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Backstepping Control Design and Its Applications to Vehicle Lateral
Control in Automated Highway Systems
by
Chieh Chen
B.Eng. (National Taiwan University, R.O.C) 1990M.Eng. (University of California, Berkeley) 1995
A dissertation submitted in partial satisfaction of the
requirements for the degree of
Doctor of Philosophy
in
Engineering-Mechanical Engineering
in the
GRADUATE DIVISION
of the
UNIVERSITY of CALIFORNIA at BERKELEY
Committee in charge:
Professor Masayoshi Tomizuka, ChairProfessor Karl HedrickProfessor Pravin Varaiya
1996
The dissertation of Chieh Chen is approved:
Chair Date
Date
Date
University of California at Berkeley
1996
Backstepping Control Design and Its Applications to Vehicle Lateral
Control in Automated Highway Systems
Copyright 1996
by
Chieh Chen
1
Abstract
Backstepping Control Design and Its Applications to Vehicle Lateral Control in
Automated Highway Systems
by
Chieh Chen
Doctor of Philosophy in Engineering-Mechanical Engineering
University of California at Berkeley
Professor Masayoshi Tomizuka, Chair
In this dissertation the e�ort is to explore new aspects of recursive backstepping
design methodology from both theoretical and application point of view. Three main
topics are investigated in this dissertation from a backstepping perspective: control
of multivariable nonlinear systems whose vector relative degrees are not well de�ned,
steering control of light passenger vehicles on automated highways, and coordinated
steering and braking control of commercial heavy vehicles on automated highways.
For a class of a�ne multivariable nonlinear systems with an equal number of inputs
and outputs, if the decoupling matrix is singular, the vector relative degree is not
well de�ned. If the mutivariable nonlinear system is strongly invertible and strongly
accessible, the vector relative degree of the system can be achieved by adding chains
2
of integrators to the input channels. Several versions of dynamic extension algorithms
have been proposed to identify the input channels where dynamic compensators (or
integrators) are needed to achieve the nonsingularity of the decoupling matrix. Once
the vector relative degree is well de�ned by adding dynamic compensators in the
input channel, the multivariable nonlinear system can be decoupled in the input-
output sense. In this dissertation, instead of decoupling the nonlinear system by
adding chains of integrators in the input channels, we modify the dynamic extension
algorithm by incorporating backstepping design methods to partially close the loop
in each design step. The resulting control law by this new approach is a static state
feedback law. Although the �nal closed loop form of the nonlinear system is not
decoupled, each output is controlled to the desired value asymptotically.
Backstepping design methodology is utilized for lateral control of light passenger
vehicles and commercial heavy vehicles in Automated Highway Systems (AHS). The
steering control algorithm for light passenger vehicles is designed by utilizing the
robust backstepping technique, whereas the coordinated steering and braking control
algorithm for commercial heavy vehicles is designed by applying the backstepping
technique for multivariable nonlinear systems without vector relative degrees.
For lateral control of light passenger vehicles, the lateral tracking error is a�ected
by the relative yaw angle of the vehicle with respect to the road centerline. Then
the relative yaw angle is controlled by the front wheel steering command. Intuitively,
this backstepping control procedure resembles the human driver behavior. Mathe-
3
matically, there is no internal dynamics in this design; i.e., both the lateral and the
yaw dynamics are under control. Another advantage of the backstepping controller is
that the road disturbance, which does not satisfy the matching condition, can be at-
tenuated e�ectively. Thus the backstepping design e�ectively utilizes the feedforward
information of the road curvature to generate the feedforward part of the steering
command. To satisfy both the ride comfort and safety requirements, we introduce
a nonlinear spring term (nonlinear position feedback) which exhibits lower gains at
small tracking errors and higher gains at larger tracking errors. Furthermore, to cope
with nonsmoothness of the road disturbance, robust backstepping control methodol-
ogy will be utilized.
For lateral control of commercial heavy vehicles, a control oriented dynamic mod-
eling approach for articulated vehicles is proposed. A generalized coordinate system
is introduced to describe the kinematics of the vehicle. Equations of motion of a
tractor-semitrailer vehicle are derived based on the Lagrange mechanics. Experi-
mental studies are conducted to validate the e�ectiveness of this modeling approach.
Two lateral control algorithms are designed for a tractor-semitrailer vehicle. The
baseline steering control algorithm is designed utilizing input-output linearization,
whereas the coordinated steering and braking control algorithm is designed based on
the multivariable backstepping technique.
4
Professor Masayoshi TomizukaDissertation Committee Chair
iii
To
My Loving Parents ,Wife and Daughter
iv
Contents
List of Figures vii
List of Tables ix
1 Introduction 1
1.1 Motivations, previous work, and objectives of this dissertation . . . . . . 11.2 Contributions of this Dissertation . . . . . . . . . . . . . . . . . . . . . 111.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Backstepping 15
2.1 Integrator Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Backstepping for Strict-feedback Systems . . . . . . . . . . . . . . . . . 192.3 Adaptive Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4 Robust Backstepping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.5 Sliding Control via Backstepping . . . . . . . . . . . . . . . . . . . . . 282.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Backstepping Control Design of a Class of Multivariable Nonlinear
Systems without Vector Relative Degrees 33
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Dynamic Extension Algorithm . . . . . . . . . . . . . . . . . . . . . . . 353.3 Combined Dynamic Extension and Backstepping Algorithm . . . . . . . 453.4 Design example : Planar Vehicle . . . . . . . . . . . . . . . . . . . . . 58
3.4.1 Decoupling Control by Dynamic Extension . . . . . . . . . . . . 593.4.2 Control by Backstepping Design . . . . . . . . . . . . . . . . . . 60
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 Lateral Control of Light Passenger Vehicles in Automated Highway
Systems 65
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
v
4.2 Vehicle Dynamics and Control Model . . . . . . . . . . . . . . . . . . . 664.3 Lateral Control of Light Passenger Vehicles . . . . . . . . . . . . . . . . 70
4.3.1 Road Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.3.2 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5 Dynamic Modeling of Tractor-Semitrailer Vehicles for Automated
Highway Systems 86
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.2 De�nition of Coordinate System . . . . . . . . . . . . . . . . . . . . . . 88
5.2.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . 885.2.2 Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . 915.2.3 Transformation between the inertial reference frame and the un-
sprung mass reference frame . . . . . . . . . . . . . . . . . . . . 935.3 Vehicle Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3.1 Tractor Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . 975.3.2 Trailer Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . 1005.3.3 Kinetic Energy and Potential Energy . . . . . . . . . . . . . . . 102
5.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.5 Generalized Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.6 Subsystems : Tire Model and Suspension Model . . . . . . . . . . . . . 121
5.6.1 Tire Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.6.2 Suspension Model . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.7 Model Veri�cation: Simulation and Experimental Results . . . . . . . . . 1265.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6 Lateral Control of Tractor-Semitrailer Vehicles on Automated High-
ways 134
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.2 Road reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.3 Steering Control Model (SIM1) . . . . . . . . . . . . . . . . . . . . . . 141
6.3.1 Model Simpli�cation . . . . . . . . . . . . . . . . . . . . . . . . 1416.3.2 Control Model with respect to the Road Reference Frame . . . . 1456.3.3 Linear Analysis of the Control Model . . . . . . . . . . . . . . . 147
6.4 Steering Control of Tractor-Semitrailer Vehicles . . . . . . . . . . . . . . 1486.4.1 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . 1486.4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.5 Steering and Braking Control Model (SIM2) . . . . . . . . . . . . . . . 1536.6 Coordinated Steering and Independent Braking Control . . . . . . . . . . 156
6.6.1 Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . 1566.6.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 161
vi
6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7 Conclusions and Future Research 166
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667.2 Suggested Future Research . . . . . . . . . . . . . . . . . . . . . . . . 168
Bibliography 171
vii
List of Figures
3.1 Block diagram of the MIMO nonlinear system after recursive staticstate feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 Dynamic extensions of the MIMO nonlinear system by adding chainsof integrators to the appropriate input channel . . . . . . . . . . . . . 45
3.3 First Step of the Backstepping Designs . . . . . . . . . . . . . . . . . 513.4 Backstepping designs of the MIMO nonlinear system . . . . . . . . . 573.5 Planar Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1 Ideal mass distribution for passenger car . . . . . . . . . . . . . . . . 684.2 Block diagram of the lateral dynamics . . . . . . . . . . . . . . . . . 704.3 De�nition of the desired yaw rate . . . . . . . . . . . . . . . . . . . . 724.4 A smooth road model . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.5 Simulation scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.6 Simulation results of backstepping controller at longitudinal velocity
= 30 MPH, solid line : lateral position at C.G., dashdot line : lateralposition at rear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.7 Simulation results of backstepping controller at longitudinal velocity= 60 MPH, solid line : lateral position at C.G., dashdot line : lateralposition at rear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.1 Coordinate System to Describe the Vehicle Motion . . . . . . . . . . 895.2 Three Reference Coordinates . . . . . . . . . . . . . . . . . . . . . . . 925.3 Inertial and Unsprung Mass Reference Frames . . . . . . . . . . . . . 945.4 Schematic Diagram of Complex Vehicle Model . . . . . . . . . . . . . 965.5 De�nition of Tire Force in the Cartesian Coordinate . . . . . . . . . . 1165.6 Tire Force Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225.7 Comprehensive Tire Model (Baraket and Fancher) . . . . . . . . . . . 1245.8 Suspension Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.9 Step input response with the longitudinal vehicle speed 30 MPH, . . 1295.10 Step input response with the longitudinal vehicle speed 35 MPH . . . 130
viii
5.11 Step input response with the longitudinal vehicle speed 40 MPH . . . 1315.12 Step input response with the longitudinal vehicle speed 46 MPH . . . 132
6.1 Unsprung Mass and Road Reference Coordinates . . . . . . . . . . . 1366.2 Simulation Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.3 Input/Output Linearization Control . . . . . . . . . . . . . . . . . . . 1526.4 Wheel Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1566.5 Input/Output Linearization Control with Trailer Independent Braking 1626.6 Input/Output Linearization Control with Trailer Independent Braking 1636.7 Comparison of Input/Output Linearization Control with (solid line)
and without (dashdot line) Trailer Independent Braking . . . . . . . . 164
ix
List of Tables
4.1 Notations of the Simpli�ed Control Model . . . . . . . . . . . . . . . 674.2 Parameters of a Passenger Car . . . . . . . . . . . . . . . . . . . . . . 82
5.1 Parameters of Complex Vehicle Model . . . . . . . . . . . . . . . . . 975.2 Parameters for a Tractor-Semitrailer Vehicle . . . . . . . . . . . . . . 1285.3 Suspension Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.4 Tire and Wheel Parameters . . . . . . . . . . . . . . . . . . . . . . . 128
6.1 Nomenclature of Control Models . . . . . . . . . . . . . . . . . . . . . 142
x
Acknowledgements
I would like to express my sincere thank to my advisor, Professor Masayoshi Tomizuka,
for his timely guidance, encouragement and support. His judicious suggestions made
the successful completion of this dissertation possible. He has been the best possible
mentor in every sense of the word. I would like to thank Professor Kameshwar Poolla
and Professor Wei Ren for their invaluable comments as members of my dissertation
committee.
Special thanks to my colleagues and friends with whom I had the greatest time:
Dr. Tsu-Chih Chiu, Dr. Liang-Jung Huang, Dr. Tom Hessburg, Dr. Tony Phillips,
Dr. Bin Yao, Victor Chu, Chieh Chen, Weiguang Niu, Li Yi, Carlos Osorio, Matt
White, Craig Smith, Mohammed Al-Majed, Wonshik Chee, Hyeoncheol Lee, Pushkar
Hingwe and Sujit Saraf.
I give my special appreciation to Dr. Ho Seong Lee and Mohammed Al-Majed
for their help in setting up the simulations and experiments and Rob Bickel, Craig
Smith and Matt White for careful proof-reading of my dissertation.
I would like to thank all my family members. My parents You Ying and Yuen Hua
deserve special recognition for their continuous encouragement and sel ess support.
My wife Eva should be specially honored for her constant encouragement and sacri�ce.
I am very grateful to my daughter Stephanie. Her cheerful smile and lovely voice made
my hard moments bearable. Lastly, and most importantly, I thank my Lord, Jesus
Christ, for his grace and love.
1
Chapter 1
Introduction
1.1 Motivations, previous work, and objectives of this disser-
tation
In this dissertation the e�ort is to explore new aspects, from both theoretical
and application point of view, in the design of the backstepping control systems [35]
with applications to vehicle lateral control in Automated Highway System (AHS).
Three main topics are investigated in this dissertation from a backstepping perspec-
tive: control of multivariable nonlinear systems whose vector relative degrees are not
well de�ned, steering control of light passenger vehicles on automated highways, and
coordinated steering and braking control of commercial heavy vehicles on automated
highways.
2
Vehicle Lateral Control in Automated Highway Systems. AHS technologies
have attracted growing attention among researchers throughout the world in the past
several years [3, 6, 17, 23, 32, 51, 58, 62, 67]. The principal motivation for an AHS is
to increase highway capacity. Potential bene�ts for AHS include:
� a substantial increase in lane capacity and therefore tra�c throughput
� improvement in driving safety on highways
� a decrease in travel time and therefore reduction in air pollution
Due to the complexities of AHS, a hierarchy of system structure is proposed in [6, 62].
According to spatial and temporal scale, control tasks in this AHS architecture are
organized into �ve layers: network layer, link layer, coordination layer, regulation
layer, and physical layer. Control tasks in the network layer and the link layer are
executed in roadside systems, whereas the control tasks in the coordination layer
and the regulation layer are performed in vehicle on-board computers. The physical
layer represents the vehicle dynamics which is controlled by the regulation layer. By
this classi�cation, vehicle lateral control sits in the regulation layer and is one of the
critical components in the framework of AHS.
Lateral control in AHS consists of two maneuvers: lane following and lane change.
The objective of lateral motion control for lane following is to achieve accurate track-
ing of a reference lane while maintaining an acceptable level of passenger comfort in
the presence of disturbances and over a wide range of operating conditions. In this
3
dissertation, lateral motion control for lane following maneuvers is investigated for
both light passenger cars and commercial heavy vehicles.
In the area of lateral motion control of light passenger vehicles, previous studies
have been conducted by utilizing both the linear control theory and the nonlinear
control theory. Among the linear control strategies, Fenton et al. [17] designed a
feedback steering controller by using the lead/lag compensator and root locus theory,
where no preview information on the road curvature is used in calculating the steering
command. Peng and Tomizuka [51] synthesized a Frequency Shaped Linear Quadratic
(FSLQ) controller with preview. The main appeal of the linear FSLQ controller is that
it provides a quantitative description of the trade-o� between passenger ride-comfort
and tracking performance. The FSLQ controller also possesses moderate modeling
robustness properties. However, the longitudinal velocity is assumed to be constant
in the linear control approach, and hence gain scheduling with respect to the longitu-
dinal velocity is required to cover the full operating range of the vehicle. Among the
nonlinear control strategies, Guldner et al. [22] and Pham et al. [52] developed steer-
ing controllers independently based on Sliding Mode Control (SMC) theory. Hingwe
and Tomizuka further pointed out that the major di�erence between these two SMC
controllers is the location of an integrator within the feedback loop[26]. Simulations
and experimental studies conducted in the California PATH program have shown that
the SMC controllers possess superior lateral tracking performance [27]. Since SMC
explicitly takes into account parameter uncertainties in the design, the controller is
4
robust with respect to parametric uncertainties. Furthermore, the longitudinal veloc-
ity is regarded as a known time-varying parameter in the SMC approach, thus gain
scheduling with respect to the longitudinal velocity is not necessary. The drawback
of this approach is that the ride quality is not explicitly considered in the design and
the stability of vehicle yaw dynamics (internal dynamics) is not guaranteed.
Lane change maneuvers have been studied using magnetic referencing system in
[8], where a sliding mode controller using �ltered errors has been proposed as the
tracking control algorithm and a reduced order Kalman �lter is designed as the state
estimator. Preliminary experimental results conducted in the California PATH pro-
gram showed the e�ectiveness of this controller at speeds up to 32 km/hr.
In contrast to light passenger vehicles, less attention has been paid to control issues
of commercial heavy vehicles for automated highway systems. The study of heavy
vehicles for AHS applications has gained interest only recently [4, 10, 16, 32, 67, 75].
The study of lateral guidance of heavy-duty vehicles is important for several reasons.
In 1993, the share of the highway miles accounted for by truck tra�c was around
28% [25]. This is a signi�cant percentage of the total highway miles traveled by
all the vehicles in US. According to Motor Vehicles Facts and Figures [42], the total
number of registered trucks (light, commercial and truck-trailer combinations) formed
approximately 10% of the national �gures in 1991 and 30.9% of the highway taxes
came from heavy vehicles. Also, due to several economic and policy issues, heavy
vehicles have the potential of becoming the main bene�ciaries of automated guidance
5
[32]. The main reasons are:
� On average, a truck travels six times the miles as compared to a passenger
vehicle. Possible reduction in the number of drivers will reduce the operating
cost substantially.
� Relative equipment cost for automating heavy vehicles is far less than for pas-
senger vehicles.
� Automation of heavy vehicles will have a signi�cant impact on the overall safety
of the automated guidance system. Trucking is a tedious job and automation
will contribute positively to reducing driving stress and thereby increase safety.
Thus commercial heavy vehicles will gain signi�cant bene�t from Advenced Vehicle
Control Systems (AVCS), and may actually become automated earlier than passenger
vehicles due to economical considerations.
Backstepping. Backstepping [31, 35, 34] is a recursive procedure which breaks a
design problem for the full system into a sequence of design problems for lower order
systems. The idea of breaking a dynamic system into subsystems is not unusual in
the design of nonlinear controllers. Sliding control [60, 61] is such an example. The
design of a sliding controller involves two steps: 1) design a stable sliding surface to
achieve the control objective, and 2) make the sliding surface attractive by pushing
system states toward the surface. To facilitate the synthesis of a sliding controller,
6
the sliding surface is designed in such a way that the relative degree from the control
input to the sliding surface variable is one; i.e., a �rst order system for the sliding
variable. Therefore, it is easier to control system states toward the sliding surface than
it is to control the original dynamic system, even in the face of plant nonlinearity and
modeling uncertainty. However, robust sliding control for uncertain nonlinear systems
requires that the matching condition be satis�ed; that is, the uncertain terms enter
the state equation at the same point as the control input. One of the advantages
of backstepping design is that the matching condition can be relaxed for a class of
nonlinear systems satisfying the so called strict feedback form. Control design for this
class of nonlinear systems can be achieved by recursive designs of scalar (�rst order)
subsystems. By exploiting the extra exibility that exists with the scalar systems,
the matching condition is not required.
Another feature of backstepping designs is that they do not force the designed
system to appear linear, which can avoid cancellations of useful nonlinearities. Fur-
thermore, additional nonlinear damping terms can be introduced in the feedback loop
to enhance robustness.
Depending on the structure of the lower order subsystems, this recursive design
procedure can be categorized as integrator backstepping, backstepping, and block
backstepping, which will be studied in this dissertation as design tools for vehicle lat-
eral control and for control of multivariable nonlinear systems without vector relative
degrees.
7
Steering Control of Light Passenger Vehicles via Backstepping. In this
dissertation, we will apply the backstepping technique to the design of the steering
control algorithm for light passenger vehicles on automated highways. The challenges
in designing the vehicle lateral controller include:
� The lateral controller requires good road tracking performance as well as passen-
ger ride quality and system safety. Small tracking errors are tolerable especially
if passenger comfort can be achieved by avoiding unnecessarily small and high
frequency steering command adjustments for small tracking errors when the
lateral tracking error is under a safety range. On the other hand, lateral track-
ing errors can not be too large because: 1). larger tracking errors may cause
a vehicle to collide with adjacent vehicles, and 2). larger tracking errors may
cause the automated vehicle to run out of the lateral sensor range and become
uncontrolled.
� Lateral dynamics strongly depends on the longitudinal velocity. It is known that
the damping of lateral dynamics is inversely proportional to the longitudinal
velocity. On the other hand, the desired yaw rate (the tracking signal for the
car) is proportional to the longitudinal velocity for the same road curvature.
Thus, it is more di�cult to control a car negotiating curved sections at high
speeds than at low speeds.
8
� In the formulation of the vehicle lateral control problem, the desired yaw rate,
_�d, which depends on the road curvature and the longitudinal speed, appears as
a disturbance input for both the lateral dynamics and the yaw error dynamics.
We shall call the �d term the road disturbance. In the automated highway
scenario, the road curvature is previewable, thus the road disturbance is known.
However, the matching condition for the road disturbance is not satis�ed since
there is only one control input for front wheel steered vehicles. This imposes
di�culties in designing nonlinear controllers.
� Vehicle parameters, especially the vehicle mass and the tire cornering sti�ness,
exhibit large uncertainties.
In this dissertation, we propose to use the backstepping technique to take into account
these challenges in the design of vehicle lateral controllers. In this approach, the
lateral tracking error is a�ected by the relative yaw angle of the vehicle with respect
to the road centerline. Then the relative yaw angle is controlled by the front wheel
steering command. Intuitively, this backstepping control procedure resembles the
human driver behavior. Mathematically, there is no internal dynamics in this design;
i.e., both the lateral and the yaw dynamics are under control. Another advantage
of the backstepping controller is that the road disturbance, which does not satisfy
the matching condition, can be attenuated e�ectively. Thus the backstepping design
e�ectively utilizes the feedforward information of the road curvature to generate the
feedforward part of the steering command. To satisfy both the ride comfort and safety
9
requirements, we introduce a nonlinear spring term (nonlinear position feedback)
which exhibits lower gains at small tracking errors and higher gains at larger tracking
errors. The use of such nonlinear action has been applied to the active suspension
controller by Lin and Kanellakopoulos [37] to optimize ride quality and suspension
travel. A velocity-dependent nonlinear damping term can be easily incorporated
in this nonlinear controller to cover the full envelope of operations. Furthermore,
to cope with nonsmoothness of the road disturbance, robust backstepping control
methodology [68, 70] will be utilized.
Coordinated Steering and Independent Braking Control of Commercial
Heavy Vehicles via Backstepping. For the lateral control of articulated heavy
vehicles, two kinds of control inputs will be used: the steering angle and braking forces
of the wheels. We will primarily rely on the front wheel steering angle. However, the
braking on trailer units will also be investigated to enhance the stability of lateral
motion. Speci�cally, braking forces can be independently distributed over the inner
and outer tires of the trailer so that the relative yaw errors between the tractor and
the trailer are reduced.
In designing the coordinated steering and braking control algorithm, we observe
that the so called decoupling matrix for this system is singular; in other words, the
vector relative degree is not well de�ned [30, 46]. Speci�cally, when di�erentiating
the outputs the steering input appears \earlier" than the braking torque input. To
overcome this di�culty, we use the braking force generated at the tire/ground inter-
10
face as a virtual control input and then backstep to determine the real braking torque
applied at the wheel.
Control of Multivariable Nonlinear Systems without Vector Relative De-
grees via Backstepping. Motivated by the steering and braking control via back-
stepping, control of a class of multivariable nonlinear systems without vector relative
degrees is investigated from the backstepping perspective. A popular control approach
for multivariable systems is to make one input control one output independent from
other inputs and outputs, i.e., decoupling control or noninteraction control in the
input/output sense. Decoupling of multivariable systems has been an active research
subject in the past two or three decades. Morgan [41] gave a su�cient condition
for decoupling of multivariable linear systems in 1964. Falb and Wolovich [15] intro-
duced the decoupling matrix which is used to characterize necessary and su�cient
conditions for decoupling of multivariable systems by static state feedback. Gilbert
[20] further classi�ed three types of coupling for multivariable linear systems, that
is, strong inherent coupling, no inherent coupling and weak inherent coupling. For a
strong inherent coupling multivariable system, no control law can e�ect decoupling;
for a no inherent coupling system, it can be decoupled by static state feedback; and
for a weak inherent coupling system, it can not be decoupled by static state feedback,
yet decoupling can be achieved by dynamic state feedback. Wang [65] developed a
recursive algorithm to design a precompensator for the weak inherent coupling mul-
tivariable system. Static and dynamic state feedback control has also been applied
11
to achieve decoupling of nonlinear systems [30, 46] and several versions of dynamic
extension algorithms [11, 30, 45, 46, 73] have been proposed to identify the input
channels where dynamic compensators (or integrators) are needed to achieve the
nonsingularity of the decoupling matrix for the extended system. Once the vector
relative degree is well de�ned by adding the integrators in the input channel, the
extended multivariable nonlinear system can be decoupled in the input-output sense.
In this dissertation, control of multivariable nonlinear systems will be studied via
the block backstepping approach.
1.2 Contributions of this Dissertation
The contributions of this dissertation are summarized as follows.
Backstepping control design of a class of multivariable nonlinear systems.
A recursive algorithm is developed to control a class of square multivariable nonlinear
systems whose decoupling matrices are singular. Past research on control of this class
of systems emphasizes the decoupling or noninteraction control by adding integrators
to the appropriate input channels; i.e., decoupling control by dynamic extension.
We provide an alternative approach from the backstepping perspective to control this
class of nonlinear systems. Speci�cally, this new control procedure is developed based
on the dynamic extension algorithm. Instead of adding integrators in input channels,
we incorporate backstepping design methods to partially close the loop in each of the
12
design steps. The resulting control law obtained by this new approach is static state
feedback.
Steering Control of Light Passenger Vehicles. A backstepping controller is
designed for lateral guidance of the passenger car in automated highway systems. In
this design, the vehicle lateral displacement is a�ected by the relative yaw angle of
the car with respect to the road centerline, and the relative yaw angle is controlled
by the vehicle's front wheel steering angle. The main features of this nonlinear design
are that the stability of both lateral and yaw error dynamics is ensured, and closed
loop performance can be speci�ed simultaneously. Furthermore, nonlinear position
feedback, which acts as low gain control at small tracking errors and high gain control
at larger tracking errors, is introduced as a trade-o� between passenger ride comfort
and tracking accuracy.
Dynamic Modeling of Articulated Commercial Vehicles. In the literature,
various mathematical models of heavy-duty vehicles have been proposed for computer
simulations in [38, 57, 64], where the goal is to develop a tool for predicting and
evaluating the longitudinal and directional response of heavy-duty vehicles. Most of
the mathematical models published in the literature adopt the Newtonian mechanics
approach to describe the body dynamics of heavy-duty vehicles. In this dissertation,
a control oriented dynamic modeling approach is proposed for articulated vehicles.
A generalized coordinate system is de�ned in this approach to precisely describe the
13
kinematics of a vehicle. Equations of motion are derived based on the Lagrange
mechanics. This modeling approach is validated by comparing �eld test data of a
class 8 tractor-semitrailer type articulated vehicle and the simulation results of the
computer model.
Coordinated Steering and Independent Braking Control of Tractor-Trailer
Vehicles. Independent braking control has been investigated [44, 53, 72] as a safety
augmented system for light passenger vehicles. However, independent braking control
of articulated vehicles has not been seriously studied. In this dissertation, a steering
control algorithm for tractor-semitrailer vehicles is designed as a baseline controller
for lane following maneuver in AHS. To enhance safety, a coordinated steering and
braking control algorithm is designed.
1.3 Dissertation Outline
The outline of the remainder of this dissertation is as follows.
� In chapter 2, the control system design based on backstepping is reviewed.
� Chapter 3 presents a recursive control algorithm for a class of multivariable non-
linear systems whose vector relative degrees are not well de�ned. The controller
design is based on both the dynamic extension algorithm and the backsteppping
control algorithm.
14
� A backstepping procedure for lateral control of passenger vehicles is developed
in chapter 4.
� In chapter 5 a control oriented dynamic modeling approach for articulated ve-
hicles is proposed. A generalized coordinate system is introduced to describe
the kinematics of the vehicle. Equations of motion of a tractor-semitrailer ve-
hicle are derived based on the Lagrange mechanics. Experimental studies are
conducted to validate the e�ectiveness of this modeling approach.
� Chapter 6 presents two lateral control algorithms for a tractor-semitrailer vehi-
cle. The baseline steering control algorithm is designed utilizing input-output
linearization, whereas the coordinated steering and braking control algorithm is
designed based on the multivariable backstepping technique presented in chap-
ter 3.
� In chapter 7 the main results of this dissertation are summarized and recom-
mendations for future research are provided.
15
Chapter 2
Backstepping
In this chapter, backstepping design [19, 31, 34, 35, 36, 71] of nonlinear systems is
reviewed. The recursive backstepping design methodology is originally introduced in
adaptive control theory to systematically construct the feedback control law, the pa-
rameter adaptation law and the associated Lyapunov function for a class of nonlinear
systems satisfying certain structured properties. In this chapter, various backstepping
design techniques, including integrator backstepping, backstepping for strict-feedback
systems, adaptive backstepping and robust backstepping, will be reviewed. For a more
complete presentation of the adaptive backstepping, refer to a recent book by Krstic,
Kanellakopoulos and Kokotovic [35]. Robust backstepping can be found in [71]. As
we mentioned in chapter 1, backstepping is a recursive procedure which breaks a de-
sign problem for the full system into a sequence of design problems for lower order
systems, and sliding control is such an example. We will give an interpretation of
16
sliding control from the backstepping perspective.
The organization of this chapter is as follows. In section 2.1, integrator backstep-
ping is presented. Backstepping for a more general class of nonlinear systems is given
in section 2.2. Adaptive and robust versions of backstepping designs are presented in
sections 2.3 and 2.4, respectively. Sliding control is interpreted in section 2.5 from a
backstepping perspective. Conclusions of this chapter are drawn in the last section.
2.1 Integrator Backstepping
Let us start the integrator backstepping by considering the second order system
_x = x2 � x3 + �
_� = u
(2.1)
The design objective is that x(t)! 0 as t!1. The control law can be synthesized
in two steps. We regard � as a real control �rst. By choosing the Lyapunov function
candidate
V1 =1
2x2
and the control law
�des = �x2 � k1x � �(x)
the control objective will be achieved. Nevertheless, � is a state and can not be set
to �des. We de�ne the variable
z = � � �des
17
as the deviation of � from its desired value �des. With the de�nition of the error
variable, we have
_z = _� � _�des
= u� (2x+ k1)(k1x+ x3 � z)
Now the Lyapunov function candidate can be augmented as
V2 = V1 +1
2z2
It's time derivative is
_V2 = x(�x3 � k1x+ z) + z(u� (2x+ k1)(k1x+ x3 � z))
To make _V2 negative de�nite, we choose the control law
u = �x+ (2x+ k1)(k1x+ x3 � z)� k2z
Then we obtain
_V2 = �x4 � k1x2 � k2z
2
which is negative de�nite. This implies that x ! 0 and � ! �des asymptotically.
In this example, � is called a virtual control, and its desired value �(x) is called
a stabilizing function. We notice that the second order system (2.1) can also be
stabilized by a linearizing control law
u = �(2x� 3x2) _x� k1 _x� k2x (2.2)
However, the �x3 term, which helps stabilizing Eq. (2.1), is canceled by the lineariz-
ing control law (2.2). Backstepping design can avoid cancellation of useful nonlinear-
ities.
18
The result of integrator backstepping is summarized in the following lemma.
Lemma 1 (Integrator Backstepping)
Consider the system
_x = f(x) + g(x)�
_� = u
(2.3)
where f(0) = 0. If there exists a stabilizing function � = �(x) and a positive de�nite,
radially unbounded function V : Rn ! R such that
@V
@x(f(x) + g(x)�(x)) < 0;
then the control
u = �c(� � �(x)) +@�
@x(f(x) + g(x)�) �
@V
@xg(x); c > 0 (2.4)
asymptotically stabilizes the equilibrium point of (2.3).
Proof: This can be easily veri�ed by computing the derivative of the Lyapunov
function candidate
Va = V +1
2(� � �(x))2
along the system trajectory (2.3) using the control law (2.4).
19
2.2 Backstepping for Strict-feedback Systems
By recursively applying the integrator backstepping technique, a systematic design
can be obtained for the strict-feedback system:
_x = f(x) + g(x)�1
_�1 = f1(x; �1) + g1(x; �1)�2
_�2 = f2(x; �1; �2) + g2(x; �1; �2)�3
...
_�k = fk(x; �1; � � � ; �k) + gk(x; �1; � � � ; �k)u
(2.5)
where x 2 Rn and �1; � � � ; �k 2 R. The Lyapunov function and the control law will be
constructed in a recursive manner.
Step 0
Design a continuously di�erentiable stabilizing function �1 = �(x) for the x sub-
system; i.e., construct a positive de�nite, radially unbounded function V (x) such that,
with this control law, its time derivative
@V
@x(f(x) + g(x)�(x)) < �W (x)
where W (x) is positive de�nite.
Step 1
We start our backstepping procedure by considering the following subsystem
_x = f(x) + g(x)�1
_�1 = f1(x; �1) + g1(x; �1)�2
(2.6)
20
In step 0, we assume �1 is a virtual control and the control law
�1 = �(x)
stabilizes the x subsystem. To take into account the deviation of the state variable
�1 from the stabilizing function �1(x), we de�ne the error variable
z1 = �1 � �(x)
Then
_z1 = _�1 �@�(x)@x
_x
= f1(x; �1) + g1(x; �1)�2 �@�(x)@x
(f(x) + g(x)(�(x) + z1))
(2.7)
We proceed in the same way as in integrator backstepping by augmenting the Lya-
punov function
V1 = V (x) +1
2z21
We want to design a stabilizing function �2 = �1(x; z1) such that the time derivative
of the Lyapunov function V1 is negative de�nite.
_V1 = _V (x) + z1 _z1
= @V (x)@x
(f(x) + g(x)(�(x) + z1)) + z1 _z1
< �W (x) + @V (x)@x
g(x)z1 + z1 _z1
(2.8)
Substituting _z1 in (2.7) into (2.8), we obtain
_V1 < �W (x) + @V (x)@x
g(x)z1
+z1ff1(x; �1) + g1(x; �1)�2 �@�(x)@x
(f(x) + g(x)(�(x) + z1))g
(2.9)
21
It is clear that, if g1(x; �1) 6= 0, by choosing the stabilizing function for the virtual
control �2 as
�2 = �1(x; z1)
= 1g1
n�k1z1 �
@V@x(x)g(x)� f1(x; �1) +
@�(x)@x
(f(x) + g(x)(�(x) + z1))o
the derivative of the Lyapunov function in (2.9) becomes
_V1 < �W (x)� k1z21 (2.10)
Step 2
In this step, we will consider the subsystem
_x = f(x) + g(x)�1
_�1 = f1(x; �1) + g1(x; �1)�2
_�2 = f2(x; �1; �2) + g2(x; �1; �2)�3
(2.11)
We observe that this subsystem can be written as
_X1 = F1(X1) +G1(X1)�2
_�2 = f2(X1; �2) + g2(X1; �2)�3
(2.12)
where X1 =
0BBB@
x
�1
1CCCA, F1(X1) =
0BBB@
f(x) + g(x)�1
f1(x; �1)
1CCCA, and G1(X1) =
0BBB@
0
g1(x; �1)
1CCCA. In
this notation, the structure of the subsystem (2.12) is identical to that of step 1 (2.6).
Similarly, we de�ne the error variable
z2 = �2 � �1(X1)
22
We proceed in the same way as in step 1 by augmenting the Lyapunov function
V2 = V1(X1) +1
2z22
We can design a stabilizing function �3 = �2(X1; z2) such that the time derivative of
the Lyapunov function V2 is negative de�nite.
This recursive procedure will terminate at the k�th step, where the actual control
law for u will be designed.
2.3 Adaptive Backstepping
In the previous two sections, we consider backstepping designs for nonlinear sys-
tems satisfying certain structured properties. In this section, we will present the idea
of adaptive backstepping design procedure for a class of nonlinear systems with un-
known parameters. The design procedure will be illustrated by an example. Consider
_x1 = x2 + ��(x1)
_x2 = u
(2.13)
where � is an known constant parameter.
Step 1
We regard x2 as a control input �rst. Denote �̂ as the estimated value for the
parameter � and the estimation error � � �̂ as ~�. Choose the Lyapunov function
candidate
V1(x1; ~�) =1
2x21 +
1
2 ~�2 (2.14)
23
It is easy to see that with the control law
x2 = �k1x1 � �̂�(x1)
� �1(x1; �̂)
(2.15)
and the adaptation law
_̂� = �(x1)x1
� �1
(2.16)
the derivative of the Lyapunov function candidate (2.14) is
_V1 = �k1x21 � 0 (2.17)
The function �1 in (2.15) is called a stabilizing function for x2, and �1 in (2.16) is
called a tuning function.
Step 2
Since x2 is not the control, we de�ne the deviation of x2 from the desired stabilizing
function �1 as
z = x2 � �1(x1; �̂)
With this new error variable z, the system (2.13) can be rewritten as
_x1 = �k1x1 + ~��(x1) + z
_z = _x2 � _�1 = u� _�1
(2.18)
and the derivative of the Lyapunov function V1 is
_V1 = x1 _x1 + ~� _~�
= �k1x21 + x1z + ~�(�1 �
1
_̂�)
(2.19)
24
Further, the dynamics of the error variable is
_z = _x2 � _�1
= u� @�1@x1
(x2 + ��(x1))�@�1@�̂
_̂�
= u� @�1@x1
x2 �@�1@�̂
_̂� � � @�1
@x1�(x1)
= u� @�1@x1
x2 �@�1@�̂
_̂� � �̂ @�1
@x1�(x1)� ~� @�1
@x1�(x1)
(2.20)
Augment the Lyapunov function by adding the error variable
V2(x1; z; ~�) = V1(x1; ~�) +1
2z2 (2.21)
By noting (2.19) and (2.20), the derivative of V2 can be computed as
_V2 = _V1 + z _z
= �k1x21 + x1z + ~�(�1 �
1
_̂�)
+z(u� @�1@x1
x2 �@�1@�̂
_̂� � �̂ @�1
@x1�(x1)� ~� @�1
@x1�(x1))
(2.22)
Grouping similar terms, we obtain
_V2 = �k1x21 +
~�(�1 �@�1@x1
�(x1)z �1
_̂�)
+z(u+ x1 �@�1@x1
x2 �@�1@�̂
_̂� � �̂ @�1
@x1�(x1))
(2.23)
To make _V2 in (2.23) nonpositive, we can choose the control law
u = �k2z � x1 +@�1
@x1x2 +
@�1
@�̂
_̂� + �̂
@�1
@x1�(x1) (2.24)
and the parameter adaptation law
_̂� = (�1 �
@�1@x1
�(x1)z)
= (�(x1)z �@�@x1
�(x1)z)
(2.25)
25
Then the derivative of V2 becomes
_V2 = �k1x21 � k2z
2 � 0 (2.26)
This implies that x1 ! 0 and z ! 0 asymptotically.
2.4 Robust Backstepping
In this section, robust backstepping design [71] is illustrated by cnsidering the
second order system
_x1 = f(x1) + x2 + ~�(x1; t)
_x2 = u
(2.27)
where ~�(x1; t) is an unknown nonlinear function bounded by h1(x1; t), i.e., j ~�(x1; t)j <
h1(x1; t). We observe that the uncertainty term ~�(x1; t) in (2.27) enters the system
dynamics one integrator prior than the control input u does.
Since the stabilizing function �i in each backstepping step is required to be con-
tinuously di�erentiable, we can not use a discontinuous sign function for a stabilizing
function. Therefore, a smooth approximation is used in the development of the con-
trol law. The following lemma quanti�es the approximation error of a sgn(�) function
by a hyperbolic tanh(�) function.
Lemma 2 Given any � > 0, the following inequality holds
0 � h � x sgn(x)� h � x � tanh(�hx
�) � �
26
where � = 0:2785 and h is any positive number.
The proof of this lemma can be found in [54] or [71].
The control algorithm for the system (2.27) is designed in two steps.
Step 1
We regard x2 as a control input in this step. Choosing the Lyapunov function candi-
date
V1 =1
2x21 (2.28)
and the stabilizing function
x2 = �k1x1 � f(x1)� h1(x1; t) sgn(x1) (2.29)
then we have
_V1 = �k1x21 + x1 ~�(x1; t)� x1h1(x1; t) sgn(x1) < 0 (2.30)
This implies that the x1 dynamics will be stabilized asymptotically, even in the face
of uncertain nonlinearity ~�(x1; t). To avoid using discontinuous stabilizing function,
a smooth approximation for (2.29) is
x2 = �k1x1 � f(x1)� h1(x1; t) tanh(�h1(x1;t)x1
�1)
� �1(x1; t)
(2.31)
where �1 in the argument of tanh(�) is a free design parameter. With this modi�ed
27
stabilizing function (2.31), the derivative of V1 can be recalculated as
_V1 = �k1x21 + x1 ~�(x1; t)� x1h1(x1; t) tanh(
�h1(x1;t)x1�1
)
� �k1x21 + x1h1(x1; t)sgn(x1)� x1h1(x1; t) tanh(
�h1(x1;t)x1�1
)
� �k1x21 + �1
(2.32)
where the last inequality is obtained by lemma 2. Thus the state variable x1 will
converge to a ball whose size depends on the freely adjusted parameter �1.
Step 2
Since x2 is not an actual control input, we de�ne an error variable
z = x2 � �1(x1; t)
Then the dynamic equation (2.27) becomes
_x1 + k1x1 + h1 tanh(�h1x1�1
) = z + ~�(x; t) (2.33)
and the derivative of V1 is
_V1 � �k1x21 + �1 + x1z (2.34)
Further, the dynamics of the error variable is
_z = _x2 � _�1(x1; t)
= u� (@�1@x1
_x1 +@�1@t)
= u� @�1@x1
(�k1x1 � h1tanh(�h1x1�1
) + z + ~�(x; t))� @�1@t
(2.35)
28
Augment the Lyapunov function V1 by including the error variable
V2 = V1 +1
2z2
Then
_V2 = _V1 + z _z
� �k1x21 + �1 + x1z + z _z
= �k1x21 + �1 + x1z + z(u� @�1
@x1(�k1x1 � h1tanh(
�h1x1�1
) + z + ~�(x1; t))�@�1@t)
(2.36)
Assume there exists a smooth function h2(x1; t) such that
j@�1
@x1~�(x1; t)j � h2(x1; t) (2.37)
It is clear that by choosing
u = �k2z � x1 +@�1
@x1(�k1x1 � h1tanh(
�h1x1�1
) + z) +@�1
@t� h2 tanh(
�h2z
�2) (2.38)
The inequality (2.36) becomes
_V2 � �k1x21 � k2z
2 + �1 � z @�1@x1
~�(x1; t)� zh2 tanh(�h2z�2
)
� �k1x21 � k2z
2 + �1 + �2
(2.39)
which implies that the control law (2.38) renders x(t) globally uniformly bounded.
2.5 Sliding Control via Backstepping
Consider an n � th order nonlinear system
x(n) = f(x) + u (2.40)
29
where u is the control input, x is the output of interest, x = [x; _x; � � � ; xn�1]T is the
state vector, and the dynamics f(x) is not exactly known, but estimated as f̂(x). The
estimation error on f(x) is assumed to be bounded by some known function F (x),
that is,
jf̂ � f j � F
The design of a sliding controller involves two steps: 1) design a stable sliding sur-
face to achieve the control objective, and 2) make the sliding surface attractive by
pushing system states toward the surface. These two steps can be interpreted as a
backstepping procedure.
Let ~x = x � xd be the tracking error. For simplicity, the sliding surface for the
system (2.40) can be chosen as
s = (d
dt+ �)n�1~x (2.41)
It is easy to see that s � 0 implies ~x! 0, or x! xd asymptotically.
Step 1
Assume s is the control input of the equation:
(d
dt+ �)n�1~x = s (2.42)
Eq. (2.42) can be written in state space form as
d
dt~x = A~x+Bs (2.43)
30
where ~x = [~x; _~x; � � � ; ~x(n�2)]T ,
A =
0BBBBBBBBBBBBB@
0 1 0 � � � 0
...
0 0 � � � 0 1
��n�1 �n � �n�2 � � � �n � � ��
1CCCCCCCCCCCCCA
2 R(n�1)�(n�1)
and
B =
0BBBBBBBBBBBBB@
0
...
0
1
1CCCCCCCCCCCCCA
2 R(n�1)�1
Since matrix A is stable, given any positive de�nite matrix Q 2 R(n�1)�(n�1), there
exists a positive matrix P 2 R(n�1)�(n�1) satisfying the Lyapunov equation
ATP + PA = �Q
Choose the Lyapunov function candidate
V1 = ~xTP ~x
By the de�nition of the sliding surface, the stabilizing function for s can simply chosen
as 0, i.e.,
s = �1(~x) (2.44)
which will achieve the control objective. With this stabilizing function, the time
derivative of V1 is
_V1 = �~xTQ~x
31
Step 2
Since s is not the real control, s can not be set to 0 all the time. Yet s can be adjusted
by the real control input u. We augment the Lyapunov function V1 as
V2 = V1 + s2 = ~xTP ~x+1
2s2 (2.45)
Its time derivative can be calculated as
_V2 = �~xTQ~x+ 2BTP ~xs+ s _s (2.46)
Furthermore, the derivative of s is
_s = ~xn + �~xn�1 + � � �+ �n _~x
= x(n) � x(n)d + �~xn�1 + � � � + �n _~x
= f(x) + u� x(n)d + �~xn�1 + � � �+ �n _~x
(2.47)
We choose the control law
u = �k1s� f̂(x) + x(n)d � �~xn�1 � � � � � �n _~x� F (x)sgn(s)� 2BTP ~x (2.48)
Then
_V2 = �~xTQ~x� k1s2 + s � (f(x)� f̂(x))� F (x)sgn(s)
� �~xTQ~x� k1s2 < 0
(2.49)
This implies that ~x and s converges to 0 asymptotically.
Furthermore, to avoid chattering due to the switching function sgn(s) in (2.48),
a smooth approximation is
u = �k1s� f̂(x) + x(n)d � �~xn�1 � � � � � �n _~x�F (x) tanh(
�F (x)s
�)� 2BTP ~x (2.50)
32
where � in the argument of the tanh(�) term is a free design parameter. With this
smooth control law and lemma 2, _V2 becomes
_V2 = �~xTQ~x� k1s2 + s � (f(x)� f̂(x))� F (x) tanh(�F (x)s
�)
� �~xTQ~x� k1s2 + �
(2.51)
Thus ~x will converge to a ball whose size depends on the freely adjusted parameter
�; i.e., global uniform boundedness is achieved.
For the traditional sliding control law, once the sliding surface, s, is designed,
the control objective becomes s ! 0 in �nite time. However, during the transition
phase that the state have not reached the sliding surface, the behavior of the state
x(t) is not guaranteed. We observe that the control law in (2.50) feedbacks one more
term, �2BTP ~x, than the traditional sliding control law. This term will ensure that
the tracking error ~x is still decreasing even during the transition phase of the sliding
control.
2.6 Conclusions
Backstepping design of nonlinear system was reviewed in this chapter. backstep-
ping is a recursive procedure which breaks a design problem for the full system into
a sequence of design problems for lower order systems. Various backstepping design
techniques, including integrator backstepping, backstepping for strict-feedback sys-
tems, adaptive backstepping and robust backstepping, were presented. Sliding control
was illustrated from the backstepping perspective.
33
Chapter 3
Backstepping Control Design of a
Class of Multivariable Nonlinear
Systems without Vector Relative
Degrees
3.1 Introduction
This chapter is concerned with the control of a class of a�ne multivariable non-
linear systems with an equal number of inputs and outputs. When the decoupling
matrix is singular, the vector relative degree is not well de�ned [30, 46] . Further-
more, if the multivariable nonlinear system is strongly invertible [28] and strongly
34
accessible [46], the vector relative degree of the system can be achieved by adding
chains of integrators to the input channels [11]. Several versions of dynamic extension
algorithms [11, 45, 65, 73] have been proposed to identify the input channels where
dynamic compensators (or integrators) are needed to achieve the nonsingularity of
the decoupling matrix. Once the vector relative degree is well de�ned by adding dy-
namic compensators in the input channel, the multivariable nonlinear system can be
decoupled in the input-output sense.
In this chapter, we are concerned with the class of strongly invertible and strongly
accessible multivariable nonlinear systems whose decoupling matrix is singular. Since
the decoupling matrix is singular, no static state feedback control law can cause the
decoupling. However, it has been shown that decoupling can always be achieved by
dynamic compensation for the strongly invertible and strongly accessible multivari-
able nonlinear system [11], where the invertibility of multivariable nonlinear systems
is given by Hirschorn [28]. Instead of attempting to decouple the nonlinear system by
adding chains of integrators in the input channels, we provide an alternative approach
to control the multivariable nonlinear systems. Based on the dynamic extension algo-
rithm in [45], we utilize the backstepping design methodology [31, 34, 35] to partially
close the loop in each design step. In this study, we shall use the dynamic extension
algorithm to identify the input channels where controls appear \too early" in the
input-output sense, and then we apply the backstepping control algorithm to par-
tially close those loops. The resulting control law proposed in this chapter is static
35
state feedback. Even though the �nal closed loop form of the nonlinear system is not
decoupled, each output is controlled to the desired value asymptotically.
The organization of this chapter is as follows. The dynamic extension algorithm
proposed in [45] is reviewed in section 3.2. The combined dynamic extension and back-
stepping design algorithm for multivariable nonlinear systems is presented in section
3.3. Controls of a planar vehicle, which illustrate and contrast both approaches, are
designed in section 3.4. Conclusions of this chapter are given in the last section.
3.2 Dynamic Extension Algorithm
In this section, we will summarize some de�nitions and results from the geomet-
ric nonlinear control theory [30, 46] and review the dynamic extension algorithm
presented in [45].
De�nition 1 Consider an a�ne nonlinear system having m inputs and m outputs
_x = f(x) +Pm
i=1 gi(x)ui (3.1)
and
y1 = h1(x)
� � �
ym = hm(x)
(3.2)
where x 2 Rn, f(x) and gi(x) are n� vectors. The nonlinear system (3.1) and (3.2) is
called input-output decoupled if, after a possible relabeling of the inputs, the following
36
two properties hold.
1. For each i, 1 � i � m, the output yi is invariant under the inputs uj, j 6= i.
2. The output yi is not invariant with respect to the input ui, 1 � i � m.
De�nition 2 (Lie Derivative)
Let h : Rn ! R be a smooth scalar function, and f : Rn ! Rn be a smooth vector
�eld on Rn, then the Lie derivative of h with respect to f is a scalar function de�ned
by Lfh = @h@x� f:
Repeated Lie derivatives can be de�ned recursively
L0fh = h
Lifh = Lf (L
i�1f h) for i = 1; 2; :::
Similarly, if g is another vector �eld, then the scalar function LgLfh(x) is
LgLfh = Lg(Lfh)
With the notation of Lie derivative, the vector relative degree for a multivariable
system is de�ned as follows.
De�nition 3 The system (3.1) and (3.2) is said to have a vector relative degree
fr1; :::; rmg at a point x0 if
1. for each i, 1 � i � m,
(Lg1Lkfhi(x); � � � ; LgmL
kfhi(x)) = (0; � � � ; 0)
for all k < ri � 1, and for all x in a neighborhood of x0,
37
2. the m�m matrix
A(x) =
0BBBBBBBBBB@
Lg1Lr1�1f h1(x) � � � LgmL
r1�1f h1(x)
Lg1Lr2�1f h2(x) � � � LgmL
r2�1f h2(x)
� � � � � � � � �
Lg1Lrm�1f hm(x) � � � LgmL
rm�1f hm(x)
1CCCCCCCCCCA
(3.3)
is nonsingular at x = x0.
Remark 1 If the vector relative degree is well de�ned for the multivariable nonlinear
system (3.1) and (3.2), we have0BBBBBBBBBBBBBB@
y(r1)1
y(r2)2
�
�
y(rm )m
1CCCCCCCCCCCCCCA
=
0BBBBBBBBBBBBBB@
Lr1f h1(x)
Lr2f h2(x)
�
�
Lrmf hm(x)
1CCCCCCCCCCCCCCA
+
0BBBBBBBBBBBBBB@
Lg1Lr1�1f h1(x) � � � LgmL
r1�1f h1(x)
Lg1Lr2�1f h2(x) � � � LgmL
r2�1f h2(x)
� � � � � � � � �
� � � � � � � � �
Lg1Lrm�1f hm(x) � � � LgmL
rm�1f hm(x)
1CCCCCCCCCCCCCCA
0BBBBBBBBBBBBBB@
u1
u2
�
�
um
1CCCCCCCCCCCCCCA
� B(x) + A(x)u
(3.4)
One may readily choose the static feedback law
u = A�1(x)(�B(x) + v) (3.5)
which achieves decoupling of the input/output dynamics. This shows that a well
de�ned vector relative degree for a multivariable nonlinear system is a su�cient con-
dition to achieve decoupling of the closed loop system.
The following theorem shows that a well de�ned vector relative degree for a mul-
tivariable nonlinear system is not only su�cient but also necessary for decoupling
control.
38
Theorem 1 [30] Consider the multivariable nonlinear system (3.1) and (3.2) with
m inputs and m outputs. Suppose for each i, 1 � i � m,
(Lg1Lkfhi(x); � � � ; LgmL
kfhi(x)) = (0; � � � ; 0)
for all k < ri � 1 and for all x in a neighborhood of x0, and
(Lg1Lri�1f hi(x0); � � � ; LgmL
ri�1f hi(x0)) 6= (0; � � � ; 0) (3.6)
Then the decoupling control problem is solvable by static state feedback if and only
if the matrix A(x0) is nonsingular, i.e., if the system has a vector relative degree
fr1; :::; rmg at x0.
Remark 2 The matrix A(x) is referred to as the decoupling matrix of the system.
If the rank of the decoupling matrix is less than m, by Theorem 1, there does
not exist a static state feedback control law which decouples the system. However,
if a system is strongly invertible and strongly accessible, it is possible to design a
decoupling control law by dynamic state feedback of the form [11]
u = (�; x) + �(�; x)v
_� = (�; x) + �(�; x)v
(3.7)
The dynamic extension algorithm provides a systematic way to insert integrators in
appropriate input channels such that the decoupling matrix of the extended system is
nonsingular. In the remainder of this section, we will brie y summarize the dynamic
39
extension algorithm [45] upon which our new algorithm is based. First, we present the
following lemma, which will be used to construct the dynamic extension algorithm.
Lemma 1 Let A(x) be a p � m matrix, and b(x) a p�vector, with x in some
neighborhood U of a point x0 2 Rn. If rank A(x) = k for every x 2 U , then
1. there exist a neighborhood V � U of x0 and a smooth map : V ! Rm such
that
A(x) (x)� b(x) =
0BBB@
0
�(x)
1CCCA ; x 2 V (3.8)
for some (p � k)-vector �(x).
2. there exist a neighborhood V � U of x0 and a smooth map � : V ! Gl(m) (with
Gl(m) the set of invertible m�m matrices) such that
A(x)�(x) =
0BBB@
Ik 0
�(x) 0
1CCCA ; x 2 V (3.9)
where �(x) is a (p� k)� k matrix.
Dynamic Extension Algorithm (Nijmeijer and Respondek [45])
Consider the a�ne multivariable nonlinear system (3.1) and (3.2),
Step 1
De�ne the integers r11; :::; r1m as the smallest numbers such that r1i � th derivative of
the output yi explicitly depends on u. We have
40
0BBBBBBBB@
y(r11)
1
...
y(r1m)
m
1CCCCCCCCA= E1(x) +D1(x)u (3.10)
where E1(x) is an m-vector and D1(x) an m�m matrix. Let
rank D1(x) = k1(x)
Re-order the output functions h1; h2; :::; hm such that the �rst k1 rows of the matrix
D1 are linearly independent. Denote the reordered output function
y =
0BBB@
y1
�y1
1CCCA
where y1 = (h1; � � � ; hk1)T and �y1 = (hk1+1; � � � ; hm)
T . Correspondingly, denote r1
= (r11; r12; � � � ; r
1k1)T and �r1 = (r1k1+1; r
1k1+2
; � � � ; r1m)T . With this vector notation, Eq.
(3.10) can be rewritten as
0BBB@
(y1)(r1)
(�y1)(�r1)
1CCCA = H1(x) + J1(x)u (3.11)
where matrices H1(x) and J1(x) are obtained by the corresponding row operations
of matrices D1(x) and E1(x), respectively. From lemma 1, we can choose an m-
vector 1(x) and an invertiblem�m matrix �1(x) such that, after applying the state
feedback law
u = 1(x) + �1(x)
0BBB@
u1
�u1
1CCCA ; (3.12)
41
with u1 = (u11; :::; u1k1)T and �u1 = (u1k1+1; :::; u
1m)
T , we obtain
0BBB@
(y1)r1
(�y1)�r1
1CCCA =
0BBB@
0
�1(x)
1CCCA+
0BBB@
Ik1 0
�1(x) 0
1CCCA
0BBB@
u1
�u1
1CCCA (3.13)
De�ne the modi�ed vector �elds
f1(x) = f(x) + g(x) 1(x) (3.14)
g1(x) = g(x)�1(x) (3.15)
Then the system dynamics after applying the static state feedback law (3.12) is
_x = f1(x) + g1(x)
0BBB@
u1
�u1
1CCCA (3.16)
Note that in this step the �rst k1 channels of the input-output dynamics (3.13) are
rendered linear and decoupled and that the r1 � th derivatives of the outputs y1 and
the �r1 � th derivatives of �y1 do not explicitly depend on any control input from �u1.
Step 2
In the second step, we di�erentiate the outputs �y1 = �h1 de�ned in step 1 with
respect to the modi�ed dynamics (3.16) until some input components from �u1 appear.
For i = k1+1; � � � ;m; let r2i be the smallest integer such that the r2i -th time derivative
of yi explicitly depends on �u1. Observe from Eq. (3.13) that such a time derivative
42
may also depend on the components of u1 and their time derivatives, which are viewed
as independent variables at this moment. Speci�cally, we have
0BBBBBBBB@
y(r2k1+1
)
k1+1
...
y(r2m)
m
1CCCCCCCCA= E2(x; ~U1) +D2(x; ~U1)�u1 (3.17)
where E2(x; ~U1) is an m� k1 vector, D2(x; ~U1) an (m� k1)� (m� k1) matrix, and
~U1 consists of all components of u1 and their time derivatives u(j)i with i = 1; :::; k1
and 0 � j.
If the matrix D2(x; ~U1) is nonsingular, we are done. If not, let rank D2(x; ~U1) =
k2. Reorder the output function �h1 such that the �rst k2 rows of the matrix D2(x; ~U1)
are linearly independent. Denote
�y1 =
0BBB@
y2
�y2
1CCCA (3.18)
where y2 = (hk1+1; � � � ; hk1+k2)T and �y2 = (hk1+k2+1; � � � ; hm)
T : Correspondingly, de-
note
r2 = (r2k1+1; � � � ; r2k1+k2)
T and �r2 = (r2k1+k2+1; � � � ; r2m)
T . As in step 1, we can choose
a static state feedback law which rendered the �rst k2 channels of the input-output
dynamics (3.17) decoupled; i.e., there exists
�u1 = 2(x; ~U1) + �2(x; ~U
1)
0BBB@
u2
�u2
1CCCA (3.19)
43
where 2(x; ~U1) is an (m� k1)-vector, �2(x; ~U1) is an invertible (m� k1)� (m� k1)
matrix, u2 is equal to (u2k1+1; � � � ; u2k1+k2
)T , and �u2 is equal to (u2k1+k2+1; � � � ; u2m)
T ,
such that the input-output dynamics (3.17) become
0BBB@
(y2)(r2)
(�y2)(�r2)
1CCCA =
0BBB@
0
�2(x; ~U1)
1CCCA+
0BBB@
Ik2 0
�2(x; ~U1) 0
1CCCA
0BBB@
u2
�u2
1CCCA (3.20)
De�ne the modi�ed vector �elds
f2(x; ~U1) = f1(x) + g1(x)
0BBB@
0
2(x; ~U1)
1CCCA (3.21)
and
g2(x; ~U1) = g1(x; ~U1)
0BBB@
Ik1 0
0 �2(x; ~U1)
1CCCA (3.22)
Then the modi�ed dynamics after step 1 and 2 becomes
_x = f2(x; ~U1) + g2(x; ~U1)
0BBB@
U2
�u2
1CCCA (3.23)
where
U2 =
0BBB@
u1
u2
1CCCA
Step 3
�
�
44
�
Step L
We proceed as in step 1 and step 2 until all controls appear ; i.e., k1+k2+� � �+kL =
m. The block diagram of the resulting structure is shown in Fig.3.1, where
~U i = (u1; _u1; �u1; � � � ; ui�1; _ui�1; �ui�1 � � � ; � � �ui; _ui; �ui; :::)T :
The higher derivatives of control inputs, shown in Fig. 3.2 are regarded as new states
of the extended system, and the dynamic extension algorithm terminates at the L�th
step.
x, U
u
u
u
y
y
x
-
1
2
L
~L-1x, U
~ 2 1x, U
~
Figure 3.1: Block diagram of the MIMO nonlinear system after recursive static statefeedback control
The main result of the dynamic decoupling is summarized as follows.
Theorem 2 The system (3.1) and (3.2) is locally input-output decouplable by
precompensation and feedback if and only if k1 + k2 + � � � + kL = m
45
1
y
y
x~~
u
u
u
____________ 111SSS
______ 11SS
-
L
1(r1)
2(r2)
x, U~L-1 x, U2
x, U
Figure 3.2: Dynamic extensions of the MIMO nonlinear system by adding chains ofintegrators to the appropriate input channel
3.3 Combined Dynamic Extension and Backstepping Algorithm
In this section, a new algorithm which modi�es the dynamic extension algorithm
by incorporating the backstepping design algorithm will be developed. Before pre-
senting this new algorithm, let us �rst consider the following nonlinear system in
Nijmeijer and van der Schaft [46].
Consider the nonlinear system
_x1 = u1
_x2 = ex1u1 + x3
_x3 = u2
(3.24)
with outputs
y1 = x1
y2 = x2
(3.25)
The input-output dynamics can be calculated as
46
d
dt
0BBB@
y1
y2
1CCCA =
0BBB@
0
x3
1CCCA+
0BBB@
1 0
ex1 0
1CCCA
0BBB@
u1
u2
1CCCA (3.26)
Since the decoupling matrix of (3.26) is singular, we cannot �nd a static state feedback
law which decouples the system. However, by de�ning
_u1 = w1
u2 = w2
we obtain 0BBB@
�y1
�y2
1CCCA =
0BBB@
0
ex1u21
1CCCA+
0BBB@
1 0
ex1 1
1CCCA
0BBB@
w1
w2
1CCCA (3.27)
This shows that by adding an integrator in the �rst input channel, we obtain an
extended system whose decoupling matrix is nonsingular. Thus Eq. (3.27) can be
decoupled by the control law
0BBB@
w1
w2
1CCCA =
0BBB@
1 0
ex1 1
1CCCA
�1 0BBB@�
0BBB@
0
ex1u21
1CCCA+
0BBB@
v1
v2
1CCCA
1CCCA (3.28)
where u1 is considered as a new state variable. We will provide an alternative approach
to control the system (3.24) and (3.25). Observe that the input-output dynamics can
be rewritten as
0BBB@
_y1
_y2
1CCCA =
0BBB@
1 0
ex1 1
1CCCA
0BBB@
u1
x3
1CCCA (3.29)
and
_x3 = u2 (3.30)
47
In this formulation, we will regard x3 as a virtual control in (3.29); then we 'back-
step' to decide u2 in (3.30). This backstepping control procedure for a multivariable
nonlinear system without a vector relative degree is formalized as follows.
Combined Dynamic Extension and Backstepping Algorithm
Consider the a�ne multivariable nonlinear system (3.1) and (3.2),
Step 1
1. Denote the output error vector as
�1 = y � yd (3.31)
where yd 2 Rm is the desired output vector. De�ne the integers �11; � � � ; �1m as
the smallest numbers such that y(�1i )i depends explicitly on u. We have0
BBBBBBBB@
�11(�11)
...
�1m(�1m)
1CCCCCCCCA=
8>>>>>>>><>>>>>>>>:E1(x)�
0BBBBBBBB@
y(�11)
d1
...
y(�1m)dm
1CCCCCCCCA
9>>>>>>>>=>>>>>>>>;+D1(x)u (3.32)
where E1(x) is an m-vector and D1(x) an m�m matrix. Let
rank D1(x) = k1(x)
Reorder the output error functions �11 ; �12; � � � ; �
1m such that the �rst k1 rows of
the matrix D1(x) are linearly independent. Denote the ordered output errors
as
�1 =
0BBB@
e1
�e1
1CCCA
48
where e1 = (�11; � � � ; �1k1)T and �e1 = (�1k1+1; � � � ; �
1m)
T . Similarly, denote �1 =
(�11; � � � ; �1k1)T and ��1 = (�1k1+1; � � � ; �
1m)
T . Thus Eq.(3.32) can be written as
0BBB@
(e1)(�1)
(�e1)(��1)
1CCCA = (H1(x)�
0BBBBBBBB@
y(�11)d1 (t)
...
y(�1m)dm (t)
1CCCCCCCCA) + J1(x)u (3.33)
where matrices H1(x) and J1(x) are obtained by corresponding row operations
of matrices D1(x) and E1(x), respectively.
2. From lemma 1, there exists a state feedback
u = 1(x; t) + �1(x; t)
0BBB@
u1
�u1
1CCCA (3.34)
where 1(x; t) is an m-vector, �1(x; t) an invertible m � m matrix, and u1 =
(u11; � � � ; u1k1)T and �u1 = (u1k1+1; � � � ; u
1m)
T , such that the input/output dynamics
(3.33) become0BBB@
(e1)(�1)
(�e1)(��1)
1CCCA =
0BBB@
0
�1(x; t)
1CCCA+
0BBB@
Ik1 0
�1(x; t) 0
1CCCA
0BBB@
u1
�u1
1CCCA (3.35)
3. Denote the lower block of the right hand side of the matrix equation (3.35) by
�v1; i.e.,
�v1 = �1(x; t) + �1(x; t)u1 (3.36)
Then Eq. (3.35) can be written as0BBB@
(e1)(�1)
(�e1)(��1)
1CCCA =
0BBB@
u1
�v1
1CCCA (3.37)
49
where �v1 is treated as an input term and is called the synthetic input. Further-
more, Eq. (3.37) can be represented in state space form as
d
dtz1 = A1 � z1 +B1
0BBB@
u1
�v1
1CCCA (3.38)
where (A1; B1) is controllable and z1 2 Rn1 is the state of (3.37) with n1 =
�11 + �12 + � � �+ �1m.
4. Since Eq. (3.38) is linear and controllable, it is easy to design a control law
0BBB@
u1
�v1
1CCCA = K1 � z1 =
0BBB@
K11
K12
1CCCA � z1 (3.39)
such that
d
dtz1 = (A1 +B1 �K
1) � z1 (3.40)
is stable. Here the controls
u1 = �1(z1) = K11 � z1 (3.41)
and
�v1 = ��1(z1) = K12 � z1 (3.42)
render the z1 subsystem stable.
5. Actually, �v1 was originally de�ned by (3.36) and cannot be set to ��1(z1). Thus
we de�ne the deviation of �v1 from ��1(z1) as �2; i.e.,
�2 = �v1 � ��1(z1) (3.43)
50
With this, the state space equation (3.38) after closing the loop by (3.39) be-
comes
ddtz1 = A1 � z1 +B1
0BBB@
u1
�v1
1CCCA
= A1 � z1 +B1
0BBB@
�1(z1)
��1(z1) + �2
1CCCA
= (A1 +B1K1) � z1 +B
0
1�2
(3.44)
where B0
1 is the right m� k1 columns of B1.
6. De�ne the modi�ed vector �elds
f1(x; t) = f(x) + g(x) 1(x; t) (3.45)
g1(x; t) = g(x)�1(x; t) (3.46)
Then the system dynamics after applying the static state feedback laws (3.34)
and (3.41) is
_x = f1(x; t) + g1(x; t)
0BBB@
�1(z1)
�u1
1CCCA
� p1(x; z1; t) + q1(x; z1; t)�u1
(3.47)
where p1(x; z1; t) is an n-vector and q1(x; z1; t) an n � (m� k1) matrix.
In this step, we have designed a control law u1 = �1(z1) which closes the �rst k1
channels of the system (See Fig. 3.3).
51
+
ξ1
Z 1ξ2
αα
1
1
Figure 3.3: First Step of the Backstepping Designs
Step 2
1. For i = 1; � � � ;m� k1, let �2i be the smallest integer such that the �2i � th time
derivative of �2i with respect to (3.47) explicitly depends on �u1. We obtain
0BBBBBBBB@
(��21)(�21)
...
(��2m�k1)(�
2m�k1
)
1CCCCCCCCA= E
02(x; z1; t) +D
02(x; z1; t)�u
1 (3.48)
where E02(x; z1; t) is an (m�k1)- vector and D
02(x; z1; t) is an (m�k1)�(m�k1)
matrix.
2. If the matrix D02(x; z1; t) is singular, let k2 denotes the rank of D
02(x; z1; t).
Reorder the vector �2 such that the �rst k2 rows of the matrix D02(x; z1; t) is
linearly independent. Denote
�2 =
0BBB@
e2
�e2
1CCCA
52
where e2 = (�21; � � � ; �2k2)T and �e2 = (�2k2+1; � � � ; �
2m�k1
)T . Correspondingly, denote
�2 = (�21; � � � ; �2k2)T and ��2 = (�2k2+1; � � � ; �
2m�k1
)T . From lemma 1, we can choose
a static state feedback which rendered the �rst k2 channels of the input-output
dynamics (3.48) decoupled; i.e., there exists
�u1 = 2(x; z1; t) + �2(x; z1; t)
0BBB@
u2
�u2
1CCCA (3.49)
such that0BBBBBBBB@
(��21)(�21)
...
(��2m�k1)(�
2m�k1
)
1CCCCCCCCA=
0BBB@
(e2)(�2)
(�e2)(��2)
1CCCA =
0BBB@
0
�2(x; z1; t)
1CCCA+
0BBB@
Ik2 0
�2(x; z1; t) 0
1CCCA
0BBB@
u2
�u2
1CCCA
(3.50)
3. Denote the lower block of the right hand side of the matrix Eq.(3.50) as the
synthetic input �v2; i.e.,
�v2 = �2(x; z1; t) + �2(x; z1; t)u2 (3.51)
then Eq. (3.50) becomes
0BBB@
(e2)(�2)
(�e2)(��2)
1CCCA =
0BBB@
u2
�v2
1CCCA (3.52)
Once again, Eq. (3.52) can be represented in state space form as
d
dtz2 = A2 � z2 +B2
0BBB@
u2
�v2
1CCCA (3.53)
53
where (A2; B2) is controllable and z2 2 Rn2 is the state of (3.52) with n2 =
�21 + � � �+ �2m�k1.
4. Since Eq. (3.53) is linear and controllable, we can �nd a state feedback control
law 0BBB@
u2
�v2
1CCCA = K2 � z2 =
0BBB@
K21
K22
1CCCA � z2 (3.54)
such that
d
dtz2 = (A2 +B2 �K
2) � z2 (3.55)
is stable. We notice that
u2 = �2(z2) = K21 � z2 (3.56)
�v2 = ��2(z2) = K22 � z2 (3.57)
render the z2 subsystem stable.
5. Recall that �v2 is not the real control. Thus �v2 cannot be set to ��2(z2). De�ne
the deviation of �v2 from ��2(z2) as �3; i.e.,
�3 = �v2 � ��2(z2) (3.58)
54
Then the state space equation (3.53) after the closing the loop by (3.54) becomes
ddtz2 = A2 � z2 +B2
0BBB@
u2
�v2
1CCCA
= A2 � z2 +B2
0BBB@
�2(z2)
��2(z2) + �3
1CCCA
= (A2 +B2 �K2)z2 +B
0
2�3
(3.59)
Combining (3.44) and (3.59), we obtain
d
dt
0BBB@
z1
z2
1CCCA =
0BBB@
A1 +B1 �K1 0
0 A2 +B2 �K2
1CCCA
0BBB@
z1
z2
1CCCA+
0BBB@
B0
1 � �2
B0
2 � �3
1CCCA (3.60)
Observe from (3.53) that �2 2 z2, the above equation can be written as
d
dt
0BBB@
z1
z2
1CCCA =
0BBB@
A1 +B1 �K1 xxx
0 A2 +B2 �K2
1CCCA
0BBB@
z1
z2
1CCCA+
0BBB@
0
B0
2 � �3
1CCCA (3.61)
6. De�ne the modi�ed vector �elds
f2(x; z1; t) = p1(x; z1; t) + q1(x; z1; t) 2(x; z1; t) (3.62)
g2(x; z1; t) = q1(x; z1; t)�2(x; z1; t) (3.63)
Then the system dynamics after applying the static state feedback laws (3.49)
and (3.56) become
_x = f2(x; z1; t) + g2(x; z1; t)
0BBB@
�2(z2)
�u2
1CCCA
� p2(x; z1; z2; t) + q2(x; z1; z2; t)�u2
(3.64)
where p2(x; z1; z2; t) is an n-vector and q2(x; z1; z2; t) an n�(m�k1�k2) matrix.
55
In step 2, we design a control law u2 = �2(z2) which closes the second k2 channels of
the system.
Step 3
�
�
�
Step L
1. For i = 1; � � � ; (m�k1�� � ��kL�1), let �Li be the smallest integer such that the
�Li � th time derivative of �Li explicitly depends on �uL�1. We obtain
0BBBBBB@
(�L1 )(�L
1)
...
(�Lm�k1�����kL�1)(�Lm�k1�����kL�1
)
1CCCCCCA
= E0L(x; z1; � � � ; zL�1; t) +D
0L(x; z1; � � � ; zL�1; t)�u
L�1
(3.65)
whereE0L(x; z1; � � � ; zL�1; t) is an (m�k1�� � ��kL�1)-vector andD
0L(x; z1; � � � ; zL�1; t)
is an (m� k1 � � � � � kL�1)� (m� k1 � � � � � kL�1) matrix.
2. Assume the decoupling matrix DL(x; z1; � � � ; zL�1; t) is full rank. We can choose
a state feedback law
�uL�1 = L(x; z1; z2; :::; zL�1; t) + �L(x; z1; z2; :::; zL�1; t)uL (3.66)
56
such that 0BBBBBBBB@
(�L1 )(�L1)
...
(�Lm�k1�����kL�1)(�Lm�k1�����kL�1
)
1CCCCCCCCA= uL (3.67)
3. Again, Eq. (3.67) can be represented in state space form as
d
dtzL = AL � zL +BLu
L (3.68)
where (AL; BL) is controllable and zL 2 RnL is the state of the (3.67) with
nL = �L1 + � � �+ �Lm�k1�����kL�1.
4. Since Eq. (3.68) is linear and controllable, we can �nd a state feedback control
law
uL = KL � zL (3.69)
such that
d
dtzL = (AL +BL �K
L) � zL (3.70)
is stable.
5. The closed loop error dynamics in z coordinate is
57
ddt
0BBBBBBBBBBBBBB@
z1
z2
�
�
zL
1CCCCCCCCCCCCCCA
=
0BBBBBBBBBBBBBB@
A1 + B1 �K1 xxx 0 � � � 0
0 A2 + B2 �K2 xxx � � � �
� 0 � � � � 0
� � � � � � xxx
0 0 0 � � 0 AL + BL �KL
1CCCCCCCCCCCCCCA
0BBBBBBBBBBBBBB@
z1
z2
�
�
zL
1CCCCCCCCCCCCCCA
� A � z
(3.71)
Since
�(A) = �(A1 +B1 �K1) [ �(A2 +B2 �K
2) [ � � � [ �(AL +BL �KL); (3.72)
where �(�) stands for the spectrum of (�). This shows that the system in z
coordinate is stable; i.e., y ! yd asymptotically. The block diagram of the
backstepping design is shown in Fig.3.4.
L
1α
α
α
2
ξ 1
LZZZZ 123
Figure 3.4: Backstepping designs of the MIMO nonlinear system
Remarks
� It is interesting to observe that the decoupling control by dynamic extension
delays the 'early appeared' controls by adding integrators in appropriate in-
58
put channels; backstepping design 'backsteps' to determine the 'late appeared'
controls.
� The backstepping control partially closes the loop at each design step; decou-
pling control by dynamic extension �rst decouples the system, and then closes
the loop simultaneously.
� The resulting control law based on the backstepping design is static state feed-
back; the decoupling control law based on the dynamic extension is dynamic
state feedback.
� For the sake of simplicity, we choose a linearizing control strategy in the back-
stepping design process. In general, �0
is and ��0
is are not necessarily to be linear.
� Notice that at most (n1 + n2 + � � � + nL) states have been controlled in the
backstepping procedure. There remains an (n � n1 � n2 � � � � � nL)-th order
internal dynamics. The e�ectiveness of this procedure hinges upon the stability
of the internal dynamics.
3.4 Design example : Planar Vehicle
Consider a vehicle moving in the horizontal plane shown in Fig.3.5, where (x1; x2)
denotes the position and x3 denotes the orientation of the vehicle in the inertial
coordinate. The control inputs are the longitudinal velocity u1 and the yaw rate u2.
59
X
u
u
X
x
1
21
2(x1 , x2 )
3
Figure 3.5: Planar Vehicle
The system equations are
0BBBBBBBB@
_x1
_x2
_x3
1CCCCCCCCA=
0BBBBBBBB@
cos x3
sin x3
0
1CCCCCCCCAu1 +
0BBBBBBBB@
0
0
1
1CCCCCCCCAu2 (3.73)
and the output functions are
y1 = x1
y2 = x2
(3.74)
We will use this system to illustrate both the decoupling control by dynamic extension
and the control by backstepping design.
3.4.1 Decoupling Control by Dynamic Extension
Since the input-output dynamics can be obtained as
0BBB@
_y1
_y2
1CCCA =
0BBB@
cos x3 0
sin x3 0
1CCCA
0BBB@
u1
u2
1CCCA (3.75)
60
the decoupling matrix is singular. We observe that the �rst control input u1 appears
too early. We may add an integrator in the �rst input channel such that the extended
system has a well de�ned vector relative degree. Set
z1 = u1
_z1 = w1
u2 = w2
(3.76)
Now di�erentiating the outputs till the new inputs w1; w2 to appear, we obtain
0BBB@
�y1
�y2
1CCCA =
0BBB@
cosx3 �z1sinx3
sinx3 z1cosx3
1CCCA
0BBB@
w1
w2
1CCCA (3.77)
If z1 = u1 is bounded away from zero, we can choose
0BBB@
w1
w2
1CCCA =
0BBB@
cosx3 sinx3
� sinx3z1
cosx3z1
1CCCA
0BBB@
v1
v2
1CCCA (3.78)
to decouple the system.
3.4.2 Control by Backstepping Design
Let 0BBB@
y1d(t)
y2d(t)
1CCCA
be the desired output vector. The control procedure which achieves asymptotically
tracking of the output vector is designed in the following two steps.
Step 1
61
1. De�ne the output error vector
�1 = y � yd
Then we have 0BBB@
_�11
_�12
1CCCA =
0BBB@
cos x3 0
sin x3 0
1CCCA
0BBB@
u1
u2
1CCCA�
0BBB@
_y1d
_y2d
1CCCA (3.79)
The rank of the decoupling matrix in Eq.(3.79) is 1. Thus we partition �1 as
�1 =
0BBB@
e1
�e1
1CCCA
where �11 = e1 and �12 = �e1.
2. By lemma 1, we choose a state feedback law0BBB@
u1
u2
1CCCA =
0BBB@
1cos x3
(u1 + _y1d)
�u1
1CCCA (3.80)
which decouples the �rst channel of the Eq. (3.79). We obtain0BBB@
_e1
_�e1
1CCCA =
0BBB@
0
tan x3 � _y1d � _y2d
1CCCA +
0BBB@
1 0
tan x3 0
1CCCA
0BBB@
u1
�u1
1CCCA (3.81)
3. De�ne the synthetic input
�v1 = tan x3 � ( _y1d + u1)� _y2d (3.82)
Then the input/output dynamics (3.81) becomes0BBB@
_e1
_�e1
1CCCA =
0BBB@
u1
�v1
1CCCA (3.83)
62
4. Choose the control law
0BBB@
u1
�v1
1CCCA =
0BBB@
�1(e1)
��1(e2)
1CCCA =
0BBB@�k1 � e
1
�k2 � �e1
1CCCA (3.84)
5. De�ne �2 as the di�erence between �v1 and ��1; i.e.,
�2 = �v1 � ��1 (3.85)
Then we have 0BBB@
_e1
_�e1
1CCCA+
0BBB@
k1 0
0 k2
1CCCA
0BBB@
e1
�e1
1CCCA =
0BBB@
0
�2
1CCCA (3.86)
Step 2
1. Recall in step 1 that
�2 = �v1 � ��1 = tan x3 � ( _y1d � k1 � e1)� _y2d + k2 � �e
1 (3.87)
Di�erentiating Eq. (3.87), we obtain
_�2 = sec2 x3�( _y1d�k1�e1)��u1+tan x3�(�y1d+k1�k1�e
1)��y2d�k2�k2�e1+k2��
2 (3.88)
2. By choosing the control law
�u1 =1
sec2x3 � ( _y1d � k1 � e1)�(�tanx3�(�y1d+k1�k1�e
1)+�y2d+k2�k2��e1+k2��
2�k3��2)
(3.89)
we obtain
_�2 = �k3 � �2 (3.90)
63
Recall that
u1 =1
cosx3� (�k1 � e
1 + _y1d) (3.91)
Eq. (3.89) can be written as
�u1 =cosx3u1
� (tanx3 � (�y1d� k1 � k1 � e1)� �y2d� k2 � k2 � �e
1� k2 � �2+ k3 � �
2) (3.92)
Thus if u1(t) 6= 0, the control (3.92) is well de�ned.
3. The closed loop error dynamics become
d
dt
0BBBBBBBB@
e1
�e1
�2
1CCCCCCCCA=
0BBBBBBBB@
�k1 0 0
0 �k2 1
0 0 �k3
1CCCCCCCCA
0BBBBBBBB@
e1
�e1
�2
1CCCCCCCCA
(3.93)
Remark : Both approaches assume that u1 6= 0. This is because u1 is the longitudinal
velocity input, and once u1 = 0, the system becomes uncontrollable.
3.5 Conclusions
Two design approaches for a class of multivariable nonlinear systems without vec-
tor relative degrees have been presented. Decoupling control of this class of nonlinear
systems results in adding chains of integrators in the input channels. We provide an
alternative approach to control the multivariable nonlinear systems by using the dy-
namic extension algorithm to identify the input channels where controls appear \too
early" in the input-output sense, then applying the backstepping control algorithm to
64
partially close those \early appeared" loops. The proposed control procedure results
in static state feedback. Even though the �nal closed loop form of the nonlinear sys-
tem is not decoupled, each output is controlled to the desired value asymptotically.
A planar vehicle is used to illustrate and contrast both approaches.
65
Chapter 4
Lateral Control of Light Passenger
Vehicles in Automated Highway
Systems
4.1 Introduction
In this chapter, a backstepping controller is designed for lateral guidance of light
passenger cars in automated highway systems. In this design, the vehicle lateral
displacement is a�ected by the relative yaw angle of the car with respect to the
road centerline, and the relative yaw angle is controlled by the vehicle's front wheel
steering angle. One feature of this design is that the closed loop performance of
lateral and yaw dynamics can be speci�ed and guaranteed simultaneously. Since ride
66
quality is an important factor in designing the lateral control algorithm for passenger
cars, a smooth road model is used in calculating the feedforward part of the steering
command. Furthermore, nonlinear position feedback, which acts as low gain control
at small tracking errors and high gain control at larger tracking errors, is introduced
in this design as trade-o� between passenger ride comfort and tracking accuracy.
Simulation using the complex vehicle model will be conducted to verify e�ectiveness
of the backstepping design approach.
The organization of this chapter is as follows. In section 4.2, the lateral control
model is presented. The development of the backstepping controller is shown in
section 4.3. Simulations using the complex vehicle model are conducted in section
4.4. Conclusions of this chapter are given in the last section.
4.2 Vehicle Dynamics and Control Model
Two types of dynamic models are developed in [50] for the design and analysis
of lateral controllers for passenger cars. A complex simulation model represents the
vehicle dynamics as realistically as possible. A simpli�ed model retains only the
signi�cant dynamics (lateral and yaw dynamics). The former is used in the simulation
study for evaluating the e�ectiveness of the controller, and the latter in the design of
the controller. The simpli�ed fourth order control model can be represented as
67
m�yCG + 2V(C�f + C�r) _yCG + 2
V(lfC�f � lrC�r)�_�� 2(C�f + C�r)��
= 2C�f� + (�1V� 2(lfC�f � lrC�r)�mV ) _�d
(4.1)
I���+ 2V(lfC�f � lrC�r) _yCG + 2
V(l2fC�f + l2rC�r)�_�� 2(lfC�f � lrC�r)��
= 2lfC�f� �2V(l2fC�f + l2rC�r) _�d
(4.2)
where the variables and symbols are de�ned and listed in Table 4.1. This control
model can be regarded as a linear system with the time-varying longitudinal velocity,
V , appearing in system parameters and the road disturbance, _�d, a�ecting both the
lateral dynamics (Eq.(4.1)) and the yaw error dynamics (Eq.(4.2)). For designing the
backstepping lateral controller, we will reformulate this control model by making the
following ideal mass assumption [1] and rede�ning the system output.
variable description
� front wheel steering angleV longitudinal velocityyCG lateral displacement of C.G. w.r.t. road coordinate_yCG lateral velocity of C.G. w.r.t. road coordinate�� relative yaw angle of the vehicle w.r.t. road coordinate�_� relative yaw rate of the vehicle w.r.t. road coordinate_�d angular velocity of the road coordinate w.r.t. the inertial coordinate
Table 4.1: Notations of the Simpli�ed Control Model
Assumption 1 (Ideal Mass Assumption)
68
������������
������������
mm��������
��������
m rfr
l lf
Figure 4.1: Ideal mass distribution for passenger car
We assume that the mass distribution of a vehicle can be idealized as two concentrated
masses (Fig.4.1) located at the vehicle's front wheel and the rear wheel, respectively.
It can be easily seen that
m = mf +mr
lfmf = lrmr
From the above two equations, the front and rear masses can be calculated as
mf =lr
lf + lrm
mr =lf
lf + lrm
Thus the moment of inertia can be expressed as
I = mf l2f +mrl
2r = mlf lr (4.3)
Notice that the ideal mass assumption results in a simple relationship between the
mass, m, and the moment of inertia, I, which will be used in the reformulation of the
control model. The lateral dynamics can be reformulated as a cascade system after
some algebraic manipulation and rede�nition of the system output. Subtraction of 1lf
69
� Eq.(4.2) from Eq.(4.1) yields
m�yCG �Ilf���+ 2
VC�r(1 +
lrlf) _yCG �
2VC�rlr(1 +
lrlf)�_�
= 2C�r(1 +lrlf)��+ ( 1
V� 2lrC�r(1 +
lrlf)�mV ) _�d
(4.4)
By the ideal mass assumption,
I
lf= m� lr (4.5)
From Eqs (4) and (5), we obtain
m(�yCG � lr���) + 2VC�r(1 +
lrlf)( _yCG � lr�_�)
= 2C�r(1 +lrlf)��+ ( 1
V� 2lrC�r(1 +
lrlf)�mV ) _�d
(4.6)
Rede�ning the new output
yo = yCG � lr��
and from Eq. (6) and (2), the reformulated control model can be represented by the
following two equations
m�yo +2VC�r(1 +
lrlf) _yo = 2C�r(1 +
lrlf)��+ ( 1
V� 2lrC�r(1 +
lrlf) �mV ) _�d (4.7)
I���+ 2V(lfC�f � lrC�r)( _yo + lr�_�) + 2
V(l2fC�f + l2rC�r)�_�� 2(lfC�f � lrC�r)��
= 2C�f lf� �2V(l2fC�f + l2rC�r) _�d
(4.8)
The block diagram of this model becomes as shown in Fig. 4.2. In this control model,
the rede�ned output, yo, is the lateral displacement of the real axle of the car relative
70
to the road centerline. As yo converges to zero, yCG converges to lr � ��, which is
small in normal highway maneuvers and will be shown in simulations.
yδ
ε
∆ εo
o
1f 2f
d
.
yo , yo.
∆ ε, ∆ ε.y.
Figure 4.2: Block diagram of the lateral dynamics
4.3 Lateral Control of Light Passenger Vehicles
The backstepping design technique is utilized to design the lateral controller in
this study. Firstly, we will regard that the relative yaw angle �� be the control input
in Eq.(7) (see Fig.4.2). Then, we 'backstep' to determine the steering angle, �, in Eq.
(8), which controls �� to the desired value in the �rst step. This control procedure
resembles the human driver engaged in lane following maneuvers. Mathematically,
the backstepping control can solve the road disturbance rejection problem where the
matching condition is not satis�ed. Another advantage of the backstepping design is
that the nonlinear spring e�ects can be easily introduced in the closed loop [35]. The
nonlinear spring terms can be used to compromise the design speci�cations such as
71
passenger comfort, tracking errors and safety.
4.3.1 Road Model
In the framework of AHS, the road curvature is previewable. This information
can be used to generate the feedforward part of the steering command. Recall from
Eqs.(4.7) and (4.8) that the road disturbance, _�d, a�ects both the lateral and yaw
error dynamics. As shown in Fig.4.3, _�d is de�ned as the angular velocity of the
road reference coordinate, OrXrYr, relative to the inertial �xed coordinate, OnXnYn.
Thus _�d is determined by the longitudinal velocity and the road curvature. The road
curvature and the road disturbance may not be smooth in roadways. In such cases,
we approximate the road curvature by a smooth function w(t), which is referred to
as the road model. The deviations of w(t) with respect to _�d will be treated as a
disturbance and its e�ect will be attenuated by the controller.
The road model is shown in Fig.4.4. In this example, _�d is given by a trapezoidal
function, and given h > 0, we can �nd a su�ciently smooth function w(t) such that
jw(t)� _�d(t)j � h
i.e., there exists a ~�(t) s.t.
_�d(t) = w(t) + ~�(t)
and
j ~�(t)j � h
72
O X
Y O
X
Y
O
X
dY
n n
n r
r
r
u
u
u
Road Centerline
εε.
.
Figure 4.3: De�nition of the desired yaw rate
When _�d is a rectangular function, it is not possible to approximate it by a smooth
function with an arbitrarily error, and _�d needs to be rede�ned appropriately.
(t)
t
ε.
d ω
Figure 4.4: A smooth road model
4.3.2 Controller Design
Since each step of the backstepping control procedure requires the derivative of the
'virtual control law' designed in the previous step, a smooth control law design is
73
essential. In this chapter, a smooth approximation of the sgn(�) function by the
hyperbolic tanh(�) function shall be used [54, 68] and is summarized in the following
lemma.
Lemma 1 Given any e > 0, the following inequality holds
0 � h � x � sgn(x)� h � x � tanh(�hx
e) � e
where � = 0:2785 and h is any positive number.
The development of the lateral controller proceeds in the following three steps. In the
�rst step, the lateral dynamics (4.7) is stabilized by using �� as the control input.
In step 2, the actual �� is made to converge to the desired value by using �_� as
the virtual control input. In step 3, the �nal steering command is determined such
that the actual �_� converges to the desired value in step II and the globally uniform
boundedness of the lateral and yaw dynamics is achieved.
Step 1
We will view �� as the virtual control input in this step and design a control law
�� = �1
such that the lateral dynamics (Eq. (4.7)) is global uniformly bounded and the state
variables converge to a ball with an arbitrary prescribed size. To facilitate designing
the control law, an intermediate variable
s = _yo + �1yo
74
is introduced. Eq. (4.7) is rewritten as
m _s+ 2VC�r(1 +
lrlf)s�m�1 _yo �
2VC�r(1 +
lrlf)�1yo
= 2C�r(1 +lrlf)��+ ( 2
VlrC�r(1 +
lrlf)�mV )(w(t) + ~�(t))
(4.9)
Consider the Lyapunov function candidate
V1 =1
2ms2 +
1
42C�r(1 +
lrlf)�3y
4o
Then the time derivative of the Lyapunov function candidate V1 is
_V1 = ms _s+ 2C�r(1 +lrlf)�3y3o _yo
= ms _s+ 2C�r(1 +lrlf)�3y3os� 2C�r(1 +
lrlf)�1�3y4o
(4.10)
Substituting Eq. (4.9) into (4.10), we obtain
_V1 = s(� 2VC�r(1 +
lrlf)s+m�1 _yo +
2VC�r(1 +
lrlf)�1yo + 2C�r(1 +
lrlf)��
+( 2VlrC�r(1 +
lrlf)�mV )(w(t) + ~�(t)))
+2C�r(1 +lrlf)�3y3os� 2C�r(1 +
lrlf)�1�3y4o
(4.11)
We choose the control law
�� = �1 = ��1Vyo �
m�12C�r(1+
lrlf)_yo � �3y
3o � (�2 �
1V)s+ (� lr
V+ mV
2C�r(1+lrlf))w(t)
� h12C�r(1+
lrlf)� tanh(�h1s
e1)
(4.12)
where e1 is a positive but arbitrary small design parameter, and h1 satis�es
(2
VlrC�r(1 +
lrlf)�mV ) � ~�(t) � h1 (4.13)
75
With this control law, the time derivative of the Lyapunov function becomes
_V1 = �2�2C�r(1 +lrlf)s2 � 2�1�3C�r(1 +
lrlf)y4o
�h1 � s � tanh(�h1se1
) + ( 2VlrC�r(1 +
lrlf)�mV ) � s � ~�(t)
� �2�2C�r(1 +lrlf)s2 � 2�1�3C�r(1 +
lrlf)y4o
�h1 � s � tanh(�h1se1
) + h1jsj
� �2�2C�r(1 +lrlf)s2 � 2�1�3C�r(1 +
lrlf)y4o + e1
(4.14)
where the �rst and second inequalities have been obtained by noting Eq.(4.13) and
lemma 1, respectively. Since e1 can be adjusted arbitrary small, the lateral dynamics
is stable and the output yo exponentially converges to a ball whose size depends on
e1.
It is interesting to notice that with the control law de�ned by Eq.(4.12), the lateral
dynamics (4.9) become
m _s+ 2�2C�r(1 +lrlf)s+ 2C�r(1 +
lrlf)�3y3o
= �h1 � tanh(�h1se1
) + ( 2VlrC�r(1 +
lrlf)�mV ) ~�(t)
(4.15)
where the third term on the left hand side of the equation is the nonlinear spring
e�ect in the closed loop and the tanh term on the right hand side of the equation
attenuates the disturbance e�ect caused by ~�(t).
Step 2
In step 1, we assumed that the relative yaw angle, ��, was the control input and design
the control law, �1. In this step, we will 'backstep' once and regard the relative yaw
76
rate, �_�, as the control input. First, we de�ne the deviations of �� from �1 as z1,
i.e.,
z1 = ��� �1 (4.16)
or
�� = �1 + z1 (4.17)
By (4.17), _V1 in (4.14) can be recalculated as
_V1 � �2�2C�r(1 +lrlf)s2 � 2�1�3C�r(1 +
lrlf)y4o + e1 + 2C�r(1 +
lrlf)sz1 (4.18)
Since we want the error variable z1 small, we augment the Lyapunov function as
V2 = V1 + 12z21 =
1
2ms2 +
1
42C�r(1 +
lrlf)�3y
4o +
12z21 (4.19)
By (4.18), the time derivative of V2 is given by
_V2 = _V1 + 1z1 _z1
� �2�2C�r(1 +lrlf)s2 � 2�1�3C�r(1 +
lrlf)y4o + e1 + 2C�r(1 +
lrlf)sz1 + 1z1 _z1
(4.20)
Notice that
_z1 = �_�� _�1
= �_�+ �1V_yo +
m�12C�r(1+
lrlf)�yo + 3�3y2o _yo + (�2 �
1V) _s+ ( lr
V� mV
2C�r(1+lrlf)) _w(t)
+ h12C�r(1+
lrlf)�h1e1sech2(�h1s
e1) _s
(4.21)
and
_s = � 2m�2C�r(1 +
lrlf)s� 2
mC�r(1 +
lrlf)�3y3o �
h1mtanh(�h1s
e1)
+ 1m( 2VlrC�r(1 +
lrlf)�mV ) ~�(t) + 1
m2C�r(1 +
lrlf)z1
(4.22)
77
�yo = _s� �1 _yo (4.23)
Substituting Eqs. (4.22) and (4.23) into (4.21), we obtain
_z1 = �_�+ �1V
_yo
�( m�1
2C�r(1+lrlf
)+ �2 �
1V+ �h2
1
2C�r(1+lrlf
)e1sech2(�h1s
e1)) � ( 2
m�2C�r(1 +
lrlf)s+ 2
mC�r(1 +
lrlf)�3y3o
+h1mtanh(�h1s
e1)� 2
mC�r(1 +
lrlf)z1) �
m�21
2C�r(1+lrlf
)_yo + 3�3y2o _yo + ( lr
V�
mV
2C�r(1+lrlf
)) _w(t)
+( m�1
2C�r(1+lrlf
)+ �2 �
1V+
�h21
2C�r(1+lrlf
)e1sech2(�h1s
e1)) � 1
m( 2VlrC�r(1 +
lrlf) �mV ) ~�(t)
(4.24)
Substitute (4.24) into (4.20) and choose the control law
�_� = �2
= � 2 1C�r(1 +
lrlf)s � k1z1 �
�1V_yo + ( m�1
2C�r(1+lrlf
)+ �2 �
1V+ �h2
1
2C�r(1+lrlf
)e1sech2(�h1s
e1))�
( 2m�2C�r(1 +
lrlf)s + 2
mC�r(1 +
lrlf)�3y3o +
h1mtanh(�h1s
e1)� 2
mC�r(1 +
lrlf)z1) +
m�21
2C�r(1+lrlf
)_yo
�3�3y2o _yo � ( lrV� mV
2C�r(1+lrlf
)) _w(t) � h2 � tanh(
�h2z1e2
)
(4.25)
Notice that the disturbance ~�(t) coming into _z1 is attenuated by the last term h2 �
tanh(�h2z1e2
) in (4.25), where e2 is a positive but arbitrarily small design parameter,
and h2 satis�es
(m�1
2C�r(1 +lrlf)+�2�
1
V+
�h21
2C�r(1 +lrlf)e1
sech2(�h1s
e1))
1
m(2
VlrC�r(1+
lr
lf)�mV ) ~�(t) � h2
78
The time derivative of the Lyapunov function V2 in (4.20) is
_V2 � �2�2C�r(1 +lrlf)s2 � 2�1�3C�r(1 +
lrlf)�3y4o � 1k1z
21
�h1 � s � tanh(�h1se1
) + ( 2VlrC�r(1 +
lrlf)�mV )s ~�(t)
� 1 � h2 � z1 � tanh(�h2z2e2
)
+ 1 � (m�1
2C�r(1+lrlf
)+ �2 �
1V+
�h21
2C�r(1+lrlf
)sech2(�h1s
e1)) 1
m( 2VlrC�r(1 +
lrlf) �mV )z1 ~�(t)
� �2�2C�r(1 +lrlf)s2 � 2�1�3C�r(1 +
lrlf)�3y4o � k1 1z
21 + e1 + 1e2
(4.26)
If �� is the real control, by Lyapunov stability theorem and (4.26) the state variables
yo, s and z1 will converge to a ball whose size depends on e1 + 1e2.
Step 3
Similar to the development in step 2, we de�ne the deviations of �_� from �2 as z2;
i.e.,
z2 = �_�� �2 (4.27)
or
�_� = �2 + z2 (4.28)
By (4.28), _V2 in (4.20) can be recalculated as
_V2 � �2�2C�r(1 +lrlf)s2 � 2�1�3C�r(1 +
lrlf)�3y4o
�k1 1z21 + e1 + 1e2 + 1z1z2
(4.29)
Since we would like the error z2 small, we augment the Lyapunov function
V3 = V2 + 22Iz22
= 12ms2 + 1
42C�r(1 +lrlf)�3y4o +
12 z
21 +
22 Iz
22
(4.30)
79
Then its time derivative is given by
_V3 = _V2 + Iz2 _z2
� �2�2C�r(1 +lrlf)s2 � 2�1�3C�r(1 +
lrlf)y4o � k1 1z
21
+e1 + 1e2 + 1z1z2 + 2Iz2 _z2
(4.31)
Notice that
I _z2 = I���� I _�2
= I���+ 2 1C�r(1 +
lrlf)I _s + k1I _z1 +
�1VI�yo
+( m�1
2C�r(1+lrlf
)+ �2 �
1V+
�h21
2C�r(1+lrlf
)e1sech2(�h1s
e1))I(� 2
m�2C�r(1 +
lrlf) _s
� 6mC�r(1 +
lrlf)�3y2o _yo �
�h21
me1sech2(�h1s
e1) _s + 2
mC�r(1 +
lrlf) _z1)
+(�m�2
1
2C�r(1+lrlf
)+ 3�3y2o)I�yo + 6�3Iyo _y2o + ( lr
V�
mV
2C�r(1+lrlf
))I �w(t)
= � 2V(lfC�f � lrC�r)( _yo + lr�_�) � 2
V(l2fC�f + l2rC�r)�_�+ 2(lfC�f � lrC�r)��
+2lfC�f� �2V(l2fC�f + l2rC�r)w(t) �
2V(l2fC�f + l2rC�r) ~�(t)
+ 2 1C�r(1 +
lrlf)I _s + k1I _z1 +
�1VI�yo
+( m�1
2C�r(1+lrlf
)+ �2 �
1V+
�h21
2C�r(1+lrlf
)e1sech2(�h1s
e1))I(� 2
m�2C�r(1 +
lrlf) _s
� 6mC�r(1 +
lrlf)�3y2o _yo �
�h21
me1sech2(�h1s
e1) _s + 2
mC�r(1 +
lrlf) _z1)
+(� m�21
2C�r(1+lrlf
)+ 3�3y2o)I�yo + 6�3Iyo _y2o + ( lr
V� mV
2C�r(1+lrlf
))I �w(t)
(4.32)
where _s, �yo and _z1 are given in Eqs. (4.22) (4.23) and (4.24), respectively. From
80
(4.31) and (4.32), the control law for front wheel steering can be chosen as
� = 12lfC�f
(� 1 2z1 � k2z2 +
2V (lfC�f � lrC�r)( _yo + lr�_�) + 2
V (l2fC�f + l2rC�r)�_�
�2(lfC�f � lrC�r)��+ 2V (l
2fC�f + l2rC�r)w(t)� h3 � tanh(
�h3z2e3
)
� 2 1C�r(1 +
lrlf)I _s� k1I _z1 �
�1V I �yo
�( m�12C�r(1+
lrlf)+ �2 �
1V +
�h21
2C�r(1+lrlf)e1
sech2(�h1se1))I(� 2
m�2C�r(1 +lrlf) _s
�6mC�r(1 +
lrlf)�3y
2o _yo �
�h21
me1sech2(�h1se1
) _s+ 2mC�r(1 +
lrlf) _z1)
�(�m�2
1
2C�r(1+lrlf)+ 3�3y
2o)I �yo � 6�3Iyo _y
2o � ( lrV � mV
2C�r(1+lrlf))I �w(t))
(4.33)
where e3 is a positive but arbitrarily small design parameter, and h3 satis�es
2
V(l2fC�f + l2rC�r) ~�(t) � h3
With this control law, the derivative of the �nal Lyapunov function, V3, is given by
_V3 = �2�2C�r(1 +lrlf)s2 � 2�1�3C�r(1 +
lrlf)y4o � 1k1z
21 � 2k2z
22
�h1 � s � tanh(�h1se1
) + ( 2V lrC�r(1 +lrlf)�mV )s ~�(t)
� 1 � h2 � z1 � tanh(�h2z1e2
) + 1 � (m�1
2C�r(1+lrlf)+ �2 �
1V ) 1
m( 2V lrC�r(1 +lrlf)�mV )z1 ~�(t)
� 2 � h3 � z2 � tanh(�h3z2e3
)� 22V (l
2fC�f + l2rC�r)z2 ~�(t)
� �2�2C�r(1 +lrlf)s2 � 2�1�3C�r(1 +
lrlf)y4o � k1 1z
21 � k2 2z
22 + e1 + 1e2 + 2e3
Because the numbers e1, e2 and e3 can be made arbitrarily small, the closed loop
system is stable and the output tracking error exponentially converges to a ball whose
size depends on e1 + 1e2 + 2e3.
Remark:
The idea of exploiting cascaded control in the design of the steering controller was
81
�rst presented by Guldner et al.[22], where the controller was designed in two steps.
Firstly, the vehicle yaw rate is considered as a �ctitious control input and is used to
control the lateral displacement. Secondly, the actual yaw rate is controlled to the
desired value by the steering rate. However, there are two major di�erences between
[22] and this study.
� In [22], the relative degree from the yaw rate to the lateral displacement, yCG,
is one. In this study, the relative degree from the yaw rate to the lateral dis-
placement, yo, is three. There is a second order internal dynamics in [22].
� The deviations of the yaw rate from the desired value is not considered in
[22]; instead, a hierarchy of time scales (fast or slow dynamics) is maintained
by choosing suitably the gains of the cascaded control loops. In this study, a
systematic backstepping design is utilized, where the deviations are quanti�ed
(z1 and z2) and fed back in the closed loop.
4.4 Simulation Results
The simulations are conducted using the complex vehicle model [50]. and the
vehicle parameters are listed in Table 4.2. The simulation scenario we used is depicted
in Fig. 4.5. The car travels along a straight roadway with initial lateral displacement
20 cm and enters a curved section with radius of curvature 500 m at time t = 5 sec.
Fig. 4.6 and Fig. 4.7 show the simulation results of the backstepping controller at
82
vehicle speeds 30 MPH and 60 MPH, respectively. The peak lateral displacement is
about 2 cm at longitudinal speed V = 26.4 m/s (60 MPH).
parameter unit value
mass (m) Kg 1485moment of inertia (Ix) Kg �m2 479.6moment of inertia (Iy) Kg �m2 2594.3moment of inertia (Iz) Kg �m2 2782.7
Cornering Sti�ness (C�f) N=rad 43,000Cornering Sti�ness (C�r) N=rad 43,000Front wheel to C.G.(lf) m 1.034Rear wheel to C.G.(lr) m 1.491
Front wheel track width (Twf) m 1.45Rear wheel track width (Twf) m 1.45
Table 4.2: Parameters of a Passenger Car
ρ = 500 m
t=5 sec.
t=8 sec.
t=15 sec.
Figure 4.5: Simulation scenario
83
0 5 10 15−10
0
10
20
Time (s)
Lat.
dis.
(cm
)
0 5 10 15−2
−1
0
1
Time (s)
Rol
l ang
le (
deg)
0 5 10 15−2
−1
0
1
Time (s)
Yaw
ang
le (
deg)
0 5 10 15−5
0
5
Time (s)
Yaw
rat
e (d
eg/s
)
0 5 10 15−2
−1
0
1
Time (s)
Ste
erin
g(de
g)
Figure 4.6: Simulation results of backstepping controller at longitudinal velocity =30 MPH, solid line : lateral position at C.G., dashdot line : lateral position at rear
4.5 Conclusions
A backstepping controller was designed for lateral guidance of the passenger car
in automated highway systems. In this design, the vehicle lateral displacement is
a�ected by the relative yaw angle of the car with respect to the road centerline, and
the relative yaw angle is controlled by the vehicle's front wheel steering angle. The
closed loop performance of lateral and yaw dynamics can be speci�ed and guaranteed
simultaneously in this approach. Furthermore, a smooth road model was used in the
84
0 5 10 15−10
0
10
20
Time (s)
Lat.
dis.
(cm
)
0 5 10 15−4
−2
0
2
Time (s)
Rol
l ang
le (
deg)
0 5 10 15−1
−0.5
0
0.5
Time (s)
Yaw
ang
le (
deg)
0 5 10 15−4
−2
0
2
Time (s)
Yaw
rat
e (d
eg/s
)
0 5 10 15−2
−1
0
1
Time (s)
Ste
erin
g(de
g)
Figure 4.7: Simulation results of backstepping controller at longitudinal velocity =60 MPH, solid line : lateral position at C.G., dashdot line : lateral position at rear
design of the lateral controller. The advantage for introducing the smooth road model
is that a smooth feedforward information is included in the steering command, which
results in a better passenger ride quality when the car is negotiating a turn. In real
time implementation of the lateral controller, the lateral errors are measured using
the magnetometors. Once the vehicle runs out of the sensor range, it becomes uncon-
trolled. To overcome this, a nonlinear spring e�ect was introduced in the feedback
loop to enhance the safety while without sacri�cing the ride qualitye. Simulation re-
sults validated the e�ectiveness of this controller in compromising the tracking error
85
and ride quality. The experimental study of this controller will be conducted in the
near future.
86
Chapter 5
Dynamic Modeling of
Tractor-Semitrailer Vehicles for
Automated Highway Systems
5.1 Introduction
Lateral control of commercial heavy-duty vehicles is studied in this and the next
chapters. Due to the popularity of the tractor-semitrailer type commercial heavy-duty
vehicle, we will use it as the benchmark vehicle in our study. Two types of dynamic
models are developed in the study of lateral control of tractor-semitrailer vehicles
in AHS: a complex simulation model and two simpli�ed control models. In this
chapter a nonlinear complex model is developed to simulate the dynamic responses
87
of tractor-semitrailer vehicles and will be exploited to evaluate the e�ectiveness of
lateral control algorithms, which will be synthesized in the next chapter. This sim-
ulation model consists of three main components: the vehicle sprung mass (body)
dynamics, a tire model and a suspension model. The main distinction between this
complex model and those in the literatures is that the vehicle sprung mass dynamics
is derived by applying Lagrangian mechanics. This approach has an advantage over
a Newtonian mechanics formulation in that this modeling approach eliminates the
holonomic constraint at the �fth wheel of the tractor-semitrailer vehicle by choosing
the articulation angle as the generalized coordinate. Since there is no constraint in-
volved in the model, it is easier for both designing control algorithms and solving the
di�erential equations numerically. Other con�gurations of articulated vehicles, for
example the tractor/three trailer combination, can also be modeled with the same
approach.
The second type of dynamic models are represented by two simpli�ed lateral
control models: one for steering control and the other for coordinated steering and
di�erential braking control. These lateral control models, which are simpli�ed from
the complex model, will be developed in the next chapter.
The organization of this chapter is as follows. In section 5.2, a coordinate system
is introduced to describe the motion of the tractor-semitrailer type commercial heavy-
duty vehicles. Based on the coordinate system in section 5.2, the kinetic energy and
the potential energy are calculated in section 5.3. In section 5.4, a set of equations
88
describing the sprung mass dynamics are derived by using Lagrange's mechanics. In
conjunction with the equations of the unsprung mass dynamics, the expression of
the generalized force corresponding to each coordinate is obtained in section 5.5. To
complete the development of the complex model, we present the tire model by Baraket
and Fancher [2] and a simpli�ed suspension model in section 5.6. E�ectiveness of this
modeling approach is shown in section 5.7 by comparing the open loop experimental
results of a tractor-semitrailer vehicle and the simulation results from the complex
model. Conclusions of this chapter are given in the last section.
5.2 De�nition of Coordinate System
5.2.1 Coordinate System
A coordinate system is de�ned to characterize the motion of a tractor-semitrailer
type of articulated vehicle. As shown in Fig. 5.1, XnYnZn is the globally �xed inertial
reference coordinate. We will obtain the expressions of vehicle kinetic and potential
energies with respect to this reference coordinate. XuYuZu is the tractor's unsprung
mass coordinate, which has the same orientation as the tractor. The Zu axis passes
through the tractor's C.G. The translational motion of the tractor in the Xn � Yn
plane and the yaw motion of the tractor along the Zn axis can be described by the
relative motion of the XuYuZu coordinate with respect to XnYnZn. Xs1Ys1Zs1 is
the tractor's sprung mass coordinate, which is body-�xed at the tractor's center of
gravity. Coordinate Xs1Ys1Zs1 has roll motion relative to coordinate XuYuZu. The
89
Zs2
Xu
YuZu
Xs
Ys
Zs
Xn
Yn
Zn
1
1
1Xs2
Ys2
Figure 5.1: Coordinate System to Describe the Vehicle Motion
trailer's motion can be characterized by describing the articulation angle between the
tractor and the trailer, or the relativemotion of the trailer's unsprung mass coordinate
Xs2Ys2Zs2 with respect to the tractor's unsprung mass coordinate Xs1Ys1Zs1. Having
de�ned the coordinate systems, a set of state variables for the tractor-semitrailer
vehicle can be introduced as
� xn : position of the tractor C.G. in Xn direction of the inertial coordinate XnYnZn
� _xn : velocity of the tractor C.G. in Xn direction of the inertial coordinate XnYnZn
� yn : position of the tractor C.G. in Yn direction of the inertial coordinate XnYnZn
90
� _yn : velocity of the tractor C.G. in Yn direction of the inertial coordinate XnYnZn
� �1 : tractor yaw angle with respect to inertial coordinate XnYnZn
� _�1 : tractor yaw rate with respect to inertial coordinate XnYnZn
� � : tractor roll angle
� _� : rate of change of tractor roll angle
� �f : articulation angle between the tractor and the trailer
� _�f : rate of change of the articulation angle between the tractor and the trailer
With the de�nition of the state variables, we can calculate the transformation matrices
between those coordinates. The transformation matrices will be used to obtain the
kinematics of the vehicle. The transformation matrices between the inertial reference
frame and the unsprung mass coordinate are
0BBBBBBBB@
in
jn
kn
1CCCCCCCCA=
0BBBBBBBB@
cos�1 �sin�1 0
sin�1 cos�1 0
0 0 1
1CCCCCCCCA
0BBBBBBBB@
iu
ju
ku
1CCCCCCCCA
(5.1)
and 0BBBBBBBB@
iu
ju
ku
1CCCCCCCCA=
0BBBBBBBB@
cos�1 sin�1 0
�sin�1 cos�1 0
0 0 1
1CCCCCCCCA
0BBBBBBBB@
in
jn
kn
1CCCCCCCCA: (5.2)
91
The transformation matrices between the unsprung mass coordinate and the tractor's
sprung mass coordinate are
0BBBBBBBB@
iu
ju
ku
1CCCCCCCCA=
0BBBBBBBB@
1 0 0
0 1 ��
0 � 1
1CCCCCCCCA
0BBBBBBBB@
is1
js1
ks1
1CCCCCCCCA
(5.3)
and 0BBBBBBBB@
is1
js1
ks1
1CCCCCCCCA=
0BBBBBBBB@
1 0 0
0 1 �
0 �� 1
1CCCCCCCCA
0BBBBBBBB@
iu
ju
ku
1CCCCCCCCA: (5.4)
The transformation matrices between the tractor's sprung mass coordinate and the
trailer's sprung mass coordinate are
0BBBBBBBB@
is1
js1
ks1
1CCCCCCCCA=
0BBBBBBBB@
cos�f �sin�f 0
sin�f cos�f 0
0 0 1
1CCCCCCCCA
0BBBBBBBB@
is2
js2
ks2
1CCCCCCCCA
(5.5)
and 0BBBBBBBB@
is2
js2
ks2
1CCCCCCCCA=
0BBBBBBBB@
cos�f sin�f 0
�sin�f cos�f 0
0 0 1
1CCCCCCCCA
0BBBBBBBB@
is1
js1
ks1
1CCCCCCCCA: (5.6)
5.2.2 Reference Frame
As shown in Fig.5.2, three types of reference frames are used to describe the
translational and rotational motion of vehicles. They are inertial reference frame
92
O X
Y O
X
Y
O
X
dY
n n
n r
r
r
u
u
u
Road Centerline
εε.
.
Figure 5.2: Three Reference Coordinates
XnYnZn, vehicle unsprung mass reference frame XuYuZu and road reference frame
XrYrZr. In this chapter the complex vehicle model is �rst derived with respect to
the inertial reference frame. However, state variables, such as the position and the
orientation, of a vehicle with respect to the inertial reference frame are not what
we are concerned with. The complex model is transformed so that it depends only
on state variables with respect to the unsprung mass reference frame. The vehicle
model relative to the unsprung mass reference frame does not depend explicitly on the
position and the orientation of the vehicle. This is widely used in vehicle dynamics
to predict and analyze vehicle handling response, since the side slip angle, yaw rate
93
and lateral acceleration of the vehicle are naturally de�ned in this reference frame.
Further, for the lane following manever in automated highway systems, the road
reference coordinate OrXrY r in Fig.5.2 is naturally introduced to describe tracking
errors of the vehicle with respect to the road centerline. The road reference coordinate
OrXrY r is such de�ned that the Xr axis is tangent to the road centerline and the
Yr axis passes through the center of gravity of the vehicle. By studing the kinematics
with respect to di�erent reference frames, transformations from the inertial reference
frame to the unsprung mass reference frame and from the unsprung mass reference
frame to the road reference frame will be obtained. In the following section, the
transformation between the inertial reference frame and the unsprung mass reference
frame will be studied. It will be used in section 5.4 to obtain the complex vehicle
model. The transformation between the unsprung mass reference frame and the road
reference frame will be studied in the next chapter to obtain control models.
5.2.3 Transformation between the inertial reference frame and the un-
sprung mass reference frame
From Fig. 5.3, the vehicle velocity at C.G. with respect to the inertial reference
frame is
VCG = _xnin + _ynjn (5.7)
where _xn is the component of the vehicle velocity along the Xn axis and _yn is the
component of the vehicle velocity along the Yn axis. The vehicle acceleration at C.G.
94
O X
Y
n
.
n
n
Xu
Yu
Ou
ε1
ε1
Figure 5.3: Inertial and Unsprung Mass Reference Frames
can be obtained by di�erentiating Eq. (5.7),
aCG = �xnin + �ynjn: (5.8)
On the other hand, the vehicle velocity at C.G. can also be denoted as
VCG = _xuiu + _yuju; (5.9)
where _xu is the velocity component along theXu axis of the unsprung mass coordinate
and _yu is the velocity component along the Yu axis of the unsprung mass coordinate.
Then the vehicle acceleration at C.G. can be obtained by di�erentiating Eq. (5.9),
aCG = �xuiu + _xuddtiu + �yuju + _yu
ddtju
= (�xu � _yu _�1)iu + (�yu + _xu _�1)ju;
(5.10)
where
d
dtiu = _�1ju
95
and
d
dtju = � _�1iu
are used in Eq. (5.10). By equating Eqs. (5.7) and (5.9) and noting the transforma-
tion matrix 0BBB@in
jn
1CCCA =
0BBB@
cos �1 �sin �1
sin �1 cos �1
1CCCA
0BBB@iu
ju
1CCCA ; (5.11)
we have
( _xu _yu)
0BBB@iu
ju
1CCCA = ( _xn _yn)
0BBB@in
jn
1CCCA
= ( _xn _yn)
0BBB@
cos �1 �sin �1
sin �1 cos �1
1CCCA
0BBB@iu
ju
1CCCA
(5.12)
or
_xn cos �1 + _yn sin �1 = _xu (5.13)
� _xn sin �1 + _yn cos �1 = _yu: (5.14)
Similarly, by equating Eqs. (5.8) and (5.10) and using the transformation matrix
(5.11), we obtain
(�xu � _yu _�1 �yu + _xu _�1)
0BBB@iu
ju
1CCCA = (�xn �yn)
0BBB@in
jn
1CCCA
= (�xn �yn)
0BBB@
cos �1 �sin �1
sin �1 cos �1
1CCCA
0BBB@iu
ju
1CCCA ;
(5.15)
96
or
�xncos �1 + �ynsin �1 = �xu � _yu _�1 (5.16)
��xnsin �1 + �yncos �1 = �yu + _xu _�1: (5.17)
Eqs. (5.13), (5.14), (5.16) and (5.17) will be used to transform equations of motion
from the inertial reference frame to the unsprung mass reference frame.
5.3 Vehicle Kinematics
In this section, translational and rotational velocities will be calculated for both
the tractor and the trailer. Then the expressions of kinetic energy and potential
energy that will be used to derive the vehicle model by applying Lagrange's equations
in section 5.4, are given. Vehicle parameters are depicted in Fig. 5.4 and listed in
Table 5.1.
2
Tw1
C.G
C.G
C.GRoll Center
Fifth Wheel
Z0
l
d1
2
l 3
d3
d2d4
l 1
h
Figure 5.4: Schematic Diagram of Complex Vehicle Model
97
parameter description
m1 tractor's massIx1; Iy1; Iz1 tractor's moment of inertia
m2 semitrailer's massIx2; Iy2; Iz2 semitrailer's moment of inertia
l1 distance between tractor C.G. and front wheel axlel2 distance between tractor C.G. and real wheel axlel3 distance between joint (�fth wheel) and trailer real wheel axle
d1; d2 relative position between tractor's C.G. to �fth wheeld3; d4 relative position between semitrailer's C.G. to �fth wheelTw1 tractor front axle track widthTw2 tractor rear axle track widthTw3 semitrailer rear axle track widthh2 distance from tractor roll center to C.G.C�f cornering sti�ness of tractor front wheelC�r cornering sti�ness of tractor rear wheelC�t cornering sti�ness of semitrailer rear wheelSxf longitudinal sti�ness of tractor front wheelSxr longitudinal sti�ness of tractor rear wheelSxt longitudinal sti�ness of semitrailer rear wheel
Table 5.1: Parameters of Complex Vehicle Model
5.3.1 Tractor Kinematics
To facilitate the calculation of the tractor translational velocity, several identities
of time derivatives of unit vectors in each coordinate frame will be established in this
section. These identities include the time derivatives of the unit vectors along the
X;Y; and Z axis of the unsprung mass coordinate and the sprung mass coordinate.
Recall that the angular velocity of the tractor's unsprung mass coordinate is
!u=n = _�1ku: (5.18)
98
Thus the derivatives of the unit vectors in the unsprung mass coordinate are
d
dtiu = !u=n � iu = _�1ju (5.19)
d
dtju = !u=n � ju = � _�1iu (5.20)
and
d
dtku = !u=n � ku = 0; (5.21)
respectively. Since the sprung mass coordinate has relative roll motion with respect
to the unsprung mass coordinate, the angular velocity of the sprung mass coordinate
Xs1Ys1Zs1 is
!s1=n = !s1=u + !u=n
= _�is1 + _�1ku
= _�is1 + � _�1js1 + _�1ks1;
(5.22)
where
ku = �js1 + ks1 (5.23)
is used in (5.22). Then the derivatives of the unit vectors in the sprung mass coordi-
nate are
d
dtis1 = !s1=n � is1 = _�1js1 � � _�1ks1 (5.24)
d
dtjs1 = !s1=n � js1 = � _�1is1 + _�ks1 (5.25)
and
d
dtks1 = !s1=n � ks1 = � _�1is1 � _�js1; (5.26)
99
respectively. Eqs. (5.19), (5.20), (5.21), (5.24), (5.25) and (5.26) will be used in the
following to calculate translational velocity of the tractor.
Translational Velocity at Tractor C.G.
Fron Fig. 5.4 the position of the tractor C.G can be expressed as
rCG1=n = rCG1=u + ru=n
= z0ku + h2ks1 + xnin + ynjn:
(5.27)
By di�erentiating (5.27), the velocity of the tractor C.G. is obtained as
vCG1=n = _xnin + _ynjn + h2ddtks1
= _xnin + _ynjn + h2� _�1is1 � h2 _�js1
= ( _xncos�1 + _ynsin�1)iu + (� _xnsin�1 + _yncos�1)ju + h2� _�1is1 � h2 _�js1
= ( _xncos�1 + _ynsin�1 + h2� _�1)is1
+(� _xnsin�1 + _yncos�1 � h2 _�)js1
+( _xnsin�1 � _yncos�1)�ks1;
(5.28)
where coordinate transformations are used in (5.28).
100
5.3.2 Trailer Kinematics
The angular velocity of trailer unsprung mass coordinate Xs2Ys2Zs2 is
!s2=n = !s1 + _�fks2
= _�is1 + � _�1js1 + _�1ks1 + _�fks2
= ( _�cos�f + � _�1sin�f)is2
+ (� _�sin�f + � _�1cos�f )js2
+ (_�1 + _�f )ks2
(5.29)
Thus the time derivatives of the unit vectors along the X, Y , and Z axis are
d
dtis2 = !s2=n � is2 = (_�1 + _�f )js2 + ( _�sin�f � � _�1cos�f )ks2 (5.30)
d
dtjs2 = !s2=n � js2 = �( _�1 + _�f )is2 + ( _�cos�f + � _�1sin�f)ks2 (5.31)
and
d
dtks2 = !s2=n � ks2 = (� _�sin�f + � _�1cos�f )is2 � ( _�cos�f + � _�1sin�f)js2; (5.32)
respectively. Eqs. (5.30), (5.31) and (5.32) will be used in the following to calculate
the translational velocity at the trailer C.G.
Translational Velocity at the Trailer C.G.
From Fig. 5.4, it is easy to see that the position vector of the trailer C.G. can be
decomposed into three components as
rCG2=n = rCG1=n + rfw=CG1 + rCG2=fw (5.33)
101
where rCG1=n is the position vector of the tractor C.G., rfw=CG1 is the position vector
from the tractor C.G. to the �fth wheel, and rCG2=fw is the position vector from
the �fth wheel to the trailer C.G. By substituting vehicle geometric parameters into
(5.33), we obtain
rCG2=n = rCG1=n � d1is1 � d2ks1 � d3is2 + d4ks2: (5.34)
Consequently, the velocity vector at trailer C.G. can be obtained by di�erentiating
(5.34),
vCG2=n = vCG1=n � d1d
dtis1 � d2
d
dtks1 � d3
d
dtis2 + d4
d
dtks2 (5.35)
Substituting identities of derivatives of unit vectors (5.24), (5.26), (5.30) and (5.32)
into (5.35), we obtain
vCG2=n = ( _xncos�1 + _ynsin�1 + h2� _�1)is1 + (� _xnsin�1 + _yncos�1 � h2 _�)js1
+( _xnsin�1 � _yncos�1)�ks1 � d1 _�1js1 + d1� _�1ks1 � d2� _�1is1 + d2 _�js1
�d3( _�1 + _�f)js2 � d3( _�sin�f � � _�1cos�f )ks2 + d4(� _�sin�f + � _�1cos�f )is2
�d4( _�cos�f + � _�1sin�f )js2
= ( _xncos�1 + _ynsin�1 + h2� _�1 � d2� _�1)is1
+(� _xnsin�1 + _yncos�1 � h2 _�� d1 _�1 + d2 _�)js1
+( _xnsin�1 � _yncos�1 + d1 _�1)�ks1
+d4(� _�sin�f + � _�1cos�f)is2
�(d3( _�1 + _�f) + d4( _�cos�f + � _�1sin�f))js2
�d3( _�sin�f � � _�1cos�f)ks2:
(5.36)
102
5.3.3 Kinetic Energy and Potential Energy
Kinetic Energy
The kinetic energy of the tractor-semitrailer vehicle can be obtained by adding
the kinetic energy component of the tractor and that of the trailer. The kinetic
energy of the tractor, which is denoted as T1, can be calculated from the translational
velocity of the tractor at C.G. and the angular velocity of the tractor's unsprung mass
coordinate,
T1 =1
2m1vCG1 � vCG1 +
1
2!s1 � I1 � !s1 (5.37)
By substituting vCG1 in (5.28) and !s1 in (5.22) into (5.37), we obtain
T1 =12m1( _xncos�1 + _ynsin�1 + h2� _�1)2
+12m1(� _xnsin�1 + _yncos�1 � h2 _�)
2
+12m1(( _xnsin�1 � _yncos�1)�)2
+12Ix1(
_�)2 + 12Iy1(� _�1)
2 + 12Iz1( _�1)
2:
(5.38)
Similarly, the kinetic energy of the trailer, denoted as T2, can be obtained from the
translational velocity at the trailer C.G. and the angular velocity of the trailer's
sprung mass coordinate, or
T2 =1
2m2vCG2 � vCG2 +
1
2!s2 � I2 � !s2 (5.39)
103
By substituting vCG2 in (5.36) and !s2 in (5.29) into (5.39), we obtain
T2 =12m2( _xncos�1 + _ynsin�1 + h2� _�1 � d2� _�1)
2
+12m2(� _xnsin�1 + _yncos�1 � h2 _�� d1 _�1 + d2 _�)
2
+12m2( _xnsin�1 � _yncos�1 + d1 _�1)2�2
+12m2d
24(�
_�sin�f + � _�1cos�f )2
+12m2(d3( _�1 + _�f ) + d4( _�cos�f + � _�1sin�f ))
2
+12m2d
23(
_�sin�f � � _�1cos�f )2
+m2d4( _xncos�1 + _ynsin�1 + h2� _�1 � d2� _�1)(� _�sin�f + � _�1cos�f )is1 � is2
�m2( _xncos�1 + _ynsin�1 + h2� _�1 � d2� _�1)(d3( _�1 + _�f ) + d4( _�cos�f + � _�1sin�f ))is1 � js2
+m2d4(� _xnsin�1 + _yncos�1 � h2 _�� d1 _�1 + d2 _�)(� _�sin�f + � _�1cos�f )js1 � is2
�m2(� _xnsin�1 + _yncos�1 � h2 _�� d1 _�1 + d2 _�)(d3( _�1 + _�f ) + d4( _�cos�f + � _�1sin�f ))js1 � js2
�m2d3( _xnsin�1 � _yncos�1 + d1 _�1)�( _�sin�f � � _�1cos�f )ks1 � ks2
+12Ix2(
_�cos�f + � _�1sin�f )2 + 1
2Iy2(�_�sin�f + � _�1cos�f )
2 + 12Iz2( _�1 + _�f )
2
(5.40)
Recall from the de�nition of the coordinate system that
is1 � is2 = cos �f (5.41)
is1 � js2 = �sin �f (5.42)
js1 � is2 = sin �f (5.43)
js1 � js2 = cos �f (5.44)
and
ks1 � ks2 = 1 (5.45)
104
Substituting Eqs. (5.41), (5.42), (5.43), (5.44) and (5.45) into (5.40), we obtain
T2 =12m2( _xncos�1 + _ynsin�1 + h2� _�1 � d2� _�1)2
+12m2(� _xnsin�1 + _yncos�1 � h2 _�� d1 _�1 + d2 _�)2
+12m2( _xnsin�1 � _yncos�1 + d1 _�1)2�2
+12m2d
24(�
_�sin�f + � _�1cos�f )2
+12m2(d3( _�1 + _�f) + d4( _�cos�f + � _�1sin�f))2
+12m2d
23(_�sin�f � � _�1cos�f )
2
+m2d4cos�f ( _xncos�1 + _ynsin�1 + h2� _�1 � d2� _�1)(� _�sin�f + � _�1cos�f )
+m2sin�f( _xncos�1 + _ynsin�1 + h2� _�1 � d2� _�1)(d3( _�1 + _�f) + d4( _�cos�f + � _�1sin�f))
+m2d4sin�f(� _xnsin�1 + _yncos�1 � h2 _�� d1 _�1 + d2 _�)(� _�sin�f + � _�1cos�f )
�m2cos�f (� _xnsin�1 + _yncos�1 � h2 _�� d1 _�1 + d2 _�)(d3( _�1 + _�f) + d4( _�cos�f + � _�1sin�f))
�m2d3( _xnsin�1 � _yncos�1 + d1 _�1)�( _�sin�f � � _�1cos�f )
+12Ix2( _�cos�f + � _�1sin�f)
2
+12Iy2(�
_�sin�f + � _�1cos�f )2
+12Iz2( _�1 + _�f )2
(5.46)
Potential Energy
The change of the potential energy for the tractor of a tractor-semitrailer vehicle
is primarily due to the roll motion. However, the change of potential energy for
the semitrailer is a�ected by both the roll and pitch motion at the linking joint (�fth
105
wheel). Furthermore, the compliance at the �fth wheel will be signi�cant in describing
the roll motion of the trailer. For simplicity, these complicated coupling will not be
modeled. Instead, the roll motion is approximated as if the articulation angle is zero,
that is, the truck is in straight con�guration. This approximation for roll motion will
be examined by comparing simulation results and experimental data in section 5.7.
Thus the potential energy can be obtained as
V = m1gh2(cos�� 1) +m2g((h2 � d2 + d4)(cos�� 1)
The Lagrangian, L, is de�ned as
L = T1 + T2 � V (5.47)
and will be used to derive vehicle body dynamics in the next section.
5.4 Equations of Motion
In this section a set of �ve second-order ordinary di�erential equations governing
the vehicle sprung mass will be obtained in two steps. First, the vehicle sprung
mass dynamics with respect to the inertial reference frame XnYnZn will be derived
by utilizing Lagrange's equation. Second, since the equations of motion with respect
to the unsprung mass coordinate are more meaningful, we will transform the vehicle
dynamics from the inertial reference frame to the unsprung mass reference frame.
Step 1 Vehicle Body Dynamics with respect to Inertial Reference Frame
106
In section 5.3 we obtain the kinetic energy and potential energy of the tractor-
semitrailer vehicle, and the Lagrangian is de�ned as
L = T1 + T2 � V
By using Lagrange's equation,
d
dt
@L
@ _xn�
@L
@xn= Fgxn (5.48)
we obtain the �rst dynamic equation
(m1 +m2)(1 + �2sin2�1)�xn � (m1 +m2)�2sin�1cos�1�yn
+(m1h2 +m2(h2 � d2 + d4)�m2d3�sin�f )sin�1 ��
+((m1h2 +m2(h2 � d2 + d4))�cos�1 +m2d1sin�1 +m2d3sin�f cos�1 +m2d3cos�f sin�1
+m2(d1 + d3cos�f )sin�1�2)��1
+m2d3(sin�1cos�f + cos�1sin�f )��f
+2(m1 +m2)�sin2�1 _xn _�+ 2(m1 +m2)�2sin�1cos�1 _xn _�1 � 2(m1 +m2)�sin�1cos�1 _yn _�
�(m1 +m2)�2(cos2�1 � sin2�1) _yn _�1 + (2m1h2 + 2m2(h2 � d2 + d4)�m2d3�sin�f )cos�1 _� _�1
�m2d3sin�1sin�f _�2 + 2(m2d1sin�1 +m2d3sin�1cos�f )� _� _�1 �m2d3�sin�1cos�f _� _�f
�(m1h2 +m2(h2 � d2 + d4))�sin�1 _�21 +m2d1cos�1 _�
21 +m2d3(cos�1cos�f � sin�1sin�f ) _�
21
+m2(d1 + d3cos�f )cos�1�2 _�21 + 2m2d3(cos�1cos�f � sin�1sin�f ) _�1 _�f
�m2d3sin�1sin�f�2 _�1 _�f +m2d3(cos�1cos�f � sin�1sin�f ) _�
2f = Fgxn
(5.49)
where Fgxn is the generalized force corresponding to the generalized coordinate xn.
By using Lagrange's equation
d
dt
@L
@ _yn�
@L
@yn= Fgyn (5.50)
107
we obtain the second dynamic equation
�(m1 +m2)�2sin�1cos�1�xn + (m1 +m2)(1 + �2cos2�1)�yn
�(m1h2 +m2(h2 � d2 + d4)�m2d3�sin�f )cos�1 ��
+((m1h2 +m2(h2 � d2 + d4))�sin�1 �m2(d1 + d3cos�f )cos�1 +m2d3sin�f sin�1
�m2(d1 + d3cos�f )cos�1�2)��1
+m2d3(sin�1sin�f � cos�1cos�f )��f
�2(m1 +m2)�sin�1cos�1 _xn _�� (m1 +m2)�2(cos2�1 � sin2�1) _xn _�1 + 2(m1 +m2)�cos2�1 _yn _�
�2(m1 +m2)�2sin�1cos�1 _yn _�1 +m2d3sin�f cos�1 _�2
+(2m1h2 + 2m2(h2 � d2 + d4)�m2d3�sin�f )sin�1 _� _�1 � 2m2(d1 + d3cos�f )cos�1� _� _�1
+m2d3�cos�1cos�f _� _�f + (m1h2 +m2(h2 � d2 + d4))�cos�1 _�21 +m2d1sin�1 _�
21
+m2d3(sin�1cos�f + cos�1sin�f ) _�21 +m2(d1 + d3cos�f )�
2sin�1 _�21
+2m2d3(cos�1sin�f + sin�1cos�f ) _�1 _�f +m2d3cos�1sin�f�2 _�1 _�f
+m2d3(sin�1cos�f + cos�1sin�f ) _�2f = Fgyn
(5.51)
where Fgyn is the generalized force corresponding to the generalized coordinate yn.
We proceed by using Lagrange's equation
d
dt
@L
@ _��@L
@�= Fg� (5.52)
108
to obtain the third dynamic equation
(m1h2 +m2(h2 � d2 + d4) �m2d3�sin�f )sin�1�xn
�(m1h2 +m2(h2 � d2 + d4)�m2d3�sin�f )cos�1�yn
(Ix1 +m1h22 + Ix2cos
2�f +m2(h2 � d2 + d4)2 + (Iy2 +m2d23)sin
2�f )��
(m2(d1 + d3cos�f )(h2 � d2 + d4) + (Ix2 � Iy2 �m2d23)�sin�f cos�f �m2d1d3�sin�f )��1
+m2d3(h2 � d2 + d4)cos�f ��f
�2m2d3sin�1sin�f _xn _��m2d3�(cos�1sin�f + 2sin�1cos�f ) _xn _�1 �m2d3�sin�1cos�f _xn _�f
+2m2d3cos�1sin�f _yn _�+m2d3�(�sin�1sin�f + 2cos�1cos�f ) _yn _�1 +m2d3�cos�1cos�f _yn _�f
+2(Iy2 +m2d23 � Ix2)sin�f cos�f _� _�f
+((Ix2 � Iy2 �m2d23)�(cos
2�f � sin2�f )�m2d1d3�cos�f � 2m2d3(h2 � d2 + d4)sin�f ) _�1 _�f
�m2d3(h2 � d2 + d4)sin�f _�2f
�(Iy1 +m1h22 + Ix2sin
2�f + (Iy2 +m2d23)cos
2�f +m2d21 +m2(h2 � d2 + d4)
2 + 2m2d1d3cos�f )� _�21
�(m1 +m2)�( _xnsin�1 � _yncos�1)2 � 2m2d1� _�1( _xnsin�1 � _yncos�1)
�m2d3(h2 � d2 + d4)sin�f _�21 = Fg�
(5.53)
where Fg� is the generalized force corresponding to the generalized coordinate �.
From Lagrange's equation
d
dt
@L
@ _�1�
@L
@�1= Fg�1 (5.54)
109
the fourth equation is obtained as
(m1h2�cos�1 +m2(h2 � d2 + d4)�cos�1 +m2d3(sin�1cos�f + cos�1sin�f )
+m2d1sin�1 +m2(d1 + d3cos�f )�2sin�1)�xn
(m1h2�sin�1 +m2(h2 � d2 + d4)�sin�1 �m2d3(cos�1cos�f � sin�1sin�f )
�m2d1cos�1 �m2(d1 + d3cos�f )�2cos�1)�yn
(m2(d1 + d3cos�f )(h2 � d2 + d4)�m2d1d3�sin�f + (Ix2 � Iy2 �m2d23)�sin�f cos�f )��
(Iz1 + Iz2 +m2(d1 + d3cos�f )2 +m2(d3sin�f + (h2 � d2 + d4)�)2
+(Iy1 +m1h22 +m2(d1 + d3cos�f )2 + Ix2sin
2�f + Iy2cos2�f )�2)��1
(Iz2 +m2d23 +m2d1d3cos�f +m2d3(h2 � d2 + d4)�sin�f )��f
+2m2(d1 + d3cos�f )� _�( _xnsin�1 � _yncos�1) �m2d3�2sin�f _�f ( _xnsin�1 � _yncos�1)
�(m1 +m2)�2( _xncos�1 + _ynsin�1)( _xnsin�1 � _yncos�1) +m2d3�sin�f _�( _xncos�1 + _ynsin�1)
+((Ix2 � Iy2 �m2d23)sin�f cos�f �m2d1d3sin�f ) _�
2 + 2m2d3(h2 � d2 + d4)sin�f _� _�1
+2(Iy1 +m1h22 +m2(d1 + d3cos�f )2 + Ix2sin
2�f + Iy2cos2�f )� _� _�1
�(m2d1d3cos�f + (Ix2 � Iy2 �m2d23)(sin
2�f � cos2�f ))� _� _�f
+(�2m2d3sin�f (d1 + d3cos�f )(1 + �2) + 2m2d3cos�f (d3sin�f + (h2 � d2 + d4)�)
+2(Ix2 � Iy2)sin�f cos�f�2) _�1 _�f
(m2d3(h2 � d2 + d4)�cos�f �m2d1d3sin�f ) _�2f = Fg�1
(5.55)
where Fg�1 is the generalized force corresponding to the generalized coordinate �1.
The last dynamic equation is obtained by using Lagrange's equation
d
dt
@L
@ _�f�
@L
@�f= Fg�f (5.56)
110
Thus we have
m2d3(sin�1cos�f + cos�1sin�f )�xn
�m2d3(cos�1cos�f � sin�1sin�f )�yn
+m2d3(h2 � d2 + d4)cos�f ��
+(Iz2 +m2d23 +m2d1d3cos�f +m2d3(h2 � d2 + d4)�sin�f )��1
+(Iz2 +m2d23)��f
+2m2d3(h2 � d2 + d4)sin�f _� _�1 �m2d3(h2 � d2 + d4)�cos�f _�21 +m2d1d3sin�f _�
21
+2m2d4sin�f cos�f _�( _xnsin�1 � _yncos�1 + (h2 � d2) _�+ d1 _�1)
+m2d3�cos�f _�( _xnsin�1 � _yncos�1 + d1 _�1) +m2d3�2sin�f _�1( _xnsin�1 � _yncos�1 + d1 _�1)
+(Ix2 � Iy2 �m2d23)sin�f cos�f _�2 + (Ix2 � Iy2 �m2d
23)(sin
2�f � cos2�f )� _� _�1
�(Ix2 � Iy2 �m2d23)sin�f cos�f�
2 _�21 = Fg�f
(5.57)
where Fg�f is the generalized force corresponding to the generalized coordinate �f .
Eqs. (5.49), (5.51), (5.53), (5.55) and (5.57) describe the dynamic behavior of the
tractor-semitrailer vehicle seen from the inertial reference frame XnYnZn.
Step 2 Vehicle Body Dynamics with respect to the Unsprung Mass Reference Frame
In this step, the vehicle model will be transformed from the inertial reference frame
to the unsprung mass reference frame. Recall in section 5.2 that the transformations
can be conducted by using
_xn cos �1 + _yn sin �1 = _xu (5.58)
� _xn sin �1 + _yn cos �1 = _yu (5.59)
�xncos �1 + �ynsin �1 = �xu � _yu _�1 (5.60)
111
and
��xnsin �1 + �yncos �1 = �yu + _xu _�1; (5.61)
where _xn is the vehicle velocity component along the Xn axis of the inertial reference
frame, _yn is the vehicle velocity component along the Yn axis of the inertial reference
frame, _xu is the vehicle velocity component along the Xu axis of the unsprung mass
reference frame, and _yu is the vehicle velocity component along the Yu axis of the
unsprung mass reference frame.
By adding the �rst dynamic equation in the inertial reference frame (5.49) � cos�1
and the second dynamic equation (5.51) � sin�1 and using (5.58), (5.59), (5.60), and
(5.61), we obtain the �rst dynamic equation in the unsprung mass reference frame as
(m1 +m2)�xu + (m1h2�+m2(h2 � d2 + d4)�+m2d3sin�f )��1 +m2d3sin�f��f
�(m1 +m2)(1 + �2) _yu _�1 + (2m1h2 + 2m2(h2 � d2 + d4)�m2d3�sin�f ) _� _�1
+m2(d1 + d3cos�f )(1 + �2) _�21 + 2m2d3cos�f _�1 _�f +m2d3cos�f _�2f
= Fgxn � cos�1 + Fgyn � sin�1
(5.62)
Similarly, by subtracting Eq. (5.49) � sin�1 from Eq. (5.51) � cos�1 and using (5.58),
(5.59), (5.60) and (5.61), we obtain the second dynamic equation in the unsprung mass
reference frame as
(m1 +m2)(1 + �2)�yu � (m1h2 +m2(h2 � d2 + d4) �m2d3�sin�f )��
�m2(d1 + d3cos�f )(1 + �2)��1 �m2d3cos�f ��f
(m1 +m2) _xu _�1 + 2(m1 +m2)� _yu _�+m2d3sin�f _�2 � 2m2(d1 + d3cos�f )� _� _�1
+m2d3�cos�f _� _�f + (m1h2 +m2(h2 � d2 + d4))� _�21 +m2d3sin�f _�21
+2m2d3sin�f _�1 _�f +m2d3sin�f�2 _�1 _�f +m2d3sin�f _�
2f
= �Fgxn � sin�1 + Fgyn � cos�1
(5.63)
112
By substituting (5.58), (5.59), (5.60) and (5.61) into the third dynamic equation
(5.53) in the inertial reference frame, the third dynamic equation in the unsprung
mass reference frame can be obtained as
�(m1h2 +m2(h2 � d2 + d4)�m2d3�sin�f )�yu
(Ix1 +m1h22 + Ix2cos
2�f +m2(h2 � d2 + d4)2 + (Iy2 +m2d23)sin
2�f )��
(m2(d1 + d3cos�f )(h2 � d2 + d4) + (Ix2 � Iy2 �m2d23)�sin�f cos�f �m2d1d3�sin�f )��1
+m2d3(h2 � d2 + d4)cos�f ��f
�(m1h2 +m2(h2 � d2 + d4)) _xu _�1 + 2m2d3�cos�f _yu _�1 +m2d3�cos�f _yu _�f
+2m2d3sin�f _yu _�+ 2(Iy2 +m2d23 � Ix2)sin�f cos�f _� _�f
�(2m2d3(h2 � d2 + d4)sin�f +m2d1d3�cos�f ) _�1 _�f
�(Ix2 � Iy2 �m2d23)�(sin
2 �f � cos2 �f ) _�1 _�f �m2d3(h2 � d2 + d4)sin�f _�2f
�(m1h22 + Iy1 +m2(h2 � d2 + d4)
2 + Ix2sin2�f + (Iy2 +m2d
23)cos
2�f )� _�21
+(m2d21 + 2m2d1d3cos�f )� _�
21 � (m1 +m2)� _y
2u + 2m2d1� _yu _�1
�m2d3(h2 � d2 + d4)sin�f _�21 = Fg�
(5.64)
The fourth and �fth dynamic equations can also be obtained from (5.55) and (5.57)
113
as
(m1h2�+m2(h2 � d2 + d4)�+m2d3sin�f )�xu
�m2(d1 + d3cos�f )(1 + �2)�yu
(m2(d1 + d3cos�f )(h2 � d2 + d4) + (Ix2 � Iy2 �m2d23)�sin�f cos�f �m2d1d3�sin�f )��
(Iz1 + Iz2 +m2(d1 + d3cos�f )2 +m2(d3sin�f + (h2 � d2 + d4)�)2
+(Iy1 +m1h22 +m2(d1 + d3cos�f )2 + Ix2sin
2�f + Iy2cos2�f )�2)��1
(Iz2 +m2d23 +m2d1d3cos�f +m2d3(h2 � d2 + d4)�sin�f )��f
�(m2d1 +m2d3cos�f )(1 + �2) _xu _�1 � (m1h2 +m2(h2 � d2 + d4))� _yu _�1 �m2d3sin�f _yu _�1
�2(m2d1 +m2d3cos�f )� _yu _�+m2d3�2sin�f _yu _�f + (m1 +m2)�2 _xu _yu
+m2d3�sin�f _xu _�+ ((Ix2 � Iy2 �m2d23)sin�f cos �f �m2d1d3sin�f ) _�2
+2m2(h2 � d2 + d4)(d3sin�f + (h2 � d2 + d4)�) _� _�1
+2(m1h22 + Iy1 +m2(d1 + d3cos�f )(d1 + d3cos�f ) + Ix2sin
2�f + Iy2cos2�f )� _� _�1
�(m2d1d3cos�f + (Ix2 � Iy2 �m2d23)(sin
2 �f � cos2 �f ))� _� _�f
+(�2m2d3sin�f (d1 + d3cos�f ) + 2m2d3cos�f (d3sin�f + (h2 � d2 + d4)�)) _�1 _�f
+(�2m2d3sin�f (d1 + d3cos�f )�2 + 2(Ix2 � Iy2)sin�f cos �f�2) _�1 _�f
+m2d3((h2 � d2 + d4)�cos�f � d1sin�f ) _�2f = Fg�1;
(5.65)
114
and
m2d3sin�f �xn �m2d3cos�f �yn +m2d3(h2 � d2 + d4)cos�f ��
(Iz2 +m2d23 +m2d1d3cos�f +m2d3(h2 � d2 + d4)�sin�f )��1
(Iz2 +m2d23)��f
�m2d3cos�f _xu _�1 �m2d3sin�f _yu _�1 + 2m2d3(h2 � d2 + d4)sin�f _� _�1
�m2d3(h2 � d2 + d4)�cos�f _�21 +m2d1d3sin�f _�21 + 2m2d4sin�f cos �f _�(� _yu + (h2 � d2) _�+ d1 _�1)
+m2d3�cos�f _�(� _yu + d1 _�1) +m2d3�2sin�f _�1(� _yu + d1 _�1)
+(Ix2 � Iy2 �m2d23)sin�f cos �f _�2 + (Ix2 � Iy2 �m2d
23)(sin
2 �f � cos2 �f )� _� _�1
+(Iy2 +m2d23 � Ix2)�2sin�f cos �f _�21 = Fg�f ;
(5.66)
respectively. Eqs. (5.62), (5.63), (5.64), (5.65) and (5.66) constitute the �rst major
component of the complex model for the tractor-semitrailer vehicle. The generalized
forces on the right hand side of (5.62), (5.63), (5.64), (5.65) and (5.66) are the other
major component of the complex model and will be explored in the next two sections.
5.5 Generalized Forces
We have seen in the previous section that deriving the generalized forces is an
important part of the modeling. In this and the next sections, we will show how
to obtain generalized forces, which appear on the right hand side of the dynamic
equations (5.62), (5.63), (5.64), (5.65) and (5.66). We notice that the external forces
acting on the vehicle body are from the tire/road interface and suspensions. Thus
to calculate the generalized forces, we will derive the expressions for the generalized
115
forces in terms of the longitudinal and lateral components of tire forces and the
vertical suspension forces. The process of calculating generalized forces are derived
from the principle of virtual work. Interested readers are refererred to the books
by Greenwood [21] and Rosenberg [55]. In the next section, we will show how to
obtain the longitudinal and lateral components of tire forces from the tire model and
suspension forces from the suspension model. To derive the expressions of generalized
forces in terms of the tire forces and suspension forces, we de�ne the sign conventions,
shown in Fig. 5.5, of tire forces, where Fai is the longitudinal tire force and Fbi is
the lateral tire force. The suspension force at the i� th tire is denoted as Fpi, whose
direction is perpendicular to both Fai and Fbi.
From Fig. 5.5, the component of the tire force along the Xn axis is
Fxni = Fai � cos�1 � Fbisin�1 (5.67)
for i = 1; � � � ; 4, and is
Fxni = Fai � cos(�1 + �f)� Fbisin(�1 + �f) (5.68)
for i = 5; 6. Similarly, the component of the tire force along the Yn axis is
Fyni = Fai � sin�1 + Fbicos�1 (5.69)
for i = 1; � � � ; 4, and is
Fyni = Fai � sin(�1 + �f) + Fbicos(�1 + �f) (5.70)
116
Fb5
Xn
Yn
XuYu
Fa1Fb1
Fa2
Fb2
Fb4
Fb3
Fa3
Fa4
Fa6
Fb6
Fa5
Figure 5.5: De�nition of Tire Force in the Cartesian Coordinate
for i = 5; 6. The position vector of the location, where the external forces Fxn1, Fyn1
and Fp1 are acting, can be obtained as
rt1 = rCG1 + l1is1 +Tw12js1 � z0ks1
= xnin + ynjn + z0kn + l1is1 +Tw12 js1 � z0ks1
(5.71)
By substituting the transformation matrices in section 5.2, we obtain the position
117
vector rt1 in the inertial reference coordinate,
rt1 = (xn + l1cos�1 � (Tw12cos� + z0sin�)sin�1)in
+(yn + l1sin�1 + (Tw12 cos�+ z0sin�)cos�1)jn
+(z0 +Tw12 sin�� z0cos�)kn
� rxt1in + ryt1jn + rzt1kn:
(5.72)
Locations of other tire and suspension forces can be similarly obtained as
rt2 = rCG1 + l1is1 �Tw12 js1 � z0ks1
= xnin + ynjn + z0kn + l1is1 �Tw12js1 � z0ks1
= (xn + l1cos�1 + (Tw12cos�� z0sin�)sin�1)in
+(yn + l1sin�1 � (Tw12 cos�� z0sin�)cos�1)jn
+(z0 �Tw12 sin�� z0cos�)kn
� rxt2in + ryt2jn + rzt2kn;
(5.73)
rt3 = rCG1 � l2is1 +Tw22js1 � z0ks1
= xnin + ynjn + z0kn � l2is1 +Tw22 js1 � z0ks1
= (xn � l2cos�1 � (Tw22 cos�+ z0sin�)sin�1)in
+(yn � l2sin�1 + (Tw22 cos� + z0sin�)cos�1)jn
+(z0 +Tw22 sin�� z0cos�)kn
� rxt3in + ryt3jn + rzt3kn;
(5.74)
118
rt4 = rCG1 � l2is1 �Tw22 js1 � z0ks1
= xnin + ynjn + z0kn � l2is1 �Tw22js1 � z0ks1
= (xn � l2cos�1 + (Tw22cos� � z0sin�)sin�1)in
+(yn � l2sin�1 � (Tw12 cos� � z0sin�)cos�1)jn
+(z0 �Tw22sin�� z0cos�)kn
� rxt4in + ryt4jn + rzt4kn;
(5.75)
rt5 = rCG1 � d1is1 � l3is2 +Tw32js2 � z0ks2
= xnin + ynjn + z0kn � d1is1 � l3is2 +Tw32 js2 � z0ks2
= (xn � (d1 + l3cos�f +Tw32 sin�f)cos�1
�((�l3sin�f +Tw32 cos�f )cos�+ z0sin�)sin�1)in
+(yn � (d1 + l3cos�f +Tw32sin�f)sin�1
+((�l3sin�f +Tw32cos�f )cos�+ z0sin�)cos�1)jn
+(z0 + (�l3sin�f +Tw32 cos�f )sin�� z0cos�)kn
� rxt5in + ryt5jn + rzt5kn;
(5.76)
119
and
rt6 = rCG1 � d1is1 � l3is2 �Tw32 js2 � z0ks2
= xnin + ynjn + z0kn � d1is1 � l3is2 �Tw32js2 � z0ks2
= (xn � (d1 + l3cos�f �Tw32sin�f)cos�1
�((�l3sin�f �Tw32 cos�f )cos�+ z0sin�)sin�1)in
+(yn � (d1 + l3cos�f �Tw32sin�f)sin�1
+((�l3sin�f �Tw32cos�f )cos�+ z0sin�)cos�1)jn
+(z0 + (�l3sin�f �Tw32 cos�f )sin�� z0cos�)kn
� rxt6in + ryt6jn + rzt6kn;
(5.77)
respectively. So far we have obtained the position vectors for the external forces.
Thus the generalized force Fgxn is
Fgxn =P6
i=1 Fxni �@rxti@xn
+P6
i=1 Fyni �@ryti@xn
+P6
i=1 FPi �@rzti@xn
(5.78)
Substituting (5.72), (5.73), (5.74), (5.75), (5.76) and (5.77) into (5.78), we obtain
Fgxn = Fxn1 + Fxn2 + Fxn3 + Fxn4 + Fxn5 + Fxn6: (5.79)
The generalized force associated with the coordinate yn is
Fgyn =P6
i=1 Fxni �@rxti@yn
+P6
i=1 Fyni �@ryti@yn
+P6
i=1 FPi �@rzti@yn
(5.80)
Substituting (5.72), (5.73), (5.74), (5.75), (5.76), and (5.77) into (5.80), we obtain
Fgyn = Fyn1 + Fyn2 + Fyn3 + Fyn4 + Fyn5 + Fyn6 : (5.81)
120
The generalized force corresponding to the coordinate � is
Fg� =P6
i=1 Fxni@rxti@�
+P6
i=1 Fyni@ryti@�
+P6
i=1 FPi@rzti@�
; (5.82)
or we have
Fg� = Fb1 � (z0cos� �Tw12sin�) + Fb2 � (z0cos�+
Tw12sin�)
+Fb3 � (z0cos� �Tw22sin�) + Fb4 � (z0cos� +
Tw22sin�)
+(Fb5cos�f + Fa5sin�f) � (z0cos� + (�Tw32 cos�f + l3sin�f)sin�)
+(Fb6cos�f + Fa5sin�f) � (z0cos� + (Tw32 cos�f + l3sin�f)sin�)
+FP1 � (Tw12 cos�+ z0sin�)� FP2 � (
Tw12 cos� � z0sin�)
+FP3 � (Tw22cos�+ z0sin�)� FP4 � (
Tw22cos� � z0sin�)
+FP5 � ((Tw32cos�f � l3sin�f)cos�+ z0sin�)
�FP6 � ((Tw32 cos�f + l3sin�f)cos�� z0sin�):
(5.83)
Similarly, the generalized force for the coordinate �1 is
Fg�1 =P6
i=1 Fxni �@rxti@�1
+P6
i=1 Fyni �@ryti@�1
+P6
i=1 FPi �@rzti@�1
; (5.84)
and can be calculated as
Fg�1 = (Fb1 + Fb2)l1 � (Fb3 + Fb4)l2
�(Fb5cos�f + Fa5sin�f) � (Sw32 sin�f + l3cos�f + d1)
�(Fb6cos�f + Fa6sin�f) � (�Sw32 sin�f + l3cos�f + d1)
�Fa1 � (Tw12 cos�+ z0sin�) + Fa2 � (
Tw12 cos�� z0sin�)
�Fa3 � (Tw22 cos�+ z0sin�) + Fa4 � (
Tw22 cos�� z0sin�)
+(Fa5cos�f � Fb5sin�f) � ((�Tw32cos�f + l3sin�f)cos�� z0sin�)
+(Fa6cos�f � Fb6sin�f) � ((Tw32 cos�f + l3sin�f)cos�� z0sin�):
(5.85)
121
The generalized force for the coordinate �f is
Fg�f =P6
i=1 Fxni �@rxti@�f
+P6
i=1 Fyni �@ryti@�f
+P6
i=1 FPi �@rzti@�f
; (5.86)
which can be calculated as
Fg�f = (Fa5cos�f � Fb5sin�f) � (l3sin�f �Tw32cos�f )
+(Fa6cos�f � Fb6sin�f) � (l3sin�f +Tw32cos�f )
�(Fb5cos�f + Fa5sin�f) � (l3cos�f +Sw32sin�f)
�(Fb6cos�f + Fa6sin�f) � (l3cos�f �Sw32 sin�f):
(5.87)
Expressions for the generalized forces in (5.79), (5.81), (5.83), (5.85) and (5.87) are
the second important component for the complex model.
5.6 Subsystems : Tire Model and Suspension Model
5.6.1 Tire Model
As discussed in the previous section, the longitudinal and lateral components of
the tire forces, Fai and Fbi, and the suspension forces, Fpi, are predicted by the tire
model and the suspension model, respectively. In this section we will brie y discuss
modeling of tire forces and suspension forces. Modeling the tire/road interaction force
is itself an active area of research. For vehicle dynamic simulations purpose, given the
road condition and the operating conditions of the tire such as the longitudinal slip
ratio, the lateral slip angle and the vertical load of the tire, the tire model will predict
both traction/braking force and cornering force generated by the tire (Fig.5.6).
122
Traction/Braking Forces
Lateral Slip Angle Cornering ForcesTIRE FORCE MODEL
Road Conditons
Normal Force
Longitudinal Slip Ratio
Figure 5.6: Tire Force Model
There are two common approaches to the tire force modeling. The �rst is curve-
�tting of the experimental data. This approach can predict a more accurate force
traction �eld. However, the data depends on tire types and it is less portable. One
of the most noticeable tire models using data curve �tting techniques is proposed by
Pacejka and Bakker [47]. In [47], a set of mathematical equations, known as \magic
formulae", are proposed to predict the forces and moments at longitudinal, lateral
and camber slip conditions. These formulae and a set of tuning parameters constitute
the basis of this model. The second approach is the analytical tire model. One way
to analyze the traction �eld is to divide the tire contact patch into two zones: the
sliding zone and the adhesion zone. Shear stresses in the sliding zone of the contact
patch are determined by the frictional properties of the tire/road interface. Shear
stresses in the adhesion zone are determined by the elastic properties of the tire. For
example, the cornering sti�ness C� and longitudinal sti�ness Cs represent the �rst
order approximation of the tire force elastic properties. We adopt the second approach
at this stage of research and use the tire model by Baraket and Fancher [2] in the
simulation model. This tire model accounts for the in uences of tread depth, mean
123
texture depth and skid number on the sliding friction of truck tires The structure of
this tire model is summarized in �gure 5.7.
To use this tire force model, tire longitudinal slip ratios and lateral slip angles
in terms of vehicle states are calculated for typical tractor-semitrailer vehicles. The
longitudinal slip ratio, �i, is equal to
�i =
8>>><>>>:
!ir�VV
for braking
!ir�Vi!iri
for traction
where V is the forward velocity, !i is the angular velocity and r is the radius of the
i� th wheel. The lateral slip angle, �i, is equal to
�1 = � � tan�1�
_yu+l1 _�1_xu�
Tw12
_�1
�
�2 = � � tan�1�
_yu+l1 _�1_xu+
Tw12
_�1
�
�3 = �tan�1�
_yu�l2 _�1_xu�
Tw22
_�1
�
�4 = �tan�1�
_yu�l2 _�1_xu+
Tw22
_�1
�
�5 = �tan�1�
� _xusin�f+( _yu�d1 _�1)cos�f�l3( _�1+_�f )
_xucos�f+( _yu�d1 _�1)sin�f�Tw32
( _�1+_�f )
�
�6 = �tan�1�
� _xusin�f+( _yu�d1 _�1)cos�f�l3( _�+_�f )
_xucos�f+( _yu�d1 _�1)sin�f+Tw32
( _�1+_�f )
�
(5.88)
124
TIRE CONDITION
FRICTION MODEL
TIRE MODEL
PREDICTION OF TRACTION
SN40- Skid number
MD - Mean texture
Gd - Tread groove
depth depth
Fx - Longitudinal force
Fy - Lateral force
α
Fy
µ
α Sx
PAVEMENT INFLUENCE
µ xp
µxs
-Peak data value
- Sliding data value
SIMULATION VALUES
TIRE DATA
PROCESSING TIRE DATA
Flatbed
Fz - Vertical load
V - Vehicle speed
a/l - Pressure distribution
Cs - Longitudinal stiffness
C α - Cornering stiffness
αSx - Longitudinal Slip
- Slip angle
PAVEMENT PARAMETERS
Figure 5.7: Comprehensive Tire Model (Baraket and Fancher)
125
5.6.2 Suspension Model
By far the majority of commercial vehicle suspensions employ the leaf spring
[39] as the vertically compliant element. For the sake of simplicity, instead of using
experimental suspension data, we will adopt an analytical approach to model the
suspension as the combination of a nonlinear spring and a damper element. As
shown in Fig. 5.8, the vertical force acting on the vehicle sprung mass through the
suspension system is equal to the static equilibrium force plus the perturbation force,
which is denoted as Fs, from the spring equilibrium point. The perturbation force
can be modeled as
Fsi =
8>>>>>>>><>>>>>>>>:
Kf1ei +Kf2e5i +Df _ei for i = 1; 2
Kr1ei +Kr2e5i +Dr _ei for i = 3; 4
Kt1ei +Kt2e5i +Dt _ei for i = 5; 6
(5.89)
where Kf1 and Kf2 are parameters of the tractor front spring, Kr1 and Kr2 are
parameters of the tractor rear spring, Kt1 and Kt2 are parameters of the trailer
spring, Df , Dr and Dt are parameters for dampers, and ei is the de ection of the
126
i� th spring from its equilibrium position and is given as
e1 = �Tw12�
e2 =Tw12�
e3 = �Tw22 �
e4 =Tw12�
e5 = �(Tw32�)cos�f + (l3�)sin�f
e6 = (Tw32 �)cos�f + (l3�)sin�f
Spring
SPRUNG MASS
Damper
Figure 5.8: Suspension Model
5.7 Model Veri�cation: Simulation and Experimental Results
In this section, simulation results of the complex vehicle model will be compared
with the open loop experimental results obtained from �eld tests. The test vehicle
is a class 8 tractor-semitrailer truck. The test truck was operated under �xed speed
cruise control and a step steering command was given manually by the driver. The
127
radius of curvature of the test track is approximately 80 meters. Measured signals
for the handling tests include lateral acceleration, yaw rate, roll angle of the sprung
mass, articulation angle between the tractor and the semitrailer, and the front wheel
steering angle. In order to compare the simulation results of the complex vehicle
model with the test vehicle, the front wheel steering angle which is recorded during
experiments is used as the steering input for the simulation model. Furthermore,
simulations are performed using the test vehicle parameters listed in Tables 5.2, 5.3
and 5.4. Some of the parameters are measured values and some are estimated values.
Simulation results of the complex model and the experimental results of the test
vehicle are compared in Figs. 5.9, 5.10, 5.11 and 5.12, respectively. In general, the
predicted simulation results agree well with the �eld test data. We observe that the
predicted response of the articulation angle between the tractor and the trailer is
slower than the actual response. The discrepencies between the predicted responses
and the test results may be attributed to:
1. some unkown vehicle parameters, e.g. the moment of inertia, tire cornering
sti�ness, the height of the roll center and the height of the vertical C.G.,
2. e�ects of dual tires and tandem axes, which impose nonholonomic constraints
on the vehicle motion,
3. unmodeled dynamics, including roll steer and chassis compliance e�ect,
4. sensor calibration errors in instrumentation.
128
parameter unit value parameter unit value
m1 Kg 8444.0 m2 Kg 23472.0Ix1 Kg �m2 12446:5� Ix2 Kg �m2 35523:7�
Iy1 Kg �m2 65734:6� Iy2 Kg �m2 181565:5�
Iz1 Kg �m2 65734:6� Iy2 Kg �m2 181565:5�
l1 m 2.59 Tw1 m 2.02l2 m 3.29 Tw2 m 1.82l3 m 9.65 Tw3 m 1.82z0 m 1:20� h2 m 0:20�
d1 m 3.06 d3 m 4.20d2 m 0:60� d4 m 1:20�
Table 5.2: Parameters for a Tractor-Semitrailer VehicleParameters marked with an asterisk are estimated values
parameter unit value parameter unit value
Kf1 N=m 2:72e5� Kf2 N=m5 3:36e10�
Kr1 N=m 8:53e5� Kr2 N=m5 1:05e11�
Kt1 N=m 1:55e6� Kt2 N=m5 1:92e12�
Df N � sec=m 9080� Dr N � sec=m 9080�
Dt N � sec=m 9080�
Table 5.3: Suspension Parameters
parameter unit value parameter unit value
Iw Kg �m2 13:15� R m 0:3�
C�f N=rad 143330.0 Clf N 127120.0C�r N=rad 97610.0 � 4 Clr N 108960.0 � 4C�t N=rad 80312.0 � 4 Clt N 95340.0 � 4
Table 5.4: Tire and Wheel Parameters
129
0 5 10−0.2
0
0.2
Time (sec)
Lat.
Acc
.(g)
0 5 10−5
0
5
10
15
Time (sec)
Yaw
Rat
e (d
eg/s
)
0 5 10−2
0
2
4
6
Time (sec)
Rol
l Ang
le(d
eg)
0 5 10−10
−5
0
Time (sec)
Tra
iler
Ang
.(de
g)
0 5 10−2
0
2
4
Time (sec)
Ste
erin
g (d
eg)
Figure 5.9: Step input response with the longitudinal vehicle speed 30 MPH,
solid line: experiment, dashdot line: simulation
130
0 5 10−0.2
0
0.2
0.4
Time (sec)
Lat.
Acc
.(g)
0 5 10
0
10
20
Time (sec)Y
aw R
ate
(deg
/s)
0 5 10−2
0
2
4
6
Time (sec)
Rol
l Ang
le(d
eg)
0 5 10−10
−5
0
Time (sec)
Tra
iler
Ang
.(de
g)
0 5 10−2
0
2
4
Time (sec)
Ste
erin
g (d
eg)
Figure 5.10: Step input response with the longitudinal vehicle speed 35 MPH
solid line: experiment, dashdot line: simulation
131
0 5 10−0.2
0
0.2
0.4
Time (sec)
Lat.
Acc
.(g)
0 5 10
0
10
20
Time (sec)Y
aw R
ate
(deg
/s)
0 5 10−2
0
2
4
6
Time (sec)
Rol
l Ang
le(d
eg)
0 5 10
−8
−6
−4
−2
0
Time (sec)
Tra
iler
Ang
.(de
g)
0 5 10−2
0
2
4
Time (sec)
Ste
erin
g (d
eg)
Figure 5.11: Step input response with the longitudinal vehicle speed 40 MPH
solid line: experiment, dashdot line: simulation
132
0 5 10
−0.2
0
0.2
0.4
0.6
Time (sec)
Lat.
Acc
.(g)
0 5 10−5
0
5
10
15
Time (sec)
Yaw
Rat
e (d
eg/s
)
0 5 10−4
−2
0
2
4
6
Time (sec)
Rol
l Ang
le(d
eg)
0 5 10−10
−5
0
Time (sec)
Tra
iler
Ang
.(de
g)
0 5 10−4
−2
0
2
4
Time (sec)
Ste
erin
g (d
eg)
Figure 5.12: Step input response with the longitudinal vehicle speed 46 MPH
solid line: experiment, dashdot line: simulation
133
5.8 Conclusions
Two types of dynamic models of tractor-semitrailer vehicles are utilized for the
design and analysis of lateral controllers. The �rst type of dynamic model is a com-
plex simulation model, which has been developed in this chapter. The second type
of dynamic models are two simpli�ed control models, which will be derived from the
complex nonlinear model in the next chapter. This modeling approach utilizes La-
grangian mechanics and has an advantage over a Newtonian mechanics formulation
in that this complex model eliminates the holonomic constraint at the �fth wheel
(linking joint) by choosing the generalized coordinates. Since there is no constraint
involved in the equations of motion, it is easier to design control algorithms and
to solve the di�erential equations numerically. The e�ectiveness of this modeling
approach was shown by comparing the experimental results of a tractor-semitrailer
vehicle and the simulation results of the complex tractor-semitrailer vehicle model.
134
Chapter 6
Lateral Control of Tractor-Semitrailer
Vehicles on Automated Highways
6.1 Introduction
This chapter follows chapter 5 and is concerned with lateral control of tractor-
semitrailer vehicles in Automated Highway Systems (AHS). In chapter 5, a complex
simulation model for the tractor-semitrailer vehicle was developed that will be used
to evaluate the performance of lateral controllers. Based on this complex model,
two simpli�ed models for steering control and steering/braking control, which will
be referred to as SIM1 and SIM2, respectively, will be derived in this chapter. A
steering control algorithm using input/output linearization is designed as a baseline
controller to achieve the lane following maneuver in AHS. As safety is always of pri-
135
mary concern in AHS, a coordinated steering-independent braking control algorithm
is considered to enhance driving safety and avoid unstable trailer yaw motion. This
coordinated steering and braking control algorithm utilizes the tractor front wheel
steering and the braking force at each of the rear trailer wheels as control inputs. We
observe that the vector relative degree of the automated vehicle under steering and
braking control is not well de�ned. Thus an input/output linearization scheme is not
applicable. However, the backstepping design methodology developed in chapter 3
for the multivariable nonlinear system whose vector relative degree is not well de�ned
can be utilized to design the coordinated steering and independent braking control
algorithm. Simulation studies using the complex vehicle model will be conducted to
show the performance of the coordinated steering and independent braking control
strategy.
The organization of this chapter is as follows. In section 6.2, the transformation
relationships between the road reference coordinate and the vehicle unsprung mass
reference coordinate are explored and will be used to obtain control models. In
section 6.3, a steering control model is formulated. Based on this model, a baseline
steering Controller is designed in section 6.4. A steering and braking control model is
formulated in section 6.5 and the coordinated steering and braking control algorithm
is designed in section 6.6. Conclusions of this chapter are given in the last section.
136
O
X
Y
O
X
Y
rn
r
r
u
u
u
Road Centerline
εε.
.d
X
Y
n
Figure 6.1: Unsprung Mass and Road Reference Coordinates
6.2 Road reference frame
In chapter 5, the vehicle model was derived with respect to the unsprung mass
reference frame. Since one of the objectives for lateral control of automated vehicles is
to follow the road, the description of the relative position and the relative orientation
of the controlled vehicle with respect to the road centerline need to be given explicitly.
To this end, the road reference coordinate OrXrY r in Fig.6.1 is naturally introduced
to describe tracking errors of the vehicle with respect to the road centerline. The
road reference frame is de�ned such that the Xr axis is tangent to the road centerline
137
and the Yr axis passes through the vehicle C.G. Once the road reference frame is
de�ned, the vehicle model with respect to the road reference frame can be obtained
by state variable transformation from the vehicle model with respect to the unsprung
mass reference frame. By equating two expressions for the vehicle velocity, one in
the unsprung mass reference frame and the other in the road reference frame, we
obtain the velocity transformation equations between the unsprung mass reference
frame and th eroad reference frame. Similarly, by equating two expressions for the
vehicle acceleration, one in the unsprung mass reference frame and the other in the
road reference frame, we obtain the acceleration transformation equations between
the unsprung mass reference frame and the road reference frame.
Recall from section 5.2 that the vehicle velocity at C.G. can be expressed as
VCG = _xuiu + _yuju; (6.1)
where _xu and _yu are velocity components of the vehicle along the Xu axis and Yu
axis of the unsprung mass coordinate, respectively. The vehicle acceleration can be
expressed as
aCG = (�xu � _yu _�1)iu + (�yu + _xu _�1)ju; (6.2)
where _�1 is the yaw rate of the unsprung mass reference frame.
On the other hand, the vehicle velocity VCG and the vehicle acceleration aCG at
C.G. can also be obtained in coordinates of the road reference frame XrYrZr. From
Fig. 6.1 and by the de�nition of the road reference frame XrYrZr such that the Yr
138
axis always passes through the vehicle C.G., the position of the vehicle C.G. with
respect to the road reference frame XrYrZr is
rCG=Or = yrjr; (6.3)
then the vehicle velocity with respect to XrYrZr is
VCG=Or = _yrjr + yrd
dtjr: (6.4)
Since the angular velocity of the XrYrZr frame is _�d kr, we have
d
dtir = _�djr (6.5)
and
d
dtjr = � _�dir: (6.6)
Substituting (6.6) into (6.4), we obtain the vehicle velocity with respect to the road
reference frame as
VCG=Or = _yrjr � yr _�dir (6.7)
Since the road reference frame XrYrZr is moving with velocity
VOr = _xrir; (6.8)
the vehicle absolute velocity is
VCG = VCG=Or +VOr = ( _xr � yr _�d)ir + _yrjr; (6.9)
139
where VCG=Or and VOr are given in (6.7) and (6.8), respectively. The acceleration
in the road reference frame coordinates can be obtained by di�erentiating (6.9),
aCG = (�xr � _yr _�d � yr��d)ir + �yrjr + ( _xr � yr _�d)ddtir + _yy
ddtjr
= (�xr � 2 _yr _�d � yr��d)ir + (�yr + _xr _�d � yr _�2d)jr
(6.10)
where (6.5) and (6.6) are used in (6.10). Furthermore, the transformation matrix
from the road reference frame to the unsprung mass reference is
0BBB@ir
jr
1CCCA =
0BBB@
cos �r �sin �r
sin �r cos �r
1CCCA
0BBB@iu
ju
1CCCA : (6.11)
If the relative yaw angle �r is small, (6.11) can be approximated as
ir = cos�riu � sin�rju
' iu � �rju
(6.12)
and
jr = sin�riu + cos�rju
' �riu + ju:
(6.13)
Substituting (6.12) and (6.13) into (6.9), we obtain
VCG = ( _xr � yr _�d)ir + _yrjr
' ( _xr � yr _�d + _yr�r)iu + ( _yr � _xr�r + yr�r _�d)ju:
(6.14)
Similarly, by substituting (6.12) and (6.13) into (6.10), we obtain
aCG = (�xr � yr��d � 2 _yr _�d)ir + (�yr � yr _�2d + _xr _�d)jr
' (�xr � yr��d � 2 _yr _�d + �yr�r � yr�r _�2d + _xr�r _�d)iu
+(�yr � yr _�2d + _xr _�d � �xr�r + yr�r��d + 2 _yr _�d�r)ju:
(6.15)
140
By neglecting third and higher order terms, Eqs. (6.14) and (6.15) can be further
simpli�ed as
VCG ' ( _xr � yr _�d + _yr�r)iu + ( _yr � _xr�r)ju (6.16)
and
aCG ' (�xr � yr��d � 2 _yr _�d + �yr�r + _xr�r _�d)iu + (�yr + _xr _�d � �xr�r)ju; (6.17)
respectively. By equating Eqs. (6.1) and (6.16), we obtain
_xu ' _xr � yr _�d + _yr�r (6.18)
and
_yu ' _yr � _xr�r: (6.19)
Similarly, by equating Eqs. (6.2) and (6.17), we obtain
�xu � _yu _�1 ' �xr � yr��d � 2 _yr _�d + �yr�r + _xr�r _�d (6.20)
and
�yu � _xu _�1 ' �yr + _xr _�d � �xr�r: (6.21)
Substituting (6.19) into (6.20) and noting
_�1 = _�r + _�d (6.22)
we obtain
�xu ' �xr � yr��d � _yr _�d + _yr _�r + �yr�r � _xr�r _�r: (6.23)
141
Similarly by substituting (6.18) into (6.21) we obtain
�yu = �yr � _xr _�r � �xr�r: (6.24)
Eqs. (6.18), (6.19), (6.23) and (6.24) will be used to formulate lateral control models
in section 6.3.
6.3 Steering Control Model (SIM1)
The steering control model will be constructed in two steps. First, a 3 d.o.f. (6
states) model is simpli�ed from the complex model in chapter 5. Next, the simpli�ed
model is transformed with respect to the road reference coordinate, which is discussed
in section 6.2. For the nomenclature of the simpli�ed models in this chapter, refer to
Table 6.1.
6.3.1 Model Simpli�cation
The following assumptions are made to simplify the complex model to one with
only lateral and yaw dynamics.
� The roll motion is negligible.
� The longitudinal acceleration �xr is small.
� The relative yaw angle �r of the tractor with respect to the road centerline is
small.
142
� The relative yaw angle �f of the tractor and the trailer is small.
� Tire slip angles of the left and the right wheels are the same.
� Tire longitudinal and lateral forces are represented by the linearized tire model.
parameter description
yr lateral displacement of the tractor C.G. from the road center line�r relative yaw angle of the tractor w.r.t. road center line�f relative yaw angle of the tractor and the trailer� radius of curvature of the road_�d desired yaw rate set by the road and is equal to _x
�
� tractor front wheel steering angleF1 braking force on the trailer left wheelF2 braking force on the trailer left wheel�i longitudinal slip ratio�i lateral slip anglem1 tractor massIz1 tractor moment of inertiam2 semitrailer massIz2 semitrailer's moment of inertial1 distance between tractor C.G. and front wheel axlel2 distance between tractor C.G. and real wheel axlel3 distance between joint (�fth wheel) and trailer real wheel axleD1 relative position between tractor's C.G. to �fth wheelD3 relative position between semitrailer's C.G. to �fth wheelTw3 semitrailer rear axle track widthC�f cornering sti�ness of tractor front wheelC�r cornering sti�ness of tractor rear wheelC�t cornering sti�ness of semitrailer rear wheelSxf longitudinal sti�ness of tractor front wheelSxr longitudinal sti�ness of tractor rear wheelSxt longitudinal sti�ness of semitrailer rear wheel
Table 6.1: Nomenclature of Control Models
143
By using the above assumptions, the complex vehicle model in section 5.4 can be
simpli�ed as
(m1 +m2)�yu �m2(d1 + d3cos�f )��1 �m2d3cos�f��f
+(m1 +m2) _xu _�1 +m2d3sin�f( _�1 + _�f)2
= Fb1 + Fb2 + Fb3 + Fb4 + Fb5 + Fb6;
(6.25)
�m2(d1 + d3cos�f)�yu + (Iz1 + Iz2 +m2(d1 + d3cos�f )2)��1
+(Iz2 +m2d23 +m2d1d3cos�f )��f
�m2(d1 + d3cos�f) _xu _�1 �m2d3sin�f _yu _�1
�2m2d1d3sin�f _�1 _�f �m2d1d3sin�f _�2f
= (Fb1 + Fb2)l1 � (Fb3 + Fb4)l2 � (Fb5 + Fb6)(d1 + l3)
+(Fa2 � Fa1)Tw12 + (Fa4 � Fa3)
Tw22 + (Fa6 � Fa5)
Tw32 ;
(6.26)
and
�m2d3cos�f �yu + (Iz2 +m2d23 +m2d1d3cos�f )��1 + (Iz2 +m2d
23)��f
�m2d3cos�f _xu _�1 �m2d3sin�f _yu _�1 +m2d1d3sin�f _�21
= �(Fb5 + Fb6)l3 + (Fa6 � Fa5)Tw32 :
(6.27)
To obtain the steering control model (SIM1), we notice that longitudinal tire forces,
Fai, in (6.25), (6.26) and (6.27) are zero under no braking and lateral tire forces, Fbi,
144
can be represented by the linearized tire model,
Fbi =
8>>>>>>>><>>>>>>>>:
C�f�f for i = 1; 2
C�r�f for i = 3; 4
C�t�f for i = 5; 6
; (6.28)
where lateral slip angles �f �r and �t are
�f ' � �_yu + l1 _�1
_xu;
�r ' �_yu � l2 _�1
_xu;
and
�t ' �_yu � d1 _�1 � l3( _�1 + _�f )
_xu+ �f ;
respectively. Substituting Fbi in (6.5) into the simpli�ed vehicle model (6.25), (6.26)
and (6.27), we obtain the control model (SIM1) as
M �q + C(q; _q) +D _q +Kq = F�; (6.29)
where
q = [yu; �1; �f ]T
is the generalized coordinate vector,
M =
0BBBBBB@
m1 +m2 �m2(d1 + d3cos�f ) �m2d3cos�f
�m2(d1 + d3cos�f ) Iz1 + Iz2 +m2(d21 + d23) + 2m2d1d3cos�f Iz2 +m2d23cos�f
�m2d3cos�f Iz2 +m2d23 +m2d1d3cos�f Iz2 +m2d
23
1CCCCCCA
145
is the inertial matrix,
C(q; _q) =
0BBBBBB@
(m1 +m2) _xu +m2d3sin�f ( _�1 + _�f )2
�m2(d1 + d3cos�f ) _xu _�1 �m2d3sin�f _yu _�1 � 2m2d1d3sin�f _�1 _�f �m2d1d3sin�f _�2f
�m2d3sin�f _yu _�1 �m2d3cos�f _xu _�1 +m2d1d3sin�f _�21
1CCCCCCA
is the vector of the Coriolis and Centrifugal forces,
D =2
_x
0BBBBBB@
C�f +C�r + C�t l1C�f � l2C�r � (l3 + d1)C�t �l3C�t
l1C�f � l2C�r � (l3 + d1)C�t l21C�f + l22C�r + (l3 + d1)2C�t l3(l3 + d1)C�t
�l3C�t l3(l3 + d1)C�t l23C�t
1CCCCCCA
is the damping matrix,
K =
0BBBBBB@
0 0 �2C�t
0 0 2(l3 + d1)C�t
0 0 2l3C�t
1CCCCCCA
is the potential matrix, and the vector F 2 R3�1 is
F = 2 C�f � [1; l1; 0]T :
Eq. (6.29) represents the simpli�ed vehicle model with respect to the unsprung mass
reference coordinate.
6.3.2 Control Model with respect to the Road Reference Frame
Recall from section 6.2 that state variables with respect to the unsprung mass
reference frame can be transformed into state variables with respect to the road
reference frame by
_yu = _yr � _xr�r; (6.30)
146
�yu = �yr � _xr _�r � �xr�r; (6.31)
_�1 = _�r + _�d (6.32)
and
��1 = ��r + ��d: (6.33)
By the assumptions that the longitudinal acceleration �xr and the relative yaw angle
�r are small, their product in (6.31) can be neglected. Substituting the state vari-
able transformation equations (6.30), (6.31), (6.32) and (6.33) into the control model
(6.29), we obtain
M �qr + �(qr; _qr; _�d; ��d) = F�; (6.34)
where
qr = [yr; �r; �f ]T
is the vector of state variables with respect to road centerline and is de�ned in Table
6.1, �(qr; _qr; _�d; ��d) 2 R3�1 is the vector with its components
�(qr; _qr; _�d; ��d)(1) =0BB@
2_x((C�f +C�r + C�t)( _yr � _x�r) + (l1C�f � l2C�r � (l3 + d1)C�t)( _�r + _�d)� l3C�t _�f )
�2C�t�f +m2d3sin�f ( _�r + _�d + _�f )2 + (m1 +m2) _x _�d �m2(d1 + d3cos�f )��d
1CCA
�(qr; _qr; _�d; ��d)(2) =0BBBBBBBBBB@
2_x((l1C�f � l2C�r � (l3 + d1)C�t)( _yr � _x�r) + (l21C�f + l22C�r + (l3 + d1)2C�t)( _�r + _�d)
+l3(l3 + d1)C�t _�f ) + 2(l3 + d1)C�t�f �m2d3sin�f ( _yr � _x�r)( _�r + _�d)
�2m2d1d3sin�f ( _�r + _�d) _�f �m2d1d3sin�f _�2f �m2(d1 + d3cos�f ) _x _�d
+(Iz1 + Iz2 +m2d21 +m2d
23 + 2m2d1d3cos�f )��d
1CCCCCCCCCCA
147
and
�(qr; _qr; _�d; ��d)(3) =0BB@
2_x (�l3C�t( _yr � _x�r) + l3(l3 + d1)C�t( _�r + _�d) + l23C�t _�f ) +m2d3sin�f ( _�r + _�d)
2
+2l3C�t�f �m2d3sin�f ( _yr � _x�r)( _�r + _�d) �m2d3cos�f _x _�d + (Iz2 +m2d23 +m2d1d3cos�f )��d
1CCA :
Eq. (6.34) is the simpli�ed model which will be used to design the steering control
algorithm in section 6.4 for the lane following maneuver.
6.3.3 Linear Analysis of the Control Model
The control model (6.34) can be further linearized by approximating cos�f ' 1,
sin�f ' �f and neglecting the second order terms. Then the linearized model has
the form
M �qr +D _qr +Kqr = F� + E1 _�d + E2��d; (6.35)
where _�d and ��d are exogenous inputs representing the disturbace e�ects on curved
roads. Two interesting properties are observed from this linearized model.
1. M is a symmetric positive de�nite matrix which contains the inertial informa-
tion of the vehicle system.
2. The D matrix can be interpreted as a damping matrix. Each element of the C
matrix contains the tire cornering sti�ness. If the cornering sti�ness is small, the
vehicle system will become lightly damped and more oscillatory. For example,
if the vehicle is operated on an icy road, the vehicle stability will decrease. We
148
also see that the vehicle longitudinal velocity _x appears in the denominator of
the damping matrix. Therefore the system damping is inversely proportional to
the vehicle longitudinal velocity, which also agrees with our physical experience.
The �rst property thatM is a positive de�nite matrix will be exploited in synthesizing
the input-output linearizing controller.
6.4 Steering Control of Tractor-Semitrailer Vehicles
6.4.1 Controller Design
In this section a steering control algorithm will be designed by applying the input-
output linearization scheme [30, 46]. The steering control model developed in section
6.3 is
M �qr + �(qr; _qr; _�d) = F� (6.36)
where M is the inertial matrix and can be partition into four blocks as
M =
0BBBBBB@
m1 +m2 �m2(d1 + d3) �m2d3
�m2(d1 + d3) Iz1 + Iz2 +m2(d1 + d3)2 Iz2 +m2d
23 +m2d1d3
�m2d3 Iz2 +m2d23 +m2d1d3 Iz2 +m2d
23
1CCCCCCA�
0BB@
M11 M12
M21 M22
1CCA
Since the matrix M is positive de�nite, both M11 and M22 are also positive de�nite.
The control model in (6.36) can be divided into two subsystems:
M11 �yr +M12
0BBB@
��r
��f
1CCCA+ �1 = C�f� (6.37)
149
and
M21 �yr +M22
0BBB@
��r
��f
1CCCA+
0BBB@
�2
�3
1CCCA =
0BBB@
l1C�f
0
1CCCA �: (6.38)
Notice that the second subsystem (6.38) can be rewritten as
0BBB@
��r
��f
1CCCA = M�1
22
8>>><>>>:�M21 �yr �
0BBB@
�2
�3
1CCCA+
0BBB@
l1C�f
0
1CCCA �
9>>>=>>>;: (6.39)
Substituting Eq. (6.39) into Eq. (6.37), we obtain the input(�)-output(yr) dynamics
as
�M11 �yr + �� = �K�; (6.40)
where
�M11 = M11 �M12M�122 M21; (6.41)
�� = �1 �M12M�122
0BBB@
�2
�3
1CCCA ; (6.42)
and
�K = C�f �M12M�122
0BBB@
l1C�f
0
1CCCA : (6.43)
Note that
�M11 = T TMT (6.44)
and
T =
0BBB@
I
�M�122 M21
1CCCA (6.45)
150
which is a full rank matrix. By the facts that the matrix M is positive de�nite and
that the matrix T has a full rank, we conclude that �M11 is also positive de�nite. If
�K 6= 0, we can choose the linearizing control law
�K� = �M11v + �� (6.46)
With this linearizing control law, the subsystems (6.37) and (6.38) become
�yr = v (6.47)
and
M22
0BBB@
��r
��f
1CCCA +
0BBB@
�2
�3
1CCCA = (
0BBB@
l1C�f
0
1CCCA �M11 �M21)v +
0BBB@
l1C�f
0
1CCCA �� (6.48)
Furthermore, by choosing
v = kd _yr + kpyr (6.49)
the output yr converges to zero asymptotically.
6.4.2 Simulation Results
The simulations are conducted using the complex vehicle model developed in
chapter 5 and the vehicle parameters are listed in Table 5.2. The simulation scenario
we used is depicted in Fig. 6.2. The tractor-semitrailer vehicle travels along a straight
roadway with an initial lateral displacement of 15 cm and enters a curved section with
a radius of curvature of 450 m at time t = 5 sec. Fig. 6.3 shows the simulation results
151
of the input-output linearization controller at a vehicle speed of 60 MPH. We see that
the lateral tracking error converges to zero asymptotically while the yaw angle of the
tractor and the relative yaw angle of the trailer are small.
t=15 sec.
ρ = 450 m
t=5 sec.
t=8 sec.
Figure 6.2: Simulation Scenario
152
0 5 10 15
0
0.2
Time (s)
Late
ral D
isp.
(m
)
0 5 10 15−4
−2
0
2
Time (s)R
oll a
ngle
(deg
)
0 5 10 15−4
−2
0
2
Time (s)
Yaw
ang
le(d
eg)
0 5 10 15−2
0
2
4
Time (s)
Tra
iler
angl
e(de
g)
0 5 10 15−5
0
5
Time (s)
Ste
erin
g(de
g)
Figure 6.3: Input/Output Linearization Control
153
6.5 Steering and Braking Control Model (SIM2)
In this section, the control model developed in section 6.3 is reformulated to
include the left and right braking forces at the trailer as another two control inputs.
Recall that in section 6.3, the simpli�ed vehicle model was obtained as
(m1 +m2)�yu �m2(d1 + d3cos�f )��1 �m2d3cos�f��f
+(m1 +m2) _xu _�1 +m2d3sin�f( _�1 + _�f)2
= Fb1 + Fb2 + Fb3 + Fb4 + Fb5 + Fb6;
(6.50)
�m2(d1 + d3cos�f)�yu + (Iz1 + Iz2 +m2(d1 + d3cos�f )2)��1
+(Iz2 +m2d23 +m2d1d3cos�f )��f
�m2(d1 + d3cos�f) _xu _�1 �m2d3sin�f _yu _�1
�2m2d1d3sin�f _�1 _�f �m2d1d3sin�f _�2f
= (Fb1 + Fb2)l1 � (Fb3 + Fb4)l2 � (Fb5 + Fb6)(d1 + l3)
+(Fa2 � Fa1)Tw12 + (Fa4 � Fa3)
Tw22 + (Fa6 � Fa5)
Tw32 ;
(6.51)
and
�m2d3cos�f �yu + (Iz2 +m2d23 +m2d1d3cos�f )��1 + (Iz2 +m2d
23)��f
�m2d3cos�f _xu _�1 �m2d3sin�f _yu _�1 +m2d1d3sin�f _�21
= �(Fb5 + Fb6)l3 + (Fa6 � Fa5)Tw32 :
(6.52)
154
By substituting the linear lateral tire model
Fbi =
8>>>>>>>><>>>>>>>>:
C�f�f for i = 1; 2
C�r�f for i = 3; 4
C�t�f for i = 5; 6
into (6.50), (6.51) and (6.52) and assuming the longitudinal tire forces on the tractor
are zero, i.e., Fa1 = Fa2 = Fa3 = Fa4 = 0, we obtain the simpli�ed model as
M �q + C(q; _q) +D _q +Kq = H � U (6.53)
where M , C(q; _q), D and K are the same as in SIM1, (6.29), and H and U are
H =
0BBBBBBBB@
2C�f 0
2l1C�fTw32
0 Tw32
1CCCCCCCCA
and
U =
0BBB@
�
Fa6 � Fa5
1CCCA �
0BBB@
�
T
1CCCA (6.54)
respectively. Eq. (6.53) is the model with respect to the unsprung mass reference
frame. Parallel to the development of SIM1 in section 6.3 and by using the coordinate
transformations (6.30) (6.31) (6.32) and (6.33), the steering and braking control model
SIM2 with respect to the road reference frame is obtained
M(qr)�qr + �(qr; _qr; _�d; ��d) = H � U (6.55)
Notice that Fa5 and Fa6 in (6.54) stand for the longitudinal forces at the left and right
wheels of the trailer. Thus T is the di�erential force acting on the trailer. We denote
155
the longitudinal force Fai < 0 when it is a braking force and Fai > 0 when it is a
traction force. In fact, the control inputs Fa5 and Fa6 at the wheels of the trailer can
only be negative, i.e., we can use only braking instead of traction. This would be a
big constraint on the control inputs Fai. However, the di�erential force T can be both
positive and negative. Furthermore, the braking forces Fa5 and Fa6 are determined
by the tire force model and are functions of the tire slip ratio. Speci�cally, as shown
in Fig. 6.4, the wheel dynamics are
Iw _!i = �Fair + �i (6.56)
where !i is the angular velocity of the wheel, Fai is the braking force generated at
the tire/ground interface, and �i is the braking torque applied at the braking disk of
the wheel. The tire slip ratio is de�ned as
�i =!ir � V
V(6.57)
and the braking force is
Fai = Cl�i (6.58)
Eq. (6.55) as well as Eqs. (6.56) (6.57) and (6.58) will be used to design the coordi-
nated steering and braking control algorithm in section 6.6.
156
ω
V
Angular
ai
Velocity
τ
r Radius
F
i
i
Moment of Inertia = Iw
Figure 6.4: Wheel Dynamics
6.6 Coordinated Steering and Independent Braking Control
6.6.1 Controller Design
In this section, a coordinated steering and braking control algorithm will be de-
signed. Motivated by Matsumoto and Tomizuka [40], we propose to use not only the
tractor front wheel steering input but also the trailer unilateral tire braking to provide
the di�erential torque for directly controlling the trailer yaw motion. The control al-
gorithm will be designed in two steps. In the �rst step, we assume the di�erential force
T is control input. Then the desired steering command �d and the desired di�erential
braking force Td are determined by input/output linearization scheme. By the nature
of unilateral braking, if Td > 0, we have Fa5d = �Td and Fa6d = 0. On the other
hand, if Td < 0, we have Fa5d = 0 and Fa6d = Td. In the second step, the required
157
braking torques �5 and �6 are determined to generate the desired braking forces Fa5d
and Fa6d by utilizing backstepping design methodologies presented in chapter 3.
Step 1
First, we de�ne the �rst system output e1 as the lateral tracking error
e1 = yr (6.59)
and the second output e2 as the articulation angle between the tractor and the trailer
e2 = �f (6.60)
Di�erentiating e1 and e2 twice, we obtain0BBB@
�e1
�e2
1CCCA =
0BBB@
M�1(1)
M�1(3)
1CCCAC( _q; _�d; ��d) +
0BBB@
M�1(1)
M�1(3)
1CCCAHU (6.61)
The number i in the parenthesis M�1(i) denotes the i� th row of the M�1 matrix.
For notational simplicity, we de�ne
J =
0BBB@
M�1(1)
M�1(3)
1CCCAH (6.62)
If the matrix J is nonsingular, we can choose the control input U as
U = �J�1
0BBB@
M�1(1)
M�1(3)
1CCCAC( _q; _�d; ��d)� J�1
8>>><>>>:KD
0BBB@
_e1
_e2
1CCCA +KP
0BBB@
e1
e2
1CCCA
9>>>=>>>;
(6.63)
This control law cancels the system nonlinearities and inserts the desired error dy-
namics. Thus the closed loop system becomes0BBB@
�e1
�e2
1CCCA +KD
0BBB@
_e1
_e2
1CCCA +KP
0BBB@
e1
e2
1CCCA = 0: (6.64)
158
Step 2
In Step 1 we regard T as a real control input; then the desired steering command
and the desired di�erential braking forces Td are set in (6.63). In this step we will
'backstep' to determine the braking torques �5 and �6 on the trailer's left and right
wheels. Recall that the wheel dynamics is
Iw _!i = �Fair + �i (6.65)
and the tire force is
Fai = Clt�i (6.66)
where the slip ratio �i is de�ned as
�i =!ir � V
V(6.67)
Combining equations ( 6.65), ( 6.66) and ( 6.67), we obtain
_Fai = Clt_�i
= Clt(@�i@V
_V + @�i@!i
_!i)
= Clt(�!irV 2
_V + rIwV
(�Clt�ir + �i))
(6.68)
Thus the equations governing the vehicle dynamics and wheel dynamics are
0BBB@
�e1
�e2
1CCCA =
0BBB@
M�1(1)
M�1(3)
1CCCAC( _q; _�d; ��d) + J � U (6.69)
and
_Fai = Clt(�!irV 2
_V + rIwV
(�Clt�ir + �i)) (6.70)
159
Recall that T = Fa6 � Fa5 and both Fa5 and Fa6 are negative. In this unilateral
braking scheme, if Td determined in (6.63) is positive, we have Fa5d = �Td and
Fa6d = 0. Thus the braking controller will apply the brake torque on the trailer left
wheel. On the other hand, if Td is negative, we have Fa5d = 0 and Fa6d = Td, so the
braking controller will apply the brake torque on the trailer right wheel. From Eq.
(6.63) in step 1, the control inputs are chosen as
� = �d(qr; _qr; _�d) (6.71)
and
T = Td(qr; _qr; _�d) (6.72)
such that the error dynamics becomes
�e1 + kd1 _e1 + kp1e1 = 0 (6.73)
and
�e2 + kd2 _e2 + kp2e2 = 0: (6.74)
Note that T is determined by the braking force Fai, and that braking force Fai can be
adjusted only through equation (6.70), i.e., the braking torque �i is the actual control
input. Therefore, T cannot be simply set to Td all the time, and �i must be adjusted
so that the di�erence between Td and T is brought to zero. This is the main idea in
the backstepping procedure. We de�ne two new variables �1 and �2 as
�1 = Fa5 � Fa5d (6.75)
160
and
�2 = Fa6 � Fa6d; (6.76)
respectively. Then we have
T = Fa6 � Fa5
= Fa6d + �2 � Fa5d � �1
= Td + �2 � �1:
(6.77)
Noting
_�1 = _Fa5 � _Fa5d
= Clt(�!5rV 2
_V + rIwV
(�Clt�5r + �5))� _Fa5d
(6.78)
and
_�2 = _Fa6 � _Fa6d
= Clt(�!6rV 2
_V + rIwV
(�Clt�6r + �6))� _Fa6d;
(6.79)
we choose
�5 = Clt�5r +IwVr(�!5r
V 2_V + 1
Clt( _Fa5d � k1�1)) (6.80)
and
�6 = Clt�6r +IwVr(�!6r
V 2_V + 1
Clt( _Fa6d � k2�2)): (6.81)
Then, we obtain
�e1 + kd1 _e1 + kp1e1 + J12(�2 � �1) = 0; (6.82)
�e2 + kd2 _e2 + kp2e2 + J22(�2 � �1) = 0; (6.83)
_�1 + k1�1 = 0; (6.84)
161
and
_�2 + k2�2 = 0; (6.85)
where J12 and J22 are the (1; 2) and (2; 2) elements of the matrix J . De�ning the
state vector (x1; x2; x3; x4)T as (e1; _e1; e2; _e2)T and transforming equations (6.82) and
(6.83) to state space form, we have
ddt
0BBBBBBBBBB@
x1
x2
x3
x4
1CCCCCCCCCCA
=
0BBBBBBBBBB@
0 1 0 0
kp1 kd1 0 0
0 0 1 0
0 0 kp2 kd2
1CCCCCCCCCCA
0BBBBBBBBBB@
x1
x2
x3
x4
1CCCCCCCCCCA
+
0BBBBBBBBBB@
0 0
J12 �J12
0 0
J22 �J22
1CCCCCCCCCCA
0BB@
�1
�2
1CCA (6.86)
Then the overall system can be rewritten as
ddt
0BBBBBBBBBBBBBBBBBB@
x1
x2
x3
x4
�1
�2
1CCCCCCCCCCCCCCCCCCA
=
0BBBBBBBBBBBBBBBBBB@
0 1 0 0 0 0
�kp1 �kd1 0 0 J12 �J12
0 0 0 1 0 0
0 0 �kp2 �kd2 J22 �J22
0 0 0 0 �k1 0
0 0 0 0 0 �k2
1CCCCCCCCCCCCCCCCCCA
0BBBBBBBBBBBBBBBBBB@
x1
x2
x3
x4
�1
�2
1CCCCCCCCCCCCCCCCCCA
(6.87)
We see that the overall system matrix can be divided by four blocks and the lower
o�-diagonal block is identically zero. Thus the eigenvalues of the overall system are
the union of those of the block diagonal matrices. Since each block diagonal matrix
is asymptotically stable, the overall system is asymptotically stable.
6.6.2 Simulation Results
We use the same scenario shown in Fig.6.2. Vehicle longitudinal speed is 26.4
m/s (60 MPH). Figs. 6.5 and 6.6 show the simulation results of the coordinated
162
steering and independent braking control. Notice that in implementing this control
algorithm, we impose upper and lower bounds on the braking torque input to avoid
tire force saturation. Comparison of the steering control designed in section 6.4 and
the coordinated steering and independent braking control is shown in Fig. 6.7, from
which we see that the independent braking helps reducing not only the trailer yaw
errors but also the tractor yaw errors.
0 5 10 15
0
0.2
Time (s)
Late
ral D
isp.
(m
)
0 5 10 15−4
−2
0
2
Time (s)
Rol
l ang
le(d
eg)
0 5 10 15−2
−1
0
1
Time (s)
Yaw
ang
le(d
eg)
0 5 10 15−1
0
1
2
Time (s)
Tra
iler
angl
e(de
g)
0 5 10 15−4
−2
0
2
Time (s)
Ste
erin
g(de
g)
Figure 6.5: Input/Output Linearization Control with Trailer Independent Braking
163
0 5 10 15−4000
−2000
0
Time (s)
tau_
5 (N
−m
)
0 5 10 15−4000
−2000
0
Time (s)ta
u_6
(N−
m)
0 5 10 15−3
−2
−1
0
x 104
Time (s)
F_a
5(N
)
0 5 10 15
−4
−2
0
x 104
Time (s)
F_a
6(N
)
Figure 6.6: Input/Output Linearization Control with Trailer Independent Braking
164
0 5 10 15
0
0.2
Time (s)
Late
ral D
isp.
(m
)
0 5 10 15−4
−2
0
2
Time (s)R
oll a
ngle
(deg
)
0 5 10 15−4
−2
0
2
Time (s)
Yaw
ang
le(d
eg)
0 5 10 15−2
0
2
4
Time (s)
Tra
iler
angl
e(de
g)
0 5 10 15−5
0
5
Time (s)
Ste
erin
g(de
g)
Figure 6.7: Comparison of Input/Output Linearization Control with (solid line) andwithout (dashdot line) Trailer Independent Braking
165
6.7 Conclusions
Two control algorithms for lateral guidance of tractor-semitrailer vehicles were
designed. The �rst was a baseline steering control algorithm and the second was a
coordinated steering and independent braking control algorithm. In the design of the
second control algorithm, we utilized tractor front wheel steering angles and trailer
independent braking forces to control the tractor and the trailer motion. The mul-
tivariable backstepping design methodology presented in chapter 3 was utilized to
determine the coordinated steering angle and braking torques on the trailer wheels.
Simulations showed that both the tractor and the trailer yaw errors under coordinated
steering and independent braking force control were smaller than those without in-
dependent braking force control.
166
Chapter 7
Conclusions and Future Research
7.1 Summary
In this dissertation new aspects were explored in the design of backstepping control
systems with applications to vehicle lateral control in AHS. Three main topics were
investigated in this dissertation: control of multivariable nonlinear systems whose
vector relative degrees are not well de�ned, steering control of light passenger vehicles
on automated highways, and coordinated steering and braking control of commercial
heavy vehicles on automated highways.
Backstepping control design of a class of multivariable nonlinear systems.
A recursive algorithm is developed to control a class of square multivariable nonlin-
ear systems whose decoupling matrices are singular. Past researches on control of
this class of systems emphasize the decoupling or noninteraction control by adding
167
integrators to the appropriate input channels; i.e., decoupling control by dynamic
extension. We provide an alternative approach from the backstepping perspective to
control this class of nonlinear systems. Speci�cally, this new controller design proce-
dure for this class of systems is developed based on the dynamic extension algorithm
by incorporating backstepping design methods to partially close the loop in each of
the design step. The resulting control law obtained by this new approach is static
state feedback.
Steering Control of Light Passenger Vehicles. A backstepping controller is
designed for lateral guidance of the passenger car in automated highway systems. In
this design, the vehicle lateral displacement is a�ected by the relative yaw angle of
the car with respect to the road centerline, and the relative yaw angle is controlled
by the vehicle's front wheel steering angle. The main feature of this nonlinear design
is that the stability of both lateral and yaw error dynamics are ensured and closed
loop performance can be speci�ed simultaneously. Furthermore, nonlinear position
feedback, which acts as low gain control at small tracking errors and high gain control
at larger tracking errors, is introduced as a trade-o� between passenger ride comfort
and tracking accuracy.
DynamicModeling of Articulated Commercial Vehicles. In this dissertation,
a control oriented dynamic modeling approach is proposed for articulated vehicles.
A generalized coordinate system is de�ned in this approach to precisely describe the
168
kinematics of a vehicle. Equations of motion are derived based on the Lagrange
mechanics. This modeling approach is validated by comparing �eld test data of a
class 8 tractor-semitrailer type articulated vehicle and the simulation results of the
computer model.
Coordinated Steering and Independent Braking Control of Tractor-Semitrailer
Vehicles. In this dissertation, a steering control algorithm for tractor-semitrailer
vehicles is designed as a baseline controller for lane following maneuver in AHS. As
safety is of primary concern in AHS, a coordinated steering and independent brak-
ing control algorithm is designed. In this algorithm, braking forces are distributed
over the inner and outer tires of the trailer to a�ect the vehicle directional dynamics
directly. Simulations showed that both the tractor and the trailer yaw errors under
coordinated steering and independent braking force control were smaller than those
without independent braking force control.
7.2 Suggested Future Research
Future research topics on the theoretical and experimental developments of this
dissertation include:
� Stability of internal dynamics for the backstepping control of the multivariable
nonlinear system whose vector relative degree is not well de�ned.
169
The backstepping control algorithm presented in chapter 3 is developed to con-
trol the outputs of nonlinear system whose vector relative degree is not well
de�ned. The e�ectiveness of this algorithm hinges upon the stability of internal
dynamics. Since the backstepping approach provides only a controller design
procedure, we still have the freedom to choose the speci�c control law at each
design step. Systematic design of the control law at each step to stabilize the
internal dynamics is an worthwhile issue to be addressed.
� Adaptive/Robust backstepping control of the multivariable nonlinear system whose
vector relative degree is not well de�ned.
The control algorithm presented in chapter 3 did not explicitly take into account
of system unknown parameters and/or unmodeled dynamics. Adaptive/robust
version of this algorithm is a natural next step for further investigations.
� Experimental study of the backstepping steering control algorithm for light pas-
senger vehicles.
Even though the control model and the simulation model for light passenger
vehicles are widely used by researchers in the California PATH program, and
are generally agreed to be quite accurate, the closed loop simulation results
from chapter 4 should be veri�ed through experimental study.
� Coordination between the longitudinal and the lateral control for commercial
heavy vehicles
170
In chapter 6, an independent braking control algorithm for tractor-semitrailer
vehicles has been proposed. Simulations showed that this algorithm is e�ective
in increasing damping of lateral dynamics. On the other hand, braking forces
will also be utilized for longitudinal control. Thus coordination between the
longitudinal and the lateral control should be investigated.
� Experimental study of the steering and braking control algorithms for commercial
heavy vehicles.
The work presented in chapter 5 and chapter 6 for the lateral control of com-
mercial heavy vehicles should be followed by the closed loop experimental study.
171
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