Mahdi Mcheik
email: mahdi.mcheik@u-bordeaux.fr

Simon Villain-Guillot
email: simon.villain-guillot@u-bordeaux.fr

D Cahn-Hilliard dynamics : Transition modes and stability criteria

The Cahn-Hilliard-Oono equation describes the dynamics of phase separation process of copolymers alloys and other binary systems. This process is driven by the minimization of the free energy. However the lower energy state in rarely reached and only for particular initial states.

Misbah and Politi have proposed a dynamical criterion which leads to …nal states that doesn't correspond to the minimization of free energy. In this article we investigate possible transition modes that may describe the dynamic of "segregation and coalescence" in those systems, in order to understand the di¤erence between these criteria.

I. INTRODUCTION

In many systems such as melts of diblock copolymer, which are built by linking two hetero-polymers, the competition between opposite interactions with long and short range, may form spatially modulated patterns below a critical temperature (or Flory-Huggins parameter). Indeed, the interaction between homogeneous polymers is attractive while the interaction between heterogeneous polymers is repulsive and thus favors segregation ; the simplest dynamics after a quench tends to reduce the interfaces between homogeneous polymer domains so as to minimize the interfacial free energy. However, as diblock copolymers are attached to each other, the complete segregation cannot occur as for the usual liquid gas phase transition or in simple polymer blends, where domain sizes can become in…nite. This frustration results in the formation of spacially periodic structures. The length of the phase modulation depends on the relative strength of the two competiting forces ; in other terms, the dynamics reduces the interfacial free energy until some …nite value of wavelength for which the two forces counterbalance.

The Cahn-Hilliard-Oono's equation is particularly adapted to describe the dynamics in such system. It is a conservative modi…ed di¤usion equation for a scalar order parameter u: @u @t = (r 2 F GL (u) u ) u = r 2 ( u + u 3 r 2 u) u:

The order parameter u can correspond to a rescaled density of atoms or relative concentration of polymers, u + u 3 derives from the Landau free energy which characterizes the short ranges repulsive interactions and the entropy of mixing, and the Oono's term u describes the non-local long-range interaction which tends to homogenize the system and forbid the full segregation. The competition between the two parts of the dynamics results in the formation of a spatially peridically modulated state.

This conservative dynamics derives from the free energy :

F(u(r)) = Z F GL dr + Z F int dr (2)
F GL is the Ginzburg-Landau free energy density F GL = 1 2 (ru(r)) 2 + 1 2 u 2 (r)+ 1 4 u 4 (r) F int is the Leibler free energy density (corresponding to Oono's non-local interaction term in the dynamics),

F int = Z u(r 0 )g(r 0 ; r)u(r)dr 0 .
where g is the kernel of the long range interaction term and is such that

r 2 g(r 0 ; r) = (r 0 r).
This dynamics is driven by the minimization of the free energy (1). However direct simulations for one dimension dynamics show that the …nal solutions are not unique and they depend on the initial states. Moreover, starting with initial random conditions, the associated …nal free energies don't correspond to the lower energy state. This lower energy is found numerically only when is large (Swift-Hohenberg limit). Otherwise, for large 1D systems, and starting with initial random conditions, numerical simulations show that many di¤erent domain sizes may coexist, due to the very slow dynamics. Therefore it is important to study the distribution of those solutions.

This paper is composed of three parts; in the …rst part we focus on di¤erent transition modes for coarsening, in the second part we show two criteria that we used to de…ne stability and the last part is dedicated to the statical study of domain sizes of …nal solutions.

II. TRANSITION MODE AT COARSENING STAGE

In this section, we will focus on di¤erent transition modes that can a¤ect the dynamics of the Cahn-Hilliard equation during the coarsening regime which follows the spinodal decomposition. During this last stage of the dynamics, we believe that the upper limit of the wave number q of the steady-state solution results from the Ostwald ripening, i.e from the diffusion between domains. This process generates interface displacements in both directions, and leads to the creation of a pro…le of larger wavelength.

We have simulated the dynamics numerically, focusing on the question of the stability of stationary (or frozen) solutions obtained from direct simulation in small boxes. We have used the same pseudo-spectral algorithm used by M. Nicoli (IFABM4), as this method is known for its stability and accuracy. But we have used another criterion [START_REF] Benilov | Stability of frozen waves in the Modi…ed Cahn-Hilliard model[END_REF] to con…rm if these solutions remain absolutely stable for larger boxes when perturbed by small amplitude noise, i.e. we have studied the evolution of perturbation superimposed above these frozen patterns replicated in larger commensurate boxes .

Starting from random initial conditions, or starting close to the expected solution, the stationary solutions of CHO equation obtained for long time simulation, are, in the case of small simulation box (typically smaller than 8 lambda CH), close to elliptical sine Sn function. Therefore we make use of the expansion of Sn in terms of hyperbolic tangents, to simulate the process of coarsening by the motion of tanh interfaces.

Sn (x) = X i ( 1) i tanh(s(x i: =2))
where s is extracted from the numerical simulation looking at the the tangent for small values of the order parameter u(x) (see FIG 1) and is the known wavelength.

The third parameter needed for our expansion is the amplitude that can be adjusted easily from the numerics.

The resulting pro…les …t very closely the numerical results (as can be seen in the FIG 1), the errors (amplitude and tangent) for di¤erent value of are less than 1 % while error on energy is reasonably small for < 10 2 and > 2 CH . The coarsening process can then be simulated by looking solely at the motion of tanh interfaces whose only degree of freedom are the positions. wavelength. The two inner kinks vanish together in symmetrical way, while the outer kinks move to the center at the same velocity. Other modes were seen during visualizations of numerical simulations of CHO dynamics, for example "3!2", "5!3" and some other can be obtained with this procedure such as "5!4","7!6" and so on...

III. COALESCENCE AND MOTION

One possible scenario for coarsening is the symmetric di¤usion, where the transfer of matter between neighboring domains can be described in term of the motion of interfaces.

This symmetric process naturally conserves matter. The dynamics of these tanh interfaces is indeed very close to the numerical simulation of the full CHO equation. The correlated motion of the tanh interfaces a¤ects the shape (or parameters) of the entire pro…le, even if is the nearest interfaces are more impacted ; in that sense, hyperbolic tangent has a nonlocal e¤ect on the entire pro…le. For example, the evolution of the tangent of the pro…le at u(x) = 0 depends on the distance between interfaces ; or the amplitude increases with , this variation being more important for small as we can notice in ment of the interfaces along the "2!1" process. For > c ( ), the pro…le is thus stable with respect the studied coarsening motion of the interfaces. In other words when further segregation would cost energy, coarsening dynamics is interrupted [START_REF] Politi | When does coarsening occur in the dynamics of one-dimensional fronts?[END_REF] and the segregation remains partial.

In picture 6 we see that for = 4 , the free energy decreases as function of displacement when 2:4 10 4 , but a small potential well appears for bigger values ( = 2:6 10 4 ) so we consider that pro…les of periods 4 are instable if 2:4 10 4 . In Figure 7 we illustrate the thresholds of stable solutions of given values of for di¤erent scenarios of symmetrical segregations ( 2 ! 1, 3 ! 2 ..) FIG. 7: In red, the stable wavelengths associated with the transition mode 2-1, in black, the wavelengths associated to mode 3-2

These 2 curves are almost identical ; and we obtain identical behaviors for other modes like merging 7 patterns into 6, 9 into 8, and even 5 into 3. Therefore, we can ensure that the upper limit of q is given by the symmetric di¤usion, found analytically by Misbah and

Torcini [START_REF] Politi | Dynamics versus energetics in pase separation[END_REF]. The others symmetric modes give slightly di¤erent results. We can conclude that segregation process may be guided by di¤erent modes of transition or with mixture of mode and not only by "2! 1" symmetric di¤usion.

IV. CRITERIA FOR THE STABILITY OF STATIONARY MICROSEGREGATED PATTERNS A. Minimization of free energy:

A natural criterion for stability is the minimization of free energy since dF dt < 0, however the absolute minimum is rarely reached. In a previous work [START_REF] Villain-Guillot | Coalescence in the 1D Cahn-Hilliard model[END_REF], it has been shown that the analytic solution of the CH for a …nite system dimension can be a good approximation for the CHO equation. This hypothesis is veri…ed numerically as shown in 1 . This approximation enables to compute the free energy expression including the additional term of non-local interaction (Oono's term). The stable wavelength s in then found to scale as the power of :

( ) 1=3
Another approach uses solutions from direct simulation (with …xed ) for small system of wavelength CHO . Starting with only two domains and two interfaces, we vary in such a way that the system can dissipate the perturbation due to the small expansion and relax to a steady state. This approach allow us to follow the free energy as function of the period.

The steady state for a given corresponds to the lowest energy reached F ( ).

It is clear that for = 0, no minimum is attain. The free energy decreases as a function of the wavelength, leading to an unconditioned coalescence, i.e. full segregation. This result is veri…ed analytically since the exact solution is known; the free energy is strictly decreasing and converge to 0.25. When 0 < < 1=4 it seems that the free energy reach a unique minimum in , that corresponds to the steady state as can be seen for = 10 5 in Fig

??.
In fact, the variation of non-local term in the CHO equation (which represents the long range interaction) becomes equal to variation the short range interaction term at this precise period, which characterized the steady state.

The minimization of free energy enables to plot this period as a function of in Figure 9. Misbah and Politi have proposed a criterion for the linear stability/instability of frozen solutions [START_REF] Politi | When does coarsening occur in the dynamics of one-dimensional fronts?[END_REF]. The basic idea is to perturb the periodic steady state u 0 (x; ) = u 0 (x + ; ) for which one can de…ne the phase (x) = qx where q = 2 = , by adding some perturbation on u 0 : u(x; t) = u 0 (x) + u 1 (x; t). First they rescaled x by introducing slow space variable X = x and consequently slow time variable T = 2 t ,and a slow phase de…ned as (X; T ) = (x; t).

Keeping only …rst order terms in when developping the partial time/space derivatives in x and t in CHO equation as functions of X and T and using the fact that the stationary solution u 0 satis…es the zero order equation, they found at the …rst order the following phase di¤usion equation

@ T = D @ 2 XX
where D = q 2 @ q < q (@ u 0 ) 2 > :@ q < q w 2 > < u 2 0 >

This di¤usion coe¢ cient is proposed as criterion for dynamic stability. If the phase FIG. 9: In red asterix, stable wave length found by minimizing of the free energy with respect to the size of the …nal domains. The results scale like 1=3 perturbations vanishes (D < 0), the stationary solution u 0 is dynamically stable ; otherwise (D > 0), the perturbation grows and the system will coalesce.

V. STATISTICAL STUDY OF THE CAHN_HILLIARD EQUATION

In this section we look at the results of direct simulation of CHO dynamics in order to guide our study. Since we have shown beloow that the two criteria of stability, one based on the minization of the free energy and the other based on the dynamics, give di¤erent …nal wave lengths, it is important to determine which one is the closest of the direct simulation result.

The relevant stationary solutions can be found only if the length of the system is larger than the natural wavelength. We therefore consider a large system (256 CH ), for which the dynamics of pattern formation is followed for long time simulation. The spatial step is declared in terms of CH (256 point per CH ) , the time step is dynamically changed to get more precision and weak white noise is added to speed up the dynamics (annealing). The wave lengths of …nal solutions are found by measuring the zeros of …nal pro…les, and they are classi…ed in small intervals of wavelength. We have focused on totally random initial conditions (of low amplitude < 0:01), so we don't privilege a priori any particular solution.

In the picture 10 we show some distributions of wavelength of the …nal stable solution for di¤erent values of . We can notice that, for high values of ; there is only a unique wavelength ( we noticed that the …nal solution is obtained quickly, the …nal wave length is close to CH ). And while the distribution is becoming larger for smaller , we still notice the presence of a central pick value for the most probable wavelength.

FIG. 11: Stability diagram (q; ) plotted in log-log to be compared to Politi-Torcini results [START_REF] Politi | Dynamics versus energetics in pase separation[END_REF] :

in black the stability zone de…ned by the solution of (1 q 2 )q 2 = 0, in red wine, the stable wavelengths found using Misbah-Politi criterion, the blue-triangles and dark-cyan-triangles lines represent the stable wavelengths found respectly for transition modes 2 ! 1 and 3 ! 2; the upper and lower limits of stable solution found statically for large system using totally random initial states are showed in violet and green line-asterix.

We resume all results obtained from criteria discussed above in this picture 11

VI. CONCLUSION

We have studied numerically a system of large dimension, with the goal of …nding a good stability criterion ending Cahn-Hilliard-Oono dynamics, as solutions are not unique and wavelengths may not be the same in di¤erent spatial part of system. In the …rst part,

we have tried to study transition modes and how patterns evolve in there vicinity. We have found that pure symmetric di¤usion process between identical domains de…nes the upper limit stability of q or the smallest …nal wavelength. So we think that more complicated transition modes may be responsible for the creation of larger patterns. In the last part, we have studied statistically a large dimension system and we have compared two di¤erent approaches of stability to these statical results. We have found a region of stable wavelengths that …ts very well with the stability criterion proposed by C. Misbah and P. Politi. This criterion is more appropriate to predict the …nal stage of the dynamics than the sole free energy minimization. However it cannot explain the distribution of solution around the central line of stability.

FIG. 1 :

 1 FIG. 1: In blue numerical solution of CHO equation starting from random initial state ( = 10 4 and = 20), in red hyperbolic tangentes combination used to simulate the numerical solution.

FIG. 2 :

 2 FIG. 2: This four pictures illustrate the dynamics extracted from direct numerical simulation of CHO equation acording transition mode "2!1", when = 10 8

Fig 3 .FIG. 3 :FIG. 5 :FIG. 6 :

 3356 FIG. 3: Transition 2!1: in blue initial state (wavelength 10), in red the …nal state (1 pattern of double wavelegnth), the discontinued line shows intermediate states

FIG. 8 :

 8 FIG.8: The black curve illustrates the free energy as function of wavelength L for = 0, red curve for = 10 5 , and blue curve for = 10 4

FIG. 10 :

 10 FIG. 10: In black, unique wavelength for = 0:1, in blue, distribution of stable wavelengths for = 10 2 , in violet, for = 10 3 , in green for = 10 4 , orange for = 10 5 , and red for = 10 6


R Lenain

B Thuilot
email: benoit.thuilot@lasmea.univ-bpclermont.fr

C Cariou

P Martinet

Rejection of sliding effects in car like robot control : application to farm vehicle guidance using a single RTK GPS sensor

Keywords: mobile robots, nonlinear control systems, adaptive control laws, kinematic GPS, agriculture

A very accurate vehicle guidance is required in numerous agricultural applications, as seeding, spraying, row cropping, . . . Accuracy in vehicle localization can be obtained in realtime from a RTK GPS sensor. Several control laws, relying on this sensor, have been previously designed and provide satisfactory results as long as vehicles do not slide. However, sliding has to occur in agricultural tasks (sloping fields, curves on a wet land, . . .). The challenge addressed in this paper is to preserve vehicle guidance accuracy in such situations. A nonlinear adaptive control law is here designed. Simulation results and field experiments, demonstrating the capabilities of that control scheme, are reported and discussed.

I INTRODUCTION

RTK GPS (also named kinematic GPS) provides realtime localization with a centimeter accuracy. Since these sensors are available, many works on vehicle guidance using this technology have been designed. Numerous applications deal with agricultural tasks. The advantages of automatic guidance in this area are obvious: it reduces the tiring driving task and increases both precision and productivity in the agronomic work carried out. Moreover, the sensor appears here very suitable: since agricultural vehicles move far from buildings or trees, interruptions in GPS signal reception (which is the major concern with this sensor) do not occur. Several guidance devices have already been marketed by specialized companies (AgSystems, Integrinautics, . . .), GPS systems suppliers (Trimble, . . .) or agricultural manufacturers (John Deere, . . .) However, the above mentioned devices, as well as works reported in the literature, make use of several sensors: RTK GPS and inertial sensors in [START_REF] Nagasaka | Automated operation in paddy fields with a fiber optic gyro sensor and GPS[END_REF], multiple RTK GPS antennas in [START_REF] O'connor | Automatic steering of a farm vehicle using GPS[END_REF], RTK GPS, vision and NIR in [START_REF] Ried | Driverless tractors[END_REF], . . . Moreover, they are mainly concerned with straight lines following on even ground. In previous works, we have addressed the twofold challenge: relying on a unique RTK GPS sensor and providing vehicles with curved path fol-lowing capabilities. Our solution, reported in [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF], consists in a nonlinear control law, taking into account for the reference path curvature, and a Kalman state reconstructor, providing the whole vehicle state vector from the information delivered from the unique RTK GPS sensor. Satisfactory experiments have been carried out on dry and even grounds. We have however noticed that as soon as vehicles move in a slippery environment (sharp curves on a wet ground, sloping fields, . . .), guidance accuracy is degraded: a lateral deviation appears, and could be important, depending on the sliding conditions. In this paper, control algorithms presented in [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF] are modified, relying on adaptive control techniques, in order to reject sliding effects on the vehicle behavior. Preliminary results presented in [START_REF] Lenain | Adaptive control for car like vehicles guidance relying on RTK GPS: rejection of sliding effects in agricultural applications[END_REF] are refined, and extensive simulations and experimentations are discussed.

II STEERING LAW IN ABSENCE OF SLIDING

A Notations

The vehicle, see Figure 1, is described according to Ackermann's model, also named bicycle model: front and rear axles are simplified to two virtual wheels (one front and one rear). This model is quite common in mobile robot control theory, when non sliding is assumed, see for instance [START_REF] Zodiac | Theory of robot control[END_REF]. Our notations are now detailed (see also Figure 1) -C is the path to be followed, -O is the center of vehicle virtual rear wheel, -M is the point on C which is the closest to O. M is assumed to be unique, which is realistic when the vehicle remains quite close from C.

s is the curvilinear coordinate of point M along C, and c(s) denotes the curvature of C at that point. y and θ are respectively lateral and angular deviation of the vehicle with respect to reference path C (see Figure 1). δ is the virtual front wheel steering angle.

v is the vehicle linear velocity, considered here as a parameter, whose value may be time-varying during the vehicle evolution. -L is the vehicle wheelbase.

B Vehicle kinematic model

Vehicle configuration is completely specified when s, y and θ are given. These 3 variables can then be seen as a vehicle state vector. Under non sliding assumption, their evolution can be shown to obey the following kinematic equations (see e.g., [START_REF] Zodiac | Theory of robot control[END_REF], [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF], . . .):

       ṡ = v cos θ 1-c(s) y ẏ = v sin θ θ = v( tan δ L -c(s) cos θ 1-c(s) y ) (1)
Velocity v is here considered as a parameter: the only control variable is δ. It is assumed hereafter that c(s) y = 1 (i.e lateral deviation does not bring vehicle point O on path C curvature center): it is always satisfied during actual path following, since the vehicle remains close to C.

C Steering law design

It has been established that most of mobile robots kinematic models can be converted, without any approximation, into almost linear models named chained forms, see for instance [START_REF] Samson | Control of chained systems. Application to path following and time-varying point-stabilization of mobile robots[END_REF]. Such an approach is attractive, since control laws can then be designed according to Linear Systems Theory, while still relying upon the actual nonlinear mobile robots kinematic models. More precisely, nonlinear kinematic model is first converted into chained form via invertible state and control nonlinear transformations. A linear control law is then designed relying on the chained form. Finally, the actual control law is computed via state and control nonlinear transformations. In [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF], this technique is applied to kinematic model [START_REF] Ackermann | Robust Lateral and Yaw Control[END_REF]. It is shown that curved path following (i.e. maintaining y and θ equal to 0) can be achieved according to nonlinear control law:

δ(y, θ) = arctan L cos 3 θ (1-c(s)y) 2 d c(s) d s y tan θ -K d (1 -c(s)y) tan θ -K p y +c(s)(1 -c(s)y) tan 2 θ) + c(s) cos θ 1-c(s)y ]) (2)
where K p and K d can be interpreted as parameters of a PD controller.

D Experimental equipment

Experiments have been carried out with an ARES 640 RENAULT-Agriculture tractor, shown on Figure 2. The RTK GPS receiver is a Thales Navigation dual frequency "Aquarius 5002" unit, which supplies an absolute position with a ±2 cm accuracy, at a 10 Hz sampling frequency. The mobile antenna has been located on the top of the cabin, straight up the point O. Therefore, current value of state variable y is inferred from direct position measurements. Current value of θ can be inferred, under non sliding assumption, from point O velocity measurements, also provided by the RTK GPS sensor. However, experiments have shown that values of θ thus obtained are indeed very noisy. Therefore, raw velocity measurements are here proceeded through a Kalman state reconstructor, see [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF]. Obviously, this approach could provide data which are not always relevant for sliding detection, as it will be seen hereafter.

E Vehicle behavior in presence of sliding

Steering law (2) provides satisfactory results, as long as no sliding occurs during experiments, see [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF]. However, farm vehicles move on fields which do not always propose good adherence properties: sliding can indeed be noticed when vehicles enter into curves, or on sloping fields. Steering law ( 2) is then strongly disturbed and lateral deviation with respect to the reference path can be observed.

As an example, Figure 3 presents lateral deviation recorded during straight line following on a sloping field. Since slope, and therefore sliding conditions, are constant, lateral deviation converges (on an average) to a constant non null value. It can also be shown that heading deviation and steering angle also converge to non null values.

III MODEL INCORPORATING SLIDING EFFECTS

Sliding effects can be taken into account, either by designing a vehicle dynamic model, or by introducing perturbations inside the kinematic model previously designed under non sliding assumption. In this paper, the second 2), and on the other hand dynamic models of agricultural vehicles do not appear very tractable from the control point of view: adherence conditions are varying, and their online identification is difficult. Dynamic models are also dependent on the vehicle configuration, and especially on the implement being used. Such difficulties are related in [START_REF] Ellouze | Control of unicycle-type robots in the presence of sliding effects with only absolute longitudinal and yaw velocities measurement[END_REF], where dynamic control is investigated.

Refining kinematic models in order to account for sliding effects has also been proposed, with an approach different from ours, in [START_REF] Ackermann | Robust Lateral and Yaw Control[END_REF]. It has also been used in nautical applications, in order to account for current effects, see [START_REF] Holzhüter | Operating experience with a high-precision track controller for commercial ships[END_REF].

A Sliding description

Vehicle dynamics analysis, see [START_REF] Dormegnie | Lectures at French Institute for Advanced Mechanics[END_REF], identifies sliding effects as the superposition of two motions:

-Translation: when the four tires slide in the same direction (e.g. when the vehicle moves along a slope), a lateral motion occurs. -Rotation: in Ackermann's model (see Figure 1), the vehicle is described by a front and a rear wheel. When sliding occurs, the two virtual wheels do not slide exactly in the same way (due to mass repartition, tire structure, . . .). This difference generates a rotation motion.

It can be accounted for these two motions by introducing two sliding variables, namely Ẏp and Θp , in model ( 1) previously designed under non sliding assumption:

       ṡ = v cos θ 1-c(s) y ẏ = v sin θ + Ẏp θ = v tan δ L -c(s) cos θ 1-c(s) y + Θp (3)
Such a model is representative exclusively of lateral motion due to sliding effect. In our application, longitudinal sliding has not been integrated since it has no influence on control accuracy: on one hand, it acts only on vehicle velocity which is considered as a parameter of model ( 3), and on the other hand steering law performances (2) have been shown to be independent from v (see [START_REF] Thuilot | Automatic guidance of a farm tractor relying on a single CP-DGPS[END_REF]).

B Estimation of sliding variables

Sliding variables Ẏp and Θp can be online estimated by comparing the actual evolution of the vehicle, assumed to obey equations (3), and its theoretical evolution in absence of sliding, which can be simulated from model [START_REF] Ackermann | Robust Lateral and Yaw Control[END_REF]. This estimation approach, depicted on Figure 4, leads then to the following relations:

Ẏ P = y k -y k-1 Te -v sin θk-1 ΘP = θ k -θ k-1 Te -v tan δ k-1 L (4)
where T e denotes the sampling period, and y k , θ k and δ k stand for the values of the lateral deviation y, the absolute vehicle heading θ and the steering angle δ at sample time k T e . Relations (4) are very convenient, since sliding variables are not computed from absolute lateral and heading deviations, but from relative deviations measured for a sampling period. Sliding estimation is therefore independent from the vehicle absolute deviation. -Relevance of model ( 3) is finally investigated by comparing the actual and simulated tractor trajectories. Figure 5 shows the 3 paths that have been followed: -Path #2 : this path is built on different kinds of ground: straight line parts take place on asphalt whereas a part of the curve is drawn on lawn.

-Path #3 : the trajectory consists mainly in a straight line on a sloping wet field (slope is indicated via arrows on Figure 5). No implement is towed. Figures 6, 7 and 8 display that simulated lateral deviation in presence of sliding is consistent with actual measurements. Nevertheless, several points can be pointed out: 6 during steady curvature -from iterations 375 up to 500). These misestimations are mainly due to the Kalman filter used to evaluate angular deviation ( θ) as it was discussed in "Experimental equipment". Since the presence of a filter introduces some delay, angular sliding variable ( ΘP ) is inevitably lowered during the curve. A posteriori computations corroborate this fact : by using a shifted angular deviation information to compute sliding variables, simulation results on lateral deviation fit more precisely to actual deviations observed (±5%). Differences between simulated and actual evolution are less important on path#2 and path#3 following (see Figures 7 and8), because ground properties (lawn for path#2 and path#3) or path to be followed (straight line in path#3) make influence of θ in sliding evaluation less important than on graveled ground and/or during a curved path following. They can be related to the pendular motion undergone by the mobile GPS antenna: since it is located on the top of the tractor cabin, ground irregularities generate, via the shock absorber linking axles to fairing, a lateral motion of the antenna, which is wrongly interpreted as a sliding motion. Simulation of model (3) therefore generates lateral deviations which do not exist in the actual experiment. Perturbations due to such pendular movements are amplified when the path to be followed takes place on a non planar ground. Figure 7 displays clearly the difference when the vehicle evolutes on asphalt and on lawn. On iterations up to 270 and after 450, lateral deviation is quite smooth whereas between iterations from 270 to 450, lateral deviation is strongly noisy. This points out that sliding variables will have to be filtered out prior to their use in control law (described hereafter), even if some delay is then introduced. Nevertheless, Figures 6 to 8 show that sliding effects are satisfactorily described by model ( 3), which can be used to design an adaptive control law. 1), and neglecting second order terms in y prove that (see also [START_REF] Lenain | Adaptive control for car like vehicles guidance relying on RTK GPS: rejection of sliding effects in agricultural applications[END_REF]):

y t→∞ -→ - β + Θp v cos 3 θ α - 2 c(s) Θp v cos 3 θ ∆ = y c ( 5
)
with :

           θ t→∞ -→ -arcsin Ẏp v δ t→∞ -→ arctan L -Θp v + c(s) cos θ 1-c(s) y α = d c(s) d s tan θ + c(s) tan θ (K d -c(s) tan θ) -K p β = tan θ (c(s) tan θ -K d )
These results corroborate that θ and δ converge also to non null values, indicating that the vehicle moves crabways. Moreover, it can be observed that rotational sliding Θp acts on steering angle δ, when translational sliding Ẏp acts on heading deviation. If c(s) was also constant, it is then immediate to prove that lateral deviation can be brought back to 0 by simply shifting the objective of steering law (2): if its objective is to bring y to the constant value -y c , then y will converge to 0 in presence of constant sliding. Steering law (2) writes then:

δ = arctan L cos 3 θ (1-c(s)(y+yc) 2 d c(s) d s (y + y c ) tan θ -K d (1 -c(s)(y + y c )) tan θ -K p (y + y c ) +c(s)(1 -c(s)(y + y c )) tan 2 θ) + c(s) cos θ 1-c(s)y ]) (6)
In practical situations, sliding variables are not constant. However, if the shift in the steering law objective (i.e. y c ) could track the current sliding conditions, then steering law (6) would still improve guidance accuracy. Two approaches are hereafter proposed to determine online the value of y c : online simulation: the vehicle model incorporating sliding effects (i.e. model ( 3)), fed by the current sliding variables Ẏp and Θp (computed from ( 4)), and controlled via steering law (2), is simulated online. The state variable y of that simulator is then used as the corrective shift y c to be introduced in the actual control law (6) steering the vehicle. When Ẏp and Θp are constant, the simulated state variable y obviously converges to the expected shift value. When they are not constant, the simulated state variable y always reflects the current sliding conditions.

direct computation: the corrective shift y c to be introduced in the actual control law ( 6) is directly computed by reporting the current sliding variables Ẏp and Θp in relation [START_REF] Holzhüter | Operating experience with a high-precision track controller for commercial ships[END_REF]. When Ẏp and Θp are constant, the corrective shift is then immediatly the suitable one. This second approach may then appear more relevant. However, it is more sensitive to noise and perturbations than the first one, where the simulator acts as a natural lowpass filter. In practical situations, the corrective shift computed from relation [START_REF] Holzhüter | Operating experience with a high-precision track controller for commercial ships[END_REF] will have to be filtered out. Therefore, none of these approaches appears clearly superior to the other. The overall adaptive control scheme is depicted on Figure 9, where the adaptive module consists either in online simulation or direct computation. In the general adaptive control framework, it can be seen as a model-based adaptive control scheme.

B Simulation results

Performances of adaptive steering law (6) have first been investigated through simulations. The kernel of the sim- The overall control scheme is depicted on Figure 9. Sliding estimation encloses a subsimulator, which does not account for sliding effects (sliding estimation module is also displayed separately on Figure 4). The two adaptation modules have been investigated. The first one, named online simulation, encloses a second subsimulator, which accounts for sliding effects, as above mentioned. The lateral deviation obtained with each adaptation module is compared on Figures 10 and11 with those obtained without adaptive correction (i.e. with steering law (2)). Figure 10 shows path # 1 following. Sliding occurs as soon as the vehicle enters into the curve: it is just as if a step was applied on sliding variables Ẏp and Θp . It can be observed that adaptive control law relying on the direct computation adaptation module reacts faster than those relying on the online simulation adaptation module. This is quite natural since the cutoff frequency of the lowpass filter used in the former adaptation module is here higher than the natural cutoff frequency of the subsimulator used in the latter module. During the curve, since sliding variables are then almost constant, lateral deviation with both adaptive laws converges to 0, as expected. Finally, at the end of the curve (as at beginning), lat-eral deviation presents some non null values, pointing out some delay in the settling of the appropriate corrective shift y c . Figure 11 shows path # 3 following, i.e. straight line following on a sloping field. The slope of the actual field, and therefore the sliding variables Ẏp and Θp used in these simulations, are not perfectly constant. It can be observed that the corrective shift y c does not evolve fast enough to keep lateral deviation constantly equal to 0. Nevertheless, the mean value of lateral deviation is close to zero, which is not the case when adaptive correction is not used.

C Experimental results

These simulation results have been corroborated by experiments. Figure 12 compares the lateral deviations measured when the farm tractor achieves path #1 following with or without adaptation module. Actual deviations appear quite similar to those obtained in simulation, see Figure 10. It can just be noticed that lateral deviation overshoots, at the beginning and at the end of the curve, are larger than in simulation. They proceed from delays pointed out when discussing Figures 6 to 8. Moreover, delays introduced by low level actuators amplify also these overshoots. Nevertheless, it must be pointed that during the curve, both adaptive control laws bring back the vehicle on the reference trajectory, when the non adaptive one leads to a constant lateral deviation.

V CONCLUSION AND PROSPECTS

In this paper, an adaptive control scheme, relying on model-based techniques, is proposed to take into account for sliding effects when steering car like mobile robots. The adaptive control law nevertheless still relies on the nonlinear steering law previously designed, under non sliding assumption, using chained systems theory. All the advantages of this nonlinear control technique are then preserved. Capabilities of the proposed adaptive scheme have been validated, both in simulations and experiments. Remaining difficulties deal with delays in the sensing system and in the low level actuation. A heading GPS sensor is currently integrated, in order to reduce delays in vehicle heading measurements, and improve their robustness. Predictive control techniques are also being studied, using a priori models, to reduce overshoots observed during curved path following. Finally, another sliding model has been developed. It still relies on mobile robot chained forms designed under non sliding assumption. The capabilities of the adaptive laws designed with this second model are presently compared to those presented here.

Figure 1 :

 1 Figure 1: Vehicle modeling parameters

Figure 2 :

 2 Figure 2: Experimental vehicle

Figure 3 :

 3 Figure 3: Vehicle behavior in presence of sliding

Figure 4 :

 4 Figure 4: Sliding estimation C Experimental validation In order to evaluate both sliding modeling and estimation, the following experiments have been carried out: -Several path following have been performed on slippery grounds with steering law (2). The experimental vehicle is the farm tractor depicted on Figure 2. -The evolutions of sliding variables Ẏp and Θp during these experiments, have then been computed according to relations (4). -Next, these actual evolutions of Ẏp and Θp have been reported in vehicle model (3) in order to simulate tractor trajectories.-Relevance of model (3) is finally investigated by comparing the actual and simulated tractor trajectories. Figure5shows the 3 paths that have been followed:

Figure 5 :

 5 Figure 5: Paths to be followed

Figure 6 :

 6 Figure 6: Path#1 following with steering law (2) First, we notice that lateral deviation is slightly underestimated during a curve following (see for instance Figure6during steady curvature -from iterations 375 up to 500). These misestimations are mainly due to the Kalman filter used to evaluate angular deviation ( θ) as it was discussed in "Experimental equipment". Since the presence of a filter introduces some delay, angular sliding variable ( ΘP ) is inevitably lowered during the curve. A posteriori computations corroborate this fact : by using a shifted angular deviation information to compute sliding variables, simulation results on lateral deviation fit more precisely to actual deviations observed (±5%). Differences between simulated and actual evolution are less important on path#2 and path#3 following (see Figures7 and 8), because ground properties (lawn for path#2 and path#3) or path to be followed (straight line in path#3) make influence of θ in sliding evaluation less important than on graveled ground and/or during a curved path following.

Figure 7 :

 7 Figure 7: Path#2 following with steering law (2) Figures 7 and 8 display quite large oscillations in simu-

Figure 8 :

 8 Figure 8: Path#3 following with steering law (2)

Figure 9 :

 9 Figure 9: Overall adaptive control scheme

  .

Figure 10 :

 10 Figure 10: Simulation of path#1 following with law (6)

Figure 11 :

 11 Figure 11: Simulation of path#3 following with law (6)

Figure 12 :

 12 Figure 12: Experiments of path#1 following with law (6)