Dominique Lépingle

Boundary behavior of a constrained Brownian motion between reflecting-repellent walls

Stochastic variational inequalities provide a unified treatment for stochastic differential equations living in a closed domain with normal reflection and (or) singular repellent drift. When the domain is a polyhedron, we prove that the reflected-repelled Brownian motion does not hit the non-smooth part of the boundary. A sufficient condition for nonhitting a face of the polyhedron is derived from the one-dimensional case. A complete answer to the question of attainability of the walls of the Weyl chamber may be given for a radial Dunkl process.

Introduction

There have been many works about stochastic differential equations with reflection on the boundary of a domain. In some of them the domain is a convex polyhedron ( [START_REF] Harrison | Reflected Brownian motion on an orthant[END_REF], [START_REF] Varadhan | Brownian motion in a wedge with oblique reflection[END_REF], [START_REF] Williams | Reflected Brownian motion with skew symmetric data in a polyhedral domain[END_REF], [START_REF] Dai | Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedra[END_REF], [START_REF] Delarue | Hitting time of a corner for a reflected diffusion in the square[END_REF]]. A typical question in this setting is the following: does the continuous process hit the non-smooth part of the boundary? The answer depends on the drift and diffusion coefficients of the process and on the direction of reflection (normal or oblique). In particular, R.Williams [START_REF] Williams | Reflected Brownian motion with skew symmetric data in a polyhedral domain[END_REF] has proven that the Brownian motion with a skew symmetry condition on the direction of reflection does not touch the intersections of the faces of the polyhedron.

On the other hand there also exists an extensive literature about non-colliding Brownian particles ( [START_REF] Dyson | A Brownian-motion model for the eigenvalues of a random matrix[END_REF], [START_REF] Bru | Wishart processes[END_REF], [START_REF] Hobson | Non-colliding Brownian motion on the circle[END_REF], [START_REF] Grabiner | Brownian motion in a Weyl chamber, non-colliding particles, and random matrices[END_REF], [START_REF] O'connell | Random matrices, non-colliding processes and queues[END_REF]). Most of these works originate in the study of the eigenvalues of Gaussian matrix processes. These eigenvalues are solutions to systems of stochastic differential equations with a singular drift that prevents the particles from colliding. Extensions of these systems are Dunkl processes [START_REF] Rösler | Markov processes related with Dunkl operators[END_REF] that have recently been developed in connection with harmonic analysis on symmetric spaces. The radial part of a Dunkl process may be considered as a Brownian motion perturbed by a singular drift which forces the process to live in a cone generated by the intersection of a finite set of half-spaces ( [START_REF] Chybiryakov | Processus de Dunkl et relation de Lamperti[END_REF], [START_REF] Chybiryakov | Dunkl processes and their radial parts relative to a root system[END_REF]). Depending on the values of some parameter, the process may touch the walls of the cone or not.

Actually it is possible to unify both theories of (normal) reflection and strong repulsion within a common framework. This is the role of stochastic variational inequalities, also called multivalued stochastic differential equations (MSDE) that were mainly developed by E. Cépa ([4], [START_REF] Cépa | Problème de Skorohod multivoque[END_REF]). These equations are associated to a convex function in a domain of R d .

Depending on the boundary behavior of this function the diffusion will (normally) reflect on the boundary, hit the boundary without local time, or live in the open domain. We shall here follow this way and concentrate on a Brownian motion living in a convex polyhedral domain, bounded or unbounded. To each face of the polyhedron is associated a repelling force with normal reflection when the repulsion is not strong enough. In this setting we shall ask whether the process may hit the various faces. Our first task will be to rule out the possibility of hitting the intersection of two faces. Once this is achieved, the problem is now basically one-dimensional and we may use the ordinary scale function of real diffusions.

In several previous works ( [START_REF] Inukai | Collision or non-collision problem for interacting Brownian particles[END_REF], [START_REF] Cépa | No multiple collisions for mutually repelling Brownian particles[END_REF]), this issue has been studied in the particular case of the hyperplanes H ij := {x = (x 1 , . . . , x d ) ∈ R d : x i = x j }, i = j and presented as the problem of collisions between Brownian particles. There is a simple collision if two coordinates coincide and a multiple collision if at least three coordinates coincide at the same time. Because the d-dimensional Brownian motion does not hit the intersection of two hyperplanes, one can guess that an additional drift does not change anything. However a rigorous proof is necessary because the singularity of the drift makes useless the usual Girsanov change of probability measure. The counterexample of Bass and Pardoux [START_REF] Bass | Uniqueness for diffusions with piecewise constant coefficients[END_REF] also showed that uniform nondegeneracy of the diffusion term does not preclude multiple collisions.

As in [START_REF] Cépa | No multiple collisions for mutually repelling Brownian particles[END_REF] where the particular case of electrostatic repulsion was considered, our proof only uses basic tools from stochastic calculus, mainly McKean's martingale method [START_REF] Mckean | Stochastic integrals[END_REF] which was already used in [START_REF] Bru | Diffusions of perturbed principal component analysis[END_REF] to prove non-collision for the eigenvalues of Wishart processes. Another way could be to use the theory of Dirichlet forms as done in [START_REF] Inukai | Collision or non-collision problem for interacting Brownian particles[END_REF] where a general condition of non-collision has been obtained.

The paper is organized as follows. In Section 2 we introduce basic definitions and notations. The main features about stochastic variational inequalities are also recalled. Section 3 is devoted to non attainability of the edges of the polyhedron. In Section 4 we give a sufficient condition of non attainability of a single face. Section 5 presents some applications to Brownian particles with nearest neighbor interaction, Wishart processes and Dunkl processes.

Multivalued stochastic differential equation in a polyhedral domain

Let (Ω, F, (F t , t ≥ 0), P) be a filtered probability space endowed with the usual conditions and B = (B t ) be a (F t )-adapted d-dimensional Brownian motion starting from the origin. Let

Φ : R d → (-∞, +∞] (1)
be a lower semi-continuous convex function such that

dom(Φ) := {x : Φ(x) < ∞} (2)
has nonempty interior. Let D := Int(dom(Φ)) .

For simplicity of notation, we will assume that Φ is C 1 on D. If x ∈ ∂D, we say that the unit vector n(x) is a unit inward normal to

D at x if n(x).(x -z) ≤ 0 (4)
for any z ∈ D. Based on the results in [START_REF] Cépa | Equations différentielles stochastiques multivoques[END_REF], the following theorem has been proved in [START_REF] Cépa | Diffusing particles with electrostatic repulsion[END_REF] (see also Theorem 2.2 in [START_REF] Cépa | Brownian particles with electrostatic repulsion on the circle: Dyson's model for unitary random matrices revisited[END_REF]).

Theorem 1 For any F 0 -measurable random variable X 0 with values in D, there exist a unique continuous (F t )-adapted process X = {X t , 0 ≤ t < ∞} with values in D and a unique continuous (F t )-adapted non-decreasing process L = {L t , 0 ≤ t < ∞} such that

X t = X 0 + B t - t 0 ∇Φ(X s ) ds + t 0 n s dL s 0 ≤ t < ∞ L t = t 0 1 {Xs∈∂D} dL s 0 ≤ t < ∞ (5)
where n s is dL s -a.e. a unit inward normal to D at X s . For any 0 < T < ∞, T 0

1 {Xs∈∂D} ds = 0 (6) and T 0 |∇Φ(X s )| ds < ∞ . (7)
From now on we concentrate on a particular polyhedral setting. Let I := {1, . . . , m} where m ≥ 1. We consider a convex function Φ of the following form

Φ(x) := i∈I φ i (x.n i -a i ) (8)
where for any i ∈ I,

φ i is a convex l.s.c. function , φ i = +∞ on (-∞, 0), φ i is C 1 on (0, +∞) n i is a unit vector a i is a real number . (9)
We may assume all n i are different. Then,

∇Φ(x) = i∈I n i φ ′ i (x.n i -a i ) D = {x ∈ R d : x.n i > a i ∀i ∈ I} D = {x ∈ R d : x.n i ≥ a i ∀i ∈ I} . ( 10
)
As D is not empty, there exists a ball with center y ∈ D and radius b > 0 included in D. Let X t be the solution given by Theorem 1. For i ∈ I let

U i t := X t .n i -a i . (11)
We will need a strengthening of inequality (7) ( [START_REF] Cépa | Brownian particles with electrostatic repulsion on the circle: Dyson's model for unitary random matrices revisited[END_REF],Th.2.2).

Lemma 2 For any i ∈ I, for any 0 < t < ∞,

t 0 |φ ′ i (U i s )| ds < ∞ . (12)
Proof. This is clear if

φ ′ i (0+) > -∞. Let J := {j ∈ I : φ ′ j (0+) = -∞} (13)
and let 0 < ε < b be such that φ ′ j (u) < 0 for any j ∈ J and u ∈ (0, ε). For K ⊂ J let

A K := {x ∈ D : x.n j < a j + ε ∀j ∈ K, x.n j ≥ a j + ε ∀j ∈ J \ K} . ( 14
)
Then for t > 0

t 0 1 A K (X s )| j ∈K n j φ ′ j (U j s )|ds ≤ j ∈K t 0 1 A K (X s )|φ ′ j (U j s )|ds < ∞ . (15)
Using ( 7) we get

t 0 1 A K (X s )| j∈K n j φ ′ j (U j s )|ds < ∞ (16)
and therefore

-(b -ε) j∈K t 0 1 A K (X s )φ ′ j (U j s ) ds ≤ t 0 1 A K (X s ) j∈K (y -X s ).n j |φ ′ j (U j s )|ds ≤ t 0 1 A K (X s )|y -X s | j∈K |φ ′ j (U j s )|ds < ∞ (17)
from the continuity of X on [0, t]. Then for any j ∈ J

t 0 |φ ′ j (U j s )| ds = t 0 1 {U j s <ε} |φ ′ j (U j s )| ds + t 0 1 {U j s ≥ε} |φ ′ j (U j s )| ds = j∈K⊂J t 0 1 A K (X s ) |φ ′ j (U j s )|ds + t 0 1 {U j s ≥ε} |φ ′ j (U j s )| ds < ∞ . ( 18
)
For any J ⊂ I, J = ∅, we set

H J := {x ∈ R d : x.n j = a j ∀j ∈ J} K J := {x ∈ R d : x.n j = a j ∀j ∈ J, x.n j > a j ∀j ∈ J} σ J := inf{t > 0 : X t ∈ H J } τ J := inf{t > 0 : X t ∈ K J } . (19) Lemma 3 Let J ⊂ I and V := span{n j , j ∈ J}. If n(x) is a unit inward normal to D at x ∈ K J , then n(x) ∈ V .
Proof. Let v ⊥ V . For ε > 0 small enough,

z 1 = x + εv z 2 = x -εv satisfy z 1 .n j = a j ∀j ∈ J z 1 .n i > a i ∀i ∈ J z 2 .n j = a j ∀j ∈ J z 2 .n i > a i ∀i ∈ J . Then n(x).(x -z 1 ) ≤ 0 n(x).(x -z 2 ) ≤ 0 and therefore n(x).v = 0 .

Nonattainability of the edges

This section is devoted to the proof of the following theorem.

Theorem 4 For any J ⊂ I with card(J) ≥ 2,

P(σ J = ∞) = 1 .
Proof. a/ We first consider the initial condition X 0 . From [START_REF] Cépa | Diffusing particles with electrostatic repulsion[END_REF] we deduce that for any u > 0 there exists 0 < v < u such that X v ∈ D a.s. Using the continuity of paths and the Markov property we may and do assume that X 0 ∈ D in order to prove that σ J = ∞ a.s. b/ We will also assume that max

i∈I φ ′ i (0+) < 0 . (20)
If not we introduce for any 0 < T < ∞ the equivalent probability measure Q defined on F T by dQ dP := exp{c(B T .

i∈I n i ) - 1 2 c 2 T | i∈I n i | 2 } where c > max i∈I φ ′ i (0+) .
The continuous process

B ′ t := B t -ct i∈I n i
is a Q-Brownian motion on [0, T ] and now

dX t = dB ′ t - i∈I n i ψ ′ i (X t .n i -a i )dt -n t dL t where ψ i (u) := φ i (u) -cu i ∈ I .
If Q(σ J < T ) = 0 then P(σ J < T ) = 0 and if this is true for any T we obtain P(σ

J = ∞) = 1.
c/ We are now going to prove that σ I = τ I = ∞ a.s. (with m ≥ 2). For any J ⊂ I let

V J := span{n j , j ∈ J} q J := dim V J π J := orthogonal projection onto V J . (21)
If q I = 1, then m = 2, n 1 + n 2 = 0 and H I = K I = ∅. Assume now q I ≥ 2 and H I = ∅.

Choose some z ∈ H I and set

Z t := π I (X t -z) . (22)
Then

Z t = Z 0 + C t - i∈I t 0 n i φ ′ i (U i s )ds + t 0 n s dL s ( 23
)
where C is a q I -dimensional Brownian motion. Set

S t := |Z t | 2 .
Then

S t = S 0 + 2 t 0 Z s .dC s -2 i∈I t 0 U i s φ ′ i (U i s )ds + 2 t 0 Z s .n s dL s + q I t. (24)
From Lemma 2 we deduce that on ∂D = ∪ J⊂I K J Z s .n s = (X sz).n s = 0 and thus

t 0 Z s .n s dL s = 0 . Let 0 < T < ∞. For t < τ I ∧ T , log S t = log S 0 + 2 t 0 Z s .dC s S s -2 i∈I t 0 U i s φ ′ i (U i s ) S s ds + (q I -2) t 0 ds S s . ( 25
)
From the assumption made in b/ there exists 0 < c ≤ ∞ such that φ ′ i ≤ 0 on (0, c] and

- t 0
U i s φ ′ i (U i s ) Ss ds ≥ - t 0 U i s φ ′ i (U i s ) Ss 1 {U i s ≥c} ds ≥ -1 c T 0 |φ ′ i (U i s )| ds > -∞ . ( 26
)
We now proceed as in ( [START_REF] Mckean | Stochastic integrals[END_REF],p.47). As t → τ I ∧ T , the local martingale part in the r.h.s. of (25) either converges to a finite limit or oscillates between +∞ and -∞. Thus it does not converge to -∞ and a.s. S τ I ∧T > 0. Therefore P(τ I ≤ T ) = 0 and the conclusion follows since T is arbitrary.

d/ Let now J ⊂ I with 2 ≤ |J| ≤ m -1. We shall show by a backward induction on |J| that P(τ J = ∞) = 1. Remark that the backward induction assumption entails the equality σ J = τ J a.s.. As previously done we may assume q J ≥ 2 and K J = ∅. Select now z ∈ K J and set

Z t := π J (X t -z) = Z 0 + C t -j∈J t 0 n j φ ′ j (U j s )ds -i ∈J t 0 π J n i φ ′ i (U i s )ds + t 0 π J n s dL s ( 27
)
where C is a q J -dimensional Brownian motion. Let again S t := |Z t | 2 . For ε > 0 and r > 0 we set

τ ε := inf{t > 0 : S t + min i ∈J (U i t ) 2 ≤ 2 ε 2 } ρ r = inf{t > 0 : |X t | ≥ r} . (28)
From the induction assumption we infer that τ ε → ∞ as ε goes to 0. Let 0 < T < ∞. We introduce the equivalent probability measure Q defined on F T by

dQ dP = exp{ τε∧ρr∧T 0 i ∈J 1 {U i s ≥ε} φ ′ i (U i s ) n i .dC s -1 2 τε∧ρr∧T 0 | i ∈J 1 {U i s ≥ε} φ ′ i (U i s ) π J n i | 2 ds} . (29)
Then

D t := C t - τε∧ρr∧T 0 i ∈J 1 {U i s ≥ε} φ ′ i (U i s )π J n i ds is a q J -dimensional Q-Brownian motion on [0, T ]. For t ≤ τ ε ∧ ρ r ∧ T , S t = S 0 + 2 t 0 Z s .dD s -2 i∈I t 0 U i s φ ′ i (U i s )ds -2 i ∈J t 0 1 {U i s <ε} Z s .n i φ ′ i (U i s ) ds +2 L⊂I,L ⊂J t 0 1 K L (X s ) Z s .n s dL s + q J t (30)
and for t < σ J ∧ τ ε ∧ ρ r ∧ T , log S t = log S 0 + 2 t 0

Zs.dDs Ss

-2 j∈J t 0

U j s φ ′ j (U j s ) Ss ds -2 i ∈J t 0 1 {U i s <ε} φ ′ i (U i s ) Ss Z s .n i ds +2 L⊂I,L ⊂J t 0 1 K L (X s ) Zs.ns Ss dL s +(q J -2) t 0 ds Ss . (31)
From the induction hypothesis and the continuity of paths, if σ J < ∞ for any L ⊂ J there exists an interval (σ Jδ, σ J ] of positive length on which X s ∈ K L . Therefore -

σ J ∧τε∧ρr∧T 0 1 K L (X s ) Z s .n s S s dL s > -∞ . ( 32
)
For s < τ ε , if U i s < ε for some i ∈ J, then S s ≥ ε 2 and we obtain as well

- σ J ∧τε∧ρr∧T 0 1 {U i s <ε} φ ′ i (U i s ) S s Z s .n i ds > -∞ . ( 33
)
The other terms behave as in c/ and thus

0 = Q(σ J ≤ τ ε ∧ ρ r ∧ T ) = P(σ J ≤ τ ε ∧ ρ r ∧ T ) . ( 34
)
Letting ε go to 0, r and T to ∞ we get

P(σ J = ∞) = 1
and we are done.

Keeping off from a wall

We first recall some facts in the one-dimensional setting [START_REF] Lépingle | Equations différentielles stochastiques multivoques unidimensionnelles[END_REF]. Let φ : R → (-∞, +∞] be a convex lower semicontinuous function. Assume φ = +∞ on (-∞, 0) and C 1 on (0, +∞). Consider the one-dimensional equation

dY t = dB t -φ ′ (Y t )dt + 1 2 dL 0 t Y t ≥ 0 (35)
where L 0 is the local time of Y at 0. There are three types of boundary behavior:

repulsion φ(0) < ∞ weak: local time not zero φ(0) = ∞, 0+ exp{2φ} < ∞ middle: local time zero φ(0) = ∞, 0+ exp{2φ} = ∞ strong: boundary not hit
We shall check the behavior of the multidimensional process X accords with this classification in the neighborhood of the faces of the polyhedron. For any i ∈ I we respectively write H i , K i , σ i , τ i in place of H {i} , K {i} , σ {i} , τ {i} .

Proposition 5 For any i ∈ I such that φ i (0) = ∞ and any t > 0,

t 0 1 H i (X s ) dL s = 0 . ( 36
)
Proof. From the occupation times formula and Lemma 1 we obtain

∞ 0 L a t (U i ) |φ ′ i (a)| da = t 0 |φ ′ i (U i s )| ds < ∞ (37)
and from φ i (0) = ∞ and the continuity of a → L a t (U i ) we deduce

L 0 t (U i ) = 0 . ( 38
) Thus 0 = U i t -(U i t ) + = t 0 1 H i (X s )n i .dB s - t 0 1 H i (X s ) j∈I φ ′ j (U j s ) n i .n j ds + t 0 1 H i (X s ) n i .n s dL s = t 0 1 K i (X s ) n i .n s dL s = t 0 1 K i (X s ) dL s = t 0 1 H i (X s ) dL s . (39)
We now set for any i ∈ I and x ≥ 0

p i (x) := x 1 exp{2(φ i (u) -φ i (1))} du .
Theorem 6 For any i ∈ I such that p i (0) = -∞ or equivalently

0+ exp{2φ i } = ∞ , (40) then P
(σ i = ∞) = P(τ i = ∞) = 1.
Proof. From Ito formula and Proposition 5 we obtain

p i (U i t ) = p i (U i 0 ) + t 0 p ′ i (U i s )[dC i s - j =i n i .n j φ ′ j (U j s )ds + j =i 1 K j (X s ) n i .n j dL s ] (41)
where C i = B.n i is a one-dimensional Brownian motion. As in the proof of Theorem 2, let

τ ε := inf{t > 0 : U i t + min j =i (U j t ) ≤ 2 ε} ρ r = inf{t > 0 : |X t | ≥ r} . (42)
Let 0 < T < ∞. We again introduce the equivalent probability measure Q defined on F T by

dQ dP = exp{ τε∧ρr∧T 0 j =i 1 {U j s ≥ε} φ ′ j (U j s ) n i .n j dC i s -1 2 τε∧ρr∧T 0 | j =i 1 {U j s ≥ε} φ ′ j (U j s ) n i .n j | 2 ds} . (43)
Then

D i t := C i t - t∧τε∧ρr 0 j =i 1 {U j s ≥ε} φ ′ j (U j s ) n i .n j ds (44) is a Q-Brownian motion on [0, T ] and for t ≤ τ ε ∧ ρ r ∧ T , p i (U i t ) = p i (U i 0 ) + t 0 p ′ i (U i s )[dD i s - j =i 1 {U i s <ε} n i .n j φ ′ j (U j s )ds + j =i 1 K j (X s ) n i .n j dL s ] . (45)
As in the proof of Theorem 2, for any j = i,

- σ i ∧τε∧ρr∧T 0 1 {U i s <ε} p ′ i (U i s ) n i .n j φ ′ j (U j s ) ds > -∞ (46)
and

+ σ i ∧τε∧ρr∧T 0 1 K j (X s ) p ′ i (U i s ) n i .n j dL s > -∞ (47)
and then

0 = Q(σ i ≤ τ ε ∧ ρ r ∧ T ) = P(σ i ≤ τ ε ∧ ρ r ∧ T ) (48)
meaning that P(σ i = ∞) = 1.

Applications

Brownian particles with nearest neighbor repulsion

H.Rost and M.E.Vares [START_REF] Rost | Hydrodynamics of a one-dimensional nearest neighbor model[END_REF] have considered the following system:

dX 1 t = dB 1 t + φ ′ (X 2 t -X 1 t ) dt dX i t = dB i t + (φ ′ (X i+1 t -X i t ) -φ ′ (X i t -X i-1 t )) dt i = 2, . . . , n -1 dX n t = dB n t -φ ′ (X n t -X n-1 t ) dt (49)
where X 1 t < . . . < X n t and φ is a positive convex function on (0, ∞) satisfying

φ(0) = ∞ , φ(∞) = 0 , 1 0 (φ ′ (x)) 2 e -2φ(x) dx < ∞ . (50)
This is a MSDE where function Φ is given by [START_REF] Cépa | No multiple collisions for mutually repelling Brownian particles[END_REF] with

φ i (x) = φ( √ 2 x), n i = 1 √
2 (e i+1e i ), a i = 0 for i = 1, . . . , n -1 and e j the j-th basis vector. Condition (50) for non-collision is stronger than (40) as can be seen from Schwarz inequality:

∞ = (φ(0) -φ(1)) 2 ≤ 1 0 (φ ′ ) 2 e -2φ 1 0 e 2φ .

Wishart and Laguerre processes

Wishart processes have been introduced in [START_REF] Bru | Diffusions of perturbed principal component analysis[END_REF] and [START_REF] Bru | Wishart processes[END_REF]. If B is a n × n Brownian matrix, a Wishart process with parameters n and δ ≥ n + 1 may be obtained as a solution to the matrix-valued SDE

dS t = S t dB t + dB ′ t S t + δ I n dt . (51)
The eigenvalues process (λ 1 t , . . . , λ n t ) of {S t } satisfies

dλ i t = 2 λ i t dW i t + (δ + j =i λ i t + λ j t λ i t -λ j t ) dt 1 ≤ i ≤ n , (52)
and the square roots

r i t = λ i t dr i t = dW i t + 1 2 δ -n r i t dt + 1 2 j =i ( 1
r i t + r j t + 1
r i t -r j t ) dt (53)
where (W i , . . . , W n ) is a n-dimensional Brownian motion. N.Demni [START_REF] Demni | Radial Dunkl processes: existence and uniqueness, hitting time, beta processes and random matrices[END_REF] has remarked that this system is a MSDE with

Φ(r 1 , . . . , r n ) = - 1 2 [(δ -n) i log r i + i>j log(r i + r j ) + i>j log(r i -r j )] (54)
on {0 < r 1 < . . . < r n } and ∞ elsewhere. The system (53) has a strong solution for δ > n.

If δ = n, we must add to the right hand side of (53) a local time at 0 that disappears in (52). It has been proven in [START_REF] Bru | Wishart processes[END_REF] that the eigenvalues never collide and if moreover δ ≥ n + 1 the smallest one never vanishes. This is in accordance with Theorem 6. Laguerre processes are Hermitian versions of Wishart processes. Only constants are changed in (52), ( 53) and (54).

Reflection groups and Dunkl processes

We only give a short introduction to this topic and refer to [START_REF] Humpreys | Reflection groups and Coxeter groups[END_REF] and [START_REF] Rösler | Markov processes related with Dunkl operators[END_REF] for more details. For α ∈ R N \ {0} we denote by s α the orthogonal reflection with respect to the hyperplane H α perpendicular to α:

s α (x) = x -2 α.x |α| 2 . ( 55
) A finite subset R ⊂ R N \ {0} is called a root system if for all α ∈ R R ∩ Rα = {α, -α} ; s α (R) = R . (56)
The group W ⊆ O(N ) which is generated by the reflections {s α , α ∈ R} is called the reflection group associated with R. Each hyperplane H β := {x ∈ R N : β.x = 0} with β ∈ R N \∪ α∈R H α separates the root system R into R + and R -. Such a set R + is called a positive subsystem and defines the positive Weyl chamber C by

C := {x ∈ R N : α.x > 0 ∀α ∈ R + } . (57)
A subset S of R + is called simple if S is a vector basis for span(R). The elements of S are called simple. Such a subset exists, is unique and we actually get

C = {x ∈ R N : α.x > 0 ∀α ∈ S} . (58)
A function k : R → R on the root system is called a multiplicity function if it is invariant under the natural action of W on R. If the multiplicity function k is positive on R + , we define the radial Dunkl process X W as the C-valued continuous path Markov process whose generator is given by on C and Φ = ∞ elsewhere. It was proved in ( [START_REF] Chybiryakov | Processus de Dunkl et relation de Lamperti[END_REF], [START_REF] Chybiryakov | Dunkl processes and their radial parts relative to a root system[END_REF]) that this equation has a unique strong solution and if moreover k(α) ≥ 1/2 for any α ∈ R then the process never hits the walls H α of the Weyl chamber. In [START_REF] Demni | Radial Dunkl processes: existence and uniqueness, hitting time, beta processes and random matrices[END_REF], it is proved that if k(α) < 1/2 for a simple root α, then the process hits H α a.s. As a consequence of this result and of Theorem 6 (see also the statement at the bottom of p.117 in [START_REF] Chybiryakov | Dunkl processes and their radial parts relative to a root system[END_REF]), we are in a position to classify the boundary behavior of the radial Dunkl process in the Weyl chamber.

L W k u(x) = 1 2 ∆u(x) + α∈R + k(α) α.∇u(x) α.x (
Proposition 7 For any α ∈ R + let σ α := inf{t > 0 : X W t ∈ H α }.

• If α ∈ R + \ S, then P(σ α = ∞) = 1,

• If α ∈ S and k(α) ≥ 1/2, then P(σ α = ∞) = 1,

• If α ∈ S and k(α) < 1/2, then P(σ α < ∞) = 1.

Trigonometric and hyperbolic interactions

Others interactions have been studied in [START_REF] Cépa | Brownian particles with electrostatic repulsion on the circle: Dyson's model for unitary random matrices revisited[END_REF].

The trigonometric system ( [START_REF] Dyson | A Brownian-motion model for the eigenvalues of a random matrix[END_REF], [START_REF] Hobson | Non-colliding Brownian motion on the circle[END_REF], [START_REF] Spohn | Dyson's model of interacting Brownian motions at arbitrary coupling strength[END_REF]) reads

dX j t = dB j t + γ 2 k =j cot X j t -X k t 2 1 ≤ j ≤ n X 1 t ≤ X 2 t ≤ . . . ≤ X n t ≤ X 1 t + 2π (61)
This can be interpreted as the solution to the MSDE associated with

It has been proved in [START_REF] Cépa | Brownian particles with electrostatic repulsion on the circle: Dyson's model for unitary random matrices revisited[END_REF] there exist a.s. collisions if γ < 1/2. The hyperbolic system ( [START_REF] Norris | Brownian motions of ellipsoids[END_REF], [START_REF] Schapira | The Heckman-Opdam Markov process[END_REF]) is

dX j t = dB j t + γ k =j coth (X j t -X k t ) 1 ≤ j ≤ n X 1 t ≤ X 2 t ≤ . . . ≤ X n t . (64)
In this case Φ(x) =

  59)for u ∈ C 2 (C) with the boundary condition α.∇u(x) = 0 for x ∈ H α . Then X W may be viewed as the solution to the MSDEdY t = dB t -∇Φ(Y t ) dtwhere B is a Brownian motion and Φ(y) = α∈R + k(α) log(α.y) (60)

Φ

  collisions occur with positive probability if γ < 1/2.


Suzanne Lesecq

Sylviane Gentil
email: sylviane.gentil@gipsa-lab.inpg.fr

Cedric Berbra
email: cedric.berbra@gipsa-lab.inpg.fr

Condition monitoring based on filter bank in the presence of data loss

It is well known that Fault Detection and Isolation (FDI) can be performed using Signal Processing Techniques or Process Control Tools. In the first set of techniques, signals are directly analysed in order to detect changes in some quantities (mean, standard deviation, frequency contents, singularity appearance, etc.) related to a particular fault. In control theory, diagnostic methods are essentially based on models that are supposed to represent the behaviour of the process, either in the normal mode or in a particular faulty one. The model constitutes a prototype behaviour that is compared to the actual state, evaluated with data acquired online from the process. From the measurements and an analytical model, quantities called residuals are computed. Among all the tools used in the FDI community, observer (or filter) banks are very appealing. However, due to the introduction of networks between the sensors and the computational unit, the design of observers that can support data loss must be considered. This paper proposes an Extended Kalman Filter that takes into account the data loss that may appear during the data transfer from the sensor to the computational unit. The lost measurement is replaced with the previous available data, the corresponding variance being gradually increased. The proposed technique has been tested on real data acquired from an Inertial Measurement Unit composed of a tri-axis accelerometer, a tri-axis magnetometer and three rate gyros.

Introduction

Signal processing is a powerful tool to deal with fault diagnosis. It can be used to analyse directly the signals measured online. This approach avoids system modelling. It is based on the fact that a change in some quantity, e.g. the signal mean or the frequency contents, characterises a particular fault occurrence. Signals may be studied either with time-domain methods (including correlation, mean or standard deviation changes [START_REF] Basseville | Detection of abrupt changes[END_REF] ), with frequency methods (e.g. Fast Fourier Transform), or with more sophisticated ones including time-frequency or time-scale methods (for instance references [START_REF] Lesecq | Arcing fault detection using wavelet transform[END_REF][START_REF] Peng | Application of the wavelet transform in machine condition monitoring and fault diagnostics: a review with bibliography[END_REF] and references therein).

Safety is an engineering area at the border of control engineering and process engineering. Its study is difficult, mainly due to the increasing complexity of industrial processes. Safety is critical not only for the process itself but also for its environment and for those working or living in proximity. It is justified also by economic reasons. Consequently, monitoring the health of the installation at each time instant in order to detect incipient faults and to isolate damaged components, known as FDI (on-line Fault Detection and Isolation), has gained increasing attention in order to enhance reliability of complex systems, with many potential industrial applications.

Many different diagnostic approaches have been developed among control scientists. The most common approaches rely on a deterministic analytical model of the process to be diagnosed [START_REF] Basseville | Detection of abrupt changes[END_REF][START_REF] Patton | Observer-based fault detection and isolation: robustness and application[END_REF][START_REF] Kinnaert | Fault Diagnosis Based on analytical models for linear and nonlinear Systems-A tutorial[END_REF][START_REF] Isermann | Fault diagnosis systems[END_REF] . The model, generally quantitative and dynamic, represents the behaviour of the process, either in the normal mode or in a particular faulty one. Usually, it is the normal process behaviour that is represented by the analytical equations, which avoids modelling fault modes. This is the case that will be considered in this paper. Diagnostic algorithms check the consistency between the expected process behaviour, as computed by the model, and the real process behaviour evaluated with measurements acquired on-line. This comparison is based on numerical quantities called residuals that are (approximately) null when there is no fault affecting the system or when none of the faults in a known set of faults is affecting the system. From these residuals, FDI algorithms have to provide decisions, such as "a fault is detected" or "fault Fi is isolated". Fault detection is a two-stage process: residual generation and residual evaluation. If the detection is reliable, Fault Isolation follows naturally using a signature table.

Among the various techniques used for FDI among the control process community, observer (or filter) banks [START_REF] Frank | Analytical and qualitative model-based fault diagnosis, a survey and some new results[END_REF] are very appealing and this technique will be used in the sequel of this paper. The principle of observer banks is given in Figure 1. For each observer i, i = 1 : N, all the system outputs y but the measurement number i (y i ) are used to provide the state estimate ˆi

x . In this way, ˆi x is insensitive to any fault that may occur in the sensor S i which provides the measurement y i . One of the most significant challenges for control scientist today is that systems have been enlarged from single physical systems to large-scale complex systems that also include networks to both control and supervise these complex systems. At the same time, sensors are getting smaller, cheaper, pervasive and more powerful. Thus, everything is getting "sensed". The possibilities created by all these sensors are extraordinary: they allow real-time monitoring of large-scale systems and hardware redundancy can be considered with small extra-cost. However, these sensors may be faulty and they have to be diagnosed in order to provide reliable information on the process functioning. Moreover, the network induces drawbacks such as delays and packet losses that may influence the FDI algorithms, especially when they implement observer banks. Notice that the network protocol implemented greatly influences the performances of the network controlled system [START_REF] Frank | Analytical and qualitative model-based fault diagnosis, a survey and some new results[END_REF] and of the diagnostic algorithms.

When estimation over a network is considered, two schemes can be run. The first one supposes that a centralised computation unit is implemented. The second scheme makes use of sensor nodes with enough computation capabilities to estimate locally the state. Shi, et al. [START_REF] Shi | Estimation over wireless sensor networks: Tradeoff between communication, computation and estimation qualities[END_REF] consider both schemes, respectively centralised estimation and distributed estimation. They give a qualitative and quantitative analysis of the trade-off between estimation qualities and the communication and computation capabilities of each network node. They show that if the computation capabilities at each sensor node are limited and without packet drops, the transmission of raw data (scheme one) always provides a better estimation (measured with the state covariance matrix) than the transmission of distributed estimates (scheme two). However, in the presence of packet drops, they show that scheme one is not always better. Actually, the implementation of a filter bank can be performed with one of these two schemes. In the first case, all the raw data are transmitted to a unique computational unit, leading to a centralised observer bank. In the second case, the observers are distributed over the system. Each observer can make use only of local data or data exchanges may be necessary. When data are transmitted (first scheme, or second case of the second scheme), they can be delayed or even lost. The observer must therefore take into account this problem.

Observer implementation (and especially Kalman Filter one) under certain information constraints has become of great interest especially since Systems (NCS) are controlled through a network (NCS). In this latter context, some authors suppose that the system state is fully measured and transmitted to the controller through the network [START_REF] Xie | Stabilizing sampled-data linear systems with Markovian Packet losses and random sampling[END_REF] . Actually, this might be obtained by considering that the observer is implemented right at the sensor node. Therefore, it supposes computational capabilities at the sensor node. Kalman filtering over lossy networks is considered for instance in [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF] and [START_REF] Hespanha | Networked control systems: analysis and design[END_REF] . All these works suppose that the data loss can be modelled either by a Bernoulli binary random sequence or with a discrete-time binary Markov model.

The aim of the present paper is to consider the Kalman Filter implementation when measurements are transmitted through a network. Previous works (for instance [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF][START_REF] Hespanha | Networked control systems: analysis and design[END_REF][START_REF] Schenato | Fundations of Control and estimation over lossy networks[END_REF][START_REF] Huang | Stability of Kalman filtering with Markovian packet losses[END_REF][START_REF] Sun | Optimal linear estimation for systems with multiple packet drpouts[END_REF][START_REF] Epstein | Probabilistic performances of state estimation across a lossy network[END_REF] ) supposed that the entire output vector y k is sent in a unique packet that may be lost. Therefore, when a packet is dropt, all the measurements at time k are lost. However, depending on the topology of the system, on the protocol used etc., measurements can be sent independently (or at least in several sets) across the network. Therefore, each output i k y (or set of outputs) is likely to be lost. In this paper, each measurement i k y is sent independently and the Kalman Filter is modified in order to deal with data loss. Note that the case of Filter Banks is straightforward and is not explained here.

The rest of this paper is organised as follows. The problem under investigation is formulated in section 2. The Kalman Filter with loss of a subset of the measurements is proposed in section 3. This Kalman Filter is exemplified in section 4: it is applied to the attitude estimation problem when measurements are issued from an Inertial Measurement Unit (IMU) connected to the processor through a network. Finally, some conclusions are drawn in section 5.

Problem formulation

Consider the linear discrete-time varying stochastic system:

1 k k k k k k k k x F x w y C x v + = + = + (1)
where A C is supposed to be observable and the pair

n n k F × ∈ℜ , m n k C × ∈ℜ , x k is the system state and
1/ 2 , k k A Q ⎡ ⎤ ⎣ ⎦ controllable.
The estimation problem with "missing observations" has been previously studied for instance in [START_REF] Shi | Estimation over wireless sensor networks: Tradeoff between communication, computation and estimation qualities[END_REF][START_REF] Xie | Stabilizing sampled-data linear systems with Markovian Packet losses and random sampling[END_REF][START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF][START_REF] Schenato | Fundations of Control and estimation over lossy networks[END_REF] for discrete-time linear systems. In these papers, the output vector y is set in a unique packet. In the present work, each measurement i k y is likely to be lost. In the literature [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF][START_REF] Schenato | Fundations of Control and estimation over lossy networks[END_REF] , the packet loss process is assumed to be an i.i.d. Bernoulli binary random sequence. This choice is clearly an idealisation that is chosen by the authors for mathematical tractability. In this way, the stability of the Kalman Filter may be studied [START_REF] Schenato | Fundations of Control and estimation over lossy networks[END_REF] by a Modified Algebraic Riccati Equation (MARE). Other authors (e.g. [START_REF] Xie | Stabilizing sampled-data linear systems with Markovian Packet losses and random sampling[END_REF][START_REF] Huang | Stability of Kalman filtering with Markovian packet losses[END_REF] ) propose the use of a discrete-time binary Markov model. Contrarily to the i.i.d. model, the Markov chain allows to capture the temporal correlation of the channel variation. With this packet loss modelling, stability results for the Kalman Filter can also be achieved [START_REF] Huang | Stability of Kalman filtering with Markovian packet losses[END_REF] . The choice of the data loss model is not discussed in this paper, and the Bernoulli process is considered.

In both contexts, the Kalman Filter equations become:

1/ 1/ / ˆk k k k T k k k k k k k x F x P F P F Q + + = = + (2) 1/ 1 1/ 1 1 1/ 1/ 1 1/ 1 1 1 1/ ˆˆ( ) k k k k k k k k k k k k k k k k k k k k k k x x K y C x P P K C P γ γ + + + + + + + + + + + + + = + - = - (3)
where A T is the transpose of matrix A.

{ } 1 0,1 k γ + ∈
indicates if the measurement vector 1 k y + has been received.

1 1 1 / 1 1 1 / 1 1 T T k k k k k k k k k K P C C P C R - + + + + + + + ⎡ ⎤ = + ⎣
⎦ is the Kalman gain matrix. Note that (2) corresponds to the prediction step while (3) is the correction step that is performed only if the measurements have been received. Note that the covariance matrix P is now a random variable, because of the randomness of γ .

As stated above, some stability properties can be achieved (see for instance [START_REF] Schenato | Fundations of Control and estimation over lossy networks[END_REF][START_REF] Huang | Stability of Kalman filtering with Markovian packet losses[END_REF][START_REF] Epstein | Probabilistic performances of state estimation across a lossy network[END_REF] and references therein).

The case where only a subset of the components of y has been received for the update stage is now presented. This gives rise to a new Kalman Filter with partial data loss.

Kalman Filter with partial data loss

The estimation problem is reformulated as follows. In the present paper, the lost measurement is replaced with a null measurement, the standard deviation of the noise being arbitrarily large. Consider matrix

m m k M × ∈ℜ
defined as follows:

( , ) 0 if ( , ) 1 if the measurement is present ( , ) 1 if the measurement is absent k k k i M i j i j M i i M i i λ = ≠ = = >> (4)
The absence of measurement i (i.e. i k y has not been received at time k) corresponds to i λ → +∞ . Let k R denote the covariance matrix of the measurement noise when some measurements are missing, written as:

k k k k R M R M = (5)
As stated above, the missing data variance becomes arbitrarily large 2 2 2 i i i

σ λ σ = .The EKF formulation is now summarized. The Kalman Filter proposed here is implemented with the traditional "prediction step" and "correction step". Note that the structure of the Kalman filter is unchanged, contrarily to the work in [START_REF] Cao | An optimal control method for applications using wireless sensor/actuators networks[END_REF] where the lines in matrix C corresponding to lost data are removed.

-Initialization initial state 0

x , initial covariance matrix 0 , 1

P I λ λ = >> . -Prediction step 1/ 1/ / ˆk k Gk k T k k Gk k k Gk k x F x P F P F Q + + = = + (6)
-Computation of the gain matrix

1 1 1 / 1 1 1 / 1 1 T T k k k k k k k k k K P C C P C R - + + + + + + + ⎡ ⎤ = + ⎣ ⎦ (7)
-Correction Step

1/ 1 1/ 1 1 1/ 1/ 1 1/ 1 1 1/ ˆˆ( ) ) k k k k k k k k k k k k k k k k k x x K y C x P P K C P + + + + + + + + + + + + = + - = - (8)
Actually, equations ( 6) to (8) can also be formulated using matrix

k k k C T C = with 1 k k T M - = .
The modifications appear in the equations of matrices K and P:

1 1 1 1 1/ 1 1/ 1/ 1 1 1/ 1 1 1/ k k k T T k k k k k k k k k k k k k k k K K T P P P C C P C R C P + + + - + + + + + + + + + + = ⎡ ⎤ = - + ⎣ ⎦ (9)

Results

The Kalman filter proposed in this paper has been applied to the problem of attitude estimation of a rigid body. The filter bank that is implemented for sensor diagnosis makes use of the same ideas. The implementation of each Kalman Filter within the filter bank is straightforward and is not explained in details. Actually a set of N observer is implemented, each Kalman Filter having as measurement vector all the measurements but one. Then equations ( 6) to (8) with their modified version ( 9) are implemented for each Kalman Filter in the filter bank.

The low-cost light-weight Inertial Measurement unit (IMU) is made of nine MEMS (Micro Electro-Mechanical Systems) sensors, namely, three rate gyros mounted at right angles, a triaxis accelerometer and a triaxis magnetometer. Lost data are supposed to be issued from the magnetometers and/or from the accelerometers. The case when rate gyro measurements are lost has been studied in [START_REF] Peck | Attitude propagation with intermittent gyro measurements and singlevector observations[END_REF] . However, this latter work cannot be used here because the attitude dynamics of the system are not known.

In the present application, sensors are situated at the same location. Therefore the concept of distributed sensors does not apply. Moreover, due to power and computational limitations at the sensor node, no high level computation can be done at the sensor node and the raw data must be transmitted through the network. Therefore, scheme one in [START_REF] Shi | Estimation over wireless sensor networks: Tradeoff between communication, computation and estimation qualities[END_REF] must be implemented. In this situation, the loss of raw data might occur.

The IMU model is first proposed. Then its discrete time version is derived. The Kalman Filter with partial data loss is then applied to the attitude estimation problem. Filter banks are classical tools in FDI. Therefore, its implementation is not explained in this paper and will be described during the oral presentation. Results for different scenarios of data loss are given at the end of this section.

IMU model

Attitude estimation is necessary for various applications ranging for instance from human motion tracking to satellite control and Unmanned submarine or aerial vehicle control where the knowledge of the actual attitude is a prerequisite for attitude control. Two frames are considered to describe the dynamic equations of a rigid body in movement: the inertial frame N(x n , y n , z n ) and the body frame B(x b , y b , z b ) attached to the Rigid Body (RB) with its origin at the centre of mass of RB. The RB orientation can be parameterized by three rotation angles with respect to frame N: yaw (ψ), pitch (θ) and roll (Ф). ω ∈ ℜ 3 is the angular velocity of the RB relative to N expressed in B. Actually, the attitude can be modelled using various representations such as Euler angles, Cardan angles, etc.. Another appealing representation makes use of unitary quaternion [START_REF] Chou | Quaternion kinematics and dynamic differential equations[END_REF] 4 0

T T q q q ⎡ ⎤ = ∈ℜ ⎣ ⎦ r , 2 1 q = .
From (unit-)quaternion algebra, the rotation operator can be re-defined. If vector r r is expressed in N, its coordinates in B are:

1 ( ) or b R q r b q r q - = = ⊗ ⊗ r r ( 10
)
where ⊗ is the quaternion product, (0, )

T T r r = r , ( 0, )
T T b b = r and R(q) is the rotation matrix defined as:

2 0 0 ( ) ( ) 2( ) T T R q q q q I qq q q × ⎡ ⎤ = - + -⎣ ⎦ r r r r (11)
The kinematics equation can be expressed with q:

( ) ( )

(1/ 2) (1/ 2) q q q ω ω = Ω = Ξ & ( 12
) 3 2 3 1 0 2 1 0 0 ( ) , ( ) , 0 0 T T q q q I q ω ω ω ω ω ω ω ω ω ω ω × × × - ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ - - ⎡ ⎤ Ξ = Ω = = - ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤ - + ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ - ⎣ ⎦ ⎣ ⎦ r (13)
An IMU (MEMS technology) is embedded in the RB. It consists of a triaxis accelerometer, a triaxis magnetometer and three rate gyros mounted at right angle. The motion is supposed quasi-static so that the linear acceleration of the RB is neglected. The sensor measurements (expressed in frame B) are modelled as:

rate gyro: 3 G G ω ω η = - ∈ℜ r r (14)
accelerometers (normalized): 3 ( )

A A b R q g η = + ∈ℜ r r (15) magnetometers (normalized): 3 (
)

M M M b R q h η = + ∈ℜ r r (16)
η i , i = {G, A, M} are assumed to be independent Gaussian white noises of appropriate dimension and covariance matrices R i . g r is the normalized gravity vector measured in

N.

[ ] ,0,

M Mx Mz h h h = r
represents the normalized magnetic field measured in frame N at the 15)-( 16) are static nonlinear equations. Therefore an Extended Kalman Filter (EKF) will be implemented.

IMU model discretisation

Consider the discrete time version of the quaternion attitude kinematics equation ( 12):

1 k k k q F q + = (17)
where ( )

1 2 exp ( ) k k F h ω = Ω r
is obtained by assuming that k ω r is constant over a sampling period h. k ω r is expressed with (14). Thus matrix

F k becomes ( ) 1 2
, exp ( )

k k G k k G k G F F F F h ω = +Δ = Ω r (18)
The error matrix ΔF k can be expressed as a matrix power series. However, when the noise η Gk and the sampling time h are small enough, ΔF k is approximated by:

1 2 ( ) k k G F h η Δ = Ω (19)
Therefore, the state equation in ( 1) is rewritten as:

( ) 1 2
,

k k h k Gk k qk qk k G k G q F q q G η η η η + = + = Σ = (20)
The state noise is given by [START_REF] Lesecq | Quadrotor attitude estimation with data losses[END_REF] :

( ) ( ) ( ) [ ] 2 2 ˆˆY Y 4 4 T T k k G k k G k k h h Q q R q R P = Σ Σ + ⊗ ( 21
)
where

⊗ is the Kronecker product, Y = [ 1 ( ) e Ω r 2 ( ) e Ω r 3
( ) e Ω r ], Ω is defined in (13) and

( 1 e r 2 e r 3 e r ) is the orthonormal basis of ℜ 3 . The measurement equation y k ∈ℜ 6 is built from ( 15) and ( 16)

( ) 0 , ( ) 0 ( ) k k T k T T A M k k k k g C q b b g q C q h η η ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ = + = + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ r r r r (22)
with ,

k k T T T T k A M η η η ⎡ ⎤ = ⎣
⎦ . The measurement noise covariance matrix is:

0 0 T T T Ak k k k Mk R R E R η η ⎡ ⎤ ⎡ ⎤ = = ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ (23)

Results

The experiment reported in the present paper considers 30% of data loss for all the measurements from the accelerometers and magnetometers. The losses are supposed independent for each sensor axes, even if this hypothesis does not take into account the natural congestion that may appear in the network. Note that the attitude is given in the roll-pitch-yaw representation for the sake of clarity. The initial attitude of the Rigid Body is (-25;-30;-10) (deg) and the final one is (10;4;15) (deg). The attitude estimation filter is initialized with (-9;5;57) (deg). Moreover, a change in the attitude reference occurs at time t=3s and the new reference is (0;0;0). The real (top) and estimated (bottom) attitudes are shown in Fig. 3. Fig. 4 gives the attitude error and the data loss indicator for the accelerometer acc x : 0 means that the measurement is present for the estimation while 1 stands for "acc x has not been received". As can been seen in Fig. 3 to 4, even in the presence of 30% of independent data loss, the attitude of the RB does not exhibits strong errors (see Fig. 4 where the error is less than 10 deg). Note that for other scenarios with various loss rate and lost data, the maximum error observed on the estimated angles depends on the movements and on the lost data that are considered. Fig. 5 exhibits the trace of matrix P k/k . It shows that the confidence in the estimate q does not drastically decreases even in the intermittent data scenario.

Conclusions

This paper has proposed a Kalman Filter that aim at dealing with partial packet losses, i.e. a subset of the measurements have been received at the computational unit. The technique proposed is derived from the Kalman Filter previously published in Sinopoli et al [START_REF] Sinopoli | Kalman filtering with intermittent observations[END_REF] . In the present work the lost data is replaced with 0 while its covariance becomes arbitrarily large. The Kalman Filter proposed here is exemplified on an attitude estimation problem. In this application data acquired from an Inertial Measurement Unit are sent to the computational unit through a network. Experiments show that the estimated attitude error does not drastically increase even in the presence of 30% of data loss, which is quite high for wired network, but which can exist in wireless network such as Zigbee one. The implementation of a filter bank in order to perform sensor fault detection and isolation (FDI) is straightforward: it will be presented during the oral presentation. Ongoing work tries to obtain stability conditions for the Kalman Filter with partial data loss. The stability conditions will depend on the model that is supposed for the data loss.

Figure 1 :

 1 Figure 1: Observer bank principle.

  m k y ∈ℜ is the output (measurement) at time k. { } are two mutually independent sequences of independent and identically distributed (i.i.d.) Gaussian white noises with covariance matrices Q and R respectively. Moreover, the pair [ ] , k k

Figure 2 :

 2 Figure 2: Inertial frame N and Body frame B definition.

  experiment. Note that (

Figure 3 :Figure 4 :Figure 5 :

 345 Figure 3: Attitude error and data loss indicator.