Da-Yan Liu
email: dayan.liu@insa-cvl.fr

Gibaru Olivier
email: olivier.gibaru@ensam.eu

Perruquetti Wilfrid
email: wilfrid.perruquetti@inria.fr

Synthesis on a class of algebraic differentiators and application to nonlinear observation

Keywords: Algebraic differentiators, Noises error analysis, Nonlinear observation

). In this paper, we summarize some existing error analysis results to give a strategy on how to chose the design parameters for the Jacobi differentiator. Then, we provide new algorithms which are more robust against the numerical errors produced by negative design parameters' values. Finally, we consider an application to nonlinear observation, where we compare the Jacobi differentiator to the high gain observer and the high order sliding modes differentiator.

Introduction

The recent algebraic parametric method proposed by Fliess and Sira-Ramírez [START_REF] Fliess | An algebraic framework for linear identification[END_REF][START_REF] Fliess | Questioning some paradigms of signal processing via concrete examples[END_REF] has been extended to numerical differentiation problem in noisy environment (see, e.g. [START_REF] Fliess | Reconstructeurs d'états[END_REF][START_REF] Tian | Fast state estimation in linear time-varying systems: an algebraic approach[END_REF][START_REF] Tian | Une contribution à l'observation et à l'estimation des systèmes linéaires[END_REF][START_REF] Fliess | Compression différentielle de transitoires bruités[END_REF][START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF][START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF][START_REF] Liu | Error analysis for a class of numerical differentiator: application to state observation[END_REF][START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF][START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF][START_REF] Liu | Differentiation by integration with Jacobi polynomials[END_REF][START_REF] Liu | Error analysis of a class of derivative estimators for noisy signals and applications[END_REF]). The obtained so-called algebraic differentiators are divided into two classes: model-based differentiators and model-free differentiators. The formers were obtained by applying the algebraic method to a differential equation which defines a class of linear systems [START_REF] Fliess | Reconstructeurs d'états[END_REF][START_REF] Tian | Fast state estimation in linear time-varying systems: an algebraic approach[END_REF][START_REF] Tian | Une contribution à l'observation et à l'estimation des systèmes linéaires[END_REF]. Hence, they were mainly used for linear systems. However, the model-free differentiators can be used for nonlinear systems. The first model-free differentiator was introduced in [START_REF] Fliess | Compression différentielle de transitoires bruités[END_REF] by applying the algebraic method to the truncated Taylor series expansion of the signal to differentiate. Then, two model-free differentiators were studied in [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF][START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], where the so-called Jacobi differentiator is the most used. Moreover, it was shown that the Jacobi differentiator can also be obtained by taking the truncated Jacobi orthogonal series expansion of the signal to differentiate. Then, it was significantly improved by admitting a time-delay [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF][START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF].

Recall that the Jacobi differentiator has some advantages. It is given by an integral formula in continuous case, which can be considered as a low-pass filter and corresponds to a convolution in discrete case. Thus, estimations at different instants can be obtained using a sliding integration window of finite length. Consequently, this differentiator is algebraic and non-asymptotic. Moreover, it exhibits good robustness properties with respect to corrupting noises (see [START_REF] Fliess | Analyse non standard du bruit[END_REF] for more theoretical details). For these reasons, the Jacobi differentiator has been considered in a wide of real-time applications, such as nonlinear observation [START_REF] Yu | Observability forms for switched systems with zeno phenomenon or high switching frequency[END_REF], fault diagnosis [START_REF] Kiltz | Fault diagnosis on a magnetically supported plate[END_REF], model-free control [START_REF] Fliess | Commande sans modèle et commande à modèle restreint[END_REF], etc.

However, there was a lack of error analysis for the Jacobi differentiator. This differentiator depends on a set of design parameters, hence it is necessary to propose a strategy to chose these parameters so as to reduce estimation errors.

For this purpose, different errors for the Jacobi differentiator have been studied in [START_REF] Liu | Error analysis for a class of numerical differentiator: application to state observation[END_REF][START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF][START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF], respectively. In particular, the definition domain of certain design parameters has been extended to negative values which can reduce the estimation errors, especially the time-delay. However, the proposed algorithms with negative design parameters' values were subject to numerical errors (see [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF][START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF]).

The aim of this paper is given as follows. Firstly, by taking some existing error analysis results we summarize the design parameters' influence on each source of errors. Then, users of the Jacobi differentiator can chose these design parameters according to their need. Secondly, we provide new algorithms which are more robust against the numerical errors produced by negative design parameters' values. Thirdly, we compare the Jacobi differentiator to the high gain observer and the high order sliding modes differentiator by estimating the state variables of a nonlinear Ball and Beam system.

Synthesis on Jacobi differentiator

Consider a noise-corrupted measurement x ̟ : I → R, x ̟ (t) = x(t)+̟(t), where I is a finite time open interval of R + , x ∈ C n (I) with n ∈ N,
and ̟ is an additive corrupting noise. The objective is to estimate the n th order derivative of x using x ̟ . For this purpose, we apply a class of algebraic differentiators involving Jacobi polynomials, which were introduced in [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF][START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] using a recent algebraic parametric method (see [START_REF] Liu | Error analysis of a class of derivative estimators for noisy signals and applications[END_REF] for other algebraic differentiators).

In the next subsection, we show how to obtain these algebraic differentiators from the point of view of analysis, instead of algebra. First, for any t 0 ∈ I, we introduce the set D t0 := t ∈ R * + ; t 0 -t ∈ I .

Algebraic differentiators involving Jacobi polynomials

For any t 0 ∈ I, x can be locally given on [t 0 -T, t 0 ] with T ∈ D t0 by the following Jacobi orthogonal series expan-sion:

∀ ξ ∈ [0, 1], x(t 0 -T ξ) = i≥0 P (µ,κ) i (•), x(t 0 -T •) µ,κ P (µ,κ) i 2 µ,κ P (µ,κ) i (ξ), (1)
where

P (µ,κ) i
is the i th order shifted Jacobi orthogonal polynomial defined on [0, 1] as follows (see [START_REF] Abramowitz | editeurs, Handbook of mathematical functions[END_REF] pp. 774-775):

P (µ,κ) i (τ ) = i j=0 i + µ j i + κ i -j (τ -1) i-j τ j , with µ, κ ∈] -1, +∞[, •, • µ,κ is a L 2 ([0, 1]
) scalar product with the associated weight function w µ,κ (τ ) = (1 -τ ) µ τ κ , and the associated norm

P (µ,κ) i 2 µ,κ = 1 2i+µ+κ+1 Γ(µ+i+1)Γ(κ+i+1) Γ(µ+κ+i+1)Γ(i+1)
, where Γ(•) is the classical Gamma function (see [START_REF] Abramowitz | editeurs, Handbook of mathematical functions[END_REF] p. 255).

In order to approximate x on [t 0 -T, t 0 ], we consider the classical polynomial approximation by taking the N + 1 first terms in the Jacobi series expansion given in (1), and we denote the obtained N th order polynomial by D (0) κ,µ,T,N x(t 0 -T •). Thus, we have:

∀ ξ ∈ [0, 1], D (0) κ,µ,T,N x(t 0 -T ξ) := N i=0 P (µ,κ) i (•), x(t 0 -T •) µ,κ P (µ,κ) i 2 µ,κ P (µ,κ) i (ξ). (2)
Hence, the n th order derivative of x can be approximated by the one of D

κ,µ,T,N x(t 0 -T •). Inspired by this idea, the (causal) Jacobi differentiator is defined as follows [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF]:

∀ ξ ∈ [0, 1], D (n) κ,µ,T,q x(t 0 -T ξ) := 1 (-T ) n d n dξ n D (0) κ,µ,T,N x(t 0 -T ξ) , (3)
where q = N -n ∈ N. It is shown in [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF][START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF] that this differentiator can also be obtained by taking the q + 1 first terms in the Jacobi series expansion of x (n) , i.e. we locally approximate x (n) by a q th order polynomial on [t 0 -T, t 0 ]:

∀ξ ∈ [0, 1], D (n) κ,µ,T,q x(t 0 -T ξ) = q i=0 P (µ+n,κ+n) i (•), x (n) (t 0 -T •) µ+n,κ+n P (µ+n,κ+n) i 2 µ+n,κ+n P (µ+n,κ+n) i (ξ). (4)
Moreover, it can be given by the following integral formula:

∀ ξ ∈ [0, 1], D (n) κ,µ,T,q x(t 0 -T ξ) = 1 (-T ) n 1 0 Q κ,µ,n,q,ξ (τ ) x(t 0 -T ξ)dτ, (5)
where

C κ,µ,n,i = (µ+κ+2n+2i+1)Γ(κ+µ+2n+i+1)Γ(n+i+1) Γ(κ+n+i+1)Γ(µ+n+i+1) , Q κ,µ,n,q,ξ (τ ) = w µ,κ (τ ) q i=0 C κ,µ,n,i P (µ+n,κ+n) i (ξ)P (µ,κ) n+i (τ ).
Finally, we substitute x in (5) by x ̟ so as to obtain the Jacobi differentiator D (n) κ,µ,T,q x ̟ (t 0 -T ξ) in noisy case. Different from the existing polynomial approaches, the idea of the Jacobi differentiator is to use a sliding integration window to estimate the value of x (n) at each t 0 ∈ I by D (n) κ,µ,T,q x(t 0 -T ξ) with a fixed value of ξ ∈ [0, 1] (see [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF][START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF]). If ξ = 0, then it produces a delay of value T ξ.

It is clear that for each t 0 ∈ I, the Jacobi differentiator D (n) κ,µ,T,q x ̟ (t 0 -T ξ) depends on a set of design parameters, except for the order of the desired derivative n: • κ, µ ∈] -1, +∞[: the parameters of Jacobi polynomials, • q ∈ N: the order of truncated Jacobi series expansion, • T ∈ D t0 : the length of the sliding integration window, • ξ ∈ [0, 1]: the parameter of time-delay T ξ.

In the next subsection, we give a general idea on how to chose these design parameters by giving some error analysis results.

Error analysis in continuous case

When we estimate x (n) (t 0 ) by D (n) κ,µ,T,q x(t 0 -T ξ) with ξ ∈ [0, 1], the associated estimation error can be decomposed in continuous case as follows:

D (n) κ,µ,T,q x ̟ (t 0 -T ξ) -x (n) (t 0 ) = D (n) κ,µ,T,q x ̟ (t 0 -T ξ) -D (n) κ,µ,T,q x(t 0 -T ξ) + D (n) κ,µ,T,q x(t 0 -T ξ) -x (n) (t 0 -T ξ) + x (n) (t 0 -T ξ) -x (n) (t 0 ) . (6)
Hence, the Jacobi differentiator D (n) κ,µ,T,q x ̟ (t 0 -T ξ) is corrupted by two sources of errors:

• The noise error contribution: e ̟ (t 0 ; n, κ, µ, T, q, ξ) :=

D (n) κ,µ,T,q x ̟ (t 0 -T ξ) -D (n)
κ,µ,T,q x(t 0 -T ξ).

(7)

• The truncated term error due to the truncated Jacobi series expansion given in (4): e R (t 0 ; n, κ, µ, T, q, ξ) :=

D (n) κ,µ,T,q x(t 0 -T ξ) -x (n) (t 0 ). (8)
Then, according to (6) the truncated term error can be divided into two parts:

• The amplitude error (in the vertical sense):

e h (t 0 ; n, κ, µ, T, q, ξ) := D (n) κ,µ,T,q (t 0 -T ξ) -x (n) (t 0 -T ξ). (9)
• The delay error due to T ξ (in the horizontal sense):

e v (t 0 ; n, κ, µ, T, q, ξ) := x (n) (t 0 -T ξ) -x (n) (t 0 ). (10)
Consequently, each source of errors also depends on the design parameters. In order to show the influence of these parameters, we study the error bounds for these errors, which depend on the same parameters. For this purpose, we give a synthesis of some existing error analysis results. 1) Noise error contributions: In continuous case, we consider the noise errors due to the two following categories of noises.

• Integrable noises: In this case the noise is assumed to be a bounded and integrable function on I, which can be divided into two parts [START_REF] Fliess | Questioning some paradigms of signal processing via concrete examples[END_REF]: the first part is a (n -1) th order polynomial, considered as a structured perturbation, and the seconded part is a high frequency perturbation, considered as an unstructured noise.

An error bound based on the integral formula given in (5) was proposed in [START_REF] Liu | Error analysis of a class of derivative estimators for noisy signals and applications[END_REF] (p. 90) for this kind of noise errors. Moreover, it was shown that the Jacobi differentiator D (n) κ,µ,T,q x ̟ (t 0 -T ξ) can eliminate a (n -1) th order structured perturbation.

• Non-independent stochastic process noises: We consider in this case a class of continuous stochastic processes with finite second moments, whose mean value function and covariance kernel are continuous functions [START_REF] Parzen | Stochastic processes[END_REF], such as Brownian motion and Poisson process.

Since a stochastic process usually is not bounded, the Bienaymé-Chebyshev inequality was used to give error bounds by calculating the mean value and the variance of the associated noise error [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF].

2) Amplitude error: Using the Taylor series expansion of x at t 0 , an error bound for the amplitude error was provided in [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF], where the (n + q + 1) th order derivative of x was assumed to be bounded on I.

3) Delay error: Instead of giving an error bound for the delay error, existing studies were based on how to chose ξ and on how to reduce the delay T ξ if ξ = 0.

• On the one hand, according to [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], we can take ξ = ξ

(n) κ,µ,q in D (n) κ,µ,T,q x ̟ (t 0 -T ξ), where ξ (n) κ,µ,q is the smallest root of P (µ+n,κ+n) q+1
. This choice of ξ significantly improved the Jacobi differentiator by admitting the time-delay T ξ. Indeed, it was shown in [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF] that e h (t 0 ; n, κ, µ, T, q, ξ (n) κ,µ,q ) < e h (t 0 ; n, κ, µ, T, q + 1, 0) < e h (t 0 ; n, κ, µ, T, q, 0). Moreover, the last inequality was confirmed in [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF] using the amplitude error bound.

• On the other hand, ξ (n) κ,µ,q depends on three design parameters. The influence of q is given by the classical orthogonal polynomial theory [START_REF] Szegö | Orthogonal polynomials[END_REF]. The influence of κ et µ was studied in [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF].

• Another choice of ξ is to take ξ = 0.5 (this case corresponds to the central Jacobi differentiator, see [START_REF] Liu | Differentiation by integration with Jacobi polynomials[END_REF][START_REF] Liu | Error analysis of a class of derivative estimators for noisy signals and applications[END_REF]). It is the optimal value of ξ which minimizes the noise error contribution. However, the time-delay is equal to 0.5T . Hence, this choice of ξ is only considered for off-line applications.

Finally, by numerically 1 calculating the noise error bound, the amplitude error bound and the time-delay, we can know their behaviors with respect to different design parameters. Then, we can deduce the influence of these design parameters on each source of errors. We summarize the obtained results in Table 1 (see [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF][START_REF] Liu | Error analysis of a class of derivative estimators for noisy signals and applications[END_REF] for more details), where the notations a ↑, b ր and c ց mean that if we increase the value for the parameter a then the error b increases and the error c decreases. Consequently, it is interesting to take negative values of κ to reduce both the truncated term error and the noise error contribution (see [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF] for more details). In the next subsection, we consider the errors in noisy discrete 1 It is very difficult to analytically study the behavior of each error bound due to their complex expressions. case.

Stochastic noise error Amplitude error Time-delay

κ ↑ ր ր ր µ ↑ ր ց ց q ↑ ր ց ց T ↑ ց ր ր
Table 1: Influence of design parameters on D

(n) κ,µ,T,q x(t 0 - T ξ (n)
κ,µ,q ) in continuous case.

Error analysis in discrete case

From now on, we assume that the noisy measurement x ̟ (t j ) = x(t j ) + ̟(t j ) is given in discrete case with an equidistant sampling period T s .

Since x ̟ is a discrete measurement, a numerical integration method is needed to approximate the integral formula given in [START_REF] Tian | Une contribution à l'observation et à l'estimation des systèmes linéaires[END_REF]. Let τ i = i m and w i > 0 for i = 0, . . . , m with m = T Ts ∈ N (except for w 0 ≥ 0 and w m ≥ 0) be respectively the abscissas and the weights for a given numerical integration method. Then, we get:

D (n) κ,µ,T,q x ̟ (t 0 -T ξ) ≈ D (n),m κ,µ,T,q x ̟ (t 0 -T ξ) := 1 (-T ) n m i=0 w i m Q κ,µ,n,q,ξ (τ i ) x ̟ (t 0 -T τ i ). (11)
Consequently, the estimation error in this discrete noisy case is given by:

D (n),m κ,µ,T,q x ̟ (t 0 -T ξ) -x (n) (t 0 ) = D (n),m κ,µ,T,q x ̟ (t 0 -T ξ) -D (n),m κ,µ,T,q x(t 0 -T ξ) + D (n),m κ,µ,T,q x(t 0 -T ξ) -D (n) κ,µ,T,q x(t 0 -T ξ) + D (n) κ,µ,T,q x(t 0 -T ξ) -x (n) (t 0 ) =e m ̟ (t 0 ; n, κ, µ, T, q, ξ) + e m (t 0 ) + e R (t 0 ; n, κ, µ, T, q, ξ), (12)
where e m ̟ (t 0 ; n, κ, µ, T, q, ξ) is the discrete noise error contribution, e m (t 0 ) is the numerical error produced by a numerical integration method, and e m R (t 0 ; n, κ, µ, T, q, ξ) is the truncated term error which is studied previously.

Remark that if κ or µ is negative then the integral given in ( 5) is an improper integral due to w µ,κ (•). Hence, there can exist a singular value at τ = 0 or at τ = 1 when we apply a numerical integration method. In order to overcome this problem, the weight w 0 (resp. w m ) was set to zero in [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF][START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF] when κ (resp. µ) is negative. Based on this strategy, some error analysis was given. 1) Discrete noise error contributions: Independent stochastic process noises: From now on, we consider a class of stochastic processes satisfying the following properties: (P 1 ) : for any s, t ≥ 0, s = t, ̟(s) and ̟(t) are independent; (P 2 ) : the mean value function of {̟(t), t ∈ I} belongs to L(I); (P 3 ) : the variance function of {̟(t), t ∈ I} is bounded on I.

Note that white Gaussian noise and Poisson noise satisfy these conditions.

• The convergence in mean square of the discrete noisy error contributions with respect to the sampling period was studied in [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF]. It was shown that the noise error contributions obtained with µ > -1 2 and κ > -1 2 converge in mean square to zero when the sampling period tends to zero. Similar result was obtained in [START_REF] Fliess | Analyse non standard du bruit[END_REF] using nonstandard analysis theory.

• For a set sampling period, by calculating the mean value and the variance of the associated noise errors the Bienaymé-Chebyshev inequality was used to give error bounds [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF], so as to study the influence of the design parameters (the obtained results will be given in the next section).

2) Numerical error:

• When the values of κ and µ are positive, we can take a small sampling period such that the numerical error can be negligible.

• When κ or µ is negative, the numerical error is much larger than the one obtained with positive κ and µ. It can compensate the truncated term error such that the time-delay can be largely reduced in some cases (see [START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF][START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF][START_REF] Liu | Error analysis of a class of derivative estimators for noisy signals and applications[END_REF]). However, these results are subject to the kind of signals to differentiate.

In the next section, we propose new algorithms for the Jacobi differentiator, which are more robust against the numerical error when the values of κ and µ are negative.

3 New algorithms for Jacobi differentiator

Change of variables

It is shown in the previous section that if κ (resp. µ) is negative then there will be a singular value at τ = 0 (resp. τ = 1) in discrete case. In order to solve this problem, instead of taking w 0 = 0 (resp. w m = 0) in a numerical integration method, we apply a change of variables in [START_REF] Tian | Une contribution à l'observation et à l'estimation des systèmes linéaires[END_REF] (see [START_REF] Press | Numerical Recipes in C, Second Edition[END_REF], p. 145).

• In the case where -1 < κ < 0 and µ ≥ 0: By taking the change of variables τ → s 1 1+κ , we get:

D (n) κ,µ,T,q x(t 0 -T ξ) = 1 (-T ) n 1 0 Q κ,µ,n,q,ξ (s 1 1+κ ) x(t 0 -T s 1 1+κ ) ds (13)
where

Q κ,µ,n,q,ξ (s 1 1+κ ) = (1 -s 1 1+κ ) µ 1 + κ q i=0 C κ,µ,n,i P (µ+n,κ+n) i (ξ)P (µ,κ) n+i (s 1 1+κ
).

• In the case where -1 < µ < 0 and κ ≥ 0: We first apply the change of variables τ → 1 -τ , we get:

D (n) κ,µ,T,q x(t 0 -T ξ) = 1 (-T ) n × 1 0 Q κ,µ,n,q,ξ (1 -τ ) x(t 0 -T (1 -τ )) dτ. (14)
Then, similar to the first case, we take the change of variables τ → s 1 1+κ in [START_REF] Fliess | Analyse non standard du bruit[END_REF] to avoid the singular value at τ = 0. • In the case where -1 < µ < 0 and -1 < κ < 0:

We first decompose the integral into two parts:

D (n) κ,µ,T,q x(t 0 -T ξ) = 1 (-T ) n 1 2 0 Q κ,µ,n,q,ξ (τ ) x(t 0 -T τ )dτ + 1 (-T ) n 1 1 2 Q κ,µ,n,q,ξ (τ ) x(t 0 -T τ )dτ. (15)
Then, we respectively apply the changes of variables proposed in the two previous cases to each integral obtained in (15) so as to avoid the singular values at τ = 0 and τ = 1.

In the following proposition, we give a new algorithm to implement the Jacobi differentiator with negative values of κ in discrete noisy case.

Proposition 1 Let x ̟ (t j ) = x(t j ) + ̟(t j ) for j = 0, . . . , M , be a discrete noise-corrupted measurement given on a finite time open interval I ⊂ R + with an equidistant sampling period T s , where x ∈ C n (I) with n ∈ N, and ̟ is an additive corrupting noise. Then, for any t j ∈ I with j = m, . . . , M , the n th order derivative of x at t j is estimated by:

D (n) κ,µ,mTs,q x ̟ (t j -mT s ξ) := 1 (-mT s ) n m i=0 w i m 1+κ Q κ,µ,n,q,ξ (τ i ) x ̟ (t j-i ), ( 16
)
where τ i = i m , w i ∈ R + is the weight for a numerical integration method with numerical integration steps equal to

h i = 1 m 1+κ (i + 1) 1+κ -i 1+κ for i = 0, . . . , m -1, and Q κ,µ,n,q,ξ (τ i ) = (1 -τ i ) µ 1 + κ q i=0 C κ,µ,n,i P (µ+n,κ+n) i (ξ)P (µ,κ) n+i (τ i ), C κ,µ,n,i = (µ+κ+2n+2i+1)Γ(κ+µ+2n+i+1)Γ(n+i+1) Γ(κ+n+i+1)Γ(µ+n+i+1) , with κ ∈] -1, 0[, µ ∈ R + , M ≥ m ∈ N * , q ∈ N, and ξ ∈ [0, 1].
Proof. Since x ̟ is given in discrete case, we need to apply a numerical integration method to approximate the integral obtained in [START_REF] Liu | Error analysis of a class of derivative estimators for noisy signals and applications[END_REF]. Moreover, x ̟ is given with an equidistant sampling period, hence we need to take the abscissas s i for i = 0, . . . , m with m = T Ts , in such a way that s

1 1+κ i = i m = τ i .
Thus, the numerical integration steps become:

h i = s i+1 -s i = i + 1 m 1+κ - i m 1+κ
, for i = 0, . . . , m -1. Finally, this proof is completed by considering the associated weights. By applying a similar technique to ( 14) and ( 15) respectively, we can also obtain the algorithms of the Jacobi differentiator with negative values of µ in discrete noisy case.

Error analysis in discrete case with new algorithms

In this subsection, we use the new algorithms obtained in the previous subsection to give error analysis in discrete case for the Jacobi differentiator with negative values of κ or µ. 1) Numerical error: Since the singularity problem for the Jacobi differentiator with negative values of κ or µ is solved in the previous subsection, we can take a small sampling period such that the numerical error can be negligible.

2) Discrete noise error contributions:

• Non-independent stochastic process noises: We can take a small sampling period such that the discrete noise errors can be considered as the ones in continuous case. Hence, we consider that the results on the influence of design parameters obtained in continuous case are still available in discrete case for considered non-independent stochastic processes.

• Independent stochastic process noises: We focus on the class of stochastic processes satisfying the properties (P 1 ) -(P 3 ) given in the previous section. In order to simplify notions, we denote the discrete noise errors e m ̟ (t 0 ; n, κ, µ, T, q, ξ) produced with new algorithms by e m ̟ . Then, as done in [START_REF] Liu | Error analysis for a class of numerical differentiator: application to state observation[END_REF][START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF], sharp noise error bounds can be given using the Bienaymé-Chebyshev inequality: for any real number γ > 0,

M γ l pγ < e m ̟ pγ < M γ h , (17)
where

M γ l = E[e m ̟ ] -γ Var[e m ̟ ], M γ h = E[e m ̟ ] + γ Var[e m ̟ ]
, and a pγ < b means that the probability for a real number b to be larger than an other real number a is equal to p γ with p γ > 1 -1 γ 2 . Since the considered noises are independent, it is easy to calculate E[e m ̟ ] and Var[e m ̟ ]. Thus, we can study the design parameters' influence on the discrete noise error contributions e m ̟ using the error bounds obtained in [START_REF] Fliess | Commande sans modèle et commande à modèle restreint[END_REF]. In order to simplify this study, we assume that the noises also satisfy the following property:

(P 4 ) : ∀ t j ∈ I, E[̟(t j )] = 0 and Var[̟(t j )] = 1.
Hence, we only need to study the variance of e m ̟ . Thus, by numerically calculating Var[e m ̟ ], we can know its behavior with respect to different design parameters. Then, we can deduce these parameters' influence on e m ̟ . For instance, we show in Fig. 1

the behavior of Var[e m

̟ ] with respect to κ and µ.

Finally, we summarize in Table 2 the results in discrete case obtained in the previous section and this section (see [START_REF] Liu | Error analysis of a class of derivative estimators for noisy signals and applications[END_REF] for more details). In particular, we can see that for the proposed new algorithms, the optimal values of κ and µ which minimize the noise error due to an independent noise are near to zero and positive.

Numerical error

Numerical error with w0 or wm = 0 with new algorithms κ or µ < 0 Large Small Independent Noise error Independent Noise error with w0 or wm = 0

with new algorithm κ ↑ ր ցր (op. at ≈ 0+) µ ↑ ր ցր (op. at ≈ 0+) q ↑ ր ր T ↑ ց ց
Table 2: Influence of design parameters on D

(n) κ,µ,T,q x(t 0 -T ξ (n) κ,µ,q ) in discrete case.

Strategy on choice of design parameters

According to Table 1 Fig. 1:

Var[e m ̟ ] with n = 1, T = 1, q = 1, ξ = ξ (1)
κ,µ,1 and κ, µ ∈ [-0.5, 1.5]. take a compromise among these parameters. In this subsection, we give a general strategy on how to chose these parameters for the Jacobi differentiator. 1) Choice of ξ: as shown previously [START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF], we take ξ = ξ (n) κ,µ,q , i.e. the smallest root of P (µ+n,κ+n) q+1 (•), in order to reduce the truncated term error by admitting a time-delay.

2) Choice of q: According to Table 1 andTable 2, a large value of q can produce a large noise error contribution, hence we should take a small value for q. As done in [START_REF] Mboup | A revised look at numerical differentiation with an application to nonlinear feedback control[END_REF][START_REF] Mboup | Numerical differentiation with annihilators in noisy environment[END_REF][START_REF] Liu | Error analysis for a class of numerical differentiator: application to state observation[END_REF][START_REF] Liu | Error analysis of Jacobi derivative estimators for noisy signals[END_REF][START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF], we can take q = 0, 1, 2. On the one hand, the truncated term error produced with q = 0 is much larger than the one produced with q = 1. This undesired effect is more important than the one that the noise error contribution produced with q = 1 is larger than the noise error contribution produced with q = 0. On the other hand, the noise error contribution produced with q ≥ 2 is much larger than the one produced with q = 1. This undesired effect is more important than the one that the truncated term error produced with q = 1 is larger than the one produced with q ≥ 2. Indeed, thanks to the choice of ξ, the truncated term error produced with q = 1 is significantly improved. Consequently, we propose to take q = 1.

3) Choice of (κ, µ): For the choices of µ and κ, we distinguish the two cases depending the nature of noises:

• Independent stochastic process noises: According to Table 2, we take a positive value for µ which is near to 0 (smaller than 1) so as to avoid a large noise error contribution. Then, we chose the value of κ according to the situation of applications. On the one hand, if one considers the noise effect is more important than the time-delay, then we can take a positive value of κ and near to 0 (smaller than 1). On the other hand, if one prefers to produce a smaller time-delay, then we can take a negative value of κ.

• Non-independent stochastic process noises: According to Table 1, we take negative values for κ and µ so as to reduce the noise error contribution. Meanwhile, a negative value of κ can also reduce the truncated term error. 4) Choice of T : The optimal value of T which minimizes the amplitude error and the noise error contribution was proposed in [START_REF] Liu | Convergence Rate of the Causal Jacobi Derivative Estimator, Curves and Surfaces[END_REF]. However, this result was obtained using the bound of a higher order derivative of the considered signal and the bound of the noise, which are usually unknown in practical applications. Indeed, the choice of T depends on the characteristic of the signal to differentiate and the variance of the noise. Hence, a criterion should be considered in order to choose an appropriate value of T (see, e.g. [START_REF] Kiltz | Parametrization of algebraic numerical differentiators to achieve desired filter characteristics[END_REF]). This is beyond the scope of this paper. 5) Choice of numerical integration method: As mentioned previously, we can take a small sampling period such that the numerical error can be negligible. Then, a used numerical integration method only plays a role in the discrete noise error constructions, especially due to independent stochastic process noises. In order to show how to choose a numerical integration method, serval numerical integration methods, such as the trapezoidal rule, the rectangle rule, the Simpson's rule and the midpoint rule, have been compared in [START_REF] Liu | Error analysis of a class of derivative estimators for noisy signals and applications[END_REF]. By considering a zero-mean white Gaussian noise, the associated discrete noise error constructions were obtained by applying each method respectively. Then, the variance values of these noise errors have been calculated and compared. According to the obtained results, the trapezoidal rule is proposed.

Application to nonlinear observation

According to a new approach of nonlinear observability based on differential algebra [START_REF] Fliess | Nonlinear observability, identifiability and persistent trajectories[END_REF][START_REF] Diop | On nonlinear observability[END_REF], the numerical differentiation is an useful tool for nonlinear observation. A comparison between the Jacobi differentiator and the high order sliding modes differentiator has been done in [START_REF] Sidhom | Sur les différentiateurs en temps réel: algorithmes et applications[END_REF]. In this section, we apply the Jacobi differentiator to the observation of a nonlinear Ball and Beam system. Moreover, we consider comparisons to two well-known methods.

Ball and Beam system

Ball and beam system is one of the most enduring popular and important laboratory models for teaching control systems engineering. Its dynamics can be described as follows:

(mr 2 + J) θ + 2mr ṙ θ + mgr cos(θ) = u, mr + mg sin(θ) -mr θ2 = 0, ( 18
)
where m is the mass of the ball, J is the length of the beam, g is the gravitational constant, r is the inertia position of the ball on the beam, θ is the beam angle and u is the torque input. The state vector of this system is (r, ṙ, θ, θ) T . Assume that the output vector is (y 1 , y 2 ) T = (r, θ) T , then we can verify that this system is observable using the algebraic approach [START_REF] Fliess | Nonlinear observability, identifiability and persistent trajectories[END_REF][START_REF] Diop | On nonlinear observability[END_REF]. By taking u = (mr 2 + J)v + 2mr ṙ θ + mgr cos(θ) with v ∈ R, the Ball and Beam system can be simplified. Then, the linearization around the beam angle θ = 0 gives us the following linear approximation of the system:

θ = v, r = -gθ. (19)
In order to stabilize the system, we take: v = k3 g r (3) + k2 g r (2) + k1 g ṙ + k0 g r, where k 0 = w 2 n1 w 2 n2 , k 1 = 2α(w n1 w 2 n2 + w n2 w 2 n1 ), k 2 = w 2 n1 + w 2 n2 + 4α 2 w n1 w n2 and k 3 = 2α(w n1 + w n2 ) with α = 0.7, w n1 = 1/α and w n2 = 3w n1 .

By denoting the state vector by z = (z 1 , z 2 , z 3 , z 4 ) T , the system obtained in [START_REF] Parzen | Stochastic processes[END_REF] can be divided into the two following sub-systems:

Ṙ = AR + Bv, y 1 = r, (20)
Θ = AΘ + Φ(Θ, R), y 2 = θ, (21)
where

R = r ṙ , Θ = θ θ , A = 0 1 0 0 , B = 0 1
and Φ(Θ, R) = 0 -g sin(θ) -r θ2 . In the next subsection, we estimate (r, ṙ, θ, θ) T from the noisy output y ̟ 1 and y ̟ 2 .

Comparisons to two existing methods 4.2.1 High gain observer

During the past few years, the high gain observer played an important role in the design of nonlinear output feedback control of nonlinear systems [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF][START_REF] Deza | High gain estimation for nonlinear systems[END_REF]. It is mainly used to estimate the derivatives of the output.

By applying the high gain observer to [START_REF] Szegö | Orthogonal polynomials[END_REF] and ( 21) respectively, we get:

˙ R = A R -K r ( y 1 -y 1 ) + Bv, y 1 = r, (22)
˙ Θ = A Θ -K θ ( y 2 -y 2 ) + Φ( Θ, R), y 2 = θ, (23)
where R = r ṙ are Θ = θ θ present the estimated state variables. The gains are given by K r = S -1 ∞ (α r )C and K θ = S -1 ∞ (α θ )C, where S -1 ∞ (α) is the unique solution of the matrix equation:

αS ∞ (α) + A T S ∞ (α) + S ∞ (α)A = CC T , S ∞ (α) = S T ∞ (α),
where C = 1 0 . Consequently, we get K r = 2α r α 2 r and K θ = 2α θ α 2 θ with α r ∈ R and α θ ∈ R.

High order sliding modes differentiator

The high order sliding modes differentiator described in [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF] can be expressed in a dynamic form as follows:

               α0 = -λ n W 1 n+1 |α 0 -x| n n+1 sign(α 0 -x) + α 1 , α1 = -λ n-1 W 1 n |α 1 -α0 | n-1 n sign(α 1 -α0 ) + α 2 , . . . αn-1 = -λ 1 W 1 2 |α n-1 -αn-2 | 1 2 sign(α n-1 -αn-2 ) + α n , αn = -λ 0 W sign(α n -αn-1 ),
where x ∈ C n+1 is the signal to differentiate, and W is a constant such that x (n+1) ∞ ≤ W . It was shown in [START_REF] Levant | Higher-order sliding modes, differentiation and output-feedback control[END_REF] that, if the gains λ i for i = 0, 1, . . . , n, are chosen properly, then the differentiator converges in a finite time T . For n = 5, the gains can be chosen as λ 0 = 1.1, λ 1 = 1.5, λ 2 = 3, λ 3 = 5, λ 4 = 8 and λ 5 = 12. In order to estimate r and ṙ (resp. θ and θ), we apply this differentiator to (20) (resp. ( 21)) with n = 2 and x = y 1 (resp. x = y 2 ).

Simulation results

We first consider the high gain observer and the sliding modes differentiator. On the one hand, when using the high gain observer for the Ball and Beam system, there are two design parameters to set: α r and α θ . During the implementations of the high gain observer, we observe that it is necessary to take large values for these parameters to reduce the estimation errors in noise-free case. However, large values of these parameters can produce large noise errors in noisy case. Hence, there is also a compromise to consider. In the following simulations, we take α r = 10 and α θ = 30 so as to produce small estimation errors in noise-free case.

On the other hand, by simulating the Ball and Beam system we obtain that r (2) ≤ 2 and θ (2) ≤ 6.7. Hence, we respectively take W r = 2 and W θ = 6.7 for the sliding modes differentiator. However, these values are usually unknown in practice applications. Moreover, during the implementations of this differentiator in noise-free case, we observe that it is not robust against a "large" sampling period, especially when it is used to estimate ṙ and θ (see [START_REF] Liu | Error analysis of a class of derivative estimators for noisy signals and applications[END_REF] for more details). In order to solve this problem, we assume y 1 and y 2 are measured with an equidistant sampling period T s = 0.0001s in the following simulations. Moreover, we assume that y ̟ 1 (resp. y ̟

2 ) is the discrete output corrupted by a noise ̟ which is a zero-mean Gaussian iid sequence and whose variance is adjusted in such a way that SNR = 24.5dB (resp. SNR = 23.6dB).

For the implementation of the Jacobi differentiator, we need four sets of design parameters for each state variable. According to the previous study on the choice of the design parameters, we take q = 1 and µ = 0. For the value of κ, we consider two cases:

• Firstly, according to Fig. 1 we take κ 1 = 0.32 to reduce the noise error contribution.

• Secondly, we take κ 2 = -0.5 to reduce the time-delay and the amplitude error. Then, we take ξ = ξ (n) κi,0,1 for n = 1, 2, i = 1, 2, in the two cases respectively. Finally, the only parameter to set is the length of each sliding window. However, we need four different values of T to estimate each state.

Finally, we give the obtained estimations of θ in Fig. 2 and Fig. 3. Since the values of the time-delay produced by the Jacobi differentiator can be calculated, we can see that by shifting the corresponding estimations we get smaller estimation errors than the other two methods. Moreover, when the value of κ is negative, the time-delay is reduced, however the noise error contribution is increased. According to the simulations, we summarize the comparison results in Table 3 κ,µ,T,q x(t 0 -T ξ (n) κ,µ,q ) with n = 1, κ = 0.32, µ = 0, q = 1, ξ (n) κ,µ,q = 0.3088 and T = 0.4.

Conclusions

In this paper, first we have shown how to obtain the socalled algebraic Jacobi differentiator from the point of view of analysis. Then, we have summarized some existing error analysis results for the Jacobi differentiator. A synthesis of the design parameters' influence on each source of errors has been provided in continuous case, where we recalled that negative values of κ can reduce both the truncated term error and the noise error contributions. Error analysis in discrete case has also been recalled, where the existing algorithm was not robust against the numerical errors obtained with negative values of κ. Hence, new algorithms have been proposed to overcome this issue. Based on these new algorithms, error analysis has been given in discrete case. Then, a strategy on how to chose the design parameters has been proposed. The obtained parameters' influence on the errors was deduced from the one on the corresponding error bounds. In general, it is impossible to find a set of optimal parameters which are suitable for all situations. Users should take parameters according to their need. For instance, we can take negative values of κ (resp. positive values of κ and smaller than 1) to reduce the time-delay (resp. to reduce the noise effect due to an independent noise). The largest difficulty is to chose the parameter T (the length of the sliding integration window), which depends on the nature of the signal to differentiate and the one of noises. Hence, as shown in [START_REF] Kiltz | Parametrization of algebraic numerical differentiators to achieve desired filter characteristics[END_REF], a criterion should κ,µ,T,q x(t 0 -T ξ (n) κ,µ,q ) with n = 1, κ = -0.5, µ = 0, q = 1, ξ (n) κ,µ,q = 0.2198 and T = 0.4. be considered in order to choose an appropriate value of T . This is beyond the scope of this paper. Finally, the Jacobi differentiator has been compared to the high gain observer and the high order sliding modes differentiator by considering different criteria. It has been shown that the Jacobi differentiator can give smaller estimation errors by admitting a time-delay.

Fig. 2 :

 2 Fig. 2: D (n),m

Fig. 3 :

 3 Fig. 3: D (n),m

  and Table2, the design parameters' influence on different errors is not the same, hence we should

		1.4
	κ	-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2	0.7 3 2 9 9 0.83299 0 .8 1 .0 3 3 8 1.0338 1.2346	0.52 0.63221	0.44 0 .4 6 0.48 0.5 0.52 0 .5 4 0.54 0.56 0.58 0 .5 8 0.6 322 1 0. 7 Minimum 0.48 0 .5 0. 56 1.2 346 1.0 33 8 0.5 2 (0,0.32)	0. 54	0 .5 6 0. 83 29 9 0 .5 8 0 .6 3 2 2 1 1.83 69 0 .7 2.2 385 2.0 377 1.6 36 1 1.4 35 3
			-0.4 -0.2	0	0.2	0.4	0.6	0.8	1	1.2	1.4
							µ

  by considering other different criteria.

	Method	Convergence time	Number of parameters
	Jacobi	T		Number of state variables: 4
	High gain	unknown	Number of sub-systems: 2
	Sliding modes	unknown	Number of sub-systems: 2
	Method	Time-delay	Robustness to noise and Ts
	Jacobi	known (T ξ	(n) κ,0,2 )	good
	High gain	unknown (small)	good
	Sliding modes	unknown	bad

Table 3 :

 3 Comparison results with different criteria


Amélie Josselin-Leray

Cécile Fabre
email: cecile.fabre@univ-tlse2.fr

Josette Rebeyrolle
email: rebeyrol@univ-tlse2.fr

Aurélie Picton
email: aurelie.picton@unige.ch

Emmanuel Planas
email: emmanuel.planas@univ-nantes.fr

Good Contexts for Translators-A First Account of the Cristal Project

Keywords: CAT tools, corpus resources for translators, Knowledge-Rich Contexts

This paper questions the notion of good contexts for translators and describes an experiment which tests the usefulness of two specific kinds of contexts in a translation task, namely (1) contexts that provide conceptual information about a term, and (2) contexts that provide linguistic information about the collocational profile of this term. In the experiment, trainee translators are asked to use several types of resources, including a set of pre-annotated contexts of various types, and to identify the contexts that they consider to be the most relevant for their task. We present the first results of this experiment, which confirm our general assumption about the usefulness of such rich contexts and indicate some differences regarding the use of contexts in the source and target language. This study takes place in the CRISTAL project whose aim is to retrieve from bilingual comparable corpora the contexts that are the most relevant for translation and to provide them to users through a CAT tool.

Introduction

Even though it is widely acknowledged as being essential to the translator, the very idea of context in translation is hard to define (Baker 2006: 321) and it also "lacks a definition that can be applied in the everyday work of a professional translator" as stated by Melby & Foster (2010: 1). Therefore, when one wants to provide translators with tools that better meet their needs-such as improved CAT tools-, one should in the first place wonder about what makes a context relevant for them. In other words, what is a 'good context' for translators? This is one of the questions the CRISTAL project 1 tries

to give an answer to. The main aim of the CRISTAL project, an acronym that stands for "Knowledge-Rich Contexts for Terminological Translation" ("Contextes Riches en Connaissances pour la Traduction Terminologique" in French), is to automatically retrieve from bilingual comparable corpora the contexts that are the most relevant for translation and to provide them to users through the CAT tool developed by Lingua & Machina, the Libellex Platform.

The first part of this paper reviews the translators' needs regarding context. It seems necessary to first identify which type of information is essential for translators, to see how this information is recorded in the tools most commonly used by translators, i.e. dictionaries, term banks and corpora, and how satisfied translators are about the way that information is recorded. In order to refine the notion of "good contexts" for translators, in part 3 we investigate what a "good example" in lexicography and what a "Knowledge-Rich Context" in terminology are, and introduce the distinction between conceptually-rich and linguistically-rich concepts. Part 4 then focuses on one aspect of the methodology of the CRISTAL project: an experimentation involving trainee translators in order to refine our idea of a "good context for translators". Finally, part 4 presents the very first results of the experiment.

Some Facts about the Needs of Translators Regarding Context

As stated by Rogers & Ahmad (1998: 195), "one of the translator's prime needs is for context-sensitive information". We may wonder what the notion of context-sensitive information encompasses and what sources of information translators can rely on-or not.

What do Translators need Contextual Information for and

Where do they Find it?

Context in Translation: a Preliminary Definition

As thoroughly explained by Melby & Foster (2006), specialists in many fields (e.g. philosophy, psychology, pragmatics, and functional linguistic) have discussed the notion of context, and various definitions have been written. The three facets of context as defined by [START_REF] Halliday | The notion of Context in Langue education)[END_REF], i.e. context of situation, context of culture and co-text are all particularly relevant in translation. However, in this paper, we will only focus on what Halliday calls co-text. While both context of situation and context of culture are outside of language itself, co-text specifically pertains to language in use. It can broadly be defined as the surrounding discourse of an utterance. Therefore, our definition of "context" in this paper will 1 CRISTAL is a three-year project (2012)(2013)(2014)(2015) funded by the French National Agency for Research (ANR-12-CORD-0020). It involves four partners: a computing research team at the University of Nantes, France (LINA), a linguistics research team at the University of Toulouse, France (CLLE-ERSS), the Translation Technologies team from the Faculty of Interpreting and Translation at the University of Geneva in Switzerland, and a firm specializing in multilingual text management (Lingua et Machina).

be limited to co-text, and will rely on the definition provided by Fuchs2 :

What is called context is the linguistic environment of an element (phonetic unit, word or group of words) within an utterances; i.e. the units that precede and follow it. Thus, in the utterance "Marie est jolie comme un coeur", the element comme has as its immediate context "jolie…un coeur" and its wider context "Marie est jolie…un coeur". By extension, the word context is also applied to the utterance(s) which precede(s) and follow(s) a given utterance in discourse.3

2.1.2

What do Translators Need Context for? Following Roberts & Bosse-Andrieu (2006: 203), let us remind here that the translation problems translators have to face can be considered as source text-related (for comprehension of the source-text) or target-text related (for transfer into the target text) and classified into three main categories: encyclopedic, linguistic or textual. Encyclopedic problems encompass "general subject-related problems as well as more specific problems dealing with proper nouns-that is, a lack of familiarity with the topic of the text or with specific places or people mentioned in the text"; linguistic problems are defined as "those attached to specific words and phrases-that is, problems related to the comprehension or translation of a given word or phrase"; finally, textual problems are those concerned with text types and the internal organization or reproduction of a given text type. [START_REF] Bowker | Off the record and on the fly[END_REF][START_REF] Bowker | Meeting the needs of translators in the age of e-lexicography: Exploring the possibilities[END_REF] draws a list of those items of contextual information that can prove "useful" for the translator to solve his source-text and target-text-related problems. They can be summed up as follows: (i) information about usage; this of course includes collocations, in particular which general-language words collocate with terms (see also Roberts 1994: 56), (ii) information about the frequency of use of a particular word or term, (iii) information about lexical and conceptual relations (such as synonymy, meronymy, hyperonymy etc.) (see also [START_REF] Marshman | Helping translators manage terminological relations: Storing and using occurrences of terminological relations and lexical relation markers[END_REF][START_REF] Rogers | The Translator and the Dictionary: Beyond Words?[END_REF], (iv) pragmatic information about style, register and genre (see also Varantola 1998), (v) information about usages to avoid. It thus seems to us that the items that are not situation-linked fall into the following categories: conceptual information and linguistic information.

To solve those problems and to make decisions, translators need external help, which they typically get by consulting other human experts and conventional resources such as dictionaries and term banks (Rogers & Ahmad 1998: 198).

2.1.3

Where do Translators Find Contextual Information? As mentioned by Varantola (2006: 216), "the translator's problem-solving techniques have changed dramatically over the past decade or so". In addition to the above-mentioned conventional resources (monolingual and bilingual dictionaries-which have undergone radical changes-; term banks), translators now also partly rely on the information provided by corpora.  Dictionaries It is mostly through examples that dictionaries provide contextual information. The empirical study on scientific and technical words (i.e. terms) in general bilingual and monolingual dictionaries carried out by [START_REF] Josselin-Leray | Place et rôle des terminologies des terminologies dans les dictionnaires unilingues et bilingues. Etude d'un domaine de spécialité : volcanologie[END_REF] has shown that up to 80.3% of users turned to dictionaries to find information about how to use the term in a sentence and that bilingual dictionaries always ranked higher in that respect. Among the respondents who chose that answer, it was the « language professionals » user group (which includes translators) that was mostly represented.

 Term Banks

Term banks typically provide contextual information through the "context" section of the terminological record. The importance given to contextual information in terminological resources by translators is confirmed by the findings of the survey by Duran-Muñoz (2010): examples were considered to be "essential data" by the respondents, and among "desirable data", one found "a greater variety of examples" and "semantic information (semantic relations, frames)".

 Corpora

Although corpus data is obviously intrinsically made of contextual information, it is a resource which seems still quite scarcely used by translators, as shown by the results of the survey by Duran-Muñoz (2010): only 5.09% of the participants quoted (parallel) corpora as being a terminological resource they "used more4 when translating". However, 41.8% of the respondents to the Mellange Survey5 , which was carried out in 2005-2006 among trainee translators and professional translators, do claim they use corpora in their translation practice (the most frequent type being the corpora in the target language).

Although there seems to be a wide array of resources translators can turn to when they need contextual information, these resources do not necessarily meet the translators' needs.

The Shortcomings of Existing Resources regarding Context

2.2.1

The Dissatisfaction of Translators regarding Contextual Information in Existing Resources: a Hard Fact We found it relevant to first look at the findings of various empirical surveys on the use of conventional resources by translators.  Dictionary Use Before starting to compile the Bilingual Canadian Dictionary, Roberts (1994: 56) carried out a survey among its potential users in order to clearly identify their needs and reached the following conclusion: "Between one-third and one-half the members in each user group [of sophisticated second language users] appreciated, to varying degrees, the number of examples presented in their present most frequently used dictionaries. But between one quarter and one half of each group felt that improvement was needed in that respect". The study by [START_REF] Josselin-Leray | Place et rôle des terminologies des terminologies dans les dictionnaires unilingues et bilingues. Etude d'un domaine de spécialité : volcanologie[END_REF] reached the same conclusion: although users were overall satisfied by the examples provided by their dictionaries (between 41.3% and 67.5% of users said they were satisfied), the level of satisfaction was lower for bilingual dictionaries, and lower among the "language professionals" user group.

 Term Banks

In the survey by Duran-Muñoz (2010), translators also had the opportunity to give their opinions regarding their needs; the second most repeated argument was "include more pragmatic information about usage and tricky translations", and the fifth one was "provide examples taken from real texts". The conclusion of her findings, found in Duran-Muñoz (2012: 82) is straightforward: "we can confirm that most of the terminological resources that are currently available (especially in electronic format) do not fulfill their requirements". Why is that so?

2.2.2

Why are Translators Dissatisfied? Varantola (e.g. 1994) has written at great length about the context-free vs. context-bound dilemma faced by translators: dictionaries and term banks only provide context-free examples, i.e. examples that are perceived as prototypical and frequent, while what the translators need to find the suitable equivalent(s) is typically context-bound. Moreover, the examples/contexts provided are not varied enough. This is especially true of term banks, as underlined by Bowker (2011: 214-215) who explains the information found on those records is rather limited and usually consists in definitions and terms presented out of context, or in only a single context. She pinpoints a paradoxical situation in which the advances of research on terminology (especially the work on Knowledge-Rich Contexts, which we will introduce in 3.2) have not been integrated into the tools translators most commonly use, i.e. term banks. However, Varantola (2006: 217) says the context-free/context-bound dilemma should now be qualified since "context-free definitions of concepts within a particular domain [which] were for a long time the theoretical ideal in terminological theory […] are now replaced by less rigid, contextually relevant definitions". She ascribes it to the availability of large corpora, whose role is also now central in dictionary-compiling. Some dictionaries now even "provide access to more examples in the form of concordances from the corpus data that lie behind the dictionary". Corpora are no panacea, though. One of arguments against corpora is that they are "tools of shallow intelligence" (Varantola, 2006: 223) when they are raw and non-tagged or POS-tagged, since the user "is left to handle the manipulation, dissection and interpretation of results". In other words, compiling the corpus and analysing the corpus can be too tedious a task for translators who often work under tight constraints 6 .

The Translator's Ideal Workstation?

In 1996, Atkins already suggested (p. 526) that the dictionary of the future should "give its users the opportunity to make their own decisions about equivalences" : the users "should be able to consult as many examples as they need of words used in their various senses, each in a variety of contexts with a variety of collocate partners". More recently, Bowker (2011: 215) suggested: "it would be more helpful for translators to have access not simply to term records that provide a single 'best' term with a solitary context, but rather to information that would allow them to see all possible terms in a range of contexts and thus find the solution that works best in the target text at hand". She insists on the fact that looking at a wide range of contexts should not be considered as a waste of time, and that this has been made easier thanks to corpus-analysis tools that present information in an easy-to-read format. She goes even further by suggesting (Bowker 2012: 391) that translators have access to the whole of the information that lexicographers usually rely on when devising a dictionary entry:

In order to arrive to that entry, lexicographers have gone through a number of intermediary steps, where they learn about the various characteristics of the words and concepts being described, such as their grammatical and collocational behaviours, the different relationships that hold between words and their underlying concepts, and the characteristics that are necessary and sufficient for distinguishing one concept in an intensional definition.

However relevant that objective might be, it seems rather ambitious and difficult to achieve in the very near future, all the more so as "lexicographers'needs are very different from translators' corpus needs" (Varantola 2006: 217). Narrowing down that objective to providing more corpus-based context data in a way that is more in keeping with the actual working conditions of translators seems more feasible, which is why the main aim of the CRISTAL project is to help design a CAT tool that provides translators with customized contexts automatically retrieved from comparable corpora. In order to reach that goal-which can also sound ambitious, we first decided to refine the notion of 'good contexts for translators' by doing two things: (i) we first looked at the way lexicographers deal with examples in dictionaries and terminographers deal with "Knowledge-Rich Contexts" (part 3), (ii) we devised an experiment with trainee translators focusing on what we thought to be "good contexts" (part 4).

Dictionary Examples and Knowledge-Rich Contexts

The thoughts of lexicographers and terminologists on "good examples" or "Knowledge Rich Contexts" provide some valuable insight into what a good context for translators might be. After examining those two aspects, we give our own definition of Conceptually Rich Contexts and Linguistically Rich Contexts.

Good Dictionary Examples

Many studies have underlined the importance of the illustrative component in dictionaries as a means to provide typical contexts about a word's meaning and usage (Atkins & Rundell 2008). In monolingual as well as in bilingual dictionaries, examples are meant to help the dictionary user both in the production and the comprehension process. They have therefore diverse functions (Rey-Debove 2005, [START_REF] Roberts | Bilingual Dictionaries Prepared in Terms of Translators' Needs[END_REF][START_REF] Siepmann | Discourse markers across languages: A contrastive study of second-level discourse markers in native and non-native text with implications for general and pedagogic lexicography[END_REF]): they can provide syntagmatic information about word patterns and collocations, together with paradigmatic information about words that are semantically-related (synonyms, hyperonyms, etc.). They may also give pragmatic and stylistic indications about registers and specific uses, or be used as a more concrete and accessible complement to definitions, with an epilinguistic dimension.

Authentic examples that meet at least some of these requirements are very difficult to extract from corpora:

Finding good examples in a mass of corpus data is labour-intensive. For all sorts of reasons, a majority of corpus sentences will not be suitable as they stand, so the lexicographer must either search out the best ones or modify corpus sentences which are promising but in some way flawed (Rundell & Kilgarriff 2011). [START_REF] Kilgarriff | GDEX: Automatically finding good dictionary examples in a corpus[END_REF] have developed a method to automatically collect sentences that are good candidate dictionary examples, using two criteria: readability (judged from sentence length and average word length; it penalizes sentences with infrequent words, more than one or two non-a-z characters, or anaphora) and informativeness (judged from the density of collocates in the sentence).

Knowledge Rich Contexts (KRCs) in Terminology

Knowledge Rich Contexts play a very important role in identifying terms in specialized texts because they show conceptual relationships between terms. It is within that framework that Ingrid Meyer defined Knowledge Rich Contexts as "a context indicating at least one item of domain knowledge that could be useful for conceptual analysis." (Meyer 2001: 281). These contexts are used in order to develop knowledge extraction tools for text-based terminology and ontology building [START_REF] Condamines | Searching for and identifying conceptual relationships via a corpus-based approach to a Terminological Knowledge Base (CTKB): Method and Results[END_REF]. Rich contexts for terminologists typically contain terms that are specific to the domain together with linguistic patterns that signal the conceptual relations between these terms as illustrated below in Meyer ( 2001):

(1) Compost is an organic material deliberately assembled for fast decomposition.

(2) Compost contains nutrients, nitrogen, potassium and phosphorus.

This type of information helps building networks of terms, generally focusing on hyperonyms (example 1) and meronyms (example 2).

KRCs for Translators: Conceptually vs. Linguistically Rich Contexts

Based on the type of information needed by translators as described in 2.1.2, and the type of information provided by dictionary examples and Knowledge-Rich Contexts as detailed in 3.1 and 3.2, we decided to extend the notion of KRC in our experiment, considering under this category two types of contexts: those that provide 'conceptual' information about a given term-called "Conceptually Rich Contexts" (CRCs) in our study, and those that provide 'linguistic' information about that term "Linguistically Rich Contexts" (LRC). In our experiment, the contexts which are neither conceptual nor linguistic are considered as poor (cf. [START_REF] Reimerink | Contextual information in terminological knowledge bases: a multimodal approach[END_REF]Reimerink et al. : 1934)).

An Experiment Centered on "Good Contexts" within the CRISTAL Project

The main aims of the experiment were (i) to check that rich contexts extracted from corpora are useful to translators, (ii) to identify which types of rich contexts (CRCs or LRCs) are the most useful to them. A pilot study was carried out in December 2013 at the University of Geneva (Switzerland). This allowed us to test the protocol and to make the necessary adjustments for the two experiments we conducted in March 2014: one at the Université Catholique de l'Ouest (Angers, France), and one at the University of Toulouse le Mirail (Toulouse, France).

We will now describe the main aspects of the protocol designed for the experimentation.

Protocol

Participants

For both the pilot study and the two experiments, participants were all trainee translators7 .7 students from the Faculty of Translation and Interpretation of the University of Geneva took part in the pilot study. There were 4 Masters' students and 3 PhD students. As for the experiments in Angers and Toulouse, the participants (42 in total) were students in their final year of a Master's Translation program.

4.1.2

Translation Ta sk The participants were asked to translate a text from English into French (i.e. from L2 into L1 for most students). The text is around 150 words long, it is a popular-science text on volcanology entitled "Cinder Cones"8 . It was chosen because it is well-structured (the two phases of the building of a cinder cone are described), because it contains a certain number of terms whose translation might be complex for a translator, even if they are not highly specialized (basalt cinder cone, fountaining stage…), and a number of syntactic patterns or collocations that are particularly tricky to translate (e.g. bubble off; transitive use of the verb erupt). Only one group out of the two was already a little familiar with the field of volcanology. The participants were allocated around 2 hours to translate the text, choose the relevant contexts and fill out an online questionnaire about the main translation difficulties and the use and usefulness of conventional resources and KRCs. Then several group interviews and a couple of one-to-one interviews were conducted.

4.1.3

Resources Since we wanted the conditions of the experiment to be as close to a real-life context as possible for translators,9 the participants had at their disposal the same type of resources as the ones they usually have when they translate in professional environment, i.e. various dictionaries and term banks. What made the experiment specific is that we added an extra resource, i.e. shortlisted contexts.

4.1.4

The Argos Interface The participants used a customized interface, Argos, with four different windows (see figure 1): (i) one for the source-text, (ii) one for the target-text, (iii) one with several icons allowing access to term banks (Termium, le Grand Dictionnaire Terminologique), a specialized bilingual dictionary of earth sciences, a general bilingual dictionary, (iv) one with a list of shortlisted contexts. The Contexts Participants were provided with contexts in the source language, and contexts in the target language, which were presented in random order. These contexts had been carefully chosen beforehand according to a classification devised by the team of linguists: details about the selection of contexts and the type of contexts provided will be discussed in the next section (4.2). During the translation process, the participants had to choose the contexts that had been most useful to them when translating by clicking on them. Once the translator had typed in a word (in the source language or the target language) in the search window for contexts, the target text window was blocked in order to ensure the translator chose at least one context, or explicitly chose that none was useful.

4.1.6

Extra Data Compiled In addition to the final translations themselves and the answers to the online questionnaires, the data compiled comprise the following:  screen-recordings performed through specific software10  logs: all keyboard activity, as well as change of windows shifts, was recorded  audio recordings of the two types of interviews11 during which the participants were asked to give more detail about the usefulness of some given contexts.

The List of Contexts

The list of contexts provided to the participants was created with two principles in mind: we wanted to compile a large enough set of contexts, in order to limit the chance that the translator would look for contextual information on one particular word and get no results. At the same time, considering that the selection of contexts is a labor-intensive task, we wanted to limit ourselves to a reasonable number of words, and to keep only words for which our definition of Rich and Poor Contexts applies (see 3.3 above): in particular, the notion of Conceptually Rich Contexts (CRCs) is irrelevant for very familiar words that are not characteristic of the field of volcanology from which the text is drawn. This compromise was difficult to reach, so we took advantage of the pilot study to adjust and complement the list of terms that we had initially created.

Term Selection

For the pilot study in Geneva, we compiled a first set of contexts illustrating the use of:  7 words (noun, verbs and adjectives) in the source language, namely some lemmas from the text to be translated that we regarded as terminological units, or at least as words related to the field of volcanology (basalt magma, blobs, cinder, cinder cone, fountaining, scoria, vesicles)  11 words in the target language, selected among the possible equivalents of the corresponding source words.

We considered both simple and multiword units. For example, contexts are provided for the word in bold type in the following sentence:

(3) As the cinders fall back to the Earth, they form layers that pile up into a cone-shaped hill.

One outcome of the pilot study was that the initial list of words proved to be very insufficient, especially in the target language: the logs compiled thanks to Argos (cf. 4.1.6) provided us with a much larger list corresponding to words that had actually been typed in the search window by the participants in order to get contexts. We thus decided to complement the first list with words that have been searched for by at least 2 participants. The final list for the experiments in Angers and Toulouse contains contexts for 22 English words and 41 French words. In the same sentence as the one mentioned above, contexts were provided for four words (in bold type) instead of just one (i.e. 2 nouns, 1 verb, 1 adjective):

(4) As the cinders fall back to the Earth, they form layers that pile up into a cone-shaped hill.

4.2.2

Context Selection The contexts were selected from several sources:  we preferentially used a 800,000 word corpus of volcanology composed of specialized and popular science texts, which is used for the CRISTAL project as a whole,  this source was complemented by a variety of web sites, giving priority where possible to texts dedicated to the presentation of volcanology to a wide audience.

We chose not to test the readability dimension of contexts (3.1.), so we only selected contexts that meet the criteria of readability (well-formed, not too long, with no anaphora elements, etc.). Contexts are one or two sentences long. As explained before, our aim is to test whether the opposition between rich and poor contexts as defined in section 3 is relevant for the translation task. As a consequence, we annotated contexts according to this dimension. In figure 1, we give 3 examples illustrating (1) a linguistically rich context for the word basalt (with the presence of the term basalt lava),

(2) a conceptually rich context providing a definition of the term, (3) a poor context. Note that linguistic and conceptual richness can combine in some contexts, which is not the case here. When possible, the conceptually rich contexts were classified into the following subcategories: definition, meronymy, hyponymy and co-hyponymy. We intended to balance the number of poor and rich contexts for each word. This proved impossible to achieve in many cases, since the great majority of contexts exhibit at least one relevant collocate. We collected 10 contexts per word, including no less than 2 poor contexts, totalling 222 contexts in English and 441 contexts in French.

First Results

We report here some preliminary observations about the results.

For the source language, 48% of the available contexts were chosen by at least one participant (108 contexts) versus 36% for the target language (152 contexts). This is an indication that the contexts are perceived as helpful, but the data are very dispersed, since about 40% of the selected contexts were chosen by only one participant in either language. Some terms are found several times in this list (fountain, cinder cone, basalt magma, and their French counterpart), showing specific translation problems.

To get a first picture of the results, we have chosen to focus on the 20 contexts that were selected most in either language. Each context was chosen between 4 and 14 times. The following examples show two of the most frequent ones.

(5) Hawaiian Eruptions are types of volcanoes and types of eruptions wherein basaltic lava is normally thrown up the air in jets. This process is called fountaining. [START_REF] Atkins | Bilingual Dictionaries: Past, Present and Future[END_REF] During an eruption of gas-rich magma, small blobs of magma are ejected.

Example 5 is a conceptually rich context, more specifically a definition. Example 6, which contains several collocations (blobs of magma, blobs ejected), is a linguistically-rich context. 2 makes a comparison between this subset of contexts and all the contexts that were made available. First, this shows that the great majority of contexts that are considered as helpful by the participants are rich contexts (90%). Second, participants show a strong preference for conceptually-rich contexts, mainly definitions, as opposed to linguistically-rich contexts. Yet we can observe that if this overall pattern applies to both languages, there are some differences: the distribution between LRCs and CRCs is different when the users are exploring source (English) and target (French) contexts. They seem to give a higher priority to CRCs and definitions in the source language. This is consistent with the idea that the CRCs should provide help for the comprehension of terms and LRCs should be more useful when checking the usage of the words in the target language. These are encouraging results: they confirm our assumption that rich contexts are seen as helpful by the participants, and they suggest differences in the way the translator uses these contexts in the source and target language. However, these first observations must be confirmed and complemented by the analysis of the entire set of data and the analysis of the replies to the questionnaires.

Rich contexts

Conclusion

The notion of context is where lexicography, terminology and translation meet. Even though the specific needs of translators regarding their resources now seem quite clearly identified, addressing them still seems quite challenging. We hope the findings of the CRISTAL project will help tailor the tools according to the translator's profiles in one aspect, that of contextual information.

To fulfil that objective, we plan to explore in detail the considerable amount of data we have collected (around 60 hours of screen recordings, and just as much structured translation logs). The evidence gathered will enable us to answer the following questions:  apart from definitions, do some sub-categories of KRCs play a specific role in the translation process?  in which precise situations do translators give preference to KRCs over conventional resources such as monolingual or bilingual dictionaries?  what is the impact of the use of KRCs and the other resources on the translation quality of the 49 final translations The main challenge the CRISTAL project plans to address in the near future is to devise a method to automatically retrieve the 'good contexts' whose main features will have then been identified.

Figure

  Figure 1: the Argos interface 4.1.5The Contexts Participants were provided with contexts in the source language, and contexts in the target language, which were

Figure 1 :

 1 Figure 1: Examples of contexts for the word basalt.

Table 2 :

 2 Distribution of the contextsTable

	CRCs (definitions)	LRCs

http://www.universalis.fr/encyclopedie/concept/. Consulted on April 2, 2014.

We translated the quotation.

We added the italics.

http://mellange.eila.jussieu.fr/Mellange-Results-1.pdf. Consulted on April 10, 2014.

Another point worth mentioning is that, in the survey carried out for the Mellange, even though 94.4% of the respondents said they used Google to research terminology, 10.2% found that Google was limited for finding information on language use because the "search results [did] not provide enough context to be useful".

This is the case in many empirical studies on translation: see for example[START_REF] Bowker | Using Specialized Monolingual Native-Language Corpora as a Translation Resource: a Pilot Study[END_REF][START_REF] Künzli | Experts vs. novices. L'utilisation de sources d'information pendant le processus de traduction[END_REF][START_REF] Varantola | Translators and their Use of Dictionaries[END_REF]

It was taken from What's so hot about volcanoes? by Wendell A. Duffield (2011), Mountain Press.

This is what Ehrensberger-Dow & Massey (2008) call "ecological validity".

BBFlashback Express.

In the one-to-one interview, the participant viewed part of the screen recording (the extracts corresponding to his translation of two terms for which contexts were provided: cinder and fountaining stage) and was asked to verbalize what he was doing, following the methodology used by Ehrenbersger-Dow and Massey (2008).


Juan Carlos

Rodríguez Esquivel

Amilcar Meneses
email: amilcar.meneses@google.com

Nicolas Perry
email: n.perry@i2m.u-bordeaux1.fr

Gestures for interaction between the software CATIA and the human via Microsoft Kinect

Keywords: Kinect, Gestures, Interaction, Cognitive Load

CAD systems are very useful tools in the industry, because they help design processes and new product development. However, it requires new mechanisms of interaction to provide greater usability of these tools. This usability can be provided through new interaction devices such as Kinect. We describe a methodology to find a series of gestures makes by the hands for control a CAD system with Kinect sensor of Microsoft. These gestures must be easy to remember for the user of the system and they don't should represent a cognitive load to make the design's experience most interactive.

Introduction

One of the most demanding works in the world is the mechanical designer because it requires being in front of a computer for many hours several days a week. The mechanical designer uses the CAD systems for accomplish his work. One of the most popular CAD systems in companies such as automotive and aviation is the software CATIA, develop by Dassault Systèmes. CATIA help us to design and develop new products [START_REF] Lee | Principles of CAD/CAM/CAE Systems[END_REF][4]. But this system can cause problems by the use of a keyboard or a mouse after a workday and is also known that sitting in front of the computer could harm us by the lack of body movements. And also, sometimes this system may not be as easy to use for novice users.

The main idea in this document is to look for the right gestures to use in MOCAP system, in this case the Kinect sensor, to help users of CATIA in the design and management of parts and assemblies. These gestures will make correspond a task or a command in software, reducing the cognitive load when the user learns and remember the gestures.

Gestures for communication

The gestures define the communication between humans. Finding a series of gestures to define an action or task is difficult because in the world exist several ways to say or do a thing, depending on the language and the culture of the people. We can have an infinite number of gestures to interact, but many of them may not be identified or remembered [START_REF] Von Hardenberg | Bare-Hand Human-Computer Interaction[END_REF][6] [START_REF] Jacko | Human-Computer Interaction Handbook: Fundamentals, Evolving Technologies, and Emerging Applications[END_REF].

One important task is to find the bridge in communication between the humans to perform gestures and then, translate these gestures to a Human-Computer communication, so that they can be expressed naturally and can be remembered easily [START_REF] Miller | The Magical Number Seven, Plus or Minus Two, Some Limits on Our Capacity for Processing Information[END_REF][3] [START_REF] Von Hardenberg | Bare-Hand Human-Computer Interaction[END_REF]. Also these gestures can't confuse the computer or do something that has not referred to do in the development of the tasks. Computer-Human interaction is also handled by the feedback that should exist with an Augmented Reality system and human for have a more natural interaction.

The augmented reality help us to supplement the interaction giving the feeling of manipulating a physical piece and not a virtual piece in 2D like the conventional systems.

Gestures definition

In a study conducted in 1955 by George A. Miller, a professor at Harvard University, talks about the limits of human ability to process information. Miller found that the ability of humans to store items in your short term memory is between 5 and 9 elements, ie, 7, plus or minus two, because these processes are provided by our biological limitations delivering the nervous system characteristic of every human being, because every human being processed information differently. Thus one of the main objectives in the design of interfaces that the number of gestures used by the user are less than nine.

The proposed methodology to the system is the use to the state machines, based on the proposed work by Carrino, S., Caon, M et al. [START_REF] Carrino | Functional Gestures for Human Environment Interaction 4[END_REF]. The idea is to find the gestures in a series of actions to be taken should take some order. Carrino proposes using finite state automata to study the changes of states in the system in which the interaction take place [START_REF] Carrino | Functional Gestures for Human Environment Interaction 4[END_REF]. The states of the automaton are the states of the system. Transitions in the automaton are given by gestures that the user uses to interact with the system. Thus, the interface designer can verify that the nine gestures are not exceeded in every state of the system.

Then to define gestures in CATIA, we must first define their states and the set of actions and tasks that change the system state.

States / interactive entities defined for the project: ─ This is the gesture that will change status , this is due to the different steps that have to be done to open a document and look between the user's files . This will depend on what state we are and shall consist of the combination of some past functional gestures.


Table 1. Set of states and functional gestures for deterministic finite-state automaton.

We found that the deterministic finite automata which describes the use of CATIA has 5 different states and 8 funcional gestures (alphabet). The set of states Q = {a, b, c, d, e} and the alphabet  Table 1 shows the description of the states and of the elements of the alphabet. The transition function is expressed in the figure 1.

The number of functional gestures taken based on the work of Miller on the ability to process information and learning these. In what states gestures and giving us a total of 8, within the limits of human capacity are proposed. The functional gestures were defined using surveys to find frequently used commands under the CATIA software environment.

To find the gestures we are developed two principal ideas:  The gestures should be easy to remember  The gestures should be natural, i.e. these gestures should be easy to do, the user will don't make any effort.

The gestures were found on the observation of daily life of people and their way of doing routine things because in everyday life that can make for some aspects similar to the worked within the CATIA environment.

Move

Left hand or right hand takes control of the mouse to scroll the screen, as if it was moving the hand to grasp an object.

Undo

Both hands are downward as if you were crushing or removing something opposite.

Select

Place the mouse pointer with the hand in the point / piece desired and leave the hand in this position for 3 seconds.

Drag

Do the gesture to select, and then, with the piece selected, down both wands to activate the movement to the piece.

Zoom +

Joining both hands at the stomach, and then hands them away horizontally.

Zoom -

Put your hands up to the stomach (hands should be away), and then bring them closer horizontally.

Rotation

We started holding the piece (place both hands at the stomach and make a gesture of pressure), then move one hand to the point where you want to rotate and wait 3 seconds. Will now rotate with one hand.

Conclusions

You can make the state characterization of the CAD systems such as CATIA. This allows knowing the transitions between them, ie actions or tasks that they change state. With the characterization of finite automaton system is possible to identify the number of actions and tasks for state changes. These actions and gestures em tasks are mapped in the case of MOCAP devices, such as Kinect. You can have the control of the state transitions to prevent over nine gestures are associated with a system status. It has been observed that these gestures do not handle all tasks in CATIA, so you can opt for multimodal gestures.

  Program closed: the CATIA software is closed and we are in the Windows environment.  Open program, no file display: the CATIA program is open, but we are in the welcome screen without any open file.  Viewing and editing: we have an open file or a new file for the part to make or change  Close piece, Save: we are in the save screen.  Close Program: shutdown screen and return to the Windows environment. The functional gestures that were handled in the project are:  Move: Move the cursor around the screen.  Rotation: Rotate Part / Assembly in CATIA.  Zoom +: zoom in the part/assembly.  Zoom -: zoom out the part/assembly.  Select/left click: choose a piece/action.  Drag: move a piece around the screen.  Undo.  Special case functional gesture 8

Fig. 1 .

 1 Fig. 1. Finite States Machine of the project, in which we can see the 5 states a,b,c,d,e and we can find the functional gestures defined by the numbers

Table 2 .

 2 Gestures -functions relationship

	Function	Gesture