Second Order Sliding Mode Observer for Estimation of Velocities, Wheel Sleep, Radius and Stiffness N.K. M’sirdi1 , A. Rabhi1 , L. Fridman2 , J. Davila2 and Y. Delanne3 1 LSIS, CNRS UMR 6168. Dom. Univ. St J´erˆome, 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 3 LCPC Nantes: Division ESAR BP 44341 44 Bouguenais Cedex

Abstract— This paper uses second-order sliding mode observers to build up an estimation scheme allowing to identify the tire longitudinal equivalent stiffness and the effective wheel radius using the existing ABS angular sensors. This estimation strategy, based on use of the proposed observer could be used with data acquired experimentally to identify the longitudinal stiffness and effective radius of vehicle tires. The actual results show effectiveness and robustness of the proposed method.

I. INTRODUCTION 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 has 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). Tire forces and road friction are difficult 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 [1]– [2]. We focus our work, as presented in this paper, to the on-line estimation of the tires slip, adherence, stiffness and effective radius. We estimate the vehicle state and identify tire forces [3]. The main contribution is the robust online estimation of the tire effective radius, wheel slip and velocities by using only simple low cost sensors. Recently, many analytical and experimental studies have been performed on estimation of the frictions and contact forces between tires and road [4],[5]. 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.) [2],[6] [7]. This makes on line estimation of forces and parameters difficult for vehicle control applications and detection and diagnosis for driving monitoring and surveillance. In [8],[3], application of sliding mode control is proposed. Observers based on the sliding mode approach have been also used in [9]. In [4] an estimation based on least squares method and

Kalman filtering is applied for estimation of contact forces. In [5] presented a tire/road friction estimation method based on Kalman filter to give a relevant estimates of the slope of µ versus slip (λs ), that is, the relative difference in wheel velocity. Robust observers with unknown inputs are efficient for estimation of road profile and for estimation of the contact forces [3],[9]. 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 [3],[9]. From the other hand it is necessary to remark that observers for mechanical systems with unknown inputs based on standard first order sliding mode approach has the following disadvantages [10],[11],[12],[13]: •for observation of the velocity a filtering is needed due to noise. This introduces a time delay and bias; •the need of filtering (converges asymptotically) destroys the finite time convergence property. The design must take care about observation and control(separation principle); •for uncertainties and parameter identification a second filtering is necessary. This leads to a bigger corruption of results. In the last two decades the 2d order sliding-mode algorithms has been designed and applied to some practical needs ( see [11][14][15] and reference therein). In this paper, a nominal modeling of the vehicle is considered in the objective of an on line estimation using the supertwisting based robust exact observer [13] for estimation of velocities, of stiffness and effective radius identification. The super-twisting algorithm proposed in this paper allows •to make the velocity observation without filtering; •provide the finite convergence to the exact value of derivative, ensuring separation principle; •to identify the uncertainties with just one filtering only; •to use least square method for parameters identifications. 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 observer has been used to reconstruct the global system state components and then to estimate the tires forces [3], [9]. The use of sliding mode approach has been motivated by its robustness with respect to the parameters and modeling

errors and has been shown to cope well with this problem.

[6]: (

Fig. 1.

R

ω

λs = vefx − 1 if vx > Ref ω (braking) R ω λs = 1 − vefx if vx < Ref ω (traction)

(4)

Wheel dynamics and the ABS system

We present first, a method to estimate the wheel angular velocities by considering the wheel angular position measurements (produced by an ABS variable reluctance sensor as shown in figure 1). Then in a second step, we estimate the longitudinal stiffness and wheel effective radius. The developed estimations can be used to detect critical driving situations and then improve the security. It can be used also in several vehicle control systems such as Antilook Brake Systems (ABS), Traction Control System (TCS), diagnosis systems, etc... The main characteristics of the vehicle longitudinal dynamics were taken into account in the developed model and used to design robust observer and estimations. The estimations are produced using only the angular wheel position measurements. The proposed method of estimation is verified 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. II. PROBLEM STATEMENT Let us consider the simplified 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. Applying Newton’s law to wheel and vehicle dynamics gives us the equations of nominal motion. q˙ = ω J ω˙ = J q¨ = Tf − Re Fx mv˙ x = Fx

(1) (2) (3)

where m is the vehicle mass, J and Re are the inertia and effective radius of the tire, respectively. vx is the linear velocity of the vehicle, ω is the angular velocity of the considered wheel, Tf 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 an opposed direction to relative motion between the tire and road surface. This relative motion determines the tire slip properties. The wheel - slip is due to deflection in the contact patch. The longitudinal wheel-slip λs is generally called the slip ratio and can be described by a kinematic relation like

Fig. 2. Wheel slip - forces steady state characteristics and 4 Sensors used by ABS 29dot/2π

During ordinary driving, however, the tire slip rarely exceeds 5%. By linearizing the model in small region (around origin), the force slip relation can be approximated as follows vx − Re ω Fx ∼ ) (5) = Cx ( vx Where Fx and Cx are, respectively, the force and the longitudinal stiffness of the tire(s) (with same units because λs is a dimensionless ratio). The dynamic equation of the whole system can be written in state space form by defining the following state variables. The angular position y = x1 = θ is measured by the ABS sensor. The angular velocity x2 = θ˙ = ω is not measured and can be obtained by observer application. The vehicle velocity x3 = vx is assumed measurable in this case. Note that these expressions assume that velocity is non zero by definition. We can write x˙ 1 = x2 (6) Re Cx Re2 Cx x2 Tf − + (7) x˙ 2 = J J J x3 Cx Cx Re x2 x˙ 3 = − (8) m m x3 Now, assuming that J and m are known, we can rewrite the system as follows x˙ 1 = x2 x˙ 2 = a1 u + θ1 ϕ1 x˙ 3 = θ2 ϕ2

(9)

with the regression vectors and parameters to be estimated defined as follows    −1  1 m J and ϕ (x) = ϕ1 (x) = x2 x2 2 − mx Jx3 3     2 θ1 = Re Cx Re Cx and θ2 = Cx Re Cx (10) and with a1 = J1 , u = Tf . In this form, we need knowledge of the system states x1 , x2 , x3 to construct the regressors ϕ1 , ϕ2 in order to estimate the parameters θ1 , θ − 2. The regressor form [18] can’t be obtained directly. However, using second order sliding modes, we can develop a robust state estimation of state x2 . Then, as this observation converges in finite time, by means of perturbation retrieval, we use an adaptive estimator based on recursive least squares.

Owing to the fact that x3 = vx is measurable, the subsystem x˙ 3 = θ2 ϕ2 (z) can be considered separately and we can use a first order Sliding Mode Observer as has already been done in [9],[3]. Note also that θ22 = θ11 can be estimated in two ways which can be combined. Let us consider first the main subsystem with as state variables x1 = q, x2 = q, ˙ and the control input u = U (t, x1 , x2 ) (may be computed in function of the system states or their estimates), this submodel (6) can be rewritten in the state space form as follows: x˙ 1 = x2 , x˙ 2 = f (t, x1 , x2 , u) + ξ(t, x1 , x2 , u) y = x1 ,

(11)

where the nominal part of the system dynamics is represented by the function f (t, x1 , x2 , u) = a1 u containing the known nominal functions, while uncertainties are concentrated in the term ξ(t, x1 , x2 , u) = θ1 ϕ1 (z). The system (11), understood in Filippov’s sense [16] is assumed such that the functions f (t, x1 , x2 , U (t, x1 , x2 )) and the perturbation ξ(t, x1 , x2 , U (t, x1 , x2 )) are Lebesgue-measurable and uniformly bounded in any compact region of the state space. Our task is then to design a finite-time convergent observer of the velocity x2 = θ˙ assuming that only the position x1 = θ and nominal model are available. Only the scalar case x1 , x2 ∈ R is considered for simplicity. In general case the observers are constructed in exactly the same way for each wheel position variable x1j in parallel. III. OBSERVER DESIGN The proposed super-twisting observer has the form (for the two first states) [13] x ˆ˙ 1 = x ˆ 2 + z1 x ˆ˙ 2 = f (t, x1 , x ˆ2 , u) + z2

(12)

where x ˆ1 and x ˆ2 are the state estimations, and the correction variables z1 and z2 are calculated by the super-twisting algorithm z1 = λ|x1 − x ˆ1 |1/2 sign(x1 − x ˆ1 ) z2 = α sign(x1 − x ˆ1 ).

(13)

It is taken for ensures observer convergence that at the initial moment x ˆ1 = x1 and x ˆ2 = 0. In order to estimate the state vector x = (x1 , x2 , x3 ) , Cx and Ref we propose the following sliding mode observer: . p b1 |sign(x1 − x b1 ) (14) x b1 = x b2 + λ |x1 − x .

x b2 = a1 u + αsign(x1 − x b1 )

(15)

x b3 = βsign(x3 − x b3 )

(16)

.

where x bi represent the observed state vector and λi is the gain. Taking x ˜1 = x1 − x ˆ1 and x ˜2 = x2 − x ˆ2 we obtain the equations for the error x ˜˙ 1 = x ˜2 − λ|˜ x1 |1/2 sign(˜ x1 ) x ˜˙ 2 = F (t, x1 , x2 , x ˆ2 ) − α sign(˜ x1 ) x ˜˙ 3 = θ2 ϕ2 − β sign(˜ x3 )

(17)

where F (t, x1 , x2 , x ˆ2 ) = f (t, x1 , x2 , U (t, x1 , x2 )) − f (t, x1 , x ˆ2 , U (t, x1 , x2 )) + ξ(t, x1 , x2 , U (t, x1 , x2 )). In our case, the system states are bounded, then the existence of a constant f + is ensured such that |F (t, x1 , x2 , x ˆ2 )| < f +

(18)

holds for any possible t, x1 , x2 and |ˆ x2 | ≤ 2 sup |x2 |. The state boundedness is true, because the system (11) is BIBS stable, and the control input u = U (t, x1 , x2 ) is bounded. The maximal possible acceleration in the system is a priori known and it coincides with the bound f + . Let α and λ satisfy the following inequalities, where p is some chosen constant, 0 < p < 1 λ>

+ q α>f ,+

(α+f )(1+p) 2 , α−f + (1−p)

(19)

Theorem 1: The observer (12),(13) for the system (11) ensures the finite time convergence of estimated states to the real states, i.e. (ˆ x1 , x ˆ2 ) → (x1 , x2 ). Proof: To prove the convergence of the state estimates we follow the same way as in [13]. First, we show the convergence of x ˜1 and x ˜˙ 1 to zero. Assume that (18) holds all the time (it will be proved further). As follows from (17), (18), estimation errors x ˜1 and x ˜2 satisfy the differential inclusion (in the Filippov sense), x ˜˙ 1 = x ˜2 − λ|˜ x1 |1/2 sign(˜ x1 ), + ˙x ˜2 ∈ [−f , +f + ] − α sign(˜ x1 ).

(20)

which means that the right hand side is enlarged in some points in order to satisfy the upper semicontinuity property [16]. In particular the second formula of (20) turns into ˜1 = 0. Note that the solutions x ˜˙ 2 ∈ [−α−f + , α+f + ] with x of (20) exist for any initial condition and are infinitely extendible in time [16]. Computing the derivative of x ˜˙ 1 with x ˜1 6= 0 obtain ˜˙ 1 ¨˜1 ∈ [−f + , f + ] − ( 1 λ x x + α sign x ˜1 ). 2 |˜ x1 |1/2

(21)

d The trivial identity dt |x| = x˙ sign x is used here. Note that at the initial moment x ˜1 = 0 and x ˜2 = x2 − 0 = x2 . The trajectory enters the half-plane x ˜1 > 0 with a positive initial value of x2 and the half-plane x ˜1 < 0 otherwise. Let x ˜1 > 0 then with x ˜˙ 1 > 0 the trajectory is confined between the axis x ˜1 = 0, x ˜˙ 1 = 0 and the trajectory of the + ¨ equation x ˜1 = −(α−f ). Let x ˜1M be the intersection of this curve with the axis x ˜˙ 1 = 0. Obviously, 2(α−f + )˜ x1M = x ˜˙ 210 , ˙ ˙ where x ˜10 > 0 is the value of x ˜1 with x ˜1 = 0. It is easy to see that for x ˜1 > 0, x ˜˙ 1 > 0

1 x ˜˙ 1 ¨˜1 ≤ f + − α sign x x ˜1 − λ < 0. 2 |˜ x1 |1/2 Thus, the trajectory approaches the axis x ˜˙ 1 = 0. ˙ The majorant curve for x ˜1 > 0, x ˜1 ≥ 0 is described by the equation x ˜˙ 21 = 2(α − f + )(˜ x1M − x ˜1 ) with x ˜˙ 1 > 0.

The majorant curve for x ˜1 > 0, x ˜˙ 1 ≤ 0 consists of two parts. In the first part the point instantly drops down from 1/2 (˜ x1M , 0) to the point (˜ x1M , − λ2 (f + + α)˜ x1M ), where, in the ”worst case”, the right hand side of inclusion (21) is equal to zero. The second part of the majorant curve is the horizontal 1/2 x1M ) = segment between the points (˜ x1M , − λ2 (f + + α)˜ (˜ x1M , x ˜˙ 1M ) and (0, x ˜˙ 1M ). |x ˜˙ | Condition (19) implies that |x˜˙1M| < 1−p 1+p < 1. 10 Let us denote as x ˜˙ 10 , x ˜˙ 1M = x ˜˙ 11 , x ˜˙ 12 , ..., x ˜˙ 1i , ... the consequent crossing points of the system (17) trajectory starting at (0, x ˜˙ 10 ) with the x ˜1 = 0 axis. Last inequality ensures the convergence of the state (0, x ˜˙ 1i ) to x ˜1 = x ˜˙ 1 = 0 ˙ and, moreover, the convergence of Σ∞ | x ˜ |. 1 0 i To prove the finite time of convergence consider the dynamics of x ˜2 . Obviously, x ˜2 = x ˜˙ 1 at the moments when x ˜1 = 0 and taking into account that x ˜˙ 2 = F (x1 , x2 , x ˆ2 , u) − α sign x ˜1 obtain that

It was assumed that the term z2 changes at a high (infinite) frequency. However, in reality, various imperfections make the state oscillate in some vicinity of the intersection and components of z2 are switched at finite frequency, this oscillations have high and slow frequency components. The high frequency term z2 is filtered out and the motion in the sliding mode is determined by the slow components [17]. It is reasonable to assume that the equivalent control is close to the slow component of the real control which may be derived by filtering out the high-frequency component using low pass filter. The filter time constant should be sufficiently small to preserve the slow components undistorted but large enough to eliminate the high frequency component. Thus the conditions τ → 0 where τ is the filter time constant, and δ/τ → 0, where δ is the sample interval, fulfilled to extract the slow component equal to the equivalent control and to filter out the high frequency component. In order to solve the parameters identification problem, system (11) can be written as

0 < α − f + ≤ |x ˜˙ 2 | ≤ α + f + x˙ 1 = x2 x˙ 2 = a1 u + θ1 ϕ1 + ξ

holds in a small vicinity of the origin. Thus |x ˜˙ 1i | ≥ (α − f + )ti , where ti is the time interval between the successive intersection of the trajectory with the axis x ˜1 = 0. Hence ti ≤

|x ˜˙ 1i | (α − f + )

where θ1 is a vector of parameters, ϕ1 is a vector of known functions and ξ is a perturbation term. Introducing the new notations, the super-twisting observer is given by x ˆ˙ 1 = x ˆ2 + λ|˜ x1 |1/2 sign(˜ x1 ) ˙x ¯ ˆ2 = a1 u + θ1 ϕ1 (t, x1 , x ˆ2 ) + α sign(˜ x1 )

and the total convergence time is given by T ≤

X

where θ¯1 are the nominal values of the vector θ1 . The finite time convergence of x1 and x2 allows to write the equation (22) in the form:

|x ˜˙ 1i | (α − f + )

So T is finite and the estimated states converge to the real states in finite time. The proof was based on inequality (18). As follows from the above consideration, sufficiently large f + provides for |˜ x2 | ≤ |˜ x2,0 | = |x2 (t0 )|, where t0 is the initial time. It implies that |ˆ x2 | ≤ |˜ x2,0 | + |x2 | ≤ 2 sup |x2 |. Hence, the suggested choice of f + is valid. Remark The only requirement for its implementation is the boundedness of the function F (t, x1 , x2 , x ˆ2 , u).

where ∆θ = θ1 − θ¯1 , ϕ1 (t, x1 , x2 , u) is a nonlinear vector containing all the functions of the states and z¯2 is the low pass filtered version of z2 . If ξ = 0 and only nominal values of the system parameters are known, the value of z¯2 is given by h(t) = z¯2 = ∆θϕ1 (t, x1 , x2 , u)

IV. PARAMETER IDENTIFICATION To proceed we will consider, for clarity of presentation only, the estimation procedures in two steps, one for x2 and one for x3 in order to estimate respectively θ1 and then θ2 . A. Identification of θ1 The convergence of x2 in a finite time ensures that the equality x ˜˙ 2 = g(t, x1 , x2 , x ˆ2 , u) − z2 = 0 holds after some finite time T . Then z2 = f (t, x1 , x2 , u) − f (t, x1 , x ˆ2 , u) + ξ(t, x1 , x2 )

z¯2 = α sign(˜ x1 ) = ∆θϕ1 (t, x1 , x2 , u) + ξ

(22)

(23)

We can then apply the Recursive Least Square (RLS) identification algorithm to estimate the parameter vector with the knowledge of z¯2 the regression vector deduced from the measurements and observations of ϕ1 . The model structure for the linear regression [18],[13] can be written as in equation (23) where h(t) is a measurable quantity, ϕ1 (t) is a regression vector made of known quantities and ∆θ is the unknown parameters vector (difference to the nominal parameters). The application of linear regression algorithms like the RLS (Recursive Least Squares) parameter estimation algorithm which can be written

V. EXPERIMENTAL RESULTS ˆ h(t)

c 1 (t, x1 , x2 , u) = z¯2 = ∆θϕ ˆ = ∆θϕ f 1 (t, x1 , x2 , u) ε(t) = h(t) − h(t) σ ˙ c f ∆θ = Γt ϕ1 (t, x1 , x2 , u)ε(t) = Γt ϕ1 ϕT1 ∆θ γt σ Γ˙ t = − Γt ϕT1 (t)ϕ1 (t)Γt γt

(24) (25) (26) (27)

c is the estimation of ∆θ the parameters vector where ∆θ ˆ and h(t) the prediction of the signal h(t). In general γt is a normalization term γt = 1+ϕ1 (t)Γt ϕT1 (t) and σ ∈ [0.9, 1] a forgetting factor. The initial conditions of the RLS algorithm c = ∆θ d0 initial are Γ0 = ρ−1 I initial gain matrix and ∆θ parameters values. Theorem 2: for the system (24) using the RLS algorithm (24) ensures the following properties: T

f −1 f (i) ∆θ function and we have

2

Γ t2 ∆θ is a non increasing

f λ min(ΓO ) f

∆θ0

∆θt ≤

In this section, we present some experimental results to validate our approach. Several trials have been done with a vehicle (P406 of LCPC) equipped with sensors for wheels angular position measurement. Measures have been acquired with the vehicle rolling at several speeds. The experimental data used here are those of the rear wheel drive. The installed sensors at each wheel are the variable reluctance ones of the Antilock Braking System (ABS see figure ). Their resolution is 29 dot per revolution (ie θ(i) = 2πn(i) 29 rad). An additional encoder (with 1000dot per turn ie θ(i) = 2πn(i) 1000 rad) have been installed for angular position measurements control and validation. Data are sampled at 1kHz frequency and several trials have been considered at different running speeds (40 Km/h, 60 Km/h, 80 Km/h, 100 Km/h and varying velocity) with and without using the ABS system. The velocities can be deduced by several ways from the displacement measurements. Here we compare three of them:

λ max(Γt )

ˆ ∈ L2 (ii) ε˜(t) = ( γσt )1/2 (h(t) − h(t)) Remark 3: The use of equations (24) ensures the asympc to ∆θ under the persistent excitation totic convergence of ∆θ condition [18]. Remark 4: In application, we have considered the delta operator for approximation of the derivation. B. Identification of θ2 Let us consider know the third equation of the observation ˜23 error dynamics (17). The Lyapunov function V (˜ x3 ) = 12 x has as time derivative V˙ (˜ x3 ) = x ˜3 x ˜˙ 3 = x ˜3 (θ2 ϕ2 − β sign(˜ x3 ))

In the figure (4) we can see that estimation of velocity signal derivation needs a filtering to reduce the noise effect. In figure (3) corresponding to low resolution encoders the problem is worse and amplitude of noise has a higher level. Filtering this data will affect the measurement precision.

(28)

If β is chosen such that β > max → (θ2 ϕ2 ) then V˙ (˜ x3 ) < 0 this shows that x ˜3 goes to zero. It can be shown that it converges in finite time also. Then, the low frequency components of the signal z3 = β sign(˜ x3 ) (or in average or by low pass filtering of z3 ) we will have x3 ) an then x ˜˙ 3 = 0 = θ2 ϕ2 − βsign(˜ x3 ) = θ2 ϕ2 = z¯3 = βsign(˜

θ(i) − θ(i − 1) T θ(i + 1) − θ(i − 1) ˙ θ(i) = 2T ˙ x ˆ2 = observed(θ)

˙ θ(i) =

Fig. 3.

Estimated velocities using the high sensors (1000dot/t)

θ21 1 x2 − θ22 m m x3

Remark 5: In the same way θ21 , assuming θ22 known or already estimated, can be identified using the Recursive Least Squares algorithm (RLS). Remark 6: Note also that both parameters in θ2 can be estimated by RLS at this step. This correspond to estimating twice θ22 , assuming at this step as previous estimation the value produced by the previous step. Remark 7: Note also that depending on the expression formulated for the forces and wheel slip in (5), (4) several ther variables can be estimated like adherence or longitudinal forces.

Fig. 4.

Estimated velocities using the ABS sensors (29dot/t)

We remark that when using the proposed observer (bottom left curve in the two figures) that the estimation remain precise despite the bad resolution of the sensors used by ABS. the observed and reconstructed velocities are compared to the measure provided by a high resolution encoder. These curves show the robustness of our observer based on second order sliding modes and super twisting algorithm versus measurement noise and additional perturbations. Recall that

Fig. 5.

Estimated velocity using the ABS sensors

has been tested on experimental data (acquired with a P406 vehicle) and shown to be very efficient using only standard sensors. The actual results prove effectiveness and robustness of the proposed method. In our further investigations we consider also the case of complete vehicle in a road with changing adherence. The estimations produced on line will be used to define a predictive control to enhance the safety. ACKNOWLEDGMENTS

the term ξ(t, x1 , x2 , u) = ϕ1 (z)θ1 is not known and correspond to a perturbation to be rejected in a first step; thank to the finite time convergence. In a second step (after the convergence time) this perturbation is retrieved by use of a low pass filtering and then the parameters θ1 can be estimated. The second step estimations are the wheel radius and its longitudinal equivalent stiffness. These estimations are shown in Fig.(5 and 6)) The estimated parameters are

J. Davila and L. Fridman gratefully acknowledge the financial 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 a collaboration managed by members of the LSIS inside the GTAA a research group supported by the CNRS. Thanks are addressed to LCPC of Nantes for experimental data and the trials with their vehicle. R EFERENCES

Fig. 6. Estimated wheel Stiffness and Radius using the ABS sensors and observations

quite good and the algorithm is very easy to apply and is not difficult to tune its parameters. VI. CONCLUSION The super-twisting second-order sliding-mode algorithm is modified in order to design a velocity observer for vehicle using only the ABS sensors already placed in standard vehicles nowadays. The finite time convergence of the observer is proved and consequently the separation principle can be considered as avoided. The gains of the proposed observer are chosen very easily ignoring the system parameters. This observer is compared, using experimental data, to classical derivation methods and is proven robust despite the bad resolution of the encoders. Its robustness combined with a sliding mode estimation of the vehicle velocity allow us to reconstruct the wheel slip. In this way, the observability problems are avoided by means of cascaded finite time converging observer instead of additional sensors. It can be shown that contact forces can also be estimated by this way. The finite time convergence of state observations in the same time as robustness and perturbation rejection allows to solve the problem of parameter identification using the equivalent control method (by retrieval of the rejected signal). The use of the equivalent control, which provides a linear regression model, allows to apply the classical parameter identification methods (RLS) to estimate the systems dynamic parameters like the tire longitudinal equivalent stiffness and the effective wheel radius. The estimation scheme built by using a Second Order Sliding Mode observers and a Sliding Mode velocity estimator

[1] B. Samadi and K.Y. Nikravesh, ”A sliding mode controller for wheel slip control,” Tehran, Iran, May 1999 [2] P. F. H. Dugoff and L. Segel. An analysis of tire traction properties and their influence on vehicle dynamic performance. SAE Transaction, vol 3, pp. 1219-1243, 1970. [3] N.K. M’sirdi, A. Rabhi, N. Zbiri and Y. Delanne. VRIM: Vehicle Road Interaction Modelling for Estimation of Contact Forces. MVDA 04. 3rd Int. Tyre Colloquium Tyre Models For Vehicle Dynamics Analysis August 30-31, 2004 University of Technology Vienna, Austria [4] Laura Ray. Nonlinear state and tire force estimation for advanced vehicle control. IEEE T on control systems technology, V3, N1, pp117-124, march 1995, [5] F. Gustafsson, ”Slip-based tire-road friction estimation”, Automatica, vol 33, no. 6, pp. 1087-1097, 1997. [6] Pacejka, H.B., Besseling, I.: Magic Formula Tyre Modelwith Transient Properties. 2nd International Tyre Colloquium on Tyre Models for Vehicle Dynamic Analysis, Berlin, 1997. Swets-Zeitlinger. [7] C. L. Clover, and J. E. Bernard, ”Longitudinal Tire Dynamics,” Vehicle System Dynamics, Vol. 29, pp. 231-259, 1998. [8] C.Canudas de Wit, P.Tsiotras, E.Velenis, M.Basset, G.Gissinger. Dynamic Friction Models for Road/Tire Longitudinal Interaction. Vehicle Syst. Dynamics 2003. V39, N3, pp 189-226. [9] A. Rabhi, H. Imine, N. M’ Sirdi and Y. Delanne. Observers With Unknown Inputs to Estimate Contact Forces and Road Profile AVCS’04 International Conference on Advances in Vehicle Control and Safety Genova -Italy, October 28-31 2004 [10] J.P. Barbot, M. Djemai, and T. Boukhobza. Sliding Mode Observers; in Sliding Mode Control in Engineering, ser.Control Engineering, no. 11, W. Perruquetti and J.P. Barbot, Marcel Dekker: New York 2002. [11] G. Bartolini, A. Pisano, E. Punta, and E. Usai. ”A survey of applications of second-order sliding mode control to mechanical systems”, International Journal of Control, vol. 76, 2003, pp. 875-892. [12] A. Pisano, and E. Usai, ”Output-feedback control of an underwater vehicle prototype by higher-order sliding modes ”, Automatica, vol. 40, 2004, pp. 1525-1531. [13] J. Davila and L. Fridman. “Observation and Identification of Mechanical Systems via Second Order Sliding Modes”, 8th. International Workshop on Variable Structure Systems,September 2004, Espana [14] A. Levant, ”Sliding order and sliding accuracy in sliding mode control,” International Journal of Control, vol. 58, 1993, pp 1247-1263. [15] Y.B. Shtessel, I.A. Shkolnikov, and M.D.J. Brown, ”An asymptotic second-order smooth sliding mode control”, Asian Journal of Control, vol. 5(4), 2003, pp 498-504. [16] A.F. Filippov, Differential Equations with Discontinuous Right-hand Sides, Dordrecht, Netherlands: Kluwer Academic Publishers. [17] V. Utkin, J. Guldner, J. Shi, Sliding Mode Control in Electromechanical Systems, London, UK:Taylor & Francis; 1999. [18] T. Soderstrom, P. Stoica, System Identification, Cambridge, Great Britain:Prentice Hall International; 1989.