N K M" Sirdi

A Rabhi

A Naamane

A Nominal Model for Vehicle Dynamics and Estimation of Input Forces and Tire Friction

Keywords: Vehicle dynamics, Sliding Modes observer, Robust nonlinear observers, tire forces estimation

In this paper a 16 DoF vehicle model is developped and discussed. Then some parial models are considered and justified for the design of robust estiamtors using sliding mode approach in order to identify the tire-road friction or input variables.

I. INTRODUCTION

In recent years, increase of safety demand in vehicles motivated research and developement in the field of active safety. More and more new safety systems are installed on vehicle for real-time monitoring and controlling the dynamic stability (EBS, ABS, ESP). Car accidents may occur for several reasons which involve either the driver or vehicle components or environment. One of the important factors determining vehicle dynamics including safety is road friction and evolution of the contact forces. Thus for vehicles and road safety analysis, it is necessary to take into account the contact force characteristics. However, tire forces and road friction are difficult to measure directly. In literature, their values are often deduced by some experimentally approximated models [START_REF] Ackermann | Robust control prevents car skidding[END_REF][9] [START_REF] Gustafsson | Slip-based tire-road friction estimation[END_REF]. Tire forces are represented by nonlinear functions of wheel slip.Generally the partial and approximated models used are not fully justified and their validity is often limited.This makes the forces and parameters difficult to estimate on line for vehicle control applications and detection and diagnosis for driving monitoring and surveillance. In this work we try to highlight approximations made and give details allowing to evaluate what is really neglected. Robust observers looking forward are based on the physics of interacting systems (the vehicle, the driver and the road).

Recently, many analytical and experimental studies have been performed on estimation of the frictions and contact forces between tires and road [START_REF] Rabhi | VRIM: Vehicle Road Interaction Modelling for Estimation of Contact Forces[END_REF] [START_REF] Canudas De Wit | Dynamic Friction Models for Road/Tire Longitudinal Interaction[END_REF]. In [START_REF] Drakunov | ABS control using optimum search via sliding modes[END_REF] [START_REF] Canudas De Wit | Dynamic Friction Models for Road/Tire Longitudinal Interaction[END_REF], application of sliding mode control is proposed. Observers based on the sliding mode approach have been also used in [START_REF] Rabhi | Observers With Unknown Inputs to Estimate Contact Forces and Road Profile[END_REF]. In [START_REF] Ray | Nonlinear state and tire force estimation for advanced vehicle control[END_REF] an estimation based using the least squares method and Kalman filtering is applied for estimation of contact forces. Gustafsson in [START_REF] Gustafsson | Slip-based tire-road friction estimation[END_REF] presented a tire/road friction estimation methode based on Kalman filter to give a relevant estimates of the slope of µ versus slip (λ), that is, the relative differnence in wheel velocity. Carlson in [START_REF] Christopher | Estimation With Applications for Automobile Dead Reckoning and Control[END_REF] presented an estimator for longitudinal stiffness and wheel effective raduis using stok vehicle sensors and GPS for low values of slip. Robust observers with unknown inputs are efficient for estimation of road profile and for estimation of the contact forces [START_REF] Rabhi | Observers With Unknown Inputs to Estimate Contact Forces and Road Profile[END_REF]. Robut observers with unknown inputs have been shown to be efficient for estimation of road profile [12][?] and for estimation of the contact forces. 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. [START_REF] Rabhi | VRIM: Vehicle Road Interaction Modelling for Estimation of Contact Forces[END_REF]. This enhances the road safety leading better vehicle adherence and maneuvers ability. The vehicle controllability in its environment along the road admissible trajectories remain an important open research problem [START_REF] Nacer | Observateurs robustes et estimateurs pour l'estimation de la dynamique des véhicules et du contact pneu -route[END_REF].

In this paper, modelling of the contact forces and interactions between a vehicle and road is revisited in the objective of on line force estimation by means of robust observers coupled with a robust and adaptive estimation for contact forces. We propose an observer to estimate the vehicle state and an adaptive estimator for tire forces identification. The designed observer is based on the sliding mode approach. The main contribution is the on-line estimation of the tire force needed for control.

We focus our work, as presented in this paper, first on modeling and second on on-line estimation of the tires forces. We estimate the vehicle state and identify tire forces. The main contribution is the emphisize of the rational behind partial approximated models and the on-line estimation of the tire force needed for control. Tire forces can be represented by the nonlinear (stochastic) functions of wheel slip. The deterministic tire models encountered are complex and depend on several factors (as load, tire pressure, environmental characteristics, etc.). The proposed estimation procedure has to be robust enough to avoid model complexity. It can then be used to detect some critical driving situations in order to improve the security. This approach can be used also in several vehicle control systems such as Anti-look Brake Systems (ABS), traction control system (TCS), diagnosis systems, etc... The estimations are produced using only the angular wheel position as measurement by the specially designed robust observer based on the super-twisting second-order sliding mode. 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. VEHICLE MODELING 1) Mechanical Models: 4 DOF system: In order to give an idea for the system modeling, we start with a simple example. In this part we develop a 4 Degrees Of Freedom (DOF) model. This means that we consider only the vertical motion (along Oz axis) and the 3 rotations. Let us consider for simplicity a table moving in 4 DOF, made of a rigid body (see figure ??) with a mass M , as length L and wide l. The table thikness is h. We first consider the reference frame R 1 attached to the table and the absolute reference R 0 .

The table is only excited by 4 forces F 1 , F 2 , F 3 and F 4 du to the springs attached at the points:P F T 1 /R 1 .; P F T 2 /R 1 for the rear right and left and P F T 3 /R 1 . P F T 4 /R 1 for the front right and left respectively. Each point is refered by its cartesian coordinates (x, y, z) (distances to the centre of gravity (CDG) / relative to R 1 ). There are no lateral nor longitudinal forces F 5 = F 6 = 0. The inertia tensor is I = diag (Ixx, Iyy, Izz). The table model is obtained applying the Lagrange formalism, considering the generalized coordinate vector q = [z, ψ, ϕ, θ] T ∈ 4 and Γ a the external forces vector. M (q)q + C(q, q) q + G(q) + V (q, q) = Γ a [START_REF] Gustafsson | Slip-based tire-road friction estimation[END_REF] z is the vertical displacement of the gravity center, ψ the roll angle, ϕ the pitch angle, θ the yaw angle. q is the velocity vector and q the accelerations one. The inertia matrix is M (q) and C(q, q) represent the coriolis and centrifugal effects. G(q) = [gM, 0, 0, 0] T is the gravitation effect. Frictions can be considered and are noted V (q, q).The system is passive, then we have the following property: Ṁ (q) -2C(q, q) is an antisymétric matrix.

The external forces applied to the table are expressed in the cartesian space (F T i : longitudinal force, F Li : lateral force, F N i : normal force):

F = [F T 1 , F L1 , F N 1 , F T 2 , F L2 , F N 2 , F T 3 , F L3 , F N 3 , F T 4 , F L4 , F N 4 ] T .
The transversal and lateral exerted forces are assumed nul.

Γ a = J T (q)F (2)
with J(q) the (12 × 4) jacobian matrix defined at the 4 contact points. This describes the fact that contact points are the connections betwen environment (in the absolute reference frame) and the system's body (in its refernce frame).

The applied forces vector is then reduced to only normal force, for simplicity of presentation (this will not be the case of the vehicle, here we have then

F = [F N 1 , F N 2 , F N 3 , F N 4 ] T and then J T (q) =     1 1 1 1 0 0 0 0 J 3,3 J 3,6 J 3,9 J 3,12 J 4,3 J 4,6 J 4,9 J 4,12     (3) M (q) =     M 0 0 0 0 M 2,2 M 2,3 M 2,4 0 M 3,2 M 3,3 0 0 M 4,2 0 M 4,4     (4)
Please note that the inertia matrix is not diagonal despite considering only the chassis alone and that different frames must be considered with geometric and kinematic models relating cartesian and operational spaces.

a) The Coriolis and centrifugal effects matrix C(q, q)::

C(q, q) =     0 0 0 0 0 C 2,2 C 2,3 C 2,4 0 C 3,2 C 3,3 C 3,4 0 C 4,2 C 4,3 0    
With as exemple we give only diagonal coefficients:

C 2,2 = 1/8 φ Iyy sin(2 ϕ+2 θ) + 1/8 φ Iyy sin(2 ϕ-2 θ) -1/4 φ Iyy sin(2 ϕ) -1/8 φ Izz sin(2 ϕ+2 θ) -1/8 φ Izz sin(2 ϕ-2 θ) -1/4 φ Izz sin(2 ϕ) +1/2 φ Ixx sin(2 ϕ) + 1/8 θ Iyy sin(2 ϕ+2 θ) -1/8 θ Iyy sin(2 ϕ-2 θ) +1/4 θ Iyy sin(2 θ) -1/8 θ Izz sin(2 ϕ+2 θ) + 1/8 θ Izz sin(2 ϕ-2 θ) -1/4 θ Izz sin(2 θ) C 3,3 = -1/2 sin(2 θ) (Iyy -Izz ) θ b)
The Transpose Jacobian matrix:: The applied forces vector is then reduced to

F = [F N 1 , F N 2 , F N 3 , F N 4 ] T J =     0 0 1 0 0 1 0 0 1 0 0 1 J 2,1 J 2,2 0 J 2,4 J 2,5 0 J 2,7 J 2,8 0 J 2,10 J 2,11 0 J 3,1 J 3,2 J 3,3 J 3,4 J 3,5 J 3,6 J 3,7 J 3,8 J 3,9 J 3,10 J 3,11 J 3,12 J 4,1 J 4,2 J 4,3 J 4,4 J 4,5 J 4,6 J 4,7 J 4,8 J 4,9 J 4,10 J 4,11 J 4,12    
This 4 DOF model gives only an idea of how much complex is the nominal model of vehicle dynamics. In what follows we restart the procédure to compute the vehicle model assuming it is composed with a body (chassis) with 6 DOF and four wheels attached to this body by four suspensions.

A. Complete 16 DoF model

q T = [x, y, z, θ z , θ y , θ x , z 1 , z 2 , z 3 , z 4 , δ 3 , δ 4 , ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 ]
where x, y, et z represent displacements in longitudinal, laterl and vertical direction. angles of roll, pitch and yaw are θ x , θ y et θ z respectively. The suspensions elongations are noted z i : (i = 1..4). δ i : stands for the steering angles (for wheels numbered as i = 3, 4), finally ϕ i : are angles wheels rotations (i = 1..4.). Vectors q, q ∈ R 16 are respectively velocities and corresponding accelerations. M (q) is the inertia matrix and C(q, q) q are coriolis and centrifugal forces. The gravity term is G . Suspensions forces are V (q, q) = K v q + K p q with respevtively damping and stifness matrices K v , K p . We can define as dynamic equations of the vehicle by applying the principles fundamental of the dynamics (see [START_REF] Beurier | Modélisation et la commande de système[END_REF]):

Γ + J T F = M .. q + C(q, . q) . q + Kq + G (5)
with as parameters only to give an idea

M =       M1,1 M1,2 M1,3 0 0 M2,1 M2,2 M2,3 M2,4 M2,5 M3,1 M3,2 M3,3 0 0 0 M4,2 0 M4,4 0 0 M5,2 0 0 M5,5       C =       0 C12 C13 0 0 0 C22 C23 C24 C25 0 C32 C33 0 0 0 C42 0 0 0 0 C52 0 0 C55       and J T =         J 1,1 J 1,2 J 1,3 J 1,4 J 2,1 J 2,2 J 2,3 J 2,4 J 3,1 J 3,2 J 3,3 0 0 0 0 J 4,4 0 0 0 0 0 0 0 0        
. This is just to show that we can decompose our system as coupled subsystems. Let us say five coupled subsystems, that we have considered in our previous works. This has been computed using a symbolic computation software considering 16 generalized variables: 6 for position and orientation of body, 4 as suspensions ones, 2 for front wheels steering and 4 as wheels rotations. The matrices M , C and K are of dimensions 16 × 16. F is input forces vector acting on wheels, it has 12 components(3 forces ( longitudinal, lateral and normal) × 4 wheels), Γ represent extra inputs for perturbations.. In the following application this model has been reduced and simplified assuming as nominal behaviour a normal driving situation [START_REF] Nacer | Observateurs robustes et estimateurs pour l'estimation de la dynamique des véhicules et du contact pneu -route[END_REF].

B. Coupled sub models

We can split the previous model, without approximations, in five parts as follows. This leads us to Rotations and orientation motions of the body:

  J 4F J 5F J 6F   = M T 12   .. q 1 .. q 2
..

q 3   + M 22   .. q 4 .. q 5
..

q 6   + + M 23     .. q 11
..

q 21 .. q 31 .. q 41     + M 24   .. q 12 .. q 22
..

q 32   + M 25   .. q 42 .. q 33
..

q 43   + C22   . q 4 . q 5 . q 6   + C23     . q 11 . q 21 . q 31 . q 41     + + C24   . q 12 . q 22 . q 32   + C25   . q 42 . q 33 . q 43  
The body's translations dynamics is given by:

  F LT F T T F N T   = M11   .. x .. y .. z   + M 12   .. q 4 .. q 5
..

q 6   + (6)
+ M13     .. q 11
..

q 21 .. q 31 .. q 41     + C12   . q 4 . q 5 . q 6   ( 7
)
Suspensions dynamics is written:

    J 7F J 8F J 9F J 10F     = M1,3   .. q 1 .. q 2
..

q 3   + M 2,3   .. q 4 .. q 5
..

q 6   + + M3,3     .. q 11
..

q 21 .. q 31 .. q 41     + C32   . q 4 . q 5 . q 6   + C33     . q 11 . q 21 . q 31 . q 41     + +     R 11 R 21 R 31 R 41         q 11 q 21 q 31 q 41     +     G 7 G 8 G 9 G 10    
The previous 16 DoF model is then equivalent to the following one:

      F T F 2 F 3 0 U 5       = M        ξ1 ξ2 ξ3 ξ4 ξ5        +C        ξ1 ξ2 ξ3 ξ4 ξ5        +       0 0 R 33 ξ 3 0 0       +       G 1 G 2 G 3 0 0      
In the last expression, we can remark that splitting the model can be realized and this model is helpful, when using reduced models, to identify what is neglected regard to our proposed nominal model with 16 DoF. The dynamic equations can be reduced, in case where we assume that motion is normal driving in a normal straith road, to translations and rotations of the body, and wheels plus supension motions. For translations we find often in literature:

m . v x = F x m . v y = F y m . v z = F z (8)
where m is the total mass of the vehicle and v = [v x , v y , v z ] T describe the vehicle velocities along x, y, z. In the left hand side of this approximate model are the forces F x , F y and F z applied in directions of x, y and z and the balance of the moments ( M x , M y , M z ), give rotations following the three directions x, y and z, is given by:

J    .. θ .. φ .. ψ    =   M x M y M z   (9)
The wheel angular motions can be written:

.

ω f l = 1 I ω (C f l -R ω F xf 1 ) (10)
.

ω f r = 1 I ω (C f r -R ω F xf 2 )
.

ω rl = 1 I ω (C rl -R ω F xr1 )
.

ω rr = 1 I ω (C rr -R ω F xr2 )
with ω f and ω r are the rotation velocities of the front and rear wheel, C mi is the motor couple applied at wheel i and T i is the braking couple applied at wheel i. Let r 1 be the distance between the center of gravity and the front axis and r 2 the distance between the center of gravity and the rear axis.

C. Partial models

The complete model is difficult to use in control applications. It involves several variables which are not available for measurement or not observable. The most part of applications deal with simplified and partial models. Let us consider, for our robust observer, the simplified motion dynamics of a quarter-vehicle model, capturing only nominal behavior [3] [6]. This model retains the main characteristics useful for the longitudinal dynamic. For a global application, this method can be easily extended to the complete vehicle and involve the four coupled wheels. The amount of neglected parts in the modeling can be considered to evaluate robustness of proposed estimators.

Applying Newton's law to one isolated wheel gives:.

m . v x = F x J r . ω = T -rF x
where m is the vehicle mass and J r , r are the inertia and effective radius of the tire respectively. v s is the linear velocity of the vehicle, ω is the angular velocity of the considered wheel. T is the accelerating (or braking) torque, and F x 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 pneumatic tire, has opposite direction of relative motion between the tire and road surface. This relative motion exhibits the tire slip properties. The wheel-slip is due to deflection in the contact patch. The longitudinal wheel slip is generally called the slip ratio and is described by a kinematic relation as [START_REF] Christopher | Estimation With Applications for Automobile Dead Reckoning and Control[END_REF].

λ = |v r -v x | max(v r , v x ) ( 11
)
where v r is the wheel velocity. Representing the adhesion coefficient as a function of the wheel slip yields the adhesion characteristic µ(λ), which depends on the road surfaces as shown in the following figure 2.ftbphFU3.5959in2.0773in0ptW slipWheel slipimfqzl00.wmf

The figure 2 shows the relations between coefficient of road adhesion µ and longitudinal slip λ for different road surface conditions. It can be observed that all curves µ(λ) start at µ = 0 for zero slip, which corresponds to the non-braking and non accelerating, free rolling wheel. With a linear increasing slip ratio from 3% to 20%. Beyond this maximum value the slope of the adhesion characteristic is maximum and then slope becomes negative. At a slip ratio of 100% the wheel is completely skidding, which 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 performance 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 can expressed theses forces as follows, where F z is the vertical force of the wheel.

F x = µF z (12)
The verticla 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:

F z = m 2(l f + l r ) (gl r -h . v x ) ( 13
)
where h is the height of the center of gravity, l f is the distance between the center of gravity and the front axis center of gravityand l r is the distance between the center of gravity and the rear axis center of gravity.

III. OBSERVER DESIGN

The sliding mode technique is an attractive approach [START_REF] Davila | Observation and Identification of Mechanical Systems via Second Order Sliding Modes[END_REF]. The primary characterestic of SMC is that the feedback signal is discontinuous, switching on one or several manifolds in the state-space. 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)

In this part we will use a Robust Differentiation Estimator (RDE) to deduce our estimations. Consider a smooth dynamics function, s(x) ∈ R. The system containing this variable may be closed by some possibly-dynamical 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 t-sliding point set is non-empty and consist locally of Filippov trajectories.

s = . s = .. s = ... = s (r-1) = 0 (14)
is non-empty and consists locally of Filippov trajectories. The motion on set [START_REF] Filippov | Differential Equations with Discontinuous Right-hand Sides[END_REF][17] is called r-sliding mode (rth-order sliding mode) [18][19].

The HOSM dynamics converge toward the origin of surface coordinates in finite 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. To estimate the derivatives s 1 and s 2 without its direct calculations of derivatives, we will use the 2 nd -order exact robust differentiator of the form [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF] .

z 0 = v 0 = z 1 -λ 0 |z 0 -s ω | 2 3 sign(z 0 -s ω ) . z 1 = v 1 = -λ 1 sign(z 1 -v 0 ) 1 2 sign(z 1 -v 0 ) + z 2 . z 2 = -λ 2 sign(z 2 -v 1 )
where z 0 , z 1 and z 2 are the estimate of s ω , s 1 and s 2 , respectively, λ i > 0, i = 0, 1, 2. Under condition λ 0 > λ 1 > λ 2 the third order sliding mode motion will be established in a finite time. The obtained estimates are

z 1 = s 1 = . s ω and z 2 = s 2 = ..
s ω then they can be used in the estimation of the state variables and also in the control.

B. Cascaded Observers -Estimators

In this section we use the previous approach to build an estimation scheme allowing to identify the tire road friction. The estimations will be produced in three steps, as cascaded observers and estimator, reconstruction of informations and system states step by step. This approach allow us to avoid the observability problems dealing with inappropriate use of the complete modeling equations. For vehicle systems it is very hard to build up a complete and appropriate model for global observation of all the system states in one step. Thus in our work, we avoid this problem by means of use of simple and cascaded models suitable for robust observers design.

The first step produces estimations of velocities. The second one estimate the tire forces (vertical and longitudinal ones) and the last step reconstruct the friction coefficient.

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 v x , are assumed available for measurements. The previous Robust Estimator is useful for retrieval of the velocities and accelerations.

1 st Step:

.

θ = v 0 = ω -λ 0 θ -θ 2 3 sign(θ -θ) . ω = v 1 = . ω -λ 1 sign( ω -v 0 ) 1 2 sign( ω -v 0 ) .. ω = -λ 2 sign( . ω -v 1 )
The convergence of these estimates is garanteed in finite time t 0 .

nd

Step: In the second step we can estimate the forces F x and F z . Then to estimate F x we use the following equation, J

.

ω = T -R ef F x (15)
In the simplest way, assuming the input torques known, we can reconstract F x as follows:

F x = (T -J . ω) R ef ( 16
)
.

ω is produced by the Robust Estimator (RE). Note that any estimator with output error can also be used to enhance robustness versus noise. In our work, in progress actually, the torque T wil be also estimated by means of use of additional equation from engine behaviour related to accelrating inputs.

After those estimations, their use in the same time with the system equations allow us to retrieve de vertical forces F z as follows. To estimate F z we use the following equation

F z = m 2(l f + l r ) (gl r -h. . v x ) (17)
v x is produced by the RE. 3 rd Step: At this step it only remains to estimate the adherence or friction coefficient. To this end we assume the vehicle rolling in a normal or steady state situation in order to be able to approximate this coefficient by the following formula

µ = F x F z (18)
IV. SIMULATION AND EXPERIMENTAL RESULTS

In this section, we give somme realistic simulation results in order to test and validate our approach and the proposed observer. In simulation, the state and forces are generated by use of a car simulator called VEDYNA [20]. In this simulator the model involved is more complex than the one of 16 DoF presented in the first part of the paper. Comparind the simplified model to the 16 DoF one, let us evaluate the robustnes of estimation The VeDyna simulated braque torque is shown in figure 3.ftbphFU3.6296in1.9847in0ptBraking torquetorqueFigure Figure 4 showes the measured and estimated wheel angular position. This signal is used to estimat velocities and accelerations.ftbphFU3.5198in1.8844in0ptAngular displacementsAngular displacementsFigure In this work we have tryed to highlight all approximations made in general when using simplified models and this paper gives some details allowing to evaluate what is really neglected. In second part od this paper, we have proposed an effecient and robust estimator baser on the second order sliding mode differentiator. This is used to build an estimation sheme allowing to identify the tire road frictions and input forces which are non observable when using the complete model and standard sensors. The estimations produced finite time converging measurements of model inputs, in three steps by cascaded observers and estimators. This method shows very good performances in simulations conducted using a more complex model (than the 16 DoF one) involved in VeDyna car simulator. Tire forces (vertical and longitudinal ones) are also estimated correctly. Simulation results are presented to illustrate the ability of this approach to give estimation of both vehicle states and tire forces. The robustness versus uncertainties on model parameters and neglected dynamics has also been emphasized in simulations. Application of this approach with inclusion of torque estimation using a simplified model for the engine behaviour, is in progress.

  Figure ?? shows the estimated wheel velocity.ftbphFU3.1868in1.5264in0ptangular velocityAngular velocity-imedgf08.wmfIn the figure ??, we represente the estimation of vehicle velocity. The figure shows the good convergence to the actual vehicle velocity.ftbphFU3.4082in1.6276in0ptVehicle velocityVelocityFigure Figure ?? shows the obtained vehicle acceleration. The observer allows a good estimation of angular velocity and acceleration.ftbphFU3.3676in1.5463in0ptEstimated and measured acceleration AccelerationFigure The last step gives us the estimated longitudinal forces F x and normal forces F z wich are presented in figure 8 and 9.ftbphFU3.2552in1.4529in0ptLongitudinal forceLongitudinal forceFigureftbphFU3.122in1.6094in0ptNormal force F z Normal forceimedi30f.wmf Finally road friction coefficient is deduced and presented in 10.ftbphFU3.1946in1.8481in0ptRoad frictionRoad frictionimedi30g.wmf V. CONCLUSIONS

VI. APPENDIX A

Definition of the matrices involved.in the model.


Lala H Rajaoarisoa
email: lala.rajaoarisoa@lsis.org

Nacer K M'sirdi
email: nacer.msirdi@lsis.org

LIMIT CYCLES STABILITY FOR COMPLEX SYSTEM

Keywords: Limit Cycles, Stability, Multiple Lyapunov Function, Hybrid System, Variable Structure System, Discrete Events System

Limit cycles are common in physical complex systems. However the nonsmooth dynamics of such systems makes stability analysis difficult. This paper gives an extension of Discrete Events System formalism to show the statibilty of Limit cycles using multiple lyapunov function.

INTRODUCTION

Many real systems are characterized by interactions between continuous (smooth or none) dynamics and discrete events. Such systems are common across a diverse range of application areas. Examples include a robotic assembly (McCarragher96 and al.), manufacturing (Pettersson99) and power system (Hiskens00 and al.).

Dynamics of several physical systems can be well described by use of ordinary differential equations despite the fact that some times nonlinearities are introduced and can be taken into account. However, there is as of time this class of systems exhibit a periodic behavior. Discrete events, such as saturation limits, can act to trap evolving system state within a constrained region of state space.

Therefore even when the underlying continuous system are unstable, the discrete events can induce a stable limit set. Limit cycles (periodic behaviour) are created in this way. Limit cycles can be stable (attracting), unstable (repelling) or non-stable(saddle). The stability of periodic behaviour can be defined also as the stability of the fixed point.

So, in this paper we want to study the problems of stability of some class of hybrid systems and how to deal with observability and controllability of such systems. Despite an amount of theory and proposed methods to test observability, it seems to be rather difficult to find an optimal way and efficient rules which cope with some class of hybrid non-linear systems. This is why we have chosen some simple mechanical systems with different phases when operating, or commutation of structures (contact and non contact situations).

MODEL

Background

We choose as illustrative example a mass spring system with a simple structure. But this system has a structure which is near to a vehicle one or a jumping robot.

Definition of a class of systems having variable structures, commutations in their dynamic behavior, non linearities (hard or smooth), non stationarity, varying parameters and other non standard features, is difficult to be done in general. So we can restrict our case to some simple situations with known involved physical phenomena.

The discrete events taken into account and the choice of their effect may be driven by some higher level or simply selected according of optimization of some criterion or performance index (Giambiasi02 and al.), (Allur94).

The switchings and commutations often appear abruptly but changes from one representation to an-other one may be very smooth or not. In another hand we must note also that such systems representation is not unique and differences can appear between the behavioral representation and physical system description or modeling for diagnosis and control.

Discrete Events System (Zeigler76) defines a way to specify systems which states change either upon the reception of an input event or due to the expiration of a time delay.

The system equation can then be written in following form:

   ẋ = f m i (x, u,t) m i : {S, I, O, δ int , δ ext , λ,t αi } (1)
where:

• x is the set of continuous states of system

• u is the set of input controls • S is the set of sequential states • I is the set of external events • O is the set of internal events • δ int : S → S is the internal state transition function • δ ext : S -× I → S + is the external state transition function • λ : S → O is the output function • t αi is the time advance function

Limit Cycles Analysis

In the sequel, we adopted following definitions to approach the stability in dynamical of the system given by the relation (1) (MSirdi98 and al.). We considered the system behavioural in the vicinity of one cyclic trajectory in the goal to analysis the orbital stability.

Limit Cycles: for a second order system with state equation defined by χ = f (χ,t), we define a positive limited set invarying for a boundedness trajectory

χ(t)( χ(t) ) < µ, ∀t > 0 by: -= {p ∈ ℜ n , ∀ξ > 0, ∃t k such that p -χ(t k ) < ξ, ∀k ∈ N} where t k is the time sequencial, with, lim k → ∞ t k = ∞
Orbital Stability: Trajectory of the system in the phase plane ℜ 2 is an orbit stable if:

-

∀ξ > 0, ∃ρ > 0 such that x 0 --(z 0 ) < ρ ⇒ in f x(t) -p < ξ, ∀t > t 0 with p ∈ ℜ n
Asymptotic Orbital Stability: Trajectory of the system in the phase plane ℜ 2 is an asypmtotic orbit

stable if: -x 0 --(.) < ρ ⇒ lim k → ∞ in f x(t) -p = 0 with p ∈ ℜ n 3 System Description

Mass Spring Model

Let us consider the mass-spring system of figure(4), with mass M, stiffness constant k and z 0 the original length of massless spring. The environment is assumed infinitely rigid: k e >> k. If this is not the case let k r be the stiffness of the spring and k the equivalent stiffness of interaction with the ground

(k ∼ = k r k e k r +k e ).
The position of the mass M, in a frame attached to the ground, is noted z. The gravity constant is g = 9.81ms -2 . Let v d = żd > 0 be the lift off velocity and v c = żc < 0 the touch down velocity.

The dynamic interaction with the ground is composed by two phases: flying and stance phases [16][17]. This system is composed by interconnectionof three subsystems (mass, spring and ground) and energy evolutions are: Potential (g) → Kinetic → potential accumulation → potential restitution → Kinetic and so on. We assume the landing without rebounds and no energy loss. Note that the values used in simulations are estimations of equivalent coefficients for out robot SAP [15] (M ∼ = 2.6kg; k = 1100). • Contact Phases. In these phases the controls are active when the springs are in contact.

-The two springs are in contact:

ẋ = f 1 (x, u,t) t ∈ R + and x(t) ∈ D 1 ⊂ R 4    ẋ4 = z = -k l M (z l -z l0 -u l ) -k r M (z r -z r0 -u r ) -g ẋ5 = θ = lk l I (z l -z l0 -u l ) -lk r I (z r -z r0 -u r ) ẋ6 = φ = lk l I (z l -z l0 -u l ) + lk r I (z r -z r0 -u r ) (2) -The right spring is in contact: ẋ = f 2 (x, u,t) t ∈ R + and x(t) ∈ D 2 ⊂ R 4    ẋ4 = z = -k r M (z r -z r0 -u r ) -g ẋ5 = θ = -lk r I (z r -z r0 -u r ) ẋ6 = φ = lk r I (z r -z r0 -u r ) (3) -The left spring is in contact: ẋ = f 3 (x, u,t) t ∈ R + and x(t) ∈ D 3 ⊂ R 4    ẋ4 = z = -k l M (z l -z l0 -u l ) -g ẋ5 = θ = lk l I (z l -z l0 -u l ) ẋ6 = φ = lk l I (z l -z l0 -u l ) (4) • Flying Phase: ẋ = f 4 (x, u,t) t ∈ R + and x(t) ∈ D 4 ⊂ R 4          ẋ1 = x 3 ẋ2 = x 4 ẋ4 = z = -g ẋ5 = θ = 0 ẋ6 = φ = 0 (5)
z r et z l can be expressed in function of z if we assume θ and φ small:

     z l1 = z -l sin θ -d sin φ = z -lθ -dφ z l2 = z + l sin θ -d sin φ = z + lθ -dφ z r1 = z + l sin θ -d sin φ = z + lθ -dφ z r2 = z + l sin θ + d sin φ = z + lθ + dφ

Switching control and supervisor model

As we have said above, our system is composed of two sub-system. One hand with continuous state and in other hand discrete behavioural state.

For this last, we considerate the following phases (Figure2) :

-All springs are in contact with ground -All springs are in flying phase

-O c = Select 1 = Contact model = 1 -O r1C = Select 2 = Right1Contact model = 2 -O l1C = Select 3 = Le f t1Contact model = 3 -O f = Select 4 = Flight model = 4 -O r2C = Select 2 = Right2Contact model = 5 -O l2C = Select 3 = Le f t2Contact model = 6 -O r1l1C = Select 4 = Right1Le f t1Contact model = 7 -O r2l2C = Select 4 = Right2Le f t2Contact model = 8
For this application let us consider the functions ξ r (z r , u r ) and ξ l (z l , u l ) as the external state transition function. So we can write:

δ ext : S -× I → S + = ξ τ (z τ , u τ ) (6)
with τ = l, r which defined the Left or Right spring

This function can be equal to 0 in the flying phase and 1 when there is contact with the corresponding spring. We want give a periodic motion for our system i.e. to damp any rotational motion and maintain hopping along z axis. For obtain this goal, let us consider the following Lyapunov function (energy of the system):

ξ r (z r , u r ) = 1 2 (1 -sign(z r -z 0 -u r )) (7) ξ l (z l , u l ) = 1 2 (1 -sign(z l -z 0 -u l )) (8)
V = 1 2 ż2 + gz + I 2M θ2 + I 2M φ2 (9) V = V LC +V T (1) +V T (2) (10) whit ≡ V LC = 1 2 ż2 + gz V T (1) = I 2M θ2 V T (2) = I 2M φ2 (11)
Energy is splitted in two parts: V LC the energy corresponding to the desired periodic hopping motion and V T (.) the transverse motion energy. It is clear that one of this energy has to be regulated to some level and the other must be damped. The right an left control inputs are

       ξ r1 u r1 = 1 2 u LC -u T (1) -u T (2) ξ l1 u l1 = 1 2 u LC + u T (1) -u T (2) ξ r2 u r2 = 1 2 u LC -u T (1) + u T (2) ξ l2 u l2 = 1 2 u LC + u T (1) + u T (2) ( 12
)
u T is the control which has to damp transverse energy V T and then rotional motions. This has as consequence to keep the system state in the plane (z, ż) with θ = θ = 0 and φ = φ = 0. The u LC has to stabilise priodic cycle (cyclic motion). The two control inputs u T and u LC have to be applyed in the time period where the corresponding spring is in contact with ground. This is made by displacement of the springs attach points u r and u l .

Convergence and stability of the periodic motion

The transverse motion and its energy V T have to be damped. Let us use as Lypunov candidate function

V 1 V 1 = 1 2 V 2 T (1) + 1 2 V 2 T (2) (13)
Its time derivative is:

V1 = V T (1) VT(1) +V T (2) VT(2) (14) VT(1) = I M θθ (15) VT(2) = I M φφ ( 16
)
using expression (??) and ( 16), lead:

VT = lk M (ξ l (z l -z l0 ) -ξ r (z r -z r0 ) (17) + (ξ r u r -ξ l u l ) θ + lk M (ξ l (z l -z l0 ) -ξ r (z r -z r0 ) + (ξ r u r -ξ l u l ) φ
Substitutind controls ξ r u r , ξ l u l by equation ( 12) in VT , we have:

VT = lk M (ξ l (z l -z l0 ) -ξ r (z r -z r0 ) + u T ) θ + lk M (ξ l (z l -z l0 ) -ξ r (z r -z r0 ) + u T ) φ (20)
We propose a transverse control input u T as follows:

u T = ξ r u r -ξ l u l = -Γ 11 ψ(V T (1) ) θ -Γ 12 ψ(V T (2) ) φ (21) -ξ l (z l -z l0 ) + ξ r (z r -z r0 )
ψ is a positive function can be sign or saturation function ( 21). we then obtain:

VT = - lk M Γ 11 ψ(V T (1) ) θ2 - lk M Γ 12 ψ(V T (2) ) φ2 (22) V1 = V T (1) VT(1) +V T (2) VT(2) (23) = -Γ 11 V T (1) ψ(V T (1) ) θ2 (24) -Γ 12 V T (2) ψ(V T (2) ) φ2 ≤ 0 (25)
V T (.) VT(.) is negative then the transverse energy V T converges to zero. We can conclude that

∀ε 1 > 0, ∃t 1 ≥ 0, such as | V T |< ε 1 , ∀t > t 1
Convergence of V LC desired reference V * LC . Let us choose another Lyapunov candidate function

V 2 = 1 2 V 2 T + 1 2 (V LC -V * LC ) 2 (26)
V * LC the constant reference energy is defined at the lft off pont żd or at the maximal desired height z max :

V * LC (0, z d ) = 1 2 ż2 d or V * LC (z max , 0) = gz max when t > t 1 we have (V T , VT ) = (0, 0), then V2 = (V LC -V * LC ) VLC ∀t > t 1 with VLC = (z + g) ż ∀t > t 1 (27) VLC = -ξ l k M (z l -z l0 ) -ξ r k M (z r -z r0 ) (28) + k M (ξ r u r + ξ l u l )ż (29) VLC = -ξ l k M (z l -z l0 ) -ξ r (30) k M (z r -z r0 ) + k M u LC ż (31)
we propose as control input :

u LC = ξ r u r + ξ l u l = -Γ 2 ψ(V -V * LC )ż (32) + ξ l (z l -z l0 ) + ξ r (z r -z r0 )
such as the derivative VLC will be negative:

VLC = - k M Γ 2 ψ(V -V * LC )ż 2 (33) Then V2 = (V LC -V * LC ) VLC (34) = - k M Γ 2 (V LC -V * LC )ψ(V LC -V * LC )ż 2 ≤ 0
The second Lyapunov function ensures global asymptoticconvergence of the system trajectories z to the orbit Ω θ(t), θ(t) converge to zero with the following control:

   u T = -Γ 11 ψ(V T (1) ) θ -Γ 12 ψ(V T (2) ) φ -ξ l (z l -z l0 ) + ξ r (z r -z r0 ) u LC = -Γ 2 ψ(V -V * LC )ż + ξ l (z l -z l0 ) + ξ r (z r -z r0 ) (35)

EXPERIMENTAL RESULTS

-First test:

We want just to show firstly when θ 0 = 0rad and φ 0 = 0rad that our system can respect the limit cycle that we have imposed with the control considered. These conditions give for the simulation in figure For the sequel, we make one test whose the goal is to show the effectiveness of our supervision

CONCLUSION

We propose of this work is to define an approach to identify and then control and supervise such class of complex systems represented by switched models. The system is composed by different sub-models. Each model switches to another instantaneously when the thresholds that define some operating points or zone, is reached by the application of the external or internal transition function. In the goal to build the best prediction of system outputs, we have to get the best switching and supervision device depending on operating point, the behavior and environment. We have illsutred the observability of this class of system with different phases when operating, or commutation of structures (contact and non contact situations). The presented experimental results emphasize efficiency of this approach for modeling, behavior analysis and prediction for such class of complex systems. We have shown, in our first results using this ap- proach, an important difference of performance of the prediction regard to the case when using fuzzy logic for estimation and supervision (Duplaix05).

In a future work, this approach will be used for modelling a vehicle in the goal to make a diagnosis, fault detection and monitoring. A diagnostic framework on this application will be considered to detect defaults and control the system. . Allur, D. L. Dill, A thoery of timed automata, theoretical computer science, ACM, vol. 126, pp.116-146, N2, 1994 .K.MSirdi, N.Manamani and N.Nadjar-Gauthier Methodology based on CLC for control of fast legged robots, Intl. Conference on intelligent robots and systems, proceedings of IEEE/RSJ, October 1998.

. Duplaix, J.F. Balmat, F. Lafont, N. Pessel, Data Analysis For Neuro-Fuzzy Model Approach, STIC-LSIS, Universite du Sud Toulon-Var, Sofa 2005.

Figure 1 :

 1 Figure 1: Mass Spring Model

  (03) good results with the desired height z m = 3.5m. The figure (04) illustrates the model commutation, here we have just cummutation of model 1 to model 4, the transverse energy V T and the cycle in vertical direction (V Lc goes to its imposed reference) and the Figure(05) shows the angle θ and φ equal to zero. -Second test: Now we consider that θ 0 = 0.5rad, φ 0 0rad. With the same conditions for simulation figure(06) shows the good results with the desired height z m = 3.5m obtain after few seconds of simulation, here t = 1.5s. Figure(07) illustrated the switching model result and the moment that corresponds to the necessary time interval to damp the transverse energy V T and stabilize the cycle in vertical direction (V Lc . The Figure (08) shows the convergence of the angle θ. -Third test:

Figure 3 :

 3 Figure 3: Cycle limite stable for θ 0 = 0rad, φ 0 = 0rad

Figure 4 :Figure 5 :Figure 6 :

 456 Figure 4: Energy of the System M when θ 0 = 0rad, φ 0 = 0rad

Figure 7 :Figure 8 :

 78 Figure 7: Energy of the System M when θ 0 = 0.5rad, φ 0 = 0rad

Figure 9 :

 9 Figure 9: Cycle limite stable for θ 0 = 1rad, φ 0 = 0.5rad

Figure 10 :Figure 11 :

 1011 Figure 10: Energy of the System M when θ 0 = 1rad, φ 0 = 0.5rad


A Rabhi

N K M Sirdi

A Elhajjaji

Estimation of Contac Forces and Tire Road Friction

Keywords: Vehicle dynamics, Sliding Modes observer, Robust nonlinear observers, tire forces, estimation

In this paper a robust differentiator via sliding mode is used 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 first produces estimations of velocities and accelerations. The second estimate the tire forces and the last reconstruct the friction coefficient. The actual results show effectiveness and robustness of the proposed method.

I. INTRODUCTION

In recent years, the increasing demand for the safety of an automobile has promoted research and developement of the technology of active safety. One of the important factors determining vehicle dynamics including safety is road friction. Advanced control of automotive vehicle systems is more and more discussed in the open literature and wiedely developed in industry. International Workshops and Journal special issues have been dedicated to automotive control.

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 parametres and variables (forces, velocities, wheel and slip) is essential to advanced vehicle control systems.

However, tire forces and road friction are diffecult 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. Recently, many analytical and experimental studies have been performed on estimation of the frictions and contact forces between tires and road.

In [START_REF] Drakunov | ABS control using optimum search via sliding modes[END_REF][1] application of sliding mode control is proposed. Observers based on the sliding mode approache have been also used in [START_REF] Rabhi | Observers With Unknown Inputs to Estimate Contact Forces[END_REF]. In [4] an estimation based on least squares method and Kalman filtering is applied for estimation of contact forces. In [START_REF] Gustafsson | Slip-based tire-road friction estimation[END_REF] presented a tire/road friction estimation method based on Kalman filter to give a relevant estimates of the slope of µ versus slip (λ), that is the relative difference in wheel velocity.

The paper [START_REF] Christopher | Estimation With Applications for Automobile Dead Reckoning and Control[END_REF] presented an estimator for longitudinal stifness and wheel effective radius using vehicle sensors and GPS for low values of slip. Robust observers with unknown inputs are efficient for estimation of road profile and for estimation of the contact forces [START_REF] Rabhi | Observers With Unknown Inputs to Estimate Contact Forces[END_REF].

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 system, etc... The main characteristics of the vehicle longitudinal dynamics are taken into account in the model used to design robust observer and estimations.

The paper is organised as follows: in section 2 we present the vehicle model and in section 3, the design of the second order sliding mode observer is presented. Some results about the states observation, tire forces and tire road friction are presented in section 4. Finally, some remarks and perspectives are given in a concluding section.

II. VEHICLE MODELING

A. Complete model

In litterature, many studies deal with vehicle modelling [START_REF] Kiencken | Automotive Control Systems[END_REF] [START_REF] Ramirez-Mendoza | Sur la modélisation et la commande des véhicules automobiles[END_REF]. These are complex and nonlinear systems constituted with many subsystems: wheels, mo-tor and system of braking, suspensions, steering, more and more electronic embarked.

The vehicle is a body topic to strengths and external moments following the three axes:

-Longitudinal, -Lateral, -Vertical, We can define as dynamic equations of the vehicle by applying the principles fundamental of the dynamics. The dynamic equations for the movements of transfer of the case are:

m . v x = P F x m . v y = P F y m . v z = P F z (1)
where: m is the total mass of the vehicle. v = [v x , v y , v z ] T : vehicle velocities along x, y, z, P F x , P F y and P F z : forces with respect to x, y and z,

The balance of the moments, following the three directions, is given by:

J    .. θ .. φ .. ψ    =   P M x P M y P M z   (2)
P M x , P M y , P M z : moments with respect to to x, y and z axies.

The wheel angular motion is given by:

.

ω fl = 1 I ω (C fl -R ω F xf 1 ) (3)
.

ω fr = 1 I ω (C fr -R ω F xf 2 )
.

ω rl = 1 I ω (C rl -R ω F xr1 ) . ω rr = 1 I ω (C rr -R ω F xr2 )
with: ω f and ω r are the rotation velocities of the front and rear wheel.

C mi is the motor couple applied at wheel i T i is the braking couple applied at wheel i r 1 is the distance between the center of gravity and the front axis r 2 is the distance between the center of gravity and the rear axis

The model is nonlinear, we can write it :

. x = f (x, u) (4)

B. Partial model

The complete models are difficult to use in control applications. The majority of applications deals with simplified and partial models. Let consider the simplified motion dynamics of a quart-vehicle model, describing only nominal behavior. This model considers 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.

. v x = F x J r . ω = T -rF x
where m is the vehicle mass and J r , r are the inertia and effective radius of the tire respectively. v s is the linear velocity of the vehicle, ω is the angular velocity of the considered wheel.

T is the accelerating (or braking) torque, and F x is the tire/road friction force.

The traction (respectively braking) force, produce at the tire/road interface when a driving (braking) torque is applied to 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 deflection in the contact patch. The longitudinal wheel slip is generally called the slip ratio and is given by a kinematic relation as [START_REF] Christopher | Estimation With Applications for Automobile Dead Reckoning and Control[END_REF].

λ = |v r -v x | max(v r , v x ) (5)
where v r is the wheel velocity.

Representing the adhesion coefficient as a function of the wheel slip yields the adhesion characteristic µ(λ), wich primarily depends on the road conditions surfaces are shown in the following figure 2. The figure 2 shows the variations between coefficient of road adhesion µ and longitudinal slip λ for different road surface conditions. It can be observed that all curves µ(λ) start at µ = 0 for zero slip, wich corresponds to the non-braked, free rolling 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:

F x = µF z (6)
where F z is the vertical force of the wheel.

The verticlal 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:

F z = m 2(l f + l r ) (gl r -h . v x ) (7)
where: h is the height of the center of gravity l f is the distance between the center of gravity and the front axis center of gravity.

l r is the distance between the center of gravity and the rear axis center of gravity.

III. OBSERVER DESIGN

The sliding mode technique is an attractive approach [START_REF] Davila | Observation and Identification of Mechanical Systems via Second Order Sliding Modes[END_REF]. The primary characterestic of SMC is that the feedback signal is discontinuous, switching on one or several manifolds in the state-space. In wat follows, we develop a second order differentiator in order to estimate the tire road friction.

A. High Order Sliding Mode Observer (HOSM)

Consider a smooth dynamics function, s(x) ∈ R.

The system containing this variable may be closed by some possibly-dynamical discontinuous feedback where the control task may be to keep the output s(x(t)) = 0. The sliding variable, the r th order sliding mode is determined by s = . s = .. s = ... = s (r-1) = 0, witch form an r-dimensional condition on the state of dynamic system [13][14].

The Levant observer will produce estimates of the successive derivatives. HOSM presents even better robust performance than traditional first order sliding mode.

HOSM dynamics converge toward the origin of surface coordinates in finite 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.

B. Robust Differentiation Estimator (RDE)

To estimate the derivatives s 1 and s 2 without its direct calculations of derivatives, we will use the 2 ndorder exact robust differentiator of the form [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF] .

z 0 = v 0 = z 1 -λ 0 |z 0 -s ω | 2 3 sign(z 0 -s ω ) . z 1 = v 1 = -λ 1 sign(z 1 -v 0 ) 1 2 sign(z 1 -v 0 ) + z 2 . z 2 = -λ 2 sign(z 2 -v 1 )
where z 0 , z 1 and z 2 are the estimate of s ω , s 1 and s 2 , respectively, λ i > 0, i = 0, 1, 2. Under condition λ 0 > λ 1 > λ 2 the third order sliding mode motion will be established in a finite time. The obtained estimates are z 1 = s 1 = .

s ω and z 2 = s 2 = .. s ω 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 first step produces estimations of velocities. The second one estimate the tire forces (vertical and longitudinal ones) and the last step reconstruct the friction coefficient.

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 v x , are assumed available for measurements. The previous Robust Differentiation Estimator is useful for retrieval of the velocities and accelerations.

1 st Step:

. b θ = v 0 = b ω -λ 0 ¯θ -b θ ¯2 3 sign(θ -b θ) . b ω = v 1 = . b ω -λ 1 sign(b ω -v 0 ) 1 2 sign(b ω -v 0 ) .. b ω = -λ 2 sign( . b ω -v 1 )
The convergence of these estimates is garanteed in finite time t 0 .

2 nd Step:

In the second step we can estimate the forces F x and F z . Then to estimate F x we use the following equation,

J . b ω = T -R ef b F x (8)
In the simplest way, assuming the input torques known, we can reconstract F x as follows:

b F x = (T -J . b ω) R ef (9)
. b ω 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 F z as follows. To estimate F z we use the following equation

b F z = m 2(l f + l r ) (gl r -h. . b v x ) (10)
b v x is produced by the RDE.

rd

Step: At this step it only remains to estimate the adherence or friction coefficient. To this end we assume the vehicle rolling in a normal or steady state situation in order to be able to approximate this coefficient by the following formula

b µ = b F x b F z (11)

IV. SIMULATION AND EXPERIMENTAL RESULTS

In this section, we give somme results in order to test and validate our approach and the proposed observer. In simulation, the state and forces are generated by use a cae simulateur VEDYNA [15].

The braque torque is shown in figure 3.

Fig. 3. Braking torque

Figure 4 showes the measured and estimated wheel angular position. This signal is used to estimat velocities and accelerations.

Figure 5 shows the estimated wheel velocity.In the figure 6, we represente the estimation of vehicle velocity. The figure shows the good convergence to the actual vehicle velocity.

Figure 7 shows the obtained vehicle acceleration. The observer allows a good estimation of angular velocity and acceleration.

The last step gives us the estimated longitudinal forces F x and normal forces F z wich are presented in figure 8 and9. Finally road friction coefficient is deduced and presented in 10. In this paper, we have proposed an effecient and robust second differentiator to build up an estimation sheme allowing to identify the tire road friction. The estimations produced in three steps by cascaded observers and estimator show good performances. Tire forces (vertical and longitudinal ones) are also. Simulation results are presented to illustrate the ability of this approach to give estimation of both vehicle states and tire forces. The robustness of the sliding mode observer versus uncertainties on model parameters has also been emphasized in simulation. Implementation of this approach to the vehicle is under investigation.

Fig. 1 .

 1 Fig. 1. Vehicle model

  m

Fig. 2 .

 2 Fig. 2. Wheel slip

Fig. 4 .Fig. 7 .

 47 Fig. 4. Angular displacements

Fig. 9 .

 9 Fig. 9. Normal force F z


A Rabhi

N K M'sirdi

Vehicle State and Parameters Estimation Using Sliding Mode Observer

Keywords: Nonlinear observer, Sliding Modes, vehicle dynamics, state estimation, slope and road profile

In this paper, we present a Robust Sliding Mode Observer for systems with unknown Inputs. The system considered is a vehicle model with unknown inputs that represent the attributes of the road (slope and banking of the road). This parameters affect the vehicle dynamic performance and behavior properties. Thus for vehicles and road safety analysis, it is necessary to take into account this parameters. However, slope and banking of road are difficult to measure directly. In this work we deal with a simple model of vehicle and we develop a method to observe slope and banking of the road.

I. INTRODUCTION

Studies of accidentologies have shown an important role of the infrastructure in road accidents. Vehicle stability control systems and state estimators commonly use lateral acceleration measurements from accelerometers to calculate lateral acceleration and sideslip angle of the vehicle [START_REF] Tseng | Development of vehicle stability control at Ford[END_REF] [START_REF] Ungoren | Experimental verification of lateral speed estimation methods[END_REF]. These acceleration measurements, however, are easily affected by disturbances such as road bank angle and suspension roll induced by suspension deflection. As a result, many researchers have pointed out that detection of the road bank angle and suspension roll is necessary to have satisfactory performance of such systems [START_REF] Nishio | Development of vehicle stability control system based on vehicle sideslip angle estimation[END_REF][1] [START_REF] Ungoren | Experimental verification of lateral speed estimation methods[END_REF]. Two common techniques for estimating these values are proposed to integrate inertial sensors directly and to use a physical vehicle model [START_REF] Nishio | Development of Vehicle Stability Control System Based on Vehicle Sideslip Angle Estimation[END_REF] [START_REF] Farrelly | Estimation of Vehicle Lateral Velocity[END_REF]. Some methods use a combination or switching between these two methods appropriately based on vehicle states [START_REF] Nishio | Development of Vehicle Stability Control System Based on Vehicle Sideslip Angle Estimation[END_REF] [START_REF] Fukada | Slip-Angle Estimation for Vehicle Stability Control[END_REF]. Direct integration methods can accumulate sensor errors and unwanted measurements from road grade and superelevation (side-slope). In addition, methods based on a physical vehicle model can be sensitive to changes in the vehicle parameters and are only reliable in the linear region.

The main objective of our work is therefore to use more available information on the environment, to drive better and to evaluate the behavior (nominal) of a vehicle in its environment. We develop in this paper a robust observers to estimate the attributes of the road (longitudinal and the transversal slopes) [START_REF] Rabhi | Estimation de la dynamique d'un véhicule en interaction avec son environnement[END_REF].

The paper is organized as follows: in section 2 we present the estimation of the slope of road and in section 3, the design of the observer for estimation of transversal slope of road is developed. Some results about the states observation, and road parameters are presented in section 4.

Finally, some remarks and perspectives are given in a concluding section.

II. ESTIMATION OF THE SLOPE OF ROAD

The slope represent an information useful for the systems of help to driving. It influences the longitudinal dynamics of the vehicle. When braking the normal loads of the wheels depend also on the slope.Among the parameters that largely influence a vehicle's performance, road slopes are the most important. It is therefore important to estimate on line the road slopes explicitly or implicity. In addition to the speed controller, many other controllers like the transmission control unit and the anti-lock brake system can benefit from these estimates.

A. Modeling

We use here the simple longitudinal model, in the case of a longitudinal slope [START_REF] Peter | SCHMIDTBAUER Bent "Road slope and vehicle mass estimation using Kalman filtering" Vehicle system dynamics[END_REF][6] [START_REF] Rabhi | Estimation de la dynamique d'un véhicule en interaction avec son environnement[END_REF].

we have:

M v . v x = F x - 1 2 C ax ρ x v 2 x -M v g sin(α) (1)
J . ω = T f -R e F x (2)
where M v is the vehicle mass and v x is the linear velocity of the vehicle. ω is the angular velocity of the considered wheel.

T is the accelerating (or braking) torque, and F x is the tire/road friction force.

C ax is the drag coefficient and ρ x is air density. α is the slope (road grade).

B. Observer

In this section we develop a robust second order differentiator [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF] to estimate the slope. The estimations will be produced in three steps as cascaded observers and estimator in order to reconstruct information and system states step by step. 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 first step produces estimations of velocities. The second one estimate the tire forces (longitudinal) and the last step reconstruct the slope.

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 v x , are assumed available for measurements. The previous Robust Differentiation Estimator is useful for retrieval of the velocities and accelerations.

1 st Step:

. b θ = v 0 = b ω -λ 0 ¯θ -b θ ¯2 3 sign(θ -b θ) . b ω = v 1 = . b ω -λ 1 sign(b ω -v 0 ) 1 2 sign(b ω -v 0 ) .. b ω = -λ 2 sign( . b ω -v 1 )
The convergence of these estimates is garanteed in a finite time t 0 .

2 nd Step:

In the second step we can estimate the forces F x . Then to estimate F x we use the following equation,

J . b ω = T -R ef b F x (3)
In the simplest way, assuming the input torques known, we can reconstruct F x as follows:

b F x = (T -J . b ω) R ef (4)
. b ω is produced by the RDE. Note that any estimator with output error can also be used for b F x to enhance robustness versus noise.

After those estimations, their use in the same time with the system equations allow us to retrieve de angle α as follows.

α = arcsin " ( b F x -1 2 C ax ρ x b v 2 x -M v . b v x ) M v g # (5)

C. Simulation results

In order to validate our approach, we compared the obtained results with the results given by the Simulator Vedyna while introducing a slope sets up of 15 degrees.

The figure 2 show the estimation of the angular position and velocity. We note that the correct estimation since the values estimated confuse themselves practically with the measures.

The last step gives us the estimated longitudinal forces F x wich are presented in figure 3.

The observer allows a good estimation of longitudinal forces F x . The difference is due to the other friction forces that are not taken into account in this model. In a third We note a good convergence of the estimates in finished time. As consequence of this good estimation, we obtain the estimation of the slope as indicated by the equation 5.

In simulations we have used the simulator Vedyna involving a complete vehicle model. We then note that satisfactory reconstruction of the variables is obtained despite that we have used a very simple model and we neglected the forces of frictions. This emphasizes the robustness of the estimator.

III. ESTIMATION OF THE ROAD BANK ANGLE

Road bank angles have a direct influence on vehicle dynamics and lateral acceleration measurement. A vehicle stability control system that knows road bank angle will have an advantageous capability in achieving desired control sensitivities for maneuvers on ice and snow, among all surfaces, while avoiding false /nuisance activation on a banked road [START_REF] Rabhi | Estimation de la dynamique d'un véhicule en interaction avec son environnement[END_REF].

Figure 6 shows a schematic diagram for a vehicle roll model with road bank angle. It is assumed that the vehicle

φ = (ϕ -ζ r ) (6)
With an accelerometer we can measure the lateral acceleration a ys :

a ys = a y + g sin(φ + ζ r ) (7)
The rotation movement following the x axis is expressed by:

J xx .. φ + C r . φ + K r φ = ma y h + mg sin(φ + ζ r ) (8)
J xx : represents the inertia moment around the x axis C r : rumble coefficient of the vehicle K r : reminder stiffness a ys , a y = lateral accelerometre measurement and bias Let us replace the equation 6 in 8 we are:

J xx .. ϕ + C r . ϕ + K r ϕ = ma ys h + J xx .. ζ r + C r . ζ r + K r ζ r (9)
we suppose that:

.. If we put x 1 = ϕ, the system can be written under the equation form of state:

   x 1 = ϕ . x 1 = x 2 . x 2 = 1 J xx (-K r x 1 -C r x 2 + ma ys h) + K r ζ r (11)
We use the robust sliding mode observer:

ẋ1 = x2 + z 1 ẋ2 = 1 J xx (-K r b x 1 -C r b x 2 + K r ζ r + ma ys h) + z 2 (12)
where x1 and x2 are estimations, and the variables z 1 and z 2 are corrections calculated by the super-twisting algorithm.

z 1 = λ|x 1 -x1 | 1/2 sign(x 1 -x1 ) z 2 = α sign(x 1 -x1 ). ( 13
)
The observer considers the unknown input that represents here the banking as disruptions, to be restored after the convergence in finished time of the estimated states.

IV. Simulation results

We give in this section some results obtained in simulations while using the robust sliding mode observer. We suppose known the different parameters of the model. We did a simulation on VeDyna in which we took the simulated banking as a constant one. The figure 7 present the comparison between the roll angle estimated and the angle given by the Simulator Vedyna. We obtain a good observation.

The estimation of the roll velocity is presented in figure (8), we note a good convergence. Nevertheless, we remarque the appearance of a chattering that comes from to the sign functions.

After the convergence of the states of the system, we restore the unknown input that presents here banking after a low pass filtering. The figure 9 shows the comparison of the estimated road bank angle and the value that we introduced in the simulator VeDyna. We remark finally that the estimated value of the road bank angle converges towards the simulated value.

V. Conclusion

In this paper, we have developed a robust observers with unknown inputs to estimate the attributes of the road (longitudinal and the transversal slopes). A Robust Sliding Mode Observer has been shown efficient for estimation of these unknown Inputs (slope and banking of the road). In this work simple models have been used to develop a robust method to observe slope and banking of the road. The Simulation results involve a complete vehicle model to emphasize performance of the method.

Fig. 1 .

 1 Fig. 1. Presentation of a vehicle in a slope

Fig. 2 .Fig. 3 .

 23 Fig. 2. Estimation of the angular positions and velocity

Fig. 4 .Fig. 5 .

 45 Fig. 4. Estimation of the linaire velocity and acceleration

Fig. 6 .

 6 Fig. 6. Vehicle Roll Model

Fig. 7 .Fig. 8 .

 78 Fig. 7. Estimation of the angle of roulis

C.R.E.A, 7 Rue du Moulin Neuf, 80000 Amiens, France, mail abdelhamid.rabhi@u-picardie.fr

LSIS, CNRS UMR 6168. Dom. Univ. St Jérôme, Av. Escadrille Normandie -Niemen, Marseille France.


N K M'sirdi
email: msirdin@lsis.org

A Boubezoul

A Rabhi

L Fridman

Sliding Modes Observers for Estimation of Performance of Heavy Vehicles

Keywords: Heavy Vehicle Modeling, Sliding Mode Observers, First and second order sliding modes, Estimation of contact forces, unknown input observer

The main objective of this work, is to improve performance handling or maneuverability, by means of the observation of vehicle dynamics in order to obtain safer and an easier driving. A nominal model is proposed to describe the nonlinear dynamic of a tractor and semi-trailer vehicle. The model is developed for the case of cornering manoeuvre at constant speed. First and second order sliding mode observers are developed to estimate the vehicle state. Input lateral forces are estimated in a last step. We focus our work to on-line observation of the system states variables and estimation of the lateral tires forces of heavy vehicles. Simulation results are compared to validate the approach.

I. INTRODUCTION

The work presented in this paper has been done in the context of the national french project ARCOS 2004. The main objective is to develop predictive procedures allowing to detect risky situations and produce alarms.

A large number of car accidents is attributed by statistic studies to increase of presence of heavy vehicles. Statistics on trucks accidents was performed to analyze the road accidents [START_REF] Desfontaines | Centre Européen d'Etudes de Sécurité et d'Analyse des Risques[END_REF]. For the accidents involving at least one truck, the truck is alone in 33 % of the cases. These accident can be divided into three types : 20 % rollover, 11 % the road departure and 2 % jackknifing. The truck structure often concerned by these accidents is the tractor vehicle and the semi trailer. This type of truck is involved for: 45 % of the trucks in the whole database, and 80 % of those involved in a rollover. [START_REF] Desfontaines | Centre Européen d'Etudes de Sécurité et d'Analyse des Risques[END_REF] Truck accidents occur for several reasons involving components of driver-vehicle-environment system. Such situation occurs when the vehicle is driven beyond the stability limits [START_REF] Sirdi | Elaboration d'alertes pour les poids lourds en situations accidentogènes. Contrôlabilité et les critères de stabilité[END_REF]. However more and more new active safety systems are developed and installed on vehicle for real-time monitoring and controlling the dynamic stability (EBS, ABS, ESP). Nevertheless, the possibility of rectifying an unstable condition can be compromised by physical limitations. Therefore, it is extremely important to detect on time a tendency towards instability [START_REF] Sirdi | Elaboration d'alertes pour les poids lourds en situations accidentogènes. Observateur par mode de glissement[END_REF]. This requires well understanding and revisiting vehicles dynamic stability [START_REF] Sirdi | Elaboration d'alertes pour les poids lourds en situations accidentogènes. Contrôlabilité et les critères de stabilité[END_REF].

In literature, several procedures have been proposed to detect instabilities in the vehicle dynamics [START_REF] Dahlberg | Commercial vehicle stability -Focusing on rollover[END_REF] [START_REF] Ervin | Two active systems for enhancing dynamic stability in heavy truck operations[END_REF] [6] [START_REF] Rakheja | Developpement of directional stability criteria for an early warning safety device[END_REF].

In general lateral slips, over steering or roll over situations are detected by means of measurements processing. Other methods use measurements combined with some dynamic model of the vehicle.

The main information needed to prevent risky situations, by efficient prediction, are the vehicle states and input contact forces. This knowlege is necessary for forward prediction of the system's behavior and preview control or safe monitoring.

In this paper, we focus our work to the on-line estimation of the lateral tires forces of heavy vehicle in a cornering manoeuver at constant speed (without breaking and acceleration).

The organization of this paper is as follows. In second section, we develop a simplified model describing the behavior of heavy vehicle. The dynamics equations are deduced by Lagrangian approach assuming a cornering manoeuvre at constant velocity. Two observers are designed in section 3. The first one is based on first order sliding mode approach and backsteppind to estimate the system state and then using the results we deduce the applied tire forces. The second observer uses the super twisting algorithm (second-order sliding mode) to observe states and then identify or estimate the tires forces. The section 4 will discuss the simulation results and validation. A conclusion is given to emphasize interest of these results for predictive diagnosis giving embedded help systems for safe driving.

II. NONLINEAR HEAVY VEHICLES NOMINAL MODEL A. Vehicle Description and motions

The type of heavy vehicle considered in this work is a tractor-semi-trailer with 5-axels (as shown by the scheme of figure 1). In order to estimate the essential dynamics in a cornering manoeuver, we adopt a simple configuration to describe our heavy vehicle [START_REF] Chen | modelling and control of articulated vehicles[END_REF]. The tractor has a body with 2axels and the attached semi-trailer is made of a body supported by 3 grouped axels.

To deduce the model of heavy vehicle, we consider the following assumptions for simplification.

•The pitch and bounce dynamics may be neglected, tractor and trailer are considered as rigid bodies. Only dynamics of two bodies (i.e. tractor and trailer's chassis) are considered.

•The total suspension motions are reduced to the roll of suspension axels only. The pitch and bounce motions are neglected.

•The essential dynamics considered here are the yaw and horizontal translation motions, the tractor roll angle and articulation angle between the tractor and trailer (see figure 2). The trailer's roll angle is measured around the tractor roll axis.

The dynamics equations of the motion of the two sprung masses is written in a coordinate reference frame The System Coordinates and reference frames R E (X E Y E Z E ) attached to the earth (see figure 1). The frames R T (X t Y t Z t ) and R ST (X st Y st Z st ) are attached to the gravity centers of the tractor and semi-trailer's sprung masses (respectively). (X u Y u Z u ) is the frame of tractor's unsprung mass (fixed at center of the front axle with Z u is parallel to Z E , see figure 2).

The relative motion of (X u Y u Z u ) with respect to the earthfixed frame (X E Y E Z E ) is the horizontal translation of the tractor and its yaw motion arround the Z E axis.

The roll motion is described by motion of (X t Y t Z t ) relative to the coordinate (X u Y u Z u ). The articulation between the tractor and trailer is described by relative motion of (X t Y t Z t ) with respect to the coordinate (X t Y t Z t ).

With this coordinate systems and description of their relative motion, we consider the following generalized coordinates:

x E : position of the tractor gravity center in R E , y E : position of the tractor gravity center in R E , ψ : yaw angle of the tractor, φ : roll angle, ψ f : angle between tractor and trailer (relative pitch).

B. Nominal dynamics equation

To obtain the dynamics equations of simplified heavy vehicle we use Lagrangian mechanics. The vehicle motion of each body can be expressed using the kinematics of the different references frames defined. The total kinetic energy (E K ) and potential energy (E P ) are expressed in the frame R E (X E Y E Z E ). The Lagrange approach leads to the following vehicle model [START_REF] Sirdi | Elaboration d'alertes pour les poids lourds en situations accidentogènes. Contrôlabilité et les critères de stabilité[END_REF]:

d dt ∂E K ∂ qi - ∂E K ∂q i + ∂E P ∂q i = F gi M (q)q + C(q, q) q + G(q) = F g (1)
where q i is the i th generalized coordinate and q is the generalized coordinate vector defined as q = [x, y, ψ, φ, ψ f ].

The matrix M (q) represent the symmetric and positive definite inertia matrix. The vector C(q, q) q gives the Coriolis and Centrifugal forces and G(q) is the gravity force vector [START_REF] Sirdi | Elaboration d'alertes pour les poids lourds en situations accidentogènes. Contrôlabilité et les critères de stabilité[END_REF]. The effects of the last tree axels may be regrouped in one equivalent.

As generalized forces, the vector F g represents the wheels -road contact forces acting on the system bodies. This vector Fig. 2. Applied forces on the tractor and semi trailer vehicle and The Motions of the system parts is made of vertical, longitudinal and lateral forces due to contact beetwen the wheels and the road (see figure 2) [START_REF] Pacejka | Magic formula tire with transient properties[END_REF]. To link these tires forces and their effects on bodies motion, an extended bicycle model is used [START_REF] Ackermann | Active steering for better safety, handing and comfort[END_REF][2] [START_REF] Rabhi | VRIM: Vehicle Road Interaction Modelling for Estimation of Contact Forces[END_REF].

The tire-road interface forces F g are related to the suspensions of each wheel through the three axles. Suspensions are modelled as a combination of a spring and a damper elements. Owing to robustness of Sliding Mode aproach, with respect to the modeling errors [START_REF] Utkin | Sliding Mode Control in Electromechanical Systems[END_REF][15] [START_REF] Slotine | Non linear state estimation using sliding observers[END_REF], we use only a simple linear nominal model for suspension.

F sfi = F 0fi + K f z fi + D f żfi F sri = F 0ri + K r z ri + D r żri F sti = F 0ti + K t z ti + D t żti for i = 1, 2 (2)
where F 0i is the static equilibrium force and z i define the deflection of the spring from its equilibrium position with K and D the suspension parameters.

For nominal model, as we consider that the suspension forces are due only to rolling motion, the deflection variables z i are given as:

z f1 = -z f2 = - w f 2 sin(φ) z r1 = -z r2 = -wr 2 sin(φ) z t1 = -wt 2 sin(φ) cos(ψ r ) + l t φ sin(ψ r ) z t2 = wt 2 sin(φ) cos(ψ r ) + l t φ sin(ψ r ) (3)
To include tire forces in the model, we consider a cornering manoeuvre realized at constant speed. Then, the longitudinal forces are assumed nulls. The total tire/road adhesion is considered torward the lateral direction (see figure 3). In this model, the unknown inputs are the lateral tire forces at the front and rear axles of the tractor and the one at the semitrailer equivalent (rear) axle. These forces will be represented by the vector F = (F f , F r , F t ). The vehicle model [START_REF] Desfontaines | Centre Européen d'Etudes de Sécurité et d'Analyse des Risques[END_REF], developed in the inertial frame, depends on the position and orientation of the vehicle in this reference. However, the measurements used generally in vehicles to analyze the dynamics are defined in the vehicle unsprung mass frame. Then, we will rewrite the vehicle model (1) (inertial reference) with respect to this reference frame (unsprung mass reference frame) using the transformation matrices between those coordinates. Then we obtain the following relations:

ẋE cos(ψ) + ẏE sin(ψ) = v x -ẋE sin(ψ) + ẏE cos(ψ) = v y ẍE cos(ψ) + ÿE sin(ψ) = vx -v y ψ -ẍ E sin(ψ) + ÿE cos(ψ) = vy -v x ψ (4)
where ẋE and ẏE are respectively the vehicle velocities in the inertial reference frame. v x and v y are respectively the vehicle velocity components along the axes X u and Y u in the unsprung mass reference frame. The transformation of the generalized forces is obtained in the same way:

F x = F gx cos(ψ) + F gy sin(ψ) F y = -F gx sin(ψ) + F gy cos(ψ) (5)
where F x and F y are the external forces respectively along the X u and Y u . They are expressed in function of lateral tire contact forces, steering angle δ and articulation angle ψ f .

III. ROBUST OBSERVERS DESIGN

To estimate lateral forces, we propose in this section to develop an observer based on the first order sliding mode approach followed by an estimator. This approach is robust versus the model and the parameters uncertainties for state estimation and is able to reject perturbations and uncertainties effects.

A. Model Parametrization

The obtained dynamics equations are written in state form in order to allow design of an observer based on the sliding mode approach [START_REF] Rabhi | Second Order Sliding-Mode Observer for Estimation of Vehicle Parameters[END_REF]. The observer is used to reconstructs the global dynamics and then we can estimate the lateral tires forces. The choice of the sliding mode approach is motivated by its robustness with respect to the parameters and modeling errors [START_REF] Sirdi | Delanne Second Order Sliding-Mode Observer for Estimation of Vehicle Parameters[END_REF]. The state variables of the model expressed in the unsprung mass reference frame are as follows:

ẋ = f (x, δ, F ) (6) x = (φ, ψ f , v x , v y , ψ, φ, ψf ) (7)
with ψ, φ, ψf to represent respectively the yaw, the roll and the rate of change of the articulation angle ψ f . Here F represent the unknown input forces and the steering angle δ represent the known system input [START_REF] Sirdi | Delanne Second Order Sliding-Mode Observer for Estimation of Vehicle Parameters[END_REF].

In our case, we assume available for measurements the roll angle φ, the angle between tractor and trailer (relative yaw at the fifth wheel) ψ f , the yaw velocity ψ and the vehicle velocities v x and v y . The unknown variables are the state components φ and ψf , and lateral tire forces F . The state vector is then split in two parts x T = [x T 1 , x T 2 ] T with:

x 1 = (φ, ψ f )
T measured and x 2 = v x , v y , ψ, φ, ψf T . The system (6) can then be written

   ẋ1 = ρ x 2 ẋ2 = f 1 (x 1 , x 2 ) + f 2 (x 1 , δ, F ) y = x 1 (8)
where ρ = 0 0 0 1 0 0 0 0 0 1 , and f 1 et f 2 are analytic functions defined in 5 .

The function f 1 (x 1 , x 2 ) may be parametrized as:

f 1 (x 1 , x 2 ) = ϕ (x 1 , x 2 , δ) .θ o + ζ
with θ o a vector of nominal system parameters (θ o the nominal values of the vector θ) and, ϕ (x 1 , x 2 , δ) a regression vector depending on well-known functions of (x 1 , x 2 , δ). The remaining term ζ is a small and bounded perturbation representing modeling errors due to use of approximations. The function f 2 (x 1 , δ, F ) may be written [START_REF] Sirdi | Elaboration d'alertes pour les poids lourds en situations accidentogènes. Contrôlabilité et les critères de stabilité[END_REF]:

f 2 (x 1 , δ, F ) = Ω (x 1 , δ) F (9) f 1 (x 1 , x 2 ) = ϕ (x 1 , x 2 , δ) .θ o + ζ (10)
Ω is a matrix in 3x5 . The vector x 2 is composed of both measured variables v x , v y and ψ, and unknown variables φ, ψf . The vector x 2 = (x 21 , x 22 ) T is made of two components, the first part x 21 = (v x , v y , ψ) T is measured and x 22 = ( φ, ψf ) T the unknown variables to be robustly observed. The model may be rewritten in an explicit triangular form with Bounded Input and finite time Bounded State (BIBS) a follows [START_REF] Sirdi | Control approach for legged robots with fast gaits: Controlled Limit Cycles[END_REF]

       ẋ1 = ρx 2 = x 22 ẋ2 = D x 21 x 22 + Ω (x 1 , δ, F ) y = x 1 (11)
The matrix D defined in R 5×5 depends on the state x and Ω is a matrix defined in R 5×3 .

B. First Order Sliding Mode Observer

1) The Backstepping Observer: To estimate both forces and velocities, starting with as measurement x 1 and x 21 , we propose the following sliding mode observer giving the estimates x1 , x22 in two steps [11][12]:

   ẋ1 = x22 + Λ 1 Sign 1 (x 1 -x1 ) ẋ2 = D x 21 x22 + Ω (x 1 , δ) F + η (12) η = Λ 21 0 0 Λ 22 Sign 2 (x 21 -x21 ) Sign 2 (x 22 -x22 ) (13)
Λ 1 , Λ 21 , Λ 22 are observer gains to be adjusted for convergence, F is an a priori estimation of the forces and Sign i is the vector of sign functions for t > t 1 . The auxiliary variable x22 is introduced for the design of the bakstepping triangular observer (see [START_REF] Sirdi | Control approach for legged robots with fast gaits: Controlled Limit Cycles[END_REF] for this observer):

x22 = x22 + Λ 1 Sign 1,moy (x 1 -x1 ) (14)
2) Finite time Convergence of the observer: For the convergence analysis, we have to express the state estimation error (x i = xi -x i ) dynamics equation. Owing to the system triangularity we can study its behavior step by step.

ẋ1 = x22 -Λ 1 Sign 1 (x 1 -x1 ) ẋ2 = ∆ + Ω (x 1 , δ) F -η (15) ∆ = D x 21 x 22 -D x 21 x22 (16)
F = F -F (17)
Step 1: Finite time convergence of x1 to x 1 in t 1 seconds: During this step the second sign function is chosen as null Sign 2 ∼ = 0 for t < t 1 . The observation error dynamic [START_REF] Utkin | Sliding mode and their application in variable structure systems[END_REF] becomes:

ẋ1 = x22 -Λ 1 Sign 1 (x 1 -x1 ) ẋ21 ẋ22 = ∆ + Ω (x 1 , δ) F ( 18
)
Let us recall that the system is BIBS and consider the following Lyapunov candidate function and compute its derivative

V 1 = xT 1 x1 2 (19) V1 = xT 1 (x 22 -Λ 1 Sign (x 1 )) (20)
If we chose Λ 1 = diag (λ 1 , λ 2 ) such as λ i > x22 (i) max for any i = 1, 2, then V1 < 0 and consequently the observation error x1 goes to zero in a finite time t 1 . After t 1 is reached we have ẋ1 = 0. Then after the Fillipov solution [START_REF] Fillipov | Differential equations with discontinuous right hand side[END_REF], we obtain in the mean average x22 (i) = λ i Sign eq (x 1 (i)). Owing to that Sign eq ∼ = Sign moy on the sliding surface (x 1 = 0), we deduce that x22 (i) = x 22 (i) and then x22 = x 22 . Note that Sign moy is the mean of Sign, this can be considered as a low pass filtering used to reduce the chattering effect in sliding modes of the first order.

Step 2 : In this step, we are interested by convergence of x22 in a finite time t 2 . Thereafter the estimation of the unknown input tire forces F can be processed.

Let us first replace the vector Sign 2 by the usual sign functions (t > t 1 )

ẋ1 = 0 = x22 -Λ 1 Sign 1 (x 1 ) ẋ2 = ∆+Ω (x 1 , δ) F -Λ 2 Sign (x 2 )
The second Lyapunov function considered is:

V 2 = xT 1 x1 2 + xT 2 x2 2 (21) V2 = xT 2 ẋ2 f or t > t 1 (22)
V 2 = xT 2 ∆ + Ω (x 1 , δ) F -Λ 2 Sign (x 2 ) (23) Knowing that F is bounded and choosing λ 2 = diag (γ 1 ...γ 5 ) with γ i large enough (γ i > |∆ + Ω (x 1 , δ)| max )
, the convergence of x2 to zero is guaranteed in a finite time t 2 > t 1 then we will have ẋ2 = 0, consequently. Then we obtain:

∆+Ω (x 1 , δ) F -Λ 2 Sign eq (x 2 ) = 0 (24)
3) Unknown Input Estimation and Conclusion: As x22 = x 22 , then as we have chosen D ≈ D and then ∆ ≈ 0.

Let us define Q = Ω T Ω and assume that it is invertible. The observation error dynamic is then reduced to:

F = Q -1 Ω T Λ 2 Sign eq (x 2 ) = F -F (25)
Now, we can define a vector F as being an estimation of forces. Furthemore, after the first and second step (for t > t 2 ) as we have x2 = x 2 , the expression of this vector F becomes:

F = F + Q -1 Ω T Λ 2 Sign moy (x 2 ) (26)
F = F +Q -1 Ω T Λ 21 0 0 Λ 22 Sign 2,moy (x 21 -x21 ) Sign 2,moy (x 22 -x22 )
After time reaches t 2 we have Sign eq (.) ∼ = Sign moy (.), during this second step the signal x2 = x 2 is reached, assuming that conditions of the first step remain valid after t 1 , we can then conclude that for any t > t 2 we have F F in the mean average.

Then the observer proposed (equations ( 12) and ( 14)) with respect to depicted conditions and the gain matrices choices (Λ 1 , Λ 2 ), gives a robust estimation of the global system state (the heavy vehicle dynamics in a cornering) converging in a finite time and the equation (26) gives reconstruction of the unknown input pneumatics tire lateral forces. We have used the robust first order sliding modes approach to estimate the system state in two steps. The robustness versus modeling errors and finite time convergence allow us to avoid knowledge of input in the first step and being able to retrieve them with a simple backstepped procedure.

C. Second Order Sliding Modes 1) Second Order SM Observer: In this subsection we propose an observer based on second-order sliding mode approach, to increase robustness versus parametric uncertainties, modelling errors and disturbances. We propose an observer following the same guidelines as in our previous work in [START_REF] Rabhi | VRIM: Vehicle Road Interaction Modelling for Estimation of Contact Forces[END_REF][13] [START_REF] Sirdi | Delanne Second Order Sliding-Mode Observer for Estimation of Vehicle Parameters[END_REF] applying the approach of [START_REF] Davila | Observation and Identification of Mechanical Systems via Second Order Sliding Mode[END_REF]. As in the previous obsrver x1 and x2 are the state estimations. Let z 1 and z 2 be vectors of observation adjustment given by the super-twisting algorithm defined as follows:

z 1 = λ 1 |x 11 -x11 | 1/2 Sign(x 11 -x11 ) λ 2 |x 12 -x12 | 1/2 Sign(x 12 -x12 ) (27) z T 2 = 0 0 0 Z 2 with Z 2 = α 1 Sign (x 11 -x11 ) α 2 Sign (x 12 -x12 )
Let us know chose as observer equation the following one where the first function (f 1 (x 1 , x 2 ) = ϕ (x 1 , x 2 , δ) θ o + ζ) is omitted like a bounded perturbation (recall that the system is BIBS) in order to be retrieved and estimated later.

ẋ1 = ρx 22 + z 1 ẋ2 = f 2 x 1 , δ, F + z 2 = Ω (x 1 , δ) F + z 2 (28)
F may be any a priori estimation of the forces (eg we can consider it as proportional to the steering angle).

2) Convergence of the Second Order Observer: The observation error dynamics is then

. x1 = ρx 22 -z 1 . x2 = f 1 (x 1 , x 2 ) + Ω (x 1 , δ) F -z 2 (29)
As the system (11 or 8) has an explicit triangular form with Bounded Input and Bounded State (BIBS in finite time) and assuming that saturation is used for the estimated force signals used by the observer, we can easily see that there exist positive constants f + j for j = 1.., 5 such that f 1 (x 1 , x 2 ) + Ω (x 1 , δ) F ≤ f + j . Then we can find α i and λ i satisfying the inequalities:

α 1 > f + 4 α 2 > f + 5 λ 1 > 2 α1-f + 4 (α1+f + 4 )(1+q1) (1-q1) λ 2 > 2 α2-f + 5 (α1+f + 5 )(1+q2) (1-q2) (30)
where i = 1, 2 and q i is some chosen constant, 0 < q i < 1,[?].

The observer (28),(27) for the system (11) ensures then a finite time converging states estimations.

3) Unknown Input forces estimation: In order to reconstruct the unknown lateral forces from the available measures and the robustly observed state we develop an estimator in this subsection. The convergence of x2 in a finite time involves the equalities (which holds in mean average or low pass filtered version):

. x2 = f 1 (x 1 , x 2 ) + Ω (x 1 , δ) F -z 2 = 0 (31) z 2 = f 1 (x 1 , x 2 ) + Ω (x 1 , δ) F (32)
By its definition (27) the term z 2 changes a very high frequency (theoretically infinite). Let us consider a low pass filtered version of this signal Z2 . Z2 = αsign (x 1 ) = f 1 (x 1 , x 2 ) + Ω (x 1 , δ) F = ϕ (x 1 , x 2 , δ) θ o + ζ + Ω (x 1 , δ) F θ o is a known vector of nominal parameters, ϕ (x 1 , x 2 , δ) is a vector of known functions of measurements or state components and ζ is a perturbation term which is rendered as small as possible by the choice of the apriori estimation θ o . We can then retrieve s the signal which will allow us to estimate the unknown input forces F . s = Z2 -θ o ϕ (x 1 , x 2 , δ) = Ω (x 1 , δ) F + ζ Ω T s = Ω (x 1 , δ)

T Ω (x 1 , δ) F + Ω T ζ

Ω T s = Q F + Ω T ζ F = F -F = Q -1 Ω T s -Q -1 Ω T ζ
As Q = Ω T Ω is invertible, the input force expression can be retrieved and we can write :

F = F + Q -1 Ω T Z2 -θ o ϕ (x 1 , x 2 , δ) -Q -1 Ω T ζ (33)
Since after in finite time we have an estimation of the forces

F = F + Q -1 Ω T Z2 -θ o ϕ (x 1 , x 2 , δ) .

IV. SIMULATION RESULTS

In this section, we give some results in order to test and validate our approach an the proposed observers. In simulation, the forces are generated by use of the Magic Formula tire model [START_REF] Pacejka | Magic formula tire with transient properties[END_REF]. The input (Steering angle) of model applied is shown in (4).

Fig. 1 .

 1 Fig. 1. Tractor and semi-trailer vehicle (components);

Fig. 3 .

 3 Fig. 3. The extended Bicycle model

Fig. 4 .

 4 Fig. 4. Steering angle

The performance of this estimation approach is satisfactory since the estimation error is minimal for state variables. So, the unknown parameters converge to their actual values.

V. CONCLUSIONS

In this paper, we have presented a new observation and estimation approach suitable for heavy vehicle. We estimate the lateral forces using observer based first and secondorder sliding mode algorithm. The finite time convergence of the observer is useful for robustness of the forces retrieval. Simulation results are presented to illustrate the ability of this approach to give estimation of both vehicle dynamics states and lateral tire forces. The robustness of the twisting algorithm versus uncertainties on the model parameters has also been emphasized in simulation.