1

1© Lotus 2009

4 wheel vehicle models in the control loop

for ESP , torque distribution and rear steer

Richard HurdwellCambridge April 2 2009

Authors

Malcolm Burgess and Robin Auckland2

© Lotus 2009

Objective and Introduction

To investigate the use of model based

controllers to enhance vehicle handling.

In particular to:

– Improve turn-in response

– Improve yaw stability

by:

– torque vectoring

– 4 wheel steering

3© Lotus 2009

Step Steer - Passive response – Yaw Rate

Yaw Rate - Passive Response

Time (sec)

0 .5 1 1.5 2 2.5 3

Yaw

Rate

(ra

d/s

)

0

.05

.10

.15

.20

4© Lotus 2009

Passive response – Lateral acceleration

Lateral Acceleration - Passive Response

Time (sec)

0 .5 1 1.5 2 2.5 3

Late

ral A

ccele

ratio

n (

g)

0

.05

.10

.15

.20

.25

.30

2

5© Lotus 2009

Feedback Control

- General principal

System

Error

Desired output

Actual output

Comparator

PID

Control variable

Conventional feedback control

6© Lotus 2009

Feedback Control

Plot

Feedback proportional terrm

Time (sec)

0 1 2 3 4 5 6 7 8 9 10

Output

0 .25

.50

.75

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

+

- 2 1 1s 2 +2*1*0.2 s +1 20

derivative2 +

+

+

+

2 derivative

10 11 s 2 +2*1*0.2 s+ 1 2 +

-

System PD controller (no Integral term)

Desired output Actual output 1

Identical to above, but with K=20 as a proportional term

2

Desired output

Actual output 2

actual output 2

actual output 1

+- +++++-

Proportional feedback

7© Lotus 2009

Feedback Control

Plot

Feedback damping terrm

Time (sec) 0 1 2 3 4 5 6 7 8 9 10

Output

0 .25

.50

.75

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

+

- 2 1 1 s 2 +2*1*0.2 s +120

derivative10 +

+

+

+

2 derivative

201 1

s 2 + 2*1*0.2 s +1 2 +

-

System PD controller (no Integral term)

Desired output Actual output 1

Identical to above, but with K=10 as a derivitive term

2

Desired output

Actual output 2

actual output 2

actual output 1

+- +++++-

Derivative feedback

8© Lotus 2009

Feedback Control

Plot

Feedback integral terrm

Time (sec)

0 1 2 3 4 5 6 7 8 9 10

Output

0

.25

.50

.75

1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

+

-2 1 1

s 2 +2*1*0.2 s +1 10

derivative2

+

+

+

+

+

2 derivative

10 11 s

2+2*1*0.2 s +1 2 +

-

System PD controller (no Integral term)

Desired outputActual output 1

Identical to above, but with K=3 as an integral term

2

Desired output

Actual output 2

actual output 2

actual output 1

1/S 3

+-++++++-

Integral feedback

3

9© Lotus 2009

Feedback Control

• Summarising the 3 terms of a PID controller:

• Proportional - Speeds up the control system response at the expense of stability

• Derivative - Is a damping term that controls the instability

but slows down the response

• Integral term - Reduces the steady state error

10© Lotus 2009

Applying feedback control to Torque vectoring

Torque Vectoring Tyre Forces

11© Lotus 2009

Torque Vectoring Tyre Forces

Maximum moment about the centre of gravity 12© Lotus 2009

Feedback Control algorithm

Vehicle Velocity

Front Steer Angle

Measured Vehicle Yaw Rate

Geometric Yaw Rate + -

K derivative

+

+

+

feedforward

Control Variable

Yaw Rate Error

r

Geometric Yaw Rate ω = Vx/r

Similar to basic rear steer

algorithm

4

13© Lotus 2009

Vehicle model

• Rigid body• 4 Unsprung corner masses

• Springs• Anti-roll bars

• Dampers• 4 Rotating wheels / inertia• Drive torque

• Pacejka tyre model (lateral and longitudinal)• Roll steer splines

• Bump steer splines• Roll centre heights• Compliance steer

• Velocity control• Steering control (steering pad)

Effectively 30 degrees of

freedom

14© Lotus 2009

Test manoeuvres

• Step steer – This is a rapid step (over 0.1 sec)

– To highlight response and stability

• Sine sweep – Increasing frequency constant amplitude steering input

– To highlight frequency response

15© Lotus 2009

Step Steer - Lateral acceleration

Lateral Acceleration

Yaw error feedback: Red - passive, Blue active

Time (sec)

0 .5 1 1.5 2 2.5 3

Late

ral A

cce

lera

tion (

g)

0

.1

.2

.3

Step steer response with feedback control16

© Lotus 2009

Step Steer – Yaw rate

Yaw Rate

Yaw error feedback: Red - passive, Blue active

Time (sec)

0 .5 1 1.5 2 2.5 3

Yaw

Rate

(ra

d/s

)

0

.05

.10

.15

.20

Step steer yaw response with feedback control

5

17© Lotus 2009

Step Steer – Normalised Lateral acceleration

Step steer lateral acceleration response with feedback control

Lateral Acceleration

Yaw error feedback: Red - passive, Blue active

Time (sec)

0 .5 1 1.5 2 2.5 3

Late

ral A

ccele

ratio

n (

g)

0

.1

.2

.3

18© Lotus 2009

Step Steer – Normalised Yaw rate

Step steer yaw response with feedback control

Yaw Rate

Yaw error feedback: Red - passive, Blue active

Time (sec)

0 .5 1 1.5 2 2.5 3

Yaw

Rate

(ra

d/s

)

0

.05

.10

.15

.20

19© Lotus 2009

Frequency response - General

Lateral Acceleration

Time (sec)

0 20 40 60 80 100

Late

ral A

ccele

ratio

n (

g)

-.3

-.1

.1

.3

Sine sweep steer Time History

20© Lotus 2009

Frequency response

Passive Feedback Control

Yaw Rate

Passive

Hz

0 .5 1 1.5 2 2.5 3 3.5 4

Yaw

Rate

(ra

d/s

)

-.2

-.1

0

.1

.2

Lateral Acceleration

Passive

Hz

0 .5 1 1.5 2 2.5 3 3.5 4

Late

ral A

ccele

ratio

n (

g)

-.3

-.1

.1

.3

Yaw Rate

Yaw Feedback Torque Vectoring

Hz

0 .5 1 1.5 2 2.5 3 3.5 4

Yaw

Rate

(ra

d/s

)

-.2

-.1

0

.1

.2

Lateral Acceleration

Yaw Feedback Torque Vectoring

Hz

0 .5 1 1.5 2 2.5 3 3.5 4

Late

ral A

ccele

ratio

n (

g)

-.3

-.1

.1

.3

6

21© Lotus 2009

General Open Loop Control

Mathematical model

Actual output? Desired output Actual system

22© Lotus 2009

Inverse model control

Actual systemDesired output inverse model of system Actual output

Mathematical model

23© Lotus 2009

Example - Inverse model control

e (̂x)ln(x)

SystemInverse Model

Desired Output Actual Output

1Desired output Actual output

24© Lotus 2009

Model based control – example system

Mass500 kg

Damper100 Ns/m

Spring

1000 N/m

x1, v1

x2, v2

aMvvCxxK ×=−+− )()( 1212 aMvvCxxK ×=−+− )()( 1212

aMvvCxxK ×=−+− )()( 1212

7

25© Lotus 2009

Mass, Spring, Damper - Passive response

1/500

Damper

Spring

Force Acceleration

1/S 1/S

Velocity

Displacement

100

1000

+

+

-

+

-

+

derivative2 11

0.1s + 1

Plot

Output - Position

Time (sec)

0 2 4 6 8 100

1

2

3

4

Input Displacement Input Velocity

Required output

Plot

Desired position

Time (sec)

0 2 4 6 8 100

.5 1.0

1.5

2.0

2.5

3.0

1/ mass

+ + - +- +

Desired step input to X1

Actual X1

response

26© Lotus 2009

Mass, Spring, Damper - inverse model

11212 )()( aMvvCxxK ×=−+−

Expanding:

11212 aMCvCvKxKx ×=−+−

Writing as differential: 2

11212 sxMsCxsCxKxKx ×=−+−

Re-arranging for x2:

sCK

sCxKxsxMx

⋅+

++×=

)(11

2

12

Input acceleration

derivative

Input VelocityInput Displacement

derivativeRequired output 500

100

1000

+

+

+

Plot

Control variable

Time (sec) 0 2 4 6 8 10

-100 -50

0

50

100

1 1 100 s +1000

Input 2

Mass

Damper

Spring

1/(Cs+K)

Calculated input needed to achieve demanded output

27© Lotus 2009

Mass, Spring, Damper- Model based control

Input Velocity Input Displacement

Plot Output response

Time (sec) 0 2 4 6 8 10

Position

0 .5

1.0 1.5 2.0 2.5 3.0

derivative

- +

- +

++

1000

100

Displacement

1/S 1/S F

Spring

Damper

1/500

Input 2

1/ mass

Velocity Acceleration Force

- +- ++ +

Calculated input needed to achieve demanded output

Actual output

28© Lotus 2009

Drive / brake torque inverse model

Inverse model

Actual system

Driving / braking torquesDesired yaw rate

Acual yaw rate

Yaw rate mappingVelocty

Steer

Steer angle

Steer angle

Feed Forward Term

Passive Response

8

29© Lotus 2009

Results – Model based control

Yaw rate

Yaw Rate

Model based control: Red - passive, Blue active

Time (sec)

0 .5 1 1.5 2 2.5 3

Yaw

Rate

(ra

d/s

)

0

.05

.10

.15

.20

Step steer yaw rate response with model based control

30© Lotus 2009

Results – Model based control

Lateral acceleration

Lateral Acceleration

Model based control: Red - passive, Blue active

Time (sec)

0 .5 1 1.5 2 2.5 3

Late

ral A

ccele

ratio

n (

g)

0

.1

.2

.3

.4

Step steer lateral acceleration response with model based control

31© Lotus 2009

Normalised response

Yaw rate

Yaw Rate

Model based control: Red - passive, Blue active

Time (sec)

0 .5 1 1.5 2 2.5 3

Yaw

Rate

(ra

d/s

)

0

.05

.10

.15

.20

Step steer Yaw Rate response with model based control - Normalised

32© Lotus 2009

Normalised response

Lateral accelerationLateral Acceleration

Model based control: Red - passive, Blue active

Time (sec)

0 .5 1 1.5 2 2.5 3

Late

ral A

ccele

ratio

n (

g)

0

.1

.2

.3

Step steer lateral acceleration response with model based control - Normalised

9

33© Lotus 2009

Frequency response

feedback control model based control

Yaw Rate

Model based control

Hz

0 .5 1 1.5 2 2.5 3 3.5 4

Ya

w R

ate

(ra

d/s

)

-.2

-.1

0

.1

.2

Lateral Acceleration

Model based control

Hz

0 .5 1 1.5 2 2.5 3 3.5 4

La

tera

l Ac

ce

lera

tion

(g

)

-.25

0

.25

.50

Yaw Rate

Yaw Feedback Torque Vectoring

Hz

0 .5 1 1.5 2 2.5 3 3.5 4

Yaw

Rate

(ra

d/s

)

-.2

-.1

0

.1

.2

Lateral Acceleration

Yaw Feedback Torque Vectoring

Hz

0 .5 1 1.5 2 2.5 3 3.5 4

Late

ral A

ccele

ratio

n (

g)

-.3

-.1

.1

.3

34© Lotus 2009

Compete system – including feedback

mapping.

steer input

Forward velocity

Des ited yaw rate (or Lateral acceleration) +

- Yaw rate

Actual Vehicle.

Lateral acceleration

Inverse model.

Drive / brake torques

Body slip

etc

Yaw error

+-

Feedback termto correct for small

inaccuracies in the model

35© Lotus 2009

Torque Vectoring Conclusions

• Feedback control shows some improvements in vehicle response to the step steer and sine sweep tests.

• Model based control shows a dramatic improvement in step steer response, with yaw rate in phase with steering input.

• An ideal response may be a detuned version of this model based control mapped to driver expectations.

• The model based control assumes a perfect inverse model of the system; in practice there will be a mismatch. To compensate for this the model based system should have some feedback and could be adaptive.

36© Lotus 2009

Latac and Yaw control

Four wheel steer inverse model

Body slip error feedback

Predicted body slip angle

PID

Modif ied Lateral acceleration

+

-

+

-

Desired body slip angle

etc.

Actual yaw rateDesired yaw rate

Front and Rear steering angles

Actual systemInverse model

Desired Lateral acceleration

Actual Lateral acceleration

+-+-

10

37© Lotus 2009

Four wheel steer inverse model

Inputs

Desired Yaw Rate and Lateral acceleration

Output

Front and Rear Steer

Lateral Acceleration

Time (sec)

0 .5 1 1.5 2 2.5 3 3.5 4

La

tera

l A

cce

lera

tio

n (

g)

0

.125

.250

.375

.500

Yaw Rate- Red / Actual - Blue / requested

Time (sec)

0 .5 1 1.5 2 2.5 3 3.5 4

Ya

w R

ate

(ra

d/s

)

0

.1

.2

.3

Steer Angle- Red / Front - Blue / Rear

Time (sec)

0 .5 1 1.5 2 2.5 3 3.5 4

Ste

er

An

gle

(°)

-10

-5

0

5

10

Model based control –Creating an example of a

desired response

38© Lotus 2009

Thank you