Parameter Estimation Techniques for Determining Safe Vehicle
Speeds in UGVs
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.
Dustin L. Edwards
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
Parameter Estimation Techniques for Determining Safe Vehicle
Speeds in UGVs
Dustin L. Edwards
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
May 10, 2008
Parameter Estimation Techniques for Determining Safe Vehicle
Speeds in UGVs
Dustin L. Edwards
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
Dustin was born February 9, 1982, the third child of James and Kathy Ed-
wards. Born and raised in Pisgah, Alabama, he attended Rosalie Elementary
School and Pisgah High School, after which, he decided to begin his college career
at Northeast Alabama Community College and later Auburn University, resulting
in many wonderful friends, fond memories, and a Bachelor of Science degree in
Mechanical Engineering. As an undergraduate, Dustin graduated Summa Cum
Laude as one of the top students in his class. 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
Parameter Estimation Techniques for Determining Safe Vehicle
Speeds in UGVs
Dustin L. Edwards
Master of Science, May 10, 2008
(B.S.M.E., Auburn University, 2005)
125 Typed Pages
Directed by David M. Bevly
This thesis develops simplified equations to predict a velocity in which vehicle
rollover or tire saturation occur. These equations are functions of different vehicle
parameters that are important to vehicle handling characteristics. Therefore, var-
ious algorithms are developed to estimate parameters such as vehicle tire stiffness,
peak tire force, and center of gravity position on-line. A number of vehicle control
systems have been developed in order to reduce rollover and help maintain vehicle
stability. However, many of these control systems do not take into account chang-
ing vehicle parameters. Therefore using the on-line estimates of these parameters,
the control systems could be more effective in decreasing the number of vehicle
accidents.
The thesis first explains the fundamentals of lateral vehicle dynamics. Basic
vehicle dynamic models are derived and validated to show the effectiveness and
shortcomings of the different models. Many assumptions are used to simplify the
v
models. The assumptions lead to simplified equations that predict a velocity in
which vehicle rollover or tire saturation occur. An equation to predict the vehicle
stopping distance is also derived. Experiments are run to control the vehicle speed
to the predictive velocity. This velocity is updated with the identified parameters
from the estimation algorithms. By providing the updated velocity to the steering
controller, a vehicle is able to transverse a maneuver at a safe speed.
vi
Acknowledgments
Without the love and support from many friends and my family, this thesis
would have been an impossible project. First and foremost, I must give thanks to
God who has always been there for me during good and bad times. He has also
blessed me and given me everything and everyone I have needed to succeed.
I would also like to thank my parents for the all the continuing support that
they have given me during the last 26 years. They have taught me many things
growing up including hard work, determination, and many other values. I would
also like to think my brother and sister for always being there for me when I needed
them.
I would like to thank my graduate advisor, Dr. Bevly. He has continually
challenged me throughout my college career an provided technical and financial
support needed to make this thesis possible. And a special thanks to all those in
the GPS and Vehicle Dynamics Lab who have helped me with the many questions
I have asked during my research.
Finally, I would like to thank my best friend and wife, Jenny, who has given
me all the love and support needed over the last two and a half years. I will be
forever indebted to her for the sacrifices she has made during the completion of
my graduate work.
vii
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.
viii
Table of Contents
List of Figures xii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Background and Literature Review . . . . . . . . . . . . . . . . . . 4
1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Vehicle Modeling 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Lateral Kinematic Model . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Lateral Bicycle Model . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Understeer Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4.1 Neutral Steer . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.2 Understeer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.3 Oversteer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 Tire Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5.1 Fiala Tire Model . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.2 Dugoff Tire Model . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Roll Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.7 Vehicle Model Validation . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8 Conclusions on Vehicle Modeling . . . . . . . . . . . . . . . . . . . 31
3 Predictive Velocity Calculations 32
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Predictive Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 Zero-Sideslip Velocity . . . . . . . . . . . . . . . . . . . . . . 33
3.2.2 Dugoff Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.3 Rollover Velocity . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Stopping / Following Distance . . . . . . . . . . . . . . . . . . . . . 38
3.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.1 Dugoff Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4.2 Rollover Equation . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4.3 Stopping Distance . . . . . . . . . . . . . . . . . . . . . . . 45
ix
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Estimation Algorithm Development 48
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Tire Parameter Estimates . . . . . . . . . . . . . . . . . . . . . . . 49
4.2.1 Tire Stiffness and Peak Tire Force Estimation with Dugoff?s
Tire Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.2 Estimation with Fiala?s Tire Model . . . . . . . . . . . . . . 55
4.3 Weight Split Estimation . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4 CG Height Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 Experiments and Validation of Estimation Algorithms . . . . . . . . 64
4.5.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . 64
4.5.2 Tire Parameter Estimation Experiments . . . . . . . . . . . 64
4.5.3 Weight Split Estimation Experiments . . . . . . . . . . . . . 74
4.5.4 CG Height Estimate Experiments . . . . . . . . . . . . . . . 76
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5 Updated Predictive Velocity Experiments 80
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2 Critical Velocity with Parameter Updates . . . . . . . . . . . . . . . 80
5.2.1 Zero Sideslip Velocity . . . . . . . . . . . . . . . . . . . . . . 81
5.2.2 Dugoff Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2.3 Rollover Velocity . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6 Conclusions 93
6.1 Overall Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.2 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.3 Recommendations for Future Work . . . . . . . . . . . . . . . . . . 94
Bibliography 96
Appendices 100
A Vehicle Nomenclature 101
B Vehicle Properties 103
C Tire Parameter and Weight Split Estimates using Data from
Carsim 105
C.1 Tire Parameter Estimator Validation with Carsim?s G35 Sedan . . . 105
x
C.1.1 Tire Parameter Estimator Testing during Lateral Maneuver 105
C.1.2 Tire Parameter Estimator Testing during Longitudinal Ma-
neuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
C.2 Weight Split Estimator Validation with Carsim?s G35 Sedan . . . . 109
xi
List of Figures
2.1 Vehicle coordinates defined by the SAE [22] . . . . . . . . . . . . . 10
2.2 Kinematic Bicycle Model . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Bicycle Model FBD . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Basic Understeer Gradient Plot . . . . . . . . . . . . . . . . . . . . 16
2.5 Generic Tire Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Circle of Friction for Tire Forces . . . . . . . . . . . . . . . . . . . . 19
2.7 Vehicle Roll FBD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.8 Comparison of the Kinematic and Bicycle Model during Slow-Speed
Turning in the G35 Sedan . . . . . . . . . . . . . . . . . . . . . . . 27
2.9 Comparison of the Kinematic and Bicycle Model during High-Speed
Cornering in the G35 Sedan . . . . . . . . . . . . . . . . . . . . . . 28
2.10 Comparison of the Bicycle Model with Linear and Non-linear Tire
Models during High-Speed Sliding Experiments in the G35 Sedan . 29
2.11 Roll Plane Model Validation of a Double Lane Change Maneuver in
Carsim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1 Roll Equation FBD . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 Safe Following Distance . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Longitudinal Free Body Diagram . . . . . . . . . . . . . . . . . . . 39
3.4 Predicting Slidout at Radiusof 152.5and 400meters with the Dugoff
Velocity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5 Predicting Slidout at Radius of 152.5 and 400 meters with the new
Dugoff Velocity Equation . . . . . . . . . . . . . . . . . . . . . . . . 44
xii
4.1 Tire Force Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Lateral Experimental Data in the G35 Sedan used in the Tire Pa-
rameter Estimator [8] . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Lateral Tire Stiffness and Peak Tire Force Estimate from G35 Data
during a Lateral Slalom [8] . . . . . . . . . . . . . . . . . . . . . . . 70
4.4 Longitudinal Acceleration in the G35 Sedan used in the Tire Esti-
mator [8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.5 Longitudinal Stiffness and Peak Tire force Estimate from G35 Data
during Acceleration and Braking[8] . . . . . . . . . . . . . . . . . . 72
4.6 Accelerations and Yaw Rate Measurements for the Combined Lat-
eral/Longitudinal Estimation [8] . . . . . . . . . . . . . . . . . . . . 73
4.7 Tire Stiffness and Peak Tire Force Estimate during Lateral and
Longitudinal Excitation of the G35 Sedan [8] . . . . . . . . . . . . . 74
4.8 Cornering Data in the G35 Sedan at NCAT Test Track . . . . . . . 75
4.9 Weight Split, Iz Estimate During Cornering in the G35 Sedan . . . 76
4.10 Double Lane Change with Added Noise in Carsim?s Large SUV . . 77
4.11 RLS Estimation of Large SUV?s CG Height, Roll Damping, and Roll
Mass Moment of Inertia during a Double Lane Change in Carsim . 78
5.1 Desired Path for Vehicle Controlled at Zero Sideslip Velocity . . . . 82
5.2 Weight Split Estimate in a Lane Change Maneuver at Zero Sideslip
Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.3 True and Desired Velocity Calculated from the Zero Sideslip Veloc-
ity in a Lane Change Maneuver in Carsim . . . . . . . . . . . . . . 84
5.4 Sideslip with and without Desired Updated Velocity . . . . . . . . . 85
5.5 Coefficient of Friction Update during 400 Meter Radius Curve in
Carsim?s G35 Sedan . . . . . . . . . . . . . . . . . . . . . . . . . . 86
xiii
5.6 True and Desired Velocity Calculated from the Dugoff Velocity in a
400 m Radius turn in Carsim . . . . . . . . . . . . . . . . . . . . . 87
5.7 Tire Slip Angles in a 400 m Radius Turn with a Drop in Road
Friction Coefficient while Controlling Speed at the Dugoff Velocity . 88
5.8 Estimated CG Height during a 400 m Radius Turn using data from
Carsim?s Large SUV . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.9 True and Desired Velocity Calculated from the Rollover Velocity in
a 200 m Radius turn in Carsim . . . . . . . . . . . . . . . . . . . . 91
C.1 Slalom in Carsim?s G35 Sedan . . . . . . . . . . . . . . . . . . . . . 106
C.2 Estimation of Tire Parameters during Slalom Maneuver . . . . . . . 107
C.3 Acceleration / Braking in Carsim?s G35 Sedan . . . . . . . . . . . . 108
C.4 Estimation of Tire Parameters during Acceleration / Braking Ma-
neuver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
C.5 Double Lane Change with Added Noise in Carsim . . . . . . . . . . 110
C.6 Weight Split, Iz Estimate . . . . . . . . . . . . . . . . . . . . . . . . 111
xiv
Chapter 1
Introduction
This thesis develops simplified equations to predict a velocity in which vehicle
rollover or tire saturation occur. These equations are functions of different vehi-
cle parameters that are important to vehicle handling characteristics. Therefore,
various algorithms are developed to estimate parameters such as vehicle tire stiff-
ness, peak tire force, and center of gravity position on-line. A number of vehicle
control systems have been developed in order to reduce rollover and help maintain
vehicle stability. However, many of these control systems do not take into account
changes in the vehicle parameters. Therefore using the on-line estimates of these
parameters, the control systems could be more effective in decreasing the number
of vehicle accidents.
The thesis first explains the fundamentals of lateral vehicle dynamics. Basic
vehicle dynamic models are derived and validated to show the effectiveness and
shortcomings of the different models. Many assumptions are used to simplify
the models. The assumptions lead to simplified equations that predict a velocity
in which vehicle rollover or tire saturation occur. An equation to predict the
vehicle stopping distance is also derived. Experiments are conducted to control
the vehicle speed to the predictive velocity. This velocity is updated with the
identified parameters from the estimation algorithms. By providing the updated
1
velocity to the speed controller, a vehicle is able to transverse a maneuver at a safe
velocity.
1.1 Motivation
With an increase of vehicles on the road, vehicle safety is becoming more im-
portant each day. Considerable work has been conducted in the past to decrease
the number of vehicle collisions. There is also a rising demand in research of au-
tonomous vehicles. With a rising demand of autonomous vehicles and an increase
of emphasis on vehicle safety, many different vehicle controls systems have been
developed and implemented in today?s cars to assist vehicle safety development.
However, many of these control systems do not take into account changing vehi-
cle parameters. With knowledge of important vehicle parameters, these control
systems could be more effective in decreasing the number of vehicle crashes.
Electronic Stability Control (ESC) is one control method to help reduce the
numbers of crashes on the road. It limits the lateral acceleration, yaw rate, or
sideslip angle by individual wheel braking, steering input, etc. With the results of
early ESC system, NHTSA has required ESC to be installed in all new vehicles by
the 2012 model year [21]. The limits on the ESC systems are generally constant,
however the ESC systems could possibly be improved by updating these limits
based off of important vehicle parameters. When vehicle parameters change, so
does the handling of the car. By taking the changing parameters into account
when setting the ESC limits, the system should show improvement.
2
Many of the active safety systems also need a measurement of sideslip. This
state has proven to be very difficult to estimate. Many researchers have devel-
oped methods to obtain this state using different sensors and estimators. While
most sensors are too expensive to incorporate into every vehicle, one method in-
tegrates noisy and biased sensors to estimate this state [15]. Other researchers
have used model based estimators along with inertial sensors to estimate sideslip
[10]. For methods that used model based estimators to provide an estimate of the
states, vehicle parameters must be known for these systems to provide an accurate
estimate. This thesis presents different methods to estimate critical vehicle pa-
rameters that could be used to update vehicle parameters in different model based
state estimators.
In addition to possibly providing estimated parameters for better accuracy
of vehicle control systems and model based estimators, this thesis also presents
a method to determine safe vehicle speed before it enters a tough maneuver. To
accomplish this, critical velocity equations are developed to provide a safe speed
to enter a turn. The value of speed calculated is then sent to the vehicle?s control
system as the maximum speed. Because the vehicle?s handling depends on certain
vehicle parameters, estimates from the parameter estimation algorithms are sent
to update the critical velocity equations. By controlling the vehicle speed below a
calculated maximum speed, the vehicle will likely already be under the ESC limits
when it gets into the maneuver.
3
1.2 Background and Literature Review
Because vehicle safety is so important, many researchers have developed differ-
ent methods to increase the vehicle?s safety performance. However, many of these
methods rely on accurate vehicle parameters to perform satisfactory. Recently, re-
searchers studied the effects different vehicle parameters had on ESC systems [21].
This study proved that different center of gravity positions could effect the perfor-
mance of the ESC systems and may not always prevent rollover or keep the vehicle
sliding off the road. For this reason, many researchers have studied methods to
estimate these critical parameters.
Since many lateral stability control systems need a measurement of sideslip,
there has been much work done in this field. Many systems to estimate this state
use model based estimators. For these to be successful, it is important for the
vehicle parameters to be accurate. One researcher used estimates of tire cornering
stiffness to improve estimation of vehicle states in a model based estimator [1]. Ryu
proposed a method to estimate vehicle parameters that could be used for model
based state estimation during periods of GPS outages [30]. When GPS signals
are available, a simple kinematic filter could be used to update the states and
also estimate important vehicle parameters [30]. When the GPS signal is lost, a
model based approach is used to estimate the vehicle states, in which the estimated
parameters are used in the model. This thesis develops methods to estimate these
needed parameters.
4
Many methods have also been developed to estimate different vehicle proper-
ties. Recent work has estimated different tire properties. While some researcher
have used lateral vehicle models to achieve an estimate of tire cornering stiffness
[30, 32], others have used lateral vehicle models along with non-linear tire mod-
els to estimate cornering stiffness and tire road friction simultaneously [6, 16].
Other methods developed by researchers include measurements of the steering
wheel torque to achieve estimates of these two important parameters [17, 20, 23].
By using a non-linear tire model that takes into account both lateral and longi-
tudinal tire models, a method to estimate both the lateral and longitudinal tire
stiffness, as well as peak tire force has been developed, previously published in
[8]. By using this method, the peak tire force can be estimated during periods
of longitudinal or lateral force generation providing more opportunity to estimate
the tire parameters. By knowing these parameters, it is possible to have a better
understanding of the vehicle?s limits and provides many uses to further increase
safety in today?s automobiles.
Other important parameters that a vehicle?s control system should take into
account include the vehicle?s center of gravity, as this heavily influences rollover
and other handling aspects of the vehicle. Past researchers have developed methods
to estimate this parameter. For example, one researcher developed a method that
uses multiple models and switching to estimate the vehicle?s CG position[33, 34].
One estimator, previously published in [9], uses a non-linear estimator to estimate
5
the lateral CG position [9]. To estimate the height of the center of gravity, a
recursive least squares algorithm can be used.
1.3 Contributions
Since vehicle parameters heavily effect the handling and limits of an automo-
bile, they must be taken into account to maximize safety of the vehicle control
systems. This research attempts to develop different methods to obtain important
vehicle parameters on-line and in real-time. By studying different vehicle models,
it is apparent these parameters can be estimated with current sensors. This re-
search develops a method to obtain estimates of peak tire force and lateral and
longitudinal tire stiffness in either a longitudinal or lateral maneuver. This will
greatly increase the chances of obtaining an estimate of the peak tire force, which
highly effects when the vehicle may slide off the road. This thesis also develops a
method to estimate the vehicle?s CG position.
To help increase the safety of regular and autonomous vehicles, a method is
introduced to limit the vehicle?s velocity during certain driving maneuvers. This
method uses predictive velocities, calculated from simplified vehicle models, to
warn the driver or even update the controlled velocity in autonomous vehicles.
These predictive velocities are based off parameters that mostly influence the han-
dling characteristics of the vehicle. When a parameter changes, the parameter
estimation algorithms will recognize this and update the predictive velocity with
a new value. This will help the vehicle?s controllers by limiting the speed of the
6
vehicle before the turn instead of trying to limit it when the vehicle has already
reached the ESC limits. In summary, this thesis will attempt to assist current
vehicle safety technology by providing the control system with estimated vehi-
cle parameters on-line and also providing a max speed in which vehicle failure is
eminent.
1.4 Contributions
This thesis develops methods to estimate critical vehicle parameters using
model based estimators. The estimated parameters are then used to determine a
safe traveling speed for the vehicle. In development of the algorithms, the following
contributions were performed:
? An estimator was developed to estimate peak tire force and tire stiffness
during lateral, longitudinal, and combined tire force generation increasing
the chances of getting an estimate of these parameters.
? A method was developed to estimate the vehicles center of gravity position
using an Extended Kalman Filter (EKF)
? A simplified velocity equation was derived to predict tire saturation.
? Predictive velocity equations and parameter estimation algorithms were used
in conjunction to send a desired velocity to the speed controller, creating a
safe traveling speed for the vehicle.
7
1.5 Thesis Organization
The remaining chapters in this thesis are organized as follows:
Chapter 2 Vehicle Modeling. Chapter 2 will lay out the basic vehicle models
used to estimate important vehicle parameters in this thesis. Validation plots are
presented to show the effectiveness and shortcomings of the different models.
Chapter 3 Critical Velocity Calculations. In Chapter 3, certain assumptions
are made to the vehicle models to simplify them. From the simplifications, critical
velocity equations are derived to minimize the sideslip, predict rollover, and to
prevent the tire from sliding.
Chapter 4 Estimation Algorithm Development. Chapter 4 develops methods
to estimate parameters that effect vehicle rollover and sliding. The algorithms are
then validated in simulation.
Chapter 5 Experiments and Validation. Chapter 5 is devoted to testing and
validation of the algorithms developed in Chapters 3 and 4. The critical velocities
are updated with the parameter estimates and sent to the velocity controller. By
using the critical velocities in different scenarios, the vehicle control system is
supplied with a safe speed to enter a turn or lane change.
Chapter 6 Conclusions. Chapter 6 will discuss the contributions and findings
of this work. Some suggestions on future work will also be discussed.
8
Chapter 2
Vehicle Modeling
2.1 Introduction
In order to develop parameter estimation schemes for controller updates, ac-
curate vehicle models must be developed. Vehicle models are the primary source
of understanding vehicle dynamics. In this chapter, vehicle models are developed
and studied in depth to fully understand how the vehicle reacts to different vehi-
cle inputs. The coordinate system used in this thesis is defined by the Society of
Automotive Engineers (SAE) and is shown in Figure 2.1. This coordinate system
defines the longitudinal axis of the vehicle as ?x?, the lateral axis to be ?y?, and the
vertical axis, ?z?, points toward the ground. The coordinate system also defines the
direction of roll rate ?p?, pitch rate ?q?, and yaw rate ?r?. All models in this chapter
will be based off the basic ?bicycle model.? Other models are developed to capture
dynamics the bicycle model is unable to describe. Also, simplifications are made to
the bicycle model that is valid during certain steady-state maneuvers. The models
developed in this chapter is important in later sections when developing parame-
ter estimation algorithms. Therefore, it is important to know the limitations and
accuracy of each model. The nomenclature used in this chapter can be viewed in
Appendix A.
9
Figure 2.1: Vehicle coordinates defined by the SAE [22]
2.2 Lateral Kinematic Model
Under certain assumptions, lateral motion of a vehicle can be described by
certain geometric relationships. These relationships allow for a mathematical de-
scription of the vehicle motion during certain maneuvers. One downside to this
model is the relationships are very simplistic and may produce large errors from
that of the true states during dynamic maneuvers. However, the relationships
described below are very helpful in understanding the lateral motions of a vehicle.
To derive the kinematic relationships of the vehicle, consider the free body
diagram (FBD) shown in Figure 2.2. One assumption of the bicycle model is the
inner and outer tires are represented by one tire at the center of the vehicle?s axle.
This is true for both the front and rear axle. The steering angle of the vehicle
is represented by ?. The slip angle of the vehicle is denoted as ? and describes
the angle between the velocity vector and the longitudinal axis of the vehicle. For
10
this section, the slip angle and steer angle are assumed to be small. By neglecting
slip angles at the tires, the velocity vector at each tire is assumed to be in the
direction of the respective tire. Note that these assumptions are reasonable for
vehicles traveling at slower speeds.
Figure 2.2: Kinematic Bicycle Model
At low speeds with no lateral tire slip, it can be shown that the perpendicular
line from each tire passes through the same point which is called the center of the
turn. If the steering angle of the front tire reaches zero, the radius of curvature (R)
goes to infinity. If the front wheels are not zero the steering angle can be described
by Equation (2.1), known as the Ackerman Angle.
? = tan?1
parenleftbiggL
R
parenrightbigg
?
parenleftbiggL
R
parenrightbigg
(2.1)
11
By using simple kinematics, the vehicle?s velocity (V) can be described as the yaw
rate (r) of the vehicle times the radius of curvature (R). Also with no lateral sliding,
the lateral acceleration, ay, of the vehicle is simply the centripetal acceleration
developed during the turn.
V = Rr (2.2)
ay = ?Vy = Rr2 = V
2
R = Vr (2.3)
Substituting Equation (2.2) into (2.1) results in the expected yaw rate of the
kinematic model, given a steer input and velocity.
r =
parenleftbiggV
L
parenrightbigg
tan(?) ?
parenleftbiggV
L
parenrightbigg
? (2.4)
2.3 Lateral Bicycle Model
The Bicycle Model is used widely in vehicle dynamics to mathematically de-
scribe the motion of a vehicle [12, 29, 22] . It is a simple yet accurate way to
estimate lateral vehicle states for vehicles that develop small roll angles. This
model neglects pitch, weight transfer, and, in this section, longitudinal dynamics.
12
Another important, although accurate, assumption is the inner and outer slip an-
gles and steer angles are lumped into one tire at the center of the axle similar to
the previous model.
To show a visual picture of the bicycle model, a free body diagram is shown in
Figure 2.3. The front and rear tire slip angles are denoted as?f and?r respectively.
By summing the forces and moments on the free body diagram, a simple set of
dynamic equations can be derived to describe the vehicle?s lateral motion.
?Fy = m?y = FyF +FyR (2.5)
?M = Iz? = aFyF +bFyR (2.6)
Figure 2.3: Bicycle Model FBD
To describe the front and rear tire forces (FyF,R), a linear tire model will be
used. This model assumes the tire forces remain in the linear region of the tire
and are proportional to the tire?s respective slip angle times the tire?s cornering
stiffness (C?), shown in Equation (2.7) and (2.8).
13
FyF = ?C?f?f (2.7)
FyR = ?C?r?r (2.8)
Because the tire sometimes leaves the linear region, more information on non-linear
tire modeling will be presented in a later section.
Substituting Equations (2.7) and (2.8) into (2.5) and (2.6), a state space
representation of this model can be developed shown by Equation (2.9).
?
?? ay
?r
?
??=
?
?? ? C0mVx ? C1mVx
? C1IzVx ? C2IzVx
?
??
?
?? Vy
r
?
??+
?
?? C?fm
C?f
Iz
?
??? (2.9)
where,
C0 = C?f +C?r
C1 = aC?f ?bC?r
C2 = a2C?f +b2C?r
(2.10)
With a state space representation, this model can be configured for control or
estimation purposes.
In order to calculate the steady state tire slip, Equations (2.5) and (2.6) are
simplified by assuming yaw acceleration is equal to zero and lateral acceleration is
equal to the centripetal acceleration shown by Equation (2.3). This results in the
following simplified equations for the lateral tire forces.
14
mV
2
R = FyF +FyR (2.11)
0 = aFyF +bFyR (2.12)
Substituting Equations (2.7-2.8) into the above equations, the steady-state tire slip
can be solved. This results in Equations (2.13-2.14).
?f = WfV
2
C?fgR (2.13)
?r = WrV
2
C?rgR (2.14)
2.4 Understeer Gradient
In order to develop a better understand of the turning response of a vehicle,
the understeer gradient of the vehicle is defined. Using the steady-state bicycle
model, the understeer gradient can be determined from the weight distribution
and the cornering stiffness [30]. By including slip angles into Figure 2.2, a simple
kinematic equation between the steer angle and tire slip angles can be developed.
? ? LR +?f ??r (2.15)
15
Substituting Equations (2.13 - 2.14) into the above equation gives:
? ? LR + (WfC
?f
? WrC
?r
)V
2
gR ?
L
R +Kusay (2.16)
From the above equation, the understeer gradient is labeled as Kus. The under-
steer gradient determines both the magnitude and the direction of the steering
inputs required for a given lateral acceleration [12]. The understeer gradient also
determines if the vehicle is neutral steer, oversteer, or understeer. Figure 2.4 shows
the basic principles with Kus being the slope of each line.
Figure 2.4: Basic Understeer Gradient Plot
2.4.1 Neutral Steer
Neutral steer occurs when the understeer gradient is zero, which results in
the front and rear steady state tire slip angles being equivalent. By studying
16
Equation (2.16), during neutral steer the steer angle required to make the turn is
approximately the Ackerman angle.
2.4.2 Understeer
Understeer occurs when Kus is greater than zero causing larger slip angles to
develop in the front tire than the rear. Because there is more slip at the front
tire, the steer angle must increase to maintain the radius of the curve. During this
condition, the steer angle increases linearly with the speed squared or the lateral
acceleration.
2.4.3 Oversteer
Oversteer is the opposite of understeer. During oversteer, Kus is less than
zero causing the rear tire slip angle to be greater than the front. Because the rear
is sliding more than the front, less steer angle is required to navigate the turn.
2.5 Tire Models
With the exception of aerodynamic forces, all external forces on the vehicle
are developed at the tire?s contact patch. Therefore it is necessary to have full
understanding of the relationship between the tire?s contact patch and the surface
the vehicle is on. The tire serves three basic functions:
1) It supports the vertical load, while cushioning against road shocks.
2) It develops longitudinal forces for acceleration and braking.
17
3) It develops lateral forces for cornering.
Figure 2.5 shows typical characteristics of a tire under lateral force generation
modeled by the Fiala tire model. More information on this model will be discussed
later. As shown in the plot, the lateral tire force remains linear with slip angle,
as modeled by Equations (2.7) and (2.8), until the tire becomes saturated. This
model relates peak tire force to the tire-road friction (?) times the normal force
(Fz), known as the peak tire force. Therefore, the peak tire force increases with
a rise in normal force. The longitudinal tire curve looks similar to the lateral tire
model but instead is linear with the longitudinal tire slip. For this reason, it is
very important to have an accurate estimate of ? in order to reasonably predict
the onset of sliding.
?60 ?40 ?20 0 20 40 60?8000
?6000
?4000
?2000
0
2000
4000
6000
8000
Tire Slip Angle (deg)
Tire Lateral Force (N)
Fz=3KN
Fz=4KN
Fz=5KN
Fz=6KN
Fz=7KN
Figure 2.5: Generic Tire Curve
18
Vertical forces on the tire are not only important for ride characteristics, but
also help to describe the max longitudinal and lateral forces developed by the
tire. As shown in Figure 2.6, the magnitude of lateral and longitudinal tire force
cannot exceed the peak tire force. When the magnitude of tire force reaches this
point, sliding occurs. By studying the figure, it is obvious that the available drive
force decreases with an increase in lateral force. Because of this effect, both forces
must be taken into account during combined lateral and longitudinal tire force
generation to develop an accurate vehicle model.
Figure 2.6: Circle of Friction for Tire Forces
Several researchers have developed models todescribe the generated tire forces.
One of the most well known models, called the Magic Formula tire model, was de-
veloped by Pacejka [27, 26, 25]. This model is an empirical formula capable of
calculating lateral and longitudinal tire forces. Alternatively, the two models used
in this paper are the Fiala and Dugoff tire model. Both models have their pros
and cons and will be discussed in more detail below.
19
2.5.1 Fiala Tire Model
The Fiala tire model was originally developed to estimate lateral tire force gen-
eration only [11]. The model was however transformed to take into account both
lateral and longitudinal forces [24]. One assumption to accomplish this transfor-
mation was lateral and longitudinal tire stiffness (C?,C?) are equal. This is not
always true however. The total slip (?) for this model is simply the magnitude of
the lateral and longitudinal slip (?y,?x), shown by Equation (2.17).
? =
radicalBig
?2y +?2x (2.17)
To calculate the total slip, the individual values for slip must be known. Both the
longitudinal and lateral slip are found using Equations (2.18) and (2.19) below.
?x = reff?w?Vxr
eff?w
during acceleration
?x = reff?w?VxVx during braking
(2.18)
?y = Vxr
eff?w
tan(?) (2.19)
Both of these values may be calculated using different sensors described in
Chapter 4. By assuming a parabolic pressure distribution on the tire?s contact
patch, Equation (2.20) is used to describe the magnitude of force on the tire, using
the Fiala tire model.
20
Ft =
??
?
??
?Fz[3??? 13(3??)2 + 127(3??)3] if ? ??m
F = ?Fz if ? ??m
(2.20)
The variable, ?m, is the value of total slip where sliding occurs in the Fiala
tire model. As described by the circle of friction, sliding is assumed to begin when
the maximum tire force is equal to ?Fz.
?m = 1? = 3?FzC
?/?
(2.21)
The individual values of lateral and longitudinal tire force (Fy,Fx) can be
obtained by breaking up the force magnitude (Ft). This is done by multiplying the
force magnitude by the ratio of total slip to each forces respective slip, as shown
in Equation (2.22) and (2.23).
Fx = ?x? F (2.22)
Fy = ?y? F (2.23)
In the case of pure lateral slip, set ?y = tan(?) and ?x = 0 in the Fiala tire
model. In case of pure longitudinal slip, set ?y = 0 [29]. By reducing the combined
force generation model to either lateral or longitudinal force generation, simpler
calculations can be obtained by reducing the amount of noisy measurements.
21
2.5.2 Dugoff Tire Model
The Dugoff model is similar to the Fiala model in that is allows for tire force
estimates during combined tire force generation. The main difference is the Dugoff
tire model assumes a uniform vertical pressure distribution on the tire?s contact
patch [7]. This is a simplification from the Fiala?s tire model, but it allows for
individual values of lateral and longitudinal tire stiffness which is shown to be
advantageous in Chapter 4. The longitudinal and lateral tire forces are given by
Equations (2.24) and (2.25), respectively.
Fx = C? ?x1+?
x
f(?) (2.24)
Fy = C?tan(?)1 +?
x
f(?) (2.25)
where,
? = ?Fz(1 +?x)2[(C
??x)2 + (C? tan(?))2]
1
2
(2.26)
f(?) =
??
?
??
(2??)? if ?< 1
1 if ?? 1
(2.27)
Similar to the Fiala tire model, this model has a transition that occurs when
? = 1. This transition occurs when the tire leaves the linear region and begins the
non-linear region. If the tire is experiencing lateral slip only, the model may be
22
reduced by setting ?x = 0 or for pure longitudinal force generation simply set ? =
0. This helps to simplify the model during driving conditions where only lateral
or longitudinal forces are generated.
2.6 Roll Model
In this section, different vehicle roll models will be described and studied.
It is very important to understand vehicle roll and rollover. Many researchers
have developed models to describe the roll dynamics of vehicles during cornering.
Some models are fairly simple while others are very in depth and require more
parameters. The simpler roll models do not include the springs and dampers of
the suspension and therefore assumes the sprung mass is stationary with the axle.
Other high-fidelity models take into account forces produced by the springs and
dampers.
In order to produce a reliable roll model, a free body diagram (FBD) must
be developed. The FBD in Figure 2.7 shows a two state roll plane model [33, 34].
Three important parameters used in this model include the CG height (hcg), roll
stiffness (K?), and roll damping coefficient (C?). This model lumps the entire
vehicle mass into the sprung mass. This assumption allows a simplified equation
for the spring and damper torques, shown in Equations (2.28-2.29).
23
Tspring = K?? (2.28)
Tdamper = C? ?? (2.29)
Notice that both equations also assume the spring and damper torques are linear
with roll (?) and roll rate ( ??), respectively.
Figure 2.7: Vehicle Roll FBD
By summing the moments about the roll center on Figure 2.7, a simple equa-
tion is derived to describe the roll dynamics of the vehicle. Equation (2.30) assumes
24
the vehicle?s sprung mass rotates about a fixed point at the centerline of the lateral
axis on the ground.
Jeff ??+C? ??+K?? = mhcg(aycos(?)+gsin(?)) (2.30)
By assuming a steady-state turn and small angles, Equation (2.30) can be
simplified to solve for the roll angle with knowledge of the CG height and the
spring roll stiffness.
? = mhcgayK
? ?mhg
= mhcgV
2
R(K? ?mhcgg) (2.31)
Equation (2.30) may be transformed into a state space representation. The state
space representation is shown in Equation (2.32).
?
?? ??
??
?
??=
?
?? 0 1
?K??mghcgJ
eff
? C?J
eff
?
??
?
?? ?
??
?
??+
?
?? 0
mhcg
Jeff
?
??a
y (2.32)
Many other models have also been used to analyze roll dynamics. Some models
developed do not assume the vehicle?s roll center is located at ground height. One
model assumes the roll center is not at ground level and the imaginary roll center
also produces reactionary forces was developed in [36, 37].
25
2.7 Vehicle Model Validation
To show the accuracy and limitations of the models, each model is validated
with experimental data. The data is from a G35 sedan at the National Center for
Asphalt Technology (NCAT) test track. More details from the sensor implementa-
tion is discussed in Chapter 4. Carsim, a high-fidelity vehicle simulation software,
is also used throughout this thesis.
With the data gathered at NCAT test track, the kinematic and bicycle model
are validated in MATLAB. The parameter values used in the simulations of the
G35 sedan is listed in Appendix B. By using MATLAB to simulate the dynamic
equations presented in this chapter, Figure 2.8 shows that both the kinematic and
bicycle model matches the recorded data at 2m/s, as would be expected. However
for larger slip angles, the assumptions of the kinematic model break down causing
the model to perform poorly. In the data logged at NCAT, slip angles remained
small enough for both the bicycle and kinematic model to hold true. When vehicles
reach higher speeds, as shown in the next experiment, the simplistic kinematic
model is not the best choice.
26
0 50 100 150 200 250?40
?20
0
20
40
Steering
Angle (deg)
0 50 100 150 200 250?10
?5
0
5
10
Datapoints
Yaw Rate (deg/sec)
Measurement
Kinematic Model
Bicycle Model
Figure 2.8: Comparison of the Kinematic and Bicycle Model during Slow-Speed
Turning in the G35 Sedan
To show the shortcomings of the kinematic model, data was logged in a G35
sedan at higher speeds around NCAT test track. The inputs were run through both
models in MATLAB and the results are shown in Figure 2.9. Notice the difference
in the kinematic and bicycle models prediction of yaw rate. While cornering at
high speeds with large slip angles, the kinematic model can not accurately predict
the vehicle?s dynamics.
27
0 10 20 30 40 50 60 70 80 90 100?4
?3
?2
?1
0
1
Steering
Angle (deg)
0 10 20 30 40 50 60 70 80 90 100?50
?40
?30
?20
?10
0
10
Time
Yaw Rate (deg/sec)
Measurement
Bicycle Model
Kinematic Model
Figure 2.9: Comparison of the Kinematic and Bicycle Model during High-Speed
Cornering in the G35 Sedan
To illustrate the shortcomings of the bicycle model with a linear tire model,
a maneuver is conducted which saturates the tires enough for the vehicle to begin
sliding. This is a very hard maneuver and is conducted to show a linear tire model
without saturation can not describe the vehicle motion at the limits of handling.
The more advanced model uses the Dugoff tire model to calculate the lateral tire
forces. Figure 2.10 shows the bicycle model with a linear tire failing to match the
data when the vehicle looses control. However, the bicycle model with a non-linear
tire model matches the data fairly well, although there is still some mismatch at
the highest peak. This is most likely due to the fact that the model does not take
into account vehicle roll dynamics.
28
0 5 10 15 20 25 30 35 40 45 50
?30
?20
?10
0
10
20
Steering
Angle (deg)
0 5 10 15 20 25 30 35 40 45 50
?60
?40
?20
0
20
40
Time
Yaw Rate (deg/sec) Measurement
Bicycle Model
Non?linear Tire Model
Figure 2.10: Comparison of the Bicycle Model with Linear and Non-linear Tire
Models during High-Speed Sliding Experiments in the G35 Sedan
Finally, the two state roll plane model is tested. The vehicle used in this
simulation is a large SUV from Carsim. Carsim is a high fidelity vehicle simulation
tool that can be used to validate simplified vehicle models. Carsim is choose in
this experiment because a vehicle is needed that produces large roll angles, unlike
the G35 sedan used in previous experiments. Appendix B provides the parameter
values used in the roll plane model during this simulation. The large SUV attempts
a double lane change in order to induce large roll rates and angles. The data from
Carsim is used to compare with the simple two state roll plane model, which is
simulated in MATLAB. The data from Carsim matches up well with the simple
roll plane model, given in Equation (2.30), as shown in Figure 2.11.
29
0 5 10 15 20?20
?10
0
10
20
Roll Rate(deg/s)
0 5 10 15 20?10
?5
0
5
10
Roll(deg)
Time
Roll Plane Model
Carsim
Figure 2.11: Roll Plane Model Validation of a Double Lane Change Maneuver in
Carsim
One difference in this model and the Carsim high-fidelity model is the springs
used in the Carsim model are highly non-linear. Shown by Equation (2.28), the
simple roll plane model assumes linear springs. Now that the roll plane has been
validated it can be used to estimate certain unknown parameters that may change
such as the roll height. This parameter is important to know as it is one of the
main factors in vehicle rollover [21].
30
2.8 Conclusions on Vehicle Modeling
In this chapter, vehicle models were developed and implemented into Matlab
in order to develop a better understanding of vehicle dynamics. A simplistic kine-
matic model was developed and shown to match the vehicle dynamics at lower
speeds in the Infiniti G35 sedan. For cornering at faster speeds, the kinematic
lateral vehicle model failed to accurately describe this vehicle?s dynamics. How-
ever, the bicycle model developed in Section 2.3 accurately described the vehicle?s
lateral motion. The pros and cons of using a linear tire model versus a non-linear
tire model in the bicycle model was also discussed in this chapter. For model-
ing the roll dynamics of a vehicle, a simple roll plane model was derived. This
model was validated with simulated data in Carsim from a large SUV. Carsim was
used to provide a vehicle with a higher center of gravity position than the G35
sedan. Overall, this chapter showed the effectiveness of describing lateral and roll
dynamics of a vehicle, while also describing the shortcomings of each models.
31
Chapter 3
Predictive Velocity Calculations
3.1 Introduction
In this chapter, equations are derived to find a steady-state speed that a
vehicle can safely transverse a curve. The velocities calculated take into account
radius of curvature, rollover, small sideslip angles, and tire-road friction. Several
assumptions are used to simplify equations of motion of the vehicle during different
maneuvers. It is also assumed that the radius of curvature of the road is known
from the path planner or a map database. These velocities are used in Chapter
5 to update the controlled velocity during different maneuvers. Also, a following
distance is calculated for vehicles driving in platoons by taking into account the
maximum braking forces available at the vehicle?s tires. These equations could be
beneficial to autonomous vehicles when traveling alone or in platoons by providing
information about what speeds to travel or how far away to follow another vehicle.
3.2 Predictive Velocity
In order to develop the equations discussed above, several assumptions are
made. Many of the equations used in this section are based off a simplified version
of the bicycle model. The simplified version of the the bicycle model assumes the
vehicle is in steady-state. The steady state sideslip is then calculated. Both the
32
zero-sideslip and Dugoff velocity rely on the steady-state bicycle model. Similarly,
the rollover velocity is derived from a simplification of the roll model. To view the
nomenclature used in this chapter, view Appendix A.
3.2.1 Zero-Sideslip Velocity
With the lateral tire slip known at the tires, certain criteria is set to calculate
a look-ahead velocity. One method used by Gillespie [12] is to find the velocity
with zero sideslip at the vehicles center of gravity (CG). As discussed in Chapter 2,
sideslip is simply the angle between the velocity vector and the vehicles longitudinal
axis and can be calculated by Equation (3.1).
? = ?r + bR (3.1)
To represent Equation (3.1) as a function of velocity, the above equation is
combined with Equation (2.14). Then by setting the sideslip equal to zero, the
zero sideslip velocity can be calculated with Equation (3.2).
V?=0 =
radicalbigg
bgC?rW
r
(3.2)
The above equation provides for a safe speed in an autonomous vehicle during
cornering or lane change maneuvers. However, this equation is not a function of
33
the turning radius and may provide speeds much lower than desired around larger
radius turns. It also doesnt take into account the friction limits of the driving
surface. Instead this velocity is a only function of the weight split and cornering
stiffness.
3.2.2 Dugoff Velocity
In order to calculate a velocity based off tire-road friction and radius of cur-
vature, the Dugoff tire model is used. This model was discussed more in depth
in Chapter 2. The purpose of using this model is it has a transition when the
tire leaves the tires circle of friction. The basic equations used in this subsection
were described previously by Equations (2.24-2.26). However, for this section the
Dugoff tire model is simplified to lateral force generation only. By making this
assumption, the Dugoff tire model reduces down to the form shown in Equations
(3.3-3.5).
Fy = C? tan(?)f(?) (3.3)
where,
? = ?Fz2C
? tan(?)
(3.4)
f(?) =
?
??
??
(2??)? if ?< 1
1 if ?? 1
(3.5)
34
Gunter and Sankar developed a friction circle interpretation of the Dugoff
model [13]. It proves that if ?> 1, the tire?s operation point is inside the friction
circle and if ? ? 1, the tires operation point is outside the friction circle. Since
longitudinal dynamics are ignored, if the lateral tire force is less than the friction
coefficient times the load on the tire, the tire remains in the circle of friction. This
is used as the criteria to calculate a critical velocity from the Dugoff tire model.
Assuming small angles, the non-equality equation, in Equation (3.6), is derived to
keep the lateral tire force inside the friction circle assuming no longitudinal force
generation and by setting ? = 1 in Equation (3.4).
?Fz
2C?? > 1 (3.6)
Substituting Equation (2.13) into Equation (3.6), the velocity that ensures the tire
remains inside the friction circle during steady-state turns can be calculated as
shown below.
VDug <
radicalbigg
?Rg
2 (3.7)
The above velocity can be easily determined if the radius of curvature and the
friction coefficient is known. During turning around a large radius of curvature,
this velocity would allow high speed turning without loosing control or sliding
35
out. If the tire-road friction coefficient changes during the turn, the vehicle could
however begin to slide out. For this reason, a method to estimate the road?s friction
coefficient will be developed in Chapter 4.
3.2.3 Rollover Velocity
To provide safe speeds for any vehicle, rollover must be taken into account.
This is especially true for vehicles with a a high center of gravity, as these vehicles
are more prone to rollover. Instead of sliding out like the above section describes, a
larger vehicle tends to rollover. Many previous researchers have worked on rollover
prediction formulas [28, 19, 14].
To obtain knowledge of when rollover will occur, a simple rollover prediction
formula is developed previously discussed here [21]. Rollover can be defined as the
point when all the vehicle?s normal force is shifted to one side [21]. By neglecting
the suspension effects, the vehicle can be modeled as a solid mass and transients
from the suspension are ignored. A simple FBD in Figure 3.1 is used to describe
this assumption.
36
Figure 3.1: Roll Equation FBD
By summing the vertical and lateral forces as well as the moment on Figure
3.1, the following equations are developed.
summationdisplay
Fy = m?ay = Fy (3.8)
summationdisplay
Fz = maz = mg?Fz = 0 (3.9)
summationdisplay
MCG = T2Fz ?hcgFy = 0 (3.10)
By simplifying the above equations, the rollover prediction formula is derived. This
equation is widely used and is shown in Equation (3.11) [14, 12, 21].
VRollover = Tg2rh
cg
=
radicalBigg
TRg
2hcg (3.11)
37
Lambert [21] recently developed a method to incorporate weight split and sus-
pension effect into the rollover prediction formula. By studying trends in rollover
data, a scale factor based on the understeer gradient of the vehicle was added to
the formula to take into account weight split. An additional constant (?) for the
suspension effects was also added as shown in the equation below.
VRollover = ? Tg2rh
cg
(1 +Kus) = ?
radicalBigg
TRg
2hcg(1 +Kus) (3.12)
3.3 Stopping / Following Distance
In order for multiple vehicles to travel together in a fleet, a safe following
distance must be known in order to prevent vehicle accidents. Figure 3.2 shows an
example of a vehicle driving too close and a vehicle at a safe following distance.
Note that the definition of ?too close? is based on braking capability of the vehicle
and the road surface.
Figure 3.2: Safe Following Distance
38
For an autonomous vehicle in a fleet, a safe following distance provides many
advantages over a vehicle without any knowledge of safe distance from the vehicle
in front. Many sensors, such as a lidar, can provide a measurement of distance
from one vehicle to the next for control purposes. If a safe following distance is
known, rear-end collisions could be prevented.
In order to calculate a proper following distance, a simple longitudinal vehicle
model is used. Figure 3.3 shows a very simplistic FBD of a longitudinal vehicle
during braking. By summing the forces in the longitudinal direction, a simple
equation is developed to describe the longitudinal motion of the vehicle.
?Fx = max = ?External Forces = ?Fxt (3.13)
Where Fxt is the braking force.
Figure 3.3: Longitudinal Free Body Diagram
39
In actuality, many forces effect the longitudinal dynamics, such as road slope,
aerodynamic forces, braking and acceleration forces at the tire, and rolling resis-
tance. However, to calculate a stopping distance, in this thesis some forces are
ignored. When the equation is simplified to include only the braking force (Fxt),
the linear deacceleration (Dx) can be described by Equation (3.14).
Dx = ?dVdt = Fxtm (3.14)
Assuming that the braking force is constant during the deacceleration, Equation
(3.14) can be integrated to find the time (ts) for a certain change in velocity [12].
ts = (Vo ?VF) mF
xt
(3.15)
In order to solve for stopping distance, the relation dt = dxV can be substituted
into Equation (3.14) for dt. After integrating the equation and assuming the final
velocity is zero, the stopping distance can be determined by Equation (3.16) [12].
SD = mV
2
o
2Fxt (3.16)
40
The minimum stopping distance occurs when the braking force equals the
peak longitudinal tire force, discussed in Chapter 2. Figure 2.6 shows that the
max available braking force is equal to the friction coefficient times the normal
force when neglecting lateral dynamics. By substituting ?Fz in for Fxt, a final
equation for stopping distance is created.
SD = mV
2
o
2?Fz =
V2o
2?g (3.17)
If the applied braking force to the tires is known, Equation (3.16) can be
used to solve for the braking distance. However, if the braking force exceeds the
peak tire force then Equation (3.17) must be used. This is because the tire-road
interaction force can saturate and the tire slides at the peak tire force.
If the vehicle is stopping on a sloped road, the equation can be modified to
take into account the road grade (?). Assuming road grade and braking are the
only external forces on the vehicle, the deacceleration of the vehicle is described
by Equation (3.18).
Dx = ?dVdt = Fxtm +gsin? (3.18)
Assuming the road slope does not change with the distance, the relation dt = dxV
can be substituted into Equation (3.18). By integrating the above equation, a
41
stopping distance formula that depends on braking and road slope is developed,
as shown in Equation (3.19).
SD = V
2
o
2(Fxtm +gsin?) =
V2o
2g(?+sin?) (3.19)
By implementing this equation during travel on a hillside, a safer following distance
will be possible, assuming the road slope is known or measured.
3.4 Experiments
To test the critical velocity and stopping distance equations, a series of tests
were run in Carsim. These test were run to determine slideout, rollover, and
stopping distance of a vehicle and how well the simplified equations match up to
the values from a high fidelity simulation in Carsim. Parameters for the G35 sedan
and the large SUV used in the following simulations are found in Appendix B. The
G35 sedan is used in the Dugoff velocity and zero sideslip velocity simulations,
while the rollover experiments are conducted with the large SUV.
3.4.1 Dugoff Velocity
The Dugoff velocity is developed to keep the vehicle?s tires inside the friction
circle. It may also be noted that this equation does not take into account vehicle
roll so during high amounts of roll the vehicle may slide out quicker than the value
calculated from Equation (3.7).
42
To test the Dugoff velocity equation, two simulations were run in Carsim.
The simulations consisted of a G35 sedan slowly accelerating around two circles,
one with a radius of 152.4 meters and another with radius of 400 meters. The
experiments consisted of four different simulations per circle. Each simulation
had a different value for tire-road friction to fully test the equation?s accuracy
to predict a slide out. Figure 3.4 shows both the simulations with the asterisks
being the Dugoff velocity calculated from the friction coefficient and the radius
of curvature. Remember, this velocity is used to predict at what speed the tire
reaches its limits.
0 20 40?0.5
?0.4
?0.3
?0.2
?0.1
0
0.1
Radius = 154.2m
Velocity (m/s)
Lateral Error (m)
mu=.25
mu=.5
mu=.75
mu=1
0 20 40 60?0.5
?0.4
?0.3
?0.2
?0.1
0
0.1
Radius = 400m
Velocity (m/s)
Lateral Error (m)
Figure 3.4: Predicting Slidout at Radius of 152.5 and 400 meters with the Dugoff
Velocity Equation
Notice as the friction coefficient rises, the accuracy of the equations decrease.
The reason the accuracy is decreasing could likely be from the effects of roll and
weight transfer. During the simulation with a value of tire road friction of 1, nearly
80 percent of the weight was transfered to the outer tire. As the friction coefficient
43
rises, more weight is transfered to the outer tire before the vehicle reaches the limit
of adhesion, causing all the vehicle?s weight to be present on one tire instead of
both. By assuming all the weight is on either the inner or outer tire, the steady-
state tire slip angle can be recalculated resulting in a new Dugoff velocity equation.
VDug <
radicalbigg
?Rg
4 (3.20)
This is a very simple change but will provide for safer results during periods
of high roll angles. The experiments were repeated for the new Dugoff velocity
and the results are shown in Figure 3.5 with the circles being the previous values
using Equation (3.7).
0 20 40?0.5
?0.4
?0.3
?0.2
?0.1
0
0.1
Radius = 154.2m
Velocity (m/s)
Lateral Error (m)
mu=.25
mu=.5
mu=.75
mu=1
0 20 40 60?0.5
?0.4
?0.3
?0.2
?0.1
0
0.1
Velocity (m/s)
Lateral Error (m)
Radius = 400m
VDug(Eqn(3.7))
VDug(Eq(3.20))
Figure 3.5: Predicting Slidout at Radius of 152.5 and 400 meters with the new
Dugoff Velocity Equation
44
3.4.2 Rollover Equation
The rollover prediction formula was developed to help predict rollover during
turning maneuvers. In this section, the formula is tested with different values of
CG height on a 100 meter radius turn. The rollover prediction formula is a function
of additional factors that have been evaluated by Lambert [21].
The first test run in this section is a simulation of a large SUV in Carsim
slowly accelerating around a curve until rollover occurs. The values of this vehicle?s
parameters are shown in Appendix B. It is assumed rollover occurs when the all
the weight of the vehicle is on the outer tires. Table 3.1 shows the simulated
rollover speed versus the calculated rollover speed with each run having a different
CG height. The results of this test show good correlation between the rollover
prediction formula and the simulation?s actual rollover.
Table 3.1 - Rollover Velocity on 100m Radius Turn
CGHeigth VRollover from Sim. (ms ) VRollover from Eq. (ms ) % Diff.
1.2 23.06 23.16 0.4 %
1.0 25.72 25.37 1.4 %
0.8 29.08 28.36 2.5 %
0.6 32.92 32.75 0.5 %
3.4.3 Stopping Distance
This section performs braking simulations in Carsim to test the effectiveness
of the minimum stopping distance equation. The simulations consisted of multiple
runs on roads with different tire-road friction (?) values. All runs will use Carsim?s
45
ABS system to assure quicker braking response and to also simulate how braking
would occur in a real vehicle. Table 3.2 below shows simulated and calculated
stopping distances (SD) for roads with different values of ?.
Table 3.2 - Stopping Distance at Vo=30ms
? SD from Sim. (m) SD from Eq. (m) % Diff.
0.25 189.3 183.5 3.06 %
0.5 94.3 91.7 2.76 %
0.75 60.9 61.2 0.49 %
1.0 45.4 45.9 1.10 %
Next, a set of simulations on a -15 degree incline were conducted. The simu-
lations for this test consist of multiple runs on an incline with different ? values.
Table 3.3 shows the predicted and actual value of stopping distances for these sim-
ulations. The simulation with a coefficient of friction of 0.25 is listed as unknown.
This is because the value ofsin(?) is greater than the value of?causing the vehicle
to slide down the hill without slowing down. The simulation shows similar results
with the vehicle never actually coming to a stop. Therefore, the stopping distance
equation can be used to predict if the road slope is to steep for the vehicle to stop
on. These simulations proves the effectiveness of this simple equation. Therefore,
in platoons of autonomous vehicles these equations would be effective in deciding
a safe following distance from the lead vehicle with a known value of ?. In order
to fully utilize the SD equations, The next chapter will focus on estimation of ?.
46
Table 3.3 - Stopping Distance at Vo=30ms ?=-15 deg
? SD from Sim. (m) SD from Eq. (m) % Diff.
0.25 unknown unknown unknown %
0.5 197.3 190.2 3.60 %
0.75 91.9 93.4 1.63 %
1.0 60.1 61.9 2.99 %
3.5 Conclusions
This chapter developed predicted velocities that proved to be effective at pre-
dicting rollover and sliding out even though many simplification were made to
obtain the equations. An equation was also derived to calculate a speed that min-
imizes the siedslip angle at the vehicle?s center of gravity. In order to calculate
the predictive velocities, many important parameters must be known, such as the
friction coefficient and the vehicle?s center of gravity position. Stopping distance
equations were derived and proved very effective at predicting the stopping dis-
tance of the vehicle assuming certain parameters are known, such as ? and ?. The
predictive velocities were then validated with a series of simulations performed in
Carsim. The results of those test showed very good correlation between the true
speed of rollover and sliding out and the predicted values calculated by the pre-
dictive velocity equations derived in this chapter. In the next chapter, algorithms
will be developed to obtain parameters needed for these equations.
47
Chapter 4
Estimation Algorithm Development
4.1 Introduction
This chapter develops parameter estimation algorithms based off of the vehi-
cle models shown in Chapter 2. The most important and difficult parameter to
estimate may be tire-road friction, as it is used for the dugoff velocity and the
stopping distance equations. To estimate this parameter, a non-linear tire model
must be used that describes the tire?s saturation. This chapter uses the Fiala and
Dugoff tire model with a non-linear estimator to obtain an estimate of peak tire
force. Other parameters that need to be estimated include the CG height and
weight split. The weight split is estimated using the bicycle model in a non-linear
estimator. To estimate CG height, a recursive least squares estimator is used.
Estimates of these important parameters are necessary to update the predictive
velocities and stopping distance equations developed in Chapter 3. Validation of
the tire parameter and weight split estimation algorithms is conducted in MAT-
LAB with data from the G35 sedan, discussed in Chapter 2. Simulations with
Carsim?s large SUV are used to validate the CG height estimator since a vehicle
with a taller CG is desired.
48
4.2 Tire Parameter Estimates
Many researchers have developed methods to estimate tire parameters. While
some researcher have used lateral vehicle models to achieve an estimate of cornering
stiffness [32, 30], others have used lateral vehicle models along with non-linear
tire models to estimate cornering stiffness and tire road friction simultaneously
[6, 16]. Another method developed by previous researchers includes measurements
of the steering wheel torque to achieve estimates of these two important parameters
[17, 20, 23]. Some researchers have used longitudinal vehicle models to estimate
the tire-road friction [4, 5]. In this section, a method will be developed to estimate
lateral and longitudinal tire parameters using non-linear tire models, previously
published in [8].
To estimate lateral and longitudinal tire stiffness as well as the peak tire force,
some type of estimator must be chosen. Since non-linear tire models are used in
this estimation, an extended Kalman filter (EKF) was selected. A basic Kalman
filter is a recursive algorithm that optimizes the estimate by statistically weighting
the accuracy of each measurement. The extended Kalman filter, shown in Equa-
tions (4.1-4.7),is similar to the basic Kalman filter except the state transition and
observation models need to only be differentiable functions of the state, instead of
linear functions.
49
Time Update
?xk|k?1 = f(?xk?1|k?1,uk) (4.1)
Pk|k?1 = FkPk?1|k?1FTk +Qk (4.2)
Measurement Update
?yk = zk ?h(?xk|k?1) (4.3)
Sk = HkPk|k?1HTk +Rk (4.4)
Kk = Pk|k?1HTk S?1k (4.5)
?xk|k = ?xk|k?1 +Kk ?yk (4.6)
Pk = (I ?KkHk)Pk|k?1 (4.7)
Because the system used in this section is non-linear, the H and F matrix will
simply be the Jacobian of the output(h) and state transition model(f) respectively,
as shown below.
Hk = ?h?x|?xk|k?1 (4.8)
Fk = ?f?x|?xk|k?1,uk (4.9)
50
Using the EKF shown above, it is possible to estimate the three states, lateral,
longitudinal tire stiffness, and peak tire force, discussed earlier by using the Fiala
or Dugoff tire model. The EKF uses a non-linear tire model as the output matrix
meaning the tire forces will need to be used as a measurement to compare against
the model. Therefore, the measurement matrix (z) will consist of the lateral and
longitudinal tire forces per axle. To get these measurements, a simple measurement
transition matrix can be developed from the FBD in Figure 4.1.
Figure 4.1: Tire Force Estimates
This figure is similar to the bicycle model earlier, except it also includes the
longitudinal forces at the tires. All other assumptions used to develop the bicycle
model in Chapter 2 are applicable for the force measurements derivation, shown
in Equation (4.10) and (4.11).
?
??
??
??
??
?
max
may
Iz ?r
0
?
??
??
??
??
?
=
?
??
??
??
??
?
1 cos(?) 0 ?sin(?)
0 sin(?) 1 cos(?)
0 asin(?) ?b acos(?)
0 1 0 0
?
??
??
??
??
?
?
??
??
??
??
?
Fxr
Fxf
Fyr
Fyf
?
??
??
??
??
?
during acceleration (4.10)
51
?
??
??
??
??
?
max
may
Iz ?r
0
?
??
??
??
??
?
=
?
??
??
??
??
?
1 cos(?) 0 ?sin(?)
0 sin(?) 1 cos(?)
0 asin(?) ?b acos(?)
?1 1 0 0
?
??
??
??
??
?
?
??
??
??
??
?
Fxr
Fxf
Fyr
Fyf
?
??
??
??
??
?
during braking (4.11)
Inspection of the above equations show, the front longitudinal force is assumed
to be zero during acceleration since the vehicle is rear wheel drive. Another as-
sumption made is the front and rear longitudinal forces are equal during braking,
although this may not always be true if the vehicle has a non 50-50 brake distri-
bution. With an equation to calculate the tire forces, the performance of the EKF
with the Dugoff or Fiala tire model can be further analyzed. The pros and cons of
each tire model in conjunction with the EKF will also be explained in the following
sections.
4.2.1 Tire Stiffness and Peak Tire Force Estimation with Dugoff?s Tire
Model
To develop an extended Kalman filter using the Dugoff Tire Model to estimate
lateral and longitudinal tire stiffness and peak tire force, Equations (2.24-2.27) will
be needed. As shown in the last section, the z matrix must be composed of the
lateral and longitudinal tire forces calculated with the transformation matrix.
52
z =
?
?? Fx
Fy
?
?? (4.12)
The state matrix (x) is composed of the states that are to be identified.
x =
?
??
??
??
C?
C?
?Fz
?
??
??
?? (4.13)
The measurement matrix in Equation (4.12) can be used for the front or rear
axle depending on which tire?s properties need to be estimated. Since the state
matrix is not changing as a function of time, the extended Kalman filter can be
simplified similarly to the one used by Daily [6]. Because the system is non-linear
with the states, the state transition matrix is simply the Jacobian of the tire force
equations with respect to the states, shown by Equation (4.14).
H =
?
?? ?Fx?C? ?Fx?C? ?Fx?(?Fz)
?Fy
?C?
?Fy
?C?
?Fy
?(?Fz)
?
?? (4.14)
The Dugoff tire model has a transition when ? = 1, therefore two separate
state matrices must be created. If ? < 1 in the Dugoff tire model, the equations
for the H matrix are listed below in Equations (4.15-4.20).
53
?Fx
?C? =
?Fz?C2?tan2(?)
((C??)2 + (C?tan(?))2)3/2 ?
?(1 + ?)(?Fz)2(?(C??)2 + (C?tan(?))2)
4((C??)2 + (C?tan(?))2)2 (4.15)
?Fx
?C? = ?
?Fz?C?C?tan2(?)
((C??)2 + (C?tan(?))2)3/2 +
C??(1 + ?)(?Fz)2C?tan2(?)
2((C??)2 + (C?tan(?))2)2 (4.16)
?Fx
?(?Fz) =
C??
((C??)2 + (C?tan(?))2)1/2 ?
C??(1 + ?)?Fz
2((C??)2 + (C?tan(?))2) (4.17)
?Fy
?C? = ?
?Fz?2C?C?tan(?)
((C??)2 + (C?tan(?))2)3/2 +
C??2(1 + ?)(?Fz)2C?tan(?)
2((C??)2 + (C?tan(?))2)2 (4.18)
?Fy
?C? =
?Fz?2C?tan(?)
((C??)2 + (C?tan(?))2)3/2 ?
tan(?)(1 + ?)(?Fz)2(?(C??)2 + (C?tan(?))2)
4((C??)2 + (C?tan(?))2)2 (4.19)
?Fy
?(?Fz) =
C?tan(?)
((C??)2 + (C?tan(?))2)1/2 ?
C?tan(?)(1 + ?)?Fz
2((C??)2 + (C?tan(?))2) (4.20)
When the Dugoff Model transitions at ? = 1, the H matrix must be modified.
The equations for the H matrix when ?? 1 are listed in Equations (4.21-4.26).
?Fx
?C? =
?
1+? (4.21)
?Fx
?C? = 0 (4.22)
?Fx
?(?Fz) = 0 (4.23)
?Fy
?C? = 0 (4.24)
?Fy
?C? =
tan(?)
1 +? (4.25)
?Fy
?(?Fz) = 0 (4.26)
Although the H matrix is very complex when the tire is saturated (? < 1),
all states are observable. In contrast when the tire is in the linear region (?? 1),
54
the peak tire force is unobservable and cannot be estimated. It can also be seen
that the lateral tire stiffness does not depend on the measurement of longitudinal
tire force and the longitudinal tire stiffness is independent of the lateral tire force
measurement. Because the peak tire force is unobservable, the EKF estimate of
this parameter remains constant when ? ? 1. This actually makes sense because
the Dugoff tire model does not depend on the peak tire force to calculate the
tire forces until the tire becomes saturated. Ideally, the peak tire force would be
estimated before the tire becomes saturated to help limit the vehicle from sliding,
but this is not possible using the Dugoff tire model. This leads to one advantage
of using the Fiala tire model, as will be shown in the next section.
4.2.2 Estimation with Fiala?s Tire Model
In addition to the Dugoff tire model, the Fiala model can also be set up for
estimation of certain tire properties. The measurement matrix remains the same
as the one shown in Equation (4.12). Since the Fiala tire model assumes lateral
and longitudinal tire stiffnesses are equal, the state matrix will be composed of
only two states, peak tire force and tire stiffness depending on the maneuver. This
simplification is shown in Equation (4.27).
x =
?
?? C?/?
?Fz
?
?? (4.27)
55
The H matrix must also be calculated by taking the partial of the Fiala tire
model (given previously in Equations (2.20-2.23)) with respect to the states in the
state matrix. Like the Dugoff model, the Fiala model also has a model transition
requiring two different H matrices depending on the value of ?. Equations (4.28-
4.31) describe the H matrix if the tire is not saturated (? ??m).
?Fx
?C?/? =
?x[9(?Fz)2 ?6C?/??Fz?t ?C2?/??2t]
9(?Fz)2 (4.28)
?Fx
?(?Fz) =
?t?xC2?/?[9?Fz ?2C?/??t]
27(?Fz)3 (4.29)
?Fy
?C?/? =
?y[9(?Fz)2 ?6C?/??Fz?t ?C2?/??2t]
9(?Fz)2 (4.30)
?Fy
?(?Fz) =
?t?yC2?/?[9?Fz ?2C?/??t]
27(?Fz)3 (4.31)
However, if the tire begins sliding (? ??m) the H matrix will be composed of
the following equations.
?Fx
?C?/? = 0 (4.32)
?Fx
?(?Fz) =
?x
?t (4.33)
?Fy
?C?/? = 0 (4.34)
?Fy
?(?Fz) =
?y
?t (4.35)
56
Unlike estimation with the Dugoff model, peak tire force can be estimated
before and after the tire begins to slide. This is advantageous because the estimator
may be able to predict the peak tire force before the tire begins to slide. However,
the tire stiffness is unobservable when the tire begins to slide.
To estimate the tire stiffness and peak tire force with both the Fiala and
Dugoff tire models, the Q and R matrix must be identified. The Q matrix is
the process noise covariance matrix and is primarily composed of the standard
deviation of the process noises. In this section the Q matrix acts like a forgetting
factor in a recursive least squares algorithm [6] . The R matrix is the measurement
noise covariance matrix and is composed of values of the noise covariances of the
measurements. In this research, the true value for the process and measurement
noise covariance matrices can not be determined analytically. Therefore, the values
in the matrices were hand chosen to provide a quick rate of convergence. The
matrices for estimation with the Dugoff model can be seen in Equations (4.36) and
(4.37).
Q = 1e?8
?
??
??
??
(80,000N)2 0 0
0 (80,000N/rad)2 0
0 0 (12,000N)2
?
??
??
?? (4.36)
R =
?
?? (5,000N)2 0
0 (5,000N/rad)2
?
?? (4.37)
57
Since estimation with the Fiala model only has two states, theQ matrix is reduced
to:
Q = 1e?8
?
?? (80,000N/rad)2 0
0 (12,000N)2
?
?? (4.38)
During periods of small excitation, the estimator may perform badly produc-
ing incorrect estimates of the states. One way to account for this is by adjusting
the H matrix. If the vehicle is driving straight and not producing any lateral or
longitudinal excitation, the H matrix can be set to zero. This will effectively hold
all the prior estimates constant. Also during periods of longitudinal excitation
only the elements of the H matrix that effect lateral dynamics can be set to zero to
avoid a bad estimate of lateral tire stiffness, but still provide an accurate estimate
of longitudinal tire stiffness and peak tire force.
4.3 Weight Split Estimation
In order to calculate the zero-sideslip velocity, the distance from the rear axle
to the vehicle?s center of gravity must be known. One method that may provide an
accurate estimate would be using the bicycle model, shown in Equation (2.9), in a
non-linear estimator. Ryu [30] used this model with a total least squares algorithm
to estimate cornering stiffness and yaw moment of inertia simultaneously. The
particular research also attempted to estimate cornering stiffness and weight split
simultaneously using this method but was unsuccessful at converging to the correct
values unless highly excited. A technique, previously published in [9] which uses
58
the bicycle model and an EKF, is used in this thesis to update the zero-sideslip
velocity with the weight split estimate.
By augmenting the state space representation of the bicycle model, shown in
Equation (2.9), in an estimator the weight split can be estimated. As discussed
earlier, the bicycle model assumes no weight transfer and the tires are assumed to
be in the linear region. The importance of this assumption will be discussed later.
Because the system is non-linear with respect to the weight split and yaw moment
of inertia, the extended Kalman filter is used. The EKF was earlier presented
in Equations (4.1-4.7). In this algorithm, the states (parameter estimates) are
assumed to be constant; therefore, the differential equations used to describe the
system (f) is simply equal to the process noise (w), shown by Equation (4.39).
f =
?
?? ?a
?Iz
?
??= w (4.39)
By studying the above equation, the state matrix (x) is composed of the distance
from the front axle to the vehicle?s center of gravity (a) and the yaw moment of
inertia (Iz). The state matrix could also include the distance from the vehicle?s
CG to the rear axle (b), but for simplification purposes this state is replaced by
L?a. To describe the output of the system, Equation 4.40 is used.
z =
?
?? ay
?r
?
??= H
?
?? a
Iz
?
??+v (4.40)
59
The measurements included in this equation is the lateral acceleration (ay) and yaw
angular acceleration (?r). The measurements are equal to the observation matrix
(H) times the states plus the measurement noise. The H matrix comes from the
linearization of the bicycle model with respect to the states being estimated. The
observation matrix is shown below in Equation (4.41).
H =
?
?? ?ay?a ?ay?Iz
??r
?a
??r
?Iz
?
?? (4.41)
where;
?ay
?a =
?(C?f + C?r)r
mVx (4.42)
?ay
?Iz = 0 (4.43)
? ?r
?a =
?(C?f + C?r)Vy
IzVx +
(?2aC?f + 2(L?a)C?r)r
IzVx +
C?f?
Iz (4.44)
??r
?Iz = ?
(?aC?f + (L?a)C?r)Vy
I2zVx ?
(?a2C?f ?(L?a)2C?r)r
I2zVx ?
aC?f?
I2z (4.45)
Since the estimator is based off of the bicycle model, this model must accu-
rately capture the actual dynamics in order for the estimated states to converge
to the correct estimate. Therefore, knowledge of all the assumptions used in the
bicycle model becomes critical. If the vehicle develops large roll angles or the ve-
hicle generates large tire slip angles, the bicycle model cannot sufficiently capture
the vehicle motion which will lead to errors in the estimated parameters.
60
4.4 CG Height Estimation
One important parameter in rollover prevention is the height of the vehicle?s
center of gravity. With added weight on a vehicle, the CG height can shift up,
increasing the chance of rollover. A recent researcher has investigated methods
to estimate vehicle CG height using multiple models and switching and compared
this method with recursive least squares [33, 34]. For this thesis, a recursive least
squares algorithm is used to estimate the parameter, similar to [34], using the roll
planer model as a measurement comparison. This will provide an estimate of CG
height to update the predicted rollover velocity equation.
To begin the estimation algorithm development, recursive least squares (RLS)
is briefly discussed. One author compared different recursive least squares tech-
niques to estimate road grade and vehicle mass [35]. The methods used include
recursive least squares, recursive least squares with forgetting factors, and recur-
sive least squares with multiple forgetting. Since the parameters estimated in this
section are assumed constant, the basic recursive least squares is used. If the
parameters were changing, a forgetting factor could be used to forget old measure-
ments. The basic least squares provides parameter estimates by minimizing the
sum of the squares of the difference between the actually observed and the com-
puted values [2]. To estimate the parameters on-line, the RLS Equations shown
below are used.
61
??k = ??k?1 +Lk(yk ??Tk ??k?1) (4.46)
Lk = Pk?k = Pk?1?k(1 +?TkPk?1?k) (4.47)
Pk = (I ?Lk?Tk)Pk?1 (4.48)
Equation (4.46) updates the estimate (??k) by adding the previous estimate
to an update gain (Lk) times the error between the model output and the mea-
surement (yk). This method is similar to the Kalman filter, but it does not utilize
a time update thereby assuming the states remain constant. The variable, Pk, is
known as the state estimation error covariance matrix and is usually initialized
with a large value, due to uncertainty in the states. More information on the
derivation and convergence of recursive least squares can be found in [35].
To estimate the CG height, the roll plane model, shown in Equations (2.30),is
used. In order to utilize recursive least squares, the model equations must be
linear. By assuming small angles the roll plane model can be rewritten, shown by
Equation (4.49) [34].
ameasy = ay +g?= 1mh
cg
[Jeff ??+C? ??+K??] (4.49)
62
Using a lateral acceleration measurement as the model input results in the
state and regression vectors shown in Equations (4.50) and (4.51) respectively.
? =
?
??
??
??
??
??
?
?
??
??
?? (4.50)
?? =
?
??
??
??
Jeff
mhcg
C?
mhcg
K?
mhcg
?
??
??
?? (4.51)
With a known value of roll mass moment of inertia (Jeff), the parameters
hcg, C?, and K? can easily be solved using simple algebra. The CG height will
most likely change with added weight which will cause Jeff to also change. A
better approach may be to estimate the value of Jeff and use a known value of
K? to solve for the CG height since K? is less likely to change with added weight.
Using this technique requires the roll dynamics to be excited and therefore must be
conducted during a turning maneuver such as a lane change. Otherwise performing
the estimation during straight driving could result in faulty estimates. The mass
also needs to be known, as stated earlier. The mass can be estimated during
acceleration or braking by [35], but it is assumed to be a known parameter in this
thesis.
63
4.5 Experiments and Validation of Estimation Algorithms
To test the estimation algorithms, different experiments were performed in
the Infiniti G35 Sedan and in Carsim. The tests performed help identify the
effectiveness of the parameter estimation algorithms. If the algorithms are effective
at providing accurate estimates, the algorithms can be used to update the vehicle
controllers maximum speed around a curve or possibly even update control gains.
4.5.1 Experimental Setup
As discussed earlier, Carsim and an Infiniti G35 is used to validate the pa-
rameter estimation algorithms. Carsim is used to get realistic measurements to
perform the estimation algorithms. The Infiniti G35 sedan is implemented with a
dual antenna GPS unit, wheel speed sensor, a 6 DOF inertial measurement unit
(IMU), and a optical encoder for steer angle measurements. Testing was performed
at the National Center for Asphalt Technology (NCAT) test track in Opelika, AL.
The IMU and wheel speed sensors provided measurements at 33 Hz, while the dual
antenna GPS provided other necessary measurements at 5 Hz. Each subsection
discusses in further detail the measurements that are important for each estimation
algorithm.
4.5.2 Tire Parameter Estimation Experiments
To validate the tire estimation algorithms developed in Section 4.2, data was
collected on the Infiniti G35 at the NCAT test track. To test the algorithm,
64
maneuvers consisted of pure lateral force generation, pure longitudinal acceleration,
and a maneuver to achieve combined lateral and longitudinal force generation. The
maneuvers are meant to fully saturate the tire to the point of sliding.
To collect the measurements for the measurement matrix, Equation (4.10)
or (4.11) is used depending upon whether the vehicle is accelerating or braking.
Therefore, both the lateral and longitudinal acceleration needs to be measured.
These measurements are taken directly from the 6 DOF IMU. Another measure-
ment needed for the calculation of tire forces is the yaw angular acceleration. This
measurement is obtained by numerically differentiating the yaw rate gyro. Note
that numerical differentiation increases the amount of noise already present in the
yaw gyro.
For components of the H matrix, the lateral and longitudinal tire slips needs to
be calculated. With measurements of wheel speed from the wheel speed sensors and
velocity from the GPS, the longitudinal slip is simply calculated with Equations
(2.18). When the vehicle is experiencing combined longitudinal and lateral force
generation, the vehicle sideslip must be used to divide the velocity measurement
from GPS into the lateral and longitudinal components. There have been many
different studies to estimate the sideslip angle. The sideslip is defined as the
difference between the vehicle heading (?) and the vehicle course (?) or direction
of travel, as given below.
65
? = ? ?? (4.52)
One method to obtain an estimate of sideslip is with a yaw gyro and GPS
receiver. This method integrates the yaw gyro during turning to obtain an esti-
mate of heading, while the yaw gyro bias is estimated during periods of straight
driving [3]. With the estimate of heading, sideslip is easily calculated with Equa-
tion 4.52 if a course measurement is available. The course measurement can be
easily obtained from GPS measurements. This is accomplished by comparing con-
secutive carrier measurements to provide a three dimensional velocity. Note that
comparing consecutive measurements introduces a half sample delay that must be
accounted for when processing the sideslip. Many errors arise from this method
including: integration of a noisy gyro, scale factor errors, and incorrect bias esti-
mates. Another author uses steering torque information to estimate the sideslip
and is valuable during periods when GPS is unavailable [38]. The method used
in this thesis uses a dual GPS antenna to get a measurement of heading. The
dual GPS antenna provides this measurement by comparing carrier measurements
at each antenna [31]. The heading measurement is then be subtracted from the
course measurement, as discussed in the previous method.
66
With the sideslip measurement known at the GPS antenna (A), the velocity
magnitude (VGPS) is broken up into lateral and longitudinal components.
VGPSx = VGPScos(?GPS) (4.53)
VGPSy = VGPSsin(?GPS) (4.54)
Now the lateral and longitudinal velocities at the antenna are set up in vector form
(?VA), This vector is easily transferred to the tires using Equation 4.55, where (?Vt)
is the velocity vector at the tire.
?Vt = ?VA + ??? ?rA/t (4.55)
In this equation, ?? includes roll, pitch, and yaw angular velocities. The po-
sition vector, rA/t, is defined from the antenna to the front or rear axle. With
the above measurement of velocity at the tires, the longitudinal slip can then be
calculated. Since the tire force calculation needs a value of wheel slip at the center
of the axle, the average from the inner and outer tire is used. This assumption
could be ineffective when the inner tire is experiencing high slip and the outer is
hardly slipping. The lateral slip angle at the front and rear axle is then calculated
from the lateral and longitudinal components of the tire velocity calculation in
Equation (4.55).
67
?f = tan?1
parenleftBigg
Vfy
Vfx
parenrightBigg
?? (4.56)
?r = tan?1
parenleftbiggVr
y
Vrx
parenrightbigg
(4.57)
In this equation, Vfy is the lateral velocity at the center of the front axle and
Vfx is the longitudinal velocity at the front axle. The subscript, r, denotes the
center of the rear axle.
Estimation during Lateral Dynamics Only Maneuver
The first experiment conducted in the Infiniti G35 sedan includes lateral dy-
namics only. This test determines the effectiveness of using the Fiala and Dugoff
tire models in an EKF by estimating the lateral tire stiffness and the peak tire force
during periods of lateral excitation only. This maneuver consists of high speed cor-
nering and a slalom at speeds ranging from 20 to 30 m/s. Figure 4.2 shows the
experimental data input from the steering encoder and the response from yaw rate
gyroscope.
68
0 10 20 30 40 50 60 70 80 90 100
?40
?20
0
20
40
60
Yaw Rate (deg/s)
0 10 20 30 40 50 60 70 80 90 100
?5
0
5
10
Time
Steer Angle (deg)
Figure 4.2: Lateral Experimental Data in the G35 Sedan used in the Tire Param-
eter Estimator [8]
Parameter estimation could be conducted using lateral force estimates only
because the longitudinal dynamics are assumed to be negligible. However, this sec-
tion is used to test the full algorithm and accounts for longitudinal measurements
also. By simplifying the Dugoff or Fiala tire model to account for only lateral
dynamics, the H matrix becomes much more simple.
With the experimental data at hand, MATLAB is used to conduct the esti-
mation off-line. However, note that the estimation technique could be performed
in real-time. Figure 4.3 shows the lateral tire stiffness and the peak tire force
estimate. The longitudinal tire stiffness is not shown because the maneuver fails
to sufficiently excite the longitudinal dynamics. The values shown are from the
vehicle?s rear axle. The peak tire force settles out in about 20 seconds. Notice for
69
estimation with the Dugoff tire model, the estimator does not update the peak tire
force estimate until around 10 seconds. This is because the state is unobservable
until the tire begins to saturate (i.e. ?< 1). As explained earlier, the estimate re-
mains constant until the tire becomes saturated. The lateral tire stiffness estimate
settles out a little faster for estimation with the Fiala tire model, although both
estimates converge closely to the same value. Appendix C shows additional tests
using Carsim to validate the estimator used in this section.
0 20 40 60 80 1000.5
1
1.5
2
2.5
3x 10
5
Lateral
Stiffness (N/rad)
Dugoff
Fiala
0 10 20 30 40 50 60 70 80 90 1000
0.5
1
1.5
2x 10
4
Time
Peak Tire Force (N)
Figure 4.3: Lateral Tire Stiffness and Peak Tire Force Estimate from G35 Data
during a Lateral Slalom [8]
Estimation during Longitudinal Dynamics Only Maneuver
The EKF will now tested for a maneuver that consists of heavy acceleration
and braking in the test vehicle. The excessive accelerations can be seen in Figure
70
4.4 and are meant to reach the peak of the tire curve. Similar to the lateral only
maneuver, this test is also conducted on an asphalt surface. On a lower friction
surface, the maneuver would not need to be as severe to reach the limits of the
tire.
0 1 2 3 4 5 6 7 8 9 10?15
?10
?5
0
5
10
Time
Longitudinal
Acceleration (m/s
2 )
Figure 4.4: Longitudinal Acceleration in the G35 Sedan used in the Tire Estimator
[8]
Lateral measurements are also included into the estimation, since the lateral
dynamics are not excited the measurements contain mostly noise. Once again, the
lateral dynamics could be left out by reducing the model to longitudinal forces
only, but this section will test to determine if the noisy lateral measurements will
effect the estimates.
As shown in Figure 4.5, the longitudinal tire stiffness settles out rather quickly
and shows similar results using either the Fiala and Dugoff tire models in the tire
71
estimation algorithm. However, the estimate of longitudinal tire stiffness is not
the same as the estimate for lateral tire stiffness. Recall that the Fiala tire model
assumes the lateral and longitudinal stiffnesses are equal, therefore this will lead to
errors in a combined maneuver. The peak tire force settles out slightly higher than
the value estimated during lateral experiments in the previous section. Although
the peak tire force for the longitudinal and lateral directions are assumed to be
equal in this thesis, they could in fact be slightly different. Appendix C shows
additional tests using Carsim to validate the estimator used in this section.
0 1 2 3 4 5 6 7 8 9 10
0.5
1
1.5
2
2.5
3
3.5
4x 10
5
Time
Longitudinal
Stiffness (N/rad)
Dugoff
Fiala
0 1 2 3 4 5 6 7 8 9 100.5
1
1.5
2x 10
4
Time
Peak Tire Force (N)
Dugoff
Fiala
Figure 4.5: Longitudinal Stiffness and Peak Tire force Estimate from G35 Data
during Acceleration and Braking[8]
72
Estimation during Combined Lateral/Longitudinal Dynamics
The last maneuver conducted in the test vehicle was to test tire parameter
estimation during a combined lateral/longitudinal maneuver. This test was done
on a skid pad where the vehicle would brake and accelerate through turns. It
highly excited both lateral and longitudinal dynamics. Figure 4.6 shows the output
from the accelerometer and the yaw gyro during the combined lateral/longitudinal
maneuver.
0 5 10 15 20 25 30 35 40 45 50?15
?10
?5
0
5
Acceleration (m/s
2 ) LongitudinalLateral
0 5 10 15 20 25 30 35 40 45 50?50
?40
?30
?20
?10
0
10
Yaw Rate (deg/sec)
Time
Figure 4.6: Accelerations and Yaw Rate Measurements for the Combined Lat-
eral/Longitudinal Estimation [8]
As shown in the previous section, the lateral and longitudinal tire stiffness are
not equal. Therefore, the tire estimation technique using the Dugoff tire model is
used in this section. Figure 4.7 shows good correlation with previous plots with the
values of lateral tire stiffness and peak tire force. However, the longitudinal tire
73
stiffness estimated shows a value much lower than that of the previous estimate.
This could be due to the inner and outer tires slipping at different rates.
0 5 10 15 20 25 30 35 40 45 501
2
3
4
5x 10
4
Longitudinal Stiffness (N)
Combined Lateral / Longitudinal
0 5 10 15 20 25 30 35 40 45 50
0.5
1
1.5
2x 10
5
Lateral
Stiffness (N/rad)
0 5 10 15 20 25 30 35 40 45 504000
6000
8000
10000
12000
Time
Peak Tire Force (N)
Figure 4.7: Tire Stiffness and Peak Tire Force Estimate during Lateral and Lon-
gitudinal Excitation of the G35 Sedan [8]
4.5.3 Weight Split Estimation Experiments
The weight split estimator is validated in this section by using data collected
at NCAT test track in the Infiniti G35 sedan. Parameter values from this vehicle
are shown in Appendix B. The measurements needed for this experiment are steer
angle at the tire, lateral acceleration, yaw acceleration, and lateral and longitu-
dinal velocity. These measurements are taken from this vehicle with a dual GPS
receiver, IMU, and a steering encoder. Data was collected for validation of the
weight split estimation algorithm during a 40 m/s turn in the test vehicle. The
74
lateral acceleration and yaw rate measurements for the maneuver used to test the
algorithm is shown in Figure 4.8. These measurements are used as inputs to the
estimator, presented in Section 4.3, to produce estimates of the weight split (a or
b) and yaw moment of inertia (Iz).
0 5 10 15 20?20
?10
0
10
r (deg/s)
0 5 10 15 20?10
?5
0
5
10
a y
(m/s
2 )
Time(s)
Figure 4.8: Cornering Data in the G35 Sedan at NCAT Test Track
Figure 4.9 shows the distance from the vehicle?s CG to the front axle converges
closely to the values from the vehicle?s factory values of weight split. The yaw
moment of inertia never completely settles out, possibly because the vehicle is
performing a steady state turning maneuver with constant speeds and radius of
turn providing little excitation. However, during maneuvers where the vehicle
produces large slip angles, the weight split and yaw moment of inertia will be
highly biased. This is due to the assumptions used to develop the bicycle model,
which is used in the a and Iz estimation algorithm. For example, if the vehicle
produces a large amount of roll the vehicle model will not match the data as well
75
as this example. Also if the tires saturate and begin to slide the assumption of a
linear tire model breaks down. However, vehicles generally operate in the linear
region of the tire with limited amounts of weight transfer, such that the algorithm
should perform satisfactorily under normal driving conditions. More tests were
conducted to test the algorithms performance using simulations in Carsim. These
test are shown in Appendix C.
0 5 10 15 200
1
2
a (m) EstimateTrue
0 5 10 15 201
2
3
b (m)
0 5 10 15 20
2000
4000
6000
Iz (kg m
2 )
Time
Figure 4.9: Weight Split, Iz Estimate During Cornering in the G35 Sedan
4.5.4 CG Height Estimate Experiments
Some experiments estimating the CG height has already been published by
[34]. The main difference in the CG height estimation algorithm used in this thesis
is it assumes a known value for roll stiffness (K?) where the other work relies on
the roll moment of inertia. One reason K? was chosen is because it will not change
with added weight like the roll moment of inertia will. To test this algorithm,
76
Carsim is used to provide a vehicle with a larger CG height. The simulations uses
a large SUV that produces larger roll angles, as opposed to the G35 sedan. Values
of the vehicle?s parameters used can be seen in Appendix B.
The Carsim output of lateral acceleration, roll, and roll velocity with the
simulated sensor noise is shown in Figure 4.10. This data is used in the recursive
least squares estimator to estimate the CG height, roll damping, and roll mass
moment of inertia from the state matrix give previously in Equation 4.51. The
vehicle performs a double lane change as can be seen in the figure.
0 5 10 15 20?10
0
10
?
(d
eg)
0 5 10 15 20
?20
0
20
? ?de
g s
0 5 10 15 20?10
0
10
ay
(m s
)
Time
Figure 4.10: Double Lane Change with Added Noise in Carsim?s Large SUV
After the simulated data is generated in Carsim, Matlab is used to add artifi-
cial noise. Noise values can be found in Appendix B in the simulation parameter
table. The Carsim data, with added noise, is then used as an input to the RLS
algorithm given in Section 4.4. The plots in Figure 4.11 show the results of the
77
RLS algorithm including estimates of the CG height, roll damping, and the roll
moment of inertia. The CG height estimate is slightly larger than the true value,
however without injected noise in the simulation the estimate lies right on the
true value. With increased noise levels, the estimate will rise slightly above the
previous value. This is no surprise since it is well known that sensor noise can lead
to biased estimates using RLS [39]. Although the estimate may be slightly off, the
estimated value in this simulation always lies above the true value. Therefore, this
acts as a safety factor when combined with the predictive rollover velocity, actually
predict rollover at a slower speed. Although the roll mass moment of inertia does
not converge to the correct value it is not used in the rollover prediction formula.
The roll damping estimate converges closely to the true value but also is not used
in the roll prediction velocity.
0 5 10 15 200
1
2
Time
hc
g(
m)
0 5 10 15 20
1
2
3x 10
4
Time
C?
(N
m
s
ra
d
)
0 5 10 15 20
1000
2000
3000
Time
Je
ff
(k
gm
2 )
Figure 4.11: RLS Estimation of Large SUV?s CG Height, Roll Damping, and Roll
Mass Moment of Inertia during a Double Lane Change in Carsim
78
4.6 Conclusion
This chapter described three different algorithms to assist in the estimation
of important vehicle parameters. First, an extended Kalman filter was used to
estimate the tire stiffness and peak tire force. The extended Kalman filter was
chosen because the system to describe the saturation of the tire was non-linear.
Two different non-linear tire models, the Fiala and Dugoff tire model, were im-
plemented into the EKF and the pros and cons of estimating the tire parameters
were discussed with each model. Next, an algorithm to estimate the vehicle?s CG
position and yaw moment of inertia was developed. This algorithm implemented
the bicycle model into an EKF to perform the parameter estimation. The param-
eters were assumed to be constant in the derivation of the weight split estimation
algorithm. The last algorithm developed in this chapter estimated CG height, roll
damping, and roll mass moment of inertia. The roll dynamics that described the
system were linear, therefore recursive least squares was chosen. To validate the
tire parameter estimator data was collected in a G35 sedan at the NCAT test track.
The tire parameter estimation algorithm was validated during a lateral, longitu-
dinal, and combined vehicle dynamic maneuver. The weight split estimator was
also validated with data from the G35 sedan, while the CG height estimator was
validated using simulated data from Carsim. The limitations of each algorithm
were also discussed in this chapter.
79
Chapter 5
Updated Predictive Velocity Experiments
5.1 Introduction
In this chapter, the predictive velocities are updated with parameter estimates
discussed in the previous chapter in real-time. The experiments conducted provide
safe look-ahead velocities to prevent rollover, sliding out, and also a minimum
sideslip velocity for more dynamic maneuvers such as dodging something in the
road. If the vehicle is following a leader, the parameter estimates could also be
used to update a safe following distance. These experiments will use Carsim and
Simulink together to simulate a true vehicle. This chapter is meant to show the
usefullness of the parameter estimates in real-time. The parameter estimates could
also be used for many other things such as updating control gains or limiting yaw
rates, but in this chapter it will be used to limit the vehicle?s velocity.
5.2 Critical Velocity with Parameter Updates
The experimental setup uses data from Carsim that is exported in real-time
to Simulink. The input into the Carsim model is the vehicle?s desired velocity.
This value is calculated in different sections depending on the maneuver and ve-
hicle. This chapter also uses Carsim?s velocity and steering controllers, but the
desired values are sent directly to the velocity controller based upon the estimated
80
parameters and the turning radius. The turning radius is assumed to be known
from the vehicle?s path planner.
5.2.1 Zero Sideslip Velocity
The Zero Sideslip Velocity was derived in Chapter 3 and attempts to keep
the value of sideslip to a minimum. This section will update this velocity with
the estimation of weight split. This velocity is used during maneuvers with abrupt
changes such as a lane change or dodging an object. The same measurements are
needed as described in Section 4.5.3 and are obtainable with real sensors, but in
this chapter the measurements are taken from Carsim on-line.
The first test conducted was a series of lane changes simulating a vehicle
dodging an object. Figure 5.1 shows the path the vehicle attempts to follow at
the desired speed calculated from Equation (3.7). Shown by the plot, the radius of
curvature varies highly throughout the maneuver. The rollover velocity and Dugoff
velocity is a function of radius of curvature, therefore may not be reasonable in this
maneuver, this creates a need for the zero sideslip velocity. As the vehicle begins
the lane change, the weight split, initially set at 50/50, is updated to speed up or
slow down the vehicle. By doing this, the vehicle?s sideslip is kept to a minimum.
The derivation of the zero sideslip velocity assumes steady-state conditions, but
that is not the case for this maneuver, therefore the sideslip cannot be constrained
to zero.
81
0 20 40 60 80 100 1200
0.5
1
1.5
2
2.5
3
3.5
4
X (m)
Y (m)
Figure 5.1: Desired Path for Vehicle Controlled at Zero Sideslip Velocity
Figure 5.2 shows the updated weight split for the vehicle. The estimates stay
constant for about 1 second. This is because there is no input into the system, as
the steering angle is initially zero until the maneuver begins. With this estimate
of weight split, the desired velocity is updated.
82
0 2 4 6 8 100.5
1
1.5
2
a (m)
Estimate
True
0 2 4 6 8 10
1.4
1.6
1.8
2
b (m)
Time (s)
Figure 5.2: Weight Split Estimate in a Lane Change Maneuver at Zero Sideslip
Velocity
Figure 5.3 shows the adaptation of the desired zero sideslip velocity, as it is
updated with the estimate of b. The desired velocity is initially calculated to be
14.12 m/s with a 50/50 weight split. As soon as the vehicle enters the turn, the
weight split estimator begins to estimate b. The updated value of b is used to
update the desired velocity, plotted in 5.3. Carsim?s velocity controller is used to
drive the true speed to the desired speed. There is a little overshoot from the
velocity controller but the vehicle quickly settles out the the desired speed.
83
0 2 4 6 8 1048
50
52
54
56
58
60
Velocity (m/s)
Time (s)
Desired Velocity
True Velcoity
Figure 5.3: True and Desired Velocity Calculated from the Zero Sideslip Velocity
in a Lane Change Maneuver in Carsim
Since the objective of this predictive velocity is to keep sideslip to a minimum,
Figure 5.4 shows the values of the vehicle?s sideslip controlled to the updated
speed, shown in Figure 5.3, versus the sideslip from a vehicle without the weight
split update. As discussed earlier, the vehicle is initially assumed to have a 50/50
weight split and travels at a constant speed of 14.12 m/s without the update of
b. The above figure shows that the sideslip is in fact minimized by updating the
zero sideslip velocity with an estimate of b. The sideslip in the plot is actually
very similar until the velocity controller begins to settle out. This plot shows the
effectiveness of this algorithm to minimize the sideslip during maneuvers such as
a lane change.
84
0 2 4 6 8 10?0.4
?0.3
?0.2
?0.1
0
0.1
0.2
0.3
0.4
Time (s)
Sideslip (deg)
Updated Velocity
Non?updated Velocity
Figure 5.4: Sideslip with and without Desired Updated Velocity
5.2.2 Dugoff Velocity
The Dugoff velocity, derived in Equation (3.7), is used in this section to update
the controlled velocity when the radius of curvature is not as variable as in the
lane change. To do this, a method to estimate tire-road friction is used, developed
in [29], to update the Dugoff velocity. The test is conducted in Carsim on a 400
m radius circle with a change in tire road friction. The friction coefficient drops
from .85 to .5 over 100 meters. The updated and true coefficient of friction using
the estimator is shown in Figure 5.5
85
0 10 20 30 40 50 600.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
?
Time (s)
True ?
Estimated ?
Figure 5.5: Coefficient of Friction Update during 400 Meter Radius Curve in Car-
sim?s G35 Sedan
Notice that the estimate of the coefficient of friction is slightly lower than
the true value. The reason for this may be that the system is not persistently
excited. The maneuver the vehicle performs is at a steady state speed on a constant
radius, therefore the inputs and outputs are very close to the same value until the
coefficient of friction changes. Another problem with this method may be when
the friction coefficient goes from a lower value to a higher value. This will be hard
to detect because the vehicle will be traveling at a lower speed when the coefficient
of friction changes, causing the tires to produce slip angles too small to estimate
the friction coefficient.
As the friction coefficient is estimated, its value is used to calculate a desired
speed from the Dugoff velocity equation. As stated in Section 3.2.2, the Dugoff
86
velocity equation is developed to ensure the tires remain inside the friction of
circle during a steady state maneuver. Figure 5.6 provides the desired velocity to
be controlled to and the vehicle?s true velocity. Carsim?s internal velocity controller
is used to drive the true velocity to the desired value.
0 10 20 30 40 50 6030
32
34
36
38
40
42
Velocity (m/s)
Time (s)
Desired Velocity
True Velocity
Figure 5.6: True and Desired Velocity Calculated from the Dugoff Velocity in a
400 m Radius turn in Carsim
Controlling the vehicle at the Dugoff velocity shows good results and provides
sufficient excitation of the tire slip angles to provide an estimate of ?. Figure
5.7 shows the values of the slip at the tires during this maneuver. Notice as the
friction coefficient changes from a high to low value, the steady-state value of tire
slip angle decreases. This is because with a lower value ?, the roll dynamics are
not effected as much as with the higher value. The effects of vehicle roll on the
Dugoff velocity equation was discussed more in depth in Section 3.4.1. If the newer
version of the Dugoff velocity equation is used, shown in Equation (3.20), the slip
87
angles produced at the tires are too small to accurately identify the value of the
friction coefficient in this maneuver. Since it is generally not desired to produce
large slip angles, the zero sideslip velocity could be used but the vehicle would be
traveling at much slower speeds. For vehicle?s with a large CG height, the rollover
velocity must also be taken into account.
0 10 20 30 40 50 60?2.5
?2
?1.5
?1
?0.5
0
0.5
Time (s)
Tire Slip Angle (deg)
Front Tire Slip Angle
Rear Tire Slip Angle
Figure 5.7: Tire Slip Angles in a 400 m Radius Turn with a Drop in Road Friction
Coefficient while Controlling Speed at the Dugoff Velocity
With a varying radius of curvature, the velocity would oscillate depending on
how much the radius is changing. During this, the smallest radius of curvature
could be used to calculate a desired velocity. However, this would not ensure high
enough slip angles for an accurate estimate of tire-road friction if the radius was
changing from a high to low value.
88
5.2.3 Rollover Velocity
To help prevent rollover, it is beneficial to know minimum safe speeds for a
vehicle to enter a turn. By updating the rollover velocity prediction formula with
an estimate of CG height, this can be accomplished. Since the rollover formula
predicts what velocity the vehicle will rollover, the limit velocity can be determined,
assuming the radius of curvature is known. To test the method of updating the
vehicle?s velocity with a CG height estimate, a large SUV in Carsim will accelerate
around a 200 meter radius turn. The vehicle will accelerate up to 90 % of the
minimum rollover velocity calculated by Equation (3.12). If the vehicle is controlled
to the minimum rollover velocity the vehicle will likely rollover with any controller
overshoot. To reduce the number of parameters needed, the understeer gradient
will be assumed to be equal to 0 (i.e. neutral steer vehicle) and the parameter to
adjust for suspension effects, ?, will be set to .9. While the vehicle is accelerating
up to the rollover velocity, the CG height will be estimated to update the minimum
safe velocity. The algorithm used to estimate the CG height was shown in Section
4.4. Figure 5.8 shows the estimated CG height for this test.
89
0 10 20 30 40 50 600
0.2
0.4
0.6
0.8
1
1.2
1.4
CG Height (m)
Time (s)
Estimate
True
Figure 5.8: Estimated CG Height during a 400 m Radius Turn using data from
Carsim?s Large SUV
In Figure 5.9, the desired and true vehicle velocities are shown. As shown by
the plot, the velocity controller causes the true velocity to overshoot the desired
velocity. If the minimum rollover velocity was used in this simulation, the controller
overshoot would likely cause the vehicle to rollover. However, this simulation scaled
down the rollover predictive velocity to 90 % of its true value to account for this.
On roads with small variations in curvature, the roll dynamics would be excited
more, causing a better estimate. In steady-state turns the roll dynamics are not
highly excited.
90
0 10 20 30 40 50 6020
30
40
50
60
70
80
Velocity (m/s)
Time (s)
Desired Velocity
True Velocity
Figure 5.9: True and Desired Velocity Calculated from the Rollover Velocity in a
200 m Radius turn in Carsim
5.3 Conclusions
In this chapter, different experiments were conducted to show the usefulness
of estimating important vehicle parameters in real-time. These test are mostly
simple tests and could be conducted during different maneuvers or be used in
different combinations, but are mostly used to show the importance of on-line
parameter estimation. Tests performed included updating a desired velocity based
on information from the path-planner, the vehicle, and the updated parameters.
To be truly effective, a combination of the velocities would need to be included into
the controller. For instance, in a larger vehicle the Dugoff and rollover predictive
velocities would be calculated with the lower velocity being sent to the controller
91
as the desired velocity. Many other test could be performed to show the usefulness
of these on-line estimates and provide an avenue for future work.
92
Chapter 6
Conclusions
6.1 Overall Contributions
In this thesis, vehicle models were developed to describe the relationship be-
tween the inputs and outputs of the vehicle. These models were initially validated
to show a match between the predicted outputs and true outputs. The models were
then simplified to predict a velocity in which rollover and sliding occurs. A velocity
equation was also developed to minimize the sideslip in a steady-state turn. By
changing critical vehicle parameters in Carsim, the velocity equations were tested
for accuracy and proved to be effective at predicting rollover and slideout, as well
as minimizing sideslip. Additionally, experiments demonstrated that certain crit-
ical parameters highly effects the speeds at which a vehicle can maneuver a turn
or lane change.
Because certain vehicle parameters have a large effect on vehicle handling,
parameter estimation algorithms were developed. To estimate the tire-road fric-
tion coefficient and tire stiffness, a non-linear tire model was used in an extended
Kalman filter. This method provided an estimate of the friction coefficient dur-
ing periods of longitudinal and lateral tire force generation. Other parameters
estimated included the CG position of the vehicle. Validation of the algorithms
were also performed using different measurements which can be obtained from real
sensors.
93
After the parameters were estimated, the critical velocity equations were up-
dated to provide a safe look-ahead traveling speed. These parameters could also
be used to update ESC systems. Simulations were ran to test the algorithms effec-
tiveness during certain maneuvers in the vehicle. These simulations showed that
by adjusting the vehicle speed to stay below the predictive velocity equations could
be very beneficial to the safety of driven or autonomous vehicles.
6.2 Limitations
Several limitations were identified in this thesis. For example, parameters
can only be estimated when the vehicle?s dynamics are being excited. Therefore,
only during certain maneuvers are the predictive velocities able to be updated.
However, during a maneuver that requires a velocity update, such as a turn or
lane change, the vehicle is usually excited enough to get an estimate of various
parameters. Also many of these estimates depend on measurements from a GPS
receiver. These measurements can falter during periods of low satellite visibility.
6.3 Recommendations for Future Work
During the research, it was realized that many other vehicle control systems
could benefit from knowledge of unknown parameters. Certain vehicle control
systems could set limits on vehicle states such as lateral velocity and yaw rate to
reduce sliding out and rollover. With knowledge of changing vehicle parameters,
these limits could be adjusted to take into account these states. By doing this,
94
many lives could be saved by reducing the number of accidents due to rollover
and sliding off the road. The control systems also need accurate measurements of
vehicle states such as sideslip. Since sideslip is expensive and difficult to measure,
many researchers have developed model based systems to provide an estimate.
By updating the model based systems with vehicle parameters using methods in
this thesis, the model based state estimators could prove to be more effective in
providing an accurate estimate.
The parameter estimation algorithms presented in this thesis could also be
useful to Unmanned Ground Vehicles (UGVs). When traveling in platoons, it
may be necessary to use the minimum stopping distance, calculated by Equation
(3.19), to set the following distance or control the velocity. Some researchers have
previously implemented throttle and braking controllers to maintain a constant
following distance for vehicles in platoons [18]. By using the minimum stopping
distance equation, the following distance in the platoon could be updated during
maneuvers, such as a braking or turning, that could provide an estimate of tire
road friction.
95
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99
Appendices
100
Appendix A
Vehicle Nomenclature
Chapter 2
a Length between CG and Front Contact Patch
ay Lateral Acceleration
? Tire Slip Angles
b Length between CG and Rear Contact Patch
? Sideslip Angle
C Roll Damping
CG Center of Gravity
C? Lateral Tire Stiffness
C? Longitudinal Tire Stiffness
Dx Deacceleration
? Steer Angle
Fy Tire Lateral Force (F, R)
Fz Tire Vertical Force (F, R)
g Acceleration due to Gravity
hCG CG Height
Iz Yaw Moment of Inertia
Jeff Roll Moment of Inertia
K Roll Stiffness
Kus Understeer Gradient
L Length from Front Axle to Rear Axle
m Vehicle Mass
r Yaw Rate
R Radius of Curvature
T Track Width
Tspring Torque from Springs
Tdamper Torque from Dampers
101
Chapter 2 cont.
V Vehicle Velocity
Vx Vehicle Longitudinal Velocity
Vy Vehicle Lateral Velocity
? Tire Road Friction Coefficient
? Total Vehicle Slip
?m Total Vehicle Slip in which sliding occurs
? Vehicle Roll Angle
f = front r = rear L = Left R = Right
Chapter 3
Dx Deacceleration
SD Stopping Distance
VDug The Velocity the tire leaves the tire-friction circle
VRollover The Velocity rollover occurs
V?=0 The Velocity to minimize sideslip
? Road Slope
Chapter 4
Hk Jacobian of Output Matix
Kk Kalman Update Gain Matrix
Lk Recursive Least Squares Update Gain
P Covariance Matrix
R Measurement Noise Covariance Matrix
Q Process Noise Covariance Matrix
x State Vector in EKF
z Measurement Matrix in EKF
?k Recursive Least Squares State Matrix
? Vehicle Course
? Vehicle Heading
102
Appendix B
Vehicle Properties
Properties of the G35 Sedan
Wheelbase: L 2.8498 m
Vehicle Mass: m 1528.2 kg
Dist. from CG to Front Contact Patch a 1.3679 m
Dist. from CG to Rear Contact Patch b 1.4819 m
Yaw Moment of Inertia Iz 2400 kg?m2
Front Tire Cornering Stiffness C?f 91674 N/rad
Rear Tire Cornering Stiffness C?r 152788 N/rad
Road Friction Coefficient ? .85
Properties of Carsim?s Large SUV
Vehicle Mass: m 2450 kg
CG Height (Sprung Mass): hCG 1.1 m
Roll Mass Moment of Inertia Jeff 1243 kg?m2
Roll Stiffness K? 2527.9 N?mdeg
Roll Damping C? 152.05 N?m?sdeg
Track Width T 1.62 m
Injected Noise Value on ay ?2ay (.1 m/s2)2
Injected Noise Value on ? ?2? (.01 rad)2
Injected Noise Value on ?? ?2?? (.005 rad/s)2
103
Properties of Carsim?s G35 Sedan
Wheelbase: L 2.85 m
Vehicle Mass: m 940 kg
Dist. from CG to Front Contact Patch a 1.019 m
Dist. from CG to Rear Contact Patch b 1.831 m
Yaw Moment of Inertia Iz 1530 kg?m2
Front Tire Cornering Stiffness C?f 78311 N/rad
Rear Tire Cornering Stiffness C?r 47033 N/rad
Injected Noise Value on ay ?2ay (.1 m/s2)2
Injected Noise Value on r ?2r (.02 rad/s)2
Injected Noise Value on ? ?2? (.02 rad)2
Injected Noise Value on V ?2V (.01 m/s)2
104
Appendix C
Tire Parameter and Weight Split Estimates using Data from
Carsim
C.1 Tire Parameter Estimator Validation with Carsim?s G35 Sedan
The estimator developed in Section 4.2 is shown in this section to test the its
effectiveness with data from Carsim. In Carsim, it is possible to know the true
values of the vehicle parameters unlike the parameters in the Infiniti G35 Sedan
used earlier to test with. Both the Fiala and Dugoff tire models are once again
used to estimate the tire parameters
C.1.1 Tire Parameter Estimator Testing during Lateral Maneuver
A simulation is ran in Carsim to test the tire parameter estimation algorithm
during a slalom in a G35 sedan. The lateral acceleration and yaw rate from the
simulation is shown in Figure C.1. The vehicle reaches excessive lateral acceleration
in this maneuver, enough so to saturate the tire enough for an estimate of peak
tire force.
105
0 2 4 6 8 10?10
?5
0
5
10
Lateral
Accelearation (m/s
2 )
0 2 4 6 8 10?50
0
50
Yaw Rate (deg/s)
Time (s)
Figure C.1: Slalom in Carsim?s G35 Sedan
The data collected from Carsim is ran through the tire parameter estimator
discussed in Section 4.2 to test the effectiveness of estimating lateral tire stiffness
and peak tire force. Artificial zero mean random noise is added into the measure-
ments to try to produce as real of a situation as possible. Values of the artificial
zero mean random noise used in this simulation is shown in Appendix B along
with the vehicle?s parameters. The true value of lateral tire stiffness is obtained
by interpolating the experimental tire curve used by Carsim. Note that the value
produced depends on what part of Carsim?s tire curve is assumed to be in the
linear region and may fluctuate depending on different models. Since the Fiala
and Dugoff tire models assume saturation of the tire occurs at different points on
the tire curve their values may be different for tire stiffness. The values from from
the estimator are shown in Figure C.2.
106
0 2 4 6 8 104
6
8
10x 10
4
Lateral Tire
Stiffness (N/rad)
0 2 4 6 8 100
5000
10000
Peak Tire Force (N)
Time (s)
Dugoff Model
Fiala Model
True Value
Figure C.2: Estimation of Tire Parameters during Slalom Maneuver
Once again the estimator with the Dugoff tire model does not begin estimating
peak tire force until the tire begins to saturate. When the estimate begins to
converge it lies slightly above the true value of peak tire force. It may also be
noted that the Fiala model does not accurately converge until the tire is saturated,
similarly to the test performed in Section 4.5.2.
C.1.2 Tire Parameter Estimator Testing during Longitudinal Maneu-
ver
The EKF will now tested for a maneuver that consists of heavy acceleration
and braking in Carsim?s G35 sedan. The excessive accelerations can be seen in
Figure C.3 and are meant to reach the peak of the tire curve. Similar to the
lateral only maneuver, this test is also conducted on an asphalt surface. On a
107
lower friction surface, the maneuver would not need to be as severe to reach the
limits of the tire.
0 5 10 15 20?10
?8
?6
?4
?2
0
2
4
6
8
10
Time (s)
Longitudinal
Acceleration (m/s
2 )
Figure C.3: Acceleration / Braking in Carsim?s G35 Sedan
As shown in Figure C.4, the longitudinal tire stiffness settles out rather quickly
and shows similar results using either the Fiala and Dugoff tire models in the tire
estimation algorithm. However, the estimate of longitudinal tire stiffness is not
the same as the estimate for lateral tire stiffness. Recall that the Fiala tire model
assumes the lateral and longitudinal stiffnesses are equal, therefore this will lead to
errors in a combined maneuver. The peak tire force settles out slightly lower than
the value estimated during lateral experiments in the previous section. Although
the peak tire force for the longitudinal and lateral directions are assumed to be
equal in this thesis, they could in fact be slightly different.
108
0 5 10 15 200
1
2
3
4x 10
4
Lateral Tire
Stiffness (N/rad)
0 5 10 15 200
2000
4000
6000
8000
10000
12000
Peak Tire Force (N)
Time (s)
Dugoff Model
Fiala Model
True Value
Figure C.4: Estimation of Tire Parameters during Acceleration / Braking Maneu-
ver
C.2 Weight Split Estimator Validation with Carsim?s G35 Sedan
The weight split estimator will be validated in this section by using data
of a G35 sedan from Carsim. Parameter values from this vehicle are shown in
Appendix B. Simulated experiments are conducted to show the effectiveness of
the estimator to provide information on the weight split, and the shortcomings of
this algorithm are also discussed. The measurements needed for this experiment
are steer angle at the tire, lateral acceleration, yaw acceleration, and lateral and
longitudinal velocity. These measurements can be taken from a real vehicle with
a dual GPS receiver, IMU, and a steering encoder. However in this experiment,
the data is taken directly from Carsim. The maneuver performed in the vehicle is
a simple double lane change that would be performed during obstacle avoidance.
The steering input, lateral acceleration, and yaw rate measurements for the double
109
lane change is shown in Figure C.5. Artificial zero mean random noise is added
into the measurements to try to produce as real of a situation as possible. Values of
the artificial zero mean random noise used in this simulation is shown in Appendix
B along with the vehicle?s parameters.
0 2 4 6 8 10?5
0
5
? (deg)
0 2 4 6 8 10?5
0
5
ay (m/s
2 )
0 2 4 6 8 10
?10
0
10
Time (s)
r (deg/s)
Figure C.5: Double Lane Change with Added Noise in Carsim
These measurements are used as inputs to the estimator, presented in Section
4.3, to produce estimates of the weight split (aorb) and yaw moment of inertia (Iz).
Figure C.6 shows the distance from the vehicle?s CG to the front axle converges
to within .02 % of the actual CG location. Although the yaw moment of inertia
never completely settles out, it does provide a good estimate of the parameter.
110
0 2 4 6 8 100
1
2
a (m)
Estimate
True
0 2 4 6 8 101
2
3
b (m)
0 2 4 6 8 101000
2000
3000
Iz (kg m
2 )
Time (s)
Figure C.6: Weight Split, Iz Estimate
111