dept. of mechanical and nuclear engineering, penn state university vehicle dynamic modeling for the...

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

A Comparative, Experimental Study of Model Suitability to Describe Vehicle Rollover Dynamics for Control Design

John T. CameronPennsylvania State University

Dr. Sean BrennanPennsylvania State University


2/23 Vehicle Dynamic Modeling for the Prediction and Prevention of Vehicle Rollover

OutlineOutline

1. Goals2. Analytical Vehicle Models3. Experimental Model Validation4. Conclusions



GoalsGoals

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



Analytical Vehicle ModelsAnalytical Vehicle Models

Standard SAE sign convention




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

1cos

sin

1cos

tiretire CF




Model 1 – 2DOF Bicycle Model

fFuKqqDqM

y

q

ff

rrr

fff

r

f Cr

V

U

Cl

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CU

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U

C

F

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0

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frfzz F

Fll

y

r

VmU

r

V

I

m

00

22

22

000

000

000

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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

frf

ssxxxzs

xzzz

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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

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Model 3 (continued) As a result of the assumption of roll steer, the external

forces acting on the vehicle change accordingly

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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

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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



Model Fitting ProceduresModel 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



Time Domain Fit ResultsTime Domain Fit Results

101

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

)

101

-100

-50

0

w (rad/s)

Pha

se (

deg)

MeasuredModel 1Model 2Model 3Model 4

101

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

)

101

0

50

100

150

w (rad/s)

Pha

se (

deg)




Model Fitting ResultsModel Fitting Results

Results for Steering Input to Lateral Acceleration

101

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

)

101

0

50

100

150

w (rad/s)

Pha

se (

deg)


Freq. Domain Fit




Results for Steering Input to Yaw Rate

101

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

)

101

-100

-50

0

w (rad/s)

Pha

se (

deg)


Freq. Domain Fit




Results for Steering Input to Roll Rate

101

-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

)

101

-100

-50

0

50

100

w (rad/s)

Pha

se (

deg)

MeasuredModel 2Model 3Model 4

Freq. Domain Fit




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

100

101

-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

)

100

101

-500

-450

-400

-350

-300

-250

w (rad/s)

Pha

se (

deg)

measuredModel 2Model 3Model 4



Remarks on Model ValidationRemarks 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



ConclusionsConclusions

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



Time Response TestsTime 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)

Ste

erin

g A

ngle

(ra

d)

1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

Yaw Rate vs. Time

Time(s)

Yaw

Rat

e (r

ad/s

)MeasuredModel 1Model 2Model 3Model 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.

Acc

el.

(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)

Ang

le (

rad)

2.5 3 3.5 4

0

0.05

0.1

Roll Rate vs. Time

Time (s)

Rol

l Rat

e (r

ad/s

)




Pseudo-Step Response, 8.9 m/s, 0.09 rad amplitude, TR Params

1 1.5 2 2.50

0.02

0.04

0.06

0.08

0.1Step, Steering vs. Time

Time (s)

Ste

erin

g A

ngle

(rad

)

1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

0.3

Yaw Rate vs. Time

Time(s)

Yaw

Rat

e (ra

d/s)


1 1.5 2 2.5

0

0.5

1


Time (s)

Lat.

Acc

el. (

m/s

2 )

2.5 3 3.5 40

0.02

0.04

0.06

0.08

0.1

Steering vs. Time

Time (s)

Ang

le (

rad)

2.5 3 3.5 4

-0.02

0

0.02

0.04

0.06

0.08

0.1Roll Rate vs. Time

Time (s)

Rol

l Rat

e (r

ad/s

)




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

0.04Lane Change, Steering Angle vs. Time

Time (s)

Ang

le (

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

Rat

e (r

ad/s

)

2 4 6 8

-0.5

0

0.5


Time (s)

Lat.

Acc

el.

(m/s

2 )


0 2 4 6 8

-0.04

-0.02

0

0.02

0.04

Steering Angle vs. Time

Time (s)

Ang

le (

rad)

0 2 4 6 8

-0.1

-0.05

0

0.05

0.1

Roll Rate vs. Time

Time (s)

Rol

l Rat

e (r

ad/s

)




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

0.04Lane Change, Steering Angle vs. Time

Time (s)

Ang

le (r

ad)

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

Rat

e (ra

d/s)

0 2 4 6 8

-0.5

0

0.5


Time (s)

Lat.

Acc

el. (

m/s

2 )


0 2 4 6 8

-0.04

-0.02

0

0.02

0.04

Steering Angle vs. Time

Time (s)

Ang

le (

rad)

0 2 4 6 8

-0.1

-0.05

0

0.05

0.1

Roll Rate vs. Time

Time (s)

Rol

l Rat

e (r

ad/s

)



Experiments PerformedExperiments 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

rus C

W

C

WK

22

r

f

f

rus C

W

C

WK

22 fusr

ffr CKW

CWC

2



Model Fitting ProcedureModel 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

R2 = 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)

Ad

dit

ion

al A

ng

le (

rad

)




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 TechnologyRosam, 1997, Development and simulation of a novel roll… ? No model or Free Body Diagram Given University of BathDarling, 1998, An Experimental Study of a Prototype… ? No model or Free Body Diagram Given University of BathFeng, 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 PATHKrishnaswami, 1998, A Regularization Approach To Robust… 2DOF Not enough information given UMTRIWielenga, 1999, A Method for Reducing On Road Rollover… 3DOF Model formulation not given DynomotiveChen, 1999, A Real Time Rollover ThreatIndex For SUV's coupled 2DOF Decoupled approach UMTRIChen, 2001, Differential Braking Based Rollover Prevention… 3DOF Parameters difficult to obtain UMTRIKitajima, 2000, Control For Integrated Side Slip Roll 8DOF*, 3DOF Equations complex, not enough information given UMTRIEger, 2003, Modeling of rollover sequences 2DOF Covers tripped rollovers University of Karlsruhe, GermanyKueperkoch, 2003, Novel Stability Control Using SBW… 3DOF Not relevant to our study Bosch CorporationRossetter, 2003, A Gentle Nudge Towards Safety… 2DOF Not relevant to our study StanfordTakano, 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 TechnologyChen, C, 1998, Steering Control of High-Speed Vehicles 2DOF Not relevant to our study PATHMammar, 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 NottinghamHyun, 2000, Vehicle Modeling And Prediction Of… NL 8DOF Not relevant to our study Texas A&MIkenaga, 2000, Active Suspension Control Of Ground… 7DOF No description of lateral dynamics Texas ArlingtonManning, 2000, Coordination Of Chassis Control Systems NL 5DOF Not enough information given University of Leeds, UKKim, 2003, Investigation Of Robust Roll Motion Control… 3DOF Clean presentation, parameters given, model worked Samchok University, South KoreaSprague, 2002, Automated stability analysis of a vehicle… 6DOF Model formulation not given Exponent Failure Analysis AssociatesHuh, 2002, Monitoring System Design For Estimating... 4DOF No roll dynamics included, only lateral weight transfer Samchok University, South KoreaCarlson, 2003, Optimal rollover prevention with SBW and diff… NL4DOF, L3DOFAll work done in simulation Stanford

Models With Experimental Validation

Models Not Experimentally Validated

dept. of mechanical and nuclear engineering, penn state university vehicle dynamic modeling for the...

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