ISSUES IN CONTROL AND MONITORING OF INTELLIGENT VEHICLES Li Li

ISSUES IN CONTROL AND MONITORING OF INTELLIGENT VEHICLES Li Li
ISSUES IN CONTROL AND MONITORING OF INTELLIGENT
VEHICLES
by
Li Li
_________________________
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF SYSTEMS AND INDUSTRIAL ENGINEERING
In Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UNIVERSITY OF ARIZONA
2005
2
THE UNIVERSITY OF ARIZONA
GRADUATE COLLEGE
As members of the Dissertation Committee, we certify that we have read the
dissertation
prepared by Li Li
entitled Issues in Control and Monitoring of Intelligent Vehicles
and recommend that it be accepted as fulfilling the dissertation requirement for the
Degree of Doctor of Philosophy
_______________________________________________________________________
Date: (Nov.30, 2004)
Fei-Yue Wang
_______________________________________________________________________
Date: (Nov.30, 2004)
Ferenc Szidarovszky
_______________________________________________________________________
Date: (Nov.30, 2004)
J. Cole Smith
Final approval and acceptance of this dissertation is contingent upon the candidate’s
submission of the final copies of the dissertation to the Graduate College.
I hereby certify that I have read this dissertation prepared under my direction and
recommend that it be accepted as fulfilling the dissertation requirement.
________________________________________________ Date: (Nov.30, 2004)
Dissertation Director: Fei-Yue Wang
3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirement for an
advanced degree at The University of Arizona and is deposited in the University
Library to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission,
provided that accurate acknowledgement of source is made. Requests for permission
for extended quotation from or reproduction of this manuscript in whole or in part
may be granted by the head of the major department or the Dean of the Graduate
College when in his or her judgment the proposed use of the material is in the
interests of scholarship. In all other instances, however, permission must be obtained
from the copyright holder.
SIGNED: _______________________________
Li Li
4
ACKNOWLEDGMENTS
I would like to thank my advisor, Dr. Fei-Yue Wang. Without his guidance and
kindly support, I might not have the opportunity to carry out research in the field of
Automated Highway Systems (AHS) and Intelligent Vehicles (IV). I would like to
thank members of my committee, Dr. Szidarovszky, Dr. Zeng, Dr. Smith, and Dr. Jin.
I appreciate their help in both my course work and research project.
I would like to thank all the professors of the Systems and Industrial Engineering
Department. It was their guidance and support that helped me to finish these four
years of tedious work. I would like to thank my friends in the University of Arizona
and Tucson. Their kind encouragement and generous help makes my achievement
possible and life enjoyable. I will never forget these special four years here.
Thousands of thanks to my dear parents! Their love, understanding, support and
encouragement are the major motivation for me in working towards the Ph. D. degree.
5
TABLE OF CONTENTS
LIST OF FIGURES ………………………………………………………………………………...7
LIST OF TABLES ..……………………………………………………………………………….10
ABSTRACT....................................................................................................................................11
CHAPTER 1
INTRODUCTION...............................................................................................14
1.1
Research Progress of Intelligent Vehicles ...................................................................14
1.2
Our Efforts in Intelligent Vehicle Field .......................................................................23
1.3
Outline of this Dissertation..........................................................................................25
CHAPTER 2
RESEARCH ADVANCES IN VEHICLE LATERAL MOTION CONTROL....28
2.1
Introduction .................................................................................................................28
2.2
Advance in Steering Control Devices .........................................................................30
2.2.1
Steer-by-Wire ..................................................................................................30
2.2.2
Steering Related Sensory.................................................................................33
2.3
Vehicle Lateral Motion Model and Estimation ...........................................................35
2.3.1
Bicycle Model .................................................................................................35
2.3.2
Other Vehicle Lateral Motion Models .............................................................40
2.3.3
Lateral Motion Estimation...............................................................................42
2.4
Vehicle Lateral Motion Control...................................................................................44
2.4.1
Frequency Domain Robust Steering Controller ..............................................44
2.4.2
Sliding Mode Steering Controller....................................................................49
2.4.3
Adaptive Steering Controller...........................................................................53
2.4.4
Fuzzy Steering Controller................................................................................54
2.4.5
Other Steering Controller ................................................................................56
2.5
Remarks.......................................................................................................................57
CHAPTER 3
A ROBUST OBSERVER DESIGNED FOR VEHICLE LATERAL MOTION
ESTIMATION
59
3.1
Introduction .................................................................................................................59
3.2
Sensor Sets and System Observability ........................................................................62
3.3
Robust Observer Design..............................................................................................63
3.4
Simulation Results.......................................................................................................66
CHAPTER 4
AN LMI APPROACH TO ROBUST VEHICLE STEERING CONTROLLER
DESIGN 71
4.1
Introduction .................................................................................................................71
4.2
System Controllability and Observability ...................................................................75
4.3
Robust Steering Controller Design..............................................................................78
4.3.1
Feedback Controller ........................................................................................78
4.3.2
Robustness Analysis Considering Model Uncertainty ....................................84
4.3.3
Higher Order Controller Design......................................................................85
4.3.4
Robustness Analysis Considering Time Delay................................................87
4.3.5
Simulation Results...........................................................................................89
6
TABLE OF CONTENTS (CONTINUED)
4.4
Speed Limit Estimation and Guideline Planning ........................................................90
4.4.1
Speed Limit Estimation for Steering and Lane Change ..................................90
4.4.2
Optimal Guideline/Trajectory Planning ..........................................................94
4.4.3
Graphical User Interface Design .....................................................................96
CHAPTER 5
COOPERATIVE DRIVING AT BLIND CROSSINGS USING
INTER-VEHICLE COMMUNICATIONS .....................................................................................98
5.1
Introduction .................................................................................................................99
5.2
Problem Description..................................................................................................101
5.2.1
Driving at Blind Crossings ............................................................................101
5.2.2
Inter-Vehicle Communication .......................................................................103
5.3
Cooperative Driving Schedule ..................................................................................107
5.3.1
Collision Free Driving Represented by Safe Patterns ...................................107
5.3.2
Solution Tree Generation and Labeling.........................................................110
5.4
Cooperative Trajectory Planning...............................................................................114
5.4.1
Trajectory Planning .......................................................................................114
5.4.2
Best Driving Plan Search ..............................................................................123
5.5
Simulation Results.....................................................................................................124
5.5.1
Simulation Results for Trajectory Planning...................................................124
5.5.2
Simulation Results for Inter-Vehicle Communication...................................131
5.6
Discussions................................................................................................................133
CHAPTER 6
TIRE FAULT OBSERVER BASED ON ESTIMATION OF TIRE/ROAD
FRICTION CONDITIONS...........................................................................................................136
6.1
Introduction ...............................................................................................................136
6.2
Problem Formulation.................................................................................................139
6.3
Observer Design........................................................................................................143
6.4
Simulation Results.....................................................................................................147
REFERENCES …………………………………………………………………………………..152
7
LIST OF FIGURES
Figure 1.1 "Klaus," the driving robot developed by German
carmaker Volkswagen, drives a VW Multi-van on
a test track (from http://www.vw-personal.de/).......................................................18
Figure 1.2 Vehicle Intelligent Sensors [19]…………………………………………………..20
Figure 1.3 Intelligent vehicle navigation algorithm [39]…………………………………….23
Figure 1.4 Research directions demonstrated in tree forms:
(a) reports related to advanced vehicle motion control,
(b) reports related to intelligent vehicles…………………………………………..24
Figure 2.1 Classification of vehicle lateral motion control models………………………….30
Figure 2.2 Diagram of steering systems: (a) conventional steering system,
(b) steer-by-wire system [7]……………………………………………………….33
Figure 2.3 "Bicycle" steering model: front steering (a) and full steering (b)………………..36
Figure 2.4 Diagram of system architecture with add-on disturbance observer [50]…………47
Figure 2.5 Diagram of system architecture with add-on disturbance observer [50]…………48
Figure 2.6 Feedback control architecture considering actuator saturation [47]……………...49
Figure 2.7 Diagram of a typical sliding mode steering controller [45]………………………52
Figure 3.1 Tracking result using the linear Luenberger observer……………………………69
Figure 3.2 Tracking result using the proposed robust observer……………………………...70
Figure 3.3 Smoothed tracking result using the proposed robust observer…………………...70
8
LIST OF FIGURES (CONTINUED)
Figure 4.1 Control outputs of y f using different controllers……………………………..90
Figure 4.2 Bounds of real steering trajectory. The green dash line is
the guideline. The two red dash lines are bounds (envelope)
curves. The blue curve denotes the real trajectory. ………………………………..91
Figure 4.3 Relation between vehicle velocity limit and road curvature
(denoted by radius)………………………………………………………………...93
Figure 4.4 Comparison of two different guidelines and corresponding
steering behaviors…………………………………………………………………95
Figure 4.5 A graphical user interface device for steering guidance……………………….....97
Figure 5.1 Vehicle grouping at crossings…………………………………………………...103
Figure 5.2 Vehicle groups and group inter-vehicle communication………………………..105
Figure 5.3 Diagram of two-lane junctions………………………………………………….108
Figure 5.4 A four-vehicle driving scenario for a two-lane junction………………………...109
Figure 5.5 A schedule tree stemmed from the driving scenario shown
in Fig.5.4. The shadow nodes represent invalid driving schedule……………….113
Figure 5.6 Trajectory generation considering one virtual vehicle…………………………..115
Figure 5.7 The cooperative driving planning framework…………………………………..124
Figure 5.8 A three-vehicle driving scenario for a two-lane junction………………………..125
9
LIST OF FIGURES (CONTINUED)
Figure 5.9 Speed profiles for vehicle B (a), virtual vehicle B (b), vehicle C (c),
vehicle D (d) and vehicle A (e)…………………………………………………..130
Figure 5.10 Average communication time with respect to different
group sizes (forbidding time length = 1)…………………………………………132
Figure 5.11 Average communication time with respect to different
forbidding time lengths (group size = 3)…………………………………………133
Figure 5.12 Handling an emergency case…………………………………………………..134
Figure 6.1 Typical tire/road friction profiles for: (a) vehicle running on
different road surface conditions with velocity 20m/h,
(b) vehicle running on dry asphalt road with different vehicle velocities [4]……138
Figure 6.2 Variation of y − yˆ = ω − ωˆ when the jump error occurs. ……………………149
Figure 6.3 Variation of
θ and θˆ when the jump error occurs…………………………..149
Figure 6.4 Variation of y − yˆ = ω − ωˆ when the shift error occurs……………………...150
Figure 6.5 Variation of
θ and θˆ when the shift error occurs……………………….......151
10
LIST OF TABLES
Table 2.1 Parameters and Their Typical Values.......................................................................37
Table 5.1 Comparison for Different Driving Plans ...............................................................130
11
ABSTRACT
Inspired by the recent developments, we studied some recent developments and
research trends in intelligent vehicle sensing and control tasks. We emphasize on
advanced vehicle motion control techniques and intelligent tires. The main research
motivation is to improve drivers/passengers’ comfort and safety as well as highway
capacity and efficiency.
In Chapter 2, we presents a review of recent developments and research trends in
vehicle lateral (steering) control tasks. It is an attempt to provide a bigger picture of
the very diverse, detailed and highly multidisciplinary research in this area. Based on
diversely selected research, this chapter explains the initiatives and techniques for
vehicle lateral (steering) control with a specific emphasis on disturbance rejection,
time delay, system dynamic variation tolerance and controller saturation handling.
Besides, some other related topics including vehicle lateral motion sensory and
observer (virtual sensor) design are also discussed.
In Chapter 3, Lateral control of vehicles on automated highways often requires
estimation of sideslip angle, yaw rate and lateral velocity, which are difficult to
measure directly. Thus, several observers (virtual sensors) were developed in the last
decade. To solve the unhandled estimation problem caused by dynamic model
uncertainty, a robust observer using H ∞ design method is proposed in this chapter. It
maintains the good disturbance rejection property that derived form previous research,
12
and simultaneously provides acceptable tolerance to model variance. Specially, effects
of displacements of sensory, dynamics variance caused by mass/velocity/tire-road
frictions change or nonlinear characteristic are studied. Simulations demonstrate the
usefulness of the proposed observer.
Being parallel to frequency domain robust steering controller designs, time
domain robust steering controller designs attract continuous interest in the last decade.
Based on previous research results, a systematic time design framework is proposed
in Chapter 4. The design task is constructed as a multi-objective optimization problem
which simultaneously considers system stabilization, disturbance rejection, actuator
saturation and time delay. A mixed L1 / H ∞ robust controller is finally obtained by
solving a set of linear matrix equalities (LMI) that guarantee the performance
requirements regarding these mentioned factors. Simulations show the effectiveness
of the proposed design method.
Cooperative driving technology with inter-vehicle communication attracts
increasing intentions recently. It aims to improve driving safety and efficiency using
appropriate motion scheduling of all the encountered vehicles. Under cooperative
driving control, the motion of individual vehicles can be conducted in a safe,
deterministic and smooth manner. This is particularly useful to heavy duty vehicles,
since their acceleration/deceleration capacity is relatively low. Specifically in Chapter
5, cooperative driving at blind crossings (crossings without traffic lights) is studied. A
concept of safety driving patterns is proposed to represent the collision free
13
movements of vehicles at crossings. The solution space of all allowable movement
schedules is described by a spanning tree in terms of safety driving patterns; four
trajectory planning algorithms are formulated to determine the driving plans with least
execution times using schedule trees. The corresponding group communication
strategy for inter- vehicle networks is also analyzed. Finally, simulation studies have
been conducted and results demonstrate the potentiality and usefulness of the
proposed algorithms for cooperative driving at blind crossings.
Many tire fault monitors are designed nowadays because tire failure is proved to
be one of the main causes of traffic accidences. However, most of them are high in
manufacturing cost and unreliable. Thus Chapter 6 is devoted to solve this problem
and a new practical tire fault observer is proposed. Based on the new introduced
dynamic tire/road friction model that considers external disturbances, the observer
estimate and track the changes of tire/road friction conditions using only vehicle track
forces and wheel angular velocity information. Tire fault diagnosis is carried out as
follows. Since the wheel speed sensor is one basic component of normal anti-lock
brake system (ABS), the observer proposed could be easily realized in low cost within
an anti-lock brake system.
14
CHAPTER 1 INTRODUCTION
1.1
Research Progress of Intelligent Vehicles
During the last twenty years, road traffic increases significantly, which results in
frequent traffic congestions and a gradually decrease in driving safety. To handle this
problem, the concepts of Intelligent Transportation Systems (ITS) had been proposed
in the late 1980s and beginning of 1990s, [1]-[51].
As one component of ITS, intelligent vehicles (IV) use sensing and intelligent
algorithms to “understand” environment situation around the vehicle, and either
assists the driver in vehicle operations (driver assistance) or fully controls the vehicle
(automation), see Larsen 1995, Shladover 1995, Ashley 1998, Martin, Marini and
Tosunoglu 1999a, 1999b, Bishop 2000.
Following the success of current research, intelligent vehicle is currently viewed
as the "next wave" for ITS. It is widely expected to function as the basic element to
construct the whole transportation system and help it operate more effectively. On the
other side, as pointed out by Vahidi and Eskandarian 2003, due to financial and
practical limitations, the short-term tendency has switched from AHS to IVI, since the
driver assist systems that can independently be implemented in today’s generation of
cars without the costly modifications in the infrastructure. Such assist systems provide
15
the driver with information, warning and operational support. ACC, stop and go
cruise, collision warning and collision avoidance systems are being developed in this
context. When each vehicle acts the desired operations, the performance of vehicle
platoon can then be improved.
Currently, this field has attracted great interests worldwide in different programs,
i.e. California PATH program (i.e. Parsons and Zhang 1989, Patwardhan, Tan and
Guldner 1997), Ohio State University Center for Intelligent Transportation Research
(i.e. Fenton, Melocik and Olson 1976, Hatipoglu, Ozguner and Redmill 2003), The
Robotics Institute of Carnegie Mellon University (i.e. Jochem, Pomerleau and Kumar
et. al 1995) and Advanced Traffic & Logistics Algorithms & Systems Lab of
University of Arizona (i.e. Mirchandani and Head 2001, Nobe and Wang 2001) in the
United
States;
Daimler–Benz
and
MAN
Project
(i.e.
Darenberg
1987),
PROMETHEUS Program (i.e. Fritz 1995), ARGO Car project (i.e. Broggi, Bertozzi
and Fascioli 1999, Broggi, Cellario and Lombardi, et. al 2003) and Project EZAuto
(i.e. Sohnitz and Schwarze 1999, Simon and Becker 1999) in Europe; some programs
under ITS Japan (i.e. Tsumura 1994, Tsugawa, Aoki and Hosaka et. al 1996); and a
number of growing projects in China (i.e. Wang, Wang and Li et. al 2003, Wang,
Tang and Sui et. al 2003, Zheng, Tang and Cheng et. al 2004).
There is no common conclusion on what functions are essential of an intelligent
vehicle. In other words, researchers now argue about what vehicles can be called
intelligent vehicles.
16
Bishop 2000 categorized the intelligent vehicles into three different levels: a) the
systems which provide an advisory/warning to the driver (collision warning systems);
b) the systems which take partial control of the vehicle, either for steady-state driver
assistance or as an emergency intervention to avoid a collision (collision avoidance); c)
the systems which take full control of vehicle operation (vehicle automation).
Generally, most researchers agree that a so called intelligent vehicle should be
capable to implement function level a), because of the importance of collision
avoidance. Dravidam and Tosunoglu 1999 estimated that 15% to 20% of the reported
accidents involve rear-end collisions. And the National Highway Traffic Safety
Administration (NHTSA) estimates that about 88% of rear-end collisions in the
United States are caused by driver inattention or by vehicles following too closely.
Therefore, collision warning systems is believed to be the fundamental function of an
intelligent vehicle.
However, as Bishop 2000 summarized, the collision warning system itself may
includes several different functions such as forward collision warning, blind spot
warning, lane departure warning, lane change/merge warning, intersection collision
warning, pedestrian detection and warning, backup warning, rear impact warning, and
rollover warning for heavy vehicles. And to monitor the driver in case of his/her
drowsy is also viewed as a special category of collision warning system. Till now,
most experiment vehicles can only reach parts of the functions listed above. Thus, the
strictly defined intelligent vehicle is yet to be developed.
17
Some researchers believed that cars won’t drive themselves. For example, Jones
2002 presented his idea as follows:
“Though cars will soon be capable of doing much of what drivers do when
guiding cars down a road, a car that operates without a driver’s input may never see
commercial production.
‘New technologies are meant to complement rather than replace the driver,’ said
Daniel McGehee, director of the human factors research program at the University of
Iowa’s Public Policy Center in Iowa City. Giving the car total control, he said, raises a
tangle of complex legal questions like, ‘Who is responsible in the event of an accident
- the driver or the carmaker? ’
Other vehicle-sensing and human factors researchers, speaking under condition of
anonymity, stressed the potential for crippling liability claims against auto
manufacturers and makers of smart car systems.
In view of that deterrent, movie- and TV-inspired visions of a future in which
cars ask only that their driver select the destination will remain confined to celluloid.”
However, more other researchers aimed to build totally automatons vehicle
within the next twenty years, see Simon and Becker 1999, Simon, Sohnitz and Becker
et. al 2000.
Fig.1.1 shows a great commercial example designed by Technical University
Braunschweig. This driving robot “Klaus” is a practical driving robot developed by
German carmaker Volkswagen, which is driving a VW Multi-van on a test track. But
18
the cars will not be able to drive and navigate only with the robot-driver. The
sophisticated car control systems and sensor technology were also used to identify the
surroundings and calculate the desired course.
Figure 1.1 "Klaus," the driving robot developed by German carmaker Volkswagen,
drives a VW Multi-van on a test track (from http://www.vw-personal.de/)
Besides the classification method of Bishop 2000, to detect and warn of
dangerous engine status, abnormal tire pressure or other similar faults is taken as
another important function of intelligent vehicle recently. For example, Pohl, Steindl
and Reindl 1999, Wang, Shan and Ding et. al 2002 showed that many traffic
accidences, extremely some really bad ones, were caused by tire failure.
Beyond the driving safety, the driving performance including comfort and time
optimality achieved increasing research interests too; see Li and Wang 2002, Weber
and Weisbrod 2003. For example, Metz and Williams 1989 investigated how to get
shortest driving time for racing cars. And in the design of Li and Wang 2002, the
19
driver/passengers’ feeling of vehicle speed are quantified and analyzed to guide the
vehicle speed adjusting.
There are some other requirements of the next generation intelligent vehicles, i.e.
save oil and pollution free; see Menig and Coverdill 1999. Since this topic has little to
do with intelligent vehicle motion control and sensing, we will not examine them
within this dissertation.
Here, we think that a general intelligent vehicle research should cover the
following fields including:
(1)
hardware: smart sensory, new control devices and intelligent center;
(2)
software: advanced control algorithms, and intelligent decisions.
First, a vehicle is a highly complex system comprising a large number of
mechanical, electronic, and electromechanical elements. To describe all the
movements of the vehicle, numerous measurements and mathematical model are
required. Besides real sensors, many virtual sensors (online observer or real-time
estimation procedure) are proposed too.
20
Figure 1.2 Vehicle Intelligent Sensors [19]
With the developing of sensor technology, sensor fusion of intelligent vehicle
becomes a new research direction in this field. For instance, Jones 2002 arranged the
intelligent sensors into several independent groups as shown in Fig.1.2. In his plan,
the input data is managed from myriad sensors and used for make split-second
21
decisions that may involve taking control from the driver. As an example, when
forward collision warning senses that a crash is imminent, data from body mass and
position sensors in the cabin instantly adjust the amount of force with which air bags
are deployed and seat belts are tightened. Noting that this is still a novel research
direction, we will not discuss sensor fusion within this dissertation
Second, increasing requirements of safe and comfortable driving have led vehicle
manufacturers and suppliers to actively pursue development programs in the so called
"by-wire" subsystems. These computer-controlled subsystems include steer-by-wire,
brake-by-wire, drive-by-wire and etc, which are connected through in-vehicle
computer networks. A steer-by-wire system replaces the traditional mechanical
linkage between the steering wheel and the road wheel actuator (e.g., a rack and
pinion steering system) with an electronic connection. Because it removes direct
kinematical relationship between the steering and road wheels, it enables control
algorithms to help enhance driver steering command, see Claesson, Poledna and
Soderberg 1998, Huh, Seo and Kim et. al 1999, Hayama, Nishizaki and Nakano et. al
2000, Yih, Ryu and Gerdes 2003.
If we take the intelligent sensors as human’s eyes and ears, the intelligent center
and the intelligent controller should be considered as human’s brain and arms
respectively. The intelligent center is usually a microcomputer that links all the
sensors and control devices. It uses the information that is collected from the
environment near the vehicle though intelligent sensors, and makes decisions of the
22
vehicle motion based on pre-stored control algorithms. Apparently, they are the most
important electronic devices in an intelligent vehicle, see Chowanietz 1995a, 1995b,
Ronald 1999, Schoner and Hille 2000, Kiencke 2001, Ribbens 2002 and Hori 2002.
Advanced control algorithms and intelligent decisions are essential for an
intelligent vehicle, since it needs to help the driver to find the optimal driving plan,
translate the driving demand into actual, usually complex, mechanical or electronic
operations and execute them. It is also expected to be capable to detect any error
driving behavior when the driver makes wrong actions, or any fault of
sensors/actuators.
For example, when the intelligent center “see” a moving obstacle on the road in
front of vehicle, it would invoke certain procedures to determine whether this
obstacles will hinder the motion of it or not. If it decided that this obstacle would not
bother its motion, it would continue its original motion. Otherwise it would either stop
or steer this obstacle. Such simple control algorithms may collaborate with each other
so as to implement certain complex actions. As a typical example, Fig.1.3 shows the
flowchart of an intelligent vehicle’s navigation algorithm that was proposed by Simon
and Becker 1999. We can see that the complex obstacle avoidance function is divided
and studied as several individual and relatively simpler functions. Due to cost
consideration, most researcher focus on how to implement such individual operations.
In this dissertation, we will address the research advance of these simpler functions,
23
while the control synthesis is neglected since it is still premature and under further
discussions.
Figure 1.3 Intelligent vehicle navigation algorithm [39]
1.2
Our Efforts in Intelligent Vehicle Field
Inspired by the recent developments, we studied some recent developments and
research trends in intelligent vehicle sensing and control tasks. Among the very
diverse, detailed and highly multidisciplinary research in this area, we emphasize on
advanced vehicle motion control techniques and intelligent tires. The main research
motivation is to improve drivers/passengers’ comfort and safety as well as highway
24
capacity and efficiency. Some research results are listed in Fig.1.4 below.
(a)
(b)
Figure 1.4 Research directions demonstrated in tree forms: (a) reports related to
advanced vehicle motion control, (b) reports related to intelligent tires.
During developing process of intelligent traffic and intelligent vehicles, varied
25
problems emerge, which motivate an increasing amount of research on related topics.
Although this dissertation tires to provide a broad overview of all the corresponding
cutting-edge techniques and applications in intelligent vehicle research, we cannot
discuss everything encountered in details. Thus, we mainly addressed our
contributions in the rest of this dissertation.
1.3
Outline of this Dissertation
The following five chapters of this dissertation are arranged as follows.
Chapter 2 presents a review of recent developments and research trends in vehicle
lateral (steering) control tasks. It is an attempt to provide a bigger picture of the very
diverse, detailed and highly multidisciplinary research in this area. Based on diversely
selected research, this chapter explains the initiatives and techniques for vehicle
lateral (steering) control with a specific emphasis on disturbance rejection, time delay,
system dynamic variation tolerance and controller saturation handling. Besides, some
other related topics including vehicle lateral motion sensory and observer (virtual
sensor) design are also discussed.
In Chapter 3, Lateral control of vehicles on automated highways often requires
estimation of sideslip angle, yaw rate and lateral velocity, which are difficult to
measure directly. Thus, several observers (virtual sensors) were developed in the last
decade. To solve the unhandled estimation problem caused by dynamic model
uncertainty, a robust observer using H ∞ design method is proposed in this chapter. It
26
maintains the good disturbance rejection property that derived form previous research,
and simultaneously provides acceptable tolerance to model variance. Specially, effects
of displacements of sensory, dynamics variance caused by mass/velocity/tire-road
frictions change or nonlinear characteristic are studied. Simulations demonstrate the
usefulness of the proposed observer.
Being parallel to frequency domain robust steering controller designs, time
domain robust steering controller designs attract continuous interest in the last decade.
Based on previous research results, a systematic time design framework is proposed
in Chapter 4. The design task is constructed as a multi-objective optimization problem
which simultaneously considers system stabilization, disturbance rejection, actuator
saturation and time delay. A mixed L1 / H ∞ robust controller is finally obtained by
solving a set of linear matrix equalities (LMI) that guarantee the performance
requirements regarding these mentioned factors. Simulations show the effectiveness
of the proposed design method.
Cooperative driving technology with inter-vehicle communication attracts
increasing intentions recently. It aims to improve driving safety and efficiency using
appropriate motion scheduling of all the encountered vehicles. Under cooperative
driving control, the motion of individual vehicles can be conducted in a safe,
deterministic and smooth manner. This is particularly useful to heavy duty vehicles,
since their acceleration/deceleration capacity is relatively low. Specifically in Chapter
5, cooperative driving at blind crossings (crossings without traffic lights) is studied. A
27
concept of safety driving patterns is proposed to represent the collision free
movements of vehicles at crossings. The solution space of all allowable movement
schedules is described by a spanning tree in terms of safety driving patterns; four
trajectory planning algorithms are formulated to determine the driving plans with least
execution times using schedule trees. The corresponding group communication
strategy for inter- vehicle networks is also analyzed. Finally, simulation studies have
been conducted and results demonstrate the potentiality and usefulness of the
proposed algorithms for cooperative driving at blind crossings.
Many tire fault monitors are designed nowadays because tire failure is proved to
be one of the main causes of traffic accidences. However, most of them are high in
manufacturing cost and unreliable. Thus Chapter 6 is devoted to solve this problem
and a new practical tire fault observer is proposed. Based on the new introduced
dynamic tire/road friction model that considers external disturbances, the observer
estimate and track the changes of tire/road friction conditions using only vehicle track
forces and wheel angular velocity information. Tire fault diagnosis is carried out as
follows. Since the wheel speed sensor is one basic component of normal anti-lock
brake system (ABS), the observer proposed could be easily realized in low cost within
an anti-lock brake system.
28
CHAPTER 2 RESEARCH ADVANCES IN VEHICLE LATERAL
MOTION CONTROL
This chapter presents a review of recent developments and research trends in
vehicle lateral (steering) control tasks. It is an attempt to provide a bigger picture of
the very diverse, detailed and highly multidisciplinary research in this area. Based on
diversely selected research, this chapter explains the initiatives and techniques for
vehicle lateral (steering) control with a specific emphasis on disturbance rejection,
time delay, system dynamic variation tolerance and controller saturation handling.
Besides, some other related topics including vehicle lateral motion sensory and
observer (virtual sensor) design are also discussed.
2.1
Introduction
Increasing traffic congestion and accidents inspired the concepts of Automated
Highway Systems (AHS) and Intelligent Vehicle (IV) more than twenty years ago.
Among a variety of techniques that had been introduced, the concept of Advanced
Vehicle Control Systems (AVCS) gets significant interests world-widely [53]-[55].
Advanced vehicle control systems help drivers by taking control of the steering,
brakes and/or throttle to maneuver the vehicle in a safe sate. Thus, it is also called
driver aid systems and intelligent (autonomous) driving systems in some literals. The
29
related technologies include smart cruise control, collision avoidance systems, and
vehicle platooning, etc. On the aspects of motion control, corresponding research can
be divided into three directions: lateral control, longitudinal control and their
combinations.
The main task of longitudinal control is vehicle following/ tracking. It requires
that an appropriate headway should be maintained between the lead vehicle and the
controlled vehicle to avoid collision. The lateral control usually refers to vehicle
steering control. Its prime task is path (road) following, or plainly, to keep the vehicle
on the road.
From the viewpoint of steering, vehicles can be divided into two kinds: single
track vehicles and track-trailers, see Fig.2.1. Track-trailer systems consist of a steering
tractor and one (sometimes more than one) passive trailer(s) linked with rigid free
joints. Single track vehicles usually refer to passenger cars or car-like vehicles/robots
which can be viewed as a steering tractor. Single track vehicles can be further
classified into two types: front steering vehicles (2WS) in which only the two front
tires can be steered and full steering vehicles (4WS) of which the front and rear tires
can be steered independently. Since the discussion of track-trailers steering control
requires a dedicated paper due to its broad range, it is not addressed here.
30
Figure 2.1 Classification of vehicle lateral motion control models.
In this chapter, we sequentially look into the following areas of vehicle lateral
motion control: 1) control devices and sensory; 2) lateral motion model; 3) lateral
motion observer; and 4) lateral motion controller. Only the last topic is extended and
discussed in details. Specially, robust controller, sliding mode controller, adaptive
controller and fuzzy controller are addressed. While it is impossible to cover the large
number of publications in this area, the key findings and trends of research are
included. The focus is on more recent literature since good reviews already exist on
the relatively long history of the subject [77], [79], [98], [141].
2.2
Advance in Steering Control Devices
2.2.1
Steer-by-Wire
Increasing requirements of safe and comfortable driving have led vehicle
31
manufacturers and suppliers to actively pursue development programs in the so called
"by-wire" subsystems. These computer-controlled subsystems include steer-by-wire,
brake-by-wire, drive-by-wire and etc, which are connected through in-vehicle
computer networks.
A steer-by-wire system replaces the traditional mechanical linkage between the
steering wheel and the road wheel actuator (e.g., a rack and pinion steering system)
with an electronic connection. Because it removes direct kinematical relationship
between the steering and road wheels, it enables control algorithms to help enhance
driver steering command [56]-[59].
For instance, Fig.2.2(b) shows a production model which was modified for full
steer-by-wire capability by replacing the steering shaft with a brushless DC
servomotor actuator [59]. A rotary position sensor measures the lower steering shaft
angle, which is equal to the front wheel steer angle scaled by the steering ratio. An
identical sensor attached to the upper steering shaft measures the hand wheel angle.
The servomotor actuator specifications are chosen based on the maximum torque and
speed necessary to steer the vehicle under typical driving conditions including
moderate emergency maneuvers.
It was reported in [56]-[59] that the response in steer-by-wire system is more
quickly and accurately than that in conventional steering system. This improves
vehicle stability and provides a basis for fault detection. Moreover, it was showed in
[57] and [59] that by effectively changing front cornering stiffness, the same vehicle
32
can be made to handle differently. Therefore, it is possible to maintain consistent
handling
characteristics
under
variable
operating
conditions.
Nevertheless,
steer-by-wire also simplifies assembly and reduces vehicle’s mass. For example, in
the steer-by-wire system proposed in [59], only the stock hydraulic power assist unit
and rack and pinion mechanism is retained. This therefore allows flexibility in
packaging.
In general, steer-by-wire system is expected to provide a better operation platform
of the lateral motion controllers.
(a)
33
(b)
Figure 2.2 Diagram of steering systems: (a) conventional steering system, (b)
steer-by-wire system [59].
2.2.2
Steering Related Sensory
Since a vehicle is a highly complex system that constitutes of varied mechanical,
electronic and electromechanical elements, numerous sensors were designed and
applied to measure the movement information. Concentrated on vehicle lateral motion
control, position and movement information of a vehicle need to be accurately
measured.
In [69], sensors for lateral control are categorized into two types:
infrastructure-based
and
infrastructure-independent.
Examples
of
the
infrastructure-based systems include discrete magnetic reference markers and
continuous magnetic tape [67]-[69]. The infrastructure-independent methods use, for
example, global positioning system (GPS), inertia position system (INS), or vision
system for sensing [60]-[76], [80]. However, these infrastructure-independent systems
34
still rely on infrastructure in the sense of reliable roadway markings in the former case,
and a reliable and accurate roadway geographical information system (GIS) database.
Global positioning system (GPS), inertial navigation system (INS) and their
combinations attract great interest in the last decade. The position and velocity of a
vehicle can be directly measured by using global position systems [60]-[61]. It had
been proven in several literals including [62]-[64] that sideslip angle, yaw rate,
heading angle and position displacements can be indirectly estimated with cooperated
inertial navigation system and global position systems. The newly developed fiber
optic gyroscopes (FOP) are capable to measure sideslip angle, yaw rate, heading angle
straightforward with high accuracy [65]-[66]. But current FOPs usually require
considerable installing and maintaining cost, which prevents their widely application
in the near future.
Magnetic sensing is another promising technology that has been developed
recently for the purposes of vehicle position measurement and guidance. By using
either magnetic tape or magnetic markers, vehicle position displacement can be gotten
as well some other useful information [67]-[69]. Besides, vision sensors can also be
employed to measure displacement. For example, the offset between the vehicle and
curve can be accurately obtained by using the laser sensor proposed in [70]. However,
measure performance of these two methods is more vulnerable to environment
disturbances than that of the above two techniques, i.e. the laser sensor is sensitive to
fog.
35
In the rest of this chapter, we assume all the information needed has already been
accurately measured.
2.3
Vehicle Lateral Motion Model and Estimation
2.3.1
Bicycle Model
Lateral vehicle dynamics has been studied since the 1950s [71]-[72]. In 1956,
Segel presented a vehicle model with three degrees of freedom in order to describe
lateral movements including roll and yaw. If roll movement is neglected, a simple
model known as "bicycle model" is obtained. This model is widely used for studies of
lateral vehicle dynamics (yaw and sideslip) now. The following discussions will be
primly carried out based on this model.
Suppose the vehicle is moving on a flat surface. By lumping the four wheels into
one virtual wheel in the centerline of the vehicle, we can have front steering and full
steering models as shown in Fig.2.3(a) and Fig.2.3(b) respectively [73]-[75].
Here Reference point CG is chosen to represent the center of gravity for vehicle
body, where vehicle velocity v is defined. Symbol A and B denote the positions of
front and rear tire/road interfaces respectively. Heading angle ψ is the angle from
the guideline to the longitudinal axis of vehicle body AB. Slide-slip angle β is the
angle from the longitudinal axis of vehicle body
to the direction of the vehicle
velocity. δ f is the front tires steering angle. δ r is the rear tires steering angle. Yaw
rate is denoted as r . f f and f r are the front and rear tire forces which are
36
perpendicular to the directions of tire movements, respectively. f w is the wind forces
acting on the aerodynamic center of the side surface and l w denotes the distance
between CG and aero-dynamical center of the side surface. l fs and l rs denotes the
distances from the front and rear sensor “looking at” points to CG, respectively. y f
and y r represent the displacements from the front and rear “looking at” points to the
guideline. Other variables are given in Table.2.I, in which the values are set for a city
bus O 305 based on IFAC benchmark example [74]. Here, c f and c r denote the
cornering stiffness of front and rear tires respectively, which we will use in the
following Eq. (2.7)-(2.8).
(a)
(b)
Figure 2.3 "Bicycle" steering model: front steering (a) and full steering (b).
37
Table 2.1 Parameters and Their Typical Values
Symbols
Mass of the vehicle m
Initial moment around z-axis I z
Distance from A and CG l f
Distance from B and CG l r
Typical values
[9950, 16000]kg
[10.85, 21.7]Ns/rad
3.67m
1.93m
6.12m
198000N/rad
470000N/rad
l fs
Stiffness coefficients of front tire c f
Stiffness coefficients of rear tire c r
Assuming that vehicle has a constant velocity, front steering model with
nonlinear tire force characteristics can be described by the differential equations
d
dt
⎛β
⎜⎜
⎝r
f f + fr
⎛
⎞
⎜
−r ⎟
⎞ ⎜
mv
⎟
⎟⎟ =
l f − lr f r
⎟
⎠ ⎜ f f
cos β ⎟
⎜
Iz
⎝
⎠
(2.1)
Using the famous "magic" formulas of tire/road friction given in [76]-[77], we
have
{
( [
{
(
]
f f = D f sin C f tan −1 B f 1 − E f α f + E f tan −1 (B f α
f
)}
f r = D r sin C r tan −1 B r [1 − E r ]α r + E r tan −1 (B r α r )
⎧
l
⎞
−1 ⎛ f
⎪α f = β + tan ⎜⎜ ⋅ r cos β ⎟⎟ − δ
⎪
⎝ v
⎠
⎨
⎪α = β − tan −1 ⎛⎜ l f ⋅ r cos β ⎞⎟
⎜ v
⎟
⎪ r
⎝
⎠
⎩
))}
f
(2.2)
(2.3)
(2.4)
where α f is the slip angle of front tires, α r is the slip angle of rear tires. The
coefficients B j , C j , D j and E j ( j = f , r ) in the models can be calculated in
practice.
In [78], Ono, Hosoe and Tuan et. al analyzed the bifurcation phenomena in above
38
model (1)-(4). The vehicle unstabilization was shown to be caused by a saddle-node
bifurcation which depends heavily on the rear tire side force saturation. By
approximating the nonlinearities with
cos β ≅ 1 , α
f
≅β +
lf
v
⋅r +δ f , αr ≅ β −
lf
v
⋅r
(2.5)
They approximated the Jacobian matrix of system model d ⎛⎜ β ⎞⎟ = F ( β , r , δ f ) at
dt ⎜ r ⎟
⎝
⎠
equilibrium point χ 0 by the matrix
⎡
c ∗f + c r∗
⎢ −
mv
=⎢
∗
∗
⎢ l f c f − lr cr
−
⎢
Iz
⎣
Aχ 0
l f c ∗f − l r c r∗ ⎤
⎥
mv 2
⎥
2 ∗
2 ∗
l f c f + lr cr ⎥
−
⎥
I zv
⎦
−1−
(2.6)
where c ∗f and c r∗ are the tangents to slopes of front and rear side force
characteristics at equilibrium point χ 0 respectively. The bifurcation situation around
point χ 0
has been checked. It showed that the linear system should be stable at a
relatively large neighborhood of this equilibrium point.
If cornering stiffness c ∗f and c r∗ is taken to be constant, we can write the linear
dynamic model for front steering vehicle as
⎡ β&
⎢
⎢ r&
⎢ ψ&
⎢
⎢ y& f
⎢ y&
⎣ r
⎤ ⎡ a 11
⎥ ⎢
⎥ ⎢ a 21
⎥=⎢ 0
⎥ ⎢
⎥ ⎢ v
⎥ ⎢− v
⎦ ⎣
a 12
a 22
1
0
0
0
0
0
0
l fs
− l rs
v
v
0
0
0⎤ ⎡ β
0 ⎥⎥ ⎢⎢ r
0⎥ ⎢ ψ
⎥⎢
0⎥ ⎢ y f
0 ⎥⎦ ⎢⎣ y r
⎤ ⎡ b11
⎥ ⎢b
⎥ ⎢ 21
⎥+⎢ 0
⎥ ⎢
⎥ ⎢ 0
⎥⎦ ⎢⎣ 0
0
0
−v
− vl fs
vl rs
d1 ⎤
d 2 ⎥⎥ ⎡ δ f
0 ⎥ ⎢⎢ ρ ref
⎥
0 ⎥ ⎢⎣ f w
0 ⎥⎦
⎤
⎥
⎥
⎥⎦
(2.7)
Similarly, the linear dynamic model of full steering vehicle is written as
⎡ β&
⎢
⎢ r&
⎢ ψ&
⎢
⎢ y& f
⎢ y&
⎣ r
⎤ ⎡ a 11
⎥ ⎢
⎥ ⎢ a 21
⎥=⎢ 0
⎥ ⎢
⎥ ⎢ v
⎥ ⎢− v
⎦ ⎣
a 12
0
0
a 22
0
0
1
l fs
0
v
0
0
− l rs
v
0
0⎤ ⎡ β
0 ⎥⎥ ⎢⎢ r
0⎥ ⎢ ψ
⎥⎢
0⎥ ⎢ y f
0 ⎥⎦ ⎢⎣ y r
⎤
⎥
⎥
⎥+
⎥
⎥
⎥⎦
⎡ b11
⎢b
⎢ 21
⎢ 0
⎢
⎢ 0
⎢⎣ 0
b12
0
b 22
0
0
0
−v
− vl fs
0
vl rs
d1 ⎤
⎡δf
d 2 ⎥⎥ ⎢
δr
0 ⎥⎢
⎢
ρ
⎥
0 ⎥ ⎢ ref
f
0 ⎥⎦ ⎣ w
⎤
⎥
⎥
⎥
⎥
⎦
(2.8)
39
where
~ v , a = −1 + (c l − c l ) / m
~v 2
a11 = − ( c r + c f ) / m
12
r r
f f
~
~
a 21 = ( c r l r − c f l f ) / I z , a 22 = − ( c r l r2 + c f l 2f ) / I z v
~ v , b = c l / I~
~v , b = c / m
b11 = c f / m
12
r
21
f f
z
~
b 22 = − c r l r / I z , d 1 = 1 / mv , d 2 = l w / I z
Here m~ = m / µ and I~z = I z / µ are the normalized mass and inertia respectively,
in which µ is common road adhesion factor. ρ ref is the curvature of the guideline.
The contribution of ρ ref to y f or y r is sometimes neglected, since it is small.
It should be pointed out that cornering stiffness varied with several factors. One
factor is that it increases with tire pressure. When the car turns, the mass transfer onto
the external wheels increases tire pressure, which can lead to notable variations in
cornering stiffness. Stephant, Charara and Meizel showed in [79] that such variations
are normally less than 10% and still tolerant for most robust steering controllers.
Full steering vehicles outperform front steering vehicles in handling and stability
significantly [145]. Usually, when the vehicle enters the curved path, the rear wheel
will first steer in the opposite direction to the front wheel in order to generate
sufficient yaw motion to follow the desired yaw rate. After that, the rear wheel steers
will synchronize with the front wheel to keep the yaw rate with desired value and also
control the lateral motion for path tracking. Since most controller design methods can
be applied to both situations without extreme modifications, we will not emphasis the
difference between front steering vehicles and full steering vehicles in the rest of this
40
chapter.
The design specifications of steering controller are primarily given in terms of
maximal displacement from the guideline and maximal steering angle and steering
angle rate. For instance, the benchmark problem mentioned in [74] mainly requires
1)
the steering angle is limited as
δ f ≤ 40 deg
2)
the steering angle rate is limited as
δ& f ≤ 28 deg/ s
3)
(2.9)
(2.10)
The displacement from the guideline must not exceed 0.15m in transient
state and 0.02m in steady state;
4)
The lateral acceleration must not exceed 2m/s for passengers comfort.
The ultimate limit is 4m/s.
Generally, "bicycle" model grasps the prime characteristics of vehicle steering
movement and yields a relatively simple linear model for analyzing. Thus it was
widely used in steering controller design in the last decade.
2.3.2
Other Vehicle Lateral Motion Models
Besides the "bicycle" model, there were some other dynamic models that had
been proposed and analyzed. These models usually set up the direct relationships of
41
vehicle longitudinal and lateral speeds in terms of steering angle. Many of them also
considered some unhandled factors that are left untouched in "bicycle" model. For
instance, the vehicle dynamics in [80] tried to incorporate vehicle aerodynamics into
lateral motion model. It was represented by the following set of nonlinear systems
equations
vy + l f r
⎧
1
δ f + v x2 ( fk 1 − k 2 )]
⎪ v& x = [T + mv y r − mfg + c f
m
v
x
⎪
⎪⎪
vy
1
r
− (l f c f − l r c r + mv x2 ) ]
⎨ v& y = [( c f + T )δ f − ( c f + c r )
m
v
v
x
x
⎪
⎪
v
1
r
y
⎪ r& =
[( l f c f + T )δ f − (l f c f − l r c r )
− ( l r2 c r + l 2f c f ) ]
⎪⎩
Iz
vx
vx
(2.11)
where v x and v y are vehicle’s longitudinal velocity and lateral velocity in vehicle
coordinate respectively. T is the traction and/or braking force. f is the rotation
coefficients. k 1 and k 2 are lift and drag parameters from aerodynamics respectively.
Besides, the following mapping function (12) describes the relationship between
vehicle velocities denoted in orthogonal coordinates and that denoted in world
coordinates
⎧ x& = v x cos( ψ ) − v y sin( ψ )
⎪
⎨ y& = v x sin( ψ ) + v y cos( ψ )
⎪&
⎩ψ = r
(2.12)
where ( x , y ) denote vehicle center gravity’s position in world coordinates. The
displacement driving requirements were brought in terms of ( x , y ) straightforward.
There has been considerable theoretical and experimental research on developing
vehicle models of different levels of complexity. Thorough discussions requirement a
dedicated publication and are not discussed in this chapter.
42
2.3.3
Lateral Motion Estimation
In many recent approaches, not all the vehicle characteristics are directly
measured due to high cost or some other reasons. Instead, several special observers
are used to reconstruct the needed information. In literals, these observers were also
called virtual sensors.
For example, knowledge of sideslip angle, yaw rate and lateral velocity is
essential in vehicle control, but is difficult to obtain directly. In 1997 and 1999,
Kiencke etc. proposed a linear observer and a nonlinear observer using reduced order
bicycle model in [81] and [82]. Soon after that, Venhovens and Naab used a Kalman
filter in [83] for a linear vehicle model in 1999. In [84], Huh et. al. constructed the
monitoring system based on KFMEC (Scaled Kalman Filter with Model Error
Compensator) technique to improve the robustness of ordinary Kalman filters. In [85],
Zhang, Xu and Rachid showed the feasibility of the sliding mode observer for vehicle
lateral motion. Similar conclusions were reached by Perruquetti and Barbot in [86].
To filter out the unexpected effect of disturbances from the observer output,
different robust design methods had been introduced in Luenberger observer
construction. In [87]-[89], [92], H ∞ filer theory was employed to design the optimal
observers to resist disturbance for reduced order bicycle model. In [90], H ∞ loop
shaping was used for observer design of the linearized lateral motion model of a
single-unit HDV (tractor-semitrailer type vehicle). In [91], H ∞ LMI design method
was applied in Luenberger observer and fault detection filter design.
43
Most above observers utilized the accurate dynamic model using nominal values
including tire concerning stiffness, vehicle mass and moment of inertia and distances
between center of mass and tires. Thus, these observers depend on an accurate
knowledge of these parameters, and are affected by variations in them. For instance,
Stephant, Charara and Meizel pointed out that the speed of center of gravity is not an
indispensable variable. One method to solve this problem is to choose the estimation
method without utilizing the vehicle dynamic model, i.e. the observers proposed in
[93] and [94]. However, the non model-based observers are naturally hard to be
applied along with the steering control system. A more reasonable method is to design
robust observer that is not sensitive to system parameter changes, or adaptive observer
that can change itself according to parameter change. In [95], a robust observer was
developed by including an extra term and adopting the Lyapunov stability theorem. It
maintains the good disturbance rejection property that derived form [87]-[92], while
provides tolerance to model variance as the observer too.
Recently, Stephant, Charara and Meizel carefully compared four observers
including linear Luenberger observer and three nonlinear observers: extended
Luenberger observer, extended Kalman filter and sliding-mode observer in [89].
Based on simulation results and practical experiments, they showed that all four
observers can yield acceptable estimation results if the observer’s parameters are
appropriately assigned.
44
2.4
Vehicle Lateral Motion Control
As early as 1969, Kasselmann and Keranen [71] developed an active steering
system based on feedback from a yaw rate sensor. With continuous efforts, people
gradually realized that the difficulties of steering control mainly lie in the following
five aspects:
1)
how to avoid skidding during steering which is one frequently
encountered hazardous situation for drivers;
2)
how to reject the disturbance caused by wind or some other reasons;
3)
how to deal with vehicle dynamics uncertainty and variation;
4)
how to handle actuator rate limits during steering;
5)
how to handle time delay exists in feedback block during steering.
To answer these five questions, numerous designed methods had been proposed
in the last two decades. In this paper, we will address robust controller, sliding model
controller, fuzzy controller and adaptive controller. Some other controllers are briefly
mentioned, too.
2.4.1
Frequency Domain Robust Steering Controller
Originated in the later 80s, frequency domain robust design techniques soon
became and remained as one of the most important techniques in field of vehicle
45
lateral motion control. It had been proven to be a practical and efficient approach by
lots of literature [73]-[75], [96]-[100].
One direct idea to avoid skidding is to remove the influence of r on the lateral
acceleration. The lateral and yaw motions of a car with active steering is decoupled by
Ackermann in [73]. It was proved that for an ideal longitudinal mass distribution, the
decoupling by yaw rate feedback is robust with respect to uncertain nonlinear tire side
force characteristic, velocity and vehicle mass. But Ackermann later showed that this
was not a simple and cheap control system since it requires measuring longitudinal
velocity of vehicle v x ≈ v sin β , yaw rate r and its derivative, slip angle β
simultaneously. Thus a practical controller was proposed in [75], in which only v x
needs to be measured. This simplified controller was proved to have similar
steady-state behavior to a car.
In [92], it was further shown that additional feedback of the yaw rate r leads to
a significant reduction of the deviation from the guideline in nearly all driving
maneuvers compared to earlier controller designs which used solely feedback of the
deviation y f . Therefore, a feedback controller with respect to both r and y f were
proposed. It is written as
δ& f = u f − k r ⋅ r
(2.13)
where u f was a determined as
u f (s)
y f (s)
= ω c2
k DD s 2 + k D s + k P
s 2 + 2 D ω c s + ω c2
(2.14)
where inside the bandwidth ω c , k P denotes a proportional part, k D denotes a
46
differential part and k DD denotes the double differential part.
As what revealed in [97] and [98], using careful poles and zeros assignment, the
system can be well stabilized. Besides it, the redundancy of the design parameters can
be used to count off the variance of vehicle dynamics. For instance, the Γ -stability
boundaries was analyzed for parameters ( k D , k DD ) in [99]. Moreover, a generic
control law for robust decoupling of lateral and yaw motion by yaw-rate feedback to
front-wheel steering was derived in [108]. It showed that ideal steering dynamics
were able to be achieved by velocity scheduled lateral acceleration feedback to
front-wheel steering. For robust yaw stabilization a velocity-scheduled yaw-rate
feedback to rear-wheel steering is given, by which the linearized system gets
velocity-independent yaw eigenvalues. In [99], it was further proven that decoupling
by yaw rate feedback is robust with respect to uncertain nonlinear tire side force
characteristic, velocity and vehicle mass, if we assumed an ideal longitudinal mass
distribution.
In the last 90’s, H ∞ robust analysis method was introduced into steering
controller design field to reduce the unexpected effect of wind disturbance. H ∞
theory is constructed to handle the deterministic disturbance model consisting of
bounded energy (square-integrable) L 2 signals and allows controller design for
narrow-band disturbance rejection, see Francis 1987 [100] and Zames 1981 [101]. In
[102], Guvenc, Bunte and Odenthal etc. designed a disturbance observer, whose
model regulation capability allows the specification and achievement of desired yaw
47
dynamics. The proposed integrated control model is shown in Fig.2.4, where G ref
was the transfer function from disturbance torque ρ ref to yaw rate r . G~ u is the
nominal system model and G u is the un-modeled dynamics.
Figure 2.4 Diagram of system architecture with add-on disturbance observer [102].
It was shown that the model regulation and disturbance rejection property of this
proposed observer can be considered as a special H ∞ loop shaping for path
following. The filtering effect is chosen to satisfy the classical loop shaping constraint
W s S + W T T < 1 , for ∀ ω
where W s and W T are the sensitivity function weight and the
(2.15)
complementary
sensitivity function weight respectively. S and T denote the sensitivity function
and transfer function, respectively.
Equivalently, Mammar etc. developed several two degree of freedom (2DOF)
48
steering controller in [103]-[105] using the H ∞ loop shaping technique, see Fig.2.5.
The design task is directly assigned as finding a robust feedback control K~ to
guarantee the stabilizing of system and minimize energy bound of the transfer
function from ρ ref to pre-selected measurement output z 1 , z 2 and z 3 . Based on
works of Kuzuya and Shin that was reported in [106], these robust 2DOF steering
controller can be easily digital implemented.
Figure 2.5 Diagram of system architecture with add-on disturbance observer [102].
In [109] and [107], the saturation properties of the steering actuator were studied.
The actual feedback control architecture considering steering actuator rate limits are
shown in Fig.2.6. Simulations and experiments pointed out that the steering angle rate
actuator saturation forms a major limitation of performance. In [107], the undesirable
limit cycles caused by saturations were analyzed by a describing function approach in
combination with the representation of limit-cycle-free regions in a parameter plane
49
of velocity and road/tire friction coefficient. The results were formulated in terms of
required actuator bandwidth that achieves robustness in the entire operating range. It
turned out that the use of a fading integrator can reduce the required actuator
bandwidth. Based on similar ideas, a compensator with high order was investigated in
[99] to achieve better performance.
Figure 2.6 Feedback control architecture considering actuator saturation [99].
Some other approaches based on H ∞ theory can be found in [110]-[116]. Most
of them were mainly devoted to find an optimal compromise point between steering
performance and wind disturbance rejection. Specially, Brennan and Alleyne
analyzed the effect of time delay in [116]. Constrained by the length, we will not
discuss them here.
2.4.2
Sliding Mode Steering Controller
Sliding mode steering controller is another frequently used steering controller.
Generally speaking, the basic idea of sliding mode control is to restrict the state space
50
trajectories of the dynamic system to a manifold called “sliding manifold” which is
usually denoted by S = 0 . This is achieved by directing the system trajectories
towards this manifold “from both sides”.
In [97], the sliding manifold was chosen as
S = c ∆ r + ∆ r&
(2.16)
where c > 0 is a constant gain that determines system behavior once the motion of
system (7) or (8) has been restricted to the neighborhood of the manifold S = 0 .
The structure of the proposed controller was shown in Fig.2.7, in which the
system ideal feedback control strategy is written as
rd = −
1
[v ( β + ψ ) + Ky ]
ls
where K > 0 determines the desired rate of decay of
(2.17)
y
.
51
Figure 2.7 Diagram of a typical sliding mode steering controller [97].
Notice both the states β and ψ are unknown, the following observer (18) and
(19) was introduced for estimating these two state variables
yˆ& = qˆ + l s r + l1 y , l1 > 0
(2.18)
q&ˆ = l 2 y , l 2 > 0
(2.19)
52
Thus the actual feedback control was obtained as
rd = −
1
[qˆ + K yˆ ]
ls
(2.20)
and the feedback control was chosen as
u f = − M u sign ( S )
(2.21)
where M u is the available steering angle rate.
To further improve passenger comfort, it was proved to be advantageous to
replace linear term K yˆ in (20) by a saturation function
rd = −
1 ⎡
⎢ qˆ + λ
ls ⎢
⎣
⎤
⎥, λ > 0, ε > 0
2
yˆ + ε ⎥⎦
yˆ
(2.22)
and the feedback control can be substituted by a continuous approximation as
u f = −M u
S
(2.23)
S + 0 . 0001
2
It was proven in [97] that this sliding mode controller yields smaller deviations
from the guideline and it has a more oscillatory behavior that shows up particularly in
the lateral acceleration and in the steering angle rate, but not in the derivations from
the guideline. Regarding settling times, there are no significant differences between
the two controllers.
There were several other sliding mode steering controllers which chose different
sliding manifold [117]-[122]. [117] studied the lateral and longitudinal control of
vehicle using a PID typed sliding surfaces, whose stability was proven using
Lyapunov theory. In [118], a velocity related sliding mode controller was proposed to
deal the input couple problem. Another special sliding mode integral action controller
53
and the corresponding sliding mode observer are used to enhance vehicle stability in a
split- µ maneuver in [112]. In [120], it was shown that the sliding mode controller
can also be used to deal with the nonlinear front steering model considering track
force control. Moreover, sliding mode steering controller is an important approach in
tractor-trailer vehicles lateral motion control [121].
2.4.3
Adaptive Steering Controller
Because of its capability of handling model uncertainty and parameter variation,
adaptive steering controllers achieve continuous interest in the last twenty years.
Generally, there are two different approaches: model concentrated approaches and
non-model concentrated approaches.
In [123], Brennan and Alleyne proposed a steering controller based on model
reference control (MRC) with a modification based on rejection of known disturbance
dynamics. Model reference control was initialed by Astrom and Wittenmark in 1997
[124]. It was proven in [123] that this method was effective for steering control
systems consist of dynamic uncertainty and disturbance. In [125], a special adaptive
observer was proposed to deal with system parameters variations. In [126], an
adaptive rule was proposed to make the controller flexible with velocity change. In
[144], a self-tuning regulator was proposed, which was claimed to have really nice
performance. However, the authors also admitted that that adaptive steering controller
is relatively complicated to be implemented in real life. Generally, the compromise for
54
a adaptive steering controller between precision and facility of implementation needs
to be clarified in real applications.
In [127]-[129], the concepts of Selected Adaptive Critic (AC) and Dual Heuristic
Programming (DHP) were used to design steering controller. Selected adaptive critic
methods are known to be capable of designing (approximately) optimal control
policies for non-linear plants (in the sense of approximating Bellman Dynamic
Programming). The present research focuses on an AC method known as dual
heuristic programming. There were lots of issues related to the pragmatics of
successfully applying the AC methods.
In [127], a straight forward utility function to capture these requirements would
take the following form:
δ f = − Ay 2f
error
− B y& 2f error − Cv y2error − D v& y2error
(2.24)
where A , B , C and D were determined from programming to indicate the human
designer’s judgement about the relative importance of each term, according to desired
plant response characteristics (e.g., the derivative terms encourage more “damped”
responses). Slightly different from [127], the control coefficients are achieved through
self-learning using neural networks in [128]. In [129], it was further shown that DHP
can be employed to optimize fuzzy steering controllers.
2.4.4
Fuzzy Steering Controller
Fuzzy set theory and Fuzzy inference was first presented by Zadeh in 1965.
55
Recently, some new approaches take advantage of Fuzzy inference to avoid
addressing complex vehicle dynamics. Moreover, these approaches were proven to
able to incorporate and utilize human steering skills to improve the automatic driving
performance [130]-[138].
For example, a direct Fuzzy control strategy was proposed by Brown and Hung
1994 for the above 4WS car model. The corresponding Fuzzy control rule is
something as
"
IF
Yaw_Rate_Error is Negative_Large (NL) AND
Front_Slip is Negative_Large (NL)
THEN
Command_Front_Steering_Angke is Positive_Large (PL)
"
The front steering command and the rear steering command were designed to
cooperate appropriately to avoid skipping in [130]. Assigning appropriate fuzzy
membership function and rule table, Brown and Hung claimed that the 4WS car using
this Fuzzy controller was quite robust to wind guests and road perturbations.
There were several other directly fuzzy controllers reported in [131]-[135].
However, it should be remarked that, in spite of the well functioning of these fuzzy
controllers, they were still heuristic controllers. Since no stability proof had was
56
presented, and then they were less reliable.
In some recent literals, the proposed fuzzy controllers were constructed as
follows. First, an optimal steering controller was constructed for each local model,
which indeed constructed a part of the fuzzy model. Then, local controllers were
combined using fuzzy rules to form a fuzzy logic controller. Therefore, the
performance of the fuzzy controller can be analyzed using linear matrix inequalities
(LMI) or algebra Riccati equation (ARE). In [136]-[138], these methods were shown
to be effective with both theoretical analysis and simulations.
2.4.5
Other Steering Controller
Recently, time domain robust design technique was used in vehicle lateral control
too [139]-[140]. Although time domain robust design is intrinsically equivalent to
frequency domain robust design, the obtained controllers differ in many aspects. A
detailed discussion in this research direction is presented in our coming paper [141].
There were also some special nonlinear steering controllers proposed in [142][143]. Usually their stability was guaranteed by Lyapunov theory. However, most
such approaches did not carefully consider robustness and actuator saturation.
In [144], a simple proportional controller was compared with H ∞ robust
controllers, fuzzy controller and adaptive controller. This proportional feedback was
found to yield the largest offset with respect to other controller, although it was only
slightly affected by the wind force. On the other hand, the self-tuning regulator
57
presents the smallest errors. The responses of H ∞ and fuzzy controllers are
comparable in most tests regarding to self-tuning regulator. Although these
conclusions were made for special controllers, they were widely accepted to take
valid in general situations.
2.5
Remarks
Some important contents leave untouched in this chapter due to length limits and
their premature. For instance, we do not mention the following studies here.
1)
the relationship between vehicle longitudinal and lateral motion should
be further analyzed, although it was proven that these two motions can
be roughly decoupled [146]-[147];
2)
the effect of sensor installment and measurement error needs further
discussions besides [148];
3)
how to make intelligent steering controller cooperate appropriately with
driver command needs to be carefully studied. Some previous works
[149]-[152] had shown that this problem is quite complex, in which both
drivers’ characteristics, feelings, and driving status needs to be
monitored and analyzed;
4)
rollover avoidance using special steering control is gaining more
attentions. Some promising results had been reported in [161]-[166].
58
5)
Moreover, fault tolerant steering control and corresponding fault
detection are obtaining more and more considerations recently
[153]-[161]. With rapidly increasing demands on driving safety, a boom
in this research field is expected in the near future.
59
CHAPTER 3 A ROBUST OBSERVER DESIGNED FOR VEHICLE
LATERAL MOTION ESTIMATION
Lateral control of vehicles on automated highways often requires estimation of
sideslip angle, yaw rate and lateral velocity, which are difficult to measure directly.
Thus, several observers (virtual sensors) were developed in the last decade. To solve
the unhandled estimation problem caused by dynamic model uncertainty, a robust
observer using H ∞ design method is proposed in this chapter. It maintains the good
disturbance
rejection
property
that
derived
form
previous
research,
and
simultaneously provides acceptable tolerance to model variance. Specially, effects of
displacements of sensory, dynamics variance caused by mass/velocity/tire-road
frictions change or nonlinear characteristic are studied. Simulations demonstrate the
usefulness of the proposed observer, [167]-[176].
3.1
Introduction
Since a vehicle is a highly complex system that constitutes of varied mechanical,
electronic and electromechanical elements, numerous sensors were designed and
applied to measure the movement information. But not all vehicle characteristics are
measured directly due to high cost or other reasons. Instead, several special observers
are used to reconstruct the needed information. In literals, these observers were also
60
called virtual sensors [176].
For example, knowledge of sideslip angle, yaw rate and lateral velocity is
essential in vehicle control, but is difficult to obtain directly. In 1997 and 1999,
Kiencke etc. proposed a linear observer and a nonlinear observer using reduced order
bicycle model in [170] and [171]. Soon after that, Venhovens and Naab used a
Kalman filter in [172] for a linear vehicle model in 1999. In [173], Huh et. al.
constructed the monitoring system based on KFMEC (Scaled Kalman Filter with
Model Error Compensator) technique to improve the robustness of ordinary Kalman
filters. In [174], Zhang, Xu and Rachid showed the feasibility of the sliding mode
observer for vehicle lateral motion. Another sliding mode observer was designed by
Perruquetti and Barbot in [175]. Recently, Stephant, Charara and Meizel carefully
compared four observers including linear Luenberger observer and three nonlinear
observers: extended Luenberger observer, extended Kalman filter and sliding-mode
observer in [176]. Based on simulation results and practical experiments, they showed
that all four observers can yield acceptable estimation results if the observer’s
parameters are appropriately assigned.
To filter out the unexpected effect of disturbances from the observer output,
different robust design methods had been introduced in Luenberger observer
construction. In [177]-[179], [182], H ∞ filer theory was employed to design the
optimal observers so as to resist disturbance. In [180], H ∞ loop shaping was used for
observer design of the linearized lateral motion model of a single-unit HDV
61
(tractor-semitrailer type vehicle). In [181], H ∞ LMI design method was applied in
Luenberger observer and fault detection filter design.
However, almost all above observers utilized the accurate dynamic model using
nominal values including tire concerning stiffness, vehicle mass and moment of
inertia and distances between center of mass and tires. Thus, these observers depend
on an accurate knowledge of these parameters, and are affected by variations in them.
For instance, Stephant, Charara and Meizel pointed out that the speed of center of
gravity is not an indispensable variable. One method to solve this problem is to
choose the estimation method without utilizing the vehicle dynamic model, i.e. the
observer proposed in [179]. However, the non model-based observers are naturally
hard to cooperate with the advanced control systems. A more reasonable method is to
design a robust observer that is not sensitive to system parameter changes, or adopt an
adaptive observer that can change itself according to dynamic change.
The robustness analysis for uncertain systems has been the focus of much
research in recent years [192], [194]-[196]. In [192], a robust observer was developed
by including an extra term. Combing the skills in [181]-[182], [194]-[196], a novel
robust observer is designed in this chapter. It maintains the good disturbance rejection
property that derived form [177]-[182], while provides tolerance to model variance as
the observer in [194]-[196], too.
62
3.2
Sensor Sets and System Observability
The linear dynamic models (2.7) and (2.8) can be written into canonical form as
x& = Ax + Bu + Ew
where x = [β
r ψ
input, and w = [ρ ref
yf
fw
yr
]
T
]
T
is the state variable, u = [δ f
(3.1)
δ r ]T
is the control
is the disturbance. A , B and E are the corresponding
system matrices.
The measurement output y is usually formulated as
y = Cx + Du
(3.2)
Here the structure of measurement matrices C and D are determined by the
sensors applied. For example, by using the laser sensors proposed in [188],
displacements y f and y s can be accurately obtained. It was also proven in [189]
and [190] that Global Position Systems (GPS) can also be applied to approximately
get heading angle ψ and displacements y f and y s simultaneously. With the Fiber
Optic Gyroscopes (FOP) described in [191], heading angle ψ and yaw rate r are
precisely measured. However, it requires considerable install and maintain cost.
Since the linear dynamic model (2.7) and (2.8) highly depend on system velocity,
the system dynamics should be modified as below considering velocity variance and
nonlinear properties
x& = ( A + ∆ A ) x + Bu + Ew
(3.3)
where ∆ A denotes the variance matrix that is determined by variance of mass,
63
velocity, tire-road friction coefficients and nonlinear characteristics.
In this chapter, the norm of ∆ A is bounded as
∆A
2
≤ε
(3.4)
Simulation reveals that ε mainly dependent on estimation error of velocity v
and road adhesion factor µ .
It can be easily proven that the system is observable if either displacements y f
or y r is measured. The system is unobservable if only ψ is measured. These
conclusions take valid for both front steering vehicle and full steering vehicle.
Simulation also shows that to measure both y f and y r may increase the
robustness of the observer. But many vehicles only equip front sensor because of cost
consideration.
3.3
Robust Observer Design
In this chapter, the proposed robust observer for the systems (3.3)-(3.4) is chosen
as
⎧
⎪ xˆ& = A xˆ + Bu − L ( y − yˆ ) + τα
⎪⎪
⎨ yˆ = C xˆ + Du
⎪
ε 2 xˆ T xˆ
⎪α =
P −1C T ( y − yˆ )
⎪⎩
( y − yˆ ) T ( y − yˆ )
(3.5)
where x̂ denotes observer state, ŷ denotes observer output. L is the observer
matrix. τ is a positive real scalar that needs to be determined.
Based on (3.3)-(3.5), we have the dynamics of observer error e = x − xˆ as
64
e& = ( A + LC ) e + ∆ Ax + Ew − α = ( A + ∆ A + LC ) e + ∆ A xˆ + Ew − τα
(3.6)
Obviously, the design task here is to find the optimal matrix L to filter w from
e.
One frequently used performance index is chosen as:
Minimize the H ∞ norm of the transfer function matrix from w to e .
This naturally leads to the well-known H ∞ design problem, which was solved
using a linear matrix inequality (LMI) problem by Boyd, Ghaoui, and E. Feron et. al
in [192], or equivalently solved using a set of Algebraic Riccati Equations (ARE)
shown by Nagpal and Khargonekar in [193].
In this chapter, we choose the energy constraint of the observer error as the
design objective. It is written as:
Choose the smallest γ > 0 such that
+∞
T
∫ e edt ≤ γ
0
+∞
2
∫w
T
wdt
(3.7)
0
Therefore, we can reach the main result as follows.
Theorem: H ∞ constraints (3.7) will be held if there exist a symmetry positive
matrix P and two positive real numbers τ and λ that satisfy
65
⎡ P[ A + LC ] + (1 + λε 2 ) I + [ A + LC ]T P + (1 / λ + 1 / τ ) 2 P 2
⎢
⎢− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −
⎢
ET P
⎣
|
+
|
PE ⎤
⎥
− − −⎥ < 0
− γ 2 ⎥⎦
(3.8)
Proof: Let’s define a Lyapunov like function V (t ) = e T Pe , where P is a
symmetry positive matrix. We have the time derivative of V (t ) along the trajectories
of (3.6) as
V& = e T P[ A + ∆ A + LC ]e + e T Ew + e T ∆ A xˆ + e T [ A + ∆ A + LC ] T Pe
+ w T E T e + xˆ T ∆ A T e − τ e T P α − τα T Pe
(3.9)
Thus, the design objective can be transformed as finding the smallest γ 2 that
satisfies:
T = V& + e T e − γ 2 w T w ≤ 0
(3.10)
Substituting (3.9) into (3.10) yields
T
T ⎡ P [ A + LC ] + I + [ A + LC ] P
⎡e⎤ ⎢
T = ⎢ ⎥ ⎢− − − − − − − − − − − − − − − − −
⎣w⎦ ⎢
ET P
⎣
+ 2 e T P ∆ Ae + 2 e T P ∆ A xˆ − 2τ e T P α ≤ 0
|
+
|
PE ⎤
⎥⎡ e ⎤
− − −⎥⎢ ⎥
w
− γ 2 ⎥⎦ ⎣ ⎦
(3.11)
Note that
e T P ∆ Ae ≤
e T P ∆ A xˆ ≤
1
λ
1
τ
e T P 2 e + λe T ε 2 e
(3.12)
e T P 2 e + τ xˆ T ε 2 xˆ
(3.13)
xˆ T ε 2 xˆ ≤ e T P α
(3.14)
we have (3.8) is a sufficient condition for (3.11).
Based on this Theorem, the robust observer design problem can be formulated as
66
below:
Design Task 3.I
Min γ
with γ > 0 , λ > 0 and moderate τ > 0 such that
⎡ PA + A T P + XC + C T X T + (1 + λε 2 ) I + (1 / λ + 1 / τ ) 2 P 2
⎢
⎢− − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −
⎢
ET P
⎣
|
+
|
PE ⎤
⎥
− − −⎥ < 0
− γ 2 ⎥⎦
(3.15)
and the observer matrix is
L = P −1 X
(3.16)
From the discussions in [194]-[196], (3.6) should be a forced, nonautonomous,
dynamic system, and contains an unknown, perturbation term. Fortunately, we find
that this condition can be naturally met. Besides, τ should not be too large due to the
constraints on implementation of the observer. In the practice, τ
should be
determined by compromising all the requirements together. But simulations prove that
constraints on τ will directly limit the range of γ at the same time, too.
3.4
Simulation Results
To demonstrate the feasibility of the proposed observer, its tracking performance
is compared with a linear Luenberger observer that is only optimized using H ∞
method to reject disturbance.
67
Using similar transformation, the linear Luenberger observer is formulated as
⎧ x&ˆ = A xˆ + Bu − L ′( y − yˆ )
⎨
⎩ yˆ = C xˆ + Du
(3.17)
Its performance index is also chosen as
Minimize the H ∞ norm of the transfer function matrix from w to e .
Thus its design task should be
Design Task 3.II
Min γ
with γ > 0 , µ > 0 and moderate τ > 0 such that
⎡ P[ A + L ′C ] + [ A + L ′C ]T P + I
⎢
⎢− − − − − − − − − − − − − − − − −
⎢
ET P
⎣
|
+
|
PE ⎤
⎥
− − −⎥ < 0
− γ 2 ⎥⎦
⎡ PA + A T P + X ′C + C T X ′ T + I
⎢
⎢− − − − − − − − − − − − − − − − −
⎢
ET P
⎣
|
+
|
PE ⎤
⎥
− − −⎥ < 0
− γ 2 ⎥⎦
(3.18)
or equivalently
(3.19)
with the observer matrix
L ′ = P −1 X ′
(3.20)
Suppose the vehicle is front steering and only the front sensor is applied. Thus
68
C = [0
0
0 1] , D = [0
0
0
0]
(3.21)
The parameters are assigned as:
~
m
= 1500 kg, I~z =2500 kg.m.m, l f = 1.2 m, l r =1.5 m, l fs = 5 m, l w = 0.5 m, c f
= c r = 80000kN/rad.
Here τ is chosen 10. The variance bound of velocity is ∆ v = 1 m/s, ∆ m~ = 100
kg, ∆ µ = 0.1. We can choose
ε = 5
Suppose v is incorrectly estimated as 16m/s when it equals to 15m/s and µ is
misestimated as 0.7 instead of 0.8 during the simulation.
The Luenberger observer matrix is
L ′ = [ 0.2869
- 3.6024
- 0.1984
- 0.0019 ]T × 10 4
- 0.2345
- 0.0025 ]T × 10 4
The robust observer matrix is
L = [ 0.3687
- 5.4516
and symmetry matrix P is chosen as
⎡ 3.1771
⎢ 0.3082
P=⎢
⎢ - 2.1728
⎢
⎣ 0.4347
0.3082
0.0443
- 2.1728
- 0.5471
- 0.5471
0.1093
9.3211
- 1.8620
0.4347 ⎤
0.1093 ⎥⎥
× 10 10
- 1.8620 ⎥
⎥
0.3823 ⎦
The gain τ is chosen as
τ = 10
Fig.3.1 shows the tracking result of slideslip angle β
using the linear
Luenberger observer optimized by solving Design Task 3.II. Fig.3.2 shows the direct
tracking output of β using the proposed observer optimized by solving Design Task
69
3.I. Fig.3.3 shows the smoothed tracking result of β using a 10 points average filter.
Here, the red lines represent the actual value while the blue lines denote the
estimation results. The maximum estimation error of the linear Luenberger observer is
around 12%, while the estimation error of the proposed robust observer is around 2%.
It is apparent that the proposed observer will yield better tracking performance.
Simulation also reveals that there is not a global optimal observer matrix that can
deal with all the velocity settings. Thus, the observer matrix L and nonlinear term
τα should change with velocity v to maintain good tracking result. Fortunately,
drivers will not change the vehicle’s speed too much when he/she is steering. Thus the
change of feedback matrix will not violate the observer system’s stability.
Figure 3.1 Tracking result using the linear Luenberger observer.
70
Figure 3.2 Tracking result using the proposed robust observer.
Figure 3.3 Smoothed tracking result using the proposed robust observer.
71
CHAPTER 4 AN LMI APPROACH TO ROBUST VEHICLE
STEERING CONTROLLER DESIGN
Being parallel to frequency domain robust steering controller designs, time
domain robust steering controller designs attract continuous interest in the last decade.
Based on previous research results, a systematic time design framework is proposed
in this chapter. The design task is constructed as a multi-objective optimization
problem which simultaneously considers system stabilization, disturbance rejection,
actuator saturation and time delay. A mixed L1 / H ∞ robust controller is finally
obtained by solving a set of linear matrix equalities (LMI) that guarantee the
performance requirements regarding these mentioned factors. Simulations show the
effectiveness of the proposed design method.
4.1
Introduction
Advanced vehicle control systems help drivers by taking control of the steering,
brakes and/or throttle to maneuver the vehicle in a safe sate. Thus, it is also called
driver aid systems and intelligent (autonomous) driving systems in some literals. The
related technologies include smart cruise control, collision avoidance systems, and
vehicle platooning, etc. By focusing on motion control aspects, corresponding
research can be divided into three directions: lateral control, longitudinal control and
72
their combinations.
The main task of longitudinal control is vehicle following/ tracking. It requires
that an appropriate headway should be maintained between the lead vehicle and the
controlled vehicle to avoid collision. The lateral control usually refers to vehicle
steering control. Its prime task is path (road) following, or plainly, to keep the vehicle
on the road. Based on previous research [197]-[211], the difficulties of steering
control mainly lie in the following five aspects:
1)
how to avoid skidding during steering;
2)
how to reject the disturbance caused by wind or some other reasons;
3)
how to deal with vehicle dynamics uncertainty;
4)
how to handle actuator rate limits during steering;
5)
what is the vehicle’s maximum speed limit for a certain curve with given
curvature.
To answer these five questions, numerous designed methods had been proposed
in the last two decades.
Originated in the later 80s, classic frequency domain design techniques remained
as the most important technique in field of vehicle steering control. It had been proven
to be a practical and efficient approach in a great number of literals [197]-[211]. To
avoid skidding, one direct idea of it is to remove the influence of yaw on lateral
73
acceleration. In [200]-[203], Ackermann etc. decoupled the lateral and yaw motions
of a car with active steering. Moreover, they proved that for an ideal longitudinal
mass distribution, the decoupling by yaw rate feedback is robust with respect to
uncertain nonlinear tire side force characteristic, velocity and vehicle mass. In [204],
Ackermann showed that a simplified controller that has acceptable steady-state
behavior. In [205], it was further revealed that additional feedback of the yaw rate can
lead to a significant reduction of the deviation from the guideline in nearly all driving
maneuvers compared to earlier controller designs which used solely feedback of the
deviation. Besides, the redundancy of the design parameters can be used to count off
the variance of vehicle dynamics. For example, the Γ -stability boundaries for mass
and velocity variance were analyzed by Ackermann and Sienel in [206].
In the last 90’s, H ∞ robust analysis method was introduced into steering
controller design field. H ∞ theory is constructed to handle the deterministic
disturbance model consisting of bounded energy (square-integrable) L 2 signals and
allows controller design for narrow-band disturbance rejection, see Francis 1987 [207]
and Zames 1981 [208]. In [209], Guvenc, Bunte and Odenthal etc. designed a
disturbance observer, whose model regulation capability allows the specification and
achievement of desired yaw dynamics. Actually, the model regulation and disturbance
rejection property of this proposed observer could be considered as a special H ∞
loop shaping for path following. Similarly in [210], Mammar, Baghdassarian and
Nouveliere presented another robust two degree of freedom controllers (2DOF)
74
controller using the H ∞ loop shaping technique described in [211]. Kuzuya and Shin
further discussed implementation of a robust 2DOF steering controller in [212].
In [209] and [213], the saturation properties of the steering actuator were studied.
Simulations and experiments pointed out that the steering angle rate actuator
saturation forms a major limitation of performance. In [213], the undesirable limit
cycles caused by saturations were analyzed by a describing function approach in
combination with the representation of limit-cycle- free regions in a parameter plane
of velocity and road/tire friction coefficient. The results were formulated in terms of
required actuator bandwidth that achieves robustness in the entire operating range. It
turned out that the use of a fading integrator can reduce the required actuator
bandwidth. Based on similar ideas, a compensator with high order was designed using
H∞
loop shaping in [209] to achieve better performance.
There are several other vehicle steering controllers including varied sliding mode
steering controllers presented in [202] and [214]-[216]; some fuzzy controllers
descried in [217]-[219]; and several adaptive controllers proposed in [220]-[221]. In
[221], four special controllers, H ∞ , adaptive, fuzzy and PID controllers, were
compared by simulations over a test track circuit. It is remarked clearly that the
simple proportional controller is the one that makes the largest errors, while the
self-tuning regulator yields the best response. H ∞ and fuzzy controllers have
equivalent responses regarding adaptive controller.
However, most previous research did not study the vehicle’s worst displacement
75
from guideline. This leads to two important problems. The first is that some
controllers may temporally drive the vehicle off the lane although the final offsets are
zero. The second is what the maximum speed limit is not clearly pointed out for a
certain curve with given curvature. To answer these two problems, this chapter
addresses point-wise-in-time error bound of vehicle offset, using L1 theory
originated by Vidyasagar in 1986 [222]. L1 theory was developed to capture
worst-case peak amplitude response due to bounded amplitude persistent L ∞
disturbances, which more precisely fits road curvature disturbance. Further analysis in
this chapter shows that actuator saturation can be alternatively handled by control the
upper input-output bound of steering angle and road curvature.
Therefore, a synthesized design framework is proposed for steering controller in
this chapter. It formulates the design task into a multi-objective optimization problem
which considers both system quadratic stabilization and disturbance rejection with
constraints on actuator saturation. All the constraints are transformed into a set of
linear matrix equalities (LMI) which can be easily solved by recently developed
toolbox [237]-[238]. A mixed L1 / H ∞ robust controller is finally generated by
solving the optimization problem.
4.2
System Controllability and Observability
The design specifications are primarily given in terms of maximal displacement
from the guideline and maximal steering angle and steering angle rate. Here, we
76
assume
δ f ≤ δ max , δ& f ≤ δ&max , δ r ≤ δ max , δ&r ≤ δ&max
y f ≤ y max ,
(4.1)
y r ≤ y max
(4.2)
In the rest of this chapter, we will consider front steering model with front
vision sensor only, and all the conclusions below can be directly applied to other
situations.
Considering the control input, we can rewrite system as
⎡ β&
⎢
⎢ r&
⎢ ψ&
⎢
⎢ y& f
⎢δ&
⎣ f
⎤ ⎡ a 11
⎥ ⎢
⎥ ⎢ a 21
⎥=⎢ 0
⎥ ⎢
⎥ ⎢ v
⎥ ⎢ 0
⎦ ⎣
a 12
a 22
0
0
0
0
1
0
0
l fs
v
0
0
0
0
b11 ⎤ ⎡ β
⎢
b 21 ⎥⎥ ⎢ r
0 ⎥⎢ ψ
⎥⎢
0 ⎥⎢ y f
0 ⎥⎦ ⎢⎣δ f
⎤ ⎡0 ⎤
⎡ 0
⎥ ⎢ ⎥
⎢ 0
0
⎥ ⎢ ⎥
⎢
⎥ + ⎢0 ⎥ u f + ⎢ − v
⎥ ⎢ ⎥
⎢
⎥ ⎢0 ⎥
⎢ − vl fs
⎥ ⎢⎣ 1 ⎥⎦
⎢⎣ 0
⎦
d1 ⎤
d 2 ⎥⎥
⎡ ρ ref ⎤
0 ⎥⎢
⎥
⎥⎣ f w ⎦
0⎥
0 ⎥⎦
(4.3)
which can be further written into canonical form as
x& = Ax + Bu + Ew
where x = [β
[
w = ρ ref
fw
]
T
r ψ
δf
yf
]
T
is state variable, u = u f
(4.4)
is control input, and
is taken as disturbance. A , B and E are the corresponding system
matrices chosen as
⎡ a11
⎢a
⎢ 21
A=⎢ 0
⎢
⎢ v
⎢⎣ 0
a12
a 22
0
0
0
0
1
l fs
0
0
v
0
0
0
0
b11 ⎤
⎡ 0
⎡0 ⎤
⎢
⎢0 ⎥
b 21 ⎥⎥
⎡ ρ ref ⎤ ,
⎢ 0
⎢ ⎥,
,
0 ⎥ B = ⎢0 ⎥ w = ⎢ f ⎥ E = ⎢ − v
⎣ w ⎦
⎢
⎢ ⎥
⎥
0 ⎥
0
⎢ − vl fs
⎢ ⎥
⎢ 0
⎢
⎥
⎥
0 ⎦
⎣1 ⎦
⎣
~
d1 ⎤
~ ⎥
d2 ⎥
0⎥
⎥
0⎥
0 ⎥⎦
Here the wind disturbance is normalized to the same amplitude level to road
curvature with
~
~
d1 = σd1 , d 2 = σd 2
77
where τ > 0 is a scale number which is determined by
σ = ρ ref
max
/ fw
max
It is easily to check that ( A, B ) is controllable pair for most vehicle parameter
setting. In the rest of this chapter, we assume the system is controllable even under
model uncertainty.
The measurement output y is normally chosen as
y = Cx
(4.5)
where the structure of measurement matrix C is determined by the sensors applied.
For example, displacements y f and y s can be accurately obtained by using the
laser sensors proposed in [224]. With the Fiber Optic Gyroscopes (FOP) described in
[225], heading angle ψ and yaw rate r could be precisely measured, although it
requires considerable cost. It was also shown in [226] and [227] that Global Position
Systems (GPS) can also be applied to approximately estimate heading angle ψ and
displacements y f and y s simultaneously.
There were several approaches of vehicle sideslip angle and lateral forces that
had been reported in the last decade. In 1997 and 1999, Kiencke etc. proposed a linear
observer and a nonlinear observer using reduced order bicycle model in [228] and
[229]. Soon after that, Venhovens and Naab used a Kalman filter in [230] for a linear
vehicle model in 1999. In [231], Huh et. al. constructed the monitoring system based
on KFMEC (Scaled Kalman Filter with Model Error Compensator) technique to
improve the robustness of ordinary Kalman filters. Besides Kalman filter, sliding
78
mode observer were designed, too. In [232], Zhang, Xu and Rachid showed the
feasibility of the sliding mode observer for vehicle lateral motion. Similar conclusions
were reached by Perruquetti and Barbot in [233]. Recently, Stephant, Charara and
Meizel carefully compared four observers including linear Luenberger observer and
three nonlinear observers: extended Luenberger observer, extended Kalman filter and
sliding-mode observer in [234]. Based on simulation results and practical experiments,
they showed that all four observers can yield acceptable estimation results if the
observer’s parameters are appropriately assigned.
It can be easily proven that the system is observable if either displacements y f
or y r is measured. This conclusion takes valid for both front steering vehicle and full
steering vehicle.
In the rest of this chapter, we assume displacement y f and yaw rate r are
measured, which is similar to the situation considered in [200]-[206]. Obviously,
( A, C )
4.3
yields an observable pair for almost all vehicle parameter settings.
Robust Steering Controller Design
4.3.1
Feedback Controller
In general, the system with disturbance w ∈ C n , saturation control input and
w
bounded state can be written as:
79
⎧ x& = Ax + B u ′ + Ew
⎪z = C x
⎪ ∞
∞
⎨
=
z
C
x
1
1
⎪
⎪⎩ z u = u ′ = sat ( Kx )
where z ∞ ∈ C n
z∞
(4.6)
is the H ∞ -performance output, z 1 ∈ C n
z1
is the L1 -performance
output, z u ∈ C n is the auxiliary performance output for bounded control input. C ∞
u
and C 1 are measurement matrices. Here, we set C 1 = ⎡⎢ 0 0 0 1 0 ⎤⎥ and C ∞ = I .
⎣0 0 0 0 1 ⎦
The design objective is to design a feedback control
⎧⎪ Kx ,
u ′ = sat ( Kx ) = ⎨
⎪⎩ sign ( Kx ) * u lim ,
Kx ≤ u lim
Kx > u lim
(4.7)
to achieve the following three objectives:
Design Objective 4.I
4.I.1)
guarantee the quadratic stability of system (6);
4.I.2)
minimize the H ∞ norm of the transfer function matrix from w to
z∞
4.I.3)
so as to reject disturbance;
keep z1 bounded to satisfy the offset constraints.
The first two objectives naturally leads to the famous H ∞ design problem, which
was solved using a linear matrix inequality (LMI) problem by Boyd, Ghaoui, and E.
Feron et. al in [223], or equivalently solved using a set of Algebraic Riccati Equations
(ARE).
In [233], Abedor, Nagpal and Poolla addressed the actuator saturation by the
80
star-norm approach proposed in [234], which estimates the upper bound of the
induced L1 -norm by over bounding the reachable set with an ellipsoid. In [235]-[236],
Nguyen and Jabbari further showed that the saturation limits could be explicitly taken
into account by constraining the linear feedback within the upper bound. This new
method has the following main properties: 1) controllers are designed so that actuators
are used at or near capacities and 2) the guaranteed performance bound is a function
of the actuator capacity. This method is similar to the H ∞ loop shaping proposed in
[209], since both of them tried to keep the linear feedback within saturation limits.
However, L1 -norm is more suitable than H ∞ -norm to describe the time-domain
point-wise-in-time bound of actuator saturation and peek offset.
Extending the conclusions in [235]-[236], we further add the L1 -norm constraints
on system state and formulate a mixed L1 / H ∞ robust controller as follows. The
system dynamics is rewritten as:
⎧ x& = Ax + Bu + Ew
⎪z = C x
⎪ ∞
∞
⎨
z
C
=
1x
⎪ 1
⎪⎩ z u′ = u = Kx
(4.8)
and the design objective is to design a feedback control u = Kx to achieve the
following four objectives
Design Objective 4.II
4.II.1)-3)
same as 4.I.1)-4.I.3);
4.II.4)
keep z u′ bounded to guarantee the inputs to remain less than or
81
equal to the saturation limits.
Before we enter the main part of this chapter, let’s introduce two lemmas in
which only one disturbance input is considered.
Lemma 4.1 [223]: Consider system ⎧⎪⎨ χ = Aχ χ + E χ ω with A χ stable. If there
⎪⎩ z χ = C χ χ
&
exists a constant symmetry matrix Q χ > 0 , for some scalar α > 0 , such that the
A χ Q χ + Q χT A χ + α Q χ + α − 1 E χ E χT ≤ 0
(4.9)
2
then the reachable set Χ is contained inside the ellipsoid {χ : χ T Q χ−1 χ ≤ ω max
},
and max z χ ≤ C χT Q χ−1C χ
t ≥0
1/ 2
.
Lemma 4.2 [223]: Consider system (10) with z ui = u i for i = 1,..., n u . Given some
desired level of performance γ ∞ > 0
and γ u ,i > 0 associated with each input, a
observer-based state feedback controller
⎧ x&ˆ = A xˆ + Bu + LC ( y − C xˆ )
⎪
⎨
λ T −1
⎪u i = − B i Q xˆ
2
⎩
(4.10)
guarantees the quadratic stability of (8) with L 2 -gain γ ∞ from w to z ∞ and
bounded control input if there exist a symmetry constant matrix Q > 0 and a scalar
λ > 0 for some α > 0 such that the following LMIs are feasible
⎡ AQ + QA T − λ BB T
⎢
E
⎢
⎢
C∞Q
⎣
E
− γ ∞2
0
⎡ AQ + QA T + α Q − λ BB T
⎢
ET
⎣
QC ∞T ⎤
⎥
0 ⎥<0
− I ⎥⎦
E ⎤
⎥≤0
−α ⎦
(4.11)
(4.12)
82
⎡ 4Q
⎢λB T
⎣ i
λBi ⎤
>0
γ u2,i ⎥⎦
(4.13)
where B i is the ith column of the input matrix B . Moreover, the control input is in
2
the ellipsoid ξ F = {x : x T Q −1 x ≤ w max
}, or equivalently u i
∞
≤ γ u ,i w max .
Here L is an
appropriately chosen observer matrix. The design of L can be found in [228], [232],
[234]-[236]. Here, we chose w max = ρ ref
.
max
Extending the conclusions in Lemma 2, multiple disturbance inputs can be treated
as follows. First determining the largest disturbance amplitude w max = max {w k }, then
k ∈n
w
assuming all the disturbance inputs to have the same peak amplitude w max . Since the
L1 -norm
of w is defined to be the supremum over all time of the two-norm of w at
each time instant, the multiple disturbance inputs case can then be considered by
simply over bounding w in Lemma 2 with w ≤ n w w max .
Based on these two lemmas, we can reach the main result as follows.
Theorem 4.1: Consider system (4.8) with z ui = u i for i = 1,..., n u and j = 1,..., n z .
1
Given some desired level of performance γ ∞ > 0 , γ u ,i > 0 and γ 1, j > 0 associated
with each input, the state feedback controller (4.10) guarantees the quadratic stability
of (4.8) with L 2 -gain γ ∞ from w to z ∞ and bounded control input if there exist a
symmetry constant matrix Q > 0 and a scalar λ > 0 for some α > 0 such that the
LMIs (4.11)-(4.13) are feasible
⎡ Q
⎢
⎢⎣C 1, j Q
QC 1T, j ⎤
⎥>0
γ 12, j ⎥⎦
where C 1, j is the jth row of matrix C 1 . Moreover, we can have u i
(4.14)
∞
≤
n w γ u ,i w max
83
and z1, j
∞
n w γ 1, j w max .
≤
Proof: Indeed, the controller given in (4.10) is the simplest controller that can be
obtained from (4.8), see [223]. Consider a more generic control feedback matrix will
lead to a bilinear matrix inequality design problem that has not been thoroughly
solved yet.
LMI (4.11) is necessary and sufficient for the existence of a state feedback
controller
+∞
∫z
T
zdt ≤ γ ∞2
for
system
+∞
0
∫w
T
wdt
(4.8)
which
guarantees
quadratic
stability
with
.
0
Inequalities in (4.12) and (4.13) are the necessary and sufficient conditions for
ui
∞
zj
∞
≤
n w γ u ,i w max .
Similarly to the proof for Lemma 2, the sufficiency of (4.13) for
≤
n w γ 1, j w max
can be directly obtained from (4.12) and Lemma 1.
Based on Theorem 1, the robust controller design problem is formulated as
below:
Design Task 4.III
Min γ ∞
with pre-selected α > 0 , γ u ,i > 0 and γ 1, j > 0 for i = 1,..., n u and j = 1,..., n z
1
such that LMIs (4.11)-(4.14) holds for system (4.8), which satisfies limits (4.1)-(4.2)
as
84
u1
≤ γ u ,1 w max ≤ δ& f max
(4.15)
≤
2γ 1,1 w max ≤ δ
f max
(4.16)
≤
2γ 1, 2 w max ≤ y f max
(4.17)
∞
z 1,1
∞
z 1, 2
∞
Grid search can be used to find global optimal parameter α , γ u ,i and γ 1, j .
However, it was found that variation of α does not significantly change the
performance γ ∞ . γ u ,i and γ 1, j are usually chosen to be smaller that maximum
allowable values to counterbalance uncertainty.
4.3.2
Robustness Analysis Considering Model Uncertainty
The system uncertainties are mainly caused by vehicle mass, velocity and
tire-road-friction variation. Simulations show that this variation is highly structured.
Thus, some frequently used robust analysis techniques cannot be applied since they
are too conservative in this situation.
In this chapter, the robustness of the proposed controller is directly studied by
checking the roots function (21) for the closed-loop system (10)-(12) in terms of m ,
v
and µ .
λ
⎡
⎤
T
h ( m , v , µ ) = det ⎢ sI − A ( m , v , µ ) + B ( m , v , µ ) B nominal
Q −1 ⎥
2
⎣
⎦
= s n + c n −1 ( m , v , µ ) s n −1 + ... + c 0 ( m , v , µ ) = 0
(4.18)
Using Routh-Hurwitz criteria, the stability bound of m , v and µ can be
determined. Simulations reveal that the obtained feedback controller can guarantee
system’s quadratic stability in most cases with variation limits as
85
∆ m ≤ 200 kg , ∆ v ≤ 5 m / s , ∆ µ ≤ 0 . 1
(4.19)
Simulation reveals that bound of actuator input and offset may be violated in
some situations. So the feedback controller should change with vehicle velocity to
maintain good steering performance. Fortunately, drivers will not change the vehicle’s
speed too much when he/she is steering, and changing feedback matrix will not
violate system’s stability, if drivers do not steer.
4.3.3
Higher Order Controller Design
Comparing to frequency domain approaches, one common shortcoming of time
domain robust controller design is that it fits for fixed order controller design only.
Although feedback controllers using information of x = [β
r ψ
yf
δf
]
T
can
satisfy safety requirements for many vehicles, a higher order controller is still
required.
In [202], [206], the feedback control rule was formulated by
δ f = 0 .89 ⋅ r − ω c2
k DD s 2 + k D s + k P
yf
s 2 + 2 D ω c s + ω c2
(4.20)
where inside the bandwidth ω c , k P denotes a proportional part, k D denotes a
differential part and k DD denotes the double differential part. It showed in [202],
[206] that this filtering feedback of y f can be sued to improve driving performance.
From the viewpoint of H ∞ filtering, Eq.(4.20) can be viewed as a special higher
feedback. By introducing an intermediate state ζ and an auxiliary control u aux , we
can extend the original feedback mode (4.10) into a high order controller, which is
86
approximately equivalent to what was proposed in [202], [216].
Discarding the contribution of ρ ref to y f , we can have
y& f ≈ v ⋅ β + l fs ⋅ r + v ⋅ψ
(4.21)
Thus, the kth derivative of y (f k ) should only be determined by β , r , δ f and
its second order derivatives as
y (f k ) ≈ g k ,1 ⋅ β + g k , 2 ⋅ r + g k , 3 ⋅ δ f + g k , 4 ⋅ δ& f
(4.22)
where those higher order derivatives of δ f are discarded since they are small. Here,
g k ,i
are recursively derived from Eq.(4.8) and (4.22). For example, we have
g 2 ,1 = v ⋅ a11 + l fs ⋅ a 21
and g 3,1 = g 2 ,1 ⋅ a11 + g 2 , 2 ⋅ a 21 .
Thus, a higher order controller can be formulated as
⎡ β& ⎤
⎢
⎥
⎢ r& ⎥
⎢ ψ& ⎥
⎢
⎥
⎢ y& f ⎥
⎢ ... ⎥
⎢ (n f ) ⎥ =
⎢y f ⎥
⎢ δ& ⎥
⎢ f ⎥
⎢ ζ& ⎥
⎢
⎥
⎢ ... ⎥
⎢⎣ζ ( n s ) ⎥⎦
⎡ 0
⎢ 0
⎢
⎢ 0
⎢
⎢ 0
⎢ ...
+⎢
⎢ g n f ,4
⎢ 1
⎢
⎢ 0
⎢ ...
⎢
⎢⎣ 0
⎡ a11
⎢a
⎢ 21
⎢ 0
⎢
⎢ v
⎢ ...
⎢
⎢ g n s ,1
⎢ 0
⎢
⎢ 0
⎢ ...
⎢
⎢⎣ 0
a12
0
0
...
0
b11
0
...
a 22
1
0
0
0
0
...
...
0
0
b 21
0
0
0
...
...
l fs
...
v 0 ... 0
... ... ... ...
0
...
0 ...
... ...
g ns , 2
0
0
0
0
0
...
...
0
0
g ns ,3
0
0
0
...
...
0
...
0 0 ... 0
... ... ... ...
0
...
0 ...
... ...
0
0
0
0
0⎤
0 ⎥⎥
0⎥
⎥
0⎥
... ⎥ ⎡ u f ⎤
+
⎥
0 ⎥ ⎢⎣u aux ⎥⎦
0⎥
⎥
0⎥
... ⎥
⎥
1 ⎥⎦
⎡ 0
⎢ 0
⎢
⎢ −v
⎢
⎢ − vl fs
⎢ ...
⎢
⎢ 0
⎢ 0
⎢
⎢ 0
⎢ ...
⎢
⎢⎣ 0
0
...
0
d1 ⎤
d 2 ⎥⎥
0⎥
⎥
0⎥
... ⎥ ⎡ ρ ref ⎤
⎥
0 ⎥ ⎢⎣ f w ⎥⎦
0⎥
⎥
0⎥
... ⎥
⎥
0 ⎥⎦
...
0 ⎤⎡ β ⎤
0 ⎥⎥ ⎢⎢ r ⎥⎥
0 ⎥⎢ ψ ⎥
⎥⎢
⎥
0 ⎥⎢ y f ⎥
... ⎥ ⎢ ... ⎥
⎥ ⎢ ( n −1 ) ⎥
0 ⎥⎢ y f f ⎥
0 ⎥⎢ δ f ⎥
⎥⎢
⎥
0 ⎥⎢ ζ ⎥
... ⎥ ⎢ ... ⎥
⎥⎢
⎥
0 ⎥⎦ ⎢⎣ζ ( n s −1) ⎥⎦
(4.23)
where n f and n s denotes the maximum derivative order of displacement y f and
87
intermediate state ζ . Normally, we keep n f < n s .
The special filter is constructed by determining a proper auxiliary control
ζ
(n f )
= F~
x
(4.24)
where ~x is the state vector appeared in (4.23), and F is the auxiliary feedback
matrix. It is obvious that the feedback rule (4.23) can be approximately realized with
an appropriately chosen F .
Theorem 4.1 still holds for extended system (4.23). Simulations prove that higher
order controller can yields better performance especially smaller displacement and
better robustness to model uncertainty. However, additional implementation cost
should also be considered when a practical control device is designed.
4.3.4
Robustness Analysis Considering Time Delay
Considering time delay exists in the observer, the feedback control should
consists of two parts: r , y f and δ f which can be instantly measured, β and ψ
which need to be estimated from observer. Taking time delay during estimation, the
actual feedback can be rewritten as
⎛ ⎡ 0 ⎤ ⎡ β (t − τ ) ⎤ ⎞
⎜⎢
⎥
⎥ ⎟⎟
0
⎜ ⎢ r ( t ) ⎥ ⎢⎢
⎥⎟
⎜
u (t ) = K ⋅ ⎜ ⎢ 0 ⎥ + ⎢ ψ (t ) ⎥ ⎟ = K 1 x (t ) + K 2 x (t − τ )
⎢
⎥
⎥⎟
⎜ ⎢ y f (t ) ⎥ ⎢⎢
0
⎥⎟
⎜⎜ ⎢
⎥ ⎢⎣
⎥⎦ ⎟
δ
t
(
)
0
f
⎦
⎝⎣
⎠
(4.25)
where τ denotes the delay cause by observer estimation. It is a constant determined
by the applied observation algorithm and implementation method. Usually, we can
88
assume
τ ≤ 0 .1 s
(4.26)
The robustness of the proposed controller can be checked using the following
theorem based on [239]-[240].
Theorem 4.2: if there exits three symmetry positive matrix P1 , P2 and P3 such
that
⎛ [ A + BK ]T P1 + P1 [ A + BK ] + τ [ P2 + P3 ]
⎜
τ [ A + BK 1 ]T [ BK 2 ]T P1
H =⎜
⎜
τ [ BK 2 ]T [ BK 2 ]T P1
⎝
∗
− P2
0
∗ ⎞
⎟
0 ⎟<0
− P3 ⎟⎠
(4.27)
then the system is asymptotically stable for delay τ > 0 .
Proof: Define Lyapunov function
τ
V ( x (t )) = x T ( t ) P1 x (t ) + ∫
t
∫x
T
( z ) P2 x ( z ) dz ds
0 t−s
we can easily check that
V& ( x (t )) ≤ x T (t ) Hx (t ) < 0
thus, it comes the conclusion.
It should be pointed out that bound limit on displacement and actuator saturation
may be violated under this situation even LMIs (4.12) holds. It simply leads to a
bilinear matrix inequality design problem if we try to consider time delay controller
(4.25) simultaneously. However, simulations show that bound limit on y f is
naturally satisfied with controller (4.10) in many cases if we appropriately choose γ u ,i
and γ 1, j .
89
4.3.5
Simulation Results
To demonstrate the feasibility of the proposed controller, its performance is
compared with a robust controller that is only optimized to reject disturbance. Its
design task is written as
Design Task 4.IV
Min γ
with γ ∞ > 0 such that LMIs (4.11) holds for system (4.8).
Suppose the vehicle is front steering and only the front sensor is applied. The
parameters are assigned as:
~
m
= 1500 kg, I~z =2500 kg.m.m, l f = 1.2 m, l r =1.5 m,
l fs
= 5 m, l w = 0.5 m, c f = c r = 80000kN/rad.
δ max = 0.7 rad, δ&max = 0.4 rad/s, w max = 0.02.
The variance bound of velocity is ∆ v = 1 m/s, ∆ m~ = 100 kg, ∆ µ = 0.1.
Suppose v is incorrectly estimated as 16m/s when it equals to 15m/s and µ is
misestimated as 0.7 instead of 0.8 during the simulation.
The feedback matrices are chosen as
⎡ - 48.4976
⎢ - 40.4636
⎢
K = ⎢ - 60.4330
⎢
⎢ - 7.8951
⎢⎣ - 92.6678
⎤
⎥
⎥
⎥
⎥
⎥
⎥⎦
and
⎡ - 9.3907
⎢ - 0.1934
K′ = ⎢
⎢ - 16.9858
⎢
⎣ - 1.7986
⎤
⎥
⎥
⎥
⎥
⎦
90
which are obtained in Task I and II respectively. Checking LMIs (26), we have the
system is still stable for τ ≤ 0 .2 s .
Calculation shows that the maximum offset can be kept less than 0.2. The read
line in Fig.4.1 denotes the control output of y f using K ′ , while the blue line
denotes the control output of y f using K . It is clear that the proposed controller
yields better performance.
Figure 4.1 Control outputs of y f using different controllers.
4.4
Speed Limit Estimation and Guideline Planning
4.4.1
Speed Limit Estimation for Steering and Lane Change
Obviously, we can pick up appropriate Q to satisfy the actuator saturation as
91
uf
max
≤
2γ uf w max ≤ δ& f lim , δ
f
max
2γ 1,1 w max ≤ δ
≤
f lim
(4.28)
Simultaneously, the maximum offset can be determined by
yf
max
≤
2γ 1, 2 w max
(4.29)
Therefore, considering (6), we can have the safe constraints for a steering
guideline with ρ ref
as
max
h min ( x ) ≤ − 2γ 1, 4 w max + g ( x ) ≤ − y f
y ≤ yf
max
+ g ( x) ≤
max
+ g ( x) ≤ y ,
2γ 1, 4 w max + g ( x ) ≤ h max ( x )
(4.30)
where h min ( x ) and hmax ( x ) represent the road boundary respectively. As shown in
Fig.4.2, the real steering trajectory will be restricted within two envelope curves
determined by the guideline and maximum offset y f
. If the two envelope curves
max
do not interfere with the road boundaries, then the controller can guarantee the safety
of the steering process.
Figure 4.2 Bounds of real steering trajectory. The green dash line is the guideline. The
two red dash lines are bounds (envelope) curves. The blue curve denotes the real
trajectory.
92
Thus, we can have a rough estimation of the safe speed limit regarding curvature,
if we simply choose the middle curve of a road curve as the guideline. The estimation
can be formulated as follow:
Planning Task 4.V
Max v
under given bound w max = ρ ref
and road width D ;
max
with certain Q > 0 , λ > 0 and α > 0 such that (13)-(17) holds as well as
yf
max
≤
2γ 1, 2 w max ≤ D / 2
(4.31)
Normally, we will use grid search technique to check the existence of a feasible
controller over the solution space of v . Fig.4.3 shows that the speed space is divided
into two parts. The area above represents the estimated unsafe driving scenarios.
Normally, the estimated speed limit is smaller than the actual maximum allowable
speed.
93
Figure 4.3 Relation between vehicle velocity limit and road curvature (denoted by
radius).
The speed limit for frequency domain steering controller (2.13)-(2.14) can be
obtained by frequency L1 design theory as
yf
max
≤
y f (s)
ρ ref ( s )
w max
(4.32)
1
Moreover, there were several other steering controllers such as sliding mode
controllers, fuzzy controllers and etc.. In [221] Chaib, Netto and Mammar compared
the performances of a linear robust controller, a fuzzy controller and an adaptive
controller. They showed that a linear steering controller is usually outperformed by
other nonlinear steering controllers in several aspects including maximum offset value.
Thus, the safe speed limit obtained here can also be viewed as an acceptable yet not
very accurate estimation for the other nonlinear steering controllers.
For the lane changing trajectory planning problem, similar method can also be
applied except the steering guideline is substituted by lane changing path. Since the
94
vehicle speed is often considered to be varied during the whole process, we just
choose the largest speed to solve LMI (4.11)-(4.13) as the in estimation process for
simplicity.
4.4.2
Optimal Guideline/Trajectory Planning
In the above analysis, we just simply chose the middle curve of the road as the
guideline. However, further analysis shows that we can pick up other guidelines to
improve ride safety, comfort, and some other demands of the driver/passengers.
Consider the two different guidelines and corresponding steering behaviors given
in Fig.4.4. The red guideline leads the driver to start steering at point A; while the
blue guideline starts at point O. Since point A lags point O, the peek value of the
desired curvature for the red guideline is higher than that of the blue guideline.
Therefore, the steering process using the blue guideline yield smaller offset. Moreover,
it allows faster speed to pass this curve. However, since the curvature of the blue
guideline is relatively flatter, it is closer to the road boundary too. This may be
dangerous although the maximum offset is small. So we cannot make the guideline as
flat as possible since the vehicle may bump off the road.
95
Figure 4.4 Comparison of two different guidelines and corresponding steering
behaviors.
Besides the safety consideration, the driver/passengers’ feelings and characteristic
demands should not be neglected. As pointed out in [214]-[216], some drivers want to
drive as fast as possible on the highway to save time, while some drivers want to
make the acceleration/deceleration of the vehicle as small as possible to make the
long trip comfortable. Thus, some optimal performance indices should be introduced
associated with the trajectory generation problem.
Based on our previous discussions [214]-[16], we just consider two objectives
there: least time consuming and least jerk during steering.
The time T used for steering can be approximately written as:
T ≈
length of the guideline
v
(4.33)
The maximum jerk during steering can be roughly estimated as:
J ≈ &y& f
max
≈ v ⋅ β& + v ⋅ r& + v ⋅ ψ&
(4.34)
max
≈ v ⋅ ( a 11 β + a 12 r + b11 δ f ) + l s ⋅ ( a 21 β + a 22 r + b 21 δ f ) + v ⋅ r
max
96
Thus, we can introduce an auxiliary L1 -performance output, by setting the
measurement matrix as
0
⎡
⎢
′
C1 = ⎢
0
⎢⎣ v ⋅ a 11 + l s ⋅ a 21
v ⋅ a 12
0
0
+ l s ⋅ a 21 + v
0
0
0
1
0
0
0
⎤
⎥
1
⎥
b11 + b 21 ⎥⎦
and get the bound of jerk as
J
max
≤
2γ 1, 3 w max
(4.35)
Thus, the optimal steering guideline planning problem can be formulated as
Planning Task II:
Min H = q (T , J
max
)
with certain steering guideline to satisfy safety requirement. Here q (T , J
max
)
is
a certain evaluation function that takes compromising between time and comfort
considerations.
Normally, we can express the steering guideline with a five point Bezier curve
and search optimal the optimal Bezier I interpolation points. Constrained by the
length of this chapter, we will not discuss the details here.
4.4.3
Graphical User Interface Design
The graphical user interface for driver guidance received constant consideration
recently. In steering, an appropriate aid system should notify the driver about the
97
desired steer angle and the actual steer angle.
For instance, a graphical user interface device is shown in Fig.4.5 below. The
blue curves indicate the road boundaries. The green arrow in the right top corner
indicates the optimal front steering angle, and the red arrow shows the actual front
steering angle. Thus, the driver will try to adapt the right steering police by making
the red arrow overlap the green arrow. This process is quite straightforward and
drivers can easily get accustomed with it.
Figure 4.5 A graphical user interface device for steering guidance.
98
CHAPTER 5 COOPERATIVE DRIVING AT BLIND CROSSINGS
USING INTER-VEHICLE COMMUNICATIONS
Cooperative driving technology with inter-vehicle communication attracts
increasing intentions recently. It aims to improve driving safety and efficiency using
appropriate motion scheduling of all the encountered vehicles. Under cooperative
driving control, the motion of individual vehicles can be conducted in a safe,
deterministic and smooth manner. This is particularly useful to heavy duty vehicles,
since their acceleration/deceleration capacity is relatively low. Specifically in this
chapter, cooperative driving at blind crossings (crossings without traffic lights) is
studied. A concept of safety driving patterns is proposed to represent the collision free
movements of vehicles at crossings. The solution space of all allowable movement
schedules is described by a spanning tree in terms of safety driving patterns; four
trajectory planning algorithms are formulated to determine the driving plans with least
execution times using schedule trees. The corresponding group communication
strategy for inter- vehicle networks is also analyzed. Finally, simulation studies have
been conducted and results demonstrate the potentiality and usefulness of the
proposed algorithms for cooperative driving at blind crossings.
99
5.1
Introduction
A variety of techniques had been introduced to increase the capacity and safety of
the existing highway systems. Among these techniques, the technology of cooperative
driving with inter-vehicle communications is now considered as a potential solution to
alleviate traffic jam and reduce collisions.
The concept of cooperative driving was first presented by JSK (Association of
Electronic Technology for Automobile Traffic and Driving) in Japan in the early
1990s [244]. It was originally used as flexible platooning of automated vehicles with a
short inter-vehicle distance over a couple of lanes. At that time, it was known as super
smart vehicle systems (SSVS) [244]-[246]. Using appropriate inter-vehicle
communications to link vehicles, cooperative driving enables vehicles to perform safe
and efficient lane changing and merging, and thus improve the traffic control
performance. Since then, the feasibility and benefits of cooperative driving have been
further discussed and examined world-widely, i.e. in California PATH project in USA
[247][248], Chauffeur project in EU [249], and Demo 2000 Cooperative Driving
System in Japan [250].
Generally, all these approaches focus on two questions: how to exchange the
information among vehicles and how to guide vehicles using the obtained information.
The answer to the former question is inter-vehicle communications [247]-[254]. It
enables the vehicles to share information about their driving status and goals, which
greatly extend the horizon of drivers or intelligent driving systems. The latter question
100
is answered by using cooperative trajectory planning [247]-[254], [255]-[258].
The conventional implementation of inter-vehicle communication links vehicles
through a remote service station (i.e. the communication tower mentioned in [251]
and [252]). The driving information of a vehicle will be first transformed to service
station and then broadcasted to other related vehicles. Or, the vehicles inquiry service
station to locate other vehicles. These approaches increase considerably the cost for
building and maintaining remote service stations.
Different from the methods above, many new designs use the peer-to-peer ad-hoc
network to achieve vehicular information sharing. As shown in [253], peer-to-peer
ad-hoc communication networks integrate four valuable features: ad-hoc connectivity,
local peer-to-peer networking, short-range and inter-personal communication.
Currently, inter-vehicle communication networks are still under discussions due to
varied driving behaviors and high mobility [254].
From the control systems perspective, cooperative driving can be used to enhance
the ride safety and quality. Cooperative trajectory planning is one frequently used
control method in this field. It designs time varying velocity profiles for the
encountered vehicles to perform a designed maneuver. Using these methods, the
motion of individual vehicles can be performed in a safe, smooth and deterministic
manner. This is particularly useful to heavy duty vehicles, because their
acceleration/deceleration capacity is quite low [255]-[258].
In this chapter, we use the idea of the cooperative driving to study vehicle
101
collision avoidance at road junctions, which is more complex than longitudinal
control of vehicle platoons. In [259], a simple case of two vehicles had been analyzed
using game theory. The differential game approaches assume that all the encountered
vehicles cannot know each others’ decision [259]-[260]. Therefore, a cooperative
driving method should naturally outperform it, especially when simultaneously
dealing with multi-vehicles.
The difficulties of cooperative driving at junctions lie in two aspects. The first
one is to quickly determine whether a special driving plan is safe. In longitudinal
platoon control, all vehicles are moving in the same direction, so it is much more
complex to guarantee the ride safety for crossing at junctions, because vehicles may
steer to totally different directions. The second one is to pick up the one with the least
time cost from hundreds of possible schedules for alleviating congestions, an
objective not considered in platoon control.
5.2
Problem Description
5.2.1
Driving at Blind Crossings
One basic method towards collision free and fast movement of vehicles is to
install traffic light systems. However, because of cost and some reasons such as in
construction sites and military zones, many road junctions have no traffic lights. We
will focus on vehicle driving scenarios at such blind crossings.
Usually, vehicle flows are assumed to arrive continuously at a junction area.
102
However, at a particular time, we only need to consider a few vehicles which are
moving in the vicinity of the crossing. Under this consideration, the continuous traffic
flow can be truncated intro small segments, which simplifies the problem greatly.
The simple grouping algorithm used here is to label the vehicles by the times they
enter the virtual circle centered at the junction point. As shown in Fig.5.1, the four
shadow vehicles inside the cycle will be considered as a group to take part in
cooperative driving; while the other three vehicles will not be considered temperately.
The radius of this virtual circle should be determined appropriately by inter-vehicle
communication protocol that has been selected for this application.
It should be pointed out that normally the blind crossings are not busy.
Simulation shows that this grouping algorithm does well for such kind of crossings.
Moreover, it is also assumed that all vehicles have relatively low speeds when
approaching the junction and they will not change lanes after entering the virtual
circle since we consider it is too close to safely change lanes within the virtual circle.
Slowing down makes the vehicles have more time to negotiate with each others and
prepare for suddenly emerged pedestrians.
103
Figure 5.1 Vehicle grouping at crossings.
5.2.2
Inter-Vehicle Communication
The inter-vehicle communications plays an important role in cooperative driving,
since the necessary driving information of other vehicles has to be transmitted to the
vehicle that makes the final driving decision.
There were many discussions on designing, implementing and testing of the
inter-vehicle communications, i.e. COCAIN (Cooperative Optimized Channel Access
for INter-vehicle communication) proposed by Kaltwasser and Kassubek in [261],
TELCO (Telecommunication Network for Cooperative Driving) proposed by
Verdone in [262], DOLPHIN (Dedicated Omni-purpose inter-vehicle communication
Linkage Protocol for HIghway automatioN) proposed by Tokuda, Akiyama and Fujii
in [263].
In blind crossing driving scenario, the encountered vehicles need to share the
104
following information among each other:
(a)
the desired moving lane number of each vehicle before the cooperative
driving plan is made;
(b)
the driving plan of each vehicle after the cooperative driving plan is
made;
(c)
the real-time position (represented as lane number and distance from the
crossings) and speed of each vehicle;
(d)
the emergency signal if needed.
In order to efficiently exchange information, we propose a new model based on
the hierarchical group message-deliver models discussed in [263]-[268].
When approaching the crossing areas, the original vehicle platoon splits into
several independent groups. The maximum allowable size of such a group is set as
three in this chapter. The initial groups are constructed by randomly assigning several
temporary dominant vehicles and letting them pick up its neighbor vehicles into its
group. The groups contain more than three vehicles will reject the tail vehicles to
meet the limits. The rejected vehicles will try to merge into other groups or formulate
a new group.
In this mode, the vehicles are supposed to be self-organized into several small
groups as shown in Fig.5.2, before they enter the virtual circle.
105
Figure 5.2 Vehicle groups and group inter-vehicle communication.
A vehicle is assumed to store the real-time positions, speeds and desired driving
lane information of all vehicles in the same group. It communicates with each other
periodically to update position and speed information. Simulations presented in
Section 5 show that three is an appropriate size for an independent group for blind
crossing scenarios.
When vehicle A and B in two different groups communicate with each other,
vehicle A will transfer the driving information of all vehicles in the same group to B.
The received driving information from B will be soon delivered to other vehicles in
the group of vehicle A. So does vehicle B to A.
It is also assumed that all vehicles have constant speed before the cooperative
driving plan being set. Thus, one vehicle can predicate the movements of other
106
vehicles in the near future based on received driving information.
Although the data transmission rate is constrained by several factors including
media and communication protocol, we assume that the vehicles within the virtual
cycle can properly acquire timely the necessary driving information of each other.
Based on the experiment results in [268]-[270], such assumptions are valid for most
blind crossings, because the radius of the virtual circle and the number of encountered
vehicles are both limited in those driving scenarios.
The inter-vehicle communication network will use tags to label the
groups/vehicles encountered. If all the vehicles’ moving information has been
collected by one vehicle, it will proceed the trajectory planning. It will simultaneously
send messages to block off other vehicles for taking this job.
After the cooperative driving plan is made, the schedule will be properly
delivered. Constrained by the communication rate, the driving plan should be
represented concisely. A detailed description of the representation method will be
presented in Section 4.
When one vehicle leaves the virtual circle, it will leave the current
communication group and join a new platoon.
Emergency signals will be broadcasted to all vehicles with the highest priority.
107
5.3
Cooperative Driving Schedule
5.3.1
Collision Free Driving Represented by Safe Patterns
In order to guarantee the safety of driving, the concept of safe driving patterns is
introduced first.
Generally, the collision may occur when
(i)
two vehicles move along the same lane, and the lag one runs into the
lead one;
(ii)
two vehicles moves on different lane but pass the same point
simultaneously.
The most frequently used control strategy for blind crossings is the zone blocking
strategy. It divides the crossing zone into server sub zones and only one vehicle is
allowed to enter a zone at any time to avoid collisions.
Based on a similar idea, a simplified strategy is proposed here, which only allows
safe driving patterns (vehicle pairs) to pass the junctions at one particular time. It is
similar to the concept of phase of traffic flows in traffic lights control.
108
Figure 5.3 Diagram of two-lane junctions.
For instance, consider the two-lane road junctions shown in Fig.5.3. Taking
symmetry into account, there only exist three driving patterns that allow two vehicles
cross junction area safely and simultaneously:
(a) one vehicle drives from lane 1 to lane 8 while the other vehicle drives from
lane 5 to lane 2;
(b) one vehicle drives from lane 1 to lane 8 while the other vehicle drives from
lane 5 to lane 4;
(c) one vehicle drives from lane 1 to lane 6 while the other vehicle drives from
lane 5 to lane 2.
In terms of safe driving patterns, we can represent a driving schedule as an
109
ordered series of safe patterns. To illustrate this idea, consider a typical driving
scenario shown in Fig.5.4, in which vehicle A needs to move from lane 7 to lane 4,
vehicle B from lane 7 to lane 6, vehicle C from lane 1 to lane 8, vehicle D from lane 1
to lane 4.
One possible driving schedule for this scenario is to let vehicle A pass the
junction first; then let vehicle B and C pass the junction at the same time; and finally
let vehicle D pass. Apparently, we can represent this schedule as the following
sequence
A | B, C | D
(5.1)
where “|” is a separator symbol that divides the sequence into three subsets: A, B and
C, D. B and C are in one subset indicating they will pass the junctions at the same
time.
Figure 5.4 A four-vehicle driving scenario for a two-lane junction.
110
Generally, consider N vehicles
i1 , i 2 ,
… , i N moving towards the junction.
Their order of passing the junction area, i.e. the cooperative driving schedule is
specified as
i1 , i 5 , …, | i 2 , i 4 , …, | i3 , …, i N
(5.2)
When one particular driving schedule is determined, the corresponding trajectory
planning process for each vehicle will then be carried out. Notice that different
driving schedules lead to different passing times. An optimal cooperative driving plan
needs to find the schedule that completes the total driving process with the least time.
In the rest of this section, we will discuss how to generate all allowable driving
schedules. The optimal schedule search process will be described in Section 5.4.
5.3.2
Solution Tree Generation and Labeling
In general, searching the allowable driving schedule will yield a tree in which
each node represents a particular driving plan (sequence) except for the root node.
One basic algorithm to generate such a tree is as follows:
Basic Solution Tree Generation Algorithm.
Suppose there are N vehicles under consideration.
1.
Generate the root node of the tree;
2.
Generate N children for the root node which represent all the possible
111
permutation orders of the vehicle sequence without any separator
symbols. It is apparently that all these orders can be enumerated through
basic permutation algorithm.
3.
For each node in the second level of the tree, generate N − 1 children by
inserting only one separator into the driving sequence that is represented
by it, since there are only N − 1 positions that a separator can be
inserted.
4.
For each node in the third level of the tree, generate N − 2 children by
inserting only one separator into the driving sequence that is represented
by it, since there are only N − 2 positions that a separator can be
inserted.
…
N.
For each node in the N − 1 level of the tree, generate one child by
inserting only one separator into the driving sequence that is represented
by it, since there is only one position that a separator can be inserted.
However, a great number of nodes in this tree can be discarded since they
represent invalid driving schedules, for which no trajectory planning is needed.
One apparent fact is that lead vehicles will always pass the junction areas earlier
than lag vehicles in the platoon, because we assume that the lag vehicle will not
change lanes. For instance, in the driving scenario shown in Fig.5.4, vehicle A should
112
always pass the crossing earlier than vehicle B. Thus the node B A C D and all its
children should be invalid.
Based on this fact, we can modify the above algorithm as
Modified Solution Tree Generation Algorithm.
Suppose there are N vehicles under consideration.
1.
Generate the root node of the tree;
2.
Generate N children for the root node which represent all the possible
permutation orders of the vehicle sequence without considering separator
symbols. Then, prune all the obtained nodes that represent invalid order
of driving.
3.-N.
Same as Basic Solution Tree Generation Algorithm.
To guarantee a collision free movement, the safety of the simultaneously moving
subnets in a driving sequence has to be checked. It is apparent that each subset in a
valid driving schedule should constitute of a safe driving pattern.
Note that M vehicles can safely cross the junctions at the same time if and only
if every pair of these vehicles is a safety pattern. Therefore, the labeling algorithm is
given by:
Safety Pair Labeling Algorithm.
113
If there are M ( M ≥ 2 ) vehicle j1 , j 2 , …, j M
pass the crossing the junctions
simultaneously, check the M ( M − 1) / 2 distinct vehicle combinations ( j1 , j 2 ), ( j1 ,
j 3 ),
…, ( j M
−1
, j M ) to see whether they are all safety patterns. If not, label the
corresponding node as an unsafe node.
Using these two considerations, most unsafe driving schedules can be pruned
from the solution tree. For examples, the driving plan tree stemmed from scenarios in
Fig.5.4 only has a few valid nodes after pruning and labeling, i.e. there are only two
valid driving schedules for the ordered sequence A, B, C, D, see Fig.5.5. The first
schedule is specified by Eq. (1) and the second is simply let vehicle A, B, C, D pass
the junction area sequentially.
Figure 5.5 A schedule tree stemmed from the driving scenario shown in Fig.5.4. The
shadow nodes represent invalid driving schedule.
114
After labeling, trajectory planning will be executed for every valid node with
respect to its driving schedule. Some driving plans will be discarded since they are
implicitly forbidden by the vehicle dynamics constraints.
5.4
Cooperative Trajectory Planning
5.4.1
Trajectory Planning
In general, a vehicle’s trajectory crossing the junctions can be divided into three
sequential stages:
(a)
approach the junction. It should avoid collision between the lead vehicle
and itself. Moreover, it should also avoid collision between itself and the
last vehicle that had passed the junctions before it but does not form a
safety pair with it;
(b)
cross the junction, which is considered as a decelerate- accelerate
process;
(c)
leave the junction. It should avoid collision between the new lead vehicle
and itself.
In order to solve these trajectory planning problems, the virtual vehicle mapping
technique was proposed and employed by Uno, Sakaguchi and Tsugawa in [271] and
[272]. The concept of a virtual vehicle is to map a vehicle on a lane onto an object
115
lane; then the interested vehicle can be controlled with respect to the virtual vehicle to
guarantee safety.
Considering the risk of failure, an algorithm similar to what proposed in [271]
and [272] is applied here. The only difference is that the lead vehicle is not mapped
onto the symmetry position of the desired lane. Actually, it will be mapped into a
position that lags off the mirror point to compensate the communication delay.
For example, as shown in Fig.5.6, vehicle A moves form lane 7 to lane 6 first;
then vehicle B moves from lane 1 to lane 6. Since vehicle B passes the junction area
right after vehicle A, it must make enough headway to avoid collision. Therefore, it
generates a virtual vehicle A' by mapping vehicle A into its own lane using the data
transmitted via the inter-vehicle communication. Classical longitudinal control will
then be performed between virtual vehicle A' and vehicle B.
Figure 5.6 Trajectory generation considering one virtual vehicle.
116
The lengths of the two headways should be determined by the dynamic properties
of encountered vehicles and the applied inter-communication protocols.
The trajectory profile of each vehicle in the driving schedule will be generated
one by one with respect to the corresponding driving order passing the junction. Every
vehicle can ‘know’ the trajectories of the other related vehicles that need to map onto
its lane.
Since we need to improve traffic efficiency, the driving plan should keep the
headway between the potential lead/virtual vehicle and itself to the minimum safe
distance.
Generally, we can classify various driving scenarios into the following four cases.
Case A)
there is neither a lead vehicle moving in the same lane nor a virtual
vehicle that needs to map before the planning vehicle enters the
junction area;
Case B)
there is a lead vehicle moving in the same lane but no virtual
vehicles that need to be mapped before the planning vehicle enters
the junction areas. Indeed, it equivalently means that the planning
vehicle will pass the junction area right after the lead vehicle;
Case C)
there is not a lead vehicle moving in the same lane but a virtual
vehicle that needs to be
enters the junction area;
mapped before the planning vehicle
117
Case D)
there is a lead vehicle moving in the same lane, and there is also a
virtual vehicle that needs to be mapped before the planning vehicle
enters the junction area. Actually, it means that the planning vehicle
will first follow the lead vehicle and then follow the virtual vehicle
before it enters the junction.
Suppose the position of the planning vehicle is denoted by x p and u is the
control input to the vehicle. Their dynamics are constrained by
u min ≤ u ≤ u max
(5.3)
0 ≤ x& p ≤ v max
(5.4)
where u min and u max are the bounds of control input u .
v max
is the maximum
allowable safe speed.
If the planning vehicle need to steer directions, there is an additional constraint
x& p ( t 3 ) ≤ v s max
(5.5)
where v s max is the maximum allowable speed for steering, and t 3 is the time when
the planning vehicle enters the junction area.
The corresponding trajectory planning algorithms can be formulated as follows.
Algorithm 5.I. Trajectory Planning for Case A.
5.I.1.
Solve the time optimal trajectory planning problem before the
planning vehicle enters the junction areas.
118
Obviously, the objective is
min t 3
(5.6)
u
constrained by the given boundary conditions
x p ( t 0 ) = x p 0 , x p ( t 3 ) = x ps
(5.7)
where x p 0 is the start position and x p 2 is the final position beside
junction areas;
5.I.2.
t0
is the time that planning starts.
There is no planning at this stage. We simply assume that the time
consumed during crossing and the vehicle speed leaves the junction
areas only depend on vehicles speed.
The time that the planning vehicle leaves the junction areas should
be
t 4 = t s + f 1 ( x& p (t 3 ))
(5.8)
and the speed should be
x& p (t 4 ) = f 2 ( x& p (t 3 ))
(5.9)
where t 4 is the time that the planning vehicle leaves junction areas.
5.I.3.
Solve the tracking problem for the new lead vehicle before the
virtual vehicle leaves the junction area.
Suppose the trajectory of the new lead vehicle is given as
&x&n = f 3 ( t )
(5.10)
where x n is the position of the new lead vehicle and f 3 (t ) is its
119
movement descriptive function in terms of time.
Thus, the error e np in headway from the pre-selected safe distance
L3
can be written as
e np = x n − x p − L 3
(5.11)
Obviously, the objective is
t4 +T
min
u
∫e
lp
dt
(5.12)
t = t4
constrained by (5.6), (5.7) and boundary condition (5.10). Here T
is a predetermined time span that is long enough to appropriately
describe the planning vehicle’s movement after it leaves the junction
areas.
Algorithm II. Trajectory Planning for Case B.
5.II.1.
Solve the lead vehicle tracking problem before the lead vehicle
leaves the junction area.
Suppose the trajectory of the lead vehicle is given by
&x&l = f 4 ( t )
(5.13)
where x l is the position of the lead vehicle and f 4 (t ) is its
movement descriptive function in terms of time.
Thus, the error e lp in headway from the pre-selected safe distance
L1
can be written as
120
e lp = x l − x p − L1
(5.14)
Obviously, the objective is
t1
min
u
∫e
lp
dt
(5.15)
t = t0
constrained by the given initial conditions
x p (t 0 ) = x p 0
(5.16)
(5.6), (5.7) and boundary condition (5.10), where t1 is the time that
the lead vehicle enters the junction areas.
5.II.2.
same as Algorithm 5.I.1 expect the initial condition is given by
x p (t1 ) = x p1
(5.17)
where x p1 is the planning vehicle’s position at time t1 .
5.II.3.
same as Algorithm 5.I.2.
5.II.4.
same as Algorithm 5.I.3.
Algorithm III. Trajectory Planning for Case C.
5.III.1.
Solve the virtual vehicle tracking problem before the virtual vehicle
leaves the junction area.
Assume the trajectory of the virtual vehicle be
&x&v = f 5 (t )
(5.18)
where x v is the position of the virtual vehicle and f 5 (t ) is its
movement descriptive function in terms of time.
121
Thus, the error e vp in headway from the pre-selected safe distance
L2
can be written as
e vp = x v − x p − L 2
(5.19)
Obviously, the objective is
t2
min
u
∫e
lp
dt
(5.20)
t = t0
constrained by the given initial conditions (5.17), and dynamic
constraints (5.6) and (5.7), where t 2 is the time that the virtual
vehicle leaves the junction areas.
5.III.2.
same as Algorithm 5.I.1 expect the initial condition is given as
x p (t 2 ) = x p 2
(5.22)
where x p 2 is the planning vehicle’s position at time t 2 .
5.III.3.
same as Algorithm I.2.
5.III.4.
same as Algorithm I.3.
Algorithm IV. Trajectory Planning for Case D.
5.IV.1.
5.IV.2.
same as Algorithm II.1.
same as Algorithm III.1 expect the initial condition is given as (5.18)
and t1 is the time that the lead vehicle enters the junction areas.
5.IV.3.
same as Algorithm I.1 expect the initial condition is given as (5.22)
and t 2 is the time that the virtual vehicle leaves the junction areas.
122
5.IV.4.
same as Algorithm I.2.
5.IV.5.
same as Algorithm I.3.
It should be pointed out that the choice of vehicle dynamic models and/or
longitudinal driving controllers will not vary the feasibility of the proposed planning
framework. Furthermore, some other driving performance index such as ride
comfortableness can be formulated and applied here. Some related discussion on
objective choice of vehicle trajectory planning can be found in our previous works
[273]-[276].
However, there is still one important issue left. Limited by the communication
rate, the cooperative driving plan should not be too complicated. Otherwise, the
cooperative driving plan cannot be correctly delivered to all the encountered vehicles.
Here, we further constrain the trajectory of one vehicle constitutes of at most five
steady acceleration/deceleration process. Thus, one vehicle’s trajectory will be
determined by at most seven dataset as follows:
<star time t 0 , velocity x& p (t 0 ) >, <velocity change time t1 , velocity x& p (t 0 ) >, ……,
<end time t 6 , velocity x& p (t 6 ) >
(5.22)
The velocity between time t i and t i +1 can be written as
x& p ( t ) = x& p ( t i ) + [ x& p (t i +1 ) − x& p ( t i )] ⋅
t − ti
t i +1 − t i
(5.23)
123
This method greatly reduces the computation costs of the above trajectory
planning problems without losing too much generality. Moreover, this technique also
relieves the burden of the inter-vehicle communication network, since there are only
twelve variables in (23) need to be encoded and delivered to the involved vehicle. The
vehicle will resolve the control inputs from this simplified velocity profile based on
its own dynamic equation.
5.4.2
Best Driving Plan Search
The total time cost of trajectory is countered from the time when the cooperative
driving process begins to the time when the last vehicle leaves the junction area. All
the un-discarded driving plans will be compared, and the one with least time cost will
be chosen as the actual driving plan.
The diagram of this cooperative driving planning framework is shown in Fig.5.7.
To summarize, the safety of the framework is guaranteed by selecting safe driving
patterns; and the efficiency is achieved by adopting the time optimal driving plan.
124
Figure 5.7 The cooperative driving planning framework.
In the searching process of optimal driving plan, the upper level nodes in the
schedule tree will be examined earlier with respect to the lower level plans. If one
plan is proven to be valid, then all its children nodes will be omitted, because a valid
node always represents the plan that uses less time than that of its children plans. For
example, if solution A | B
C | D in Fig.5.5 is a valid plan, then its child node A | B |
C | D need no analysis.
5.5
Simulation Results
5.5.1
Simulation Results for Trajectory Planning
A demonstration simulation case shown in Fig.5.8 is studied here. In this scenario,
125
vehicle A moves from lane 7 to lane 6, vehicle B from lane 3 to lane 6, vehicles C and
D from lane 1 to lane 6.
Figure 5.8 A three-vehicle driving scenario for a two-lane junction.
The speeds of the four vehicles are all 10m/s at the beginning time. The current
distances between vehicle A, B, C, D and the junction areas is 20m, 10m, 15m and
30m respectively.
Suppose the four vehicles have identical dynamic properties. The maximum
navigation speeds for them are all 10m/s before they enter the junction and 20m/s
after they leave. The maximum steering speeds are all 5m/s.
The maximum control input is constrained by
− 10 m / s 2 ≤ u ≤ 10 m / s 2
(5.24)
The length of each vehicle is assumed as 4m. The safety headway L1 defined in
126
(15) is chosen as 5m, while the safety headway L 2 defined in (20) is 7m.
Apparently, there exist several allowable driving plans. Let’s take the trajectory
planning process for the driving plan B | C | D | A as an example. It lets vehicle B, C,
D and A pass the junction sequentially. The trajectory planning will be carried out for
vehicle B, C, D and A sequentially.
1)
the trajectory planning that vehicle B should adopt is Algorithm I.
First, it will carry out a time optimal trajectory until it enters the junction
areas. Obviously, this trajectory should begin at t = 0 s and end at
t = 1 . 125
s. In the first 0.625s, vehicle B keeps speed at 10m/s, and then
slows down with u = − 10 m / s 2 in the next 0 .5 s.
Suppose that vehicle B passes the junction areas within 1 .5 s, and its speed
becomes 4m/s when it leaves the junction areas. Then, it accelerates to 20m/s with
u = 10 m / s 2
in the next 1 .6 s. It will keep this speed in the rest of the planning. Thus,
its speed profile should be what shown in Fig.5.9(a).
2)
vehicle C will pass the junction areas right after vehicle B. Apparently, it
should use planning Algorithm III.
Here, we divide its trajectory into three parts:
127
2.1)
from t = 0 s to t = 2 .625 s, it carries out a decelerate process to enlarge
the headway between the virtual vehicle and itself;
2.2)
it passes the junction areas after vehicle B;
2.3)
it tracks the new head vehicle B.
The initial distance between vehicle C and virtual vehicle B is -2m. Mapping
vehicle B into the lane of vehicle C, we can have the virtual vehicle’s speed trajectory
as shown in Fig.5.9(b).
To accurately solve the minimum tracking error problem (21) is difficult and
unnecessary. Here, we adopt the following simple driving trajectory: first, it
continuously decelerates with u = − 8 m / s 2 from t = 0 s to t = 0 .8 s. Then, it keeps
constant speed at 2m/s from t = 0 .8 s to t = 2 .625 s.
It should be pointed out that this speed profile is clear and easy to adopt for
human drivers. Moreover, it is approximately optimal too.
Suppose vehicle C pass the junction areas in 1 s with speed keeping at 2m/s. It
accelerates to 20m/s with u = 10 m / s 2 in the next 1 .8 s and keeps this speed in the rest
of our planning. Thus, the whole speed profile for vehicle C can be shown as
Fig.5.9(c).
3)
the trajectory of vehicle D follows Algorithm II.
128
It can be depicted as
3.1)
decelerate with u = − 8 m / s 2 from t = 0 s to t = 0 .8 s;
3.2)
keeps speed at 2m/s from t = 0 .8 s to t = 4 .625 s;
3.3)
after crossing the junction areas, it accelerates to 20m/s with u = 10 m / s 2
in the next 1 .8 s and keeps this speed in the rest of our planning.
Fig.5.9(d) shows the speed profile for it.
4)
finally, let’s apply Algorithm III for vehicle A.
Similarly, we can get a simple speed profile as
4.1)
decelerate to 0m/s with u = − 2 .5 m / s 2 from t = 0 s to t = 4 s and stop in
the next 0.625 s;
4.2)
cross the junction in 1 s and simultaneously accelerate to 2m/s.
4.3)
accelerates to 20m/s with u = 10 m / s 2 in the next 1 .8 s and keeps this
speed in the rest of our planning. The entire process is shown in
Fig.5.9(e).
129
(a)
(b)
(c)
(d)
130
(e)
Figure 5.9 Speed profiles for vehicle B (a), virtual vehicle B (b), vehicle C (c), vehicle
D (d) and vehicle A (e).
Thus, the total time consumed for driving plan B | C | D | A is 5.625 s. The time
costs for other driving plans are also listed in Table.5.I. It is clear that the optimal
driving plan may significantly save time for the entire traffic flow. In this case, the
final optimal driving schedule is B | C | D | A.
Table 5.1 Comparison for Different Driving Plans
Driving Plan
Time Consumed
B|D|C|A
5.63s
B |A| D | C
6.55s
A| B | D | C
6.72s
A| D | C | B
6.83s
B | D |A| C
7.55s
……
……
131
5.5.2
Simulation Results for Inter-Vehicle Communication
To test our task managing strategy for communications in multi-car at blind
crossings, a simple time division event-driven multi-agents communication model is
set up and simulated.
There are four parameters in this model:
1.
the communication group size;
2.
the total number of vehicle;
3.
the contacting rate. It is a probability that is used to describe how easy
one “free” vehicle can get connected with another “free” vehicle. Here,
“free” means that vehicles are not engaged in a conversation. The higher
the contacting rate, the easier one free vehicle sets up a conversation
with another free vehicle. This parameter is determined by the media,
protocol and burden of the inter-vehicle communication networks.
Apparently, a high burden will lead to a low contacting rate.
4.
the forbidding time length. It is a number to describe how long one free
vehicle should wait before it communicates again with the same vehicle
that it has “talked” before. Notice that one vehicle prefers to receive new
information other than obtain old information again and again from the
same vehicle.
132
It is assumed that the data information will be transferred in a constant speed.
When one free vehicle carries out conversation with another free vehicle, they will
become occupied and do not response to other request.
By choosing different values of these parameters, we compare the effect of
communication group size and forbidding time length. Fig.5.10 shows the average
communication time with respect to different group sizes while the forbidding time
length is chosen to be 1. Obviously, three is an appropriate size for an independent
group in this case. Similar conclusions were reached for different forbidding time
lengths.
Figure 5.10 Average communication time with respect to different group sizes
(forbidding time length = 1).
Fig.5.11 shows the average communication time with respect to different
forbidding time length while group size is chosen as 3. It reveals that the forbidding
time length should be mediate. However, there is not a simple rule to choose an
133
appropriate the forbidding time length for different group sizes and the total number
of vehicles.
Figure 5.11 Average communication time with respect to different forbidding time
lengths (group size = 3).
5.6
Discussions
Although our algorithms work well in many scenarios of driving at junctions in
simulation, there are still several problems that need to be further addressed:
1)
The complexity of this cooperative driving planning will increase
quickly with the number of vehicles. How to generate the solution space
more efficiently has to be further analyzed before the proposed algorithm
applied in real applications. Besides, our simulations reveals that a
periodic-turned traffic light system is really a simple yet effective
solution for busy road crossings, i.e. if there are more than ten vehicles
move toward the junction area simultaneously.
2)
The actual driving scenario at junctions might be more complex than
what we discussed here. One important case is how to deal with
134
emergencies, especially the sudden appearance of pedestrians. One
potential answer is to share each vehicle’s sensor information through
the inter-vehicle communication as what has been discussed in [248]. If
an emergence happens, the first ‘known’ vehicle will soon stop all the
related vehicles synchronically. For example, as shown in Fig.5.12, an
emergent halt signal sent out from vehicle A quickly stops vehicle B that
intends to move to the blocked lane, when it suddenly ‘sees’ a bicycle on
that lane. The cooperative driving plan will be regenerated when the
detected obstacles moved away.
Figure 5.12 Handling an emergency case.
3)
In this chapter, the speed and position of all the vehicles are assumed to
be accurately measured. In case of colliding with the lead vehicle, the
135
headway for the steering direction and communication delay is often
enlarged to counterbalance the measurement error. However, if the
sensors on vehicle fail and send out wrong data, it may still cause severe
accidents. Our framework does not provide a mechanism to detect such
faults yet. One possible way to solve this problem is to introduce
vehicle-to-road communication and let a road monitor system to check
the potential faults.
4)
The geometry constraints of vehicles and junction areas have not been
well discussed in this chapter. However, they should not be neglected in
real conditions. For instance, if a van truck steers from lane 1 to lane 8 as
depicted in Fig.5.3, it may block off lane 2 at the same time. This
problem can be solved by judging whether two vehicles form a safe
driving pattern or not by considering both their moving directions and
geometry characteristics.
136
CHAPTER 6 TIRE FAULT OBSERVER BASED ON ESTIMATION
OF TIRE/ROAD FRICTION CONDITIONS
Many tire fault monitors are designed nowadays because tire failure is proved to
be one of the main causes of traffic accidences. However, most of them are high in
manufacturing cost and unreliable. This chapter is devoted to solve this problem and a
new practical tire fault observer is proposed. Based on the new introduced dynamic
tire/road friction model that considers external disturbances, the observer estimate and
track the changes of tire/road friction conditions using only vehicle track forces and
wheel angular velocity information. Tire fault diagnosis is carried out as follows.
Since the wheel speed sensor is one basic component of normal anti-lock brake
system (ABS), the observer proposed could be easily realized in low cost within an
anti-lock brake system.
6.1
Introduction
Nowadays, people find out that many traffic accidences, extremely some really
bad ones, are caused by tire failure. Therefore, to measure and monitor the tire and to
determine the road surface friction in a cruising state well within the safe limit is quite
important. In the last decade, there are numerous literatures on it. One basics way to
solve this problem is to model tire/road friction and to estimate the related parameters
online so as to monitor the state of the tire indirectly. The most often used description
137
of the tire/road model is a set of curves of the wheel slip with which the normalized
friction force is defined. And based on it, the tire/road friction condition could be
analyzed [277][278].
In this field, the two analytical models presented by Bakker et al. [279] are often
used by researchers. In these two models, µ = F / Fn (where F denotes the Friction
force and Fn denotes the normalize force) is mainly determined by the wheel slip s
with regard to some other parameters such as speed. Here s is defined as
⎧ rω
⎪⎪1 − v , if v > rω , v ≠ 0, braking
s=⎨
⎪1 − v , if v < rω , ω ≠ 0, driving
⎪⎩ rω
in which r is the radius of the wheel, ω is the angular velocity, v is the
longitudinal velocity.
The two curves shown in the following Fig.6.1, which is obtained by Harned et al.
[280], demonstrates the typical relation between µ and s under different road
conditions. However, this model lacks a physical interpretation and it’s difficult to put
into use for too many underdetermined parameters.
138
(a)
(b)
Figure 6.1 Typical tire/road friction profiles for: (a) vehicle running on different road
surface conditions with velocity 20m/h, (b) vehicle running on dry asphalt road with
different vehicle velocities [280].
139
Recently, dynamic friction models, such as the one presented by Candudas de Wit
et al. in [281], have been proposed successfully to identify and compensate the
friction in mechanical systems. In [282] Candudas de Wit et al. use the so called
LuGre model to estimate a parameter in the model that reflects changes in the
tire/road characteristics for the first time. In this chapter, both the angular and
longitudinal velocity is assumed to be measurable. Since in some applications, the
longitudinal velocity cannot be obtained, it’s necessary to design an observer to
estimate the slip ratio using the angular information only. In [283] and [284],
Candudas de Wit, Jingang Yi and et al. formulate the model together with a new
nonlinear observer to solve this problem.
This chapter is devoted to extend the approach of [285] and [286] in two ways.
First, the inevitable disturbance of the model parameter is considered in the model
since the actual road condition parameter should be varied within a certain range for a
certain time. Second, a proper fault detection rule should be set associated with the
system when it’s used as a fault observer. In order to achieve these two goals, a
modified model is proposed and a novel fault observer for the tire/road contact
friction is constructed as following in this chapter based on strong practical stable
theory.
6.2
Problem Formulation
A potential advantage of such models is their ability to describe closely some of
140
the physical phenomena found in road/tire friction (i.e. hysteresis loops, pre-sliding
displacement, etc), and to depend on a parameter directly related with the phenomena
to be observed, like for instance the change on the road characteristics (i.e. dry or wet).
Dynamic models can be formulated as a lumped or distributed ones.
The lumped friction model proposed in Canudas-de-Wit and Tsiotras [282] is
based on a similar dynamic friction model for contact-point friction problems
developed previously, which is called LuGre model. The model given below has been
proved to be a good approximation of distributed tire/road friction models that are
able to represent the typical characteristics as the ones displayed by Fig.6.1.
z& = −vr − θ
σ 0 vr
g (v r )
z
F = (σ 0 z + σ 1 z& − σ 2 vr ) Fn
with g (vr ) = µ c + ( µ s − µ c )e
− vr / v s 1 / 2
(6.1)
(6.2)
.
Here σ 0 is the rubber longitudinal stiffness, σ 1 is the rubber longitudinal
damping, σ 2 is the viscous relative damping, µ s is the normalized static friction
coefficient, and µ c is the normalized Coulomb friction. vs is the Stribeck relative
velocity, Fn is the normal force, and vr = v − rω is the relative velocity. z is an
introduced parameter denotes the internal friction state. Surfaces are very irregular at
the microscopic level and two surfaces therefore make contact at a number of
asperities. It could be viewed as two rigid bodies that make contact though elastic
bristles. z could be viewed as the average deflection of those bristles.
141
And θ is an undetermined parameter of the tire/road condition on which we
focus in this chapter. It will change with the variation of the road condition if the tire
condition doesn’t change. Under normal tire pressure, some typical data of it is: for
dray asphalt conditions, θ = 1 ; for wet asphalt conditions, θ = 2.5 and for snow
conditions, θ = 4 . The abnormal tire pressure will lead to significant variation of θ .
In most of the previous works, no disturbance is considered in the modeling
process. However, the road condition could not be ideal and should be varied within a
certain range. In this chatper, we model all the disturbance of into the variation of θ
as following
θ& = ε
(6.3)
where ε represent the certain stable stochastic disturbance process. Since the
highway road condition doesn’t vary abruptly, without lose any generality, we could
assume this disturbance satisfies:
1)
ε is bounded as ε ≤ τ , τ > 0 ;
t
2)
θ is bounded, i.e. θ − θ = θ 0 + ∫ εdτ − θ ≤ ∆θ max , where θ 0 denotes
0
the initial value of θ and θ denotes the expectation of θ .
The value of τ and ∆θ max should be determined by the actual environment.
From the practical tests, we could see that these values are normally small. So
142
estimating and monitoring the value of θ , we could detect a fault if the value of θ
becomes especially abnormal. In the following parts, we will see how to design a fault
observer for θ and how to get a fault detection rule. Before further discussion, let’s
review some basic properties of this model.
Remark 6.1: the model has the following important properties:
(i)
1 ≥ µ s ≥ g (vr ) ≥ µ c ≥ 0 , ∀vr ∈ R ;
(ii)
f (v r ) =
vr
g (v r )
0 ≤ f (vr ) ≤ ρ1 ,
is positive and bounded and f ′(vr ) is also bounded, i.e.
0 ≤ f ′(vr ) ≤ ρ 2 , ∀vr ∈ R .
Considering the lumped tire/road friction model together with velocity dynamics,
the estimation model can be formulated as following
mv& = 4 F − σ v mgv
(6.4)
Jω& = −rF − uT
(6.5)
where J and m is the inertia and ¼ mass of the wheel. The term σ v is the rolling
resistance coefficient and g is the gravity constant.
Assume that only the variable ω is measurable, and the lumped friction
parameters with θ = 1 has been identified offline (see table 1 for sample data from
Triangle Tire Corporation).
Table 6.I Sample data of the off-line identification for LuGre Model
Parameter Value Unit
143
σ0
σ1
σ2
µc
µs
25
5
0.1
0.5
0.9
12.5
vs
6.3
1/m
s/m
s/m
m/s
Observer Design
Let’s introduce the same change of coordinates in [280]: x1 = σ 0 z , x2 = v ,
x3 = vr = v − rω . From which we can get:
x&1 = σ 0 z& = −σ 0 x3 − σ 0θ
x& 2 = g[ x1 + σ 1 (− x3 − θ
x3
g ( x3 )
x&3 = v& − rω& = α [ x1 + σ 1 (− x3 − θ
where α = g (1 +
x1
(6.6)
x1 ) − σ 2 x3 ] − gσ v x2
(6.7)
x3
g ( x3 )
g ( x3 )
x1 ) − σ 2 x3 ] − gσ v x2 + uT
(6.8)
mr 2
).
4J
Define x , y respectively as x = [x1
introduce ϕ ( x) =
where
x3
⎡ 0
A = ⎢⎢− g
⎢⎣ α
x3
g ( x3 )
x2
x3 ] , y =
T
1
( x2 − x3 ) = ω , and
r
x1 , we could rewrite the system as
0
− gσ v
− gσ v
x& = Ax + Bθϕ ( x) + EuT
(6.9)
y = Cx
(6.10)
−σ0
⎤
− g (σ 1 + σ 2 ) ⎥⎥ ,
− α (σ 1 + σ 2 )⎥⎦
⎡ −σ0 ⎤
B = ⎢⎢− gσ 1 ⎥⎥ ,
⎢⎣ − ασ 1 ⎥⎦
1
⎡
C = ⎢0
r
⎣
1⎤
− ⎥ ,
r⎦
144
⎡0 ⎤
E = ⎢⎢0⎥⎥ .
⎢⎣1⎥⎦
Notice that ( A, C ) is a observer pair, we can introduce the observer feedback
matrix L and obtain the following observer structure which are like the ones
proposed in [282] and [284]:
x&ˆ = Axˆ + Bθˆϕ ( xˆ ) − L ( y − yˆ ) + kB sgn( y − yˆ ) + EuT
&
(6.11)
θˆ = γ 1ϕ ( xˆ )( y − yˆ )
(6.12)
yˆ = Cxˆ
(6.13)
where k = max{x : 2θ max ϕ ( x) } . From above, we have k < ∞ . And γ 1 is a positive
x
real number that is introduced to adjust the variation rate of θˆ .
From the above observer, we can get the dynamics of the estimated error as
x& e = x& − xˆ& = [ A + LC ]xe + B[θϕ ( x) − θˆϕ ( xˆ )] − kB sgn( ye )
&
(6.14)
ˆ ye
θ&e = θ& − θˆ = ε − γ 1ϕ ( x)
(6.15)
ye = y − yˆ = Cxe
(6.16)
Before we reach the main results, let’s introduce the definition of the strong
practical stable and a useful lemma.
⎧ x& = f (t , x)
Definition:if the solution of the dynamic system ⎨
exits for any
⎩ x(t0 ) = x0
initial value x(t0 ) = x0 during (t 0 , ∞) ,and there exits a positive real number d
and a time t d > 0 such that the solution satisfies x(t ) ≤ d , for any t ≥ t d , this
system is said to be strong practical stable with respect to d . Here x denotes
145
ess. sup .{ x } .
Lemma 1 (see [287][288]): The dynamic system described in definition 1 is
−1
strong practical stable with respect to d 0 = (r1 ⋅ r2 )(c) , if there exits a first-order
derivable positive function V (•) : R n × R → R + , a continuous positive infinity
function ri (•) : R + → R + , i = 1,2 , and a positive infinity function r3 (•) : R + → R +
satisfies: a) there exits a positive real number c such that r3 ( x ) > 0 for any x > c
and
r3 ( c ) = 0
;
b)
r1 ( x ) ≤ V ( x, t ) ≤ r2 ( x )
;
c)
∂V ( x, t )
= ∇ xV ( x, t ) ⋅ f ( x, t ) ≤ − r3 ( x ) .
∂t
With this representation we shall now verify the following Theorem under the
assumed condition holds.
Theorem 1: The observer error state system designed above is strong practical
λmax ( P) ⋅ 4 θ e max τ + 4 θ e max γ 2
stable with respect to d =
⋅ , when there exit positive
λmin ( P) ⋅ λmin (Q )
γ1
2
symmetric matrix P and Q satisfy the following equations
− Q = [ A + LC ]T P + P[ A + LC ]
(6.17)
PB = γ 2 C T
(6.18)
where γ 2 is a positive weighted number.
Proof: Considering the Lyapunov function V (t , x) = xeT Pxe +
γ2 2
θ e , Since
γ1
θϕ ( x) − θˆϕ ( xˆ ) = [θϕ ( x) − θϕ ( xˆ )] + [θϕ ( xˆ ) − θˆϕ ( xˆ )] = θ [ϕ ( x) − ϕ ( xˆ )] + θ eϕ ( xˆ )
We should have
146
V& = xeT {[ A + LC ]T P + P[ A + LC ]}xe − kxeT PB sgn( y e ) − k[ B sgn( y e )]T Pxe
γ
+ xeT PB[θϕ ( x ) − θˆϕ ( xˆ )] + [θϕ ( x) − θˆϕ ( xˆ )]T B T Pxe + 2 2 θ e [ε − γ 1ϕ ( xˆ ) ye ]
γ1
= x {[ A + LC ] P + P[ A + LC ]}xe − 2kγ 2 ye sgn( y e )
T
e
T
γ
+ 2γ 2 ye [θϕ ( x) − θˆϕ ( xˆ )] + 2 2 θ e [ε − γ 1ϕ ( xˆ ) y e ]
γ1
= xeT {[ A + LC ]T P + P[ A + LC ]}xe + 2γ 2 ye {[θϕ ( x) − θˆϕ ( xˆ )] + θ eϕ ( xˆ )}
− 2kγ 2 y e sgn( ye ) − 2γ 2θ eϕ ( xˆ ) ye + 2
γ2
θ eε
γ1
= xeT {[ A + LC ]T P + P[ A + LC ]}xe + 2γ 2θ [ϕ ( x) − ϕ ( xˆ )] ye
− 2kγ 2 y e sgn( ye ) + 2
γ2
θ eε
γ1
Based on the assumption, we should have
θ [ϕ ( x) − ϕ ( xˆ )] ≤ θ max [ ϕ ( x) + ϕ ( xˆ ) ] ≤ 2k ,
Thus
2γ 2θ [ϕ ( x) − ϕ ( xˆ )] ye − 2kγ 2 ye sgn( ye ) ≤ 0
And since ε ≤ τ , we can have
γ
V& ≤ xeT {[ A + LC ]T P + P[ A + LC ]}xe + 4 2 θ e
γ1
max
τ
Notice (20), we should have
V& ≤ −λmin (Q ) xe
2
+4
γ2
θe
γ1
max
τ
If we choose
r1 ( xe ) = λmin ( P) xe , r2 ( xe ) = λmax ( P ) xe
2
r3 ( xe ) = λmin (Q ) xe
2
−4
γ2
θe
γ1
2
max
+4
γ2
θe
γ1
2
max
,
τ
Then based on lemma 2, the observer error state system is strong practical stable
147
with respect to
λmax ( P) ⋅ 4 θ e max τ + 4 θ e max γ 2
⋅
d =
λmin ( P) ⋅ λmin (Q)
γ1
2
Based on this result, if the tire in the steady state, and the observer output error
should be bounded as
y − yˆ =
1
1
2
[( x2 − xˆ 2 ) − ( x3 − xˆ 3 )] ≤ ( x2 − xˆ 2 + x3 − xˆ 3 ) ≤ d
r
r
r
Thus two fault detection rules can be formulated as:
1)
2
trigger the fault alarm should if y − yˆ > d ;
r
2)
trigger the fault alarm should if θˆ − θ > ∆θ max .
The estimated value of θˆ will indicate the approximate tendency of θ from
which we can judge what type of error occurs. Generally, θˆ will be abnormally big
expected if the tire pressure is too high; θˆ will be abnormally small if the tire
pressure is too low. In the practice, τ
and ∆θ max could be determined by
measurement. And a helpful Matlab toolbox that could be used to solve this LMI
problem could be found in [286].
6.4
Simulation Results
Using the data given in Table 6.I and assuming
r = 0.2m , m = 5kg , J = 0.25kgm 2 kg, Fn = 15 Kgm 2 / s 2
we should have
148
0
− 20 ⎤
⎡ 0
⎡ − 20 ⎤
⎥
⎢
A = ⎢ − 9.8 − 4.9 − 50.96⎥ , B = ⎢ − 49 ⎥ , C = [0 4 − 4]
⎢
⎥
⎢⎣12.86 − 4.9 − 66.88⎥⎦
⎢⎣− 64.21⎥⎦
Choose
0.1
− 0.3872⎤
7.65 ⎤
⎡ 1.0
⎡− 11.92 − 7.14
⎡5 ⎤
⎢
⎥
⎢
⎢
⎥
P = ⎢ 0.1
1.0
− 0.8552⎥ , L = 0 , Q = ⎢ − 7.14 − 31.63 28.39 ⎥⎥
⎢ ⎥
⎢⎣− 0.3872 − 0.8552 0.8342 ⎥⎦
⎢⎣ 7.65
⎢⎣5⎥⎦
28.39 − 26.82⎥⎦
we can have
λmax ( P) = 1.9 , λmin ( P ) = 0.006 , λ min (Q ) = 0.67
Assume
θe
max
= 0.2 , τ = 0.1 , γ 1 = 20000 , γ 2 = 1
we will have
d = 0.062 and y − yˆ = 0.495 .
In the simulation process, the value of θ changes from 1 to 3 at t = 5 . The
following Fig.6.2 and Fig.6.3 show the value of y − yˆ = ω − ωˆ , θ and θˆ
respectively. From the Fig.6.2, we can find that y − yˆ = ω − ωˆ exceeds the threshold
0.495 immediately after t = 5 . θˆ reaches the new value 3 soon after t = 5 , and
reaches 3 at t = 7 . We can also see that the statistic characteristics of the error output
changes, too. It’s obvious that the fault alarm will be trigger in time.
149
Figure 6.2 Variation of y − yˆ = ω − ωˆ when the jump error occurs.
Figure 6.3 Variation of θ and θˆ when the jump error occurs
150
Assume in the following simulation process, the value of θ shifts from 1 to 3
during t = 5 and t = 10 . The following Fig.6.4 and Fig.6.5 show the value of
y − yˆ = ω − ωˆ , θ and θˆ respectively. From Fig.6.4 we can find that y − yˆ = ω − ωˆ
does not exceed the threshold 0.495. But θˆ keeps following θ all the time. It’s
obvious that the fault alarm will be trigger in time, too.
Figure 6.4 Variation of y − yˆ = ω − ωˆ when the shift error occurs.
151
Figure 6.5 Variation of θ and θˆ when the shift error occurs.
152
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