Dynamic tire friction models for vehicle traction control - Decision ...

Proceedings of tbe 38*Conference on Decision & ControlThP07 17:20Phoenix, Arizona USA December 1999Dynamic Tire Friction Models for Vehicle Traction ControlCARLOS CANUDAS DE WIT^Laboratoire d’Automatique de Grenoble, UMR CNRS 5528ENSIEG-INPG, B.P. 46, 38 40.2 ST. Martin d’lieres, FRANCEPANAGIOTIS TSIOTRAS2School of Aerospace EngineeringGeorgia Institute of Technology, Atlanta, GA 30332-01 50, USAAbstractIn this paper we derive a dynamic friction force modelfor road/tire interaction for ground vehicles. The modelis based on a similar dynamic friction model for contactdeveloped previously for contact-point friction problems,called the LuGre model [4]. We show that thedynamic LuGre friction model is able to accurately capturevelocity and roadlsurface dependence of the tirefriction force.1 IntroductionThe problem of traction control for ground vehicles is ofenormous importance to automotive industry. Tractioncontrol systems reduce or eliminate excessive slippingor sliding during vehicle acceleration and thus enhancethe controllability and maneuverability of the vehicle.Proper traction control design will have a paramounteffect on safety and handling qualities for future passengervehicles. Traction control aims to achieve maximumtorque transfer from the wheel axle to forwardacceleration. The friction force in the tirelroad interfaceis the main mechanism for converting wheel angularacceleration (due to the motor torque) to forwardacceleration (longitudinal force). Therefore, the studyof friction force characteristics at the roadltire interfacehas received a great deal of attention in the automotiveliterature.A common assumption in most of tire friction modelsis that the normalized tire friction pFp=-=F,,Friction forceNormal forceis a nonlinear function of the normalized relative velocitybetween the road and the tire (slip coefficient s)with a distinct maximum; see Fig. 1. It is also understoodthat p depends also on the velocity of the vehicleand road surface conditions, among other factors (see[3] and [lo]). The curves shown in Fig. 1 illustrate howthese factors influence the shape of p.‘Directeur de Recherche. Corresponding author. Email:canudasQlag.ensieg.inpg.fr‘Associate Professor. Email: p. tsiotrasaae. gatech.edu0-7803-5250-5/99/$10.00 0 1999 IEEE 3746The static model shown in Fig. 1 is derived empiricallybased solely on steady-state experimental data [lo, 11.Under steady-state conditions, experimental data seemto support the force vs. slip curves of Fig. 1. Nevertheless,the development of friction force at the tirelroadinterface is very much a dynamic phenomenon. In otherwords, the friction force does not reach its steady-stateinstantaneously, but rather exhibits significant transientbehavior which may differ significantly from itssteady-state value. Experiments performed in commercialvehicles, have shown that the tirelroad forces donot vary along the curves shown Fig. 1, but “jump”from one value to an other when these forces are displayedin the p - s plane [15].In this paper, we develop new, speed-dependent, dynamicfriction models that can be used to describe thetirelroad interaction. These models have the advantagethat are developed starting from first principlesand are based on simple contact (punctual) dynamicfriction models [4]. Thus, the parameters entering themodels have a physical significance which allows thedesigner to tune the model parameters based on experimentaldata. The models are also speed-dependent,which agrees with experimental observations. A simpleparameter in the model can also be used to capturethe road surface characteristics. Finally, our model isshown to be well-defined everywhere and hence, is appropriatefor control law design.2 Tirelroad friction modelsIn this study we consider a system of the form:where m is 114 of the vehicle mass and J, T are theinertia and radius of the wheel, respectively. v is thelinear velocity, w is the angular velocity, U is the accelerating(or braking) torque, and F is the tirelroadfriction force. For the sake of simplicity, only longitudinalmotion will be considered. The dynamics of thebraking and driving actuators are also neglected.

Wheel withlumped friction PWheel withdistributed friction FFigure 2: One-wheel system with: lumped friction (left),distributed friction (right)particular conditions of constant linear and angular velocity.The Pacejka model has the formF(s) = c1 sin(c2 arctan(c3s - cq(c3s - arctan(ss))>),otui 4Figure 1: Typical variations of the tire/road friction profilesfor: different road surface conditions (top),different vehicle velocities (bottom). Curves obtainedby Harned in et al [lo].2.1 Slip/Force mapsThe most common tire friction models used in the literatureare those of slip/force maps. They are definedas oneto-one (memory-less) maps between the frictionF, and the longitudinal slip rate s, defined as:1 - if v > TW, v # 0 brakings={ (3)1 - & if v < TW,W # 0 drivingThe slip rate results from the reduction of the effectivecircumference of the tire (consequence of the treaddeformation due to the elasticity of the tire rubber),which implies that the ground velocity will not be equalto II = rw. The slip rate is defined in the interval [0,1].When s = 0 there is no sliding (pure rolling), whereass = 1 indicates full sliding.The slip/force models aim at describing the shapesshown in Fig. 1 via static maps F(s) : s c+ F. Theymay also depend on the vehicle velocity U, i.e. F(s, U),and vary when the road characteristics change.One of the most well-known models of this type is Pacejka'smodel (see, Pacejka and Sharp [13] ), also knownas the "magic formula". This model has been shownto suitably match experimental data, obtained under3747where the 4s are the parameters characterizing thismodel. These parameters can be identified by matchingexperimental data, as shown in Bakker et al. [l].The parameters ci depend on the tire characteristics(such as compound, tread type, tread depth, inflationpressure, temperature), on the road conditions (such astype of surface, texture, drainage, capacity, temperature,lubricant, i.e., water or snow), and on the vehicleoperational conditions (velocity, load); see Pasterkampand Pacejka [12].As an alternative to the static F(s) maps, dynamicmodels based on the dynamic friction models of Dah1["I1, can be adapted to suitably describe the road-tirecontact friction. Dynamic models can be formulated asa lumped or distributed models, as shown in Fig. 2.This distinction will be discussed next.2.2 Lumped modelsA lumped friction model assumes punctual tire-roadfriction contact. An example of such a model can bederived from the LuGre model2 (see Canudas et al,[4]), i.e.'Dahl's models lead to a friction displacement relation thatbears much resemblance with stress-strain relations proposed inclassical solid mechanics.'This model differs from the one in [4] in the way that the func-1tion g(v) is defined. Here we propose to use the term e-Iwp/wa12instead the term e-(u~/u~)2 as in the LuGre model in order tobetter match the pseudo-stationary characteristic of this model(map s ct F(s) ) with the shape of the Pacejka's model, as it willbe shown later.(4)

where 00 is the normalized rubber longitudinal lumpedstiffness, 01 the normalized rubber longitudinal lumpeddamping, 02 the normalized viscous relative damping,pc the normalized Coulomb friction, ps the normalizedStatic friction,(pc < ps E [0, l]), vs the Stribeckrelative velocity, F, the normal force, v, = (rw- U) therelative velocity, and a the internal friction state.Dynamic friction models specifically for tires have beenreported in the work of Clover and Bernard [6], wherethey develop a differential equation for the slip coefficient,starting from a simple relationship of the relativereflections of the tire elements in the tire contactpatch. They still use the semi-empirical static force/slipmodels, however, to compute the corresponding frictionforce. In that respect, such models can be bestdescribed as quasi-dynamic models.2.3 Distributed modelsDistributed models assume the existence of an area ofcontact (or patch) between the tire and the road, asshown in Fig. 2. This patch represents the projectionof the part of the tire that is in contact with the road.The contact patch is associated to the frame 0,, withC as the axis coordinate. The patch length is L.Distributed dynamical models, have been studied previously,for example, in the works of Bliman et al. [2].In these kinds of models, the contact patch area is discretizedto a series of elements, and the microscopicdeformation effects are studied in detail. In particular,Bliman at al. characterize the elastic and Coulombfriction forces at each point of the contact patch, butthen they give the aggregate effect of these distributedforces by integrating over the whole patch area. Theypropose a second order rate-independent model (similarto Dahl’s model), and show that, under constant v andw, there exist a choice of parameters that closely matcha curve similar to the one characterizing the magic formula.Similar results can be obtained by using a model basedin the first-order LuGre friction model, i.e.Nevertheless, it is also possible to include different normalforce distribution if necessary, i.e. 6Fn = f(c).Note that Eq. (6) describes a partial differential equation(PDE), i.e.that should be solved in both: time and space.2.4 Relation between distributed model and themagic formulaThe linear motion of the differential bF in the patchframe 0, is C = rw, for positive w, and [ = -rw, fornegative w (the frame origin changes location when thewheel velocity reverses). Hence ( = rlwl. We can thusrewrite (6) in the [ coordinates as:where s = v,./wr = 1 - v/wr. Assuming that U, andw are constant (hence also U,, and s), the above equationdescribes a linear space-invariant system havingthe sign of the relative velocity as its input.The solution of the above equation over the spaceinterval [[(to), [(t~)], or equivalent over [CO,[I], withbz([O) = (0 = 0 isIntroducing this solution together(7), and integrating, we obtainand using (8) we obtain\-,with Eq. (9) in Eq.Finally, we have that F(s), is given aswith g(vr) defined as before andwhere, dF is the differential friction force, SFn = F,/Lthe differential normal force, V p = (rw - v) the relativevelocity, and 6z the differential internal friction state.This model assumes that:0 the normal force is uniformly distributed, and0 the contact velocity of each differential state ele-. ment is equal to U,.3748+ F,O~TWSwith 7 = 1 - O1lVrl/g(S) andg(s) = pc + (ps - pc) e-Ipw5/va14for some constant w, and s E [0, 11.Uncertainty in the knowledge of the function g(vr), canbe modeled by introducing the parameter 0, as8(vr) = eg(vr) ,where g(v,) is the nominal known function. Computationof the function F(s, e), from Eq. (12) as a function

~ N-1~ N-1Parameter Value Umtsdzi(t) = 6zi. Similarly, we have that0.0018 [s/m]PS 0.9' vs 12.5 [m/s]Table 1: Data used for the plot shown in Fig. 3of 8, gives the curves shown in Fig. 3. These curvesmatch reasonably well the experimental data shown inFig. 1-(a), for different coefficient of road adhesion usingthe parameters shown in Table 1. Hence, the parameter8, suitably describes the changes in the roadcharacteristics.where each of the can be approximated using forwarddifferences, as:6zi+1-62;-={ d dzi 7 i=0,1, ... N-24 0 i=N-1Hence, for each i-th equation we haveSimilarly,with AF,,i = F, fL, Vi, and A< = LfN, Fcan be approximated as:N -1 N-1F = AFi = (a06zi + ~ 16ii) AFn,iAC+~2vrFni=Oi=Owhich simplifies to:Introducing Z, as the mean value of all the 6zi, i.e.z=- - 6ZiNi=Owe have, from Eq. (14), thatFigure 3: Static view of the distributed LuGre model, un- . 1 N-l Iv Ider different values for l/O. Braking case, withv = 20m/s = 72Km/h. These curves show thenormalized friction p = F(s) fFn, as a functionof the slip rate s.P = -- (6~i+l - 6zi) TW + U, - ~o;-dZ (15)L i=OdVP)Noticing that E&' (Szi+l - dzi) = 6x0, and taking620 = 0, we have thatNote that the steady-state representation of Eq. (12)can be used to identify the model parameters by feedingthis model to experimental data. These parameters canalso be used in the simpler lumped model, which can beshown to suitably approximate the solution of the PDEdescribed by Eqs. (6) and (7). This approximation isdiscussed next.2.5 From distributed to lumped modelsUnder the assumptions given in subsection 2.3 we canapproximate the PDE in Eqs. (6)-(7) by a set of nordinary differential equations via a spatial discretization.To this end, let's divide the contact patch intoN equally spaced discrete points, to each we associatethe "discrete" average displacement 6zi, i.e. dzi =6z(iL/N, t), Vi = 0,1,. ,. N - 1. The space/time scalardz(C, t) is thus approximated by the N-dimensionaltime vector, 6z = [dzo, 6z1,. .., 6zN-I]T where, for thesake of simplicity of the notation, we have written3749F = (002 + O I + ~ CzV,) Fn (17)These equations describe the approximate behaviour ofthe PDE, in terms of the mean variable P. When comparedto Eqs. (4)-(5), they indicate that the lumpedmodel can be used as a suitable approximation of thedistributed one. Therefore, parameters identified fromthe stationary behaviour shown in Fig. 3, can be usedin the lumped model (4)-(5).3 Traction ControlWe consider the one-wheel model with the tirelroadfriction described in Eqs. (1)-(2). Using the pseudcstatic(or steady-state) force friction point of view, thefriction force is given as an algebraic (static) function of

the slip coefficient. Typical friction force vs. slip coefficientcurves are shown in Fig. 1. This figure suggests asimple way to achieve maximum traction between theroad and the wheel tire. Namely, to operate at the maximumpoint of the friction/slip curve. This “extremumseeking” control strategy requires the a priori knowledgeof the optimal target slip. With the exceptionof [8], where the authors present a control algorithmwhich does not require the a priori knowledge of theoptimal slip, current literature does not seem to haveadequately dealt with this problem. Nonetheless, slipand friction estimation algorithms have been proposedand verified experimentally in [12].A simple traction control law using this idea, and basedon sliding mode techniques, is given in [9]. For thesimplified one-wheel friction model of Eqs. (1)-(2) thiscontrol law is given byr T 1+T F-ksgn(S) (18)U= [m(l-sd) Iwhere Sd is the desired slip coefficient, S is given bys = (9 - Sd) TW andIndeed, simple calculation shows thats = (1 - Sd) TdJ - v (20)Substituting Eq. (18) into Eq. (20) and using Eqs. (1)-(2), one obtainsS = -qsgn(S) (21)This implies that S + 0 after S(O)/q seconds. Themajor drawback of the control law in Eq. (18) is thatis highly oscillatory due to the zero order sliding modeS = 0. One can reduce the chattering by smoothingthe discontinuity of sgn(.) via low-pass filtering. In thesmoothed implementation of the previous ccntrol law,the term ksgn(S) is replaced by the term ksat(S/@),where = XQ, and where sat(.) is the saturation function.For more details on the previous control law, theinterested reader is referred to [9].4 Numerical exampleIn this section we use the traction control law of the previoussection on two different friction models. In particular,we are interested in differences between staticand dynamic friction models. We consider the onewheelmodel with the values shown in Table 1, and:m = 500 Kg, J = 0.2344 Kgm2, T = 0.25m, F,, = mg.The first simulation was performed using the steadystateLuGre model from Eq. (12). The relevant parametersof the LuGre friction model are shown in Table 1.The maximum traction is achieved for sd = 0.15. Theresults of the simulations, using the smoothed versionof the traction control law presented in the previoussection, are shown in Fig. 4. The second simulation3750x 10‘II5-:-:;yi-2 ‘I0 05 01 0 15 02 0 25 03Time8 3000l?2 100000.05 0.1 0.15 0.2 025 0.3TimO11 , , I0‘ I0 01 01 0 15 02 0 25 03TWFigure 4: Static friction model,was performed using the dynamic friction model givenin Eqs. (4)-(5) with the values shown in Table 1. Theresults of the simulation are shown in Fig. 5. In bothcases, the initial applied torque for both static and dynamiccases was U = 10000Nm. The history profiles ofx lo‘II4’I0.05 0.1 0.15 0.2 025 0.3Tim150001 II0.05 0.1 0.15 0.2 025 0.3Tim0 0 05 01 0 15 02 025 03TIMFigure 5: Dynamic friction model.the applied torque and the slip coefficient are very similar.The most serious discrepancy between Figs. 4 and5 is the actual friction force developed between the tireand the ground for the two cases. These figures showclearly that the maximum friction force predicted usingthe dynamic friction model is more than three timesthe maximum of the friction force predicted using thestatic friction model during the initial transient. Thesteady-state value of the friction force for both cases isalmost the same. Since the main mechanism for transferringthe axle torque to forward movement is frictionforce, these results suggest that new traction controlalgorithms using dynamic friction models may have anII

advantage over traditional control laws based on trackingthe optimal slip coefficient. Finally, Fig. 6 showsthe distances traveled by the wheel for each case, alongwith the path of a point at the circumference of thewheel. A complete circle indicates complete slipping(the wheel spins without moving forward) whereas acycloid indicates that the relative velocity of the contactpoint is zero. Because of the higher friction forcedeveloped in the dynamic friction model, the wheel hastraveled a longer distance than for the static frictioncase.AcknowledgementsThe LuGre version of the dynamic friction model was developedduring the visit of the first author at the Schoolof Aerospace Engineering at the Georgia Institute of Technologyduring December 1998, as part of the CNRS/NSFcollaboration project (NSF award no. INT-9726621/INT-9996096). The first author would like to thank M. Sorineand P.A. Bliman for interesting discussions on distributedfriction models.(a) Wheel trajectory with static friction model.46 44 6 2 0 02 04 06 08 1 12 14(b) Wheel trajectory with dynamic frictionmodel.Figure 6: Comparison of wheel trajectories using staticand dynamic tire friction models (note differentstopping points).5 ConclusionIn this paper, we have derived a new dynamic, speedandsurface-dependent tire friction model for use in vehicletraction control design. This model captures veryaccurately most of the main characteristics that havebeen discovered via experimental data. It was alsoshown that distributed models can collapse to a lumpedmodel, which is rich enough to capture the main dynamiccharacteristics. Since the main mechanism fortransferring the axle torque to forward movement isfriction force, these results suggest that new tractioncontrol algorithms using dynamic friction models mayhave an advantage over traditional control laws basedon simple, static friction models. Although for the sakeof brevity the discussion has been restricted to tractioncontrol, the results of the paper have an immediate applicationto the design of ABS systems.3751References[l] Bakker, E., L. Nyborg, and H. Pacejka, “Tyre Modellingfor Use in Vehicle Dynamic Studies,” Society of AutomotiveEngineers Paper # 870421, 1987.[2] Bliman, P.A., T. Bonald, and M. Sorine, “HysteresisOperators and Tire Friction Models: Application to vehicledynamic Simulator,” Prof. of ICIAM. 95, Hamburg, Germany,3-7 July, 1995.[3] Burckhardt, M., Fahnuerktechnik: Radschlupfregelsysteme.Vogel-Verlag, Germany, 1993.[4] Canudas de Wit, C., H. Olson, K. J. Astrom, andP. Lischinsky, “A New Model for Control of Systems withFriction,” IEEE TAG, Vol. 40, No. 3, pp. 419-425, 1995.[5] Canudas de Wit, C., R. Horowitz, R. and P. Tsiotras,“Model-Based Observers for Tire/Road Contact FrictionPrediction,” In New Directions in Nonlinear ObserverDesign, Nijmeijer, H. and T.1 Fossen (Eds), Springer Verlag,Lectures Notes in Control and Information Science,May 1999.[6] Clover, C. L. and J. E. Bernard, “Longitudinal TireDynamics,” Vehicle System Dynamics, Vol. 29, pp. 231-259,1998.[7] Dahl, P. R., “Solid Friction Damping of MechanicalVibrations,” AIAA Journal, Vol. 14, No. 12, pp. 1675-1682,1976.[8] Drakunov, S., U. Ozgiiner, P. Dix, and B. Ashrafi,“ABS Control Using Optimum Search via Sliding Modes,”IEEE Pansactions on Control Systems Technology, Vol. 3,NO. 1, pp. 79-85, 1995.(91 Fan, Z., Y. Koren, and D. Wehe, A Simple TractionControl for Tracked Vehicles. In Proceedings of the AmericanControl Conference, pp. 1176-1177, Seatlle, WA, 1995.[lo] Harned, J., L. Johnston, and G. Scharpf, Measurementof Tire Brake Force Characteristics as Related toWheel Slip (Antilock) Control System Design. SAE Zhnsactions,Vol. 78 (Paper # 690214), pp. 909-925. 1969.Ill] Liu, Y. and J. Sun, “Target Slip Tracking Using Gain-Scheduling for Antilock Braking Systems,” In The AmericanControl Conference, pp. 1178-82, Seattle, WA, 1995.[12] Pasterkamp, W. R. and H. B. Pacejka, “The Tire asa Sensor to Estimate Friction,” Vehicle Systems Dynamics,Vol. 29, pp. 409-422, 1997.[13] Pacejka, H. B. and R. S. Sharp, “Shear Force Developmentsby Psneumatic tires in Steady-State Conditions: AReview of Modeling Aspects,” Vehicle Systems Dynamics,Vol. 20, pp. 121-176, 1991.[14] Wong J. Y., Theory of Ground Vehicles. John Wiley& Sons, Inc., New York, 1993.[15] van Zanten, A., W. D. Ruf, and A. Lutz, “Measurementand Simulation of Transient Tire Forces,” in InternationalCongress and Ezposition, (Detroit, MI). SAE TechnicalPaper # 890640, 1989.