Arnau Padrol
email: arnau.padrol@ub.edu

Eva Philippe
email: eva.philippe@imj-prg.fr

Francisco Santos
email: francisco.santos@unican.es

Many regular triangulations and many polytopes

We show that for fixed d > 3 and n growing to infinity there are at least (n!) d-2±o(1) different labeled combinatorial types of d-polytopes with n vertices. This is about the square of the previous best lower bounds. As an intermediate step, we show that certain neighborly polytopes (such as particular realizations of cyclic polytopes) have at least (n!) (d-1)/2 ±o(1) regular triangulations.

Introduction

A polytope is the convex hull of a finite set of points in a real Euclidean space. Its combinatorial type is given by its poset of faces (subsets of the polytope maximized by linear functionals, ordered by inclusion). In the preface of his now classical book in polytope theory [START_REF] Grünbaum | Convex polytopes[END_REF], Grünbaum traces the problem of enumerating the number of combinatorial types of polytopes back to Euler, and cites its difficulty as one of the main reasons for the "decline in the interest in convex polytopes" at the beginning of the XXth century.

These efforts were concentrated in the case of 3-dimensional polytopes, starting with many contributions by Cayley and Kirkman, according to Grünbaum's historical remarks in [START_REF] Grünbaum | Convex polytopes[END_REF]Chapter 13.6]. Thanks to Steinitz's Theorem, which gives a correspondence between combinatorial types of 3-dimensional polytopes and 3connected planar graphs, nowadays we have quite precise knowledge on the number of 3-polytopes with n vertices [START_REF] Bender | The number of rooted convex polyhedra[END_REF][START_REF] Richmond | The asymptotic number of convex polyhedra[END_REF] and the distribution of many combinatorial parameters [START_REF] Bender | Submaps of maps I General 0-1 laws[END_REF].

In contrast, for higher-dimensional polytopes the problem is still very far from being solved. One of the main difficulties lies in the lack of a combinatorial characterization of face lattices of polytopes. Mnëv's Universality Theorem [START_REF] Mnëv | The universality theorems on the classification problem of configuration varieties and convex polytopes varieties[END_REF] and its extension by Richter-Gebert [START_REF] Richter-Gebert | Realization Spaces of Polytopes[END_REF], imply that deciding whether a poset is the face lattice of a 4-dimensional polytope is computationally hard (∃R-complete). It seems thus that a simple combinatorial characterization is impossible, which is one of the intrinsic difficulties of the enumeration problem. The problem remains hard even when restricting to the "generic" case of simplicial polytopes, where all faces except the whole polytope are simplices (equivalently, polytopes whose combinatorial type does not change when the vertices are perturbed), see [START_REF] Adiprasito | The universality theorem for neighborly polytopes[END_REF].

However, the mere number of polytopes is relatively small. In 1986 Goodman and Pollack [START_REF] Goodman | Upper bounds for configurations and polytopes in R d[END_REF] showed that the number of (labeled) combinatorially different simplicial d-polytopes with n vertices is bounded by (n!) c d for some constant c d depending solely on d, and Alon [START_REF] Alon | The number of polytopes, configurations and real matroids[END_REF] proved that this upper bound is valid for non-necessarily simplicial polytopes too. This contrasts with the number of combinatorially different simplicial (d -1)-spheres with n vertices, which grows at least as e (n d/2 ) [12,[START_REF] Nevo | Many triangulated odd-dimensional spheres[END_REF].

In 1982 Shemer [START_REF] Shemer | Neighborly polytopes[END_REF] had devised constructions producing about (n!) 1 2 ±o(1) different simplicial polytopes. This matches the upper bound, except for the fact that the constant c d in the upper bound of Goodman and Pollack and Alon is d 2 ± o [START_REF] Adiprasito | The universality theorem for neighborly polytopes[END_REF], much bigger than the 1/2 obtained by Shemer. The construction was greatly improved by Padrol [START_REF] Padrol | Many neighborly polytopes and oriented matroids[END_REF] (see also [START_REF] Gonska | Neighborly inscribed polytopes and Delaunay triangulations[END_REF]) who showed that there are at least (n!) d/2 ±o(1) (labeled) neighborly polytopes. There are alternative constructions that give these many different combinatorial types of polytopes, which led Nevo and Padrol to ask whether the number of d-dimensional polytopes with n vertices and m facets was bounded above by m n+o(n) (unpublished). As the maximal number of facets of a d-polytope with n vertices is O(n d/2 ) by the Upper Bound Theorem [START_REF] Mcmullen | The maximum numbers of faces of a convex polytope[END_REF], this would imply that the bound of (n!) d/2 ±o (1) is asymptotically tight.

The main result in this paper gives a negative answer to this question, by essentially doubling the exponent of n! in the construction of Padrol: Theorem 1.1 The number of different labeled combinatorial types of d-polytopes with n vertices for fixed d > 3 and n growing to infinity is at least (n!) d-2±o (1) .

All the polytopes that we construct are (d -1)/2 -neighborly. That is, they are neighborly for odd d, but only d 2 -1 -neighborly if d is even. In fact, for even d the number of neighborly polytopes in our family is at most the same as in the family constructed by Padrol. See Remark 4.12 for more details.

123

Enumerating polytopes is intimately tied to enumerating regular triangulations of point configurations; that is, triangulations arising as lower envelopes of polytopes of one dimension more. In fact, the number of (combinatorial types of) simplicial d-polytopes with n vertices coincides with that of (d -1)-dimensional regular triangulations with n -1 vertices. See the beginning of Sect. 4 for details on this relation. In the same vein, counting all triangulations, regular or not, is related to counting simplicial spheres.

In particular, the Goodman-Pollack bound implies the same upper bound of (n!) d 2 ±o (1) for the number of regular triangulations, while the construction of Kalai [12] can be adapted to derive that the cyclic d-polytope with n vertices has at least e (n d/2 ) triangulations in total [START_REF] Jesús | Triangulations: Structures for Algorithms and Applications[END_REF]Theorem 6.1.2].

Observe that the upper bound is for the total number of (combinatorially different) regular triangulations of all polytopes (for fixed parameters n and d), while the construction of Kalai counts triangulations of a single polytope. For regular triangulations of a single polytope, it is shown in [6, Theorem 7.2.10] that the Cartesian product of a cyclic 3-polytope with n vertices and a segment has at least (n/2)! = (n!) 1/2±o (1) regular triangulations. The second result in this paper is a significant improvement of this lower bound, showing for example that: Theorem 1.2 For fixed d ≥ 3 and n going to infinity, there are realizations of the cyclic d-polytope with n vertices having at least

(n!) d-1 2 ±o(1)
regular triangulations. It has to be noted that the total number of triangulations of a polytope (or point configuration) depends only on its oriented matroid (another combinatorial invariant that is finer than the combinatorial type), while the number of regular triangulations varies for different realizations of the same oriented matroid.

Apart of its intrinsic interest, Theorem 1.2 is an intermediate step for Theorem 1.1; the proof of Theorem 1.1 consists in showing that all of the many polytopes constructed by Padrol [START_REF] Padrol | Many neighborly polytopes and oriented matroids[END_REF] admit realizations with the many regular triangulations stated in Theorem 1.2.

This makes our proof of Theorem 1.1 more geometric, as opposed to combinatorial, than previous constructions of "many" polytopes. In fact, the combinatorial types of polytopes obtained with our method may depend on choices made along the construction, for example via the choice of realizations used for the Padrol polytopes, which affects what triangulations of them are regular, or via the particular lifting vectors used for the regular triangulations.

A point configuration is an ordered sequence P = ( p 1 , . . . , p n ) ∈ R d×n . We formally consider P a sequence rather than a set since the ordering of the points p i is sometimes important, but we will slightly abuse notation and write things like p i ∈ P, or call the points p i the elements of P. In this paper we will usually assume that the points in P are distinct and in convex position, where the latter means that all of them are vertices of the polytope conv(P). We say that P is k-neighborly if any subset of k points is the vertex set of a face of conv(P), and just neighborly if it is d 2 -neighborly; the latter makes sense since the simplex is the only d-polytope that is more than d 2 -neighborly. A triangulation T of P is a simplicial complex on a subset of [n] such that

(i) F∈T conv({ p i |i ∈ F}) = conv(P), (ii) for all F, F ∈ T , conv({ p i |i ∈ F}) ∩ conv p i |i ∈ F is a common face of conv({ p i |i ∈ F}) and conv p i |i ∈ F .
More generally, a subdivision of P is a collection T of subsets of [n], closed under taking faces (if F ∈ T then the set of indices of points in P in a face of conv({ p i |i ∈ F}) must be in T too), that fulfills the two conditions above. See [6, Sec. 2.3] for details.

A subdivision T of P is regular if there is a lifting vector w ∈ R [n] such that for any

F ∈ T , conv({( p i , w(i))|i ∈ F}) is a lower face of conv({( p i , w(i))|i ∈ [n]}).
Here, we call a face F of a polytope P in R d+1 lower if its outer normal cone contains a vector with last coordinate negative. That is, if there is a functional c ∈ R d+1 with c d+1 < 0 that is maximized on F. We denote T (P) the set of regular triangulations of P. For a triangulation T , we call cells its maximal faces. (These are sometimes called facets, but we reserve the word facet for facets of a polytope).

A point q / ∈ P is said to be in general position with respect to P if no hyperplane spanned by points of P contains q, and in very general position with respect to P if moreover no small perturbation of q changes T (P ∪ {q}). An argument similar to that in [3, Part 2] shows that configurations in very general position form a dense open subset of the space of all point configurations.

Two points p i , p j ∈ P are said to be triangulation-inseparable in P if we have that (i) T (P \ {p i }) = T (P \ {p j }) up to relabeling j to i, and (ii) for any T ∈ T (P \ {p i }) there is a lifting vector w ∈ R [n] which restricted to both P \{ p i } and P \{ p j } produces T as a regular triangulation (up to relabeling j to i).

Let p be a vertex of conv(P). We define P/ p to be any point configuration obtained as the intersection of the half-lines positively spanned by p -p| p ∈ P\{ p} with an affine hyperplane that does not contain p and intersects all these half-lines. Following [6, Definition 4.2.9] we call P/ p the contraction of P at the point p. All the configurations that can be obtained as P/ p have the same triangulations and the same regular triangulations. In fact, regular triangulations of P/ p are exactly the links at p of regular triangulations of P [START_REF] Jesús | Triangulations: Structures for Algorithms and Applications[END_REF]Lemmas 4.2.20 and 4.2.22]. Here, the link of a triangulation T at a point p i , which we denote T / p i , is defined as

T / p i := {F ⊂ [n] \ {i}|F ∪ {i} ∈ T } .

3 Many regular triangulations

The main idea of our construction of configurations with a large number of regular triangulations is to split a point into two triangulation-inseparable points and to estimate the number of regular triangulations generated after this operation. This is inspired by the study of triangulations of cyclic polytopes done in [START_REF] Rambau | Triangulations of cyclic polytopes and higher Bruhat orders[END_REF][START_REF] Rambau | Francisco: The generalized Baues problem for cyclic polytopes[END_REF].

First, we show that we can indeed obtain triangulation-inseparable pairs by such a splitting. Lemma 3.1 Let P be a point configuration in R d and p ∈ P in very general position with respect to P \ {p}. Then there is an ε > 0 such that p and p are triangulationinseparable in P ∪ {p } for any p ∈ B( p, ε) in very general position with respect to P. Here B( p, ε) denotes the ball of radius ε centered at p. Proof Up to relabeling, we can assume that P = ( p 1 , . . . , p n ) and p = p n . By definition of being in very general position with respect to P \ {p n }, there exists some η > 0 such that T (P) = T (P \ {p n } ∪ {p }) for all p ∈ B( p n , η). In particular, any such a p fulfills the first condition for being triangulation-inseparable with p.

For each regular triangulation T ∈ T (P) we can choose a specific lifting vector w T ∈ R [n] that induces T , and choose it so that the point ( p n , w T (n)) is still in general position with respect to the lifted configuration

{( p i , w T (i))|i ∈ [n -1]}. Hence there is some 0 < ε T < η such that {( p i , w T (i))|i ∈ [n]} and {( p i , w T (i))|i ∈ [n -1]} ∪ {( p , w T (n))} have the same faces, for all p ∈ B( p n , ε T ,). This means that w T induces T as a regular triangulation of P \ {p n } ∪ {p } for all p ∈ B( p n , ε T ).
If we take ε = min T ∈T (P) ε T , we obtain that p and p are triangulation-inseparable in P ∪ {p } for all p ∈ B( p, ε).

The following result is our main technical lemma, which provides lower bounds for the number of triangulations under the presence of triangulation-inseparable points. The main ideas are illustrated in Example 3.3.

Lemma 3.2 Let P be a point configuration in R d and let p ∈ P be a vertex of conv(P) that is in very general position with respect to P \ {p}. We denote C the minimum number of cells in a regular triangulation of P/ p.

Let p be such that p and p are triangulation-inseparable in P ∪ { p }, p is in very general position with respect to P, and p is a vertex of conv(P ∪ { p }). Then we have

|T (P ∪ {p })| ≥ |T (P)| × (C + 1).
Proof We denote P the point configuration P \ {p} ∪ {p }.

Let us call a regular triangulation T of P ∪ P good if there is a regular triangulation T of P such that T and T coincide when restricted to P \ {p}. Since a regular triangulation of P is determined by its restriction to P \ {p}, this definition implicitly gives a map φ : { good triangulations of P ∪ P } → T (P).

123

We claim that for every T ∈ T (P) we have

|φ -1 (T )| ≥ c(T / p) + 1 ≥ C + 1,
where we denote by c(L) (and call size of L) the number of cells of a pure polyhedral complex L. This formula implies the statement.

Let T be a regular triangulation of P. To avoid confusion we denote by T the triangulation T but considered as a triangulation of P . Let w ∈ R P∪P be a lifting vector producing T and T when restricted to P and P , which exists because p and p are triangulation-inseparable. We will assume moreover a genericity condition on w that will be detailed later at item [START_REF] Gonska | Neighborly inscribed polytopes and Delaunay triangulations[END_REF].

For each t ∈ R we consider the following lifting vector w t ∈ R P∪P , which varies continuously with t:

• For q ∈ P \ {p}, w t (q) := w(q) is independent of t. • If t ≤ 0 then w t ( p) := w( p) and w t ( p ) := w( p ) -t. • If t ≥ 0 then w t ( p) := w( p) + t and w t ( p ) := w( p ).
Let T t be the regular subdivision of P ∪ P produced by w t . We have that:

(1) The restriction of T t to P \ {p} coincides with the restriction of T : Indeed, if σ is a face of T contained in P \ {p} then w, and hence any w t , sends all of P ∪ P \ σ above some supporting hyperplane of the lift of σ ; hence, σ is a face in T t . Conversely, suppose σ is a cell in T t for some t that is contained in P \{ p}.

If t ≤ 0 then w t , and hence w, sends P \ σ above the hyperplane. Hence, σ is a cell of T . If t ≥ 0 then w t , and hence w, sends P \ σ above the hyperplane. Hence, σ is a cell of T , which restricted to P \ {p} coincides with T . (2) If t ≤ 0 then for every cell σ ∈ T t , σ \ {p } is a face in T : This is because for t ≤ 0 we have that w t restricted to P equals w, which produces T as a regular triangulation of P. (3) If t ≥ 0 then for every cell σ ∈ T t , σ \ {p} is a face in T : Same proof. (4) If T t is not a triangulation for a certain t then every non-simplicial cell is of the form τ ∪ {p, p } where τ is a cell of T / p. Let σ be a non-simplicial cell of T t . By claims (2) and (3), σ uses both of p and p and either

σ \ {p } is in T or σ \ {p} is in T . Hence, σ \ {p, p } is in T / p = T / p . ( 5
ε > 0 is small enough, then c(L t+ε ) = c(L t-ε )- 1 and c(L t+ε ) = c(L t-ε ) + 1.
There is a single cell of T t of the form τ ∪ { p, p }.

If s is in the neighborhood of t, all the cells of T s not contained in τ ∪ {p, p } remain unchanged because they are defined by an open condition on s. For s < t, we have that τ ∪ {p} is a cell of T s but τ ∪ {p } is not, because p is above the hyperplane spanned by τ ∪ {p}. Similarly, for s > t, we have that τ ∪ {p} is not a cell of T s but τ ∪ {p } is.

Claim [START_REF] Adiprasito | The universality theorem for neighborly polytopes[END_REF] says that whenever T t is a triangulation it is a good triangulation and it lies in the preimage of T . As we move t continuously from -∞ to +∞ there are finitely many values of t where T t is not a triangulation, by claims (4) and [START_REF] Bender | The number of rooted convex polyhedra[END_REF]. Of course, outside those values the triangulation T t is constant, and claim [START_REF] Gonska | Neighborly inscribed polytopes and Delaunay triangulations[END_REF] says that (if p is in general position and w is generic) at those values the change in the triangulation is a geometric bistellar flip in a cell of the form τ ∪ {p, p }. This flip changes the numbers of cells of T / p contained in L t and in L t by one unit, increasing L t and decreasing L t as t increases, by [START_REF] González | Associahedra minimize f -vectors of secondary polytopes of planar point sets[END_REF]. By property [START_REF] Tamal | On counting triangulations in d dimensions[END_REF] the size of L t grows from zero to c(T / p) as t goes from -∞ to +∞, so we encounter at least c(T / p) + 1 different good triangulations in the preimage of T along the process.

Example 3.3

To illustrate Lemma 3.2, we use a two-dimensional example (reminiscent of some classical proofs of the recurrence relation for Catalan numbers). It has the advantage of clarity, as it can be easily depicted, see Fig. 1.

When a point p is split into { p, p }, the facets of the polytope that were incident to p are divided into two families, those that remain facets after the splitting, and those that replace p by p . Moreover, new facets containing both p and p are created. Similarly, the cells incident to p in a triangulation are divided into two families, those containing p and those containing p , and new cells containing both p and p are created. This can be read in the link T / p, which is divided into two parts, L t and L t , without full Fig. 1 A two-dimensional illustration of the proof of Lemma 3.2. In the left picture, a triangulation of a hexagon. In the rest, the vertex p is split into { p, p }, and increasing values of the parameter t induce different good triangulations of the heptagon P ∪ P dimensional intersection: We have that

F ∪ { p} ∈ T t whenever F ∈ L t , F ∪ { p } ∈ T t whenever F ∈ L t , and that F ∪ {p, p } ∈ T t whenever F ∈ L t ∩ L t .
When t = -∞, we have that L t ∩ L t = L t , and it coincides with the boundary faces of T / p that are incident to p in conv(P ∪ {p }). As the values of t increase, L t ∩ L t flips successively through each of the simplices of T / p, giving rise to different triangulations. At the end, when t = ∞, we have that L t ∩ L t = L t , and it coincides with the boundary faces of T / p that are incident to p in conv(P ∪ {p }). The number of different triangulations thus created is therefore one more than the number of cells in the link T / p.

There are some important differences that only appear in higher dimensions. First of all, all triangulations of a polygon are regular, while starting in dimension 3 there are polytopes with non-regular triangulations. Moreover, in dimension two there is only one way to go from T -∞ to T ∞ since the link T / p is one-dimensional, while in higher dimensions there are usually several different paths between these two triangulations. The c(T / p) + 1 triangulations T t that we see as t ranges from -∞ to ∞ will depend on the relative position of p and p and the choice of the lifting vector w. Finally, in a polygon, the vertex figure is just a segment (with interior points) and the number C in the statement is always 1, so the lemma does not give an interesting bound in that case.

Without any further constraint this lemma is not very useful, as conv(P/ p) could be a simplex and C = 1. However, a lower bound on C can be proved if we have knowledge on the neighborliness of P/ p, thanks to the following lemma.

Recall that for a pure d-dimensional simplicial complex C and 0 ≤ j ≤ d + 1 we denote

h j (C) = j k=0 (-1) j-k d + 1 -k d + 1 -j f k-1 (C), where f k (C)
h k (∂ Q) = n -d -1 + k k .
Proof Let T be a triangulation of Q. We want a bound on f d (T ).

We

≤ j ≤ d, h j (∂ Q) -h j-1 (∂ Q) = h j (T ) -h d+1-j (T ),
where ∂ Q is the boundary simplicial complex of Q, of dimension d -1, and we take

h -1 (∂ Q) = 0.
Then we have:

f d (T ) = d+1 l=0 h l (T ) = h k (∂ Q) -h -1 (∂ Q) + k j=0 h d+1-j (T ) + d+1 l=k+1 h l (T ) ≥ h k (∂ Q).
The first h-coefficients of neighborly polytopes are well known, as they achieve the maximum allowed by the Upper Bound Theorem ( [14, Lemma 2], see also [START_REF] Ziegler | Lectures on Polytopes[END_REF]Lemma 8.26]). In particular we have:

h k (∂ Q) = n -d -1 + k k .
Remark 3.5 From the proof one derives that for 1 ≤ k ≤ d 2 , a triangulation T has exactly h k (∂(Q)) cells if, and only if, h j (T ) = 0 for every j ≥ k + 1. This, in turn, is equivalent to all interior cells of T having dimension at least dk.

As a consequence of the previous two lemmas we have: Theorem 3.6 Let P = ( p 1 , . . . , p n-1 , q) be a configuration of n points in very general convex position in R d such that: (i) for every d + 1 ≤ i ≤ n -1, p i and q are triangulation-inseparable in P i := ( p 1 , . . . , p i , q), and (ii) the point configuration P/q is k-neighborly.

Then

|T (P)| ≥ n-1 m=d m -d + k k ,
which is of order (n!) k±o (1) for fixed k and d.

Proof For k = 0 the statement is void, therefore we assume that k ≥ 1. We proceed by induction on n. In the base case n = d + 1 we have that P = P d is a simplex, with only one regular triangulation, so the result is trivial.

Assume that the theorem is true for n = m -1. Note that P m-1 satisfies the hypotheses of the theorem. Indeed, the first condition is automatic and the second follows because P m /q is a subset of P/q, and a subset of a k-neighborly point configuration is still k-neighborly. Now, since p m and q are triangulation-inseparable in P m by the first hypothesis, we can apply Lemma 3.2 to deduce that

|T (P m )| ≥ |T (P m-1 )| × (C m + 1),
where C m is the minimum number of cells in a regular triangulation of P m /q = (P/q)\{ p m+1 , . . . , p n-1 }. And since P m /q is a k-neighborly (d-1)-dimensional simplicial polytope on m vertices (all points are vertices since it is at least 1-neighborly), Lemma 3.4 implies that C m ≥ m-d+k k . At the end, using the induction hypothesis we conclude that:

|T (P)| ≥ n-1 m=d m -d + k k ≥ n-1 m=d m -d + k k k = ((n -d -1 + k)!) k × 1 (k -1)!k (n-d) k = exp(kn log n + o(n log n)).
The combination of these results provides Theorem 1.2: a lower bound of order

(n!) d-1 2
±o (1) for the number of regular triangulations of cyclic polytopes in certain realizations. Recall that the cyclic d-polytope with n vertices is a neighborly simplicial polytope that can be realized as the convex hull of n arbitrary points p 1 , . . . , p n along the moment curve (t, t 2 , . . . , t d ) ∈ R d |t ∈ R . See for example [START_REF] Ziegler | Lectures on Polytopes[END_REF] for details. (i) q is a point in the halfspace x d+1 > 0, (ii) for 1 ≤ i ≤ n, the point p i lies in the half-line from q through ( p i , 0), (iii) for d + 2 ≤ i ≤ n, and for every hyperplane H spanned by d + 1 points taken among { p 1 , . . . , p i-1 }, the points q and p i lie on the same side of H .

Remark 4.3

Positive lexicographic liftings exist for every point configuration, and are a special case of the lexicographic liftings produced with a sign vector in {+, -} n , as defined e.g. in [START_REF] Gonska | Neighborly inscribed polytopes and Delaunay triangulations[END_REF]Def. 4.1]. One way to construct a positive lexicographic lifting is to choose q arbitrarily with x d+1 > 0 and then take p i := (1

-ε i ) q + ε i ( p i , 0) for constants 0 < ε n ε n-1 • • • ε 1 . See Fig. 2.
The faces of conv( P) that do not contain q give a particular subdivision of P that is called the placing, or pushing, triangulation. We refer the reader to Section 4.3.1 of [START_REF] Jesús | Triangulations: Structures for Algorithms and Applications[END_REF] for more details.

Definition 4.4 A face F of a polytope Q is visible from a point p ∈ R d if
there is an affine functional that is zero on F, strictly positive on p and strictly negative on Q \ F. F is hidden from p if there is an affine functional that is zero on F and strictly negative both on p and on Q\F. Note that a face that is not a facet can be both visible and hidden from p, and if p is in general position with respect to Q and p / ∈ Q, then any face of Q (even facets) is either visible or hidden from p.

Let P = ( p 1 , . . . , p n ) be a point configuration in general position in R d . We denote P i := ( p 1 , . . . , p i ). The placing triangulation T n of P n is defined iteratively by taking for T 1 the singleton {1} and for T i the union of the faces of T i-1 with all simplices of the form F ∪ {i} where F gives a face of conv(P i-1 ) that is visible from p i . T i is the only triangulation of P i that contains T i-1 . The pulling triangulation of P is the union of all simplices that give proper faces of conv(P) and all F ∪ {n} where F ⊆ [n -1] gives a proper face of conv(P). (Proper faces are those different from the whole polytope). Lemma 4.5 Let P = ( p 1 , . . . , p n , q) be a positive lexicographic lifting of the point configuration P = ( p 1 , . . . , p n ) ⊂ R d in convex position. For i ∈ [n] we denote P i := ( p 1 , . . . , p i ) and P i := ( p 1 , . . . , p i ). Then:

(i) The faces of conv( P n ) that are hidden from q are exactly the liftings of faces of the placing triangulation of P n . (ii) The faces of conv( P n ) that are visible from q are exactly the liftings of faces of the pulling triangulation of P n . (iii) For i ∈ [n -1], the faces of conv( P i ) that are hidden, resp. visible, from p i+1 coincide with the faces that are hidden, resp. visible, from q. (iv) The faces of conv( P) are exactly the faces of conv( P n ) that are hidden from q, which are the liftings of faces of the placing triangulation of P n , and all conv({ p i |i ∈ F} ∪ { q}) where F gives a face of conv(P n ).

Proof Items (i) and (ii) are reformulations of [START_REF] Jesús | Triangulations: Structures for Algorithms and Applications[END_REF]Lemma 4.3.4] and [START_REF] Jesús | Triangulations: Structures for Algorithms and Applications[END_REF]Lemma 4.3.6], which correspond to the case where the point q is "at infinity". In that case, the faces of conv( P n ) hidden from q correspond to the lower faces of conv( P n ), thus to faces of the corresponding induced regular subdivision of P n . The faces of conv(P n ) visible from q correspond to the lower faces of the lifting of P n induced by the opposite (negative) heights. Then, in both [6, Lemma 4.3.4] and [START_REF] Jesús | Triangulations: Structures for Algorithms and Applications[END_REF]Lemma 4.3.6] where we take the opposite heights, the condition on the constant c 0 and the heights amounts to asking that the lifting is a positive lexicographic lifting. Item (iii) follows from the definitions and the fact that a face of a polytope Q is hidden, resp. visible, from a point p if and only if it is contained in a facet of Q that is hidden, resp. visible from p.

For (iv), notice that the faces of conv( P) that do not contain q are exactly the faces of conv( P n ) hidden from q. The same argument as before shows that they form the placing triangulation of P n . If F ⊆ [n] is such that F ∪ { q} gives a face of P, let h be a supporting hyperplane of this face. Then the intersection of h with the hyperplane w d+1 = 0 is a supporting hyperplane of the face given by F for conv(P n ) in R d × {0}. Corollary 4.6 Let P = ( p 1 , . . . , p n ) ⊂ R d be a point configuration in convex position. Let P = ( p 1 , . . . , p n , p n+1 ) be a positive lexicographic lifting of P and let P = ( p 1 , . . . , p n , p n+1 , p n+2 ) be a positive lexicographic lifting of P, with respect to the same order. Then the combinatorial type of conv( P) is completely determined by (the oriented matroid of) the point configuration P.

Proof According to Lemma 4.5 (iv), the faces of conv( P) are the liftings of faces of the placing triangulation of P n+1 and all conv( p i |i ∈ F ∪{ p n+2 }) where F gives a face of conv( P n+1 ). The definition of the placing triangulation and Lemma 4.5 (i), (ii), (iii) imply that the placing triangulation of P n+1 (and thus also the faces of conv( P n+1 )) is determined by the placing and pulling triangulations of the P i .

If one starts with a 0-dimensional point configuration (that is a point repeated multiple times), and then perfoms a sequence of positive lexicographic liftings always with respect to the same order, then one obtains a cyclic polytope. If the order is altered at each step, then many combinatorial types of polytopes are obtained, but not necessarily neighborly. Moreover, different lifting orders might give rise to equivalent polytopes. However, if one restricts to changing the order of the lifting only every two dimensions, then neighborliness is preserved and the combinatorial type can be controlled. This is used in [START_REF] Padrol | Many neighborly polytopes and oriented matroids[END_REF] to construct many neighborly polytopes. The original presentation in [START_REF] Padrol | Many neighborly polytopes and oriented matroids[END_REF] is in terms of lexicographic extensions of the Gale dual, but we refer to the following primal version for liftings taken from [START_REF] Gonska | Neighborly inscribed polytopes and Delaunay triangulations[END_REF]. We repeat the main ideas of that proof for the reader's convenience. Theorem 4.7 [START_REF] Gonska | Neighborly inscribed polytopes and Delaunay triangulations[END_REF]Theorem 5.5(i)] Let P = ( p 1 , . . . , p n ) ⊂ R d be a k-neighborly point configuration in general position. Let P = ( p 1 , . . . , p n , p n+1 ) be a positive lexicographic lifting of P and let P = ( p 1 , . . . , p n , p n+1 , p n+2 ) be a positive lexicographic lifting of P, with respect to the same order. Then P is (k + 1)-neighborly.

Proof Let S be a subset of [n] of size k or k -1. Then { p i |i ∈ S} is the vertex set of a face of conv(P). Hence, it follows from Lemma 4.5 (iv) that { p i |i ∈ S} ∪ { p n+1 } is the vertex set of a face of conv( P). The same reasoning shows that any subset of P of size k + 1 that contains p n+2 or p n+1 is the vertex set of a face of P.

For the remaining cases, let S be a subset of [n] of size k + 1. We want to show that p i |i ∈ S is the vertex set of a face of conv( P). Let m ≤ n be the largest element of S. Denote P m = ( p 1 , . . . , p m ) and P m = ( p 1 , . . . , p m ). We have that S\{m} gives a face of conv(P m-1 ) by neighborliness, thus S \ {m} gives a face of the pulling triangulation of P m-1 , thus S \ {m} gives a face of P m-1 visible from p m by Lemma 4.5 (ii), thus S gives a face of the placing triangulation of P m , and thus S gives a face of the placing triangulation of P. It follows from Lemma 4.5 (iv) that S gives a face of P.

The following lemma allows us to prove that the combinatorial type can be controlled without explicitly using the rigidity of neighborly oriented matroids of odd rank as it was originally done in [START_REF] Padrol | Many neighborly polytopes and oriented matroids[END_REF]Proposition 6.7]. Lemma 4.8 Let P = ( p 1 , . . . , p n ) be an r -neighborly point configuration in even dimension d = 2r such that n > d +2. Let P = ( p 1 , . . . , p n , p n+1 ) be a lexicographic lifting of P and let P = ( p 1 , . . . , p n , p n+1 , p n+2 ) be a positive lexicographic lifting of P, with respect to the same order. Then n is the only index k ∈ [n] such that the double contraction P/{ p n+1 , p k } is r -neighborly.

Proof For r ≥ 1, we know that P is 2-neighborly, so all pairs {n +1, k} for k ∈ [n] give edges of P, and all points of the configuration P/{ p n+1 } are vertices. This justifies that the double contraction P/{ p n+1 , p k } is well-defined. If d = r = 0, P/{ p n+1 } is a 1-dimensional configuration of points ordered linearly n, n -1, . . . , 2, 1, n + 2. Thus, the double contraction is well-defind only for k = n + 2 and k = n and we already have the result of the lemma.

Note that P is a realization of P/{ p n+2 , p n+1 }. It follows from the definition of contraction that a set S ⊆ [n]\{k} gives a face of conv( P/{ p n+1 , p k }) if and only if S ∪ {n + 1, k} gives a face of conv( P).

We denote P i := ( p 1 , . . . , p i ) for i ∈ [n].

We first show that P/{ p n+1 , p n } is r -neighborly. Let S ⊆ [n + 2] \ {n + 1, n} be a subset of cardinality r . If S contains n + 2, we define S := S ∪ {n}\{n + 2}. S is a subset of [n] of cardinality r , hence it defines a face of conv(P) and S ∪ {n + 1, n} = S ∪ {n + 1, n + 2} indeed defines a face of conv( P). If S does not contain n + 2, then it is a subset of [n] of cardinality r and hence it gives a face of conv(P). Thus, S ∪ {n} gives a face of the pulling triangulation of P, and by Lemma 4.5(ii) a face of conv( P n ) that is visible from p n+1 . Therefore, S ∪ {n, n + 1} gives a face of the placing triangulation of conv( P), thus a face of conv( P). Now, let k be an element of [n -1]. To show that P/{ p n+1 , p k } is not r -neighborly, we will exhibit a subset S ⊆ [n] \ {n + 1, k} of cardinality r such that S ∪ {n + 1, k} does not give a face of conv( P). Since n > d + 2, we can find a subset W of [n] of cardinality d + 2 = 2(r + 1) that contains k but not n. Radon's theorem implies that there is a partition of W into two subsets W 1 and W 2 such that conv({

p i |i ∈ W 1 }) ∩ conv( p j | j ∈ W 2 ) = ∅.
In particular, W 1 and W 2 do not give faces of conv(P). Since P is r -neighborly, W 1 and W 2 necessarily have at least r + 1 elements, so they are both exactly of cardinality r + 1. (This is where the assumption of even dimension is used). We define T to be the W i that contains k, and S := T \ {k}. Since T does not give a face of conv(P) and does not contain n, it does not give a face of P n that is visible from p n+1 . Hence, T ∪ {n + 1} = S ∪ {n + 1, k} does not give a face of the placing triangulation of P. However, all faces of P not containing n + 2 must be faces of the placing triangulation of P by Lemma 4.5(iv). Thus, S ∪ {n + 1, k} does not give a face of P. Corollary 4.9 [18, Proposition 6.1] and [8, Lemma 6.1] Let P = ( p 1 , . . . , p n ) be an r -neighborly point configuration in even dimension d = 2r . Then there are at least n! (d+2)! distinct labeled combinatorial types of (d + 2)-polytopes with n + 2 vertices obtained by the following construction:

• Choose a permutation σ of n.

• Define the point configuration P σ = ( p σ (1) , . . . , p σ (n) ).

• Let P σ be a positive lexicographic lifting of P σ and let P σ = ( p σ (1) , . . . , p σ (n) , p n+1 , p n+2 ) be a positive lexicographic lifting of P σ . • Define P = ( p 1 , . . . , p n , p n+1 , p n+2 ).

• Take the convex hull conv( P). Remark 4.10 In fact, [START_REF] Gonska | Neighborly inscribed polytopes and Delaunay triangulations[END_REF]Lemma 6.1] gives a bound improved by a factor n + 1, but this does not change the asymptotics of the bound on the total number of polytopes.

Proof Let P = ( p 1 , . . . , p n , p n+1 , p n+2 ) be a point configuration in R d+2 obtained as in the statement, with a permutation σ that we do not know. We will show that we can recover σ (n), σ (n -1), . . . , σ (d +3) from P and the face lattice of conv( P). This implies that distinct choices for σ (n), σ (n -1), . . . , σ (d + 3) give distinct labeled combinatorial types conv( P), and there are n! (d+2)! such choices. We will consecutively recover the values of σ (m) starting from m = n until m = d +3. Suppose that we have already recovered σ (n), σ (n -1), . . . , σ (m +1) for some d + 3 ≤ m ≤ n. We consider the point configuration P m := P\{ p σ (n) , . . . , p σ (m+1) } (where we abuse notation for the labels but the only important thing is to record the last two points). It follows from Corollary 4.6 that its combinatorial type is well defined, because it is obtained as the relabeling of the double lifting of the point configuration ( p σ (1) , . . . , p σ (m) ) (with the two additional points p n+1 and p n+2 ). Moreover, since the point configuration ( p σ (1) , . . . , p σ (m) ) is r -neighborly, it follows from Lemma 4.8 that we can recover σ (m) as the only index k ∈ [m] such that P m /{ p n+1 , p k } is r -neighborly.

Construction of many polytopes

We will use the following slight variation of the construction used in [START_REF] Padrol | Many neighborly polytopes and oriented matroids[END_REF] to give a lower bound for the number of polytopes. Proof We build iteratively sets P 2k that contain realizations of distinct labeled combinatorial types of neighborly polytopes of dimension 2k with nd + 2k vertices.

We define P 0 to be the singleton with the degenerate configuration of nd labeled points in the 0-dimensional space.

Suppose that we have constructed P 2k for some 0 ≤ k < d 2 . Let P 2k+2 be the union over all configurations P ∈ P 2k of the distinct labeled point configurations obtained from P by relabelings and two positive lexicographic liftings in the same order, as in Corollary 4.9. This union is disjoint because if P is a double lifting of P, we can recover the combinatorial type of P by taking P/{ p n-d+2k+2 , p n-d+2k+1 }. Hence, Corollary 4.9 gives that |P 2k+2 | ≥ |P 2k | × (n-d+2k)! (2k+2)! . Theorem 4.7 ensures that the point configurations in P 2k+2 are neighborly.

For k = d 2 we obtain that:

|P 2 d 2 | ≥ d 2 -1 k=0 (n -d + 2k)! (2k + 2)!

Fig. 2 A 1 4. 1

 211 Fig. 2 A positive lexicographic lifting P ⊂ R 2 of a configuration P ⊂ R 1

Theorem 4 .

 4 11 [18, Theorem 6.8] The number of labeled combinatorial types of neighborly d-polytopes with n > d vertices obtained from a 0-dimensional point configuration by a sequence of positive lexicographic liftings (with orders that might change along each step of the sequence) is at least (n!) d 2 ±o(1) .

)

  Each such τ ∪ {p, p } appears as a cell for at most one value of t: If t ≤ 0 then the value of t is fixed by the fact that w / p needs to be joined to one and only one of p and p to give a cell of T t . (7) In the limit when t → -∞ we have that L t = T / p (and hence L t is lowerdimensional) and in the limit t → +∞ we have that L t = T / p (and hence L t is lower-dimensional). In these limits, T t equals the triangulation obtained by placing point p (respectively p) in T (respectively in T ). This implies L t = T / p (respectively L t = T / p).[START_REF] Gonska | Neighborly inscribed polytopes and Delaunay triangulations[END_REF] We can take w sufficiently generic so that no T t contains two different nonsimplicial cells. Suppose that (τ 1 , τ 2 ) is a pair of cells in T / p such that τ 1 ∪{ p, p } and τ 2 ∪ {p, p } are in T t for the same value of t. Let H 1 and H 2 be the two hyperplanes in R d+1 spanned by the lifts of τ 1 ∪ {p} and τ 2 ∪ {p} for that t. Our hypothesis implies that H 1 and H 2 intersect the vertical line { p }×R at the same height (namely, at height w t ( p )). If this happens for a sufficiently generic choice of w then the intersection of H 1 with { p } × R does not change when slightly perturbing the heights of all points in τ 1 \ τ 2 : This implies that this intersection point lies in the affine span of (the lifted) configuration (τ 1 ∩ τ 2 ) ∪ {p}. Hence, p lies in the affine span of (the original) (τ 1 ∩ τ 2 ) ∪ {p}, and p is not in general position. (9) If T t is not a triangulation, and

t ( p ) = w( p )t equals the height at which the lifted hyperplane containing τ ∪ {p} meets the vertical line { p } × R; same, changing p and p , if t ≥ 0. Thus, we at most have one value of t in (-∞, 0] and one in [0, ∞). Moreover, there cannot be two values, one negative and one positive. Indeed, if τ ∪ {p, p } is a cell of T t for t < 0, then the point ( p , w(p )) is below the lifted hyperplane containing τ ∪{ p}, so ( p, w(p)) is above the lifted hyperplane containing τ ∪ {p } and there is no t > 0 such that τ ∪ {p, p } is a cell of T t . (

6

) Assuming T t is a triangulation, let L t := T t / p\ p and L t := T t / p \ p. L t and L t are contained in T / p and they are complementary in the sense that their union equals T / p and their intersection is lower dimensional. L t and L t are contained in T / p = T / p by properties (2) and (3). They are complementary because every cell τ ∈ T

  use the following result from McMullen and Walkup [15, Thm. 2], cited in a modern version in [6, Thm. 2.6.11]. For any 0

Work of Padrol and Philippe is supported by grants ANR-17-CE40-0018 and ANR-21-CE48-0020 of the French National Research Agency ANR (projects CAPPS and PAGCAP). Work of Padrol and Santos is supported by grant PID2019-106188GB-I00 of MCIN/AEI/10.13039/501100011033 and project CLaPPo (21.SI03.64658) of Universidad de Cantabria and Banco Santander.

Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.

Proof of Theorem 1. [START_REF] Alon | The number of polytopes, configurations and real matroids[END_REF] We first fix the last vertex q = p n on the moment curve and then define the points p 1 , . . . , p n-1 consecutively. At step i, we slide the point p i along the moment curve until it is close enough to p n so that Lemma 3.1 implies them to be triangulation-inseparable, after a perturbation of p i into very general position if needed. For d ≥ 3, the contraction of the last vertex in a cyclic polytope with n vertices is a (d -1)-dimensional cyclic polytope with n -1 vertices, and in particular d-1 2

-neighborly. Hence Theorem 3.6 gives the result.

It is not clear to us whether cyclic polytopes (or neighborly polytopes in general) do indeed have more triangulations than "typical" simplicial polytopes of the same dimension and number of vertices. In fact, in dimension two quite the opposite is true: the convex n-gon minimizes the number of triangulations and of regular triangulations among point configurations of n points in general position [START_REF] González | Associahedra minimize f -vectors of secondary polytopes of planar point sets[END_REF][START_REF] Kupavskii | Minimum number of partial triangulations[END_REF].

Many polytopes

Let us call polytopal (simplicial) d-ball any (labeled) simplicial complex that can be realized as a regular triangulation of a configuration of points in dimension d. By adding a point "at infinity" to a polytopal d-ball one obtains a polytopal d-sphere with one more vertex, and viceversa. Thus, the number of combinatorially different labeled polytopal d-balls with n vertices coincides with the number of combinatorially different labeled simplicial (d + 1)-polytopes with n + 1 vertices.

On the other hand, if two simplicial polytopes are combinatorially different then no triangulation of the first can be combinatorially equal to one of the second, because we can recover the boundary complex of a simplicial polytope from any of its triangulations. Hence: In this section we show that not only cyclic polytopes but all the Gale sewn polytopes introduced in [START_REF] Padrol | Many neighborly polytopes and oriented matroids[END_REF] fulfill (in certain realizations) the conditions of Theorem 3.6. This provides us with a large family of polytopes with many regular triangulations, to which we can apply Lemma 4.1 and obtain even more polytopes.

In order to have a self-contained presentation, we give in Sect. 4.1 all the definitions and lemmas that are used in the proofs of the constructions in the Sect. 4.2. Most of the contents of the latter can be traced back to [START_REF] Gonska | Neighborly inscribed polytopes and Delaunay triangulations[END_REF][START_REF] Padrol | Many neighborly polytopes and oriented matroids[END_REF], but observe that the presentation in [START_REF] Padrol | Many neighborly polytopes and oriented matroids[END_REF] is formulated in the Gale dual setting of extensions while ours, and the one in [START_REF] Gonska | Neighborly inscribed polytopes and Delaunay triangulations[END_REF], is already formulated in a primal setting of liftings.

If d is odd, instead of taking a pyramid as in [18, Corollary 6.10], we do one last positive lexicographic lifting on all the elements of P The combination of these constructions allows us to prove Theorem 1.1: The number of different labeled combinatorial types of d-polytopes with n vertices for fixed d > 3 and n growing to infinity is at least (n!) d-2±o (1) .

Proof of Theorem 1.1

We start by applying Theorem 4.11 in dimension d -1. The last step of the construction of the many (d -1)-polytopes in that theorem is a positive lexicographic lifting P = ( p 1 , . . . , p n-1 , q) from a d-2 2 -neighborly (d -2)-polytope P.

Lemma 3.1 ensures that we can do this lifting step by step so that for every i from d to n -1, p i and q are triangulation-inseparable in ( p 1 , . . . , p i , q). Indeed, the value of ε i in Remark 4.3 can be taken arbitrarily small. While very general position is not guaranteed by the construction, note that these configurations are in general position, and hence we can do a small perturbation into very general position if needed without changing the combinatorial type.

Moreover, note that by construction P is the contraction P/ q, and that similarly ( p 1 , . . . , p i ) = ( p 1 , . . . , p i , q)/ q. These contractions are thus d-2 Proof This is a direct consequence of the observation after [7, Lemma 3.1], see also [START_REF] Jesús | Triangulations: Structures for Algorithms and Applications[END_REF]Lemma 8.4.1]: a triangulation of a d-polytope is completely determined by its Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Antoine Lenat
email: antoine.lenat@ls2n.fr

Olivier Cheminat
email: olivier.cheminat@cetim.fr

Damien Chablat
email: damien.chablat@ls2n.fr

Camilo Charron
email: camilo.charron@ls2n.fr

Improving Welding Robotization via Operator Skill Identification, Modeling, and Human-Machine Collaboration: Experimental Protocol Implementation

Keywords: Skills, Scheme, Cognition, Welding, Robotics

The industry of the future, also known as Industry 5.0, aims to modernize production tools, digitize workshops, and cultivate the invaluable human capital within the company. Industry 5.0 can't be done without fostering a workforce that is not only technologically adept but also has enhanced skills and knowledge. Specifically, collaborative robotics plays a key role in automating strenuous or repetitive tasks, enabling human cognitive functions to contribute to quality and innovation. In manual manufacturing, however, some of these tasks remain challenging to automate without sacrificing quality. In certain situations, these tasks require operators to dynamically organize their mental, perceptual, and gestural activities. In other words, skills that are not yet adequately explained and digitally modeled to allow a machine in an industrial context to reproduce them, even in an approximate manner. Some tasks in welding serve as a perfect example. Drawing from the knowledge of cognitive and developmental psychology, professional didactics, and collaborative robotics research, our work aims to find a way to digitally model manual manufacturing skills to enhance the automation of tasks that are still challenging to robotize. Using welding as an example, we seek to develop, test, and deploy a methodology transferable to other domains. The purpose of this article is to present the experimental setup used to achieve these objectives.

theory [27, p.88]. Skills are based on goals, rules to generate activity, operational invariants, and inference. When an operator, as a welder, is resolving a situation he is anticipating the evolution of the situation with and without his intervention, this is a goal. To interact with the situation, he does some action, gathers information, and controls, these rules generate an activity. He has to decide which information is pertinent and/or true, this is based on beliefs and implicit knowledge. These is operational invariants. All of this needs a computational activity from the brain to adapt to the situation, which we will name inferences. When some of these pillars are consistent from one situation to another, we obtain a scheme. Schemes are, in Vergnaud own words, "the invariant organization of activity for a certain class of situations"(p. 88). Hoarau [START_REF] Hoarau | Revealing misleading schemes through operator activity analysis: A factory case study[END_REF] gives examples of activity analyses made with scheme framework, where strategies used by operators are revealed by identifying consistent control loops from one situation to another. Moreover, the scheme can be adapted for specific fields like mathematics [START_REF] Charron | Conceptualization of fractions and categorization of problems for adolescents[END_REF].

The four elements of the scheme require consistent regulation to facilitate adaptation by the operator. Welders, for instance, must actively gather information on the current situation, enabling them to compare it with the predicted scenario. Any mismatch will trigger a need for immediate adaptive measures; cognitive or physical ones. We can observe that an operator evolves in a situation to resolve a problem, having a productive impact. At the same time, the ability to adapt to various situations while performing tasks contributes to a valuable experiential learning process, making constructive growth for the individual. These regulations can operate in a closed loop for action or a longer loop in planning, for example. Finally, a scheme can be a good way to understand skills, giving an opportunity to have a modus operandi transferable from one field to another. This modus operandi began by understanding the profession's analysis. In doing so, we can propose a set of variables and situations to analyze the four fields of the scheme. When we have measured variables in these situations, we can search for invariant organization that are schemes. These schemes are a good way to analyze and describe skills, abilities and competencies. Each of this terms describe one part of the schemes while the latter include dynamical organization and adaptation to a specific class of situations. This is why we will use the term of schemes in a way that incorporate skills, abilities and competencies.

Welding Variables

Studying welding revealed many parameters, around sixty, that can directly influence the bead or have an impact on the welder. We propose here a classification of these variables into the following categories: Human factors, Physico-chemical forces, Process control parameters (electrical and shielding gas), Workpiece parameters, Manipulation parameters, Filler metal parameters, and Ergonomic parameters.

The choice of current, a critical parameter in welding, significantly influences welding energy, alongside welding speed and voltage. Manually selected by the welder, it becomes our first variable to consider. However, this setting is subject to variation based on the distance between the electrode (torch) and the workpiece to be assembled, which we refer to as stick out. Similarly, the torch angle impacts the distribution of this energy. Therefore, we need to have an estimation of the torch's position and orientation concerning the workpiece.

This orientation and position are directly influenced by the welder's choices and manipulation, notably via sick-out. Estimating how the latter is precisely carried out is crucial to understanding the human-specific aspect. That's why we need to focus on the complete body movement of the welder to estimate the welding speed and the variations that may be imposed. Moreover, good welding can be obtained with several combinations of parameters since they synergize.

A major difference exists between the two welding processes we intend to study (GMAW and GTAW): the filler material deposition. In the first one, the filler metal is supplied by a melting electrode. Therefore, deposition quantity is directly linked to the welding current, nature of the filler material, and wire diameter. In GTAW, the welder manually introduces the filler metal with a filler rod: hence, they must manage the feed rate, deposition quantity, and the location of deposition (in the weld pool, in the arc, etc.).

Other choices are left to welders, although these choices are guided by standards regarding the dimensions of the parts and other constants. We particularly consider the choice of the number of passes, stops (stopping the bead and then restarting), length of filler metal exiting the GMAW nozzle before welding, and the precision applied to the preparation of parts like chamfering and grinding.

The human factor is thus crucial: an analysis of it seems relevant. We can explore the degree of investment, physical or cognitive fatigue, adaptation to discomfort related to posture or welding fumes, eye position, and reaction time.

The piece to be assembled will govern the welding parameters; therefore, the dimensions and nature of the piece materials must be specified. Among these parameters, we can also look into shielding gas, especially the nature of the gas and its flow rate. The welding to be performed will depend on the position of the two pieces to be assembled. If welding vertically, we must consider the variation of forces at play, including conduction in the metal pieces and gravity on the weld pool. All these variables are shown in figure 1.

The next section will introduce the measurement tools that we are considering for the study of these parameters.

Measuring instruments

After defining relevant parameters, we must choose measurements guided by the question: how can a welder achieve a weld that a robot cannot?

Among the various measurement systems, we find those associated with electrical parameters. These parameters govern the supply of energy necessary for Welding relies on the supply of energy and, in some cases, on the supply of metal. To analyze the volume of metal supplied in GTAW, we plan to graduate the metal rods to have reference points when filming. The dimensions of the GTAW filler rod must also be documented, especially its diameter. The analysis of the filler metal can thus be carried out using: (i) Welding camera (Cavitar ©1 ), (ii) Measuring wire-feed roller, and (iii) Ruler.

Next, we have a set of measurements related to the workpieces to be assembled; the dimensions of these are taken into account to define the parameters. Additionally, we can verify pre-welding modifications (chamfering, grinding), and finishing touches: (i) Surface condition (roughness), (ii) Angle protractor or Goniometer, and (iii) Ruler/caliper.

Measurements of the human body can be used for two purposes: Tracking movement and analyzing physiological functions : (i) Inertial sensors, (ii) Stopwatch (for welding speed), (iii) EEG (Electroencephalogram), (iv) ECG (Electrocardiography, (v) EMG (Electromyography), (vi) Eye-tracking, (vii) Optical motion capture, and (viii) VO2 (Oxygen consumption).

Finally, we can analyze the weld pool during welding to obtain, notably, its dimensions (width and/or thickness). Sound analysis of welding is also relevant: (i) Welding camera (Cavitar © ), (ii) Laser, (iii) Microphone / acoustic sensor, and (i) Ultrasound.

Video capture can be valuable for several reasons, particularly to easily visualize information in a qualitative manner. This allows us to analyze the addition of filler metal in GTAW [START_REF] Manorathna | Human behaviour capturing in manual tungsten inert gas welding for intelligent automation[END_REF], observe variations in torch manipulation, identify body segments involved during welding, estimate arc length, and more.

To complete these measurements, a survey or interview can be developed to understand the welder's choices and gain insights into how they justify their actions and decisions.

Analysis of Measurement Tools

We do not intend to delve into an analysis involving machine learning or neural networks. These techniques require extensive training and, thus, a large database that we do not possess. Conversely, we prefer to focus on a detailed understanding of a small sample and describe their skills within the schemes framework.

Electrical Parameters: Process Control

Arc welding is governed by its electrical parameters, particularly current and voltage. These two parameters, combined with welding speed, allow the calculation of welding energy, an interesting indicator to understand the process evolution.

The set current is directly adjusted by the operator in the generator; however, this value fluctuates with the distance between the welding torch and the workpiece. Some welding machines offer the capability to measure this physical quantity and store it in memory, enabling precise post-analysis. This measurement can be made with a multimeter, Hall effect [START_REF] Manorathna | Human behaviour capturing in manual tungsten inert gas welding for intelligent automation[END_REF] or WeldQAS.

Voltage is in synergy with current and also varies depending on the distance between the torch and the workpiece (stick-out). Thus, this distance can be used as a control parameter on voltage by the welder during the process. Multimeter and WeldQAS can also be used to measure these quantities, especially voltage [START_REF] Manorathna | Human behaviour capturing in manual tungsten inert gas welding for intelligent automation[END_REF].

Filler Metal: A Critical Process Difference

For GMAW welding, using a consumable electrode, the filler metal's direct dependence on current ensures consistent wire feed rate. Analyzing it involves using a measuring roller integrated with the generator's unwinding rollers to derive the feed rate. Coupled with wire diameter, this yields feed volume. However, data gaps persist, notably the wire feed angle, obtainable through motion capture leveraging IMUs (Inertial Measurement Unit). Zhang [START_REF] Zhang | Measurement of three-dimensional welding torch orientation for manual arc welding process[END_REF] used IMUs to measure the inclination of GTAW's torch, but the same idea can be used in GMAW for torch and filler metal.

In GTAW, the welder manually holds the filler rod, rendering measuring roller use impractical. Some motion capture methods are disqualified as they disrupt the filler rod's weight and the welder, presenting challenges for analysis. However, nearby cameras can effectively capture filler metal angle and frequency. Although the uniform filler rod makes added metal volume inconspicuous, graduation is considered for visual assistance in video, aiding feed volume calculation.

We will use a welding camera to measure these angles based on [START_REF] Manorathna | Human behaviour capturing in manual tungsten inert gas welding for intelligent automation[END_REF] and more parameters such as frequency or transfer mode [START_REF] Gonçalves E Silva | Effect of dynamic wire in the GTAW process[END_REF][START_REF] Silwal | Effect of vibration and hot-wire gas tungsten arc (GTA) on the geometric shape[END_REF].

Sound: A Characteristic Emission

The welding process produces noise during the operation, dependent on numerous parameters, both in the audible range and inaudible acoustic emissions, as reported by [START_REF] Wang | Analysis of arc sound characteristics for gas tungsten argon welding[END_REF]. Some welders seem to use this sound as an indicator [START_REF] Manorathna | Human behaviour capturing in manual tungsten inert gas welding for intelligent automation[END_REF][START_REF] Saini | An investigation of gas metal arc welding sound signature for on-line quality control[END_REF][START_REF] Wang | Feature extraction in welding penetration monitoring with arc sound signals[END_REF]. Although the sound may vary significantly based on the nature and thickness of the workpieces to be welded, the welder may adapt and use it as an indicator to regulate the process. [START_REF] Wang | Analysis of arc sound characteristics for gas tungsten argon welding[END_REF], particularly caution on the microphone angle and gas flow. Therefore, we consider the study of sound relevant and will use a microphone during our experiments. The parameters studied will include the frequencies and amplitudes of the sound signal; frequency domain analysis using a Fourier transform is also relevant, as shown by [START_REF] Saini | An investigation of gas metal arc welding sound signature for on-line quality control[END_REF][START_REF] Wang | Analysis of arc sound characteristics for gas tungsten argon welding[END_REF].

Some people propose estimating the depth (penetration) of the weld pool based on sound, as demonstrated by [START_REF] Pal | Monitoring of Weld Penetration Using Arc Acoustics[END_REF]. Sound appears to be an interesting possibility for real-time control of the welding process [START_REF] Lv | Investigation on Acoustic Signals for On-line Monitoring of Welding[END_REF]. Sound offers valuable characteristics for studying the welder's schemes, dimensions of the weld pool, and has a high potential for automating certain tasks currently not feasible for robots. Therefore, sound analysis is highly recommended.

Temperature: Anticipating Cooling

The workpiece temperature, coupled with the cooling time, governs the metallurgical structure of the weld bead; these details can be found in phase diagrams. These diagrams predict the phases obtained after thermodynamic equilibrium based on temperature and the chemical composition of the alloy. Temperature is a critical parameter depending on the welding energy supplied, material nature, and the environment.

The measurement of temperature in welding is described in the standard ISO 13916 [START_REF]Soudage -Mesurage de la température de préchauffage, de la température entre passes et de la température de maintien du préchauffage[END_REF], which mandates measuring the temperature at a specific distance from the weld bead. Various measurement tools are available with different levels of precision, including: (i) Thermosensitive products (e.g., pencils or paints) (TS), (ii) Contact thermometer (CT), (iii) Thermocouple (TE), and (iv) Optical or electrical devices for non-contact measurements (TB).

In our case, the electric arc in the GTAW process reaches 3000 °C in a room temperature environment; metals will rise to around 300 °C. With such a wide measurement range, we cannot expect high precision. We can still measure the temperature of the bead for some information with optical measurement. Temperature can, indeed, be a way for an operator to analyze the welding process (directly or indirectly).

Weld pool Dimensions: A Visible Factor for the Welder

The weld pool represents the liquid part of the workpieces being assembled, with a very high light intensity; it is the most visible element, perhaps the only one, that the welder sees during the process. Its dimensions directly influence the shape of the weld bead since it's forming during its cooling.

Dimensions include the visible surface and the penetration depth. Applying too much energy in one area risks melting the piece throughout its thickness and creating a hole. It can be an interesting element to measure. However, the strong contrasts imposed by the weld pool and ambient lighting make video measurements challenging. We will explore different approaches.

Laser Measurements

Laser techniques analyze light diffraction to measure the surface dimensions of a weld pool [START_REF] Alvarez Bestard | Measurement and estimation of the weld bead geometry in arc welding processes: the last 50 years of development[END_REF][START_REF] Kovacevic | Sensing Free Surface of Arc Weld Pool Using Specular Reflection: Principle and Analysis[END_REF]. However, articles often neglect weld pool penetration analysis. Ensuring data continuity, particularly in-depth, is challenging; analyzing weld bead thickness is feasible, but penetration rate assessment remains elusive. The technique lacks portability and exhibits variable performance. This method is expensive, complex to implement, and appears irrelevant to our study, considering its drawbacks.

Ultrasonic Measurements

Ultrasonics can be used to determine weld pool dimensions, especially its depth [START_REF] Hardt | others: Ultrasonic measurement of weld penetration[END_REF][START_REF] Alvarez Bestard | Measurement and estimation of the weld bead geometry in arc welding processes: the last 50 years of development[END_REF] mention several articles using this technique, particularly to obtain weld pool depth. However, the heating of the workpiece and sensors limits the use for real-time (in-line) process control. Ultrasonics are effective in determining boundaries between states of matter in a welding process. Ultrasonics can also be coupled with a laser system described earlier [START_REF] Mi | Real-Time Weld Penetration Depth Monitoring With Laser Ultrasonic Sensing System[END_REF], or with data such as intensity, voltage, or sound. Finally, studies on weld beads using ultrasonic techniques are generally proposed to detect common defects.

Camera Measurements

Cameras are very sensitive to strong contrasts, as mentioned; however, it is interesting to note that image processing can be used to obtain weld pool images. We mentioned Weez-U Welding © , which use a camera mounted on the robot to capture multiple successive images with different intensity filters before compiling them. Some authors use high-speed cameras with significant magnification to observe weld pool solidification [START_REF] Chiocca | Etude de l'influence des écoulements dans le bain de fusion sur les mécanismes de solidification en soudage sur[END_REF][START_REF] Delapp | Quantitative observations of surface flow and solidification on autogenous GTA weld pools[END_REF]. Some cameras offer direct filtering to observe through the arc radiation; this is notably the case with the Cavitar © , which can help measure the dimensions of the weld pool. Since we can use Cavitar to measure the angle of the filler rod and torch with the workpiece, we can use it to measure the dimensions of the weld pool.

Operator Motion Capture

We aim to compare two elements: what the welder precisely executes and what justifies their actions. This involves studying the welder's movements to understand the welding speed, selected intensity, regularity of the motion, etc. These details are then compared with the reasons behind the actions, exploring why a variation in movement occurs at a specific moment and what information is perceived. Analyzing the operator's beliefs and knowledge is crucial, considering both implicit and explicit dimensions.

Among the various measurement instruments, we have particularly focused on motion capture. The welder's movements reflect decisions, personality, and the ability to interact with the environment to adapt to a situation. In our case, the situation involves performing an electric arc weld, a specialized process where the quality of the result is not directly observable. Moreover, the electric arc produces intense visible radiation and electromagnetic disturbances. It is essential to keep this in mind when analyzing measurement tools.

To capture human body movement, we need the orientation and position of different body segments to estimate joint angles. Tools considered for this purpose include:

-Optical motion capture: the reflection of infrared light (wavelength greater than 800 nm) on passive markers to calculate the spatial position of markers. This system ensures high precision.

-IMU: a system consisting of an accelerometer and a gyroscope to measure acceleration and angular velocity, respectively. These data are then used to calculate the spatial position and orientation of a segment to which the IMU is attached. They are known for their high acquisition frequency and reasonable cost. -RGB video: standard color video (Red Green Blue) that can be used to define the outlines of a subject. It can be coupled with an infrared camera to provide more information on the distance between a subject and the camera; this is then referred to as a depth camera. It combines ease of implementation and reasonable cost. -Depth camera: estimation of the distance to an object using stereographic vision (double camera). Some depth cameras use infrared radiation, similar to optical motion capture. It allows reproducing movement in a threedimensional environment.

Optical Motion Capture

Motion capture involves a set of cameras (transmitter/receiver) emitting light in near-infrared (IR) (wavelength 820 nm). The arc welding radiation spectrum is primarily in the visible range and extends into the UV. However, there are emissions in the IR range, with a peak observed at 817 nm, which may introduce noise into our tools. More information on radiation is available in articles by [START_REF] Gourzoulidis | Photobiological hazards in shielded metal arc welding[END_REF][START_REF] Tenkate | Optical Radiation Hazards of Welding Arcs[END_REF], [START_REF] Weglowski | Investigation on the electric arc light emission in TIG welding[END_REF].

Optical motion capture is currently used in welding with virtual or augmented reality environments [START_REF] Mueller | Intuitive Welding Robot Programming via Motion Capture and Augmented Reality[END_REF][START_REF] Ong | Augmented realityassisted robot programming system for industrial applications[END_REF]. The limitation we can highlight is the virtual aspect of welding, relying on models that may not be as precise as reality. An article by [START_REF] Erden | Identifying welding skills for training and assistance with robot[END_REF] proposes performing this measurement by creating occlusion between the cameras and the arc using an opaque box based on measurements made by [START_REF] Van Essen | Identifying Welding Skills for Robot Assistance[END_REF]. However, these measurements suffer from data loss and noise due to radiation.

We conducted experiments to observe the impact of the welding process on optical motion capture measurements with a SmartTrack 3/M from ART. This tool is a portable motion capture camera, easily transportable in a workshop. The major drawback of this tool is its limit to 4 markers.

We tried to estimate the noise made by the electric arc on motion and IMU. Therefore, we decided to place two markers as fixed references on a linear support on the table. These two markers will be named body #0 and #1. Another one was used for measure and get placed on the Weez-U © robot arm. This one will be named body #2. The experiment involved linear motion with torch ignition at the midpoint and a constant speed. The aim was to confirm data linearity by monitoring any mid-displacement variations, ensuring no interference from the electric arc. Moreover, our fixed reference (body #0 and body #1) should be measured during the whole welding process. Figure 2 illustrates the position according to a Cartesian coordinate system located at the center of the camera for the three present references. The trials measure the robot's positioning throughout the trajectory, unaffected by torch ignition. However, for the two fixed bodies, we observe a loss of vision (characterized by the return of positions to 0). This effect is not systematic during the trials.

Fig. 2: Welding arc noise test measurements on optical motion capture

In conclusion, these trials indicate good measurement stability after arc ignition if detection is ensured. However, some markers may experience occlusion if close to the axis between the camera and the electric arc, with the arc's radiation hindering marker detection. The solution appears viable, but careful attention should be paid to the camera's placement relative to the welder.

A new trial was conducted to measure the welder's arm movements. We placed a marker on the forearm and another on the same arm of the operator. We calibrated a new sensor, placing it out of the field of vision to check for possible spurious detection.

Fig. 3: Results of capturing the movement of a welding arm

The results are available in Figure 3. The absent calibrated body is body #1; we observe its detection due to radiation produced by the electric arc. This emphasizes the need to pay attention to the camera's positioning relative to the welder and the workpiece.

The other two bodies (body #0 and body #2) are generally well-detected, but body #0 is temporarily undetected (approximately 5 seconds). Thus, the arc introduces a significant source of noise.

Combined with the limit of four markers, this solution, despite its considerable advantages in precision and portability, seems incongruous with our objectives. However, potential avenues can be explored: 1. To address the low sensor count, acquiring an additional SmartTrack camera and synchronizing data acquisition could be considered. 2. To mitigate the disturbance from the electric arc, placing a band-pass filter in front of the lenses is an option. However, this may introduce optical path drift and data blurring. Additionally, as the arc and camera radiation are close, the band-pass filter doesn't guarantee satisfactory results.

Inertial Measurement Units (IMUs)

Inertial Measurement Units employ a gyroscope and an accelerometer to provide angular velocity and acceleration of a body. This theoretically enables obtaining a segment's position through double integration of accelerometer data and its orientation through gyroscope data integration. However, the high sensibility of IMU makes acceleration data subject to noise. IMU are also used to obtain angle by measuring the acceleration of gravitation and coupling with data from the gyroscope by signal processing. Some researchers utilize these measurement tools in welding, including [START_REF] Zhang | Measurement of three-dimensional welding torch orientation for manual arc welding process[END_REF][START_REF] Zhang | Welding torch attitude-based study of human welder interactive behavior with weld pool in GTAW[END_REF]. To assess IMU immunity to the measurement environment, we coupled the SmartTrack experiment with an inertial sensor attached to the Weez-U © arm. The sensor is positioned on a segment further from the torch for ease of implementation using an existing strap.

Similar to the initial observation with optical motion capture, no variations are observed during arc ignition. Although the trajectory is coded for constant speed, we note jerks attributed to the robot's architecture. Fig. 4: Inertial measurements with acceleration, angular velocity, and magnetic field according to a Cartesian reference.

On figure 4 we observe significant angular velocity and acceleration noise when the torch ignites, initially attributed to the magnetic field, but now believed to be the robot's vibrations during welding.

A follow-up experiment with the IMU on both a table and the floor aimed to confirm this hypothesis. The goal was to check if displacement was measured while the sensor was stationary. On the table, during arc ignition (identified by the magnetometer), the accelerometer displayed a highly noisy signal. On the floor, these signals were absent, demonstrating the sensor's immunity to disturbances from the electric arc.

Inertial Measurement Units appear promising for capturing human body movement. For a complete body, 15 sensors are necessary (head, neck, lumbar, two arms, two forearms, two hands, two thighs, two shins, two feet). Manufacturers offer suits providing joint data directly from these measurements, such as Movella © (with Xsens IMUs) or Teslasuit © , the latter potentially suitable for comprehensive testing.

RBG video

Traditional camera video is qualitatively valuable and can be used quantitatively by outlining the operator's contour for posture analysis, especially with a depth camera. However, this application is usually used in classification more than precise measurement of articulation's angle. Moreover, the intense light interference from the electric arc complicates this task. Cameras can provide additional insights into filler metal management or weld pool dimensions. Despite challenges, some industries use videos with extensive processing, capturing successive images with varying intensity filters and combining them to retain relevant portions, reducing significant contrast. Video capture remains beneficial for qualitative aspects, facilitating easy action visualization and data processing validation through comparisons with obtained data graphics. They can be used to take all the working scenes or be placed on the operator to have his field of view.

Depth camera

The depth camera was explored to capture an operator's contours and movements. This involves substantial data processing and multiple cameras to estimate the three-dimensional positions of all segments, along with the risk of occlusions. Tests were conducted with an Intel RealSense D400 camera using stereoscopy. The two images in Figure 5 show depth capture without an electric arc on the left and with an electric arc on the right. While the quality is good, distinguishing the workpiece, the background robot, the operator, and the torch, the image significantly distorts when an electric arc is lit. Despite its interest, this technology lacks precision and is affected by electric arc noise, rendering it unsuitable for our study.

Measurement by force sensors/pressure plate

A force plate is a tool in the form of a thin platform placed on the ground. It provides information on the distribution of forces and moments applied to support surfaces, offering an estimate of the subject's center of gravity changes while moving on it. Mainly used in gait analysis, these plates show the temporal Fig. 5: Depth camera scan difference with (left) and without (right) electric arc evolution of vertical forces, providing insights into gesture deviation, direction, and balance. While redundant if we have access to the welder's complete motion, these plates can offer valuable information about the welder's balance and positional changes during welding but the welder needs to get on it during the process.

Summary of Motion Capture Measurements

Motion capture is dominated by optical techniques with passive markers. While highly precise, this technology requires large spaces with numerous cameras. The arc's radiation interferes with measurement wavelengths, introducing noise and occluding markers despite precautions [START_REF] Erden | Identifying welding skills for training and assistance with robot[END_REF][START_REF] Van Essen | Identifying Welding Skills for Robot Assistance[END_REF]. We utilized a portable Smart-Track 3 motion capture; despite its precision, the limitation to four markers is a significant drawback. If access to a complete room becomes available, we may reconsider this solution; currently, it is not feasible. Inertial sensors are costeffective and offer high acquisition frequency and sensitivity. Though susceptible to noise, they provide precise data, making them relevant for our study [START_REF] Ahmad | Reviews on various inertial measurement unit (IMU) sensor applications[END_REF][START_REF] Gu | IMU-based motion capture system for rehabilitation applications: A systematic review[END_REF]. Additional force plates can complement the setup. These techniques can also yield information on the torch's orientation and position.

Physiological measurements: The body's response

Physiological measurements encompass signals from the body's electrical activity (muscles -EMG, heart -ECG, brain -EEG), eye movements (gaze tracking), oxygen consumption, emotional responses, etc. While some are valuable for user interface development, in our context, most won't provide significant information.

Measurements of physiological electrical signals

The analysis of muscular activity helps determine the muscles used in a gesture, providing a more detailed analysis of the movement and revealing associated muscle fatigue. Brain activity reveals cognitive load and the regions involved during an action, highlighting the preferential processing of one piece of information over another (approximately). Heart activity provides information on a person's stress; being also linked to physical activity, this complex data requires correlation with other information. Although it offers relevant details, it is not essential.

These three activities can be measured using the electrical signals produced by the human body: (i) EEG: Electroencephalogram for the brain, (ii) ECG: Electrocardiogram for the heart, and (iii) EMG: Electromyogram for the muscles.

However, the generated intensities are low and highly influenced by the sensor's proximity to the skin. Additionally, electromagnetic disturbances from the welding arc can lead to significant interference. Due to the limited information provided by physiological measurements, their sensitivity to noise, and the substantial data processing required, we believe their use is not necessary.

Gaze tracking

The gaze tracking system, utilizing infrared oculometry, recalibrates the gaze direction on a video obtained from a camera located on the nose's edge. Some glasses supplement these data with a gyroscope to provide head orientation. Gaze tracking can offer valuable insights into an operator's behavior, revealing visual cues in the environment and providing information on cognitive information processing through fixation duration and gaze scanning rate. In our case, we aim to determine the gaze position concerning the weld pool's location, understanding whether the operator looks before or after the pool, fixates on it, or maintains a constant scan. However, the protective hood against the welding arc's light poses a challenge to using this technology, limiting its application in our study. Despite having oculometric data, we lack a detailed image, only obtaining a luminous point surrounded by darkness.

An attempt was made using Tobii Proglasses 2, provided by the University of Nantes. The camera quality remains good until the welding hood is worn. At that moment, the video becomes less usable and completely unworkable when the electric arc is initiated. Several possibilities could enable the use of this technique:

1. Modify the protective glass: Creating a protective glass with two different filter intensities, a weaker filter in front of the camera and a standard one for the eyes. 2. Utilize the gyroscope: Before initiating the arc, the trajectory can be identified based on the gap between the parts to be welded. Head tilt variations can be used to align the trajectory, considering the weld pool at the center of this line. 3. Move the camera: Placing the camera on the other side of the filtering glass may provide a clearer image; additional video processing might make the video usable.

The first hypothesis poses safety risks if not executed correctly, and video quality might not guarantee to observe part positions. The second hypothesis is sensitive to gyroscope data drift. The third hypothesis involves challenges such as the difference in the field of view between the camera and eyes, along with data processing difficulties. Eye tracking is used in welding, especially in conjunction with virtual reality. Despite the potential benefits of eye-tracking measurements, ensuring effective utilization of this measurement tool cannot be guaranteed.

Summary and conclusion of the measurement tools

In summary, considering all the measurement tools mentioned here and their potential use for studying the welder's schemes, we propose a matrix of possible utilization (see Table 1). This matrix indicates whether a measurement tool is applicable (yes or no) in our case. If a tool is considered feasible under certain conditions, we may note "perhaps" with an explanation.

Within these parameters, we can identify which one matches with a cognitive aspect of skills, meaning one pillar of the scheme. Consider, for instance, the electric parameters that an operator can set, embodying a prescribed rule of action. By examining the correlation between a torch manipulation reaction and the weld pool dimension, we identify an inference mechanism at play. Time reaction being unknown, we have to be careful when we analyze data. Through collaborative discussions with the operator and meticulous comparisons with other parameters, we can find and derive goals and operational invariants. This intricate analysis provides valuable insights into the interplay of cognitive skills and their manifestation within the defined framework.

Survey and interview: A subjective measure of the rationale of the operator

To gather the operator's insights, especially to understand the rationale behind their actions, we can utilize a survey or conduct an interview. However, a drawback of using a survey is its restrictive nature, providing limited freedom for the operator's responses and potentially directing them in a particular direction. This limitation could subsequently constrain the depth of the analysis we can perform.

Several interview techniques are available. Our objective is to comprehend the reasons behind specific actions, and for this purpose, Pierre Vermersch's explicitation interview appears most suitable [START_REF] Vermersch | L'entretien d'explicitation. Psychologies & psychothérapies[END_REF]. However, this method is acknowledged for its complexity in implementation, requiring training in interview techniques. Another equally interesting and easier-to-implement method is autoconfrontation.

Autoconfrontation, a commonly used technique, involves using traces, including videos, to place an operator back into the action situation, allowing them to 1: Summary of the measurement tools applicability in our study explain their gestures [START_REF] Theureau | Les entretiens d'autoconfrontation et de remise en situation par les traces matérielles et le programme de recherche cours d'action[END_REF]. Another method is the double instruction, aiming to prompt individuals to assume the role of an instructor to explain their profession [START_REF] Lainé | Chapitre 9. -L'instruction au sosie[END_REF].

Interview techniques focus the interviewee's attention on specific aspects, such as action realization with video support in autoconfrontation, precise and non-generalized actions in the explicitation interview, and pedagogical debate in the case of double instruction. The importance of pauses remains central in these techniques. As this report concentrates on objective measurements, we won't delve further into these interview techniques.

Proposed experimental protocol

We propose creating a reference piece for a group of welders. This piece should encompass various situations, easily adaptable into a few variants (thickness and material), and representative of many mechanized welding production processes.

To achieve this, we present the piece in Figure 6. The welder will have at his disposal all the necessary sheets and tubes accompanied by the technical drawing of this reference part. We propose to manufacture this piece with three different thicknesses: 2 mm, 4 mm, and 6 mm. A second piece may be produced in a subsequent trial to analyze edge-to-edge welds, especially in GTAW welding, to observe a task that is challenging for automation.

Before carrying out the welds, we will equip the welders with a set of inertial sensors to obtain the orientation of body segments (arms, forearms, torso, etc.). These inertial sensors can be incorporated into a suit (Teslasuit© or Movella©, or using IMUs with custom coding). We will focus on joint amplitudes, particularly those of the upper limbs, directing our attention to torch manipulation speed, pause times, and back-and-forth movements.

We will add a set of sensors to the work environment: (i) Non-contact thermometer (infrared) to measure temperature, (ii) A microphone to analyze sound, (iii) A welding camera (Cavitar © ) to analyze the fusion pool with filler metal and stick-out, (iv) An RGB camera to qualitatively analyze the complete body movement and provide a record of the activity, (v) Inertial measurement units to capture movements.

After completing the weld, we suggest conducting an exchange with the welder in the form of an interview to analyze the reasons behind their actions and understand their activity.

This article underscores the challenges and opportunities related to assessing welders' schemes in a welding environment. The interference of the electric arc (light, electromagnetic field, etc.) on most measurement tools justifies comparing objective indicators, such as sound and temperature, alongside subjective evaluations through self-confrontation interviews [START_REF] Theureau | Les entretiens d'autoconfrontation et de remise en situation par les traces matérielles et le programme de recherche cours d'action[END_REF]. Our proposed approach involves analyzing welders' movements and reactions using objective measures like sound and temperature in various representational situations provided in a reference workpiece (see Figure 6). Our analysis will focus on identifying the control loop made by welders to operate a regulation during the welding process. The goal is to identify invariant organizational aspects across multiple situations, reflecting their schemes. By examining welders' movements and reactions with objective data, we aim to gain a comprehensive understanding of their schemes and develop a model encompassing all facets of their expertise. This model aspires to address current technical limitations faced by robots in specific tasks by incorporating human gestures and cognitive aspects of skills.

Conclusions and future work

The workforce in industries, including welders, constitutes the core of Industry 5.0, yet understanding the principles behind their skills poses challenges. In this article, we propose a set of measures to explore the reasoning behind each decision made by welders. Acknowledging the numerous variables influencing the welding process, we aim to identify critical ones used by welders, recognizing the presence of implicit skills, described within schemes framework.

Measuring a welder's movement is crucial but is a time-consuming step. Inertial Measurement Units (IMUs) offer a suitable compromise, despite requiring signal processing. While the weld pool dimensions are critical, measuring them poses challenges.

This article primarily centers on welding, yet the methodology employed here can be applicable to various professions. In summary, individuals seeking to analyze schemes must comprehend the intricacies of the job to determine multiple representative situations. Within these scenarios, it is imperative to select variables relevant to the operator. Then we have to measure them. Throughout the data analysis process, emphasis should be placed on identifying control loops that connect perception and action. Upon identification of these loops, it is crucial to verify their consistency across different situations. The organizational constants revealed through this process are indicative of the skills.

Fig. 1 :

 1 Fig. 1: Mind map of welding parameters to study

Fig. 6 :

 6 Fig. 6: Reference piece to be used in the experimental protocol This part incorporates the following situations (positions according to ISO 6947 [13]: 1. Angle at the ceiling -plate against plate (position PD) 2. Flat angle -plate against plate (position PB) 3. Flat angle -tube against plate (position PB) 4. Vertical ascending angle -tube against plate (position PF) 5. Flat angle -tube against tube (position PB -external)

Table

	Measurement	Applicable in	Under	Not	Comments
	Tool	our study	conditions	applicable
	Welding display	X
	Cavitar ©	X
	Non-contact	X
	thermometer
	ECG			X
	EEG			X
	EMG			X
	Eye tracking		X		Maybe with adaptation
					of the protective glass
	Measuring	X			Applicable in GMAW,
	roller				not in GTAW
	IMU	X
	Laser		X		Expensive
	Micro/acoustic	X			Noise and risk of
	sensor				occlusion
	Motion capture		X
	Multimeter	X
	Oscilloscope	X
	Pressure		X		Accurate measurements
	platform				would require coverage
					over a substantial area.
	Thermosensitive	X			Not very precise
	product
	Survey	X
	Ruler/caliper	X
	Thermocouple		X		Can be done if
					pre-welded
	Contact		X		Does not allow real-time
	thermometer				study
	Ultrasonics		X
	Video	X			Caution about what is
					intended to be measured
	Welding speed	X
	VO2			X	Little interest

https://www.cavitar.com/applications/welding-imaging/