Development and Improvement of Active Vehicle

Development and Improvement of Active Vehicle
Development and Improvement of Active Vehicle Safety Systems by
Means of Smart Tire Technology
Mustafa Ali Arat
Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Mechanical Engineering
Saied Taheri, Chair
Mehdi Ahmadian
John Ferris
Robert West
Hesham Rakha
August 15th, 2013
Blacksburg, Virginia
Keywords: Active Vehicle Safety Systems, Smart/Intelligent Tire, Adaptive Control,
Optimization, Nonlinear Systems
c 2013, Mustafa Ali Arat
Copyright Development and Improvement of Active Vehicle Safety Systems by Means
of Smart Tire Technology
Mustafa Ali Arat
ABSTRACT
The dynamic behavior of a vehicle is predominantly controlled by the forces and moments
generated at the contact patch between the tire and the road surface. As a result, tire
characteristics can dramatically change vehicle response, especially during maneuvers that
yields the tires to reach to the limits of its adhesion capacity. To assist the driver in such
cases and to prevent other possible instability scenarios, various vehicle control systems
e.g. anti-lock brakes (ABS), stability controllers (ESP, ESC) or rollover mitigation schemes
are introduced, which are generally known as active vehicle safety systems. Based on the
above facts, one can easily come to the conclusion that to improve upon the current control
algorithms developed for the technology in use; a vehicle control system design requires
accurate knowledge of the tire states. This study proposes the use of a smart tire system
that can provide information on momentary variation of tire features through the sensor units
attached directly on the tire and develops control algorithms based on this information to
assure the match-up between tire and controller dynamics. A prototype smart tire system
was developed for field testing and for detailed analysis of its potential. Based on the
collected prototype data, novel observer and controller schemes were developed to obtain
dynamic tire state information and to improve vehicle handling performance. The proposed
R
algorithms were implemented and evaluated using numerical analysis in Matlab/Simulink
environment. For a more realistic simulation environment, vehicle models were integrated
R
from Mechanical Simulations CarSim
software suite.
This work was supported in part by the Turkish National Ministry of Education and the
NSF I/UCRC Center for Tire Research.
Acknowledgments
I am infinitely grateful to many individuals for their support, collaboration, and friendship
that inspired and bolstered me through this journey. The demanding route through the
doctorate would have never come to an end without their contributions. First and foremost
I would like to thank to my advisor, Prof. Saied Taheri, for his constant presence and continuous encouragement. He has always been there for guidance as well as for motivation to
keep me moving forward. I would also like to sincerely thank Prof. Mehdi Ahmadian, Prof.
John Ferris, Prof. Robert West and Prof. Hesham Rakha for their invaluable support and
feedback on my studies and for all their help while serving in my doctorate committee.
Special thanks go to all of my colleagues at CenTire and CVeSS for the academic and
non-academic discussions we shared. I want to especially thank Mr. Bharat Singh, Dr.
Nenggen Ding, Dr. Jian Zhao, Dr. Clement Nagode and Mr. Micheal Craft who have not
hesitated to spare their time on all the tedious questions I kept coming up with. I also owe
many thanks to my dear friends, Dr. Onur Bilgen, Mr. Matt Oremland, Dr. Kyonghoon
Cho, Mr. Umut Atalay and many others I could not list here, who always kept my spirits up
whenever the pressure peaked. Of course, none of these would have been possible without
the unconditional love and infinite support of my parents, and without my teacher-mother
Mrs. Serpil Arat. Finally, I would like to thank to my wife Seda for her care, her endless
love and for being there at every step.
iii
Dedication
...to my beloved Grandfather, my lifelong mentor,
Mustafa Ali Altıntaş
iv
Contents
1 Introduction
1.1
1
Smart Tire Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.1
State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.1.2
Studies at the Intelligent Transportation Laboratory (ITL) . . . . . .
8
1.2
Objectives & Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.3
Document Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.4
List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2 Mathematical Modeling and Simulations
16
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2
Vehicle Models used in Control System Development . . . . . . . . . . . . .
17
2.2.1
Quarter Car Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2.2
Bicycle (Linear 2 DoF) Model . . . . . . . . . . . . . . . . . . . . . .
20
Models used in Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . .
25
2.3.1
Nonlinear Vehicle Model . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.3.2
R
CarSim
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
Tire Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.4.1
32
2.3
2.4
A Modified Dugoff Tire Model for Estimation Studies . . . . . . . . .
v
2.4.2
Tire Models Used in Simulation Studies: . . . . . . . . . . . . . . . .
3 Adaptive Vehicle Stability Control
34
39
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.2
Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.3
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.4
Tire Slip-Angle Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3.4.1
Smart Tire based Estimation . . . . . . . . . . . . . . . . . . . . . .
47
3.4.2
Model based Observer Derivation . . . . . . . . . . . . . . . . . . . .
48
3.5
Tire Slip-angle based Stability Control . . . . . . . . . . . . . . . . . . . . .
55
3.6
System Validation using Simulation . . . . . . . . . . . . . . . . . . . . . . .
59
3.7
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
4 Advanced Anti-lock Braking
68
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
4.2
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.3
Estimation of Surface Friction Condition . . . . . . . . . . . . . . . . . . . .
73
4.3.1
Smart Tire Based Surface Classification . . . . . . . . . . . . . . . . .
73
4.3.2
Model Based Surface Friction Observer . . . . . . . . . . . . . . . . .
76
4.4
A Self-Tuning Anti-lock Brake System Algorithm . . . . . . . . . . . . . . .
80
4.5
System Validation using Simulation . . . . . . . . . . . . . . . . . . . . . . .
86
4.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
5 Integrated Vehicle Control Systems
100
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.2
Integrated Stability Control based on Lyapunov Direct Method
vi
. . . . . . . 102
5.3
5.4
5.2.1
Control Allocation Problem . . . . . . . . . . . . . . . . . . . . . . . 102
5.2.2
System Validation using Simulation . . . . . . . . . . . . . . . . . . . 108
L1 Adaptive Control Method
. . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.3.1
Derivation of the Control Algorithm
. . . . . . . . . . . . . . . . . . 114
5.3.2
System Validation using Simulation . . . . . . . . . . . . . . . . . . . 120
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 Conclusion
130
6.1
Summary and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.2
Future Extensions and Impact . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Bibliography
135
A Projection Operator
150
B Case Study - L1 Adaptive Control of an Active Suspension System
158
B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
B.2 Mathematical Model and Control Algorithm . . . . . . . . . . . . . . . . . . 159
B.3 System Validation in Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 164
B.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
vii
List of Figures
1.1
Common vehicle chassis control systems available in modern vehicles . . . .
2
1.2
Proposed role of the smart tire in active safety systems . . . . . . . . . . . .
3
1.3
Diagram for the potential instrumentation and outputs of the smart tire
. .
4
1.4
Interaction of the smart tire with the vehicle and highway infrastructure . .
5
1.5
The prototype smart tire system developed at the ITL . . . . . . . . . . . .
8
1.6
Testing of the developed smart tire prototype . . . . . . . . . . . . . . . . .
9
1.7
Tire vibration waveform generation. . . . . . . . . . . . . . . . . . . . . . . .
10
1.8
Results of the signal processing algorithms using the smart tire prototype
developed in ITL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
2.1
The quarter-car model illustration . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2
Stability analysis for a quarter car model . . . . . . . . . . . . . . . . . . . .
19
2.3
Single track vehicle model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.4
Vehicle steady-state response under US, NS and OS characteristics . . . . . .
24
2.5
Vehicle transient response to the given pulse steering input . . . . . . . . . .
25
2.6
R
CarSim
features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.7
Simplified tire contact patch geometry (adopted from [1]) . . . . . . . . . . .
33
2.8
Simplified bristle geometry for the Brush tire model . . . . . . . . . . . . . .
35
viii
3.1
The functioning of stability control systems
. . . . . . . . . . . . . . . . . .
40
3.2
Observer designs for lateral velocity estimation . . . . . . . . . . . . . . . . .
45
3.3
Observer designs for lateral velocity estimation . . . . . . . . . . . . . . . . .
46
3.4
Comparison of vehicle state response time . . . . . . . . . . . . . . . . . . .
47
3.5
Variation in the instantaneous amplitude of the lateral and radial acceleration
signal power regarding slip angle and wheel load . . . . . . . . . . . . . . . .
3.6
49
Variation in instantaneous amplitude of the lateral acceleration signal as function of the tire-slip angle under different loading conditions . . . . . . . . . .
50
3.7
Results for the proposed SMO scheme for longitudinal tire force estimation .
51
3.8
Results for the proposed SMO scheme for lateral tire force estimation . . . .
53
3.9
Tire slip-angle estimation performance . . . . . . . . . . . . . . . . . . . . .
55
3.10 Control signals for high-µ condition . . . . . . . . . . . . . . . . . . . . . . .
60
3.11 Vehicle response comparison on high-µ condition . . . . . . . . . . . . . . . .
61
3.12 Front and rear tire-slip angle values for the controlled and uncontrolled vehicles compared to the desired values . . . . . . . . . . . . . . . . . . . . . . .
61
3.13 Vehicle and wheel speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.14 Adaptation of the front and rear axle cornering stiffness values . . . . . . . .
63
3.15 Low-µ surface test results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
3.16 Control signals for low-µ surface testing . . . . . . . . . . . . . . . . . . . . .
64
3.17 Front and rear tire-slip angle values for the controlled and uncontrolled vehicles compared to the ideal values . . . . . . . . . . . . . . . . . . . . . . . .
64
3.18 Wheel and vehicle velocity variation during low friction surface test . . . . .
65
3.19 Adaptation of the front and rear axle cornering stiffness values during low
friction surface test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3.20 Comparison of control efforts for slip-angle and yaw rate based control algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
66
4.1
Tire force-moment characteristics as a function of road surface conditions . .
70
4.2
Frequency response analysis of the smart tire signals
. . . . . . . . . . . . .
74
4.3
PSD analysis of pre-trailing and post-trailing signal domains . . . . . . . . .
74
4.4
Computation of the vibration ratio (R) on different surface conditions . . . .
75
4.5
Fuzzy logic algorithm to classify the current surface condition . . . . . . . .
75
4.6
Variation in the vibration levels under low and high slip conditions . . . . .
76
4.7
Smart tire based wheel load estimation performance. . . . . . . . . . . . . .
77
4.8
Fuzzy logic algorithm to classify the current surface condition . . . . . . . .
78
4.9
Model based surface friction estimation performance. . . . . . . . . . . . . .
80
4.10 Tire force-slip curve and slip-angle relation. . . . . . . . . . . . . . . . . . .
81
4.11 System constraints depicted on phase plane. K1 < Ta < K2 K3 < λa < K4 .
83
4.12 FSM description of the ABS state switching logic . . . . . . . . . . . . . . .
87
4.13 Results for the threshold adaptation and system dynamics on high-µ surface
condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
4.14 Braking performance of the proposed algorithm on high-µ surface . . . . . .
89
4.15 Results for the threshold adaptation and system dynamics on low-µ surface
condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
4.16 Braking performance of the proposed algorithm on low-µ surface . . . . . . .
91
4.17 Braking performance of the conventional ABS algorithm on straight line braking 92
4.18 Phase plane comparison of adaptive threshold and conventional ABS algorithms in high-µ and low-µ tests . . . . . . . . . . . . . . . . . . . . . . . . .
93
4.19 Jump-µ test condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
4.20 Results for the threshold adaptation and system dynamics on jump-µ surface
condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
4.21 Results for the threshold adaptation and system dynamics on split-µ surface
condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
95
4.22 Results for the conventional ABS algorithm on jump-µ surface condition . .
95
4.23 Phase plane comparison of adaptive threshold and conventional ABS algorithms in jump-µ test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
4.24 Phase plane comparison of adaptive threshold and conventional ABS algorithms in split-µ test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
4.25 Comparison of algorithm performances in jump-µ surface test . . . . . . . .
97
4.26 Comparison of algorithm performances in split-µ surface test . . . . . . . . .
97
5.1
The active-set algorithm for numerically solving the control allocation problem
using weighted-least squares. . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2
Integrated Chassis Control (ICC) system block diagram. . . . . . . . . . . . 109
5.3
Control signals assigned by the ICC scheme. . . . . . . . . . . . . . . . . . . 109
5.4
Wheel brake forces distribution by dynamic control allocation and rule-based
method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.5
Wheel slip variations by dynamic control allocation and rule-based method. . 111
5.6
Tire force response in ICC with dynamic and rule-based control allocation
methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.7
The L1 norm with respect to varying low-pass and control gain. . . . . . . . 121
5.8
Given ramp-steer maneuver. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.9
(a) Resulting control signals, and (b) yaw-rate values with and without the
control intervention on differing surface friction conditions. . . . . . . . . . . 123
5.10 Control signals by the L1 and Lyapunov adaptive methods (a) steering input,
and (b) desired yaw-moment on high-mu. . . . . . . . . . . . . . . . . . . . . 124
5.11 (a) Tire slip-angle variations, and (b) corresponding tire lateral force utilization.125
5.12 (a) Vehicle CG trajectory in comparison with an open-loop system and with
Lyapunov based control method implemented (b) Comparison of vehicle yawrate with L1 control and Lyapunov methods. . . . . . . . . . . . . . . . . . . 126
xi
5.13 Control signals by the L1 and Lyapunov adaptive methods (a) steering input,
and (b) desired yaw-moment on low-mu. . . . . . . . . . . . . . . . . . . . . 126
5.14 (a) Tire slip-angle variations, and (b) corresponding tire lateral force utilization.127
5.15 (a) Vehicle CG trajectory in comparison with an open-loop system and with
Lyapunov based control method implemented (b) Comparison of vehicle yawrate with L1 control and Lyapunov methods. . . . . . . . . . . . . . . . . . . 128
A.1 (a) Convex (b) non-convex sets. . . . . . . . . . . . . . . . . . . . . . . . . . 151
A.2 Convex function f (x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
A.3 Graphical illustration of the projection operator in R2 . . . . . . . . . . . . . 154
B.1 Quarter car suspension model with single point tire-road contact . . . . . . . 160
B.2 Frequency response comparison for the linearized QC plant with integral adaptive and L1 adaptive methods . . . . . . . . . . . . . . . . . . . . . . . . . . 163
B.3 Closed loop block diagram of the L1 control scheme . . . . . . . . . . . . . . 164
B.4 L1 controller performance with bump profile. . . . . . . . . . . . . . . . . . . 165
B.5 L1 controller performance with bump and ditch profiles. . . . . . . . . . . . 166
B.6 Comparison of the controller performance for H∞ and L1 control methods . 167
B.7 Test bed for the proposed adaptive active suspension control algorithm . . . 168
B.8 Vehicle response to the road profile . . . . . . . . . . . . . . . . . . . . . . . 168
B.9 Acceleration thresholds specified by ISO-2631 . . . . . . . . . . . . . . . . . 169
xii
List of Tables
2.1
Classification of the common tire models used in research . . . . . . . . . . .
32
3.1
Brake rules for the DYC implementation . . . . . . . . . . . . . . . . . . . .
59
4.1
Triggering signals and threshold values for the baseline ABS . . . . . . . . .
91
4.2
Summary of the system evaluation using numerical analysis . . . . . . . . . .
98
B.1 Quarter vehicle parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
B.2 Body Acceleration RMS for H∞ and L1 strategies . . . . . . . . . . . . . . . 166
B.3 Body Acceleration RMS in CarSim Simulations . . . . . . . . . . . . . . . . 170
xiii
Chapter 1
Introduction
Vehicle active chassis safety systems, which are designed to help the driver especially during emergency maneuvers, have been the topic of considerable research over the past three
decades. These systems are tuned based on the vehicle dynamic characteristics and its interaction with the road surface which is initiated at the contact patch of the tires. This small
patch dictates the resulting motion of the vehicle and is a major governing factor of the vehicle’s stability and control, especially under severe maneuvers. This relationship makes it very
desirable to track the interaction between the tire and the road surface and use the information to enhance the performance of the active safety systems. Furthermore, these dynamics
are not intuitive for the daily driver and are not perceivable as long as the vehicle progresses
in the intended path. On the other hand, for an average driver, it becomes too difficult to
stabilize a vehicle that is not responding to his/her commands leading to an unstable vehicle.
To improve vehicle stability during such events, active safety systems (Figure 1.1) have been
introduced which will intervene once certain thresholds are reached. A considerable number
of studies have been conducted on the effects of such systems on highway safety and majority of them have concluded positively. A highly referred study which was completed at the
National Highway Transportation Safety Administration (NHTSA) in 2007 indicates 14% to
28% reduction in overall fatal crash involvements in passenger cars and Light Trucks [2]. In
[3], Farmer reports the results of a risk analysis study completed in seven states in the U.S.
over two years. The study is carried out using otherwise identical vehicles with and without
stability controllers and concludes that the availability of stability controller yields to a 34%
1
Chapter 1. Introduction
2
Electronic Stability
Control
Complexity
Adaptive Cruise
Control
Anti-lock Brake System
Collision
Mitigation System
Rollover
Prevention
Active Torque Distribution
Advanced Driver
Assistance System
Integrated Vehicle
Control
Active Steering System
Active Suspension System
Figure 1.1: Common vehicle chassis control systems available in modern vehicles
reduction on the overall fatal crash involvement risk. Another similar study was conducted
by Lie et.al. [4], this time in Sweden where the authors analyzed crash data from 1998 to
2004 on various surface conditions and concluded that stability control systems show around
13 overall effectiveness and a 35% effectiveness on low friction surfaces. A more recent study
by Ferguson [5] shows results from investigations conducted world-wide with the conclusions
similar to the ones mentioned above. As a consequence of these and many other similar
studies, the European Union decided in 2009 to make stability control systems mandatory
so that all new models will be equipped with a stability control system starting November
1 of 2011 and old models without a controller will have to be withdrawn from traffic by the
end of 2013 [6]. A similar act also took place in the U.S. via NHTSA requiring all vehicles
carrying passengers with a gross vehicle weight rating of 4, 536 Kg (10, 000 pounds) or less to
be equipped with a stability control system by 2012. By this act, NHTSA estimates 5, 300
to 9, 600 fatalities to be prevented once all passenger vehicles are instrumented with the
stability control system [7].
These reports underline the fact that the current safety systems have undoubtedly become a
life-saving technology. Nevertheless; although these systems have advanced in many aspects,
there are still many areas that they can be improved. As mentioned above, the tire-road
Chapter 1. Introduction
3
Environment/Road Conditions
Surface Irregulatories
Smart
Tire
Control Unit
- Performance
- Handling
+/-
- Ride
Throttle/Brake
- Stability
+/-
Steering
Aerodynamic Inputs
Figure 1.2: Proposed role of the smart tire in active safety systems
contact conceals very valuable information about vehicle’s momentary action on the road.
Regarding their positions, it is easy to imagine possible improvements for vehicle handling
and comfort characteristics that could be achieved by receiving information directly from the
tires [8]. Present methods for estimating tire-road parameters make use of information from
chassis based sensor systems and highly rely on vehicle kinematic relations. In addition, this
indirect approach introduces additional uncertainties to the control procedure and may result
in inaccuracies such as integration errors or time lags which might be crucial for the driver
and driver assist systems to react to an emergency. The advances in on-board electronics
and sensor technology have given rise to a new concept, namely smart tire or intelligent
tire technology, as a potential solution to this problem (Figure 1.2). Smart tire technology
basically proposes instrumenting the tire and monitoring its momentary conditions, which
provides important information about the tire-road contact (Figure 1.3). Furthermore it has
the potential to significantly reduce or completely remove the aforementioned inaccuracies
by reducing the need for indirect means.
This potential forms the basis and the motivation behind this research for exploring the development and implementation of new active safety systems based on smart tire concept. In
the light of the above arguments, the main goal of this study is to develop control algorithms
Chapter 1. Introduction
4
Sensor Set
Microprocessor
Power Supply
Tire-Road Contact
Smart Tire System
Tire characteristics (tire
pressure, temperature,
wear condition, etc.)
Tire dynamics (tire slip
ratio, slip-angle, tire
forces, aligning moment,
etc.)
Road conditions (surface
friction, road roughness,
etc.)
Figure 1.3: Diagram for the potential instrumentation and outputs of the smart tire
based on the information made available by smart tire for further improvements of vehicle
and highway safety. A strong background on the state of the art of smart tire technology
is an important prerequisite before promoting to the development stage. For that purpose,
the rest of this chapter provides a summary of the advances in the smart tire technology
along with the initial studies completed on a prototype system developed in the Intelligent
Transportation Laboratory (ITL). Following that the objectives and contributions of this
study are provided and an outline of this documentation is given.
1.1
1.1.1
Smart Tire Technology
State of the Art
The potential of the smart tire technology in the development and improvement of safety
systems can be better judged by the possible amount of information that can be obtained
about the tire in terms of the sensor outputs. With this concept, the tire will not only be
responsible for generating the forces and moments to drive and steer the vehicle, but also
for supplying the control/safety systems with valuable information regarding the tire-road
contact characteristics (Figure 1.4). From the application perspective, the range of possible
solutions and products comprises various direct and indirect systems as well as simple and/or
complex means for the relevant driver information systems. Some of the proposed automotive
applications of a smart tire system can be listed as follows:
• providing real time information about the tire condition to the driver [9],
Chapter 1. Introduction
5
• providing existing electronic control units (ECUs) with load, friction and slip angle
information [10],
• using this information for providing services for external users (e.g. vehicle-to-vehicle
(V2V) communication or vehicle-to-infrastructure (V2I) communication) [11].
A standardized technology found in almost every vehicle today, the Tire Pressure Monitoring Systems (TPMS) are considered to be the first smart tire product introduced into the
market. Today, with the advances in sensor technology, the interest has also spread into tire
temperature monitoring [12], which is also highly related to and therefore can be utilized
in observing tire wear. Furthermore, the effects of such features on tire-road dynamics (tire
forces, tire-road friction, etc.) are expected to help promoting the studies in the development
of smart tire based active safety systems.
Safety / Control Systems
Chassis based
Sensors
Vehicle Network
Gateway
Driver Interface
Smart Tire System
V-2-V / V-2-I Systems
Figure 1.4: Interaction of the smart tire with the vehicle and highway infrastructure
The first patent for the TPMS appeared in 1985 [13] which has been followed by many
others along with a slight shift in the development activities towards new aspects such as
self-sufficient power supply or wireless means for data transmission/handling. As such, these
technologies are also very important for an advanced smart tire system. Nonetheless, despite
the new level of safety TPMS provide, they lack essential information regarding the state of
the tire road contact characteristics. With the aim of obtaining more information for further
optimization of vehicle and highway safety, research in this field has focused on developing
Chapter 1. Introduction
6
more sophisticated and competent intelligent tire systems. One of the earliest studies for a
prototype system was carried out in the Darsmstadt University of Technology in Germany.
The resulting system used a magnet placed inside a single tire tread block and the movements
of this magnet was monitored by a Hall-effect sensor which allowed the tire deformation to
be measured [14]. Another innovative application in the field was the system developed by
the Continental Tire, namely the Side Wall Torsion (SWT) sensor, in 1999 [15]. The SWT
sensor allowed measurement of the tire side wall deformation and thereby estimation of the
forces acting at the tire-road contact. Another mainstream in this initial phase of research
was using acoustic wave based systems (i.e. Surface Acoustic Wave Sensors) [9], which was
very advantageous in terms of power requirements as the sensor system did not require an
additional power source.
The success of these pioneering studies triggered the Apollo and Friction programs [16, 17], a
collaboration of several Europe based research institutes and private companies. The Apollo
program started in 2002 in the lead of VTT (Technical Research Center of Finland) and
concluded in 2005 with quite valuable results for the intelligent tire research in general.
A thorough feasibility analysis had been conducted and different approaches apart from
hall-effect sensors had been tested. Basically a 3-in-1 approach was employed where three
different sensor systems (a strain sensor, an optical based sensor and an accelerometer) were
placed inside a single tire. Out of these three major sensor systems, accelerometers came out
to be the most viable solution due to their durability and ease of utilization inside the tire.
The results stated a firm proof for that the possibility exists to predict tire characteristics
based on sensors embedded in the tire and solidified the possibility of the benefits of an
intelligent tire concept.
The Apollo program was followed by the Friction project to complete the task of measuring
the surface friction condition. The project was majorly aimed at proving the possibility of
quantifying the friction condition by using a sensor fusion approach that integrates tire based
sensors with currently available cost-effective chassis based systems. A specific subtitle of
the project was dedicated to further analysis of the optical based sensor developed in the
Apollo program. The potential of the approach was studied by employing a detailed finite
elements analysis (FEA) and simulations. The optical sensor was reported to provide quite
accurate information about tire’s momentary deflection which was utilized to estimate the
Chapter 1. Introduction
7
generated tire forces and eventually the surface friction [18].
The results of these initial studies have made a substantial headway in the field, but also
yielded to a number of other key issues waiting to be resolved such as the signal-to-noise ratio
in sensor readings, difficulties in decomposition of coupled deflections and measurements or
transmission rate of the collected information for real-time active safety control systems. A
number of individual studies attempted to come up with feasible solutions to these problems.
In [19] Umeno studied vibrations of a free-rolling tire and proposed an algorithm to estimate
tire friction and detect hydroplaning. In another study, Zhang et.al. [20], investigated design
considerations for a sensor system to be implemented inside the tire, specifically a SAW based
sensor, and commented on the required environment and signal conditions. Seki et.al. in
[21] introduced a new approach by using tire sound to analyze tire characteristics via wavelet
analysis which yields to the estimation of tire tread patterns. Matsuzaki et.al. proposed the
use of strain sensors in smart tire systems in [22] and further investigated the potential of such
system in [23, 24]. In [25], Ohori et.al. introduced another approach to the smart tire concept
by instrumenting the rim and investigated measurement of tire forces and moments by means
of this new system. Another exciting study was carried out by Braghin et.al. [26] where the
authors investigated the use of accelerometer inside the tire as well as its advantages in the
development of vehicle control systems. This approach has been adopted by many others
including Brusarosco et.al. [27] and Savaresi et.al. [28], who further investigated the use of
acceleration based sensors inside tire; and has become a mainstream in smart tire research
per se. Upon the advances in the sensor research in the subject, recent studies focus more
on its implementation. Audisio et.al. [29] reports the initial studies and introduces their
road-map about a commercial product based on smart tire technology specifically designed
for use in safety systems. Ergen et.al. [30] investigates a top-down design procedure for
the circuitry to be implemented inside the tire along with an application specific wireless
transmission protocol. Another investigation on the application of smart tire systems in
vehicle control is done by Cheli et.al. [31] where the authors specifically review possible
improvements on the anti-lock brake systems (ABSs) by assuming wheel loads and frictions
are available through smart tires. Next, Erdogan et.al. [32] proposed a method to estimate
tire-road friction through a smart tire based mechanism. Finally, in [33] and [34] same
authors investigate further use of smart tires in ABS and stability control systems. As for
the industrial interest about this technology, many private companies already exposed their
Chapter 1. Introduction
8
R
initial prototypes in various events, such as the Intelligent Tire by Continental, CyberTire
by Pirelli and Contact Area Information Sensor (CAIS) by Bridgestone, and a commercial
product is expected to be released in a near future.
1.1.2
Studies at the Intelligent Transportation Laboratory (ITL)
Perceiving the soaring interest in the subject, an industrial partner has approached ITL to
collaborate in the development of a prototype smart tire system and to further study possible benefits of it. Initial studies have been carried out on the details of the mechanisms
through which tires generate forces and moments. These studies helped with determining
the intended tire and surface characteristics using the sensory outputs. In what follows,
various sensor candidates were reviewed to be implemented inside the tire and regarding
the results of the studies summarized above, micro tri-axial accelerometers were selected for
instrumentation (Figure 1.5).
Instrumented Tire
Signal Conditioner
Contact Patch
Data Logging
Figure 1.5: The prototype smart tire system developed at the ITL
Testing with multiple sensors inside the tire proved to be impractical due to wiring issues
in the rapid rotating frame of the tire. As a result, a single micro tri-axial accelerometer
was mounted on the center of the innerliner of the tire. For the prototype system all data
and power transmission were managed via wiring inside the tire through a high-speed rated
slip ring. A signal conditioner powered the accelerometer and the data was logged on a PC
for further analysis. After proper instrumentation of the tire, the system was taken out for
Chapter 1. Introduction
9
testing using the in-house tire testing trailer (Figure 1.6).
(a) Prototype mounted on the testing trailer
(b) The in-house tire testing trailer at ITL
Figure 1.6: Testing of the developed smart tire prototype
The collected data indicated that as a tire rolls and deforms during operation under load,
the tire produces a signature waveform that can be interpreted into information regarding
the tire and road surface conditions. Basically the four major zones of curvature within an
inflated tire, as shown in Figure 1.7a, shape the waveform per revolution. These four zones
include the contact patch which varies with braking and acceleration, the zones just before
and just after the contact patch relating to the leading and trailing edges, respectively, and
finally the area of the tire that is largely unaffected by loading. Through implementation of
a multi-axis accelerometer, it was possible to measure the acceleration and deformation of
the tire in the longitudinal, lateral and vertical directions (Figure 1.7b). The data collected
from the accelerometer was processed using various methods including neural networks and
power spectrum densities (PSD).
The prototype was extensively tested on different surface conditions to collect sufficient
amount of data for further analysis. Finally, the studies were concluded by the development
of two novel signal processing algorithms [35] that can compute the dynamic wheel load (Figure 1.8a) and tire slip-angle (Figure 1.8b) variations. Furthermore, based on the wheel load
information, a surface condition classification method has been developed and implemented
for the development of a new ABS control algorithm [36].
Chapter 1. Introduction
(a) The four zones of curvature within an inflated
tire
10
(b) Sample vibration data logged for a full
revolution
Figure 1.7: Tire vibration waveform generation.
(a) The wheel load algorithm is based on contact
patch characteristics.
(b) Tire slip-angle estimation is based on lateral
acceleration signals.
Figure 1.8: Results of the signal processing algorithms using the smart tire prototype
developed in ITL
1.2
Objectives & Contributions
As summarized above, the tire-road characteristic plays a vital role in the development and
performance of the vehicle control systems. The current methods for obtaining information
on tire-road characteristics heavily rely on indirect estimation mechanisms and are quite
Chapter 1. Introduction
11
open for improvements. The literature proves that smart tire technology is a promising path
for reducing the dependency on indirect estimation and for attaining considerable improvements in the estimation of dynamic tire states. The initial studies completed at the ITL
also concluded with very encouraging results including a prototype system and algorithms
to obtain valuable information from the data collected by an accelerometer mounted inside
the tire. In addition, various discussions between the ITL and our industry partner during
the development of the prototype system indicated that there is a considerable interest in
the market for such a technology, which proclaimed its implementation as an open field to
explore. Based on these arguments, this study aims to develop novel vehicle control systems
that can make efficient use of the smart tire technology and takes full advantage of being
able to obtain information about tire-road characteristics.
The active safety systems available on vehicles today are based on the longitudinal and lateral
stability of the vehicle chassis as well as its rolling stability. Regarding these groundwork on
the subject, the objectives of this study can be expanded as follows:
• Investigating points in vehicle stability and control where the smart tire becomes most
beneficial.
• Developing algorithms for primary vehicle chassis control systems (e.g. ABS, stability
control, active steering, and torque distribution) with additional information from the
smart tire
• Developing novel control algorithms that can coordinate various primary chassis control
systems to achieve better vehicle stability and handling performance.
• Finally, developing integrated vehicle control algorithms that can inherently account
for the actuation dynamics as well as adapting to possible variations in environmental
conditions, aiming to compute a set of control laws guaranteeing optimal actuator
coordination and performance.
The study was initiated by conducting a brief parametric study based on mathematical
vehicle models, which revealed the points where the smart tire will be most useful in vehicle
control systems. The results were then utilized to channel the information from the smart
tire into novel control algorithms for improved vehicle safety and performance. Based on
these endeavors, the major contributions of this research can be listed as follows:
Chapter 1. Introduction
12
• Contributed in the development of a prototype smart tire system that is capable of
providing dynamic wheel load and tire slip-angle information in addition to classifying
current road surface condition, and developed novel signal processing algorithms to
estimate corresponding tire-road states.
• Developed two adaptive stability control algorithms primarily based on the tire slipangle and force information provided by smart tire, which integrates active steering
and yaw control schemes to improve vehicle safety and performance.
• Developed a self-tuning ABS algorithm based on contemporary rule-based commercial
systems that can update the pre-defined rule set to provide best possible braking
performance irrespective of the surface conditions.
• An Integrated Chassis Control scheme is introduced to accommodate the previously
developed stability and braking control algorithms. A dynamic control allocation strategy is implemented that can optimally distribute wheel brake forces while successfully
avoiding actuator saturation or over-rating conditions.
• Finally, evaluated the effectiveness of the developed control algorithms using validated
nonlinear vehicle models and a commercial software (CarSim) under various surface
and driving conditions.
Regarding these outcomes, this study is a frontier in a new phase of developments on vehicle
control systems by allowing effective use of information about the tire-road interaction.
The smart tire technology brings in the possibility of acquiring very detailed and accurate
knowledge of the dynamical variations of the tire as well as the contact patch between the
tire and the road, which in return opens up a new lane for advancements in the field. In what
follows, the results of this research set down a strong foundation in terms of the integration
of this novel technology in vehicle controls and to extend it to further advance in vehicle
safety and performance.
1.3
Document Outline
The rest of this document is organized as follows:
Chapter 1. Introduction
13
Chapter 2: Mathematical Modeling & Simulation details the vehicle dynamics and related
mathematical models utilized in the development of the control algorithms as well as in
their simulations. Also a brief analysis of system stability is conducted to figure out which
parameters might impel most the improvement of vehicle control systems.
Chapter 3: Adaptive Vehicle Stability Control first briefly reports the studies on the development of vehicle stability systems so far. Next, the sensor fusion approach to estimate the
tire slip-angle is outlined. Finally the work on a new adaptive control algorithm for vehicle
stability is introduced, which utilizes the sensor fusion concept to obtain dynamic information about tire slip angle and provides assistance to the driver to maintain stability of the
vehicle.
Chapter 4: Advanced Anti-lock Braking briefly summarizes the literature on handling and
braking studies based on ABS algorithms up to dat. In what follows the integrated estimation scheme for obtaining surface friction condition with the use of the smart tire and
a model based observer is explained. Finally, the chapter introduces an ABS algorithm
integrated with the surface friction estimation to improve performance. Based on the estimated friction condition, the algorithm is capable of updating the predefined limit criteria
for applying/releasing the brakes, thereby providing best performance irrespective of any
variations on the surface conditions.
Chapter 5: Integrated Vehicle Control Systems introduces the notion of integrated chassis
control and a dynamic control allocation strategy. The control allocation strategy implements optimal tire force distribution to articulate the stability and braking control schemes.
The resulting lower level controller aims to signal the brakes for the optimum brake forces
with respect to the yaw moment requirement to maintain vehicle stability. In what follows,
a new adaptive control strategy, namely the L1 adaptive control method, is presented that
proposes improved adaptation capabilities.
Chapter 6: Conclusion shortly summarizes the outcomes of the above studies based on the
results and the current state-of-the-art. Finally, the chapter reflects on possible extensions
of this research for further development.
Chapter 1. Introduction
1.4
14
List of Publications
This research so far has led to a series of peer-reviewed conference and journal publications
as listed below:
Peer Reviewed Abstracts:
• Arat, M.A., Taheri, S., “Application of Smart Tire Technology in Vehicle Chassis
Control Systems”, 17th International Conference of ISTVS, Blacksburg, VA, 2011
• Arat, M.A., Singh, K.B., Taheri, S., “Application of a Smart Tire System in Improving the Performance of Advanced Vehicle Control Systems”, 31st Annual Meeting and
Conference on Tire Science and Technology, Cleveland, Ohio, 2012
• Arat, M.A., Taheri, S., “Development and Improvement of Active Vehicle Safety
Systems by Means of Smart Tire Technology”, SIAM Conference on Control and Its
Applications, San Diego, CA, 2013
Peer Reviewed Conference Papers:
• Arat, M.A., Singh, K.B., Taheri, S., “An Intelligent Tire based Adaptive Vehicle
Stability Control System”, International Conference on Advanced Vehicle Technologies
and Integration (AVTI), Changchun, China, 2012
• Arat, M.A., Singh, K.B., Taheri, S., “An Adaptive Vehicle Stability Control Algorithm
based on Tire Slip-Angle Estimation”, SAE Commercial Vehicle Engineering Congress,
Rosemont, Illinois, 2012
• Arat, M.A., Singh, K.B., Taheri, S., “Adaptive Vehicle Stability Control with Optimal
Tire Force Allocation”, ASME 2012 International Mechanical Engineering Congress
and Exposition, Houston, Texas, 2012
• Singh, K. B., Arat, M. A., and Taheri, S., “Adaptive Control of Antilock Braking
System Using an Intelligent Tire Based Tire-Vehicle State Estimator ”, FISITA 2012
World Automotive Congress, Beijing, China, 2012
• Singh, K. B., Arat, M. A., and Taheri, S., “Enhancement of Collision Mitigation
Braking System Performance Through Real-Time Estimation of Tire-Road Friction
Chapter 1. Introduction
15
Coefficient by Means of Smart Tires”, SAE Commercial Vehicle Engineering Congress,
Rosemont, Illinois, 2012
• Singh, K. B., Arat, M. A., and Taheri, S., “Development of a Smart Tire System
and its use in Improving the Performance of a Collision Mitigation Braking System”,
ASME 2012 International Mechanical Engineering Congress and Exposition, Houston,
Texas, 2012
Peer Reviewed Journal Papers:
• Arat, M.A., Singh, K.B., Taheri, S., “An Intelligent Tire based Adaptive Vehicle Sta-
bility Controller ”, International Journal of Vehicle Design - Special Issue on Advanced
Developments in Tire Modeling, Analysis and Dynamics (in print)
• Arat, M.A., Singh, K.B., Taheri, S., “Optimal Tire Force Allocation by Means of
Smart Tire Technology”, SAE Int. J. Passeng. Cars - Mech. Syst. 6(1):2013.
• Singh, K., Arat, M.A., and Taheri, S.,“Enhancement of Collision Mitigation Braking
System Performance Through Real-Time Estimation of Tire-road Friction Coefficient
by Means of Smart Tires”, SAE Int. J. Passeng. Cars - Electron. Electr. Syst.
5(2):607-624, 2012
• Singh, K.B., Arat, M.A., Taheri, S., “An Intelligent Tire Based Tire-Road Friction
Estimation Technique and Adaptive Wheel Slip Controller for Anti-lock Brake System”,
Journal of Dynamic Systems, Measurement and Control, 135, 0310032, 2013.
Chapter 2
Mathematical Modeling and
Simulations
2.1
Introduction
To be able to analyze the interplay of forces and motions acting on the vehicle, one needs
an integrated treatment of all system components interacting with each other. The basis of
such a theoretical analysis of vehicle systems is an appropriate mathematical model adapted
to the given task. This underlying model has to be sufficiently detailed to represent the
essential properties of the vehicle system and, at the same time, as simple as possible to
allow for reasonable simulation capabilities. Obtaining a balance between the reality and
abstraction of the model is most important as it dictates the quality of the results achievable
on a system’s dynamical behavior. These conflicting requirements depict the challenge in
selecting the appropriate model for different engineering tasks.
In this study, linearized vehicle models are utilized for control system development and
their evaluations are done on more realistic non-linear models. The following subsections
detail these models. The handling and braking studies are based on a two-degrees-of-freedom
quarter car model. Another linearized two-degrees-of-freedom model (bicycle model) is used
in stability studies to represent the vehicle lateral dynamics. The initial evaluations of the
developed control algorithms are executed using an eight-degrees-of-freedom nonlinear vehicle model which represents the lateral, longitudinal, roll and yaw dynamics as well as the
16
Chapter 2. Mathematical Modeling and Simulations
17
R
rotational dynamics of each of the four wheels on the vehicle. Finally CarSim
commercial
software is utilized as a more realistic test bed.
2.2
Vehicle Models used in Control System Development
2.2.1
Quarter Car Model
The most simplistic yet very useful model in vehicle dynamics analysis is the quarter-car
model (a.k.a. single corner model) which as its name implies is derived by taking only a
single wheel of the vehicle into account. This model, despite its simplicity, can quite successfully represent the longitudinal and vertical dynamics on a single wheel with the proper
selection of required parameters. Although based on a single wheel, a quarter car model can
also be developed in a non-linear fashion and with higher degrees of freedom by including
suspension dynamics and/or road profile variations if needed. Therefore it is a widely used
and accepted model in suspension as well as braking studies. In this study, the quarter car
model is utilized to define the vehicle longitudinal dynamics in the development of a new
anti-lock braking system (ABS) algorithm (4).
Considering the essential features for brake dynamics (e.g. wheel angular speed, vehicle
speed, wheel load and the net torque acting on the wheel), Figure 2.1 represents the free
body diagram of the single wheel which is the basis for the model.
Fz = mg
ω
Tb
v
Fx = −ma
Figure 2.1: The quarter-car model illustration
The model basically consists of a single wheel attached to the quarter vehicle with mass (m).
Chapter 2. Mathematical Modeling and Simulations
18
The respective equations of motion for this system are given as below:
ma = −Fx
(2.1)
J ω̇ = Fx Rw − Tb
(2.2)
For a quarter car system, the longitudinal tire force (Fx ) is the key variable that dominates
the wheel dynamics. It can be defined as a non-linear function of wheel load (Fz ), tire
longitudinal slip (λ), tire slip angle (α) and the maximum tire-road friction coefficient (µmax ):
Fx = Fz µ(α, λ, µmax )
When studying the straight line braking, the tire slip angle (α), is generally omitted assuming
negligible steering activity. As for the tire slip ratio (wheel slip), it can be defined by the
ratio of the relative velocity between the tire and the road to the maximum of the two, which
is mathematically defined as below:
λ=
v − Rw ω
max(v, Rw ω)
(2.3)
It is worth to note that zero slip characterizes the free rolling when no friction force applies
(Fx = 0) on the wheel, and unit slip (λ = 1) indicates complete wheel lock-up. The quarter
car model can reveal very valuable details about the vehicle’s stability characteristics during
braking. The rest of this section aims to provide a brief analysis to underline the important
parameters in the development of a successful ABS algorithm. The key element in such an
analysis is the accuracy of the selected friction or tire model. The Burckhardt model is one of
the most widely used models as it defines the utilized friction (µ) in terms of wheel slip and
constant coefficients (C1 , C2 and C3 ) that are pre-defined w.r.t. maximum surface friction:
µ(λ) = C1 (1 − e1−C2 λ ) − C3 λ
(2.4)
The constant coefficients help to adjust the model for different surface conditions. In what
follows, the wheel slip (λ) dynamics can be determined by taking the time derivative of
equation 2.3 and substituting equations 2.1, 2.2 and 2.4 for the appropriate terms, which
yields to the below differential equation:
λ̇ =
Rw (Tb − Te )
Jv
(2.5)
Chapter 2. Mathematical Modeling and Simulations
where
19
(1 − λ)J
Te = Rw µ(λ)Fz 1 +
2
mRw
is the equilibrium torque. It is important to note that the rate of change of slip drops to
zero when equilibrium torque equals to the brake torque. Next equation 2.5 is augmented
with the brake torque dynamics by assuming a constant brake torque rate:
T˙b = U
(2.6)
The phase plane of the resulting system can provide an insight on the stability characteristics of a braking vehicle as depicted in Figure 2.2a. This simple analysis shows the
relation between the applied brake torque, or brake pressure as they can be assumed linearly
proportional, and the wheel lock-up condition which defines the stability of the vehicle.
1800
λ’ vs. λ
1400
Prediction
1200
Reselection
b
Brake Torque (T ) [Nm]
1600
Stable equilibria
Unstable equilibria
Guaranteed Wheel
Lock−up
1
0
Increasing Tb
1000
800
600
400
200
0
0
Stable Braking
0.2
0.4
0.6
0.8
1
Slip
(a) Phase-plane for the wheel slip dynamics with
constant brake torque rate
(b) System trajectory for the quarter-car with
simple rule-based ABS algorithm implemented
Figure 2.2: Stability analysis for a quarter car model
As these results indicate, heedlessly increasing the brake torque results in wheel lock-up which
dramatically reduces the braking performance and even worse drives the vehicle unstable. To
avert such failure scenarios, ABS algorithms are proposed that allows brakes to be applied in
a cyclic way to adjust the applied brake torque/pressure. Figure 2.2b illustrates the system
trajectory (Tb vs. λ) of the system defined in eq. 2.5 and 2.6 with a simple rule-based ABS
algorithm at work. The algorithm basically prevents constant increase in the applied brake
torque by releasing and reapplying the brakes w.r.t. specified limits, namely the Prediction
and Reselection rules. Equation 2.5 suggests that, aside from the applied brake torque, the
behavior of the system trajectory also highly depends on the surface friction condition, which
Chapter 2. Mathematical Modeling and Simulations
20
defines the tire force characteristics and thereby also the equilibrium points. Therefore a set
of Prediction and Reselection rules defined for a certain surface condition will not yield to the
same performance, or will not even guarantee averting wheel lock-up on a different surface.
A viable solution is adapting the specified rule-set to the varying surface conditions where
the involvement of a smart tire would be essential for dynamically observing the tire-road
conditions. These results motivated the development of a smart tire based ABS algorithm
detailed in Chapter 4.
2.2.2
Bicycle (Linear 2 DoF) Model
The vehicle bicycle model is a linearized model basically for vehicle’s lateral dynamics. It
provides a mathematical description of the vehicle motion without considering the forces that
affect the pitch and roll motion assuming that these forces are canceled due to symmetry
w.r.t. the imaginary single track passing through the middle of the vehicle, hence the other
name for the model as single track. The equations for steering and turning motions are
based purely on geometric relationships governing the system.
𝜶𝒇
𝑽
𝜹𝒇
𝑿
𝝍
𝒖
𝑭𝒚
𝒍𝒇
𝒇𝒓𝒐𝒏𝒕
𝑽
𝒀
𝒗
𝑪. 𝑮.
𝒍𝒓
𝑹
𝑭𝒚
𝜹𝒓
𝒓𝒆𝒂𝒓
𝜶𝒓
Figure 2.3: Single track vehicle model
Figure 2.3 shows the free body diagram for the bicycle model. As mentioned above the
Chapter 2. Mathematical Modeling and Simulations
21
left and right side wheels are represented on a single track with front and rear wheels only.
Furthermore in this study the rear steer (δr ) is neglected assuming a front-steer vehicle. In
its simplest form the lateral dynamics of the vehicle chassis can be defined using Newton’s
law. The sum of the lateral forces acting on the vehicle chassis defines the lateral acceleration
and the total moment acting at the vehicle c.g. is defined by the lateral forces at the front
and rear axles:
mv̇ = Fyf ront + Fyrear
(2.7)
Iz ṙ = lf Fyf ront − lr Fyrear
(2.8)
where Iz is the vehicle chassis’ moment of inertia w.r.t. the Z axis, r stands for the rate of
change of the yaw angle and Fyf ront and Fyrear represent the sum of the lateral tire forces at
the front and rear axles. Tires have intrinsic nonlinear characteristics (Section 2.3), including
the relation between the generated tire forces and the excitation parameters (e.g. tire slip
ratio and slip angle); however a linear approximation is acceptable at lower excitation rates
(i.e. |α| ≤ ±10◦ ) [37]. Assuming small tire slip-angles, the tire forces can be defined by the
following linear relations:
Fyf ront = 2Cαf αf
(2.9)
Fyrear = 2Cαr αr
(2.10)
where Cαf and Cαr are the front and rear tire cornering stiffness values. The same small angle
approach allows defining the front and rear tire slip-angles as below by utilizing kinematic
relations from the model geometry depicted in Figure 2.3.
v − lf r
u
v − lr r
= −
u
αf = δ −
(2.11)
αr
(2.12)
In what follows, equations 2.9-2.12 are substituted back into equation 2.7 and 2.8 which
yields to define what is generally known as the vehicle bicycle model:
2Cf + 2Cr
2Cf lf − 2Cr lr
Cf
v̇ = −
v−
+u r+
δ
mu
mu
m
2Cf lf2 + 2Cr lr2
2Cf lf − 2Cr lr
Cf lf
ṙ = −
v−
r+
δ
Iz u
Iz u
Iz
(2.13)
(2.14)
Chapter 2. Mathematical Modeling and Simulations
22
Chassis side-slip angle (β) is considered as another important indicator for vehicle lateral
performance and can be defined as the proportion between lateral and longitudinal vehicle
speeds:
β=
v
u
(2.15)
Derivating equation 2.15 w.r.t. time and substituting for v̇ in equation 2.13 yields to another
common representation used in vehicle stability studies:
2Cf + 2Cr
β−
β̇ = −
mu
2Cf lf − 2Cr lr
Cf
δ
+1 r+
2
mu
mu
(2.16)
Equations 2.13 and 2.16 can be used in place of each other depending on the purpose of the
analysis or control strategy to be accomplished. The bicycle (single-track) model provides
an invaluable tool for vehicle lateral stability analysis and control design. A very important
performance metric in lateral stability analysis is the understeer gradient of a vehicle which
basically indicates the responsiveness of the vehicle to the given steering inputs. Assuming
the single-track model undergoing a constant radius (R) turn, the understeer gradient can
be derived as follows. First the steering angle during a constant radius turn is approximated
as:
δ=
L
+ αf − αr
R
(2.17)
Next, using the linear tire model in equations 2.9 and 2.10 and the approximation for lateral
acceleration as ay = V 2 /R, equation 2.17 is rewritten as:
L
δ= +
R
mr
mf
−
Cαf
Cαr
ay
L
= + Kay
2
R
(2.18)
where mf = m(lr /L) and mr = m(lf /L) and K stands for the understeer gradient. A significant point for this study is found by comparing equation 2.17 and 2.18 which demonstrates
the direct relation between the understeer gradient and tire slip-angles. As mentioned before, this metric defines the vehicle’s dynamic behavior for a given steering input, and due
to the direct relation, same interpretation can be made using the tire slip-angles as well.
Using equation 2.17 and 2.18 one can validate the relation of the difference between the
front and rear slip-angles and the vehicle’s understeer (steering angle needs to be increased
to negotiate a turn of constant radius), neutralsteer (steering angle should remain constant)
and oversteer (steering angle needs to be decreased) behavior. Furthermore, equations 2.13,
2.14 and 2.16 can be solved under steady-state conditions (ṙ = 0, v̇ = 0, β̇ = 0) to observe
Chapter 2. Mathematical Modeling and Simulations
23
the variation of these characteristics w.r.t. tire slip-angles.
r =
v =
V /R
δ
m Cαf lf − Cαr lr 2
L
−
V
R 2LR
Cαf Cαr
lf mu2
V /R
2lr −
L Cαr
m Cαf lf − Cαr lr 2
L
V
−
R 2LR
Cαf Cαr
(2.19)
(2.20)
It is important to note that in these definitions:
m Cαf lf − Cαr lr 2
V =
2LR
Cαf Cαr
mf
mr
−
Cαf
Cαr
V2
= αf − αr
2R
(2.21)
Figure 2.4 depicts the variation of the turning radius, vehicle yaw rate and lateral speed w.r.t.
the vehicle speed for the oversteer (OS), neutralsteer (NS) and understeer (US) vehicle, which
are basically defined by the difference between the tire slip-angles.
The transient response of the single-track model can be studied using the characteristics
equation of the system defined by eq. 2.14 and 2.16:
2
s +
2m(Cαf lf2 + Cαr lr2 ) + 2Iz (Cαf + Cαr )
mIz V
s+
4Cαf Cαr l2 2(Cαf lf − Cαr lr )
−
mIz V 2
Iz
=0
(2.22)
the roots of which are given as:
λ1,2 = −
where
(Cαf lf2 + Cαr lr2 ) ± 2σ
Iz V
(2.23)
1
Iz uCαf Cαr L
m Cαf lf − Cαr lr 2 2
2
2 2
σ = (Cαf lf + Cαr lr ) −
L−
V
m
2L
Cαf Cαr
and where the understeer gradient appears inside the square root as in the form given in
eq. 2.21. The qualitative evaluation of this analysis is done by brief simulations, results of
which are shown in Figure 2.5. Similar to the steady-state case, the results show the distinct
characteristics of an US, NS and OS vehicle under the same steering input.
This brief analysis summarizes some fundamental results about vehicle stability and also
Chapter 2. Mathematical Modeling and Simulations
24
70
60
αf − αr < 0
Turning radius [m]
50
40
αf − αr = 0
30
20
10
αf − αr > 0
0
−10
0
5
10
15
20
25
Vehicle speed [m/s]
Vc
30
(a) Position response
2
αf − αr < 0
1.8
Vc
Yaw rate [rad/s]
1.6
1.4
1.2
αf − αr = 0
1
0.8
0.6
αf − αr > 0
0.4
0.2
0
0
5
10
15
20
25
30
Vehicle speed [m/s]
(b) Yaw-rate response
10
5
Lateral velocity [m/s]
αf − αr < 0
0
αf − αr = 0
−5
−10
αf − αr > 0
Vc
−15
−20
5
10
15
20
25
Longitudinal velocity [m/s]
(c) Lateral velocity response
Figure 2.4: Vehicle steady-state response under US, NS and OS characteristics
reveals the importance of tire parameters such as cornering stiffness or slip-angle in defining
a vehicles stability characteristics. Furthermore, the results suggest that observing the tire
slip-angles allows one to comment on the momentary lateral chassis dynamics. Based on
Chapter 2. Mathematical Modeling and Simulations
25
Steering (rad)
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.5
Time [s]
0.6
0.7
Side−slip Angle (rad)
0.8
0.9
1
Yaw Rate (rad/s)
0.1
US
NS
OS
0.05
US
NS
OS
0.8
0.6
0
0.4
0.2
−0.05
0
−0.1
0
0.2
0.4
0.6
Time [s]
0.8
1
−0.2
0
0.2
0.4
0.6
Time [s]
0.8
1
Vehicle Trajectory
10
Y [m]
8
6
4
US
NS
OS
2
0
0
5
10
15
X [m]
20
25
30
Figure 2.5: Vehicle transient response to the given pulse steering input
these deductions, the stability control investigations focus on the estimation and use of tire
slip-angle in control algorithms.
2.3
2.3.1
Models used in Simulation Studies
Nonlinear Vehicle Model
A four degree-of-freedom (DoF) chassis model is adopted from [38], that accounts for vehicle’s
longitudinal, lateral, yaw and roll dynamics, and is integrated with rotational dynamics for
each of the four wheels. The resulting model is a nonlinear eight DoF vehicle model that also
utilizes combined slip tire model to compute tire forces. The chassis dynamics are derived
Chapter 2. Mathematical Modeling and Simulations
26
using Lagrange’s method based on the principle of conservation of energy:
d ∂T
∂T
∂U
−
+
= Qi
dt ∂ q̇i ∂qi ∂qi
(2.24)
where T stands for system’s kinetic energy, U for potential, Qi refers to the external generalized forces and moments acting on the system and qi represents the generalized coordinates
to describe system motion. The longitudinal (u), lateral (v) and yaw (r) velocities and the
roll angle (φ) of the chassis are taken as the generalized coordinates. Following from [?], the
generalized forces and moments are found to be:
Qu = Fxf l cosδ − Fyf l sinδ + Fxf r
Qv = Fxf l sinδ + Fyf l cosδ + Fyf r
Qr = lf Fxf l sinδ + lf Fyf l cosδ + Mzf l − bFyf r + Mzf r
Qφ = −(kφf l + kφf r )φ̇
(2.25)
where Qu is the sum of the forces in longitudinal direction, Qv is the sum of forces in lateral
direction, Qr is the sum of the moments in yaw direction and Qφ is the sum if the moments
in roll direction. Defining the kinetic and potential energy of the system as:
h
0
2
0
2
T = 1/2m (u − h φr) + (v + h φ̇)
i
+ 1/2Ix φ̇2 + 1/2Iy (φr)2
(2.26)
+1/2Iz (r2 − φ2 r2 + 2θr rφ̇) − Ixz rφ̇
U = 1/2(Cφf l + Cφf r )φ2 − 1/2mgh0 φ2
(2.27)
yields to the equations of motion for chassis as below:
m(u̇ − rv − h0 φṙ − 2h0 rφ̇) = Qu
m(v̇ + ru + h0 φ̈ − h0 r2 φ) = Qv
Iz ṙ + (Iz θr − Ixz )φ̈ − mh0 (u̇ − rv)φ = Qr
(Ix + mh02 )φ̈ + mh0 (v̇ + ru) + (Iz θr − Ixz )ṙ − (mh02 + Iy − Iz )r2 φ
+(Cφf l + Cφf r − mgh0 )φ = Qφ
(2.28)
The wheel dynamics are adopted from the quarter car dynamics in equation 2.2, with the
addition of and engine torque term (Te ) to account for the possible throttle input from the
Chapter 2. Mathematical Modeling and Simulations
27
driver:
J ω̇i = (Tei − Tbi ) − Fxi Rwi
where i denotes the wheel location (e.g f l, f r, rl, rr). Finally the tire forces (Fxi
(2.29)
& F yi )
are computed using a combined slip tire model which is further detailed in section 2.4
2.3.2
R
CarSim
Developed in 1991 at the University of Michigan Transportation Research Institute [39],
R
CarSim
software package provides realistic vehicle models with as many degrees-of-freedoms
(DoF) for various simulation studies and for many test applications including the evaluaR
tion of control systems [40, 41, 42]. In this study, the models provided by CarSim
are
downloaded into the MATLAB/Simulink environment, where the proposed control algorithms are implemented and the resulting vehicle characteristics are evaluated under given
conditions. The vehicle models run with 15 mechanical DoF which account for 6 DoF for
the sprung mass rigid body, 2 DoF for each suspension, single DoF on each wheel and a 1
DoF steering system; in addition to other dynamic DoF such as tire response (2 DoF lagged
response), brake hydraulics and throttle lags [43]. Various chassis types are offered (Figure
2.6) based on standard European chassis styles such as B,C,D & E-classes as well as sports
utility vehicles (SUVs). In this study specifically the D and E class sedan passenger cars
and SUVs are utilized. Furthermore many test conditions can be modified including driver
models, surface conditions (e.g. friction, bank angle, etc.), environmental conditions (e.g.
gust wind, obstacle, etc.) or timing options for braking/throttling (Figure 2.6).
The vehicle model in CarSim is defined by fully nonlinear equations of motion as derived
using first principles. The equations are in the form of a system of ordinary differential
equations (ODEs). In CarSim vernacular, the program contains several built-in solvers. A
solver in CarSim represents a complete set of equations of motion defining a particular combination of front and rear suspension types and a trailer (if included). Following paragraphs
provide brief descriptions of modeling details on the individual elements of the vehicle and
the environmental factors.
Suspension: Each wheel on the vehicle has vertical travel and rolling rotation as degrees
of freedom. The front suspension is always modeled as fully independent, while rear suspensions may be independent, twist axle, or solid axle at the option of the user. Each
Chapter 2. Mathematical Modeling and Simulations
D-Class/E-Class SUV
28
D-Class/E-Class/F-Class Sedan
Full Size SUV
B-Class/C-Class Hatchback
(a) Utilized vehicle models
(b) User interface
R
Figure 2.6: CarSim
features
suspension contains full compliance effects in lateral / longitudinal motion and angular motion, and friction (hysteresis) effects in the suspension springs. All kinematics (camber, toe,
etc.), spring stiffnesses, damping, and compliance can be defined by the user through lookup
tables / curves, which can be fully nonlinear. For this reason, the suspension type need not
be specified beyond “independent”, as the solver uses kinematic curves (the consequence of
whatever geometry may be present) rather than the geometry itself. The effects of roll and
jacking forces are derived from the kinematic curves and the compliance effects, rather than
as a consequence of conventional geometric analyses.
Steering: Complete vehicle steering mechanics are included in the CarSim model. The virtual driver can steer the steering wheel to any angle up to a user-definable maximum angle.
The steer angle at the wheels is found by multiplying the steer angle through the steering
rack ratio (user-defined tabular data). The actual angle at the wheel/tire is augmented by
the angular effects of the compliance in the suspension. The compliance deflection is found
as a result of the sum of moments in the suspension about the kingpin. Steering torque at
the steering wheel is calculated by multiplying the total moment about the kingpin back
through the steering system ratio. The driver steer angle can be defined as an open-loop
function of time in the form of tabular data, or can be controlled by a virtual driver. The
virtual driver will always try to follow a user defined driving path. A level of realism is added
using a nominal response time lag, and a preview time. Preview time is the delay between
Chapter 2. Mathematical Modeling and Simulations
29
when the driver sees an upcoming event, such as a curve in the road, and when the vehicle
must react to it.
Brakes: For the braking system in CarSim the input pressure (at the brake pedal) is controlled by either an open-loop control or the virtual driver. The pressure is proportioned to
each brake by the master cylinder according to user-definable curves, which can be controlled
independently for each wheel. Furthermore, the user can define braking torque as a function
of pressure for each brake individually, allowing large amounts of flexibility in brake system
control. Dynamics in the hydraulic brake fluid are modeled by a first order transient lag and
a constant time delay. Both the “fluid dynamics time constant” and the “transport delay”
can be defined for each brake individually. When the brakes are applied to a spinning wheel,
the supplied torque causes the wheel to decelerate. This allows the braking torque to be calculated using Newtons second law. When the wheel locks up, however, the brake no longer
does any work. Still, the braking torque does not go to zero, but rather supplies just enough
torque to keep the wheel from spinning. These two braking situations represent two unique
mathematical cases. Thus, when lockup is detected the solver will switch to a locked-up
braking model. The brake system is replaced by a torsional spring and damper that winds
up the wheel to resist the braking loads at the tire contact patch. The model switch occurs
when the wheel reaches a rotation speed equivalent to a low user-defined forward speed,
and will stay locked up until the torque in the torsional spring/damper is greater than the
supplied braking torque.
Powertrain: CarSim uses a detailed powertrain model that includes engine torque output
and fuel consumption, and detailed transmission effects. Torque and fuel consumption are
both calculated as functions of engine RPM and throttle angle using user-defined tabular
data. Torque is transmitted to either an automatic or a manual transmission that include
the effects of the torque-converter or clutch, as well as efficiency and inertia that can be
specified for each gear. Torque is transmitted to the driven wheels through the differential
(final drive) ratio or alternatively, through an all-wheel drive system with user-defined front
to rear torque-split. As with steering and braking, the throttle, shifting, and clutch can
be controlled by the end user as open-loop functions of time. Alternatively the clutch (if
applicable) and shift timing can be controlled by a shift schedule, and the throttle controlled
by the virtual driver model.
Chapter 2. Mathematical Modeling and Simulations
30
Tires: CarSim allows the user to model tire forces using a default internal model (with
or without nonlinear camber and overturning moment effects), the Pacejka 5.2 Magic Formula model, or any external model that can be programmed in C code or Simulink. The
CarSim Internal Tire Model uses nonlinear tabular data of longitudinal force, lateral force,
aligning moment, and overturning moment as functions of longitudinal slip, lateral slip angle,
load, and camber angle. The required data represents typical quantities that are obtainable
during experimental tire testing. The vertical load in the tire is found by treating the tire
as a linear spring in the vertical direction, and so a stiffness coefficient must be supplied.
Tire models such as the Magic Formula model represent methods of curve-fitting tire test
data. Thus, it is presumed that by inputting tire test data directly into CarSim (which is
designed to interpret and use it directly) the results should compare well with those found
using equation based tire models.
Aerodynamics: Aerodynamic effects on the vehicle are included. Three forces and three
moments are applied to the sprung mass at a point known as the aerodynamic reference
point. Users may define the aerodynamic properties of the vehicle model by inputting six
coefficients of drag (as functions of aerodynamic slip angle), vehicle reference length, location
of the aerodynamic reference point, and vehicle frontal area. The user may also define the
wind amplitude, wind heading, and air density for a given simulation using nonlinear tabular
data, as a function of time.
Solver Method
The equations of motion of the CarSim vehicle system are in the form of a set of ODEs. The
CarSim solver integrates the differential equations using a second-order implicit Runge-Kutta
algorithm known as RK2. Second-order Runge-Kutta is best understood as a refinement
of Euler’s Method. Given the present value of the dependent variable y(t), Euler’s method
calculates the derivative k1 = dy/dt|t0 and estimates the next value of y(t) to be yt+h =
yt + hk1 . It is therefore considered a linear ones step method. It is simple to implement, but
accuracy is low and the time step h must be made very small in order to improve accuracy.
Runge-Kutta integration methods sacrifice the linearity of Eulers method, but retain the one
step format. Other integration methods exist which retain linearity but move into a multistep format. With second order Runge-Kutta (RK2), a second estimate of the derivative
is used to increase the accuracy. Knowing yt the derivative at the current time is found as
Chapter 2. Mathematical Modeling and Simulations
31
∗
k1 = dy/dt|t . The derivative k1 is sed to make an initial estimate for yt+h called yt+h
and
∗
the derivative at t + h is estimated using yt+h
as k2 = dy/dt|t+h . The new value for y, called
y ¯∗ , is found to be y ¯∗ = yt + h[(k1 + k2 )/2]. Thus, the average of the two estimated derivative
was used to estimate the new value of y. The RK2 method is summarized as Equation 2.30
and provides better results than the simplest form of Euler’s method.
k1
k2
yt+h
y
dy =
h
dt t
y+k h
dy 1
=
h
dt t+h
k1 + k2
= y+
h
2
(2.30)
Because Runge-Kutta methods are future-looking and do not rely on information from previous time-steps, it is easy to adapt the size of the time-step to increase or decrease accuracy as
required. However, CarSim implements a fixed time-step in order to facilitate co-simulation
abilities with external software packages, compatibility with experimentally measured tabular data, and ability to match constant step size test data. As a consequence, the solver
in CarSim has no internal error-checking, and the end user must be cautious. It is recommended that simulations are carried out at several time-steps to verify the insensitivity of
the model to further time-step refinement [ref].
2.4
Tire Models
Tires basically define the amount of forces and moments acting on a vehicle’s chassis during
its motion, which emphasizes the role of the utilized tire models in any study. There are
numerous tire models developed so far to serve for different purposes and provides results at
different accuracy levels. One can select a model depending on whether the problem in hand
allows for the use of computational power required for a more theoretical model or requires
faster processing that can be achieved with a more empirical model. Similarly, models have
been developed to feature transient dynamics of the tire or its steady state characteristics
(Table 2.1).
In this research, different models are used basically for two purposes; for estimation and
control studies which provide analytical expressions to be substituted in the derivation of
Chapter 2. Mathematical Modeling and Simulations
32
Table 2.1: Classification of the common tire models used in research
Steady State
Transient
Empirical Models
Magic Formula, NicholasComstock
SWIFT, SIMON, TM-Easy,
SMAC
Theoretical Models
Dugoff, Brush, String, Burckhardt, Kiencke, Unitire
LuGre, Modified NicholasComstock
various observer dynamics, and for simulations studies where the developed control and
estimation algorithms are tested.
2.4.1
A Modified Dugoff Tire Model for Estimation Studies
A control algorithm basically depends on the feedback of system states, which are rarely
available for direct measurement. Most of the time, it requires an estimation scheme that
takes in the system outputs (sensor measurements, etc.) and computes the system states
for the control algorithm. This estimation scheme, similar to the control algorithm, requires
a computationally lightweight but sufficiently accurate mathematical model of the system.
Therefore tire models, which provide such analytical expressions, essentially provide the solution.
First model utilized in the estimation studies is the so-called Dugoff tire model. Dugoff
et.al. published number of studies about the analytical modeling of tire-road interaction
(circa 1970) [1, 44], which eventually yielded to this widely known model today. The model
basically assumes a simplified contact patch geometry (Figure 2.7) where the line segment
0 − 1 − 2 represents the centerline during a turning maneuver, hence the indicated slip angle
(α). Point 1 stands for the sliding boundary below which (line segment 0 − 1) the tire tread
is assumed to adhere to the ground perfectly. At point 1 the elastic stresses due to tread
deformation (α) starts to become saturated which yields the rubber to begin sliding relative
to the ground until the stress values drop to zero at the tread liftoff point 2.
Assuming uniform pressure distribution over the contact patch (which is found to be a weak
Chapter 2. Mathematical Modeling and Simulations
33
4
𝜻
2
1
𝜼
3
0
𝛼
Figure 2.7: Simplified tire contact patch geometry (adopted from [1])
assumption, nevertheless yields to reasonable results), the resultant tire forces are given as
a function of tire slip and surface friction as bellow:
λ
f (ζ)
1−λ
tan(α)
F y = Cα
f (ζ)
1−λ
F x = Cλ
where
and
µFz (1 + λ)
ζ= p
2 (Cλ λ)2 + (Cα tan(α))2
f (ζ) =

(2 − ζ)ζ, if ζ < 1
1,
if ζ ≥ 1
(2.31)
(2.32)
Chapter 2. Mathematical Modeling and Simulations
2.4.2
34
Tire Models Used in Simulation Studies:
As mentioned earlier the forces and moments acting on the vehicle body are generated at
the tire-road contact patch. Therefore a successful simulation of the vehicle motion during a
certain maneuver requires highly accurate modeling of the tire characteristics. Increasing the
number of degrees-of-freedom and further parameterization might seem to be a solution at
first; however such models (finite element models) require proportionally high computational
power. The Brush model is an idealized representation of the tire in the tire-road contact
region [38] which is assumed to be realized through a number of massless, elastic elements,
the so-called bristles. Figure 2.8 depicts such a tire representation with the tire slip-angle
(α) and the forces acting on it. The base point of each bristle is attached to a circular belt,
while the tip adheres to the road surface. The bristles are subjected to a level of stress
due to the wheel load and surface friction which leads to their deformation. Brush model
basically states that integrating these stresses on the bristles along the contact patch yields
to the expressions for forces and moments generated on the tire. A major assumption is
the parabolic stress distribution (as opposed to the Dugoff with uniform distribution) over
the bristles. Furthermore, the tire carcass is assumed to be completely stiff neglecting the
carcass deflections and the bristles are assumed to be linearly elastic disregarding the nature
of the rubber characteristics.
Following the above assumptions the resulting tire forces can be given as follows. In the case
of pure longitudinal slip (α = 0), the longitudinal tire force becomes:
Fx =








Cλ
λ
1+λ






µF sgn(λ),
z

 1 Cλ
−
3
2

3 
λ λ 
λ
3
Cλ
1 + λ 1 + λ  
1
1+λ 

−
 , for |λ| ≤ |λs |


2
µFz
(µFz )

 27
for |λ| > |λs |
(2.33)
where λs stands for the slip condition where the available frictional force becomes short to
keep the tip of the tread element attached anymore and it phases into full slip condition.
As for the case of pure side-slip (λ = 0), the lateral tire force and the aligning moment are
Chapter 2. Mathematical Modeling and Simulations
35
𝑽
𝝎𝒘
𝑭𝒛
𝜶
Slide
-𝑎
Adhesion
𝑥𝑠
0
𝑎
Parabolic pressure distribution yields:
𝑭𝒙
𝑞𝑧 =
3𝐹𝑧
𝑥
1−
4𝑎
𝑎
2
𝑭𝒛
𝑭𝒚
Figure 2.8: Simplified bristle geometry for the Brush tire model
defined as:
Fy
Mz

2
3
3
1
C
|tan(α)|tan(α)
1
C
tan
(α)

α
α
−Cα tan(α) +
−
, for |α| ≤ |αs |
3
µFz
27 (µFz )2
=
(2.34)

−µF sgn(α),
for |α| > |αs |
z

3

 Cα tan(α)α 1 − Cα tan(α) , for |α| ≤ |αs |
3µFz 3
(2.35)
=

0,
for |α| > |αs |
where αs stands for the slip-angle where, similar to the pure longitudinal case, the available
frictional force becomes short to keep the tread element following a straight line anymore
and a is the half of the contact patch length. Brush model is also modified for the case
of a combined lateral and longitudinal slip. In that case the bristles will be deformed in a
direction proportional to the magnitude of the stress levels in each direction. Nonetheless
the method of computing the forces remains the same as integrating the stress expressions
over the contact patch. Assuming isotropic tire properties (e.g. Cλ = Cα = Cs ) the resulting
equations for longitudinal and lateral tire forces along with the aligning moment are defined
Chapter 2. Mathematical Modeling and Simulations
36
respectively as below:
where
σx
Fx = F (λ, α, µ, µmax ) p 2
σx + σy2
σy
Fy = F (λ, α, µ, µmax ) p 2
σx + σy2
p
!
C σ2 + σ2 3
s
x
y
2α 1 − 3µFz
Mz = p
2
p
C σ2 + σ2 C σ2 + σ2 s
s
x
y
x
y
− 3
+3
3µFz
3µFz
(2.36)
(2.37)
(2.38)






µFz 1 −
and
!3 
p 2
2
p
Cs σx + σy
 , for σx2 + σy2 ≤ 3µFz Cs 3µFz
F (λ, α, µ, µmax ) =

3µFz 
p

2
2

σx + σy > for
µFz sgn(α),
Cs σx =
λ
,
λ+1
σy =
tan(α)
λ+1
Another practical solution has come out to be as developing relatively lightweight analytical
expressions with constants that are to be found empirically. A very well-known such model is
the so-called Magic Formula (MF) developed by Hans Pacejka [45], which is also implemented
in the simulation studies carried out for this research. The MF basically defines the lateral
and longitudinal tire forces in addition to the tire aligning moment as follows:
Fx = Dx sin (Cλ arctan (Bx λ − Ex (Bx λ − arctan(Bx λ)))) Svx
(2.39)
Fy = Dy sin (Cα arctan (By α − Ey (By α − arctan(By α)))) Svy
(2.40)
Mz = Dt sin (Ct arctan (Bt α − Et (Bt α − arctan(Bt α)))) Svt
(2.41)
where B represents the effect of the tire stiffness, C defines the shape of the trigonotmetric
sine function, D stands for the peak values as the sine function can take maximum value of
unity, E is known as the curvature factor which helps with better defining any possible local
deformations, and the final terms Svy , Svx and Svt represent the amount of the resulting
force and moment curve’s shift from the origin as the tire might behave differently during
Chapter 2. Mathematical Modeling and Simulations
37
braking and acceleration even under the same conditions. On a side note, Ct represents the
torsional stiffness of the tire. These parameters can be obtained by testing the tire under
the conditions to be simulated.
The tire models described above characterize the tire at its steady-state, in other words
they ignore any variations with respect to time, which largely holds true under constant surface friction conditions. However, if the surface friction is varying considerably during the
maneuver, the transient dynamics might become effective enough to invalidate the steadystate assumptions. A collaboration between the Lund and Grenoble universities yielded to
the development of a new tire model, named as LuGre model after the names of the collaborating institutes, to account for the tire’s transient dynamics. Based on a similar simplified
tire representation using bristles as in the Brush model, the LuGre model accounts for the
variations of the bristle deflections with respect to both time and position, hence allowing to
simulate transient behavior. A lumped and distributed model is developed first as a proof
of concept and for computational convenience. Next a more complex distributed model is
developed which, as opposed to the point contact assumption in the lumped model, assumes
a contact area on which different pressure distribution schemes can be applied depending on
the type of application. In this study an average lumped model is utilized which offers similar
results to a detailed distributed model by keeping the number of states that describes the
friction minimum. This is accomplished by defining a mean friction state as shown below:
z̄˙ = vr −
where

σ0 |vr |
+ κωw z̄
g(vr )
g(vr ) = µc + (µs −
(2.42)
vr α 
−
v 
µc )e s
µmax
and µc represents the normalized Coulomb friction, µs is the normalized static friction, µmax
is the maximum road surface friction, vs is the Stribeck relative velocity details about which
is provided in [?] and is omitted here as it is out of the scope of this research and vr is the
relative velocity of wheels to the vehicle chassis (vr = Rωw − v). The term κ stands for the
pressure distribution effect which in this study is taken as a constant by assuming a uniform
distribution scheme. Using these equations the resulting tire force is expressed as:
F (t) = (σ0 z̄(t) + σ1 z̄˙ + σ2 vr )Fz
(2.43)
Chapter 2. Mathematical Modeling and Simulations
38
where σ0 , σ1 and σ2 are constants that represent rubber longitudinal stiffness, rubber longitudinal damping and viscous relative damping respectively.
Chapter 3
Adaptive Vehicle Stability Control
3.1
Introduction
Stability control systems have drawn great interest since their introduction in the late 1980s
[46, 47]. They have evolved dramatically especially over the past decade thanks to the rapid
advancements in the computational power of the on-board electronics. The principle idea of
a stability control system is to assist the driver during a severe steering maneuver and prevent
any possible spin or drift out, sometimes even by intervening the driver commands (Figure
3.1a). Three major methods have been proposed so far for these systems to interact with the
vehicle dynamics: differential braking, active steering and active torque distribution (Figure
3.1b). In addition to these, active suspensions are becoming available with the emerging
technology which is used in a number of integrated control schemes.
Systems which use differential braking utilize the on board anti-lock braking system (ABS)
and apply the individual wheel brakes according to a known strategy to generate a correcting
yaw moment on the vehicle body. Their performance heavily relies on the performance of
the onboard ABS. Active steering systems allow the stability controller to intervene with
the driver’s steer command and implement any correction when necessary. A well-studied
active steering system is steer-by-wire. It is adapted from the aeronautic term fly-by-wire
which basically refers to the idea of replacing the manual control mechanisms with electronic
(wired) based equivalents. However steer-by-wire systems are not considered for commercial
applications yet, mostly due to concerns about reliability and capital costs. As a result,
39
Chapter 3. Adaptive Vehicle Stability Control
(a) Role of stability control
40
(b) Conventional stability control systems
Figure 3.1: The functioning of stability control systems
mechanical based active front steering systems have recently been introduced into the market.
Active torque distribution is another popular method for stability control. It makes use of
an advanced differential mechanism in the driveline to distribute the engine torque onto
the wheels in a way that it can control the resulting yaw moment on the vehicle body. As
summarized in Chapter 1, the effectiveness of such stability control systems has allowed them
to become almost ubiquitous in modern vehicles. They have already become a requirement
and almost standardized by many countries. This naturally triggers further research on the
subject to further push their performance and to find new avenues for improved safety.
3.2
Literature Review
The earliest introduction of a commercial stability control system was done by MercedesBenz and Robert Bosch GmbH where they introduced the Electronic Stability Program
R
(ESP
). Since then a considerable number of studies have been done on various control
schemes based on the three aforementioned systems as well as on their combinations. These
studies can be grouped into a fewer number of titles which would form the mainstreams
in the subject. One of these mainstreams is about utilizing individual control systems to
their utmost possible performance and mostly refers to the initial studies. Among these
Chapter 3. Adaptive Vehicle Stability Control
41
individual systems, differential braking based ones (e.g. ESP) attracted more attention due
to the widely used ABSs on board. A number of commercial systems followed ESP, such
as BMW’s Dynamic Stability Control (DSC) [48], GMs Stabilitrak [49] or Active Handling
System [50] and Toyotas Vehicle Stability Control (VSC) system [51], all based on active
braking of individual wheels. On the other hand, valuable academic studies lead the way to
the improvement of these commercialized systems. In [52], Pilutti et.al. investigated three
different control designs based on a linearized two degree-of-freedom vehicle model integrated
with brake dynamics and the authors concluded that saturation of the brake actuators play
a cardinal role in the effectiveness of the active braking based stability controllers. Another
important study was done by Ghoneim et.al. [53] where they studied the performance of
two controllers, a yaw-rate feedback system and a full-state (yaw-rate and sideslip) feedback system. The authors developed a Luenberger observer to be able to obtain sideslip
information and concluded that the additional information from the observer improved the
controller performance; however it is also critical to accurately estimate the variable which
might easily introduce abrupt errors into the system. A similar work is done by Fukada
[54] where he investigates combining an observer based method with numerical integration
to find chassis sideslip while dealing with tire nonlinearities and disturbances such as road
slant. Nishio et.al. [55] follow the same path and reports the results of an application of
the estimator on a differential braking based control system. A more recent study by Baffet
et.al. [56] employs an extended Kalman filter and estimates lateral tire forces and friction
in addition to chassis sideslip.
Active steering based controllers have drawn great interest from academia until recently
commercial systems were released into the market which still do not use complete steer-bywire technology. The essential contribution in this area is done by Ackerman [57, 58, 59, 60].
In these studies Ackerman principally analyzed decoupling of yaw rate which defines the
rotational dynamics of the vehicle body from lateral acceleration which is directly related
to the slip of the vehicle body in lateral direction. As a result, the control designs were
not affected by the lateral disturbances such as side-winds or road slants. In [61] Isermann
introduced a development methodology for a fault-tolerant steer-by-wire system and used
a brake-by-wire application as a proof of concept. Mammar et.al. [62] and You et.al. [63]
took this one step further and analyzed H control schemes for active steering based stability
control systems. Another noteworthy study was done by Hsu and Gerdes in [64] where,
different from the other studies, they investigated a steer-by-wire vehicle at the limits of
Chapter 3. Adaptive Vehicle Stability Control
42
handling and proposed a control algorithm using feedback linearization.
The third general method in stability control is active drive torque distribution which is
based on distributing the engine torque on the right and left side wheels (rather than front
and rear as in the traction control systems) to generate a yaw moment on the chassis which
would eventually stabilize the vehicle when needed. A major advantage of these systems is
that they do not reduce vehicle’s acceleration as opposed to the active braking based systems.
An introductory investigation was done by Ikushima and Sawase [65] which reported the results of a torque distribution based yaw-feedback control application on an actual vehicle.
Huchtkoetter and Gassmann [66] studied an advanced differential mechanism that would aid
in various commercialized control algorithms. Similarly Osborn and Shim [67] study the application of an advanced differential mechanism which would allow individual wheel torque
control together with a PI stability control scheme and report significant improvement. Finally, recent studies focus more on advanced control schemes such as [68] where the authors
employ internal mode control for robustness.
Another mainstream in the subject deals with the coordination of the individual control
systems from an upper level controller point of view. These decentralized systems have been
developed under different names such as Integrated Chassis Control (ICC), Global Chassis
Control (GCC) or Unified Chassis Control (UCC). Trachtler [69] details such a system, so
called Vehicle Dynamics Management (VDM), which suggests an upper-level control strategy based on the yaw-rate feedback to handle the active braking, steering and suspension
systems. Another analysis is done by Hac and Bodie [70] where the authors investigate the
stability problem on a simplified vehicle model, the stability effects of small perturbations
in the tire-forces characteristics and of chassis control subsystems (lower lever controllers),
and eventually the performance of an upper level controller (UCC) that integrates an active brake and an active suspension system. Similar studies investigating the integration of
active braking and suspension systems are also done by Smakman [71, 72], and by Duda
and Berkner [73] where the authors focused more on reducing the activation of the braking
system to increase the ride comfort together with stability. In [74] and [75] Hwang et.al.
investigate the performance of an integrated active steering and active braking system. The
authors study two methods for integration, a supervisory algorithm which controls individual
systems and a so-called unified algorithm which requires combining the physical control units
onboard and they report very close performance results for both algorithms. Also in [76]
Chapter 3. Adaptive Vehicle Stability Control
43
and [77] Nagai et.al. analyze integrating an active braking system based on model matching
control method with active rear and front steering systems respectively. Selby et.al. in [78]
report the improvements that can be achieved by coordinating active steering with differential braking. A common conclusion of these studies states that despite the improvements
in the system performance one needs to be careful about the saturation regions of the individual controllers as the braking system might run the tires into saturation while the active
steering system still tries to stabilize the vehicle. A good analysis of these saturation regions
is done by He et.al. in [79] at the end of which the authors propose a metric to determine
the vehicle operating ranges and thereby the desired individual control system to employ.
Active steering and active suspension combination has drawn less interest than the other
two among research groups. An investigation of such an integration is done by March and
Shim in [80] where the authors focus on the handling ability of the system. Chen et.al. in
[81] study the system from a broader point of view and report improvements for ride comfort
as well as handling.
The integrated control schemes have an indisputable improvement over the individual controllers. Nevertheless adopting a decentralized integration methodology introduces additional complications such as saturation of one system disregarding the other(s) or integrating
a system that has indirect effect on stability (eg. suspension system). A solution to these
issues is found by deriving a centralized control algorithm that can command on multiple
controllers in a unified fashion. Due to its top-down design strategy, a centralized control
structure can be developed more comprehensively by taking all systems into account and
thus it is expected to provide better performance. The last mainstream to be discussed in
this study considers such stand-alone systems. An early investigation is done by Matsumato
and Tomizuka in [82] where the authors develop an optimal control law using linear quadratic
regulator (LQR) theory and report performance variations by switching between front and
rear wheel steering and traction control. Similar studies followed again using optimal control
techniques [83, 84] or robust control techniques such as Hinf analysis [85] or µ-synthesis [86].
An interesting subtitle under this mainstream is the multilayer structures. The systems
developed using such structure implements the control law which is still derived using unified approach through a lower level loop-control design. Although such a strategy reminds
the decentralized controllers, a major difference between them is that the lower level loopcontrol in this method that implements the upper-level control law is developed considering
the coupled dynamics of the actuators unlike the decentralized systems which commands the
Chapter 3. Adaptive Vehicle Stability Control
44
individual lower level controllers separately. Optimal control is the most popular technique
for lower level loop-control design. Good examples of such systems are given by Mokhiamar
and Abe in [87] where a sliding mode control (SMC) is proposed for upper level control,
by Hattori et.al. in [88] or by Shen et.al. in [89, 90, 91, 92] where again SMC strategy is
adopted for upper level control.
3.3
Motivation
A common need in the vehicle stability algorithms summarized above is to obtain information
about system states (e.g. lateral velocity, chassis side-slip) or axle forces, which are not
directly available from sensors on board a vehicle due to cost associated with them. Therefore
observer schemes are required to estimate these values which are in general based on chassis
kinematics and linear approximations to tire force and moment properties. Nevertheless,
due to the nonlinearities in tire characteristics and intrinsic deviations in estimation schemes
(such as integral errors, or deviations due to simplifications in system dynamics), the resulting
estimates might become overwhelmed with cumulating inaccuracies which yields the control
signals to deteriorate. Figure 3.3 shows results of an example simulation study designed for
lateral velocity estimation scheme. Given the double lane change maneuver in Fig. 3.3a, a
R
D-class sedan vehicle model is simulated using CarSim
software to obtain vehicle dynamics
parameters. Based on the bicycle vehicle mode summarized in Chapter 2, multiple observers
(Direct, Luenberger, Kalman) are designed and implemented to have a better judgment on
possible performance variations.
The results indicate that other than the direct estimation method which is taken as a reference model and is actually not real-time implementable [93], the common observers yield to
significant overshoots especially around the turn points during the given maneuver where the
vehicle approaches neutral steer characteristics. Similar issues develop in the estimation of
other vehicle states as well, such as vehicle side-slip or lateral axle forces. Various solutions
have been proposed to address this problem, mostly by integrating additional sensor inputs
into the estimation scheme such as inputs from inertial navigation system (INS) and/or
global positioning system (GPS) [94, 95], or inputs from optical sensors [96]. Nevertheless
these solutions come with costly sensor units and high computational power requirements;
and moreover they still employ indirect methods to estimate the parameters of interest.
Chapter 3. Adaptive Vehicle Stability Control
Sensor
Data
45
Plant
+
-
+
L
-
B
+
+

+
C
A
K
(a) Block diagram expression for Luenberger observer
Sensor
Data
+
Plant
-
Observer
Noise
+
+
K_g
B
+
+

+
+
-
C
A
K
(b) Block diagram expression for Kalman filter
Figure 3.2: Observer designs for lateral velocity estimation
This study proposes the use of a sensor fusion approach which integrates a smart tire system
with a model based observer to obtain information about tire slip-angle variations. The
smart tire system provides information about tire slip-angle by directly monitoring tire-road
contact which alleviates possible intrinsic estimation errors of the model based observer.
In what follows an adaptive stability control algorithm is developed based on the tire slipangles. Using the tire slip-angle as the parameter of interest provides two major advantages.
One is that the strong correlation between tire slip angle and vehicle drive characteristics as
summarized in Chapter 2 is utilized. The typical stability controllers are generally based on
a yaw-rate feedback which might retard the controller performance due to the intrinsic time
lag in the generation of yaw motion on the vehicle chassis. This could be better explained
by analyzing the transients of the system states during a lateral maneuver. Figure 3.4 shows
the results of the tire slip-angle response and chassis yaw-rate for a sinusoidal steering input.
The results depict a noticeable lag between the two responses. This phenomenon can be
explained by looking into the dynamics of lateral force at the tire-road contact patch. As
Chapter 3. Adaptive Vehicle Stability Control
46
Lateral Distance [m]
4
3
2
1
0
0
50
100
150
200
250
Longitudinal Distance [m]
(a) DLC Maneuver
1.5
1.5
Estimated
Actual
0.5
0
−0.5
−1
−1.5
−2
0
0.5
0
−0.5
−1
−1.5
2
4
6
8
Time [s]
(b) Direct estimation results
10
−2
0
Estimated
Actual
1
Lateral Velocity [km/h]
1
Lateral Velocity [km/h]
Lateral Velocity [km/h]
1
1.5
Estimated
Actual
0.5
0
−0.5
−1
−1.5
2
4
6
8
Time [s]
10
−2
0
2
4
6
8
10
Time [s]
(c) Kalman estimation results (d) Luenberger estimation results
Figure 3.3: Observer designs for lateral velocity estimation
summarized in [97] the tire slip-angle is initiated by the generation of the lateral tire force
after a given steering input, which is then followed by the generation of chassis side-slip and
yaw motions on the vehicle chassis. The other advantage is the presence of tire slip-angles
in the linearized vehicle lateral dynamics model that gives way to manipulate this model
to present the tire slip-angles as its states in the development of the control laws which is
directly based on the stability characteristics of a vehicle.
The proposed control algorithm follows the steps of the latter mainstream by integrating
Active Front Steering (AFS) and Direct Yaw Control (DYC) schemes. The smart tire system
provides successful dynamic tire slip-angle information in the linear region of tire force curve
and the model based observer meets the default by estimating the tire slip-angle in the
nonlinear region when necessary. With the addition of vehicle yaw rate and steering wheel
angle signals from chassis based sensors, the control algorithm computes corrective steering
and yaw moment values which are to be implemented through AFS and DYC units. To
compute these inputs, Lyapunov’s direct stability method is followed which carries out an
adaptation law along the way. As a result the control system becomes more robust against
variations in driving conditions such as surface friction. The steering input is implemented
the tire slip-angle response and chassis yaw-rate for a sinusoidal steering input. The results depict a noticeable lag
between the two responses. This phenomenon can be explained by looking into the dynamics of lateral force at the
tire-road contact patch. As summarized in [98] the tire slip-angle is initiated by the generation of the lateral tire force
after a given steering input, which is then followed by the generation of chassis side-slip and yaw motions on the
vehicle chassis. The other advantage is the presence of tire slip-angles in the linearized vehicle lateral dynamics
Chapter
3. Adaptive
Control
model that
gives way to Vehicle
manipulateStability
this model to
present the tire slip-angles as its states in the development of the 47
control laws which is directly based on the stability characteristics of a vehicle.
1
0.9
0.8
SWA
Yaw Rate
Tire SA
Normalized
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
Time [s]
Figure 3-4 Comparison of vehicle state response time
Figure 3.4: Comparison of vehicle state response time
The proposed control algorithm follows the steps of the latter mainstream by integrating Active Front Steering
(AFS) and Direct Yaw Control (DYC) schemes. The smart tire system provides successful dynamic tire slip-angle
as an auxiliary input to the drivers whereas for the DYC command a lower level optimal
32
force distribution algorithm is developed. Based on the corrective yaw moment, this lower
level control algorithm optimizes the wheel brakes forces to be implemented by utilizing
current tire force information from the developed model based observer. The rest of this
chapter goes into the details of the proposed algorithms for these tasks. Section 3.4 explains
the sensor fusion approach with smart tire based tire slip-angle estimation method and the
model based observer scheme. Sections 3.5 and 3.6 summarize the derivation of the control
laws as well as the results from the simulations carried out. The chapter is concluded by
discussing the results and benefits of the proposed systems.
3.4
Tire Slip-Angle Estimation
The sensor fusion approach followed for tire slip-angle estimation integrates the smart tire
system with a model based observer scheme that uses a Sliding Mode Observer (SMO) to
estimate tire forces and a Luenberger Observer to obtain tire slip-angles.
3.4.1
Smart Tire based Estimation
This section presents an implementation strategy for the dynamic estimation of the tire-slip
angle using the developed smart tire system prototype. The algorithm development for the
Chapter 3. Adaptive Vehicle Stability Control
48
smart tire system starts by logging and analyzing data from the system under different test
conditions. The analyses aim to find the synthetic parameters that are the most sensitive
to the variations in tire slip-angle values. Through extensive parametric studies, a strong
interdependence appeared between the magnitude of the lateral acceleration signal in time
domain and the tire slip-angle variations. The magnitude of the lateral acceleration signal
was characterized by estimating the signal power (absolute amplitude of the signal) on a
per revolution basis as depicted in Figure 3.5a. Another key parameter is identified as the
tire loading condition which basically defines the proportion between the signal power and
the rate of change of variation in the slip-angle values. The tire load information can also
be obtained from the smart tire system using the radial acceleration signals. Similar to
the tire slip-angle, the radial signals are correlated to the instantaneous vertical load by
computing the signal power which strongly correlates to the applied vertical load. Figure
3.5b indicates the variation in radial signal power under different load conditions and the
distinct boundaries demarcating the change in the instantaneous load.
Considering these two synthetic parameters, a line fitting algorithm is employed to estimate
the tire slip angles. The proposed method provides successful estimate of tire-slip angles
in the linear range of operation. Figure 3.6 shows the results of a sample application using
smart tire signals during a steer sweep motion. Despite this limitation the smart tire method
provides sufficient information for the proposed control algorithm because the algorithm, as
in most other controllers based on linearized mathematical models, aims to maintain the
system trajectory in the linear region. Nevertheless, to ensure robust performance and guarantee system convergence even during the vehicle maneuvers which show highly nonlinear
tire characteristics, a sensor fusion approach, combining the smart tire system with a model
based observer is used. The model based estimation methodology uses a tire force estimator
in conjunction with a Luenberger observer to make dynamics estimates of the tire slip-angle.
Details regarding the methodology adopted to estimate the tire slip angle using an observer
based approach are presented in the following section.
3.4.2
Model based Observer Derivation
The proposed observer scheme makes use of lateral and longitudinal force information obtained from SMO using a random walk model and utilizes a Luenberger observer to output
the tire slip-angle. As their dynamics differ, the derivations for longitudinal and lateral force
Chapter 3. Adaptive Vehicle Stability Control
49
(a) Lateral acceleration signal vs. slip-angle variations
(b) Radial acceleration signal vs. wheel load variations
Figure 3.5: Variation in the instantaneous amplitude of the lateral and radial acceleration
signal power regarding slip angle and wheel load
estimation are considered in separate subsections.
Longitudinal Force Estimation
The first approach to estimate the tire longitudinal force is based on a simplified wheel
dynamics model. The dynamic equation for the angular motion of the wheel is given as:
Jw ω̇i = Twi − Tbi − Fxi Rwi − Frri Rwi
(3.1)
Chapter 3. Adaptive Vehicle Stability Control
50
60
60% Rated Load
80% Rated Load
100% Rated Load
Normalized Signal Power [dB]
50
40
30
20
10
0
-4
-3
-2
-1
0
1
2
3
4
Slipangle [ ]
Figure 3-6
Variation
in instantaneous
amplitude of the lateral
acceleration
as function
of the tire-slip angle
underas
Figure
3.6:
Variation
in instantaneous
amplitude
of signal
the lateral
acceleration
signal
different loading conditions
function of the tire-slip angle under different loading conditions
3.4.2 Model Based Observer Derivation:
38B
where
subscript
i = f l,makes
f r, rl,
is used
to separately
representobtained
the four
ofa the
The the
proposed
observer scheme
use rr
of lateral
and longitudinal
force information
from wheels
SMO using
randomTwalk
model
a Luenberger
observer tothe
output
the tire
slip-angle.
their dynamics
differ,
vehicle.
Frri and
Fxi represent
drive
and
brake As
torque
delivered
totheeach
wi , T
bi , Rand
wi ,utilizes
derivations
for longitudinal
and lateral
force
considered
separate subsections.
specific
wheel,
the effective
radius
of estimation
the tire,arethe
rollinginresistance
force and the longitudinal
tire force of that specific wheel respectively. Rearranging equation 3.1 yields an expression
Longitudinal Force Estimator: The first approach to estimate the tire longitudinal force is based on a simplified
for the longitudinal force as:
wheel dynamics model. The dynamic equation for the angular motion of the wheel is given as:
F xi =
Twi − Tbi − Jw ω̇i
− Frri
Rwi
(3.1)
(3.2)
where
the
wheel drive, torque
can
be estimated
theofengine
torque,
wi to
where
the subscript
, , isT
used
separately
represent theusing
four wheels
the vehicle.
,
,the , engine
angular
and
theandwheel
angular
velocity
[98].
It is
assumed
thatradius
the brake
pressure
represent the
drive
brake torque
delivered
to each
specific
wheel,
the effective
of the tire,
the
and velocity,
of each
is an
signal. tire
Therefore,
brake
torque
Tbi can
be computed
by the
rollingwheel
resistance
forceavailable
and the longitudinal
force of thatthe
specific
wheel
respectively.
Rearranging
equation (3.1)
brake
gain
(Kbf , Kfor
wheelforce
rolling
yields
an expression
theThe
longitudinal
as: resistance force, Frri , is given by the expression:
br ).
Frri = 0.005 + 3.24 · 10−2 (Rwi ωi )2
(3.2)
(3.3)
Even though equation 3.2 presents a relatively simple (open-loop) method to estimate the
where the wheel drive torque
can be estimated using the engine torque, the engine angular velocity, and the
longitudinal
tire force, it is not
advisable to use this approach, since in realworld conditions,
wheel angular velocity [99]. It is assumed that the brake pressure of each wheel is an available signal. Therefore, the
finding
the time derivative of angular wheel speed signals (ω̇i ) can pose some challenges. To
brake torque
can be computed by the brake gain (
,
). The wheel rolling resistance force,
, is given by
avoid the need of taking derivatives, a sliding mode observer (SMO) based estimation scheme
the expression:
is proposed. Using the sliding mode structure, the state estimates (ω̂i ) evolve according to
the wheel dynamics model in eq. 3.1, the force model, Ḟxi = 0 (i.e. tire forces modeled35as a
MO) based estimation scheme is proposed. Using the sliding mode structure, the state estimates (̂
) evolve
0 (i.e. tire forces modeled as a random
cording to the wheel dynamics model (eq. (3.1)), the force model,
alk model) and the sign of the measurement estimation error (difference between actual (
) and estimated (̂
ngular wheel speed) as:
Chapter 3. Adaptive Vehicle Stability Control
)
51
random walk model) and the sign of the measurement estimation error (difference between
(3.4)
actual ωi and estimated ω̂i angular wheel speed) as:
Jw ω̂˙ i = (Twi − Tbi ) − F̂xi Rwi − Frri Rwi + k1 sgn(ωi − ω̂i )
˙
F̂xi = k2 sgn(ωi − ω̂i )
here
(3.4)
(3.5)
(3.5)
k1 and
k2 are gains
the observer
sgn(.)
denotes
signum function.
The(Figure
results 3-7)
(Fig-show that the
and where
are the
observer
and gains
. and
denotes
signum
function.
The results
ure 3.7) show that the estimated longitudinal force using the proposed observer formulated
timated longitudinal force using the proposed observer formulated using equations (3.4) and (3.5) ensures stable
using equations 3.4 and 3.5 ensures stable estimation of the tire longitudinal force.
5
0.06
Slip Ratio [ ]
2
Long Acceleration [m/s ]
timation of the tire longitudinal force.
0
-5
-10
0
5
10
0.04
0.02
0
-0.02
-0.04
0
5
Longitudinal Force (Fx fl) [N]
Time [sec]
10
Time [sec]
Estimator Performance
6000
4000
2000
0
-2000
-4000
0
Actual (CarSim)
Estimated (SMC Observer Based)
1
2
3
4
5
6
7
8
9
10
Time [sec]
Figure 3.7: Results for the proposed SMO scheme for longitudinal tire force estimation
Figure 3-7 Results for the proposed SMO scheme for longitudinal tire force estimation
ateral Force Estimator: The same SMO structure is utilized for lateral forces as well. A four wheel vehicle model
considered to derive the vehicle rigid body dynamics:
(3.6)
36
Chapter 3. Adaptive Vehicle Stability Control
52
Lateral Force Estimator:
The same SMO structure is utilized for lateral forces as well. A four wheel vehicle model is
considered to derive the vehicle rigid body dynamics:
Fxf l + Fxf r + Fxrl + Fxrr
+ rv
m
Fyf ront + Fyrear
v̇ =
− ru
m
Fxf r + Fxrr (tr /2) − Fxf l + Fxrl (tr /2) lf Fyf ront − lr Fyrear
ṙ =
+
Iz
Iz
u̇ =
(3.6)
(3.7)
(3.8)
The tire lateral forces are modeled as the sum of on the sides of the chassis for front and
rear axles (Fyf ront , Fyrear ) and again with a random walk model:
Ḟyf ront = 0
Ḟyrear = 0
(3.9)
The vehicle dynamics are described by the following state and measurement equations:
X = [x1 , x2 , x3 ] = [Fyf ront , Fyrear , r]
Y
= [y1 , y2 ] = [ay , r]
(3.10)
Vectors X̂ and Ŷ represent the state estimations and the measurement estimations where
the measurements are modeled as:
ŷ1 =
x̂1 + x̂2
,
m
ŷ2 = x̂2
(3.11)
where m is the vehicle mass. The estimation errors for states and measurements are denoted
respectively as:
ex = [x1 − x̂1 , x2 − x̂2 , x3 − x̂3 ]
ey = [y1 − ŷ1 , y2 − ŷ2 ]
(3.12)
Finally the resulting state estimates evolve according to the four-wheel vehicle model given in
eq.3.6-3.8, to the random walk force model given in eq. 3.9 and the sign of the measurement
Chapter 3. Adaptive Vehicle Stability Control
53
estimation errors, which can be written in open form as:
where
,
1
x̂˙ 1 = k11 sgn(ey1 ) + k12 sgn(ey2 )
˙x̂2 = k21 sgn(ey ) + k2
22 sgn(ey2 )
1
1
t
r
+ lf x̂2 − lr x̂1 + k31 sgn(ey2 )
x̂˙ 3 =
Fxf r + Fxrr − Fxf l + Fxrl
Iz 2
(3.13)
(3.13)
where
k21 , the
k22 ,observer
k31 are gains.
the observer
Fxij is theforce
longitudinal
estimate
,
, k11 , ,k12 , are
is gains.
the longitudinal
estimate force
for the
individual wheels
for the individual wheels obtained using the estimation scheme proposed above. A complete
obtained using the estimation scheme proposed above. A complete study for the convergence of the SMO is
study for the convergence of the SMO is presented in [99]. The performance of the observer
presented in [100].
The performance
of the
observer
waswhich
evaluated
for the
a sweep
steering
can run the tire
was evaluated
for a sweep
steering
input
can run
tire forces
intoinput
the which
saturation
forces into theregion.
saturation
region.
Figure 3-8
indicates
that the
observer
can the
cope
withextreme
the given
extreme maneuver
Figure
3.8 indicates
that
the observer
can
cope with
given
maneuver
and provide
robust
estimates
of thevalues.
lateral force values.
and provide robust
estimates
of the
lateral force
4
1
x 10
Total Lateral Tire Force - Front Axle
Actual
Estimated
[N ]
0.5
0
-0.5
-1
0
2
4
1
x 10
4
6
8
Actual
Estimated
0.5
[N ]
10
Time [S]
Total Lateral Tire Force - Rear Axle
0
-0.5
-1
0
2
4
6
8
10
Time [S]
Figure3.8:
3-8 Results
scheme
lateral
tiretire
force
estimation
Figure
Results for
forthe
theproposed
proposedSMO
SMO
schemeforfor
lateral
force
estimation
Luenberger Observer: To finally estimate the tire slip angle, an update equation for the front slip angle is derived as
a function of the tire forces. Using the kinematic relation for front and rear slip angles (eq. 2.11and 2.12) the update
equation for the front slip angle can be derived by taking their time derivatives.
(3.14)
Chapter 3. Adaptive Vehicle Stability Control
54
Luenberger Observer
To finally estimate the tire slip angle, an update equation for the front slip angle is derived
as a function of the tire forces. Using the kinematic relation for front and rear slip angles
(eq. 2.11 and 2.12) the update equation for the front slip angle can be derived by taking
their time derivatives.
v̇ + lf ṙ
u
lr ṙ − v̇
=
u
α̇f = δ̇ −
(3.14)
α̇r
(3.15)
Substituting expression for vy and r from equations (3.7) and (3.8) respectively into equation
(3.14) yields:
α̇f = δ̇ + r −
−
lf2
1
+
mu Iz u
(Fyf l + Fyf r ) −
lf lr
1
−
mu Iz u
tr lf (Fxf r + Fxrr ) − (Fxf l + Fxrl )
2Iz u
(Fyrl + Fyrr )
(3.16)
Thus the following equation is integrated to update αf :
α̂˙ f = δ̇ + r −
lf2
1
+
mu Iz u
(Fyf l + Fyf r ) −
tr lf (Fxf r + Fxrr ) − (Fxf l + Fxrl )
2Iz u
+k(may − Fyf l − Fyf r − Fyrl − Fyrr )
lf lr
1
−
mu Iz u
(Fyrl + Fyrr )
−
(3.17)
where k is the observer feedback gain and ay is the measured lateral acceleration. The update
law for the rear tire slip-angle αr is obtained using the same approach:
α̂˙ r
2
1
lr
1
lf lr
=
−
(Fyf l + Fyf r ) −
−
(Fyrl + Fyrr )
Iz u mu
Iz u mu
tr lf (Fxf r + Fxrr ) − (Fxf l + Fxrl ) − r
−
2Iz u
(3.18)
Using equations (3.18) and (3.18) tire slip-angle values are successfully estimated provided
that the tire forces are made available. Figure (3.9) shows the performance of the algorithm
under double lane change and fishhook maneuvers.
Chapter 3. Adaptive Vehicle Stability Control
55
Slipangle Estimator Performance
2.5
Slipangle Estimator Performance
Slipangle Estimator
Performance
Actual (CarSim)
Estimated (Luenberger observer based)
2.5
25
Actual (CarSim)
Estimated (Luenberger observer based)
2
2
25
20
1.5
Estimated (Luenberger observer based)
20
1.5
15
1
15
Tire
Slipangle[
Tire
Slipangle[
] ]
1
Tire Slipangle[  ]
Tire Slipangle[  ]
Slipangle Estimator
Performance
Actual
(CarSim)
Estimated (Luenberger observer based)
Actual (CarSim)
0.5
0.5
0
-0.5
0
-0.5
10
10
5
5
0
0
-5
-5
-1
-1
-10
-10
-1.5
-1.5
-2
0
-2
01
12
2 3
3 4
5 5 6
Time
[S] [S]
Time
4
6
7 7
8 8
99
-15
-15
0
0
1010
11
(a)(a)
Double
lane
change
maneuver
Double
lane
change
maneuver
(a)
Double
lane
change
maneuver
22
33
44
5 5
6 6
Time
Time
[S][S]
7
7
8
8 9
9 10
10
(b)
maneuver
(b)
Fishhook
maneuver
(b)Fishook
Fishhook
maneuver
Figure
3.9:3-9.
Tire
estimation
performance
Figure
3-9.
Tireslip-angle
slip-angle estimation
estimation
performance
Figure
Tire
slip-angle
performance
3.5
Tire-Slip
Angle
Based
StabilityControl:
Control:
3.5 T
ire-Slip
Angle
Based
Stability
18B
3.5
18B
Tire Slip-angle based Stability Control
The derivation of the control algorithm is based on the bicycle model summarized in Chapter 2. An auxiliary yaw
The derivation of the control algorithm is based on the bicycle model summarized in Chapter 2. An auxiliary yaw
moment is augmented into the equations of motion as a control input:
moment
is augmented
into the equations
of motion
as a control
input:
The
direct relationship
between
the tire-slip
and
vehicle lateral dynamics provides a good
2
2
(3.19)
2
2
measure of the possible improvements in stability
control.
This section details the derivation
(3.19)
2
2
(3.20)
of an adaptive chassis stability control
algorithm
2 based on direct Lyapunov method.
2
(3.20) As
This auxiliary input (
) represents the moment to be applied on the vehicle chassis for maintaining stability.
mentioned
the developed
control
benefits
andfor
DYC
schemes.
The
This auxiliary
inputbefore,
( ) represents
the moment
to system
be applied
on the from
vehicleAFS
chassis
maintaining
stability.
Defining the lateral dynamics, the kinematic definitions for tire-slip angles (eq. 2.11-2.12) are used to redefine the
control
laws dynamics,
make usethe
ofkinematic
the tire-slip
angle for
information
provided
by smart
The
Defining
the lateral
definitions
tire-slip angles
(eq. 2.11-2.12)
aretire
usedsystem.
to redefine
the
governing equations with the front and rear tire-slip angles as the states of the system:
derivation
ofwith
the the
control
algorithm
is based
thestates
bicycle
model
as summarized in Chapter
governing
equations
front and
rear tire-slip
angleson
as the
of the
system:
(3.21)
2. An auxiliary yaw moment is augmented into the equations of motion as a control
input:
(3.21)
(3.22)
m(v̇ + ur) = 2Cf αf + 2Cr αr
where
where
(3.19)
(3.22)
Iz ṙ = lf (2Cf αf ) − lr (2Cr αr ) + Mb
2
2
(3.20)
2
2
The auxiliary input (Mb ) represents the moment applied on the vehicle chassis for maintain2
ing stability. Based on the tire slip-angle dynamics as defined 2in equations 3.14-3.15) the
2
2
governing equations
of the conventional bicycle model can be redefined
with the front and
The steering term
in equations (3.21) and (3.22) accounts for the sum of the driver steering input and the
additional input from the AFS controller:
The steering term
in equations (3.21) and (3.22) accounts for the sum of the driver steering input and the
additional input from the AFS controller:
Next manipulating these governing equations and rewriting them in the state-space form yields:
0
Next manipulating these governing
0 equations and rewriting them in the state-space form yields:
0
0
(3.23)
(3.23)40
Chapter 3. Adaptive Vehicle Stability Control
56
rear tire-slip angles as the states of the system:
Iz α˙f = −(a11 )αf − (a12 )αr + Iz (δ̇ + r) − (lf /u)Mb
(3.21)
Iz α˙r = −(a21 )αf − (a22 )αr + Iz r + (lr /u)Mb
(3.22)
where
a11
a21
2Cf (Iz + lf2 m)
=
mu
2Cf (Iz − lf lr m)
=
mu
a12
a22
2Cr (Iz − lf lr m)
=
mu
2Cr (Iz + lr2 m)
=
mu
The steering term δ in equations (3.21) and (3.22) accounts for the sum of the driver steering
input and the additional input from the AFS controller:
δ = δd + δc
Next manipulating these governing equations and rewriting them in the state-space form
yields:











 U1




 Iz 0   α˙f   a11 a12   αf   Iz (δ̇d + r) 


+

−
=


 

 
 


 U2 

0 Iz
α˙r
a21 a22
αr
Iz r
(3.23)
where U1 and U2 represent the control law:



 U1





 Iz
=

 

 U2 

0
Choosing the state vector x = [αf


 δ̇c
−(lf /u)  



(lr /u)  Mb







αr ]T , the system can be rewritten in short form as
Aẋ + Bx + C = U . Following that a candidate Lyapunov function (eq 3.24) is defined
based on the system dynamics (A and B) and by including a set of adaptation parameters.
A major advantage in using the Lyapunov method is that it allows computing control law
for guaranteed asymptotic stability without the need for the analytical solutions of system
Chapter 3. Adaptive Vehicle Stability Control
equations.
1 T
V (x, t) =
x̃ Ax̃ + p̃T Γp̃ +
2
57
Z
x̃T B x̃dt
(3.24)
where Γ and p define the adaptation law which ensures system stability in case of parameter variations. In this study the adaptation parameters are selected as the front and rear
cornering stiffness values (Cf and Cr ) to be able to adapt the controller to the variations
in the road surface conditions. As the Lyapunov’s criteria state [100], being able to define
a positive definite candidate function and demonstrating that its rate of change in time always decreases (negative definite) would allow one to comment about system’s stability. The
positive definiteness of equation 3.24 can be guaranteed by selecting a positive diagonal Γ
matrix. In what follows differentiating the candidate function gives:
V̇ (x, t) = x̃T Ax̃˙ + p̃T Γp̃˙ + x̃T B x̃
(3.25)
where x̃ = x − xd defines the error. Expanding the terms in equation (3.25) and using the
symmetry of matrices A and Γ it can be rewritten as:
V̇ (x, t) = x̃T (−Bx − C − U − Aẋd + Bx − Bxd ) + p̃T Γp̃˙
(3.26)
where the control law can be defined as U = Âẋd + B̂xd + Ĉ − Dx̃ with D as the control
gain andˆdenoting the estimated values using the adaptation law which is defined by further
simplification of equation (3.26). Letting the general form of the adaptation law defined by:
H p̃ = Ãẋd + B̃xd + C̃
where p = [C̃f
(3.27)
C̃r ]T and substituting into equation 3.26 yieds:
V̇ (x, t) = −x̃T Dx̃ + p̃T Γp̃˙ + H T x̃
(3.28)
Asymptotic stability of the system is ensured by satisfying V̇ (x, t) < 0 which can be realized
by selecting a positive definite controller gain matrix D and zeroing the terms inside the
parentheses which also define the adaptation parameters as:
p̃˙ = −Γ−1 H T x̃
Z
p = p0 + (−Γ−1 H T x̃)dt
(3.29)
Chapter 3. Adaptive Vehicle Stability Control
58
Finding out the form of the adaptation parameters, the control laws can finally be written:
Iz α̇rd + â21 αfd + â22 αrd − Iz ψ̇ − D22 (αr − αrd ) (u/lr )
= Iz α̇fd + â11 αfd + â12 αrd − Iz (δ̇d + ψ̇) − D11 (αf − αfd ) (1/Iz )
Mb =
δ̇c
(3.30)
(3.31)
+(lf /Iz u)Mb
The desired slip-angle values (αfd and αrd ) are derived by using the definitions for steady
state steering to be able to negotiate a turn with radius R and for the lateral force acting
on the front and rear axles during the turn:
δss
F yf
L
mf
mr u2
=
+
−
R
2Cf
2Cr R
2
= mf u /R = 2Cf αf
Fyr = mr u2 /R = 2Cr αr
(3.32)
(3.33)
where mf = lr m/L and mr = lf m/L. Using equations 3.32 and 3.33 the desired slip-angle
values can be written as:
αfd
mf δu2
=
mr
mf
−
2Cf L +
u2
2Cf
2Cr
αrd
mr δu2
=
mr
mf
−
2Cr L +
u2
2Cf
2Cr
(3.34)
The control laws in the above equations can be implemented in various ways. The steering
input is relatively easier as an Active Front Steering (AFS) mechanism can be assumed
available so that the steering can be simply added to the driver’s input. Whereas for the
yaw moment signal a full lower level control algorithm is needed since there is no direct
method to implement the resulting value above. This problem of implementing a correcting
yaw moment on the vehicle chassis is generally known as Direct Yaw Control (DYC). The
DYC can be implemented through different methods on the vehicle chassis (e.g. torque
distribution, differential braking). In this study a differential braking method is utilized
which administers individual brakes to generate the required yaw moment on the chassis.
Initially, a rule based algorithm is developed to select appropriate wheel(s) to brake with
respect to the vehicle’s momentary understeer/oversteer characteristics and the direction of
turn. The decisions about these conditions are made by monitoring the desired (eq. 3.34)
and actual slip angle values and the direction of the turn. Table 3.1 summarizes the rules
Chapter 3. Adaptive Vehicle Stability Control
59
used for such a differential braking method. Finding which wheel to brake, the required
brake torque is simply calculated by using:
Tb =
Mb Rw
tr /2
(3.35)
Table 3.1: Brake rules for the DYC implementation
3.6
αactual
αactual − αdesired
Mb
Wheel to brake
-
actual > desired
+
FR
-
actual > desired
-
RL
-
actual < desired
+
-
actual < desired
-
+
actual < desired
+
RR
+
actual < desired
-
FL
+
actual > desired
+
+
actual > desired
-
System Validation using Simulation
The developed control algorithm is implemented and evaluated using numerical analysis.
The algorithm is implemented in the Matlab/Simulink environment and the vehicle dynamics are simulated using the CarSim commercial software. A D-class sedan vehicle model
is downloaded into Simulink environment and integrated with the control algorithm. The
system is then evaluated under an evasive double lane change (DLC) maneuver at a given
initial speed which is kept constant by the driver model.
The desired path for the evasive DLC is selected as given in Figure 3.3a, the system is first
tested on a high-friction (µ = 0.8) surface with an initial speed of 30m/s. The given speed
easily runs an uncontrolled vehicle unstable, whereas the controlled vehicle seems to cope
with it and successfully negotiates the turn. Figure 3.10a and 3.10b represent the AFS and
Chapter 3. Adaptive Vehicle Stability Control
60
DYC commands computed using the Lyapunov’s direct stability method. Compared with
the driver’s steering on Figure 3.10a, the AFS command seems in reverse which suggests
that it applies a corrective counter-steer which cannot easily be done by the inexperienced
daily-driver but is achieved through AFS without disturbing the driver. Figure 3.11a shows
the trajectory of vehicle’s CG and Figure 3.11b shows the yaw rate of the vehicle, which
also comply with the above inference that the control algorithm can successfully stabilize
the vehicle under these challenging conditions. Shown in Figure 3.12 are the traction performances for the front and rear tire-slip angle values compared to the desired values as in
equation 3.34. Slight variations in the rear slip-angles can be accounted by the time lag in
the slip-angle generation at the rear tires as summarized in section 3.3. Another point to
note is that the controller maintains the slip-angle values in the linear range of the tire forcemoment curve where the smart tire is capable of providing successful slip-angle estimations.
The results also indicate that the controller helps to maintain an understeer behavior during
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0
0
-0.5
-1
-1
-1.5
-1.5
2
-2
0
4
2
6
Time [s]
4
8
6
10 8
x 10
4
2
x 10
4
DYC command
DYC command
Steering command
Steering command
Driver input Driver input
0.5
-0.5
-2
0
2
10
1.5
1.5
1
1
Mb [Nm]
3
Mb [Nm]
3
c [deg]
c [deg]
the turn as desired.
0.5
0
0.5
0
-0.5
-0.5
-1
-1
-1.5
0
-1.5
2 0
4
2
6
Time [s]
Time [s]
(a) Steering
( ) input,
(a)input,
Steering
(a)
Steering
input( )
4
8
6
10 8
10
Time [s]
(b)Corrective
Corrective
moment,
)
(b)yaw
Corrective
(
(b)
moment( moment,
input
)
Figure 3-11.Figure
Control
signals
computed
the algorithm
high- surface
3-11.
Control
signals by
computed
by theon
algorithm
on high- surface
6
6
4
2
0
-2
-4
4
2
0
-2
-4
60
60
Desired path
Desired
path
Controlled
Figure 3.13 presents the wheel and
vehicle
speed
variations and the applied brake torque
Controlled
Controlled
Uncontrolled
40
Uncontrolled
40
Uncontrolled
values during the turn to help evaluating the lower level (optimal force allocation) controller
20
20
0
0
performance.
-20
-40
-60
Yaw rate [deg/s]
8
Yaw rate [deg/s]
8
ateral distance of the path [m]
ateral distance of the path [m]
Figure 3.10: Control signals for high-µ condition
-20
-40
-60
Controlled
Uncontrolled
-0.5
-0.5
-0.5-0.5
-1 -1
-1 -1
-1.5
-1.5
-2 -2
0 0
2 2
4 4
6 6
8 8
-1.5-1.5
0 0
10 10
2 2
4 4
6 6
8 8
10 10
Time
Time
[s] [s]
Time
Time
[s][s]
Steering
input,
() )
(a)(a)
Steering
input,
( Stability
Chapter 3. Adaptive
Vehicle
Control
Corrective
moment,
(b)(b)
Corrective
moment,
( ( ) )
61
6
6
4
4
2
2
0
0
60 60
Desired
Desired
path path
Controlled
Controlled
Uncontrolled
Uncontrolled
20 20
-2 -2
-4 -4
0
0
-20 -20
-40 -40
-60 -60
-6 -6
-80 -80
-8 -8
-100-100
-10 -10
0 0
Controlled
Controlled
Uncontrolled
Uncontrolled
40 40
Yaw rate [deg/s]
8
Yaw rate [deg/s]
8
Lateral distance of the path [m]
Lateral distance of the path [m]
Figure
3-11.
Control
signals
computed
algorithm
high-surface
surface
Figure
3-11.
Control
signals
computed
byby
thethe
algorithm
onon
high-
50 50
100100
150150
200200
250 250
-120-120
0 0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10 10
Time
Time
[s] [s]
Travel
distance
along
Travel
distance
along
the the
pathpath
[m] [m]
(a)(a)
Trajectory
followed
byby
vehicle
CGCG
Trajectory
followed
vehicle
(a)
Vehicle
CG
path
(b)(b)
Yaw
rate
Yaw
rate
(b) Chassis
yaw-rate
response
Figure
3-12
. Performance
of of
a controlled
and
uncontrolled
vehicle
negotiating
thethe
DLC
maneuver
onon
highFigure
3-12
. Performance
a controlled
and
uncontrolled
vehicle
negotiating
DLC
maneuver
high-surface
surface
Figure 3.11: Vehicle response comparison on high-µ condition
Figure
3-14
presents
thethe
wheel
and
vehicle
speed
variations
and
thethe
applied
brake
torque
values
during
thethe
turn
to to
Figure
3-14
presents
wheel
and
vehicle
speed
variations
and
applied
brake
torque
values
during
turn
help
evaluating
thethe
lower
level
(optimal
force
allocation)
controller
performance.
help
evaluating
lower
level
(optimal
force
allocation)
controller
performance.
αf [deg]
5
Desired
Controlled
Uncontrolled
0
−5
0
5
αr [deg]
1
2
3
4
5
6
7
8
9
10
48 48
Desired
Controlled
Uncontrolled
0
−5
0
1
2
3
4
5
6
7
8
9
10
Time [s]
Figure 3.12: Front and rear tire-slip angle values for the controlled and uncontrolled
vehicles compared to the desired values
Finally Figure 3.13 shows the variations of the adaptation laws for the front and rear axle
cornering stiffness values. Also shown are the design and nominal values. The design value
corresponds to the cornering stiffness of the tire under static load and on a unity friction
Chapter 3. Adaptive Vehicle Stability Control
62
surface whereas the nominal value varies with the surface friction. The front and rear axle
design values are taken as 120, 000N/rad and for the high-friction surface (µ = 0.8) the
nominal value is expected to become 80% of the design value. As can be deduced from
equation 3.29 the adaptation law is directly related to the integration of the error between
the desired and actual slip angles. As a result, the front axle value in Figure 3.14 converges
to the design value due to the minimal error in the front slip angle whereas the rear axle
value converges to the nominal value.
120
2000
Front left
Front right
Rear left
Rear right
1800
110
Brake Torque [Nm]
Velocity [kph]
1600
100
90
80
front left wheel
front right wheel
rear left wheel
rear right wheel
vehicle
70
60
0
2
4
6
8
1400
1200
1000
800
600
400
200
10
0
0
2
Time [s]
(a)
4
6
8
10
Time [sec]
(b)
Figure 3.13: Vehicle and wheel speeds
In what follows the system is tested on a low friction surface (µ = 0.25). Figure 3.15a shows
the CG trajectory at a reduced initial speed (11m/s) and indicates that the controlled vehicle
again can initiate the turn whereas the uncontrolled vehicle goes unstable very early on.
Figure 3.16 represents the controller commands for AFS and DYC systems during the low
friction maneuver. The AFS control implements higher reverse steering which is again very
significant in stabilizing the vehicle. Due to the lower force capacity from the tires on a low
friction surface, the brake system would not be as effective, hence the lower magnitudes in
the DYC command.
Figure 3.17 is also good indicator for the performance of the controller on the low friction
surface. The desired front and rear tire-slip values are tracked satisfactorily and the variations
of the wheel and vehicle speeds are still in a quite reasonable range. Finally, shown in Figure
3.19 are the adaptation values of the front and rear axle cornering stiffness’. Both converge
very close to the nominal value of 26, 000N m, which is about 20% of the design value,
Chapter 3. Adaptive Vehicle Stability Control
2
x 10
5
Design Value
Nominal Value
1.5
Cf [N/rad]
63
1
0.5
0
0
x 10
1
3
4
5
6
7
8
9
10
5
6
7
8
9
10
4
Design Value
15
Cr [N/rad]
2
Nominal Value
10
5
0
0
1
2
3
4
Time [s]
Figure 3.14: Adaptation of the front and rear axle cornering stiffness values
8
Desired path
Controlled
Uncontrolled
Controlled
Uncontrolled
80
60
4
2
Yaw rate [deg/s]
Lateral distance of the path [m]
6
0
-2
-4
-6
40
20
0
-20
-8
-40
-10
0
50
100
150
200
250
0
5
(a) Trajectory followed by vehicle’s CG
10
15
20
25
Time [s]
Travel distance along the path [m]
(b) Yaw rate compared for controlled and
uncontrolled systems
Figure 3.15: Low-µ surface test results
120, 000N m.
To conclude the simulation efforts, a comparison study is conducted to have a better judgment on the performance of the proposed algorithm. Another stability control algorithm
based on vehicle yaw-rate feedback is implemented in the same environment and executed
Chapter 3. Adaptive Vehicle Stability Control
64
10
8000
Steering command
Driver input
8
DYC command
6000
6
4000
4
M b [Nm]
 [deg]
2000
2
0
0
-2000
-2
-4000
-4
-6000
-6
-8
0
5
10
15
20
-8000
0
25
5
10
15
20
25
Time [s]
Time [s]
(a) Steering input (δc )
(b) Corrective yaw moment (Mb )
Figure 3.16: Control signals for low-µ surface testing
6
Desired
Controlled
Uncontrolled
 f [deg]
4
2
0
-2
-4
-6
0
5
10
15
20
25
15
20
25
6
Desired
Controlled
Uncontrolled
 r [deg]
4
2
0
-2
-4
-6
0
5
10
Time [s]
Figure 3.17: Front and rear tire-slip angle values for the controlled and uncontrolled
vehicles compared to the ideal values
the same DLC maneuver. Figure 3.20 shows the comparison of the resulting control laws
Chapter 3. Adaptive Vehicle Stability Control
65
45
40
35
Velocity [kph]
30
25
20
15
front left wheel
front right wheel
rear left wheel
rear right wheel
vehicle
10
5
0
0
5
10
15
20
25
Time [s]
Figure 3.18: Wheel and vehicle velocity variation during low friction surface test
2
x 10
5
Design Value
Cf [N/rad]
1.5
1
Nominal Value
0.5
0
0
15
x 10
5
10
15
20
25
4
Cr [N/rad]
Design Value
10
Nominal Value
5
0
0
5
10
15
20
25
Time [s]
Figure 3.19: Adaptation of the front and rear axle cornering stiffness values during low
friction surface test
for each algorithm. As the results indicate, the slip-angle based control algorithm yields to
a considerable reduction in the control signals, which eventually impose less loads on the
actuators and lower levels of interference with the driver’s inputs. In addition, Figure 3.20b
Chapter 3. Adaptive Vehicle Stability Control
66
indicates that the proposed algorithm results in lower levels of desired yaw moment values,
as a result lower braking action is required.
4
0.04
1.5
x 10
SlipAngle based
YawRate based
0.03
SlipAngle based
YawRate based
1
DYC signal [Nm]
Steering [rad]
0.02
0.01
0
-0.01
0.5
0
-0.5
-0.02
-1
-0.03
-0.04
0
2
4
6
8
Time [sec]
(a) Steering input (δc ) comparison
10
-1.5
0
2
4
6
8
10
Time [sec]
(b) Corrective yaw moment (Mb ) comparison
Figure 3.20: Comparison of control efforts for slip-angle and yaw rate based control
algorithms
3.7
Conclusion
This chapter details the derivation of an adaptive vehicle stability control algorithm based
on tire slip-angle. The first part summarizes the implemented tire slip-angle estimation
methodology that utilizes the so-called sensor fusion approach. Integrating the smart tire
technology with a model based observer scheme, the method provides robust tire slip-angle
information in both linear and saturation regions of the tire force-slip curve. In what follows,
the derivation of the control algorithm is presented. The resulting control laws are implemented by integrating active front steering (AFS) and direct yaw control (DYC) methods
to intervene with the driver’s inputs. The DYC part is implemented by designing another
lower level control algorithm that optimally distributes the wheel brake forces.
The algorithms are evaluated using numerical analysis in Matlab/Simulink environment
and using vehicle model from CarSim software. The results of simulation studies indicate
that the algorithm successfully intervenes with the steering and braking mechanisms and stabilize the system during the given evasive lane change maneuver. Choosing the adaptation
Chapter 3. Adaptive Vehicle Stability Control
67
parameters as the tire cornering stiffness, the algorithm is also capable of running on a lower
friction surface condition. A comparative study provides a better measure of improvements
and advantages using the proposed algorithm. Same simulations executed for the yaw-rate
based control algorithm indicate that proposed algorithm impose less oscillations on the AFS
system, which also yields to lower levels of interference with the driver inputs. Similarly the
proposed algorithm requires lower levels of desired yaw moment values, which leads to fewer
loads on the brake/traction dynamics. Another merit of this algorithm is that it eliminates
the need for the lateral velocity variable which would require a state estimator and would
introduce more uncertainty into the system.
Chapter 4
Advanced Anti-lock Braking
4.1
Introduction
Conventional anti-lock braking systems (ABS) provide vehicles the ability to achieve shorter
stopping distance and also mediate to maintain directional control and stability. The main
objective of the ABS algorithm is to maximize the tire longitudinal traction by preventing
the wheels from being locked during braking. This also helps to maintain control on the
steering as the tires do not pass the limit for saturation and allows steering input to remain
effective. The generic ABS control approach is based on wheel slip and wheel angular acceleration control [101, 102]. The principle of the operation in a typical ABS is simply formed on
limited cycling of longitudinal wheel slip in a desired range [103]. The algorithm sets certain
bounds for wheel angular acceleration and wheel angular speed and uses a complex rule set
to decide for the pressure mode of the actuator. Nevertheless, these conventional ABSs do
not take the varying tire force-moment characteristics on different types of road surfaces into
consideration, which naturally have their own appropriate operating conditions. The forces
and moments generated at the tire-road contact patch are altered by driver input based on
various handling maneuvers. Vehicle dynamic behavior is primarily controlled through three
driver inputs, and these driver inputs indirectly control the vehicle motion by affecting the
tire forces. The forces experienced through the tires are actually the primary forces affecting
vehicle handling dynamics, and the magnitude of these forces that can be transferred from
the tire to the driving surface are limited by the contact patch area, the vertical load on the
tire and the coefficient of friction between the tire and road surface. Moreover, as stated by
68
Chapter 4. Advanced Anti-lock Braking
69
the friction ellipse theory [104], the total friction force acting between the tire and the road
cannot exceed the maximum value determined by the friction coefficient (µ) and the vertical
load Fz . A decrease in the lateral force is expected as the longitudinal force increases, and
vice versa. This ultimately leads to the conclusion of inefficient braking performance as well
as unexpected consequences when the vehicle encounters a road surface that is not defined
in the ABSs default settings.
Therefore, knowledge about the road surface friction condition is essential for propelling
new designs in vehicle control systems. On the other hand, lack of knowledge about friction leads to developing over-conservative design rules which might result in reduced driving
performance. In normal driving conditions the frictional force is not fully utilized and the
developed tire force will be somewhere in the interior of the friction circle. With the force
generated at the tire-road contact, a relative motion arises between the tire structure and
the road surface, referred as the tire slip The relation between the resulting tire force and
this slip motion depends on many factors, namely, tire inflation pressure, vertical load, tire
wear, temperature, etc., and also contains information about the maximum surface friction
condition. When the tire is exposed to excitation with high utilization, beyond the point
corresponding to the maximum available friction force, the tire is sliding and the resulting
tire force directly corresponds to the friction coefficient. This interdependence allows for
estimation of the surface friction condition using the smart tire technology, which is then to
be utilized in improvement of the ABS logic to adapt itself to the most appropriate operating
points for the corresponding surface condition.
The goal of this chapter is to present an implementation strategy for a new adaptive ABS
control algorithm that makes use of a surface friction estimation algorithm to adapt itself
to possible variations in friction condition. The estimation algorithm integrates two methods for satisfactory results: a smart tire based method with a model based observer. The
smart tire based estimation uses a method that characterizes the terrain using the measured
frequency response of the tire vibrations whereas the model based observer makes use of a
Brush model together with tire force estimations to obtain the maximum surface friction. In
what follows, a new ABS algorithm is developed that can adapt an initially defined rule-set
to the variations on the surface friction conditions. As mentioned earlier the conventional
ABS algorithms define multiple rule-sets to cope with such variations. The proposed algorithm allows simplifying the rule-set by only defining rules for the applied brake torque and
Chapter 4. Advanced Anti-lock Braking
70
wheel slip in advance and adapting them dynamically to the varying surface conditions.
4.2
Background
The ABS is one of the earliest developed active safety systems together with the traction
controllers. As mentioned earlier, ABS is originally developed to prevent wheels from locking
up for better braking and stability performance. Modern ABSs, on the other hand, also strive
to maximize the wheel brake forces by controlling the tire slip ratio to operate around an
optimum value, which varies w.r.t. the road surface characteristics as depicted in Figure 4.1.
Friction coefficient as a function of slip ratio for different surfaces
1.4
Dry asphalt
Wet asphalt
Dry concrete
Cobble wet
Cobble dry
Snow
Ice
Coefficient of Friction [  ]
1.2
1
0.8
0.6
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Slip Ratio [ ]
Figure 4.1: Tire force-moment characteristics as a function of road surface conditions
The basic function of an ABS is to either hold or release the braking pressure on the wheels
if there is a danger of the wheels locking. At the same time, the ABS needs to re-permit application of the brakes again once the danger of locking has been averted. It could also hold
or release the braking pressure; all in order to keep the slip ratio at the wheel from exceeding the given optimum value. Depending on the number of wheels the ABS is expected to
control, it can be implemented with four channels four sensors, three channels three sensors
or one channel one sensor. Each of these channels is to be controlled by a solenoid valve,
and depending on the state of these valves, brake pressure on the wheel can be held even,
released or increased. Open valve allows the brake to be controlled by the driver by allowing
the amount of brake pressure desired by the driver to be applied to the brake. Closing the
Chapter 4. Advanced Anti-lock Braking
71
valve isolates that brake from the master cylinder which yields to holding the brake pressure
and preventing it from increasing even if the driver pushes the brake pedal harder. Finally
when the valve is in the release position, the pressure from the brake is released. In this
position, not only is the brake isolated from any further braking actions of the driver, but
the amount of braking pressure on the wheel is actively reduced. These three states allow
adjusting the amount of brake pressure desired by the driver to be applied to the brake by
avoiding any wheel lock-up situation.
ABS algorithms typically utilize wheel speed measurements to predict if the wheels will
lock and if the danger of locking has been averted. The process of determining whether or
not the wheel is going to lock is called prediction. Prediction point slip is defined as the
wheel slip at the instant the control unit predicts for the first time in a brake cycle that the
wheel is going to lock. The process of determining whether or not the danger of locking has
been averted is called reselection. Reselection point slip is defined as the wheel slip at the
instant it is predicted for the first time in a brake cycle that the danger of locking is averted.
A number of other factors influence the working of an ABS in addition to surface friction. One major factor is the rate of the brake torque application which depends on the
brake pedal action during the first cycle, and depends on the pressure build characteristics
of the modulator in the subsequent cycles. Another factor is the initial longitudinal velocity
of the vehicle as it determines how quickly the vehicle can come to a stop. The brake effort
distribution from front to rear also makes a significant difference. Reul et.al. in [105] present
results of an analysis that summarizes two major methods for the improvement of vehicle
brake systems in general. One method relates to maximization of the available surface friction which requires adjusting the tire and surface physical properties and is not discussed in
this research. After quoting the strong dependence between slip oscillations and brake performance [106], the second method proposes controlling the oscillations of the operating slip
point for improved braking. Two synthetic parameters surface to accomplish this task; the
tire load which helps to maximize the tire’s road holding capacity and the wheel brake force
which aims to maximize the tire grip level and which is the focus of this study. Numerous
logic based ABS control systems have been developed and reported in literature to address
the use of these methods and to improve brake performance in the presence of the above
variations.
Chapter 4. Advanced Anti-lock Braking
72
A pioneering and widely cited study for the application of anti-lock brakes in automobiles
is by Guntur and Ouwerkerk [107, 108], which contain a good discussion of various implementation strategies with different prediction and reselection rule sets. Other more recent
studies work on different control methodologies and other benefits from ABS. In [109], Taheri
studies performance of a new sliding mode based nonlinear control algorithm integrated with
four wheel steering. Yeh et.al. in [110, 111] introduce a four-phase ABS control algorithm
with the addition of a high and low pressure hold states. They also investigate an analytical
derivation method for a stability guaranteed rule-set in [112]. Sliding mode control (SMC)
constitutes a large portion in ABS algorithm studies especially due to its robust nature
against uncertainties in the brake hydraulics. Another widely cited study that uses SMC is
by Drakunov et.al. [113] where the authors take a simplified brake hydraulics model as well
as the uncertainties in the optimal slip value corresponding to the current road surface into
account. A similar study that uses a SMC scheme is by Unsal and Kachroo [114] where the
authors study a sliding mode observer (SMO) scheme and develop a real-time implementable
control law based on slip ratio estimations from the SMO. Another mainstream in the development of brake controllers is the development and use of the LuGre tire model which allows
accounting for the tire transient characteristics. A number of studies have been conducted
specifically under the California PATH program [115, 116, 117], where the authors investigated the integration of LuGre model in brake dynamics and proposed control algorithms
again using a SMC scheme. Due to its coherent nature with rule-based control, Fuzzy logic
algorithms constitute another large part in this research. In [118] Keshmiri et.al. study
the performance of fuzzy based controllers on varying road surface conditions and conclude
that fuzzy logic by itself does not always provide best braking action. Habibi and Shahri
propose the integration of SMC and Fuzzy logic in [119] to overcome this problem. More
recent studies in the subject take advantage of the advancements in the onboard electronics
and propose using control algorithms of higher complexity. Petersen et.al. study a gain
scheduling scheme based on slip optimization in [120, 121]. A robust control algorithm that
uses linear matrix inequalities (LMI) scheme to gain schedule the brake torque is proposed
by Baslamisli et.al. in [122]. Finally Dae Keun et.al study an implementation strategy for
model predictive control scheme based on nonlinear brake dynamics in [123]. A common
assumption in these recent studies is the availability of an electronic brake system (brakeby-wire) replacing the hydraulic brakes that might yield to crucial time lags.
A common practical problem in the above summarized ABS systems is the availability of
Chapter 4. Advanced Anti-lock Braking
73
wheel slip information which cannot be measured with any feasible sensor system available
on current vehicles in the market. Often the only measurements available to the ABS system
are measurements of the individual wheel speeds at the four wheels.
4.3
Estimation of Surface Friction Condition
This section provides a brief overview of the friction estimation method utilized for in the
development of the proposed ABS algorithm. A so called sensor fusion approach is followed
which basically integrates a model based observer scheme with the smart tire based estimations of the surface friction condition. An integral approach is followed due to the argument
that each method provides robust results at different levels of excitation on the tire. The
level of excitation here basically refers to the range of slip ratio the tire experiences. The
smart tire is capable of providing successful estimations in the linear slip range whereas the
model based observer can cope with the nonlinear dynamics if the tire is driven into the
saturation region. The summary of each method is given below in substantial details.
4.3.1
Smart Tire Based Surface Classification
The smart tire based approach characterizes the road surface using the frequency response of
the tire vibrations logged using the prototype system. The effect of various test conditions
(tire load, translational speed, tire pressure) on the tire vibration spectra are studied by
varying each of these parameters during testing of the instrumented tire. More details of
the procedure are provided in [35]. The power spectrum of each accelerometer signal from
these tests is computed using Welchs averaged modified periodogram method [124]. The
results as shown in Figure 4.2 indicate a marked difference especially in the concentration
of the higher frequencies on the spectrum of the circumferential acceleration signals. These
variations present an opportunity to characterize the road condition using the tire vibration
pattern information.
Another significant demarcation is seen in the level variations in signal power between the
pre-trailing and post-trailing domains of the signal. Due to the presence of more prominent
variations, the analysis is pursued by only using the pre-trailing portion of the vibration
signals. In what follows, the PSD content of the signal is executed through band-pass filters
Chapter 4. Advanced Anti-lock Braking
(a) Circumferential Acceleration Signal
74
(b) Power spectrum density (PSD)
Figure 4.2: Frequency response analysis of the smart tire signals
to distinguish between the low frequency (e.g. 10 − 500Hz band) and high frequency (e.g.
600Hz to 2500Hz) contents.
(a) Pre-trailing domain PSD
(b) Post-trailing domain PSD
Figure 4.3: PSD analysis of pre-trailing and post-trailing signal domains
After filtering the frequency response of the smart tire signal, the change in the vibration
level ratio (R) between the two frequency contents is measured. It is evident from results
that the vibration level ratio (R) increases when the tire was tested on the wet road surface
relative to when the tire was tested on the dry road surface. This change in the vibration level
Chapter 4. Advanced Anti-lock Braking
75
ratio can be attributed to the increased slippage of the tire, and thus it has been confirmed
that the slipperiness of a road surface can be decided by setting a proper threshold value.
For this purpose, a fuzzy logic classification approach [35, 125] is utilized to classify among
different surface conditions w.r.t. the given vibration ratios. Based on the interdependence
of the given test conditions (tire load, translational speed, and tire pressure) and the way
they affect the vibration spectra of a tire, a set of linguistic rules are developed. The classifier
performance was validated on smooth asphalt, regular asphalt, rough asphalt and wet asphalt
(Figure 4.5).
(a) Distinction of high and low frequency
domains
(b) Vibration ratio on dry and wet
surface conditions
Figure 4.4: Computation of the vibration ratio (R) on different surface conditions
Translational Speed
Fuzzy Rules
Tire Pressure
Vibration Ratio
Figure 4.5: Fuzzy logic algorithm to classify the current surface condition
As mentioned earlier, the smart tire based classification algorithm results in robust estimation
Chapter 4. Advanced Anti-lock Braking
76
of surface condition in the range of wheel slips that correspond to the linear tire force curve.
Nevertheless, higher levels of excitation on the tire triggers additional modes of vibration
which disrupts the correlation summarized above. The increasing misclassification rate under
high slip conditions are attributed to the increased vibration levels in the circumferential
acceleration signal due to the stick/slip phenomenon linked to the tread block vibration
modes (Figure 4.6). Therefore a model based approach is followed to estimate road surface
friction under high-slip conditions.
(a) Circumferential acceleration under low slip conditions
(b) Circumferential acceleration under high slip conditions
Figure 4.6: Variation in the vibration levels under low and high slip conditions
4.3.2
Model Based Surface Friction Observer
To develop a robust road surface friction coefficient (µm ax) estimation technique, a tire
model based observer scheme is used. The observer scheme basically requires the tire force
(Fx ), wheel load (Fz ) and slip (λ) information and in turn estimates the surface friction
condition. The details of providing each of the listed parameters are given as follows. The
sliding mode observer scheme detailed in Chapter 3 is utilized for tire longitudinal force
estimation.
Chapter 4. Advanced Anti-lock Braking
77
Numerous methods have been proposed to estimate the dynamic wheel load information,
recursive
least squares
(RLS) [126,
127] or
Inpparameters
this study,
and such
in turn
nasestimates
the surface
frictio
on condition.
The
T details
of Kalman
pproviding filters
eachh of [128].
the listed
aree a
given
smart. The
tiresliding
basedmode
is taken
as the basis.
Thefor
method
utilizes
a novel
sigas follows.
mestimation
observermethod
scheme
s
detaile
d in Chapter
3 is utilized
tire longitudin
nal force
estima
ation.
nal processing algorithm using the intelligent tire radial acceleration signal. The algorithm
is based on two synthetic parameters, the contact patch length (CPL) and the radial ac-
Numerous methods havee been proposeed to estimate the dynamic wheel load innformation, succh as recursivee least
celeration signal amplitude. In addition, a strong dependence of both these parameters is
squares (R
RLS) [129, 130] or Kalman filters [131]. In this study, a sm
mart tire basedd estimation m
method is taken as the
observed on the tire rolling speed and inflation pressure. Having identified parameters that
basis. The method utilizes a novel sign
nal processing
g algorithm usiing the intelliggent tire radiall acceleration ssignal.
are sensitive to the tire normal load, a relationship between these system inputs/parameters
The algoriithm is based on two syntheetic parameters, the contact patch length (CPL) and thee radial acceleeration
(rolling speed, inflation pressure, contact patch length and signal amplitude) and the system
signal amp
plitude. In addiition, a strong dependence
d
off both these parrameters is obsserved on the ttire rolling speeed and
output (tire normal load) is established using an explicit artificial neural network (ANN)
inflation pressure.
p
Havin
ng identified parameters
p
thatt are sensitive to the tire noormal load, a rrelationship beetween
[129] based formulation. In what follows, the selected ANN is trained for various learning
these systeem inputs/param
meters (rolling
g speed, inflatio
on pressure, coontact patch lenngth and signaal amplitude) aand the
rate and termination criteria [130] and modeled to make highly complex, nonlinear and mul-
system ou
utput (tire norm
mal load) is established
e
usiing an expliciit artificial neuural network (ANN) [132] based
tidimensional associations between selected input parameters and output. Further details on
formulation
n. In what follo
ows, the selectted ANN is trained for variouus learning ratee and terminatiion criteria [133] and
the selected ANN are provided in [35]. The results indicate acceptable degree of accuracy in
modeled to
o make highly
y complex, non
nlinear and mu
ultidimensionall associations bbetween selectted input param
meters
the predictions of the tire load across the full range of the tire operating conditions (Figure
and outputt. Further detaails on the seleected ANN arre provided in [32]. The ressults indicate aacceptable deggree of
4.7).
accuracy in
n the prediction
ns of the tire lo
oad across the full
f range of thhe tire operatinng conditions (F
Figure 4-7).
Figure Figure
4.7: Smart
load
estimation
performance.
4-7 Smart
S tire
tirebased
based
d wheel
wheel load
es
stimation
perforrmance
Next parameter of interest for the model
m
based observer is thhe tire longituudinal stiffnesss ( ). Satisffactory
Next parameter of interest for the model based observer is the tire longitudinal stiffness (Cλ ).
performancce of the wheeel dynamics-based observeer in the smal
all slip (|λ| < 3%) region pprovides us w
with an
Satisfactory performance of the wheel dynamics-based observer in the small slip (|λ| < 3%)
of
o the tire using
an on-line paarameter estim
m. Equation (18) can
region provides us with an opportunity
togadaptively
estimate Cλmation
of thealgorithm
tire using
an on-line
opportunity
y to adaptively
y estimate
be rewritte
en into estimation
a standaard parameter
identification
form
fo (18)
as follows
s: rewritten into a standard parameter
parameter
algorithm.
Equation
can be
(4.1)
where
parameter and
is the system outpu
ut (from the wheel
w
dynamiccs-based obseerver),
is the measu
ured slip ratio. The unknow
wn parameter
is the unkknown
can be iddentified in reaal-time
using the parameter iden
ntification app
proach. The RLS
R
algorithm as summarizeed in [134] prrovides a methhod to
known parametter at each sam
mpling time to m
minimize the ssum of the squuares of the moodeling
iteratively update the unk
Chapter 4. Advanced Anti-lock Braking
78
identification form as follows:
y(t) = φT (t)θ(t)
(4.1)
where y(t) = Fx is the system output (from the wheel dynamics-based observer), (t) = Cλ is
the unknown parameter and φT (t) = λ is the measured slip ratio. The unknown parameter
(t) can be identified in real-time using the parameter identification approach. The RLS
algorithm as summarized in [131] provides a method to iteratively update the unknown
parameter at each sampling time to minimize the sum of the squares of the modeling error
using the past data contained within the regression vector, φ(t). The performance of the RLS
algorithm is evaluated with simulations where the road surface is designed to have sudden
friction coefficient changes and the vehicle maneuver is straight driving with intermittent
vehicle maneuver
is straight
driving
intermittent
pedalestimates
presses. Figure
4-8the
indicates
theinresulting
estimates
gas pedal
presses.
Figurewith
4.8 indicates
the gas
resulting
can track
variations
the
friction condition
of the terrain.
can track the variations
in the friction
condition of the terrain.
[
]
T ire-Road Friction Coefficient
1
[ ]
0.8
0.6
0.4
0.2
0
2
4
6
8
10
Longitudinal stiffness (C x )
T ime [sec]
4
9
Parameter Estimation Results
x 10
8
7
Actual
Estimated
6
5
4
0
2
4
6
8
10
Time [sec]
Figure
4-8 algorithm
Results of to
theclassify
estimation
algorithm
Figure 4.8: Fuzzy
logic
the current
surface condition
The last parameter required for friction estimation is the wheel slip ratio which can be obtained from the wheel and
The last parameter required for friction estimation is the wheel slip ratio which can be ob-
vehicle speedtained
information.
A field tested algorithm, which is also followed in this study, is provided by Savaresi and
from the wheel and vehicle speed information. A field tested algorithm, which is
Tanelli in [135].
maininidea
the proposed
algorithm
is toand
estimate
speed
differently
also The
followed
thisofstudy,
is provided
by Savaresi
Tanellithe
in vehicle
[132]. The
main
idea of according to
the proposed
algorithm
is to estimate
the vehicle
speed conditions.
differently according
to the
the current vehicle
status, so
as to account
for the different
motion
Specifically,
thecurrent
status of the vehicle
vehicle status, so as to account for the different motion conditions. Specifically, the status of
is used to model the condition in which the vehicle speed is very low, constant, accelerating or braking condition
the vehicle is used to model the condition in which the vehicle speed is very low, constant,
Once the vehicle status is set, the estimated vehicle speed is computed based on the following observations. When
the vehicle has very low or constant speed, the estimated vehicle speed can be obtained as the average of the four
wheel speeds, as basically the slip is equal to zero for all wheels. When the vehicle is accelerating, instead, as the
Chapter 4. Advanced Anti-lock Braking
79
accelerating or braking condition. Once the vehicle status is set, the estimated vehicle speed
is computed based on the following observations. When the vehicle has very low or constant
speed, the estimated vehicle speed can be obtained as the average of the four wheel speeds,
as basically the slip is equal to zero for all wheels. When the vehicle is accelerating, instead,
as the driving wheels have a non-zero longitudinal slip due to traction force, the estimated
vehicle speed must be obtained as the average of the non-driving wheels. Finally, the braking condition is considered as the most critical, and it requires an appropriate integration
procedure of the accelerometer signal.
Having identified the required parameters, a tire model-based closed-loop feedback observer
is utilized. The estimator concept used here relies on the Dugoff tire model which has been
summarized in Chapter 2. In its simplest formulation, the model describes the relationship
between the tire force (Fx ) and the slip (λ) as a function of tire stiffness (Cλ ) and the surface
friction coefficient (µmax ). The above estimations yield the friction coefficient as the only
unknown in the model, hence executing the inverse model provides expressions for µmax as
the following. The intermittent function f (ξ) is given in terms of the tire force Fx :
f (ξ) =
Fx (1 − λ)
Cλ λ
(4.2)
Next. the piecewise function to define ζ can be defined:
ξ=

p
1 − 1 − f (ξ), f (ξ) < 1
1,
(4.3)
f (ξ) ≥ 1
Finally the surface friction can be computed from the inverse expression as below:
µmax =
2(Cλ λ)ζ
Fz (1 − λ)(1 − As Vs )
(4.4)
whereAs is a constant for the friction reduction factor and Vx represents the slip linearized
speed parameter (Vs = u(λ)). Figure 4.9 shows the estimator performance evaluated on a
jump-µ condition during braking. The results indicate that the observer successfully converges to the actual values, nevertheless with an initial time lag. This time lag is mainly
caused by the lack of excitation as mentioned earlier. Hence the sensor fusion approach
by integrating the model based observer with the smart tire estimations yield to the suc-
Chapter 4. Advanced Anti-lock Braking
80
cessful estimation of the surface friction in a substantially wide range of tire excitation levels.
Next section introduces an adaptive rule-based wheel slip control algorithm proposed to
improve the ABS performance. The algorithm aims to gain-schedule a simplified rule-set to
the variations in surface conditions estimations provided by the methods summarized above.
1
0.9
Surface Friction Coeff.
0.8
0.7
0.6
0.5
0.4
0.3
0.2
Model based Observer
Actual (CarSim)
0.1
0
0
5
10
15
20
25
Time [s]
Figure 4.9: Model based surface friction estimation performance.
4.4
A Self-Tuning Anti-lock Brake System Algorithm
In a vehicle brake system, the applied brake torque can be basically considered as a function
of longitudinal tire slip and tire force. Current ABS algorithms define rule-sets that are based
on vehicle and/or wheel speed and acceleration to regulate the brake pressure. These limits
are meant to keep the system trajectory bounded around a peak point that corresponds to
the maximum applicable brake force as long as possible for the shortest stopping distance.
The principle of the proposed algorithm is to define such rules that will define adaptive limits
w.r.t. possible variations on the surface friction conditions, which will confine the system
trajectory to a stable limit cycle.
The control algorithm is based on the quarter car model summarized in Chapter 2. If
the motion of the wheel is extended to two dimensions, then the effect of tire slip-angle
on the force-moment curve also must be considered. As mentioned earlier, there exists a
strong dependency between the lateral and longitudinal tire force-moment characteristics
Chapter 4. Advanced Anti-lock Braking
81
defined by the friction circle phenomena. Nevertheless, the range of the slip-angle variation
in a daily driving situation, which generally remains in the stable region, yields only to a
reasonable trade-off as indicated in Figure 4.10. Furthermore stability control algorithms,
as summarized in Chapter 3, works to confine the slip-angle in the stable region to maintain
driver’s control abilities and vehicle stability. Therefore, the following derivations in this
study omit the effect tire slip-angle on the tire force-slip curve.
1
0.9
0.8
0.7
()
0.6
0.5
 =0o
0.4
 =2o
0.3
 =4o
0.2
 =6o
0.1
 =8o
 =10o
0
0
0.2
0.4
 [%]
0.6
0.8
1
Figure 4.10: Tire force-slip curve and slip-angle relation.
In what follows, the wheel slip dynamics can be defined as below using the quarter car model
dynamics and the expression for wheel slip which is reduced from eq. 2.3 by only considering
the braking case (max(v, Rw ω) → v):
d v − Rw ω
λ̇ =
dt
v
Rw ω̇ Rw ω v̇
= −
+
v
v v
(4.5)
The wheel dynamics can be defined by substituting for the wheel and vehicle acceleration
Chapter 4. Advanced Anti-lock Braking
82
terms above from the quarter car model detailed in Chapter 2
1
v̇ = −
v
Rw
=
Jv
1
R2
(1 − λ) + w
m
J

Fz µ +
1 Rw
Tb
v J


J(1 − λ) 


Tb − Rw Fz µ 1 +

2
mRw


|
{z
}
(4.6)
Te
where Te is referred as the equilibrium torque. It is worth to note that Te defines the isoclines
in the phase plane Tb vs. λ as it zeros the rate of change of the slip when equal to the applied
brake torque Tb . The wheel slip control problem is essentially to regulate the value of the
longitudinal slip λ to a given setpoint λs and apply optimal braking force that corresponds
to the maximum value along the tire force-slip curve as long as possible until the vehicle
reaches a safe speed. The slip set-point can be obtained from a set of constants as in the
conventional rule-based ABS algorithms or it can also be commanded from a higher-level
control system such as ESP. In either case, the controller must be robust with respect to uncertainties in the tire characteristics, the brake pads/discs, the variations in the road surface
conditions, the load on the vehicle, etc. In this study an adaptation scheme is developed with
a four-phase rule based algorithm to remove steady-state error due to model inaccuracies
and in particular due to the surface friction coefficient µmax . The proposed control algorithm
can be seen as a minimum-seek algorithm to optimize the threshold values in a four-phase
rule set (prediction, hold-high, reselection and hold-low) w.r.t. the variations in the surface
friction condition.
The optimized rule set is to be devised as an appropriate switching logic to confine the
trajectory of the wheel slip dynamics (4.5) inside the adapted thresholds as well as to induce
a limit cycle on the system. The evolution of limit cycles in piecewise linear functions such
as the brake dynamics has been studied extensively. Two good examples are by Wellstead
and Pettit in [133] and [134] using graphical methods (Bond graphs) and by Goncalves in
[135] where the author analyzes stability of piecewise linear systems using linear matrix inequalities (LMI). Similarly, Tanelli et.al. in [136, 137] study limit cycles specifically in ABS
and compose conditions for the existence of a limit cycle in an ABS integrated wheel dynamics. The given conditions basically define the interrelations between the defined thresholds
among themselves and between the system isocline (Te ), and can be summarized as follows.
Chapter 4. Advanced Anti-lock Braking
83
Condition 1 states that the upper threshold for the maximum allowable brake torque (prediction line) should not be allowed to intersect with the isocline. Condition 2 states the
threshold for the lower threshold for minimum allowable brake torque (reselection line) has
to intersect with the isocline. Finally Condition 3 indicates that the slip condition of the
system trajectory that corresponds to when the trajectory intersects with the reselection
line has to be lower than the slip condition at the intersection of the reselection line and the
isocline in the saturation region of the tire force-slip curve. These conditions are utilized in
this study to develop the constraints expressions so that they will be expected to guarantee
to induce a limit cycle when set. The rule-set are defined by the following notations
1800
K
Brake Torque (T b) [Nm]
1600
→ Minimum allowable br
→ Maximum allowable b
→ Minimum allowable w
→ Maximum allowable w
2
1400
1200
K
1000
800
1
T
e
600
400
200
0
0
K
K
3
4
0.2

'
eq

0.4
eq
0.6
0.8
1
Slip
Figure 4.11: System constraints depicted on phase plane.
K1 < Ta < K2
K3 < λa < K4
Based on these conditions and the above notation, the inequality constraints to guarantee
the existence of a limit cycle are defined.
C1 → K2 > Te (µmax , λs )
C2 → K1 > Te (µmax , 1)
C3 → K1 < Te (µmax , λs )
C4 → λ0eq < λeq
C5 → K3 < λ s
(4.7)
Chapter 4. Advanced Anti-lock Braking
84
where λeq defines the slip condition when the isocline crosses K1 threshold, and λ0eq states
the slip conditions when the system trajectory crosses the same threshold. To carry out
the optimization of this rule-set, a cost function is structured by the system state (λ),
control inputs (U) and the system constraints. Let the wheel slip dynamics (4.6) have the
equilibrium point (Tbs , λs ) The proposed cost function can be written as:
J=
Z
1
(λ − λs )Q(λ − λs ) + (U − Us )T R(U − Us ) + ξ(λ̇ − λ̇sensor ) + η T C
2
(4.8)
where Q ∈ R and R ∈ R6x6 refer to control gain values, λ̇sensor is the rate of change of wheel
slip as measured or computed by sensor readings, C ∈ R5 states the constraints, ξ stands for
the costate and η T ∈ R5 refers to the Lagrange multipliers. The inequality constraints for
the rule-set and control input are reformulated as equalities to be able to implement them
into the cost function, so that the solution can be executed in a single run. To be able to
accomplish this transformation proxy inputs are augmented into the constraint equations.
The proxy inputs do not actually affect the system dynamics however indicate the level of
offset of the system trajectory from the designated thresholds.
2
J(1 − λs )
− Rw Fz µmax 1 +
2
mRw
K12 − (Rw Fz µmax )2
2
J(1 − λs )
− K12
Rw Fz µmax 1 +
2
mRw
λ2eq − (λ0eq )2
K22
= U12 ≥ 0
= U22 ≥ 0
= U32 ≥ 0
= U42 ≥ 0
λ2s − K32 = U52 ≥ 0
(4.9)
The equality constraints are then given as:
C1 =
C2 =
C3 =
C4 =
2
J(1 − λs )
−
+ Rw Fz µmax 1 +
2
mRw
U22 − K12 + (Rw Fz µmax )2
2
J(1 − λs )
2
2
U3 + K1 − Rw Fz µmax 1 +
2
mRw
U42 − λ2eq + (λ0eq )2
U12
K22
C5 = U52 − λ2s + K32
(4.10)
Chapter 4. Advanced Anti-lock Braking
85
Substituting eq. 4.10 into eq. 4.7 completes the definition of the cost function. Analytically
solving the optimization problem for this cost function is neither likely nor feasible; therefore
numerical methods are utilized to compute the solutions. The minimization of the proposed
cost function is accomplished by solving for its differential when it converges to zero (dJ = 0).
In what follows, the differential of the cost function is given as:
dJ =
∂J
∂J
∂J
dλ +
dU
+
dη
∂λ
∂UT
∂η T
(4.11)
The solution can be obtained by equating each partial derivative to zero which provides a
system of equations to be solved for the adaptive threshold values. The partial derivative of
the cost function w.r.t. the system state λ provides a single equation which can be solved
to find the system costate ξ/
∂J
dλ =
∂λ
∂ λ̇
Q(λ − λs ) + ξ
∂λ
→ Q(λ − λs ) + ξ
!
dλ = 0
∂ λ̇
=0
∂λ
(4.12)
To be able to compute ∂ λ̇/∂λ, the tire forces Fx can be approximated by the widely accepted
Burckhardt model (eq. 2.4) as in the stability analysis completed in Chapter 2. Having the
surface friction information, the tire force-slip curve can be reconstructed to find corresponding model coefficients by using a linear regression algorithm. After calculating the costate ξ
that zeros the first partial derivative, the partials of the cost function w.r.t. the inputs and
the Lagrange multipliers are computed. The partial of the cost function w.r.t. the input
vector augmented with the proxy inputs U = [U1 − U6
Tb ]T provides a set of 6 equations.
∂J ∂J ∂J ∂J ∂J ∂J ∂J
dU(6x1)
∂U1 ∂U2 ∂U3 ∂U4 ∂U5 ∂U6 ∂Tb
∂J ∂J ∂J ∂J ∂J ∂J ∂J
→
=0
∂U1 ∂U2 ∂U3 ∂U4 ∂U5 ∂U6 ∂Tb
∂J
dU =
∂UT
(4.13)
In the computation of the above set of equations, the input control gain R is taken as a
positive semi-definite diagonal weight matrix that individually penalizes the input offsets.
Chapter 4. Advanced Anti-lock Braking
86
The set of equations can be reformulated as:
∂J
dη = (CT )(1x5) dη(5x1)
∂η T
→ C = 0(5x1)
(4.14)
The resulting 11 equations can be solved simultaneously by numerical analysis for the unknown thresholds and the Lagrange multipliers. To be able to find the threshold values, the
proxy inputs U1−5 are assumed to be zero as it is the goal of the constructed optimization
scheme. The equations are then implemented in Matlab/Simulink environment and solved
numerically utilizing the Levenberg-Marquardt [138] algorithm (LM). The LM algorithm is a
very widely used optimization algorithm based on nonlinear least squares minimization. To
briefly summarize, the algorithm initially approximates the equations by their linearization.
A new estimate for the subsequent iteration is assumed by adding an infinitesimal value ∂ˇ
on the predefined initial guess. In what follows, the differentiation of the system is solved for
this infinitesimal value. If either the parameter vector at the calculated step or the reduction
of sum of squares from the latest parameter vector fall below predefined limits, the algorithm
stops and the last parameter vector is considered to be the solution. Further details of the
algorithm is omitted here not to distract off of the course of this research but can be found
in [139]. A disadvantage of the LM algorithm is its sensitivity to the initial guess for the
parameters to be solved. Depending on the level of nonlinearity and the number of minima
of the given set of equations, the algorithm might deviate from a feasible result. Therefore
extra attention must be paid to the initial estimates of the sought parameters. On the other
hand an advantage of this algorithm is that it does not require the system of equations to
be square which fits to the problem in hand. The following section provides the results of
the evaluation of the proposed algorithm using numerical analysis.
4.5
System Validation using Simulation
The proposed algorithm above is evaluated using a nonlinear 8 degree-of-freedom (DOF)
vehicle model detailed in Chapter 2, which captures the rotational dynamics of the four
wheels in addition to the vehicles longitudinal, lateral, roll and yaw motions. A finite state
machine (FSM) is implemented in Simulink to select the control actions according to the
threshold values computed in the proposed algorithm. The FSM is composed of four discrete
Chapter 4. Advanced Anti-lock Braking
87
states, e.g. rise, hold-high, release, hold-low, each of which has an associated control action
as their names imply. The transition between these states takes place when the system
trajectory hits the adaptive threshold value.
[> max]
2
Hold_d
entry: u=0;
1
[ min]
[TbTbmin]
1
Increase
entry: u=10000;
[Tb>Tbmin] Decrease
entry: u=-10000;
2
[Tb<Tbmax]
2
1
[ max]
Hold_i
entry: u=0;
[TbTbmax]
1
[< max]
2
Figure 4.12: FSM description of the ABS state switching logic
The brake torque input is assumed to be implemented by a constant actuator rate at a
nominal value of 10kN m/s which is proven to be a feasible value by Petruccelli et.al. in
[140]. The wheel & vehicle longitudinal speed and the control of the longitudinal slip ratio
over time are known as the two key parameters in quantifying ABS performance. Each of
these key parameters will be presented to show the system performance for both a highµ (µmax = 0.85) surface, representative of dry asphalt and a low-µ (µmax = 0.3) surface,
representative of an icy surface. A series of straight line braking maneuvers are executed
on different surface conditions. The dynamics of the adaptive threshold values are shown in
the following figures. The results indicate that the solution algorithm allows the threshold
values to converge to effective values before the ABS starts to operate; therefore yields to
stable brake behavior.
Figure 4.13 shows the results for the adaptation of the threshold values K1 to K4 and the
respective system parameters, Tb and λ. The front tires seem to experience less slip oscillations and as a result can handle higher brake torque values. This can be grounded on
the load transfer towards the front wheels which increases the tires road holding capacity
and accordingly the increases the applicable brake force capacity. Whereas the rear tires
experience a drop in the wheel loads which yields to the opposite effect and yields to higher
levels of slip oscillations. Another point worth to note is that the brake torque for the front
tires maintain almost constant around the maximum applicable value, which will not be
possible due to the leaks and lags in a hydraulic brake system. In this numerical analysis,
Chapter 4. Advanced Anti-lock Braking
88
Front
Rear
1
1
Brake Torque [Nm]
0.5
1
1.5
2
2.5
Slip [%]
0.5
0
0
K4
Slip
K3
2000
1500
K2
1000
Tb
500
0
0
K1
0.5
1
1.5
Time [s]
2
2.5
3
Slip
K3
0.5
0
0
3
Brake Torque [Nm]
Slip [%]
K4
0.5
1
1.5
2
2.5
3
2000
K2
1500
Tb
1000
K1
500
0
0
0.5
1
1.5
2
2.5
3
Time [s]
(a) Threshold evolution and system dynamics at (b) Threshold evolution and system dynamics at
front wheel
rear wheel
Figure 4.13: Results for the threshold adaptation and system dynamics on high-µ surface
condition
only the mechanical lag is introduced into the model. Given an initial velocity of 25m/s, the
algorithm yields to a satisfactory braking performance on the high-µ condition. The vehicle
comes to a complete stop after 2.91s at around 40.12m. Figure 4.14 also shows the wheel
and vehicle speeds variations.
Figure 4.15 shows the adaptation characteristics of the threshold values on the low-µ surface.
As the surface friction level decreases, the tires road grip capability minimizes which also
yields to higher levels of slip oscillations. Nevertheless the prediction and reselection rules
successfully maintain stability of the brake system.
Figure 4.15 depicts the braking performance of the adaprive control algorithm in the low-µ
surface. Given the same initial speed (u0 = 25m/s), the vehicle comes to a complete stop
after 6.44s at about 87.05m.
Both tests show satisfactory braking performance under challenging conditions. The algorithm can confine the system specifically inside the prediction and reselection lines while
maintaining the slip ratio within a close proximity to the desired slip (Figure 4.16). The
increasing levels of slip oscillations towards the end of the braking occurs due to the decreasing velocity which yields to infinitely fast open loop wheel slip dynamics which the solution
Chapter 4. Advanced Anti-lock Braking
89
30
Vehicle Speed
Wheel Speeds
25
Stopping Distance [m] = 40.12
Stopping Time [s] = 2.91
Speed [m/s]
20
15
10
5
0
0
0.5
1
1.5
2
2.5
3
Time [s]
Figure 4.14: Braking performance of the proposed algorithm on high-µ surface
algorithm cannot cope with. Furthermore uncertainties such as load transfer or road bank
might also cause increasing levels of slip oscillations.
To get a better measure of the improvements in the performance of the proposed intelligent
tire based ABS algorithm, it is compared to a conventional ABS algorithm used in modern day vehicles. The details of the algorithm are provided by Day and Roberts in [103].
It utilizes a rule-based method to maintain longitudinal wheel slip in a certain range on
the information provided by angular wheel speed and acceleration sensors. The following
methodology explains the adopted selection criteria for the acceleration/deceleration thresholds.
• Wheel deceleration threshold (−a): The relationship between the wheel-deceleration
and slip curves for different initial braking velocities (10−40m/s) was studied. Clearly,
for high velocities, the deceleration threshold should be set at higher values. In contrast
for low initial velocities the threshold should be decreased. A reasonable selection for
this threshold could be the 10% slip value. Such a selection will be able to provide
10% slip for high initial braking velocities and slip exceeding 10%, but still lower than
20% for lower velocities.
• Wheel acceleration threshold (+a): This threshold is used to mark a positive tendency
in the wheel acceleration. Appropriate threshold values are 3 − 6m/s2 .
• Wheel acceleration threshold (+A): This threshold denotes a recovery from the pre-
Chapter 4. Advanced Anti-lock Braking
90
Front
1
Slip [%]
K4
0
0
Brake Torque [Nm]
Slip
K3
0.5
1
2
3
4
5
6
7
1500
K2
Tb
1000
K1
500
0
0
1
2
3
4
5
6
7
Time [s]
(a) Threshold evolution and system dynamics at front
wheel
Rear
1
Slip [%]
K4
0
0
Brake Torque [Nm]
Slip
K3
0.5
1
2
3
4
5
6
7
1500
K2
Tb
1000
K1
500
0
0
1
2
3
4
5
6
7
Time [s]
(b) Threshold evolution and system dynamics at rear
wheel
Figure 4.15: Results for the threshold adaptation and system dynamics on low-µ surface
condition
dicted lock and is higher than (+a). Selection of too high values of (+A) threshold
results in higher recovered wheel speed and lower slip values. Suitable threshold values
are 9 − 12m/s2 .
Chapter 4. Advanced Anti-lock Braking
91
30
Vehicle Speed
Wheel Speeds
25
Stopping Distance [m] = 87.05
Stopping Time [s] = 6.44
Speed [m/s]
20
15
10
5
0
0
1
2
3
4
5
6
7
Time [s]
Figure 4.16: Braking performance of the proposed algorithm on low-µ surface
The summary of the triggering signals and threshold values are given in Table 4-1.
Table 4.1: Triggering signals and threshold values for the baseline ABS
Command
Triggering Signal
Threshold Value
Hold
−a
−50m/s2
Decrease
> slip
20%
Stop Decrease
+a
4m/s2
Increase
+A
10m/s2
Figure 4.17 a and b show the performance of the baseline ABS model on high-µ and low-µ
surfaces, respectively. Given an initial velocity of 25m/s, the results show that the vehicle
requires 44.86m to come to a complete stop on the high-µ surface, whereas on the lowµ surface this distance is 108.25m. Clearly, executing the described adaptive threshold
based ABS control algorithm shows an improved braking performance respective to the
conventional algorithm. The longitudinal slip quickly averages around the calculated desired
slip ratio after brake application and oscillates about this point for the remainder of the
braking maneuver with the help of the adaptive threshold values, until the ABS cut- off
speed is reached. The result of staying within these close boundaries is an increase in the
average longitudinal braking force available to the tire.
Chapter 4. Advanced Anti-lock Braking
92
30
30
Vehicle Speed
Wheel Speeds
25
Vehicle Speed
Wheel Speeds
25
Stopping Distance [m] = 44.86
Stopping Time [s] = 3.59
20
Speed [m/s]
Speed [m/s]
20
15
15
10
10
5
5
0
0
Stopping Distance [m] = 108.25
Stopping Time [s] = 8.54
0.5
1
1.5
2
2.5
3
3.5
Time [s]
(a) High-µ surface testing
4
0
0
2
4
6
8
10
Time [s]
(b) Low-µ surface testing
Figure 4.17: Braking performance of the conventional ABS algorithm on straight line
braking
The comparative phase plane analysis shown in Figure 4.18 depicts the system trajectory
behavior using the adaptive threshold and conventional ABS algorithms. As the figures
indicate, the adaptive threshold algorithm significantly reduces the variations in the applied
brake torque without excessive disturbance in the slip oscillations that might cause instability
as it is guaranteed intrinsically by the predefined constraints. Consequently, this reduction
in the brake torque variations yields to a shorter stopping distance which might distinguish
between a crash or safe brake scenario.
Finally the proposed algorithm is evaluated on surface conditions with varying friction levels.
First a jump-µ condition is implemented where the surface friction coefficient is assumed
to jump from high to low at 15m and then back to high after 30m (Figure 4.19). This
maneuver can be very harsh for many conventional rule-based ABS logics as these sudden
changes cannot be easily apprehended due to the lack of surface friction information and
might lead to instabilities. On the other hand the proposed algorithm adapts the ruleset
definitions with respect to these variations and thereby maximizes the brake force utilization.
Figure 4.20 depicts the adaptation of the threshold values and the resulting brake torque
and slip variations. Obtaining the surface friction from the estimation scheme, the algorithm
computes the optimal threshold values for improved braking performance and guaranteed
stability as well. Next the system is put on test under slip-µ condition where the left wheel
are kept on the same jump-µ surface while the right wheels are driven on high friction. Figure
Chapter 4. Advanced Anti-lock Braking
93
2200
1500
Adaptive Threshold Alg.
Conventional Alg.
2000
Adaptive Threshold Alg.
Conventional Alg.
1600
Brake Torque [Nm]
Brake Torque [Nm]
1800
1400
1200
1000
800
600
1000
500
400
200
0
0
0.2
0.4
0.6
0.8
0
0
1
0.2
0.4
Slip [%]
0.6
0.8
1
Slip [%]
(a) Front tire phase-plane on high-µ surface
(b) Front tire phase-plane on low-µ surface
2200
1500
Adaptive Threshold Alg.
Conventional Alg.
2000
Adaptive Threshold Alg.
Conventional Alg.
1600
Brake Torque [Nm]
Brake Torque [Nm]
1800
1400
1200
1000
800
600
1000
500
400
200
0
0
0.2
0.4
0.6
0.8
1
Slip [%]
(c) Rear tire phase-plane on high-µ surface
0
0
0.2
0.4
0.6
0.8
1
Slip [%]
(d) Rear tire phase-plane on low-µ surface
Figure 4.18: Phase plane comparison of adaptive threshold and conventional ABS
algorithms in high-µ and low-µ tests
4.22 summarizes the results in the same manner to underline the adaptation of thresholds
on the left wheel brakes while maintaining an optimal level for avoiding any lock-up scenario
on the right hand side. The phase-plane of the system similarly proves guaranteed stability
while providing admirable braking performance.
It is clear from the executed simulations that the effect of the varying friction coefficient is
significant, but the proposed algorithm handles these unexpected changes in the road surface
conditions quite well. With the momentary surface condition information from the smart tire
integrated estimation scheme the proposed algorithm copes with the variations by adapting
the threshold values. The comparison between the time histories of tire slip (Figure 4.20 and
Chapter 4. Advanced Anti-lock Braking
94
Figure 4.19: Jump-µ test condition
Front
1
Rear
1
Brake Torque [Nm]
0.5
1
1.5
2
2.5
3
K2
Tb
K1
1000
0
0
0.5
1
1.5
2
2.5
3
3.5
Slip
K3
0.5
0
0
3.5
3000
2000
Slip [%]
0.5
0
0
K4
Slip
K3
Brake Torque [Nm]
Slip [%]
K4
0.5
1
1.5
2
2.5
3
3.5
1500
K2
Tb
1000
K1
500
0
0
Time [s]
0.5
1
1.5
2
2.5
3
3.5
Time [s]
(a) Threshold evolution and system dynamics at (b) Threshold evolution and system dynamics at
front wheel
rear wheel
Figure 4.20: Results for the threshold adaptation and system dynamics on jump-µ surface
condition
4.22) clearly shows how the instantaneous information about the change of friction coefficient
allows the ABS controller to prevent wheel lock-up during the jump from high friction to
low friction. This can be attributed to the fact that the adaptive threshold algorithm yields
to a significant reduction in both frequency and amplitude of the oscillations in the applied
brake torque as well as in slip. Figure 4.23 is a good indicator for this reduction, which
compares the resulting phase plane (Tb vs. λ) of each algorithm at front and rear tires. As
summarized in the introduction, the reduced oscillation levels helps to maximize the tire
grip level in which manner it also allows higher brake forces to be applied. As a result,
the adaptive algorithm yields to considerable improvement in the braking performance by
decreasing the stopping distance significantly. Figure 4.25 shows the vehicle and wheel speed
variations in comparison between the two algorithms.
Chapter 4. Advanced Anti-lock Braking
95
Left
Right
1
1
0.5
1
1.5
2
2.5
3
Tb
K1
1000
0
0
0.5
1
1.5
2
Time [s]
2.5
3
Slip
K3
0.5
0
0
3.5
K2
2000
K4
Slip [%]
0.5
0
0
3000
Brake Torque [Nm]
Slip
K3
0.5
1
1.5
2
2.5
3
3.5
3000
Brake Torque [Nm]
Slip [%]
K4
K2
Tb
2000
K1
1000
0
0
3.5
0.5
1
1.5
2
2.5
3
3.5
Time [s]
(a) Threshold evolution and system dynamics at (b) Threshold evolution and system dynamics at
left wheel
right wheel
Figure 4.21: Results for the threshold adaptation and system dynamics on split-µ surface
condition
Front
Rear
Slip [%]
0.5
0
0
Brake Torque [Nm]
1
1
2
3
2000
1000
0
0
1
2
Time [s]
3
4
0.5
0
0
4
Brake Torque [Nm]
Slip [%]
1
1
2
3
4
2
3
4
2000
1000
0
0
1
Time [s]
(a) Brake torque and slip performance for front (b) Brake torque and slip performance for rear tire
tire
Figure 4.22: Results for the conventional ABS algorithm on jump-µ surface condition
Chapter 4. Advanced Anti-lock Braking
96
2500
1500
Adaptive Threshold Alg.
Conventional ABS Alg.
Adaptive Threshold Alg.
Conventional ABS Alg
Brake Torque [Nm]
Brake Torque [Nm]
2000
1500
1000
1000
500
500
0
0
0.2
0.4
0.6
0.8
0
0
1
0.2
Slip [%]
0.4
0.6
0.8
1
Slip [%]
(a) Front tire phase plane on jump-µ
(b) Rear tire phase plane on jump-µ
Figure 4.23: Phase plane comparison of adaptive threshold and conventional ABS
algorithms in jump-µ test
3000
3000
Adaptive Threshold Alg.
Conventional ABS Alg.
2500
Brake Torque [Nm]
Brake Torque [Nm]
2500
Adaptive Threshold Alg.
Conventional ABS Alg.
2000
1500
1000
500
2000
1500
1000
500
0
0
0.2
0.4
0.6
0.8
Slip
(a) Front tire phase plane on split-µ
1
0
0
0.2
0.4
0.6
0.8
1
Slip
(b) Rear tire phase plane on split-µ
Figure 4.24: Phase plane comparison of adaptive threshold and conventional ABS
algorithms in split-µ test
Chapter 4. Advanced Anti-lock Braking
97
30
30
Vehicle Speed
Wheel Speeds
25
Vehicle Speed
Wheel Speeds
25
Stopping Distance [m] = 53.91
Stopping Time [s] = 4.01
20
Speed [m/s]
Speed [m/s]
20
15
15
10
10
5
5
0
0
Stopping Distance [m] = 48.67
Stopping Time [s] = 3.34
1
2
3
4
0
0
5
0.5
1
Time [s]
1.5
2
2.5
3
3.5
Time [s]
(a) Conventional ABS algorithm performance
(b) Adaptive threshold algorithm performance
Figure 4.25: Comparison of algorithm performances in jump-µ surface test
30
30
Vehicle speed
Wheel speeds
25
Stopping Distance [m] = 52.01
Stopping Time [s] = 4.11
20
Speed [m/s]
Speed [m/s]
25
15
10
5
0
0
Vehicle speed
Wheel speeds
Stopping Distance [m] = 47.76
Stopping Time [s] = 3.35
20
15
10
5
1
2
3
4
Time [s]
(a) Conventional ABS algorithm performance
5
0
0
0.5
1
1.5
2
2.5
3
Time [s]
(b) Adaptive threshold algorithm performance
Figure 4.26: Comparison of algorithm performances in split-µ surface test
3.5
Chapter 4. Advanced Anti-lock Braking
4.6
98
Conclusion
This chapter presents a road surface condition classification system through the integration of
a smart tire system with a model based observer, which is later employed in an adaptive rulebased ABS algorithm developed based on the optimization of brake thresholds. Simulations
are carried out on a series of braking maneuvers to examine the possible improvements
with the proposed ABS algorithm in braking performance, assuming that the additional
information concerning road surface condition could be provided by the sensor fusion method.
The results reveal that the proposed algorithm significantly reduces the oscillations in the
applied brake and slip values which yields to an increase in the tire road grip levels and
thereby in the applied brake force. Impressive improvements are obtained for the executed
straight line braking tests the results from the numerical analyses. Table 4.2 compares and
summarizes the results for both algorithms.
Table 4.2: Summary of the system evaluation using numerical analysis
High-µ surface
Low-µ surface
Stopping
Distance [m]
Improvement
over baseline
[%]
Stopping
Distance [m]
Improvement
over baseline
[%]
Conventional
ABS
44.86
-
108.25
-
Adaptive
Threshold
40.12
10.56
87.05
19.58
Jump-µ surface
Split-µ surface
Stopping
Distance [m]
Improvement
over baseline
[%]
Stopping
Distance [m]
Improvement
over baseline
[%]
Conventional
ABS
53.91
-
52.01
-
Adaptive
Threshold
48.67
9.72
47.76
8.17
In the light of these results, the following conclusions can be drawn. The actual road surface
Chapter 4. Advanced Anti-lock Braking
99
condition information provided by an integrated estimation approach using smart tire technology can be used to develop an adaptive rulebased ABS algorithm. The adaptive nature
of the proposed algorithm provides significant performance improvements with less slip and
brake torque oscillations which underline that knowing the actual road surface condition can
be quite favorable for enhancing the current ABS models and reducing the vehicle stopping
distance significantly. In addition, following the same approach (rule-based) as in the conventional ABS algorithms allows the proposed algorithm to be implemented on the currently
available braking systems without the requirement of major modifications.
Chapter 5
Integrated Vehicle Control Systems
5.1
Introduction
The studies detailed in Chapters 3 and 4 provide strong arguments on the utilization and
advantages of the smart tire technology in vehicle control applications. The proposed control
algorithms yield to considerable performance improvements in the addressed vehicle states,
namely the stability and braking dynamics. Nevertheless the daily driving conditions might
easily pose more complicated operation conditions which require multiple of such controllers
to activate simultaneously. On the other hand, the co-existence of several control subsystems
without coordination can cause various drawbacks. For example, a major practical issue is
that the design of software and hardware becomes more complicated due to the dramatically increased number of sensors and signal transmission requirements. Another critical
issue is that, because of the possible function overlapping among these systems, there will
be inevitable conflicts in their control objectives and control actions. If not coordinated, the
performance of these systems might get worse than that of individual systems or even worse
than a passive system without any active control. For example, anti-lock braking systems
(ABS) and electronic stability programs (ESP) are both based on the tire slip ratio control,
however the primary objective of an ABS is to maintain the slip ratio around the optimal
value (normally corresponding to peak longitudinal friction coefficient), while the objective
of ESP is to control wheel slip ratios properly for vehicle stability improvement via braking
the selected wheels, requiring optimum use of the available lateral force.
100
Chapter 5. Integrated Vehicle Control Systems
101
As a result, the ensuing challenge becomes successful integration of these control systems
to cope with such conditions while maintaining a comfortable ride, and based on the above
summary, the two key problems of integrated vehicle dynamics control to be solved are:
• to avoid the conflicts and interventions among different subsystems;
• to exploit the potentials of each subsystems by communication and coordination among
the sub-systems since different systems have different action domains.
The synthesis of such integrated control schemes has been studied extensively; Chapter 3
provided an exhaustive literature review focusing on the stability control applications. Common aliases for these methods are Integrated Vehicle Dynamics Control or Integrated Chassis
Control (ICC). Despite the abundant number of publications, the definition for ICC systems
is not standardized. Nevertheless, the expected attributes from such a system are definitive,
i.e. they need to coordinate multiple subsystems systematically according to control objectives and actions in terms of both software and hardware, rather than simply putting the
subsystems altogether.
Inspired by this challenge, this chapter describes the derivation of an ICC algorithm for vehicle stability control. The first application takes the algorithms derived in Chapters 3 and
4 as the basis and employs a dynamic control allocation algorithm to successfully integrate
the two. The control allocation task intends to deploy the wheel brake forces to generate the
desired yaw-moment on the vehicle chassis and it is accomplished through an optimization
procedure utilizing a weighted least squares method. The resulting control signals are then
implemented via the ABS to avoid any possible wheel lock-up scenarios while providing sufficient braking performance. As a result, combined with the additional steering input, the
system maintains vehicle yaw stability even during a challenging evasive double lane change
maneuver as given in Chapter 3. The next focal point of the chapter is the introduction
of a new adaptive control methodology, namely the L1 adaptive control method, to replace
the upper-level stability control algorithm. The considerable advantages this new method
introduces are: it proposes an architecture that yields to robust performance while allowing
fast adaptation so long as the actuator dynamics concede; and it extends the adaptation of
system dynamics from selected parameters to non-linear functions of time and states. The
subsequent sections detail these applications and present results of simulation studies for
evaluating the proposed systems’ performances.
Chapter 5. Integrated Vehicle Control Systems
5.2
102
Integrated Stability Control based on Lyapunov
Direct Method
As summarized above, a decentralized implementation of active safety systems might easily
impair the vehicle performance and even destabilize the system due to objective conflicts.
A major challenge in the integration of control systems is the inherently coupled vehicle
dynamics states. To tackle with such issues, the control algorithm needs to be derived
considering a so-called centralized or multi-layered structure where the control inputs should
be implementable considering lower level control and actuator dynamics; and any possible
objective conflict is taken care at an upper-level algorithm. Based on these terms, this section
combines the studies in Chapters 3 and 4 and forms an integrated vehicle control system by
dynamically allocating the required brake forces with respect to the applied yaw-moment
and actuator dynamics.
5.2.1
Control Allocation Problem
Control allocation deals with the problem of distributing a given control demand among an
available set of actuators. Most existing methods are static in the sense that the resulting
control distribution depends only on the current control demand. In this study a method for
dynamic control allocation is proposed, in which the resulting control distribution utilizes
time varying system dynamics as well as the distribution in the previous sampling instant.
As a result, the method extends regular quadratic programming control allocation by also
penalizing the actuator rates. This also leads to a frequency dependent control distribution
which can be designed to account for different actuator bandwidths. The control allocation
problem is posed as a constrained quadratic program which provides automatic redistribution of the control effort when one actuator saturates in position or in rate. When no
saturation occurs, the resulting control distribution coincides with the control demand fed
through a linear filter.
The existence of a redundancy in the set of existing actuators set (e.g. four wheel brakes)
allows for more than one combination of actuator activities to yield to the same generalized
control action, and hence give the same overall system behavior. This design freedom is often
used in the advantage of the control allocation process, e.g. the priority among the multiple
Chapter 5. Integrated Vehicle Control Systems
103
set of actuator combinations or among the actuators can be determined by optimizing a
static performance index. This can also be thought of as affecting the distribution of control
effect in magnitude among the actuators. Regardless of the selected method (optimization
based allocation [141, 142], daisy chain allocation [143], direct allocation [144] etc.), the resulting mapping from the generalized or virtual control command, v(t), to the upper-level
control input, u(t), can be written as a static relationship:
u(t) = h (v(t))
(5.1)
A possibility that has been little explored is to also affect the distribution of the control
effect in the frequency domain, and use the redundancy to have different actuators operate
in different parts of the frequency spectrum. This requires the mapping from v to u to
depend also on previous values of u and v, hence:
u(t) = h (v(t), u(t − T ), v(t − T ), u(t − 2T ), v(t − 2T ), ...)
(5.2)
where T is the sampling interval. This equation results in the dynamic control allocation procedure. In this study, the method for handling the dynamic control allocation is implemented
through the so-called quadratic programming scheme as in the following form:
u(t) = arg min J(u(t)) + γ 2 kWv (Au(t) − v(t))k2
u<u<u
(5.3)
The first term J(u(t)) represents a sub-cost for minimal actuator activity while the second
term represents the constraint on tracking the virtual control command v(t) computed by
the upper level algorithm. In this study, the method is extended by an extra term added to
the sub-cost optimization criterion to also penalize the actuator rates as explained above.
The derivation for the brake-force allocation application is initiated by stating the relation
between the yaw moment generated on the chassis and the wheel brake forces:
tr
Fxf l + Fxrl − Fxf r − Fxrr
2
+ lf ∆Fyf l + ∆Fyf r − lr (∆Fyrl + ∆Fyrr )
∆Mb =
(5.4)
where ∆ indicates the change in value. As well known, utilizing longitudinal forces will yield
to a change in the lateral force due to the limited force generation capacity of tires which
can be explained by the friction circle phenomena. In this study the Dugoff tire model (as
Chapter 5. Integrated Vehicle Control Systems
104
detailed in Ch. 2) is utilized to come up with an expression for the change in the lateral tire
forces with respect to the longitudinal tire forces (friction ellipse concept). In what follows,
the rate of change in lateral force can be expressed as:
∆Fyi =
∂Fyi
Fx
∂Fxi i
(5.5)
where i = (f r, f l, rr, rl). Using this expression the yaw moment can be rewritten as:
tr
Fxf l + Fxrl − Fxf r − Fxrr
2
∂Fyf l
∂Fyf r
∂Fyrr
∂Fyrl
+ lf
Fx
Fx +
Fx
Fx +
− lr
∂Fxf l f l ∂Fxf r f r
∂Fxrl rl ∂Fxrr rr
∆Mb =
(5.6)
In addition, the rate of change of the lateral force with respect to longitudinal can be written
using the analytical expression of the Dugoff model:
Therefore:
∂Fyi
∂Fxi
=
∂Fyi ∂Fxi
/
∂λi ∂λi
∂Fyi
∂λi
= Cα tan(αi )
∂Fxi
λi
= Cλ
f˙(ξi )(1 − λi ) + f (ξi )
(1 − λi )2
f˙(ξi )λ + (1 + λ)f (ξi )
(1 − λi )2
Cα tan(αi )(f˙(ξi )(1 − λi ) + f (ξi ))
∂Fyi
=
∂Fxi
Cλ (f˙(ξi )λ + (1 + λ)f (ξi ))
(5.7)
(5.8)
(5.9)
(5.10)
The differentiation of the function f (ξ) with respect to λ are given as:

µFz (λ − 1)


f1 − f2 − f3 , if − 2 < p 2 2
λ Cλ + Cα2 tan(α)2
f˙(ξ) =
µFz (λ − 1)

0,
if − 2 ≥ p

λ2 Cλ2 + Cα2 tan(α)2
(5.11)
Chapter 5. Integrated Vehicle Control Systems
105
where
f1 =
f2
f3
λµCλ2 Fz (λ − 1)
µFz (λ − 1)
p
+2
2 λ2 Cλ2 + Cα2 tan(α)2
!
(5.12)
3
2 (λ2 Cλ2 + Cα2 tan(α)2 ) 2
µFz
µFz (λ − 1)
p
µFz (λ − 1)
−
3
2
2
2
2
2 λ Cλ + Cα tan(α)
2(λ2 Cλ2 + Cα2 tan(α)2 ) 2
p
=
2 λ2 Cλ2 + Cα2 tan(α)2
!
µFz (λ − 1)
p
µFz
+2
2 λ2 Cλ2 + Cα2 tan(α)2
p
=
2 λ2 Cλ2 + Cα2 tan(α)2
!
(5.13)
(5.14)
Based on the above results, the dynamic control allocation problem is formulated that computes the actuator control signals with minimum possible activation while being subjected
to minimal error between the implemented and desired yaw moment. Consequently, the implemented yaw moment value as given in equation 5.6 can be rewritten by the multiplication
of a coefficient matrix that includes the above differentiations and the wheel brake forces as
the lower level control signals:
Mb = Blower u(t)
(5.15)
where the coefficient matrix Blower is defined as:
Blower =
∂Fyf l
tr
+ lf
2
∂Fxf l
∂Fyf r
tr
− + lf
2
∂Fxf r
∂Fyrl
tr
− lr
2
∂Fxrl
∂Fyrr
tr
− − lr
2
∂Fxrr
(5.16)
and the control signal vector u(t) is formed by the wheel brake forces:
u(t) = Fxf l
F xf r
Fxrl
Fxrr
T
(5.17)
This expression provides the constraint for the force allocation problem in terms of the
longitudinal tire forces that will successfully implement the desired yaw moment computed
by the VSC algorithm. Substituting the above results into the quadratic programming
problem in equation 5.3, it can be rewritten as follows:
u(t) = arg min
u<u<u
kWu1 u(t)k2 + kWu2 u(t − T )k2 + γ 2 kWv (Blower u − Mb )k2
(5.18)
Chapter 5. Integrated Vehicle Control Systems
106
where u(t − T ) stands for the control signal at the previous time step. The weighting
coefficient (Wv ), which becomes a scalar in this case because Mb ∈ R, scales the upper level
control input, and u and u define the lower and upper bounds respectively for the actuators.
The upper bound can be defined using the friction circle phenomena:
u=
q
µ2 Fz2i − Fy2i ≥ Fxi
(5.19)
As the DYC in this study is based on differential braking action, the lower bound for control
can be defined as zero (u = 0 < Fxi ). The sub-cost function J(u(t)) is formed by respectively
scaling the L2 -norm of the current and previous time step control signals. The last scale
factor (γ) in the proposed cost function defines the emphasis on the minimization of the
error in force allocation to meet the requirement from the upper level control.
This linearly-constrained quadratic problem is solved numerically using a weighted-leastsquare (WLS) approach and computes the optimal wheel brake forces (Fxi ) at each sample.
For computational efficiency the algorithm initializes with a reformulation as in the below
form:



2 




γWv Blower 
 γWv Mb  

 u(t) − 

u(t) = arg min 


 
u<u<u 



Wu
Wu u(t − T )

= arg min kAopt u(t) − Bopt k2
(5.20)
u<u<u
For this reformulation to hold true without losing generality the weighing matrices Wu1 and
Wu2 are to be assumed symmetric and such that:
Wu = Wu21 + Wu22
1/2
(5.21)
The Active Set Algorithm (ASA) is utilized to numerically solve this problem and implemented in the Matlab/Simulink environment for evaluation. The active set refers to the set
of solutions that satisfies the constraints the optimization problem is subjected to; hence ASA
is a widely preferred method in solving problems with equality and/or inequality constraints.
The algorithm basically approximates to the optimal solutions by repetitively solving the
given equality problems and finalizes at the allowable rate of convergence in the defined active
Chapter 5. Integrated Vehicle Control Systems
107
set [145]. Figure 5.1 details the algorithm implemented in numerical analysis software.
Initialize
Compute Residue
iter++
Eliminate
saturated actuators
(if any)
Solve for u(t) using
reduced
parameters
Yes
u(t) feasible?
Yes
No
u(t) optimal?
No
Find primary
bounding
constraints
Bail
out
Remove constraint
from the working
set
Add constraints to
the working set
Figure 5.1: The active-set algorithm for numerically solving the control allocation problem
using weighted-least squares.
After initializing the problem into the form as given in equation 5.21, the first process becomes computing the residue between desired and current states. The saturated actuators
can be provided and accounted for if information is available before entering the iterative
loop. Otherwise, the loop starts by reducing any saturated actuators out of the system
Chapter 5. Integrated Vehicle Control Systems
108
dynamics (Aopt → A0opt ) if there is any after each iteration. In what follows, the problem is
solved for the reduced system (ut (t) = (A0opt )−1 r) which at the same time removes perturbations due to active constraints. The solution in hand is then examined if it remains in the
feasible set by comparing to the given upper and lower bounds which are updated with respect to the actuator rates as detailed above. If the solutions are not feasible, the algorithm
computes the primary bounding constraints that are at the closest distance to the resulting
u(t) and updates the working set by adding the constraints at the minimum distance to
the results. Next the algorithm proceeds to the loop for another iteration. Whereas, if the
solution turns out feasible, it is examined for optimality (ATopt rupd ≤ ), and affirmative
results yield to the termination of the loop. On the other hand, if the optimality condition is
not satisfied, the working set is updated by removing the constraints that are at the farthest
distance to the results this time, and algorithm proceeds to the loop again. The execution
of the algorithm is limited by the number of iterations, and in this study a heuristic limit
is assigned for the simulation purposes, which can be further adjusted systemically for the
application in hand.
The resulting control signals (u(t)) are then implemented through the ABS logic introduced
in Chapter 4. The brake signal is determined by means of a proportional regulator that
triggers the system according to the current and desired brake torque values. Next section
presents the results of the proposed ICC with the control allocation scheme in simulation
studies.
5.2.2
System Validation using Simulation
The control allocation algorithm detailed in the previous section is implemented in Matlab/Simulink
environment. Figure 5.2 illustrates the block diagram of the complete ICC including the upper level VSC integrated with the dynamic control allocation and lower level ABS algorithms.
The system is evaluated using the same evasive double lane change maneuver as in the VSC
study. The resulting corrective yaw-moment is fed into the control allocation algorithm
which distributes the wheel brake forces with respect to the actuator rates and saturation
while generating the required yaw-moment on the chassis. The following figures summarize
the results of the studied simulations. Figure 5.3 compares the control signals assigned by the
proposed control allocation theme to the results of the rule-based method to underline some
Chapter 5. Integrated Vehicle Control Systems
109
Upper-level Control
Driver
Inputs
Lower-level Control
Adaptive Vehicle
Stability Control
Dynamic Control
Allocation
Power
Steering Unit
Anti-lock Braking
Algorithm
Hydraulic
Control Unit
Vehicle Dynamics
Vehicle
Feedback
Figure 5.2: Integrated Chassis Control (ICC) system block diagram.
of the advantages of optimal brake distribution. The steering commands (Fig. 5.3a) are also
given with respect to the driver’s input, which is to be added to the controller signals. In
the control allocation algorithm, the sampling rate T for calculating actuator rate is taken
as 100Hz, which is also a common practice in most vehicle controller area network (CAN)
applications.
×104
1.5
0.06
Control Alloc.
0.04
Rule-based
Yaw Moment [Nm]
0.02
Steering [rad]
Control Alloc. Applied
1
Driver Input
0
−0.02
Rule-based
0.5
0
−0.5
−1
−0.04
−0.06
Control Alloc. Desired
0
2
4
6
Time [sec]
(a)
8
10
−1.5
0
2
4
6
Time [s]
8
10
(b)
Figure 5.3: Control signals assigned by the ICC scheme.
Figure 5.4 presents the allocated wheel brake torque values on each wheel using the proposed
algorithm in comparison to the rule-based method in Chapter 3. The control allocation
algorithm evidently reduces the peaks on each wheel as well as the abrupt precipitations.
This, as indicated in Figure 5.5, yields to significantly reduced oscillations in wheel slip simply
because of the feasible demand from the braking system rather than instinctive regulation.
The upper and lower limits both for saturation and actuator rates set in the control allocation
Chapter 5. Integrated Vehicle Control Systems
110
Front Left
Front Right
3000
3000
Rule-based
Control Alloc.
2000
1500
1000
500
0
Rule-based
Control Alloc.
2500
Brake Torque [Nm]
Brake Torque [Nm]
2500
2000
1500
1000
500
0
2
4
6
Time [s]
8
0
10
0
2
(a)
10
Rear Right
Rear Left
3000
Rule-based
Control Alloc.
Rule-based
Control Alloc.
2500
Brake Torque [Nm]
2500
Brake Torque [Nm]
8
(b)
3000
2000
1500
1000
2000
1500
1000
500
500
0
4
6
Time [s]
0
2
4
6
Time [s]
(c)
8
10
0
0
2
4
6
Time [s]
8
10
(d)
Figure 5.4: Wheel brake forces distribution by dynamic control allocation and rule-based
method.
algorithm helps with this smoother control action, which is also apparent in Figure 5.6 which
shows the tire force response against the slip variations. As the results indicate, the control
allocation algorithm helps with maintaining the tire forces at the possible peak that leads
to a significantly more efficient use of tire forces, whereas the rule based algorithm, though
does not saturate, but easily drives the tires above the feasible slip values.
Chapter 5. Integrated Vehicle Control Systems
111
Front Right
Front Left
Control Alloc.
Rule-based
0.8
0.6
Slip
Slip
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
Control Alloc.
Rule-based
0.8
0
2
4
6
Time [s]
8
−0.2
10
0
2
(a)
Control Alloc.
Rule-based
Control Alloc.
Rule-based
0.8
0.6
Slip
Slip
10
Rear Right
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
8
(b)
Rear Left
0.8
4
6
Time [s]
0
2
4
6
Time [s]
(c)
8
10
−0.2
0
2
4
6
Time [s]
8
10
(d)
Figure 5.5: Wheel slip variations by dynamic control allocation and rule-based method.
Chapter 5. Integrated Vehicle Control Systems
112
Front Right
5000
4000
4000
3000
3000
Force [N]
Force [N]
Front Left
5000
2000
1000
0
1000
0
Control Alloc.
Rule-based
−1000
−2000
−1
2000
−0.5
0
Slip
Control Alloc.
Rule-based
−1000
0.5
−2000
−1
1
−0.5
(a)
1
Rear Right
Rear Left
5000
Control Alloc.
Rule-based
4000
Control Alloc.
Rule-based
4000
3000
Force [N]
Force [N]
0.5
(b)
5000
2000
3000
2000
1000
1000
0
0
Slip
0
0.2
0.4
0.6
Slip
(c)
0.8
1
0
0
0.2
0.4
0.6
0.8
1
Slip
(d)
Figure 5.6: Tire force response in ICC with dynamic and rule-based control allocation
methods.
Chapter 5. Integrated Vehicle Control Systems
5.3
113
L1 Adaptive Control Method
The trend in adaptive control theory has taken the path of defining larger systems that can
be proven asymptotically stable via the well-known Lyapunov methods. Questions such as
at which location should the uncertainty appear, what should be the degree of mismatch,
how should the adaptive law be modified, etc., to get a negative definite or semidefinite
derivative of the associated candidate Lyapunov function for a new class of systems, can be
found in almost every study, including this one, addressing the current and next stage of
development in the theory of adaptive control. A very compelling archive is available on the
efforts for various adaptation techniques utilized over the years, counting from relaxation of
the matching conditions in strict-parametric feedback and feedforward systems [146, 147],
analysis of robustness of these schemes into unmodeled dynamics [148, 149], extensions to
global or semiglobal output feedback stabilization [150], systems with time-varying parameters [151, 152], nonminimum phase systems [153] and finally to relaxation of the relative
degree requirement via input-filtering [154].
These fundamental results provide sufficient conditions on the bounds of uncertainties and
initial conditions, nevertheless, in practical applications with nonlinear feedback systems
boundedness, ultimate boundedness, or asymptotic convergence might fall weak. Such applications require sufficient quantification of unmodeled dynamics, latencies and noise while
satisfying performance requirements; and both of these requirements play a cardinal role
in the adaptation process. Ideally, an adaptive controller would correctly respond to all
the changes in the system’s initial conditions, reference inputs, and uncertainties by quickly
identifying a set of control parameters that would provide a satisfactory system response,
and such satisfactory results are generally achievable with fast estimation schemes that demand high adaptation gains. Although theoretically possible, these high adaptation rates
in practice can create high frequencies in the control signals and increased sensitivity to
time delays. Therefore, an elementary matter with adaptive controllers is system robustness
in the presence of fast adaptation. A considerable number of studies in literature report
consistently limited rate of variation of uncertainties, by providing examples of destabilization due to fast adaptation [153]. The L1 adaptive control method attempts to address this
question of robustness with sufficiently fast adaptation by setting a architecture in place for
which adaptation is decoupled from robustness. This decoupled architecture allows satisfying
transient performance and robustness requirements by avoiding the need of gain scheduling,
Chapter 5. Integrated Vehicle Control Systems
114
persistence of excitation or resorting to unfeasibly high feedback gain. The other equally
important advantage of the L1 method in this application is the extended scope of adapta-
tion with the selected control structure. Rather than the adaptation of selected parameters,
the proposed L1 method allows for adaptation to a nonlinear function of time and system
states. Finally these features of L1 adaptive control theory have been verified in a compelling
number of publications and practical applications [155, 156, 157, 158].
The rest of this section facilitates the development of the L1 adaptive control method and
applies it to develop a novel VSC algorithm, which replaces the Lyapunov based algorithm
in the previous exercise. The section is concluded by the validation of the algorithm in the
same ICC scheme in simulation studies.
5.3.1
Derivation of the Control Algorithm
The control architecture developed in this section considers a semi-linear multi-input-multioutput (MIMO) system with uncertain nonlinearities as a function of system states and time.
The derivation aims to show that, subject to certain assumptions satisfied, the system can
be transformed to a linear system with time varying parameters and disturbances, and the
controller objective is to ensure the output of the system to track given reference signals
while satisfying semi-global performance criteria. To start with, the modified bicycle model
derived in Chapter 3 is reconsidered and extended. First form of extension is by adding
second order steering dynamics with the required steering column torque as the input to the
system. By this extension, it is aimed to involve actuator dynamics, partially if not entirely,
and move a step closer to practical application. Second form of extension is by considering
a nonlinear tire model rather than a linear approximation that can capture the saturation
Chapter 5. Integrated Vehicle Control Systems
115
effect of the tire force response. The resulting equations are rewritten as below:
2Cf (Iz + lf2 m)
u
2Cr (Iz − lf lr m
u
α̇f = −
+
αf −
−
αr
Iz mu
lf + lr
Iz mu
lf + lr
lf
u
δ + δ̇ −
Mb + θ1 (t, x)
+
lf + lr
Iz u
2Cf (Iz lf lr m)
u
2Cr (Iz + lr2 m)
u
α̇r = −
+
αf −
−
αr
mu
lf + lr
mu
lf + lr
lr
u
δ+
Mb + θ2 (t, x)
+
lf + lr
Iz u
bs
ks
δ̈ =
δ̇ +
δ + τc
Js
Js
(5.22)
where
θ(t, x) = [θ1

θ 2 ]T
2
n

 a11 x + a12 x + · · · + a1n x 

= 


a21 x + a22 x2 + · · · + a2n xn
(5.23)
represents the nonlinearities and saturation in tire response by complementing the linear
Cαi αi approximation by a polynomial function. Numerical analysis suggests that one can
successfully near to results of a MF tire model by an eight order (n = 8) polynomial. The
set of equations in 5.22 can be rewritten in state-space form as follows:
ẋ(t) = Ax(t) + Bu(t) + Θ(t, x)
(5.24)
where the coefficient matrices are given as:







A=





2Cf (Iz + lf2 m)
u
2Cr (Iz − lf lr m
u
u
+
−
−
−
−
Iz mu
lf + lr Iz mu
lf + lr
lf + lr
2Cf (Iz − lf lr m)
u
2Cr (Iz + lr2 m
u
u
−
+
−
−
−
Iz mu
lf + lr
Iz mu
lf + lr
lf + lr
0
0
0
0
0
−
ks
Js

−1 



0 



1 


cs 
−
Js
Chapter 5. Integrated Vehicle Control Systems

lf
 0 Iu
z


 0 − lr

Iz u
B=


0
 0


1
0






,





116

 θ1 (t, x)


 θ (t, x)
 2
Θ(t, x) = 


0



0












In what follows, the system dynamics are re-considered as in the following MIMO system:
ẋ = Ax(t) + B̄ (µ(t)x(t) + ωu(t)) + ξ(t)
y = Cx(t),
x(0) = x0
(5.25)
where the polynomial approximation functions θi (t, x) are evaluated as the summation
B̄ (µ(t)x(t) + ωu(t)) + ξ(t), ω represents an unknown constant input gain and the output
y(t) is given only in terms of the states. Provided that the linear part of the system dynamics (A ∈ Rn×n ) is Hurwitz and the input and output coefficients are the same rank
(B̄, C T ∈ Rn×m ), the control objective is to ensure that the outputs y(t) track given bounded
reference signals r(t) ∈ Rm both in transient and steady state. In the given form of system in
equation 5.25, the unknown parameters (ω) and nonlinearities are assumed to be bounded;
furthermore the nonlinearities are expected to be differentiable with bounded derivatives. In
the subsequent analysis, the terms and their Laplace transforms are used interchangeably
where s denotes the Laplace operator, mainly for brevity. The next step in the derivation is
to introduce a so-called passive identifier or state predictor with the estimated unknowns:
ˆ
x̂˙ = Ax̂(t) + B̄ (µ̂(t)x(t) + ω̂u(t)) + ξ(t),
x̂(0) = 0
(5.26)
where the adaptive estimates are given using the Projection operator:
˙
ω̂(t)
= ΓP roj ω̂(t), −(x̃T (t)P B̄)T uT (t)
˙
µ̂(t)
= ΓP roj θ̂(t), −(x̃T (t)P B̄)T xT (t)
˙
T
T
ˆ
ˆ
ξ(t) = ΓP roj ξ(t), −(x̃ (t)P )
(5.27)
The Projection operator augments robustness of the state estimates by enabling a tradeoff
with system performance. A major advantage of using this method in estimation is the
Chapter 5. Integrated Vehicle Control Systems
117
guaranteed boundedness of the estimates. In mathematical terminology, Projection operator
refers to a linear transformation between a given and a desired vector space. By using a
bounded space to project the given parameters, the operator avoids a divergence scenario.
Further details on the design of this operator are provided in Appendix A. It is also important
to note that the predictor as given above makes use of the system states (x(t)) which are
assumed to be known and the tracking error is given by x̃(t) = x̂(t) − x(t). The rest of the
parameters are defined as follows: Γ stands for the positive definite adaptation gain and P
represents the solution of the algebraic Lyapunov equation (AT P + P A = −Q < 0) which
guarantees the convergence of the predictor. Having defined the unknown parameters and
nonlinearities by the help of the adaptive state predictor, the control signal can be generated
by using these estimates, the given reference signal, system states and a low-pass gain filter:
u(s) − KCLP (s)r̄(s)
(5.28)
where K ∈ Rm×m is the constant control gain and the regulated reference signal is given as:
r̄(t) = µ̂(t)x(t) + ω̂u(t) + ξ(t) − Kg r(t)
(5.29)
The pre-filter gain Kg aims decoupling of the hypothetical reference model dynamics as given
below, which leads to trackable reference signals:
yr (s) = C(sI4×4 − A)−1 B̄Kg r(s)
= Gr (s)Kg r(s)
¯ operates on the estimate of the nonlinearities:
The other regulated signal ξ(t)
¯
ˆ
ξ(s)
= inv(B̄)ξ(s)
ˆ
= (Gr (s))−1 C(sI4×4 − A)−1 ξ(s)
(5.30)
The choice of the low-pass gain CLP (s) and the control gain K are to be made based on the
listed criteria being satisfied on the following transfer function:
Tf (s) = ωK(I + CLP (s)ωK)−1 CLP (s)
• Tf (s) is stable with Tf (0) = I and strictly proper,
Chapter 5. Integrated Vehicle Control Systems
118
• Tf (s)Gr (s)−1 is stable and strictly proper,
• The L1 norm condition should be satisfied, ||Ḡ(s)||L1 L < 1, where L represnets them
maximum L1 norm of µ(t)(L = max||µ(t)||L1 ) and the transfer function is given by:
Ḡ(s) = (sI − A)−1 B (I − Tf (s))
The transfer function Tf (s) is formed by considering the system stability as well as guaranteed
boundedness of the system inputs, that is selection of CLP (s) and K satisfying these criteria
will prevent any unrestrained increment of states with arbitrary u(t). Let’s further detail the
procedure to find the gain values satisfying these criteria. A strictly proper selection of the
low-pass gain CLP (s) also yields to strictly proper transfer function. The stability of Tf (s)
can be addressed by choosing CLP (s) and K to render −ωKCLP (s) Hurwitz. Similarly, the
multiplication Tf (s)Gr (s)−1 can be made proper by the choice of the low-pass gain Gr (s),
on the other hand its stability follows from the assumption that the poles of Gr (s)−1 reside
in the left-half plane. Securing the third criterion first requires the stability of Tf (s). Next,
based on the description of L1 norm, the following inequality holds true:
||Ḡ(s)||L1 ≤ ||sI − A)−1 B̄||L1 ||I − Tf (s)||L1
and one can deduce that forming the transfer function Tf (s) with its maximum eigenvalue
(max (Tf )) arbitrarily small yields to:
lim ||I − Tf (s)||L1 = 0
λ→−∞
As the eigenvalues of the transfer function Tf (s) are established by the selected gain values, it is shown that the last criterion can always be satisfied by fair choices of CLP (s) and K.
The given control architecture in equations 5.26-5.30 yields to satisfactory traction performance provided that the unknown parameters and nonlinearities are successfully predicted
to compute the control signal. Therefore the convergence of the predicted values poses an
important question of the practicality of the method. Next part summarizes a brief analysis
on the convergence of the state estimations and shows that they are strictly bounded, which
also indicates that the unknown parameters converge to expected values. Lyapunov analysis
allows estimating the bound on the prediction error of the system while proving stability.
Chapter 5. Integrated Vehicle Control Systems
119
The analysis is initiated by employing the following candidate function:
˜ = x̃T (t)P x̃(t)
V (t, x̃, ω̃, µ̃, ξ)
X
1
T
T
T
˜
˜
+
tr(ω̃ (t)ω̃(t)) +
µ̃ (t)µ̃(t) + ξ (t)ξ(t)
Γ
(5.31)
Assuming negligible variation of the input gain (ω̇ = 0), the time derivative of this function
is found as:
V̇ (t) = −x̃T (t)Qx̃(t) +
2 X T
˙
µ̃ (t)µ̇(t) + ξ˜T (t)ξ(t)
Γ
(5.32)
Since the unknown parameters and nonlinearities are to be bounded, the Projection operator
ensures that:
X
˜
max tr(ω̃ T (t)ω̃(t)) +
µ̃T (t)µ̃(t) + ξ˜T (t)ξ(t)
≤ 4 max(tr(ω T ω)) + M 2 + Ξ2
(5.33)
where ω, M and Ξ represent the bounds of the unknowns. Further assuming the bounds on
the time derivatives:
||µ̇(t)||∞ < dµ
˙
||ξ(t)||
∞ < dξ
yields to the following inequality:
max
X
˙ ≤ 2(M 2 dµ + Ξ2 dξ )
µ̃T (t)µ̇(t) + ξ˜T (t)ξ(t)
(5.34)
Next, consider the operator
λmax (P )
γ = 4 max(tr(ω T ω)) + M 2 + Ξ2 + 4
(M 2 dµ + Ξ2 dξ )
λmin (Q)
(5.35)
The positive definite matrices P and Q satisfy
P ≤ λmax (P )I,
λmin (Q)I ≤ Q
which then allows writing
x̃T Qx̃ ≥
λmin (Q) T
4
x̃ (λmax (P )I)x̃ ≥ (M 2 dµ + Ξ2 dξ )
λmax (P )
Γ
(5.36)
Chapter 5. Integrated Vehicle Control Systems
120
Using the inequality 5.34, it concludes that:
V̇ (t) < 0
(5.37)
which restates the stability. In what follows, from the initial conditions x̂(0) = x(0), one can
also deduce that:
V (0) ≤
γ
X
1
˜
tr(ω̃ T (t)ω̃(t)) +
µ̃T (t)µ̃(t) + ξ˜T (t)ξ(t)
<
Γ
Γ
(5.38)
and the continuity of the candidate function dictates the result:
V (t) ≤
γ
,
Γ
∀t ≥ 0
(5.39)
Finally, this result, together with the fact that λmin ||x̃(t)||2 ≤ x̃T (t)P x̃(t) ≤ V (t) yields to:
||x̃(t)|| ≤
r
γ
λmin (P )Γ
(5.40)
The inequality in 5.40 states the bound on the prediction error and together with 5.37
indicates the convergence of it. Proving that the predicted values converge as expected, the
control objective reduces to designing the gain values K and CLP (s) for desired performance.
5.3.2
System Validation using Simulation
The algorithm as summarized above is implemented in Matlab/Simulink environment for
numerical analysis and validation in simulation studies. Before moving forward with the
maneuver experiments, the stability criteria are evaluated so that an estimated set of acceptable control gains is formed. For the initial analysis, a relatively simpler low-pass gain
is selected as a symmetric matrix of first order transfer functions as in the below form:

a
 s+a
CLP (s) = 

0
0
a
s+a




by which the transfer function Tf (s) is also rendered stable for a similarly symmetric positive
definite gain matrix K (e.g. K × I2×2 ). In what follows, the L1 norm condition is evaluated
Chapter 5. Integrated Vehicle Control Systems
121
for varying cut-off frequencies on the selected low-pass gain (a). Setting the bound on the
uncertainty µ(t) as 1 (i.e. representing the maximum possible variation on surface friction),
numerical values of the norm condition can be computed. Figure 5.7 presents the resulting
norm values against the cut-off frequencies.
3
K=10
K=25
K=50
K=75
K=100
||G(s)||L1 L
2.5
2
1.5
1
0.5
0
0
10
20
30
Cut-off frequency (a)
40
Figure 5.7: The L1 norm with respect to varying low-pass and control gain.
The results imply that there exists a trade-off between the gain K and the cut-off frequency,
however it is possible to find a range of cut-off (a) values that renders the system stable
irrespective of the selected K. On the other hand, simulation results indicate that the cutoff frequency also affects the oscillations on the resulting control signals, and based on the
results and on the range found above, one can tune the low-pass gain for desired performance.
A sample study has been conducted by initiating a ramp steer input to the system. The
vehicle dynamics are simulated using the 8-degree-of-freedom model as derived in Chapter
2. The system is tested at 60kph speed and on high and low friction conditions (0.85 and
0.25 respectively). In what follows, the controller is expected to allow tracking a desired
yaw-rate while following the steering input that are taken as the system outputs.
This initial simulation indicates the adaptation capability of the algorithm, specifically to
the variations in the tire responses. The open-loop yaw rate variations indicate the difference of the vehicle response due to the changing surface friction and corresponding tire force
variations. Yet the system with the control intervention remains in a very reasonable vicinity
of the desired yaw-rate signal. Next, the system is put on test with a D-class vehicle model
from CarSim software for an evasive double lane change maneuver (DLC) as before. Two
Chapter 5. Integrated Vehicle Control Systems
122
4
Steering [deg]
3
2
1
0
0
1
2
Time [s]
3
4
Figure 5.8: Given ramp-steer maneuver.
sets of simulations are carried out for high and low friction conditions at respectively varying
speeds. The first test is run on high friction surface to negotiate the given DLC at a speed
of 33m/s (120kph). Following figures summarize the results. The results on the high friction condition indicate promising improvement of the control performance. Comparing the
control signal between the two methods, one difference can be seen in the continuity of the
steering input. The addition of the second order steering dynamics actually allows computing
a steering torque desired at the steering column as a more practical way of implementation,
and furthermore yields to the reduced oscillations and peaks on the resulting steering angle
degrees. A visible influence as seen on Figure 5.10 and Figure 5.11 is on the utilization of
the lateral tire forces, which can be attributed to the change in the adaptation strategy. The
proposed L1 method estimates and adapts to the nonlinearities in the system as a function
of states and time rather than to only selected parameters. This provides an advantage to
the L1 method, and furthermore the decoupled structure of adaptation law allows increasing
the adaptation gain values, considering a practical bound, to have better estimates of the
nonlinearities. Finally, the proposed method yields to a reduction in yaw-rate of the vehicle
of almost 0.10rad/s (or 5deg/s) in its peak. The previous section summarizes the variation
in the driver/passenger comfort levels with respect to yaw-rate which steps up or down at
about 0.12rad/s (or 7deg/s), which therefore indicates that the L1 method further improves
the drive comfort in addition to the control allocation while maintaining satisfactory stability.
Next, the vehicle is tested on the low friction surface (µnom = 0.25) with the same DLC
Steering [deg]
Chapter 5. Integrated Vehicle Control Systems
123
0
−1
High-mu
Low-mu
−2
0
0.5
1
1.5
0
0.5
1
1.5
2
2.5
3
3.5
4
2
2.5
Time [s]
3
3.5
4
Yaw moment [Nm]
3000
2000
1000
0
−1000
(a)
0.7
0.6
Yaw rate [rad/s]
0.5
0.4
0.3
0.2
Desired
Open-loop (high-mu)
Controlled (high-mu)
Open-loop (low-mu)
Controlled (low-mu)
0.1
0
−0.1
0
0.5
1
1.5
2
2.5
Time [s]
3
3.5
4
(b)
Figure 5.9: (a) Resulting control signals, and (b) yaw-rate values with and without the
control intervention on differing surface friction conditions.
maneuver and at a reduced speed of 22m/s (or 80kph). Following are the simulation results
in comparison to the Lyapunov based control algorithm performance.
Chapter 5. Integrated Vehicle Control Systems
124
×104
1
0.06
Steering [rad]
0.02
DYC Signal [Nm]
Driver
L1 Control
Lyap. Method
0.04
0
−0.02
−0.04
−0.06
0
2
4
6
Time [s]
8
10
L1 Control
LDM based
0.5
0
−0.5
−1
0
2
(a)
4
6
Time [s]
8
(b)
Figure 5.10: Control signals by the L1 and Lyapunov adaptive methods (a) steering input,
and (b) desired yaw-moment on high-mu.
The results indicate that the proposed L1 method can also satisfactorily adapt to vary-
ing surface conditions, and the corresponding change in the system dynamics. Similar to
the previous results, the control algorithm yields to extended utilization of the lateral tire
forces without leading to any instabilities or saturation conditions (Figure 5.14). The most
significant improvement can be seen on the vehicle yaw-rate response (Figure 5.15b). The
Lyapunov based algorithm successfully stabilizes the vehicle and allows negotiating the maneuver by avoiding any crash scenario. Nevertheless, the L1 method allows for further
reducing the yaw-rate variations while providing the same level of stability conditions, which
provides an increased feeling of safety to the driver and passenger(s).
10
Chapter 5. Integrated Vehicle Control Systems
125
Front
10
αf [deg]
5
0
L1 Control
Lyap. Method
−5
−10
2
0
4
6
8
10
6
8
10
Rear
10
αr [deg]
5
0
−5
−10
0
2
4
Time [s]
(a)
Front
Ff [N]
5000
0
−5000
−8
L1 Control
Lyap. Method
−6
−4
−2
0
Rear
2
4
6
8
−6
−4
−2
0
Time [s]
2
4
6
8
Fr [N]
5000
0
−5000
−8
(b)
Figure 5.11: (a) Tire slip-angle variations, and (b) corresponding tire lateral force
utilization.
Chapter 5. Integrated Vehicle Control Systems
Desired path
Open-loop
Lyap. Method
L1 Control
lateral distance [m]
6
4
2
0
0.4
−2
−4
0
50
100
150
200
longitudinal distance [m]
Lyap. Method
L1 Control
0.2
Yaw rate [rad/s]
8
126
0
−0.2
−0.4
250
0
2
(a)
4
6
Time [s]
8
10
(b)
Figure 5.12: (a) Vehicle CG trajectory in comparison with an open-loop system and with
Lyapunov based control method implemented (b) Comparison of vehicle yaw-rate with L1
control and Lyapunov methods.
×104
1
0.15
Steering [rad]
0.05
Yaw Moment [Nm]
L1 Control
Desired
Lyapunov
0.1
0
−0.05
−0.1
0
2
4
6
Time [s]
(a)
8
10
L1 Control
Lyapunov
0.5
0
−0.5
−1
0
2
4
6
Time [s]
8
(b)
Figure 5.13: Control signals by the L1 and Lyapunov adaptive methods (a) steering input,
and (b) desired yaw-moment on low-mu.
10
Chapter 5. Integrated Vehicle Control Systems
127
Front
αf [deg]
5
0
−5
L1 Control
Lyapunov
0
2
4
6
8
10
6
8
10
Rear
αr [deg]
5
0
−5
2
0
4
Time [s]
(a)
Ff [N]
2000
0
L1 Control
Lyapunov
−2000
−8
−6
−4
−2
0
αf [deg]
2
4
6
8
−6
−4
−2
0
αr [deg]
2
4
6
8
Fr [N]
2000
0
−2000
−8
(b)
Figure 5.14: (a) Tire slip-angle variations, and (b) corresponding tire lateral force
utilization.
Chapter 5. Integrated Vehicle Control Systems
Desired
L1 Control
Lyapunov
Open-loop
5
lateral distance [m]
128
0
−5
0
50
100
150
200
longitudinal distance [m]
250
(a)
1
L1 Control
Lyapunov
Yaw rate [rad/s]
0.5
0
−0.5
−1
0
2
4
6
Time [s]
8
10
(b)
Figure 5.15: (a) Vehicle CG trajectory in comparison with an open-loop system and with
Lyapunov based control method implemented (b) Comparison of vehicle yaw-rate with L1
control and Lyapunov methods.
Chapter 5. Integrated Vehicle Control Systems
5.4
129
Conclusion
This chapter introduces the concept of integrated vehicle control systems (ICC) that structure a centralized control strategy to allow employment of multiple safety systems and actuators focusing on distinct parts of the vehicle without diminishing each other’s performances.
A dynamic control allocation strategy is presented that is first used to integrate the stability
and anti-lock brake algorithms. The control allocation strategy optimally distributes the
wheel brake forces while avoiding any saturation or actuator over-rate scenarios. In what
follows, a new adaptive control method, the so-called L1 adaptive control, is introduced and
implemented to replace the upper-level control algorithm based Lyapunov’s direct method.
This new control strategy proposes adaptation to disturbances and nonlinearities in the
form of a function of time and states, which are not required to be known a priori. The
system dynamics introduced in Chapter 2 are then manipulated to combine the nonlinearities due to the tire force response on a time varying function, and the friction is taken as
a time varying disturbance, and this new form is used in the derivation of the control signals.
The new method is evaluated first with an open-loop structure only providing a ramp-steer
input on two different friction conditions. This maneuver is expected to drive the tire force
into the saturation/nonlinear region and the controller responds satisfactorily both to the
nonlinearities in the tire as well as the dynamics variations due to the change in the friction
conditions. Next, an evasive double lane change maneuver is executed as in the previous
experimentation. The results are presented in comparison with the first adaptive control
application. The performance of the new algorithm suggests further improvements over the
first method, mainly due to the extended adaptation capability. As a result, the proposed
ICC with the L1 adaptive control and gain-scheduling ABS proposes reliable safety levels
while minimizing the compromise in the drive comfort.
Chapter 6
Conclusion
6.1
Summary and Comments
As summarized in Chapters 1 and 2, the dynamics at the tire road contact have an immense effect on the vehicle’s handling and stability characteristics as the majority of the
forces and moments acting on the vehicle chassis are generated at the tire contact patch.
Sudden changes in the dynamics at this contact patch results in abrupt variations in vehicle characteristics which may lead to lose of control for the inexperienced driver. These
types of instabilities underline the importance of monitoring and utilization of the dynamic
variations of the tires and tire-road contact for improved vehicle controls and pose it as a
novel engineering problem. The active safety systems available today seek to prevent such
unintended vehicle behavior by assisting drivers in maintaining control of their vehicles.
Nevertheless, the lack of knowledge about the tire-road interactions highly limits their effectiveness. Recent studies in the field introduced the smart tire technology as a solution
for dynamic (real-time) tire monitoring. The smart tire technology suggests instrumenting
the tire itself for obtaining information on the momentary variations on tire conditions and
tire-road interactions.
Motivated by this opportunity and necessity in the field, this study develops tire parameter estimation algorithms and adaptive control strategies to improve vehicle handling and
braking performances. The algorithms use the so-called sensor fusion approach that integrates the smart tire technology and model based nonlinear observers to provide information
130
Chapter 6. Conclusion
131
on tire forces, slip-angle and surface friction conditions. The proposed control and observer
algorithms are evaluated using numerical analysis under challenging conditions. To get a
better measure of possible improvements in vehicle performance, the tests are executed together with baseline algorithms inspired by conventional systems available today. The results
demonstrate that the proposed algorithms can successfully negotiate the given tasks as well
as promising considerable improvements over the baseline systems.
The proposed stability algorithm takes advantage of the strong interdependency between
the tire slip-angles and the vehicle drive characteristics. Since the algorithm is based on a
modified bicycle model that replaces the states with front and rear tire slip-angles, it removes
the need for lateral velocity estimation which increases system complexity and introduces
intrinsic estimation errors (e.g. integral errors). Furthermore, due to the issues with lateral
velocity estimations, a common practice today is to deprive it out of the derivation by assuming zero as desired value. However, depending on the road conditions, especially on bank
and graded roads, this might result in undesirable deviations from stable behavior. Another
preferable outcome of the tire slip-angle based controller is the reduction in the control load
that minimizes the interference with the driver’s inputs while maintaining stability as well
as lessening the computational power requirements. The algorithm is also finalized with a
lower level optimal force distribution algorithm which further improves the performance by
minimizing the vehicle velocity variations which is a common issue with differential braking
based stability control systems.
The proposed adaptive ABS wheel slip controller utilizes a surface classification method
based on the integration of the smart tire technology with a model based observer scheme.
Motivated by the initial studies that proved utilization of this information is veritably favorable for enhancing the vehicle’s braking performance, this algorithm proposes an adaptation
scheme that optimizes a set of threshold values to switch between four phases that basically
increases, holds or decreases the applied brake torque/pressure value. The algorithm demonstrates significant improvements in braking distances under different surface conditions over
the baseline system. Another advantage of the proposed algorithm is that it follows the same
rule-based approach as the conventional ABS algorithms, which makes it easier to implement
on currently available brake systems on board.
Finally, an Integrated Chassis Control strategy is introduced by implementing a dynamic
Chapter 6. Conclusion
132
control allocation algorithm that bridges between the stability and brake controllers. Integrated control aims to improve the overall vehicle dynamics performance by derivation of
control laws in a unified fashion, based on dynamic models that account for all of the primary
directions (e.g. lateral, longitudinal, vertical) of interest for control. The main advantages
sought by the use of this approach are as follows.
• To avoid the conflicts and interventions among different subsystems.
• To exploit the potentials of the available processor units at their most feasible by
handling the communication and coordination among the sub-controls in the derivation
step.
This reduces the computational load on the electronic control unit and yields to the utilization of them only to compute and distribute the control signals. Taking care of the coordination task in the derivation step also assures dispelling of any possible conflicts among the
sub-control domains.
To summarize, the conclusions of this study are:
• New tire and vehicle mathematical models were developed based on smart tire technology,
• New Estimation algorithms for processing the smart tire accelerometer signal were
developed and used to estimate the tire slip-angle, dynamic wheel load variations and
road surface friction condition,
• The estimated parameters were included in the new control algorithms developed for
improving vehicle handling and stability and reducing stopping distance,
• New control vehicle control algorithms were developed using a slip angle based model
and it was shown that these algorithms can improve vehicle stability in severe driving
scenarios,
• A new ABS control algorithm was developed based on adaptation of the brake pressure
with respect to surface friction condition, which was shown to significantly improve the
vehicle braking performance.
Chapter 6. Conclusion
133
• The stability and braking controllers are integrated by introducing a dynamic control
allocation strategy that yields to an Integrated Chassis Control scheme.
All in all, this study provides a very promising first step in the introduction of smart tire technology into the active vehicle safety systems. With the introduction and wider use of smart
tire technology on vehicles in near future, these algorithms will be readily implementable.
To the author’s best knowledge, the technology is expected to be released into the market in
recent future starting with high-end sports cars. It is hoped that this study underlines the
potential of the integration of this new technology with the vehicle control systems, so that
current and future vehicle control systems would benefit from the smart tire technology at
its most.
The proposed algorithms provide a substantial base on the grand-scheme of integrating
smart tires and vehicle control systems. Regarding this base study, the course of this research is directed to investigate for further idealization and improvement in feasibility of the
control strategies and to study their evaluations by including hardware implementations.
The following section summarizes possible ideas for improvements and lists respective future
studies.
6.2
Future Extensions and Impact
The final remarks in terms of the future works of this study are about the implementation of
the proposed control algorithms. The studied concepts have been proven primarily through
numerical analysis. Although numerical methods are quite viable and widely accepted, they
require the system components such as sensor noise or measurement uncertainties to be
modeled, which, depending on the physical system in hand might not be a straightforward
process. Even with the inclusion of such components, the real world applications might introduce further unexpected and/or unmodeled uncertainties which might yield to unintended
performance alterations. In addition, these variations with the real world and simulation
implementation of the algorithms might delude with the improvement results in the comparison studies. To avoid these issues, the so-called Hardware-in-the-Loop (HiL) systems are
utilized, which relieves the cost of testing by not requiring a complete vehicle in the very cost
sensitive research environment; at the same time provides a reliable test-bed to imitate real
Chapter 6. Conclusion
134
world conditions. Therefore, a follow-up step of this study might be taken by implementing
and evaluating the proposed algorithms on a HiL system. An initial challenge concerning
the practicality of this research is then the adjustment of the algorithms for real-time application. The available coding for numerical analysis, including the optimization procedures,
might require further alteration of the selected solution methods specific for the hardware in
use. Furthermore, specific real-time strategies (i.e. hard, firm or soft) need to be determined
for respective sections of each algorithm.
The successful implementation of the algorithms on hardware will take the study to the
next state where they can be migrated to production vehicles for specific tuning and marketing. Based on the analysis as reported in the preceding chapters, these control systems
are expected to immensely improve individual vehicle safety as well as the traffic network
as a whole. The proposed study also holds a foundation to greatly benefit the Intelligent
Transportation Systems (ITS) that are subject to substantial interest today and anticipated
to form the future of ground transportation. The smart tire technology will play a central
role in binding all members of the transportation network from road surface to the central
supervisory offices, which ultimately will yield to an integrated ITS scheme. It will also
allow providing extensive information to form databases about various system states such as
road condition, congestion, speed limits, etc. which are crucial in the further development of
active safety systems but are not available today through any means of sensory equipment
in use.
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Appendix A
Projection Operator
As summarized in Chapter 5, the adaptive control strategies are being utilized in an expanding rate. Although being introduced in the midst of the 20th century, the lack of theoretical
background suspended the wide-spread of the adaptive techniques until late 1990s. A significant advantage of adaptive controllers is that they can account for system uncertainties
without requiring explicit system identification structures, which is particularly noticeable
on highly nonlinear systems such as automobile dynamics as addressed in this study. A key
assumption that needs to be persuaded though is that the estimates of these uncertainties
provided for the control signal computation are to be bounded to be able to guarantee the
so-called feedback stabilizability. The Projection Operator is introduced as a remedy in satisfying this condition In the context of the adaptation law, it basically allows transforming
the estimated parameters, say p, onto a convex and bounded set Π, which is assumed to define the upper and lower bounds of the expected parameters values, so that the estimations
are confined into a known span avoiding parameter drift. This appendix is aimed to provide
the formal mathematical description of the operator.
First, the notions of a convex set and convex function need to be defined as they appear
extensively through the following definitions.
Definition A set Ω ⊂ Rn is convex if for all x, y ∈ Ω the following holds true:
λx + (1 − λ)y ∈ Ω,
as illustrated below.
150
∀λ ∈ [0 1]
Appendix A. Projection Operator
151
(a)
(b)
Figure A.1: (a) Convex (b) non-convex sets.
Definition A function f : Rn → R is convex if for all x, y ∈ Rn the following holds true:
f (λx1 + (1 − λ)x1 ) ≤ λf (x1 ) + (1 − λ)f (x2 ),
∀λ ∈ [0 1]
An illustration is shown below.
f (x)
f (x)
x1
x2
x
Figure A.2: Convex function f (x).
Lemma A.0.1. Letting f : Rn → R be a convex function, the set (also known as the sublevel
set) Ωδ , θ ∈ Rn |f (θ) ≤ δ is convex for any arbitrary constant δ.
Appendix A. Projection Operator
152
The proof can be given as follows. Let θ1 , θ2 ∈ Ωδ , then f (θ1 ) ≤ δ and f (θ2 ) ≤ δ. Since f (x)
is convex ∀λ ∈ [0 1], one can find
f (λθ1 + (1 − λ)θ2 ) ≤ λf (θ1 ) +(1 − λ) f (θ2 ) ≤ λδ + (1 − λ)δ = δ
| {z }
| {z }
≤δ
≤δ
therefore f (θ) ≤ δ, thus θ ∈ Ωδ .
Lemma A.0.2. Let f (θ) : Rn → R be a continuously differentiable convex function, choose
an arbitrary constant δ and consider the convex set Ωδ , θ ∈ Rn |f (θ) ≤ δ. Also let θ∗ be an
interior point of Ωδ , i.e. f (θ∗ ) < δ. Finally choose θb as a boundary point so that f (θb ) = δ.
Then the following holds true:
where ∇f (θb ) =
∂f (θ)
∂θ1
. . . ∂f∂θ(θ)
k
T
(θ∗ − θb )T ∇f (θb ) ≤ 0
is the gradient vector of f (.) evaluated at θb .
The proof can be given as follows. Since f (θ) is assumed convex, one can write,
f (λθ∗ + (1 − λ)θb ) ≤ λf (θ∗ ) + (1 − λ)f (θb )
which equivalently leads
f (θb + λ(θ∗ − θb )) ≤ f (θb ) + λ(f (θ∗ ) − f (θb ))
so for any 0 < λ ≤ 1
f (θb ) + λ(θ∗ − θb )) − f (θb )
≤ f (θ∗ ) − f (θb ) ≤ δ − δ = 0
λ
and hence taking the limit λ → 0 yields to the given lemma.
In what follows the projection operator can be introduced as below.
Definition Consider a convex bounded set with smooth boundary given as by
Ωc , {θ ∈ Rn |f (θ) ≤ c} ,
∀c ∈ [0 1]
Appendix A. Projection Operator
153
where f : Rn → R is the following smooth convex function
f (θ) =
2
(θ + 1)θT θ − θmax
2
θ θmax
with θmax being the norm bound imposed on the vector θ, and θ > 0 is the projection
tolerance bound of choice. the projection operator is defined as:



y,
if f (θ) < 0




if f (θ) ≥ 0 and ∇f T y ≤ 0
f (n) = y,



∇f (θ)(∇f (θ))T


yf (θ), if f (θ) ≥ 0 and ∇f T y > 0.
y −
2
||∇f (θ) ||
where ∇f (θ) is the function gradient evaluated at θ.
A geometrical interpretation of the operator in R2 can be given as follows to further clarify.
Lets define a convex set Ω0 as:
Ω0 , θ ∈ R2 |f (θ) ≤ 0
and let Ω1 represent another convex set such that
Ω1 , θ ∈ R2 |f (θ) ≤ 1
which also yields to Ω0 ⊂ Ω1 . Finally let ΩA , {θ|0 < f (θ) ≤ 1} represent an annulus region
on R2 . Within ΩA the projection algorithm subtracts a scaled component of y that is normal
to boundary {θ|f (θ) = λ}. When λ = 0, the scaled normal component us 0, and when λ = 1,
the component of y that is normal to the boundary Ω! is entirely subtracted from y, so that
P roj(θ, y) is tangent to the boundary {|f (θ) = 1}.
Based on this example, a significant property of the projection operator can be given as
follows:
(θ − θ∗ )T (P roj(θ, y) − y) ≤ 0
where θ∗ ∈ Ω0 . The proof starts by noticing
(θ − θ∗ )T (P roj(θ, y) − y) = (θ∗ − θ)T (y − P roj(θ, y))
(A.1)
modified when θ ∈ Ω0 . Let
ΩA , Ω1 \Ω0 = θ|0 < f (θ) ≤ 1
represent an annulus region. Within ΩA the
projection algorithm subtracts a scaled component of y that is normal to boundary θ|f (θ) = λ}. When λ = 0, the scaled normal
component is 0, and when λ = 1, the component of y that is normal to the boundary
Ω1 is
Appendix
Projection
Operator
entirelyA.subtracted
from
y, so that Proj(θ, y, f ) is tangent to the boundary θ|f (θ) = 1 .
This discussion is visualized in Figure 1.
y
Proj(θ, y)
{θ|f (θ) = 1}
ΩA
∇f (θ)
θ
θ∗
154
{θ|f (θ) = 0}
Figure
1: Visualization
of Projection
Operator inoperator
R2 .
Figure A.3:
Graphical
illustration
of the projection
in R2 .
Remark. Note that (∇f (θ))T Proj(θ, y) = 0∀θ when f (θ) = 1 and that the general structure
which
> 0 and
∇f
(θ)T y > 0, equals to
of if
thef (θ)
algorithm
is as
follows
(θ − θ) (y − P roj(θ, y)) = (θ − θ) y − y −
∗
T
Proj(θ, y) = y − α(t)∇f (θ) ∇f (θ)(∇f (θ))T
∗
T
2
yf (θ)
(5)
||∇f (θ) ||the left hand side
for some time varying α when the modification is triggered. Multiplying
of the equation by (∇f (θ))T and solving for α one finds that
and 0 otherwise. Finally, following the second Lemma
yields
α(t) = (∇f (θ))T ∇f (θ)
≤0
z the form
}|
and thus the algorithm takes
∗
T
−1
>0
(∇f (θ))T y
(6)
≥0
{ z }| { z}|{
(θ − θ) ∇f (θ) (∇f (θ))T y f (θ)
−1 ≤ 0 T
(∇f (θ)) yf (θ)
Proj(θ, y) = y − ∇f (θ)||∇f
(∇f(θ)||
(θ))2T ∇f (θ)
(7)
the modification is active. Notice that the f (θ) has been added to the definition,
so inwhere
tabular
form
making (7) continuous.

∗
Lemma 6. One important
property
of the projection operator follows.
0,
if f Given
(θ) < θ0 ∈ Ω0 ,






(θ − θ ∗ )T (Proj(θ, y, f ) − y) ≤ 0.
T
0,
if f (θ) ≥ 0 and ∇f(8)
y≤0
∗
T
(θ − θ) (y − P roj(θ, y)) =
≤0
>0
≥0


}|
{ z }| { z}|{
z


∗
T
T

(θ − θ) ∇f (θ) (∇f (θ)) y f (θ)



, if f (θ) ≥ 0 and ∇f T y > 0
2
||∇f (θ)||
which leads to the above stated property.
The use of the projection operator can be presented in an example Model Reference Adaptive
Control (MRAC) scheme as follows. Consider the first order system
ẋ = Ap x + Bp u,
y = Cx
x(0) = x0
Appendix A. Projection Operator
155
where Ap and Bp are constant plant parameters. The MRAC scheme proposes computing a
control input u that allows this system to track the dynamics of a desired reference system
drive by a reference signal given as
ẋm = Am xm + Bm r
where Am and Bm are constant reference parameters and r is a bounded external reference
signal. Attention must be paid to the stability of the reference model (i.e. Am is a known
Hurwitz matrix). The MRAC control law is given by:
u = −kx (t)x + kg r
where kx and kg are the feedback gain values. The closed loop dynamics then becomes:
ẋ = (Ap − Bp kx )x + Bp kg r
which allows the possibility of perfect reference model matching by the nominal feedback
gains:
Ap − Bp kx∗ = Am ,
Bp kg∗ = Bm
An interesting choice of kg is as follows which leads to zero steady-state error:
kg =
1
CA−1
m Bp
However, it is usually the case that the plant parameters Ap and Bp are unknown to us.
Therefore an adaptation law is introduced to compute the ideal parameters that is expected
to approximate to the nominal gain values as given above:
˙
k̂(t) = −ΓxeT P Bp ,
k̂x (0) = kx0
where Γ is the adaptation gain, P = P T > 0 is the solution for the algebraic Lyapunov
equation ATm P + P Am = −Q for arbitrary Q = QT > 0, and the error signal e(t) proceeds
by the following dynamics:
ė = ẋ − ẋm
= −am (x − xm ) + (am − ap + bp âx )x + (bp âr − bm )r
Appendix A. Projection Operator
156
The convergence of this error dynamics can be easily proven by Lyapunovs direct method.
The following candidate function:
V (t) = eT P e +
1
(k̂x − kx∗ )T (k̂x − kx∗ )
Γ
Defining k̃x = k̂x − kx∗ leads to the derivative:
2
V̇ (t) = −eT Qe + 2eP Bp (k̂x − kx∗ )T x + k̃xT k̃˙ x
Γ
1˙
= −eT Qe + 2k̃xT
k̂x + xeT P Bp
Γ
T
= −e Qe ≤ 0
which proposes stability. As for the asymptotic convergence, one can resort to the Barabalats
lemma, which proposes that a differentiable function V (t) has a finite limit as t → ∞ and
uniformly continuous, then one can conclude that V̇ (t) → 0 as t → ∞. It is clear that the
candidate function satisfies the finite limit condition. For uniform continuity, the second
derivative can be checked
V̈ (t) = −2eT Qė
As the given form of the error dynamics ė(t) is assumed to be bounded, we can conclude
that V̇ (e) is uniformly continuous and thus its limit converges to zeros as t tends to infinity,
which also proves that e(t) → 0 as t tends to infinity. A key assumption in this derivation,
as mentioned earlier, is the boundedness of the error signal which is directly dependent on
the boundedness of the adaptation law results. Therefore, the projection operator can be
utilized to theoretically guarantee this condition. Lets consider the adaptation law above
implemented by the projection operator as follows
˙
k̂ = ΓP roj(kx , −xeT P Bp ),
kx (0) − kx0
and follow the same candidate function by simply replacing the adaptation law which yields
to
V̇ (t) = −eT Qe + 2k̃x P roj(kx , −xeT P Bp ) + xeT P Bp
Then, from the property B-1 of the projection operator as proven previously, one can write
k̃x P roj(kx , −xeT P Bp ) + xeT P Bp ≤ 0
Appendix A. Projection Operator
157
which conveniently yields to
V̇ (t) ≤ −eT Qe ≤ 0
This, as detailed above, proves the stability of the system and asymptotic convergence of
the error signals, while the projection operator ensuring the boundedness of the adaptation
law.
Appendix B
Case Study - L1 Adaptive Control of
an Active Suspension System
This study investigates the implementation of an adaptive control strategy in active suspension controls. The derivation of the control algorithm and the execution of initial simulations
are carried out using a quarter-car model. Road profiles are generated for ISO-D and E grade
surface conditions using standard power spectra that served as the system input. To further
test the adaptation of the control algorithm and the system response, additional undulations
(e.g. bump and ditch) are included on the profile. The dimensions of these additional challenges are adjusted as the physical limitations of the considered suspension system allows.
The results of the simulation studies indicate that the adopted control algorithm yields to
very promising results in terms of attenuating any excess body acceleration and body travel
response that might deteriorate ride comfort. Based on the acceleration margins suggested
by the ISO 2631 standards, the proposed control algorithm successfully leads the actuators
to isolate the vibrations excited due to the road profile. Finally, the controller is tested
using a full-vehicle model of a D-class sedan, implemented in CarSim software. The vehicle
is assumed to be equipped with four fully active suspensions. A D-grade rough asphalt with
two standard cleats is used as a road profile to excite the system in the frequency range of
interest, and the results indicate that the proposed adaptive control strategy significantly
improves the isolation capabilities of the suspension system.
158
Appendix B. L1 Adaptive Control of an Active Suspension System
B.1
159
Introduction
The design of an effective suspension system requires a number of conflicting requirements to
be met. The suspension setup has to ensure a comfortable ride while proving good road holding and cornering characteristics. Also, optimal contact between tires and the road surface
is needed in various driving conditions in order to maximize braking capacity and thereby
safety. The conventional suspension designs in use today come with preset characteristics
and aim to achieve vibration isolation through passive means, which might not be able to
resolve the trade-off between such conflicting performance requirements.
This study focuses on the application of an adaptive control methodology for an active
suspension aiming driver comfort as well as vehicle stability. Specifically, the use of L1
adaptive control methodology (Hovakimyan and Cao, 2006) is investigated, which proposes
to improve robustness of the overall control strategy as well as guaranteeing stability in the
case of parameter variations. Analysis on a quarter-car model shows that the proposed adaptive control methodology is superior to a robust H∞ control algorithm that is most generally
used in conventional systems. Further simulations using a full vehicle model using CarSim
software shows the proposed method is capable of reducing the sprung mass vibrations below
comfort levels (ISO-2631) as well as guaranteeing transverse stability.
B.2
Mathematical Model and Control Algorithm
Consider a quarter car model of suspension system as shown in Figure B-1. The sprung
mass ms represents the constant mass of the car body, frame, internal components that are
supported by the suspension. The sprung mass might vary according to the loading of the
car and the adaptive control method is expected to overcome such variations. The unsprung
mass mus is the mass of the assembly of the axle and the wheel, ks and bs are respectively
the spring and damper coefficients of the passive components of the suspension system. The
suspension spring constant ks comprises of a linear stiffness coefficient however the option
to combine with nonlinear components is left open. The coefficient ks and bs are assumed
to have known initial values. The coefficient kt is the linear tire radial spring constant.
The control force generated by the active actuator connected between sprung and unsprung
masses is denoted by fs while r denotes the road disturbance input acting on the unsprung
Appendix B. L1 Adaptive Control of an Active Suspension System
160
mass. The vertical displacements of the sprung and unsprung masses with respect to their
undeformed suspension positions are denoted by xs and xus respectively.
Using the above parameters, the dynamic model of the suspension system is given as follows:
ms ẍs = −bs (ẋs − ẋus ) − ks (xs − xus ) + fs
mus ẍus = bs (ẋs − ẋus ) + ks (xs − xus ) − kt (xus − r) − fs
(B.1)
From equation 1 the state space representation is described as follows:
Figure B.1: Quarter car suspension model with single point tire-road contact

1
0
0
 0

 k
bs
ks
bs

s
−
 −

ms
ms
ms
A =  ms

 0
0
0
1


 ks
bs
(ks + kt )
bs
−
−
mus mus
mus
mus







,











B1 = 





0
1
ms
0
−
1
mus
ẋ(t) = Ax(t) + B1 u(t) + B2 r(t)






,






 0


 0

B2 = 

 0


 k
t
mus












Appendix B. L1 Adaptive Control of an Active Suspension System
161
where x ∈ R4 , u = fs ∈ R.
Table B.1: Quarter vehicle parameters
Sprung Mass
ms
300kg
Unsprung Mass
mus
60kg
Damper Coefficient
bs
N
1, 000 m/s
Spring Coefficient
ks
16, 000 N
m
Tire Stiffness
kt
190, 000 N
m
For the control application, the output y(t) is taken as the travel of the sprung mass (xb ),
hence C = [1 0 0 0], D = 0. With this choice of output, the system dynamics can be
re-written in the following format:
ẋ(t) = Ax(t) + b(wu(t) + r(t))
y(t) = Cx(t),
x(0) = x0
(B.2)
Furthermore we can assume a symmetric set of upper and lower bounds for the disturbance
r(t):
|r(t)| ≤ ∆
(B.3)
where ∆ is a known conservative L∞ bound of r(t), which in this application becomes the
limits on the road undulations. We further assume that the road disturbance, r(t), is in the
form of a continuously differentiable signal with bounded derivatives. The control objective
with the L1 adaptive method is to design a full-state feedback controller to ensure that the
system output, y(t), tracks a given bounded reference signal, rt (t), both in transient and
steady state, while all other error signals remain bounded. With this definition in hand,
we consider the following state estimator for the quarter-car system dynamics with possible
nonlinearities:
˙
x̂(t)
= Ax̂(t) + b(wu(t) + θ̂(t)x(t) + r̂(t))
ŷ(t) = C x̂(t),
x̂(0) = x0
(B.4)
which replaces the unknown nonlinearities in the form of θ(t)x(t) with θ(t) as any nonlinear
function of t and unknown disturbances (r(t)) with an adaptive estimation governed by the
Appendix B. L1 Adaptive Control of an Active Suspension System
162
below equations:
˙
r̂(t)
= Γr P roj r̂(t), −x̃T P b
˙
T
θ̂(t) = Γθ P roj θ̂(t), −x(t)x̃ P b
(B.5)
In equation B.5, x̃ defines the error signal which is given as x̃(t) = x̂(t)−x(t). The adaptation
gains are given by [Γθ , Γr ] ∈ R+ and P is the solution of the algebraic Lyapunov equation
AT P + P A = −Q, Q > 0. After defining the system dynamics and state estimator, the
control signal can be generated in the following form:
u(s) = C(s)usub (s)
(B.6)
where the sub-control usub is given as:
kg rt (t) − θ̂T (t)x(t) − r̂(t)
w
1
=
CA−1 b
usub =
kg
(B.7)
(B.8)
In this definition, C(s) is a strictly proper, stable transfer function with low-pass gain C(0) =
1. The stability of the closed loop system is guaranteed if one can design C(s) to satisfy the
L1 gain stability requirement:
||G(s)||L1 L < 1,
(B.9)
where G(s) = (sI − A)−1 b(1 − C(s)). Implementing C(s) with a low-pass filter effect allows the proposed control methodology to modify the loop transfer function such that the
phase-margin of the system is expected to improve with increasing adaptation gain [ref].
As well known, with conventional adaptive control methods, increasing adaptation gains
actually yields to better tracking performance but at a cost of observing higher gain crossover frequencies which naturally reduces the phase margin and weakens the relative stability/robustness of the system. Whereas careful selection of the gain transfer function C(s)
allows for satisfactory tracking as well as improving/guaranteeing relative stability. A sample case is studied using a linear approximation to the proposed controller implemented on
the quarter-car dynamics. The system is trimmed down to a SISO transfer function with
control force (fs ) as input and body travel (xb ) as the output. This linear approximation
allows the use of tools from classical controls. Figure 1.2 shows the frequency response of
Appendix B. L1 Adaptive Control of an Active Suspension System
163
the plant (quarter-car) and the controller-plant loop transfer functions with increasing adaptation gains, (Γ). Comparing the results for an adaptive integral control and L1 methods,
one can observe that L1 method manages to limit the shift of the gain cross-over frequency
with increasing Γ and consequently allows pulling up the phase margin. The complete L1
Adaptive Control
Magnitude (dB)
100
0
−100
−200
10−1
100
101
Frequency (rad/s)
102
103
L1 Method
Magnitude (dB)
100
0
−100
−200
10−2
100
Frequency (rad/s)
102
Figure B.2: Frequency response comparison for the linearized QC plant with integral
adaptive and L1 adaptive methods
adaptive controller consists of equations B.4-B.6 subject to L1 gain stability requirement in
equation B.9. The closed loop system is illustrated in Figure B.2.
Appendix B. L1 Adaptive Control of an Active Suspension System
u
State Estimation
Quarter Car
164
x̂
x
+
−
Adaptive Law
rt
θ̂, σ̂
kg rt − θ̂T x − σ̂ /ω
C(s)
Control Law
Figure B.3: Closed loop block diagram of the L1 control scheme
B.3
System Validation in Simulation
The efficiency of the proposed L1 control scheme is tested through a series of computer simulations. The road profile is given as a disturbance and the body travel, body acceleration
and the suspension deflection are plotted as the outputs. First couple of road profiles are
given as simple obstacles to test the controller performance. Figure ?? shows the response of
the proposed active suspension system in comparison to a passive counterpart. The results
indicate quite satisfactory reduction of sprung mass oscillation within reasonable actuation
requirements.
Next the proposed controller is compared to a widely used robust control
methodology, namely the H∞ control. The details of the adopted derivation methodology
is given by Ezzine and Tedesco [] in further details and skipped here for bravity. In this
application, the algorithm is specifically modified to minimize body acceleration (ẍs ) while
avoiding resonance at the tire-hop and rattlespace frequencies. The road disturbance is taken
from FHWA Long-term Pavement Performance (LTTP) database with max. 0.05m undulation. The system is introduced with a nonlinear spring coefficient for the suspension and
the tire stiffness in the form of k(1 − sin(2πfr t)). Furthermore, for more realistic simulation
results, the dynamics for a hydraulic actuator are implemented as an addition to the above
described spring nonlinearities. The dynamics are taken as a first order transfer function
(Ga (s) = 45/(s + 45)) which leads to 0̃.015s time delay in the application of the control
Appendix B. L1 Adaptive Control of an Active Suspension System
Road Disturbance
165
Control force
0.15
500
0
0.1
Y [m]
fs(t) [N]
−500
−1000
0.05
−1500
−2000
0
0
1
2
3
4
5
−2500
0
1
2
Time [s]
3
4
5
Time [s]
Body acceleration
Body Travel
5
0.1
L1 Control
Open Loop
L1 Control
Open Loop
xb’ [m/s2]
xb [m]
0.05
0
0
−0.05
−0.1
0
1
2
3
Time [s]
4
5
−5
0
1
2
3
4
5
Time [s]
Figure B.4: L1 controller performance with bump profile.
signal. Figure B.4 summarizes both controller performances for the same road disturbance.
The results show that the H∞ control method successfully rejects any possible resonance
response and noticeably reduces oscillations. The body acceleration is effectively reduced as
the design objective of the algorithm requires. Nevertheless the L1 control method yields
to superior performance in terms of both selected indicators. Although this improved performance comes at a cost of higher actuator loading by the L1 method, the peak values
still remain in a reasonable range. Finally, Table B.2 shows the root-mean square (RMS)
of body acceleration values which is a standard indicator for ride comfort. As the results
suggest, both controllers can cope with the nonlinearities and actuator dynamics, and reduce
the body accelerations remarkably, but the proposed adaptive method provides exceeding
improvements.
Appendix B. L1 Adaptive Control of an Active Suspension System
Road Disturbance
166
Control force
0.15
2000
1500
0.1
1000
500
fs(t) [N]
Y [m]
0.05
0
−0.05
0
−500
−1000
−1500
−0.1
−2000
0
5
10
15
−2500
0
5
Time [s]
10
15
Time [s]
Body acceleration
Body Travel
8
0.15
L1 Control
Open Loop
0.1
L1 Control
Open Loop
6
4
xb’ [m/s2]
xb [m]
0.05
0
2
0
−2
−0.05
−4
−0.1
0
−6
5
10
15
−8
0
5
10
15
Time [s]
Time [s]
Figure B.5: L1 controller performance with bump and ditch profiles.
Table B.2: Body Acceleration RMS for H∞ and L1 strategies
Open-loop
0.830 m/s2
H∞ Control
0.502 m/s2
L1 Control
0.210 m/s2
In what follows, the control algorithm is implemented on a full vehicle model in CarSim software. The vehicle (Figure B-7a) is assumed to be equipped with active-suspension actuators
on the front and rear wheels, and control signal is computed for each actuator as the control
force, fs . The vehicle is tested on an ISO-D grade (rough asphalt) road profile on which two
standard cleats (1200 × 200 ) are placed at 30m distance (Figure B-7b). The response for the
passive system and the control application are summarized on Figure B-8.
Appendix B. L1 Adaptive Control of an Active Suspension System
Control force
0.04
800
0.03
600
0.02
400
0.01
200
fs (N)
Y (m)
Road Distrubance
0
−200
−0.02
−400
−0.03
−600
2
4
6
8
10
H∞ Control
L1 Control
0
−0.01
−0.04
0
−800
0
2
4
Time (s)
6
8
10
Time (s)
Body acceleration
Body travel
4
0.04
Open−loop
H∞ Control
3
Open−loop
H∞ Control
L1 Control
2
L1 Control
0.02
x’b (m/s2)
0.03
0.01
xb (m)
167
0
1
0
−0.01
−1
−0.02
−2
−0.03
−3
−0.04
0
2
4
6
Time (s)
8
10
−4
0
2
4
6
8
10
Time (s)
Figure B.6: Comparison of the controller performance for H∞ and L1 control methods
The performance of the control algorithm is evaluated based on the thresholds provided
by the ISO 2631 standards. Figure B-9 shows weighted acceleration values for the duration
of exposure that is suggested by the ISO and the weighing factors for the range of frequencies (0.5 − 80Hz) for which the human body is expected to be most sensitive. ISO-2631
standard suggests that these acceleration thresholds to be increased by 6dB (doubled) for
fatigue-limited proficiency boundaries and to be decreased by 10dB (divide three-fold) for
comfort boundaries.
The selected road profile excites the vehicle body at lower frequen-
cies (fr < 3.5Hz) which are more often related to comfort/fatigue related problems. The
duration of exposure is taken as 8hr. in general as given the standard maximum driving
period, and based on the thresholds as given in Figure B 9 the fatigue limit for the frequency
Appendix B. L1 Adaptive Control of an Active Suspension System
168
Road Disturbance
0.05
0.04
Elevation [m]
D-Class/E-Class/F-Class Sedan
0.03
0.02
0.01
0
−0.01
0
10
20
30
40
50
60
Travel [m]
(a) CarSim D-Class Sedan Vehicle
(b) Given road profile
Figure B.7: B-Class/C-Class
Test bedHatchback
for the proposed adaptive active suspension control algorithm
Control force
Body Acceleration
Body Travel
8
0.05
Open loop
L1 Control
5000
Open loop
L1 Control
6
0
2
fs [N]
xs’ [m/s2]
xb [m]
4
0
0
−2
−4
−6
−0.05
0
1
2
3
4
−8
0
5
1
2
3
4
−5000
0
5
1
2
3
4
5
4
5
Time [s]
Time [s]
Time [s]
(a) Front
Control force
Body Acceleration
Body Travel
5000
8
0.05
Open loop
L1 Control
Open loop
L1 Control
6
0
2
fs [N]
xs’ [m/s2]
xb [m]
4
0
0
−2
−4
−6
−0.05
0
1
2
3
Time [s]
4
5
−8
0
1
2
3
4
5
−5000
0
1
Time [s]
2
3
Time [s]
(b) Rear
Figure B.8: Vehicle response to the road profile
range of interest is found as 1̃.5m/s2 and the corresponding comfort limit can be found as
0̃.37m/s2 . The resulting weighted accelerations are then evaluated with respect to these
Appendix B. L1 Adaptive Control of an Active Suspension System
169
2
Wieghted Accel. (m/s )
28
18
10
6.5
4
2.5
1.5
1
0.6
0.3
0.15
0.1
1 Hz.
2.5 Hz.
8 Hz.
25 Hz.
50 Hz.
80 Hz.
0.1
0.5
1
2
4
8
16 24
Exposure Duration (h)
(a) Vibration limits as a function of exposure times
Weighing Factor (dB)
5
0
−5
−10
−15
−20
1
2
4
8
16
31.5
63
Frequency,f (Hz)
(b) Weighting factors for longitudinal vibrations
Figure B.9: Acceleration thresholds specified by ISO-2631
values as shown in Table B.3. The summarized acceleration values indicate that the passive
suspension system is actually a good design that can remain the body inertance vibrations
below the specified fatigue level. Adding the actuators with the proposed adaptive control
on the other hand takes the vibration levels down to the comfort levels which is especially
significant in driving for long durations.
Appendix B. L1 Adaptive Control of an Active Suspension System
170
Table B.3: Body Acceleration RMS in CarSim Simulations
B.4
Passive System
Active System
Front Suspension
0.83m/s2
0.39m/s2
Rear Suspension
0.82m/s2
0.40m/s2
Conclusion
Active suspension systems are becoming available and feasible in a considerably increasing
rate. They are designed not only to improve the ride quality and comfort but also the safety
of the vehicle. As extreme cornering may be required to remain on the road or to avoid
an obstacle, implementing the active suspension system are capable of significantly reducing roll-over index. Moreover, as the active part of the suspension would be taking care
of realizing good cornering behavior and of static load variations, the primary suspension
springs can be tuned purely for optimizing comfort and road holding. Simulations show that
the required force and energy for leveling the car even on extreme bounce conditions are
reachable, so it can be concluded that the active suspension system is able to reasonably
level the car.
The proposed adaptive algorithm is also compared to other commonly utilized controllers
in conventional systems. To obtain a better measure of the algorithm’s capabilities it is
compared to a widely used robust algorithm, namely the H∞ method. The results indicate
that the oscillation levels reduce at a very close rate for both methods, nevertheless the
proposed algorithm is capable of better improving the ride comfort by significantly reducing
the vehicle body motions, as the algorithm is derived based on that objective. Finally the
algorithm is put into examination on a more realistic test-bed by using a vehicle model from
the CarSim software. The suspension design for the selected D-class sedan vehicle is taken as
the base model for the comparison study. The active system yields to superior performance
in both courses of evaluation by reducing the peak body travel and the body acceleration rms
almost about half in comparison to the passive suspension. All in all, the performance of the
proposed algorithm is found very promising for utilizing in active suspensions and dampers
which require high adaptability and robustness at the same time due to the rapidly varying
conditions they operate under. The L1 adaptive control method provides such features by
allowing one to modify the controller to both satisfy stability and robustness while providing
Appendix B. L1 Adaptive Control of an Active Suspension System
171
adaptability. As a follow-up of this study, further development of trivially selected low-pass
gain could be considered which would improve the control response on parts of the given
road profile with higher-frequency contents as well as helping with regulating the control
efforts.
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