Thomas Chevet
email: thomas.chevet@l2s.centralesupelec.fr

Cristina Stoica Maniu
email: cristina.stoica@l2s.centralesupelec.fr

Cristina Vlad
email: cristina.vlad@l2s.centralesupelec.fr

Youmin Zhang
email: ymzhang@encs.concordia.ca

Eduardo Fernandez Camacho

Eduardo F Camacho
email: efcamacho@us.es

Chance-Constrained MPC for Voronoi-based Multi-Agent System Deployment

Keywords: Multi-agent systems, Decentralized control, Model predictive control

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

INTRODUCTION

To carry out monitoring or search missions, a multiagent system (MAS) composed of unmanned vehicles can be deployed over a bounded convex area [START_REF] Cortes | Coverage control for mobile sensing networks[END_REF]. Several solutions exist for the deployment problem [START_REF] Schwager | Unifying geometric, probabilistic, and potential field approaches to multi-robot deployment[END_REF] but often rely on a dynamic Voronoi tessellation [START_REF] Voronoï | Nouvelles applications des paramètres continus à la théorie des formes quadratiques[END_REF] of the area of interest. The objective is usually to drive the MAS into a centroidal Voronoi configuration where the objective of each agent is to reach the center of mass of the Voronoi cell [START_REF] Sharifi | Cooperative multi-vehicle search and coverage problem in an uncertain environment[END_REF] or into a Chebyshev configuration where the objective is the Chebyshev center [START_REF] Nguyen | Optimization-based control for multi-agent deployment via dynamic Voronoi partition[END_REF][START_REF] Hatleskog | Voronoibased deployment of multi-agent system[END_REF][START_REF] Chevet | Decentralized MPC for UAVs formation deployment and reconfiguration with multiple outgoing agents[END_REF] of the cell, which is often less complex to compute than the center of mass. In this context, the Voronoi cells and objective points are time-varying due to the movement of the agents which are constrained to remain inside their cells. The control algorithm then has to drive each agent to its objective under this time-varying movement constraint. Generally, the literature deals with the nominal case, i.e. uncertainties/perturbations are not considered. However, real systems are subject to perturbations (e.g. wind gusts) and noises (e.g. measurement noise from sensors). The perturbations acting on the system can be modeled using deterministic or stochastic approaches. The deterministic approach requires a model of the perturbation while the stochastic approach only requires a mean and a covariance. In order to deal with these stochastic perturbations, several control methods have been proposed for a single agent, such as H ∞ [START_REF] Ugrinovskii | Robust H infinity control in the presence of stochastic uncertainty[END_REF] or optimal control [START_REF] Petersen | Minimax optimal control of stochastic uncertain systems with relative entropy constraints[END_REF], and for several agents, such as consensus [START_REF] Ma | Consensus control of stochastic multi-agent systems: a survey[END_REF] or predictive control [START_REF] Dai | Distributed stochastic MPC of linear systems with additive uncertainty and coupled probabilistic constraints[END_REF].

Model predictive control (MPC) is used for a wide variety of systems dealing with constraints, uncertainties, etc. MPC can also be used for the control of a multi-agent system either in a centralized, distributed or decentralized way. If the system is subject to bounded perturbations, robust MPC strategies have been developed [START_REF] Limon | Robust tube-based MPC for tracking of constrained linear systems with additive disturbances[END_REF] in order to further guarantee the feasibility and stability properties for a simple system. However, if these perturbations have a stochastic nature and an unbounded support such as normally distributed noises, the classical deterministic robust MPC strategies will be too conservative. To handle this situation, several stochastic MPC algorithms have been proposed to control a single system [START_REF] Farina | Stochastic linear model predictive control with chance constraints -A review[END_REF] or a multi-agent system [START_REF] Dai | Distributed stochastic MPC of linear systems with additive uncertainty and coupled probabilistic constraints[END_REF] subject to perturbations and probabilistic constraints. However, these methods often rely on a known bound for the perturbation and fixed constraints over time.

In this paper, a MAS composed of several unmanned vehicles is deployed inside a convex bounded two-dimensional polyhedral area. Each agent is subject to modeling errors and perturbations, which are represented respectively by the covariance and the mean of a process noise, as well as measurement noise, both with unknown bounds. At each time instant, each agent computes its Voronoi cell based on the knowledge of the other agents' positions. Each agent is also constrained to remain inside its cell, which can be formulated as an optimization problem subject to timevarying constraints. In this way, the agents are subject to time-varying constraints. In [START_REF] Gavilan | Chance-constrained model predictive control for spacecraft rendezvous with disturbance estimation[END_REF], the authors propose a chance-constrained MPC (CCMPC) algorithm able to deal with time-varying constraints and process noise with unknown bounds for a single system. The current paper extends the result of [START_REF] Gavilan | Chance-constrained model predictive control for spacecraft rendezvous with disturbance estimation[END_REF] to the Voronoi-based deployment of a multi-agent system subject to process and measurement noise, where the full state information is not available. Due to the noises considered in the agents' dynamical model, stochastic constraints appear in the optimization problem. These stochastic constraints are transformed into algebraic constraints by computing a bound ensuring that the constraints are satisfied for almost all possible Gaussian noise signals. Each agent uses the CCMPC algorithm to track the Chebyshev center of its Voronoi cell and the MAS converges to a static Chebyshev configuration. The contribution of this paper covers: 1) the extension of the method proposed in [START_REF] Gavilan | Chance-constrained model predictive control for spacecraft rendezvous with disturbance estimation[END_REF] when the estimation of the state has to be performed; 2) the application to the Voronoi-based deployment of a MAS subject to process and measurement noise; 3) the validation of the proposed chance-constrained MPC with a simulated fleet of quadrotor unmanned aerial vehicles (UAVs).

Notation. In the following, R is the set of the real numbers. The matrices 0 n×m and 1 n×m are the matrices of size n×m filled with zeros and ones respectively. The matrix I n is the identity matrix of size n × n. The transpose of the matrix A is denoted by A . A matrix diag (a 1 , . . . , a n ) is the diagonal matrix with diagonal elements a 1 , . . . , a n . The vector |x| contains the absolute value of each element of x. The Euclidean norm of the vector x is x 2 2 = x x. The notation A 0 (resp. A 0) means that A is a positive semidefinite (resp. positive definite) matrix. The matrix A ⊗ B is the Kronecker product of the matrices A and B. If r is a random variable, then E(r) is the expected value of r and P (r ≤ α) is the probability of the event r ≤ α, µ r and Σ r are respectively the mean and covariance of r. The set of all integers from n to m is denoted by n, m.

PROBLEM FORMULATION

In the context of a homogeneous multi-agent system1 composed of N agents, this section presents the general model of an agent i ∈ 1, N subject to process noise and measurement noise. Then, it introduces the control objective of the considered multi-agent system.

System model

An agent i ∈ 1, N obeys the continuous-time linear time invariant dynamics:

ẋi (t) = A C x i (t) + B C u i (t) + d i (t) y i (t) = Cx i (t) + γ i (t) (1)
with x i ∈ R n the state vector, u i ∈ R m the input vector, y i ∈ R p the output vector, d i ∈ R n the process noise, γ i ∈ R p the measurement noise and A C , B C and C matrices of appropriate dimensions. The pairs (A C , B C ) and (A C , C) are respectively controllable and observable.

The system (1) is discretized with the sampling period T s :

x i (k + 1) = Ax i (k) + Bu i (k) + δ i (k) y i (k) = Cx i (k) + γ i (k) (2) with A = exp (A C • T s ), B = Ts 0 exp (A C • τ ) Bdτ , δ i (k) = Ts 0 exp(A C • τ )d i ((k + 1)T s -τ )dτ . Assumption 1. The process noise d i (t) = p(t) + w i (t),
where p(t) is an external perturbation and w i (t) represents the modeling errors, and the measurement noise γ i (k) are independent and normally distributed with means µ d (t) = p(t) and µ γ (k) = 0 p×1 and covariance matrices Σ d = Σ w 0 and Σ γ 0, respectively2 . The value of p(t) is known at each time instant.

For all the agents, the pair (A, C) is assumed to be observable. Let L ∈ R n×m be the gain of a Luenberger observer such that A L = A -LC is stable. The dynamics of the estimated state x i (k) is:

x i (k + 1) = A x i (k) + Bu i (k) + L (y i (k) -y i (k)) y i (k) = C x i (k) (3)
The estimation error

ε i (k) = x i (k) -x i (k) has for dynamics: ε i (k + 1) = A L ε i (k) + δ i (k) -Lγ i (k) (4
) with the covariance matrix of the initial estimation error chosen such that Σ ε (0) 0. By introducing ε i (k) in (3): [START_REF] Franklin | Digital Control of Dynamic Systems[END_REF]. The matrix exp(A C • t) has full rank for all t [START_REF] Horn | Matrix analysis[END_REF]) and, by Assumption 1, Σ d 0, thus Σ δ 0 [START_REF] Horn | Matrix analysis[END_REF]. At time k > 0, the random vectors [START_REF] Horn | Matrix analysis[END_REF]. Since Σ δ 0, Σ ε (k) 0. 2

x i (k + 1) = A x i (k) + Bu i (k) + LCε i (k) + Lγ i (k). (5) Proposition 1. The covariance matrix Σ δ of δ i (k) is posi- tive definite and the covariance matrix Σ ε (k) of the esti- mation error ε i (k) is positive definite ∀k > 0. Proof. The covariance matrix Σ δ is such that Σ δ = Ts 0 exp(A C • τ )Σ d exp(A C • τ )dτ
ε i (k -1), δ i (k -1) and γ i (k -1) are independent. From (4), the covariance matrix of ε i (k) is Σ ε (k) = A L Σ ε (k -1)A L + Σ δ + LΣ γ L 0 (

Control objective

The state vector of an agent is

x i = x i x 1,i y i x 2,i with [x i y i ] the position of the agent in the two-dimensional plane R 2 , x 1,i ∈ R n1 and x 2,i ∈ R n2 possible additional
states (e.g. speed) such that n 1 +n 2 +2 = n. Denote by W ⊂ R 2 the convex bounded deployment region. The position of each agent belongs to W, i.e. the restriction

x r i = [x i y i ] of x i is such that x r i ∈ W. This region is divided into N Voronoi cells, one for each agent, such that W = N i=1 V i (k) and V i (k) ∩ V j (k) = ∅, ∀i, j ∈ 1, N , i = j. The Voronoi cell of an agent i is the convex bounded polygon defined as V i (k) = x ∈ W x r i (k) -x 2 ≤ x r j (k) -x 2
, ∀i = j where x r i (k) and x r j (k) are the restrictions of the state vectors x i (k), x j (k) ∈ R n of agents i and j. As a convex bounded polygon, the Voronoi cell of the agent i is the set:

V i (k) = {x ∈ W|H i,W (k)x ≤ θ i (k)} (6)
where H i,W (k) ∈ R s×2 and θ i (k) ∈ R s , with s the number of sides of the cell. Assumption 2. Each agent is equipped with sensors allowing it to know the other agents positions and is then able to compute its own Voronoi cell.

The Chebyshev center c i (k) of the cell V i (k) is the center of the largest disc lying in V i (k). This center is obtained by a linear optimization problem maximizing the value of the disc's radius such that the radius is positive and each point of the disc is in the cell [START_REF] Chevet | Decentralized MPC for UAVs formation deployment and reconfiguration with multiple outgoing agents[END_REF].

The objective is then the deployment of the multi-agent system inside a bounded convex polygon W into a static configuration such that the agents' positions are at the Chebyshev centers of their Voronoi cells despite the considered process noise and measurement noise acting on the MAS. Each agent computes its own control signal and is driven towards an objective point

x 0 i (k), u 0 i (k) solution of x 0 i (k) = Ax 0 i (k)+Bu 0 i (k) such that the restriction of x 0 i to the position in the plane R 2 is equal to the Chebyshev center of its Voronoi cell. Moreover, an extended Voronoi cell V i,x (k) = {x ∈ R n |H i (k)x ≤ θ i (k)} can be defined with H i (k) ∈ R s×n .
Given the shape of the state vector, the matrix H i is obtained by adding n 1 columns of zeros after the first column of H i,W and n 2 columns of zeros after the second column of H i,W . The matrix H i and the vector θ i will then be used as a polyhedral constraint on the state vector for the MPC problem3 .

Since the positions of the agents evolve over time, the Voronoi cells and their associated Chebyshev centers are time-varying. With a state-feedback control law, the nominal multi-agent system has been shown to converge towards a static Chebyshev configuration [START_REF] Hatleskog | Voronoibased deployment of multi-agent system[END_REF]. Nevertheless, in the presence of process noise modeled as additive Gaussian noise, a robust control algorithm is required in order to ensure the noise rejection and the multiagent system convergence towards a static configuration. The next section then presents a decentralized robust MPC algorithm based on a chance-constrained approach.

MAIN RESULT

In this section, a decentralized chance-constrained MPC algorithm is formulated based on the estimation of the agent's state. The useful matrices for the definition of the optimization problem are defined. Then, the cost function and constraints of the optimization problem are expressed with the previously defined matrices. However, due to the presence of stochastic perturbations in the agent's model, the agent is subject to probabilistic constraints. A way to transform these probabilistic constraints into algebraic constraints is finally introduced.

State prediction

Over a prediction horizon N p , the estimated state (5) and the estimation error (4) become respectively:

X i (k) = F x i (k) + G u U i (k) + G ε E i (k) + G γ Γ i (k) (7) E i (k + 1) = F ε ε i (k) + G ε δ ∆ i (k) -G ε γ Γ i (k) (8) with X i (k) = x i (k + 1) • • • x i (k + N p ) , U i (k) = u i (k) • • • u i (k + N p -1)
and E i (k) (resp. Γ i (k) and ∆ i (k)) defined the same way as U i (k) by replacing u i by ε i (resp. γ i and δ i ). The matrix F is such that

F = A • • • A Np
and the matrix F ε is obtained by replacing A by A L in F . The matrix G u is such that:

G u =       B 0 n×m • • • 0 n×m . . . . . . . . . . . . . . . . . . 0 n×m A Np-1 B • • • • • • B       , ( 9
)
the matrices G ε and G γ are obtained by replacing B in (9) respectively by LC and L, and the matrices G ε δ and G ε γ are obtained by replacing A by A L and B respectively by I n and L.

Chance-constrained MPC problem

Based on [START_REF] Gavilan | Chance-constrained model predictive control for spacecraft rendezvous with disturbance estimation[END_REF], a decentralized robust chance-constrained model predictive controller for an agent i ∈ 1, N following the dynamics (2) is expressed as follows:

min

U i (k) J(k) (10a) s.t. x i (k + l) ∈ V i,x (k), l ∈ 1, N p , ( 10b
) U i (k) ∈ U i , (10c)
with

J(k) = E X obj i (k) Q X obj i (k) + U i (k) RU i (k) . The weights Q, R 0 are diagonal matrices and X obj i (k) = X i (k) -X 0 i , where X 0 i = 1 Np ⊗ x 0 i (k) with x 0 i (
k) the objective state vector defined in Section 2.2. For a vehicle, u 0 i is often the null vector, thus it is not considered here.

If u 0 i = 0 m×1 , then U i (k) is replaced in (10a) by U i (k) - U 0 i (k), where U 0 i (k) = 1 Np ⊗ u 0 i (k).
Due to stochastic process and measurement noise, solving the optimization problem (10) requires to find a bound for the unknown terms appearing in the constraint (10b) as detailed hereafter. In Section 3.3, this bound is the solution of an optimization problem under probabilistic constraints.

Given the expression of X i (k) from ( 7), the cost function (10a) is rewritten by removing the terms that do not depend on U i (k) and have no influence on the solution of (10):

J(k) = U i (k) G u QG u + R U i (k) + 2(F x i (k) + G ε µ E (k) -X 0 i ) QG u U i (k) (11)
with µ E (k) = E (E i (k)). Since µ γ (k) = 0 p×1 , the mean of Γ i (k) is the null vector and it does not appear in the expression of the cost function J(k).

The extended Voronoi cell is such that

V i,x (k) = {x ∈ R n |H i (k)x ≤ θ i (k)} as detailed in Section 2.2.
To predict the evolution of the Voronoi cells over the prediction horizon, each agent would have to know at each time instant the input sequence of all the other agents. With Assumption 2, there is no need of communication between the agents. The extended Voronoi cell of each agent is then considered constant over the prediction horizon. Thus, the constraint (10b) can be rewritten over the prediction horizon by using (7) and replacing X(k) by its expression:

H i,u (k)G u U i (k) ≤ Θ i,u (k) -H i,u (k)F x i (k) -H i,u (k) (G ε E i (k) + G γ Γ i (k)) (12)
where

H i,u = I Np ⊗ H i and Θ i,u = 1 Np ⊗ θ i .
In the constraint (10c), U i is a set of polyhedral constraints such that

U i = u ∈ R mNp |u| ≤ U max , where U max ∈
R mNp with only positive components. With this formulation, the constraint (12) depends on the knowledge of the process noise and the measurement noise over the prediction horizon. However, the value of these signals is not known and the only available information are the mean and covariance of these noises. The constraint (12) then needs to be modified to depend only on the available information in order to solve the problem (10).

Noise bound for robust satisfaction of constraints

In ( 12), a bound for the uncertain term

H i,u (k) G ε E i (k)+ G γ Γ i (k) , which is also written H i,u (k) [G ε G γ ] Ξ i (k), with Ξ i (k) = E i (k) Γ i (k)
the disturbance vector, has to be found in order to express the constraint (12) only based on known information.

If the disturbance vector Ξ i (k) is normally distributed with mean µ Ξ (k) and covariance matrix Σ Ξ (k), then: 13) follows a chi-squared law with (n + p)N p degrees of freedom [START_REF] Flury | A first course in multivariate statistics[END_REF], where (n + p)N p is the size of Ξ i (k). Then, finding the solution α of the chance constraint P χ 2 ((n + p)N p ) ≤ α = P ensures that Y i (k) is bounded by α with a probability P [START_REF] Gavilan | Chance-constrained model predictive control for spacecraft rendezvous with disturbance estimation[END_REF].

Y i (k) = (Ξ i (k) -µ Ξ (k)) Σ -1 Ξ (k) (Ξ i (k) -µ Ξ (k)) (
The objective is then to find a bound b

i (k) such that b i (k) ≤ -M i Ξ i (k), where M i = H i,u (k) [G ε G γ ],
with a probability P close to one. With this bound, the constraint ( 12) is satisfied for almost all disturbances. This bound is:

(b i (k)) l = min Ξ i (k) -(M i ) l Ξ i (k) s.t. Y i (k) ≤ α (14)
where (M i ) l and (b i (k)) l are respectively the l-th row of M i and the l-th element of b i (k) and Y i (k) defined in (13).

To guarantee the existence of the bound, Ξ i (k) has to be normally distributed, thus Σ Ξ has to be invertible. Since ε i (k -1), ∆ i (k -1) and Γ i (k -1) are independent, from the expression of E i (k) in ( 8), this covariance is:

Σ Ξ (k) = Σ E (k) -G ε γ Σ Γ,L -Σ Γ,L G ε γ Σ Γ ( 15
)
where:

Σ E (k) = F ε Σ ε (k-1)F ε +G ε δ Σ ∆ G ε δ +G ε γ Σ Γ G ε γ , ( 16
)
with Σ Γ = I Np ⊗ Σ γ , Σ ∆ = I Np ⊗ Σ δ and Σ Γ,L a block lower diagonal matrix with Σ γ the element at line l and column l -1, where l > 1.

Proposition 2. The covariance matrix Σ Ξ (k) is positive definite ∀k > 0.

Proof. By taking the Schur complement [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF] in ( 15), Σ Ξ (k) 0 if and only if Σ Γ 0 (which is always true) and S(k)

0 with S(k) = Σ E (k) - G ε γ Σ Γ,L Σ Γ -1 Σ Γ,L G ε γ . Since Σ γ 0, Σ γ is invertible and Σ Γ -1 = I Np ⊗ Σ γ -1 . Using (16), S(k) becomes S(k) = F ε Σ ε (k -1)F ε + G ε δ Σ ∆ G ε δ + G ε γ Σ Γ,1 G ε γ , with Σ Γ,1 = diag Σ γ , 0 (p-1)Np×(p-1)Np . To have S(k)
0, at least one of the matrices defining S(k) has to be positive definite while the others only need to be positive semidefinite. Proposition 1 gives Σ ε (k -1) 0 thus [START_REF] Horn | Matrix analysis[END_REF]. Since

F ε Σ ε (k -1)F ε 0 (
Σ Γ,1 0, it is verified that G ε γ Σ Γ,1 G ε γ 0. It remains to prove that G ε δ Σ ∆ G ε δ 0. The matrix G ε δ defined in Section 3.1 is G ε δ = I nNp + T where T is a strictly lower triangular matrix. Thus, G ε δ Σ ∆ G ε δ = Σ ∆ + T Σ ∆ T + T Σ ∆ +Σ ∆ T ,
with the last two terms positive semidefinite because Σ ∆ is a block diagonal matrix and T is strictly lower triangular, thus their eigenvalues are all 0. In addition, T Σ ∆ T 0 and

Σ ∆ 0 thus Σ Ξ (k) 0. 2 With Proposition 2, Σ Ξ (k) is invertible.
By applying a similar procedure to the one presented in [START_REF] Gavilan | Chance-constrained model predictive control for spacecraft rendezvous with disturbance estimation[END_REF], the optimization problem ( 14) is explicitly solved and the l-th element of b i (k) is:

(b i (k)) l = -(M i ) l αΣ Ξ (k) (M i ) l -(M i ) l µ Ξ (k). ( 17
)
Then, by replacing the constraint (10b) by ( 12) with the bound defined in (17), it is now possible to find a solution to the proposed decentralized CCMPC (10).

SIMULATIONS

In this paper, the considered multi-agent system is a fleet of four quadrotor UAVs. The nonlinear continuous-time state-space model of a UAV agent is presented in [START_REF] Chevet | Decentralized MPC for UAVs formation deployment and reconfiguration with multiple outgoing agents[END_REF]. This model is the one used to simulate the behavior of an agent. The control architecture for this system is as follows: a position reference is provided to the position controller which computes pitch and roll references for the orientation controller. The yaw reference is constant during the flight. Then, the orientation and the altitude controllers provide torque and upward thrust inputs to the nonlinear UAV model. The chance-constrained MPC algorithm described in Section 3 is used for the position controller, while the orientation and altitude controllers are obtained by feedback linearization as in [START_REF] Voos | Nonlinear control of a quadrotor microuav using feedback-linearization[END_REF].

Position subsystem model

To design the position controller, the position subsystem is obtained from the linearized and discretized state-space model [START_REF] Chevet | Decentralized MPC for UAVs formation deployment and reconfiguration with multiple outgoing agents[END_REF] to fit the system (2). For one UAV agent, the state vector is

x i = [x i v x,i y i v y,i ] ,
with i ∈ 1, 4, x i , y i the agent's position and v x,i , v y,i the components of its horizontal speed. The input vector is

u i = [φ i θ i ]
, with φ i and θ i the agent's roll and pitch angles. The yaw reference remains equal to 0 • during the flight, thus the state matrices defined in (2) are:

A = I 2 ⊗ 1 T s 0 1 B = 1 2 gT s 0 1 -1 0 ⊗ T s 2 , C = I 2 ⊗ [1 0] (18)
with g = 9.81 m • s -2 the gravitational acceleration and T s the sampling period. The noise covariances are Σ d = diag(0.001, 0.025, 0.001, 0.025) and Σ γ = 0.0011 • I 2 and µ d will be described in Section 4.4.

Position controller

The chance-constrained model predictive controller runs at T out s = 0.2 s with a prediction horizon N p = 10, while the inner loop (the orientation controller) runs at T in s = 0.001 s. Thus, to accurately estimate the system's state, the structure from Fig. 1 is used. On a real UAV, the variation rate of the angle reference is limited to avoid important variations on the actuators' inputs. Thus, here, the variation rate of the angle reference is limited to ±100 • /s. Then, this reference

φ ref i (k) θ ref i (k)
is sent to the inner loop controller and an observer, both running at T in s as shown in Fig. 1. This observer is meant to provide an accurate value of the state to the outer loop controller and thus runs at the same rate as the inner loop controller. The state estimate is then sent to the position controller every T out s seconds.

With this structure, the prediction model (3) considered for the chance-constrained MPC design uses the matrices in (18) with T s = T out s , while the second observer's model is obtained from the matrices in (18) with T s = T in s . The observer gain L in (3) is a linear quadratic estimator obtained with the weights Q = 100I 4 and R = I 2 . The observer in Fig. 1 is also a linear quadratic estimator obtained with the weights Q = 10I 4 and R = I 2 . The objective point x 0 i (k) is a vector with four elements: the coordinates of the Chebyshev center of the cell as first and third elements and zeros as second and fourth elements. Moreover, the weights Q and R for the cost function (10a) are chosen to be Q = I Np ⊗ diag(10, 1, 10, 1) and R = 100I 2Np . The angle reference is limited to ±30

• such that U i = u ∈ R 2Np |u| ≤ π/6 • 1 2N p .
Finally, with the considered dimensions for the problem, the chi-squared variable presented in Section 3.3 has 60 degrees of freedom. Thus, if P is chosen to be 0.999, nearly all the disturbances are bounded by α = 99.607 in (14).

Orientation and altitude controllers

The following orientation and altitude controllers are obtained as presented in [START_REF] Voos | Nonlinear control of a quadrotor microuav using feedback-linearization[END_REF]. is out of the scope of this paper and these are given for reproducibility of the results. The torque on the x axis is computed as τ

x,i (k) = (I z -I y ) ω y,i (k)ω z,i (k) + I x K 1 ω x,i (k) + K 2 φ ref i (k) -φ i (k)
where ω x,i , ω y,i and ω z,i are the angular speeds I x = I y = 0.03 kg • m 2 and I z = 0.04 kg • m 2 are the moments of inertia along the three axes and K 1 and K 2 are parameters chosen to obtain the desired dynamics for the inner loop. Here, the values K 1 = -200 and K 2 = 100 are considered for the parameters. The torques τ y,i (k) and τ z,i (k) are obtained with a cyclic permutation of x, y and z in the expression of τ x,i (k) and by replacing φ by θ and ψ (the yaw angle) respectively, with ψ ref i (k) = 0. With the same approach, the total upward thrust is

F t,i (k) = m • (g + K 3 v z,i (k) + K 4 z ref i (k) -z i (k) )/(cos φ i (k) cos θ i (k)) -1
where z i and v z,i are the altitude and vertical speed, z ref i is the altitude reference, m = 1.4 kg is the mass of the UAV and K 3 and K 4 are parameters chosen to obtain the desired dynamics for the altitude loop. Here, the values K 3 = -3.6 and K 4 = 4 are considered for the parameters.

Simulation results

A fleet of four quadrotor UAVs is deployed inside a square of 20 m of side length: W = x ∈ R 2 |x| ≤ 10 • 1 2 . They all start from a hovering position with z i (0) = 5 m, with i ∈ 1, 4, at

x 1 (0) = [1 0 1 0] , x 2 (0) = [1 0 -1 0] , x 3 (0) = [-1 0 1 0] , x 4 (0) = [-1 0 -1 0] .
At t = 0 s, they start tracking the Chebyshev center of their associated Voronoi cell and are subject to a process noise and a measurement noise having a mean µ d (t) = µ γ (k) = 0 4×1 and the covariances Σ δ and Σ γ described in Section 4.1. From t = 12 s to t = 20 s, each UAV is subject to a wind gust of 2 m/s on each axis such that µ d (t) = [0 2 0 2] which is assumed to be known by the agents as stated in Assumption 1.

The first part of the movement, from t = 0 s to t = 9.8 s, is shown in Fig. 2. The initial positions [x i (0) y i (0)] are represented by discs (red discs). The evolution of the real positions [x i y i ] (blue circles) as well as the evolution of the estimated positions [ x i y i ] (red circles) and the Chebyshev centers c i (black asterisks) of the four agents every 0.2 s are presented inside the Voronoi partition at t = 9.8 s. The position of each agent converges to its Chebyshev center as expected from [START_REF] Hatleskog | Voronoibased deployment of multi-agent system[END_REF], while the estimated position converges quickly to the real position. In the following, since the formation is symmetric, only the results for the agent 2 are provided, the results for the other agents being similar.

Figure 3 presents the distance between the real position [x 2 y 2 ] of agent 2 and its Chebyshev center c 2 and the distance between the real position and the estimated position [ x 2 y 2 ] of agent 2. The estimated position is initialized to the agent's real position. The bottom plot of Fig. 3 shows that the estimation error is relatively small ( ε 2 (k) 2 < 6 • 10 -3 ) during the entire movement.

The top plot of Fig. 3 shows that the agent converges to its Chebyshev center since the distance between the real position of agent 2 and c 2 is close to 0 m after 2 s. However, from t = 12 s to t = 20 s, the system is subject to wind perturbations. Because there is no integral action in the proposed control algorithm, a static error appears while the system is perturbed. Without perturbation, the agent converges back to its Chebyshev center.

CONCLUSION

This paper presents a decentralized chance-constrained model predictive control algorithm with state estimation meant to drive a multi-agent system subject to process noise and measurement noise into a static configuration when deployed inside a bounded convex two-dimensional area. At each time instant, each agent computes a Voronoi cell in which it will evolve and the Chebyshev center of this Voronoi cell. Then, with the proposed approach, the system converges towards a static configuration in which each agent lies on its Chebyshev center despite model mismatch and perturbations (represented respectively by the covariance and the mean of the process noise) and measurement noise. Simulation results show the efficiency of the proposed approach. Future work will extend this result to the case of biased measurements corresponding, for example, to sensor faults.

Fig. 1 .

 1 Fig. 1. Structure of the position controller.

Fig. 2 .

 2 Fig. 2. Evolution of the agents' real and estimated positionand Chebyshev center from t = 0 s to t = 9.8 s.

Fig. 3 .

 3 Fig. 3. Distance between the agent 2 real position and itsChebyshev center (top) and between the agent 2 real position and its estimated position (bottom).

A homogeneous MAS is considered to simplify the reading of the paper. However, the results can be extended to the case of heterogeneous MAS.

These means and covariances are considered to be the same for all the agents. However, the result can be generalized to the heterogeneous case where the means and covariances are different for each agent.

Such constraints cause a coupling between the agents, thus the need for a decentralized control algorithm.

The authors acknowledge the LIA of the CNRS on Information, Learning and Control, MEECOD project ("Moderniser l'Enseignement par l'Expérimentation sur la COordination de Drones") supported by UPSaclay, the Natural Sciences and Engineering Research Council of Canada, MEyC Spain (contract DPI2016-76493-C3-1-R), the European Research Council and the UE ERDF (Advanced Grant OCONTSOLAR, Project ID: 789051).


Jessica Kojadinovic
email: jessica.kojadinovic@univ-nantes.fr

Michel Potier

Matthieu Le Corre

Richard P Cosson

Paco Bustamante

Mercury content in commercial pelagic fish and its risk assessment in the Western Indian Ocean

Keywords: Mercury, Impregnation levels, Pelagic fish, Reunion Island, Mozambique Channel, Health standards

As top predators of pelagic food webs, large fish naturally bioaccumulate mercury (Hg). Determining Hg burdens in commercialized fish is essential considering the concern about effects of contaminants on human health and the legal thresholds that are therefore set for local consumption and/or exportation. Total Hg levels were measured in the muscular tissue of 183 fish of five commercially important species from the tropical zone of the Western Indian Ocean. All individuals were measured and sexed in order to study the impregnation of Hg with size and sex within each species. Values of Hg found in this part of the Indian Ocean were comparable to Hg in muscular tissue of the same species studied in other areas. The highest Hg levels were noted in Swordfish (Xiphias gladius) caught in waters surrounding Reunion Island (3.97 ± 2.67 µg.g -1 dry weight). Following the Swordfish, in decreasing order of Hg content, were the Yellowfin Tuna (Thunnus albacares) and the Skipjack (Katsuwonus pelamis), then the Common Dolphinfish (Coryphaena hippurus) and the Wahoo (Acanthocybium solandri). In the North of the Mozambique Channel, Swordfish had higher Hg levels than Yellowfin Tunas, and Dolphinfish exhibited intermediate Hg levels. The size of a fish was a determinating factor of its Hg burden, as was the species. Differences in size-normalized Hg levels were observed between the two study zones for Swordfish and Common Dolphinfish. Sex, in contrast, did not influence Hg levels suggesting that females and males have similar feeding habits.

The muscular Hg levels presented here suggest that consumers of fish originating from the Western Indian Ocean should limit themselves to one Swordfish based meal per week, or one fish meal a day if they choose to eat tuna or Common Dolphinfish.

Introduction

Historically, mercury (Hg) has been used by many cultures for a variety of symbolic and useful purposes, such as in good luck charms, to ward off evil, as material for ceremony objects (Egyptians), as colorant and as cosmetics [START_REF] Boudou | Recherche en écotoxicologie expérimentale sur les processus de bioaccumulation et de transfert des dérivés du mercure dans les systèmes aquatiques continentaux[END_REF]Mercury Task Force, 2002). Mercury played a predominant role in alchemy, and was thought to have medicinal uses such as curing syphilis in the 19th century [START_REF] Dracobly | Theoretical change and therapeutic innovation in the treat-ment of syphilis in mid-nineteenth-century France[END_REF]. The large scale accidents of Minamata in the 1950's, and in Iraq in the early 1970's have, however, caught the world's attention on the potential toxicity of this metal. Much research has been conducted since, and health and ecological hazards are relatively well assessed today. The major anthropogenic Hg sources stem from its use in industries such as the manufacture of plastic, chlorine, caustic soda (sodium hydroxide), caustic potash (potassium hydroxide) and antifouling paint. Fossil fuel burning, base metal smelting, waste incinerators and Hg based fungicides used in agriculture are also important input sources of Hg in the environment. Natural sources of Hg include volcanic emissions, degassing from soils, and volatilization from the ocean. As a natural element, Hg is present as Hg vapors in the atmosphere, as inorganic Hg in the geosphere (to a very large extent encountered as mercuric sulphide cinnabar) and as inorganic or organic Hg in the hydrosphere. Among organic Hg species, methylmercury (MeHg) is the most abundant, and is bioaccumulated by aquatic organisms and biomagnified through the food chain.

Because of MeHg's trophic transfer property, organisms occupying high trophic positions and characterized by long lifespans, such as tuna fish and billfish are susceptible to contain high Hg burdens. Indeed, Hg found in fish is chiefly present as MeHg [START_REF] Cappon | Mercury and selenium content and chemical form in fish muscle[END_REF][START_REF] Bloom | A survey of size-specific mercury concentrations in game fish from Maryland fresh and estuarine waters[END_REF][START_REF] Wagemann | Percent methylmercury and organic mercury in tissues of marine mammals and fish using different experimental and calculation methods[END_REF][START_REF] Kehrig | Total and methylmercury in a Brazilian estuary, Rio de Janeiro[END_REF]. The study of Hg impregnation in top predators is crucial as it serves as evidence for environmental exposure levels. Furthermore Hg burdens in large predatory fish are of particular interest because fish represent the major source of human exposure [START_REF] Svensson | Fish as a source of exposure to mercury and selenium[END_REF].

To the best of our knowledge, there have been very few published studies on Hg in fish from the Western Indian Ocean [START_REF] Kureishy | Total mercury content in some marine fish from the Indian Ocean[END_REF]. Fisheries of the Western Indian Ocean represent 5% of worldwide catches. Tunas and associated species amount to approximately 30% of the catches in the Western Indian Ocean and 17% of worldwide tuna catches, which sets it in second place after the Pacific [START_REF] Pianet | Etat des stocks de thonidés dans l'Océan Indien[END_REF]. Among tunas and associated species, the major tunas, which include the Yellowfin (Thunnus albacares), the Skipjack (Katsuwonus pelamis), the Bigeye (Thunnus obesus), the Albacore (Thunnus alalunga) and the Southern Bluefin (Thunnus maccoyii ), are of particular importance for fisheries since they correspond to 70% of all catches in the Indian Ocean. Swordfish (Xiphias gladius) also stand out in fishery reports. Longliners based in Reunion Island have been specializing in Swordfish since the early 1990's, with catches amounting to 1 741 tons per year in 2000. Total catches in the Western Indian Ocean exceed 20 000 tons per year nowadays [START_REF] Pianet | Etat des stocks de thonidés dans l'Océan Indien[END_REF]. Furthermore, other species such as Wahoos (Acanthocybium solandri ) and Common Dolphinfish (Coryphaena hippurus) are targeted by sport fisherman.

This paper provides new data for Hg levels in muscle of Swordfish, Yellowfin tunas, Skipjacks, Wahoos and Common Dolphinfish originating from the Southwestern Indian Ocean. Potential differences in Hg bioaccumulation between sexes were tested. Because Hg levels almost invariably increase with the size of the fish (e.g. [START_REF] Mackay | Selenium and heavy metals in black marlin[END_REF][START_REF] Monteiro | Mercury content of Swordfish, Xiphias gladius, in relation to length, weight, age and sex[END_REF][START_REF] Bloom | A survey of size-specific mercury concentrations in game fish from Maryland fresh and estuarine waters[END_REF]Dietz et al., 1996;[START_REF] Gilmour | A survey of size-specific mercury concentrations in game fish from Maryland fresh and estuarine waters[END_REF][START_REF] Stafford | Mercury contamination and growth rate in two piscivore populations[END_REF], length-adjusted Hg levels were used to compare the Hg impregnation of each species found in Reunion Island waters and in the North of the Mozambique Channel. Species Hg impregnations were also compared within study zones. Considering the nutritional value of fish and the large quantities consumed in this part of the world, our results were discussed in the light of international guidelines for Hg intake, and of potential health risk for the populations which feed on these fish.

Materials and methods

Study site and species

The samples of the five pelagic species studied during this investigation originated from two geographically distant sites in the Western Indian Ocean. The first site corresponds to waters surrounding Reunion Island, a 2 512 km 2 French territory located 700 km east of Madagascar (21 • 7'S ; 55 • 33'E). The second sampling zone was situated in the northern part of the Mozambique Channel, delimited by the following coordinates : 10 • 34'S and 17 • 07'S, and 41 • 11'E and 47 • 19'E (Figure 1). These two sites correspond to two major fishing (seine and longline) areas of the Western Indian Ocean [START_REF] Weenarain | Impact économique des activité thonières industrielles[END_REF].

The Swordfish is the largest of the sampled species averaging 165 cm LJLF (Lower Jaw Fork Length) in body length [START_REF] Fao | Fisheries global information system, Species identification sheets[END_REF]. Females have a higher growth rate than males and therefore, reach greater lengths. In the Southwestern Indian Ocean, females may reach up to 250 cm LJLF, and males, up to 210 cm LJLF [START_REF] Vanpouille | Etude de la croissance de l'espadon (Xiphias gladius). In: Etude des caractéristiques biologiques de l'espadon (Xiphias gladius) dans le sud-ouest de[END_REF]. The Swordfish's lifespan is thought to be under 25 years [START_REF] Fao | Fisheries global information system, Species identification sheets[END_REF]. Thirty-five thousand tons are fished in the Indian Ocean every year [START_REF] Vanpouille | Etude de la croissance de l'espadon (Xiphias gladius). In: Etude des caractéristiques biologiques de l'espadon (Xiphias gladius) dans le sud-ouest de[END_REF].

Yellowfin tunas measure around 155 cm FL (Fork Length), but they can grow up to 240 cm FL and weigh up to 175 kg (NOAA fisheries, Oct. 2005). The maximum reported age is 8 years [START_REF] Fao | Fisheries global information system, Species identification sheets[END_REF]). This species is the second most coveted by industrial and artisanal fisheries. The Western Indian Ocean basin is characterized by some of the largest quantities of Yellowfin tunas captures by seines and longliners in the world (around 300 000 tons per year in 1994) [START_REF] Pianet | Etat des stocks de thonidés dans l'Océan Indien[END_REF].

The Skipjack is the smallest tuna in the Indian Ocean. It usually weighs less than 5 kg and measures around 60 cm FL, although it can reach up to 30 kg and 100 cm FL in length [START_REF] Stéquert | La pêche de surface des thonidés tropicaux dans l'Océan Indien[END_REF][START_REF] Quéro | Les poissons de mer des pêches françaises, identification, inventaire et répartition de 209 espèces[END_REF]. Estimates of longevity vary between 8 and 12 years [START_REF] Fao | Fisheries global information system, Species identification sheets[END_REF]). This species is the most commercialized tuna with more than 1 000 000 tons fished per year in the world, of which 200 000 tons come from the Western Indian Ocean [START_REF] Muus | Guide des poissons de mer et pêche[END_REF][START_REF] Opic | Poissons commerciaux du sud-ouest de l'Océan Indien[END_REF].

The maximum reported size for the Wahoo is 250 cm FL total length, with a maximum weight of 83 kg. Typically, individuals attain a size of 100-170 cm. It is believed that Wahoo lives up to 5-6 years of age [START_REF] Fao | Fisheries global information system, Species identification sheets[END_REF]. Wahoo and other Seerfish annual catches approach 120 000 tons in the Indian Ocean [START_REF] Pianet | Etat des stocks de thonidés dans l'Océan Indien[END_REF].

The Common Dolphinfish is known to reach a maximum of 200 cm FL in length and 30 kg in weight, but it more commonly measures 50-100 cm FL and weighs 14 kg. Longevity is estimated around 4 years [START_REF] Fao | Fisheries global information system, Species identification sheets[END_REF]. Although they are not subject to specific fishing from longliners in the Western Indian Ocean, Common Dolphinfish commercialization is non negligible [START_REF] Gaertner | Analyse des interactions entre les captures de grands pélagiques de la flotille palangrière réunionnaise et les conditions de pêche (caractéristiques techniques, environnement)[END_REF]. This fish is also highly prized as a gamefish.

Since 1988, fish aggregation devices (FADs) have multiplied around Reunion Island, representing 85% of coastal fishing by 1994 [START_REF] Conand | Les DCP à l'île de La Réunion: histoire, évolution, influence sur les prises et l'activité de la pêche côtière[END_REF]. FADs characteristically concentrate young Yellowfin Tunas, Skipjacks, Wahoos and Dolphinfish.

Fish sampling and mercury analysis

A total of 65 fish (3 species) were caught with a traw line in the northern part of the Mozambique Channel and 118 fish (5 species) were collected by sport fishermen in Reunion waters. Each fish was measured, and weighed when possible. The Lower Jaw Fork Length (LJFL, from the tip of the lower jaw to the fork of the caudal fin) was measured on Swordfish whereas the Fork Length (FL, from the tip of the snout to the fork of the caudal fin) was noted for the others species. Individuals were sexed during dissection by the examination of the gonads. For practical reasons, white muscle was sampled for analysis in the abdominal area above the vent of the fish. We considered that Hg is uniformly distributed in fish edible muscle as it has been shown for Swordfish (Freeman and Home, 1973). The sampled muscle was conserved frozen at -20 • C. It was then blended, dried in an oven at 50 • C to constant mass and grounded to a fine powder. Glass and plastic utensils were washed with detergent, plunged in a bath of mixed nitric acid (35 ml.l -1 ) and chlorhydric acid (50 ml.l -1 ) for a minimum of 24 h, rinsed 3 times in deionized (Milli-Q quality) water and dried in an oven at 50 • C before use.

Total Hg analyzes were carried out with an Advanced Mercury Analyzer (combustion analyzer ALTEC 254) on aliquots ranging from 7 to 24 mg of dried sample weighed to the nearest 0.01 mg. Accuracy of the preparation was tested by preparing replicate TORT-2 reference standards and blanks along with each set of samples. The detection limits and recovery rates of Hg were respectively equal to 0.0007 µg.g -1 and 101±3.5%. Reproducibility was estimated as the closeness of 2 replicate measurements. The coefficient of variation given by relative standard deviations was always lower than 10%. Element levels are expressed in µg.g -1 of dry weight (d.w.), and the mean moisture content of each tissue is provided in Table 1. These values were used for the conversion of our results to wet weight (w.w.) basis for comparison with other studies.

Table 1 Percentage of water in the white muscle of pelagic fish from Reunion Island and the Mozambique Channel (mean ± standard deviation).

Data analysis

Prior to statistical analysis, bivariate plots of Hg levels against fish's lengths were visually examined for outliers. Statistical analysis were performed using the GNU R statistical system (R Development Core Team, 2005). All statistical samples submitted to tests were first checked for normality by means of Shapiro-Wilk tests. In the case of non-departure from normality, parametric tests were used in the subsequent analyzes, otherwise, non-parametric analogues were used.

The validity of the replicates of the level measures was tested by means of t-tests for paired samples. The influence of species on elemental levels was tested by means Kruskal-Wallis tests followed by Wilcoxon tests for independent samples using Bonferroni's p-value correction. The influence of sex and sampling location on Hg levels were tested by means of t-tests or Wilcoxon tests, after having checked samples for normality by means of Shapiro-Wilk tests.

Simple linear regression models (RMs) were estimated to explain the relationship between Hg and size. The best determination coefficients of Hg content against fish length regressions were obtained using logarithmic transformations of both variables. The RMs were thus realized on log Hg values (µg.g -1 d.w.) and log lengths (cm). RMs were also used to estimate the average size above which Hg content may exceed tissue levels that were used to set consumption advisories.

In most fish, muscle Hg content increases with the size of the individual [START_REF] Mackay | Selenium and heavy metals in black marlin[END_REF][START_REF] Monteiro | Mercury content of Swordfish, Xiphias gladius, in relation to length, weight, age and sex[END_REF][START_REF] Bloom | A survey of size-specific mercury concentrations in game fish from Maryland fresh and estuarine waters[END_REF]Dietz et al., 1996;[START_REF] Gilmour | A survey of size-specific mercury concentrations in game fish from Maryland fresh and estuarine waters[END_REF][START_REF] Stafford | Mercury contamination and growth rate in two piscivore populations[END_REF]. The difference in the average length of fish is thus a bias when comparing subsamples using comparison tests such as those described above. In order to truly measure the effect of other potential influencing factors such as the sampling location or the species, tested subgroups should contain fish of the same mean size. To test the influence of the location, an analysis of covariance (ANCOVA) was Table 2 Mean Hg levels (µg.g -1 d.w.), inter-specific and inter-locational comparisons results in pelagic fish from two areas of the tropical Western Indian Ocean. SD stands for standard deviation and CV for coefficient of variation. For each species, the significance of the Hg level differences among locations is given in the last column. In case of significant difference, the letters a and b are used to indicate which subgroups differ. Subgroups sharing the same letter do not differ significantly. The significance of the Hg level differences between species are given in the bottom line. In case of significant difference, the letters X, Y and Z, are used to indicate which subgroups differ.

performed for each species, in order to adjust the Hg levels in both sampling sites (Mozambique Channel and Reunion Island) using fish length as a covariate. The ANCOVA was also used to adjust the Hg levels in each species, using fish length as a covariate, to test whether size was the only explanatory factor of the differences observed between species. In the Mozambique Channel, Dolphinfish were excluded from the ANCOVA because of their very small sample size once the outliers were removed. The ANCOVAs were realized on log Hg values and log lengths. Residuals were checked for normality by means of Shapiro tests, and for homocedaticity by plotting fitted values vs. residuals [START_REF] Venables | Statistics and computing. Modern applied statistics with S. Springers[END_REF][START_REF] Faraway | Sampling the edible muscleof the swordfish (Xiphias gladius) for the total mercury content[END_REF].

In an effort to facilitate the interpretation of the results we have adopted classical conventions throughout this paper. Kruskal-Wallis, ANCOVA, Wilcoxon and t tests will be represented, respectively, by the following acronyms : KW , AOCV , W and t.

Results

Summary statistics for Hg levels in the white muscle of each species caught in the Mozambique Channel and Reunion waters are presented in Table 2. Males and females were pooled since no influence of sex was revealed within the species-location subgroups which were tested (Table 3).

Size-specific Hg levels

The large individual variations in Hg levels within species-location subgroups (high CVs in Table 2) can be explained by the large range of sizes of the fish in each subgroup. Indeed, Hg levels were found to be positively correlated with the length of the fish (Figure 2). The relationships between the Hg levels ([Hg]) and the length (L) were best fitted by the a regression of the form:

[Hg] = a.L b , where a and b are equation parameters estimated from the data and given for each species-location subgroup in Table 4. These regressions models were significantly better than constant models in all but one cases, Reunion Swordfish, probably because of a too small number of individuals. Besides, the model linking Hg and length in Mozambique Channel Dolphinfish is not given because of the excessively small sample size when ignoring the outliers.

Site differences

From the results given in Table 2 it appears as though the levels of muscular Hg differed between Swordfish and Common Dolphinfish caught in Reunion waters and those caught in the Mozambique Channel. However, the average size of the individuals collected in both zones were not identical (Table 2), and as it has been discussed in Section 3.1, Hg levels vary with the fish's length. ANCOVAs were realized, for Swordfish, in order to assess whether fish length differences alone explained the Hg levels variation noted between both sampling areas. The ANCOVA results indicated that the regression slopes of muscular Hg levels against fish length were not significantly different between areas (p = 0.548), but that the intercepts were significantly different (p < 0.001). We may thus conclude that, for Swordfish, muscular Hg accumulates at the same rate with respect to length in both locations, but that Hg levels were different, with higher levels in Reunion Swordfish.

Despite the different ranges of sizes of Yellowfin Tunas sampled in the Reunion and Mozambique areas, Hg levels did not significantly differ between these two groups (Table 2). Non-significantly different regression slopes of Hg on fish length (AOCV : p = 0.152) and non-significantly different intercepts (AOCV : p = 0.051) between areas confirmed that there were no Hg level differences between Mozambique Channel and Reunion Island fish, and that Hg was bioaccumulated at the same rate with respect to size in Yellowfin Tunas of both zones.

Mercury level variations among species

In both study areas, the Swordfish showed the highest Hg levels of all species. In Reunion, it was followed by the Yellowfin Tuna, the Skipjack, the Common Dolphinfish and the Wahoo in order of decreasing Hg burdens (Table 2). In the Mozambique Channel, the large sizes of the sampled Dolphinfish resulted in higher muscular Hg levels than in Yellowfin Tunas. A more extensive sampling of Dolphinfish in the Mozambique Channel would be necessary to better estimate Hg levels in this species.

ANCOVAs were realized, for both sites, to set aside the effect of length in order to assess whether other factors than length were responsible for the variations in Hg levels among species. In other words, would there still be differences in Hg burden among species if they all had the same average size? In both sites, the answer was positive, since the intercepts of the regression of Hg burdens against length were significantly different between species (p < 0.001). We may conclude that factors inherent to the species had an influence on Hg levels in these fish. Furthermore, the regression slopes of Hg burdens against length were the same for Swordfish and Yellowfin Tunas from the Mozambique Channel (p = 0.212) and for the five species from Reunion waters (p = 0.619). Hence, Hg levels in muscle seemed to be bioaccumulated at the same rate with respect to fish length in all species.

Discussion

Mercury, size, sex, species and location

The size of a fish is known as a determining factor of its Hg burden [START_REF] Mackay | Selenium and heavy metals in black marlin[END_REF][START_REF] Jaffar | Selected trace metal concentrations in different tissues of fish from coastal waters of Pakistan (Arabian Sea)[END_REF][START_REF] Bloom | A survey of size-specific mercury concentrations in game fish from Maryland fresh and estuarine waters[END_REF]Dietz et al., 1996;[START_REF] Gilmour | A survey of size-specific mercury concentrations in game fish from Maryland fresh and estuarine waters[END_REF][START_REF] Stafford | Mercury contamination and growth rate in two piscivore populations[END_REF][START_REF] Kraepiel | Sources and variations of mercury in tuna[END_REF], and has been confirmed here.

For Swordfish larger than 125 cm LJFL, females' growth rate is higher than that of male [START_REF] Kume | Size composition and sexual maturity of billfish caught by the Japanese longline fishery in the Pacific Ocean est of 130 • W. Bull[END_REF][START_REF] Quéro | Les poissons de mer des pêches françaises, identification, inventaire et répartition de 209 espèces[END_REF][START_REF] De Martini | Biological research in support of swordfish stock assessment[END_REF]NOAA fisheries, Oct. 2005), and more generally, females are proportionally heavier than males of equal length [START_REF] Ward | Broadbill swordfish, status of world fisheries[END_REF]Elscot, 2000, Skilman &Young 1974 in). It can be suspected that these physiological differences may affect length-Hg relations. In this study, no difference in Hg levels was observed in any of the five species. The influence of sex-differentiated growth rates between male and female Swordfish for individuals larger than 125 cm was probably diluted since very few fish were above this length: two females and two males among Reunion samples, and in the Mozambique Channel lot, only one female measured more than 125 cm in length.

According to the average sizes of the five teleost species investigated here (Section 2.1), the Swordfish is the largest followed by the Yellowfin Tuna, the Wahoo, the Common Dolphinfish and the Skipjack in decreasing order. In the Mozambique sample, the Common Dolphinfish had the same mean length as the Yellowfin Tunas, in consequence, their muscular Hg levels were not significantly different (Table 2 & Figure 3). Among Reunion fish, Skipjacks had high Hg levels with respect to their size when compared to the other species (Figure 3). The opposite was observed for Wahoos. For example, Skipjacks caught around Reunion Island were smaller than Common Dolphinfish but exhibited higher Hg values, and Wahoos, which were in average larger than both of the latter, had the lowest Hg levels (Table 2). These interspecific differences in Hg levels were probably linked to differences in each species physiology, feeding rate, growth rate, lifespan, migratory patterns, foraging habits and/or diet. For instance, the low Hg levels in Wahoos could be explained by the surface foraging habits of this species (0-12 m) which contrasts with the others which feed from the surface to great depths (up to 800 m for Swordfish) where Hg methylation, in these poorly oxygenated waters, enhances its bioaccumulation [START_REF] Monteiro | Mercury concentrations in prey fish indicate enhanced bioaccumulation in mesopelagic environments[END_REF][START_REF] Kraepiel | Sources and variations of mercury in tuna[END_REF]. A clear example of differential vertical distribution of Hg levels in fish is given by [START_REF] Monteiro | Mercury concentrations in prey fish indicate enhanced bioaccumulation in mesopelagic environments[END_REF], who found a 4-fold increase in Hg levels from epipelagic (100-150 m) to mesopelagic (300-1 200 m) lanternfish species in the sub-tropical mid-North Atlantic. Locational differences between the North of the Mozambique Channel and Reunion waters were confirmed for Swordfish. More data on Dolphinfish from the Mozambique Channel are needed to verify whether these fish have higher Hg burdens than those collected in Reunion waters, irrespectively of their size. If the latter is confirmed, it would go against the results obtained for Swordfish. Consequently, it could be hypothesized that the differences in fish Hg impregnation were due to differences in their dietary prey composition and/or in the trophic position they occupy in both areas, rather than differences in environmental Hg levels. Conversely, the intense volcanic activity on Reunion Island could account for higher Hg levels in Swordfish caught in the waters surrounding this island. This second hypothesis would be more likely if Dolphinfish did not show higher levels in the Mozambique Channel.

Comparison with published data

The Hg levels detected in pelagic fishes during this study were quite similar to phylogenetically related species from oceans around the world as shown in Table 5, especially given that the fish compared here vary in sizes. On the Southwestern Indian Ocean scale, Seychelles waters seem to be richer in Hg than the Mozambique Channel and Reunion waters, since Yellowfin Tunas, Skipjacks and Wahoos caught there had higher Hg levels than those analyzed in this study. These values are however relatively low in comparison with Hg levels that can be found in the muscular tissue of fish analyzed in polluted coastal areas, or in deep-water benthopelagic fish. For example, the small piscivorous Acestrorhynchus guianensis from the Amazon basin, which is polluted by goldmining, exibits Hg levels as high as 4.3 µg.g -1 d.w. [START_REF] Durrieu | Goldmining and mercury contamination of the piscivorous fish Hoplias aimara in French Guiana (Amazon basin)[END_REF]. Furthermore, median Hg levels of 13 µg.g -1 d.w. were reported for the small deep-sea Smooth Grenadier, Nezumia aequalis, found in the 500-1 750 m bathymetric zones of the Rockall Trough, North West of the British Isles [START_REF] Mormede | Trace elements in deep-water fish species from the Rockall Trough[END_REF]. The record of highest Hg levels in teleost Table 5 Mercury levels (Mean ± Standard Deviation, µg.g -1 wet weight) in muscle of marine fish from various geographical areas. * indicates the locations for which data were transformed from dry weight to wet weight basis.

species is also noteworthy since it was found in Black Marlins from the apparently unpolluted waters off North-Eastern Australia, with 24 µg.g -1 d.w. reported in its muscle tissue [START_REF] Mackay | Selenium and heavy metals in black marlin[END_REF].

Risk to the fish and their predators

Mercury in fish can pose health risks to the fish themselves and to their predators, including humans. Mercury levels of 5 µg.g -1 w.w. in fish muscle can be associated with loss of appetite, emaciation, decrease coordination and mortality in fish itself [START_REF] Eisler | Mercury hazards to fish, wildlife, and invertebrates: a synoptic review[END_REF], while levels of 15 µg.g -1 w.w. are required for provoking adverse effects in their predators [START_REF] Spry | Metal bioavailability and toxicity to fish in low-alkalinity lakes: A critical review[END_REF]. In this study, the highest Hg level recorded was 2.68 µg.g -1 w.w. in a Swordfish, suggesting that Hg is not a threat to the fish themselves, nor to their predators.

Since the large scale MeHg poisoning accident of Minamata in Japan in 1957, Hg levels in fish have been thoroughly examined in view of human consumption. In 1979, the Food and Drug Administration in the United States (US FDA) established an action level of 1 µg.g -1 w.w. to regulate MeHg content in commercial fish. Although guidance or standards for Hg in fish tissues are not always uniform (Burger and Gochfeld, In Press), most international standards have, since then, adopted an action limit of 0.5 µg.g -1 w.w., except for predatory fish (such as billfish, tunas and other high-Hg level fish) for which the allowed level is 1 µg.g -1 w.w. (IPCS, 1987;[START_REF] Epa | Consultation technique sur les cadre juridique et les politiques économiques à adopter pour promouvoir une aquaculture commerciale durable en afrique subsaharienne[END_REF]. It is recommended that the population restrains from consuming, on a regular basis, species exceeding these values. Based on this, Hg is considered as a potential safety hazard for Skipjacks, Halibuts, the Spanish Mackerel (Scomberomorous maculatus), the King Mackerel (S. cavalla), Marlins, Sharks, the Swordfish and the Bluefin Tuna [START_REF] Mendez | Total mercury content -fish weight relationship in swordfish (Xiphias gladius) caught in the Southwest Atlantic Ocean[END_REF][START_REF] Dua | Mercury toxicology as assessed through fish scales[END_REF].

Because a large percentage of Hg is present as MeHg in the edible portions of fish [START_REF] Cappon | Mercury and selenium content and chemical form in fish muscle[END_REF][START_REF] Bloom | A survey of size-specific mercury concentrations in game fish from Maryland fresh and estuarine waters[END_REF][START_REF] Wagemann | Percent methylmercury and organic mercury in tissues of marine mammals and fish using different experimental and calculation methods[END_REF], we assumed that concentration of total Hg (T-Hg) was equal to that of MeHg in the muscle. None of the Common Dolphinfish or Skipjacks considered in this study were above the 1 µg.g -1 guideline. Yellowfin Tunas caught in the Mozambique Channel were also under this limit whereas, one of the 20 Yellowfin Tunas fished around Reunion Island exceeded the provisional maximum permissible level. In the Mozambique Channel and Reunion Island, 2.6% and 42.9% of the swordfish respectively, were above the authorized limits. This last result on Reunion Swordfish should however be taken very cautiously and not generalized owing to the very small sample size (7 fish). Further analyzes should be conducted to provide a truly valid percentage. Furthermore, the body length of these fish, which is a major variation factor of Hg levels (Section 3.1), is to be considered when addressing the problem of potential health risks as the consumption of smaller fish reduces the amount of Hg ingested. Moreover, the thresholds established for Hg in fish muscle destinated to human consumption do not take into account the amount of fish ingested by consumers. To evaluate the potential health risk to people through consumption of these fish, Hg intake rates were estimated on the basis of the Hg levels in fish muscle (our results transformed into wet weight basis) and daily fish consumption found in the literature [START_REF] Antoine | Données actuelles des ichtyosarcotoxismes à la Réunion[END_REF]FAO, 2001;Mauritius Central Statistics Office, 2002). The daily human exposure to Hg from fish consumption can be calculated with the following formula: µg Hg . day -1 = µg Hg . g f ish -1 × g f ish . day -1 Different Acceptable Daily Intake (ADI) limits have been established by national and international instances. The ADI set by the World Health Organization (WHO) for T-Hg is 0.71 µg.day -1 .kg -1 body weight, and restricted to 0.35 µg.day -1 .kg -1 body weight for pregnant women because foetus are more [START_REF] Antoine | Données actuelles des ichtyosarcotoxismes à la Réunion[END_REF], Mauritius Central Statistics Office 2002[START_REF] Fao | Fisheries global information system, Species identification sheets[END_REF].

sensitive to Hg toxicity, for nursing mothers, and for children less than 10 years [START_REF] Chan | Metal accumulation and detoxification in humans[END_REF][START_REF] Dhhs | Lead, cadmium, mercury and selenium in Greenland marine animals[END_REF]. The French (AFSSA) and the Canadian health agencies follow the same guidelines as the WHO, whereas the U.S. FDA and U.S. Environmental Protection Agency (EPA) have set more restrictive ADI limits for MeHg (0.4 and 0.1 µg.day -1 .kg -1 body weight respectively for all the population) [START_REF] Hirsch | Avis de l'agence française de sécurité sanitaire des aliments relatif à l'évaluation des risques sanitaires liés à l'exposition au mercure des femmes enceintes et allaitantes et des jeunes enfants[END_REF].

Considering an average adult body weight of 70 kg (EPA, 1994), the T-Hg WHO ADI can be approximated as 50 µg.day -1 .adult -1 . This health risk limit was compared to the estimated daily consumption of Hg in various countries where fish originating from the two study zones are susceptible to be eaten (Table 6). The T-Hg values exceeding the most restrictive guidlines set by the EPA (equivalent to 10.5 µg T-Hg.day -1 .adult -1 ) are also indicated in Table 6, but will not be discussed. Only 4 of the 7 locations seemed confronted with potential health risks due to Hg, i.e. Reunion and Coastal Mainland France. Furthermore, these conclusions are valid only if the population's fish intake is restricted to Swordfish caught in Reunion waters. This obviously rarely represents the reality. For example, inhabitants of the Comoros and Creoles from Reunion Island, chiefly eat demersal fish (Ministère de la production et de l 'environnement, 2000). French living on Mainland France do favor pelagic fish such as Tuna, Marlin and Swordfish, but also eat local demersal fish [START_REF] Antoine | Données actuelles des ichtyosarcotoxismes à la Réunion[END_REF]. Moreover, the risk for human health can be minimized if the consumption of Swordfish is limited to a certain size. For example, given the limited data in this study, Hg levels seem to exceed the permissible level only in individuals measuring more than 132 cm in length (Figure 2). For smaller Swordfish the health risk is considered as null, with average daily Hg Table 7 Safety limits (WHO), expressed as the frequency of meals for which fish is the main element, based on an adult standard portion size of 230 g. These estimations do not apply for pregnant woman and children.

intakes per capita of 28.78 µg in Reunion and coastal France, 36.45 µg in Comoros and 38.37 µg in Mauritius. In the same way, Western Indian Ocean Yellowfin Tunas measuring less than 224 cm should be under the 1 µg.g -1 advisory level, as would be Skipjacks, Wahoos and Common Dolphinfish of any size. There is, however, a non negligible drawback in encouraging the consumption of small Swordfish since most of these individuals are immature: the smallest mature female recorded in the Western Indian Ocean measured 127 cm LJF L [START_REF] Taquet | Dorade coryphène commune (Coryphaena hippurus)[END_REF]. Targeting immatures can pose a great threat to the entire Swordfish stock.

In order to better appreciate the amounts of fish safe for consumption, exposure limits can be expressed as the number of meals that an adult can eat per day, or per week, and be under the 50 µg Hg.day -1 ADI (hence 350 µg Hg.week -1 ). In risk assessment, the standard portion size of uncooked fish eaten by an average adult is estimated to be 230 g [START_REF] Epa | Consultation technique sur les cadre juridique et les politiques économiques à adopter pour promouvoir une aquaculture commerciale durable en afrique subsaharienne[END_REF][START_REF] Brodberg | Evaluation of potential health effects of eating fish from Black Butte reservoir (Glenn and Tahama counties): guidelines for sport fish consumption[END_REF]. Safety limits, expressed as the frequency of meals for which fish is the main element, based on the Hg levels reported in this study are given in Table 7. For individuals weighing more, or less, than 70 kg, it is assumed that their consumption rates will be proportionally higher or lower, respectively, yielding an overall similar exposure level following the consumption of equally contaminated fish [START_REF] Brodberg | Evaluation of potential health effects of eating fish from Black Butte reservoir (Glenn and Tahama counties): guidelines for sport fish consumption[END_REF].

Nevertheless, consumers should bear in mind that standards have a safety margin. A concrete example of the limitations of the guidelines for Hg is given by two decades of investigation on the neighboring Seychellois population. During the Seychelles Child Development study, 779 children from the Seychelles whose mothers ate great amounts of Hg-contaminated fish (0.3 µg.g -1 w.w. mean content) during their pregnancies (on average 12 meals per week) were followed from birth to adulthood and showed no significant evidence of verbal, motor, memory or reasoning skills deficits [START_REF] Matthews | Mercury content of commercial important fish of the Seychelles, and hair mercury levels of a selected part of the population[END_REF][START_REF] Myers | Prenatal methylmercury exposure from ocean fish consumption in the Seychelles child development study[END_REF][START_REF] Shamlaye | The Seychelles child development study: two decades of collaboration[END_REF]. Furthermore, a dietary survey carried out with 26 adult women from a fish-eating community in the Brazilian Amazon indicated that eating tropical fruit reduced Hg exposure from fish consumption [START_REF] Passos | Eating tropical fruit reduces mercury exposure from fish consumption in the Brazilian Amazon[END_REF].

Conclusion

This study confirms that Swordfish has high Hg levels in comparison with most other pelagic fish. The large size of this species is one of the main explanatory factor of the high Hg burdens as they increases with fish length. Besides, the Swordfish was the only species for which clear differences in Hg levels were noted between fish from both study areas.

As far as human safety is concerned, the results of this study suggest that consumers of fish originating from the South Western Indian Ocean should limit themselves to one meal a day if they consume tunas, Wahoo or Common Dolphinfish, or one meal a week if they eat Swordfish. Pregnant women, nursing mothers and children should limit their consumption of the largest pelagic species. Mercury level determination would be of interest in other edible tissues such as liver and gonads where Hg is known to accumulate [START_REF] Cizdziel | Distribution of mercury in the tissues of five species of freshwater fish from Lake Mead, USA[END_REF][START_REF] Bustamante | Distribution of trace elements in tissues of benthic and pelagic fish from the Kerguelen Islands[END_REF]. Further investigations should also include other large commercialized teleost species such as Marlins. In any case, consumers should bear in mind that standards have a margin of safety and that fish provides essential nutrients such as Omega-3 fatty acids, phosphorus, calcium, and various vitamins and minerals. Moreover, fish constitutes an important source of protein, and is often the main livelihood for many people throughout the world. For these reasons, it is important to adapt Hg thresholds to local population's true exposure levels.

Fig. 1 .

 1 Fig. 1. Map of the study area with sampling zones materialized by shaded circles.

Fig. 2 .

 2 Fig.2. Mercury levels (µg.g -1 d.w.) against fish length (cm). Mozambique and Reunion data are represented by and • respectively, and the corresponding regression curves by full and dotted lines respectively. The gray horizontal line materializes the threshold level for human consumption (1 µg.g -1 ) set by WHO.

Fig. 3 .

 3 Fig. 3. Position of species in relation to their mean length and mean Hg levels in both study zones. Error bars represent the standard deviation.

Table 3 P

 3 -values of pairwise t-tests or Wilcoxon tests for the comparison of Hg level in males and females of each species, in the Mozambique Channel and Reunion Island. (-) indicates subgroups for which samples had too few values to be tested.

	Table 4
	Regression models linking Hg levels ([Hg]) to length (L): [Hg] = a.L b .

Table 6

 6 Estimated daily consumption of T-Hg (µg), in various countries in which people are likely to eat fish captured in the Mozambique Channel and Reunion waters. The bold values indicate the most probable consumer countries. The shadowed values are above the ADI set by the WHO. The values in dotted boxes are above the ADI set by the EPA. The countries' average yearly fish consumption per capita (kg f resh weight) is given under each country considered

Acknowledgments

This research was supported by the Conseil Général de La Réunion. J.K. also benefited from support of the Conseil Régional de La Réunion and the European Social Fund through a PhD grant. The authors would like to thank J. Bourjea and M. Taquet (Ifremer -La Réunion), as well as Maeva Pêche au Gros, Réunion Fishing Club, L. & P. Berthier, and other anonymous Reunionese fisherman for their assistance in the sampling effort. We would also like thank C. Churlaud, (Centre Commun d'Analyse -La Rochelle), E. Robert, M. Rouquette, P. Grondin and N. Ghanem (University of La Réunion) for their help in the sample preparation and analysis. We would also like to thank the referees for helpful comments on the manuscript.