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LoughboroughUniversity
Kalman Filter based Observer Design for Handling Dynamics :
The Sideslip Estimation Problem
Matt Best
Department of Aeronautical and Automotive EngineeringLoughborough University
LoughboroughUniversity
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
LoughboroughUniversity
Kalman Filter Operation
Kalman FilterModel
K
Inputs (steer angle)
Σ
Sensor measurements
Sensor estimates
-
+
State Correction
KalmanGain
(Source model)
LoughboroughUniversity
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
LoughboroughUniversity
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
LoughboroughUniversity
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
LoughboroughUniversity
0 2 4 6 8 10-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
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
LoughboroughUniversity
0 2 4 6 8 10-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 1060
80
100
120
140
160
180
200
0 2 4 6 8 10-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
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
LoughboroughUniversity
0 2 4 6 8 10-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 2 4 6 8 1060
80
100
120
140
160
180
200
Front CorneringStiffness
Sideslip Velocity
Errors due to under-determined model :
}
}
Study 1 results
−−−−=
−−=
brvubrvu
CCcrvcrv
CC rrff ˆˆ ,
ˆˆ
δδ
αααα
LoughboroughUniversity
0 2 4 6 8 10-20
-15
-10
-5
0
5
10
15
20
0 2 4 6 8 10-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 2 4 6 8 10-0.5
0
0.5
Lat. AccelSensor
Yaw Rate
Sideslip Velocity
Source dataLinear K FilterFast adapting
Performance under high sensor noise condition
Study 1 results
LoughboroughUniversity
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
LoughboroughUniversity
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 −−++
++=&
LoughboroughUniversity
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
&
&
&
& )(
LoughboroughUniversity
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
0 2 4 6 8 10-2
-1.5
-1
-0.5
0
0.5
0 2 4 6 8 10
0
2
4
6
8
10
Front CorneringStiffness
Long. AccelSensor
Sideslip Velocity
Source dataLinear K filterExtended KF
Lat. AccelSensor
Study 2 results
LoughboroughUniversity
0 2 4 6 8 10-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0 2 4 6 8 10-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0 2 4 6 8 1021
22
23
24
25
0 2 4 6 8 10-0.1
0
0.1
0.2
0.3
0.4
0.5
ForwardVelocity
Roll Rate
Roll AngleYaw Rate
Source dataLinear K filterExtended KF
Study 2 results
LoughboroughUniversity
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
0.2
0.3
0.4
0.5
Yaw Rate
Source dataExtendedExtended Fast
Front CorneringStiffness
Sideslip Velocity
Study 2 results
LoughboroughUniversity
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
?
LoughboroughUniversity
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
,,
,,)(
&
&&
LoughboroughUniversity
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
0 1 2 3 4 5-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
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
LoughboroughUniversity
0 1 2 3 4 530
35
40
45
50
55
0 1 2 3 4 5-0.6
-0.4
-0.2
0
0.2
0.4
0 2 4 6 8 100
0.2
0.4
0.6
0.8
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
LoughboroughUniversity
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