Andrey V Kudryavtsev
email: andrey.kudryavtsev@femto-st.fr

Guillaume J Laurent
email: guillaume.laurent@ens2m.fr

Cédric Clévy
email: cedric.clevy@femto-st.fr

Brahim Tamadazte

Philippe Lutz

Characterization of Model-based Visual Tracking Techniques for MOEMS using a New Block Set for MATLAB/Simulink

Microassembly is an innovative alternative to the microfabrication process of MOEMS which is quite complex. It usually implies the usage of microrobots controlled by an operator. The reliability of this approach has been already confirmed for the micro-optical technologies. However, the characterization of assemblies has shown that the operator is the main source of inaccuracies in the teleoperated microassembly, so there is a great interest in automating the microassembly process. One of the constraints of automation in microscale is the lack of high precision sensors capable to provide the full information about the object position. Thus, the usage of visualbased feedback represents a very promising approach allowing to automate the microassembly process. The purpose of this paper is to characterize the techniques of object position estimation based on the visual data, i.e. visual tracking techniques from the ViSP library. These algorithms allows to get the 3D object pose using a single view of the scene and the CAD model of the object. The performance of three main types of modelbased trackers is analyzed and quantified: edge-based, texturebased and hybrid tracker. The problems of visual tracking in microscale are discussed. The control of the micromanipulation station used in the framework of our project is performed using a new Simulink block set. Experimental results are shown and demonstrate the possibility to obtain the repeatability below 1 µm.

I. INTRODUCTION

The needs for highly integrated, complex and smart MOEMS keep increasing especially for instrumentation and biomedical tools application fields [START_REF] Tolfree | Commercializing micro-nanotechnology products[END_REF]- [START_REF] Syms | Micro-opto-electro-mechanical systems alignment stages with vernier latch mechanisms[END_REF]. The main lock for these developments relies in microfabrication process that are quite complex. To tackle this lock, several innovative alternatives are emerging, among them the use of microrobots to achieve robotized microassembling of elementary blocks.

Several works already established the viability of this approach and notably show that the key feature relies in the capability of the system to achieve modular and highly accurate assembling, i.e. typically smaller than 5 µm (maximum acceptable error) [START_REF] Dechev | Microassembly of 3-d microstructures using a compliant, passive microgripper[END_REF], [START_REF] Agnus | Robotic microassembly and micromanipulation at femto-st[END_REF]. Several concepts of microoptical benches to be assembled have also been proposed [START_REF] Bargiel | Towards micro-assembly of hybrid moems components on a reconfigurable silicon free-space micro-optical bench[END_REF]- [START_REF] Aljasem | Scanning and tunable micro-optics for endoscopic optical coherence tomography[END_REF]. In [START_REF] Clévy | Positioning accuracy characterization of assembled microscale components for micro-optical benches[END_REF], it is also established that a positioning accuracy smaller than 1 µm can be achieved in teleoperated mode. The operator being the main source of inaccuracies [START_REF] Clévy | Positioning accuracy characterization of assembled microscale components for micro-optical benches[END_REF], there is a great interest in automating the micro-assembly process: increasing the yields throughput but also quantifying the main sources of inaccuracies which is of great interest for the design of MOEMS blocks and microrobots, the clean room fabrication and assembly strategies. Fig. 1: a) Microcomponent picking using the microgripper; b) example of assembled microoptical bench [START_REF] Clévy | Positioning accuracy characterization of assembled microscale components for micro-optical benches[END_REF].

The main objective of the described works is to achieve automated assemblies of MOEMS through visual servoing approach and than study and characterize its precision, repeatability, etc. To achieve this goal, a strategy based on high level closed-loop vision control will be implemented. The studied methods are model-based visual tracking algorithms from the ViSP library, which is able to directly provide the 3D object pose using a single view of the scene [START_REF] Marchand | Visp for visual servoing: a generic software platform with a wide class of robot control skills[END_REF].

The components to be assembled are displayed in Fig. 1. The assembly consists in insertion of holder in the V-groove guiding rails of silicon baseplate using a micromanipulation station that will be described in Section II.

Previous works on the field of automatic microassembly demonstrate the possibility of model-based visual trackers application [START_REF] Tamadazte | Cad model-based tracking and 3d visual-based control for mems microassembly[END_REF]. However, these strategies can not be employed directly in our case, because of several constraints. Firstly, all objects of the scene are made of silicon which causes the reflections. Secondly, the object contains the deformable parts. Finally, the ratio between length and thickness is very high. Thus, the goal of this paper is to implement CAD-based visual tracking techniques and to quantify its performances for further automation of microassembly. For this, first of all, we should discriminate the noises : related to the algorithm, to the environmental factors and to the robot structure. Than, we will quantify the tracker precision by using a 3D trajectory. To simplify the analysis of microassemblies, the software parts (micromanipulation control, visual tracking) will be developed in the form of Simulink blocks in C++ language.

The remainder of this paper is organized as follows : Section II presents the used equipment, micromanipulation station in particular. In Section III, we present the software architecture allowing to interface different equipment such as translation and rotation stages, camera, etc. Sections IV to VI of this article describe the steps of trackers analysis and performances quantification. Finally, conclusions and prospects are discussed at the end.

II. EXPERIMENTAL SETUP

For the precise control of position and the alignment of optical path, we use the 3D microassembly station that comprises a serial robot of 4 degrees of freedom (XYZΘ) with a microgripper and a vision system (Fig. 2). The whole system is mounted on an antivibration table. The characteristics of the robot and vision system are represented in Tables I and II respectively.

One of the most important parameters, that was considered during the choice of the vision system, is a depth of field, because it is indispensable to have a focused image in the whole area of object displacement. The depth of field of presented vision system is about 2 mm.

The position of the camera in relation to the robot is represented in Fig. 3.

III. SIMULINK BLOCKS DEVELOPMENT

Simulink is a data flow graphical programming language tool for modeling, simulating and analyzing multidomain dynamic systems. Its primary interface is a graphical block diagramming tool and a customizable set of block libraries. It is widely used in automatic control, aerospace engineering, signal processing and computer engineering applications [START_REF] Xue | System Simulation Techniques with MATLAB and Simulink[END_REF]. In many fields, MATLAB/Simulink has already become the number one simulation language. Furthermore, the creation of custom Simulink blocks is possible using a MATLAB Sfunction. Custom code may be written in M, in C or in Fortran languages.

Camera Robot X w Z w Y w Θ w X c Y c Z c
The development of software in the form of Simulink blocks has several advantages. Firstly, it allows creating one block which exercises one function, thus, the system becomes modular. Secondly, the fact of working in a Simulink environment allows the usage of all already existing blocks, such as mathematical functions, scopes, advanced control laws, filters, etc. Thirdly, it became possible to use the dedicated functions such as "Robotic toolbox" developed by Peter Corke; it is a software package that allows a MATLAB user to readily create and manipulate data types fundamental to robotics such as homogeneous transformations, quaternions and trajectories [START_REF] Corke | A robotics toolbox for matlab[END_REF]. Finally, as MATLAB/Simulink is widely used in a field of engineering, the developed blocks can be easily integrated in other projects. For all these reasons, we have chosen to develop the Simulink blocks for robot control and vision.

B. Blocks for Robot Control

In order to integrate the possibility of robot control in MATLAB/Simulink, a custom block set has been developed. It contains the blocks allowing the entire control of SmarAct and PI Mercury stages [START_REF]Smaract official website[END_REF] [16]:

• two types of blocks for a stage : one block for position control and one block of speed control. There is a possibility to impose the displacement limits for every movement which allows to secure the station, in particular fragile components such as microgripper;

• one block for microgripper control. It allows to control the two fingers of microgripper either simultaneously or separately;

• one block which provides access to a joystick. It provides a possibility of teleoperation control of the station. The code of this block is based on SDL 2.0 (Simple DirectMedia Layer) library [START_REF] Lantinga | Simple directmedia layer[END_REF].

Fig. 4 represents an example of control of SmarAct stage.

The block set contains also the blocks of 3D model-based tracking that will be described in the following section. All of the blocks of the library are represented in Fig. 5.

C. Blocks for Tracking

In order to implement the tracking techniques, the C++ ViSP library was used. ViSP standing for Visual Servoing Platform is a modular cross platform library created by Inria Lagadic team that allows prototyping and developing applications using visual tracking and visual servoing techniques [START_REF] Marchand | Visp for visual servoing: a generic software platform with a wide class of robot control skills[END_REF].

The tracking techniques studied in this article use a 3D model of the tracked object to provide 3D information of the object position from a monocular image. They can be divided into three main classes described in [START_REF] Pressigout | Real-time hybrid tracking using edge and texture information[END_REF]: • Edge-based tracking. It relies on the high spatial gradients outlining the contour of the object or some geometrical features;

• Texture-based tracking, which uses texture information for object tracking;

• Hybrid tracking. It is appropriate to track textured objects with visible edges.

For every type of tracker one block was created, thus, three blocks.

IV. TRACKERS ANALYSIS IN A STILL FRAME

The first stage of pose estimation analysis consisted in tracking of the object in a still frame (Fig. 6). It should be noted that even with a still frame, neither the edge-based tracking nor the hybrid method are not perfectly stable. The tracking results obtained using the blocks displayed in previous section are represented in the form of standard deviation in Table III.

0 • 0.0194 • pitch 0.0068 • 0 • 0.0152 • yaw 0.0014 • 0 • 0.0252 •
The noise of X and Y translation coordinates does not exceed 120 nm which represents an error of less than 30% of the image pixel size. This noise level represents a maximal limit of tracker performance for our application. The predominant noise on the Z coordinate can be explained by the fact that the focal distance is considerably higher than the size of the sensor, and the pinhole projection model becomes close to a parallel one, thus, it is more difficult to estimate the depth coordinate. The problem of depth coordinate estimation with long focal distance has also been established for SEM environments where assumption of parallel beams is close to reality [START_REF] Cui | Calibration of scanning electron microscope using a multi-images non-linear minimization process[END_REF] [START_REF] Cui | 6-dof automatic micropositioning using photometric information[END_REF]. In this section, the object pose is recorded while the robot is not moving. This experiment allows to quantify the influence of the environmental factors such as luminosity and temperature change during the experiment, vibrations of axes of robot and of camera support, human presence in the room, etc.

The results (Table IV) were obtained from the video stream with the object fixed in the microgripper and the pose is recorded over 1000 images with 30 frames per second. The results confirm the observations made before concerning the imprecision of the measurements along the Z axis of camera frame. The standard deviation of measured coordinates X and Y has become bigger comparing to the results with a still frame. However, this level of noise is not suprising. The order of magnitude is similar to the results of noise characterization in millimeter sized micromanipulation systems presented in [START_REF] Boudaoud | Noise characterization in millimeter sized micromanipulation systems[END_REF]: the vibration at the free end of the cantilever of 30 mm subject to the environmental noise on an anti-vibration table and with human activity reaches 123.7 nm. Thus, since the distance between the microgripper tip and the point of robot mount is about 20 cm in our system, it may explain that the noise level can reach 700 nm.

VI. TRAJECTORY TRACKING

Next step of tracking analysis consists in comparing the measurements between proprioceptive robot sensors and the visual sensor. In this section we will use the hybrid tracker to estimate the position of the object, which gave the best results during the previous experiments. The robot control is performed by applying a position set-point to the robotic axis and in order to simplify the analysis, the rotations are not taken into account. The comparison is performed by using the trajectory represented in Fig. 7, where the curves are represented in the world frame.

Thereafter, we use the following notations :

• i -image number;

• R c -camera frame;

• R w -world frame;

• t c (i) = {x c y c z c } T -object translations in the camera frame obtained with the tracker;

• t w (i) -object translations in the world frame (proprioceptive robot sensors);

• c t w (i) = {x w y w z w } T -
object translations in the world frame transformed to the camera frame;

In order to be able to compare the results, it is necessary to transform object coordinates in the world frame t w (i) to the camera frame c t w (i). This transformation can be represented by a matrix c M w which contains the information about frame rotations and translations:

c t w (i) = c M w t w (i) (1)
The c M w matrix, which represents the extrinsic parameters of the camera, is not known. In order to estimate it, an optimization algorithm was used. The goal of optimization consists in minimization of the distance d between the coordinates obtained from the tracker t c and the sensor values of robot axis c t w . The optimization criteria is then defined as a sum of squared distance between two curves:

d(i) = t c (i) -c M w t w (i) (2)
J = n i=1 d T (i) A d(i) (3)
The previous analysis of obtained coordinates deviation between two curves allows to notice that Z c coordinate estimated by the tracker is not only affected by noise but contains no real information about the object position, which means that any filter would be useless. So, in order to minimize its influence, the coefficient of 0.001 was applied for the Z coordinate.

A = 1 0 0 0 1 0 0 0 0.001
The used optimization algorithm is a Levenberg-Marquardt algorithm that is implemented in the Mathworks Optimization Toolbox [START_REF] Prajna | Sostools: sum of squares optimization toolbox for matlab-users guide[END_REF] which can be used directly due to the fact that we are working on MATLAB/Simulink environment. It was programmed to take the best fit on 20 optimizations from random initial transforms. The results are represented in Fig. 8 and in Table V.

The standard deviations of errors between the tracker and the robot joint coordinates attain now 2.8 µm along X and 4 µm along Y axis. These deviations include intrinsic robot positioning errors which can typically reach several µm of amplitude. For example, M-111-DG translation stages have 150 µrad (Physik Instrumente company datasheet) yaw deviation which in our case (20 cm long robot end-effector) may induce an error of 3 µm. shown that this kind of robot have typical positioning accuracy of a few µm and that the main sources of positioning errors are backlash, perpendicularity errors and yaw deviations [START_REF] Tan | Characterization and compensation of xy micropositioning robots using vision and pseudo-periodic encoded patterns[END_REF].

VII. CONCLUSION

The creation of Simulink blocks in a standard programming language C++ allows to simplify the work with the equipment : program becomes graphic and modular and all already existing tools of MATLAB/Simulink can be used.

The problems of visual tracking in a micro-scale were studied through this paper. The analysis of trackers in different situations allowed to estimate the instability of tracking algorithm, environmental factors influence, precision of robot axis. In our setup, the performance of the algorithm can attain 0.12 µm, the noise level due the environment is about 0.7 µm. Finally, the robot axis imperfections add about 2 µm of uncertainty. Thus, the sum gives a value of 2.82 µm that corresponds to the results in Table V. Obtained results prove that it is possible to have a precision better than 1 µm for X and Y coordinates using the visual tracking techniques presented in this paper. It meets the specifications declared for assembly accuracy and represents a great interest for multi-Dof position measurements at the microscale. These vision tracking methods of object position estimation can be directly applied to many kinds of applications such as in the MEMS field. For example, they could be used to measure the position of the micromirror presented in [START_REF] Espinosa | Piston motion performance analysis of a 3dof electrothermal mems scanner for medical applications[END_REF] where sensor integration is strongly limited by available space constraints.

The trackers performance may be improved by using a texture or a periodic structure on the object sides.

Through experimental results, we also identified a nonvalidity of data obtained from the tracker concerning the Z c axis of camera which is due to the fact that the focal distance is considerably higher than the size of a sensor. There are several approaches allowing to compensate this effect. First of all, it is possible to use robot sensors and reconstruct the Z c coordinate, however, the estimation will be limited by stages precision. Secondly, a second camera may be used. However, the stereoscopic vision setup must be calibrated and for threedimensional reconstruction to be possible, the location and orientation of the cameras at different capture instants must be accurately known [START_REF] Roth | Robust object pose estimation from feature-based stereo[END_REF]. Finally, the system may be completed by unidirectional sensors capable to estimate Z c coordinate such as laser sensors.

ACKNOWLEDGMENT

These works have been funded by the Franche-Comté region, partially supported by the Labex ACTION project (contract "ANR-11-LABX-01-01") and by the French RENATECH network through its FEMTO-ST technological facility.

Fig. 2 :Fig. 3 :

 23 Fig. 2: Configuration of the micromanipulation station with four DOF.

Fig. 4 :

 4 Fig. 4: Usage example of a block of rotation stage angle control

Fig. 6 :

 6 Fig. 6: Different types of tracking : a) edge-based; b) texturebased; c) hybrid.

Fig. 5 :

 5 Fig. 5: Overview of the blocks of simulink library developed to control the micromanipulation station in teleoperated or automated mode

Fig. 7 :

 7 Fig. 7: 3D Trajectory in the world frame to be tracked (in mm).

Fig. 8 :

 8 Fig.8: Experimental comparison between the results from the visual sensor t c (red) and from robot sensors c t w (blue) in R c using hybrid tracker;

TABLE I :

 I Characteristics of the stages comprised in the robot used in a micromanipulation station.

		Product reference	Specifications
	Translation stages :
	XY	M-111-DG	Stroke : 15 mm
		PI Mercury	Backlash : 2 µm
			Min. Inc. Motion : 0.05 µm
			Unidir. repeatability : 0.1 µm
	Z	M-121-DG	Stroke : 25 mm
		PI Mercury	Backlash : 2 µm
			Min. Inc. Motion : 0.05 µm
			Unidir. repeatability : 0.1 µm
	Rotation stage :
	Θ	SR3610S	Stroke : 360 •
		SmarAct	Resolution : < 10µ •

TABLE II :

 II Characteristics of the vision system.

		Product reference	Specifications
	Camera :
	IDS uEye	UI-3480CP	CMOS Rolling Shutter
		Aptina	Pixelpitch : 2.2 µm
			Pixel Class : 5 Megapixel
			Resolution (h x v) : 2560x1920
	Objectif	CVO GM10HR35028	Class : high resolution
			Focal distance : 50 mm

TABLE III :

 III Standard deviation of position measurement for different trackers in a still frame for 1000 iterations in the camera frame.

	Coordinates	Edge-based	Texture-based	Hybrid
	X	0.031 µm	0 µm	0.117 µm
	Y	0.059 µm	0 µm	0.010 µm
	Z	3.204 µm	0 µm	0.760 µm
	roll	0.0071 •

TABLE IV :

 IV Standard deviation of position measurement for different trackers with a static object in the camera frame.

	Coordinates	Edge-based	Texture-based	Hybrid
	X	0.2245 µm	0.7861 µm	0.2551 µm
	Y	0.7026 µm	0.8650 µm	0.6207 µm
	Z	24.3304 µm	44.1736 µm	14.8640 µm
	roll	0.0859 •	0.1607 •	0.0610 •
	pitch	0.0577 •	0.1254 •	0.0539 •
	yaw	0.0608 •	0.0984 •	0.0461 •

TABLE V :

 V Standard deviation error obtained between visual hybrid tracker and robot sensors in R c

	Coordinates	Standard deviation error
	∆x	2.8165 µm
	∆y	4.0918 µm
	∆z	215.8632 µm

  Tan et al. have also recently

		0.6
		0.4
	X (mm)	0 0.2
		-0.2
		0 -0.4	200	400 Image(t) 600	800	1000


Xin Xu

Joël Agnus

Micky Rakotondrabe

Development and characterization of a new silicone/platine based 2-DoF sensorized end-effector for micromanipulators

This paper reports the development and the characterization of a new silicone end-effector with integrated sensor based on a set of platine piezoresistive gage. Used as interface between a micromanipulator or actuator and a manipulated small object, the features of the end-effector are: 1) its compactness (length between 750µm and 2mm, width=40µm, thickness=25µm), 2) the fully integrated force measurement system thanks to microfabricated platine piezoresistive gage, 3) the 2-dof (degrees of freedom: along y and z axis) measurement capability, 4) the high sensitivity of measurment provided for each axis (0.6% to 0.8% for the range of measured force of 1mN), 4) and the relatively high gauge factor (G=6). This paper reports the principle, the development, the microfabrication and the characterization of the 2-dof sensorized end-effector. silicone/platine (Si/Pt) end-effector, sensorized end-effector, Platine strain gage, micromanipulation, microfabrication, 2 degrees of freedom, force measurement.

-INTRODUCTION

In micromanipulation and microassembly applications and in precise positioning in general, the development of microactuators and of integrated actuated microsystems and miniaturized systems has been raising since more than a decade. These existing systems permit to position small objects with a micrometric or even a submicrometric resolution. Among the most used systems are piezoelectric materials based microgrippers [START_REF] Driesen | Micro manipulating by adhesion with two collaborating mobile micro robots[END_REF][START_REF] Kemper | Development of a tactile low-cost microgripper with integrated force sensor[END_REF][START_REF] Menciassi | Force feedback-based microinstrument for measuring tissue properties and pulse in microsurgery[END_REF][START_REF] Rakotondrabe | Development and Force/Position Control of a New Hybrid Thermo-Piezoelectric microGripper dedicated to micromanipulation tasks[END_REF] . These piezoelectric systems are well appreciated due to the high bandwidth (up to several hundreds of Hz) and the very high resolution of positioning (submicrons) they can offer. Furthermore, the fact that their power is electrical makes their design easier. Finally, piezoelectric materials can also be used as sensors, a property which has been used to develop actuators with selfmeasurement capability (called self-sensing) in [START_REF] Rakotondrabe | Simultaneous displacement and force self-sensing in piezoelectric actuators and applications to robust control of the displacement[END_REF] .

Unfortunately, piezoelectric based systems are prone to nonlinearities (hysteresis and creep) which make them finally lose the accuracy although their high resolution. Furthermore, due to the high Q-factor of most of piezoelectric structures and actuators, the microsystems that enclose them exhibit badly damped oscillation which compromise their stability. All these characteristics result in unsuccessufl micromanipulation and microassembly tasks. In order to improve the performances of these tasks, closed-loop control techniques have been carried out and successfully applied. These techniques include standard PID (proportional integral derivative) structure or advanced methods such as H-inf, passivity, interval based synthesis and adaptive scheme (see [START_REF] Khadraoui | Interval Modeling and Robust Control of Piezoelectric Microactuators[END_REF][START_REF] Leang | Hysteresis, creep, and vibration compensation[END_REF][START_REF] Devasia | A survey of control issues in nanopositioning[END_REF] and references herein). However, these control techniques dealt with positioning and force control is not yet well settled. In fact, while control of positioning in mono and multiple degrees of freedom (DoF) is extensively studied, control of force is not, due to the lack of convenient sensors. Visual feedback technique can be used to estimate the force applied by an actuator acting on a manipulated object but this requires a precise knowledge of the property of the object or of the manipulator in order to have a precise estimate of the force. Another technique consists in using integrated gage on the actuator itself but if an end-effector is used for the manipulation, the measurement from the gage could be erroneous if the end-effector's model is not accounted for. External force sensors based on capacitive structures have also been developed [START_REF]The femtotool company[END_REF] . While their measurement resolution can be very high (up to tens of NanoNewton), the actual versions are hardly utilizable when the actuators are manipulatinig objects. Furthermore, the above mentionned force measurement techniques utilized in piezoelectric actuators are mostly adapted for one-DoF measurement, i.e. measurement in one axis.

In order to go towards the automation of these actuated microsystems and then to improve the reliability of the above applications, it is necessary that both the position and the force signals brought into play be measured for the feedback. While the study and development of position sensors completely integrated within the actuated microsystems have been emerging these last years [START_REF] Rakotondrabe | Characterization of a 2-dof electrothermal MEMS with integrated actuation and sensing capability[END_REF][START_REF] Chow | Displacement sensing of a microelectrothermal actuator using a monolithically integrated thermal sensor[END_REF] , that of integrated force sensors is not really enrolled. In our previous work [START_REF] Agnus | Sensorized silicone finger tip[END_REF] , we have demonstrated that combining silicone (Si) as structures and platine (Pt) materials as gage permitted to develop sensorized end-effectors with force sensing that could yield sub-microNewton resolution. The sensorized end-effector is utilizable as mechanical amplifier and adapter between a actuator and a manipulated object as pictured in Fig. 1 which depicts an example of one axis force measurement. However, these preliminary results were valuable for one DoF measurement. This paper is motivated by the need to extend the previous Si/Pt sensorized end-effector to 2-DoF in order to permit the measurement of force in two directions during micromanipulation or microassembly applications with sub-microNewton resolution. Several 2-DoF sensorized end-effectors with different dimensions have therefore been microfabricated, with this paper reporting the principle of functioning, the steps of microfabrication and their characterization. The remainder of this paper is organized as follows. Section-2 presents the principle of the sensorized end-effector. In section-3, after quickly reporting the microfabrication process, the fabricated structures with different dimensions are presented. Section-4 is devoted to the characterization of the fabricated sensorized end-effectors. Finally, conslusions and perspective works are provided in section-5.

-PRINCIPLE OF THE Si/Pt SENSORIZED END-EFFECTOR

Fig. 1-a shows the CAD-scheme of the Si end-effector equipped with two embedded Pt strain gage and placed onto an actuator. In our applications, the actuator is a piezoelectric cantilevered actuator capable of bending along two axis (Y and Z). The end-effector serves as interface between the actuator and the manipulated object in order to adapt with the shape of the latter (spheric, parallelepiped,...). When an object is manipulated, both the object and the end-effector are (more or less) deformed. According to the types of manipulated objects, to the type of the actuators and to the range of force to be measured, it is important to well choose the compliance of the end-effector. Thanks to the embedded strain gage sensors, this end-effector can measure with a high resolution the manipulation force both along Y-axis and Z-axis. Fig. 1.b pictures a principle scheme of the emplacement of the measurement gage. Two pairs of Pt gage are proposed: the two gage of each pair are distanced of d and c from the central axis of the end-effector and are connected in parallel allowing a high sensitivity measurement along Z-axis (out-of-plane measurement). On the other hand, the two pairs of gage are placed as far as possible from the central axis in order to maximize the sensitivity measurement along Y-axis (in-plane measurement). More precisely, when a force is applied by the end-effector to an object along the Y-axis, it is deformed in the same direction. Then, the two gage in the left side of the X-axis (see Fig. 1.b) are compressed while the two gage in the right side are expanded. By utilizing the difference of resistivity variation of the left and right sides, the force along Y-axis can be estimated. On the other hand, a force along the Z-axis results in the deformation of the endeffector along the same axis, and then a simultaneous compression or simultaneous expansion of the four gage. Hence, the summation of the four resistivity variations provides an estimation of the force along Z-axis. Notice that the four platine pizoresistive sensing elements are placed on the maximum stress area of the deformable silicone structure, i.e. near its embedded extremity.

-MICROFABRICATION OF SENSORIZED END-EFFECTORS

-Microfabrication process

Fig. 3 summarizes the microfabrication process which is splitted into four steps summarized as follows. The first step consists in the settling of the platine gage on the upper surface of the silicone wafer by using Pt sputtering and lift-off techniques. In step-2, the electrodes are settled by using Alu sputtering and lift-off techniques. Finally, step-3 and step-4 consist in structuring the Si end-effector by etching (DRIE) successively the back side and the front side of the wafer.

-The fabricated end-effectors

Fig. 4 (left) pictures samples of fabricated sensorized end-effectors. They possess width of w=40µm, thickness of T=25µm and four different lengths (L=750µm, 1mm, 1.5mm and 2mm). Fig. 4 (right) pictures a zoom of the gage on one of the end-effectors. Two tlengths of Pt gage have been microfabricated and deposited (l=100µm to 500µm).

-CHARACTERIZATION

To characterize the sensitivity of the sensorized end-effector and therefore its performances to measure force, we use the following setup. The tip of a force sensor from Femtotools company [START_REF]The femtotool company[END_REF] is made in contact with the tip of the end-effector in order to stimulate manipulation of object. This is done by pushing the Femtotools force sensor towards the end-effector thanks to a manual micropositioning table. In the meantime, the Femtotools force sensor is used to measure the manipulation force and therefore can be used to characterize and to validate the measurement from the gage of the end-effector. The resistance variation of the gage are measured thanks to a precise digital multimeter. Notice that the different sensorized end-effectors with different lengths of gage were characterized successively.

In Fig. 5.a is depicted the resistances evolution of the gage when a force along the Y-axis (in-plane) ranging between 0 and 1mN is applied at the tip of the longest sensorized end-effector ( L=2mm) equipped with the smallest Pt gage (l=100µm). The circle-line curve corresponds to the resistance of the two gage of the left side while the triangle-line curve corresponds to the resistance of the two gage of the right side. In Fig. 5.b is depicted the resistances evolution when a force along the Z-axis (out-of-plane) is applied to the same sensorized end-effector. From these curves, we observe that the left gage and the right gage have almost the same behavior and are all linear. These figures yield that the sensitivity (

ΔR R

) of the sensorized end-effector is of about 0.21% for a range of 1mN in the Yaxis and of about 0.76% for a range of 1mN in the Z-axis. The corresponding gauge factor ( G= ΔR Rε , where ε is the strain of the gage) is about G=6. The characterization of the different dimensions of end-effectors with different dimensions of strain gage demonstrate that the sensitivity can reach 0.6% and 0.8% along Y-axis and along Z-axis respectively for the range of measurement (1mN). These characteristics are in good accordance with the requirements of micromanipulation and microassembly applications.

-CONCLUSION AND PERSPECTIVES

This paper presented the development, the microfabrication process and the characterization of a 2-DoF sensorized end-effector. Based on silicon structure, the end-effector is equipped with four well placed platine gage with a high gauge factor. The sensorized end-effector is therefore able to measure micromanipulation force along two directions (Y-axis and Z-axis). The characterization demonstrates that the sensitivity of the sensors can reach 0.6% and 0.8% for a range of 1mN along the Y-axis and along the Z-axis respectively, with a gauge factor up to G=6, which are in agreement with the requirements for micromanipulation carried out at FEMTO-ST Institute. The integration of the end-effectors onto multiple DoF piezoelectric microgrippers are ongoing and tests on real micromanipulation tasks will be carried out.

Fig. 1

 1 Fig.1Principle of a piezoelectric actuator with a sensorized end-effector utilized as adapter between the actuator and the manipulated object. The end-effector is equipped with strain gage used to measure the manipulation force.

Fig. 2 .

 2 Fig.2. (a): CAD-scheme of a sensorized 2-DoF end-effector. (b): emplacement of the two pairs of strain gage permitting to measure force in two directions.

FIg. 3

 3 FIg.3 Microfabrication process of the sensorized end-effector based on Si structure and Pt gage.

Fig. 4

 4 Fig.4 Samples of microfabricated sensorized end-effector (left). Zoom on the Pt gage deposited on one end-effector (right).

Fig. 5 .

 5 Fig.5. (a): resistances evolution of the Pt gage versus the force along Y-axis. (b): resistances evolution of the Pt gage versus the force along the 2-axis.

AKNOWLEDGMENTS

This work was supported by the national ANR-Emergence MYMESYS-project (ANR-11-EMMA-006: High Performances Embedded Measurement Systems for multiDegrees of Freedom Microsystems). We acknowledge the the Labex ACTION project (contract ANR-11-LABEX-01-01), the Equipex ROBOTEX project (contract ANR-10-EQPX-44-01) and the French RENATECH network through its FEMTO-ST technological facility (MIMENTO clean room).


J Escareno

J Abadie

M Rakotondrabe

E Piat

H ∞ -based Position Control of a 2DOF Piezocantilever Using Magnetic Sensors

The article addresses the position control problem of a 2 degrees of freedom (DOF) piezoelectric cantilever by means of an embedded magnetic-based position sensor. The active part of the piezocantilever used in the experimental setup is similar to cantilevers previously developed and already used for low-frequency micro-actuators in microrobotics devices. The contribution relies on the estimation of the biaxial displacement of the piezocantilever via conventional Hall-effect (HE) sensors, reducing the mechanical complexity and cost aspects. The actual sensing approach is validated via the implementation of a real-time position control based on the H∞ scheme. In comparison with high resolution sensors, as laser or confocal chromatic (high-cost) or capacitive displacement (bulky), the actual sensorcontrol system is provides a satisfactory performance to cope with traditional micro-positioning tasks requiring a micrometer resolution. The performance of the embedded magnetic-based position sensor is evaluated, in open-and closed-loop, with respect the measurements provided by a Keyence laser sensors.

I. Introduction

Piezoelectric materials are well appreciated for the development of precise positioners. This recognition is mainly due to the high resolution, the high bandwidth and the high stiffness they can offer. Furthermore, the fact that they are powered electrically makes them easy to setup. Finally, their inherent property of physical reversibility makes them usable for sensors, or actuators or both (self-sensing). PZT materials (lead zirconate titanate) which are ceramics are the most used piezoelectric materials because they are widely available for a low cost, and their coupling coefficient is high relative to almost other piezoelectric materials. However, these materials are typified by nonlinearities (hysteresis and creep) that result in a loss of the overall accuracy of the positioners. Moreover, many structures of positioners (cantilever structures) yield badly damped oscillation at their response which may compromise their stability. To overcome these unwanted effects, different control techniques of the positioners have been proposed. While open-loop (feedforward) control techniques were able to reject the nonlinearities and to damp the oscillation with a high packageability level of the overall system [START_REF] Kuhnen | Inverse feedforwrad controller for complex hysteretic nonlinearities in smart-materials systems[END_REF][4] [START_REF] Jung | New open-loop actuating method of piezoelectric actuators for removing hysteresis and creep[END_REF][6] [START_REF] Rakotondrabe | Bouc-Wen modeling and inverse multiplicative structure to compensate hysteresis nonlinearity in piezoelectric actuators[END_REF][8] [START_REF] Janaideh | Inverse rate-dependent Prandtl-Ishlinskii model for feedforward compensation of hysteresis in a piezomicropositioning actuator[END_REF], their main limitation is the lack of 1 J. Escareno is with the Polytechnic Institute of Advanced Sciences, 7-9 rue M. Grandcoing, 94200 Ivry-sur-Seine, France 2 AS2M department, FEMTO-ST Institute, UFC-CNRS-ENSMM-UTBM, Besançon, FRANCE robustness face to model uncertainties or to external disturbance. Contrary to feedforward techniques, closedloop control techniques (feedback) can offer better robustness by rejecting the effects of the above mentionned nonlinearities and oscillation. They also permit to reach substantial overall performances (accuracy, repeatability, external disturbance rejection). Different feedback techniques have been used: PID structures with simple tuning, passivity technique, robust H ∞ technique, robust interval techniques... [10][11][12] [START_REF] Khadraoui | Combining Hinf approach and interval tools to design a low order and robust controller for systems with parametric uncertainties: application to piezoelectric actuators[END_REF]. However, feedback control of miniaturized piezoelectric based systems and positioners are limited to the lack of convenient sensors as we will detail in the next paragraph. In addition, the control of piezoelectric actuators with multiple degrees of freedom (DOF) is not well settled. The main challenge in such system is the couplings between the different axis which should be considered during the control design since they may cause an unstability of the actuator [14][15]. This article addresses the motion control problem of a 2 degrees of freedom (DOF) piezoelectric cantilevered actuator (piezocantilever) by means of an embedded magnetic-based position sensors. While Hall effect is proposed as a basis for the displacement sensors, which permits high embeddability at low cost, a H ∞ control is proposed for the control in order to maintain a performances robustness.

In closed-loop, the position of the cantilever must be measured. This can be done with several sensing approaches. Non-contact external sensors can be used. very high resolution is not critical, these sensors are usually based on distance measurement with laser-based approaches [START_REF] Rakotondrabe | Complete open loop control of hysteretic, creeped, and oscillating piezoelectric cantilevers[END_REF], [START_REF] Rakotondrabe | Nonlinear modeling and estimation of force in a piezoelectric cantilever[END_REF] or with confocal chromatic-based approaches [START_REF] Piat | Nanoforce estimation based on Kalman filtering and applied to a force sensor using diamagnetic levitation[END_REF], [START_REF] Abadie | Modeling and experimentation of a passive low frequency nanoforce sensor based on diamagnetic levitation[END_REF], [START_REF] Cherry | Analysis of a passive microforce sensor based on magnetic springs and upthrust buoyancy[END_REF]. Measurements based on the observation of periodic silicium nano-patterns with optical microscopy [START_REF] Galeano | Twin-scale Vernier Micro-pattern for Visual Measurement of 1-D in-plane Absolute Displacements with Increased Range-to-Resolution Ratio[END_REF] are also possible when large displacements have to be measured with high resolution. If very high resolution is mandatory, interferometers have to be implemented [START_REF] Boudaoud | Modeling and optimal force control of a nonlinear electrostatic microgripper[END_REF], [START_REF] Boudaoud | Noise characterization in millimeter sized micromanipulation systems[END_REF]. In vacuum environnement, electronic vision based on a Scanning Electron Microscope (SEM) is also a way to estimate displacements for nanorobotics tasks [START_REF] Marturi | Visual servoing-based approach for efficient autofocusing in scanning electron microscope[END_REF], [START_REF] Dahmen | Tracking of objects in motion-distorted scanning electron microscope images[END_REF]. The main drawback of external sensors is the size, which is significantly big compared with the micro-actuator, this fact prevents the implementation/design of compact and/or embedded control systems. The difficulty increases for applications that require multi-axial measurements. In order to overcome such limitation, sensors are integrated micro-actuators must be implemented (e.g. capacitive sensors) [START_REF] Enikov | Three-dimensional microfabrication for a multi-degree-of-freedom capacitive force sensor using fibre-chip coupling[END_REF], [START_REF] Muntwyler | Three-axis microforce sensor with tunable force range and sub-micronewton measurement uncertainty[END_REF]. Nevertheless, the resulting microsystem architecture is generally complex and difficult to tune.

The novelty and contribution of the paper consists in the estimation measurement of the piezocantilever tip displacement acquired via a magnetic-based sensing (MBS) system having embedded low-cost Hall-effect (HE) sensors. The results presented throughout the paper demonstrate that the MBS provide satisfactory performances for classical microrobotics applications. It is worth to highlight the MBS system's embeddability and low-cost. The MBS system is evaluated trough an application in robust control based on a H ∞ scheme. The experimental results show the effectiveness of the proposed sensing approach using low-cost magnetic sensors and H ∞ controller to regulate static and to track slow time-varying 2D reference signals.

The paper is organized as follows. In section-II, the experimental setup is presented. Section-III details the estimation approach used to obtain the actuator's displacements from the raw data provided by the magnetic sensors. While in the section-IV is presented the closedloop control of the piezoelectric cantilever actuator using the H ∞ robust technique. Finally section-V provides concluding remarks and perspective works.

II. Experimental Setup

The experimental setup (Fig. 1) is composed of a 2-DOF piezocantilever, two embedded magnetic sensors (Fig. 2) and two external laser sensors which are used to validate the performances of the magnetic sensors. A dSPACE and computer real-time system is used to acquire the different signals and to implement the estimator and the controller. The piezoelectric cantilever actuator is based on 36 piezoelectric layers glued themselves which permit to work at low input voltage. This cantilever is controlled by two inputs U y and U z each one limited to ±10 volts. One extremity of the cantilever is clamped while the other can bend in the 2D plane ( δ) according to the input U , as depicted on Fig. 3 and such that:

δ = δ y δ z U = U y U z (1)
The scalar input U y (resp. U z ) generates a displacement δ y along axis y (resp. δ z ). However, unwanted displacement Z (resp. Y ) due to the coupling is also observed. The behavior of the cantilever's displacement, yielded by the input U , features couplings, hysteresis and creep [START_REF] Rakotondrabe | Complete open loop control of hysteretic, creeped, and oscillating piezoelectric cantilevers[END_REF].

The set of sensors used to perform the study are: i) Two external laser sensors (LC2420 from Keyence) pointing at the tip of the piezocantilever in order displacement measurement along the y and z axes (see Fig. acquire the magnetic field. Such sensors are orthogonally arranged in order to measure the magnetic field vector B m arising from the 2D displacement of a magnet attached to the piezocantilever's tip (Fig. 2). The distance between this magnet and the HE sensors is therefore an image of the actuator's deflections. An estimated displacement ˆ δ is computed through an analytic function f ( B m ), described in section III, such that:

δ = δy δz = f (B m ) with B m = B y B z (2)
The measurement of L m and B m are simultaneously obtained by a dSpace DS1005 acquisition system. The measurements performed by the Keyence sensors will be considered as reference measurements, which means that they are supposed to be the representation of the actual piezocantilever deflections, called position or displacement in this paper. This assumption leads to:

L m = δ (3)
The characterization in open-loop of the piezoelectric actuator is shown in Fig. 4 and in Fig. 5. The characterization consists in applying a circular (U y , U z ) inputs (using sinusoidal and cosinusoidal signals) of frequency equal to 0.1Hz, corresponding to the blue solid-line in Fig. 4, while the black dot-line of Fig. 4 corresponds to the actual actuator displacement. One can notice the influence of the nonlinearities (hysteresis and creep) and the couplings arising from the simultaneous motion of the cantilever. Fig. 5 shows that magnetic field B m , resulting from the actuator's motion, follows a proportional behavior.

III. Estimation of the Piezocantilever's Displacements from Magnetic Field

The magnetic field B m measured by the HE sensors is a function of the cantilever position δ:

B m = f ( δ) (4)
This can be written as a system of two equations:

B y = g(δ y , δ z ) B z = h(δ y , δ z ) (5)
where g(δ y , δ z ) and h(δ y , δ z ) are functions described later. In [START_REF] Bancel | Three-dimensional analytical optimization of permanent magnets alternated structure Magnetics[END_REF] and [START_REF] Ravaud | Magnetic field produced by a parallelepipedic magnet of various and uniform polarization Progress[END_REF], Bancel and Ravaud propose a three dimensional analytical expression to compute the magnetic field B generated by a magnet. Such method provides an accurate estimation of B but the main drawback is that the expressions can not be analytically inverted. In our case, in order to retrieve the position of the magnet, the equations system Equ.5 is inverted such that:

δy = g (B y , B z ) δz = h (B y , B z ) (6)
In order to simplify the analytic inversion, it is convenient to use simplified expressions for g(δ y , δ z ) and h(δ y , δ z ).

We propose the following expressions:

B y (δ y , δ z ) = Aδ y + Bδ z + Cδ y δ z + D B z (δ y , δ z ) = Eδ y + F δ z + Gδ y δ z + H (7)
where A, B, . . . ,H are constant scalars to be numerically calculated during the calibration process of the HE sensors described in the next section. The inverted system giving the cantiveler position knowing the magnetic field measured by the HE sensors is:

δy (B y , B z ) = α1 C √ α2+α3 -B C δz (B y , B z ) = √ α4+α5 2(B.G-C.F ) (8)

A. Calibration of the Magnetic Sensor

The system described by Equ.8 requires a prior calibration before implementation. The calibration process consists in determining the scalar coefficients A, B, . . . ,H. The input is defined so that the cantilever's position follows a parametrized coordinates pattern covering the entire working space (Fig. 7). The frequency of the reference coordinates is low in order to avoid dynamic errors in the calibration. Then, the experimental displacement data δ y , δ z , B y and B z are acquired using the Keyence laser sensors and the HE sensors. The mean value of a 20 data set is computed for coordinate point of the pattern reducing the measurement noise in δ y , δ z , B y and B z . In order to obtain the function relating magnetic field and position it is applied a nonlinear least-squares curve fitting using the equations system Equ. 7 and the actual measured magnetic field. Fig. 6 shows the magnetic field from the identified parameters and Equ. 7 compared with its experimental measurement. This validates the calibration of the magnetic sensors. Having the parameters of Equ. 7, the inverse model (estimator) described by Equ.8 is yielded. Hence, the estimate displacements δy and δz are now available.

IV. H ∞ -based Control of the Piezo-cantilever

In this section, we use the estimate measurement δy and δz in a feedback control application. The H ∞ control technique is used for that. This permits to maintain the stability of the actuator and to improve its performances although the presence of the couplings and the nonlinearities (hysteresis and creep). The hysteresis nonlinearity yields uncertainties on the approximate linear model while the creep nonlinearity and the couplings can be considered as disturbances (see for instance [START_REF] Rakotondrabe | Modeling and robust H-inf control of a nonlinear and oscillating 2-dof multimorph cantilevered piezoelectric actuator', a chapter in 'Smart materials-based actuators at the micro/nano-scale: characterization, control and applications[END_REF]).

A. Model of the 2-DOF actuator

Assuming that the output displacements of the actuator is now available (thanks to the magnetic sensors), the 2-DOF piezoelectric actuator behavior can be described as follows [START_REF] Rakotondrabe | Modeling and robust H-inf control of a nonlinear and oscillating 2-dof multimorph cantilevered piezoelectric actuator', a chapter in 'Smart materials-based actuators at the micro/nano-scale: characterization, control and applications[END_REF]:

δ y = k y D y (s)U y + b y = G y (s)U y + b y δ z = k z D z (s)U z + b z = G z (s)U z + b z (9)
where k i is the static gain, D i (s) (such as D i (s = 0) = 1) is the dynamics and b i is the disturbance, such that i ∈ {y, z}. Due to the hysteresis, the gain k i (decribed in µm) is subjected to uncertainties. The disturbance b i includes the creep, the couplings and a part of the hysteresis also. More details on their characterization can be found in [START_REF] Rakotondrabe | Modeling and robust H-inf control of a nonlinear and oscillating 2-dof multimorph cantilevered piezoelectric actuator', a chapter in 'Smart materials-based actuators at the micro/nano-scale: characterization, control and applications[END_REF]. The identification shows that: According to the model in Equ.9, we have two linear models each one with an with external disturbance. In this section, we calculate two controllers C y (s) and C z (s) for these two systems based on the standard H ∞ technique. With this technique, it is possible to a priori account these disturbances such that performances will still be maintained in their presence (performances robustness). The calculation of the two controllers is done with the same way. Hence, to ease the reading, let us denote C(s) any of the controllers C y (s) and C z (s) and G(s) = kD(s)U any of the systems G y (s) and G z (s). Fig. 8 pictures the block diagram of the closed-loop in which δ r indicates the desired displacement.

G y (s) = -0.
1) Definition of the Specifications: The following specifications will be used for the calculation of the controller.

• Tracking performances specifications:

For any of the displacements δ y and δ z , i.e. for δ of Fig. 8, we impose the following requirements:

a settling time no more than 30ms, zero overshoot, and a statical error less than 1%.

• Disturbance rejection: It is wanted that the maximal error due to the disturbance b remain less than 2%. • Command moderation:

Finally, we also introduce a limitation of the input control U (i.e. U y and U z ) such that it does not exceed ±10V for the operating range of δ r . This operating range is set equal to ±35µm for the Yaxis and ±25µm for the Z-axis. In fact, these values are taken from Fig. 4 which provided the output displacements when applying inputs of ±10V .

2) Standard form and standard H ∞ problem: From the above specifications, three weighting functions denoted W 1 , W 2 and W 3 are systematically introduced: the first one is intended to weight the error signal ε(s) in order to account the tracking performances, the second is to account the command moderation and the third one is to weight the input disturbance signal b in order to account its rejection specifications.

Fig. 9 pictures the corresponding weighted closed-loop in which O 1 is the weighted output error signal and O 2 stands for the weighted control output, while I is the new disturbance accounting the weighting function. From this scheme, the standard scheme used for the controller synthesis is derived as depicted on Fig. the interconnection between an augmented system P and the controller C(s) to be synthesized. The input of the interconnection are composed of the exogenous signals δ r and I, while the output is composed of the weighted signals O 1 and O 2 .

The standard H ∞ problem consists in finding an optimal value γ > 0 and the controller C(s) stabilizing the interconnection in Fig. 10 and guaranteeing the following inequality:

F l (P(s), C(s)) ∞ < γ (11)
where F l (P η (s), C η (s)) is the lower linear fractionnar transformation between P(s) andC(s) defined here as follows:

O 1 O 2 = F l (P(s), C(s)) δ r I (12)
From Fig. 9, we have:

O 1 = W 1 Sδ r -W 1 SW 3 I O 2 = W 2 Cδ r -W 2 W 3 SCI (13)
where S η = (I + GC) -1 is the sensitivity function.

Using (Inequa. 11) and (Equ. 13), the standard H ∞ problem becomes into finding C(s) and an optimal value of γ such that:

       W 1 S ∞ < γ W 1 W 3 S ∞ < γ W 2 SC ∞ < γ W 3 W 2 SC ∞ < γ (14)
which is satisfied if we find a controller ensuring the following inequalities:

               |S| = σS < γ 1 W1 |S| = σS < γ 1 W1W3 |SC| = σSC < γ 1 W2 |SC| = σSC < γ 1 W2W3 (15)
where σS and σSC are the upper singular values of S and of SC respectively. To solve the problem in (Equ. 15), we use the Glover-Doyle algorithm which is based on the Riccati equations [START_REF] Glover | State-space formulae for all stabilizing controllers that satisfy an H∞-norm bound and relations to risk sensivity[END_REF] [START_REF] Doyle | Statespace solutions to standard H2 and H∞ control problems[END_REF]. The transfers are calculated from the specifications in Section. IV-B as presented in the next subsection.

3) Derivation of the weighting functions: To obtain the weighting function 1 W1(s) , the specifications of tracking performances are used. To account the statical error ε s , the settling time t r and the no-overshoot transient part, we use the weighting function as follows:

1 W 1 = K dt s + 3 εs t s + 3 tr t ( 16
)
where t rt = 30ms, ε st = 0.01 and k dt = 1. Concerning the gabarit 1 W1W3 , the disturbances rejection specifications are used. We use following gabarit for that:

1 W 1 W 3 = K d b s + 3 εs b s + 3 tr b (17)
where t r b = 10ms, ε s b = 0.02 and k d b = 1. Finally, the garabit corresponding to control moderation is written as

1 W 2 = δ max U max (18)
where U max = 10V , and δ max = 35µm for Y-axis and δ max = 20µm for the Z-axis.

C. Experimental Results

The controllers C y (s) and C z (s) have been calculated. We obtained controllers order equal to 7. They have been implemented in Matlab-Simulink. A circular reference input has been applied to the closed-loop in order to check its efficiency to track complex trajectory at slow frequencies. Figures Fig. 11 and Fig. 12 depict the results obtained with a radius of 25µm, and a frequency of 0.1Hz and 1Hz respectively. The figures show the reference to be tracked, the outputs from the magnetic sensors that are used for the feedback, and the actual outputs measured by the optical sensors. As we can see, the measurements from the used magnetic sensors well track the actual displacements. Furthermore, the closed-loop conveniently tracks the trajectory which demonstrates that the magnetic sensors can be successfully used for feedback applications. It is implicit that position regulation control is successfully assured. The experimental results demonstrate the viability of these proposed magnetic-based measuring sensors to deal with motion control of the 2-DOF piezocantilever.

V. CONCLUSION

The present paper has addressed the position control problem for a 2-DOF piezoelectric actuator using an alternative position sensing approach sensors based on magnetic field. To deal with such problem, we have implemented an embedded module using low-cost Hall-effect sensors and a magnet mounted on the actuator's tip. It is presented in detail the estimation-calibration process to obtain the position of the cantilever using the magnetic field row data. A H ∞ control algorithm is designed to overcome the non-linearities (creep, hysteresis and couplings) arising from the 2D motion of the piezocantilever. The proposed sensing approach results more attractive in terms of mechanical complexity in comparison with current position measurement techniques traduced in bulky and expensive sensors. The experimental results with closed-loop control of the actuator demonstrated the interest of the magnetic sensors in feedback applications.

Fig. 1 .

 1 Fig. 1. Experimental setup: (1) Sensor's conditioning circuit, (2) Hall-effect sensors and (3) Keyence sensors.

  1-Fig.3). The Keyence sensors provide the measurement of δ, denoted as L m . They have 10nm resolution, 100µm to 200µm of precision and more than 1kHz bandwidth. ii) Two hall-effect (HE) sensors (Micronas HAL401) embedded in the piezoelectric cantilever are used to

Fig. 2 .Fig. 3 .

 23 Fig. 2. Magnetic-based sensing system elements: piezoelectric cantilever actuator, hall-effect sensors (dimension: 4.5x2.5x1.2 mm 3 ) and magnet (dimension:1x1x1 mm 3 )

Fig. 4 .

 4 Fig. 4. Open loop response of the cantilever to a circular input U of frequency 0.1 Hz.

Fig. 5 .

 5 Fig. 5. Response Bm of the hall sensors to a circular input U of frequency 0.1 Hz.

with α 1 2 - 4 (

 124 =B 2 .E+C(B.Bz-2 By.F -B.H+2 D.F ) -B(Bz.C-C.H-2 A.B.G C +A.F +B.E-By.G+D.G) -A.B.F +B.By.G-B.D.G α 2 =(Bz.C-A.F +B.E-By.G-C.H+D.G) 2 -(4 B.G-4 C.F ).(A.Bz-By.E-A.H+D.E) α 3 =-Bz.C 2 +C 2 .H+2 A.B.G-A.C.F -B.C.E+By.C.G-C.D.G and α 4 =(Bz.C-A.F +B.E-By.G-C.H+D.G) B.G-C.F )(A.Bz-By.E-A.H+D.E) α 5 =-Bz.C+A.F -B.E+By.G+C.H-D.G

Fig. 6 .

 6 Fig.[START_REF] Croft | Creep, hysteresis and vibration compensation for piezoactuators: atomic force microscopy application[END_REF]. Analytical estimation of the measured magnetic field Bm on the entire working space of the cantilever.

Fig. 7 .

 7 Fig. 7. Parametrized motion pattern to calibrate/estimate the cantilever displacement based on the magnetic field Bm throughout the entire working space.

Fig. 8 .

 8 Fig. 8. The closed-loop system.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Weighted closed loop system.

Fig. 11 .

 11 Fig. 11. Performance of cantiliver position control while tracking a circular reference at 0.1 Hz

Fig. 12 .

 12 Fig. 12. Performance of cantiliver position control while tracking a circular reference at 1 Hz

  4329(s -6863)(s -30)(s 2 + 1.19s + 4.96 × 10 7 )

(s + 2913)(s + 27)(s 2 + 457s + 1.5 × 10 7 ) G z (s) = 1.78(s + 6838)(s + 39)(s 2 -3304s + 2.17 × 10 7 ) (s + 1063)(s + 35)(s 2 + 92s + 1.5 × 10 7 ) (10) B. Standard H ∞ robust control

ACKNOWLEDGMENT

This work was supported by the national ANR-Emergence MYMESYS-project (ANR-11-EMMA-006: High Performances Embedded Measurement Systems for multiDegrees of Freedom Microsystems), the CNRSproject MiM-HaC and the Equipex ROBOTEX project (contract "ANR-10-EQPX-44-01").


Mohamed Dkhil
email: mohamed.dkhil@femto-st.fr

Aude Bolopion

Stéphane Régnier

Michaël Gauthier

Optimization of the

come

Optimization of the size of a magnetic microrobot for high throughput handling of micro-objects Mohamed Dkhil (1,2) , Aude Bolopion 1 , Stéphane Régnier 2 , Member, IEEE, Michaël Gauthier 1 , Member, IEEE Abstract-One of the greatest challenges in microrobotic is to handle individually a large number of objects in a short time, for applications such as cell sorting and assembly of microcomponents. This ability to handle a large number of microobjects is directly related to the size of the microrobot. This paper proposes a theoretical study of the size of a magnetic microrobot maximizing its capacity of displacement. It demonstrates that there is an optimal size can be obtained, due to a trade-off between the inertial and the viscous effects. Analytical expressions of the optimal size and the related frequency of motion are derived from a simplified model to highlight the influence of the geometrical and the physical parameters of the magnetic manipulation system such as the viscosity of the liquid and the size of the workspace. A numerical simulation validates the analytical analysis and demonstrates a high displacement capacity of the microrobot (around 100 back and forth motions per second for a robot of around 20 µm in water).

I. INTRODUCTION

Magnetic actuation is used to control robots for a wide range of applications, such as minimally invasive medical procedures [START_REF] Nelson | Microrobots for minimally invasive medicine[END_REF], cells manipulation [START_REF] Kawahara | On-chip manipulation and sensing of microorganisms by magnetically driven microtools with a force sensing structure[END_REF], [START_REF] Uvet | On-chip single particle loading and dispensing[END_REF], [START_REF] Inomata | Onchip magnetically driven micro-robot for enucleation of oocyte[END_REF] and parallel assembly at the microscale [START_REF] Donald | Planar microassembly by parallel actuation of mems microrobots[END_REF]. They are based on the use of an artificial magnetic structure called microrobot that can be powered and controlled by a magnetic source [START_REF] Nelson | Microrobots for minimally invasive medicine[END_REF], [START_REF] Ivan | First experiments on magpier: a planar wireless magnetic and piezoelectric microrobot[END_REF], [START_REF] Pawashe | Two-dimensional autonomous microparticle manipulation strategies for magnetic microrobots in fluidic environments[END_REF]. The size of the microrobots ranges from a few micrometers to a few millimeters.

The magnetic microrobots are promising solution to manipulate micro-objects with high precision and rapidity. Indeed, thanks to the development of microfabrication techniques, many complex forms are manufactured (star shape, U shape, rectangular shape, etc.) [START_REF] Ivan | First experiments on magpier: a planar wireless magnetic and piezoelectric microrobot[END_REF], [START_REF] Pawashe | Two-dimensional autonomous microparticle manipulation strategies for magnetic microrobots in fluidic environments[END_REF] and investigated on the precise positioning of objects [START_REF] Sakar | Wireless manipulation of single cells using magnetic microtransporters[END_REF]. In addition, the ability to handle (position, move, sort, characterize, etc.) a large number of micro-objects is crucial. For example, biologists would greatly benefit from a microrobotic system able to characterize and to sort each cell of a whole population individually according to its mechanical and/or electrical properties. The number of cells in a population can reach several millions. Sorting a whole population thus requires to perform a large number of elementary operations, such as positioning and moving, per second. This is directly related to the ability of the microrobot to perform a large number of back-and-forth motions per second in its workspace.

No matter its geometrical shape, this dynamic capacity of displacement depends on the size of the microrobot. Indeed, the behavior of a small microrobot is ruled by viscous forces, whereas inertial effects limit the velocity of large microrobots. Therefore, there is an optimal size maximizing the dynamic capacity of displacement.

In this paper the size of the microrobot maximizing the dynamic capacity of displacement is determined. The optimization method can be applied for different shapes of microrobot, but in this work the method is illustrated for a spherical geometry of microrobot. An analytical expression of the optimal size and its related capacity of displacement are derived based on a simplified model. It highlights the influence of the geometrical and physical parameters of the non contact magnetic system on the dynamic capacity of displacement. A numerical simulation is performed to solve the dynamic model of motion without these assumptions to validate the analytical results. It shows good agreement with the analytical expressions. This paper demonstrates that a magnetic microrobot of a size of a few tens of micrometers can perform more than 100 back-and-forth motions per second in its workspace. This work will enable to develop magnetic devices with high capacity of displacement that can be used in the future for any application requiring the handling of a large number of micro-objects. This paper is organized as follows. In Section II the motion of a ferromagnetic microrobot driven by a magnetic force is modeled. In Section III the analytical expression of the optimal size of the robot is derived based on simplifying assumptions. A numerical simulation which does not require such assumptions is performed in Section IV to validate the analytical approach. Conclusions and perspectives of this work are discussed in Section V.

II. DYNAMIC MODELING A. Non contact magnetic actuation system

Most of the magnetic actuation systems present similar designs [START_REF] Bouchebout | An overview of multiple dof magnetic actuated micro-robots[END_REF] based on several coils. The optimization method proposed in this paper is illustrated on a magnetic device based on four coils using direct propulsion (the movement of the microrobot is produced by the magnetic force). The two pairs of coils enable controlled in-plane displacements (see Figure 1). The microrobot is composed of a ferromagnetic material since its magnetization is higher than paramagnetic or diamagnetic materials, leading to higher magnetic forces and thus higher velocities [START_REF] Ivan | First experiments on magpier: a planar wireless magnetic and piezoelectric microrobot[END_REF]. It is placed in a workspace filled with a liquid. An ambient environment would present a smaller viscosity but the adhesion force between the robot and the substrate would be high, leading to a poor reproducibility of the control. In addition most of the current systems are in liquid environments since it enables biomedical applications [START_REF] Sakar | Wireless manipulation of single cells using magnetic microtransporters[END_REF], [START_REF] Pawashe | Two-dimensional autonomous microparticle manipulation strategies for magnetic microrobots in fluidic environments[END_REF], [START_REF] Kim | Threedimensional control of engineered motile cellular microrobots[END_REF]. The workspace is located in the center of the magnetic actuation device. A non uniform magnetic field is created by applying a current in the coils. The size of the workspace is closely related to the size of the robot. At the microscale practical considerations also limit the size of the workspace since imaging small particles necessitate a high magnification lens which presents a small field of view. In that work it will be considered that the size of the workspace is proportional to the size of the robot.

Workspace Coil

Mobile support

Fig. 1. Non contact magnetic actuation device: four coils create a magnetic force to control the position of the robot. They are placed on a mobile support to adapt the gap between the coils to the size of the workspace.

To determine the size of the microrobot that enables the highest dynamic capacity of displacement a reference trajectory along one axis will be considered (see Figure 2). The microrobot is placed at one extremity of the workspace, on the coil axis (o, x) (position A). It is first moved to position B, at the other extremity of the workspace, using the coil 1. Then it is moved back to position A using coil 2. It is considered that the same time T is needed to perform each trajectory A → B and B → A. The total trajectory is thus performed in a time 2T .

The microrobot used is spherical (radius r) and composed of ferromagnetic material. The workspace is located at a distance d from the coils. Its size w is proportional to the one of the microrobot: w = ar. Two criteria are used to evaluate the performance of the microrobot: (i) the average velocity V m needed to perform the round trip A → B → A, (ii) the related frequency of motion f = 1 2T :

     V m = x(T ) T = ar T , f = 1 2T = V m 2ar . (1)
A dynamic model will be derive to determine these two criteria with respect to the radius r.

B. Dynamic model

Since the displacement A → B and B → A is symmetric the analysis will be focused on the motion created by one coil only. Figure 3 shows the configuration of the system when the coil 1 is switched on.

The forces applied to the microrobots are: (i) the magnetic force produced by the coils, (ii) the drag force, (iii) the gravitational force compensated by normal force of the substrate. It is assumed that the adhesion force between the microrobot and the substrate is negligible since it is in liquid environments.

The magnetic force F m applied to a ferromagnetic spherical microrobot, is given by [START_REF] Yesin | Modeling and control of untethered biomicrorobots in a fluidic environment using electromagnetic fields[END_REF]:

F m = 4 3 πr 3 ( M.∇) B, ( 2
)
where B is the external magnetic field vector which depends on the position of the microrobot (x, y, z) and the current applied in the coils, M is the magnetization vector and r is the radius of the microrobot. Soft magnetic materials (e.g: nickel and iron) are characterized by their low coercivity (H c < 10 3 A/m). They can be easily magnetized by external fields [START_REF] Yesin | Modeling and control of untethered biomicrorobots in a fluidic environment using electromagnetic fields[END_REF]. It is thus assumed in this work that the magnetic field produced by the coils induces values of external excitation H higher than H c , and thus that the ferromagnetic material is saturated. This hypothesis will be discussed in the numerical analysis.

The drag force applied to a spherical microrobot depends on the density of the fluid ρ f and its dynamic viscosity η. For Reynolds numbers smaller than 1000 it is defined by the Schiller-Naumann model [START_REF] Zeng | Wall-induced forces on a rigid sphere at finite reynolds number[END_REF]:

F d = -k 1 V -k 2 V 1.687 V V , (3)
where k 1 = 6πrη, k 2 = 1.448πr 1.687 η 0.313 ρ 0.687 f and t V = ( ẋ, ẏ, ż) is the vector representing the velocity of the microrobot.

Using the second Newton's law and a projection along the (o, x) axis, the motion the microrobot is given by:

m ẍ = F m (x, i) + F d ( ẋ) , ( 4
)
where m is the mass of the microrobot, ẍ is its acceleration and i is the current applied in the coil. Based on equations ( 2), ( 3) and ( 4) the motion of the microrobot is ruled by: ẍ + 1

τ 1 ẋ + 1 τ 2 ẋ1.687 = M s ρ p ∂B(x, i) ∂x , (5)
where

τ n = m k n , n = (1, 2)
, ρ p is the density of the microrobot and M s is the value of the saturated magnetization of the microrobot.

The behavior of the microrobot is given by equation ( 5). In order to study the impact of the microrobot size on the dynamic behavior, analytical resolution of the equation ( 5) will be performed under some assumptions in Section III.

III. ANALYTICAL ANALYSIS OF THE EQUATION OF

MOTION

In order to derive an analytical expression of the average velocity V m and of the motion frequency f defined in equation ( 1) from the dynamic model ( 5) some assumptions must be made:

1) The fluid flow is considered as a Stokes flow, thus the drag force is given by :

F d ( ẋ) = -6πrη ẋ = -k 1 ẋ. (6)
2) The magnetic field gradient is considered linear in function of the input current i [START_REF] Kummer | Octomag: An electromagnetic system for 5-dof wireless micromanipulation[END_REF]. Thus the variation of the magnetic field B as a function of the position x of the microrobot can be expressed using the Maxwell equations as follows:

∂B(x, i) ∂x = i.g(x), (7)
where g(x) is a function that characterizes the non linearity of the magnetic field gradient ∇B with respect to the position x.

3) The magnetic field gradient is considered to be linear around the operating point (x 0 = d + ar, i = i 0 ). Thus, based on a Taylor expansion, the linearized expression of ( 7) is given by:

∂B(x, i) ∂x (x -x 0 ) ∂(i.g(x)) ∂x | (x 0 , i 0 ) + i.g(x 0 ) (8) = (x -x 0 ) ∂ 2 B(x, i) ∂x 2 | (x 0 , i 0 ) + i i 0 ∂B(x, i) ∂x | (x 0 , i 0 ) . (9)
These assumptions will be discussed in Section IV. The analytical results will be compared to the numerical simulations of the equation of motion performed without making these assumptions. Based on the above equations the simplified dynamic model obtained is expressed as follows:

∆ ẍ + 1 τ 1 ∆ ẋ -α∆x = βI, (10)
where:

                       α = M s ρ p ∂ 2 B(x, i) ∂x 2 | (x 0 ,i 0 ) > 0, β = M s ρ p ∂B(x, i) ∂x | (x 0 ,i 0 ) < 0, I = i i 0 and ∆x = x -x 0 . (11)
The parameters α and β define the intensity of the magnetic force which depends on the operating point. In order to create the magnetic force, the current in the coils is controlled. Several signals of current can be applied to perform the trajectory A → B. In this work a constant value of current is used to illustrate the optimization method. Using the Laplace transform then its inverse properties, the position x(t) when the input current is i = i 0 is given by:

x(t) = x 0 + β λ + -λ - exp(λ + t) -1 λ + - exp(λ -t) -1 λ - , (12)
where λ (+,-) = -1∓ 1 + 4ατ 1 2 2τ 1 are the poles of system.

Equation [START_REF] Zeng | Wall-induced forces on a rigid sphere at finite reynolds number[END_REF] presents two behaviors for spheres of small or large radius detailed in the following paragraphs. The optimal radius r op can be approximated by the limit between these two behaviors.

A. First behavior -small radius

When the radius r of the microrobot tends to 0, the position x(t) can be simplified as follows:

x(t) = x 0 + β 0 α 0 (exp(α 0 τ 1 t) -1) , (13)
where α 0 and β 0 are respectively the values of α and β when x 0 = d. This expression represents the solution of the dynamic model given by: 1

τ 1 ∆ ẋ -α 0 ∆x = β 0 I. (14)
This equation could also be obtained from the general dynamic model [START_REF] Kim | Threedimensional control of engineered motile cellular microrobots[END_REF] by neglecting the inertial term. It can be concluded that the inertial force is negligible for small microrobots.

Based on the Taylor expansion exp(h) 1 + h and the expression of τ 1 and β 0 , the average velocity and the frequency of the motion can be expressed as follows:

V m -β 0 τ 1 = - 2ρ p β 0 9η r 2 and f - β 0 ρ p 9ηa r. ( 15
)
Note that β 0 < 0 ( see [START_REF] Yesin | Modeling and control of untethered biomicrorobots in a fluidic environment using electromagnetic fields[END_REF]), V m and f are positive functions. The modification of the workspace size (parameter a) does not change the average velocity. This is due to the high dynamic behavior: the microrobot reaches a permanent velocity instantaneously because the inertial force is negligible. The frequency of motion f decreases while the workspace size increases, and it increases linearly with respect to the radius of the microrobot.

B. Second behavior-large radius

Using an asymptotic expansion of the poles λ + and λ - for a high radius, x(t) can be expressed as follows:

x(t) = x 0 + β α cosh( √ αt) -1 . ( 16
)
This expression represents the solution of the dynamic model given by:

∆ ẍ -α∆x = βI. (17)
This model can also be obtained from the general dynamic model [START_REF] Kim | Threedimensional control of engineered motile cellular microrobots[END_REF] if the viscous force is neglected. It can be concluded that the viscous force is negligible for the spheres with a large radius.

Based on the Taylor expansion cosh(h) 1 +

h 2 2 in (16) the average velocity and the frequency of motion can be expressed as:

V m - βar 2 and f - β 8ar . ( 18
)
Note that β < 0 (see [START_REF] Yesin | Modeling and control of untethered biomicrorobots in a fluidic environment using electromagnetic fields[END_REF]), V m and f are positive functions. The average velocity (resp. the frequency of motion) increases (resp. decreases) with respect to the workspace size and the microrobot radius.

C. Optimal radius

Two different behaviors can thus be derived from equations ( 15) and ( 18): for small radius microrobots the frequency of motion f increases when the radius r increases (equation ( 15)) while for large ones the frequency decreases when the radius increases (equation ( 18)). The maximum frequency of motion is obtained for an optimal radius r op which is approximated by the intersection of the curves of f for the small and large radius. Based on ( 15) and ( 18), the values of this optimal radius can be computed by solving the equation:

8(β 0 ρ p ) 2 r 3 + 81η 2 aβ = 0. ( 19
)
To get an analytical solution of this equation ( 19), the approximation that the magnetic field gradient is constant is made, with β = β 0 . This approximation will be discussed in the next section. Under these conditions the optimal radius can be expressed by:

r op = - 81aη 2 8β 0 ρ 2 p 1 3 . ( 20
)

TABLE I NUMERICAL VALUES USED FOR THE SIMULATIONS

Microrobot Medium

Material: nickel -Liquid: water

ρ p = 8902 (kg/m 3 ) ρ f = 1000 (kg/m 3 ) M s = 4.7710 5 (A/m) η f = 10 -3 Pa.s -Liquid: blood ρ f = 1066 (kg/m 3 ) η f = 6.10 -3 Pa.s
Using the parameters presented in Table I and for a workspace size that is ten times the dimension of the microrobot (a = 10), placed at a distance d = 4 mm of the coil and for a magnetic field gradient ∇B = 3 T/m (which represents β 0 = -160 mATkg -1 from ( 10)), the optimal radius is 20 µm in water and 67 µm in blood. This radius of a few tens of micrometers offer the possibility to manipulate objects in microscale, for different applications such as cell sorting or conveying of artificial micro-objects.

IV. NUMERICAL STUDY OF MOTION

The analytical analysis has highlighted two typical behaviors, for small and large microrobots. The influence of the geometrical and physical parameters of the magnetic actuation system has been derived, and the expression of the optimal radius has been determined. However several assumptions and simplifications have been made. This section presents the numerical analysis of the dynamic model defined in [START_REF] Donald | Planar microassembly by parallel actuation of mems microrobots[END_REF], taking into account the actual magnetic field gradient and the Schiller Naumann model of the flow, to evaluate the relevance of the analytical results.

A. Experimental identification of the magnetic field

In order to perform the numerical simulation, the magnetic field gradient must be determined. Most of the magnetic manipulation systems use ferromagnetic core coils for which there is no analytical expression of the magnetic field. An identification of experimental measurements of the magnetic field is performed in this paper. Alternatively finite element modeling could be used.

Figure 4 shows the experimental system used to measure the magnetic field along the axis of the coil. Measurements were made using a teslameter (F.W Bell, Model 7010) which is based on the use of an Hall probe to measure the intensity of the magnetic field. The distance between the coil and the probe is controlled by a 3-axis manual manipulator (Newport Corporation). The current applied to the coil is set to 0.4 A which is a typical value for the setup. Figure 4 Fig. 4.

Experimental setup used for the magnetic field measurement along the coil axis: an Hall sensor measures the magnetic field. A 3 axis manipulator controls the distance between the coil and the magnetic probe.

shows the experimental measurements. The minimum value of the magnetic field is measured at a distance of 10 mm from the coil and is about 2.9 mT. Therefore, the minimum external induction is about 2308 A/m which is greater than the value of the coercive induction for a nickel sample [START_REF] Graham | Demagnetizing factors for disk samples of nonideal soft magnetic materials[END_REF], [START_REF] Mulyokov | Magnetic hysteretic properties of submicron grained nickel and their variations upon annealing[END_REF]. The magnetization of the nickel sphere is thus saturated in the whole workspace. To calculate the magnetic gradient field from the magnetic field measurements the experimental measurements are approximated by a fourth order polynomial, which is then differentiated.

B. Simulation of the motion of the microrobot

The motion of the microrobot is analysed using a numerical simulation software (Matlab) based on the dynamic model [START_REF] Donald | Planar microassembly by parallel actuation of mems microrobots[END_REF]. The numerical values used are given in Table I. Figure 5 represents the frequency of motion f simulated for a scale factor a = 10 and an offset distance d = 4 mm. This numerical solution is compared to the analytical solutions [START_REF] Mulyokov | Magnetic hysteretic properties of submicron grained nickel and their variations upon annealing[END_REF] and (18) developed in Section III also represented in Figure 5.

The trends of the analytical solution derived for small and large radius microrobots are similar to the numerical simulation. The main results of Section III are thus confirmed: (i) for small microrobots the inertial force is negligible, and (ii) for large microrobots the viscous force is negligible. Several reasons can explain the differences between the analytical and the numerical solutions. The first one is that the magnetic field gradient has been considered as constant to derive the analytical solution. The second one is that the inertia of the liquid is neglected in the simplified dynamic model. The model can be improved by considering a converted mass of the microrobot in (17).

The optimal radius obtained by the numerical simulation is equal to 22 µm in water and corresponds to a maximum frequency f max = 127 Hz. A ferromagnetic microrobot of 22 µm size will be able to perform 127 cycles of back-andforth motion per second in its workspace, and thus to handle a large number of micro-objects individually in a short time, for applications such as assembly of microcomponents and cell transportation. Figure 6 shows the analytical curves and the numerical values of the optimal radius calculated for different geometries of the workspace (parameters a and d). Based on the analytical expressions ( 15), ( 18) and (20) the increase of the scale factor a decreases the maximum frequency of motion but increases the value of the optimal radius. If the distance d between the coil and the workspace increases the microrobot is subject to a lower magnetic field gradient (β and β 0 decreases when d increases). According to ( 15), ( 18) and (20) the frequency f will decrease while the optimal radius will increase. The numerical simulation confirms these trends. However, the difference between the analytical curves and the simulated values of the optimal radius is important for large scale factors. Indeed, the analytical results are obtained assuming small displacement around the initial position (x = d + ar), which holds only for small scale factors.

The use of different liquid environments changes the behavior of the microrobots due to the modification of the dynamic viscosity η. Figure 7 illustrates this result. For viscous environments the response time of the system increases (the acceleration decreases) which corresponds to slower motions. Thus, the average velocity V m and the motion frequency f decrease. The simulated results show that the optimal radius for a blood environment is equal to three times the one for a water environment. This is in good agreement with the analytical expression (20) since a factor of 3.3 was predicted by the term η 2 3 . The numerical simulation has thus confirmed the trends of the analytical expressions obtained from a simplified model. These expressions can be used to derive rules to develop microrobots for high throughput handling of micro-objects.

V. CONCLUSION

In this work a general approach has been proposed to optimize the size of a ferromagnetic microrobot aiming to 7. Impact of the liquid medium viscosity on the frequency of motion f . For a water medium the maximum frequency is f max = 127 Hz for an optimal radius r op = 22 µm. For a blood medium f max = 84 Hz and r op = 68 µm. The scale factor a is 10 and the offset d is 4mm. maximize its dynamic capacity of displacement. The optimization method presented in this paper can be applied for different shapes of microrobot, but it has been illustrated for a spherical microrobot. The analytical approach demonstrates that there is a trade off between the viscous forces ruling the behavior of small microrobots and the inertial effects limiting the motion of large ones. An optimal radius of a few tens of micrometers approximated by the limit between the two behaviors has been derived. The frequency of motion obtained for the microrobots of optimal size is more than 100 Hz, which is highly promising for applications that necessitate to handle a large number of micro-objects. Good agreement between the analytical approach and the simulation results (that do not assume neither a constant gradient magnetic field nor a Stokes flow) is highlighted, which validates the approximation made in the analytical analysis. The influence of the scale factor, the offset distance and the dynamic viscosity has been studied. This theoretical work shows a general method that can be applied to identify the best microrobot size when developing a wireless magnetic device. Therefore, the magnetic device can be used for any application requiring to handle a large number of objects, such as cells sorting for example.

This modeling and simulation work is a preliminary work, which will be validated experimentally in future works using microrobots with sizes ranging from a few micrometers to several hundred micrometers with different shapes.

Fig. 2 .

 2 Fig. 2. Side view of the magnetic device: A reference trajectory along one axis is considered to evaluate the performance of the microrobot. The microrobot is placed at one extremity of the workspace (position A) and it is moved to position B, at the other extremity of the workspace, using the coil 1. It is then moved back to A using coil 2. a: scale factor, d: offset distance, r: radius of the microrobot, T : time needed to perform the trajectory A → B.

Fig. 3 .

 3 Fig. 3. Configuration of the system when the coil 1 is actuated : the workspace is located at a distance d from coil 1 and the microrobot moves under the action of the magnetic field B to cover the distance w = ar from A to B. F m and F d are the magnetic and the drag force respectively.

Fig. 5 .

 5 Fig.5. Frequency of motion f derived by the analytical and the numerical approaches. The maximal frequency derived by the analytical approach f max = 306 Hz is obtained for an optimal radius r op = 20 µm. The numerical simulation gives a maximal frequency f max = 127 Hz for an optimal radius r op = 22 µm. Trends of the curves and order of magnitude of the frequency and of the optimal radius are thus similar.

Fig. 6 .

 6 Fig.6.Optimal radius r op computed as a function of the workspace size a for different offsets d obtained by the analytical and the numerical approaches (A.S: analytical solution, N.S: numerical solution).

  Fig.7. Impact of the liquid medium viscosity on the frequency of motion f . For a water medium the maximum frequency is f max = 127 Hz for an optimal radius r op = 22 µm. For a blood medium f max = 84 Hz and r op = 68 µm. The scale factor a is 10 and the offset d is 4mm.

Acknowledgments

This work has been supported by the French Agence Nationale de la Recherche, through the LEMA project (contract "ANR 12 BS03 007 01"), by the Labex AC-TION project (contract "ANR-11-LABX-01-01") and by the "Région Franche-comté".