Hicham Hachadi

And El

Hassan Youssfi
email: el-hassan.youssfi@univ-amu.fr

M R C N R S 7373

THE POLYANALYTIC REPRODUCING KERNELS

Keywords: Reproducing kernel, Polyanalytic function, Bergman space, Fock space I2M, U

Let ν be a rotation invariant Borel probability measure on the complex plane having moments of all orders. Given a positive integer q, it is proved that the space of ν-square integrable q-analytic functions is the closure of q-analytic polynomials, and in particular it is a Hilbert space. We establish a general formula for the corresponding polyanalytic reproducing kernel. New examples are given and all known examples, including those of the analytic case are covered. In particular, weighted Bergman and Fock type spaces of polyanalytic functions are introduced. Our results have a higher dimensional generalization for measure on C p which are in rotation invariant with respect to each coordinate.

INTRODUCTION

In their recent work A. Haimi and H. Hedenmalm [START_REF] Haimi | The polyanaytic Ginibre ensembles[END_REF]- [START_REF] Haimi | Asymptotic expansion of polyanalytic Bergman kernels[END_REF], established asymptotics for the Bergman-Fock type space of polyanalytic functions with respect to a given weight and mentioned that in general finding explicit formula for these kernels is difficult ( [START_REF] Haimi | Asymptotic expansion of polyanalytic Bergman kernels[END_REF], p. 4668). This problem was also addressed by D. Alpay ( [3], p. 479). The main goal of this paper is to answer this question in the general context of rotation invariant Borel probability measure on the complex plane having moments of all orders. More precisely, given a positive integer q, we shall establish the formula for the reproducing kernel for Hilbert spaces of square q-analytic functions with respect to a rotation invariant Borel probability measure. The reduction of our formula to the unit disc D gives an explicit formula for the weighted Bergman spaces of polyanalytic functions on D, which in turn, which reduces to the result of Koshelv [START_REF] Koshelev | The kernel function of a Hilbert space of functions that are polyanalytic in the disc[END_REF] when the weight is trivial. We point out that the result of Koshelev is proved by a very specific method based on integration by parts which does not work for the weighted case. Other applications are given to provide new results on other Bargmann-Fock type spaces of polyanalytic functions on C and related projections.

We recall that a function f (z) is called a polyanalytic function of order q (or just q-analytic) in the domain Ω ⊆ C if in this domain it satisfies the generalized Cauchy-Riemann equation (1.1)

∂ q f ∂ zq = 0. Polyanalytic functions inherit some of the properties of analytic functions and the simplest case is the so-called bianalytic functions. However, as in the theory of several complex variables, many of the properties break down once we leave the analytic setting. They are naturally related to polyharmonic functions see [START_REF] Balk | Polyanalytic functions and their generalizations, Complex analysis I[END_REF], [START_REF] Hayman | Representation and uniqueness theorems for polyharmonic functions[END_REF] and [START_REF] Render | Real Bargmann spaces, Fischer decompositions, and sets of uniqueness for polyharmonic functions[END_REF] for further results.

The properties of these functions was studied by several authors see Balk and Zuev [START_REF] Balk | On polyanalytic functions[END_REF], Balk [START_REF] Balk | On the Hayman uniqueness problem for polyharmonic functions[END_REF] and Dzhuraev [START_REF] Dzhuraev | Multikernel functions of a domain, kernel operators, singular integral operators[END_REF] and the references therein. It is well known that any q-analytic function in the domain Ω can be uniquely expressed as (1.2) f (z) = q-1 j=0 z j φ j (z) .

where the φ j (z) are holomorphic in Ω. This representation was used to study the boundary behavior and integral representation of polyanalytic functions. Hilbert spaces of polyanalytic functions and related projections were considered for the case of the unit disc by Koshelev [START_REF] Koshelev | The kernel function of a Hilbert space of functions that are polyanalytic in the disc[END_REF] and later by Vasin [START_REF] Vasin | Projections onto L p -spaces of polyanalytic functions[END_REF] and A. K. Ramazanov [START_REF] Ramazanov | Representation of the Space of Polyanalytic Functions as a Direct Sum of Orthogonal Subspaces. Application to Rational Approximations[END_REF] and [START_REF] Ramazanov | On the structure of spaces of polyanalytic functions[END_REF]. In the latter reference a representation of the space of polyanalytic functions as direct sum of orthogonal subspaces is given and applied to rational approximation. The case of the Bargmann-Fock space of polyanalytic functions was studied by N. L. Vasilevski [START_REF] Vasilevski | On the structure of Bergman and poly-Bergman spaces[END_REF]- [START_REF] Vasilevski | Poly-Fock spaces, Operator theory advances and applications[END_REF] and later by L. D. Abreu [START_REF] Abreu | Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions[END_REF]- [START_REF] Abreu | On the structure of Gabor and super Gabor spaces[END_REF] in connection with Gabor and time-frequency analysis. A deep study of the general case of weighted Bargmann-Fock space of polyanalytic functions was considered by A. Haimi and H. Hedenmalm [START_REF] Haimi | The polyanaytic Ginibre ensembles[END_REF]- [START_REF] Haimi | Asymptotic expansion of polyanalytic Bergman kernels[END_REF], where they obtain the asymptotic expansion of the polyanalytic Bergman kernel as well as the asymptotic behavior of the generating kernel and the asymptotic in the bulk for the q-analytic Bergman spaces in the setting of the weights e -2mQ (see [START_REF] Haimi | Asymptotic expansion of polyanalytic Bergman kernels[END_REF]). Their approach relies on the study of polyanalytic Ginibre ensembles and appeals to the connection with random normal matrix theory and Landau levels.

Polyanalytic functions of several variables were considered by Avanissian and Traoré [START_REF] Avanissian | Sur les fonctions polyanalytiques de plusieurs variables[END_REF] and [START_REF] Avanissian | Extension des théorèmes de Hartogs et de Lindelöf aux fonctions polyanalytiques de plusieurs variables[END_REF]. They are defined in an anlogous way. Namely, a function f (z) is called a polyanalytic function of order q = (q 1 , • • • , q p ) ∈ N p 0 (or just q-analytic) in the domain Ω ⊂ C p if in this domain it satisfies the generalized Cauchy-Riemann equation

(1.3) ∂ q 1 +•••qp f ∂ zq 1 1 • • • ∂ zqp p = 0.
They can be uniquely expressed as

(1.4) f (z) = (q 1 -1,...,qp-1) j=(0,••• ,0) z j φ j (z)
where the φ j (z) are holomorphic in Ω where for j = (j

1 , • • • , j p ), k = (k 1 , . . . , k p ) ∈ N p 0 and z = (z 1 , • • • z p ) ∈ C p , the inequality j ≤ k means that j l ≤ k l for all l = 1, • • • , p and z j := z j 1 1 • • • z jp p .
However, there are very few results are available in this case.

STATEMENTS OF THE MAIN RESULTS

In this section we will state the main results in the one dimensional case. The higher dimensional analogs will be stated at the end of the paper.

The setting is the following. We recall that a sequence s = (s d ), d ∈ N 0 , is said to be a Stieltjes moment sequence if it has the form

s d = +∞ 0 t d dµ(t),
where µ is a non-negative measure on [0, +∞[, called a representing measure for s. These sequences have been characterized by Stieltjes [START_REF] Stieltjes | Recherches sur les fractions continues[END_REF] in terms of some positive definiteness conditions. We denote by S the set of such sequences and if s ∈ S we let M(s) the convex cone of the representing measures of s. It follows from the above integral representation that each s ∈ S is either non-vanishing; that is, s d > 0 for all d, or else s d = δ 0 for all d. We denote by S * q the set of all non-vanishing elements of S having a representing measure µ with support containing at least q strictly positive elements. Fix an element s = (s d ) ∈ S * q and let µ ∈ M(s). It is known [START_REF] Bommier-Hato | Hankel operators and the Stieltjes moment problem[END_REF] that the sequence (s d 1 2d ) converges to limit R s ∈]0, +∞], where R s is the supremum over all t > 0 such that t is in the support of µ. We denote by D s the disc in C centered at the origin with radius R s with the understanding that D s = C when R s = +∞.

For each pair of non-negative integers (d, n) such that n ≤ q -1, let P n (µ) be the subspace of the Hilbert L 2 (x d dµ(x)) consisting of all polynomials with degree at most n furnished with the real inner product

f, g := +∞ 0 f (x)g(x)x d dµ(x), f, g ∈ P n (µ)
and denote by Q d,n : (0, +∞) × (0, +∞) → C the corresponding reproducing kernel.

Consider the following function

F q,s (λ, x, y) := +∞ d=0 λ d Q d,q-1 (x, y) + q-1 d=1 λd Q d,q-1-d (x, y). (2.1)
where λ is a complex number and (x, y) ∈ [0, +∞[×[0, +∞[. Our first result is the following: Theorem A. For all fixed non-negative real numbers x and y, the series λ → F q,s (λ, x, y) converges uniformly on compact subsets of the disc centered at 0 with radius R 2 s . Next, let µ ∈ M(s) and denote by ν denote the image measure on C of µ ⊗ σ under the map (t, ξ) → √ tξ from [0, +∞[×T onto C, where σ is the rotation invariant probability measure on the unit circle T in C. Then ν is rotation invariant. Conversely, it is known [START_REF] Berg | Rotation invariant moment problems[END_REF] that any rotation invariant Borel probabiltiy ν on C is of this form. Since µ is supported in the interval [0, +R s ], it follows that the support of ν is contained in closure D s of the open disc D s .

We consider the Hilbert space L 2 (ν) of square integrable complex-valued functions in D s with respect to the measure ν. We denote by A 2 ν,q the space of those q-analytic functions on D s which are square integrable with respect to ν. The natural inner product inherited from that of L 2 (ν) turns A 2 ν,q into a pre-Hilbert space. We are now prepared to state our second main result. Theorem B. The space A 2 ν,q is a Hilbert space which coincides with the closure of the q-analytic polynomials in L 2 (ν). Moreover, for each set compact K ⊂ D s we have that

sup z∈K |f (z)| ≤ C f L 2 (ν)
for all q-analytic polynomials f ∈ L 2 (ν), where

C = C(K) := sup z∈K F q,s (|z| 2 , |z| 2 , |z| 2 ).
Furthermore, the reproducing kernel of A 2 ν,q is given by

R ν,q (z, w) = F q,s (z w, |z| 2 , |w| 2 ), z, w ∈ D s .
Remark C . When the measure µ has a finite support and the number points of the support of µ is q, then Theorem B gives Cauchy type formula for polyanalytic functions. Such an example can be obtained using the Kroutchouk measure and related orthogonal polynomials.

A first application of our results provides the weighted polyanalytic Bergman kernel of the unit disc {z ∈ C : |z| < 1}. More precisely, for α > -1, we consider the space A 2 α,q of all square integrable of q-analytic functions with respect to the measure dν α (z) :

(1 -|z| 2 ) α dA(z)
π , where dA(z) is the Lebesgue measure on D. We will prove the following Theorem D. The space A 2 α,q is a Hilbert space which coincides with the closure of the q-analytic polynomials in L 2 (ν α ) and its reproducing kernel is given by

K α,q (z, w) = q α+q-1 α (1-zw) q-1 (1-z w) α+q+1 q-1 j=0 (-1) j q-1 j α+q+j α+q-1 |z-w| 2j |1-z w| 2j . for all z, w ∈ D.
We point out that when α = 0, this result was established by Koshelev [START_REF] Koshelev | The kernel function of a Hilbert space of functions that are polyanalytic in the disc[END_REF] by different method limited to the case α = 0, but does not work for α = 0.

A third application of our results provides the weighted polyanalytic Bergman kernel for the weighted Fock space. Namely, let α > 0, and denote by F α,q (C) the space A 2 α,q of all square integrable of q-analytic functions with respect to the measure dν α (z) : |z| 2α e -|z| 2 dA(z) π , where dA(z) is the Lebesgue measure on C We will establish the following Theorem E. The space F α,q is Hilbert space which coincides with the closure of the q-analytic polynomials in L 2 (ν α ) and its reproducing kernel is given by K α,q (z, w) = e z wL α+1 q-1 (|z -w| 2 ) , for all z, w ∈ C where L α q-1 is classical weighted Laguerre polynomials of degree q -1 and weight α. We point out that when α = 0, this result was established by Haimi and Hedenmalm [START_REF] Haimi | The polyanaytic Ginibre ensembles[END_REF] using a different method which does not go trough for α = 0.

PRELIMINARY RESULTS

We collect a few preliminary results from the refrences [START_REF] Dunkl | Orthogonal Polynomials of Several Variables[END_REF] or [START_REF] Ismail | Classical and Quantum Orthogonal Polynomials in One Variable[END_REF]. Let s = (s n ) be a Stieltjes moment sequence and µ ∈ S representing measure of s. We assume in this section that the support of µ has N (µ) ≥ q elements. For each non-negative integers n ≤ N (µ) -1 set

D µ,n := s 0 s 1 ... s n s 1 s 2 ... s n+1 . . . . . . • • • . . . s n s n+1 • • • s 2n and for x ∈ C, let D µ,n (x) := s 0 s 1 ... s n s 1 s 2 ... s n+1 . . . . . . • • • . . . s n-1 s n • • • s 2n-1 1 x • • • x n
It is well-known that the sequence (P µ,n )

N (µ)-1 n=0
of orthogonal polynomials with respect to the measure dµ(x) is given by

P µ,n (x) = D µ,n (x) D µ,n-1 D µ,n . (3.1)
so that the reproducing kernel Q µ,n is given by

Q µ,n (x, y) = n j=0 D µ,j (x)D µ,j (y) D µ,j-1 D µ,j . (3.2)
We recall the following classical theorem of Heine a proof of which can be found in [START_REF] Dunkl | Orthogonal Polynomials of Several Variables[END_REF] Lemma 3.1. The determinants D n and D n (x) have the integral representations

D n = 1 (n + 1)! [0,+∞[ n+1 1≤j<k≤n+1 (x j -x k ) 2 dµ(x 1 ) • • • dµ(x n+1 ) (3.3) D n (x) = 1 n! [0,+∞[ n n i=1 (x -x i ) 1≤j<k≤n (x j -x k ) 2 dµ(x 1 ) • • • dµ(x n ) (3.4)
In what follows we shall fix the measure µ, and for each d ∈ N 0 , we consider determinants and orthogonal polynomials with respect to the measure x d dµ(x). Then we simply set

D x d µ(x),n := D d,n and D x d µ(x),n (x) := D d,n (x). (3.5)
The sequence of orthogonal polynomials with respect to the measure x d dµ(x) will be then denote by (P d,m )

N (µ)-1 m=0
and it is given by

P d,n (x) = D d,n (x) D d,n-1 D d,n . (3.6) so that the corresponding reproducing kernel Q d,n is given by Q d,n (x, y) = n j=0 D d,j (x)D d,j (y) D d,j-1 D d,j . (3.7)
Lemma 3.2. Suppose that the support of µ is unbounded. Then for any positive integer n and x ∈ [0, +∞[, there exist t x > 0 and a constant C x > 0 such that

|P d,n (x)|µ ([t x , +∞[) ≤ Cx t d/2 (3.8) for all t ≥ t x .
Proof. In view of Lemma 3.1 by Cauchy-Schwarz inequality we see that

|D d,n (x)| 2 ≤ D d,n-1 [0,+∞[ n n i=1 (x -x i ) 2 1≤j<k≤n (x j -x k ) 2 (x 1 • • • x n ) d dµ(x 1 ) • • • dµ(x n ) Since the degree n of the polynomial D d,n (x) is positive, there is t x > 0 such that |D d,n (x)| ≤ |D d,n (t)|, for all t ≥ t x
and thus by Lemma 3.1 we get

|D d,n (x)| 2 D d,n-1 +∞ tx x d n+1 dµ(x n+1 ) ≤ +∞ tx |D d,n (x n+1 )| 2 x d n+1 dµ(x n+1 ) ≤ (n + 1)!D d,n
Taking C = (n + 1)! and using (3.6) completes the proof.

ORTHOGONAL POLYNOMIALS WITH RESPECT TO ROTATION INVARIANT MEASURES

Throughout this section, fix an element s = (s d ) ∈ S * q and let µ ∈ M(s). For each pair of non-negative integers (d, n), with d arbitrary and n ≤ q -1, let (P d,k ), k ∈ {0, • • • , n} be a sequence of orthonormal polynomials of Hilbert space P n (x d µ) equipped the L 2 (x d dµ(x)) inner product. For all integers m, n ∈ N 0 , set m ∧ n := min(m, n). By the change of variables formula, we see that

Ds H m,n (z)H m ,n (z)dν(z) = +∞ 0 T H m,n (r 1/2 ξ)H m ,n (r 1/2 ξ)dσ(ξ)dµ(r) = +∞ 0 H m,n (r 1/2 )H m ,n (r 1/2 )dµ(r) T ξ m+n ξ m +n dσ(ξ) = δ m+n ,m +n δ m∧n,m ∧n = δ (m,n),(m ,n ) .
This completes the proof.

Lemma 4.2. Let n and d be positive integers such that n ≤ q -1. Consider a polynomial f in n-

variables such that f (x 1 , • • • , x n ) > 0 for all pairwise distinct elements x 1 , • • • , x n of [0, +∞[, and set
γ d,n (f ) := +∞ 0 • • • +∞ 0 f (x 1 , • • • , x n )(x 1 • • • x n ) d dµ(x 1 ) • • • dµ(x n ) (4.3) Then lim d→+∞ γ d+1,n (f ) γ d,n (f ) = lim d→+∞ s d+1 s d n (4.4) Proof. Let η be the image of the measure on [0, +∞[ of f (x 1 , • • • , x n )dµ(x 1 ) • • • dµ(x n ) under the map (x 1 , • • • , x n ) → x 1 • • • x n . Then γ d,n (f ) := +∞ 0 x d dη(x) (4.5)
so that by [START_REF] Bommier-Hato | Hankel operators and the Stieltjes moment problem[END_REF] we see that

lim d→+∞ γ d+1,n (f ) γ d,n (f ) = R (4.6)
where R is the supremum over all t > 0 such that t is in the support of η. Moreover, it can be easily checked that if t > 0, then t is in the support of η if and only if t 1 n is in the support of µ. This completes the proof.

Now we prove Theorem A

Proof of Theorem A. We only need prove that for all non-negative real numbers x and y, the series

S s,q (λ) = q-1 n=1 +∞ m=0 λ m P m,n (x)P m,n (y), (4.7)
converges uniformly on compact sets of D s . We shall distinguish two cases. First, assume that the support of µ is bounded; that is R s is finite. In view (3.6), the latter series can be written if the form

S s,q (λ) := q-1 n=1 +∞ m=0 λ m D m,n (x)D m,n (y) D m,n-1 D m,n .
Using the integral expressions (3.3) and (3.4) with respect to the measure r m dµ(r) instead of dµ(r), we see that D m,n (x) is a finite sums of terms of the form x j γ m,n (f ) where j ∈ N 0 and f is a function of the form

f (x) = x k 1 1 • • • x kn n 1≤j<k≤n (x j -x k ) 2 . (4.8)
The same holds for D m,n (y) with y instead of x. Finally, we observe that D m,n = γ m,n (g), and D m,n-1 = γ m,n-1 (g) where

g(x) := 1≤j<k≤n (x j -x k ) 2 . (4.9)
Therefore the series S s,q is a linear combination of series of the form S s,q,j,l (λ) :=

q-1 n=1 +∞ m=0 λ m x j y l γ m,n (f )γ m,n (h) γ m,n (g)γ m,n-1 (g)
where f and h are of the form (4.8) and g is given by (4.8). Appealing to Lemma 4.2 and using D'Alembert's rule yields that the series S s,q,j,l (λ) converges as long as |λ| < R s . From this it is also clear that the series converges uniformly on compact sets of D s .

Next, suppose that R s = +∞. Let x, y be arbitrary non-negative real numbers. Then by Lemma 3.2, there t x,y such that |P m,n (x)P m,n (y)| ≤ (n + 1)! t m , for all t ≥ t x,y . This proves that the series (4.7) convergence absolutely.

Finally, the inequality in Theorem A follows by Cauchy-Schwarz inequality. The remaining equality in the theorem is straightforward. This completes the proof.

Next, we denote by A 2 (s) the subspace of L 2 (ν) consisting of all functions of the form

f (z) = q-1 n=0 +∞ |m|=0 a m,n H m,n (z) on D s that satisfy q-1 n=1 +∞ m=0 |a m,n | 2 < +∞.
We equip the space A 2 (s) with the natural inner product

f, g s := q-1 n=0 +∞ m=0 a m,n b m,n , (4.10) for all members f (z) = q-1 n=0 +∞ m=0 a m,n H m,n (z) and g(z) = q-1 n=0 +∞ m=0 b m,n H m,n (z) of A 2 (s).
It is standard that this is a Hilbert space which contains all q-analytic polynomials, which is contained in L 2 (ν) and its inner product coincides with the scalar product inherited from the scalar product of L 2 (ν). Indeed, we have Theorem 4.3. The space A 2 (s) consists of q-analytic functions and its reproducing K s,q kernel is given by

K s,q (z, w) = F s,q (z w, |z| 2 , |w| 2 ), z, w ∈ D s ,
where F s,q is the function defined by (2.1).

Proof. By virtue of Theorem A, the series

K s,q (z, w) = q-1 n=0 +∞ m=0 H m,n (z)H m,n (w). (4.11)
converges uniformly for z w lying in a compact subset of D s . Since the system (H m,n ), m, n ∈ N 0 , n ≤ q -1 forms an orthonormal basis of A 2 (s), a little computing shows that

K s,q (z, w) = q-1 n=0 +∞ m=0 H m,n (z)H m,n (w) = q-1 n=0 +∞ m=0 (z w) m P m,n (|z| 2 )P m,n (|w| 2 ) + q-1 n=1 n-1 m=0 (z w) m P n-m,m (|z| 2 )P m,n (|w| 2 )
When q = 1, we are in the analytic case. Since P m,0 is constant, the latter sum gives

K s,1 (z, w) = +∞ m=0 (z w) m P m,0 P m,0 .
However, , when q ≥ 2, we have

K s,q (z, w) = +∞ m=0 (z w) m q-1 n=0 P m,n (|z| 2 )P m,n (|w| 2 ) + q-1 n=1 n-1 m=0 (zw) n-m P n-m,m (|z| 2 )P n-m,m (|w| 2 ) = F s,q (z w, |z| 2 , |w| 2 ). Now each element f of A 2 (s) admits a unique representation f (z) = q-1 n=0 +∞ m=0 a m,n H m,n (w).
By Cauchy-Schwarz inequality, it follows that this series converges uniformly on compact sets of D s and hence it defines a q-analytic function. Moreover, it can be easily checked

f (z) = f, K s,q (•, z) .
for all z ∈ D s . This completes the proof.

Now we are ready to prove Theorem B

Proof of Theorem B. It suffices to show that each q-analytic function which belongs to L 2 (ν) is an element of the space A 2 (s). Now let f be q-analytic function which belongs to L 2 (ν). By (1.4) we know that f has a unique representation of the form

f (z) = q-1 n=0 z n f n (z), z ∈ D s ,
where the functions f n are analytic on D s . Therefore, f can written in the form

f (z) = q-1 n=0 +∞ m=0 z n f m (z), z ∈ D s ,
where f m are anlytic polynomials and the series converges uniformly on compact sets of D s . In view of Theorem A, we see that f admits a unique representation of the

f (z) = q-1 n=0 +∞ m=0 c m,n H m,n , z ∈ D s ,
where c m,n are complex coefficients and the series converging uniformly on compact sets of

D s . Since f is in L 2 (ν) it follows that q-1 n=0 +∞ m=0 |c m,n | 2 < +∞ showing that f ∈ A 2 (s).
The proof is now complete.

THE POLYANALYTIC BERGMAN SPACE ON THE UNIT DISC

In this section apply our approach to different classes of orthogonal polynomials to produces natural examples of Hilbert spaces of polyanalytic functions.

We start with the weighted polyanalytic Bergman space on D. Consider the weighted Lebesgue measure on D given by

dA α (z) := 1 -|z| 2 α dA(z) π , α > -1,
where dA(z) is the Lebesgue measure on D. We denote by A α q (D), the weighted q-polyanalytic Bergman space on D where q ∈ N 0 and α > -1. This is the space of all q-polyanalytic functions f on D which are square integrable with respect to dA α (z).

It can be easily checked that the measure ν is the image measure in D of µ ⊗ σ under the map

(t, ξ) → √ tξ from [0, 1[×T onto D where µ is the measure [0, 1[ given by dµ(t) := (1 -t) α dt.
The corresponding moment sequence is (5.1)

s d = 1 0 t d (1 -t) α dt = Γ (d + 1) Γ (α + 1) Γ (d + α + 2) .
Lemma 5.1. Suppose that ϕ is an automorphism of the unit disc. The the Bergman kernel K q,α of A α q (D) follows the transformation rule

K q,α (z, ξ) = ϕ (z)ϕ (ξ) (α+q+1)/2 ϕ (z)ϕ (ξ) (q-1)/2 K q,α (ϕ(z), ϕ(ξ)) (5.2)
for all z, ξ ∈ D.

Proof. It is sufficient to assume that ϕ • ϕ(z) = z, for all z ∈ D. We recall the that the measure dA(z)

(1-|z| 2 ) 2 is invariant under the action of the automorphism group of the unit disc. We also observe that for any fixed ξ ∈ D, the function z

→ (ϕ )(z)) (α+q+1)/2 (ϕ (z)) (q-1)/2 K q,α (ϕ(z), ξ) is an element of A α q (D)
. By the reproducing property and change of variables formula we see that

(ϕ (z)) (α+q+1)/2 ϕ (z) (q-1)/2 K q,α (ϕ(z), ξ) = D (ϕ (w)) (α+q+1)/2 ϕ (w) (q-1)/2 K q,α (ϕ(w), ξ)K q,α (z, w)dA α (w) = D ϕ (w) (α+q+1)/2 (ϕ (w)) (q-1)/2 K q,α (w, ξ)K q,α (z, ϕ(w))dA α (w) = D (ϕ (w)) (α+q+1)/2 ϕ (w) (q-1)/2 K q,α (ξ, w)K q,α (ϕ(w), z)dA α (w) = ϕ (ξ) (α+q+1)/2
(ϕ (ξ)) (q-1)/2 K q,α (z, ϕ(ξ))

Replacing ξ by ϕ(ξ) the latter equalities yield

ϕ (z)ϕ (ξ) (α+q+1)/2 ϕ (z)ϕ (ξ) (q-1)/2 K q,α (ϕ(z), ϕ(ξ)) = K q,α (z, ξ)
This completes the proof.

We shall make use of the classical Jacobi polynomials P (α,d) n with parameters (α, d) and degree n. An explicit formula for these polynomials is given by

P (α,d) n (x) = 1 2 n n k=0 α + n k d + n n -k (x -1) n-k (x + 1) k . (5.3)
It is well-known by formula (3.96) in ( [START_REF] Shen | Spectral methods: Algorithms, Analysis and Applications[END_REF], p. 71) that these polynomials verify the equality

P (α,d) n (1 -2x) = Γ(n + α + 1) n!Γ(n + α + d + 1) n j=0 (-1) j n j Γ(n + j + α + d + 1) Γ(j + α + 1) x j . (5.4)
The Jacobi polynomials satisfy the orthogonality condition and hence for each non-negative integer d, the reproducing kernel of space of polynomials of degree at most q -1 with respect to the L 2 -inner product associated to the measure t d dµ(t) is then

Q d,q-1 (x, y) = q-1 n=0 P α,d n (2x -1)P α,d n (2y -1) h α,d n = q-1 n=0 P d,α n (1 -2x)P d,α n (1 -2y) h α,d n .
By the identity (3.114) in ( [START_REF] Shen | Spectral methods: Algorithms, Analysis and Applications[END_REF], p. 75) we see that Q 0,q-1 (x, 0) = Γ(q+α+1)) (q-1)!Γ(α+1) P 1,α q-1 (1 -2x) (5.7) so that by (5.4) we obtain F q,s (0, x, 0) = Q 0,q-1 (x, 0) = q α + q -1 α q-1 j=0 (-1) j q -1 j α + q + j α + q -1 x j

We observe that if z ∈ D, then K q,α (z, 0) = F q,s (0, |z| 2 , 0). Let z, w ∈ D let ϕ w (z) := z -w 1 -z w . By Lemma 5.1, we have

K q,α (z, w) = ϕ w (z)ϕ w (w) (α+q+1)/2 ϕ w (z)ϕ w (w) (q-1)/2 K q,α (ϕ w (z), 0) Since ϕ w (z)ϕ w (w) (α+q+1)/2 ϕ w (z)ϕ w (w) (q-1)/2 = (1 -zw) q-1 (1 -z w) α+q+1 and |ϕ w (z)| 2j = [z -w| 2j |1 -z w| 2j it follows that K q,α (z, w) = q α+q-1 α (1-zw) q-1 (1-z w) α+q+1 q-1 j=0 (-1) j q-1 j α+q+j α+q-1 |z-w| 2j
|1-z w| 2j .

WEIGHTED POLYANALYTIC FOCK SPACES

The second example is the weighted measure defined on C by

dν (z) := |z| 2α e -|z| 2 dA (z) , α > -1,
where dA (z) is the normalized Lebesgue measure on C. We denote by A α q (C) the weighted q-polyanalytic Fock space on C where q is a positive integer. This is the space of all q-analytic functions f on C which are square integrable with respect to dν(z). The measure ν is the image measure in C of µ ⊗ σ under the map (t, ξ) → t 1/2 ξ from [0, +∞[×T onto C where µ is the measure on [0, +∞[ given by dµ(t) := 1 Γ(α + 1)

t α e -t dt.

The corresponding moment sequence is (6.1)

s d = +∞ 0
t d e -t dt = Γ (α + d + 1) Γ(α + 1) .

We will use the classical weighted Laguerre polynomials L α n of degree n and weight α. These polynomials satisfy, (6.2)

+∞ 0 L d+α n (x) L d+α n (x) x d+α e -x dx = Γ (d + n + 1) n! δ n,n .
They have the following explicit representation

(6.3) L d+α n (x) = n r=0 (n + d + α)! r!Γ (n + d + α + 1 -r) (-x) n-r (n -r)! .
We point out some useful formulas, the first one is due to Bailey [START_REF] Bailey | On the product of two Legendre polynomials with different arguments[END_REF] (6.4)

L d+α n (x) L d+α n (y) = Γ (d + α + n + 1) n! n l=0 (xy) n-l L d+α+2n+-2l l (x + y) (n -l)!Γ (d + α + n + 1 -l) .
The others are (6.5)

L d+α n (x -y) = n r=0 y r r! L d+α+r n-r (x) . (6.6) L d+α n (x -y) = e -y +∞ r=0 y r r! L d+α+r n (x)
which are easy consequences of the definition of L d+α n (x). To compute the series F q,s (λ, x, y) in this case, it is sufficient to calculate the following expression (6.7)

S α,q (λ) = q-1 n=0 +∞ d=-n n!λ d Γ (n + d + α + 1) L d+α n (x) L d+α n (y) .
using first (6.4) and then (6.5) and (6.6) we have

S α,q (λ) = q-1 n=0 λ -n n! +∞ d=n λ d+n L d+α n (x) L d+α n (y) Γ (d + α + n + 1) = q-1 n=0 λ -n n! +∞ d=0 λ d Γ (d + α + 1) L d+α-n n (x) L d+α-n n (y) = q-1 n=0 λ -n +∞ d=0 λ d n r=0 (xy) n-r L d+α+n-2r r (x + y) (n -r)!Γ (d + α -r + 1) = q-1 n=0 λ -n n r=0 (xy) n-r λ r (n -r)! +∞ d=r λ d-r Γ (d + α -r + 1)! L d+α+n-2r r (x + y) = q-1 n=0 λ -n n r=0 (xy) n-r λ r (n -r)! +∞ d=0 λ d Γ (d + α + 1) L d+α+n-r r (x + y) = q-1 n=0 λ -n n r=0 (xy) n-r λ r (n -r)! e λ L α+n-r r (x + y -λ) = q-1 n=0 e λ L α n x + y -λ - xy λ
and since q-1 r=0 L β r = L β+1 q-1 we deduce that (6.8) F q,s (λ, x, y) = e λ L α+1 q-1 x + y -λ -xy λ .

THE HIGHER DIMENSIONAL CASE

Consider p Stieltjes moment sequences s(1), • • • , s(p) ∈ S * q and for each j let µ j ∈ M(s(j)) and denote by ν j denote the image measure on C of µ j ⊗ σ under the map (t, ξ) → √ tξ from [0, +∞[×T onto C, where σ is the rotation invariant probability measure on the unit circle T in C. Then the support of each ν j is contained in the closure of the disc D j centred at 0 with radius R s(j) . Then we set ν := ν 1 ⊗ • • • ⊗ ν p and consider the Hilbert space L 2 (ν) of square integrable complex-valued functions in D 1 × • • • × D p with respect to the measure ν. We denote by A 2 ν,q the space of those q-analytic functions on D 1 × • • • × D p which are square integrable with respect to ν. The natural inner product inherited from that of L 2 (ν) turns A 2 ν,q into a pre-Hilbert space. We are now prepared to state the higher dimensional analog of Theorem B.

Theorem B'. The space A 2 ν,q is Hilbert space which coincides with the closure of the q-analytic polynomials in L 2 (ν). Moreover, for each set compact K ⊂ D s we have that

sup z∈K |f (z)| ≤ C f L 2 (ν)
for all q-analytic polynomials f ∈ L 2 (ν), where

C = C(K) := sup z∈K n j=1 F q,s(j) (|z j | 2 , |z j | 2 , |z j | 2 ).
Furthermore, the reproducing kernel of A 2 ν is given by R ν,q (z, w) = p j=1 F q,s(j) (z j wj , |z j | 2 , |w j | 2 ), z, w ∈ D s .

Remark C'. As in Remark C, in the one variable case, when the measures µ j have a finite support with exacly q elements, provides the polyanalytic Cauchy type kernel of the unit polydisc

D n := {z = (z 1 , • • • , z n ) ∈ C : max j=1,••• ,n |z j | < 1}.
In a similar manner, we obtain the weighted polyanalytic Bergman kernel of the unit polydisc {z = (z 1 , • • • , z n ) ∈ C : max j=1,••• ,n |z j | < 1}. More precisely, for α > -1, we consider the space A 2 α,n,q of all square integrable of q-analytic functions with respect to the measure dν α,n (z) :

1 π n n j=1
(1 -|z j | 2 ) α dV (z), where dV (z) is the Lebesgue measure on C n . We will obtain the following Theorem D'. The space A 2 α,n,q is Hilbert space which coincides with the closure of the q-analytic polynomials in L 2 (ν α ) and its reproducing kernel is given by K α,n,q (z, w) = n j=1 K α,q (z j , w j ) for all z = (z 1 , • • • , z n ), w = (w 1 , • • • , w n ) ∈ D n , where K α,q is the weighted polyanalytic Bergman kernel of the unit disc.

We also obtain by similar arguments the weighted polyanalytic Bergman kernel for the weighted Fock space in C n . Namely, let α > 0, and denote by F α,q (C) the space A 2 α,q of all square integrable of q-analytic functions with respect to the measure dν α (z) : |z| 2α e -|z| 2 dV (z)

π n , α > -1 where dA(z) is the Lebesgue measure on C n We will establish the following Theorem E'. The space F α,q is Hilbert space which coincides with the closure of the q-analytic polynomials in L 2 (ν α ) and its reproducing kernel is given by K α,n,q (z, w) = e z w n j=1 L α+1 q-1 (|z j -w j | 2 )

for all z = (z 1 , • • • , z n ), w = (w 1 , • • • , w n ) ∈ C n .

. 2 ) 4 . 1 .

 241 (4.1) and for all z = rξ ∈ C, r ≥ 0, |ξ| = 1,H m,n (z) := r |m-n| ξ m ξn P |m-n|,m∧n (r 2 ). (4Lemma The family (H m,n ) forms an orthogonal system in L 2 (ν). Proof. Let (m, n), (m , n ) ∈ N 2 0 .We first observe that for m + n = m + n, we have +∞ 0 H m,n (r 1/2 )H m ,n (r 1/2 )dµ(r) = +∞ 0 r |m-n| P |m-n|,m∧n (r)P |m-n|,m ∧n (r)dµ(r) = δ m∧n,m ∧n

  1)x d (1 -x) α dx = δ n,n h α,+ n + 1)) Γ (d + n + 1) Γ (α + d + n + 1) (α + d + 2n + 1).(5.6)


Saheb Maghsoodloo

Baptiste Bouillot

Jérome Douzet

Son Ho-Van

Jean-Michel Herri

Crystalliza

Crystallization of mixed gas hydrates under equilibrium and non-equilibrium conditions: PVTx measurements and thermodynamic modeling

Introduction

  MAGHSOODLOO Saheb*, BOUILLOT Baptiste, Douzet Jérôme, HO-VAN Son, HERRI Jean-Michel Ecole des Mines de Saint-Etienne, SPIN, CNRS 5307, LGF, F-42023 Saint-Etienne, France * Corresponding author: saheb.m@emse.fr What are the gas hydrates? Experimental set-up and procedures Research and problematics Schematic diagram of the experimental procedures: a) Quick crystallization process. b) volume) do gas hydrates form? How is the guest distribution in hydrate phase? How does the rate of crystallization affect the final state? have a significant impact on clathrate hydrate applications which thermodynamic equilibrium is essentially taken into account such as energy storage and transportation or CCS. Different hydrate composition and pressure at final state


Jim Mainprice
email: jmainpri@laas.fr

Mamoun Gharbi
email: magharbi@laas.fr

Thierry Siméon

Rachid Alami
email: rachid@laas.fr

Sharing effort in planning human-robot handover tasks

For a versatile human-assisting mobilemanipulating robot such as the PR2, handing over objects to humans in possibly cluttered workspaces is a key capability. In this paper we investigate the motion planning of handovers while accounting for the human mobility. We treat the human motion as part of the planning problem thus enabling to find broader type of handing strategies. We formalize the problem and propose an algorithmic solution taking into account the HRI constraints induced by the human receiver presence. Simulation results with the PR2 robot illustrate the efficacy of the approach.

I. INTRODUCTION

Robots and humans working together in cooperation can accomplish more sophisticated tasks beneficing from the combined power/precision of the robot and the reasoning of the human. This relationship brings new problems and challenges to robotics research.

In this paper we focus on the particular problem of finding good object handover configurations, that is formulated as a special instance of the motion planning problem [START_REF] Choset | Principles of robot motion: theory, algorithms, and implementation[END_REF], [START_REF] Latombe | Robot motion planning[END_REF], [START_REF] Lavalle | Planning algorithms[END_REF]. We consider robots such as the PR2, with navigation and manipulation capabilities and introduce the notion of "shared effort" in the handover plan.

Human-aware motion planning is a rather active area of research [START_REF] Kulic | Physiological and subjective responses to articulated robot motion[END_REF], [START_REF] Lam | Human-Centered Robot Navigation, Towards a Harmoniously Human-Robot Coexisting Environment[END_REF], [START_REF] Mainprice | Planning Human-aware motions using a sampling-based costmap planner[END_REF], [START_REF] Scandolo | An Anthropomorphic Navigation Scheme for Dynamic Scenarios[END_REF], [START_REF] Sisbot | Human aware mobile robot motion planner[END_REF]. These works generally use a costmap approach in order to account for the human preferences according to the proxemics theory [START_REF] Hall | A system for the notation of proxemic behavior[END_REF]. Other work aim at making humanoid robots motions more human-like [START_REF] Asfour | Human-like motion of a humanoid robot arm based on closed-form solution of the inverse kinematics problem[END_REF], [START_REF] Sentis | A whole-body control framework for humanoids operating in human environments[END_REF], [START_REF] Toussaint | Optimization of sequential attractor-based movement for compact behaviour generation[END_REF], [START_REF] Zacharias | Making planned paths look more humanlike in humanoid robot manipulation planning[END_REF] to increase the robot-motion legibility. However we are not aware of motion planners that account for the human motions for generating more comfortable robot behaviors.

Human-robot handovers have been studied through several aspects such as relative placement [START_REF] Koay | Exploratory study of a robot approaching a person in the context of handling over an object[END_REF], arm motion trajectory and dynamics [START_REF] Shibata | Experimental analysis of handing over[END_REF], coordination and signaling [START_REF] Huber | Humanrobot interaction in handing-over tasks[END_REF], human safety, acceptance or comfort. Recently, Edsinger and Kemp, demonstrated in [START_REF] Edsinger | Human-robot interaction for cooperative manipulation: Handing objects to one another[END_REF] that humans tend to adapt to the robot shape and its limited grasping capacities by selecting handover positions to ease the transfer of the object to be exchanged. In the paper they discuss the importance of geometrical features to be taken into account such as the notion of private space [START_REF] Hall | A system for the notation of proxemic behavior[END_REF] to generate the reaching gestures. They claim that taking into account such constraints would result in more intuitive, more comfortable, and more efficient handovers. Cakmak et al [START_REF] Cakmak | Using spatial and temporal contrast for fluent robot-human hand-overs[END_REF] have introduced the notion of contrast to design the handover posture. This was motivated by observing that robot poses are often not conveying the Fig. 1. A young person that is in a hurry to get the object will express more comfort getting the object above the cluttered table while an older person will take the object from the side even with a longer wait.

intent because of the ambiguous boundary between carry and handover postures.

In this work we propose to consider the human motion while planning for robot-human handovers in a workspace possibly cluttered by obstacles. Our aim is to better account for human preferences such as eagerness to get the object or physical capacities. We introduce a number of criteria for the exchange to be safe, legible and fluent, and propose a formulation of the underlying planning problem. We also introduce a new parameter named mobility to balance between "shared effort" and comfort. We have developed an efficient algorithmic solutions to this planning problem. Figure 1 presents an example of a handover task solved by our planner with different settings of the mobility parameter.

The paper is organized as follows. Section II gives a formal definition to the handover planning problem. Section III introduces a simple but yet computationally efficient algorithm based on a combination of grid-based and samplingbased methods. Section IV presents the simulation and experimental results obtained using this approach.

II. THE HUMAN-ROBOT HANDOVER PLANNING PROBLEM

In this section we propose a formal definition of the handover planning problem. We first introduce the input and output of the problem, then we define the search space along with the feasibility and interaction constraints to be taken into account.

A. Inputs and outputs

In order to account for the human's motion we consider motion plans that are composed of two paths: the giver's and the receiver's paths that take them from their initial posture to a feasible handover goal posture.

Formally the inputs of the problem are the initial configurations of the robot q init r and of the human q init h . The problem also takes as input their kinematic model and the representation of the workspace W . The output plan will consist of the two paths τ r and τ h represented as parametric curves in their respective configuration space.

Next, we define the handover configuration space as the space of all feasible handover postures and further present a set of properties that are used to assess the quality of the output plan in terms of interaction constraints.

B. The handover configuration space

Let us consider the configuration space formed by the cartesian product between the robot configuration space C r and the human configuration space C h :

C = C r × C h
The configuration space C contains all configurations allowed by the kinematics of both the robot and the human. Thus solving the handover problem implies to find a handover configuration q hand = (q r , q h ) ∈ C. The configuration q hand belongs to a feasible subspace C f eas ∈ C. This subspace is a restriction of C regarding the constraints listed below: a) Collision free: at the handing configuration q hand , the robot and the human have to be collision free regarding self collision, collision with obstacles and with each other. This subspace is named C f ree .

b) Reachability: at the handing configuration q hand , the object to be exchanged has to be reacheable by both partners i.e. the gripper of the robot and the hand of the human must grasp the object. This subspace is named C reach . c) Stability: at the handing configuration q hand , the robot and the human have to be stable regarding newton law of mechanics. This subspace is named C stab .

d) Accessibility: this constraint corresponds to the existence of a collision free path between the handing posture q hand and the initial configuration q init = (q init r , q init h ). The set of handing configuration accessible from q init defines the subspace named C access .

The set of all feasible handover configurations C f eas is the intersection of these four subspaces:

C f eas = C f ree ∩ C reach ∩ C stab ∩ C access .
C f eas only restricts the configuration space to feasible handover motions where the robot and the human can meet in a handing posture. In this possibly large subspace of C many configurations may be undesirable because they do not respect social protocols or do not consider explicitly human preferences. In the next subsection we detail some important properties that have to be accounted for in order to generate valid human-robot handovers regarding intuitive social rules and the more formal proxemics theory.

C. HRI constraints

In order to account for the safety of the interaction and the legibility of the robot's intentions, we have introduced in [START_REF] Sisbot | Human aware mobile robot motion planner[END_REF] and [START_REF] Mainprice | Planning Human-aware motions using a sampling-based costmap planner[END_REF] a set of HRI constraints that rely on notions from the proxemics theory [START_REF] Hall | A system for the notation of proxemic behavior[END_REF] as well as user studies such as [START_REF] Koay | Exploratory study of a robot approaching a person in the context of handling over an object[END_REF]. These constraints have been introduced and used in [START_REF] Sisbot | Human aware mobile robot motion planner[END_REF] to generate navigation human-aware motions. They have also been used in [START_REF] Mainprice | Planning Human-aware motions using a sampling-based costmap planner[END_REF] to generate comfortable handover motions in cluttered environments. Each interaction constraint is associated to a function that assesses the constraint. These constraints can be summarized in three sets.

First, we consider comfort constraints that prevent generating discomfortable configurations. They integrate criteria such as robot proximity [START_REF] Hall | A system for the notation of proxemic behavior[END_REF], robot visibility [START_REF] Koay | Exploratory study of a robot approaching a person in the context of handling over an object[END_REF] to limit the effect of surprise and musculoskeletal comfort of a given posture [START_REF] Marler | A new discomfort function for optimization-based posture prediction[END_REF]. We referred to the corresponding cost as c comf in the rest of the paper.

Second, we consider motion constraints to generate reasonable effort on the human side. These constraints depend on two parameters: displacement, and standing. The length of the path τ h that the human will have to walk and wether the human needs to stand up are used to assess this constraint. We will refer to the corresponding cost as c mot .

Finally, we also account for fluency constraints, limiting the total duration of the handover, and favoring efficient plans. For this, we consider a cost c length related to the maximum value of the time taken by either the robot or the human to reach the handover configuration.

Some of these desired proprieties such as the human displacement and the action duration may contradict one another. To balance the impact of the different properties on the output plan, we introduce a mobility parameter reflecting the giver's physical capabilities and his eagerness to obtain the object.

Indeed, the handover duration may generate discomfort if it does not match with the human possible eagerness or urgency to get the object. High mobility values will balance motion and comfort constraints to favor quicker plans, resulting in the final cost defined as :

c = (c mot + c conf ) * (1 -m) + c length * m
where m ∈ [0 1] is the mobility factor. As explained in next section these interaction constraints and their corresponding cost functions are evaluated during the planning process and are combined together according to the human preferences modeled by the mobility parameter.

III. ALGORITHM

This section presents the handover planner that was developed to compute human-robot handovers while accounting for the interaction constraints introduced above. The approach relies on a combination of grid-based and samplingbased algorithms that consider the workspace obstacles and the kinematics models of both the human and the robot. After some grid based preprocessing, the method consists of iteratively sampling feasible handover configurations, evaluating their cost and finally returning the minimal cost plan obtained.

The main steps of the handover planner are sketched in Algorithm 1. An initialization phase, called initGrids, computes the accessibility of the human and the robot in the plane which are stored in two planar grids. These grids also contain the navigation distance to the robot and human initial position (see Figure 2). In this phase, two preselected sets of human-robot handover configurations are also loaded for a standing and sitting human (see Figure 3).

After the initialization phase, each iteration consists of computing a handover configuration q hand that encodes for the robot and the human DoFs.

The navigation parameters p = (x, y, θ) of the human at q hand , are first generated in the SampleHumanPos function by sampling a random position in the human accessible space. The navigation path of the human τ h to reach this position is computed by using a standard technique [START_REF] Latombe | Robot motion planning[END_REF] consisting of descending the distance gradient in the preprocessed human grid. Note that the robot path τ r is computed similarly by descending the distance gradient in the robot grid.

The position p is then transformed into to a fully specified handover configuration q hand in the BestFeasibleConf function that iterates through the set of preselected humanrobot handing configurations loaded in the initialization phase. Then if no feasible robot path τ r is found between the robot initial position and the handover configuration the algorithm returns to step one. Otherwise the cost of the solution plan is evaluated and stored if it improves the minimum cost solution found by previously computed handover plans.

The method loops over these steps until the stopping criterion is satisfied. In the current implementation the stoping conditions combines two stoping criteria : maximum time or a minimal improvement of the best current solution. The final robot path τ r consists of a set of way points corresponding to the trasversed cells centers interpolated by straight lines. The orientation θ along τ r is selected implicitly by facing the robot to the next way point.

In the next subsections we further detail the processing done during the initialization phase and the three steps of the algorithm. We also describe some additional pre-processing that can be done to speed-up the sampling of constrained handover solutions.

A. Distance propagation and initialization

In order to facilitate the computation of feasible handover configurations and the cost evaluation of those solutions, the method integrates a precomputing phase in which two dimensional grids are constructed and processed. Two grids, depicted on Figure 2, one for the robot and one for the human, provide an approximation of the free-space and the navigation distance to the initial position. This enables to determine which region of the workspace are accessible to the human and the robot at a very low computational cost.

The free-space grids are computed using bounding cylinders of the robot and the human in resting postures. The resting postures, also depicted in Figure 2, correspond to navigation configuration of the arms. A cell is marked as free if when placed at its center, the corresponding bounding cylinder does not overlap with the workspace obstacles.

The accessible space and the navigation distance to the initial position of a cell are simultaneously computed with a standard wave propagation technique. Figure 2 shows the propagated distance of the robot and the human from their initial position, where green cells are close to the initial position and red cells are far.

The initialization phase also loads a set of predefined handing configurations illustrated in Figure 3. These humanrobot handover configurations are named Q HR in the rest of the paper. They are selected offline and do not depend on the workspace nor on the absolute position of the human and the robot. Thus each configuration is defined relatively to the human position and consists of the human's and the robot's arm DoFs. The selection of these configurations can be done by sampling relative human-robot positions and using them to compute the associated handover configurations.

B. Sampling human positions

The first step of each iteration consists of sampling a human position and orientation p = (x, y, θ) inside the accessible space stored in the pre-processed grid. In order to sample this triplet a cell is selected then a point is sampled inside the cell and finally an orientation is randomly sampled.

Since each sampled position p leads to one and one only best handover plan, it is important to sample the positions that yield better solutions. We provide in Section III-D, two enhancements of the preprocessing phase to bias both the selection of the cell and the orientation of the human.

C. Returning the best feasible configuration

The configurations Q HR illustrated in Figure 3 are sorted according to the c comf cost (see Section II-C). When searching for the best feasible handover configuration at human position p, the first feasible configuration is selected (i.e. collision free and accessible to the robot).

This process enables to find constrained handover configurations, e.g. a small slot in a wall that separates disconnected parts of the workspace. Moreover, since the configurations are sorted, the best configuration regarding comfort cost is quickly determined without recomputing cost.

D. Reduce the search space and biasing sampling

When the human and the robot lie in separated parts of the workspace, such depicted in Figure 4, the algorithm samples the human accessible space without considering the robot accessibility and reaching capacity. The human direction is also sampled randomly which is often away from the robot approaching direction.

In order to speed-up and ameliorate the generation of feasible handover configurations we propose two enhancements to the basic version. First, we construct a smaller fused grid which contains only human cells that can lead to potentially valid handovers by discarding cells that are unreachable to the robot. Secondly, a value which approximates the total cost of the candidate handover solution is precomputed and stored in the grid. Human position sampling is then biased to low cost regions of the accessible space stored in the fused. The sampling of the human direction is also biased to an estimate of the robot approaching direction.

1) Fused grid: To generate the fused grid of the human and robot accessible space the minimal and maximal humanrobot distances in the preselected set of handing configuration Q HR are computed. These min and max values are used to define a crown region around the human such as illustrated in Figure 5. Then each accessible cell in the human grid which crown region does not overlap any cell in the robot grid (i.e. that can not yield a valid handover configuration) are suppressed from the human grid.

Hence, by discarding the cells that are not within reaching distance from the robot accessible space a smaller fused grid is obtain, such as depicted in Figure 4.c. Note that the fused grid does not account for 3D collision checking which is performed in the exploration phase.

2) Bias sampling: In order to bias the sampling of human position to candidates that will yield better handover configurations, each cell in the fused grid is filled with path length cost and motion length cost as follows:

c B = c mot * (1 -m) + c length * m
where m is the mobility parameter, this value c B approximates the final c presented in Section II-C that is used evaluate the quality of the candidate handover solution.

In order to bias the human direction sampling, the valid cell that minimizes the robot motion from its initial position to the crown region (in red in Figure 5) is stored in the fused grid. When sampling θ, directions facing this cell are favored.

Next section provides simulation results of this algorithm with different settings of the m parameter. We also present a user study that has been conducted to validate this approach.

IV. RESULTS

In this section we report the capability of the algorithm to find handover plans in workspaces containing home furniture such as tables and shelves. We report the strategies produced by the planner using different values of m and the convergence rate of the algorithm when using different pre-processing variants and their sampling schemes. We also discuss discretization issues.

In order to assess its performance, the algorithm has been implemented, along with test environments, into the path planning software Move3D [START_REF] Simeon | Move3D: a generic platform for path planning[END_REF] and simulation were performed on a 2.26GHz INTEL processor.

A. Influence of the mobility parameter

Figure 6 shows three handover strategies that have been computed for the same problem using three values of the m parameter. For low values of m, the human is supposedly less involved, asked as little effort as possible, on the contrary high values of m supposedly require more effort and participation of the human resulting in quicker handover strategies.

• m = 0 : Generates a long path for the robot to reach the handover position and the human does not move.

• m = 0.35: A shorter robot path to a feasible handover position over the table is allowed by small displacement of the human.

• m = 1: Shared effort between robot and human enables a constrained handover position through the shelves. Note that the planner is able to find different feasible handover positions for the same initial position of the human and the robot in the same environment. The resulting plan accounts for the feasibility of the handover position and motion using the 3D models of the human and the robot even though planning of navigation motion is performed in 2D cartesian space. The simplest case (m=0) is not shown in a figure since all variants converge to the displayed solution after a couple of iterations. However for the two more complex cases the basic pre-processing shows difficulty to find handover over the table (m=0.35) or through shelves (m=1). In order to generate more direct (and lower cost) handovers the basic version requires much more iterations than the other variants. The basic method keeps sampling handover configurations in the free space similar to the ones shown in Figure 6.a and 6.d, but placed elsewhere in the free-space.

Thanks to fused grids the algorithm generates more samples on the boundary of the free space (close to the tables) and thanks to bias it discovers more easily the solution through the shelves. Figure 8 shows similar results averaged over 300 runs of the planner which confirms these results. In average a good quality solution is obtained in less than 1 sec with the fused grid and bias sampling preprocessing.

C. Influence of discretization

Figure 9 shows for the case m=1 that the grid size has little influence on the performance of the handover sampling stage. The influence is limited to the preprocessing stage for which distance propagation and fused grid computation depend on the resolution. Note that the preprocessing is problem dependent (initial position of the robot and the human) and has to be performed for each query.

In terms of preprocessing time, the basic preprocessing requires 200msec (resp. 660msec) for a resolution of 20cm (resp. 10cm) and the additional cost of computing the fused grid is 68msec (resp. 1109msec). Thus, using a resolution of 20cm, the preprocessing stays below 1sec, however it may become higher for discretization bellow 10cm. The performance above tend to indicate that this approach could be used to dynamically adapt the computed handover motion plan during execution in order to account for possible human motions or to adapt the mobility factor if the human motion does not look compatible with the proposed handover location.

V. IMPLEMENTATION AND USER STUDY

We have conducted an HRI experiment confronting the participants to choices of handover configurations that are outputs of our planner: the shortest-time feasible plan at the cost of substantial effort asked to the human, or the plan that minimizes the human effort, at the cost of low global performance.

The handover planner has been implemented on the PR2 robot. The trajectories of the arm were planned within Move3D and executed with a Sof tM otion controller [START_REF] Broquère | Soft motion trajectory planner for service manipulator robot[END_REF] limiting the motion in jerk, accelerations and velocity. The moving objects were localized with a tag detection module based on ARToolKit [START_REF] Prince | 3d live: Real time captured content for mixed reality[END_REF]. We made use of an external Kinect to localize humans in the scene we also controlled the head of the robot and diffused sound on the robot speakers.

Figure 10 shows two participants being handed the object in two distinct settings of the m parameter. The 34 subjects recruited at the LAAS-CNRS in Toulouse, France were faced with one handover only. At their arrival, the subjects were briefed before the interaction, the handover was video recorded and the participants were finally handed a survey.

We expected that accounting for the human mobility resulted in more fluent and more efficient hand-overs. We also expected that the mobility of the human receiver depended on the task and intrinsic parameters. Hence we hypothesized that appropriate matching and tuning of the mobility parameter regarding the human preferences would result in a smoother and a more comfortable human-robot interaction.

The environment was built such that to put the planner in a "substantially difficult" situation: save time at the cost of human effort and unease to perform the task. The robot was able too build a shared plan where it hands over the object through a hole that is not large enough to allow the robot nor the human to cross.

The participants were assigned to four distinct cases. • Shortest hand-over plan with chronometer • Shortest hand-over plan with 'sudoku'

• Less effort hand-over plan with chronometer • Less effort hand-over plan with 'sudoku' Where sudoku and chronometer corresponded to two distinct tasks, that supposedly increased or decreased the mobility of the participant. Objective and subjective measures (timing and survey) were used to validate the hypothesis. The results lent support to the hypothesized role of the mobility parameters and also corroborated the proxemics theory which has been the basis of this work.

VI. CONCLUSIONS AND FUTURE WORKS

This paper was motivated by the need for robot to plan for handover motions in possibly cluttered scenes while accounting for the human safety and comfort. Hence we have introduced a new type of path planning problem that we call the handover planning problem. The novelty of this approach is to account for human motions for planning the robot motion and the associated handover configuration. We have formalized problem and proposed a simple but efficient algorithm. This algorithm is based on a combination of simple sampling-based and grid-based techniques providing a time efficient solution. In order to account for the human motion we introduce a new parameter to qualify the human state we call mobility.

We have used this framework to compute handover solution while considering the human static. However the fast convergence times indicate that it can be used to dynamically adapt handover configuration while the human is moving. In future work we also aim to further formalize and generalize this approach to new human-robot interactions problems. We also plan to perform a more realistic user-study accounting for the human motion during execution of planned motion.

Fig. 2 .

 2 Fig. 2. The distance propagation (a) is human centered and (b) is robot centered. the green cells correspond to nearest positions, and the red the farthest.

Fig. 3 .

 3 Fig. 3. The preselected configurations of the robot relative to the human standing (similar configurations are generated for the human sitting).

Fig. 4 .Fig. 5 .

 45 Fig. 4. The human and the robot lie in separated part of a workspace, parted in two by a table. The accessible space of the robot (a) and human (b) and the fused grid (c) are computed.

Fig. 6 .

 6 Fig.6. Three values of the mobility parameters are used to generate three handover startegies. The first three pictures depict the resulting trajectories while the three bottom pictures show the final handover configuration that accounts for the 3D obstacles.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Unique convergence curve, with two settings of m, using the three pre-processing variants yielding three sampling strategies over the scenario depicted in Figure 6 (m=0.35 right, m=1 left)

Fig. 9 .

 9 Fig. 9. Averaged convergence over 300 runs on the scenario of Figure 6 with m=1. Three distinct grid sizes 10, 15 and 20 cm are superposed

Fig. 10 .

 10 Fig. 10. Two participants of the HRI handover experiment. The first one is handed the object with a low mobility and the second one with a high mobility.

Algorithm 1 :

 1 Computing handover plans input: Human initial position : p h Robot initial position : Mobility of the human : m output : The human and robot handover conf : q hand Human path : τ h

	Robot path : τr
	begin
	cost best ← ∞
	G ← initGrids(p h , pr, m);
	while StopCondition() do
	p ← SampleHumanPos()
	τ h ← DescendOnHumanGrid(p)
	q hand ← BestFeasibleConf(p)
	if q hand == N U LL then
	continue
	p rob ← GetRobotPos(q hand )
	τr ← DescendOnRobotGrid(p

rob ) if not τr == N U LL then continue cost ← ComputeCost(m,τ h ,τr,q hand ) if cost > cost best then continue else cost best ← cost StoreBest(q hand ,τ h ,τr) return end

VII. ACKNOWLEDGMENTS

This work has been conducted within the EU SAPHARI project (http://www.saphari.eu/) funded by the E.C. Division FP7-IST under Contract ICT-287513.