A Mohand Ousaid

A Bolopion

S Haliyo

S Régnier

V Hayward

Stability and Transparency Analysis of a Teleoperation Chain For Microscale Interaction

Keywords: Bilateral control, microscale teleoperation, passivity, transparency, stability

Microscale teleoperation with haptic feedback requires scaling gains in the order of 10 4 -10 7 . These high gains impose a trade-off between stability and transparency. Due to the conservative approach used in most designs, transparency is reduced since damping is added to the system to guarantee stability. Starting from the fact that series, negative feedback and parallel connection of passive systems is a passive system, a new approach is addressed in this work. We propose here a complete teleoperation chain designed from the ground up for full transparency and stability, including a novel self-sensing probe and a high fidelity force-feedback haptic interface. By guaranteeing the passivity of each device and assuming that the human operator and the environment are passive systems, a homothetic direct coupling can be used without jeopardizing the stability and provides best transparency. The system is experimentally demonstrated in the complex case of a probe interacting with a water droplet under human control, while accurately transcribing the interaction back to operator.

I. INTRODUCTION

Human interaction with the microscale remains an important challenge for several applications in nanotechnology, biology and physics such as characterization and analysis of microscale phenomena. In fact, at this scale the operators cannot interact directly with objects because of the scale and especially of severe differences in physics: mechanical behavior is no longer dictated by gravity, and short range forces such as electrostatic, capillary and van der Waals forces dominate [START_REF] Régnier | Microrobotics for Micromanipulation[END_REF]. Bilateral micro teleoperation in this case appears to be an interesting solution to supplement human perception and assist the operator in his/her task [START_REF] Hollis | Toward a tele-nanorobotic manipulation system with atomic scale force feedback and motion resolution[END_REF], [START_REF] Hatamura | Direct coupling system between nanometer world and human world[END_REF]. Two issues arise in this case: stability and transparency. The first is a necessary condition for the usability of the system, while the second relates to the ability of the setup to provide the operator with an accurate feeling of the microscale interaction [START_REF] Lawrence | Stability and transparency in bilateral teleoperation[END_REF], [START_REF] Raju | Design issues in 2port network models of bilateral remote teleoperation[END_REF]. Since 1990, several micro teleoperation systems have been proposed to provide a sense of touch and to enhance manipulation capabilities at the microscale. As an early example, Hamatura and Morishita proposed a nanomanipulation system to allow human operators to experience the "nanometer world" [START_REF] Hatamura | Direct coupling system between nanometer world and human world[END_REF]. Hollis introduced haptic feedback for the first time using a Scanning * are with Institut des Systèmes Intelligents et de Robotique, Université Pierre et Marie Curie, CNRS UMR 7222, 4 Place Jussieu, 75252 Paris Cedex, France {mohand_ ousaid,haliyo,regnier,hayward}@isir.upmc.fr + is with AS2M department -FEMTO-ST Institute, CNRS UMR 6174, UFC, ENSMM, UTBM aude.bolopion@femto-st.fr Tunneling Microscope (STM) [START_REF] Hollis | Toward a tele-nanorobotic manipulation system with atomic scale force feedback and motion resolution[END_REF]. Lawrence proposed tools for quantifying performances of general teleoperation system architecture, stability and transparency [START_REF] Lawrence | Stability and transparency in bilateral teleoperation[END_REF]. Following those works, Sitti and Hashimoto connected an haptic interface with an Atomic Force Microscope (AFM) to feel and manipulate small objects, using homothetic gains to adapt micro and macro scales [START_REF] Sitti | Tele-nanorobotics using atomic force microscope[END_REF]. Ryu and Hannaford also proposed a control scheme to ensure stable teleoperation under a wide variety of environments and operating speeds based on passivity control [START_REF] Ryu | Stable teleoperation with time-domain passivity control[END_REF]. In [START_REF] Boukhnifer | Passivity based control for microteleoperation system[END_REF], a bilateral controller based on [START_REF] Anderson | Bilateral control of teleoperation with time delay[END_REF] was presented using passivity approaches to deal with time delay for microscale operation. In recent literature, various approaches (passivity, oblique coordinates, observers...) are used to improve performances of bilateral micro teleoperation systems, especially in terms of stability [START_REF] Venture | Force-feedback coupling for micro-handling applications[END_REF]- [START_REF] Ni | Asynchronous event-based visual shape tracking for stable haptic feedback in microrobotics[END_REF] However, no work achieved in the same time stability and transparency of the micro teleoperation systems. All of the above-mentioned systems are subjected to the classical engineering trade-off between stability and transparency. Due to the conservative approaches used in those designs, transparency is reduced since damping is added to the system to guarantee stability. On the other hand, a micro to macro teleoperation system requires homothetic gains in the order of 10 4 to 10 7 . Those scaling factors greatly fragilize stability, especially in the presence of uncertainties like disturbances or sensor noise. Recently, a study shows that unlike indirect coupling, the direct coupling (Direct Force Feedback control scheme) that uses only homothetic gains is transparent but it suffers from a trade-off between force amplification and workspace limitation [START_REF] Bolopion | Analysis of stability and transparency for nanoscale force feedback in bilateral coupling[END_REF]. Furthermore, transparency is generally affected by the mechanical properties of the haptic interface. For example, small forces are badly transmitted due to problems of friction, inertia and hysteresis that tend to deteriorate the feeling provided by such systems To overcome those limitations and this trade-off, we propose in this paper a new approach. Instead of using available devices with a dedicated bilateral control scheme to ensure the overall stability, we propose a modular micro teleoperation system using custom components designed with passivity and transparency constraints. The stability is then guaranteed as the serial connection of passive systems yield a passive system [START_REF] Peter | Bilateral teleoperation : An historical survey[END_REF]. The used haptic interface is presented in [START_REF] Millet | The ultimate haptic device: First step[END_REF]. Its design provides performances covering human haptic perception and avoids the mechanical degradation of the feedback signal. Its transparency is shown on a large bandwidth [START_REF] Mohand-Ousaid | Haptic interface transparency achieved through viscous coupling[END_REF]. The microscale probe and the associated force sensor are described in detail in [START_REF] Mohand-Ousaid | Microforce sensor by active control of a comb-drive[END_REF]. A direct coupling scheme is proposed here to provide a stable and transparent microscale teleoperation chain using these components. This connection of passive components overcomes the need to sacrifice the transparency for stability. Tools based on passivity and impedance analysis are introduced to evaluate the stability and the transparency. An experimental case study is presented, where a human operator probes a water droplet. The haptic feedback is a faithful replica of the microscale interaction, due to the accuracy of the force sensor and the haptic interface, and the used direct coupling scheme.

This paper is composed of six sections. Passivity theory is addressed in section II. In section III and IV, the master system and slave system used in this work are presented respectively. The stability and transparency of the bilateral teleoperation system are analysed and the validity of the proposed approach is confirmed by experiments in section V. In the last section, the work is summarized.

II. PASSIVITY THEORY

Theoretical proof of the stability of any interconnected system is quite difficult, if not impossible, to provide. The passivity concept has consequently attracted great interests because it gives an easier way to prove the stability [START_REF] Zhao | A notion of passivity for switched systems with state-dependent switching[END_REF].

Passivity is defined such that a given system with state x, input u and output y is said passive if there exists a storage function V (x) ≥ 0, with an initial stored energy V (0), and a real-valued rate W (u, y), such that the inequality (1) holds ∀t ≥ 0.

In other words, the stored energy of the system is always less than or equal to the sum of the initial stored energy and the energy provided to the system. A passive system is internally stable [START_REF] Willems | Dissipative dynamical systems part I : General theory[END_REF]- [START_REF] Popov | Hiperstabilitatea sistemilor automate[END_REF].

(V (x(t)) -V (x(0))) t 0 W (u(τ), y(τ))dτ (1)
The passivity concept provides a strong tool to study the stability of systems and interconnected systems [START_REF] Calcev | Passivity approach to fussy control systems[END_REF]. Any serial, parallel or negative feedback combination of two passive subsystems is also passive [START_REF] Peter | Bilateral teleoperation : An historical survey[END_REF], [START_REF] Gudrun | Kinaesthetic feedback and enhanced sensitivity in robotic endoscopic telesurgery[END_REF]. According to the Parseval's Theorem [START_REF] Riley | Mathematical methods for the physical sciences[END_REF], [START_REF] Brogliato | Dissipative systems analysis and control : Theory and applications[END_REF] the passivity of a LTI (Linear Time Invariant) system with a rational transfer function H is guaranteed if (i) for the continuous time case -all poles of H(s) have real parts less than zero; -and H( jw) + H * ( jw) 0 for all w ∈ [-∞ ∞] (ii) for the discrete time case -all poles of H(z) have a module less than one; -and H(e jθ ) + H * (e -jθ ) 0 for all θ ∈ [0 2π] where the superscript * denotes complex conjugate transposition. If a LTI system is passive, the system is stable when coupled to an arbitrary passive system which is itself passive [START_REF] Richard | Stable haptic interaction with virtual environment[END_REF].

These criteria are used to verify the passivity of slave and master devices. In this work no distinction is made between the continuous and the discrete case, since the sampling frequency is ten times higher than the bandwidth of the system ( f e = 1 T e =1kHz).

III. MASTER SYSTEM: DUAL STAGE HAPTIC INTERFACE

The haptic interface with one degree of freedom [START_REF] Millet | The ultimate haptic device: First step[END_REF] used in this work is shown in Fig. 1. Haptic interfaces using electromagnetic drives are subjected to an inherent inertiatorque trade-off which limits transparency : the higher the torque, the higher the inertia. Consequently the transparency is degraded with inertia.

The proposed interface is based on a dual-stage architecture. It is comprised of a large motor coupled to a small one, also carrying the handle, through a passive viscous clutch based on eddy currents. This clutch mechanism accurately transforms slip velocity of the large motor into torque and exhibits a linear behavior on a large bandwidth (for brevity, no distinction is made in the following between forces and torques). This first order relationship between the velocity control and the handle eliminates the dynamics of the large motor from the user experience. A feed-forward path is provided through the small motor to fill in the transients or fast variations of the input signal that the large motor could not provide. As this second motor has negligible friction and inertia, unwanted forces could be kept below human detection threshold achieving quasi-perfect transparency.

A. Haptic interface control scheme

Referring to Fig. 2, the system has two torque commands and one torque output, which can potentially provide a number of control options. Here, the primary control objective is regulation and tracking of the output torque τ out .

The reference torque τ d , is compared to the torque produced by the clutch, τ coupler , which is proportional to the relative velocity between the large and small motors. The large motor is enslaved by the compensator C(z) to the torque error. The small motor is controlled to reduce the error without compensation since it is needed to transiently compensate for the slower response of the large motor. If the regulation is good enough, when the handle is backdriven, the power required to move the large motor must be entirely supplied by the power amplifier, not by the user. A polynomial controller is used to achieve this objective.

Figure 3 represents the response of the interface to a null reference. The controller compensates the inertia of the large motor. The dynamic range of the interface covers from 5.2 mN to 5.7 N. Furthermore, the control scheme doesn't change the passivity of the haptic interface according to the properties of passive interconnected systems described in section II.

B. Interface performances

User tests were carried out to evaluate the performances of the interface in [START_REF] Mohand-Ousaid | Haptic interface transparency achieved through viscous coupling[END_REF]. The aim of the experiment was to validate the high degree of transparency of the device. The operator had to perform a task of detecting a small amplitude stimulus, which can be assimilated to scanning a planar surface at the microscale with a localized minor defect. The result of the tests shows that the interface achieves one order of magnitude of improvement in transparency over existing designs, i.e ten times smaller stimuli can be accurately detected. The inertial load seen by a finger is only around one gram.

IV. SLAVE SYSTEM : ACTIVE MICRO FORCE SENSOR

The slave device performs the physical interaction at the microscale and provides force sensing. The force sensor is presented in depth in [START_REF] Mohand-Ousaid | Microforce sensor by active control of a comb-drive[END_REF]. The sensor uses electrostatic energy instead of elastic energy to perform the measurement. It is designed around a linear comb-drive actuator. Its working principle is to keep the probe on its equilibrium position by a force to cancel out external forces. As the comb-drive actuator is perfectly known and controlled, the balancing force is equal to the external force and directly provides the measure (Fig. 4).

Based on the position feedback of the probe y(t) an Hinfinity controller generates an opposite signal u(t) in order to balance the external applied force and to maintain the probe at its equilibrium position, as shown in Fig. 5. Instead of measuring the interaction using the signal coming from the displacement of the probe, it is computed as being proportional to the value of the opposite voltage applied by the controller generating the opposite force F act . In this case, F act -F ext . This technique which uses the closedloop control principle allows maintaining the sensitive part of the sensor in the linear range, improving precision, increasing the sensor dynamic range and improving the robustness and the stability of the sensor. Thus, the sensor probe can track the integrality of the tip-sample interaction curves. It especially avoids several problems inherent to scanning probe microscopy using elastic probes, such as pull-in, pull-off and hysteresis.

Small motor

Large motor

The LTI transfer function H(s) linking F act to F ext is:

H(s) = G(s)K(s) 1 + G(s)K(s) (2)
where G(s) and K(s) are respectively the system plant and the H ∞ controller [START_REF] Mohand-Ousaid | Microforce sensor by active control of a comb-drive[END_REF]:

Controller

Mechanical probe

G(s) = k n w 2 n s 2 + 2ζ w n s + w 2 n , (3)
where k n =0.032, ζ =0.0047 and w n =64 rad/s and K(s) = 78.4s 3 + 1.56810 8 s 2 + 7.46810 7 s + 3.86110 11 s 4 + 7516s 3 + 2.57110 6 s 2 + 3.12410 8 s + 1.09910 8 (4) All the poles of the function H(s) have a negative real part and the real part of the transfer function is positive over the entire bandwidth. This verifies the LTI passivity statements given above. The system thus remains passive over the entire bandwidth.

V. BILATERAL MICRO TELEOPERATION SYSTEM

A direct force feedback control scheme is used for the bilateral coupling. This scheme is to be the most intuitive formulation to accurately provide amplified forces to the operator. It is achieved by two homothetic gains α f and α d , respectively the force and the position scaling factors. These gains are necessary to adjust the scale between the macro and micro environment. As this coupling is solely comprised of those linear gains, the transparency is ideal, as shown in [START_REF] Bolopion | Tuning the gains of haptic couplings to improve force feedback stability in nanorobotics[END_REF]. The gain α f is chosen according to the dynamic range of the sensor and of the haptic interface. Its value in this case is 0.05 10 6 . The gain α d is chosen according to the displacement range of the sample holder and the handle. For a good trade-off between precision and manipulation comfort, the value of 1/α d =140 10 -4 is used.

As shown in Fig. 8, the position of the handle is scaled down and used as position set point to the transducer carrying the sample holder. Its transfer function M p (s)(M p (z)) can be considered as unity, M p (s) = M p (z) 1. The force measured by the sensor is scaled up as force set point to the haptic interface.

A. Stability analysis

The whole system can be presented as a series connection of sub-systems (blocks) as depicted in Fig. 6. The system is composed of the haptic interface connected to the human operator and to the slave device (considered as the sum of the transducer of the sample holder and the force sensor) through direct haptic coupling. The slave device is connected to the environment.

Assuming that the user will not voluntarily destabilize the system, his/her hand is considered as a passive system [START_REF] Colgate | Robust impedance shaping telemanipulation[END_REF]. The environment is also assumed to be passive as there is no time delay. As the direct coupling itself is passive, the passivity of the whole system is then guaranteed since the series connection of passive systems yield a passive system as detailed previously. As shown in [START_REF] Richard | Stable haptic interaction with virtual environment[END_REF], if a LTI system is passive, the system is stable when coupled to arbitrary passive system which is itself passive. This provides a sufficient condition for the stability of the micro teleoperation system presented in this paper.

B. Transparency analysis

To analyze the transparency of the system, the impedance criterion is used in this paper. The operator side impedance Z op = F op X h is compared with the environment side impedance Z e = F e X e . The scaling gains are also taken into account. The perfect transparency is achieved [START_REF] Lawrence | Stability and transparency in bilateral teleoperation[END_REF] if :

Z op = α f α d Z e (5)
The transparency investigation is carried out for two specific cases: a slave in non-contact mode (Z e → 0, no applied force) and a slave in contact mode. For the first mode (non-contact mode), the operator-side impedance depends only on the characteristics of the haptic interface's small motor. As shown in [START_REF] Millet | The ultimate haptic device: First step[END_REF], the choice of this motor is made to ensure minimal inertia. Thus, the ideal transparency is achieved for this mode.

For the second mode (contact mode), the environment is modeled as a sample with a stiffness k e . Figure 7 represents a Bode diagram of Z op and of

α f α d Z e
for three values of k e representing typical stiffness values of soft matter. The result shows that the impedance of the micro environment is transmitted to the human operators for k e < 1 N/m. The bandwidth of the sensor currently limits the stiffness of the samples that can be sensed up to 1 N/m. By increasing the sensor bandwidth, the system could interact with stiffer samples.

C. System validation 1) Experimental bench:

The experimental setup is depicted in Fig. 8(b). The sensor is placed on an anti-vibration table to avoid external mechanical perturbations. An optical microscope provides a visual feedback of the sample (droplet of water) and of the probe. The whole system is connected to a controller PC running RTAI/Linux real time OS. This system is adapted to numeric control implementation.

2) Tests and results: The experiment is performed in a clean room to limit the influence of environmental conditions. The system is tested on a complex case for microscale force sensing, which consists in measuring the time-course interaction of a thin glass probe with a water droplet under direct human control. The task is split in four main phases, namely 'approach', 'contact', 'retract', and 'contact break'.

A droplet with a diameter of 0.8 mm is pipetted on the tip of a needle. The droplet is approached towards the probe using the position of the interface handle as a set point for the sample holder. The droplet is contacted and then retracted in approach-retract cycle. The interaction force is measured by the sensor and fed to the user through the haptic interface. Figures 9 and10 represent respectively the force measured by the sensor over a cycle of approach-retract and the force transmitted to the operator using a probe with a diameter of 140 µm. The resulting curves highlight the high level of transparency of the system since the output force of the haptic interface is a linear function of the measured interaction.

The system remains stable over the experiments and achieves perfect transparency. In fact, the force felt by the user through the interface handle is exactly the force measured by the sensor amplified by α f . Hence, this investigation shows the performances and the importance of the micro teleoperation system applications. The system has also been tested by several users with no experience in microscale phenomena. Various interaction forces, like pull-in and pulloff forces, are correctly rendered to the operators as shown above.

VI. CONCLUSION

The design of a new micro teleoperation system that is not subjected to the stability/transparency trade-off is presented. It is based on the coupling of an active force sensor with a high-fidelity haptic device through a direct bilateral control scheme. The system provides a natural and intuitive interac-tion with the microscale, owing to the particularities of each device. First, the force sensor provides a force measurement which is not based on a structural stiffness but on a force balancing principle. Hence its output accurately reflects the physical interaction, independently of the probe's mechanical properties. The haptic device is based on a dual stage design for a single degree of freedom. Its performances are very close to human haptic perception range. The direct bilateral coupling allows for linear transformations between microand macroscales. As a result, the system transposes successfully the sense of touch into the microscale. The system is validated by measuring the interaction of a glass probe with a droplet of water involving adhesion and capillary forces. This complex interaction is accurately rendered to the operator with unprecedented haptic clarity.

Future work would improve fabrication of the sensor using MEMS fabrication technique, while retaining the same principle. The bandwidth of the sensor is then expected to reach the kHz range. Another improvement would transform the rotation degree freedom of the haptic interface to translation freedom, in order to improve the system ergonomic.

Fig. 1 .

 1 Fig. 1. Dual stage haptic interface. A large and small motor are coupled through a magnetic passive clutch based on eddy currents. The output torque is a linear function of the differential velocity of both motors summed to the torque provided by the small motor.

Fig. 2 .

 2 Fig.2. Torque control scheme of the haptic interface.

Fig.

  Fig. Haptic interface response to a null reference. As the controller compensates the inertia of the large motor, the torque felt by the user is null.Rigid probe Sample

Fig. 4 .

 4 Fig. 4. Force balance or compensation principle. (a) The sensor probe is attracted by the sample with a force F ext , while (b) the controller maintains the probe at the equilibrium position by applying a balance force F act to compensate the external force.

Fig. 5 .Fig. 6 .

 56 Fig.5. Active force sensing control scheme. The controller keeps the probe at its equilibrium position in order to balance external force F ext . This principle provides directly the measure of the force.

Fig. 7 .

 7 Fig. 7. Impedance bode diagram, contact mode (k e = 0.01, 0.1 and 1 N/m).

Fig. 8 .

 8 Fig. 8. Global view of the micro teleoperation system. (a) System architecture, (b) Experimental setup.

Fig. 9 . 10 .

 910 Fig. 9. Interaction force measured by the sensor over a cycle of approachretract vs the relative position between the sample and the probe.

ACKNOWLEDGEMENTS

This work was supported by the French Agence Nationale de la Recherche, through NANOROBUST project, under Contract ANR 2011 NANO 006.


Didier El Baz
email: elbaz@laas.fr

Vincent Boyer
email: vboyer@laas.fr

Julien Bourgeois
email: julien.bourgeois@univ-fcomte.fr

Eugen Dedu
email: eugen.dedu@univ-fcomte.fr

Kahina Boutoustous
email: kahina.boutoustous@univ-fcomte.fr

Distributed Discrete State Acquisition and Concurrent Pattern Recognition in a MEMS-based Smart Surface#

Keywords: MEMS, smart surface, state acquisition, pattern recognition, distributed algorithms, asynchronous algorithms, communication management, multithreading, multicores

A distributed smart surface based on MEMS technologies is considered. We lay down the mathematical foundations of distributed discrete state acquisition. Distributed state acquisition algorithms and concurrent pattern recognition methods are proposed. A multithreaded Java smart surface simulator which runs on multicore machines is presented. A first series of computational results is displayed and analyzed.

I. INTRODUCTION

Micro-Electro Mechanical Systems (MEMS) actuator arrays with embedded intelligence, which are often referred to as smart surfaces, have gained attention recently, e.g. see [START_REF] Biegelsen | Airjet paper mover[END_REF] and [START_REF] Fukuta | Design fabrication and control of MEMS-based actuator arrays for air-flow distributed micromanipulation[END_REF].

We consider here a smart surface designed for conveying positioning or sorting micro parts (see [START_REF]ANR PSIRob 2006 project[END_REF] and [START_REF]Smart Surface[END_REF]). The MEMS-based distributed micro robotics system is an array of fully integrated micro modules, the so-called cells. Each cell contains sensor, processing unit and actuator (see [START_REF] Boutoustous | A framework to calibrate a MEMS sensor network[END_REF] and [START_REF]ANR PSIRob 2006 project[END_REF]). Cells are connected via a communication network. Each cell has at most 4 neighbors (see Fig. 1). Distributed computing (e.g. distributed state acquisition and concurrent pattern recognition) on dedicated architectures like a smart surface is a source of a rich problematic mainly due to the scarcity of resources like number of sensors (each part will recover a small number of sensors), memory size and computing power or the presence of faults. To the best of our knowledge, the literature on sorting and positioning micro parts in a low resolution context is almost nonexistent.

In this paper, we propose techniques for distributed discrete state acquisition, communication management and pattern recognition. In particular, we give a mathematical model of discrete state acquisition and propose several distributed iterative algorithms; i.e. we consider synchronous and asynchronous state acquisition methods. We propose also simple initial points and give convergence results for distributed algorithms. We propose stopping criteria in the synchronous case and in the asynchronous case.

We derive also several techniques for concurrent pattern recognition. These techniques are particularly interesting when cells present faults or when parts are initially positioned any manner on the smart surface. Finally, we introduce SSS, a multi threaded Java smart surface simulator that has permitted us to evaluate and validate experimentally our distributed algorithms on multicore machines.

Section 2 gives more details on the smart surface. Section 3 deals with distributed state acquisition. Pattern recognition techniques are presented in Section 4. In Section 5, we introduce SSS, a multithreaded Java smart surface simulator. Computational results related to pattern recognition are displayed and analyzed in Section 6. Conclusions and future works are presented in Section 7.

II. THE SMART SURFACE

Assembly line workstations need to be fed with wellpositioned and well-oriented parts. These parts are often jumbled and they need to be sorted and conveyed to the right workstation. To do so, the following operations must be performed on parts: identification, sorting, orienting, positioning and feeding. Among the most promising solutions to perform these tasks, is the combination of MEMS in order to form a micro robotics array.

There have been numerous projects on MEMS actuator arrays in the past and more particularly in the 1990's. These pioneering research works have developed different types of MEMS arrays that are based on actuators which are either pneumatic (see [START_REF] Fujita | Group work of microactuators[END_REF] and [START_REF] Pister | A planar air levitated electrostatic actuator system[END_REF]), magnetic or t electrostatic or servoed roller wheels (see research works have been conducted in o sensors and to add intelligence to MEMS ac these works either failed to propose solutions or that are fully integrated (e.g. see [START_REF] Fukuta | Design fabrication and control of MEMS-based actuator arrays for air-flow distributed micromanipulation[END_REF]).

The objective of the Smart Surface proje [START_REF] Matignon | Design of semidecentralized control laws for distributed-airjet micromanipulators by reinforcement learning[END_REF]) is to design a distributed and in manipulator based on an array of micro mod realize an automated positioning and conveyi micro module will be composed of a micro a sensor and a processing unit. The coope modules thanks to an integrated network will to recognize the parts and to control micro actu move and position accurately the parts on th We consider small parts that cover a small n modules and that are moved via air nozzle rectangular holes on the front-side of Fig. 2). through a micro valve in the back-side of the passes through the nozzle. The advantage of th the micro actuators, the most fragile part of protected. Circle holes (see Fig. 2) are use sensors to detect the presence of the part on the

III. DISTRIBUTED STATE ACQUISIT

In this Section, we lay down the mathema of discrete state acquisition of the smart sur derive and study several distributed sy asynchronous algorithms. For each cell, th obtain global knowledge of the state of the sma know precisely where a part is situated.

A. Mathematical model of distributed state ac

Without loss of generality, we assume that part on the smart surface and one sensor acquisition can be modeled as the follow problem: find 0,1 such that fixed point which satisfies:

, where is the number of cells of the smart su mapping from into . A vector 0 thermo bimorph, e [START_REF] Bohringer | Algorithms for sensorless manipulation using a vibrating surface[END_REF]). Recent order to include ctuator arrays but s at a micro-scale ect (see [START_REF] Fort-Piat | Smart Surface based on autonomous distributed micro robotic systems for robust and adaptive micromanipulation[END_REF]

where the operations and ∑ let , 0,1 , 0, otherwise; similarly, ∑ , and ∑ , 0 otherwise. Definition 1: Distributed synchrono described via the following success (this method is also referred to as iteration on a product of finite sets):

,

Where the initial approximation , , bal state of the smart surface ges they receive from their the set of all neighbors of ecomposed the same way as

1, … , , , 1, … , , 1, … , , ,
are defined as follows: 0 0,

ous state acquisition can be sive approximation method s a discrete iteration or an 0,1, … is chosen as follows: The reader is referred to Robert [START_REF] Robert | Discrete Iterations, a Metric Study[END_REF] for a study on mathematical and algorithmic aspects of discrete iterations.

1, … , , , 1, … , , , 1, … ,
Theorem 1: The mapping is monotone.

Proof: from the definition of the mapping , we clearly have , , ; Q.E.D.

Theorem 2: The distributed discrete iteration converges to .

Proof: from the definition of mapping , we have .

Moreover, we have , 1,2, … , 1, 2, …
Since the mapping is monotone, and . We denote by a discrete distance on 0, 1 . We have: ,,,,,,, , ,

Let be the Manhattan distance on , we

We note that , is finite for any , . As a consequence, the distributed discrete algorithm converges monotonically to in finite number of iterations; Q.E.D.

Let us denote now by the largest Manhattan distance of the smart surface.

Theorem 3: The number of iterations of the discrete algorithm is bounded by 1.

Proof: The distributed discrete iteration starting from , generates a monotone sequence of vectors of . Time after time, each cell makes acquisition of the augmented local state of its neighbors via messages it receives from them and combines these augmented states with its own augmented state in order to produce an updated augmented state, i.e. a more accurate vision of the state of the mart surface. At each new iteration, cells gain a more accurate vision of the actual discrete state of the smart surface by a unit of distance. This corresponds to a link between two cells along each direction. Finally, the sequence converges to the fixed point which is such that 1. Nevertheless, this test permits a cell to stop computation at global convergence since if a component of is equal to one, then the cell "knows" that there is a part on the smart surface and if the augmented local state does not change from an iteration to another, then the shape of this part is stationary. We recall that the "horizon" of a cell increases by one unit of distance along each direction at each iteration. Let , denote the set of active neighbors of cell at iteration , i.e. the set of neighbors for which the improved local stopping test presented in Definition 2 is not satisfied at iteration . Let , denotes the th active neighbor of cell at iteration . The behavior of the improved distributed synchronous discrete algorithm can be represented as follows:

For from 1 to n do While local stopping test is not satisfied 1 derive , :

For j from 1 to

, do send to ,

End do

For j from 1 to , do receive from ,

End do

End While

End do

Improved distributed synchronous discrete algorithm

B. Distributed asynchronous algorithms

In the above subsection we have presented a first model of distributed state acquisition in the synchronous case. We present now a mathematical model in the more general asynchronous context where each cell can perform updating phases at its own pace, i.e. computation can be done without order nor synchronization. We derive convergence results by using the general convergence theorem of Bertsekas (see [START_REF] Bertsekas | Parallel and Distributed Computation, Numerical Methods[END_REF]). We propose also a stopping method.

We assume that there is a set of times 0,1,2, … at which one or more sub vectors , 1, … , , of vector are updated by some cells. We denote by the subset of times at which the sub vector is updated. Let , , … , , be the subset of labels used during the updating phases of cell with: 0 , , , 1, … , n , .

We assume that lim , ∞ , , 1, … , , this assumption guarantees that new values of the components of the sub vectors are used as computations go on. We also assume that the sets , 1, … , , are infinite; this assumption guarantees that no component of the iterate vector is abandoned forever. In particular, cells will not stop their computations before convergence. We denote by the set of labels used during the computations performed by the different cells. Definition 3: Distributed asynchronous state acquisition can be described via the following successive approximation method denoted by , , , , where is the initial approximation defined in Section II.A.

, , … , , , , , , Theorem 4:
The distributed asynchronous discrete iteration , , , converges to .

Proof: In order to show convergence of the asynchronous algorithm , , , we build a sequence of level sets which satisfies the conditions of the general asynchronous convergence theorem of Bertsekas, see page 431 in [START_REF] Bertsekas | Parallel and Distributed Computation, Numerical Methods[END_REF].

Let

, we define the sets .

The so called synchronous convergence condition of Bertsekas , , , and every limit point of is a fixed point of if , , is satisfied. This result follows from Theorem 2, the monotone property of mapping and the fact that is the smallest vector such that .

The so-called box condition of Bertsekas, i.e.: … , 0,1,2, … is also satisfied since the level sets are Cartesian products of subsets 0, 1 . As a consequence, the general asynchronous convergence theorem of Bertsekas applies. Q.E.D.

We note that the sequence of nonempty subsets satisfies: , and .

The reader is also referred to Radid [START_REF] Radid | Itérations Booléennes et sur des ensembles de cardinal fini[END_REF] for various results related to asynchronous discrete iterations.

Among the many interests of distributed asynchronous iterations, one can quote the better efficiency of the algorithms since each cell goes at its own pace and there is no waiting time for synchronization. This is particularly true in the case of monotone convergence, where the use of last updates permits always one to improve the iterate vector. One can quote also fault tolerance since distributed asynchronous iterations tolerate some messages losses (see [START_REF] Bertsekas | Parallel and Distributed Computation, Numerical Methods[END_REF]).

In the case of a cell fault, a distributed asynchronous algorithm may end with a vector slightly different from , however, there will be no deadlock with such an algorithm.

Implementation

In this subsection, we show how asynchronous algorithms have been implemented. We consider also convergence detection of asynchronous algorithms. Several procedures can be used in order to detect convergence of distributed asynchronous discrete iterations. One can use for example the Dijkstra and Scholten procedure [START_REF] Dijkstra | Termination detection for diffusing computation[END_REF] (see also [START_REF] Bertsekas | Parallel and distributed iterative algorithms: a selective survey[END_REF] and [START_REF] Baz | An efficient termination method for asynchronous iterative algorithms on message passing architecture[END_REF]). The reader is also referred to El Baz [START_REF] Baz | A method of terminating asynchronous iterative algorithms on message passing systems[END_REF] for a method based on level sets. The procedure in [START_REF] Dijkstra | Termination detection for diffusing computation[END_REF] relies on generation of activity graph and acknowledgement of messages. Initially, only one cell is active, i.e. the so-called root that is denoted by . The cell starts computation and sends messages to its neighbors; these messages activate the neighbors that become the so-called sons of and so on. All cells become eventually active. All messages are acknowledged at once but activation messages of father that are acknowledged only when a son becomes inactive. The activity graph moves on according to the messages received and satisfaction of the conditions:

. A cell sends messages to its neighbors if and only if it is active and the above condition is not satisfied. Finally, the algorithm stops when the cell R stops; i.e. all local stopping criteria are satisfied and there is no message in transit in the system. This type of convergence detection method is quite natural in the context of discrete iterations since it is not necessary to modify the distributed asynchronous iterative algorithm so that it converges in finite time.

Let us denote by Active( ) the logical variable that stores the behavior of the th cell: if Active( ) is True, then the th cell performs computation. If Active( ) is False, then the th cell does nothing. Initially, all cells are inactive, but . A cell becomes active when receiving a message. A cell becomes inactive when the following extended local asynchronous stopping criterion is satisfied. Definition 4: The extended asynchronous local stopping test is given by: and all cells activated by cell are inactive.

All cells can be activated many times but the root, , which is active only once. Finally, the algorithm stops when becomes inactive. In the sequel, we shall denote by isend and ireceive, respectively, nonblocking send and receive, respectively. These communication primitives permit one to implement asynchronous communication. Formal proofs of validity for this type of algorithm including convergence detection method have been established in [START_REF] Bertsekas | Parallel and Distributed Computation, Numerical Methods[END_REF] and [START_REF] Baz | An efficient termination method for asynchronous iterative algorithms on message passing architecture[END_REF].

IV. CONCURRENT PATTERN RECOGNITION

Once distributed state acquisition has been performed, the concurrent pattern recognition phase can begin. We propose a simple concurrent pattern recognition method whereby cells do not communicate. Basically, cells compute concurrently several contour based differentiation criteria like number of components of vector with value 1 such that there exists 0, or region-based criteria like number of components of vector with value 1, i.e. surface like criteria, or maximum length between 1 of the part. The criteria considered in this paper are detailed in [START_REF] Boutoustous | An exhaustive comparison framework for distributed shape differentiation in a MEMS sensor actuator array[END_REF]. Finally, we note that to the best of our knowledge, there are relatively few papers in the literature on pattern recognition methods in the low resolution context. We mainly refer to Ishida [START_REF] Ishida | Recognition of low resolution characters by a generative learning method[END_REF], for different approaches related to low resolution character recognition and to Tabbone [START_REF] Tabbone | A new shape descriptor defined on the Radon transform[END_REF], for a novel approach based on the Radon transform for complex shapes identification.

The value of the different criteria can vary according to the orientation and position of the part on the smart surface. In the case of part rotation, for example, we have noted that the surface of 3x3 square, where the unit of distance is the length of a cell, can vary from 9 to 13, according to the orientation of the part on the smart surface, see Fig. 3 obtained with SSS, a multithreaded smart surface simulator that will be detailed in the next section. In the sequel, we present two new approaches for part differentiation.

The first approach relies on the use of a single reference position for each part. Each cell compares computed values of the criteria for the current position of the part on the smart surface with values of criteria of each part obtained for a single reference position (those values are stored in a database of registered parts). We propose to compute gaps between the measured criteria and the criteria value of registered parts for differentiation purpose. This method is particularly interesting when some cells present faults.

Another particularity of the first approach is to consider only a subset of well known criteria like surface or perimeter of the part. The criteria that amplify tiny differences between parts, like product of the differences between consecutive columns and consecutive line are discarded. In the sequel, the number of criteria considered will be denoted by . Let denote the number of different parts. Let denote the reference value of th criterion of the th part (those values are computed offline). Let denote the value of the th criterion of the part on the smart surface. Each cell computes the following gaps:

1 1 , 1, … , .
The second approach relies on the use of a set of reference positions for the different parts on the smart surface. We take into account rotations of parts with one degree increments. Without loss of generality and for symmetry reason, only rotations from 1 up to 45 degrees can be considered. Let 1, … , 45 . We denote by the reference value of th criterion of the th registered part rotated by degrees; those values are also computed offline. We denote by the set of all reference values for part , , … , ,

. Each cell computes the following gaps.

1 1 , 1, … , .
We note that the former gap, , presents the advantage to require a limited amount of memory and a small computing time, while the latter gap, , permits one to expect better differentiation of parts, particularly in the case where parts can have any orientation on the smart surface.

Decision making concerning pattern recognition at each cell relies on the respective values of the gaps. The pattern that is chosen corresponds to the gap or that is the closest to zero. We note that all cells make the same computation concurrently and thus take the same decision.

Concurrent decision making based on gaps is particularly interesting with respect to fault that may occur on the smart surface, i.e. sensor failures or parts positioned any manner on the smart surface. The use of gaps is also interesting with respect to recognition of pattern slightly modified, i.e. parts that present tiny faults.

V. SSS, A MULTITHREADED SMART SURFACE SIMULATOR

We present now SSS, a smart surface simulator developed at LAAS-CNRS. The simulator SSS has permitted us to evaluate distributed synchronous and asynchronous state acquisition algorithms and concurrent pattern recognition methods. SSS is a multithreaded Java code that runs on multicore machines.

SSS has permitted us to validate experimentally the distributed algorithms and to study in detail communications between cells, stopping criteria and the efficiency of the proposed methods. The reader is referred to the site [START_REF]Smart Surface[END_REF] for some demos with SSS. SSS permits one to build a smart surface that has any size and different patterns like squares, rectangles, L shapes, I shapes and so on, that will become references or that will correspond to a given part. SSS permits one also to place the generated patterns everywhere on the smart surface. It is possible to rotate shapes on the smart surface and to introduce sensors faults (see Fig. 3, for a square shape). SSS allows one to choose a synchronous or asynchronous distributed state acquisition algorithm, to carry it out and to display dynamically the augmented local state of any cell (see right window of Fig. 4 that corresponds to iteration 2). One can also have a dynamic view of the activity graph of the smart surface. One can choose criteria and a differentiation method, e.g. gaps based methods or differentiation methods studied in [START_REF] Boutoustous | An exhaustive comparison framework for distributed shape differentiation in a MEMS sensor actuator array[END_REF]. Finally, one can display the results of the pattern recognition phase for the different criteria selected. SSS permits one also to display some statistics.

VI. TESTS

The multithreaded Java smart surface simulator SSS has been carried out in parallel on a multicore machine with Intel Quadro Xeon 3.0 GHz processor.

In this section, we compare a first series of results obtained with SSS for the gap methods proposed in Section IV and the total differentiation method presented in [START_REF] Boutoustous | An exhaustive comparison framework for distributed shape differentiation in a MEMS sensor actuator array[END_REF]. We have considered three parts: a square, the so-called Sq part, an L shaped part and a I shaped part. All parts have been placed randomly 200 times, leading to 200 draws with SSS. We classified the criteria according to their performance. For each draw, we have computed the values of 17 criteria and we have applied the total differentiation method which gives the differentiation rate. The criteria were afterwards classified according to the differentiation rate and only the criteria with the best differentiation rates were selected. Table I displays results for the best two criteria: S and A, respectively, were S denotes the surface and A the product of angles of type "V", respectively (see [START_REF] Boutoustous | An exhaustive comparison framework for distributed shape differentiation in a MEMS sensor actuator array[END_REF]). For 200 draws, we note that the criterion S has correctly differentiated the part Sq in 37% of cases. In the second test, we are interested in part differentiation via a combination of the criteria S and A. Table II shows the differentiation rate obtained by using the criteria combination. We note that the part Sq was correctly differentiated in 59% of the cases; which is better than with S or A alone. The average differentiation rate was increased from 48,76% (for S alone) and 40.83% (for A alone) up to 67.16% for the combination of A and S. We compare now the results of the above differentiation method with those obtained with the methods based on gaps introduced in Section IV of this paper. These results show that the methods based on gaps g and g' improve the differentiation rate in the following cases: -with a single criterion, e.g. the criterion S, the differentiation rate is improved from 48.76% (see first row of Table I) to 99% for g (see first row of Table 3) and 99% for g' (see first row of Table IV).

. -with several criteria, the combination of criteria also improves the differentiation rate from 67.16% (see first row of Table 2), to 98.83% (see third row of Table III) and 92,83% (see third row of Table IV), respectively, for g and g', respectively.

We conclude that the differentiation methods based on gaps give better results than the total differentiation method presented in [START_REF] Boutoustous | An exhaustive comparison framework for distributed shape differentiation in a MEMS sensor actuator array[END_REF] when parts can have any orientation on the smart surface. Additionally, one criterion alone may be sufficient to reach almost 100% differentiation rate.

VII. CONCLUSIONS AND PERSPECTIVES

In this paper, we have considered a smart surface for conveying, positioning and sorting micro parts. We have laid down the mathematical foundations of smart surface state acquisition. We have proposed stopping criteria and several distributed synchronous and asynchronous algorithms and we have established convergence results for the studied methods. We have also proposed a concurrent pattern recognition method based on gaps. Finally, we have presented SSS, a multithreaded Java smart surface simulator that we have developed in order to evaluate and validate distributed algorithms and we have displayed and analyzed a first series of results for randomly generated instances with SSS.

Future directions of research concern fault detection and solutions that propose degraded but everlasting behavior particularly in the synchronous case.

We shall consider concurrent pattern recognition methods that exploit smart surface natural parallelism, e.g. each cell can apply a mapping to and compute criteria. Typical mappings can be some rotations. This kind of preconditioning can enrich pattern recognition.

We shall also study combined pattern recognition and part motion; it may be efficient to recognize a part while moving it on the smart surface.

It may also be interesting to derive pattern recognition techniques that are not criteria based and which exploit directly the part code.

Finally, actual implementation on the distributed smart surface must be made in order to complete the study.

We believe MEMS-based smart surfaces to have great industrial impact for manipulating micro parts in many areas like semiconductor industry and micromechanics.

Figure 1 .

 1 Figure 1. Communication network of the smart surface

Figure 2 .

 2 Figure 2. General overview of the Smart

For from 1

 1 to n do While Active(R) = True If Active(i) = True then 1

Figure 3 .

 3 Figure 3. Example of different surface values for the same square with different orientations (SSS screenshots)

Figure 4 .

 4 Figure 4. SSS smart surface window (left) and "extended" local state window of a given cell at iteration 2 (right)

  . being considered component wise), is a good starting point in order to converge to the solution of the fixed point problem that makes sense in term of state acquisition.

	only fixed point for . It is
	generated by a discrete
	remains stationary at a
	order relation is component … , . A simple illustration
	such that , have: 1, 1, … ,	1, , . However, derived
	tual local state of the cell of
	vers completely the surface.

choosing a good initial to a fixed point that makes We shall see in the sequel r , which is in fact a sub solution, i.e. which satisfies (the inequality

  ImplementationFor simplicity of notation, we denote in the sequel by the set of neighbors of node . The th neighbor of node is denoted by . The behavior of the distributed synchronous discrete algorithm can be represented as follows.

			For i from 1 to n do
			For k from 1 to	1 do
			:
			For j from 1 to	do
			send	to
			End do
			For j from 1 to	do
			receive	from
			End do
			End do
			End do
			Distributed synchronous discrete algorithm
			Definition 2: The local stopping test stops the computation of
			cell at iteration if .	1, … ,	such that	,	1 and
			The above stopping test is only based on local data, i.e. a
			comparison of the augmented local state at two consecutive
			iterations and
	,	, , ,	; this shows that at
	convergence, all augmented local states are similar and that all
	cells have the same vision of the global state of the smart
	surface). Q.E.D.
	For a rectangular smart surface with size 1 1 sequences of communications are necessary , at most to make acquisition of local states and at most 1 1 1 iterations are necessary to obtain the solution.

The distributed algorithm converges in finite time. It is nevertheless possible to derive simple stopping tests in order to reduce the number of iterations.

Improved local stopping test

We present now a distributed stopping test designed in order to stop iterations at global convergence by using only local data.

TABLE I .

 I DIFFERENTIATION RATES FOR THE CRITERIA A AND S

	Criteria	Sq	I	L	Average
	S	37.00%	52.00%	57.30%	48.76%
	A	33.50%	40.50%	48.50%	40.83%

TABLE II .

 II

		DIFFERENTIATION RATES FOR CRITERIA COMBINATION
	Criteria	Sq	I	L	Average
	A and S	59,50%	74,00%	68,00%	67,16%

  Table III displays the results obtained with the gaps g.

TABLE III .

 III

		DIFFERENTIATION RATES WITH THE FIRST GAP
	Criteria	Sq	I	L	Average
	S	100%	99.00%	98.00%	99.00%
	A	100%	99.00%	78.00%	92.33%
	A and S	100%	100%	96.50%	98.83%
	Table IV gives the results obtained with the gaps g'.

TABLE IV .

 IV

		DIFFERENTIATION WITH RATES WITH THE SECOND GAP
	Criteria	Sq	I	L	Average
	S	100%	99.00%	98.00%	99.00%
	A	100%	99.00%	78.00%	92.33%
	A and S	100%	99.50%	79.00%	92.83%

# This work has been funded by ANR grant PSIRob 2006 (project Smart Surface, ANR 06 ROBO 0009 03)


Jonathan El

Gilles Stupfler

Extreme versions of Wang risk measures and their estimation for heavy-tailed distributions

Keywords: Asymptotic Normality, Conditional Tail Moment, Distortion Risk Measure, Extreme-Value Statistics, Heavy-Tailed Distribution

Among the many possible ways to study the right tail of a real-valued random variable, a particularly general one is given by considering the family of its Wang distortion risk measures. This class of risk measures encompasses various interesting indicators, such as the widely used Value-at-Risk and Tail Value-at-Risk, which are especially popular in actuarial science, for instance. In this communication, we first build simple extreme analogues of Wang distortion risk measures and we show how this makes it possible to consider many standard measures of extreme risk, including the usual extreme Value-at-Risk or Tail-Value-at-Risk, as well as the recently introduced extreme Conditional Tail Moment, in a unified framework. We then introduce adapted estimators when the random variable of interest has a heavy-tailed distribution and we prove their asymptotic normality. The finite sample performance of our estimators is assessed on a simulation study and we showcase our techniques on an actuarial data set.

Introduction

Understanding the extremes of a random phenomenon is a major question in various areas of statistical application. A stimulating topic comes from the fact that extreme phenomena may have strong adverse effects on financial institutions or insurance companies, and the investigation of those effects on financial returns makes up a large part of the recent extreme value literature.

The most used risk measure is the Value-at-Risk (VaR) which corresponds to the quantile of the loss distribution. Of course, the estimation of a single extreme quantile only gives incomplete information on the extremes of a random variable. To put it differently, it may well be the case that a light-tailed distribution (e.g. a Gaussian distribution) and a heavy-tailed distribution share a quantile at some common level, although they clearly do not have the same behavior in their extremes. Besides, the VaR is not a coherent risk measure in the sense of [START_REF] Artzner | Coherent measures of risk[END_REF], which is a undesirable feature from the financial point of view. This is why other quantities, which take into account the whole right tail of the random variable of interest, were developed and studied such as the the Conditional Tail Expectation (CTE).

One may then wonder if such risk measures may be encompassed in a single, unified class. An answer, in our opinion, lies in considering Wang distortion risk measures (DRMs), introduced by [START_REF] Wang | Premium calculation by transforming the layer premium density[END_REF]. The aforementioned VaR and CTE actually are particular cases of Wang DRMs, and so are many other interesting risk measures such as for example the Wang transform [START_REF] Wang | A class of distortion operators for pricing financial and insurance risks[END_REF] which is very popular in finance.

Wang risk measures

Let X be a positive random variable. [START_REF] Wang | Premium calculation by transforming the layer premium density[END_REF] introduced a family of risk measures called distortion risk measures (DRMs) by the concept of a distortion function: a function g : [0, 1] → [0, 1] is a distortion function if it is nondecreasing with g(0) = 0 and g(1) = 1. The Wang DRM of X with distortion function g is then defined by:

R g (X) := ∞ 0 g(1 -F (x))dx
where F is the cumulative distribution function (cdf) of X. Denote by q the quantile function of X, namely q(α) = inf{x ≥ 0 | F (x) ≥ α} for all α ∈ (0, 1). Noticing that F (x) = inf{α ∈ (0, 1) | q(α) > x} and thus F is the right-continuous inverse of q, a classical change-of-variables formula and an integration by parts then entail that R g (X), provided it is finite, can be written as a Lebesgue-Stieltjes integral:

R g (X) = 1 0 g(α)dq(1 -α) = 1 0 q(1 -α)dg(α).
A Wang DRM can thus be understood as a weighted version of the expectation of the random variable X.

Extreme versions of Wang risk measures may be obtained as follows. Let g be a distortion function and for every β ∈ (0, 1), consider the function g β (y) which is defined by:

g β (y) :=      g y 1 -β if y ≤ 1 -β 1 otherwise = g min 1, y 1 -β .
We can then obtain extreme versions of Wang risk measures as

R g,β (X) = 1 0 q(1 -α)dg β (α).
The distortion measure dg β of this risk measure is concentrated on [0, 1 -β], so that it only takes into account those (high) quantiles of X whose order lies in

[β, 1]. It is actually straightforward to obtain R g,β (X) = 1 0 q(1 -(1 -β)s)dg(s).
(1)

Estimation using an asymptotic equivalent of a Wang DRM

We now give a first idea to estimate this type of extreme risk measure. Let (X 1 , . . . , X n ) be a sample of independent and identically distributed copies of a random variable X having cdf F , and let (β n ) be a nondecreasing sequence of real numbers belonging to (0, 1), which converges to 1.

Assume X has a heavy-tailed distribution. Recall that a function f is said to be regularly varying at infinity with index b ∈ R if f is nonnegative and for any x > 0, f (tx)/f (t) → x b as t → ∞;

the distribution of X is then said to be heavy-tailed when 1 -F is regularly varying with index -1/γ < 0, the parameter γ being the so-called tail index of the cdf F . This condition, which is a usual restriction in extremes, essentially says that 1 -F (x) is in some sense close to x -1/γ when x is large. In this case it holds that R g,βn (X) = q(β n )

1 0 s -γ dg(s)(1 + o(1)) as n → ∞ provided 1 0 s -γ-η dg(s) < ∞ for some η > 0.
This suggests that the above idea for the construction of the estimator can still be used provided n is large enough. Specifically, if F n denotes the empirical cdf related to this sample and q n denotes the related empirical quantile function:

F n (x) = 1 n n i=1 I{X i ≤ x} and q n (α) = inf{t ∈ R | F n (t) ≥ α} = X nα ,n in which X 1,n ≤ • • • ≤ X n,n
are the order statistics of the sample (X 1 , . . . , X n ) and • is the ceiling function, we set

R AE g,βn (X) := X nβn ,n 1 0 s -γn dg(s) (2)
where γ n is any consistent estimator of γ (see for example [START_REF] Hill | A simple general approach to inference about the tail of a distribution[END_REF].

Estimation using a functional plug-in estimator

The two successive approximations

q(1 -(1 -β n )s) ≈ q(β n )s -γ ≈ X nβn ,n s -γn ,
which are at the heart of the construction of the AE estimator, may both introduce substantial errors. Our idea now is to introduce an alternative estimator obtained by making a single approximation, which we can expect to perform better than the AE estimator. We consider the statistic obtained by replacing the function

s → q(1 -(1 -β n )s) by its empirical counterpart s → q n (1 -(1 -β n )s) = X n(1-(1-βn)s) ,n in (1)
. This yields the estimator

R P L g,βn (X) = 1 0 q n (1 -(1 -β n )s)dg(s) (3)
which we call the PL estimator. Contrary to the AE estimator, the PL estimator is well-defined and finite with probability 1, and does not require an external estimator of γ. If g is further assumed to be continuous on [0, 1], this L-statistic takes the simpler form

R P L g,βn (X) = X nβn+1,n + n(1-βn)-1 i=1 g i n(1 -β n ) [X n-i+1,n -X n-i,n ].
In our first main result we prove the asymptotic normality of our two estimators (AE and PL) which are obtained under the restriction n(1 -β n ) → ∞. Thus, it only ensures that our estimators consistently estimates so-called intermediate (i.e. not "too extreme") Wang DRMs, in the sense that the order of the smallest quantile that it takes into account must converge sufficiently slowly to 1. In other words, our estimators should only be used to estimate those risk measures above a lower threshold q(β n ) that belongs to the range covered by the available data. The ultimate goal of our study is to remove this restriction by showing how our extreme-value framework and the expression of our estimators make it possible to use the extrapolation methodology of [START_REF] Weissman | Estimation of parameters and large quantiles based on the k largest observations[END_REF] in order to estimate proper extreme Wang DRMs. Our extrapolated estimators are based on the same idea: to estimate the quantity of interest at an arbitrarily extreme level, this quantity is estimated first at an intermediate level where an estimator is known to be consistent, and then multiplied by an extrapolation factor which depends on an external estimate of the tail index γ.

Our second main result examines the asymptotic distribution of this type of estimator.

Simulation study

The finite-sample performance of our estimators is illustrated on a simulation study, where we consider a couple of classical heavy-tailed distributions (the Fréchet distribution and the Burr distribution) and the following three different distortion functions :