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ROBSUST ESTIMATION OF TIRE-ROAD FRICTIONS AND CONTACT FORCES A. Rabhi1 , N.K. M'sirdi1 , M. Ouladsine1 and Fridman2 1 LSIS, CNRS UMR 6168. Dom. Univ. St Jérôme, Av. Escadrille Normandie - Niemen

13397. Marseille Cedex 20. France. e-mail: [email protected] 2 UNAM Dept of Control, Division of Electrical Engineering,Faculty of Engineering, Ciudad Universitaria,

Universidad Nacional Autonoma de Mexico, 04510, Mexico, D.F., Mexico [email protected]

Abstract— This paper uses a robust second order differentiator to build up an estimation scheme allowing to identify the tire road friction. The estimations are produced in three steps as cascaded observers and estimator. The rst produces estimations of velocities. The second estimate the tire forces (vertical and longitudinal ones) and the last reconstruct the friction coef cient. The actual results show effectiveness and robustness of the proposed method. Index Terms— Sliding modes, nonlinear observers, high order sliding modes, robust state and tire forces estimation.

I. INTRODUCTION More and more new active safety systems are developed and installed on vehicles for real-time monitoring and controlling the dynamic stability (EBS, ABS, ESP). Car accidents occur for several reasons which may involve the driver or components of the vehicle or environment. Such situations appears when the vehicle is driven beyond the adherence or stability limits. Nevertheless, the possibility of rectifying an unstable condition can be compromised by physical limits. Therefore, it is extremely important to detect (on time) a tendency towards instability. This requires then robust observers looking forward based on the physics of interacting systems (the vehicle, the driver and the road). The active safety becomes more important in recent research on intelligent transportation systems (ITS) technology. This have to be done without adding expensive sensors and then requires quite robust observers looking forward based on the physics of interacting systems (the vehicle, the driver and the road). The tire forces properties affect the vehicle dynamic performance. The control of ground - vehicle interactions becomes important. The design of traction controller is based on the assumption that vehicle and wheel angular velocities are both available on-line by direct measurements and/or estimations. Thus the knowledge of tire parameters and variables (stiffness, forces, velocities, wheel slip and radius) is essential to advanced vehicle control systems [1]–[3]. However, tire forces and road friction are dif cult to measure directly and to represent precisely by some deterministic model equations. In the literature, their values are often deduced by some experimentally approximated models [6][4]. Recently, many analytical and experimental studies have been performed on estimation of the frictions and contact

forces between tires and road [9],[27],[10]. Tire forces can be represented by the nonlinear (stochastic) functions of wheel slip. The deterministic tire models encountered are complicated and depend on several factors (as load, tire pressure, environmental characteristics, etc.) [4],[5][24]. This makes on line estimation of forces and parameters dif cult for vehicle control applications and detection and diagnosis for driving monitoring and surveillance [2]. In [25],[23]„ application of sliding mode control is proposed. Observers based on the sliding mode approach have been also used in [8]. In [9] an estimation based on least squares method and Kalman ltering is applied for estimation of contact forces. In [10] presented a tire/road friction estimation method based on Kalman lter to give a relevant estimates of the slope of versus slip ( ), that is, the relative difference in wheel velocity. The paper [28] presented an estimator for longitudinal stiffness and wheel effective radius using vehicle sensors and GPS for low values of slip. Robust observers with unknown inputs are ef cient for estimation of road pro le and for estimation of the contact forces ,[8]. Tracking and braking control reduce wheel slip. This can be done also by means of its regulation while using sliding mode approach for observation and control [2],[8]. This enhances the road safety leading better vehicle adherence and maneuvers ability but the vehicle controllability in its environment along the road admissible trajectories still remain an important open problem. From the other hand it is necessary to remark that observers for mechanical systems with unknown inputs based on standard rst order sliding mode approach has the following disadvantages [11],[12],[13]: for observation of the velocity a ltering is needed corrupting the results ; the need of ltering destroys the nite time convergence property and design must take care about observation and control(separation principle); for uncertainties and parameter identi cation a second ltering is necessary. This leads to a bigger corruption of results. In the last two decades the second-order sliding-mode algorithms has been designed and applied to some practical needs ( see [12][15][17][18] and reference therein). A robust exact differentiator [16] based on super twisting algorithm ([15]) ensures a nite time convergence to the values

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of the corresponding derivatives and provides the best possible accuracy of the derivatives for the given value even considering deterministic noise, sampling step and in the case of discrete measurements. In this paper, a nominal modelling of the vehicle is considered in the objective of an on line estimation using the super-twisting based robust exact observer [13] for estimation of velocities stiffness and effective radius identi cation. The modi cation of the super-twisting algorithm proposed in this paper allowing to make the velocity observation without ltering; provide the nite convergence to the exact value of derivative, ensuring separation principle; to identify the uncertainties with just one ltering only; to use least square method for parameters identi cations. In this work, we deal with a simple vehicle model coupled with wheel - road contact. We describe a vehicle model in the objective of on line estimation using robust observers. The obtained dynamics equations may be written in a state space form which allowed us to de ne an observer based on the sliding mode approach (as presented in [2],[8]). The observer has been used to reconstruct the global system state components and then to estimate the tires forces [2], [8]. The use of sliding mode approach has been motivated by its robustness with respect to the parameters and modelling errors and has been shown to cope well with this problem. We focus our work, as presented in this paper, to the on-line estimation of the tires sleep, adherence, stiffness and effective radius. We estimate the vehicle state and identify tire forces [3]. The main contribution is the robust on-line estimation of the tire effective radius, wheel sleep and velocities needed for a control by using only simple low cost sensors (ABS sensors). In this work, we deal with a simple vehicle model in order to estimate of tire-road friction. This estimation can be used to detect a critical driving situation to improve the security. It can be used also in several vehicle control systems such as Anti-look brake systems (ABS), traction control system (TCS), diagnosis systems, etc...The main characteristics of the vehicle longitudinal dynamics are taken into account in the model used to design robust observer and estimations. The proposed method of estimation is veri ed through one- wheel simulation model with a "Magic formula" tire model and then application results (on a Peugeot 406) show an excellent reconstruction of the velocities, tire forces and radius estimation.

Fig. 1.

Weel dynamics

Applying Newton's law to wheel and vehicle dynamics gives us the equations of nominal dynamics in the motion. :

mv x : J!

= Fx = T Ref Fx

(1) (2)

where m is the vehicle mass and J,Ref are the inertia and effective radius of the tire, respectively. vx is the linear velocity of the vehicle, ! is the angular velocity of the considered wheel, T is the accelerating (or braking) torque, and Fx is the tire/road friction force. The tractive (respectively braking) force, produced at the tire/road interface when a driving (braking) torque is applied to a pneumatic tire, has opposed the direction of relative motion between the tire and road surface. This relative motion exhibits the tire slip properties. The wheel - slip is due to de ection in the contact patch. The longitudinal wheel-slip is generally called the slip ratio and is described by a kinematic relation as [5]: ( R ! = vefx 1 if vx > Ref ! (braking) (3) Ref ! =1 if vx < Ref ! (traction) vx Representing the adhesion coef cient as a function of the wheel slip yields the adhesion characteristic ( ); wich primarily depends on the runway surfaces are shown in the following gure (2).

II. P HYSICAL PROPERTIES AND M ODEL In the literature, many studies deal with vehicle modelling [21][27] These are complex and nonlinear systems. The complete models are dif cult to use in control applications. The most part of applications deal with simpli ed and partial models [9][22]. Let us consider the simpli ed motion dynamics of a quarter-vehicle model, capturing only nominal behavior. This model retains the main characteristics of the longitudinal dynamic. For a global application, this method can be easily extended to the complete vehicle and involve the four coupled wheels.

Fig. 2.

Wheel slip

The gure 2 shows the variations between coef cient of road adhesion and longitudinal slip s for different road surface conditions. It can be observed that all curves ( ) start at = 0 for zero slip, wich corresponds to the non-braked, free rolling

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wheel. With increasing slip ratio of 3% and 20%: Beyond this maximum value the slope of the adhesion characteristic is negative. At a slip ratio of 100% the wheel is completely skidding, wich corresponds to the locking of the wheel. The adhesion characteristic plays an essential role for both the design and the validation of ABS. Overall, to improve the functioning of an ABS it is desirable to have some real-time information about the adhesion characteristic. By assuming that the longitudinal forces are proportional to the transversal ones, we expressed theses forces as follows: (4)

Fx = Fz

where Fz is the vertical force of the weel. The vertical forces that we use in our model are function of the longitudinal acceleration and the height of the center of gravity. The vertical force can be represented as: m (glr Fz = 2(lf + lr )

:

h:v x )

(5)

where: h is the height of the center of gravity lf is the distance between the center of gravity and the front axis center of gravity. lr is the distance between the center of gravity and the rear axis center of gravity. III. OBSERVER DESIGN In what follows, we develop a second order differentiator in order to obtain estimates of the tire road friction.

A. High Order Sliding Mode Observer (HOSM) Consider a smooth dynamics function, s(x) 2 R. The system containing this variable may be closed by some possiblydynamical discontinuous feedback where the control task may be to keep the output s(x(t)) 0. Then, provided that : :: successive total time derivatives s; s; s:::; s(r 1) are continuous functions of the closed system state space variables, and the r-sliding point set :

::

s = s = s = ::: = s(r

1)

=0

B. Robust Differentiation Estimator (RDE) To estimate the derivatives s1 and s2 without its direct calculations of derivatives, we will use the 2nd -order exact robust differentiator of the form [29] :

z0 :

0

2

s! j 3 sign(z0

jz0

s! )

1 2

= v1 = v0 ) sign(z1 1 sign(z1 = v1 ) 2 sign(z2

z1 : z2

v0 ) + z2

where z0 , z1 and z2 are the estimate of s! ; s1 and s2 , respectively, i > 0; i = 0; 1; 2. Under condition 0 > 1 > 2 the third order sliding mode motion will be established in : a nite time. The obtained estimates are z1 = s1 = s! and :: z2 = s2 = s! then they can be used in the estimation of the state variables and also in the control. C. Cascaded Observers - Estimators In this section we develop a robust second order differentiator to build up an estimation scheme allowing to identify the tire road friction. The estimations will be produced in three steps as cascaded observers and estimator in order to reconstruct information and system states step by step. This approach allow us to avoid the observability problems dealing with inappropriate use of modeling equations. For vehicle systems it is very hard to build up a complete and appropriate model for observation of all the system states. Thus in our work, we avoid this problem by means of use of simple and cascaded models suitable for robust observers design. The rst step produces estimations of velocities. The second one estimate the tire forces (vertical and longitudinal ones) and the last step reconstruct the friction coef cient. The robust differentiation observer is used for estimation of the velocities and accelerations of the wheels. The wheels angular positions and the velocity of the vehicles body vx , are assumed available for measurements. The previous Robust Differentiation Estimator is useful for retrieval of the velocities and accelerations. :

b = v0 = ! b :

:

! b

= v1 = ! b

::

! b

=

b

0

:

2 3

sign( 1

! 1 sign(b

b 2 sign(!

b)

v0 ) 2 sign(b !

v0 )

v1 )

Then to estimate Fx we use the following equation,

(6)

:

J! b=T

is non-empty and consists locally of Filippov trajectories. The motion on set [14][20] is called r-sliding mode (rth-order sliding mode) [17][29]. The Levant observer will produce estimates of the successive derivatives. HOSM presents even better robust performance than traditional rst order sliding mode. HOSM dynamics converge toward the origin of surface coordinates in nite time always that the order of the sliding controller is equal or bigger than the sum of a relative degree of the plant and the actuator.

= v0 = z1

:

(T Fbx =

Ref Fbx

(7)

:

J! b)

Ref

(8)

! b is produced by the RDE. Note that any estimator with output error can also be used to enhance robustness versus noise. After those estimations, their use in the same time with the system equations allow us to retrieve de vertical forces Fz as follows. To estimate Fz we use the following equation : m (glr h:vbx ) (9) Fbz = 2(lf + lr )

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vbx is produced by the RDE. At this step it only remains to estimate the adherence or friction coef cient. To this end we assume the vehicle rolling in a normal or steady state situation in order to be able to approximate this coef cient by the following formula b=

Fbx Fbz

(10)

IV. SIMULATION AND EXPERIMENTAL RESULTS In this part, we present simulations results in order to test and validate the proposed approach. Different measures are obtained by simulation using the Vedyna software pakage 2 with the simulated vehicle rolling at several speeds. Figure 4 shows the measured and estimated wheel angular position. This signal is used to estimate velocities and accelerations. We can remark that the observations and simulated measures coincide well. Figure 5 shows the estimated wheel velocity. This results shows that there is no additive noise or chattering in the estimations. Figure 6 presents the estimation of the vehicle velocity and shows the good convergence to the actual one. Figure 7 shows the obtained estimation of the vehicle acceleration. The last step gives us the estimated longitudinal forces Fx and normal forces Fz which are presented in gure 8 and 7. Finally road friction coef cent is deduced and presented in gure 10.

Fig. 5.

angular velocity

Fig. 6.

Vehicle velocity

V. CONCLUSION

Fig. 3.

Vedyna

The super twisting second order sliding mode algorithm is modi ed in order to design a velocity and acceleration in board estimator for vehicle dynamics. The nite time convergence allows us to develop a cascad of observers and estimators proceeding step by step in order to avoid observability problems. We have proposed an ef cient and robust second order differentiator to build up an estimation scheme allowing to identify the tire road friction. The estimations, produced in three steps by cascaded observers and estimators, show good performances. Tire forces (vertical and longitudinal ones) are also well estimated. The results show effectiveness and robustness of the proposed method. Implementation of this approach to the test vehicle (P406) is under investigation. ACKNOWLEDGMENTS

Fig. 4.

Angular displacements

J. Davila and L. Fridman gratefully acknowledge the nancial support of the Mexican CONACyT (Consejo Nacional de Ciencia y Tecnologia), grant no. 43807-Y, and of the Programa de Apoyo a Proyectos de Investigacion e Innovacion Tecnolgica (PAPIIT) UNAM, grant no. 117103. This work has been done in the context of a project managed by members of the LSIS inside the GTAA (Groupe Thématique Automatique et Automobile). The GTAA is a research group supported by the CNRS.

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Fig. 7.

Fig. 8.

Estimated and measured acceleration Fig. 9.

Normal force Fz

Fig. 10.

Road friction

Longitudinal force

Many thanks are addressed by the authors to the LCPC of Nantes and to Y. Delanne for experimental data and the trials with their vehicle Peugeot 406. R EFERENCES [1] A. Rabhi, H. Imine, N.K. M'Sirdi, et Yves Delanne « Observers With Unknown Inputs to Estimate Contact Forces and Road Pro le» pp 188193 International Conference on Advances in Vehicle Control and Safety 2004 Italy. [2] N.K. M'sirdi, A. Rabhi, N. Zbiri and Y. Delanne. VRIM: Vehicle Road Interaction Modelling for Estimation of Contact Forces. Accepted for TMVDA 04. 3rd Int. Tyre Colloquium Tyre Models For Vehicle Dynamics Analysis August 30-31, 2004 University of Technology Vienna, Austria [3] Nacer K. M'Sirdi. Observateurs robustes et estimateurs pour l'estimation de la dynamique des véhicules et du contact pneu - route. JAA. Bordeaux, 5-6 Nov 2003. [4] P. F. H. Dugoff and L. Segel. An analysis of tire traction properties and their in uence on vehicle dynamic performance. SAE Transaction, vol 3, pp. 1219-1243, 1970. [5] Pacejka, H.B., Besseling, I.: Magic Formula Tyre Modelwith Transient Properties. 2nd International Tyre Colloquium on Tyre Models for Vehicle Dynamic Analysis, Berlin, Germany (1997). Swets and Zeitlinger. [6] B. Samadi and K.Y. Nikravesh, "A sliding mode controller for wheel slip control," Tehran, Iran, May 1999 [7] R. Kazemi, M. Kabganian and M. R. Modir Zaare, " A new (ABS)," Proceeding of the 2000 SAE automotive dynamics and stability conference, Troy, Michigan, pp. 263-270, May 2000 [8] A. Rabhi, H. Imine, N. M' Sirdi and Y. Delanne. Observers With Unknown Inputs to Estimate Contact Forces and Road Pro le AVCS'04 International Conference on Advances in Vehicle Control and Safety Genova -Italy, October 28-31 2004

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