Ihab S Mohamed
email: ihab.mohamed@inria.fr

Guillaume Allibert

Philippe Martinet
email: philippe.martinet@inria.fr

Model Predictive Path Integral Control Framework for Partially Observable Navigation: A Quadrotor Case Study

Recently, Model Predictive Path Integral (MPPI) control algorithm has been extensively applied to autonomous navigation tasks, where the cost map is mostly assumed to be known and the 2D navigation tasks are only performed. In this paper, we propose a generic MPPI control framework that can be used for 2D or 3D autonomous navigation tasks in either fully or partially observable environments, which are the most prevalent in robotics applications. This framework exploits directly the 3D-voxel grid acquired from an on-board sensing system for performing collision-free navigation. We test the framework, in realistic RotorS-based simulation, on goal-oriented quadrotor navigation tasks in a cluttered environment, for both fully and partially observable scenarios. Preliminary results demonstrate that the proposed framework works perfectly, under partial observability, in 2D and 3D cluttered environments.

I. INTRODUCTION

Having a safe and reliable system for autonomous navigation of robotic systems such as Unmanned Aerial Vehicles (UAVs) is a highly challenging and partially-solved problem for robotics communities, especially for cluttered and GPSdenied environments such as dense forests, crowded offices, corridors, and warehouses. Such a problem is very important for solving many complex applications, such as surveillance, search-and-rescue, and environmental mapping. To do so, UAVs should be able to navigate with complete autonomy while avoiding all kinds of obstacles in real-time. To this end, they must be able to (i) perceive their environment, (ii) understand the situation they are in, and (iii) react appropriately.

Obviously enough, this problem has been already addressed in the literature, particularly those works related to dynamics and control, motion planning, and trajectory generation in unstructured environments with obstacles [START_REF] Mellinger | Trajectory generation and control for precise aggressive maneuvers with quadrotors[END_REF], [START_REF] González | A review of motion planning techniques for automated vehicles[END_REF], [START_REF] Mohta | Fast, autonomous flight in GPS-denied and cluttered environments[END_REF], [START_REF] Baca | Model predictive trajectory tracking and collision avoidance for reliable outdoor deployment of unmanned aerial vehicles[END_REF], [START_REF] Ryll | Efficient trajectory planning for high speed flight in unknown environments[END_REF]. Moreover, the applications of the path-integral control theory have recently become more prevalent. One of the most noteworthy works is Williams's iterative path integral method, namely MPPI control framework [START_REF] Williams | Model predictive path integral control: From theory to parallel computation[END_REF]. In this method, the control sequence is iteratively updated to obtain the optimal solution on the basis of importance sampling of trajectories. In [START_REF] Williams | Information-theoretic model predictive control: Theory and applications to autonomous driving[END_REF], authors derived a different iterative method in which the control-and noise-affine dynamics constraints, on the original MPPI framework, are eliminated. This framework is mainly based on the information-theoretic interpretation of optimal control using KL-divergence and free energy, while it was previously based on the linearization of Hamilton-Jacob Bellman (HJB) equation and application of Feynman-Kac lemma. Although different methods are adopted to derive the MPPI framework, they are practically equivalent and theoretically related 1 . An extension to Williams's information-theoretic-based approach is presented in [START_REF] Kusumoto | Informed information theoretic model predictive control[END_REF], where a learned model is used to generate informed sampling distributions.

The attractive features of MPPI controller, over alternative methods, can be summarized as: (i) a derivative-free optimization method, i.e. no need for derivative information to find the optimal solution; (ii) no need for approximating the system dynamics and cost functions with linear and quadratic forms, i.e., non-linear and non-convex functions can be naturally employed, even that dynamics and cost models can be easily represented using neural networks; (iii) planning and execution steps are combined into a single step, providing an elegant control framework for autonomous vehicles. However, one drawback of MPPI is that its convergence rate is empirically slow, which has been addressed in [START_REF] Okada | Acceleration of gradient-based path integral method for efficient optimal and inverse optimal control[END_REF].

In the context of autonomous navigation, it is observed that the MPPI controller has been mainly applied to the tasks of aggressive driving and UAVs navigation in 2D cluttered environments. To do so, MPPI requires a cost map, as an environment representation, to drive the autonomous vehicle. Concerning autonomous driving, the cost map is obtained either off-line [START_REF] Williams | Information-theoretic model predictive control: Theory and applications to autonomous driving[END_REF] or from an on-board monocular camera using deep learning approaches [START_REF] Drews | Aggressive deep driving: Combining convolutional neural networks and model predictive control[END_REF], [START_REF] Buyval | Model predictive path integral control for car driving with autogenerated cost map based on prior map and camera image[END_REF]. Regarding UAV navigation in cluttered environments, the obstacle (i.e., cost) map is assumed to be available, and only static 2D floormaps are used. Conversely, in practice, the real environments are often partially observable, with dynamic obstacles. Moreover, it is noteworthy that only 2D navigation tasks are performed so far, which limits the applicability of the control framework. For this reason, this paper focuses on MPPI for 2D and 3D navigation tasks in a previously unseen and dynamic environment. In particular, the main contributions of our work can be summarized as follows:

1) We propose a generic MPPI framework for autonomous navigation in cluttered 2D and 3D environments, which are inherently uncertain and partially observable. To the best of our knowledge, this point has not been reported in the literature, which opens up new directions for research. 2) We demonstrate this framework on a set of simulated quadrotor navigation tasks using RotorS simulator and Gazebo [START_REF] Furrer | Robot Operating System (ROS): The Complete Reference[END_REF], assuming that: (i) there is a priori knowledge about the environment (namely, fully observable case); (ii) there is no a priori information (namely, partially observable case). In this case, the robot is building and updating the map, which represents the environment, online as it goes along. This allows the opportunity to navigate in dynamic environments. 3) To ensure a realistic simulation, our proposed framework is evaluated taking into account the modelling errors, noisy sensors, and windy environments.

This paper is organized as follows. Section II describes the quadrotor dynamics model which represents our case study for the framework validation, whereas in Section III the real-time MPPI control strategy is explained. Our proposed framework is then described in Section IV and evaluated in Section V. Finally, concluding remarks are provided in Section VI.

II. QUADROTOR DYNAMICS MODEL

Considering a quadrotor vehicle model illustrated in Fig. 1, the dynamics model can be defined by assigning a fixed inertial frame W and body frame B attached to the vehicle. The origin of the body frame, B, is located at the center of mass of the quadrotor, where x B and y B lie in the quadrotor plane defined by the centers of the four rotors, and z B is perpendicular to this plane and points upward. The inertial reference frame, W, is defined by x W , y W , and z W , with z W pointing upward. We assume that the first rotor (i.e., along the +x B axis) and the third rotor rotate counter clockwise (namely, +1), whilst the second (i.e., along the +y B axis) and fourth rotors rotate clockwise (namely, -1). Here, the vehicle is only subject to: (i) a gravitational acceleration g in -z W direction; (ii) the sum of the forces generated by each rotor, F = 4 i=1 F i , acts along the +z B direction. Furthermore, the Euler angles, with ZXY transformation sequence, are used to model the rotation of the quadrotor in frame W, where the roll φ, pitch θ, and yaw ψ angles refer to a rotation about x B , y B , and z B axis, respectively. The rotation matrix from B to W is accordingly expressed as

F 1 F 2 F 3 F 4 x B y B z B {B} x W y W z W {W} -gz W
W R B =   c ψ c θ -s φ s ψ s θ -c φ s ψ c ψ s θ + c θ s φ s ψ c θ s ψ + c ψ s φ s θ c φ c ψ s ψ s θ -c ψ c θ s φ -c φ s θ s φ c φ c θ   ,
where s x = sin(x) and c x = cos(x) ∀ x ∈ {φ, θ, ψ}. The transformation matrix from Euler angular velocities, Φ, to body frame angular velocity, Ω, is given by

T =   c θ 0 -c φ s θ 0 1 s φ s θ 0 c φ c θ   .
Given all considerations above and according to the welldescribed models in [START_REF] Mellinger | Trajectory generation and control for precise aggressive maneuvers with quadrotors[END_REF] and [START_REF] Kai | Nonlinear feedback control of quadrotors exploiting first-order drag effects[END_REF], the dynamics of the position ξ, linear velocity v, orientation locally defined by Euler angles Φ, and body angular rates Ω can be written as

ξ = v, m v = -mge 3 + W R B F e 3 , Φ = T -1 Ω, J Ω = Γ -Ω × JΩ, ( 1
)
where m is the mass of the quadrotor and J its inertia matrix expressed in the body frame

B, ξ = [x, y, z] T , v = [v x , v y , v z ] T , Φ = [φ, θ, ψ] T , Ω = [p, q, r] T , e 3 = [0, 0, 1] T ∈ R 3 , F ∈ R +
is the accumulated force (i.e., thrust) generated by all rotors and constitutes the first control input to the system, Γ

= [τ x , τ y , τ z ] T = [L(F 2 -F 4 ), L(F 3 - F 1 ), (M 1 -M 2 + M 3 -M 4 )] T ∈ R 3
is the total torque applied to the vehicle with its components expressed in B which represents the second control input, and L ∈ R + is the distance from the center of the vehicle to the axis of rotation of each rotor. In this paper, the state of the system is defined as x = [x, y, z, φ, θ, ψ, v x , v y , v z , p, q, r] T ∈ R 12 . Each rotor produces a vertical force, F i , and moment, M i , according to

F i = k F ω 2 i and M i = k M ω 2 i
, where ω i is the angular velocity of the i th rotor, k F is the rotor force constant, and k M is the rotor moment constant. Accordingly, the mapping between the control inputs, namely F and Γ, and the system's input, i.e., ω i , in order to control the quadrotor, can be expressed as

    F τ x τ y τ z     =     k F k F k F k F 0 k F L 0 -k F L -k F L 0 k F L 0 k M -k M k M -k M         ω 2 1 ω 2 2 ω 2 3 ω 2 4     . (2)

III. MPPI CONTROL STRATEGY

The MPPI controller is a stochastic Model Predictive Control (MPC) method, which can be applied to nonlinear dynamics and non-convex cost objectives. So, it is a sampling-based and derivative-free approach. The key idea of MPPI is to sample thousands of trajectories, based on Monte-Carlo simulation, in real-time from the system dynamics. Each trajectory is then evaluated according to a predefined cost function. Consequently, the optimal control sequence is updated over all trajectories. This can be easily done, in realtime, by taking advantage of the parallel nature of sampling and using a Graphics Processing Unit (GPU).

Let assume that the discrete-time stochastic dynamical system has a form of

x t+1 = f (x t , u t + δu t ) , (3)
where x t ∈ R n is the state vector of the system at time t, u t ∈ R m denotes a control input for the system, and δu t is a zero-mean Gaussian noise vector with a variance of Σ u , i.e., δu t ∼ N (0, Σ u ), which represents the control input updates. Given a finite time-horizon t ∈ {0, 1, 2, • • • , T -1}, the objective of the stochastic optimal control problem is to find a control sequence, u = (u 0 , u 1 , . . . u T -1 ) ∈ R m×T , which minimizes the expectations over all generated trajectories taken with respect to (3), i.e., J = min u E [S (τ )], where

S (τ ) ∈ R is the state-dependent cost-to-go of a trajectory τ = {x 0 , u 0 , x 1 , • • • , u T -1 , x T }.
Accordingly, the optimal problem can be formulated as

J = min u E φ (x T ) + T -1 t=0 q (x t ) + 1 2 u T t Ru t , (4)
where φ (x T ), q (x t ), and R ∈ R m×m are a final terminal cost, a state-dependent running cost, and a positive definite control weight matrix, respectively. To solve this optimization problem, we consider the iterative update law derived in [START_REF] Williams | Model predictive path integral control: From theory to parallel computation[END_REF], in which MPPI algorithm updates the control sequence, from t onward, as

u t ← u t + K k=1 exp -(1/λ) S (τ t,k ) δu t,k K k=1 exp -(1/λ) S (τ t,k ) , (5)
where K is the number of random samples (namely, rollouts), λ ∈ R + is a hyper-parameter so-called the inverse temperature, and

S (τ t,k ) = φ (x T ) + T -1 t=0 q (x t , u t , δu t ) is the modified cost-to-go of the k th rollout from time t onward.
In this work, the modified running cost q (x, u, δu) is defined as

q = q(x) + 1 2 u T Ru + 1 -ν -1 2 δu T Rδu + u T Rδu, (6)
where ν ∈ R + refers to the exploration noise which determines how aggressively the state-space is explored. It is noteworthy that the low values of ν result in the rejection of many sampled trajectories because their cost is too high, while too large values result in that the controller produces control inputs with significant chatter.

The real-time control cycle of MPPI algorithm is described in Algorithm 1 with more detail. At each time-step ∆t, the system current state is estimated, and a K × T random control variations are generated on a GPU using CUDA's random number generation library (lines 2 : 3). Then, based on the parallel nature of sampling, all trajectory samples are executed individually in parallel. For each trajectory, the dynamics are predicted forward and its expected cost is computed (lines 5 : 12), bearing in mind that the cost of each Algorithm 1 Real-Time MPPI Control Scheme [START_REF] Williams | Model predictive path integral control: From theory to parallel computation[END_REF] Given: K, T : Number of rollouts (samples) & timesteps (u 0 , u 1 , . . . , u T -1 ) ≡ u: Initial control sequence f, ∆t: Dynamics & time-step size φ, q, λ, ν, Σ u , R: Cost functions/hyper-parameters SGF: Savitzky-Galoy (SG) convolutional filter 1: while task not completed do 2:

x 0 ← StateEstimator(), x 0 ∈ R n 3: δu ← RandomNoiseGenerator(), δu ∈ R K×T 4: S (τ k ) ← TrajectoryCostInitializer(), S (τ k ) ∈ R K 5:
for k ← 0 to K -1 do 6:

x ← x 0 7:

for t ← 0 to T -1 do 8:

x t+1 ← x t + f (x t , u t + δu t,k ) ∆t 9: S (τ t+1,k ) ← S (τ t,k ) + q 10:
end for 11:

S (τ k ) ← S (τ t+1,k ) + φ (x T ), ∀t = T -1 12:
end for 13:

Smin ← min k [ S (τ k )] 14:
for t ← 0 to T -1 do 15:

u t ← u t + K 1 k=0 exp 1 λ [ S(τ t,k )-Smin] δu t,k K 1 k=0 exp 1 λ [ S(τ t,k )-Smin]
16:

end for 17:

u ← SGF(u)

18:

u 0 ← SendToActuators(u) 19: for t ← 1 to T -1 do 20: u t-1 ← u t 21:
end for

22: u T -1 ← ControlSequenceInitializer(u T -1 )
23:

Check for task completion 24: end while trajectory is zero-initialized (line 4). The control sequence is then updated (lines 14 : 16), taking into account the minimum sampled cost Smin (line 13). Due to the stochastic nature of the sampling procedure which leads to significant chattering in the resulting control, the control sequence is then smoothed using a Savitzky-Galoy (SG) filter (line 17). Finally, the first control is executed (line 18), while the remaining sequence of length T -1 is slid down to be used at next time-step ∆t (lines 19 : 22).

IV. GENERIC MPPI FRAMEWORK

In this section, we present a generic and elegant MPPI framework, as illustrated in Fig. 2, in order to not only navigate autonomously in previously unseen 2D or 3D environments while avoiding collisions with obstacles but also to explore and map them. Moreover, we describe how the environment is represented to be used by MPPI for partially and fully observable navigation tasks. In Fig. 2, we present the block diagram of our proposed control framework integrated into the Robot Operating System (ROS). The individual components of our framework are described in detail in the following sections.

A. Environment Representation

To clarify how MPPI can be used for 2D or 3D navigation in unseen environments, we assume that the environment is discretized into a 2D or 3D grid M g (see Fig. 2), where each cell is labeled as free, occupied, or unknown, i.e., M g = M free ∪ M occ ∪ M unk . In practice, this labeling can be acquired from depth sensors using OctoMap [START_REF] Hornung | OctoMap: An efficient probabilistic 3D mapping framework based on octrees[END_REF]. As the 3D grid is a grid of cubic volumes of equal size called voxels, we can represent the environment by a set of layers N along z W direction, where N = ( z r ), z refers to the real environment's height, and r is the voxel size. Accordingly, each layer represents a 2D occupancy grid, producing intotal N 2D grids for a given environment, as illustrated in Fig. 3. Since this work is mainly concerned with the control framework, the perception problem is not presently covered. Thus, the perception is here imitated by the so-called 2D or 3D mask F f ov , which represents the sensor's field of view (FoV) as the robot cannot normally perceive the entire environment. For the sake of simplicity, we assume that the

1 st Layer 1 2 nd Layer 2 3 rd Layer 3 4 th Layer 4 N th Layer N M unk ≡ -1 M occ ≡ 1 M free ≡ 0 3D F f ov , Dim: (3 × 3 × 3),
where robot is placed in 3 , and 2 : 4 are updated.

Robot at (0, 0, 3 ) 3D F f ov has dimensions f x × f y × f z and an orientation f θ , where f x and f y represent the number of cells in the 2D grid along x W and y W axes. While f z represents the number of layers along z W , assuming that the robot's vertical FoV

f N is between [(int( -fz 2 ) + n ) : (int( fz 2 ) + n )]
where the robot is located in layer n (see Fig. 3). Note that the 3D F f ov is equivalent to f N 2D F f ov .

B. Fully/Partially Observable 2D/3D Navigation Tasks

Let assume that M g represents the global map representation of a given environment. This map is initialized using a priori knowledge about the environment. Let M mppi be the local map of MPPI, which has the same size as M g and each cell is initialized with -1 referring to unknown cells. This local map is continuously updated using an onboard sensing system, as depicted in Fig. 2.

1) Fully Observable Case: In a fully observable case, MPPI is directly provided with a global map M g to compute the trajectories' cost for avoiding obstacles, taking into account the current robot position obtained from a state estimator. For a 2D navigation task, a 2D floor grid is sufficient; only the first layer, 1 , of 3D voxel-grid will be accessed by MPPI. While, for the 3D case, the 3D voxel-grid is required for performing collision-free navigation. Clearly, given the robot's position, particularly its z-component, the corresponding layer, n , to z-component is used for evaluating the cost-to-go of each sampled trajectory (see Fig. 3). At the moment, the main limitation of the fully observable case is that our framework is only able to handle static environments; this is an important motivation for defining a partially observable navigation task that is prevalent in robotics applications.

2) Partially Observable Case: In this case, the robot is located in a previously unseen and dynamic environment and must navigate to: (i) a predefined goal, (ii) or explore and map that area. As the environment is unknown and M g is not directly accessible by MPPI, the robot must perceive its environment (in our case, through the predefined mask F f ov ) and react appropriately. Here, the MPPI map M mppi , which is initialized with -1, is fed directly into the control framework as a local map for avoiding obstacles. This map is continuously updated as the robot moves around. For the sake of clarity, let us consider the map in Fig. 3 as M g , where the robot is located in (0, 0, 3 ) and F f ov has dimensions of 3 × 3 × 3. As a consequence, based on the intersection between M g and F f ov , the layers from 2 to 4 in M mppi are updated, while other layers have remained constant.

Algorithm 2 Real-Time Generic MPPI Framework

Given: M g , M mppi : Global and local map of MPPI F f ov : Sensor's FoV and its parameters 1: while task not completed do 2:

x 0 ← StateEstimator(), x 0 ∈ R n 3: M mppi ← MapUpdate(M g ∩ F f ov ) 4: u 0 ← MPPIController(M mppi ) 5:
Check for task completion 6: end while Generally speaking, the real-time implementation of our proposed generic MPPI framework for 2D or 3D navigation in cluttered environments is better described in Algorithm 2, which employs the MPPI control scheme described above. At each time-step ∆t, the current state is estimated (line 2). The local map of MPPI M mppi is then updated accordingly, given the global map M g (line 3). As discussed previously, M g in conjunction with F f ov is used for updating the local map M mppi , as the perception modules have not been considered in the current work. Thus, in practice, the robot must be equipped with an on-board sensing system, e.g. depth camera or laser scanner, with a maximum FoV F f ov . This currently sensed data is used to obtain a 2D or 3D occupancy map, which is continuously updated, i.e., M mppi . For instance, a quadrotor-based exploration algorithm is proposed in [START_REF] Cieslewski | Rapid exploration with multi-rotors: A frontier selection method for high speed flight[END_REF] to build a real-time 3D map. Finally, this map enables the controller to find the optimal control u 0 to be executed, resulting in collision-free navigation (line 4).

V. SIMULATION DETAILS AND RESULTS

In this section, we describe how we evaluate our approach in terms of simulation scenarios and performance metrics. Moreover, we conduct realistic simulations to evaluate and demonstrate the performance of the proposed framework.

A. Simulation Setup, Scenarios, and Metrics 1) Simulation Setup: In order to evaluate the performance of our proposed MPPI framework in a previously unseen and cluttered environment, simulation studies have been performed using RotorS simulator and Gazebo [START_REF] Furrer | Robot Operating System (ROS): The Complete Reference[END_REF]. The parameters of the real simulated quadrotor (namely, Humminbird quadrotor) are tabulated in Table I. To ensure a realistic RotorS-based simulation, all navigation tasks are carried out by (i) using noisy sensors such as GPS and IMU, (ii) adding external disturbances such as continuous wind with changing speed and direction, as proposed in [START_REF] Tezenas Du | BOARR: A benchmark for quadrotor obstacle avoidance based on ROS and RotorS[END_REF], and (iii) considering ±10% of modelling errors in the real values of mass m and inertia J of the prediction model given in [START_REF] Mellinger | Trajectory generation and control for precise aggressive maneuvers with quadrotors[END_REF]. The MPPI controller has a time horizon t p of 3 s, a control frequency of 50 Hz, and generates 2700 samples each timestep ∆t. The rest of its hyper-parameters are also listed in Table I, where the 2.5 value in Σ u represents the noise in the thrust input F and R = λΣ -1 u . For the SG filter, we set the length of filter window and order of the polynomial function to 51 and 3, respectively. The real-time execution of MPPI is performed on a GeForce GTX 1080 Ti, where all algorithms are written in Python and have been implemented using ROS. 0.17

T 150 g [m s -2 ] 9.81 K 2700 k F [N rpm -2 ] 8.55 × 10 -6 λ 0.02 k M [N rpm -2 ] 1.6 × 10 -2 ν 1000 J [kg m 2 ] Diag 7 × 10 -3 , 7 × 10 -3 , 12 × 10 -3 Σu Diag 2.5, 5 × 10 -3 , 5 × 10 -3 , 5 × 10 -3 R Diag 8 × 10 -3 , 4, 4, 4
In our RotorS-based simulations, the full-state information, x = [x, y, z, φ, θ, ψ, v x , v y , v z , p, q, r] T , is directly provided, using an odometry estimator based on an Extended Kalman Filter (EKF), as an input to the controller. While, since the actual control signal of the quadrotor is the angular velocity of each rotor ω i , the controller outputs F and Γ are directly converted into ω i using (2), to be sent to the quadrotor as Actuators message. In summary, the closed-loop of MPPI in ROS, at each ∆t, can be summarized as: (i) MPPI first receives the odometry message and the updated map M mppi obtained from the on-board sensor; (ii) the control action is then computed and published. As mentioned before, the whole process of our proposed framework is summarized in Fig. 2.

Since we are interested in goal-oriented quadrotor navigation tasks in cluttered environments, the state-dependent cost function is defined as

q(x) = (x -x des ) T Q(x -x des ) + 10 8 C 1 + 10 5 C 2 ,
where:

C 1 = {x : (z < 0) or (M mppi [x, y, z] = 1)}, C 2 = {x : ( v > v max ) or (((|cos φ|) or (|cos θ|)) < 0.1)
or (z > 8.5)},

x des = (x des , y des , z des , 0, 0, ψ des , zeros(1, 6)), Q= Diag(2.5, 2.5, 5, 1, 1, 50, zeros(1, 6)), ∀v max ≤ 1.5 m s , Diag (5, 5, 15, 30, 30, 50, zeros(1, 6)), Otherwise. The first term C 1 indicates the collision with ground or obstacles, while C 2 prevents the quadrotor from (i) going too fast, (ii) using too aggressive roll and pitch angles, or (iii) colliding with the ceiling, where C 1 and C 2 are boolean variables. x des refers to the desired position to be reached and its orientation, while Q is a weighing matrix.

2) Simulation Scenarios: Two different scenarios are considered for evaluating the proposed framework: 2D scenario and 3D scenario. The 2D scenario refers to a 40×40×8.5 m windy forest of cylindrical obstacles placed in a 2D grid pattern, where each cylinder has a radius of 0.16 m with equal spacing of 4 m apart. The 3D scenario refers to the same environment described in the 2D scenario, while we have added two horizontal layers of cylinders at z = 3 m and 6 m with the same spacing each, as shown in Fig. 4. The voxel size r is set to 0.2 m, to ensure high accuracy and to meet the reality when a 3D occupancy grid is involved. As a result, the 3D voxel grid M g has 43 layers, each layer represents a 40 × 40 2D grid. 2 The 3D scenario is used for performing fully and partially observable 3D navigation tasks, while the former is used for the 2D tasks. For both scenarios, two different cases are considered: (i) Fully Observable Case (FOC), in which an a priori knowledge about the environment is used to initialize the global map M g ; in FOC, M mppi is exactly M g ; and (ii) Partially Observable Case (POC), where it is assumed that there is no a priori information about the environment; for this reason, the local map M mppi is discovered and built online as the robot moves around. 3) Performance Metrics: We define two metrics for evaluation. First, the performance of MPPI in POC is compared to its performance in FOC, considering the latter case as a baseline. To achieve a fair comparison, in all simulations, the robot navigates to the same specified goals with a maximum velocity v max of 1.5 m s -1 . The predefined goals (in [m]) in order are: G 1 = (23, 38, 1.5), G 2 = (40, 23, 4), G 3 = (22, 0, 8), G 4 = (0, 22, 5), then the UAV will land. While, at each ∆t, the desired yaw angle ψ des is updated, letting the front camera points towards the next goal. For both cases (FOC/POC), we use a number of indicators, as a second metric, to describe the general performance such as the number of collisions N col , task completion percentage t comp , average completion time t av , average flying distance d av , average flying speed v av , and average energy consumption E av (for more details, we refer to [START_REF] Tezenas Du | BOARR: A benchmark for quadrotor obstacle avoidance based on ROS and RotorS[END_REF]). In all simulations, the task is considered to be terminated if the quadrotor reached the given goals (i.e., t comp = 100%) or crashed into an obstacle (i.e., N col = 1).

B. Simulation Results

The performance of our proposed framework is validated for both fully and partially observable 2D/3D quadrotor navigation tasks, considering the predefined goals. In all partially observable tasks, we set F f ov to 5 × 5 × 3 m, while f θ represents the angle between the current and next goal. To test whether the quadrotor is able to navigate successfully through the cluttered environment, we performed 5 trials for both FOC and POC. The reader is invited to watch the whole simulation results at https://bit.ly/2PAbESO. 1) 2D Navigation Results: Figure 5 shows an example of a final trajectory generated by MPPI in the case of (i) using a 2D global map M g for FOC, or (ii) using only a 2D local map M mppi for POC. The 2D-floor M g (with grey circles) and its updated M mppi (with blue circles, representing the obstacles within the robot's F f ov ), for given goals, are shown in Fig. 6, including the generated trajectories in both cases and the robot's FoV F f ov . In both 2D FOC and POC, it can be seen that MPPI is able to safely navigate through the windy obstacle field, in spite of the presence of modelling errors and measurements noise. 2) 3D Navigation Results: Another example of successfully generated trajectories for fully and partially observable 3D navigation tasks in a 3D scenario environment is depicted in Fig. 7. In this scenario, since M mppi consists of 43 2D grids, it is very difficult to visualize clearly in this paper the updates over all layers and insure that the robot performs collision-free navigation. So, first, we show only the 11 th layer 11 , which has been mainly updated while the robot was moving towards G 1 and while it was landing after task completion, as an example of how the 3D M mppi is updated (see Fig. 8). Second, it is highly recommended to use the indicators described previously, especially for 3D cases. 3) Overall Performance: Table II shows the performance statistics for the five trials in 2D and 3D scenarios, considering the parameters of the controller tabulated in Table I. In all trials (i.e., 20 trials in total), we can observe that the quadrotor navigates autonomously (i.e., N col = 0) while avoiding obstacles with an average flying speed v av of about 1.2 m s -1 , which is closer to the maximum specified velocity, i.e., v max = 1.5 m s -1 , regardless of the limited FoV of the sensor in both 2D and 3D scenarios. Note that these 20 trials are equivalent to more than 3 km of autonomous navigation. Moreover, it can be seen that the average execution time of MPPI per iteration, t mppi , in the 3D scenario is approximately equal to that in the 2D scenario (with very low standard timesteps T (which is related to the time horizon t p and control frequency) and the exploration variance ν (which is related to the number of sampled trajectories K) play an important role in determining the behavior of MPPI. Therefore, we tested six different tuning cases, where different values of T and ν have been considered, as shown in Table III. In the first three tuning cases, we study the influence of changing T where ν remains constant (i.e., ν = 1000 as defined before in Table I). While the effect of changing ν is studied in the last three cases, where ν was varied between 300 and 800.

For the sake of simplicity, each tuning case was tested by letting the quadrotor navigates only to G 1 = (23, 38, 1.5) then lands. In each tuning case, we conducted 3 trials. As a consequence, the total trials for both FOC and POC cases are 72. For Tuning Case 1, where a short time horizon t p is chosen, it can be clearly noticed that the MPPI controller is unable to complete the trials at a satisfactory rate, where the success rate t comp varies from 36.3% to a maximum of 57.4% in both 2D FOC and POC (1/3 successful trials).

For other tuning cases (namely, from case 2 to 6), we can observe that the controller performs perfectly and is able to successfully complete all trials while avoiding obstacles. We can also notice that as T and ν increase, the performance of the controller improves. Clearly, for Tuning Case 3 and 6 where high values of T and ν are chosen, we can see that the average flying distance d av (for both 2D and 3D scenarios) is the shortest (compared to other successful cases) with low standard deviation values, which means that the quadrotor is taking a more direct route leading to the goal. However, having too long time horizons increase the computational effort dramatically because each trajectory takes more time to simulate. While, if the exploration variance ν is too large, the controller produces control inputs with significant chatter. We also notice during our simulations that higher values of control weight matrix R leads to empirically slow convergence of MPPI towards the goal and fluctuated motion. For this reason, in all experiments, we set T , ν, and R to the values given in Table I, where MPPI performs very consistently for all given goals.

VI. CONCLUSIONS AND FUTURE WORK

Within this work, we proposed an extension to the classical MPPI framework that enables the robot to navigate autonomously in 2D or 3D environments while avoiding collisions with obstacles. The key point of our proposed framework is to provide MPPI with a 2D or 3D grid representing the real world to perform collision-free navigation. This framework has been successfully tested on realistic simulations using quadrotor, considering both fully and partially observable cases. The current simulations illustrate the efficiency and robustness of the proposed controller for 2D and 3D navigation tasks. Although the theoretical stability of MPPI has not been addressed and proofed in the literature, its practical stability can be achieved by setting the MPPI parameters carefully as we described previously. We will explore the possible methods that may allow in the future to study the theoretical stability of MPPI. Our future work will also include the implementation of the framework in practical applications.

Fig. 1 .

 1 Fig. 1. Schematic of the considered quadrotor model in conjunction with the coordinate systems and forces acting on a vehicle frame.

  Quadrotor

Fig. 2 .

 2 Fig. 2. The global architecture of our proposed framework.

Fig. 3 .

 3 Fig. 3. 3D environment representation of a generic MPPI.

Fig. 4 .

 4 Fig. 4. The 3D cluttered environment used for 3D tasks.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Comparison of 3D trajectories in a 2D scenario.

Fig. 7 .

 7 Fig. 7. Comparison of 3D trajectories in a 3D scenario.

Fig. 8 .

 8 Fig. 8. The update of the 11 th layer 11 in the 3D M mppi , showing the generated trajectory in POC.

TABLE II GENERAL

 II PERFORMANCE OF MPPI IN 2D AND 3D SCENARIOS FORBOTH FOC AND POC In fact, this is not surprising as the 2D-or 3D-voxel grids are stored and directly used by MPPI on the GPU. For other indicators, we can observe that the performance of MPPI in POC cases is slightly different than its performance in FOC cases, although we have set the same parameters for all simulations. The reason behind this difference is that the control variations δu are generated randomly, each time-step ∆t, on the GPU. The number of

	Indicators	2D Scenario FOC POC	3D Scenario FOC POC
	Ncol(tcomp[%]) 0 (100%) 0 (100%) 0 (100%) 0 (100%) 122.1 ± 1.2 124.5 ± 3.0 128.2 ± 3.0 129.5 ± 1.4 tav [s] dav [m] 150.3 ± 0.8 151.1 ± 1.9 152.9 ± 1.8 153.1 ± 0.87
	vav [m s -1 ] Eav [W h] tmppi [ms]	1.23 ± 0.02 1.22 ± 0.03 1.19 ± 0.02 1.18 ± 0.01 6.63 ± 0.07 6.76 ± 0.16 6.96 ± 0.16 7.01 ± 0.07 18.4 ± 0.51 18.8 ± 0.42 18.2 ± 0.21 18.8 ± 0.67
	deviation values), showing the superiority of the proposed framework even with 3D-voxel grids and its applicability to be used for 3D navigation tasks.

TABLE III EFFECT

 III OF CHANGING T AND ν ON THE BEHAVIOR OF THE PROPOSED

		CONTROLLER
	Indicators	2D Scenario FOC POC	3D Scenario FOC POC
	Tuning Case 1: T = 75 (i.e., tp = 1.5 s) and ν = 1000 N col (tcomp[%]) 1 (57.4±30) 1 (57.4±30) 1 (40.3±6) 1 (36.3±9)
	Tuning Case 2: T = 100 (i.e., tp = 2 s) and ν = 1000 N col (tcomp[%]) 0 (100%) 0 (100%) 0 (100%) 0 (100%)
	dav [m]	51.7±0.79	51.7±1.1	51.1±0.16 52.1±0.68
	N col (tcomp[%]) dav [m]	0 (100%) 50.4±0.86	0 (100%) 50.7±0.25	0 (100%) 50.2±0.49 50.5±0.36 0 (100%)
	N col (tcomp[%])	0 (100%)	0 (100%)	0 (100%)	0 (100%)
	dav [m]	52.4±0.78	52.5±0.36	52.8±0.74 53.0±0.45
	Tuning Case 5: T = 150 and ν = 500 N col (tcomp[%]) 0 (100%) 0 (100%) 0 (100%)	0 (100%)
	dav [m]	51.9±0.59	51.5±0.85	52.1±0.25 52.6±0.36
	Tuning Case 6: T = 150 and ν = 800 0 (100%) 0 (100%) 0 (100%) N col (tcomp[%]) dav [m] 50.9±0.25 51.0±0.46 51.5±0.93 51.9±0.33 0 (100%)

Tuning Case 3: T = 125 (i.e., tp = 2.5 s) and ν = 1000 Tuning Case 4: T = 150 (i.e., tp = 3 s) and ν = 300

Guillaume Allibert is with the Université Côte d'Azur, CNRS, I3S, France. allibert@unice.fr

In the sense that the method in[START_REF] Williams | Information-theoretic model predictive control: Theory and applications to autonomous driving[END_REF] can exactly recover that in[START_REF] Williams | Model predictive path integral control: From theory to parallel computation[END_REF] if dynamics is considered to be affine in control. In other words, the iterative method in[START_REF] Williams | Information-theoretic model predictive control: Theory and applications to autonomous driving[END_REF] can be seen as the generalization of the latter method.

The cell size of 2D grid is 1 m (i.e., 1:1 scale), since all obstacles, in our case, are placed in a 2D grid pattern with integer numbers.

The supplementary video attached to this work is available at: https://bit.ly/2PAbESO

This research was supported by the ANR CLARA project (ANR-18-CE33-0004).


Bastien Duval
email: bastien.duval@univ-pau.fr

Ainara Gredilla

Silvia Fdez-Ortiz De Vallejuelo

Emmanuel Tessier

David Amouroux

Alberto De Diego

A simple determination of trace mercury concentrations in natural waters using Dispersive Micro-Solid Phase Extraction preconcentration based on functionalized graphene nanosheets

Keywords: Mercury, Preconcentration, Ultra-trace analysis, Graphene nanosheets, Dispersive Micro-Solid Phase Extraction (DMSPE), Water sample

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

Introduction

Among other pollutants, mercury (Hg) is of major concern in aquatic environments. While mercury is present at low concentration in natural waters [START_REF] Gworek | Mercury in Marine and Oceanic Waters-a Review[END_REF], its specific biogeochemical cycling allows bioaccumulation and biomagnification in the food chain leading to serious consequences for animals and higher predators, including humans [START_REF] Anual | Mercury and risk assessment from consumption of crustaceans, cephalopods and fish from West Peninsular Malaysia[END_REF][START_REF] Arcagni | Species-and habitat-specific bioaccumulation of total mercury and methylmercury in the food web of a deep oligotrophic lake[END_REF]. Consequences of mercury contamination are not only noticeable in terms of people's health [START_REF] Aaseth | Mercury exposure and health impacts in dental personnel[END_REF][START_REF] Khan | The relationship between mercury exposure and epigenetic alterations regarding human health, risk assessment and diagnostic strategies[END_REF] but also in the economic sphere [6,7] [9,10].

Thereby, mercury concentrations in topsoil, atmosphere and oceanic surface waters have increased respectively by a factor of 1.2, 4-6 and 3 since the pre-industrial period [START_REF] Driscoll | Mercury as a Global Pollutant: Sources, Pathways, and Effects[END_REF][START_REF] Lamborg | A global ocean inventory of anthropogenic mercury based on water column measurements[END_REF]. Besides, the numerous ecological studies about mercury show that primary anthropogenic mercury emissions are still increasing [START_REF] Driscoll | Mercury as a Global Pollutant: Sources, Pathways, and Effects[END_REF][START_REF]Global Mercury Assessment 2013: Sources, Emissions, Releases and Environmental Transport[END_REF]. Therefore, the scenarios for the future are quite pessimistic, predicting a change in mercury emission of -4 % to 96 % [START_REF] Streets | Projections of Global Mercury Emissions in 2050[END_REF].

Monitoring natural concentrations of total mercury in aquatic systems remains a difficult challenge since Hg is mainly occurring at ultra-trace levels: 0.3 to 8 ng L -1 in waters free from local sources of mercury, either anthropogenic or natural, and 10 to 40 ng L -1 for waters influenced by mercury mining and/or industrial pollution [START_REF] Wiener | Ecotoxicology Of Mercury[END_REF]. Indeed, a wide range of study has highlighted such levels of mercury in natural waters either seawater [START_REF] Gworek | Mercury in Marine and Oceanic Waters-a Review[END_REF][START_REF] Lamborg | A global ocean inventory of anthropogenic mercury based on water column measurements[END_REF] or freshwater [START_REF] Cavalheiro | Assessment of background concentrations of organometallic compounds (methylmercury, ethyllead and butyl-and phenyltin) in French aquatic environments[END_REF].

The development of low cost and easy handling analytical methods is then a necessity and it is required to fulfil some of the objectives of the Minamata Convention on Mercury, a global treaty to protect human health and the environment from the adverse effects of mercury. In that sense, Solid Phase Extraction (SPE) procedures to preconcentrate Hg before its analysis have shown a great interest in the scientific community due to its efficiency and simplicity. Most of the studies focus on the preparation of micro-column filled with specific adsorbents for Hg preconcentration. Polymers such as zwitterion-functionalized polymer microspheres (ZPMs) [START_REF] Jia | Zwitterion-functionalized polymer microspheres-based solid phase extraction method on-line combined with HPLC-ICP-MS for mercury speciation[END_REF] or ion-imprinted polymer (IIP) especially designed for Hg adsorption [START_REF] Soleimani | Highly selective solid phase extraction of mercury ion based on novel ion imprinted polymer and its application to water and fish samples[END_REF] can be successfully used for rapid enrichment of mercury species.

More simple in its preparation yet more expensive, the use of aminated Amberlite XAD-4 resin [START_REF] Çaylak | Use of an aminated Amberlite XAD-4 column coupled to flow injection cold vapour generation atomic absorption spectrometry for mercury speciation in water and fish tissue samples[END_REF] or silica reversed-phase (RP-C 18 ) [START_REF] Muñoz | Speciation of Organometallic Compounds in Environmetal Samples by Gas Chromatography after Flow Preconcentration on Fullerenes and Nanotubes[END_REF] respectively followed by FI-CVG-AAS and GC/MS analysis allows determination of Hg species, but its applicability to real samples remains difficult due to high limits of detection.

The investigation of new materials for Hg selective preconcentration made a great step forward in recent years. Indeed, due to their specific properties, nanomaterials have attracted increasing interest in the development of preconcentration methods for contaminants [START_REF] Kaur | A review on applications of nanoparticles for the preconcentration of environmental pollutants[END_REF], and, more recently, Hg analysis. Then, the micro-column used to preconcentrate Hg can be packed with nanocomposites, previously functionalized for specific adsorption of Hg: magnetite (Fe 3 O 4 ) nanoparticles [START_REF] Tadjarodi | A magnetic nanocomposite prepared from chelator-modified magnetite (Fe3O4) and HKUST-1 (MOF-199) for separation and preconcentration of mercury(II)[END_REF], graphene oxide (GO) [START_REF] Sotolongo | An easily prepared graphene oxide-ionic liquid hybrid nanomaterial for micro-solid phase extraction and preconcentration of Hg in water samples[END_REF], multiwalled carbon nanotubes (MWCNTs) [START_REF] Muñoz | Speciation of Organometallic Compounds in Environmetal Samples by Gas Chromatography after Flow Preconcentration on Fullerenes and Nanotubes[END_REF][START_REF] Labutin | Femtosecond laser-induced breakdown spectroscopy[END_REF], carbon nanotube sponges (CNT sponges) [START_REF] Wang | Simultaneous determination of copper, cobalt, and mercury ions in water samples by solid-phase extraction using carbon nanotube sponges as adsorbent after chelating with sodium diethyldithiocarbamate prior to high performance liquid chromatography[END_REF]. Hence, all these methodologies include synthesis of the adsorbent, preparation of the column, elution of the Hg species directly followed by analysis, preventing direct application on the field. Moreover, multiplication of the preparation steps, from collection to analysis, may be problematic considering the potential contaminations or losses of Hg [START_REF] Stoichev | Speciation Analysis of Mercury in Aquatic Environment[END_REF].

The use of field techniques reduces travelling and storage times of the samples, and facilitate the acquisition of results in developing countries and/or remote areas. In that sense, K. Leopold et al. [START_REF] Zierhut | Analysis of total dissolved mercury in waters after on-line preconcentration on an active gold column[END_REF][START_REF] Huber | Nanogold-Decorated Silica Monoliths as Highly Efficient Solid-Phase Adsorbent for Ultratrace Mercury Analysis in Natural Waters[END_REF][START_REF] Schlathauer | Sustainable and reagent-free mercury trace determination in natural waters using nanogold dipsticks[END_REF] have conducted various studies in the last decade to develop a passive sampler for reagentfree on-site Hg preconcentration: nanogold-coated dipsticks. Thermal desorption of trapped Hg followed by AFS detection allows determination of Hg contents lower than the ppt level (ng L -1

), and applicability to real samples, either seawater or freshwater, have been demonstrated. Nevertheless, implementation of this methodology in developing countries remains difficult because of the heavy preparation of the dipsticks (cleaning, coating, and functionalization) and their high cost (reagents and chemicals): sampling campaigns would need numerous of this passive sampler. Also, dipsticks have to be regenerated by thermal annealing at 600 °C for 20 min before each use, which greatly lengthens the preparation time of a sampling campaign. Finally, reproducible production of the nanogold-coated dipsticks remains difficult. As an alternative to expensive gold materials, Tavares et al. [START_REF] Tavares | Reliable quantification of mercury in natural waters using surface modified magnetite nanoparticles[END_REF] have produced magnetite nanoparticles using iron oxide nanoparticles coated with silica shells functionalized with dithiocarbamate groups in order to take advantage of the strong affinities between Hg and thiol group (R-SH) [START_REF] Li | Combined experimental and theoretical investigation on selective removal of mercury ions by metal organic frameworks modified with thiol groups[END_REF][START_REF] Huang | Effective removal of inorganic mercury and methylmercury from aqueous solution using novel thiol-functionalized graphene oxide/Fe-Mn composite[END_REF] to preconcentrate Hg prior to its analysis by thermal decomposition AAS with gold amalgamation. Again, synthesis of the nanoparticles includes numerous steps and reagents, and requires important scientific knowledge. Besides, extraction of Hg using these nanoparticles with recovery close to 100% necessitates at least 24 hours. Finally, the limit of detection of 1.8 ng L -1

, reached with this procedure prevents its application in pristine areas.

Several studies conducted by Sitko et al. have already been made using modified graphene nanomaterials for Dispersive Micro-solid Phase Extraction (DMSPE) such as carbon nanotubes [START_REF] Zawisza | Ni, Cu, Zn and Pb prior to wavelength-dispersive X-ray fluorescence spectrometry[END_REF],

multiwalled carbon nanotubes (MWCNTs) [START_REF] Kocot | Dispersive micro solidphase extraction using multiwalled carbon nanotubes combined with portable total-reflection X-ray fluorescence spectrometry for the determination of trace amounts of Pb and Cd in water samples[END_REF], graphene oxide (GO) [START_REF] Zawisza | Graphene oxide as a solid sorbent for the preconcentration of cobalt, nickel, copper, zinc and lead prior to determination by energydispersive X-ray fluorescence spectrometry[END_REF] or graphene nanosheets [START_REF] Kocot | Trace and ultratrace determination of heavy metal ions by energydispersive X-ray fluorescence spectrometry using graphene as solid sorbent in dispersive micro solid-phase extraction[END_REF] in order to preconcentrate various elements (Mn, Fe, Co, Ni, Cu, Zn, Se, Pb, Cd). Nevertheless, none of these works deals with Hg, probably because of the high limit of detection of the apparatus (wavelength-dispersive X-ray fluorescence WDXRF, total-reflection X-ray fluorescence spectrometry TXRF, or energy-dispersive X-ray fluorescence spectrometry EDXRF) and the difficulties of working with such element (contaminations or loses).

In this work, we have used the specific properties of graphene nanosheets, a cheap material, to develop an innovative analytical method. Firstly, graphene is flexible so it can be easily attached to a support, which makes it a worthwhile candidate in the use of Micro-solid Phase Extraction (MSPE). It is as well hydrophobic and non-polar with a strong affinity for carbon-based ring compound due to the hexagonal arrays of carbon atoms in its structure. Graphene shows also a huge specific surface (2630

m 2 g -1
) [START_REF] Stoller | Graphene-Based Ultracapacitors[END_REF], which encourages its use as a sorbent in MSPE. It should be noted that this amazing material could be synthesized easily and affordably in most of the laboratories all around the world, including those of developing countries. Among graphene-based material, graphene nanosheets display great adsorption capacities. Indeed, both surfaces of a planar sheet of graphene are accessible to adsorption of analytes [START_REF] Sitko | Graphene as a new sorbent in analytical chemistry[END_REF]. Also, nanostructured material as graphene nanosheets allows the adsorption of the organic compound via non-covalent interactions [START_REF] Valcárcel | Carbon nanostructures as sorbent materials in analytical processes[END_REF].

In the proposed method, based on Kocot et Sitko [START_REF] Kocot | Trace and ultratrace determination of heavy metal ions by energydispersive X-ray fluorescence spectrometry using graphene as solid sorbent in dispersive micro solid-phase extraction[END_REF], dissolved Hg in water samples is first preconcentrated using a nanoparticles suspension consisting of graphene nanosheets as an adsorbent, Ammonium Pyrrolidine DithioCarbamate (APDC) as a chelating agent and Triton-X-100 as a surfactant. To obtain high recovery of the Hg on the nanoparticles, the procedure is optimized for various analytical parameters such as the amount of nanoparticles, sonication time and sample volume. The Hg-rich nanoparticles are then collected in a Polyethersulfone (PES) filter, whose stability at different storage conditions is also checked. The potential existence of matrix effects due to the presence of chloride salts (estuarine or seawater), organic matter (continental water) and metals (calcium) able to compete with mercury for the binding sites of the sorbent is investigated, and the applicability of the whole method is finally verified by the analysis of real samples (seawater and lake water).

Together with the in-field preconcentration of a large volume of water on a small and light membrane filter and the direct analysis of this filter back to the lab greatly enhance the simplicity of all the analytical process for a precise and accurate determination of mercury content in water samples: from the sampling to the result!

Material and methods

Preparation of nanoparticles suspension

The principle of our method is based on the adsorption of mercury on graphene nanosheets.

Nevertheless, metal ions in the form of hydrophilic complexes, including dissolved mercury, cannot be adsorbed directly on the graphene surface with a great efficiency [START_REF] Kocot | Trace and ultratrace determination of heavy metal ions by energydispersive X-ray fluorescence spectrometry using graphene as solid sorbent in dispersive micro solid-phase extraction[END_REF]. Therefore, it is necessary to functionalize the graphene nanosheets using a chelating agent that will be adsorbed on the graphene nanosheets via non-covalent interactions (π-π stacking) [START_REF] Sitko | Graphene as a new sorbent in analytical chemistry[END_REF]. Ammonium Pyrrolidine DithioCarbamate (APDC) was chosen as a chelating agent due to some specificity in its structure. Indeed, strong affinities between APDC and mercury, mainly due to the presence of thiol groups (R-SH) in the structure of APDC, lead to the chelation of mercury by APDC (covalent bonds) [START_REF] Kocot | Trace and ultratrace determination of heavy metal ions by energydispersive X-ray fluorescence spectrometry using graphene as solid sorbent in dispersive micro solid-phase extraction[END_REF]. -SH groups have already shown one of the strongest binding affinity either for inorganic mercury (sulphur-containing organic matter or polymers with Hg(II)) or organic mercury (sulphur-containing organo-metallic compounds or thiolate-like species with MeHg) [START_REF] Huang | Effective removal of inorganic mercury and methylmercury from aqueous solution using novel thiol-functionalized graphene oxide/Fe-Mn composite[END_REF]. We also need to add some Triton-X-100 in the nanoparticles suspension to increase the viscosity of the solution, hence improving the buoyancy of graphene nanosheets and, in this way, to obtain a real suspension of nanoparticles.

Then, following the recommendations of Kocot and Sitko [START_REF] Kocot | Trace and ultratrace determination of heavy metal ions by energydispersive X-ray fluorescence spectrometry using graphene as solid sorbent in dispersive micro solid-phase extraction[END_REF], that applied DMSPE for the preconcentration of Co, Ni, Cu and Pb, the graphene/APDC/Triton-X-100 nanoparticles suspension was prepared as follows: 25 mg of graphene nanosheets (~5 µm diameter, 1-5 nm thickness, Green Stone Swiss Co.), 80 mg of APDC (Purum p.a. ≥ 98.0 %, Sigma-Aldrich) and 10 mg of Triton-X-100

(Laboratory-grade, Sigma-Aldrich) in 20 mL of high purity water (18.2 MΩ cm), sonicated for 60 min, to obtain concentrations of 1.25 g L -1

; 4 g L -1 and 0.5 g L -1 for graphene, APDC and Triton-X-100, respectively. To avoid the aggregation of the graphene and to ensure its homogeneity within the nanoparticles suspension, the solution was sonicated 5 min before each use.

Procedure for mercury preconcentration (optimum conditions)

The chelation between APDC and mercury can effectively occur only with a pH between 4 and 5, so initially, the pH of the water sample containing mercury was adjusted by the addition of ultrapure HCl and/or NH 3 solutions (Optima grade, Fisher Scientific). In the second step, 0.5 mL of nanoparticles suspension was added to either 20 mL or 200 mL of the aqueous solution to be analysed. Then, the solution containing the complex formed by Hg, APDC and graphene was sonicated for 5 min and collected on a membrane filter (Polyethersulfone PES, ø 25 mm, 0.2 µm, Pall Corporation) by filtration (250 mL Erlenmeyer glass flask) under vacuum. Fig. 1 shows Transmission Electron Microscopy (TEM) images of water samples containing (a) graphene alone, (b) graphene and APDC, and (c) Hg, APDC and graphene. This figure really highlights an increase of the size in the observed particles from graphene alone (67 nm) to the complex Hg/APDC/Graphene (700 nm to 1.89 µm). Consequently, the size of this complex is larger than the membrane filter pores and is quantitatively retained by the filtering PES membrane. We noticed that the size of the complex Hg/APDC/Graphene remains lower than the certified commercial size of graphene nanosheets (~5 µm diameter, 1-5 nm thickness). In our study, we assume that both the time in the ultrasonic bath (about 60 min) and the use of Triton-X-100 during the preparation of the nanoparticles suspension would promote the dislocation of aggregates, hence reducing the size in comparison with the raw graphene nanosheets [START_REF] Gigault | A new analytical approach based on asymmetrical flow field-flow fractionation coupled to ultraviolet spectrometry and light scattering detection for SWCNT aqueous dispersion studies[END_REF].

Finally, the membrane filter obtained is dried for 5 min in a laminar flow hood (Class 100) and kept into polycarbonate filters holder (Petri slides) at low temperature (-7°C) before its analysis.

Instrumentation and method for mercury analysis

Direct pyrolysis gold amalgamation and atomic absorption spectroscopy (AAS), using an Advanced Mercury Analyser (Altec AMA-254, SymaLab), allows the quantification of Hg in solid samples. The whole membrane filter is dropped on a nickel boat especially designed for this instrument (1000 µL, SymaLab). To increase analytical performances, nickel boats used for the analysis of the PES filters were previously cleaned by burning at 750 °C daily, using the AMA-254. This step has been repeated until a relative standard deviation under 10 % between two successive burnings is obtained. After the introduction of the sample into the analyser using the nickel boat, an increase in temperature up to 750 °C under oxygen atmosphere allows first the drying of the matrix, and then the decomposition of the sample and its dry mineralization. Decomposition products and mercury are driven by the oxygen stream through a catalytic tube continuously heated at 550 °C. This catalytic tube allows decomposition and reduction of methylmercury to gaseous elemental mercury Hg 0 and the removal of halogens. Then, total mercury vapour is trapped in a quartz tube filled with gold-coated sand that allows the preconcentration of mercury, and so improve the sensitivity of the instrument. A few seconds later, the gold amalgamator is heated at about 950 °C. The quantification of this released mercury is carried out using two absorption cells (1 cm and 10 cm lengths, wavelength = 253.65 nm), both of them thermostatically controlled at 125 °C, that allows sequential quantification of mercury at two different ranks (Rank 1 < 25 ng Hg, and Rank 2 > 25 ng Hg).

All instrumental parameters have been optimized for analysis of PES filters as follow: drying step of 150 sec, decomposition step of 150 sec and waiting step of 45 sec. Accuracy and reproducibility of the AMA-254 were assessed by using the Certified Reference Material BCR-320R (channel sediment 0.85 ± 0.09 mg kg -1 of total Hg), analysed regularly all along the measurement process according to the EPA method 7473 (SW-846).

Calibration for mercury quantification

To quantify the accumulated mass of mercury on the PES filter, the AMA-254 was calibrated using an external matrix-matched calibration curve in the absolute mass of mercury ( .

Coefficients of determination R 2 of the four different calibration curves are all above 0.99947, suggesting validated good linear relationship. Nevertheless, taking into account the standard deviations, the slopes given by these models are significantly different at a 95 % confidence level. This could be explained by the relative difficulty to weigh correctly the mass of Hg(II) standard solution added directly in the nickel boat in the case of matrix-matched calibration. Indeed, in practice, when the operator drops the solution into the nickel boat, the balance is not stable either due to adsorption on the nickel boat surface and/or evaporation. Therefore, in this work, we decided to use the calibration curves obtained according to the procedure developed in this study under optimized conditions for data treatment.

Sampling of natural waters

To check the suitability of the proposed method to the analysis of natural waters, real samples were collected in April 2017 at two locations of the Pyrénées-Atlantiques (Nouvelle-Aquitaine, France).

Freshwater was collected from Lac Des Carolins (43°20'15'' N, 0°24'23'' W), at a depth of 40 cm using acid cleaned polyethylene bottles (2 L) that were rinsed three times with the sampling water. Important wildlife in the surroundings of the lake has a great influence on its biogeochemical characteristics, especially concerning the relatively high content of organic matter measured in the sample ([DOC] = 5.6 mg L -1

). Coastal seawater was collected in the same way out of the Bay of St Jean de Luz (43°23'37'' N, 1°39'44'' W, Bay of Biscay, North-East Atlantic Ocean). Time collection of the samples corresponds to high tide, and the seawater is characterized by a lower amount of organic matter

([DOC] = 0.90 mg L -1
) and a high salinity (35.6 PSU). All water samples were filtered through a PES membrane (0.2 µm), acidified with ultra-pure hydrochloric acid (1 % v/v) and stored at 4 °C for no longer than two days before further manipulation and analysis.

Results and Discussion

Optimisation of the Hg preconcentration method

Amount of nanoparticles

The amount of nanoparticles used in the preconcentration step will have a great influence on the efficiency of the proposed procedure. Enough nanoparticles must be available to chelate all the mercury present in the sample. Furthermore, an extra amount of nanoparticles will i) ensure that the possible presence of other competing metals will not negatively affect the effectiveness of Hg sequestration and ii) allow the use of a larger volume of sample to eventually improve the detection limit of the method. In this work, the influence of the amount of nanoparticles has been studied. High purity water samples of 25 mL (n=4) were spiked with 100 ng of mercury (Hg(II) standard solution), and various amounts of nanoparticles have been added (expressed as volume of nanoparticles suspension added, from 100 µL to 2000 µL). Then, all the samples were subjected to the preconcentration procedure with a sonication time of 5 min. Blanks (n=4) were also produced for the various amounts of nanoparticles, following the same procedure with unspiked high purity water. The results are shown in Fig. 2a. which provides the mercury recovery obtained as a function of the amount of nanoparticles. No significant change was noticed in the recovery but the best condition selected was for a volume of nanoparticles suspension of 500 µL, to ensure that all the mercury in the sample would be trapped even in the presence of a high amount of other competing metals. Indeed, a large amount of nanoparticles suspension leads to an increase in the blanks signal. The graphene shows similar properties to activated carbon so it can adsorb the gaseous species of mercury present in the air during the preparation of the nanoparticles suspension [START_REF] Amouroux | Sampling and probing volatile metal(loid) species in natural waters by in-situ purge and cryogenic trapping followed by gas chromatography and inductively coupled plasma mass spectrometry (P-CT-GC-ICP/MS)[END_REF][START_REF] Wang | Adsorption of gaseous elemental mercury with activated carbon impregnated with ferric chloride[END_REF][START_REF] Lusilao-Makiese | Analytical Performances of Nanostructured Gold Supported on Metal Oxide Sorbents for the Determination of Gaseous Mercury[END_REF]. This could lead to the contamination in the blanks, making necessary to work under a clean atmosphere and to minimize the amount of nanoparticles suspension.

Sonication time

In previous works about preconcentration of metals using graphene nanoparticles as a sorbent, it is shown that the contact time between the nanoparticles suspension and the water sample does not appear as a critical variable as sorption occurs immediately [START_REF] Zawisza | Ni, Cu, Zn and Pb prior to wavelength-dispersive X-ray fluorescence spectrometry[END_REF][START_REF] Pereira | Development of a dispersive liquid-liquid microextraction method for iron extraction and preconcentration in water samples with different salinities[END_REF]. Nevertheless, this parameter has been investigated here (Fig. 2b.) within the time range of 1-15 min in the ultrasonic bath. So, high purity water samples of 25 mL (n=4) were spiked with 100 ng of mercury, and preconcentrated using 500 µL of nanoparticles suspension with various sonication times (1-15 min). Blanks (n=4) were also performed for each condition, following the same procedure with unspiked high purity water. Results depicted in Fig. 2b. show that sorption of mercury is significantly less effective when the time in the ultrasonic bath is lower than 1 min. Besides, for 2 min of sonication, the procedure shows a lower reproducibility (two replicates with about 87% of recovery and two replicates with about 100% recovery) confirming that the sorption of mercury does not occur immediately. Finally, longer sonication times result in poorer reproducibility. This might be due to the influence of the ultrasonic bath that could degrade the non-covalent interactions (π-π stacking) between graphene nanosheets and APDC. Therefore, sonication time of 5 min allows an effective and reproducible activation of the binding sites. For in-field direct application, we can imagine that the use of the ultrasonic bath can be easily replaced by a simple shaking, with longer time.

Sample volume

The volume of the water sample used during the method of preconcentration is one of the critical steps in the analytical performance. Indeed, the choice of a larger volume of samples leads to an increase in the preconcentration factor and, hence, to a decrease in the detection limits, provided that the nanoparticles present (same amount, lower concentration) can sequestrate quantitatively all the mercury that can be much more diluted in the medium. Calibration curves were obtained as described in Section 2.4 using two different volumes of ultrapure water, i.e. 20 and 200 mL. Fig. 2c andd. show the absorbance obtained after the analysis of the filters produced according to the procedure described in Section 2.2, as a function of the theoretical amount of mercury added to the sample.

Blanks (n=3 for 20 mL and n=2 for 200 mL of the sample) have been carried out for the two volumes, following the same procedure with unspiked high purity water. From 1.6 ng to 210 ng of mercury added to the test solution, no significant change can be observed in the efficiency of the preconcentration when working with 20 or 200 mL. A larger volume could have been tested, nevertheless, increasing the volume above 200 mL will be more time consuming for the operator in charge of the procedure because the filtration step will be much longer, especially with samples containing a high content of organic matter and/or suspended material. For future use of this method to quantify mercury in an aquatic environment, it is then important to adapt the sample volume to the specification of the water sample, especially when the concentration of Hg is very low.

Optimization of the storage conditions

The possible application of this preconcentration method directly in the field, in remote areas, implies the transportation of the filters from the sampling site back to the laboratory for their analysis. Thus, storage conditions of the filters is an important factor to take into account to minimize sample contamination as well as potential losses of mercury by evaporation. On the one hand, elemental mercury (Hg 0 ) is present at trace levels in the ambient air. Typically, Total Gaseous Mercury (TGM) concentrations averages about 1.5 ng m 3 in background ambient air throughout the world [42]. In our laboratory, TGM concentrations have been reported by Lusilao-Makiese et al. [START_REF] Lusilao-Makiese | Analytical Performances of Nanostructured Gold Supported on Metal Oxide Sorbents for the Determination of Gaseous Mercury[END_REF] (6.3 ± 1.6 ng m 3 ), and are higher than the background ambient air levels due to the activities that take place in the laboratory (analysis and development of methodologies for metals and metalloids). This concentration, though, remains quite low for a laboratory building located in an urban area. Then, as Hg 0 is subject to adsorption on all solid surfaces, it could eventually be adsorbed on the Hg-enriched filters produced in the preconcentration step and lead to an overestimation of the Hg content. On the other hand, the complex formed by Hg, APDC and graphene may be sensitive to degradation by light, temperature, oxygen, humidity etc… Then the exposure of the filters to various environmental conditions could lead to desorption of the mercury from the filters.

To avoid such problems, the stability of Hg-enriched filters has been evaluated using various storage conditions according to an isochronous design [START_REF] Lamberty | The study of the stability of reference materials by isochronous measurements[END_REF][START_REF] Linsinger | Planning and combining of isochronous stability studies of CRMs[END_REF][START_REF] Gawlik | Testing sample stability in short-term isochronous stability studies for EU-wide monitoring surveys of polar organic contaminants in water[END_REF]. It is based on a storage design of the samples at various temperatures for different time intervals allowing all measurements to be done at the same time i.e. at the end of the study. In classical stability studies, measurements of the samples are achieved throughout the study i.e. at a different time so any drift of the measurement system over time can lead to incorrect conclusions. Thus, isochronous measurements only require repeatability conditions whereas classical stability studies require also long-term reproducibility.

The storage design set up in this study is shown in Table 1. Hg-enriched filters were obtained according to the present procedure under optimized conditions (500 µL of nanoparticles suspension, 5 min sonication time), using 25 mL of high purity water samples spiked with 100 ng of mercury (n=3).

Then, the filters were kept into polycarbonate filter holders (Petri slides) and Zip-lock bags at the corresponding temperature for the required time. It has been assumed that mercury is stable on the filters at low temperature (-7 °C). Thus, this reference temperature (i.e. -7 °C in a freezer) is the temperature at which the samples were always transported or kept before the analysis and the testing temperatures (i.e. 4 °C in a cool room and 21 °C in a flow hood at ambient temperature) were the conditions at which test samples were stored for a selected period of time, before returning to the reference temperature. The total amount of Hg in all the filters was measured together at the end of the experiment.

To distinguish the various storage conditions of the filters and their efficiency, a Kruskal-Wallis test (nonparametric test), followed by a Conover and Iman test has been performed. Fig. 3 summarizes the data collected in the isochronous measurement experiment along with the results of the Kruska-Wallis test. On the one hand, after eight weeks of storage, the ratio between the recovery of the testing temperature T1 (21 °C) and the recovery of the reference temperature decreases significantly down to 84.7 ± 1.9 %. On the other hand, after eight weeks, the decrease of this ratio with the testing temperature T2 (4 °C) is less important (93.2 ± 1.4 %) but still significant. The only storage condition that does not show a significant difference with the results from the reference temperature is as follows: one week of storage at 4 °C. This allows the in-field application of the mercury preconcentration method as it is usually possible to keep the filters at low temperature (4 °C) in a portable fridge before coming back to the lab where the filters can be stored at low temperature (< -7 °C). If filters must be stored for a longer time, it is recommended to do it at freezing temperature, and even in that case, possible loss of mercury should be expected.

Potential interferences: matrix effects

The potential occurrence of matrix effects due to the presence of interfering compounds in the sample was also investigated. For the sake of simplicity, the possible effect of salinity, organic matter and competing metals were modelled with NaCl, natural organic matter (NOM) and Ca 2+ , respectively.

Then, applicability to real samples with high organic matter content, and high salinity was demonstrated.

Organic matter

According to various studies about the adsorption of dissolved mercury and the influence of organic matter on this process, the efficiency of our new methodology could be eventually reduced by the presence of organic matter [START_REF] Yang | Effect of dissolved organic matter on adsorption and desorption of mercury by soils[END_REF][START_REF] Wu | Adsorption Characteristics and the Effect of Dissolved Organic Matter on Mercury(II) Adsorption of Various Soils in China, Soil and Sediment Contamination[END_REF][START_REF] Wang | Soil dissolved organic matter affects mercury immobilization by biogenic selenium nanoparticles[END_REF]. The reference material Suwanee River natural organic matter (SRNOM), purchased from International Humic Substances Society (IHSS, USA) was used to model organic matter. A solution with NOM = 500 mg L -1 was prepared by dissolving 12.5 mg of NOM in 25 mL of high purity water. Firstly, high purity water samples of 25 mL (n=4) were spiked with 100 ng of mercury, and various amounts of natural organic matter solution were added ([NOM] = 0 -18.4 mg L - 1

). Blanks (n=4) were also produced in the same extent and under the same conditions with unspiked high purity water. The whole samples were preconcentrated using 500 µL of nanoparticles suspension and 5 min sonication time. Recoveries in Fig. 4a., calculated using the theoretical amount of Hg in the water sample and the amount of Hg obtained from the analysis of the Hg-enriched filter, show us that for NOM < 5 mg L -1 there is no significant influence of the organic matter on the preconcentration of mercury on the nanoparticles (98±2 % < R < 105±1 %). Nevertheless, for NOM > 5 mg L -1 , the recovery decreases steadily down to 84±3 % for NOM = 18.4

The potential influence of organic matter was also tested with real samples from Lac des Carolins ([DOC] = 5.6 mg L -1 ). 200 mL of water sample from this lake (n=2) spiked with a known amount of mercury (1.6 -208 ng of mercury), and control solutions (n=2) with the same amount of mercury in 200 mL of high purity water were simultaneously prepared. Then, samples were stored at room temperature, protected from light, in a laminar flow hood at least for 12 hours (equilibrium step). The samples were then treated according to the procedure described in Section 2.2. The percentage of controls, calculated as the ratio between mercury found in the real sample and mercury found in the controls for the same amount of spiked mercury (Fig. 4c) demonstrates that our new methodology for mercury preconcentration can be applied successfully to samples with a relatively high content of organic matter.

Bravo et al. [START_REF] Bravo | The interplay between total mercury, methylmercury and dissolved organic matter in fluvial systems: A latitudinal study across Europe[END_REF], have studied the influence of organic matter on mercury species concentrations within 29 streams across Europe. In this study, total organic carbon (TOC) values range from 0.6 to 22.2 mg L -1 and 76 % of the streams show TOC < 5 mg L -1 , suggesting that our method is applicable to many freshwater systems.

Salinity

By analogy with organic matter, it has been shown that the presence of chloride salts might reduce the efficiency of the adsorption because of the formation of the resistant complex HgCl 4 -leading to an inhibition of the mercury adsorption on other sorbents [START_REF] Green-Ruiz | Effect of salinity and temperature on the adsorption of Hg(II) from aqueous solutions by a Ca-montmorillonite[END_REF][START_REF] Huang | Design and synthesis of core-shell Fe 3 O 4 @PTMT composite magnetic microspheres for adsorption of heavy metals from high salinity wastewater[END_REF]. The presence of chloride salts and their influence on the efficiency of the preconcentration procedure have been studied to determine applicability to seawater samples. Firstly, sodium chloride (NaCl, Merck KGaA, Denmark) has been added to high purity water samples of 25 mL (NaCl, 0 -36 g L -1 ), also spiked with 100 ng of mercury.

Replicates (n=4) of these synthesized solutions allowed us to produce Hg-enriched filters that have been analysed by AMA-254. Recoveries in Fig. 4b., calculated using the theoretical amount of Hg in the water sample and the amount of Hg obtained from the analysis of the Hg-enriched filter, show us that for NaCl < 22 g L -1 there is no significant influence of sodium chloride on the preconcentration of mercury on nanoparticles (100±1 % < R < 107±1 %). Nevertheless, the recovery decreases slightly until 93±2 % for NaCl = 36 g L -1

. For most natural waters, including groundwater, continental and estuarine waters, the presence of moderate amounts of salt should not lead to significant errors derived from the occurrence of matrix effects. Even in the case of high salinity seawaters, the effect should not be very pronounced (about 7 % suppression of the analytical signal).

To validate this hypothesis, we decided to apply the method of preconcentration to real samples with high salinity. 200 mL of water sample (n=2) from St Jean de Luz were spiked with a known amount of mercury (1.9 -208 ng of mercury). Control solutions (n=2) with the same amount of mercury in 200 mL of high purity water samples were simultaneously prepared. After an equilibration step of 12 hours, Hg-enriched filters were produced according to the preconcentration procedure. The percentage of control, depicted in Fig. 4d., demonstrates that our new methodology for mercury preconcentration can be applied successfully to samples with high salinity. Elsewhere, even if seawater contains high amounts of potential sources of exhaustion of the binding sites (sulphate, chloride, sodium, potassium etc…), the successful application of our new methodology to real seawater samples underlines the absence of competition of these ions with Hg for binding to the APDC.

Competing metals

With the variety of metal ions that could bind competitively with the chelating agent (APDC), calcium (Ca 2+ ) is usually the major species present in water samples, and so it has been selected to investigate the competition with mercury for the chelation step. High purity water samples of 25 mL containing various amount of calcium (Ca 2+ = 0 -18.5 mg L -1 ) were spiked with 100 ng of mercury.

The ratio between results obtained for the reference filter (without calcium) and other conditions (R = 102 -109 %) confirms that mercury is predominately bond to the APDC in all the conditions tested.

Considering the strong binding affinity between -SH groups from APDC and Hg, a high competition or influence from the presence of other ions than Ca 2+ is not expectable.

Method blank levels and optimization

Working under clean conditions is also a crucial consideration for successful analysis at very low concentration levels, in particular with mercury. In that sense, as mentioned in previous sections, for each change in a parameter during the preconcentration procedure (sample volume, nanoparticles suspension, sonication time, and amount of Ca 2+ , NaCl and organic matter in the sample) replicates of high purity water were submitted to the same protocol. Low levels and good reproducibility for these blank samples lead to a very good limit of detection.

Firstly, to check any contamination before their use, unused PES membrane filters have been analysed using the AMA-254. The absorbance obtained was 0.0025±0.0006 (n=15) and this value is not significantly different (t-test, p < 0.05) from the value given by the analysis of empty nickel boats, 0.0018±0.0002 (n=9), suggesting that no cleaning step is necessary for the PES filters.

Secondly, most of the experiments have been conducted in a trace metals laboratory whereas the last experiments, about real samples, took place in a clean laboratory, dedicated to ultra-trace mercury analysis. In the first cases, plastic flasks of 25 mL, cleaned in a 10% HNO 3 bath, were used as a container and the blanks levels reached 1.9±0.2 ng Hg per filter. In the clean laboratory, special attention was paid to the procedure for cleaning all vessels used for sampling (polyethylene vials) and sample preparation (glass vials, glass filtration system, and filter holders) with successive acid baths (10% HNO 3 , 10% HNO 3 and 10% HCl), reducing the blank levels down to 0.60±0.03 ng Hg per filter.

Finally, it has been shown that graphene on its own can adsorb volatile Hg species, so it is probably responsible for the Hg content in the blank filters. This hypothesis was confirmed by the analysis of graphene directly by AMA-254: 0.82±0.01 mg Hg kg -1 . One possibility to reduce such contamination, and therefore to reduce the blank levels, should be to buy new graphene nanosheets that will be managed only in a glove box under argon flux. Another solution could be to find a purification process not affecting graphene nanosheets structure and properties.

Analytical performances

The analytical performances of the method developed in this work are summarized in Table 2 together with other methods for determination of mercury species at trace and ultra-trace levels. Traditional and/or newly developed methods highlight various problems preventing their application in-field and/or in developing countries: heavy sample preparation, high cost due to reagents and/or apparatus, and high limits of detection. Indeed, very sensitive analytical methods commonly used for the determination of Hg species, CV-AFS (LOD < 0.06 ng L -1 ) [START_REF] Lescord | Factors affecting biotic mercury concentrations and biomagnification through lake food webs in the Canadian high Arctic[END_REF][START_REF] Eklöf | Spatial and temporal variation of THg concentrations in run-off water from 19 boreal catchments, 2000-2010[END_REF] and GC-ICP-MS (LOD < 0.04 ng L -1 )

[13], require a strong knowledge in analytical chemistry, and maintenance. Moreover, the use of these techniques implies a significant economic cost, due to either the equipment or the reagents. The use of a commercial mercury analyser for the analysis of the filters produced according to our new procedure greatly simplifies all the analytical process. Most of the newly developed methodologies focus on preconcentration of the Hg species followed by an elution step [START_REF] Soleimani | Highly selective solid phase extraction of mercury ion based on novel ion imprinted polymer and its application to water and fish samples[END_REF][START_REF] Muñoz | Speciation of Organometallic Compounds in Environmetal Samples by Gas Chromatography after Flow Preconcentration on Fullerenes and Nanotubes[END_REF][START_REF] Sotolongo | An easily prepared graphene oxide-ionic liquid hybrid nanomaterial for micro-solid phase extraction and preconcentration of Hg in water samples[END_REF][START_REF] Shirkhanloo | Ultrasound assisted-dispersive-ionic liquid-micro-solid phase extraction based on carboxyl-functionalized nanoporous graphene for speciation and determination of trace inorganic and organic mercury species in water and[END_REF]55], making their application in the field laborious, if not impossible, which is not the case in our new method with the direct analysis of the filter. Finally, the most interesting published analytical methods for in-field Hg preconcentration need also strong knowledge in analytical chemistry for the preparation of the nanogold-coated passive sampler [START_REF] Schlathauer | Sustainable and reagent-free mercury trace determination in natural waters using nanogold dipsticks[END_REF] and the magnetite nanoparticles [START_REF] Tavares | Reliable quantification of mercury in natural waters using surface modified magnetite nanoparticles[END_REF] while the functionalization of the graphene nanosheets in our study appears much simpler. Moreover, in this work, just a few minutes are necessary to recover the whole Hg species on the functionalized graphene nanosheets whereas Tavares et al. [START_REF] Tavares | Reliable quantification of mercury in natural waters using surface modified magnetite nanoparticles[END_REF] need 24 hours. Overall, the novelty and the superiority of our work lie mainly on its simplicity all along its process: sampling (easily transportable filter), preconcentration (quickly and effortlessly producible adsorbent) and analysis (direct analysis of the solid without any elution step needed using a cheap equipment).

This new method for quantification of mercury by DMSPE using graphene nanosheets and direct analysis by pyrolysis gold amalgamation and atomic absorption spectroscopy (AAS) resolves many of these limitations and can be still improved. It provides a LOD as low as 0.38 ng L -1 for 200 mL of samples, with a large linear range (0.38 -1038 ng L -1

). As concentration levels of total mercury in water samples (sea, estuarine, fresh) range in the ng L -1

, in most of the cases, our method is applicable, only with 600 mL of water sample for triplicate analysis. In the case of water samples presenting smaller mercury content, usually coming from pristine areas, such as ocean and many remote water bodies, the matrix is not complex, making the filtration step easier to manage. Therefore, for these particular cases, increasing the volume up to 1 L should allow a reduction of the LOD to match the Hg concentration in the sample.

A

  simple determination of trace mercury concentrations in natural waters using dispersive Micro-Solid phase extraction preconcentration based on functionalized graphene nanosheets Bastien Duval, Ainara Gredilla, Silvia Fdez-Ortiz de Vallejuelo, Emmanuel Tessier, David Amouroux, Alberto de Diego To cite this version: Bastien Duval, Ainara Gredilla, Silvia Fdez-Ortiz de Vallejuelo, Emmanuel Tessier, David Amouroux, et al.. A simple determination of trace mercury concentrations in natural waters using dispersive Micro-Solid phase extraction preconcentration based on functionalized graphene nanosheets. Microchemical Journal, 2020, 154, pp.104549. 10.1016/j.microc.2019.104549. hal-02429200

  2.3 -207 ng Hg). Indeed, matrix effects can occur during the analysis of the PES filters and there is no existing certified reference material with a similar matrix. The Hg(II) standard solution (Strem Chemicals, USA) was prepared in 1% HCl with a concentration of 400 µg Hg L -1 and various amounts were directly dropped into a nickel boat containing a PES filter resulting in absolute Hg masses of 2, 4, 10, 21, 31, 42, 53, 85, 144 and 207 ng. Here, we used Hg(II) considering the low fraction of MeHg in natural waters (< 10 % of total mercury) and assuming there would not be any difference in the adsorption process using

	Rank1 (R 2 = 0.99974) and Rank2 (R 2 = 0.99971), with slopes of respectively (38.6 ± 0.1) and (576 ±
	3) ng Hg Absorbance Unit	-1

either Hg(II) and/or MeHg (Section 2.1.). The absorbance (A) resulting from the analysis of unused filters was used for blank correction. The calibration functions obtained provide a reliable way to determine Hg content in the PES filters for both absorption cells, Rank1 (R 2 = 0.99947) and Rank2 (R 2 = 0.99996), with slopes of respectively (36.4 ± 0.3) and (560 ± 1) ng Hg AbsorbanceUnit -1 .

The calibration curves according to the present procedure under the optimized conditions were also set up in the absolute mass of mercury (1.9 -210 ng). Various samples containing 200 mL of high purity water were spiked with known amounts of Hg(II) standard solution ([Hg] = 400 µg L -1 ) resulting in absolute Hg masses of 2, 4, 8, 13, 19, 22, 43, 85, 148 and 211 ng. These samples were subjected to the preconcentration process previously described (Section 2.2.). The obtained calibration functions provide a reliable way to determine Hg content in the PES filters for both absorption cells,

Table 1 : Storage design for isochronous measurements to evaluate stability of the filters

 1 ://doi.org/10.1016/j.microc.2016.09.012.[55] S. Yang, D. Zhang, H. Cheng, Y. Wang, J. Liu, Graphene oxide as an efficient adsorbent of solid-phase extraction for online preconcentration of inorganic and organic mercurials in freshwater followed by HPLC-ICP-MS determination, Analytica Chimica Acta. 1074 (2019) 54-61. https://doi.org/10.1016/j.aca.2019.04.066.

	caprine	blood	samples,	Microchemical	Journal.	130	(2017)	245-254.

httpsAnalysis × Storage at temperature T Storage at very low temperature (T REF )

Table 2 : Methods for analysis of mercury species in natural waters

 2

	Method	Volume (L)	Water type	Hg species	Quantification	RSD (%)	LOD (ng.L -1 )	Note	Reference
	Procedure resented in this work	0.2 (5min extraction)	Fresh and seawater	THg	External calibration	4.2 (Blanks) 3.0 (Standard) 4.5 (Seawater) 3.2 (Fresh water)	0.4	APDC/Graphene Nanosheets -DMSPE, filtration, detection and double gold amalgamation before AAS	This work
	GC-ICP-MS	0.1	Fresh and seawater	Hg 2+ MeHg +	Isotopic Dilution (equilibration time 12h)	2.5 3.4	0.04 0.01	Alkyl derivatization to produce volatile species, extraction in GC organic solvent	[13]
	CV-AFS (EPA 245.7)	0.25	Fresh water	THg	External calibration	n.d.	0.04	KBr oxidation, SnCl2 reduction, purge of Hg(0), and AFS detection	[53]
	CV-AFS (EPA 1631)	0.1	Fresh water	THg	External calibration	~5	0.06	BrCl oxidation, SnCl2 reduction, purge of Hg(0), and double gold amalgamation before AFS detection	[52]
	IIP-CV-AAS	0.1	Fresh and seawater	Hg 2+	External calibration	2.4	0.5	Preparation of the IIP for Hg 2+ specific adsorption detection IIP -SPE column, elution by EDTA, and AAS	[15]
	WCNTs-GC-MS	0.025	Fresh and seawater	Hg 2+ MeHg + EtHg +	Internal standard quantification	6.2 7.2 6.8	4 3 3	Complexation of Hg species by NaDDC, MWCNTs -alkyl derivatization SPE column, elution by ethyl acetate with online	[17]
								Preparation of the IL-GO hybrid nanomaterial for Hg
	IL-GO-ETAAS	0.005	Fresh water	THg	External calibration	3.9	14	adsorption IL-GO -SPE column, elution by 20% HNO3 and	[20]
								ETAAS detection
	G-COOH-FI-CV-AAS	0.010	Fresh water	Hg 2+ EtHg + MeHg +	External calibration	<3	9.8	NG-COOH as solid-phase sorbent for US-D-IL-µ-Elution by HNO3 and analysis by FI-CV-AAS SPE	[54]

Acknowledgements: This work has been partially supported (65%) by the FEDER funds through the INTERREG V-A Spain-France-Andorra (POCTEFA 2014-2020) (REPLIM project, Ref. EFA056/15), and by the Basque Government through the Consolidated Research Group Program 2013-2018 (Ref.

IT-742-13). Bastien Duval is grateful to the University of the Basque Country UPV/EHU for his predoctoral fellowship.

The authors would like to thank S. Mallet (SymaLab) for his technical assistance and UT2A Laboratory for access to the AMA-254 instrument. Special thanks to J. Beney, English teacher, for proofreading this manuscript.