Dept. Of Mechanical and Nuclear Engineering,
Penn State University
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
A Comparative, Experimental Study of Model
Suitability to Describe Vehicle Rollover
Dynamics for Control Design
John T. Cameron
Pennsylvania State University
Dr. Sean Brennan
Pennsylvania State University
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
2
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Outline
1.
Goals
2.
Analytical Vehicle Models
3.
Experimental Model Validation
4.
Conclusions
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
3
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Goals
Examine various vehicle models to determine the
effect that different assumptions have on:
Model order
Model complexity
Number and type of parameters required
Experimentally validate the models to:
Determine model accuracy
Relate modeling accuracy to assumptions made
Determine the simplest model that accurately represents a
vehicles planar and roll dynamics
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
4
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Analytical Vehicle Models
Standard SAE sign convention
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
5
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Analytical Vehicle Models
Basic Assumptions Common to All Models
All models are linear
Result:
•
Small angles are assumed making cos(
θ
)≈1, sin(
θ
)≈0
•
Constant longitudinal velocity (along the x

axis)
•
The lateral force acting on a tire is directly proportional to slip
angle
•
Longitudinal forces ignored
•
Tire forces symmetric right

to

left
sin
1
cos
sin
1
cos
tire
tire
C
F
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
6
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Analytical Vehicle Models
Model 1
–
2DOF Bicycle Model
f
Fu
Kq
q
D
q
M
y
q
f
f
r
r
r
f
f
f
r
f
C
r
V
U
C
l
U
C
U
C
l
U
C
F
F
0
0
0
r
f
r
f
zz
F
F
l
l
y
r
V
mU
r
V
I
m
0
0
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
7
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Analytical Vehicle Models
Model 2
–
3DOF Roll Model
Assumes the existence of a sprung mass
No x

z planar symmetry
Originally presented by Mammar et. al., National Institute of Research
on the Transportations and their Security (INRETS), Versailles, France
in 1999
r
f
r
f
s
s
xx
xz
s
xz
zz
s
F
F
l
l
y
gh
m
K
r
V
D
hU
m
mU
r
V
I
I
h
m
I
I
h
m
m
0
0
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
r
f
r
f
zz
F
F
l
l
y
r
V
mU
r
V
I
m
0
0
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
r
f
r
f
zz
F
F
l
l
y
r
V
mU
r
V
I
m
0
0
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
8
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Analytical Vehicle Models
Model 3
–
3DOF Roll Model
Assumes the existence of a sprung mass
x

z planar symmetry
Roll

steer influence
Originally presented by Kim and Park, Samchok University, South
Korea, 2003
0 0 0 0 0 0 2 2
0 0 0 0 0 0 0 0 2 2
0 0 0 0 0 0
s
f
zz f r
r
s xx s s
m m h V mU V y
F
I r r l l
F
m h I m hU D K m gh
r
f
r
f
zz
F
F
l
l
y
r
V
mU
r
V
I
m
0
0
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
9
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Analytical Vehicle Models
Model 3 (continued)
As a result of the assumption of roll steer, the external forces
acting on the vehicle change accordingly
f
f
r
r
r
f
f
f
r
f
C
r
V
U
C
l
U
C
U
C
l
U
C
F
F
0
0
0
f
f
r
r
r
r
f
f
f
f
r
f
C
r
V
U
C
l
U
C
U
C
l
U
C
F
F
0
*
*
3
,
3
,
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
10
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Analytical Vehicle Models
Model 4
–
3DOF Roll Model
Assumes a sprung mass suspended upon a massless frame
x

z planar symmetry
No roll steer influence
Originally presented by Carlson and Gerdes, Stanford University,
2003
r
f
r
f
xx
zz
F
F
h
h
l
l
y
mgh
K
r
V
D
mU
r
V
I
I
mh
m
2
2
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
r
f
r
f
zz
F
F
l
l
y
r
V
mU
r
V
I
m
0
0
2
2
2
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
11
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Analytical Vehicle Models
Effect of assuming force equivalence
Slightly changes plant description (i.e. eigenvalues)
Additionally, causes a higher gain in roll response from the
massless frame assumption
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
12
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Model Fitting Procedures
1.
Experimentally determine the understeer gradient to find the
relationship between front and rear cornering stiffness values.
Considering both frequency and time domains*:
2.
Determine estimates on cornering stiffness values by fitting of
the 2DOF Bicycle Model (Model 1).
3.
Determine estimates on roll stiffness and damping by fitting of
Models 2
–
4.
*

Time domain maneuvers were a lane change and a pseudo

step
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
13
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Time Domain Fit Results
10
1
0
5
10
15
Frequency Response, Steering Input to Yaw Rate, Mercury Tracer
U =16.5 Cf =22750 Cr =19958.4561 K =38000 D =5000
w (rad/s)
Mag (dB)
10
1
100
50
0
w (rad/s)
Phase (deg)
Measured
Model 1
Model 2
Model 3
Model 4
10
1
15
20
25
30
35
40
Frequency Response, Steering Input to Lateral Acceleration, Mercury Tracer
U =16.5 Cf =22750 Cr =19958.4561 K =38000 D =5000
w (rad/s)
Mag (dB)
10
1
0
50
100
150
w (rad/s)
Phase (deg)
Measured
Model 1
Model 2
Model 3
Model 4
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
14
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Model Fitting Results
Results for Steering Input to Lateral Acceleration
10
1
15
20
25
30
35
40
Frequency Response, Steering Input to Lateral Acceleration, Mercury Tracer
U =16.5 Cf =45500 Cr =75562.5 K =53000 D =6000
w (rad/s)
Mag (dB)
10
1
0
50
100
150
w (rad/s)
Phase (deg)
Measured
Model 1
Model 2
Model 3
Model 4
Freq. Domain Fit
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
15
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Model Fitting Results
Results for Steering Input to Yaw Rate
10
1
0
5
10
15
Frequency Response, Steering Input to Yaw Rate, Mercury Tracer
U =16.5 Cf =45500 Cr =75562.5 K =53000 D =6000
w (rad/s)
Mag (dB)
10
1
100
50
0
w (rad/s)
Phase (deg)
Measured
Model 1
Model 2
Model 3
Model 4
Freq. Domain Fit
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
16
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Model Fitting Results
Results for Steering Input to Roll Rate
10
1
5
0
5
10
15
Frequency Response, Steering Input to Roll Rate, Mercury Tracer
U =16.5 Cf =45500 Cr =75562.5 K =53000 D =6000
w (rad/s)
Mag (dB)
10
1
100
50
0
50
100
w (rad/s)
Phase (deg)
Measured
Model 2
Model 3
Model 4
Freq. Domain Fit
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
17
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Model Fitting Results
Inconsistency in roll rate measured response does not appear at
lower speeds
Better sensors are required to clarify inconsistencies in data
–
especially lateral acceleration and roll rate
10
0
10
1
10
0
10
Frequency Response, Steering Input to Roll Rate
U =8.9 Cf =45500 Cr =75560 K =53000 D =6000
w (rad/s)
Mag (dB)
10
0
10
1
500
450
400
350
300
250
w (rad/s)
Phase (deg)
measured
Model 2
Model 3
Model 4
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
18
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Remarks on Model Validation
As a result of overall accuracy and simplicity, Model 3
was chosen for further investigation. This entails:
The development of model

based predictive algorithms for
rollover propensity
The development of control algorithms for rollover
mitigation
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
19
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Conclusions
A relatively simple dynamic model is capable of
modeling both the planar and roll dynamics of a
vehicle well under constant speed conditions.
Relatively accurate measurements may be taken with
inexpensive sensors
The dynamics are seen even with commercial grade sensors
Important for industry because such sensors are typically
found in production vehicles
Extra care should be taken when model fitting in the
time domain
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
20
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Time Response Tests
Pseudo

Step Response, 8.9 m/s, 0.09 rad amplitude, FR Params
1
1.5
2
2.5
0.02
0.04
0.06
0.08
0.1
Step, Steering vs. Time
Time (s)
Steering Angle (rad)
1
1.5
2
2.5
0
0.05
0.1
0.15
0.2
0.25
0.3
Yaw Rate vs. Time
Time(s)
Yaw Rate (rad/s)
Measured
Model 1
Model 2
Model 3
Model 4
1
1.5
2
2.5
0
0.2
0.4
0.6
0.8
1
1.2
Lat. Accel. vs. Time
Time (s)
Lat. Accel. (m/s
2
)
2.5
3
3.5
4
0
0.02
0.04
0.06
0.08
0.1
Steering vs. Time
Time (s)
Angle (rad)
2.5
3
3.5
4
0
0.05
0.1
Roll Rate vs. Time
Time (s)
Roll Rate (rad/s)
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
21
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Time Response Tests
Pseudo

Step Response, 8.9 m/s, 0.09 rad amplitude, TR Params
1
1.5
2
2.5
0
0.02
0.04
0.06
0.08
0.1
Step, Steering vs. Time
Time (s)
Steering Angle (rad)
1
1.5
2
2.5
0
0.05
0.1
0.15
0.2
0.25
0.3
Yaw Rate vs. Time
Time(s)
Yaw Rate (rad/s)
Measured
Model 1
Model 2
Model 3
Model 4
1
1.5
2
2.5
0
0.5
1
Lat. Accel. vs. Time
Time (s)
Lat. Accel. (m/s
2
)
2.5
3
3.5
4
0
0.02
0.04
0.06
0.08
0.1
Steering vs. Time
Time (s)
Angle (rad)
2.5
3
3.5
4
0.02
0
0.02
0.04
0.06
0.08
0.1
Roll Rate vs. Time
Time (s)
Roll Rate (rad/s)
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
22
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Time Response Tests
Lane Change Maneuver, 17.8 m/s, Right

to

Left, then Left

to

Right, FR
0
2
4
6
8
0.04
0.02
0
0.02
0.04
Lane Change, Steering Angle vs. Time
Time (s)
Angle (rad)
0
2
4
6
8
0.15
0.1
0.05
0
0.05
0.1
0.15
Yaw Rate vs. Time
Time(s)
Yaw Rate (rad/s)
2
4
6
8
0.5
0
0.5
Lat. Accel. vs. Time
Time (s)
Lat. Accel. (m/s
2
)
Measured
Model 1
Model 2
Model 3
Model 4
0
2
4
6
8
0.04
0.02
0
0.02
0.04
Steering Angle vs. Time
Time (s)
Angle (rad)
0
2
4
6
8
0.1
0.05
0
0.05
0.1
Roll Rate vs. Time
Time (s)
Roll Rate (rad/s)
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
23
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Time Response Tests
Lane Change Maneuver, 17.8 m/s, Right

to

Left, then Left

to

Right, Time
0
2
4
6
8
0.04
0.02
0
0.02
0.04
Lane Change, Steering Angle vs. Time
Time (s)
Angle (rad)
0
2
4
6
8
0.15
0.1
0.05
0
0.05
0.1
0.15
Yaw Rate vs. Time
Time(s)
Yaw Rate (rad/s)
0
2
4
6
8
0.5
0
0.5
Lat. Accel. vs. Time
Time (s)
Lat. Accel. (m/s
2
)
Measured
Model 1
Model 2
Model 3
Model 4
0
2
4
6
8
0.04
0.02
0
0.02
0.04
Steering Angle vs. Time
Time (s)
Angle (rad)
0
2
4
6
8
0.1
0.05
0
0.05
0.1
Roll Rate vs. Time
Time (s)
Roll Rate (rad/s)
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
24
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Experiments Performed
Determination of Understeer Gradient
Understeer gradient is a constant indicating the additional amount
of steering necessary to maintain a steady

state turn per g of
lateral acceleration (e.g. units are rad/g)
Provides a relationship between the front and rear cornering
stiffness‘
Lateral acceleration was measured on a 30.5 m radius circle at 6.7,
8.9, and 11.2 m/s
r
f
f
r
us
C
W
C
W
K
2
2
r
f
f
r
us
C
W
C
W
K
2
2
f
us
r
f
f
r
C
K
W
C
W
C
2
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
25
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Model Fitting Procedure
Step 1
–
Determine understeer gradient
Plotting additional steering angle vs. lateral acceleration, the
understeer gradient is simply the slope of the line
y = 0.045x + 0.018
R
2
= 0.9965
0.024
0.026
0.028
0.03
0.032
0.034
0.036
0.125
0.175
0.225
0.275
0.325
0.375
0.425
Lat. Accel (g's)
Additional Angle (rad)
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
26
/23
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Analytical Vehicle Models
Paper
Model Order
Method of validation
Who are they with
Williams, 1995, Nonlinear control of roll moment distribution…
NL 2DOF
No roll dynamics included, only a "roll moment factor"
Georgia Institute of Technology
Rosam, 1997, Development and simulation of a novel roll…
?
No model or Free Body Diagram Given
University of Bath
Darling, 1998, An Experimental Study of a Prototype…
?
No model or Free Body Diagram Given
University of Bath
Feng, 1998, Automatic Steering Control of Vehicle Lateral...
2 & 3DOF
Errors in published formulation
PATH
Feng, 2000, Decoupling Steering Control For Vehicles…
2 & 3DOF
Errors in published formulation
PATH
Krishnaswami, 1998, A Regularization Approach To Robust…
2DOF
Not enough information given
UMTRI
Wielenga, 1999, A Method for Reducing On Road Rollover…
3DOF
Model formulation not given
Dynomotive
Chen, 1999, A Real Time Rollover ThreatIndex For SUV's
coupled 2DOF
Decoupled approach
UMTRI
Chen, 2001, Differential Braking Based Rollover Prevention…
3DOF
Parameters difficult to obtain
UMTRI
Kitajima, 2000, Control For Integrated Side Slip Roll
8DOF*, 3DOF
Equations complex, not enough information given
UMTRI
Eger, 2003, Modeling of rollover sequences
2DOF
Covers tripped rollovers
University of Karlsruhe, Germany
Kueperkoch, 2003, Novel Stability Control Using SBW…
3DOF
Not relevant to our study
Bosch Corporation
Rossetter, 2003, A Gentle Nudge Towards Safety…
2DOF
Not relevant to our study
Stanford
Takano, 2003, Study on a vehicle dynamics model for…
3DOF
Errors in published information
Tokyo University of Ag. and Tech.
Oh, 2004, The Design of a Controller for the SBW System
9DOF
Model formulation not given
Hyundai/Hanyang University
Paper
Model Order
Comments
Who are they with
Sharp, 1993, On the design of an active control system for a…
3DOF
Complex formulation, parameters are difficult to obtain
Cranfield Institute of Technology
Chen, C, 1998, Steering Control of HighSpeed Vehicles
2DOF
Not relevant to our study
PATH
Mammar, 1999, Speed Scheduled Vehicle Lateral Control
3DOF
Nicely derived, but no experimental validation. Includes a
Evry University, France
mathematical proof on its model matching abilities.
Cole, 2000, Evaluation Of Design Alternatives For Roll Control…
3DOF
Model is developed through a software package
University of Nottingham
Hyun, 2000, Vehicle Modeling And Prediction Of…
NL 8DOF
Not relevant to our study
Texas A&M
Ikenaga, 2000, Active Suspension Control Of Ground…
7DOF
No description of lateral dynamics
Texas Arlington
Manning, 2000, Coordination Of Chassis Control Systems
NL 5DOF
Not enough information given
University of Leeds, UK
Kim, 2003, Investigation Of Robust Roll Motion Control…
3DOF
Clean presentation, parameters given, model worked
Samchok University, South Korea
Sprague, 2002, Automated stability analysis of a vehicle…
6DOF
Model formulation not given
Exponent Failure Analysis Associates
Huh, 2002, Monitoring System Design For Estimating...
4DOF
No roll dynamics included, only lateral weight transfer
Samchok University, South Korea
Carlson, 2003, Optimal rollover prevention with SBW and diff…
NL4DOF, L3DOF
All work done in simulation
Stanford
Models With Experimental Validation
Models Not Experimentally Validated
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