope bias error (rad/s),
third gyroscope bias error (rad/s),
relative error in first gyroscope scale factor (unitless),
relative error in second gyroscope scale factor (unitless),
relative error in third gyroscope scale factor (unitless),
residual receiver clock bias error (m),
residual receiver clock drift rate error (m/s),
first satellite pseudorange bias error (m),
second satellite pseudorange bias error (m),
= last satellite pseudorange bias error (m).
The variables δab 1 , . . . , δaS 1 , . . . , δgb 1 , . . . , δgS 1 , . . . are sensor calibration coefficient errors. They are numbered 1–3 because the sensor input axis directions
are not fixed in RPY coordinates (as they would be for noncarouseled strapdown
systems) or NED coordinates (as they would be for noncarouseled gimbaled
systems).
A rotation vector ρ representing attitude has been introduced in Chapter 9 for
coning error correction. In that case and in this case, the representation should
work because the magnitude |ρ| π, where the rotation vector model behaves
badly.
State Vector Dynamics The first-order differential equations defining the state
dynamic model are as follows:
δvN
d
,
δlat =
dt
RN (φlat ) + hlat
d
δvE
δlon =
,
[RE (φlat ) + halt ] / cos φlat
dt
d
δalt = −δvD ,
dt
d
δvN = δaN − 2 sin φlat δvE + wa1N (t) + wa2N (t) + wa3N (t),
dt
(10.52)
(10.53)
(10.54)
(10.55)
TIGHTLY COUPLED INTEGRATION
d
δvE = δaE + 2 sin φlat δvN + 2 cos φlat δvD + wa1E (t)
dt
+ wa2E (t) + wa3E (t),
d
δvD = δaD − 2 cos φlat δvN + wa1D (t)
dt
+ wa2D (t) + wa3D (t),
d
δaN = 0,
dt
d
δaE = 0,
dt
d
δaD = 0,
dt
/
0
d
1
1
1 ψ sin (ψ)
δρN = ρD ωN − ρN ωD + 1 −
dt
2
2
2 1 − cos (ψ)
0
/
ρD (−ρD ωE + ρE ωD ) − ρN (−ρE ωN + ρN ωE )
,
×
ψ2
&
%
d
1
1
1 ψ sin (ψ)
δρE = ρD ωN − ρN ωD + 1 −
dt
2
2
2 1 − cos (ψ)
0
/
ρD (−ρD ωE + ρE ωD ) − ρN (−ρE ωN + ρN ωE )
,
×
ψ2
&
%
d
1
1
1 ψ sin (ψ)
δρD = − ρE ωN + ρN ωE + 1 −
dt
2
2
2 1 − cos (ψ)
0
/
−ρE (−ρD ωE + ρE ωD ) + ρN (ρD ωN − ρN ωD )
,
×
ψ2
!
def
ψ = ρN2 + ρE2 + ρD2 ,
d
δωN
dt
d
δωE
dt
d
δωD
dt
d
δab 1
dt
d
δab 2
dt
421
(10.56)
(10.57)
(10.58)
(10.59)
(10.60)
(10.61)
(10.62)
(10.63)
(10.64)
= wω1 (t),
(10.65)
= wω2 (t),
(10.66)
= wω3 (t),
(10.67)
= wba 1 (t),
(10.68)
= wba 2 (t),
(10.69)
422
GNSS/INS INTEGRATION
d
δab 3
dt
d
δaS 1
dt
d
δaS 2
dt
d
δaS 3
dt
d
δgb 1
dt
d
δgb 2
dt
d
δgb 3
dt
d
δgS 1
dt
d
δgS 2
dt
d
δgS 3
dt
d
δclock b
dt
d
δclock d
dt
d
δPRN 1
dt
d
δPRN 2
dt
= wba 3 (t),
(10.70)
= wSa 1 (t),
(10.71)
= wSa 2 (t),
(10.72)
= wSa 3 (t),
(10.73)
= wbg 1 (t),
(10.74)
= wbg 2 (t),
(10.75)
= wbg 3 (t),
(10.76)
= wSg 1 (t),
(10.77)
= wSg 2 (t),
(10.78)
= wSg 3 (t),
(10.79)
= wb clock (t),
(10.80)
= wb clock (t),
(10.81)
= wPRN 1 (t),
(10.82)
= wPRN 2 (t),
(10.83)
..
.
d
δPRN 29 = wPRN 29 (t),
dt
(10.84)
where is the earth rotation rate in radians per second, RN (φlat ) is the meridional radius of curvature of the geoid model at latitude φlat , and RE (φlat ) is the
transverse radius of curvature.
Equations 10.61–10.63 are from Eq. 9.40, the Bortz model for rotational
dynamics. These can be integrated numerically for propagating the estimated
state vector, and the related covariance matrix of estimation uncertainty can be
423
FUTURE DEVELOPMENTS
propagated using the first-order approximation
d
Pρ = Fρ Pρ + Pρ FTρ + Q,
dt
∂fBortz
,
Fρ =
∂ρ
(10.85)
(10.86)
where fBortz is as defined by Eq. 9.40. The function fBortz (required for numerical integration of ρ) is implemented in the MATLAB m-file fBortz.m, and
the matrix Fρ is implement in the MATLAB m-file BortzF.m —both on the
accompanying CD-ROM.
Measurement Variables Measurements include outputs of the inertial sensors
and the pseudoranges output from the GNSS receiver:
1
2
3
4
5
6
7
8
a1
a2
a3
ω1
ω2
ω3
ρPRN 1
ρPRN 2
35 ρPRN 29
=
=
=
=
=
=
=
=
..
.
first accelerometer output,
second accelerometer output,
third accelerometer output,
first gyroscope output
second gyroscope output,
third gyroscope output,
pseudorange from first GNSS satellite,
pseudorange from second GNSS satellite,
= pseudorange from last GNSS satellite,
Only those pseudoranges from acquired satellites need be considered, however,
so the actual measurements count may be more like half of 35.
Model Implementation The further derivation, development and evaluation of
this integration model is left as an exercise for the reader.
10.5 FUTURE DEVELOPMENTS
GPS and INS were both developed for worldwide navigation capability, and
together they have taken that capability to new levels of performance that neither
approach could achieve on its own.
The payoff in military costs and capabilities had driven development of GPS
by the Department of Defense of the United States of America. However, early
pioneers in GPS development had already foreseen many of the markets for
GNSS/INS integration, including such applications as automating field equipment operations for farming and grading. These automated control applications
require inertial sensors for precise and reliable operation under dynamic conditions, and integration with GPS has brought the costs and capabilities of the
424
GNSS/INS INTEGRATION
resulting systems to very practical levels. The results of integrating inertial systems with GPS has made enormous improvements in achievable operational speed
and efficiency.
GNSS systems architectures continue to change with the addition of more
systems, more satellites, more signal channels and more aiding systems, and
integration-compatible inertial systems are also likely to continue to improve
as the market expands, driving hardware costs further downward. It is a part
of the “silicon revolution,” harnessing the enormous power and low cost of
electronic systems to make our lives more enjoyable and efficient. As costs
continue downward, the potential applications market continues to expand.
We have only begun to explore applications for GNSS/INS. We are limited
only by our ability to imagine them. The opportunities are there for you to shape
the future.
APPENDIX A
SOFTWARE
A.1 SOFTWARE SOURCES
The MATLAB m-files on the accompanying CD-ROM are the implementations
used to produce many of the examples and figures illustrating GNSS/INS implementation methods and performance evaluation methods. These are intended to
demonstrate to the reader how these methods work. This is not “commercial
grade” software, and it is not intended to be used as part of any commercial
design process or product implementation software. The authors and publisher do
not claim that this software meets any standards of mercantibility, and we cannot
assume any responsibility for the results if they are used for such purposes.
There is better, more reliable commercial software available for GNSS and
INS analysis, implementation and integration. We have used the MATLAB INS
and GPS toolboxes from GPSoft to generate some of the figures, and there are
other commercial products available for these purposes, as well. Many of the
providers of such software maintain internet websites describing their products
and services, and the interested user is encouraged to search the internet to shop
for suitable sources.
The following sections contain short descriptions of the MATLAB m-files
on the accompanying CD-ROM, organized by they chapters in which they are
mentioned.
Global Positioning Systems, Inertial Navigation, and Integration, Second Edition, by M. S. Grewal, L. R. Weill, and A. P. Andrews
Copyright © 2007 John Wiley & Sons, Inc.
425
426
SOFTWARE
A.2 SOFTWARE FOR CHAPTER 3
The MATLAB script ephemeris.m calculates a GPS satellite position in ECEF
coordinates from its ephemeris parameters. The ephemeris parameters comprise
a set of Keplerian orbital parameters and describe the satellite orbit during a
particular time interval. From these parameters, ECEF coordinates are calculated
using the equations from the text. Note that time t is the GPS time at transmission
and tk tk in the script) is the total time different between time t and the epoch
time toe (toe). Kepler’s equation for eccentric anomaly is nonlinear in Ek (Ek)
and is solved numerically using the Newton–Raphson method.
The following MATLAB script calculates satellite position for 24 h using
almanac data stored on the CD:
GPS position(PRN#) plots satellite position using PRNs one at a time for
all satellites.
GPS position 3D plots satellite position for all PRNs in three dimensions.
Use rotate option in MATLAB to see the satellite positions from the equator,
north pole, south pole, and so on.
GPS el az (PRN#, 33.8825, -117.8833) plots satellite trajectory for a
PRN from Fullerton, California (GPS laboratory located at California State
University, Fullerton).
GPS el az all (33.8825, -117.8833) plots satellite trajectories for all
satellites from Fullerton, California (GPS laboratory located at California
State University, Fullerton).
GPS el az one time (14.00, 33.8825, -117.8833) plots the location
of satellites at 2 p.m. (14:00 h) for visible satellites from GPS laboratory
located at California State University, Fullerton.
A.3 SOFTWARE FOR CHAPTER 5
A.3.1 Ionospheric Delays
The following MATLAB scripts compute and plot ionospheric delays using
Klobuchar models:
Klobuchar fix plots the ionospheric delays for GEO stationary satellites for
24 h, such as AOR-W, POR, GEO 3, and GEO 4.
Klobuchar (PRN#) plots the ionospheric delays for a satellite specified by
the argument PRN, when that satellite is visible.
Iono delay (PRN#) plots the ionospheric delays for a PRN using dualfrequency data, when a satellite is visible. It uses the pseudorange carrier
phase data for L1 and L2 signals. Plots are overlaid for comparison.
SOFTWARE FOR CHAPTER 9
427
A.4 SOFTWARE FOR CHAPTER 8
osc ekf.m demonstrates an extended Kalman filter tracking the phase, ampli-
tude, frequency, and damping factor of a harmonic oscillator with randomly
time-varying parameters.
GPS perf.m performs covariance analysis of expected performance of a GPS
receiver using a Kalman filter.
init var initializes parameters and variables for GPS perf.m.
choose sat chooses satellite set or use default for GPS perf.m.
gps init initializes GPS satellites for GPS perf.m.
calcH calculates H matrix for GPS perf.m.
gdop calculates GDOP for chosen constellation for GPS perf.m.
covar solves Riccati equation for GPS perf.m.
plot covar plots results from GPS perf.m.
SchmidtKalmanTest.m compares Schmidt–Kalman filter and Kalman filter
for GPS navigation with time-correlated pseudorange errors.
shootout.m compares performance of several square root covariance filtering
methods on an ill conditioned problem from P. Dyer and S. McReynolds,
“Extension of Square-Root Filtering to Include Process Noise,” Journal of
Optimization Theory and Applications 3, 444–458 (1969).
joseph called by shootout.m to implement “Joseph stabilized” Kalman
filter.
josephb called by shootout.m to implement “Joseph–Bierman” Kalman
filter.
josephdv called by shootout.m to implement “Joseph–DeVries” Kalman
filter.
potter called by shootout.m to implement Potter square-root filter.
carlson called by shootout.m to implement Carlson square-root filter.
bierman called by shootout.m to implement Bierman square-root filter.
A.5 SOFTWARE FOR CHAPTER 9
ConingMovie.m generates a MATLAB movie of coning motion, showing
how the body-fixed coordinate axes move relative to inertial coordinates.
(The built-in MATLAB function movie2avi can convert this to an avi-file.)
VertChanErr.m implements the Riccati equations for the INS vertical channel error covariance with accelerometer
noise levels in the order of 10−2 ,
√
−1
10 , 1, 10, and 100 m/s/ h, with and without aiding by a barometric
altimeter. Generates the plot shown in Fig. 9.32.
Euler2CTMat converts from Euler angles to coordinate transformation matrices.
CTMat2Euler converts from coordinate transformation matrices to Euler angles.
428
SOFTWARE
RotVec2Quat converts from rotation vectors to quaternions.
Quat2RotVec converts from quaternions to rotation vectors.
Quat2CTMat converts from quaternions to coordinate transformation matrices.
CTMat2Quat converts from coordinate transformation matrices to quaternions.
RotVec2CTMat converts from rotation vectors to coordinate transformation
matrices.
CTMat2RotVec converts from coordinate transformation matrices to rotation
vectors.
fBortz.m computes the nonlinear function fBortz for integrating the Bortz
“noncommutative” attitude integration formula.
FBortz(rho,omega) computes the dynamic coefficient matrix for integrating
the Riccati equation for rotation rates.
AccBiasCarousel.m simulates the propagation of accelerometer bias error
in an inertial navigation system and creates the plot shown in Fig. 9.33.
A.6 SOFTWARE FOR CHAPTER 10
HSatSim.m generates measurement sensitivity matrix H for GPS satellite sim-
ulation.
Damp2eval.m evaluates DAMP2 GPS position tracking filters for a range of
host vehicle RMS accelerations and acceleration correlation times.
YUMAdata loads GPS almanac data from www.navcen.uscg.gov - /ftp/GPS/
almanacs/yuma/ for Wednesday, March 08, 2006 10:48 AM, converts to
arrays of right ascension and phase angles for 29 satellites. (Used by
Damp2eval.m)
Damp2Params.m solves transcendental equation for alternative parameters in
DAMP2 GPS tracking filter.
Damp3Params.m solves transcendental equation for alternative parameters in
DAMP3 GPS tracking filter.
GPSTrackingDemo.m applies the GPS vehicle tracking filters TYPE2,
DAMP2, DAMP3 and FIG8 to the same problem (tracking a vehicle moving
on a figure-8 test track).
Fig8TrackDemo.m generates a series of plots and statistics of the simulated
figure-8 test track trajectory.
HorizINSperfModel.m calculates INS error model parameters as a function
of CEP rate.
Fig8Mod1D.m simulates trajectory of a vehicle going around a figure-8 test
track.
GPSINSwGPSpos.m simulates GPS/INS loosely-coupled integration, using
only standard GPS and INS output position values.
GPSINSwPRs.m simulates GPS/INS tightly-coupled integration, using GPS
pseudoranges and INS position outputs.
APPENDIX B
VECTORS AND MATRICES
The “S” in “GPS” and in “INS” stands for “system,” and “systems science” for
modeling, analysis, design, and integration of such systems is based largely on
linear algebra and matrix theory. Matrices model the ways that components of
systems interact dynamically and how overall system performance depends on
characteristics of components and subsystems and on the ways they are used
within the system.
This appendix presents an overview of matrix theory used for GPS/INS integration and the matrix notation used in this book. The level of presentation is
intended for readers who are already somewhat familiar with vectors and matrices. A more thorough treatment can be found in most college-level textbooks on
linear algebra and matrix theory.
B.1 SCALARS
Vectors and matrices are arrays composed of scalars, which we will assume to
be real numbers. Unless constrained by other conventions, we represent scalars
by italic lowercase letters.
In computer implementations, these real numbers will be approximated by
floating-point numbers, which are but a finite subset of the rational numbers. The
default MATLAB representation for real numbers on 32-bit personal computers
is in 64-bit ANSI standard floating point, with a 52-bit mantissa.
Global Positioning Systems, Inertial Navigation, and Integration, Second Edition, by M. S. Grewal, L. R. Weill, and A. P. Andrews
Copyright © 2007 John Wiley & Sons, Inc.
429
430
VECTORS AND MATRICES
B.2 VECTORS
B.2.1 Vector Notation
Vectors are arrays of scalars, either column vectors,
⎡ ⎤
v1
⎢ v2 ⎥
⎢ ⎥
⎢ ⎥
v = ⎢ v3 ⎥ ,
⎢ .. ⎥
⎣.⎦
vn
or row vectors,
y = [y1 , y2 , y3 , . . . , ym ].
Unless specified otherwise, vectors can be assumed to be column vectors.
The scalars vk or yk are called the components of v or y, respectively. The
number of components of a vector (rows in a column vector or columns in a row
vector) is called its dimension. The dimension of v shown above is the integer n
and the dimension of y is m. An n-dimensional vector is also called an n-vector.
Vectors are represented by boldface lowercase letters, and the corresponding
italic lowercase letters with subscripts represent the scalar components of the
associated vector.
B.2.2 Unit Vectors
A unit vector (i.e., a vector with magnitude equal to 1) is represented by the
symbol 1.
B.2.3 Subvectors
Vectors can be partitioned and represented in block form as a vector of subvectors:
⎡ ⎤
x1
⎢x2 ⎥
⎢ ⎥
⎢ ⎥
x = ⎢x3 ⎥ ,
⎢ .. ⎥
⎣.⎦
x
where each subvector xk is also a vector, as indicated by boldfacing.
431
VECTORS
B.2.4 Transpose of a Vector
Vector transposition,, represented by the post-superscript T transforms row vectors to column vectors, and vice versa:
⎡ ⎤
y1
⎢ y2 ⎥
⎢ ⎥
⎢ ⎥
vT = [v1 , v2 , v3 , . . . , vn ],
yT = ⎢ y3 ⎥ .
⎢ .. ⎥
⎣ . ⎦
ym
In MATLAB, the transpose of vector v is written as v
.
B.2.5 Vector Inner Product
The inner product or dot product of two m-vectors is the sum of the products of
their corresponding components:
xT y
or
def
x·y =
m
.
xk yk .
k=1
B.2.6 Orthogonal Vectors
Vectors x and y are called orthogonal or normal if their inner product is zero.
B.2.7 Magnitude of a Vector
The magnitude of a vector is the root-sum-squared of its components, denoted
by | · | and defined as
√
vvT (row vector)
5
6 n
6.
=7
v2,
def
|v| =
k
k=1
yT y (column vector)
5
6 m
6.
=7
y2.
def
|y| =
k
k=1
B.2.8 Unit Vectors and Orthonormal Vectors
A unit vector has magnitude equal to 1, and a pair or set of mutually orthogonal
unit vectors is called orthonormal.
432
VECTORS AND MATRICES
B.2.9 Vector Norms
The magnitude of a column n-vector x is also called its Euclidean norm. This
is but one of a class of norms called “Hölder norms,”1 “lp norms,” or simply
“p-norms”:
2 n
31/p
.
def
xp =
|xi |p
,
i=1
and in the limit (as p → ∞) as the sup2 norm, or ∞ norm:
def
x∞ = max |xi |.
i
These norms satisfy the Hölder inequality:
|x T y| ≤ xp yq
for
1
1
+ = 1.
p q
They are also related by inequalities such as
x∞ ≤ xE ≤ x1 ≤ nx∞ .
The Euclidean norm (Hölder 2-norm) is the default norm for vectors. When
no other norm is specified, the implied norm is the Euclidean norm.
B.2.10 Vector Cross-product
Vector cross-products are only defined for vectors with three components (i.e., 3vectors). For any two 3-vectors x and y, their vector cross-products are defined as
⎡
⎤
x2 y3 − x3 y2
def
x ⊗ y = ⎣x3 y1 − x1 y3 ⎦ ,
x1 y2 − x2 y1
which has the properties
x ⊗ y = −y ⊗ x,
x ⊗ x = 0,
|x ⊗ y| = sin(θ )|xy|,
where θ is the angle between the vectors x and y.
1
Named for the German mathematician Otto Ludwig Hölder (1859–1937).
“Sup” (sounds like “soup”) stands for supremum, a mathematical term for the least upper bound of
a set of real numbers. The maximum (max) is the supremum over a finite set.
2
433
MATRICES
B.2.11 Right-Handed Coordinate Systems
A Cartesian coordinate system in three dimensions is considered “right handed”
if its three coordinate axes are numbered consecutively such that the unit vectors
1k along its respective coordinate axes satisfy the cross-product rules
11 ⊗ 12 = 13 ,
(B.1)
12 ⊗ 13 = 11 ,
(B.2)
13 ⊗ 11 = 12 .
(B.3)
B.2.12 Vector Outer Product
The vector outer product of two column vectors
⎡ ⎤
⎡ ⎤
y1
x1
⎢ y2 ⎥
⎢x2 ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
y = ⎢ y3 ⎥
x = ⎢x3 ⎥ ,
⎢ .. ⎥
⎢ .. ⎥
⎣ . ⎦
⎣.⎦
xn
ym
is defined as the n × m array
⎡
x1 y1
⎢x2 y1
⎢
def ⎢
xyT = ⎢x3 y1
⎢ ..
⎣ .
x1 y2
x2 y2
x3 y2
..
.
x1 y2
x2 y2
x3 y2
..
.
xn y1
xn y2
xn y2
⎤
. . . x1 ym
. . . x2 ym ⎥
⎥
. . . x3 ym ⎥
⎥,
.. ⎥
..
.
. ⎦
. . . xn ym
a matrix.
B.3 MATRICES
B.3.1 Matrix Notation
For positive integers m and n, an m-by-n real matrix A is a two-dimensional
rectangular array of scalars, designated by the subscript notation aij , and usually
displayed in the following format:
⎡
⎤
a11 a12 a13 . . . a1n
⎢ a21 a22 a23 . . . a2n ⎥
⎢
⎥
⎢
⎥
A = ⎢ a31 a32 a33 . . . a3n ⎥ .
⎢ ..
..
..
.. ⎥
..
⎣ .
.
.
.
. ⎦
am1
am2
am3
. . . amn
434
VECTORS AND MATRICES
The scalars aij are called the elements of A. Uppercase bolded letters are used
for matrices, with the corresponding lowercase letter denoting scalar elements of
the associated matrices.
Row and Column Subscripts The first subscript (i) on the element aij refers to
the row in which the element occurs, and the second subscript (j ) refers to the
column in which aij occurs in this format. The integers i and j in this notation
are also called indices of the elements. The first index is called the row index,
and the second index is called the column index of the element. The term “(ij)th
position” in the matrix A refers to the position of aij , and aij is called the “(ij)th
element” of A:
←
→
columns
1st
↓
a11
a21
a31
..
.
2nd
↓
a12
a22
a32
..
.
3rd
↓
a13
a23
a33
..
.
...
...
...
...
..
.
nth
↓
a1n
a2n
a3n
..
.
am1
am2
am3
...
amn
rows
←
←
←
..
.
←
1st
2nd
3rd
..
.
mth
If juxtaposition of subscripts leads to confusion, they may be separated by
commas. The element in the eleventh row and first column of the matrix A
would then be denoted by a11,1 , not a111 .
Dimensions The positive integers m and n are called the dimensions of a matrix
A: m is called the row dimension of A and n is called the column dimension of A.
The dimensions of A may also be represented as “m × n,” which is to be read as
“m by n.” The symbol “×” in this notation does not indicate multiplication. (The
number of elements in the matrix A equals the product mn, however, and this is
important for determining memory requirements for data structures to hold A.)
B.3.2 Special Matrix Forms
Square Matrices A matrix is called square if it has the same row and column
dimensions. The main diagonal of a square matrix A is the set of elements aij for
which i = j . The other elements are called off-diagonal. If all the off-diagonal
elements of a square matrix A are zero, A is called a diagonal matrix. This and
other special forms of square matrices are illustrated in Fig. B.1.
Sparse and Dense Matrices A matrix with a “significant fraction” (typically,
half or more) of zero elements is called sparse. Matrices that are decidedly
not sparse are called dense, although both sparsity and density are matters of
degree. All the forms except symmetric shown in Fig. B.1 are sparse, although
435
MATRICES
...
0
0
0
0
0
0
0
0
0
..
.
... 0
Strictly
lower triangular
0
0
0 ... 0
0
... 0
a21
1
0 ... 0
a21
0
0 ... 0
1 ... 0
a31 a32
an1 an2 an3 . . . ann
Diagonal
.
an1 an2 an3 . . . 1
..
.
...
..
0 ... 0
...
...
.
...
..
a31 a32
...
0 ... 0
...
0
...
1
...
...
... 1
.
. . . a3n
... 0
a31 a32 a33 . . . 0
...
0
0
...
a21 a22
..
Unit
lower triangular
...
0
0
. . . ann
.
Lower triangular
a11
0
...
...
..
. . . a3n
1
...
0
a22 . . . a2n
...
0
0
...
a33 . . . a3n
0
...
0
0 a12 a13 . . . a1n
a22 . . . a2n
...
1
...
0
...
0 a21 a22 . . . a2n
0
Strictly
upper triangular
1 a12 a13 . . . a1n
...
a11 a12 a13 . . . a1n
Unit
upper triangular
...
Upper triangular
an1 an2 an3 . . . 0
Identity
Symmetric
a13 a23 a33 . . . a3n
...
0
0 0 . . . dn
Fig. B.1
0 0 0 . . .1
..
.
...
0 0 1 . . .0
...
0 d3 . . . 0
0
...
a12 a22 a23 . . . a2n
...
0 1 0 . . .0
...
0 d2 0 . . . 0
...
...
...
.
..
a11 a12 a13 . . . a1n
...
.
..
...
1 0 0 . . .0
...
d1 0 0 . . . 0
a1n a2n a3n . . . ann
Special forms of square matrices.
sparse matrices do not have to be square. Sparsity is an important characteristic
for implementation of matrix methods, because it can be exploited to reduce
computer memory and computational requirements.
Zero Matrices The ultimate sparse matrix is a matrix in which all elements
are 0 (zero). It is called a zero matrix, and it is represented by the symbol “0”
(zero). The equation A = 0 indicates that A is a zero matrix. Whenever it is
necessary to specify the dimensions of a zero matrix, they may be indicated by
subscripting: 0m×n will indicate an m × n zero matrix. If the matrix is square,
only one subscript will be used: 0n will mean an n × n zero matrix.
436
VECTORS AND MATRICES
Identity Matrices The identity matrix will be represented by the symbol I. If
it is necessary to denote the dimension of I explicitly, it will be indicated by
subscripting the symbol: In denotes the n × n identity matrix.
B.4 MATRIX OPERATIONS
B.4.1 Matrix Transposition
The transpose of A is the matrix AT (with the superscript “T” denoting the
transpose operation), obtained from A by interchanging rows and columns:
⎡
a11
⎢ a21
⎢
⎢ a31
⎢
⎢ ..
⎣ .
a12
a22
a32
..
.
a13
a23
a33
..
.
am1
am2
am3
⎤T ⎡
a11
a1n
⎢a12
a2n ⎥
⎢
⎥
⎢
a3n ⎥
⎥ = ⎢a13
⎢
.. ⎥
⎣. . .
. ⎦
a1n
. . . amn
...
...
...
..
.
a21
a22
a23
a31
a32
a33
...
a2n
...
a3n
⎤
. . . am1
. . . am2 ⎥
⎥
. . . am3 ⎥
⎥.
.. ⎥
..
.
. ⎦
. . . amn
The transpose of an m × n matrix is an n × m matrix.
The transpose of the matrix M in MATLAB is written as M
.
Symmetric Matrices A matrix A is called symmetric if AT = A and skew symmetric (or anti-symmetric) if AT = −A. Only square matrices can be symmetric
or skew symmetric. Therefore, whenever a matrix is said to be symmetric or
skew-symmetric, it is implied that it is a square matrix. Any square matrix A can
be expressed as a sum of its symmetric and antisymmetric parts:
A = 12 (A + AT ) + 12 (A − AT ) .
symmetric
antisymmetric
Cross-Product Matrices The vector cross-product ρ ⊗ α can also be expressed
in matrix form as
⎡ ⎤ ⎡ ⎤
ρ1
α1
ρ ⊗ ρ = ⎣ρ2 ⎦ ⊗ ⎣α2 ⎦
(B.4)
ρ3
α3
⎡
⎤
ρ2 α3 − ρ3 α2
= ⎣ρ3 α1 − ρ1 α3 ⎦
(B.5)
ρ1 α2 − ρ2 α1
= [ρ⊗]α
⎡
0
−ρ3
0
= ⎣ ρ3
−ρ2 ρ1
⎤⎡ ⎤
ρ2
α1
−ρ1 ⎦ ⎣α2 ⎦ ,
0
α3
(B.6)
(B.7)
437
MATRIX OPERATIONS
where the “cross-product matrix”
⎡
0
def
[ρ⊗] = ⎣ ρ3
−ρ2
−ρ3
0
ρ1
⎤
ρ2
−ρ1 ⎦
0
(B.8)
is skew-symmetric.
B.4.2 Subscripted Matrix Expressions
Subscripts represent an operation on a matrix that extracts the designated matrix
element. Subscripts may also be applied to matrix expressions. The element in
the (ij)th position of a matrix expression can be indicated by subscripting the
expression, as in
{AT }ij = aij .
Here, we have used braces {} to indicate the scope of the expression to which
the subscripting applies. This is a handy device for defining matrix operations.
B.4.3 Multiplication of Matrices by Scalars
Multiplication of a matrix A by a scalar s is equivalent to multiplying every
element of A by s:
{As}ij = {sA}ij = saij .
B.4.4 Addition and Multiplication of Matrices
Addition of Matrices Is Associative and Commutative Matrices can be added
together if and only if they share the same dimensions. If A and B have the same
dimensions, then addition is defined by adding corresponding elements:
{A + B}ij = aij + bij .
Addition of matrices is commutative and associative. That is, A + B = B + A
and A + (B + C) = (A + B) + C.
Additive Inverse of a Matrix The product of a matrix A by the scalar −1 yields
its additive inverse −A:
(−1)A = −A,
A + (−A) = A − A = 0.
Here, we have followed the not uncommon practice of using the symbol “−”
both as a unary (additive inverse) and binary (subtraction) operator. Subtraction
of a matrix A from a matrix B is equivalent to adding the additive inverse of A
to B:
B − A = B + (−A).
438
VECTORS AND MATRICES
Multiplication of Matrices is Associative but Not Commutative Multiplication
of an m × n matrix A by a matrix B on the right-hand side of A, as in the
matrix product AB, is defined only if the row dimension of B equals the column
dimension of A. That is, we can multiply an m × n matrix A by a p × q matrix
B in this order only if n = p. In that case, the matrices A and B are said to be
conformable for multiplication in that order, and the matrix product is defined
element by element by
n
.
def
{AB}ij =
aik bkj ,
k=1
the result of which is an m × q matrix. Whenever matrices appear as a product
in an expression, it is implied that they are conformable for multiplication.
Products with Identity Matrices Multiplication of any m × n matrix A by a
conformable identity matrix yields the original matrix A as the product:
AIn = A,
Im A = A.
B.4.5 Powers of Square Matrices
Square matrices can always be multiplied by themselves, and the resulting matrix
products are again conformable for multiplication. Consequently, one can define
the pth power of a square matrix A as
Ap = A
× · · · × A .
×A×A
p elements
B.4.6 Matrix Inversion
If A and B are square matrices of the same dimension, and such that their product
AB = I,
then B is the matrix inverse of A and A is the matrix inverse of B. (It turns out
that BA = AB = I in this case.) The inverse of a matrix A is unique, if it exists,
and is denoted by A−1 . Not all matrices have inverses. Matrix inversion is the
process of finding a matrix inverse, if it exists. If the inverse of a matrix A does
not exist, A is called singular. Otherwise, it is called non-singular.
B.4.7 Generalized Matrix Inversion
Even nonsquare and/or singular matrices can have generalized inverses. The
Moore-Penrose generalized inverse of an m × n matrix A is the n × m matrix
A+ such that
439
BLOCK MATRIX FORMULAS
AA+ A = A,
A+ AA+ = A+ ,
(AA+ )T = AA+ ,
(A+ A)T = A+ A.
B.4.8 Orthogonal Matrices
A square matrix A is called orthogonal if AT = A−1 . Orthogonal matrices have
several useful properties:
•
•
•
Orthogonality of a matrix A implies that the row vectors of A are jointly
orthonormal vectors, and the column vectors of A are also jointly orthonormal vectors.
The dot products of vectors are invariant under multiplication by a conformable orthogonal matrix. That is, if A is orthogonal, then xT y =
(Ax)T (Ay) for all conformable x and y.
Products and inverses of orthogonal matrices are orthogonal.
As a rule, multiplications by orthogonal matrices tend to be numerically
well conditioned, compared to general matrix multiplications. (The inversion
of orthogonal matrices is obviously extremely well conditioned.)
B.5 BLOCK MATRIX FORMULAS
B.5.1 Submatrices, Partitioned Matrices, and Blocks
For any m × n matrix A and any subset Srows ⊆ {1, 2, 3, . . . , m} of the row
indices and subset Scols ⊆ {1, 2, 3, . . . , n} of the column indices, the subset of
elements
A = {aij |i ∈ Srows , j ∈ Scols }
is called a submatrix of A.
A partitioning of an integer n is an exhaustive collection of contiguous subsets
Sk of the form
S1
S2
Sp
1, 2, 3, . . . , 1 , (1 + 1), . . . , 2 , . . . , (p−1 + 1), . . . , n .
The collection of submatrices formed by partitionings of the row and column
dimensions of a matrix is called a partitioning of the matrix, and the matrix is
said to be partitioned by that partitioning. Each submatrix of a partitioned matrix
440
VECTORS AND MATRICES
A is called a partitioned submatrix, partition, submatrix block, subblock, or block
of A. Each block of a partitioned matrix A can be represented by a conformable
matrix expression, and A can be displayed as a block matrix:
⎤
⎡
B C D ... F
⎢ G H J . . . L⎥
⎥
⎢
⎥
⎢
A = ⎢M N P . . . R⎥
⎢ ..
..
.. . .
.. ⎥
⎣.
. .⎦
.
.
V W X ... Z
where B, C, D, . . . stand for matrix expressions. Whenever a matrix is displayed
as a block matrix, it is implied that all block submatrices in the same row have
the same row dimension and that all block submatrices in the same column have
the same column dimension.
A block matrix of the form
⎤
⎡
A 0 0 ... 0
⎢0 B 0 . . . 0 ⎥
⎥
⎢
⎢0 0 C . . . 0 ⎥
⎢
⎥,
⎢ .. .. .. . .
.⎥
⎣. . .
. .. ⎦
0
0
0
... M
in which the off-diagonal block submatrices are zero matrices, is called a block
diagonal matrix, and a block matrix in which the block submatrices on one side
of the diagonal are zero matrices is called a block triangular matrix.
Columns and Rows as Blocks There are two special partitionings of matrices
in which the block submatrices are vectors. The column vectors of an m × n
matrix A are the block submatrices of the partitioning of A for which all column
dimensions are 1 and all row dimensions are m. The row vectors of A are the
block submatrices of the partitioning for which all row dimensions are 1 and all
column dimensions are n. All column vectors of an m × n matrix are m-vectors,
and all row vectors are n-vectors.
B.5.2 Rank and Linear Dependence
A linear combination of a finite set of n-vectors {vi } is a summation
4
4 of the sort
a
v
for
some
set
of
scalars
{a
}.
If
some
linear
combination
ai vi = 0 and
i
i i i
at least one coefficient ai = 0, the set of vectors {vi } is 4
called linearly dependent.
Conversely, if the only linear combination for which
ai vi = 0 is the one for
which all the ai = 0, then the set of vectors {vi } is called linearly independent.
The rank of a n × m matrix A equals the size of the largest collection of its
column vectors that is linearly independent. Note that any such linear combination
can be expressed in the form Aa, where the nonzero elements of the column mvector A are the associated scalars of the linear combination, and the number of
441
BLOCK MATRIX FORMULAS
nonzero components of A is the size of the collection of column vectors in the
linear combination. The same value for the rank of a matrix is obtained if the
test is applied to its row vectors, where any linear combination of row vectors
can be expressed in the form aT A for some column n-vector A.
An n × n matrix is nonsingular if and only if its rank equals its dimension n.
B.5.3 Conformable Block Operations
Block matrices with conformable partitionings may be transposed, added, subtracted, and multiplied in block format. For example,
/
0T / T
0
A B
A CT
,
=
C D
B T DT
/
0 /
0 /
0
A B
E F
A+E B+F
+
=
,
C D
G H
C+G D+H
/
0 /
0 /
0
A B
E F
AE + BG AF + BH
×
=
.
C D
G H
CE + DG CF + DH
B.5.4 Block Matrix Inversion Formula
The inverse of a partitioned matrix with square diagonal blocks may be represented in block form as [53]
0−1 /
/
0
A B
E F
=
,
C D
G H
where
E = A−1 + A−1 BHCA−1 ,
F = −A−1 BH,
G = −HCA−1 ,
H = [D − CA−1 B]−1 .
This formula can be proved by multiplying the original matrix times its alleged
inverse and verifying that the result is the identity matrix.
B.5.5 Inversion Formulas for Matrix Expressions
Sherman—Morrison Formula A “rank 1” modification of a square matrix A
is a sum of the form A + bcT , where b and c are conformable column vectors.
Its inverse is given by the formula
[A + bcT ]−1 = A−1 −
A−1 bcT A−1
.
1 + cT A−1 b
442
VECTORS AND MATRICES
Sherman—Morrison—Woodbury Formula This is the generalization of the
above formula for conformable matrices in place of vectors:
[A + BCT ]−1 = A−1 − A−1 B[I + CT A−1 B]−1 CT A−1 .
Hemes Inversion Formula A further generalization of this formula (used in
the derivation of the Kalman filter equations) includes an additional conformable
square matrix factor in the modification:
[A + BC−1 DT ]−1 = A−1 − A−1 B[C + DT A−1 B]−1 DT A−1 .
(B.9)
B.6 FUNCTIONS OF SQUARE MATRICES
B.6.1 Determinants and Characteristic Values
Elementary Permutation Matrices An elementary permutation matrix is formed
by interchanging rows or columns of an identity matrix In :
i
1 ... 0 ...
⎜ ... . . . ...
⎜
i ⎜
⎜ 0. . . . 0. . . .
⎜
.. . . .
= ⎜ ..
⎜
j ⎜0 ... 1 ...
⎜.
..
⎝ ..
.
0 ... 0 ...
⎛
P[ij ]
j
0
..
.
1
..
.
0
..
.
0
⎞
... 0
.. ⎟
.⎟
... 0⎟
.. ⎟
⎟
. ⎟.
⎟
... 0⎟
.⎟
..
. .. ⎠
... 1
Multiplication of a vector x by P[ij ] permutes the ith and j th elements of x. Note
that P[ij ] is an orthogonal matrix and that P[ii] = In , the identity matrix.
Determinants of Elementary Permutation Matrices The determinant of an
elementary permutation matrix P[ij ] is defined to be −1, unless i = j (i.e.,
P[ij ] = In ):
def
det(P[ij ] ) =
<
−1, i = j,
+1, i = j.
Permutation Matrices A permutation matrix is any product of elementary permutation matrices. These are also orthogonal matrices. Let Pn denote the set of
all distinct n × n permutation matrices. There are n! = 1 × 2 × 3 × · · · × n of
them, corresponding to the n! permutations of n indices.
443
FUNCTIONS OF SQUARE MATRICES
Determinants of Permutation Matrices The determinant of a permutation
matrix can be defined by the rule that the determinant of a product of matrices
is the product of the determinants:
det(AB) = det(A) det(B).
Therefore, the determinant of a permutation matrix will be either +1 or −1. A
permutation matrix is called “even” if its determinant is +1 and “odd” if its
determinant equals −1.
Determinants of Square Matrices The determinant of any n × n matrix A can
be defined as follows:
def
det(A) =
.
det(P)
n
C
{AP}ii .
i=1
P∈Pn
This formula has O(n × n!) computational complexity (for a sum over n! products of n elements each).
Characteristic Values of Square Matrices For a free variable λ, the polynomial
def
pA (λ) = det[A − λI] =
n
.
ai λi
i=0
is called the characteristic polynomial of A. The roots of pA (λ) are called the
characteristic values (or eigenvalues) of A. The determinant of A equals the product of its characteristic values, with each characteristic value occurring as many
times in the product as the multiplicity of the associated root of the characteristic
polynomial.
Definiteness of Symmetric Matrices If A is symmetric, all its characteristic
values are real numbers, which implies that they can be ordered. They are usually
expressed in descending order:
λ1 (A) ≥ λ2 (A) ≥ λ3 (A) ≥ · · · ≥ λn (A).
A real square symmetric matrix A is called
positive definite
non-negative definite
indefinite
non-positive definite
negative definite
if
if
if
if
if
λn (A) > 0,
λn (A) ≥ 0,
λ1 (A) > 0 and λn (A) < 0,
λ1 (A) ≤ 0, and
λ1 (A) < 0.
Non-negative definite matrices are also called positive semidefinite, and nonpositive definite matrices are also called negative semidefinite.
444
VECTORS AND MATRICES
Characteristic Vectors For each real characteristic value λi (A) of a real symmetric A, there is a corresponding characteristic vector (or eigenvector) ei (A)
such that ei (A) = 0 and Aei (A) = λi (A)ei (A). The characteristic vectors corresponding to distinct characteristic values are mutually orthogonal.
B.6.2 B.6.2 The Matrix Trace
The trace of a square matrix is the sum of its diagonal elements. It also equals
the sum of the characteristic values and has the property that the trace of the
product of conformable matrices is independent of the order of multiplication-a
very useful attribute:
.
trace(AB) =
{AB}ii
(B.10)
i
=
=
..
i
j
j
i
..
Aij Bj i
(B.11)
Bj i Aij
(B.12)
= trace(BA).
(B.13)
Note the product AB is conformable for the trace function only if it is a square
matrix, which requires that A and BT have the same dimensions. If they are
m × n (or n × m), then the computation of the trace of their product requires mn
multiplications, whereas the product itself would require m2 n (or mn2 ) multiplications.
B.6.3 Algebraic Functions of Matrices
An algebraic function may be defined by an expression in which the independent
variable (a matrix) is a free variable, such as the truncated power series
f (A) =
n
.
B k Ak ,
k=−n
where the negative power A−p = {A−1 }p = {Ap }−1 . In this representation, the
matrix A is the independent (free) variable and the other matrix parameters (Bk )
are assumed to be known and fixed.
B.6.4 Analytic Functions of Matrices
An analytic function is defined in terms of a convergent power series. It is
necessary that the power series converge to a limit, and the matrix norms defined
in Section B.1.7 must be used to define and prove convergence of a power series.
This level of rigor is beyond the scope of this book, but we do need to use one
particular analytic function, the exponential function.
445
FUNCTIONS OF SQUARE MATRICES
Matrix Exponential Function The power series
e
∞
.
1 k
A ,
=
k!
A def
(B.14)
k=0
def
k! = 1 × 2 × 3 · · · × k,
(B.15)
does converge3 for all square matrices A. It defines the exponential function of
the matrix A. This definition is sufficient to prove some elementary properties of
the exponential function for matrices, such as
•
•
•
•
•
•
•
e0n = In for 0n , the n × n zero matrix.
eIn = eIn for In , the n × n identity matrix.
T
eA = {eA }T .
(d/dt)eAt = AeAt = eAt A.
The exponential of a skew-symmetric matrix is an orthogonal matrix.
The characteristic vectors of A are also the characteristic vectors of eA .
If λ is a characteristic value of A, then eλ is a characteristic value of eA .
Powers and Exponentials of Cross-product Matrices The fact that exponential functions of skew-symmetric matrices are orthogonal matrices will have
important consequences for coordinate transformations (Appendix C), because the
matrices transforming vectors from one right-handed coordinate system (defined
in Section B.1.2.11) to another can can be represented as the exponentials of
cross-product matrices (defined in Eq. B.9). We show here how to represent the
exponential of a cross-product matrix
⎡
⎤
0 −ρ
ρ2
0 −ρ1 ⎦
[ρ⊗] = ⎣ ρ3
−ρ2 ρ1
0
in closed form. The first few powers can be calculated by hand, as
[ρ⊗]0 = I3 ,
[ρ]1 = [ρ⊗],
⎡ 2
−ρ3 − ρ22
2
⎣
[ρ⊗] =
ρ2 ρ1
ρ3 ρ1
ρ2 ρ1
−ρ32 − ρ12
ρ3 ρ2
⎤
ρ3 ρ1
ρ3 ρ2 ⎦
−ρ22 − ρ12
= ρρT − |ρ|2 I3 ,
3
However, convergence is not fast enough to make this a reasonable general-purpose formula for
approximating the exponential of A. More reliable and efficient methods can be found, e.g., in [41].
446
VECTORS AND MATRICES
[ρ⊗]3 = [ρ⊗][ρ⊗]2
= [ρ⊗][ρρT − |ρ|2 I3 ]2
= −|ρ|2 [ρ⊗],
[ρ⊗]4 = −|ρ|2 [ρ⊗]2 ,
..
.
[ρ⊗]2k+1 = (−1)k |ρ|2k [ρ⊗],
[ρ⊗]2k+2 = (−1)k |ρ|2k [ρ⊗]2 ,
so that the exponential expansion
exp([ρ⊗]) =
+∞
.
1
[ρ⊗]l
!
l=1
A +∞
B
. (−1)k |ρ|2k+1
1
[ρ⊗]
= [ρ⊗]0 +
|ρ|
(2k + 1)!
k=0
B
A +∞
1 . (−1)k |ρ|2k+2
+ 2
[ρ⊗]2
|ρ|
(2k + 2)!
k=0
⎡
⎤
0
−ρ
ρ
2
1 − cos(|ρ|) T sin(|ρ|) ⎣
ρ3
0 −ρ1 ⎦ ,
= cos(|ρ|)I2 +
ρρ +
|ρ|2
|ρ|
−ρ2 ρ1
0
(B.16)
where ! denots the factorial function (defined in Eq. B.16).
B.6.5 Similarity Transformations and Analytic Functions
For any n × n nonsingular matrix A, the transform X → A−1 X A is called a
similarity transformation of the n × n matrix X . It is a useful transformation for
analytic functions of matrices
f (X ) =
∞
.
ak X k ,
k=0
because
f (A−1 X A) =
∞
.
k=0
ak (A−1 X A)k
447
NORMS
∞
.
= A−1 (
ak X k )A
k=0
= A−1 f (X )A.
If the characteristic values of X are distinct, then the similarity transform performed with the characteristic vectors of X as the column vectors of A will
diagonalize X with its characteristic values along the main diagonal:
A−1 X A = diag {λ },
f (A−1 X A) = diag {F(λl )},
f (X ) = Adiag {F(λ )}A−1 .
(Although this is a useful analytical approach for demonstrating functional dependencies, it is not considered a robust numerical method.)
B.7 NORMS
B.7.1 Normed Linear Spaces
Vectors and matrices can be considered as elements of linear spaces, in that they
can be added and multiplied by scalars. A norm is any nonnegative real-valued
function · defined on a linear space such that, for any scalar s and elements
x and y of the linear space (vectors or matrices),
x = 0
x > 0
sx
x + y
iff
iff
=
≤
x = 0,
x = 0,
|s|x,
x + y,
where iff stands for “if and only if.” These constraints are rather loose, and many
possible norms can be defined for a particular linear space. A linear space with a
specified norm is called a normed linear space. The norm induces a topology on
the linear space, which is used to define continuity and convergence. Norms are
also used in numerical analysis for establishing error bounds and in sensitivity
analysis for bounding sensitivities. The multiplicity of norms is useful in these
applications, because the user is free to pick the one that works best for her or
his particular problem.
We define here many of the more popular norms, some of which are known
by more than one name.
B.7.2 Matrix Norms
Many norms have been defined for matrices. Two general types are presented
here. Both are derived from vector norms, but by different means.
448
VECTORS AND MATRICES
Generalized Vector Norms Vector norms can be generalized to matrices by
treating the matrix like a doubly-subscripted vector. For example, the Hölder
norms for vectors can be generalized to matrices as
A(p) =
⎧
m .
n
⎨.
⎩
i=1 j =1
|ai,j |p
⎫1/p
⎬
⎭
.
The matrix (2)-norm defined in this way is also called the Euclidean norm, Schur
norm, or Frobenius norm. We will use the notation · F in place of · (2) for
the Frobenius norm.
The reason for putting the parentheses around the subscript p in the above
definition is that there is another way that the vector p-norms are used to define
matrix norms, and it is with this alternative definition that they are usually allowed
to wear an unadorned p subscript. These alternative norms also have the following
desirable properties.
Desirable Multiplicative Properties of Matrix Norms Because matrices can be
multiplied, one could also apply the additional constraint that
ABM ≤ AM BM
for conformable matrices A and B and a matrix norm · M . This is a good
property to have for some applications. One might also insist on a similar property
with respect to multiplication by vector x, for which a norm · V1 may already
be defined:
|AxV2 ≤ AM xV1 .
This property is called compatibility between the matrix norm || · ||M and the
vector norms || · ||V1 and || · ||V2 . (Note that there can be two distinct vector
norms associated with a matrix norm: one in the normed linear space containing
x and one in the space containing Ax.)
Matrix Norms Subordinate to Vector Hölder Norms There is a family of alternative matrix “p-norms” [but not (p)-norms] defined by the formula
def
||A||p =
sup ||Ax||p
,
||x|| = 0 ||x||p
where the norms on the right-hand side are the vector Hölder norms and the
induced matrix norms on the left are called subordinate to the corresponding
Hölder norms. The 2-norm defined in this way is also called the spectral norm
of A. It has the properties:
||diagi {λi }||2 = max |λi |
i
and ||Ax||2 ≤ ||A||2 ||x||2 .
449
FACTORIZATIONS AND DECOMPOSITIONS
The first of these properties implies that ||I||2 = 1. The second property is compatibility between the spectral norm and the vector Euclidean norm. (Subordinate
matrix norms are guaranteed to be compatible with the vector norms used to
define them.) All matrix norms subordinate to vector norms also have the property
that ||I|| = 1.
Computation of Matrix Hölder Norms The following formulas may be used
in computing 1-norms and ∞-norms of m × n matrices A:
B
A m
.
||A||1 = max
|aij | ,
i≤j ≤n
||A||∞ = max
i=1
⎧
n
⎨.
1≤i≤m ⎩
j =1
⎫
⎬
|aij | .
⎭
The norm ||A||2 can be computed as the square root of the largest characteristic
value of AT A, which takes considerably more effort.
Default Matrix Norm When the type of norm applied to a matrix is not specified
(by an appropriate subscript), the default will be the spectral norm (Hölder matrix
2-norm). It satisfies the following bounds with respect to the Frobenius norm and
the other matrix Hölder norms for m × n matrices A:
√
||A||2 ≤ ||A||F ≤
n||A||2 ,
√
1
√ ||A||1 ≤ ||A||2
≤
n||A||1 ,
m
√
1
√ ||A||∞ ≤ ||A||2
≤
m||A||∞ ,
n
√
max |aij | ≤ ||A||F ≤
mn max |aij |.
1≤i≤m
1≤j ≤n
1≤i≤m
1≤j ≤n
B.8 FACTORIZATIONS AND DECOMPOSITIONS
Decompositions are also called factorizations of matrices. These are generally
represented by algorithms or formulas for representing a matrix as a product of
matrix factors with useful properties. The two factorization algorithms described
here have either triangular or diagonal factors in addition to orthogonal factors.
Decomposition methods are algorithms for computing the factors, given the
matrix to be “decomposed.”
B.8.1 Cholesky Decomposition
This decomposition is named after André Louis Cholesky [9], who was perhaps
not the first discoverer of the method for factoring a symmetric, positive-definite
matrix P as a product of triangular factors.
450
VECTORS AND MATRICES
Cholesky Factors A Cholesky factor of a symmetric positive-definite matrix P
is a matrix C such that
CCT = P.
(B.17)
Note that it does not matter whether we write this equation in the alternative
form FT F = P, because the two solutions are related by F = CT .
Cholesky factors are not unique, however. If C is a Cholesky factor of P, then
for any conformable orthogonal matrix M, the matrix
def
A = CM
satisfies the equation
AAT = CM(CM)T
= CMMT CT
= CCT
(B.18)
= P.
That is, A is also a legitimate Cholesky factor. The ability to transform one
Cholesky factor into another using orthogonal matrices will turn out to be very
important in square-root filtering (in Section 8.1.6).
Cholesky Factoring Algorithms There are two possible forms of the Cholesky
factorization algorithm, corresponding to two possible forms of the defining
equation:
P = L1 LT1 = UT1 U1
= U2 UT2 = LT2 L2 ,
(B.19)
(B.20)
where the Cholesky factors U1 , U2 are upper triangular and their respective transposes L1 , L2 are lower triangular.
The first of these is implemented by the built-in MATLAB function chol
(P), with argument P a symmetric positive-definite matrix. The call chol (P)
returns an upper triangular matrix U1 satisfying Eq. B.20. The MATLAB m-file
cho12.m on the accompanying diskette implements the solution to Eq. B.21.
The call chol2 (P) returns an upper triangular matrix U2 satisfying Eq. B.21.
Modified Cholesky Factorization The algorithm for Cholesky factorization of
a matrix requires taking square roots, which can be avoided by using a modified
Cholesky factorization in the form
P = UDUT ,
(B.21)
where D is a diagonal matrix with positive diagonal elements and U is a unit
triangular matrix (i.e., U has 1’s along its main diagonal). This algorithm is
implemented in the m-file modchol.m on the accompanying diskette.
FACTORIZATIONS AND DECOMPOSITIONS
451
B.8.2 QR Decomposition (Triangularization)
The QR decomposition of a matrix A is a representation in the form
A = QR,
where Q is an orthogonal matrix and R is a triangular matrix. Numerical methods
for QR decomposition are also called “triangularization” methods. Some of these
methods are an integral part of square-root Kalman filtering and are presented in
Section 8.1.6.3.
B.8.3 Singular-Value Decomposition
The singular-value decomposition of an m × n matrix A is a representation in
the form A = Tm DTn , where Tm and Tn are orthogonal matrices (with square
dimensions as specified by their subscripts) and D is an m × n matrix filled with
zeros everywhere except along the main diagonal of its maximal upper left square
submatrix. This decomposition will have either of three forms:
depending on the relative values of m and n. The middle matrix D has the block
form
⎧
[diagi {σi }|0m×(n−m) ] if m < n,
⎪
⎪
⎨
{σi }
if m = n,
D = diag
i
⎪
⎪
diag
{σ
}
i i
⎩
if m > n,
0
(m−n)×n
σ1 ≥ σ2 ≥ σ3 ≥ · · · ≥ σp ≥ 0,
p = min(m, n).
That is, the diagonal nonzero elements of D are in descending order, and nonnegative. These are called the singular values of A. For a proof that this decomposition
exists, and an algorithm for computing it, see the book by Golub and Van
Loan [41].
The singular values of a matrix characterize many useful matrix properties,
such as
||A||2 = σ1 (A),
452
VECTORS AND MATRICES
rank (A) = r such that σr > 0 and either σr+1 = 0 or r = p (the rank of a
matrix is defined in Section B.1.5.2), and
the condition number of A equals σ1 /σp .
The condition number of the matrix A in the linear equation Ax = b bounds the
sensitivity of the solution x to variations in b and the sensitivity of the solution
to roundoff errors in determining it. The singular-value decomposition may also
be used to define the “pseudorank” of A as the smallest singular value σi such
that σi > εσ1 , where ε is a processor- and precision-dependent constant such that
0 < ε 1 and 1 + ε ≡ 1 in machine precision.
These relationships are useful for the analysis of state transition matrices of Kalman filters, which can be singular or close enough to being singular that
numerical roundoff can cause the product P T to be essentially singular.
B.8.4 Eigenvalue–Eigenvector Decompositions of Symmetric Matrices
Symmetric QR Decomposition The so-called “symmetric QR” decomposition
of an n × n symmetric real matrix A has the special form A = T DT T , where the
right orthogonal matrix is the transposed left orthogonal matrix and the diagonal
matrix
D = diagi {λi }.
That is, the diagonal elements are the characteristic values of the symmetric
matrix. Furthermore, the column vectors of the orthogonal matrix T are the
associated characteristic vectors ei of A:
A = T DT T
=
n
.
λi ei eTi ,
i=1
T = [e1
e2
e3
...
en ].
These relationships are useful for the analysis of covariance matrices, which are
constrained to have nonnegative characteristic values, although their numerical
values may stray enough in practice (due to computer roundoff errors) to develop
negative characteristic values.
B.9 QUADRATIC FORMS
Bilinear and Quadratic Forms For a matrix A and all conformable column
vectors x and y, the functional mapping (x, y) → x T Ay is called a bilinear
form. As a function of x and y, it is linear in both x and y and hence bilinear.
In the case that x = y, the functional mapping x → x T Ax is called a quadratic
form. The matrix A of a quadratic form is always a square matrix.
453
DERIVATIVES OF MATRICES
B.9.1 Symmetric Decomposition of Quadratic Forms
Any square matrix A can be represented uniquely as the sum of a symmetric
matrix and a skew-symmetric matrix:
A = 12 (A + AT ) + 12 (A − AT ),
where 12 (A + AT ) is called the symmetric part of A and 12 (A − AT ) is called
the skew-symmetric part of A. The quadratic form x T Ax depends only on the
symmetric part of A:
x T Ax = x T { 12 (A + AT )}x.
Therefore, one can always assume that the matrix of a quadratic form is symmetric, and one can express the quadratic form in summation form as
x T Ax =
n .
n
.
aij , xi xj =
.
i=1 j =1
=
n
.
i=j
aii xi2 + 2
i=1
aij xi xj +
.
.
aij xi xj
i=j
aij xi xj
i<j
for symmetric A.
Ranges of Quadratic Forms The domain of a quadratic form for an n × n
matrix is n-dimensional Euclidean space, and the range is in (−∞, +∞), the
real line. In the case that x = 0,
if A is positive def inite,
the range of x → x T Ax is
(0, +∞);
if A is non-negative def inite,
the range of x → x T Ax is
[0, +∞);
if A is indef inite,
the range of x → x T Ax is
(−∞, +∞);
if A is non-positive def inite,
the range of x → x T Ax is
(−∞, 0];
if A is negative def inite,
the range of x → x T Ax is
(−∞, 0).
If x T x = 1, then λn (A) ≤ x T Ax ≤ λ1 (A). That is, the quadratic form maps the
unit n-sphere onto the closed interval [λn (A), λ1 (A)].
B.10 DERIVATIVES OF MATRICES
B.10.1 Derivatives of Matrix-Valued Functions
The derivative of a matrix with respect to a scalar is the matrix of derivatives of
its elements:
454
VECTORS AND MATRICES
⎡
f11 (t) f12 (t) f13 (t)
···
⎢ f21 (t) f22 (t) f23 (t)
···
⎢
⎢
f
(t)
f
(t)
f
(t)
···
31
32
33
F(t) = ⎢
⎢
⎢ ..
..
..
⎣ .
.
.
fm1 (t) fm2 (t) fm3 (t) · · · fmn (t)
⎡
d/dtf11 (t) d/dtf12 (t) d/dtf13 (t)
⎢ d/dtf21 (t) d/dtf22 (t) d/dtf23 (t)
⎢
⎢
d/dtf31 (t) d/dtf32 (t) d/dtf33 (t)
d/dtF(t) = ⎢
⎢
⎢
..
..
..
⎣
.
.
.
d/dtfm1 (t) d/dtfm2 (t) d/dtfm3 (t)
⎤
f1n (t)
f2n (t)⎥
⎥
⎥
f3n (t)⎥ ,
⎥
.. ⎥
. ⎦
···
···
···
⎤
d/dtf1n (t)
d/dtf2n (t) ⎥
⎥
⎥
d/dtf3n (t) ⎥ .
⎥
⎥
..
⎦
.
. . . d/dtfmn (t)
The rule for the derivative of a product applies also to matrix products:
d/dt[A(t)B(t)] = [d/dtA(t)]B(t) + A(t)[d/dtB(t)],
provided that the order of the factors is preserved.
Derivative of Matrix Inverse If F(t) is square and nonsingular, then F(t)F−1 (t)
= I, a constant. As a consequence, its derivative will be zero. This fact can be
used to derive the formula for the derivative of a matrix inverse:
0 = d/dtI
= d/dt[F(t)F−1 (t)]
= [d/dtF(t)]F−1 (t) + F(t)[d/dtF−1 (t)],
d/dtF−1 (t) = −F−1 [d/dtF(t)]F−1 .
(B.22)
Derivative of Orthogonal Matrix If the F(t) is orthogonal, its inverse F−1 (t) =
FT (t), its transpose, and because
d/dtFT (t) = [d/dtF(t)]T = ḞT ,
one can show that orthogonal matrices satisfy matrix differential equations with
antisymmetric dynamic coefficient matrices:
d
I
dt
d
= [F(t)FT (t)]
dt
= Ḟ(t)FT (t) + F(t)ḞT (t),
0=
d
I
dt
d
= [FT (t)F(t)]
dt
= ḞT (t)F(t) + FT (t)Ḟ(t),
0=
455
DERIVATIVES OF MATRICES
Ḟ(t)FT (t) = −[F(t) · FT (t)]
FT (t)Ḟ(t) = −[Ḟ(t)F(t)f(t)]
= −[FT (t)Ḟ(t)]T
= antisymmetric matrix
= right ,
= −[Ḟ(t)FT (t)]T
= antisymmetric matrix
= left ,
Ḟ(t) = lef t F(t),
Ḟ(t) = F(t)
right .
That is, all time-differentiable orthogonal matrices F(t) satisfy dynamic equations
with antisymmetric coefficient matrices, which can be either left- or right-side
coefficient matrices.
B.10.2 Gradients of Quadratic Forms
If f (x) is a differentiable scalar-valued function of an n-vector x, then the vector
0
/
∂f
∂f ∂f ∂f
∂f T
=
,
,
,···,
∂x
∂x1 ∂x2 ∂x3
∂xn
is called the gradient of f with respect to x. In the case that f is a quadratic
form with symmetric matrix A, then the ith component of its gradient will be
⎛
⎞
/
0
.
∂ T
∂f ⎝.
(x Ax) =
ajj xj2 + 2
aj k xj xk ⎠
∂x
∂x
i
i
j
j <k
⎛
= ⎝2aii xi + 2
.
aik xk + 2
i<k
⎛
= ⎝2aii xi + 2
.
⎞
aik xk ⎠
i=k
=2
n
.
aik xk
k=1
= [2Ax]i .
That is, the gradient vector can be expressed as
∂ T
(x Ax) = 2Ax.
∂x
.
j <i
⎞
aj i xj ⎠
APPENDIX C
COORDINATE TRANSFORMATIONS
C.1 NOTATION
We use the notation Cfrom
to denote a coordinate transformation matrix from one
to
coordinate frame (designated by “from”) to another coordinated frame (designated
by “to”). For example,
CECI
ENU denotes the coordinate transformation matrix from earth-centered inertial
(ECI) coordinates to earth-fixed east-north-up (ENU) local coordinates and
CRPY
NED denotes the coordinate transformation matrix from vehicle body-fixed
roll-pitch-yaw (RPY) coordinates to earth-fixed north-east-down (NED)
coordinates.
Coordinate transformation matrices satisfy the composition rule
A
CBC CA
B = CC ,
where A, B, and C represent different coordinate frames.
What we mean by a coordinate transformation matrix is that if a vector v has
the representation
⎡ ⎤
vx
v = ⎣vy ⎦
(C.1)
vz
Global Positioning Systems, Inertial Navigation, and Integration, Second Edition, by M. S. Grewal, L. R. Weill, and A. P. Andrews
Copyright © 2007 John Wiley & Sons, Inc.
456
457
NOTATION
in XYZ coordinates and the same vector v has the alternative representation
⎡
⎤
vu
v = ⎣ vv ⎦
vw
(C.2)
⎡ ⎤
⎡ ⎤
vu
vx
VW ⎣
⎣vy ⎦ = CU
vv ⎦ ,
XY W
vz
vw
(C.3)
in UVW coordinates, then
where “XYZ” and “UVW” stand for any two Cartesian coordinate systems in
three-dimensional space.
The components of a vector in either coordinate system can be expressed
in terms of the vector components along unit vectors parallel to the respective
coordinate axes. For example, if one set of coordinate axes is labeled X, Y and
Z, and the other set of coordinate axes are labeled U , V , and W , then the same
vector v can be expressed in either coordinate frame as
v = vx 1x + vy 1y + vz 1z
(C.4)
= vu 1u + vv 1v + vw 1w ,
(C.5)
where
•
•
•
•
the unit vectors 1x , 1y , and 1z are along the XYZ axes;
the scalars vx , vy , and vz are the respective components of v along the XYZ
axes;
the unit vectors 1u , 1v , and 1w are along the UVW axes; and
the scalars vu , vv , and vw are the respective components of v along the
UVW axes.
The respective components can also be represented in terms of dot products
of v with the various unit vectors,
vx = 1Tx v = vu 1Tx 1u + vv 1Tx 1v + vw 1Tx 1w ,
(C.6)
vy =
vw 1Ty 1w ,
(C.7)
vz = 1Tz v = vu 1Tz 1u + vv 1Tz 1v + vw 1Tz 1w ,
(C.8)
1Ty v
=
vu 1Ty 1u
+
vv 1Ty 1v
+
458
COORDINATE TRANSFORMATIONS
which can be represented in matrix form as
⎡ ⎤ ⎡ T
1x 1u
vx
⎢ ⎥ ⎢ T
⎣vy ⎦ = ⎣1y 1u
1Tx 1v
1Ty 1v
⎥⎢ ⎥
1Ty 1w ⎦ ⎣ vv ⎦
1Tz 1u
1Tz 1v
1Tz 1w
vz
def
=
VW
CU
XY Z
⎡
⎤
1Tx 1w
⎤⎡
vu
⎤
(C.9)
vw
vu
⎢ ⎥
⎣ vv ⎦ ,
(C.10)
vw
which defines the coordinate transformation matrix CUVW
XYZ from UVW to XYZ
coordinates in terms of the dot products of unit vectors. However, dot products
of unit vectors also satisfy the cosine rule (defined in Section B.1.2.5)
1Ta 1b = cos(θab ),
(C.11)
where θab is the angle between the unit vectors 1a and 1b . As a consequence,
the coordinate transformation matrix can also be written in the form
⎡
UVW
CXYZ
⎤
cos(θxu ) cos(θxv ) cos(θxw )
⎢
⎥
= ⎣ cos(θyu cos(θyv ) cos(θyw )⎦ ,
cos(θzu )
cos(θzv )
(C.12)
cos(θzw )
which is why coordinate transformation matrices are also called “direction cosines
matrices.”
Navigation makes use of coordinates that are natural to the problem at hand:
inertial coordinates for inertial navigation, orbital coordinates for GPS navigation,
and earth-fixed coordinates for representing locations on the earth.
The principal coordinate systems used in navigation, and the transformations
between these different coordinate systems, are summarized in this appendix.
These are primarily Cartesian (orthogonal) coordinates, and the transformations
between them can be represented by orthogonal matrices. However, the coordinate transformations can also be represented by rotation vectors or quaternions,
and all representations are used in the derivations and implementation of GPS/INS
integration.
C.2 INERTIAL REFERENCE DIRECTIONS
C.2.1 Vernal Equinox
The equinoxes are those times of year when the length of the day equals the
length of the night (the meaning of “equinox”), which only happens when the
INERTIAL REFERENCE DIRECTIONS
459
sun is over the equator. This happens twice a year: when the sun is passing from
the Southern Hemisphere to the Northern Hemisphere (vernal equinox) and again
when it is passing from the Northern Hemisphere to the Southern Hemisphere
(autumnal equinox). The time of the vernal equinox defines the beginning of
spring (the meaning of “vernal”) in the Northern Hemisphere, which usually
occurs around March 21–23.
The direction from the earth to the sun at the instant of the vernal equinox is
used as a “quasi-inertial” direction in some navigation coordinates. This direction is defined by the intersection of the equatorial plane of the earth with the
ecliptic (earth-sun plane). These two planes are inclined at about 23.45◦ , as illustrated in Fig. C.1. The inertial direction of the vernal equinox is changing ever
so slowly, on the order of 5 arc seconds per year, but the departure from truly
inertial directions is neglible over the time periods of most navigation problems.
The vernal equinox was in the constellation Pisces in the year 2000. It was in the
constellation Aries at the time of Hipparchus (190-120 BCE) and is sometimes
still called “the first point of Aries.”
C.2.2 Polar Axis of Earth
The one inertial reference direction that remains invariant in earth-fixed coordinates as the earth rotates is its polar axis, and that direction is used as a reference
direction in inertial coordinates. Because the polar axis is (by definition) orthogonal to the earth’s equatorial plane and the vernal equinox is (by definition) in
the earth’s equatorial plane, the earth’s polar axis will always be orthogonal to
the vernal equinox.
A third orthogonal axis can then be defined (by their cross-product) such
that the three axes define a right-handed (defined in Section B.2.11) orthogonal
coordinate system.
Fig. C.1 Direction of vernal equinox.
460
COORDINATE TRANSFORMATIONS
C.3 COORDINATE SYSTEMS
Although we are concerned exclusively with coordinate systems in the three
dimensions of the observable world, there are many ways of representing a location in that world by a set of coordinates. The coordinates presented here are
those used in navigation with GPS and/or INS.
C.3.1 Cartesian and Polar Coordinates
René Descartes (1596–1650) introduced the idea of representing points in threedimensional space by a triplet of coordinates, called “Cartesian coordinates” in
his honor. They are also called “Euclidean coordinates,” but not because Euclid
discovered them first. The Cartesian coordinates (x, y, z ) and polar coordinates
(θ, φ, r) of a common reference point, as illustrated in Fig. C.2, are related by
the equations
x = r cos(θ ) cos(φ),
(C.13)
y = r sin(θ ) cos(φ),
(C.14)
z = r sin(φ),
r = x 2 + y 2 + z2 ,
%
&
z
1
1
− π ≤φ≤+ π ,
φ = arcsin
r
2
2
y (−π < θ ≤ +π),
θ = arctan
x
(C.15)
with the angle θ (in radians) undefined if φ = ± 12 π.
Fig. C.2
Cartesian and polar coordinates.
(C.16)
(C.17)
(C.18)
461
COORDINATE SYSTEMS
Fig. C.3 Celestial coordinates.
C.3.2 Celestial Coordinates
The “celestial sphere” is a system for inertial directions referenced to the polar
axis of the earth and the vernal equinox. The polar axis of these celestial coordinates is parallel to the polar axis of the earth and its prime meridian is fixed to
the vernal equinox. Polar celestial coordinates are right ascension (the celestial
analog of longitude, measured eastward from the vernal equinox) and declination
(the celestial analog of latitude), as illustrated in Fig. C.3. Because the celestial
sphere is used primarily as a reference for direction, no origin need be specified.
Right ascension is zero at the vernal equinox and increases eastward (in the
direction the earth turns). The units of right ascension (RA) can be radians,
degrees, or hours (with 15 deg/h as the conversion factor).
By convention, declination is zero in the equatorial plane and increases toward
the north pole, with the result that celestial objects in the Northern Hemisphere
have positive declinations. Its units can be degrees or radians.
C.3.3 Satellite Orbit Coordinates
Johannes Kepler (1571–1630) discovered the geometric shapes of the orbits of
planets and the minimum number of parameters necessary to specify an orbit
(called “Keplerian” parameters). Keplerian parameters used to specify GPS satellite orbits in terms of their orientations relative to the equatorial plane and the
vernal equinox (defined in Section C.2.1 and illustrated in Fig. C.1) include the
following:
•
Right ascension of the ascending node and orbit inclination, specifying the
orientation of the orbital plane with respect to the vernal equinox and equatorial plane, is illustrated in Fig. C.4.
(a) Right ascension is defined in the previous section and is shown in
Fig. C.3.
462
COORDINATE TRANSFORMATIONS
Fig. C.4
•
•
•
1
Keplerian parameters for satellite orbit.
(b) The intersection of the orbital plane of a satellite with the equatorial
plane is called its “line of nodes,” where the “nodes” are the two
intersections of the satellite orbit with this line. The two nodes are
dubbed “ascending”1 (i.e., ascending from the Southern Hemisphere to
the Northern Hemisphere) and “descending”. The right ascension of the
ascending node (RAAN) is the angle in the equatorial plane from the
vernal equinox to the ascending node, measured counterclockwise as
seen looking down from the north pole direction.
(c) Orbital inclination is the dihedral angle between the orbital plane and
the equatorial plane. It ranges from zero (orbit in equatorial plane) to
90◦ (polar orbit).
Semimajor axis a and semiminor axis b (defined in Section C.3.5.2 and illustrated in Fig. C.6) specify the size and shape of the elliptical orbit within
the orbital plane.
Orientation of the ellipse within its orbital plane, specified in terms of the
“argument of perigee,” the angle between the ascending node and the perigee
of the orbit (closest approach to earth), is illustrated in Fig. C.4.
Position of the satellite relative to perigee of the elliptical orbit, specified in
terms of the angle from perigee, called the “argument of latitude” or “true
anomaly,” is illustrated in Fig. C.4.
The astronomical symbol for the ascending node is
, often read as “earphones.”
463
COORDINATE SYSTEMS
For computer simulation demonstrations, GPS satellite orbits can usually be
assumed to be circular with radius a = b = R = 26,560 km and inclined at 55◦
to the equatorial plane. This eliminates the need to specify the orientation of the
elliptical orbit within the orbital plane. (The argument of perigee becomes overly
sensitive to orbit perturbations when eccentricity is close to zero.)
C.3.4 ECI Coordinates
Earth-centered inertial (ECI) coordinates are the favored inertial coordinates in
the near-earth environment. The origin of ECI coordinates is at the center of
gravity of the earth, with (Fig. C.5)
1. axis in the direction of the vernal equinox,
2. axis direction parallel to the rotation axis (north polar axis) of the earth,
and
3. an additional axis to make this a right-handed orthogonal coordinate system,
with the polar axis as the third axis (hence the numbering).
The equatorial plane of the earth is also the equatorial plane of ECI coordinates, but the earth itself is rotating relative to the vernal equinox at its sidereal
rotation rate of about 7,292,115,167 × 10−14 rad/s, or about 15.04109 deg/h, as
illustrated in Fig. C.5.
C.3.5 ECEF Coordinates
Earth-centered, earth-fixed (ECEF) coordinates have the same origin (earth center) and third (polar) axis as ECI coordinates but rotate with the earth, as shown
Fig. C.5 ECI and ECEF Coordinates.
464
COORDINATE TRANSFORMATIONS
in Fig. C.5. As a consequence, ECI and ECEF longitudes differ only by a linear
function of time.
Longitude in ECEF coordinates is measured east (+) and west (−) from the
prime meridian passing through the principal transit instrument at the observatory
at Greenwich, UK, a convention adopted by 41 representatives of 25 nations at the
International Meridian Conference, held in Washington, DC, in October of 1884.
Latitudes are measured with respect to the equatorial plane, but there is more
than one kind of “latitude.” Geocentric latitude would be measured as the angle
between the equatorial plane and a line from the reference point to the center
of the earth, but this angle could not be determined accurately (before GPS)
without running a transit survey over vast distances. The angle between the pole
star and the local vertical direction could be measured more readily, and that
angle is more closely approximated as geodetic latitude. There is yet a third
latitude (parametric latitude) that is useful in analysis. The latter two latitudes
are defined in the following subsections.
C.3.5.1 Ellipsoidal Earth Models Geodesy is the study of the size and shape
of the earth and the establishment of physical control points defining the origin
and orientation of coordinate systems for mapping the earth. Earth shape models
are very important for navigation using either GPS or INS, or both. INS alignment
is with respect to the local vertical, which does not generally pass through the
center of the earth. That is because the earth is not spherical.
At different times in history, the earth has been regarded as being flat (firstorder approximation), spherical (second-order), and ellipsoidal (third-order). The
third-order model is an ellipsoid of revolution, with its shorter radius at the poles
and its longer radius at the equator.
C.3.5.2 Parametric Latitude For geoids based on ellipsoids of revolution,
every meridian is an ellipse with equatorial radius a (also called “semimajor
axis”) and polar radius b (also called “semiminor axis”). If we let z be the Cartesian coordinate in the polar direction and xmeridional be the equatorial coordinate in
the meridional plane, as illustrated in Fig. C.6, then the equation for this ellipse
will be
2
xmeridional
z2
+
=1
a2
b2
(C.19)
= cos2 (φparametric ) + sin2 (φparametric )
=
=
[b sin(φparametric )]2
[a cos(φparametric )]2
+
.
a2
b2
+
b2
(C.20)
2
a 2 cos2 (φparametric )
a2
sin (φparametric )
b2
(C.21)
(C.22)
465
COORDINATE SYSTEMS
Fig. C.6
Geocentric, parametric, and geodetic latitudes in meridional plane.
That is, a parametric solution for the ellipse is
xmeridional = a cos(φparametric ),
(C.23)
z = b sin(φparametric ),
(C.24)
as illustrated in Fig. C.6. Although the parametric latitude φparametric has no physical significance, it is quite useful for relating geocentric and geodetic latitude,
which do have physical significance.
C.3.5.3 Geodetic Latitude Geodetic latitude is defined as the elevation angle
above (+) or below (−) the equatorial plane of the normal to the ellipsoidal
surface. This direction can be defined in terms of the parametric latitude, because
it is orthogonal to the meridional tangential direction.
The vector tangential to the meridian will be in the direction of the derivative
to the elliptical equation solution with respect to parametric latitude:
/
0
∂
a cos(φparametric )
vtangential ∝
(C.25)
∂φparametric b sin(φparametric )
/
0
−a sin(φparametric )
=
,
(C.26)
b cos(φparametric )
and the meridional normal direction will be orthogonal to it, or
0
/
b cos(φparametric )
vnormal ∝
,
a sin(φparametric )
as illustrated in Fig. C.6.
(C.27)
466
COORDINATE TRANSFORMATIONS
The tangent of geodetic latitude is then the ratio of the z- and x-components
of the surface normal vector, or
a sin(φparametric )
b cos(φparametric )
a
= tan(φparametric ),
b
tan(φgeodetic ) =
(C.28)
(C.29)
from which, using some standard trigonometric identities,
tan(φgeodetic )
sin(φgeodetic ) = !
1 + tan2 (φgeodetic )
a sin(φparametric )
=!
,
a 2 sin2 (φparametric ) + b2 cos2 (φparametric )
1
cos(φgeodetic ) = !
1 + tan2 (φgeodetic )
b cos(φparametric )
=!
.
a 2 sin2 (φparametric ) + b2 cos2 (φparametric )
(C.30)
(C.31)
(C.32)
(C.33)
The inverse relationship is
tan(φparametric ) =
b
tan(φgeodetic ),
a
(C.34)
from which, using the same trigonometric identities as before,
tan(φparametric )
sin(φparametric ) = !
1 + tan2 (φparametric )
b sin(φgeodetic )
=!
,
a 2 cos2 (φgeodetic ) + b2 sin2 (φgeodetic )
1
cos(φparametric ) = !
1 + tan2 (φparametric )
a cos(φgeodetic )
=!
,
a 2 cos2 (φgeodetic ) + b2 sin2 (φgeodetic )
(C.35)
(C.36)
(C.37)
(C.38)
467
COORDINATE SYSTEMS
and the two-dimensional X -Z Cartesian coordinates in the meridional plane of a
point on the geoid surface will
xmeridional = a cos(φparametric )
(C.39)
a 2 cos(φgeodetic )
,
=!
a 2 cos2 (φgeodetic ) + b2 sin2 (φgeodetic )
(C.40)
z = b sin(φparametric )
(C.41)
2
b sin(φgeodetic )
=!
a 2 cos2 (φgeodetic ) + b2 sin2 (φgeodetic )
(C.42)
in terms of geodetic latitude.
Equations C.40 and C.42 apply only to points on the geoid surface. Orthometric height h above (+) or below (−) the geoid surface is measured along
the surface normal, so that the X-Z coordinates for a point with altitude h
will be
xmeridional = cos(φgeodetic )
⎛
⎞
2
a
⎟
⎜
× ⎝h + !
⎠,
a 2 cos2 (φgeodetic ) + b2 sin2 (φgeodetic )
z = sin(φgeodetic )
⎛
(C.43)
⎞
2
b
⎟
⎜
× ⎝h + !
⎠.
2
2
2
2
a cos (φgeodetic ) + b sin (φgeodetic )
(C.44)
In three-dimensional ECEF coordinates, with the X-axis passing through the
equator at the prime meridian (at which longitude θ = 0),
xECEF = cos(θ )xmeridional
(C.45)
= cos(θ ) cos(φgeodetic )
⎛
⎞
2
a
⎟
⎜
× ⎝h + !
⎠,
2
a 2 cos2 (φgeodetic ) + b2 sin (φgeodetic )
yECEF = sin(θ )xmeridional
(C.46)
(C.47)
468
COORDINATE TRANSFORMATIONS
= sin(θ ) cos(φgeodetic )
⎛
⎞
2
a
⎟
⎜
× ⎝h + !
⎠,
2
2
2
2
a cos (φgeodetic ) + b sin (φgeodetic )
zECEF = sin(φgeodetic )
⎛
(C.48)
⎞
2
b
⎟
⎜
× ⎝h + !
⎠,
a 2 cos2 (φgeodetic ) + b2 sin2 (φgeodetic )
(C.49)
in terms of geodetic latitude φgeodetic , longitude θ , and orthometric altitude h with
respect to the reference geoid.
The inverse transformation, from ECEF XYZ to geodetic longitude–latitude–altitude coordinates, is
θ = atan2(yECEF , xECEF ),
=
>
e2 a 2 sin3 (ζ )
2
3
φgeodetic = atan2 zECEF +
, ξ − e a cos (ζ ) ,
b
h=
ξ
− rT ,
cos(φ)
(C.50)
(C.51)
(C.52)
where atan2 is the four-quadrant arctangent function in MATLAB and
ζ = atan2(azECEF , bξ ),
!
2
2
+ yECEF
,
ξ = xECEF
a
,
rT = 1 − e2 sin(φ)
(C.53)
(C.54)
(C.55)
where rT is the transverse radius of curvature on the ellipsoid, a is the equatorial
radius, b is the polar radius, and e is elliptical eccentricity.
C.3.5.4 Geocentric Latitude For points on the geoid surface, the tangent
of geocentric latitude is the ratio of distance above (+) or below (−) the
equator [z = b sin(φparametric )] to the distance from the polar axis [(xmeridional =
a cos(φparametric )], or
tan(φGEOCENTRIC ) =
b sin(φparametric )
a cos(φparametric )
(C.56)
=
b
tan(φparametric )
a
(C.57)
=
b2
tan(φgeodetic ),
a2
(C.58)
469
COORDINATE SYSTEMS
from which, using the same trigonometric identities as were used for geodetic
latitude,
tan(φgeocentric )
sin(φgeocentric ) = !
1 + tan2 (φgeocentric )
(C.59)
b sin(φparametric )
=!
a 2 cos2 (φparametric ) + b2 sin2 (φparametric )
(C.60)
b2 sin(φgeodetic )
,
=!
a 4 cos2 (φgeodetric ) + b4 sin2 (φgeodetic )
(C.61)
1
cos(φgeocentric ) = !
1 + tan2 (φgeocentric )
(C.62)
a cos(φparametric )
=!
a 2 cos2 (φparametric ) + b2 sin2 (φparametric )
(C.63)
a 2 cos(φgeodetic )
=!
.
a 4 cos2 (φgeodetic ) + b4 sin2 (φgeodetic )
(C.64)
The inverse relationships are
tan(φparametric ) =
tan(φgeodetic ) =
a
tan(φgeocentric ),
b
(C.65)
a2
tan(φgeocentric ),
b2
(C.66)
from which, using the same trigonometric identities again,
tan(φparametric )
sin(φparametric ) = !
1 + tan2 (φparametric )
(C.67)
a sin(φgeocentric )
=!
,
a 2 sin2 (φgeocentric ) + b2 cos2 (φgeocentric )
(C.68)
a 2 sin(φgeocentric )
sin(φgeodetic ) = !
,
a 4 sin2 (φgeocentric ) + b4 cos2 (φgeocentric )
(C.69)
1
cos(φparametric ) = !
1 + tan2 (φparametric )
(C.70)
470
COORDINATE TRANSFORMATIONS
b cos(φgeocentric )
,
=!
2
a 2 sin (φgeocentric ) + b2 cos2 (φgeocentric )
(C.71)
b2 cos(φgeocentric )
cos(φgeodetic ) = !
.
a 4 sin2 (φgeocentric ) + b4 cos2 (φgeocentric )
(C.72)
C.3.6 LTP Coordinates
Local tangent plane (LTP) coordinates, also called “locally level coordinates,”
are a return to the first-order model of the earth as being flat, where they serve
as local reference directions for representing vehicle attitude and velocity for
operation on or near the surface of the earth. A common orientation for LTP
coordinates has one horizontal axis (the north axis) in the direction of increasing
latitude and the other horizontal axis (the east axis) in the direction of increasing
longitude, as illustrated in Fig. C.7. Horizontal location components in this local
coordinate frame are called “relative northing” and “relative easting.”
C.3.6.1 Alpha Wander Coordinates Maintaining east–north orientation was
a problem for some INSs at the poles, where north and east directions change
by 180◦ . Early gimbaled inertial systems could not slew the platform axes fast
enough for near-polar operation. This problem was solved by letting the platform
axes “wander” from north but keeping track of the angle α between north and a
reference platform axis, as shown in Fig. C.8. This LTP orientation came to be
called “alpha wander.”’
C.3.6.2 ENU/NED Coordinates East–north–up (ENU) and north–east–
down (NED) are two common right-handed LTP coordinate systems. ENU coordinates may be preferred to NED coordinates because altitude increases in the
upward direction. But NED coordinates may also be preferred over ENU coordinates because the direction of a right (clockwise) turn is in the positive direction
Fig. C.7
ENU coordinates.
471
COORDINATE SYSTEMS
Fig. C.8 Alpha wander.
with respect to a downward axis, and NED coordinate axes coincide with vehiclefixed roll–pitch–yaw (RPY) coordinates (Section C.3.7) when the vehicle is level
and headed north.
The coordinate transformation matrix CENU
NED from ENU to NED coordinates
and the transformation matrix CNED
ENU from NED to ENU coordinates are one and
the same:
⎡
⎤
0 1
0
NED
⎣
0⎦ .
(C.73)
CENU
NED = CENU = 1 0
0 0 −1
C.3.6.3 ENU/ECEF Coordinates The unit vectors in local east, north, and
up directions, as expressed in ECEF Cartesian coordinates, will be
⎡
⎤
− sin(θ )
1E = ⎣ cos(θ )⎦ ,
(C.74)
0
⎡
⎤
− cos(θ ) sin(φgeodetic )
1N = ⎣ − sin(θ ) sin(φgeodetic ) ⎦ ,
(C.75)
cos(φgeodetic )
⎡
⎤
cos(θ ) cos(φgeodetic )
1U = ⎣ sin(θ ) cos(φgeodetic ) ⎦ ,
(C.76)
sin(φgeodetic )
and the unit vectors in the ECEF X, Y, and Z directions, as expressed in ENU
coordinates, will be
⎡
⎤
− sin(θ )
(C.77)
1X = ⎣− cos(θ ) sin(φgeodetic )⎦ ,
cos(θ ) cos(φgeodetic )
472
COORDINATE TRANSFORMATIONS
⎡
⎤
cos(θ )
1Y = ⎣− sin(θ ) sin(φgeodetic )⎦ ,
sin(θ ) cos(φgeodetic )
⎡
⎤
0
1Z = ⎣cos(φgeodetic )⎦ .
sin(φgeodetic )
(C.78)
(C.79)
C.3.6.4 NED/ECEF Coordinates It is more natural in some applications to
use NED directions for locally level coordinates. This coordinate system coincides with vehicle-body-fixed RPY coordinates (shown in Fig. C.9) when the
vehicle is level headed north. The unit vectors in local north, east, and down
directions, as expressed in ECEF Cartesian coordinates, will be
⎡
⎤
− cos(θ ) sin(φgeodetic )
1N = ⎣ − sin(θ ) sin(φgeodetic ) ⎦ ,
cos(φgeodetic )
⎡
⎤
− sin(θ )
1E = ⎣ cos(θ )⎦ ,
0
⎡
⎤
− cos(θ ) cos(φgeodetic )
1D = ⎣ − sin(θ ) cos(φgeodetic ) ⎦ ,
− sin(φgeodetic )
(C.80)
(C.81)
(C.82)
and the unit vectors in the ECEF X, Y, and Z directions, as expressed in NED
coordinates, will be
⎡
⎤
− cos(θ ) sin(φgeodetic )
⎦,
− sin(θ )
1X = ⎣
(C.83)
− cos(θ ) cos(φgeodetic )
Fig. C.9
Roll-pitch-yaw axes.
473
COORDINATE SYSTEMS
⎡
⎤
− sin(θ ) sin(φgeodetic )
⎦,
cos(θ )
1Y = ⎣
− sin(θ ) cos(φgeodetic )
⎡
⎤
cos(φgeodetic )
⎦,
0
1Z = ⎣
− sin(φgeodetic )
(C.84)
(C.85)
C.3.7 RPY Coordinates
The RPY coordinates are vehicle fixed, with the roll axis in the nominal direction
of motion of the vehicle, the pitch axis out the right-hand side, and the yaw axis
such that turning to the right is positive, as illustrated in Fig.C.9. The same
orientations of vehicle-fixed coordinates are used for surface ships and ground
vehicles. They are also called “SAE coordinates,” because they are the standard
body-fixed coordinates used by the Society of Automotive Engineers.
For rocket boosters with their roll axes vertical at lift-off, the pitch axis is
typically defined to be orthogonal to the plane of the boost trajectory (also called
the “pitch plane” or “ascent plane”).
C.3.8 Vehicle Attitude Euler Angles
The attitude of the vehicle body with respect to local coordinates can be specified
in terms of rotations about the vehicle roll, pitch, and yaw axes, starting with
these axes aligned with NED coordinates. The angles of rotation about each of
these axes are called Euler angles, named for the Swiss mathematician Leonard
Euler (1707–1783). It is always necessary to specify the order of rotations when
specifying Euler (pronounced “oiler”) angles.
A fairly common convention for vehicle attitude Euler angles is illustrated in
Fig. C.10, where, starting with the vehicle level with roll axis pointed north:
1. Yaw/Heading. Rotate through the yaw angle (Y ) about the vehicle yaw
axis to the intended azimuth (heading) of the vehicle roll axis. Azimuth is
measured clockwise (east) from north.
Fig. C.10 Vehicle Euler angles.
474
COORDINATE TRANSFORMATIONS
2. Pitch. Rotate through the pitch angle (P ) about the vehicle pitch axis to
bring the vehicle roll axis to its intended elevation. Elevation is measured
positive upward from the local horizontal plane.
3. Roll. Rotate through the roll angle (R) about the vehicle roll axis to bring
the vehicle attitude to the specified orientation.
Euler angles are redundant for vehicle attitudes with 90◦ pitch, in which case
the roll axis is vertical. In that attitude, heading changes also rotate the vehicle
about the roll axis. This is the attitude of most rocket boosters at lift-off. Some
boosters can be seen making a roll maneuver immediately after lift-off to align
their yaw axes with the launch azimuth in the ascent plane. This maneuver may
be required to correct for launch delays on missions for which launch azimuth
is a function of launch time.
C.3.8.1 RPY/ENU Coordinates With vehicle attitude specified by yaw angle
(Y ), pitch angle (P ), and roll angle (R) as specified above, the resulting unit
vectors of the roll, pitch, and yaw axes in ENU coordinates will be
⎡
⎤
sin(Y ) cos(P )
1R = ⎣cos(Y ) cos(P )⎦ ,
sin(P )
⎡
⎤
cos(R) cos(Y ) + sin(R) sin(Y ) sin(P )
1P = ⎣− cos(R) sin(Y ) + sin(R) cos(Y ) sin(P )⎦ ,
− sin(R) cos(P )
⎡
⎤
− sin(R) cos(Y ) + cos(R) sin(Y ) sin(P )
1Y = ⎣ sin(R) sin(Y ) + cos(R) cos(Y ) sin(P )⎦ ;
− cos(R) cos(P )
(C.86)
(C.87)
(C.88)
the unit vectors of the east, north, and up axes in RPY coordinates will be
⎡
⎤
sin(Y ) cos(P )
1E = ⎣ cos(R) cos(Y ) + sin(R) sin(Y ) sin(P )⎦ ,
− sin(R) cos(Y ) + cos(R) sin(Y ) sin(P )
⎡
⎤
cos(Y ) cos(P )
1N = ⎣− cos(R) sin(Y ) + sin(R) cos(Y ) sin(P )⎦ ,
sin(R) sin(Y ) + cos(R) cos(Y ) sin(P )
⎡
⎤
sin(P )
1U = ⎣ − sin(R) cos(P ) ⎦ ;
− cos(R) cos(P )
(C.89)
(C.90)
(C.91)
475
COORDINATE SYSTEMS
and the coordinate transformation matrix from RPY coordinates to ENU coordinates will be
⎡
RPY
= [1R
CENU
1P
1TE
⎤
⎢ ⎥
1Y ] = ⎣1TN ⎦
1TU
⎡
SY CP
= ⎣CY CP
SP
(C.92)
CR CY + SR SY SP
−CR SY + SR CY SP
−SR CP
⎤
−SR CY + CR SY SP
SR SY + CR CY SP ⎦ , (C.93)
−CR CP
SR = sin(R),
(C.94)
CR = cos(R),
(C.95)
SP = sin(P ),
(C.96)
CP = cos(P ),
(C.97)
SY = sin(Y ),
(C.98)
CY = cos(Y ).
(C.99)
where
C.3.9 GPS Coordinates
The parameter in Fig. C.12 is the RAAN, which is the ECI longitude where
the orbital plane intersects the equatorial plane as the satellite crosses from the
Southern Hemisphere to the Northern Hemisphere. The orbital plane is specified
by and α, the inclination of the orbit plane with respect to the equatorial plane
(α ≈ 55◦ for GPS satellite orbits). The θ parameter represents the location of
the satellite within the orbit plane, as the angular phase in the circular orbit with
respect to ascending node.
For GPS satellite orbits, the angle θ changes at a nearly constant rate of about
1.4584 × 10−4 rad/s and a period of about 43,082s (half a day).
The nominal satellite position in ECEF coordinates is then given as
x = R[cos θ cos − sin θ sin cos α],
(C.100)
y = R[cos θ cos + sin θ sin cos α],
(C.101)
z = R sin θ sin α,
(C.102)
360
deg,
43, 082
360
= 0 − (t − t0 )
deg,
86, 164
R = 26,560,000 m.
θ = θ0 + (t − t0 )
(C.103)
(C.104)
(C.105)
476
COORDINATE TRANSFORMATIONS
GPS satellite positions in the transmitted navigation message are specified in
the ECEF coordinate system of WGS 84. A locally level x 1 , y 1 , z1 reference
coordinate system (described in Section C.3.6) is used by an observer location
on the earth, where the x 1 − y 1 plane is tangential to the surface of the earth,
x 1 pointing east, y 1 pointing north, and z1 normal to the plane. See Fig. C.11.
Here,
XENU = CECEF
ENU XECEF + S,
CECEF
ENU = coordinate transformation matrix from ECEF to ENU,
S = coordinate origin shift vector from ECEF to local reference,
⎤
⎡
− sin θ
cos θ
0
⎥
⎢
CECEF
ENU = ⎣− sin φ cos θ − sin φ sin θ cos φ ⎦ ,
cos φ cos θ
cos φ sin θ
sin φ
⎡
⎤
XU sin θ − YU cos θ
⎢
⎥
S = ⎣ XU sin φ cos θ − YU sin φ sin θ − ZU cos φ ⎦ ,
−XU cos φ cos θ − YU cos φ sin θ − ZU sin φ
Fig. C.11
Pseudorange.
COORDINATE TRANSFORMATION MODELS
Fig. C.12
477
Satellite coordinates.
XU , YU , ZU = user’s position,
θ = local reference longitude,
φ = local geometric latitude.
C.4 COORDINATE TRANSFORMATION MODELS
Coordinate transformations are methods for transforming a vector represented in
one coordinate system into the appropriate representation in another coordinate
system. These coordinate transformations can be represented in a number of
different ways, each with its advantages and disadvantages.
These transformations generally involve translations (for coordinate systems
with different origins) and rotations (for Cartesian coordinate systems with different axis directions) or transcendental transformations (between Cartesian and
polar or geodetic coordinates). The transformations between Cartesian and polar
coordinates have already been discussed in Section C.3.1 and translations are
rather obvious, so we will concentrate on the rotations.
C.4.1 Euler Angles
Euler angles were used for defining vehicle attitude in Section C.3.8, and vehicle
attitude representation is a common use of Euler angles in navigation.
Euler angles are used to define a coordinate transformation in terms of a
set of three angular rotations, performed in a specified sequence about three
specified orthogonal axes, to bring one coordinate frame to coincide with another.
478
COORDINATE TRANSFORMATIONS
The coordinate transformation from RPY coordinates to NED coordinates, for
example, can be composed from three Euler rotation matrices:
⎡
Yaw
−SY
CY
0
CY
RPY
CNED
= ⎣ SY
0
⎡
CY PP
⎢ SY CP
=⎢
⎣ −SP
(rollaxis)
Pitch
Roll
⎤ ⎡
⎤ ⎡
⎤
0
CP 0 SP
1 0
0
0⎦ ⎣ 0
1 0 ⎦ ⎣0 CR −SR ⎦
(C.106)
1
−SP 0 CP
0 SR CR
⎤
−SY CR + CY SP SR
SY SR + CY SP CR
CY CR + SY SP SR −CY SR + SY SP CR ⎥
⎥, (C.107)
⎦
CP SR
CP CR
(pitchaxis)
(yawaxis)
in NED coordinates
where the matrix elements are defined in Eqs. C.94–C.99. This matrix also rotates
the NED coordinate axes to coincide with RPY coordinate axes. (Compare this
with the transformation from RPY to ENU coordinates in Eq. C.93.)
For example, the coordinate transformation for nominal booster rocket launch
attitude (roll axis straight up) would be given by Eq. with pitch angle P = 12 π
(CP = 0, SP = 1), which becomes
⎡
⎤
0 sin(R − Y )
cos(R − Y )
⎣
⎦
CRPY
NED = 0 cos(R − Y ) − sin(R − Y ) .
1
10
0
That is, the coordinate transformation in this attitude depends only on the difference between roll angle (R) and yaw angle (Y ). Euler angles are a concise
representation for vehicle attitude. They are handy for driving cockpit displays
such as compass cards (using Y ) and artificial horizon indicators (using R and
P ), but they are not particularly handy for representing vehicle attitude dynamics.
The reasons for the latter include the following:
•
•
Euler angles have discontinuities analogous to “gimbal lock”
(Section 6.4.1.2) when the vehicle roll axis is pointed upward, as it
is for launch of many rockets. In that orientation, tiny changes in vehicle
pitch or yaw cause ±180◦ changes in heading angle. For aircraft, this
creates a slewing rate problem for electromechanical compass card displays.
The relationships between sensed body rates and Euler angle rates are mathematically complicated.
C.4.2 Rotation Vectors
All right-handed orthogonal coordinate systems with the same origins in three
dimensions can be transformed one onto another by single rotations about fixed
axes. The corresponding rotation vectors relating two coordinate systems are
COORDINATE TRANSFORMATION MODELS
479
defined by the direction (rotation axis) and magnitude (rotation angle) of that
transformation.
For example, the rotation vector for rotating ENU coordinates to NED coordinates (and vice versa) is
⎡ √ ⎤
π/√2
ENU
⎣
ρNED = π/ 2⎦ ,
(C.108)
0
◦
which has magnitude |ρENU
NED | = π(180 ) and direction north—east, as illustrated
in Fig. C.13. (For illustrative purposes only, NED coordinates are shown as
being translated from ENU coordinates in Fig. C.13. In practice, rotation vectors
represent pure rotations, without any translation.)
The rotation vector is another minimal representation of a coordinate transformation, along with Euler angles. Like Euler angles, rotation vectors are concise
but also have some drawbacks:
1. It is not a unique representation, in that adding multiples of ±2π to the magnitude of a rotation vector has no effect on the transformation it represents.
2. It is a nonlinear and rather complicated representation, in that the result of
one rotation followed by another is a third rotation, the rotation vector for
which is a fairly complicated function of the first two rotation vectors.
But, unlike Euler angles, rotation vector models do not exhibit “gimbal lock.”
C.4.2.1 Rotation Vector to Matrix The rotation represented by a rotation
vector
⎡ ⎤
ρ1
ρ = ⎣ρ 2 ⎦
(C.109)
ρ3
Fig. C.13
Rotation from ENU to NED coordinates.
480
COORDINATE TRANSFORMATIONS
can be implemented as multiplication by the matrix
def
C(ρ) = exp(ρ⊗)
⎛⎡
0
= exp ⎝⎣ ρ3
−ρ2
def
−ρ3
0
ρ1
(C.110)
⎤⎞
ρ2
−ρ1 ⎦⎠
0
(C.111)
⎡
⎤
ρ2
1 − cos(|ρ|) T sin(|ρ|) ⎣ 0 −ρ3
ρ3
0 −ρ1 ⎦ (C.112)
ρρ +
= cos(|ρ|)I3 +
|ρ|2
|ρ|
−ρ2
ρ1
0
⎡
⎤
0 −u3
u2
0 −u1 ⎦ , (C.113)
= cos(θ )I3 + (1 − cos(θ ))1ρ 1Tρ + sin(θ ) ⎣ u3
−u2
u1
0
def
θ = |ρ|,
def ρ
1ρ = ,
ρ|
(C.114)
(C.115)
which was derived in Eq. B.17. That is, for any three-rowed column vector v,
C(ρ)v rotates it through an angle of |ρ| radians about the vector ρ.
The form of the matrix in Eq. C.1132 is better suited for computation when
θ ≈ 0, but the form of the matrix in Eq. C.112 is useful for computing sensitivities using partial derivatives (used in Chapter 8).
ENU
in Eq. C.108 transforming between
For example, the rotation vector ρNED
ENU and NED has magnitude and direction
θ =π
[sin(θ ) = 0, cos(θ ) = −1],
⎡ √ ⎤
1/√2
1ρ = ⎣1/ 2⎦ ,
0
respectively, and the corresponding rotation matrix
⎡
CENU
NED
0
= cos(π)I3 + [1 − cos(π)]1ρ 1ρ + sin(π) ⎣ u3
−u2
−u3
0
u1
⎤
u2
−u1 ⎦
0
= −I3 + 21ρ 1Tρ + 0
2
Linear combinations of the sort a1 I3×3 + a2 [1ρ ⊗] + a3 1ρ 1Tρ , where 1 is a unit vector, form a
subalgebra of 3 × 3 matrices with relatively simple rules for multiplication, inversion, etc.
481
COORDINATE TRANSFORMATION MODELS
⎡
⎤ ⎡
⎤
−1
0
0
1 1 0
0⎦ + ⎣1 1 0⎦
= ⎣ 0 −1
0
0 −1
0 0 0
⎡
⎤
0 1
0
0⎦
= ⎣1 0
0 0 −1
transforms from ENU to NED coordinates. (Compare this result to Eq. C.73.)
Because coordinate transformation matrices are orthogonal matrices and the
ENU
matrix CENU
NED is also symmetric, CNED is its own inverse. That is,
NED
CENU
NED = CENU .
(C.116)
C.4.2.2 Matrix to Rotation Vector Although there is a unique coordinate
transformation matrix for each rotation vector, the converse is not true. Adding
multiples of 2π to the magnitude of a rotation vector has no effect on the resulting coordinate transformation matrix. The following approach yields a unique
rotation vector with magnitude |ρ| ≤ π.
The trace tr(C) of a square matrix M is the sum of its diagonal values. For
the coordinate transformation matrix of Eq. C.112,
tr[C(ρ)] = 1 + 2 cos(θ ),
(C.117)
from which the rotation angle
|ρ| = θ
= arcos
%
(C.118)
&
tr[C(ρ)] − 1
,
2
(C.119)
a formula that will yield a result in the range 0 < θ < π, but with poor fidelity
near where the derivative of the cosine equals zero at θ = 0 and θ = π.
The values of θ near θ = 0 and θ = π can be better estimated using the sine
of θ , which can be recovered using the antisymmetric part of C(ρ),
⎡
0
A = ⎣ a21
−a13
def 1
= 2 [C(ρ)
−a21
0
a32
⎤
a13
−a32 ⎦
0
− CT (ρ)]
⎡
sin(θ ) ⎣ 0 −ρ3
ρ3
0
=
θ
−ρ2
ρ1
(C.120)
⎤
ρ2
−ρ1 ⎦ ,
0
(C.121)
(C.122)
482
COORDINATE TRANSFORMATIONS
from which the vector
⎡
⎤
a32
⎣a13 ⎦ = sin(θ ) 1 ρ
|ρ|
a21
(C.123)
will have magnitude
!
2
2
2
a32
+ a13
+ a21
= sin(θ )
(C.124)
and the same direction as ρ. As a consequence, one can recover the magnitude
θ of ρ from
&
%!
2
2
2 tr[C(ρ)] − 1
a32 + a13 + a21 ,
θ = atan2
2
(C.125)
using the MATLAB function atan2, and then the rotation vector ρ as
⎡
⎤
a
32
θ ⎣ ⎦
a13
ρ=
sin(θ ) a
21
(C.126)
when 0 < θ < π.
C.4.2.3 Special Cases for sin(θ ) ≈ 0 For θ ≈ 0, ρ ≈ 0, although Eq. C.126
may still work adequately for θ > 10−6 , say.
For θ ≈ π, the symmetric part of C(ρ),
⎡
s11
S = ⎣ s12
s13
s12
s22
s23
def 1
= 2 [C(ρ)
+ CT (ρ)]
= cos(θ )I3 +
≈ −I3 +
⎤
s13
s23 ⎦
s33
1 − cos(θ ) T
ρρ
θ2
2 T
ρρ
θ2
(C.127)
(C.128)
(C.129)
(C.130)
and the unit vector
def
1ρ =
1
ρ
θ
(C.131)
483
COORDINATE TRANSFORMATION MODELS
⎡
satisfies
2u21 − 1
⎢
S ≈ ⎣ 2u1 u2
2u1 u3
2u1 u2
2u22 − 1
2u2 u3
2u1 u3
⎤
⎥
2u2 u3 ⎦ ,
2u23 − 1
(C.132)
which can be solved for a unique u by assigning uk > 0 for
⎤⎞
s11
k = argmax ⎝⎣ s22 ⎦⎠ ,
s33
!
uk = 12 (skk + 1)
⎛⎡
(C.133)
(C.134)
then, depending on whether k = 1, k = 2, or k = 3,
k=1
s11 + 1
u1 ≈
2
u2 ≈
s12
2u1
u3 ≈
s13
2u1
k=2
k=3
s12
2u2
s13
2u3
s22 + 1
2
s23
2u2
s23
2u2
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
s11 + 1 ⎪
⎪
⎭
2
⎤
u1
ρ = θ ⎣ u2 ⎦ .
u3
(C.135)
⎡
and
(C.136)
C.4.2.4 Time Derivatives of Rotation Vectors The mathematical relationships
between rotation rates ωk and the time derivatives of the corresponding rotation
vector ρ are fairly complicated, but they can be derived from Eq. C.221 for the
dynamics of coordinate transformation matrices.
Let ρENU be the rotation vector represented in earth-fixed ENU coordinates
that rotates earth-fixed ENU coordinate axes into vehicle body-fixed RPY axes,
and let C(ρ) be the corresponding rotation matrix, so that, in ENU coordinates,
1E = [ 1 0 0 ]T ,
C(ρENU )1E = 1R ,
1N = [ 0 1 0 ]T ,
C(ρENU )IN = 1P ,
1U = [ 0 0 1 ]T ,
C(ρENU )1U = 1Y ,
484
COORDINATE TRANSFORMATIONS
CRPY
ENU = [ 1R
1P
= [C(ρENU )1E
1Y ],
C(ρENU )1N
C(ρENU )1U ]
= C(ρENU )[ 1E 1N 1U ]
⎡
⎤
1 0 0
= C(ρENU ) ⎣ 0 1 0 ⎦
0 0 1
CRPY
ENU = C(ρENU ).
(C.137)
(C.138)
That is, C(ρENU ) is the coordinate transformation matrix from RPY coordinates
to ENU coordinates. As a consequence, from Eq. C.221,
d
d
C(ρENU ) = CRPY
dt
dt ENU
⎡
⎤
0
ωU −ωN
0
ωE ⎦ CRPY
= ⎣ −ωU
ENU
ωN −ωE
0
⎡
⎤
ωP
0
−ωY
⎣ ωY
0
−ωR ⎦ ,
+ CRPY
ENU
−ωP
ωR
0
⎡
⎤
0
ωU −ωN
d
⎣
−ω
0
ωE ⎦ C(ρENU )
C(ρENU ) =
U
dt
ωN −ωE
0
⎡
⎤
0 −ωY
ωP
0 −ωR ⎦ ,
+ C(ρENU ) ⎣ ωY
−ωP
ωR
0
(C.139)
(C.140)
(C.141)
⎡
where
ωRPY
⎤
ωR
= ⎣ ωP ⎦
ωY
(C.142)
is the vector of inertial rotation rates of the vehicle body, expressed in RPY
coordinates, and
⎡
ωENU
⎤
ωE
= ⎣ ωN ⎦
ωU
(C.143)
is the vector of inertial rotation rates of the ENU coordinate frame, expressed in
ENU coordinates.
COORDINATE TRANSFORMATION MODELS
485
The 3 × 3 matrix equation C.141 is equivalent to nine scalar equations:
∂c11
ρ̇E
∂ρE
∂c12
ρ̇E
∂ρE
∂c13
ρ̇E
∂ρE
∂c21
ρ̇E
∂ρE
∂c22
ρ̇E
∂ρE
∂c23
ρ̇E
∂ρE
∂c31
ρ̇E
∂ρE
∂c32
ρ̇E
∂ρE
∂c33
ρ̇E
∂ρE
+
+
+
+
+
+
+
+
+
∂c11
ρ̇N
∂ρN
∂c12
ρ̇N
∂ρN
∂c13
ρ̇N
∂ρN
∂c21
ρ̇N
∂ρN
∂c22
ρ̇N
∂ρN
∂c23
ρ̇N
∂ρN
∂c31
ρ̇N
∂ρN
∂c32
ρ̇N
∂ρN
∂c33
ρ̇N
∂ρN
+
+
+
+
+
+
+
+
+
∂c11
ρ̇U
∂ρU
∂c12
ρ̇U
∂ρU
∂c13
ρ̇U
∂ρU
∂c21
ρ̇U
∂ρU
∂c22
ρ̇U
∂ρU
∂c23
ρ̇U
∂ρU
∂c31
ρ̇U
∂ρU
∂c32
ρ̇U
∂ρU
∂c33
ρ̇U
∂ρU
= −c1,3 ωP + c1,2 ωY − c3,1 ωN + c2,1 ωU ,
= c1,3 ωR − c1,1 ωY − c3,2 ωN + c2,2 ωU ,
= −c1,2 ωR + c1,1 ωP − c3,3 ωN + c2,3 ωU ,
= −c2,3 ωP + c2,2 ωY + c3,1 ωE − c1,1 ωU ,
= c2,3 ωR − c2,1 ωY + c3,2 ωE − c1,2 ωU ,
= −c2,2 ωR + c2,1 ωP + c3,3 ωE − c1,3 ωU ,
= −c3,3 ωP + c3,2 ωY − c2,1 ωE + c1,1 ωN ,
= c3,3 ωR − c3,1 ωY − c2,2 ωE + c1,2 ωN ,
= −c3,2 ωR + c3,1 ωP − c2,3 ωE + c1,3 ωN ,
where
⎡
c11
⎣ c21
c31
c12
c22
c32
⎤
c13
def
c23 ⎦ = C(ρENU )
c33
and the partial derivatives
uE (1 − u2E ){2[1 − cos(θ )] − θ sin(θ )}
∂c11
,
=
∂ρE
θ
uN {−2u2E [1 − cos(θ )] − θ sin(θ )(1 − u2E )}
∂c11
,
=
∂ρN
θ
uU {−2u2E [1 − cos(θ )] − θ sin(θ )(1 − u2E )}
∂c11
,
=
∂ρU
θ
uN (1 − 2u2E )[1 − cos(θ )] + uE uU sin(θ ) − θ uE uU cos(θ ) + θ uN u2E sin(θ )
∂c12
,
=
∂ρE
θ
uE (1 − 2u2N )[1 − cos(θ )] + uU uN sin(θ ) − θ uN uU cos(θ ) + θ uE u2N sin(θ )
∂c12
,
=
∂ρN
θ
486
COORDINATE TRANSFORMATIONS
−2uE uN uU [1 − cos(θ )] − (1 − u2U ) sin(θ ) − θ u2U cos(θ ) + θ uU uN uE sin(θ )
∂c12
,
=
∂ρU
θ
uU (1 − 2u2E )[1 − cos(θ )] − uE uN sin(θ ) + θ uE uN cos(θ ) + θ uU u2E sin(θ )
∂c13
,
=
∂ρE
θ
−2uE uN uU [1 − cos(θ )] + (1 − u2N ) sin(θ ) + θ u2N cos(θ ) + θ uU uN uE sin(θ )
∂c13
,
=
∂ρN
θ
uE (1 − 2u2U )[1 − cos(θ )] − uU uN sin(θ ) + θ uN uU cos(θ ) + θ uE u2U sin(θ )
∂c13
,
=
∂ρU
θ
uN (1 − 2u2E )[1 − cos(θ )] − uE uU sin(θ ) + θ uE uU cos(θ ) + θ uN u2E sin(θ )
∂c21
,
=
∂ρE
θ
uE (1 − 2u2N )[1 − cos(θ )] − uU uN sin(θ ) + θ uN uU cos(θ ) + θ uE u2N sin(θ )
∂c21
,
=
∂ρN
θ
−2uE uN uU [1 − cos(θ )] + sin(θ )(1 − u2U ) + θ u2U cos(θ ) + θ uU uN uE sin(θ )
∂c21
,
=
∂ρU
θ
uE {−2u2N [1 − cos(θ )] − θ (1 − u2N ) sin(θ )}
∂c22
,
=
∂ρE
θ
uN (1 − u2N ){2[1 − cos(θ )] − θ sin(θ )}
∂c22
,
=
∂ρN
θ
uU {−2u2N [1 − cos(θ )] − θ (1 − u2N ) sin(θ )}
∂c22
,
=
∂ρU
θ
−2uE uN uU [1 − cos(θ )] − (1 − u2E ) sin(θ ) − θ u2E cos(θ ) + θ uE uN uU sin(θ )
∂c23
,
=
∂ρE
θ
uU (1 − 2u2N )[1 − cos(θ )] + uE uN sin(θ ) − θ uE uN cos(θ ) + θ u2N uU sin(θ )
∂c23
,
=
∂ρN
θ
uN (1 − 2u2U )[1 − cos(θ )] + uE uU sin(θ ) − θ uE uU cos(θ ) + θ u2U uN sin(θ )
∂c23
,
=
∂ρU
θ
uU (1 − 2u2E )[1 − cos(θ )] + uE uN sin(θ ) − θ uE uN cos(θ ) + θ uU u2E sin(θ )
∂c31
,
=
∂ρE
θ
−2uE uN uU [1 − cos(θ )] − (1 − u2N ) sin(θ ) − θ u2N cos(θ ) + θ uU uN uE sin(θ )
∂c31
,
=
∂ρN
θ
uE (1 − 2u2U )[1 − cos(θ )] + uU uN sin(θ ) − θ uN uU cos(θ ) + θ uE u2U sin(θ )
∂c31
,
=
∂ρU
θ
−2uE uN uU [1 − cos(θ )] + (1 − u2E ) sin(θ ) + θ u2E cos(θ ) + θ uU uN uE sin(θ )
∂c32
,
=
∂ρE
θ
uU (1 − 2u2N )[1 − cos(θ )] − uE uN sin(θ ) + θ uE uN cos(θ ) + θ u2N uU sin(θ )
∂c32
,
=
∂ρN
θ
487
COORDINATE TRANSFORMATION MODELS
uN (1 − 2u2U )[1 − cos(θ )] − uE uU sin(θ ) + θ uE uU cos(θ ) + θ u2U uN sin(θ )
∂c32
,
=
∂ρU
θ
uE {−2u2U [1 − cos(θ )] − θ sin(θ )(1 + u2U )}
∂c33
,
=
∂ρE
θ
uN {−2u2U [1 − cos(θ )] − θ sin(θ )(1 + u2U )}
∂c33
,
=
∂ρN
θ
uU (1 − u2U ){2[1 − cos(θ )] − θ sin(θ )}
∂c33
=
∂ρU
θ
for
def
θ = |ρENU |,
def ρE
def ρN
uE =
,
uN =
,
θ
θ
def
uu =
ρU
.
θ
These nine scalar linear equations can be put into matrix form and solved in least
squares fashion as
⎡
⎡
⎤
⎢
ρ̇E
⎢
⎢
L ⎣ ρ̇N ⎦ = R ⎢
⎢
ρ̇U
⎣
⎡
ωR
ωP
ωY
ωE
ωN
ωU
⎤
⎥
⎥
⎥
⎥,
⎥
⎦
⎤
/
0
ρ̇E
⎣ ρ̇N ⎦ = [LT L]\[LT R] ωRPY .
ωENU
ρ̇U
∂ ρ̇/∂ω
(C.144)
(C.145)
The matrix product LT L will always be invertible because its determinant
det[LT L] = 32
[1 − cos(θ )]2
,
θ4
lim det[LT L] = 8,
θ→0
and the resulting equation for ρENU can be put into the form
0
/ 0/
∂ ρ̇
ωRPY
.
ρ̇ENU =
ωENU
∂ω
The 3 × 6 matrix ∂ρ/∂ω can be partitioned as
/ 0 /
0
∂ ρ̇
∂ ρ̇ ∂ ρ̇
=
∂ω
∂ωRPY ∂ωENU
(C.146)
(C.147)
(C.148)
(C.149)
488
COORDINATE TRANSFORMATIONS
with 3 × 3 submatrices
0
/
∂ρ
1
1
sin(|ρ|)
|ρ| sin(|ρ|)
ρρT +
I + [ρ⊗]
=
−
∂ωRPY
|ρ|2
2|ρ|[1 − cos(|ρ|)]
2[1 − cos(|ρ|)]
2
(C.150)
= 1ρ 1Tρ +
lim |ρ
P →0
θ sin(θ )
θ
[I − 1ρ 1ρ T ] + [1ρ ⊗],
2[1 − cos(θ )]
2
P ∂ ρ̇
= I,
∂ωRPY
(C.151)
(C.152)
0
/
1
sin(|ρ|)
∂ ρ̇
ρρT
=−
−
∂ωENU
|ρ|2
2|ρ|[1 − cos(|ρ|)]
−
1
|ρ| sin(|ρ|)
I + [ρ⊗]4
2[1 − cos(|ρ|)]
2
= −1ρ 1Tρ −
(C.153)
θ sin(θ )
θ
[I − 1ρ 1Tρ ] + [1ρ ⊗].
2[1 − cos(θ )]
2
∂ ρ̇
= −I.
|ρ|→0 ∂ωENU
(C.154)
(C.155)
lim
For locally leveled gimbaled systems, ωRPH = 0. That is, the gimbals normally
keep the accelerometer axes aligned to the ENU or NED coordinate axes, a
process modeled by ωENU alone.
C.4.2.5 Time Derivatives of Matrix Expressions The Kalman filter implementation for integrating GPS with a strapdown INS in Chapter 8 will require
derivatives with respect to time of the matrices
∂ ρ̇ENU
∂ωRPH
(Eq.C.150)
and
∂ ρ̇ENU
∂ωENU
(Eq. C.153).
We derive here a general-purpose formula for taking such derivatives and then
apply it to these two cases.
General Formulas There is a general-purpose formula for taking the time
derivatives (d/dt)M(ρ) of matrix expressions of the sort
M(ρ) = M(s1 (ρ), s2 (ρ), s3 (ρ))
= s1 (ρ)I3 + s2 (ρ)[ρ⊗] + s3 (ρ)ρρT ,
(C.156)
(C.157)
that is, as linear combinations of I3 , ρ⊗, and ρρT with scalar functions of ρ as
the coefficients.
489
COORDINATE TRANSFORMATION MODELS
The derivation uses the time derivatives of the basis matrices,
d
I3 = 03 ,
dt
d
[ρ⊗] = [ρ̇⊗],
dt
d T
ρρ = ρ̇ρT + ρρ̇T ,
dt
where the vector
ρ̇ =
d
ρ,
dt
(C.158)
(C.159)
(C.160)
(C.161)
and then uses the chain rule for differentiation to obtain the general formula
d
∂s1 (ρ)
∂s2 (ρ)
M(ρ) =
ρ̇I3 +
ρ̇[ρ⊗] + s2 (ρ)[ρ̇⊗],
dt
∂ρ
∂ρ
+
∂s3 (ρ)
ρ̇[ρρT ] + s3 (ρ)[ρ̇ρT + ρρ̇T ],
∂ρ
(C.162)
where the gradients ∂si (ρ)/∂ρ are to be computed as row vectors and the inner
products [∂si (ρ)/∂ρ]ρ̇ will be scalars.
Equation C.162 is the general-purpose formula for the matrix forms of interest,
which differ only in their scalar functions si (ρ). These scalar functions si (ρ) are
generally rational functions of the following scalar functions (shown in terms of
their gradients):
∂
|ρ|p = p|ρ|p−2 ρT ,
∂ρ
(C.163)
∂
sin(|ρ|) = cos(|ρ|)|ρ|−1 ρT ,
∂ρ
(C.164)
∂
cos(|ρ|) = − sin(|ρ|)|ρ|−1 ρT
∂ρ
(C.165)
Time Derivative of ∂ ρ̇ENU /∂ωRPY In this case (Eq. C.150).
s1 (ρ) =
|ρ| sin(|ρ|)
,
2[1 − cos(|ρ|)]
∂s1 (ρ)
1 − |ρ|−1 sin(|ρ|) T
=−
ρ ,
∂ρ
2[1 − cos(|ρ|)]
s2 (ρ) = 12 ,
(C.166)
(C.167)
(C.168)
490
COORDINATE TRANSFORMATIONS
∂s2
= 01×3 ,
∂ρ
0
/
1
sin(|ρ|)
,
−
s3 (ρ) =
|ρ|2
2|ρ|[1 − cos(|ρ|)]
∂s3 (ρ)
1 + |ρ|−1 sin(|ρ|) − 4|ρ|−2 [1 − cos(|ρ|)] T
=
ρ ,
∂ρ
2|ρ|2 [1 − cos(|ρ|)]
(C.169)
(C.170)
(C.171)
∂s1 (ρ)
∂s2 (ρ)
d ∂ ρ̇ENU
ρ̇I3 +
ρ̇[ρ⊗] + s2 (ρ)[ρ̇⊗]
=
dt ∂ωRPY
∂ρ
∂ρ
∂s3 (ρ)
ρ̇[ρρT ] + s3 (ρ)[ρ̇ρT + ρρ̇T ],
(C.172)
∂ρ
%
&
1 − |ρ|−1 sin(|ρ|)
1
=−
(ρT ρ̇)I3 + [ρ̇⊗],
2[1 − cos(|ρ|)]
2
%
&
1 + |ρ|−1 sin(|ρ|) − 4|ρ|−2 [1 − cos(|ρ|)]
+
× (ρT ρ̇)[ρρT ],
2|ρ|2 [1 − cos(|ρ|)]
&
%
sin(|ρ|)
1
[ρ̇ρT + ρρ̇T ].
−
(C.173)
+
|ρ|2
2|ρ|[1 − cos(|ρ|)]
+
Time Derivative of ∂ ρ̇ENU /∂ωENU In this case (Eq. C.153),
s1 (ρ) = −
|ρ| sin(|ρ|)
,
2[1 − cos(|ρ|)]
∂s1 (ρ)
1 − |ρ|−1 sin(|ρ|) T
=
ρ ,
∂ρ
2[1 − cos(|ρ|)]
s2 (ρ) = 12 ,
∂s2
= 01×3 ,
∂ρ
0
/
1
sin(|ρ|)
,
(ρ)
=
−
−
s3
|ρ|2
2|ρ|[1 − cos(|ρ|)]
∂s3 (ρ)
1 + |ρ|−1 sin(|ρ|) − 4|ρ|−2 [1 − cos(|ρ|)] T
=−
ρ ,
∂ρ
2|ρ|2 [1 − cos(|ρ|)]
(C.174)
(C.175)
(C.176)
(C.177)
(C.178)
(C.179)
d ∂ ρ̇ENU
∂s1 (ρ)
∂s2 (ρ)
ρ̇I3 +
ρ̇[ρ⊗] + s2 (ρ)[ρ̇⊗],
=
dt ∂ωENU
∂ρ
∂ρ
+
∂s3 (ρ)
ρ̇[ρρT ] + s3 (ρ)[ρ̇ρT + ρρ̇T ]
∂ρ
(C.180)
491
COORDINATE TRANSFORMATION MODELS
%
&
1 − |ρ|−1 sin(|ρ|)
1
=
(ρT ρ̇)I3 + (ρ)[ρ̇⊗],
2[1 − cos(|ρ|)]
2
%
&
1 + |ρ|−1 sin(|ρ|) − 4|ρ|−2 [1 − cos(|ρ|)]
−
× (ρT ρ̇)[ρρT ],
2|ρ|2 [1 − cos(|ρ|)]
&
%
sin(|ρ|)
1
[ρ̇ρT + ρρ̇T ].
−
(C.181)
−
|ρ|2
2|ρ|[1 − cos(|ρ|)]
C.4.2.6 Partial Derivatives with Respect to Rotation Vectors Calculation of
the dynamic coefficient matrices F and measurement sensitivity matrices H in
linearized or extended Kalman filtering with rotation vectors ρENU as part of the
system model state vector requires taking derivatives with respect to ρENU of
associated vector-valued f- or h-functions, as
F=
∂f(ρENU , v)
,
∂ρENU
(C.182)
H=
∂h(ρENU , v)
,
∂ρENU
(C.183)
where the vector-valued functions will have the general form
f(ρENU , v) or h(ρENU , v)
= {s0 (ρENU )I3 + s1 (ρENU )[ρENU ⊗] + s2 (ρENU )ρENU ρTENU }v, (C.184)
and s0 , s1 , s2 are scalar-valued functions of ρENU and v is a vector that does not
depend on ρENU . We will derive here the general formulas that can be used for
taking the partial derivatives ∂f(ρENU , v)/∂ρENU or ∂h(ρENU , v)/∂ρENU . These
formulas can all be derived by calculating the derivatives of the different factors
in the functional forms and then using the chain rule for differentiation to obtain
the final result.
Derivatives of Scalars The derivatives of the scalar factors s0 , s1 , s2 are
0
/
∂si (ρENU ) ∂si (ρENU ) ∂si (ρENU )
∂
,
(C.185)
si (ρENU ) =
∂ρENU
∂ρE
∂ρN
∂ρU
a row vector. Consequently, for any vector-valued function g(ρENU ) by the chain
rule, the derivatives of the vector-valued product si (ρENU )g(ρENU ) are
∂si (ρENU )
∂g(ρENU )
∂{si (ρENU )g(ρENU )}
= g(ρENU )
+si (ρENU )
,
∂ρENU
∂ρENU
∂ρENU
3×3matrix
(C.186)
3×3matrix
the result of which will be the 3 × 3 Jacobian matrix of that subexpression in f
or h.
492
COORDINATE TRANSFORMATIONS
Derivatives of Vectors The three potential forms of the vector-valued function
g in Eq. C.186 are
⎧
⎨ Iv = v,
ρENU ⊗ v,
g(ρENU ) =
⎩
ρENU ρTENU v,
(C.187)
each of which is considered independently:
∂v
= 03×3 ,
∂ρENU
(C.188)
∂ρENU ⊗ v
∂[−v ⊗ ρENU ]
=
,
∂ρENU
∂ρENU
= −[v⊗],
⎡
= −⎣
0
v3
−v2
−v3
0
v1
(C.189)
(C.190)
⎤
v2
−v1 ⎦ ,
0
(C.191)
∂ρT v
∂ρENU ρTENU v
∂ρENU
= (ρTENU v)
+ ρENU ENU ,
∂ρENU
∂ρENU
∂ρENU
(C.192)
= (ρTENU v)I3×3 + ρENU vT .
(C.193)
General Formula Combining the above formulas for the different parts, one
can obtain the following general-purpose formula:
∂
{s0 (ρENU )I3 + s1 (ρENU )[ρENU ⊗] + s2 (ρENU )ρENU ρTENU }v
∂ρENU
/
0
∂s0 (ρENU ) ∂s0 (ρENU ) ∂s0 (ρENU )
=v
∂ρE
∂ρN
∂ρU
0
/
∂s1 (ρENU ) ∂s1 (ρENU ) ∂s1 (ρENU )
+ [ρENU ⊗ v] ccc
∂ρE
∂ρN
∂ρU
− s1 (ρENU )[v⊗]
+ (ρTENU v)ρENU
/
∂s2 (ρENU ) ∂s2 (ρENU ) ∂s2 (ρENU )
∂ρE
∂ρN
∂ρU
+ s2 (ρENU )[(ρTENU v)I3×3 + ρENU vT ],
applicable for any differentiable scalar functions s0 , s1 , s2 .
0
(C.194)
493
COORDINATE TRANSFORMATION MODELS
C.4.3 Direction Cosines Matrix
We have demonstrated in Eq.C.12 that the coordinate transformation matrix
between one orthogonal coordinate system and another is a matrix of direction
cosines between the unit axis vectors of the two coordinate systems,
⎡
VW
CU
XY Z
⎤
cos(θXU ) cos(θXV ) cos(θXW )
= ⎣ cos(θY U ) cos(θY V ) cos(θY W ) ⎦ .
cos(θZU ) cos(θZV ) cos(θZW )
(C.195)
Because the angles do not depend on the order of the direction vectors (i.e.,
θab = θba ), the inverse transformation matrix
⎡
Z
CXY
UV W
cos(θU X ) cos(θU Y )
= ⎣ cos(θV X ) cos(θV Y )
cos(θW X ) cos(θW Y )
⎡
cos(θXU ) cos(θXV )
= ⎣ cos(θY U ) cos(θY V )
cos(θZU ) cos(θZV )
⎤
cos(θU Z )
cos(θV Z ) ⎦ ,
cos(θW X )
⎤T
cos(θXW )
cos(θY W ) ⎦ ,
cos(θZW )
VW T
= (CU
XY Z ) .
(C.196)
(C.197)
(C.198)
That is, the inverse coordinate transformation matrix is the transpose of the
forward coordinate transformation matrix. This implies that the coordinate transformation matrices are orthogonal matrices.
C.4.3.1 Rotating Coordinates Let “rot” denote a set of rotating coordinates,
with axes Xrot , Yrot , Zrot , and let “non” represent a set of non-rotating (i.e.,
inertial) coordinates, with axes Xnon , Ynon , Znon , as illustrated in Fig. C.14.
Any vector vrot in rotating coordinates can be represented in terms of its
nonrotating components and unit vectors parallel to the nonrotating axes, as
vrot = vx,non 1x,non + vy,non 1y,non + vz,non 1z,non
⎡
⎤
vx,non
= [ 1x,non 1y,non 1z,non ] ⎣ vy,non ⎦
vz,non
= Cnon
rot vnon ,
(C.199)
(C.200)
(C.201)
where vx,non , vy,non , vz,non are nonrotating components of the vector, 1x,non , 1y,non ,
1z,non = unit vectors along Xnon , Ynon , Znon axes, as expressed in rotating coordinates
vrot = vector v expressed in RPY coordinates
494
COORDINATE TRANSFORMATIONS
Fig. C.14
Rotating coordinates.
vnon = vector v expressed in ECI coordinates,
Cnon
rot = coordinate transformation matrix from nonrotating coordinates to rotating coordinates
and
Cnon
rot = [ 1x,non
1y,non
1z,non ].
(C.202)
The time derivative of Cnon
rot , as viewed from the non-rotating coordinate frame,
can be derived in terms of the dynamics of the unit vectors 1x,non , 1y,non and
1z,non in rotating coordinates.
As seen by an observer fixed with respect to the nonrotating coordinates, the
nonrotating coordinate directions will appear to remain fixed, but the external
inertial reference directions will appear to be changing, as illustrated in Fig. C.14.
Gyroscopes fixed in the rotating coordinates would measure three components of
the inertial rotation rate vector
⎡
⎤
ωx,rot
ωrot = ⎣ ωy,rot ⎦
(C.203)
ωz,rot
in rotating coordinates, but the non-rotating unit vectors, as viewed in rotating
coordinates, appear to be changing in the opposite sense, as
d
1x,non = −ωrot ⊗ 1x,non ,
dt
d
1y,non = −ωrot ⊗ 1y,non ,
dt
d
1z,non = −ωrot ⊗ 1z,non ,
dt
(C.204)
(C.205)
(C.206)
495
COORDINATE TRANSFORMATION MODELS
as illustrated in Fig. C.14. The time-derivative of the coordinate transformation
represented in Eq. C.202 will then be
d non
C =
dt rot
/
d
1x,non
dt
d
1y,non
dt
= [−ωrot ⊗ 1x,non
d
1z,non
dt
0
− ωrot ⊗ 1y,non
= −[ωrot ⊗][1x,non ]
1y,non
= −[ωrot ⊗]Cnon
rot ,
⎡
0
−ωz,rot
def
0
[ωrot ⊗] = ⎣ ωz,rot
−ωy,rot ωx,rot
(C.207)
− ωrot ⊗ 1z,non ]
1z,non ]
⎤
ωy,rot
−ωx,rot ⎦ .
0
(C.208)
(C.209)
The inverse coordinate transformation
non −1
Crot
non = (Crot )
(C.210)
T
= (Cnon
rot ) ,
(C.211)
the transpose of Cnon
rot , and its derivative
d rot
d
C = (Cnon )T
dt non dt rot
%
&
d non T
=
C
dt rot
(C.212)
(C.213)
T
= (−[ωrot ⊗]Cnon
rot )
(C.214)
T
T
= −(Cnon
rot ) [ωrot ⊗] ,
(C.215)
=
Crot
non [ωrot ⊗].
(C.216)
In the case that “rot” is “RPY” (roll-pitch-yaw coordinates) and “non” is “ECI”
(earth centered inertial coordinates), Eq. C.216 becomes
d RPY
= CRPY
C
ECI [ωRPY ⊗],
dt ECI
(C.217)
and in the case that “rot” is “ENU” (east-north-up coordinates) and “non” is
“ECI” (earth centered inertial coordinates), Eq. C.208 becomes
d ECI
= −[ωENU ⊗]CECI
C
ENU ,
dt ENU
(C.218)
496
COORDINATE TRANSFORMATIONS
and the derivative of their product
ECI
RPY
CRPY
ENU = CENU CECI ,
0
/
0
/
d RPY
d ECI
d RPY
RPY
ECI
C
=
+ CENU
C
C
C
dt ENU
dt ENU ECI
dt ECI
(C.219)
(C.220)
RPY
ECI
RPY
= [[−ωENU ⊗]CECI
ENU ]CECI + CENU [CECI [ωRPY ⊗]]
= [−ωENU ⊗] CECI
CRPY + CECI CRPY [ωRPY ⊗],
ENU ECI ENU ECI
CRPY
ENU
CRPY
ENU
d RPY
RPY
C
= −[ωENU ⊗]CRPY
ENU + CENU [ωRPY ⊗].
dt ENU
(C.221)
Equation C.221 was originally used for maintaining vehicle attitude information in strapdown INS implementations, where the variables
ωRPY = vector of inertial rates measured by the gyroscopes, (C.222)
ωENU = ωearthrate + ωvE + ωvN ,
⎡
⎤
0
ω⊕ = ω⊕ ⎣ cos(φgeodetic ) ⎦ ,
sin(φgeodetic )
⎡ ⎤
vE ⎣ 0 ⎦
1 ,
ωvE =
rT + h 0
(C.223)
(C.224)
⎡
ωvN
⎤
−1
vN ⎣
0 ⎦,
=
rM + h
0
(C.225)
and
where
ω⊕ = earth rotation rate
φgeodetic = geodetic latitude
vE = the east component of velocity with respect to the surface of the
earth
rT = transverse radius of curvature of the ellipsoid (Eq. 6.41)
vN = north component of velocity with respect to the surface of the earth
rM = meridional radius of curvature of the ellipsoid (Eq. 6.38)
h = altitude above (+) or below (−) the reference ellipsoid surface
(≈mean sea level)
Unfortunately, Eq. C.221 was found to be not particularly well suited for
accurate integration in finite-precision arithmetic. This integration problem was
eventually solved using quaternions.
COORDINATE TRANSFORMATION MODELS
497
C.4.4 Quaternions
The term quaternions is used in several contexts to refer to sets of four. In
mathematics, it refers to an algebra in four dimensions discovered by the Irish
physicist and mathematician Sir William Rowan Hamilton (1805–1865). The
utility of quaternions for representing rotations (as points on a sphere in four
dimensions) was known before strapdown systems, they soon became the standard representation of coordinate transforms in strapdown systems, and they have
since been applied to computer animation.
C.4.4.1 Quaternion Matrices For people already familiar with matrix algebra,
the algebra of quaternions can be defined by using an isomorphism between 4 × 1
quaternion vectors q and real 4 × 4 quaternion matrices Q:
⎡
⎡
⎤
⎤
q1
q1 −q2 −q3 −q4 ,
⎢ q2 ⎥
⎢ q2
q1 −q4
q3 , ⎥
⎢
⎥
⎥
q=⎢
(C.226)
⎣ q3 ⎦ ↔ Q = ⎣ q3
q4
q1 −q2 , ⎦
q4 −q3
q4
q2
q1
Q1
Q2
Q3
Q4
= q1 Q1 + q2 Q2 + q3 Q3 + q4 Q4 ,
⎤
⎡
1 0 0 0,
0 1 0 0, ⎥
def ⎢
⎥
=⎢
⎣ 0 0 1 0, ⎦ ,
0 0 0 1
⎤
⎡
0 −1 0
0
1
0 0
0 ⎥
def ⎢
⎥,
=⎢
⎣ 0
0 0 −1 ⎦
0
0 1
0
⎤
⎡
0
0 −1 0
0
0
0 1 ⎥
def ⎢
⎥,
=⎢
⎣ 1
0
0 0 ⎦
0 −1
0 0
⎤
⎡
0 0
0 −1
0 0 −1
0 ⎥
def ⎢
⎥,
=⎢
⎣ 0 1
0
0 ⎦
1 0
0
0
(C.227)
(C.228)
(C.229)
(C.230)
(C.231)
in terms of four 4 × 4 quaternion basis matrices, Q1 , Q2 , Q3 , Q4 , the first of
which is an identity matrix and the rest of which are antisymmetric.
C.4.4.2 Addition and Multiplication Addition of quaternion vectors is the
same as that for ordinary vectors. Multiplication is defined by the usual rules for
matrix multiplication applied to the four quaternion basis matrices, the multiplication table for which is given in Table C.1. Note that, like matrix multiplication,
498
COORDINATE TRANSFORMATIONS
TABLE C.1. Multiplication of Quaternion Basis
Matrices
First Factor
Second Factor
Q1
Q2
Q3
Q4
Q1
Q2
Q3
Q4
Q1
Q2
Q3
Q4
Q2
−Q1
−Q4
Q3
Q3
Q4
−Q1
−Q2
Q4
−Q3
Q2
−Q1
quaternion multiplication is noncommutative. That is, the result depends on the
order of multiplication.
Using the quaternion basis matrix multiplication Table (C.1), the ordered product AB of two quaternion matrices
A = a1 Q1 + a2 Q2 + a3 Q3 + a4 Q4 ,
(C.232)
B = b1 Q1 + b2 Q2 + b3 Q3 + b4 Q4
(C.233)
can be shown to be
AB = (a1 b1 − a2 b2 − a3 b3 − a4 b4 )Q1
+(a2 b1 + a1 b2 − a4 b3 + a3 b4 )Q2
(C.234)
+(a3 b1 + a4 b2 + a1 b3 − a2 b4 )Q3
+(a4 b1 − a3 b2 + a2 b3 + a1 b4 )Q4
in terms of the coefficients ak , bk and the quaternion basis matrices.
C.4.4.3 Conjugation Conjugation of quaternions is a unary operation analogous to conjugation of complex numbers, in that the real part (the first component
of a quaternion) is unchanged and the other parts change sign. For quaternions,
this is equivalent to transposition of the associated quaternion matrix
Q = q1 Q1 + q2 Q2 + q3 Q3 + q4 Q4 ,
(C.235)
QT = q1 Q1 − q2 Q2 − q3 Q3 − q4 Q4
(C.236)
so that
↔ q∗ ,
Q Q =
T
(q12
(C.237)
+
q22
+
↔ q∗ q = |q|2 .
q32
+
q42 )Q1
(C.238)
(C.239)
499
COORDINATE TRANSFORMATION MODELS
C.4.4.4 Representing Rotations The problem with rotation vectors as representations for rotations is that the rotation vector representing successive rotations
ρ1 , ρ2 , ρ3 , . . . , ρn is not a simple function of the respective rotation vectors.
This representation problem is solved rather elegantly using quaternions, such
that the quaternion representation of the successive rotations is represented by the
quaternion product qn ×qn−1 × · · · ×q3 ×q2 ×q1 . That is, each successive rotation
can be implemented by a single quaternion product.
The quaternion equivalent of the rotation vector ρ with |ρ| = θ,
⎡
⎡
⎤
⎤
ρ1
u1
def
def
ρ = ⎣ ρ2 ⎦ = θ ⎣ u2 ⎦
ρ3
u3
(C.240)
(i.e., where u is a unit vector), is
⎡
⎢
⎢
⎢
⎢
def ⎢
q(ρ) = ⎢
⎢
⎢
⎢
⎣
% &
θ
cos
2
ρ1 sin(θ/2)
θ
ρ2 sin(θ/2)
θ
ρ3 sin(θ/2)
θ
% &
θ
cos
⎥ ⎢
%2 &
⎥ ⎢
θ
⎥ ⎢
u1 sin
⎥ ⎢
⎥ ⎢
%2&
⎥=⎢
θ
⎥ ⎢
u2 sin
⎥ ⎢
⎢
⎥ ⎢
%2&
⎦ ⎣
θ
u3 sin
2
⎡
⎤
⎤
⎥
⎥
⎥
⎥
⎥
⎥,
⎥
⎥
⎥
⎥
⎦
(C.241)
and the vector w resulting from the rotation of any three-dimensional vector
⎡
⎤
v1
v = ⎣ v2 ⎦
v3
def
through the angle θ about the unit vector u is implemented by the quaternion
product
q(w) = q(ρ)q(v)q∗ (ρ)
% &
⎡
θ
cos
⎢
⎢
%2 &
⎢
⎢ u1 sin θ
def ⎢
%2&
=⎢
⎢
⎢ u2 sin θ
⎢
⎢
%2&
⎣
θ
u3 sin
2
def
⎡
⎤
⎥ ⎡
⎥
⎥
⎥ ⎢
⎥ ⎢
⎥×⎢
⎥ ⎣
⎥
⎥
⎥
⎦
⎢
⎢
⎢
⎢
v1 ⎥
⎥ ⎢
⎥×⎢
v2 ⎦ ⎢
⎢
⎢
⎢
v3
⎣
0
⎤
% &
θ
cos
2% &
θ
−u1 sin
%2&
θ
−u2 sin
%2&
θ
−u3 sin
2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
(C.242)
(C.243)
500
COORDINATE TRANSFORMATIONS
⎡
⎤
0
⎢ w ⎥
⎢ 1 ⎥
=⎢
⎥,
⎣ w2 ⎦
(C.244)
w3
w1 = cos(θ )v1 + [1 − cos(θ )][u1 (u1 v1 + u2 v2 + u3 v3 )]
+ sin(θ )(u2 v3 − u3 v2 ),
(C.245)
w2 = cos(θ )v2 + [1 − cos(θ )][u2 (u1 v1 + u2 v2 + u3 v3 )]
+ sin(θ )(u3 v1 − u1 v3 ),
(C.246)
w3 = cos(θ )v3 + [1 − cos(θ )][u3 (u1 v1 + u2 v2 + u3 v3 )]
+ sin(θ )(u1 v2 − u2 v1 ),
⎤
⎡
⎤
v1
w1
⎣ w2 ⎦ = C(ρ) ⎣ v2 ⎦ ,
w3
v3
(C.247)
⎡
or
(C.248)
where the rotation matrix C(ρ) is defined in Eq. C.113 and Eq. C.242 implements
the same rotation of v as the matrix product C(ρ)v. Moreover, if
def
q(wk ) = v
(C.249)
q(wk ) = q(ρk )q(wk−1 )q∗ (ρk )
(C.250)
and
def
for k = 1, 2, 3, . . . , n, then the nested quaternion product
q(wn ) = q(ρn ) · · · q(ρ2 )q(ρ1 )q(v)q∗ (ρ1 )q∗ (ρ2 ) · · · q∗ (ρn )
(C.251)
implements the succession of rotations represented by the rotation vectors ρ1 , ρ2 ,
ρ3 , . . . , ρn , and the single quaternion
def
q[n] = q(ρn )q(ρn−1 ) · · · q(ρ3 )q(ρ2 )q(ρ1 )
= q(ρn )q[n−1]
(C.252)
(C.253)
then represents the net effect of the successive rotations as
q(wn ) = q[n] q(w0 )q∗[n] .
(C.254)
COORDINATE TRANSFORMATION MODELS
501
The initial value q[0] for the rotation quaternion will depend upon the inital
orientation of the two coordinate systems. The initial value
⎡ ⎤
1
0 ⎥
def ⎢
⎥
(C.255)
q[0] = ⎢
⎣ 0 ⎦
0
applies to the case that the two coordinate systems are aligned. In strapdown
system applications, the initial value q[0] is determined during the INS alignment
procedure.
Equation C.252 is the much-used quaternion representation for successive
rotations, and Eq. C.254 is how it is used to perform coordinate transformations
of any vector w0 .
This representation uses the four components of a unit quaternion to maintain
the transformation from one coordinate frame to another through a succession
of rotations. In practice, computer roundoff may tend to alter the magnitude of
the alegedly unit quaternion, but it can easily be rescaled to a unit quaternion by
dividing by its magnitude.
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