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Dipl.-Ing. Alfred PrucknerDipl.-Ing. Sven Fischer

15th ADAMS USER CONFERENCEROM, 15. - 16. November 2000

INSTITUT FÜRKRAFTFAHRWESENAACHEN

Vehicle DynamicsControl for a4 Wheel SteeringPrototype Car

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- Control Strategy

- Development Method

- Side Slip Angle Observation

- Side Slip Angle Control

- Control Strategy

- Outlook

Overview

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4 Wheel Steering Control Strategy

β

Vehicle Side Slip Angle ββββ

Steady State Cornering

-4

-3

-2

-1

0

1

2

3

0 1 2 3 4 5 6 7 8 9lateral acceleration [m/s²]

v

ehic

le s

ide

slip

ang

le [

° ]

Steady State Cornering

-4

-3

-2

-1

0

1

2

3

0 1 2 3 4 5 6 7 8 9lateral acceleration [m/s²]

v

ehic

le s

ide

slip

ang

le [

° ]

dry road

icy road

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Smaller Wheel Base at low velocity

4 Wheel Steering Control Strategy

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Larger Wheel Base at high velocity

4 Wheel Steering Control Strategy

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4 Wheel Steering Control Strategy

β

Minimize the vehicle side slip angle β

Side Slip Angleestimation

Side Slip Anglecontrol

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- Control Strategy

- Development Method

- Side Slip Angle Observation

- Side Slip Angle Control

Overview

- Outlook

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Virtual Prototyping

Measurement

Control information

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MATRIXxControl algorithm

MATRIXxControl algorithm

ADAMSFull Vehicle Model

ADAMSFull Vehicle Model

C-CodeControl algorithm

C-CodeControl algorithm

FORTRAN-filesVARSUBTIRSUBTIRSUB

Subroutine Template

Interface ADAMS und MATRIXx

discrete time interval

Virtual Prototyping

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- Control Strategy

- Development Method

- Side Slip Angle Observation

- Side Slip Angle Control

Overview

- Outlook

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Side Slip Angle Observation

- linear tire characteristics- small angles- centre of gravity on the street

FF

FR

lR

lF

ay

ψψψψ.

δv

v ψBM.

βBM�

���

�

δδ

�����

����

�

�

−+���

���

�

ψβ

�����

����

�

�

−−−

−+−

+−

=���

���

�

ψβ

h

v

z

shh

z

svv

shsv

z

sh2hsv

2v

z

svvshh

2shhsvvshsv

*

Jc*l

Jc*l

v*mc

v*mc

*

v*Jc*lc*l

Jc*lc*l

v*mc*lc*l

1v*mcc

���

�

Linear Bicycle Model

uBxAx ⋅+⋅=�

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Side Slip Angle Observation

A

B

u y

x���� x

Linear Bicycle Model

uBxAx ⋅+⋅=�

steering angleyaw velocity

lateral acceleration

yaw velocityside slip angle

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Side Slip AngleObservation

Dry Road

ADAMS

Bicycle Model

Double Lane Change ManeuvreDry Road

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

4 5 6 7 8 9 10 11 12time [s]

ve

hicl

e si

de s

lip a

ngle

[ ° ]

ADAMS

Bicycle Model

Double Lane Change ManeuvreDry Road

-40

-30

-20

-10

0

10

20

30

4 5 6 7 8 9 10 11 12time [s]

yaw

vel

ocity

[°/s

]

ADAMS

Bicycle Model

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Side Slip AngleObservation

Icy Road

ADAMS

Bicycle Model

Double Lane Change ManeuvreIcy Road

-40

-30

-20

-10

0

10

20

4 5 6 7 8 9 10 11 12time [s]

v

ehic

le s

ide

slip

ang

le [

° ]

ADAMS

Bicycle Model

Double Lane Change ManeuvreIcy Road

-80

-60

-40

-20

0

20

40

60

80

4 5 6 7 8 9 10 11 12time [s]

yaw

vel

ocity

[°/s

]

ADAMS

Bicycle Model

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A

B

u y

x���� x

�x Ax Bu= +

Linear Observer

Cy +

-

L

( )yy LuBxAx −+⋅+⋅=�

( )[ ] yy LuBxAuBxAxxx~ −+⋅+⋅−⋅+⋅=−= ���

( ) x~ LCAx~ −=� ( )( ) ( )∏ −=−−=

n

1iiλ1 LCA I det

Side Slip Angle Observation

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Non-Linear Observeru y

C

L

x(((( )))) (((( ))))(((( ))))yyu,xLu,xfx −−−−−−−−====����

+-y

( ) ( ) ( ) ( )xxu,xxfu,xfxu,xfx −⋅

∂∂+== ��

( ) ( ) ( ) ( )yyu,xLxxu,xxfxxx~ −⋅−−⋅

∂∂=−= ��

�

( ) ����

�

λλ

=����

���

�

�

ψ∂∂

β∂∂⋅��

����

�−

�����

����

�

�

ψ∂ψ∂

β∂ψ∂

ψ∂β∂

β∂β∂

=2

1yy

2221

1211

00aa

10

llll

u,xF�

�

����

�

��

x

Side Slip Angle Observation

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Side Slip AngleObservation

Icy Road

ADAMS

Bicycle Model

LuenbergerObserver

Double Lane Change ManeuvreIcy Road

-40

-30

-20

-10

0

10

20

4 5 6 7 8 9 10 11 12time [s]

v

ehic

le s

ide

slip

ang

le [

° ]

ADAMS

Bicycle Model

Luenberger Observer

Double Lane Change ManeuvreIcy Road

-80

-60

-40

-20

0

20

40

60

80

4 5 6 7 8 9 10 11 12time [s]

yaw

vel

ocity

[°/s

]

ADAMSBicycle ModelLuenberger Observer

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Side Slip AngleObservation

Dry RoadWet Road

ADAMS

Bicycle Model

LuenbergerObserver

Double Lane Change ManeuvreDry Road

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

4 5 6 7 8 9 10 11 12time [s]

v

ehic

le s

ide

slip

ang

le [

° ]

ADAMS

Bicycle ModelLuenberger Observer

Double Lane Change Maneuvre Wet Road

-15

-10

-5

0

5

10

15

4 5 6 7 8 9 10 11 12time [s]

v

ehic

le s

ide

slip

ang

le [

° ]

ADAMS

Bicycle ModelLuenberger Observer

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Side Slip AngleObservation

Dry RoadIcy Road

ADAMS

Bicycle Model

LuenbergerObserver

Steady State Cornering Dry Road

-4

-3

-2

-1

0

1

2

3

0 1 2 3 4 5 6 7 8 9lateral acceleration [m/s²]

v

ehic

le s

ide

slip

ang

le [

° ]

ADAMS

Bicycle Model

Luenberger Observer

Steady State Cornering Dry Road

-4

-3

-2

-1

0

1

2

3

0 1 2 3 4 5 6 7 8 9lateral acceleration [m/s²]

v

ehic

le s

ide

slip

ang

le [

° ]

ADAMS

Bicycle ModelLuenberger Observer

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- Control Strategy

- Development Method

- Side Slip Angle Observation

- Side Slip Angle Control

Overview

- Outlook

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Side Slip Angle Control

Bicycle Model Controller

A

B

u y

x���� x

uBxAx ⋅+⋅=� kββββu

1

Dstat_

V

H

Ti1Ti1kk

⋅⋅⋅⋅ωωωω⋅⋅⋅⋅++++⋅⋅⋅⋅ωωωω⋅⋅⋅⋅++++⋅⋅⋅⋅====

δδδδδδδδ==== ββββββββ

Static Transfer Function

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0 1 2 3 4 5 6 7 8 9

lateral acceleration [m/s²]

Stee

ring

Rat

io

r /

f

smaller wheel base at low velocity

larger wheel base

at high velocity

characteristic velocity

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Side Slip Angle Control

State Space Controlleru y

C

L

x(((( )))) (((( ))))(((( ))))yyu,xLu,xfx −−−−−−−−====����

+-y

( )u,xfx =�

x

K+

-

xwanted

uBxAuufx

xfx ∆⋅+∆⋅=∆⋅��

�

�

∂∂+∆⋅���

�

�

∂∂=∆�

Ricatti Equation

(((( )))) (((( )))) (((( )))) (((( )))) QkAkPkA1kP 0T* ++++⋅⋅⋅⋅⋅⋅⋅⋅====++++

(((( )))) (((( )))) (((( )))) (((( )))) (((( )))) (((( )))){{{{ }}}} 1*T*T RkB1kPkBkB1kP1kK −−−−++++⋅⋅⋅⋅++++⋅⋅⋅⋅⋅⋅⋅⋅++++====++++

(((( )))) (((( )))) (((( )))) (((( )))) (((( ))))1kPkB1kK1kP1kP *TT*0 ++++⋅⋅⋅⋅⋅⋅⋅⋅++++−−−−++++====++++

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Side Slip AngleControl

Dry Road

Without 4WS

Bicycle ModelController

State SpaceController

Steady State CorneringDry Road

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

0 1 2 3 4 5 6 7 8lateral acceleration [m/s²]

side

slip

ang

le [°

]

without 4WS

Bicycle Model

State Space Controller

Steady State CorneringDry Road

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 1 2 3 4 5 6 7 8lateral acceleration [m/s²]

rear

ste

erin

g an

gle

[°]

without 4WS

Bicycle Model

State Space Controller

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Side Slip AngleControl

Icy Road

Without 4WS

Bicycle ModelController

State SpaceController

Steady State CorneringIcy Road

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

0 1 2 3 4 5 6 7 8lateral acceleration [m/s²]

side

slip

ang

le [°

]

without 4WS

Bicycle Model

State Space Controller

Steady State CorneringIcy Road

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 1 2 3 4 5 6 7 8lateral acceleration [m/s²]

rear

ste

erin

g an

gle

[°]

without 4WS

Bicycle Model

State Space Controller

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Side Slip AngleControl

Icy Road

Without 4WS

Bicycle ModelController

State SpaceController

Double Lane Change Maneuvredry road, icy road

-6

-4

-2

0

2

4

6

0 10 20 30 40 50 60 70 80 90 100x-path [m]

y-pa

th [m

]

icy road, state space modelicy road, bicycle modelicy road, without 4WSdry road

Double Lane Change Maneuvredry road, icy road

-10

-5

0

5

10

15

20

0 10 20 30 40 50 60 70 80 90 100x-path [m]

side

slip

ang

le [°

]

icy road, state space modelicy road, bicycle modelicy road, without 4WSdry road

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- Control Strategy

- Development Method

- Side Slip Angle Observation

- Side Slip Angle Control

Overview

- Outlook

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Outlook

Virtual PrototypingMeasurement

Control information C - Code

���� Software Rapid Prototyping

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Dipl.-Ing. Alfred PrucknerDipl.-Ing. Sven Fischer

15th ADAMS USER CONFERENCEROM, 15. - 16. November 2000

INSTITUT FÜRKRAFTFAHRWESENAACHEN

Vehicle DynamicsControl for a4 Wheel SteeringPrototype Car