Transcript Document 7626058

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
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
Outline
1.
2.
3.
4.
Goals
Analytical Vehicle Models
Experimental Model Validation
Conclusions
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
2/23
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
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
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
3/23
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
Analytical Vehicle Models
 Standard SAE sign convention
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
4/23
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
cos   1
sin    
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
Ftire  Ctire
• Longitudinal forces ignored
• Tire forces symmetric right-to-left
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
5/23
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
Analytical Vehicle Models
 Model 1 – 2DOF Bicycle Model
Mq  Dq  Kq  Fu f
0
 m
 0 I
zz

 0
0
0 V  0  mU
 
0  r   0
0
0  0
0
 Cf
 F f   U
F    C
 r   r
 U
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
 y
q   
  
0 V  0 0 0  y   2
0  r   0 0 0    2l f
0   0 0 0    0

lf Cf
U
l r Cr
U
6/23
2 
F f 

 2l r   
Fr 

0 

0 V  C 
f
 
  r     f
0
0    

Dept. Of Mechanical and Nuclear Engineering,
Penn State University
0
 m
 0 I
zz

 0
0
0 V  0  mU
 
0  r   0
0
0  0
0
0 V  0 0 0  y   2
0  r   0 0 0    2l f
0   0 0 0     0
2 
F 
 2l r   f 
F
0   r 
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
0
 m
 0 I
zz

 0
0
0
 m
 0 I
zz

ms h I xz
0 V  0  mU
 
0  r   0
0
0  0
0
 ms h  V  0  mU
  
 I xz   r   0
0
I xx   0 ms hU
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
0 V  0 0 0  y   2
0  r   0 0 0    2l f
0   0 0 0     0
2 
F 
 2l r   f 
F
0   r 
 y   2
0  V  0 0
0


  
0   r   0 0
0
    2l f
D    0 0 K  ms gh      0
7/23
2 
F 
 2lr   f 
F
0   r 
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
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
 m
 0 I
zz

 0
0
0
 m
 0
 I zz

 ms h 0
0 V  0  mU
 
0  r   0
0
0  0
0
ms h  V  0 mU
  
0   r   0
0
I xx    0 ms hU
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
0 V  0 0 0  y   2
0  r   0 0 0    2l f
0   0 0 0     0
2 
F 
 2l r   f 
F
0   r 
 y  2
0  V  0 0
0


  
0   r   0 0
0
     2l f
D     0 0 K  ms gh      0
8/23
2 
F 
2lr   f 
F
0   r 
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
Analytical Vehicle Models
 Model 3 (continued)
 As a result of the assumption of roll steer, the external forces acting
on the vehicle change accordingly
 Cf
 F f   U
F    C
 r   r
 U
 Cf

 F f ,3   U
F   
 r ,3    C r
 U

Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover

lf Cf
U
l r Cr
U

lfCf
U
lr Cr
U
9/23

0 V  C 
f
 
  r     f
0
0    

 *f 
 V  C 
   r   f 
f
 r*     0 
 

 
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
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
0
 m
 0 I
zz

 0
0
0
 m
 0 I
zz

 0
0
0 V  0  mU
 
0  r   0
0
0  0
0
 mh V  0  mU
  
0   r   0
0
I xx   0
0
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
0 V  0 0 0  y   2
0  r   0 0 0    2l f
0   0 0 0     0
2 
F 
 2l r   f 
F
0   r 
 y   2
0  V  0 0
0


  
0   r   0 0
0
    2l f
D    0 0 K   mgh    2h
10/23
2 
F f 

 2l r   
F
2h   r 
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
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
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
11/23
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
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
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
12/23
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
Time Domain Fit Results
Frequency Response, Steering Input to Yaw Rate, Mercury Tracer
U =16.5 Cf =-22750 Cr =-19958.4561 K =38000 D =5000
Frequency Response, Steering Input to Lateral Acceleration, Mercury Tracer
U =16.5 Cf =-22750 Cr =-19958.4561 K =38000 D =5000
40
15
Mag (dB)
Mag (dB)
35
30
25
10
5
20
15
0
1
1
10
10
w (rad/s)
w (rad/s)
0
100
50
0
Phase (deg)
Phase (deg)
150
Measured
Model 1
Model 2
Model 3
Model 4
-50
-100
Measured
Model 1
Model 2
Model 3
Model 4
1
1
10
10
w (rad/s)
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
w (rad/s)
13/23
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
Model Fitting Results
 Results for Steering Input to Lateral Acceleration
Frequency Response, Steering Input to Lateral Acceleration, Mercury Tracer
U =16.5 Cf =-45500 Cr =-75562.5 K =53000 D =6000
40
Mag (dB)
35
30
25
20
15
1
10
w (rad/s)
Phase (deg)
150
100
50
0
Measured
Model 1
Model 2
Model 3
Model 4
1
10
w (rad/s)
 Freq. Domain Fit
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
14/23
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
Model Fitting Results
 Results for Steering Input to Yaw Rate
Frequency Response, Steering Input to Yaw Rate, Mercury Tracer
U =16.5 Cf =-45500 Cr =-75562.5 K =53000 D =6000
Mag (dB)
15
10
5
0
1
10
w (rad/s)
Phase (deg)
0
-50
-100
Measured
Model 1
Model 2
Model 3
Model 4
1
10
w (rad/s)
 Freq. Domain Fit
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
15/23
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
Model Fitting Results
 Results for Steering Input to Roll Rate
Frequency Response, Steering Input to Roll Rate, Mercury Tracer
U =16.5 Cf =-45500 Cr =-75562.5 K =53000 D =6000
15
Mag (dB)
10
5
0
-5
1
10
w (rad/s)
Phase (deg)
100
50
0
-50
Measured
Model 2
Model 3
Model 4
-100
1
10
w (rad/s)
 Freq. Domain Fit
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
16/23
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
Model Fitting Results
 Inconsistency in roll rate measured response does not appear at
lower speeds
Frequency Response, Steering Input to Roll Rate
U =8.9 Cf =-45500 Cr =-75560 K =53000 D =6000
Mag (dB)
10
0
-10
0
1
10
10
w (rad/s)
-250
Phase (deg)
-300
-350
measured
Model 2
Model 3
Model 4
-400
-450
-500
0
1
10
10
w (rad/s)
 Better sensors are required to clarify inconsistencies in data –
especially lateral acceleration and roll rate
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
17/23
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
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
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
18/23
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
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
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
19/23
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
Time Response Tests
 Pseudo-Step Response, 8.9 m/s, 0.09 rad amplitude, FR Params
Step, Steering vs. Time
Yaw Rate vs. Time
Lat. Accel. vs. Time
0.3
1.2
0.08
0.06
0.04
0.2
0.15
Measured
Model 1
Model 2
Model 3
Model 4
0.1
0.05
0.02
0
1
1.5
Time (s)
2
Lat. Accel. (m/s 2)
0.25
Yaw Rate (rad/s)
Steering Angle (rad)
0.1
2.5
1
1.5
Time(s)
2
Steering vs. Time
1
0.8
0.6
0.4
0.2
0
2.5
1
1.5
Time (s)
2
2.5
Roll Rate vs. Time
0.1
0.1
Roll Rate (rad/s)
Angle (rad)
0.08
0.06
0.04
0.02
0.05
0
0
2.5
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
3
Time (s)
3.5
4
2.5
20/23
3
Time (s)
3.5
4
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
Time Response Tests
 Pseudo-Step Response, 8.9 m/s, 0.09 rad amplitude, TR Params
Step, Steering vs. Time
Yaw Rate vs. Time
Lat. Accel. vs. Time
0.3
1
0.25
0.06
0.04
0.02
0.2
0.15
Measured
Model 1
Model 2
Model 3
Model 4
0.1
0.05
0
0
1
1.5
Time (s)
2
2.5
Lat. Accel. (m/s 2)
0.08
Yaw Rate (rad/s)
Steering Angle (rad)
0.1
1
1.5
Time(s)
2
Steering vs. Time
0.5
0
2.5
1
1.5
Time (s)
2
2.5
Roll Rate vs. Time
0.1
0.1
0.08
Roll Rate (rad/s)
Angle (rad)
0.08
0.06
0.04
0.02
0.06
0.04
0.02
0
-0.02
0
2.5
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
3
3.5
Time (s)
4
21/23
2.5
3
3.5
Time (s)
4
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
Time Response Tests
 Lane Change Maneuver, 17.8 m/s, Right-to-Left, then Left-to-Right, FR
Lane Change, Steering Angle vs. Time
Yaw Rate vs. Time
Lat. Accel. vs. Time
0.04
0.15
0
-0.02
0.1
Lat. Accel. (m/s 2)
Yaw Rate (rad/s)
Angle (rad)
0.02
0.5
0.05
0
-0.05
-0.1
-0.15
-0.04
0
2
4
6
Time (s)
8
0
2
4
6
Time(s)
8
-0.5
Measured
Model 1
Model 2
Model 3
Model 4
2
4
6
Time (s)
8
Roll Rate vs. Time
0.04
0.1
0.02
0.05
Roll Rate (rad/s)
Angle (rad)
Steering Angle vs. Time
0
0
-0.02
-0.04
0
-0.05
-0.1
0
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
2
4
6
Time (s)
8
0
22/23
2
4
6
Time (s)
8
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
Time Response Tests
 Lane Change Maneuver, 17.8 m/s, Right-to-Left, then Left-to-Right, Time
Lane Change, Steering Angle vs. Time
Yaw Rate vs. Time
Lat. Accel. vs. Time
0.04
0.15
0
-0.02
Lat. Accel. (m/s 2)
Yaw Rate (rad/s)
0.02
Angle (rad)
0.5
0.1
0.05
0
-0.05
-0.1
-0.15
-0.04
0
2
4
6
Time (s)
8
0
2
4
6
Time(s)
8
-0.5
Measured
Model 1
Model0 2
Model 3
Model 4
2
4
6
Time (s)
8
Roll Rate vs. Time
0.04
0.1
0.02
0.05
Roll Rate (rad/s)
Angle (rad)
Steering Angle vs. Time
0
0
-0.02
-0.04
0
-0.05
-0.1
0
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
2
4
6
Time (s)
8
0
23/23
2
4
6
Time (s)
8
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
K us 
Wf
Wr

2  C f 2  Cr
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‘
Wf
Wr
K us 

2  C f 2  Cr
Cr 
Wf C f
Wr  2  K us  C f
 Lateral acceleration was measured on a 30.5 m radius circle at 6.7,
8.9, and 11.2 m/s
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
24/23
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
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
0.036
y = 0.045x + 0.018
2
R = 0.9965
Additional Angle (rad)
0.034
0.032
0.03
0.028
0.026
0.024
0.125
0.175
0.225
0.275
0.325
0.375
0.425
Lat. Accel (g's)
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
25/23
Dept. Of Mechanical and Nuclear Engineering,
Penn State University
Analytical Vehicle Models
Paper
Williams, 1995, Nonlinear control of roll moment distribution…
Rosam, 1997, Development and simulation of a novel roll…
Darling, 1998, An Experimental Study of a Prototype…
Feng, 1998, Automatic Steering Control of Vehicle Lateral...
Feng, 2000, Decoupling Steering Control For Vehicles…
Krishnaswami, 1998, A Regularization Approach To Robust…
Wielenga, 1999, A Method for Reducing On Road Rollover…
Chen, 1999, A Real Time Rollover ThreatIndex For SUV's
Chen, 2001, Differential Braking Based Rollover Prevention…
Kitajima, 2000, Control For Integrated Side Slip Roll
Eger, 2003, Modeling of rollover sequences
Kueperkoch, 2003, Novel Stability Control Using SBW…
Rossetter, 2003, A Gentle Nudge Towards Safety…
Takano, 2003, Study on a vehicle dynamics model for…
Oh, 2004, The Design of a Controller for the SBW System
Models With Experimental Validation
Model Order
Method of validation
NL 2DOF
No roll dynamics included, only a "roll moment factor"
?
No model or Free Body Diagram Given
?
No model or Free Body Diagram Given
2 & 3DOF
Errors in published formulation
2 & 3DOF
2DOF
3DOF
coupled 2DOF
3DOF
8DOF*, 3DOF
2DOF
3DOF
2DOF
3DOF
9DOF
Errors in published formulation
Not enough information given
Model formulation not given
Decoupled approach
Parameters difficult to obtain
Equations complex, not enough information given
Covers tripped rollovers
Not relevant to our study
Not relevant to our study
Errors in published information
Model formulation not given
Models Not Experimentally Validated
Model Order
Comments
3DOF
Complex formulation, parameters are difficult to obtain
2DOF
Not relevant to our study
3DOF
Nicely derived, but no experimental validation. Includes a
mathematical proof on its model matching abilities.
Cole, 2000, Evaluation Of Design Alternatives For Roll Control…
3DOF
Model is developed through a software package
Hyun, 2000, Vehicle Modeling And Prediction Of…
NL 8DOF
Not relevant to our study
Ikenaga, 2000, Active Suspension Control Of Ground…
7DOF
No description of lateral dynamics
Manning, 2000, Coordination Of Chassis Control Systems
NL 5DOF
Not enough information given
Kim, 2003, Investigation Of Robust Roll Motion Control…
3DOF
Clean presentation, parameters given, model worked
Sprague, 2002, Automated stability analysis of a vehicle…
6DOF
Model formulation not given
Huh, 2002, Monitoring System Design For Estimating...
4DOF
No roll dynamics included, only lateral weight transfer
Carlson, 2003, Optimal rollover prevention with SBW and diff… NL4DOF, L3DOFAll work done in simulation
Paper
Sharp, 1993, On the design of an active control system for a…
Chen, C, 1998, Steering Control of High-Speed Vehicles
Mammar, 1999, Speed Scheduled Vehicle Lateral Control
Vehicle Dynamic Modeling for the Prediction
and Prevention of Vehicle Rollover
26/23
Who are they with
Georgia Institute of Technology
University of Bath
University of Bath
PATH
PATH
UMTRI
Dynomotive
UMTRI
UMTRI
UMTRI
University of Karlsruhe, Germany
Bosch Corporation
Stanford
Tokyo University of Ag. and Tech.
Hyundai/Hanyang University
Who are they with
Cranfield Institute of Technology
PATH
Evry University, France
University of Nottingham
Texas A&M
Texas Arlington
University of Leeds, UK
Samchok University, South Korea
Exponent Failure Analysis Associates
Samchok University, South Korea
Stanford
Dept. Of Mechanical and Nuclear Engineering,
Penn State University