G Khodja

˚c Fiter

˚l Hetel

˚t Floquet

Local practical stabilization for a class of discrete-time switched affine systems

Keywords: Switched affine systems, switching control, local practical stabilization

This paper deals with the study of discrete-time switched affine systems. While generally the literature is focused on the global (practical) stabilization problem, there are classes of switched affine systems that can be stabilized only locally. In this paper we deal with such a class of switched affine systems. More precisely, we consider switched affine systems with decoupled switching in the state matrix and the affine input term. For this particular class of systems, a switching control law is designed and conditions ensuring the system local practical stability are provided. A qualitative approach is given based on the existence of a stabilizing linear state feedback. These results are illustrated using a numerical example.

INTRODUCTION

Switched systems [START_REF] Liberzon | Switching in systems and control[END_REF]) represent a class of hybrid dynamical systems [START_REF] Goebel | Hybrid dynamical systems[END_REF]). They consist on a family of subsystems and a switching law that orchestrates the switching among them. A particular important class of these systems is the class of switched affine systems [START_REF] Seatzu | Optimal control of continuous-time switched affine systems[END_REF]), [START_REF] Bolzern | Quadratic stabilization of a switched affine system about a nonequilibrium point[END_REF]), [START_REF] Hetel | Local stabilization of switched affine systems[END_REF]) that have several practical applications, particularly in the domain of power converters [START_REF] Deaecto | Switched affine systems control design with application to dc-dc converters[END_REF]), [START_REF] Corona | Stabilization of switched affine systems: An application to the buck-boost converter[END_REF]), [START_REF] Albea | Hybrid dynamic modeling and control of switched affine systems: application to dc-dc converters[END_REF]), [START_REF] Serieye | Freematrices min-projection control for high frequency dc-dc converters[END_REF]). In this paper, we are interested in the design of locally stabilizing switching laws for switched affine systems. This means finding conditions to select the active subsystem according to the system state so that stability is ensured. The presence of the affine terms makes that each subsystem has its own equilibrium point. However, the stabilization of the overall system is generally looked towards a desired equilibrium point which is usually different than the equilibria of the subsystems. The problem, is even more challenging in the discretetime case [START_REF] Deaecto | Practical stability of discrete-time switched affine systems[END_REF]), [START_REF] Albea Sanchez | Robust switching control design for uncertain discrete-time switched affine systems[END_REF]), which is the main object of study of this paper. In general, for discrete-time switched affine systems, the stabilization cannot be addressed to an equilibrium point but only to a neighborhood containing it, or to a limit cycle. Such discrete-time switched systems have been studied for example in [START_REF] Deaecto | Stability analysis and control design of discrete-time switched affine systems[END_REF]), [START_REF] Egidio | Novel practical stability conditions for discrete-time switched affine systems[END_REF]), [START_REF] Serieye | Robust stabilization to limit cycles of switching discrete-time affine systems using control lyapunov functions[END_REF]) and [START_REF] Serieye | Stabilization of switched affine systems via multiple shifted lyapunov functions[END_REF]). In [START_REF] Deaecto | Stability analysis and control design of discrete-time switched affine systems[END_REF]), a stabilizing min-type switching state feedback function and a characterization of the attractive invariant set is provided based on general quadratic Lyapunov functions. In [START_REF] Egidio | Novel practical stability conditions for discrete-time switched affine systems[END_REF]), stability conditions are given based on Lyapunov-Metzler inequalities in order to ensure practical stability of an equilibrium point for discrete-time switched affine systems. In [START_REF] Serieye | Stabilization of switched affine systems via multiple shifted lyapunov functions[END_REF]) conditions for designing a stabilizing switching function are also given based on Lyapunov-Metzler conditions. However differently from [START_REF] Egidio | Novel practical stability conditions for discrete-time switched affine systems[END_REF]), the positive invariant set is characterized by the union of several potentially disjoint ellipsoids. LMI conditions for the existence of the stabilizing switching law are provided based on the existence of multiple shifted Lyapunov functions. In [START_REF] Egidio | Global asymptotic stability of limit cycle and h2/h performance of discrete-time switched affine systems[END_REF]), conditions for stabilization towards a limit cycle are given. The extension to the case of uncertain switched affine systems has been recently addressed in [START_REF] Serieye | Robust stabilization to limit cycles of switching discrete-time affine systems using control lyapunov functions[END_REF]).

Note that all these results provide global stabilization conditions. However, there are important classes of switched affine systems that can be only locally stabilized. Consider for instance a scalar switched affine system x k`1 " ax k `b where the state matrix a may take two values in the set A " t1.02, 1.03u and where the affine term b switches among B={0.05, -0.05}. Clearly, since all the values of the state matrix are greater than 1, the system cannot be globally stabilized. No matter what the switching law is, if

|x k | ą max ␣ 5 2 , 5 3 ( then |x k`1 | ą |x k |.
However, one may easily check in simulations that when |x 0 | ă min ␣ 5 2 , 5 3 ( the solutions of the system

x k`1 " " 1.02x k ´0.05, when x k ě 0, 1.03x k `0.05, when x k ă 0, (1)
k | ą 0.05 1`1.02 then |x k`1 | ă |x k |
, that is the system can be locally practically stabilized. The local practical stabilization is clearly ensured by the affine terms b. The existing literature cannot address the example under study. For this reason, in this work, we are interested in the local practical stabilization problem. More specifically, we consider a special class of discrete-time switched affine systems with decoupled switchings in the state matrix and the affine term b. Methods for designing a switching control law that locally practically stabilizes the system to some neighborhood of the origin are given. A qualitative approach is provided, based on the existence of a stabilizing linear state feedback. Numerical examples are given to illustrate the approach. The research in this work is related to studies on sampled data controllers for switched affine systems [START_REF] Hetel | Robust sampleddata control of switched affine systems[END_REF]), [START_REF] Sanchez | Practical stabilization of switched affine systems with dwell-time guarantees[END_REF]) and on relay systems in [START_REF] Govindaswamy | On relay control for discrete time systems using linear matrix inequalities[END_REF]). This article is organized as follows: in Section 2, the system under study is presented with related assumptions. In Section 3 some necessary preliminary results needed for the synthesis of the control are given. The main theorem of the article is given in Section 4. Numerical examples are then presented in Section 5.2. Proofs are in the appendix.

Notations. The following notations will be used all along the paper. For a given symmetric matrix P , P ą 0 ( resp. P ă 0) indicates it is positive definite (resp. negative definite), and the notation EpP, γq corresponds to the ellipsoid EpP, γq " tx P R n : x T P x ď γu for some γ ą 0. Bpϵ, 0q, for some ϵ ą 0 describes the space ball centered at the origin with radius ? ϵ. convpVq denotes the convex hull of a set V and intpVq its interior. ∥X∥ is the Euclidean norm either for a vector X P R n or a matrix X P R nˆn . For a finite set of vectors V " tv 1 , v 2 , . . . , v N u, where v i P R m , @i P t1, ¨¨¨, N u and a function f : R m Ñ R we denote argmin vPV f pvq " tv P V : f pvq ď f pwq, @w P Vu and argmin vPV f pvq " v i with i " mintj P t1, ¨¨¨, N u : v j P argmin wPV f pwqu.

PROBLEM STATEMENT

Consider the following discrete time system:

x k`1 " A λ k x k `Bu k , x 0 P R n , k P N (2) where x k is the system state. A i P R nˆn , i P Λ " t1, ¨¨¨, Lu are L known matrices, λ k P Λ is the index of the active subsystem at time k and B P R nˆm . u k P R m is the control input such that it takes values in a finite set of vectors V " tv 1 , v 2 , ¨¨¨, v N u, v i P R m , @i P t1, ¨¨¨, N u.
With respect to the classical literature on switched affine systems, system (2) can be represented in the form:

x k`1 " A λ k x k `bσ k , x 0 P R n , k P N (3)
where σ k P t1, ¨¨¨, N u and b i " Bv i , i P t1, ¨¨¨, N u. σ k is the classical switching control, i.e. σ k " i when u k " v i for some i P t1, ¨¨¨, N u. However, this switched affine system has the particularity that the state matrix switches according to another parameter λ k which is arbitrarily switching. In this sense, we say that system (3) is a switched affine system with decoupled switching in the state matrix and the affine term. The main reason for studying this system is the need to emphasize the role played by the affine terms b i " Bv i in the local stabilization. Here we assume that λ k is arbitrarily varying to show the importance of the affine terms. However, we may always formalize another problem statement where λ k is also a control parameter (maybe also related to σ k ).

All along the article the following Assumptions are considered:

Assumption 2.1. The set V is non empty and 0 P int pconvpVqq.

Assumption 2.2. At every time k, the active index λ k is known.

Assumption 2.3. There exist K P R mˆn , 0 ă β ă 1 and a symmetric positive definite matrix P " P T ą 0 such that the function V pzq "

z T P z , V : R n Ñ R `, satisfies the condition V ´pA i `BKq x ¯ă p1 ´βqV pxq, (4)
@x P R n \t0u, @i P Λ.

Assumption 2.3 implies that there exists a linear static state feedback gain K which ensures the quadratic stability of the system x k`1 " pA λ k `BKqx k uniformly with respect to the switching law λ k P Λ.

The objective is to design the control law u k " Πpx k , λ k q, Π : R n ˆΛ Ñ V such that the closed loop system is locally practically stable for any switching sequence tλ k u kPN .

PRELIMINARY RESULTS

Before presenting our main results, some useful preliminary results are given: Proposition 1. Given K P R mˆn , P " P T ą 0 and a finite set

V " tv 1 , ¨¨¨, v N u Ă R m , such that Assumption 2.1 holds, then Dγ ą 0 : x T P x ă γ ùñ Kx P convpVq, @x P R n . (5) Proof. See Appendix A.
Proposition 1 indicates that for any given linear state feedback Kx, there exists a sufficiently small neighborhood of the state space such that Kx belongs to the convex polytope described by the switching vectors v i , i P t1, ¨¨¨, N u.

The following theorem from [START_REF] Ziegler | Lectures on polytopes[END_REF]) -Fundamental Theorem of Polytopes -establishes the link between the vertex and affine representations of a convex polytope. Theorem 2. [START_REF] Ziegler | Lectures on polytopes[END_REF]) Consider the convex polytope conv pVq whose vertices are the vectors v i , i P t1, ¨¨¨, N u. There exist vectors

h i P R m , i P t1, ¨¨¨, n h u, such that conv pVq " ␣ y P R m : h T i y ď 1, @i P t1, ¨¨¨, n h u ( . (6)
Using Theorem 2, we may compute the largest level set EpP, γq of the quadratic Lyapunov Function V pxq " x T P x where Kx belongs to the convex polytope described by the controls v i . Lemma 3. Consider the convex polytope convpVq convpVq "

␣ y P R m : h T i y ď 1, @i P t1, ¨¨¨, n h u ( , (7)
and a matrix K P R mˆn . Then for any positive γ ˚such that γ

˚ď 1 h T i KP ´1K T h i
, @i P t1, ¨¨¨, n h u,

x P EpP, γ ˚q implies Kx P convpVq.

Proof. See Appendix B.

The convexity relation described in Lemma 3 will be further used in the following section in order to design a stabilizing switching control.

MAIN RESULT

In this section we present our main results. The following theorem provides methods for designing a locally (practically) stabilizing switching law based on the existence of a stabilizing linear state feedback (Assumption 2.3). In addition, an estimation of the domain of attraction and of the attractor is provided. Theorem 4. Consider system (2) such that Assumptions 2.2 and 2.1 hold. Let h i P R m , i P t1, ¨¨¨, n h u be vectors such that conv pVq " ␣ y P R m : h T i y ď 1, @i P t1, ¨¨¨, n h u

( . (8)
Assume that there exist P " P T ą 0, β ą 0 and K such that Assumption 2.3 holds. Let

γ " min iPt1,¨¨¨,n h u 1 h T i KP ´1K T h i , (9)
δ " max

vPV v T B T P Bv β ´ϵ , with 0 ă ϵ ă β ă 1 (10) ρ " max iPΛ,vPV ˜∥A i ∥ d δ λ min pP q `∥Bv∥ ¸2 λ max pP q. ( 11
)
If ρ ă γ, then system (2) with the control law

u k " argmin vPV px T k A T λ k P Bvq (12)
is locally practically stabilizable from EpP, γq to EpP, ρq, that is x 0 P EpP, γq ùñ lim kÑ8

x k P EpP, ρq.

Proof. The proof of this theorem follows from results of Propositions 5 and 6 given in the Appendix C. Proposition 5 shows that whenever x k P EpP, γq\EpP, δq we have ∆V px k q ă 0, which means that trajectories starting in EpP, γq\EpP, δq are all decreasing toward the ellipsoid EpP, δq. However, a trajectory that is inside this ellipsoid might leave it. Proposition 6 give an estimation of EpP, ρq that contains the trajectories leaving EpP, δq.

Remark. Theorem 4 shows that when Assumption 2.3 is satisfied, the switching law (12) ensures the local practical stabilization of system (2). Any solution with initial condition in the ellipsoid EpP, γq (the estimate of the domain of attraction) with γ given in (9) converges towards the ellipsoid EpP, ρq (the attractor) with ρ given in (11) (provided that ρ ă γ).

Remark 2. The method presented here can be adapted for the case where λ k is also a control parameter. Considering the same conditions as in Theorem 4, a pair of state dependent switching laws θ : R n Ñ Λ and Π : R n Ñ V can be derived such that system (2) with λ k " θpx k q and u k " Πpx k q is locally practically stable. More precisely the following switching functions are obtained:

pθpxq, Πpxqq " argmin λPΛ,vPV `xT A T λ P A λ x `2x T A T λ P Bv (14)
This pair of switching laws can be derived using the same developments used in Theorem 4 (in the steps given in (C.9)-(C.11) we choose the controls v and λ which minimize the expression M px, λ, vq).

NUMERICAL EXAMPLES

Example 1

Consider the motivating example given in the Introduction. For this example, the vectors h 1 , h 2 in (8) are t 1 0.05 , ´1 0.05 u . The conditions of Theorem 4 are satisfied for β " 0.01, P " 0.4904, K " ´0.0350.

(15) With these parameters we obtain γ " 1 and ρ " 0.1566. This means that trajectories with initial condition |x 0 | ă 1.4280 converge to the set r´0.5650, 0.5650s.

Example 2

Consider a system of the form (2) such that A λ k switches arbitrarily between the two matrices

A 1 " " 1 0.01 0 1 ȷ , A 2 " " 1.02 0 ´0.01 0.99 ȷ ( 16
)
and B is defined by B " " 0.005 0 0 0.005 ȷ . Both A 1 and A 2 have eigenvalues outside the unit circle. The control u k takes values in the set

V " "" 1 1 ȷ , " 1 ´1ȷ , " ´1 ´1ȷ , " ´1 1 ȷ* . ( 17
)
For this example the vectors h i , i " 1, . . . , 4, are given by "" 2 together with the trajectories of x 1 and x 2 for the initial condition x 0 " r0.1866 0.1312s T .

1 0 ȷ , " ´1 0 ȷ , " 0 1 ȷ , " 0 ´1ȷ* . ( 18
Fig. 2. Domain of attraction EpP, γq and attractor EpP, ρq for the system in Example 5.2. The black lines delimit the region in the state space for which Kx P convpVq.

As we can see in the simulations, the estimation of the attractor EpP, ρq is quite conservative. This leaves space for improvement in the estimation of ρ.

CONCLUSION

This work investigated the local stabilization properties of discrete-time switched affine systems. A particular class of these systems, characterized by a decoupling in the state matrix and the affine term has been treated. A stabilizing switching law design has been proposed for the affine term whatever the switching of the state matrix is. Sufficient conditions for the existence of a locally practically stabilizing switching law have been given. In addition, a method for estimating the domain of attraction and the attractor has been given. These results have been tested on numerical examples. In the future, we intend to provide methods for simultaneously design switching laws for the switching matrix and the affine terms in order to improve the local stabilization properties.

Appendix A. PROOF OF PROPOSITION 1

Proof. V is a non empty subset of R m containing 0 in its interior, which means that there exists an R m ball Bpϵ 1 , 0q of sufficiently small radius ? ϵ 1 , centered in 0, contained in the convex hull of V. The function Kx, K P R mˆn is linear and continuous from R n to R m . Then there exists an R n ball Bpϵ 2 , 0q centered in 0 with radius ? ϵ 2 , such that

x P Bpϵ 2 , 0q ùñ Kx P Bpϵ 1 , 0q. (A.1)
Since Bpϵ 2 , 0q Ă R n exists and is non empty, then an ellipsoid EpP, γq can be found inside it, with a γ that satisfies γ ď ϵ 2 λ min pP q. This comes from the fact that for all x P EpP, γq we have x T x ď γ λminpP q and for all x P Bpϵ 2 , 0q we have x T x ď ϵ 2 . Then, in order to have EpP, γq Ď Bpϵ 2 , 0q, γ λminpP q ď ϵ 2 needs to hold.

Appendix B. PROOF OF LEMMA 3

Proof. The proof of this lemma is adopted from [START_REF] Hindi | Analysis of linear systems with saturation using convex optimization[END_REF]). We give it in what follows for the sake of self-containment. From equation ( 6), the condition Kx P convpVq is equivalent to the existence of n h ą 0 vectors h i from R m such that h T i Kx ď 1 , @i P t1, ¨¨¨, n h u.

We want to show that whenever x P EpP, γq, h T i Kx ď 1, @i P t1, ¨¨¨, n h u. Note that this is implied by the assumption

x T K T h i h T i Kx ď 1 whenever x T P γ x ď 1, (B.1) @i P t1, ¨¨¨, n h u. Let us remark that if K T h i h T i K ď P γ , @i P t1, ¨¨¨, n h u then condition (B.1) is satisfied. Rewriting K T h i h T i K ď P γ as P ´KT h i γh T i K ě 0 (B.2)
and using the Schur Complement Lemma, we can show that this is equivalent to:

» - P K T h i h T i K 1 γ fi fl ě 0. (B.3)
Then using the reverse formula of the Schur Complement, gives:

1 γ ´hT i KP ´1K T h i ě 0, (B.4)
which is equivalent to:

γ ď 1 h T i KP ´1K T h i , @i P t1, ¨¨¨, n h u. (B.5)
Therefore, we have shown that if 0 ă γ ď 1 h T i KP ´1K T hi , @i P t1, ¨¨¨, n h u, then (B.1) holds, and therefore the Proposition holds.

Appendix C

Proposition 5. Consider system (2) such that Assumptions 2.2 and 2.1 hold. Let h i , i P t1, ¨¨¨, n h u be R m vectors such that: conv pVq " ␣ y P R m : h T i y ď 1 @i P t1, ¨¨¨, n h u ( . (C.1) Assume that there exist P " P T ą 0, β ą 0 and K such that Assumption 2.3 holds. Let:

γ ď 1 h T i KP ´1K T h i , @i P t1, ¨¨¨, n h u (C.2) δ ą max vPV " v T B T P Bv β ´ϵ * , 0 ă ϵ ă β ă 1 (C.3)
If δ ă γ, then for system (2) with the switching control

u k " Πpx k , λ k q " argmin vPV px T k A T λ k P Bvq
we have V pA λ k x k `Bu k q ă V px k q, @λ k P Λ, (C.4) whenever

x k P EpP, γq\EpP, δq, @k P N.

Proof. Consider K P R mˆn , 0 ă β ă 1 and P " P T ą 0 such that Assumption 2.3 holds. Then we have:

pA λ k x k `BKx k q T P pA λ k x k `BKx k q ´xT k P x k ă ´βx T k P x k , @λ k P Λ, x k ‰ 0. (C.5)
Proposition 1 states that when convpVq is a nonempty subset of R m and that 0 P intpconvpVqq, then Dγ ą 0 such that for any x P EpP, γq , Kx P convpVq. The latter means that @x k P EpP, γq, Dα j px k q ě 0, @j P t1, ¨¨¨, N u, satisfying

N ÿ j"1 α j px k q " 1, such that Kx k " N ÿ j"1 α j px k qv j . (C.6) Inequality (C.5) implies that x T k A T λ k P A λ k x k `2x T k A T λ k P BKx k `xT k K T B T P BKx k ´p1 ´βqx T k P x k ă 0, @λ k P Λ, @x k ‰ 0. (C.7)
Then taking into account (C.6) leads to

N ÿ j"1 α j px k q `xT k A T λ k P A λ k x k `xT k K T B T P BKx k ´p1 ´βqx T k P x k `2x T k A T λ k P Bv j ˘ă 0, (C.8)
For all λ k P Λ, x k P EpP, γq. This is the same as N ÿ j"1 α j px k qM px k , λ k , v j q ă 0, @λ k P Λ, @x k P EpP, γq,

(C.9) with M px k , λ k , v j q " x T k A T λ k P A λ k x k `xT k K T B T P BKx k ´p1 ´βqx T k P x k `2x T k A T λ k P Bv j .
Since all α j pxq ě 0, then there exists at least one index i ˚P t1, ¨¨¨, N u such that M px k , λ k , v i ˚q ă 0. This means that the minimum of M px k , λ k , vq is always negative for v P V. Let us remark that argmin vPV M px k , λ k , vq " argmin vPV px T k A T λ k P Bvq, @λ k P Λ (C.10) with matrix A λ k corresponding to the active state matrix at instant k. Therefore the following is satisfied:

M px k , λ k , v ˚q ă 0, @x k P EpP, γq, @λ k P Λ (C.11) with v ˚" argmin vPV px T k A T λ k P Bvq " u k . Let us recall that u k is a function of x k and λ k , i.e. u k " Πpx k , λ k q " argmin vPV px T k A T λ k P Bvq. Using the fact that ∆V x k " V px k`1 q ´V px k q " V pA λ k x k `BΠpx k , λ k qq ´V px k q " pA λ k x k `BΠpx k , λ k qq T P pA λ k x k `BΠpx k , λ k qq ´xT k P x k " x T k A T λ k P A λ k x k `2x T k A T λ k P BΠpx k , λ k q `Πpx k , λ k q T B T P BΠpx k , λ k q ´xT k P x k and adding and subtracting Π T px k , λ k qB T P BΠpx k , λ k q to M px k , λ k , Πpx k , λ k qq lead to the following @x k P EpP, γq: M px k , λ k , Πpx k , λ k qq " ∆V px k q `xT k K T B T P BKx k `βx T k P x k ´Πpx k , λ k q T B T P BΠpx k , λ k q ă 0, (C.12)
In what follows we use (C.12) in order to characterize the set of states x k such that ∆V px k q ă 0. More precisely, we try to find an ellipsoid EpP, δq inside EpP, γq such that ∆V px k q ă 0 when x k P EpP, γq\EpP, δq. A sufficient condition to prove that ∆V px k q ă 0, @x P EpP, γq\EpP, δq is to show that x T k `KT B T P BK `βP ˘xk ´vT i B T P Bv i ą 0, for all i P t1, ¨¨¨, N u whenever x P EpP, γq\EpP, δq.

Using the S-procedure, we can show that the condition

x T k `KT B T P BK `βP ˘xk ´vT i B T P Bv i ą 0, @i P t1, ¨¨¨, N u whenever x Let us take c " β ´ϵ, for some ϵ ą 0 sufficiently small such that 0 ă ϵ ă β ă 1. Then a feasible δ is found such that cδ ´vT i B T P Bv i ą 0, @i P t1, ¨¨¨, N u which means that δ ą v T i B T P Bv i c , @i P t1, ¨¨¨, N u.

(C.16)

In order to take into account all possible switches, δ should be chosen such that δ ą max vPV v T B T P Bv β´ϵ . Proposition 6. Consider system (2) with u k P V, @k P N.

Given P " P T ą 0 and δ ą 0. If x k P EpP, δq, then x k`1 P EpP, ρq with ρ " max iPΛ vj PV ˜∥A i ∥ d δ λ min pP q `∥Bv j ∥ ¸2 λ max pP q.

(C.17)

Proof. Let x k P EpP, δq, which means that x T k P x k ă δ. Since P is symmetric and positive definite: λ min pP q∥x k ∥ 2 ď x T k P x k ď λ max pP q∥x k ∥ 2 (C.18) where λ min pP q and λ max pP q are the minimum and maximum eigenvalues of P . The last equation leads to

||x k || 2 ă δ λ min pP q . (C.19)
We are looking for a lower bound of ρ ą 0 such that:

x T k`1 P x k`1 ă ρ, @x k P EpP, δq, @λ k P Λ. (C.20) We know that :

x `∥Bv∥ ¸2 λ max pP q.

(C.26) Then, it is clear that x k`1 P EpP, ρq with ρ " max iPΛ,vPV ˜∥A i ∥ d δ λ min pP q `∥Bv∥ ¸2 λ max pP q.

Fig. 1 .

 1 Fig. 1. Behaviour of the system depending on different initial conditions

  `KT B T P BK `pβ ´cqP ˘xk `cδ ´vT i B T P Bv i ą 0 (C.15) Let us choose c ă β in order to havex T

		T k P x k ą δ,	(C.13)
	is satisfied if and only if there exists c ą 0 such that
	x T k	`KT B T P BK `βP ˘xk ´cpx T k P x k ´δq ą 0. ´vT i B T P Bv i	(C.14)
	The latter is the same as
	x T k

k

`KT B T P BK `βP ´cP ˘xk ą 0.

  k`1 " A λ k x k `Bv, (C.25) using the fact that λ min pP q∥x k`1 ∥ 2 ď x T k`1 P x k`1 ď λ max pP q∥x k`1 ∥ 2 , equation (C.25) gives:

	which gives :
	∥x k`1 ∥ 2 ă max iPΛ,vPV `∥Bv∥ ¸2 , x T ˜∥A i ∥ d δ λ min pP q k`1 P x k`1 ă
	max iPΛ,vPV	d ˜∥A i ∥	δ λ min pP q
					λ k P Λ,	v P V.	(C.21)
			It follows
			∥x k`1 ∥ ď max iPΛ,vPV	∥A i ∥∥x k ∥ `∥Bv∥,	(C.22)
			∥x k`1 ∥ 2 ď max iPΛ,vPV	p∥A i ∥∥x k ∥ `∥Bv∥q	2 .	(C.23)
			From (C.19) we can have :
					d
			∥x k ∥ ă	δ λ min pP q	,	(C.24)


Anthony Quenehen

Nathalie Klement

Amine Mohamed Abdeljaouad

Lionel Roucoules

Olivier Gibaru

Economic and ergonomic performance enhancement in

Introduction

Background

Implementation of Collaborative Robotics represents a potential source of improvement for assembly processes in terms of operational efficiency. Their ability for safe integration in presence of human makes them valuable coworkers. Since there is no need for physical separation (along with appropriate safety measures), one may design smooth hybrid processes for human and collaborative robot, where allocation to either resource can be done at task level. If such configurations open way for economical performance enhancement, they should however be considered along-with some limitations for robot motion speed and ability to perform complex tasks.

Shifting from a purely economical performance perspective, ergonomic benefit may raise the attractiveness of hybrid processes. Delegating appropriate tasks to the robot may relieve the operator of awkward postures or fatigue from the repetition of load handling. In those conditions, designing such a process would result in solving a multiobjective task allocation problem among human and robot.

Proposal

Pushing further the use of collaborative robot technology, one could consider not only allocating tasks either to CONTACT Anthony Quenehen anthony.quenehen@ensam.eu Arts et Métiers Institute of Technology, LISPEN, HESAM Université, F-59000Lille, France operator or robot but also proposing a simultaneous joint action of robot and operator, splitting then the ergonomic load, while maintaining the dexterity and speed of the operator. This third possible type of allocation offers advantages for both economical and ergonomic performance: the overall task execution time is not extended (using operator time as a reference) since the operator speed can still be at use, and by splitting the task content appropriately, the ergonomic load for the operator may be significantly reduced. As a drawback, this option engages both resources, preventing possible parallelisation of another task during this sequence. Interestingly, the consideration of those two collaboration modes offers a three way trade-off (operator, robot, cooperation of both) where none of the selected mode dominates the two others on both performance objectives, as summarised in Table 1.

In this paper, the authors propose a set of solutions for this problem, focusing on the advantages of incorporating various collaboration modes. With a concern for real-life application, the proposed solutions will also be tested through real use cases, to confirm the relevance of the proposed model, hence offering valuable tools for practitioners.

The rest of this article is structured as follows: Section 2 will review existing literature on comparable problems and related resolution methodologies. Section 3 will define a model and detail the strategy of resolution. Section 4 will review the obtained results, and finally Section 5 will conclude this paper and open on further possible work.

Literature review

Through the description of the considered problem, two main issues emerged:

• Interaction between human and robot • Assignment of tasks between those resources to improve a double criteria: economic and ergonomic.

In the current section, a literature review about these issues is done.

Human-robot interaction

Several types of safe interactions can be foreseen in the context of assembly operations, ranging from physical separation between operators and robot to simultaneous action on the same work piece. Classifications are based on spatial and temporal separation, see [START_REF] Thiemermann | Direkte Mensch-Roboter-Kooperation in der Kleinteilemontage mit einem SCARA-Roboter[END_REF], [START_REF] Matheson | Human-Robot Collaboration in Manufacturing Applications: A Review[END_REF], describing comparable interactions, despite use of different names, see Figure 1 for Thiemermann representation. A more refined model has been proposed by [START_REF] Aaltonen | Refining Levels of Collaboration to Support the Design and Evaluation of Human-robot Interaction in the Manufacturing Industry[END_REF], which includes additional factors such as goal sharing or possibility of physical contact between operator and cobot. In the present work, the model presented in Figure 1 delivers sufficient accuracy and its terminology will be used throughout this paper.

In terms of assembly process, the so-called synchronised mode indicates that operator and robot are working on their respective task in parallel, providing their selection satisfies related constraints, typically precedence of tasks, or feasibility in case of robot allocation. In such case the ergonomic load is fully removed from the operator, but at the expense of a longer completion time, since it is generally observed that robot motion is reduced when sharing working space with human, impairing then economical performance (cycle time or makespan). Regarding actual cooperation, the resulting ergonomic gain depends on the way the work elements of the task have been split, with regard to the ergonomic model selected. Due to the intentional proximity between operator and robot, a special attention has to be given to safety conditions during the process design phase. As stated in ISO 15066, Robot motion speed has to be controlled to comply with energy limitation in case of collision.

Various activity scopes can be seen when designing a collaborative workplace, depending on the extent of the influencing factors considered, as stated by [START_REF] Simões | Designing Human-Robot Collaboration (HRC) Workspaces in Industrial Settings: A Systemic Literature Review[END_REF]. Most complex scenarios include human and social factors, whereas the simplest ones focus solely on the constituents (operator or cobot). Intermediate levels focus on process performance and therefore incorporate several factors such as task allocation strategy, ergonomic cost and task duration and feasibility by each resource. Multi-objective approaches exist, following a sequential process starting from breaking down the work content in tasks, whose feasibility by each resource is then assessed, based on complexity, ergonomic cost, or requirement for specific equipment [START_REF] Gjeldum | Collaborative Robot Task Allocation on an Assembly Line Using the Decision Support System[END_REF][START_REF] Pini | A Systematic Approach to the Engineering Design of a HRC Workcell for Bio-Medical Product Assembly[END_REF]. [START_REF] Malik | Collaborative Robots in Assembly: A Practical Approach for Tasks Distribution[END_REF] propose a more gradual approach where tasks rather have an affinity score for allocation towards operator or robot, based on the above-mentioned factors. Generally, tasks are then allocated to the idle resource [START_REF] Tsarouchi | On a Human-Robot Collaboration in An Assembly Cell[END_REF][START_REF] Faber | Human-robot Cooperation in Future Production Systems: Analysis of Requirements for Designing an Ergonomic Work System[END_REF]. In brief, design of collaborative workplace may encompass several aspects with different levels of details. When considering process performance aspects, the proposed strategies are mainly sequential and offer little possibility of optimisation for allocation and sequencing, especially considering joint usage of both resources (cooperation mode).

The gap we intent to address is to take the best advantage of multiple collaboration modes to design an optimal process in terms of economical performance and ergonomic risk. In that purpose, allocation problems will also be reviewed, as close situation this work, to identify transferable optimisation strategies in comparable models.

Allocation problems

The studied problem in the current paper is an allocation problem: the main objective is to allocate every task to some resource (worker, robot or both), respecting precedence constraints and minimising costs (economic and ergonomic). Both scheduling and line balancing problems deal with allocation consideration.

Scheduling

Scheduling problems have been widely studied in the literature for many years. In the following, the case of Unrelated Machines and Resource Constrained Project Scheduling Problems (RCPSP) are detailed among all scheduling problems, because these two consider several different resources, as in the problem studied in this paper.

Unrelated Parallel Machines (PMS)

. connect to the problem studied in this paper since the processing time of a task depends on the machine to which it is allocated. Although this problem has been less investigated than identical parallel machine scheduling, several methods have been proposed to solve it, ranging from exact algorithms [START_REF] Rocha | Exact Algorithms for a Scheduling Problem with Unrelated Parallel Machines and Sequence and Machine-Dependent Setup Times[END_REF][START_REF] Wang | Exact Methods for Order Acceptance and Scheduling on Unrelated Parallel Machines[END_REF], to heuristics [START_REF] Herrmann | Heuristics for Unrelated Machine Scheduling with Precedence Constraints[END_REF][START_REF] Fanjul-Peyro | Size-reduction Heuristics for the Unrelated Parallel Machines Scheduling Problem[END_REF][START_REF] Wang | Effective Heuristic for Large-Scale Unrelated Parallel Machines Scheduling Problems[END_REF] and metaheuristics like ant colony [START_REF] Lin | Unrelated Parallel Machines Scheduling Problem Using An Ant Colony Optimization Approach[END_REF], simulated annealing [START_REF] Kim | Unrelated Parallel Machine Scheduling with Setup Times Using Simulated Annealing[END_REF][START_REF] Lin | A Multi-Point Simulated Annealing Heuristic for Solving Multiple Objective Unrelated Parallel Machine Scheduling Problems[END_REF] or genetic algorithms [START_REF] Nikabadi | A Hybrid Algorithm for Unrelated Parallel Machines Scheduling[END_REF][START_REF] Vallada | A Genetic Algorithm for the Unrelated Parallel Machine Scheduling Problem with Sequence Dependent Setup Times[END_REF].

In parallel machine scheduling problems, the most widely studied objective is the makespan. Some papers also have multi-objective considerations where jobs' tardiness is taken into account [START_REF] Pfund | A Survey of Algorithms for Single and Multi-objective Unrelated Parallel-machine Deterministic Scheduling Problems[END_REF]. However, to the best of our knowledge, the ergonomic aspect has never been dealt with in this issue. Moreover, there is a major difference between our problem and parallel machine scheduling: in our problem, a task can be performed simultaneously by both of the operator and the robot, while in parallel machine scheduling a job cannot be by allocated to more than one machine. We are thus facing additional constraints and challenges.

RCPSP. represents scheduling problems with resources

allocation. There are precedence constraints between tasks. Resources in a given quantity are needed to process a task. Usual hypothesis about RCPSP are the following.

• Resources are renewable: resources are returned at the end of the execution of a task and thus become available again to realise other task. • Resources can be of several types, they are available in a given quantity over the entire time horizon. • Tasks are non-preemptive, once started they cannot be interrupted. • Tasks are subject to precedence constraints.

• Tasks require the same amount of resources throughout their execution time.

Several RCPSP models exists. The considered case study in the current paper can be seen as a Multi-Mode RCPSP (MRCPSP). Mode defines the execution mode for each task: an execution time plus a quantity of resources for each type of resource. In MRCPSP, several modes can be used to process each task. Tasks require a variable number of resources, and execution time depends on the number of assigned resources. MRCPSP has been introduced by [START_REF] Elmaghraby | Activity Networks: Project Planning and Control by Network Models[END_REF]. Its resolution has drawn attention from researchers (Van Peteghem and Vanhoucke 2010), extensive literature reviews have been realised on this problem by [START_REF] Yang | Resource-Constrained Project Scheduling: Past Work and New Directions[END_REF] and [START_REF] Hartmann | A Survey of Variants and Extensions of the Resource-constrained Project Scheduling Problem[END_REF]. [START_REF] Kolisch | PSPLIB-a Project Scheduling Problem Library: OR Software-ORSEP Operations Research Software Exchange Program[END_REF] propose a mathematical model to model MRCPSP.

Most of the papers about MRCPSP deal with economic objectives (makespan minimisation). None of them considers the ergonomic objective, in the case of allocation of tasks to worker or robot or both of them.

Line balancing

Assembly Line Balancing Problems (ALBP) and their multiple extensions cover a wide range of situations, but share common characteristics that enable them to be described and classified in several categories [START_REF] Battaïa | A Taxonomy of Line Balancing Problems and their Solutionapproaches[END_REF], [START_REF] Boysen | A Classification of Assembly Line Balancing Problems[END_REF]. The purpose of this section is to identify potential similarities between the studied problem and previous research, to help building an appropriate modelling and collect insights on applicable resolution methods.

Although it may not seem initially appropriate to include line balancing problems in the context of single station, one may notice that in the case of synchronised mode (see Figure 1, two following products can be transformed simultaneously by the operator and cobot in two different workplaces. Therefore, this condition is comparable to a 2-station line where workload has to be best balanced between operator and cobot, to minimise the overall cycle time. Nevertheless, this possibility of overlapping between following products affects the calculation of the economical performance, which will be reflected in the following section.

The studied environment, featuring a single station, is not fit for SALBP type-1 resolution. Type 2 resolution may be suitable, as cycle time is considered as the economic target to be minimised. Characterising this problem from an ALBP perspective leads to consider the following elements :

• Resources are heterogeneous: Robot cannot be assigned to every task, and its completion time for a given time is higher than operator. • Most tasks can be performed by simultaneous usage of both resources (cooperation), which affects their completion time and ergonomic cost. • Both ergonomic and economic performances are set as objectives.

Implementing robots in place of operators at workstations has been reflected in RALBP (Robot Assembly Line Balancing Problem), where Task Times are dependant of the selected robot. Robots do not necessarily have the ability to perform every task [START_REF] Rubinovitz | RALB -A Heuristic Algorithm for Design and Balancing of Robotic Assembly Lines[END_REF]. Such work may be seen as a particular case of heterogeneous manpower, which has been further studied in ALWABP (Assembly Line Worker Assignment Balancing Problem), with application towards Sheltered Work Centres for Disabled [START_REF] Moreira | Robust Assembly Line Balancing with Heterogeneous Workers[END_REF][START_REF] Borba | A Heuristic and a Branch-and-Bound Algorithm for the Assembly Line Worker Assignment and Balancing Problem[END_REF][START_REF] Moreira | Simple Heuristics for the Assembly Line Worker Assignment and Balancing Problem[END_REF][START_REF] Miralles | Advantages of Assembly Lines in Sheltered Work Centres for Disabled. A Case Study[END_REF]). Operator and robot may be considered indeed as heterogeneous manpower, having distinct abilities and task times. However, robot being slower that operator, the only economical benefit in its introduction is based on parallelisation of task, and its ability to cooperate with operator. Parallelisation of tasks in a single workstation has been studied in ALBP literature, as in doublesided lines or Variable Workplace Assembly Line Balancing Problem (VWALBP). such models are used for large size products where several operators (workplaces) can be affected to the same workstation [START_REF] Becker | Balancing Assembly Lines with Variable Parallel Workplaces: Problem Definition and Effective Solution Procedure[END_REF]. In such conditions, task assignment is constrained by their related mounting position on the product, including the necessity to avoid obstruction between operators. Therefore simultaneous cooperation on same task is not considered.

Ergonomics has also been studied as an objective of line balancing problems, as a possible continuation of ALWABP, when used to describe environment with disabled workforce. Several methods issued by practitioners have been reflected by [START_REF] Otto | Incorporating Ergonomic Risks into Assembly Line Balancing[END_REF] to evaluate the ergonomic risk for an operator based on posture, effort or carried load during the studied process. Such problems -named ErgoALBP -tend to minimise the ergonomic risk across a determined number of stations. One of the difficulties when trying to balance ergonomic risk with economic cost is to estimate an impact of the former on the latter. Part of the answer has been brought by the Predetermined Motion Energy Systems (PMES) [START_REF] Battini | Ergonomics in Assembly Line Balancing Based on Energy Expenditure: a Multi-objective Model[END_REF]. In this model, ergonomic impacts of tasks are expressed as relaxation time, hence of the same dimension of the economic cost. This methodology relies on the calculation of the energy spent by the operator to perform necessary movements, maintain his posture, and carry out process tasks, considering the associated load due to components handling. In case of single workstation, operator remains static, so focus will be put on achievement of fatigue reduction through optimisation of components handling. Such model has been used by [START_REF] Weckenborg | Assembly Line Balancing with Collaborative Robots Under Consideration of Ergonomics: A Cost-Oriented Approach[END_REF] to study economic and ergonomic benefit of introducing collaborative robots on a manual line. In a comparable approach, although using different ergonomic model, [START_REF] Pearce | Optimizing Makespan and Ergonomics in Integrating Collaborative Robots into Manufacturing Processes[END_REF] suggested both economic cost and ergonomic risk could be reduced in a real-life assembly process, by sharing task appropriately between operator and robot. Both works considering task could be affected to operator or robot, but not the possibility of them cooperating on a same task.

In conclusion, it appears that the cooperation mode between operator and robot has been scarcely studied, especially in conjunction with its inherent benefits towards prevention of ergonomic risk while maintaining potentially high economical performance, as it can be seen in Table 2.

Proposition of resolution method

In the previous section, it has been shown that the studied problem is an extension of MRCPSP or SALBP (with one station). These classical problems are NP-Hard problems, it does not exist any algorithms to solve these problems in a polynomial time. Our problem is an extension as it considers ergonomic aspect in addition to classical economic criteria. Thus the considered problem is also NP-Hard. In the following, proposed approximate method is presented.

Objective function

The proposed method will be used on the twelve generated instances. The application of a method allows to define a solution X: the schedule of tasks with their assignment to resources (operator, robot, both). An assignment solution X provides operator's and robot's respective cycles times CT O (X) and CT C (X) (defined as 'the time when the operator, resp the robot, finishes his The results are compared through the following objective function, defined in (2), where α ∈ [0, 1[ stands for a given trade-off between ergonomic and economic cost. More explanations are given by [START_REF] Quenehen | Assembly Process Design: Performance Evaluation Under Ergonomics Consideration Using Several Robot Collaboration Modes[END_REF].

C eco (X) = max {CT O (X), CT C (X)} C ergo (X) = max {0, R tot (X) -(C eco (X) -CT O (X))} (1) H α (X) = (1 -α).C eco (X) + α.C ergo (X) (2)
C eco corresponds to the makespan required for assembly operation, and is solely depending on which resource finishes last. C ergo , however, is affected by the value of R tot (X) and the gap between CT O (X) and CT C (X). This may generates three possible cases as detailed in Figure 2.

Determination of economic and ergonomic costs in real case study

Several ergonomic models may apply to assembly processes [START_REF] Otto | Incorporating Ergonomic Risks into Assembly Line Balancing[END_REF], they consider have in common considerations for posture, extreme joint positions, load carried, movement and vibration, and applied force as main criteria. Intensity and frequency of exposure to those risks are also impacting the ergonomic evaluation.

Output result may be expressed as an index or a score, as per RULA (Rapid Upper Limb Assessment) (McAtamney and Nigel Corlett 1993), or as a relaxation time at the end of the assembly cycle as proposed by PMES [START_REF] Battini | Ergonomics in Assembly Line Balancing Based on Energy Expenditure: a Multi-objective Model[END_REF][START_REF] Battini | Preventing Ergonomic Risks with Integrated Planning on Assembly Line Balancing and Parts Feeding[END_REF]. In the present work, this latter model will be considered to assess the case study, since it delivers the ergonomic cost in the same dimension as economic cost. This feature may have significant impact when using or not the time available at end of cycle for rest or starting up the next assembly.

In the present work, PMES model has therefore been selected and used on a real case study. With a fully manual process as a start point, process has been run until total cycle time became stable (learning curve of operator). Then five consecutive cycles are video recorded and broken down into elementary operations, for each of them the mean duration value is kept as a reference. Likewise for ergonomic cost, each movement from each elementary operation is assessed according to the tables provided by [START_REF] Battini | Ergonomics in Assembly Line Balancing Based on Energy Expenditure: a Multi-objective Model[END_REF], to determine the value of the associated relaxation time, if applicable. 1 Tasks are then generated, as grouping of consecutive elementary operations that practically needs to be done by the same resource (e.g. picking and setting of screw). To populate Task Times for robots, a multiplying factor is applied from the manual task times, according to the experimental findings of Quenehen, Pocachard, and Klement (2019) in a comparable case study. In case the task cannot be completed by the robot, the time value is set to ∞. To determine both task times and relaxation time for the cooperation mode, each individual task is performed using both robot and operator, leaving the handling of the heaviest component to the robot (as a rule to minimise ergonomic cost). Then the same video capture process as per manual task is applied. If the task has no benefit from cooperation mode (e.g. picking of a single component), its task and relaxation times are set to ∞. Results for the case study can been seen in Figure 6.

Impact of overlap on objective function

As mentioned in Section 2.2.2, the usage of synchronised mode at the end or beginning of the assembly cycle will create an opportunity for the idle resource to kick off the next product to assemble, which may naturally be used in that purpose in real life, see Figure 3. The operator idle time at the end of the cycle will be named Operator Waiting Time (OWT), respectively Cobot Waiting Time (CWT) for the cobot. The time available at the beginning of the cycle due to the late start of the cobot will be referred to as Cobot Lag (CL), respectively, Operator Lag (OL) for the operator. Depending on the respective values of those variables, C eco and C ergo may be affected, resulting in new cost values: C eco and C ergo . Different configurations and their impact are illustrated in Figure 4.

Approximate method development

For such range of instance, heuristic methods have proven efficient to propose solutions. Usual heuristics affect tasks to the first available resource, which is not suitable to test the cooperation mode since it may only be triggered in the unlikely eventuality of both resources being available at the exact same time. To visualise the effects of the three considered modes, a probabilistic allocation has been selected. For each task, a number R is randomly generated between 0 and 1, and two limit values L1 and L2 are set. If R < L1, task is assigned to the operator, if L1 ≤ R < L2, task is affected to cobot, and if R ≥ L2, the task is allocated to the cooperation mode, i.e. cobot and operator simultaneously. This heuristic L is given by Algorithm 1. Tasks are assigned according to precedence's constraints.

By nature, a heuristic gives an approximate solution. By coupling it to a metaheuristic, better solutions may be found, since a set of solution will be tested, instead of a single one. This principle has first been proposed by [START_REF] Gourgand | An Analogy Between Bin Packing Problem and Permutation Problem: A New Encoding Scheme[END_REF] and more recently applied to several industrial applications in Klement and Silva (2020). The principle is illustrated in Figure 5.

The metaheuristic goes through the space of solutions, considering lists Y of tasks, using a neighbourhood system V. Each considered list Y of tasks gives an assignment solution X thanks to the list algorithm L, whose cost is assessed using the overlap function described in Figure 4. Then, this solution is evaluated thanks to an objective function H. The metaheuristic then generates a new list Y from the current one, using the neighbourhood system V. In the developed method, a swap between two tasks is used as a neighbourhood system. The tasks of this new list Y , the neighbour, are then Put this task at the end of the list 12 assigned thanks to the heuristic L, to give a solution X . This solution X is then evaluated using the objective function H. Finally both solutions (the current X and the neighbour X ) are compared using the principle of the chosen metaheuristic. Algorithm 2 illustrates this hybridisation using the simulated annealing as a metaheuristic.

Instance generation

A set of instances is necessary to test both algorithm capability and benefit of multiple mode usage. Therefore 12 random test instances have been generated by a special

∈ [0.1; 0.3] ∈ [0.3; 0.5] SosSt1 BosSt1 [15-20] tasks SosSt2 BosSt2 SosSt3 BosSt3 SosBt1 BosBt1 [20-25] tasks SosBt1 BosBt1 SosBt1 BosBt1
purpose algorithm, as extensions of the original real case study represented in Figure 6. Problem is meant for single station, so the number of tasks should be limited (14 in case study for instance). Therefore two intervals have been used, [15,20] and [20,25], to generate the number of tasks with a uniform distribution. Each task has approximately four to six predecessors at most, which represents a realistic assembly process. These precedence constraints also feature a final nod.

The task times of the operator are generated with a uniform distribution in the interval between 1 and 15 seconds. The k pace ratio between operator and robot for each task is derived from observed values in case studies and is picked randomly in the interval [4,6]. Likewise for the determination of task time of cooperative mode, values will be randomly picked in the interval [0.7, 1] of the corresponding manual task time.

Around 28% and 21% of the generated tasks are respectively not feasible by robot, or no fit for the cooperative mode. These percentages are centred on average value from real life examples. For these tasks, the processing time with robot or cooperation will be set to infinite value.

The order strength and the maximum order are also recorded for each generated instance. In total four families of instances, based on their number of tasks and order strength, will be used to evaluate the proposed algorithm performances. In each family, three randomly generated instances are tested.

Table 3 summarises the 12 generated instances, with their respective name. These instances will then be used to experiment the proposed method.

Results

In this section, experimentation is detailed. First, the proposed approximate method is applied. Then, results will be checked by programming the sequence using collaborative robot and operator, to test the relevance of the proposed model and solving tool in real life situation.

Application on test instances

Validation tests are necessary to prove both the capability of the proposed algorithm and the benefit of using all collaboration modes. To this purpose, test instances have been generated as extension of observed real case. Due to its probabilistic nature for task allocation, the algorithm has to be run multiple times to reach any conclusion. Moreover, the task allocation may be influenced by parameters L1 and L2, therefore several pairs of L1&L2 are to be tested (0.5&0.75, 0.25&0.5, 0.33&0.66, 0.6&0.7 have been used). Only the best solutions of the multiple runs for every pair are recorded. Similar procedure is used to test solutions without cooperation mode, by setting L2 value to 1. No dominant pair L1&L2 has been identified for all test instances and all values of α. Each result has been computed in less than 10 seconds using a 2.4 Ghz processor. The used metaheuristic is the simulated annealing presented by Algorithm 2, hybridised with heuristic given by Algorithm 1, with IterMax = 100000. Parameters T temperature and β decreasing factor have been generated using algorithms from [START_REF] Van Laarhoven | Simulated Annealing[END_REF]. T is computed around some hundreds, and β ≈ 0.999. Tested instances with all three collaboration modes only rarely achieved repetition of the best solution, nevertheless the range of the best three solutions was consistently below 1% of the target value, which is an acceptable value for real-life application considering natural fluctuation of the process execution. When tested with only two collaboration modes (without cooperation), similar range could be found, and best solution could be achieved multiple times by the algorithm in 45% of cases. Full results are visible in Table 4.

Overall results show little benefit of cooperation mode at low α values, which is natural since this mode offers little time saving compared to manual and prevents parallelisation of tasks. However, for α = 0.5, introduction of cooperation mode delivers better results, with an average improvement of 4% across the test instances, which brings significant benefit for economic and ergonomic trade-off. For greater α values, cooperation also delivers improved results, which is also foreseeable result since ergonomic focus will lead to use collaborative robot as a 'third hand' across the whole process (see Figure 7).

Application on real case study

Similar trend could be observed on real case study, with significantly higher benefit (up to 29.4% reduction for the objective function at α = 0.5), see Table 5. Algorithm results for the real case study have been physically applied in a working environment (see Figure 8) for better grounded assessment of the proposed method. Video captures of several representative configurations can be found here . 2 Several observations could be made on the relevance of the model and solution.

Handling of transitions between tasks

Throughout the implementation of proposed solutions, it has been necessary to add or modify several robot activation (through button pressing) to secure absence of unwanted interference between operator and cobot. Indeed, even though task times are deterministic, in practice operator pace may fluctuate. As long as this concerns a succession of manual tasks, this is little concern. But at the process points where interaction may happen between operator and robot -i.e. when there is an allocation change between two consecutive tasks, some measures have to be taken to avoid safety risk for the operator. As an example, if the operator is delayed on a manual task on a main piece, it must be secured that the robot will be waiting until this task is completed before to execute its own next task if it involves interacting with the same piece, as this situation may lead to potential collision or trapping of operator hands. Likewise, in case of opposite switch from robot or cooperative task towards manual task, which may involve releasing a component or sub assembly from robot gripper, it is mandatory to wait until operator is ready to receive it.

Such situations may be handled through high level technological solution that can observe operator movement to anticipate the moment he or she will be ready to interact, which may not require any alteration in the process design. Nevertheless for real-life application with limited investment, simple and robust solutions have to be considered, where operator has to give positive confirmation of his readiness to move to the next step. Such confirmation action may have a noticeable effect on the obtained cycle time, since they are generally in the range of a second each. Such confirmation actions are comparable in appearance to the setup times, which are already studied in multiple balancing problems [START_REF] Sternatz | Enhanced Multi-Hoffmann Heuristic for Efficiently Solving Real-world Assembly Line Balancing Problems in Automotive Industry[END_REF][START_REF] Vallada | A Genetic Algorithm for the Unrelated Parallel Machine Scheduling Problem with Sequence Dependent Setup Times[END_REF][START_REF] Rocha | Exact Algorithms for a Scheduling Problem with Unrelated Parallel Machines and Sequence and Machine-Dependent Setup Times[END_REF].

However, setup times are generally dependant on the sequence of tasks (necessity for tooling changeover for instance) and can be represented in a matrix form, where the value of the setup time can be found at the coordinates representing the two task numbers involved. In the studied case, those setup times, or rather transition times, are dependant on the respective allocations of the tasks involved: a transition has be added if at least one of the direct predecessors of the considered task has a different allocation than the task itself.

Conclusion and further works

In this paper, an assignment problem between different resources is studied. Several tasks are needed to make one product. There are precedence constraints between some of the tasks. The considered resources are an operator and a robot. A third allocation option is considered: both operator and robot are used at the same time to process the task (cooperation mode). The task time to process a task is different according to the mode: worker, robot or cooperation. In average, robot needs k more times than the worker to make the same activity. According to the mode, relaxation times are defined. Thus each task is defined by three process times and relaxation times. The objective of the considered problem is to assign all the tasks to resources, considering two objectives: economic and ergonomic performance. An approximate method has been developed to identify approximate solutions, highlighting some benefits in using concurrently several collaboration modes to balance ergonomic and economic performance. This method and tools maybe easily generalised towards other ergonomic models involving relaxation times. With concern for real-life application, the objective function and solving algorithm have incorporated cyclic production impact. Results on several test instances show significant performance leverage when both economic and ergonomic aspects are considered, offering new possibilities for designing better quality collaborative process. Application on real case study could confirm the theoretical findings and also highlighted the need for modelling transition impact when different collaboration modes are being used on consecutive tasks. A specific study has been carried out on the minimisation of fatigue accumulated through cycles using PMES. Other ergonomic models can be tested, covering other aspects of ergonomic assessment, as force application, extreme joint position, or even awkward posture. As long as an ergonomic score can be associated to a given task, the algorithm developed in this paper may be used, providing the objective function is amended to reflect the requirement of the associated model.

Nevertheless, several limitations and potential extensions could be considered. Video recording methodology used in this study has limited accuracy and ability to capture whole body movement. Extension of this type of study into broader environment, as a workcell for instance, would benefit from technology like motion capture, as proposed by [START_REF] Daria | Integrating Mocap System and Immersive Reality for Efficient Human-centred Workstation Design[END_REF]. Moreover, the ergonomic risk has been assessed through a single operator, designing an improved solution that may not benefit equally from the changes implemented. The alleged reconfigurability of the collaborative robot could be studied further to develop processes custom solutions for operator of various age or presenting different anthropometric characteristics, as suggested by [START_REF] Calzavara | Ageing Workforce Management in Manufacturing Systems: State of the Art and Future Research Agenda[END_REF] and [START_REF] Katiraee | Consideration of Workers' Differences in Production Systems Modelling and Design: State of the Art and Directions for Future Research[END_REF]. Furthermore, introducing changes to an existing process, using an algorithm based on a inevitably simplified model, may overlook several human aspects, which may undermine the resulting solution [START_REF] Neumann | Industry 4.0 and the Human Factor-A Systems Framework and Analysis Methodology for Successful Development[END_REF]). As a comprehensive strategy for implementation of collaborative robots towards improved work condition and performance, human factors should also be considered as a key contribution. Notes 1. To trigger relaxation time in the presented case study, some component weights have been increased, within the capability range of the selected collaborative robot 2. https://www.youtube.com/channel/UCTe59h3sT9fxOv6U

BvzsnGg/videos

Figure 1 .

 1 Figure 1. Human robot interaction categories, adapted from Thiemermann (2005).

Figure 2 .

 2 Figure 2. Possible formulations for C ergo based on process configuration.

Figure 3 .

 3 Figure 3. illustration of overlap between production cycles.

Algorithm 1 :

 1 Heuristic L to assign task to resources Data: List of task, precedence matrix, task time and recovery time according to the resource, L1 and L2 while All tasks are not assigned do

Algorithm 2 :2

 2 Principle algorithm of the hybridisation between a list algorithm and the simulated annealing Data: Temperature T 0 , decreasing factor β, Maximum number of iterations IterMax, Initial solution Y ∈ iter := 0, T := T 0 1 X := L(Y): apply the list algorithm to the list Y Record solution RY := Y, RX := X Y and X := X with the probability 14 e -H(X )-H(X) T iter := iter + 1 15 Generate a new temperature T := β × T 16

Figure 4 .

 4 Figure 4. Alteration of C eco and C ergo according to process configuration.

Figure 5 .

 5 Figure 5. Hybridisation metaheuristic -List algorithm.

Figure 6 .

 6 Figure 6. Precedence graph of our problem, with Task Time (TT) and Relaxation Time (RT) in seconds for each type of task assignment.

Figure 7 .

 7 Figure 7. Relative improvement brought by introduction of cooperation mode for various α values.

Figure 8 .

 8 Figure 8. Experimental setup for real case application (here using cooperation mode).

Table 1 .

 1 Qualitative comparison of collaboration modes under performance criteria.

		Operator	Robot	Cooperation
	Eco. performance	High	Low	High
	Ergo. performance	Low	High	Intermediate
	Parallelisation of tasks	High	High	Low

Table 2 .

 2 Summary of literature review.

	Constraints

last task'), and the required relaxation time R tot (X) for the operator. Two criteria are considered: C eco the economic cost and C ergo the ergonomic cost, given in Equations (1).

Table 3 .

 3 Test instances.

	Order strength

Table 4 .

 4 Results for test instances, IR stands for Improvement Ratio.

	Alpha		0			0.25			0.5			0.75			0.9
	modes	3	2	IR	3	2	IR	3	2	IR	3	2	IR	3	2	IR
	Instance								BosBt1
	H	123.3 121.9 0.0% 112.75	114.1	1.2%	89.8	94.00	4.5%	45.975	46.875	1.9%	18.45	19.01	2.9%
	C eco	123.3 121.9		125.7	125.2		172.8	184.50		183.9	186.9		184.5	190.1
	C ergo	90.8	92.2		73.9	80.8		6.8	3.50		0	0.2		0	0
	Instance								BosBt2
	H	127.5 126.4 0.0% 121.4	121.7	0.2% 100.8	104.45	3.5%	53.75	53.075	0.0%	20.22	21.5	6.0%
	C eco	127.5 126.4		129.8	126.4		171.9	208.60		215	206.6		202.2	215
	C ergo	102.2 107.6		96.2	107.6		29.7	0.30		0	1.9		0	0
	Instance								BosBt3
	H	139	138.2 0.0% 119.6	128.625 7.0%	99.35 105.55	5.9%	49.75	52.925	6.0%	20.69	21.17	2.3%
	C eco	139	138.2		135.7	139.1		155.9	211.10		199	211.7		206.9	211.7
	C ergo	84.3 100.5		71.3	97.2		42.8	0.00		0	0		0	0
	Instance								BosSt1
	H	105.8 105.8 0.0% 104.125 104.15	0.0%	86.35	93.35	7.5%	43.05	46.075	6.6%	17.23	18.7	7.9%
	C eco	105.8 105.8		108.2	106		172.7	186.70		172.2	184.3		172.3	187
	C ergo	107.6 107.6		91.9	98.6		0	0.00		0	0		0	0
	Instance								BosSt2
	H	112.1 113.1 0.9%	99.75	102.575 2.8%	79.75	84.15	5.2%	40.05	42.475	5.7%	16.05	17.1	6.1%
	C eco	112.1 113.1		116.9	113.9		159.5	160.60		157.8	167.5		160.5	171
	C ergo	77.5	73.5		48.3	68.6		0	7.70		0.8	0.8		0	0
	Instance								BosSt3
	H	124.6 123.6 0.0% 119.825 119.225 0.0%	98.05 103.00	4.8%	50.975	51.5	1.0%	19.61	20.6	4.8%
	C eco	124.6 123.6		127	125		196.1	206.00		203.9	206		196.1	206
	C ergo	120.3 110		98.3	101.9		0	0.00		0	0		0	0
	Instance								SosBt1
	H	158.4 151.2 0.0% 143.75	138.875 0.0% 115.7	111.90	0.0%	60.075	56.775	0.0%	23.83	22.65	0.0%
	C eco	158.4 151.2		164	151.2		207.7	223.80		240.3	227.1		238.3	226.5
	C ergo	94.1 101.9		83	101.9		23.7	0.00		0	0		0	0
	Instance								SosBt2
	H	125.1 124.1 0.0% 110.625 114.925 3.7%	93.25	96.65	3.5%	46.5	47.8	2.7%	18.57	19.56	5.1%
	C eco	125.1 124.1		123.1	124.1		158.9	189.40		186	191.2		185.7	195.6
	C ergo	90.1	87.4		73.2	87.4		27.6	3.90		0	0		0	0
	Instance								SosBt3
	H	161.9 155	0.0% 148.15	148.425 0.2% 132.5	129.95	0.0%	63.025	75.325 16.3%	27.9	40.11 30.4%
	C eco	161.9 155		161.3	156.8		168.4	224.30		252.1	245.8		279	392.1
	C ergo	117.6 119.7		108.7	123.3		96.6	35.60		0	18.5		0	1
	Instance								SosSt1
	H	102.9 100.4 0.0%	94.6	91.65	0.0%	75.45	70.55	0.0%	37.45	37.475	0.1%	14.97	14.86	0.0%
	C eco	102.9 100.4		106	100.5		150.2	139.90		149.8	149.9		149.7	148.6
	C ergo	61.6	75.7		60.4	65.1		0.7	1.20		0	0		0	0
	Instance								SosSt2
	H	134.5 132.8 0.0% 124.05	125.6	1.2% 101.6	104.30	2.6%	50.625	53.575	5.5%	20.48	21.43	4.4%
	C eco	134.5 132.8		134.4	132.8		201.7	191.40		198.6	214.3		204.8	214.3
	C ergo	99.4 104		93	104		1.5	17.20		1.3	0		0	0
	Instance								SosSt3
	H	98.6 102.2 3.5%	90.775	97.9	7.3%	70.85	79.60 11.0%	35.1	40.8	14.0%	14.96	16.32	8.3%
	C eco	98.6 102.2		101.5	101.8		136.7	153.50		140.4	163.2		149.6	163.2
	C ergo	71.6	88.1		58.6	86.2		5	5.70		0	0		0	0

Table 5 .

 5 Summary results for real case study.

	α		0			0.25			0.5			0.75			0.9
	Modes	3	2	IR	3	2	IR	3	2	IR	3	2	IR	3	2	IR
	Instance								Real case study
	H'	77.3	81.3	4.9%	66.8	75.6	11.6%	49.0	69.4	29.4%	24.7	44.7	44.9%	9.9	18.8	47.4%
	C eco	77.3	81.3		80.2	81.3		95.0	97.9		98.6	117.0		98.6	138.8
	C ergo	49.6	67.8		26.5	53.8		3.0	30.9		0.0	16.5		0.0	2.9

Data availability statement

The data that support the findings of this study are available from the corresponding author, AQ, upon reasonable request.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes on contributors

Anthony Quenehen is an associate professor of Arts et Métiers Institute ofTechnology, Lille campus, where he received in 2022 his PhD degree in collaborative robot implementation in production systems. Prior to his academic career, he had a 20-year experience in automotive industry with Toyota Motors as a senior manager in Quality Control. His research interests cover Lean Manufacturing techniques, Advanced Manufacturing Technologies, and their possible interactions, as well as education science and design based research.

Nathalie Klement is currently an assistant professor in industrial engineering and operational research at Arts et Métiers, Lille in the LISPEN laboratory. She works on optimisation problems using approximate methods, used to solve logistic and industrial problems at all decision levels: system sizing at a strategic level, planning and resource allocation at a tactical level, and scheduling at an operational level. All these problems are solved in today's 4.0 context, for example reconfigurable manufacturing systems to cope with production variability in number and type. Lionel Roucoules is currently professor at Arts et Métiers Institute (ENSAM -France -www.ensam.eu) on the campus of Aix en Provence. He has been promoted in September 2008. He was previously associate professor at the Université de Technologie de Troyes (UTT, France). He has been graduated from the Ecole Normale Supérieure (agrégation) in 1994. He received his PhD in 1999 from the National Polytechnic Institute of Grenoble in collaboration (European label) with the Polytechnic University of Valencia (Spain). Afterwards he spent on year of post-Doctoral study at the University of Twente (Netherlands). The context of his research globally concerns System Engineering process treated as collaborative and integrated activities in PLM context. That provides a new approach for design and decisionmaking rationale, knowledge synthesis embedded in agile by least commitment design thinking approach. This approach is supported by IT Model Based System Engineering to provide Interoperable and Reconfigurable IT system in ever changing industrial context. He is member of the international IFIP WP5.1 and associate member of the CIRP.

Mohamed Amine

Olivier