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LRV: Laboratoire de Robotique de Versailles
VRIM: Vehicle Road Interaction Modelling for Estimation of Contact Forces
N. K. M'SIRDI¹, A. RABHI¹, N. ZBIRI¹ and Y. DELANNE²
¹LRV, FRE 2659 CNRS, Université de Versailles St Quentin, France² LCPC: Division ESAR; (Nantes) BP 44341 44 Bouguenais cedex
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LRV: Laboratoire de Robotique de Versailles
Outline
Problematic for on line estimation
Contact models (static & dynamic ones)
Vehicle Dynamics an Estimation model
Design of a nonlinear robust observer
Simulations results
Conclusion
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LRV: Laboratoire de Robotique de Versailles
Need of On line Estimation of contact forces
The knowledge of the tire/road contact is necessary for vehicle control, road safety, ...
Dynamics: Use of the “Relaxation Length” leads to dynamic equation of the longitudinal tire force.
Appropriate formulation of the model to permit the on-line estimation of tire forces.
– Stochastic behaviour (not completely deterministic)– Nonstationary processus (time varying)
Speed Vx
Re
brakeforce
braketorque
IntroductionProblematic for on line estimation
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LRV: Laboratoire de Robotique de Versailles
Braking and Tractive forces at given Slip Angles vs. Slip Ratio
Slip Ratio vs. Lateral Force at given Slip Angles
100
Fx à 50 km/h sol sec MXT 175 R14
-9000
-8000
-7000
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
-100 -80 -60 -40 -20 0 20 40
60 80
700 daN500 daN300 daN
Longitudinal Forces in function of Fz at
given Velocity
Various intereting Contact Models Exist
s
2a
k
i
p
2bkis
Braking Vx
Vx
”still no internal dynamics”
Contact models (static or steady state)
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LRV: Laboratoire de Robotique de Versailles
« Coefficient longitudinal » influence of Velocity
1020
3040
5060
7080
90100
90
80
70
60
50
400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
µ
% glissem ent
vitesse km /h
Relation µx = f(%glissement, vitesse)
Longitudinal Models
0 10 20 30 40 50 60 70 80 90 1000
0.5
1
1.5
Glissement (%)
Mu
µxmax
Kx
µxbloq
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
5000
6000
-10 -8 -6 -4 -2 0 2 4 6 8 10
drift
eff
ort
Y
Slip: 0
Influence of Load
7000 N5000 N
3000 N
carrossage: 0
pressure : 2.5 bars
Transverse Forces in function of Fz
Cannot be reduced to y(
”still no internal dynamics”
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LRV: Laboratoire de Robotique de Versailles
Contact Models
PhysicalProperties
- adhesion/Slipping- Pressure distribution- Stiffness Kx et Ky
Assume - constant Velocity, slip angle, - invariant Stifness Kx,Ky, Fz constant,…Uniformity of behaviour
Dugoff, Sakai, Gim, Guo, Lee, Brush Model
Mechanical Properties
- Elasticity theory
Pacejka, Fiala , …
Friction Models
LuGre, Bliman, …
- Relaxation length- contact dynamics…
has internal dynamics
Assumptions: ponctual, never lost, Stationary pressure distribution, symmetry,
perfect rotation, road curvature invariant, …
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LRV: Laboratoire de Robotique de Versailles
One-wheel dynamics
One-wheel dynamics
rFfTI w 2vCFvm x
where : angular wheel velocity, v : vehicle velocityF : tire force, T : applied torques : wheel-slipI : wheel inertia, r : Wheel radium, m : vehicle masseCx : aerodynamic drag, fw : friction coefficient
L o n g i tu d in a l w h e e l s l ip0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1
0
5 0 0
1 0 0 0
1 5 0 0
2 0 0 0
2 5 0 0
3 0 0 0
3 5 0 0
4 0 0 0
4 5 0 0
Lon
gitu
dina
l ti
re f
orce
(N
)
F 0
s
FC
0
ss
FC
Slip-Tire force characteristic
)(sfF
kinematics relationship of wheel-slip
phaseon accelarati during
phase braking during
rv
=s
v
v=s
s
s
vs represents the slip velocity: vs=v-r
Tire equations
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LRV: Laboratoire de Robotique de Versailles
Tire equations
The wheel-slip can be presented by a first order relaxation length :
phaseon acceleratiω
phase braking
s
s
vsrsσ
vvssσ
dt
ds
s
F
dt
dF
Ffs )(1
))(( 1*sκ vsfvCFσ
)(1 Ffsκ s
FC
)(sfF
with
sκvCF-F-vFσ 0*
Tire differential equation ( when s<sc, sc is the critical slip)
Locally we can write
Modelling of Tire Contact
( )F VF C V r
0( ) ( )F V F F C V r
Model has internal dynam
icsO
r mem
ory from on state to the next
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LRV: Laboratoire de Robotique de Versailles
Vehicle dynamics
cos( ) sin( )
sin( ) cos( )
sin( ) cos( )
x xf F yf F xr
y xf F yf F yr
z xf f F yf f F r yr
mV F F F
mV F F F
J F l F l l F
1
1
f f f xff
r r r xrr
T rFJ
T r FJ
+ expression of the 4 forces
4 dynamic equations
, , ,xf yf xr yrF F F F
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LRV: Laboratoire de Robotique de Versailles
The model can be written in the state space form
1 2
2 3
3 2 3
( )
( , )
x x
x u x Bu
x x x
1 ,( , , , )f rx x y 2 ,( , , , )x y f rx V V 3 ( , , , )xf xr yf yrx F F F F
Position vectorVelocity vectorForces vector
With State variable:
Unknown parameters: 1 2 3 4 5 6 7 8( , , , , , , , )T
1 3 5 71 1 1 1
; ; ;f r f fl l t t
0 0 0 02 4 6 8; ; ;
xf xr yf yr
f r f f
F F F F
l l t t
x f(x) Bu
y h(x)
State space form:
η
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LRV: Laboratoire de Robotique de Versailles
Adaptive Estimation of Tire forces
Robust Observer
2 3 2 2 2
3 2 3 3 2 2
ˆ ˆ ˆ( ) ( )
ˆˆ ˆ ˆ ˆ( , ) ( )
x u x Bu H sign x x
x x x H sign x x
2 3 2 2
3 2 3 2 3 3 2
( ) ( )
ˆˆ ˆ( , ) ( , ) ( )
x u x H sign x
x x x x x H sign x
The dynamics of the estimation errors
The system is linear with regard to the unknown parameters
Adaptive and robust sliding mode observer design
x̂xx~ θ̂θθ
~
))θxΨ(x)(Ψθ)xΨ(θ)xΨ(Ψ(x)θ ˆ(~
ˆˆˆ
Vehicle
Tire/road interface
Observer
Input x x̂
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LRV: Laboratoire de Robotique de Versailles
2 2 21
2TV x x
2 2 3 2 2 2( ) ( ) 0T TV x u x x H sign x
2 2 0x x
Convergence analysis
The system power is limited, then Forces are bounded,The a priori estimation is also bounded.
Then
3x
2H
2 0x 0t t
First step : convergence of 2x
2 0S x the sliding surface S is attractive
gives
Consequently 3 2 2( ) ( )equivx u H sign x [n]
The second step consider the reduced sliding dynamics, xr=(x3)
2 3, H H
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LRV: Laboratoire de Robotique de Versailles
According to equation ( n)
13 2 2 2 3 2 3 3 2ˆˆ ˆ( ( )) ( ( , ) ( , ) ( ))TV H sign x x x x x H sign x
2 3 2 3 ˆˆ ˆ( , ) ( , )x x x x
3 0V
By considering the choice of gain H3>>β we finally obtain the convergence of force estimation:
12 3 3 2ˆ ( , ) ( )x x H sign x
3 2 2( ) ( )x u H sign x
3 3 31
2TV x x
Second step : reduced sliding dynamics, xr=(x3)
Convergence analysis
Now, let us consider a second Lyapunov function:
Note also that the parameters values con also be retrieved
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LRV: Laboratoire de Robotique de Versailles
SimulationsThe parameters of simlation model
Parameter
Value Units
MJzFz
Jf,Jrrf,rr
1600301516000
0.70.27
KgKg.m2
NKg.m2
m
0 1 2 3 4 5 6 7 8 9 10-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
t(s)
Steering Angle
radH2 =
10 0 0 0 0
0 4 0 0 0
0 0 35 0 0
0 0 0 40 0
0 0 0 0 40
104 0 0 30 0
10 40 0 0 0
0 0 500 0 20
0 0 0 140 0
H3 =
Gains and parameters of observer
Vehicle
Tire/road interface
Observer
Input x x̂
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LRV: Laboratoire de Robotique de Versailles
0 5 100
0.5
1
1.5
2Vy
t(s)
m/s
0 5 10-0.02
-0.01
0
0.01
0.02
0.03psip
t(s)
rad/s
0 5 1036
38
40
42
44
46wf
t(s)
rad/s
0 5 1030
35
40
45
50wr
t(s)
rad/s
0 5 10
11
11.5
12
12.5
13
13.5
14Vx
t(s)m
/s Velocities
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LRV: Laboratoire de Robotique de Versailles
0 5 10-500
0
500
1000
1500
2000
2500Fxf
t(s)
N
0 5 10-2000
-1500
-1000
-500
0
500
1000Fxr
t(s)
N
0 5 10-400
-200
0
200
400
600Fyf
t(s)
N
0 5 10-300
-200
-100
0
100
200
300Fyr
t(s)
N
Forces
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LRV: Laboratoire de Robotique de Versailles
Conclusion
An appropriate Model for on line state estimation (can be extended for more than 5 Degres Of Freedom)
Robust Observer for on-line tire force estimation (using concept of relaxation length / local linearization)
The sliding mode technique is used to be robust with respect to uncertainties on the model, and unknown events (finite time convergence)
Possibility to quantify parameters of the tire/road friction.
The simulation result illustrate the ability of this approach to give efficient tire force estimation.
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LRV: Laboratoire de Robotique de Versailles
0 2 4 6 8-0.5
0
0.5
1
1.5
2Vy
t(s)
m/s
0 2 4 6 8-0.1
-0.05
0
0.05
0.1psip
t(s)
rad/
s0 2 4 6 8
35
40
45
50
55wf
t(s)
rad/
s
0 2 4 6 830
35
40
45
50
55wr
t(s)ra
d/s
0 2 4 6 811
12
13
14
15
16Vx
t(s)
m/s
Steering Angle Velocities
0 1 2 3 4 5 6 7 8-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
t(s)
rad
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LRV: Laboratoire de Robotique de Versailles
0 2 4 6 80
100
200
300
400
500
600Fxf
t(s)
N
0 2 4 6 8-1000
0
1000
2000
3000
4000Fxr
t(s)
N
0 2 4 6 8-500
0
500
1000
1500Fyf
t(s)
N
0 2 4 6 8-600
-400
-200
0
200
400
600
800Fyr
t(s)
N
Forces
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LRV: Laboratoire de Robotique de Versailles
0 5 10 15 2010
12
14
16
18
20Vx
t(s)
m/s
0 5 10 15 20-2
0
2
4
6
8Vy
t(s)
m/s
0 5 10 15 20-0.02
0
0.02
0.04psip
t(s)
rad/s
0 5 10 15 2030
40
50
60
70wr
t(s)
rad/s
0 2 4 6 8 10 12 14 16 18 20-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
t(s)
rad
Steering angle
0 100 200 300 4000
20
40
60
80
100trajectory
Velocities
Steering Angle
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LRV: Laboratoire de Robotique de Versailles
0 5 10 15 20-1000
0
1000
2000
3000
4000Fxf
t(s)
N
0 5 10 15 20-2000
-1500
-1000
-500
0
500
1000Fxr
t(s)
N
0 5 10 15 20-1000
-500
0
500
1000Fyf
t(s)
N
0 5 10 15 20-1000
-500
0
500
1000Fyr
t(s)
N
Forces