R Franco

H Ríos

D Efimov

W Perruquetti
email: wilfrid.perruquetti@ec-lille.fr

Adaptive Estimation for Uncertain Nonlinear Systems: A Sliding-Mode Observer Approach

Keywords: Adaptive Observer, Nonlinear Systems, Sliding-Modes

This paper deals with the problem of adaptive estimation, i.e. the simultaneous estimation of the state and parameters, for a class of uncertain nonlinear systems. A nonlinear adaptive sliding-mode observer is proposed based on a nonlinear parameter estimation algorithm. The nonlinear parameter estimation algorithm provides a rate of convergence faster than exponential while the sliding-mode observer ensures ultimate boundness for the state estimation error attenuating the effects of the external disturbances. Linear matrix inequalities (LMIs) are provided for the synthesis of the adaptive observer and some simulation results show the feasibility of the proposed approach.

I. INTRODUCTION

D ISTURBANCES and uncertainties widely exist in al- most all physical systems in the real world, in the form of external perturbations, unknown system dynamics and/or unknown parameters. One of the main problems for the robust control design is the existence of such disturbances and the fact that the whole state is not always available for measuring. A useful tool to deal with this problem has been studied during the last decades, i.e. the adaptive control theory. This theory has received a great deal of attention due to the need of adaptive control design. This area has grown to turn into one of the widest in terms of algorithms, analytical tools and techniques for design (see, for instance [START_REF] Astolfi | Nonlinear and Adaptive Control with Applications[END_REF]). Particularly, the design of observers estimating simultaneously the whole state and the parameters of the system by some on-line adaptation law is an important problem in the adaptive control area (see, e.g. [START_REF] Besançon | Nonlinear Observers and Application[END_REF]).

In this framework, a class of adaptive extended state observers is proposed in [START_REF] Pu | A class of adaptive extended state observers for nonlinear disturbed systems[END_REF] that significantly increases the applications of extended state observers to nonlinear disturbed systems. In [START_REF] Ekramian | Adaptive state observer for Lipschitz nonlinear systems[END_REF] the problem of adaptive observer synthesis for Lipschitz nonlinear systems is addressed. The sufficient conditions are given in terms of LMIs that ensure the state estimation error convergence to zero with known parameters. Also, an adaptive observer form is presented which is used to achieve the adaptive estimation under additional constraints. In [START_REF] Ahmed-Ali | Adaptive observer for a class of parabolic PDEs[END_REF] an adaptive observer for a class of parabolic PDEs is proposed, the observer is designed to provide online estimates of the system state and unknown parameters based on sampled data and also sufficient conditions for the observer to be exponentially convergent are established. On the other hand, a hybrid adaptive observer is designed in [START_REF] Folin | Sampled-data adaptive observer for a class of state-affine output-injection nonlinear systems[END_REF] and it is shown that the state estimation error is exponentially convergent if the sampling period is small enough and a persistent excitation condition holds. In this vein, in [START_REF] Farza | Adaptive observers for nonlinear parameterized class of nonlinear systems[END_REF] an adaptive observer was proposed to exponentially estimate the state and the unknown parameters under a persistent excitation condition for uniformly observable Multiple-Input-Multiple-Output (MIMO) nonlinear systems. The authors also proposed an adaptive observer for a class of uniformly observable MIMO nonlinear systems with general nonlinear parametrization which provides exponential convergence to zero of the adaptive estimation error. Adaptive observers, using delays are proposed by [START_REF] Oyvind | Redesign of adaptive for improved parameter identification in nonlinear systems[END_REF]. The authors provided a method for redesigning adaptive observers for nonlinear systems that increases the computational effort but provides better parameter estimations and some robustness properties. In the context of nonlinear parametrization, very few results are available in the literature. For instance, in [START_REF] Kojic | Adaptive control of nonlinearly parameterized systems with a triangular structure[END_REF] an adaptive control is proposed for a class of nonlinear systems with a triangular structure and nonlinear parametrization. This approach provides boundedness for regulation and tracking tasks. In [START_REF] Tyukin | Adaptive observers and parameter estimation for a class of systems nonlinear in the parameters[END_REF] adaptive observers are proposed for a class of systems with nonlinearities in the parameters. These observers can reconstruct asymptotically the unknown state and the parameters.

In the sliding-mode area, different adaptive observers have been proposed. In [START_REF] Zhang | Adaptive sliding-mode observer design for a selective catalytic reduction system of ground-vehicle diesel engines[END_REF] an adaptive sliding-mode observer is designed for the selective catalytic reduction system in diesel-engine after-treatment systems achieving better performance with respect to a Luenberger-like observer. In the fault detection problem, an adaptive sliding-mode observer is proposed by [START_REF] Rahme | Adaptive sliding-mode observer for sensor fault diagnosis of an industrial gas turbine[END_REF] to solve the problem of sensor fault diagnosis in an industrial gas turbine. Such an adaptive observer provides simultaneous fault detection and identifi-cation despite the presence of noise. In [START_REF] Laghrouche | Adaptive second-order sliding-mode observer-based fault reconstrucion for PEM fuel cell air-feed system[END_REF] an adaptivegain second-order sliding-mode observer is developed for estimating the system states and the uncertain parameters. The performance of the proposed observer is illustrated in a fuel cell air-feed system using a hardware-in-loop emulator. An adaptive sliding-mode observer is proposed by [START_REF] Yan | Adaptive sliding-mode-observer-based fault reconstruction for nonlineary systems with parametric unvertainties[END_REF] for a class of nonlinear systems with unknown parameters and faults. Based on the main properties of the sliding-mode observers, an asymptotic fault reconstruction is given taking into account that the relative degree of the output, with respect to the fault, is equal to one. It is important to highlight that most of the proposed adaptive observers are based on linear parameter estimation algorithms. Additionally, most of these works do not consider external disturbances and the convergence rate of the state and parameter estimation errors is asymptotic or exponential. In this context, in [START_REF] Efimov | Design of impulsive adaptive observers for improvement of persistency of excitation[END_REF] it was shown that improvements of convergence rate, for the parameter estimation error, cannot be achieved by simply increasing the observer gains but changing the structure of the adaptive observers.

In this paper a nonlinear adaptive sliding-mode observer is proposed, based on a nonlinear parameter identification algorithm, for uncertain nonlinear systems. The proposed nonlinear adaptive observer is an extension of the one presented by [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF]. This extension tries to increase the class of nonlinear systems for which the adaptive observer can be applied allowing to have a parameter distribution matrix and nonlinear terms depending on the full state and the input. In [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF], such functions were considered just depending on the measurable output and input. The proposed parameter identification algorithm provides an ultimate bound for the state and parameter estimation error and attenuating the effects of the external disturbances. The synthesis of the observer is given in terms of LMIs. The proofs are developed based on the Lyapunov function approach and the input-tostate stability theory. Some simulation results illustrate the efficiency of the proposed nonlinear adaptive sliding-mode observer.

This paper is organized as follows. The problem statement is given in Section II. The preliminaries are discussed in SectionIII . The adaptive observer and the convergence properties are presented in Section IV. Simulation results are shown in Section V and conclusions in Section VI. Notation: The Euclidean norm of a vector q ∈ R n is denoted by q . For a matrix Q ∈ R m×n , denote its smallest singular value σ min (Q) = λ min (Q T Q) and its induced norm as

Q := λ max (Q T Q) = σ max (Q)
, where λ max is the maximum eigenvalue and λ min is the minimum one, σ max is the largest singular value. For a Lebesgue measurable function u : R ≥0 → R m , define the norm u (t0,t1) := ess sup t∈(t0,t1) u(t) , then u ∞ := u (0,+∞) and the set of functions u with the property u ∞ < +∞ is denoted as

L ∞ . For a matrix Q : R ≥0 → R m×n , denote Q ∞ := Q (0,+∞) . A continuous function α : R ≥0 → R ≥0 belongs to class K if it is strictly increasing and α(0) = 0; it belongs to class K ∞ if it is also unbounded. A continuous function β : R ≥0 × R ≥0 → R ≥0 belongs to class KL if,
for each fixed s, β(r, s) ∈ K with respect to r, and for each fixed r, β(r, s) is decreasing to zero with respect to s. The term ∇V (x)f (x) denotes the directional derivative of a continuously differentiable function V with respect to the vector field f evaluated at any point x.

II. PROBLEM STATEMENT

Consider the following class of uncertain nonlinear systems,

ẋ = Ax + φ(x, u) + G(t, x, u)θ + Dw(t), (1)
y = Cx, (2)
where x ∈ R n is the state vector, y ∈ R p is the measurable output vector, u ∈ R m is the control input vector, θ ∈ R q is a vector of unknown constant parameters, and w ∈ R l is a vector of external disturbances. The matrices A, C and D are known, they have corresponding dimensions, and the pair (A,C) is detectable. The functions φ : R n × R m → R n and G : R ≥0 ×R n ×R m → R n×q are also known and they ensure uniqueness and existence of solutions for system (1) for all admissible disturbances. It is important to highlight that in [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF] the functions φ(x, u) and G(t, x, u) were considered depending only on the measurable output y. The aim of this paper is to provide estimations of the state and parameter vectors, i.e. x and θ, respectively; only using the information of the output y and attenuating as much as possible the effects of the external disturbances w.

Let us consider that system (1)-( 2) satisfies following assumptions.

Assumption 1. ||x|| ∞ < +∞, ||u|| ∞ < +∞, ||ω|| ∞ < +∞, and ||G(t, x(t), u(t)|| ∞ < +∞ for all t ≥ 0.
Assumption 2. The functions φ(u, x) and G(t, x, u) are globally Lipschitz with respect to x; and the parameter vector θ is bounded, i.e. for all t ≥ 0, all x, x ∈ R n and all u ∈ R m , the following conditions are satisfied

||φ(x, u) -φ(x, u)|| ≤ L φ ||x -x||, ||G(t, x, u) -G(t, x, u)|| ≤ L g ||x -x||, ||θ|| ≤ θ + ,
with L φ , L φ , θ + > 0 being some known positive constants.

III. PRELIMINARIES

Consider the following nonlinear system,

ẋ = f (x, w), (3)
where x ∈ R n is the state, w ∈ R l is the external disturbances, and f : R n × R l → R n is a locally Lipschitz function. For an initial condition x 0 ∈ R n and an external disturbance w ∈ L ∞ , denote the solution by x(t, x 0 , w) for any t ≥ 0 for which the solution exists.

The following stability properties for system (3) are introduced (for more details see [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF], [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF], [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF], [START_REF] Sontag | The ISS philosophy as a unifying framework for stabilitylike behavior[END_REF] and [START_REF] Sontag | On the input-to-state stability property[END_REF]). Definition 1. [START_REF] Bernuau | Verification of ISS, iISS and IOSS properties applying weighted homogeneity[END_REF]. The system (3) is said to be Input-to-State Stable (ISS) if for any w ∈ L ∞ and any x 0 ∈ R n there exist some functions β ∈ KL, γ ∈ K such that

||x(t, x 0 , w)|| ≤ β(||x 0 ||, t) + γ(||w|| ∞ ), ∀t ≥ 0.
Lemma 1. [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF]. Let V : R n → R ≥0 be a smooth function. If there exist some positive constants ψ 1 , ψ 2 , ψ 3 , ψ 4 > 0 and γ ∈ (0, 1] such that

ψ 1 ||x|| 2 ≤ V (x) ≤ ψ 2 ||x|| 2 , ∇V (x)f (x, ω) ≤ -ψ 3 V γ (x), ∀||x|| ≥ µ := ψ 4 ||ω|| ∞ ,
then system (3) is ISS with respect to the input ω. Moreover, the following bounds are satisfied:

||x(t, xo, ω)|| ≤ ψ(t) ψ 1 , ∀γ ∈ (0, 1), ∀t ≤ T (x0), e -ψ 3 2 t ψ 2 ψ 1 ||x0||, γ = 1, ∀t ≤ T (x0), ||x(t, xo, ω)|| ≤ ψ2 ψ1 µ, ∀t > T (x0),
where

ψ(t) = (ψ 1-γ 2 ||x 0 || 2(1-γ) -ψ 3 (1 -γ)t) 1 2(1-γ) and T (x 0 ) ≤      max 0, ψ 1-γ 2 ||x 0 || 2(1-γ) -ψ 1-γ 1 µ 2(1-γ) ψ 3 (1-γ)
, ∀γ ∈ (0, 1), max 0,

2[ln(||x 0 ||)-ln(µ)] ψ 3 , γ = 1.
Let us consider the following interconnected nonlinear system

ẋ1 =f 1 (x 1 , x 2 , ω), ( 4
) ẋ2 =f 2 (x 1 , x 2 , ω), (5)
where

x i ∈ R ni , ω ∈ R l , and f i : R n1 × R n2 × R l → R ni
ensures existence of the system solutions at least locally, for i = 1, 2. Suppose that there exist ISS Lyapunov functions V 1 and V 2 , for (4) and ( 5), respectively; and some functions

ψ i1 , ψ i2 , ψ i3 ∈ K ∞ , γ i , χ i ∈ K, such that, for all x i ∈ R ni
and any w ∈ L ∞ , the following hold:

ψi1(||xi||) ≤ Vi(xi) ≤ ψi2(||xi||), i = 1, 2, (6)
V1(x1) ≥ max[χ1(V2(x2)), γ1(||w||)] ⇒ ∇V1(x1)f1(x1, x2, ω) ≤ -ψ13(V1(x1)), (7) V2(x2) ≥ max[χ2(V1(x1)), γ2(||w||)] ⇒ ∇V2(x2)f2(x1, x2, ω) ≤ -ψ23(V2(x2)). (8)
Thus, the following nonlinear small-gain result is introduced, in terms of ISS Lyapunov properties, for the interconnected system (4)-( 5). Theorem 1. [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF]. Suppose that the interconnected system (4)-( 5) has ISS Lyapunov functions V 1 and V 2 satisfying the conditions (6)- [START_REF] Oyvind | Redesign of adaptive for improved parameter identification in nonlinear systems[END_REF]. Then, the system (4)-( 5) is ISS if

χ 1 • χ 2 (r) < r, ∀r > 0. (9)

IV. ADAPTIVE SLIDING-MODE OBSERVER

Let us introduce the following adaptive observer

Ω = (A -LC)Ω + G(t, x, u), (10) θ
= ΓΩ T C T y -C x α , ( 11
) ẋ = Ax + φ(x, u) + G(t, x, u) θ + L(y -C x) + kDsign[F (y -C x)] + Ω θ, (12)
Define the function a α := |a| α sign(a), for any α ∈ [0, 1)

and any a ∈ R; and the function sign[q] := q ||q|| , for any q ∈ R m , also θ ∈ R q and x ∈ R n are the estimations for θ and x, respectively; and Ω ∈ R n×q is an auxiliary variable. If the signal G is persistently exciting (PE), then, due to the filtering property of the variable Ω, the variable CΩ is also PE. Note that the function y -C x α in ( 11) is understood in the component-wise sense. The observer matrix gain L ∈ R n×p has to be selected such that (A -LC) is Hurwitz,

Γ = Γ T > 0 ∈ R q×q , while F ∈ R l×p , k ≥ 0 are designed later.
Let us consider that the following assumption holds.

Assumption 3. Let 0 < min ≤ σ min (CΩ(t)) for all t ≥ 0 and ||CΩ|| ∞ ≤ max < +∞.
The existence of max is guaranteed by Assumption 1 and the fact that matrix (A -LC) is Hurwitz; while if the matrix CΩ has a full column rank then the existence of min is also assured. The values of max and min can be evaluated numerically during experiments.

A. Nonlinear Parameter Identification Algorithm

Define θ := θ -θ and δ := x + Ω θ with x := x -x. Then, the error dynamics are given by:

θ = -ΓΩ T C T CΩ θ -Cδ α , ( 13
) δ = (A -LC)δ + D(ω -ksign[F (y -C x)]) + ∆ φ (x, x, u) + ∆ g (t, x, x, u)θ, (14)
where ∆ φ (x, x, u) := φ(x, u)-φ(x, u) and ∆ g (t, x, x, u) := G(t, x, u)-G(t, x, u). The ISS convergence properties of the error dynamics [START_REF] Laghrouche | Adaptive second-order sliding-mode observer-based fault reconstrucion for PEM fuel cell air-feed system[END_REF] with respect to the input δ for α ∈ [0, 1) are given by the following result.

Lemma 2. [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF]. Let Assumption 3 be satisfied. Then, the error dynamics (13), with α ∈ [0, 1) and Γ = Γ T > 0, is ISS with respect to δ and the trajectories satisfy:

|| θ(t)|| ≤ √ r1 r α-1 2 2 || θ0|| 1-α + ψθt 1 1-α , ∀t ≤ Tθ 0 , ( 15
) || θ(t)|| ≤ r1 r2 µθ, ∀t > Tθ 0 , (16)
where θ0 := θ(0),

ψ θ := 2 α-1 α 2 α+1 min (2r 2 ) α+1 2 /2(1 + α), r 1 := 2λ max (Γ), r 2 := 2λ min (Γ) and µθ = p 1 α+1 [(1 + α) α + 2 2-α α α-1 ] 1 α+1 min(1 + α) α α+1 ||C||||δ||∞, Tθ 0 ≤ max   0, 2 1-α (1 + α) || θ0|| 1-α -µ 1-α θ α 2 α+1 min r2   ,
for any θ0 ∈ R q , and p the dimension of the output.

The results of Lemma 2 establish that the trajectories of the error dynamics [START_REF] Laghrouche | Adaptive second-order sliding-mode observer-based fault reconstrucion for PEM fuel cell air-feed system[END_REF], i.e. θ(t), enter into the bound r 1 /r 2 µ θ in a finite time for any α ∈ [0, 1). It is clear that the size of such a bound depends on α. In the same vein, the following lemma illustrates the ISS convergence properties of the error dynamics ( 14) with respect to the inputs θ and ω. Lemma 3. Let Assumption 1 and 2 hold. Assume that k = ω ∞ and the following matrix inequalities

P D ≤ C T F T , ( 17
)     ζ β 2 I n P P -Λ -1 0 0 -Λ 0 -ε η 2 2 I n     ≤ 0, (18)
ζ := (A -LC) T P + P (A -LC) + η 1 P + σC T C, are feasible for 0 < P T = P ∈ R n×n , F ∈ R l×p , L ∈ R n×p , 0 < Λ T = Λ ∈ R n×n , η 1 := 2(L φ + L g θ + ), η 2 := η 1 ||Ω|| ∞ ,
and some positive constants β, ε, σ > 0. Thus, system (14) is ISS with respect to the inputs θ and ω. Furthermore, the trajectories satisfy:

||δ(t)|| ≤e -ψ δ 2 t λ max (P ) λ min (P ) ||δ 0 ||, ∀t ≤ T δ0 , ( 19
)
||δ(t)|| ≤ λ max (P ) λ min (P ) µ δ , ∀t > T δ0 , ( 20
)
where δ0 := δ(0), ψ δ := α δ (1 -ρ)/λ max (P ), α δ := βλ min (P ) + σ C 2 /2, ρ ∈ (0, 1) and

µ δ = 2ε + σ 2 max 2ρα δ || θ||∞ + 8||F || 2 ρα δ σ ||w||∞, T δ 0 ≤ max 0, 2[ln(||δ0||) -ln(µ δ )] ψ δ ,
for any δ 0 ∈ R n .

All the proofs of the proposed results are omitted due to the lack of space. It is clear that the size of µ δ could be minimized in order to attenuate the effects of w. On the other hand, note that condition (17) introduces some structural restrictions. Particularly, one has that the triple (A, D, C) must not have invariant zeros, and the relative degree of the output y with respect to the input w must be equal to one. The feasibility of ( 18) can be ensured, for small enough η 1 and η 2 , due to the fact that the pair (A, C) is detectable. In this sense, it is worth highlighting that the matrix inequality (20) can be easily rewritten to obtain an LMI with respect to the matrices P and Y := P L fixing the values for β, σ, ε and Λ.

B. Convergence of the Adaptive Observer

In order to show the convergence properties of the adaptive observer ( 10)-( 12), the statements provided by Lemmas 2, 3, and Theorem 1 are applied. The following result shows that the interconnected error system ( 13)-( 14) is ISS with respect to the external disturbances ω, for any α ∈ [0, 1). Theorem 2. Let Assumptions 1, 2 and 3 hold. Let the matrix inequalities (17), [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF] and

(pκ δ ) 2 α+1 C 2 (2ε + σ 2 max )λmin(P ) 2 min (1 + α) 2α α+1 λmax(P )ρα δ < 1, (21)
be feasible for 0

< P T = P ∈ R n×n , F ∈ R l×p , L ∈ R n×p , 0 < Λ T = Λ ∈ R n×n , 0 < Γ T = Γ ∈ R q×q , constants α δ = βλ min (P ) + σ C 2 /2, η 1 = 2(L φ + L g θ + ), η 2 = η 1 ||Ω|| ∞ , κ δ := (1 + α) α + 2 2-α α α-1 , β, ε, σ > 0, α ∈ [0, 1), r 1 = 2λ max (Γ), ρ ∈ (0, 1)
and p the dimension of the output. Then, the interconnected error system (13)-( 14) is ISS with respect to the input w.

Theorem 2 implies that the estimation error e := x -x = δ +Ω θ is also ISS with respect to ω, for any α ∈ [0, 1), since

||e(t)|| ≤ (1 + ||Ω||) θ(t) δ(t)
, ∀t ≥ 0.

In the ideal case, i.e. w ≡ 0 and k = 0, the estimations θ and x converge to their real values and the rate of convergence for θ is faster than exponential for any α ∈ [0, 1). For the perturbed case, i.e. w ≡ 0 with k = ||w|| ∞ , the effect of the external disturbance w is completely attenuated.

V. SIMULATION RESULTS

Let us consider a robotic system depicted by Fig. 1. The dynamics of such a system is given as follows J 1 q1 + M gl sin(q 1 ) + k(q 1 -q 2 ) = w(t),

J m q2 + c f q2 -k(q 1 -q 2 ) = u(t),
where q 1 and q 2 are the angular position of the link and motor, respectively; J 1 and J m denote the inertia of the link and the motor, M is the mass of the link, g is the gravitational acceleration, l represents the length of the link and c f = 1 is the friction coefficient of the motor which is the unknown parameter to estimate. The numerical values of the parameters of the system are shown in Table I.

Parameters

Value

J 1 0.8[Kgm 2 ] Jm 1[Kgm 2 ] g 9.8 m/s 2 l 0.6 [m] k 20 [N m/rad] Define ζ 1 = q1 , ζ 2 = q2 , x = [x 1 , x 2 , x 3 , x 4 ] T := [q 1 , ζ 1 , q 2 , ζ 2 ]
T and θ = c f . Thus, the system in state-space form can be written as follows

ẋ =     0 1 0 0 -k J 1 0 k J 1 0 0 0 0 1 k Jm 0 -k Jm 0     A +     0 -M gl J 1 sin(x1) 0 1 Jm u     φ(x,u) +    0 0 0 -1 Jm x4    G(t,x,u) θ +    0 1 0 0    D w(t), y = 1 0 0 0 0 1 0 0 x.
It can be shown that such a system satisfies Assumptions 1 and 2. Let us consider that the initial conditions for the system are x(0) = [-2, -2, -1.5, 2] T while the initial conditions for the adaptive observer (10)-( 12) are given as Ω(0) = 0, θ(0) = 0 and x(0) = 0. For simulation purposes, the input is given as u(t) = sin(0.2t) + 0.5 sin(3t) + 2 cos(50t) while the disturbance term is taken as w(t) = 0.3 cos(2t) + 0.1.

Fixing Γ = 630, k = 1 and Λ = I 4 , one can use SeDuMi solver among Yalmip in Matlab to find a solution for the matrix inequalities ( 17) and ( 18), and thus, verify (21). The following feasible solution is found with In order to better illustrate that the size of the convergence neighborhood can be improved with respect to the selection of α, the following error index is proposed: , where ∆T = 2 is a time window width in which the corresponding signal is evaluated. The results are depicted by Fig. 5. In this case, it is clear that for α = 0.5 provides the better result.

L φ1 = 7, L g1 = 2, Ω ∞ = 0.492, P =   
e RM S (

VI. CONCLUSIONS

This paper contributes to a nonlinear adaptive slidingmode observer based on a nonlinear parameter identification algorithm for uncertain nonlinear systems. The proposed nonlinear adaptive observer is an extension of the one presented by [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF]. This extends the class of nonlinear systems for which the adaptive observer can be applied allowing to have a parameter distribution matrix and nonlinear terms depending on the full state and the input. In [START_REF] Ríos | An adaptive sliding-mode observer for a class of uncertain nonlinear systems[END_REF], such functions were considered just depending on the measurable output and input. The proposed parameter identification algorithm provides an ultimate bound for the state and parameter estimation error and attenuating the effects of the external disturbances. The synthesis of the observer is given in terms of LMIs. The proofs are developed based on the Lyapunov function approach and the input-to-state stability theory. Some simulation results illustrate the efficiency of the proposed nonlinear adaptive sliding-mode observer.

Figure 1 .

 1 Figure 1. Model of a single-link flexible-joint robot manipulator.

Figure 2 .α = 1 Figure 3 .

 213 Figure 2. Estimation of x 3 .

Figure 4 .

 4 Figure 4. Estimation of θ.

Figure 5 .

 5 Figure 5. Estimation Error Index e RM S (t).

Table I PARAMETERS

 I OF THE ROBOT MANIPULATOR.

  The corresponding simulations have been done in Matlab with Euler explicit discretization method and sampling time equal to 0.001. Based on some simulations, one can show that Assumption 3 holds with max = 0.008, min = 0.001 and ρ = 0.9. Taken all the corresponding values, one can show that and thus, the conditions of Theorem 2 are satisfied. The simulation results for different values of α ∈ [0, 1) are depicted by Figs.2, 3 and 4. It is clear that due to the disturbances, the corresponding estimations converge to a neighborhood of the real value but faster than the linear algorithm, i.e. when α = 1.

						  (pκ δ )	2 α+1 C 2 (2ε + σ 2 max )λ min (P )r 1 min (1 + α) α+1 λmax(P )ρα δ σ 2 2α	 	α=0	=0.619,
						  (pκ δ )	2 α+1 C 2 (2ε + σ 2 max )λ min (P )r 1 min (1 + α) α+1 λmax(P )ρα δ σ 2 2α	 	α=0.5	=0.169,
						  (pκ δ )	2 α+1 C 2 (2ε + σ 2 max )λ min (P )r 1 min (1 + α) α+1 λmax(P )ρα δ σ 2 2α	 	α=1	=0.038,
	14.3132 -0.2447	0.1127	0.0013	
	-0.2447 21.0065 -8.7811 -0.1673 0.1127 -8.7811 12.1135 -0.3947   ,
	0.0013		-0.1673 -0.3947	0.0395
			30.2427	0.4684	
	L =	  -25.0825 -0.3414	93.6232 127.9623	  ,
			18.4329	1412.0191
	F = 293.5014 322.8523 , β = 0.6324, ε = 1023.4,
			σ = 281.3023.


B N Nichols

D Keator

S Ghosh

G Flandin

T Nichols

K J Gorgolewski

Y O Halchenko

M Hanke

C Haselgrove

K G Helmer

D Marcus

R Poldrack

J Turner

D Kennedy

J B Poline

K M Pohl

Application of the Neuroimaging Data Model to Represent and Exchange Primary and Derived Data

Introduction

Access to the provenance of primary and derived data is increasingly recognized as an essential aspect of reproducibility in biomedical research [START_REF] Collins | Policy: NIH plans to enhance reproducibility[END_REF]. While data sharing is the norm in some biomedical communities, neuroimaging has lagged in open data and provenance reporting. The overarching goal of this effort is to address neuroimaging data sharing barriers [START_REF] Poline | Data sharing in neuroimaging research[END_REF] and continue developing a fundamentally new, granular data exchange standard, called the Neuroimaging Data Model (NIDM), which incorporates provenance as a primitive to document cognitive neuroimaging workflow [START_REF] Keator | Towards structured sharing of raw and derived neuroimaging data across existing resources[END_REF].

NIDM enhances W3C PROV [START_REF] Moreau | Provenance: An Introduction to PROV[END_REF] to provide a language that communicates provenance by representing primary data, workflow, and derived data as linked Agent, Activity, and Entity objects. Similar to the way a sentence conveys a standalone thought, NIDM enables provenance statements that express the way a given piece of data was produced. This abstract presents our preliminary work to represent primary and derived experimental results with three uses cases that aim to 1) automate OpenfMRI dataset analyses, 2) integrate multi-modal measurements of the National Consortium on Alcohol & Neurodevelopment in Adolescence (NCANDA), and 3) capture derived data from a cross species Conte center project.

Methods

NIDM is developed using a community-driven process that engages stakeholders to participate in the identification of use-cases that drive development. In-person workshops and weekly video conferences are used to maintain communication while example NIDM documents, specifications, and software are developed by the INCF NIDASH task force members.

NIDM Experiment captures details about an investigation using the modeling pattern depicted in Figure 1. This pattern models metadata from neuroimaging data management systems at three levels from XCEDE [START_REF] Gadde | XCEDE: an extensible schema for biomedical data[END_REF]: Project, Study, and Acquisition. The Project level captures administrative information, while the Study and Acquisition levels model MRI sessions and neuropsychological (NP) tests.

Results

Use Case 1: OpenfMRI Database

• Task-based fMRI datasets with standardized organization and metadata • NIDM Experiment models were developed using an example dataset [START_REF] Hanke | A high-resolution 7-Tesla fMRI dataset from complex natural stimulation with an audio movie[END_REF] to capture demographics, primary data, and metadata (e.g., DICOM, Cognitive Atlas [7] terms) • Models were designed to enable automated analyses Use Case 2: NCANDA Data Integration

• A multi-site study investigating the effect of alcohol use on the structure and function of developing adolescent brains • The system architecture developed by the data analysis core of the consortium integrates clinical, NP tests, and survey data [START_REF] Rohlfing | N-CANDA data integration: anatomy of an asynchronous infrastructure for multi-site, multi-instrument longitudinal data capture[END_REF] • To enable reporting across these systems, NIDM models were developed to facilitate report generation within the consortium and enhance the metadata available during future data releases Use Case 3: CONTE Center Cross-Species Modeling • The Conte center at UC Irvine is studying the effect of early life fragmented maternal care on cognitive vulnerabilities in adolescence across species • The overall informatics goal is to provide center investigators with inter-project data, including models for maternal and fetal heart rate, fetal movement, maternal blood oxygenations levels, derived DTI and fMRI measurements, and brain connectivity graphs • A web application was developed to query and visualize data from NIDM object model graphs

Conclusions

NIDM Experiment supports a variety of use cases focused on representing primary and derived data organization with the intent of simplifying data exchange, integration, and sharing. These preliminary results indicate the flexibility of a community-driven data modeling approach and that provenance-based data exchange is being incorporated into a number of projects.

Figure 1 .

 1 Figure 1. The NIDM Experiment core structure that is extended to model Project, Study, and Acquisition information.

Figure 2 .

 2 Figure 2. Example of an abbreviated NIDM Experiment Acquisition-level model to capture demographic information and a T1-weighted anatomical MRI scan.

Acknowledgments

We acknowledge the work of all INCF task force members and many other colleagues who have helped in this effort. Further we acknowledge the BIRN coordinating center's (U24RR025736-01) long-standing support and the support by the NCANDA Consortium (U01AA021697-01).


Mohammad Y Abualhoul

Edgar Talavera Munoz

Fawzi Nashashibi
email: fawzi.nashashibi@inria.fr

The Use of Lane-Centering to Ensure the Visible Light Communication Connectivity for a Platoon of Autonomous Vehicles

Keywords: VLC, Lane-Centering, Platoon, ITS, Road safety applications

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

I. INTRODUCTION Nowadays, Intelligent Transportation Systems (ITS)

promise to provide safer and more secure autonomous vehicular mobility by combining and utilizing vehicular sensors, control systems, different sorts of vehicular communication methods, and even specific driving formations. Moreover, by enabling the Cooperative ITS (C-ITS) mode, the cooperation between autonomous vehicles can provide the necessary remote and local data to perceive the vehicle's surrounding environment and raise the independent vehicles awareness. The proper information exchange between vehicles in convoybased applications can further improve both road traffic safety and string stability, especially when deploying fully autonomous applications [START_REF] Tsugawa | Inter-vehicle communications and their applications to intelligent vehicles: an overview[END_REF], [START_REF] Abualhoul | Visible Light and Radio Communication for Cooperative Autonomous Driving: applied to vehicle convoy[END_REF].

Due to the vast deployment of C-ITS services for vehicular applications, well-known wireless technologies such as the Dedicated Short-Range (DSR) communications for ITS [START_REF] Jiang | Ieee 802.11p: Towards an international standard for wireless access in vehicular environments[END_REF], [START_REF] Anaya | Vulnerable road users detection using v2x communications[END_REF] are extensively used. These standardized vehicular communication technologies, including the widely deployed IEEE 802.11p, are pushed lately toward an unstable performance for wireless networks data access, with a remarkable increase in both latency and channel congestion levels. This instability introduces more usage constraints for hard-safety ITS applications, for instance, the convoy-based applications [START_REF] Abualhoul | Visible Light and Radio Communication for Cooperative Autonomous Driving: applied to vehicle convoy[END_REF], [START_REF] Wang | Reliability evaluation of ieee 802.11p-based vehicle-to-vehicle communication in an urban expressway[END_REF].

New communication solutions using an independent medium are being proposed for ITS applications, such as VLC, which was recently proved to be a suitable supportive solution for radio-based communication in vehicular environments. In addition to the many advantages of deploying optical-based communication solutions, VLC can realize both low-cost implementation and dual functionality (communication signaling and lightning) [START_REF] Abualhoul | Visible Light and Radio Communication for Cooperative Autonomous Driving: applied to vehicle convoy[END_REF], [START_REF] Abualhoul | Platooning control using visible light communications: A feasibility study[END_REF], [START_REF] Béchadergue | Experimental comparison of pulse-amplitude and spatial modulations for vehicleto-vehicle visible light communication in platoon configurations[END_REF]. However, the ambient noise (signal interference and optical noise) and the Field-of-View (FOV) (mobility issues) limitations are two main challenges of any VLC deployment for ITS applications. Various solutions have been proposed to deal with such constraints [START_REF] Abualhoul | Enhancing the field of view limitation of visible light communication-based platoon[END_REF], [START_REF] Cilean | Current challenges for visible light communications usage in vehicle applications: A survey[END_REF]. Nevertheless, the Field-of-View (FOV) limitation, which is expressed as the maximum angular size of the Photo-diode (PD), remains as the primary mobility challenge when VLC-based platoon members face slight changes in driving course, or lane changing.

On the other hand, the lane detection technique has always been considered as a crucial cost-efficient and reliable enabler for many driving-assistance autonomous systems [START_REF] Wang | Lane detection and tracking using b-snake[END_REF]. The concept of Lane-Centering (LC) is simplistic and depends on performing real-time processing of the lane detection data and restoring the autonomous vehicle's position by locating itself referring to the side lanes (no cooperation is required). Such a technique can be beneficial in tunnels scenarios or when the alignment of convoy-based vehicles is essential. In this paper, we propose the use of the LC technique to enhance the VLC inter-connectivity between the members of a platoon of autonomous vehicles, as depicted in Fig. 1. By combining both solutions (VLC and LC), the dependency on exchanging vehicular status and GPS positions will not be essential to relocate platoon members and ensure direct Lineof-Sight (LoS). Therefore, the main focus of our proposal is on preserving the alignment between platoon members with the minimum possible error and without requiring any information exchange between convoy members. Such a deployment is expected to allow the autonomous platoon members to return to the aligned status even during the VLC disturbance, which is likely to occur. In this paper, we make the following contributions:

• Suggesting a new architecture to enhance the platoon performance by utilizing both LC and VLC. • Defining the limitations and performance of both technologies.

• Evaluating the overall system performance. The remainder of this paper is organized as follows. Section II presents related work and a review of the literature. The over-all system proposal for platooning application enhancing VLC connectivity using LC is further explained in Section III. We report on the system's performance and our findings in Section IV. Finally, the conclusion and future work are given in Section V.

II. RELATED WORK

The convoy-based autonomous vehicles formation concept has been widely investigated and proposed to improve traffic fluidity by increasing road capacity while reducing fossil fuel consumption and CO 2 levels. The leading PATH project in California [START_REF] Shladover | Path at 20 -history and major milestones[END_REF] and PRAXITELE project in France [START_REF] Daviet | Longitudinal and lateral servoing of vehicles in a platoon[END_REF] were the pioneering projects demonstrating the platoon application. Other projects such as Auto21 CDS [START_REF] Hall | Car platoons simulated as a multiagent system[END_REF] are dedicated to advancing smooth merging/leaving for a platooning-enabled system. Also, the SARTRE project [START_REF] Tom Robinson | Operating platoons on public motorways: An introduction to the sartre platooning programme[END_REF] has successfully demonstrated the fully autonomous platoon.

At the vehicular control level, the most straightforward requirement to form a platoon is usually the longitudinal control, as described in [START_REF] Oncu | Cooperative adaptive cruise control: Network-aware analysis of string stability[END_REF], [START_REF] Milanés | Cooperative adaptive cruise control in real traffic situations[END_REF]. Simple approaches such as Adaptive Cruise Control (ACC) with a constant time headway policy has been a universal solution to ensure a safe inter-distance between platoon members [START_REF] Milanés | Cooperative adaptive cruise control in real traffic situations[END_REF].

Both conceptual and experimental studies on the usecase of VLC for outdoor applications have been conducted and presented for convoy-based ITS applications [START_REF] Cailean | Visible light communications: Application to cooperation between vehicles and road infrastructures[END_REF][START_REF] Abualhoul | Visible light intervehicle communication for platooning of autonomous vehicles[END_REF][START_REF] Masini | Vehicular visible light networks with full duplex communications[END_REF]. The studies and system evaluation have tackled the main technology limitations and use-cases showing that VLC can satisfy the stringent reachability requirements for Inter-Vehicle Communication (IVC) in dense vehicle traffic conditions [START_REF] Demir | Cooperative adaptive cruise control using visible light communication[END_REF]. We at INRIA have also implemented our VLC prototype to evaluate and test the performance of VLC technology for platoon applications and examining real outdoor conditions [START_REF] Abualhoul | Study and evaluation of laser-based perception and light communication for a platoon of autonomous vehicles[END_REF].

Our previously conducted study [START_REF] Abualhoul | Study and evaluation of laser-based perception and light communication for a platoon of autonomous vehicles[END_REF] dedicated to evaluating the deployment of Laser-Range-Finder (LRF) and VLC for the same platoon application. The evaluation and system implementation were intended to provide a reliable backup system to switch the autonomous driving modes between ACC and Cooperative-ACC (C-ACC) to even making an emergency stop when failure scenario is detected.

Moreover, laser detection use-cases and deployment to improve vehicle awareness using different approaches, including the use of laser detection to implement obstacle detection applications, remains costly solution and requires a high computational process compared to Camera-based vision detection [START_REF] Guo | A novel obstacle detection method based on distortion of laser pattern[END_REF].

By using vision-detection, lane detection approach can offer either alternative or supportive solution to the laserbased-sensor systems compared with the previous proposal that uses Laser Range Finder (LRF) alongside the VLC in [START_REF] Abualhoul | Study and evaluation of laser-based perception and light communication for a platoon of autonomous vehicles[END_REF]. Further challenges to deploying vision-based lane detection for autonomous driving have also been discussed in the DARPA project [START_REF] Behringer | The darpa grand challengedevelopment of an autonomous vehicle[END_REF].

Further contributions such as Wang in [START_REF] Wang | Lane detection and tracking using b-snake[END_REF] propose the use of a B-Snake algorithm to improve lane detection models. Others, such as Hafner in [START_REF] Hafner | Localization tools for benchmarking adas control systems[END_REF] focus on the limitations of lane detection deployment for ITS and provide interesting mathematical solutions to overcome some computational and decision barriers. Lee has also presented a unique lane centering and control algorithm based on future predictions for autonomous driving maneuvers [START_REF] Lee | A unified framework of the automated lane centering/changing control for motion smoothness adaptation[END_REF].

III. PLATOON CONTROL USING BOTH VLC AND LANE

CENTERING

In this section, we first present the classic kinematic longitudinal and lateral control model that governs the motion of an autonomous vehicle, and which was used to drive platoons in our previous studies [START_REF] Abualhoul | Platooning control using visible light communications: A feasibility study[END_REF], [START_REF] Abualhoul | Study and evaluation of laser-based perception and light communication for a platoon of autonomous vehicles[END_REF]. Afterward, we present the proposed LC control mechanism to apply longitudinalcorrection controller based on vision-based lane detection.

A. Communication-based Platoon Control Model

For simplicity, the deployed vehicle kinematics model was an approximation of the Ackerman steering mechanism (bicycle model). The bicycle model is defined by three main parameters, δ , l, and φ , which respectively represents the steering angle, wheelbase, and the orientation of the vehicle. Therefore, the kinematic model which governs the platoon members vehicle's motion was detailed in [START_REF] Abualhoul | Platooning control using visible light communications: A feasibility study[END_REF] using a constant steering angle δ , with a circular vehicle's trajectory and a radius R such as:

         l = R • tan(δ ) φ = atan ∆y ∆x -θ ∆x = X L -X F . ∆y = Y L -Y F (1)
where the steering angle δ is given by:

δ = atan 2 • l • sin(φ ) ∆x 2 + ∆y 2 . ( 2
)
Figure 2 depicts the general horizontal plan of two vehicles forming a platoon. The longitudinal and lateral distances ∆x and ∆y represent the projection of the inter-distance d. To estimate the value d, the controller must be aware and updated about (∆x, ∆y, δ , φ ) in real-time, which will, therefore, allow proper longitudinal and lateral control corrections with some margin of error. The GPS location, vehicle orientation, and vehicle's relative speed are passed over one-hop to every preceding member in platoon chain (from preceding to follower vehicle). Such requirements make it essential to have continuous and reliable communication between platoon members, and any disconnection or even data exchange latency can result in hazardous situations with potentially dangerous driving consequences.

B. Lane Centering Model

The LC concept can be expressed as a lateral control adjustment forcing the autonomous vehicle to center its position by continuously referring to the side lanes, and without the need for any further cooperative information.

Such lateral correction can be highly useful for convoybased applications. For platooning formations, ensuring members' centering can even remarkably improve the overall system string stability [START_REF] Naranjo | Lane-change fuzzy control in autonomous vehicles for the overtaking maneuver[END_REF]. Furthermore, it becomes even more interesting when an optical-based communication system (VLC) is used to exchange information between platoon members. This approach allows every individual vehicle in convoy formation to locally change its behavior (orientation) to ensure a fixed position all along the driving path, even when communication is interrupted. Estimating the lateral error (ε) as defined by Equation 3is the first step in executing correction, where C position , represents the reference point (lane-center) obtained directly from the artificial vision sensor (MOBIL-EYE Camera in our validation [28]).

ε = Dright -Cposition , i f Cposition≤ 0 Dleft -Cposition , otherwise (3)
To ensure proper autonomous driving orientation to position the vehicle on the lane-center; we use Equation 4to calculate the exact displacement projection using the vehicle local GPS coordinations (x, y), where δ x and δ y represent the error in both x and y coordinates.

δ x = cos(θ ) * ε * σ north δ y = sin(θ ) * ε * σ east (4)
where the calculation of θ depends on (ε) direction and the azimuth angle ( α) as follows:

For (ε < 0): θ =          α, 0 • ≤ α < 90 • α -90, 90 • ≤ α < 180 • α -180, 180 • ≤ α < 270 • α -270, otherwise (5)
For (ε ≥ 0):

θ =          90 -α, 0 • ≤ α < 90 • 180 -α, 90 • ≤ α < 180 • 270 -α, 180 • ≤ α < 270 • 360 -α, otherwise (6)
As orientation adjustments are required to apply the right correction (front vehicle direction), both the azimuth angle ( α) and the correction direction (σ ) are calculated as follows:

σ north = -1, i f 270 > α ≥ 90 +1, otherwise (7)
σ east = -1, i f α ≤ 180 +1, otherwise (8)
The azimuth angle ( α) is a crucial element to calculate all necessary lateral control corrections. An accurate estimation of α can dramatically improve overall system performance. Therefore, a higher resolution ( 4-point method) shown in Fig. 4 is applied to stabilize the estimated values further and avoid any GPS errors compared to the conventional 2-point method.

Instead of using only 2 GPS points, we use 4 points to successfully calculate 3 full consecutive azimuth angle updates and apply a mean to the result after distributing empirically optimized weights using the following formula. The platooning lane detection evaluations have used Mobileye [28] with the detailed parameters in Table I. The Mobileye refresh rate was tuned to limit vision to ~4m to avoid any conflict with other platoon members or road users. A further sample image of the real-time detection is depicted in Fig. 5.

C. Visible Light Communication Model

The definition of the VLC channel characteristics is strongly affected by several physical parameters such as the LED Lambertian emission pattern and both transmitter and receiver orientation. From Fig. 6, the channel path-loss can be presented as:

P loss = (m+1)A ph 2πd n cos m (ϕ)cos (ψ) , 0 < ψ, ϕ < θ c 0 elsewhere , (10)
where θ c is the maximum incidence and irradiance angles max(ψ, ϕ). A ph is the physical area of the PD and m = -ln 2 ln(cos φ ) is the Lambertian emission order, which is a key parameter specifying the directivity of the transmitter [START_REF] Abualhoul | Platooning control using visible light communications: A feasibility study[END_REF]. As mentioned earlier, we conducted an experimental system evaluation using a VLC prototype for platoon applications in [START_REF] Abualhoul | Visible light intervehicle communication for platooning of autonomous vehicles[END_REF]. The evaluation confirmed the inverse square proportional relationship between inter-vehicle distance d and the received power for a link range of up to 30 meters in outdoor conditions.

The path-loss exponent n found to be 2 for various weather situations and also sensitive to any misalignment which is the exact reason to deploy assisting control application such as the LC in this work. For more configuration details on both the physical and packet level see Table II.

IV. SYSTEM EVALUATION

The performance of the proposed platoon system deploying both VLC and LC is evaluated by independently defining the following:

• The 90% of the received power defining the lateral area at the receiver side over different inter-vehicle distances, which represents the reliable performance of the VLC link with better immunity against ambient noise. • The lateral control error before and after using the LC application, and how accurately the system can reposition the platoon members on the lane center. To validate and test the proposed LC system limitations, two vehicles (Mitsubishi IMIEV) where equipped with an LC (using Mobileye camera). For initial positioning, one vehicle was located at the far left edge of the lane, while the second was located on the far right side of the same lane, as depicted Fig. 6.

The experimentally acquired and processed data are of two vehicles driven in a platoon mode. Figure 7 represents sample results of the second vehicle's deviation variation from lanecenter. The deviation variation over the time interval (0 to 400 seconds) represents the case when the LC system is inactive; this can explain the remarkable misalignment which can reach up to 1-meter lateral error over 10 meters of intervehicle distance, which is outside the VLC lateral coverage limitations for reliable connection as shown in Fig. 8.

On the other hand, over the time interval (400 to 100 seconds), the deviation is reduced to less than 20 cm on average just after the LC is activated. To further understand how the activation of the LC system can ensure the VLC connectivity, we provide empirical results of the VLC link range tests using an actual VLC prototype. Figure 8 map the VLC link optical beam spot (spot diameter represent the lateral covering area) over different inter-vehicle distances. In Fig. 8, and for a vehicle distance of 10 meters, the misalignment is limited to 90 cm from lane center to guarantee a detectable signal level with an adequately received signal over noise ratio (SNR). Other scenarios were experimentally validated as depicted in Fig. 9, which represents the same second platoon member's deviation data but for an autonomous driving path of 60 km as depicted in Fig. 10. The obtained data are part of collaborative work for AUTOCC-ITS project for urban nodes interoperability tests [29]. The second driving path was meant to evaluate the LC system with the introduction of different detection confusion situations, where lanes-marking could be deficient, disconnected, or covered by on-road materials. The results depicted in Fig. 9 record the spikes 0 and 1, which represent a confusion when the camera detects an adjacent multi-lane scenario due the relatively long intervehicle distance. Spikes 0 and 1 are still tolerable to achieve a reliable VLC communication over 10 meters of intervehicle distance (see Fig. 8). Other spikes (2,3,4) can exceed the tolerable VLC lateral error (1 meter) which was the direct result of a temporary lane-marking deficiency along the driving path.

V. CONCLUSION AND FUTURE WORK

In this paper, we studied and evaluated the deployment of the LC application to enhance the (VLC) optical-based communication connectivity for a platoon of autonomous vehicles. The results obtained confirmed the reduction in the misalignment between platoon members and ensured the VLC link stability independently of any cooperation requirements when the road lanes are detectable. Other detection confusion or temporary lane-marking deficiency scenarios can be tolerated and might introduce a temporary VLC interruption. Compared to communication-based platoon and LRF -VLC systems, we believe that the presented approach of applying the correction locally by using LC makes the system more resilient and able to rapidly recover any communication interruptions. However, the presented work in this paper is an experiment-based investigation meant for concept validation. We conclude that the study needs to further extend the investigation cases by including different trajectory scenarios, weather conditions, and more accurate platoon controllers.

Fig. 1 .

 1 Fig. 1. Lane Detection and VLC down-link for platooning application.

øFig. 2 .

 2 Fig. 2.Plan view of two vehicles platoon Leader (Proceeding) and Follower, which have the longitudinal (∆x) and lateral (∆y) displacements, inter-vehicle distance (d).

Fig. 3 .

 3 Fig. 3. System overview for Lane-Centering mechanism depicting the lateral error left and right values (D le f t , D right ) obtained by the artificial vision system.

Figure 3

 3 Figure 3 illustrates the LC over-all detection mechanism and the applied lateral correction to compensate for the misalignment and reposition the desired vehicle (red vehicle) onto its desired location (lane center). The location error of the red vehicle represents the current position in UTM (Universal Transverse Mercator) coordinates x and y; where both D le f t and D right represent the distance from the vehicle center point to the edges of the lanes. By estimating the values of D le f t and D right , and comparing them with the local vehicle position, the vehicle will have an initial awareness of the lateral error ε and perform

Fig. 4 .

 4 Fig. 4. The 4-point Azimuth angle calculation.

Fig. 5 .

 5 Fig. 5. A sample of the real-time artificial vision camera with detected objects for autonomous driving using Mobileye.

Fig. 6 .

 6 Fig. 6. Two-vehicle VLC-based platoon, with an inter-vehicle distance d showing the incidence and irradiance angles (ψ, ϕ) used to calculate the path-loss.

Fig. 7 .

 7 Fig. 7. Data Sample of a platoon member deviation from lane-center before and after activating the LC application.

Fig. 8 .

 8 Fig.8. The three main zones where the VLC receiver was placed to evaluate the received signal level. The experiment was conducted with respect to variations in direct distance changes and out-door conditions.

Fig. 9 .

 9 Fig. 9. Real-time lateral error sample data obtained during the 60km tour on the M40 highway-Madrid. The data demonstrate the undetectable lane marks on the road, cases 2, 3, and 4.

Fig. 10 .

 10 Fig. 10. 60 km of Lane centering (LC) driving route on the M40 highway Madrid.

  VI. ACKNOWLEDGEMENT This work has been supported by the European project AUTOC-ITS 2015-EU-TM-0243-S, where the obtained results are part of collaborative work between UPM-Madrid and INRIA-Paris during the urban nodes autonomous driving and interoperability validations tests.

TABLE I LANE

 I CENTERING CAMERA (MOBIL-EYE) PARAMETERS

	Artificial Vision Camera Parameter	Value
	Model	Mobileye 5 Series
	Transmission data Rate	10 Hz
	Focus range	4m to infinity
	Angle of view	35
	Pixel size	6.0µm x 6.0µm