08-16-2017, 09:19 PM
ABSTRACT
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VEHICLE HANDLING, STABILITY,AND BIFURCATION ANALYSIS FOR NONLINEAR VEHICLE MODELS
Vehicle handling, stability, and bifurcation of equilibrium conditions were studied
using a state vector approach. The research provided a framework for an
improved method of vehicle handling assessment that included non-linear regions
of performance and transient behavior. Vehicle models under pure lateral slip,
constant velocity, and constant front steer were developed. Four-wheel, two-axle
vehicle models were evolved from simpler models and were extended to include vehicle
roll dynamics and lateral load transfer effects. Nonlinearities stem from tire
force characteristics that include tire force saturation. Bifurcations were studied
by quasi-static variations of vehicle speed and front steer angle. System models
were expanded, assessing overall stability, including vehicle behavior outside
normal operating ranges. Nonlinear models of understeering, oversteering, and
neutral steering vehicles were created and analyzed. Domains of attraction for stable
equilibrium were discussed along with physical interpretations of results from
the system analysis.
VEHICLE HANDLING, STABILITY,
AND BIFURCATION ANALYSIS
FOR NONLINEAR VEHICLE MODELS
by
Vincent Nguyen
Thesis submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Master of Science
2005
Advisory Committee:
Dr. Gregory A. Schultz, Chairman/Advisor
Professor Balakumar Balachandran
Professor David Holloway
c Copyright by
Vincent Nguyen
2005
DEDICATION
To my father, who always allowed me to find my own path. Thank
you for your support in all of my endeavors.
ii
ACKNOWLEDGEMENTS
Thanks to: Dr. Greg Schultz for providing me with this amazing opportunity
for research. And for his guidance, support and especially
his enthusiasm throughout the entire process. Dr. Balakumar Balachandran,
for his essential technical assistance, and for finding the
time to meet with me every week. His supervision and guidance made
this research possible. Ivan Tong at Aberdeen Test Center (ATC) for
his assistance in discussing and developing this work, and in particular
his vital intuitions that led to the expanded model. Kevin Kefauver
and the Roadway Simulator group at Aberdeen Test Center for their
assistance and data that supported this research.
Also thanks to Dr. David Holloway who introduced me to the exciting
field of automotive engineering which has not only led me to this work,
but has predominated my research, academic and personal interests for
the last four years.
Lastly, I d like to thank Nicole Craver who has been by my side, my
best friend, for the last seven years. She has supported me in all things,
this work included. I could not have done it without her.
ii
TABLE OF CONTENTS
List of Tables vii
List of Figures vii
1 Introduction 1
1.1 SAE vehicle model . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Steady-state vehicle handling classification . . . . . . . . . . . . . . 5
1.3 State space, system stability, and phase portraits . . . . . . . . . . 10
1.4 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.1 Handling classifications . . . . . . . . . . . . . . . . . . . . . 15
1.4.2 Stability notions . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4.3 State space approaches . . . . . . . . . . . . . . . . . . . . . 17
1.5 Contributions and thesis organization . . . . . . . . . . . . . . . . . 17
2 Development of theoretical models 20
2.1 Bicycle model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.1 Bifurcation in steer . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.2 Bifurcation in velocity . . . . . . . . . . . . . . . . . . . . . 29
2.2 Tandem-axle model . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Four-wheel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
iv
3 LTV model 45
3.1 Tire Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.1 The Magic Tire Formula . . . . . . . . . . . . . . . . . . . . 46
3.1.2 The genetic optimization algorithm . . . . . . . . . . . . . . 49
3.1.3 Results of the algorithm . . . . . . . . . . . . . . . . . . . . 53
3.1.4 Modified formulation . . . . . . . . . . . . . . . . . . . . . . 55
3.2 LTV four-wheel model results . . . . . . . . . . . . . . . . . . . . . 67
3.3 Expanded model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.1 High-slip tire force model . . . . . . . . . . . . . . . . . . . 71
3.3.2 Expanded model homoclinic orbit generation . . . . . . . . . 74
3.3.3 Expanded model results . . . . . . . . . . . . . . . . . . . . 74
3.4 Lateral load transfer model . . . . . . . . . . . . . . . . . . . . . . . 78
3.4.1 Additional states . . . . . . . . . . . . . . . . . . . . . . . . 78
3.5 Lateral load model results . . . . . . . . . . . . . . . . . . . . . . . 82
4 Analysis and observations 86
4.1 Phase portraits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.2 Equilibrium points . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3 Bifurcation diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.4 Understeer and neutral steer . . . . . . . . . . . . . . . . . . . . . . 93
4.4.1 Understeer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.4.2 Neutral steer . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4.3 US/OS/NS bifurcation diagrams . . . . . . . . . . . . . . . 97
4.5 Expanded model results . . . . . . . . . . . . . . . . . . . . . . . . 102
4.5.1 General stability versus practical stability . . . . . . . . . . 102
4.5.2 Practicality of constant velocity assumptions . . . . . . . . . 108
v
4.6 Lateral load model . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.7 Nonlinear steady-state handling classification . . . . . . . . . . . . . 109
5 Summary and recommendations for future work 117
5.1 Recommendations for future work . . . . . . . . . . . . . . . . . . . 118
Bibliography 120
vi
LIST OF TABLES
2.1 Bicycle model tire parameters. . . . . . . . . . . . . . . . . . . . . . 23
2.2 Tandem-axle bicycle model tire parameters. . . . . . . . . . . . . . 34
2.3 4 wheel model tire parameters. . . . . . . . . . . . . . . . . . . . . . 41
3.1 LTV lateral tire force data @ 20 psi. . . . . . . . . . . . . . . . . . 56
3.2 LTV lateral tire force data @ 35 psi. . . . . . . . . . . . . . . . . . 57
3.3 LTV lateral tire force data @ 50psi. . . . . . . . . . . . . . . . . . . 58
3.4 LTV GA coefficient results. . . . . . . . . . . . . . . . . . . . . . . 66
vii
LIST OF FIGURES
1.1 SAE vehicle coordinate orientations. . . . . . . . . . . . . . . . . . 4
1.2 SAE four-wheel vehicle parameters. The blue circle represents the
vehicle CG, _ represents the vehicle sideslip, and _f represents the
front steer angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Simple bicycle model of a two-axle, four-wheel vehicle. . . . . . . . 6
1.4 Diagram of low-speed cornering with bicycle model. . . . . . . . . . 7
1.5 Tire force orientations for linear model used in [8] for classification
of US/OS/NS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Handling diagram for US/OS/NS using linear tire model. . . . . . . 10
1.7 Handling diagram for nonlinear tire forces. . . . . . . . . . . . . . . 11
1.8 Spring-mass-damper system. . . . . . . . . . . . . . . . . . . . . . . 11
1.9 Phase portrait for simple spring-mass-damper system with m = 1,
k = 1, c = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1 Bicycle model presented in [18]. . . . . . . . . . . . . . . . . . . . . 21
2.2 SAE representation of bicycle model. . . . . . . . . . . . . . . . . . 22
2.3 Tire force diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Tire force versus slip angle for front and rear tires in bicycle model. 24
2.5 Phase portrait for bicycle model at _f = 0 radians and V = 20 m/s. 26
vii
2.6 Phase portrait for bicycle model at _f = 0.015 radians and V = 20
m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7 Phase portrait for bicycle model at _f = 0.030 radians and V = 20
m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.8 Bicycle model bifurcation diagram. Equilibrium values of _ versus
steer angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.9 Bicycle model bifurcation diagram. Equilibrium values of r versus
steer angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.10 Phase portrait for bicycle model at V = 10 m/s and _f = 0.015
radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.11 Phase portrait for bicycle model at V = 20 m/s and _f = 0.015
radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.12 Phase portrait for bicycle model at V = 20 m/s and _f = 0.015
radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.13 Bicycle model bifurcation diagram. Equilibrium values of _ versus
velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.14 Bicycle model bifurcation diagram. Equilibrium values of r versus
velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.15 SAE representation of bicycle model for a tandem-axle vehicle. . . . 33
2.16 Phase portrait for tandem-axle model at V = 10 m/s and _f = 0.015
radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.17 Phase portrait for tandem-axle model at V = 25 m/s and _f = 0.015
radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.18 Phase portrait for tandem-axle model at V = 40 m/s and _f = 0.015
radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
ix
2.19 Bicycle model bifurcation diagram. Equilibrium values of _ versus
front steer angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.20 Bicycle model bifurcation diagram. Equilibrium values of r versus
front steer angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.21 SAE representation of the 4 wheel model. . . . . . . . . . . . . . . . 38
2.22 Phase portrait for 4 wheel model at V = 10 m/s and _f = 0.015
radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.23 Phase portrait for 4 wheel model at V = 20 m/s and _f = 0.015
radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.24 Phase portrait for 4 wheel model at V = 30 m/s and _f = 0.015
radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.25 Bicycle model bifurcation diagram. Equilibrium values of _ versus
speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.26 Bicycle model bifurcation diagram. Equilibrium values of r versus
speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1 General Magic Tire Formula slip curve [19]. . . . . . . . . . . . . . 47
3.2 GA flowchart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3 Genetic algorithm mean square error vs. iterations. . . . . . . . . . 54
3.4 Tire force vs. slip angle, GA results. . . . . . . . . . . . . . . . . . 54
3.5 LTV tire lateral force vs. slip angle at 20 psi. . . . . . . . . . . . . 56
3.6 LTV tire lateral force vs. slip angle at 35 psi. . . . . . . . . . . . . 57
3.7 LTV tire lateral force vs. slip angle at 50 psi. . . . . . . . . . . . . 58
3.8 Original GA results for lateral force at the highest vertical load
tested, performed for 20 runs at each tire pressure. . . . . . . . . . 60
x
3.9 GA results when holding C=1.3 for lateral force at the highest vertical
load tested, performed for 20 runs at each tire pressure. . . . . 60
3.10 Example of undesirable GA solution for 35 psi, C=1.3, E allowed
to float according to full tire model formulation. . . . . . . . . . . . 61
3.11 Example of undesirable GA solution for 50 psi, C=1.3, E allowed
to float according to full tire model formulation. . . . . . . . . . . . 61
3.12 Lateral force vs. vertical load for undesirable solution at 35 psi. . . 62
3.13 Lateral force vs. vertical load for undesirable solution at 50 psi. . . 62
3.14 Example of a desirable GA solution for 50 psi, C=1.3, E=optmized
constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.15 Lateral force vs. vertical load for desirable solution at 50 psi. . . . . 64
3.16 Lateral force vs. slip angle for 20 psi data. . . . . . . . . . . . . . . 65
3.17 Lateral force vs. slip angle for 35 psi data. . . . . . . . . . . . . . . 65
3.18 Phase portrait for LTV 4 wheel model at V = 10 m/s and _f = 0.015
radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.19 Phase portrait for LTV 4 wheel model at V = 27 m/s and _f = 0.015
radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.20 Phase portrait for LTV 4 wheel model at V = 36 m/s and _f = 0.015
radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.21 Bifurcation diagram for LTV 4 wheel model, equilibrium values of
_ versus speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.22 Bifurcation diagram for LTV 4 wheel model, equilibrium values of
r versus speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.23 Tire force for small tire slip angle with force resolved perpendicular
to velocity direction. . . . . . . . . . . . . . . . . . . . . . . . . 72
xi
3.24 Tire force direction for 90_ tire slip angle with force resolved perpendicular
to velocity direction. . . . . . . . . . . . . . . . . . . . . 73
3.25 Tire force direction for 90_ tire slip angle with force resolved perpendicular
to tire heading. . . . . . . . . . . . . . . . . . . . . . . . 73
3.26 Wrapped tire lateral force function. . . . . . . . . . . . . . . . . . . 73
3.27 Expanded phase portrait for LTV 4 wheel model at V = 10 m/s
and _f = 0.015 radians. . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.28 Expanded phase portrait for LTV 4 wheel model at V = 27 m/s
and _f = 0.015 radians. . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.29 Expanded phase portrait for LTV 4 wheel model at V = 36 m/s
and _f = 0.015 radians. . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.30 Vehicle roll axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.31 Unsprung mass free body diagram. . . . . . . . . . . . . . . . . . . 79
3.32 Sprung mass free body diagram. . . . . . . . . . . . . . . . . . . . . 80
3.33 Phase portrait for lateral load transfer model at V = 10 m/s and
_f = 0.015 radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.34 Phase portrait for lateral load transfer model at V = 25 m/s and
_f = 0.015 radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.35 Phase portrait for lateral load transfer model at V = 40 m/s and
_f = 0.015 radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.1 Vehicle orientation plot for a drift equilibrium point. . . . . . . . . 90
4.2 Vehicle orientation plot for equilibrium points of LTV model at 10
m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3 Vehicle orientation plot for equilibrium points of LTV model at 27
m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
xii
4.4 Zoomed in view of bifurcation diagram for LTV 4 wheel model,
equilibrium values of _ versus speed. . . . . . . . . . . . . . . . . . 93
4.5 Low speed, understeering phase plot. . . . . . . . . . . . . . . . . . 95
4.6 High speed, understeering phase plot. . . . . . . . . . . . . . . . . . 95
4.7 Low speed, neutral steering phase plot. . . . . . . . . . . . . . . . . 96
4.8 High speed, neutral steering phase plot. . . . . . . . . . . . . . . . . 97
4.9 Bifurcation diagram for LTV 4 wheel model, equilibrium values of
_ versus speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.10 Bifurcation diagram for LTV 4 wheel model, equilibrium values of
r versus speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.11 _ Speed bifurcation diagram for understeering model. . . . . . . . . 99
4.12 r Speed bifurcation diagram for understeering model. . . . . . . . . 100
4.13 _ Speed bifurcation diagram for neutral steering model. . . . . . . . 100
4.14 r Speed bifurcation diagram for neutral steering model. . . . . . . . 101
4.15 Phase portrait for extended LTV model at V = 27 m/s. . . . . . . 103
4.16 Orientation plot for trajectory highlighted in V = 27 m/s phase plot.104
4.17 Phase portrait for extended LTV model at V = 36 m/s. . . . . . . 105
4.18 Orientation plot for trajectory highlighted in V = 36 m/s phase plot.106
4.19 Orientation plot for a rapidly spinning vehicle. . . . . . . . . . . . . 107
4.20 Normalized tire force for bicycle model. . . . . . . . . . . . . . . . . 114
4.21 Zoomed in view of normalized tire force forbicycle model. . . . . . . 114
4.22 (_r - _f ) versus normalized tire force for theoretical bicycle model. 115
4.23 Normalized tire force for LTV four-wheel model. . . . . . . . . . . . 115
4.24 (_r - _f ) versus normalized tire force for LTV four-wheel model. . . 116
xii
5.1 Phase portrait with overlaying plot of stable equilibrium at several
other steer angles. Provided by ATC. . . . . . . . . . . . . . . . . . 118
xiv
Chapter 1
Introduction
Vehicle dynamics and stability have been of considerable interest to automotive
engineers, automobile manufacturers, the government, public safety groups, and
the general public for a number of years. The obvious dilemma is that people
naturally desire to drive faster and faster on the roads and highways, yet they
expect their vehicles to be infinitely stable and safe during all normal and emergency
maneuvers. For the most part, people pay little attention to the limited
handling potential of their vehicles until some unusual behavior is observed that
often results in fatality. Extreme examples of this are the handling behavior of
the Chevrolet Corvair in the 1960s and the recent rollovers experienced with the
Ford Explorer in the 1990s. Although there was much confusion about the exact
cause of the Explorer rollovers, since they seemed to in part be linked to a model
of Firestone tires, it is interesting that Ford soon lengthened the wheelbase of the
vehicle. Nonetheless, significant incidents occurred, resulting in public outcry for
improvement in safety. Note that the rates of speed at which drivers travel is rarely
mentioned by the public as a root cause.
The fundamentals of the physics of vehicle handling began to be explored in
earnest in the 1930s and 1940s by Olley et al. [15]. This work began by exploring
1
the basic behavior of pneumatic tires, which at the time were bias ply constructions.
Radial tires began to gain widespread use in the 1970s. These new tires behaved
somewhat differently, affecting vehicle behavior, and led to a rapid development of
speed-rated tires. Better tires made it more comfortable for drivers to travel even
faster. Consequently, interest in vehicle handling continued.
Recent efforts to better understand vehicle handling have demonstrated that
much is still to be learned and developed in this field as vehicles continue to
evolve. These efforts include cooperative work done by the major automobile
manufacturers through the Alliance of Automobile Manufacturers [2], rule-making
work and studies conducted by the National Highway Traffic Safety Administration
(NHTSA), inspired by the rollover problems experienced with popular sport
utility vehicles (SUVs), and Light Tactical Vehicle (LTV) handling studies being
conducted by the U.S. Army.
The rapid success of sport utility vehicles in the U.S. has heightened interest
in related rollover problems. Though most of the rollovers were tripped by leaving
the roadway or hitting an obstacle, approximately 10% are unexplained and likely
related to vehicle handling behavior [5]. The relatively high center-of-gravity of
SUVs make them highly susceptible to rollover for any number of reasons. The
introduction of stability control systems in American cars has opened up many
new and exciting opportunities for vehicle dynamicists and controls engineers in
the field of vehicle handling and stability research. New questions have arisen,
such as how to identify a spin-out while it is happening, what to do to control the
behavior, and how to control the behavior without creating an additional safety
hazard, such as making the vehicle completely unresponsive. In any regard, the
field of vehicle handling and stability is perhaps more exciting and full of problems
2
to solve than ever.
Historically, vehicle handling has been studied predominately by first quantifying
the steady-state behavior of vehicles and then trying to relate steady-state
principles to transient dynamics. This is so because steady-state behavior is much
easier to visualize than transient dynamics, which are much more difficult to describe,
let alone visualize. Performance within the linear region of modern tires,
usually from 0.3 to 0.4 g of lateral acceleration, is well understood and predictable
for steady-state maneuvers, and also, to some extent, in the transient case. However,
tires are very non-linear beyond 0.4 g and eventually saturate with subsequent
degradation in lateral force capability. Combining complicated tire characteristics
with lateral weight transfer, differences in front/rear roll stiffness, suspension and
steering kinematics and compliance, and other factors make transient behavior
very difficult to describe and predict. The differential equations that describe
vehicle motion can be written in terms of a few key variables and parameters as
linear time-invariant systems. However, the variables and parameters used in these
equations are often highly non-linear.
This research, presents a way to visualize transient behavior over a broad set
of vehicle operating states. The work presented here helps bridge the gap between
steady-state handling principles and transient dynamics, and provides an interesting
way to visualize handling behavior, understand how changes in vehicle set-up
affect stability, and provide a better way to teach vehicle dynamics. Ideas for future
work to extend this research to possibly better characterize transient behavior
will be introduced. In this chapter, the basics of steady-state vehicle dynamics will
be presented, followed by discussion of how the system equations are typically developed.
Finally, a brief literature review and introduction of the current research
3
will be given.
1.1 SAE vehicle model
Figure 1.1: SAE vehicle coordinate orientations.
Unless otherwise noted, this paper uses the standard Society of Automotive
Engineers (SAE) coordinate system shown in Figure 1.1 [8]. The vehicle s positive
x axis is defined to be along the forward direction of the vehicle s longitudinal axis.
The y axis is defined to be towards the right-hand side of the vehicle (while facing
forward) and the z axis points in the downward direction. A two-axle, four-wheel
vehicle with front wheel steering making a right-hand turn is shown in Figure 1.2.
Also shown are the orientations of key vehicle parameters used in this research.
The green line denotes the path of the vehicle center of gravity (CG), shown as
a blue circle. The vehicle s instantaneous velocity, V , is shown tangent to the
vehicle path. Vehicle sideslip, _, is defined as the angle between the vehicle x axis
and the velocity vector at the CG, with positive sideslip defined with the vehicle
axis oriented to the left of velocity. Front steer angle, _f , is the angle between the
centerline of the front tires and the vehicle x axis. Positive steer is achieved with
4
the wheels steered to the right. Vehicle yaw rate, r is defined as a positive rotation
about the vehicle z axis.
Figure 1.2: SAE four-wheel vehicle parameters. The blue circle represents the
vehicle CG, _ represents the vehicle sideslip, and _f represents the front steer
angle.
1.2 Steady-state vehicle handling classification
Vehicle handling behavior is predominantly classified using the so-called understeer
(US), oversteer (OS), and neutral steer (NS) conditions. These terms are traditionally
applied to steady-state handling conditions. Steady-state handling can
be defined as a maneuver in which there are constant vehicle parameters (steer
angle, velocity, roll angle, etc.) and the vehicle motion is constant (constant yaw
5
rate, constant sideslip). Physically, this refers to a vehicle travelling at a constant
velocity along a constant radius turn.
A two-axle, four-wheeled vehicle can be simplified using the so-called bicycle
model [8], where each axle can be approximated as a single tire in line with the CG
of the vehicle. The bicycle model representation of a four-wheeled vehicle is shown
in Figure 1.3, and a diagram of a bicycle model vehicle under low-speed cornering
conditions is shown in Figure 1.4. During low-speed cornering, it is assumed that
the tires have not yet developed any lateral slip and are rolling in the direction
of the velocity. Under this assumption, the front steer can be estimated as L/R,
where L is the wheelbase of the vehicle and R is the radius of the turn. This steer
angle is sometimes referred to as the Ackerman steer angle and is expressed in
radians.
Figure 1.3: Simple bicycle model of a two-axle, four-wheel vehicle.
Under high-speed cornering conditions, where lateral tire slip has developed,
the relationship between the actual steer angle and the Ackerman steer angle is
6
Figure 1.4: Diagram of low-speed cornering with bicycle model.
typically used to classify US/OS/NS for steady-state handling. For a high-speed
right-hand turn, an understeering vehicle will have a front steer angle that is greater
than the Ackerman steer angle. An oversteering vehicle will exhibit a lower steer
angle than the Ackerman steer angle, and a neutral steering vehicle maintains the
Ackerman steer angle through the high speed turns.
US/OS/NS can be described analytically. The tire force orientation for the
US/OS/NS classification used by Gillespie [8] is shown in Figure 1.5. Tire lateral
force is labelled as F and the tire slip angle, _, is defined as the angle between
the velocity at the tire and the heading of the tire. Using the bicycle model
under steady-state cornering in the positive yaw direction (right-hand turn), an
expression for front steer angle can be developed (Eq. (1.1)). The terms _f and
_r represent the front and rear tire slip angles. If in the maneuver, the front slip
angle is greater than the rear, the subsequent front steer angle is greater than
the Ackerman steer angle and the vehicle exhibits understeer. Conversely, if the
rear slip angle is greater than the front, the steer angle is less than the Ackerman
7
steer angle and the vehicle exhibits oversteer. If the slip angles are equal, then the
vehicle is steering at the Ackerman steer angle and is exhibiting neutral steer.
_f =
L
R
+ _f - _r (1.1)
Figure 1.5: Tire force orientations for linear model used in [8] for classification of
US/OS/NS.
Gillespie takes the analysis further with the use of a linear tire force model.
Tire lateral force is assumed to be a linear function of the slip angle, and F = C_ _
describes the relationship. The coefficient C_ is called the tire-cornering stiffness
and is a property of the tire. The tire model can then be applied along with a force
balance equation to provide another expression for front steer angle, Eq. (1.2) [8].
Wf and Wr are the vehicle weights at the front and rear axles. In this expression,
if
_
Wf
C_f
- Wr
C_r
_
is equal to 0, the vehicle is always neutral steering. However, if
_
Wf
C_f
- Wr
C_r
_
is positive, front steer angle can be expected to be greater than the
Ackerman steer angle (at any given positive speed). In addition, the steer angle
can be expected to continue to increase with respect to increasing vehicle speed.
This means that the vehicle not only exhibits understeer, but it exhibits terminal
understeer (increasing steer angle) as the vehicle speed is increased. Similarly
8
when
_
Wf
C_f
- Wr
C_r
_
is negative, the front steer angle will always be less than the
Ackerman angle (for positive velocity), and it will continue to decrease as vehicle
speed is increased. This vehicle not only exhibits oversteer, but exhibits terminal
oversteer as the velocity is increased.
_f =
L
R
+
Wf
C_f
-
Wr
C_r
!
V 2
gR
(1.2)
A handling diagram for the linear tire model and vehicle driving on a constant
radius is provided in Figure 1.6. Steer angle is plotted against lateral acceleration
(V 2/Rg). A neutral steering vehicle will maintain a constant steer angle (the
Ackerman steer angle). An understeering vehicle produces steer angles greater
than the Ackerman steer angle for nonzero velocity and will continue to increase
steer angle at a rate proportional to the lateral acceleration. An oversteering vehicle
operates at a steer angle less than the Ackerman steer angle and will decrease steer
angle at a rate proportional to the lateral acceleration.
In practice, during a maneuver, the operator has no notion of the Ackerman
steer angle. Instead, the operator perceives a change in steer angle as velocity
is increased or decreased. In addition, because of nonlinear tire responses, some
vehicles initially understeer, but as lateral acceleration is increased, a transition
to neutral steer and eventually oversteer occurs. Consequently, it may be more
practical from a driver s point of view to think of the onset of neutral steer and
oversteer as occurring when the required steer angle to negotiate the turn begins
to decrease (as speed is increased).
As such, the commonly accepted relationship between the actual steer angle
and the Ackerman steer angle (greater than or less than) may not be descriptive
enough. For practical use, US/OS/NS should be defined by the slope of the steer
angle/acceleration curve rather than just the value of the steer angle (as compared
9
to the Ackerman steer angle). With the linear tire model, the relationship to the
no-slip (Ackerman) steer angle coincides with rate of change of the steer angle (if
_ > L/R then _ is always increasing and vice versa), so there is no distinction
between the two definitions. With nonlinear tire models, this is not necessarily the
case and a distinction must be made. In Figure 1.7, a typical handling diagram
for a heavy truck is shown. Notice how the vehicle transitions from understeer
to oversteer. The transition point occurs well before the steer angle drops below
the Ackerman angle. US/OS/NS within the nonlinear tire force regions will be
discussed later in this thesis.
Figure 1.6: Handling diagram for US/OS/NS using linear tire model.
1.3 State space, system stability, and phase portraits
The models used in this research are presented in state-space format. System states
are the essential parameters required to describe the system dynamics. Furthermore,
all of the systems presented are autonomous, meaning that the governing
10
Figure 1.7: Handling diagram for nonlinear tire forces.
differential equations are completely a function of the current state. Autonomous
system equations can be put into the form x? = F(x;M), where x is the state
vector, M is a vector of system parameters and F is a vector function.
Figure 1.8: Spring-mass-damper system.
In the spring mass system depicted in Figure 1.8, system operation can be
defined by the one-dimensional position and velocity of the mass (y and y?). The
11
state vector x can be defined as in Eq. (1.3). The governing equation for the
unforced system is shown in Eq. (1.4). This represents an autonomous system,
since the governing equation is expressed in state space as in Eq. (1.5), where M
is a vector of the system parameters (m, c, and k), and the vector function F is
defined according to Eq. (1.6).
x =
0
BB@
y
y?
1
CCA
(1.3)
my + cy? + ky = 0 (1.4)
x? =
0
BB@
y?
y
1
CCA
= F(x;M) (1.5)
F(x;M) =
0
BB@
y?
(-ky - cy?)/m
1
CCA
(1.6)
Equilibrium solutions in state space refer to solutions where the states hold
steady through time. In analytic terms, an equilibrium solution, x0, is a state
where the rate of change of the state vector x? 0 = F(x0;M) = 0. The local
stability of an equilibrium solution can be determined by observing the behavior
of the linearized vector function F at the equilibrium solution [17]. If the state
vector x has n dimensions, x can be expressed as x = [x1, x2, x3, , xn] and F
as F = [F1, F2, F3, , Fn], where Fi = x? i 8 i 2 Z, 1 _ i _ n. Linearization
is accomplished by first determining the Jacobian matrix, A, which is defined as
A = DxF(x;M) at x = x0, where DxF(x;M) is defined by Eq. (1.7). If the
eigenvalues of the Jacobian matrix A have all nonzero real parts, the equilibrium
point, x0 is considered to be hyperbolic. For a hyperbolic equilibrium point, if all
the eigenvalues have negative real parts, then all local perturbations away from
the equilibrium solution, x0, decay with time and the solution is stable. If one or
12
more of the eigenvalues of A have positive real parts, then perturbations in some
directions away from x0 will increase with time and the solution is unstable. If
an unstable point has some eigenvalues with negative real parts and the rest with
positive real parts, perturbations away from x0 in certain directions will decay
while perturbations in other directions will increase, and the solution is called a
saddle point [17].
For the simple spring-mass-damper example, the equilibrium solution can be
found by solving F(x;M) = 0. The only equilibrium solution is y = 0, y? = 0.
The Jacobian matrix can be defined according to Eq. (1.8). The eigenvalues at
x0 =
0
BB@
0
0
1
CCA
can be shown to always be negative for positive values of m, k, and
c. Therefore, x0 can be considered locally stable (and globally stable since the
system is linear). Another way to visualize this is to look at a phase portrait of
the system. Figure 1.9 is a phase portrait of the spring-mass-damper system with
m = 1, k = 1, and c = 1. The phase portrait is a graphical representation of
the state space, with the abscissa as the velocity value (y?) and the ordinate as
the position value (y). The trajectories shown in blue represent the evolution of
the states from the initial conditions (represented by the red x s) for velocity and
position. The phase portrait clearly shows that trajectories initiated throughout
the phase plane are attracted to the equilibrium point at (0, 0). This demonstrates
the stability of the solution, since perturbations away from the equilibrium point
will propagate back toward the equilibrium point. Phase portraits can be used
as visualization tools to describe system state progression and assess qualitative
13
stability, and will be used throughout this research.
DxF(x;M) =
0
BB@
@F1
@x1
@F1
@x2
@F1
@xn
@F2
@x1
@F2
@x2
@F2
@xn
..
..
. . .
..
@Fn
@x1
@Fn
@x2
@Fn
@xn
1
CCA
(1.7)
A = DxF(x;M) =
0
BB@
0 1
-k/m -c/m
1
CCA
(1.8)
Figure 1.9: Phase portrait for simple spring-mass-damper system with m = 1,
k = 1, c = 1.
14
1.4 Literature review
1.4.1 Handling classifications
Several vehicle dynamic handling classifications have been developed. Most commonly,
the classifications refer to steady state handling conditions and US/OS/NS.
Detailed discussions can be found in the text books [8] [15]. One such treatment
was presented in Gillespie [8] and accounted for several different influencing factors,
including tire cornering stiffnesses, camber thrust, roll steer, lateral force compliance
steer, aligning torque, and lateral load transfer. The analysis addressed
many different parameters that affect the handling classification, but treatment
of individual terms was somewhat simplified. For instance, tire forces were only
considered in the linear range, and lateral load transfer effects were simplified at
each axle using a second order polynomial function.
Milliken also provided some discussion about US/OS/NS for the linear tire
case, but recognized that the linear model only applied up to about 0.3 g of lateral
acceleration [15]. Nonlinear tire and vehicle characteristics were addressed using
the Milliken Research Associates (MRA) Moment Method. This method characterized
steady-state handling by examining the charts of vehicle yaw moment
versus lateral acceleration. The analysis is somewhat obscure since it deals with
quantities not directly linked to physical parameters.
Gillespie and Milliken also discussed the idea of critical speed for oversteering
vehicles. The critical speed is the speed (for a constant radius test) where the steer
angle has decreased back to zero. In the handling diagrams, this is the speed that
corresponds to the lateral acceleration where the plot crosses the x-axis (Figure
1.6).
15
Karnopp briefly tackled the issue of nonlinear tire forces in the US/OS/NS
classification [10]. A similar method is presented later in this thesis. Karnopp also
mentioned the capability of a vehicle to exhibit different US/OS behavior at or
near limit conditions, depending on the tire saturation rates [10].
1.4.2 Stability notions
Handling classifications allow for stability limit definitions. For instance, US/OS/NS
can be quantified using understeer and oversteer gradients (slope of handling diagram)
[8] [15], and quantifiable limits can be defined. Gillespie uses critical speed
(where the vehicle has turned back to zero steer) for oversteering vehicles as a
stability limit [8].
Milliken provided some additional stability discussions using the linear vehicle
model. The stability of steady-state operation was evaluated for US/OS/NS vehicles
with respect to step sideslip inputs (at 0 steer). Also, steady-state stable
operating conditions were calculated using the linear bicycle model for particular
steer angles and sideslip values. This type of analysis is relatively simple for a
linear system, since steer angle and sideslip can be directly superimposed to define
overall tire forces. In the nonlinear case, Milliken s Moment Method was developed
to determine steady-state operating conditions given a particular steer angle and
sideslip.
A numeric bifurcation analysis is presented in [4] that studies the hunting motions
of rail vehicles. Hopf bifurcations [17] and limit cycle stability are examined
for railcars with four- and two-axle bogies, resulting in a simple bifurcation model
that relates the onset of stable limit cycles (hunting motions) to vehicle speed.
Stability limits are also required to define loss of vehicle control during tran-
16
sient field testing procedures. Forkenbrock, in [6], presented a NHTSA-developed
standard to define a spinout during a sine steer maneuver. Spinout or loss of vehicle
control is defined using yaw rate drop-off following a maneuver. After the steer
maneuver, if the vehicle yaw rate is not reduced to a percentage of the maximum
within a certain time, a loss of control is determined.
1.4.3 State space approaches
Interestingly enough, little work has been done with vehicle stability using a state
space system. For constant vehicle parameters, the state space vehicle models are
time invariant systems. Karnopp used a state space approach to study the stability
of a linear vehicle system (bicycle model with linear tire forces) [10]. Steadystate
dynamic equilibrium solutions were calculated and the stability of dynamic
equilibrium solutions were also assessed directly by linear stability methods.
Ono et al. presented a similar state space model in [18]. This model instead
used nonlinear tire forces. Stability was briefly assessed, and changes in the stable
solutions with respect to steer angle were studied. A front steer controller was
also proposed that intended to keep the nonlinear system stable while maneuvering.
Nevertheless, Ono s work was fairly brief and simplified in terms of stability
analysis, since the focus of the work was on control.
1.5 Contributions and thesis organization
The main contributions of this thesis are the following:
The nonlinear bicycle model for stability analysis presented in [18] was extended
to include tandem-axle vehicle dynamics and independent four-wheel
17
dynamics.
Bifurcations of equilibria were shown to occur with respect to vehicle velocity,
in addition to steer angle.
A Light Tactical Vehicle (LTV) four-wheel model was created, which included
the development of a nonlinear tire model generated from limited
experimental tire data.
The vehicle model was extended to study operations beyond normal operating
limits. This allowed analysis of overall system stability characteristics.
A lateral load transfer model was also presented. This model included roll
dynamics and tire force propagation.
A detailed discussion about the physical insights and practical applications
of the analysis are provided.
A presentation of US/OS/NS for nonlinear tire models was created and is
presented along with analysis results for US/OS/NS vehicles.
The rest of this thesis is organized as follows:
Chapter 2: In this chapter, all vehicle models are presented, beginning with
the original model provided in [18]. Bifurcations, or qualitative changes in the
phase portraits, are shown as front steer angle is varied. Similar bifurcations
involving the loss of stable equilibrium solutions are demonstrated with respect to
velocity as the control parameter. The model is then extended for use with tandemaxle
vehicles. Model parameters were adjusted to provide similar results as with
the two-axle bicycle model. An independent four-wheel (non-bicycle) model is
18
then presented for a two-axle vehicle. Advantages are discussed, and results for
the four-axle case are shown.
Chapter 3: In this chapter, the creation of a vehicle-based LTV model is
outlined. A tire model is first developed using a semi-empirical formulation along
with a genetic optimization algorithm. Four-wheel LTV model results obtained
with the new tire model are presented. The model is then expanded to allow for
accurate results at broader operating ranges. Domains of attraction for stable
points are also studied. Lateral load and roll dynamics are discussed, and a lateral
load LTV model is developed and results are presented.
Chapter 4: In this chapter, the physical insights gained from the analysis
are discussed. The practical meanings of the phase portraits and the equilibrium
solutions are discussed, as well as the domains of attraction for the stable points.
Bifurcation diagrams with respect to steer angle and the velocity are investigated
and tied to physical behavior. The expanded phase portraits are discussed in terms
of practical stability and analytical system stability. US/OS/NS classifications are
presented for nonlinear vehicle models, and US/NS/OS vehicles are studied using
the nonlinear analysis. Lateral load model results are also examined.
Chapter 5: Concluding remarks along with suggestions for future work are
collected and presented in this chapter.
19
Chapter 2
Development of theoretical models
In this chapter, an effort is made to systematically describe the basic concepts and
stability and bifurcation analysis techniques presented in this thesis. First, earlier
work by Ono et al. [18] is reproduced for a bicycle model of a two-axle vehicle to
explain and develop the basic concepts. Then, the simple bicycle model is extended
to tandem rear-axle vehicles. Finally, a four-wheel vehicle model and associated
analysis is presented, neglecting lateral weight transfer and roll dynamics. In
Chapter 3, a case study is presented for a four-wheel general purpose utility vehicle,
using real tire data. This new model is further developed to include lateral weight
transfer and roll dynamics. A detailed discussion of physical insights and the utility
of this stability approach is given in Chapter 4.
2.1 Bicycle model
The bifurcation analysis presented by Ono et al. [18] was based on a simple
bicycle model. In this approach, the actual forces acting on the vehicle are
approximated by modelling each axle as a single unit (single wheel). As such,
individual tire slip angles or individual tire loading during cornering, were not
20
account for. Ono s model was setup in non-SAE standard coordinates, as shown
in Figure 2.1. The variables af and ar represented the distance from the front
and rear axles to the center of gravity (cg) of the vehicle. Positive front steer was
defined in the left direction, and positive vehicle sideslip, _, was defined to the left
of the vehicle center line. The same model in standard SAE coordinates is shown in
Figure 2.2. Due to the orientation of the original system and the symmetry of the
bicycle model approach, both system coordinate representations are equivalent
in terms of the system equations, Eq. (2.3).
Figure 2.1: Bicycle model presented in [18].
mv(
d
dt
_ + r) = Ff + Fr (2.1)
Iz
d
dt
r = (afFf - arFr)cos(_) (2.2)
0
BB@
?_
r?
1
CCA
=
0
BB@
Ff+Fr
mv - r
(afFf-arFr)cos(_)
Iz
1
CCA
(2.3)
Only constant vehicle characteristics were considered in this analysis, meaning
21
Figure 2.2: SAE representation of bicycle model.
there was no acceleration in the direction of the velocity and the front steer angle,
_f , remained constant. Motion was described with two states: _, representing
vehicle sideslip and r, representing vehicle yaw rate.
In this formulation, the overall axle forces (Ff and Fr) are defined in a direction
perpendicular to vehicle velocity at the cg. The bicycle model approach treated
each axle as a single tire with a single slip angle. Axle forces were calculated
using an empirical tire formula, which was a simplified general form of the well
known magic tire formula [19]. The tire equation used by Ono et al. is shown in
Eq. (2.4). The slip angle, _, and the direction of the force were defined according
to Figure 2.3. The parameters B, C, D, and E were all constant parameters based
on empirical tire data, with slip angles as the only variable. The parameters used
for the front and rear tires were different, accounting for the tires themselves as
well as suspension setup and the weight distribution effects. A list the coefficients
22
used are shown in Table 2.1. In Figure 2.4 the tire force versus slip angle diagrams
are shown for the front and rear tires. Note that the nonlinear characteristics of
the tires prior to, during and after saturation were represented.
F = D sin
_
C tan-1
n
B _ - E
_
B x - tan-1(B _)
_o_
(2.4)
Figure 2.3: Tire force diagram.
Table 2.1: Bicycle model tire parameters.
B C D E
Front 11.275 1.56 -2574.7 -1.999
Rear 18.631 1.56 -1749.7 -1.7908
Slip angles at each of the tires were calculated by resolving the i and j components
of the velocity vectors based on the vehicle coordinate system, as shown
in Eqs. (2.5) and (2.6). The vehicle was assumed to be rigid in the yaw direction.
Eqs. (2.7), (2.8), (2.9), and (2.10) give the front and rear slip angles of the states.
23
Figure 2.4: Tire force versus slip angle for front and rear tires in bicycle model.
The terms _f and _r are the sideslips at each of the wheels, and _f and _r are the
associated slip angles.
Vf = V cos(_) i + (V sin(_) + arr) j (2.5)
Vr = V cos(_) i + (V sin(_) - arr) j (2.6)
_f = tan-1
V sin(_) - af r
V cos(_)
!
= _ - tan-1
af rcos(_)
V
!
(2.7)
_r = tan-1
V sin(_) - arr
V cos(_)
!
= _ - tan-1
arrcos(_)
V
!
(2.8)
_f = _f - _f (2.9)
_r = _r (2.10)
At any given state of _ and r, front and rear tire slip angles were calculated
using Eqs. (2.7) through (2.10). The slip angles were then used in the tire force
model, Eq. (2.4), to generate the tire forces, Ff and Fr. These forces were then
24
used in the state equations, Eq. (2.3), to determine the time gradients of both state
variables. Numerical integration was preformed to determine the propagation of
the state variables from any given initial condition.
2.1.1 Bifurcation in steer
Using the vehicle parameters presented in the literature [18], specifically af = 1.2m,
ar = 1.3m, Iz = 3000kgm2, V = 20 m/s, and m = 1500kg, the phase portrait
at 0 radians of front steer angle is shown in Figure 2.5. These phase portraits
describe the propagation of the states for a relatively wide range of initial states.
Each trajectory begins at the blue x s (the initial state) and propagates through
time. The red o s represent solutions for equilibrium points. Equilibrium solutions
are the roots of the state space equations. In other words, they are states where
Eq. (2.3) equals zero (?_ = r? = 0). In this portrait, a stable equilibrium solution
at _ = 0 and r = 0 clearly exists. Stability of this solution can be qualitatively
determined as multiple trajectories propagate toward this point. There also exists
two saddle point equilibrium solutions.
Figure 2.6 shows the phase portrait at 0.015 radians of front steer angle. The
stable equilibrium point has migrated towards a positive yaw rate and a negative
vehicle sideslip. All three equilibrium points are still present. At 0.030 radians
of front steer (Figure 2.7), the stable equilibrium point and one saddle point have
disappeared, leaving only the other saddle point. This represents a bifurcation with
respect to steer angle somewhere between 0.015 and 0.030 radians. A bifurcation
is a qualitative change in the system with respect to a certain variable. In this
case, the qualitative change was a loss of two equilibrium solutions. Physical
interpretations of the bifurcations are discussed in Chapter 4. All trajectories
25
were solved using an ODE solver in Matlab.
Figures 2.8 and 2.9 are the bifurcation diagrams for front steer angle as the
varying parameter. Figure 2.8 shows the values of sideslip, _, for each of the
equilibrium points in the phase portrait as _f is varied. Figure 2.9 shows the corresponding
diagram for yaw rate, r, as _f is varied. Stable equilibrium solutions are
identified in blue, and non-stable points (saddle points in this case) are identified
in red. At a front steer angle of 0 radians, there are three equilibrium points. As
front steer angle is increased the stable point and one saddle point converge and
disappear forming a saddle node bifurcation [17]. As front steer angle is increased
past the bifurcation point, only one saddle point remains.
Figure 2.5: Phase portrait for bicycle model at _f = 0 radians and V = 20 m/s.
26
Figure 2.6: Phase portrait for bicycle model at _f = 0.015 radians and V = 20
m/s.
Figure 2.7: Phase portrait for bicycle model at _f = 0.030 radians and V = 20
m/s.
27
Figure 2.8: Bicycle model bifurcation diagram. Equilibrium values of _ versus
steer angle.
Figure 2.9: Bicycle model bifurcation diagram. Equilibrium values of r versus
steer angle.
28
2.1.2 Bifurcation in velocity
A similar bifurcation analysis was done in this research with velocity as the control
parameter. Setting _f to 0.015, the velocity (V ) was varied. Figure 2.10 shows
the phase portrait at V = 10 m/s. As before, trajectories begin at the blue x s
and propagate through time. Equilibrium points are designated as red o s. At
_f = 0 and V = 10 m/s there exists the three equilibrium points, as before. At the
same steer angle with velocity increased to 20 m/s, the equilibrium points begin to
migrate (Figure 2.11). As velocity is increased further, a bifurcation (similar to the
one seen with increased steer angle) occurs. Figure 2.12 shows the phase portrait
at V = 30 m/s. The stable and one saddle node equilibrium point converged
and disappeared, leaving only the other saddle node equilibrium point. Figures
2.13 and 2.14, are the bifurcation diagrams with velocity as the control parameter.
Again the red points designate unstable equilibrium points and the blue points are
stable.
29
Figure 2.10: Phase portrait for bicycle model at V = 10 m/s and _f = 0.015
radians.
Figure 2.11: Phase portrait for bicycle model at V = 20 m/s and _f = 0.015
radians.
30
Figure 2.12: Phase portrait for bicycle model at V = 20 m/s and _f = 0.015
radians.
Figure 2.13: Bicycle model bifurcation diagram. Equilibrium values of _ versus
velocity.
31
Figure 2.14: Bicycle model bifurcation diagram. Equilibrium values of r versus
velocity.
32
2.2 Tandem-axle model
Figure 2.15: SAE representation of bicycle model for a tandem-axle vehicle.
The previous analysis was extended in this research to different vehicle configurations,
including tandem-axle vehicles. Tandem axles are normally found on
heavier trucks to distribute the rear load over multiple axles. Figure 2.15 shows
the bicycle model of a tandem-axle vehicle using SAE coordinates.
Again, all tire forces were oriented perpendicular to vehicle velocity. The representative
state equations are given in Eq. (2.11). All tire slip angles were resolved
in the same fashion as the two-axle bicycle model. Eq. (2.12) gives the slip angle
for the rearmost axle. Rear tire force, Frr was determined using the same tire force
model as in the two-axle case.
0
BB@
?_
r?
1
CCA
=
0
BB@
Ff+Fr+Frr
mv - r
(afFf-arFr-arrFrr)cos(_)
Iz
1
CCA
(2.11)
_rr = _ - tan-1
arrcos(_)
V
!
(2.12)
33
For the numerical analysis, the tandem-axle vehicle was based on the previous
two-axle model in terms of vehicle parameters. This was done for comparison
and validation of the model results. The distance from the center of gravity to the
rearmost axle, arr, was set to 1.6m. Initially, the rear tire force parameters from the
two-axle model were used for both tandem axles. However, rear force saturation
was not evident, since effective rear force was doubled (the rear stabilizing moment
was more than doubled). Consequently, the original tire data was altered by
halving the scaling term D for both the rear axles. This allowed overall magnitude
for each tire to be scaled down while maintaining curve shape. Table 2.2 shows
the new parameters used for the tires in the tandem-axle model.
Table 2.2: Tandem-axle bicycle model tire parameters.
B C D E
Front 11.275 1.56 -2574.7 -1.999
Rear 18.631 1.56 -874.85 -1.7908
Holding all other vehicle parameters the same, and setting _f = 0.015 rad,
Figures 2.16, 2.17, and 2.18 show the phase portraits of the system at three different
speeds. Even though the total rear forces were about the same as in the two-axle
bicycle model, the bifurcation point changed. This was because the system now
had an additional rear axle which produced a slightly better stabilizing moment.
As compared to the original system, higher velocities were achieved before the
phase portrait showed a qualitative change from three equilibrium points to a
single point.
Bifurcation diagrams were also generated for the tandem-axle case. Figures 2.19
and 2.20 show the bifurcation diagrams with steer angle as the control parameter.
34
Velocity was set to 35 m/s. The bifurcation diagrams are characteristically similar
to the original system.
Figure 2.16: Phase portrait for tandem-axle model at V = 10 m/s and _f = 0.015
radians.
35
Figure 2.17: Phase portrait for tandem-axle model at V = 25 m/s and _f = 0.015
radians.
Figure 2.18: Phase portrait for tandem-axle model at V = 40 m/s and _f = 0.015
radians.
36
Figure 2.19: Bicycle model bifurcation diagram. Equilibrium values of _ versus
front steer angle.
Figure 2.20: Bicycle model bifurcation diagram. Equilibrium values of r versus
front steer angle.
37
2.3 Four-wheel model
Figure 2.21: SAE representation of the 4 wheel model.
The original bicycle model was also extended to a four-wheel model, as a first
step in the development of the vehicle model used in Chapter 3. In the bicycle
model, errors were induced by characterizing the tire forces at each axle based
on average slip angles. In the four-wheel model, individual wheel velocities and
directions were calculated, allowing individual wheel slip angles to be used to
calculate individual tire forces. The tire forces were then applied at the true tire
location, accounting for the full geometry of the vehicle.
The four-wheel case does not share the same symmetry characteristics as the
bicycle model, therefore equivalence did not exist between the SAE standard coordinate
system and the coordinates used by Ono. Consequently, the coordinates
were defined according to the SAE standard (Figure 2.21). Two additional vehicle
38
parameters were introduced, tf and tr, the front and rear track widths. Individual
tire forces, sideslips and slip angles are distinguished with the subscripts fl ,fr,
rl, and rr, referring to the front left, front right, rear left, and rear right corners.
Eqs. (2.13), (2.14), (2.15), and (2.16) show the formulation of the individual
sideslips. Eqs. (2.17), (2.18), (2.19), and (2.20) define the slip angles at each the
wheel. Tire force was again based solely on slip angle and was determined using
Eq. (2.4). Tire forces were taken as perpendicular to the velocity at each of the
individual tires, rather than perpendicular to the velocity at the CG (as was done
in earlier). This change removed a small yaw rate ® assumption used in the
bicycle models. Note that real-world tire force data is measured perpendicular to
the tire velocity.
_fr = tan-1
V sin(_) + af r
V cos(_) - tf
2 r
!
(2.13)
_fl = tan-1
V sin(_) + af r
V cos(_) + tf
2 r
!
(2.14)
_rr = tan-1
V sin(_) - arr
V cos(_) - tr
2 r
!
(2.15)
_rl = tan-1
V sin(_) - arr
V cos(_) + tr
2 r
!
(2.16)
_fr = tan-1
V sin(_) + af r
V cos(_) - tf
2 r
!
- _f (2.17)
_fl = tan-1
V sin(_) + af r
V cos(_) + tf
2 r
!
- _f (2.18)
_rr = tan-1
V sin(_) - arr
V cos(_) - tr
2 r
!
(2.19)
_rl = tan-1
V sin(_) - arr
V cos(_) + tr
2 r
!
(2.20)
Removing the small r assumption and resolving the individual forces required
more complicated system equations, given by Eqs. (2.21) and (2.22). The four-
39
wheel model parameters were again based on the two-axle bicycle model. As with
the tandem-axle model, the four-wheel model was initially constructed for direct
comparison to the original bicycle model by keeping all of the parameters the
same (or at least roughly equivalent). This was done to help validate the fourwheel
model results. The front or rear tire force coefficients were used for both
tires on each axle, with the scaling factor D halved. This created approximately
the same total axle force for front and rear axles while accounting for the different
slip angles for the left and right sides. Table 2.3 shows the tire parameters used
for the four-wheel model.
mv
d
dt
_ + r
!
= Ffrcos(_fr - _) + Fflcos(_fl - _)
+Frrcos(_rr - _) + Frlcos(_rl - _) (2.21)
Iz
d
dt
r = Ffr
0
@
s
a2f
+ (
tf
2
)2sin
_
2
- _fr + tan-1
tf/2
af
!!1
A (2.22)
+Ffl
0
@
s
a2f
+ (
tf
2
)2sin
_
2
- _fl - tan-1
tf/2
af
!!1
A
-Frr
0
@
s
a2r
+ (
tr
2
)2sin
_
2
+ _rr + tan-1
tr/2
ar
!!1
A
-Frl
0
@
s
a2r
+ (
tr
2
)2sin
_
2
+ _rl - tan-1
tr/2
ar
!!1
A
Keeping all vehicle parameters the same and setting tf = tr = 1.2m, _f =
0.0154 rad and varying V , Figures 2.22, 2.23, and 2.24 show the four-wheel model
phase portraits. Figures 2.25 and 2.26 show the associated bifurcation diagrams.
As before, there exists a saddle node bifurcation as velocity is increased.
The results were very similar to the two-axle bicycle model. The four-wheel
model does, however, offer several advantages over the standard bicycle model. Individual
tire slip angles are generated, thereby reducing errors that may be caused
40
Table 2.3: 4 wheel model tire parameters.
B C D E
Front 11.275 1.56 -1287.4 -1.999
Rear 18.631 1.56 -874.85 -1.7908
by averaging the slip angles when using nonlinear tire force functions. Forces are
now taken in directions perpendicular to the velocity at the tire, removing the small
r assumption. Tire forces are also taken at their individual locations accounting
for vehicle track width geometry. Axle force models (as used in the bicycle model)
may also be difficult to obtain in relation to a single slip angle, and are at best
approximations based on individual tire data. The four-wheel model lends itself
to more practical application, since it allows for individual tire force generation.
Also, individual tire data is more commonly available. This four-wheel model was
the basis for the more advanced real-world models presented in the next chapter.
41
Figure 2.22: Phase portrait for 4 wheel model at V = 10 m/s and _f = 0.015
radians.
Figure 2.23: Phase portrait for 4 wheel model at V = 20 m/s and _f = 0.015
radians.
42
Figure 2.24: Phase portrait for 4 wheel model at V = 30 m/s and _f = 0.015
radians.
Figure 2.25: Bicycle model bifurcation diagram. Equilibrium values of _ versus
speed.
43
Figure 2.26: Bicycle model bifurcation diagram. Equilibrium values of r versus
speed.
44
Chapter 3
LTV model
The four-wheeled model introduced in Chapter 2 removed some of the limitations
of the original bicycle model, in that the tire data for the bicycle model needed the
effects of weight transfer and differences in right and left tire slip angles embedded
in the data. In practice, this is not easy to achieve. Because the four-wheel model
uses true tire data (as tested on a tire test rig), the four-wheel model is more easily
and accurately applied to real-world vehicles.
This chapter applies the four-wheel model to a Light-Duty Utility Military
Tactical Vehicle (LTV), and extends the analysis over a broad range of the state
space. The LTV was selected because extensive handling tests of over-weighted
LTVs were recently conducted at the U.S. Army Aberdeen Test Center (ATC),
which provided key vehicle and tire data. It was anticipated that this work might
support and help explain the findings of the testing at ATC. In addition, development
of an accurate and easily generated math model allows vehicle stability to be
evaluated at all conceivable payload conditions, without the need for extensive and
potentially dangerous field tests at the limits of performance. Lastly, a four-wheel
model with lateral weight transfer is presented and briefly discussed.
45
3.1 Tire Model
The first step in creating a vehicle model is developing a realistic tire model. The
tire model used by Ono et al. [18] was a very general form of a semi-empirical
tire model commonly refereed to as The Magic Tire Formula. A more advanced
version of this formulation was used in the current LTV model. The new formulation
allowed tire force to be characterized by both slip angle and vertical load (as
opposed to slip angle alone) from a limited set of tire data obtained under specific
loading conditions. The improved tire model also allowed for the creation of more
complicated vehicle models that included roll motions and dynamic lateral loading
conditions of the tires.
The following sections of the paper present the general formulation of the Magic
Tire formula, a genetic algorithm for coefficient optimization, and a means to overcome
shortfalls in the range of the available test data. More specifically, guidelines
are presented to extend tire data limited below saturation to regions beyond saturation.
3.1.1 The Magic Tire Formula
The initial tire model considered for the LTV model used the full version of the
magic tire formula [19]. This semi-empirical formula is regarded as the foremost
tire force model for vehicle dynamic simulations to date [20], and has been shown to
very accurately represent tire data [13] [3] [19]. The model used for this simulation
was a pure slip model for tire lateral force, whereby tire lateral force was defined
as a function of normal load and slip angle. Camber and combined longitudinal
and lateral slip effects were neglected.
The Magic Tire Formula is re-written in general terms in Eq. (3.1), where X can
46
represent either longitudinal slip ratio or lateral slip angle, and Y represents th