i

Vehicle Stability through Integrated

Active Steering and Differential Braking

by

Byeongcho Lee

A thesis

presented to the University of Waterloo

in fulfillment of the

thesis requirement for the degree of

Master of Applied Science

in

Mechanical Engineering

Waterloo, Ontario, Canada, 2005

©Byeongcho Lee, 2005

ii

I hereby declare that I am the sole author of this thesis.

I authorize the University of Waterloo to lend this thesis to other institutions or individuals for the

purpose of scholarly research

Byeongcho Lee

I further authorize the University of Waterloo to reproduce this thesis by photocopying or by other

means, in total or in part, at the request of other institutions or individuals for the purpose of scholarly

research.

Byeongcho Lee

iii

Acknowledgements

1 I love you, LORD; you are my strength. 2The LORD is my rock, my fortress, and my

savior; my God is my rock, in whom I find protection. He is my shield, the strength of my

salvation, and my stronghold.

Psalms 18:1-2

This thesis would not have been possible without the patient guidance and great

encouragement of my supervisors Dr. Amir Khajepour and Dr. Kamran Behdinan. Dr.

Khajepour has taught me a great academic inspiration whenever I encountered an academic

problem. Dr. Behdinan gave me many helpful feedbacks to continue my thesis.

My sincere gratitude also goes to Mr. Shim who was my former professional supervisor in

Mando Corporation. Director Mr. Shim has taught me that I never give up my efforts to solve

a problem. He always gave me an advice to keep my vision.

Most importantly, I would like to express my sincere gratitude to my whole families. I

wish to thank my mother who has given an encouragement and love. I especially thank my

wife who has given a great love and patience. She always gave me a grate understanding

whenever I turned my path.

I wish to thank the readers of my thesis, Professor A and B for their helpful feedback and

suggestions.

iv

Abstract

This thesis proposes a vehicle performance/safety method using combined active steering

and differential braking to achieve yaw stability and rollover avoidance. Yaw stability

control is a continuing action, while rollover avoidance control is emergency action. Each

controller gives the correction steering angle and correction moment, produced by braking

force, to the steering and braking actuators.

Active steering is shown to have an immediate effect on the vehicle‟s yaw and roll motion;

however, it also causes a large trajectory deviation in the desired course. Therefore, in a high

speed cornering situation, steering control is not the best way to achieve yaw stability and

rollover avoidance.

Differential braking has less influence on the vehicle‟s yaw and roll dynamics under

normal driving conditions; however, it can reduce the vehicle‟s yaw on a low friction road

using differential braking force. In the case of a high speed steady-state cornering situation,

braking control is a very efficient way to reduce the risk of rollover occurrence.

A four degree-of-freedom vehicle model is used to study rollover and yaw motion. This

simplified model is sufficient for understanding the effects of differential braking and active

steering on vehicle stability. In order to use differential braking and active steering in the

vehicle‟s yaw stability and rollover avoidance, the models of the braking and steering

systems are also derived. For simplicity, the steering and brake subsystem dynamics are

considered as a second and first order transfer function, respectively.

The implemented SIMULINK model shows the advantages and disadvantages of steering

and braking control methods through a variety of input signals, such as J-turn, sinusoidal, and

Fishhook inputs. Also, the nonlinear model of the vehicle using ADAMS software is built

and the road profile is included to evaluate and compare the yaw stability and rollover

avoidance with the linear 4 DOF model.

The integrated active steering and differential braking control are shown to be most

efficient in achieving yaw stability and rollover avoidance, while active steering and

v

differential braking control has been shown to improve the vehicle performance and safety

only in yaw stability and rollover avoidance, respectively.

vi

Table of Contents

Acknowledgements ............................................................................................................................... iii

Abstract ................................................................................................................................................. iv

Table of Contents .................................................................................................................................. vi

List of Figures ..................................................................................................................................... viii

List of Tables ....................................................................................................................................... xii

Nomenclature ...................................................................................................................................... xiii

Chapter 1 Introduction ........................................................................................................................... 1

Chapter 2 Literature Review .................................................................................................................. 5

2.1 Steer-by-Wire ............................................................................................................................... 5

2.2 Active Steering ............................................................................................................................. 7

2.3 Differential Braking ..................................................................................................................... 9

Chapter 3 Sport Utility Vehicle Model ................................................................................................ 12

3.1 Four Degree of Freedom Vehicle Model ................................................................................... 12

3.2 Steering System Model .............................................................................................................. 20

3.3 Brake System Model .................................................................................................................. 23

3.4 Control Parameters ..................................................................................................................... 24

3.4.1 Desired Yaw Rate ............................................................................................................... 25

3.4.2 Rollover Coefficient ............................................................................................................ 27

Chapter 4 Controller ............................................................................................................................ 29

4.1 Yaw Stability Control ................................................................................................................ 29

4.1.1 Active Steering Controller .................................................................................................. 29

4.1.2 Differential Braking Controller ........................................................................................... 30

4.1.3 Integrated Controller ........................................................................................................... 31

4.2 Rollover Avoidance Control ...................................................................................................... 32

4.3 Integrated Yaw Stability and Rollover Avoidance Controller ................................................... 33

Chapter 5 Simulation Results ............................................................................................................... 35

5.1 Test Maneuvers and Vehicle Parameters ................................................................................... 35

5.1.1 Test Maneuvers ................................................................................................................... 35

5.1.2 Vehicle Parameters ............................................................................................................. 39

5.2 Simulation Results of Linear Model .......................................................................................... 40

5.2.1 Simulation Results of J-turn Maneuver............................................................................... 40

vii

5.2.2 Simulation Results of Sinusoidal Maneuver ....................................................................... 45

5.3 Simulation Results of Nonlinear Model ..................................................................................... 50

5.3.1 Simulation Results of J-turn Maneuver ............................................................................... 50

5.3.2 Simulation Results of Sinusoidal Maneuver ....................................................................... 56

5.4 Comparison of Linear and Nonlinear Simulation Results .......................................................... 61

5.4.1 Simulation Results of J-turn Maneuver ............................................................................... 61

5.4.2 Simulation Results of Sinusoidal Maneuver ....................................................................... 66

5.5 Advantage of Integration Control ............................................................................................... 71

5.5.1 Simulation Results for J-turn Maneuver at High Speed ...................................................... 71

5.5.2 Simulation Results for Fishhook Maneuver at High Speed................................................. 75

Chapter 6 ADAMS Model and Evaluation .......................................................................................... 80

6.1 ADAMS Model Building ........................................................................................................... 80

6.2 Simulation Results for ADAMS Model ..................................................................................... 83

6.2.1 Simulation Results for J-Turn Input .................................................................................... 84

6.2.2 Simulation Results for Sinusoidal Input .............................................................................. 87

Chapter 7 Conclusions and Discussion ................................................................................................ 90

7.1 Discussion .................................................................................................................................. 90

7.1.1 Advantages and Disadvantages of Steering Control ........................................................... 90

7.1.2 Advantages and Disadvantages of Braking Control ............................................................ 90

7.1.3 Summary of Discussion ....................................................... Error! Bookmark not defined.

7.2 Conclusions ................................................................................................................................ 92

7.3 Future Works .............................................................................. Error! Bookmark not defined.

Bibliography ......................................................................................................................................... 94

Appendix A Kinematics of Yaw and Roll Motions ............................................................................. 97

Appendix B Magic Formula of Tire ................................................................................................... 109

Appendix C MATLAB and SIMULINK Model ................................................................................ 113

Appendix D ADAMS MODEL .......................................................................................................... 120

viii

List of Figures

Figure 2-1 Comparison of Conventional Power Steering System and Steer-by-Wire System [27] ..... 5

Figure 2-2 Architecture of Steer-by-Wire [6] ....................................................................................... 6

Figure 2-3 Concept of Additional Steering Angle Actuation Principle [13] ........................................ 8

Figure 2-4 Yaw Moment Change by Braking Force for Each Wheel [24] .......................................... 10

Figure 3-1 The Free Body Diagram about Yaw Motion of a SUV ...................................................... 13

Figure 3-2 The Free Body Diagram about Roll Motion of a SUV ..................................................... 13

Figure 3-3 Schematic of Tire Operating at a Slip Angle ..................................................................... 16

Figure 3-4 Schematic of a Steer-By-Wire System .............................................................................. 21

Figure 3-5 Input and Output Relationship between Vehicle and Steering/Brake Model ..................... 24

Figure 3-6 Ackermann Steering Geometry for 4 Wheels (dotted) and 2 Wheels (dashed) Vehicle .... 26

Figure 4-1 Yaw Stability Controller Structure for Active Steering ..................................................... 30

Figure 4-2 Yaw Stability Controller Structure for Differential Braking .............................................. 31

Figure 4-3 Yaw Stability Controller Structure for Integrated Control ................................................. 32

Figure 4-4 Rollover Avoidance Controller Structure for Integrated Control....................................... 33

Figure 4-5 Integrated Yaw and Rollover Controller Structure for Reference Matching Control ........ 34

Figure 5-1 Test Road Profile for J-Turn .............................................................................................. 36

Figure 5-2 J-Turn Input vs. Time ......................................................................................................... 37

Figure 5-3 Sinusoidal Input vs. Time................................................................................................... 37

Figure 5-4 Fishhook Maneuver vs. Time ............................................................................................. 38

Figure 5-5 Vehicle Trajectory at 60 km/h (Linear) .............................................................................. 41

Figure 5-6 Magnified Vehicle Trajectory at 60 km/h (Linear) ............................................................ 42

Figure 5-7 Tire Turning Angle vs. Time (Linear)............................................................................... 42

Figure 5-8 Moment Input vs. Time (Linear) ........................................................................................ 43

Figure 5-9 Slip Angle vs. Time (Linear) .............................................................................................. 43

Figure 5-10 Yaw Rate vs. Time (Linear) ............................................................................................. 44

Figure 5-11 Roll Angle of Sprung Mass vs. Time (Linear) ................................................................. 44

Figure 5-12 Rollover Coefficient vs. Time (Linear) ............................................................................ 45

Figure 5-13 Vehicle Trajectory at 60 km/h (Linear) ............................................................................ 46

Figure 5-14 Magnified Vehicle Trajectory at 60 km/h (Linear) .......................................................... 47

Figure 5-15 Tire Turning Angle vs. Time (Linear)............................................................................. 47

Figure 5-16 Moment Input vs. Time (Linear) ...................................................................................... 48

ix

Figure 5-17 Slip Angle vs. Time (Linear) ............................................................................................ 48

Figure 5-18 Yaw Rate vs. time (Linear) ............................................................................................... 49

Figure 5-19 Roll Angle vs. Time (Linear) ............................................................................................ 49

Figure 5-20 Rollover Coefficient vs. Time (Linear) ............................................................................ 50

Figure 5-21 Vehicle Trajectory at 60 km/h (Nonlinear) ....................................................................... 52

Figure 5-22 Magnified Vehicle Trajectory at 60 km/h (Nonlinear) ..................................................... 53

Figure 5-23 Handwheel Input and Tire Turning Angle vs. Time (Nonlinear) .................................... 53

Figure 5-24 Moment Input vs. Time (Nonlinear) ................................................................................. 54

Figure 5-25 Slip Angle vs. Time (Nonlinear)....................................................................................... 54

Figure 5-26 Yaw Rate vs. Time (Nonlinear) ........................................................................................ 55

Figure 5-27 Roll Angle vs. Time (Nonlinear) ...................................................................................... 55

Figure 5-28 Rollover Coefficient vs. Time (Nonlinear) ....................................................................... 56

Figure 5-29 Vehicle Trajectory at 60 km/h (Nonlinear) ....................................................................... 57

Figure 5-30 Magnified Vehicle Trajectory at 60 km/h (Nonlinear) ..................................................... 58

Figure 5-31 Tire Turning Angle vs. Time (Nonlinear) ....................................................................... 58

Figure 5-32 Moment Input vs. Time (Nonlinear) ................................................................................. 59

Figure 5-33 Slip Angle vs. Time (Nonlinear)....................................................................................... 59

Figure 5-34 Yaw Rate vs. Time (Nonlinear) ........................................................................................ 60

Figure 5-35 Roll Angle vs. Time (Nonlinear) ...................................................................................... 60

Figure 5-36 Rollover Coefficient vs. Time (Nonlinear) ....................................................................... 61

Figure 5-37 Vehicle Trajectory at 60 km/h (Nonlinear vs. Linear) ...................................................... 62

Figure 5-38 Magnified Vehicle Trajectory at 60 km/h (Nonlinear vs. Linear) .................................... 63

Figure 5-39 Tire Turning Angle vs. Time (Nonlinear vs. Linear) ....................................................... 63

Figure 5-40 Moment Input vs. Time (Nonlinear vs. Linear) ................................................................ 64

Figure 5-41 Slip Angle and Yaw Rate vs. Time (Nonlinear vs. Linear) .............................................. 64

Figure 5-42 Yaw Rate vs. Time (Nonlinear vs. Linear) ....................................................................... 65

Figure 5-43 Roll Angle vs. Time (Nonlinear vs. Linear) ..................................................................... 65

Figure 5-44 Rollover Coefficient vs. Time (Nonlinear vs. Linear) ...................................................... 66

Figure 5-45 Vehicle Trajectory at 60 km/h (Nonlinear vs. Linear) ...................................................... 67

Figure 5-46 Magnified Vehicle Trajectory at 60 km/h (Nonlinear vs. Linear) .................................... 67

Figure 5-47 Tire Turning Angle vs. Time (Nonlinear vs. Linear) ....................................................... 68

Figure 5-48 Moment Input vs. Time (Nonlinear vs. Linear) ................................................................ 68

x

Figure 5-49 Yaw Rate vs. Time (Nonlinear vs. Linear) ...................................................................... 69

Figure 5-50 Yaw Rate vs. Time (Nonlinear vs. Linear) ...................................................................... 69

Figure 5-51 Rollover Coefficient vs. Time (Nonlinear vs. Linear) ..................................................... 70

Figure 5-52 Rollover Coefficient vs. Time (Nonlinear vs. Linear) ..................................................... 70

Figure 5-53 Vehicle Trajectory at 100 km/h in the Dry Asphalt ......................................................... 72

Figure 5-54 Rollover Coefficient vs. Time .......................................................................................... 72

Figure 5-55 Tire Turning Angle vs. Time ............................................................................................ 73

Figure 5-56 Moment Input vs. Time .................................................................................................... 73

Figure 5-57 Slip Angle vs. Time .......................................................................................................... 74

Figure 5-58 Yaw Rate vs. Time ........................................................................................................... 74

Figure 5-59 Roll Angle of Sprung Mass vs. Time ............................................................................... 75

Figure 5-60 Vehicle Trajectory at 100 km/h in the Dry Asphalt ......................................................... 76

Figure 5-61 Rollover Coefficient vs. Time .......................................................................................... 77

Figure 5-62 Tire Turning Angle vs. Time ............................................................................................ 77

Figure 5-63 Moment Input vs. Time .................................................................................................... 78

Figure 5-64 Slip Angle vs. Time .......................................................................................................... 78

Figure 5-65 Yaw Rate vs. Time ........................................................................................................... 79

Figure 5-66 Roll Angle of Sprung Mass vs. Time ............................................................................... 79

Figure 6-1 ADAMS Wire Frame Model .............................................................................................. 81

Figure 6-2 Corner View of ADAMS Model ........................................................................................ 82

Figure 6-3 ADAMS road profile .......................................................................................................... 83

Figure 6-4 Longitudinal Trajectory and Velocity vs. Time ................................................................. 84

Figure 6-5 Trajectory for uncontrolled and controlled vehicle according to J-turn Input ................... 85

Figure 6-6 Trajectory for controlled vehicle according to J-turn Input ............................................... 86

Figure 6-7 Trajectory for uncontrolled vehicle according to J-turn Input ........................................... 87

Figure 6-8 Trajectory for uncontrolled and controlled vehicle according to Sinusoidal Input ............ 88

Figure 6-9 Trajectory for controlled vehicle according to Sinusoidal Input ........................................ 88

Figure 6-10 Trajectory for uncontrolled vehicle according to Sinusoidal Input .................................. 89

Figure A-1 Kinematics of Yaw-Roll Motion ....................................................................................... 97

Figure A-2 Kinematics About the Center of Curvature ..................................................................... 106

Figure A-3 Lateral Force vs. Slip angle of Tire ................................................................................. 110

Figure A-4 Aligning Moment vs. Slip Angle of Tire......................................................................... 111

xi

Figure A-5 Longitudinal Force vs. Longitudinal Percent Slip ........................................................... 112

Figure A-6 SIMULINK Model for Uncontrolled and Active Steering Controlled ............................ 114

Figure A-7 SIMULINK Model for Uncontrolled and Differential Braking Controlled..................... 115

Figure A-8 SIMULINK Model for Uncontrolled and Integrated Controlled ..................................... 116

Figure A-9 SIMULINK Model for Vehicle Plant .............................................................................. 117

Figure A-10 SIMULINK Model for Active Steering Controller ....................................................... 118

Figure A-11 SIMULINK Model for Differential Braking Controller (1) .......................................... 118

Figure A-12 SIMULINK Model for Differential Braking Controller (2) .......................................... 119

Figure A-13 ADAMS Plant Input Variable ........................................................................................ 120

Figure A-14 ADAMS Plant Output Variable ..................................................................................... 120

Figure A-15 ADAMS/Controls Plant Export to MATLAB ............................................................... 121

Figure A-16 ADAMS Sub Block Diagram ........................................................................................ 121

Figure A-17 SIMULINK Model for ADAMS/View Control ............................................................ 122

xii

List of Tables

Table 5-1 Vehicle Parameters……………………………………………………………………...…39

Table 7-1 Summary of Advantage and Disadvantage for Steering and Braking Control…………….91

xiii

Nomenclature

Symbol Explanation

a distance between COM of unsprung mass and front axle;

01a acceleration velocity vector of sprung mass with respect to initial frame;

12a acceleration vector of unsprung mass with respect to body fixed frame of sprung mass;

02a acceleration vector of unsprung mass with respect to initial frame;

b distance between COM of unsprung mass and rear axle;

amb actuator motor damping;

rpb rack and tire damping about steering axis;

fC front cornering stiffness;

rC rear cornering stiffness;

C roll stiffness of passive suspension;

d half of track width;

pD roll damping of passive suspension;

BF longitudinal brake force;

lfxF , longitudinal force at left front tire;

rfxF , longitudinal force at right front tire;

lrxF , longitudinal force at left rear tire;

rrxF , longitudinal force at right rear tire;

fxF , total longitudinal force at front tire;

rxF , total longitudinal force at rear tire;

COMyF , lateral external force acting on COM of unsprung mass;

lfyF , lateral force at left front tire;

rfyF , lateral force at right front tire;

lryF , lateral force at left rear tire;

rryF , lateral force at right rear tire;

fyF , total lateral force at front tire;

ryF , total lateral force at rear tire;

ltyF , total left lateral tire force;

rtyF , total right lateral tire force;

lfzF , normal force at left front tire;

rfzF , normal force at right front tire;

lrzF , normal force at left rear tire;

xiv

rrzF , normal force at right rear tire;

g acceleration due to gravity;

h height of the COM of sprung mass above the roll axis;

Rh roll center height over ground;

amI steering actuator motor operating current;

rpJ lumped inertia of steering rack and wheel about steering axis

sxJ , sprung mass moment of inertia around roll axis;

szJ , sprung mass moment of inertia around yaw axis;

uszJ , unsprung mass moment of inertia around yaw axis;

wJ tire wheel moment of inertia around wheel roll axis;

amk steering actuator motor current constant;

aemfk steering actuator motor back-EMF constant;

Bk brake scale factor;

tk steering actuator scale factor;

zk scale factor for the tire self-aligning moment

sl steering arm length;

L distance between front and rear axle;

m total mass of vehicle;

usm unsprung mass;

sm sprung mass;

dM direct yaw moment;

zM tire self-aligning moment;

rcxM , roll moment acting on roll center;

susxM , moment due to passive suspension;

COMzM , yaw moment acting on COM of unsprung mass;

p roll rate;

hydP hydraulic pressure command;

hydP resulting braking pressure;

r actual yaw rate;

desr desired yaw rate;

errr error yaw rate;

gr total gear ratio;

amsr gear reduction ratio for steering actuator motor;

R radius of curvature;

xv

amR steering actuator motor resistance;

wR tire wheel radius;

01r position vector of sprung mass with respect to initial frame;

12r position vector of unsprung mass with respect to sprung mass body fixed frame;

02r position vector of unsprung mass with respect to initial frame;

aT steering actuator output torque;

amT steering actuator motor torque;

tT torsion bar torque;

01T rotational matrix of sprung mass with respect to initial frame;

12T rotational matrix of unsprung mass with respect to sprung mass body fixed frame;

02T rotational matrix of unsprung mass with respect to initial frame;

v magnitude of vehicle velocity;

lfv tire velocity along with wheel centerline at left front tire;

rfv tire velocity along with wheel centerline at right front tire;

lrv tire velocity along with wheel centerline at left rear tire;

rrv tire velocity along with wheel centerline at right rear tire;

xv longitudinal velocity;

yv lateral velocity;

amV steering actuator motor operating voltage;

01v velocity vector of sprung mass with respect to initial frame;

12v velocity vector of unsprung mass with respect to sprung mass body fixed frame;

02v velocity vector of unsprung mass with respect to initial frame;

lfα slip angle at left front tire;

rfα slip angle at right front tire;

lrα slip angle at left rear tire;

rrα slip angle at right rear tire;

01α angular acceleration vector of sprung mass with respect to initial frame;

12α angular acceleration vector of unsprung mass with respect to sprung mass body fixed

frame;

02α angular acceleration vector of unsprung mass with respect to initial frame;

β sideslip angle of at COM;

lfβ side slip angle at left front tire;

rfβ side slip angle at right front tire;

lrβ side slip angle at left rear tire;

xvi

rrβ side slip angle at right rear tire;

δ steer angle of front tire;

iδ steer angle of inner tire;

oδ steer angle of outer tire;

amη gear head efficiency for steering actuator motor;

hθ hand wheel angle;

pθ pinion rotation angle;

tθ tire turning angle;

sΘ inertia matrix of sprung mass;

usΘ inertia matrix of unsprung mass;

μ road adhesion coefficient;

roll angle;

lfσ percent longitudinal slip angle at left front tire;

rfσ percent longitudinal slip angle at right front tire;

lrσ percent longitudinal slip angle at left rear tire;

rrσ percent longitudinal slip angle at right rear tire;

lfτ driving moment for left front tire;

rfτ driving moment for right front tire;

lrτ driving moment for left rear tire;

rrτ driving moment for right rear tire;

ψ yaw angle;

amω steering actuator motor angular velocity;

01ω angular velocity vector of sprung mass with respect to initial frame;

12ω angular velocity vector of unsprung mass with respect to sprung mass body fixed

frame;

02ω angular velocity vector of unsprung mass with respect to initial frame;

1

Chapter 1

Introduction

While significant progress has been made in optimizing the performance and function of

ground vehicles, an active safety technology that can enhance a vehicle‟s ability to respond

to and avoid potentially dangerous situations has not been received much attention. In a

typical driving situation, such as a slow lane change on a dry road, the controllability of the

vehicle does not depend on the driver‟s experience because the driver can easily control the

vehicle‟s behavior. However, in a more severe or unexpected situation, such as a quick lane

change to avoid an obstacle on an icy road or highway, the driver is forced to react quickly

and control the vehicle through his/her experience. In these extreme situations, many drivers,

despite their level of experience, can lose control of the vehicle, causing in motor vehicle

accidents, such as rollover or collision, and often resulting in human fatalities.

To reduce the danger of motor vehicle accidents, two major avenues of safety measures

have been proposed: active safety and passive safety. Active safety represents preventive

measures to reduce the possibility of accidents or crashes, while passive safety means

geometric adjustment on the suspension system or reactive measures, such as air bags, to

reduce the severity of injuries. In the last three decades, passive safety measures have been

favored over active safety measures due to the high cost and difficulty in implementation.

Active safety is now being considered for two major reasons. First, new trends in motor

vehicle construction, such as weight reduction, and the popularity of SUVs (Sport Utility

Vehicles) have reduced the viability and effectiveness of passive safety measures. Second,

human error, the largest contributing factor in 75% of motor vehicle accidents, cannot be

mitigated by passive safety techniques [1].

In considering types of active controls to fulfill the need for active safety measures, two

technologies have been identified. On the one hand, active braking control has been

suggested. For instance, ABS/TCS (Antilock Brake System/Traction Control System) control

is capable of maintaining tire braking and traction forces near their maximum value to

control longitudinal motion. Furthermore, VDC/VSC (Vehicle Dynamic Control/Vehicle

2

Stability Control) is mainly considered to control the yaw moment by generating differential

longitudinal forces on left and right tires. From tire dynamics, longitudinal force usually has

a margin to its saturation, even when lateral forces are near saturation. Under such

conditions, controlling longitudinal forces is an effective way to influence vehicle yaw

characteristics, thereby influencing the lateral motion of the vehicle, which can cause

skidding on a curved or slippery road [2]. On the other hand, active steering control has also

been proposed. For example, the main purpose of 4WS (4 Wheel Steering System), which

was introduced in early 1980s, is to minimize the vehicle side slip and the yaw rate.

However, 4WS has only been implemented in small volumes because, despite its useful

performance, the increased vehicle cost did not appeal to general customers. Recently, active

steering, which generates a compensating torque for yaw disturbances, was proposed as

driver assistance system for vehicle dynamics. The active steering system can reduce an

unexpected deviation between the desired yaw rate and the actual yaw rate by compensating

a small angle for the steering wheel angle, which is generated by drivers based on experience

[3].

Most recently, drive-by-wire (or x-by-wire) has been considered a state-of-the-art

technology to achieve active safety demands. This concept, as with many vehicles‟ active

technologies, comes from airplane technology (i.e. fly-by-wire) [4, 5]. Drive-by-wire, in

terms of braking, steering, throttling and other vehicle functions, is performed without the

usual brake booster, steering shaft, rack and pinion gear, throttle cable and linkages, etc.

Mechanical linkages will be replaced with a small electric-mechanical module to generate the

reaction force or torque. These systems communicate by electrical wires or via wireless

transmission technology. Information from sensors measured throughout the vehicle is

passed to an electronic control module. This control unit provides the input signal to operate

an actuator, which performs the mechanical function of the system.

One of the most important applications of x-by-wire is the SBW (steer-by-wire) system, in

which the conventional steering system using the traditional mechanical linkages and

hydraulics is replaced with an electrical equivalent, such as sensors, actuators, and a digital

controller. SBW provides car manufacturers more flexibility in designing the vehicle interior

3

due to extra space available by removing the linkages. This will provide a safer passenger

compartment in the event of an accident. The advantages of SBW are: (a) Increased safety,

(b) Increased design flexibility, (c) Reduced labour and inventory costs due to eliminating

mechanical linkages and pipes, and (d) Reduced lead time to develop due to application of

software instead of hardware [6, 7]. On account of these potential benefits, SBW has

received considerable attention from both the automotive industry and academic research

institutions.

The U.S National Highway Traffic Safety Administration (NHTSA) reported that

approximately 10% of fatal accidents were the result of non-collision crashes and that among

these rollover was involved in approximately 90% of non-collision fatal crashes.

Furthermore, the average percentage of rollover occurrence in fatal accidents was

significantly higher than in other type of accidents. In comparison to other types of vehicles,

SUVs had the highest rollover rates because they are constructed with higher ground

clearance. A report by NHTSA showed in 1999 that the probability of fatality is 2-4 times

greater when a car is struck by an SUV or light truck than when struck by a passenger car.

Due to the increasing popularity of SUVs, the percentage of fatal rollover crashes is also

significantly increasing. To avoid these fatal rollover accidents, differential braking is

considered the most effective way to manipulate tire forces to reduce the lateral acceleration

of the vehicle. Differential braking can also reduce the forward speed that contributes to the

lateral acceleration of the vehicle, which cannot be reduced by steering control [8].

Based on a review of background information, integrated SBW and differential braking to

improve handling performance and prevent rollover occurrence in SUVs is proposed in this

thesis. SBW and differential braking are viable measures because of their potential benefits.

A reliable and robust controller must be designed that is able to overcome the present

hurdles, such as anxiety due to lack of mechanical linkage and malfunctions. With regard to

the controlled region, the yaw motions that cause skidding and the roll motions that cause

rollover need significant consideration.

4

The ultimate goal of this thesis is to develop a new vehicle performance/safety method

using combined active steering and differential braking in order to achieve the yaw and roll

motion stability in uncertain environments. Active steering has an immediate effect on the

vehicle‟s yaw and roll dynamics; however, it also causes a large trajectory deviation from the

desired course. Differential braking has less influence on the vehicle‟s yaw and roll dynamics

under normal driving conditions; however, it can reduce not only the vehicle‟s yaw on a low

friction road, but also the vehicle‟s rollover by using reduced longitudinal velocity. The

integrated yaw stability and rollover avoidance using active steering and differential braking

control was proposed and the proposed control method can improve the vehicle stability and

reduce the risk of accidents by taking advantages of active steering/differential braking and

complementing the disadvantages of them. Due to the increasing popularity of SUVs, which

have the highest rollover rates, the proposed control method will be verified in a SUV. The

integrated active steering and differential braking control are shown to be most efficient in

achieving yaw stability and rollover avoidance, while individual control (active steering or

differential braking control) or individual goal (yaw stability or rollover avoidance) has been

shown only to improve a partial and limited portion of vehicle performance and safety.

The remainder of this thesis is organized as follows: a review of SBW technology and yaw

and roll control using active steering and differential braking is presented in Chapter 2. In

Chapter 3, a four degree-of-freedom model of an SUV model including yaw and roll motions

is defined and presented. The control and controller design are presented in Chapter 4. In this

chapter, the kinematic tire model and the rollover coefficient are used to eliminate the yaw

rate error and reduce the risk of rollover. The yaw stability, the rollover avoidance, and the

combined yaw stability and rollover control are presented. The simulation results and

discussions are presented in Chapter 5. In this chapter, both linear and nonlinear simulation

results are reviewed based on several maneuvers, such as J-turn, sinusoidal, and fishhook. In

Chapter 6, the ADAMS model is evaluated by the proposed controller. Finally, the

advantages and disadvantages of steering control and differential braking control are

discussed and concluded in Chapter 7.

5

Chapter 2

Literature Review

2.1 Steer-by-Wire

Drive-by-wire is a rapidly developing technology in the field of vehicle dynamics control for

active safety and driving comfort. This technology will dramatically change the lifestyle of

most people who rely on ground vehicles. As major subsystems of drive-by-wire, steering,

braking, throttling and other functions can be performed by on-board computer and in-

vehicle networks instead of by traditional mechanical linkages. One of the most important

subsystems of drive-by-wire is steer-by-wire, in which the conventional steering system

using traditional mechanical linkages and hydraulics is replaced with an electrical equivalent.

Figure 2-1 shows a conventional power steering system and steer-by-wire system.

Figure 2-1 Comparison of Conventional Power Steering System and Steer-by-Wire System [27]

Figure 2-2 represents a conceptual design for a steer-by-wire system. The system can be

subdivided into three major parts: a controller, a hand wheel subsystem, and a road wheel

subsystem. An actuator in the hand wheel system provides road feedback to the driver. This

6

is also commanded by the controller and is based on information provided by a sensor in the

road wheel system [9].

Figure 2-2 Architecture of Steer-by-Wire [6]

The essential function of the conventional steering system of an automobile is manual

positioning, which means the driver should feel the road conditions and act through a linkage

mechanism. The driver must sense all demands for steering actions such as keeping within

driver‟s lane and avoiding obstacles and must apply all input steering commands to the

steering wheel. However, it may be difficult for the driver to sense and control unexpected

deviations from the established path due to external lateral forces e.g. wind gusts or road

irregularities. In addition to driver control, a restoring torque that acts to keep the steering

wheel on a straight path is generated by geometrical effects. However, this has nonlinearity

issues due to various parameters such as road adhesion coefficient and irregularity of road

surface. One of the main issues that need to be addressed is how to supply control input to

electronic module through sensors despite the lack of connection.

7

Due to these difficulties and nonlinearities, the design of the controller is focused on

robustness and adaptive control. In the view of robustness and stability analysis, a generic

controller with bi-directional position feedback was proposed. The design goals are to match

the dynamic of hydraulic or electric power steering, which may notionally be subdivided into

a manual and an assistance steering part [10].

Most recently, an artificial neural network controller was developed for steer-by-wire of an

intelligent vehicle. This was shown to be necessary to appreciate and understand the

modelling difficulties and complexities inherent in modelling such steering systems. These

difficulties and complexities were also shown to underpin the benefits and efficacy of neural-

network control techniques for steer-by-wire system. By considering the nonlinearity of

steer-by-wire systems, a neural-network model-reference adaptive control technique was

proposed [11].

2.2 Active Steering

In 1969, Kasselmann and Keranen [12] proposed an active steering system that measures the

yaw rate by a gyroscope and uses a proportional feedback controller to generate an additive

steering input for the front wheels. Figure 2-3 illustrates the concept of an additional steering

angle actuation principle as an active control. This early study never made it to a product

testing. Some of its ideas are, however, still relevant for actual and future active steering

systems. Actuators for adding a feedback controlled steering angle to the driver commanded

steering angle might be placed in the rotational motion of the steering column or in the lateral

motion of the steering linkage.

In the early 1980s, studies by Ackermann contributed a robust steering control law, where

robustness refers to variations of operating conditions. Ackermann also suggested that the

robustness and tracking accuracy could be drastically improved by additional feedback of the

yaw rate to the steering actuator [14, 15]. During the 1980s and early 90s, 4WS became a hot

topic in steering control dynamics. Furukawa et al [16] reviewed from the viewpoint of

vehicle dynamics and control. Typically, the front wheel steering was unchanged and a

hydraulic or electric actuator for additional rear-wheel steering was used. Initially, only feed

8

forward control laws (some with gains scheduling) were employed; later, feedback from

vehicle dynamics sensors was also introduced in order to reduce the effect of disturbance

torques and parameter uncertainty, an example is the work by Hirano and Fukatani [17].

Figure 2-3 Concept of Additional Steering Angle Actuation Principle [13]

In 1990, Ackermann [18] proposed a concept for feedback of the yaw rate to active front

and rear wheel steering. His design goal was a clear separation of the track following task of

the driver from the automatic stabilization that balances disturbance torques around a vertical

axis. This goal is achieved in a robust way, i.e. independent of the operating conditions. The

robust decoupling effect is obtained by integral feedback of the yaw rate to front-wheel

steering.

Recently, Huh and Kim [19] proposed an active steering control method based on the

estimated lateral tire forces. In order to estimate the lateral force, a monitoring model using

the vehicle dynamic model and roll motion is utilized. The estimated force is compared with

the optimal reference force and is compensated by a controller operated by a fuzzy logic

controller. Front wheel active steering leads to additional lateral forces, which can be used in

order to reject yaw and roll torque disturbances that a rise due to slippery roads or from

9

asymmetric braking and wind forces. This is effective even on decreased road adhesion

conditions [20].

Active steering also has applications in rollover avoidance for vehicles with an elevated

center of gravity, as with the growing market of SUVs. In addition to the driver‟s steering

angle, a small auxiliary steering angle is set by an actuator. The control law is based on

proportional feedback of the roll rate and roll acceleration. Feedback of the roll rate and the

roll acceleration was shown to help reducing the transient rollover risk. Track Width Ratio

(TWR is the ratio of the half track width to the height of the center of gravity) was used in

order to decide whether or not a small auxiliary angle was needed [21].

2.3 Differential Braking

To reduce accidents, braking (or stopping) distance is considered as a key role to prevent

accidents during pure braking condition on low μ road. ABS that can avoid wheel lock-up

and use maximum brake force was suggested as a solution in the 1950s. Eventually, most

vehicles were equipped with ABS in the late 1980s. However, under a cornering situation on

an icy road, the vehicle is involved in a more complicated situation. If the inner brake force is

greater than the outer brake force, then the vehicle spins to inward direction. Alternatively, if

the outer brake force is greater than the inner brake force, then the vehicle drifts in an

outward direction. In these situations, an SUV with an elevated center of gravity of sprung

mass could be subjected to an increased chance of rollover accidents. Differential braking

control was proposed as the most viable means to avoid rollover accidents.

In the late 1980s, an independently controlled braking force between inner and outer rear

wheels was proposed [22]. Yaw acceleration patterns were examined in terms of peaks and

troughs based on experiments during low and high deceleration braking. In the case of spin,

they controlled rear brake force to reduce yaw moments by braking force distribution

method, which reduced inner brake force and increased outer brake force. In a drift out

situation, yaw was reduced by preventing the lock-up phenomenon of the front wheel by

using ABS.

10

Extended to distribution control, Matsumoto et al [23] introduced four wheel braking

distribution control. A target yaw rate is set based on the steering wheel angle and vehicle

speed, and yaw rate feedback control was performed to modulate the distribution of braking

force. These investigations showed that the left-right distribution control is more effective

than the front-rear distribution control in order to control yaw moment.

Furthermore, Vehicle Stability Control (VSC) was proposed by active brake control in

limit cornering [24]. They showed that the change in vehicle yaw moment was caused by the

application of braking force to each wheel, as shown in Figure 2-4. It was claimed that the

application of a large outward yaw moment is effective for the case where the vehicle

becomes unstable with the sudden increase of side slip angle. It was also suggested that the

application of a proper inward yaw moment while decelerating the vehicle is effective for the

case where the course trace becomes difficult due to the saturation of the front wheel

cornering force.

Ffr

Ffl

Frl

Frr

Ya

w M

om

en

td

rift

ou

ts

pin Frl Frr

Ffr

Ffl

Braking Force

Figure 2-4 Yaw Moment Change by Braking Force for Each Wheel [24]

Anti-rollover braking for vans was proposed in [25]. A rollover was examined and

categorized from first to fourth turns in general. They found that a rollover accident could

11

occur if lateral acceleration is over a threshold. At this point, differential braking instead of

full braking will reduce the lateral acceleration and prevent rollover accidents.

More recently, differential-braking-based rollover prevention for SUVs was proposed [8].

In contrast to most existing rollover warning systems, which are based on signal threshold

techniques, they introduced time-to-rollover matrices, which are measures that estimate time

just before rollover. The root-locus technique was used to design the feedback control of

differential braking. The lateral acceleration was selected because of its consistent root-locus

pattern in wide range of speed.

Active steering has been considered an efficient means to influence the vehicle‟s yaw

dynamics because it has an immediate effect on the vehicle‟s yaw dynamics. For example,

4WS was invented to make it possible to control yaw rate and designed based on the concept

of zero-side slip. Driver assistance systems, which produce a compensating torque for yaw

disturbances, were also suggested and commercialized. Active steering has also been

considered as an effective way to reduce the risk of rollover. However, it also causes a large

trajectory deviation from the desired courses because the desired steering angle will be

changed. Differential braking can also reduce the risk of rollover by using reduced

longitudinal velocity. However, it has less influence on the vehicle‟s yaw and roll dynamics

under normal driving conditions.

Active steering and differential braking control both have advantages and disadvantages:

Active steering is more efficient but it causes trajectory deviation. Differential braking on the

other hand causes less trajectory deviation but not as effective as active steering. With the

increasing popularity of SUVs, a new vehicle performance/safety control method using

combined advantages of active steering and differential braking control is needed to improve

the vehicle‟s stability in terms of yaw and roll dynamics.

12

Chapter 3

Sport Utility Vehicle Model

A four degrees-of-freedom is considered for the vehicle to study rollover and yaw motion.

This simplified model will be sufficient for understanding the affects of differential braking

and active steering on the vehicle stability. In order to use differential braking and active

steering in the vehicle stability, the models of the braking and steering systems are also

derived.

3.1 Four Degree of Freedom Vehicle Model

The model used in this proposal is similar to the one introduced in [28]. Figure 3-1 and

Figure 3-2 illustrate the free body diagrams of a SUV for the yaw and roll motions,

respectively. The yaw motion can be presented as XY plane as shown in Figure 3-1. The roll

motion can be represented as YZ plane as shown in Figure 3-2. The vehicle model consists of

two rigid bodies; the unsprung mass and the sprung mass. The unsprung mass is a

composition of the front and rear axles, the four tire wheels and the frame. The sprung mass

is a composition of the chassis and the body. The sprung mass is linked to the unsprung mass

with a one degree-of-freedom joint, where the axis of rotation is the roll axis. The roll axis is

assumed to be a fixed axis parallel to the ground in the vehicle‟s longitudinal direction. The

roll movement of the sprung mass is damped and sprung by a passive suspension system,

which is represented as the rotational spring and damper. The CG of unsprung mass is

assumed to be in the road plane, since the contribution to the roll movement is considered to

be negligible. The four wheel planar model [29] is taken for the unsprung mass in order to

represent the main features of vehicle steering dynamics in a horizontal plane. The multibody

system describes the vehicle‟s longitudinal, lateral, yaw and roll dynamics.

13

O0X0

Y0

X1F

y,lf

Fx,lf

Fx,rf

Fy,rf

Fy,rr

Fy,lr

Fx,rr

Fx,lr

2d

ba

V

f

r

r

O1

Y1

f

r

L

Figure 3-1 The Free Body Diagram about Yaw Motion of a SUV

Y1

Z1

O1

Y2

Z2

O2

msg

musg

hR

h

Fy,l

Fy,r

Fz,l

Fz,r

Mx,sus

Roll Axis

msa

y,s

2d

Figure 3-2 The Free Body Diagram about Roll Motion of a SUV

14

Appling the Newton‟s second law to the free body diagram in Figure 3-1and Figure 3-2,

the nonlinear equations of vehicle motions can be obtained. The equations are written based

on the coordinates attached to the vehicle. Defining:

Tyx ψz (1)

Tprvv yxz (2)

where x and y are the displacement components in longitudinal and lateral direction, ψ

and r are the yaw angle and yaw rate of unsprung mass, and p are the roll angle and roll

rate of sprung mass, respectively.

The nonlinear second order differential equations of the vehicle can be written as:

),,(),()( uzzQzzkzzm (3)

where 44)( xzm is symmetric positive definite mass matrix, 14),( x

zzk is

generalized gyroscope and centrifugal vector, 14),,( xuzzQ is generalized active force

vector. )(zm , ),( zzk and ),,( uzzQ are:

ssxs

s

ssus

ssus

mhJhm

hm

hmmm

hmmm

2

,

)3,3(

cos0

0sin

cos00

0sin0

)(m

zm (4)

where: 22

,,,)3,3( sin)(cos ssxszusz mhJJJm

sin(cos

cos(2sin

sin)()(

cos2)(

),(

2

,,

2

,,

2

rmhJJvhmr

pmhJJvhmr

rphmrvmm

prhmrvmm

sszsxxs

sszsxys

sxsus

sysus

zzk (5)

15

x,suss

y,fy,rx,ltx,rt

y,ry,f

x,rx,f

Mhm

aFbF)Fd(F

FF

FF

sin

),,(

g

uzzQ (6)

where x,fF is total longitudinal force at front tire, x,rF is total longitudinal force at rear tire,

fyF , is total lateral force at front tire and ryF , is total lateral force at rear tire. ltyF , is total

left lateral tire force, rtyF , is total right lateral tire force, and susxM , is moment due to passive

suspension. They can be written as:

δ)F(Fδ)F(FF y,rfy,lfx,rfx,lfx,f sincos (7)

)F(FF x,rrx,lrx,r (8)

δ)F(Fδ)F(FF x,rfx,lfy,rfy,lfy,f sincos (9)

)F(FF y,rry,lry,r (10)

δFδFFF y,lfx,lfx,lrx,lt sincos (11)

δFδFFF y,rfx,rfx,rrx,rt sincos (12)

pDCM px,sus (13)

The forces can be expressed as jkiF , , where subscript i represents the longitudinal force

when i = x and the lateral force when i = y. Furthermore, subscript j represents the left tire

force when j = l and the right tire force when j = r and subscript k represents the front tire

force when k = f and the rear tire force when k = r. For example, lfxF , represents the

longitudinal left front tire force. Furthermore, C is roll stiffness of passive suspension and

pD is roll damping of passive suspension.

Also, mus is unsprung mass, ms is sprung mass, Jz,us is unsprung mass moment of inertia

around yaw axis, Jz,s is sprung mass moment of inertia around yaw axis, Jx,s is sprung mass

16

moment of inertia around roll axis, h is height of the center of mass (COM) of sprung mass

above the roll axis, respectively. The details of the derivation of the equation are also given

in Appendix A.

In order to estimate the tire forces, a linear tire model is used. A rolling tire travels straight

along the wheel plane if no side forces occur. During cornering, however, the tire contacts

slip laterally while rolling such that its motion is no longer in the straight direction. The angle

between its direction of motion and the wheel plane is referred to as the slip angle. This slip

angle generates a lateral force at the tire-ground interface and an aligning moment because

the force acts slightly behind the center of the wheel. Figure 3-3 illustrates schematic of

tire operating at a slip angle.

Slip Angle( )

Velocity

Side Force ( F y )

Brake Force ( F x )

Self Aligning Torque ( M z )

Figure 3-3 Schematic of Tire Operating at a Slip Angle

The lateral tire force can be assumed as a function of cornering stiffness and slip angle of

the tire. The tire self-aligning moment can be approximated as a function of slip angle. This

approximation is valid for small slip angles and steady-state conditions. These can be

represented as follows:

fffyrfylfy C αμ22 ,,, FFF (14)

rrryrrylry C αμ22 ,,, FFF (15)

17

zz kM (16)

where μ is road adhesion coefficient, fC is cornering stiffness of front tire, rC is cornering

stiffness of rear tire, fα is slip angle of front tire, rα is slip angle of rear tire. The lateral

forces in the left and right tires are assumed the same. zk is the scale factor for the tire self-

aligning moment. The self-aligning moment is assumed to be small in vehicle dynamics and

can be neglected because it can not affect the vehicle behavior. However, this moment should

be considered in steering system.

A more simplified model of slip angle can be driven from the bicycle model. The slip

angle can be linearized as follows:

βδα f (17)

βα r (18)

For the sake of simplicity, the longitudinal percentage slip is neglected. However, the brake

force produced by a brake system should be considered. The total brake force, which can be

produce by the left or right brake system, is assumed as acting on the center of mass of

unsprung mass.

There have been many efforts to describe the tire‟s behavior beyond the linear region. The

Fiala and University of Arizona models are simple nonlinear models calculating the tire

forces based on basic tire properties. The Smithers and the Delft (“Magic Formula”) models

are more complex models calculating the tire forces based on coefficients from experiments.

The “Magic Formula”, widely recognized for its accuracy, is based on empirical data-fitting

method developed by H. Pacejka [31, 32] at the Netherlands‟ Delft University of Technology.

He showed that the lateral force and aligning moment are functions of slip angle and

longitudinal force is a function of longitudinal slip. In this study, the linear tire model is used

to derive at a complete linear model of the vehicle and the “Magic Formula” is used to

nonlinear simulation. A detailed investigation of the Magic Formula is explained in

Appendix B.

18

Using the linear tire model and assuming small angle as sincos , the system

Equation (3) can be simplified and linearized. Since main purpose of this study is roll and

yaw control, the state variables of Equation (3) can be simplified to:

TT

prxxxx β4321x (19)

where xy vvβ is vehicle slip angle, r is yaw rate, p is roll rate and is roll angle,

respectively. Using Equations (19) and (3), the linearized state space equation of the vehicle

becomes the descriptor state space equation as follows:

2211 uu uu GGFxxE (20)

where

1000

00

000

00

2

,

,,

ssxxs

szusz

sx

mhJvhm

JJ

hmmv

E (21)

0100

0

00/)(μ)(μ

00/)(μ)(μ22

hmCDvhm

vbCaCbCaC

mvvbCaCCC

spxs

xrfrf

xxrfrf

gF (22)

T

ffu aCC 001G (23)

T

u 00102G (24)

δ1u is steering angle as the first input and dMu2 is the direct yaw moment input due to

braking force between the right and left brake as the second input. Equation (20) can be also

written as state space presentation as follows:

2211 uu uu BBAxx (25)

where

19

0100

34333231

24232221

14131211

1

aaaa

aaaa

aaaa

FEA (26)

T

uu bbbb 413121111

1

1 GEB (27)

T

uu bbbb 423222122

1

2 GEB (28)

All elements of 21 ,, uu BBA as a function of vehicle inertia and dimension properties are

given as follows:

T

xsssx

ss

xsssx

ps

sssx

sssx

xsssx

rfssx

xsssx

rfssx

T

vmhmmhmJ

hmChm

vmhmmhmJ

Dhm

mhmmhmJ

mhmmhmJ

vmhmmhmJ

bCaCmhJ

vmhmmhmJ

CCmhJ

a

a

a

a

)(

)(

)(

)(

)(μ)(

)(

)(μ)(

222

,

222

,

222

,

222

,

2222

,

2

,

222

,

2

,

14

13

12

11

g

(29)

T

xszusz

rf

szusz

rfT

vJJ

CbCa

JJ

bCaC

a

a

a

a

0

0

)(

)(μ

)(μ

,,

22

,,

24

23

22

21

(30)

20

T

xsssx

s

xsssx

p

xsssx

rfs

sssx

rfs

T

vmhmmhmJ

hmCm

vmhmmhmJ

mD

vmhmmhmJ

bCaChm

mhmmhmJ

CChm

a

a

a

a

)(

)(

)(

)(

)(μ

)(

)(μ

222

,

222

,

222

,

222

,

34

33

32

31

g

(31)

0

)(

μ

μ

)(

μ)(

222

,

,,

222

,

2

,

41

31

21

11

sssx

fs

szusz

f

xsssx

fssx

mhmmhmJ

Chm

JJ

aC

vmhmmhmJ

CmhJ

b

b

b

b

(32)

0

0

10

,,

42

32

22

12

szusz JJ

b

b

b

b

(33)

3.2 Steering System Model

As we discussed early in Section 2.1, the conventional steering system with the traditional

mechanical linkages and hydraulics is replaced with an electrical system with sensors,

actuators, and a controller. Figure 3-4 shows the schematic diagram of SBW system, which

comprises three major subsystems: a controller, a hand wheel subsystem, and a road wheel

subsystem. The basic mechanism of steering system is a rack and pinion configuration with

an electrical actuator.

21

p,

Mz

Jh

Handwheel

Handwheel

Feedback Motor

Angle sensor

Angle sensor

Steering Actuator

Motor

Controller

t

h

Jrp

brp

Tam

Trm

h

rm

am

p

Torque sensor

Jrm

Jam

kam

Figure 3-4 Schematic of a Steer-By-Wire System

In the hand wheel subsystem, the key component is the handwheel feedback motor. The

purpose of the handwheel feedback motor is to communicate to the driver via tactile means

the direction and the level of forces acting between the tires and the road. A by-product of

these forces is the self-centering effect that occur when the driver releases the steering wheel

while existing a turn. Both the self-centering effect and torque feedback are important

characteristics that a driver expects to feel as same as a conventional steering system. The

force feedback actuator is assumed by a brushless DC motor. The dynamics between

22

handwheel and tire can be neglected because no mechanical linkage between handwheel and

tire exists.

Controller receives the angle of handwheel/pinion and pinion torque and gives current in

order to generate the realistic torque and assist torque. There can be used two kinds of sensor;

angle sensor and torque sensor. Two rotary position sensors – one on the steering column and

the other one on the pinion – provide absolute measurements of both angles. One torque

sensor, which is attached at the pinion, measures the torque as a feedback to the handwheel.

This torque signal is a basis how much torque the system should be supplied. However, a

controller and hand wheel subsystem dynamics are not considered because this thesis focuses

on the overall vehicle dynamics.

The dynamics between the steering actuator and tire can be considered as:

zprpprp MTbJ a (34)

where p is the pinion angle, rpJ is the total moment of inertia of the steering system, rpb is

the viscous damping of the steering system, aT is the actuator torque, and zM is the tire self-

aligning moment, respectively. The nonlinear terms of the differential equation of the

steering dynamics are the tire self-aligning moment because of the tire nonlinear

characteristic. The tire self-aligning moment can be linearized as described in Equation (16).

This approximation is valid for small slip angles and steady-state conditions. Furthermore,

p can be expressed as the tire angle as:

gp r (35)

where gr is the total gear ratio.

Substituting Equation (16)) and (35) into Equation (34) and arranging the equation yields

the 2nd

order ordinary differential equation of the steering system:

agtgrpgrp TrkrbrJ (36)

23

3.3 Brake System Model

The most common way to implement the differential braking technique is to employ the

existing ABS in vehicles. There are three major factors in a hydraulic ABS system: 1) the

saturation effect of the ABS in which brake pressure to the wheel is limited to prescribed

wheel slip and acceleration, 2) the overall brake gain from hydraulic pressure to brake force,

and 3) a dynamic lag term introduced to represent the hydraulic system response to an input

signal. Rotational wheel effects are not considered.

The saturation effect of the ABS is the result of seeking a control performance where the

maximum longitudinal braking force is imparted to the road without excessive longitudinal

slip and wheel acceleration. The saturation threshold can be assumed to be constant for a

given tire load and road adhesion coefficient.

The overall brake gain represents a scalar value based on the physical dimension of the

hydraulic system with the assumption that disk brakes are used. The brake gain describes the

steady state gain from the desired hydraulic brake pressure in the disk brake caliper cylinder

to an ideal longitudinal braking force applied at the tire/road interface. This ideal brake force

may not be attainable due to road surface conditions and vertical tire loading.

The hydraulic system response is modeled as a first order lag with time constant 0.2 [8, 32]

such that

hydhyd PP

2.0

1 (37)

where hydP is hydraulic pressure command and is hydP the resulting braking pressure. The

model can be interpreted as the dynamics lag between a hydraulic pressure command and the

resulting brake pressure. Since the focus of this thesis is on the overall stability of a vehicle

and the time constant of the electronics part of an ABS system hydraulic dynamics, the

details of the ABS control can be omitted. Therefore, the resulting braking force can be

obtained from following equation.

hydBB PkF (38)

24

where BF is the longitudinal brake force and Bk is the brake scale factor.

Equation (37) and (38) are used individually for each tire. Therefore, the direct yaw moment,

which is the second input signal ( dMu2 ), can be calculated from as follows:

)( LXRXd FFdM (39)

where RXF and LXF are respective total right and left longitudinal tire forces.

The state space equations for the vehicle model, Equation (25), are obtained using the 2nd

order ordinary differential equation for the steering system model, Equation (37), and the 1st

order ordinary differential equation for the brake system model, Equation (37). The input

parameters of steering and brake system are independent of the variable of state space

equation. The output parameters of two systems can be just considered input signals to the

vehicle plant model. Therefore, the two ordinary differential equations can be solved

separately through the transfer function of differential equations. Figure 3-5 shows that the

input and output relationship between vehicle and steering/brake model.

Vehicle System

Steering

System

Brake

System

u2 = M

d

u1 =

Brake

Command

Steering

Command

Figure 3-5 Input and Output Relationship between Vehicle and Steering/Brake Model

3.4 Control Parameters

The stability of a vehicle depends on yaw rate and rollover coefficient. The yaw rate

error between actual yaw rate and desired yaw rate gives information about how much the

vehicle has the risk of spin or course deviation. The rollover coefficient gives information

about how much the vehicle has the risk of rollover.

25

3.4.1 Desired Yaw Rate

When a vehicle drives through a curve in an ideal case, the wheels only move in tangential

direction at low lateral acceleration in order to hold the vehicle. The speed component of the

contact point in the tire‟s lateral direction then vanishes

0yy evv (40)

where v is the vehicle velocity vector.

This kinematic constraint equation can be used for the vehicle‟s trajectory. Within the

validity of the kinematic tire model the necessary steering angle of the front wheels can be

constructed via given instantaneous turning center as shown in Figure 3-6. At the low speed

of vehicle the steering angle of inner and outer tire can be calculated from the geometry as

follows:

dR

baitan (41)

dR

baotan (42)

When the turning radius is large, i.e. dR , the steering angle equation can be calculated

approximately

R

baoi

1tan (43)

For the sake of simplicity, a 4-wheel vehicle can be considered as a 2-wheel steering

geometry and the velocity at the center of mass can be assumed as equal to the velocity at the

rear axle. According to the kinematic tire model as described in Equation (40), the velocity at

the rear axle can only have a component in the vehicle‟s longitudinal direction

T

xr v 00v (44)

26

d

ba

R

i

i

o

o

M

path

v

Figure 3-6 Ackermann Steering Geometry for 4 Wheels (dotted) and 2 Wheels (dashed) Vehicle

The velocity at the front axis can be obtained from kinematics

frrfrf // rωvv (45)

where fv is the velocity at the front axle and fr /r is the position vector from rear axle to

front axle and rf /ω is the angular velocity of front axle with respect to rear axle. The

velocity at the front axle becomes

0

)(

0

00

0

0

0// rba

vba

r

v xx

frrfrf rωvv (46)

27

The unit vector for the lateral direction at the front axle can be defined as follows:

T

ye 0cossin (47)

According to Equation (40) the velocity component lateral to the wheel must vanish,

0)(cossin rbavevv xyy (48)

From Equation (48) a first order differential equation for the yaw angle is obtained. This yaw

rate can be considered the desired yaw rate because it was came up with not the road or tire

conditions but the kinematic geometry.

tan)( ba

vr x

des (49)

3.4.2 Rollover Coefficient

To reduce the risk of rollover, the exact vehicle status about roll movement should be known.

One measure for rollover is the rollover coefficient suggested in [21]. The threshold of

rollover can be regarded as the rollover coefficient, which represents the balance moment by

the vertical force acting on the left and right tires; i.e. gravitation forces of sprung mass and

unsprung mass and tire normal loads lzF , and rzF , , and balance of moment with respect to

the center of mass on the planar plane. As shown in Figure 3-2, the Rollover coefficient can

be defined as [21]:

lzrz

lzrz

cFF

FFR

,,

,, (50)

where lzF , and rzF , are the normal force of left and right tire, respectively. If rzlz FF ,, ,

cR becomes zero. If lzF , or rzF , is zero, i.e. the left or right tire is about to lift and hence,

cR is ± 1. In order to avoid the rollover, cR should be less than 1.

28

The denominator can be obtained from the balance of tire normal forces and vehicle weight.

The numerator can be calculated using the moment with respect to the center O1 as shown in

Figure 3-2

0,, gmFFF rzlzz (51)

0cossin ,,, sysRsrzlzx amhhhmFFdM g (52)

where sincos 2

, rhhrvva xysy as described in Appendix A. (A.19)

Substituting (51) and (52) into (50) and arranging yields:

sincos,

,,

,,h

ahh

md

m

FF

FFR

sy

R

s

rzlz

rzlz

cg

(53)

The linearized rollover coefficient can be obtained by assuming sin,mms ,

Equation(53) yields:

d

ha

d

hhR

syRc

g

, (54)

where hrvva xysy

, (55)

The second term of Equation (54) can be also neglected because of its magnitude. Finally,

the rollover coefficient becomes:

g

syRc

a

d

hhR

, (56)

From this coefficient, we see that the rollover factor depends on the lateral acceleration,

which mainly relies on the three components shown in Equation (53).

29

Chapter 4

Controller

A PD controller is considered for the yaw stability and rollover avoidance. The main purpose

of the controller is to stabilize the vehicle‟s yaw motion by reducing yaw rate error and to

avoid the vehicle‟s rollover occurrence. The yaw stability is the primary control mission and

the rollover avoidance is emergency control mission in the viewpoint of controller. To fulfill

these missions, active steering, differential braking and integration of active steering and

differential braking control cases are considered to compare the advantages and

disadvantages of each controlled case.

4.1 Yaw Stability Control

The yaw motion of the vehicle is one of the most important motions from the viewpoint of

vehicle dynamics; especially on slippery roads. The driver can usually control when the

vehicle shows the neutral steer characteristic because the yaw rate is proportional to velocity.

This neutral steering provides good stability control in a cornering situation, such as entry

and exit on and off the express way during winter driving conditions.

4.1.1 Active Steering Controller

Figure 4-1 shows the assumed controller structure for active steering. The driver makes a

turn and gives the handwheel, hδ , to the steering system. The first input signal, δ1u , is

calculated by the steering dynamics through Equation (37), which was explained in Section

3.2 and is given to the vehicle model and the kinematic tire model. The kinematic tire model

produces the desired yaw rate, which was described in Section 3.4.1. The state space

equation also produces four state values, T

prβx through Equation (25).

Subtracting desired yaw rate from actual yaw rate, the yaw rate error, deserr rrr , is

obtained. The purpose of the controller is to reduce the yaw rate error to achieve the vehicle

stability through neutral steer characteristic. The controller makes a small correction angle,

30

cδ , based on the yaw rate error. If the vehicle is understeer condition, the correction angle is

positive. If the vehicle is oversteer condition, the correction angle is negative.

Controller

Vehicle System

Steering

Dynamics

c

+

-Kinematic

Tire Model

Desired yaw rate

rdes

Actual yaw rate

r

rerr

= r - rdes

+

u1 =

c

h

+

Figure 4-1 Yaw Stability Controller Structure for Active Steering

4.1.2 Differential Braking Controller

Figure 4-2 shows the assumed controller structure for differential braking. The mechanism of

the handwheel, hδ , and the first input signal, δ1u , are the same as the active steering

controller. The kinematic tire model and vehicle plant model produce the desired yaw rate

and actual yaw rate, respectively. The yaw rate error calculated from subtracting desired yaw

rate from actual yaw rate. However, the first input signal, δ1u , is not be effected because

we focus on the pure braking condition without the steering action. Instead of the correction

angle, cδ , the differential braking controller gives the commanded hydraulic pressure to the

brake system, which was explained in Section 3.3. Through Equations (37, 39), the direct

yaw moment as the second input, dMu2 , is given to the vehicle plant to achieve the

vehicle stability through neutral steer characteristic.

In comparison to the steering dynamics, the brake dynamics is more complicated. The

brake system has four brake actuators: left front brake, left rear brake, right front brake, and

right rear brake, while the steering system has only one actuator. As seen in Figure 2-4, the

31

moment produced by these four brakes make a different movement of the vehicle, such as

spin or drift out situations.

Brake

Dynamics

Vehicle System

Steering

Dynamics

Controller

+

-Kinematic

Tire Model

Desired yaw rate

rdes

Actual yaw rate

r

u1 =

u2 = M

d

h

Phyd

rerr

= r - rdes

Figure 4-2 Yaw Stability Controller Structure for Differential Braking

4.1.3 Integrated Controller

Figure 4-3 presents the yaw stability controller structure for integrated control. The active

steering and differential braking controllers are combined using the yaw rate error as the

feedback signal to achieve the yaw stability. Controller gives the correction angle and the

direct yaw moment to the steering actuator and braking actuator, respectively. The individual

mechanism of controller is the same as the active steering controller and differential braking

controller.

32

Controller

Brake

Dynamics

Vehicle Syste

Steering

Dynamics

Controller

+

-Kinematic

Tire Model

Desired yaw rate

rdes

Actual yaw rate

r

+

u2 = M

d

h

Phyd

u1 =

c

c

rerr

= r - rdes

+

Figure 4-3 Yaw Stability Controller Structure for Integrated Control

4.2 Rollover Avoidance Control

In comparison to yaw stability control, the rollover avoidance control is not a continuing

operation because the roll motion of vehicle does not affect the planar motion of the vehicle,

such as the trajectory or course deviation, as long as the rollover coefficient, which was

explained in Section 3.4.2, lies under a stable region. Rollover avoidance using steering and

braking control is considered as an emergency control method using the dead zone. The

individual controller for rollover avoidance control is omitted because its method can be

merged into the integrated controller.

The integrated controller as shown in Figure 4-4 presents the PD controller structure in

case of rollover avoidance for the integrated control. Rollover coefficient can be calculated

from Equation (50). In comparison to yaw stability, these PD controllers will only be

activated if the rollover coefficient is over the threshold. PD gains were obtained by trial-

and-error method.

33

Vehicle System

Steering

Dynamics

PD

Controller

Rollover

Coefficient

h

y

Brake

Dynamics

PD

Controller

c-

+

Rc

-1 1R

c

- Rc

u2 = M

d

u1 = -

c

Figure 4-4 Rollover Avoidance Controller Structure for Integrated Control

4.3 Integrated Yaw Stability and Rollover Avoidance Controller

The integrated yaw stability and rollover avoidance controller is combined with the

individual control techniques; reference matching controller and PD controller. In normal

conditions, the yaw stability controller works as the main controller. However, in emergency

conditions, such as a quick lane change to avoid an obstacle, the rollover avoidance

controller works as a supplemental controller. The basic controller is the same as the yaw

stability and rollover avoidance controller. In normal conditions, the yaw stability controller

gives the correction steering angle and direct yaw moment. In emergency conditions, the

rollover avoidance controller gives the supplement correction angle and supplement yaw

moment. The final controlled value will be added as follows:

cRrc (57)

cRrd MMM (58)

where c and dM are the total correction angle and total direct yaw moment by generating

steering and braking actuators, respectively. The subscript r represents yaw stability and Rc

represents rollover avoidance. Figure 4-5 presents the integrated yaw stability and rollover

avoidance controller structure.

34

Vehicle

System

Steering

Dynamics

h -

+Kinematic

Tire Model

Desired yaw rate

rdes

Actual yaw rate

r

rerr

= r - rdes

+

Rollover coefficient

Rc

Brake

Dynamcis

r

Rc

Mr

M Rc

+

+

Rc

-1 1R

c

- Rc

Yaw Stability

Braking

Controller

Steering

Controller

Rollover Avoidance

Braking

PD Controller

Steering

PD Controller

Brake

Dynamics

+

+

c

u2 = M

d

u1 = +

c

+

Figure 4-5 Integrated Yaw and Rollover Controller Structure for Reference Matching Control

35

Chapter 5

Simulation Results

Simulations are performed for both the linear and nonlinear 4 DOF model developed in

MATLAB and SIMULINK to evaluate the performance of the proposed active steering,

differential braking and integration control. Three kinds of input including J-turn, sinusoidal

and fishhook are used. To evaluate and compare the proposed integration control developed

for the simplified 4 DOF model, the ADAMS model of the same SUV is developed and

examined. Simulation results of the 4 DOF and ADAMS models are presented and examined

and it is shown that the 4 DOF model is sufficient for the evaluation and yaw and rollover

control design of the SUV. The MATLAB and SIMULINK model are described in Appendix

C and the ADAMS model is described in Appendix D.

5.1 Test Maneuvers and Vehicle Parameters

5.1.1 Test Maneuvers

To evaluate the performance of the vehicle with/without control, several standard test

maneuvers have been proposed in NHTSA report [26]. The maneuvers selected in this thesis

are J-turn, sinusoidal, and fishhook maneuvers. J-turn and sinusoidal maneuver are

performed in case of yaw stability. To evaluate rollover avoidance, fishhook maneuver is

selected because it will excite the vehicle‟s roll motion.

The first maneuver is a J-turn input to follow road profile as shown in Figure 5-1 in order

to compare with ADAMS simulation results. The length of entry road and exit road are 50 m

and the radius of curvature is 50 m. The J-turn maneuver excites vehicle roll and yaw

motions, which can occur in a sudden turn such as on a „cloverleaf‟ ramp. It is assumed the

vehicle is initially traveling in a straight line. After following the entry road, the driver turns

the handwheel from 0 to 57.46 degree within 0.33 seconds. The driver holds the steering

angle during the curvature section and the driver returns the handwheel from 57.46 to

0 degree within 0.33 seconds. Figure 5-2 shows the steering angle patterns at 60 km/h. To

36

compare the performance of the vehicle with/without a controller the worst scenario, μ-split

road is selected as shown in Figure 5-1. The icy surface, i.e. μ=0.2, lies in the middle of

curvature at the right tire side for about 15 m and all other roads are assumed as dry asphalt,

i.e. μ=1.0. When the vehicle goes through this surface the driver holds the steering wheel at

the same angle and different tire forces occur. These forces cause the yaw disturbance at the

center of mass.

The second maneuver is sinusoidal input as shown in Figure 5-3. With investigating step

response, consideration of sinusoidal response is also necessary; as a sinusoidal input has

very similar effects in real driving condition like lane change situation. The worst scenario is

similar with μ-split road of J-turn. For the sake of simplicity, the μ-split road is assumed from

3.5 seconds to 4.5 seconds during 1 second. The yaw disturbance is implemented as the cause

of different tire forces for the left and right side.

Figure 5-1 Test Road Profile for J-Turn

37

0 1 2 3 4 5 6 7 8 9 10

0

10

20

30

40

50

60

Time (sec)

Ste

ering a

ngle

(deg)

J-turn Input vs. Time

Figure 5-2 J-Turn Input vs. Time

0 1 2 3 4 5 6 7 8 9 10-50

-40

-30

-20

-10

0

10

20

30

40

50Sinusoidal Input vs. Time

Time (sec)

Ste

ering a

ngle

(deg)

Figure 5-3 Sinusoidal Input vs. Time

38

The third maneuver is a fishhook input as shown in Figure 5-4. The fishhook maneuver

attempts to induce two-wheel liftoff by suddenly making a drastic turn in one direction and

then turning back even further in the opposite direction. As shown in Figure 5-4, the driver

steers the handwheel from 0 to -12 degrees during 0.25 seconds. After maintaining the

steering angle for 0.5 seconds, the driver turns the handwheel back to 0 degrees within 0.20

seconds. The driver then turns the handwheel to 57 degrees within 1.2 second and maintains

the angle for the remainder of the maneuver.

The simulations are performed and compared for four different cases: uncontrolled case,

pure active steering control case, pure differential braking control case, and

integration control case including active steering and differential braking control.

Individual controllers were explained in Figure 4-1 , Figure 4-2 and Figure 4-3.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-20

-10

0

10

20

30

40

50

60

time (s)

ste

ering a

ngle

(deg)

Fishkhook Input vs. Time

Figure 5-4 Fishhook Maneuver vs. Time

39

5.1.2 Vehicle Parameters

The parameters used in the simulations are listed in Table 5-1 and they correspond to a 1997

Jeep Cherokee [8]. The ADAMS model uses the same parameters as in Table 5-1.

Table 5-1 Vehicle Parameters

No Symbol Definition Values

1 a distance between COM of unsprung mass and

front axle

1.147 m

2 b distance between COM of unsprung mass and rear

axle

1.431 m

3 fC front cornering stiffness 63946 N/rad

4 rC rear cornering stiffness 55040 N/rad

5 C roll stiffness of passive suspension 51879 Nm/rad

6 d half of track width 0.7335 m

7 D roll damping of passive suspension 4269.4 Nms/rad

8 h height of the COM of sprung mass above the roll

axis

0.2818 m

9 Rh roll center height over ground 0.4835 m

10 usm unsprung mass 1338.07 kg

11 sm sprung mass 324.93 kg

12 wR tire wheel radius 0.337 m

13 μ road adhesion coefficient 1

14 sxJ , sprung mass moment of inertia around roll axis 602.8 kgm2

15 szJ , sprung mass moment of inertia around yaw axis 2163.4 kgm2

16 uszJ , unsprung mass moment of inertia around yaw axis 540 kgm2

40

5.2 Simulation Results of Linear Model

The Linear simulation results are performed for the linear 4 DOF mode as described in

Equation (25) and for the steering and brake system dynamics as described in Equation (37)

and (Error! Reference source not found.), respectively. All initial conditions were assumed

to be zero, except a constant velocity.

5.2.1 Simulation Results of J-turn Maneuver

Figure 5-5 shows the vehicle trajectory at 60 km/h in the μ-split road. The thick solid line

represents the desired trajectory calculated from Equation (43) and (49) and the thin solid

line represents the uncontrolled case. The dashed and dotted lines represent active steering

controlled case and differential braking controlled case, respectively. Finally, the dashed-

dotted line illustrates the integration of active steering and differential braking controller,

operating together. Each controlled case follows well to the desired path, while uncontrolled

case does not follow the desired trajectory. In order to investigate the vehicle trajectory, the

rectangle in Figure 5-5 is magnified as shown in Figure 5-6. In comparisons three controlled

cases, the active steering control results in a slightly larger deviation, while the differential

braking control results in a slightly smaller deviation. The integration controlled case lies in

the middle of both cases.

Figure 5-7 shows the tire turning angle versus time according to the handwheel input as

shown in Figure 5-2. The active steering controller increases the tire turning angle for a short

time to eliminate the yaw rate error. This modified angle could be the cause of increasing

deviation from desired course. However, the differential braking did not affect the steering

angle. Instead of reducing tire angle, moment input, which is produced by the controller, is

applied as shown in Figure 5-8 to eliminate the yaw rate error. In the integration controlled

case, the increased steering angle and the added moment are less than both of the individual

control cases. In comparisons the steering controlled case and the braking controlled case, the

active steering controller affects the early stage of response and the differential braking

controller affects in a slightly late stage of response in all the results. The advantages of

steering control are therefore shown to have a faster response time than the brake control.

41

This graph indicates that steering control has more direct influence. While this represents a

significant advantage in some situations, it could also be disadvantageous. At high speeds,

yaw and roll motion could be amplified by even a small correction angle trough steering

control. The most effective control is therefore using the integrated control method.

Figure 5-9 and Figure 5-10 show the slip angle and yaw rate versus time, respectively. In

spite of yaw disturbance, the yaw rate of controlled case is slightly changed and follows well

the desired yaw rate. Figure 5-11 and Figure 5-12 present the roll angle and rollover

coefficient versus time. As we can see, both patterns are very similar. The rollover

coefficient, which was described in Section 3.4.2, is well defined.

0 20 40 60 80 100 120 1400

10

20

30

40

50

60

70

80

90

100

x position (m)

y p

ositio

n (

m)

Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-5 Vehicle Trajectory at 60 km/h (Linear)

42

80 85 90 95 100 105 110 115 12020

25

30

35

40

45

50

55

60

65

70

x position (m)

y p

ositio

n (

m)

Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-6 Magnified Vehicle Trajectory at 60 km/h (Linear)

0 1 2 3 4 5 6 7 8 9 10-2

-1

0

1

2

3

4

5

6

time (s)

tire

turn

ing a

ngle

(deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-7 Tire Turning Angle vs. Time (Linear)

43

0 1 2 3 4 5 6 7 8 9 10-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

time (s)

mom

ent

input

(Nm

)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-8 Moment Input vs. Time (Linear)

0 1 2 3 4 5 6 7 8 9 10-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

time (s)

slip

angle

(deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-9 Slip Angle vs. Time (Linear)

44

0 1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

time (s)

yaw

rate

(deg/s

)

Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-10 Yaw Rate vs. Time (Linear)

0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

1.5

2

2.5

3

time (s)

roll

angle

of

spru

ng m

ass (

deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-11 Roll Angle of Sprung Mass vs. Time (Linear)

45

0 1 2 3 4 5 6 7 8 9 10-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

time (s)

rollo

ver

coeff

icie

nt

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-12 Rollover Coefficient vs. Time (Linear)

5.2.2 Simulation Results of Sinusoidal Maneuver

Figure 5-13 shows the vehicle trajectory at 60 km/h in the μ-split road. The thick solid line

and thin solid line represent the desired trajectory and uncontrolled trajectory of the vehicle.

Each controlled case follows well to the desired path; as the dashed line for steering control,

the dotted line for braking control and the dashed-dotted line for integration control. Figure

5-14 shows the magnified trajectory of the vehicle in order to compare three controlled cases.

The active steering control results in a slightly larger deviation, while the differential braking

control results in a slightly small deviation. The integration controlled case lies in the middle

of both cases.

Figure 5-15 presents the tire turning angle versus time according to the sinusoidal response

of handwheel angle. Braking control could not act on the tire turning angle; however, the tire

turning angle was slightly increased. Figure 5-16 shows moment input as second input. In

uncontrolled case, the yaw disturbance according to μ-split road is shown. In this simulation,

46

steering control could affect moment input, which shows that moment input is supplied

through left or right brake.

Figure 5-17 and Figure 5-18 show the slip angle and yaw rate versus time, respectively. In

spite of yaw disturbance, the yaw rate of controlled case is slightly changed and follows well

the desired yaw rate. Figure 5-19 and Figure 5-20 present the roll angle and rollover

coefficient versus time.

0 20 40 60 80 100 120 140 1600

5

10

15

20

25

30

35

x position (m)

y p

ositio

n (

m)

Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-13 Vehicle Trajectory at 60 km/h (Linear)

47

70 75 80 85 90 95 100 105 110 115 12015

20

25

30

x position (m)

y p

ositio

n (

m)

Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-14 Magnified Vehicle Trajectory at 60 km/h (Linear)

0 1 2 3 4 5 6 7 8 9 10-4

-3

-2

-1

0

1

2

3

4

time (s)

tire

turn

ing a

ngle

(deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-15 Tire Turning Angle vs. Time (Linear)

48

0 1 2 3 4 5 6 7 8 9 10

-1500

-1000

-500

0

500

1000

1500

time (s)

mom

ent

input

(Nm

)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-16 Moment Input vs. Time (Linear)

0 1 2 3 4 5 6 7 8 9 10-3

-2

-1

0

1

2

3

4

time (s)

slip

angle

(deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-17 Slip Angle vs. Time (Linear)

49

0 1 2 3 4 5 6 7 8 9 10-25

-20

-15

-10

-5

0

5

10

15

20

time (s)

yaw

rate

(deg/s

)

Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-18 Yaw Rate vs. time (Linear)

0 1 2 3 4 5 6 7 8 9 10-3

-2

-1

0

1

2

3

time (s)

roll

angle

of

spru

ng m

ass (

deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-19 Roll Angle vs. Time (Linear)

50

0 1 2 3 4 5 6 7 8 9 10

-0.4

-0.2

0

0.2

0.4

0.6

time (s)

rollo

ver

coeff

icie

nt

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-20 Rollover Coefficient vs. Time (Linear)

5.3 Simulation Results of Nonlinear Model

The Nonlinear simulation results are performed for the nonlinear 4 DOF mode as described

in Equation (3). The steering and brake system dynamics as described in Equation (37) and

(Error! Reference source not found.) are the same as the linear simulations. To compare

both linear and nonlinear, the parameters are the same as those of the linear including a

constant velocity of v=16.67m/s or 60 km/h for the vehicle. In the nonlinear model, two

kinds of inputs including J-turn and sinusoidal are also performed. The important differences

between linear and nonlinear model simulations are that trigonometric function cannot be

neglected and tire forces will not be linearized and are calculated based on the nonlinear tire

model described in Appendix B.

5.3.1 Simulation Results of J-turn Maneuver

Figure 5-21 to Figure 5-28 present the simulation results for the nonlinear model in case of J-

turn input at 60 km/h in the μ-split road. These results are similar to the simulation results of

51

the linear model. The legend is the same as the simulation results of linear model. Each

controlled case follows well to the desired path, while uncontrolled case does not follow the

desired trajectory. In comparisons the simulation results of linear model, the active steering

control results in a slightly larger trajectory deviation, while the differential braking control

results in a slightly smaller trajectory deviation. The integration controlled case lies in the

middle of both cases.

Figure 5-23 shows the tire turning angle versus time according to the handwheel input. The

active steering controller quickly increases the tire turning angle for a short time to eliminate

the yaw rate error. This modified angle could be the cause of increasing deviation from

desired course. However, the integration controller gives only a slightly increased the tire

turning angle. Figure 5-24 presents the moment input as the second input and the direct yaw

disturbance. The braking and integration controller are well acted to eliminate the

disturbance.

52

0 20 40 60 80 100 120 1400

10

20

30

40

50

60

70

80

90

100

x position (m)

y p

ositio

n (

m)

Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-21 Vehicle Trajectory at 60 km/h (Nonlinear)

53

80 85 90 95 100 105 110 115 12020

25

30

35

40

45

50

55

60

65

70

x position (m)

y p

ositio

n (

m)

Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-22 Magnified Vehicle Trajectory at 60 km/h (Nonlinear)

0 1 2 3 4 5 6 7 8 9 10-1

0

1

2

3

4

5

time (s)

tire

turn

ing a

ngle

(deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-23 Handwheel Input and Tire Turning Angle vs. Time (Nonlinear)

54

0 1 2 3 4 5 6 7 8 9 10-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

time (s)

mom

ent

input

(Nm

)Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-24 Moment Input vs. Time (Nonlinear)

0 1 2 3 4 5 6 7 8 9 10-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

time (s)

slip

angle

(deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-25 Slip Angle vs. Time (Nonlinear)

55

0 1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

time (s)

yaw

rate

(deg/s

)

Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-26 Yaw Rate vs. Time (Nonlinear)

0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

1.5

2

2.5

3

3.5

time (s)

roll

angle

of

spru

ng m

ass (

deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-27 Roll Angle vs. Time (Nonlinear)

56

0 1 2 3 4 5 6 7 8 9 10-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time (s)

rollo

ver

coeff

icie

nt

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-28 Rollover Coefficient vs. Time (Nonlinear)

Figure 5-25 and Figure 5-26 show the slip angle and yaw rate versus time. In spite of yaw

disturbance, the yaw rate of controlled case is slightly changed and follows well the desired

yaw rate. Figure 5-27 and Figure 5-28 present the roll angle and rollover coefficient versus

time, respectively.

5.3.2 Simulation Results of Sinusoidal Maneuver

Figure 5-29 to Figure 5-35 present the simulation results for the nonlinear model in case of J-

turn input at 80 km/h in the μ-split road. These results are also similar to the simulation

results of the linear model. The legend is the same as the simulation results of linear model.

Each controlled case follows well to the desired path, while uncontrolled case does not

follow the desired trajectory. Figure 5-31 shows the tire turning angle and the handwheel

input versus time. The active steering controller slightly increases the tire turning angle for a

short time to eliminate the yaw rate error. Figure 5-32 presents the moment input as the

second input and the direct yaw disturbance. The braking and integration controller are well

57

acted to eliminate the disturbance. Figure 5-33 shows the slip angle and yaw rate versus time.

In spite of yaw disturbance, the yaw rate of controlled case is slightly changed and follows

well the desired yaw rate, while the uncontrolled case shows the large deviation from the

desired yaw rate.

Figure 5-33 and Figure 5-34 show the slip angle and yaw rate versus time. In spite of yaw

disturbance, the yaw rate of controlled case is slightly changed and follows well the desired

yaw rate. Figure 5-35 and Figure 5-36 present the roll angle and rollover coefficient versus

time, respectively.

0 20 40 60 80 100 120 140 1600

5

10

15

20

25

30

35

x position (m)

y p

ositio

n (

m)

Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-29 Vehicle Trajectory at 60 km/h (Nonlinear)

58

70 75 80 85 90 95 100 105 110 115 12015

20

25

30

x position (m)

y p

ositio

n (

m)

Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-30 Magnified Vehicle Trajectory at 60 km/h (Nonlinear)

0 1 2 3 4 5 6 7 8 9 10-4

-3

-2

-1

0

1

2

3

4

time (s)

tire

turn

ing a

ngle

(deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-31 Tire Turning Angle vs. Time (Nonlinear)

59

0 1 2 3 4 5 6 7 8 9 10-1500

-1000

-500

0

500

1000

1500

time (s)

mom

ent

input

(Nm

)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-32 Moment Input vs. Time (Nonlinear)

0 1 2 3 4 5 6 7 8 9 10-3

-2

-1

0

1

2

3

time (s)

slip

angle

(deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-33 Slip Angle vs. Time (Nonlinear)

60

0 1 2 3 4 5 6 7 8 9 10-20

-15

-10

-5

0

5

10

15

20

time (s)

yaw

rate

(deg/s

)Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-34 Yaw Rate vs. Time (Nonlinear)

0 1 2 3 4 5 6 7 8 9 10-3

-2

-1

0

1

2

3

time (s)

roll

angle

of

spru

ng m

ass (

deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-35 Roll Angle vs. Time (Nonlinear)

61

0 1 2 3 4 5 6 7 8 9 10

-0.4

-0.2

0

0.2

0.4

0.6

time (s)

rollo

ver

coeff

icie

nt

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-36 Rollover Coefficient vs. Time (Nonlinear)

5.4 Comparison of Linear and Nonlinear Simulation Results

To evaluate the linear 4 DOF model, a comparison of the results for the linear and nonlinear

model is needed. For the sake of simplicity, the integration control for the linear model and

nonlinear model is compared with the uncontrolled case. The J-turn and sinusoidal responses

are used.

5.4.1 Simulation Results of J-turn Maneuver

Figure 5-37 and Figure 5-38 show the vehicle trajectory at 60 km/h in the μ-split road for the

integration control in case of J-turn maneuver. The thick solid line and thin solid line

represent the desired trajectory and uncontrolled trajectory of the vehicle. As we can see, the

integration control case follows well to the desired path. It is hard to find the difference of

both nonlinear and linear model. According to this fact, the linearized 4 DOF model should

be efficient to discuss the yaw and roll dynamics of the vehicle.

62

Figure 5-39 and Figure 5-40 present the tire turning angle versus time according to the

handwheel input and the direct yaw moment input for step input. As we can see, the different

between linear and nonlinear simulation results cannot be distinguished. The slip angle of the

vehicle is a little different, which is caused by the nonlinearities, as shown in Figure 5-41.

However, the yaw rate error is minimized as shown in the same figure. Figure 5-42 shows the

yaw rate versus time. The simulation results of linear and linear model are almost the same.

Figure 5-43 and Figure 5-44 show the roll angle and rollover coefficient of the vehicle. The

rising and falling patterns are slightly different because of the nonlinearities.

0 20 40 60 80 100 120 1400

10

20

30

40

50

60

70

80

90

100

x position (m)

y p

ositio

n (

m)

Desired

Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-37 Vehicle Trajectory at 60 km/h (Nonlinear vs. Linear)

63

80 85 90 95 100 105 110 115 12020

25

30

35

40

45

50

55

60

65

70

x position (m)

y p

ositio

n (

m)

Desired

Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-38 Magnified Vehicle Trajectory at 60 km/h (Nonlinear vs. Linear)

0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

time (s)

tire

turn

ing a

ngle

(deg)

Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-39 Tire Turning Angle vs. Time (Nonlinear vs. Linear)

64

0 1 2 3 4 5 6 7 8 9 10-3000

-2000

-1000

0

1000

2000

3000

time (s)

mom

ent

input

(Nm

)Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-40 Moment Input vs. Time (Nonlinear vs. Linear)

0 1 2 3 4 5 6 7 8 9 10-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

time (s)

slip

angle

(deg)

Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-41 Slip Angle and Yaw Rate vs. Time (Nonlinear vs. Linear)

65

0 1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

time (s)

yaw

rate

(deg/s

)

Desired

Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-42 Yaw Rate vs. Time (Nonlinear vs. Linear)

0 1 2 3 4 5 6 7 8 9 10-0.5

0

0.5

1

1.5

2

2.5

3

time (s)

roll

angle

of

spru

ng m

ass (

deg)

Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-43 Roll Angle vs. Time (Nonlinear vs. Linear)

66

0 1 2 3 4 5 6 7 8 9 10-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

time (s)

rollo

ver

coeff

icie

nt

Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-44 Rollover Coefficient vs. Time (Nonlinear vs. Linear)

5.4.2 Simulation Results of Sinusoidal Maneuver

Figure 5-45 and Figure 5-46 show the vehicle trajectory at 60 km/h in the μ-split road for the

integration control in case of sinusoidal maneuver. The thick solid line and thin solid line

represent the desired trajectory and uncontrolled trajectory of the vehicle. As we can see,

both controlled cases follow well to the desired path, while uncontrolled case has a large

deviation.

Figure 5-47 and Figure 5-48 present that the handwheel input versus tire turning angle and

the direct yaw moment input for sinusoidal input. The rising pattern and value of tire turning

angle are almost the same. However, the direct yaw moment input shows slight deviation.

The nonlinearity causes a little delay in braking control. The difference between linear and

nonlinear model cannot be distinguished.

67

0 20 40 60 80 100 120 140 1600

5

10

15

20

25

30

35

x position (m)

y p

ositio

n (

m)

Desired

Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-45 Vehicle Trajectory at 60 km/h (Nonlinear vs. Linear)

70 75 80 85 90 95 100 105 110 115 12015

20

25

30

x position (m)

y p

ositio

n (

m)

Desired

Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-46 Magnified Vehicle Trajectory at 60 km/h (Nonlinear vs. Linear)

68

0 1 2 3 4 5 6 7 8 9 10-3

-2

-1

0

1

2

3

time (s)

tire

turn

ing a

ngle

(deg)

Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-47 Tire Turning Angle vs. Time (Nonlinear vs. Linear)

0 1 2 3 4 5 6 7 8 9 10-1000

-800

-600

-400

-200

0

200

400

600

800

1000

time (s)

mom

ent

input

(Nm

)

Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-48 Moment Input vs. Time (Nonlinear vs. Linear)

69

0 1 2 3 4 5 6 7 8 9 10-3

-2

-1

0

1

2

3

time (s)

slip

angle

(deg)

Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-49 Yaw Rate vs. Time (Nonlinear vs. Linear)

0 1 2 3 4 5 6 7 8 9 10-20

-15

-10

-5

0

5

10

15

20

time (s)

yaw

rate

(deg/s

)

Desired

Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-50 Yaw Rate vs. Time (Nonlinear vs. Linear)

70

0 1 2 3 4 5 6 7 8 9 10-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

time (s)

roll

angle

of

spru

ng m

ass (

deg)

Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-51 Rollover Coefficient vs. Time (Nonlinear vs. Linear)

0 1 2 3 4 5 6 7 8 9 10

-0.4

-0.2

0

0.2

0.4

0.6

time (s)

rollo

ver

coeff

icie

nt

Uncontrolled

Nonlinear Intergation

Linear Integration

Figure 5-52 Rollover Coefficient vs. Time (Nonlinear vs. Linear)

71

5.5 Advantage of Integration Control

The advantages of integrated control comparing with active steering and differential braking

control are hardly found in Section 5.2 and 5.3. In order to distinguish the advantages of

integrated control, the vehicle speed is changed from 80 km/h to 100 km/h to follow the same

road profile. In high speed velocity, the rollover often occurs. As explained in Section 3.4.2,

the rollover phenomenon can be judged by the magnitude of rollover coefficient.

5.5.1 Simulation Results for J-turn Maneuver at High Speed

Figure 5-53 presents the vehicle trajectory at 100 km/h in the dry asphalt road. The integrated

control and steering controlled case can be prevented the rollover occurrence, while the

uncontrolled case and braking controlled case cannot be reduced the risk of rollover

occurrence as shown in Figure 5-54.

The tire turning angle versus time is shown in Figure 5-55. In order to eliminate the yaw

rate error the tire turning angle is quickly increased. When the rollover coefficient across the

threshold, the tire turning angle is quickly reduced. After this action, the tire turning angle

moves an oscillation by a controller. The integrated control gives the same results in early

stage, then the tire turning angle remains by differential braking force.

Figure 5-58 shows the yaw rate versus time. As we can see, the yaw rate error is very small

in the early stage. However, in case of emergency maneuver, the yaw rate error is no longer

the control parameter. The rollover coefficient becomes the primary control parameter. The

integrated control shows the best results without rollover and yaw rate oscillation. The

vehicle yaw stability and rollover avoidance are identified through Figure 5-53 to Figure

5-54.

72

0 20 40 60 80 100 120 1400

10

20

30

40

50

60

70

80

90

100

x position (m)

y p

ositio

n (

m)

Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Rollover Occurrence

Figure 5-53 Vehicle Trajectory at 100 km/h in the Dry Asphalt

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

time (s)

rollo

ver

coeff

icie

nt

Uncontrolled

Steering controlled

Braking controlled

Integration controlledRollover Occurrence

Figure 5-54 Rollover Coefficient vs. Time

73

0 1 2 3 4 5 6-3

-2

-1

0

1

2

3

4

5

6

7

time (s)

tire

turn

ing a

ngle

(deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-55 Tire Turning Angle vs. Time

0 1 2 3 4 5 6-6000

-4000

-2000

0

2000

4000

6000

time (s)

mom

ent

input

(Nm

)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-56 Moment Input vs. Time

74

0 1 2 3 4 5 6-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

time (s)

slip

angle

(deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-57 Slip Angle vs. Time

0 1 2 3 4 5 6-5

0

5

10

15

20

25

30

35

time (s)

yaw

rate

(deg/s

)

Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-58 Yaw Rate vs. Time

75

0 1 2 3 4 5 60

1

2

3

4

5

6

time (s)

roll

angle

of

spru

ng m

ass (

deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-59 Roll Angle of Sprung Mass vs. Time

5.5.2 Simulation Results for Fishhook Maneuver at High Speed

In order to verify the advantage of integrated control, a fishhook maneuver is chosen

fishhook instead of sinusoidal maneuver. Figure 5-60 presents the vehicle trajectory at 100

km/h in the dry asphalt road. The integrated control and steering controlled case can be

prevented the rollover occurrence, while the uncontrolled case and braking controlled case

cannot be reduced the risk of rollover occurrence as shown in Figure 5-61.

The tire turning angle versus time is shown in Figure 5-62. The tire turning angle is

quickly increased in order to reduce the yaw rate error. When the rollover coefficient across

the threshold at around 2 seconds, the tire turning angle is quickly reduced. After this action,

the tire turning angle moves an oscillation. The integrated control gives the same results in

early stage, then the tire turning angle remains at around 2.5 seconds by differential braking

force.

76

Figure 5-65 shows the yaw rate versus time. As we can see, the yaw rate error is very small

in the early stage. However, in case of emergency maneuver at around 2 seconds, the yaw

rate error is no longer the control parameter. The rollover coefficient becomes the primary

control parameter. The integrated control shows the best results without rollover and yaw

rate oscillation.

0 20 40 60 80 100 120-10

0

10

20

30

40

50

60

x position (m)

y p

ositio

n (

m)

Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Rollover Occurrence

Figure 5-60 Vehicle Trajectory at 100 km/h in the Dry Asphalt

77

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

time (s)

rollo

ver

coeff

icie

nt Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-61 Rollover Coefficient vs. Time

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2

-1

0

1

2

3

4

5

time (s)

tire

turn

ing a

ngle

(deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-62 Tire Turning Angle vs. Time

78

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2000

-1000

0

1000

2000

3000

4000

5000

time (s)

mom

ent

input

(Nm

)Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-63 Moment Input vs. Time

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-10

-8

-6

-4

-2

0

2

time (s)

slip

angle

(deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-64 Slip Angle vs. Time

79

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-10

-5

0

5

10

15

20

25

30

35

time (s)

yaw

rate

(deg/s

)

Desired

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-65 Yaw Rate vs. Time

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-2

-1

0

1

2

3

4

5

6

time (s)

roll

angle

of

spru

ng m

ass (

deg)

Uncontrolled

Steering controlled

Braking controlled

Integration controlled

Figure 5-66 Roll Angle of Sprung Mass vs. Time

80

Chapter 6

ADAMS Model and Evaluation

6.1 ADAMS Model Building

To evaluate the proposed controller on a more realistic car model, ADAMS software was

used. Figure 6-1 presents the wire frame of a Jeep Cherokee based on the given data in Table

5.1. The coordinates of model are shown at the left lower corner. The x-axis follows the

longitudinal direction of the vehicle and the y and z-axis follow the right hand rule. The wire

frame model consists of three major parts: body, front axle with steering revolute joint and

rear axle, and suspension. The body‟s geometry data are obtained from the vehicle‟s

brochure. The body is assumed as homogeneous material to meet the vehicle‟s weight. Two

axles are assumed a simple cylinder. The front axle is combined three cylinders to make the

steering mechanism. Two side cylinders are constrained to the revolute joint with the center

cylinder. The suspension is assumed as a simple translational spring/damper. The geometry

of suspensions is assumed by a simple vertical mechanism. There are two center columns in

the middle of axles. The roles of center column give the vehicle two motions: the roll motion

and the vertical motion. The upper part of center columns is connected by a fixed joint to a

body. The lower part of center column is constrained by a revolute joint, which has a pin

along with the longitudinal direction of the vehicle, to axles. This revolute joint gives the

vehicle the roll motion. These upper and lower cylinders are constrained by a translational

joint, which gives the vehicle the vertical motion.

For the sake of simplicity, a steering linkage mechanism is assumed the revolute joint as

shown in Figure 6-2 in order to give only rotational movement with respect to the front axle,

i.e. z-axis. The differential braking actuators are not implemented by four individual brake

mechanisms attached to the tire. Instead, the direct yaw moment, which is produced by

differential braking force, acting on the center of mass of the vehicle is assumed to

implement the differential braking control.

81

Figure 6-1 ADAMS Wire Frame Model

82

Figure 6-2 Corner View of ADAMS Model

Figure 6-3 shows ADAMS road profile which is the same as the MATLAB road profile to

be able to compare the simulation results. The geometry is the same except for the entry

length. The road profile is made by 45 nodes and 56 triangle elements. The adhesion

coefficients are assumed to be μ = 1.

83

Figure 6-3 ADAMS road profile

6.2 Simulation Results for ADAMS Model

To compare the simulations results for both MATLAB and ADAMS, the same inputs, a J-

turn and a sinusoidal input, are used at 60 km/h. The simulations were performed by

ADAMS/Solver through communication between SIMULINK and ADAMS software. Figure

6-4 shows the longitudinal trajectory and velocity versus time. The solid line represents the

longitudinal trajectory and the dotted line represents the longitudinal velocity. The vehicle

reaches at 16.67 m/s or 60 km/h after 6 seconds. When the vehicle moves 100m, the

longitudinal velocity remains steady.

84

Figure 6-4 Longitudinal Trajectory and Velocity vs. Time

6.2.1 Simulation Results for J-Turn Input

Figure 6-5 shows the vehicle trajectory in uncontrolled and integration controlled case. The

solid line represents integrated control case and the dotted line represents uncontrolled case.

The vertical axis is opposite direction because the positive z-axis is selected as shown in

Figure 6-1. The simulation results are very similar to Figure 5-37. Figure 6-6 and Figure 6-7

present the top view of controlled and uncontrolled case. In the middle of curvature, the

trajectory of the vehicle has a slight deviation from the desired course.

85

Figure 6-5 Trajectory for uncontrolled and controlled vehicle according to J-turn Input

86

Figure 6-6 Trajectory for controlled vehicle according to J-turn Input

87

Figure 6-7 Trajectory for uncontrolled vehicle according to J-turn Input

6.2.2 Simulation Results for Sinusoidal Input

Figure 6-8 shows the vehicle trajectory for uncontrolled and integration controlled cases. The

solid line represents integrated control case and the dotted line represents uncontrolled case.

The simulation results are very similar to Figure 5-45. In order to compare the vehicle‟s

trajectory, Figure 6-9 and Figure 6-10 present the top view of controlled and uncontrolled

case.

88

Figure 6-8 Trajectory for uncontrolled and controlled vehicle according to Sinusoidal Input

Figure 6-9 Trajectory for controlled vehicle according to Sinusoidal Input

89

Figure 6-10 Trajectory for uncontrolled vehicle according to Sinusoidal Input

90

Chapter 7

Conclusions and Discussion

7.1 Discussions

Advantages and Disadvantages of Steering Control

Advantages

One of the most important advantages of active steering is having an immediate effect

on the vehicle‟s yaw and roll motion.

It is more energy efficient. Active steering requires only one-fourth of the front tire

force needed for braking [3].

Disadvantages

Active steering control on a μ-split road could produce yaw disturbance moment due

to different longitudinal forces

The active steering causes a large trajectory deviation in the desired course. In a low

speed driving condition, this deviation may have no effect on the vehicle. However,

in a high speed cornering situation, a small change in tire angle may cause larger

deviation.

Advantages and Disadvantages of Braking Control

Advantages

The steering control is effective in linear region, in which tire forces are proportional

to tire slip angle. ABS and VDC provide active safety under the critical limit

conditions in driving and braking directions. In the cornering situation, however, roll

stiffness distribution is needed; without this, the vehicle could be subjected to a

rollover situation.

91

The trajectory of the vehicle and deviation from the reference path is smaller with

braking control than with steering control.

Implementation of braking control is easier than steering control because

commercialized systems, such as ABS and VDC, can be used with small

modifications to the software.

In case of high speed steady-state cornering situation, braking control is very efficient

way to reduce the risk of rollover occurrence.

Disadvantages

Under a strict limitation, braking control is less efficient than steering control.

The contribution through deceleration by braking involves delay times; therefore

braking control dose not have an immediate effect on the vehicle‟s yaw and roll

dynamics.

The advantages and disadvantages of steering control and braking control are summarized in

Table 7-1.

Table 7-1 Summary of Advantage and Disadvantage for Steering and Braking Control

Steering Controlled Braking Controlled

1 Efficiency More Less

2 Trajectory deviation More Less

3 Rollover Avoidance More More

4 Implementation Hard Easy

92

7.2 Conclusions

Many chassis control components have been suggested and developed in the past three

decades. Some of them have already been commercialized in a wide range of vehicles. The

ultimate goal of these controls is to achieve active safety, such as the yaw stability and

rollover avoidance. Active steering is one of the most important active safety technologies

and will potentially be one of the leading trends in the near future. Differential brake control

has also been introduced and has been applied since 1950s. In the early stages of differential

braking control, longitudinal direction was only considered to be control; for example, ABS.

However, an independently controlled braking control, such as braking force distribution

method, was considered to reduce yaw moment under a cornering situation on an icy road.

The four degrees-of-freedom model of a vehicle that has unsprung and sprung mass was

reviewed and the nonlinear equations were developed. For the sake of simplicity, the steering

dynamics and the brake dynamics were assumed to be the second order and first order

differential equations, respectively.

To test the efficiency of steering control and brake control, both linear and nonlinear model

were simulated using SIMULINK. Also, an ADAMS model, including road profile, was built

and the proposed controllers were evaluated. The results of the linear and nonlinear model

simulations were then compared.

The active steering can affect the vehicle‟s yaw and roll dynamics immediately. This fast

response can reduce the risk of rollover and establish yaw stability. In case of middle speed

around 60 km/h, the active steering control gives a good result to follow the desired course.

According to the simulation results of J-turn input, rollover does not occur at 60 km/h. In this

case, the merit of integration control cannot be verified. To evaluate the benefit of integration

control, the vehicle speed was changed from 60 km/h to 100 km/h for the same road profile.

Then, the rollover avoidance controller as an emergency controller shows its effects. The

rollover occurrence is avoided but the trajectory has a larger deviation.

93

The differential braking control has less influence than steering control with regard to the

performance of trajectory. At 100 km/h for J-turn maneuver, the braking control cannot

prevent the rollover even if it was shown a smaller deviation.

Based on the advantages and disadvantages of steering and braking control, an integration

of steering and braking is the best way to control the vehicle in wide range of speed. The

integration method, which combines active steering and differential braking controls, is the

most efficient control in terms of vehicle yaw stability and rollover avoidance. The proposed

controllers for the active steering, differential braking, and integration control for the yaw

stability and rollover avoidance were verified in a SUV, which have the highest rollover

rates.

94

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conference, 1999

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22-24, Ann Arbor, Michigan, 2000, pp.187-194

[3] J. Ackermann, T. Bunte and D. Odenthal, “Advantages of active steering for vehicle dynamics

control”, Proc. 32nd

Int‟l Symposium on Automotive Technology and Automation, Vienna, 1999

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Automation in Vehicles”, Safety Science, v. 24, n. 1, 1996, pp. 35-49

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[6] S. Amberkar, J.G. D‟Ambrosio, et al, “A System-Safety Process For By-Wire Automotive

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Requirements”, Society of Automotive Engineers Technical Paper Series No. 2000-01-0822,

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[8] B.C. Chen and H. Peng, “Differential-Braking-Based Rollover Prevention for Sport Utility

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wire system for intelligent vehicles”, Heavy Vehicle Systems, v 9, n 1, 2002, pp 1-26

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[13] J. Ackermann, “Robust car steering by yaw rate control”, In proceedings IEEE Conference

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steering studies from the viewpoint of vehicle dynamics and control”, Vehicle System Dynamics,

vol. 18, pp. 151-186, 1989.

[17] Y. Hirano and K. Fukatani, “Development of robust active rear control”, In Proceedings

International Symposium on Advanced Vehicle Control, pp. 359-375, 1996

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Decision and Control, Honolulu, pp. 2033-2034, 1990.

[19] K. Huh and J. Kim, “Active Steering Control based on the estimated tire forces”, Journal of

Dynamic Systems, Measurement, and Control, vol. 123, pp.505-511, 2001

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Control”, 32nd International Symposium on Automotive Technology and Automation, Vienna,

Austria, pp. 263-270, 1999

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Vehicles with Elevated Center of gravity”, International Conference on Advanced in Vehicle

Control and Safety, Amiens, France, pp.118-123, 1998.

[22] H. Nakazato, K. Iwata and Y. Yoshioka, “A new system for independently controlling braking

force between inner and outer real wheel”, SAE paper 890835

[23] S. Matsumoto, H. Yamaguchi, H. Inoue and Y. Yasuno, “Improvement of vehicle dynamics

through braking distribution control” SAE paper 920645

96

[24] K. Koibuchi, M. Yamamoto, Y.Fukada, and S. Inagaki, “Vehicle Stability Control in limit

cornering by Active Brake”, SAE paper 960487

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SAE paper 2000-01-1642

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[27] http://www.mobis.co.kr/eng/html/rnd/rd01_6.html

[28] L. Segal, “Theoretical prediction and experimental substantiation of the response of the

automobile to steering control”, in IMechE, pp. 310-330, 1956-1957

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with environmental interaction”, In Proceedings of the 2000 American control Conference,

Chicago, 2000

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SAE 870421

[31] H. Pacejka, E.Bakker and L.Nidner, “A New Tire Model with an Application in Vehicle

Dynamic Studies”, SAE 890087

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Braking”, Proceedings of the Amereican Control Conference, Seattle, Washington,

pp. 1667-1671, June 1995

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behavior”, In Proceedings of AVEC 2000, 6th International Symposium on Advanced Vehicle

Control, Hiroshima, Japan, 2002

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1992

97

Appendix A

Kinematics of Yaw and Roll Motion

This appendix presents a detailed investigation of the kinematics of yaw-roll motion to find

the motion of the vehicle. Figure A-1 illustrates the kinematics relationship between

unsprung mass and sprung mass with respect to the inertial frame. Coordinates C0, C1 and C2

represent inertial frame, unsprung mass body fixed frame at the centre of mass (COM), point

O1, and sprung mass body fixed frame at the COM, point O2, respectively.

Figure A-1 Kinematics of Yaw-Roll Motion

All translation and rotation vectors based on kinematics are needed in order to derive the

governing equation of the vehicle.

1) Coordinate C1 with respect to Coordinate C0:

a) Position:

Transformation matrix from C0 to C1

98

100

0ψcosψsin

0ψsinψcos

01T (A.1)

Connection vector from C0 to C1

Tyx 001r (A.2)

b) Velocity:

Angular velocity vector from C1 with respect to C0

Tr0001ω (A.3)

Velocity vector from C1 with respect to C0

T

yx vv 001v (A.4)

c) Acceleration:

Angular acceleration vector from C1 with respect to C0

Tr 000101 ωα (A.5)

Acceleration vector from C1 with respect to C0

T

xyyx rvvrvv 001010101 vωva (A.6)

2) Coordinate C2 with respect to Coordinate C1:

a) Position:

Transformation matrix from C1 to C2

cossin0

sincos0

001

12T (A.7)

Connection vector from C1 to C2

99

T

R

TT

R hhhh cossin000000 1212 Tr (A.8)

b) Velocity:

Angular velocity vector from C2 with respect to C1

Tp 0012ω (A.9)

Velocity vector from C2 with respect to C1

Thphp sincos01212 rv (A.10)

c) Acceleration:

Angular acceleration vector from C 2 with respect to C1

Tp 001212ωα (A.11)

Acceleration vector from C2 with respect to C1

T

pphpph sincos(cossin(01212va (A.12)

3) Coordinate C2 with respect to Coordinate C0:

a) Position:

Transformation matrix from C0 to C2

cossin0

sinψcoscosψcosψsin

sinψsincosψsinψcos

120102 TTT (A.13)

Connection vector from C0 to C2

T

R hhhyx cossin120102 rrr (A.14)

b) Velocity:

Angular velocity vector from C2 with respect to C0

100

Trp 0120102 ωωω (A.15)

Velocity vector from C2 with respect to C0

cos

cos

sin

1201120102

hp

hpv

hrv

y

x

rωvvv (A.16)

c) Acceleration:

Angular acceleration vector from C 2 with respect to C0

Trrpp 1201120102 ωωααα (A.17)

Acceleration vector from C2 with respect to C0

)(2 12010112011201120102 rωωvωrαaaa (A.18)

sincos(

)sin(cos(

cos2sin(2

pph

rpphrvv

rprhrvv

xy

yx

(A.19)

where 01a is absolute acceleration, 12a is relative acceleration , 1201 rα is Euler

acceleration , 12012 vω is Coriolis acceleration and )( 120101 rωω is Centrifugal

acceleration, respectively. They can be calculated as follows:

T

rh sin1201rα (A.20)

T

hrpcos22 1201 vω (A.21)

T

hr sin0)( 2

120101 rωω (A.22)

All forces and moments for unsprung mass and sprung mass can be shown as in Figure 3-1

and Figure 3-2.

4) Forces:

101

Force matrix acting on sprung mass

0

y,ry,f

x,rx,f

s FF

FF

F (A.23)

Force matrix acting on unsprung mass

gus

us

-m

0

0

F (A.24)

where x,fF is total longitudinal force at front tire, x,rF is total longitudinal force at rear tire, fyF ,

is total lateral force at front tire, and ryF , is total lateral force at rear tire, respectively. They can

be calculated as follows:

δ)F(Fδ)F(FF y,rfy,lfx,rfx,lfx,f sincos (A.25)

)F(FF x,rrx,lrx,r (A.26)

δ)F(Fδ)F(FF x,rfx,lfy,rfy,lfy,f sincos (A.27)

)F(FF y,rry,lry,r (A.28)

where, if forces can be expressed as jkiF , , subscript i represents the longitudinal force when

i = x and the lateral force when i = y. Furthermore, subscript j represents the left tire force

when j = l and the right tire force when j = r and subscript k represents the front tire force

when k = f and the rear tire force when k = r. For example, lfxF , represents the longitudinal

left front tire force.

5) Moments:

Moment matrix acting on sprung mass

102

y,fy,rx,ltx,rt

x,sus

s

aFbF)Fd(F

M

0M (A.29)

Moment matrix acting on unsprung mass

0

0

x,susx,COM

us

MM

M (A.30)

where a is distance between COM of unsprung mass and front axle, b is distance between

COM of unsprung mass and rear axle, d is half of track width, COMxM , is roll moment acting

on roll axis, susxM , is moment due to passive suspension, ltyF , is total left lateral tire force,

and rtyF , is total right lateral tire force, respectively. They can be calculated as follows:

δFδFFF y,lfx,lfx,lrx,lt sincos (A.31)

δFδFFF y,rfx,rfx,rrx,rt sincos (A.32)

DCM x,sus (A.33)

where C is roll stiffness of passive suspension and D is roll damping of passive

suspension.

6) Newton-Euler Equation

Newton‟s 2nd

Law

)( c2

1

a2

1

0 i

i

i

i

iim FFa (A.34)

where ia

F is the resultant active force and ic

F is the resultant constraint force.

Euler‟s Equation

103

) (2

1

2

1

000 ic

i

ia

i

iiiii MMωΘωαΘ (A.35)

where ia

M is the resultant active momentum and ic

M is the resultant constraint

momentum. Product inertia of sprung mass and unsprung mass can be negligible due to the

simplicity. They can be expressed as follows:

Inertia matrix of sprung mass

usz

usy

usx

us

J

J

J

,

,

,

00

00

00

Θ (A.36)

Inertia matrix of unsprung mass

sz

sy

sx

s

J

J

J

,

,

,

12

00

00

00

TΘ (A.37)

The notations of sprung and unsprung mass are shown below in order to fit general

notation.

sus mmmm 21 ,

uss FFFF 21 , (A.38)

uss MMMM 21 ,

uss ΘΘΘΘ 21 ,

7) Nonlinear 2nd Order Differential Equations

The proper choice of minimal coordinates and minimal velocities respectively as:

Tyx ψz

Tprvv yxz (A.39)

104

The angular velocity and linear velocity can be determined in terms of Jacobian Matrices.

zzJzzωzzJzzω )(),()(),(21 R02R01 (A.40)

zzJzzvzzJzzv )(),()(),(21 T02T01

where:

0100

0000

0000

1RJ

0100

0000

1000

2RJ

0000

0010

0001

1TJ

sin000

cos010

0sin01

2T

h

h

h

J

In addition, the angular acceleration and linear acceleration can be determined in terms of

Jacobian Matrices.

),()(),,(),()(),,( 2R021R01 21zzαzzJzzzαzzαzzJzzzα (A.41)

),()(),,(),()(),,( 2R021T01 21zzazzJzzzazzazzJzzza

where: T0001α Tr 002α

0

1 x

y

rv

rv

a

cos

sin)(

cos22

2

hp

rphrv

hrprv

x

y

a

Nonlinear second order differential equation can be represented as follows by minimal

form:

),,(),()( uzzQzzkzzm (A.42)

105

where: 2

1

R

T

RT

T

T iiii)(

i

iim JΘJJJzm is symmetric positive definite mass matrix,

2

1

T

R

T

T )(),(ii

i

iiiiiiim ωΘωαΘJaJzzk is generalized gyroscope and

centrifugal matrix, and 2

1

T

R

T

T ii),,(

i

ii MJFJuzzQ is generalized active force matrix.

All Matrices can be represented as follows in details:

ssxs

s

ssus

ssus

mhJhm

hm

hmmm

hmmm

2,cos0

0)3,3(sin

cos00

0sin0

)(m

zm (A.43)

where: 22,,, sin)(cos)3,3( ssxszusz mhJJJm

sin(cos

cos(2sin

sin)()(

cos2)(

),(

2

,,

2

,,

2

rmhJJvhmr

pmhJJvhmr

rphmrvmm

prhmrvmm

sszsxxs

sszsxys

sxsus

sysus

zzk (A.44)

sin

),,(

gsx,susx,COM

z,COMy,fy,rx,ltx,rt

y,COMy,ry,f

x,rx,f

hmMM

MaFbF)Fd(F

FFF

FF

uzzQ (A.45)

8) Relationship between the lateral tire force and the side slip angle

The lateral force can be represented as tire side slip and normal force and calculated from the

tire model, as described in Appendix B.

)F,α(F)F,α(F rfz,rfy,lfz,lfy, rflf ff (A.46)

)F,α(F)F,α(F rrz,rry,lrz,lry, rrlr ff

106

To calculate the lateral force, the tire slip angle of each tire is need. This is a function of the

steering angle and the side slip angle. The steering angle is known value. The side slip angle

can be calculated from kinematics about instantaneous center of curvature, as shown in

Figure A-2.

2d

b a

lr

lf

Center of Curvature

Vlf

Vlr

rCC Y1

X1

V

lf

lr

r ij

Figure A-2 Kinematics About the Center of Curvature

From kinematics we can obtain as follows:

ijij rωvv 0101 (A.47)

where ijv is the wheel velocity of each tire and ijr is the position vector from COM of

unsprung mass to each tire center. The wheel velocity can be expressed as follows:

00

0

0

0

0101 arv

drv

d

a

r

v

v

y

x

y

x

lflf rωvv (A.48)

Similarly,

T

yxrf arvdrv 0v (A.49)

107

T

yxlr brvdrv 0v (A.50)

T

yxrr brvdrv 0v (A.51)

From the kinematics, the side slip of each tire can be calculated as follows:

ij

ijij

v

vβ

of components-x

of components-ytan 1 (A.52)

Therefore, the slip angle can be obtained as follows:

lfflf βδα rffrf βδα (A.53)

lrlr βα rrrr βα

where:

drv

arv

x

ylf

1tanβ drv

arv

x

yrf

1tanβ (A.54)

drv

brv

x

ylr

1tanβ drv

brv

x

y

rr

1tanβ

9) Relationship between the longitudinal tire force and the longitudinal slip ratio

The longitudinal force can also be represented as a function of longitudinal slip and normal

force and calculated from tire model, as described in Appendix B.

)F,σ(F)F,σ(F rfz,rfx,lfz,lfx, rflf ff (A.55)

)F,σ(F)F,σ(F rrz,rrx,lrz,lrx, rrlr ff

To calculate the longitudinal force, the longitudinal slip ratio is needed. The longitudinal slip

can be represented as the ratio of tire velocity to tire rotational velocity. They can be

calculated as follows:

108

rft

rfrf

lft

lflf

R

v

R

v

ω1100σ

ω1100σ (A.56)

rrt

rrrr

lrt

lrlr

R

v

R

v

ω1100σ

ω1100σ

where: sinδδcosδsinδcos arvdrvvarvdrvv yxrfyxlf

drvvdrvv xrrxlr

Also, tire rotational velocity can be obtained as follows:

wrfxrfrftwlfxlflft RFJRFJ ,, τωτω (A.57)

wrrxrrrrtwlrxlrlrt RFJRFJ ,, τωτω

109

Appendix B

Magic Formula of Tire

“Magic formula” of tire describes the characteristics of side force, brake force and self

aligning torque in terms of slip angle and longitudinal slip. This mathematical representation

forms the basis for a model describing tire behavior during combined braking and cornering.

Hans B. Pacejka [31,32] proposed that the best way to represent measured tire data is

formula method containing special functions. This method makes use of formula with

coefficients which describe some of the typifying quantities of a tire, such as slip stiffness at

zero slip and force and torque peak values. This method would make it possible to investigate

the effect of changes of these quantities upon the handling and stability properties of the

vehicle. “Magic Formula” of tire was proposed as below:

The lateral force can be expressed as follows:

vyyyyyy SBCDF ))(tan(sin 1 (B.1)

))α((tan)α)(1( 1hyy

y

yhyyy SB

B

ESE (B.2)

where: 30.1yC (B.3)

zzy FFD 10111.222

(B.4)

707.0354.0 zy FE (B.5)

)γ022.01()))208.0(tan82.1(sin1078 1

yy

zy

DC

FB (B.6)

028.0hyS zvy FS 8.14 (B.7)

where yC is shape factor coefficient, yD is peak factor coefficient, yE is curvature factor

coefficient, yB is stiffness factor coefficient, hyS is horizontal shift coefficient and vyS is

110

vertical shift coefficient, respectively. hyS and vyS depend on camber angle. The values for

the coefficient were calculated from empirical data-fitting method. Figure A-3 shows lateral

force vs. slip angle at normal force 4kN.

Figure A-3 Lateral Force vs. Slip angle of Tire

The aligning moment can be expressed as follows:

vzzzzzz SBCDM ))(tan(sin 1 (B.8)

))α((tan)α)(1( 1hzz

z

zhzzz SB

B

ESE (B.9)

where: 40.2zC (B.10)

zzz FFD 28.272.22

(B.11)

)γ030.01()72.286.1(

11.02

zz

Fzz

zDC

eFFB

z

(B.12)

111

070.01

04.4643.0070.02

zzz

FFE (B.13)

015.0hzS )945.0066.0(2

zzvz FFS (B.14)

where zC is shape factor coefficient, zD is peak factor coefficient, zE is curvature factor

coefficient, zB is stiffness factor coefficient, hzS is horizontal shift coefficient and vzS is

vertical shift coefficient, respectively. hzS and vzS depend on camber angle. Figure A-4

shows aligning moment vs. slip angle at normal force 4kN.

Figure A-4 Aligning Moment vs. Slip Angle of Tire

The longitudinal force can be expressed as follows:

))(tan(sin 1

xxxxx BCDF (B.15)

)σ(tan)σ)(1( 1

x

x

xxx B

B

EE (B.16)

112

where: 65.1xC (B.17)

zzx FFD 11443.212

(B.18)

zz

Fzz

xDC

eFFB

z069.02)2266.49(

(B.19)

486.0056.0006.02

zzx FFE (B.20)

where xC is shape factor coefficient, xD is peak factor coefficient, xE is curvature factor

coefficient and xB is stiffness factor coefficient, respectively. There is no horizontal shift

coefficient and vertical shift coefficient because these coefficient depend on camber angle.

Figure A-5 shows longitudinal force vs. longitudinal percent slip at normal force 4kN.

Figure A-5 Longitudinal Force vs. Longitudinal Percent Slip

113

Appendix C

MATLAB and SIMULINK Model

114

Figure A-6 SIMULINK Model for Uncontrolled and Active Steering Controlled

Uncontr

olle

d

Active

Ste

ering c

ontr

olle

d

Kin

em

atic T

ire M

odel

tire

_in

pu

t_ra

te2

tire

_in

pu

t_ra

te2

tire

_in

pu

t_ra

te1

tire

_in

pu

t_ra

te1

tire

_in

pu

t2

tire

_in

pu

t2

tire

_in

pu

t1

tire

_in

pu

t1

tim

e

tim

e

ste

eri

ng

_ra

te

ste

eri

ng

_ra

te1

ste

eri

ng

_in

pu

t

ste

eri

ng

_in

pu

t1

sta

tes_

ou

t1

sta

tes_

ou

t1

sta

tes_

ou

t2

sta

tes_

ou

t

du

/dt

rate

4

du

/dt

rate

1

du

/dt

rate

MA

TL

AB

Fu

ncti

on

ha

nd

wh

ee

l

inp

ut

de

sire

d_

sta

te

de

sire

d_

sta

te

sta

tes

rollo

ver_

coeff

ca

lcu

lati

on

blo

ck2

wa

^2

s +

2*z

eta

*wa

s+w

a^2

2 actu

ato

r 2

nd

_o

rde

r2

wa

^2

s +

2*z

eta

*wa

s+w

a^2

2 actu

ato

r 2

nd

_o

rde

r1

yaw

rate

err

or

rollo

ver

coeff

corr

ection a

ngle

acti

ve

_st

ee

rin

g C

on

tro

lle

r2

tire

angle

Fx

desired y

aw

rate

slip

angle

yaw

rate

rollr

ate

roll

angle

long.

velo

city

mom

ent

long.

forc

e

yaw

rate

err

or

Ve

hic

le P

lan

t2

tire

angle

Fx

slip

angle

yaw

rate

rollr

ate

roll

angle

long.

velo

city

mom

ent

long.

forc

e

Ve

hic

le P

lan

t1

tire

angle

desired_y

aw

rate

x_positio

n

y_positio

n

Kin

em

ati

c T

ire

Mo

de

l

-K-

Ga

in4

-K-

Ga

in1

-K-

Ga

in

Clo

ck

115

Figure A-7 SIMULINK Model for Uncontrolled and Differential Braking Controlled

Uncontr

olle

d

Kin

em

atic T

ire M

odel

D

iffe

rential B

rake c

ontr

olle

d

tire

_in

pu

t_ra

te

tire

_in

pu

t_ra

te3

tire

_in

pu

t_ra

te1

tire

_in

pu

t_ra

te1

tire

_in

pu

t

tire

_in

pu

t3

tire

_in

pu

t1

tire

_in

pu

t1

tim

e

tim

e

ste

eri

ng

_ra

te

ste

eri

ng

_ra

te1

ste

eri

ng

_in

pu

t

ste

eri

ng

_in

pu

t1

sta

tes_

ou

t

sta

tes_

ou

t3

sta

tes_

ou

t1

sta

tes_

ou

t1

du

/dt

rate

4

du

/dt

rate

2

du

/dt

rate

1

MA

TL

AB

Fu

ncti

on

ha

nd

wh

ee

l

inp

ut

lon

g.

forc

e

rollo

ve

r_co

eff

ya

w_

rate

_e

rro

r

Fx

dif

fere

nti

al_

bra

kin

g_

co

ntr

oll

er3

de

sire

d_

sta

te

de

sire

d_

sta

te

sta

tes

rollo

ve

r_co

eff

ca

lcu

lati

on

blo

ck3

wa

^2

s +

2*z

eta

*wa

s+w

a^2

2 actu

ato

r 2

nd

_o

rde

r3

wa

^2

s +

2*z

eta

*wa

s+w

a^2

2 actu

ato

r 2

nd

_o

rde

r1

tire

an

gle

Fx

de

sire

d y

aw

ra

te

slip

an

gle

ya

wra

te

rollr

ate

roll

an

gle

lon

g.

ve

locity

mo

me

nt

lon

g.

forc

e

ya

w r

ate

err

or

Ve

hic

le P

lan

t3

tire

an

gle

Fx

slip

an

gle

ya

wra

te

rollr

ate

roll

an

gle

lon

g.

ve

locity

mo

me

nt

lon

g.

forc

e

Ve

hic

le P

lan

t1

tire

an

gle

de

sire

d_

ya

wra

te

x_

po

sitio

n

y_

po

sitio

n

Kin

em

ati

c T

ire

Mo

de

l

-K-

Ga

in4

-K-

Ga

in2

-K-

Ga

in

Clo

ck

116

Figure A-8 SIMULINK Model for Uncontrolled and Integrated Controlled

Uncontr

olle

d

Kin

em

atic T

ire M

odel

Inte

gra

tion C

ontr

ol

tire

_in

pu

t_ra

te2

tire

_in

pu

t_ra

te4

tire

_in

pu

t_ra

te1

tire

_in

pu

t_ra

te1

tire

_in

pu

t2

tire

_in

pu

t4

tire

_in

pu

t1

tire

_in

pu

t1

tim

e

tim

e

ste

eri

ng

_ra

te

ste

eri

ng

_ra

te1

ste

eri

ng

_in

pu

t

ste

eri

ng

_in

pu

t1

sta

tes_

ou

t2

sta

tes_

ou

t2

sta

tes_

ou

t1

sta

tes_

ou

t1

du

/dt

rate

4

du

/dt

rate

3

du

/dt

rate

1

MA

TL

AB

Fu

ncti

on

ha

nd

wh

ee

l

inp

ut

lon

g.

forc

e

rollo

ve

r_co

eff

ya

w_

rate

_e

rro

r

Fx

dif

fere

nti

al_

bra

kin

g_

co

ntr

oll

er2

de

sire

d_

sta

te

de

sire

d_

sta

te

sta

tes

rollo

ve

r_co

eff

ca

lcu

lati

on

blo

ck4

wa

^2

s +

2*z

eta

*wa

s+w

a^2

2 actu

ato

r 2

nd

_o

rde

r4

wa

^2

s +

2*z

eta

*wa

s+w

a^2

2 actu

ato

r 2

nd

_o

rde

r1

yaw

ra

te e

rro

r

rollo

ve

r co

eff

corr

ectio

n a

ng

le

acti

ve

_st

ee

rin

g C

on

tro

lle

r1

tire

ang

le

Fx

desire

d y

aw

ra

te

slip

an

gle

ya

wra

te

rollr

ate

roll

an

gle

long

. v

elo

city

mo

me

nt

lon

g.

forc

e

yaw

ra

te e

rro

r

Ve

hic

le P

lan

t4

tire

an

gle

Fx

slip

an

gle

ya

wra

te

rollr

ate

roll

an

gle

lon

g.

ve

locity

mo

me

nt

lon

g.

forc

e

Ve

hic

le P

lan

t1

tire

an

gle

desire

d_

ya

wra

te

x_

po

sitio

n

y_

po

sitio

n

Kin

em

ati

c T

ire

Mo

de

l

-K-

Ga

in4

-K-

Ga

in3

-K-

Ga

in

Clo

ck

117

Figure A-9 SIMULINK Model for Vehicle Plant

u(1

)=d

elt

a

u(2

)=x

u(3

)=y

u(4

)=ya

w

u(5

)=ro

ll

u(6

)=V

x

u(7

)=V

y

u(8

)=r

u(9

)=ro

ll_

do

t

u(1

0)=

Vx_

do

t

u(1

1)=

Vy_

do

t

u(1

2)=

r_d

ot

u(1

3)=

roll

_d

ot_

do

t

Vx_

do

t

Vy_

do

t

r_d

ot

roll

_d

ot_

do

t

Vx

Vy r

roll

_d

ot

x y

ya

w

roll

alp

ha

_lf

alp

ha

_lr

alp

ha

_rf

alp

ha

_rr si

gm

a_

lf

sig

ma

_lr

sig

ma

_rf

sig

ma

_rr

om

eg

a_

lf

om

eg

a_

lr

om

eg

a_

rf

om

eg

a_

rr

Fxlf

= u

(1);

Fxlr

= u

(2);

Fxrf

= u

(3);

Fxrr

= u

(4);

Tw

_lf

= u

(5);

Tw

_lr

= u

(6);

Tw

_rf

= u

(7);

Tw

_rr

= u

(8);

8

ya

w r

ate

err

or

7

lon

g.

forc

e

6

mo

me

nt

5

lon

g.

ve

locit

y

4

roll

an

gle

3

roll

rate

2

ya

wra

te

1

slip

an

gle

u(8

)

ya

wra

te1

MA

TL

AB

Fu

ncti

on

ya

w_

dis

turb

an

ce

wh

ee

l_ve

l4

wh

ee

l_ve

l3

MA

TL

AB

Fu

ncti

on

wh

ee

l_ve

l

wh

ee

l_sp

ee

d4

wh

ee

l_sp

ee

d4

MA

TL

AB

Fu

ncti

on

wh

ee

l_m

od

el

MA

TL

AB

Fu

ncti

on

ve

h_

sta

tes

MA

TL

AB

Fu

ncti

on

ve

h_

mo

de

l

slip

_an

gle

Fz

long

_slip

tire

_to

rqu

e

tire

_la

t

tire

_lo

ng

tire

_fo

rce

s3

1 s

spe

ed

slip

_o

ut4

slip

_o

ut4

MA

TL

AB

Fu

ncti

on

slip

_a

ng

le

f(u

)

slip

an

gle

2

u(9

)

roll

rate

1

u(5

)

roll

an

gle

1

d*(

u(1

)+u

(2)-

u(3

)-u

(4))

mo

me

nt1

MA

TL

AB

Fu

ncti

on

lon

g_

slip

Vx_

i

lon

g.

ve

locit

y1

1 s

inte

gra

tor2

1 s

inte

gra

tor1

1 s

inte

gra

tor

forc

es_

ou

t4

forc

es_

ou

t4

Tw

Tw

_rr

Tw

Tw

_rf

Tw

Tw

_lr

Tw

Tw

_lf

Fz_

r

Fz_

rr

Fz_

f

Fz_

rf

Fz_

r

Fz_

lr

Fz_

f

Fz_

lf

Fx_

rr

Fx_

rr

Fx_

rf

Fx_

rf

Fx_

lr

Fx_

lr

Fx_

lf

Fx_

lf

De

mu

x

De

mu

x

De

mu

x

Clo

ck

3

de

sire

d y

aw

ra

te

2 Fx

1

tire

an

gle

118

1

correction angle

MATLAB

Function

active_steering_controller_rollPID

PID Controller1

PID

PID Controller

2

rollover coeff

1

yaw rate error

Figure A-10 SIMULINK Model for Active Steering Controller

4

right rear brake

3

right front brake

2

left rear brake

1

left front brake

MATLAB

Function

differential_braking_controller_roll

1

0.2s+1

brake actuator3

1

0.2s+1

brake actuator2

1

0.2s+1

brake actuator1

1

0.2s+1

brake actuator

PID

PID Controller

Demux1

rollover_coeff

Figure A-11 SIMULINK Model for Differential Braking Controller (1)

119

1

Fx

y aw_rate_error

lef t f ront brake

lef t rear brake

right f ront brake

right rear brake

differential_braking_controller_yaw

rollov er_coef f

lef t f ront brake

lef t rear brake

right f ront brake

right rear brake

differential_braking_controller_roll

Signal 1

Signal Builder4

Signal 1

Signal Builder3

Signal 1

Signal Builder2

Signal 1

Signal Builder1

-K-

Gain4

-K-

Gain3

-K-

Gain2

-K-

Gain1

-1

Gain

Demux

3

yaw_rate_error

2

rollover_coeff

1

long. force

Figure A-12 SIMULINK Model for Differential Braking Controller (2)

120

Appendix D

ADAMS MODEL

Figure A-13 ADAMS Plant Input Variable

Figure A-14 ADAMS Plant Output Variable

121

Figure A-15 ADAMS/Controls Plant Export to MATLAB

1

yaw_rate

ADAMS_yout

Y To Workspace

ADAMS_uout

U To Workspace

ADAMS_tout

T To Workspace

Mux

Mux

Demux

Demux

Clock

Mechanical

Dynamics

ADAMS Plant

2

direct_moment

1

tire_angle

Figure A-16 ADAMS Sub Block Diagram

122

Figure A-17 SIMULINK Model for ADAMS/View Control

Kin

em

atice T

ire M

odel

ya

w_

rate

tire

_in

pu

t_ra

te2

tire

_in

pu

t_ra

te2

tire

_in

pu

t2

tire

_in

pu

t2

tim

e

tim

e

ste

eri

ng

_ra

te

ste

eri

ng

_ra

te1

ste

eri

ng

_in

pu

t

ste

eri

ng

_in

pu

t

du

/dt

rate

4

du

/dt

rate

de

sire

d_

sta

te

de

sire

d_

sta

te

MA

TL

AB

Fu

ncti

on

cu

rve

_in

pu

t

ad

am

s_su

b

wa

^2

s +

2*z

eta

*wa

s+w

a^2

2 actu

ato

r 2

nd

_o

rde

r2

PID

PID

Co

ntr

oll

er1

PID

PID

Co

ntr

oll

er

tire

angle

desired_y

aw

rate

x_positio

n

y_positio

n

Kin

em

ati

c T

ire

Mo

de

l

-K-

Ga

in4

-K-

Ga

in1

Clo

ck