Aurélien Deya
email: aurelien.deya@univ-lorraine.fr

Massimiliano Gubinelli
email: gubinelli@iam.uni-bonn.de

Martina Hofmanova
email: hofmanov@math.tu-berlin.de

Samy Tindel
email: stindel@purdue.edu

Élien Aur

Massimiliano Deya

Martina Gubinelli

AND Samy Hofmanov Á

Tindel

ONE-DIMENSIONAL REFLECTED ROUGH DIFFERENTIAL EQUATIONS

come

Introduction

In its original formulation [START_REF] Lyons | Differential equations driven by rough signals[END_REF], Lyons' rough paths theory aimed at the study of the standard differential model

dy t = f (y t ) dx t , y 0 = a ∈ R d , t ∈ [0, T ] , (1.1)
where f : R d → L(R N , R d ) is a smooth enough application and x, y : [0, T ] → R m are (typically non-differentiable) continuous paths. In order to deal with the lack of regularity one has to drop both the classical differential or integral formulation of the problem and turn to a description of the motion on arbitrarily small, but finite scales. Eq. (1.1) can be interpreted as the requirement that increments of y should behave locally as some "germ" given by a Taylor-like polynomial approximation of the right hand side. A rough path X constructed above the irregular signal x is the given of the appropriate monomials with which such a local approximation is constructed. One of the key results of the rough path theory is that, under appropriate conditions, only one continuous function y can satisfy all these local constraints. In this case we say that the path y satisfies the rough differential equations (RDEs) (1.1).

While the approach of Lyons [START_REF] Lyons | Differential equations driven by rough signals[END_REF][START_REF] Lyons | System control and rough paths[END_REF][START_REF] Lyons | Differential equations driven by rough paths[END_REF] stresses more the control theoretic sides of the theory, and in particular the mapping from rough paths over x to rough path over y, it has been Davie [START_REF] Davie | Differential Equations Driven by Rough Paths: An Approach via Discrete Approximation[END_REF] who observed the usefulness of these local expansions. Following Davie's insight, one of the author of the present paper [START_REF] Gubinelli | Controlling rough paths[END_REF] introduced a suitable Banach space where these local expansions can be studied efficiently. The work of Friz and Victoir [START_REF] Friz | Multidimensional Stochastic Processes As Rough Paths: Theory and Applications[END_REF] showed also how to systematically generate and analyse the local expansions for (1.1) leading to a very complete theory for RDEs.

It later turned out that these principles, or at least some adaptation of them, remain valid for other -less standard -differential models, such as delay [START_REF] Neuenkirch | Delay equations driven by rough paths[END_REF] or Volterra [START_REF] Deya | Rough Volterra equations 2: Convolutional generalized integrals[END_REF] rough equations and homogeneisation of fast/slow systems [START_REF] Kelly | Deterministic Homogenization for Fast-Slow Systems with Chaotic Noise[END_REF]. The basic idea of local coherent expansions as effective description of rough dynamical systems has been developed more recently in numerous PDE settings (see e.g. [START_REF] Gubinelli | Paracontrolled distributions and singular PDEs[END_REF][START_REF] Hairer | Rough stochastic PDEs[END_REF][START_REF] Hairer | Solving the KPZ equation[END_REF], to mention but a few spin-offs amongst a flourishing literature) leading to the development of the general framework of regularity structures by Hairer [START_REF] Hairer | A theory of regularity structures[END_REF], which allows to handle local expansions of a large class of distributions. For a recent nice introduction to rough path theory and some applications see [START_REF] Friz | A Course on Rough Paths: With an Introduction to Regularity Structures[END_REF]. This being said, in the vast majority of the situations so far covered by rough paths analysis, and especially in all the above quoted references, the success of the method lies in an essential Date: October 24, 2016. Financial support by the DFG via Research Unit FOR 2402 is gratefully acknowledged.

way on fix-point and contraction mapping methods to establish existence and more importantly uniqueness of the object under consideration. Unfortunately, the existence of such a contraction property is not known in the case of the reflected rough differential equation, which we propose to study in this paper. To be more precise, we will focus on the one-dimensional RDE reflected at 0, which can be described as follows: given a time T > 0, a smooth function f : R → L(R N ; R) and a p-variation N -dimensional rough path X with 2 p < 3 (see Definition 1), find an R 0 -valued path y ∈ V p 1 ([0, T ]) and an R 0 -valued increasing function (or "reflection measure")

m ∈ V 1 1 ([0, T ]) that together satisfy dy(t) = f (y(t))dX t + dm t , y t dm t = 0 . (1.2)
Thus, the idea morally is to exhibit a path y that somehow follows the dynamics in (1.1), but is also forced to stay positive thanks to the intervention of some regular "local time" m at 0. Of course, at this point, it is not exactly clear how to understand the right hand side of (1.2), and we shall later give a more specific interpretation of the system, based on rough paths principles (see Definition 2).

The stochastic counterpart of (1.2), where X is a standard N -dimensional Brownian motion and the right hand side is interpreted as an Itô integral, has been receiving a lot of attention since the 60's (see e.g. [START_REF] Lions | Stochastic differential equations with reflecting boundary conditions[END_REF][START_REF] Saisho | Stochastic differential equations for multi-dimensional domain with reflecting boundary[END_REF][START_REF] Skorohod | Stochastic equations for diffusion processes in a bounded region 1, 2[END_REF][START_REF] Tanaka | Stochastic differential equations with reflecting boundary condition in convex regions[END_REF]), with several successive generalizations regarding the (possibly multidimensional) containment domain of y. This Brownian reflected equation has also been investigated more recently through the exhibition of Wong-Zakai-type approximation algorithms [START_REF] Aida | Wong-Zakai approximation of solutions to reflecting stochastic differential equations on domains in Euclidean spaces[END_REF].

When 1 p < 2, Problem (1.
2) can be naturally interpreted and analyzed by means of Young integration techniques. This situation was first considered by Ferrante and Rovira in [START_REF] Ferrante | Stochastic Differential Equations with Non-Negativity Constraints Driven by Fractional Brownian Motion[END_REF] for the d-dimensional positive domain R d ≥0 , with exhibition of an existence result therein. Using some sharp p-variation estimates for the Skorohod map, Falkowski and Slominski [START_REF] Falkowski | Stochastic differential equations with constraints driven by processes with bounded p-variation[END_REF][START_REF] Falkowski | Sweeping processes with stochastic perturbations generated by a fractional Brownian motion[END_REF] have recently provided a full treatment of the Young case (at least when considering reflection on hyperplanes), by proving both existence and uniqueness of the solution.

The more complex rough (or step-2) version of (1.2), which somehow extends the Brownian model, has been first considered by Aida in [START_REF] Aida | Reflected rough differential equations[END_REF], and further analysed by the same author in [START_REF] Aida | Rough differential equations containing path-dependent bounded variation terms[END_REF] for more general multidimensional domains. Nevertheless, in these two references, only existence of a solution to (1.2) can be established and the uniqueness issue is left open. The lack of regularity of the Skorohod map clearly appears as the main obstacle towards a uniqueness result in the approach followed in [START_REF] Aida | Reflected rough differential equations[END_REF][START_REF] Aida | Rough differential equations containing path-dependent bounded variation terms[END_REF].

Our aim in this study is to complete the above picture in the one-dimensional situation, that is to prove uniqueness of a solution to the problem (1.2). Actually, for the reader's convenience, we will also provide a detailed proof of the existence of a solution in this setting, and simplify at the same time some of the arguments used by Aida in [START_REF] Aida | Reflected rough differential equations[END_REF][START_REF] Aida | Rough differential equations containing path-dependent bounded variation terms[END_REF]. The subsequent analysis accordingly offers a thorough -and totally self-contained -proof of well-posedness of the problem (1.2).

The strategy is inspired by the recent results on rough conservation laws [START_REF] Deya | A priori estimates for rough PDEs with application to rough conservation laws[END_REF]. Indeed, there is an analogy between (1.2) and the kinetic formulation of conservation laws where the so-called kinetic measure appears. As for (1.2), this measure is unknown and becomes part of the solution which brings significant difficulties, especially in the proof of uniqueness. The latter is then based on a tensorization-type argument, also known as doubling of variables, and subsequent estimation of the difference of two solutions.

In the case of (1.2), we put forward a fairly simple proof of uniqueness based on a direct estimation of a difference of two solutions. In particular, in this finite dimensional setting no technical tensorization method is needed. The existence is then derived from a compactness result, starting from a smooth approximation of the rough path X. In both cases, the key of the procedure consists in deriving sharp estimates for the remainder term which measures the difference between the (explicit) local expansion and the unknown of the problem. The strategy thus heavily relies on the so-called sewing lemma at the core of the rough paths machinery (see Lemma 1). The estimates on the remainder are then converted via a rough Gronwall lemma (see Lemma 2) into estimates for the unknown (resp. for some function thereof) in order to establish existence (resp. uniqueness).

The paper is organized along a very simple division. In Section 2, we start with a few reminders on the rough paths setting and topologies, which allows us to give a rigorous interpretation of the problem (1.2), as well as the statement of our well-posedness result (Theorem 4). We also introduce the two main technical ingredients of our analysis therein, namely the above-mentioned sewing and Gronwall lemmas, with statements borrowed from [START_REF] Deya | A priori estimates for rough PDEs with application to rough conservation laws[END_REF]. Section 3 is then devoted to the proof of uniqueness, while Section 4 closes the study with the proof of existence. Regarding the latter existence issue, we will first provide an exhaustive treatment of the problem in the one-dimensional situation (the main topic of the paper), and then give a few details on possible extensions of our arguments to more general multidimensional domains (Section 4.2).

Setting and main result

To settle our analysis, we will need the following notations and definitions taken from rough paths theory. First of all, let us recall the definition of the increment operator, denoted by δ. If g is a path defined on [0, T ] and s, t ∈ [0, T ] then δg st := g t -g s , if g is a 2-index map defined on [0, T ] 2 then δg sut := g st -g su -g ut . For g : [0, T ] → E and ϕ : E → F (with E, F two Banach spaces), we will also use the convenient notations ϕ (g) st := Observe in particular that if ϕ is a smooth enough mapping, then δϕ(g) st = ∇ϕ (g) st δg st and ϕ (g) st -ϕ(g s ) = ∇ϕ (g) st .

(2.2)

In the sequel, given an interval I we call a control on I (and denote it by ω) any superadditive map on S I := {(s, t) ∈ I 2 : s t}, that is, any map ω : S I → [0, ∞[ such that, ω(s, u) + ω(u, t) ω(s, t), s u t.

We will say that a control ω is regular if lim |t-s|→0 ω(s, t) = 0. Also, given a control ω on a time interval I = [a, b], we will use the notation ω(I) := ω(a, b). Now, given a time interval I, a parameter p > 0, a Banach space E and a function g : S I → E, we define the p-variation norm of g as g V p 2 (I;E) := sup

(t i )∈P(I) i |g t i t i+1 | p 1 p
, where P(I) denotes the set of all partitions of the interval I, and we denote by V p 2 (I; E) the set of maps g : S I → E for which this quantity is finite. In this case,

ω g (s, t) := g p V p 2 ([s,t];E)
defines a control on I, and we denote by V p 2 (I; E) the set of elements g ∈ V p 2 (I; E) for which ω g is regular on I. We then denote by V p 1 (I; E), resp. V p 1 (I; E), the set of paths g :

I → E such that δg ∈ V p 2 (I; E), resp. δg ∈ V p 2 (I; E)
. Finally, we define the space V p 2,loc (I; E) of maps g : S I → E such that there exists a countable covering {I k } k of I satisfying g ∈ V p 2 (I k ; E) for every k. We write g ∈ V p 2,loc (I; E) if the related controls can be chosen regular. Definition 1. Fix a time T > 0 and let N 1, 2 p < 3. Then we call a continuous N -dimensional p-variation rough path on [0, T ] any pair

X = (X 1 , X 2 ) ∈ V p 2 ([0, T ]; R d ) × V p/2 2 ([0, T ]; R d,d ) (2.3)
that satisfies the relation

δX 2;ij sut = X 1,i su X 1,j ut , s < u < t ∈ [0, T ] , i, j ∈ {1, . . . , d} . (2.4)
Such a rough path X is said to be geometric if it can be obtained as the limit, for the p-variation topology involved in ( 2.3), of a sequence of smooth rough paths

(X ε ) ε>0 , that is with X ε = (X ε,1 , X ε,2 ) explicitly defined as X ε,1,i st := δx ε,i st , X ε,2,ij st := t s δx ε,i su dx ε,j u ,
for some smooth path

x ε : [0, T ] → R N .
We are now in a position to provide a clear interpretation of the problem (1.2).

Definition 2. Given a time T > 0, a real a ≥ 0, a differentiable function f : R → L(R N ; R) and a p-variation N -dimensional rough path X with 2 p < 3, a pair (y, m) ∈ V p 1 ([0, T ]; R 0 ) × V 1 1 ([0, T ]; R 0 ) is said to solve the problem ( 1.2) on [0, T ] with initial condition a if there exists a 2-index map y ∈ V p/3 2,loc ([0, T ]; R) such that for all s, t ∈ [0, T ], we have      δy st = f i (y s )X 1,i st + f 2,ij (y s )X 2,ij st + δm st + y st y 0 = a and m t = t 0 1 {yu=0} dm u , (2.5)
where we have set f 2,ij (ξ) := f i (ξ)f j (ξ) and m([0, t]) := m t .

Remark 3. Eq. (2.5) should be read as the given of a local expansion of the function y: it says that around each time point s the function can be approximated by the germ

t → y s + f (y s )X 1 st + f 2 (y s )X 2
st + δm st up to terms of order ω(s, t) p/3 where ω is a control. The term δm st is characteristic for this reflected problem: the measure m increases only at times u where y u = 0 effectively "kicking" the path y away from the negative axis. In some sense it can be considered as a Lagrange multiplier enforcing the constraint

y u ≥ 0 for all u ∈ [0, T ].
With this interpretation in hand, our well-posedness result reads as follows:

Theorem 4. Let T > 0 and a > 0. If f ∈ C 3 b (R; L(R N , R)), that is if f is 3-time differentiable,
bounded with bounded derivatives, and if X is a continuous geometric N -dimensional p-variation rough path, then Problem (1.2) admits a unique solution (y, m) on [0, T ] with initial condition a.

Let us conclude this preliminary section with a presentation of the two main technical results that will be used in our analysis, and the proofs of which are elementary and can be found e.g. in [START_REF] Deya | A priori estimates for rough PDEs with application to rough conservation laws[END_REF].

Lemma 1 (Sewing lemma). Fix an interval I, a Banach space E and a parameter ζ > 1. Consider a map G : I 3 → E such that G ∈ Im δ and for every s < u < t ∈ I,

|G sut | ω(s, t) ζ ,
for some regular control ω on I. Then there exists a unique element

ΛG ∈ V 1/ζ 2 (I; E) such that δ(ΛG) = G and for every s < t ∈ I, |(ΛG) st | C ζ ω(s, t) ζ , (2.6)
for some universal constant C ζ .

Lemma 2 (Rough Gronwall Lemma). Fix a time horizon T > 0 and let g : [0, T ] → [0, ∞) be a path such that for some constants C, L > 0, κ 1 and some controls ω 1 , ω 2 on [0, T ] with ω 1 being regular, one has δg st C( sup

0 r t g r ) ω 1 (s, t) 1 κ + ω 2 (s, t), (2.7)
for every s < t ∈ [0, T ] satisfying ω 1 (s, t) L. Then it holds

sup 0 t T g t 2e c L,κ ω 1 (0,T ) g 0 + sup 0 t T (ω 2 (0, t)e -c L,κ ω 1 (0,t)) ,
where c L,κ is defined as

c L,κ = sup 1 L , (2Ce 2 ) κ . (2.8) 3. Uniqueness Theorem 5. Let T > 0 and a > 0. If f ∈ C 3 b (R; L(R N , R)) and X is an N -dimensional p-variation rough path, then Problem (1.
2) admits at most one solution (y, m) on [0, T ] with initial condition a.

Proof. Let (y, µ) and (z, ν) be two solutions for (2.5). Set Y := (y, z) ∈ V p 1 ([0, T ]; R 2 ) and with decomposition (2.5) in mind, write

δY st = F i (Y s )X 1,i st + F 2,ij (Y s )X 2,ij st + δM st + Y st , 0 s t T. (3.1)
where we use the shorthands

F i (Y ) := (f i (y), f i (z)), F 2,ij (Y ) := (f 2,ij (y), f 2,ij (z)), M := (µ, ν) ∈ V 1 1 ([0, T ]; R 2 ) and Y := (y , z ) ∈ V p/3 2,loc ([0, T ]; R 2 ).
From now non and until the end of the proof, we fix an interval

I ⊂ [0, T ] such that Y ∈ V p/3
2 (I; R 2 ) and consider the following controls on

I: ω Y (s, t) := Y p V p 1 ([s,t]) , ω Y, (s, t) := Y p/3 V p/3 2 ([s,t]) , ω ∆ (s, t) := y -z p V p 1 ([s,t]) , ω ∆, (s, t) := y -z p/3 V p/3 2 ([s,t]) , ω M (s, t) := M V 1 1 ([s,t]) = µ V 1 1 ([s,t]) + ν V 1 1 ([s,t]) .
Without loss of generality, we will assume that ω X (I) 1, where ω X is a fixed control such that

|X 1 s,t | + |X 2 s,t | 1/2 ω X (s, t) 1/p , 0 s t T .
Now, consider a smooth function ϕ : R → R 0 and set h(x 1 , x 2 ) := ϕ(x 1 -x 2 ) for all x 1 , x 2 ∈ R.

A direct computation via Taylor expansion, combined with (3.1), shows that

δh(Y ) st = ∇h (Y ) st δY st = H i (Y s )X 1,i st + H 2,ij (Y s )X 2,ij st + t s ϕ (y u -z u )(dµ u -dν u ) + h st (3.2)
where h is a map in V p/3 2 (I; R), and where we have set, for all Y = (y, z) ∈ R 2 ,

H i (Y ) := ∇h(Y )F i (Y ) = ϕ (y -z)(f i (y) -f i (z)) (3.3) H 2,ij (Y ) := ∇H i (Y )F j (Y ) = ϕ (y -z)(f 2,ij (y) -f 2,ij (z)) + ϕ (y -z)(f i (y) -f i (z))(f j (y) -f j (z)) . (3.4)
Step 1: A general estimate on h . Given that h is a remainder term, we wish to use the sewing map to estimate it. Applying δ to eq. (3.2) and using (2.4), we get, for 0 s u t T :

δh sut = δH i (Y ) su X 1,i ut -H 2,ij (Y s )X 1,j su X 1,i ut + δH 2,ij (Y ) su X 2,ij ut = δH i (Y ) su -H 2,ij (Y s )X 1,j su X 1,i ut + δH 2,ij (Y ) su X 2,ij ut . (3.5)
We need to expand the quantity δH i (Y ) su -H 2,ij (Y s )X 1,j su in (3.5), in order to show that h is suitably small and depends in a very precise way on ϕ and on the difference ∆ := y -z. In fact, by Taylor expansion and using (3.1) we get

δH i (Y ) su -H 2,ij (Y s )X 1,j su = ∇H i (Y ) su δY su -H 2,ij (Y s )X 1,j su = ∇H i (Y ) su F j (Y s )X 1,j su + ∇H i (Y ) su F 2,jk (Y s )X 2,jk su + ∇H i (Y ) su Y su + ∇H i (Y ) su δM su -H 2,ij (Y s )X 1,j su = ( ∇H i (Y ) su -∇H i (Y s ))F j (Y s )X 1,j su + ∇H i (Y ) su F 2,jk (Y s )X 2,jk su + ∇H i (Y ) su Y su + ∇H i (Y ) su δM su , since H 2,ij (Y ) = ∇H i (Y )F j (Y ).
Plugging this identity back into equation (3.5) and neglecting to write down explicitly the time indexes, we end up with:

δh = ( ∇H i (Y ) -∇H i (Y ))F j (Y )X 1,j X 1,i + ∇H i (Y )F 2,jk (Y )X 2,jk X 1,i + ∇H i (Y )Y X 1,i + ∇H i (Y ) δM X 1,i + δH 2,ij (Y )X 2,ij . (3.6)
Using elementary algebraic manipulations, as well as the relation H 2,ij (Y ) = ∇H i (Y )F j (Y ), we obtain:

( ∇H i (Y ) su -∇H i (Y s ))F j (Y s ) = ( H 2,ij (Y ) su -H 2,ij (Y s )) + ( ∇H i (Y ) su F j (Y s ) -H 2,ij (Y ) su ) = ∇H 2,ij (Y ) su δY su -∇H i (•) ∇F j (•) (Y ) su δY su ,
where the identity

H 2,ij (Y ) su -H 2,ij (Y s ) = ∇H 2,ij (
Y ) su δY su directly stems from (2.2), and where we define:

∇H i (•) ∇F j (•) (Y ) su := 1 0 ∇H i (Y s + τ δY su ) τ 0 ∇F j (Y s + σδY su ) dσdτ .
Therefore, we can rewrite eq. (3.6) as

δh = ∇H 2,ij (Y )δY X 1,j X 1,i -∇H i (•) ∇F j (•) (Y )δY X 1,j X 1,i + ∇H i (Y )F 2,jk (Y )X 2,jk X 1,i + ∇H i (Y )Y X 1,i + ∇H i (Y )δM X 1,i + ∇H 2,ij (Y )δY X 2,ij . (3.7)
In order to further evaluate the rhs of this relation in terms of the test function ϕ, let us write explicit expressions for the gradients ∇H i (Y ) and

∇H 2,ij (Y ) computed at (a, b) ∈ R 2 : ∇H i (Y )(a, b) = ϕ (y -z)(f i (y) -f i (z))(a -b) + ϕ (y -z)(f i (y) -f i (z))a +ϕ (y -z)f i (z)(a -b)
and

∇H 2,ij (Y )(a, b) = ϕ (y -z)(f 2,ij (y) -f 2,ij (z))(a -b) + ϕ (y -z)(f 2,ij (y)a -f 2,ij (z)b) +ϕ (y -z)(f i (y) -f i (z))(f j (y) -f j (z))(a -b) +2ϕ (y -z)(f i (y) -f i (z))(f j (y)a -f j (z)b) .
At this point, consider the quantity

|||ϕ||| := sup y,z∈R (|ϕ (y -z)| + |y -z||ϕ (y -z)| + |y -z| 2 |ϕ (y -z)|) . (3.8)
Then, denoting by C f any quantity that only depends on f , we have for all 1 ≤ i, j, k ≤ N and s < t ∈ I,

| ∇H i (Y ) st | + | ∇H 2,ij (Y ) st | C f |||ϕ||| | ∇H i (Y ) st F 2,jk (Y s )| C f |||ϕ||| y -z ∞;[s,t] | ∇H i (•) ∇F j (•) (Y ) st δY st | C f |||ϕ||| (ω ∆ (s, t) 1/p + y -z ∞;[s,t] ω Y (s, t) 1/p ) | ∇H 2,ij (Y ) st δY st | C f |||ϕ||| (ω ∆ (s, t) 1/p + y -z ∞;[s,t] ω Y (s, t) 1/p ) | ∇H i (Y ) st Y st | C f |||ϕ||| (ω ∆, (s, t) 3/p + y -z ∞;[s,t] ω Y, (s, t) 3/p ) .
Going back to (3.7), we get that for all s < u < t ∈ I,

|δh sut | C f |||ϕ||| ω (s, t) + ω X (s, t) 2/3 ω ∆ (s, t) 1/3 + ω X (s, t) 1/3 ω ∆, (s, t) 3/p
where ω is the control on I given for every s < t ∈ I by

ω (s, t) := ω M (s, t) p/3 ω X (s, t) 1/3 + y -z p/3 ∞;[s,t] ω X,Y (s, t) ,
with ω X,Y (s, t) := ω X (s, t) + ω Y (s, t) 1/3 ω X (s, t) 2/3 + ω Y, (s, t) .

We are therefore in a position to apply the sewing lemma and conclude that for all s < t ∈ I,

|h st | C f,p |||ϕ||| ω (s, t)
+ ω X (s, t) 2/3 ω ∆ (s, t) 1/3 + ω X (s, t) 1/3 ω ∆, (s, t) 3/p (3.9)

for some quantity C f,p that only depends on f and p.

Step 2: A first application. Our aim now is to apply the previous bound to the non-smooth function ϕ(ξ) = ϕ 0 (ξ) := |ξ|. To this end, we will rely on the smooth approximation ϕ ε defined for ε > 0 as ϕ ε (ξ) = ε 2 + |ξ| 2 for all ξ ∈ R. Let us denote the associated objects with

h ε , h ε , H ε,i , H ε,2,ij ,.... In this case |ϕ ε (ξ)| 1 , |ϕ ε (ξ)| 1/ ε 2 + |ξ| 2 , |ϕ ε (ξ)| 3/(ε 2 + |ξ| 2 )
and so, with the notation (3.8), we have the uniform estimate |||ϕ ε ||| 3. Plugging this estimate into (3.9), we get: where the first two limits are simple limits of functions, and the last one is understood in the weak sense. Notice that we also use the convention sign(0) = 0 above.

|h ε,st | C f,p ω (s, t) + ω X (s, t) 2/3 ω ∆ (s, t) 1/3 + ω X (s, t) 1/3 ω ∆, (s, t)
With those preliminaries in mind, let us take limits in (3.2). To begin with, as ε → 0, a standard dominated convergence argument and relation (3.11) yield:

t s ϕ ε (y u -z u )d(µ u -ν u ) → t s sign(y u -z u )d(µ u -ν u ).
(3.12)

In addition, owing to the fact that y t 0, z t 0, we have

t s sign(y u -z u )d(µ u -ν u ) = t s 1 {yu>zu 0} dµ u - t s 1 {zu>yu 0} dµ u - t s 1 {yu>zu 0} dν u + t s 1 {zu>yu 0} dν u .
Hence, using the conditions µ t = t 0 1 {yu=0} dµ u , ν t = t 0 1 {zu=0} dν u , we end up with:

t s sign(y u -z u )d(µ u -ν u ) = - t s 1 {zu>yu 0} dµ u + t s 1 {yu>zu 0} dν u = - t s 1 {zu>yu 0} d(µ u + ν u ) + t s 1 {yu>zu 0} d(µ u + ν u ) = - t s 1 {yu =zu} d(µ u + ν u ) = -ω M (s, t) + t s 1 {yu=zu} d(µ u + ν u ) . ( 3
H ε,i (Y ) = Ψ i (Y ) , and lim ε→0 H ε,2,ij (Y ) = Ψ 2,ij (Y ) , (3.14)
where the limits are simple limits of functions and where we have:

Ψ i (Y ) := sgn(y -z)(f i (y) -f i (z)) , Ψ 2,ij (Y ) := sgn(y -z)(f 2,ij (y) -f 2,ij (z)) .
Taking relations (3.12), (3.13) and (3.14) into account, we can now take limits as ε → 0 in (3.2). This ensures the convergence of the quantity h ε,st to some limit Φ st (for all s < t ∈ I), and using (3.9) we get that the path Φ(Y ) := |y -z| satisfies the following equation:

δΦ(Y ) st = Ψ i (Y s )X 1,i st + Ψ 2,ij (Y s )X 2,ij st -ω M (s, t) + t s 1 {yu=zu} d(µ u + ν u ) + Φ st . (3.15)
Moreover, invoking relation (3.10), we have for all s < t ∈ I:

|Φ st | C f,p ω (s, t) + ω X (s, t) 2/3 ω ∆ (s, t) 1/3 + ω X (s, t) 1/3 ω ∆, (s, t) 3/p . (3.16)
Here and in the sequel, we denote by C f,p any quantity that only depends on f and p.

Step 3: Bounds for ω ∆ and ω ∆, . Let us now estimate ω ∆ and ω ∆, in terms of ω . To this end, we can first use the fact that the path ∆ := y -z is (obviously) given by h(Y ) with the choice ϕ(ξ) = ψ(ξ) := ξ. In this case h = y -z , |||ψ||| = 1, so that (3.9) becomes |y st -z st | C f,p ω (s, t) + ω X (s, t) 2/3 ω ∆ (s, t) 1/3 + ω X (s, t) 1/3 ω ∆, (s, t) 3/p for all s < t ∈ I, and accordingly we have

ω ∆, (s, t) C (1)
f,p ω (s, t) + ω X (s, t) 2/3 ω ∆ (s, t) 1/3 + ω X (s, t) 1/3 ω ∆, (s, t) for some fixed constant C

f,p . As a result, for any interval I 0 ⊂ I satisfying

C (1) f,p ω X (I 0 ) 1/3 1/2 , (3.17)
and for all s < t ∈ I 0 , we have ω ∆, (s, t) 2C

f,p ω (s, t) + ω X (s, t) 2/3 ω ∆ (s, t) 1/3 .

(3.18)

Besides, going back to the equation satisfied by ∆ (again, take ϕ(ξ) = ξ in (3.2)), we easily obtain that for all s < t ∈ I,

|δ∆ st | C f,p y -z p ∞;[s,t] ω X (s, t) + ω M (s, t) p + ω ∆, (s, t) 3 1/p , so ω ∆ (s, t) C f,p y -z p ∞;[s,t] ω X (s, t) + ω M (s, t) p + ω ∆, (s, t) 3
and for any interval I 0 ⊂ I satisfying (3.17), we get by (3.18)

ω ∆ (s, t) C (2) f,p y -z p ∞;[s,t] ω X (s, t) + ω M (s, t) p + ω (s, t) 3 + ω X (s, t) 2 ω ∆ (s, t) ,
for some constant C

(2)

f,p . Finally, for any interval I 0 ⊂ I satisfying both (3.17) and

C (2) f,p ω X (I 0 ) 2 1/2 , (3.19)
and for all s < t ∈ I 0 , we have

ω ∆ (s, t) 2C (2) f,p y -z p ∞;[s,t] ω X (s, t) + ω M (s, t) p + ω (s, t) 3 .
(3.20)

Step 4: Conclusion. By injecting (3.18) and (3.20) into (3.16), we can derive the following assertion: for any interval I 0 ⊂ I satisfying (3.17) and (3.19), and all s < t ∈ I 0 , it holds that

|Φ st | C f,p ω (s, t) 3/p + y -z ∞;[s,t] ω X (s, t) 3/p + ω M (s, t)ω X (s, t) 2/p ,
which, by the definition of ω , gives

|Φ st | C f,p y -z ∞;[s,t] ω X,Y (s, t) 3/p + ω M (s, t)ω X (s, t) 1/p .
Going back to eq. (3.15) and observing that y -z ∞;[s,t] = sup [s,t] Φ(Y ), we obtain that for any such interval I 0 and for all s < t ∈ I 0 ,

δΦ(Y ) st +ω M (s, t) C f,p sup [s,t] Φ(Y )+ω M (s, t) ω X (s, t)+ω X,Y (s, t) 3 1/p + t s 1 {yu=zu} (dµ u +dν u ) .
We are finally in a position to apply the Rough Gronwall Lemma 2 with ω 1 := ω X + ω 3 X,Y and ω 2 (s, t) := t s 1 {yu=zu} (dµ u + dν u ), and assert that for every 0 s < t T , sup

[s,t] Φ(Y ) + ω M (s, t) C f,p,X,Y Φ(Y s ) + t s 1 {yu=zu} (dµ u + dν u ) , that is sup r∈[s,t] |y r -z r | + ω M (s, t) C f,p,X,Y |y s -z s | + t s 1 {yu=zu} (dµ u + dν u ) ,
for some constant C f,p,X,Y .

Assume now that [s, t] is an interval where y = z in (s, t) but y(s) = z(s). Then Just as in [START_REF] Aida | Reflected rough differential equations[END_REF][START_REF] Aida | Rough differential equations containing path-dependent bounded variation terms[END_REF], our strategy towards existence will appeal to some a priori bound on the measure term of the (approximated) equation. The result more generally applies to the so-called Skorohod problem and it can be read as follows in the one-dimensional case. Lemma 3. Let g be a continuous R-valued path defined on some interval I = [ 1 , 2 ], and consider a solution

(y, m) ∈ C(I; R 0 ) × V 1 1 (I; R 0 ) of the Skorohod problem associated with g in the domain R 0 , that is (y, m) satisfies for all s < t ∈ I      δy st = δg st + δm st , y 1 = g 1 , m t = t 0 1 {yu=0} dm u .
Then for all s < t ∈ I it holds that

δm st 8 g 0,[s,t] , (4.1)
where g 0,[s,t] := sup s u<v t |δg uv |.

The proof of (4.1) can be easily derived from the arguments of the proof of [3, Lemma 2.3] (namely, the same arguments as those leading to the forthcoming general Lemma 4). Let us provide some details though, not least to give the non-initiated reader an insight on how the specific constraints of the reflecting problem can be exploited.

Proof of Lemma 3. For all s < t ∈ I, one has

|δy st | 2 = |δg st | 2 + |δm st | 2 + 2δg st δm st = |δg st | 2 + 2 t s δm su dm u + 2 t s δg st dm u = |δg st | 2 + 2 t s δy su dm u + 2 t s δg ut dm u ,
where we have just used the fact that δm su = δy su -δg su for the last identity. Moreover, since t s y u dm u = t s y u 1 {yu=0} dm u = 0 and y s 0, we get:

|δy st | 2 |δg st | 2 + 2 t s δg ut dm u .
Therefore,

|δy st | 2 g 2 0,[s,t] + 2 g 0,[s,t] δm st 5 g 2 0,[s,t] + 1 4 |δm st | 2 ,
and so y 0,[s,t] 3 g 0,[s,t] + 1 2 δm st . Finally,

δm st y 0,[s,t] + g 0,[s,t] 4 g 0,[s,t] + 1 2 δm st ,
and the result follows.

Proof of Theorem 6. We start from a sequence of smooth rough paths X ε converging to X as ε → 0, in the space of continuous p-variation geometric rough paths. We can then find a regular control ω X such that, for all s, t ∈ [0, T ],

|X 1 st | + |X 2 st | 1/2 ω X (s, t) 1/p , sup ε>0 (|X ε,1 st | + |X ε,2 st | 1/2 ) ω X (s, t) 1/p .
For every ε > 0, let X ε be the path which corresponds to X ε and consider the solution y ε to reflected ODEs starting from y 0 :

     dy ε t = f (y ε t ) dX ε t + dm ε t y ε 0 = y 0 and m ε t = t 0 1 {y ε u =0} dm ε u .
Recall that the existence (and uniqueness) of such a solution is a standard result, based on the Lipschitz regularity of the Skorohod map with respect to the supremum norm. Then by Taylor expansion it is not difficult to show that these solutions correspond to rough solutions (y ε , m ε ) in the sense of (2.5)

δy ε st = f i (y ε s )X ε,1,i st + f 2,ij (y ε s )X ε,2,ij st + δm ε st + y ε, st s, t ∈ [0, T ] (4.2)
where y ε, ∈ V p/3

2 ([0, T ]; R). Let us set from now on

ω y ε (s, t) := y ε p V p 1 ([s,t];E) , ω ε, (s, t) := y ε, p/3 V p/3 2 ([s,t];E) , ω m ε (s, t) := m ε V 1 1 ([s,t];E) = δm ε st = m ε ([s, t]
) , and observe that from eq. ( 4.2) we have

|δy ε st | C f (ω X (s, t) 1/p + ω X (s, t) 2/p ) + ω m ε (s, t) + ω ε, (s, t) 3/p . (4.3)
Here and in the sequel, we denote by C f , resp. C f,p , any quantity that only depends on f , resp. (f, p).

Step 1: Bounds on the approximate solutions. We would like to pass to the limit in ε and obtain solutions of the limiting problem. In order to do so we need uniform estimates for y ε, st . They are obtained via an application of the sewing map.

To this end, one can proceed as in the proof of Theorem 5, Step 1. Specifically, we can just replace Y by y, H by f and H 2 by f 2 in relation (3.5). We then repeat all the steps up to relation (3.7), which yields the following relation for δy ε, (for more simplicity, we neglect to write down the time indexes explicitly):

δy ε, = ∇f 2,ij (y ε )δy ε X ε,1,j X ε,1,i -∇f i (•) ∇f j (•) (y ε )δy ε X ε,1,j X ε,1,i + ∇f i (y ε )f 2,jk (y ε )X ε,2,jk X ε,1,i + ∇f i (y ε )y ε, X ε,1,i + ∇f i (y ε )δm ε X ε,1,i + ∇f 2,ij (y ε )δy ε X ε,2,ij . (4.4)
Combining this expansion with (4.3), we get, for every interval I ⊂ [0, T ] such that ω X (I) 1 and all s < u < t ∈ I,

|δy ε, sut |
C f ω y ε (s, t) 1/p ω X (s, t) 2/p + (ω X (s, t) 2/p + ω m ε (s, t) + ω ε, (s, t) 3/p )ω X (s, t) 1/p C f,p ω X (s, t) + ω X (s, t) 1/3 ω m ε (s, t) p/3 + ω X (s, t) 1/3 ω ε, (s, t) 3/p .

We are therefore in a position to apply the sewing lemma and assert that for every interval I ⊂ [0, T ] such that ω X (I) 1 and all s < t ∈ I, we have |y ε, st | C f,p ω X (s, t) + ω X (s, t) 1/3 ω m ε (s, t) p/3 + ω X (s, t) 1/3 ω ε, (s, t) 3/p , which immediately entails that ω ε, (s, t) C

(1)

f,p ω X (s, t) + ω X (s, t) 1/3 ω m ε (s, t) p/3 + ω X (I) 1/3 ω ε, (s, t) , for some constant C

f,p . As a result, for every interval I ⊂ [0, T ] such that ω X (I) 1 and C (1) f,p ω X (I) 1/3 1/2 , (

one has ω ε, (s, t) 2C

(1)

f,p ω X (s, t) + ω X (s, t) 1/3 ω m ε (s, t) p/3 , s < t ∈ I. (4.6)
Step 2: Control of the approximate measures. Consider the path g ε : [0, T ] → R defined as g ε t := y ε t -m ε t , and observe that (y ε , m ε ) is then a solution of the Skorohod problem in R 0 associated with g ε , in the sense of Lemma 3. Therefore, by (4.1), it holds that

ω m ε (s, t) 8 g ε 0,[s,t] . (4.7)
On the other hand, from eq. ( 4.2), we have

δg ε st = f i (y ε s )X ε,1,i st + f 2,ij (y ε s )X ε,2,ij st + y ε, st , 0 s t T ,
and so

g ε 0,[s,t] C f ω X (s, t) 1/p + ω X (s, t) 2/p + ω ε, (s, t)) 3/p . (4.8)
Injecting successively (4.8) and (4.6) into (4.7) yields that for every interval I satisfying the conditions in (4.5) and every s < t ∈ I,

ω m ε (s, t) C (2)
f,p ω X (s, t) 1/p + ω X (I) 1/p ω m ε (s, t) , for some constant C

f,p , and so, if we assume in addition that

C (2) f,p ω X (I) 1/p 1/2 , (4.9)
we obtain ω m ε (s, t) 2C

f,p ω X (s, t) 1/p , s < t ∈ I . (4.10)

From here we can easily conclude that

ω m ε ([0, T ]) C f,p,X (4.11)
for some quantity C f,p,X independent from ε.

Step 3: Passage to the limit for the measure. With all the bounds in place we can now pass to the limit as ε → 0 via subsequences. We start with the measure. Using (4.11) we can assert that there exists a weakly convergent subsequence of measures (m ε(k) ) k 1 on [0, T ], and we will denote by m their limit. Then it holds that

m([0, t]) = lim k m ε(k) ([0, t]) t ∈ C (4.12)
where C ⊆ [0, T ] is the (dense) set of continuity points of the function t → m([0, t]). Now consider any interval I satisfying both the conditions in (4.5) and in (4.9), and for s < t ∈ I, introduce a sequence s , resp. t , of points in C decreasing to s, resp. increasing to t, and such that s k < t k . Using (4.10), we have

m(]s , t ]) = lim k m ε(k) (]s , t ]) C f,p ω X (s, t) 1/p ,
and so m([s, t]) C f,p ω X (s, t) 1/p , which proves that the function m t := m([0, t[) is continuous and accordingly that m ∈ V 1 1 ([0, T ]; R 0 ), as expected.

Step 4: Passage to the limit for the path. Consider the subsequence (y ε(k) , m ε(k) ) k as defined in the previous step. Using (4.3) we have, for all s, t ∈ [0, T ], lim sup

k |δy ε(k) st | C f (ω X (s, t) 1/p + ω X (s, t) 2/p ) + ω m (s, t) + lim sup k ω ε(k), (s, t) 3/p ,
and for every interval I satisfying both the conditions in (4.5) and in (4.9) (we denote J the family of such intervals), we have lim sup

k ω ε(k), (s, t) C f,p ω X (s, t) + ω X (s, t) 1/3 ω m (s, t) p/3 , s < t ∈ I .
From this bound we can choose a further subsequence, still called (y ε(k) , m ε(k) ) k so that y ε(k) → y in C([0, T ]; R 0 ). It is easy now to pass to the limit in eq. ( 4.2) and conclude that there exists a map y : ∆

[0,T ] → R such that δy = f i (y)X 1,i + f 2,ij (y)X 2,ij + δm + y , and
|y st | C f,p ω X (s, t) + ω X (s, t) 1/3 ω m (s, t) p/3 3/p , s < t ∈ I ∈ J .
The fact that m t = t 0 1 {yu=0} dm u (for all t) follows immediately from the relation m ε t = t 0 1 {y ε u =0} dm ε u , and finally the pair (y, m) does define a solution to the RRDE (2.5).

4.2.

Generalization to multidimensional domains. We conclude this study with a few details on possible extensions of the previous arguments (towards existence) to more general multidimensional domains. Together, these results will thus offer a simplification of some of the arguments and topologies used in [START_REF] Aida | Reflected rough differential equations[END_REF][START_REF] Aida | Rough differential equations containing path-dependent bounded variation terms[END_REF].

Let us first extend Definition 2 of a reflected rough solution to more general settings, along the classical approach of the reflected problem. Let D ⊂ R d be a connected domain and for every x ∈ ∂D, denote by N x the set of inward unit normal vectors at x, that is

N x := ∪ r>0 N x,r , N x,r := {n ∈ R d : |n| = 1, B(x -rn, r) ∩ D = ∅} where B(z, r) := {y ∈ R d : |y -z| < r}, for z ∈ R d and r > 0. Definition 7. Given a time T > 0, an element a ∈ D, a differentiable function f : R d → L(R N ; R d ) and a p-variation N -dimensional rough path X with 2 p < 3, a pair (y, m) ∈ V p 1 ([0, T ]; D) × V 1 1 ([0, T ]; R d ) is said to solve the reflected rough equation in D with initial con- dition a if there exists a 2-index map y ∈ V p/3 2,loc ([0, T ]; R d ) such that for all s, t ∈ [0, T ], we have      δy st = f i (y s )X 1,i st + f 2,ij (y s )X 2,ij st + δm st + y st y 0 = a and m t = t 0 1 {yu∈∂D} n yu d|m| u , (4.13)
where we have set

f 2,ij (ξ) := ∇f i (ξ)f j (ξ), |m| t := m V 1 1 ([0,t];R d ) and for each y ∈ ∂D, n y ∈ N y .
The existence of a solution for (4.13) can actually be derived from the same arguments as in the one-dimensional situation. The only step of the procedure needing for a revision is the so-called

Step 2, since it involves the a priori bound (4.1) which is specific to the one-dimensional Skorohod problem. To this end, we shall exploit the following (sophisticated) substitute, borrowed from [1, Lemma 2.2]. Lemma 4. Let D ⊂ R d be connected domain that satisfies the two following assumptions:

(A) There exists a constant r 0 > 0 such that N x = N x,r 0 = ∅ for any x ∈ ∂D ;

(B) There exist constants δ 0 > 0 and β 1 satisfying: for every x ∈ ∂D, there exists a unit vector l x such that l x , n 1/β for every n ∈ ∪ y∈B(x,δ 0 )∩∂D N y .

Let g ∈ V p 1 (I; R d ), for some interval I = [ 1 , 2 ], such that g 1 ∈ D, and consider a solution Remark 9. Of course, Theorem 6 can retrospectively be obtained as a particular application of Theorem 8. Nevertheless, we have found it important, for pedagogical reasons, to first provide a full and self-contained treatment of the one-dimensional situation.

(y, m) ∈ C(I; D) × V 1 1 (I; R d )
Proof. As mentionned above, and apart from minor changes of notation due to the vectorial character of the equation, Steps 1, 3 and 4 of the proof of Theorem 6 can be readily transposed to this setting, and thus we only need to focus on the extension of Step 2.

In fact, with the same notations as in the one-dimensional proof and considering only those intervals I = [s 0 , t 0 ] satisfying the two conditions in (4.5), we have by (4.14), (4.8) and (4.6) that for all s < t ∈ I, ω m ε (s, t) Ψ(ω g ε ,ε (s, t)) Ψ(C f,p (ω X (s, t) + ω X (s, t)ω m ε (s, t) p )), (4.15) where Ψ(λ) := C 1 [e pC 2 (1+λ 1/p ) λ + 1](e C 2 (1+λ 1/p ) + 1)λ 1/p and C f,p is a fixed constant. Eq. (4.15) implies in particular that the control ω m ε is regular if ω X is regular, which is our case. Let G I be the function G I (λ) := Ψ(C f,p (1 + ω X (I)λ p )). By choosing t 0 near to s 0 we can have both (4.5) and G I (3G I (0)) 2G I (0), since ω X (I) → 0 as t 0 ↓ s 0 . This choice of t 0 depends only on ω X and G I (0) (which is actually independent of I). Now eq. (4.15) implies also that ω m ε (s 0 , t) G I (ω m ε (s 0 , t)) , t ∈ I .

We want to establish that ω m ε (I) 2G I (0) and to this end we can apply the method of continuity. Let A ⊆ I be the set of t ∈ I such that the property ω m ε (s 0 , t) 2G I (0) is true. Note that [s 0 , s 0 + δ] ⊆ A for δ small enough by the continuity of the control ω m ε . Moreover A is closed in I since if (t n ) n ⊆ A is a sequence converging to t * then, again by regularity of ω m ε we have ω m ε (s 0 , t * ) = lim n ω m ε (s 0 , t n ) 2G I (0). Finally A is also open in I since if t * ∈ A then for δ small enough ω m ε (s 0 , t) 3G I (0) for all t ∈ (t * -δ, t * + δ) ∩ I. But then our choice of I guarantee that ω m ε (s 0 , t) G I (ω m ε (s 0 , t)) G I (3G I (0)) 2G I (0) , t ∈ (t * -δ, t * + δ) ∩ I , from which we see that (t * -δ, t * + δ) ∩ I ⊆ A and that A is open in I. We can then conclude that A = I, namely that ω m ε (I) 2G I (0). Now we can reason in this way for any nonempty interval I t,δ = (t-δ, t+δ)∩[0, T ] by choosing δ = δ(t) > 0 small enough to satisfy our conditions. In this way we construct an open covering ∪ t I t,δ(t) of [0, T ] from which we can extract a finite covering (I k ) k independent of ε and such that

ω m ε (I k ) 2G I (0)
for all I k in the covering. This bound provides us with the expected substitute for (4.11), and we can then follow Steps 3 and 4 of the proof of Theorem 6 to get the conclusion.

1 0

 1 dτ ϕ(g s + τ (δg) st ) , ϕ (g) st := g s + σ(δg) st ) .(2.1)

. Existence 4 . 1 .Theorem 6 .

 416 sup r∈[s,t]|y r -z r | + ω M (s, t) 0 which implies that sup r∈[s,t] |y r -z r | = 0 everywhere so we find a contradiction and such interval cannot exist. This concludes the proof of uniqueness. 4The one-dimensional case. Let T > 0 and a > 0. If f ∈ C 2 b (R; L(R N , R)) and X is a geometric N -dimensional p-variation rough path, then Problem (1.2) admits at least one solution (y, m) on [0, T ] with initial condition a.

Theorem 8 . 4 .

 84 Let D ⊂ R d be a connected domain satisfying Conditions (A) and (B) of Lemma Then there exists at least one solution (y, m) to the reflection problem (4.13) in D.

  of the Skorohod problem associated with g in the domain D, that V 1 1 ([0,t];R d ) and for each y ∈ ∂D, n y ∈ N y . Then for all s < t ∈ I it holds thatm V 1 1 ([s,t]) C 1 [e pC 2 (1+ g 0,[s,t] ) g V p 1 ([s,t]) + 1](e C 2 (1+ g 0,[s,t] ) + 1) g 0,[s,t] ,(4.14)where C 1 , C 2 are constants depending only on the domain and g 0,[s,t] := sup s u<v t |δg uv |.

	is (y, m) satisfies for all s < t ∈ I
	
	 	δy
	 

st = δg st + δm st , y 1 = g 1 , m t = t 0 1 {yu=0} n yu d|m| u ,

where |m| t := m


Anas Salmi
email: anas.salmi@grenoble-inp.fr

Pierre David
email: pierre.david@grenoble-inp.fr

Eric Blanco
email: eric.blanco@grenoble-inp.fr

Joshua D Summers
email: jsummer@clemson.edu

STANDARDIZED VOCABULARIES FOR ASSEMBLY SYSTEMS MODELLING AND AUTOMATION ALTERNATIVES DESCRIPTION

The aim of this research is to support assembly lines designers in conceiving new processes with optimal automation levels selection. Several alternatives with various automation options may exist. Graphic representations and analyses of the different designs are needed. The finality is to offer a quick, exhaustive, and reliable way of modelling alternatives based on a given product design. In this sense we propose a new assembly tasks vocabulary to be combined to an existing lower layer vocabulary of elementary motions and a graphic modelling language. These developments deal with an existing automation decision approach as an extension allowing to overcome identified gaps and to ease its implementation and computerization. The proposal facilitates assembly systems alternatives generation with automation options consideration based on an initial representation. The generated alternatives are then subject to further analyses with regard to automation criteria and performance indicators considering planned production targets.

INTRODUCTION

Deciding about assembly process automation during the early phase of the system design is a complex problem. This complexity is basically caused by the multiple criteria involved in the decision: parts features, their impact on assembly complexity, cost, quality, volume to produce, ergonomics, etc. The question is to decide where to automate or not, and to what extent for an optimal process configuration with regard to the criteria and planned production targets. Before reaching the stage of the final decision, multiple analyses and comparisons of alternatives should be conducted. A first concept of the system must be available as a basis or starting point of the analyses. This concept could be ideally graphical so that it can be easily analyzed and discussed between the stakeholders involved in the decision. It should also enable providing different alternatives of resources allocation with possible degrees of automation, generally labelled Levels of Automation (LoA) in this research area. The LoA options are treated using a LoA scale going from completely manual (low LoA) to high automated or robotized systems (high LoA).

The paper is organized as follows: In section 2, a literature background is presented. In section 3, a focus is performed on the related existing automation decision method to which the paper contributes. The different lags are then identified. In section 4, the proposal is presented. The way to use the vocabulary is then tackled in the automation decision issue in section 5. A simple example is given in section 6. Finally, the paper is concluded in section 7 with openings and future works proposals.

LITERATURE IN AUTOMATION DECISION

The purpose of this research is to select the right automation level for a future assembly production. Yet, the literature in the field of automation is predominantly technical and seeks principally to improve processes performance, technology, autonomy, adaptability, and productivity. It is less devoted to find the most appropriate system configuration to a given future production considering manufacturer's preferences and strategy [START_REF] Hill | Manufacturing Strategy: Text and Cases, 3rd edition[END_REF]. In fact, the literature focusing on deciding the suitable automation level is not abundant and a need to an efficient method to support the decision has been identified [START_REF] Lindström | Aligning manufacturing strategy and levels of automation: A case study[END_REF]. Existing literature methods tailored to support automation decision making were reviewed in (Salmi, David, Blanco, & Summers, 2015.b). Eleven decision methods were found and analyzed. Multiple critics were addressed to these methods revealed as not fully useful to be applied for early phase automation decision making. In fact, after requirements definition, no method is revealed to be completely satisfactory or fulfilling all of the requirements. These requirements are basically related to: the applicability to an early phase of new systems design, the objectivity of the method applicability, the analytic reasoning according to the method's granularity level, partial automation possibilities offering, the profitability consideration, the manufacturer's preferences involving, and the traceability of the decision. This led to a new decision approach proposal in (Salmi, David, Blanco, & Summers, 2015.b) addressing these different requirements. The proposed method which consequently consists in the most recent and promising one represents a basis for this work. Our purpose here is to concretize its implementation and ease its computerization after gaps identification. These gaps are presented in next subsection with a thorough review of the method.

THE AUTOMATION DECISION APPROACH: A REVIEW AND GAPS IDENTIFICATION

The purpose of this section is to describe the literature LoA decision method representing the basis of this work via a review and gaps identification.

THE APPROACH REVIEW

The method defined in (Salmi, David, Blanco, & Summers, 2015.b) is presented in Figure 1 by a simplified outline. The method is defined in such a way so that the whole requirements previously presented in section 2 can be satisfied. To do so, as shown in Figure 1, the idea is to support systems alternatives models definition based on a generic initial design model. The initial model has to be defined based on the product design. The systems alternatives are generated considering this initial model and other information: planned production information (plant environment, cadence, manufacturer's strategy and preferences, quality level, etc) and other possible standardized data or historical information according to the experience or previous projects. Automation options that can be allowed according to the different gathered inputs are selected in every alternative with resources LoAs combinations and dimensioning: duplication if the cadence should be increased while the resource LoA is not productive enough or assignment to multiple tasks to maximize resources occupation (Salmi, David, Blanco, & Summers, 2015.a). Once possible alternatives available, evaluations can be conducted with regard to multiple dimensions (e.g. cycle time, cost, and ergonomics). By exploring the different possible alternatives, a best process with regard to considered dimensions can be found. The focus of this research is on the second and third bloc of the outline of Figure 1, respectively the initial generic model definition and the generation of systems alternatives. The second bloc consists in the initial generic model definition. This is based on the product design and parts features (representing then the first bloc of the diagram of Figure 1) involving complexity to assemble and resulting time estimates. The method defines this initial model using assembly motions and their sequencing using a standardized vocabulary of elementary motions from (Renu, Mocko, & Koneru, 2013). This vocabulary, inspired from "Methods Time Measurements" (MTM) [START_REF] Maynard | Methods Time Measurements[END_REF] and "Design For Assembly" (DFA) methods [START_REF] Boothroyd | Product Design for Manufacture and Assembly -Third Edition[END_REF], is associated in the decision method to a dedicated modeling language labeled ASML defined in (Salmi, David, Summers, & Blanco, 2014). The combination ASML and vocabulary allows assembly process modeling by elementary motions definition based on a given assembly sequence of parts assembly. Annex Figure 1 shows an assembly process modeling using in ASML (Annex Figure 1.b) for a product design assembly sequence shown in Annex Figure 1.a. The next step in the decision procedure, located in bloc 3 of the outline, is an iterative procedure of alternatives of processes generation. This is basically conducted by resources allocation to modeled motions of the initial generic model with consideration of the previously discussed inputs if the planned production. As an example, Annex Figure 2 shows an alternative of resources allocations of the model of Annex Figure 1 considering a unique resource labeled R1. Lags are then mandatory following ASML rules defined in (Salmi, David, Summers, & Blanco, 2014). Such a scenario of resources assignment can be conducted in the case of a very low volume to be produced so that a unique resource can be satisfactory. For the automation perspective, to every allocated resources a LoA is associated. These LoA selections should be performed with respect to LoA criteria (plant environment, company strategy, quality level, ergonomics, etc...), manufacturer's preferences and best practices, and a pre-balancing performing to allow an enough productive system according to required volume to be produced. The consideration of criteria can allow, impose, or forbid certain LoAs choices to some assembly motions. For example, manual can be forbidden for motions related to welding, or robotic can be imposed for painting or automotive windshield installing throughout the whole assembly process. Possible process resources automation configurations are then subject to deeper analyses considering performance indicators (time, cost, volume, etc…). The saturation of alternatives generation and evaluations by the loop described in figure 1 should lead to an optimal alternative keeping: the best deal with regard to multiple criteria. This optimal one goes in the detailed design process. If no satisfactory alternative is found, a feedback is then provided to the product designers to try to improve its design and the easiness or cost to assemble it using approaches such as DFA. It can be seen that a key element of the method is the vocabulary of motions allowing modeling the initial model. A focus is performed in next sub-section on this vocabulary and possibility of extending it.

THE VOCABULARY OF MOTIONS: A POSSIBLE EXTENSION

As previously discussed, the method uses a vocabulary of elementary assembly motions from (Renu, Mocko, & Koneru, 2013). The vocabulary is associated to ASML to model processes. After multiple analyses of this vocabulary, we think it can be extended by adding some new motions to be more exhaustive. Table 1 shows a complete list containing the original vocabulary with some additional motions we propose to add for more standardization and to cover more operations. These extra motions are highlighted in bold in Table 1. The grey colored cells in Table 1 contain the core motions representing value added motions. The remaining motions represent unavoidable extra motions that should be done to perform the value added ones. With regard to core motion, the extra-motions can be in upstream (for example preparing the parts to be assembled, such as "identify" or "get") or in downstream (releasing the parts, such as "place", "move", or "lay"). The obtained list of Figure 1 can still be extendable by eventually adding other motions that can be particular to a specific context assembly field, such as PCB (Printed Circuit Boards), semi-conductors assembly, or other particular assembly fields.

THE IDENTIFIED METHOD'S GAPS

The placement of the decision approach, just in between the finalization of the product design and the early phase of assembly system design, is challenging from the utilized data point of view, basically for data availability issues, product design maturity, and important decisions not yet taken and that should be fixed. One of the important inputs for assembly system design is the product to be assembled design and parts features. With a CAD model analysis, the obtained data on the assembly are expressed through alignments of components, insertion between one and another, or elementary actions as getting, insertion, align, tighten, etc. On the other hand, the needed information for assembly sequence description or for assembly resources allocation is different in terms of granularity level. Even if the vocabulary described in section 3.2 can be of interest for the automation possibilities analyses and LoA criteria involving (e.g. motions analyses with regard to ergonomics, repetitive motions as good signs for possible automation, quality, etc..), a too detailed granularity level does not seem to be the most suitable for the step of initial generic model building for multiple reasons. These reasons are discussed through the following sub-sections followed by a proposed solution in next section to address the different gaps.

THE GENERICITY OF THE INITIAL ASSEMBLY PROCESS MODEL

As described in ASML modelling language (Salmi, David, Summers, & Blanco, 2014) and in the automation decision method (Salmi, David, Blanco, & Summers, 2015.b), the first representation of the assembly process is performed using a standardized vocabulary, the one shown in Table 1. Annex Figure 1.b shows an example of an ASML modelled process for a product design of Annex Figure 1. To represent such an assembly process based on product design using this kind of motions can be a long work with a high risk of error. Also, this first ASML definition should be independent from the assembly technology so that automation alternatives with different LoAs consideration can be later considered. This cannot be possible using the discussed vocabulary of motions. For example, an operator needs to "get" a tool (LoA=2) while it is already mounted for an automatic system (LoA=3). A second example can be the description in motions of the screwing task of two parts (P1) and (P2) by a resource R shown in the appendix table. It can be observed that the description depends on the selected LoA for the 4 different LoAs. The number of motions is also significant for a simple task as screwing. Such a low level with as much detail does not seem to be the most appropriate to handle assembly system design and automation decision making problem.

DEPENDENCIES BETWEEN ELEMENTARY MOTIONS FROM RESOURCE POINT OF VIEW

Another issue is to be able to assign, in the initial model assumed here to be generic, separate resources (e.g. workstations) to the different process activities. As explained, resources can be then assigned to assembly motions in the initial model using the possibility that offers the ASML language (Salmi, David, Summers, & Blanco, 2014).. Yet, the way to systematically generate alternatives of resources assignment to motions cannot be easily performed. We realize that dependencies exist between elementary assembly motions and a resource executing them, whatever is its automation level. For example, when there is an 'Align' of parts made by a resource, an 'Insert' action that follows should be executed by the same resource. This makes the problem complex when managing independent motions that should be grouped by an executing resource, and when considering a high number of motions and different possibilities of LoAs.

MOTIONS COMPATIBILITY

To solve the motions dependencies discussed in 3.3.2, one can try to allocate as much motions as possible to the same resource. Yet, some sequential motions can be incompatible between them because of their natures or resource selected LoA. They can belong to different assembly activities (some of the motion belongs to screwing, others to soldering, riveting, etc.). A same resource can be consequently technically not able to execute those heterogeneous activities if assigned to them, especially when inflexible (dedicated machines, specific tools required, etc). By contrast a more flexible resource as a manual can be able to execute them sequentially. Meaningful group of elementary motions can be useful to define before the resources assignment step. This meaning is expressed by the main activity or purpose of a group formed by complementary or dependent motions that can be deduced from core motions figuring in that group (grey colored cells of Table 1 motions).

ANALYZING PRODUCTION RATE

To conduct the LoA decision, every motion of the assembly process is time estimated regarding different possible LoAs and the parts design features. The resources time estimation is mandatory to design assembly system so that it can reach the required production rate. Every resource should then run at the production cadence. The time estimation on the level of every assigned resource considering its selected automation level should satisfy the planned production takt-time. The assembly sequence representation must then enable the description of timely measurable assembly actions once an LoA is associated to given actions. When using low layer granularity level motions vocabulary, an assembly motion on the product is described with a different sequence for dissimilar LoAs. As a result, the assignments of motions to resources for selected LoAs considering required productivity target will be also different. The motions to be assigned to a resource will depend then on the selected LoA. This increases the problem resolution complexity especially for a systematic generation of alternatives. Addressing these difficulties using a low layer granularity seems to be unmanageable.

THE COMBINATORY EXPLOSION OF ASSEMBLY SYSTEMS ALTERNATIVES

In the described automation decision method, assembly systems alternatives with automation options are generated with consideration of elementary motions with resources assignments and automation possibilities combination analyses. It can be obvious that the total number of assembly elementary motions (Table 1) used to model the initial process for a complete real product assembly can be huge. As consequences, iterating all the possible alternatives of resources assignments to these elementary motions with all feasible automation options to scan can be unfeasible because of the combinatory explosion to perform, with additionally the previously discussed feasibility constraints to manage. The low layer vocabulary of assembly motions is interesting regarding its offered details, associated time estimates and analytic possibilities of automation criteria that can be considered (ergonomic motions, repetitively of actions, etc), but revealed seriously penalizing to be implemented and applied for the problem resolution. The challenge is to keep the advantages offered by the low layer vocabulary. At a same time, the multiple identified gaps have to be addressed in order to make the approach easily implementable and to be able to propose computerizable resolution issues with feasible computational time in such NPhard problems where resolution time is generally exponential. We propose in next section a solution addressing the different encountered difficulties.

A VOCABULARY OF ASSEMBLY TASKS PROPOSAL

The aim of this section is to propose a solution to address the different identified weaknesses. The solution we are proposing is to build and use a standardized vocabulary of a higher layer granularity: the one of tasks. Then, to preserve the discussed benefits of the original low layer vocabulary, the proposed vocabulary should match with this original one. These features are detailed through this section.

THE VOCABULARY OF ASSEMBLY TASKS

In this proposal, we define a task as an activity of the assembly sequence that can be labelled in the same way, whatever the LoA employed. A task has same or equivalent results when performed by a resource independently from its LoA (e.g. screwing). We define an assembly tasks vocabulary following the same principles of MTM [START_REF] Maynard | Methods Time Measurements[END_REF], DFA [START_REF] Boothroyd | Product Design for Manufacture and Assembly -Third Edition[END_REF], and others for assembly time analyses and standardizations [START_REF] Miller | Reasoning: Installation Process Step Instructions as an Automated Assembly Time Estimation Tool[END_REF] (Renu, Mocko, & Koneru, 2013). In these works a low granularity vocabulary proposed for analytic studies, basically time estimation issues. For the actual issue of assembly systems design and automation decision issues, as previously discussed, a higher layer vocabulary can be more appropriate, at least during the first steps to ease process modelling and resources assignments. The use of the lower layer can be still interesting too later in the system design for the further analyses (time estimates, repetitive motions identification, ergonomics, complex motions, etc..). This invites to create a link tasks to motions to preserve their advantages. The vocabulary of assembly tasks includes all conventional tasks that can be encountered in assembly manufactories, consisting mostly in available assembly techniques. Based on the literature and on industrial observations, we identified 20 assembly tasks. The proposed list of tasks is given in Table 2. It can be observed that major tasks match with the previously discussed core motions. This preliminary list can be refined or extended. The focus of this paper is to show that this vocabulary will help making automation decision more organized, more standardized, and more rigorously applicable within the methodology presented in 3.1. It is tailored for assembly systems modelling and automation deciding issues. To allow lower layer analyses, the vocabulary of tasks is linked to the previously presented low layer vocabulary. This is tackled in next section 4.2.

THE LINK OF ASSEMBLY TASKS TO ASSEMBLY MOTIONS

An important point to support the LoA decision process is to be able to move from one granularity level to another (higher to lower and vice-versa). The proposed high layer vocabulary of tasks matches with the level of elementary motions. Every task is then defined by a kinematic decomposition as a succession of elementary motions for the different LoA levels. This allows converting the representation to a more detailed one when it is needed (e.g. ergonomics analysis, time estimation). Depending on the task and resource automation level, the required motions can differ. For some levels, the task can be impossible to perform. For example: if a task necessitates an automated tool to be performed (such as for soldering), a low LoA defined by the only use of the worker's physical strength (full manual, LoA=1) is not valid. Also, for a same task and a same selected automation level, it can be possible to have multiple representations in motions if types in this technique can exist (tools or machines types, used technology or energy, parts features, materials, etc,..). Here we talk about involved parameters to enter for every couple (task, LoA) in order to have the appropriate task type and consequently the corresponding representation in motions. For example for the task "riveting" for a selected automation level equals to 2 (manual with automated tool), different types of rivets (dimensions) can exist, or different types of riveting tools or their energy can exit (pressure, temperature, etc..). Some of the tasks can be used to be repeated 2, 3, or n times, e.g, a cleaning or welding tasks, can be single pass or multiple pass. In this case the number of passes can be managed as a parameter in the motions decomposition table or otherwise as a multiplier of the resulting motions number, and consequently the resulting time estimates. Another example is shown in the Appendix table representing possible decompositions of the "Screwing" task to the required motions to perform it for a 2 parts assembly "P1" and "P2". Every part Pi represents an object with attributes consisting in parts features impacting the time estimates and the complexity to handle or assemble the part, such as the thickness, surface, symmetry, easiness to handle, etc... The decompositions may be usable in both directions: to deduce the detailed motions for a given task, but also to identify tasks corresponding to a detailed sequence of elementary motions.

It can be seen that the proposed vocabulary offers a higher layer of abstraction, but allows at a same time to keep the benefits of the lower vocabulary layer thanks to the possibility of converting every modelled task to its corresponding motions according to appropriate tasks parameters. In the next section 5, the way to use this vocabulary in assembly modelling and automation decision is more concretely presented followed by an application example in section 6.

THE USE OF VOCABULARIES IN ASSEMBLY MODELLING AND AUTOMATION DECISION

As mentioned, the proposed standardized vocabulary is tailored to support early phase assembly system design. This includes modelling of the assembly, analyses of automation possibilities, and optimization by alternatives evaluations and comparisons with regard to criteria and performance indicators. We follow the approach defined in (Salmi, David, Blanco, & Summers, 2015.b) described in section 3. To tackle the different identified weaknesses, we propose a little adjustment of the approach: we recommend using the new vocabulary of tasks presented in section 4.1 for the generic initial representation modelling. This is detailed in section 5.1. Then, the alternatives can be generated as explained in section 5.2 through 3 steps thanks to the link tasks to motions previously proposed in section 4.2.

A GENERIC INTIAL ASSEMBLY TASKS MODEL

The first graphic representation of the process is defined in tasks using ASML and the proposed tasks vocabulary instead of motions. Such a representation corresponds to one of the multiple possible sequences to assemble a product [START_REF] De Morello | Representation of Mechanical Assembly Sequences[END_REF]. Based on this selected sequence and the product design, tasks are defined based on how the parts are supposed to be assembled such as: screwed, soldered, riveted, snap fitted, etc. As in ASML, motions are graphically modelled in rectangles, we recommend using different features for tasks representation (such as hexagons) so that the models, the motions-based and tasks-based models, can be differentiated and confusion avoided. For automation technical feasibility and to support selecting a relevant sequence, it is recommended to schedule as much as possible similar tasks in succession to make tasks grouping easier (detailed in section 4.2). Defining this first ASML model considering the succession of similar tasks and with respect to the sequence of parts assembly can be performed using an AND/OR graph [START_REF] De Morello | Representation of Mechanical Assembly Sequences[END_REF]. We propose figuring the assembly techniques (tasks of Table 2) on every arc in the AND/OR graph to ease this process. For example, if according to the AND/OR graph, we have: screw(P1,P2), then: solder(P1+P2,P3) OR screw(P1+P2,P4). It is preferable to schedule the assembly so that the two screwing will be in succession. In this case this will lead to the following schedule: screw(P1,P2), screw(P1+P2,P4), and finally solder(P1+P2,P3).

ASSEMBLY SYSTEMS ALTERNATIVES GENERATION

Based on the tasks-based ASML model, alternatives of assembly systems can be generated. An alternative results from resources allocating to ASML tasks with different LoA options selecting. The assignments have to be performed through 3 steps as follows:

STEP 1: TASK LOA SELECTION AND MOTIONS DEDUCTION

Because of the possibility to decompose tasks to associated motions once an LoA is selected to a given task as described in section 3.2, it can be possible to convert a tasks-based model to a lower layer motions-based model. This aims at tasks time estimating in order to allow correct resources allocation to satisfy the required cadence as explained in Step 2. But also, motions representation can be useful to detect phenomena that can be good signs to automation, such as repetitive motions, or involve other criteria such as ergonomics (e.g. handling heavy parts) or security aspects (e.g. warm parts assembly or unhealthy environments).

STEP 2: TASKS TIME ESTIMATES DETERMINATION FOR SELECTED LOA

For every task, when converted to motions, a time estimate can be calculated considering the motions architecture: a sum of motions time estimates in the case of sequential, the max of the values if parallel (Salmi, David, Blanco, & Summers, 2015.a). This is enabled by the fact that every elementary motion is time estimated depending on the LoAs and the task parameter that can involve multiple features: part complexity, surface, number of times to perform the motion, etc. A table of tasks time estimates in the different selectable LoAs and appropriate parameters according to the given assembly application can be useful to be performed first in order to decrease the computation time when dealing with several alternatives generation instead of using heavy complete time standards databases.

STEP 3: RESOURCES ASSIGING TO TASKS

Once the tasks time estimates are obtainable in the different LoAs thanks to the conversion to motions, resources assigning to tasks can be performed. The higher layer can then be regained: the one of tasks. The resources assignment is performed according to productivity requirement and to obtained tasks time estimates for selected LoAs. As the process cadence is obviously given by the slowest resource of the production process, we identify 3 possible cases for optimal resources assignment:

• A resource to be assigned to a unique task A first scenario is to obtain, for a selected LoA to considered tasks in the process, a time estimate approximately equal to the required takt time. This means that a resource of selected LoA will be enough productive compared to the required cadence. In this case the resource is well dimensioned. It is yet recommended to keep a certain margin to prevent resources failures especially for high production volumes by resource duplicating following the second scenario principles.

• Multiple duplicated resources for a same task

A second scenario is to obtain, for a selected LoA to considered tasks in the process, a higher value of time estimate compared to the required takt time. This means that a resource of selected LoA will be slow if assigned to this task compared to required cadence. In this case n duplicated resources of a same type are necessary to achieve the cadence target. The factor n by which the same resource is duplicated will multiply the task cadence n times until reaching the required production cadence.

• A unique resource for multiple tasks

The last possibility is to obtain for a selected LoA, a lower time estimate value compared to required planned production takt time. This means that a resource of the selected LoA will be too fast compared to the required cadence if assigned to only that task. This resource should be moderated for multiple objectives: maximizing the resource workload, minimizing the total number of resources, and consequently minimizing production costs. It is then obviously recommended that every resource performs as much as possible of tasks. Yet, this can be feasible only when concerned tasks are technically compatible. The compatibility between tasks can be related to resources or to tasks natures:

-

Compatibility with regard to resources

The concept of resources flexibility is here involved. We mean by resource flexibility the ability for a given LoA to handle different tasks with different natures. For example, for a 4 LoA scale, a worker using only his physical strength (LoA=1), using a tool (LoA=2), or similarly an industrial robot (LoA=4) are generally supposed to be flexible. Consequently, more than a task can be sequentially performed by a same selected flexible resource if the cadence requirement is still fulfilled. This is less commonly used to be applicable for inflexible resources, such as a dedicated automatic machine (LoA=3). Such a resource is not designed to execute tasks of completely different natures.

-Compatibility with regard to tasks

In this case, compatibility is more related to the tasks. Tasks are generally compatible if they are of the same nature or if they use similar technique. For tasks natures' compatibility, a compatibility matrix including all the given assembly process modelled tasks can be useful to be build. This matrix will represent for every assembly task, the compatibility with regard to all the other tasks. To be compatible, tasks can be simply of a same type, such as screwing. Or they can be similar from required motions point of view, such as screwing and bolting. Two tasks of a same nature can be incompatible considering the parts features to assemble, such as their thickness, weight, etc. For example, screwing very thin parts can be incompatible with screwing thick parts (tools issue). These compatibilities can be more related to the given application itself or can depend on the method user analysis, reasoning, or own preferences.

To handle both compatibilities we propose a compatibility matrix taking into account both of compatibilities: with regard to resources and to tasks natures. This matrix should show the compatibilities of each task with regard to all the other model's tasks in the different possible resources LoAs. The matrix will then be a 3 dimensional matrix × × where n is the number of tasks in the assembly model and m the number of possible automation levels in the used LoA scale.

EXAMPLE

As an application example, we reconsider the previous example of product assembly shown in Annex Figure 1. The assembly process was modelled using motions in Annex Figure 1.b for the corresponding product design assembly shown in Annex Figure 1.a. Then resources should be assigned to these motions with automation options possibilities, with all relevant combinations generation and analysis. One assignment example was shown in Annex Figure 2 defined intuitively using a unique resource. Yet, this consists in only one of the several possible ways of system alternatives that can be defined. To scan all possible alternatives, alternatives should be generated based on Annex Figure 1.b following the original method. This model contains 12 motions. For a 4 LoA scale, this gives 16777216 possible alternatives (4 12 ) of tasks LoA combinations, and consequently of resources possible LoAs. Most of these combinations are meaningless and unfeasible because of the different reasons previously explained in section 3.3 (e.g. dependencies between motions from resource point of view).

If we model the process following the proposed approach of standardized vocabulary, we obtain following Step 1 of section 5.2.1, only 3 tasks as shown in Annex Figure 3. The number of possible LoA combinations decreases then to only 64 for the same example and a 4 LoA scale. This confirms the significant reduction of the combinatory explosion. Considering this LoA scale, a compatibility matrix of tasks × × (36 elements) should be here filled versus a × × (576 elements) of the previous motions management approach. Moreover, motions are still deducible once LoAs are selected for tasks. We show in Annex Figure 4 one possible combination with tasks and selected resources and corresponding motions representation. The proposed configuration in Annex Figure 4 is as follows: Resource R1 selected as a manual resource (LoA=1) is executing tasks T1. The resource is duplicated 3 times to achieve the assumed planned production rate. A second resource R2 of a LoA=4 (robotic) is assigned to task T2. Finally, task T3 is executed by an automatic dedicated machine (LoA=4). Motions of every task in selected LoA are shown inside the blocks. In this example, Tasks T2 and T3 can be considered as compatible (both of them are screwing). They can be eventually allocated to a unique resource if the this can be possible considering obtainable tasks time estimates based on corresponding motions estimates (Step 2 in section 5.2.2) compared to the required cadence (Step 3 in section 5.2.3).

CONCLUSION & FUTURE WORKS

In this work a high layer vocabulary of assembly tasks is proposed. The proposal aims to support assembly system design and deals with a literature automation decision method.

A solution is proposed to handle multiple identified gaps and to ease the systematic use and implementation of the decision method. The proposed vocabulary seeks to provide a more appropriate granularity level to describe the element of the assembly sequence. This enhances the creation of the first generic ASML model that should not depend on the resource LoA to use. Obtaining a lower layer representation of motions for selected LoA is still possible for specific analyses, such as for repetitive motions identifying, other signs to automation as ergonomic ones (handling of heavy parts), or tasks time estimating. This is facilitated by the link tasks to motions which allow a possible quick conversion of the tasks-based model to a motions-based one. Tasks time estimates obtaining enables efficient resources allocation considering automation options with respect of productivity targets. Managing tasks rather than motions hides the issue of dependencies of motions from executing resource point of view implicitly managed by tasks described in motion for the various LoAs. It also reduces considerably the combinatory explosion of system alternatives where the number of elements to manage is much lower. The approach is also flexible and generic because of the standardized vocabularies and the standardized modelling language ASML. The list of standardized vocabulary of tasks, also the one of motions, can be eventually more extended by new tasks definition and their decomposition to motions.

The whole approach can be also transposed to completely other applications, such as for manufacturing or logistics, by development of vocabularies of tasks and motions to be combined to ASML with evaluation criteria defining. Applications to Lean manufacturing by focus on core or valueadded motions, on non-value-added motions, or on their time estimates, can represent one of interesting future orientations.

Another opening can consist in automatic generation of works instructions for workers especially for low automation levels (LoA=1 and 2) to assist operators in assembly or their trainings. Time to be spent on worksheet producing can be then saved by such automatic generation based on vocabularies conversions. For automated (LoA=3 and 4), it can be useful for designing machines, control algorithms of machines and robots, identifying required sensors, and dimensioning effectors according to parts features and resulting generated motions from tasks.

As future works in this field of assembly modelling and automation decision, one of the challenges consists in developing methods and algorithms to compute the generation of assembly system alternatives and their evaluation with regard to LoA criteria. The purpose will be then the optimal assembly system design automated search. The way of translating automatic generated instructions from a CAD design tools to our vocabularies represents also for us one of future research axes. This can open the possibility to automate the whole design process from product design to corresponding assembly system design with optimal resources allocations and automation options selection through a same integrated computer aided support tool.

Figure 1 :

 1 Figure 1: The automation decision making method outline (reproduced from (Salmi, David, Blanco, & Summers, 2015.b))

Table 1 :

 1 The standardized vocabulary of assembly motions (Adapted from(Renu, Mocko, & Koneru, 2013))

	Align	Exchange	Inspect	Press	Scan
	Apply	Get	Lay	Read	Solder
	Clean	Handstart	Move	Remove	Tighten
	Connect	Hit[Hammer]	Open	Restock	Unscrew
	Disengage	Identify	Operate	Restrict	Wait
	Engage	Insert	Place	Rivet	Walk

Table 2 :

 2 Proposed Vocabulary of Assembly Tasks

	Bolting	Clinching	Inspecting	Scanning
	Bracket fixing	Clipping	Pinning	Screwing
	Brazing	Feeding	Placing	Setting-up
	Clamping	Gluing	Press fitting	Snap fitting
	Cleaning	Hooping	Riveting	Welding

Conferences & Computers and Information in

Engineering Conference, 12-15 August 2012. Chicago, IL. Renu, R., Mocko, G., & Koneru, A. (2013). Use of Big Data and Knowledge Discovery to Create Data. Procedia Computer Science, 20, 446-453. Salmi, A., David, P., Blanco, E., & Summers, J. (2015.a).

Assembly Modelling and Time estimating during the early phase of Assembly Systems Design. Proceedings of the 15th IFAC INCOM Symposium on Information Control Problems in Manufacturing, May 11-13, 2015. Ottawa. Salmi, A., David, P., Blanco, E., & Summers, J. (2015.b).

Deciding the level of automation during the design of assembly systems: literature review of decision methods and a new approach proposal. Proceedings of the CIE45 Conference on Computers & Industrial Engineering, October 28-30, 2015. Metz. Salmi, A., David, P., Summers, J., & Blanco, E. (2014). A modelling language for assembly sequences representation, scheduling and analyses. International Journal of Production Research,52(13), 3986-4006. Annex Figure 4: An example of resources automation alternatives description using proposed approach of tasks vocabulary Annex Table : The Screwing task decomposing to required elementary motions in a 4 levels automation scale

ANNEX

Complementary figures and tables