Boris Andreianov
email: boris.andreianov@univ-tours.fr

Massimiliano D Rosini

Microscopic selection of solutions to scalar conservation laws with discontinuous flux in the context of vehicular traffic

Keywords: Conservation laws, traffic flow, discontinuous flux, Riemann solvers, first order follow-the-leader model, order-preservation, point constraint

In the context of road traffic modeling we consider a scalar hyperbolic conservation law with the flux (fundamental diagram) which is discontinuous at x = 0, featuring variable velocity limitation. The flow maximization criterion for selection of a unique admissible weak solution is generally admitted in the literature, however justification for its use can be traced back to the irrelevant vanishing viscosity approximation. We seek to assess the use of this criterion on the basis of modeling proper to the traffic context. We start from a first order microscopic follow-the-leader (FTL) model deduced from basic interaction rules between cars. We run numerical simulations of FTL model with large number of agents on truncated Riemann data, and observe convergence to the flow-maximizing Riemann solver. As an obstacle towards rigorous convergence analysis, we point out the lack of order-preservation of the FTL semigroup.

Introduction

Discontinuous-flux scalar conservation law

The main object of study in this paper is the first order macroscopic model for vehicular traffic

∂ t ρ + ∂ x f (ρ, x) = 0, f (ρ, x) . = f -(ρ) if x < 0, f + (ρ) if x 0, (1)
where ρ = ρ(t, x) ∈ [0, 1] is the (normalized) density at time t 0 and position x ∈ R, while f -0 and f + 0 are the fundamental diagrams (fluxes) corresponding to the two sections x < 0 and x > 0 of the road separated by x = 0, called "interface" in the sequel. We assume that f -and f + are convex and satisfy the matching conditions

f -(0) = f + (0), f -(1) = f + (1).
Our concern is with admissibility of solution discontinuities at the interface location x = 0, which is a question asked also for other models of vehicular traffic with point singularities, such as [START_REF] Benyahia | A macroscopic traffic model with phase transitions and local point constraints on the flow[END_REF][START_REF] Garavello | The Aw-Rascle traffic model with locally constrained flow[END_REF][START_REF] Villa | Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model[END_REF]. Our paper shares this primary motivation with the recent work [START_REF] Shen | Traveling wave profiles for a follow-the-leader model for traffic flow with rough road condition[END_REF]. The author of [START_REF] Shen | Traveling wave profiles for a follow-the-leader model for traffic flow with rough road condition[END_REF] studies admissible jumps at the flux discontinuity location by analyzing (both analytically and numerically) stationary wave profiles. We pursue (numerically only, at the present stage) the complementary line of investigation, by looking at solutions of the Riemann problems and interpreting our results in terms of global properties of the flow (we put forward the flux maximization principle). Equation ( 1) is a scalar hyperbolic conservation law in one space dimension. It arises in road traffic but also in many other application problems, such as two-phase flows through porous media [START_REF] Andreianov | Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks' medium[END_REF][START_REF] Gimse | Solution of the Cauchy problem for a conservation law with a discontinuous flux function[END_REF], continuous sedimentation in a clarifier-thickener unit [START_REF] Bürger | Monotone difference approximations for the simulation of clarifier-thickener units[END_REF][START_REF] Diehl | Continuous sedimentation of multi-component particles[END_REF] or ion etching in the semiconductor industry [START_REF] Ross | Two new moving boundary problems for scalar conservation laws[END_REF].

The derivatives in [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF] are interpreted in the weak sense. Indeed, even for smooth data classical solutions may not exist globally in time since discontinuities can arise in finite time. Yet weak solutions are in general not unique. This motivates the introduction of additional admissibility criteria able to single out a unique solution. In principle, these conditions depend on the physical phenomena under consideration. However it turns out that in the setting of continuous flux (the case f -≡ f + in (1)) the same notion of entropy admissible solution ( [START_REF] Kruzhkov | First order quasilinear equations with several independent variables[END_REF], see also [START_REF] Lax | Shock waves and entropy[END_REF][START_REF] Oleȋnik | Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation[END_REF][START_REF]Spaces BV and quasilinear equations[END_REF]) is relevant for all traditional applications. In different fluid mechanics applications the widely accepted justification for the choice of this solution notion is based upon the vanishing viscosity method that can be traced back as far as [START_REF] Hopf | The partial differential equation ut + uux = µuxx[END_REF][START_REF] Rayleigh | Aerial plane waves of finite amplitude[END_REF]. The use of the same solution notion in the road traffic (the Lighthill-Whitham-Richards model [START_REF] Lighthill | On kinematic waves. II. A theory of traffic flow on long crowded roads[END_REF][START_REF] Richards | Shock waves on the highway[END_REF], LWR model in the sequel) is rather based upon the coincidence of admissible discontinuities: those observed in the traffic context turn out to be the admissible jumps in the entropy sense [START_REF] Kruzhkov | First order quasilinear equations with several independent variables[END_REF][START_REF] Oleȋnik | Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation[END_REF].

The situation is drastically different in the discontinuous-flux setting of [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF]. Coexistence of infinitely many admissibility criteria -equally consistent from the pure mathematical standpoint -has been discovered in [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF] and further developed e.g. in [START_REF] Andreianov | Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks' medium[END_REF][START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF][START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF][START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF][START_REF] Garavello | Conservation laws with discontinuous flux[END_REF]. To be precise, there exist infinitely many L 1 -contractive semigroups of solutions to the formal problem (1), moreover, different semigroups correspond to different modeling contexts or different model parameters. It is highlighted in the recent survey paper [START_REF] Andreianov | New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux[END_REF] that choosing admissibility condition for (1) at the interface x = 0 means prescribing an interface coupling condition; this is very much similar to prescribing different boundary conditions (Dirichlet, Robin, Neumann, obstacle, etc.) for classical elliptic or parabolic PDEs. As a matter of fact, interface coupling conditions are part of the model and have to be prescribed independently from the differential relations applied in open space-time domains. E.g. in the context of flows in "two-rocks" porous media, the physically relevant coupling depends on additional nonlinearities φ -, φ + accounting for capillary pressures that are not even present in the formal PDE formulation [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF], see [START_REF] Andreianov | Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks' medium[END_REF]. Thus, considering (1) without precise indication of the desired interface coupling means working with an incomplete model, similar to speaking of "Laplacian operator on a bounded domain" without specific boundary conditions!

In the context of hyperbolic models of road traffic, many nontrivial models are postulated by prescribing Riemann solvers. The choice of admissibility conditions at x = 0, i.e. the prescription of precise interface coupling conditions, is equivalent to the choice of a Riemann solver at the interface location [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF][START_REF] Garavello | Conservation laws with discontinuous flux[END_REF]. Making this choice is not always straightforward. In the traffic context, the lack of an established method to single out physically reasonable discontinuities leads to the introduction of different notions of solutions for the same problem (sometimes in the same paper!), see for instance [START_REF] Benyahia | A macroscopic traffic model with phase transitions and local point constraints on the flow[END_REF][START_REF] Garavello | The Aw-Rascle traffic model with locally constrained flow[END_REF][START_REF] Villa | Moving bottlenecks for the Aw-Rascle-Zhang traffic flow model[END_REF]. In [START_REF] Garavello | Conservation laws with discontinuous flux[END_REF], it is even stated that "there is no a priori preferable physical solution". In this respect, let us stress that the vanishing viscosity (or, more generally, vanishing capillarity, cf. [START_REF] Andreianov | Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks' medium[END_REF]) admissibility criterion is successful in fluid dynamics applications, but it is not recognized as instrumental in the context of traffic modeling.

The flow maximization principle, models with a point constraint on the flux and their justification

There is a number of works on trafic modeling where the jump admissibility criterion is taken according to the principle of flow maximization: we refer, for instance, to [START_REF] Bürger | On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux[END_REF][START_REF] Colombo | Phase transition model for traffic at a junction[END_REF][START_REF] Delle Monache | Priority-based Riemann solver for traffic flow on networks[END_REF][START_REF] Kolb | Pareto-optimal coupling conditions for the Aw-Rascle-Zhang traffic flow model at junctions[END_REF][START_REF] Moutari | Modelling road traffic accidents using macroscopic second-order models of traffic flow[END_REF]. It also appeared in modeling of flows in porous media [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF][START_REF] Andreianov | Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks' medium[END_REF][START_REF] Cancès | Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution[END_REF][START_REF] Kaasschieter | Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium[END_REF], the resulting solution being referred to as the "optimal entropy solution" and justified via the vanishing capillarity approach in specific flux configurations. The flow maximization principle is appealing but not always justified; as a matter of fact, quite often it is implicitly assumed. This naturally occurs when one takes a successful -e.g., having a well-developed well-posedness theorysolution notion from the related literature, even if it may actually address a quite different application context. A further motivation stems from the fact that one and the same notion of solution was recognized as the correct one for very different applications, at least in the classical continuous-flux case of (1). However, we have seen that somewhat unexpectedly, the discontinuous-flux setting reveals much more diversity. From the accurate modeling perspective, in traffic applications the flow maximization should result as a collective behavior emerging from individual-driver behavior. However, flow maximization does not always occur in practice, as it is evident in the capacity drop phenomenon at crossroads [START_REF] Srivastava | Empirical observations of capacity drop in freeway merges with ramp control and integration in a first-order model[END_REF] and in the Braess paradox phenomenon [START_REF] Braess | Über ein paradoxon aus der verkehrsplanung[END_REF]. In particular, it was demonstrated in [START_REF] Andreianov | Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks[END_REF] that the capacity drop and the Braess paradox can be reproduced starting from the concept of flux limitation (as opposed to the flow maximization) introduced in [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF][START_REF] Colombo | Pedestrian flows and non-classical shocks[END_REF].

Let us briefly focus on the flux limitation model of [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF]. It formally takes the form (1) with continuous flux (i.e., f -= f + . = f ) and with explicit (though formal) interface condition f (ρ(t, 0 ± )) F (t) for some map F given beforehand. This model is introduced essentially in terms of the intuitively appealing Riemann solver at the interface x = 0; but the authors of [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] also provide a global entropy formulation and deduce it from the asymptotic analysis involving a discontinuous-flux conservation law similar to (1), but with two flux discontinuities situated at x = ±ε, with ε → 0 + . Doing so, they rely upon the global entropy formulation for (1) appearing in the influential works for the case where the graphs f -, f + can be ordered, such as [START_REF] Karlsen | L 1 stability for entropy solutions of nonlinear degenerate parabolic convectiondiffusion equations with discontinuous coefficients[END_REF][START_REF] Seguin | Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients[END_REF][START_REF] Towers | Convergence of a difference scheme for conservation laws with a discontinuous flux[END_REF]. However, the admissibility formulation in use in [START_REF] Karlsen | L 1 stability for entropy solutions of nonlinear degenerate parabolic convectiondiffusion equations with discontinuous coefficients[END_REF][START_REF] Seguin | Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients[END_REF][START_REF] Towers | Convergence of a difference scheme for conservation laws with a discontinuous flux[END_REF] can be traced back to -or connected to, see e.g. [START_REF] Andreianov | Entropy conditions for scalar conservation laws with discontinuous flux revisited[END_REF][START_REF] Diehl | A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients[END_REF] -the vanishing viscosity approximation, not suitable for road traffic applications. Another source of justification is the "minimal jump" condition of [START_REF] Gimse | Solution of the Cauchy problem for a conservation law with a discontinuous flux function[END_REF], which relies on a mathematical rather than on a modeling selection principle. As a matter of fact, a byproduct of our analysis is a more realistic justification of the model introduced in [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF].

We point out that at the present stage our justification is not rigorous because our conclusions are based only upon accurate modeling and numerical experiments lacking, at the present stage, the full convergence proof.

1.3 Microscopic-to-macroscopic limit in traffic modeling. The purpose and the contents of the paper Our main goal is to propose a method to select an appropriate solution. We rely on the paradigm of passage to the limit from a well-assessed microscopic model to a macroscopic model, similar to taking the hydrodynamical limit of Boltzmann equations. For the LWR traffic model, such passage to the limit was fully realized for the first time in [START_REF] Di Francesco | Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit[END_REF]. The main advantage of this approach is that it requires to set assumptions on the interacting behavior of the cars only at the microscopic level. Since traffic dynamics are essentially microscopic, it is much easier to physically motivate the microscopic rather than some macroscopic assumptions. Thus, we first encode the interacting rules in a (first order) microscopic follow-the-leader (FTL) approximation of (1). We then obtain a macroscopic Riemann solver as many particle limit by applying, at the level of numerical simulations and for carefully identified sets of data, the approximation procedure adapted from [START_REF] Di Francesco | Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit[END_REF]. It turns out that from this procedure, we indeed observe the emergence of the Riemann solver which maximizes the flow at x = 0. Let us stress that we do not require this property at the microscopic level, as we only prescribe elementary car interaction rules. Therefore our results further assess the relevance of the flow maximization admissibility criterion, at least in the present setting of discontinuous-flux LWR model [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF], in the related LWR model with point constraint on the flow and its variants [START_REF] Andreianov | Analysis and approximation of one-dimensional scalar conservation laws with general point constraints on the flux[END_REF][START_REF] Andreianov | Crowd dynamics and conservation laws with nonlocal constraints and capacity drop[END_REF][START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF]. A closely related investigation of the FTL approximation of the discontinuous-flux LWR model, with a focus on emerging standingwave profiles (which is a way of understanding admissibility of discontinuities), is conducted in [START_REF] Shen | Traveling wave profiles for a follow-the-leader model for traffic flow with rough road condition[END_REF]; our results are complementary of those of [START_REF] Shen | Traveling wave profiles for a follow-the-leader model for traffic flow with rough road condition[END_REF].

Further, we identify specific difficulties on the way of rigorously proving the observed numerical convergence results. First, the techniques of [START_REF] Di Francesco | Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit[END_REF] and subsequent works [START_REF] Di Francesco | Deterministic particle approximation of scalar conservation laws[END_REF][START_REF] Di Francesco | Follow-the-leader approximations of macroscopic models for vehicular and pedestrian flows[END_REF][START_REF] Di Francesco | Deterministic particle approximation of the Hughes model in one space dimension[END_REF][START_REF] Di Francesco | Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic[END_REF] on the same subject do not extend readily to discontinuous-flux LWR model [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF]. Second, note that while dealing with discontinuous-flux scalar conservation laws, common techniques of passage to the limit arose that are equally applicable to vanishing viscosity (or vanishing capillarity), Finite Volume and Wave-Front Tracking approximations. These techniques are essentially based upon identification of a small number of explicit limits that take the form of stationary solutions, possibly discontinuous at x = 0 (such discontinuities are therefore claimed admissible at the limit), and on the L 1 -contraction property valid already at the level of approximations. We refer to [START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF] for a formalization of this technique. Note that the latter contraction principle is equivalent to the order-preservation of the solution semigroup, by virtue of the conservativity. We raise the question whether the FTL approximation of car densities fulfills the order-preservation property, and answer this question by a counterexample. This assesses the necessity of developing a non-standard approach to the limit analysis of the FTL approximations. Moreover, while one expects on heuristic grounds that the admissibility at x = 0 can be disconnected from admissibility (in the standard entropy sense) in the regions {x > 0}, {x < 0}, the localization argument based on the finite speed of propagation in LWR models does not apply neither since the FTL model possess an infinite speed of propagation. We illustrate and discuss these two properties of the FTL semigroup, which make it qualitatively different from its expected limit which is the LWR model ( 1) with interface coupling prescribed by the flow maximization principle.

Outline of the paper

The paper is organized as follows. In the next section we introduce the model and the definition of admissible solutions corresponding to a Riemann solver in Definition 4. In Section 3 we recall the definition of the Riemann solver RS M which maximizes the flow at x = 0 and then show how it can be deduced from a FTL model. In Section 4 we show that, differently from the macroscopic model, the microscopic model does not satisfy the order-preservation property and the finite speed of propagation. In the last section we explain how the Riemann solver introduced in [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] for a scalar conservation law with point constraint on the flow can be deduced from a FTL model.

The model and general definition of solutions

Consider a road network parametrized by x ∈ R and composed of two single lane roads R - . = (-∞, 0) and R + . = [0, +∞) connected together at the junction x = 0. Assume that each driver has the same behavior, all cars are identical and have maximal speed V max > 0. Let V -, V + > 0 be respectively the speed limits in R -and R + , with V -, V + V max . The natural adaptation of the Lighthill, Whitham [START_REF] Lighthill | On kinematic waves. II. A theory of traffic flow on long crowded roads[END_REF] and Richards [START_REF] Richards | Shock waves on the highway[END_REF] (LWR) model to the present case is expressed by the Cauchy problem for a one-dimensional scalar conservation law

∂ t ρ + ∂ x f (ρ, x) = 0, x ∈ R, t > 0, ρ(0, x) = ρ(x), x ∈ R, (2)
where ρ = ρ(t, x) ∈ [0, 1] is the (normalized) density of cars at time t > 0 and position x ∈ R, ρ ∈ L 1 loc (R; [0, 1]) is the initial density and the flux f

: [0, 1] × R → [0, +∞) is defined by f (r, x) . = f -(r) if x ∈ R -, f + (r) if x ∈ R + , where f -(r) . = r v -(r), f + (r) . = r v + (r).
Here v -, v + : [0, 1] → [0, V max ] are the velocity functions. We assume that v -and v + are continuous non-increasing functions with

v -(0) = V -, v + (0) = V + , v -(1) = 0 = v + (1),
and such that f -and f + are bell-shaped. Notice that f (r, x) = r v(r, x), where v :

[0, 1] × R → [0, V max ] is defined by v(r, x) . = v -(r) if x ∈ R -, v + (r) if x ∈ R + .
Typical choice for v -and v + is

v -(r) . = min V -, V max (1 -r) , v + (r) . = min V + , V max (1 -r) . (3)
Since ( 2) is a conservation law, it is necessary to consider solutions in the sense of distributions, the so-called weak solutions. Define R + . = [0, +∞).

Definition 1. We say that ρ ∈ C 0 R + ; BV(R; [0, 1]
) is a weak solution to Cauchy problem (2) if the following conditions are satisfied:

(w.1) For any test function ϕ ∈ C ∞ c (R + × R; R) with compact support in x < 0, we have R+×R- ρ ∂ t ϕ + f -(ρ) ∂ x ϕ dx dt + R- ρ(x) ϕ(0, x) dx = 0. (w.2) For any test function ϕ ∈ C ∞ c (R + × R; R) with compact support in x > 0, we have R+×R+ ρ ∂ t ϕ + f + (ρ) ∂ x ϕ dx dt + R+ ρ(x) ϕ(0, x) dx = 0.
(w.

3) The Rankine-Hugoniot condition is satisfied along x = 0, namely

f -ρ(t, 0 -) = f + ρ(t, 0 + ) , for a.e. t > 0. (4)
We stress that coupling condition (4) guarantees the conservation of the total number of cars. Weak solutions are in general not unique. A well known criterion to select a unique weak solution was proposed by Kruzhkov [START_REF] Kruzhkov | First order quasilinear equations with several independent variables[END_REF]. However Kruzhkov solutions are not always physically reasonable [START_REF] Lefloch | Hyperbolic systems of conservation laws[END_REF]. This is evident, for instance, when dealing with traffic flows through crossroads [START_REF] Colombo | Phase transition model for traffic at a junction[END_REF][START_REF] Dal Santo | Representation of capacity drop at a road merge via point constraints in a first order traffic model[END_REF][START_REF] Delle Monache | Priority-based Riemann solver for traffic flow on networks[END_REF][START_REF] Garavello | Conservation laws with discontinuous flux[END_REF] or point-wise bottlenecks [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF]. However, we stick to the notion of Kruzhkov admissibility away from the flux discontinuity location x = 0. Namely, following [START_REF] Garavello | Conservation laws with discontinuous flux[END_REF] we postulate Definition 2. We say that ρ ∈ C 0 R + ; BV(R; [0, 1]) is an admissible solution to Cauchy problem (2) if it satisfies the following conditions:

(a.1) ρ is a weak solution to Cauchy problem (2) in the sense of Definition 1. (a.2) For any test function ϕ ∈ C ∞ c (0, +∞) × (-∞, 0); R with ϕ 0 and κ ∈ [0, 1], we have

R+×R- |ρ -κ| ∂ t ϕ + sign(ρ -κ) f -(ρ) -f -(κ) ∂ x ϕ dx dt 0. (a.3) For any test function ϕ ∈ C ∞ c (0, +∞) × (0, +∞); R with ϕ 0 and κ ∈ [0, 1], we have R+×R+ |ρ -κ| ∂ t ϕ + sign(ρ -κ) f + (ρ) -f + (κ) ∂ x ϕ dx dt 0.
Conditions listed in Definition 2 do not select a unique weak solution. However, to do so it is sufficient to uniquely characterize the physically reasonable discontinuities along x = 0. This can be achieved by choosing a Riemann solver.

Let 1 A be the indicator function of the set A.

Definition 3. We say that RS :

[0, 1] 2 → BV(R; [0, 1]) is a Riemann solver if for any (ρ L , ρ R ) ∈ [0, 1] 2 the self- similar function ρ(t, x) . = RS[ρ L , ρ R ](x/t) (5)
together with its traces along x = 0

ρ -. = ρ(t, 0 -) = RS[ρ L , ρ R ](0 -), ρ + . = ρ(t, 0 + ) = RS[ρ L , ρ R ](0 + ),
satisfy the following conditions:

(r.1) ρ is an admissible solution to Cauchy problem (2) with initial density ρ(x) .

= ρ L 1 R-(x) + ρ R 1 R+ (x). (6)
(r.

2) The self-similar function

r -(t, x) . = ρ(t, x) if x < 0, t 0, ρ - if x 0, t 0,
is the Kruzhkov solution to the Riemann problem

∂ t r -+ ∂ x f -(r -) = 0, x ∈ R, t > 0, r -(0, x) = ρ L 1 R-(x) + ρ -1 R+ (x), x ∈ R, namely for any test function ϕ ∈ C ∞ c (R + × R; R) with ϕ 0 and κ ∈ [0, 1] R+×R |r --κ| ∂ t ϕ + sign(r --κ) f -(r -) -f -(κ) ∂ x ϕ dx dt + ρ L R- ϕ(0, x) dx + ρ - R+ ϕ(0, x) dx 0.
(r.

3) The self-similar function

r + (t, x) . = ρ + if x < 0, t 0, ρ(t, x) if x 0, t 0,
is the Kruzhkov solution to the Riemann problem

∂ t r + + ∂ x f + (r + ) = 0, x ∈ R, t > 0, r + (0, x) = ρ + 1 R-(x) + ρ R 1 R+ (x), x ∈ R, namely for any test function ϕ ∈ C ∞ c (R + × R; R) with ϕ 0 and κ ∈ [0, 1] R+×R |r + -κ| ∂ t ϕ + sign(r + -κ) f + (r + ) -f + (κ) ∂ x ϕ dx dt + ρ + R- ϕ(0, x) dx + ρ R R+ ϕ(0, x) dx 0. (r.4) We have RS[ρ -, ρ + ](ξ) = ρ -if x < 0, ρ + if x 0. ( 7
) (r.5) The function (ρ L , ρ R ) → f -(ρ -), f + (ρ + )) is continuous.
Some comments on the conditions listed in Definition 3 are in order. Condition (r.1) ensures that a Riemann solver associates to any Riemann datum (6) a self-similar function ρ through [START_REF] Andreianov | Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks' medium[END_REF], which is an admissible solution to Riemann problem (2), [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF]. In particular it satisfies coupling condition (4)

f -(ρ -) = f + (ρ + ). (8)
Conditions (r.2), (r.3) can be rephrased as follows: r -and r + are the solutions to the initial-boundary value problems [START_REF] Amadori | Initial-boundary value problems for nonlinear systems of conservation laws[END_REF][START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF][START_REF] Colombo | Well posedness of balance laws with boundary[END_REF]]

     ∂ t r -+ ∂ x f -(r -) = 0, x < 0, t > 0, r -(0, x) = ρ L , x < 0, r -(t, 0) = ρ -, x < 0, t > 0,      ∂ t r + + ∂ x f + (r + ) = 0, x > 0, t > 0, r + (0, x) = ρ R , x > 0, r + (t, 0) = ρ + ,
x > 0, t > 0, coupled through condition [START_REF] Andreianov | Analysis and approximation of one-dimensional scalar conservation laws with general point constraints on the flux[END_REF]. Notice that r -has only waves generated at x = 0 and with negative speed, whereas r + has only waves generated at x = 0 and with positive speed. By (r.4) the ordered pair of the traces of the admissible solution selected by RS is in a sense a fixed point of RS by [START_REF] Andreianov | Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks[END_REF]. It is a minimal requirement to develop a numerical scheme with a time discretization based on RS, see [START_REF] Corli | Coherence and chattering of a one-way valve[END_REF][START_REF] Dal Santo | Representation of capacity drop at a road merge via point constraints in a first order traffic model[END_REF]. Condition (r.4) is therefore a stability condition. Condition (r.5) is a regularity property for RS.

We are now in the position to give the definition of admissible solutions to Cauchy problem (2) associated to a Riemann solver RS. Definition 4. Fix a Riemann solver RS in the sense of Definition 3. We say that ρ ∈ C 0 R + ; BV(R; [0, 1]) is an RS-admissible solution to Cauchy problem (2) if it is an admissible solution in the sense of Definition 2 and

RS[ρ(t, 0 -), ρ(t, 0 + )](ξ) = ρ(t, 0 -) if ξ < 0, ρ(t, 0 + ) if ξ 0, for a.e. t > 0.
In [START_REF] Garavello | Conservation laws with discontinuous flux[END_REF] it is proved that the above definition selects a unique admissible solution, given a fixed Riemann solver. We are therefore left to define a physically reasonable Riemann solver. This is achieved in the next section.

3 Microscopic deduction of the Riemann solver RS M

In this section we select a Riemann solver. The choice is motivated by basic well established physically reasonable rules on the microscopic interactions between the cars. We first deduce from the microscopic rules a FTL model. We then obtain an empirical Riemann solver as a many particle limit in analogy to what is done in [START_REF] Di Francesco | Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit[END_REF]. It turns out that the obtained Riemann solver coincides with the Riemann solver RS M that maximizes the flow at x = 0, even if we do not require any flow maximization property for the FTL model. For completeness we recall in the next subsection the definition of RS M . We then introduce the FTL model underlying the simulations which are used to show that the corresponding Riemann solver is indeed RS M . In Fig. s 1 and 2 we present the results of some selected simulations.

Riemann solver RS M

In this subsection we recall the Riemann solver RS M which maximizes the flow at x = 0. This means that if RS is a Riemann solver, then by ( 8)

f -RS[ρ L , ρ R ](0 -) = f + RS[ρ L , ρ R ](0 + ) f -RS M [ρ L , ρ R ](0 -) = f + RS M [ρ L , ρ R ](0 + ) .
We introduce some notations. Let L -, L + : [0, 1] 2 → BV(R; [0, 1]) be the Lax Riemann solvers associated to the fluxes f -and

f + . Notice that L -[ρ L , ρ R ] ≡ L + [ρ L , ρ R ] if and only if ρ L , ρ R max{R -, R + }, where R -, R + ∈ (0, 1) are implicitly defined by v(R -) = V -, v(R + ) = V + .
Let M -and M + be the densities at which f -and f + reach their maxima. Let

Q -(ρ L ) . = f -(ρ L ) if ρ L < M -, f -(M -) if ρ L M -, Q + (ρ R ) . = f + (M + ) if ρ R < M + , f + (ρ R ) if ρ R M + , be the maximal flows at x = 0 attained by L -[ρ L , ρ] and L + [ρ, ρ R ] for ρ varying in [0, 1]. Let then ρ(ρ L , ρ R ) ∈ [M -, 1] and ρ(ρ L , ρ R ) ∈ [0, M + ] be implicitly defined by f -ρ(ρ L , ρ R ) = min Q -(ρ L ), Q + (ρ R ) = f + ρ(ρ L , ρ R ) .
Then RS M : [0, 1] 2 → BV(R; [0, 1]) is defined by

RS M [ρ L , ρ R ](ν) . = L -[ρ L , ρ(ρ L , ρ R )](ν) if ν < 0, L + [ρ(ρ L , ρ R ), ρ R ](ν) if ν 0.

FTL model

In this subsection we introduce the FTL model corresponding to Riemann problem (2), [START_REF] Andreianov | On interface transmission conditions for conservation laws with discontinuous flux of general shape[END_REF]. Fix (ρ L , ρ R ) ∈ [0, 1] 2 and consider the approximate Riemann problem

∂ t ρ + ∂ x f (ρ, x) = 0, x ∈ R, t > 0, ρ(0, x) = ρ L 1 [-δ,0) (x) + ρ R 1 [0,δ] (x), x ∈ R. (9)
Here δ > 0 is introduced so that the traffic has finite total length of cars, which is δ (ρ L + ρ R ). The case ρ L = 0 = ρ R is trivial; assume therefore that ρ L + ρ R = 0. Let [x min , x max ] be the support of the initial datum, namely

x min . = -δ if ρ L = 0, 0 if ρ L = 0, x max . = δ if ρ R = 0, 0 if ρ R = 0.
In analogy with [START_REF] Di Francesco | Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit[END_REF], for any fixed n ∈ N, we let . = δ (ρ L + ρ R )/n and consider the approximate discrete density

r(t, x) . = n i=1 x i (t) -x i+1 (t) 1 [xi+1(t),xi(t)) (x), (10)
where x i are the solutions to the FTL model

       ẋ1 = V + , t > 0, ẋi = v xi-1-xi , x i , t > 0, i ∈ {2, . . . , n + 1}, x i (0) = xi , t > 0, i ∈ {1, . . . , n + 1}. ( 11
)
Above x1 . = x max , xn+1 . = x min and for i ∈ {2, . . . , n} we define

ρ L = 0 =⇒ x 0 i+1 = -δ + (n -i) ρ L 0, i ρ R ρ L + ρ R n , ρ R = 0 =⇒ x 0 i+1 = δ -i ρ R 0, i ρ R ρ L + ρ R n .
Notice that by definition xi-1 xi ρ(x) dx = and for any t 0 we have

r(t) L 1 (R) = δ (ρ L + ρ R ), x 1 (t) = x max + V + t.
We recall that ( 11) can be interpreted as a Lagrangian many particle approximation of [START_REF] Andreianov | Crowd dynamics and conservation laws with nonlocal constraints and capacity drop[END_REF]. Moreover [START_REF] Andreianov | Entropy conditions for scalar conservation laws with discontinuous flux revisited[END_REF] corresponds to the following basic rules:

-Cars respect the speed limits.

-Cars do not overtake each other.

-Each car adjusts instantaneously its velocity according to the distance from its predecessor.

-Higher velocities correspond to higher distances from the predecessor.

In [START_REF] Di Francesco | Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit[END_REF] it is proved that in the case V -= V + = V max > 0 the discretized density r given in (10) converges a.e. in L 1 loc (R + × R) to the unique Kruzhkov [START_REF] Kruzhkov | First order quasilinear equations with several independent variables[END_REF] solution ρ to [START_REF] Andreianov | Crowd dynamics and conservation laws with nonlocal constraints and capacity drop[END_REF]. The rigorous proof for the convergence of the discretized density r in the general case V -= V + is beyond the purposes of the present paper and is left to a future work. Here we are just interested in empirically deducing a Riemann solver from ad hoc computer assisted numerical simulations of ( 10), [START_REF] Andreianov | Entropy conditions for scalar conservation laws with discontinuous flux revisited[END_REF]. It should be stressed that the simulations appear indeed as stable and convergent in all simulations we have run.

We are not interested in the solutions of ( 9) in the whole of R, but only in the vicinity of x = 0. For this reason we do not investigate the nature of the waves created at x = ±δ and their interactions with the waves starting from x = 0.

Computer assisted numerical simulations allow to describe the limit solutions for any Riemann problem (2), (6). Our choice for the initial data (ρ L , ρ R ) ∈ [0, 1] 2 used for the simulations allows to encompass all the cases necessary to deduce a Riemann solver, which turns out to coincide with RS M . In Fig. s 1 and 2 we present the outputs of the numerical simulations in the case V -, V + ∈ (0, 1/2); the remaining cases are analogous and are not furnished here not to overload the paper. In

. = ρ L , f -(ρ L ) and u R . = ρ R , f + (ρ R
) in the fundamental diagram and below it we show the profile of the corresponding approximate discrete density r at a time t > 0 of interest. For each simulation we take n = 1000, V max = 1 and v -, v + given by (3) with

V -, V + ∈ { 2- √ 2 4 , 1 4 }.

Qualitative comparison of LWR and FTL models

In this section we consider a homogeneous road and assume V -= V + = V max . In [START_REF] Di Francesco | Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit[END_REF] it is proved that in this case the unique entropy solution to the LWR model is the limit of discrete densities corresponding to the FTL model. Despite this link between the two models, there are deep differences between them. We show below that the order-preservation property and the finite speed of propagation, well known for the LWR model, are not shared by the FTL model. In a sense, these properties are gained only at the limit n → +∞. We stress that the examples given below can be easily adapted to the case of inhomogeneous roads: it is enough to place all the cars on the right of x = 0. In fact, properties of the discontinous-flux semigroup cannot be better than those of the continuous-flux one.

For simplicity consider a single lane road with neither entrances nor exits and parametrized by x ∈ R. Let > 0 be the length of each car. The evolution of the traffic can be then described at the macroscopic level by the LWR model and at the microscopic level by the FTL model. The LWR model is expressed by the conservation law

∂ t ρ + ∂ x f (ρ) = 0, x ∈ R, t > 0, (12)
where ρ = ρ(t, x) ∈ [0, 1] is the (normalized) density at time t and position x, and f (ρ) . = ρ v(ρ) is the flux corresponding to the velocity v(ρ) .

= V max (1 -ρ).
The FTL model is given by the system of ordinary differential equations

ẋ1 = V max , t > 0, ẋi = v xi-1-xi , t > 0, i ∈ {2, . . . , n}, (13)
Ρ f R R F R F 1 u L u R Ρ f R R F R F 1 u L u R Ρ f R R F R F 1 u L u R Ρ f R R F R F 1 u L u R Ρ f R R F R F 1 u L u R Ρ f u R R F R F 1 u L u R Ρ f u R R F R F 1 u L u R Ρ f u R R F R F 1 u L u R Ρ f u R R F R F 1 u L u R Ρ f R R F R F 1 u L u R Fig. 1. Case V-< V+ < 1/2.
where x i = x i (t) ∈ R is the position at time t of the front bumper of the i-th car labeled starting from the right and n is the total number of cars. The corresponding discrete density is

r(t, x) . = n-1 i=1 x i (t) -x i+1 (t) • 1 [xi+1(t),xi(t)) (x). (14)
Notice that the total length of the traffic is n and is linked to the discrete density r and the solution ρ to LWR model [START_REF] Andreianov | Well-posedness of general boundary-value problems for scalar conservation laws[END_REF] as follows:

R r(t, x) dx + = n = R ρ(t, x) dx, t 0.

Finite speed of propagation

It is well known that in the LWR model, information propagates with speed bounded by V max . On the contrary, according to the FTL model the propagation of information can be instantaneous. The next example shows that this drawback of the FTL model is responsible for an unrealistic behavior of the cars.

Example 1. Consider a traffic light placed at x = 0, which at time t = 0 turns from red to green. Assume that at time t = 0 the cars in x 0 are bumper to bumper while in x > 0 no car is present. The corresponding FTL model ( 13) is

       ẋ1 = V max , t > 0, ẋi = v xi-1-xi , t > 0, i ∈ {2, . . . , n}, x i (0) = -(i -1) , t > 0, i ∈ {1, . . . , n}. (15)
We obviously have

x 1 (t) = V max t, t > 0, x i (t) > -(i -1) , t > 0, i ∈ {2, . . . , n}. ( 16
) Ρ f R R F R F 1 u L u R Ρ f R R F R F 1 u L u R Ρ f R R F R F 1 u L u R Ρ f R R F R F 1 u L u R Ρ f u R R F R F 1 u L u R Ρ f u R R F R F 1 u L u R Ρ f u R R F R F 1 u L u R Ρ f R R F R F 1 u L u R Ρ f R R F R F 1 u L u R Ρ f R R F R F 1 u L u R Fig. 2. Case V+ < V-< 1/2.
The corresponding LWR model ( 12) is

     ∂ t ρ + ∂ x f (ρ) = 0, x ∈ R, t > 0, ρ(0, x) = 1 if x ∈ [-n , 0), 0 otherwise, x ∈ R. (17)
In the time interval t ∈ 0, -n /f (1) , its entropy solution ρ is the juxtaposition of a stationary shock S(0, 1) starting from x = -n and a rarefaction R(1, 0) starting from x = 0, namely

ρ(t, x) =      1 if -n x/t < f (1) (f ) -1 (x/t) if f (1) x/t < f (0) 0 otherwise =        1 if -n x/t < -V max , 1 2 1 -ξ Vmax if -V max x/t < V max , 0 otherwise, (18)
see Fig. 3. From [START_REF] Bürger | On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux[END_REF] we can deduce the corresponding trajectories of the cars. Indeed, if x = X i (t) is the trajectory

x f (0) t f (1) t -n 1 ρ(t, •)
x t Fig. 3. Solution (18) for t ∈ 0, -n /f (1) . On the right, darker (x, t)-regions correspond to higher densities.

It is easily seen that the lack of order-preservation for the homogeneous FTL model underlying [START_REF] Andreianov | Well-posedness of general boundary-value problems for scalar conservation laws[END_REF] is inherited by the FTL model underlying the discontinuous-flux équation (1): indeed, its is enough to consider initial data supported in R + . Remark 3. In the context of (1), preservation by various approximation schemes of specific steady states is the cornerstone of the proofs of convergence of vanishing (or adapted) viscosity approximations, of Godunov Finite Volume approximations, or of the Wave-Front Tracking approximations to [START_REF] Adimurthi | Optimal entropy solutions for conservation laws with discontinuous flux-functions[END_REF], see [START_REF] Andreianov | New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux[END_REF][START_REF] Andreianov | A theory of L 1 -dissipative solvers for scalar conservation laws with discontinuous flux[END_REF]. In the context of fluxes of kind (3), it is not difficult to show that the FTL approximation does preserve the key steady state corresponding to the maximization of the interface flux. Yet, the absence of L 1 -contraction at the FTL level precludes us from exploiting the convergence paradigm suitable for above mentioned order-preserving approximations. This leaves space to future work on rigorous justification of the convergence behavior reported in the present paper.

Conservation laws with point constraint on the flow

In this section we show how the above arguments justify the Riemann solver RS F introduced in [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] for the scalar conservation law with point constraint on the flow

     ∂ t ρ + ∂ x f (ρ) = 0, x ∈ R, t > 0, ρ(0, x) = ρ(x), x ∈ R, f ρ(t, 0 ± ) F, x ∈ R, t > 0. (21)
Above ρ ∈ [0, 1] is again the (normalized) density of cars, f : [0, 1] → [0, f max ] is the flux and F ∈ (0, f max ) is the capacity of a pointwise bottleneck localized at x = 0, e.g., a toll gate. We assume that the flux is bell shaped, attains its maximal value f max > 0 at R max ∈ (0, 1) and takes the form f (ρ) = ρ v(ρ), where the velocity function v : [0, 1] → [0, V max ] is a continuous non-increasing function such that v(0) = V max and v(1) = 0. Since we are interested only in deducing a Riemann solver, we consider only Riemann initial conditions ρ(x) .

= ρ L 1 x<0 (x) + ρ R 1 x 0 (x). (22)
As in [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] we introduce an approximate Riemann problem

∂ t ρ + ∂ x f ε (ρ, x) = 0, x ∈ R, t > 0, ρ(0, x) = ρ(x), x ∈ R, (23)
where

f ε : [0, 1] × R → [0, f max ], ε > 0, is defined by f ε (ρ, x) . = f (ρ) if |x| ε, f F (ρ) if |x| < ε, (24)
where f F (ρ) . = ρ min{V F , v(ρ)}, with V F ∈ (0, V max ) implicitly given by

V F = v(R F ), R F V F = F, R F > R max .
In words, we zoom at x = 0, extend the location of the bottleneck to the interval (-ε, ε), consider there a flux f F having F as maximum value and corresponding to a velocity function of the form (3). We then apply RS M at the discontinuity points x = ±ε of f ε , construct the approximate solution ρ ε , let ε go to zero and obtain the limit function ρ. It is easy to seen then that ρ ≡ RS F [ρ L , ρ R ]. For clarity, we consider in the next example one case.

Example 3. Fix 0 < V < V max and consider the flux f (r) . = r v(r) with v(r) . = min{V, V max (1 -r)}, see Fig. 6. By definition R max = 1 -V Vmax . Fix ρ L , ρ R ∈ (0, R max ) and assume that f (ρ L ) > F . The solution to ( 22), (23) can be constructed as follows. Introduce the following notation

u L . = ρ L , f (ρ L ) , u R . = ρ R , f (ρ R ) , U F . = (R F , F ), u ε L . = ρ L , f F (ρ L ) , u ε R . = ρ R , f F (ρ R ) , ǔL . = ρL , f (ρ L ) , ǔR . = ρR , f (ρ R ) , ǓF . = ŘF , f ( ŘF ) ,
where ρL , ρR , ŘF ∈ (0, R max ) are implicitly defined by We first apply RS M at the discontinuity points x = ±ε of the flux f ε . At x = -ε we obtain the shock S(u L , U F ) with negative speed (of propagation) and the contact discontinuity CD(U F , u ε L ) with speed V F . At x = ε we obtain the (stationary) non-classical shock NS(u ε R , ǔR ) and CD(ǔ R , u r ) with speed V . At x = 0 we apply the Lax Riemann solver associated to the flux f F and obtain CD(u ε L , u ε R ) with speed V F . Once CD(u ε L , u ε R ) reaches x = ε, we apply RS M and obtain NS(u ε L , ǔL ) and CD(ǔ L , ǔR ) with speed V . Analogously, when CD(U F , u ε L ) reaches x = ε, we apply again RS M and obtain NS(U F , ǓF ) and CD( ǓF , ǔL ) with speed V . At last, by letting ε → 0 + we obtain = ρ L 1 [-δ,0) (x) + ρ R 1 [0,δ] (x).

f (ρ L ) = f F (ρ L ), f (ρ R ) = f F (ρ R ), f ( ŘF ) = f (R F ) = F.
RS F [ρ L , ρ R ](ξ) =            ρ L if ξ < f (ρ L )-F ρ L -R F , R F if f (ρ L )-F ρ L -R F ξ < 0, ŘF if 0 ξ < V, ρ R if ξ V.
By letting then n → +∞, δ → +∞ and ε → 0 + we obtain the solution selected by RS F for Riemann problem ( 21), [START_REF] Colombo | Pedestrian flows and non-classical shocks[END_REF].

Remark 5. We underline that in [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] it is considered the discontinuous flux [START_REF] Corli | Coherence and chattering of a one-way valve[END_REF] with f F (ρ) . = F fmax f (ρ). Even if it differs from our choice, at the limit we obtain the same result. The reason is simply because in any cases we obtain at the limit the Riemann solver RS M which maximizes the flow. Remark 6. Note that a yet different justification of the model of [START_REF] Colombo | A well posed conservation law with a variable unilateral constraint[END_REF] can be given in terms of a velocity constraint; roughly speaking, prescribing limited velocity results in prescribing a specific flux limitation, due to the boundary layer phenomena proper to hyperbolic scalar conservation laws [START_REF] Amadori | Initial-boundary value problems for nonlinear systems of conservation laws[END_REF][START_REF] Andreianov | New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux[END_REF][START_REF] Andreianov | Well-posedness of general boundary-value problems for scalar conservation laws[END_REF][START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF][START_REF] Colombo | Well posedness of balance laws with boundary[END_REF]. However this line of justification is beyond the scope of the present paper.

Fig.s 1

 1 and 2 we show u L

Fig. 6 .

 6 Fig. 6. The approximate solution constructed in Example 3.

Remark 4 .

 4 Another possible way to deduce RS F is to consider the microscopic FTL model corresponding to the Cauchy problem for[START_REF] Colombo | Well posedness of balance laws with boundary[END_REF] with truncated Riemann datum ρ(x) .

MDR is member of GNAMPA. MDR acknowledges the support of the National Science Centre, Poland, Project "Mathematics of multi-scale approaches in life and social sciences" No. 2017/25/B/ST1/00051, by the INdAM-GNAMPA Project 2019 "Equazioni alle derivate parziali di tipo iperbolico o non locale ed applicazioni" and by University of Ferrara, FIR Project 2019 "Leggi di conservazione di tipo iperbolico: teoria ed applicazioni".

of the i-th car, then Ẋi = v ρ(t, X i ) , t > 0, i ∈ {1, . . . , n}, X i (0) = -(i -1) , t > 0, i ∈ {1, . . . , n}.

We obviously have X 1 (t) = V max t, t > 0, X i (t) = -i , t ∈ 0, -(i -1) /f (1) , i ∈ {2, . . . , n}.

By comparing [START_REF] Bürger | Monotone difference approximations for the simulation of clarifier-thickener units[END_REF] and [START_REF] Colombo | Phase transition model for traffic at a junction[END_REF] it is clear that the trajectories corresponding to the microscopic [START_REF] Braess | Über ein paradoxon aus der verkehrsplanung[END_REF] and macroscopic [START_REF] Bürger | An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections[END_REF] descriptions do not coincide. Indeed, according to the microscopic description each car start to move at time t = 0 + with strictly positive speed. On the contrary, according to the macroscopic description, with exception for the rightmost car, all the other cars need some time before starting to move. In Fig. 4 we plot the trajectories corresponding to the two levels of descriptions. Remark 1. We stress that real life experience shows that, at least in the case considered in Example 1, the macroscopic description is more appropriate than the microscopic one. Indeed, everyday experience shows that the cars don't start to move all together immediately after t = 0.

Order-preservation property

In this subsection we consider the order-preservation property. It is well known that the LWR model satisfies if ρ 1 and ρ 2 are solutions to [START_REF] Andreianov | Well-posedness of general boundary-value problems for scalar conservation laws[END_REF] corresponding to the initial data ρ1 and ρ2 , and ρ1 ρ2 , then also ρ 1 ρ 2 . The next example shows that FTL model ( 13) is not order-preserving.

1 as in Fig. 5 with

By considering x1 3 , x1 2 , x1 1 and x2 4 , . . . , x2 1 as initial positions for FTL model [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF], we obtain the discrete densities r 1 and r 2 via [START_REF] Benyahia | A macroscopic traffic model with phase transitions and local point constraints on the flow[END_REF]. By construction, at the initial time we have r 1 (0, •) r 2 (0, •). Clearly ẋ1 3 (0) = ẋ2 4 (0) and ẋ1 2 (0) = 0 < ẋ2 3 (0); thus, at least for t > 0 sufficiently small, we have Remark 2. It should be stressed that order-preservation implies L 1 -contractivity by virtue of the classical Crandall-Tartar lemma. The semigroups of solutions of (1) considered in the present paper are L 1 -contractive (cf. [START_REF] Andreianov | The semigroup approach to conservation laws with discontinuous flux[END_REF]), governed by accretive operators on L 1 . As we can now see, the FTL semigroup (defined on densities) is conservative but not order-preserving: the FTL semigroup is not governed by an accretive operator on L 1 . However, since the above example concerns the case with space-homogeneous flux, the convergence results of, e.g., [START_REF] Di Francesco | Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit[END_REF] apply. Thus we observe that the approximation of (1), although convergent and motivated by modeling considerations, fails to share the key mathematical structure of the limit problem.


Elwan Héry

Philippe Xu

Philippe Bonnifait

Distributed asynchronous cooperative localization with inaccurate GNSS positions

Localization remains a major issue for autonomous vehicles. Accurate localization relative to the road and other vehicles is essential for many navigation tasks. When vehicles cooperate and exchange information through wireless communications, they can improve mutually their localization. This paper presents a distributed cooperative localization method based on the exchange of Local Dynamic Maps (LDMs). Every LDM contains dynamic information on the pose and kinematic of all the cooperating agents. Different sources of information such as dead-reckoning from the CAN bus, inaccurate (i.e. biased) GNSS positions, LiDAR and road border detections are merged using an asynchronous Kalman filter strategy. The LDMs received from the other vehicles are merged using a Covariance Intersection Filter to avoid data incest. Experimental results are evaluated on platooning scenarios. They show the importance of estimating GNSS biases and having accurate relative measurements to improve the absolute localization process. These results also illustrate that the relative localization between vehicles is improved in every LDMs even for vehicles not able to perceive surrounding vehicles but which are instead perceived by others.

I. INTRODUCTION

Cooperative localization is a well-known issue in mobile robotics. Many works describe this problem as a rigid structure [START_REF] Ahmad | Cooperative robot localization and target tracking based on least squares minimization[END_REF] where the different robots are the nodes and the bearing [START_REF] Zelazo | Rigidity Theory in SE(2) for Unscaled Relative Position Estimation using only Bearing Measurements[END_REF] or range [START_REF] Zelazo | Decentralized Rigidity Maintenance Control with Range Measurements for Multi-Robot Systems[END_REF] measurements are the links. This formulation can be used for robots in a swarm formation. Nevertheless, intelligent vehicles can also use rangeonly measurements with some sensors like Ultra Wideband technology [START_REF] Hoangt | Cooperative Localization in VANETs: An Experimental Proof-of-Concept Combining GPS, IR-UWB Ranging and V2v Communications[END_REF]. These vehicles are often following each other and do not always see other vehicles on their sides. In this situation, few vehicles are visible and the cooperative localization must be done with less observation than in a swarm formation.

Different formulations exist to manage the fusion of the localization information of the vehicles. A simple Extended Kalman Filter (EKF) can be applied. However, this method can generate data incest issues because of the redundant use of the poses of the vehicles. Other formulations exist to avoid this problem. Each vehicle may communicate the raw measurements from sensors [START_REF] Lassoued | Mutual Localization and Positioning of Vehicles Sharing GNSS Pseudoranges: Sequential Bayesian Approach and Experiments[END_REF] instead of a fused pose. Another solution is to use two EKF [START_REF] Karam | Localization of a Group of Communicating Vehicles by State Exchange[END_REF], [START_REF] Aufrère | A state exchange approach in real conditions for multi-robot cooperative localization[END_REF], the first one without the information from the other using only the sensors 1 The authors are with Sorbonne universités, Université de technologie de Compiègne, CNRS, Heudiasyc UMR 7253, France firstname.surname@hds.utc.fr of the vehicle. This first filter estimates the pose of the vehicle for the other, this pose can be communicated. The second filter uses the sensor information of the vehicle and the information of other vehicles to estimate the pose. This pose is not communicated. Other data fusion filters exist to avoid data incest. The covariance intersection filter (CIF) can solve this problem [START_REF] Julier | A non-divergent estimation algorithm in the presence of unknown correlations[END_REF]. The covariance matrix of the pose is nevertheless often pessimist. The split covariance intersection filter (SCIF) can give a better estimation of the covariance matrix [START_REF] Julier | General Decentralized Data Fusion with Covariance Intersection (CI)[END_REF], [START_REF] Li | Cooperative Multi-Vehicle Localization Using Split Covariance Intersection Filter[END_REF], [START_REF] Wanasinghe | Decentralized Cooperative Localization for Heterogeneous Multi-robot System Using Split Covariance Intersection Filter[END_REF].

In this paper, we combine different sources of information for localization. The most straightforward way is to use a GNSS receiver to compute the vehicle position. However, many low-cost GNSS receivers are inaccurate, that is they suffer from non-centered errors that act as a bias. This bias needs to be compensated in order to have a consistent pose estimate [START_REF] Tao | Mapping and localization using GPS, lane markings and proprioceptive sensors[END_REF]. On possible way to observe this bias is to perceive geo-referenced features in a map such as using a camera for lane markings detection [START_REF] Tao | Mapping and localization using GPS, lane markings and proprioceptive sensors[END_REF]. Another commonly used sensor is the LiDAR. It is often used to detect other vehicles in the surrounding [START_REF] Zhang | Efficient L-shape fitting for vehicle detection using laser scanners[END_REF]. Cameras can also be used for vehicle detection [START_REF] Tekeli | A Fusion of a Monocular Camera and Vehicle-to-Vehicle Communication for Vehicle Tracking: An Experimental Study[END_REF]. This relative measurement between vehicles can also be used to improve the global localization of all the vehicles [START_REF] Peker | Vehicle localization enhancement with VANETs[END_REF]. Adding also proprioceptive information from the CAN bus and wireless communication (see Fig. 1), we propose a real-time distributed asynchronous cooperative localization with GNSS biases estimation. The main contribution of this work is to present experimental results on how cooperative localization can improve cooperative navigation by estimating accurate and consistent relative localization between vehicles. This work also shows that accurate localization can be propagated to improve inaccurate vehicles pose estimates.

In Sec. II, the distributed asynchronous data fusion is presented and the evolution and the different observations are described in detail. In Sec. III the experimental platform is presented and the results are analyzed.

II. DISTRIBUTED ASYNCHRONOUS DATA FUSION

For navigation purpose, an autonomous vehicle needs to be able to localize itself and the surrounding vehicles as shown in Fig. 2. The knowledge of its own absolute localization is important for autonomous driving, but can be inaccurate. To be able to stay on the road and avoid obstacles, the perception of the carriageway borders and obstacles is needed. The proposed data fusion algorithm uses a Local Dynamical Map (LDM) [START_REF] Eiter | Towards a Semantically Enriched Local Dynamic Map[END_REF] to represent the other traffic participants in interaction. This map contains its own state and the states of the other agents. In this paper, the agents are only cars. The proposed method uses a low cost GNSS receiver along with lane markings and carriageway borders detection to improve localization. Therefore, the pose (position and heading) of each agent is described in a working frame by a state vector with an associated covariance matrix. This working frame is defined with its origin near the area of driving of the vehicles and two axes oriented toward the east (x axes) and the north (y axes). The relative poses between the agents with their covariance matrices can be computed from the LDM.

The estimated state X = X 1 , . . . , X N T of the LDM contains the states of N agents in cooperation. Each state

X i = q i , b i , u i T contains the pose of the agent q i = x i , y i , θ i T , the bias b i = b xi , b yi T
of the GNSS receiver of this agent and the kinematic information u i = ( v i , ω i )

T . We use the longitudinal speed v i and the yaw rate ω i to extrapolate the pose at any given time. Therefore, the state X i is a 7 dimensional vector. Fig. 3 shows the asynchronous data fusion algorithm. At the center, the state of the LDM is represented in green. When an observation (in yellow) is sent by a source of information (in orange) the state is updated by the module in blue. The state is firstly extrapolated at the observation time using the evolution model before using the observation model. The evolution model represents the motion of the vehicles and the observation function links the state to the observation. As each observation update is implemented in a thread, a mutex protects the state while it is processed. This method is used to avoid the loss of an observation if multiple updates are computed in parallel.

This algorithm is fully distributed: each vehicle executes the same algorithm and uses communication to send and receive LDMs. The cooperative localization is improved by this information exchange.

A. Evolution stage

When a new information (sensor measurement or LDM) is received, the state is extrapolated at the time of this new information before doing a Kalman update.

The evolution model uses the kinematic information of each agent i of the LDM :

Xi [t|t-1] = Xi [t-1|t-1] +          vi [t-1|t-1] ∆t cos θi [t-1|t-1] + ωi [t-1|t-1] ∆t 2 vi [t-1|t-1] ∆t sin θi [t-1|t-1] + ωi [t-1|t-1] ∆t 2 ωi [t-1|t-1] ∆t 0 0 0 0          (1)
where ∆t = t -t -1 is the elapsed time between the observation time t and the last estimated time t -1 of the LDM.

Let

X [t|t -1 ] = evo (X [t -1 |t -1 ]
, ∆t) the evolution model and Σ model the covariance matrix of the model noise. We suppose that the covariance matrix of the model noise contains the variance of the bias noise and the longitudinal speed and yaw rate noise which are supposed to be constant. The model on the pose is supposed to be very accurate.

The covariance matrix estimate of the state X [t|t -1 ] is approximated by a first order Taylor expansion:

Σ X [t|t -1 ] = JΣ X [t -1 |t -1 ] J T + Σ model ∆t 2 (2)
where J is the Jacobian matrix of the evolution model

J = ∂evo ∂X [t -1 |t -1 ] (3)

B. LDM update with ego-measurements

Once the state is predicted at exactly the same time as the observation, it can be updated. Two update methods are used depending on whether the information comes from the vehicle's sensors or from other vehicles.

1) Extended Kalman Filter: A classical Extended Kalman Filter update is used for internal information. The innovation Y [t] is computed from the observation Z [t], the predicted one using the observation function obs () and the state

X [t|t -1 ]: Y [t] = Z [t] -obs (X [t|t -1 ]) (4)
The Jacobian matrix of the observation function is then computed as

H = ∂obs ∂X [t|t -1 ] (5)
The updated state with its covariance matrix is classically computed as follows:

Σ Y [t] = HΣ X [t|t -1 ] H T + Σ Z [t] (6) K = Σ X [t|t -1 ] H T Σ -1 Y [t] (7) X [t|t] = X [t|t -1 ] + KY [t] (8) Σ X [t|t] = (I -KH) Σ X [t|t -1 ] (I -KH) T (9) +KΣ Z [t] K T
All the observation functions required to implement the update of the Kalman filter are presented in the following.

2) Dead-reckoning observation function: The observation model of the dead-reckoning (DR) data is simply the identity for the kinematic part of the state:

obs CAN (X [t|t -1 ]) = u e [t|t -1 ] (10)
In practice, the high frequency of the DR sensors (e.g. 50 Hz) guarantees frequent updates of the LDM.

3) GNSS observation function: One can observe experimentally the presence of a bias in the GNSS measurements. Fig. 4 pictures the error distribution for two vehicles following each other during a 10-minute test.

The GNSS observation function involves the bias on the position and not on the heading:

obs GN SS (X [t|t -1 ]) = q e [t|t -1 ] + b e [t|t -1 ] 0 (11)
This formulation with the bias b e [t|t -1 ] adds a degree of freedom to the position. Therefore, this bias is estimated from the difference between the position estimate and the GNSS observation of this position.

4) Camera observation function:

As the GNSS measurements are biased, an observation of the environment which is compared to a high-definition (HD)map is needed to observe the GNSS bias. The problem of changing coordinates between the HD map and the current GNSS system must be handled with great care to avoid introducing errors and maintain centimeter accuracy.

We suppose that each vehicle is equipped with a camera able to detect and localize the carriageway borders. This kind of sensor gives the signed distance directly between the vehicle and the center of the lane.

The observation model of this observation needs to mapmatch the pose of the ego vehicle q e [t|t -1 ] included in the state of the LDM with the known map M :

p f renet , - → t f renet , - → n f renet = map_match (q e [t|t -1 ] , M ) (12)
The map matching algorithm gives the closest Frenet frame from the position

p e [t|t -1 ] = x e [t|t -1 ] y e [t|t -1 ] .
This frame is composed of its origin p f renet , the unit tangent -→ t f renet and the unit normal -→ n f renet to the center of the lane.

To compute the signed lateral distance n e [t|t -1 ], one can project the position p e [t|t -1 ] into the normal -→ n f renet :

obs CAM (X [t|t -1 ]) = (p e [t|t -1 ] -p f renet ) T • - → n f renet (13) 5)
Relative pose observation function: The relative pose between agent i and the ego vehicle e is computed from their absolute poses q i and q e :

obs LiDAR (X [t|t -1 ]) =   cos (θ e [t|t -1 ]) sin (θ e [t|t -1 ]) 0 -sin (θ e [t|t -1 ]) cos (θ e [t|t -1 ]) 0 0 0 1   ∆q ( 14
)
where ∆q = q i [t|t -1 ]-q e [t|t -1 ]. More information is given in [START_REF] Héry | Pose and covariance matrix propagation issues in cooperative localization with LiDAR perception[END_REF].

C. LDM update with communicated information

When the LDM of another vehicle is received, the ego-LDM is first extrapolated at the time instant of the received one with the same evolution stage which aligns the LDM timestamps and compensates for the communication latency. Please note that a Kalman update cannot be applied. Indeed, as the agents are exchanging their states, some information cannot be considered as independent since they can be used several times in the data fusion process. A covariance intersection filter is used to avoid data-incest issues.

1) Covariance Intersection Filter: The application of this algorithm is very similar to the EKF. The difference is that Σ X [t|t -1 ] and Σ Z [t] are weighted by 1/ω and 1/ (1 -ω), respectively. These weights are used to find the new covariance matrix Σ X,ωmin [t|t] which gives the smallest uncertainty ellipsoid including the intersection of the uncertainty ellipsoid of the two data to be fused. This means that even if the two uncertainty ellipsoids are correlated, the estimate stays consistent as the intersection is not smaller. For example, if the two uncertainty ellipsoids are the same, the resulting uncertainty ellipsoid remains this ellipsoid and not a smaller one as would give a Kalman filter update.

For a given weight ω, compute (using the Joseph's form to improve the numerical stability):

Σ Y,ω [t] = H 1 ω Σ X [t|t -1 ] H T + 1 1 -ω Σ Z [t] (15)
K ω = 1 ω Σ X [t|t -1 ] H T Σ -1 Y,ω [t] (16)
Σ X,ω [t|t] = (I -K ω H) 1 ω Σ X [t|t -1 ] (I -K ω H) T +K ω 1 1 -ω Σ Z [t] K T ω ( 17
)
Find ω which minimizes a given norm of the covariance matrix Σ X,ω [t|t], often done using the determinant or the trace:

ω min = arg min ω∈[0,1] (norm (Σ X,ω [t|t])) (18)
The updated state with its covariance matrix is given as follows:

X [t|t] = X [t|t -1 ] + K ωmin Y [t] (19) Σ X [t|t] = Σ X,ωmin [t|t]
2) Communication observation function: Let us suppose for simplification that the agents are organized in the same order in the different LDMs. The observation model in this case is simply given by the identity function:

obs COM (X [t|t -1 ]) = X [t|t -1 ] . (20)
This update with this observation function and the covariance intersection filter is the key stage for cooperative localization. In an experiment in which two vehicles were driving in a platoon, we have observed that the biases of the two vehicles were close to each other but did not result to the same value (see Fig. 4). Estimating the correlation between the two biases is complex. In [START_REF] Bento | Cooperative GNSS positioning aided by road-features measurements[END_REF], the biases are supposed to be the same for all the vehicles. In this work, we choose to model the bias of each agent as an independent bias on its own state and estimate it during the filtering. In practice it is not necessary to include the bias in the state exchange, as we have assumed the biases as uncorrelated. For simplification we exchange the whole state in our experiment.

III. EXPERIMENTAL RESULTS

A. Experimental platform

Two experimental Renault Zoe cars drove in a platoon on a test road composed of two roundabouts linked by a straight lane. The two vehicles run nine laps of this test track and data were recorded. Both vehicles were equipped with a low cost, multi-constellation (GPS, GLONASS) GNSS receiver Ublox 8. The pose of the vehicle was provided by the Ublox receiver using its internal automotive model. Only loose coupling is presented in this work. The ground truth of the poses of both vehicles were obtained by an RTK GNSS receiver / IMU Novatel Span CPT. The dead-reckoning information (longitudinal speed and yaw rate) was retrieved from the CAN bus. These observations were measured by the encoders and the gyrometer of the vehicles. The follower was equipped with a SICK LD-MRS LiDAR. One horizontal layer of this sensor was used to estimate the relative pose of the leader vehicle using the algorithm described in [START_REF] Héry | LiDAR based relative pose and covariance estimation for communicating vehicles exchanging a polygonal model of their shape[END_REF], [START_REF] Low | Linear Least-Squares Optimization for Point-to-Plane ICP Surface Registration[END_REF]. Finally, the carriageway borders detections were simulated from the observation model applied on the ground truth poses. A Gaussian noise with a standard deviation of 20 cm was added to the signed lateral distance. For map matching, we used the Lanelet projection described in [START_REF] Bender | Lanelets: Efficient map representation for autonomous driving[END_REF], [START_REF] Héry | Map-based curvilinear coordinates for autonomous vehicles[END_REF].

The data fusion was implemented with the LiDAR perception and the map-matching in python using the ROS (Robot Operating System) framework. These algorithms run with the dataset replay at the original speed. The proposed solution is fast enough for real-time applications on a classical computer.

B. Metrics

In this work, both vehicles have their own LDM in which the leader vehicle pose q l and the follower vehicle pose q f are represented. From these two absolute poses, the relative pose of the ego vehicle in the frame of the other can be computed using Eq. 14. In our case, the pose of the leader in the frame of the follower f q l for the LDM of the follower and the pose of the follower in the frame of the leader l q f for the LDM of the leader (q k = (x k , y k , θ k )

T is the pose of the vehicle at sample k and q * k denotes the ground truth). e p is the mean of the position error for the K samples:

e p = 1 K K k=1 x * k y * k - x k y k ( 21
)
e θ is the mean of the absolute heading error for the K samples: Finally, the consistency is computed for a probability error of 5% with a χ 2 test. The consistency C corresponds to:

e θ = 1 K K k=1 |θ * k -θ k | (22)
C = P r (q * k -q k ) T Σ q * k + Σ q k -1 (q * k -q k ) < χ 2 3;0.05 (23
) where χ 2 3;0.05 = 7.815 corresponds to the quantile of a χ 2 distribution with 3 degrees of freedom and a confidence of 95%. Σ q * k is the covariance matrix of the ground truth of q k as this ground truth is not certain even with RTK correction. This uncertainty is not negligible when accurate LiDAR measurements are used and must be taken into account. In practice, C is measured as the percentage of samples verifying this test.

C. Biases estimation

When carriageway borders are detected and map-matched with a high-definition map, the GNSS bias can be estimated. Fig. 5 shows the GNSS errors and the biases estimated in the LDM of the follower for the follower for the blue and black curves and in the LDM of the leader for the leader for the magenta and red curves. One can see that the biases are well estimated and stay close to the averages of the errors like low-pass filter. Even if some correlation can be seen between the GNSS errors of the two Ublox receivers, the biases are not the same. These estimated biases are different between the vehicles and change also during the 10 min of the experiment. One can see that the filters are able to estimate quite accurately the biases in real-time.

Using the estimated biases, the GNSS errors can be corrected. Fig. 6 presents these unbiased errors with estimated 95% confidence intervals. The horizontal uncertainty provided by the Ublox 8 is assimilated as the standard deviation which leads to the same confidence intervals on x and y directions. One can see on Fig. 6 that the error bounds are quite pessimistic as the 5864 samples of the dataset are all in the 95% confidence intervals. One can also see in this figure, the periodicity of the confidence intervals. Some trees and buildings are near the test track and therefore the visibility of the satellites is not the same everywhere in Fig. 6. GNSS errors with biases correction. The 95% confidence intervals have been computed using the uncertainty provided by the Ublox receiver, the uncertainty of the Span CPT and the estimated standard deviation of the biases. x is in blue and y in black for the follower and x is in magenta and y in red for the leader.

(a) f q l (b) l q f Fig. 7. Stanford diagrams of the relatives position errors from the relative poses f q l and l q f using a CIF with (in green) and without (in red) LiDAR measurements.

this road. Nevertheless this visibility is similar for the two vehicles but with a small offset as the vehicles are following each other. The nine laps are clearly visible in this pattern. Tab. I shows the position and heading errors and the consistency when carriageway borders are not available for both vehicles and therefore biases cannot be estimated. The position error drops when the bias can be estimated and the consistency becomes also pessimistic instead of being inconsistent. This consistency is similar as the one of the GNSS error when biases are compensated (see Fig. 6).

D. Cooperative localization accuracy and consistency

Without LiDAR observation both vehicles poses can be considered as uncorrelated. Even if it seems that the state does not improve much when LiDAR measurements are used (see Tab. II), the relative poses computed from this state are much more accurate. The relative position error decreases from 0.54m and 0.66m to 0.15m and 0.43m and the relative heading error from 3.45°and 4.02°to 1.83°a nd 2.17°. This result is illustrated by (simplified) Stanford diagrams in Fig. 7 where one can see the confidence bound in

Without bias estimation

With bias estimation LDM of f LDM of l LDM of f LDM of l q f q l f q l q l q f l q f q f q l f q l q l q f l q f ep (m) q l f q l q l q f l q f q f q l f q l q l q f l q f q f q l f q l q l q f l q f ep (m) 0. function of the relative position error. The confidence bound was computed with a confidence of 95% using a Rayleigh distribution as an approximation of the norm of the error. The use of the LiDAR detection (in green) improves greatly the accuracy while staying consistent as the dots are above the black lines for more than 95% of the samples (the green cloud is much more compact, closer to the vertical axis and the confidence bounds are much smaller). One can see that the relative poses computed from the state of the leader are improved, even if this vehicle is not equipped with a LiDAR and, therefore, not able to localize itself relatively to the other vehicle. The communication improves the states of the LDM of every agent if each agent is linked to others by one relative pose only. Therefore, bidirectional perceptions are not essential in this distributed data fusion scheme even if the vehicle with the perception sensor, the follower in this case, can compute a more accurate relative pose, as illustrated in Fig. 7. Errors stay consistent in the different scenarios. A good estimate of the uncertainty is very important in this cooperative localization process, not only for reliability but also to obtain accurate estimates of the poses. Indeed, the bias gives a degree of freedom to the filter. If the uncertainty is badly estimated, the bias can have difficulties to properly converge. The communication update has been also tested using EKF instead of CIF. We have noticed that the data incest issues increase the rigidity of the bias which cannot converge properly. The results are in this case inaccurate and inconsistent.

A tight field of view is simulated for the LiDAR. In this scenario, the relative pose is not available in the curved parts of the road. One can see that in this case, when the LiDAR is able to detect other vehicles only on the straight lane, the relative pose is less accurate. Indeed, the perception algorithm gives better results on curved roads when the back and one side of the vehicle are in the field of view of the LiDAR ("L-shape") instead of only the back (in this case, the heading of the other vehicle is difficult to estimate). This configuration gives more accurate relative poses. One can also see that the poses are more consistent when the relative pose is not available on the curved roads. Indeed, the lateral part of this pose becomes very accurate and can become sometimes in conflict with the carriageway borders detections. In these two configurations, it can be noticed that the use of the perception presents the advantage to be used with only two vehicles minimum. In many other works using only ranges or bearings [START_REF] Zelazo | Rigidity Theory in SE(2) for Unscaled Relative Position Estimation using only Bearing Measurements[END_REF], [START_REF] Zelazo | Decentralized Rigidity Maintenance Control with Range Measurements for Multi-Robot Systems[END_REF] more vehicles are needed to be able to build the rigid structure [START_REF] Ahmad | Cooperative robot localization and target tracking based on least squares minimization[END_REF] from these relative constraints.

E. Unavailability of carriageway borders

Two other scenarios are presented to test the robustness of this cooperative localization when perception of carriageway borders is not available.

One can see in Tab. III that even if one vehicle cannot detect the carriageway borders (and therefore estimate by itself its own GNSS bias), this bias can be estimated using the pose of the other vehicle and the relative pose between the two vehicles. The consistencies given when the carriageway borders are not available for the follower are less consistent than for the leader. Indeed the transformation from the leader frame to the follower frame presents more nonlinearity issues than from the follower frame to the leader frame. Nevertheless, the relative poses stay consistent in both scenarios.

IV. CONCLUSION

This work has presented a cooperative localization architecture by state exchange of traffic participants. It is based on EKF and CIF to manage the correlation between the state of the different vehicles. The state of each vehicle is included in the LDM which is essential for its own navigation tasks. In this process, multiple sources of information are fused: deadreckoning from the CAN bus, GNSS pose, relative poses between vehicles computed from LiDAR scans, carriageway

Carriageway borders available

Carriageway borders not available Carriageway borders not available for both vehicles for the follower for the leader LDM of f LDM of l LDM of f LDM of l LDM of f LDM of l q f q l f q l q l q f l q f q f q l f q l q l q f l q f q f q l f q l q l q f l q f ep (m) 0. In future works, this cooperative localization will be tested with more than two vehicles.

Fig. 1 .

 1 Fig. 1. Platooning of the two Renault Zoés on the test track Seville in the innovation campus of UTC in Compiègne, France. The two vehicles use GNSS localization from a Ublox 8 receiver. The direct Wifi 802.11p communication (in yellow) is simulated in this work, as the carriageway borders detections (in blue). A CAN bus gateway is used to get the deadreckoning information (in green). Finally, the follower is equipped with an LDMRS LiDAR (in red).

Fig. 2 .

 2 Fig. 2. Pose of the ego vehicle qe (in blue), of the cooperative vehicle qc (in green) and relative pose of the cooperative vehicle in the frame of the ego vehicle e qc (in red). The signed distance to the center of the lane ne is in light blue.

Fig. 3 .

 3 Fig.3. Asynchronous data fusion. The LDM X is shared between the different processing modules and sent to the other agents in cooperation.

Fig. 4 .

 4 Fig. 4. GNSS errors of two vehicles driving in platoon. Biases are clearly visible.

Fig. 5 .

 5 Fig.5. GNSS errors with bias estimates (in bold) for the follower on x (in blue) and y (in black) and for the leader on x (in magenta) and y (in red).

TABLE I POSITION

 I ep AND HEADING e θ ERROR AND CONSISTENCY C OF THE COOPERATIVE LOCALIZATION WITH AND WITHOUT BIAS ESTIMATION AND WITHOUT LIDAR MEASUREMENTS.

TABLE II POSITION

 II ep AND HEADING e θ ERROR AND CONSISTENCY C OF THE COOPERATIVE LOCALIZATION WITH A CIF, WITH AND WITHOUT LIDAR MEASUREMENTS AND WHEN THE FIELD OF VIEW OF THE LIDAR IS TIGHT.

		2.11 1.99 0.63 1.99 2.11 0.59 0.22 0.24 0.55 0.23 0.22 0.73
	e θ (°)	1.67 1.36 1.93 1.37 1.67 1.95 2.20 3.37 4.04 3.66 2.10 4.27
	C (%)	37.1 77.3 100 75.4 37.8 100	98.4 99.7 100 99.8 98.8 99.6
	With LiDAR	Without LiDAR	Tight field of view
	LDM of f	LDM of l	LDM of f	LDM of l	LDM of f	LDM of l
	q f

TABLE III POSITION

 III ep AND HEADING e θ ERROR AND CONSISTENCY C WHEN CARRIAGEWAY BORDERS ARE NOT AVAILABLE FOR THE FOLLOWER OR FOR THE LEADER.

  borders measured by a camera and finally the LDMs of other vehicles received by the communication. Experimental results have shown that the use of a LiDAR to link the poses of the different vehicles improves the relative poses computed from the states of the LMDs. Therefore, platoon navigation is more accurate and safer. We have also shown in this work than, even if the carriageway borders are not available for one of the vehicles, the cooperation gives it the capacity to estimate its GNSS bias and improve its own localization.

		20 0.26 0.15 0.25 0.20 0.42	0.23 0.28 0.15 0.26 0.24 0.43	0.20 0.39 0.15 0.39 0.21 0.32
	e θ (°)	1.23 2.38 1.83 2.47 1.15 2.17	1.35 2.47 1.82 2.55 1.26 2.54	1.54 2.58 1.79 2.26 1.47 1.66
	C (%)	96.8 95.5 99.4 94.9 97.2 98.4	90.4 93.7 99.3 93.2 90.6 97.9	96.4 94.0 99.5 93.5 96.7 98.5

ACKNOWLEDGMENT

This work was carried out in the framework of the Equipex ROBOTEX (ANR-10-EQPX-44-01) and Labex MS2T (ANR-11-IDEX-0004-02) projects. It was also carried out within SIVALab, a shared laboratory between Renault and Heudiasyc UMR UTC/CNRS.