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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