Studies of the stability of the Dispersion Free Steering ...
Vehicle Stability through Integrated Active Steering and … · vehicle‟s yaw stability and...
Transcript of Vehicle Stability through Integrated Active Steering and … · vehicle‟s yaw stability and...
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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
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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
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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.
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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
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differential braking control has been shown to improve the vehicle performance and safety
only in yaw stability and rollover avoidance, respectively.
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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
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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
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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
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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
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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
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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
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List of Tables
Table 5-1 Vehicle Parameters……………………………………………………………………...…39
Table 7-1 Summary of Advantage and Disadvantage for Steering and Braking Control…………….91
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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;
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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;
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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;
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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;
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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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96
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SAE 870421
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Dynamic Studies”, SAE 890087
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Braking”, Proceedings of the Amereican Control Conference, Seattle, Washington,
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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