Solving Dynamic Controllability Problem of Multi-Agent Plans with Uncertainty

Guillaume Casanova 1 and Cédric Pralet 1 and Charles Lesire 1 and Thierry Vidal2

Abstract.

Executing multi-agent missions requires managing the uncertainty about uncontrollable events. When communications are intermittent, it additionally requires for each agent to act only based on its local view of the problem, that is independently of events which are controlled or observed by the other agents. In this paper, we propose a new framework for dealing with such contexts, with a focus on mission plans involving temporal constraints. This framework, called Multi-agent Simple Temporal Network with Uncertainty (MaSTNU), is a combination between Multi-agent Simple Temporal Network (MaSTN) and Simple Temporal Network with Uncertainty (STNU). We define the dynamic controllability property for MaSTNU, and a method for computing offline valid execution strategies which are then dispatched between agents. This method is based on a mixed-integer linear programming formulation and can also be used to optimize criteria such as the temporal flexibility of multi-agent plans.

Introduction

In robotic applications such as the autonomous exploration of large and hazardous areas, better performances can be obtained by using multiple robots. This can indeed lead to a faster achievement of the mission due to parallel realizations of tasks, and bring redundancy for continuing the mission in case of robot failures. One difficulty to overcome in this context is that the tasks allocated to robots must be coordinated, since there may exist precedence or synchronization constraints between tasks, or more generally constraints on the minimum/maximum temporal distances between tasks. To handle these multi-agent temporal constraints, Multi-agent Simple Temporal Networks (MaSTNs [START_REF] Boerkoel | Distributed Algorithms for Incrementally Maintaining Multiagent Simple Temporal Networks[END_REF]) were recently introduced, with techniques for computing, in a distributed way, allowed distances between timepoints involved in plans [START_REF] Boerkoel | Distributed Algorithms for Incrementally Maintaining Multiagent Simple Temporal Networks[END_REF], or earliest/latest occurrence times of time-points [START_REF] Casanova | Managing dynamic multi-agent simple temporal network[END_REF].

However, one issue when using MaSTN for robotic missions is that MaSTN are not designed for obtaining decision strategies which are robust to the uncertainty about the occurrence time of uncontrollable time-points. For instance, they are not adapted to obtain plans which are feasible whatever the exact duration of tasks turn out to be. Along this line, they are not as expressive as the framework of Simple Temporal Network with Uncertainty (STNU [START_REF] Morris | Dynamic Control Of Plans With Temporal Uncertainty[END_REF]), which makes an explicit distinction between executable time-points, which can be directly controlled, and contingent time-points, which cannot. In STNU, robust execution strategies describe a way to set the occurrence time of executable time-points depending on timepoint occurences observed so far, and these strategies are built under the strong assumption that the realization of every time-point in the temporal network is instantaneously observed. Such an assumption is often violated for multi-robot systems, since each event might be observable only from a restricted set of geographical positions.

This is why we propose a new framework for managing temporal constraints over multi-agent systems. This framework, called Multiagent Simple Temporal Network with Uncertainty (MaSTNU), can be seen as an attempt to combine MaSTN and STNU. It is equipped with algorithms to compute robust execution strategies which respect every temporal constraint of the multi-agent plan despite the uncertainty about the occurrence time of contingent time-points, and which are applicable even in constrained environments featuring intermittent communications between agents. Such distributed execution strategies are obtained using a centralized offline procedure based on a Mixed Integer Programming (MIP) formulation of what we call the multi-agent dynamic controllability problem. This procedure is run at the mission center before triggering the coordinated deployment of the agents on the field.

The paper is organized as follows. Sect. 2 introduces some background on STN, MaSTN, STNU, and dynamic controllability checking. Sect. 3 presents the MaSTNU framework. Sect. 4 details our MIP approach for dealing with multi-agent dynamic controllability. Sect. 5 provides experimental results and discusses several ways to optimize execution strategies.

Background

Simple Temporal Network

A standard framework for reasoning about temporal constraints is the framework of Simple Temporal Problems (STPs [START_REF] Dechter | Temporal Constraint Networks[END_REF]). Basically, an STP is a pair S = (V, E) defined by a set V = {v1, . . . , vn} of time-points representing event occurrence times, and a set E of temporal constraints between these time-points. Each constraint e ∈ E takes the form vjvi ∈ [Lij, Uij], where Lij ∈ R ∪ {-∞} and Uij ∈ R ∪ {+∞} respectively specify a minimum and a maximum temporal distance between vi and vj. A specific time-point v0 called the reference-point is usually added to V for representing a reference temporal position, and unary temporal constraints such as vi ∈ [L0i, U0i] can then be easily expressed as distance constraints with regards to this reference-point (constraints vi-v0 ∈ [L0i, U0i]).

Using Mixed Integer Linear Programming

STPs have a natural graphical representation called Simple Temporal Networks (STNs), which contain one vertex per time-point in V and one edge vi → vj labeled by [Lij, Uij] per temporal constraint in E. STNs are appealing in practice to deal with temporal aspects because several problems that can be formulated on STNs are solvable in polytime [START_REF] Dechter | Temporal Constraint Networks[END_REF], such as determining whether there exists an assignment of time-points satisfying all temporal constraints.

STNs were extended to deal with multi-agent problems on one hand and with uncertain temporal durations on the other hand. Thereafter, we give some background on these two distinct extensions.

Multi-agent Simple Temporal Network

STNs were extended to a multi-agent context, where time-points are not controlled by a single agent but are instead partitioned among a set of agents A. This extension is called MaSTN for Multiagent Simple Temporal Network [START_REF] Boerkoel | Distributed Algorithms for Incrementally Maintaining Multiagent Simple Temporal Networks[END_REF]. Formally, an MaSTN is defined by:

• a set of local STNs (one per agent a ∈ A); the local STN associated with agent a, denoted by S a L , is defined by V a , the set of local time-points owned by a, and E a L , the set of local edges which connect only time-points in V a ; • a set of external edges EX , each of which constrains the temporal distance between two time-points belonging to distinct agents; apart from its local edges in E a L , each agent a is aware of the subset of external constraints which hold on one of its local vertices.

v A 1 to v A 6 (resp. v B 1 to v B 8 and v C 1 to v C 8 ). Edge (v A 1 , v A 2 ) is an example of a local edge for agent A. Edge (v A 3 , v B 3
) is an example of an external edge in EX . It enforces some synchronization between agents A and B.

MaSTN algorithms were defined to compute, in a distributed way, possible temporal distance between pairs of time-points (distributed partial path-consistency algorithms [START_REF] Boerkoel | Distributed Algorithms for Incrementally Maintaining Multiagent Simple Temporal Networks[END_REF]), as well as earliest/latest dates associated with time-points (distributed arc-consistency algorithms [START_REF] Casanova | Managing dynamic multi-agent simple temporal network[END_REF]).

v 0 v B 3 acq2 charge receive swOn swOff [3, 5] v B 6 v B 7 v B 8 v B 5 [2, 10] [4, 10] v B 1 v B 2 [1, 2] [6, 20] v B 4 [8, ∞] [0, ∞] [0, ∞] [0, ∞] [0, 10] [0, 0] [0, 0] v A 1 [5, 8] acq3 acq4 sendAcq3 [3, ∞] [1, 3] [0, 16] [0, 16] v C 4 v C 1 [1, 2] v C 2 v C 3 [3, 4] [4, 4] sendAcq4 v C 6 v C 7 [0, ∞] v C 8 [4, 7] v C 5 [0, 30] [0, 0] [0, 0] acq1 recharge [2, 4] [2, 5] [0, ∞] v A 4 v A 2 v A 3 [5, ∞] v A 5 v A 6 [1, 1] maintenance Figure 1.
Example of a Multi-agent STN involving 3 agents A, B, C (one line per agent)

Simple Temporal Network with Uncertainty

In another direction, STNs were extended to Simple Temporal Networks with Uncertainty (STNUs [START_REF] Morris | Dynamic Control Of Plans With Temporal Uncertainty[END_REF]) in order to represent uncertain durations, that is durations whose value is fixed by an external process rather than by the planning agent itself.

Formally, an STNU is a triple (V, E, C), where V is a set of timepoints, E is a set of requirement links, and C is a set of contingent links. Each requirement link is defined as in standard STN. Each contingent link is defined by a pair of time-points (vi, vj) and by a temporal interval [Lij, Uij] with 0 < Lij < Uij < ∞. The duration of such a contingent link, that is distance vjvi, is known to be between Lij and Uij, but its precise value is not controlled. In this case, vi and vj are respectively called the activation time-point and the contingent time-point. Last, a time-point cannot be the contingent time-point of two distinct contingent links. Any time-point which is not a contingent time-point for some contingent link is called an executable time-point. In the following, we denote by VE the set of executable time-points and by VC the set of contingent time-points.

Fig. 2 gives an example of an STNU involving six time-points plus the reference time-point v0. Requirement links such as v B 2v B 1 ∈ [START_REF] Dechter | Temporal Constraint Networks[END_REF][START_REF] Morris | A structural characterization of temporal dynamic controllability[END_REF] are depicted using continuous lines, while contingent links such as [START_REF] Dechter | Temporal Constraint Networks[END_REF] are depicted using dashed lines. In this STNU, the set of executable time-points is

v B 1 -v A 1 ∈ [1,
VE = {v A 1 , v A 2 , v B 2 , v B 3 } and the set of contingent time-points is VC = {v B 1 , v A 3 }.
The fundamental problem associated with an STNU is to determine whether it is dynamically controllable, which informally means that there exists a way to dynamically assign values to executable time-points depending on observations collected, so that all requirement links are satisfied whatever the precise values of contingent links turn out to be at execution time.

More formally, dynamic controllability over STNU can be defined as follows. First, a projection of an STNU is an STN obtained by replacing each contingent link vj

-vi ∈ [Lij, Uij] by a deterministic link vj -vi ∈ [d, d] with d ∈ [Lij, Uij].
A schedule is an assignment of values to all time-points. An execution strategy R can then be defined as a mapping from projections to schedules. An execution strategy R is said to be valid when for every projection p, schedule R(p) satisfies all requirement links. An execution strategy R is said to be dynamic iff for every executable time-point v and every projections p1, p2 of the STNU, if the assignment of all time-points scheduled before v are the same in R(p1) and R(p2), then the values assigned to v in R(p1) and R(p2) are the same. In other words, the execution time of v can only depend on the information gathered before executing v. Last, an STNU is Dynamically Controllable (DC) iff there exists an execution strategy with is both valid and dynamic.

The STNU provided in Fig. 2 is dynamically controllable, and a valid dynamic execution strategy can be: (a

) execute v A 1 at time 0, (b) wait for v B 1 to happen, (c) execute v B 2 at time v B 1 + 4, (d) execute v A 2 at time v B 2 + 6, (e) execute v B 3 at time v B 2 + 6, (f) wait for v A 3 to happen. v 0 v A 1 [0, 5] v A 2 v A 3 [1, 10] v B 1 v B 2 [4, 6] v B 3 [6, 12] [1, 4] [1, 5] [6, 8]

Checking Dynamic Controllability using MIP

Several algorithms do exist for checking dynamic controllability [START_REF] Morris | Dynamic Control Of Plans With Temporal Uncertainty[END_REF][START_REF] Morris | Dynamic controllability and dispatchability relationships[END_REF][START_REF] Hunsberger | Efficient execution of dynamically controllable simple temporal networks with uncertainty[END_REF]. One of them consists in using graph-based algorithms for com-puting mandatory wait constraints on requirement links. A wait constraint (v k , w ijk ) on a requirement link (vi, vj) ∈ E means that vj can only be executed either after v k is executed, or after w ijk time units since the execution of vi.

In a completely different direction, DC checking on STNUs can also be formulated as a Mixed-Integer linear Program (MIP) [START_REF] Cui | Optimising bounds in simple temporal networks with uncertainty under dynamic controllability constraints[END_REF]. One advantage of such a MIP formulation is that it can be adapted for answering more general queries, such as minimally updating time bounds on contingent links such that a non-DC STNU becomes DC.

Fig. 3 gives the disjunctive linear model introduced in [START_REF] Cui | Optimising bounds in simple temporal networks with uncertainty under dynamic controllability constraints[END_REF], from which a MIP model can be obtained using some linearization steps. Roughly speaking, the model contains two continuous decision variables lij and uij for each pair of time-points (vi, vj). Variables lij and uij respectively represent the lower and upper bounds imposed on the distance vj -vi between time-points vi and vj. The model also contains a set of continuous wait variables w ijk (one variable per triple of time-points (vi, vj, v k ) such that v k is a contingent timepoint). These variables have the same meaning as seen previously. Discrete decision variables are present in the MIP model after the linearization process. If a solution is found to the problem, then the STNU is DC. See [START_REF] Cui | Optimising bounds in simple temporal networks with uncertainty under dynamic controllability constraints[END_REF] for details concerning the correctness of the modeling and the linearization process.

An optimization function fopt can easily be added to the model, for instance to maximize the flexibility of solutions by using fopt 3 Multi-agent Simple Temporal Network with Uncertainty (MaSTNU)

= i<j (uij -lij). ∀(vi, vj) ∈ E, Lij ≤ lij ≤ uij ≤ Uij (1) ∀(vi, vj) ∈ C, (lij = Lij) ∧ (uij = Uij) (2) ∀vi, vj, v k ∈ V,        l ik ≤ uij + l jk ≤ u ik l ik ≤ lij + u jk ≤ u ik u ik ≤ uij + u jk lij + l jk ≤ l ik (3)
∀(vi, v k ) ∈ C, ∀vj ∈ VE, (l jk < 0) ∨ uij ≤ l ik -l jk lij ≥ u ik -u jk (4)
∀(vi, v k ) ∈ C, ∀vj ∈ VE, u ik -u jk ≤ w ijk (5) ∀(vi, vj) ∈ E, ∀v k ∈ VC , min(l ik , w ijk ) ≤ lij (6) ∀(vi, v k ), (vm, vj) ∈ C 2 , (w ijk < 0) ∨ (w ijk -lmj ≤ w imk ) (7) ∀(vi, v k ) ∈ C, ∀vm, vj ∈ V, w ijk -umj ≤ w imk (8)

Framework Definition

As explained in the introduction, STNU cannot be directly reused in a multi-agent setting, where each agent only controls a subset of the executable time-points and only observes the occurrence of a subset of the contingent time-points. This is why we introduce Multi-agent STNU (MaSTNU).

Formally, an MaSTNU is a quadruplet (A, V, E, C), with A a set of agents and (V, E, C) an STNU (V denotes the set of executable and contingent time-points, E the set of requirement links, and C the set of contingent links). Additionally, as in MaSTN, time-points in V are partitioned among A, that is for every time-point v ∈ V there exists a unique agent a ∈ A which owns v, denoted by owner (v) = a. Semantically speaking, if v is an executable time-point, then the owner of v is the agent which controls the execution of the event associated with v. If v is a contingent time-point, the owner of v is the unique agent which is assumed to instantaneously observe the realization of v. Time-points owned by other agents are not supposed to be directly observed, however information about their realization can be obtained thanks to external contingent links. Reference-point v 0 represents a clock synchronized between agent and is considered to be simultaneously owned by all agents.

In the following, for each agent a ∈ A, V a denotes the set of time-points owned by a, called the local time-points of a. We denote by E a L (resp. C a L ) the set of local requirement links (resp. contingent links), which hold only on time-points owned by a. Analogously to MaSTN, we also define EX (resp. CX ) as the set of external requirement links (resp. contingent links), which connect two time-points owned by different agents.

Fig. 4 gives an example of an MaSTNU involving two agents A and B, which respectively own time-points

V A = {v A 1 , v A 2 , v A 3 } and V B = {v B 1 , v B 2 , v B 3 }. The link from v A 1 to v B 1 is an external contingent link, the link from v B 2 to v A 2 is an external requirement link, the link from v B 1 to v B
2 is a local requirement link, and there is no local contingent link. Semantically speaking, external contingent links model observations received by an agent, the source of these observations being owned by other agents. For instance, the source of a contingent link might be the start of a data transmission process triggered by one agent, and the target of this link might be the end of this data transmission process, observed by the receiving agent. As in MaSTN, external requirement links correspond to synchronization constraints between agents. For instance, they can serve to express that there must not be more than 10 time units between successive surveillances of a given area by two distinct agents.

v 0 v 0 v A 1 [0, 5] v A 2 v A 3 [1, 10] v B 1 v B 2 [4, 6] v B 3 [6, 12] [1, 4] [1, 5] [6, 8] Agent A's STNU Agent B's STNU

Dynamic Controllability Revisited

The multi-agent nature of MaSTNU requires an adaptation of the dynamic controllability property. Indeed, given an MaSTNU (A, V, E, C), computing a valid dynamic execution strategy for STNU (V, E, C) does not necessarily give an applicable multi-agent strategy. As an illustration, consider the MaSTNU provided in Fig. 4. The STNU associated with it is the STNU previously shown in Fig. 2. By considering the execution strategy seen for this STNU and by partitioning it between agents, we obtain the following strategy:

• for agent A: (a) execute v A 1 at time 0, (b) execute v A 2 at time v B 2 + 6, (c) wait for v A 3 to happen; • for agent B: (a) wait for v B 1 to happen, (b) execute v B 2 at time v B 1 + 4, (c) execute v B 3 at time v B 2 + 6.
The issue with such a strategy is that agent A has no guarantee to be able to execute it, because it might not observe external time-point v B 2 owned by agent B. This is why we introduce a new definition of dynamic controllability which is adapted to MaSTNU. Let (A, V, E, C) be an MaSTNU and let R be an execution strategy for the associated STNU (V, E, C). Execution strategy R is said to be distributed iff for every projections p, p ′ (that is for every two possible assignments of the duration of contingent links) and for every agent a ∈ A, if schedule R(p) and schedule R(p ′ ) assign the same value to all contingent time-points owned by a, then they also assign the same value to all executable time-points owned by a. In other words, each agent only acts based on its own immediate observations, which means that the execution strategy is robust to the missing observations of external time-points.

An MaSTNU (A, V, E, C) is then said to be dynamically controllable iff STNU (V, E, C) admits an execution strategy which is valid (it induces schedules which satisfy all requirement links), dynamic (decisions are made only based on past information), and distributed (previous definition).

For the MaSTNU given in Fig. 4, an example of a valid, dynamic and distributed execution strategy is:

• for agent A: (a) execute v A 1 at time 4, (b) execute v A 2 at time 19, (c) wait for v A 3 to happen; • for agent B: (a) wait for v B 1 to happen, (b) execute v B 2 at time v B 1 + 6 if v B 1 ≤ 7 and at time v B 1 + 4 otherwise, (c) execute v B 3 at time v B 2 + 8.
In the following, we introduce techniques for checking DC for MaSTNU and for automatically computing distributed strategies.

Dynamic Controllability Check and Computation of Execution Strategies

To check DC for an MaSTNU (A, V, E, C), we first check DC for STNU (V, E, C). If this STNU is not DC, then the MaSTNU is not DC either, because acceptable execution strategies for MaSTNU are more restricted than acceptable execution strategies for STNU. Otherwise, if STNU (V, E, C) is DC, we perform additional operations to determine whether the original MaSTNU is DC.

From one MaSTNU to a set of local STNUs

The key idea in our method is to transform the original MaSTNU S into a distributed MaSTNU, which contains no external link between agents, and then to partition this distributed MaSTNU into a set of local STNUs {S a | a ∈ A}. Fig. 5 shows an example of such a process. The reason why we consider distributed MaSTNUs as the transformation target is that if each agent a ∈ A uses a valid dynamic execution strategy R a for its own local STNU S a , then the global strategy obtained by joining strategies R a is valid and dynamic, and it is also distributed because we are sure that in R a , each agent acts only based on the observations it is supposed to get at execution (no possible occurrence of external time-points in the execution strategy thanks to the partitioning). In other words, the set of local execution strategies {R a | a ∈ A} allows to dynamically control the MaSTNU. Globally, to transform the original MaSTNU into a set of local STNUs, we need to perform two kinds of operations:

1. to replace external requirement links of the original MaSTNU by requirement links which are local to agents, as done in Fig. 5 for

external constraint v A 2 -v B 2 ∈ [6, 8
] which will necessarily be satisfied thanks to the two internal requirements

v A 2 -v0 ∈ [19, 19] and v B 2 -v0 ∈ [11, 13
] which are present in the partitioned MaSTNU; the introduced local requirement links can be stronger than in the initial MaSTNU, and they are implicitly used to coordinate agent actions; 2. to remove external contingent links and to replace them by local contingent links, as done in Fig. 5 for

(v A 1 , v B 1 )
which is replaced by (v0, v B 1 ); more generally, the external source v of a contingent link (v, w) must be replaced by a local source u contained in the agent which owns w.

The way these two operations are realized is presented in the two following sections. Compared to DC reasoning on STNU, it is worth mentioning that the transformation of the original MaSTNU into several local STNUs is a combinatorial decision problem, because for instance there is not necessarily a unique way of distributing/sharing the satisfaction of external requirement links among agents, or a unique way of reassigning the contingency source of a contingent time-point. The associated decision problem is formalized using a MIP model, which allows us to reuse elements from the existing MIP model given in Sect. 2.4 for standard STNUs. In the MIP model built, we capture several constraints guaranteeing the satisfaction of the external requirement links of the original MaSTNU, and several constraints guaranteeing that the local contingency assumptions made in the distributed MaSTNU are not restrictive with regards to the set of possible scenarios covered by the external contingent links of the original MaSTNU. By adding a linear optimization function, MIP solvers can then be used to find an optimal distribution of temporal constraints such that all local STNUs are DC.

In the following, as in the MIP model of DC for STNU, we use, for every i < j, variables l ij and uij to represent the lower and upper bounds imposed on the distance vj -vi between vi and vj. Moreover, for i < j, we also use uji as a substitute for -lij.

Internalization of external requirement links

Let us consider an external requirement link e = (vi, vj), with owner (vi) = a, owner (vj) = b, a = b. Initially, e is labeled by [Lij, Uij]. The main issue is that a and b might not have enough information during execution to ensure that e is respected. For example, if b waits to observes vi before executing vj then it might fail at respecting e if the delay for observing vi is greater than Uij. Similarly, a has no information about when it should be executing vi in a way such that b can execute vj and respect e.

To make sure that the upper bound of e is respected during execution without using any communication between a and b, it suffices to 1. each link (pi, pi+1) involved in p is either an internal link (owner (pi) = owner (pi+1)), or an external contingent link ((pi, pi+1) ∈ CX or (pi+1, pi) ∈ CX ); 2. path p defines a path from vi to vj which is shorter than Uij, so that if all constraints between time-points in p are satisfied, then e is also satisfied.

v 0 v A 1 [0, 5] v A 2 v A 3 [1, 10] v B 1 v B 2 [4, 6] v B 3 [6, 12] [1, 4] [1, 5] [6, 8] Original MaSTNU v 0 v A 1 [4, 5] v A 2 [19, 19] v A 3 [1, 10] A's local STNU v 0 v B 1 [5, 9]
Such paths are called distributed paths. A contingent link in a distributed path is necessarily satisfied at execution by definition. An internal requirement link in a distributed path is satisfiable by the agent holding it at execution as long as its local STNU is DC. Therefore, if all local STNUs are DC, then all constraints in path p are satisfiable at execution, and therefore the original external requirement link is satisfiable as well. Similar distributed paths must be found to justify that the lower bound of e is satisfied.

In order to find such paths, it actually suffices to decide on the sequence of contingent links to use for justifying the satisfaction of bound u ij , because successive links which are internal to a single agent can be harmlessly collapsed into a single internal link thanks to the path consistency property over STNU. See Figure 6 for an illustration of a path p = [vi, v k 1 , v l 1 , v k 2 , v l 2 , . . . , v kn , v ln , vj] covering the satisfaction of requirement link (vi, vj). On this example, the satisfaction of requirement link (vi, vj) is covered by the satisfaction of some requirement over (vi, v k 1 ) and (v l 1 , vj); the satisfaction of (v l 1 , vj) is itself covered by the satisfaction of some requirement over (v l 1 , v k 2 ) and (v l 2 , vj)... and so on until there is no more external link to satisfy. Informally speaking, the path built use a sequence of quadrilaterals, and it introduces some new temporal distance constraints with regards to time-points which are correlated through contingent links. Exploiting these correlations is the only way to be robust to the absence of communication. Also, thanks to the path consistency property again, by imposing that local STNUs must be DC, it suffices to search for distributed paths which cross each agent at most once.

In order to formalize such a process, we define two sets:

• set EX which contains all possible external links which are not contingent links, and therefore might require justification; this set is given by

EX = {(vi, vj) ∈ V 2 | (owner (vi) = owner (vj)) ∧ ((vi, vj) / ∈ CX )) ∧ ((vi, vj) / ∈ CX ))}; v k1 v l1 v k2 v l2 v k3 v ln v kn v ln-1 c1 cn v i c2 v j Figure 6
. Justification path for an external requirement link between v i and v j

• set Q which contains all quadruplets (vi, vj, v k , v l ) such that -(vi, vj) ∈ EX is an external requirement link whose satisfaction which might have to be justified when building justification paths;

-(v k , v l
) is associated with a contingent link which can be used in the justification for (vi, vj), which means that (1) either (v k , v l ) = (v0, v0) (case in which the upper bound uij over vjvi is justified by a path through the reference timepoint), ( 2) or (owner

(v k ) = owner (vi)) ∧ ((v k , v l ) ∈ CX ∨ (v l , v k ) ∈ CX ).
Note that the size of Q is at most cubic in the number of timepoints, since v k and v l are related by a contingent link and because a contingent time-point can only have a unique contingent link pointing to it.

To model the requirement to cover external requirement links by distributed paths, we introduce a MIP modeling which uses the following variables:

• ∀(vi, vj) ∈ EX , bij ∈ {0, 1} is a boolean decision variable encoding that we need to justify external requirement link vjvi ≤ uij; for an illustration of what could happen without preventing cycles in justifications; it also helps bounding the search process since it suffices to consider distributed paths which cross each agent at most once and each contingent link at most once, which explains the value chosen for the upper bound of hij. We impose several linear constraints for representing the satisfaction of the external requirements by distributed paths. First, the lower and upper bounds associated with contingent links of the original MaSTNU cannot be shrinked:

• ∀(vi, vj, v k , v l ) ∈ Q, z ijkl ∈ {0, 1}
v A 1 v A 2 [2, 2] v B 1 v C 1 [2, 2] [0, 1] [0, 1]
∀(vi, vj) ∈ CX , (lij ≤ Lij) ∧ (uij ≥ Uij) (9)
Next, every external requirement link in the initial MaSTNU must be justified:

∀(vi, vj) ∈ EX s.t. Uij = +∞, bij = 1 (10) ∀(vi, vj) ∈ EX s.t. Lij = -∞, bji = 1 (11)
If an external requirement link (original or intermediate) must be justified, then there exists a unique contingent link justifying it:

∀(i, j) ∈ EX , bij = (v i ,v j ,v k ,v l )∈Q z ijkl (12)
An external requirement link is justified if there exists a shorter distributed path:

∀(i, j, k, l) ∈ Q, uij ≥ u ik + u kl + u lj + (z ijkl -1)M (13)
In the previous equation, M is a large constant equal to Lij -U ik -U kl -U lj , so that the constraint is always satisfied when z ijkl = 0.

Then, every external requirement link used in a justification must also be justified (again, see Fig. 6 for an illustration):

∀(vi, vj, v k , v l ) ∈ Q s.t. (v l , vj) ∈ EX , z ijkl ≤ b lj (14)
Finally, we are preventing cycles in justifications thanks to the following set of constraints (in the following equation, H is the maximum value of hij variables):

∀(vi, vj, v k , v l ) ∈ Q s.t. (v l , vj) ∈ EX hij + (1 -z ijkl )(H + 1) ≥ h lj + 1 (15)
Fig. 8 shows a representation of the MaSTNU obtained after the internalization process of external requirement links. In this example, A and B tighten internal constraints (v0,

v A 1 ), (v0, v A 2 ), (v0, v B 2 ) and (v0, v B 3 ) so that external requirement link (v B 2 , v A 2
) is satisfied at execution. Contrarily to the mono-agent STNU solution seen in Fig. 2,v A 2 has no temporal flexibility anymore.

v 0 v A 1 [4, 5] v A 2 [19, 19] v A 3 [1, 10] v B 1 v B 2 [4, 6] [11, 13] v B 3 [6, 12] [1, 4] [1, 5]

Internalization of external contingent links

External contingent links also have to be internalized, otherwise external time-points might appear in execution strategies, which would invalidate the distributivity of these strategies. Globally, the idea in the internalization of links in CX is that every potential situation which may be encountered owing to the original MaSTNU must be covered by scenarios considered at the level of local STNUs.

To illustrate the transformation proposed, let us consider an external contingent link c = (vi, vj), with owner

(vi) = a, owner(vj) = b, a = b. c is labeled by [Lij, Uij], with Lij > 0.
Any execution strategy directly using the fact that "vj occurs necessary between Lij and Uij units of time after vi" cannot be sound as b does not directly observe vi. This is why we need to explicitly erase c from the MaSTNU representation while keeping the uncontrollable status of vj. The only solution to do this is to replace link (vi, vj) by an internal contingent link (v k , vj) in the set of local contingent constraints of agent b. In this case, we say that we use substitution triangle (vi, vj, v k ). In the following, we define the set of candidate substitution triangles by T = {(vi, vj, v k ) | (vi, vj) ∈ CX , v k ∈ V owner (v j ) \ {vj}}. For every external contingent link (vi, vj), as there is a freedom in the local time-point v k chosen for activating vj, we add in the MIP model the following set of decision variables:

• ∀(vi, vj, v k ) ∈ T, c kj ∈ {0, 1} is a boolean decision variable
encoding that we substitute external contingent link (vi, vj) by a new internal contingent link (v k , vj).

Several constraints are imposed over these variables. First, every external contingent link must be substituted by exactly one internal contingent link:

∀(vi, vj) ∈ CX , v k | (v i ,v j ,v k )∈Q c kj = 1 (16)
If an external contingent link (vi, vj) is substituted by an internal contingent link (v k , vj), then the bounds specified by (vi, vj) must not be less restrictive than the bounds given by path vi → v k → vj:

∀(vi, vj, v k ) ∈ T, u kj ≥ u ki + uij + (c kj -1)M ′ 0 < l kj ≤ l ki + lij + (1 -c kj )M ′ (17)
with M ′ a large constant.

If external contingent link (vi, vj) is substituted by internal contingent link (v k , vj), then the associated requirement link over (vi, v k ) must be justified:

∀(vi, vj, v k ) ∈ T, (vi, v k ) ∈ EX : c kj ≤ b ik (v k , vi) ∈ EX : c kj ≤ b ki (18)
Fig. 9 shows the distributed MaSTN obtained after internalizing both external requirement links and external contingent links. Compared to the previous example, (v0, v B

3 ) had to be constrained further. Moreover, (v0, v B 1 ) and (v A 2 , v A 3 ) are now contingent links.

v 0 v A 1 [4, 5] v A 2 [19, 19] v A 3 [1, 10] v B 1 [5, 9] v B 2

Dynamic Controllability of Local STNUs

Finally, we have to express that the local STNU associated which each agent must be dynamically controllable. This is expresses by adapting the model provided in Section 2.4. An adaptation is required because due to the choice in the internalization of external contingent links, the set of local contingent link is not fixed. This implies for instance that the constraint given in Eq. 8 must be replaced by:

∀v k ∈ CX , ∀vi ∈ V owner (v k ) \ {v k }, ∀vm, vj ∈ V, w ijk -umj ≤ w imk + (1 -c ik )M ′′ (19)
with M ′′ a large constant. Similar transformations must be applied for Eq. 4, 5, and 7.

Distributing local STNUs

If a solution to the global MIP problem exists, this solution describes a distributed MaSTNU. The latter can be partitioned into a set of N local STNUs, one for each agent, while ignoring external constraints.

Local STNUs can be dispatched between agents, and the mission can start.

Discussion

Completeness The techniques defined for checking DC are sound but not complete, essentially because of the internationalization of external contingent links, which can make lose some information on the correlation between time-points. More precisely, by transforming an MaSTNU into a set of local STNUs, we do not represent some correlations between contingent time-points. For instance, consider an MaSTNU involving one contingent time-point x belonging to agent A, two contingent time-points y, z belonging to agent B, and two contingent links yx ∈ [2, 3] and zx ∈ [START_REF] Casanova | Managing dynamic multi-agent simple temporal network[END_REF][START_REF] Cui | Optimising bounds in simple temporal networks with uncertainty under dynamic controllability constraints[END_REF]. In this case, the temporal constraint zy ∈ [-1, 1] necessarily holds (y and x are correlated). With our approach, we cannot represent it by a contingent link between y and z (negative lower bound), and we cannot add a new time-point t in B pointing to both y and z since t would have to be observable by B. We believe that all these points are more STNU related issues (representing non-causal uncertainty).

MIP versus propagation techniques for STNU As mentioned previously, searching for robust execution strategies is a combinatorial task and we cannot directly reuse polynomial DC checking techniques available in the literature [START_REF] Morris | Dynamic Control Of Plans With Temporal Uncertainty[END_REF][START_REF] Morris | Dynamic controllability and dispatchability relationships[END_REF][START_REF] Hunsberger | Efficient execution of dynamically controllable simple temporal networks with uncertainty[END_REF]. Such a combinatorial aspect is also present in [START_REF] Cui | Optimising bounds in simple temporal networks with uncertainty under dynamic controllability constraints[END_REF] for building an STNU which is DC from an initial STNU which is not.

Case without contingent links The techniques defined can also be used in the particular case where there is no contingent link, that is where the MaSTNU is actually an MaSTN. With regards to existing work on MaSTN, one contribution is that the model introduced allows to compute robust executiong strategies which can be executed independently by the agents. We are not aware of previous works on this point for MaSTN. Also, when there is no contingent link, it can be shown that there is actually no boolean decision variable in the model and the MIP becomes a linear program solvable in polytime.

Objective functions Our method can optimize the set of local STNUs by maximizing objective function f opt = i<j (uijlij).

Many other metrics can be used. For example, Fig 10 shows the local STNUs obtained when minimizing the latest execution time of the last time-point, to finish the mission as soon as possible (minimization of fopt = maxi u0i). With this new objective function, the maximum mission completion time is reduced from 29 time units to 24 time units. Objective functions can also be used to balance the flexibility of solutions between agents, in order to avoid overly constrained agents. In this case, we define the normalized flexibility metrics f :

v 0 v 0 v A 1 [0, 1] v A 2 [15, 15] v A 3 [1, 9] v B 1 [1, 5] v B 2 [4, 6] [7, 9] v B 3
∀a ∈ A, f (a) = 1 |V A | v i ,v j ∈V A (uij -lij).
The corresponding objective function to maximize is then fopt = mina∈A f (a). This is particularly useful when the number of agents is high, since in this case the maximum global flexibility can often be reached by constraining as much as possible an unique agent. Another option can be to keep the initial objective function and constrain the problem such that each agent achieves a minimal threshold t of normalized flexibility: ∀a ∈ A, f (a) ≥ t.

Running Times We tested our MIP approach using the CPLEX solver on 500 instances of randomly-generated MaSTNUs, ranging homogeneously from 10 to 40 nodes. All experiments were run on 3.0GHz Intel cores and 4GB memory. Finding the optimal solution to the MIP problem typically takes between 0.2 seconds for the 6nodes example used in this paper, and 1200 seconds for a 4-agents and 40-nodes example containing 4 external contingent links and 10 external requirement links. However, in this last case a first solution, that strictly improves the solution found by removing external contingent links, is found in 80 seconds, at the expense of a drop in the flexibility of 75% compared to the optimal solution.

It must be noted that our current implementation of DC for local STNU (section2.4) is based on the O(N 5 )-time DC-checking techniques from [START_REF] Morris | Dynamic Control Of Plans With Temporal Uncertainty[END_REF], and is the primary cause of the scalability performances. A more efficient algorithm in O(N 4 )-time can be found in [START_REF] Morris | A structural characterization of temporal dynamic controllability[END_REF], however this algorithm is more complex to translate into a MIP formalism and is beyond the scope of this paper.

Impact of Observations on Temporal Flexibility

For the sake of consistency with real life applications vocabulary, in the remainder of this paper we will refer to external contingent links as observations.

We want to measure the improvements made by our method on the "quality" of resulting execution strategies. To this end, we define the Relative temporal flexibility of a solution as being the ratio of the value of the objective function of solution to the value of the objective function of the corresponding STNU:

Relative Temporal Flexibility =

f opt(M aST N U ) f opt(ST N U )
. This metric gives us a strong indicator of the "quality" of a solution compared to best and worst cases. At 100% it means the solution is as much flexible as the corresponding STNU, at 0% it means that the solution found is totally rigid with no tolerance for execution error.

We also assume that external contingent links take values in [0, x], i.e observations of events by other agents are made within x unit of times. We compare x to the temporal flexibility of external requirement links in order to obtain the relative delay of observations:

Relative delay of observations = |E X | |C X | • (v i ,v j )∈C X (u ij )
(v i ,v j )∈E X (u ij -l ij ) . This ratio measure the "quality" of observations (lower ratio means better quality), which may be translated as cost in real applications.

Fig 11 shows the impact of the number of observations (i.e number of external contingent links) and their quality (i.e their immediacy) on the relative flexibility of the solution found by our method. The parameters of the generated MaSTNUs are as follow: 4 agents of 5 nodes each, 4 external requirement links (connecting two randomlychosen nodes from distinct agents). The number of internal requirement (resp. contigent) links for each agent were randomly drawn from 4 to 6 (resp. from 0 to 2). Additional nodes and external contingent links, representing observations, were then added depending on the experiments.

Each point on a curve represents the mean of the relative flexibility of the solution found over 10 MaSTNUs randomly-generated with the corresponding set of parameters. We display the results for two limit cases, the "Full observations" and the "No observations" cases, and an intermediary one, the "Half observations" case.

The "No observation" curve corresponds to MaSTNUs without any external contingent links: C X = ∅. In this case, agents cannot receive any informations during execution, so the flexibility of the solution is minimal: agents must agree on a rigid schedule before The "Full observations" curve corresponds to MaSTNUs wherein each event is observed by each other agent: ∀v ∈ V, ∀a ∈ A, a = owner(v), ∃w ∈ V a s.t. (v, w) ∈ CX . In the extreme case with no delays of observation, the resulting solution has the same quality than if the MaSTNU were considered as a STNU. In the opposite extreme case with high delays of observation, no useful informations can be received from the observations, and the agents must act on their own as in the no observation case.

The "Half observations" curve corresponds to MaSTNUs identical to the "Full observations" set, except half of vertices in V are not observed by any other agent. The quality of the solution actually depends on which vertices are observed: for instance observations of external vertices are more likely to be useful than observations of internal vertices.

Higher numbers of observations lead to higher flexibility of solutions, at the expense of increased computational costs and potentially increased workload during the mission if the observations were not initially scheduled.

Conclusion

Dynamic controllability is an important property for temporal plans with uncertainty as it improves the odds of success of a mission. In this paper we showed how to handle uncertain temporal constraints in multi-agent temporal plans thanks to Multi-agent Simple Temporal Network with Uncertainty, and how to use a MIP model to get executable plans which are dispatched between the agents. There are several future work directions for improving the management of MaSTNU, such as solving the MIP incrementally to repair infeasible MaSTNUs by adding observations one by one while optimizing computation times, or distributing the MIP solving process in order to reoptimize temporal plans during the mission, or taking into account the existence of communication rendez-vous [START_REF] Morris | Managing temporal uncertainty through waypoint controllability[END_REF].

Fig. 1

 1 Fig. 1 gives an example of an MaSTN involving three agents A, B, C. Agent A (resp. B and C) owns variables v A 1 to v A 6 (resp. v B

Figure 2 .

 2 Figure 2. Example of dynamically controllable STNU

Figure 3 .

 3 Figure3. Disjunctive linear model for encoding DC on STNU[START_REF] Cui | Optimising bounds in simple temporal networks with uncertainty under dynamic controllability constraints[END_REF]

Figure 4 .

 4 Figure 4. Example of an MaSTNU

Figure 5 .

 5 Figure 5. Original MaSTNU and its partitioning

  is a boolean decision variable encoding that contingent link between v k and v l (either link(v k , v l ) or link (v l , v k )) is used to justify the satisfaction of vjvi ≤ uij; • ∀(vi, vj) ∈ EX , hij ∈ [0,H] are decision variables encoding the height of the justification of the satisfaction of vjvi ≤ uij, with H a constant equal to max(|A| -2, |CX |); these height variables are used to avoid cycles, that is to avoid cases in which the satisfaction of the upper bound on an external link e is justified by the upper bound associated with an external link e ′ , and in which the upper bound of e ′ is justified by the upper bound of e; see Fig. 7

Figure 7 .

 7 Figure 7. Cycles in justifications without the use of heights variables: the satisfaction of external requirement v C 1v A 2 ≤ 1 can be justified by the satisfaction of external requirement v C 1v B 1 ≤ 1, and reciprocally

Figure 8 .

 8 Figure 8. MaSTNU after internalization of external requirement links

Figure 9 .

 9 Figure 9. MaSTNU after internalization of external requirement and contingent links

Figure 10 .

 10 Figure 10. Mission duration optimization

Figure 11 .

 11 Figure 11. Influence of numbers of observations and delays

Onera -The French Aerospace Lab, F-31055, Toulouse, France, email: name.surname@onera.fr

Ecole Nationale dIngenieurs de Tarbes, Tarbes, France, email: name.surname@enit.fr


Thuan Quang

Tran

Thi Tinh Phung

Quang Trung Nguyen

Truong Giang Le

Corinne Lagrost
email: corinne.lagrost@univ-rennes1.fr

Highly sensitive and rapid determination of sunset yellow in drinks using a low-cost carbon material-based electrochemical sensor

Keywords: Graphene oxide, electrochemical deposition, electrochemical sensor, sunset yellow detection, sensitivity, reusability

Starting from simple graphite flakes, an electrochemical sensor for sunset yellow monitoring is developed by using a very simple and effective strategy. The direct electrochemical reduction of a suspension of exfoliated graphene oxide (GO) onto a glassy carbon electrode (GCE) surface leads to the electrodeposition of electrochemically reduced oxide at the surface, obtaining GCE/ERGO-modified electrodes. They are characterized by cyclic voltammetry measurements (CV) and field emission scanning electron spectroscopy (FE-SEM). The GCE/ERGO electrode has a high electrochemically active surface allowing efficient adsorption of SY. Using differential pulse voltammetry (DPV) technique with only 2 min accumulation, the GCE/ERGO sensor exhibits good performance to SY detection with a good linear calibration for concentration range varying 50-1000 nM (R 2 = 0.996) and limit of detection (LOD) estimated to 19.2 nM (equivalent to 8.9 µg.L -1 ). The developed sensor possesses a very high sensitivity of 9 µA/µM while fabricated with only one component. This electrochemical sensor also displays a good reliability with RSD value of 2.13% (n = 7) and excellent reusability (signal response change < 3.5% after 6 measuring/cleaning cycles). The GCE/ERGO demonstrates a successful practical application for determination of sunset yellow in commercial soft drinks.

Introduction

Synthetic dyes are usually added to foodstuff and soft drinks not only for improving appearance and texture but also for maintaining natural colors during process or storage. Generally, they exhibit higher performances than natural dyes owing to excellent water solubility, charming color uniformity, high stability to light, oxygen and pH, low microbiological contamination and relatively low production costs. They are then largely used in food, but also in cosmetics or drugs, as additives to enhance consumer acceptance. Many of these dyes are based on -N=N-diazo functional group combined with aromatic rings. However, an excessive consumption may cause dangerous pathogenic effects including contact urticaria and anaphylaxis, immunosuppression, asthma, eczema, anxiety migraines [START_REF] Thune | Provocation tests with antiphlogistica and food additives in recurrent urticarial[END_REF][START_REF] Shari | Contact anaphylaxis: a review[END_REF][START_REF] Koutsogeorgopoulou | Immunological aspects of the common food colorants, amaranth and tartrazine[END_REF][START_REF] Yadav | In vitro studies on immunotoxic potential of orange II in splenocytes[END_REF], bladder cancer in men and hepatic carcinoma in mice [START_REF] Srivinivasan | Biodegradation of carcinogetic textile azo dye using bacterial isolates of mangrove sediment[END_REF]. As early as 1920s, they were already related to hyperactivity, hypersensitivities learning problems [START_REF] Arnold | Artificial food colors and attention-deficit/hyperactivity symptoms: conclusions to dye for[END_REF]. Sunset yellow represents one of the most frequently artificial dyes applied into food products. Undoubtedly, the use is controlled by regulations, for instance, an accepted daily intake (ADI) of 4 mg/kg body/day has been established by the European Food Safety Authority (EFSA) [START_REF]Reconsideration of the temporary ADI and refined exposure assessment for Sunset Yellow FCF (E 110)[END_REF]. Some European countries (Finland, Sweden) have even banned its use. In this context, a convenient and effective method is needed for sunset yellow detection in order to control food quality and thus providing safety assurances for consumers. Up-to-date, some methods including UV-VIS [START_REF] Llamas | Second order advantage in the determination of amaranth, sunset yellow FCF and tartrazine by UV-vis and multivariate curve resolution-alternating least squares[END_REF] fluorescence [START_REF] Yuan | Determination of sunset yellow in soft drinks based on fluorescence quenching of carbon dot[END_REF], spectrophotometric [START_REF] Zeynali | Simultaneous spectrophotometric determination of sunset yellow and quinoline yellow in a single step[END_REF] and Raman spectroscopy [START_REF] Zhu | Designing of the functional paper-based surface-enhanced raman spectroscopy substrates for colorants detection[END_REF], high performance liquid chromatography [START_REF] Alves | Determination of synthetic dyes in selected foodstuffs by high performance liquid chromatography with UV-DAD detection[END_REF][START_REF] Sha | Determination of Sunset Yellow and Tartrazine in Food Samples by Combining Ionic Liquid-Based Aqueous Two-Phase System with High Performance Liquid Chromatography[END_REF][START_REF] Branch | Sunset yellow, tartrazine and sodium benzoate in orange juice distributed in Iranian market and subsequent exposure assessment[END_REF][START_REF] Alp | Simultaneous Determination of Sunset Yellow FCF, Allura Red AC, Quinoline Yellow WS, and Tartrazine in Food Samples by RP-HPLC[END_REF], high performance liquid chromatography-mass spectrometry [START_REF] Zou | Determination of seven synthetic dyes in animal feeds and meat by high performance liquid chromatography with diode array and tandem mass detectors[END_REF] have been applied to the determination of sunset yellow in foods. However, these methods generally require costly equipment, specially trained operators, tedious sample preparation and /or may have a poor limit of detection. Recently, electrochemical detection method has attracted much attention as a promising approach for the determination of the electrochemically active substances thanks to numerous advantages including rapid response, simple-touse, excellent sensitivity, time efficiency, portable monitoring and relatively low production cost [START_REF] Niu | Highly Sensitive and Selective Nonenzymatic Detection of Glucose Using Three-Dimensional Porous Nickel Nanostructures[END_REF][START_REF] Moghimi | FePt Alloy Nanoparticles for Biosensing: Enhancement of Vitamin C Sensor Performance and Selectivity by Nanoalloying[END_REF]. Sunset yellow is an electrochemically-active compound and could be practically determined by this way [START_REF] Vladislav | Electroanalytical Methods for Determination of Sunset Yellow -a Review[END_REF]. Some modified A c c e p t e d m a n u s c r i p t glassy carbon electrodes were developed for detection of sunset yellow such as poly-L-cysteine modified GCE [START_REF] Zhang | Highly sensitive determination of Sunset Yellow in drink using a poly (lcysteine) modified glassy carbon electrode[END_REF], polypyrrole/single wall carbon nanotube composite modified GCE [START_REF] Wang | Determination of Sunset yellow in foods based on a facile electrochemical sensor[END_REF], bismuth film modified GCE [START_REF] Królicka | Bismuth film electrodes for adsorptive stripping voltammetric determination of sunset yellow FCF in soft drinks[END_REF], functionalized montmorillonite modified GCE [START_REF] Songyang | Highlysensitive and rapid determination of sunset yellow using functionalized montmorillonite-modified electrode[END_REF], ZnO nano-flower modified GCE [START_REF] Ya | A zinc oxide nanoflower-based electrochemical sensor for trace detection of sunset yellow[END_REF]. Briefly, use of the nanomaterials remarkably enhance sensitivity of electrochemical (bio)sensors owing to their beneficial properties such as high surface-to-area ratio, exceptional physiochemical behaviors, high adsorption and reactive capacity and thus it was extensively applied to foods and environmental analysis [START_REF] Perez-Lopez | Nanomaterials based biosensors for food analysis applications[END_REF][START_REF] Marty | Advantages of Carbon Nanomaterials in Electrochemical Aptasensors for Food Analysis[END_REF][START_REF] Mishra | Food Safety Analysis Using Electrochemical Biosensors[END_REF][START_REF] Maduraiveeran | Nanomaterials based electrochemical sensor and biosensor platforms for A c c e p t e d m a n u s c r i p t environmental applications Trends in Environmental Anal[END_REF].

Graphene which is comprised of an atomically thin 2D carbonaceous materials represents an archetypal supporting materials owing to its large specific surface area, excellent conductivity, high catalytic activity, exceptional mechanical properties. Logically, it has attracted interests for sensing application [START_REF] Shao | Graphene based electrochemical sensors and biosensors: A review[END_REF][START_REF] Rowley-Neale | An overview of recent applications of reduced graphene oxide as a basis of electroanalytical sensing platforms[END_REF]. Two main routes could be introduced for the synthesis of graphene materials-based sensors. Firstly, graphene film-based sensors could be achieved by a simple drop-casting of a dispersion of graphene in solution onto conducting surfaces. Such a dispersed solution was commonly prepared from graphene oxide (GO) using chemical reduction [START_REF] Wang | Application of graphene-modified electrode for selective detection of dopamine[END_REF][START_REF] Chan | Fabrication of graphene/gold-modified screenprinted electrode for detection of carcinoembryonic antigen[END_REF][START_REF] Pumera | Graphene for electrochemical sensing and biosensing[END_REF][START_REF] Wang | Facile synthesis of Au supported on ionic liquid functionalized reduced graphene oxide for simultaneous determination of Sunset yellow and Tartrazine in drinks[END_REF]. However, this preparation method has some intrinsic limitations such as lack of the film thickness control and irreversible agglomerates formation due to strong planar stacking of the 2D graphene nanosheets. Thus, an alternative method for preparation of more structured graphene films is preferable. Electrochemical reduction from a graphene oxide (GO) dispersed in solution represents a highly potential method to this purpose. This approach could be achieved via two steps, namely: (i) assembling highly dispersed GO onto electrode surface by solution deposition and ii) carrying an electrochemical reduction of deposited GO layers into electrochemically reduced graphene oxide (ERGO) [START_REF] Chan | Fabrication of graphene/gold-modified screenprinted electrode for detection of carcinoembryonic antigen[END_REF][START_REF] Zhang | Electrochemical deposition of nickel nanoparticles on reduced graphene oxide film for nonenzymatic glucose sensing[END_REF][START_REF] Zhou | Controlled synthesis of large-area and patterned electrochemically reduced graphene oxide films[END_REF][START_REF] Shao | Facile and controllable electrochemical reduction of graphene oxide and its applications[END_REF][START_REF] Guo | A Green Approach to the Synthesis of graphene nanosheet[END_REF][START_REF] Báez | Reduced Graphene Oxides: Influence of the Reduction Method on the Electrocatalytic Effect towards Nucleic Acid Oxidation[END_REF][START_REF] Wang | A facile electrochemical sensor based on reduced graphene oxide and Au nanoplates modified glassy carbon electrode for simultaneous detection of ascorbic acid, dopamine and uric acid[END_REF]. Recently, a much more straightforward process was proposed using a direct electrochemical reduction of GO solutions into ERGO which is (electro-)deposited onto the electrode surfaces [START_REF] Chen | Direct electrodeposition of reduced graphene oxide on glassy carbon electrode and its electrochemical application[END_REF][START_REF] Asadian | Glassy carbon electrode modified with 3D graphene-carbon nanotube network for sensitive electrochemical determination of methotrexate[END_REF][START_REF] Zhang | An Easily Fabricated Electrochemical Sensor Based on a Graphene-Modified Glassy Carbon Electrode for Determination of Octopamine and Tyramine[END_REF][START_REF] Xue | Direct electrodeposition of well dispersed electrochemical reduction graphene oxide assembled with nickel oxide nanocomposite and its improved electrocatalytic activity toward 2, 4, 6-Trinitrophenol[END_REF]. The successfully prepared ERGO electrochemical sensors could be applied for determination of various substances such as phenolic compounds [START_REF] Chen | Direct electrodeposition of reduced graphene oxide on glassy carbon electrode and its electrochemical application[END_REF][START_REF] Xue | Direct electrodeposition of well dispersed electrochemical reduction graphene oxide assembled with nickel oxide nanocomposite and its improved electrocatalytic activity toward 2, 4, 6-Trinitrophenol[END_REF], biogenic amines [START_REF] Zhang | An Easily Fabricated Electrochemical Sensor Based on a Graphene-Modified Glassy Carbon Electrode for Determination of Octopamine and Tyramine[END_REF], dopamine, ascorbic and uric acid [START_REF] Yang | Direct electrodeposition of reduced graphene oxide on carbon fiber electrode for simultaneous determination of ascorbic acid, dopamine and uric acid[END_REF][START_REF] Lee | One-Step Electrochemical Fabrication of Reduced Graphene Oxide/Gold Nanoparticles Nanocomposite-Modified Electrode for Simultaneous Detection of Dopamine, Ascorbic Acid, and Uric Acid[END_REF], glucose [START_REF] Yang | Nickel nanoparticle-chitosan-reduced graphene oxidemodified screen-printed electrodes for enzyme-free glucose sensing in portable microfluidic devices[END_REF]. However, relatively few graphenebased electrochemical sensors have been developed for electrochemical determination of sunset yellow. Some graphene materials based electrochemical sensors for this purpose could be quoted such as graphene-Ni modified GCE [START_REF] Gan | Graphene decorated with nickel nanoparticles as a sensitive substrate for simultaneous determination of sunset yellow and tartrazine in food samples[END_REF], graphene-PTA film modified GCE [START_REF] Gan | One-step electrochemical approach for the preparation of graphene wrapped-phosphotungstic acid hybrid and its application for simultaneous determination of sunset yellow and tartrazine[END_REF], graphene/AuNPs modified GCE [START_REF] Wang | Highly sensitive electrochemical determination of Sunset Yellow based on gold nanoparticles/graphene electrode[END_REF]. Most of these sensors require multi-steps and sophisticated preparation, and very often incorporate composite materials or metallic nanomaterials.

Herein, we report the preparation of ERGO modified glassy carbon surface as an electrochemical sensor for monitoring sunset yellow (SY). The preparation strategy is based on a simple and costly-effective process, starting directly from low-cost graphite flakes. The GCE/ERGO electrodes are obtained from the simple electrochemical deposition route, hence demonstrating in full a very robust procedure. Optimization of the preparation of GCE/ERGO electrode along with optimization of the analytical experimental parameters for SY titration are thoroughly performed. The electrochemical behaviour of SY at the GCE/ERGO platform is studied in depth. Finally, the GCE/ERGO is applied to real sample analysis on commercial soft drinks. The obtaining GCE/ERGO show great potential as accurate, low cost, rapid and highly sensitive portable sensor, only based on single component graphene.

Material and methods

Chemicals and instrumentation

Chemicals of analytical grade were used without any further purification. Graphite flakes (99%, 325 mesh particle size), lithium perchlorate (LiClO4 99%), potassium hexacyanoferrate (II) trihydate (K4Fe(CN)6.3H2O 98%) and hexacyanoferrate (III) (K3Fe(CN)6 99 %) were obtained from Sigma Aldrich. Potassium permanganate (KMnO4), sulfuric acid (H2SO4 98%), phosphoric acid (H3PO4 85%), boric acid (H3BO3, 99%) hydrochloric acid (HCl 37%), sodium hydrophosphate hydrate (Na2HPO4.7H2O 99%), sodium dihydrophosphate (NaH2PO4 99%), sodium acetate (CH3COONa 99%), acetic acid (CH3COOH 100%), and solvents (acetone CH3COCH3 and diethyl ether Et2O) were purchased from Merck. The sunset yellow (90.5%) standard compound was obtained from LGC standard, Dr. Ehrenstorfer GmbH, German. Ultrapure water (18.2Ω) was obtained from Milli-Q Integral 3 purification system. Soft drinks (Samurai, Sting, Number 1 and Mirinda) were purchased from a local Hanoi supermarket.

Electrochemical experiments (cyclic voltammetry (CV) and differential pulse voltammetry (DPV) were carried out using a CIMPS electrochemical workstation (Zahnner) with a conventional three-electrode system. The working electrode is either a glassy carbon electrode (GCE of 3 mm diameter), or a GCE modified with ERGO for sensing experiments, or an ITO plate (2 x 1 cm 2 ) for SEM analyses. A Pt wire and a saturated calomel reference electrode serve as counter and reference electrodes, respectively.

Morphology of ERGO layer deposited onto ITO surface was characterized using Field Emission scanning electron A c c e p t e d m a n u s c r i p t microscopy (FE-SEM, Hitachi S4800). Sunset yellow detection by chromatography method was performed on µHPLC instrumentation (Ultimate 3000, Thermo Scientific) equipped with an UV detector.

Graphene oxide (GO) preparation

Graphene oxide (GO) was synthesized from natural graphite flakes following an adapted Hummer method [START_REF] Marcano | Improved Synthesis of Graphene Oxide[END_REF][START_REF] Yu | High-efficient Synthesis of Graphene Oxide Based on Improved Hummers Method[END_REF]. Briefly, graphite flakes (1.0 g) and boric acid (0.010 g) were first dispersed into 60 mL of concentrated H2SO4/H3PO4 mixture (9/1 = v/v) maintained in ice-cooling bath with a magnetically stirring. KMnO4 (3.0 g, 3 wt. equivalent) was slowly added in small portions in the mixture maintained at less than 5ᵒC, followed by stirring for 1.5 h. Another amount of the KMnO4 (3.0 g, 3 wt equivalent) was slowly added at less than 5ᵒC. The mixture was then heated up to 35ᵒC while stirring for 3 hours. Deionized water (60 mL) was gradually added to the mixture reaction which was heated up to 98ᵒC and maintained at this temperature for 15 min. After cooling at room temperature, H2O2 aqueous solution (3 mL, 30%) was slowly added to quench the reaction. A color change from dark brown to bright yellow occurred. Adding deionized water (200 mL) afforded a yellow suspension which was decanted through a centrifugation process at 10000 rpm for 6 min. A yellow paste was obtained and washed with aqueous 1 M HCl (100 mL) through repeated centrifugations at 7000 rpm for 6 minutes (3 times). A brown solid was obtained, and dried at 60ᵒC for 7 hours. Finally, it was re-dispersed in acetone (20 mL) and precipitated with diethyl ether (50 mL). The obtained precipitate was filtered using cellulose acetate membrane (0.45 µm) under vacuum and thoroughly rinsed with diethyl ether. After drying at 60ᵒC for 7h, the GO was obtained as a brown solid (1.84 g).

Electrochemical preparation of the ERGO modified electrodes (GCE/ERGO)

Before modification, glassy carbon disk electrode was thoroughly polished using DP-Nap polishing paper 1 µm (Struers) and Al2O3 slurry 0.3 µm (Struers) to mirror-like surface and then ultra-sonicated for 5 min in acetone and water, respectively. The polished electrodes were further activated in aqueous 0.5 M H2SO4 by cycling 5 successive potential scans between 0 and 1.8 V at the scan rate v = 50 mV.s -1 . The GO solid powder was firstly dispersed in deionized water (8.0 mL, 1 mg.mL-1) by ultra-sonication for 1 h. This suspension was subsequently centrifuged at 500 rpm for 20 min and decanted for removing unreacted graphite flakes (see Electronic Supplementary Material Fig. S1). The LiClO4 salt was then added to the supernatant, leading to a solution of GO containing 0.2 M LiClO4 which was sonicated for 15 minutes before electrochemical deposition. The GCE was then immersed in the GO solution under magnetic stirring and Ar bubbling. Successive potential cycles between 0.6 to -1.5 V/SCE were applied for electrodeposition of electrochemically reduced graphene oxide (ERGO).

Sample preparation and Detection

Sunset yellow (SY) standard solution (10 -4 M) were daily prepared from stock solution in deionized water. The electrolyte solutions exhibiting different pH values were prepared from 0.1 M acetate buffer for pH = 3, 4 and 5 and 0.1 M phosphate buffer for pH = 6, 7 and 8. Before use, the soft drink samples were filtered on KX Nylon membrane syringe (13 mm, 0.22 µm, Kinesis), being simply diluted in the buffer or deionized water for electrochemical or chromatography analyses, respectively. The DPV signals were recorded from 0 to 0.9 V/SCE after 2 min of accumulation at 0 V/SCE. Optimal DPV experimental conditions were determined to be 10.5 mV step potential, 20 mV pulse amplitude, 120 ms pulse width and scan rate 35 mV.s -1 . Besides, soft drink samples were also determined by HPLC method using RP-C18 column (endcapped 5 µm, Purospher@ STAR, Hibar @RT 250-4.6 mm) for comparison (see ESI).

Results and discussion

Preparation and characterization of the GCE/ERGO sensing electrodes

The GCE/ERGO electrode was prepared through the electrochemical reduction of GO by employing cyclic voltammetry (CV), following previously described procedure [START_REF] Chen | Direct electrodeposition of reduced graphene oxide on glassy carbon electrode and its electrochemical application[END_REF][START_REF] Asadian | Glassy carbon electrode modified with 3D graphene-carbon nanotube network for sensitive electrochemical determination of methotrexate[END_REF][START_REF] Zhang | An Easily Fabricated Electrochemical Sensor Based on a Graphene-Modified Glassy Carbon Electrode for Determination of Octopamine and Tyramine[END_REF][START_REF] Xue | Direct electrodeposition of well dispersed electrochemical reduction graphene oxide assembled with nickel oxide nanocomposite and its improved electrocatalytic activity toward 2, 4, 6-Trinitrophenol[END_REF]. The corresponding voltammograms display the characteristic signals for the electrodeposition of electrochemically reduced graphene oxide (ERGO) at the GCE surface (see Electronic Supplementary Material Fig. S2) [START_REF] Chen | Direct electrodeposition of reduced graphene oxide on glassy carbon electrode and its electrochemical application[END_REF]. This very simple deposition method can be applied to any conducting surface. The electrodeposition of the ERGO film on surface by CV could be directly confirmed by FE-SEM image as shown in Fig. 1. The surface morphology presents the typical crumpled lamellar structure of graphene nanosheets. Practically, the number of the potential cycles is an important parameter in view of current sensitivity as it directly correlates with the thickness of graphene film. A true advantage of the electrodeposition technique is the possibility to easily control the thickness of graphene film thanks to the number of potential cycles. In the following, the effect of CVs cycles, namely 5, 10 and 15 scans, was investigated with respect to the electrochemical properties of GCE/ERGO modified electrodes as well as to their sensitivity towards sunset yellow detection.

A c c e p t e d m a n u s c r i p t

The electrochemically active area of the modified GCE/ERGO electrodes were determined by recording CV of equimolar concentration of K3Fe(CN)6/K4Fe(CN)6 in aqueous KCl compared to bare GCE. The voltamogramms display a well-defined and reversible redox system at both GCE/ERGO and GCE electrodes (see Electronic Supplementary Material Fig. S3). Interestingly, the peak-topeak separation (ΔEp = Epc-Epa, where Epc and Epa are the cathodic and anodic peak potentials, respectively) and the peak current intensities (Ip) were monotonically affected by the surface modification, depending on the number of scans (see Electronic Supplementary Material Fig. S4). A significant increase of the current intensities along with a decrease of ΔEp is observed at GCE/ERGO compared to GCE, and the amplitude of these effects correlates with the number of scans performed during the electrodeposition process. Such an observation falls in line with the modification of GCE surface during reduction in GO solution. The peak-to-peak separation at 50 mV.s -1 is found equal to 110 mV at GCE electrode, 94 mV, 80 and 72 mV for GCE/ERGO electrodes prepared with 5, 10, and 15 scans respectively (see Electronic Supplementary Material Table S1), indicating a strong improvement of the electron transfer kinetics at the GCE/ERGO modified surfaces, and especially at the ones obtained with more than 10 scans.

The variation of the intensity of the anodic peak current (Ipa) is linear as a function of the square root of the scan rate (v 1/2 ) (Electronic Supplementary Material Fig. S4). From the slope of the line, the electrochemically active surface area (A) could be calculated from Randles-Sevcik equation (Eq. 1):

Ipa = 2.69 × 10 5 .n 3/2 A.D 1/2 .v 1/2 .C (Eq. 1)
where Ipa is the peak current intensity (A); n is the number of electrons exchanged, here n=1, C is the concentration (mol.cm -3 ) of the redox probe in bulk solution, D (7.6 x 10 -6 cm 2 .s -1 ) is the diffusion coefficient, v the scan rate v (V.s -1 ); and A the surface area A (cm 2 ). Bare GCE electrode has an electrochemically active surface area or ESCA value of 0.069 cm 2 , in good agreement with the geometric value of 0.071 cm 2 . As expected, GCE/ERGO surfaces display higher ECSA values, equal to 0.083, 0.097 and 0.115 cm 2 corresponding to electrodes prepared with 5; 10 and 15 CV scans, respectively (see Electronic Supplementary Material Table S1). Thus, GCE/ERGO modified surfaces allow increasing the ECSA value up to 1.67 folds using 15 scans modification compared to GCE.

Electrochemical response of sunset yellow (SY) at different GCE/ERGO electrodes were examined using CV and DPV techniques (Fig. 2). The CVs at the GCE/ERGO electrodes show a well-defined and reversible signal at 0.70 V/ECS with a peak-to-peak separation ΔEp = 15-17 mV while a quasireversible signal is usually obtained at bare GCE (Fig. 2A). In addition, the current intensity at bare GCE is much lower (Ipa = 0.4 µA) than those obtained at GCE/ERGO (Ipa = 5.6 µA), indicating an enhanced sensitivity for SY detection. Note that the capacitive currents were gradually increased following the number of CV scans, leading to an almost similar value of the faradic peak current intensity for all the ERGO-modified electrodes. Since DPV measurements are not sensitive to capacitive currents, the DPV signal intensity of SY increases following the number of CV scans in the electrodeposition process of ERGO, 5 CVs < 10 CVs < 15 CVs (Fig. 2B) in agreement with the increase of ECSA as shown above.

Based on these results, it could be concluded that using 15 CVs for electrodeposition process has led to GCE/ERGO

Electrochemical properties of the SY onto GCE/ERGO surface

Effect of the pH value

Effect of the pH value on the electrochemical answer of SY was firstly investigated by varying the pH from 3 to 8 using both acetate and phosphate buffer solutions. As shown in Fig. 3., the intensity of the peak currents (Ipa and Ipc) gradually increases from pH 3 to 5 and decreases from 5 to 8 (Fig. 3A). Therefore, a maximum Ipa (and Ipc) value is obtained with pH 5 acetate buffer solution, which is selected for further investigation. In addition, both anodic and cathodic peak potentials are negatively shifted with increase of pH, suggesting that the oxidation mechanism of SY involves protons exchange along with electron transfers. A linear relationship is found between the anodic peak potential Epa and pH with a slope equal to -0.029 V per pH unit (Fig. 3B). Similar behaviors have been reported in the literature data [START_REF] Zhang | Highly sensitive determination of Sunset Yellow in drink using a poly (lcysteine) modified glassy carbon electrode[END_REF][START_REF] Wang | Determination of Sunset yellow in foods based on a facile electrochemical sensor[END_REF][START_REF] Wang | Facile synthesis of Au supported on ionic liquid functionalized reduced graphene oxide for simultaneous determination of Sunset yellow and Tartrazine in drinks[END_REF][START_REF] Gan | Graphene decorated with nickel nanoparticles as a sensitive substrate for simultaneous determination of sunset yellow and tartrazine in food samples[END_REF][START_REF] Wang | Highly sensitive electrochemical determination of Sunset Yellow based on gold nanoparticles/graphene electrode[END_REF][START_REF] Wang | Highly sensitive electrochemical determination of Sunset Yellow based on gold nanoparticles/graphene electrode[END_REF][START_REF] Majidi | Carbon Nanotube -Ionic Liquid (CNT-IL ) Nanocamposite Modified Sol-Gel Derived Carbon-Ceramic Electrode for Simultaneous Determination of Sunset Yellow and Tartrazine in Food Samples, Food Anal[END_REF][START_REF] Wang | Highly sensitive electrochemical determination of sunset yellow based on the ultrafine Au-Pd and reduced graphene oxide nanocomposites[END_REF][START_REF] Li | Construction of novel electrochemical sensors based on bimetallic nanoparticle functionalized graphene for determination of sunset yellow in soft drink[END_REF][START_REF] Deng | Simultaneous detection of sunset yellow and tartrazine using the nanohybrid of gold nanorods decorated graphene oxide[END_REF][START_REF] Pogacean | Graphene-based materials produced by graphite electrochemical exfoliation in acidic solutions : application to sunset yellow voltammetric detection[END_REF]. Such a value is half of the theoretical Nernstian value of 0.059 mV.pH -1 and indicates that the ratio of protons to electrons involved in the oxidation pathway is equal to 0.5.

Effect of the scan rate

To further investigate the electrochemical oxidation process of SY at GCE/ERGO, a series of CVs was recorded in a pH 5 acetate buffer solution with different scan rates (v) in the range 0.02-1.0 V.s -1 (Fig. 4Error! Reference source not found.A). The intensities of the anodic and the cathodic peak current are linearly proportional to the scan rates in the studied range, showing that the oxidation of SY is an adsorption-controlled process (Fig. 4B). Upon increasing the scan rate, the anodic and cathodic peak potentials start to broaden and shift to positive and negative directions with respect to formal apparent potential E°'. This behavior is connected to the kinetics of the electrochemical process. Indeed, the application of a large overpotential is required to keep a sufficient flux of electrons to be transferred to the electrode since the timescale of the CVs experiments (RT/Fv) decreases to a point to become comparable to the electron transfer kinetics. By using the Laviron's formalism, which is based on the classical Butler-Volmer theory, 58 the peak potential values were plotted against the logarithm of the scan rate (lnv), leading to two linear regions for each cathodic and anodic potentials, respectively (Fig. 4C). The intercept of these lines (vi) allows a determination of the apparent electron transfer rate kinetic rate constant (kET) following Eq. 2.

kET = (1-α)nFvi /RT (Eq. 2) where n is the number of exchanged electrons, F is Faraday' constant, R the universal gas constant, T is temperature (293K) and vi is the intercept of the linear regions in the Ep-lnv plots.

The slopes for the linear region at higher scan rates are equal to RT/(1-α)nF and -RT/αnF for the anodic and cathodic processes, respectively. From these slopes values, we could estimate that α and n are equal to 0.5 and 2, respectively. It comes a value of 6.3 s -1 for kET.

The oxidation mechanism of SY at GCE/ERGO involves one proton and two electrons. This result falls in line with a recent paper that thoroughly investigates the electrochemical oxidation pathway of SY by combining theoretical calculations and spectroelectrochemical analyses [START_REF] Sierra-Rosales | Experimental and theoretical insights into the electrooxidation pathway of azo-colorants on glassy carbon electrode[END_REF].

Optimisation of DPV parameters for SY detection

DPV technique is proposed for SY determination due to its high sensitivity and ease of operation. The DPV parameters including Es (step potential) ts (pulse width), Ep (pulse amplitude), tp (sample period) and Idt (integration time) were firstly investigated using 0.5 µM SY in a pH 5 acetate buffer Since the electrochemical process involves the adsorption of SY onto GCE/ERGO surfaces, the effect of accumulation time along with the effect of the potential value employed for the accumulation step need to be investigated. The DPV measurements show that SY response was significantly increased following the accumulation times from 0 to 3 min (Fig. 5A and see Electronic Supplementary Material Fig. S8). The SY signal was increased by 140; 200 and 265 % corresponding to one; two and three min of accumulation. This result suggests that GCE/ERGO surfaces exhibited a high adsorption capacity. Finally, we consider that 2 min of accumulation time is an acceptable duration, leading to a good response and thus was used for further measurements. Considering the effect of potential values, SY response was found to be almost independent to accumulation potential as shown in Fig. 5B. Therefore, the potential at 0 V was used as an accumulation potential for SY determination.

Calibration curve and detection limit for SY detection

DPV measurements using the GCE/ERGO electrodes were performed at different SY concentrations from 50 to 1000 nM in acetate buffer (pH = 5) (Fig. 6A). It was observed that SY response regularly increases as a function of SY concentration, following a linear equation Ip(µA) = 0.0083*C(nM) -0.40 (R 2 = 0.994) (Fig. 6B). The limit of detection (LOD) was estimated to be 19.2 nM from linear regression in a limited range from 50 to 350 nM following Eq. 3 [START_REF] Shrivastava | Methods for the determination of limit of detection and limit of quantitation of the analytical methods[END_REF].

LOD = 3Sy/b (Eq. 3)
where Sy is the standard deviation of the response and b is the slope of the calibration curve for a limited range from 50 to 100 nM. The comparison of GCE/ERGO with some other type of materials modified GCE electrodes for SY determination are typically summarized in Table 1. It could be observed that the GCE/ERGO electrochemical sensor developed in this work exhibits good performance for SY detection with an acceptable detection limit when compared to other previously described systems. For practical applications, low detection limit might not be a critical point for real sample analysis since the typical amount of SY in food sample, notably in beverages is rather high. However, considering that human health and food safety regulations tend to be more stringent, the detection of SY in a more sensitive manner could become of importance. [START_REF] Pogacean | Graphene-based materials produced by graphite electrochemical exfoliation in acidic solutions : application to sunset yellow voltammetric detection[END_REF][START_REF] Jampasa | Electrochemically reduced graphene oxide-modified screenprinted carbon electrodes for a simple and highly sensitive electrochemical detection of synthetic colorants in beverages[END_REF]. Interestingly, the performance our GCE/ERGO compares well with these two examples but exhibits a higher sensitivity equal to 9 µA/µM compared to 0.79 µA/µM, [START_REF] Jampasa | Electrochemically reduced graphene oxide-modified screenprinted carbon electrodes for a simple and highly sensitive electrochemical detection of synthetic colorants in beverages[END_REF] and 0.017 µA/µM, [START_REF] Pogacean | Graphene-based materials produced by graphite electrochemical exfoliation in acidic solutions : application to sunset yellow voltammetric detection[END_REF] respectively. Actually, the sensitivity toward SY of the sensor developed in this work is of the highest reported ones [START_REF] Wang | Determination of Sunset yellow in foods based on a facile electrochemical sensor[END_REF][START_REF] Gan | Graphene decorated with nickel nanoparticles as a sensitive substrate for simultaneous determination of sunset yellow and tartrazine in food samples[END_REF][START_REF] Gan | Electrochemical sensor based on graphene and mesoporous TiO2 for the simultaneous determination of trace colorants in food[END_REF]. The LOD shows very good result compared to those estimated in references, [START_REF] Pogacean | Graphene-based materials produced by graphite electrochemical exfoliation in acidic solutions : application to sunset yellow voltammetric detection[END_REF][START_REF] Jampasa | Electrochemically reduced graphene oxide-modified screenprinted carbon electrodes for a simple and highly sensitive electrochemical detection of synthetic colorants in beverages[END_REF] especially since LOD at ERGO/SPE might be probably overestimated regarding reported detection DPV curves. [START_REF] Jampasa | Electrochemically reduced graphene oxide-modified screenprinted carbon electrodes for a simple and highly sensitive electrochemical detection of synthetic colorants in beverages[END_REF]

Repeatability, storage stability and reusability

The reliability of this sensor is evaluated by repetitive experiments (n = 7) in pH = 5 acetate buffer solution containing 0.5 µM SY (see Electronic Supplementary Material Fig. S9). The RSD value is calculated to be 2.13 %, demonstrating the high reliability of the assay. The electrochemical sensor was then stored in pH 5 buffer at room temperature conditions for stability measurements. The DPV measurements showed no change of the signal after 24 h and only a slight decrease of 15 % after 48 hours (see Electronic Supplementary Material Fig. S10). The electrochemical sensor could be efficiently regenerated through a simple potential cycling (5 cycles) in 0.5 M aqueous H2SO4 solution. The response of the electrochemical sensor exhibited no significant change (below of 3.5 %) after 5 successive measurement/cleaning cycles (see Electronic Supplementary Material Fig. S11), showing its high reusability.

Interference

The anti-interference ability of the GCE/ERGO sensors was also evaluated taking into account of the potential presence of ingredients or food additives such as sweeteners or preservatives. The DPV responses of a standard solution of SY (0.5 µM in pH 5 acetate buffer) were studied after addition 100-fold concentration of KCl, NaCl, CaCl2, glucose, sodium citrate and ascorbic acid (AA). There is no obvious influence of these interfering species since the DPV signal variations were below 5% of the initial SY response without interferences (Fig. 7).

SY

Real Sample analysis

For evaluating its potential application, the ERGO/GCE sensor was applied to the detection of SY in real samples of soft drinks collected from a local market, namely Sting, Samurai, Number 1 and Mirinda. We carried out DPV measurements under optimized experimental conditions. A very small volume (less than 15 % of total volume of solution) of sample (60 µL of Samurai, 16 µL of Mirinda, 8 µL of Number 1 and 40 µL of Sting) was added directly to the pH 5 acetate buffer solution (5 mL) (Fig. 8). To evaluate the accuracy of the sensor, known amounts of SY were spiked into samples, leading to an increase of the intensity of the DPV response. These observations clearly demonstrate that such sensor could enable identification of the presence of SY in the drink. Each sample undergoes 7 detection rounds and results are gathered in Table 2. The RSD is acceptable, indicating that the precision of the sensor is good. The recoveries were then tested, and values in the range 93.35 to 107.7% indicate that the method possesses good accuracy.

HPLC method was also used to determine the concentration of SY in the soft drink samples and the results are given in Table 2. From the comparison of HPLC method and the ERGO/GCE electrochemical detection, it can be seen that the two methods lead to very similar results, with a small relative variation. Thus the fabricated ERGO/GCE sensor has a very good potential for monitoring SY in real samples with a low cost, simple preparation and good accuracy.

Conclusion

Accurate, rapid and highly sensitive titration of Sunset Yellow could be performed at a GCE electrode modified with graphene materials. This work proposes a simple and effective route for preparing GCE/ERGO sensors directly from graphite flakes, allowing the development of low-cost electrochemical devices. Electrochemical reduction of exfoliated graphite enables reduction of GO in to reduced GO and deposition onto GCE surface. The result also indicates that thickness and electron transfer properties of the ERGO layer could be easily controlled by means of the CV scan numbers. The modified GCE/ERGO preparing from 15 CVs scans exhibited higher surface area and larger sensitivity toward SY detection than those prepared with 5 or 10 scans. The resulting sensor demonstrated a high performance for SY determination with a low detection limit of 19.2 nM and a high sensitivity of 9 µA/µM, high reliability and a successful References practical applicability for SY analysis in real samples. Furthermore, this sensor could be easily cleaned and reused even after measurement in real samples. The ERGO/GCE sensing platform exhibits a high potential for developing a portable, reliable, rapid and low-cost on-site sensors for monitoring food colorants in foods or in environmental effluents.

A c c e p t e d m a n u s c r i p t

Fig. 1 .

 1 Fig. 1. FE-SEM images of (A) bare pristine ITO and (B) ERGO film electrodeposited onto the ITO surface after 10 potential scanning cycles.

AFig. 2 .

 2 Fig. 2. Electrochemical response of 10 µM SY at GCE/ERGO in pH = 5 acetate buffer (A) CVs at v = 50 mV.s -1 (B) DPV (step potential Es = 5 mV, pulse potential Ep = 40 mV, pulse width tp = 200 ms, scan rate 10 mV s -1 , integration time Idt = 90 ms; accumulation times of 3 minutes at 0 V). GCE/ERGO prepared with 5 (red line), 10 (green line) and 15 (blue line) CV scans. In (A), signal obtained at bare GCE (black line) for comparison purposes.

Fig. 3 .

 3 Fig. 3. (A) CVs at v = 50 mV.s -1 of 10 µM SY at a GCE/ERGO electrode in acetate or phosphate buffer

A c c e

  p t e d m a n u s c r i p t solution while an accumulation time of 2 min at a potential of 0 V/SCE was maintained. The results demonstrated that the maximum DPV response could be obtained with optimized experimental parameters including scan rate v = 35 mV.s -1 ; Es = 10.5 mV; ts = 300 ms; Ep = 20 mV, tp = 120 ms; time of integration Idt = 60 (ms) (see Electronic Supplementary Material Fig.S5, S6 and S7).

Fig. 4 .

 4 Fig. 4. (A) CVs of 10 µM SY in a pH 5 acetate buffer solution at the GCE/ERGO surface at scan rates varying from 20 to 1000 mV s -1 ; (B) Plot of the intensity of the anodic (•) and cathodic (◼) peak currents Ip against scan rate v (V.s -1 ); (C) Plot of the anodic (◼) and cathodic (•) peak potential Ep (V) against lnv (v, V.s -1 ).

Fig. 5 .

 5 Fig. 5. (A) Relative intensity of DPV signal with an accumulation potential at 0 V for different accumulation times. Intensity of the signal with no accumulation time is taken as a reference. (B) Relative intensity of DPV signal using 2 minutes of accumulation for different potential values. Intensity of the signal at 0 V is taken as a reference. Error bars represent the standard deviation for triple measurements.

Fig. 6 .

 6 Fig. 6. (A) DPV of SY at different concentrations from 50 -1000 nM, namely 50, 100, 200, 350, 500, 700, 1000 nM. (B) Plot of the current intensity in the DPV response towards SY concentration. Optimized DPV conditions: scan rate v = 35 mV.s-1, Es = 10.5 mV, ts = 300 ms, tp = 120 ms, Idt = 60 ms and Ep = 20 mV, accumulation time of 2 minutes at 0 V/SCE.

Fig. 7 .

 7 Fig. 7. Response of the sensor towards SY detection (0.5 µM) in pH 5 buffer with the addition of 100-folds concentration of interferences. Error bar represents the SD of triple measurements.

Fig. 8 .

 8 Fig. 8. DPV at GCE/ERGO electrochemical sensor for analysis of Samurai sample: DPV conditions: Es = 10.5 mV; ts = 300 ms; Ep = 20 mV; tp = 120 ms; Idt = 60 ms; accumulation time of 2 minutes at 0 V/SCE. Before (black line) and after addition of 60 µL of Sting (red line) followed by addition of 150 nM of SY (green line), of 350 nM of SY (blue line).

Table 1 .

 1 Performance of different modified electrodes for SY detection

	Electrode	Linear range (nM)	Detection limit (nM)	Reference
	Au/rGO/GCE	2-110000	2	50
	PLPA/GCE	400-14010	40	61
	Poly(L-	8-700	4	20
	Cysteine)/GCE
	Au-	686-	1.5	54
	Pd/rGO/GCE	331000
	MWCNT/GCE	55-11000	22	62
	CTAB-	80-10000	4.2	63
	Gr/Pt/GCE
	CTAB-MMT-	2.5-200	0.7	23
	Ca/GCE
	GN-Ni/GCE	7.37-442	2.16	48
	Chitosan-	200-	66	64
	graphene/GCE	100000
	Gr/GCE	1000-	1800	57
		100000
	ERGO/SPE	10-20000	0.5	65
	ERGO/GCE	50-1000	19.2	This
				work
	GCE: glassy carbon electrode; rGO: reduced graphene
	oxide; PLPA: poly-L-phenylalanine; MWCNT: multi-walled
	carbon nanotube; CTAB-Gr: hexadecyltrimethyl bromide
	functionalized	graphene;	CTAB-MMT-Ca:
	hexadecyltrimethyl bromide functionalized montmorillonite;
	ERGO: electrochemically reduced graphene oxide; GN:
	graphene; SPE:screen-printed carbon electrode
	An important point is the fact that our GCE/ERGO sensor
	which is prepared through a very simple strategy incorporates
	only one component, in sharp contrast with many other
	systems. In this connection, we found two other interesting
	sensing examples for SY detection only based on GCE
	modified with ERGO

Table 2 .

 2 Determination of SY in soft drinks samples by electrochemical detection at ERGO/GCE and by HPLC

	Sample	Added	Expected	Found	RSD	Recovery	By HPLC	Relative error
		(nM)	(nM)	(nM)	(%)	(%)	(nM)	(%)
	Samurai	-	-	292.65	5.63	-	307.56	-4.85
		150	457.56	439.85	0.88	96.13
		200	657.56	613.85	2.05	93.35
	Mirinda	-	-	353.77	4.85	-	406.36	-12.94
		150	556.36	538.34	6.47	96.76
		200	756.36	814.64	4.54	107.70
	Sting	-	-	264.99	4.37	-	292.06	-9.27
	Number 1	-	-	161.85	5.24		171.5	-5.63

Acknowledgement:

The authors would like to thank Vietnam Academy of Science and Technology (VAST.CTG.08/16-18) the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 104.06-2016.42 for funding support. We thank also Mr. Van Nhan LE for performing HPLC measurements and fruitful discussions on sunset yellow determination.

Conflict of interest:

The authors declare that they have no conflict of interest.