Aude Bolopion
email: bolopion@isir.fr

Barthélemy Cagneau
email: cagneau@isir.fr

D Sinan Haliyo

Stéphane Régnier
email: regnier@isir.fr

D S Haliyo
email: haliyo@isir.fr

Analysis of stability and transparency for nanoscale force feedback in bilateral coupling

Keywords: Telenanorobotics, Force feedback, Haptic coupling, Bilateral control, Nanomanipulation

This paper deals with the problem of finding a compromise between stability and transparency for bilateral haptic control in nanorobotics. While manipulating objects with an AFM, real time visual feedback is not available. Force feedback is used to compensate for this lack of visual information. The structure of the control scheme and the value of the controller gains are critical issues for stability, transparency, and ease of manipulation. Two common control schemes are analyzed for submicron scale interactions. Based on stability and transparency criteria, the influence of each of the controllers' gains is derived. The applications for which the bilateral couplings are best suited, as well as their intrinsic limitations are discussed. The theoretical analysis is validated with an experiment composed of several phases with high dynamic phenomena.

Introduction

Handling of objects at micro or nanoscales is still a challenge especially due to unavailable real time visual feedback while manipulating objects with an AFM, and the difficulty to design accurate grippers and sensors [START_REF] Sitti | Microscale and nanoscale robotics systems [grand challenges of robotics[END_REF]. Haptics appeared to be an interesting solution to deal with these objects [START_REF] Ferreira | Virtual reality and haptics for nanorobotics[END_REF], after R.L. Hollis developped the first system to feel the substrate's topology using a master arm [START_REF] Hollis | Toward a telenanorobotic manipulation system with atomic scale force feedback and motion resolution[END_REF]. Specific problems arise while dealing with haptics for nanoscale applications. Indeed, scaling factors are needed to set up a bilateral control. We will note A f (resp. A d ) the force (resp. motion) amplification factor. Based on dimensional analysis, [START_REF] Goldfarb | Dimensional analysis and selective distorsion in scaled bilateral telemanipulation[END_REF] presents a method to select these coefficients in order to minimize the environment's distortion. They should be chosen such that A f = A d for surface dominated interactions and A f = A 2 d for structurally dominated interactions. However, practical limitations, such as the device workspace or the forces that can be felt by the user, will prevent to use these relations. Scaling factors, as well as time delayed communications or discretization of signals may lead to instability. Tools to cope with such classical problems have been developped for macroscale systems. They include scattering variables and wave variables to deal with time delayed communications ( [START_REF] Niemeyer | Stable adaptive teleoperation[END_REF], [START_REF] Anderson | Bilateral control of teleoperators with time delay[END_REF]). Another common approach based on the passivity theory is to use observers to monitor the power flows in the system. Damping is added by controllers to dissipate the excess of energy when needed [START_REF] Ryu | Sampled-and continuous-time passivity and stability of virtual environment[END_REF]. These tools designed for macroscales have been applied in nanorobotics. For example, wave variables and passivity controllers have been used in simulated environments ( [START_REF] Daunay | 6 DOF haptic feedback for molecular docking using wave variables[END_REF], [START_REF] Kim | Task-based and stable telenanomanipulation in a nanoscale virtual environment[END_REF]). Recent works, including [START_REF] Boukhnifer | H-infinity loop shaping bilateral controller for a two-fingered tele-micromanipulation system[END_REF], used H ∞ theory to get a robust stability against time delays and scaling factors. [START_REF] Boukhnifer | Wave-based passive control for transparent micro-teleoperation system[END_REF] uses wave variables to ensure stability and also focuses on transparency degradation.

Indeed, the trade-off between stability and transparency is particulary difficult to deal with in nanorobotics. In [START_REF] Kaneko | Impedance shaping based on force feedback bilateral control in macro-micro teleoperation system[END_REF], the authors highlight the equivalent resultant impedance felt by the user compared to that of the environment. However, it does not deal with sta-bility. [START_REF] Onal | A scaled bilateral control system for experimental one-dimensional teleoperated nanomanipulation[END_REF] applies a passivity controller to nanomanipulation system. However, some limitations in terms of transparency are pointed out since the pull-in, which is an active phenomenon, is smoothed by the controller. Special care has to be taken while applying this controller to nanoscale applications.

Transparency and stability are the two criteria suitable to evaluate bilateral couplings' performances. The main idea of this paper is to use them to adapt the couplings from the macroscale to the nanoscale. Reminding that scaling factors strongly influence stability, our work will be based on classical controllers well known for macroscale applications in order to study the impact of such factors. The choice of these bilateral couplings is made according to the available inputs and outputs of the nano environment. Hereafter, we only use proportional and proportional-integral controllers so that an analytical analysis can be carried out, to understand the influence of each gain on the system stability and transparency. All the results will be validated through experiments with high dynamic forces. Compared to [START_REF] Venture | Forcefeedback micromanipulation with unconditionally stable coupling[END_REF], a transparency analysis is undertaken, and a comparison between two control schemes is made.

This paper is organized as follows: in section 2, we present the experiments and the experimental setup that we will use to validate the control schemes; then, the most intuitive control scheme, the Direct Force Feedback is presented in section 3. We will show its characteristics for the particular case of nanorobotics. Section 4 introduces another coupling, the Force-Position control scheme. An analysis of its stability and transparency properties is carried out, and will be used to choose the controller gains. The application fields of these couplings are also discussed.

Experimental setup

Coupling validation protocol

To compare the performance of the different couplings and the influence of the controller gains, we will perform a one-dimensional manipulation. It consists in approaching the substrate with the probe, applying a force and finally retracting from the substrate. The force applied on the cantilever as a function of the distance from the substrate is depicted in figure 1 Fig. 1 Force applied on a cantilever while performing an approach-retract experiment the instants when the cantilever is attracted (pullin) and released (pull-off) from the substrate. -Two types of interacting forces are present: the very first (B) and the last (D) are attractive (after the pull-in and before the pull-off), while those in between are repulsive (C). -The force magnitude variation is high between pullin and pull-off phenomena (factor 10 to 100). -The pull-in force magnitude is in the order of tenth of nanonewtons.

The bilateral coupling must render to the user these phenomena present at nanoscales. As its structure depends on the used devices, the experimental platform will be detailed before performing any analysis. The experimental platform used to validate the coupling is depicted in figure 2(a). It is composed of a nanotranslator (manufactured by Physik Instrumente)1 with a motion range of 12µm and a resolution of 1.83nm. It moves the substrate along the vertical direction, to approach or retract it from a fixed cantilever.

The force applied by the substrate on the cantilever (F e ) is measured with a laser. The deflection of the probe is measured using a beam focused on the cantilever, which is reflected onto a photodiode. [START_REF] Tian | 3D nano forces sensing for an AFM based nanomanipulator[END_REF] gives more details on laser optics. Then the normal force F e can be computed from equation [START_REF] Sitti | Microscale and nanoscale robotics systems [grand challenges of robotics[END_REF]:

F e = k c d = k c S V (1)
where:

-k c : stiffness of the cantilever (from a hundredth up to several dozen N • m -1 ) -calibrated as it was demonstrated in [START_REF] Xie | Calibration of lateral force measurements in atomic force microscopy[END_REF]. Note that µN • m -1 or nN • m -1 are also consistent units, more adapted to describing nanoscale phenomena -d: deflection of the cantilever -S: sensitivity of the calibrated photodiode -V : measured output voltage As highlighted in equation ( 1), the force depends on the stiffness of the cantilever. This parameter may vary significantly depending on the performed task. Manipulation operations require stiff probes to interact precisely with objects, whereas soft ones can measure smaller forces. Therefore they are suited for exploration or teaching purposes. The bilateral coupling analysis should take into account the cantilever stiffness.

The haptic device used is a 3 degrees of freedom of force feedback Virtuose, manufactured by Haption2 (figure 2(b)).

Power flow

Figure 3 summarizes the power flow between the different subsystems (operator, haptic interface, coupling, slave device and environment), and the inputs and outputs of the subsystems. F m represents the force fed back to the user, while V s is the desired velocity set to the nanotranslator.

The bilateral coupling must be designed accordingly to the subsystem's inputs and outputs (master and slave devices' characteristics, sensors available in the environment). However, it is not really restrictive since many teleoperation systems for nanoscale applications presented in the literature fulfill these requirements ( [START_REF] Onal | A scaled bilateral control system for experimental 1-D teleoperated nanomanipulation applications[END_REF], [START_REF] Ammi | Robotic assisted micromanipulation system using virtual fixtures and metaphors[END_REF], [START_REF] Ando | Micro teleoperation with haptic interface[END_REF]).

Objectives

The study of the haptic coupling properties will be based on the classical theory of automation. Our objective is to determine the influence of the different controller gains on the coupling performances. We will use stability criteria, as Routh-Hurwitz or Llewelyn, and Bode analysis of the transfer functions. The second step will be to find the parameters that will lead to the most transparent, but still stable, control scheme. This will be done according to the conclusions about the influence of each gain on the system. This proposed tuning must be robust with respect to the environment, and especially the cantilever's stiffness. Then we will be able to determine the applications for which the control schemes are the best suited, considering their intrinsic structure and properties. All these conclusions will be validated by experiments based on the protocol described in paragraph 2.1.

Direct Force Feedback

In this section the first control scheme, namely Direct Force Feedback (DFF ) is introduced and analysed. Stability (using the Routh-Hurwitz criterion) and transparency issues are considered to derive its specificities concerning nanoscale applications. Approach-retract experiments are conducted using different control parameters to verify the analysis.

Control scheme structure

This control scheme, depicted in figure 4, is the most intuitive formulation to provide amplified forces to the operator [START_REF] Goethals | Accurate haptic teleoperation on soft tissues through slave friction compensation by impedance reflection[END_REF]. Basically, the user operates a haptic device in the macro-world to impose the displacements of the slave device in the nano-world. The blocks and power flows defined in figure 3 Fe: force applied by the substrate on the cantilever Vc: velocity of the cantilever in the operational space Fop: force applied by the operator on the master arm Vm: velocity of the master arm in the operational space Fig. 3 Power flow between the subsystems

The master arm is modeled by a rigid body for which inertia and damping are respectively M v and B v . The slave robot's transfer function is modeled by a second order function with two time constants τ 1 and τ 2 :

V (s) = [(B v + M v s)] -1 N (s) = [(1 + τ 1 s)(1 + τ 2 s)] -1 (2)
where s is the Laplace variable. Numerical values of the parameters have been identified as:

M v = 0.4kg ; B v = 0.1N • s • m -1 τ 1 = 1.35 • 10 -3 s ; τ 2 = 0.57 • 10 -3 s (3)

Stability

The slave device interacts with a remote environment, which must be considered for the stability analysis. Two approaches can be used. The first is to consider a method that is applicable for any passive environments (e.g., methods based on passivity analysis). However, they are more conservative since stability is guaranteed for any environment as long as it is passive. Therefore, they are not useful for our concern to point out the limitations induced by the cantilever's stiffness or the environment's characteristics.

The second approach is to model the environment and to consider stability with respect to the specificities of both the coupling and the environment. It will be shown theoretically and experimentally that this control scheme is stable in a given context. To model the environment, assumption (1) is made:

Assumption 1
The slave device is linked with its environment through a spring constant of stiffness k c (the cantilever). The substrate is modeled as a spring k s . These two serial springs are linked such that the equivalent stiffness k eq is:

1 k eq = 1 k s + 1 k c (4)
Hertz's theory is widely used to model the contact between a cantilever and the substrate [START_REF] Maugis | Contact, adhesion and rupture of elastic solids[END_REF]. In the case of a contact between a sphere (which can be the extremity of a cantilever's tip) and a plane, it states that the contact stiffness k s is:

k s = 3 2 Ka ( 5
)
The variables are listed below, numerical values are given for a silicon cantilever and a glass substrate:

-a: contact area between the sphere and the plane a 3 = RtFe K -K: equivalent Young's modulus of the sphere and the plane K =

1 3 4 " 1-ν 2 1 E 1 + 1-ν 2 2 E 2
« -E 1,2 : Young's modulus for the cantilever and substrate (respectively E 1 = 150GPa and E 2 = 69GPa) -ν 1,2 : Poisson's ratio for the cantilever and substrate (respectively ν 1 = 0.17 and ν 2 = 0.25) -R t : sphere's radius of curvature R t = 10nm Even if the assumption (1) is not very restrictive, it is sufficient to point out the inherent problems of the proposed control scheme in the nano-world. Considering this assumption, and the control scheme depicted in figure 4, the transfer function between F op and V m can be derived. Since the system is linear-time invariant (LTI ), the Routh-Hurwitz criterion can be applied to this transfer function. A necessary and sufficient condition of stability is:

R = A f A d < γ k eq = R max (6)
where γ =

Bv(τ1+τ2)[M 2 v +MvBv(τ1+τ2)+B 2 v τ1τ2] [Mv(τ1+τ2)+Bvτ1τ2] 2
As γ only depends on the systems' parameters, for a given environment, according to equation ( 6) the system's stability only depends on the ratio

A f A d .
The worst case for the issue of stability is when the equivalent stiffness is the highest. Using equation ( 4), this corresponds to k eq = k c . In the following sections we will use this approximation.

Transparency

Transparency of haptic couplings is defined in [START_REF] Lawrence | Stability and transparency in bilateral teleoperation[END_REF], [START_REF] Hokayem | Bilateral teleoperation: an historical survey[END_REF]. It is based on the comparison between the impedance of the environment Z e = -F e /V c and that felt by the operator Z op = F op /V m . Ideal transparency is achieved when:

Z op = Z e (7)
However, for submicron scales, equation ( 7) does not make any sense. It is necessary to consider the scaling factors A f and A d such that the impedances can be compared. In our context, perfect transparency will be achieved if:

F op V m = -A f F e A d V c ⇔ Z op = A f A d Z e (8)
It has to be noted that the force sensor used modifies the profile of the measured forces, and therefore the operator's feeling. However, we will not deal with that issue, as we will focus only on the influence of the coupling and the haptic device.

Using the control scheme depicted in figure 4, the impedance felt by the operator is derived:

Z op = F op V m = A f A d Z e N (s) + 1 V (s) (9)

Contact

While in contact, the impedance felt by the operator will be that of equation [START_REF] Kim | Task-based and stable telenanomanipulation in a nanoscale virtual environment[END_REF]. According to equation [START_REF] Daunay | 6 DOF haptic feedback for molecular docking using wave variables[END_REF], this corresponds to the impedance he should ideally feel (

A f A d Z e
) modulated by the nanotranslator dynamic. It is also influenced by the haptic device characteristics.

In the frequency domain (s = jω), equation ( 9) can be rewritten as:

Z op = A f A d Z e 1 τ 1 τ 2 ω 2 + (τ 1 + τ 2 )jω + 1 + (M v jω + B v ) (10)
For low frequencies, the impedance Z DF F op,LF can be approximated by:

Z DF F op,LF ≈ ω<<1 A f A d Z e + B v (11)
The user feels the environmental impedance, as well as the viscosity of the haptic interface. However, B v can be set aside compared to

A f A d Z e .
For high frequencies, the impedance Z DF F op,HF is:

Z DF F op,HF ≈ ω>>1 M v jω ( 12
)
This result is valid as far as the stiffness of Z e is a finite value. Since Z e can be computed as Z e = keq jω , and according to assumption [START_REF] Sitti | Microscale and nanoscale robotics systems [grand challenges of robotics[END_REF]

k eq = k c , Z e ≈ ω>>1 0.
According to equation [START_REF] Kaneko | Impedance shaping based on force feedback bilateral control in macro-micro teleoperation system[END_REF], the transparency of the coupling is only affected by the inertia of the haptic device for high frequencies. The Bode's diagram corresponding to these contact impedances is plotted in figure 5. (these values will be used to perform experiments in the following sections). However, the same conclusions would have been derived for any values of these parameters.

Figure 5 confirms that the system is perfectly transparent for low frequencies (and the viscosity of the haptic interface can indeed be set aside), and affected by the haptic device's inertia for high frequencies.

Non-contact

When no forces are applied to the cantilever, the environment's impedance is Z e = 0, which, according to equation [START_REF] Boukhnifer | H-infinity loop shaping bilateral controller for a two-fingered tele-micromanipulation system[END_REF], leads to:

Z op = M v jω + B v (13)
At low frequencies, the operator will mainly feel the viscosity of the haptic device. At high frequencies, the inertia of the master arm will be predominant. This can indeed be verified in the Bode's diagram depicted in figure 6.

Using the DFF control scheme, the rendering will only depend on the haptic device characteristics, for both contact and non contact modes.

Determination of the control scheme parameters

As seen in the previous stability and transparency analyses, the control scheme parameters determines the performances of the system. Based on the previous paragraph conclusions, this section will highlight the influence of the gains on the coupling. For this control scheme, only two parameters must be tuned: A f and A d . As seen in section 3.3, the system transparency only depends on the master and slave devices, but not on the gain values. Therefore, only stability considerations must be considered. Figure 7(a) allows the user to select A f and A d taking into account the equation ( 6), for a particular value of k eq . The figure 7(b) represents the region of stability for different values of k eq . As the stiffness increases, the condition of stability becomes stricter.

A f A d keq = 2.4N • m -1 A f > γ k e q A d ⇒ i n s t a b i l i t y A f < γ k e q A d
A f A d keq = 0.05N • m -1 keq = 2.4N • m -1 keq = 48N • m -1
(b) Evolution of the stability and instability regions for different keq Fig. 7 System's stability with respect to A f and A d .

It should also be noted that A f and A d are scaling factors, and must also be chosen according to amplification requirements.

Experimental results

Contact and pull-off forces

A cantilever with a spring constant k c = 2.4N • m -1 is used. According to equation ( 6) and figure 7(a), the maximum value allowed for the ratio R to assure a stable system is:

R max = 21.7 ( 14
)
A d is chosen according to the master and slave motion ranges. To get a good compromise between easiness and accuracy of manipulation, the value A d = 0.05 • 10 6 is selected (the master motion range corresponds then to a displacement of 5µm for the slave). A f is selected so that the system remains stable, i.e. so that the Routh-Hurwitz criterion given in equation ( 6) is satisfied. The value A f = 0.12 • 10 6 will be used.

The obtained results (forces sent back to the user) are depicted in figure 8. During the first stage (C), the user applies forces on the substrate (the maximum is 0.30N). While retracting, the user must counterbalance the forces resulting from the adhesion effects (D). A force equal to -0.13N is necessary to release the cantilever from the substrate (E). The system remains stable and the forces sent back to the user are equal to those measured by the photodiode scaled by A f (F m = A f F e according to the control scheme's design). Therefore, the operator indeed feels phenomena happening in the environment. However, using our haptic interface, forces fed back remain too low to increase the accuracy of the performed task. The force scaling factor has to be increased in order to provide better force feedback.

During this second experiment, the value of A d remained the same, but A f was increased: A f = 6.0 • 10 6 .

With these values:

R = 120 > R max (15)
The Routh-Hurwitz criterion is no longer satisfied. Consequently, the system is predicted to be unstable. The results are plotted in figure 9. Remarkably, the forces have been amplified compared to the forces in figure 8. It is then easier for the user to detect repulsive and attractive forces (C, D) and the high variation of the forces due to the pulloff (E). Although force reflection has been improved by modifying the value of A f , the system's stability is affected. When the cantilever establishes contact with the substrate, it creates high amplitude oscillations in the system (O). This is very disturbing for the user who has to act like a damper to absorb the excessive energy responsible for the instability.

For a given velocity-scaling factor, the DFF control scheme suffers from a trade-off between stability and force amplification.

Pull-in, contact and pull-off forces

To render the pull-in force to the user, it is necessary to increase A f . However, as seen in equation ( 6) and demonstrated in figure 9, if A d is kept constant, the system is unstable. Thus, it should be increased to ensure stability. Since A d represents the velocity-scaling factor, this implies that the user will need to move the haptic handle over longer distances than in the experiment depicted in figure 8 to perform the same displacement. This makes the experiment very long, as it will be demonstrated later.

According to equation ( 6), the cantilever stiffness must also be considered for stability. Since F m = A f F e and F e = k c d where d is the cantilever deflection, equation (6) becomes:

A d ≥ 1 d F m γ (16)
For the same value F m fed back to the user, A d is proportional to the inverse of the deflection. Considering the pull-in phenomenon, the deflection of soft cantilevers is bigger than that of stiff ones. Consequently, the jump into contact will happen when the probe is higher from the substrate. Therefore, the condition on A d to assure stability is stricter for stiff cantilevers. We will therefore choose a softer cantilever (k c = 0.05N • m -1 ).

To be able to feel the pull-in force, we chose A f = 6700 • 10 6 and A d = 50 • 10 6 . For this cantilever, R max = 1040. Therefore, for these scaling factors, the system is predicted to be stable.

The results obtained are presented in figure 10. The forces that should have been sent back to the user, as well as those actually felt by the operator are shown (i.e., after saturation). As previously explained, the pullin phenomenon is between 10 to 100 times weaker than the pull-off. Therefore, if the scaling factors are kept constant, the force corresponding to the pull-off is strongly amplified and neither the haptic device nor the operator can cope with such forces. That is why forces are saturated to 5N, so that the manipulation remains comfortable for the user. The pull-in effect can clearly be felt since the peak amplitude is around 2N.

However, since A d has been increased in order to keep the system stable, the experiment is time consuming (one and a half minutes whereas the cantilever was soft and initially set very close to the contact point). The DFF control scheme is highly transparent, since both pull-in and pull-off phenomena were felt. However, it suffers from a trade-off between force amplification and velocity-scaling, which can result either in instabilities or time consuming experiments. Therefore, for easier manipulations, it is necessary to modify the structure of the control scheme to add some damping on the system. This can avoid instabilities that appear in figure 9 while reaching the contact.

In section 4.1, a second well-known control scheme, which takes into account these necessary conditions to avoid the trade-off pointed out above, will be compared to the DFF in the special context of micro and nanorobotics.

Force-Position control

In this section, the Force-Position (FP ) control scheme is studied. Stability analysis is conducted using discrete time variable z since it involves numerical integrations which make the system very sensitive to the sampling period. Transparency analysis is carried out in the continuous-time domain in order to compare the results with those obtained for the DFF control scheme. Based on the conclusions, we will choose the gains of the FP controller, and we will define what the applications of such a coupling are.

Control scheme structure

As for the DFF, the inputs of the Force-Position control scheme are the velocity of the haptic device handle and the force applied by the environment on the cantilever (figure 11). The outputs are the velocity used as the desired reference for the nanotranslator and the force that will be fed back to the user by the haptic device. As previously, A f and A d are respectively the force and velocity scaling factors. Discrete time formulation is used to take into account effects of the sampling period T e . z represents the discrete time variable. The star superscript is for discrete parameters. It was shown in section 3 that the DFF control scheme presents limitations in terms of stability, depending on the desired scaling factors. To improve this issue, two controllers are added in the FP coupling.

V * m V (z) F * op F * m C(z)(B f p , K f p ) Gn A d 1/A f V * c V * s N (
G n and C(z) are respectively a proportional (P ) and a proportional-integral (PI ) controller. C(z) has been discretized using Tustin's approximation:

C(z) = B f p + K f p T e (z + 1) 2(z -1) (17)
In the following, V (z) (resp. N (z)) will refer to discrete time transfer functions corresponding to V (s) (resp. N (s)). They are computed using the Z-transform function Z {.}:

V (z) = (1 -z -1 )Z V (s) s = 1 B v 1 -δ z -δ (18)
where δ = e -Bv Te Mv .

N (z) = 1 -α 1 z -1 z -e -Te τ 1 -α 2 z -1 z -e -Te τ 2 (19)
where

α 1 = 1 1- τ 2 τ 1 and α 2 = 1 1- τ 1 τ 2 .
Compared to the DFF control scheme, the feedback force F m , is computed with the PI controller. The integral gain K f p and the proportional gain B f p can be used to modify the stiffness and damping of the rendered force. The gain G n is used to compute the desired velocity of the slave device.

Stability

To derive the stability conditions for the FP control scheme, a first approach is to verify the Routh-Hurwitz criterion. This method was previously used in section 3. However, it has been noticed that the environment must be modeled. Considering that three gains and two scaling factors are used in the control scheme depicted in figure 11, it is obvious that the relationships between these parameters will be complex and will not allow to highlight the influence of each of the gains on stability. Therefore, the Routh-Hurwitz criterion is not appropriated to analyze the stability of this bilateral coupling. To assure the system's stability without modeling the environment, two main approaches are currently used. The first is passivity [START_REF] Ryu | Sampled-and continuous-time passivity and stability of virtual environment[END_REF], which deals with energy flow in the system. The second is absolute stability [START_REF] Adams | Stable haptic interaction with virtual environments[END_REF]. As for passivity, it ensures that if the control scheme is connected to passive blocks (in our case the environment and the operator which can be considered passive as in [START_REF] Hogan | Controlling impedance at the man/machine interface[END_REF]), the system will remain stable. Both of these criteria lead to sufficient but not necessary conditions. Since absolute stability is less conservative, we will use this criterion, based on Llewelyn's theorem.

The admittance matrix P of the coupling is defined as:

V * m V * c = P(z) • F * op F * e = p 11 (z) p 12 (z) p 21 (z) p 22 (z) • F * op F * e ( 20
)
where:

p 11 (z) = V (z) [A f + A d G n C(z)]/D(z) p 12 (z) = [A f A d G n C(z) V (z)]/D(z) p 21 (z) = [G n V (z) C(z)]/D(z) p 22 (z) = A f G n [1 + C(z) V (z)]/D(z) D(z) = A f + A d G n C(z) + A f V (z) C(z)
Theorem 1 (Llewelyn [START_REF] Llewellyn | Some fundamental properties of transmission systems[END_REF])

A system represented by the admittance matrix P is unconditionally stable if and only if the following conditions hold:

C 1 = Re(p 11 ) ≥ 0 (21) C 2 = Re(p 22 ) ≥ 0 (22) C 3 = 2Re(p 11 )Re(p 22 ) -|p 12 p 21 | -Re(p 12 p 21 ) ≥ 0 (23)
These inequalities will be used to verify if the chosen gains meet stability requirements.

Transparency

In order to compare the results to those obtained for the DFF, transparency was studied in the continuous time domain. Using the control scheme depicted in figure 11, the impedance felt by the operator Z op = F op /V m can be computed:

Z op = n ze Z e + A f C(s)V (s) + A f + A d G n C(s) d ze Z e + A f V (s) + A d G n C(s)V (s) ( 24
)
where:

n ze = A f G n C(s)V (s)N (s) + A f G n N (s) ( 25
)
d ze = A f G n V (s)N (s) (26)

Contact

When the contact is established, Z F P op,LF can be computed for low frequencies:

Z F P op,LF ≈ ω<<1 A f A d + A f keq K f p Z e ( 27
)
As K f p increases, the impedance felt by the operator tends to the ideal impedance

A f A d Z e .
For high frequencies, the impedance Z F P op,HF felt by the user is:

Z F P op,HF ≈ ω>>1 M v jω (28)
For the same reasons as the ones exposed in section 3.3.1, the user only feels the inertia of the haptic device for ω >> 1.

Bode's diagram represented in figure 12 is useful to illustrate the analytical results, and to compare them to those obtained for the DFF control scheme.

k eq = 2.4N • m -1 , A f = 6 • 10 6 , A d = 0.05 • 10 6 (
the same values as for figure 5),

K f p = 100N • m -1 , B f p = 2N • s • m -1 and G n = 48.0m • N -1 • s -1 .
As already underscored in remark 1, these values are only defined to illustrate the commentary, but do not change the conclusions.

For low frequencies, the variations in the Bode's magnitude are the same for the simulated contact impedance and for the environment. The real and felt impedances only differ by a static gain which can be reduced by increasing K f p . Consequently, when the user reaches the contact point, he is able to detect the variations of the forces involved during the process. Although the system is not as perfectly transparent as DFF, it is well suited for manipulation tasks.

Non-contact

When no force is acting on the cantilever, the operator feels the impedance:

Z op = A f (K f p + B f p jω) A f jω + A d G n (K f p + B f p jω) +M v jω+B v (29)
For low frequencies, the impedance Z F P op,LF can be approximated from:

Z F P op,LF ≈ ω<<1 A f A d G n + B v (30)
To minimize the impedance felt by the operator, without affecting the scaling factors, G n is the only parameter that can be tuned. The higher it is, the better the transparency will be. However, whatever the values of the control scheme's gains, the user will feel the viscosity of the haptic interface (as was the case for the DFF control scheme).

At higher frequencies, he feels the impedance Z F P op,HF :

Z F P op,HF ≈ ω>>1 M v jω (31)
As for the DFF, the operator will feel the inertia of the haptic device.

The Bode's diagram (figure 13) confirms the validity of the approximations made in equations (30) and (31). When compared to the Bode's diagram for the DFF control scheme (figure 6) for non-contact, it highlights the lack of transparency for low frequencies. However, this difference can be reduced by increasing G n . As highlighted in table 1 which summarizes the approximated impedance for the DFF and FP control schemes, the FP coupling is less transparent than the

Contact low frequencies

A f A d Ze + Bv A f K f p A d K f p +A f keq Ze high frequencies Mvjω Mvjω
DFF. To obtain the same transparency K f p has to be high enough to ensure a stiff contact, while G n will influence the non-contact behavior of the coupling. The higher it is, the less viscous the feeling for the operator will be.

Determination of the control scheme parameters

The performance of this bilateral coupling highly depends on the controller parameters and the scaling factors (see paragraphs 4.2 and 4.3). However, relations derived in the previous sections do not allow to easily choose the gains since each of them is composed of many parameters. It will be useful to consider particular cases to derive simple necessary conditions of stability. Using these relations, and transparency considerations, gains will be chosen. Then, Llewelyn criterion will be used to see if the gains meet sufficient stability conditions.

Scaling factors

As in the first control scheme, A f and A d depend on user's requirements. Indeed, one might want a precise positioning, and/or important force feedback, according to the manipulation tasks. As in section 3, we will choose A d = 0.05 • 10 6 . A f will be such that the forces sent back to the user are high enough for an untrained user to distinguish the different phenomena encountered during the experiment.

Proportional controller G n

Problems of stability can be due to control schemes but also to numerical computation. Indeed, the force F * m at time k + 1 is determined using information of positions and velocities at time k (see figure 11):

F * m (k + 1) = B f p ∆V (k) + K f p ∆X(k) (32)
where:

∆V (k) = A d V * s (k) -V * m (k) ∆X(k) = A d X * s (k) -X * m (k) (33)
For similar reasons, the expression of V * s is:

V * s (k + 1) = G n F * e (k) - 1 A f F * m (k) (34)
Considering ( 32) and (34) and the fact that the position X * s is computed using Tustin's discretization, when the probe is well above the substrate (no force applied on it, i.e. F * e = 0), F * m is given by:

F * m (k + 1) = λ 1 F * m (k -1) + λ 2 V * m (k) + (λ 3 + λ 4 ) K f p (35)
where:

λ 1 = -G n B f p A d + K f p A d Te 2 A f λ 2 = -B f p + K f p T e 2 λ 3 = A d X * s (k -1) + V * s (k -1)T e 2 λ 4 = -X * m (k -1) + V * m (k -1
)T e 2 Avoiding numerical instabilities leads to an upper bound on G n , a necessary condition for stability: In [START_REF] Gil | Stability analysis of a 1 DOF haptic interface using the Routh-Hurwitz criterion[END_REF] a relation between B f p , K f p and T e is derived to assure the stability of the system while in contact with a rigid environment. In that work, the authors apply the Routh-Hurwitz criterion to a control scheme similar to ours. However, the output of the coupling is the position X m instead of the velocity V m and the backward difference is used to determine the discrete controller C(z) (instead of Tustin). We will use the same methodology applied to our system. As for the DFF control scheme, the system considered is LTI.

|λ 1 | < 1 ⇔ G n < A f A d K f p Te 2 + A d B f p = G n lim (
The discrete time transfer function is:

V * m F * op = V (z) 1 + V (z)C(z) (37)
Before applying the Routh-Hurwitz criterion on the characteristic equation, a bilinear transformation (z = 2+wTe 2-wTe ) should be made. It leads to:

b 2 w 2 + b 1 w + b 0 = 0 ( 38
)
where:

b 2 = 4B v (1 + δ) -4(1 -δ)B f p (39) b 1 = (1 -δ)(4B v + 4B f p -2K f p T e ) (40) b 0 = 2K f p T e (1 -δ) (41)
The Routh-Hurwitz criterion is achieved if and only if b 0 , b 1 and b 2 have the same sign. Since δ < 1, (41) is always positive. Therefore, the system will be stable if and only if b 1 > 0 and b 2 > 0, which implies:

B v + B f p > K f p T e 2 (42)
B v 1 + δ 1 -δ > B f p (43)
(42) is the same condition as that found in [START_REF] Gil | Stability analysis of a 1 DOF haptic interface using the Routh-Hurwitz criterion[END_REF] and states that the stiffness of the coupling is bounded by the inherent damping of the haptic interface and that added by the coupling. Moreover, if the sampling period increases, K f p must decrease for the same amount of damping to guarantee stability. Using the first order Taylor development of x → exp(x) in the region of 0 ( BvTe Mv → 0 since T e → 0), δ ≈ 1 -BvTe Mv . ( 43) can be approximated by:

2M v T e > B v + B f p (44)
This highlights that the maximum damping (and therefore, according to (42), the maximum stiffness) admissible is limited by the inertia of the master arm and is inversely proportional to the sampling period T e . This is a convincing argument of the importance of the sampling period for stable haptic feedback.

Summary

The relations derived in section 4.3 as well as in paragraphs 4.4.2 and 4.4.3 are summarized in the table 2.

The minimum and maximum values each gain can take to ensure necessary stability conditions are given, as well as transparency considerations.

As seen in paragraph 4.4.2, G n has to be high to allow a transparent non contact behavior. According to (36), for stability reasons, G n must be smaller than

A f A d K f p Te 2 +A d B f p
. Therefore, smaller values for B f p and K f p allow for a higher G n lim , and increase the non contact transparency.

Concerning K f p , a maximum value to verify the Routh-Hurwitz criterion is derived. However, as explained above, for transparency reasons K f p should be low enough to limit the viscosity when there is no contact.

Experimental results

To compare the results with the ones obtained with the DFF control scheme we use the same cantilever as in section 3.

K f p Te 2 -B v = 0.15N • s • m -1 and non contact trans- parency is still good. G n is chosen such that G n = 0.90G n lim = 48.0m • N -1 • s -1 .
The selected parameters must satisfy ( 21), ( 22) and ( 23) to ensure that the system will remain stable whatever the environment. In order to check such conditions, the values of C 1 , C 2 and C 3 are plotted in the frequency domain. The results are given in figure 14. For pulsations lower than ω c = 35rad • s -1 , C 1 , C 2 , and C 3 are positive: Llewelyn's criterion is verified. According to the system bandwidth, pulsations greater than ω c will be attenuated, therefore the system will remain stable with the chosen values.

To verify experimentally that the gains we chose are adapted to the stated problem, we performed the same approach-retract operation as in section 3. The results obtained are plotted in figure 15. The forces felt by the operator, as well as those measured by the force sensor (and scaled by A f ) are represented. They must be compared to those in figure 9. The system remained stable during this experiment, contrary to the DFF control scheme. Moreover, even if the force sent back to the user is computed through the control scheme (and not directly fed back), the feeling that the operator got reflects what happened in the remote environment since F m and A f F e plots are similar. The oscillations that can be seen in F m 's plot were induced by the virtual coupling. Since the bandwidth of the haptic device is limited, they were not disturbing for the user. Therefore, although this control scheme is less transparent than the DFF, the feeling is good enough to allow the operator to feel the pull-off phenomenon with a peak of amplitude 1N. Consequently, the analysis performed in this paper allows for an efficient tuning of the controller presented in section 4.1. The pull-in phenomena is not visible on figure 15. Indeed, to avoid time-consuming manipulations, we chose A d = 0.05 • 10 6 (i.e., such that a velocity of 1cm • s -1 of the haptic handle represents 0.2µm • s -1 for the slave device). With this value of A d , the velocity of the nanotranslator was too high compared to the dynamics of the pull-in effect and thus could not be reflected to the user.

To prove the robustness of our approach with respect to the environment's stiffness, the same experiment was performed using different cantilevers. The results are presented in figure 16. The same velocityscaling factor was used for the three probes. The force amplification was chosen so that the user could clearly feel the contact (about 5N were fed back via the haptic device), for a cantilever's deflection smaller than 5µm. This lead to A f = 200 • 10 6 for the cantilever of stiffness k c = 0.05N • m -1 , and A f = 0.2 • 10 6 for stiffness k c = 48N • m -1 . Other gains were chosen using table 2. It should be noted that these experiments were performed in a non-controlled environment. Conditions of humidity and temperature may have changed between the experiments. However, the pull-off was indeed greater for cantilevers with a low stiffness. As noted above, the oscillations on the plots of figure 16 are induced by the virtual coupling, but are not disturbing for the operator due to the limited bandwidth of the haptic device.

The FP control scheme is adapted to nanomanipulations, especially because the system is stable, even when in contact with the substrate, for cantilevers with stiffnesses of 0.05N • m -1 up to 48N • m -1 . Moreover, the analysis is validated experimentally. Forces of 10nN (pull-off for the cantilever of stiffness k c = 0.05N • m -1 ) were felt by the user.

Conclusion

Two different control schemes have been analysed in this paper, in the particular context of nanorobotic applications. The Direct Force Feedback control scheme suffers from a trade-off between stability and force amplification if time consuming manipulations have to be avoided. However, transparency is high as proven analytically, and underscored by the fact that we were able to feel both pull-in (forces of 0.5nN) and pull-off phenomena. Using Force-Position control scheme, higher force amplification without suffering from the duration of the manipulation can be achieved. Forces of 10nN were felt by the user and stable contact for cantilevers of stiffnesses from 0.05N • m -1 to 48N • m -1 was demonstrated. The choice of the control scheme will therefore depend on the relevant application. For a highly transparent rendering, DFF is appropriate, however the experiments will be time-consuming. For a more complex task, implying high displacements, the FP control scheme should be chosen. It will greatly improve the operator's ability by providing him or her with force feedback. Using the analysis carried out, the influence of each gain on the bilateral coupling is highlighted. These results will therefore help to realize real-time adaptation of the gains.

Notations

  (a) (arrows indicate the chronological path). The corresponding plot of the force applied on the cantilever as a function of time is given in figure 1(b). That is what the operator should ideally feel. Several points should be noted: -Two discontinuous phenomena (referenced as A and E on the plots), are respectively representative of Force as a function of the time

2. 2 Fig. 2

 22 Fig. 2 Experimental setup

Fig. 5

 5 Fig. 5 Bode's diagram for contact impedances, DFF control scheme

Fig. 6

 6 Fig. 6 Bode's diagram for non contact impedances, DFF control scheme

⇒

  s t a b i l i t y (a) Stability and instability regions for keq = 2.4N • m -1

Fig. 8

 8 Fig.8Approach-retract curve for a cantilever of stiffness kc = 2.4N • m -1 , using A d = 0.05 • 10 6 , A f = 0.12 • 10 6 . R < Rmax. DFF control scheme.

Fig. 9

 9 Fig.9Approach-retract curve for a cantilever of stiffness kc = 2.4N • m -1 , using A d = 0.05 • 10 6 , A f = 6 • 10 6 . R > Rmax. DFF control scheme.

Fig. 10

 10 Fig.10Approach-retract curve with pull-in for a cantilever of stiffness kc = 0.05N • m -1 , A d = 50 • 10 6 , A f = 6700 • 10 6 . R < Rmax. DFF control scheme.

Fig. 11

 11 Fig. 11 Force-Position control scheme

  Fig. 12 Bode's diagram for contact impedances, FP control scheme

Fig. 13

 13 Fig. 13 Bode's diagram for non contact impedances, FP control scheme

36) 4 . 4 . 3

 443 Proportional integral controller B f p and K f p

1 C 2 C 3 Fig. 14

 12314 Fig. 14 Values of C 1 , C 2 and C 3 with respect to ω.

Fig. 15

 15 Fig.15Approach-retract curve for a cantilever of stiffness kc = 2.4N • m -1 , using A d = 0.05 • 10 6 , A f = 6 • 10 6 . FP control scheme.

• m - 1 kc = 2 .4N • m - 1 kc = 48N • m - 1 Fig. 16

 121116 Fig.16Approach-retract curve for cantilevers of stiffnesseskc = 0.05N • m -1 (K f p = 100N • m -1 , B f p = 1.2N • s/m, Gn = 2329m/N • s), kc = 2.4N • m -1 (K f p = 100N • m -1 , B f p = 2.0N • s • m -1 , Gn = 48.0m • N -1 • s -1 ), and kc = 48N • m -1 (K f p = 100N • m -1 , B f p = 1.5N • s • m -1 , Gn = 1.98m • N -1 • s -1). FP control scheme.

  are clearly identified. The controller scales down the motions provided by the user by a coefficient A d , and magnifies environmental forces by a factor A f to provide force feedback.

	Fop	Fm	Fe				Fe
	Vm	Vm	Vs				Vc
	Operator	Master device	Coupling	Slave device	Environment
		Inputs				Outputs
			Fop	+	+	Fm	A f	Fe
				V (s)
			Vm				1/A d	Vs	N (s)	Vc
			Fig. 4 Direct Force Feedback control scheme

Table 1

 1 Approximated values of Fop for DFF and FP control schemes

	DFF	FP

Table 2

 2 Valid range of values for FP controller gains.

		Minimum value	Maximum value
		high for a stiff
	K f p B f p	contact feeling (sec. 4.3.1) (42)→ K f p Te -Bv 2 high for non	(42), (44) → 4Mv T 2 e low for non contact transparency (sec. 4.3.2)
	Gn	contact transparency (sec.	(36) →	A d K f p	A f Te 2 +A d B f p
		4.3.2)

  [START_REF] Niemeyer | Stable adaptive teleoperation[END_REF].1 (k c = 2.4N • m -1 ), with the same scaling factors (A d = 0.05 • 10 6 , A f = 6 • 10 6 ). Other parameters are chosen according to the results of the transparency and stability analysis (table 2).A good compromise for transparency in contact and non contact mode is found if K f p = 100N • m -1 . This value is indeed smaller than the maximum allowed value

	4Mv T 2 e	. Enough damping is introduced by choosing B f p =
	2.0N • s • m -1 . It is greater than the minimum value

http://www.physikinstrumente.com/

http://www.haption.com/

Acknowledgements This work was supported by the French National Agency of Research, through the PACMAN project.


Aude Bolopion

Member, IEEE Hui Xie

D Sinan Haliyo

Stéphane Régnier

3D Haptic Handling of Microspheres

In this paper, a fully teleoperated 3D micro assembly task with haptic feedback is presented. Microspheres (diameter: 4 -6µm) are manipulated by pick-and-place. The setup is composed of a dual-tip gripper controlled through a haptic interface. To grasp the spheres, the tips must be correctly positioned with respect to the objects. The approach proposed to align the gripper is based on a user-driven exploration of the tobe-manipulated object. During this step, the haptic feedback is based on amplitude measurements from cantilevers in dynamic mode. Hence, the operator perceives the contact while freely exploring the manipulation area. A virtual guide is generated to pull the user to the optimum contact point, allowing correct positioning of dual tips. For the pick-and-place operation, the haptic feedback provides the user with information about the microscale interactions occurring during the operation. As experimental validation, a two-layer pyramid composed of four nylon microspheres is built in ambient conditions.

I. INTRODUCTION

Manipulation of objects of less than ten micrometers is a challenging issue as it shares many difficulties with nanomanipulation. Among them, the visual feedback under optical microscope is limited and do not enable the accurate positioning of tools and objects. Due to the scale reduction, adhesion forces become predominant over gravitational ones [START_REF] Fearing | Survey of sticking effects for micro parts handling[END_REF]. Tools to manipulate these objects must be carefully designed [START_REF] Menciassi | From "macro" to "micro" manipulation: models and experiments[END_REF]. Fully automated micromanipulation is difficult to achieve, due to the high influence of environmental parameters, the lack of repetability and mainly the lack of intuitiveness of the overall system. Haptic feedback appears as a promising solution to provide assistance to the operators [START_REF] Bukusoglu | Haptic manipulation of microspheres using optical tweezers under the guidance of artificial force fields[END_REF], in particular for AFM based manipulation [START_REF] Guthold | Controlled manipulation of molecular samples with the nanomanipulator[END_REF], [START_REF] Ferreira | Virtual reality and haptics for nanorobotics[END_REF]. Solutions to assist micromanipulations through haptic feedback are first steps towards intuitive nanomanipulations.

Early examples of teleoperation with force feedback at this scale deal only with feeling the shape of a substrate or objects [START_REF] Hollis | Toward a tele-nanorobotic manipulation system with atomic scale force feedback and motion resolution[END_REF]. [START_REF] Marliere | Multi-sensorial interaction with a nano-scale phenomenon: The force curve[END_REF] reports a haptic implementation of a approach/retract task of an AFM probe. First remote tasks inducing a modification of the sample are indentations (e.g. direct patterning on a substrate [START_REF] Li | Videolized atomic force microscopy for interactive nanomanipulation and nanoassembly[END_REF]). Tasks involving pushing/pulling or cutting objects are also of primary interest [START_REF] Guthold | Investigation and modification of molecular structures with the nanomanipulator[END_REF], [START_REF] Sitti | Teleoperated touch feedback from the surfaces at the nanoscale: modeling and experiments[END_REF]. As only two measurements are directly available from an AFM cantilever (bending and torsion), 3D haptic feedback of nanoscale interactions between the tool and the object can only be achieved by the use of contact mechanics models. Such models are used for 3D haptic feedback in surface indentation and touching micro-objects [START_REF] Li | Development of augmented reality system for afm-based nanomanipulation[END_REF], [START_REF] Vogl | Augmented reality user interface for an atomic force microscope-based nanorobotic system[END_REF], [START_REF] Onal | Teleoperated 3-D force feedback from the nanoscale with an atomic force microscope[END_REF]. However, so far, no manipulation tasks are reported.

The authors are with Institut des Systèmes Intelligents et de Robotique, Université Pierre et Marie Curie -Paris 6, CNRS UMR 7222, 4

Place Jussieu, 75005 Paris, France. {bolopion, xie, haliyo, regnier}@isir.upmc.fr All the above mentioned works use a single AFM cantilever and static measurements. Consequently, only one controllable contact point is available to the user. Using the AFM in contact mode implies also some limitations compared to dynamic mode: the dynamic mode where the force measurement is obtained through variations on the amplitude or frequency of a vibrating probe is generally considered of finer quality than static contact mode, where the measured force is directly proportional to the deflection of the probe [START_REF] Sitti | Survey of nanomanipulation systems[END_REF]. In addition, on the above mentioned works, a frequent time-consuming factor is the necessity of a preliminary scan, especially in the case where vision quality is poor (for objects smaller than few micrometers), or contact mechanics models are used for 3D feedback, or path planning is required to implement virtual guides.

We report here a complex fully teleoperated 3D microassembly task, consisting of building a two layer pyramidal structure from four ∅5 ± 1µm microspheres. It is based on the analysis of haptic coupling schemes we did previously [START_REF] Bolopion | Analysis of stability and transparency for nanoscale force feedback in bilateral coupling[END_REF]. First results of 2D teleoperation of ∅50µm microspheres with haptic feedback using rolling were obtained in [START_REF] Bolopion | 2D micro teleoperation with force feedback[END_REF]. This current report deals with 3D microassembly in ambient conditions of microspheres ten times smaller. The system uses two independent AFM probes to collaboratively grasp and position each object, as reported in [START_REF] Xie | Three-dimensional automated micromanipulation using a nanotip gripper with multi-feedback[END_REF]. Teleoperation through haptic feedback is extensively used in every step of the operation and is based on the dynamic mode AFM. As the proposed approach does not aim for full automation but instead relies on the operator to increase the flexibility of the system, the pre-scan step is avoided and is replaced by a user guided initial exploration. This exploration allows for an in-line calculation of virtual guides, helping the operator to correctly align the dual tip gripper with respect to the manipulated object, even in the case of poor visual accuracy. Additionally, a haptic feedback scheme is presented for pick-and-place of microspheres. This paper is organized as follows. The experimental setup and the manipulation protocol are described in Section II and Section III respectively. Haptic feedback based on dynamic mode measurements and in-line construction of virtual guides to accurately align the grippers to microspheres are discussed in Section IV. Section V depicts pick-and-place experiments and their detailed analysis.

II. EXPERIMENTAL SETUP

Detailed specifications of the manipulation setup are discussed in [START_REF] Xie | Three-dimensional automated micromanipulation using a nanotip gripper with multi-feedback[END_REF]. A brief summary is given here. The micromanipulation platform is depicted in Fig. 1. The AFM gripper is equipped with an optical microscope, and two sets of nanopositioning devices and optical levers to coordinate two AFM cantilevers with protruding tips (namely, Tip I and Tip II, Nanosensors ATEC-FM) facing each other, forming a dual-tip gripper. Tip I is fixed on a X-Y -Z motorized micropositioning stage (resolution: 50 nm), while Tip II is actuated by an open-loop X-Y -Z piezotube (sub-nm resolution) mounted on a X-Y -Z manual stage (resolution: 0.5 µm). A closed-loop X-Y -Z nanostage (resolution: 0.1 nm) is used to actuate the sample holder during microassembly. Coarse alignment of the Tip I and Tip II is achieved under the optical microscope with large displacements of motorized and manual stages.

Each cantilever disposes of its own optical lever, comprised of a laser source and a four-quadrant photodiode. Data acquisition occurs at 500-800 Hz for static force sampling and at 600 kHz for amplitude through a NI 6289 ADC.

Cantilevers can be used in two different modes: tapping and static. For the tapping mode, a piezoceramic excites each probe at its natural frequency. The amplitude of resulting oscillations is measured through the variations of the voltage output on the photodiode:

A = β • ∆V
where A is the amplitude measurement, β = 10 -6 m • V -1 is a calibrated conversion factor and ∆V is the differential voltage response of the photodiode.

In static mode, the normal force applied on the cantilever F is measured directly from the output voltage of the photodiode:

F = k n S n ∆V (1)
where k n = 2.8N • m -1 is the normal stiffness of the cantilever, and

S n = 8 • 10 -7 m • V -1
is the sensitivity of the optical levers.

An Omega haptic interface, manufactured by Force Dimension 1 is provided for intuitive user control of the manipulator. This master arm is a 3 degrees of freedom device. The user manipulates the handle and the resulting position p h is scaled down to be used to control the actuators (nanostage and piezotube). The haptic force F h sent to the user through the haptic interface is based on measurements from the two photodiodes (V I and V II ). As represented by the switches S 1 and S 2 in Fig. 1, different translators and feedbacks are used at each step of the microassembly. Next sections detail the use of the haptic interface to interactivly perform a microassembly task.

III. 3D MICROASSEMBLY PROTOCOL

The manipulation area and a coarse positioning of the tips is determined using the optical microscope. The operator then places sequentially each tip on both sides of the object using the Omega, controlling respectively the sample holder through the nanostage and Tip II through the piezotube. In both steps the haptic feedback is provided based on the amplitude variations of each tip in tapping mode. Once the object is hold between two tips, the lift-off and release are achieved by haptic control of the sample holder. Contact mode measurement are used to provide the force feedback.

The approach proposed here is based on a user-driven exploration of the manipulated object. The haptic feedback allows the operator to feel when he/she touches the object while freely exploring the manipulation area. Note that during this operation the vertical position of the probes are constrained to a few micrometers above the substrate and the operator controls only the horizontal motion. The data recorded during this exploration is processed in-line and generates a virtual guide to pull the user to the optimum contact point. User "feels" and sequentially adjusts the contact force for both tips, assuring an adequate grip on the object. In the third phase of the manipulation, both grippers are immobilized on both sides of the object and the operator controls the motion of the sample holder (this method ensures that the displacement of both of the tips with respect to the substrate is identical), while still receiving haptic feedback calculated from the output of two probes.

Tip-alignment phases, including the haptic feedback and virtual guide generation and pick-and-place phases with force feedback are detailed in the following.

IV. ASSISTED GRIPPER ALIGNMENT

The alignment of each tip is a user-driven process. The operator moves the tip while receiving haptic feedback derived from amplitude measurements of the AFM probe. During the initial exploration and prior to the generation of virtual guides, the haptic feedback is only on the x axis (Fig. 2(a)). As the operator scans manually the surface of the tobe-manipulated object, the data is recorded to reconstruct its shape and create the virtual guide. This virtual guide generates the haptic feedback along y axis, pulling the tip to the calculated grasp line y 0 , parallel to the x axis and crossing the sphere's center.

A. Tapping Mode Measurements

In tapping mode, each probe is excited at its natural frequency. At constant height from the substrate and away from objects, this results in oscillations on a constant amplitude, noted A 0 . While approaching an object, starting from a few hundreds of nanometers, the tip contacts the object intermittently and the amplitude A t decreases until a minimum value A CP is reached at full-contact between the tip and the object.

Fig. 2 illustrates the principle of object detection from amplitude variations. The tip is first set to a given height from the substrate h 0 . This step is achieved in an initial phase and the user controls the motion only in the (x, y) plane parallel to the substrate. While the tip moves on the grasp direction of the gripper, the x axis, amplitude decreases until contact (Fig. 2(b)). On the y axis, perpendicularly to the grasp direction, both tips must be aligned with the center of the sphere. This matches the minimum of amplitude along the y axis, at a fixed x position (Fig. 2(c)).

B. Haptic Feedback for Tip Alignment

To align the gripper with the sphere and bring it to contact the visual feedback from the optical microscope does not provide sufficient resolution. Haptic feedback aims to compensate for this lack of visual feedback. The haptic coupling used is depicted in Fig. 3. Each tip is sequentially aligned on the grasp line and brought to contact.

1) x axis: The haptic feedback along the x axis should provide following information:

• R 1 : force null when the tip is far from the object • R 2 : increasing force as the tip approaches the object • R 3 : increasing force as the tip applies a force on the object According to the variation of amplitude described previously, the following haptic feedback F hx is proposed and satisfies requirements R 1 -R 3 :

F hx = -α a (A -A 0 ) if A > A CP -α a (A -A 0 ) + k x (x -x CP ) else
where α a is a scaling factor. The amplitude A 0 is measured at the beginning of the experiment while the tip oscillates at its natural frequency. Two cases are distinguished:

• before contact (first equation): as the amplitude is decreasing while the tip approaches the object, an increasing repulsive force is sent to the user so that he or she is aware of the presence of the object • while in contact (second equation): a spring k x between the position of the contact point x CP and the current position of the tip x is added to the feedback of the first equation. It simulates the force applied by the tip to the sphere. The contact point location x CP is acquired and set when the amplitude measurement reaches A CP .

2) y axis: The force perceived along the y axis must enable the user to align the tip with respect to the sphere, on the grasp line. The haptic feedback along the y axis is not available before all the points have been recorded and the computation of virtual guide is achieved. During this exploration in search of the y 0 position, the x axis haptic feedback is provided to the user so that he or she perceives the sphere's location. When y 0 is computed, a haptic feedback F hy simulating a spring k y between y 0 and the current position y of the tip is sent to the user:

F hy = k y (y -y 0 ) (2)

C. Virtual Guide Generation

During the initial exploration in tapping mode, n contact points (x i t , y i t ), with their matching amplitudes A i t (i = 1..n) are collected. The contact position data is acquired only if the actual amplitude A t is in [15%A 0 , 70%A 0 ] interval to avoid inaccurate measurements. In order to define the z i t coordinate for each contact point (x i t , y i t ), the approximation

z i t = A i t /2
is proposed. This is a relative position since the cantilever is oscillating around the height h 0 set manually. The calculation of the z 0 coordinate of the sphere is thus relative to h 0 , and is not accurately known. However, as the only parameter useful for haptic feedback is the y 0 coordinate, this approximation is acceptable. n points (t 1 , ...t n ) are recorded during the exploration process. These points are used to reconstruct the shape of the manipulated sphere, calculate the grasp line and provide the haptic feedback along y axis. With prior knowledge of the shape of the object, and the n recorded points, the sphere can be reconstructed from the surface equation:

(x -x 0 ) 2 + (y -y 0 ) 2 + (z -z 0 ) 2 = R 2 ( 3
)
where R is the radius of the sphere, and x 0 , y 0 and z 0 are the coordinates of its center.

A least mean square algorithm is used to compute the parameters of the sphere (center and radius) to best fit the n recorded points. The position y 0 of the grasping point along the y axis is then known. Note that as all the points are on the same side of the sphere along x axis (x < 0 for Tip I and inversely for Tip II), the calculated x 0 coordinate may be inaccurate. However, as stated above, the only parameter used for virtual guide is y 0 .

D. Experimental Validation of Tip Alignment

Manipulated objects are nylon microspheres, with a diameter of 4 -6µm. Both tips are first positioned manually at the correct height (around 500 -600nm) above the substrate, using dynamic mode detection of the sample. [START_REF] Bolopion | Analysis of stability and transparency for nanoscale force feedback in bilateral coupling[END_REF] proposes a haptic feedback solution for this step. Each tip is then sequentially positioned by the operator at each side of the object.

1) x axis haptic heedback: Experimental results acquired while moving the tip along x axis and contacting the microsphere are depicted in Fig. 4. The position of the tip is represented in Fig. 4(a), the amplitude measurement is depicted in Fig. 4(b) and the haptic feedback in Fig. 4(c). In area 1, the tip is away from the sphere and the feedback is null. In area 2, the user distinctly perceives the haptic feedback as the tip approaches the sphere and intermittent contact starts. An additional feedback is transmitted when an effort is applied by the cantilever on the sphere in area 3, increasing the sensation of stiffness.

Compared to using direct force measurement from a cantilever in static mode, tapping mode amplitude measurement enables a better sensitivity on x axis. In static mode, as the measurement direction is almost aligned with the probes' length, the equivalent stiffness is extremely high compared to k n on z axis. Hence, a static detection on x axis would only occur when a quite important force is already applied on the object. On the contrary, the use of tapping mode allows an earlier detection of the object as only intermittent contact with the object produces a detectable signal. This allows the user to be aware of the object's presence and avoids to push it involuntarily. 2) y axis haptic feedback: The accuracy of the virtual guide depends on the number of recorded points and their relative position. The influence of two parameters, the number of points n and the minimum distance between two points (noted d) are empirically optimized. Used values are n = 12 and d = 0.3µm. y axis feedback is effective as soon as the virtual guide is generated. Its value is calculated using Equation (2). Fig. 5 represents the force perceived by the user. The position of the tip as well as y 0 are also given.

The haptic feedback on the y axis helps the user to align the tip with respect to the sphere as y 0 is at the equilibrium point of the virtual spring. Precise positioning is achieved since contact information is transmitted to the user through the x axis of the haptic device F hx , and alignment is ensured thanks to the haptic force F hy .

V. PICK-AND-PLACE WITH HAPTIC FEEDBACK Fig. 6.

Haptic coupling for pick-and-place. The user manipulates the nanostage with respect to the two tips by setting the position of the haptic device. Haptic feedback is derived from force measurement from the sum of the two photodiodes' outputs.

After the object is grasped between both tips, it will be lifted from substrate, transported to the target location and released. Haptic feedback is proposed for this task to render to the user the forces measured by both probes, with proper scaling. As depicted in Fig. 6 the Omega haptic device is used for position control of the sample holder through the nanostage, while the 2 tips holding the microsphere are immobilized. The force data is obtained in static mode, from the deflection of each probe as measured directly on photodiodes using Equation (1). The manipulation is carried out in ambient conditions, at 20 • C and a relative humidity of 48%.

A. Haptic Feedback of Nanoscale Interactions

The haptic feedback is designed to return to the user the nanoscale interactions of the pick-and-place operations as faithfully as possible. It is synthesized from force responses of Tip I and Tip II. As detailed in [START_REF] Xie | Three-dimensional automated micromanipulation using a nanotip gripper with multi-feedback[END_REF], adhesion forces F ao between the sphere and the substrate can be estimated as:

F ao = F I + F II ( 4
)
where F I (resp. F II ) is the force applied to the cantilever I (resp. II). Hence, the haptic force rendered to the user is computed as:

F hz = α f [(F I -F I 0 ) + (F II -F II 0 )] (5)
where F 0 = F I 0 + F II 0 is the force measured when the tips are holding the sphere before lift-off and it is naturally proportional to the grasping force applied by the tips to the object. Removing this offset allows the user to discard the grasping force which is not useful for pick-and-place. Moreover, in the case where the grasped object is lost hazardously during the lift-off, the measured forces F I and F II will fall back to zero and Eq. ( 5) will give a negative value, pulling back the probes to the substrate.

α f is a force amplification factor used to scale the measured forces and the haptic force sent to the user. the nominal value used here is α f = 2.0 • 10 6 . This coefficient is set considering the magnitude of nanoscale interactions that should be felt by the user (in particular the pull-off force). Detailed discussion on the definition of this parameter can be found in [START_REF] Bolopion | Analysis of stability and transparency for nanoscale force feedback in bilateral coupling[END_REF]. Fig. 7 represents the haptic feedback during a pick-and-place operation of a 5µm sphere from a glass substrate and in the insert the forces measured from probes. The curve's starting point is the contact state between the microsphere and the substrate. As the nanostage moves down (hence the object held by the tips is lifted), probes are bent down measuring negative forces (inset i). During the pickup, when the nanostage position reaches around -900nm , the microsphere pulls off the substrate with a minimum force of -1125nN overcoming the adhesion. Note that after the pull-off, the measured force falls to -550nN, and not to the pre-pick-up null value (inset ii). Actually, as the tip/object contact points are in the lower hemisphere, during the liftoff the object slides slightly down, increasing the grasping force. During the transport phase, a change on the force can be noted. This is again due to the sliding of the sphere in the gripper. This hypothesis is backed-up by the approx. 0.2µm difference seen between pick-up and touch down positions along z axis.

During the release operation the microsphere snaps-in the substrate (inset iii). As the object is pushed to the substrate, between (iii) and (iv) the contact force compensates the grasping force, until the tips pull off the sphere (inset iv) and slide down from the object to the substrate (inset v). At this point, it is sufficient to move apart both tips along y direction to release the sphere from the gripper and achieve the operation. Note that as the contact area at object/tip interface is much smaller than at the object/substrate interface due to the sharp tips used, the problem of the object adhering to a probe is limited.

B. Construction of a Two-layer Pyramid

In order to validate 3-D manipulation capabilities of the haptic system, four nylon microspheres with diameter of 4-6µm were used to build a two layer pyramid microstructure [START_REF] Miyazaki | Pick and place shape forming of threedimensional micro structures from fine particles[END_REF]. Microspheres were deposited on a freshly cleaned glass substrate. An area of interest for the experiments was selected under an optical microscope with 20× optical magnification. The 3D micropyramid is built using the teleoperated system described in previous sections. The result of this assembly is depicted in Fig. 8.

VI. CONCLUSION

A teleoperated 3D microassembly task has been successfully performed thanks to the use of haptic feedback. Although a complex tool composed of two cantilevers as a dual-tip gripper has to be used to perform accurate grasping and pick-and-place at this scale, users unfamiliar with the setup were able to carry out the task as haptic feedback ensures a high intuitiveness of the setup. The use of the cantilever in both static and dynamic mode enabled a complex manipulation using an AFM tool. A two layer pyramid is built from four ∅5 ± 1µm spheres in ambient conditions for experimental validation of the dexterity and usability of the overall setup.

The methodology presented in this paper can be applied directly to other type of objects, such as carbon nanotubes or nanowires, except for y axis virtual guides which are specific to spherical objects but can be further extended to nonspherical objects by taking the object geometry in account for in-line generation.

Fig. 1 .

 1 Fig. 1. AFM gripper based telemicromanipulation system. (left) The haptic device providing a user interface to control the three-dimensional microassembly with real time haptic feedback. (right) The dual-probe gripper comprised of two AFM cantilevers with protruding tips for pickand-place micromanipulation.

Fig. 2 .

 2 Fig. 2. A schematic diagram of haptic exploration by local of the lower semi-microsphere using a oscillating cantilever. (a) Top view of desired grasp configuration (b) Front view shows the tip tapping the microsphere while approaching on x-axis. (c) Side view shows the tip tapping the microsphere when scanning on y-axis with a fixed x position.

Fig. 3 .

 3 Photo I Photo IIA I

Fig. 4 .

 4 Fig. 4. Haptic feedback along the x axis while exploring the to-bemanipulated sphere; (a) position of the tip, (b) amplitude measurement, and (c) force sent to the user. For the haptic feedback, the coefficients are set to: α a = 3 • 10 6 N.m -1 , k x = 10 • 10 6 N.m -1 and a CP = 0.1µm.

Fig. 5 .

 5 Fig. 5. Haptic feedback along the y axis while aligning the tip with the sphere: position of the tip and force sent to the user. k y = 10 • 10 6 N.m -1

Fig. 7 .

 7 Fig. 7. Synthesized normal force responses from both microcantilevers during the pick-and-place manipulation of a microsphere: (i) pick-up; (ii) The microsphere is detached from substrate after the pull-off; (iii) The microsphere snaps in the substrate; (iv) gripper/microsphere pull-off; (v) The gripper snaps in the substrate; (vi) Manipulation is ended with slight bending of the microcantilevers.

Fig. 8 .

 8 Fig. 8. Teleoperated 3-D microassembly results in a micropyramid. The photos are captured under magnification of 20× and 100× respectively.

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VII. ACKNOWLEDGMENTS

This work was supported by the ANR (French Agency of Research) through the PACMAN Project, and the NANOROL Project under Grant No. PSIROB07-184846.