Sofiane Khadraoui

Member, IEEE Micky Rakotondrabe

Member, IEEE Philippe Lutz

Optimal design of piezoelectric cantilevered actuators with guaranteed performances by using interval techniques

Keywords: Piezoelectric actuators, Design and development, Interval analysis, Bounded parameters, Set-inversion problem, Guaranteed and optimal design, micro and nano-scale

Optimal design of piezoelectric cantilevered actuators

with guaranteed performances by using interval techniques.

I. INTRODUCTION

I N traditional robotics, robots and systems use joints and (DC) motors as principal elements of motion. However, inherent friction in the articulated mechanisms of these robots strongly limits their use in applications that require high accuracy and high resolution such as microrobotic and micromaniplation applications. Consequently, microrobots are generally designed with active or smart materials and deformable structures instead. The advantage is that the resolution of the yielded displacement is highly increased in these structures as the friction is minimized. Additionally to that, the manufacturing of smaller and miniaturized actuators is easier with smart materials and deformable structures than with DC-motors and classical joints. Piezoelectric materials are one of the most recognized smart materials used to develop microrobots and systems for micro-nano positioning (precise positioning). This recognition is thanks to the high resolution (up to nanometric level), the high bandwidth (up to tens of kiloHertz) and the high force density they can offer. Furthermore, the ease of power supply (electrical energy) makes their use more generalized. Finally, piezoelectric materials can also be used as sensors [START_REF] Lin | Diagnostic lamb waves in an integrated piezoelectric sensor/actuator plate: analytical and experimental studies[END_REF]- [START_REF] Costa Branco | On the electromechanics of a piezoelectric transducer using a bimorph cantilever undergoing asymmetric sensing and actuation[END_REF] or as simultaneous sensors and actuators (called self-sensing) for applications where the integrating external sensors is impossible [START_REF] Rakotondrabe | Dynamic displacement self-sensing and robust control of cantilevered piezoelectric actuators dedicated to microassembly tasks[END_REF], [START_REF] Ma | Piezoelectric self-sensing actuator for beam vibration control based on blind source seperation[END_REF]. Although the good resolution and the high bandwidth of piezoelectric actuators, one of their limitations is the restricted range of displacement offered (about 0.1% strain) which is a great disadvantage for some of the above mentionned applications. A technique to overcome this limitation was to employ stepper piezoelectric actuators (stick-slip, inch-worm,...) [START_REF] Rakotondrabe | Development, modeling and control of a micro/nanopositioning 2-dof stick-slip device[END_REF], [START_REF] Zhang | A load-lock-compatible nanomanipulation system for scanning electron[END_REF], [START_REF] Zhang | A load-lock-compatible nanomanipulation system for scanning electron[END_REF]- [START_REF] Peng | Modeling of piezoelectric-driven stick-slip actuators[END_REF]. Stepper piezoelectric actuators are well known to provide a large range of displacements. However, as they result from the assembly of several components, these actuators are more complex to develop and their miniaturization is still limited. Therefore, they are generally employed for tasks that really require very large strokes and where there is enough space to place them. Notable complementary systems to stepper piezoelectric actuators are piezoelectric microgrippers which can perform a fine positioning with a very high resolution and a very high speed [START_REF] Edamana | Modeling and Optimal Low-Power on-off Control of Thin-Film Piezoelectric Rotational Actuators[END_REF]- [START_REF] Haddab | A microgripper using smart piezoelectric actuators[END_REF]. These microgrippers can be utilized as endeffector of the stepper actuators or can work independently [START_REF] Rakotondrabe | Development and Force/Position Control of a New Hybrid Thermo-Piezoelectric microGripper dedicated to micromanipulation tasks[END_REF], [START_REF] Rakotondrabe | Modelling and robust position/force control of a piezoelectric microgripper[END_REF].

A piezoelectric microgripper is generally composed of two piezoelectric actuators that have cantilevered structures called piezocantilevers. Most of the applications utilize one piezocantilever as the displacement actuator while the second one as the force actuator in order to ensure the precise positioning by controlling the manipulation force [START_REF] Rakotondrabe | Development and Force/Position Control of a New Hybrid Thermo-Piezoelectric microGripper dedicated to micromanipulation tasks[END_REF], [START_REF] Rakotondrabe | Modelling and robust position/force control of a piezoelectric microgripper[END_REF]. The design of piezoelectric cantilevered structures devoted to micro/nano positioning, microassembly and micromanipulation applications has been addressed in the past [START_REF] Grossard | Mechanical and Control-Oriented Design of a Monolithic Piezoelectric Microgripper Using a New Topological Optimization Method[END_REF]- [START_REF] Ouyang | Design of a new compliant mechanical amplifier[END_REF]. These works proposed methodologies to design optimized compliant mechanisms with piezoelectric actuation in order to obtain maximized displacement. These approaches are efficient but find their limitation when the design should account specifications on the structures or on the dimensions. These approaches yielded complex structures in that case. In this paper, we propose a new approach to design piezoelectric actuators that also account the structures and the dimensions in the specifications. The approach consists in imposing a priori some wanted performances and some constraints on the geometrical dimensions of the actuators. By combining these specifications with the physical model of the actuator, the proposed design problem is formulated as a set-inversion problem which is solvable using interval techniques. The main advantage of the proposed approach is that if a design solution exists, the specified performances are guaranteed. In fact, such guaranteed performances are inherited from the properties of interval tools and techniques that are used to handle and to solve the problem in this paper. The approach permits therefore to find a set of geometrical sizes within a priori given intervals such that the designed actuator satisfies the specified performances. To demonstrate the approach, an illustrative example of design is proposed. The example consists in redesigning an existing piezocantilever actuator into another one which is more optimized. The designed and optimized actuator will have better performances but smaller dimensions relative to the initial actuator. A prototype is fabricated and experimental characterization on this was carried out, which confirmed the theoretical results.

Interval techniques and related arithmetics have been used in several applications in the past, except for the design of mechanical systems which we intend to propose in this work. According to the history, the first apparition of intervals was in 1897 by Archimedes when he tried to compute the lower and the upper bounds of π. Further, the idea of using intervals for calculation was proposed in 1924 by Burkill and in 1931 by Young. But, Interval arithmetics became really popular just after the appearance of the R.E. Moore's book in 1966 [START_REF] Moore | Interval Analysis[END_REF]. This later provided a general method and some formalization of intervals and related arithmetics with an application of automated error analysis. Nowadays, several applications of intervals are devoted to:

• guaranteed estimation [START_REF] Jaulin | Set inversion via interval analysis for nonlinear bounded-error estimation[END_REF], [START_REF] Jaulin | Interval constraint propagation with application to boundederror estimation[END_REF];

• stability analysis of uncertain systems [START_REF] Jaulin | Applied Interval Analysis[END_REF]- [START_REF] Kharitonov | Asymptotic stability of an equilibrium position of a family of systems of linear differential equations[END_REF];

• robust controllers synthesis [START_REF] Khadraoui | Interval Modeling and Robust Control of Piezoelectric Microactuators[END_REF]- [START_REF] Rakotondrabe | Performances inclusion of stable interval systems[END_REF]. The main interests of intervals and related arithmetics are: the simple and natural way to represent parametric uncertainties by just bounding and their ability to predict a guaranteed set of solution (or non-solution) of design or of controllers that will satsify (or not satisfy) some specifications. This paper utilizes these properties to design piezoelectric actuators such that a more optimized structure (in term of geometrical sizes) than an existing actuator will also provide better performances. The approach can be applied to broader types of actuators but the example carried out in this paper is the design of unimorph piezocantilevers. This permits to show the methodology with an illustrative example.

The paper is organized as follows. Section-II is dedicated to brief preliminaries on intervals. In section-III, we give the analytical model of multimorph piezocantilevers by using points (i.e. not intervals). The model of unimorph piezocantilevers is derived in the same section. In section-IV, we introduce interval analysis to describe the design problem of unimorph piezoelectric actuators. This design problem is formulated as a set-inversion problem and is solved using algorithms and arithmetics of intervals. Finally, section-V is dedicated to a prototype fabrication and experimental verifications.

II. BRIEF PRELIMINARIES ON INTERVALS AND THEIR ARITHMETICS

A. Basic Terms and Concepts on intervals

More details on the preliminaries given here can be found in [START_REF] Moore | Interval Analysis[END_REF], [START_REF] Jaulin | Applied Interval Analysis[END_REF] or [START_REF] Moore | Cloud 'Introduction to Interval Analysis[END_REF].

A closed interval denoted by [x], is the set of real numbers given by:

[x] = [x -, x + ] = x ∈ R/x -≤ x ≤ x + (1)
The endpoints x -and x + are respectively the left and right endpoint of [x]. We say that [x] is degenerate if x -= x + . By convention, a degenerate interval [a, a] can be described with the number a. A degenerate interval number is also called a "point number".

The width of an interval [x] is given by: w([x]) = x + -x -. The midpoint of [x] is given by: mid(

[x]) = x + +x - 2 . The radius of [x] is defined by: rad([x]) = x + -x - 2

B. Operations on intervals

If we have two intervals [x] = [x -, x + ] and [y] = [y -, y + ] and a law • ∈ {+, -, ., /} , we can write:

[x] • [y] = {x • y |x ∈ [x], y ∈ [y] } (2)
Table I gives the details of the above interval operations.

TABLE I: Arithmetic operations on intervals [START_REF] Moore | Interval Analysis[END_REF], [START_REF] Jaulin | Applied Interval Analysis[END_REF].

Operation Definition + [x] + [y] = [x -+ y -, x + + y + ] - [x] -[y] = [x --y + , x + -y -] * [x] * [y] = [min{x - * y -, x + * y -, x - * y + , x + * y + }, max{x - * y -, x + * y -, x - * y + , x + * y + }] / [x]/[y] = [x] * [1/y + , 1/y -], 0 / ∈ [y]
The intersection of two intervals [x] ∩ [y] is as follows.

1-If y + < x -or x + < y -the intersection is the empty set:

[x] ∩ [y] = ∅ (3)
2-Otherwise:

[x] ∩ [y] = [max x -, y -, min x + , y + ] (4)
In the latter case, the union of [x] and [y] is also an interval:

[x] ∪ [y] = [min x -, y -, max x + , y + ] (5)
When [x] ∩ [y] = ∅, the union of the two intervals is not an interval. For that, the interval hull is defined:

[x]∪[y] = [min x -, y -, max x + , y + ] (6)
An interval hull is therefore the result of the union of two non-connected intervals.

It is verified that:

[x] ∪ [y] ⊆ [x]∪[y] for any two intervals [x] and [y] .

III. ANALYTICAL (POINTS) MODELS OF

PIEZOCANTILEVERS

In this section, we first present the linear static and dynamic models of multimorph piezocantilevers by utilizing the approach of Ballas [START_REF] Ballas | Piezoelectric multilayer beam bending actuators : Static and dynamic behavior and aspects of sensor integration[END_REF]. From that, we derive the static and dynamic models of unimorph piezocantilevers. These models will be used in the next sections for the design of a new unimorph piezocantilever with smaller sizes and that will be able to provide better performances than an existing unimorph. The choice of studying unimorph piezocantilevers as illustrative example is related to their simplicity of fabrication and their wide use in microsystems. Although, the proposed design methodology can be applied to other types of actuators subjected that their models are available. Piezoelectric materials, in particular piezoelectric ceramics, exhibit nonlinearities (hysteresis and creep). These nonlinearities will not be tackled in this paper. In fact, this paper deals with the design of piezoelectric cantilevered actuators that will be able to achieve some desired performances in term of output range for a given maximal input voltage (static characteristic) and in term of first resonant frequency (dynamic characteristic). The effect of the hysteresis is negligible on such performances because only the extremum values of the output are important to evaluate the range. A linear model is sufficient to understand the maximal furnishable range of deflection and a more complicated and nonlinear model would not bring additional information. We therefore propose in the paper to use the standard and traditional linear model. In a control point of view however (feedforward of feedback), introducing the nonlinearities in the model is essential in order to achieve other performances like accuracy or stability.

A. Static and dynamic models of multimorph piezocantilevers

A multimorph, also called a multilayered, piezocantilever is a cantilever made up of several layers. At least, one of these layers is piezoelectric. The piezoelectric layers are called active layers while the non-piezoelectric layers are called passive layers. Passive layers also serve as electrodes. When an electric voltage is applied to the piezoelectric layers, the whole cantilever bends. The static and dynamic behavior and models of piezocantilevers have been studied in depth in [START_REF] Ballas | Piezoelectric multilayer beam bending actuators : Static and dynamic behavior and aspects of sensor integration[END_REF]- [START_REF] Weinberg | Working equations for piezoelectric actuators and sensors[END_REF]. The static behavior is described by the deflection of the piezocantilever versus the applied voltage, while the dynamic behavior mainly concerns the resonant frequencies. In this paper, we adopt the description of multimorph piezocantilevers proposed by Ballas [START_REF] Ballas | Piezoelectric multilayer beam bending actuators : Static and dynamic behavior and aspects of sensor integration[END_REF]. The main advantage of Ballas's description is that the deflection at any point along the cantilever can be calculated.

Consider a clamped-free multimorph piezocantilever as depicted in Fig. 1. This piezocantilever is composed of n piezoelectric and passive layers glued themselves. The width and thickness of the i th layer are denoted by w i and h i respectively, while the total length L of all layers is supposed to be similar.

If U denotes the voltage applied to the piezocantilever, the deflection δ(x) at any point x along the piezocantilever is given by [START_REF] Ballas | Piezoelectric multilayer beam bending actuators : Static and dynamic behavior and aspects of sensor integration[END_REF]:

δ(x) = m piezo x 2 2C U (7)
where m piezo and C are given as follows:

m piezo = 1 2 n i=1 w i d 31,i s 11,i h i 2zh i -2h i i j=1 h j + h 2 i C = 1 3 n i=1 w i s 11,i 3h i z - i j=1 h j z - i-1 j=1 h j + h 3 i (8)
Parameters s 11,i represent the piezoelectric or passive compliances at a constant electric field while d 31,i represent the piezoelectric constants. z in [START_REF] Bergander | Mobile cm3-microbots with tools for nanoscale imaging and micromanipulation[END_REF] represents the distance between the neutral axis and the lower surface of the piezocantilever and is given by:

z = - n i=1 w i s 11,i h 2 i -2 n i=1 w i s 11,i h i i j=1 h j 2 n i=1 w i s 11,i h i (9)
Relation [START_REF] Zhang | A load-lock-compatible nanomanipulation system for scanning electron[END_REF] describes the static behavior of the multimorph piezocantilever. The resonant frequency, which describes the dynamics of the actuator, is given by the following expression [START_REF] Ballas | Piezoelectric multilayer beam bending actuators : Static and dynamic behavior and aspects of sensor integration[END_REF]:

f = (kL) 2 2πL 2 C n i=1 ρ i h i w i (10)
where ρ i is the density of the i th layer and k is a constant that satisfies the following equation:

1 + cos(kL)cosh(kL) = 0 (11)
An analytic solution of equation ( 11) is not possible. Numerical solution of (11) leads to m distinct roots k m L (m = 1, 2, 3, ...∞). The subscript m corresponds physically to the number of the appropriate vibratory mode. Table II presents the first five solution kL of (11) [START_REF] Tang | Numerical evaluation of uniform beam modes[END_REF], [START_REF] Soedel | Vibrations of shells and plates[END_REF].

To summarize, the resonant frequencies associated to the modes of the piezocantilever are obtained by means of the

f m = (k m L) 2 2πL 2 C n i=1 ρ i h i w i (12)
k m L being defined in Table II.

Many cases only require the first resonant frequency f 1 (first mode). This first resonant frequency is given by:

f 1 = (1.8751) 2 2πL 2 C n i=1 ρ i h i w i (13)
When the number of piezoelectric layers in a multimorph actuator is high, a high range of deflection can be obtained at low voltage. However, the fabrication of the actuator and the wire connection are complex. For the illustrative example of this paper, we propose to restrict our study to the design of unimorph piezocantilevers, only composed of one passive layer and one piezoelectric layer. This does not take off the possibility to generalize the proposed guaranteed design method for other kinds of piezoelectric actuators, or even for other kinds of actuators.

B. Derivation of the static and dynamic models of unimorph piezocantilevers

A unimorph piezocantilever is made up of one active layer and one passive layer (Fig. 2). The length L and width w are the same for both layers. The thickness of the piezoelectric layer (active layer) is h p , while that of the passive layer is h mp . In the sequel, we denote: After calculation, the static model in [START_REF] Zhang | A load-lock-compatible nanomanipulation system for scanning electron[END_REF] becomes the following model for unimorph piezocantilevers:

δ(x) = N um δ Den δ (14)
with: and the resonant frequencies in [START_REF] Edamana | Modeling and Optimal Low-Power on-off Control of Thin-Film Piezoelectric Rotational Actuators[END_REF] becomes:

N um δ = -
f m = (k m L) 2 2πL 2 Den δ 12s mp 11 s p 11 (s mp 11 h p + s p 11 h mp ) (ρ mp h mp + ρ p h p ) (15
) Remark 1: These models show that the static and the dynamics beahviors of the piezocantilevered actuator (unimorph and multimorph) strongly depend on the geometrical sizes and on the physical properties. By using these models, it is possible to find convenients dimensions of the actuators that would satisfy some specified performances in term of range of deflection and in term of resonant frequencies. That will be the aim of the next section.

IV. COMBINING INTERVALS AND THE ANALYTICAL MODEL TO DESIGN PIEZOCANTILEVERS

The previous section was devoted to the modeling of multimorph piezocantilevers with derivation of the models of a unimorph actuator. In this section, we will use these models to design the actuator such that some predefined static and dynamic performances are satisfied. More precisely, the problem consists in finding a set of unimorph with smaller sizes and better performances than a given and existing unimorph. For that, we propose to combine the above analytical models with interval techniques which permits to transform the problem into a set-inversion problem solvable with interval tools. The design is said with guaranteed performances because: if a design solution exists, it is guaranteed that the specifications will be reached.

A. General objective

Consider an existing unimorph actuator A u1 as pictured in Fig. 3-a. This unimorph has a length L, a width w and layers thicknesses h p and h mp . The total thickness is h = h p + h mp . Let δ and f 1 denote the range of deflection obtained with a voltage U and the first resonant frequency of this unimorph. The problem consists in redesigning this actuator into another unimorph A u2 with smaller dimensions (except for the width) denoted by L d × w × h d (subscript d means desired) and with better performances for the range of deflection δ d and for the first resonant frequency f 1d (see Fig. 3-b). The total thickness h d of the unimorph A u2 is calculated with the thickness h dp of the active layer (piezoelectric) and the thickness h dmp of the passive layer:

h d = h dp + h dmp .
A quick analysis of the models of a unimorph piezocantilever presented in the previous section permits the following remarks: • the resonant frequency f m is conversely proportional to the square of its length. Indeed, since k m L is constant (see Table II), the resonant frequency described by ( 15)) becomes proportional to 1 L 2 ; • the deflection is directly proportional to the square of its length. In fact, the deflection, which is described by ( 14)), is proportional to x 2 . Hence, if we are interested on the deflection at the tip of the cantilever, we have δ(x = L) proportional to L 2 . Therefore decreasing the length will yield an increase of the resonant frequency. However, the range of deflection will be reduced. A compensation for this range reduction without disrupting the resonant frequency may be accomplished by some setting on the thicknesses of the piezoelectric and passive layers. Such problem cannot be solved manually but in this paper, this is accounted automatically.

B. Specifications for the design

Let δ(L) and δ(L d ) be the deflections at the tip of the unimorph actuators A u1 and A u2 respectively. Remind that f 1 and f d1 denote their first resonant frequencies. The specifications for the design consist in finding suitable dimensions (L d , h dp and h dmp ) of the unimorph A u2 for which the following requirements hold:

• the same materials (piezoelectric and passivle) than those of the existing unimorph A u1 are used of the unimorph A u2 ; • the length of the unimorph A u2 must be smaller than that of the unimorph A u1 . We choose

L d = L α
, where α > 1 is a given scalar (α ∈ R);

• the total thickness of the unimorph A u2 must be lower or equal to that of the unimorph A u1 :

h d = h dp + h dmp ≤ h = h p + h mp ;
• the deflection at the tip of the unimorph A u2 must be higher or equal to that of the unimorph A u1 when the same voltage is applied: δ d (L d ) ≥ δ(L); • and the first resonant frequency of the unimorph A u2 must be higher or equal to that of the unimorph A u1 : f d1 ≥ f 1 . From the models in ( 14) and ( 15), we see that the width w does not influent on the range and on the resonant frequencies. This parameter will not therefore be introduced in the design problem. Instead, we will impose the width of A u2 be equal to the width of A u1 . The remaining parameters to be sought for are the dimensions L d , h dp and h dmp .

C. Problem formulation using inequalities

If we fix the parameter α in L d = L α , the number of variables (parameters) to be sought for during the further computation is reduced into two: the thicknesses h dp and h dmp . If required, it is still possible to let L d as also an unknown parameter.

Based on the static and dynamic modeling of unimorph piezocantilevers presented previously, the specifications, requirements and objectives given in Subsection IV-B can be mathematically transcribed into the following problem.

Problem: Find h dp and h dmp such that:

                                                       m piezo (h dp , h dmp ) L α 2 2C(h dp , h dmp ) U ≥ m piezo (h p , h mp )L 2 2C(h p , h mp ) U (1.8751) 2 2π   L α   2 C(h dp , h dmp ) w(ρ mp h dmp + ρ p h dp ) ≥ (1.8751) 2 2πL 2 C(h p , h mp ) w(ρ mp h mp + ρ p h p ) h dp + h dmp ≤ h ( 16
) which are equivalent to the following inequalities:

                           m piezo (h dp , h dmp ) α 2 C(h dp , h dmp ) ≥ m piezo (h p , h mp ) C(h p , h mp ) α 4 C(h dp , h dmp ) w(ρ mp h dmp + ρ p h dp ) ≥ C(h p , h mp ) w(ρ mp h mp + ρ p h p ) h dp + h dmp ≤ h (17)

D. Problem formulation using intervals

The set of thickness h d = h dp + h dmp that satisfies the inequality h d ≤ h can be expressed by an interval

[h d ] = [0, h].
In other words, the thickness of each layer of the unimorph A u2 (i.e. h dp and h dmp ) is bounded by the total thickness h. This allows to represent the parameters h dp and h dmp by the intervals [h dp ] and [h dmp ] respectively. When interval parameters are used, the inequalities can be transformed as a system of inclusions. Thus, the design problem defined by inequalities in ( 17) can be reformulated as follows.

Consider [θ] = [[h dp ], [h dmp ]
] as a box (vector of intervals) with elements: [h dp ] and [h dmp ]. Let Θ be the set of parameters [h dp ] and [h dmp ] that satisfies the inequalities [START_REF] Haddab | A microgripper using smart piezoelectric actuators[END_REF]. So, the problem, when formulated with intervals, consists in finding the suitable values of [θ] such that:

Θ := {θ ∈ D |[H](θ) ⊆ [Y] } ( 18
)
where D is the domain of definition of θ.

[H](θ) and [Y] are defined as follows:

[H](θ) =           m piezo ([h dp ], [h dmp ]) α 2 C([h dp ], [h dmp ]) - m piezo (h p , h mp ) C(h p , h mp ) α 4 C([h dp ], [h dmp ]) (ρ mp [h dmp ] + ρ p [h dp ]) - C(h p , h mp ) (ρ mp h mp + ρ p h p ) h -[h dp ] -[h dmp ]           (19) and [Y] =   [0, +∞] [0, +∞] [0, +∞]   ( 20
)
The problem of finding the set parameter [θ] that ensures ( 18) is a set-inversion problem and can be solved using interval techniques. The set-inversion operation consists to compute the reciprocal image of a compact set called subpaving. The setinversion algorithm SIVIA (Set Inversion Algorithm Via Interval Analysis [START_REF] Jaulin | Set inversion via interval analysis for nonlinear bounded-error estimation[END_REF], [START_REF] Jaulin | Applied Interval Analysis[END_REF]) allows to approximate with subpavings the set solution Θ described by [START_REF] Rakotondrabe | Development and Force/Position Control of a New Hybrid Thermo-Piezoelectric microGripper dedicated to micromanipulation tasks[END_REF]. This approximation is realized with an inner and outer subpavings, denoted by Θ and Θ respectively, such that Θ ⊂ Θ ⊂ Θ. The subpaving Θ corresponds to the parameter vector for which the problem (18) holds. If Θ = ∅, then it is guaranteed that no solution exists for [START_REF] Rakotondrabe | Development and Force/Position Control of a New Hybrid Thermo-Piezoelectric microGripper dedicated to micromanipulation tasks[END_REF]. If a solution Θ exists, it is guaranteed that any parameters ([h dp ] and [h dmp ]) inside this set will satisfy the desired specifications in [START_REF] Haddab | A microgripper using smart piezoelectric actuators[END_REF].

In many problems (signal estimation, control design,...) [START_REF] Jaulin | Set inversion via interval analysis for nonlinear bounded-error estimation[END_REF], [START_REF] Jaulin | Interval constraint propagation with application to boundederror estimation[END_REF] [30]- [START_REF] Rakotondrabe | Performances inclusion of stable interval systems[END_REF], interval tools, in particular the SIVIA algorithm, have been demonstrated theoretically and with experiments to guarantee the efficiency of the solution, if this exists. This is one of the major advantages of interval techniques and tools. In our case, if the solution Θ exists, it is guaranteed that the unimorph actuator with parameters within this solution will satisfy the specifications.

We give in Table III the recursive SIVIA algorithm used to solve the design problem of piezocantilevers [START_REF] Rakotondrabe | Development and Force/Position Control of a New Hybrid Thermo-Piezoelectric microGripper dedicated to micromanipulation tasks[END_REF] with guaranteed solution and guaranteed non-solution. SIVIA algorithm requires a search box [θ](0) (possibly very large). The search box is also called initial box and Θ may belong to this. The inner and outer subpavings (Θ and Θ) are initially empty. represents the wanted accuracy of computation.

In the most cases, we are interested to compute an inner subpaving Θ because it is guaranteed that Θ is enclosed in the Θ, i.e. Θ ⊂ Θ. When no inner subpaving exists i.e. Θ = ∅, it is possible to choose parameters inside the outer subpaving, i.e. choose θ ∈ Θ.

TABLE III: SIVIA Algorithm [START_REF] Jaulin | Set inversion via interval analysis for nonlinear bounded-error estimation[END_REF], [START_REF] Jaulin | Applied Interval Analysis[END_REF] which is modified to fit with the set-inversion problem described by [START_REF] Rakotondrabe | Development and Force/Position Control of a New Hybrid Thermo-Piezoelectric microGripper dedicated to micromanipulation tasks[END_REF].

SIVIA(in:

[H], [Y], [θ], ; out: Θ, Θ ) 1 if [H](θ) [Y] = ∅ return; 2 if [H](θ) ⊆ [Y] then; Θ := Θ [θ] ; Θ := Θ [θ] return; 4 if width([θ]) < then {Θ := Θ [θ]}; return; 5 bisect [θ] into L([θ]) and R([θ]); 6 SIVIA([H], [Y], L([θ]), ; Θ, Θ); SIVIA([H], [Y], R([θ]
), ; Θ, Θ).

V. NUMERICAL CALCULATION, PROTOTYPE FABRICATION AND EXPERIMENTAL VERIFICATIONS

This section is devoted to the numerical calculation, the fabrication of a prototype and the experimental validation.

A. Materials

First, we present here the materials that compose the layers of the existing unimorph. The piezoelectric material used is based on PZT (lead zirconate titanate) commercially available (from Physike Instrumente (PI) company). The passive layer is based on Nickel material from Goodfellow company. The same materials will be employed to develop the designed unimorph A u2 . The product of reference "EPO-TEK H22" from PI is used to glue the piezoelectric and passive layers and to form the new PZT-Nickel unimorph piezocantilever. This is a thermal glue with which the gluing is performed at 120 o C during 12h. With this procedure, we have a very thin film and a robust interface between the two materials. If the glue layer is not thin enough, its thickness will affect the behavior (resonant frequency and range of deflection) of the unimorph structure. Table IV summarizes some physical properties of the PZT and of the Nickel materials. The unimorph A u1 that is initially available and to be redesigned into another unimorph A u2 has the following characteristics.

• length: L = 20mm;

• thicknesses: h p = 450µm and h mp = 200µm;

• width: w = 1mm. For the example of redesign, we set α = 2 such that the desired length L d is twice smaller than the initial length L:

L d = L α = 20mm 2 = 10mm (21)
Finally, the width of the unimorph A u2 , which is considered equal to the width of the unimorph A u1 , is: w = 1mm. Now, we solve the problem [START_REF] Rakotondrabe | Development and Force/Position Control of a New Hybrid Thermo-Piezoelectric microGripper dedicated to micromanipulation tasks[END_REF] with the two remaining unknown parameters [h dp ] and [h dmp ].

B. Solving the set-inversion problem

The problem (18), using the above numerical values, has been solved using an initial box [h dp ] 0 ×[h dmp ] 0 = [10, 640]× [10, 640] and an accuracy = 1µm. The solving has been done with the SIVIA algorithm (see Table IV). Fig. 4 pictures the results from the algorithm.

In this figure, the area S 1 corresponds to the guaranteed solution (inner subpaving Θ), i.e. the set h dp × h dmp of the wanted unimorph A u2 that satisfies the specified performances. Any choice of h dp and h dmp inside the subpaving Θ ensures the inclusions given in [START_REF] Rakotondrabe | Development and Force/Position Control of a New Hybrid Thermo-Piezoelectric microGripper dedicated to micromanipulation tasks[END_REF]. The area S 2 corresponds to ∆Θ and contains the boxes for which no decision can be taken, such that: Θ = Θ ∪ ∆Θ. ∆Θ can be minimized by increasing the computation accuracy. Finally, the area in S 3 corresponds to the parameters [h dp ] and [h dmp ] for which it is guaranteed that the inclusions (18) do not hold. C. Fabrication of the unimorph and experimental verifications S 1 region in Fig. 4 depicts a set of solution of h dp × h dmp with which a unimorph piezocantilever will satisfy the specifications in Subsection IV-B. In order to demonstrate the efficiency of the approach, a prototype of unimorph having dimensions within this set solution is fabricated. We choose:

                   L d = 10mm h dp = 200µm h dmp = 100µm w = 1mm (22)
Fig. 5-a presents a photography of the fabricated unimorph piezocantilever.

To check the performances, a static and a harmonic analysis have been carried out with the fabricated unimorph A u2 and • a unimorph piezocantilever. Both the existing unimorph A u1 and the designed unimorph A u2 are characterized with this same setup; • an optical sensor (from Keyence company) with resolution of 10nm and which is used to measure the deflection of the piezocantilevers; • and a dSPACE acquisition board and a computer to generate the input voltage and to acquire the measurements.

The first experiment consists in evaluating the range of deflection (static characteristic) of the actuators. For that, a sine input voltage U is applied to the actuators and the resulting output deflection δ is reported. The range can be evaluated from the plot of δ versus U . As this experiment concerns the static characteristic, the frequency f of the sine input voltage should be low. In fact, if we increase the frequency, the dynamics of the actuators will affect the resulting (U, δ)curve and then the static characteristic can not be anymore evaluated due to the phase-lag. For the considered actuators, a sine voltage of frequency f = 2Hz was convenient. The amplitude is U = 40V. This sine voltage was applied to the unimorph A u1 , and then to the unimorph A u2 . Fig. 6-a depicts the deflection of the designed unimorph A u2 versus the voltage U .

Then, the second experiments consist in performing a harmonic analysis. The aim is to characterize the first resonant frequencies of the unimorphs. Fig. 6-b depicts the results obtained with the designed unimorph A u2 .

Table V summarizes the results of characterization of both unimorph piezocantilevers. In the table, both simulation and experimental results are reported. From this table, the maximum deflection of the designed unimorph A u2 (in excess of 10µm) and its resonant frequency (in excess of 1600Hz) are higher than those of the unimorph A u1 (about 8µm and 960Hz respectively). Also, the geometrical dimensions of the unimorph A u2 are strictly less than those of the unimorph A u1 . This confirms that the specifications are ensured. We can see from the table a slight difference between the simulation results and the experimental results, in particular for the designed actuator A u2 . This difference may be due to the thickness of the glue used to interface the piezoelectric material and the passive material. Although this slight difference, the experimental results confirm and demonstrate the efficiency of the approach. These experimental and simulation results confirm the proposed approach to design the piezoelectric actuator. Remark 2: It is possible to design other kinds of piezoelectric actuators with the proposed approach, like bimorph piezocantilevers. A bimorph piezocantilever provides a larger deflection compared to that of a unimorph when the same voltage is applied. However, bimorph piezocantilevers present more complex electric connections and are difficult to fabricate. The fabrication simplicity of unimorph piezocantilevers makes them widely used in the precise positioning. This is why the example carried out in this paper concerns unimorph actuators.

Remark 3: The technique proposed in this paper can also be used to other kinds of actuators (different from piezoelectric actuators), or even to non-actuated structures. As long as a physical and geometrical model is available, the technique consists in combining the model with interval techniques which can be afterwards solved thanks to a set-inversion algorithm.

Remark 4: In this paper, the performances used for the design were the range of displacement and the first resonant frequency. It is however possible to apply the proposed design methodology with other kinds of performances, for instance with the accuracy or with the resolution. Finally, an interesting problem consists in designing an actuator yet in a feedback control scheme having a given controller. In this case, the problem comes back to the design of the actuator such that the closed-loop possesses some desired performances like tracking accuracy, limited input voltage, settling time, overshoots...

VI. CONCLUSION

This paper presented the design of piezoelectric actuators by using interval techniques. The main goal was to redesign an existing piezoelectric actuator (unimorph) in order to have a new unimorph actuator having smaller sizes but that would provide better performances. The problem was formulated as a set-inversion problem that was solved using interval techniques. The main advantage of the proposed approach is the guarantee of the performances if solution exists. Fabrication of a prototype and experimental characterization confirmed the efficiency of the proposed approach. The paper described as illustrative example the design of unimorph piezoelectric actuators, but the approach can be applied to other kinds of actuators subjected that their physical/geometrical models are available.

Fig. 1 :

 1 Fig. 1: A clamped-free multimorph piezocantilever.

• s p 11 and s mp 11 ,Fig. 2 :

 112 Fig.2: A unimorph piezocantilever.

Fig. 3 :

 3 Fig. 3: a: an existing unimorph piezocantilever A u1 . b: the desired unimorph piezocantilever A u2 .

Fig. 4 :

 4 Fig. 4: Set solution Θ corresponding to the parameters h dp and h dmp .

Fig. 5 :

 5 Fig. 5: a: photography of unimorph piezocantilevers. b: experimental setup.

Fig. 6 :

 6 Fig. 6: a: static characteristic of the fabricated prototype unimorph A u2 . b: frequency response (magnitude) of the fabricated prototype unimorph A u2 .

TABLE II :

 II Solutions of the equation[START_REF] Peng | Modeling of piezoelectric-driven stick-slip actuators[END_REF]. characteristic roots k m L (m = 1, 2, ...) given in TableII. The resonant frequency of the m th mode is given by:

	m	1	2	3	4	5	• • •
	kmL	1.8751	4.6941	7.8548	10.9955	14.137	• • •

  3d 31 s mp 11 s p 11 h mp (h p + h mp ) x 2 U

	Den δ = (s p 11 ) 2 h 4 mp + (s mp 11 ) 2 h 4 p + s mp 11 s p 11 p h 3 mp + 6h 2 p h 2 mp + 4h mp h 3 p )

TABLE IV :

 IV Physical characteristics of PZT and Nickel materials.

	materials	compliance	piezoelectric	density ρ
		s 11	constant d 31
	PZT	15 × 10 -12 m 2 /N	-210 × 10 -12 m/V	7800kg/m 3
	Nickel	5 × 10 -12 m 2 /N	0	8900kg/m 3

TABLE V :

 V Comparison of the obtained results.

					maximum	maximum	resonant	first resonant
	piezocantilever	length of the	thickness of	thickness of	deflection	deflection	frequency	frequency
		piezocantilever	the piezo-layer	the passive layer	(experimental)	(model simulation	(experimental)	(model simulation
			the piezo-layer			with the dimensions)		with the dimensions)
	unimorph A u1	L = 20mm	hp = 450µm	hmp = 200µm	|δ| ≈ 8.5µm	|δ| = 8.48µm	f 1 ≈ 951.67Hz	f 1 = 952.12Hz
	unimorph A u2	L d = 10mm	h dp = 200µm	h dmp = 100µm	|δ d | ≈ 12µm	|δ| = 10.45µm	f 1 d ≈ 1641Hz	f 1 = 1755.5Hz


Alexis Michaud
email: alexis.michaud@univ-paris3.fr

Replicating in Naxi (Tibeto-Burman) an Experiment Designed for Yorùbá: An Approach To 'Prominence-Sensitive Prosody' vs. 'Calculated Prosody'

The starting-point of this study is the hypothesis (suggested by an overview of typologically varied languages) that it may be useful to characterise prosodic systems in terms of the degree to which they rely on the calculation of tone sequences. Each language could be placed at a certain position along a typological continuum between two types of prosodic organisation: (i) 'calculated prosody', in languages such as Yorùbá (in which tone serves complex morphophonological functions), whose prosodic structure hinges on the calculation of a tone sequence, by categorical processes such as the association of lexical tones and/or boundary tones, reassociation/'tone floating', and downstep; and (ii) 'prominence-sensitive prosody', typologically more common, found in languages such as Chinese, which have fewer elements of categorical tonal calculation, and in which intonation appears to reflect phrasing and informational structure in a largely noncategorical way. In an effort to test (and refine) this hypothesis, an experiment used for Yorùbá [8, 9] is adapted to Naxi, a Sino-Tibetan language which, like Yorùbá, has three lexical tones (High, Mid and Low), but which is hypothesised to be closer to the 'prominencesensitive prosody' type, whereas Yorùbá would be closer to the 'calculated prosody' type. This pilot study on sentences in which all syllables bear the same tone does bring out differences between the two languages in terms of phenomena of phrasing and of prominence.

A typological proposal: 'prominencesensitive prosody' vs. 'calculated prosody'

Some definitions

As a preliminary to the proposed typological overview, some indications on the terms used may be useful. Prosody is here understood in its broadest sense: it includes lexically distinctive suprasegmentals, intonation, the expression of attitudes and emotions, and performance factors: rhythm and speech rate [15]. Lexically distinctive suprasegmentals may be, depending on the language, lexical stress, lexical tone, lexical pitch accent... Intonation is an abstract structure, not to be confused with the parameters whereby it manifests itself (in particular, it is a different notion from F 0 , which is a physical parameter, and pitch, which is a percept). Intonation can be conveniently divided into two components: (i) intonational phrasing, sometimes called syntactic intonation because of its strong (though not bijective) link to syntax, (ii) pragmatic intonation, determined by the information structure of the utterance. The term intonational group is used here loosely as a cover term for the various levels of intonational phrasing.

An overview of typologically varied tone languages

Typologically, there appears to be considerably more diversity among prosodic systems than labels such as 'tone languages', 'pitch accent languages ' and 'stress languages' suggest [7, page 1]. 'Tone languages' do not make up a unified typological pole. An overview suggests that some prosodic systems, more than others, go by a calculation of tone sequences. The prosody of Mambila (Niger-Congo, Bantoid) seems to be highly constrained by its four level lexical tones. Ngamambo (Bantoid) [1] also appears highly constrained, as a five-term paradigm is open after a High-tone syllable: High, Mid, Low, downstepped Low (!L), or flat Low (L°). On the other hand, in Hausa, an Afro-Asiatic language which only has two tones, speakers apparently play to a larger extent on intonation to convey degrees of informational prominence (A. Rialland, p.c.). It thus seems that the degree of constraint on the margin of freedom left to the speaker depends on the number of level tones and of categorical processes of tone modification.

Formulation proposed

The proposal put forward here is that prosodic systems could be placed along a typological continuum between (i) prosodic systems that hinge on the calculation of a tone sequence (by categorical processes of association of lexical tones and/or boundary tones, reassociation/'tone floating', and downstep), and (ii) prosodic systems in which phonetic realisation is sensitive to the prominence of individual items within the utterance, i.e. where intonation reflects informational structure (with such typical phenomena as contrastive emphasis). 'Prominence-sensitive prosody' can be illustrated by languages such as Mandarin Chinese. Despite having four lexical tones, Mandarin allows local intonational modifications which carry a communicative load comparable to the one they have in a language such as English (as has been known since [3]): the phonetic realisation of tones depends on the degree of prominence of individual syllables, and on intonational phrasing. Rather than unambiguously categorical processes such as downstep and tonal reassociation, Mandarin Chinese shows many noncategorical modifications of lexical tones, and few categorical modifications. In the study of Mandarin tones, much attention has been directed to the sandhi process whereby a sequence of two tone-3 syllables yields a tone 2+tone 3 sequence; on the other hand, the many cases of "minor sandhi" that have been proposed for Mandarin (e.g. by Chao Yuen-ren) are not clearly categorical, so much so that "It is not clear whether it is desirable or even possible to segregate tonal coarticulation [i.e. a noncategorical modification] from tone sandhi proper [i.e. a categorical modification]" [4, page 25].

It seems that 'calculated-prosody' systems would lend themselves especially well to the automatic synthesis of F 0 curves, on the basis of tone sequences alone, whereas for 'prominence-sensitive' prosodic systems, other parameters have to be taken into account: for instance, a system of synthesis proposed for Mandarin assigns a coefficient of strength to each individual syllable [14].

Section 2 aims to provide some experimental evidence to help refine these tentative typological reflections.

A pilot study comparing two widely different prosodic systems (Naxi and Yorùbá)

using a similar experimental setup

Overall presentation of the experiment

The pilot study reported below consists in adapting to Naxi, a Sino-Tibetan language, an experiment originally designed to investigate Yorùbá, a language of the Niger-Congo family [8,9]. Both languages have High, Mid and Low lexical tones; unlike in Naxi, however, tone plays complex morphophonological functions in Yorùbá (see, e.g., the discussion by [2] of the subject marker as a floating High tone); and the Mid tone of Yorùbá can be considered as the absence of tone [6 and references therein], whereas the Mid tone of Naxi is a tone in its own right. In some respects, the prosodic system of Naxi (investigated first-hand [10,11]) appears closer to that of neighbouring tone languages (such as Mandarin Chinese [3,4,14]) than to that of Yorùbá. In the terms proposed in section 1, Yorùbá is closer to the 'calculated prosody' type, whereas Naxi is closer to the 'prominence-sensitive prosody' type (though it possesses some elements of tonal calculation, namely the addition of a High tone for intensification, and some cases of 'floating' High tone [11]). Using a similar method for both languages appeared as an interesting means to gain a perspective on the differences between these two types of prosodic organisation.

Method

The data reported here consist in sentences in which all syllables bear the same lexical tone, as in [9]:

( "I'm crawling into [it]." (The final particle was /mɤ/ or /se/, at the language consultant's choice.)

Five native speakers of (Western) Naxi recorded these sentences (their Chinese equivalent was used as a prompt). (For ease of cross-reference with other publications based on data from the same language consultants, they are referred to using the same codes as in [10]: F2, M4, M5, M7, M9.)

In addition to its H, M and L tones, Naxi possesses a rising LH contour [11]; due to its rarity on lexical items, constructing an utterance such as (4a) is difficult; it appeared useful to record it nonetheless. The recordings took place in a quiet room in Lijiang Prefecture, China. An electroglottographic signal was recorded simultaneously with the audio, in order to obtain a precise measurement of F 0 and indications on voice quality. The Chinese translation of the sentences was used as a prompt (sentences 1-4 were arranged in random order). The speakers were instructed to say the sentence twice, with a deep breath in-between repetitions. Due to some repetitions, a total of 54 utterances were finally retained (10 by F2, 17 by M4, 11 by M5, 8 by M7, 7 by M9). The beginning and endpoint of each syllable rhyme (i.e. excluding initial consonants) were annotated manually on the basis of inspection of the audio signal supplemented by auditory impression. F 0 and the open quotient (hereafter O q ; it offers an indication on the degree of vocal fold abduction [5 and references therein]) were estimated using a method set out in [5, 10].

Results

Figures 1 and 2 (on the last page) show average curves of F 0 and O q for tones H (sentence 1) and LH (sentence 4a) by speaker M5, superimposing the F 0 curves of all the syllables of each sentence. It was chosen not to average the data across speakers due to differences in overall F 0 register which might have biased the results. The curves are resampled at 20 equally spaced time points. Due to the small number of repetitions averaged (two on fig. 1, four on fig. 2), it did not appear useful to indicate standard deviation. These figures can be proposed as representative of the behaviour of the four speakers (crossspeakers differences will not be discussed in detail; graph 1 shows the cross-speaker differences in declination over the utterance).

On figure 1, the third syllable, which corresponds to an intonational juncture, is markedly longer than the seven others; this third syllable and the last syllable (the 8 th ) both have a more strongly decreasing F 0 than the others (the decrease is on the order of 3 musical tones), and, contrary to all other syllables, have a decreasing O q , indicating a tensing of the vocal folds, as opposed to the relaxing of the vocal folds at the end of the other six syllables. On figure 2, the longest syllable is, again, the one located at an intonational juncture (syllable 2). The curves in figure 2 do not show a strict layering as in figure 1, where the mean F 0 of successive syllables decreased from the first to the last: in figure 2, syllables 4 and 7 have a higher mean F 0 than the syllable that preceeds them. The extent of the F 0 rise also differs across syllables.

These piecemeal observations, by themselves, do not have demonstrative value; they nonetheless illustrate the general fact that, in Naxi, grammatical words are typically weaker (which does not appear to be equally true in Yorùbá), e.g. /nɯ̀ /, 'and', the fourth syllable, is realised with flat rather than rising F 0 , in the continuity of the preceding syllable (its starting-point by and large corresponds with the F 0 endpoint of the preceding syllable). The adverb 'very', /gjà / (6 th syllable), likewise a grammatical word, is also weak-weaker than the adjective /hò / (syll. 7), which is realised slightly higher, and with a more clearly rising F 0 .

These facts have a bearing on the representation of F 0 curves. In the study of Yorùbá that inspired the present experiment, F 0 curves over individual syllables are simplified to two data points, "a point towards the beginning of the vowel portion of the syllable and another towards the end" [8, page 60]. (This is an 'impressionistic' choice, in that it is does not rest on evidence from perception tests.) Such an approach does not seem to commend itself for Naxi, because of the effect of intonational junctures, manifested by lengthening and a downward tilt in F 0 . Also, the data points retained for Yorùbá are plotted at an equal distance, which amounts to normalising for duration, as on figure 3. This stylisation of the F 0 curves may be adequate for Yorùbá: two data points per syllable can capture an upward or downward movement, and the literature on this language suggests that this information is sufficient to represent prosodic events; on the other hand, such a stylisation does not appear adequate for the Naxi data. As shown by figures 1 and 2, considerable differences in length emerge (beyond the thresholds indicated by To sum up briefly the observations: in Naxi, all the syllables within an intonational group are realised at a similar height (modulo lexical tones, intrinsic F 0 , and a slight declination from one syllable to the next), and the last syllable of the group is typically lengthened, with final lowering (a downward shift in F 0 register, or a decrease in F 0 in the course of the syllable, or again a combination of both). The syllable which follows the intonational juncture (i.e. the first syllable of the next group) typically starts from a somewhat raised height (partial resetting of the declination line). For reasons of space, individual strategies in the realisation of intonational junctures cannot be set out in detail; to take one example, in M9's careful, deliberate style of delivery, group-final (and utterance-final) syllables stand out very clearly by a length about double that of the other syllables, whereas final lengthening is minimal.

In view of these observations, normalising the data for duration does not appear adequate for Naxi. Moreover, it does not appear enlightening to approximate declination over the utterance by comparing the average F 0 of the first and last syllables (as was done by [8,9] for Yorùbá), because this procedure overlooks the division of the utterance into intonational groups. The most apparent downward trend in Naxi appears to be due to group-and utterance-final lowering, rather than to (continuous) declination. With a view to comparison with the Yorùbá data, declination was nonetheless estimated as a ratio of the average F 0 of the first and last syllable. The results are shown on graph 1: values on the order of -1.6 musical tone over the utterance for tone H and tone M, -1.75 musical tone for tone L. (The instruction that the speakers should take a deep breath between sentences may have contributed to this relatively steep declination.) F 0 decrease in the course of the last syllable is more frequent than downward shift in F 0 register. Graph 1 brings out the variability observed across speakers: in Naxi, even under controlled, 'laboratory' conditions (uttering sentences that have been prepared beforehand), local modifications are of such amplitude that the declination line cannot be recovered from the observed F 0 curve in a straightforward way.

General discussion and conclusion

Though very slender indeed (so much so that no statistical treatment was attempted), and open to some criticisms, the experimental results set out in section 2 reflect the considerable structural difference between the prosody of Naxi and of Yorùbá. The following differences emerge: (i) Final lowering is observed in Yorùbá [8, page 17] as well as in Naxi; in Naxi, however, it is usually accompanied by lengthening.

(ii) The marking of intonational junctures (by lengthening and a final dip in F 0 ) is very salient in Naxi prosody. (iii) Grammatical words, as opposed to lexical words, are typically realised (all other things being equal) with slightly lower F 0 in Naxi, whereas it seems to be a general observation that grammatical words are NOT realised as 'weaker' or with lower F 0 than lexical words in Yorùbá (and in other languages of the Niger-Congo family; A. Rialland, p.c.). (iv) Monosyllables elicited in a carrier phrase in Naxi are much longer (average over 600 syllables: 260 ms [10]) than would be expected for Yorùbá; I consider this lengthening as evidence of the prominence-sensitive realisation of Naxi words (a syllable placed inside a carrier sentence is in focal/prominent position), as opposed to the more calculated realisation of tone in Yorùbá, whereby F 0 realisations do not usually depart considerably from what can be predicted on the basis of the tone sequence of the utterance.

Unlike in Yorùbá, where it appears possible, in read speech, to elicit sequences of like tones that come out as regular, gradually lowered F 0 curves, it appears that in Naxi, under a similar experimental setup, at least two factors disrupt (to a certain extent) the regularity of the tone sequences: (i) junctures between intonational groups are clearly marked by lengthening and F 0 decrease; and (ii) the informational importance (in the broad sense) of individual words influences their prosodic realisation. (These factors are even more salient in continuous/'spontaneous' speech.)

Concerning perspectives for future research, the typological distinction between 'calculated prosody' and 'prominence-sensitive prosody' could arguably be extended to languages that do not possess lexical tones: a language such as Wolof, which signals information structure by morphosyntactic means (a verb form may be 'subjectemphatic', 'object-emphatic' or 'verb-emphatic'), appears to have a prosodic system very unlike 'prominence-sensitive' systems, despite not having any lexical tones [12].

  [13]) at the junctures indicated below by the symbols [|] (optional) and [ǁ] (realised by all five speakers): (1) /kó só sé ǁ ŋv̩ ́ kó | ʂɤ́ jɤt ʰɑ/ (2) /tʰɯ̄ | ʈʂʰɯ̄ ŋi | kʰɯ̄ ʂɯ̄ mɤ̄ ndzɯ./ (3) /ŋɤ̀ | kʰv̩ ̀ kyk ò | tʰè bṿ mɤ/ (4) /tʰɑ̀ ́ lỳ ́ ǁ nɑ̀ ́ nɯ̀ ́ ŋɑ̀ ́ | gjà ́ hò /. (The presence or absence of the optional breaks [|] apparently depends on tempo-a variable which was not controlled by explicit instructions given to the speakers.)

  (4a) tʰɑ̀ ́ lỳ ́ nɑ̀ ́ nɯ̀ ́ ŋɑ̀ ́ gjà ́ hò ́ 3 rd sg. see+TOP. 2 nd sg. and 1 st sg. very correct, adequate "He thinks that you and I [getting together] is quite right./ He thinks that our getting together is a very good thing." Informants F2, M7 and M9 chose to record a variant (4b) that they considered more satisfactory. (4b) tʰɑ̀ ́ nɑ̀ ́ nɯ̀ ́ ŋɑ̀ ́ gjà ́ hò ́ mɑ̀ ́ 3 rd sg. 2 nd sg. and 1 st sg. very correct final part.

"He, you and I [getting together] is quite right, isn't it."


Julien Alexandre
email: alexandre@ensta.fr

Sandretto

Alexandre Chapoutot
email: chapoutot@ensta.fr

DynIBEX: a Differential Constraint Library for Studying Dynamical Systems

Keywords: CCS Concepts, Mathematics of computing → Solvers, Interval arithmetic, •Theory of computation → Timed and hybrid models, Interval constraints, Validated numerical integration, Runge-Kutta methods

Interval analysis tools have been used in many techniques and methods to study dynamical systems. Nevertheless, custom homemade libraries are usually developed for these purposes. DynIBEX is a free open-source library combining validated numerical integration methods with a constraint programming library named IBEX. Its purpose is to provide all the basic interval operators which may be used into more complex algorithms for studying dynamical systems.

INTRODUCTION

An important amount of research deals with cyber-physical systems to prove safety or study stability properties. One of the mathematical foundation of cyber-physical systems are (hybrid) dynamical systems which are complex objects to study, due to their non trivial temporal evolutions.

Studying dynamical systems to prove safety or stability properties refers to consider the set of all the trajectories. Interval analysis is an appealing approach because it is tractable for high dimensional systems and combining with space decomposition it is possible to represent complex geometric forms. Interval analysis tools are found in many work such as for verification methods based on simulation or on reachable set computation, [START_REF] Henzinger | Beyond HYTECH: Hybrid systems analysis using interval numerical methods[END_REF][START_REF] Chen | Taylor model flowpipe construction for non-linear hybrid systems[END_REF][START_REF] Bouissou | Set-based simulation for design and verification of simulink models[END_REF][START_REF] Bouissou | Simulation and verification of hybrid systems using hyson[END_REF], or in verification techniques based on SMT solver, mainly to define decision procedures over the reals, as in [START_REF] Gao | dReal: An SMT solver for nonlinear theories over the reals[END_REF] or in SMT modulo ODE [START_REF] Eggers | Improving the SAT modulo ODE approach to hybrid systems analysis by combining different enclosure methods[END_REF]. Control synthesis problem have also been considered in [START_REF] Fribourg | Stability controllers for sampled switched systems[END_REF][START_REF] Dit Sandretto | Tuning PI controller in non-linear uncertain closed-loop systems with interval analysis[END_REF].

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. In all the previous cited works a common set of interval operators are used which could be gathered into a library. DynIBEX1 is such a library providing operator to deal with interval constraints (IBEX library2 ) and differential equations. Note that DynIBEX follows the same purpose than CORA [START_REF] Althoff | An introduction to CORA 2015[END_REF] with an emphasis on the interval constraint programming approach. We report the main features of DynIBEX in this poster.

HSCC '16 Vienna, Austria

FEATURES OF DYNIBEX

Features coming from IBEX library

The most basic notion used in interval analysis is the in-

clusion function [f ] : IR n → IR k for a function f : R n → R k which satisfies for all [x] ∈ IR n , f ([x]) = {f (x) | x ∈ [x]} ⊆ [f ] ([x]
) . R stands for the set of real values while IR stands for the set of interval values. The natural inclusion function is the simplest to obtain: all occurrences of the real variables are replaced by their interval counterpart and all arithmetic operations are evaluated using interval arithmetic [START_REF] Jaulin | Applied Interval Analysis[END_REF]. To avoid some limitations with interval arithmetic, other kinds of arithmetic may bu used such as affine arithmetic [START_REF] Bouissou | Simulation and verification of hybrid systems using hyson[END_REF]. Both arithmetic are available in IBEX.

Consider some function g : R n → R k and some set Z ⊂ R k . A contractor Cc : IR n → IR n associated to the generic constraint c ≡ g(x) ∈ Z is a function taking a box [x] as input and returning a box Cc (

[x]) satisfying i) Cc ([x]) ⊆ [x] and ii) g ([x]) ∩ Z = g (Cc ([x])) ∩ Z. Hence, Cc provides a box containing the set {x ∈ [x] | g(x) ∈ Z} of solutions of c included in [x]: i) ensures that the returned box is included in [x] and ii) ensures that no solution of g(x) ∈ Z in [x] is lost. Standard contractor operators are defined in IBEX.
Paving is a common method in interval analysis, it is related to the construction of a set S ⊂ R n as a list of nonoverlapping boxes [x i ] ⊂ S with a non null width. This approach can be used to describe a set, by a list of inner boxes ([x i ] ⊂ S), of outer boxes ([x i ] ⊂ S), and the frontier, i.e., a list of boxes for which we cannot conclude of the membership to S in an acceptable computation time. For example, if the set S describes a ring such as

S = {(x, y) | x 2 + y 2 ∈ [1, 2]}, the paving of S in the box [-2, 2] × [-2, 2
] is given on the left.

Validated numerical integration methods

In DynIBEX different classes of differential equations can be treated with interval analysis tools: Ordinary Differential Equations (ODE) defining a function y(t) from f : R × R n × → R n such that it is solution of ẏ = f (t, y), where ẏ is the time derivative of y(t), and Differential Algebraic Equations (DAE) of index 1 defined two functions y(t) and x(t) from f : R × R n × R m → R n and g : R × R n × R m → R m such that they are solutions of ẏ = f (t, x, y) ∧ 0 = g(t, x, y) for all t. Note that x(t) is implicitly defined by g.

DynIBEX can solve these kinds of equations, with interval initial values and/or interval parameters, using various validated numerical integration schemes based on explicit and implicit Runge-Kutta methods among Heun and Midpoint methods, Radau-IIA (order 3), classical Runge-Kutta, Lobatto-IIIA (order 4), Lobatto-IIIC (order 4).

The combination of validated numerical integration methods for ODEs and DAEs with features of IBEX offers additional capabilities as considering embedded constraints to ODEs and DAEs. Such constraints may represent some physics laws such as energy preservation along trajectories. Basically, DynIBEX can consider constraints of the form c ≡ ∀t, 0 = h(t, y, p) with y the solution of an ODE or a DAE and p stands for a vector of bounded uncertain parameters.

For example, DynIBEX can solve the following DAE while checking the consistency of the initial values of the state and algebraic variables. It produces at time t = 0.5 the enclosure y ∈ ([1.8732, 1.8733], [1.3651, 1.3653], [5.0552, 5.0558]).

Function ydot = Function ( y , x , Return ( y [ 1 ] + x [ 0 ] , y [ 0 ] -y [ 1 ] * x [ 0 ] , y [ 0 ] * y [ 2 ] -x [ 1 ] ) ) ; Function g = Function ( y , x , Return ( y [ 0 ] -x [ 1 ] , y [ 1 ] -2 * x [ 0 ] ) ) ;
ivp dae h1 problem = ivp dae h1 ( ydot , g , 0 . 0 , y i n i t , x i n i t ) ; simulation simu = simulation(&problem , 0 . 5 , RADAU3 DAE, 1 e -1 4 ) ; simu . r u n s i m u l a t i o n ( ) ;

IntervalVector s a f e ( 3 ) ; s a f e [ 0 ] = I n t e r v a l ( 0 , 9 ) ; s a f e [ 1 ] = I n t e r v a l ( 0 , 5 ) ; s a f e [ 2 ] = I n t e r v a l ( 0 , 3 ) ; bool f l a g = simu . s t a y e d i n ( s a f e ) ; c o u t << "simu s t a y e d i n s a f e : " ; c o u t << f l a g ? " t r u e " : " f a l s e " << e n d l ; return 0 ; }

Satisfaction problems

Additionally in DynIBEX, a set of operators have been defined to handle constraints of the form ∀t, y(t) ∈ S or ∃t, y(t) ∩ U = ∅. In the above example, the method stayed in is used to check if the trajectory y(t) stay in a given box for all t. This feature, added to the contractors and paving capabilities, allow to solve satisfaction problems based on differential equations.

CONCLUSION

A quick overview of DynIBEX library has been presented, showing the main features to study dynamical systems. Note also that it is straightforward to simulate or study switched systems such as [START_REF] Fribourg | Stability controllers for sampled switched systems[END_REF]. Further work has still to be done to take into account more differential problems such as delay differential equations.

ACKNOWLEDGMENTS

This research benefited from the support of the "Chair Complex Systems Engineering -École polytechnique, THALES, DGA, FX, DASSAULT AVIATION, DCNS Research, ENSTA ParisTech, Télécom ParisTech, Fondation ParisTech, FDO ENSTA"

  c 2016 ACM. ISBN 978-1-4503-2138-9. DOI: 10.1145/1235

  i n t main ( ) { Variable y ( 3 ) ; // S t a t e v a r i a b l e s IntervalVector y i n i t ( 3 ) ; y i n i t [ 0 ] = I n t e r v a l ( 1 . 0 ) ; y i n i t [ 1 ] = 1 . 0 ; y i n i t [ 2 ] = 3 . 0 ; Variable x ( 2 ) ; // A l g e b r a i c v a r i a b l e s IntervalVector x i n i t ( 2 ) ; x i n i t [ 0 ] = I n t e r v a l ( 0 . 5 ) ; x i n i t [ 1 ] = 1 . 0 ;

http://perso.ensta-paristech.fr/˜chapoutot/dynibex/

http://www.ibex-lib.org