1

Sliding mode based controller / observer to regulate Vehicle wheelslip A. El Hadri+ , J.C. Cadiou+ , N.K. M’Sirdi+ and Y. Delanne◦ + University of Versailles Laboratoire de Robotique de Paris 10-12 Avenue de L’europe, 78140 V´elizy, FRANCE ◦ Laboratoire Central des Ponts et chauss´ ees, Centre de Nantes. Phone: (33) 1.39.25.49.70 Email [email protected]

Abstract— This paper presents a nonlinear observer and controller based on sliding mode appraoch for vehicle traction control. The concept of relaxation length is used to describe the wheel-slip variation (first order model of wheel-slip). Index Terms— Nonlinear observer, state estimation, sliding mode, tire forces, wheelslip, relaxation length.

I. I NTRODUCTION The active safety is becoming important due to recent research on Intelligent Transportation Systems (ITS) technology. Several safety systems has been developed for cars (ABS, TCS, Collision warning/avoidance control system, etc...). The knowledge of the characteristics of the tire/road adhesion are a most important factor to achieve these control systems. It was shown that tire forces influence the vehicle dynamic performance [2]. However, tire forces or tire/road adhesion (µ) are difficult to measure directly. Their values are often deduced by some approximate dynamic tire models. The tire models proposed in literature are complex and depend on several factors (load, tire pressure, environmental characteristics, etc.), (see references [3], [4], [5], [6], [7], [8]). These models has been developed particularly for vehicle dynamic simulations and analysis. They depend on several parameters and are generally derived from experimental data based on machine tests. For vehicle control systems, the knowledge of tire forces can be done through estimation methods (see references [9], [10], [12]), thus many hard-to-measure signals have to be observed or estimated. The vehicle traction control is realized to obtaining a desired vehicle motion, maintaining vehicle stability and steerability, ensure safety, etc. Many advanced researches in this field was developed by using some recent approaches of control with respect to the conventional controllers which are table-driven [?], [?], [?]. These works use for control the kinematic relationship of wheel-slip. In this paper we deals only the pure longitudinal dynamic of vehicle and we develope a method for longitudinal tire force estimation. The method is based on the sliding mode approach [14], [15] known to be robust to parametric uncertainties and perturbations. A simplified longitudinal vehicle model is developed in this study to design observer and for computer

simulations. The state variables considered in this model are angular wheel velocity, vehicle velocity and tractive/braking force. The equations of the two first states are determined by Newton’s law. The last state is derived from the relaxation length concept [16], [17], [18]. A first order slip model based relaxation length concept is used to establish a differential equation of the tactive/braking force. In the next section, we describe the system dynamics and gives a state form of system. The system observability is study in section III. In section IV a design of the sliding mode observer is proposed, and observer stability analysis and observation error convergence conditions are presented. Simulation results are given in section V. Conclusion and perspective of this work are presented in section VI. II. S YSTEM DYNAMICS In this section, we describe the model for vehicle longitudinal dynamic. This model will then be used for system analysis, observer design and computer simulations. The model describe in this study retains the main characteristics of the longitudinal dynamic. The application of Newton’s law to wheel and vehicle dynamics gives the equations of motion. The concept of the relaxation length is used to deduce the differentials equations of wheelslip and tractive/braking force The input signal is the torque applied to the wheel. The dynamic equation for the angular motion of the wheel is I ω˙ = T − fw ω − rF (1) where I is the moment of inertia of the wheel, fw is the viscous rotational friction and r is the radius of the wheel. The applied torque T results from the difference between the shaft torque from the engine and the break torque. F is the tire tractive/braking force which result from the deformation of the tire at the tire/ground contact patch. The vehicle motion is governed by the following equation mv˙ = F − cx v 2

(2)

where cx is the aerodynamic drag coefficient and m is the vehicle mass.

2

The tractive (braking) force, produced on the tire/road interface when a driving (braking) torque is applied to a pneumatic tire, oppose the direction of relative motion between the tire and road surface. This relative motion determines the tire slip properties. The slip is due to deflection in the contact patch ([2], [6]). Thus longitudinal slip (s) is associated with the development of tractive or braking force and we can express F as: F = f (s) (3) The longitudinal wheelslip is generally called the slip ratio and it is describe by a kinematic relationship as1 :  s¯ = vvs during braking phase (4) vs s¯ = rω during traction phase where vs = v − rω, represent the slip velocity in the contact patch. Recently many advanced studies ([19], [20]) anlyse the behavior of the tire properties in rapid transient maneuvers such as cornering on uneven roads, braking torque variation and oscillatory steering. This studies deals with transients in tire force and use the concept of the relaxation length to account the deformation of the carcass in the contact patch that is responsible for the lag in the response to lateral and longitudinal slip. The motivation for this studies is to improve understanding of the tire behavior with respect to experimental results and then include it in vehicle dynamic simulations. The concept of relaxation length has been formulated particularly for the lateral dynamic to model transient tire behavior (see references: [21], [22], [17]). This concept has been used also for longitudinal dynamic. In [23], [24] the authors have used the relaxation length concept to describe the longitudinal and lateral forces to studies the tire dynamic behavior which is represented by a rigid ring model. In [16], [18] the authors gives a review to the concept of relaxation length and presents a formulation for both the longitudinal slip and slip angle as state variables which well be used with any semi-empirical tire model. The pure longitudinal slip can be presented by a first order relaxation length as:  σ s˙ + vs = vs during braking (5) σ s˙ + rωs = vs during traction The steady-state solution of equations (5) equals the normal definition of longitudinal slip given by (4). The relaxation length (σ) equals the slip stiffness (Cκ) divided by the longitudinal stiffness in the contact patch (κx = 2kx l2 ): Cκ σ= κx

(6)

where l is the half contact length and kx is the stiffness per unit of length of the tread. The slip stiffness is defined as the local derivative of the stationary tire force-slip characteristic (see Fig. 1): Cκ =

∂F ∂s

(7)

Fig. 1.

Force-Slip characteristic

According to equations (3) and (7) we can suggest a formulation of the time derivative of F as: F˙ = Cκ s˙ During acceleration phase, the equation (8) become: σ F˙ = Cκ (−rωs + vs )

(9)

However at small slip, we can write:

with

F = Cκ s

(10)

∂F Cκ = = 2kx l2 ∂s s=0

(11)

Then equation (9) can be written as: σ F˙ = −rωF + Cκ vs

(12)

Note that for small slip, the obtained equation (11) is the same as the one in ( [25]) used to generate longitudinal force with a rigid ring model. So locally, we can extended the equation (12) to the large slip case and we propose a time derivative of F as: σ F˙ = −rω(F − F0 ) + Cκ vs (13) the parameter2 F0 is the intersection of the slop F-axis (see Fig. 1).

∂F ∂s

and the

III. C ONTROL FORMULATION For wheelslip regulation, a passivity control approach with sliding mode technique is used. Recall the slip dynamic equation during acceleration phase: σ s˙ + rωs = vs

(14)

and let us define sd as the desired wheelslip. So, the desired slip velocity vsd can be defined by: vsd = rωsd

(15)

Let es and evs denote respectivelly the control errors of wheelslip and slip velocity, (es = s − sd , evs = vs − vsd ). Then the wheelslip error dynamic equation is given by: σ e˙ s + rωes = evs

(16)

So, both σ and ω are strictly positive in acceleration phase then equation (16) is exponentially stable and we can prove that if evs → 0 when t → ∞ then es → 0 when t → ∞. In order to study the convergence of evs , we need to rewrite the system dynamics in state space form.This is possible by choosing convenient state variables. These are: x1 = ω,

x2 = vs ,

x3 = F

The unknown parameters of sytem are: θ1 =

1 the upper bar is introduced here to denote the steady-state of slip with respect to transient slip

(8)

2 At

small slip F0 = 0.

F0 , σ

θ2 =

1 σ

3

Then, the state space form of system is:  x = f (x) + bu y = h(x)

V. S LIDING MODE OBSERVER BASED CONTROL (17)

with  − fIw x1 − Ir x3 2 1 f (x) =  − cmx (rx1 + x2 )2 + rfIw x1 + ( m + rI )x3  κx x2 + rθ1 x2 − rθ2 x2 x3  1    x1 I b =  − Ir  , h(x) =  0  , u=T 0 0 

To achieve the above objectives, we establish the equation of the error dynamic evs as follow: e˙ vs = x˙ 2 − rsd x˙ 1 r e˙ vs = g(x) − (1 + sd ) u I

(18)

with g(x) = (1 + sd )

fw 1 r2 x1 + ( + (1 + sd ) )x3 I m I

Cx (rx1 + x2 )2 (19) m Then, the controller based on sliding mode approach can be designed as: −

u=

I (ˆ g − ksign(evs )) r(1 + sd )

(20)

where k is a positive sliding gain to be defined later and gˆ is an estimation of function g. The implementation of this control law and stability analysis need the knowledge of full state variables. But only on angular velocity is available by measurement. The remaining state will then be estimated by an observer. IV. S YSTEM O BSERVABILITY The observability of the system given in (17), is shown by use rank criterion ([26]) based on the Lie derivative, (this concept gives a local observability). The Lie derivative of the output h along the field vectors f is given by: ∂h .f (x) (21) ∂x The construction of the observability matrix is given by repeated Lie derivative as follows:   dh  d(Lf (h))    O(x) =  (22)  ..   . Lf h(x) =

dn−1 (Lf (h))

where n is the dimension of the system state space. For (17), we obtain the following observability matrix:   1 0 0 f  0 − Ir O(x) =  − Iw 2 fw r(Kx +rθ1 −rθ2 x3 ) r(fw +Irθ2 x2 ) − I2 I I2 (23) The observability matrix is full rank for any state, then system (17) is locally observable everywhere in the phase space.

From (17) we can see that nonlinear system is linear with regard the unknown parameters and system (17) can be rewritten as:  x˙ = ξ(x) + Ψ(x)θ + gu (24) y = h(x) with  − fIw x1 − Ir x3 2 1 ξ(x) =  − cmx (rx1 + x2 )2 + rfIw x1 + ( m + rI )x3  κx2     0 0 θ1   0 0 Ψ(x) = , θ= θ2 rx2 −rx2 x3 

Then the unknown parameters can be estimated. To estimate both the system state an the unknown parameters we propose the following adaptative observer based on sliding mode approach:  ·  x ˆ = ξˆ + Ψ(ˆ x)θˆ + gu + Λs (25) ·  ˆ θ=η with  − fIw y − Ir x ˆ3 2 1 ξˆ =  − cmx (ry + x ˆ2 )2 + rfIw y + ( m + rI )ˆ x3  κx x ˆ2     λ1 0 0  0 Λs =  λ2  · sign(x1 − x ˆ1 ), Ψ(ˆ x) =  0 λ3 rˆ x2 −rˆ x2 x ˆ3     θˆ1 η1 ˆ θ= ˆ , η= η2 θ2 

where x ˆT = (ˆ x1 , x ˆ2 , x ˆ3 ) represents the estimation of state variables and (θˆ1 , θˆ2 ) are the parameters estimation. The observer gains λi and variables (η1 , η2 ) will be defined hereafter. Recall that the single measurement available is x1 . The dynamics of the state observation and parameters estimation errors x ˜=x−x ˆ and θ˜ = θ − θˆ are then:  ·  x ˜ = ξ˜ + Ψ(x)θ − Ψ(ˆ x)θˆ − Λs (26) ·  ˜ θ = −η with − Ir x ˜3 2c 1 x ˜ ξ =  − m (ry + x ˆ2 )˜ x2 + ( m + Ir )˜ x3 − κx x ˜2





cx 2 ˜2 mx



and where Ψ(x)θ − Ψ(ˆ x)θˆ = Ψ(ˆ x)θ˜ + (Ψ(x) − Ψ(ˆ x))θ

(27)

Now, consider the sliding surface S = {˜ x1 = 0}. The sliding condition that guarantees the attractivity of S is given by: ·

x ˜1 x ˜1 < 0 which can be rewritten as: r x ˜1 (− x ˜3 − λ1 sign(˜ x1 )) < 0 I

(28)

(29)

4

The time derivative of V is given by:

The inequality (29) is verified if λ1 is chosen such that λ1 >

r |˜ x3 |max I

with |˜ x3 |max is the maximum value of x ˜3 at any time, then x ˆ1 converges to x1 in finite time and stay equal to the actual value for all t > t0 . Moreover, we have also: ·

x ˜1 = 0

·

·

∀t > t0

˜r + evs e˙ vs + θ˜T P −1 θ˜ (36) V˙ = x ˜Tr x T ˜ T ˜ T ¯T −1 ˜ ˜ ˜ ˙ V = −ξ Ωξ − ξ Θξ + θ (Ψ (˜ xr )˜ xr − Q η) − k |evs | (37) where   Ω= h

On the sliding syraface, we can consider the mean value of the signal generated by the “sign” function: r sign(˜ x1 ) , − x ˜3 Iλ1

χ(ˆ xr ) 0

1 −( m

i

Cx m

0 Cx ˜2 m x



2 sd ) rI )

+ (1 +

0 0





d

(r(2 + s )y + x ˆ2 )

and 

(30)

By substituting the equivalent form of “sign” in equations (26), by developing and using equations (27), we obtain the following reduced sliding dynamics:



Θ=

Θ 0

Now we define the parameters adaptation law η as: ¯ xr )˜ η = QΨ(ˆ xr

(38)

·

ε)θ˜ − Θ˜ xr x ˜r = −χ(ˆ xr )˜ xr + Ψ(ˆ

(31)

For implementation, we can develope η as:

·

θ˜ = −η with "

2cx m (ry



cx ˜2 mx rθ2 x ˆ3 − rθ1

χ(ˆ xr ) = Θ=

+x ˆ2 ) κx

1 1 +λ2 ) −( m + r(λIλ 1 rλ3 − Iλ 1

0

#

rθ2 (ˆ x2 + x ˜2 )

VI. S TABILITY ANALYSIS IN CLOSED - LOOP The closed-loop analysis is performed on the basis of the reduced system (31) and control error equation (18) with gˆ is defined by: fw Cx 2 1 r2 gˆ = (1 + sd ) y − r2 (1 + sd )2 y + ( + (1 + sd ) )ˆ x3 I m m I (33) Now define ξ = (˜ xTr , evs )T to be the augmented state vector which has as equation: ·

x ˜r = −χ(ˆ xr )˜ xr + Ψ(ˆ ε)θ˜ − Θ˜ xr (34) 2 1 r Cx x3 − (r(2 + sd )y + x ˆ2 )evs e˙ vs = ( + (1 + sd ) )˜ m I m Cx − x ˜2 evs − ksign(evs ) (35) m Let us, consider the following Lyapunov function: 1 T 1 1 x ˜ x ˜r + e2vs + θ˜T Q−1 θ˜ 2 r 2 2

where Q = diag(q1 , q2 ) is a positive diagonal matrix.

(39)

where Γ is a unit vector. Note that on sliding surface we have: Iλ1 sign(˜ x1 ) r

(40)

Iλ1 ¯ T P Ψ (ˆ xr )Γsign(˜ x1 ) r

(41)

V˙ = −ξ˜T Ωξ˜ − ξ˜T Θξ˜ − k |evs |

(42)

x ˜3 = −



¯ is a where xr = (x2 , x3 )T is the reduced state and Ψ submatrix of Ψ composed by the two last rows. The stability analysis of the obtained reduced dynamics of observation errors will be studied in closed-loop with controller.

˜ = V (ξ, θ)

¯ T (ˆ η = QΨ xr )Γ˜ x3

(32)

thus η can rewritten as: η=− we obtain

To guarantee the stability in closed-loop, the matrix Ω must be positive definite. Let us define p1 , p2 , p3 to be the eigenvalues. d ˆ2 ) is positive, then if we impose So, p3 = Cx m (r(2 + s )y + x p1 , p2 > 0, we found the gains λ1 and λ2 as: λ2 =?? λ3 =?? Now define ρ0 to be the minimum eigenvalue of the matrix Θ: o nc x ρ0 = λmin (Θ) = min x ˜2 , θ2 (ˆ x2 + x ˜2 ) m Then, the time derivative of V can be bounded as follows:

2

V˙ 6 −(pm + ρ0 ) ξ˜ − k |evs | where pm , is the minimum of {p1 , p2 , p3 }. So, by appropriate choice of p1 , p2 , p3 and k thus V˙ is strictly negative. Consequently, x ˜r and evs tend asymptotically to zero. However, the estimation error of the unknown parameters is only bounded. Finally the control error es converge exponentially to zero.

 

5

Parameter m J r Cx

Value 400 1.3 0.3 0.3

Units Kg Kgm2 m N/m2 s2

TABLE I T HE PARAMETERS OF THE TYRE MODEL .

VII. S IMULATION RESULTS To illustrate the performance of the proposed approach of estimation we consider a one-wheel model (eqautions (1) and (2)) with “Magic formula” tire model. The model parameters are listed in table (I). Figure (??) presents the µ − slip characteristic used to generate the tire force to be estimated. The equation (5) is used to calculate the value of the slip ratio for model simulation with σ = 0.09m. The total torque applied to the wheel during a braking phase is shown in figure (??)The observer gains are: λ1 = 140, q1 = 2.5, q2 = 3.5 · 10−5 , p1 = 500 and p2 = 300. We have taken an error of 25% on the different parameter listed in table (I). The value of the longitudinal stiffness in the contact patch κx is chosing by taking the typical values for kx and l (kx = 17 · 106 N/m2 and l = 0.05m for a load Fz = 4000N ). The unknown parameters θ1 and θ2 are initialized respectively -1.5·104 N m−1 and 15m−1 . The initial conditions are x1 (0) = y(0) = 91rad/s, x2 (0) = 30m/s and x3 (0) = −103 N . Figure (??) show the convergence of the angular wheel velocity estimation to actual value in finite time and estimation error will be zero due to the fact that angular wheel velocity is measured. In figures (?? and ??) we show respectively the asymptotic convergence of the vehicle velocity and longitudinal tire force to actual values. In simulation, the actual tire force is generated by “Magic formula” tire model. The performance of this estimation approach is satisfactory since the estimation error is minimal for state variables. The unknown parameters can not converge necessarily to their actual values. They are be good for adaptation.

VIII. C ONCLUSION In this paper, we have presented a new estimation method for vehicle longitudinal dynamics based sliding mode observer. The main contribution is the proposed differential equation of the tractive/braking force derived from the relaxation length concept. The interest of this method is to be able to estimate the tire force on-line by using only angular wheel velocity measurement and applied torque which can be deduced by engine RPM sensors and Throttle Position Sensor (TPS) and also the braking pressure. Simulation results was illustrated the ability of this approach to give well estimation of both vehicle velocity and longitudinal tire force. The robustness of the sliding mode observer versus uncertainties on the model parameters has also been showed in simulation. The extended work is to applying this method to an instrumented vehicle.

R EFERENCES [1] D. L. Y. Delanne and G. Schaefer, “Analysis of influence of road factors on single vehicle run-off-road accidents: use of a vehicle/road model, road characteristics and safety,” in 9th Int. Conf. Road Safety in Europe, 1998. [2] P. F. H. Dugoff and L. Segel, “An analysis of tire traction properties and their influence on vehicle dynamic performance,” SAE Transaction, vol. 3, pp. 1219–1243, 1970. [3] L. S. J.E. Bernard and R. Wild, “Tire shear force generation during combined steering and braking maneuvers,” SAE Transaction, vol. 8, pp. 2935–2969, 1977. [4] A. B. K.M. Captain and D. Wormley, “Aalytical tire models for dynamic vehicle simulation,” Vehilce System Dynamic, vol. 8, pp. 1–32, 1990. [5] H. P. E. Bakker and L. Linder, “A new tire model with an application in vehicle dynamics studies,” SAE Transaction, vol. 98, no. 6, pp. 101–113, 1989. [6] G. Gim and P. Nikravesh, “An analytical model of pneumatic tyres for vehicle dynamic simulations part1: Pure slips,” Int J. Vehicle Design, vol. 11, no. 6, pp. 589–618, 1990. [7] G. Gim and P. Nikravesh, “An analytical model of pneumatic tyres for vehicle dynamic simulations part2: Comprehensive slips,” Int J. Vehicle Design, vol. 12, no. 1, pp. 19–38, 1991. [8] H. Pacejka, “The tyre as a vehicle component,” in proceedings of the 26th FISITA congress ’96: Engineering challenge human friendly vehicles, Prague, 1996. [9] L. Ray, “Nolinear tire force estimation and road friction identification: simulation and experiments,” Automatica, vol. 33, no. 10, pp. 1819– 1833, 1997. [10] F. Gustafsson, “Slip-based tire-road friction estimation,” Automatica, vol. 33, no. 6, pp. 1087–1997, 1997. [11] K. H. K. Yi and S. Lee, “Estimation of tire-road friction usinf observer based identifiers,” Vehicule System Dynamics, vol. 31, pp. 233–261, 1999. [12] C. C. de Wit and P. Tsiotras, “Dynamic tire friction models for vehicle traction,” in 38th IEE-CDC, 1999. [13] C. C. de Wit and R. Horowitz, “Obsever for tire/road contact friction using only wheel angular velocity information,” in 38th CDC’99, 1999. [14] V. Utkin, Sliding Mode and their Application in Variable Structure Systems. Mir, 1977. [15] J. H. J.J.E. Slotine and E. Misawa, “Nonlinear state estimation using sliding observers,” in IEEE Conf. on Decision and Control, Athen, Greece, pp. 332–339, 1986. [16] J. Bernard and C. Clover, “Tire modeling for low-speed and high-speed calculations,” in SAE Paper No. 950311, 1995. [17] J. Maurice and H. Pacejka, “Relaxation length behavior of tyres,” Vehicule System Dynamics Supplement, vol. 27, pp. 239–242, 1997. [18] C. Clover and J. Bernard, “Longitudinal tire dynamics,” Vehicule System Dynamics, vol. 29, pp. 231–259, 1998. [19] H. Pacejka and I. . Besselink, “Magic formula tyre with transient properties,” Vehicule System Dynamics Supplement, vol. 27, pp. 234– 249, 1997. [20] W. R. A. Van Zanten and A. Lutz, “Measurement and simulation of transient tire forces,” in SAE Paper No. 890604, 1989. [21] H. Pacejka, “In-plane and out-of-plane dynamics of pneumatic tyres,” Vehicule System Dynamics, vol. 10, pp. 221–251, 1981. [22] H. P. J.P. Maurice, “Dynamic tyre response to yaw angle variations,” in proceedings of the international symposium on advanced vehiclecontrol : AVEC’98, Nagoya Japan. SAE, Tokyo 1998, 1998. [23] P. Z. J.P. Maurice and H. Pacejka, “The influence of belt dynamics on cornering and braking properties of tyres,” in 15th IAVSD Symposium Dynamics of vehilces on roads and tracks, Budapest, Hungary, 1997. [24] P. Zegelaar and H. Pacejka, “Dynamic tyre responses to brake torque variations,” Vehicule System Dynamics Supplement, vol. 27, pp. 65–79, 1997. [25] P. Zegelaar and H. Pacejka, “In-plane dynamics of tyres on uneven roads,” [26] R. Hermann and A. Kerner, “Nonlinear controllability and observability,” IEEE Transaction on Automatic Control, vol. 22, pp. 728–740, 1977.