/smash/get/diva2:617438/FULLTEXT01.pdf

/smash/get/diva2:617438/FULLTEXT01.pdf
Linköping Studies in Science and Technology. Dissertations
No. 950
Collision Avoidance Theory
with Application to Automotive Collision Mitigation
Jonas Jansson
Department of Electrical Engineering
Linköping University, SE–581 83 Linköping, Sweden
Linköping 2005
Collision Avoidance Theory with Application to Automotive Collision
Mitigation
c 2005 Jonas Jansson
[email protected]
http://www.control.isy.liu.se
Division of Automatic Control
Department of Electrical Engineering
Linköping University
SE–581 83 Linköping
Sweden
ISBN 91-85299-45-6
ISSN 0345-7524
Printed by UniTryck, Linköping, Sweden 2005.
To Eva, Alexander, and Alice
Abstract
Avoiding collisions is a crucial issue in most transportation systems as well as in
many other applications. The task of a collision avoidance system is to track objects
of potential collision risk and determine any action to avoid or mitigate a collision.
This thesis presents theory for tracking and decision making in collision avoidance
systems. The main focus is how to make decisions based on uncertain estimates
and in the presence of multiple obstacles. A general framework for dealing with
nonlinear dynamic systems and arbitrary noise distributions in collision avoidance
decision making is proposed. Some novel decision functions are also suggested.
Furthermore, performance evaluations using simulated and experimental data are
presented. Most examples in this thesis are from automotive applications.
A driving application for the work presented in this thesis is an automotive
emergency braking system. This system is called a collision mitigation by braking
(CMbB) system. It aims at mitigating the consequences of an accident by applying
the brakes once a collision becomes unavoidable. A CMbB system providing a
maximum collision speed reduction of 15 km/h and an average speed reduction of
7.5 km/h is estimated to reduce all injuries, classified as anything between moderate
and fatal, for rear-end collisions by 16 %. Since rear-end collision correspond to
approximately 30 % of all accidents this corresponds to a 5 % reduction for all
accidents.
The evaluation includes results from simulations as well as two demonstrator
vehicles, with different sensor setups and different decision logic, that perform
autonomous emergency braking.
i
Acknowledgments
For me the project leading up to this thesis has been a long (roughly 60000 km)
and fantastic journey. To help me on this journey I have had two excellent guides
Professor Fredrik Gustafsson at Linköping University and Jonas Ekmark Volvo
Car Corporation. You have helped me stay on a collision free path throughout the
project and I have greatly enjoyed working with both of you. Another guide on my
journey has been Robert Hansson. Your support of my project has been invaluable
and I feel privileged having worked under your leadership.
Several people helped me with proof reading this thesis. I am specially grateful to
Dr. Rickard Karlsson, Gustaf Hendeby, and Andreas Eidehall who helped me with
the illustrations. You really went out of your way to help me during the final stage
of my writing. For this I am very grateful. I would also like to thank Ulla Salaneck, Dr. Ragnar Wallin, Michael Shanahan, and Daniel Levin you all contributed
significantly to improve my work. Other people who deserve to be thanked for
helping me during different phases of the project are Magdalena Lindman VCSC
for providing accident statistics. Lars Nilsson VTEC, Fredrik Lundholm VCC,
and Lena Westervall VCC who all have helped performing various tests with the
demonstrator vehicles.
I also express my gratitude to Professor Lennart Ljung, Dr. Ahmed El-Bahrawy,
and Raymond Johansson who together with Robert Hansson recruited me for this
project. I would also like to acknowledge all my colleagues at the Chassi & Vehicle Dynamic department at Volvo and at the Control & Communication group at
Linköping University, it has been a privilege working with all of you.
This project was supported by the Swedish foundation for strategic research through
the Excellence Center in Computer Science and Systems Engineering in Linköping
(ECSEL) and Volvo Car Corporation’s PhD program, for this I am very grateful.
Finally I would like to thank my family, Eva for enduring my endless travels between Linköping and Göteborg, Alexander for helping me design the cover of the
thesis, and Alice for keeping me awake with cheerful shouts of support while writing
these acknowledgments at 2:00 am in the morning.
Jonas Jansson
Linköping, April 2005
iii
Contents
I
Introduction
1 Preliminaries
1.1 Background . .
1.2 Thesis Outline
1.3 Contributions .
1.4 Publications . .
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2 Collision Avoidance Overview
2.1 Collision Avoidance Applications . . . . . . . . . . . . . . . . .
2.1.1 Automotive Collision Avoidance . . . . . . . . . . . . .
2.1.2 Aerospace Applications . . . . . . . . . . . . . . . . . .
2.1.3 Marine Applications . . . . . . . . . . . . . . . . . . . .
2.1.4 Industrial Robot Manipulators . . . . . . . . . . . . . .
2.1.5 Unmanned Autonomous Vehicles . . . . . . . . . . . . .
2.2 Automotive Safety . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Passive Safety . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Active Safety . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Forward Collision Mitigation . . . . . . . . . . . . . . .
2.2.4 Ideal Collision Mitigation by Braking Performance . . .
2.2.5 Automotive Safety History . . . . . . . . . . . . . . . .
2.3 Accident Statistics . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Accident Types . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Effectiveness of Collision Mitigation by Braking Systems
2.3.3 Weather Conditions . . . . . . . . . . . . . . . . . . . .
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Tracking
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3 Automotive Tracking Sensors
3.1 The Radar Sensor . . . . . . . . . . . .
3.2 The Lidar Sensor . . . . . . . . . . . . .
3.3 Vision Systems for Obstacle Recognition
3.4 Infrared Vision Sensors . . . . . . . . . .
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Contents
3.5
3.6
Inter-Vehicle Communication . . . . . . . . . . . . . . . . . . . . . . 32
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Tracking for Collision Avoidance Applications
4.1 The Kalman Filter . . . . . . . . . . . . . . . . . . . . . .
4.1.1 The Stationary Kalman Filter . . . . . . . . . . . .
4.1.2 The Time-Varying Kalman Filter . . . . . . . . . .
4.1.3 The Extended Kalman Filter . . . . . . . . . . . .
4.2 The Point Mass Filter . . . . . . . . . . . . . . . . . . . .
4.3 The Particle Filter . . . . . . . . . . . . . . . . . . . . . .
4.4 Measurement Association . . . . . . . . . . . . . . . . . .
4.4.1 Gating . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Measurement Association Methods and Techniques
4.5 Ranking Measurements . . . . . . . . . . . . . . . . . . .
4.5.1 Ranking Measurements by Range Only . . . . . .
4.5.2 Ranking by Range Rate and Time-to-Collision . .
4.5.3 Ranking Objects by their Azimuth Angle . . . . .
4.6 Sensor Fusion . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Models for Tracking and Navigation . . . . . . . . . . . .
4.7.1 The Constant Velocity Model . . . . . . . . . . . .
4.7.2 The Constant Acceleration Model . . . . . . . . .
4.7.3 The Coordinated Turn Model . . . . . . . . . . . .
4.7.4 The Nearly Coordinated Turn Model . . . . . . . .
4.8 Navigation . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8.1 Bicycle Model . . . . . . . . . . . . . . . . . . . . .
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Decision Making
5 Deterministic Decision Making
5.1 Collision Avoidance Decision Strategy . . . . . . . . . . . . . . . . .
5.2 Static Analysis of Collision Mitigation . . . . . . . . . . . . . . . . .
5.3 Collision Avoidance Decision Functions . . . . . . . . . . . . . . . .
5.3.1 Headway and Headway Time . . . . . . . . . . . . . . . . . .
5.3.2 Time to Collision . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.3 Closest Point of Approach . . . . . . . . . . . . . . . . . . . .
5.3.4 Potential Force . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.5 Required Longitudinal Acceleration . . . . . . . . . . . . . .
5.3.6 Required Longitudinal Acceleration for Piecewise Constant
Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.7 Required Longitudinal Acceleration with Actuator Dynamics
5.3.8 Required Lateral Acceleration . . . . . . . . . . . . . . . . . .
5.3.9 Required Centripetal Acceleration . . . . . . . . . . . . . . .
5.3.10 Multi-Dimensional Acceleration to Avoid Collision . . . . . .
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Contents
vii
6 Multiple Obstacles Decision Making
6.1 Constant Control . . . . . . . . . . . . . . . . . . . .
6.1.1 Multiple Objects Constant Acceleration . . .
6.1.2 Summary of the Constant Control Approach
6.2 Heuristic Control . . . . . . . . . . . . . . . . . . . .
6.3 Exhaustive Search . . . . . . . . . . . . . . . . . . .
7 Statistical Decision Making
7.1 Numerical Approximation . . . . . . . . .
7.2 Analytical Approximation . . . . . . . . .
7.2.1 Estimating Statistical Properties of
7.3 Probability of Collision . . . . . . . . . . .
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the Decision Function
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Evaluation
8 Collision Mitigation by Braking System Testing
8.1 Simulation Environment . . . . . . . . . . . . . . . . . . . . . .
8.2 The Demonstrator Vehicles . . . . . . . . . . . . . . . . . . . .
8.2.1 Demonstrator Vehicle Tracking Sensors . . . . . . . . .
8.2.2 The Braking System of the First Demonstrator Vehicle
8.3 Demonstrator 1 — Probabilistic Decision Making . . . . . . . .
8.3.1 Collision Scenarios . . . . . . . . . . . . . . . . . . . . .
8.3.2 Scenarios to Provoke Faulty Interventions . . . . . . . .
8.3.3 Test Results . . . . . . . . . . . . . . . . . . . . . . . . .
8.4 Demonstrator 2 — Multiple Object Decision Making . . . . . .
8.4.1 Test Results . . . . . . . . . . . . . . . . . . . . . . . . .
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9 Supplementary Simulation Studies
9.1 Comparison of Multiple Object Algorithms . . . . . . . . . . . .
9.1.1 Test Method and Specifications . . . . . . . . . . . . . . .
9.1.2 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Statistical Decision Making by Stochastic Numerical Integration
9.2.1 Tracking Model . . . . . . . . . . . . . . . . . . . . . . . .
9.2.2 Estimation Approach . . . . . . . . . . . . . . . . . . . .
9.2.3 State Noise Model . . . . . . . . . . . . . . . . . . . . . .
9.2.4 Measurement Noise Model . . . . . . . . . . . . . . . . . .
9.2.5 Extended Kalman Filter . . . . . . . . . . . . . . . . . . .
9.2.6 Simulations and Tests . . . . . . . . . . . . . . . . . . . .
9.2.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . .
9.2.8 Hardware and Real-Time Issues . . . . . . . . . . . . . . .
9.2.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 Statistical Decision Making Using a Gaussian Approximation . .
9.3.1 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . .
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viii
Contents
10 Summary
V
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Appendices
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A Coordinate Systems
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B Notation
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C Safety Systems Introduced in Volvo Cars
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Bibliography
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Part I
Introduction
1
Chapter 1
Preliminaries
Safe and collision-free travel is vital in today’s society. It is also an important issue
in many industrial processes. In aerospace and naval applications, radar based
support systems to avoid collisions have been used for several decades. Currently,
collision avoidance (CA) systems are starting to emerge in automotive applications. The challenge in designing a CA system is in balancing the effectiveness
of avoiding collisions versus the risk of false alarms. Automotive applications in
particular present several challenges; dense traffic causing complex scenarios with
many moving objects; low-cost sensors and computational units have to be used.
Furthermore, the dynamic capabilities of a vehicle may change rapidly, e.g. tire-toroad friction may change significantly from one moment to the next.
This thesis discusses general theory for CA decision making and its application
in automotive systems. The main focus is dealing with uncertainties in the decision
making process and how to handle complex multiple obstacle scenarios. Specifically, a framework for dealing with uncertainty is introduced. In this framework
stochastic numerical integration is used to evaluate the confidence of each decision. Furthermore, algorithms for decision making in multiple obstacle scenarios
are presented. The proposed algorithms use different strategies to search the set
of feasible avoidance maneuvers, to find an escape path if it exists. Some novel
collision avoidance decision functions are also introduced. These functions address
different issues such as brake system characteristics, finding the optimal avoidance maneuver for a constant acceleration motion model, and changing obstacle
dynamics when the obstacle comes to a stop.
1.1
Background
The work on this thesis has been carried out in cooperation between Volvo Car
Corporation and the Control and Communication group at Linköping university.
A driving application for the project has been an emergency braking system that
tries to reduce the collision speed by autonomous braking. Thus, most of the
3
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Chapter 1
Preliminaries
examples in this thesis come from automotive applications.
The initial research question was to find methods and algorithms for tracking
and decision making for automotive collision mitigation systems that ensure a positive and noticeable impact on accident statistics, while minimizing the risk of false
alarms.
1.2
Thesis Outline
This thesis presents theory for tracking and decision making in collision avoidance
systems. The main application example is from automotive collision avoidance.
Part I of the thesis contains an introduction and gives an overview of different
collision avoidance applications and automotive safety. The remaining thesis is
divided into three parts.
In Part II existing tracking and positioning theory is reviewed. In particular
a Bayesian approach to tracking is emphasized. In the Bayesian approach each
parameter of interest is considered to be a stochastic variable. The goal is to find
the entire probability density function for each variable. Furthermore, an overview
of the most common tracking sensors is presented. Typical performance of sensors
used in automotive applications is also discussed.
In Part III theory for collision avoidance decision making is discussed: most
of the research contribution is in this part. Several novel collision risk metrics
are introduced and analyzed in Section 5.3. Algorithms for multiple obstacles are
discussed in Chapter 6. In Chapter 7 a framework based on statistical decision
making is introduced.
In Part IV evaluation of several automotive emergency braking systems is performed, using theory from Part I and II. Tests with two different demonstrator
vehicles and simulation studies are presented. In the first demonstrator vehicle,
probabilistic decision making and a sensor fusion concept is tested. The second
demonstrator vehicle was used to test multiple object decision making. The simulation studies evaluate different approaches to calculating the probability of a
collision and an extensive comparison of multiple object CA algorithms.
1.3
Contributions
The main contributions of this thesis are:
• A framework for collision avoidance decision making based on statistical decision making, in Chapter 7.
• Three novel algorithms for multiple object collision avoidance decision making, in Chapter 6.
• Results of computational complexity bounds of a tree-search algorithm for
multiple object collision avoidance decision making, in Chapter 6.
Section 1.4
Publications
5
• A comparison of two alternative methods for calculating the acceleration
required to avoid a collision in two-dimensional motion, in Section 5.3.
• Novel alternative expressions to calculate the longitudinal acceleration required to avoid a collision, in Section 5.3.
• Results from field tests with two demonstrator vehicles, Chapter 8.
• A comparison of three alternative multiple object algorithms in a comprehensive simulation study, in Section 9.1.
• A real-time implementation of a probabilistic CA tracking and decision algorithm based on particle filtering methods and stochastic numerical integration, in Section 9.2.
1.4
Publications
This sections presents the publications, patents, and patent applications written
or co-authored by the author, relating to this thesis.
Publications:
F. Gustafsson, F. Gunnarsson, N. Bergman, U. Forssell, J. Jansson,
R. Karlsson, and P-J Nordlund. Particle filters for positioning, navigation and tracking. IEEE Transactions on Signal Processing, 50:425–437,
February 2002.
J. Jansson, J. Ekmark, and F. Gustafsson. Decision making for collision
avoidance systems. In SAE technical paper 2002-01-0403, volume SP1662, Detroit, MI, USA, March 2002. Society of Automotive Engineers
2002 World Congress.
J. Jansson and F. Gustafsson. A probabilistic approach to collision risk
estimation for passenger vehicles. In Proceedings of the 15th International Federation of Automatic Control World Congress, pages T–Th–
M05:Model–based fault diagnosis of automotive systems/Area code 8b :
Automotive Control, Barcelona, Spain, July 2002.
J. Jansson. Tracking and Decision Making for Automotive Collision
Avoidance. Linköping Studies in Science and Technology. Licentiate
Thesis No. 965, Linköping University, Linköping, Sweden, September
2002.
R. Karlsson, J. Jansson, and F. Gustafsson. Model-based statistical
tracking and decision making for collision avoidance application. In
Proceedings of the American Control Conference, Boston, MA, USA,
July 2004.
6
Chapter 1
Preliminaries
J. Jansson. Advanced Microsystems for Automotive Applications 2004,
chapter Dealing with Uncertainty in Automotive Collision Avoidance,
pages 165–180. Springer, Berlin, Germany, March 2004. ISBN 3-54020586-1.
J. Jansson and F. Gustafsson. Multiple object collision avoidance decision making. Submitted to International Federation of Automatic Control Journal: Control engineering practice, 2005.
J. Jansson and F. Gustafsson. A framework and overview of collision
avoidance decision making. Submitted to International Federation of
Automatic Control Journal: Automatica, 2005.
Patents:
J. Jansson, J. Bond, J. Engelman, J. Johansson, N. Tarabishy, and
L. Tellis. Collision mitigation by braking system. U.S. Patent No.
6607255, 2001.
J. Jansson, J. Bond, J. Engelman, J. Johansson, N. Tarabishy, and
L. Tellis. Accelerator actuated emergency braking system. U.S. Patent
No. 6655749, 2001.
J. Jansson, J. Bond, J. Engelman, J. Johansson, N. Tarabishy, and
L. Tellis. System for exerting an amplified braking force. U.S. Patent
No. 6659572, 2001.
J. Jansson, J. Bond, J. Engelman, J. Johansson, N. Tarabishy, and
L. Tellis. Radar-guided collision mitigation system with a passive safety
feature. U.S. Patent No. 6617172, 2001.
J. Jansson and J. Johansson. A probabilistic framework for decision
making in collision avoidance. Swedish/German/UK Patent No. 273930,
2001.
Patent applications:
J. Jansson and J. Ekmark. A method and a computer readable storage
device for estimating tire-to-road friction. U.S. Application No. 10/
858896, patent pending, 2004.
J. Jansson and W. Birk. An alternative brake criterion for collision
avoidance. Submitted to the European patent authority, patent pending, 2005.
J. Jansson and J. Ekmark. A multiple object collision avoidance algorithm. Submitted to the European patent authority, patent pending,
2005.
Chapter 2
Collision Avoidance Overview
Avoiding collisions is a crucial issue in most transportation systems as well as many
other applications. Detecting and avoiding a possible collision have been studied
for several different fields of application such as air traffic control (ATC), automotive collision avoidance, robot manipulator control etc. The task of any collision
avoidance system is ultimately to avoid two or more objects from colliding. However, in this thesis the notion of a collision avoidance system will be extended to
also include systems trying to reduce the consequence of an imminent collision i.e.,
collision mitigation (CM) systems. CA systems avoid collisions either by performing an autonomous avoidance maneuver or by issuing a warning to an operator. To
mitigate collision consequences other actions might be taken, e.g. in an automotive
application this might be to pretension the seat-belts and inflate the airbags; in
a fighter aircraft one could consider ejecting the pilot if a collision is unavoidable.
Any action performed by a CA system will be called an intervention. Depending
on application and the type of intervention considered, the metric for measuring
collision threat and the decision making algorithm might vary significantly. This
chapter presents an overview of common collision avoidance applications, it also
discuses passenger car safety, which is the main motivation for the work in this
thesis. The last part of this chapter is devoted to accident statistic and a brief
analysis of the potential benefit of a specific automatic emergency braking system.
2.1
Collision Avoidance Applications
CA systems are being used in a wide range of different areas and under very
different circumstances. Typical sensors used to detect obstacles are radar, lidar
or vision sensors. This section presents an overview of some of the important areas
of application and highlights important issues.
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Chapter 2
Collision Avoidance Overview
2.1.1 Automotive Collision Avoidance
Traffic accidents are one of the major causes of death and injuries in todays society.
Automotive manufactures have started to introduce more and more driver support
systems to help prevent accidents. The first step in CA systems for automotive
applications is adaptive cruise control (ACC), which is currently available as an
option for several car models. ACC systems adapt the speed to any in-path vehicle,
should it travel slower than the set speed of the host vehicle. The cruise control
system is only allowed to exert limited deceleration (typically −3 m/s2 ); some
systems also issue a warning to the driver when this acceleration is not sufficient
to avoid collision. Current ACC systems are sold as a comfort system and can
be switched on and off by the driver; they also disengage at low speeds (below
40 km/h). The next step in automotive CA is to introduce systems that are always
active and perform autonomous braking and/or give warning when a collision is
imminent. Such systems are starting to emerge on the market, and are described
in [10, 20, 61, 77, 85, 119]. A big challenge in automotive collision avoidance is that
even in normal driving, the traffic situation might be very complex from a sensing
view point, with numerous obstacles to be detected and classified. In addition
to this, sensors and computational resources must be low-cost. Another issue is
that often the tire-to-road friction is unknown and might change rapidly. A more
thorough introduction to automotive CA is given in Section 2.2.
2.1.2 Aerospace Applications
Radar-based air traffic control (ATC) systems have been in use for several decades.
Traffic alert and collision avoidance system (TCAS) has been used on board US
transport aircraft since the beginning of 1990s. These systems typically aim at
helping pilots and air traffic controllers in keeping a regulated minimum separation
between any two aircraft. A breach of this distance is referred to as a conflict,
[125]. In [81] an extensive review of methods for collision avoidance in ATC systems is given. This area becomes ever more interesting as the airspace surrounding
large airports becomes more crowded. There exist different types of systems both
in cockpit and on the ground to provide decision-support to pilots and air traffic
controllers. More references to ATC applications can be found [80, 104, 109, 132].
Other applications of airborne collision avoidance is mid-air collision avoidance for
fighter aircrafts. Here, the CA system also attempt to maintain a regulated minimum separation between aircrafts. Apart from issuing a warning the system may
also be authorized to take over steering controls to perform an evasive maneuver
to avoid collision. Such a system is described in [116].
2.1.3 Marine Applications
In marine applications, radar systems are commonly used to detect other vessels
in the vicinity. For larger ships automatic radar plotting aid (ARPA) [17] is widely
used. These systems provide valuable help in adverse weather conditions, when
other vessels might go unnoticed, and also at large ports with high vessel density.
Section 2.2
Automotive Safety
9
Large ships normally have very limited maneuvering capabilities. Therefore it is
crucial that a possible collision is detected early so that an avoidance maneuver
can be initiated before the collision becomes unavoidable [40]. As with ATC, CA
is a growing issue as the waters close to large ports are becoming more crowded.
2.1.4 Industrial Robot Manipulators
Industrial robotic control is also a discipline where CA plays an important role.
These applications differ from the ones discussed previously in that normally no
human operator is involved. Instead, one is interested in finding a control law
which guarantees reaching a goal state without colliding. Thus, the CA system is
an integral part of a control system that has other mission objectives. Common
approaches to this problem are potential field algorithms, which was introduced in
[76] and path planning methods [74, 84, 117]. Manipulator arms are often described
by a kinematic chain and can be quite complex. This often results in quite complex
kinematic models. Hence, one often talks about configuration space or joint space of
the robot. Further references that deal with nonholonomic kinematics, uncertainty
and multiple moving obstacles are given in [34, 95, 96, 129].
2.1.5 Unmanned Autonomous Vehicles
There are numerous examples of applications with unmanned autonomous vehicles.
Similarly to robot manipulator applications, one is here normally not interested in
a system that takes over or warns a human operator. Instead, one looks for a
controller that tries to reach the goal in an effective manner and at the same time
avoid collisions with any obstacle. Often similar CA algorithms that are applied
for robot manipulators can also be used for autonomous vehicles. Apart from CA,
for autonomous vehicles, simultaneous localization and map-building is often also
an issue [21, 29, 30]. For detailed discussions of autonomous vehicles see [18].
2.2
Automotive Safety
Passenger car safety is an issue that has received increasing attention over the
past few decades. Over the last decades, automobile crashworthiness has improved
drastically. The risk for a fatal injury in a new car has decreased by 90% compared
to a car from the early 80’s, according to [32]. Other evidence of improvements
in crash performance can be found from the results of crash tests carried out by
organizations all over the world (see Section 2.2.1). Despite this progress, many
people are still killed or injured in traffic accidents. In the USA, there were 37409
accidents reported in FARS (a database recording all fatal car crashes reported in
the USA [28]) for year 2000. In the reported accidents, a total of 41821 people were
killed. The number of non-fatal crashes and non-fatal injuries the same year is of
course much greater. Despite the improved crashworthiness of modern cars it is
clear that there is still a need to significantly reduce traffic related injuries. Having
evolved rapidly, existing passive safety technologies have become quite mature.
10
Chapter 2
Collision Avoidance Overview
Figure 2.1: Example of a CM system. The system uses radar to measure distance and
speed relative to the vehicle ahead. If there is danger of colliding, the system warns the
driver by a flashing red light projected on the windshield.
Therefore, a significant reduction of injuries by passive safety technologies becomes
technically infeasible as well as too costly.
In recent years, people in the automotive industry have increased their efforts
in trying to find other ways of reducing the number of people injured in traffic.
One alternative way to avoid dangers is to try to prevent accidents from happening
by means of driver support systems. One of the first such systems was the anti
blockier system (ABS) braking system (see [3] for a description), which helps the
driver maintain steerability during hard braking maneuvers. One active safety
system which have proved very effective in reducing the risk for a injury is the yaw
control which will be described below. In [32] this system is claimed to reduce the
risk for a injury by up to 50% depending on the road condition.
In PartIV of this thesis prototype vehicles hosting a collision mitigation systems
will be studied.
The CM system uses sensors to observe the environment directly in front of the
host vehicle. Based on information from the forward-looking sensors, a decision can
be made to deploy countermeasures to avoid imminent frontal collisions. Typical
interventions are warning signals (audible, visible or haptic/tactile), braking and
steering. Figure 2.1 shows an example of a CM system that warns the driver of an
imminent threat by a visible display. When vehicle safety is discussed many people
normally think of what will be referred to as passive safety. However, in modern
cars, several other aspects have to be considered. Systems such as ABS brakes, yaw
control etc. are introduced to enhance driver and passenger safety. These systems
are sometimes called active safety systems. Conversely, a car with good handling
Section 2.2
Automotive Safety
11
capability is sometimes also said to have good active safety. In Sections 2.2.1 and
2.2.2 an account of different aspects of vehicle safety is given.
In broad terms the safety properties of a car can be divided into three groups:
• Security
• Passive safety
• Active safety
Security deals with the car’s capacity to protect against theft and vandalism and
possibly also attacks on the owner. This will not be discussed further in this thesis.
The definition of passive and active safety will be discussed in more detail in the
following sections.
2.2.1 Passive Safety
When a vehicle’s passive safety properties are discussed, this refer to the ability
of the car to protect passengers from injuries in a collision. The ability of the
car to protect its passengers is sometimes called crashworthiness. In this thesis a
car’s crashworthiness or passive safety properties is defined as its ability to protect
passengers from injury from the instant that the collision occurs and afterwards.
One way to achieve good passive safety is to optimize the structure of the car’s
body in such a way that it absorbs crash energy whilst keeping the crash pulse
(momentaneous acceleration) experienced by the passengers as low as possible.
Some systems that help to protect the passengers in a collision are seat belts,
airbags, belt pretensioners, and belt load limiters. The interior of the car should
also be designed in such a way that panels, switches, etc. cause minimal injury.
To achieve good passive safety properties, quite a few areas of the car’s design
must be considered. The crashworthiness of automobiles is tested in collision tests.
Organizations that perform such tests are EuroNCAP [101], ANCAP [100], JNCAP
[68], and NHTSA [102]). In the tests performed by these organizations, passive
safety properties are measured in terms of acceleration and forces experienced by
a crash test dummy in specific collision scenarios. The crashworthiness of cars is
also evaluated by insurance institutes (for example, Folksam [31] and IIHS [33])
which rate the performance for different vehicles in real-life accidents.
2.2.2 Active Safety
Active safety also comprises several different aspects of the car. Here, they are
divided into three groups: preventive, dynamic and collision mitigation. A brief
account of these groups is given below:
1. Preventive — The preventive aspect of active safety is about the driver seeing threats and being seen himself. If threats are detected early the driver can
take precautionary action before the situation becomes dangerous. Several
factors play a role in this:
12
Chapter 2
Collision Avoidance Overview
One safety technology, aimed to improve the drivers perception, presented in
the Volvo safety concept car was a see-through A-pillar.
Figure 2.2:
• Detecting threats: To improve the driver’s perceptive ability the traditional approach is through improving headlight performance, headlight
cleaning, windshield cleaning, minimizing glare in the windows etc. Examples of new innovations that improve driver perception are night vision systems and see-through A-pillars. A night vision system uses an
IR camera and a driver display unit to present the IR image to the
driver. This enables the driver to detect warm objects further away
than is possible using conventional headlights. In [82, 90, 118] night vision systems for automotive applications are discussed. The see-through
A pillar, which was presented in the Volvo Safety Concept Car (SCC),
improves the driver’s perception by allowing him or her to see through
the A-pillar, see Figure 2.2.
• Visibility: Visibility of the driver’s own vehicle is another important
issue in preventive active safety. In short this means that the car should
be designed in such a way that it is easily observed by other road-users.
Examples of features to improve visibility are daytime running lights
and high-positioned brake lights.
• Driver information: Driver information also plays an important role in
the driver’s ability to detect dangers. Through GPS technology and
digital maps, the driver can be provided with geographically-dependent
information. Examples of other systems that could provide the driver
with essential information to avoid collisions are, road friction monitoring systems [47] and tire pressure monitoring systems [106, 107]. It is
important to remember that the driver’s cognitive ability is limited, and
Section 2.2
Automotive Safety
13
information overload can cause the driver to fail to detect, or react erroneously to, imminent threats. Today, there are systems that prioritize
what information to display to the driver depending upon the situation,
e.g. the intelligent driver information system (IDIS) introduced on the
Volvo S40 and V50.
• The driver: To ensure that the driver’s perception of the traffic environment is adequate, it might be sensible to monitor the driver himself. A
driver monitoring system can, for instance, be used to warn the driver
if he falls asleep behind the steering wheel.
2. Dynamic — From a safety perspective, a car should be designed to have safe
handling and ride characteristics. By this we mean that it should be easy for
the driver to keep control of the vehicle in all road conditions, in any traffic
situation and during all types of maneuvers. Examples:
• It should be easy to perform avoidance maneuvers without losing control
of the vehicle.
• The car should be minimally influenced by side wind.
In short, it should be easy for the driver to make the car follow his intended
path. To achieve these properties, quite a few areas of the vehicle’s design
come into play. Important factors are steering and brake system characteristics. There are already several active systems in existence today that improve
a vehicle’s dynamic properties. Here “active systems” refers to the use of sensors and actuators to perform some kind of closed loop control to help the
driver in controlling the vehicle. Some common systems are listed below:
• ABS brake systems prevent the wheels from locking during hard braking.
This enables the driver to achieve fair deceleration whilst maintaining
steerability of the vehicle.
• Yaw control systems monitor steering angle, yaw rate and lateral acceleration. Brake force is applied to individual wheels to aid the driver
in keeping control of the vehicle in situations where the lateral traction
limit is reached.
• Traction control systems help maintain traction and stability by reducing engine torque and applying brakes if the wheels are spinning.
• Roll stability systems apply individual brakes to decrease the risk of
rolling over when the car experiences high roll angle rates and roll angles.
3. Collision Mitigation — CM systems are given some perception of the environment surrounding the vehicle. Based on this perception the system takes
steps to avoid or mitigate imminent collisions. The main scope of this thesis
falls in this field. It must be pointed out that these systems are often called
Collision Avoidance systems. The term “avoidance” might imply that accidents should be completely avoided by these systems. For several reasons,
14
Chapter 2
Collision Avoidance Overview
systems that avoid all collisions are unattainable for automotive applications.
This thesis will therefore talk about CM systems. Generally, a CM system
will try to reduce the severity of the accident as much as possible under some
constraints. In the best case, accidents might be avoided altogether whilst
in the worst cases the systems have no positive effect at all. The perception
of a CM system can come from several sources. The environment may be
perceived with radar sensors, laser radar, vision sensors, ultrasonic sensors,
GPS sensors and inter-vehicle communication. Properties of individual sensors will be discussed in Chapter 3. Based on the information acquired from
the sensors, the vehicle itself acts to prevent or mitigate collisions. Typical
actions the systems can take to mitigate a collision are issuing a warning to
the driver, applying the brakes, and changing the course of the vehicle by applying torque to the steering wheel. Other possible countermeasures might be
activating the brake lights to avoid being hit from behind, early airbag inflation or adjusting the vehicle’s height to increase crash compatibility. Several
systems exist or have been proposed with CM functionality:
• Lane-keeping aid systems [75, 89] monitor the lane markings. Using the
observations of the lane markings an estimate of the vehicle’s position
in the lane can be obtained. Should the vehicle swerve out of the lane a
warning can be issued or a steering intervention executed.
• Lane change aid systems monitor the blind spot and some distance behind the car. The system can then warn or intervene by adding steering
wheel torque to avoid a collision when switching lane.
• Forward collision mitigation systems [10, 20, 26, 119] monitor what is in
front of the host vehicle and intervene to prevent or mitigate a frontal
collision. A more detailed introduction to CM systems is given in Section
2.2.3.
• Adaptive cruise control works like normal cruise control, but adapts the
speed to the vehicle in front; if the driver is closing in on a vehicle in
the same lane. ACC is really a CM system. The reason that it is dealt
with separately here is that it is marketed as a comfort system that
is switched on and off by the driver examples of ACC system on the
market are: Toyota’s Radar cruise control which is based on a lidar
and was introduced in 1997; Mercede’s Distronic which is based on a
mm-radar and was introduced in 1998; Jaguar’s Adaptive cruise control
which is based on a mm-radar and was introduced in 1999, and BMW’s
Active Cruise Control which is based on a mm-radar and was introduced
in 2002 ). ACC systems are described in [130].
2.2.3 Forward Collision Mitigation
Forward looking CM systems mainly try to avoid or mitigate frontal collisions. The
countermeasure that is discussed mainly in the application examples of this thesis is
braking, and to some extent warning. Hereafter, forward collision warning systems
Section 2.2
Automotive Safety
15
will be referred to as FCW systems and forward collision mitigation by braking
systems as CMbB systems. Although FCW and CMbB systems are similar in
many ways, they also present different challenges. A FCW system, must rely on
the driver to react quickly and correctly to the warning. However, this might not
always be the case. A warning might actually startle the driver and cause his
performance to degrade in a critical situation. A great deal of work is going on to
determine how to warn the driver in the best way (e.g. see [131]). The timing in
FCW and CMbB systems is also quite different. In a FCW system, driver reaction
time must be taken into account. In [85] it is stated that 85 % of all drivers
are able to react to a warning within 1.18 seconds. Apart from reaction time, one
must also take into account that the driver might not perform the optimal avoidance
maneuver. Taking this into consideration means that a FCW system has to predict
future states several seconds. For CMbB systems we assume that braking will be
initiated at the last second, i.e., when a collision is becoming unavoidable or very
close to this point. The reason for this is that a driver may in some cases drive in
such a way that a collision is “spatially close” without the situation being critical.
A FCW system therefore presents the problem of making correct predictions over a
longer time horizon. In a CMbB system on the other hand the tolerance for faulty
decisions is lower. In a FCW system some faulty interventions are tolerated, whilst
in CMbB systems practically no faulty interventions are allowed.
2.2.4 Ideal Collision Mitigation by Braking Performance
In this section a study of what performance to expect from the best possible CMbB
system, for a head-on collision scenario, using a detailed handling model for the host
vehicle will be made. The assumption here is that it is possible to determine exactly
when a collision becomes unavoidable and immediately commence with automatic
braking. The distance needed to avoid collision (by means of steering or braking) for
the head-on collision scenario in is plotted in Figure 5.4. Assuming that there exists
no better maneuver than pure steering or braking, the distance at which a collision
becomes unavoidable is given by the minimum of the plotted braking distance and
the steering avoidance distance in Figure 5.4. The steering avoidance distance and
braking distance intersects at approximately 37 km/h, indicating that, to avoid
a collision, braking is more efficient at low speeds and steering is more efficient
at high speeds. To generate the escape path, a 15 degree-of-freedom handling
model that has been verified against measured data from a Volvo S80 was used.
The brake and steering system has a delay of 0.2 second and the brake system
rise time is 0.3 seconds. From Figure 2.3 it can be noted that for low speed the
dynamics is no longer important to determine the steering avoidance distance.
Here, instead, the turning radius of the vehicle limits the possibility to avoid a
object by means of steering. If it would be possible to start braking at the distance
where a collision becomes unavoidable (by pure braking or steering) the collision
speed would be as plotted in Figure 2.4. Thus, at relative speeds below 37 km/h,
ideally, a collision could be completely avoided. At speeds above 37 km/h the speed
should be reduced by 22 km/h. However, it should be stated that the maneuver
16
Chapter 2
Collision Avoidance Overview
Distance neede to avoid collision by braking and steering
120
brakedist
steerdist
100
Distance [m]
80
60
40
20
0
0
50
100
150
Speed [km/h]
Figure 2.3: Distances required to avoid a stationary object collision by braking and by
steering. This plot shows the result from simulation with a 15 degree-of-freedom handling
model. Both the object and the host vehicle have a width of 2 meters. The steering
maneuver is optimized for each velocity so that the lateral force is maximized.
that has been simulated here is a simple one where the driver quickly turns the
steering wheel (at 700 ◦ /second) either to the left or the right. The maneuver
was optimized in the sense that the steering wheel is turned to the angle which
optimizes lateral force for each velocity. It has not been studied if this actually
is the most efficient avoidance maneuver (by steering). Furthermore, avoidance
at even shorter distances might be possible by a combined steering and braking
maneuver.
2.2.5 Automotive Safety History
Ever since automobiles were introduced to the mass market in the beginning of
the century, safety has been an issue. During the first half of the decade, the
focus was mainly on energy-absorbing structures, reinforced steel passenger cages
and visibility issues. In 1959 Volvo introduced the first three-point seat belt. In
the 1980s the first driver support system was introduced on a large scale. This
was the well-known ABS brake system. In the 1990s more driver support systems
became common. Examples of such systems are traction control systems, yawcontrol systems etc. The first ACC system was introduced in 1999 by Mercedes. In
2002 Nissan introduced the first lane-keeping assist system and also the first CMbB
system. Another major trend during the last two decades is the increased use of
airbag systems. This is one of the major reasons for the improved crashworthiness
of modern cars. In Appendix C a list of different safety designs introduced in Volvo
Section 2.3
17
Accident Statistics
Collision Speed
Collision Speed [km/h]
150
100
50
With CMbB
No CMbB
0
0
50
100
Initial Speed [km/h]
150
Figure 2.4: Speed at which the CMbB equipped vehicle would hit the stationary obstacle
for different initial speeds. The distance at which braking starts is given by the minimum
distance for avoidance in Figure 2.3. The width of both the obstacle and the vehicle is
assumed to be 2 m. For high closing velocities the speed reduction is 22 km/h.
cars is given. However, Volvo was not first with all of these designs.
2.3
Accident Statistics
To investigate the possible effects of forward looking CM systems, some accident
statistics will be examined. A thorough investigation of potential effects of FCW
and CMbB systems with respect to accident statistics is presented in [133], other
works discussing the potential effects of forward looking CM systems are [19, 77].
In this section data from three American databases is used. The general estimates
system (GES), [36], is a sample database of all types of accidents. The national
automotive sampling system NASS, [99], database is composed of two data-bases:
the crashworthiness data system (CDS), [22], and the GES. The fatality analysis reporting system (FARS) database, [28], records all car accidents with fatal
outcome.
2.3.1 Accident Types
Given that the field of view (FOV) of CM systems is often limited, it is clear
that not all accident types can be mitigated. Typical accidents where forward
looking CM systems are ineffective are those where the principal other vehicle
(POV) approaches the host vehicle from the side, so that the POV appears in the
18
Chapter 2
5%
4%
11%
Collision Avoidance Overview
Other
Single vehicle hitting nonfixed object
Single vehicle hitting fixed object
Sideswipe opposite direction
Rear end
Head on
Angle
Sideswipe same direction
35%
16%
1%
2%
26%
The diagram show how common different types of accident are. The data is
from the GES database year 1998.
Figure 2.5:
sensors’ FOV late or not at all. Figure 2.5 shows how common different types of
accidents were in the USA in the year 1998. The statistics presented in Figure
2.5 come from the GES database. Categories which are very likely to be affected
by a forward looking CM system are rear-end (26 %) and single vehicle hit fixed
(16 %) and non-fixed object (11 %) collisions. The reason for this is that in many
of these accidents the struck object has been in the sensor’s FOV for a long time
(several seconds). The statistics indicate that potentially more than 50 % of all
accidents might be affected by a forward looking CM system (assuming that all
rear-end and single vehicle accidents can be affected). A very important aspect
when considering the effect of an active safety function is to know what the driver
did prior to the accident. In [126] this is investigated for rear-end collisions. Rearend collisions were divided into five groups. It is reported that in more than 68 %
of the accidents the driver makes no avoidance maneuver (braking or steering) for
all the five groups.
2.3.2 Eectiveness of Collision Mitigation by Braking Systems
It is clear that there is a connection between collision speed and the severity of
passenger injury. The kinetic energy of a vehicle is given by
Ekin =
mv 2
,
2
(2.1)
Section 2.3
19
Accident Statistics
Fatality risk
1
Joksch,1993
O‘Day and Flora, 1982
0.9
0.8
p(F |∆V )
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
Figure 2.6:
20
40
60
80
Change in speed, ∆V, [km/h]
100
120
Fatality risk versus collision-induced speed change.
where m is the mass of the vehicle and v is the speed. In a (completely non-elastic)
collision with a fixed barrier, a CMbB halving the collision speed, thus, cause a
decrease in the collision energy by a factor of 4. Clearly, reducing the collision
speed has a large potential in injury reduction. Accident research has found that
the collision induced speed change (∆V ) and the probability of being injured are
strongly correlated. In [69] and [105] the relationship between fatality risk and ∆V
is investigated. A good approximation according to [69] and [105] is to describe
the relationship between the probability of fatality for a given ∆V , p(F |∆V ), up
to some maximum speed, according to
p(F |∆V ) ∝ (∆V )4 ,
(2.2)
which is illustrated in Figure 2.6. In real-life accidents the speed change of a
vehicle in a collision between two vehicles can be calculated by the equation for
conservation of momentum
m1 v0,1 + m2 v0,2 = m1 v1 + m2 v2 ,
(2.3)
where v0,i and vi denote the velocity before and after the collision respectively.
The speed change for vehicle i is given by
∆V1 = v0,i − vi .
(2.4)
Example 2.1
Consider a collision where one car hits another stationary car; assume that both
vehicles have equal mass. If the collision is completely non-elastic we can calculate
20
Chapter 2
Collision Avoidance Overview
∆V for the two vehicles using (2.3) and (2.4). In this case ∆V for both the vehicles
will be
v0,1
.
(2.5)
∆V =
2
The effectiveness of CMbB systems with respect to reducing the collision speed is
one of the main issues of this thesis. In this section a simple assumption that the
probability density function (PDF) of the collision speed reduction, p(vr |CM bB),
from the CMbB system, has a uniform distribution,
p(vr |CM bB) ∈ U(−15, 0),
(2.6)
where vr denotes the CMbB system induced speed reduction. Hence, the system
can reduce the collision speed by up to 15 km/h, with an average expected speed
reduction of 7.5 km/h.
To study the effectiveness of the CMbB system the probability of sustaining
injury in frontal collision will be studied. Injury severity is often quantified using
the abbreviated injury scale (AIS), which range from 1 (minor) to 6 (unsurvivable),
sometimes 0 is used to indicate no injury. An AIS2 injury is a moderate injury
and AIS2+ denote injuries which are classified as AIS2 or higher. In Figure 2.7 the
probability of sustaining a AIS2+ injury, p(AIS2+|ṽ), in relation to the relative
speed between the vehicles ṽ is illustrated. The data is from the NASS database (year 1998-2000) and only includes frontal collisions of cars with similar size
and crash performance as Volvo cars. To the data a fourth order polynomial was
fitted. Note, that for high and low velocities the polynomial fit is poor, it is
only intended to be used from 5 km/h to 105 km/h. For higher velocities there
are relatively few accidents and the probability of an injury is relatively constant
(≈ 1). For low velocities the probability of a injury is low (≈ 0). Therefore,
these accidents are expected to have little effect on the analysis in this section.
The probability distribution of rear-end collisions, p(ṽ), with respect to relative
velocity ṽ, is illustrated in Figure 2.8. This data is from the GES data-base (year
1998). The effect of the CMbB system is also shown in Figure 2.8. The CMbB
system effect on the collision speed is calculated according to
∞
p(ṽ − vr )p(vr |CMbB) dvr ,
(2.7)
p(ṽ|CMbB) =
−∞
where p(r|CM bB) ∈ U(−15, 0) denotes the PDF for reducing ṽ. The joint probability density of sustaining AIS2+ injury, p(AIS2+, ṽ), in a rear-end collision can
be calculated according to
p(AIS2+, ṽ) = p(AIS2+|ṽ)p(ṽ).
(2.8)
The probability of an injury with and without the CMbB system according to (2.8)
is illustrated in Figure 2.9. The effect of the CMbB system is calculated according
Section 2.3
21
Accident Statistics
Probability of a AIS2+ injury
1
p(AIS2+|ṽ)
0.8
0.6
0.4
NASS data
Poly. fit
0.2
0
0
20
40
60
Relative velocity [km/h]
80
100
Figure 2.7: Probability of sustaining a AIS2+ injury versus relative velocity. The accident
data is from the NASS data-base. Only cars with similar size and collision performance
as Volvo cars are included. A fourth order polynomial was fitted to the data, using
least-squares minimization.
Distribution of rear−end collisions for different velocitys
GES data
Poly. fit
With CMbB
0.02
p(ṽ)
0.015
0.01
0.005
0
0
20
40
60
80
Relative velocity [km/h]
100
120
Figure 2.8: Distribution of rear-end collisions from GES 1998. Again a fourth order
polynomial was fitted to the data. The effect of the CMbB system is also shown. The
speed reduction of the CMbB system is assumed to be uniformly distributed according
to U(0, −15).
22
Chapter 2
Collision Avoidance Overview
Risk of AIS2+ at a certain relative velocity
No CMbB
With CMbB
p(AIS2+, ṽ)
0.02
0.015
0.01
0.005
0
0
20
40
60
Relative velocity [km/h]
80
100
Probability of a AIS2+ injury in a rear-end collision with respect to ∆V . The
CMbB system reduces the probability of a fatal rear-end collision by 8.5%̇. Note, that
for higher values the polynomial fit becomes negative. The probability should only be
considered up to this point. For higher values the number of accidents is relatively small
and the probability of injury is relatively constant (≈ 1).
Figure 2.9:
to
105
0
p(AIS2+, ṽ|CMbB) dṽ
.
105
p(AIS2+, ṽ) dṽ
0
(2.9)
The CMbB system reduces the probability of an AIS2+ injury by approximately
16 % for all rear-end collisions. Given that rear-end collisions constitute approximately 30 % of all accidents a CMbB system, with an average speed reduction of
7.5 km/h, have the potential of reducing the risk for AIS2+ injuries for all accidents
by approximately 5 %.
2.3.3 Weather Conditions
Different sensors are not affected in the same way by weather and light conditions.
Some sensors are more robust to bad weather conditions while other sensors might
be more sensitive. However, the sensors that do not cope well with bad weather
or poor light conditions, might have other advantages. When designing a CM
system it becomes interesting to examine in what conditions accidents occur. The
statistics in Table 2.1 and 2.2 come from the GES and the FARS year 1998 and
were presented in [133]. It is worth noting that most accidents occur in the day
time (68.6 %) and under no adverse weather conditions (82.7 %). It is also clear
that for fatal accidents a much larger share occurs during dark conditions.
Section 2.3
23
Accident Statistics
Table 2.1:
Accident frequency for different weather conditions.
Weather conditions
FARS [%]
GES [%]
No adverse weather condition
87.26
82.72
Rain
8.80
13.22
Sleet (hail)
0.39
0.24
Snow
1.26
1.71
Fog
1.50
0.42
Rain and fog
0.16
0.03
Sleet and fog
0.02
0.01
Other (smog, smoke, dust)
0.19
0.29
Unknown
0.44
1.35
Table 2.2:
Accident frequency for different light conditions.
Light conditions
FARS [%]
GES [%]
Day time
50.55
68.55
Dark
30.00
11.49
Dark but illuminated
14.92
11.76
Dawn
1.96
1.41
Dusk
2.30
2.29
Unknown
0.28
1.49
24
Chapter 2
Collision Avoidance Overview
Part II
Tracking
25
Chapter 3
Automotive Tracking Sensors
In order to track objects and predict collisions the movement of the host vehicle
and the movement of the surrounding objects need to be monitored. This chapter
will discuss sensors used to detect and observe other objects in the surrounding
environment of the host vehicle. These sensors are referred to as “tracking sensors”.
The applications in this thesis are mainly from automotive examples. Thus,
this chapter discusses automotive tracking sensors. The general description of each
sensor should be valid for any application, but the performance will be typical
for sensors that are commercially available for automotive applications. For automotive applications in particular, one very important requirement is the price of
the sensor. In general, there may exist sensors with better performance and more
capabilities than the ones discussed below.
3.1
The Radar Sensor
Radio detection and ranging (Radar), is traditionally the most common tracking
sensor. Radar has been used in military applications for several decades. Today
radar is common in for example aircraft and missile tracking, air traffic control,
and naval applications. In recent years, radar has been introduced in automotive
ACC systems. The most important measurements provided by the radar sensor
are range (r), range rate (ṙ), azimuth angle (ϕ), and elevation angle. In Figure
3.1, range and azimuth angle measurements are illustrated. The radar sensor is
an active sensor in the sense that it emits electro-magnetic radiation to illuminate
targets. Emitting and receiving radar waves is typically achieved by the same antenna, thus, the sensor is constantly switching between the sending and receiving
mode. The emitted energy has a particular waveform. By examining the echo of
the transmitted signal, information on the environment is gathered. Most long
range automotive radars use frequencies in the region 76 − 77 GHz. By their wavelength they are often called millimeter-wavelength radars (hereafter referred to as
mm-radars). Other automotive radar sensors operate in the 5 GHz and 24 GHz
27
28
Chapter 3
Automotive Tracking Sensors
Figure 3.1: Measurements provided by the radar sensor are range, range rate, and azimuth angle. Due to the relative wide main lobe of the emitted radar pulse, the azimuth
resolution is poor. The lobe width is determined by the size of the antenna.
region. In [114] the basic theory of radar systems is described. The shape of the
radiation pattern is determined by the antenna design. In automotive radars, the
antenna is designed so that the main lobe has a conical shape. The beam width
is normally 1 − 4◦ both in the azimuth and the elevation direction. To provide
azimuth angle information, the sensor either mechanically sweeps the antenna over
a range or electronically switches between different emission angles. The total field
of view (FOV) (see Figure 3.1 for an example) is normally 10 − 15◦ for available
ACC radar sensors. The techniques that are used to provide azimuth resolution
can of course also be used to provide elevation angle information. However, most
existing automotive radars do not provide elevation measurements (scanning is only
one-dimensional). The radar sensor has both advantages and disadvantages.
Advantages:
• Bad-weather performance: Frequencies at 30 − 300 GHz are generally lineof-sight and experience attenuation due to absorption in atmospheric gases
(0.3 − 0.5 dB/km [1]). The sensor does, however, have the ability to detect
objects in darkness, haze, rain and snow for the short distances required
for automotive applications (where the range is normally less than 200 m).
Automotive radar sensors are also insensitive to dirt deposits (i.e., mud and
dirt from the road).
• Range and range rate: Automotive radars provide accurate range measurements, in contrast to passive sensors. Some radars also measure the range
rate using the Doppler effect.
Disadvantages:
• Clutter and spurious reflections: A number of objects in the environment
reflect emitted radar signals. Unwanted reflections (often called clutter) from
the road surface for instance, might give “ghost” obstacles (i.e., obstacles
that do not exist). Multipath propagation might also cause such phenomena.
Section 3.2
29
The Lidar Sensor
• Resolution: Because of the relatively wide lobe, the spatial resolution of the
radar is poor. Its ability to measure spatial properties of the observed object
is therefore limited.
• Elevation: Since automotive radars have no resolution in elevation and a wide
beam, it might be hard to discriminate obstacles from low overhead objects
(road signs above the road, bridges, tunnels etc).
Most available sensors have an update frequency of 10 Hz, i.e., one sweep is completed in 100 ms. In Table 3.1 some basic sensor properties from different suppliers
are displayed. Range accuracy is often better than 1 m and angular resolution is
normally better than 1◦ (for radars with scanning antennas). In [1, 41, 113, 127]
mm-radars for automotive purposes are discussed. A thorough description of radar
sensors is given in [114].
Table 3.1: Properties of Radar Sensors. Note, that this table was compiled in 2002.
Automotive radar sensors are under rapid development.
Radar Supplier
Fujitsu Ten
Mitsubishi
Denso
Nec
Hitachi
A.D.C.
Bosch
Waveform
FM-CW
FM-Pulse
Delphi
>120 m
150 m
FOV
16
◦
12-16
◦
Mechanical 8 beams
◦
Mechanical 8 beams
?
150 m
20
120 m
16◦
Synchronized 9 beams
FSK
120 m
16◦
Monopulse
150 m
10
◦
150 m
◦
FM-CW
FSK
FM-CW
150 m
150 m
8
?
switched 3 beams
synchronized 3 beams
12
◦
Monopulse
16
◦
Mechanical
◦
Monopulse
Monopulse
Eaton
FSK
150 m
12
Visteon
FSK
150 m
12◦
3.2
Scanning
FM-CW
FM-CW
Autocruise
Range
The Lidar Sensor
A laser radar (lidar) is similar in its basic function to a mm-radar. Lidars provide
range, range rate, azimuth and elevation measurements. While most automotive
radar sensors provide no elevation measurements, there are some lidars that do.
The lidar sensor is also an active sensor, i.e., it uses a laser diode to illuminate obstacles. Lidars operate in the infrared frequency region. Wavelengths are around 850
nm (just above visible light). The wave propagation properties of lidars are similar
to visible light. Lidars often consist of a one dimensional (distance) measurement
device combined with a mechanical beam deflection system (e.g. a rotating mirror)
30
Chapter 3
Automotive Tracking Sensors
to provide spatial measurements. Distance is measured by observing time of flight
between received and transmitted signals. Lidar systems for automotive purposes
are discussed in [120], [113] and [112]. In [127] a comparison between mm-radar
and lidar for ACC purposes is presented. Compared to mm-radars, lidars have
advantages and disadvantages:
Advantages:
• Resolution: The beam of a lidar is often quite narrow. There are lidars that
measure at several hundred azimuth angles and for several angles of elevation
in one sweep (normally 100 ms). They thus provide a pixel map which could
potentially give much more detailed information of objects than a mm-radar.
• Clutter: Due to the narrow beam and detection, the lidar does not experience
clutter and spurious reflections to the same extent as a mm-radar. Note that
some lidars still have a wide angle beam and detector, and may also experience
multipath problems.
• Gray scale: The photo detector can easily detect reflected intensity, thus
providing a gray scale “picture”. This may be used to monitor lane markings.
• Light conditions: Lidars are relatively insensitive to light conditions.
• Cost: Lidars are less expensive compared to mm-radars.
Disadvantages:
• Bad weather: In hard rain, fog or snow, the lidars experience performance
degradation.
• Dirt deposits: Lidars are sensitive to dirt deposits on the lens. Dirt deposits
on the tracked vehicle may also cause altered reflectivity and thus problems
in detection.
3.3
Vision Systems for Obstacle Recognition
Vision systems use one or several cameras together with a microprocessor to perform image processing. Since they operate in the visible light region, their capabilities are similar to that of our own eyes. The two main types of system are:
• Single camera systems - using either a monochrome or a color camera. One
use in automotive applications for single camera systems is to monitor the
lane markings in lane-keeping aid systems, [75, 89].
• Stereo camera systems - A stereo camera system provides a 3D image by combining the images from two (or more) cameras. In such a system, range can
be measured through triangulation. Because of the 3D information obstacle
Section 3.3
31
Vision Systems for Obstacle Recognition
Table 3.2: Properties of Lidar Sensors. Note, that this table was compiled in 2002.
Automotive lidar sensors are under rapid development.
Radar Supplier
Mitsubishi
Denso
Denso (gen.II)
Nec
Omron
Omron (gen II)
Range [m]
Vertical range
FOV
scanning
130
4◦
12◦
1-D
16
◦
2-D
40
◦
2-D
20
◦
1-D
4.4
◦
120
4.4
◦
100
◦
120
150
150
3
3.3
◦
6.5
◦
◦
Kansei
120
3.5
A.D.C.
150
?
10.5
◦
20 − 30
?
◦
2-D
12
◦
1-D
17
◦
?
detection is easier. Generally, shape or pattern recognition is not needed to
the same extent as for a single camera system. Stereo vision systems for
automotive applications are discussed in [13, 51]
The performance of a vision system depends on among other thing the optics, the
size of the sensor, number of pixels, and the dynamic range. The update frequency
of many vision sensors is 25 Hz. However, image processing can be very computationally demanding. Therefore, the output from a vision sensor (after image
processing) is normally at a lower data rate. Vision system performance in automotive systems is discussed in [23], [14] and [50]. In the VITA II project [121]
extensive tests were performed using several vision systems to perform automated
driving.
Advantages:
• High resolution: A pixel array with 640 × 480 pixels is common. This resolution makes it possible to measure spatial properties of the obstacle with good
accuracy. Advanced target classification is also possible from the detailed
images.
Disadvantages:
• Sensitivity to light conditions: A vision system can suffer severe performance
degradation in certain light conditions (for example thick fog, heavy rain, or
wet roadway combined with backlight).
• Sensitivity to dirt: Dirt deposits on/in front of the lens can cause problems
to a vision system.
• Computational demands: The image processing algorithms are computationally intensive.
32
Chapter 3
Figure 3.2:
3.4
Automotive Tracking Sensors
Image displayed by a night vision system.
Infrared Vision Sensors
Infrared (IR), or thermal, cameras basically give the same information as any
normal camera. In many ways an IR vision system thus has the same properties as
normal vision systems discussed in the previous chapter. The difference is that the
IR camera is sensitive to wavelengths typical to heat radiation (the same region
that lidars normally operate in). In automotive systems, IR cameras have been
introduced for night vision systems. Night vision systems are used to enhance
the driver’s perceptive abilities (Figure 3.2). These sensors are most sensitive to
wavelengths corresponding to the normal body temperature of humans and large
animals. So far, IR cameras have not been used in automotive collision mitigation
(CM) systems. The ability to measure the temperature of objects is unique to the
IR sensor. IR cameras could potentially be quite useful in classifying objects and
providing accurate angle measurements. For example, on some cars the exhaust
system is visible and easily observed in a heat sensitive camera. Humans and
animals are also often easily detected in an IR-image. A disadvantage with the IR
sensor is that it costs more than a normal vision sensor for visible light.
3.5
Inter-Vehicle Communication
Today, vehicles are more frequently being equipped with navigation systems consisting of GPS receivers and digital maps. If cars were equipped with communica-
Section 3.6
Summary
33
tion systems it would be possible to transmit the navigation data to other vehicles.
This data could then be used by the tracking system of the other vehicle. Potentially, this could yield very accurate measurements of kinematic properties as
well as an accurate object classification (for example make, model and performance
specification can be transmitted to tracking vehicles).
3.6
Summary
The two main tasks for tracking sensors are:
1. To detect obstacles and provide measurements for the tracking system.
2. To provide data for object classification.
In a CM system, objects must be classified into at least one of the following groups:
• Obstacles which the system should react to, i.e., obstacles to be avoided.
• Obstacle which the system should ignore, i.e., false obstacles or obstacles that
are not harmful to collide with.
Today, commercial radars, lidars and mono-camera sensors for automotive use
exist. These sensors have different strengths and weaknesses. To be able to provide
both accurate measurements and correct object classification for a CM system, one
sensor might not be sufficient. This is due to the fact that a sensor might report
a target that does not really exist (a false positive). A sensor might also miss
detecting an object (false negative). By using more than one sensor the risk for
erroneous decision can be reduced. Furthermore, to be able to monitor the function
of the sensors themselves (i.e., diagnosis) it might also be necessary to include more
than one tracking sensor. How to merge data from several sensors will be discussed
further in Section 4.6.
34
Chapter 3
Automotive Tracking Sensors
Chapter 4
Tracking for Collision
Avoidance Applications
The task of a target tracking system is to detect objects and estimate kinematic
quantities of these objects. Estimating kinematic quantities (states) of an object
is often called filtering or state estimation. Example of quantities that are of interest to estimate in collision avoidance (CA) applications are position, speed,
acceleration and turn rates. For automotive and naval applications it is often sufficient to look at motion in the ground plane (i.e., motion in two dimensions only).
For aerospace, robotics and underwater application three-dimensional motion often needs to be considered. The coordinate systems to be used in this thesis are
defined in Appendix A. A thorough introduction to tracking theory is found in
[7, 8, 15, 16].
This chapter will present the most common tracking filters that appear in the
tracking literature. Particular emphasis is on the Kalman filter (KF), the extended Kalman filter (EKF) and the particle filter (PF) which will be used later
in Part IV. A brief description of common tracking models is also given. In a CA
system, several objects usually have to be tracked simultaneously. In this thesis the
tracking system’s representation of one tracked object is referred to as a track. Another common name for this in the tracking literature is a trace. In a multi-target
tracking (MTT) system one must determine which measurement belongs to which
track. This task is called measurement or data association and will be discussed
in Section 4.4. For observing the targets, a CA system can have several tracking
sensors. The task of merging the measurement from the different sensors will be
addressed in Section 4.6. An example of the architecture of a tracking system is
shown in Figure 4.1. In general, all the states, xt , that are of interest cannot be
measured directly with the tracking sensors. Instead, they have to be estimated
from the measurements denoted by y. The measurements are often corrupted by
measurement noise. Estimating states that cannot be directly measured from noisy
measurements is a well-studied problem. In control theory literature, an algorithm
35
36
Chapter 4
Tracking for Collision Avoidance Applications
Figure 4.1: Schematic illustration of a tracking system providing state estimates to a
decision and control system.
that estimates or reconstructs the non-measurable states from the measurements
is called an observer. State estimation is described in [5, 46, 67, 70]. All state
estimation is model-based. The model describes the dynamics of the system to be
estimated and how the measurements relate to the states of interest. A general
model for a dynamic system is given by (4.1). Since the sensors and processing units
work with discrete time, discrete time models is used for all tracking algorithms.
xt+T = f (xt , νt ).
(4.1)
In (4.1), xt is the state vector at time t and νt is the process noise (also called
system disturbance). In the vehicle tracking examples, the main contributions to
νt come from the human or controller input commands to the tracked object. Thus,
the driver or controller is modeled as a stochastic process, whose objective is often
unknown. The measurement relation is given by (4.2).
yt = h(xt , et ),
(4.2)
were yt is the measurement vector and et is the measurement noise. It is assumed
that the process noise distribution pν (·) and the measurement noise distribution
pe (·) are known. There are two main approaches that can be taken to state estimation:
1. The deterministic (Fisher) approach: In this approach, it is assumed that
there exists a true but unknown value of the states x, and the goal of the
estimation is to find this true value. The connection between the stochastic
observation vector y and the states x is given by the probability density function (PDF) p(y|x). A popular choice of state estimate x̂, is to take x̂ as the
value that maximizes the likelihood of the observation (4.3).
x̂t = arg max p(Yt |xt ),
xt
(4.3)
where Yt = {y1 , y2 , · · · , yt }. Another common approach is to choose x̂ so
that the square of the expected prediction error is minimized:
x̂t = arg min(yt − h(x̂t|t−T ))2 .
x̂
(4.4)
2. The Bayesian Approach: The observation vector yt is assumed to have a
known probability density function p(yt |xt ) which can depend on the states.
Section 4.1
37
The Kalman Filter
In the Bayesian approach the state vector xt is considered to be a stochastic
variable. It is assumed that the random vector xt has a known prior density function p(xt ). The prior distribution encompasses the knowledge of xt
before any observation has been made. When a measurement is made, this
observation alters our knowledge of xt . The distribution after an observation
has been made is called the posterior density function p(xt |Yt ). The posterior
density function can be expressed using Bayes’ rule:
p(xt |Yt ) =
p(Yt |xt )p(xt )
.
p(Yt )
(4.5)
In (4.5), p(Yt ) is a scalar constant that can be found through marginalization:
p(Yt |xt )p(xt ) dx.
(4.6)
p(Yt ) =
Rn
The posterior density p(x|Y) is a very general solution to the tracking problem. It can be used to calculate the probability of any characteristics of
states. As will be shown in Chapter 7, this approach is very suitable for decision making in CA systems. A motivation for favoring the Bayesian approach
is that when you make a decision in a CA system, all possible scenarios should
be considered. Therefore, a point estimate of x is not sufficient. The general
Bayesian solution p(x|Y), on the other hand, contains all information that
can be known of the state vector, given the measurements. In many applications, however, one is interested in finding a point estimate of state vector
x̂. Popular choices are the least mean square estimate given by (4.7) or the
maximum a posteriori estimate given by (4.8)
xp(x|y) dx,
(4.7)
x̂MMS =
Rn
x̂MAP = arg max p(x|Y).
x
(4.8)
In the following Sections 4.1 – 4.3 filtering methods that provide a solution to the
Bayesian state estimation problem are examined.
4.1
The Kalman Filter
The well known KF, first introduced in [71], is widely used in tracking applications,
and there are numerous examples of its implementations.
4.1.1 The Stationary Kalman Filter
This section reviews the results for the stationary Kalman filter. For the stationary
case the linear time discrete dynamic model and the measurement equation can be
38
Chapter 4
Tracking for Collision Avoidance Applications
written according to
xt+T = Axt + Bνt ,
yt = Cxt + et ,
νt ∈ N (0, Q),
et ∈ N (0, R).
(4.9)
(4.10)
Thus, in (4.9) and (4.10), A, B and C are constant matrices. Q and R are the
covariance matrices of the process noise, νt , and the measurement noise et respectively. The sampling time of the system is T . The subscript t denotes the sample
time at sample number n, thus t = nT and t + T = (n + 1)T . A straightforward
estimation of xt is given by
x̂t|t = Ax̂t|t−T + Kεt ,
(4.11)
εt = yt − C x̂t|t−T .
(4.12)
where
In (4.11) the choice of K will of course affect the filter. If K is chosen as in (4.13)
this gives the time invariant Kalman filter. In Kalman filtering, K is often referred
to as the Kalman gain,
(4.13)
K = P C T (CP C T + R)−1 .
In (4.13) the covariance matrix P is given by the solution to the stationary Riccati
Equation (4.14) according to
P = AP AT − AP C T (CP C T + R)−1 CP AT + BQB T .
(4.14)
The KF gives the Bayesian solution p(xt |yt ). Under the assumption that νt ∈
N (0, Q) and et ∈ N (0, R), then x ∈ N (0, P ). The PDF is given by (4.15).
T
−1
1
p(xt |Yt ) = e−0.5(x−x̂t ) P (x−x̂t ) .
p
(2π) (det P )
(4.15)
The state estimation error covariance P = cov(xt − x̂t ) = E(xt − x̂t )(xt − x̂t )T is
obtained by solving (4.14). Note, that in the stationary case the Kalman gain K
and state covariance matrix can be calculated off-line (i.e., they do not depend on
the measurements).
4.1.2 The Time-Varying Kalman Filter
For the time varying KF a linear time-varying model given by
xt+T = At xt + Bt νt ,
νt ∈ N (0, Qt ),
(4.16)
yt = Ct xt + et ,
et ∈ N (0, Rt ),
(4.17)
is considered. The subscript t of the system matrices A, B and C indicates that
the dynamics of the system, and also the measurement equation, vary over time.
Also the process noise covariance Qt and measurement noise covariance Rt may
vary with time. Due to the time variability of the system the state covariance P
Section 4.1
39
The Kalman Filter
will now vary with time. The Kalman gain thus needs to be calculated at each
time instant. The time-varying Kalman filter is given by
x̂t|t = x̂t|t−T + Pt|t−T CtT (Ct Pt|t−T CtT − Rt )−1 (yt − Ct x̂t|t−T ),
Pt|t = Pt|t−T +
x̂t+T |t = At x̂t|t ,
Pt|t−T CtT (Ct Pt|t−T CtT
−1
− Rt )
Ct Pt|t−T ,
Pt+T |t = At Pt ATt + Bt Qt BtT .
(4.18a)
(4.18b)
(4.18c)
(4.18d)
The Equations (4.18a) – (4.18b) are called the measurement update and (4.18c)
– (4.18d) are called the time update. In the measurement update, the state and
covariance estimate is revised according to the latest measurement. In the time
update step a prediction is made. In a CA system, predictions for several future
time instances might be required. Prediction for time t + 2, t + 3, t + 4, · · · are easily
obtained by iterating the time update equations. As in the stationary case, the a
posteriori density is known if νt ∈ N (0, Qt ) and et ∈ N (0, Rt ). Then p(xt+T |yt ) is
given by (4.15), where the state estimate x̂t+T |t and cov(xt+T − x̂t+T |t ) = Pt+T |t
are given by the time update (4.18c) and (4.18d). For a deeper study of Kalman
filtering, [5, 46, 70] are recommended.
4.1.3 The Extended Kalman Filter
In many cases the dynamic model and the measurement equations are nonlinear.
This is, for example, the case for many tracking models. The assumption of Gaussian noise might not be fulfilled either. In such cases it is hard to find an analytical
solution to the estimation problem, i.e., finding p(x|Y). Consider a system that
can be written as
xt+T = f (xt ) + νt ,
yt = h(xt ) + et ,
νt ∈ N (0, Qt ),
et ∈ N (0, Rt ).
(4.19a)
(4.19b)
The idea of extended Kalman filtering (EKF) is to solve an approximation of the
original problem by linearizing the equations and approximating the noise distributions with Gaussian distributions. There are different approaches that can be
taken; we start with discussing how to deal with a non-linear measurement equation. Linearizing (4.19b) by Taylor expansion and neglecting higher order terms
we get
∂h(x) h(xt ) = h(x̂t|t−T ) +
(xt − x̂t|t−T ),
(4.20)
∂x x=x̂t|t−T
= Ht . A new measurement equation
where the Jacobian matrix ∂h(x)
∂x x=x̂t|t−T
(4.21) is formed using the expansion (4.20),
ȳt = yt − h(x̂t|t−T ) + Ht x̂t|t−T ≈ Ht x̂t|t−T + et .
(4.21)
The right hand side of (4.21) looks as in the linear case (4.17), and the Kalman
filter measurement update (4.18a),(4.18b) can be applied.
40
Chapter 4
Tracking for Collision Avoidance Applications
Another approach to treat a nonlinear measurement equation is to make a nonlinear transformation of the measurements. The aim of the transformation is for
the transformed measurements to yield a linear measurement equation. One such
transformation is to take ȳ = h−1 (y). In example 4.1, a transformation of nonlinear
measurements from a radar sensor is shown.
Example 4.1
A radar sensor measuring range and azimuth angle has the following measurement
equation
y=
r
ϕ
.
(4.22)
In a tracking system using a Cartesian coordinate system, position measurements
in Cartesian coordinates (px and py ) would yield a linear measurement equation.
To obtain this we perform a nonlinear transformation of the range and angle measurement from a radar; the transformed measurement equation is given by
r cos(ϕ)
px
=
(4.23)
ȳ =
r sin(ϕ)
py
The problem with this approach is that the measurement covariance R must also be
transformed. This can only be done approximately, and Gaussianity is in general
not preserved.
To be able to use the Kalman filter time update equations (4.18c) and (4.18d),
f (xt ) must also be linearized. Starting with a nonlinear continuous time model,
there are two choices for how to do this. Either by using discretized linearization
or linearized discretization. Both these methods are described in [46]. If a discrete
time nonlinear function exists as in (4.19a), a Taylor expansion for each component
(i) of the state vector yields the estimate
1
(i)
xt+T ≈ f (i) (x̂t )+ (f (i) ) (x̂t )(xt − x̂t )+ (xt − x̂t )T (f (i) ) (x̂t )(xt − x̂t )+ νi,t , (4.24)
2
where (f (i) ) denotes a row vector consisting of partial derivatives with respect to
all states. Using this expression the time update of the EKF will be
x̂t+T |t = f (x̂t|t )
(4.25)
T
Pt+T |t = f (x̂t|t )Pt|t f (x̂t|t ) + Qt ,
where
(4.26)


(f (1) )


..

f = 
.


(f (n) )
(4.27)
Section 4.2
41
The Point Mass Filter
The EKF will give a sub-optimal solution since the dynamics and measurement
equations of the system are an approximation of the true system. The true process
and measurement noise might also not be Gaussian, in which case the posteriori
distribution given by the EKF is only an approximation of the true density. For
highly nonlinear systems or when the PDFs are not well approximated with Gaussian distributions, the EKF is not sufficient. To improve tracking performance
several techniques using multiple models have been proposed.
4.2
The Point Mass Filter
So far we have only discussed the use of linearized models and Gaussian noise
processes to deal with nonlinear and non-Gaussian systems. We will now look
at methods where we do not linearize the system, but instead we try to solve
the estimation problem for the “true” system, [67, 79, 115]. The systems are
nonlinear with additive noise, as described by (4.19a) and (4.19b). The process
and measurement noise distributions (pν (·) and pe (·)) are assumed to be known,
but they are not necessarily Gaussian. As already stated, it is often not possible to
find an analytical solution to the Bayesian tracking problem. The idea is to solve
the problems by numerical approximation. A recursive solution to the Bayesian
estimation problem is given by (4.28) – (4.30).
p(yt |xt )p(xt |Yt−T )
,
p(yt |Yt−T )
p(xt+T |Yt ) =
p(xt+T |xt )p(xt |Yt ) dxt ,
p(xt |Yt ) =
(4.28)
(4.29)
Rn
where
p(yt |Yt−T ) =
Rn
p(yt |xt )p(xt |Yt−T ) dxt ,
(4.30)
and the recursion is initiated by p(x0 |Y−T ) = p(x0 ). Inserting the measurement
equation (4.19b) into (4.28) we get
p(xt |Yt ) = α−1
t pet (yt − h(xt )) p(xt |Yt−T ),
(4.31)
where
αt = p(yt |Yt−T ) =
Rn
pet (yt − h(xt )) p(xt |Yt−T ) dxt .
(4.32)
The time update is given by
p(xt+T |Yt ) =
Rn
pνt (xt+T − f (xt )) p(xt |Yt ) dxt .
(4.33)
One approach to approximately solve the recursive estimation problem is to evaluate the recursion (4.31) – (4.33) for several points around the working point (i.e.,
the current value of xt ). These points are then updated at each time step through
42
Chapter 4
Tracking for Collision Avoidance Applications
the recursion. We thus have a quantization of the state space. Each point is assigned a weight or mass corresponding to the probability that the states assume
that particular value. This approach has sometimes been called the point mass
filter (PMF), essentially it means that the PDF is approximated with a finite number of samples from the distribution and the integrals are approximated with a
Riemann sum. Assume that a grid consisting of N points is chosen to approximate
the density function p(xt |Yt ). These N points in the state space will be denoted
xt (k), k = 1, 2, · · · , N . The associated probability mass for each point is denoted
p(xt (k)|Yt ), k = 1, 2, · · · , N . Assuming that the grid is uniform with the distance δ
between the points, an integral over a region in the state space can be approximated
by a sum
N
p(xt ) dxt ≈
p (xt (k)) δ n .
(4.34)
Rn
k=1
Applying the PMF approximation yields the recursion:
p(xt (k)|Yt ) = α−1
t pet (yt − h (xt (k))) p(xt (k)|Yt−T ),
xt+T (k) = f (xt (k)), k = 1, 2, · · · , N
p(xt+T (k)|Yt ) =
N
pνt (xt+T (k) − f (xt (n)) p(xt (n)|Yt−T )) δ n .
(4.35a)
(4.35b)
(4.35c)
n=1
In order for the PMF to work, the grid-points must be adapted to the support of
the conditional density. One way to do this is to remove points with low mass.
Doing this will of course decrease the number of points, so when the number of
grid points falls below a lower bound one should increase the number of points. A
thorough account of the PMF, with tracking and navigation examples, is given in
[11].
4.3
The Particle Filter
One of the main problems with the PMF approach is the computational complexity.
The relative error of brute force techniques for approximating the integral (4.29)
is shown in [11] to be of the complexity order O(L−1/n ) (where L depends on the
distance between grid points and n is the state dimension). For high dimensional
problems, the PMF approach thus becomes intractable.
The PF approach bears many similarities to the PMF. This also aim at finding a
numerical approximation of the recursive Bayesian solution to the tracking problem
in (4.28) – (4.30). The PF approach also uses a set of points to represent the
posteriori distribution of the states. Instead of discretizing the state space, the PF
depends on the assumption that it is possible to draw samples from the a priori
distribution p(xt−T |Yt−T ). Here only a brief description of the theory is given.
For more details on PF see [12, 25, 39, 48, 72, 91, 123]. The Bayesian solution
given by the particle filter method provides an approximative solution to (4.5)
by approximating the probability density p(xt |Yt ) by a large set of N particles
Section 4.4
43
Measurement Association
{xt }N
i=1 , where each particle has an assigned relative weight, γt , such that all
weights sum to unity. The location and weight of each particle reflect the value
of the density in the region of the state space. When the measurement relation is
according to
(4.36)
yt = h(xt ) + et .
(i)
(i)
The likelihood p(yt |xt ) can be calculated as:
γt = p(yt |xt ) = pet (yt − h(xt )).
(4.37)
Each sample from the prior distribution can then be simulated according to the
dynamic model. The resulting swarm of particles represents the posterior density.
However, as time evolves the particle cloud is spread over an ever increasing region of the state space. This eventually leads to a poor representation of the true
distribution and divergence of the PF. By introducing a resampling step, as proposed in [39], problems with divergence can be alleviated. This is referred to as
sampling importance resampling (SIR), and is summarized in Algorithm 4.1. The
PF approach is often referred to as simulation-based, because of the fact that it
uses Monte Carlo simulations to approximate the solution.
1. Generate N samples {x0 }N
i=1 from p(x0 ).
(i)
(i)
(i)
2. Compute γt = pe (yt |xt ) and normalize, i.e.,
(i)
(i) N
(j)
γ̄t = γt / j=1 γt , i = 1, . . . , N .
3. Generate a new set {xt }N
i=1 by resampling with replacement N times from
(i)
(j)
(i)
(j)
{xt }N
,
with
probability
γ̄t = P r{xt = xt }.
i=1
(i)
(i)
(i)
4. Prediction: xt+T = f (xt
izations
(i)
, ut , νt ), i = 1, . . . , N using different noise real-
(i)
νt .
5. Increase t and continue from step 2.
Algorithm 4.1: Particle filter.
4.4
Measurement Association
A Multi target tracking (MTT) system’s task is to track several objects simultaneously. The system thus supervises several targets (represented by their estimated
(i)
state vector xt , i = 1, 2, · · · , J, where J is the maximum number of tracks) at
any given time. The tracking sensors will then generally provide measurements
(i)
originating from several objects, yt , i = 1, 2, · · · , M . Before updating the state
estimates of current tracks with new measurements, the system needs to determine
which measurement belongs to which track. The task of assigning measurements
44
Chapter 4
Tracking for Collision Avoidance Applications
Gating
5
4.5
y
5
4
h(x̂(3) )
y−coordinate [m]
3.5
3
2.5
y3
2
y2
1.5
y1
1
h(x̂(2) )
y
4
(1)
h(x̂
)
0.5
0
−1
Figure 4.2:
0
1
2
x−coordinate [m]
3
4
Gating example in a Cartesian coordinate system.
to tracks is called measurement association. Some reference where measurement
association is discussed are [7, 9, 16, 49]. Given a dense target environment, noisy
measurements, multiple reflections on each target and the fact that the sensors
might erroneously detect object caused by multi path propagation or spurious reflections, this is not always an easy task.
4.4.1 Gating
The first step in assigning measurements to tracks is called gating. Gating is used to
eliminate unlikely observation-to-track pairings, and if no conflicts exist, to assign
the measurement. For each track, a subset G(i) is defined in the measurement
coordinate space, G(i) ⊂ Rm . If a measurement falls within the gate of a track
(j)
yt ∈ G(p) , then that measurement is considered a candidate for updating that
track. Figure 4.2 shows an example of a gating procedure in a 2-D Cartesian space.
Clearly there are conflicts: There are several measurements falling within the same
gate (i.e., {y1 , y2 } ∈ G(1) and ({y2 , y3 , y4 } ∈ G(2) ) and there are also measurements
falling within more than one gate (y2 ∈ G(1) and y2 ∈ G(2) ). If a measurement
falls in one gate only and there are no other measurements that can be associated
to that gate, then that measurement can be directly associated with the track to
which the gate belongs. If there are conflicts, on the other hand, further correlation
logic is needed. If a measurement does not fall into any gate this measurement is
Section 4.4
Measurement Association
45
considered a candidate to initiate a new track.
4.4.2 Measurement Association Methods and Techniques
In the previous section it was shown that after the gating procedure there might still
be conflicts regarding which measurement to assign to which track. In this section
common techniques to resolve these conflicts is reviewed. In tracking literature,
the process of solving conflicting assignments is referred to as data or measurement
association.
The Nearest Neighbor Approach
The nearest neighbor (NN) approach looks for a unique pairing (i.e., one observation
can only be used to update one track). To decide which observation to associate
with which track, a distance measure gij is formed. gij denotes the distance from
observation i to track j. The idea is then to choose the assignments in such a way
that the total distance is minimized. One popular choice of distance measure is
the likelihood function associated with the assignment of observation j to track i.
In this case the idea is to maximize the total likelihood of all assignments.
The Multiple Hypothesis Approach
In the multiple hypothesis tracking (MHT) approach, a track is only updated with
one measurement, as in the NN approach. If there are conflicts, however, the tracks
are divided into several tracks and all possible associations are made. That is, one
measurement can now be used to update several tracks. The fact that a measurement could also be used to initiate a new track or may be a false measurement is
also considered.
Example 4.2
In Figure 4.2, there are three tracks (x̂(1) ,x̂(2) ,x̂(3) ) before the measurement association. In the association step track x̂(1) is updated with y1 and y2 . x̂(1) is thus
divided into two tracks x̂(11) and x̂(12) . Similarly x̂(2) splits into three tracks and
x̂(3) is not split.
In the MHT approach it is of course possible that the number of tracks becomes
large. This has to be dealt with in order for the solution to be tractable. One way
to handle the problem is calculating the probability of the different tracks, and
terminating tracks with low probability.
Clustering Techniques
In some applications it is common to receive multiple measurement from the same
obstacle. In this case, it is possible to initiate and maintain several tracks that
correspond to the same entity. Having several tracks for the same entity might
cause confusion in further analysis and waste computational effort. Possibly, the
46
Chapter 4
Tracking for Collision Avoidance Applications
Clustering
5
4.5
y−coordinate [m]
4
A
3.5
3
C
2.5
2
B
1.5
0
0.5
1
1.5
2
2.5
x−coordinate [m]
3
3.5
Figure 4.3: Example of measurement clustering in a Cartesian coordinate system. In this
case all tracks that are within a minimum distance from others are clustered together.
This result in tree clusters of tracks.
erroneous allocation of tracks could lead to discrimination of interesting targets.
One way to handle this is to cluster tracks or merge tracks that are likely to represent the same target. Figure 4.3 shows an example where several tracks (denoted
by crosses) in a Cartesian coordinate system are clustered into three groups A, B
and C. How to cluster objects is not clear and will differ for different applications.
In CA applications one will typical try to estimate the physical size of the object,
this estimate can be used to determine if several tracks actually represent the same
object. Clustering is sometimes used to track troop movements in military applications. Here the entire troop is considered to be the tracked entity. An introduction
and further references to clustering techniques are given in [49]. Another approach
to deal with the problem of multiple measurement from the same object is to allow
tracks to be updated with several measurements, i.e., several measurements are
associated with the same track. One reason for using clustering is to save computational power; in Example 4.3 it is discussed why this might be desirable for CA
applications.
Section 4.5
Ranking Measurements
47
Traffic scenario where the vehicle ahead of the host vehicle (object 2) has
come to a stop. Depending on what manoeuvre the driver of the host vehicle decides on,
object 1, 2, 3, and the stationary object, 4, on the side of the road can all constitute a
threat.
Figure 4.4:
Example 4.3
The computational power of a commercial CA system will of course be limited.
This means that the maximum number of live tracks at any one time will be
limited. Here a system that is able to track four different objects simultaneously is
examined. In Figure 4.4, there are four objects that can become a threat depending
on how the host vehicle driver chooses to avoid the stopped vehicle ahead (object 2).
Figure 4.5 shows measurements from a commercial radar sensor. The measurement
was performed on a large open tarmac area with only one stationary target present
(see Figure 4.6). Clearly, in Figure 4.5 the radar detects two targets which are
coming from the same physical object. In this case it is certainly possible to initiate
two tracks from the two measurements. If this would happen in the scenario from
Figure 4.4 two tracks would be used for the stopped vehicle (object 2), and since
the maximum number of tracked objects is four, one of the other potential threats
will not be tracked.
4.5
Ranking Measurements
It has already been stated that there is an upper limit to the number of objects
that can be tracked simultaneously (because of limited computational power). The
sensor, however, might possibly detect more objects than can be dealt with by the
48
Range [m]
Chapter 4
Multiple Detection of a Stationary Target − Range and Azimuth Angle
40
Target 1
Target 2
30
20
10
0
−0.5
Azimuth Angle [degrees]
Tracking for Collision Avoidance Applications
0
0.5
1
1.5
Time [s]
2
2.5
3
5
Target 1
Target 2
0
−5
−10
−15
−0.5
0
0.5
1
1.5
Time [s]
2
2.5
3
Range and azimuth angle measurements from a commercial radar sensor. The
host vehicle is approaching one lone stationary obstacle at 40 km/h. Two obstacles at the
same range and an angular spacing of 3 − 4◦ are detected (between time 0.5 and 1.75),
however, only one physical object is present, see Figure 4.6. This is a clear case of multiple
reflections from one object. Note, that the tracking and measurement association is done
internally in the radar sensor and the algorithm is not fully known.
Figure 4.5:
tracking system. To make the total system as effective as possible, it is desirable
to use the most relevant measurements and to remove irrelevant measurements.
In this thesis the process of selecting relevant measurements will be called ranking. This section will discuss how ranking of measurements can be done in a CA
system. Assume that the sensor whose measurements should be ranked, measures
a maximum of M different obstacles. The measurements are given in a vector
Y = (y (1) , y (2) , · · · , y (M) ). Because of computational limits only i ≤ M measurements are allowed to be passed on to the tracking system. The idea is now to
reorder the measurements of Y to get Ỹ = (ỹ (1) , ỹ (2) , · · · , ỹ (M) ), where ỹ (1) is the
most relevant measurement and so on. How to perform the ranking in general will
of course depend on the type of application considered. Here considerations when
ranking the importance of a measurement for different CA systems are given. Since
the aim of the ranking process is to decrease the computational burden, it must not
be computationally demanding itself. The sensors normally measure range, range
rate and azimuth angle, so the ranking will be based on these quantities.
Section 4.5
Ranking Measurements
49
The test area where the measurement of Figure 4.5 was performed, there are
no other objects close to the stationary obstacle.
Figure 4.6:
4.5.1 Ranking Measurements by Range Only
For CA systems, it might be wise to prioritize the measurements according to
range, since objects close to the host vehicle in many cases constitute the greatest
threat. For some systems, it is at least possible to say that measurements with a
range above a certain value are irrelevant (see Example 4.4).
Example 4.4
Consider a CMbB system which performs braking when a collision becomes unavoidable. The collision unavoidable point is closer than 30 m for stationary targets
and speeds up to 150 km/h. Assuming that other vehicles have similar handling
characteristics, one can say that measurements where the range is much larger
than 60 m are irrelevant. For such a system it would be wise to rank measurements
below 60 m as more relevant than those above 60 m.
The danger in ranking measurements on range alone is that one might miss objects
that are serious threats because of a high relative speed.
4.5.2 Ranking by Range Rate and Time-to-Collision
Some sensors measure range rate directly. For such sensors it is possible to prioritize
objects according to their oncoming speed, which is approximated by the measured
Ṙ. An estimate using both range and range rate is time-to-collision (tTTC ) given
50
Chapter 4
Tracking for Collision Avoidance Applications
by
R
, Ṙ < 0.
(4.38)
Ṙ
An alternative is to prioritize measurements with a short tTTC or high closing
velocity Ṙ, reasoning that they present a large threat because of their high relative
speed. The danger here is of possible discrimination of objects with a short range
that are traveling at the same speed or faster then the host vehicle. Such an object
can very quickly become a threat if the driver of that vehicle hits the brakes hard.
The tracking sensors often pick up a lot of stationary road-side objects which make
such discriminations likely. The tTTC for road-side objects is often short but often
they constitute no threat.
tTTC ≈ −
4.5.3 Ranking Objects by their Azimuth Angle
Vision sensors often only measure angle to the object ahead. For such a sensor
the measurements need to be ranked based on the angular position only. A simple
approach would be to consider objects straight ahead as more dangerous. This
approach, of course, also has the risk of missing some threats. For example, an
object that is dead ahead but far away and has low oncoming speed will be considered more dangerous than an object nearby with a high relative speed but azimuth
angle not equal to zero. In general one should also consider the vehicle’s own turn
rate when one uses the azimuth angle to prioritize threats. If the vehicle is turning,
objects with an azimuth angle in that direction are probably a greater threat than
those with azimuth angle equal to zero.
It is clear that real threats will sometimes be missed if one chooses to prioritize
objects only after range, tTTC or azimuth angle. A more appropriate and general
approach is to use all of the quantities to rank the relevance of each measurement.
One suggestion on how this could be done is given in [38]. Of course when designing
the ranking algorithm one should always keep in mind that this step is used to
decrease computational load. It is therefore desirable to keep the algorithm as
computationally simple as possible.
All the theory needed to build a tracking system has now been presented. Algorithm 4.2 summarizes the tracking algorithm that will be used later in demonstrator
vehicle 1.
4.6
Sensor Fusion
In tracking systems, it is often the case that there are several tracking sensors. For
example, in a CMbB system it is desirable to have more than one tracking sensor
to increase the reliability of the system, i.e., to avoid making faulty interventions
caused by invalid obstacles or spurious target observations at one sensor. Now the
question arises how to fuse the data from the different sensors. This task is often
called sensor or data fusion. A thorough treatment of sensor fusion problems is
given in [49]. To fuse independent estimates of the same quantity is straightforward.
Section 4.6
51
Sensor Fusion
1. Association: The purpose of the association is to assign measurement to
the current tracks. This is done for each tracking sensor (in this case the
measurements come from a radar sensor). The following steps are performed:
(a) Cluster measurements.
Measurements from the same sensor that are very close to each other
are clustered.
(b) Rank measurements.
Measurements are reordered (range, tTTC and azimuth angle are here
used to determine the importance of each measurement). The ordered
measurements are denoted Y = (y (1) , y (2) , · · · , y (M) ).
(c) Select the most relevant measurements.
(d) Gating.
The measurements Y are compared with h(x̂(j) ) to check if they fall in
any of the gates G(j) . Here a rectangular gate according to:


|r(i) − h1 (x̂(j) )| < rgate


y (i) − h(x̂(j) ) ∈ G(j) →  |ṙ(i) − h2 (x̂(j) )| < ṙgate 
(4.39)
|ϕ(i) − h3 (x̂(j) )| < ϕgate
is used.
(e) Measurement association.
If there exist conflicts after the gating procedure these are resolved using a NN approach. Here a geometric distance measure is used. The
measurement i that is associated with track j is denoted y i,j .
2. Filtering: Update state estimates for track j ∈ {1, 2, 3, 4} with associated
measurement y i,j .
x̂t|t = x̂t|t−T + Pt|t−T CtT (Ct Pt|t−T CtT + Rt )−1 (yt
(j)
(j)
(j)
(j)
(i,j)
(j)
− Ct x̂t|t−T ),
Pt|t = Pt|t−T + Pt|t−T CtT (Ct Pt|t−T CtT + Rt )−1 Ct Pt|t−T .
(j)
(j)
(j)
(j)
(j)
3. Prediction:
(a) Predict future states.
(j)
(j)
x̂t+T |t = At x̂t|t ,
Pt+T |t = At Pt ATt + Bt Qt BtT .
(j)
(j)
(b) Iterate the previous step to obtain predictions for as many time instants
as desired.
Algorithm 4.2: Tracking algorithm used in demonstrator vehicle 1.
52
Chapter 4
Tracking for Collision Avoidance Applications
Figure 4.7: Pixel/data level fusion. In this approach the information from each sensor is
associated with current tracks and then jointly fused to extract features of the measured
object.
If X 1 and X 2 are two independent estimates of the same quantity with covariance
matrices P 1 and P 2 , Equation (4.40a) will yield the optimal fused estimate, in the
sense that the covariance of X fused is minimized.
X fused = P ((P 1 )−1 X 1 + (P 2 )−1 X 2 ),
1 −1
P = ((P )
2 −1 −1
+ (P )
)
.
(4.40a)
(4.40b)
The problem of fusing sensor information in tracking systems is that the sensors
measure widely different quantities. For example, a mm-radar measures reflected
radar energy at the antenna, a vision sensor measures light intensity at each pixel,
and a GPS receiver measures the flight time of coded transmissions from satellites. In many applications one is not only interested in tracking kinetic quantity
of detected objects; one might also be interested in inferring a number of different
features of the object. These features as well as the kinetic quantities can then be
used to classify the object. When designing a system one must decide on what level
fusion of sensor data should be performed. Figures 4.7 and 4.8 schematically illustrates two extreme approaches to fusion of sensor information. In the pixel level
fusion illustrated in Figure 4.7, one tries to fuse the data at the lowest level, i.e.,
fusing the raw sensor output. This approach is the most general and promises the
best results. It is also the computationally most demanding approach. For large
systems, pixel level fusion is often infeasible because of the computational cost as
well as system complexity issues. The other extreme is the approach illustrated in
Figure 4.8. Here the data from each sensor is treated separately and has its own
tracking and feature extraction. The estimated states and features are then fused
at the top level. This approach has the advantage that it is robust against failure
of one sensor node. Should one sensor fail there will still be a state estimate from
the other sensors. From a system engineering point of view such a system is also
easier to understand and maintain. For example, switching to another sensor will
not require a total redesign of the entire system. These two examples are, of course,
extremes and any combination between these two is possible. It is also possible to
have feedback in the system for sensor management and diagnosis. For example,
to decrease the computational load in one sensor information from other sensors
Section 4.7
Models for Tracking and Navigation
53
Declaration level fusion. Here each sensor extracts features and maintains its
own objects. The tracked objects from each sensor are then fused in the last stage.
Figure 4.8:
can be used. An example of such a system is given in the example below.
Example 4.5
In a CA system it is desirable to determine not only the speed and distance of
objects close to the host vehicle. One also wants to determine its size and type.
Figure 4.9 shows a system that fuses data from a mm-radar and a camera sensor
to achieve this. In this system the radar is used to detect obstacles and determine
the range, range rate and azimuth angle. This information is then fed back to the
camera sensor, which performs template matching to determine if the object is a
car or not. It also provides a width measurement and a more accurate azimuth
angle measurement of the obstacle.
In CA systems, faulty interventions are not allowed. Since available tracking sensors all have weaknesses that cause them to sometimes report false targets, the
sensor fusion is a crucial part of the system. In CA systems, sensor information
should be fused in such a way that the risk of passing on false targets as valid ones
to the decision making algorithm is minimized.
4.7
Models for Tracking and Navigation
It has already been stated that most tracking is model-based. In this section common models that describes the tracked object’s motion will be discussed. Tracking
models are further discussed in [86, 87]. The performance of a tracking system will,
to a large extent, depend upon how well the dynamics of the true system is described by the model. Generally speaking, a better model yields a better tracking
performance. For CA it is of interest to find the Bayesian solution p(xt+T |Yt ), so
the better the model, the closer the posterior distribution will be to the true distribution of the vehicle’s future states. However, it is also desirable to use as simple a
model as possible, both for computational issues and for understanding the system.
Modeling vehicle dynamics is a well-explored area. Very detailed dynamic models exist for cars, aircraft, robot manipulators etc. These models all break down
the system into several subsystems and model the dynamics of each subsystem;
54
Chapter 4
Tracking for Collision Avoidance Applications
Figure 4.9: Fusion of a radar and a camera sensor. The radar is used to detect obstacles
and measure kinematic quantities. The detected obstacles (indicated by the points in the
sensor image) are fed back to the camera sensor that performs template matching in the
pixel image. After a positive matching spatial properties such as width and height can
then be measured in the pixel image.
thus, providing a very accurate description of the system. A good introduction to
automotive vehicle dynamics is given in [37, 128]. Such models are often too complex for tracking purposes. Often they are computationally too demanding and,
more importantly, since the tracking sensors cannot measure all the quantities in
the models the final estimation result may not be much better than for a simpler
model. Instead, a model that captures the major characteristics of the vehicle’s
dynamics should be used. Apart from deciding which tracking model to choose,
it is also necessary to think about which coordinate system to track in. In [4], it
is suggested that modified polar coordinates are better for angle-only tracking. In
this thesis, Cartesian coordinate systems will mainly be used, these are described in
Appendix A. The host vehicle and other objects are distinguished by a superscript
host or object, e.g. host and object position in the x-direction are denoted Pxhost
and Pxobj . ω denotes the vehicle turn/yaw rate. Below common models to describe
the dynamics of tracked objects are presented for the two-dimensional case. The
constant velocity and constant acceleration models are straightforward to extend
to higher dimensions.
4.7.1 The Constant Velocity Model
One of the simplest models for tracking is given in (4.41). In this model the
motion of the vehicle is modeled as having piecewise constant velocity in the x
and y direction. The transitions between different velocity are modeled by white
Section 4.7
Models for Tracking and Navigation
55
random noise, and there is no correlation between motion in the x and y directions.
T
x = Px Py Vx Vy
,
(4.41)




1 0 T 0
T 2 /2
0




2
 0 1 0 T 
 0
T /2 
 xt + 
 νt .
(4.42)
xt+T = 




0 
 0 0 1 0 
 T
0 0 0 1
0
T
Assuming that the noise is Gaussian distributed and that the covariance matrix is
given by
0
σV2 x
Qt =
.
(4.43)
0
σV2 y
An assumption of constant velocity means that the motion is described by a straight
line between two measurement updates.
4.7.2 The Constant Acceleration Model
A model that assumes that tracked objects have constant velocity is, of course, a
very simple one. For tracking of maneuvering targets one often wants to model the
dynamics better. A simple extension of the model is to add the acceleration as a
state, and assume that the acceleration is piecewise constant. This is often called
the constant acceleration model (4.44)−(4.46)
T
,
(4.44)
x = Px Py Vx Vy Ax Ay




2
0
0
1 0 T 0 T2
T 3 /6



2 
3
 0 1 0 T 0 T 
 0
T /6 
2 




 2


0 
0 
 0 0 1 0 T
 T /2
xt+T = 
(4.45)
 xt + 
 νt ,
 0 0 0 1 0
 0
T 
T 2 /2 








0 
0 
 0 0 0 0 1
 T
0 0 0 0 0
1
0
T
2
0
σA
x
Qt =
.
(4.46)
2
0
σA
y
A constant acceleration will model the motion as straight lines or parabolic trajectories.
4.7.3 The Coordinated Turn Model
The constant velocity and constant acceleration models are very nice to use in the
sense that they are both linear models. When modeling the motion of a vehicle,
56
Chapter 4
Tracking for Collision Avoidance Applications
there are more aspects that could be considered. A simple assumption is to say
that a vehicle can generate forces either longitudinally (in its heading direction)
or laterally (orthogonal to the heading direction). If one assumes that the velocity
is constant and that a constant lateral force is applied, this results in centripetal
motion, with constant turn rate. Thus the motion is described by straight lines
and circle segments.
T
,
(4.47)
x = Px Py Vx Vy




1−cos(ωT )
sin(ωT )
1 0
T 2 /2
0
ω
ω




1−cos(ωT
)
sin(ωT
)
2
 0 1 −(


0
T /2 
) − ω
ω



 νt , (4.48)
xt+T = 
 xt + 

cos(ωT )
sin(ωT ) 
0 
 0 0
 T
0 0
− cos(ωT )
sin(ωT )
0
T
0
σV2 x
Qν =
.
(4.49)
0
σV2 y
In the coordinated turn model ω denote the vehicle turn rate. Such a model was
proposed in [94], and is often called the coordinated turn model. In this model it is
assumed that the turn rate of the observed object is known and piecewise constant.
In reality, the turn rate is often not known and has to be estimated.
4.7.4 The Nearly Coordinated Turn Model
The model, which assumes piecewise constant turn rate and includes ω as a part
of the state estimation process, is called the nearly coordinated turn model. The
discrete-time nearly coordinated turn model is given by (4.50) – (4.51).
T
x = Px Py Vx Vy ω
(4.50)




sin(ωT )
1−cos(ωT )
2
1 0
T
0
/2
0
0
ω
ω




)
)
 0 1 −( 1−cos(ωT
 0
) − sin(ωT
0 
T 2 /2 0 
ω
ω





xt+T = 
cos(ωT )
sin(ωT ) 0 
0
0 
 xt +  T
 νt
 0 0




 0 0
 0
− cos(ωT )
sin(ωT ) 0 
T
0 
0


Qν = 
0
0
σV2 x
0
0
σV2 y
0
0
0
0


0 
1
0
0
T
(4.51)
(4.52)
σω2
A comparison between the constant velocity model (piecewise straight line motion),
the constant acceleration model (parabolic trajectories), and the nearly coordinated
turn model (motion on circle segments), in a tracking application, is given in Example 4.6.
Section 4.7
57
Models for Tracking and Navigation
Tracked vehicle path
30
Global position [m]
20
10
0
t1
t3
t2
t4
t5 t6
t7
−10
−20
−30
−40
0
20
40
60
80
Global position [m]
100
120
Figure 4.10:
Trajectory of a tracked vehicle performing: acceleration–lane change–
deceleration–right turn. Acceleration starts at point t1 , lane change at point t3 , deceleration at t5 and right turn starts at t6 .
Example 4.6
In Figure 4.11 a comparison between the constant velocity, the constant acceleration model and the nearly coordinated turn model applied in an automotive-like
estimation problem is illustrated. In the comparison the manoeuvre described in
Figure 4.10 is used. A bicycle model described in Section 4.8.1 was used to generate
the true trajectory. For each model, 1000 Monte Carlo simulations are performed
(the same measurement noise realizations are used for each model). The EKF parameters (θ) were tuned to give the best possible tracking performance according
to
N
Pt − P̂t .
(4.53)
θ = arg min
θ
t=1
The measurements are given by




 
r
r
er




 
y = h(xt ) + et =  ṙ  + et =  ṙ  +  eṙ  ,
ϕ
ϕ
(4.54)
eϕ
where all the ei :s are independent white Gaussian random variables, with σr =
0.5 m, σṙ = 1 m/s and σϕ = 0.5π/180 rad. The measurement rate and tracking
system update frequency in this example is 10 Hz. By looking at Figure 4.11,
it appears that the constant acceleration model and the nearly coordinated turn
model have similar performance. Looking at the mean square predicted error and
the maximum predicted error in Table 4.1, it is clear that the coordinated turn
model performs better for both aspects. For all the models an EKF was used, since
the measurement equations are nonlinear. The nearly coordinated turn model is
58
Chapter 4
Tracking for Collision Avoidance Applications
Tracked path for 3 tracking models (average path from 1000 MC simulations)
4
2
Global position [m]
0
−2
−4
−6
−8
True Path
Constant Acceleration
Constant Velocity
Coordinated Turn
−10
−12
−14
−16
0
50
100
Global position [m]
150
Tracked trajectory for the constant velocity model, the constant acceleration
model and the nearly coordinated turn model. The figure illustrates the average path from
1000 MC simulations for each model.
Figure 4.11:
nonlinear and thus the model needs to be linearized each time step. This is not
the case for the constant velocity and the constant acceleration model. The filter
for the nearly coordinated turn model will thus be slightly more computationally
demanding. In the Monte Carlo simulations performed here, the required CPU time
for the nearly coordinated turn model was 2.8 times more than for the constant
velocity model. The constant acceleration model required 1.2 times more CPU
time than the constant velocity model.
4.8
Navigation
This chapter has mainly discussed the tracking of other vehicles’ dynamic states.
The tracking of the host vehicle’s states is sometimes called navigation or positioning. In this thesis the term navigation will be used for the task of estimating the
states of the CA host vehicle. Basically the same filtering techniques and vehicle
models that are used for tracking other vehicles can be used for this purpose. The
only difference is that other sensors are used to provide information of the kinematic
properties of the vehicle. The input to navigation systems comes from accelerometers, gyroscopes, wheel speed sensors, air/water flow sensors and GPS systems etc.
Example 4.7 illustrate the potential of using the information from a steering wheel
angle sensor, for navigation in a automotive application. The navigation sensors
Section 4.8
59
Navigation
Table 4.1: Results from a comparison of three tracking models. For each model 1000
Monte Carlo simulations were performed, where the same noise realizations were used
for all models. Both the average error in column two and the maximum error in column
three are measures of the spatial distance between the true position and the prediction
position. The CPU time does not say anything about the actual time to execute each
algorithm and should only be used for comparing the algorithms with each other
Tracking model
RMSE [m]
Max error [m]
CPU time
Constant velocity
0.88
5.2
1
Constant acceleration
0.65
4.7
1.2
Nearly coordinated turn
0.56
3.5
2.8
often have a lower noise level than the tracking sensors. Furthermore, quantities
such as acceleration and turn rates can be measured directly. Therefore, it might
make sense to use a more accurate model for navigation than for tracking other
vehicles. How should a more accurate vehicle model be constructed? Often one
tries to describe how the forces affecting the motion are being generated.
Example 4.7
Sometimes cars are equipped with a steering wheel position sensor and a brake
pedal force and position sensor. These sensors can be used to increase the tracking
and prediction performance of the host vehicle. Figure 4.12 shows the relationship
between steering wheel angle and yaw rate of a vehicle traveling at 50 km/h. There
is a delay in the order of several hundred ms between the steering wheel angel and
the yaw rate. Compared with the use of yaw rate measurements only, the use of
steering wheel measurements will tell you much earlier when the vehicle will turn.
In this case it will clearly make sense to model the dynamics between steering wheel
and yaw rate in the navigation system. The performance of a CA system using
this information would be significantly improved by the “preview” offered by the
use of the steering wheel sensor.
In some sense this is very similar for cars, aircraft and boats. The forces are generated when the control actuators are moved out of the heading direction of the
vehicle. This creates a slip angle between e.g. the wheels or rudder and the heading
direction. The force of friction between tire-to-road, rudder-to-air or water generates forces that are used to control the vehicle. It is important to note that the
forces that can be generated in this fashion are limited. Thus, there is a fundamental nonlinear constraint to the motion of the vehicle. In the next section a simple
handling model for the motion of a car is given.
4.8.1 Bicycle Model
A more accurate model for modeling handling properties of a car is the well known
bicycle model. In this model the effect of both front and both rear wheels are
60
Chapter 4
Tracking for Collision Avoidance Applications
Yaw rate and steering wheel angle for a lane change maneuver
150
Yaw rate
Steering wheel angle
Yaw rate [deg/s]
100
50
0
−50
−100
−150
0
1
2
3
4
Time [s]
5
6
7
8
Figure 4.12: The yaw rate and steering wheel angle for a vehicle traveling at 50 km/h
performing a lane change manoeuvre. The steering wheel angle has been scaled to be of
the same size as the yaw rate. The time constant between steering wheel angle and yaw
rate is approximately 500 ms.
merged to one front and one rear wheel (thus the vehicle is modeled as a bicycle).
The motion of the vehicle is then approximated by calculating the forces generated
at the front and rear wheel. Figure 4.13 shows a schematic image of this model. In
the figure only the lateral wheel forces are indicated. The longitudinal and lateral
forces on the wheels are given as a function of the longitudinal and lateral slip,
respectively. The longitudinal slip is given by the slip coefficient
s=
rwh ωwh − vx
,
rwh ωwh
(4.55)
where rwh denote the wheel radius and ωw the wheel turning rate. The relationship between slip and force is a nonlinear function; a model for this is given by
Pacejkas’ magic formula [6] which is plotted in Figure 4.14. This kind of model can
explain phenomena such as skidding. Therefore it is often used, for example, as a
reference model in yaw control systems. For tracking other vehicles, this model is
probably too detailed. For tracking the host vehicle (navigation) this model might
be appropriate. In fact, in vehicles equipped with a yaw control system, a bicycle
model is used as a reference model to detect when the vehicle is skidding. The yaw
acceleration is calculated from the torque balance equation
Jz ω̇ = cos(δ)Ff a − Fr b,
(4.56)
where in this case Jz is the moment of inertia and longitudinal wheel forces, pitching
motion, roll motion, wind forces, rolling resistance, and suspension dynamics are
Section 4.8
61
Navigation
A schematic image of the bicycle model, here only the lateral forces on each
tire are indicated. The longitudinal forces work along the wheel direction normal to the
lateral forces.
Figure 4.13:
Force as a function of slip
1
0.8
Normalized force µ (F/N)
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.5
0
Slip
0.5
1
Figure 4.14: Relationship between slip and force between tire and road surface according
to Pacejkas’ magic formula. Here the value is normalized with the normal force on the
wheel. The values correspond to warm and dry asphalt.
62
Chapter 4
Tracking for Collision Avoidance Applications
neglected. The lateral wheel forces are calculated from the slip angle for the front
wheel, αf , and rear wheel, αr , which can be solved from
aω + v sin(β)
tan(δ − αf ) =
(4.57)
v cos(β) ,
bω − v sin(β)
tan(αr ) = (4.58)
v cos(β) ,
The bicycle model is used to model the host vehicle dynamics in the simulation
environments for the algorithms used in the demonstrator vehicle presented in
Section 8.2. A derivation of the dynamic equations for the bicycle model is given
in [103], where it is referred to as the single track model.
Part III
Decision Making
63
Chapter 5
Deterministic Decision Making
By definition the goal of a collision avoidance (CA) system is to avoid or mitigate
collisions by an autonomous action or warning. Here the task of deciding on an
action is called decision making. The problem of deciding when and how the system
should act is complicated by several factors. Among these factors are measurement
and process noise. Anther factor is that simplified motion models must be used for
computational reasons.
A CA decision is based on the current state estimate of the host vehicle and
those of other objects. The estimates are provided by the tracking system, and are
used in one or more decision function to determine the appropriate collision avoidance action. Because of the measurement and process noise, the state estimates
are uncertain.
In this chapter the state estimate uncertainty is neglected. The focus is on
how to determine the threat of a collision given that the state of other objects is
known. CA strategies in general and various collision avoidance decision functions
will be discussed in Section 5.1 and 5.3 respectively. The task of dealing with
multiple obstacles will be addressed in Chapter 6. To deal with uncertainty in the
decision process, a probabilistic approach, based on statistical decision making, is
proposed in Chapter 7, where a general framework for dealing with nonlinear and
non-Gaussian models is presented, with different alternatives for how to calculate
statistical decision making rules.
5.1
Collision Avoidance Decision Strategy
The two main desired properties of a CA system are that it should avoid all collisions and that no faulty interventions should occur. The problem with choosing
a CA intervention strategy is that these properties are often contradictory. In
order to analyze possible strategies, five different states will be defined. In Figure 5.1, these states are visualized for an automotive scenario in which a vehicle is
approaching a stationary obstacle. A short description of each state is given below:
65
66
Chapter 5
Normal
operation
Collision
avoidable
Deterministic Decision Making
Collision
unavoidable
Collision
Post
collision
Figure 5.1: The figure shows an example of the five possible states of a vehicle approaching
a stationary obstacle. At first the obstacle is far away and presents no imminent threat,
this is called normal operation. As the vehicle comes closer to the obstacle it enters
the collision avoidable state, here the obstacle is a threat but there still exists an escape
path that can be followed by the appropriate driver maneuver. At some point it becomes
impossible to avoid the collision; this is called the collision unavoidable state. The collision
state is when the vehicle collides. The state after the collision is called the post collision
state.
1. Normal operation: In this state there is no imminent threat or risk of a
collision.
2. Collision avoidable: Here there is a threat to the vehicle. A non-negligible
risk exists that a collision will occur. In this state it is still possible to avoid
the imminent collision by an appropriate avoidance maneuver. Typically this
state is perceived by a human as dangerous. What is perceived by a CA
system as a dangerous situation will of course vary depending upon the CA
decision function used. It is normally in this state that collision warning
systems are activated. Any system intended to avoid collision has to act in
this state.
3. Collision unavoidable: In this state a collision is imminent, and cannot be
avoided by any maneuver. Although the collision cannot be avoided, it might
still be possible to significantly reduce its severity by reducing the collision
speed and by taking other mitigating actions.
4. Collision: This is the state when a collision occurs.
5. Post collision: The state when a collision has occurred. If the CA system is
still operational, actions to avoid secondary collisions could be considered.
It should be noted, since information of the surrounding environment is limited, it is
in many cases possible to go directly from the normal operating state to the collision
unavoidable state. A common example of such an event is a multiple collision (pileup) on a free-way. In this case a following vehicle is in the normal operating state
until the lead vehicle suddenly collides. If the traffic is dense and the time gap to
Section 5.1
Collision Avoidance Decision Strategy
67
H
(a) Overtaking situation, here two vehicles are on a collision course. The appropriate maneuver for the overtaking vehicle is to accelerate and then turn back into the right hand
lane to avoid a collision.
H
(b) Meeting on a narrow road, the two vehicles might be on a collision course. Normally
both drivers will steer their vehicle to the right side of the road just before they pass
each other.
Figure 5.2: Two traffic situation examples where the two vehicles involved are on a
collision course. Even though the situations are threatening in some sense, there is no
danger except if the driver has misjudged the situation (e.g. there is not enough time to
finish the overtaking, or the road is too narrow for the two vehicles to pass).
the lead vehicle is small the collision instantaneously becomes unavoidable. The
collision unavoidable state is clearly defined by the laws of physics. The collision
avoidable state on the other hand is harder to define. Despite there being many
situations in normal driving where a collision is close, these situations are not
always perceived as threatening (or at least not as situations where a CA system
should intervene). Two examples of such situations, are a overtaking situation and
meeting on a narrow road, illustrated in Figure 5.2. The strategy for when to
perform an autonomous CA intervention will inevitably be a trade-off between a
system’s effectiveness in avoiding collisions and faulty intervention frequency. The
tolerance for faulty intervention will, of course, depend on the type of intervention.
For example, an erroneous warning may not be as bad as an erroneous avoidance
maneuver.
In collision warning (CW) systems, human reaction time and non-ideal performance of the driver must be accounted for. Thus, the decision boundary for a
CW system must be in the collision avoidable state; with sufficient time to react
before the collision unavoidable state is reached. For an inattentive driver to be
able to avoid an imminent collision, the warning normally has to come more than
one second before the collision unavoidable state is entered (see [85]).
68
Chapter 5
Deterministic Decision Making
Avoidance maneuver along a circular path
20
15
host
r
y
P [m]
10
5
0
−5
−5
whost
0
wobj
5
10
15
20
Px [m]
Figure 5.3: A steering avoidance maneuver approximated by constant lateral acceleration
(9.82 m/s2 ).
5.2
Static Analysis of Collision Mitigation
To analyze the potential of a system which intervenes only when the collision
becomes unavoidable, the ideal performance of an automotive CMbB system is
studied in Section 2.2.4. In this section the same analysis is performed with a simplified model of the host vehicles motion. The scenario that is studied is the head-on
against a stationary obstacle illustrated in Figure 5.1. Both the vehicle and the
obstacle are assumed to be 2 meters wide. Furthermore, it is assumed that maximum deceleration and centripetal acceleration can be achieved instantaneously.
Thus, the steering maneuver follows a circular path according to Figure 5.3. The
coordinate system used here is described in Appendix A. The braking distance and
the distance needed to avoid the obstacle by a steering maneuver (with constant
centripetal acceleration) are plotted in Figure 5.4. The radius rhost of the circle
for the host vehicle traveling with a constant centripetal acceleration ay (here it is
assumed that ay = 9.82 m/s2 can be achieved) is given by
rhost =
vx2
.
ay
(5.1)
Section 5.2
69
Static Analysis of Collision Mitigation
Distance required to avoid a collision
50
45
Brake distance
Steer distance
40
Distance [m]
35
30
25
20
15
10
5
0
0
20
40
60
80
Initial velocity [km/h]
100
120
Distance required to avoid a collision by braking and by steering. The assumption here is that the vehicle can instantaneously achieve 9.82 m/s2 deceleration or
9.82 m/s2 lateral acceleration.
Figure 5.4:
The distance to avoid an obstacle of width wobj , assuming that the host vehicles
width is whost , is given by the trajectory swept by the right side of the vehicle
according to
(rhost + whost /2)2 = Px2 + (Py − rhost )2 .
(5.2)
This is illustrated in Figure 5.3. Setting the longitudinal position, Py , equal to
wobj /2 and solving for Px (the longitudinal distance) yields
(whost )2
(wobj )2
+ rhost wobj −
rhost whost +
4
4
v2
(whost )2 − (wobj )2
= ± ( x )(whost + wobj ) +
ay
4
Px = ±
= Px,steer ,
(5.3)
(5.4)
(5.5)
where, in this case, the positive solution is of interest. The distance traveled during
constant acceleration is given by
Px (t) = Px,0 +
t
Vx,0 +
t0
τ
a ds
dτ,
(5.6)
t0
1
= Px,0 + Vx,0 t + Ax,0 t2 .
2
(5.7)
70
Chapter 5
Deterministic Decision Making
Collision velocity
120
With braking
Without braking
Collision velocity [Km/h]
100
80
60
40
20
0
0
20
40
60
80
Initial velocity [km/h]
100
120
Figure 5.5: The collision speed for the head-on scenario with and without autonomous
braking. The assumption is that a deceleration of 9.82 m/s2 can be applied instantaneously when the collision unavoidable state is entered. This deceleration is applied at
the minimum of the braking and steering avoidance distance, which are plotted in Figure 5.4.
Similarly the velocity is given by
Vx (t) = Vx,0 +
t
a dτ,
(5.8)
t0
= Vx,0 + Ax,0 t.
(5.9)
By setting the final velocity Vx (t) to 0, t can be eliminated and the braking distance
can be calculated from
2
vx,0
.
(5.10)
Px,brake =
2ax
Thus, the braking distance and steering avoidance distance are
2
vx,0
,
2a
x
v2
(whost )2 − (wobj )2
.
= ( x )(whost + wobj ) +
ay
4
Px,brake =
(5.11)
Px,steer
(5.12)
As can be seen from both the equations and the graph, the braking distance is
proportional to the square of the velocity, whilst the steering distance is basically
linearly proportional to the velocity. Furthermore, for low speeds (below 40 km/h),
Section 5.3
Collision Avoidance Decision Functions
71
braking is more efficient and for high speeds (above 40 km/h), steering is more
efficient to avoid a collision. If braking is applied exactly at the collision unavoidable
boundary, the collision speed for this scenario would be as illustrated in Figure 5.5.
For scenarios where the initial speed is below 40 km/h the accident is completely
avoided. For scenarios where the initial speed is above 40 km/h, the collision speed
is reduced by 25 km/h. The results from the analysis carried out in this section
is thus very similar to the once presented in Section 2.2.4, which are based on a
detailed handling model.
5.3
Collision Avoidance Decision Functions
As already stated, a CA decision function, g(X), that measures the threat other
objects constitute is required to make any decision. The two main factors to consider when deriving a decision function are what dynamic model to use for the
host vehicle and other objects and the assumption of future actions of the host and
obstacle. The model describing dynamic properties and future action is here called
the decision model,
x̃ = xobj − xhost ,
x̃˙ = f (x̃),
(5.13a)
(5.13b)
where x̃ is the relative state vector and f (·) describes the dynamics of the relative
motion. Note, that while the tracking model is naturally in discrete time, because of
the discrete time measurement updates, the decision model is given in continuous
time. The decision model is used to calculate the future trajectory (or possibly
trajectories) according
t
f (x̃)dτ + x̃(t0 ).
(5.14)
x̃(t) =
t0
Often the same models that are used for tracking are also used for decision making.
Below the time continuous constant velocity, constant acceleration and coordinated
turn model are summarized. Note, that the constant acceleration and the constant
velocity also hold for relative states.
• Constant velocity:
T
x = PT VT
,
0 I
ẋ =
xt .
0 0
(5.15a)
(5.15b)
The solution for the position and velocity state is given by
P(t) = P0 + V0 t,
(5.16a)
V(t) = V0 .
(5.16b)
72
Chapter 5
Deterministic Decision Making
• Constant acceleration:
T
x = PT VT AT
.


0 I 0


ẋ = 0 0 I  xt .
0
0
(5.17a)
(5.17b)
0
The solution for the position and velocity state is given by
P(t) = P0 + V0 t +
1
A0 t2 ,
2
V(t) = V0 + A0 t.
(5.18a)
(5.18b)
• Coordinated turn (two dimensions):
x = Px

0

0

ẋ = 
0

0
0
1
0
0
0
0
0
1
−ω
0
ω
0
T
ω ,

0

0

0
 xt .

0
0 0
0
0
0
Py
Vx
Vy
(5.19a)
(5.19b)
The solution for the position and velocity state is given by
sin(ωt)
1 − cos(ωt)
Vx,0 +
Vy,0 ,
ω
ω
1 − cos(ωt)
sin(ωt)
Vx,0 −
Vy,0 ,
Py (t) = Py,0 −
ω
ω
Vx (t) = cos(ωt)Vx,0 + sin(ωt)Vy,0 ,
Px (t) = Px,0 +
Vy (t) = − cos(ωt)Vx,0 + sin(ωt)Vy,0 .
(5.20a)
(5.20b)
(5.20c)
(5.20d)
Given the trajectory of each object a function to measure the threat is formulated. Below several CA decision function are derived for the most commonly
assumed decision models. Also novel alternatives to deal with piecewise constant motion, actuator dynamics, constant centripetal acceleration, and general
2-dimensional acceleration are presented.
5.3.1 Headway and Headway Time
The headway (pHW ) is the distance to the obstacle in the path ahead. However,
this does not always correspond to the radial distance (r) often measured by the
sensors. One example when they differ is when the vehicle is traveling along a
Section 5.3
73
Collision Avoidance Decision Functions
curved path. A thorough treatment of curved coordinate systems is given in [27].
The headway time tHW is given by
tHW = solt {Phost (t + t0 ) = Pobj (t0 )},
(5.21)
where solt {} denotes the solution with respect to t such that the expression within
the brackets is fulfilled. Assuming both vehicles travel at equal and constant speed,
v host = v obj , ahost = aobj = 0,
(5.22)
the headway time (tHW ) is given by
tHW =
pHW
.
v0host
(5.23)
This metric is often used to evaluate the hazard level for how a car is being driven
with respect to the vehicles in front of it, see e.g., [124].
5.3.2 Time to Collision
The time to collision (tTTC ) is defined as follows
tTTC = solt {p̃x (t) = 0}.
(5.24)
When calculating time to collision the relative acceleration is often assumed to be
constant according to (5.17b). With a constant acceleration decision model of the
vehicles motion the solution to (5.24) reduces to solving
0 = p̃x,0 + ṽx,0 t +
The solution is given by

p̃

− ṽx,0
,


x,0
√


2 −2p̃
ṽx,0

x,0 ãx,0
ṽ
x,0

,

− ãx,0 − √
ãx,0
2 −2p̃
ṽ
ã
x,0 x,0
tTTC = − ṽx,0 +
x,0
,

ãx,0
ãx,0




Undefined




Undefined
ãx,0 t2
.
2
(5.25)
ṽx,0 < 0
and ãx,0 = 0,
ṽx,0 < 0
and ãx,0 = 0,
ṽx,0 ≥ 0
and ãx,0 < 0,
ṽx,0 ≥ 0
and ãx,0 ≥ 0,
2
ṽx,0
(5.26)
− 2p̃x,0 ã < 0.
2
− 2p̃x,0 ãx,0 < 0 or
Thus, tTTC is given by the smallest positive solution. If ṽx,0
(ṽx,0 ≥ 0 and ãx,0 ≥ 0) no solution exist. This means that the velocity and
acceleration is such that no collision occurs.
5.3.3 Closest Point of Approach
The headway and time to collision decision functions deal with motion in one
dimension only (obstacles that are determined to be in the path of host vehicle).
74
Chapter 5
Deterministic Decision Making
The closest point of approach (pCP A ) is often used in two-dimensional motion (often
for marine or ATC applications). It denotes the predicted minimum separation
between two vehicles,
PCP A = mint>t0 (P̃(t)).
(5.27)
Often an assumption of constant velocity according to (5.15b) is assumed when
deriving the PCP A , in this case PCP A (for 2D motion) can be derived as follows.
The separation is given by
"
P̃(t) = P̃ 2 (t) + P̃ 2 (t),
x
y
(5.28)
where the relative distance in each direction, P̃i (t), is given by (5.16a). For constant
velocity the relative separation becomes
"
P̃(t) = (P̃x,0 + Ṽx,0 t)2 + (P̃y,0 + Ṽy,0 t)2 .
(5.29)
The time for the closest point of approach (tCP A ) is obtained by solving
d|P̃(t)|2
= 0.
dt
(5.30)
This yields
P̃x,0 Ṽx,0 + P̃y,0 Ṽy,0
.
2 + Ṽ 2
Ṽx,0
y,0
(5.31)
(P̃y,0 Ṽx,0 − P̃x,0 Ṽy,0 )2
.
2 + Ṽ 2
Ṽx,0
y,0
(5.32)
tCP A = −
Thus
PCP A =
5.3.4 Potential Force
In potential field algorithms an artificial potential force acting on the host vehicle
is calculated. This force is used to generate a control input to avoid any obstacle.
This potential force can also be viewed as a collision decision function. A common
and simple expression to calculate the potential force is given by
c
F = − i P̃,
P̃
(5.33)
where c is a constant, and i ∈ N. Potential field algorithms are common in robotic
and autonomous vehicle applications. This approach was first introduced in [76],
references to different applications and more sophisticated potential field algorithms
than (5.33) are [17, 35, 42, 43, 92, 93, 95, 97, 98, 122].
Section 5.3
75
Collision Avoidance Decision Functions
5.3.5 Required Longitudinal Acceleration
The required longitudinal acceleration, ax , measures the longitudinal acceleration
required to bring the relative velocity to zero at the time of the collision
#
T T $
host
= 0 0
ax,req = solahost
,
(5.34)
p̃x (t) ṽx (t)
x
{· · · } denotes the solution with respect to the host vehicle acceleration
where solahost
x
ahost
where
p̃
x (t) ≥ 0 for all t. With the assumption of constant acceleration the
x
solution to (5.34) is given by solving the equation system
%
0=
0=
ṽx,0 + ãx t,
2
p̃x,0 + ṽx,0 t + ãx2t ,
(5.35)
This yields the constant acceleration required to avoid a collision, according to
obj
obj
ahost
x,req = ax,0 − ãx = ax,0 −
|ṽx,0 |ṽx,0
.
2p̃x,0
(5.36)
5.3.6 Required Longitudinal Acceleration for Piecewise Constant
Motion
In some cases a better motion model is to assume that all vehicles will remain
stationary when zero velocity is reached. Thus, instead of a constant acceleration,
a piecewise constant acceleration is assumed. For a situation where the obstacle
comes to a stop before the collision occurs solving (5.34) means solving the equation
system

0 = v host + ahost t,
x,0
x
(5.37)
ahost
t2
p̃
host
x
=
v
t
+
,
x,final
x,0
2
where
p̃x,final = p̃x,0 −
obj 2
(vx,0
)
aobj
x,0
.
(5.38)
With the assumption of a constant host acceleration the required acceleration to
avoid a collision is given by
host 2
)
(vx,0
.
ahost
=
−
x,req
obj 2
(vx,0
)
2 p̃x,0 − 2aobj
(5.39)
x,0
This CA decision function is also discussed for automotive application in [54].
76
Chapter 5
Deterministic Decision Making
5.3.7 Required Longitudinal Acceleration with Actuator Dynamics
The equations (5.36) and (5.39) were derived assuming that the maximum deceleration can be achieved momentaneously. In most vehicles, it will take some time
from a control input to a system response. A better CA decision function also considers the actuator dynamics. A simple model is to describe the actuator dynamic
(in this case the brake system) by a first order model Laplace domain transfer
function according to
Gbrake (s) =
k1
.
s/k1 + 1
(5.40)
Hence, instead of assuming that the acceleration can be changed instantaneously,
the assumption is that the reference acceleration aref , which is the input to the
dynamical system (5.40), can be instantaneously changed. The step response for
this system yields the following expression for the acceleration
ã(t) = aref − aref e−k1 t
(5.41)
where aref is the input requested acceleration. With the acceleration according to
(5.41) the velocity and distance traveled during deceleration can be calculated by
aref
aref e−k1 t
+ aref t +
,
k1
k1
aref e−k1 t
aref
aref t2
aref
−
t+
+ 2 .
p̃(t) = ṽ0 t −
2
k1
2
k1
k1
ṽ(t) = ṽ0 −
(5.42)
(5.43)
The time to bring the velocity to zero is given by the solution to (5.42), which
yields
tstop
−ṽ0 k1 + aref + L(−e
=
aref k1
(ṽ0 k1 −aref )
aref
)aref
,
(5.44)
where w = L(x) denotes the solution to wew = x. By combining (5.44) and
(5.43) the stopping distance to be calculated. A decision function accounting for
actuator dynamics is given by the stopping distance calculated for aref = −aa ,
where aa denotes the maximum possible acceleration. This can be compared to
the measured distance, to determine any action. In Figure 5.6 a comparison of the
braking distance according to (5.36) and (5.43) is shown.
Example 5.1: Brake System Model
To calculate the stopping distance according to the model above, the brake system
characteristics need to be known. A complete brake system model is very complex.
However, looking at overall vehicle performance, a simple yet accurate description
can be obtained for the vehicle’s acceleration by the first order system in (5.40).
Figure 5.7 shows measurement data from ten brake maneuvers with a Volvo V70
Section 5.3
77
Collision Avoidance Decision Functions
Brake distance
Alarm delay
60
0.14
(5.39)
(5.43)
0.12
40
0.1
Alarm delay [s]
Brake distance [m]
50
30
0.08
20
0.06
10
0.04
0
0
50
100
150
Initial velocity [km/h]
0.02
0
50
100
150
Initial velocity [km/h]
Figure 5.6: The left figure shows the braking distance according to (5.36) and (5.43),
respectively. The maximum acceleration is assumed to be 9.82 m/s2 and the rise time
for first order brake system is 0.3 seconds. If the actuator dynamics is neglected in the
decision making, as in (5.36), the CA action will be delayed compared to a criterion that
accounts for the actuator dynamics ,(5.43). This delay is illustrated in the right figure.
Measured and simulated braking system
Simulated
Measured
2
Acceleration [m/s ]
0
−5
−10
−15
0
0.5
1
Time [s]
1.5
2
Distance [m]
20
10
5
0
0
Figure 5.7:
Simulated
Measured
15
0.5
1
Time [s]
1.5
2
Data from a hard braking maneuver, 110–0 km/h for a Volvo V70.
78
Chapter 5
Deterministic Decision Making
and a first order approximation. The measurements were performed on warm, dry
asphalt. For the simulated data a first order system was used with transfer function
Gbrake (s) =
k1
,
s/k1 + 1
(5.45)
with k1 = 7 and aref = 11 identified such that the system rise time is 0.3 s and
the stationary value is 11 m/s2 , Comparing distance traveled after 2 seconds the
difference is less than 1 m, which is sufficient for CA purposes. More information
on typical brake system behavior can be found in [3].
5.3.8 Required Lateral Acceleration
Similar to ahost
it is possible to derive the lateral acceleration to avoid a collision.
x
This means looking for a steering maneuver that makes the host vehicle pass either
to the right or to the left of the obstacle. This is given by
%
Ahost
y,i
= solAhost
y
&
p̃x (t)
0
=
,
whost /2 + wobj /2
|p̃y (t)|
(5.46)
Here constant acceleration (in the global coordinate system) is assumed, according
to (5.17b). The lateral position with constant acceleration is given by (5.18a).
The time to collision is given by (5.26). Hence, solution to (5.46) is obtained from
solving
±(
Ãy t2TTC
wobj
whost
+
) = P̃y,0 + Ṽy,0 tTTC +
,
2
2
2
(5.47)
which yields
obj
Ahost
y,i = Ay − Ãy
host
obj
2 −P̃y,0 ± ( w 2 + w 2 ) − Ṽy,0 tTTC
= Aobj
.
y,0 −
t2TTC
host
(5.48)
obj
The sign in front of ( w 2 + w 2 ) determines the acceleration required to pass to
host
the left (Ahost
y,left ) and right (Ay,right ) of the obstacle, respectively. The minimum
lateral acceleration to avoid a collision is given by
host
host
Ahost
y,req = min(|Ay,left |, |Ay,right |).
(5.49)
Note that the acceleration is kept constant in the ground fixed coordinate system.
Thus, the resulting vehicle trajectory is parabolic.
Section 5.3
79
Collision Avoidance Decision Functions
5.3.9 Required Centripetal Acceleration
An alternative way to calculate the acceleration required to avoid a collision by
some steering maneuver is to assume a constant lateral acceleration in a vehicle
fixed coordinate system. This means finding a solution to (5.46) for the case where
the host vehicle’s motion is now described by a circular path according to the
coordinated turn model (constant radial velocity and centripetal acceleration) see
Figure 5.3. Finding a solution to (5.20a) can be difficult so here an approximation
is used. First the time to collision is calculated according to (5.26). Furthermore,
the obstacle moves with constant acceleration according to (5.17b). For t = tTTC
the position of the obstacle’s right or left edge is given by
2
Aobj
x tTTC
,
2
2
Aobj
wobj
y tTTC
±
.
= P̃y,0 + Vyobj tTTC +
2
2
obj
Px,TTC
= P̃x,0 + Vxobj tTTC +
(5.50)
obj
Py,TTC
(5.51)
Now the static reasoning that was used in Chapter 5.2 can be used. The radius of
the circle is given by
(v host )2
(5.52)
rhost = xhost .
ay
Assuming the center of the vehicle passes through the origin, the trajectory swept
by the right side of the vehicle is given by
(rhost + whost /2)2 = Px2 + (Py − rhost )2 .
(5.53)
Thus the circle radius is given by
host 2
r
host
Px2 + Py2 − (w 4
=
whost + 2Py
)
.
(5.54)
Inserting (5.52) into (5.54) and solving for the acceleration yields the required
centripetal acceleration to pass the obstacle on the left side.
ahost
y,l =
(vxhost )2 (whost + 2Py,TTC )
2
2
Px,TTC
+ Py,TTC
−
(w host )2
4
.
(5.55)
For the right maneuver the acceleration is instead given by
ahost
y,r =
(vxhost )2 (whost − 2Py,TTC )
2
2
Px,TTC
+ Py,TTC
−
(w host )2
4
.
(5.56)
For the special case that the obstacle is stationary and has the same width as the
host vehicle (5.55) and (5.48) yield the same expression.
80
Chapter 5
Deterministic Decision Making
5.3.10 Multi-Dimensional Acceleration to Avoid Collision
The required accelerations (5.48) and (5.55) are restricted to work only in one
predefined direction (longitudinal or lateral). A simple way to extend these onedimensional metrics to consider both longitudinal and lateral maneuvers, is simply
to take
|, |Ahost
|).
Areq = min(|Ahost
x
y
(5.57)
A CA decision algorithm based on (5.57) will be called a single obstacle constant
control (SOCC) algorithm, since it only consider one obstacle and escape maneuvers with a constant acceleration. Note that for the SOCC algorithm an avoidance
maneuver is restricted to be only in the lateral or longitudinal direction. Furthermore, there is no distinction between different directions i.e., it is assumed that
the bound for lateral and longitudinal acceleration is the same. A tire’s ability to
generate forces in an arbitrary direction is normally described by an ellipse, the
friction ellipse (Ef ), see e.g. [37, 103, 128]. An alternative to thresholding (5.57) is
to determine if the acceleration vector in the vehicles longitudinal direction, Ahost
x,req ,
or in the lateral direction, Ahost
,
lies
within
the
friction
ellipse,
y,req
Ahost
x,req ∈ Ef
or Ahost
y,req ∈ Ef .
(5.58)
In order to evaluate (5.58) the coefficient of friction, µ, must be known. The
estimation of tire-to-road friction is discussed in [44, 45].
For general two-dimensional motion, it is possible to calculate, analytically, the
minimum constant acceleration vector (for an arbitrary direction) required to avoid
a collision. The value of the acceleration vector can be calculated from
2
2
2
(Ahost
req ) = Ax + Ay,i ,
(5.59)
where Ay,i , (5.48), can be expressed as a function of Ax , where tTTC is calculated
according to (5.26). Thus, (5.59) is differentiated with respect to Ax
2
∂(A2x + A2y (Ax ))
∂(Ahost
req ) (Ax )
=
,
∂Ax
∂Ax
and the min and max points are found by the solution to
2
∂(Ahost
req ) (Ax )
= 0.
∂Ax
(5.60)
(5.61)
The solution yields min and max points for the acceleration vector Ahost
req . Note
that the solution has to be calculated both for passing to the left and to the right
of the object. The special case with only braking (given by (5.36)) should of course
also be considered. A collision can be avoided if
Ahost
req ∈ Ef .
(5.62)
Finally it should be noted that a solution to (5.61) is easily obtained by any symbolic interpreting equation solver. The expression is however complicated and not
easily written down. Figure 5.8 shows the difference in alarm distance between
(5.57) and the optimal solution from (5.61).
Section 5.3
81
Collision Avoidance Decision Functions
Alarm distance
Alarm delay
18
16
0.16
(5.57)
(5.61)
0.14
0.12
Alarm delay [s]
Alarm distance [m]
14
12
10
8
0.1
0.08
0.06
6
0.04
4
0.02
2
0
50
100
Initial velocity [km/h]
0
0
50
100
Initial velocity [km/h]
Figure 5.8: Braking distance according to (5.57) and (5.61). The maximum acceleration
is assumed to be 9.82 m/s2 laterally and longitudinally. The right plot shows how much
later an alarm is issued according to (5.61) compared to (5.57) if these functions are used
for a CA decision.
82
Chapter 5
Deterministic Decision Making
Chapter 6
Multiple Obstacles Decision
Making
How is the threat of a collision in a multiple object scenario evaluated? In this
section this question is simplified. The alternative question is: can a conflict be
avoided or not? The set of input signals that avoid conflicts will be called the set
of admissible maneuvers. For the multiple object case the admissible maneuvers
are given by
(6.1)
Ut = Ut1 ∩ Ut2 ∩ · · · ∩ Utn ,
An
for n observed objects. If Ut = ∅, the empty set, a conflict is unavoidable.
alarm is issued if too much control effort is required (minUt (maxt Ut ) > umax )
or no admissible control sequence is found, Ut = ∅. In this section three different
approaches to find Ut , with increasing computational complexity are proposed.
Perfect state knowledge is assumed, i.e., process and measurement noise will be
neglected. It is stressed that the main objective in this section is not to actually
control the vehicle, but merely to determine if a conflict can be avoided or not.
6.1
Constant Control
The first approach will be called the constant control method. The idea here is to
try to find a constant control signal uhost
= uhost
that avoids conflicts. Allowing
t
t0
only constant avoidance maneuvers is of course a severe restriction but it may allow
the derivation of an analytical decision function. The decision functions derived in
Section 5.3 used an assumption of constant control (constant velocity or constant
acceleration). The closest point of approach (5.32), which is common in marine
and ATC applications [17, 83], was derived under the assumption of a constant
velocity. The longitudinal acceleration required to avoid a collision (5.36), which is
often used in automotive applications [2, 20, 24, 119, 130], was derived under the
assumption of constant acceleration.
83
84
Chapter 6
Multiple Obstacles Decision Making
ay
E
F2
F
F1
Figure 6.1: The host vehicle is approaching two stationary obstacles. Only constant
avoidance maneuvers according to (5.48) or (5.36) are considered. Thus, all considered
escape paths are parabolic trajectories or possibly straight lines. Each obstacle defines
a set of “forbidden” lateral accelerations. The union of all sets (Fi ) defines the set F of
maneuvers leading to a collision.
6.1.1 Multiple Objects Constant Acceleration
In this section the notion of constant longitudinal and lateral acceleration to avoid
collision (5.57) to deal with multiple object scenarios. To do this a set of “forbidden” lateral accelerations is defined according to
host
,
(6.2)
Fj = ahost
,
a
j,y,left j,y,right
for each obstacle j = 1, · · · , n. A new set is then formed
F = F1 ∪ · · · ∪ Fn .
(6.3)
/ F,
Ut ∈
(6.4)
Furthermore
denote all lateral accelerations that avoid collision and
E = [−aa , aa ],
(6.5)
denotes the set of all feasible lateral accelerations. If
Ut ∩ E = ∅,
(6.6)
there exists no (constant) lateral acceleration that avoids collision. A multiple
object scenario and the formation of F is illustrated in Figure 6.1. The lateral
Section 6.2
85
Heuristic Control
acceleration to avoid collision in the multiple object case is given by
ay,multi = min(|Ut |).
(6.7)
The required longitudinal acceleration in the multiple object case is given by
ax,multi = min(aj,x ),
(6.8)
where {j : 0 ∈ Fj }. The multiple object acceleration to avoid a collision is thus
amulti = min(|ax,multi|, ay,multi ).
(6.9)
The underlying motion model for this criterion is (4.44). Because of the assumption
of constant acceleration (either in the longitudinal or lateral direction) the escape
paths are always parabolic trajectories or straight lines. A decision algorithm based
on (6.9) will be called a multiple obstacle constant control (MOCC) algorithm in
this thesis. The MOCC algorithm is also described in [60].
6.1.2 Summary of the Constant Control Approach
Summarizing the constant control approach, its strength is that a closed form
decision function such as (6.9) may be found. The drawback of the approach is
that if a more complex maneuver is required to avoid collision (see for instance
Figure 6.2) the algorithm will fail to find this maneuver.
6.2
Heuristic Control
In the second approach, the heuristic control method, a control sequence Ut that
avoids collision is searched for by simulating the future movements and using a
heuristic control scheme ut = J(Xt ) to control the host vehicle. For each time
instance, proximity to obstacles is checked. If there is a conflict an alarm is issued.
An outline of a general heuristic control algorithm is given in Algorithm 6.1. There
are several areas of application where a controller that avoids collision has to be
designed. Robot manipulators and autonomous moving robots are applications
where much work has been devoted to finding control strategies that avoid collision.
In these applications, reactive control strategies [43, 92, 93, 95, 97, 98, 122] and
path planning schemes [74, 84] have been extensively researched. Note that here
the controller is only used in a simulation to see if a conflict occurs, not to actually
control the vehicle. Here an alternative strategy is proposed. The strategy uses
the acceleration required to avoid collision according to (5.57) from the previous
section. This algorithm will be referred to as the multiple obstacles heuristic control
(MOHC) algorithm. The heuristic control approach offers the advantage in that it
can find complex maneuvers that avoid collision. The disadvantage on the other
hand is that it requires more computations than the constant control approach
(i.e., simulating the whole event), and there is no guarantee that an existing feasible
avoidance maneuver will be found. This means that just as for the constant control
approach the algorithm may issue a warning even though a maneuver that avoids
conflicts exists.
86
Chapter 6
Multiple Obstacles Decision Making
In this scenario the relative velocity is too high for braking to avoid collision.
Just steering will cause the host vehicle to collide with the vehicles in the adjacent lanes.
However, a hard braking maneuver followed by a left steering maneuver will avoid the
collision. Thus, the constant control strategy will fail (i.e., give an alarm even though the
collision is not unavoidable). The algorithm from Example 6.2, on the other hand, detects
that as the oncoming vehicle moves forward, steering will eventually be a possibility to
avoid the collision.
Figure 6.2:
T
1. Calculate the control input ut = J(Xt ); Xt = xhost , xobj1 , · · · , xobjn .
2. Simulate the time updates Xt+T , using ut according to the decision model
(5.13).
3. Check proximity between the host and obstacles. If there is a conflict discontinue the simulation and issue an alarm.
4. If t = t0 + N T the simulation is discontinued. A conflict free path has been
found.
5. Go back to 1.
Algorithm 6.1: A general heuristic control algorithm.
Section 6.3
Exhaustive Search
87
The basic idea is to choose an obvious escape path (constant acceleration) if it
exists. Otherwise, full braking is applied until an obvious escape path appears or
the vehicle collides. Here is an outline of the algorithm.
1. Calculate the acceleration required to avoid collision according to (5.57). If it
is below the threshold for a collision being unavoidable an escape path exists
and the simulation can be aborted.
= −aa .
2. Calculate the time updates Xt+T , using uhost
x
3. Check the proximity to obstacles, dispatch an alarm if a conflict is detected.
4. If t = t0 + N T the simulation is discontinued without an alarm.
5. Otherwise go back to 1.
Algorithm 6.2: Multiple Obstacles Heuristic Control
6.3
Exhaustive Search
It is desirable to have a method that guarantees finding any existing avoidance
maneuver. This is not achieved by the previous two approaches. However, finding
a conflict-free escape path is clearly not a convex problem. Thus, gradient search
methods may not be efficient. As a third alternative an exhaustive search over all
possible maneuvers is therefore proposed. This will theoretically guarantee finding
a solution, if it exists. In practice though, both time and the control space have to
be discretized. This means that ut is approximated with a finite number of control
sequences ūt ∈ J ∗ = {(ut0 , · · · , utN )|uti ∈ {u(1) , · · · , u(M) }} ⊂ J, where u(i) are
the possible values of the discretized control signal. Thus, only a finite set of M
possible control inputs is used at each time instant, and the time interval (from
t = t0 to t = tN ), is divided into N segments of length T . This means that the
exhaustive search may also fail to find an existing solution for specific scenarios
if the discretization is made too sparse. How should N and M be chosen? To
answer this we need to know how well an arbitrary trajectory generated with ut ∈
(k)
[umin , umax ] can be described, by the best approximating control sequence Ut ,
n
where k = 1 · · · M . Below, results for this are given for a constant acceleration
model according to (4.44) and where uti ∈ {−aa , 0, aa }. The tree of possible paths
to approximate the true vehicle trajectory with this model is shown in Figure 6.3.
The error at time tN = N T is given by pN (u) − pN (ū) = pN − p̄N = ∆pN where
88
Chapter 6
Multiple Obstacles Decision Making
Tree of possible paths
0.8
Leaf
0.6
Node
Position [m]
0.4
0.2
Branch
umax
u0
0
u
min
−0.2
−0.4
−0.6
−0.8
0
0.05
0.1
0.15
0.2
0.25
Time [s]
0.3
0.35
0.4
Figure 6.3: Tree of possible vehicle path approximations (1-dimensional motion). At each
time instant three possible maneuvers are allowed ut ∈ {umin , 0, umax }
.
NT
p̄N = p̄0 +
v̄0 +
0
τ
ā(s) ds
dτ
0
= /p̄0 = 0, v̄0 = 0/
N T τ
ā(s) ds dτ
=
0
0
= /a(s) piecewise constant/
τ
N
−1 (i+1)T ā(s) ds dτ
=
v̄i +
i=0
=
N
−1
i=0
=
N
−1
i=0
=


iT
(i+1)T
iT


iT
i
āj T +
j=0

2
a
¯
T
i


āj T 2 +
2
j=0

(N − 1 − i)āi T 2 +
i=0
=
N
−1
i=0
(2N − 2i − 1)
i=0


ā(s) ds dτ 
iT
i
N
−1
N
−1
τ
a¯i T 2
2
2
āi T
.
2
(6.10)
Section 6.3
89
Exhaustive Search
Assume a constant acceleration model according to (4.44), where
motion is independent for each direction. The acceleration is determined by the
control input u(t) ∈ [−aa , aa ], where aa is the maximum available acceleration. An
arbitrary trajectory generated by u(t) can then be approximated with a piecewise
constant input ū(t) = {ū0 , · · · , ūN −1 }, ui ∈ {−aa , 0, aa } for time t = t0 to tN ,
ti+1 − ti = T . ūi , i = 0, · · · , N − 1 can be chosen such that the error at time tN
Theorem 6.1
tN
0
t
u(s) dsdt −
∆pN =
0
0
tN
t
ū(s) dsdt ≤
0
aa T 2
.
4
(6.11)
Proof of Theorem 6.1 Follows directly from (6.10), which gives all possible
positions of the nodes in the tree. The nodes are equidistant with a separation of
aa T 2
2 , as shown in Figure 6.3.
Assume a constant acceleration model according to (4.44), where
motion is independent for each direction. The acceleration is determined by the
control input u(t) ∈ [−aa , aa ], where aa is the maximum available acceleration. An
arbitrary trajectory generated by u(t) can then be approximated with a piecewise
constant input ū(t) = {ū0 , · · · , ūN −1 }, ui ∈ {−aa , 0, aa } for time t = t0 to tN ,
ti+1 − ti = T . ūi , i = 0, · · · , N − 1 can be chosen such that the maximum position
error for all i = 0 · · · N is
Theorem 6.2
aa (tN − t0 )2
≤
.
(
max
(∆p
))
min
i
ūi ∈{−aa ,0,aa } 1≤i≤N
2N
(6.12)
Proof of Theorem 6.2 It is always possible to choose
a strategy ui so that
t
|∆vi | < aa T /2, ∀i, where ∆vi = vi − v̄i and v̄i = t0i (ū(t)). This is shown by
induction
1. ∆v0 = 0
2. Suppose 0 ≤ |∆vn | ≤ aa T /2.
3. Then it is true that ūn can be chosen so that 0 ≤ |∆vn+1 | ≤ aa T /2. ∆vn+1 =
∆vn + ∆un T = ∆vn + (un − ūn )T . We have |∆vn + un T | ≤ |aa T /2| + |aaT | =
3aa T /2 thus, there are three cases:
(a) aa T /2 ≤ ∆vn + un T ≤ 3aa T /2. In this case choose ūn = −aa ⇒
|∆vn+1 | ≤ aa T /2.
(b) −aa T /2 < ∆vn +un T < aa T /2. In this case choose ūn = 0 ⇒ |∆vn+1 | <
aa T /2.
(c) −3aa T /2 ≤ ∆vn + un T ≤ −aa T /2. In this case choose ūn = aa ⇒
|∆vn+1 | ≤ aa T /2.
90
Chapter 6
Multiple Obstacles Decision Making
Thus, with this choice of ū, one has
ti
∆v(t)dt| ≤
|∆pi | = |
ti
|∆v(t)|dt
(6.13)
aa T /2dt
(6.14)
= N aa T 2 /2 = aa (tN − t0 )2 /2N.
(6.15)
≤
t0
tN
t0
|∆v(t)|dt ≤
t0
tN
t0
Proposition The bound given in Theorem 6.2 may actually be reduced a factor 2
by choosing a smarter strategy that not only tries to limit the absolute value of ∆v
but also makes sure that ∆v changes sign (if necessary) so that |∆p| first increases
and then decreases.
An exhaustive search quickly becomes computationally expensive as the discretization is made finer. Thus, this method may not be useful for on-line applications. However, it may be useful for off-line applications as a reference for
automatic testing of other decision making algorithms. In particular for some applications (such as CMbB) it is sufficient to find one escape path, i.e., the entire
tree of possible manoeuvres does not need to be searched. When a escape path has
been found the search can be discontinued. In this case an efficient strategy for
searching the tree of possible manoeuvres may provide significant reduction to the
average time of traversing the tree. In Section 9.1 a comparison between the three
approaches, to finding a manoeuvre that avoids collision, is presented. A general
discussion on the computational complexity of finding collision free paths can be
found in [111].
An alternative approach to multiple obstacle collision avoidance decision making
is to view the problem as a path planning problem [74, 84, 95, 122]. A difference
between a typical path planning problem and the CA decision problem (as posed
in this thesis) is that for the path planning problem there is a well defined goal
state while for the CA problem the goal state is not of importance.
Chapter 7
Statistical Decision Making
The decision making algorithm of a CA system bases its decision on the output
from the tracking algorithm. This output is associated with some uncertainty.
Thus, the CA decision is based on uncertain parameters. Furthermore, in general
several actions a could be considered. The CA strategy can be described by a rule
data-base (DB), that for each possible state determines what action to take. Here
deterministic decision making functions as described in Section 5.3 are often used.
Let the rule database be denoted by the nonlinear function
(7.1)
r(Xt ) = a,
T
obj1
obj2
objn
where Xt = xhost
. Each state is nonlinearly mapped
x
x
·
·
·
x
t
t
t
t
to one action a. The action a can be continuous or discrete. In the sequel it is
assumed to be discrete and take on discrete values ak . The nonlinear function
r(Xt ) will be referred to as the CA strategy. Typically, the CA strategy is based
on thresholding one or more decision function g(Xt ) to determine on an action,
e.g.



g(Xt ) < g1
⇒ a1 ,




g ≤ g(X ) < g ⇒ a2 ,
1
t
2
r(Xt ) =
(7.2)

3

≤
g(X
)
<
g
⇒
a
,
g
 2
t
3



g ≤ g(X )
⇒ a4 ,
t
3
Where gi denotes threshold values chosen to get the appropriate CA action. The
problem is that the state estimates are uncertain causing the decision to be uncertain. An overview of the decision problem is given in Figure 7.1. In Example 7.1
a decision making problem based on uncertain state estimates is described.
Example 7.1
In this example a simple one-dimensional collision avoidance problem is considered.
The distance required to avoid a collision with a stationary object, using a deceleration of 1 g ≈ 9.82 m/s2 , for different initial velocities is shown in Figure 7.2.
91
92
Chapter 7
Sensors
yt
✲ Filter
)
p(xt |Yt✲
?
Statistical Decision Making
x✲
Action✲akt
t
CA decision
The problem: how to interface uncertain state estimates with deterministic
decision making?
Figure 7.1:
Braking distance versus relative speed. The elliptical region represents the
uncertain state estimate and the shaded region defines the collision unavoidable state,
which should lead to an automatic braking action.
Figure 7.2:
The CA system is supposed to perform an autonomous action when a deceleration
(according to (5.36) larger than 9.82 m/s2 is required to avoid the collision. Hence,
the two CA actions, a1 and a2 , are braking and no braking. Suppose an object has
been detected close to the decision region. An uncertainty region for that object is
calculated as indicated in Figure 7.2. Should the system perform an autonomous
action in this case? This is a strategic decision for the CA system designer. If the
tolerance for false alarms is low, one should require a high confidence of the braking
action to be correct. Thus, in this case one should not take any action. Conversely,
if one wants to maximize safety one should take action even when there is only a
slight possibility that the collision is unavoidable. If the uncertainty region is time
varying the probability of making the correct decision will vary. In this case decision strategy r(Xt ) needs to be augmented to deal with the uncertainty associated
with the decision.
To calculate the probability for each action Prob(ak |Yt ), given the measurements
Section 7.1
93
Numerical Approximation
Yt , one needs to evaluate
Prob(ak |Yt ) =
Ir(Xt )=ak p(Xt |Yt ) dx.,
where Ir(Xt )=ak is an indicator function according to

1 if r(X ) = ak ,
t
Ir(Xt )=ak =
0 otherwise.
(7.3)
(7.4)
Probabilistic decision making for automotive CA systems is discussed [52, 53, 61,
62, 65, 73], for airspace applications it is discussed in [108, 109] and in [81] which
give a excellent overview of methods used airspace applications, in [97] a probabilistic potential field algorithm is discussed, and in [96] path planning with a
probabilistic description of future positions is presented.
7.1
Numerical Approximation
In most cases an analytical solution for (7.3) cannot be found. Using stochastic
integration the integral (7.3) can be approximated by
k
Prob(a |Yt ) = Ir(Xt )=ak p(Xt |Yt ) dx
≈
N
1 I
k,
N i=1 r(Xt )=a
xi ∈ p(Xt |Yt ),
(7.5)
where the xi ’s are samples of the posterior distribution. In Example 7.2 an approximation of the PDF for the decision function time to collision (5.24) is generated by
drawing a large number of samples from the state vector distribution. By evaluating (7.5) a specific action can be performed with a predefined confidence according
to
Prob(ak ) > 1 − α,
(7.6)
where α determines the confidence level. Thus, one may now decide to take each action with a certain confidence, as shown in Figure 7.3. An example where stochastic
numerical integration is used to calculate the probability of a collision avoidance
action is given in Section 9.2 and is also described in [66].
Example 7.2
In many cases it is impossible to calculate the PDF of the decision function analytically. In this example the time to collision tTTC according to (5.26) will be
studied. Assume that the states are independent and

  
p̃x
N (10, 0.52 )

  
x̃ = ṽx  ∈ N (−3, 0.52) .
ãx
N (−4, 22 )
94
Chapter 7
Statistical Decision Making
p(xt |Yt✲
Action ai with confidence✲
)
pi
✲Supervisor
yt
Sensors
Figure 7.3:
✲ Filter
✲
Rule
{xkt }N
k=1
DB
{akt }N
k=1
The solution: a supervisory algorithm that assesses confidence in action
decisions.
The probability density for the tTTC can be approximated by drawing a large
number of samples from the PDF of x̃ and for each sample calculate the tTTC .
This is shown in Figure 7.4. In this case, the resulting PDF is not Gaussian,
and it is also skewed so that the mean value is tTTC = 1.70 s while the nominal
tTTC = 1.61 s.
PDF for tTTC
1.6
1.4
Probability density
1.2
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
Time−to−collision [s]
3
3.5
4
The PDF for the tTTC (5.26). p̃x ∈ N (10, 0.52 ), ṽx ∈ N (−3, 0.52 ), ãx ∈
N (−4, 2 ). The resulting PDF is not Gaussian and it is skewed toward higher tTTC ’s.
The mean value is tTTC = 1.70 s while the true tTTC = 1.61 s.
Figure 7.4:
2
Section 7.2
7.2
95
Analytical Approximation
Analytical Approximation
In the previous section statistical decision making based on stochastic numerical
integration was discussed. The reason for the numerical approach is that an analytical solution for (7.6) is often impossible to find. The disadvantage of the
general solution, provided by the numerical approach (7.5), is that the resulting
algorithm is computationally expensive. In this section an alternative decision rule
is proposed. A simple CA strategy with only one CA action according to
(g(X̂t ) > gth ) ⇒ a,
(7.7)
is considered. Accounting for the uncertainty leads to the following CA strategy
Prob(g(X
(7.8)
t ) > gth ) > 1 − α ⇒ a,
where α determines the confidence. A CA strategy according to (7.7) is computationally cheap, but if p(Xt ) varies a lot the probability of making the right decision
might also vary significantly. A CA strategy according to (7.8) is, on the other
hand, computationally expensive. An alternative CA strategy is given by
g(X̂t ) − C1 B̂ (g(Xt )) > gth + C2 D̂ (g(Xt )) ⇒ a.
(7.9)
Here B(·) and D(·) denote bias and standard deviation respectively, and Ci are
design parameters. A CA strategy according to (7.9) relies on an assumption that
the distribution p(g(x̂t )) is well approximated by a Gaussian distribution. If g(x̂t )
truly is Gaussian, (7.9) corresponds to evaluating (7.8), and the decision will be
made with a certain confidence, which is determined by C1 and C2 .
7.2.1 Estimating Statistical Properties of the Decision Function
How are the bias B (g(Xt )) and the standard deviation D (g(Xt )) calculated? The
bias and standard deviation of (7.9) is defined in the following way
B (g(X)) = E (g(X)) − g(X),
"
2
D(g(Xt )) = E (g 2 (Xt )) − E (g(Xt )) ,
(7.10)
(7.11)
'
(
where E(·) denotes the expectation value. E g j (Xt ) is known as the j:th order
zero point moment of g(Xt ), which is defined by
j
E(g (Xt )) =
g j (Xt )p(g(Xt )) dXt ,
(7.12)
Rn
where p(g(Xt )) is the PDF of g(Xt ). Considering that g(·) can be a quite complicated function, the integral (7.12) or even the PDF p (g(Xt )) might be hard
to compute analytically and computationally costly to approximate by numerical
integration. Hence, an approximation of the expected value in (7.12) will be used,
96
Chapter 7
Statistical Decision Making
where an assumption of a Gaussian distributed Xt is used. The approximation is
obtained from the expectation value of a Taylor expansion of g j (Xt ) given by
g j (Xt ) = g j (X̂t ) +
n
n n
∂g j
∂ 2gj
X̄i +
X̄i X̄j + · · · ,
∂Xi
∂Xi ∂Xj
i=1
i=1 j=1
(7.13)
where X̄i = Xi − X̂i and Xi denotes the i:th element of the state vector Xt . An
approximation of the expectation E(g j (Xt )) can thus be found from the expression
E(g j (X)) = g j (X̂) +
n
n n
∂g j
∂ 2gj
E(X̄i ) +
E(X̄i X̄k ) + · · · .
∂Xi
∂Xi ∂Xk
i=1
i=1
(7.14)
k=1
By choosing how many terms of the series expansion to consider the accuracy and
computational burden can be traded off. If terms of the second order and higher is
neglected, this results in the well known Gauss approximation formula. To evaluate
(7.14) it is clear that higher order moments of Xi need to be known. In the case
Xt ∈ N (X̂t , Pt ) these can be calculated from the characteristic function
T
Ψ(t) = eb
X̂+ X
T PX
2
.
(7.15)
T
and P is the covariance matrix of X. The j:th order
Here b = b1 b2 · · · bn
moments of Xi are given by
E(Xij ) =
∂ j Ψ(t) ∂bji
b=0
.
(7.16)
Thus, inserting the approximation from (7.14) into (7.10) and (7.11) an approximation of B (g(Xt )) and D (g(Xt )) can be calculated. The same reasoning can
of course also be used to estimate other interesting statistical properties such as
skewness and kurtosis. In Section 9.3 the application of (7.7) and (7.9) to the
longitudinal acceleration to avoid a collision, (5.36), is compared.
7.3
Probability of Collision
In the previous section a method for calculating the probability for each possible
CA decision was proposed. The decision is based on a decision function where
each action can be made with a predefined confidence. Sometimes an appropriate
decision function cannot be described by an analytic expression. This may for
instance be the case if one is interested in using a complex dynamic model for the
vehicles involved. In this case an alternative is to simulate the system for future
time instants and calculate the probability for a collision in each time instant. Thus,
p(xt+T |Yt ), · · · ,p(xt+T N |Yt ) must be calculated for each vehicle. The number of
predictions N is determined by how fine time is discretized and how far in the
Section 7.3
97
Probability of Collision
H
p̃ = p̃x
T
p̃y
D
The set D corresponds to all possible configurations between the two vehicles
so that their bodies intersect, i.e., they collide. The area indicated in the figure is not
actually an area since it not only depends on the relative position between the two vehicles,
but also on their heading angles. So the indicated area is actually a 3-dimensional volume
of the vehicle’s relative position and orientation. It is over this volume the PDF should
be integrated to obtain the collision probability.
Figure 7.5:
future one wish to check for collisions. The probability of a collision can then be
calculated as
Prob(collision) = max (Prob(xt+iT ∈ D)) ,
i
i = 1, · · · , N,
(7.17)
where D denote the state space region corresponding to a collision. Below an
example is presented for calculating the probability of collision in a two dimensional
case.
The probability of collision for each time instant can be calculated according to
Prob(collision at time t + T ) = Prob(p̃x,t+T , p̃y,t+T , ψ̃t+T ∈ D)
(7.18)
=
pt+T (p̃x , p̃y , ψ̃|Yt ) dxdydψ,
p̃x ,p̃y ,ψ̃∈D
where D (Figure 7.5) is the set of all possible vehicle configurations corresponding to
a collision, p̃x is the relative longitudinal position, p̃y is the relative lateral position,
and ψ̃ is the relative heading angle. The probability density pt+T (p̃x , p̃y , ψ̃|Yt ) of
the vehicles’ relative position is obtained directly from the Bayesian solution to the
tracking problem, if relative coordinates are chosen. If ground fixed coordinates are
used, the density of the relative position has to be calculated before the collision
probability can be calculated. Assuming xhost and xobj are independent, the density
of x̃ = xhost − xobj is given by the convolution (7.19)
∞
pxobj (x)px̃host (x̃ − x) dx.
(7.19)
pX̃ (x̃) =
−∞
98
Chapter 7
Statistical Decision Making
PDF for a vehicle traveling at 60 km/h
6
5
4
3
2
1
0
4
2
0
−2
−4
global y coordinate [m]
90
100
98
96
94
92
102
104
global x coordinate [m]
Figure 7.6: Probability density of a car traveling at 60 km/h, where Qt has been chosen
to correspond to typical handling properties of the car. The plot shows the PDF at time
t+0.7, t+0.8, t+0.9 and t+1.0 seconds respectively.
In general, calculating (7.19) and (7.18) may be computationally demanding. For
the special but common case where xhost and xobj are Gaussian and independent
the relative states are also Gaussian, x̂host − x̂obj ∈ N (xhost − xobj , P host + P obj ) =
N (x̃, P̃ ). An example of how the probability density can evolve over time for
a Kalman filter estimate is shown in Figure 7.6. If the relative heading angle
is disregarded, D can be approximated with a rectangle, the width of which is
whost + ŵobj and length is lhost . Here w denotes the width of the object and lhost
denotes the host vehicle’s length. The length of the obstacle is neglected since it
may not be known. With the assumption that movement in the x and y direction
is independent the probability of a collision can be calculated according to
pt+iT (p̃x , p̃y |Yt ) dxdy
Prob(collision at time t + i) =
p̃∈D
=
whost +ŵobj
2
−(whost +ŵobj )
2
w
= (Φ(
where
−
host
+ŵ
2
obj
σy
(y−p̃y )2
e 2Π̃yy
"
dy
2π Π̃yy
− py
2
0
−lhost
) − Φ(−
− (x−p̃x )
e 2Π̃xx
dx
2π Π̃xx
w host +ŵ obj
2
σy
Π̃xx
Π̃ =
0
− py
))(Φ(
0 − px
lhost − px
) − Φ(−
)),
σx
σx
(7.20)
0
Π̃yy
(7.21)
Section 7.3
99
Probability of Collision
denotes the state error covariance matrix for the relative position. Algorithm 7.1
summarizes decision making using probability of collision to determine if the system should be activated. In the algorithm an assumption that movement in the x
and y direction is independent is made. The area D is approximated with a rectangular area. This algorithm will be used in the first demonstrator vehicle which
(j)
(j)
1. x̂host
x̂t P̂thost and P̂t where there are j = 1, · · · , n objects and t =
t
T, · · · , N T , is calculated using Algorithm 4.2.
2. Transform the position and its covariance matrix to a host vehicle fixed coordinate system (if relative coordinates are used this
step is not needed).
sin(ψ
)
−
cos(ψ
)
t
t
host
); Γt =
p̃ = Γt (p̂obj
t − p̂t
cos(ψt )
sin(ψt )
Π̃t = Γt (Πobj
+ Πhost
)ΓTt
t
t
3. Calculate the collision probability for each time, t, and each obstacle, j.
Prob(collision
at time t + iT ) =
whost +ŵobj
whost +ŵobj
−py,t
−py,t
0−px,t
lhost −px,t
2
2
Φ(
= Φ(
)
−
Φ(−
)
)
−
Φ(−
)
,
σy,t
σy,t
σx,t
σx,t
where Φ is the cumulative function of the Gaussian distribution.
4. Calculate maximum probability for all objects and prediction times.
Prob(collision) = maxi,j (Probt+i,obstaclej (collision))
5. Threshold the probability to determine if the system should activate.
If(Prob(collision) > Threshold)
Activate system
Algorithm 7.1: Probability of Collision
is presented in Section 8.3.
100
Chapter 7
Statistical Decision Making
Part IV
Evaluation
101
Chapter 8
Collision Mitigation by Braking
System Testing
In this chapter two collision mitigation by braking systems will be described. One
system uses the probability of collision, according to Algorithm 7.1, to decide when
to perform braking interventions. It only considers each detected object individually and not in relation to other objects. The other system has a multiple obstacle
decision algorithm according to (6.9). The strategy for both systems is to reduce
collision speed as much as possible under the constraint that no faulty interventions
are allowed. This means that the system should engage the brakes as soon as the
collision unavoidable driving state is entered.
Performance of the systems has been evaluated both by simulation and by field
tests with demonstrator vehicles. A schematic overview of the simulation environment is presented in Figure 8.1. The simulation environment has been designed to
[Obstacles]
Environment
Driver_Command
Driver_in
Obstacles
Traking Sensor Measurements
Obstacles
Predictions
Driver Inputs
Collision
Requested Brake
Vehicle state
Virtual Driver/Chassi control
Vehicle Kinematic States
Virtual_in
Host Vehicle
Navigation Sensor Measurements
Sensor Model
In
Collision Probability
Host Vehicle
Tracking System
Decision Making
Host Vehicle Model
Simulation model for a CMbB system, real time code is directly generated
from the same code used in the tracking and decision block of the model.
Figure 8.1:
bear as much resemblance as possible to the demonstrator vehicle and its sensors.
103
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This enables the use of the same code (for the tracking system and decision making
system) in both the simulation environment and in the demonstrator vehicle.
8.1
Simulation Environment
The simulation environment in Figure 8.1 has been implemented using Matlab
and Simulink. From the Simulink code it is possible to auto-generate real time code.
Thus the tracking system and decision making blocks can be directly implemented
in the demonstrator vehicle, using Real-time Workshop to produce the real time
code. The main purpose of a simulation environment is to test performance of the
algorithm in different scenarios and to see whether or not it is possible to provoke
a faulty intervention, especially for high speed collisions, where physical testing
with a demonstrator vehicle might be too dangerous or too costly. Here is a short
description of the blocks that the model consists of (corresponding to the blocks of
Figure 8.1):
Driver Inputs: The driver inputs are described by variables that specify the
driver’s steering wheel, brake, and accelerator commands. These are all specified before the simulation starts, and are not affected by the CMbB system’s
brake intervention. The driver will thus continue the same maneuver as he
was attempting before the system was activated. The CMbB system will of
course (when activated) override the drivers accelerator and braking commands.
Environment: The environment describes the traffic environment of the host vehicle. It is specified by variables describing the objects surrounding the host
vehicle. These are also specified before the simulation starts and are not
affected by each other or by the host vehicle movement. Together with the
driver inputs, the environment variables define a specific scenario. Typical
objects are other vehicles and stationary objects. There is currently no information about lane geometries, changes in road conditions or infrastructure.
Host Vehicle Model: The host vehicle model is a dynamic handling model that
describes the physical limitations of the handling performance of the host
vehicle. It takes driver commands such as steering wheel angle, brake and
accelerator commands as input. The outputs are position, velocity, acceleration and turn rate (yaw rate) of the host vehicle. For most simulations, a
bicycle model (as described in Section 4.8.1) was used. This model has no
actuator dynamics except for the pressure build-up in the braking system.
The pressure build-up is modeled as a ramp which has a rise time of 300 ms
to achieve maximum brake force. To model tire to road friction, Pacejkas
magic formula, [6], was used. For the simulations in Section 2.2.4, a more
detailed model was used. This model has 15 degrees of freedom and has been
verified against a Volvo S80.
Section 8.2
105
The Demonstrator Vehicles
Sensor model: The sensor model simulates the host vehicle’s tracking and navigation sensors. It takes inputs from the environment block and from the host
vehicle. It calculates range, range rate, and azimuth angle to the surrounding
objects. To these “ideal” measurements, noise is added according to


r


(8.1)
y =  ṙ  + et .
ϕ
The additive sensor noise et is assumed to be white Gaussian and independent. The form of the sensor output is the same in that from the sensors
in the demonstrator vehicles. This block also adds noise to the navigation
sensor measurement (yaw rate and speed of the host vehicle)
Tracking system: The tracking systems are quite different for each of the two vehicles. In the first vehicle the tracking and fusion algorithms were developed
within this research project. The tracking system uses nearest-neighbor measurement association, an EKF tracking filter and a nearly coordinated turn
model for the vehicles dynamics. In the second vehicle an existing system
partially developed by a sensor supplier was used. Thus, the exact design of
the tracking filters is unknown.
Decision Making: The decision making algorithm takes, as input, data from the
tracking system and outputs a binary value (true or false) to indicate whether
or not the state for autonomous braking has been reached. The first system
uses a probabilistic decision criteria. The second system has a multiple object
decision making algorithm.
Virtual Driver/Chassis Control: This block performs braking when the decision making system determines that the collision risk is too high. In the
prototype systems discussed in this chapter, maximum braking is always performed when the intervention threshold is reached. The rise time of the brake
system is 300 ms, and the maximum deceleration is 8 m/s2 . There is also a
system delay of 100 ms from the time that the threshold value is exceeded and
brake actuation is performed. This performance is similar to the performance
of the brake system of demonstrator vehicle 1.
8.2
The Demonstrator Vehicles
The demonstrator vehicles in both cases are standard Volvo V70s. They have
been equipped with two tracking sensors and one dSPACE Autobox. The Autobox
(Figure 8.2) is used to perform necessary computations for the tracking and decision
making system and is also used to actuate the brakes. The processor of the Autobox
is a 400 MHz Power PC. To evaluate the system’s performance in collisions, an
inflatable car (Figure 8.3) is used. For safety and cost reasons, collision tests have
been made at speeds up to 70 km/h only.
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The dSPACE unit is installed under the floor of the luggage compartment
Figure 8.3:
Inflatable car used for collision test, maximum collision speeds 70 km/h.
Section 8.2
107
The Demonstrator Vehicles
8.2.1 Demonstrator Vehicle Tracking Sensors
The tracking sensors in the demonstrators are a mm-radar installed in the front
bumper (see Figure 8.4) in combination with, a lidar for the first demonstrator and
Figure 8.4:
The mm-radar installed in the front bumper of the vehicle.
a vision sensor for the second demonstrator, installed behind the windshield close to
the rear view mirror (Figure 8.5). The radar and lidar sensors are production-like
units that are used for ACC systems. The measurement vector (for one obstacle)
that is used in the tracking algorithm from each sensors is given by

yradar
r


r


ṙ


=  ϕleft

 ϕcenter






ṙ
r




 , ylidar = 

 , yfused =  ϕleft
ṙ




 ϕcenter
ϕcenter
ϕright
ϕright




,



(8.2)
where ϕleft is the azimuth angle to the left edge of the detected object, ϕcenter is the
azimuth angle to the center of the detected object, ϕright is the azimuth angle to the
right edge of the detected object. For the second demonstrator only a fused observation of the radar and vision sensor was used, this is indicated by the measurement
vector yfused. The lidar and radar sensors also provide additional measurement on
reflected energy etc. This information is, however, not used. Both the radar and
lidar can detect obstacles at ranges beyond 150 m and have an update frequency
of 10 Hz. The field of view (FOV) is 13.6◦ for the mm-radar, 14◦ for the laser
radar and 48◦ for the vision sensor. The narrow FOV provided by radar and lidar
is often sufficient for ACC systems. For the first demonstrator the performance of
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Figure 8.5:
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The laser radar installed behind the windshield of the demonstrator vehicle.
the CMbB system will degrade because of the narrow FOV. To check the range
accuracy of the radar and lidar sensors in the first demonstrator vehicle, stationary
measurements were performed. In these tests, the distance to a parked vehicle was
measured when the host vehicle was standing still. Both the laser and the radar
measured the true distance to an accuracy within 0.1 m. However this is not the
case when the host vehicle is moving. Figure 8.6 shows the range measurements
from the laser and the mm-radar as a stationary target is approached head-on at
40 km/h. There is a clear discrepancy between the range measurements from the
laser and the mm-radar (the difference is approximately two meters). In dynamic
scenarios, the measurements for each sensor is consistent but the discrepancy between the sensors will cause problems when fusing the sensor data (in the tracking
algorithm this is solved by using large gates in the measurement association step).
This problem is believed to be caused by an unknown and varying delay in the
sensors. There was no time stamp of a measurement or any way to synchronize
the sensors. Another problem with the lidar and radar used here is that none
of them provides accurate measurements over short distances. In most cases the
sensor “loses” sight of objects when the range is shorter than 10 meters (see e.g.
Figure 8.6 at time 28). Furthermore, as the measured object moves out of the FOV
of the mm-radar the azimuth angle measurement deteriorates. This means that in
many cases the azimuth angle measurement is erroneous when the entire object is
not in the FOV of the sensor. Figure 8.7 shows typical performance for the sensor
in a head-on scenario. At time 3 the range is approximately 10 m and the entire
object is not within the sensors FOV; the azimuth angle grows larger until the time
when the object is lost by the sensor (at time 3.8). In this scenario it looks as if the
measured object moves out of the path of the host vehicle, even though it really
Section 8.2
109
The Demonstrator Vehicles
Range measurements
45
Lidar
mm−radar
40
35
Range [km/h]
30
25
20
15
10
5
0
−5
0
5
10
15
20
Time [s]
25
30
35
40
Figure 8.6: In dynamic scenarios there is often a discrepancy between range measurements
from the lidar and the mm-radar. In this scenario an inflatable car is approached head-on
at 40 km/h. The collision occurs at time 33 s. At time 28 s the laser radar loses track of
the obstacle. The difference in the range measurements is approximately two meters. For
other test runs under the same conditions the difference is different.
was stationary and was hit head-on by the host vehicle. The behavior illustrated in
Figure 8.7 causes problems in head-on collisions at low speed (less than 30 km/h)
where the distance for intervention is much shorter than 10 m. Clearly there is a
lower limit at which collision tests can be performed with reliable results. This
limit lies around 30 km/h. In many of the tests with the demonstrator vehicles,
an inflatable car was used as an obstacle. It is therefore interesting to examine if
its radar cross section differs from that of ordinary vehicles. To investigate this,
stationary tests were performed. The measurements were performed as illustrated
in Figure 8.8 (picture taken from the viewpoint of the host vehicle). Here both the
host vehicle and the measured objects are stationary. From these measurements it
was concluded that both obstacles were detected with the same accuracy and that
the reflected signal strength from the rear end of a Volvo V70 and from the rear
end of the inflatable car were very similar.
8.2.2 The Braking System of the First Demonstrator Vehicle
The braking system is of great importance for the kind of CMbB system that is
considered here (intervention only when the collision unavoidable state has been
entered). This is due to the fact that an intervention is issued very late. In many
cases less than 1 second before collision. Because of this short time, it is crucial that
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Degeneration of Azimuth Angle Measurement
50
Range [m]
40
30
20
10
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
2.5
3
3.5
4
4.5
Time [s]
Azimuth Angle [Degrees]
0
−2
−4
−6
−8
−10
−12
−14
0
0.5
1
1.5
2
Time [s]
Figure 8.7: Range and azimuth angle measurements provided by the mm-radar, from
a stationary obstacle straight ahead. The obstacle is approached head-on without any
steering maneuvers before the collision occurs, at time 3.9 s. At time 3 the range to
the obstacle is approximately 10 m. After this point the azimuth measurement clearly
deteriorates.
Figure 8.8: In measuring the radar response from the back of a Volvo V70 and from the
inflatable car, it was found that the radar echo was as strong from the inflatable car as
from the Volvo in this particular set up.
Section 8.2
111
The Demonstrator Vehicles
the brake system quickly reaches maximum deceleration. The first demonstrator
vehicle is equipped with a production ABS system. Brake actuation (when CMbB
system initiated) is achieved by opening a solenoid valve in the brake booster.
There is also a vacuum pump that can be manually switched on and off to ensure
vacuum in the brake booster at intervention. Performance of the brake system is
plotted in Figure 8.9. The plot shows a CMbB system induced brake maneuver,
i.e., the driver does not touch the pedal. The notch in the acceleration curve between time 15 s and time 16 s comes from activation of the ABS system. Both the
speed and acceleration are derived from the wheel speed sensor. The rise time for
the CMbB system actuated braking maneuver is typically more than 500 ms. The
maximum deceleration achieved in the test, illustrated in Figure 8.9, is approximately 6.3 m/s2 . The condition when this test was performed was wet asphalt, the
Velocity [km/h]
Velocity and acceleration under a CMBB system induced brake manouver
60
40
20
0
13
14
15
16
17
Time [s]
18
19
20
14
15
16
17
Time [s]
18
19
20
2
Acceleration [m/s ]
5
0
−5
−10
13
The speed and acceleration of the host vehicle when the system intervenes
(initial speed 50 km/h). The brakes are applied by opening a solenoid valve in the brake
booster, maximum deceleration achieved is 6.3 m/s2 .
Figure 8.9:
temperature was between 0◦ and 5◦ and the vehicle was equipped with non-studded
winter tires. This explains the low maximum deceleration reached. It should be
stressed that brake actuation through the opening of a solenoid valve in the brake
booster is specific to this demonstrator vehicle, and that better alternatives exist.
For the second demonstrator the brakes are not applied at an alarm. The reason
for this is to be able to compare performance with a single object algorithm which
typically alarms later than the multiple object algorithm.
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Overview of scenarios that are evaluated. Test results refer to test performed
with the demonstrator vehicle and an inflatable dummy obstacle. In the simulations the
same tracking and decision algorithm is used.
Table 8.1:
Scenario
Test results
Simulation results
Figure 8.11
Figures 8.10, 8.12, 8.13
Lead Vehicle Brakes Hard
-
Figure 8.15
Host vehicle cut-in
-
Figure 8.17
Obstacle cut-in
-
Figures 8.19, 8.20
Drive-by
Figure 8.23
Figure 8.22
Late avoidance
Figure 8.26
Figure 8.25
Head-on to Stationary Obstacle
8.3
Demonstrator 1 | Probabilistic Decision Making
In this section the performance of a CMbB system with a probabilistic decision
making algorithm in different traffic scenarios is studied. The evaluation of different
scenarios have been performed both with the demonstrator vehicle and inflatable
dummy car and also in simulations. The test scenarios can be divided into two
groups.
• The first group consists of scenarios where there is a collision. For these
scenarios the mitigation performance of the system is of interest, i.e., how
much the speed of the host vehicle was reduced at collision, distance between
colliding objects when the system activates, etc.
• The second group consists of scenarios where there is no collision. These
scenarios are used for designing and verifying the CMbB system, i.e., to
check if it is possible to provoke a faulty intervention.
Table 8.1 gives an overview of all the scenarios that have been evaluated The system
should only react when a collision is unavoidable, so any intervention occurring
without being followed by a collision will by definition be a faulty intervention.
The important characteristics of the tracking and decision making system for
the first demonstrator vehicle is summarized in Table 8.2.
8.3.1 Collision Scenarios
In all the simulated scenarios, the same settings for the sensors is used. It should
be noted however that the FOV of the sensors in the simulated scenarios is larger
than the FOV of the sensors in the demonstrator vehicle. The measurement noise
in the simulation environment is assumed to be white, Gaussian, and independent.
The standard deviation of the range measurement is 0.5 m, the standard deviation
of range rate is 0.5 m/s2 and the standard deviation of azimuth angle is 1◦ . These
settings are used both for the mm-radar and the lidar. The scenarios that are
Section 8.3
Table 8.2:
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Demonstrator 1 | Probabilistic Decision Making
Summary of important characteristics of the first Demonstrator vehicle
Feature/property
Value
FOV of the mm-radar
13.6◦
FOV of the lidar
14.0◦
Tracking and decision making system update rate
10 Hz
Measurement association
Track initialization
Tracking filter
Dynamic model
NN
3 detections
EKF
Nearly coord. turn
Maximum number of active tracks (tracking alg.)
4
Number of future predictions in the tracking system
20
Temporal spacing between predictions
100 ms
Number of future predictions in the tracking system
20
Std. on range measurements in mm-radar model
1m
Std. on range rate measurements in mm-radar model
0.5 m/s
Std. on azimuth angle measurements in lidar model
1◦
Std. on range measurements in lidar model
1m
Std. on range rate measurements in lidar model
Std. on azimuth angle measurements in lidar model
1 m/s
1◦
studied in this section are mainly those where a CMbB has a large potential of
significantly reducing the collision speed. Examples of scenarios where CMbB has
low potential for reducing the collision speed are angular accidents (where the angle
between colliding vehicles is large e.g. intersection collisions), rear end collisions
with a large lateral offset between the colliding vehicles, and frontal collisions where
the relative speed between the colliding vehicles is large. Note, that the since
demonstrator vehicle test was only performed up to 70 km/h little effort was made
to tune the system above this velocity. Therefore, the test results of main interest
is those between 0 and 70 km/h also for simulations.
Head-on to Stationary Obstacle
The first scenario to be studied is the head-on against a stationary object, described
by Figure 5.1. This scenario has already been discussed in Sections 5.1 and 2.2.4.
This type of scenario corresponds well to many real life accidents where the driver
of the striking vehicle is inattentive, or even falls asleep, and hits a stationary object. One will also observe this performance for accidents where the host vehicle
follows another vehicle that suddenly stops (rear end collisions), in cases where the
leading vehicle comes to a stop at a distance greater than the intervention distance.
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Another reason for studying this scenario is its simplicity, meaning that it is easy
to perform both simulations and physical testing. The intervention distance will
of course depend on the lateral offset between the host vehicle and the obstacle.
Figure 8.10 shows the range between the vehicles versus the host vehicle longitudinal speed from simulation results of the head-on scenario with zero lateral offset.
Figure 8.11 shows the same results from testing with the demonstrator vehicle. In
Range versus range rate − no lateral offset
40
35
30
Range [m]
25
20
15
10
5
0
0
20
40
60
Range rate [km/h]
80
100
Simulation test results: Head-on to stationary object scenario described
in Figure 5.1. The figure displays the average from 20 Monte Carlo simulations, for each
velocity. The width of the host vehicle is set to 1.8 m and the width of the obstacle is
1.5 m.
Figure 8.10:
Figure 8.12 the speed reduction is shown for simulation results for the head-on
scenario when there is an 0.5 m offset between the host vehicle and the stationary
obstacle. As previously mentioned, the performance of the braking system is crucial to a CMbB system. To evaluate the effect of a more efficient braking system,
the head-on scenario with zero lateral offset was simulated again and the braking
system performance was changed to powerful state-of-the-art braking systems. The
brake pressure rise time is assumed to be 100 ms and the maximum deceleration
is assumed to be 9.82 m/s2 . Simulated performance of the CMbB system with a
more powerful braking system is shown in Figure 8.13.
Rear End Collision, Lead Vehicle Brakes Hard
Rear end collision is one of the most common types of accident. Approximately
30 % of all accidents are of this type. A rear end collision is defined as a collision
between two vehicles traveling in the same direction, where the trailing vehicle
Section 8.3
115
Demonstrator 1 | Probabilistic Decision Making
Head on scenario no lateral offset − test result with protype vehicle
60
50
Range [m]
40
30
20
10
0
10
20
30
40
50
Range rate [km/h]
60
70
Figure 8.11: Test with demonstrator vehicle: Head-on to stationary obstacle described in Figure 5.1. Results from testing where the demonstrator vehicle hits the inflatable car, which is approximately 1.5 m wide.
Range versus range rate − head−on, 0.5 m lateral offset
25
Range [m]
20
15
10
5
0
20
30
40
50
60
70
Range rate [km/h]
80
90
100
Figure 8.12: Simulation test results: Head-on against a stationary object scenario
described in Figure 5.1, average results from 20 Monte Carlo simulations. In these simulations there was a 0.5 m lateral offset between the two vehicles.
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Range versus range rate − no lateral offset
30
25
Range [m]
20
15
10
5
0
−20
0
20
40
60
Range rate [km/h]
80
100
Figure 8.13: Simulation test results: Head-on stationary object scenario described in
Figure 5.1, both vehicles have a width of 2 m. Here a high performance braking system
is used when simulating the CA manoeuver. The thick line shows the average speed
from 20 Monte Carlo simulations, and the thinner lines show the performance in single
simulations.
hits the leading vehicles’ rear end with its front end (Figure 8.14). Even though
H
v
a
v
Figure 8.14: Rear-end collision scenario. Initially both vehicles travel with the same
velocity. Suddenly the leading vehicle brakes hard, achieving a deceleration of 8 m/s2 .
the injuries from rear end collisions are not life-threatening most of the time, this
type of accident is one of the most costly to society. This is partly because the
accident type is so common and partly because the whiplash injuries that are often
caused by rear end collision are hard to treat and cause victims pain over long
periods of time. The performance of the CMbB system for this type of accident
will vary with many factors such as relative speed of the vehicles, headway distance,
braking force applied by the lead vehicle, lateral offset, etc. If the head distance
is long enough, performance will be the same as in the previous scenario (head-on
against a stationary obstacle). In Figure 8.15 simulation results are illustrated for
Section 8.3
117
Demonstrator 1 | Probabilistic Decision Making
a scenario where the lead vehicle and the following vehicle initially travel at the
same speed with a head distance of 1 second. Suddenly the lead vehicle brakes
Range versus host vehicle speed − no lateral offset
25
Range [m]
20
15
10
5
0
30
40
50
60
70
80
Host vehicle speed [km/h]
90
100
Figure 8.15: Simulation test results: Rear end collision scenario described in Figure 8.14. The plot shows the average speed from 20 Monte Carlo simulations.
hard (8 m/s2 ). There is no lateral offset between the vehicles.
Cut-in Behind a Stationary Object
In this scenario the host vehicle travels straight ahead, and then makes a sudden
lane change. In the lane that is entered there is a stationary obstacle. The lane
change maneuver is executed as a sinusoid input to the steering wheel. The duration
of the maneuver is 3.14 s for speeds below 40 km/h and 1.57 s for speeds above
40 km/h. The amplitude of the input maneuver is tuned to give a 2 m lateral
displacement. The scenario is illustrated in Figure 8.16. In the simulated scenario,
H
Cut-in scenario — the host vehicle makes a sudden lane change; in the lane
that it enters there is a stationary obstacle.
Figure 8.16:
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Range versus host vehicle speed
30
25
Range [m]
20
15
10
5
0
0
20
40
60
Host vehicle speed [km/h]
80
100
Figure 8.17: Simulation test results: The driver makes a lane change maneuver (a
sinusoid input that cause a 2 m lateral offset of the host vehicle). The stationary obstacle
is situated exactly at the point where the maneuver ends in the middle of the lane as
illustrated in Figure 8.16. The plot shows the average performance from 20 Monte Carlo
simulations.
an average speed reduction of approximately 5 km/h for most speeds is attained.
For low speeds the reduction is greater, and as the speed becomes greater than
100 km/h there is no speed reduction. This type of scenario was also tested with
the demonstrator vehicle. However the result was poor. For low speeds (below
50 km/h) the obstacle will move out of the sensors’ FOV before turning in towards
the obstacle. In such cases the tracking system often fails to re-acquire the target
and react in time before the collision. The sensors also have problems detecting
the stationary target under hard maneuvers. This causes problems even for speeds
where the obstacle does not move out of the FOV.
Cut-in, in Front of the Host Vehicle
This scenario is the opposite of the previous example. Here the host vehicle drives
straight ahead and another vehicle traveling at a lower speed enters the host vehicle’s lane. Figure 8.18 gives an overview of the scenario. This scenario tests the
ability of the tracking system to track a sudden maneuver of a tracked object. The
efficiency of the CMbB system will of course depend on several factors such as relative speed between the two vehicles, how close the lead vehicle cuts in front of the
host vehicle, etc. This scenario has only been evaluated through simulations. Two
variant of the scenarios is studied. The first scenario is illustrated in Figure 8.19.
Here the relative speed is low. The other vehicle is traveling 20 km/h slower than
Section 8.3
119
Demonstrator 1 | Probabilistic Decision Making
H
Figure 8.18: Cut-in scenario, the other vehicle, that is traveling at a lower speed, enters
the same lane as the host vehicle.
Range versus host vehicle speed
30
25
Range [m]
20
15
10
5
0
20
30
40
50
60
70
Host vehicle speed [km/h]
80
90
100
Simulation test results: Cut-in scenario illustrated in Figure 8.18. The
time to collision when the lead vehicle cuts in is 1 second. The relative speed is 20 km/h,
i.e., the lead vehicle travels 20 km/h slower than the host vehicle.
Figure 8.19:
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the host vehicle, the cut-in occurs late when the time to collision is one second.
As Figure 8.19 shows for initial host speeds up to 60 km/h, the collision speed
is reduced almost 10 km/h on the average. For higher speed the tracking system
loses track of the lead vehicle as the sudden lane change is performed. In these
cases there is no intervention. In the second scenario illustrated in Figure 8.20 the
Range versus host vehicle speed
30
25
Range [m]
20
15
10
5
0
50
60
70
80
Host vehicle speed [km/h]
90
100
Simulation test results: Cut-in scenario illustrated in Figure 8.19 The
time to collision when the lead vehicle cuts in is 3 seconds. The relative speed is 50 km/h,
i.e., the lead vehicle travels 50 km/h slower than the host vehicle.
Figure 8.20:
relative speed is much greater, 50 km/h, and when the cut-in occurs, the time to
collision is 3 seconds.
8.3.2 Scenarios to Provoke Faulty Interventions
The scenarios to provoke faulty interventions are those where the host vehicle is
close to a collision. Since no collision occurs in these scenarios, any CMbB system
intervention is considered a faulty intervention. These scenarios are used for design,
i.e., for choosing the intervention threshold. The threshold for CMbB intervention
is set as low as possible without giving faulty interventions in any of these near-miss
scenarios. Testing with the demonstrator vehicle is also performed to verify that
the thresholds are correct. During this work, it has not been possible to make test
runs with the demonstrator vehicle against moving obstacles. Therefore, scenarios
where the obstacle is stationary is studied.
Section 8.3
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Demonstrator 1 | Probabilistic Decision Making
Drive-By Scenario
The first scenario to be studied is a drive-by scenario. This can be viewed as a
special case of the head-on collision scenario that was presented in Section 8.3.1.
Here the offset between the host vehicle and the obstacle is just large enough for the
host vehicle to pass the stationary object without colliding (see Figure 8.21). From
H
Figure 8.21: Drive-by scenario, the offset between the host vehicle is just large enough
for the two vehicles not to collide.
the simulation results in Figure 8.22, it is clear that this particular scenario does
not generate high collision probabilities. The threshold for an intervention is when
the probability of collision is larger than 0.7, for all the plotted speeds. However,
Probability of collision − Drive by scenario, threshold at 0.7
1
0.9
Probability of collisioon
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
Time [s]
2.5
3
3.5
4
Simulation test results: The probability of collision in a drive-by scenario
illustrated in Figure 8.21; the plotted probabilities are from 20 Monte Carlo simulations
at 30, 40, 50 and 60 km/h.
Figure 8.22:
when this test was performed with the demonstrator vehicle and the inflatable car
much greater probabilities were observed. Figure 8.23 shows the probabilities for
four drive-by tests at 50 km/h with the demonstrator vehicle and the inflatable car.
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The reason for the high probabilities in this scenario was found to be deterioration
Probability of collision for a close drive by scenario at 50 km/h
0.7
0.6
Probability
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
Time [s]
Test with demonstrator vehicle: The probability of collision in a driveby scenario illustrated in Figure 8.21. The results are from 4 tests with the demonstrator
vehicle and inflatable car at 50 km/h.
Figure 8.23:
of the sensor measurements for short ranges. This problem was already illustrated
in Figure 8.7. When examining the sensor input from these tests one finds that
in many cases the measurements (from the mm-radar) appear to be coming from
an obstacle that is moving into the path of the host vehicle. Even though this
causes much higher probabilities than from the simulations, none of the drive-by
tests caused a faulty intervention.
Head-on to Stationary Object Late Avoidance
The late avoidance maneuver examined in this section is a maneuver where the host
vehicle travels towards a stationary obstacle and avoids the obstacle by steering
away as late as possible. This maneuver is illustrated in Figure 8.24. In Figure 8.25
simulation results from this maneuver are illustrated. The estimated probability
of collision during the maneuver is plotted (for speeds 30, 40 and 50 km/h). The
first peak corresponds to simulations at 50 km/h, and the second peak corresponds
to speeds of 40 km/h. The peak for simulations at 30 km/h lies exactly between
the 40 and 50 km/h peaks, reaching a maximum value of approximately 0.2. The
intervention threshold was 0.7 for all these speeds. There is no intervention for the
simulated late avoidance scenarios. This should, of course, also be the case since
there is no collision. The test results illustrated in Figure 8.26 show probabilities
below 0.5 for the late avoidance scenario. However, during trials performing this
Section 8.3
Demonstrator 1 | Probabilistic Decision Making
123
(a) Initially, the vehicle approaches the sta- (b) At the last moment the driver makes an
tionary obstacle head-on.
evasive steering maneuver.
(c) The maneuver is sufficient to avoid the ob- (d) During the maneuver the driver turned the
stacle with the smallest possible margin.
steering wheel as fast as possible. The entire maneuver is executed at the limits of
this vehicle’s handling capabilities.
Figure 8.24: Example of a late avoidance maneuver. The driver steers away from the
obstacle at the last moment and manages to avoid a collision.
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Probability of collision − late avoiodance scenario, threshold at 0.7
1
0.9
Probability of collisioon
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
Time [s]
Figure 8.25: Simulation test results: The probability of collision for a late avoidance
maneuver scenario illustrated in Figure 8.24. The plotted probabilities are from closing
speeds of 30, 40 and 50 km/h, 20 Monte Carlo simulations for each speed are plotted.
The peak lateral acceleration during the avoidance maneuver is 4 m/s2 . The intervention
threshold is set to 0.7 for all speeds.
scenario it has been possible to get faulty interventions. A possible reason for the
faulty intervention is the deteriorated performance of the sensors at short range.
It is, however, worth noting that there were few complaints about these faulty
intervention from different test drivers. One possible explanation for this is that
the steering avoidance maneuver is so violent that the hard braking is not felt or
possibly considered to be appropriate because of the extreme maneuver.
Driving on Public Roads
Apart from testing the system in specific scenarios on a test track, the demonstrator
vehicle has also been driven on public roads in normal traffic situations. In these
tests the braking actuation of the CMbB system is shut off. Any intervention
(indicated by a LED head-up display) is considered a faulty intervention as long
as an accident does not actually occur. During driving in normal traffic it was
found that some faulty interventions do occur. Below, some typical situations that
have been identified to cause faulty intervention will be discussed. The causes and
possible solutions to avoid these faulty interventions will also be discussed.
• Low speed driving: When driving at low speed (below 20 km/h) in target-rich
environments, faulty interventions sometimes occur for no apparent reason.
Looking closer at the sensor performance and the tracking algorithm, this is
Section 8.3
125
Demonstrator 1 | Probabilistic Decision Making
Probability of collision − late avoidance scenario
0.5
Probability of collision
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
2
4
6
Time [s]
8
10
12
Figure 8.26: Test with demonstrator vehicle: The probability of collision for a late
avoidance maneuver scenario illustrated in Figure 8.24. The plotted probabilities are from
tests with the demonstrator vehicle and the inflatable car. The closing speeds range from
30 - 70 km/h.
not very strange. When traveling at low speeds there are often objects close
to the vehicle. At short distances the sensor measurements often deteriorate.
There is also an increased risk for spurious reflections, i.e., false targets, with
a lot of reflecting surfaces close to the sensors. In the current algorithm the
gates for measurement association are large (several meters). New obstacles
are also very “easily” initialized (three consecutively associated samples are
required). Due to this, the risk that the algorithm should detect faulty targets
is high. The current system could be improved in several ways to attempt to
correct this particular problem. By synchronizing the sensors, the gates for
measurement association could be kept smaller. Initialization of new objects
should possibly be made more “carefully”, i.e., require more than three consecutive measurements to initialize new obstacles. The inferior performance
at short range is specific to the sensors used here. Using other sensors might
alleviate the problems with poor measurements at short range; again it is
stressed that both sensors of the current system were designed mainly for use
in ACC systems.
• Spatial sampling: Driving close to structures that have a spatially periodic
appearance (for example fences and roadside reflector markers) sometimes
cause faulty interventions. This happens due to the sensor sampling rate and
the equidistant spacing of the reflecting objects. For example, a fence might
appear to be a target moving at the same speed as the host vehicle at certain
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speeds. Even worse, it might also look like an object moving into the path of
the host vehicle.
Example 8.1
The sensor sampling frequency is 10 Hz. If the host vehicle is traveling at
30 m/s it will travel 3 m between two consecutive samples. If a structure by
the side of the road has a spatial periodicity of 3 meters it may appear as an
object moving along with the host vehicle at the same speed.
The problem of spatial sampling might be tough to solve using radar sensors
only. Of course, by the measured range rate (via Doppler shift) the targets
can be determined as stationary and thus be ignored. This would remove all
stationary targets, which is not desirable because it decreases the effectiveness of the system. Another sensor, e.g. a vision sensor, could possibly be
used to classify this type of target as invalid. Possibly this problem will also
be diminished if the gates can be made smaller.
• Multiple Reflections on Large vehicles: The radar sensor’s image of an object
is clearly not the same as our visual image. The strongest point of reflection
from an object might vary quite a lot between two samples. For large vehicles
the risk of receiving measurements from different points on the vehicle is
particularly high. A typical scenario where this kind of phenomenon has
caused faulty interventions is when meeting a large vehicle in a right hand
curve. This problem might require sensors that have a better capability of
classifying the object than the ones currently used.
• Spurious reflections: There is always a risk that the sensor reports a ghost
target due to spurious reflections. This has mainly been a problem with the
mm-wavelength radar. An example of how spurious reflections might cause a
faulty intervention is when driving behind a vehicle. A faulty measurement
from the space between the host vehicle and the lead vehicle would look like
a hard braking maneuver from the lead vehicle. From the driving on public
roads, so far no such faulty interventions have been observed. However, for
an ACC system using the same radar sensor, speed control has been observed
when there is clearly no obstacle ahead on which to perform control. Again,
by using smaller gates the probability of faulty associations decreases. By
the use of two sensors (operating at different wavelengths) this problem is
alleviated to a great extent.
8.3.3 Test Results
In this a specific CM system that uses the probability of collision as a metric for
determining when to perform braking actuation is studied. The question is if this
metric provides correct decisions for any type of traffic situation. This has been
tested by simulation and with a demonstrator vehicle. In all the simulations and
Section 8.3
Demonstrator 1 | Probabilistic Decision Making
127
test drives, no faulty intervention was found to be caused by an erroneous decision.
However some faulty interventions did occur during tests with the demonstrator
vehicle. These interventions were found to be caused by tracking system inadequacies. It should be mentioned that the tracking system was not designed primarily
to suppress spurious reflections and faulty measurements from the sensor; it was
designed rather for fast detection of suddenly appearing targets. Also, the sensor’s
behavior when a objects is moving out of the FOV is not correctly modeled.
The design strategy to ascertain that the CMbB system avoids making faulty
decisions was to use a model of the vehicle that overestimates the handling capabilities of the vehicle. The probability threshold is then set high enough not to
make faulty decisions. Using this design method it becomes interesting to study the
effect of the system in collision scenarios, i.e., will there be any significant reduction of the collision speed. The simulation and testing in this chapter shows that
significant speed reduction is possible for certain types of accident, e.g. rear-end
collisions and head-on collisions with stationary obstacles. In many cases the speed
is reduced by around 10 km/h, and in the best case it is reduced by approximately
15 km/h. This reduction is achieved with a brake system that is far from optimal
(both in the simulation environment and in the demonstrator vehicle). Also most
of the demonstrator vehicle testing was performed in late December with temperatures between 0 and 5◦ Celsius, and on wet asphalt. There are several obvious
improvements that could be made to achieve better performance than with the
tested system. Improvements of importance is to use a state of the art braking
system; the incorporation of steering wheel and brake pedal sensors in the navigation system; synchronization of the tracking sensors and reducing the system delay
from decision to brake actuation. In simulations with a high performance braking
system (Figure 8.13) an average speed reduction of almost 20 km/h was achieved
for certain scenarios.
Reducing the collision speed is particularly effective for low relative speeds. For
high relative speeds (well above 100 km/h) the probability of serious injuries and
death will still be very high even if the collision speed is reduced by 20 km/h. Looking at the distance at which interventions occur for relative speeds below 100 km/h
one finds that the range is less than 20 m. For a CMbB system like the one studied
in this chapter, a maximum sensor range of 50 m is therefore probably sufficient.
The current sensors detect obstacles at ranges beyond 150 m, which is much more
than needed. The sensors’ narrow FOV, however, creates problems, as measurements deteriorate at short ranges. This causes bad performance (especially at low
relative speeds) and also possible faulty interventions. In some of the scenarios that
try to provoke faulty interventions, much higher collision probabilities are observed
in tests with the demonstrator vehicle than from simulations. This phenomenon
has been found to be caused mainly by inconsistent sensor performance at short
ranges.
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8.4
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Collision Mitigation by Braking System Testing
Demonstrator 2 | Multiple Object Decision Making
In this section, tests with a second demonstrator is presented. The main purpose is
to compare a multiple object algorithm with a single object algorithm on realistic
tracking input. However in this case the tracking system is provided by a sensor
supplier and not fully known.
The demonstrator vehicles are similar in that they are both a Volvo V70s, and
that they use a dual sensor tracking system. The tracking system installed in
demonstrator 2 is provided by a sensor supplier. Therefore, the exact function of
the tracking system is not known. Some important characteristics of the tracking
and decision making system of the second demonstrator vehicle are summarized in
Table 8.3.
Table 8.3:
Summary of important characteristics of the second demonstrator vehicle.
Feature/property
Value
FOV of the mm-radar
13.6◦
48◦
FOV of the camera
Tracking and decision making system update rate
10 Hz
Maximum number of active tracks (tracking alg.)
30
Decision bound for lateral acceleration
7 m/s2
Decision bound for longitudinal acceleration
10 m/s2
8.4.1 Test Results
Performance was evaluated for two simple scenarios. The first scenario is a head-on
collision scenario with three stationary obstacles (Figure 8.27) at approximately
70 km/h. Figure 8.28 and 8.29 show the relative positions and the intervention
distance for the multiple object and the single object algorithm, respectively, for
two test runs. In these two test runs the intervention is 400 and 300 ms earlier
than the single object algorithm. Using a brake system model as Example 5.1 the
collision speed is 31.0 and 35.5 km/h for the multiple object algorithm. For the
single object algorithm the collision speed is 54.7 and 50.8 km/h.
In the second test scenario that is studied, the host vehicle also travels headon against three stationary objects, but in this case it brakes hard to avoid the
collision. Figure 8.30 illustrates the range, range rate and the normalized required
acceleration to avoid colliding for both algorithms. The normalized required acceleration is referred to as the threat number (T N ), and is defined according to
T N = min(|
Ahost
Ahost
y
x
|, |
|),
Aa,x
Aa,y
(8.3)
Section 8.4
Demonstrator 2 | Multiple Object Decision Making
(a) Three obstacles
70 km/h.
are
approached
at (b) The left obstacle is a Volvo V70. The center and right obstacle are inflatable dummies.
(c) At approximately 10 m the driver brakes
to reduce the collision speed.
Figure 8.27:
129
(d) The center obstacle is struck head on.
Collision scenario seen from the vision sensor’s view. The speed is 70 km/h.
where Aa,x = 9.82 m/s2 is the maximum longitudinal acceleration and Aa,y =
7 m/s2 is the maximum lateral acceleration that the host vehicle can achieve. While
the multiple object algorithm indicate that a high acceleration is required to avoid a
collision, the single object algorithm displays very moderate threat number values.
None of the algorithms are close to 1, which is the threshold for an intervention.
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Alarm distance for multiple and single object algorithms at 71.5 km/h
5
Alarm multiple alg.
4
Obstacle1
Relative lateral position [m]
3
2
Alarm single alg.
1
Obstacle2
0
−1
Obstacle3
−2
−3
−4
−5
0
5
10
15
20
25
30
Relative longitudinal position [m]
35
40
Figure 8.28: Test results from a collision test at 70 km/h. The vertical lines indicate the
distance at which each algorithm alarms. Automatic braking at 18.2 and 10.2 meters
correspond to a collision speed of 30.9 and 54.7 km/h respectively. The faint, dashed lines
indicate the estimated right and left side of each obstacle. The tracking system loses track
of the left and right obstacle at approximately 15 meters.
Alarm distance for multiple and single object algorithms at 70 km/h
5
Alarm multiple alg.
4
Obstacle1
Relative lateral position [m]
3
2
Alarm single alg.
1
Obstacle2
0
−1
Obstacle3
−2
−3
−4
−5
0
5
10
15
20
25
30
Relative longitudinal position [m]
35
40
Figure 8.29: Test results from a collision test at 70 km/h. The vertical lines indicate
the distance at which each algorithm alarms. Automatic braking at 17 and 11.8 meters
correspond to a collision speed of 35.5 and 50.8 km/h respectively. The faint, dashed lines
indicate the estimated right and left side of each obstacle. The tracking system loses track
of the left and right obstacle at approximately 15 meters.
Section 8.4
131
Demonstrator 2 | Multiple Object Decision Making
Hard braking to avoid a collision
40
20
0
0
100
0.5
1
1.5
2
2.5
3
0.5
1
1.5
2
2.5
3
50
0
0
1
TN multiple obj.
TN single obj.
2
TN (m/s )
Rangerate [km/h]
Range [m]
60
0.5
0
0
0.5
1
1.5
Time [s]
2
2.5
3
2
Figure 8.30: Test results from a hard braking maneuver (−7.5 m/s ), close to three
stationary obstacles. An alarm is triggered when T N ≥ 1 (the required longitudinal
acceleration is less than −9.82 m/s2 and the required lateral acceleration is larger than
7 m/s2 .
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Chapter 9
Supplementary Simulation
Studies
9.1
Comparison of Multiple Object Algorithms
One problem with multiple object decision making algorithms is that they become
more complex both from a computational and an understanding perspective. For
single object decision algorithms it is normally quite easy to predict the behavior
of the algorithm. For a multiple object algorithm, on the other hand, it might
be harder to forecast the behavior. It is therefore desirable to have at method to
automatically search for errors or unexpected behavior. This section discusses an
exhaustive search method to compare CA decision algorithms. A comparison of
four decision making algorithms is presented in an extensive simulation study. The
four algorithms to be compared are
• Single obstacle constant control (SOCC) described in Section 5.3.10,
• Multiple obstacles constant control (MOCC) described in Section 6.1.1,
• Multiple obstacles heuristic control (MOHC) described in Algorithm 6.2,
• Multiple obstacles exhaustive search (MOES) described in Section 6.3.
The focus is on finding situations where the algorithms perform differently. In this
evaluation full state knowledge of all objects is assumed, i.e., no tracking system
uncertainty. A more thorough account is given in [78].
9.1.1 Test Method and Specications
The basic idea of the approach taken here is to perform an exhaustive search over
all possible situations that the decision algorithm can encounter. Each decision
making algorithm is then evaluated for each situation. Of course, this is not possible since there are an infinite number of possible situations. Instead, the possible
133
134
Chapter 9
Supplementary Simulation Studies
states for each object are discretized and then all possible combinations are formed.
Here, it is assumed that each object is described by a constant acceleration model.
The state vector and dynamical model for an object is, thus, given by (4.46). The
number of situations that can be generated in this fashion grows very rapidly. Assuming that there are n objects and that there are k possible state vectors for each
k!
different situations can be formed. However, many of the situations
object n!(k−n)!
created in this manner are uninteresting to examine and can be discarded before
the evaluation starts. Below strategies used to remove uninteresting situations are
described.
• Symmetry — This strategy relies on the decision making algorithm responding exactly the same to scenarios around the host vehicles longitudinal axis.
Thus, all symmetric scenarios can be removed.
• Deriveability — If one set of object states causes an alarm, all supersets
(adding more objects) will also cause an alarm, for the algorithms considered
here. Starting with only one object the response for all supersets, of scenarios
where there is an alarm, is known and need not to be checked. By starting
with the situation including only one object a large number of multiple object
situations is also checked. Then one proceeds with the remaining situations
where there are two objects, and so on.
• Minimum distance — If no object comes within a minimum distance of the
host vehicle one may assume that there is no alarm.
• Path crossing — If none of the objects crosses the current path of the host
vehicle there should not be an alarm since keeping that maneuver constant
will avoid collision.
Since all situations are automatically generated, there is no before hand information
whether there should be an alarm or not. In general it is not possible to say if an
alarm was correct or not. However, it is possible to compare the response from
different algorithms. By using a more advanced decision algorithm as a reference,
it is thus possible to find many (but not all) situations where there are false alarms.
In the test results discussed below four decision making algorithms (SOCC,
MOCC, MOHC and MOES) are compared. The test specification is given in Table 9.1. Initially, there is a huge number (574965720) of situations to evaluate.
By using the strategies described above, only 435 + 77642 + 12265925 had to be
evaluated. This could be done in less than 100 hours on two desktop computers.
9.1.2 Test Results
Test results are summarized in Table 9.2. There are some results which might seem
counter intuitive. First, there is a number of scenarios where MOES alarms but
MOCC does not. This seem counter-intuitive since MOES searches a large set of
different escape paths. However, since only a finite number of paths are examined
the solution has a limited spatial resolution. Thus, narrow passages are difficult
Section 9.1
135
Comparison of Multiple Object Algorithms
Specifications for automatic situation generation and testing comparing decision making algorithms. The 1512 states come from combining 84 positions, 9 velocities,
and 2 accelerations
Table 9.1:
Feature/property
Value
Maximum number of obstacles
3
Number of discrete states
1512
Number of scenarios included in the test
574965720
Obstacle velocity range
± 54 km/h
Number of trajectories searched by the MOES alg.
Table 9.2:
50625
Test results comparing decision making algorithms in 12344002 different situ-
ations.
Feature/property
Value
Situations where MOCC alarms but MOHC does not
0.21 %
Situations where MOCC alarms but MOES does not
2.98 %
Situations where MOES alarms but MOCC does not
1.37 %
Situations where MOHC alarms but MOES does not
2.80 %
Situations where SOCC alarms but MOES does not
1.86 %
to find. In this respect the MOCC and MOHC algorithms are superior since they
are based on an analytical solution and will find any existing narrow passage. The
problem is very similar to the narrow passage or narrow corridor problem for path
planing applications [74, 84, 110, 122]. Secondly, in some situations SOCC alarms
but MOES does not. The reason for this is that MOES tests escape path with
combined lateral and longitudinal acceleration. This sometimes provides a more
efficient avoidance maneuver than pure longitudinal or lateral acceleration which
is assumed in the SOCC algorithm. In fact, a large portion of all scenarios where
MOCC and MOHC alarms but MOES does not, is due to the fact that a slightly
better avoidance maneuver can be found by combining lateral and longitudinal
acceleration.
To assess the general benefit of the multiple object algorithm, a comparison
with the SOCC algorithm was performed. This comparison is conducted on a subset of the scenarios and only multiple object scenarios are included. Situations
where either only SOCC or only the multiple object algorithms alarms are considered. These situations are used as initial condition and are simulated forward and
backward in time to find the timing difference between when the algorithms first
alarm. The result is shown in Table 9.3
136
Chapter 9
Table 9.3:
Supplementary Simulation Studies
Average difference in alarm timing between SOCC and the multiple object
algorithms.
Algorithm
Alarm time difference
9.2
MOCC
MOHC
MOES
0.47 s
0.41 s
0.26 s
Statistical Decision Making by Stochastic Numerical Integration
In this section statistical decision making by stochastic numerical integration according to (7.5) is considered in a simulation study. The objective of the evaluation
is threefold. The first objective is a comparison between EKF and PF for some
different measurement noise distributions. The second objective is to evaluate
stochastic integration when using an EKF. Finally, the real time performance of a
numerical solution is considered. The comparison is made for a single scenario, the
head-on collision scenario illustrated in Figure 5.1, when traveling at 60 km/h. Furthermore, this section also presents some measurement results from an automotive
radar.
9.2.1 Tracking Model
The general tracking model takes the form of a nonlinear state space model for
the vehicle dynamics and sensor measurements (4.1) and (4.2). Here, the linear
constant acceleration model (4.46) will be used to model the dynamic properties
of all vehicles. The measurement relation (8.1) comes from radar measurements of
range, range rate and bearing. Where the measurement noise, et , typically includes
clutter, multi-path and multiple reflection points in the vehicle ahead.
9.2.2 Estimation Approach
The vehicle dynamic model (4.46) to be used in the tracking filters is linear. However, the measurement noise and the process noise (driver inputs) are not Gaussian.
Thus, a recursive estimation method that can handle this is required. Here the PF
(given in Section 4.1), that provide a general nonlinear and non-Gaussian Bayesian
solution to the estimation problem, is compared with EKF (4.25) — (4.26). In
the EKF, approach the true noise distributions are approximated with Gaussian
distributions (4.15).
9.2.3 State Noise Model
The classical assumption in target tracking is to assume a Gaussian distribution
for the process noise νt . In Figure 9.1, recorded driving data is used to make
histograms of the longitudinal component of the acceleration noise, νt . The data
is, of course, highly correlated in time, since the acceleration does not fluctuate
Section 9.2
Statistical Decision Making by Stochastic Numerical Integration
137
rapidly. This could be modeled by introducing a Markov chain based on the empirical data. However, for braking situations a distribution based on accelerations
collected close to rapid decelerations should probably be used instead. A further
Empirical distribution of vehicle acceleration
0.18
0.16
0.14
Probability
0.12
0.1
0.08
0.06
0.04
0.02
0
−6
Figure 9.1:
−4
−2
0
Acceleration [m/s2]
2
4
Acceleration histogram of a vehicle driven in urban traffic for 45 minutes.
and natural option is to switch between different distributions depending on the
driving situation. In urban traffic one distribution is used, on the highway another
one, etc. This leads to a mode dependent distribution pν (ν; ξ). Here the mode can
be determined from the host vehicle speed, it may depend on the state vector, external information from navigation and telematics systems, etc. A simple approach
is to use one mode when v < 70 km/h and another one for v > 70 km/h. In the
sequel in this section, the histogram illustrated in Figure 9.1 is used to model the
process noise in the PF. For the EKF, a Gaussian equivalent is used.
9.2.4 Measurement Noise Model
The measurement equation for the radar involves range, range rate and relative
angle measurements to the vehicle ahead (which in this case is a stationary obstacle). In many applications, Gaussian noise is used to approximate the measurement
noise. Here some other noise distributions will be examined. An experiment was
performed by towing a radar-equipped vehicle, which then measure the range to
the towing vehicle under driving conditions with an almost constant range. Data is
collected using an FM-CW radar at 40 m and constant speed 60 km/h on a Volvo
S80 sedan. The range data is correlated, probably due to some oscillations in the
rope and from internal filters in the radar sensor. A second order state space model
is fitted to data using the n4sid method in Matlab system identification toolbox,
138
Chapter 9
Supplementary Simulation Studies
[88], in order to produce de-correlated range data. This PDF range error is presented in Figure 9.2 together with a Gaussian approximation. The measurement
Range pdf, towrope: 40 [m] speed: 60[km/h]. Target vehicle: Volvo S80
25
Probability density function (pdf)
20
15
10
5
0
−0.2
−0.15
−0.1
−0.05
0
0.05
Range [m]
0.1
0.15
0.2
0.25
Measurement range noise PDF around the nominal range r = 40 m for the
target vehicle Volvo S80 sedan and an equivalent Gaussian approximation.
Figure 9.2:
noise from the experiment is collected under nearly ideal circumstances. For instance, the data was collected from behind, i.e., no aspect angle to the car, and
on a smooth test track. In many cases, due to road curvatures, other vehicles, uneven road surfaces relative lateral position and lateral movement, the measurement
range distribution might be different. Also, the exact reflection point is uncertain
since the azimuth angle is not very exact, so for medium distances the main reflection point may be located on different parts of the car. Hence, for other driving
situations the measurement noise PDF may have a larger variance and different
shape. A bi-Gaussian PDF example is presented in Figure 9.3. This measurement
noise model is motivated by modeling the vehicle by two-point reflectors, which is
a simplified way of describing both multi-path propagation and complex reflection
geometry. The exact appearance of the PDF will vary with many factors such as
target vehicle, sensor and traffic situation. Here only a bi-modal distribution with
highly separated peaks is tested. This is to cover many situations with different
aspect angles, where the density is due to multi-path propagation, and multiple
reflections points, for instance in the rear-end, wheel housing, and rear-view mirror.
More measurements and experiments are needed to establish the empirical PDF
for different driving situations, so the proposed density should just be an example
of non-Gaussian PDF influence. In the simulation study different assumptions of
the PDF are compared.
Section 9.2
Statistical Decision Making by Stochastic Numerical Integration
139
1
Histogram
Gauss sum pdf
Normal approx.
0.9
0.8
Probability
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−3
−2
−1
0
Range [m]
1
2
3
Figure 9.3: Bimodal range measurement noise and Gaussian approximation of range
density pe = 0.75N (0, 0.42 ) + 0.25N (1.6, 0.42 ).
9.2.5 Extended Kalman Filter
The Bayesian recursions in (4.28) do not in general have an analytical solution. For
the special case of linear dynamics, linear measurements and Gaussian noise, there
exists a solution, which is retrieved by the Kalman filter (Section 4.1). For many
nonlinear problems the noise assumptions are such that a linearized solution will
be a good approximation. This is the idea behind the EKF, where the model is
linearized around the previous estimate. When a Gaussian mixture measurement
PDF is used
et ∈
M
pi N (µi , σi2 ),
(9.1)
i=1
where N (µ, σ 2 ) denotes a Gaussian density with mean (µ) and covariance (σ 2 ), the
measurement update equation must be modified as
x̂t|t = x̂t|t−T + Kt (yt − h(x̂t|t−T ) − m̄),
(9.2)
M
where m̄ = i=1 pi µi . If the measurement PDF is not known analytically, the
approximation using a single Gaussian PDF introduces a bias (since m̄ = 0).
The decision criterion (9.3) is not analytically solvable when the state variables
are considered to be Gaussian. By sampling the position and velocity distribution
around the estimates, a Monte Carlo integration technique can solve the decision
criterion. Hence, part of the computational burden present in the particle filter is
introduce in this step for the EKF method.
140
Chapter 9
Table 9.4:
Supplementary Simulation Studies
Measurement noise models used in the simulation.
Range distributions (pet )
(1)
pet = 0.75N (0, 0.42 ) + 0.25N (1.6, 0.42 )
(2)
pet = 0.75N (0, 0.22 ) + 0.25N (1, 0.42 )
(3)
pet = Empirical (Figure 9.2)
9.2.6 Simulations and Tests
The simulation study compares the traditional EKF tracking filter with the particle
filter for different sensor noise assumptions. The acceleration required to avoid
collision (5.36) is used in a hypothesis test to determine when to start braking,
according to
obj
Pr{aobj
x − ãx = ax −
|ṽx |ṽx
} > 1 − α,
2p̃x,0
(9.3)
where ath is the acceleration threshold for when an accident is considered to be
imminent. The collision speed is calculated using a first-order system according
to the one described in Example 5.1 The real-time performance in a hardware
collision avoidance test platform is also tested. To simplify analysis of the system
performance for different noise distributions, a single scenario is studied. The
scenario to be studied is, again, one where the target object is not moving and the
vehicle with the CMbB system is approaching with constant velocity (60 km/h).
The motion model of the vehicles is a constant acceleration according to (4.44).
Two main models for the measurement sensor are considered. In the first the
measured range distribution from Figure 9.2, based on measurements collected from
experimental data on a Volvo S80 is used. It is concluded that the range density is
narrow and not as bimodal as expected. This may be due to the ideal circumstances
of the experiment, as described in Section 9.2.4. Other driving situations are of
course also of interested. Hence the second model of the measurement range noise
is a bi-modal Gaussian, i.e.,
et ∈ p1 N (µ1 , σ12 ) + p2 N (µ2 , σ22 ),
(9.4)
where N (µ, σ 2 ) denotes a Gaussian density with mean (µ) and covariance (σ 2 ).
A sample of et belongs to one of the distributions, with probability pi . The three
distributions used are collected in Table 9.4. The simulation results are given in are
given in Table 9.5. Here p1 = 0.75 and p2 = 0.25, since the rear-end of the vehicle is
assumed to have a larger radar cross section and constitute the main reflector. The
range rate and azimuth were found to be well approximated by Gaussian distributions, with σṙ = 0.2 and σϕ = 0.01. The longitudinal acceleration process noise,
Q ≈ 0.5/T is used to model driving behavior and model imperfections. A simple
brake system model according to (5.45) is used to calculate the collision speed. The
parameters k1 and k2 are chosen to give a brake system rise-time of 300 ms and a
Section 9.2
141
Statistical Decision Making by Stochastic Numerical Integration
Table 9.5:
Results from 1000 Monte Carlo simulations and different cases.
pet (r) RMSE (p̃) RMSE (ṽ) Coll. speed σ Coll. speed
Case
Filter
I
PF
pet
0.11
0.29
-6.59
0.51
EKF
(1)
pet
0.16
0.29
-6.61
0.53
0.08
0.29
-6.56
0.49
0.11
0.29
-6.56
0.51
0.06
0.29
-6.52
0.47
0.06
0.29
-6.52
0.47
I
II
PF
II
EKF
III
PF
III
EKF
(1)
(2)
pet
(2)
pet
(3)
pet
(3)
pet
maximum deceleration of 9.82 m/s2 . The somewhat lower maximum deceleration
(compared to Section 9.2) is due to the fact that the test track exhibited very good
frictional conditions during the brake system tests.
The brake decision is based on a hypothesis test on the expected required acceleration according to (5.36). A acceleration threshold ath = −8 m/s2 and the
confidence level to α = 0.05 have been chosen. For the PF approach, the hypothesis is evaluated for each particle. In the particle filter N = 5000 particles is used,
and the same amount of samples are drawn to perform the stochastic integration
for the EKF. 1000 Monte Carlo simulations are performed for both PF and EKF.
9.2.7 Simulation Results
The different simulations from the Monte Carlo study are summarized in Table 9.5
together with tracking and CMbB system performance. Three different cases were
considered. In cases I and II two different bi-modal range distributions were used
and in case III the empirical range distribution from Figure 9.2 was used. All
other values were the same for all the simulation cases. For non-Gaussian range
distributions, such as the bimodal Gaussian mixture proposed for the measurement
noise, the PF method increases estimation performance (position RMSE) slightly.
The mean difference in collision speed is basically not affected, however, the EKF
has a somewhat larger collision speed variance, but this is probably insignificant.
The result is under the assumptions of quite accurate range rate and azimuth
measurements and a high measurement update rate, f = 1/T = 20 Hz. Hence,
for systems that do not measure the relative speed more significant differences are
expected.
9.2.8 Hardware and Real-Time Issues
In the simulations presented in Section 4.7, the entire algorithm was implemented
in standard Matlab code. However, the ultimate goal with the particle filter based
approach is to incorporate the algorithm and test the idea in an on-line application in a collision avoidance system. The test platform uses dSPACE hardware
(equipped with a 1 GHz Power PC) together with Simulink and Real-Time Work-
142
Chapter 9
Supplementary Simulation Studies
shop (RTW). Hence, the algorithm must be written in C-code and Matlab/MEX
functions for Simulink. The entire environment is then compiled using the RTW
compiler to the dedicated hardware platform. In Figure 9.4, the Simulink diagram is shown for the underlying MEX C-code functions i.e., the particle filter and
the decision making algorithm. These are incorporated in the testbed using the
RTW-compiler. The entire algorithm runs faster than real-time on the dedicated
hardware, with T = 0.05 s and N = 5000 particles.
x
To Workspace
Scope2
Out1
inter
1
y
Measurements
z
(par)
Unit Delay
Parameters
In1
Out1
In2
Out2
Particle Filter
x
xhat
xhat
To Workspace3
In1
Out1
In2
Out2
Scope1
DecisionMaking
acc_req_est_hyp
Scope3
Scope4
y
To Workspace2
Simulink diagram of the CMbB system model used to generate RTW code
for the dedicated hardware.
Figure 9.4:
9.2.9 Conclusions
A decision rule in a CMbB system for late braking using a hypothesis test based on
estimates of the relative longitudinal dynamics have been implemented. Both an
EKF and a particle filter were evaluated for different noise assumptions. For nonGaussian range distributions such as the bimodal-Gaussian proposed for the measurement noise, the PF method increases estimation performance slightly. However,
the range error is in the order of decimeters for all tested scenarios, so the mean
difference in collision speed is basically not affected. The EKF has a somewhat
larger collision speed variance, but this is probably insignificant. The result is under the assumption of quite accurate range rate and azimuth measurements, and
a moderate process noise. Hence, a greater difference might occur if, for instance,
the range rate is not measured or if more maneuverability i.e., larger process noise
is considered.
The computational cost of the EKF method is somewhat smaller than for the
PF, but rather close. This is due to the fact that the EKF uses a stochastic
integration to calculate the probability of collision. So for a decision rule, such as
(7.5), a particle filter approach might be preferable. It should also be kept in mind
that the scenario studied here is very simple (from a tracking sense); for more
complex scenarios with maneuvering target and tracking platform the difference
between the methods may be larger. In addition, the particle filter method is more
flexible, so if a more complex state-space model is used, linearization errors in the
dynamical model can be avoided.
Section 9.3
Statistical Decision Making Using a Gaussian Approximation
143
In the simulation study the simulations for both the PF and EKF approach
were run faster than real-time on an ordinary desk-top computer. The particle filter
was also implemented in the Volvo testbed hardware (used in vehicle tests) using
the Matlab Simulink RTW compiler. The tracking filter and decision algorithm
executed faster than the real-time constraint, T = 0.05, using N = 5000 particles
for the tracking model described earlier.
9.3
Statistical Decision Making Using a Gaussian Approximation
In the previous section a CA system using statistical decision making based on
stochastic numerical integration was discussed. In this section a similar system
based on the same decision function, the longitudinal acceleration required to avoid
a collision (5.36), is studied. Two CA decision strategies compared. The first
strategy is according (7.7) which neglect the uncertainty of the state estimate.
This gives the following decision rule
|ṽˆx,0 |ṽˆx,0
obj
host
< ath1 ⇒ a,
(9.5)
ax,req = âx,0 −
2p̃ˆx,0
where the action a is full emergency braking, ath1 is the acceleration threshold,
ˆ
ˆ
âobj
x,0 , ṽx,0 , p̃x,0 denote estimated object acceleration, estimated relative velocity,
and estimated relative distance, respectively. The second CA decision strategy is
according to (7.9), yielding the following decision rule
' host
(
'
' host ((
ax,req − C1 B ahost
⇒ a.
(9.6)
x,req < ath2 + C2 D ax,req
To evaluate the two decision strategies the same head-on scenario considered in
previous sections (Figure 5.1) will be studied.
First, the probability of performing an intervention during the sample interval
when the host vehicle is exactly at the intervention boundary will be studied i.e.,
at the instant when a collision is becoming imminent. Here a collision is defined
as being imminent when a deceleration of 8 m/s2 or more is required to avoid
colliding. In Figure 9.5 the probability of an intervention for the criterion (9.5)
and (9.6) can be compared at different closing velocities; the parameters for the
calculation can be found in Summary 9.3. Note, in Figure 9.5 the probability
of an intervention is illustrated only for a certain time instant (i.e., when the
collision is becoming imminent). Next, system performance in terms of reduction
of the collision speed and the probability of making a too early intervention will
be studied. Too early an intervention is defined as a CMbB system-induced brake
maneuver before the collision imminent boundary is reached. The process noise,
ν, and measurement noise, e, are assumed to be independent and Gaussian. The
values of the noise variances were chosen to correspond to the measured PDFs
presented in the previous section and illustrated in Figures 9.1 and 9.2. The brake
144
Chapter 9
Supplementary Simulation Studies
Probability of intervention at decision boundary: Std.(p) = 0.21, Std.(v) = 0.25
0.45
Criterion (9.5)
Criterion (9.6)
0.4
Probability of intervention
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
10
20
30
40
Relative speed [km/h]
50
60
Figure 9.5: Probability of performing a CMbB intervention at the sampling interval when
the collision imminent boundary is crossed. The required acceleration is −8 m/s2 .
system model is according to the model described in Example 5.1. However, the
maximum deceleration is somewhat lower since the perfect conditions on the test
tracks are not always present on a normal road. Both decision criterion functions
are applied to the tracking system output. A summary of the simulation parameters
is given in Table 9.3. The simulation results are presented in Table 9.7 for speeds
from 5 - 60 km/h. For each speed, 2000 Monte Carlo simulations were performed,
and for each scenario both decision criteria were applied to the same tracking
data. At speeds higher than 60 km/h a steering avoidance maneuver is much more
efficient than a braking maneuver, therefore higher velocities are not as interesting
to examine. Table 9.8 presents the same simulation scenarios as Table 9.7, but here
ˆ 1) and ṽ ∈ N (ṽ,
ˆ 1)
the measurement noise was doubled, so that p̃ ∈ N (p̃,
9.3.1 Analysis
How well does the deterministic criterion (9.5) work, compared to statistical decision criterion (9.6)? In Figure 9.5 the probability of an intervention at a certain
time instant is compared for different velocities. From the Figure 9.5 it is clear
that for the criterion (9.5) the probability for an intervention varies while for (9.6)
it is kept constant (for speeds above 15 km/h). Why is it that the decision criterion (9.6) manages to maintain a constant risk level? To understand this the
criterion for intervention is studied more closely. Rewriting the estimated required
Section 9.3
Statistical Decision Making Using a Gaussian Approximation
Table 9.6:
145
Summary of important simulation parameters.
SIMULATION SUMMARY
1. Decision criterion (9.5) parameters: ath1 = −8.5 m/s2 .
2. Decision criterion (9.6) parameters: C1 = 1, C2 = 1 and
ath2 = −8 m/s2 ].
3. State noise.
ˆ 0.012 )
N (ã,
ˆ 0.252 ) and ã ∈
ˆ 0.252 ), ṽ ∈ N (ṽ,
p̃ ∈ N (p̃,
4. Sensor and tracking system update rate 10 Hz.
5. Decision making process data rate 100 Hz.
6. Measurement noise. et ∈ N (0, 0.252 ).
7. Brake system model. k1 = 7, ⇒ rise time of 300 ms.
Table 9.7: Head-on scenario simulation results. ṽ is the initial closing velocity, ṽc is the
speed at the collision instant and Prob(faulty) denotes the probability of a faulty (i.e.,
too early) intervention.
Criterion (9.5)
ṽ [km/h]
Criterion (9.6)
ṽc [km/h]
Prob(faulty)
ṽc [km/h]
Prob(faulty)
5
5
0
5
0
10
7.3
0.28
10
0
15
10.6
0.18
14.5
0.013
20
12.8
0.11
14.4
0.035
25
14.2
0.073
15.3
0.024
30
15.1
0.033
15.9
0.016
35
15.7
0.019
15.6
0.017
40
15.8
0.014
15.5
0.019
45
16.1
0.006
15.0
0.015
50
15.8
0.005
13.8
0.012
55
15.4
0.003
12.3
0.017
60
14.2
0
10.4
0.009
146
Chapter 9
Supplementary Simulation Studies
Table 9.8: Head-on scenario simulation results. Here the standard deviation of the measurement noise was doubled.
Criterion (9.5)
ṽ [km/h]
Criterion (9.6)
ṽc [km/h]
Prob(faulty)
ṽc [km/h]
Prob(faulty)
5
3.5
0.329
5.0
0
10
7.1
0.340
10.0
0
15
10.3
0.249
15.0
0
20
12.3
0.199
17.2
0.028
25
13.4
0.138
16.4
0.038
30
14.9
0.078
16.7
0.033
35
15.4
0.060
16.7
0.031
40
15.6
0.037
16.6
0.025
45
15.8
0.021
16.0
0.021
50
15.3
0.017
14.6
0.027
55
14.9
0.011
13.4
0.026
60
14.3
0.005
11.7
0.020
acceleration as
ˆ2
obj
ˆ = âobj − ṽ
âhost
− ã
x,req = â
2p̃ˆ
(ṽ + νv )2
= aobj + νa −
2(p̃ + νp )
= aobj + νa −
(ṽ 2 + 2ṽνv + νv2 )
,
2(p̃ + νp )
(9.7)
where in our case νp , νv and νa ∈ N (0, σi2 ). When p̃ and ṽ are large compared to
the noise terms νv and νp (9.7) is approximately according to
obj
âhost
+ νa −
x,req ≈ a
(ṽ 2 + 2ṽνv )
.
2p̃
(9.8)
Equation (9.8) has a Gaussian distribution since it consists of a sum of Gaussians. Thus, when p̃ and ṽ are large compared to their uncertainty, âhost
x,req ∈
ṽ 2
ṽ 2
obj
2
2
N (a + 2p̃ , σa + ( p̃ ) σv ) is a good approximation. The criterion (9.6) will thus
be equivalent to the hypothesis test according to (7.8), the parameters C1 and C2
will determine the confidence level of the test. What are the implications of this
in terms of collision mitigation performance? Looking at the simulation results of
Table 9.7, it shows that for speeds of 30 km/h and above the probability of a faulty
Section 9.3
Statistical Decision Making Using a Gaussian Approximation
147
intervention is kept constant for the criterion (9.6). For criterion (9.5), on the other
hand, the probability of a faulty intervention decreases as the closing velocity becomes higher. At velocities around 35 km/h the performance is the same for both
criteria. This is also the velocity at which they cross in Figure 9.5. Both criteria
performed badly at low speed. This is mainly due to the fact that the dynamic of
the brake system is not captured in the collision avoidance decision function (5.36).
The assumption of a constant measurement noise causes the assumptions of p̃ and
ṽ being large compared to the noise to be violated at low closing velocities. The
required acceleration (5.36) will thus not be Gaussian. This causes the probability
of intervention to vary significantly for low speeds.
9.3.2 Conclusions
In this section, statistical decision making, by approximating the distribution of
the decision function by a Gaussian distribution, applied to automotive collision
avoidance has been discussed. A criterion that explicitly uses an approximation
of the statistical properties of the CA decision function has been simulated and
analyzed. To compute the approximation, the estimated tracking noise covariance
(which was assumed to be Gaussian) was used. The main reason for this criterion
was to be able to deal with changing uncertainty of the decision function without
having to evaluate the entire PDF. The criterion is motivated by properties of the
Gaussian distribution; for Gaussian distributed variables the criterion corresponds
to a hypothesis test. It was shown that the CA decision function (5.36) studied
can be considered to be Gaussian under certain circumstances. In head-on collision
scenarios where the assumption of Gaussianity holds (v > 25 km/h), the criterion
(9.6) applied to (5.36) keeps the risk of a faulty intervention at a relatively constant
level. However, comparing the results in Tables 9.7 and 9.8 one can observe a
change in that risk level. For the criterion (9.5) the risk of a faulty intervention
is changed more significantly as the measurement noise is increased. For scenarios
where the assumptions do not hold (e.g. at low speeds) the criterion (9.6) becomes
too cautious, causing braking to be too late. Bad performance at low speeds is also
caused by the fact that the decision function used does not account for the brake
system dynamics.
148
Chapter 9
Supplementary Simulation Studies
Chapter 10
Summary
This thesis discusses theory for collision avoidance decision making and its application to automotive collision mitigation by braking. In the introduction a rough
estimation of the effect of a CMbB system, which reduces the collision speed in
rear-end collisions, is calculated. For a system with a uniformly distributed collision
speed reduction, between 0 and 15 km/h, AIS2+ classified in rear-end collisions are
estimated to be reduced by 16 %. This corresponds to a reduction of all AIS2+
injuries of approximately 5 %.
Part II deals with target tracking. Automotive tracking sensors are discussed in
Chapter 3. In Chapter 4 tracking theory is reviewed. A specific tracking algorithm
(Algorithm 4.2) which is used in the first demonstrator vehicle is presented. A comparison of different tracking models in a specific automotive scenario is presented
in Section 4.7.
In Part III, CA decision making is discussed. Different decision functions are
discussed in Chapter 5. A decision function describes the deterministic part of the
CA system’s decision i.e., how is the threat, posed by another object, quantified.
One common decision function quantifies the threat from another vehicle by the
longitudinal acceleration required to avoid collision (5.36). In situations where the
object stops before a collision occurs (5.36) gives incorrect values. To remedy this
a decision function (5.39) that assumes that any vehicle remains stationary once it
reaches zero velocity is proposed.
In Section 5.3.7 a decision function that accounts for the actuator dynamics is
proposed. According to Figure 5.6 neglecting the brake system dynamics corresponds to delaying the CA decision by up to 140 ms, assuming a maximum deceleration of 9.82 m/s2 and a brake system rise time of 300 ms.
In Section 5.3.10 it is shown how a general two dimensional acceleration vector
to avoid a collision can be found. A comparison between the general acceleration
and acceleration only in the longitudinal or lateral direction (5.58) for a head-on
collision is shown in Figure 5.8. The maximum difference is achieved at 60 km/h
where an avoidance maneuver according to the general two dimensional acceleration can be performed 160 ms later and still avoid the collision.
149
150
Chapter 10
Summary
Multiple object collision avoidance decision making is discussed in Chapter 6. The
approach presented is to search for a control input that avoids conflicts. Three
alternatives for how to find such an control is proposed:
• The first alternative (6.9) assumes that the acceleration is kept constant during the entire avoidance maneuver and therefor only explore a very limited
set of all possible avoidance manoeuvres. A CA system using this approach
is implemented in the second demonstrator vehicle. Performance for two test
runs with demonstrator 2, in a head-on collision at 70 km/h with three stationary obstacles was presented in Section 8.4. The multiple object algorithm
reduces the collision speed to 30.9 km/h and 35.5 km/h respectively for the
two test runs. The single object algorithm reduces the collision speed to
54.7 km/h and 50.8 km/h.
• The second alternative described in Algorithm 6.2 also finds escape paths
using constant acceleration but if no escape path is found at t = t0 the
system is simulated until an escape path is found or a collision occurs.
• The third alternative is to perform a search over all possible maneuvers. To
do this the set of possible maneuvers has to be discretized. Theorem 6.1 and
6.2 give a local and global bound on how well an arbitrary escape path can
be approximated depending on how fine the discretization is made.
A comparison of the three alternative multiple object decision algorithms and a
single object algorithm is presented in Section 9.1. Compared with a single object
algorithm the three multiple object algorithms (MOCC, MOHC, and MOES) are
found to alarm 0.47 s, 0.41 s, and 0.26 s earlier than the single object algorithm,
respectively.
To deal with uncertainty in the CA decision making process a statistical approach is proposed in Section 7. A general framework to deal with nonlinear
decision function and non-Gaussian noise for CA decision making based on Monte
Carlo integration is presented. The evaluation of such a system is presented in
Section 9.2. Another alternative to statistical decision making is to simulate the
motion of each object and determine the probability of a collision for each (simulated) future time instant. The first demonstrator vehicle uses this method. In
collision tests, in a head-on scenario, this system reduced the collision speed up to
15 km/h.
Part V
Appendices
151
Appendix A
Coordinate Systems
In this thesis, Cartesian coordinates according to Figure A.1 will be used. For
some models global (ground fixed) coordinates are used. This is indicated by
capital letters P , V , and A for position velocity and acceleration respectively. Host
vehicle fixed coordinates are denoted by small letters p, v, and a. The origin of
the global coordinate system is in the front bumper of the host vehicle at time
t0 as indicated in Figure A.1. Thus, at t = t0 the x-direction denotes the host
vehicles longitudinal direction and the y-direction is in the host vehicles lateral
direction, for both the global and the local coordinate system. As time evolves
the local and global coordinate systems are no longer the same (see Figure A.2),
unless the host vehicle is stationary. The heading direction of the host vehicle,
in the global coordinate system, is denoted ψ and turn rate by ω. Furthermore,
relative coordinates is denoted by P̃ = P obj − P host , Ṽ = V obj − V host , and
à = Aobj − Ahost . Bold font P, V, and A is used to indicate vector valued
position, velocity and acceleration. A summary of the coordinate system notations
is given in Table A.1
Table A.1:
Coordinate systems.
Coordinate system
Symbols
Global position, velocity and acceleration
Host fixed position, velocity and acceleration
P, V , A
p, v, a
Relative position, velocity and acceleration
P̃ , Ṽ , Ã
Vector valued position, velocity and acceleration
P, V, A
153
154
Appendix A
Coordinate Systems
wobj
p = px
T
py
x
y
H
whost
Figure A.1: Cartesian coordinate system, the x-direction is aligned with the host vehicle’s
heading direction. Note that the host is represented by a center point in the very front
of the vehicle whereas the obstacle is represented by a center point on the surface closest
to the host vehicle (which is what most of the sensors measure).
t = t 1 > t0
y
H
x
p(t1 ) = px (t1 )
P = Px
T
py (t1 )
T
Py
= p(t0 )
x
t = t0
y
Figure A.2:
aligned.
As time evolves the global and the local coordinate systems are no longer
Appendix B
Notation
Symbols
A
Matrix describing a linear dynamical system
aa
B
Maximum possible acceleration
Matrix describing how the control input affects the
states
Dhead
Dbrake
et
Headway distance to vehicle in front
Distance needed to avoid a stationary obstacle by
means of braking
Distance needed to avoid a stationary obstacle by
means of steering
Measurement noise
ê
D
Estimate of e
Residual vector, innovations
f (·)
State equation transition mapping (discrete-time)
ϕ
G(i)
Azimuth angle
Gate belonging to track i
g(·)
φ
Collision avoidance decision function
Host vehicle heading angle
Φ(·)
h(·)
Gaussian probability distribution function
Measurement relation
H
Kt
Linearized measurement relation
Kalman gain matrix at time t
J
Maximum number of tracks in the tracking system
Dsteer
155
Notation
156
lhost
Length of the host vehicle
L(.)
Lambert’s W function, the solution w = L(x) to
wew = x.
M
Maximum number of measured objects from one
tracking sensor
Mass of vehicle i
mi
n
N
Prob{·}
Number of objects
Number of grid points in a PMF or particles, samples
in a PF
Probability
p(·)
pet
Probability density function
Measurement noise probability density
pνt
px 0
Process noise probability density
Initial state probability density
px , py
p
x and y positions in a host fixed coordinate system
T
Position vector px py
P
Covariance matrix
r
r(·)
R
Radial distance, range to another object
Decision strategy function, mapping all states to a
CA action
Measurement noise covariance matrix
R
Π
The set of real numbers
Covariance matrix for p
σ
Q
Standard deviation
Process noise covariance matrix
T
Sample period
u
νt
Control input
Process noise
Yt
The cumulative set of measurements up to and including time t, {y1 , y2 , · · · , yt }
ω
∆V i
Turn rate
Speed change of vehicle i during a collision
v01
v1
Speed of vehicle 1 before collision
Speed of vehicle 1 after collision
v host
v obj
Speed of the host vehicle
Observed object’s speed
Notation
whost
157
Width of the host vehicle
obj
Width of an observed object
State vector at time t
xt
x1 , . . . , xn
(i)
States of the ith object in an MTT system
State variables
x = (x1 · · · xn )T
X
State vector
State vector for own vehicle and all observed objects
X = [xhost xobj1 ...xobjn ]
xt−T,i
Sample state variable drawn from a known prior distribution
xdes
1
xref
1
Desired value of x1
Reference value of x1
yt
Measurement at time t
w
xt
Acronyms
ABS
Anti blockier systeme/antilock braking system
ATC
CA
Air traffic control
Collision avoidance
ccd
CM
Charge coupled device
Collision mitigation
CMbB
CW
Collision mitigation by braking
Collision warning
CPU
Central processing unit
DOF
EHB
Degrees of freedom
Electro hydraulic brake system
EKF
FARS
Extended Kalman filter
Fatality analysis reporting system
FCA
FCM
Forward collision avoidance
Forward collision mitigation
FCW
FM-CW
Forward collision warning
Frequency modulated continuous wave
FM-Pulse
FOV
Frequency modulated pulsed wave
Field of view
FSK
GES
Frequency shift keying
General estimates system
Notation
158
GPS
Global Positioning System
i.i.d.
IMM
Independent identically distributed
Interacting multiple model
IR
KF
Infrared
Kalman Filter
Lidar
LMS
Light detection and ranging
Least mean square
LS
MAP
Least squares
Maximum a posteriori
MC
Monte Carlo
MHT
MHz
Multiple hypotheses tracking
Mega hertz
ML
MLE
Maximum likelihood
Maximum likelihood estimate
MMSE
MTT
Minimum mean square error
Multiple target tracking
NCAP
NN
New car assessment program
Nearest neighbor
PDF
PF
Probability density function
Particle filter
PMF
Radar
Point mass filter
Radio detection and ranging
RCS
RMSE
Radar cross section
Root mean square error
TTC
Time-to-collision
THW
Time-headway
Appendix C
Safety Systems Introduced in
Volvo Cars
1920-1960
1927
1944
1944
1954
1956
1956
1957
1958
1959
-
Safety glass
Full safety cage
Laminated windshield
Front defroster
Windshield washers
Padded dashboard
Front seat safety belt anchorage points
Padded door handles
Three-point front safety belts
1960-1970
1965
1966
1966
1966
1967
1968
1969
1969
-
Power brakes with pressure limiting valves
Dual circuit triangular split brake system
Front and rear crumple zones
Roll-over roof protection bar
Two-point rear safety belts
Front seat head restraints
Three-point inertia reel front seat belts
Heated rear window
1970-1980
1972
1972
1972
1972
-
Three-point inertia reel rear seat belts
Child safety seat
Child-proof rear door locks
Audio visual seat belt reminder
159
160
1972
1972
1973
1973
1974
1974
1974
1978
1979
Appendix C
-
Safety Systems Introduced in Volvo Cars
Hazard warning lamps
Rear head restraints
Energy absorbing bumpers
Headlamp wiper-washers
Isolated fuel tank
Multi stage impact-absorbing steering column
Stepped bore master cylinder braking system
Front and rear fog lights
Wide angle rear view mirrors
1980-1990
1982
1982
1984
1985
1986
1987
1987
-
Anti-submarining seat protection
Door warning lamps
ABS antilock braking system
ETC electronic traction control
High level brake lamp
Automatic safety belt tensioner
SRS driver side airbag
1990-2000
1990
1990
1991
1991
1992
1994
1998
1998
1998
1998
1998
1999
1999
1999
-
Three-point inertia reel rear center seat belt
Integrated rear child seat
SIPS side impact protection system
Automatic height adjusting seat belts
SRS passenger side airbag
SIPS front seat side impact airbags
ROPS roll-over protection system
Boron steel windshield A-Pillars
SIPS II front seat side impact airbags
WHIPS whiplash protection system
EBD electronic brake force distribution
DSTC dynamic stability and traction control
IC inflatable curtain airbags
HID xenon headlights
2000-
2001
2002
2002
2002
-
EBA electronic brake assist
RSC roll stability control
Boron steel reinforced roof structure
Lower compatibility cross-member
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M. Millnert: Identification and control of systems subject to abrupt changes.
Thesis no. 82, 1982. ISBN 91-7372-542-0.
A.J.M. van Overbeek: On-line structure selection for the identification of multivariable systems. Thesis no. 86, 1982. ISBN 91-7372-586-2.
B. Bengtsson: On some control problems for queues. Thesis no. 87, 1982. ISBN
91-7372-593-5.
S. Ljung: Fast algorithms for integral equations and least squares identification
problems. Thesis no. 93, 1983. ISBN 91-7372-641-9.
H. Jonson: A Newton method for solving non-linear optimal control problems
with general constraints. Thesis no. 104, 1983. ISBN 91-7372-718-0.
E. Trulsson: Adaptive control based on explicit criterion minimization. Thesis
no. 106, 1983. ISBN 91-7372-728-8.
K. Nordström: Uncertainty, robustness and sensitivity reduction in the design
of single input control systems. Thesis no. 162, 1987. ISBN 91-7870-170-8.
B. Wahlberg: On the identification and approximation of linear systems. Thesis
no. 163, 1987. ISBN 91-7870-175-9.
S. Gunnarsson: Frequency domain aspects of modeling and control in adaptive
systems. Thesis no. 194, 1988. ISBN 91-7870-380-8.
A. Isaksson: On system identification in one and two dimensions with signal
processing applications. Thesis no. 196, 1988. ISBN 91-7870-383-2.
M. Viberg: Subspace fitting concepts in sensor array processing. Thesis no. 217,
1989. ISBN 91-7870-529-0.
K. Forsman: Constructive commutative algebra in nonlinear control theory. Thesis no. 261, 1991. ISBN 91-7870-827-3.
F. Gustafsson: Estimation of discrete parameters in linear systems. Thesis no.
271, 1992. ISBN 91-7870-876-1.
P. Nagy: Tools for knowledge-based signal processing with applications to system
identification. Thesis no. 280, 1992. ISBN 91-7870-962-8.
T. Svensson: Mathematical tools and software for analysis and design of nonlinear
control systems. Thesis no. 285, 1992. ISBN 91-7870-989-X.
S. Andersson: On dimension reduction in sensor array signal processing. Thesis
no. 290, 1992. ISBN 91-7871-015-4.
H. Hjalmarsson: Aspects on incomplete modeling in system identification. Thesis no. 298, 1993. ISBN 91-7871-070-7.
I. Klein: Automatic synthesis of sequential control schemes. Thesis no. 305, 1993.
ISBN 91-7871-090-1.
J.-E. Strömberg: A mode switching modelling philosophy. Thesis no. 353, 1994.
ISBN 91-7871-430-3.
K. Wang Chen: Transformation and symbolic calculations in filtering and control. Thesis no. 361, 1994. ISBN 91-7871-467-2.
T. McKelvey: Identification of state-space models from time and frequency data.
Thesis no. 380, 1995. ISBN 91-7871-531-8.
J. Sjöberg: Non-linear system identification with neural networks. Thesis no.
381, 1995. ISBN 91-7871-534-2.
R. Germundsson: Symbolic systems – theory, computation and applications.
Thesis no. 389, 1995. ISBN 91-7871-578-4.
P. Pucar: Modeling and segmentation using multiple models. Thesis no. 405,
1995. ISBN 91-7871-627-6.
H. Fortell: Algebraic approaches to normal forms and zero dynamics. Thesis no.
407, 1995. ISBN 91-7871-629-2.
A. Helmersson: Methods for robust gain scheduling. Thesis no. 406, 1995. ISBN
91-7871-628-4.
P. Lindskog: Methods, algorithms and tools for system identification based on
prior knowledge. Thesis no. 436, 1996. ISBN 91-7871-424-8.
J. Gunnarsson: Symbolic methods and tools for discrete event dynamic systems.
Thesis no. 477, 1997. ISBN 91-7871-917-8.
M. Jirstrand: Constructive methods for inequality constraints in control. Thesis
no. 527, 1998. ISBN 91-7219-187-2.
U. Forssell: Closed-loop identification: Methods, theory, and applications. Thesis
no. 566, 1999. ISBN 91-7219-432-4.
A. Stenman: Model on demand: Algorithms, analysis and applications. Thesis
no. 571, 1999. ISBN 91-7219-450-2.
N. Bergman: Recursive Bayesian estimation: Navigation and tracking applications. Thesis no. 579, 1999. ISBN 91-7219-473-1.
K. Edström: Switched bond graphs: Simulation and analysis. Thesis no. 586,
1999. ISBN 91-7219-493-6.
M. Larsson: Behavioral and structural model based approaches to discrete diagnosis. Thesis no. 608, 1999. ISBN 91-7219-615-5.
F. Gunnarsson: Power control in cellular radio systems: Analysis, design and
estimation. Thesis no. 623, 2000. ISBN 91-7219-689-0.
V. Einarsson: Model checking methods for mode switching systems. Thesis no.
652, 2000. ISBN 91-7219-836-2.
M. Norrlöf: Iterative learning control: Analysis, design, and experiments. Thesis
no. 653, 2000. ISBN 91-7219-837-0.
F. Tjärnström: Variance expressions and model reduction in system identification. Thesis no. 730, 2002. ISBN 91-7373-253-2.
J. Löfberg: Minimax approaches to robust model predictive control. Thesis no.
812, 2003. ISBN 91-7373-622-8.
J. Roll: Local and piecewise affine approaches to system identification. Thesis no.
802, 2003. ISBN 91-7373-608-2.
J. Elbornsson: Analysis, estimation and compensation of mismatch effects in
A/D converters. Thesis no. 811, 2003. ISBN 91-7373-621-X.
O. Härkegård: Backstepping and control allocation with applications to flight
control. Thesis no. 820, 2003. ISBN 91-7373-647-3.
R. Wallin: Optimization algorithms for system analysis and identification. Thesis
no. 919, 2004. ISBN 91-85297-19-4.
D. Lindgren: Projection methods for classification and identification. Thesis no.
915, 2005. ISBN 91-85297-06-2.
R. Karlsson: Particle Filtering for Positioning and Tracking Applications. Thesis
no. 924, 2005. ISBN 91-85297-34-8.
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