A New Dynamic Multi-D.O.F. Tire-Model

Copyright @ IFAC Control in Transportation Systems, Braunschweig, Germany, 2000

A NEW DYNAMIC MULTI-D.O.F. TIRE-MODEL Michel Sorine • J oel Szymanski ••

• INRIA Rocquencourt, 78153 Le Chesnay Cedex, France •• RENA ULT, 1 av. du Golf, 78288 Guyancourt Cedex, France

Abstract: A new dynamic model of the generation of shear forces by pneumatic tires for all speed is proposed. This model is suitable for a wide range of applications from the study of steering performance of vehicle at low speed to realism enhancement for interactive vehicle simulator. The elastic deformation of the belt with respect to the rim is takeninto account, along with the elasto-plastic behavior of rubber-road contact in the contact patch. The model reproduce tire behaviors like frictional vertical torque at standstill, lateral force and vertical torque due to yaw speed, and also relaxation length behavior. Moreover, as the "Pacejka Magic Formula", it represents very well the standard stationary friction curves. Copyright (j) 2000 IFAC Keywords: friction, tire, modelling, dynamic tire model, low-speed tire simulation

transient or multi-d.oJ. situations are constructed, based on the extension of Coulomb friction model to these cases. In the first section a new general differential friction model is presented and applied to the modelisation of the rubber-road friction. In the second section a model of longitudinal and lateral shear forces and self aligning torque generated by a tire is given, first for a blocked wheel (not rotating), then for a non blocked wheel with a solid carcass, and finaly a complete model including the carcass flexibility is presented along with a comparison of Pacejka magic formula with the complete model in the stationary case.

1. INTRODUCTION

Many models of tire-road friction forces have been proposed for analysis, simulation, monitoring and control of vehicle dynamics (Pacejka, 1981), (Sakai, 1981), (Gim and Nikravesh, 1991). One of the most frequently used is probably the so-called "Pacejka Magic Formula" (Pacejka and Bakker, 1991). This model describes the steady state behavior of the tire (see also (Egbert Bakker, 1987) and more recently a simplified dynamic behavior (Pacejka and Besselink, 1996). It is sufficient for studying basic vehicle handling-performances and control laws used in classical electronic systems like ABS or VDC. With the introduction of new systems like the Electric Power Steering, tire models are now expected to describe the tire behavior at low speed or during transients or when several degrees-of-freedom (d.oJ.) are concerned. For the above example, the control engineer needs a simulation tool to design control law, and the vehicle dynamicist needs a simulation tool to evaluate and tune the steering performances. For example tire shear forces generated at low speed by tire yaw speed have to be taken into account and this is difficult with Pacejka Magic Formula. In this paper simple models for the low-speed.

2. FRICTION MODEL The positions are time functions u E Cl (0, =; Rn) associated with generalized Rn-valued forces f(u) (denoted by boldfaced symbols) with n = 2 if not otherwise stated. T.he inner and dot products, the norm in Rn are x. y, x x y and Ix I. The coordinatevector (or matrix) of x in a referential RA with unitary vectors X A , Y A , ZA, is x = (x)"R A • A central role is played by functionals u -+ f(u) that do not depend upon liLl and may exhibit a

549

memory effect: they are Rate-Independent Hysteresis operator, meaning that the shapes of hysteresis cycles do not depend upon lul. Dry friction, as a function of position, and elasto-plastic stressstrain relations are of this type. On the opposite, viscous friction is Rate-Dependent. See (Bliman and Sorine, 1993).

This QVI has a lot of solutions. It can be easily proved that a solution, Iv(u), of (3) is given by:

K-

An anisotropic 2-d.oJ. Coulomb friction model, thermodynamically consistent, is given. Denote: Fk = F[ > 0, a 2 x 2-matrix of friction forces, C = {tPllF; 1tPl SI}, a convex set of admissible friction forces, Id u) satisfying the Maximal Dissipation- Rate "Principle", i.e. the Variational Inequality (VI): and

VtP E C

i V + IFkujF;2 I V = u,

{

Id u ) =

IF::I

(1)

x:: :FJc =

.

if uiO, =C if u=O

Maximal Dissipation-Rate: I du) . u

i) .(f - tP) 20, 1(0) = 10 = {tPllF;1 tP l S 1F;1/1}

x::-1 F + IFkulJ:-Jc -2 F = Ju,

= CxyF,

I v(u)

F(O)

= Fa

(5)

= ( {3K 0

0 ) ({3-I)K ' Cry

1 2 0: / Fk (

0

(0: _

0 )

1)1/2Fk

= (1,-1)

'

J =

(1)

(6)

1

Consider a rubber slider in contact with the road as shown Fig.!. Let U be the area of the slider, P the pressure over it, and 'Rv the slider referential. The contact is characterized by K = UT, F k = PUll, 0: and {3, all supposed to be diagonal in 'Rv with the following eigenvalues: • r x (ry): stiffness along Xv (Yv), • J1.r (J1.y): friction coefficient along X V (Y V

),

• a x (ay): overshoot factor along X v (Y v), • f3x (f3y): overshoot factor along X V (Y v). During rotation (~ i 0), the friction forces generated between rubber and road result in a friction force and a torque at the center of the slider. As the rubber elements in the tire contact patch,

F k being as before, let K = KT > 0 be the 2 x 2 matrix of transient stiffness for small friction. Here u being interpreted as a strain and I V (u) as the associated stress, it is natural to consider the elastic and plastic deformations U e = K -1 I V (u) and up = u - u e . Then I V (u) . up is the dissipation rate expected to be positive and even, maximal in some sense, as for Coulomb friction. This property is obtained by taking I as a solution of the following Quasi Variational Inequality (QVI):

VtPEC(f)

(4)

2.3 Macroscopic Model of Rubber-Road Contact

= IFkUI.

A class of I-d.oJ. differential models of dry friction is proposed in (Bliman and Sorine, 1993), (Bliman and Sorine, 1995). Remark that there exists other differential models for viscous friction like LuGre model (K.J. Astrom and Lischinsky, 1995)). The rubber-road contact forces in the low-to-medium speed-range seem closer to dry friction, so the models are built using a direct parametrization of the class of dissipative hysteresis presented in (Bliman and Sorine, 1993) together with an extension to the 2-d.o.f. case. A thermomechanical description is again the base of the construction.

{

= 10

(2)

2.2 Differential Models of Dry Friction

(u - K- 1

Vlt=O

where, Fk and K are as before and commute with the weight matrices 0: > 1 and {3 > 1:

The unique multi valued solution of this VI is F2

I

This is a 2-d.oJ. generalization of the usual Dahl model (Bliman and Sorine, 1993). Its asymptotic behavior is given by the selection of le (u) having, for each time, a left limit and right continuity, and being constant during stopping intervals. The model (4), of order one, can't describe the overshoot of the friction force present for some 1d.oJ. motions. Thus it is extended to second order, as in the I-d.oJ. case (Bliman and Sorine, 1993):

2.1 A nisotropic Coulomb friction model

Vu, "It, Idu)(t) E C I c(u)(t) . u(t) 2 tP· u(t)

1

Fig.!. Rubber slider geometry and referential. never undergo simultaneously large translations and rotations, the friction torque can be described by an uncoupled unidimensional model like (5). The resulting model is then:

(3)

F = x:: (-IFkul:F;2F + Ju),

Remark that the convex set C(f) depends upon the solution I: it is a way to get regularity for I and is the origin of the term "Quasi" in QVI.

M

=S

(_I~IM;1 M+ 1~)

{ Iv(u) = CryF, 550

Tnv(1/!)

,

= ezM

(7)

Where f D and TnD are the road reaction force and torque on the slider, and where:

C\5

S -

Mk

=

z (f3z!

1)5

oz~z 0 ( 0 (oz _ l)~z ) ' C z

z)

(8)

) =( 1-1

Fig. 3. Rubber slider friction force and torque.

=

Friction forces at time t and abscissa ( are given by (9). Let X(t,() be the associated state. All the sliders are supposed unconstrained in the initial state. The total friction force F and torque M z in 0, which depend upon an empirical weighting factor CM z, are given by:

• 5z CSZ~;ry: torsional stiffness along ZD, where CS z is an empirical coefficient, • ~z = cJJz P ~y %: torsional friction torque, where cJJz is an empirical coefficient. • oz, f3z: torsional overshoot factor along Z D. Let v (uT, t); ) T and X (FT M T )T, then (7) can be rewritten as a differential system:

=

=

1/2

F(t)

. _

fD(U) ) _ + Bv, ( TnD(1/J) -

X - -N(v)AX

N( )= v

lt

Mz(t)

SMkl

U1

)

1

(X)

=

j(TnD

+ CMz(XW

x fD)(t,()d(

-1/2

C=

Behavior for small yaw angle Let v = 0, so that F and M z are constant, and 6I/J be a small yaw angle along Zw. The resulting torque oM z is,

z

'RD

(10)

Xy

For simulation, (9) is written into the moving frame RD. Notice that, n being askew-symmetric matrix,

fD(t,()d(, 1/2

2

k

j

-1/2

CX (9)

0) ,B = (1CJ0 S0) (IF 0 11/J11' ~ (C0 C0)

. (1CFwith A = 0

=

= (X)n + t};n (X)n D

D

1/2

.

iSM z = - j

(5 z o1/J

+ cM z wr ye6I/J)d(

-1/2

3. TIRE MODEL

and C z, the torsional stiffness along Zw, is then

A 2-d.oJ. generalization of the tire model initially presented in (Bliman et al., 1995) is proposed, based on the above 2-d.oJ. dry friction models.

Cz =

lw ( 2 2) -iSM #z = r Y12 cMzl + cSzw

(11)

Asymptotic torsional behavior Let V = 0 and denote Mz. the value of the asymptotic friction torque M z when 1/J tends to infinity,

3.1 Blocked wheel model System description Let Rw be the tire referential as shown Fig. 2. The yaw speed of the rim, at standstill, is 1/J along Z wand the translation speed is V VxXw + VyYw. wand I are the width and length of the contact patch. The pressure P is supposed to be uniform.

1/2

Mz. = -

j

(~zPw + ~yPw()d(

-1/2

=

Mz.

= -Pwl~:

(C!-,zw

+ I)

(12)

Simulation Fig. 4 shows the lateral friction-force distribution with time for V = 0 and a yaw speed 1t};1 lrad s-1 and -45 :S 1/J :S 45. Fig. 5 shows the resulting torsional torque.

=

Fig. 2. Contact patch geometry and referential

3.2 Rotating wheel model with a solid carcass

The description is identical to the preceding one, but now the wheel is spining with a speed iN > O.

Modelisation Suppose that the contact patch consists of independent sliders along X w, this being the case when the belt is sufficiently rigidly linked to the rim. Then R.D and R.w are identical up a translation by (X w (see Fig. 3). The speed of a slider of abscisse ( along X w is u V + 1/J(Y w .

Modelisation The belt and the rim are considered as a solid so that the behavior of the tire is uniquely defined by the belt-road behavior. Define

=

551

Stationary response In this case the partial differential equation (13) becomes an ordinary differential equation in space, 1 l -oX = -, . [-N(v)AX + Bv] X(--) = 0 (14) OC C '2

Denote, • Dx 0"

= wrx~

the slip stiffness,

= wry ~ the cornering stiffness, h I"Ignmg stl'ffness. = cM z wrY12I' tea Dy p = wry i~ the curvature force stiffness, D zp = Cszwr y w24 the curvature torque

• Dy 0 • D Zo • zs

•

0

21 2

stiffness. Fig. 4. Lateral friction force distribution. Yaw . 1 speed 1J/i1 = lrad s- and V = O.

Nt l ::: 0 and:

For small yaw speed and slip rate,

=

X w' -X w the coordinate basis of the slider in the contact patch. The model (9) is used to calculate the friction force at time t on the slider of coordinate C along X w'. The slider speed is

(ti7, -J;) T

v =

with it = V

-~X w + -J;c Y w, and

X (t, C) is the associated friction state. The state derivative in (9) is now a material derivative,

DX Dt

oX

Let Vy and

= Ux = 0 and

*::

0, then

v

= (0, -J;~, -J;f

·oX

= 7ft + C OC

(15 )

The slider enters unconstrained in the contact patch at C= - ~, so that (9) can be rewri tten,

-DX = -N(v)AX + Bv,

Dt l X(t, -"2)

= 0,

X(O,c)

3.3 Rotating wheel model with carcass flexibility

=0

System description Consider now, that the belt has three degrees of freedom: a translation T = xX W + yY wand a rotation J = 6Z w (see Fig. 6). The parameters of the belt behavior are:

1/2

F(t)

=

fic

XY '

(13)

O)X(t,C)dC

-1/2 1/2

Mz(t)

=

J(O, CMzC, Cz)X(t,C)dC

-1/2

,so

100

SOl

-

Fig. 6. System description, with carcass flexibility.

I

!- 'f

] .~'== -04

-03

-02

-01

0

If

01

m : equivalent weight of the belt, J : equivalent inertia of the belt, k x : belt-rim stiffness along X w, k y : belt-rim stiffness along Yw , ko : belt-rim stiffness along Zw, • Cx : belt-rim viscous coefficient along X w, • c y : belt-rim viscous coefficient along Yw , • CO : belt-rim viscous coefficient along Zw.

• • • • •

, 02

03

04

(rad)

Fig. 5. Tire torsional torque at standstill.

552

Modelisation The same equations as before are used to describe the friction forces in the contact patch in RA. At time t a slider speed in RA is

v

= (uT, ~+6f,

'500

with

=

Vx + i: + y~ - o[Vy + Y- ~(~ + 6)] -~, y u = Vy + Y+ o(Vx + i:) + (x - ~)(~ + 6) The model gives the resulting friction on A. Denote now Fx A , FY A and Mz A. The friction forces Fx ,Fy , and M Z at point Q in Rw are given by: Ux

'000

o

_~,----:,--c:--~_:_----:.-_:::_"~,,'--,~.~ ..' -6(')

Fig. 8. Stationary lateral friction forces simulated with MCPS and PaMaF The belt motion with respect to the rim is described by the following equations (the rim motion is supposed slow, compared to the belt motion),

= -cxi: = -cyY fJ = -C66 -

mx my

'0

PACLlKA MC"

kxx + FX A + oFYA kyY - oFx A + FYA (17) k60 + M ZA - yFXA + XFYA

Stationary response In the stationary case the deformation of the belt are stabilized, so that

-00

6

10

12

6(")

x=

FX A + oFYA FYA - OFXA kx ' Y= ky 0= M ZA - yFx A + xFYA k6

Fig. 9. Stationary self aligning torque simulated with MCPS and PaMaF

(18)

with where Fx A , FYA and MZ A are functions of 0 and x, y. For small slider deformations,

• •

D YaF = D Y k 6-k6D Zo- the sideslip stiffness, D ZaF = D Zcr k 6 k6D z", the aligning stiffness, Q

Remark that these stiffnesses are functions of the corresponding stiffnesses for a solid carcass. In th~ case of a purely longitudinal slip (Vy = 0 and 1/J = 0), the longitudinal friction force and the slip stiffness remain unchanged. Identically i.n the case of a purely yaw motion, it means Vx - ~ = 0, Vy 0, the relation for the force and torque remain unchanged.

=

, >-

'''CEJKA

~.MCPS.

'500

Simulation results Fig. 7, 8 and 9, 10, 11 show the stationary response curve for a 165/65R14 tire simulated with the models MCPS (standing for "Modele de Contact Pneu Sol") and PaMaF (for "Pacejka Magic Formula"). The uncoupled stationary characteristics simulated by both models are very similar. The coupled characteristics, see Fig. 10, are also very similar for the longitudinal and lateral friction forces. For the self aligning torque, PaMaF curves and MCPS curves are quite different. In fact, it can be shown that the influence of the longitudinal force on self aligning torque is not naturaly described by PaMaF . Fig. 11 shows the lateral force and self aligning torque generated by the trajectory curvature of the wheel. Here again, it appears that PaMaF does not describe this tire characteristics.

~ ''''1500

'000 500

6

10

12

~.(")

Fig. 7. Stationary longitudinal friction forces simulated with MCPS and PaMaF The stationary friction forces are solution of this system of three equations. In the case of a purely sideslip (meaning ~, = 0, Vx - ~ = 0), x ::= 0 and,

(20)

553

Application to friction modelling and compensation. In: European Control Conference (ECC'93, Ed.). ECCA. Bliman, P.-A and M. Sorine (1995). Easy-touse realistic dry friction models for automatic control. In: European Control Conference (ECC'95, Ed.). ECCA. Bliman, P.-A., T. Bonald and M. Sorine (1995). Hysteresis operators and tyre friction models. application to vehicle dynamic simulations. In: ICIAM95. ICIAM. Hambourg. Egbert Bakker, Lars Nyborg, H.B. Pacejka (1987). Tyre modelling for use in vehicle dynamics studies. SAE. Gim, Gwanghun and Parviz E. Nikravesh (1991). An analytical model of pneumatic tyres for vehicle dynamic simulations. part3: Validation against data.. Int. J. of vehicle Design. K.J. Astrom, C. Canudas de Wit, H. Olsson and P. Lischinsky (1995). A new model for control of systems with friction. IEEE Transactions on Automatic Control 40(3), 419-425. Pacejka, H.B. (1981). In-plane and out-of-plane dynamics of pneumatics tyres. Vehicle System Dynamics (10), 221-251. Pacejka, H.B. and Egbert Bakker (1991). The magic formula tyre model. Vehicle System Dynamics (21), 1-18. Pacejka, H.B. and I.J.M. Besselink (1996). Tyre Models for Vehicle Dynamic Analysis. Chap. Magic Formula Tyre Model with Transient Properties, pp. 234-249. Swets & Zeitlinger. Sakai, Hideo (1981). Theoretical and experimental studies on the dynamic properties of tyres, part2: Experimental insvestigation of rubber friction and deformation of a tyre. Int J. of Vehicle Design 2(2), 182-226.

.:r···~···_·" !:'J-

,.. •.•.

.

Q

••

J(l

~;,':!.-

~r.:~. J~::::~~~ '.(H)

!.t~----~ ::~~ F,(HI

Fig. 10. Stationary coupled characteristics simulated with MCPS and PaMaF for small slip angle and slip rate. Load 4000 N.

1-.. MCl'6 PAC£J.... I

_''''OL---:'0.-'~02"--0"".'---'0~'~oo-,~06"--0"'.'---'0~.'~o.O-,~

..

CIA"V.~(m·IJ·

"I allgntng tofque

°O.!"""'--:o.=-,--7:0.'''--0::':'.'---'07 .• ---:'0.';---:;';0.'''--0::':'.'---'07.,---:'o.=-,---c ~(",-IJ

Fig. 11. Stationary curvature response from MCPS and PaMaF . Load 2000 N. 4. CONCLUSION The proposed dynamic tire model has no singularity when the wheel is blocked, it can simulate the transients behaviors at low or high speed and also at standstill. Moreover, the model describes as well, the stationary behavior of the tire including coupled and yaw speed effects. The parameters of the model can be easily identified from tire manufacturers data, and its numerical complexity can be adjusted with the number of points chosen to describe tread in the contact area. Today, this model has been integrated into a full vehicle simulator and is used for ABS, VDC and Electric Power Steering simulations. It has been validated for low speed vehicle simulations. It shows very good agreement with experiments, particularly in low speed cornering situations or at standstill. These results will be published later.

5. REFERENCES Bliman, P.-A and M. Sorine (1993). A system theoretic approach of systems with hysteresis:

554