Andre Benine-Neto

Stefano Scalzi

Said Mammar

André Benine-Neto

Saïd Mammar

Vehicle lane keeping control based on piecewise affine regions

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

It can be noted that earlier work was devoted to autonomous vehicles in highway scenarios in which the experienced lateral accelerations are generally low [START_REF] Fenton | On the steering of automated vehicles: Theory and experiment[END_REF]. These scenarios have also been considered in more recent works for the development of assistance systems which replace the driver during the lane departure avoidance maneuvers [START_REF] Minoiu-Enache | Driver steering assistance for lane departure avoidance[END_REF]. Nevertheless, in order to provide more safety to the drivers, these Lane Keeping Assistance Systems (LKAS) must be able to operate in the complete domain of tire forces, including the nonlinear behavior [START_REF] Hatipoglu | Automated lane change controller design[END_REF] and [START_REF] Falcone | Predictive active steering control for autonomous vehicle systems[END_REF], as many accidents occur due to the vehicle loss of control when the tire forces are saturated.

A possibility to take into account the nonlinear behavior of the lateral tire forces is the design of a parallel distributed controller based on a Takagi-Sugeno model [START_REF] Mammar | Lane keeping automation at tire saturation[END_REF] and [START_REF] Oufroukh | Invariant set based vehicle handling improvement at tire saturation using fuzzy output feedback[END_REF].

Piecewise affine (PWA) systems also offer a possibility to model the tire forces saturation and they have been used in the development of some driver assistance systems. In [START_REF] Borrelli | A Hybrid Approach to Traction Control[END_REF] the longitudinal tire forces are approximated by PWA functions for a traction control problem. The lateral tire forces have also been considered in PWA form to design yaw rate controllers in [START_REF] Bernardini | Driveby-wire vehicle stabilization and yaw regulation: A hybrid model predictive control design[END_REF], [START_REF] Benine-Neto | Vehicle yaw rate control based on piecewise affine regions[END_REF] and [START_REF] Palmieri | Constrained robust design of lateral vehicle dynamics control systems[END_REF].

In [START_REF] Benine-Neto | Vehicle yaw rate control based on piecewise affine regions[END_REF], the authors proposed a parametrization of the vehicle dynamics in terms of yaw rate to design a PWL controller for an active steering system. The parametrization in terms of this variable is chosen because sensors to measure it are available in the currently commercialized vehicles.

In this paper, the same approaches are employed for the approximation of the lateral tire force and the parametrization of vehicle dynamics. Nevertheless, the model is extended with the positioning of the vehicle in the lane in order to design a lane keeping controller which takes into account the lateral tire force nonlinear tire behavior and is able to reject disturbances such as the road curvature.

The paper is organized as follows: Section II presents the vehicle lateral dynamics. In Section III, the PWA approximation of the nonlinear model is shown, as well as a parametrization to describe the partitions in terms of the yaw rate. The strategy used for the design of the Piecewise Linear (PWL) control is addressed in Section IV. The proof of stability for the closed-loop PWA system is presented in Section V. The results obtained by simulations are discussed in Section VI and a conclusion summarizes the improvements obtained by the proposed lane keeping controller design.

II. NONLINEAR VEHICLE MODEL

A widely used and simplified single track vehicle model, as illustrated Fig. 1, was introduced by [START_REF] Ackermann | Robust Control[END_REF] and is considered to capture the essential vehicle lateral steering dynamics:

m( vy + rv x ) = f sf cos δ f + f sr J ṙ = l f f sf cos δ f -l r f sr ( 1
)
where r is the vehicle yaw rate, v x and v y are the longitudinal and lateral vehicle speed, δ f is the steering angle, m is the vehicle mass, l f (l r ) is the distance from the front (rear) axle to the center of gravity (CG), J is the vehicle inertia with respect to the vertical axis through the CG and f sx , with x = f, r, are the front and rear lateral forces which are modeled according to the Pacejka tire model [START_REF] Pacejka | Tire and Vehicle Dynamics[END_REF].

f sx (α x ) = D sin {Catan [(1 -E) Bα x + Eatan (Bα x )]} , (2)
α f = δ f - v y + l f r v x , α r = - v y -l r r v x , (3)
where α f (α r ) is the front (rear) tire sideslip angle. The vehicle parameters for the simplified single track vehicle model [START_REF] Ackermann | Robust Control[END_REF], and the parameters from the Pacejka tire model (B, C, D and E) (2) are presented in Table I. Their numerical values have been identified from the CarSim D-Class vehicle model equipped with 225/65R17 front and rear tires.

The Pacejka tire model is depicted by the solid line of the left plot in Fig. 2.

The simple nonlinear model [START_REF] Ackermann | Robust Control[END_REF] shows, as well known in literature (see for example [START_REF] Ono | Bifurcation in vehicle dynamics and robust front wheel steering control[END_REF]), a limited stability region containing two unstable equilibrium points and a stable one (bifurcation analysis). which also depends on the driver steering wheel angle, The causes of the instability, which also depends on the driver steering angle, are due to the nonlinear tire behavior of the tire-road forces which are taken into account in the design of the lane keeping controller by the piecewise affine (PWA) model described in the next section.

l f l r α f α r δ f r v y v r v f v x v f lf f lr f sf f sr R y L β l s road centerline
α f [deg] fs f [N] X 1 X 2 X 3 α * f -α * f r * -r *

III. PIECEWISE AFFINE VEHICLE MODEL

In order to approximate system (1) by a PWA system, the lateral tire forces domain, which are function of α f and α r , can be partitioned in polyhedral sets, which are indicated by the index i, and for each of them an affine system is used to describe the corresponding linearized dynamics.

Considering that the front and rear tire forces described by [START_REF] Oufroukh | Invariant set based vehicle handling improvement at tire saturation using fuzzy output feedback[END_REF] are approximated by the following PWA functions:

f sf (α f ) = e f i + d f i α f f sr (α r ) = e ri + d ri α r (4)
where d f i , d ri , e f i and e ri depend on the front and rear tire forces for each region (i).

System (1) is linearized around uniform rectilinear motion (constant v x = v, r = 0, v y = 0, δ f = 0) taking into account the PWA functions (4) adopted to approximate the nonlinear tire forces. The vehicle dynamics can be represented by the PWA system:

ẋd = A d i x d + B d i u + a d i , (5)
and the corresponding dynamics are:

A d i =   -2 d f i +dri mv -1 -2 d f i l f -drilr mv 2 2 drilr -d f i l f J -2 d f i l 2 f +dril 2 r Jv   , (6)
B d i = 2d f i mv 2d f i l f J , a d i = 2 e f i +eri mv 2 e f i l f -erilr J . ( 7
)
where the control input is the front wheel steering angle u = δ f , while the vehicle sideslip angle β (which replaces v y since v y = v sin β) and the yaw rate r are the state variables

x d = [β, r] T .
The partitions, indicated in the left subplot of Fig. 2 by the values of front (rear) sideslip angle α * f,(r) (only front tire is depicted for sake of clarity), must be chosen adequately in order to take into account the linear and the nonlinear tire behavior of the tire forces.

Usually the partitions of PWA systems are expressed in terms of the state-space variables, therefore in [START_REF] Borrelli | A Hybrid Approach to Traction Control[END_REF] and [START_REF] Bernardini | Driveby-wire vehicle stabilization and yaw regulation: A hybrid model predictive control design[END_REF] system [START_REF] Borrelli | A Hybrid Approach to Traction Control[END_REF] is rewritten in terms of α f and α r . However, these variables are not measurable and for that estimation algorithms and expensive optical sensors are required.

In order to circumvent this drawback, the procedure proposed in [START_REF] Benine-Neto | Vehicle yaw rate control based on piecewise affine regions[END_REF] is employed in this work to parametrize the vehicles dynamics, so that the partitions of the PWA system are defined only in terms of the vehicle yaw rate, which is available for measurement at low cost. The parametrization is obtained from a steady-state analysis in which ( 1) is solved for different values of steering angles, and the steady-state values of v y and r are used in equations (3) to compute the front (rear) wheel sideslip angle, α f,(r) in steady-state tire behavior.

Values of yaw rate at steady-state, which yields the wheel sideslip angle corresponding to the partitioning of the tire force domain (represented by α * f in the right plot of Fig. 2), can be used to define the partitions of the PWA system (5) in term of state-space variables. These parametrized partitions, represented by the two hyperplanes at r = -r * and r = r * , define the operating regions denoted by X i , with i = {1, 2, 3} of the PWA system (5), as shown in Fig. 2.

For the region that includes the origin (denoted by i = 1), the commonly used linear tire forces approximation is obtained by setting e f 1 = 0, e r1 = 0, d f 1 = c f and d r1 = c r where c f and c r are related to the parameters in the Pacejka's formula as follows:

c f = B f C f D f , c r = B r C r D r . (8)
Even though the approximation of the tire force (and corresponding yaw rate parametrization) can be refined, it is shown in the simulation section that improved performance can be obtained by using this partitioning.

A. Additional Dynamics for Lane Keeping

For lane keeping purposes, the model ( 5) has to be expanded with the dynamics of the relative yaw angle and the lateral displacement with respect to the lane centerline. These measurements are provided by a video sensor. Let ψ L = ψ -ψ t be the yaw angle error which is the angle between the vehicle longitudinal axis and the tangent to the road. The road reference curvature ρ ref is defined by ( ψt = vρ ref ), and the following equality can be derived:

ψL = r -vρ ref . (9)
Denoting by l s the look-ahead distance, the equation giving the evolution of the measurement of the lateral offset y L from the centerline at sensor location is obtained by

ẏL = v(β + ψ L ) + l s r. (10)
Equations ( 9) and ( 10) are included in system (5) and the PWA system describing the vehicle dynamics and positioning is described by:

ẋp = A p i x p + B p i u + B p ρ ρ ref + a p i ( 11
)
where x p = [β, r, ψ L , y L ] T and

A p i =   A d i 0 2×2 0 1 0 0 v l s v 0   , B p i = (B d i ) T 0 0 T , B p ρ = 0 0 -v 0 T , a p i = (a d i ) T 0 0 T . (12)
The controller to ensure zero lateral offset for model [START_REF] Johansson | Piecewise Linear Control Systems[END_REF], despite the road curvature is presented in the sequel.

IV. CONTROL STRATEGY

In order to steer the vehicle to a zero lateral-offset condition, it is necessary to model the dynamics of the road curvature ρ ref considering it as a disturbance on the lateraloffset. According to [START_REF] Rajamani | Vehicle Dynamics and Control[END_REF], the road curvature can be modeled as straight, circular and clothoid sections. Clothoids ensure a smooth transition between different road curvature values (e.g. straight to circular roads) and they may cover large distances on the road. For a vehicle traveling at a constant speed, the road curvature is constant in circular sections, therefore an integral term (denoted I 1i ) is needed to reject this disturbance. In section represented by a clothoid, ρ ref is considered as increasing linearly with respect to time (as a ramp function), thus an additional integral term (denoted

I 2i
) is required to obtain zero lateral-offset. The dynamics due to the integral terms and an additional derivative term (denoted D i ) are defined as follows:

α1 = y L , α2 = α 1 , α3 = -1 τ α 3 + y L , ( 13
)
where τ is the filter time constant. Defining:

y Ld = -1 τ 2 α 3 + 1 τ y L , (14)
the P IID i regulator becomes:

δ f = -P i y L -I 1i t 0 y L dν -I 2i t 0 ν 0 y L dηdν -D i y Ld = -P i y L -I Ii α 1 -I 2i α 2 -D i y Ld , for i = 1, 2, 3. (15
) Replacing ( 15) in [START_REF] Johansson | Piecewise Linear Control Systems[END_REF], considering that u = δ f , and including the dynamics (13) the closed loop PWA system becomes:

ẋ = A i x + B ρ ρ ref + a i (16)
where x = [β, r, ψ L , y L , α 1 , α 2 , α 3 ] T and

A i =             A d i 0 -b 1i k i -b 1i I 1i -b 1i I 2i b 1i Di τ 2 0 -b 2i k i -b 2i I 1i -b 2i I 2i b 2i Di τ 2 0 1 0 0 0 0 0 v l s v 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 -1 τ             , B ρ = (B p ρ ) T 0 0 0 T , a i = (a p i ) T 0 0 0 T , ( 17
) where b 1i = 2d f i mv , b 2i = 2d f i l f J
and k i = (P i + Di τ ). Fig. 3 shows the schema for the PWL control structure. The Simulink Response Optimization toolbox was used to tune the design parameters through numerical optimization; the design requirements are expressed in terms of rise time, settling time and overshoot. The gains from each P IID i are obtained from a Sequential Quadratic Programming method as described in [START_REF] Gill | Practical optimization[END_REF]. They are computed such that each subsystem is stable, however the stability of the whole systems considering the switches between each subsystem is done by means of the PWA systems theory ( [START_REF] Johansson | Computation of piecewise quadratic lyapunov functions for hybrid systems[END_REF] and [START_REF] Johansson | Piecewise Linear Control Systems[END_REF]), as described in the next section.

V. STABILITY OF P IID i CONTROLLER The proof of stability is based on the search of a continuous piecewise quadratic Lyapunov function by means of a convex optimization problem in terms of linear matrix inequalities (LMI). As described in [START_REF] Johansson | Computation of piecewise quadratic lyapunov functions for hybrid systems[END_REF], for the convenient treatment of the affine terms, the extended state vector x is defined as:

x = [x 1] T . (18)
Similarly, a bar over a matrix indicates that it has been modified to be compatible with the extended state vector. Therefore the system (16) becomes:

ẋ = Āi x + Bρ ρ ref , ( 19
) in which Āi = A i a i 0 1×7 0 , Bρ = B ρ 0 . (20)

A. Piecewise Quadratic Lyapunov Functions

A compact matrix parametrization of continuous piecewise quadratic functions on polyhedral partitions, named Continuity matrices, Fi = [F i f i], is defined in [START_REF] Johansson | Piecewise Linear Control Systems[END_REF] as:

Fi x(t) = Fj x(t) f or x(t) ∈ X i ∩ X j , (21)
in which X i and X j are two neighbor regions of the state space.

Then the format of the continuous piecewise quadratic Lyapunov function candidate becomes:

V (x) = xT Pi x, f or x(t) ∈ X i ( 22
)
where Pi = F T i T Fi , and T is a symmetric matrix. In order to express the restriction that x ∈ X i via a linear form in x, as required in the computations, Cell Bounding matrices, Ēi = [E i e i ], are defined in [START_REF] Johansson | Piecewise Linear Control Systems[END_REF], as follows:

Ēi x ≥ 0 if x(t) ∈ X i ( 23
)
The stability of the PWA model ( 16) can be proved using the theorem proposed in [START_REF] Johansson | Computation of piecewise quadratic lyapunov functions for hybrid systems[END_REF], as follows:

Consider symmetric matrices T , U i and W i such that U i and W i have non negative entries, while

P i = F T i T F i , Pi = F T i T Fi , ( 24
)
satisfy:

A T i P i + P i A i + E T i U i E i < 0 P i -E T i W i E i > 0 f or i = 1 ( 25
)
ĀT i Pi + Pi Āi + ĒT i U i Ēi < 0 Pi -ĒT i W i Ēi > 0 f or i = 2, 3 (26)
then, every trajectory x(t) of the PWA model ( 16) with ρ ref ≡ 0 tends to zero exponentially.

Solving the LMI problem (( 25) and ( 26)) a piecewise quadratic Lyapunov function ( 22) has been computed proving the stability of ( 16), which ensures that zero lateral offset in steady state is obtained with the proposed PWL controller.

The performance of the lane keeping controller ( 15) is evaluated by simulations on the Carsim environment, as shown in the next section.

VI. SIMULATION RESULTS

Simulations have been carried out to validate the proposed control system with respect to nonlinear combined lateral and longitudinal tire forces and additional dynamics, such as pitch, roll and driver dynamics, which were neglected at the control design stage. The standard CarSim D-Class vehicle, used to analyze the responses of the controlled vehicles, has a nonlinear second order speed depending rack and pinion ratio steering system; for the active steering a realistic actuator with a bandwidth of 10Hz is considered (see [START_REF] Macadam | Application of an optimal preview control for simulation of closed-loop automobile driving[END_REF]). The simulations were performed at v = 30.6 [m/s].

The same vehicle in two different configurations has been used in the simulations. The first configuration consisted of the vehicle equipped with the designed controller in the linear region only (equation ( 15) with i = 1), named lin.-ctrl; The second configuration consisted of the vehicle equipped with the proposed PWL control strategy (P IID i ) (equation ( 15) with i = 1, 2, 3), named PWL-ctrl. Two different scenarios are presented in the sequel: response to sudden disturbances as external lateral force and yaw moment, and response to sudden direction changes (in dry and wet pavement).

A. Sudden Disturbances

To analyze the improved performance, the vehicle has been subjected to a sudden disturbances on lateral force and yaw moment as shown in the upper-left subplot of Fig. 4. The disturbances are strong enough to make the vehicle lateral dynamics enter the nonlinear region during the lane departure avoidance maneuver, as it can be seen in the bottom-left subplot of Fig. 5, showing the active regions of the PWLctrl vehicle. It is important to note that the control input of the PWL controller is reduced when the vehicle yaw rate exits the linear region (i = 1), as shown in the bottomright subplot of Fig. 5. Consequently the tire sideslip angles, depicted on the upper-right subplot of Fig. 5 do not increase as much as for the vehicle lin.-ctrl. In the upper-left subplots of Fig. 4 and Fig. 5 it is shown the vehicles trajectories and the lateral-offset respectively, The reduction in about 50% of the overshot in lateral-offset for the PWL-ctrl confirms its improved performance for lane keeping purposes.

B. Sudden Change in Direction

Several maneuvers have been performed to analyze the response of the lane keeping controller with respect to change in directions. The results presented correspond to a demanding lane change maneuver, in which the vehicles reach a lateral acceleration of 8.9 [m/s 2 ], in two different road adherence conditions, while no external disturbances are applied.

Considering firstly good adherence conditions, characterized by the road adhesion coefficient (µ = 1), the enhanced performance of the proposed control strategy for lake keeping is clear when compared to the response of lin.ctrl, for the maneuver shown in the upper-left subplot of Fig. 6. Analyzing Fig 7, it can be seen that by the switch of PWL-ctrl between the regions i = 1, 2, in the bottomleft subplot, the control action depicted on the bottom-right, is reduced when the vehicle yaw rate reaches the partition limits. Consequently the vehicle remains in a controllable condition avoiding the saturation of tire forces which can be seen by the smaller values of front wheel sideslip angle in the upper-right subplot. Moreover the lateral-offset is steered to zero faster than for the lin.-ctrl vehicle. The vehicle lateral dynamics (vehicle sideslip angle and yaw rate) are shown in the bottom subplots of Fig. 6. In order to evaluate the robustness with respect to different road adherence coefficients, the same maneuver has been performed for a deteriorated road adherence (µ = 0.7), corresponding to wet pavement. The results are shown in Fig. 8 and9. Even though the lateral-offset reaches more significant values, as the maneuver is very demanding, it can be seen that, also in this case, the proposed PWL controller performance is improved with respect to the lin.-ctrl, which is not able to follow the maneuver.

VII. CONCLUSIONS

In this paper it has been shown that a lane keeping steering control can be designed taking into account the nonlinear behavior of the tire forces. The vehicle dynamics has been parametrized with respect to the measurable yaw rate, considering the steady state behavior of the vehicle. This parametrization avoids the use of the tire sideslip angles which are not directly measurable, requiring estimation algorithms or high cost optical sensors. The use of the proposed P IID i controller, with gains depending on the parametrized vehicle dynamics, is able to track the lane despite affine disturbances and parameters uncertainties. The stability of the controlled system is proved by the quadratic piecewise Lyapunov function, computed from a LMI problem. Simulation results including disturbances rejections and step references, carried out on a standard CarSim D-Class vehicle model, were presented to show the system enhanced performance and robustness. The simulations also confirm that the proposed PWL control, can greatly improve the vehicle stability and is advantageous in very demanding maneuvers, when compared to the linear controller, consisting in the use of the single controller as designed for the linear region only.

Moreover, the simulations showed that the switching between the PWL controllers based on a yaw rate threshold value is satisfactory and that it is able to enhance the performance, avoiding refinements of the proposed partitioning. Since this controller was designed for a constant speed, a gain scheduling scheme should be proposed in order to allow larger longitudinal speed variations. Among future work, it is also envisaged to implement the system on a prototype vehicle for perform practical evaluation.

Fig. 1 .

 1 Fig. 1. Single track vehicle model

Fig. 2 .

 2 Fig. 2. Pacejka model for CarSim D-Class vehicle model equipped with 225/65R17 front and rear tires (solid line) and PWA approximations (dashed line)

2 P IID 1, 3 Fig. 3 .

 233 Fig. 3. Piecewise linear control schema.

Fig. 4 .

 4 Fig. 4. Response to a sudden disturbance on vehicles lin.-ctrl (dashed line) and PWL-ctrl (continuous line) at v = 30.6 [m/s].

Fig. 5 .

 5 Fig. 5. Response to a sudden disturbance on vehicles lin.-ctrl (dashed line) and PWL-ctrl(continuous line) at v = 30.6 [m/s].

Fig. 6 .

 6 Fig. 6. Response to a lane change on vehicles lin.-ctrl (dashed line) and PWL-ctrl (continuous line) at v = 30.6 [m/s] on dry road.

Fig. 7 .

 7 Fig. 7. Response to a lane change on vehicles lin.-ctrl (dashed line) and PWL-ctrl (continuous line) at v = 30.6 [m/s] on dry road.

Fig. 8 .

 8 Fig. 8. Response to a lane change on vehicles lin.-ctrl (dashed line) and PWL-ctrl (continuous line) at v = 30.6 [m/s] on wet road.

Fig. 9 .

 9 Fig. 9. Response to a lane change on vehicles lin.-ctrl (dashed line) and PWL-ctrl (continuous line) at v = 30.6 [m/s] on wet road.

TABLE I

 I

	VEHICLE PARAMETERS FOR THE LINEAR MODEL (5):
	m	2023	[kg]	J	6286	[kg m 2 ]
	l f	1.26	[m]	lr	1.90	[m]
	a	1.5	[m]	ls	6	[m]
	B f,r	11.4592		C f,r	1.4
	D f	6628		Dr	4557
	E f	-0.5		Er	-0.7

IFSTTAR -LIVIC, 14 Route de la Minière, 78000 Versailles, France (e-mail: andre.benine-neto@ifsttar.fr)

University of Rome Tor Vergata, Electronic Engineering Department -DIE, Via del Politecnico 1, 00133 Roma -Italy, (e-mail: scalzi@ing.uniroma2.it)

3 IBISC

-EA 4526 EUVE, 40 rue du Pelvoux CE1455, 91020, Evry, Cedex, France (e-mail: said.mammar@inrets.fr)

ACKNOWLEDGMENT

This work has been sponsored by the french ANR consortium PARTAGE.


Prisca Laure

Bogne Nyitchogna

Lydie Nouvelière

Olivier Orfila

Sébastien Glaser

Said Mammar

Impact of Full Speed Range ACC on the Traffic, the Safety and the Energy Consumption

Keywords: Driver model, ADAS model, HDM, IDM, traffic simulator. Résumé

In this paper, an assessment of the FSRACC impacts on a simultaneous combining of traffic, safety and environment is made on the basis of the simulation across a network while taking into account the penetration rate of FSRACC (from 0% to 100%). To carry out this study, it is necessary to have some indicators and models. Various variables can be used to evaluate the longitudinal ADAS impacts on these key characteristics referring to indicators of traffic, safety and environment that will therefore be used for this evaluation. Some models consider the behavior of individual drivers under the influence of vehicles in their proximity (microscopic models) or the collective flow of vehicles (macroscopic models) or the platoon of vehicles (mesoscopic models).Within the framework of this study, a simulator is developed with driver and FSRACC models. Both driver and FSRACC simulators are merged. The indicators matching each model are calculated, analyzed, interpreted and compared before and after merging according to the penetration rate of FSRACC equipped vehicles. It is shown that the proposed system can provide a natural following performance similar to a human driving. The proposed simulator can be used to evaluate other longitudinal ADAS based on accelerations models and to compare different kinds of such longitudinal ADAS.

Dans cet article, une évaluation des impacts du système FSRACC sur une combinaison simultanée de trafic, sécurité et environnement se fait sur la base de la simulation à travers un réseau, tout en tenant compte du taux de pénétration du FSRACC (de 0% à 100%). Pour mener à bien cette étude, il est nécessaire d'avoir des indicateurs et des modèles. Différentes variables peuvent être utilisées pour évaluer les impacts des ADAS longitudinales sur ces caractéristiques clés faisant référence à des indicateurs de trafic, de sécurité et d'environnement qui seront donc utilisés pour cette évaluation. Certains modèles prennent en considération le comportement des conducteurs individuels sous l'influence de véhicules dans leur proximité (modèles microscopiques) ou un flux collectif de véhicules (modèles macroscopiques) ou un peloton de véhicules (modèles mésoscopiques). Dans le cadre de cette étude, un simulateur est développé avec des modèles pour le conducteur et le FSRACC. Le conducteur et le FSRACC sont fusionnés. Les indicateurs correspondant à chaque modèle sont calculés et analysés, interprétés et comparés avant et après la fusion en fonction du taux de pénétration des véhicules équipés. Il a été démontré que le système proposé peut fournir une performance de poursuite naturelle similaire à la conduite humaine. Le simulateur de trafic proposé peut être utilisé pour évaluer d'autres ADAS longitudinales basées sur des modèles d'accélérations et pour comparer différent types de systèmes longitudinaux. Number of anticipated vehicles in the driver model T ' Reaction time for the driver model (s) V s

Relative distance error (%) r c

Inverse Time To Collision error (1/s) τ

Error correlation time (s)

Introduction

Advanced driver-assistance systems (ADAS) support a driver in his driving tasks [START_REF] Van Arem | The impact of cooperative adaptive cruise control on trafficflow characteristics[END_REF]. Road transport is related to many problems which are remaining difficult to be solved. Among others are the problems of traffic congestion, pollution, climate change, energy saving, risk of accidents. ADASs are being developed to enhance driving comfort, reduce driving errors, improve safety, increase traffic capacity and reduce energy consumption [START_REF] Xiao | A comprehensive review of the development of adaptive cruise control systems[END_REF], e.g. Adaptive Cruise Control).

In addition, ADASs can also have side effects on traffic-flow, safety and environment. Key characteristics, such as safety, traffic-flow efficiency and environmental protection are the focus of most transportation researches. Many researchers studied effects of ADAS on each of these key characteristics, or on a combination of two of them, even though there are few papers on the impacts on only both safety and environment. Furthermore, there are nearly no study on the impacts of ADAS on all these key characteristics. These researchers used some indicators symbolizing the key characteristics to assess these effects and in most cases, these indicators are linked. For example an indicator for the traffic can also be used to evaluate safety or environment. It follows that it becomes difficult to study impacts on traffic without those on safety or without those on environment. Consequently, it would be better and advantageous to study impacts of ADAS, at the same time, on all the key characteristics.

Several papers [START_REF] Wang | The impact of adaptive cruise control systems on highway safety and traffic flow[END_REF][START_REF] Davis | Effect of adaptive cruise control systems on mixed traffic flow near an on-ramp[END_REF] assessing the impact of the proportion of ADAS equipped vehicles have appeared, but not all of them were found to have favorable effects [START_REF] Davis | Effect of adaptive cruise control systems on mixed traffic flow near an on-ramp[END_REF]. Problems such as cut-in, congestion at bottlenecks or near on-ramps on freeways are remaining difficult to assess with an ADAS equipped vehicle. New ADAS have been developed or are being developed; considering current difficulties, it is important to assess their impacts on the key characteristics above-quoted to meet the needs of researchers, automakers, governments and consumers. Control of the vehicle includes longitudinal control which is responsible for the axial dynamics evolution and lateral control which is responsible for maintaining the direction. This paper will focus on the longitudinal mode of the vehicle (ISO 22178, 2009);  Full Speed Range Adaptive Cruise Control (FSRACC) systems (ISO 22179, 2009) ;  Intelligent Speed Adaptation (ISA) systems [START_REF] Young | Intelligent speed adaptationeffects and acceptance by young inexperienced drivers[END_REF];  Ecodriving (Saboohi et al., 2009).

Most of the recent ADAS researches concerns Adaptive Cruise Control (ACC) systems, which are now available on a wide range of passengers vehicles [START_REF] Auckland | Advanced driver assistance systems: Objective and subjective performance evaluation[END_REF] and are the focus of researchers, automakers, governments and consumers across the world [START_REF] Xiao | A comprehensive review of the development of adaptive cruise control systems[END_REF]. Nowadays, they are available on standard vehicles (e.g. Cars). ACC systems are being developed by many researchers and the impacts on traffic-flow and/or safety and/or energy consumption of those optimized-ACC systems have also been investigated (e.g. [START_REF] Xiao | A comprehensive review of the development of adaptive cruise control systems[END_REF], Bose et al., 2001, Touran et al., 1999). By contrast, for alike effects, few researchers investigated S&G, FSRACC, ISA systems and Ecodriving (Moon et al., 2009, Servin et al., 2006, Barth et al., 2009). However, there are almost none paper on the impacts of only S&G systems.

Our review has showed that almost none paper has been published on the impact of the above-quoted optimized-ADASs on a simultaneous mix of traffic, safety and environment. Moon et al. (2009) described the design, tuning, and evaluation of a full-range ACC system with collision avoidance (CA). The goal of the control algorithm for the full-range ACC with CA is to achieve the behavior of the subject vehicle that would seem natural to a human driver in normal-driving situations and to achieve a safe behavior in severe-braking situations in which large decelerations are necessary. To integrate the ACC and CA systems, the proposed algorithm makes the subject vehicle operate in three modes: comfort (ACC), large deceleration (ACC+CA), and severe-braking (CA). By means of simulations with real data on driving, it has been shown that the proposed control strategy can provide a natural following performance that is similar to human manual-driving at both high-speed driving and low-speed S&G situations, and can prevent the vehicle-to-vehicle distance from dropping to an unsafe level in a variety of driving conditions (safe, warning and dangerous modes). [START_REF] Zhao | Full-range adaptive cruise control based on supervised adaptive dynamic programming[END_REF] proposed a supervised adaptive dynamic programming (SADP) for a FSRACC system. The control action is based on the SADP and introduces the concept of inducing region to speed up the learning efficiency. The SADP control algorithm performs well in all simulations scenarios (normal ACC, S&G, emergency braking, cut-in and driver habits changing) and always better than more traditional controllers.

Our study is thus focused on the evaluation of the impacts of FSRACC system on the traffic, the safety and the energy consumption. After having described the system the methodology, models and crucial indicators will be presented. Then, for a given relevant scenario, we will present, analyze and understand the results of the simulations. Finally, we will conclude the evaluation of our system in the above-cited key characteristics and give a vision of future works.

Full Speed Range Adaptive Cruise Control

Definition

FSRACC system is a combination of ACC and S&G systems. The purpose of FSRACC is a partial automation of the longitudinal control of the vehicle to reduce the load of the driver. FSRACC system is intended to provide longitudinal control of equipped vehicles while traveling on highways (roads where non-motorized vehicles and pedestrians are prohibited) under free-flowing and congested traffic conditions. FSRACC provides support, from the vehicle stop to the maximum speed of the system. The system will attempt to stop behind a vehicle already followed in his vehicle deceleration capabilities and will be able to restart after the driver had sent as command to the system to resume its journey from the stop. The system is not obliged to react to stationary or slow objects. FSRACC is essentially an improvement of the ACC (Automated Cruise Control or Adaptive Cruise Control) which allows the host vehicle to follow a target vehicle at an appropriate distance by controlling the engine and/or the transmission and the brake up to the stop.

FSRACC system constraints

In our study, we have applied the ISO limitations for FSRACC system. All the vehicles have their speeds between 0 and 130 km/h (French maximum authorized speed). The range of accelerations is dependent to the values of current speed. These limitations have been applied only to the FSRACC equipped vehicles. Then, the velocity-dependent acceleration limitations, a max and a min which are respectively the upper boundary and the lower boundary of accelerations, are determined as follows:

2 m ax / 2 s m a  a min = -5m/s 2 if speed < 5m/s a min = -3.5m/s 2 if speed >20m/s
Linear interpolation of a min for 5 < speed < 20m/s Thus, the accelerations of the FSRACC vehicles is calculated based on the reference model and restricted by the acceleration limit.

Methodology for FSRACC impacts study

To estimate the impacts of our system on the traffic-flow, the safety and the energy consumption, the method used here is the modeling: driver, FSRACC and fuel consumption models are used and the effects are analyzed using a simulation scenario on highways. Simulations to be applied are either microscopic (individual vehicle) or mesoscopic (platoons of vehicles). The simulator has been developed on Matlab (Matrix Laboratory), we have supposed to have a road network, the simulator incorporates the driver and FSRACC vehicles according to their penetration rate (from 0% to 100% in the traffic). Many simulations can also be done, the simulator takes into account, the initial conditions, the data from Simulink of the leader of the platoon (first vehicle in the file), the generation of equipped and non-equipped vehicles, the choice of the type of model, the appeal of functions calculating driver and FSRACC accelerations, the mathematical equations of indicators. The simulator is designed such as to work only in a simulator incorporating the driver model, we can just turn the rate of percentage (0%), and to work on a simulator incorporating only the FSRACC vehicles, a penetration rate of 100% is chosen. The indicators matching each model can also be calculated, analyzed, interpreted and compared before and after merging according to the penetration rate of FSRACC equipped vehicles.

Models

In this section, the driver and FSRACC models used in the simulations on Matlab are presented.

FSRACC model

The model used to represent FSRACC vehicles is the Intelligent Driver Model (IDM) presented by [START_REF] Kesting | Adaptive cruise control design for active congestion avoidance[END_REF] at which we added the ISO accelerations limitations to have the desired FSRACC system. [START_REF] Kesting | Adaptive cruise control design for active congestion avoidance[END_REF] proposed an ACC-based traffic assistance system with an active jam avoidance strategy. In such a system, the driving behavior automatically adapts to different traffic situations (moving in free traffic, approaching an upstream congestion front, moving in congested traffic, leaving the downstream congestion front, and passing infrastructural bottleneck sections) by means of local information. The detection of the local traffic state can be improved by non-local information provided by infrastructure to-car and inter-vehicle communication. Traffic simulations of a three-lane freeway with an on-ramp bottleneck and a mixed flow consisting of cars and trucks showed improvement in traffic stability and dynamic road capacity with 'intelligent' ACC vehicles. Only a small percentage of ACC vehicles can significantly improve the traffic performance. An ACC equipment level of 5% improves the traffic flow quality and reduces the travel times for the drivers in a relevant way, while an equipment level of 25% ACC vehicles eliminated completely traffic congestion.

The main function of ACC (enhancement of conventional cruise control) is to control vehicle speed adaptively to a forward vehicle by using information about: (1) ranging to forward vehicles,(2) the motion of the subject (ACC equipped) vehicle and (3) driver commands. It is fundamentally intended to provide longitudinal control of equipped vehicles while traveling on highways under free-flowing traffic conditions. [START_REF] Kesting | Adaptive cruise control design for active congestion avoidance[END_REF] proved that the IDM model is appropriate for simulating ACC systems because this model meets a number of compulsory criteria (collision-free of the car-following dynamics, the dynamics should also correspond to a natural and smooth manner of driving, only a few parameters). Some car following models cannot be simulated as an ACC system (e.g. Optimal Velocity Model, Bando et al., November 1995).

The acceleration of the IDM is a function of the headway spacing, the speed and the relative velocity. Its expression combines a free acceleration term and a braking term. The IDM has five crucial parameters (desired speed v 0 , safe time gap T, maximum acceleration a, desired deceleration b, jam distance s 0 ) [START_REF] Kesting | Adaptive cruise control design for active congestion avoidance[END_REF]. This model is widely used by researchers. They have showed that the destabilizing effects of reaction times and estimation errors can essentially be compensated for by the stabilizing effects of spatial and temporal anticipation. In order to put this balance of stabilizing and destabilizing effects into a more general context, they have formulated the human-driver model (HDM) as a meta-model that can be used to extend a wide class of car-following models, where the acceleration depends only on the positions, velocities and accelerations of the own and preceding vehicles.

Driver model

Then the chosen driver model is the HDM applied to the IDM which has a built-in anticipative and smooth braking strategy, and which reaches good scores in a first independent attempt to benchmark micromodels based on real traffic data (Brockfeld et al., 2002). It is supposed that drivers are aware of their finite reaction time and anticipate the traffic situation accordingly. The future distance, the future velocity and the future relative velocity are anticipated. Predictive terms depending on the reaction time are added to the estimation errors for the anticipation of the gap, the speed and the relative speed. The new HDM acceleration (depending on the anticipative variables) is split up into a single vehicle acceleration on a nearly empty road depending on the considered vehicle only, and a braking deceleration taking into account the vehicle-vehicle interaction with the preceding vehicle. The reaction to several vehicles ahead is modeled by summing up the corresponding vehicle-vehicle pair interaction accelerations from a vehicle β to vehicle α for the n a nearest preceding vehicles β. The gap between the vehicles β and α is the sum of all net gaps between them. The model has five parameters with two deterministic parameters, namely the reaction time T' and the number n a of anticipated vehicles. The other parameters namely stochastic sources V s and r c characterize the degree of the estimation uncertainty of the drivers, while τ denotes the correlation time of errors (Treiber et al., 2006).

Fuel Consumption model

Fuel consumption models can be classified into three different approaches namely macroscopic (vehicles streams), mesoscopic and microscopic ones. Macroscopic and mesoscopic models are essentially based on the average speed of the traffic flow, they are suitable for the evaluation of the total fuel consumption on an urban traffic network. Microscopic models are based on estimating the instantaneous fuel consumption of the vehicle from the instantaneous values of different parameters related to the vehicle itself or to the conditions of operating (route type, road type, traffic conditions, road geometry). In the literature, the latter are divided into several approaches as follows :

 empirical models based entirely on data (Alçelik et al., 1989) ;  physical models, by contrast to empirical ones, they are based on physical principles [START_REF] Bosch | Mémento de Technologie Automobile[END_REF];  semi-empirical models based on physical principles like physical models and empirical data as empirical models (Wang et al., 2008). The fuel consumption model used in our evaluation is the semi-empirical model developed in our laboratory. We proposed a fuel consumption model to estimate the effects of the penetration rate of Ecodriving on fuel consumption and traffic congestion: the model is based on the mechanical energy consumed with a performance ratio function of the vehicle speed [START_REF] Orfila | Impact of the penetration rate of ecodriving on fuel consumption and traffic congestion[END_REF]. To complete the work of Orfila, the model is used here for the evaluation of the FSRACC system on the fuel consumption with the same values of model parameters.

Indicators

Traffic-flow is a part of the traffic study focused on the analysis and characterization of the evolution of vehicles on selected routes [START_REF] Buisson | Comprendre le trafic routier Methodes et calculs[END_REF]. Various variables can be used to assess the impacts of a longitudinal driver-assistance system on traffic-flow, they refer to indicators. An indicator of road traffic is one way to evaluate the performance of a ride or a network. It is calculated from macroscopic (e.g. throughput, compactness, occupancy rate, spatial mean speed, life span and length of traffic jam) or microscopic (e.g. time-gap, spacing, individual speed) variables from the sensors [START_REF] Cohen | Initiation a l'ingénierie du trafic routier[END_REF] or computed from numerical models. The measures for traffic are instantaneous position, speed, acceleration of a vehicle in the platoon, indicators can be global mean speed, global mean distance. The impacts of road transportation on environment have to deal with concerns from the greenhouse effect, by the atmospheric pollution to the rarefaction of fossil fuel resources. Some indicators are used to assess impacts of ADAS on fuel consumption and gases emissions and they can be classified in global and local indicators. Global indicators intervene in the level of a whole road network whereas local indicators intervene in the level of the ego-vehicle (the one equipped with the ADAS). For example, among them, there are consumption per vehicle (local) and total consumption (global).

Besides traffic and environmental concerns, safety issues are incontrovertible in the development of ADAS which are believed to reduce risk of accidents, improve safety and enhance performance of drivers. To assess the impacts of longitudinal ADAS on safety, it will be necessary to use some safety indicators like time to collision (TTC), time headway (TIV). Why these indicators and these measures than others? The positions of the vehicles can show if there are collisions or not, they can also show that the vehicles run with constant gap during the simulations. The positions can also show if the presence of FSRACC system affects the total travel time during traffic disturbances (Ioannou et al., 2007). Jerk analyses make it possible to identify safety critical driving behavior or "accident prone" drivers. They also facilitate the development of safety measures such as active safety systems or advanced driver assistance systems, ADAS, which could be adapted for specific groups of drivers or specific risky driving behavior [START_REF] Bagdadi | Jerky driving-An indicator of accident proneness?[END_REF]. Accelerations (Ioannou et al., 1993) are ideal to see the FSRACC system in terms of limitations, they can be compared to jerks. Speeds are ideal to see if the other vehicles mimic the leader speed profile or to evaluate the efficiency of the system [START_REF] Kesting | Adaptive cruise control design for active congestion avoidance[END_REF]. In the one hand, it would be better to see the gaps [START_REF] Wang | The impact of adaptive cruise control systems on highway safety and traffic flow[END_REF], time headway (Fancher et al., 2001), relative velocity of each vehicle for each penetration rate and each simulation to optimize our system well. In the other hand, it will be better and fast to see the results for all penetration rates in the same figure to have a good understanding of the results, so the best idea would be to calculate global and deterministic indicators.

Within the framework of this study, the relevant measures for the assessment of the impacts of FSRACC system on our key characteristics are the instantaneous jerk, acceleration, speed and position. The relevant indicators are respectively:

 global mean speed, global mean distance ;  consumption in litters per 100 kilometers of vehicles (Greenwood, 2003);  headway time, spacing, percentage of time where the time headway is under 2 seconds (Legal headway time limit in France).

Simulations Results

6.1.Scenario for the FSRACC simulation

The main function of Low Speed Following (LSF) systems is to control vehicle speed adaptively to a forward vehicle. In contrast with ACC systems, it provides automatic car-following at lower speed ranges, it doesn't provide speed regulator control and allows stopping and starting vehicle on highways under congested traffic (ISO 22178 ,2009) . The objective is to elaborate a scenario which represents ideally the FSRACC system which is a combination of LSF and ACC systems. The figure 1 illustrates the speed profile used for the leader (first vehicle in the file), it combines well the driving at lower and higher speeds on highways under free-flowing and congested traffic conditions.

Fig. 1. Speed profile of the first vehicle.

6.2.Initial conditions and models parameters

Simulations was made on a single-lane road section with 20 vehicles where the leader has the profile defined as in figure 1. The maximum length of vehicles is assumed to be 5 m and the minimum length is assumed to be 3 m, the step rate is 25% and the percentage varies from 0% to 100%. For each percentage, five simulations were performed for more data, this will allow a better understanding of the results. Each simulation lasted 10 minutes and the update time interval was 0.1 s. The maximum and minimum jerks for the FSRACC vehicles are assumed to be 1.5 m/s 3 and -3 m/s 3 respectively to keep the concept of comfort in acceleration while permitting severe braking. The initial gaps, speeds, accelerations, jerks are 10 m, 0 m/s, 0 m/s 2 and 0 m/s 3 respectively. The initial  the 2 boxplots, lower and upper, represented on either side of the box. These two whiskers delimit adjacent values which are determined from the interquartile range (Q3-Q1);  the extreme or aberrant values located beyond adjacent values are individualized. They are represented by markers (square, or star, etc.). The interests of the box plot are various: this is a quick way to include the profile of a quantitative statistical series; it allows to identify aberrant observations; it allows to highlight the symmetry or not of a distribution by identifying the position of the median in the box, and the asymmetry of mustaches; it facilitates the comparison between different distributions; it graphically represents both the parameters of central tendency (median) and the parameters of dispersion (quartile).

Results for the combination of IDM and HDM-IDM for the values of HDM parameters T ' =1.1 s, n a =5, V s =0.05 and r c =0.01/s

The reaction time and the number of anticipated vehicles are the most pertinent HDM parameters, the IDM parameters are fixed. The figure 2 shows the boxplots for some indicators, there are no aberrant values for each indicator (due to the choice of the step rate and the number of simulations). They can be understood as follows: the percentage of time where the time headway is under 2 s is represented for all the vehicles at different penetration rates (figure 2.a). We can see that the HDM-IDM vehicles drive most safely without FSRACC system. The higher the percentage, the greater the vehicles drive too close. For a 100%-FSRACC vehicles, the boxplot is only a line representing the median, it means that data are not enough, it is normal because we are in the case of the basic model IDM where there almost are no random values, so each of the five simulations have nearly the same data. The same analysis is made for the fuel consumption in liters per 100 km (figure 2.b), it appears that the fuel consumption is asymmetric regarding the percentages, for each percentage the values are increasing and the highest value of the consumption is reached for only 25%. Moreover, the values of boxplots are close. So with FSRACC the vehicles consumed not so higher than without it. The test seems to keep the fuel consumption constant when the percentage of FSRACC increases, this is due to the results obtained for the speeds in figure 2.d. The fuel consumption heavily depends on the speeds which vary slightly with the increase of the penetration rate of the FSRACC vehicles. The fuel consumption also depends on the vehicle parameters, the factors related to the condition of operational use of the vehicle (stop, kind of trip, road type, traffic conditions, road geometry) and the driver. From the figure 2.c, it appears that the increase in the percentage rate has no influence on the covered distance. Figure 2.d makes appearing that the speeds are slightly different, so the system has slightly influence on the speeds of the vehicles. For a 100%-FSRACC, Q1=M=Q3=20.43 m/s (no maximum, no minimum). For a 0%-FSRACC, the height of the boxplot for other percentage is due to the presence of drivers (many random values). Figures 3 to 6 show the accelerations, jerks, speeds and locations of all vehicles for a simulation with a 0%, 25% and 100%-FSRACC vehicles: the vehicles jerks were all initialized at zero, throughout the simulation until the end, the jerks of vehicles equipped with FSRACC decrease while those of drivers are delayed and train oscillations, due to the Wiener process which takes into account the persistence of estimation errors. The higher the percentage, the lower the values of jerks and the smaller the oscillations. We have large values of jerks for almost the first 200 seconds while from the almost 200 th s to the end, the jerks are trying to stabilize at zero. The jerks of FSRACC vehicles are limited by the minimum and maximum values of the jerks. The higher the percentage of FSRACC increases and the vehicle accelerates and decelerates more modestly. The acceleration values of the FSRACC vehicles stay within the limitations imposed by the system depending on the values of the corresponding speeds as found by [START_REF] Nouvelière | Safe Full Speed Range ACC applied to a platoon of light electric vehicles[END_REF]. The higher the penetration rates, the lower the oscillations of the accelerations. The drivers tend to follow the leader's trajectory. All the vehicles speeds are between 0 km/h and 130 km/h. Vehicles tend to reach their desired speed, the equipped vehicles are struggling to follow the trajectories of drivers. The more we increase the percentage, the more oscillations decrease. The increasing of penetration rate of FSRACC has no influence on the locations and the total travel time as also found by Ioannou et al. (2007). It appears from figure 7 that the higher the penetration rate increases, the more the gap between the trajectories of the vehicles regulates and then the vehicles increase the safety in their driving. Figure 8 shows that the time headway of the vehicles is high at the beginning and becomes lower during the simulation until the end. They are lower than the safe time headway (2 s) with the equipment. This result is due to the speed profile. It should be noted that if T ' =0, n a =1,V s =r c =0, the human-driver extension HDM-IDM is switched off and the original basic model IDM is recovered.

Conclusions and future work

A traffic simulator was presented with the fusion of a driver simulator using HDM model applied to IDM model and an ADAS model for the Full Speed Range Advanced Cruise Control (FSRACC) using the IDM model. An evaluation under a given scenario on highways was made among several indicators symbolizing safety, ) environment and traffic. The results are presented with box plots for a statistical view and with matlab simulations out of the FSRACC penetration rate in the traffic. It has been shown that the proposed FSRACC system can provide a natural following performance similar to human driving. Future work will consider other longitudinal ADAS to be evaluated in the same way, namely IDM with ISO accelerations limitations for the ACC system, IDM with ISO accelerations limitations concerning LSF system, and possibly the ISA and Ecodriving systems. These systems will be subsequently compared to one another.

  Mots-clé: Modèle de conducteur, Modèle d'ADAS, HDM, IDM, simulateur de trafic. Vehicles in interaction in the driver model n a

  The meta-model HDM (Human Driver Model) presented byTreiber et al. (2006) is used in our work to represent driver model.Treiber et al. (2006) generalizes a wide class of time-continuous microscopic traffic models to include essential aspects of driver behavior not captured by these models. Finite reaction time and errors in estimating the input variables are clearly essential factors of driver behavior affecting the performance and stability of vehicular traffic. More precisely, they have considered: estimations errors; finite reaction time (delay);  temporal anticipation;  spatial anticipation (looking several vehicles ahead).

  leader is 190 m. The parameters used for the FSRACC and HDM-IDM combination (application of the meta-model HDM to the car-following model IDM) models are given by the table 1.

Fig. 2 .Fig. 7 .

 27 Fig. 2. Boxplots for relevant indicators. From left to right : (a) percentage of time where the time headway is under 2sat the top left; (b) fuel consumption in liters per 100 kilometers; (c) Global mean distance; (d) Global mean speed.

Fig. 8 .

 8 Fig. 8. Time headway of all vehicles.

Table 1 .

 1 Values of the IDM and HDM parameters as used in the simulations.

	IDM parameter	IDM parameter value	HDM parameter	HDM parameter value
	v 0	120 km/h	T '	{1.1 s; 0 s}
	T	1.5 s	n a	{5;1}
	a	0.73 m/s 2	V s	{5%;0%}
	b	1.67 m/s 2	r c	{0.01/s;0/s}
	s 0	2 m	τ	20 s
	6.3.Results analysis
	6.3.1 Boxplots
	Some figures are represented by boxplots which is nowadays most used by researchers to represent
	schematically the distribution of a variable. The boxplot of a sample has these values:  the first quartile vale (Q1);  the median value (M);  the third quartile value (Q3);

Table 2 .

 2 Values of the boxplots for the percentage of time where the time headway is under 2 s.

	Penetration rate vs. box plot values	Min	Q1	M	Q3	Max
	0%	2.635	3.334	15.89 17.67 18.71
	25%	18.82	23.8	28.28 28.7 28.89
	50%	30.77	34.91	37.87 38.58 39.41
	75%	45.48	45.73	48.87 50.15 52.11
	100%	None	None	57.69 None None

Table 3 .

 3 Values of the boxplots for the Fuel consumption in litters per 100 km.

	Penetration rate vs. box plot values	Min	Q1	M	Q3	Max
	0%	11.05	11.11	11.3 11.47 11.51
	25%	10.88	11.28	11.77 11.9 12.23
	50%	11.01	11.1	11.17 11.68 11.86
	75%	11.42	11.44	11.6	11.65 11.72
	100%	11.89	11.9	11.95 12.04 12.06


Qi Cheng
email: qi.cheng@ifsttar.fr

Lydie Nouvelière

Olivier Orfila
email: olivier.orfila@ifsttar.fr

A PIECEWISE VEHICLE FUEL CONSUMPTION MODEL FOR ECO-DRIVING ASSISTANCE SYSTEM

This paper presents a new method to model the vehicle fuel consumption in the design of an eco-driving assistance system developed by IFSTTAR within the European Commission FP7 project eco-Driver. The ambition of the driver assistance system is to combine safety requirements and fuel consumption reduction by optimizing the vehicle speed and the gear management. The exact estimation of the vehicle fuel consumption plays a key role in the driver assistance system design. This paper extends the authors' results about the vehicle fuel consumption model. The instantaneous fuel consumption rate is modelled as a piecewise polynomial of the instantaneous engine speed and the instantaneous engine torque. The real vehicle experiments show the good performances of this piecewise fuel consumption model.

I. Introduction

Reducing the vehicle fuel consumption becomes more and more important in the automobile industry. The safety and the fuel economical efficiency are the first two important issues when the customers decide to buy a new car. The automobile manufacturers have invested huge money to improve the power train efficiency. But it is a long term process. Another short term lydie.nouveliere@ibisc.univ-evry.fr way is to change the driver's behavior which has an important influence of the vehicle fuel consumption by helping him in his driving task. The frequent accelerations and decelerations should be avoided since they cost lots of energy.

The economical-ecological way of driving (eco-driving) is a citizen driving behavior which allows to reduce the energy consumption up to 15%, limiting the GHG effect responsible for the climatic warming, and to decrease the risk of accidents (by 10 to 15%).

In the eco-driving assistance system (EDAS), an optimal profile, to be sent to the Humanmachine interface module to advise the driver in real time, can be computed. This optimal profile is obtained by an optimization process on the vehicle fuel consumption. The fuel consumption model is then one of the key points in the design of the eco-driving assistance system [START_REF] Luu | Vehicle speed control for a safety-economicalecological compromise: Development of a driver assistance system[END_REF] before the choice of the reference signal to be sent to the driver (speed, gear, ...).

In the current eco-driving assistance system, the vehicle fuel consumption model is described by a single equation. In this paper, a piecewise model is proposed in which the engine map could be separated into some sub-areas and in each sub-area, the fuel consumption is approximated by a single equation of a 2-order polynomial of the engine torque and engine speed. With this new modelling technique, the fuel consumption estimation precision could be greatly improved.

This paper is organized as follows. After a brief presentation of the EU ecoDriver project, a schematic diagram of the proposed eco-driving system is given in Section 2. Then, the simple vehicle model is described in Section 3. The details of the vehicle fuel modeling technique is given in Section 4. Some experimental results are shown is Section 5. A final analysis concludes on this EDAS.

II. Eco-driving assistance system

A. EcoDriver Project

The project ecoDriver -Supporting the driver in conserving energy and reducing emissionsan FP7 EU project -addresses the need to consider the human element when encouraging â ȂIJgreenâ Ȃİ driving, since driver behaviour is a critical element in energy efficiency. The focus of the project is on technology working with the driver. The project aims to deliver the most effective feedback to drivers on green driving by optimizing the driver-power train-environment feedback loop. It will carry out a substantial programme of work to investigate how best to win the support of the driver to obtain the most energy-efficient driving style for best energy use.

The project aims to examine driving not only with present and near-term power trains but also with a full range of future vehicles, including various types of hybrid and plug-in electric vehicles. A comprehensive evaluation will be carried out both in the laboratory (a variety of driving simulators) and in real world driving in both the private and fleet contexts. Scenarios will be developed to assess the implications for the future effectiveness of green driving support. The target of ecoDriver is to deliver a 20% improvement in energy efficiency by autonomous means alone, which opens up the possibility of greater than 20% savings in combination with cooperative systems.

B. The design of the eco-driving system

The schematic description of the model for the eco-driving system design is presented in Figure 1. The approach considers that from road and vehicle data, an optimal profile in terms of speed and gear number, is computed and sent to the Human-machine interface module to advise the driver.

The aim of the global chain schematic diagram of the full system for eco-driving system is to combine safety requirements and fuel consumption reduction by optimizing the vehicle speed and the gear management. It requires information on various signals from the vehicle (speed, position, gear number), environment (longitudinal headway spacing), road (legal speed limitation, road geometrical attributes) to compute such an optimal profile. For that, we developed a power train model that provides the vehicle state vector and the instantaneous fuel consumption model that gives the vehicle fuel consumption information. Then, an optimization procedure is developed to suggest an optimal profile in terms of speed and gear number to the driver in order to reduce the fuel consumption considering the road data and the safety requirement.

III. Vehicle model

The main output of the vehicle model is the longitudinal dynamics vehicle state: current longitudinal position, speed, acceleration, gear number, engine speed, engine torque. An illustration of the vehicle dynamics model is given in Figure 2. A common power train model is used

vx = r 2 (M gf R + 0.5ρSC x v 2 x + M g sin(θ)) M r 2 + I et N 2 f N 2 t + I dw + r(ηN f N t T e -T b ) M r 2 + I et N 2 f N 2 t + I dw , (1)
where

I et = I e + I t , I dw = I d N 2 f + I w .
The designation of the variables is given in Table I. In the Model (1), the influence of the brake torque is ignored.

IV. Fuel consumption modelling

A. The state-of-art of the fuel consumption model

The estimation and the prediction of the real fuel consumption play a key role in the ecodriving system. The eco-driving assistant system requires information on various signals from the vehicle (speed, position, gear number), environment (longitudinal headway spacing), road (legal speed limitation, road geometrical attributes) to compute such an optimal profile in terms of speed and gear number in order to minimise the vehicle fuel consumption. A large literature has been contributed to the modelling of the vehicle fuel consumption. Generally, there are 3 kinds of vehicle fuel consumption model:

• Instantaneous fuel consumption computes through the vehicle power, e.g. [START_REF] Post | Fuel consumption and emission modeling by power demand and a comparison with other models[END_REF], [START_REF] Wong | Theory of Ground Vehicle[END_REF] and [START_REF] Rakha | Virginia tech comprehensive power-based fuel consumption model: Model dvelopment and testing[END_REF].

• Instantaneous fuel consumption computes through vehicle speed and acceleration, e.g. [START_REF] Akcelik | Some results on fuel consumption modles[END_REF], [START_REF] Ahn | Estimating vehicle fuel consumption and emissions based on instantaneous speed and acceleration levels[END_REF], [START_REF] Kamal | Ecological vehicle control on roads with up-down slopes[END_REF].

• Instantaneous fuel consumption computes through engine speed and engine torque, [START_REF] Luu | Dynamic programming for fuel consumption optimization on light vehicle[END_REF], [START_REF] Saerens | Optimal Control Using Pontryagin's Maaximum Principle and Dynamic Programming[END_REF]. [START_REF] Akcelik | Some results on fuel consumption modles[END_REF] gives a simple equation to calculate the instantaneous fuel consumption based on the vehicle speed and acceleration. Then, a complicated model based a 3-order polynomial of the vehicle speed and accelaration is proposed by [START_REF] Ahn | Estimating vehicle fuel consumption and emissions based on instantaneous speed and acceleration levels[END_REF]. Along the same direction, [START_REF] Kamal | Ecological vehicle control on roads with up-down slopes[END_REF] uses only part of terms in [START_REF] Ahn | Estimating vehicle fuel consumption and emissions based on instantaneous speed and acceleration levels[END_REF] to calculate the vehicle fuel consumption.

Another way to model the vehicle fuel consumption is to use the vehicle power directly. [START_REF] Post | Fuel consumption and emission modeling by power demand and a comparison with other models[END_REF] proposes a linear relationship of the vehicle power to calculate the instantaneous fuel consumption rate. An modified fuel consumption model using the vehicle power and the engine instantaneous speed is proposed in [START_REF] Wong | Theory of Ground Vehicle[END_REF]. Recntly, the idea of using a 2-order polynomial of the vehicle power to compute the fuel consumption rate is developed in [START_REF] Rakha | Virginia tech comprehensive power-based fuel consumption model: Model dvelopment and testing[END_REF].

A new method to model the vehicle fuel consumption using the engine torque and engine speed is proposed in [START_REF] Luu | Dynamic programming for fuel consumption optimization on light vehicle[END_REF], [START_REF] Luu | Vehicle speed control for a safety-economicalecological compromise: Development of a driver assistance system[END_REF] and [START_REF] Luu | Ecological and safe driving assistance system: design and strategy[END_REF]. The instantaneous fuel consumption is calculated by a linear combination of the engine torque and engine speed. Differently, a 2-order polynomial of the engine torque and engine speed to compute the vehicle instantaneous fuel consumption is proposed in [START_REF] Saerens | Optimal Control Using Pontryagin's Maaximum Principle and Dynamic Programming[END_REF].

It is difficult to say which kind of model is better than the others, since one kind of these models can be converted into the others kind of fuel consumption models. A comparison of these models are given in [START_REF]Modélisation de la consommation en carburant d'un véhicule léger à partir de données expérimentales[END_REF].

B. A new piecewise fuel consumption model

To extend our previous results ( [START_REF]Modélisation de la consommation en carburant d'un véhicule léger à partir de données expérimentales[END_REF], [START_REF] Luu | Dynamic programming for fuel consumption optimization on light vehicle[END_REF], [START_REF] Luu | Vehicle speed control for a safety-economicalecological compromise: Development of a driver assistance system[END_REF], [START_REF] Luu | Ecological and safe driving assistance system: design and strategy[END_REF]), A new vehicle instantaneous fuel consumption is proposed in this paper. The innovation of our new model is that a piecewise model is used to describe the non linear relationship among the instantaneous fuel consumption and the engine speed ω e and engine torque T e . According to the real fuel consumption data, we find that it is difficult to calculate the fuel consumption rate by a single equation. However, the engine map could be separated into some sub-areas which can be approximated by a single equation.

In our new fuel consumption model, the instantaneous fuel consumption rate F C is cut into several pieces according to the values of the engine speed ω e and engine torque T e , see Figure 3. First, we separate the entire engine map into some sub-areas in which the fuel consumption can be expressed by a single equation; Second, we estimate the parameters of the fuel consumption model in each sub-area. In each sub-area, the vehicle instantaneous fuel consumption rate is expressed by

F C = 2 i=0 2 j=0 L i,j ω i e T j e . (2)
The key problem of this modeling technique is to set the number of the sub-area and find the brake point of each sub-area. Generally speaking, the more the number of sub-area is, the more precise the model is. The brake point of each sub-area is detected by the least squares method. The shape of some sub-areas are regular, while the one of the others are not (e.g. F C10 in Figure 3).

V. Experimental results

A. Test vehicle

A middle class manual transmission car, the RENAULT Clio 3 Eco 2 equipped with CAN fuel consumption measurements see Figure 4, is chosen in our experiments for its popularity in France. All the parameters of the powertrain model and the fuel consumption model are estimated from the real experimental data of the RENAULT Clio 3 Eco 2.

B. Test scenarios

Several basic scenarios are required for the validation of the powertrain model and the fuel consumption model to verify its coherence:

• SC1: Constant speed scenario -The driver will drive at a given steady speed v c taken on the interval 10 to 130 km/h, with a different gear ratio N .

• SC2: Reaching a desired speed v d from 30 to 130 km/h with change of every 10 km/h -Starting at a null velocity, the driver has to accelerate to reach v d then decelerates until stopping.

• SC3: Realistic scenario -On open road, the driver will travel 5 km at urban roads, interurban roads and highway.

C. Experimental validation of the model

To achieve the vehicle dynamics state, some sensors were installed in the test vehicle: Vehicle speed and acceleration sensors, engine speed sensor, acceleration pedal sensor and fuel flow meter. The speed is recorded using the vehicle own odometer. The fuel consumption is acquired from the CAN Bus channel. Its resolution of measurements is 80mm. 3 . A RTMAPS (INTEMPORA, France) platform is used to guarantee the real-time data logging phase and registration. The experimental plan consisted in travelling on a dedicated route constituted of different road profiles.

The results of one real test are shown in the Figure 5, 5, 6, 7, 8, 9 and 10. The vehicle speed v is shown in Figure 5, the unit is km/h. The road slope and the gear number are shown in Figure 5. The corresponding engine torque and engine speed are shown in Figure 7 and Figure 8, respectively.

The instantaneous fuel consumption rate F C (mL/s) calculated from Model 2 are shown in Figure 9 in blue line, while the instantaneous fuel consumption rate read from the CAN bus is plotted in green line. It is clear that the estimated instantaneous fuel consumption rate (blus line) could track its real value (green line) as a whole. The error occurs when this is a strong brake. This is because the influence of the brake torque is ignored in the Model [START_REF] Luu | Vehicle speed control for a safety-economicalecological compromise: Development of a driver assistance system[END_REF].

The cumulated fuel consumption in shown in Figure 10, the blue line indicates the results estimated from the Model 2 and the green line indicates the results from the CAN bus. In Figure 10, we can see that the cumulated error of the fuel consumption calculated from our model is not big and the performance of our new piecewise model is confirmed.

VI. Conclusions

An innovative fuel consumption modelling technique is proposed in this paper in order to develop an eco-driving assistance system. The vehicle fuel consumption is estimated by a piecewise polynomial of the instantaneous engine speed and the instantaneous engine torque. The accuracy of the model is improved through piecewise modelling technique. The key issues of this new fuel consumption model is to find the brake points of each sub-model.

Fig. 1 .

 1 Fig. 1. Schematic diagram of the power train model in the eco-driving system.

Fig. 2 .

 2 Fig. 2. The vehicle dynamics model.

Fig. 3 .

 3 Fig. 3. Sub-area division of the engine map.

Fig. 4 .

 4 Fig. 4. The test vehicle: RENAULT Clio 3 Eco 2.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Vehicle speed.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Engine torque.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Instantaneous fuel consumption rate F C.

TABLE I Nomenclature

 I

of the powertrain model to describe the vehicle longitudinal dynamics ([2]):


Qi Cheng
email: qi.cheng@ifsttar.fr

Olivier Orfila
email: olivier.orfila@ifsttar.fr

Lydie Nouvelière

A NEW ECO-DRIVING ASSISTANCE SYSTEM

FOR A LIGHT VEHICLE : ENERGY MANAGEMENT AND SPEED OPTIMIZATION Qi Cheng, Lydie Nouvelière and Olivier Orfila Abstract-This paper presents a new method to model and optimize the vehicle fuel consumption and its speed in the design of an eco-driving assistance system (EDAS) developed within the EU ecoDriver project. The main objective of this EDAS is to combine a precise fuel consumption model with a robust optimization module. An optimal speed profile is obtained to reduce the energy consumption. The gear management is also included in this procedure. The instantaneous fuel consumption rate is expressed as a piecewise polynomial of the instantaneous engine speed and engine torque. A dynamic programming technique is used to optimize the vehicle fuel consumption considering the safety requirements. The real vehicle experiments show the good performance of the piecewise model. The algorithm is implementable in a light vehicle Renault Clio 3.

I. Introduction

The significant and continuing increase of fuel costs leads companies and governments to think about the means to reduce the consumption of energy. In addition, legitimate concerns related to sustainable development and to the satisfaction of the international standards of lydie.nouveliere@ibisc.univ-evry.fr environmental management systems, encourage all vehicle users to travel in an eco-driving way to decrease CO 2 emissions.

The safety and the fuel economical efficiency are the first two important issues when the customers decide to buy a new car. Indeed ecodriving can not only bring an economy in money and ecology but it can also minimize the risk of traffic accident (by 10 to 15%) by applying a responsible driving. The economical-ecological way of driving (eco-driving) is a citizen driving behavior which allows to reduce the energy consumption up to 15% and to limit the Greenhouse Effect responsible for the climatic warming.

The principle of the proposed Eco-Driving Assistance System (EDAS) consists in providing an optimal profile sent to the Human-machine interface module on board to advise the driver in real time. This optimal profile is computed by an optimization process on the vehicle energy consumption. The fuel consumption model is the first key point in the design of the EDAS [START_REF] Luu | Vehicle speed control for a safety-economicalecological compromise: Development of a driver assistance system[END_REF] before generating the reference signal to the driver (speed, gear, ...).

In the general case, the vehicle energy consumption model is represented by a simple function of the engine speed, engine torque, engine power, vehicle acceleration and vehicle speed.

Here, a piecewise model is applied by considering the engine map as several separated subareas in each of which the fuel consumption is estimated by a single 2-order polynomial function of the engine torque and engine speed. With this technique, the precision of the fuel consumption model can be greatly improved.

Then, a dynamic programming technique is used to optimize the vehicle energy consumption considering the safety requirements. An optimal profile of the vehicle speed and gear number is hence computed to permit the eco-driving.

The paper is organized as follows. After a brief presentation of the EU ecoDriver project, a schematic diagram of the proposed eco-driving system is given in Section 2. A simple vehicle model is described in Section 3 followed by the details of the vehicle fuel modeling technique in Section 4. The dynamic programming technique is presented in Section 5. Some experimental results are shown is Section 6 regarding the energy modeling. Some conclusions are given in Section 7.

II. Eco-driving assistance system design

This work is a part of European project FP 7 ecoDriver.

A. EcoDriver Project

The project ecoDriver -Supporting the driver in conserving energy and reducing emissions -a EU FP7 project -addresses the need to consider the human element when encouraging green driving, since driver behavior is a critical element in energy efficiency. The focus of the project is to develop an eco-driving technology working with the driver. The project aims to deliver the most effective feedback to drivers on green driving by optimizing the driver-power train-environment feedback loop. It will carry out a substantial programme of work to investigate how to win the maximum support for the driver to obtain the most energy-efficient driving style in terms of energy use.

The project examines driving not only with present and near-term power trains but also with a full range of future vehicles, including various types of hybrid and plug-in electric vehicles. A comprehensive evaluation will be carried out both with driving simulators and in fleets of vehicles. The target of ecoDriver project is to deliver a 20% improvement in energy efficiency by an autonomous means (without cooperative systems).

B. Power train diagram

The Figure 1 gives a description of the power train model used in the EDAS. From road and vehicle measured data, optimal speed and gear number profiles are computed and sent to the human-machine interface module to advise the driver.

The aim of the global chain schematic diagram of the full system for the EDAS is to combine safety requirements and fuel consumption reduction by optimizing the vehicle speed and the gear management. It requires information on several measured signals from the vehicle (speed, position, gear number), environment (longitudinal headway spacing), road (legal speed limitation, road geometrical attributes) to obtain such optimal profiles. Hence a power train model that provides the vehicle state vector and the instantaneous fuel consumption model giving the vehicle fuel consumption information has been developed and experimentally validated. An optimization procedure is formulated which suggests an optimal profile in terms of speed and gear number to the driver in order to reduce the fuel consumption considering the road data and the safety requirement.

III. Vehicle model

The vehicle dynamics model used by the optimization module is as follows (

[2]) vx = r 2 (M gf R + 0.5ρSC x v 2 x + M g sin(θ)) M r 2 + I et N 2 f N 2 t + I dw + r(ηN f N t T e -T b ) M r 2 + I et N 2 f N 2 t + I dw , (1)
where

I et = I e + I t , I dw = I d N 2 f + I w .
This model gives only the longitudinal part of vehicle dynamics, the one needed for the energy consumption. The designation of the variables is given in Table I. In the Model (1), the influence of the brake torque is ignored. The relationship between the engine speed ω e and vehicle speed v x is given by

ω e = N t N f v x 2πr . (2)
IV. Fuel consumption modeling

The estimation and the prediction of the real fuel consumption play key roles in the eco-driving system. A large literature has been contributed to the modeling of the vehicle fuel consumption [START_REF] Akcelik | Some results on fuel consumption modles[END_REF], [START_REF] Post | Fuel consumption and emission modeling by power demand and a comparison with other models[END_REF], [START_REF] Wong | Theory of Ground Vehicle[END_REF], [START_REF] Ahn | Estimating vehicle fuel consumption and emissions based on instantaneous speed and acceleration levels[END_REF]. Some new methods to model the vehicle fuel consumption using the engine torque and engine speed are discussed in [START_REF] Luu | Dynamic programming for fuel consumption optimization on light vehicle[END_REF], [START_REF] Luu | Vehicle speed control for a safety-economicalecological compromise: Development of a driver assistance system[END_REF], [START_REF]Ecological and safe driving assistance system: design and strategy[END_REF], [START_REF] Saerens | Optimal Control Using Pontryagin's Maaximum Principle and Dynamic Programming[END_REF]. It is difficult to know which kind of model is better than the others, since one kind of these models can be converted into the others. A comparison of these models is given in [START_REF] Luu | Modélisation de la consommation en carburant d'un véhicule léger à partir de données expérimentales[END_REF].

A. A new piecewise fuel consumption model

Based on our previous results in [START_REF] Luu | Modélisation de la consommation en carburant d'un véhicule léger à partir de données expérimentales[END_REF], [START_REF] Luu | Dynamic programming for fuel consumption optimization on light vehicle[END_REF], [START_REF] Luu | Vehicle speed control for a safety-economicalecological compromise: Development of a driver assistance system[END_REF], [START_REF]Ecological and safe driving assistance system: design and strategy[END_REF], a new vehicle instantaneous fuel consumption model is proposed in this section. The innovation of this new model is that a piecewise model is used to describe the non linear relationship among the instantaneous fuel consumption and the engine speed ω e and engine torque T e . According to the real fuel consumption data, it is difficult to compute the fuel consumption rate by a single equation on the whole engine map. However, the engine map could be separated into some sub-areas which can be approximated by a single equation.

In our new energy consumption model, the instantaneous fuel consumption rate F C is cut into several pieces according to the values of the engine speed ω e and engine torque T e , see Figure 2. First, we separate the entire engine map into some sub-areas in which the fuel consumption can be expressed by a single equation; Secondly, the parameters of the fuel consumption model are estimated using the real experiments data in each sub-area in which the vehicle instantaneous consumption rate is expressed by

F C = 2 i=0 2 j=0 L i,j ω i e T j e . (3)
The main problem of this modeling technique is to set the number of the sub-areas and find the break point of each sub-area. Generally speaking, the more the number of sub-areas is, the more precise the model is. The break point of each sub-area is detected by a least squares method. The shape of some sub-areas is regular, while the one of the others is not (e.g. F C10 in Figure 2).

V. Optimization method

Since the fuel consumption modeled is fixed with its estimated parameters, the longitudinal model given in section 3 can be used in an optimization procedure.

A. Problem formulation

The problem of the optimal energy consumption is a classic optimal control problem. Firstly, an objective function J should be determined; Then, a method of optimization is chosen to solve this problem. The dynamic programming technique is one of the popular methods to solve this optimization problem [START_REF] Bellman | Dynamic programming[END_REF]. The discrete optimal fuel consumption problem is expressed as

                   min U N k=0 J k (X k , U k ) X k+1 = f (X k , U k , θ k ) X k ∈ [X lbk , X ubk ] U k ∈ [U lbk , U ubk ] X(0) = X 0 X(N ) = X N , (4)
where X k = [v x,k ] is the vehicle longitudinal speed, U k is the control variable, θ k is the system parameters, X lbk and X ubk are the lower and upper speed bounds, U lbk and U ubk are the lower and upper bounds of the control variables.

B. Objective function

The objective function J must be a compromise between the fuel consumption (F C) and the travel time (T ) as well as the driver feeling (SM OOT H), J = Q1 * F C + Q2 * T + Q3 * SM OOT H, [START_REF] Wong | Theory of Ground Vehicle[END_REF] where SM OOT H is computed by

SMOOTH = K-1 k=0 (A * (max(0, X k+1 -X k )) + (1 -A) * abs(min(0, X k+1 -X k ))),
where A is a penalty coefficient of the increase speed. It permits not to obtain too much oscillations between hard accelerations and hard braking phases during the trip.

F C is a function of the engine speed ω e and engine torque T e and it can be expressed by a function of vehicle speed v x by using the vehicle model in Section 3 (Equation 1, 2). The travel time T and the SM OOT H function are also the function of vehicle speed v x . In this objective function J, each term can be expressed by the vehicle speed v x .

C. Dynamic programming

Dynamic programming is used widely to solve the optimal control problem. Dynamic programming is based the Bellman's maximum principle [START_REF] Bellman | Dynamic programming[END_REF]. The basis of dynamic programming is a backward iteration. For k = N -1, N -2...0,

J N = ζ N (X N ), (6)
J k (X n ) = min U k (ζ k (X k , U k ) + J k+1 (f (X k , U k ))). (7)
Using the J k , first, we can obtain the optimal control variable U k , for k = 0, 1, ..., N -1

Ûk = arg min U k (ζ k (X k , U k ) + J k+1 (f (X k , U k ))).
(8) Then, we can rebuild the optimal trajectories of X k with the initial value X(0) by the following equation,

X k+1 = f (X k , Ûk , θ k ). (9)
Thanks to the dynamic programming technique, an optimal profile of the vehicle speed v x and gear number N is obtained in order to reduce the fuel consumption.

VI. Experimental results

A. Test vehicle

A middle class manual transmission car, the RENAULT Clio 3 equipped with CAN fuel consumption measurements (see Figure 3) is chosen in our experiments as it represents the car of a "man in the street" in France. All the parameters of the power train model and the fuel consumption model are estimated from the real experimental data of the RENAULT Clio 3.

B. Experimental validation of the energy model

To achieve the vehicle dynamics state, some sensors were installed in the test vehicle: longitudinal speed (odometer) and acceleration, engine speed, acceleration pedal and fuel flow meter. The energy consumption is acquired from the CAN channel with a 80mm. The results of one real test are shown in the Figure 4, 5, 6, 7, 8 and 9. The vehicle speed v is shown in Figure 4 in km/h. The road slope and the gear number are shown in Figure 4. The corresponding engine torque and engine speed are shown in Figure 6 and Figure 7, respectively. The instantaneous fuel consumption rate F C (mL/s) calculated from Model 3 is shown in Figure 8 compared to the instantaneous fuel consumption rate read from the CAN bus. It is clear that the estimated instantaneous fuel consumption rate (blue line) could track its real value (green line) as a whole. The error occurs when this is a strong brake because the influence of the brake torque is ignored in the Model [START_REF] Luu | Vehicle speed control for a safety-economicalecological compromise: Development of a driver assistance system[END_REF].The cumulated fuel consumption in shown in Figure 9 associated to the results estimated from the Model 3. We can see that the cumulated error of the fuel consumption calculated from our model is not big and the performance of our new piecewise model is confirmed.

Another real test results of the EADS are given in Figure 10. Our prototype car runs under the conditions that there is always another vehicle runs before our car and the speed limit in this trip is known. We set Q1 = Q2 = Q3 = 1 and A = 0.4 in the objective function J. The preceding vehicle speed is given in blue dash-dot line. The optimal speed profile given by our new methods is presented in red line. The method developed in [START_REF]Ecological and safe driving assistance system: design and strategy[END_REF] is displayed in violet dash line as a reference. It is clear that our new profile is better the the old one since it avoids the frequent accelerations and decelerations. The total fuel consumption is saved by 20% with our new profile.

VII. Conclusions

An innovative fuel consumption modeling technique is proposed in this paper in order to develop an eco-driving assistance system. The vehicle fuel consumption is estimated by a piecewise polynomial of the instantaneous engine speed and the instantaneous engine torque. The accuracy of the model is improved through piecewise modeling technique. The key issues of this new fuel consumption model is to find the break points of each sub-model.

Fig. 1 .

 1 Fig. 1. Schematic diagram of the power train model in the eco-driving system.

Fig. 2 .

 2 Fig. 2. Sub-area division of the engine map.

Fig. 3 .

 3 Fig. 3. The test vehicle: RENAULT Clio 3.

  3 resolution. A RTMAPS ( c INTEMPORA, France) platform is used to guarantee the real-time data logging phase and registration on board. The experimental plan consisted in travelling on a dedicated route constituted of different road profiles.

Fig. 4 .

 4 Fig. 4. Vehicle speed.

Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .

 5678 Fig. 5. The road slope and gear number

Fig. 9 .

 9 Fig. 9. Cumulated fuel consumption.

Fig. 10 .

 10 Fig. 10. Optimal profile.

TABLE I

 I Nomenclature of the power train model