kalman filter based observer design for handling dynamics ... · for real-time state estimation of...

LoughboroughUniversity

Kalman Filter based Observer Design for Handling Dynamics :

The Sideslip Estimation Problem

Matt Best

Department of Aeronautical and Automotive EngineeringLoughborough University


Kalman filter background

Popular method ! (174 hits for 'vehicle Kalman filter' - Compendex '95 - '00)eg vehicle handling/stability, suspension control, global positioning

Classical uses :• Real-time state reconstruction• Filtering

Specific vehicle handling goals :• Reconstruction of handling states from small sensor set (real-time ?)• . . . particularly side-slip velocity (vehicle attitude when cornering)• Tyre force prediction / modelling• Model identification


Kalman Filter Operation

Kalman FilterModel

K

Inputs (steer angle)

Σ

Sensor measurements

Sensor estimates

-

+

State Correction

KalmanGain

(Source model)


The Linear Kalman Filter

w(k) : modelling errors

v(k) : sensor model errors + measurement noise

‘balance’ the Kalman Filter(LQR design)

x(k+1) = Ad x(k) + Bd u(k) + w(k)

ys(k) = C x(k) + D u(k) + v(k)

( )

=

=RS

SQee

υω

e TTkk

k

kk E ,

x(k), u(k) : true states and inputs

ys(k) : sensor measurement

For an optimal observer, w(k) and v(k) are zero mean white noise processes

xe(k+1) = Adxe(k) + Bdu(k) + K { ys(k) - Cxe(k) - Du(k) }

K


Three Simulation Studies on Handling Observers

Study 1 : Linear and Linear Adaptive Kalman Filter

Study 2 : Nonlinear Adaptive Kalman Filter

Study 3 : Nonlinear Adaptive for states and parametersBest M.C. and Gordon T.J., ‘Combined State and Parameter Estimation of Vehicle Handling Dynamics’ proceedings from the 5th International Symposium on Advanced Vehicle Control (AVEC), Ann Arbor, USA, August 2000, pp 429-436

Best M.C., Gordon T.J. and Dixon P.J., ‘An Extended Adaptive Kalman Filter for Real-time State Estimation of Vehicle Handling Dynamics,’ Vehicle System Dynamics : International Journal of Vehicle Mechanics and Mobility, Vol 34, No 1, pp 57-75, 2000.

Best M.C. and Gordon T.J., ‘Real-Time State Estimation of Vehicle Handling Dynamics Using an Adaptive Kalman Filter’ proceedings from the 4th

International Symposium on Advanced Vehicle Control (AVEC), Nagoya, Japan, September 1998, pp 183-188


Study 1 summary

Source model :

KF model :KF algorithm :

Inputs :Sensors :

Adaptation :

Yaw / roll / sideslip with • Pacejka lateral tyre force model• lateral load transfer & inclined roll axis • constant forward speed

Linear yaw / sideslip, ‘bicycle’

Linear & Linear time varying

One or two Lateral Accels

Recursive least-squares based on estimated states

Steer angle


0 2 4 6 8 10-0.4

-0.2

0

0.2

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1

0 2 4 6 8 10-0.8

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0

0.2

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0.6

0 2 4 6 8 10-4

-2

0

2

4

6

8

10

Source dataLinear Kalman filter

Lat. AccelSensor

Yaw RateSideslip Velocity

Study 1 results


0 2 4 6 8 10-0.4

-0.2

0

0.2

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1

0 2 4 6 8 1060

80

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0 2 4 6 8 10-0.8

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0

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0 2 4 6 8 10-4

-2

0

2

4

6

8

10

Source dataLinear K filterFast adaptingSlow adapting

Front CorneringStiffness

Lat. AccelSensor

Yaw RateSideslip Velocity

Study 1 results


0 2 4 6 8 10-0.8

-0.6

-0.4

-0.2

0

0.2

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0.6

0 2 4 6 8 1060

80

100

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Sideslip Velocity

Errors due to under-determined model :

}

}

Study 1 results

−−−−=

−−=

brvubrvu

CCcrvcrv

CC rrff ˆˆ ,

ˆˆ

δδ

αααα


0 2 4 6 8 10-20

-15

-10

-5

0

5

10

15

20

0 2 4 6 8 10-0.8

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0

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0

0.5

Lat. AccelSensor

Yaw Rate

Sideslip Velocity

Source dataLinear K FilterFast adapting

Performance under high sensor noise condition

Study 1 results


Study 1 Conclusions

Good yaw rate estimation

Excellent noise rejection

Model must be more detailed (higher order ?) to avoid steady-state problems with Sideslip Velocity

ß


Study 2 plan :

Introduce longitudinal mode, so steer induced deceleration decouples Cαf and v :

x

y

δ

Flat

Flong

bc

Also introduce roll mode to expand on number of estimated states

( ) 2δµδδ

ααα

fff CuwMhrpvMr

u

Cr

u

bCuM −−++

++=&


Study 2 summary

Source model :KF model :KF algorithm :

Sensors :Adaptation :

Real-time race game model

Yaw / roll / sideslip / longitudinal

Extended (nonlinear), involving• recursive form of Ricatti equation• matrix inversion• Jacobian (derivative matrix) calcs

Longitudinal + 3 Lateral Accels

Nonlinear algorithm allows parameters to be defined as (eg slow-varying) states :

( )( )

+

=

=

(t)

(t)

(t)

(t)

(t)C

(t)C

(t)

t

r

fαα

α

α ωωωωωωωω

χχχχxf

xfx

&

&

&

& )(


0 2 4 6 8 1030

40

50

60

70

80

90

0 2 4 6 8 10-2

-1.5

-1

-0.5

0

0.5

1

1.5

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-1.5

-1

-0.5

0

0.5

0 2 4 6 8 10

0

2

4

6

8

10


Long. AccelSensor

Sideslip Velocity

Source dataLinear K filterExtended KF

Lat. AccelSensor

Study 2 results


0 2 4 6 8 10-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

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-0.08

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0

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0 2 4 6 8 1021

22

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25

0 2 4 6 8 10-0.1

0

0.1

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ForwardVelocity

Roll Rate

Roll AngleYaw Rate

Source dataLinear K filterExtended KF

Study 2 results


0 2 4 6 8 100

50

100

150

0 2 4 6 8 10-2

-1.5

-1

-0.5

0

0.5

0 2 4 6 8 10-0.1

0

0.1

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0.5

Yaw Rate

Source dataExtendedExtended Fast


Sideslip Velocity

Study 2 results


Study 2 Conclusions

Basic principle works - better Sideslip estimation

Roll mode rather noisy (noise matrix tuning ?)

Unstable if adaptation rate is too high

Why ‘re-identify’ tyre with a changing Cα

Use nonlinear tyre in KF model

ß

ß

?


Study 3

Similar models and design to study 2, but :

• KF model now includes Pacejka formula for tyre forces

• More emphasis on scope for parameter adaptation (on-line model identification), through :

( )( )

+

=

=

(t)

(t)

(t)(t)

(t)(t)

(t)

(t)t

as

a

s

a

s

ωω

uηxf

uηxf

ηxχ

,,

,,)(

&

&&


0 1 2 3 4 50

1

2

3

4

5

0 1 2 3 4 5-1.5

-1

-0.5

0

0.5

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0

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0 1 2 3 4 5-5

0

5

10

15

Yaw Rate

Sideslip Velocity

Lat. AccelSensor

Source dataLinear K filterEKF model onlyEKF

Tyre loads (kN)

Study 3 results


0 1 2 3 4 530

35

40

45

50

55

0 1 2 3 4 5-0.6

-0.4

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0

0.2

0.4

0 2 4 6 8 100

0.2

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1

1.2

1.4

Pacejka InitialCornering Stiffness

Sideslip Velocity

Simultaneous adaptation of KF model parameters (values normalised)

C of G posn Roll inertia Mass Yaw inertia

Source dataLinear K filterEKF

Study 3 results


Conclusion : Sideslip EstimationAccurate steady-state Sideslip velocity estimation is

limited by model, and particularly tyre model accuracy

KF SideslipEstimate

Friction Estimate

µµµµ

Tyre Model

(Including roll

moment distribution)

Sensors

(using separate

wheel speed algorithms)

Adaptation

speed ?

On-line

identification

kalman filter based observer design for handling dynamics ... · for real-time state estimation of...

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