A Study of Vehicle Properties That Influence Rollover
and Their Effect on Electronic Stability Controllers
Except where reference is made to the work of others, the work described in this thesis
is my own or was done in collaboration with my advisory committee. This thesis does
not include proprietary or classified information.
Kenneth D. Lambert
Certificate of Approval:
George T. Flowers
Alumni Professor
Mechanical Engineering
David M. Bevly, Chair
Associate Professor
Mechanical Engineering
David Beale
Professor
Mechanical Engineering
Joe F. Pittman
Interim Dean
Graduate School
A Study of Vehicle Properties That Influence Rollover
and Their Effect on Electronic Stability Controllers
Kenneth D. Lambert
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
December 17, 2007
A Study of Vehicle Properties That Influence Rollover
and Their Effect on Electronic Stability Controllers
Kenneth D. Lambert
Permission is granted to Auburn University to make copies of this thesis at its
discretion, upon the request of individuals or institutions and at their
expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Kenneth was born July 19, 1983, the second child of Karen and Jon Lambert and
brother to Christopher. Born and raised in Birmingham, Alabama, he attended Cherokee
Bend Elementary School, Mountain Brook Junior High School, and Mountain Brook
High School, after which, he decided to follow in his brother?s footsteps and attend
Auburn University, resulting in many wonderful friends, fond memories, and a Bachelor
of Science degree in Mechanical Engineering. As an undergraduate, Kenneth learned
from his experience designing and building mechanical components for the 2005 Solar
Car. After his undergraduate studies were completed, he continued his education by
staying at Auburn University and working on his Masters of Science degree in Mechanical
Engineering, under Dr. David Bevly in the GPS and Vehicle Dynamics Lab (Gavlab).
iv
Thesis Abstract
A Study of Vehicle Properties That Influence Rollover
and Their Effect on Electronic Stability Controllers
Kenneth D. Lambert
Master of Science, December 17, 2007
(B.S.M.E., Auburn University, 2005)
147 Typed Pages
Directed by David M. Bevly
In this thesis, the vehicle properties that most influence rollover are investigated,
and methods to improve stability are examined. Every year, vehicle rollover is the cause
of thousands of fatalities on US highways. Electronic Stability Controllers (ESC) have
been proven to reduce the incidence of rollover; however, improvement is still possible
and necessary. With the development of a detailed vehicle model that includes roll
and individual wheel dynamics, research has been done to investigate the properties
that most affect rollover. Using these key vehicle properties, equations are developed to
estimate the maximum lateral acceleration and velocity allowed before rollover. With
a good knowledge of the stability limits, ESC systems are developed in simulation, and
testing is done to investigate how these controllers can be optimized to greater ensure
stability during evasive maneuvers. Results prove that stability can be improved and
that rollover can be averted with correct execution of ESC limits and outputs.
v
Acknowledgments
Without the support of family and friends throughout the last few years, this thesis
would not have been possible. I must first give thanks to God for all that He has done
and will continue to do in my life. I feel as if the journey is just beginning.
I would like to thank my parents for the emotional and monetary support that they
have given me over the last 24 years. Without their continuous love and guidance, I
would not be the person I am today. I would also like to thank my brother Chris for
always being a phone call away when I needed a break from work.
I would like to thank Dr. David M. Bevly, my graduate adviser for challenging me
in my undergraduate years and sparking my interest in vehicle dynamics. His motivation
has helped me to realize my potential.
I would also like to thank everyone in the GAVLAB for their help with my graduate
work, especially all of the guys who I got to share the L-2 office (The Deuce) with.
Thanks for letting me by your test driver when one was needed. I now know what 1 g
of braking from 80 mph feels like.
Finally, the completion of my Masters degree would have been a daunting task
without the love and support of my best friend and soon-to-be wife Chame?e. I am
forever indebted to her for the sacrifices she has made over the last two and a half years.
vi
Style manual or journal used Journal of Approximation Theory (together with the
style known as ?aums?). Bibliography follows the IEEE Transactions format.
Computer software used The document preparation package TEX (specifically
LATEX) together with the departmental style-file aums.sty.
vii
Table of Contents
List of Figures xi
1 Introduction 1
1.1 Vehicle Rollover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.1 Rollover Crash Statistics . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 Rollover Crash Costs . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.3 Electronic Stability Controllers . . . . . . . . . . . . . . . . . . . . 3
1.3 Current Efforts for Rollover Safety . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Static Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Unmanned Ground Vehicles and Rollover . . . . . . . . . . . . . . . . . . 6
1.5 Thesis Purpose and Contributions . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Vehicle Modeling 10
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Vehicle Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Angular Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Bicycle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Simple Roll Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Independent Wheel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.7 Tire Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.7.1 Tire Forces and Slip Angles . . . . . . . . . . . . . . . . . . . . . . 21
2.7.2 Tire Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.8 Simulation Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.9 Test Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.9.1 Quasi-Static Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . 30
2.9.2 Dynamic Maneuvers . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.10 Simulations in MATLAB and CarSim . . . . . . . . . . . . . . . . . . . . 37
2.11 Conclusions on Vehicle Modeling . . . . . . . . . . . . . . . . . . . . . . . 39
3 Vehicle Rollover Factors 41
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2 Vehicle Rollover Prediction Formulas . . . . . . . . . . . . . . . . . . . . . 41
3.2.1 Static Stability Factor . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.2 Static Vehicle Rollover Formula . . . . . . . . . . . . . . . . . . . . 43
3.2.3 Inclusion of Suspension Effects . . . . . . . . . . . . . . . . . . . . 47
3.3 Properties That Most Influence Vehicle Rollover Propensity . . . . . . . . 48
viii
3.3.1 CG Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3.2 Track Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.3 Understeer Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.4 Suspension Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.5 Friction Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4 The Inclusion of Understeer Gradient Into the Prediction of Rollover . . . 71
3.4.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.4.2 Empirical Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4 Electronic Stability Controller Development 76
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 ESC Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3 ESC Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Stability Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5 Power Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.6 All-Wheel Braking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.7 Independent Wheel Braking . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.7.1 Controller Development . . . . . . . . . . . . . . . . . . . . . . . . 82
4.7.2 Controller Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.8 Active Torque Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.9 Steering Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.9.1 Steering Control with All-Wheel Braking . . . . . . . . . . . . . . 89
4.9.2 Independent Wheel Braking with Steering Control . . . . . . . . . 92
4.10 ESC with State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5 Simulation Results for ESC 96
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2 Simulation Results for Varying Vehicle Properties . . . . . . . . . . . . . . 97
5.2.1 Varying CG Height . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.2.2 Varying Weight Split . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3 Simulation Results for Optimized ESC Limits and Inputs . . . . . . . . . 103
5.3.1 Varying CG Height With Optimized ESC Controllers . . . . . . . 104
5.3.2 Varying Weight Split With Optimized ESC Controllers . . . . . . 106
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6 Conclusions 109
6.1 Overall Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.1.1 Parameters That Most Influence Rollover . . . . . . . . . . . . . . 109
6.1.2 Vehicle Rollover Prediction . . . . . . . . . . . . . . . . . . . . . . 109
6.1.3 ESC Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.1.4 Effect of Varying Vehicle Properties on ESC . . . . . . . . . . . . . 110
ix
6.2 Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.3 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . 112
Bibliography 114
Appendices 120
A Vehicle Nomenclature 121
B Vehicle Properties 122
C ESC Controller Description 123
C.1 Stability Threshold Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
C.2 ESC Types and Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
C.3 Power Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
C.4 All-Wheel Braking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
C.5 Independent Wheel Braking . . . . . . . . . . . . . . . . . . . . . . . . . . 128
C.6 Active Torque Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 129
C.7 Steering Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
C.8 Steering Modification with All-Wheel Braking . . . . . . . . . . . . . . . . 131
C.9 Independent Wheel Braking with Steering Control . . . . . . . . . . . . . 132
x
List of Figures
1.1 Static Stability Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Static Stability Factor Curve [52] . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 The rollover of a UGV at the 2004 DARPA Grand Challenge . . . . . . . 8
2.1 Vehicle coordinates defined by the SAE [38] . . . . . . . . . . . . . . . . . 10
2.2 Diagram used for the derivation of the lateral velocity and lateral accel-
eration of the bike model . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 The free body diagram for the Bicycle Model . . . . . . . . . . . . . . . . 12
2.4 Roll diagram - static . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Roll diagram - with roll . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 The free body diagram of an unsprung mass for roll equation derivation . 16
2.7 The free body diagram of a sprung mass for roll equation derivation . . . 17
2.8 The free body diagram of a tire . . . . . . . . . . . . . . . . . . . . . . . . 20
2.9 Tire curve diagram for maximum lateral forces . . . . . . . . . . . . . . . 22
2.10 Front tire slip angle diagram . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.11 The tire friction circle used for defining maximum forces allowed [60] . . . 23
2.12 Tire curve for a typical truck tire - Source: CarSim . . . . . . . . . . . . . 25
2.13 Pacejka tire curve diagram for maximum lateral forces . . . . . . . . . . . 29
2.14 The Constant Radius maneuver of a rigid vehicle performed in MATLAB 31
2.15 The vehicle behavior with a constant steer angle and gradually increasing
velocity on an understeer vehicle . . . . . . . . . . . . . . . . . . . . . . . 32
2.16 The SIS maneuver of a rigid vehicle performed in MATLAB . . . . . . . . 33
xi
2.17 The J-turn maneuver performed in MATLAB . . . . . . . . . . . . . . . . 35
2.18 The steer angle inputs for the Fishhook maneuver, defined by NHTSA . . 36
2.19 The Fishhook maneuver performed in MATLAB . . . . . . . . . . . . . . 37
2.20 Comparison of the SIS maneuver performed in MATLAB and CarSim . . 39
2.21 Comparison of the fishhook maneuver performed in MATLAB and CarSim 40
3.1 SSF Diagram - without roll . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 SSF Diagram - with roll . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Diagram used for the derivation of the roll formula . . . . . . . . . . . . . 44
3.4 Lateral Acceleration during Constant Radius maneuver with changing CG
heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5 Rollover Velocities during Constant Radius maneuver with changing CG
heights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6 Rollover Velocities during Constant Radius maneuver with changing CG
heights and weight transfer . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.7 Rollover Velocities during Fishhook maneuver with varying CG heights
and suspension effects (MATLAB simulation) . . . . . . . . . . . . . . . . 53
3.8 Lateral Acceleration during Constant Radius maneuver with a changing
track width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.9 Rollover Velocities during Constant Radius maneuver with a changing
track width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.10 Rollover Velocities during Fishhook maneuver with changing TW and
suspension effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.11 High-speed cornering with steer and slip angles . . . . . . . . . . . . . . . 58
3.12 Steer angle variations with lateral acceleration . . . . . . . . . . . . . . . 61
3.13 Example curve of the understeer test using the constant radius method . 62
3.14 Example curve of the understeer test using the constant speed method . . 63
xii
3.15 Lateral Acceleration during Constant Radius with a changing weight split 65
3.16 Rollover Velocities during Constant Radius with a changing weight split
in CarSim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.17 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.18 Lateral acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.19 The lateral acceleration of a vehicle in the SIS maneuver with changing
suspension stiffnesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.20 Paths of vehicles with varying suspension stiffnesses . . . . . . . . . . . . 69
3.21 Lateral accelerations of vehicles with varying suspension stiffnesses . . . . 69
3.22 The positions of a vehicle in the fishhook maneuver with changing friction
coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.23 Rollover velocities with varying Weight Splits . . . . . . . . . . . . . . . . 72
3.24 Rollover velocities with varying C? Values . . . . . . . . . . . . . . . . . . 72
4.1 ESC sensors diagram. Source: IIHS [53] . . . . . . . . . . . . . . . . . . . 77
4.2 The vehicle?s performance with the power reduction controller . . . . . . . 79
4.3 The vehicle?s performance with the all-wheel braking controller . . . . . . 80
4.4 The vehicle?s braking forces with the all-wheel braking controller . . . . . 81
4.5 The FBD used for the derivation of brake steer moments . . . . . . . . . . 83
4.6 ESC with independent wheel braking. Source: IIHS [53] . . . . . . . . . . 84
4.7 The vehicle?s behavior with the independent wheel braking controller . . . 85
4.8 The vehicle?s longitudinal forces with independent wheel braking . . . . . 86
4.9 The vehicle?s behavior with the added torque controller . . . . . . . . . . 87
4.10 The vehicle?s longitudinal forces with the added torque controller . . . . . 88
4.11 The vehicle?s behavior with the steering modification controller . . . . . . 89
xiii
4.12 The vehicle?s behavior with steering modification and constant braking . . 90
4.13 The vehicle?s velocity with steering modification and constant braking . . 91
4.14 The vehicle?s behavior with independent braking and steering control . . . 92
4.15 The vehicle?s velocity with independent braking and steering control . . . 93
4.16 The vehicle?s longitudinal forces with independent braking and steering
control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.1 The fishhook maneuver with changing CG height and no ESC present . . 98
5.2 The fishhook maneuver with changing CG height and all-wheel braking
and steering modification . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3 The fishhook maneuver with changing CG height and independent wheel
braking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.4 The fishhook maneuver with changing WS and no ESC present . . . . . . 101
5.5 The fishhook maneuver with changing WS and all-wheel braking and
steering modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.6 The fishhook maneuver with changing WS and independent braking . . . 103
5.7 CG height changes with an optimized all-wheel braking and steering ESC 104
5.8 CG height changes with an optimized independent braking ESC . . . . . 105
5.9 WS changes with an optimized all-wheel braking and steering ESC . . . . 106
5.10 WS changes with an optimized independent braking ESC . . . . . . . . . 107
C.1 The single stage controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
C.2 The two stage controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
C.3 The braking torques applied to simulate a power reduction . . . . . . . . 126
C.4 The braking torques applied to simulate a milti-step braking controller . . 127
C.5 The braking torques applied to simulate a variable braking controller . . . 128
C.6 The fishhook maneuver with braking times with independent wheel braking129
xiv
C.7 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
C.8 ? & Lateral Accel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
C.9 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
C.10 ? & Lateral Accel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
xv
Chapter 1
Introduction
1.1 Vehicle Rollover
Rollover of a vehicle occurs when the lateral acceleration reaches a point where two
wheels of the vehicle are lifted off of the ground and the vehicle is rotated toward one
side. Vehicle rollover occurs in two ways, either tripped or untripped. Tripped rollover
accounts for around 95% of the rollover accidents on todays highways and occurs when
a vehicle leaves the roadway and begins to slide sideways, causing the tires to dig into
soft ground or strike an object such a curb. Untripped rollovers normally occur when
a top-heavy vehicle attempts to perform an evasive maneuver that it physically cannot
handle.
Although most rollovers are tripped, the likelihood of being able to prevent rollover
once the tires have become stationary is minimal to none. Also, by examining ways
to prevent untripped rollover, stability and traction can be improved, leading to fewer
chances for untripped rollover to occur.
1.2 Motivation
With an ever increasing number of passenger vehicles on highways today, fatalities
due to vehicle rollover are becoming a larger concern. Increased driver distraction,
Sport Utility Vehicle (SUV) and pick-up truck popularity, along with increased highway
speeds all lead to more accidents. Automobile manufacturers have been developing
and implementing better safety features, improving safety standards, and creating more
1
rigorous maneuvers for the testing of new vehicles. While these have aided in crash
avoidance and severity reduction, more can be done to further save lives and money.
1.2.1 Rollover Crash Statistics
In 2005, noncollision vehicle rollovers accounted for 2.3% of all vehicle accidents on
US highways; however, this small percentage accounted for 10.9% of roadway fatalities.
Also in 2005, 4,266 lives were lost due to noncollision rollover, an increase of 5% over
2004?s 4,045 noncollision rollover fatalities [39, 40, 16].
Table 1.1 - Vehicles in Single-Vehicle Crashes: 1994, 2003, 2004
Vehicle Type - Rollover Occurrence 1994 2003 2004
Passenger Car Rolled Over 114,116 97,962 94,836
Total 1,174,709 1,036,538 980,463
Percentage 9.7 % 9.4 % 9.7 %
Van Rolled Over 9,942 11,408 11,116
Total 100,986 129,757 118,678
Percentage 9.8 % 8.8 % 9.4 %
Pickup Rolled Over 52,123 49,078 48,933
Total 276,363 291,675 292,625
Percentage 18.9 % 16.8 % 16.7 %
Sport Utility Rolled Over 18,154 57,686 56,962
Total 73,469 227,770 246,221
Percentage 24.7 % 25.3 % 23.1 %
Source: NHTSA, NCSA, GES, 1994, 2003, 2004
Table 1.1 shows the number of vehicle rollover occurrences and the total number of
accidents per year [56]. Although the percentage of vehicles that rolled in single-vehicle
accidents is generally decreasing, the number of rollovers involving vans and SUVs is
on the rise, as is the fatality rate. The National Highway Traffic Safety Administration
(NHTSA) has been monitoring rollover incidents closely in the past decade; nevertheless,
demonstrating that more needs to be done to prevent this deadly occurrence.
2
1.2.2 Rollover Crash Costs
Currently, vehicle crashes cost drivers and taxpayers billions of dollars a year. Motor
vehicle accidents not only account for vehicle repair costs, but are also responsible for
medical costs, productivity losses, rehabilitation costs, travel delays, court and legal
costs, emergency services, insurance costs, and the costs to employers. In 2000, the
cost of motor vehicle crashes totaled $230.6 billion dollars in the United States alone
[1]. Despite the fact that vehicle rollovers account for a small percentage of roadway
accidents, it?s costs are high due to the severity of rollover and the fact that most rolled
vehicles are considered by insurance companies to be a total loss.
1.2.3 Electronic Stability Controllers
Electronic stability controllers present in new vehicles have shown that rollovers can
be avoided and costs can be reduced. For over thirty years, the Insurance Institute of
Highway Safety (IIHS) has been conducting analysis of rollover data and crash statistics.
With their results, they estimate that over 10,000 fatalities could be avoided each year
if ESC was installed in all vehicles on the road [53]. ESC has been proven to reduce
fatal multi-vehicle rollover crashes by 32%, fatal single-vehicle crashes by 56%, and
total single-vehicle rollover crashes by over 40%. Susan Ferguson, Institute senior vice
president for research, states, ?The findings indicate that ESC should be standard on all
vehicles.? She continues, ?Very few safety technologies show this kind of large effect in
reducing crash deaths? [53]. With the results of early ESC system, NHTSA has required
ESC to be installed in all new vehicles by the 2012 model year.
3
1.3 Current Efforts for Rollover Safety
In addition to the IIHS?s study of vehicle accidents, the National Highway Traffic
Safety Administration (NHTSA) has been studying the causes and effects of vehicle
rollover [21, 19, 20, 22]. Test procedures have been created by NHTSA to examine a
vehicle?s rollover propensity and compare it to similar vehicles. The testing methods
include static and dynamic maneuvers created specifically by NHTSA researchers to
effectively reproduce real-life situations. Using these testing procedures, NHTSA has
collected data and produced rollover ratings for hundreds of vehicles over the last few
years.
1.3.1 Static Testing
For comparison between vehicles, NHTSA has created a method of comparing
rollover propensity by calculating a vehicle?s Static Stability Factor (SSF). This index is
calculated from a vehicle?s center of gravity (CG) height (HCG) and the lateral distance
between the wheels, or the track width (TW). Figure 1.1 shows how the two properties
are measured.
Figure 1.1: Static Stability Factor
4
Once the two vehicle properties are measured or estimated, the SSF is calculated
by dividing the track width by twice the CG height.
SSF = TW2?H
CG
(1.1)
With the vehicle?s SSF known, its value can be compared to other vehicles and a
rollover percentage can be estimated. Vehicles with a higher CG and narrower track
width have a smaller SSF value (Ex: SUVs, Vans) and have a greater chance for rollover
than those with higher SSF values (Ex: Sports Cars), according to the SSF analysis. The
SSF can also be plotted on a graph similar to Figure 1.2 and a trend can be observed.
Vehicles with a lower SSF value have a higher chance of rollover, and receive a lower star
rating from NHTSA.
Figure 1.2: Static Stability Factor Curve [52]
5
The difference in rollover likelihood can be easily shown using the SSF of a vehicle.
The derivation and background of this rollover propensity index is discussed further in
Chapter 3.
1.4 Unmanned Ground Vehicles and Rollover
Unmanned ground vehicles are beginning to alter the way military operations are
accomplished. Within the last two years, defense contractors have been awarded millions
of dollars to research and develop UGVs, including a two-year, 1 million dollar contract to
Metal Storm [50]. David Smith, Metal Storm CEO, believes that with such an emphasis
on the safety of today?s ground soldiers, it?s no wonder that UGVs are getting some well
deserved attention. Smith says, ?[UGVs] could be the first line of defense in protecting
high-value assets? [31]. John McHale, writer for Military & Aerospace Electronics, says
that ?... autonomous ground vehicles promise to be a major paradigm shift in ground
warfare? [37].
In addition to military UGV?s, the Defense Advanced Research Projects Agency
(DARPA) has had two UGV races that pushed the limits of UGV technology. In the
2004 Grand Challenge, fifteen teams attempted to navigate the 145 mile course that no
team finished. In 2005, the second Grand Challenge had four teams complete the 132
mile course in less than 10 hours. The 2007 Urban Challenge, to be held in November,
will once again push the limits of unmanned navigation of ground vehicles. The latest
race will put UGVs in a mock city environment and require numerous new technologies
to be implemented.
With such a large prize value going to the winners, mission completion is key. Past
teams have suffered from vehicle rollover, which immediately eliminates the chance for
6
2007 DARPA Urban Challenge Top Prizes:
1st Place - $2 Million
2nd Place - $1 Million
3rd Place - $500,000
a win. Figure 1.3 shows the vehicle from team ENSCO rolling during the 2004 Grand
Challenge.
Some UGVs also have the ability to be teleoperated. This is where an operator
remotely drives the vehicle with sight, normally from vehicle mounted cameras. This
method has its limitations due to the fact that the operator has no feedback of the
vehicle?s roll or lateral acceleration, so it becomes easier to roll. Therefore, there is a
need for an ESC to be implemented.
In order for these UGVs to remain stable at the requiredhigher speedswith increased
and varying payloads, an adaptive roll controller needs to be developed to allow for
variances in loading, vehicle, and road properties. New UGVs are also being required
to travel further and at higher speeds, exponentially raising the risk of rollover during
operation, which can be detrimental to a mission for larger-scale vehicles. ESC systems,
similar to those in todays passenger vehicles, will be a vital component to the control
systems of UGVs to maintain their stability as speeds increase to reduce mission times.
1.5 Thesis Purpose and Contributions
Since vehicle rollover is dangerous, costly, and detrimental, its aspects need to be
studied to find how it can be prevented. The research presented in this thesis includes
an investigation into the properties that most influence rollover, in order to analyze how
changes in these properties alter vehicle dynamics. Additionally, this thesis investigates
various ESC algorithms to see how they are affected by these changes.
7
Figure 1.3: The rollover of a UGV at the 2004 DARPA Grand Challenge
The largest contribution of this thesis is the identification of factors that are the
greatest threats to a vehicle?s stability. After identifying the key factors, the Static
Stability Factor equation is examined and a formula is created to predict when a vehicle
will be pushing the limits of stability based upon a few key parameters. The formulas
are then tested in simulation and proven to be a good estimate of vehicle rollover. This
analysis can then be used to predict roll limits and stability thresholds for varying vehicle
configurations.
8
Several types of stability controllers are derived and created in simulation, and two
of the controllers are compared and contrasted for effectiveness and adjustability to
changing vehicle properties. Although individual methods of stability control have been
analyzed, this thesis is one of the first to analyze the adaptability of the ESC systems.
1.6 Thesis Organization
Chapter 2 Vehicle Model Development - Chapter 2 derives the vehicle model used
for stability investigation, defines test procedures, and introduces simulations created in
MATLAB and CarSim.
Chapter 3 Vehicle Rollover Factors - Chapter 3 examines the factors that influence
vehicle rollover, discusses formulas used for predicting vehicle rollover, and investigates
the appropriateness of the formulas.
Chapter 4 Electronic Stability Controller Development - Chapter 4 describes the
various types of stability controllers developed in simulation.
Chapter 5 Simulation Results of ESC - Chapter 5 investigates the adaptability of
two types of ESC systems to varying vehicle parameters. It then looks at the methods
and results of ESC optimization.
Finally chapter 6 completes the comparison of ESC systems and the effectiveness of
the rollover prediction formulas.
9
Chapter 2
Vehicle Modeling
2.1 Introduction
To begin studying the aspects of vehicle rollover, an advanced model is required.
The model is based off of the basic ?Bicycle Model? that accounts for basic vehicle
properties [23, 49, 38]. The model is then modified to include vehicle roll and suspension
characteristics [60, 3, 14, 48, 47, 26, 30]. Finally, the model is extended further to include
individual wheel dynamics, braking, and acceleration forces.
2.2 Vehicle Coordinates
The coordinates used for the vehicle model are defined by the Society of Automotive
Engineers (SAE), and are described by Gillespie and Milliken [23, 38] and shown in figure
2.1.
Figure 2.1: Vehicle coordinates defined by the SAE [38]
10
The SAE convention defines the longitudinal axis of the vehicle to be x, the lateral
axis to be y, and the vertical axis to be z toward the ground. Roll is defined as rotation
about the x axis, Pitch is rotation about the y axis, and Yaw is rotation about the z
axis.
2.3 Angular Dynamics
To begin the analysis, lateral acceleration ay and velocity are calculated for a steady
state turn. figure 2.2 shows how the lateral velocity and lateral acceleration are derived.
Figure 2.2: Diagram used for the derivation of the lateral velocity and lateral acceleration
of the bike model
Since the turning is steady state, the radius of the turn (R) is constant, and ?Vy = 0,
so there is no lateral sliding. Also, the yaw rate of the vehicle (r) is equal to the angular
11
velocity of the vehicle (?), in reference to the point of rotation. The steady state angular
acceleration ay consists of only the centripetal acceleration, and can be written in terms
of velocity and yaw rate or turning radius as shown below.
Vx = R?? = R?r (2.1)
ay = ?Vy = R??2 = R?r2 = V
2x
R = V ?r (2.2)
2.4 Bicycle Model
The ?Bicycle Model? is simply a two wheeled vehicle model that is used to study
basic vehicle dynamics. In this research, the Bicycle Model is used to describe the
lateral vehicle dynamics only. Roll, pitch, weight transfer, and longitudinal wheel slip
are assumed to be negligent and are not taken into account in the Bicycle Model. Also,
steer angles and tire slip angles (?) for the left and right tires are combined into averages
of thetwo. Thefollowing figureshowsthe freebodydiagram (FBD) for thebicycle model.
Figure 2.3: The free body diagram for the Bicycle Model
In order to find the lateral dynamic properties for the Bicycle Model, summations
of the forces are taken in the lateral (y) and longitudinal (x) directions from Figure 2.3
12
as shown below.
summationdisplay
Fx = MT ?ax = FyF ?sin(?) (2.3)
summationdisplay
Fy = MT ?ay = FyR +FyF ?cos(?) (2.4)
Forces due to braking, acceleration, rolling resistance, and air drag are ignored for
now, but will be added later. MT is the total mass of the vehicle (sprung and unsprung).
In order to solve for the yaw rate (r), the moment is taken at the center of gravity.
summationdisplay
MCG = Iz ? ?r = ?b?FyR +a?FyF ?cos(?) (2.5)
By simply rearranging Equation (2.5), the vehicle yaw acceleration can be found.
?r = 1I
z
?[Fyf ?a?cos(?)?Fyr ?b] (2.6)
Vehicle sideslip (?) is the angle between the vehicle?s heading and course and is
calculated by taking the arcsine of the lateral and total velocities of the vehicle.
? = sin?1 VyV (2.7)
From Figure 2.3, the relationships for the longitudinal and lateral velocities can
be derived. At higher speeds, the vehicle slip angle (?) will play a role in the vehicle
13
dynamics. Without the effects of yaw rate, the equations are:
Vy = V ?sin(?) ?Vy = ?V ?sin(?) +V ? ?? ?cos(?) (2.8)
Vx = V ?cos(?) ?Vx = ?V ?cos(?) +V ? ???sin(?) (2.9)
With effects of yaw rate added, the equations are:
?Vy = ?V ?sin(?) +V ? ?? ?cos(?) +V ?r?sin(?) (2.10)
?Vx = ?V ?cos(?) +V ? ?? ?sin(?)?V ?r?cos(?) (2.11)
These equations will be used later in the derivation of the roll angle. In a steady-
state turn, the centripetal acceleration can be found to be:
acen = Vx ?r = V ?r?cos(?) (2.12)
By rearranging Equation (2.4) and substituting in Equation (2.12), the lateral ac-
celeration can be found.
ay = ?Vy +acen = FyR +FyF ?cos(?)MT (2.13)
?Vy = FyR +FyF ?cos(?)
MT ?V ?r?cos(?) (2.14)
Note that the above equations include the effects of centripetal acceleration and
sideslip. The lateral tire forces used in Equations (2.13) and (2.14) are derived using
various tire models. This will be discussed further in Section 2.7.
14
2.5 Simple Roll Model
The next step in increasing the complexity of the vehicle model is adding the roll
dynamics. By doing this, the vehicle model includes vertical and lateral forces for the
left and right sides of the vehicle, and lateral weight transfer is included.
The simple vehicle roll model is made by creating a two-dimensional diagram of the
sprung and unsprung masses. The sprung mass is held up by a fictitious pivot arm (d1),
from the vehicle?s roll center, the effective pivot point at which the sprung mass pivots.
Spring forces are initialized as zero after the static deflection. figures 2.4 and 2.5 show
how the roll model is derived.
Figure 2.4: Roll diagram - static Figure 2.5: Roll diagram - with roll
In order to derive the roll dynamics, the roll model is split up into Unsprung and
Sprung halves. figures 2.6 and 2.7 show the FBDs used for the derivation of the equations
of motion for the vehicle roll.
15
Figure 2.6: The free body diagram of an unsprung mass for roll equation derivation
The weight of the vehicle is accounted for with resultant forces Rz and Ry. In order
to solve for the equations of motion, forces are added and the moment is taken at the
roll center for the unsprung mass in Figure 2.6.
summationdisplay
Fy = m? ?y = 0 = Ry ?FyL ?FyR (2.15)
summationdisplay
Fz = m? ?z = 0 = FzL +FzR +FsL ?FsR ?Rz ?m?g (2.16)
summationdisplay
MRC = Ix ? ??Unsprung = 0 = hRC ?(FyL +FyR)+ TW2 ?(FzL ?FzR)
+ B ? ??+K? ??+MARB (2.17)
Here, B is the roll damping of the vehicle, K? is the roll stiffness, and MARB is
the moment applied by the anti-roll, or torsion bar. The equations are solved with
the assumption of steady-state axle dynamics. Rearranging Equations (2.15) and (2.17)
16
yields:
m? ?y = Ry ?FyL ?FyR = 0 (2.18)
TW
2 ?(FzR ?FzL) = B?
??+K? ??+MARB +hRC ?(FyL +FyR) (2.19)
By combining the previous equations with ones taken from the sprung mass, seen in
Figure 2.7, the vehicle model can be further expanded to include roll angles and weight
transfer.
Figure 2.7: The free body diagram of a sprung mass for roll equation derivation
17
The forces for the sprung mass are also added together and the moment is taken
again at the roll center.
summationdisplay
Fy = M ?ay = Ry (2.20)
summationdisplay
Fz = M ?az = Rz +FsR ?FsL ?M ?g = 0 (2.21)
summationdisplay
MRC = Ix ? ?? = M ?g?d?sin(?) +M ? ?y?d?cos(?)
? MARB ?B? ???K? ?? = 0 (2.22)
Solving for the roll acceleration (??), the following equation can be derived.
?? = 1
Ix ?[M ?g?d?sin(?) +M ? ?y?d?cos(?)?MARB ?B ?
???K? ??] (2.23)
Substituting in Equation (2.10), the roll acceleration is found.
?? = 1
Ix ?[M ?g?d?sin(?)?B ?
???K? ???MARB
+ M ?( ?V ?sin(?) +V ? ?? ?cos(?) +V ?r?sin(?))?d?cos(?)] (2.24)
The weight transfer equations can also be formed from these equations. Rearranging
Equations (2.15) and (2.20) and substituting for Ry yields:
MT ? ?y = Ry = FyL +FyR (2.25)
Substituting this into the unsprung moment, the following equation is derived:
TW
2 ?(?Fz) =
TW
2 ?(FzR ?FzL) = B ?
??+K? ??+MARB +hRC ?MT ? ?y (2.26)
18
In order to account for different weight splits, the weight transfer equation is split
into front and rear components. Also, ?y is altered to include yaw rate, resulting in the
following equation for the front and rear weight transfer.
?Fzf = 2TW
f
?[Bf ? ??f +K?f ??+MARBf
+ hRCf ?MTf ?( ?V ?sin(?) +V ? ?? ?cos(?) +V ?r?sin(?))] (2.27)
?Fzr = 2TW
r
?[Br ? ??r +K?r ??+MARBr
+ hRCr ?MTr ?( ?V ?sin(?) +V ? ?? ?cos(?) +V ?r?sin(?))] (2.28)
Vertical tire forces can then be calculated by taking the static load and subtracting
or adding the weight transferred during the maneuver.
FzfL = Fzf2 ??Fzf FzfR = Fzf2 +?Fzf
FzrL = Fzr2 ??Fzr FzrR = Fzr2 +?Fzr (2.29)
Because the tire lateral forces are a function of vertical load, these vertical forces for the
individual tires can now be used for solving the lateral tire forces. In order to find these
lateral forces, a model of the tire is required as shown in Section 2.7.
2.6 Independent Wheel Model
In order to implement independentwheel braking, the vehicle model needs to include
the dynamics of the independent wheels and tires [15]. To do this, a free body diagram
is derived for an individual wheel. Figure 2.8 shows the side view of a tire, including
braking and engine torques.
19
Figure 2.8: The free body diagram of a tire
After analysis of the tire FBD, the results in the following set of equations are
derived and used to describe the tire dynamics:
summationdisplay
Fx = M ?ax = FxBearing ?Fx (2.30)
summationdisplay
Fz = M ?az = FzBearing ?Fz = 0 (2.31)
summationdisplay
MBearing = Iw ? ?? = Reff ?Fx +?engine ??brake (2.32)
Then solving for the longitudinal force, the solution becomes:
Fx = 1R
eff
[Iw ? ?? +?brake ??engine] (2.33)
The force in Equation (2.33) is the longitudinal force applied by the vehicle to the
tire. Although this force includes the effects of engine and braking torque, tire models
20
will still limit the lateral and longitudinal forces allowed by the tires. To correctly account
for these effects, tire models must be investigated and applied to the vehicle model.
2.7 Tire Properties
In order to properly model a vehicle with pneumatic tires, a model must be im-
plemented to describe the lateral and longitudinal forces that are limited by physical
properties of a tire.
2.7.1 Tire Forces and Slip Angles
The forces that are allowed by a tire depends on several factors, however there is a
maximum friction force allowed when a non-linear model is used. This peak tire force
is dependent upon the vertical force and slip angle of the tire, and other factors such as
air pressure, surface characteristics, and temperature. Figure 2.9 depicts a typical tire
curve in which the peak force can be seen.
The tires slip angle (?) and cornering stiffness (C?) will be discussed in greater
detail later in this section.
The total horizontal tire force is divided up into lateral and longitudinal directions.
Figure 2.10 shows how the tire?s velocity components are defined.
The magnitudes of these forces are limited by physical properties that can be ex-
plained the tire friction circle [10]. Figure 2.11 demonstrates how the tire forces are
limited using the concept of the friction circle. As seen in the figure, if the vehicle is
accelerating or braking, the peak lateral force allowed by the tires is decreased.
21
Figure 2.9: Tire curve diagram for maximum lateral forces
Equation (2.34) defines how the maximum lateral and longitudinal forces allowed
by the tire are limited by the peak tire force.
Fz ?? ? Ftire =
radicalBig
Fx2 +Fy2 (2.34)
Figure 2.10: Front tire slip angle diagram
22
Figure 2.11: The tire friction circle used for defining maximum forces allowed [60]
The peak tire force is a function of the vertical tire force, and the tire-ground
friction coefficient (?). The tire-ground friction coefficient is considered to be constant in
simulation for simplicity; however, in actuality, it changes due to variations in load, tire,
and surface conditions. With the tire?s velocities broken up into lateral and longitudinal
components and the steer angle averaged from left and right tires, the tire?s slip angle
(?) can be solved. From Figure 2.10, the slip angles of the tires can be found. Equations
(2.35), (2.36), (2.37), and (2.38) show the slip angles calculated from the tire?s velocity
components.
23
?fL = tan?1
bracketleftBigg
V ?sin(?) +r?a
V ?cos(?) +r? twf2
bracketrightBigg
?? (2.35)
?fR = tan?1
bracketleftBigg
V ?sin(?) +r?a
V ?cos(?)?r? twf2
bracketrightBigg
?? (2.36)
?rL = tan?1
bracketleftBigg
V ?sin(?)?r?b
V ?cos(?) +r? twr2
bracketrightBigg
(2.37)
?rR = tan?1
bracketleftBigg
V ?sin(?)?r?b
V ?cos(?)?r? twr2
bracketrightBigg
(2.38)
One could add a rear steer component by simply adding in ?rear to account for the
addition of a rear steer angle to the bicycle model. Equations (2.35), (2.36), (2.37), and
(2.38) also include effects of the vehicle?s yaw rate, which is sometimes removed. These
effects could have been ignored; however, with the large yaw rates induced during some
rollover maneuvers, the model is more accurate when these terms are included.
2.7.2 Tire Models
Several tire models are available to capture the tire characteristics shown previously
in Figure 2.9. The most popular tire models include the linear, Dugoff, and Pacejka
tire models, and a look-up table [10]. The linear tire model is acceptable for slow speed
maneuvers; however, since vehicle roll is being studied, higher speeds are used and the
model will not accurately capture the dynamics of the tires in these ranges. The Dugoff
and Pacejka tire models have previously been proven accurate and are widely accepted,
and simulations described in this thesis use both models.
Simulations in CarSim were done using a look-up table model. Here, a vertical force
and slip angle are given, and the simulator looks up the correct lateral and longitudinal
24
forces in a table. Although this method can be very accurate, it requires data taken
from tire tests and cannot be easily modified. Figure 2.12 shows the lateral force curve
for a typical truck tire in the CarSim tire library.
Figure 2.12: Tire curve for a typical truck tire - Source: CarSim
Dugoff?s Model
The Dugoff Tire model was first published in 1969 as a method for finding the lateral
force allowed by a tire [11, 12]. Instead of the actual parabolic shape of the tire forces,
the Dugoff model assumes that a uniform pressure is distributed on the tire?s contact
patch. This simplification of the tire?s forces allows easier calculations, and requires
fewer parameters to be known. Another advantage of this model is that tire stiffness
values for the lateral and longitudinal directions can be independently defined.
The equations for the tire forces using the Dugoff tire model are as follows:
Fy = ?C? ?tan(?)?f(?) (2.39)
Fx = ?C? ? ?x1+?
x
?f(?) (2.40)
25
where C? is the tire?s cornering stiffness, C? is the longitudinal stiffness, and ?x is the
longitudinal slip ratio. The parameters F? and ? are defined as:
f(?) =
?
??
??
(2??)?? if ? < 1
1 if ? ? 1
? = ??Fz ?(1 +?x)
[(C? ??x)2 +(C? ?tan(?))2]12
where Fz is the tire?s normal force, and ? is the friction coefficient for the tire and road.
The lateral force allowed is directly proportional to the tire cornering stiffness, and the
longitudinal force allowed is proportional to the longitudinal tire stiffness.
The Pacejka Model
In the 1960s, Hans Pacejka became a lead researcher into the propertiesof pneumatic
passenger vehicle tires. His research led him to publish several papers [43, 42] and a book
[44] on this subject. His tire model is considered to be quite accurate and is widely used
in modeling of tires today.
The non-linear equations for the Pacejka begins with the general equation [49]:
Y(X) = D?sin(C ?tan?1(B ?x?E(B ?x?tan?1(B ?x)))) (2.41)
where Y(X) = y(x) + Sv and x = X ? Sh. Sv is the vertical shift and Sh is the
horizontal shift that can be included in the model. Due to the fact that this thesis is not
an investigation into tires and the shift parameters are not well known, these properties
were set to zero. Y is the output variable (Fx, Fy, or Mz) and X is the input variable
26
(slip angle ? or slip ratio ?x). Equation (2.41) is then simplified to solve for the lateral
and longitudinal forces allowed by the tires.
Fy = D?sin(C ?tan?1(By ???E(By ???tan?1(By ??)))) (2.42)
Fx = D?sin(C ?tan?1(Bx??E(Bx ???tan?1(Bx ??)))) (2.43)
The input variable is set to be the slip angle ?, and B is adjusted when solving for
lateral or longitudinal forces. To fill in the force equations, several tire properties must
be known. The first, C in Equation (2.44), is the shape factor.
C = 2pi ?sin?1
parenleftBigys
D
parenrightBig
(2.44)
This property is independent of the normal force distributed on the tire. The second
Pacejka tire property, D in equation 2.45, is the key factor in determining the maximum
lateral force on the tire curve.
D = a1 ?F2z +a2 ?Fz (2.45)
This property is dependent on only the normal force distributed on the tire and the
Pacejka parameters. The third Pacejka tire property, E in equation 2.46, is the curve
factor of the tire curve.
E = a6 ?F2z +az ?Fz +a8 (2.46)
This property is also dependent on only the normal force distributed on the tire and
the Pacejka parameters, but affects the curvature of the maximum tire force curve. The
27
fourth Pacejka tire property, B, is derived from Equations (2.47) and (2.48). For the
lateral tire forces, BCD is defined as:
ByCD = a3 ?sin(a4 ?tan?1(a5 ?Fz)) (2.47)
And for the longitudinal case:
BxCD = a3 ?F
2z +a4 ?Fz
ea5?Fz (2.48)
BCD is effectively the cornering stiffness, C?, and defines she slope of the tire curve at
small slip angles ?.
Parameters a0,a1,...,a8 are constant terms defined for each tire. These values are
found from test data and are dependent on the tire. Table 2.1 shows parameters taken
from a paper written by Pacejka in 1989 [42].
Table 2.1 - Pacejka Tire Parameters
a0=0 a5=0.208
a1=-22.1 a6=0
a2=1011 a7=-0.354
a3=1078 a8=0.707
a4=1.82
The resulting lateral and longitudinal forces are dependent upon the slip angles
of the tire, vertical force, and tire properties. Figure 2.13 shows the results for a tire
modeled using the Pacejka tire model. The tire?s cornering stiffness C? and peak force
are altered with changing normal forces; however a point is reached where increasing the
normal tire force allows little change in lateral force produced.
28
?30 ?20 ?10 0 10 20 30?6000
?4000
?2000
0
2000
4000
6000
Lateral Force vs. Slip Angle
? (deg)
F y
(N)
Fz= 1500 N
Fz= 3000 N
Fz= 4500 N
Fz= 6000 N
Figure 2.13: Pacejka tire curve diagram for maximum lateral forces
2.8 Simulation Vehicle
The vehicle modeled in this thesis is a typical SUV with properties taken from a 2000
Chevrolet Blazer. The vehicle was chosen due to previous knowledge of the parameters,
and experimental testing has previously been done to confirm their accuracy. Actual
properties used in this thesis can be found in Appendix B.
2.9 Test Maneuvers
In order to correctly compare the derived vehicle model to that in CarSim, testing
maneuvers must be created to perform validation. NHTSA has created several types of
29
maneuvers for vehicle rollover testing [22, 8]. The maneuvers are designed to test certain
aspects of the vehicle?s behavior. Quasi-static maneuvers test the vehicles likelihood
to rollover in steady-state turning, while dynamic maneuvers provoke transient vehicle
properties that are brought about by dynamic weight transfer and suspension effects.
For simulations, the friction coefficient (?) can be altered, depending on the simulated
surface.
2.9.1 Quasi-Static Maneuvers
Quasi-static rollover testing is composed of maneuvers that test a vehicle?s propen-
sity for rollover with a model that removes the effects of transients. Maneuvers that fall
under the quasi-static realm include the constant radius and the steadily increasing steer.
These maneuvers gradually increase the vehicle?s steer angle or velocity, causing a semi-
static rollover. The quasi-static testing done in this thesis begins with a rigid vehicle,
and then the suspension is added in order to investigate its effect during the quasi-
static maneuvers. Although, the quasi-static maneuvers incorporate the main factors
that affect vehicle rollover, they do not include effects of vehicle roll due to suspension
configurations and other transient properties.
2.9.1.1 Constant Radius
The constant radius maneuver consists of a vehicle going around a circular track
with a constant turning radius (R). The vehicle simulation is begun at a stop or slow
speed, and then a low constant longitudinal acceleration is applied to the vehicle. Fig-
ure 2.14 shows the vehicle?s performance during a standard constant radius simulation.
The maneuver is considered to be quasi-static due to the fact that it does not excite
30
the dynamic behavior of the vehicle until wheel lift occurs. Suspension transients are
minimal, since the roll rate is small, and the lateral acceleration is limited by the friction
coefficient of the roadway.
0 10 20 30 40
0
0.2
0.4
0.6
Lat. Accel. (g) 0 10 20 30 400
10
20
30
40
Velocity (mph)
0 10 20 30 40?4
?3
?2
?1
0
1
Time (sec)
Roll (deg)
0 10 20 30 405.2
5.3
5.4
5.5
5.6
5.7
Time (sec)
delta (deg)
?40 ?20 0 20 40
?80
?70
?60
?50
?40
?30
?20
?10
0
EAST (m)
NORTH (m)
Figure 2.14: The Constant Radius maneuver of a rigid vehicle performed in MATLAB
Obstacles to overcome in this simulation include the fact that steer angle will not
be constant to hold the constant radius. Due to the increasing velocity, the vehicle will
become more understeer, caused by the increasing slip angles. Figure 2.15 shows what
would occur if the steer angle were held constant while steadily increasing in velocity.
31
0 5 10 15 20 25 304
4.5
5
5.5
6
6.5
7
time (sec)Front Tire Angle (deg)
?40 ?20 0 20 40
?80
?70
?60
?50
?40
?30
?20
?10
0
East (m)
North (m)
Figure 2.15: The vehicle behavior with a constant steer angle and gradually increasing
velocity on an understeer vehicle
As the vehicle increases in velocity, the vehicle?s path is widened, and the radius
is not constant. Since the vehicle modeled is slightly understeer, the steer angle must
increase to hold a constant radius during the acceleration. The understeer gradient of a
vehicle is discussed in detail in Chapter 3.
32
2.9.1.2 Steadily Increasing Steer
The Steadily Increasing Steer (SIS) maneuver consists of a vehicle traveling at a
constant velocity, and uniformly increasing the steer angle. The maneuver is considered
to be quasi-static due to the fact that it does not excite the suspension dynamics, similar
to the CR test. Figure 2.16 shows how a rigid vehicle performs in the SIS Maneuver.
0 50 100 150 200
0
20
40
60
80
100
120
EAST (m)
NORTH (m)
0 5 10 15 20?8
?7
?6
?5
?4
?3
?2
?1
0
time (sec)
Front Tire Angle (deg)
0 5 10 15 20?0.7
?0.6
?0.5
?0.4
?0.3
?0.2
?0.1
0
0.1
Time (sec)
Lateral Acceleration (g)
0 5 10 15 20?0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
Vehicle Roll (deg)
Figure 2.16: The SIS maneuver of a rigid vehicle performed in MATLAB
Although the maneuver can be used to analyze a vehicle?s behavior during a quasi-
static maneuver, NHTSA uses it to define maximum steer angles in other maneuvers.
For example, when the lateral acceleration reaches 0.3 g during this maneuver, the steer
33
angle is taken, multiplied by 6.5, and used for the maximum steer angle in the fishhook
maneuver.
2.9.2 Dynamic Maneuvers
Since real life vehicle rollovers are caused by dynamic maneuvers, testing that incor-
porates the effects created by vehicle transients should be conducted. In 2004, NHTSA
began to test a vehicle?s rollover propensity with dynamic maneuvers, in order to more
accurately assess a vehicle?s rollover propensity [22]. The maneuvers defined by NHTSA
are created to excite certain vehicle dynamics that are not taken into account in the
static and quasi-static testing procedures. The maneuvers included in NHTSA?s testing
are the J-turn, the fishhook, and the double lane change. The J-turn maneuver is a re-
sult of a step steer angle applied during a constant velocity run. The fishhook maneuver
simulates a road edge recovery maneuver that emulates a sinusoidal steering input that
greatly increases the vehicle?s chances for rollover. The double lane chance maneuver
is quite similar to the fishhook maneuver, in that it requires two steer angles, but it
does not excite the vehicle dynamics as much as the fishhook, and will not be discussed
further in this thesis.
2.9.2.1 J-Turn
The J-turn maneuver is one of the simplest dynamic test that can be performed. It
consists of a vehicle driving at a constant velocity with the application of a sudden and
large steer angle. The vehicle?s behavior during the maneuver is shown in Figure 2.17.
The J-turn maneuver is a good test of how weight transfer affects a vehicle when a step
steer input is applied. It is also a highly-repeatable maneuver, although its profile is not
34
one usually seen on actual roadways. For this reason, the fishhook maneuver is a better
choice for testing real-life situations.
0 20 40 60?50
?40
?30
?20
?10
0
EAST (m)
NORTH (m)
0 5 10 15?2
0
2
4
6
8
10
12
time (sec)
Front Tire Angle (deg)
0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time (sec)
Lateral Acceleration (g) 0 5 10 15?5
?4
?3
?2
?1
0
Time (sec)
Vehicle Roll (deg)
Figure 2.17: The J-turn maneuver performed in MATLAB
2.9.2.2 Fishhook
The fishhook maneuver developed by NHTSA is a good example of a real-world
road edge recovery maneuver. The fishhook is also an excellent maneuver due to its
repeatability and its ability to excite the dynamics of a vehicle that influence rollover
[62].
The inputs of NHTSA?s fishhook maneuver are defined by a vehicle?s performance
during other maneuvers. The maximum steer angle, A, is defined from the vehicle?s
performance during the SIS maneuver at 50 mph using a continuously growing steer
35
angle input of 13.5 deg/sec at the handwheel. When the vehicle reaches 0.3 g of lateral
acceleration, the steer angle is measured, and multiplied by 6.5 to get A, the fishhook
maximum steer angle. Figure 2.18 shows the steer input for the NHTSA fishhook ma-
neuver. It is usually performed at 50 mph, although other speeds are acceptable as long
as they are compared to other vehicles tested at the same speed.
Figure 2.18: The steer angle inputs for the Fishhook maneuver, defined by NHTSA
The time constants T1 and T2 were set by NHTSA to be 0.25 and 3.0 seconds
respectively. The times are chosen for their ability to excite the vehicle?s roll rate into
a semi-sinusoidal reaction. Figure 2.19 shows the behavior of the modeled SUV in the
fishhook maneuver performed in MATLAB.
For simulations in this thesis, the fishhook maneuver is the dynamic test used due
to the fact that it is easily repeatable, simulates real-world rollover threats, and is able
to excite the vehicle?s dynamic properties.
36
0 20 40 60 80?10
0
10
20
30
40
50
EAST (m)
NORTH (m)
0 2 4 6 8 10?15
?10
?5
0
5
10
15
time (sec)
Front Tire Angle (deg)
0 2 4 6 8 10?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
Time (sec)
Lateral Acceleration (g)
0 2 4 6 8 10?4
?2
0
2
4
6
Time (sec)
Vehicle Roll (deg)
Figure 2.19: The Fishhook maneuver performed in MATLAB
2.10 Simulations in MATLAB and CarSim
The simulations in this thesis were created using MATLAB and CarSim. MAT-
LAB, an interactive programming environment, has a great ability to allow the user to
define exactly what is desired, since the model dynamics are user defined. CarSim, a
commercially available simulation package for multibody vehicle dynamics, has several
advantages in that it provides a more complex vehicle model, and already has tested ve-
hicle models built in. It allows the user to create or modify preexisting steering inputs,
velocity profiles, vehicles, and roads among other things.
Of course, simulations in both programs have their limitations. The simulation of
vehicle dynamics has been researched and published by several authors [28, 55] and the
limitations of 3-dimensional vehicle simulations have been discussed by Day and Garvey
37
[9]. MATLAB?s limitations are small, although they include the requirement that the
all of the vehicle properties must be found and inserted by the user, all equations of the
vehicle must be fully developed and the programmer must be certain that all units are
consistent. This is not the case with CarSim, since all of the vehicle dynamic equations
are built in to the software package. The limitations of CarSim include the fact that
some properties are hard to set, user inputs are limited, and equations used in the vehicle
model are not known.
In order to verify the suitability of the simulations, maneuvers were performed in
both programs and then compared with each other. The first maneuver, the SIS, is
shown in Figure 2.20. The results from the simulations are fairly uniform. A slight
difference can be seen in the yaw rate and lateral acceleration, likely due to a difference
in one of the many tire properties incorporated in the CarSim model. The difference can
be shown in the position plot, although the results were deemed close enough for testing
purposes in this thesis.
The second maneuver used for comparison is the fishhook. Figure 2.21 shows the
vehicle?s behavior during the dynamic maneuver. The results of the fishhook maneuver
comparison are similar to the SIS maneuver, although there is a slight difference in
steer angle between the maneuvers. Despite the small variation in steer angle (due to
a smoothing function in MATLAB simulation), the results for the lateral acceleration,
yaw rate, and position are actually closer to each other than in the SIS maneuver. The
only other discrepancy between the two simulations is a small oscillation in the lateral
acceleration during the CarSim simulation. This is due to minimal differences in the
spring rate and damping in the suspension or tires. Once again, for the requirements in
this thesis, the resulting discrepancy in the two simulations is acceptable.
38
0 50 100
0
20
40
60
80
100
EAST (m)
NORTH (m)
MATLAB
CarSim
0 5 10 15?8
?7
?6
?5
?4
?3
?2
?1
0
time (sec)
Front Tire Angle (deg)
0 5 10 15?25
?20
?15
?10
?5
0
Yaw Rate (deg/s)
Time (sec)
0 5 10 15?0.5
?0.4
?0.3
?0.2
?0.1
0
Time
Lateral Acceleration (g)
Figure 2.20: Comparison of the SIS maneuver performed in MATLAB and CarSim
2.11 Conclusions on Vehicle Modeling
The vehicle model developed in this chapter has been implemented in a MATLAB
simulation and has been tested for accuracy. With prior testing by others and a compar-
ison to the commercially developed vehicle-modeling software CarSim, the vehicle model
exhibits true-to-life vehicle behavior. With this vehicle model, testing with the devel-
oped quasi-static and dynamic maneuvers can now be performed in order to investigate
39
0 20 40 60
?10
0
10
20
30
40
50
EAST (m)
NORTH (m)
MATLAB
CarSim
0 5 10?20
?15
?10
?5
0
5
10
15
20
time (sec)
Front Tire Angle (deg)
0 5 10?50
0
50
Yaw Rate (deg/s)
Time (sec)
0 5 10?1
?0.5
0
0.5
1
Time
Lateral Acceleration (g)
Figure 2.21: Comparison of the fishhook maneuver performed in MATLAB and CarSim
the properties that affect vehicle rollover and ESC systems can be inserted to the vehicle
model to test how the rollover properties can accounted for in the roll controllers.
40
Chapter 3
Vehicle Rollover Factors
3.1 Introduction
Several properties affect how a vehicle will perform during evasive maneuvers. With
a good knowledge of these properties, rollover formulas can be derived and tested for
accuracy. Testing can then be done with the rollover formulas to see if the maximum
lateral acceleration and velocity before rollover are accurately predicted when vehicle
properties such as CG height, track width, and understeer gradient are altered.
3.2 Vehicle Rollover Prediction Formulas
One important aspect of studying vehicle rollover is the knowledge of when a vehicle
will rollover. In this chapter, two types of vehicle rollover formulas are to be discussed:
the Static Stability Factor and another that is derived to predict when a vehicle will
rollover, given some key vehicle properties.
3.2.1 Static Stability Factor
NHTSA?s basic measurement of vehicle rollover propensity is the Static Stability
Factor (SSF). This static test ignores all vehicle properties other than the center of
gravity height and track width (the width between the left and right tires).
The SSF equation is derived by finding what lateral force would be required to push
the vehicle over, when the outer wheel is tripped. Figures 3.1 and 3.2 show the diagrams
41
used during derivation. In Figure 3.2, there are no forces on the inner wheel since the
FBD is made at the point where the vertical force on that wheel is zero.
Figure 3.1: SSF Diagram - without roll Figure 3.2: SSF Diagram - with roll
To begin developing the SSF equation, the forces in the horizontal and lateral di-
rections are added and the moment is taken about the CG. Both are set to zero, since
the system is considered to not be accelerating (i.e. in steady state).
summationdisplay
Fy = M ?ay = Fy (3.1)
summationdisplay
Fz = M ?az = M ?g?Fz = 0 (3.2)
Rearranging Equations (3.1) and (3.2), one can solve for Fy and Fz.
M ?ay = Fy (3.3)
M ?g = Fz (3.4)
42
The moment is also taken at the center of gravity to complete analysis.
summationdisplay
MCG = TW2 ?Fz ?HCG ?Fy = 0 (3.5)
Rearranging Equation (3.5) and replacing values for Fy and Fz to get the SSF, the basic
equation can be found.
HCG ?M ?ay = TW2 ?M ?g (3.6)
The above equation can be further rearranged to solve for ay to obtain the SSF equation.
When ay is divided by g, a unitless value is created with the value that is used as the
Static Stability Factor. Equation (3.7) shows the formula for the SSF.
SSF = ayg = TW2?H
CG
(3.7)
This equation is essentially the value of lateral acceleration (in units of ?g?s) needed
to roll a vehicle when the friction coefficient is high enough to allow for tripping.
3.2.2 Static Vehicle Rollover Formula
In order to define a rollover threshold, an equation predicting the point of rollover
would be greatly beneficial. Previous research has been done on the creation of a rollover
prediction formula, however, little has been published with simulations or experimental
data validating the formula [24, 23, 45, 32].
In order to do this without too much complexity, some assumptions must be made
as well as some vehicle properties ignored. The first step is to define what is going to
be considered when a vehicle rollover is eminent. Several ways to define vehicle rollover
43
detection have been previously published [25, 60], but for the purpose of creating a
formula, rollover is to be considered eminent when a vehicle?s normal force is distributed
on only the outer tires of the vehicle (i.e. vertical forces on the inner tire become zero).
Another simplification that must be made is to eliminate the effects of suspension
characteristics. By stiffening the suspension to a point of rigidity, the whole vehicle can
be considered to be a solid mass and suspension transients is ignored. This not only
removes suspension effects, it also throws out changes in CG height and lateral distance
from the CG to the tires. Figure 3.3 shows how the forces are distributed over the
simplified vehicle model.
Figure 3.3: Diagram used for the derivation of the roll formula
With the simplified vehicle model, one can now take a summation of the forces in
the vertical and lateral directions.
summationdisplay
Fy = M ?ay = Fy (3.8)
summationdisplay
Fz = M ?az = M ?g?Rz (3.9)
44
Rearranging equations 3.8 and 3.9, one can solve for Ry and Fz.
Fy = M ?ay = M ? V
2
R = M ?V ?r (3.10)
M ?g = Rz (3.11)
The moment can also be taken at the center of gravity to further the analysis.
summationdisplay
MCG = TW2 ?Rz ?HCG ?Fy = 0 (3.12)
By plugging in Equations (3.10) and (3.11) into equation (3.12), one can get an
equation relating the track width, gravitational force, CG height, velocity, and yaw rate.
TW
2 ?M ?g?HCG ?M ?V ?r = 0 (3.13)
By simplifying and rearranging Equation (3.13), the following result is given:
TW
2?HCG = SSF =
V ?r
g =
V 2
R?g (3.14)
In Equation (3.14), the static stability factor of a vehicle can be compared to various
vehicle properties. The equation can be rearranged further to estimate critical vehicle
properties. If the CG height, track width, and yaw rate / radius of curvature are known,
one can solve for the critical velocity that rollover will occur.
VRollover = TW ?g2?r?H
CG
=
radicalbiggTW ?R?g
2?HCG (3.15)
45
If the velocity, track width, and yaw rate / radius of curvature are known, one can solve
for the critical CG height that rollover will occur.
HCGRollover = TW ?g2?r?V = TW ?R?g2?V 2 (3.16)
If the velocity, CG height, and yaw rate / radius of curvature are known, one can solve
for the critical track width that rollover will occur.
TWRollover = 2?HCG ?r?Vg = 2?HCG ?V
2
R?g (3.17)
And finally if the velocity, CG height, and track width are known, one can solve for the
critical yaw rate (rRollover) or radius of curvature (RRollover) that rollover will occur.
rRollover = TW ?g2?V ?H
CG
(3.18)
RRollover = 2?HCG ?V
2
TW ?g (3.19)
For the above equations to hold, the SSF value must me less or equal than the
friction coefficient (SSF? ?). This is a requirement since the vehicle will most likely
slide out if ? is too low. When ? is too low,the frictional forces can become less than
the lateral tire forces. Once this occurs, the tire slip angles become large and the tires
and the vehicle will experience sideslip.
46
3.2.3 Inclusion of Suspension Effects
To include suspension effects, a look at the transients involved with weight transfer
and suspension effects must be taken. When a vehicle is going around a turn, the
center of gravity is shifted laterally outward, leading to a roll angle, and a change in the
distance from CG to the contact patches on the wheels. Work has been previously done
by Carlson and Gerdes [5], as well as Gillespie [23] to show the effects of dynamics of
suspension and weight transfer in dynamic maneuvers.
To take the lateral shift of the CG into account, Gillespie uses a scalar to reduce the
value of the SSF equation [23]. The equation that used in his work for the prediction of
rollover turns into:
ay
g =
TW
2?HCG ?
bracketleftBigg
1
1+R? ?(1? HRCHCG)
bracketrightBigg
(3.20)
The scalar that he added in [23],
bracketleftbigg
1
1+R??(1?HRCHCG )
bracketrightbigg
, can be statically reduced using
some known vehicle parameters. Gillespie states that for passenger cars, HRCHCG equals
about 0.5 and the roll rate (R?) is generally around 6 degrees/g, or 0.1 radians/g.
Substituting in these values into the above equation, the scale factor becomes around
0.95. In other words, the rollover threshold is reduced by about 5% due to transients
involved with suspension dynamics.
Although the roll center and CG heights are vehicle specific, the reduction value
will be assumed to be generally universal for passenger cars. For vehicles with a larger
CG height and different suspension setups such as SUVs and trucks, a larger scale factor
may be needed.
Gillespie continues saying that the rollover formula could be also altered from the
addition of lateral tire deflection experienced in a turn. This effect could contribute to
47
another 5% reduction in possible lateral acceleration that a vehicle can handle before
rollover. Testing in this thesis incorporates stiffer than normal tires, so that tire effects
are minimal.
To include the suspension effects into the the SSF and rollover prediction formulas, a
scale factor (?) has been added. Effectively, the weight transfer and suspension geometry
effects reduce the maximum rollover velocity.
SSF = ?? TW2?H
CG
(3.21)
VRollover = ?? TW ?g2?r?H
CG
= ??
radicalbiggTW ?R?g
2?HCG (3.22)
With the knowledge acquired from Gillespie, the range of the scale factor would be
from 0.9 to 0.95 since the vehicle is a typical SUV, where weight transfer is greater than
a typical passenger car. The actual value of ? = 0.92 was chosen from an analysis of
earlier simulation data taken from the vehicle.
3.3 Properties That Most Influence Vehicle Rollover Propensity
Several properties that are vehicle and road dependent play a role in the likelihood a
vehicle will rollover during evasive maneuvers. Table 3.1 identifies the vehicle properties
thought to have the greatest influence on vehicle rollover.
Table 3.1 - Properties Studied for Influence on Rollover Propensity:
CG Height
Track Width
Understeer Gradient
- Weight Split
- Tire Cornering Stiffness ratios (front/rear)
Suspension Stiffness
Friction Coefficients
48
Simulations were created in CarSim to test the vehicle roll formula derived above.
In order to find the critical rollover velocity, a constant radius test was chosen, with
a steady longitudinal acceleration of 0.833 m/s2. This maneuver not only allows for
the precise control of the vehicle, it also removes any dynamic behaviors from vehicle
transients that could further complicate the initial analysis performed in this section.
3.3.1 CG Height
The first property that is tested is the effect of CG height of the vehicle on rollover
propensity. As shown in Figure 3.4, with all other properties held constant, the CG
height is varied from 0.5 to 0.9 meters, in increments of 0.1 meters. For this simulation,
the track width is 1.5 m, and the turn radius is 40 m. The simulation is run for each
vehicle setup, and the lateral accelerations and longitudinal velocities of the vehicle are
recorded and compared.
Figure 3.4: Lateral Acceleration during Constant Radius maneuver with changing CG
heights
49
Table 3.2 - Lateral Accelerations with Changing HCG
HCG (m) SSF Rollover ay % Diff.
0.9 0.83 0.80 3.75 %
0.8 0.94 0.91 3.30 %
0.7 1.07 1.03 3.88 %
0.6 1.25 1.20 4.17 %
0.5 1.50 1.39 7.91 %
Values of the peak lateral acceleration from the simulation are compared to the
SSF values in Table 3.2. The SSF of the vehicle provides an adequate measure of the
maximum lateral acceleration achieved before rollover when CG height is varied on the
rigid vehicle.
Figure 3.5 shows the results of the peak velocities during the maneuver.
Figure 3.5: Rollover Velocities during Constant Radius maneuver with changing CG
heights
50
With the values taken from the rollover testing, the peak velocities can be compared
to the rollover Equation (3.15). Table 3.3 shows the values taken from simulation, as
well as the solutions from Equation (3.15) and the percent difference of the two.
Table 3.3 - Critical Velocity Comparisons with Changing CG Height
HCG (m) VRollover from Sim. (km/hr) VRollover from Eq. (km/hr) % Diff.
0.9 63.4 65.1 2.6 %
0.8 66.4 69.1 4.1 %
0.7 69.8 73.8 5.7 %
0.6 74.8 79.7 6.5 %
0.5 81.2 87.3 7.5 %
One source for the discrepancy between the simulation and analytical prediction
is the fact that the peak velocity for the vehicle in simulation is somewhere between
two-wheel-lift, and a roll angle of 90?. This can be seen in Figures 3.5 and 3.6 where the
peak velocities are not always well defined. It would be difficult to correctly extract the
point of time when the vehicle?s inner wheels were both off of the ground. Also, other
sources of error could include tire deformation, slipping, and slight suspension deflection
(which is not captured in Equation (3.22)).
To test the rollover equation with the effect of suspension transients, Figure 3.6
shows that there is a reduction in the maximum rollover velocity allowed by the vehicle
in the constant radius simulation in CarSim. This trend is predicted using Equation
(3.15) with ? = 0.92 and is shown in Table 3.4. The errors are reduced with the scale
factor, and the rollover velocity predictions are adequate and acceptable.
In order to investigate the effect of CG height variations on a vehicle with transients
caused by weight transfer and suspension characteristics, simulations were created and
51
Figure 3.6: Rollover Velocities during Constant Radius maneuver with changing CG
heights and weight transfer
Table 3.4 Critical Velocity Comparisons with Changing CG Height & Transients
HCG (m) VRollover from Sim. (km/hr) VRollover from Eq. (km/hr) % Diff.
0.9 62.7 59.9 4.7 %
0.8 65.6 63.5 3.3 %
0.7 69.1 67.9 1.8 %
0.6 73.7 73.3 0.5 %
0.5 80.2 80.3 0.1 %
run in MATLAB. Figure 3.7 shows the maximum velocities allowed before rollover in
the fishhook maneuver.
The trend of lower rollover velocities for higher CG heights is the same for the
dynamic testing in MATLAB, however the percent differences between the rollover ve-
locities is larger. This is likely due to differences in suspension and tire characteristics in
MATLAB and CarSim setups. One interesting trend that can be noticed is the slightly
parabolic shape of the rollover velocities. Although the equation for SSF and rollover
52
0.4 0.5 0.6 0.7 0.8 0.950
55
60
65
70
75
80
85
CG Height [m]
Rollover Velocity [kph]
Figure 3.7: Rollover Velocities during Fishhook maneuver with varying CG heights and
suspension effects (MATLAB simulation)
velocity is linearly dependent on the CG height, the simulation results show that the re-
lationship is not exactly linear. This effect is likely due to the transients and suspension
/ tire deflection.
3.3.2 Track Width
The second property to affect rollover propensity that is tested is the track width
of the vehicle. With all other properties held constant, the track width of the vehicle
is varied from 1.2 to 1.7 meters, in increments of 0.1 meters. The simulation is again
run for each setup, and the lateral accelerations and rollover velocities of the vehicle are
recorded and compared to the other configurations. The rollover lateral accelerations are
compared to the ones calculated from the SSF formula. The predicted values perfectly
53
Table 3.5 - Lateral Accelerations with Changing TW
HCG (m) SSF Rollover ay % Diff.
1.2 1.00 1.0 0 %
1.3 1.08 1.08 0 %
1.4 1.17 1.16 0.8 %
1.5 1.25 1.26 0.8 %
1.6 1.33 1.34 0.7 %
1.7 1.42 1.42 0 %
fit to the simulated results. Figure 3.9 shows the results of the peak velocities during
the maneuver.
Figure 3.8: Lateral Acceleration during Constant Radius maneuver with a changing
track width
This time, the differences in SSF and rollover lateral acceleration are lower than with
the previous tests. Simulations are once again run to test the suitability of Equation
(3.15) to track width variations. Figure 3.9 also contains better defined peaks of velocity
54
during the test maneuvers than the CG height variations. Again, other sources of error
could include tire deformation, slipping, and slight suspension deflection.
Figure 3.9: Rollover Velocities during Constant Radius maneuver with a changing track
width
Again, the peak velocities can becompared to the rollover Equation (3.15). Table 3.6
shows the values taken from CarSim simulation, as well as the solutions from Equation
(3.15) and the percent difference of the two analysis.
Table 3.6 - Critical Velocity Comparisons with Changing TW
TW (m) VRollover from Sim. (km/hr) VRollover from Eq. (km/hr) % Diff.
1.2 69.8 71.3 2.1 %
1.3 72.4 74.2 2.5 %
1.4 74.8 77.0 2.9 %
1.5 77.2 79.7 3.2 %
1.6 79.6 82.4 3.4 %
1.7 81.6 84.9 4.0 %
Table 3.1: The rollover formula solutions versus the simulated rollover velocity
55
Testing is also done to see how suspension transients affect simulations with changes
in track width. Here, the errors in the simulated rollover velocity and the velocity
calculated from the modified rollover formula (in Table 3.7) are larger than without the
weight transfer (in Table 3.6). This discrepancy could be fixed by altering the scale factor
?; however, the scale factor is still a good fit for the overall results of the vehicle. Figure
3.3.2 shows the results when vehicle?s suspension is added and is no longer considered
rigid.
Table 3.7 - Critical Velocity Comparisons with Changing TW
TW (m) VRollover from Sim. (km/hr) VRollover from Eq. (km/hr) % Diff.
1.2 67.6 65.6 3.0 %
1.3 70.1 68.3 2.6 %
1.4 72.4 70.9 2.1 %
1.5 75.0 73.4 2.2 %
1.6 79.2 75.8 4.5 %
1.7 79.8 78.1 2.2 %
56
Simulations were again done in MATLAB to verify the results from CarSim. In the
fishhook maneuver, the change in track width once again created a linear relationship to
rollover velocity. This is to be expected, since the predicted rollover velocity is directly
proportional to the track width. The rollover velocities that occurred during the fishhook
maneuver are more linear than with the changing CG height simulations. Another inter-
esting occurrence shown in later simulations is the fact that all of the vehicle properties
(ay, ?, Vy, etc...) are similar, right up until the point of rollover.
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.870
72
74
76
78
80
82
84
86
Track Width [m]
Rollover Velocity [kph]
Figure 3.10: Rollover Velocities during Fishhook maneuver with changing TW and sus-
pension effects
3.3.3 Understeer Gradient
The understeer gradient of a vehicle can greatly influence how a vehicle handles in
a turn. The value is dependent upon vehicle and tire properties and suspension charac-
teristics. This derivation of the understeer gradient follows the discussions published by
57
Rajamani and Gillespie [49, 23]. Figure 3.11 shows a diagram of a vehicle going around
a turn at high speeds.
Figure 3.11: High-speed cornering with steer and slip angles
With the assumption that the turning radius is much larger than the wheel base of
the vehicle (R >> L), the following equation can be derived:
???F +?R = LR (3.23)
The equation can be rearranged to find the steer angle to hold the turn (similar to the
Ackerman Steering Angle):
? = LR +?F ??R (3.24)
58
Note that ? is in terms of radians. If ? and ? are in degrees, the equation becomes
? = 57.3? LR + ?F ??R. The slip angles ?F and ?R can be related to properties such
as the turning radius (R) using the similar procedures as before. Using the small angle
approximations, the following equations can be derived by summing forces in the lateral
direction and taking the moment at the CG:
summationdisplay
Fy = MT ?ay = MT ? V
2x
R = FyF +FyR (3.25)summationdisplay
MCG = FyF ?a+FyR ?b = 0 (3.26)
Equation (3.26) can be rearranged and simplified to form the relationship between
the front and rear tire forces.
FyF = FyR ?ba (3.27)
After substituting the new lateral tire force relationship equation into Equation (3.25),
the lateral force for the front tires can be put into terms of mass, velocity, and turning
radius.
FyR = MT ? aL ? V
2x
R = MR ?
V 2x
R = WR ?
V 2x
g?R (3.28)
where MR is the weight on the rear axle of the vehicle, calculated using the wheelbase
and the distance from the CG to the front axle, i.e. MR = MT ? aL. An equation for the
front lateral tire forces can be made using the same relationships as above.
FyF = MT ? bL ? V
2x
R = MF ?
V 2x
R = WF ?
V 2x
g?R (3.29)
59
where MF is the weight on the rear axle of the vehicle, calculated similarly to the method
above, i.e. MF = MT ? bL.
Next, it is assumed that slip angles are small, and a linear tire curve can be used
so that the lateral tire forces are proportional to the slip angles, such that Fy = C? ??.
The slip angles can now be solved for, in terms of mass, cornering stiffness (C?), velocity,
and turning radius as shown below.
?F = FyF2?C
?F
= WF2?C
?F
? V
2x
g?R ?R =
FyR
2?C?R =
WR
2?C?R ?
V 2x
g?R (3.30)
The slip angles can now be substituted back into the Equation (3.24) to produce
the following relationship:
? = LR +?F ??R = LR +
bracketleftbigg W
F
2?C?F ?
WR
2?C?R
bracketrightbigg
? V
2x
g?R (3.31)
? = LR +Kus ?ay (3.32)
where:
Kus = WF2?C
?F
? WR2?C
?R
(3.33)
The parameter Kus in the above equation is called the understeer gradient. The
sign of the understeer gradient defines whether a vehicle is considered to be ?Understeer?
(KUS > 0), ?Oversteer? (KUS < 0), or ?Neutral Steer? (KUS = 0). Figure 3.12 shows
how the steering angle is dependent upon the lateral acceleration of the vehicle for the
three cases.
The properties that establish the understeer gradient are the weight split (the
amount of weight on the front/rear) and the tire cornering stiffness ratio (front/rear)
60
Figure 3.12: Steer angle variations with lateral acceleration
using the above relationship. In reality, KUS is a function of several vehicle properties
since the cornering stiffness is a function of suspension, load, and other vehicle param-
eters. Gillespie defines two tests that adequately estimate the understeer gradient of a
vehicle, while mirroring normal driving situations [23].
Using a constant radius test and multiple runs at varying velocities, one can derive
the understeer gradient. By taking the derivative of Equation (3.32), one gets:
??
?ay =
?(L/R)
?ay +KUS ?
?ay
?ay (3.34)
and since the radius of turn is constant, the first term cancels out.
KUS = ???a
y
(3.35)
With the above equation, the understeer gradient is in terms of the change in steer angle
over the change in lateral acceleration. This can be easily seen in Figure 3.13, where
61
the vehicle is either understeer at all speeds (limit understeer), or is understeer at low
lateral accelerations and becomes oversteer at higher values (limit oversteer).
Figure 3.13: Example curve of the understeer test using the constant radius method
Using the constant speed method, one can derive the understeer gradient using
maneuvers that mimic normal driving. By using Equation (3.32) again and substituting
in the relationship R = V 2ay ? Vr , one gets:
? = LR +KUS ?ay = L?ayV 2 +KUS ?ay (3.36)
and the derivatives can be once again taken to get:
KUS = ???a
y
? ?(L/V
2)
?ay (3.37)
With the above equation, the understeer gradient is in terms of the change in steer angle
and the change in lateral acceleration, since speed and wheelbase are constant. Figure
3.14 shows how it is determined if a vehicle is understeer or oversteer with this method.
62
The Ackerman steer angle gradient (the last term in the previous equation) is plotted
on the figure with at a constant slope. Vehicles with ? values higher than the Ackerman
steer angle are considered understeer, while lower values portray oversteer vehicles. See
Gillespie [23] for a more detailed description of testing methods of KUS.
Figure 3.14: Example curve of the understeer test using the constant speed method
Understeer
Vehicles can be described as understeer when the understeer gradient, Kus, is greater
than zero. The understeer gradient is defined as follows:
WF
C?F >
WR
C?R =??F > ?R =? Kus > 0 (3.38)
In a constant radius turn, an understeer vehicle requires an increasing steer angle with an
increase in speed to maintain the constant radius turn. Most passenger cars are designed
to be understeer, in order to allow the everyday driver to be able to handle the vehicle
in evasive maneuvers.
63
Oversteer
Vehicles can be described as oversteer when the understeer gradient, Kus, is less
than zero.
WF
C?F <
WR
C?R =??F < ?R =? Kus < 0 (3.39)
An oversteer vehicle will want to turn more when the vehicle?s velocity is increasing (the
back end wants to slide out). Vehicles are not usually designed to be oversteer due to
driving difficulties associated with the setup. In the constant radius test, the steer angle
in an oversteer vehicle would have to be decreased as the longitudinal velocity increases
to maintain the CR turn.
Neutral Steer
Vehicles can be described as neutral steer when the understeer gradient, Kus, is
equal to zero.
WF
C?F =
WR
C?R =??F = ?R =? Kus = 0 (3.40)
A neutral steer vehicle would optimize the handling characteristics during turning; how-
ever, a slightly understeer configuration is actually desired in production vehicles due to
handling requirements.
Weight Split
Previous research has been done on the effects of the longitudinal location of the
CG [59, 60, 61]. This property, known as weight split (WS), has been proven to play
64
a role in the rollover propensity (even though it is not captured in the SSF equation),
especially in 15-passenger vans, trucks, and SUVs that have been overloaded toward the
rear. The weight split of a vehicle is usually shown as a ratio of the weight on the front
axle to the weight on the rear axle. For example, a vehicle with 60% of the weight on
the front axle is referred to as a 60/40 WS vehicle.
Simulations in CarSim are performed with changing weight split. With the CG
height set to 0.6 m and track width set to 1.5 m, the SSF is equal to 1.25. The maximum
lateral accelerations, shown in Figure 3.15, show that the weight split and understeer
gradient of a vehicle can influence rollover propensity.
Figure 3.15: Lateral Acceleration during Constant Radius with a changing weight split
The oversteer vehicles roll earlier than the understeer vehicles; however, while the
lateral accelerations for the understeer vehicles are close to the SSF predicted values,
the oversteer vehicles have more error between the simulated and predicted values. The
42.5/57.5 WS vehicle has over a 13% error lateral acceleration when compared to the
SSF value.
65
Longitudinal velocity can also be examined during the constant radius turn as well.
Figure 3.16 shows the peak velocities attained before rollover occurs when the weight
split is altered.
Figure 3.16: Rollover Velocities during Constant Radius with a changing weight split in
CarSim
Since weight split is not taken into account in the rollover velocity formula, it is
constant for all configurations of the vehicle. The results from simulation differ by only
2.6 km/hr, or 3.5 %. The inclusion of the understeer gradient into the rollover prediction
formulas is discussed in greater detain in Section 3.4.
Tire Cornering Stiffness Ratios (front/rear)
The other property that affects the understeer gradient is the tire cornering stiffness
ratio from front to rear. The cornering stiffnesses is not usually the same for vehicles
since it is a factor of tire properties and forces. Additionally, one of the principal factors
in tire cornering stiffness is inflation pressure. Due to the fact that drivers may not
correctly monitor tire pressure, the cornering stiffness can be changed when the weight
split remains neutral causing a noticeable difference in how a vehicle handles.
66
Figures 3.17 and 3.18 show how the vehicle is affected when the cornering stiffness
ratio (front/rear) is altered and the friction coefficient is 1.25.
Figure 3.17: Position Figure 3.18: Lateral acceleration
When the front cornering stiffness is much greater than the rear (oversteer), the
rear of the vehicle begins to slide out. This can be seen in Figure 3.17, where the turn
radius for the oversteer vehicle is smaller than the understeer vehicle. Also the understeer
vehicle with higher rear cornering stiffness eventually rolls, while the oversteer vehicle
does not roll. Although large changes in the ratio of cornering stiffness does not happen
often, it has been proven that it does affect vehicle rollover and handling.
3.3.4 Suspension Stiffness
Another factor that is investigated to test its influence on rollover is the suspension
stiffnesses. Although this is not directly accounted for in the SSF or rollover velocity
equations, it does affect how the weight transfer scale factor is decided. Variances in the
suspension stiffness can affect how much or how little the CG of a vehicle will move in
the lateral direction around a turn. This will in turn affect the cornering stiffnesses of
67
the vehicle, further changing the understeer gradient. With a soft suspension, the weight
transfer will increase and the vehicle will likely roll over earlier.
Figure 3.19 shows the results of the lateral accelerations of a vehicle performing the
steadily increasing steer maneuver with a varying suspension stiffness. As predicted, the
vehicle with the soft suspension rolls when the lateral acceleration reaches 1.05 g. The
vehicle with a medium suspension stiffness rolls around 1.2 g, and the vehicle with a stiff
suspension rolls around 1.27 g.
Figure 3.19: The lateral acceleration of a vehicle in the SIS maneuver with changing
suspension stiffnesses
Themaximumlateral accelerations could bepredicted with theoriginal and modified
SSF formulas. For the rigid suspension, the SSF is equal to 1.25, an error of less than
2% from the simulated results. For the vehicle with the medium suspension stiffness, the
modified SSF formula should be used with ? = 0.92. The SSF would become 1.15, giving
a 4% error. With the loose suspension, the scale factor should be altered to account for
the increased weight transfer. With ? = 0.88, the SSF is then 1.1 g, an error of around
5% when compared to the results from figure 3.19.
68
Simulations in CarSim were then performed to test the effects of suspension stiffness
during dynamic maneuvers. As expected, the vehicle with the soft suspension rolled
earlier than the other two configurations. Figures 3.20 and 3.21 show the vehicles?
performances during the fishhook maneuver.
Figure 3.20: Paths of vehicles with varying
suspension stiffnesses
Figure 3.21: Lateral accelerations of vehi-
cles with varying suspension stiffnesses
Both the vehicles with soft and medium suspension stiffnesses rolled during the
simulations. The soft suspension caused the vehicle to rollover soon after the second
steer input, while the medium suspension allowed for the vehicle to hold the turn longer.
Both vehicles that rolled had some oscillatory behavior before rollover. This could likely
be avoided by increasing the damping of the vehicles? suspensions. On the other hand,
the vehicle with the rigid suspension was able to complete the fishhook maneuver without
rolling.
3.3.5 Friction Coefficients
The last property to be analyzed for vehicle rollover tendencies is the friction coeffi-
cient between the tire and ground. It is widely known that with a slick surface, a vehicle
will slide out before rollover has a chance to take place. This behavior is due to the fact
69
that a slick surface has a low friction coefficient (?). On a wet or icy road, ? can be as
low as 0.3 (limiting the maximum lateral acceleration to 0.3 g), while a dry surface has
a coefficient of around 1. For some of the testing done in this thesis, the coefficient was
increased to 1.3 or greater in order to ensure the incidence of rollover.
Figure 3.22 shows simulations of the fishhook maneuver created in CarSim. When
the friction coefficient is 0.3 and 1.0, the vehicle tends to slide and rollover does not
occur; but when the coefficient is 1.3, the vehicle rolls right after the second steer input
is applied. This is due to the fact that the overall tire force applied is greater than the
lateral acceleration applied.
Figure 3.22: The positions of a vehicle in the fishhook maneuver with changing friction
coefficients
Using Equation (2.34) can be used to help predict the occurrence of sliding and of
rollover likelihood. Since Ftire ? Fz ??, if the SSF ? ?, there is a large chance that
rollover could occur. However, if the SSF > ?, chances of rollover are reduced due to
the fact that the vehicle will most likely slide before an untripped rollover will occur.
70
3.4 The Inclusion of Understeer Gradient Into the Prediction of Rollover
It was shown in Section 3.3.3 that the understeer gradient plays a role in vehicle
rollover propensity. To further improve the rollover prediction formulas, the effects of the
understeer gradient should be taken into account. Although it sounds simple, modifying
the rollover velocity equations into a form that includes the understeer gradient greatly
increases their complexity. In addition to seeking a simpler form of these equations, an
investigation into empirical data taken from simulation was done to look for trends in
the data that could be simply inserted into the rollover prediction equations.
3.4.1 Simulation Results
In order to further investigate the effects that understeer gradient has upon vehi-
cle rollover, more simulations were created using CarSim. The Constant Radius and
Fishhook maneuvers were chosen for their ability to analyze the vehicle quasi-static and
dynamic behaviors in rollover cases. The friction value used in the simulations was set
to 1.5, a high value chosen to ensure enough lateral force to allow the vehicle to rollover.
For the first case, four vehicle setups were chosen, with alterations only in weight
split. Figure 3.23 displays the rollover velocities for the different vehicle configurations.
For both maneuvers, the rollover velocity increased as the weight was shifted forward.
This trend will be discussed further in Subsection 3.4.2.
For the changing tire cornering stiffness case, three vehicle setups were chosen. Fig-
ure 3.24 displays the rollover velocities for the understeer, neutral steer, and oversteer
conditions. For the constant radius maneuver, the rollover velocity remains almost con-
stant as the vehicle setup changes. As before, the fishhook maneuver results in a linear
71
increase in rollover velocity as the vehicle becomes understeer. This trend will also be
discussed further in Subsection 3.4.2. The discrepancy in the CR and Fishhook ma-
neuver trends shows why it is difficult to derive an equation that predicts rollover with
changing values of understeer gradient.
3.4.2 Empirical Trends
In Figures 3.23 and 3.24, it is shown that trends in the rollover velocity as a function
of KUS show up in the data. Although the trends are not completely uniform for all of the
simulation results, their linearity shows that a scale factor based upon the understeer
gradient could improve the estimated rollover velocities. By adding the scale factor
(1 + KUS) into Equation (3.15), the velocities predicted fit the empirical data better
than the prior method.
VRollover = ?? TW ?g2?r?H
CG
?(1 +KUS) = ??
radicalbiggTW ?R?g
2?HCG ?(1 +KUS) (3.41)
Figure 3.23: Rollover velocities with vary-
ing Weight Splits
Figure 3.24: Rollover velocities with vary-
ing C? Values
72
Table 3.8 compares the critical rollover velocities from CarSim simulations to the
predicted values found using Equation (3.41). Using the derived rollover equation, the
estimated rollover velocity for all of the vehicle configurations with changing WS would
be 79.7 kph for the CR maneuver, and 49.6 kph for the Fishhook maneuver. However,
with the modified equation, the value is scaled for each setup. Although the total error
is about the same, the predicted values follow the trend shown from the CarSim results.
Table 3.8 - Critical Rollover Velocity for Changing Weight Splits (kph)
Constant Radius
WS (F/R) KUS(deg/g) CarSim Eq. (3.15) % err. Eq. (3.41) % err.
42.5 / 57.5 -0.139 74.2 79.7 7.4 % 71.6 3.6 %
47.5 / 52.5 -0.0464 74.7 79.7 6.7 % 72.8 2.5 %
52.5 / 47.5 0.0464 76.2 79.7 4.6 % 74.0 2.9 %
57.5 / 42.5 0.139 78.2 79.7 1.9 % 75.0 4.3 %
Fishhook
42.5 / 57.5 -0.139 43 49.6 15.3 % 42.9 0.2 %
47.5 / 52.5 -0.0464 46 49.6 7.8 % 44.4 3.6 %
52.5 / 47.5 0.0464 49 49.6 1.2 % 46.9 4.5 %
57.5 / 42.5 0.139 51 49.6 2.8 % 49.6 2.8 %
Somewhat similar results were found when changing values of tire cornering stiffness.
Using the basic derived equation, the critical rollover velocity for the all of the vehicle
setups is 79.7 kph for the CR maneuver, and 51.6 kph for the fishhook maneuver. Table
3.9 shows the results of the critical velocities when the changes in understeer gradient
are included. The results for Equation (3.41) are not as consistent for this case as the
previous, partially due to the data taken from the CarSim simulations. Although the
critical velocities change a good deal in the fishhook maneuver, they are almost constant
in the constant radius maneuver. The errors are however greatly reduced when the
understeer gradient is taken into account for the Fishhook maneuver.
73
Table 3.9 - Critical Rollover Velocity for Changing C? Values (kph)
Constant Radius
Config. KUS(deg/g) CarSim Eq. (3.15) % err. Eq. (3.41) % err.
Oversteer -1.82 76.6 79.7 4.1 % 71.0 7.9 %
Neutral Steer 0.0 76.8 79.7 3.8 % 73.3 4.8 %
Understeer 1.82 76.7 79.7 4.0 % 75.7 1.3 %
Fishhook
Oversteer -1.82 41 51.6 25.9 % 41.4 1.0 %
Neutral Steer 0.0 48 51.6 7.5 % 47.5 1.0 %
Understeer 1.82 59 51.6 14.3 % 56.0 5.3 %
It could be argued that the high friction coefficient could be the reason that the
trends are not completely universal for the changing understeer gradient values. In
order to examine this, the same simulations were run with different friction coefficients
(1.3, 1.25, 1.2, 1.0). In these cases, some or all of the vehicles ended up sliding excessively
and not completing the desired maneuver. Since the research presented in this thesis is
focused on vehicle rollover, an in-depth analysis into the sliding of the vehicle will not
be further discussed.
Despite the trends only being shown with empirical results, the rollover prediction
formula is improved with the understeer gradient scale factor included. Work is contin-
uing on the investigation of an improvement to Equation (3.41).
3.5 Conclusion
In this chapter, the Static Stability Factor was derived, and modified to include
the effects of transients due to weight transfer and suspension configurations. Formulas
were derived for the rollover threshold as a function of critical rollover properties, such
as velocity, track width, CG height, and radius of turn/yaw rate. Although simulations
74
were created to test the validity of the rollover velocity equation, testing could have as
well been done to examine Equations (3.16)-(3.19). The simulations created in CarSim
and MATLAB verified the formula?s accuracy for predicting the velocity before rollover.
With the knowledge of the effects of key vehicle properties on rollover, research can
now be done to see how electronic stability controllers are affected by variances these
properties.
75
Chapter 4
Electronic Stability Controller Development
4.1 Introduction
It is well known that Electronic Stability Controllers (ESC) have been saving lives
throughout the last decade [27]. In a recent mandate by NHTSA, ESC will be required
to be in all new vehicles by the 2012 model year (2011 calendar year) [41]. Despite the
new requirements, some companies have decided to make it a standard feature earlier
than the required date. For example, Ford Motor Company is making ESC standard in
all passenger vehicles by by the end of 2009 [18].
Previous research into the effectiveness of ESC systems shows that the technology
saves lives [1, 34, 33, 58]. Over the last century, the IIHS has also looked into crash data
and found that ESC has saved lives on US highways. This chapter discusses the basics
of ESC systems, types of ESC systems, and the derivation of various types that have
been explored in simulation in this thesis. The simulations in this chapter were created
in MATLAB with the vehicle model derived in Chapter 2.
4.2 ESC Basics
ESC is an extension of Anti-Lock Brake systems (ABS), using similar sensors and
actuators. In Figure 4.1, the sensors used in a typical ESC system are shown. With
the yaw rate, steer angle, lateral acceleration, and wheel speed measurements, the ESC
can estimate the vehicle?s course, and decide if the error in course (where the vehicle is
pointed) and heading (where the vehicle is traveling) is too large (i.e. sideslip or lateral
76
velocity). If the error is larger than a predefined limit, the ESC system is triggered, and
the vehicle reacts to reduce the understeer or oversteer error.
Figure 4.1: ESC sensors diagram. Source: IIHS [53]
4.3 ESC Types
Several types of ESC are currently being implemented in today?s vehicles. With the
predicted number of lives saved, as well as the proven success rate, there are obvious
reasons for ESC systems to be in every ground vehicle. Table 4.1 shows the types of
Stability Controllers tested in this thesis using the simulations created in MATLAB. The
systems begin with simple power reduction controllers, and then become more complex
with variable braking and steering control [4, 57, 63, 35, 17].
Table 4.1 - ESC Types:
Power Reduction
All-Wheel Braking
Independent Wheel Braking
Active Torque Distribution
Steering Modification
Steering Modification with All-Wheel Braking
Independent Wheel Braking with Steering Control
77
4.4 Stability Threshold
In order to keep the vehicle in a safe region, the limits on handling must be defined.
Several vehicle properties that predict rollover can be observed, but not all are easy
to measure. For example, the simplest property that would anticipate rollover is the
vertical forces on the tires. When rollover occurs, it is always preceded by the vertical
force on the inner tires of the vehicle going toward zero. In actuality, the vertical forces
on a tire are hard to measure at best. To set a limit on vehicle stability, easily measured
properties must be used.
From simulation results, it has been shown that vehicle rollover is also preceded
by higher than normal values of lateral acceleration and yaw rate. If an ESC system is
to set a limit on the maximum value of lateral acceleration and yaw rate allowed, the
controller would have an easily measurable, accurate method of stability enhancement.
It would also be possible to measure roll rate for an estimate of roll, which could in turn
be used in the ESC system. The sensors used, a lateral accelerometer and a gyroscope,
are also being installed into more production vehicles today than ever, due to lowering
production and installation costs.
4.5 Power Reduction
The most basic type of controller used for stability maintenance is the power reduc-
tion controller. When an unsafe level of lateral acceleration or yaw rate is detected, the
torque delivered from the engine is reduced on all drive wheels, slowing the vehicle to a
safer level of dynamics to prevent rollover.
78
Simulations in MATLAB show how effective the controller can be. Figure 4.2 dis-
plays how the vehicle reacts when the controller is applied during the fishhook maneuver,
causing a reduction in velocity. In this simulation, the engine torque was limited when
the vehicle reached a magnitude of 0.4 g of lateral acceleration.
0 20 40 60 80 100?10
0
10
20
30
40
50
60
EAST (m)
NORTH (m)
Without ESC
With ESC
0 2 4 6 8 1027
27.5
28
28.5
29
29.5
30
30.5
Time (sec)
Vehicle Velocity (mph)
Without ESC
With ESC
0 2 4 6 8 10?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
0.8
Time (sec)
Lateral Acceleration (g)
Without ESC
With ESC
0 2 4 6 8 10?6
?4
?2
0
2
4
6
Time (sec)
Vehicle Roll (deg)
Without ESC
With ESC
Figure 4.2: The vehicle?s performance with the power reduction controller
A negative aspect of the controller is that it is often too weak to keep the vehicle
stable during evasive maneuvers. In Figure 4.2, the lateral acceleration and roll angle are
only slightly reduced by the controller. By only limiting the power fed to the tires, the
79
vehicle may not be slowed enough to prevent rollover at higher speeds. Adding braking
torques to the engine power reduction can increase the ESC?s effectiveness.
4.6 All-Wheel Braking
The all-wheel braking controller is somewhat similar to the power reduction con-
troller; however, instead of only reducing the positive driving torques, a braking force
is applied to all four wheels. Figure 4.3 shows how the all-wheel braking controller can
keep the vehicle in the stability threshold during the evasive maneuver.
0 20 40 60 80 100?10
0
10
20
30
40
50
EAST (m)
NORTH (m)
Without ESC
With ESC
0 2 4 6 8 1024
25
26
27
28
29
30
Time (sec)
Vehicle Velocity (mph)
Without ESC
With ESC
0 2 4 6 8 10?1.5
?1
?0.5
0
0.5
1
Time (sec)
Lateral Acceleration (g)
Without ESC
With ESC
0 2 4 6 8 10?3
?2
?1
0
1
2
3
4
Time (sec)
Vehicle Roll (deg)
Without ESC
With ESC
Figure 4.3: The vehicle?s performance with the all-wheel braking controller
80
Figure 4.4 displays the braking forces applied to the vehicle during the maneuver.
The ESC implemented in this simulation contained a two stage system, where the first
stage limits the lateral acceleration to 0.3 g. Once that value is reached, a braking force
of 667 N is applied at the contact patch of every wheels (200 N-m of torque). The second
stage is triggered once the lateral acceleration reaches or exceeds 0.45 g. A braking force
of 1500 N is then applied at each tire (450 N-m of torque). These braking forces were
backed out of previous test data where hard braking occurred, and simulations were
created to justify the results.
0 2 4 6 8 10
?1500
?1000
?500
0
Longitudinal Tire Forces (in Newtons)
Fx Front Left
0 2 4 6 8 10
?1500
?1000
?500
0
Fx Front Right
0 2 4 6 8 10
?1500
?1000
?500
0
Fx Rear Left
0 2 4 6 8 10
?1500
?1000
?500
0
Fx Rear Right
Time (sec)
Without ESC
With ESC
Figure 4.4: The vehicle?s braking forces with the all-wheel braking controller
81
The all-wheel braking controller?s simplicity allows it to be added to any vehicle
with computer controlled braking. Drawbacks of the controller are similar to the power
reduction controller. Although the controller should have enough braking torque to
greatly reduce the vehicle?s velocity duringan evasive maneuver, the vehicle may have too
much weight transfer to effectively prevent rollover. The controller also adds increased
longitudinal wheel forces, which can cause tire saturation and lock-up or sliding, due to
the friction circle limits discussed previously in Chapter 2. The controller can be made
much more effective by adding individual wheel braking torques. Independent braking
requires separate brake modules for each wheel, but it?s popularity is growing.
4.7 Independent Wheel Braking
A proper vehicle ESC system should include the ability to control the speeds of
individual wheels. With the added degree of control, torques can be applied to the
wheels that would more accurately keep the vehicle in the stability region. By putting
a braking torque on specified wheels during a turn, the vehicle?s yaw rate error can be
controlled more than by applying the braking torque to all wheels. Additionally, the
vehicle?s longitudinal velocity is reduced, resulting in a reduction in lateral acceleration
and yaw rate among other things. The underlying property that is used in this controller
is brake steer [46]. By braking the wheels on one side of the vehicle, an added moment is
applied, and the vehicle?s yaw rate will be reduced or enlarged, depending on the need.
4.7.1 Controller Development
In order to understand how the vehicle will behave when certain wheels are braked,
a free body diagram is created, and the moments are taken about the CG. Figure 4.5
82
shows the FBD used for the brake steer derivation. The steer angle is assumed to be
small enough to discount its effects upon the longitudinal braking forces. With the brake
steer moment and the difference in desired and actual yaw rate, the controller has enough
information to apply braking forces to reduce the yaw error.
Figure 4.5: The FBD used for the derivation of brake steer moments
Using the FBD in Figure 4.5, and assuming that the steer angle (?) is small, the
following moment can be calculated:
MBS = twf2 ?[FxfR ?FxfL] + twr2 ?[FxrR ?FxrL] (4.1)
MBS = twf2 ?FBSf + twr2 ?FBSr (4.2)
Independent braking has been well studied for the application in ESC systems [7]. If
the vehicle is oversteer, the outer wheels will be braked in a turn to decrease the vehicle?s
yaw rate and reduce the error in the vehicle?s course and heading. Alternately, if the
vehicle is understeer, the inner wheels will be braked to increase the vehicle?s yaw rate
and reduce the error in the vehicle?s course and heading. Figure 4.6 shows how a typi-
cal independent wheel braking controller works. The application of braking torques to
83
independent wheels can be varied to increase the effectiveness of the stability controller.
Depending on the level of lateral acceleration and yaw rate, the brake pressures could
be altered in order to achieve maximum performance while turning.
Figure 4.6: ESC with independent wheel braking. Source: IIHS [53]
4.7.2 Controller Behavior
From the derived controller with independent wheel braking, one can see how the
vehicle reacts in Figure 4.7. For the understeer vehicle modeled, the controller applies
a variable braking torque to the inner wheels of the vehicle when the stability threshold
is compromised (once again set to 0.4 g of lateral acceleration). The control gains were
not optimized for the research in this thesis and therefore additional improvement could
be possible with careful study and implementation.
Although Figure 4.6 shows one wheel being braked, the simulations created in MAT-
LAB used braking on both front and rear wheels on one side of the vehicle, depending
84
0 20 40 60 80 100?10
0
10
20
30
40
50
60
EAST (m)
NORTH (m)
Without ESC
With ESC
0 2 4 6 8 1022
23
24
25
26
27
28
29
30
Time (sec)
Vehicle Velocity (mph)
Without ESC
With ESC
0 2 4 6 8 10?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
0.8
Time (sec)
Lateral Acceleration (g)
Without ESC
With ESC
0 2 4 6 8 10?6
?4
?2
0
2
4
6
Time (sec)
Vehicle Roll (deg)
Without ESC
With ESC
Figure 4.7: The vehicle?s behavior with the independent wheel braking controller
upon the understeer / oversteer conditions. Early results showed that the controller is
more effective with the added braking forces from two wheels rather than from just one
wheel. Figure 4.8 shows the braking forces applied to the wheels from the simulation
depicted in Figure 4.7. Once again the braking forces and stability limits could be op-
timized to increase the stability controllers effectiveness. Since the vehicle modeled is
slightly understeer, braking is applied to the vehicle?s inner wheels. This induces a brake
moment that slows the vehicle down while reducing the yaw rate error.
85
0 2 4 6 8 10?800
?600
?400
?200
0
200
Longitudinal Tire Forces (Newtons)
Fx Front Left
0 2 4 6 8 10?800
?600
?400
?200
0
200
Fx Front Right
0 2 4 6 8 10?800
?600
?400
?200
0
200
Fx Rear Left
0 2 4 6 8 10?800
?600
?400
?200
0
200
Fx Rear Right
Time (sec)
Without ESC
With ESC
Figure 4.8: The vehicle?s longitudinal forces with independent wheel braking
The controller used in this simulation once again has two stages. The first stage
sets limits of lateral acceleration to 0.3 g for normal driving. Once that threshold is
crossed, independent braking is applied to the desired wheels with a controller that
applies somewhere between 667 N and 1500 N of braking force, depending on the value
of lateral acceleration. The second stage of the controller applies 1500 N of braking force
to the desired wheels once 0.45 g of lateral acceleration is breached.
Since this controller is quite effective in maintaining stability and is going being
implemented in most production cars within the next few years, this controller will be
further investigated in Chapter 5 to see how effective the controller can be when it is
subjected to changing vehicle parameters.
86
4.8 Active Torque Distribution
Another method for traction and stability control is the active distribution of en-
gine torque to the outer or inner wheels using active differentials. This method has
been implemented on Acura?s RL, where it adds the engine torque distribution to the
independent wheel braking to correct for understeer and oversteer. Figure 4.9 shows the
vehicle?s behavior during the fishhook maneuver.
0 20 40 60 80 100?10
0
10
20
30
40
50
EAST (m)
NORTH (m)
Without ESC
With ESC
0 2 4 6 8 1027
27.5
28
28.5
29
29.5
30
30.5
Time (sec)
Vehicle Velocity (mph) Without ESC
With ESC
0 2 4 6 8 10?1.5
?1
?0.5
0
0.5
1
Time (sec)
Lateral Acceleration (g)
Without ESC
With ESC
0 2 4 6 8 10?3
?2
?1
0
1
2
3
4
Time (sec)
Vehicle Roll (deg)
Without ESC
With ESC
Figure 4.9: The vehicle?s behavior with the added torque controller
By adding torques to the outer wheels and braking the inner wheels of the understeer
vehicle, the vehicle?s yaw rate error is greatly reduced, but the overall lateral acceleration
and roll angle are not reduced by much. Figure 4.10 displays the braking and engine
87
forces applied at the wheels. Although the overall results are good for this method, its
application is also going to be limited due to the fact that its installation into production
vehicles is quite expensive.
0 2 4 6 8 10
?200
0
200
400
600
Longitudinal Tire Forces (Newtons)
Fx Front Left
0 2 4 6 8 10
?200
0
200
400
600
Fx Front Right
0 2 4 6 8 10
?200
0
200
400
600
Fx Rear Left
0 2 4 6 8 10
?200
0
200
400
600
Fx Rear Right
Time (sec)
Without ESC
With ESC
Figure 4.10: The vehicle?s longitudinal forces with the added torque controller
4.9 Steering Control
Another method for vehicle stability control uses active steering modification. When
a vehicle enters a curve with too much speed, the controller will limit the magnitude of
the steer angle input. Figure 4.11 shows an example of this controller. Although the yaw
rate and lateral acceleration are limited to a safe limit, the vehicle?s path is not what is
desired. This type of controller would not be acceptable for today?s highways because
88
obstacle avoidance is often required. In order for a controller with steering modification
to work, it must be combined with braking forces.
0 20 40 60 80 100 120?10
0
10
20
30
40
50
60
EAST (m)
NORTH (m)
Without ESC
With ESC
0 2 4 6 8 10?15
?10
?5
0
5
10
15
time (sec)
Front Tire Angle (deg)
Without ESC
With ESC
0 2 4 6 8 10?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
0.8
Time (sec)
Lateral Acceleration (g)
Without ESC
With ESC
0 2 4 6 8 10?6
?4
?2
0
2
4
6
Time (sec)
Vehicle Roll (deg)
Without ESC
With ESC
Figure 4.11: The vehicle?s behavior with the steering modification controller
4.9.1 Steering Control with All-Wheel Braking
By combining two of the previous controllers, another ESC system can be created.
With steering modification and all-wheel braking, the vehicle?s stability can be increased
while trying to reduce the yaw error. Figure 4.12 shows the vehicle?s performance with
89
this controller. The yaw error is reduced from the steering only controller, and the
maximum values of the roll angle and lateral acceleration are reduced.
0 20 40 60 80 100 120?10
0
10
20
30
40
50
60
EAST (m)
NORTH (m)
Without ESC
With ESC
0 2 4 6 8 10?15
?10
?5
0
5
10
15
time (sec)
Front Tire Angle (deg)
Without ESC
With ESC
0 2 4 6 8 10?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
0.8
Time (sec)
Lateral Acceleration (g)
Without ESC
With ESC
0 2 4 6 8 10?6
?4
?2
0
2
4
6
Time (sec)
Vehicle Roll (deg)
Without ESC
With ESC
Figure 4.12: The vehicle?s behavior with steering modification and constant braking
With this controller, the vehicle?s velocity is reduced by over ten percent, due to the
simple braking algorithm. The ESC limits the lateral acceleration to 0.4 g, while 1500 N
of braking force (450 N-m of torque) is applied to all of the wheels once that threshold is
crossed. Figure 4.13 displays the change in velocity of the vehicle during the maneuver.
90
0 2 4 6 8 1026.5
27
27.5
28
28.5
29
29.5
30
30.5
Time (sec)
Vehicle Velocity (mph)
Without ESC
With ESC
Figure 4.13: The vehicle?s velocity with steering modification and constant braking
This controller is similar to one used by Randal Whitehead in his research on his
masters thesis [60]. In that work, the controller was implemented on a scaled vehicle
in order to test the suitability of scaled vehicles for rollover testing. This was the most
advanced controller possible for the scaled vehicle setup that he used. For this reason,
and the fact that the implementation costs of this controller are low, this controller and
the independent braking controller will be the focus of Chapter 5 to further investigate
the performance of the two controllers to prevent rollover under various vehicle scenarios.
91
4.9.2 Independent Wheel Braking with Steering Control
By expanding the previous controller to include independent wheel braking, the
stability of the vehicle can be further guaranteed [6, 2]. Figure 4.14 shows how a vehicle
behaves with the independent wheel braking and steering modification controller. With
the initial setup of the controller, the vehicle?s stability is slightly enhanced, as the yaw
rate and lateral acceleration are reduced. Also, the yaw error is greatly reduced, and the
vehicle almost keeps the desired path.
0 20 40 60 80 100 120?10
0
10
20
30
40
50
60
EAST (m)
NORTH (m)
Without ESC
With ESC
0 2 4 6 8 10?15
?10
?5
0
5
10
15
time (sec)
Front Tire Angle (deg)
Without ESC
With ESC
0 2 4 6 8 10?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
0.8
Time (sec)
Lateral Acceleration (g)
Without ESC
With ESC
0 2 4 6 8 10?5
?4
?3
?2
?1
0
1
2
3
4
5
Time (sec)
Vehicle Roll (deg)
Without ESC
With ESC
Figure 4.14: The vehicle?s behavior with independent braking and steering control
92
Figure 4.15 shows the velocities of the vehicles. The velocity of the ESC equipped
vehicle is reduced by over 5 miles per hour. This reduction in combination with the
moment applied by brake steer allows the steering controller to keep the vehicle on the
ideal path.
0 2 4 6 8 1024
25
26
27
28
29
30
Time (sec)
Vehicle Velocity (mph)
Without ESC
With ESC
Figure 4.15: The vehicle?s velocity with independent braking and steering control
Figure 4.16 displays the longitudinal forces applied by the independent braking
controller. Although this controller seems to be somewhat ideal, its implementation
is not as simple as others. Priorities must be made in order to establish what aspect
of the controller is dominant. Therefore, this controller, along with the active torque
distribution, would be a good area of research for future work.
93
0 2 4 6 8 10?800
?600
?400
?200
0
200
Longitudinal Tire Forces (in Newtons)
Fx Front Left
0 2 4 6 8 10?800
?600
?400
?200
0
200
Fx Front Right
0 2 4 6 8 10?800
?600
?400
?200
0
200
Fx Rear Left
0 2 4 6 8 10?800
?600
?400
?200
0
200
Fx Rear Right
Time (sec)
Without ESC
With ESC
Figure 4.16: The vehicle?s longitudinal forces with independent braking and steering
control
4.10 ESC with State Estimation
Research on the estimation of vehicle mass, sideslip, and roll parameters has been
done throughout the last decade [29, 36, 51]. Dustin Edwards has been investigating
the methods of estimation of vehicle properties (tire split, tire friction, and weight split)
that could be used to optimize stability controllers [13]. Solmaz, Akar, and Shorten
have also been investigating the estimation of CG height using sliding mode controllers
[54]. With a knowledge of the CG location and other vehicle properties, ESC systems
can be adjusted to limit the maximum lateral acceleration and yaw rate to different
values, depending on the loading conditions. The implementation of state estimation,
mainly CG height and longitudinal location (WS), would be another good area for future
94
research. The benefits of the knowledge of these properties could be rather large, and
are discussed briefly in Chapter 5.
4.11 Conclusion
This chapter introduced a variety of electronic stability controllers that could be
useful in improving the handling of a vehicle and the prevention of rollover. In Chapter 5,
the all-wheel braking with steering modification and the independent braking controllers
will be specifically studied for the prevention of rollover using MATLAB simulations
with the roll model developed in Chapter 2.
95
Chapter 5
Simulation Results for ESC
5.1 Introduction
Since it has been shown in Chapter 3 that certain vehicle properties can affect
how a vehicle handles during extreme maneuvers, this chapter investigates how some
ESC systems are affected by these changing vehicle properties. For two controllers, the
independent braking and the all-wheel braking with steering modification introduced
in Chapter 4, the robustness of the controller is studied to investigate how they are
affected with changing CG height and weight split. The effect that track width has upon
stability controllers will not be discussed since track width is a fixed vehicle property.
These vehicle properties were chosen because they exhibited the ability to change the
maximum lateral acceleration and velocity allowed before rollover.
This chapter is divided into four different sections that include simulations used for
comparisons of ESC with the vehicle property variations:
- Varying CG Height
- No ESC
- All-Wheel Braking and Steering Modification
- Independent Braking
- Varying Weight Split
- No ESC
- All-Wheel Braking and Steering Modification
- Independent Braking
96
- Varying CG Height With Optimized ESC Controllers
- All-Wheel Braking and Steering Modification
- Independent Braking
- Varying Weight Split With Optimized ESC Controllers
- All-Wheel Braking and Steering Modification
- Independent Braking
5.2 Simulation Results for Varying Vehicle Properties
To investigate the effect of changing vehicle properties on the two ESC controllers,
simulations in MATLAB were created that allowed multiple test runswith varyingvehicle
parameters set by the user. The maneuver chosen was the NHTSA fishhook due to the
fact that it has been shown to most excite the rollover dynamics of the vehicle. The
velocity chosen for the testing in this section is 35 miles per hour. This velocity was
chosen due to the fact that it induced rollover in about half of the simulations when
ESC was not present. With rollover occurring in some of the uncontrolled simulations,
it is then known that critical lateral accelerations and yaw rates are achieved, and ESC
would become crucial to the vehicle?s stability in many of the maneuvers.
97
5.2.1 Varying CG Height
The first vehicle property to be examined is the CG height. Without any ESC
implemented, the vehicle will perform in a manner depicted in Figure 5.1. Without any
ESC present, the vehicle with CG heights of 0.8 and 0.9 meters roll after the second
steer input is applied. The results for the position and lateral acceleration for the other
simulations are almost identical, while the roll angle differs.
Figure 5.1: The fishhook maneuver with changing CG height and no ESC present
98
Since it is now known how the vehicle behaves with changing CG height, testing
the all-wheel braking and steering modification controller can be done. Figure 5.2 shows
how the vehicle behaves during the fishhook maneuver with the ESC. When the all-wheel
braking and steering modification controller is applied using the same stability threshold
and brake forces, all of the vehicles in simulation remain stable. As the CG height is
increased, the ESC system applies a higher brake force due to the increased yaw rates
and lateral accelerations created by the vehicle.
0 5 1040
45
50
55
Time (sec)Vehicle Velocity (kph)
Desired
0 5 10?15
?10
?5
0
5
10
15
Time (sec)Front Tire Angle (deg)
Desired
0 50 100
?20
0
20
40
EAST (m)
NORTH (m)
0 5 10?4
?2
0
2
4
Time (sec)
Vehicle Roll (deg)
0 5 10?50
0
50
Yaw Rate (deg/s)
Time (sec)
0 5 10?1
?0.5
0
0.5
1
Time (sec)Lateral Acceleration (g)
.5 m
.6 m
.7 m
.8 m
.9 m
Figure 5.2: The fishhook maneuver with changing CG height and all-wheel braking and
steering modification
99
As seen in Figure 5.3, the independent braking controller prevents the rollover when
the CG height is 0.8 m; however, rollover still occurs when it is 0.9 m. It does not reduce
the lateral acceleration and yaw rate as much as the previous controller, but it does
greatly reduce the yaw error.
Figure 5.3: The fishhook maneuver with changing CG height and independent wheel
braking
100
5.2.2 Varying Weight Split
The other vehicle property to be examined is the weight split. Without any ESC
implemented, the vehicle will perform in a manner depicted in Figure 5.4. In this simu-
lation all of the configurations rolled over except for the 57.5/42.5 configuration. With
these results, the all-wheel braking and steering modification ESC can be implemented
and compared.
Figure 5.4: The fishhook maneuver with changing WS and no ESC present
Figure 5.5 shows how the ESC system affects the vehicle. As with all of the previous
setups, the controller was able to prevent rollover. However, the controller once again
reduced the steer angle in a way that would most likely cause a collision in practice.
The methods of combining stability and path tracking could be a good avenue for future
101
research. Simulations show that there is no optimal solution and that some compromises
must be made in order to prevent rollover.
Figure 5.5: The fishhook maneuver with changing WS and all-wheel braking and steering
modification
Figure 5.6 shows the independent wheel braking controller with changing WS. This
controller once again has a greater ability to keep the desired path. The independent
wheel controller was able to prevent rollover for three out of four vehicle configurations.
The one setup that rolled (42.5/57.5 WS) is a configuration that simulates an oversteer
vehicle with a large rear payload, could not be kept stable during the maneuver. Perhaps
with knowledge of some vehicle properties, changes in the vehicle?s stability limits and
ESC outputs could prevent rollover more often as investigated in the next sections.
102
Figure 5.6: The fishhook maneuver with changing WS and independent braking
5.3 Simulation Results for Optimized ESC Limits and Inputs
As previously discussed, a knowledge of key vehicle parameters would allow ESC
systems to be altered to more adequately adjust the stability threshold and ESC outputs
in order to reduce rollover. The following sections compare the same vehicle configura-
tions in the same NHTSA fishhook maneuver. In these simulations, the values for the
maximum lateral acceleration allowed is optimized, as well as the braking forces applied
at the wheels.
103
5.3.1 Varying CG Height With Optimized ESC Controllers
With knowledge of the changing CG height, the ESC system can be adapted to
further prevent rollover. Figure 5.7 shows the same simulations performed previously in
Figure 5.2, but with an optimized controller for each vehicle setup.
0 5 10
45
50
55
Time (sec)Vehicle Velocity (kph)
Desired
0 5 10?15
?10
?5
0
5
10
15
Time (sec)Front Tire Angle (deg)
Desired
0 50 100
?20
0
20
40
EAST (m)
NORTH (m)
0 5 10
?3
?2
?1
0
1
2
3
4
Time (sec)
Vehicle Roll (deg)
0 5 10
?30
?20
?10
0
10
20
30
40
Yaw Rate (deg/s)
Time (sec)
0 5 10
?0.5
0
0.5
Time (sec)Lateral Acceleration (g)
.5 m
.6 m
.7 m
.8 m
.9 m
Figure 5.7: CG height changes with an optimized all-wheel braking and steering ESC
This controller once again prevents rollover due to the reduction in yaw rate. In
the cases where the CG height is smaller (0.5 and 0.6 m), the controller increases the
maximum lateral acceleration and yaw rate allowed before rollover and the braking forces
applied by the controller are reduced. With the adjusted stability limits and braking
forces, the vehicle is allowed to stay closer to the desired path, and the yaw error is
reduced.
104
The independent braking controller can also be adapted to take knowledge of chang-
ing CG heights into effect. Figure 5.8 displays results using the independent braking
controller.
0 5 1030
35
40
45
50
55
Time (sec)Vehicle Velocity (kph)
Desired
0 5 10?2
?1
0
1
2
TimeLateral Velocity (m/s)
0 20 40 60 80
0
20
40
EAST (m)
NORTH (m)
0 5 10
?4
?2
0
2
4
6
Time (sec)
Vehicle Roll (deg)
0 5 10?50
0
50
Yaw Rate (deg/s)
Time (sec)
0 5 10?1.5
?1
?0.5
0
0.5
1
Time (sec)Lateral Acceleration (g)
.5 m
.6 m
.7 m
.8 m
.9 m
Figure 5.8: CG height changes with an optimized independent braking ESC
The optimized controller manages to keep the vehicle with a CG height of 0.9 stable,
despite the fact that it is very close to rollover. Also, the braking forces for the more
stable vehicles are reduced to a maximum of 300 N in order to limit the effect of the
ESC on the vehicle.
105
5.3.2 Varying Weight Split With Optimized ESC Controllers
Changes in weight split have been proven to render ESC systems ineffective if the
vehicle is very oversteer and the controller gains are not correct. Figure 5.9 shows
the fishhook maneuver with the all-wheel braking and steering modification controller.
The optimized controller once again manages to keep the vehicles stable by altering the
stability limits and braking forces. The 42.5/57.5 WS vehicle is also able to keep more
to the desired path than with the standard controller simulated previously in Figure 5.6.
0 5 1040
45
50
55
Time (sec)Vehicle Velocity (kph)
Desired
0 5 10?15
?10
?5
0
5
10
15
Time (sec)Front Tire Angle (deg)
Desired
0 50 100
?20
0
20
40
EAST (m)
NORTH (m)
0 5 10?4
?2
0
2
4
Time (sec)
Vehicle Roll (deg)
0 5 10
?30
?20
?10
0
10
20
30
Yaw Rate (deg/s)
Time (sec)
0 5 10
?0.5
0
0.5
Time (sec)Lateral Acceleration (g)
42.5/57.5
47.5/52.5
52.5/47.5
57.5/42.5
Figure 5.9: WS changes with an optimized all-wheel braking and steering ESC
106
Figure 5.10 shows a simulation of the optimized independent braking controller in
action. This controller is once again able to prevent rollover in all of the cases tested.
For the 42.5/57.5 WS case, the vehicle?s path is widened and the yaw error is large.
However, with the general system applied earlier in the chapter, this vehicle configuration
experienced rollover. The knowledge of the weight split allowed the controller to increase
its brakingforces when it was necessary, and reducethe brakingforces when not, resulting
in the prevention of rollover.
0 5 10
30
35
40
45
50
55
Time (sec)Vehicle Velocity (kph)
0 5 10?5
?4
?3
?2
?1
0
1
2
TimeLateral Velocity (m/s)
0 50 100
0
20
40
EAST (m)
NORTH (m)
0 5 10
?4
?2
0
2
4
6
8
Time (sec)
Vehicle Roll (deg)
0 5 10?60
?40
?20
0
20
40
60
Yaw Rate (deg/s)
Time (sec)
0 5 10
?1
?0.5
0
0.5
1
Time (sec)Lateral Acceleration (g)
42.5/57.5
47.5/52.5
52.5/47.5
57.5/42.5
Figure 5.10: WS changes with an optimized independent braking ESC
107
5.4 Conclusions
The ESC controllers tested in this chapter were subjected to changes in CG height
and weight split. With basic limits of lateral acceleration and yaw rate and standard
braking forces applied, the stability controllers were able to reduce the occurrence of
vehicle rollover most of the time. However, when the controllers were optimized for the
vehicle setup, rollover was avoided and path following was improved.
For the all-wheel braking with steering modification controller, the vehicle?s stability
was greatly improved. However, the vehicle?s path error was usually large enough that
the implementation in a real vehicle could prove detrimental in the event of an obstacle
avoidance maneuver. Perhaps with more research and varied controller gains, the all-
wheel braking with steering modification controller could perform better at keeping the
desired path, while remaining in a stable region.
Theindependentbrakingcontroller?s ability to keep the vehicle?s path whilereducing
the vehicle?s lateral acceleration and yaw rate is promising. This type of controller, which
is going to implemented into most new production passenger vehicles, has already proven
successful on today?s highways. With a knowledge of CG parameters taken from state
estimation techniques, vehicle stability controllers can be improved and the number of
rollover incidents can be reduced. Simulation results have proven that by optimizing
ESC systems to the individual vehicle, specifically providing knowledge of the vehicle?s
CG location (height and weight split), stability and handling can be greatly improved
during evasive maneuvers.
108
Chapter 6
Conclusions
6.1 Overall Contributions
This thesis demonstrates that with knowledge of a few key vehicle properties, ESC
systems can be optimized for better improvement. In order to test the parameters that
influence vehicle rollover, a complex vehicle model was created and compared to the
CarSim, a commercially available vehicle dynamics software package.
6.1.1 Parameters That Most Influence Rollover
Chapter 3 investigates the key vehicle properties that most affect vehicle rollover. In
simulation, it was shown that CG height, track width, understeer gradient, and friction
coefficients affect how a vehicle performs during test maneuvers. Simulations in CarSim
show that as CG height, understeer gradient, and friction coefficient increase, so does
the chance of rollover. Although track width is not a variable property, it was also shown
that as the track width is reduced, the chances for rollover increase.
6.1.2 Vehicle Rollover Prediction
Using free body diagrams derived for a four-wheeled vehicle, equations were de-
rived for the prediction of vehicle rollover. Simulation results proved the validity of the
modified Static Stability Factor equation. When the CG height or understeer gradient
increased or the track width decreased, the maximum lateral acceleration before rollover
109
allowed by the vehicle was decreased. The Static Stability Factor equation was also
modified to include a scale factor for the inclusion of suspension dynamics.
Another rollover prediction formula for the rollover velocity was then created and
tested using multiple simulations. Using knowledge of CG height, track width, and
radius of turn (or yaw rate), the formula was proven to effectively predict the velocity
that a vehicle rolled over in MATLAB and CarSim simulations. Like the SSF, this
equation was modified to include a scale factor for weight transfer. The significance of
this formula is large because with a few vehicle properties, parameter estimates, and
sensor measurements, one can predict if a vehicle will roll.
6.1.3 ESC Development
Chapter 4 consists of the discussion and derivation of seven different types of Elec-
tronic Stability Controllers. Thestability controllers are modeled in MATLAB and tested
for effectiveness. Although only two are examined in Chapter 5, the active torque and
the independent braking with steering correction controllers show promise for increased
vehicle stability.
6.1.4 Effect of Varying Vehicle Properties on ESC
In Chapter 5, two of the ESC systems were chosen to be tested for robustness
to the principal changing vehicle properties. It was shown that the all-wheel braking
with steering modification controller can be problematic due to errors in heading, but
is very effective in rollover mitigation. An independent wheel braking controller was
also examined to test its effectiveness to changing vehicle parameters. This system was
110
proven in simulation to be a good choice due to its ability to ensure vehicle stability,
while remaining close to the desired path.
Chapter 5 also examines the controllers? abilities to adapt to changing vehicle prop-
erties. With a prior knowledge of the key vehicle parameters, it was shown that the ESC
systems were capable of improving the stability and handling of a vehicle with slight
modification of stability limits and ESC outputs.
6.2 Difficulties
In the development of vehicle simulations, small problems arose with the creation of
programs in MATLAB. Since every vehicle property must be known for the MATLAB
simulations, there were consistency errors when the simulations were first compared to
those in CarSim. Due to CarSim?s advanced user platform, time to become familiar
with the program was required before all of the vehicle parameters could be accurately
changed in order to compare to MATLAB simulations.
Another difficulty that arose was the lack of prior knowledge and papers on the
development and implementation of complicated ESC systems. Most of the ESC systems,
especially the more complicated algorithms, required some adjustment of controller gains
and stability limits in order to get the controllers to work properly. However, with time,
the ESC development and implementation became successful and accurate testing was
performed.
As described in Chapter 3, the effects of understeer gradient were not completely
consistent when changes in weight split and cornering stiffness were made. The discrep-
ancies of the empirical results produced some errors with the inclusion of understeer
gradient scale factor in rollover prediction formula. Also, a simple, easy to calculate
111
analytical solution for vehicle rollover velocity and lateral acceleration with the effects of
understeer gradient has not been found yet. More work is needed to create a simple, easy
to calculate rollover equation that includes the understeer gradient and weight transfer.
6.3 Recommendations for Future Work
While doing this research, it was realized that complicated ESC systems could be-
come a daunting task to take on. In order to fully understand all of the dynamics and
characteristics of the active torque distribution and the independent braking with steer-
ing controllers, a great deal research needs to be done. Both ESC systems are very
promising, some of which are in production vehicles today.
Work is continuing into the inclusion of the understeer gradient effects upon the
prediction of rollover velocity and lateral acceleration. It has been shown that the un-
dersteer gradient plays a part in the rollover propensity of a vehicle; however, a simple
analytical solution that includes its effects could improve the accuracy of the formula.
Although accuracy of the rollover velocity and lateral acceleration equations was
proven, testing can also be done to prove the accuracy of the prediction of the other
vehicle properties in Equations (3.16), (3.17), (3.18), and (3.19). The velocity and lat-
eral acceleration (SSF) equations were the only ones analyzed due to their importance;
however, the prediction of critical CG height, track width, yaw rate, or radius of turn
could prove to be useful.
Another good area of future research is state estimation in combination with ESC.
With the knowledge of the understeer gradient and the CG height and lateral location,
these parameters can be inserted to an adaptable ESC and further optimization would
be possible on actual vehicles. The implementation of the ESC algorithms onto a UGV
112
would prove beneficial and could validate the results of this thesis. To take the ESC
systems that require independent braking into account, the brake systems must be in-
dependently controlled for each wheel, with the ability to be controlled remotely by
computer.
113
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Appendices
120
Appendix A
Vehicle Nomenclature
a Length between CG and front contact patch
ay Lateral Acceleration
b Length between CG and rear contact patch
B Shock Damping (f/r)
CG Center of Gravity
d Length from rc to CG
? Steer Angle
Fb Damping Force (f/r)
Fk Spring Force (f/r)
Fy Tire Lateral Force (fL, fR, rL, rR)
Fz Tire Vertical Force (fL, fR, rL, rR)
HCG CG Height
Hrc RC Height (f/r)
KARB Anti-Roll Bar Stiffness (f/r)
Ks Spring Stiffness (f/r)
M Sprung Mass (f/r)
m Unsprung Mass (f/r)
MT Total Mass
Marb Anti-roll Bar Moment (f/r)
? Roll Angle
r Yaw Rate
rc Roll Center (f/r)
Ry Lateral Reaction Force (fL, fR, rL, rR)
Rz Vertical Reaction Force (fL, fR, rL, rR)
S Length between L and R springs/dampers (f/r)
tw Track Width (f/r)
V Vehicle Velocity
Vx Vehicle Longitudinal Velocity
Vy Vehicle Lateral Velocity
f = front r = rear L = Left R = Right
121
Appendix B
Vehicle Properties
Typical SUV Properties - Taken From a 2000 Chevrolet Blazer
Wheelbase: L 2.72 m
Front Track Width: TWf 1.45 m
Rear Track Width: TWr 1.40 m
CG Height (Sprung Mass): HCG 0.6 m
RC Height: HRC 0.4 m
Unsprung Mass Height: Hm 0.25 m
Vehicle Mass: MT 2150 kg
Sprung Mass: M 1720 kg
Unsprung Mass: m 430 kg
Steering Ratio SR 18
Standard Weight Split WS 55/45 (f/r)
Dist. from CG to Front Contact Patch a 1.22 m
Dist. from CG to Rear Contact Patch b 1.5 m
Moment of Inertia Iz 3800 kg?m2
Mass Moment of Inertia about x-axis Ix 1243 kg?m2
Tire Cornering Stiffness per Tire C? 60000 N/rad
Stiffness of Front Springs kf 72500 N/m
Stiffness of Rear Springs kr 67000 N/m
Distance between Front springs Skf 0.7747 m
Distance between Rear springs Skr 0.9906 m
Distance between Front dampers Sbf 0.7747 m
Distance between Rear dampers Sbr 0.762 m
Force from Front Anti-Roll Bar per Tire FARBf 750 N/degree
Force from Rear Anti-Roll Bar per Tire FARBr 550 N/degree
122
Appendix C
ESC Controller Description
C.1 Stability Threshold Stages
The stability thresholds defined in MATLAB simulations consist of single and two-
stage levels. The single stage controllers consist of one preset value of lateral accelera-
tion or yaw rate that limits the acceptable level of the two vehicle properties. Figure
C.1 depicts the single stage controller regulations. Once the magnitude of the lateral
acceleration or yaw rate exceeds the maximum acceptable level, the controller is applied
and stability is improved.
Figure C.1: The single stage controller
The two stage controller has more options. With two predefined levels of lateral
acceleration and yaw rate, a ?warning? and ?critical? level can be established. When
the critical level is exceeded, the controller applies the maximum amount of breaking
torque or steering modification. However, between the warning level and critical level,
123
there are several options. For example, possible choices for braking levels levels include a
step value, a linear increase from zero or another point, or even a nonlinear increase with
increasing values of lateral acceleration or yaw rate. The possibilities of the controller
types are endless. Figure C.2 depicts the double stage controller regulations.
Figure C.2: The two stage controller
C.2 ESC Types and Inputs
The controllers implemented in MATLAB were created to be inserted into the ve-
hicle model described in Chapter 2. All of the controllers modeled had inputs of lateral
acceleration and yaw rate. These vehicle properties were used because the measure-
ments are easily measured with an accelerometer and gyroscope, sensors that can be
relatively inexpensive to install. The more complex controllers also use measurements of
independent wheel velocities and steer angle.
124
Table C.1 - ESC Types:
Power Reduction
All-Wheel Braking
Independent Wheel Braking
Active Torque Distribution
Steering Modification
Steering Modification with All-Wheel Braking
Independent Wheel Braking with Steering Control
C.3 Power Reduction
With the vehicle modeled in MATLAB, the power reduction controller limits the
power applied to the wheels. Since the rolling resistance and air drag were not included
in the vehicle model, the power reduction is modeled as a slight braking force. Although
the effects of air drag and rolling resistance are not necessarily linear, the assumption is
adequate for testing purposes, since the vehicle usually finishes the maneuver before a
noticeable velocity change occurs.
The controller works as follows: if the absolute value of the lateral acceleration or
yaw rate measured is greater than the preset limit, then the controller adds a slight
braking force to all of the wheels. This braking torque was found from previous data
with a vehicle performing coast down tests. A linear fit for the deceleration was made,
and the braking force (i.e. power reduction) was found by varying the braking forces in
simulation until the behavior matched the experimental data.
Figure C.3 shows the braking torques applied for a power reduction controller. For
example, when the lateral acceleration exceeded a set limit (0.3 g), the controller simu-
lated a power reduction with a braking torque of 50 N-m.
125
Figure C.3: The braking torques applied to simulate a power reduction
C.4 All-Wheel Braking
The all-wheel braking controller is somewhat similar to the power reduction con-
troller, except for differences in the braking forces applied. Several different options were
modeled in MATLAB simulations for the braking forces applied. The constant braking
controllers are comprised of single and multi-stage limits, with varying braking forces.
The basic all-wheel controller was almost identical to the step input in the power
reduction controller; however, the braking force applied was greater. This not only added
the effects of wind and rolling resistance, but it also incorporated an actual braking force
on the vehicle.
To further improve the controller?s performance, a second stage of the controller
was added. By setting two limits of lateral acceleration and yaw rate, the controller
can apply different braking torques depending on the threat of stability loss. Figure C.4
shows an example of a multi-stage controller with two limits incorporated. When the
126
vehicle reaches a warning lateral acceleration (0.3 g), a braking torque of 200 N-m is
applied. If the lateral acceleration reaches the second defined value (0.45 g), a maximum
braking torque of 450 N-m is applied. This level was once again found using previous
test data where hard braking occurred.
Figure C.4: The braking torques applied to simulate a milti-step braking controller
In order to increase the controller?s ability to improve stability, the braking torques
can be altered using the values of lateral acceleration. Figure C.5 shows how the variable
braking torques can be applied to the all-wheel braking controller. Here the controller
applies the 200 N-m braking torque when the lower limit is breached, but once that
level is exceeded, the braking torque grows depending on the magnitude of the lateral
acceleration of the vehicle. This increase continues until the maximum defined level of
lateral acceleration or yaw rate is achieved and the maximum braking torque is applied
(450 N-m).
127
Figure C.5: The braking torques applied to simulate a variable braking controller
C.5 Independent Wheel Braking
The independent wheel braking controller modeled in MATLAB used the variable
braking torques shown in Figure C.5; however, the brakes are applied to independent
wheels (or sides) of the vehicle. The derivation of the brake steer moment is described
in Section 4.7.1. The longitudinal braking forces applied by the wheels are calculated
by dividing the braking torque by the tire?s effective radius (Fx = ?/Reff). If this force
is greater than the lateral force allowed by the tire model, the maximum lateral force
becomes that allowed by the tire model and some sliding occurs.
If the vehicle is understeer (Kus > 0) and the stability threshold is compromised,
braking torques are applied to the inner wheels. In the oversteer case (Kus < 0), outer
wheel braking is applied to increase stability. For the neutral steer case, inner wheel
braking is also applied, in order to reduce errors in yaw rate. Figure C.6 shows a vehicle
performingthe fishhookmaneuver with the independentwheelbraking controller applied.
128
When the vehicle exceeds the stability limit, the braking torques are applied to the inner
wheels since it is understeer.
0 10 20 30 40 50 60 70 80 90?5
0
5
10
15
20
25
30
35
40
EAST (m)
NORTH (m)
Vehicle Path
Braking Applied
Figure C.6: The fishhook maneuver with braking times with independent wheel braking
C.6 Active Torque Distribution
The active torque distribution controller is similar to the independent wheel braking
controller; yet, a positive longitudinal force is applied to the wheels that are not being
braked during evasive maneuvers. In the simulations presented in this thesis, the engine
torques applied were constant (200 N-m) when the stability threshold was exceeded.
This was chosen for simplicity, although the effectiveness of the controller could probably
improve with more research and modification.
129
C.7 Steering Modification
The steering modification controller required a compromise between stability and
path following. There is no perfect amount of steering modification allowed by the
controller, yet a value must be chosen that reduces the unsafe level of lateral acceleration
and yaw rate, while not deviating far from the path.
The controller modeled in MATLAB once again uses a two stage stability threshold.
Once the first stability threshold is crossed (past the warning lateral acceleration or
yaw rate), the steering controller simply holds the steer angle until the driver?s steer
angle input is reduced. If the lateral acceleration or yaw rate continues to increase past
the second stability threshold, the controller then actively reduces the steer angle by a
preset percentage. The setting this percentage was done by trial and error to see what
compromise between stability and path deviation is acceptable. Figures C.7 and C.8
show the performance of the steering modification controller when the second stage of
the controller reduces the steer angle by 0.1% for each time interval (0.01 sec). The path
of the vehicle is close to that of one without the controller, but the stability of the vehicle
is compromised.
0 20 40 60 80
0
10
20
30
40
50
EAST (m)
NORTH (m)
Without ESC
With ESC
Figure C.7: Position
0 2 4 6 8 10
?10
0
10
time (sec)
Delta (deg)
0 2 4 6 8 10
?1
?0.5
0
0.5
Time
Lat Accel (g)
Without ESC
With ESC
Figure C.8: ? & Lateral Accel.
130
Figures C.9 and C.10 show the performance of the steering modification controller
when the second stage of the controller reduces the steer angle by 0.5% for each time
interval (0.01 sec). With this controller output, the stability of the vehicle is more
guaranteed, but the vehicle?s path is not close to the desired path.
0 20 40 60 80 100
?20
?10
0
10
20
30
40
50
60
EAST (m)
NORTH (m)
Without ESC
With ESC
Figure C.9: Position
0 2 4 6 8 10
?10
0
10
time (sec)
Delta (deg)
0 2 4 6 8 10
?1
?0.5
0
0.5
Time
Lat Accel (g)
Without ESC
With ESC
Figure C.10: ? & Lateral Accel.
As seen in the previous figures, the compromise between stability and path following
is a difficult one. For the simulations in this thesis, the controller was given a steer angle
reduction of 0.3% per time interval once the second stability index was compromised.
This value was chosen because it successfully reduces the lateral acceleration and yaw
rate of the vehicle while staying somewhat close to the path.
C.8 Steering Modification with All-Wheel Braking
The steering modification with all-wheel braking controller is simply a combination
of the two previously-described controllers. Once the first stability threshold is crossed,
the steer angle is held constant and a medium brake force is applied on all of the wheels.
Similarly, if the second threshold is entered, the vehicle applies a large braking force and
the steer angle is reduced by 0.3% per time interval.
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The gains of the controller (braking torques and steering modification) could be
adjusted to possibly improve the stability of the vehicle, although the all-wheel braking
aspect of the controller is somewhat limited, since it does not induce a brake steer
moment that can reduce the vehicle?s yaw rate error.
C.9 Independent Wheel Braking with Steering Control
The independent wheel braking with steering modification controller is the most
complex of the controllers described in this thesis. This controller uses the two-stage
stability threshold, similar to the basic independent wheel braking controller; however,
if the lateral acceleration and yaw rate are less than the maximum allowed value before
the second stability threshold, the controller modifies the steer angle while individually
braking particular wheels to slow the vehicle and improve path following. This is made
possible with the knowledge of the vehicle?s yaw rate, lateral acceleration, velocity, and
driver steering input.
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