Gilles Tagne
email: gilles.tagne@hds.utc.fr

Reine Talj
email: reine.talj@hds.utc.fr

Ali Charara
email: ali.charara@hds.utc.fr

Passivity Analysis and Design of Passivity-Based Controllers for Trajectory Tracking at High Speed of Autonomous Vehicles

Autonomous intelligent vehicles are under intensive development, especially this last decade. This paper focuses on the lateral control of intelligent vehicles, with the aim of minimizing the lateral displacement of the autonomous vehicle with respect to a given reference trajectory. The control input is the steering angle and the output is the lateral error displacement. After passivity analysis of the system to establish the properties of passivity between some inputs and outputs, we present design and validation of lateral controllers based on passivity, to ensure robust stability and good performances with respect to parametric variations and uncertainties encountered in driving applications. The control strategies have been validated in closed-loop on SCANeR T M studio [1], a driving simulation engine, according to several real driving scenarios. The validation shows robustness and good performances of the proposed control approaches, and puts in evidence the improvement brought by the proposed Nested Passivity-Based Controller (PBC).

I. INTRODUCTION

 for some examples. This is an area of growing research and one of the major challenges today is to ensure autonomous driving at high speed.

Three main steps are necessary to ensure an autonomous navigation: the perception and localization, the path planning and the vehicle control. The vehicle control can be divided into two tasks: longitudinal control and lateral control. This paper focuses on the lateral control of intelligent vehicles.

Lateral control consists on handling the vehicle using the steering wheel to follow the reference trajectory. Given the high nonlinearity of the vehicle on one hand, and the uncertainties and disturbances in automotive applications on the other hand, a very important issue to be considered in the control design is the robustness. The controller should be able to reject the disturbances and deal with parameter uncertainties and variations.

In recent years, considerable research has been made to provide lateral guidance of autonomous vehicles. In literature, many control strategies have been developed. Simple PID controllers have been proposed in [START_REF] Broggi | The ARGO autonomous vehicle's vision and control systems[END_REF] and [START_REF] Zhao | Dynamic motion planning for autonomous vehicle in unknown environments[END_REF]. Moreover, other classical techniques have been used. We can cite H ∞ [START_REF] Hima | Trajectory Tracking for Highly Automated Passenger Vehicles[END_REF], state feedback [START_REF] Rajamani | Vehicle dynamics and control[END_REF], Lyapunov stability based control [START_REF] Benine-Neto | Dynamic controller for lane keeping and obstacle avoidance assistance system[END_REF], fuzzy logic [START_REF] Naranjo | Lane-Change Fuzzy Control in Autonomous Vehicles for the Overtaking Maneuver[END_REF], linear quadratic optimal predictive control [START_REF] Kim | Control strategy for high-speed autonomous driving in structured road[END_REF] and many others. On the other hand, Model Predictive Control (MPC) appears to be well suited to the trajectory following [START_REF] Levinson | Towards Fully Autonomous Driving : Systems and Algorithms[END_REF], [START_REF] Besselmann | Autonomous Vehicle Steering Using Explicit LPV-MPC[END_REF]. In [START_REF] Tagne | Higher-Order Sliding Mode Control for Lateral Dynamics of Autonomous Vehicles , with Experimental Validation[END_REF], Sliding Mode Control (SMC) has been applied. This control strategy is known for its robustness against uncertainties and its capacity to reject disturbances. However, its main drawback is the chattering.

In the literature, one can find also some comparisons between existing controllers. In [START_REF] Chaib | H inf, adaptive, PID and fuzzy control: a comparison of controllers for vehicle lane keeping[END_REF], a comparison is made between proportional, adaptive, H ∞ and fuzzy controllers. More recently, in [START_REF] Heß | Comparison of Trajectory Tracking Controllers for Emergency Situations[END_REF], a comparison of two controllers for trajectory tracking was performed. In [START_REF] Menhour | Continuous-Time and Discrete-Time Switched H inf. State Feedback Controllers : Application for a Robust Steering Vehicle Control[END_REF], continuous-time and discrete-time switched H ∞ are compared. It is difficult to make an objective classification from the literature, but it is clear that different results showed that the class of adaptive controllers represents a very promising technique for such uncertain and nonlinear application.

For the best of our knowledge, and after a large bibliographical study, the characteristics of the vehicle dynamics have never been studied in means of energetic and passivity properties. However, if such properties exist, it could allow a very well comprehension of the system and of its inner interconnections. Such result could be the cornerstone for the development of a very promising family of passivitybased controllers, well suited for this uncertain and complex nonlinear system.

Passivity is a concept that represent a very interesting stability property of some physical systems. In fact, passive systems are a class of dynamical systems in which the energy exchange plays a central role. A passive system cannot store more energy than is supplied to it, what reflects a strong stability criterion. Theory of passivity is a framework for analyzing physical systems and designing controllers using a description of the input-output relationship based on energy considerations. We study the passivity to analyze the frequency behavior, determine the passive outputs to easily control the system. We can therefore find the passivity as a way to impose robust stability by developing passivity-based controllers. This is particularly relevant in this application given the parametric variations and uncertainties (speed, curvature, road friction coefficient, etc...). In this paper, the passivity of several input-output maps of the system have been proved. Then, we present the design of two control laws based on the state feedback Passivity-Based Control (PBC). The first one is a simple PD and the second one is a Nested PBC.

To design the controllers, we consider that the vehicle is equipped with sensors and/or observers to measure yaw rate, lateral error and its derivative. To validate the control strategies, the closed-loop system with SCANeR T M studio [1], a driving simulation engine, has been simulated according to several real driving scenarios. The validation shows robustness and good performances of the proposed control approaches.

This paper is organized as follows. Section II presents the dynamical models of the vehicle, used for control design and validation. In Section III, the control problem definition is presented, then the passivity analysis is presented in Section IV. The control strategies for lateral vehicle dynamics are developed in Section V. Section VI presents results. Finally, we conclude in Section VII, with some remarks and future work directions.

II. DYNAMIC MODELS OF VEHICLE

To design the controller, a simple and widely used dynamic bicycle model [START_REF] Rajamani | Vehicle dynamics and control[END_REF] is considered. Dynamic equations in terms of slip angle and yaw rate of the bicycle model are given by:

β = - µ(C f +C r ) mV x β -1 + µ(L f C f -L r C r ) mV 2 x ψ + µC f mV x δ ψ = - µ(L f C f -L r C r ) I z β - µ(L 2 f C f +L 2 r C r ) I z V x ψ + µL f C f I z δ (1)
where β , ψ and δ represent respectively the sideslip angle, the yaw angle of the vehicle and the steering wheel angle (control input). Table I

III. CONTROL PROBLEM DEFINITION

The aim of the lateral control is to minimize the lateral displacement of the vehicle with respect to a given reference path. The lateral error dynamics at the center of gravity of the vehicle, with respect to a reference path, is given by ë = a ya y re f ; where a y and a y re f represent respectively the lateral acceleration of the vehicle, and the desired one on the reference path. Assuming that the desired lateral acceleration can be written as a y re f = V 2

x ρ, where ρ is the curvature of the road, and given that a y = V x ( β + ψ) [START_REF] Rajamani | Vehicle dynamics and control[END_REF], we have:

ë = V x ( β + ψ) -V x 2 ρ (2)
Replacing β by its expression in equation (1), we obtain:

ë = - µ(C f +C r ) m β - µ(L f C f -L r C r ) mV x ψ -V 2 x ρ + µC f m δ (3)
The new system state variables are x = (β , ψ, ė, e) and has the following dynamics:

ẋ = Ax + B 1 δ + B 2 ρ (4)
where,

A =        - µ(C f +C r ) mV x -1 - µ(L f C f -L r C r ) mV 2 x 0 0 - µ(L f C f -L r C r ) I z - µ(L 2 f C f +L 2 r C r ) I z V x 0 0 - µ(C f +C r ) m - µ(L f C f -L r C r ) mV x 0 0 0 0 1 0        B 1 =      µC f mV x µL f C f I z µC f m 0      , B 2 =     0 0 -V 2 x 0     (5)
The control input is the steering wheel angle and the lateral error is the output. The aim of the lateral control is to cancel error of the lateral displacement. Then, for a given curvature ρ and longitudinal velocity V x , the desired behavior corresponds to ė = e = 0. Hence, it is easy to prove that the desired equilibrium point is:

(β , ψ, ė, e) = (β , ψ , 0, 0) with β = (L r - L f mV 2 x µC r (L f +L r ) )ρ ψ = V x ρ (6)
At the equilibrium point, the control input is:

δ = (L f + L r )ρ + mV 2 x (L r C r -L f C f ) µC f C r (L f + L r ) ρ (7)
Hence, define the new error variables:

   β = β -β ψ = ψ -ψ δ = δ -δ (8)
The error dynamics of the system (4) having the origin as equilibrium point ( β , ψ, ė, e) = (0, 0, 0, 0) becomes:

ẋ = A x + B 1 δ (9)
where, A and B 1 have been defined above [START_REF] Zhao | Dynamic motion planning for autonomous vehicle in unknown environments[END_REF].

IV. PASSIVITY ANALYSIS

The Kalman-Yakubovich-Popov (KYP) lemma is considered to be one of the pillars for control and systems theory. To demonstrate the passivity of an output, this lemma is considered [START_REF] Lozano | Dissipative Systems Analysis and Control: Theory and Applications[END_REF].

Proposition 1: The map δ → ë is Strongly Strictly Passive.

Proof: The proof of strong strict passivity is established showing that the transfer function H 0 (s) of the map δ → ë is Strongly Strictly Positive Real (SSPR).

Assume that the road coefficient of friction µ = 1. Combining dynamic equations of the bicycle model in terms of slip angle and yaw rate (1) with the equation of lateral error dynamic (3), and after some calculations, one can find that the transfer function H 0 (s) between δ as input and ë as output is given by:

H 0 (s) = ë(s) δ (s) = as 2 + bs + c s 2 + ds + f (10)
where,

                   a = C f m b = L r C f C r (L f +L r ) mI z V x c = C f C r (L f +L r ) mI z d = (C f +C r ) mV x + (L 2 f C f +L 2 r C r ) I z V x f = C f C r (L f +L r ) 2 mI z V 2 x + (L r C r -L f C f ) I z (11)
The transfer function H 0 (s) has a relative degree equal to 0. Then, according to the KYP Lemma applied in the frequency domain, the proof of SSPR is tantamount to verifying :

ℜ[H 0 ( jω)] ≥ ζ > 0, ∀ω ∈ (-∞, +∞), (12)
with ℜ(), the operator returning the real part. Whatever the uncertainties and variations encountered, the model parameters (C f ,C r , L f , L r , m, I z ) are always positive. Thus, the coefficients a, b, c, d, f of H 0 (s) are always positive. According to the criterion of Routh-Hurwitz, zeros and poles of H 0 (s) are strictly stable, so H 0 (s) is minimum phase and Hurwitz. Setting s = jω, one can prove that:

ℜ[H 0 ( jω)] ≥ ζ > 0, ∀ω ∈ (-∞, +∞) (13)
hence, the transfer function H 0 (s) is Strongly Strictly Positive Real (SSPR), yielding to the desired result. Proposition 2: The map output δ → ė is passive.

Proof: The transfer function H 1 (s) of the output ė with respect to the input δ is given by:

H 1 (s) = 1 s H 0 (s) (14)
The transfer function H 1 (s) is a cascade connection of an integrator with the strongly strictly positive real transfer function H 0 (s). Hence, H 1 (s) is positive real, yielding the passivity of the map δ → ė, the desired result. For more details see [START_REF] Lozano | Dissipative Systems Analysis and Control: Theory and Applications[END_REF]. Proposition 3: The map δ → ψ is Strictly Passive.

Proof: The proof is established at the same manner of the previous subsections.

Proposition 4: The map δ → β is not passive.

Proof: The transfer function H 3 (s) of β , relative to the input δ is given by:

H 3 (s) = β (s) δ (s) = ks + l s 2 + ds + f (15)
where,

    
d and f defined in [START_REF] Levinson | Towards Fully Autonomous Driving : Systems and Algorithms[END_REF],

k = C f mV x l = L r C f C r (L f +L r ) mI z V 2 x - L f C f I z
The transfer function H 3 (s) has a relative degree equal to 1. The variables d, f and k are always positive, but l can be either positive or negative depending on the value of V x and other parameters. When l < 0, H 3 (s) has a negative zero, and is not positive real, what proves the proposition.

The zero of the transfer function H 3 (s) is stable if and only if l is positive, i.e.

- L f C f I z + L r C f C r (L f + L r ) mI z V 2 x > 0, hence, V x < L r C r (L f + L r ) L f m . (16)
This fact can be understood as if the map δ → β is passive when the speed is limited as in [START_REF] Menhour | Continuous-Time and Discrete-Time Switched H inf. State Feedback Controllers : Application for a Robust Steering Vehicle Control[END_REF], and this characteristic is stolen at high speeds. Using the parameters given in Table I, the speed limit of equation ( 16) corresponds to V x < 16.5m/s 60km/h. Proposition 5: The map ψ → ë is Strictly Passive and the map ψ → ė is Passive.

Proof: The proof is established at the same manner of the previous subsections.

Summarizing, considering the error system (9) with the control input δ , then the output ė is passive (P). This is an interesting property of the system that favor the design of passivity based controllers (PBC) using a feedback on this passive output.

On the other hand, the map δ → ψ is strictly passive (SP); moreover, the map ψ → ė is passive. The passivity of these two cascaded subsystems could be interesting to develop nested (PBC) with the intermediate variable ψ. The Fig. 1 can resume the passivity maps of the system. 2, where S 1 is the system and S 2 the controller. Two reasonings can be used to analyze the stability of the closed-loop system. We can seek an energy function based on the closed-loop system. This can typically be obtained by a Lyapunov function. Or we can try to interpret this as a negative dynamic interconnection of two blocks. Specifically, for systems with structural characteristics of passivity, we can try to interpret this as a negative dynamic interconnection of two passive systems then use the passivity theorem [START_REF] Hill | The stability of nonlinear dissipative systems[END_REF] to conclude on the stability of the closedloop system. Obviously, as the system has characteristics of passivity, we will use the second reasoning (equivalent closed-loop interconnections), which is more intuitive. The following corollary (corollary 4.1 in [START_REF] Lozano | Dissipative Systems Analysis and Control: Theory and Applications[END_REF]) remind a very useful property of stability of two systems on feedback interconnections.

Corollary 1: Considering the diagram of Fig. 2, this feedback system with finite gains is stable if either of the following statements is true:

• S 1 is Passive (P) and S 2 is Input Strictly Passive (ISP).

• S 1 is Output Strictly Passive (OSP) and S 2 is Passive.

A. PD controllers

The aim of the lateral control of autonomous intelligent vehicles is to minimize the lateral displacement of the vehicle with respect to a given reference path. Consider the system (9) in which δ is the input and lateral error e is the output. The PD controller applied to that system is given by the following input:

δ = -K D ė -K P e (17)
where K D and K P are state-feedback positive gains.

It is important to note that a PD with the output e is equivalent to a PI with the passive output ė. So, we can interpret the closed-loop system as the interconnection of two subsystems with respectively inputs u 1 and u 2 and outputs y 1 and y 2 as:

y 1 = u 2 = ė u 1 = -y 2 = δ ( 18
)
We have S 1 = H 1 is passive (see Section IV, Proposition 2). S 2 (s) = K P + K D s s S 2 defines an Input Strictly Passive (ISP); It is easy to show that:

ℜ [S 2 ( jω)] ≥ ε > 0, ∀ω ∈ (-∞, +∞)
So the closed-loop system is stable and describes a passive map between the input and the output.

Stability of the system is assured with the PD controller for all gains K D and K P positive. This has also been shown in [START_REF] Rajamani | Vehicle dynamics and control[END_REF]. Note that the gains are chosen taking into account practical considerations and the desired performances. Finally, the control input applied to the system (4) is:

δ PD = δ + δ = -K P e -K D ė + (L f + L r )ρ + mV 2 x (L r C r -L f C f ) µC f C r (L f +L r ) ρ (19)
With this classical PD controller, we have robust stability unfortunately the performance depends on parameters of the system. To have good performance over a wide operating range, such a controller must be adaptive gains. In [START_REF] Tagne | Immersion and Invariance Control for Reference Trajectory Tracking of Autonomous Vehicles[END_REF], the proposed controller by Immersion and Invariance (I&I) can be interpreted as a dynamic state feedback and an adaptive PD, where the gains depend on the parameters. This control law is given by:

δI&I = -mKλ C f e -m(K+λ ) C f ė + C f +C r C f β + L f C f -L r C r C f V x ψ ( 20
)
With this command, performance can be fixed in advance. Its main drawback is that, large parametric uncertainties degrade performances.

With controllers that have a PD structure, one gets a robust stability. With PI structure, robustness against uncertainties and disturbances can be improved, but the system becomes easily destabilizable. In the next paragraph, we will develop a nested controller based on the passivity properties of the system, to obtain a good compromise between robust stability, and robustness against parametric uncertainties and disturbances. Indeed, the closed-loop system preserves its passivity properties necessary for high speed driving while having good performance.

B. Nested Passive Controllers (Nested PBC)

Considering the Fig. 1 and knowing that the yaw rate dynamics is faster than the lateral error dynamics, we can decompose the lateral controller into two passive nested controllers. This is a major interest since e and ψ are controlled simultaneously. We can therefore create two separate controllers for each own dynamics based on its characteristic of passivity. In this subsection, we develop a control strategy of an autonomous vehicle using two passive controllers to ensure guidance and stability in two separate loops. The outer controller minimizes the lateral error. The inner controller minimizes the yaw rate error by providing the corresponding steering angle. The particularity of such a strategy is double: it helps designing a robust controller while preserving the passivity properties of the closed-loop system to ensure good performance. If necessary, two controllers of different natures can be used, in order to benefit from the robustness of each. It also allows an control of the lateral error and the yaw rate error. Fig. 3 scheme illustrates the proposed control strategy.

Proposition 6: Considering the diagram in Fig. 3 (where the control is achieved respectively by two PD and PI controllers), the closed-loop system is stable and passive.

Synthesis of the outer controller:

This controller is designed to cancel the lateral error e. The control input is the yaw rate error.

Subsystem ∑ 1 : ψ -→ ė is passive (P), so any ISP controller guarantee stability and passivity of the closed-loop system. In this paper, we choose a simple PI. We remind that a PI applied to the passive output ė is equivalent to a PD applied to the output e. So, the outer control input of the controller is given by:

ψ = -K D1 ė -K P1 e (21)
where K D1 and K P1 are positive gains.

Synthesis of the inner controller:

This controller is designed to cancel the yaw rate error with respect to the reference given by the outer controller. The control input is the steering angle δ .

Subsystem ∑ 2 : δ -→ ψ is strictly passive (SP), so any passive (P) controller guarantee stability and passivity of the closed-loop system. In this paper, we choose a PI. The control input of the inner controller is given by:

δ = -K I2 ψ -K P2 ψ (22)
where K I2 and K P2 are positive gains. Given the strict passivity (SP) of the output ψ for an input δ , the closed-loop system with a PI controller is stable and passive. Note that any passive controller, a simple proportional for example, would achieve the same result. The addition of integral action, well known to reject constant disturbances also has the advantage that the controller can be implemented without the knowledge of δ which depends on the uncertain model parameters.

VI. SIMULATION RESULTS

To validate our control laws, tests have been performed on the simulation environment SCANeR T M studio, a driving simulation engine, according to several real driving scenarios. For control laws, we used the following values of the gains: K P = 0.08 and K D = 0.01 for the PD controller. K P1 = 10, K D1 = 1, K P2 = 0.05 and K I2 = 0.02 for the Nested PBC, with the nominal vehicle parameters (see Table I). To highlight the improvements brought by the PBC controllers, a comparison with a previously developed I&I controller [START_REF] Tagne | Immersion and Invariance Control for Reference Trajectory Tracking of Autonomous Vehicles[END_REF] was made. The tests are performed on a road with a curvature varying between -0.02 and +0.02m -1 (see Fig. 4). This test was made with the goal of verifying the capacity of controllers to track the reference trajectory during normal driving with known nominal parameters. Fig. 5 presents different curves: the reference path and the trajectories followed by the controlled vehicles; the lateral error and the yaw angle error. All controllers ensure the reference tracking with low errors (the lateral displacement of the closed-loop system remains smaller than 15cm in this test conditions). Note that the PD controller is less robust to curvature variations. This controller requires only the measurement of the lateral error, what is not sufficient to ensure a robust tracking in highly nonlinear areas. For small changes in curvature, the errors of the three controllers are almost similar. This test shows the good performance of I&I and Nested PBC controllers during normal driving at high and varying speed, and varying curvature, when the nominal parameters are known. We will evaluate the robustness of these two controllers with respect to parametric uncertainties.

B. Robustness to vehicle parameter uncertainties

In this subsection, we evaluate the robustness of the controllers with respect to parametric uncertainties of the vehicle. It is important to note that the parametric uncertainties can be due to the fact that the parameters may vary, but are considered to be fixed for the command. This is the case of the mass of the vehicle, for example. Fig. 6 presents lateral errors for uncertainties on the value of the cornering stiffness for previous test. Fig. 7 presents the robustness of the control laws against the uncertainties on the vehicle mass. For uncertainties in the order of +/ -15%, on the value of cornering stiffness (or of +/ -10% for mass), PBC controllers are able to follow the path with almost With the I&I controller, the error depends strongly on the value of the parametric uncertainty. The passive controller (Nested PBC) is more robust to parameter uncertainties. Indeed, the inputs of the controller depend only on passive outputs (errors), which do not depend on uncertain parameters. However, the input of the I&I controller depends on these parameters. Moreover, the PBC controller also has the advantage of using less measurements than the previous one. Indeed, the I&I controller needs an estimation of the sideslip angle. Hence, these results put in evidence the improvement brought by the Nested PBC controller.

VII. CONCLUSION

In this paper, after a detailed study of the passivity properties of the model, controllers based on these properties are proposed to ensure a robust trajectory following. The design of two controllers was presented: a simple PD controller and a nested PBC controller. They guarantee robust stability (not depending on the value of the system parameters) and passivity of the closed-loop system. Results show that a greater consideration of structural features of the model during the design of a controller -namely the passivity properties-can significantly improve the robustness of the controller for autonomous driving applications. To illustrate this improvement, a comparison was made with an Immersion and Invariance controller developed previously in [START_REF] Tagne | Immersion and Invariance Control for Reference Trajectory Tracking of Autonomous Vehicles[END_REF]. The validation shows robustness and good performance of the proposed Nested PBC controller.

In this paper, the study of passivity properties has been done considering the road coefficient of friction µ = 1. An analysis for variant µ will be studied by the authors. In addition, a study of robustness based on sensitivity analysis will be performed. Moreover, an adaptive version of the Nested PBC controller is under study.

Fig. 1 .

 1 Fig. 1. Passivity maps of the system

Fig. 2 .

 2 Fig. 2. Feedback interconnection of passive systems scheme

Fig. 3 .

 3 Fig. 3. Nested PBC control strategy

Fig. 4 .

 4 Fig. 4. Road informations and longitudinal speed

Fig. 5 .

 5 Fig. 5. Trajectories; reference and control laws

Fig. 6 .Fig. 7 .

 67 Fig. 6. Robustness against uncertainties of cornering stiffness

  presents vehicle parameters and nomenclature.

This is a very active research field that has been studied since the This work was carried out in the framework of the Labex MS2T, which was funded by the French Government, through the program Investments for the future managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02).

The authors are working at Heudiasyc Laboratory, UMR CNRS 7253,


Gilles Tagne
email: gilles.tagne@hds.utc.fr

Reine Talj
email: reine.talj@hds.utc.fr

Ali Charara
email: ali.charara@hds.utc.fr

Immersion and Invariance vs Sliding Mode Control for Reference Trajectory Tracking of Autonomous Vehicles

. The second controller is based on Immersion and Invariance (I&I) principle. Given the implicit resemblance between the SMC and the I&I principle, the comparison allows us to highlight the advantages and disadvantages of each control strategy for robust lane keeping for intelligent vehicles. To validate the control strategies, the closed-loop system has been simulated on Matlab-Simulink has been made using the experimental data acquired on the vehicle DYNA of Heudiasyc laboratory, a Peugeot 308, according to several real driving scenarios. The validation shows robustness and performances of each control approach, and puts in evidence the improvement brought by the I&I controller.

I. INTRODUCTION

The development of Advanced Driver Assistance Systems (ADAS) and autonomous driving has been favored by technological advances in recent years. Three main steps are necessary to ensure an autonomous navigation: the perceptionlocalization, the path planning and the vehicle control. The vehicle control can be divided into two tasks: longitudinal control and lateral control. This paper focuses on the lateral control of intelligent vehicles. This is a very active research field that has been studied since the 1950s.

Lateral control consists on handling the vehicle using the steering wheel to follow the reference trajectory. Given the high nonlinearity of the vehicle on one hand, and the uncertainties and disturbances in automotive applications on the other hand, a very important issue to be considered in the control design is the robustness. The controller should be able to reject disturbances and deal with parameter uncertainties and variations. For example, in [START_REF] Levinson | Towards Fully Autonomous Driving : Systems and Algorithms[END_REF], a recent presentation of Junior; Stanford's autonomous research vehicle (the second at the DARPA Urban Challenge) is made for the purpose of ensuring robust autonomous driving.

In recent years, considerable research has been made to provide lateral guidance of autonomous vehicles. In literature, many control strategies have been developed. Simple PID controllers have been proposed in [START_REF] Broggi | The ARGO autonomous vehicle's vision and control systems[END_REF] and [START_REF] Zhao | Dynamic motion planning for autonomous vehicle in unknown environments[END_REF]. We also have nested one in [START_REF] Marino | Nested PID steering control for lane keeping in autonomous vehicles[END_REF]. Moreover, other classical techniques have been used. We can cite H ∞ [START_REF] Hima | Trajectory Tracking for Highly Automated Passenger Vehicles[END_REF], state feedback [START_REF] Rajamani | Vehicle dynamics and control[END_REF], Lyapunov stability based control [START_REF] Benine-Neto | Dynamic controller for lane keeping and obstacle avoidance assistance system[END_REF], fuzzy logic [START_REF] Naranjo | Lane-Change Fuzzy Control in Autonomous Vehicles for the Overtaking Maneuver[END_REF], fuzzy Takagi-Sugeno LQ [START_REF] Soualmi | Fuzzy Takagi-Sugeno LQ controller for lateral control assistance of a vehicle[END_REF], linear quadratic optimal predictive control [START_REF] Kim | Control strategy for high-speed autonomous driving in structured road[END_REF], backstepping based approach [START_REF] Nehaoua | Backstepping based approach for the combined longitudinal-lateral vehicle control[END_REF] and many others. On the other hand, Model Predictive Control (MPC) appears to be well suited to the trajectory following [START_REF] Levinson | Towards Fully Autonomous Driving : Systems and Algorithms[END_REF], [START_REF] Besselmann | Autonomous Vehicle Steering Using Explicit LPV-MPC[END_REF]. In [START_REF] Hingwe | Experimental evaluation of a chatter free sliding mode control for lateral control in AHS[END_REF] and [START_REF] Tagne | Higher-Order Sliding Mode Control for Lateral Dynamics of Autonomous Vehicles , with Experimental Validation[END_REF], Sliding Mode Control (SMC) has been applied to the lateral control. This robust control strategy is well suited to driving applications, given its robustness against uncertainties and its capacity to reject disturbances. However, its main drawback is the chattering.

A variety of different vehicle controllers was proposed for trajectory tracking. Most of the works published previously do not compare their results with other approaches. Unfortunately, comparisons between existing control concepts are very rare. In [START_REF] Chaib | H inf, adaptive, PID and fuzzy control: a comparison of controllers for vehicle lane keeping[END_REF], a comparison was made between proportional, adaptive, H ∞ and fuzzy controllers. More recently, in [START_REF] Heß | Comparison of Trajectory Tracking Controllers for Emergency Situations[END_REF], a comparison of two controllers for trajectory tracking emergency was performed. In [START_REF] Menhour | Continuous-Time and Discrete-Time Switched H inf. State Feedback Controllers : Application for a Robust Steering Vehicle Control[END_REF], continuous-time and discrete-time switched H ∞ were compared. It is therefore difficult to make an objective classification from the literature but different results showed clearly that the class of adaptive controllers represents a very promising technique for such uncertain and nonlinear application.

In [START_REF] Tagne | Higher-Order Sliding Mode Control for Lateral Dynamics of Autonomous Vehicles , with Experimental Validation[END_REF], we have developed a controller based on higherorder sliding mode control. Given the implicit resemblance between the sliding mode controller and the I&I principle, for the purpose of designing nonlinear and adaptive controllers, a controller based on the I&I approach is developed for a comparative study.

The I&I principle is a relatively new method for designing nonlinear and adaptive controllers. The method relies upon the notions of system immersion and manifold invariance. The basic idea of the I&I approach is to achieve the control objective by immersing the plant dynamics into a (possibly lower-order) target system that captures the desired behavior [START_REF] Astolfi | Nonlinear and Adaptive Control with Applications[END_REF]. This is achieved by finding a manifold in state-space that can be rendered invariant and attractive; with internal dynamics that reflect the desired closed-loop dynamics and by designing a control law that takes the state of the system towards the manifold. In others words, the I&I theory consists of defining a target dynamics and to design a control law that renders the manifold of the target dynamics attractive and invariant. Indeed, the I&I reformulation of the stabilization problem is implicit in sliding mode control, where the target dynamics are the dynamics of the system on the sliding surface, which is made attractive by a discontinuous control law.

In this paper, the design of both controllers is presented. Moreover, the I&I method allows to prove a very strong stability criterion of the closed-loop system. In fact, for all controller gains chosen to be positive, the closed-loop system is stable. To design the controller, we consider that the vehicle is equipped with sensors or observers to measure sideslip angle, yaw rate, lateral error and its derivative. Both controllers use the same signals (variables). To validate the control laws, tests are made with real data acquired on the vehicle DYNA, on the tracks and circuits of CERAM 1 . The simulation results show the performance and robustness of the proposed approaches and put in evidence the improvement brought by the I&I controller.

This paper is organized as follows. Section II presents the dynamical models of the vehicle, used for control design and validation. In Section III, the control problem definition is presented. The design of the SMC and the I&I controllers is made in Section IV. Section V presents the simulation results. Finally, we conclude in Section VI, with some remarks and future work directions.

II. DYNAMIC MODELS OF VEHICLE

In this work, we use two vehicle models. To design the controller, a simple and widely used dynamic bicycle model [START_REF] Rajamani | Vehicle dynamics and control[END_REF] is considered (see Fig. 1). This model is used to represent the lateral vehicle behavior and assumes that the vehicle is symmetrical, and tire sideslip angles on the same axle are equal. The roll and pitch dynamics are neglected and angles are assumed to be small (steering, sideslip, yaw). With a linear tire force model we obtain a Linear Parameter Varying (LPV) model, where the longitudinal velocity V x is considered as a varying parameter. Dynamic equations in terms of sideslip angle and yaw rate of the bicycle model are given by:

   β = - (C f +C r ) mV x β -(1 + L f C f -L r C r mV 2 x ) ψ + C f mV x δ ψ = - L f C f -L r C r I z β - L 2 f C f +L 2 r C r I z V x ψ + L f C f I z δ (1)
where β , ψ and δ represent respectively the sideslip angle, the yaw angle of the vehicle and the steering wheel angle (control input). Table I presents the vehicle parameters and nomenclature.

To validate in simulation the proposed controllers in closed-loop, we used a more representative model, namely

III. CONTROL PROBLEM DEFINITION

The aim of the lateral control of autonomous intelligent vehicles is to minimize the lateral displacement of the vehicle with respect to a given reference path. The lateral error dynamics at the center of gravity of the vehicle, with respect to a reference path, is given by ë = a ya y re f ; where a y and a y re f represent respectively the lateral acceleration of the vehicle, and the desired one on the reference path. Assuming that the desired lateral acceleration can be written as a y re f = V 2

x ρ, where ρ is the curvature of the road and given that a y = V x ( β + ψ) [START_REF] Rajamani | Vehicle dynamics and control[END_REF], one can find:

ë = V x ( β + ψ) -V x 2 ρ (2)
Replacing β by its expression in equation ( 1), we obtain:

ë = - C f +C r m β - L f C f -L r C r mV x ψ -V 2 x ρ + C f m δ (3)
The new system state variables are x = (β , ψ, ė, e) , corresponding to the sideslip angle, the yaw rate, the lateral error and its derivative. The system has the following dynamics:

ẋ = Ax + B 1 δ + B 2 ρ (4)
where,

A =        - (C f +C r ) mV x -1 - (L f C f -L r C r ) mV 2 x 0 0 - (L f C f -L r C r ) I z - (L 2 f C f +L 2 r C r ) I z V x 0 0 - (C f +C r ) m - (L f C f -L r C r ) mV x 0 0 0 0 1 0        B 1 =      C f mV x L f C f I z C f m 0      , B 2 =     0 0 -V 2 x 0     (5)
The aim of the lateral control is to cancel the lateral displacement error. Then, for a given curvature ρ and longitudinal velocity V x , the desired behavior corresponds to ė = e = 0. Hence, it is easy to prove that the desired equilibrium point is [START_REF] Rajamani | Vehicle dynamics and control[END_REF]:

(β , ψ, ė, e) = (β , ψ , 0, 0) with β = (L r - L f mV 2 x C r (L f +L r ) )ρ ψ = V x ρ (6)
At the equilibrium point, the control input is:

δ = (L f + L r )ρ + mV 2 x (L r C r -L f C f ) C f C r (L f + L r ) ρ (7)
We define the new error variables:

   β = β -β ψ = ψ -ψ δ = δ -δ (8)
Hence, the error dynamics of the system (4) having the origin as equilibrium point ( β , ψ, ė, e) = (0, 0, 0, 0) become:

ẋ = A x + B 1 δ (9)
where, A and B 1 have been defined above [START_REF] Marino | Nested PID steering control for lane keeping in autonomous vehicles[END_REF].

IV. SMC AND I&I CONTROLLERS DESIGN

After a brief presentation of the SMC controller, we will present the design of the I&I controller.

A. SMC controller

The SMC has been developed since 1950s and is recognized as one of the most promising techniques for robust control. The principle of SMC is to constrain the system trajectories to reach in finite time and remain on a sliding surface. However, its main drawback is the chattering. In this subsection, we recall the main lines of the design of the SMC controller. We used the super-twisting algorithm to minimize the lateral displacement. For more details, see [START_REF] Tagne | Higher-Order Sliding Mode Control for Lateral Dynamics of Autonomous Vehicles , with Experimental Validation[END_REF].

The objective is to cancel the lateral displacement error. To this end, choosing the sliding variable s as follows:

s = ė + λ e (10)
we obtain: ṡ = ë + λ ė. Replacing ë by its expression (3), we obtain:

ṡ = - C f +C r m β - L f C f -L r C r mV x ψ -V 2 x ρ + C f m δ + λ ė (11)
The variable s has a relative degree r = 1 with respect to the input δ . We can write ṡ, of the form:

ṡ(t, s) = φ (t, s) + ϕ(t, s)δ (12)
with,

φ (t, s) = - C f +C r m β - L f C f -L r C r mV x ψ -V 2 x ρ + λ ė ϕ(t, s) = C f m (13)
It is assumed that there exist positive constants s 0 , b min , b max , C 0 such that ∀x ∈ R n and |s(t, x)| < s 0 , the system satisfies the following conditions:

   |δ | ≤ δ max 0 < b min ≤ |ϕ(t, s)| ≤ b max |φ (t, s)| < C 0 (14)
So, the SMC algorithm based on super-twinting is given by:

δ ST = δ 1 + δ 2 δ 1 = -α 1 |s| 1/2 sign(s) δ2 = -α 2 sign(s) (15)
with α 1 and α 2 positive constants. The finite time convergence to the sliding surface is guaranteed for all:

   α 2 > C 0 b min α 1 ≥ 4C 0 (b max α 2 +C 0 ) b 2 min (b min α 2 -C 0 ) (16)
To avoid important peaks in transient phases, we add an equivalent command δ eq obtained by solving the equation ṡ = 0. This term has the role of a feedforward that brings the system near to the sliding surface, and is given by:

δ eq = - m C f φ (t, s) (17)
Hence, the steering angle representing the control input of the system is defined as follows:

δ SMC = δ ST + δ eq (18)

B. I&I controller

The principle of the I&I consists to achieving the control objective by immerging the system into a target dynamics that ensures the objective of the command. To design the I&I controller, some assumptions should be verified: the system and the target dynamics should both have their equilibrium at the origin respectively. The immersion of the system is achieved by an attractive and invariant manifold. So, the command consists on bringing the system state to the manifold. Astolfi et al. [START_REF] Astolfi | Nonlinear and Adaptive Control with Applications[END_REF] presents the main stabilization result of the Immersion and Invariance method. Now consider the vehicle lateral dynamical model [START_REF] Naranjo | Lane-Change Fuzzy Control in Autonomous Vehicles for the Overtaking Maneuver[END_REF]. As mentioned before, the main objective of the steering controller is to cancel the lateral error displacement with respect to a given trajectory, i.e. e = ė = 0 at the equilibrium.

The target system is consequently chosen to be the image of the system ( 9) when e = ė = 0. More precisely, the target state vector is (ξ 1 , ξ 2 ), where ξ 1 and ξ 2 represent β and ψ respectively, when e = ė = 0. Note that for e = ė = 0, we also have ë = 0. The dynamics can be expressed as follows,

ξ1 ξ2 = 0 -1 C r (L f +L r ) I z - L r C r (L f +L r ) I z V x ξ 1 ξ 2 ( 19
)
Proposition 1: The target model ( 19) has a globally asymptotically stable equilibrium at the origin (0, 0).

Proof: The dynamics of the state vector ξ = [ξ 1 , ξ 2 ] can be written in the form ξ = Cξ , with

C = 0 -1 C r (L f +L r ) I z - L r C r (L f +L r ) I z V x (20)
Then, after some simple calculations, we obtain:

det(sI -C) = s 2 + L r C r (L f + L r ) I z V x s + C r (L f + L r ) I z (21)
The matrix C verifies the Routh-Hurwitz stability criterion (V x > 0), which yields to the desired result.

Proposition 2: Consider the system (9) having the equilibrium point at the origin. Moreover, the subsystem [START_REF] Kokotovic | On vanishing stability regions in nonlinear systems with high-gain feedback[END_REF], which is the image of the system ( 9) for e = ė = 0, has a globally asymptotically stable equilibrium at the origin. Then, the system ( 9) is I&I-stabilisable with target dynamics [START_REF] Kokotovic | On vanishing stability regions in nonlinear systems with high-gain feedback[END_REF].

Proof: We define now the off-the-manifold variable

z = ė + λ e, s.t. λ > 0 (22)
Note that we choose z = s, the sliding variable defined above for the design of the SMC controller.

We have to select a control input δ such that the trajectories of the closed-loop system are bounded and z = ė + λ e converges to zero. Notice that, when z → 0, e converges exponentially to zero with the rate of convergence λ , yielding to the desired result. There is a strong resemblance with the principle of SMC which is to constrain the system trajectories to reach in finite time and remain on a sliding surface s. In the SMC, the convergence in finite time is guaranteed by a special law [START_REF] Chaib | H inf, adaptive, PID and fuzzy control: a comparison of controllers for vehicle lane keeping[END_REF] with respect to certain conditions [START_REF] Heß | Comparison of Trajectory Tracking Controllers for Emergency Situations[END_REF]. For the I&I controller, we choose an exponential convergence. To this end, let

ż = -Kz, with K > 0. ( 23
)
where K represents the rate of exponential convergence of z to zero. Replacing ż and z by their expressions in (23), and after some calculations, one can find that the corresponding control input has the following expression:

δ = - m(K + λ ) C f ė - mKλ C f e + C f +C r C f β + L f C f -L r C r C f V x ψ (24)
The closed-loop system becomes:

    β ψ ë ė     = A 11 A 12 A 21 A 22     β ψ ė e     (25)
with,

A 11 = 0 -1 C r (L f +L r ) I z - L r C r (L f +L r ) I z V x = C, A 12 = -(K+λ ) V x -Kλ V x - L f m(K+λ ) I z - L f mKλ I z , A 21 = 0 0 0 0 , A 22 = -(K + λ ) -Kλ 1 0 .
The closed-loop system can be interpreted as the interconnection of two subsystems: S 1 with the state variables ( β and ψ), and S 2 (with the state variables ė and e). Given that the interaction matrix A 21 is identically zero, hence, the dynamics of the subsystem S 2 are independent of S 1 , and can be written as follows:

ë ė = -(K + λ ) -Kλ 1 0 ė e ( 26
)
The subsystem S 2 combines and represents the interactions between the dynamics of both off-the-manifold variables z and e, which ensure the convergence of e to zero.

Proposition 3: The subsystem S 2 has a globally asymptotically stable equilibrium at the origin (0, 0).

Proof: The dynamics of the state vector ζ = [ ė, e] can be written in the form : ζ = A 22 ζ . The matrix A 22 verifies the Routh-Hurwitz stability criterion for all λ > 0 and K > 0, hence the attractivity of the manifold is ensured. Consequently, the rates of convergence K and λ of z and e respectively, have to be chosen based on other practical considerations related to the system.

The subsystem S 1 depends on S 2 via the matrix A 12 . Defining ũ = (K + λ ) ė + Kλ e, the subsystem S 1 in closedloop has the form:

β ψ = C β ψ + -1 V x - L f m I z ũ ( 27
)
Finally it is clear that when S 2 converges to (0, 0), the input ũ of the subsystem S 1 converges to zero. Then, the system S 1 converges to the target dynamics [START_REF] Kokotovic | On vanishing stability regions in nonlinear systems with high-gain feedback[END_REF], which is stable and converges to (0, 0). Indeed, we have proved previously that this target model has a globally asymptotically stable equilibrium at the origin (0, 0). So, the trajectories of the closed-loop system are bounded, yielding to the desired result. For more details on the proof of global asymptotic stability of the closed loop system, see [START_REF] Kokotovic | On vanishing stability regions in nonlinear systems with high-gain feedback[END_REF], which presents that the cascade of Global Asymptotic Stable systems (GAS) with a bounded trajectory gives an global asymptotic stable system. In other words, GAS + GAS + bounded trajectory = GAS. Stability of the system is assured with the proposed I&I controller for all its gains λ > 0 and K > 0. This is a very strong result that reveals some interesting characteristics of the system. Finally, the control input applied to the system (4) is:

δ I&I = δ + δ = -m(K+λ ) C f ė -mKλ C f e + C f +C r C f β + L f C f -L r C r C f V x ψ + mV 2 x C f ρ (28)
The design of this controller allows us to prove that the stability of the controller is guaranteed for all positive gains. This result helps us to show certain characteristics of passivity of the system. These structural features of the system help to improve the design of the controller. Moreover, the response time of the controller is entirely determined by the parameters K and λ , which is not the case for the SMC controller. Finally, the design of this controller guarantees robust stability and some a priori known performances (response time of the controller).

V. RESULTS

To validate our control laws, the used experimental data were acquired on the vehicle DYNA on the CERAM test circuits. This vehicle is equipped with different sensors or observers providing the dynamic variables of the vehicle and its traveled path. The simulations in closed-loop are performed using the real data, as a reference with the full vehicle model. For the control laws, we used the gains λ = 8 and K = 1 for the I&I controller, and λ = 8, α 1 = 0.005, α 2 = 0.002 for the SMC controller, with the nominal vehicle parameters (see Table I).

A. Test of the controllers during normal driving with nominal parameters

In this test (Fig. 3), lateral acceleration is less than 4m/s 2 . Longitudinal velocity is almost constant (13.5m/s). Fig. 3 shows the reference path and the trajectories followed by the controlled vehicles, the longitudinal speed variations and the lateral error. We observe that both controllers can Fig. 3. Longitudinal speed, trajectories and lateral error ensure the convergence of the lateral error. After the transient periods (change in curvature of the road), the SMC controller converges to 0 while the I&I controller converges to a small but nonzero value. The convergence time of the SMC controller is lower than that of the I&I controller. Both controllers have maximum comparable errors and ensure trajectory tracking (variable curvature).

Several other tests were performed, including scenarios of driving at high speed as well as situations where there is a saturation of lateral forces. So, tests show the good performance of both control strategies during normal driving at high and varying speed for nominal parameters known.

B. Robustness of controllers to vehicle parameter uncertainties

As mentioned previously, one of the major challenges for trajectory tracking is the robustness of the controller. In this subsection, we evaluate the robustness of the controllers with respect to parametric uncertainties of the vehicle. It is important to note that the parametric uncertainties can be due to the fact that the parameters may vary, but are considered to be fixed for the command, for example the mass of the vehicle.

It is difficult to estimate accurately the cornering stiffness of the tire. Moreover, this parameter varies greatly depending on the type of the road, the vertical load, the camber, etc. It is therefore important to assess the robustness of the controller over cornering stiffness variations. Fig. 4 presents lateral errors for different uncertainties on the value of the cornering stiffness for the previous test. For uncertainties in the order of +/ -30%, on the value of cornering stiffness, both controllers are able to follow the path with acceptable errors. In other words, the controllers are able to track the trajectory with a road coefficient of friction of 0.7 (cornering stiffness proportional to road coefficient of friction). Therefore, the robustness of the controller against cornering stiffness variations implicitly allows us to evaluate the robustness with respect to an unknown road coefficient of friction.

For uncertainties in the order of 10% on the value of cornering stiffness or mass, the SMC controller have similar errors (to those of nominal conditions). With the I&I controller, the error depends on the value of the parametric uncertainty. Note that the value of the parameter uncertainty strongly influences the convergence time of the SMC controller. When parametric uncertainty exceeds a certain threshold, the convergence over time is not guaranteed, and the errors of the SMC controller increases with great rates (see Fig. 4, SMC +/ -30%).

To summarize, we observe that the SMC controller provided low and almost constant errors while the condition of convergence in finite time is satisfied. When parametric uncertainty exceeds a certain threshold, the convergence in finite time is not guaranteed, and the error increases with a very large rate. It is important to note that this threshold is not known a priori and depends on the parameters, variables of the system and the amplitudes of the noises in the data. By increasing the values of the SMC controller gains to ensure acceptable errors for large parametric variations, one obtains an excessive increase in the phenomenon of chattering. This has the effect of degrading the passenger comfort. Also, it is important to note that in practice it is limited by the bandwidth of the steering actuator. In this validation, we use a steering actuator with a cut-off frequency of 10Hz. For reasons of comfort we could impose a lower frequency, which would has the effect of degrading the performance of the SMC.

With the I&I controller, the error depends on the value of the parametric uncertainty and remains acceptable even for large variations. We note that large variations in the parameters degrade the performance but stability is maintained. This clearly shows that we have a robust stability (not depending on the value of the parameters of the system). Finally, these results show robustness and performances of each control approach.

VI. CONCLUSION

In this paper, we have compared two controllers for trajectory tracking of an autonomous vehicle. The first controller is the higher order sliding mode and specifically the super-twisting algorithm. The second controller is based on the Immersion and Invariance principle. Given the great similarity between these two control strategies, we aimed to improving the sliding mode by using the Immersion and Invariance principle (reduction or elimination of chattering, reduction of the sensitivity to the dynamic of the actuator, greater taking into account of the model).

The design of both controllers has been presented. The SMC controller has shown better performance for small parameter variations, while the condition of convergence in finite time is satisfied. This condition depends on the value of the system parameters and noise. It is therefore difficult to tune this controller. The design of the I&I controller allowed us to prove that the stability of this controller is guaranteed for all positive gains (λ > 0 and K > 0). In addition, the response time of that controller is totally determined by the parameters K and λ . So, performance and tuning of this controller is easily determined. This controller is able to give acceptable errors for large parametric uncertainties and guarantees a comfortable ride (absence of chattering). Also it is more robust to perform emergency maneuvers (for example, big turns). It is clear that with a greater consideration of the model and the characteristics of passivity of the system, the control by I&I can greatly improve the performances for autonomous driving applications and ADAS.

Stability of the system is assured with the proposed I&I controller for all its gains λ > 0 and K > 0. This is a very strong result that reveals some interesting characteristics of the system. More deep and promising characteristics of the system are under study now by the authors. Moreover, we will also test these control laws on a semi-autonomous vehicleour robotized vehicle, in the laboratory Heudiasyc.

Fig. 1 .

 1 Fig. 1. Bicycle model

Fig. 2 .

 2 Fig. 2. Experimental vehicle (DYNA)

Fig. 4 .

 4 Fig. 4. Robustness against uncertainties of cornering stiffness

TABLE I VEHICLE

 I PARAMETERS AND NOMENCLATURE (BICYCLE MODEL)

	V x	Longitudinal velocity	-	[m/s]
	β	Sideslip angle	-	[rad]
	ψ	Yaw rate	-	[rad/s]
	δ	Steering wheel angle	-	[rad]
	m	Mass	1719	[kg]
	I z	Yaw moment of inertia	3300	[kgm 2 ]
	L f	Front axle-COG distance	1.195	[m]
	L r	Rear axle-COG distance	1.513	[m]
	C f	Cornering stiffness of the front tire	170550	[N/rad]
	C r	Cornering stiffness of the rear tire	137844	[N/rad]
	the 4-wheel model to represent the behavior of the vehicle
	and Dugoff's tire model for longitudinal and lateral tire
	forces.

CERAM -"Centre d'Essais et de Recherche Automobile de Mortefontaine" is an automobile testing and research center located in France.

This work was carried out in the framework of the Labex MS2T, which was funded by the French Government, through the program Investments for the future managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02).

The authors are working at Heudiasyc Laboratory, UMR CNRS 7253,