LRV: Laboratoire de Robotique de Versailles VRIM: Vehicle Road Interaction Modelling for Estimation of Contact Forces N. K. M'SIRDI¹, A. RABHI¹, N. ZBIRI¹

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


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


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


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

0100020003000400050006000700080009000

-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

ki

p

2bkis

Braking Vx

Vx

”still no internal dynamics”

Contact models (static or steady state)


« Coefficient longitudinal » influence of Velocity

1020

3040

5060

7080

90100

9080

7060

5040

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

µ

% glissementvitesse 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

0100020003000400050006000

-10 -8 -6 -4 -2 0 2 4 6 8 10

drift

effo

rt Y

Slip: 0

Influence of Load

7000 N5000 N3000 N

carrossage: 0

pressure : 2.5 bars

Transverse Forces in function of Fz

Cannot be reduced to y(

”still no internal dynamics”


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, …


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

Lo

ng

itu

din

al t

ire

forc

e (N

)

F 0

sFC

0

ssFC

Slip-Tire force characteristic

)(sfF

kinematics relationship of wheel-slip

phaseon accelarati during

phase braking during

rv=svv=s

s

s

vs represents the slip velocity: vs=v-r

Tire equations


Tire equations The wheel-slip can be presented by a first order relaxation length :

phaseon acceleratiωphase braking

s

s

vsrsσvvssσ

dtds

sF

dtdF

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


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 FJ 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 forces4 dynamic equations

, , ,xf yf xr yrF F F F


The model can be written in the state space form

1 2

2 3

3 2 3

( )( , )

x xx u x Bux 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 Fl l t t

x f(x) Buy h(x)

State space form:

η


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

ObserverInput x x̂


2 2 212

TV 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.

Then3x

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


According to equation ( n)1

3 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 3 312

TV 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


SimulationsThe parameters of simlation modelParamete

rValue 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

-0.050

0.050.1

0.150.2

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


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


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


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.


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


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


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


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

Documents

LRV: Laboratoire de Robotique de Versailles VRIM: Vehicle Road Interaction Modelling for Estimation of Contact Forces N. K. M'SIRDI¹, A. RABHI¹, N. ZBIRI¹