Design of a GPS aided guidance, navigation, and

Design of a GPS aided guidance, navigation, and
Calhoun: The NPS Institutional Archive
Theses and Dissertations
Thesis Collection
1994-03
Design of a GPS aided guidance, navigation, and
control system for trajectory control of an air vehicle
Hallberg, Eric N.
Monterey, California. Naval Postgraduate School
http://hdl.handle.net/10945/26457
DUDLEY KNOX LIBRARY
NAVAL POSTGRADUATE SCHOOL
MONTEREY CA
93943-5101
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Design of a GPS Aided
Guidance, Navigation, and Control System
for Trajectory Control of an Air Vehicle
by
Eric N. Hallberg
Lieutenant, United States
Navy
B.S. University of Pennsylvania, 1984
Submitted in partial fulfillment of the
requirements for the degree of
MASTER OF SCIENCE
IN
AERONAUTICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOL
March
Department
,
1994
of Aeronautics
u
and Astronautics
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10.
WORK
PROGRAM
TITLE (Include Security
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Design of a
GPS
UNI"!
ACCESSION
Classification)
Aided Guidance, Navigation, and Control System for Trajectory Control of an Air Vehicle
PERSONAL AUTHOR(S)
.
la.
LT. Eric N. Hallberg
14. DATE OF REPORT (Year, Month, Day)
13b. TIME COVERED
FROM 01/92 TO 03/94
March 1994
TYPE OF REPORT
Master's Thesis
SUPPLEMENTARY NOTATION
.
The views expressed
Icial policy or position of the
COSATI CODES
GROUP
FIELD
PAGE COUNT
119
15.
in this thesis are those of the author and do not reflect the
Department of Defense or the United States Government.
18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number)
SUB-GROUP
Trajectory Control,
GNC,
Linear Quadratic Regulator,
LQR, Kalman
Filt
Nonlinear Simulation, ^-Implementation, Bluebird
ABSTRACT ('Continue on reverse if necessary and identify by block number)
The advent of GPS has afforded the aerospace controls engineer a powerful, new means of controlling air vehic
tiis work explores a new method of designing and implementing controllers and guidance systems for autonomous con
l.
air vehicles utilizing
terest
when
a
GPS
integrated guidance, navigation and control system.
realizing controllers to track reference trajectories given in
iplementation,
and dynamic simulation of a
inertial reference
This
is
a subject of consider*
inertial reference frame.
an Unmanned
commands and other measured outputs (such
The
des:
Air Vehicle (Uj
precise tracking trajectory controller for
presented. This design provides a natural conversion of
)m an
an
as
GPS
sign
frame to a body-fixed reference frame. This achieves automatic recruiting of the actual
ule preserving the properties of the original design (linearization principle).
i.
*]
a.
DISTRIBUTION/AVAILABILITY OF ABSTRACT
UNCLASSIFIED/UNLIMITED DsAME AS RPT.
NAME OF RESPONSIBLE INDIVIDUAL
Isaac
I.
APR
ABSTRACT SECURITY CLASSIFICATION
UNCLASSIFIED
22b.
K aminer
83
21.
DTIC USERS
TELEPHONE (Include Area Code)
408) 656-2972
edition may be used until exhaust ed
All other editions are obsolete
22c.
OFFICE SYMBOL
AA/KA
:l a sffflcTrlo'TOFTHTSF
UNCLASSLFLED
ABSTRACT
The advent
means
of
GPS
has afforded the aerospace controls engineer a powerful, new
of controlling air vehicles. This work explores a
implementing controllers and guidance systems
utilizing
a
GPS
an
of designing
autonomous control
when
The
design, implementation,
of a precise tracking trajectory controller for
an
Unmanned
a subject
and dynamic simulation
Air Vehicle
mmands
sented. This design provides a natural conversion of co
GPS
is
realizing controllers to track reference trajectories given
inertial reference frame.
outputs (such as
and
of air vehicles
integrated guidance, navigation and control system. This
of considerable interest
in
for
new method
signals)
from an
inertia! reference
(UAV)
is
pre-
and other measured
frame to a body-fixed
refer-
ence frame. This achieves automatic recruiting of the actuators while preserving the
properties of the original design (linearization principle).
ui
6,1
TABLE OF CONTENTS
I.
INTRODUCTION
1
II.
DEVELOPMENT OF THE DYNAMIC MODEL
4
A.
III.
IV.
REFERENCE FRAMES
4
1.
Local Tangent Plane Reference Frame
4
2.
Body-Fixed Reference Frame
5
3.
Flight
Path or Wind Reference Frame
6
B.
COORDINATE TRANSFORMATIONS
7
C.
NOTATION
9
D.
RIGID
BODY EQUATIONS OF MOTION
10
1.
Linear Motion
10
2.
Angular Rotation
11
3.
External Forces and
4.
The State Space Representation
14
5.
Trim and Linearization
19
Moments
12
THE LINEAR QUADRATIC REGULATOR DESIGN
23
A.
LQR OVERVIEW
23
B.
DESIGN REQUIREMENTS
27
C.
THE SYNTHESIS MODEL
28
D.
THE DESIGN PROCESS
32
E.
LQR CONTROLLER PERFORMANCE
34
CONTROLLER IMPLEMENTATION ON THE NONLINEAR PLANT
A.
^-IMPLEMENTATION
B.
^-IMPLEMENTATION OF THE CONTROLLER
41
41
iv
IN
SIMULINK
47
MONTEREY CA
C.
GENERATION OF THE TRAJECTORY COMMANDS
48
D.
STATE FEEDBACK TO OUTPUT FEEDBACK
54
E.
V.
93943-510!
1.
Sensor Modeling
54
2.
Kalman
58
Filtering
INTEGRATION OF THE FULL NONLINEAR SIMULATION
.
.
APPLICATION TO THE CONTROL OF BLUEBIRD
A.
63
P-IMPLEMENTED CONTROLLER PERFORMANCE CHARACTERISTICS
B.
63
AN AIRPORT DEPARTURE AND ARRIVAL FLIGHT SIMULATION
VI.
60
72
CONCLUSIONS AND RECOMMENDATIONS
78
A.
CONCLUSIONS
78
B.
RECOMMENDATIONS
79
APPENDIX A
MATLAB
APPENDIX B
SIMULINK FILES
APPENDIX C
D-IMPLEMENTATION PROOFS REFERENCED
FILES
81
98
...
100
REFERENCES
106
INITIAL DISTRIBUTION LIST
107
LIST OF TABLES
3.1
EIGENVALUES OF BLUEBIRD
29
3.2
EIGENVALUES OF BLUEBIRD WITH YAW DAMPER
30
3.3
TRANSMISSION ZEROS OF SYNTHESIS MODEL
34
3.4
EIGENVALUES OF THE FEEDBACK SYSTEM
35
3.5
COMMAND AND CONTROL BANDWIDTHS
38
4.1
ACCELEROMETER CHARACTERISTICS
55
4.2
ANGULAR RATE SENSOR CHARACTERISTICS
56
4.3
INCLINOMETER AND MAGNETOMETER CHARACTERISTICS
56
VI
LIST OF FIGURES
2.1
Local Tangent Plane Coordinate System
5
2.2
Body-Fixed Coordinate System
6
2.3
Wind
2.4
SIMULINK
Nonlinear Sixteen State Dynamic Model of an Air Vehicle
20
2.5
SIMULINK
Nonlinear Nine State Dynamic Model of an Air Vehicle
21
3.1
Standard
3.2
Feedback Configuration
3.3
Synthesis and Analysis Model
31
3.4
Control-Loop Bandwidth: Elevator Channel
35
3.5
Control-Loop Bandwidth: Throttle Channel
36
3.6
Control-Loop Bandwidth: Aileron Channel
36
3.7
Command- Loop Bandwidth: X
Position Channel
37
3.8
Command-Loop Bandwidth: Y
Position Channel
37
3.9
Command-Loop Bandwidth: Z
Position Channel
38
3.10
Bank Angle and Heading Response
3.11 Pitch
or Flight Path Reference
Frame
7
LQR feedback configuration
for
.
24
Root Locus Analysis
27
to
Ramp
in
Y Command
39
Angle and Altitude Response to
Ramp
in
Z
Command
39
3.12 Trajectory Error
Due
to a Constant
Wind Disturbance
4.1
Block diagram of the nonlinear controller
4.2
V- Implementation
4.3
Commanded
4.4
Example
4.5
Generation of Trajectory
4.6
Angular Rate Sensor
of
K
(A)
of controller on Bluebird
Trajectory Logic Block
Commanded
40
46
49
50
Trajectory Revision
52
Commands
53
55
vii
SIMULINK
4.7
Sensor Models in
4.8
Frequency Response of Position Filter
59
4.9
Frequency Response of Euler Angle Filter
60
4.10
SIMULINK Diagram
62
5.1
Test Trajectory
5.2
Trajectory #1: Position Error and Euler Angles
66
5.3
Trajectory #1: Velocity Data
67
5.4
Trajectory #1: Control Activity
68
5.5
Test Trajectory
5.6
Trajectory #2: Position Error and Euler Angles
70
5.7
Trajectory #2: Velocity Data
71
5.8
Trajectory #2: Control Activity
72
5.9
Trajectory #2: Position Error for Varying Turn Rates
73
57
of the Full Nonlinear Simulation
#1
65
#2
69
5.10 Trajectory #2: Position Error for Varying
5.11 Departure
and Arrival
at
Wind
Velocities
an Airfield
5.12 Airfield Scenario: Position Error
and Euler Angles
5.13 Airfield Scenario: Control Activity
vui
74
75
76
77
ACKNO WLED GMENT
There are a few special people
would not have been completed.
It
I
would
like
mention without
whom
this project
was a true pleasure to have studied aeronautics
under Professor Isaac Kaminer and Professor Rick Howard. Their advice and professional counsel
were invaluable.
whose vision and expertise are
avionics lab
dividuals
where
who
this
I
am
work was accomplished.
best while understanding the least.
you
I
Kaminer
directly responsible for the creation of the superb
sacrificed the most.
patience; without
especially grateful to Professor Isaac
To
Finally, a special thanks to five in-
Nicole, Katie, Eric,
And
to Patricia,
and Patty, you were the
thank you
would not have been able to complete
IX
for
your exceptional
this project.
INTRODUCTION
I.
The advent
means
of
GPS
has afforded the aerospace controls engineer a powerful new
Current guidance schemes rely in some part on
of controlling air vehicles.
ground based radars, navigational
and control of the
onboard
GPS
capabilities.
integrated
The
beams,
Guidance
etc.
have been developed seperately and then combined in
air vehicle
a somewhat adhoc process.
aides, beacons, localizer
Precise tracking of inertial fixed trajectories using an
GNC
suite affords a
quantum
leap in autonomous flight
greatest impact of the proposed technology
autonomous unmanned
of trajectory tracking control for
is
expected in the area
vehicles,
and automatic
approach and landing of manned vehicles dicussed next.
Control of the commercial air
more and more demanding
to the
as
traffic
throughout the country continues to become
an increased number of vehicles vie
for limited access
major commercial aviation hubs. Sophisticated and expensive ground based
radar control
facilities
employ a
large
number
of personnel to individually instruct
the pilots of these aircraft on the trajectories they are to
aircraft trajectory such as required
by an approach into an
attention from a ground based air traffic controller.
effectively control the aircraft
equipment available and
Airfields
is
is
fly.
Precise control of the
airfield requires
Furthermore,
ATC
constant
ability to
influenced by ground based radar coverage and navaid
often negatively influenced by atmospheric conditions.
with limited resources are often unable to take-off and land aircraft requiring
instrument departures and
arrivals.
Flight patterns around major aviation hubs are, in general, inertia! based trajectories.
That
is,
an
aircraft
is
required to track a certain path over the ground while
adhering to a certian altitude schedule, irrespective of
air
mass disturbances. In some
such as on
cases,
approach to land or when the aerodrome
final
significant terrain or cultural
is
utilizing a
GPS
integrated
among
development, precise adherence to the desired trajectory
In any case,
crucial for flight safety.
situated
is
GNC
commanding an
inertial trajectory directly
system could prove to be more cost effective and
accurate than current methods. Furthermore, such a guidance scheme could open up
many more
airports to significant commercial air traffic without requiring the capital
investment and maintenance of ground based radar
Unmanned
tionally, in
in the
of
some
air vehicles
can be a cost effective means of power projection. Addi-
human physiological
cases
performance of an
paramount importance
limits
The
air vehicle.
as
facilities.
may prove
to be the limiting factor
precision delivery of munitions
becomes
weapons and weapon delivery platforms continue to
in-
crease in cost, thus limiting their numbers. All of these concerns can be addressed
by autonomous
a
air vehicles utilizing
GPS
aided guidance, navigation and control
suite.
All of these applications have a
the particulars of the vehicles
tonomous control of
may
their trajectory.
common
thread running through them. While
vary considerably, the intent
As a proof of concept,
this
is
to acheive au-
work presents a new
design process for the synthesis of a guidance, navigation, and control system for a
UAV named
jectories.
Bluebird.
Bluebird
is
Postgraduate School.
and
air
is
a
The
function of the this
UAV operated at
It
the
GNC
system
Unmanned
is
to track inertial tra-
Air Vehicle Lab at the Naval
has a 12.5 foot wingspan and a 20 pound payload capability,
currently being equipped with a
full
avionics suite, including
IMU, GPS, and
data sensors.
The design
process began with the development of a nonlinear
Bluebird implemented in
SIMULINK. A
dynamic model of
typical cruise flight condition
was chosen as
the point for linearization. After linearization of the nonlinear model, the work cen-
tered on the design of a linear controller.
approach was used since
controller within the
sign of the
it
provides an intuitive
framework of
LQR controller,
inertial to
means
synthesis
of synthesizing a multivariable
real world design constraints.
Following the de-
the challenge of implementing the linear controller on the
nonlinear plant was addressed.
from
LQR (Linear Quadratic Regulator)
A novel method of converting commands
body reference cooridinates was used
[Ref. 10].
and outputs
This method achieves
automatic recruiting of the actuators, while preserving a certain linearization property.
Next, the accuracy of the nonlinear simulation was enhanced with the addition
of high fidelity
models of sensors used onboard Bluebird. Additionally, Kalman
filters
were designed in order to provide optimal state estimates. Finally, the performance
of the controller
was evaluated
in simulations
with the
full
nonlinear model.
DEVELOPMENT OF THE DYNAMIC
MODEL
II.
The development
of an integrated guidance, navigation,
and control system
required a high fidelity nonlinear model of the aircraft dynamics.
The
discussion
begins with an explanation of nomenclature, abbreviations, and a definition of frames
of reference.
REFERENCE FRAMES
A.
Three
different reference frames are
used in this report. They are:
• Local Tangent Plane or Inertial Reference
•
Body- Fixed Reference Frame
•
Wind
1.
or Flight Path Reference
Frame
Frame
Local Tangent Plane Reference Frame
The
position of the air vehicle
must be maintained with respect to the
tangent plane coordinate system. This coordinate system
ray from the center of the earth to
its surface.
east, the positive
up. This
is
it
will
formed by extending a
A plane is attached tangent
of intersection of the ray with the Earth's surface.
our purposes here
is
While
it is
local
somewhat
to the point
arbitrary, for
be convenient to define the positive x direction as pointing
y direction as pointing north, and the positive
depicted in Figure
z direction as pointing
2.1.
For the purposes of this development, the rotation of the earth and
its
associated Coriolis' forces can be ignored and the local tangent plane reference frame
prime
meridian
equator
Figure
2.1:
Local Tangent Plane Coordinate System
can be considered to be an inertia! reference frame.
In this work, {/}
is
used to
represent the inertia! reference frame.
2.
Body-Fixed Reference Frame
The body-fixed
reference frame
is
a right hand orthogonal system with
the origin at the center of gravity of the air vehicle.
The
positive x direction points
towards the nose. The positive y direction points out the right wing and the positive z
direction points towards the
bottom
of the air vehicle.
The
velocity of the air vehicle
with respect to the inertia! reference frame, resolved along the
body-fixed reference frame, are termed
u, v,
and
x, y,
w, respectively.
and
z axis of the
The angular
rate of
rotation of the air vehicle with respect to the inertia! reference frame, resolved in the
body-fixed reference frame, are called
forces,
are
moments, angular
shown
in Figure 2.2.
reference frame.
rates,
The
and
p, q,
and
r,
respectively. Positive values for the
linear velocities in the body-fixed reference
abrieviation,
{#},
is
frame
used to represent the body-fixed
Yb
zb
q
r
components
Aerodynamic force components
u
V
w
X
Y
Z
Aerodynamic moment components
L
M
N
Moment
"»
y
•i
'yz
>«
>«r
2.2:
of inertia
Body-Fixed Coordinate System
Flight Path or
The wind
Yaw
Axis
P
about each axis
Products of inertia
3.
Pitch
Axis
Angular rates
Velocity
Figure
Roll
Axis
Wind
Reference Frame
reference frame
is
also a right
hand orthogonal system with
origin at the center of gravity, e.g., of the air vehicle.
velocity vector of the air vehicle.
The
for
a and
(3
The x
orientation of the
respect to the body-fixed reference frame
The equations
[Ref. 11]
is
axis
is
its
aligned with the
wind reference frame with
defined in terms of the angles
a and
/?.
are given below.
a=
(w/u)
(2.1)
sin- (v/V)
(2.2)
tan
and
/3
=
l
where the vectors
Figure
2.3.
The
Figure
B.
u, v, w,
and
abrieviation,
2.3:
Wind
V
are velocity
{W},
is
components of the
air vehicle defined in
used to represent the wind reference frame.
or Flight Path Reference
Frame
[Ref. 11]
COORDINATE TRANSFORMATIONS
In order to use these three coordinate systems, one
between them
freely.
The Euler
angles,
must be able
$,0, and #, termed
to transform
pitch,
roll,
and yaw,
are defined in order to express the orientation of the body-fixed reference frame with
respect to the inertial reference frame. For the purposes of this development, a 3-2-1
Euler angle transformation will suffice as
grees.
The
3-2-1 transformation
is
its
given without explanation but a good development
of Euler angle transformations in general can be found in [Ref.
the angular rotation
is
equal to 90 de-
singularity occurs at
more apparent when the transformation
6].
is
The nature
of
expressed as the
product of three rotation matricies. In the case of a 3-2-1 rotation sequence, the three
matrices in Equation 2.3 correspond to rotations about the yaw, pitch, and
of the air vehicle.
Of
course, the three matrices can be multiplied out for
result contained in a single
matrix
,
although the resulting matrix
is
roll
axes
an analytic
somewhat busy
to inspect. In any case, the transformation between a free vector resolved in the inertial reference
is
frame and the same vector resolved
in the body-fixed reference
frame
given by:
cos
l
V=
\P
- sin $
sin
cos
$
#
The
l
V
is
inverse
the inverse
Not
1
$
— sin $
cos
1
cos
sin
1
where
— sin
cos
a free vector resolved in {/} and B V
is
is
also denned.
is
the
V
$
cos $
sin
same vector resolved
Conveniently, since the transformation
is
in
transformations, however, are orthonormal.
the case of angular rotation rates.
The body-fixed
Of
{B}.
orthonormal,
simply the transpose of the rotation matrices shown in Equation
all,
(2.3)
2.3.
particular interest
is
reference frame's angular rate of
rotation with respect to the inertia! reference frame can be related to the rate of
change of the Euler angles by a transformation matrix. The development
forward and
q,
is
fully explained in [Ref. 11].
final
sin
$
tan
cos
cos
$
tan
— sin $
$
sin$ sec0
straight
transformation matrix from
r to the time rate of change of the Euler angles, $, 0, ^,
1
By
The
is
cos$ sec0
is
p,
given by:
P
(2.4)
q
r
integrating Equation 2.4, the time history of the Euler angles can be obtained.
Aerodynamic forces and moments are often calculated using
derivatives defined with respect to the
define the orientation of the
stability
and control
wind reference frame. The angles, a and
/?,
wind reference frame to the body-fixed reference frame.
8
Therefore, a transformation matrix can be obtained that relates a free vector, such as
lift
is
or drag, resolved in
expressed
{W}
to the
v=
{B}. The transformation
a cos ft — cos a sin /? — sin a
sin
sin
is
in
as:
cos
where B V
same vector resolved
cos
f3
a cos (3
/?
— sin a sin /?
a free vector resolved in {£} and
^V
cos
is
wV
(2.5)
a
the same vector resolved in
{W}.
NOTATION
C.
Some standardized
abbreviations will simplify the development of the nonlinear
kinematic model of the air vehicle. This short-hand
where multiple frames of reference are common
•
Peg represents the position vector
is
used in the
field of robotics
[Ref. 5].
from the origin of the
local tangent plane to
the center of gravity of the air vehicle.
•
Vcg
and B a cg represent the velocity and acceleration, measured
gravity of the air vehicle, with respect to {/}, resolved in {B}.
of
• I v cg
Vcg are
commonly termed
u, v,
and
at the center of
The components
w.
and I a cg represent the velocity and acceleration, measured
at the center of
gravity of the air vehicle, with respect to {/}, resolved in {/}.
•
b ub
is
the angular velocity of the
resolved in {B}.
The components
{B} coordinate system with
of
b ub are commonly termed
• i ub represents the angular velocity of the
to {/}, resolved in {/}.
respect to {/},
p, q,
and
{B} coordinate system with
r.
respect
•
gR
{B}, in {/}. The inverse
in
•
{W},
in
B F and B
{B}. The inverse
•
•
BL
I
L
is
is
represented by g^R.
{B}.
the total external inertial force and
the inertial angular
the inertial angular
its
is
momentum of
u, its derivative
the
fixed reference frame.
1.
is
denoted as
is
denoted as Jj(u)
(v)
BODY EQUATIONS OF MOTION
A strapdown IMU, as
among
motion of the
acting on the
resolved in { /}.
with respect to {B}
In general, an avionics suite on a
Therefore,
moment
the body resolved in {B}.
momentum of the body
derivative with respect to {/}
RIGID
free vector resolved
in {/}.
Given a vector
and
D.
in
*F and *N denote
body resolved
•
represented by B R
N denote the total external inertial force and moment acting on the
body resolved
•
is
yyR represents the transformation matrix used to express a
in
•
represents the transformation matrix used to express a free vector resolved
modern
air vehicle utilizes
name implies, maintains a constant
The output
other reasons,
of the sensors
it is
on the
a strapdown IMU.
orientation in the body-
IMU
are resolved in {B}.
most convenient to develop the equations of
air vehicle in the body-fixed reference
frame.
Linear Motion
An
application of Newton's
Law
the total external force applied to a body
inertial acceleration.
is
to linear
motion of a body states that
equal to the mass of the body times
This could be written in the inertial reference frame
10
as:
its
where
=
'a
'*»,
(2.6)
or in the body-fixed reference frame as follows:
B rj?
_
= __
m
B„a
= mfRti) eg
m **<.,.
=
Coriolis'
theorem can be used to
(2.7)
relate the inert ial
and body accelerations of the
air
vehicle as follows:
\
=
-\ +
where the difference in the derivatives
is
B
Wfl
x\
(2.8)
S)
explained in Chapter
II,
part C. Equation 2.8
can be substitued into Equation 2.7 in order to obtain the desired expression for the
sum
of the external inertia! forces resolved in the body-fixed reference frame.
BF
=
m{- B v
=
m
-Jl
at
2.
cg
Bv c9
+ B uB
x B v cg )
+ rn (B U)B
x B v cg ).
(2.9)
Angular Rotation
Euler's law for the conservation of angular
gravity states that:
11
momentum
at the center of
=
'Leg
where I L cg
is
the total
is
the angular
moment
momentum of
the air vehicle with respect to {/} and
N
cg
as:
BL
cg
= fR'Nc,.
(2.11)
theorem can be used again to expand B L cg obtaining:
BL
eg
=
^
BL
cg
+ B uB
x B L cg
can be shown that the angular momentum, B L cg of the
of
an
,
inertia tensor,
all
denned
as
JB and
,
(2.12)
.
It
ignored
J
applied to the air vehicle. Equation 2.10 can be written in the
body-fixed reference frame
Coriolis'
(2.10)
'Nc,,,
air vehicle is
the product
B
the body's angular velocity, uB
,
where we
spining elements. Substituting this definition of B L cg into Equation 2.12,
results in:
BL
cg
=
B
j
(JB UB)
+ %J
x
Jb b ujb
.
(2.13)
t
Recall that B L^
= f&Ncg = B N^.
Using this relationship, Equation 2.13 can be
equivalently expressed as:
B
N
cg
=
^-(JB
Bu
B)
+ B uB
x
JB B u> B
(2.14)
at
3.
External Forces and
Moments
Equation 2.9 and Equation 2.14 from the preceding sections can be compactly expressed as follows.
12
Bp
m fBv + m
cg
t
U
UB +
JBT
dt
t
By
B
B x v cg )
B U) X
U
JB BB,U B
B
(
Bu
bringing derivative terms in Equation 2.15 to the
d_
dt
hand
side,
we
obtain,
BF
- b ujb X B v cg
+
-Jg B uB x Jb b ujb + JE^8 N
v,eg
3
left
(2.15)
(2.16)
1
ub
Next, the forces and moments in Equation 2.16 can be expanded as follows:
Bp
B
B
FgRAVITATIONAL + FpBOPULSIVE + FAERODYNAMIC
b
Npropulsive + Na
'AERODYNAMIC
N
,
(2.17)
where
force due to gravity
FgRAVITATIONAL
b Ff
PROPULSIVE
BF
force due to engine's thrust
generated
by aerodynamic surfaces
force
moment generated by engine's thrust
moment generated by aerodynamic surfaces
AERODYNAMIC
^PROPULSIVE
B
AERODYNAMIC
J
N
The
gravitational force expressed in {/}
given by
is
Fgravity
=
mg
where "g"
is
the gravitational constant. Then
B
Fgravity
= jBnl
R Fgravity-
(2.18)
The expansion of the propulsive forces and moments is
ering the case of centerline thrust. In that case, no external
i.e.,
Npropulsive
=
0,
and the propulsive
forces can
simplified
moments
be expressed
by consid-
are generated,
as:
T
B
Fprop =
13
(2.19)
where
T
represents the thrust of the powerplant.
Aerodynamic
forces
and moments are commonly calculated using nondi-
mensional stability and control derivatives. These derivatives are obtained by approximating the aerodynamic forces and moments acting on the air vehicle using a Taylor
series
expansion about a given trim point. Typically, values for these derivatives are
Sometimes, a few second
available for the first order terms of the expansion only.
order terms are available, such as the terms associated with
a and
/?
[Ref.
7].
All
other higher order terms are usually ignored in this approximation.
In general, the aerodynamic forces
are
computed
and moments acting on the
The nondimensional
as follows.
stability
air vehicle
and control derivatives are
dimensionalized by multiplication by the appropriate constants, such as wing span,
dynamic pressure, chord,
etc.
The dimensional
derivative
is
then multiplied by the
perturbations of each aerodynamic variable or control deflection from
trim point.
The summation
of the forces
variables
and control
moments,
results in the total
4.
and moments due to
all
of the
nominal
aerodynamic
deflections, in addition to the trim value of the forces
aerodynamic force and moment acting on the
and
air vehicle.
The State Space Representation
In order to
implement the dynamic model
in
a state-space form suitable for
numerical simulation, the states of the model need to be chosen. This
arbitrary
its
and many choices
will
work so long as consistency
is
is
somewhat
maintained in the
approach.
As was evident from the development of the rigid body equations
the body-fixed reference frame
define the states.
The
first
is
of motion,
the most convenient coordinate system in which to
three states are defined as the inertial velocity of the air
vehicle resolved in the body-fixed reference frame.
14
These are abrieviated as
u, v,
and
w
more compactly
or
as
Bv
The
cg .
fourth through sixth states are defined as the
angular velocity of the air vehicle with respect to {/} resolved in {B}.
abrieviated as p,
q,
and
more compactly
r or
These are
b ub
as
Control inputs are represented by the vector A:
A=
where 6C 6r and 6a are the
,
6r , 6a ]
[6 e ,
elevator, rudder,
,
(2.20)
and aileron inputs,
respectively.
Control inputs for the throttle are represented by 6tr t>
Typically, the terms in the Taylor series expansion for the aerodynamic
stability derivatives are partial derivatives with respect to
where
U
is
the magnitude of the air vehicle's velocity vector and
angles defining the orientation of
variables, p,
u/U, a,
qr,
and
r,
{B} with
respect to
are states of the model.
The
{W}
first
/?,
p, q,
a and
[Ref. 3].
The
/3
and
r,
are the
last three
three can be represented as
a combination of states in the model as follows. First note that,
U
and
for small values of angles
a and
w no,,v
_
— BWr>B
"•
cgi
/?,
a
is
(2.21)
approximately equal to
w/U and
/?
is
approximately equal to v/U.
It
turns out, the stability derivatives can be placed in matrix form as
follows:
Cl v
Cyu
Cl p
Cyp
Cl
Cya
Cl p
Cyp
CL q
Cy
Co a
Cd p
Cd„
Cyt
Cd f
Ci«
^ip
Cf«
C'r
^m a ^m p ^m, ^m p
C na Gn p C n , O nr
dC
Cdu
dx'
Ciu
Cd p
Ch
^mu
^rnp
Cnu
C np
Similarily, for the control derivatives:
15
Ci
Cl t
\Cl„
Cl,t
CL,a
Cy,.
Cvlr
Cy,.
Co,.
CD
Co ta
dC
dA~
.
c«.
Ci lr
Ci Sa
Cmg e
Crns r
Crn Sa
cnge
Cn tr
Cn fa
where
x'
forces
and moments can be expressed
B
where
is
the vector composed of u/U, a,
Faebo
NAERO
M\
derivatives
q,
= qS %R
frR
and
5
ir
/?,
.
and
p, q,
r.
Now
the aerodynamic
as follows:
^
+
S M + l>i+ lfA
'*
are matrices used to dimensionalize the stability
and convert the
>'
<
2 22 >
-
and control
state vector, x, to x':
=
q
I
=
u
v
w
T
p q
r
.
]
dynamic pressure,
5 = diag{— 5, 5, —5, 56, Sc, 56},
x'
M' = diag{l/FT
,
=
M'x,
1/Vt, 1/Vt, b/2VT c/2VT b/2VT },
,
,
and
x'
M' =
=
M'x,
diag{0, C /(2VT ),6/(2yT ), 0,0,0}.
Equation 2.16 can now be further expanded using expressions of the forces
and moments derived above.
16
B
d_
dt
Vcg
'eg
B
+
- b Jb\ b »b* b Jb b ub)
UB
BF
B
Js
B
l
(2.23)
N
where
BF
B
BF
GRAV
N -{
&R
^R
{
qS {
Notice that there
is
r
+
bf
PROP
CFO + %M'x + %M>x +
Strt+
|f A
(2.24)
.
}
}
}
a state derivative term on the right hand side of Equa-
tion 2.23 due to the second order terms in the Taylor series expansion of the aerody-
namic
forces
and moments
in
Equation 2.24.
By
bringing
it
to the left
hand
side of
Equation 2.24 and combining terms, we obtain:
d_
Ucg
dt
*UB
- b ubx
1
b
-*Jb (b ubX b Jb ub)
-'"{[I
MJ^TqS^M'
b
B
B
x
'eg
+
M-{
b Fgrav
+
+
U>B
Fprop
+ ^TqS{CFo +
Stn
&A)
(2.25)
where:
rp
—
w1 _
B
ByR
&R
and
Mi =
m
BJ
B
and
X=
/e-M,-Wgf AT.
Equation 2.25 expresses the derivative of the
in
matrix form.
It is
first six
solved in the user denned
17
(2.26)
states of the nonlinear
MATLAB
Fen
model
block, state-deriv.m,
which
is
Appendix A. The physical parameters
available for inspection in
to Bluebird are stored in a
MATLAB
within the function, state-deriv.m
.m
file,
Bluebird-data. m,
and are
specific
called
from
In order to change the vehicle being modeled,
.
one simply changes the constants in Bluebird-data.m
.
Next, the Euler angles were added as the additional three states of the
model. The time history of the Euler angles
in
compact form
is
denned
Equation 2.4 and written
in
as:
A = S(A)%,,
(2.27)
where
A=
and
sin
1
5(A)
=
$
tan
cos
$
sin$ sec0
The
user defined
MATLAB
Fen
$ tan
— sin $
cos
cos$ sec0
block, eu/.m, solves Equation 2.27, the details of
which can be found in Appendix A.
Finally, the inertia! position of Bluebird
lb
r
cg
The
rotation matrix,
J
can be computed as follows.
_/
— „v cg —-ipB„
B*l Veg-
B R,
is
(2.28)
implemented
in a
MATLAB
posb2i.m, in order to convert the vector B vcg to the vector
is
I
v cg
.
then integrated to obtain a time history of the position of the
local tangent plane.
To
four actuators, levator,
of 1/12.
The
increase fidelity of the simulation, a
Kidder,
resulting nonlinear
^aileron,
^throttle,
is
first
The
block,
vector
J
Vcg
air vehicle in the
order model of the
included with a time constant
dynamic model now has sixteen
18
Fen
states
which are
computed using Equations
states
=[u
d_
dt
w
v
p
2.25, 2.27,and 2.28;
r
=
[X
B
-
fl
Ypos
Zpos
6epo ,
6Tpot
6apot
o;
sx
+
- b Jb\ b ub* b Jb b ub)
Mj^TqS^M'
'eg
]
B
UB
BF
P ROP
Strt
Fgrav
+ Mr 1
+
{
+ ^TqS{CFo + ff A)
A = S(A) B uB
}}
(2.29)
(2.30)
,
(2.31)
eg-
The SIMULINK diagram
5.
6 tpo ,
T P T Ad*.
"l A
poa
»-{[[
U>B
for clarity:
il
$ Q V Xpos
q
'eg
B
summarized here
of the nonlinear
model
is
shown
in Figure 2.4.
Trim and Linearization
For linear controller design, the nonlinear equations above must be trimmed
and linearized
available
The
for a typical cruise flight condition for Bluebird.
A SIMULINK
tool
is
which can be used to find equilibrium points of nonlinear dynamic models.
user specifies which states and control inputs are to remain fixed along with
their stationary values
and the trim routine searches
for values of the state
vector for which the derivative of the state vector equals zero.
known, another
SIMULINK
in order to find the rate of
resulting linear
model
is
tool perturbs the states
and input
With the trim condition
around the specified trim point
change of the states and control inputs (Jacobian). The
returned in state space format. Since Equation 2.28 can only
19
w+
^
w
1
Mux
*
u
>
s
w
Elevator
V
w
Rudder
w
p
Aileron
q
Throttle
r
phi
Returns Derivative
of [u
v
wpq
-
r]
theta
psi
posx
posy
posz
Returns Derivative
of Euler
Angles
elevator
rudder
aileron
throttle
Returns Derivative
of Position in
Rotate Velocity
t{1}
in
{B}to{l}
First
Order
Actuators
Figure 2.4:
Vehicle
SIMULINK Nonlinear Sixteen State Dynamic Model of an
be trimmed for v^
and
2.27,
=
0,
not a typical flight condition,
and then include Equation 2.28
We
we
will
Air
trim Equations 2.25
for linearization.
are interested in trimming the
model
in velocity. Hopefully, the deriva-
tive of the position states will never equal zero in flight.
Also, in trim, the control
inputs are equal to the actuator positions. Therefore, actuator states are also removed
from the nonlinear model. The nine state nonlinear model of Bluebird used
is
shown
in Figure 2.5.
A
typical cruise flight condition for Bluebird
20
is
given by:
for trim
u
V
w
p
Mux
q
r
phi
theta
psi
d(Euler)/dt=S(Euler)*w
Equation 1.25
Figure 2.5:
Vehicle
SIMULINK
Nonlinear Nine State Dynamic Model of an Air
• flight speed equal to 73 feet per second
• flight
path angle equal to zero
• wings level attitude
The trim
The nonlinear model depicted
in Figure 2.4
values of the nine states,
v,
control inputs, £ e
,
6r
,
6a
,
model cannot be trimmed
The
origin of {/}
is
u,
w, p,
q,
r,
was trimmed at
$, 0, and #, and the four
and 6t were returned. While the sixteen
in position,
this condition.
state nonlinear
can be linearized at an arbitrary position.
it
conveniently chosen.
These values were then used to linearize
the complete nonlinear model, including position and actuator states.
21
The
resulting
linear
model of Bluebird and numerical values
the Appendix B.
22
for the
trim condition are included in
III.
THE LINEAR QUADRATIC REGULATOR
DESIGN
The previous chapter developed a
model for a
to derive a linear
controller
is
cruise flight condition. In this chapter, a linear
The
dynamic
developed to provide trajectory tracking for the linear model.
methodology was selected to design the
intuitive
nonlinear model of Bluebird, which was used
means
following
is
of manipulating the
controller.
LQR gains
is
LQR
Based on design requirements, an
presented. See [Ref.
a brief review of the properties of an
LQR
9] for details.
controller utilized in this
design process.
A.
LQR OVERVIEW
Consider the linear system
x
g
=\
z
y
i
where x €
Suppose
#
n
,
u €
BT
C\D\ =
= Ax + Bu
= CiX + D lU
= x
(3.1)
and z £ B?.
and D\
column rank. Then,
is full
T
z z
= x T C?ClX + u T D[D x u.
Define a cost, J, as follows:
J=
and
let
Q=
CfCi, and
Assume (C\,A)
The standard
LQR
is
J (z
T z)dt
=
R = D\D
X
.
J
{x
T
C?ClX + u T DlD
Note:
Q
observable and (A,
problem
is
>0 and
B)
is
R>
u)dt
=
(3.2)
0.
controllable.
to find a controller, u
23
x
Consider Figure
3.1.
K(s)x, such that the feed
back system in Figure
3.1
is
internally stable
such controller uses a constant gain, u
and J
= —Kx,
K=
is
minimized.
It
turns out, one
where
-R- l B J P,
(3.3)
and P solves the Alegbraic Riccatti Equation:
-i D r
1
P+Q=
A 1 P + PA - PBR^B
Figure 3.1 shows the feedback interconnection of the plant
Here the inputs
w
(3.4)
Q and
the controller K.
can include commands and disturbances.
Q
ci
w
t
Q
u
z
X
K
Figure 3.1: Standard
It
turns out that the controller,
margins of no
less
LQR feedback
configuration.
K, has guaranteed simultaneous phase and gain
than 60 degrees and 3dB, respectively
[Ref. 12].
Furthermore, the
controller has asymptotic properties which are exploited in the design process
are discussed next.
Define the Hamiltonian matrix,
H
,
as:
bt
H = \a -br-AT
x
Q
Let
T
be given by:
24
and
/
T=
p
I
then:
I
r -i =
where
P
-p
I
solves the Riccatti equation, Equation 3.4.
Note,
A-BR~ B T P
l
T -i HT _
=
1
l
Therefore the eigenvalues of
det{sI-H)
-BET B T P
-AT + PBR~ B T
H are the roots of the following polynomials:
det{sI-T- x HT)
=
det(sI-A-\-BR- 1 B T P)det((sI+A T -PBR- 1 B T )
Clearly, the eigenvalues of the Hamiltonian consist of the eigenvalues of
R=
unstable reflections about the imaginary axis. Let
det(sl
It
-H) =
can be shown that the det(sl
det{sI-H)
— H)
si
-A
(l/p)R^
sl
pR\, R\
1
>
0,
Q
d and their
then;
BT
+A
can be equivalently expresses as [Ref.
9]:
= -l n det{sI-A)det{-sI-A)det{I+{l/p)R; B T {-sI-A T )- C{C
1
1
1
1
(sI-A)- B)
Let
0(a)
=
det(sl
-
A)
and
0{s)
= C (aI-A)x
l
B.
Then
det(sl
-H) =
-r<f>(s)<f>(-s)det(I
25
+
l
{l/p)R^ 6{-s)6{s)))
(3.5)
The SISO example
as p
is
varied from
demonstrate what occurs to the eigenvalues of Q d
will best
to oo. Let
e(s)
where
tp(s) are
*M/*w
the zeros of 0(s). Then,
det(sl
It
=
-H) =
-l»tf *)*(-«)(/
follows that the eigenvalues of
+ (l/pW*)1>(-')/<K*)<K-*))'
H are the roots of,
<f>(s)<f>(-s)
+ (l/pMsM-s).
Consider the feedback system shown in Figure
SISO root
and
locus analysis
its
same
as
Equation
are the eigenvalues of
as p goes to
•
Q
c/,
3.6.
standard configuration for
is
is:
+ (l/p)1>(s)1>(-3),
Since
we know that the
stable eigenvalues of
H
standard root locus techniques show that
[Ref. 9]:
p eigenvalues of Q
go to the stable zeros of C\{sl
<j
reflection of the unstable zeros of
zeros of
This
characteristic equation
<p(s)<f>(s)
exactly the
3.2.
(3.6)
C
x
{sl
- A)~
• n-p eigenvalues of
l
Qd
C\{sl
— A)~ X B,
— A)~ X B
where p
is
or the stable
the
number
of
B.
go to -oo in Butterworth patterns.
as p goes to oo
•
n eigenvalues of
Q
d go to the stable eigenvalues of
of the unstable eigenvalues of A.
26
A
and the stable
reflections
X
V
*
1/p
>
^(-a)V'(s)
Figure 3.2: Feedback Configuration for Root Locus Analysis.
DESIGN REQUIREMENTS
B.
These properties of an
to
LQR controller were utilized in
a design process in order
meet the following design requirements.
1.
Zero Steady State Error
•
Achieve zero steady state values
commands in
command
for all error variables in response to
position along the x,
y,
and
z inertial axes.
ramp
Note that a ramp
in postion corresponds to a constant heading, constant velocity
trajectory.
2.
Bandwidth Requirements
•
The input-output command response bandwidth (command-loop bandwidth) along any of the three command channels should be no greater
than
•
The
1
radian per second and no
less
than 1/10 radian per second.
control-loop bandwidth should not exceed 12 radians per second for
the elevator and aileron actuators, and 5 radians per second for the throttle
actuator.
These numbers represent 80
27
%
of the corresponding actuator
bandwidths and
shall ensure the actuators are not driven
beyond
their
linear operating range.
3.
Closed Loop Damping
•
The dominant
closed loop eigenvalues should have a
damping
ratio of at
least 0.7.
C.
THE SYNTHESIS MODEL
The
and the
model
synthesis
LQR
is
the primary interface between the control design
algorithm. At the heart of the synthesis model
of Bluebird developed in Chapter
is
a linear model
II.
Bluebird has four control inputs, namely elevator, rudder, ailerons, and
throttle.
The
elevator
and throttle are natural choices
z position in steady state.
The remaining two
to control the lateral variable (y position).
means of generating
is
control inputs could be used
Both rudder and
aileron provide
accelerations in the lateral plane. In fact, rudder
effective at generating sideslip
x and
for controlling
is
than aileron. In the linear plant, lateral position
the double integral of lateral acceleration. Subsequently, the resulting
controller will
attempt to use rudder to null out errors
using ailerons and to use rudder for turn coordination.
it is
LQR
in lateral position,
to turn the plane. However, the desired controller response
presence of wind,
more
is
to
bank
i.e.,
to turn
Furthermore, in the
desired that Bluebird fly wings level, crabbed into the
wind, rather than use a wing down, top rudder technique. For these reasons,
the rudder was removed as a control input to the linear model.
As can be seen from Table
damped. This
light
damping
3.1,
the dutch
roll
mode
of Bluebird
is
lightly
of 0.111 could pose a performance problem. Since
28
rudder
is
available
and not used
in the design of the trajectory controller, the
nonlinear model of Bluebird was modified to include a
dutch
roll
Yaw
damping.
rate
yaw damper
for
improved
was fed back to the rudder through a constant
gain block with a value of 0.55.
Additionally, the rudder was
external input. Note that Bluebird
is still
fully controllable
removed
as an
with the remaining
three control inputs.
TABLE
3.1:
EIGENVALUES OF BLUEBIRD
Mode
Frequency
Longitudinal
rad/sec
Damping
Short Period
5.9
0.735
Phugoid
0.497
0.0344
Dutch Roll
2.4
0.111
Spiral
0.0384
-1
Roll Response
-4.572
1
Lateral-Directional
The nonlinear model of Bluebird with three inputs and
was
linearized, as per
Chapter
s
II,
integral
yaw damper
returning the linear model,
= Ax + Bu
=
-{i
(3.7)
where
x
=
uvwpqrQQV
iT
Xpos
Zpos
Ypos
6epot
6apot
6tpot
and
W
=
\T
[
Sdtvaior
This linear model was used in the
^aileron
LQR
^throttle
design.
with a yaw damper are given in Table 3.2 where
29
The
it
eigenvalues of Bluebird
can be seen that the dutch
roll
mode
has been
fifteen since
TABLE
3.2:
damped
Note that the number of states decreases to
out.
the rudder actuator state was removed.
EIGENVALUES OF BLUEBIRD WITH YAW DAMPER
Mode
Frequency
Longitudinal
rad/sec
Short Period
5.9
0.735
Phugoid
0.497
0.0344
Dutch Roll
2.35
0.5
Spiral
0.1788
1
Roll Response
-4.5686
1
Damping
Lateral-Directional
Consider Figure
linear
3.3.
Here
K
is
the controller to be designed,
model of Bluebird, and the block, 5, within the dotted
line
is
G
is
the
the synthesis
model.
The
signal, iu, represents the
w=
The
[
XpOScmd
commanded
trajectory inputs:
ZpOS^d
YpOScmd
signal xi represents the linear
Qcmd
Q^nd
and angular position
<P cmd
j
states in the linear
model.
=
Xi
The
tory.
Xpos
signal e represents the errors
The
signal z
is
Zpos
#
$
iT
between the commanded and current trajec-
comprised of the outputs of the matrices, Q, and R. Since
zero steady state error
position,
Ypos
is
desired while tracking a
ramp command
two integrators were placed on each error
30
signal.
in inertia!
This also ensures
*
JL
X
Figure
3.3: Synthesis
and Analysis Model
perfect tracking of constant heading trajectories in the presence of a constant
wind disturbance. Thus
Q
was chosen to be;
'
9i
Q=
t
93
.
1
a
£31
£32
1
»2
*
*
values of CxX were chosen to place six transmission zeros from u to z at
appropriate locations.
will
£12
£21
q2
.
The
£11.
i
'
move very near
If
they are well chosen, six poles in the closed loop plant
to the placed transmisssion zeros.
With
that in mind, the
transmission zeros were chosen as appropriate target locations for the poles
added by the addition of the error
The
states.
q xx weightings are used as a
command bandwidth.
mechanism
for obtaining the desired
Increasing the value of qxx increases the relative propor-
tion of that error state in the regulated output vector z.
gain increases the
command bandwidth
controlled state to
its
commanded
value
31
The
resultant
in that channel in order to
more
quickly.
LQR
move the
The matrix R
the plant
Q
is
a constant diagonal matrix required to be
has three control inputs,
R is
full
rank. Since
of the form,
/ rn
R=\
r 22
T33
\
The elements of R
are used as a
mechanism for
selecting the control bandwidth.
An increase in rxx increases the relative proportion of that
regulated output
z.
The
resultant
LQR
actuator energy in the
gain decreases the control bandwidth
of that control input.
THE DESIGN PROCESS
D.
Design requirements given in the previous section are SISO in nature.
They
are expressed as bandwidth limitations of the individual actuators and
rise
time and damping characteristics along the command channels. Note, the
rise
time
LQR
is
inversely proportional to the
command bandwidth. The
following
design process provided a means of obtaining a multivariable solution to
achieve
SISO design
specifications.
With an appropriate
model that incorporated
linear representation of Bluebird
and a synthesis
well placed transmission zeros, the design "knobs"
were adjusted in order to meet performance requirements. The design "knobs"
are the elemental weightings, q xx and r xx , in the
Q
and
R matrices.
The
design
R
to sat-
process iterated through the following steps.
1.
Initially let qxx equal
isfy control
1.
Iteratively
determine weights for
loop bandwidth requirements. Increasing r xx decreases the control
bandwidth along that channel.
2.
With
R
from step
1,
iteratively
32
determine weights for qxx to satisfy
command
loop bandwidth requirements. Increasing qxx increases the
bandwidth along that channel and decreases the
3.
If it is
command
rise time.
required to increase the damping of a lightly
damped mode,
use
an eigenvector decomposition to determine the primary states affecting that
mode. Include a weighting on the derivative of those
Ensure that control-loop bandwidths are
4.
values of qxx in step
2.
It is
Connect the
formance
in
terms of
LQR
still
satisfactory with the
possible that all of the performance requirements
are not acheivable within control
5.
states in the output z.
bandwidth
limitations.
controller to the linear plant
command
and evaluate the per-
response and disturbance rejection.
6.
Confirm satisfaction of other design requirements, including damping.
7.
If
any step
is
appropriate changes.
deleted. Synthesis
Bode
unsatisfactory, go back to the synthesis
may need
Transmission zeros
model and make
to be added, moved, or
model outputs may need to be reevaluated.
plots were used to determine compliance with the requirements.
After five iterations through the seven step process, the following values for
and
R
Q
matrices resulted in a controller design that met design requirements.
0.5
Q=
i
02
0.01
i
(M
0.0625
0*4
0.0625
a
a2
1
1
1
1
5000
R=
1000
1000
The transmission
zeros created in the sythesis
ble 3.3.
33
model are shown
in Ta-
TABLE
E.
3.3:
TRANSMISSION ZEROS OF SYNTHESIS MODEL
Channel
Cxi
Cx2
Cx3
Freqency (rad/sec)
Damping
Xpoa
1
0.2
0.01
0.1
1
Y
1 poa
1
0.4
0.0625
0.25
0.8
Zpoa
1
0.4
0.0625
0.25
0.8
LQR CONTROLLER PERFORMANCE
The eigenvalues of the feedback interconnection of the plant and
axe given in Table 3.4.
It is
controller
apparent that the zeros created in the synthesis
model were well placed and attracted the integrators created by the addition
of the error states.
There are two
sets of lightly
present a problem because their frequency
is
damped
poles.
These do not
an order of magnitude greater
than the frequency of the eigenvalues associated with the trajectory commands.
Notice that the actuator poles did not change, indicating that the control band-
widths were slower than the actuator bandwidths. Actuator models provide a
simple means of determining
if
the control bandwidths exceeded the actuator
bandwidths.
Figures 3.4 through 3.6 depict the control-loop bandwidths for the elevator, aileron,
and
flight controls
throttle.
was so
The
cross coupling
slight that
it is
between longitudinal and
lateral
not shown due to scale.
Figures 3.7 through 3.9 depict the command-loop bandwidth for step com-
mands
in inertia! position. Notice that there
mand and Z
some coupling between
X
com-
response; the rest are essentially uncoupled.
A summary
is
is
of the resulting
presented in Table
command and
3.5.
34
control bandwidths achieved
TABLE
EIGENVALUES OF THE FEEDBACK SYSTEM
3.4:
Mode
Frequency (rad/sec)
Damping
Axis Response
0.08
0.92
Axis Response
0.21
0.77
Z Axis Response
0.21
0.77
elevator
12.1
1.0
aileron
12.3
1.0
throttle
12.4
1.0
others
0.62
0.79
1.90
0.35
2.17
1.0
2.35
0.18
2.26
0.68
4.54
1.0
5.86
0.74
X
Y
Frequency (rad/sec)
u
:
'':
S>-180
:
:
!
i
—i__^
TTT
...,;...
7
:->;
T
'
:••
\
™ -360
L
'.
i.
-540
.
i
'
'
10
:
i
i
i
i
j
;
1
:
I
I
<
i
I
1
1
i
10
10
10
Frequency (rad/sec)
10
Figure 3.4: Control-Loop Bandwidth: Elevator Channel
The response
is
the reponse to a
in the
to two types of trajectory
ramp command in
commands
Y position.
is
of interest.
The
first
This corresponds to a change
heading of the commanded trajectory. The response in terms of angle of
bank and heading
activity
is
shown
in Figure 3.10.
35
The
controller achieves the
—
throttle
Frequency (rad/sec)
:
;
:
i
:
:
i
-.J..
:
i
o
deg
CO
o
0>
o
r>
J-i
:• •
:
1
•
•
'
i
i-
Phase
i
10
10
Figure
3.5:
i
10
Frequency (rad/sec)
rm-i
10
.
— —^_
:
..
,
10
Control-Loop Bandwidth: Throttle Channel
aileron
co
O
-50
-
100-
Frequency (rad/sec)
&-180
-360
-
10'
1
10
10
Frequency (rad/sec)
10
10
Figure 3.6: Control-Loop Bandwidth: Aileron Channel
desired result of turning Bluebird to the required heading.
phase response of the heading state
The response
to a
sponse to a change in
is
due to adverse yaw.
ramp command
flight
The nonminimum
in
Z position corresponds to the
re-
path angle of the commanded trajectory. Figure 3.11
36
X
channel
100
-200
10
Frequency (rad/sec)
~Tt-i~4—-^
-t^^JTTj^^^
0g>-180
35
5
^^^^^
-360
.::
£ -540
-720
i
10
i
i
i
i
I
10
Figure 3.7:
i
:
1
1
1
;
1
1
I
;
1
1
1
1
1
10
Frequency (rad/sec)
:
:
1
L.LU.J.U1
!~
:
f
.
10
10
Command-Loop Bandwidth:
Y
Li|i[|
X
Position Channel
channel
m
100
to
o
-200-
10
Frequency (rad/sec)
10
10
10
Frequency (rad/sec)
10
10
1* -360
95
CO
£
-720
-1080-
10
-2
10
Figure 3.8:
Command-Loop Bandwidth:
command
shows the response to
this
Note that the positive
z direction
is
in
The wind
orientation
is
Position Channel
terms of pitch angle,
6,
and altitude
2.
down, hence the negative pitch angle.
Finally, the response to a constant
ure 3.12.
Y
such that
37
wind disturbance
it
is
shown
affects all three axes of
in Fig-
Bluebird
;
Z channel
.ii
.1
;
;Y
mil.
channel
P—BEgr^^T^
1
-:i
j
CD
100
-
O
X channel
j
-200
~
f
"i
10
"r
....
*
*
l
'
i'r-i"r*
-2
^
"!-;
;
,
T"ji
r
1
.
Til"!
i
i__
=-t
i
:::::!
V
T
I
_I_i
4^"
10
10
10
Frequency (rad/sec)
10
!
.,
".'_"
!
!!!!!!
|
!
'
:
1
~
CT>
S
-360
CO
£
-720
f
-1080
'
:
TABLE
3.5:
:
1
1
:
1
:
1
1
i
1
1
Command-Loop Bandwidth: Z
i
i
1
10
T~
:
1
1
10
Position Channel
COMMAND AND CONTROL BANDWIDTHS
Loop
Control Loop
Break Frequency
Elevator
4.0
Aileron
5.0
Throttle
1.0
rad/sec
Command Loop
X
Y
equally.
1
10
Frequency (rad/sec)
10
Figure 3.9:
'*
'
in
"
10
The wind
Axis
0.75
Axis
1.0
Z Axis
1.0
disturbance begins at
38
t
=
0.
0.05
0.04
1
i
i
-
w
0.03- f\
=
0.02
I
0.01 j
Heading
k_
y
a
Bank Angle
I
/\
-0.01
I
1
10
(sees)
Bank Angle and Heading Response
Figure 3.10:
—
25
30
25
20
15
Time
Ramp
to
in
Y Command
!
20
Altitude
15
10
-
5Pitch
Ang le
-
degrees
•
1
10
Time
_
_
1
Figure 3.11: Pitch Angle and Altitude Response to
39
20
15
(sees)
Ramp
in
Z
Command
40
Figure 3.12: Trajectory Error
60
Time
(sees)
Due
to a Constant
40
100
Wind Disturbance
CONTROLLER IMPLEMENTATION
ON THE NONLINEAR PLANT
IV.
In
Chapter
III,
of an air vehicle.
a linear controller was designed to control the trajectory
However,
the nonlinear plant.
It
this controller
would only be
cannot be implemented, as
effective for
commanded
is,
on
trajectories that
represent relatively small perturbations from the specific trajectory for which
it
was designed.
Intuitively,
it
depend on the heading angle
gravity
$
or 0.
is
ty.
clear that the
dynamics
of the plant
do not
Furthermore, the orientation of the force of
the only change in the dynamics of the air vehicle due to changes in
It
turns out that these issues were addressed in [Ref.
methodology
method
is
for
implementing controllers on nonlinear plants
involves differentiating
some
10],
is
where a new
proposed.
The
of the inputs to the controller, hence the
term, P-implementation.
This chapter begins with a general description of the structure of V-
implementation. Furthermore, the specifics of
linear
model
simulation
is
in
SIMULINK
are discussed.
its
implementation on the non-
Next, the fidelity of the nonlinear
improved by incorporating output feedback. This step involves
inclusion of high fidelty sensor models
and Kalman
filters.
Finally, all of the
pieces of the complete nonlinear simulation are brought together in
A.
SIMULINK.
D-IMPLEMENTATION
Using the development
in ...Chapter II, the vehicle
pressed in state space form as follows:
41
dynamics can be
ex-
B
j
v cg
= fv
(
Bv
»u>B ,u)+fR(A)g
cg
t
B
B
j u>B = U{ v
- P^ ='B R(A) B v
B
UB,u.)
cg ,
t
eg
(4.1)
where /„ and fw are continuously
of control inputs.
Xti
.
—
To condense the
where xi €
6
i? ,
1
'eg
:=
TK)9
UG7J6
denotes the vector
define
eg
/.(•)
:=
/.(•)
/-(•)
fl(.)
L(.) :=
S(.)
x 2 £ B? and L 6
as the vector of linear
With
we
A
6*6
.ft
r :=
track.
notation,
B
/»(•)
and
differentiable
.
Furthermore, we define
Pc
Ac
€#,
(4.2)
and angular position commands that Bluebird must
this notation, the
dynamics of the augmented plant can be written
as follows:
*•
Xp
Q :=<
ei
e2
j/i
I
where
y\
V2
= h{x v ,u) + f2 (x p
= Li\Xp)X v
= [/ 0](y 2 - r)
= [0 /](j/2 - r)
= /i(x v ,u)
= Xv
and y 2 are the available measurements,
errors separated into linear
t\
)
and
.
(
l
e 2 are the trajectory
and angular components. Notice that L
function of the orientation vector
A =
have not included any extra dynamics
42
[0
I]x p
.
^
'*'
is
only a
For simplicity of exposition,
for the actuators or sensors.
we
The
£
set
Q
of trajectories where the plant
is
expected to operate
given
is
by
vo
^VQ
S
{(x^Xp^ucro)
:=
r
'PO
/l(**0»^Po> u o)
where T corresponds to the
This
Bluebird.
+ /2(*po) =
set of prescribed linear
a broad definition of trimmed
is
€r},
,r
:
°
-
and angular positions of
Notice that
flight.
it
does
not preclude the presence of an inertia! acceleration due to centripetal force.
As
usual,
we
restrict the angular positions to
(— 7r/2,7r/2) x [— 7r/2,x/2],
as the inverse of L{.)
Notice, from the definition of
B
x^ G £ and
Notice that the set £
,
Xpo ,
u
,
r
)
€
£,
Xp,,,
u
,
yi
,
follows that:
by
Xp„
=
tq
6
T.
Given
M z n» uo)
yio
:=
^2o
:= Spo
6u,6yi,
xp
turbations of x„,
x^,
,
it
ir/2.
we obtain
e 2o
6x p
=
not denned at
x
constant
easily parameterized
is
of [— 7r/2,7r/2]
constant.
e l0 := [/ 0](y 2o
Let 6x„,
is
equation (4.1)
B
wb £ £ —* wb =
AG£— A=
(im
some subset
,
y 2o
u,
,
yi,
r
,
:=
6y 2
[0
,
y2,
e lo ,
/](y2o
-
r
)
:=
"
r
)
:=
0..
£r,
6ei
and 6e 2 correspond to small per-
r,
ci,
and
and
earized models associated with the rigid
43
e 2o
e2
about the nominal values
respectively.
body Q and the
The family
set
£
is
of lin-
defined as
Qi := {Gi (r
r
),
<E
T}, where
6x v
6x p
6yi
£/ (fo)
:=
£y 2
{
fei
*e 2
k
<*o
= Ai^x,, + A 2 6x p + B^u
= Z6x w + A4^x p
= C\8x v + i^i^u
= Sx p
= [/ 0](<fy2 - <5r)
= [0 TPife-Jr)
= ^(xvojrcuo),
(4.4)
and
g xf RiAotf-^Ao)
A2 =
-' R(Aq)Vq x S~ l {Ao)
b
A,=
(4.5)
Matrices
A 2 and A4 were derived using the identity in Appendix C. The intent of
this derivation is to isolate the plant
that
dynamics that are a function of Ao. Notice
A2 represents the contribution
of the force of gravity to the
dynamics of
Bluebird and A* represents the sensitivity of Bluebird's trajectory to changes
in the air vehicle's spatial orientation.
Let
r*o
€ S be
given. Define
6x v
6x p
6yi
o)=<
6y 2
6e\
6e 2
a
b
=
=
=
=
=
=
=
A\Sx v
6x v
+
C\6x v
+ A 2 6x p + B6u
A^Sxp
+ D\6u
6x p
(4.6)
[I
0](6y2
[0
I](Sy 2
-6r)
-6r)
u(a\*,r ,uo),
where
A2 =
g)<ffl(A )"
,
A4
=
-v x
(4.7)
Notice that Equation 4.6 decribes the linear model of Bluebird used for
the design of the
as the origin of
LQR
controller in the previous chapter,
where
ro
was chosen
{/} with Ao equal to zero. Note, at this condition, f R(0)
Recall, the structure of the controller developed in
44
Chapter
III is
=
given by:
/.
= Bc2 Sx c2
= [Se, 0] T
= Cc i6x cl + Cc2 6x c2 + Dcl 6y! + [Da D c2 )\6e
6x c i
fCi
=
\
.
6x c2
6u
Based on K\
nonlinear plant
Q
,
we propose
x
T
.
]
implement the following controller
to
for the
:
f
x el
ic2
fC
(4.8)
6e 2
(A) :=
= Bc2 L" (A)[e 1 0] T
= Ccix* + CaL-\A)[ei
1
+De2 L-\\)[0
i
A
u
=
=
+ D e iyi
(4.9)
[0/]y 2
+ ^oaX-^A)^!
Xea
b
Figure 4.1 shows the block diagram of
gain scheduling variable.
e2 ]
T
0]
T
fC (A).
0]
T
.
Notice that
As a function of A, L~ l serves
A
serves as a
to properly resolve
the trajectory error at the input to the controller. Note, the controller forms
the derivative of the measured outputs,
y\.
Recall, y\
is
the measurement of
the states x v and the dynamics of x v are essentially independent of the air
vehicle's spatial orientation or position in {/}.
An
integrator at the output of
the controller serves to recover the properties of the linear design.
formed using the outputs y 2 and the commanded trajectory
measurement of the
in {/}, in
It
states x p
L~ l
is
is
the
formed
turns out that the implementation of Figure 4.1 has an important prop-
we need
to
make
the following assumptions:
Dim(x c2 ) = dim(u) = dim(y 2 ).
— Ad Bc2
— Cd
CC2
si
full
Recall, y 2
serves to resolve this error, originally
A2. The matrix
has
error
{B}.
erty discussed next. First
Al.
.
r.
The
rank at
s
=
0.
45
yi
5
x c2^
n
St7
—
1
tn
\J
2/2
e
K,
s
u
i
L' 1
V
1
D*L- 1
B,c2
K =
Cd
cl
D
c,
<->c2
cl
[Da Da]
Figure 4.1: Block diagram of the nonlinear controller
A3. The matrix pair (A c i,Cc i)
Also denote by T(Qi
results
from connecting
Similarly,
T(Q
be
,
its
K,
)
we let Ti(Q
,
Qi
)
:
to Ki
JC )(ro)
6r
,
Then the
• the feedback systems
—*
£y 2 the closed loop Unear system that
and by T(Qi
K\
)(s) its transfer matrix.
denote the linearization of the closed loop system
at the equilibrium point
transfer matrix.
(A)
observable.
is
Ki
K.
%{Q
,
determined by
and
ro
let
Ti(Q
,
K
)(ro)(s)
following hold.
K
)(r
)
and T(Qi
,
K,\
)
have the same closed
loop eigenvalues;
Ti(G ,fC )(r )(s)
=
L(A )T(g lo
A =
[0
,
K
{
){a)L-\ho),
/jxpo
Thus, the eigenvalues of the linearizations at each operating point axe
preserved; furthermore, the input-output behavior of the linearized operators
46
is
preserved in a well-defined sense.
The
reader
complete discussion on approximations to
differentiation.
The proof
of this result
is
this
is
10] for a
referred to [Ref.
method that avoid using pure
contained in Appendix C.
P-EVIPLEMENTATION OF THE CONTROLLER IN SEVIULINK
B.
In general, three steps are required to implement the linear controller
developed Chapter
on the nonlinear plant.
III
form the derivative of x v as per Equation
tor
[
B Vcg, B UB]. The
first
not compute the derivative of B Vcg since
b u>b
is
Recall, x v
is
three elements of the vector, B v cg
processed acceleration outputs from the
derivative of
4.9.
First, the controller
computed by the
IMU.
it
will
equal to the vec,
are available as
Therefore, the controller need
have
it
available directly.
tion.
The
is
vector e\
The
controller.
Secondly, the controller needs to act on the vectors, ei and
position error e\
needs to
formed as the difference
in
commanded and
e2.
The
linear
current posi-
B
then multiplied by the transformation matrix R. This
is
effectively resolves the linear trajectory error in the body-fixed reference frame.
Along
angles.
this position trajectory, there exists
Recall that x v
jectories.
=
constant
is
a corresponding trajectory of Euler
one of the constraints on the
This includes a broad range of
flight conditions
set of tra-
such as steady turns,
steady pull-ups, climbing or descending turns, or constant heading.
is
natural to define the linear trajectory as a sequence of positions,
While
it is
it
more
convenient to define the derivatives of the Euler angles rather than the values of
the angles directly. Consider, for example, that for
in £, the
components of A can be described by: 6
turn rate.
many trajectories
=
0;
=
0;
and
of interest
$=
desired
Furthermore, the relationship between the rates of change of the
Euler angles and b ub
is
used as a means of resolving the angular position error
47
in
{B}. Recall,
b
= S-1 (A)A.
u>b
(4.10)
Therefore, the derivative of the Euler angle states
Euler angle rate
is
removed to form
e'2.
is
formed and the commanded
This error
is
resolved in
{B}
Equation 4.10. The integrator at the end of the controller recovers the
using
effect of
the initial differentiation.
Thirdly, the required error states are formed by integrating the rotated
Figure 4.1 indicates that integral action
linear trajectory error vector t\.
is
accomplished at the output of the controller in order to recover the original
properties of the linear design. This accounts for one of the integration steps
on the
error.
Therefore, only one additional integrator
double integration action on the trajectory error
B for
a complete description. The
required to provide
t\.
Figure 4.2 shows the P-implemented controller in
See Appendix
is
SIMULINK, file plantl.m.
LQR gain has been parsed into
several separate matrices for clarity of control action.
C.
GENERATION OF THE TRAJECTORY COMMANDS
The commanded
trajectory
is
specified with respect to the inertial refer-
ence frame. At this point,
it is
performance capabilities
known and
is
those capabilities provided there
restrictions
with respect to the
assumed that a knowledge of the
is
that the specified trajectory
no wind. The
air
air vehicle's
is
within
air vehicle has certain airspeed
mass that cannot be violated regardless of
the desired trajectory to be tracked. These restrictions typically provide lower
and upper bounds on the velocity
of the air vehicle with respect to the airmass.
48
Yaw Damper
Rudder
Gain
Proportional
Accelerations
Gain on Stales
Angular Rates
du/dt
Actuators
Positon
Gain on
Euler Angles
Euler Angle Error
du/dt
Elevator
Aileron
1st Integral Gain
on
Form
Commanded
Rotate
Position Error
Error
{1}
2nd
\m
to {B}
Trajectory
++
Throttle
Integral
Gain
Proportional
Gain on
Postion Error
Figure 4.2: D-Implementation of controller on Bluebird
An example of a
lower limit
is
the
stall
speed of the
air vehicle.
Such limits are
usually based on fundamental physical limitations of the airframe and a "flyable" trajectory can
become "unflyable" under
a logic block positioned between the
ensures that the
certain conditions.
commanded
commanded trajectory can be
Therefore,
trajectory and the controller
flown at current flight conditions
within user defined indicated airspeed limits, shown in Figure 4.3.
manded
linear trajectory enters the block as a time
inertial reference frame.
the
IMU
and
air
mass
Onboard
velocities
stamped position
The comfix in
the
sensors provide both inertia! velocities from
from the
49
pitot-static system.
Furthermore, a
dual vane device instrumented on Bluebird provides both
a and
{3
measure-
ments. Note: a close approximation to these readings could be obtained from
IMU
measurements
Portion
as outlined in
-J
diVdt
Chapter
II.
I
Schedule
Figure
With
4.3:
Commanded
Trajectory Logic Block
these measurements, the wind vector resolved in {/}
*W = 7n RB Vca
eg
is
calculated as:
w v,cg
-' B£,R
B "W" v
(4.11)
where
w Vcg =
and
V
t
is
the indicated airspeed obtained from the pitot-static system.
The commanded
indicated airspeed of the air vehicle, Vt-cmd,
50
is
calculated as:
where / u cm(f
is
the numeric derivative of the
commanded
indicated airspeed
is
trajectory
altered as follows.
The
is
9
commanded linear
trajectory. If the
not within specified limits, the
and
angles
= tan-\v x lvx
tp
commanded
are calculated as follows:
(4.12)
)
and
iJ>
= sin-
where the components of I v an d are
define the
commanded
Finally, the
1
I
(vy /\ v cmd \)
[v x ,v v ,v z ]
T
.
(4.13)
Note that the angles 6 and
velocity vector's orientation in {/}.
amount that Vt-cmd
is
outside of indicated airspeed limits
subtracted from the magnitude of I v cm i, resulting in the magnitude of the
commanded
inertia! velocity,
Vmod
where
I
v mo d
is
=
the
termed V.
file
V
is
—cos(0)sin(xl>)
atn(V')
cos{\j))
sin(6)cos(ip)
—sin(6)sin(xj>)
new commanded
that implements this logic
is
to:
—sin(0)
V
(4.14)
cos(0)
inertia! velocity.
commanded
is
new
then resolved in {/} according
cos(9)cos(tJ>)
grated and sent to the controller as the
.m
ij>
This
command
trajectory.
The
is
inte-
MATLAB
windlogic.m and can be found in Appendix
A.
The
net effect of the trajectory logic block
up against one of its airspeed
followed.
A
but change
choice
its
is
limits, the
simple.
When
Bluebird runs
commanded trajectory can no
made to maintain the
magnitude. Notice that
is
this
direction of the
longer be
commanded
method does not
affect the
velocity
turn rate
associated with the trajectory and subsequently, no processing of the angular
51
trajectory
fly
commands
is
required. In this way, Bluebird
a trajectory that would force
The performance
is initially
flying
never
commanded
to
to exceed the performance limits.
of the trajectory logic block can be seen in Figure 4.4.
The lower limit for Bluebird was
bird
it
is
arbitrarily choosen as 63 feet per second. Blue-
due north at a ground speed of 73
into 20 feet per second of
feet per second,
crabbed
wind from the west. The commanded trajectory turns
90 degrees to the east. Notice that the original trajectory would result in an
indicated airspeed of 53 fps while the revised trajectory results in a
commanded
trajectory of 63 fps.
85
100
120
-
60
seconds
80
Time
100
120
-
60
seconds
80
Time
40
20
100
t
50-
O-50
20
40
Figure 4.4: Example of
If
the
commanded
Commanded
trajectory
is
Trajectory Revision
generated using a velocity rather than
position schedule, the differentiation block in Figure 4.4 can be removed.
velocity schedule
is
specified in {/} as a sine
frequency, and phase along the x and y axis.
corresponds to the magnitude and the
wave
of appropriate magnitude,
The commanded ground speed
commanded
turn rate corresponds to
the frequency of the sine function. Constant heading flight
52
A
is
a subset of these
trajectories
where the frequency
is
zero. This
commands makes determining turning
method
of generating trajectory
As an example, con-
rate (hand) simple.
sider the case of generating the velocity schedule for a circular flight pattern
at
a ground speed of 100 fps and a turn rate of 0.1 radians per second.
derivative of the
r :=
<
commanded
by time
trajectory, r, parameterized
x axis velocity
100sin(0.1<
y axis velocity
z axis velocity
100sm(0.1<
+
+
The
is:
pi/2)
0)
desired climb or descent rate
(4.15)
6
e
0.1
The SIMULINK block
that generates the
commanded
trajectory
is
in Figure 4.5.
Velocity
Angular
Schedule
Linear
Processed
'
Trajectory
Commands
indices
A, re peed
Limit
inertia.
Velocity
|
Logic
1
I
1
I
s
00 1
63
J
lower
limit
1
u
S3 1
upper
limit
Figure
4.5:
Generation of Trajectory
53
Commands
shown
.
STATE FEEDBACK TO OUTPUT FEEDBACK
D.
Up
until this point, the
development of the tracking controller has dealt
exclusively with full state feedback. This section will detail a
the
full state
feedback
in conjunction
is
method whereby
replaced by high fidelity models of the onboard sensors
with Kalman
filters
designed to provide optimal state estimates,
using onboard sensor data.
Sensor Modeling
1.
This work builds upon sensor models developed in two prior theses.
In [Ref.
1],
a detailed model of the inertial measurement unit,
oped. In [Ref.
2],
a detailed model of the
GPS
unit
is
IMU,
developed.
is
devel-
The IMU
is
a compact, lightweight, low power unit which integrates nine sensor measure-
ments
in
one box. These sensors are three axis accelerometers, three axis rate
gyros, pitch
and
roll
inclinometers, and a magnetometer.
The accelerometers
are instrument grade, signal conditioned, and temperature compenstated. Full
scale output
is
+/
—
3
g's.
The
accelerometer's frequency response
100 Hz. However, the antialiasing inside the
of all of the sensors to 20 Hz.
to
compenstate
An
IMU
limits the effective
internal initialization
for accelerometer bias
and
rate gyros used by the
bandwidth
cross axis error. Table 4.1
IMU
past
program allows the unit
specifications of the accelerometers incorporated into the sensor
The
is flat
shows the
model
[Ref. 8]
are solid state vibrating element
angular rate sensors. This relatively new technology uses no moving parts.
piezoelectric bender element
angle.
The element
is
mounted end
fastened to the base
is
to
end but rotated
A
at a 90 degree
resonantly driven such that the
sense element swings a reciprocating arc. Under zero angular rate conditions,
the motion of the sense element due to the drive element does not produce any
54
TABLE
ACCELEROMETER CHARACTERISTICS
4.1:
Acceleration Range
±2g's
Acceleration Bandwidth
20
Acceleration Scale Factor
0.2% of Full Scale
0.2% of Full Scale
Acceleration Noise Floor
0.0005 g's
Cross Axis Sensitivity
0.5% of Full Scale
Acceleration Bias
bending of the sense element.
cause
Hz
momentum
When
a rate of rotation exists, Coriolis forces
to be transfered into the plane perpendicular to the
of the drive element, thus causing bending of the sense element.
transducer picks up a signal from the sense element
when
it
is
A
motion
pressure
bent that
is
proportional to the angular rate with a phase dependent on the direction of
rotation.
Figure 4.6 shows the configuration of two rate sensors mounted in a
"tuning fork" configuration. Table 4.2 shows specifications of the rate sensors
incorporated into the sensor model [Ref.
Figure
4.6:
8].
Angular Rate Sensor
55
TABLE
4.2:
ANGULAR RATE SENSOR CHARACTERISTICS
Rotational Rate Range
±114.6deg / sec
Rotational Rate Bandwidth
20
Rotational Rate Bias
2.0% of Full Scale
Rotational Rate Scale Factor
0.5% of Full Scale
Rotational Rate Noise Floor
0.05% of Full Scale
0.5% of Full Scale
Cross Axis Sensitivity
The inclinometers
low bandwidth sensor
to
(
utilize
a liquid crystal pendulous sensor.
approximately 0.12 radians per second) that
be integrated with the rate sensors
angular position.
The
Hz
The
fluxgate
for high
is
It is
a
meant
bandwidth measurements of
magnetometer provides heading measurements.
specifications incorporated into the Euler angle sensor
models are shown
in Table 4.3 [Ref. 8].
TABLE
4.3:
INCLINOMETER AND MAGNETOMETER CHARAC-
TERISTICS
Pitch and Roll Bandwidth
±50 deg
1/2 Hz
Pitch and Roll Accuracy
0.2 deg
Pitch and Roll Range
GPS
plane position.
Heading Range
±180 deg
Heading Accuracy
3.0 deg
Heading Repeatability
0.5 deg
Heading Linearity
0.5%
provides data in a form that can be converted to local tangent
A
brief
summary
of the errors included in the
follows.
GPS
Error Sources
Selective Availability
56
GPS model
- intentional degradation of pseudorange signal by Departmant of Defense
• Clock Differences
•
drift
and bias
GPS
in
clock
Ephemeris Error
- error introduced
in converting pseudoranges to inertia! position fix
Each of these sensor components is simulated in block diagram form
utilizing internal
sources of error.
models
is
The upper
in Figure 4.7
M—
Q
level
SIMULINK
and contained
in the
diagram of these sensor
SIMULINK
urad
BodyThnj«
Aoo*W*tora
dJCSS*
J"
ImiM
Aooitirdlon*
*l
»
In
rn
RateQyro*
out*
*m
IflflHMMMSM
-*
It
M«gno*«t«f
>
on
l»artftn
-»
SIMULINK
modeling principles based on manufacturer specifications and
known
shown
in
OaMM
Aetwttom
Figure
4.7:
Sensor Models in
57
SIMULINK
file
plant l.m.
2.
Kalman
Filtering
With the outputs
of the
modeled sensors
available,
Kalman
filters
were developed along the linear and angular position channels to provide optimal
An
estimates of the states for the controller.
approach analogous to the
LQR
design was used. Consider the linear system described by:
x
.
where v and
= Ax + Bu + Gw
= Cz + v
V
A
gain matrix, L, was found such that the
W.
x
x.
The Kalman
L=
Y
is
Kalman
filter
given by:
= Ax + Bu + L(z - Cx)
produces an optimal estimate of
where
.m
w are zero mean white noise with respective power spectral densities
of
and
/.
(4 - 16)
positive semidefinite
gain
L
(4.17)
is
calculated as follows:
YCT V~\
and solves the algebraic Riccati equation:
AY + YA T - YCT V- CY + GWGT =
X
A
synthesis
original plant.
model was formed that included the dynamics of the
The IMU used
in Bluebird incorporates
an initialization routine
that removes steady state bias from the sensors. Therefore, extra dynamics were
not required in the
Kalman filter to compensate for bias. The design
process was
primarily driven by the bandwidth limitations of the inclinometers and
The
values of
V
and
W were used as "knobs"
58
GPS.
in the iterative design process.
—
GPS
For the
was
sensor, a break frequency of 1/2 radian per second
desired.
For the Euler angle sensors, a break frequency of 1/10 radian per second was
desired.
It
was
wash out the accelerometers and rate sensor
also desired to
biases at low frequencies.
The Kalman
filter for
the position estimate blends processed ac-
GPS
position fixes. Figure 4.8 shows the frequency
celerometer outputs with the
response along the x axis of the
Kalman
position
filter.
The other two
axes have
similar dynamics.
i
i
i
i
gae^n
i
|
'^tt4^2
i
|
l
'
'
'
-T
L
"
1
J-J-LLiLL
CO
8-2
y' '
c
Acceferometer Sensor dashed Ime
(0
*
-40-
1
i
•"
;
10-
degrees
-
;;;;
io-'
;
<3PS? $0n$or: solicf
;
\
;
;
;;;;
•
;;;;;:;;
ioFrequency - rad/sec
g
8
g
4f«i
!::;:!
;
;
;
;
;
f"-i^L?f!?H
i-ffH-
o
'f?
Phase
g
ill
10'
10"
<— >• •j-j-j|j-i-:
10
io-
Figure
4.8:
:••••:•••:-•
lf4^if[-----i--l-l-H-
10
Frequency
10"
-
10'
rad/sec
Frequency Response of Position Filter
The Kalman filter for the Euler angles blends the outputs
lar rate sensors,
of the angu-
converted to Euler angle rates, with Euler angle measurements
from the inclinometers and magnetometer. Figure 4.9 shows the frequency
re-
soonse of one channel of the Euler angle filter. The remaining two angle channels
59
are similar.
CD
-5
:
g-10 "
:
-
:"
'r
3
o>-15
.
.y*.
:
f...
.'.
:
:
:
\\a"
:
:
:
:
:
:
!4lt
^vl
::
:
:::::::
i
:
i
^<v:y::-lnc1lh6riheierS^
..
'.
.;.
i
.;
,
.:.L
.
Angular ^at^^ensorr^JaBhiBdirei-V^-j- ;
ro
u«>y
-20
10
•
C-
10"
10Frequency - rad/sec
10"
10'
10
10Frequency - rad/sec
10
10'
5°
!
o
S
o
"?
o
(0
(0
£-50
10
Figure 4.0: Frequency Response of Euler Angle Filter
E.
INTEGRATION OF THE FULL NONLINEAR SIMULATION
The full
nonlinear simulation can
ter II, the nonlinear rigid
Wind
reference frame, therefore a
simulation whose output
in {/}.
If
the simulation
is
in a
Chap-
SIMULINK
expanded to include the
a moving airmass, the dynamics of Bluebird can be simulated at an
arbitrary night condition.
tial
together. Recall in
body dynamics were implemented
block labeled Equations of Motion.
effects of
now be pieced
is
is
SIMULINK
a vector
Next, the wind velocity
velocity of Bluebird,
Bv
cg .
usually referenced with respect to the iner-
The
is
wv
block, Wind,
is
included in the
full
comprised of the wind velocity resolved
resolved in {J5} and added to the inertia!
result
60
is
the velocity of Bluebird with respect
Bv
to the airmass, resolved in {B}. This velocity, rather than
computing the aerodynamic contribution to the
used in Equation 2.25.
in the
.m
A more detailed
state-deriv.m
file
is
description of
SIMULINK
the avionic sensor suite onboard Bluebird.
output
is
SIMULINK
how
this
come from a
from the
block to process the
filter
erates actuator
commands
is
and moment
accomplished
block, Sensors, that models
The output from
block,
Kalman
trajectory.
The
is
filtered
The commands
trajectory block which uses the
commanded
these sensors
filters.
directed to the V-Implemented Controller block.
the controller
manded
total external force
used when
contained in Appendix A.
All sixteen states are sent to a
appropriately processed in a
cg , is
to
measured outputs
The
controller gen-
necessary to maintain the air vehicle on the com-
trajectory, thus completing the loop.
the
SIMULINK
nonlinear simulation
the
SIMULINK
file
is
plantl.m.
61
shown
The complete top
in Figure 4.10
level
view of
and contained
in
QPS
IMU
Air
Data
Kalman
Filters
Sensors
Trajectory
Generation
Figure 4.10:
SIMULINK Diagram
62
of the Full Nonlinear Simulation
V.
APPLICATION TO THE CONTROL OF
BLUEBIRD
D-IMPLEMENTED CONTROLLER PERFORMANCE CHAR-
A.
ACTERISTICS
The performance
characteristics of the trajectory tracking controller de-
signed in Chapters III and
sensor and
filter
IV were evaluated using a two
step process. First, the
blocks were removed and the controller was connected directly
to the nonlinear equations of motion block. Utilizing pure state feedback, Blue-
bird was flown along two fundamentally different types of trajectories. These
two
trajectories served as general
examples of the
Next, the sensor and
in
Chapter IV, Equation
A
general deparure and arrival trajectory, which
4.4.
set of all trajectories defined
types of trajectories tested in the
first
step,
is
filter
blocks were added.
a combination of the two
was commanded and flown with
the controller utilizing output feedback. Data from this simulation were used
as input to a virtual prototype simulation discussed later.
The dynamic
tion.
At
trimmed
flight simulations
were started using the same
this initial condition, Bluebird
for level flight at 73 fps.
The
is
initial condi-
aligned with the positive x-axis and
positive x direction
is
considered to be
heading north. The mechanics of the dynamic simulation use a right-hand
thogonal coordinate system described in Chapter
is
pointing east and the positive z-axis
is
II.
As
such, the positive y-axis
pointing down.
venient from a computational standpoint since
it
or-
This choice
is
con-
coincides with the body-fixed
reference frame of Bluebird at the initial condition specified above. Typically,
63
however, the positive z axis
orthogonality
west.
is
is
considered to be pointing up.
If
the right-hand
maintained, the positive y-axis would then point towards the
For ease of visualization, this
the cooridinate system used to display
is
the results of the simulations.
The
in
simulations are intended to evaluate the capabilities of the controller
terms of the nature of trajectories that can be followed in a stationary
and in an
air
flight serve as
mass moving
at constant velocity.
Two
basic kinds of
the bases for the test trajectories. Each test trajectory
no- wind conditions, and then again with the wind added at
the
air
mass
trimmed
is
flown in
some point during
flight.
The simplest form of trimmed flight
is
constant velocity, constant heading.
This corresponds to a trajectory defined by a ramp
tion.
command
in inertial posi-
This was the basic trajectory that the controller was designed to track.
Figure 5.1 shows a three dimensional plot of the
first test
In this
trajectory.
case the trajectory encompasses 30 seconds of flight heading north at 73 fps
followed by a 90 degree turn to join a trajectory heading east at 73 fps while
On
climbing at 300 feet per minute.
second from the north
time
=
is
added
the second
at the
flight,
time the turn
is
a wind of 20 feet per
commanded
(elapsed
30 seconds).
Figure 5.2 contains the
first
four graphs of flight data.
The
first
graph
shows the time history of Bluebird's distance from the commanded trajectory.
The next three graphs show the time
history of the Euler angles. Consider the
baseline flight (no- wind data). Bluebird begins the turn at an elapsed time of
30 seconds and exits the turn at an elapsed time of 45 seconds. Approximately
30 seconds later the trajectory error
is
64
nearly zero. In the presence of 20 feet
commanded
position (10
second
intervals) li'o'
500 N
flight path:
400 N
©300 s
ground
tarck:
"....., .;"
1200
(0
100,
x range
y range
(ft)
Figure
5.1:
Test Trajectory
(ft)
#1
per second of wind from the north, the trajectory error also goes to zero in
about the same period of time. The graphs of the Euler angles indicate that
Bluebird
is
flying wings level,
crabbed into the cross wind, which
is
the desired
result.
Figure 5.3 shows the groundspeed and indicated airspeed during the
flights.
Notice that in both cases, the
second
is
commanded groundspeed
of 73 feet per
eventually maintained. Finally, Figure 5.4 shows the time history of
the control activity during the
Trimmed flight does
flights.
not necessarily have b u>b equal to zero. For instance,
any steady turning manuever fits the definition of trimmed flight given in Chap65
Magnitude
of Position Error
60
80
Time -seconds
Bank Angle
50
/^V
50
20
40
60
80
100
120
140
100
120
140
Time -seconds
Pitch Angle
80
60
Time -seconds
Heading
W
100
<D
»_
0)
•100
20
40
60
80
Time -seconds
Figure
5.2:
Trajectory #1: Position Error and Euler Angles
ter IV. Figure 5.5
shows the three dimensional plot of the second
In this case the test trajectory
rate
is
is
test trajectory.
a helix flown at 73 feet per second.
one revolution per minute and the helix angle
66
is
The turn
4 degrees which corre-
—
—
1
Groundspeed
78
!
j\
76
no wind:
J
^
wind:
':
"
"
1
"-.-.-
'*
\
72-
^Y^^
V
'
70
'
1
1\
/
w 74-
1
!
!
1
1
i
20
40
60
i
1
100
80
140
120
Time -seconds
Indicated Airspeed
95
-1
/
I
—
i
\
:
1
1
—
\:
90-
j
:
:
1
1
85-
r
;
":V
a
:
"
\
80i
r<
i
x
»
r v — -:-•'"
75-
:
"*^
i
70
"
40
20
Figure
"
60
80
Time -seconds
5.3:
i
i
100
120
140
Trajectory #1: Velocity Data
sponds to a climb rate for Bluebird of 300 feet per minute. For this
test,
no
indicated airspeed limits were placed on Bluebird.
Consider Figure 5.6 which shows the position error and Euler angle time
history for the helix trajectory. Notice that with no
ers Bluebird to join the
The constant
the second
commanded
controller
manuev-
trajectory with zero error in steady state.
pitch and bank angles confirm the steady state performance.
flight,
On
a wind of 20 feet per second from the east was added at the
start of the helical trajectory (elapsed
it is
wind the
clear that the
time equal to 40 seconds).
Intuitively,
bank and pitch angle must vary as Bluebird traverses the
67
Elevator Activity
.'"*!
'
no
\
.
'-.-.-.-'
wind:
h.
L.
0)
<D
V
-2
20
40
60
80
100
120
140
Time -seconds
Rudder Activity
I
1
i
i
i
i
i
i
i
i
i
i
40
60
80
0)
0)
(1)
"0
20
100
120
140
100
120
140
100
120
140
Time -seconds
Aileron Activity
W
(1)
0)
<D
T5
20
40
60
80
Time -seconds
Throttle Activity
2
c
a.
20
40
60
80
Time -seconds
Figure
helix.
5.4:
Trajectory #1: Control Activity
To an observer on Bluebird, the wind, while constant
in {/}, represents
a sinusoidal disturbance. This explains the sinusoidal nature of the position error
around the
helix.
Figure 5.7 shows the groundspeed and indicated airspeed
68
commanded
flight path;
position (10. second intervals): "d"
"__£_"''
1200
x range
y range
(ft)
Figure
5.5:
around the helix during the two
groundspeed
is
oscillation
experienced.
is
Test Trajectory
flights.
(ft)
#2
For the no- wind
maintained while for the
flight in
Finally, Figure 5.8
flight,
the
commanded
wind, a one foot per second
shows the time history of the
control activity during the flights.
Four additional
flights
were flown using the helix trajectory in an attempt
to ascertain the sensitivity of the position error to changes in
rate
and wind
velocity.
commanded
turn
Figure 5.9 shows the position error around the helix
for three different turn rates
with a constant wind of 10 feet per second. The
dashed line corresponds to a turn rate of 3 degrees per second or 2 minutes
'
Magnitude
of Position Error
10
f
no
\
:
e
5
..../',
S.\.
\"\
\
\
"
v-
\csA
50
V
V"
i
100
v
V
\
V.'
i
150
200
250
200
250
Time -seconds
Bank Angle
50
(0
o
^.
TJ
-50
50
100
150
Time -seconds
Pitch Angle
20
i
i
1
i
i
i
i
50
100
150
200
i
10
.*'"
CD
r—
ft-r
.
.
.
-
n
.
••
.
TJ
-20
250
Time -seconds
Heading
250
150
100
Time -seconds
Figure
5.6:
Trajectory #2: Position Error and Euler Angles
per revolution; the solid line corresponds to a turn rate of 6 degrees per second
or
1
minute per revolution; and the dash dot
line
corresponds to a turn rate
of 9 degrees per second or 40 seconds per revolution.
70
The
error increases with
Groundspeed
76
75741
no wind:
—
'»:v".»'
i*!
y
•
1
V
'
\
t\
X
X
§73sJi^?C72-
wind:
':
1-
->
V/
l'
-;r
71
70
50
100
250
200
150
Time -seconds
Indicated Airspeed
100
7
*
90
\
/•"TV
/;
'
:
i
:
80
/
v
v
\
/
:
Q.
70
:
\
/
i
:
:/
60
50
100
50
250
200
150
Time -seconds
Figure
5.7:
increasing turn rate.
The
state angle of
bank of
command trajectories
place an upper
Trajectory #2: Velocity Data
9 degree per second turn rate corresponds to a steady
thirty degrees,
requiring
bound on the
no wind.
more than a
It
may
not be desireable to
certain angle of
bank and
this
may
error.
Figure 5.10 shows the position error around the helix for three different
wind
velocities at a constant turn rate of 6 degrees per second.
in velocity
from 10 to 25
feet per second.
The
local
The wind
maxima
values of the
position error are proportional to the magnitude of the disturbance.
71
varies
Elevator Activity
»5
"
..^•v
..r:"'
"X***"
\
'5
!
!
ho wind:"
wind:"-.-.-.-"
/
\
50
!
•."''"
.""^s
100
'
:
150
"v
.<
\
/
200
250
200
250
200
250
Time -seconds
Rudder
Activity
w 10
4r
i_
0)
i
o
-10
100
50
150
Time -seconds
Aileron Activity
u
l
4* w
-
\x
,
*m*t
,
m+m
t
V
10
50
100
150
Time -seconds
Throttle Activity
50
100
150
200
Time -seconds
Figure
B.
5.8:
Trajectory #2: Control Activity
AN AIRPORT DEPARTURE AND ARRIVAL FLIGHT
SIM-
ULATION
many cases,
the
good example of
this
In
A
trimmed
flight
condition of an air vehicle changes often.
would be a standard instrument departure or
72
arrival
100
150
Time -seconds
Figure 5.9: Trajectory #2: Position Error for Varying Turn Rates
to
an
airfield.
These
trajectories are typically a combination of constant radius
turns and wings level flight, while often climbing and descending. Trajectory
position errors
become
critical
when the
air vehicle is
on
final
approach with a
constant heading, constant velocity trajectory.
Consider Figure 5.11.
If
an
airfield is
imagined to be located
at the origin,
then this trajectory would be indicative of a typical departure followed by a
typical arrival to that airfield.
different radii
is
The
scenario utilizes turning trajectories of three
connected by straight line trajectories. The
a constant 73 feet per second throughout.
73
Wind
commanded
velocity
zero.
Thirty
is initially
100
150
Time -seconds
Figure 5.10: Trajectory #2: Position Error for Varying
seconds into the
At 90 seconds
east.
flight,
the wind
is
added
into the flight the
wind
on
final
Finally, with Bluebird
the wind
is
is
at 10 feet per
Wind
Velocities
second from the east.
increased to 20 feet per second from the
approach tracking a 4 degree glideslope,
rapidly shifted 90 degrees to the north and decreased in magnitude
to 5 feet per second.
Figure 5.12 shows the time history of the position error, wind velocity, and
Euler angles during the
activity.
as
it is
flight.
Figure 5.13 shows the time history of the control
Note, however, the relative difficulty of analyzing data of this nature
somewhat
difficult to visualize. Flight
74
simulation data was saved to a
file
flight
path:
ground
'_
track:
600,
commanded
position (10
second
'6'
intervals):
£>400.
y range
(ft)
2000
x range
(ft)
Figure 5.11: Departure and Arrival at an Airfield
and processed
for compatiblity as
package, Designer's Workbench.
Bluebird was developed in [Ref.
and
an input
A
13].
file
for
a 3-D visualiztion software
virtual prototype of
Monterey Airport and
The simulation was then run
arrival to the virtual prototype airfield. In Designer's
as a departure
Workbench, the
flight
can be viewed from multiple perspectives and virtual prototypes of standard
cockpit displays further enhance visualization.
on video tape,
[Ref. 14].
75
One
possible result
is
captured
Magnitude
of Position Error
100
250
150
Time -seconds
Bank Angle
20
(I)
<D
<D
i_
0)
<D
"0
20
100
150
Time -seconds
Pitch
Angle
100
250
150
Time -seconds
Heading
500
<D
(D
L.
0)
(D
"0
500
50
100
150
200
250
Time -seconds
Figure 5.12: Airfield Scenario: Position Error and Euler Angles
76
Elevator Activity
^^V
to
'•
•
y*»«
.*/-•
^--
<D
T3
50
100
200
150
250
Time -seconds
Rudder
So
"0
J Vrrr*wr
V
v~
—
Activity
A-
r
Ju
v
.
100
50
200
150
250
Time -seconds
Aileron Activity
.5
£o
j hi
i
.
.
m
»
i
y—
.
V—f
^""Tfr*^
Ju
V
1
-5
50
100
150
200
250
200
250
Time -seconds
Throttle Activity
1.5
c
2
1
0)
a
0.5
50
100
150
Time -seconds
Figure 5.13: Airfield Scenario: Control Activity
77
CONCLUSIONS AND
VI.
RECOMMENDATIONS
CONCLUSIONS
A.
Based on the data presented
in this thesis, the following conclusions are
drawn.
•
SIMULINK provides an effective environment for the developement of nonlinear simulations for air vehicles.
model of the plant
at
As a
result of this
an arbitrary trim condition
development, a linear
easily obtained for
is
design purposes.
•
LQR design
techniques utilizing a synthesis model and weighting "knobs"
provide a straight forward means of obtaining satisfactory controller gains
for
MIMO
systems while meeting design requirements.
• ^-Implementation of the linear trajectory tracking controller allows the
controller to operate effectively
conditions, trajectories defined
For
in steady state.
wind disturbance
by an arbitrary
is
by an arbitrary
flight conditions
[vq, cjo]
flight
are tracked perfectly
with wind, rejection of a constant
accomplished along the family of trajectories defined
[t>o»<*>o
disturbance in {/}
on the nonlinear plant. In no- wind
is
=
Oj.
However,
for turning flight,
seen as a sinusoidal disturbance in
soidal tracking error results. For
errors are usually small.
78
a constant wind
{B} and a
moderate bank angles and turn
sinu-
rates, the
• Preprocessing of the trajectory
commands by an adaptive
filter
allows for
steady state control of the air vehicle's velocity with respect to the air mass
in the forward path, thus not affecting stability.
With
sufficient
margins
transient deviations in indicated airspeed, this would allieviate the
concern of stalling the
air vehicle
when
for
major
tracking an inertial trajectory in
wind.
•
When
and
analyzing a nonlinear plant and controller, test simulations are vital
in
some
cases the only
means
of performance evaluation.
The
three
dimensional plots and time history graphs are fine for simple trajectories,
but are
difficult to
analyze for more complex cases.
virtual prototyping software package, like Designer's
pressive.
The enhanced
situational awareness
of watching the designed controller operate
for
a
"pilot's perspective"
The
capabilities of a
Workbench, are im-
and visualization
capabilities
on a virtual prototype allow
feedback not otherwise attainable on the desk
top.
RECOMMENDATIONS
B.
Based on the conclusions presented above and the experience
of developing
the simulation package presented in this thesis, the following recommendations
are
made.
• W^hile the rigid
body equations
of
motion are nonlinear with respect to
the kinematics involved, they are completely linear with respect to the
stability
and control
derivatives.
The constant
control derivatives could be replaced by functions
is
available.
79
coefficient stability
when
and
further flight data
•
A
similar design process used for the trajectory controller could be done
using Tioo design methodology.
•
The
trajectory preprocessor could be used to convert an inertial fixed tra-
jectory into an air mass fixed trajectory.
where the
pared to
•
This might have applications
air vehicle's inertial position is of
its
performance with respect to the
Running simulations
real
secondary importance comair
mass.
time in Desiner's Workbench rather than using
batch post processed data would be the next logical step. Further work
might lead to virtual prototype visualizations based on real-time simulations or downlinks
from actual
air vehicles.
80
APPENDIX
The SIMULINK models
lowing
MATLAB
.m
files
A:
MATLAB
of Bluebird (plant
Lm &
FILES
plant2.m) use the
fol-
as user defined functions.
STATE.DERIV.M
X
X
%
Function to calculate derivative of u,v,w,p,q,r
X
%
based on
X
%
1:
kinematics
X
%
2:
gravity
X
%
3:
propulsion
X
%
4:
aerodynamics
X
X
X
y,
Variables brought from workspace:
X
X
y,
X
x
[contrl inputs, state variables (1 - 9), wind vel]
*/,
X
- (da,de,dr,dtrt,u,v,w,p q,r,phi,theta psi, wind xyz)X
X
x
%
Variables called from function "blue.data"
X
x
X
air density
X
rho
%
b = wing span
X
X
81
X
c = wing cord
'/•
X
s =
X
X
Cfo
X
Cfu = Stability derivative for control inputs
X
X
m = airplane mass
X
X
lb
X
To = Thrust scale term
%
X
Pe = Engine postion matrix
X
wing area
Steady state force term
inertia tensor matrix (body frame)
*/•
X
X
X
X
%
X
X
function accel = state_deriv(x)
XXXXXX
Function call to get the aircraft data
[uO,wO,rho,Cfx,Cfo,Cfu,Cfxdot,s,b,c,m,Pe,To,Ib] = blue.data;
XXXXXX
seperate the combined vector into seperate elements
u = [x(D; x(2); x(3)]
dtrt » x(4);
state = [x(5); x(6)
;
x(7)
;
x(8); x(9)
lambda = [x(ll); x(12); x(13)]
82
;
x(10)]
wind = Cx(14); x(15); x(16)]
y.'/.'/.'/.X'/.
calculate velocity wrt the airmass and form state vector
'/.'/.y.'/.'/.y.
that will be used to calculate the aerodynamic forces/moments
ias = u + wind
statel = [ias(l)-uO; ias(2); ias(3)-w0; x(8)
y.y.y.y.y.'/.
;
x(9)
;
x(10)]
calculate total velocity, vt
vt - (ias(l)*2 + ias(2)*2 + ias(3)~2)~ .5;
milt
calculate qbar
qbar = .5*rho*(vt"2)
XXXm
calculate Ml
Ml = diag([l/vt, 1/vt, 1/vt, (.5*b)/vt, (.5*c)/vt, (.5*b)/vt]);
lWm
calculate M2
M2 = diag([0, 0, (.5*c)/(vt~2)
m%X%
,
calculate Sprime
83
0,
0, 0]);
Sprime = diag([-l,
mm
1,
-1, b, c, b]*s);
calculate Mu
Mu = inv([eye(3)*m,zeros(3) zeros (3) ,Ib])
;
Wan
calculate Tw2b
alpha = ias(3)/vt;
beta = ias(2)/vt;
Tl = [cos(alpha), 0, -sin(alpha); 0,1,0; sin(alpha), 0, cos(alpha)];
T2 - [cos (beta), -sin(beta)
Tw2b
,
0;
sin(beta)
,
cos (beta), 0; 0,0,1];
[T1*T2, zeros (3); zeros (3), T1*T2]
calculate chi
y.y.y.y.y.y.
Chi = eye (6) - Mu*Tv2b*qbar*Sprime*Cf xdot*M2
calculate Propulsion matrix
'/.y.'/.'/X/.
Prop =
[
eye (3);
0,-Pe(3),Pe(2);
Pe(3),0,-Pe(l);
-Pe(2),Pe(l),0];
84
y.y.'/.y.'/.'/t
calculate gravity vector and rotation matrix {1} to {B}
Rot = [1, 0, -sin(lambda(2));
0,
cos(lambda(l))
0,
-sin(lambda(l)), cos(lambda(2))*cos(lambda(l))]
,
cos(lambda(2))*sin(lambda(l))
Ru2b = [Rot zeros (3) ]
;
g = [0; 0; 32.174];
FgU = m*g;
'/.'/.'/.
'/.'///.
'/.y.'/.'/t'/t'/.
put the components due to gravity; thrust; and control surface
deflections together for their contribution to x-dot
thrust = Prop*To*dtrt
gravity = Ru2b*FgU;
Ctrl
qbar*(Tw2b*(Sprime*(Cfo + (Cfu*u))));
xdotu = (Mu*(ctrl+thrust+gravity))
'/.'/.'/.y.'/.y.
calculate kinematic contribution
omegax = [0,-state(6) ,state(5) ;state(6) ,0,-state(4) ;-state(5) ,state(4),0]
¥ilb = (-inv(Ib))*(omegax*Ib)
85
;
;
;
[-omegax, zeros (3); zeros (3), wxlb]
Rot
xdotrot = Rot*state;
\IJ*hl%
state vector feedback component xdot
xdotcfx = qbar*(Mu*(Tw2b*(Sprime*(Cfx*(Ml*statel)))));
XXXXXX
add three components of xdot together and premult by inv(Chi)
xdot* inv( Chi)* (xdot rot+xdotcfx+xdotu)
XXXXXX
return xdot
accel«xdot
EULER-B2I.M
X
Transformation
[p
q r] to
X
86
lambda-dot
function ldot = eul_b2u(x)
,
,
/. /.'/.'/.7.
separate the composite vector
%%%%%
and [phi theta psi]
'x'
into [p q r]
omega = [x(l); x(2); x(3)]
phi=x(4);
theta= x(5)
psi=x(6)
%%%%%
calculate the rotation matrix {1} to {B}
y.'/.'/.y.'/.
based on euler angles
Rb2u =
[1,
sin(phi)*tan(theta)
0,
cos(phi), -sin(phi)
0,
sin(phi)*sec(theta)
VkkhU
,
,
cos (phi) *tan (theta)
;
cos (phi) *sec (theta)]
calculate lamda-dot
ldot = Rb2u*omega;
EULERJ2B.M
xmxxxmxxxxmxxmmmxmmmxxmmxmmmmmm
87
%
y.
Transformation lambda-dot
X
to
[p
X
q r]
mmmmmmmmmmmmmmmmmitxmmmm
function omegadot = eul_i2b(x)
y.'/.'/X/.
seperate the composite vector
%*/,%%%
and [phi theta psi]
'x'
into lambda-dot
.
ldot = [x(l); x(2); x(3)]
phi=x(4);
theta= x(5)
psi»x(6);
%%*/*%*/*
calculate the rotation matrix {B} to {1}
y.y.X'/.
based on euler angles
Rb2i = [1, sin(phi)*tan(theta)
,
cos (phi) *tan (theta)
0,
cos(phi), -sin(phi)
0,
sin(phi)*sec(theta), cos (phi) *sec (theta)]
y.'/.y.'/.
calculate lamda-dot
omegadot = inv(Rb2i)*ldot;
88
;
P0SITI0N-B2I.M
mmmmmmxmmxmxxmxmmxmmxmmmmxm
x
x
%
From the workspace:
X
X
X
X
1:
free vector 'u* resolved in {B} e(l:3)
X
X
2:
euler angle vector {phi,theta,psi} e(4:6)
X
X
%
%
Returns
X
:
X
X
X
1:
free vector 'u resolved in {1}
X
X
X
function ans
pos_b2i(e)
XXXXX
this will rotate the trajectory error through phi, theta, psi
XXXXX
from {b} to {i}. (3-2-1 transformation
phi=e(4);
theta=e(5)
psi-e(6);
;
;
;
;
m.psi = [cos(psi) ,sin(psi) ,0
-sin(psi) ,cos(psi) ,0
0,0,1];
m_theta = Ccos(theta) ,0,-sin(theta)
0,1,0
sin(theta) ,0, cos (that a)]
m_phi
[1,0,0
0, cos (phi)
,
sin (phi)
0,-sin(phi) ,cos(phi)]
rotb2i = inv(m_phi*m_theta*m_psi)
u = Ce(l); e(2); e(3)]
ans = rotb2i*u;
POSITIONJ2B.M
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
x
x
90
X
From the workspace:
•/.
X
1:
free vector 'u' resolved in {1} e(l:3)
X
2:
euler angle vector {phi,theta,psi} e(4:6)
1:
free vector 'u resolved in {B}
*/.
'/.
Returns
X
X
X
mmmmmmmmxmmmmmmmitmmmmiE
function ans = pos_i2b(e)
XXXXX
this will rotate through phi, theta, psi
XXXXX
from {i} to {b}. (3-2-1 transformation)
phi=e(4);
theta=e(5)
psi=e(6)
m_psi = [cos(psi) ,sin(psi) ,0
-sin(psi) ,cos(psi) ,0
0,0,1];
m_theta = [cos (theta) ,0,-sin(theta)
0,1,0
91
sin(theta) ,0,cos(theta)]
m_phi
0,
[1,0,0
cos (phi) , sin (phi)
0,-sin(phi) ,cos(phi)3
roti2b
(m_phi*m_theta*m_psi)
u = Ce(l); e(2); e(3)]
axis
;
= roti2b*u;
LIMITXOGIC.M
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
X
X
X
funtion to limit trajectory commands, if required
X
X
X
X
from workspace:
X
X
1:
commanded inertial velocity
X
X
2:
inertial wind
X
92
X
3:
lower IAS limit
%
4:
upper IAS limit
revised commanded velocity
returns:
%
x
WW
•/•/•/•/•/•/•/•/•/•/•/•/ •/•/»/•/•/ •/•/•/•/
WWWWWWWV'i
function vcom=limit(u)
seperate u
vel.i = [u(l);u(2);u(3)];
=*
[u(4),u(5),u(6)];
ll=u(7)
ul=u(8);
'/.'/•'/•'/•'/•
calculate magnitud and direction of commanded velocity
gs=sqrt (vel.i (1)~2 + vel_i(2)~2);
ang=atan2(vel_i(2) ,vel_i(l))
1*1*1*1*1*1*
'/.
y.
•/•/•/•/•/•/•/•/•/•//•/•/•/•/•/•/•/•/ */*/*/*/*/
wind.i
1*
%
%
1X1*1,1
*/.
calculate commanded IAS (steady state)
93
vt= sqrt((vel_i(l)+vind_i(l))~2 +(vel_i(2)+wind_i(2))~2 + (vel_i(3)+wind_:
y.'/.'/.'/.'/.y.
Prepare return variable (may not be limited)
vcom = vel_i;
*/.'/•'/•'/•%'/•
Check limits and revise if outside
if vt > ul;
over = vt - ul;
vcom(l) - (gs - over)*cos(ang)
vcom(2) s (gs - over)*sin(ang)
end;
if vt < 11;
under = 11 - vt;
vcom(l) = (gs + under) *cos (ang)
94
vcom(2) = (gs + under) *s in ( ang)
end;
BLUEBIRD -DATA.M
•/•/•/•/•/•/•/•/•/•/•/
WW
•/•/•/ •/•/•/•/
WW*/ •/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/•/
•/•/•/
35
Aircraft data for Bluebird
X
%
mmmmmmmmmmummmmmmmxmxmi
function [uO,wO,rho,Cfx,Cfo,Cfu,Cfxdot ,s,b,c,m,Pe,To,Ib] = blue_data
/././. 7.
trimmed flight speed and angle of attack
/.
uO = 73.3;
wO
=»
0;
M.W.W.
Density: Sea level- std day
rho = .0023769;
y.V/.V/.
derivative matrix due to state variables
95
xxxxx
rows:
[CD CY CL CI Cm Cn]
xxxxx
col:
[u v/U w/U p q r]
Cf x = [0
.188
-.31
0;
.0973;
3.94 0;
4.22
-.363 0.1;
-.0597
-1.163
-11.77
-.0481
.0487
0;
-.0452];
XXXXX
derivative matrix due to control inputs
XXXXX
rows:
[CD CY CL CI Cm Cn]
XXXXX
col:
[elev rud ail]
Cfu - [.065
0;
.0697 0;
.472 .0147 0;
.0028 .265;
-1.41
0;
-.0329 -.0347];
XXXXX
derivative matrix due to x-dot (alpha_dot k Beta.dot)
96
Cfxdot =
[000000;
0;
1.32
0;
0;
0];
steady state force vector
y.'/.'/.X'/.
Cfo = [.03; 0;
.385; 0; 0; 0]
XXXXX
physical dim.
XXXXX
WT =55 LBS.
m
1.
7095;
s = 22 .38;
b s 12 .42;
c
ss
1. 802;
engine data (4 HP motor)
XXXXX
Pe = [0; 0; 0]
To = [15 ;0;0];
inertia tensor matrix
y.y.'/.y.y.
lb «
[
10
0;
16.12
0;
7.97];
97
APPENDIX
The nine
file
state nonlinear
model
E0M-9.m and was trimmed
of bluebird
contained in the
is
SIMULINK
at a flight condition of
• flight
speed equal to 73 fps
• flight
path angle equal to zero
using the
SIMULINK FILES
B:
TRIM command. The
resulting trim values for the state vector
and
input vector are:
73.3
-0.0023
and u
=
0.2858
The
model
in
LINMOD command was
of Bluebird (contained in the
Chapter
in the
III)
MATLAB
B
synthesis
SIMULINK
file
EOM-16.m and
about this trim point. The resulting linear system
file
is
described
contained
Linear 16. mat.
The rudder was removed
of the
used to linearize the sixteen state nonlinear
as a control input (remove the second
column
matrix) and the resulting linear model was used as a basis for the
model contained
model was used
in the
to determine the
SIMULINK
LQR
98
gain.
file
synthesis.m.
The
This synthesis
synthesis model,
Q
and
R
weighting matrices, and resulting
LQR
gain
is
contained in the
MATLAB
file
LQR-dat.mat.
The full nonlinear simulation is contained in the SIMULINK file,
The
MATLAB
ables.
The
file
.m
file
simdata loads the workspace with the appropriate
simdata
trajectory schedule.
plantl.m.
calls
the .m
Any changes
file
to the
trajectory.
m
commanded
vari-
in order to generate the
trajectory or wind distur-
bance schedule can be made in trajectory, m.
A
version of the nonlinear simulation that does not use the
sensor blocks
is
contained in the
SIMULINK file
quicker than plantl.m.
99
plant2.m.
It
filter
and
runs considerably
APPENDIX C: ©-IMPLEMENTATION
PROOFS REFERENCED
Identity. Lei x
(z
B?
=
const be given. Then
d
I
(
B R(A)x)
= - BI R(A)x
x S-\A).
(C.l)
ti
and
= xxfR(A)S-
-£ (fR(A)x)
K
Proof: To derive both equations we
j
(
= B uB
B R(A))
t
and the following
will
1
(C.2)
(\).
need Poisson's Law:
x B R(A),
(C.3)
identity:
axb = -bxa
for
any vectors a and
6 of
j
(
(C.4)
compatible dimensions. Now, consider
B R(A)x)
t
= B wB
= (j
t
x B R{A)x
using equation (C.3), (C.4) and x
Next, by the chain rule
we
=
(
B R(A))x +{,
= - B R(A)x
R{A)j x
t
xB
uB
,
(C.5)
const.
get
i&RM*) = ^(^(a)x)|a
= ^(^(A)x)S(A) %,.
Equation (C.l) now follows by comparing equations (C.5) and (C.6).
100
(C.6)
To obtain equation
(C.2), consider
j(iU(A)f R(\)) =
j
B R(\))fR(\) +'B
('
R(\)j (fR(A)) =
since
B R(A)f R(A)
=
/ VA. Now, using equations (C.3) and (C.7)
d
= -f R(A) B uB
(fR(A))
dt
A\.
Suppose that assumptions A\ through
Dim{x C 2) = dim(u) =
get:
(C.8)
we
obtain:
= xxffl(A)5- 1 (A).
^(ffi(A)x)
..1
we
x
Finally, following the steps in the derivation of equation (C.l)
Theorem
(C.7)
0,
t
t
A3
hold.
<ftm(y 3 ).
A2. The matrix
si
- Ad
—Cc
has full rank at s
=
Bc3
CC2
i
0.
A3. The matrix pair (A ci,
C \)
is
c
observable.
Then for each equilibrium point of Q
in
£
the following properties are
observed:
• the feedback systems
%(Q
,
K
)(r
)
and T(Qi
,
K\
)
have the same closed
loop eigenvalues;
Ti(Q
X
)(r )(s)
=
L(A )T(Glo
Ao
101
=
[0
,
/]x w
/C,
)(s)L-\A
),
Proof: In the proof we
D& D&
set the controller matrices £) c i,
to zero. This
does not change the results but considerably simplifies the algebra.
Further-
more, we will drop explicit dependence of the controller parameters on a. Let
(x vo ,x Po
=
Pov
\
Ao
,u ,ro € S) be given. Consider the feedback interconnection
J
(Ao) and linear controller
of the linear plant Qi
AC/
.
The
F of this
state matrix
feedback system has the following form:
A\
Ai
Ai
I
F:=
BjQcx
BiCci
Aci
Bc3
(C.9)
B \C\
C
C2
Next we
K
.
linearize the feedback interconnection of the plant
However, in order to that,
troller states
first
Q and the
we must determine the
x c i and x^ along the trajectory
ro
£
S.
values of the con-
From equation
obtain:
d
—x clo = A
d
cl
x cU
+ Br,cl —y u + B&Lr
_ 1/k
1
(A
,
)e
A
~n x ao
= x c2o
wo
Notice, since along r
CciaJcio
+ Cc2 L -1((Ao)e
.
:
eo
we
=
=
0,
t/i
=
x C2
const,
—
get
£
~£
di
ci
—
A. C \X C\
= Cc \x c
102
i
.
u
controller
=
const
(4.9)
we
=
Now, using Assumption A3 we conclude that x c \
compute the
In order to
Q and
K
we must
,
first
0.
linearization of the feedback interconnection of
obtain the linearizations of equations (4.3) and (4.9)
=
about the operating points (x^^x^^uq) G £ and (x c i
0, cq
=
—
y 2o
the plant
Q
ro
is
=
=
«o,yi
=
determined by ro € S. The linearization of
0) respectively,
The
given by (4.4).
0,x c2o
linearization of the controller
K
has the
following form:
U
= A el U+Bj + Bc3 L-
£c2
=
1
(0 2
1
Cclfcl
+ Cc2 L~
($2
—
-p)
p)
(CIO)
It is
easy to verify that the state matrix
M:=
Ax
A2
L
BaCiAi
A|
B
el
dA
2
M of %{Q
,
C
)(r
B
+ B^L- 1 A
(C.ll)
BrtP\B
el
Cc
CcuLr
\
Let
0"
/
L
P:=
0/0
•
0/.
.0
Obviously,
"/
p-'
)le
M
algebra
it is
0'
L-
=
1
0/0
.000/
•
easy to show that
= P~ l MP
r
Ai
A2
i
A»
BcxCxAy
B
B \C\A + Bc3 A
22
C
C
<
2
ci
C
c\
103
) is
.
BC \C\B
Bc
Ai
A2
I
Ai
d
\
/
A\
A2
I
Aa
of the
first
M. The
A cl B \C\B
/]
C
ccl
now
part of the theorem
an observation that the matrices
and
+ £c3[0
cc2
o
The proof
B
F
follows
from Assumption A2 and
M are in the form of the matrices F
and
proof of the second part of the theorem consists of the following
steps:
1.
compute the
transfer matrix of the linearization of the controller
using equation (CIO) from controller inputs 6\,
2.
2
,
K
p to controller output
apply a similarity transformation
Pi
/
=
to the linearization of the plant
L- 1
Q (=
Q\ (ro)) given
by equation
(4.4)
derive the transfer matrix from the control inputs of the linear plant
the outputs 6\ and 6 2 using this
3
(A)
new
1
and 2 to get the
A simple computation shows
01, #2,
of the transfer matrices obtained in
that the transfer matrix from the controller inputs
is
r/
given by:
= Ccl (sl - Art)-\B#L4-
to
final result.
p to the controller output
f)(s)
rj
state-space realization;
compute the feedback interconnection
steps
and
B A{s) + Cc2 L-
l
c
l
-(b 2 (s)
-
p(s))
-(b 2 {s)-p(s))
3
= Hs)
A
Us)
L-ip(s)
104
(C.12)
where Ki(s)
is
the transfer matrix for the controller K\
.
Applying similarity transformation Pi to equation
the transfer matrix from
77
to 0\ and
"
3
Ai
I
L
is
A2 B
•
M
JU
(C.13)
L
.
where the transfer matrix
and computing
results in:
d
few
(4.4)
given in packed matrix form.
A
simple observation
shows that
where
G>&,(.s) is
Now
the transfer matrix for the plant Qi
.
routine algebra shows that the transfer matrix from p to 6 2 of the
feedback interconnection of the transfer matrices in equations (C.12) and (C.13)
is
given by:
Ti(Q
X
)(r
)M = L(A
)T(Glo
105
,
fC,
1
)(3)X" (A
).
REFERENCES
1.
"A Non- Linear Simulation For An Autonomous
Aerial Vehicle," Master's Thesis, Department Of Aeronautics,
Naval Postgraduate School, Monterey, CA, September 1993.
Kuechenmeister, David R.,
Unmanned
" Integration of Differential GPS and Inertial Navigation using Complementary Kalman Filter, "Master's Thesis, Department Of Aeronautics, Naval Postgraduate School, Monterey, CA, September 1993.
2.
Marquis, Carl,
3.
Kaminer, I.I., Khargonekar, P.P., and Robel, G., "Design of Localizer Capture and Track Modes For a Lateral Autopilot Using #«, Synthesis," IEEE
Control Systems Magazine, vol 10, pp. 13-21, 1990.
4.
Roskam, Jan, and Lan, Edward, Airplane Aerodynamics And Performance,
Roskam Aviation And Engineering
Corp., 1980.
Addison-
5.
Craig, J.J., Introduction to Robotics Mechanics and Control,
Wesley, New York, 1986.
6.
Schmidt, L.V., Class Notes for AE3340, U.S. Naval Postgraduate School,
Monterey, CA. 1992.
7.
Roskam,
J.,
Airplane Flight Dynamics and Automatic Flight Controls,
Roskam Aviation and Engineering
8.
Watson
tries,
9.
corp, Ottawa,
Industries, Techical Specifications for
KS, 1979.
IMU-600D, Watson
Indus-
Eau-Claire, WI, 1993.
Kaminer,
I.I.,
Class Notes For AE4276, U.S. Naval Postgraduate School,
Monterey, CA. 1993.
10.
Kaminer, I.I., Pascoal, A.M., and Khargonekar, P.P., "A Velocity Algorithm For Implementation Of Non-Linear Gain- Scheduled Controllers,
submitted for publication in Automatica.
'
11.
Nelson, R,C, Flight Stability and Automatic Control, McGraw-Hill, Inc.,
1989.
Optimal Control Systems, John
12.
Kwakernaak, H., and Sivan,
Wiley and Sons, 1972.
13.
Lagier,
14.
Lagier, Mark Virtual Prototype Demonstration Tape, U.S. Naval Postgraduate School, Monterey, CA. March, 1994.
R., Linear
Mark, "An Application of Virtual Prototyping to the Flight Test
and Evaluation of an Unmanned Air Vehicle, " Master's Thesis, Department
Of Aeronautics, Naval Postgraduate School, Monterey, CA, March 1994.
106
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I.
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Naval Postgraduate School
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NAVAL POSTGk.
MONTEREY CA 93*h4!
GAYLORD
S
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