Nonlinear modeling for a class of microrobotic systems using piezoelectric stick-slip actuators

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Haptic Interfaces Summary

This section has survey some of the most popular haptic interfaces used in micromanipulation. They are based on general-purposed design so as to be applied in a wide range of applications. However, being general leads to a compromise among performance properties such as the degree of freedom (DOF) in motion and in haptic space, the range and geometry of working space, the maximum rendering forces back to user, the dynamic range (ration between maximum temporary force and friction; with 1 mN as minimum value as threshold of perception), and etc. A more complete survey is summarized in the table 1.3.

Dedicated Haptic Interfaces for Micro-Manipulation

Nowadays, the available haptic interfaces are still far from being qualied with respect to the need required by micro-manipulation, which may demand a combination of the performance properties described above. For example, some advanced microrobotic applications require large working space and large frequency bandwidth.
Typical existing solutions employ cantilever structure to augment workingspace, however, the frequency bandwidth must be reduced due to the dynamics of cantilever design. Another example is some special micro-manipulation may require dedicated interface design to enhance operation intuitiveness, where general purposed haptic interfaces can hardly fulll the request.
In summary, available haptic interface solutions cannot fully address the needs from micro-manipulation. Eorts are still needed in haptic interface mechanical design and the development of related algorithms to realize intuitive and eective micro-manipulation.

Challenges

This chapter has introduced the basics of microrobotics, and the existing technologies for micro-manipulation. As discussed, there is a blank window for small batch production of complex microsystems, which demands exible solution to address the need. Under such circumstances, Percipio Robotics has proposed its semi-automatic cobotic platform Chronogrip to bridge the cap between prevailing non-exible mass automation and low-yield handmade process. However, the solution is not yet complete, there is still challenges in terms of trajectory tracking and haptic rendering.
Concretely, there are three problems to resolve:
In the human-robot interaction, control in velocity is crucial. This issue concerns the study of positioning actuator. Piezoelectric stick-slip actuator that used in Chronogrip system is promising solution to realize a great range of microrobotic applications, and many innovation companies have commercialized this kind of product with reasonable price. However, none of these companies or any research results is able to propose a complete control strategy for velocity, let alone acceleration control. The main diculty lies in the complexity of the dynamics of piezoelectric stick-slip actuators. More precisely, no research result has been reported to be able to fully describe the dynamics of such actuator. To be more general, no control strategy is available nowadays to address complete trajectory tracking for piezoelectric stick-slip actuator, and its dynamics is not yet fully understood.
In the microworld, surface forces are more dominant due to scale eect (see the section 1.1.1.1). In such situation, thermal agitation and other environmental elements cause the micro-objects to follow high dynamic motions. As a consequence, this will lead to high dynamic interaction forces between target objects and gripper tools. Taking into account the existing mechanical structure and design of the haptic interface, no option is able to render high dynamics arisen from microrobotic applications. The lack of high bandwidth haptic interface causes the interaction with the microworld non-intuitive and inecient.
In prestigious watchmaking industry, due to its exibility of models and small batch production, the manufacturing process can hardly be automated. But handmade process has saturated accuracy and productivity, which makes the watchmaking players more and more dicult to conquer their place in global competition. To address this challenge, Percipio Robotics proposes Chronogrip to realize cobotic production. However, the most signicant issue lies in the interface. A joystick or tablet is way too dierent from the conventional tools (the tweezers) used by watchmaking experts, and hence they have to be trained very hard to be familiar with the interface. Furthermore, there is no haptic feedback which makes the process nonintuitive. Nowadays, there is not yet a haptic interface based on a tweezers form available in prototype.
The combination of realizing haptics and maintaining mechanical properties of tweezers is a challenging task.

Working Principle of Piezoelectric Stick-Slip Actuator

A piezoelectric stick-slip actuator is made of (gure 2.2) a Piezoelectric Element (PE), a slider moving along a linear axis and a friction material between the PE and the slider. The slider is in charge of carrying a load (e.g. robot axis). It is guided by the deformable PE that converts electrical energy to mechanical energy.
During a slow deformation of the PE, the contact friction force Ff drives the slider to a linear motion (gure 2.2(a)). After an abrupt contraction of the PE, the slider slips and cannot fully follow the sudden motion of the PE because the inertia force becomes greater than the contact friction force. The slip phase is illustrated in gure 2.2(b). An alternate stick and slip sequence produces a displacement of the slider relative to the PE. By repeating those operations, large range of motion of the slider can be achieved. This function mode is called stepping mode. The input voltage signal applied to the PE is a sawtooth sequence so that alternate slow and abrupt deformations can be realized. If there is only stick motion without any slip, the slider can be driven with higher precision. This function mode is called scanning mode. Unlike conventional motors, motion direction cannot be changed by inverting input power. If input sawtooth sequence is given by repeating slow rise and abrupt drop, the actuator goes towards one direction, which will be referred to as forward direction; the opposite direction can be produced if sawtooth sequence is given by repeating abrupt rise and slow drop, and this direction will be referred to as backward direction, as shown in the gure 2.3.

Presentation of the microrobotic system

The microrobotic system is designed by SmarAct company. It is composed of a 6 DOF parallel robot and a 3 DOF Cartesian robot (see the gure 2.4). Each axis of the microrobot is actuated by a piezoelectric stick-slip actuator of the same reference (SLC-1720-S-HV). The maximum stroke of each axis in the Cartesian microrobot is 12 mm, the maximum amplitude of the input driving voltage is 100 V and the scanning resolution is in submicron range. The study is mainly concerned with the dynamic modeling of the stick-slip actuator. The actuator of the Y axis in the Cartesian structure is used for the experimental validation. The motion of this actuator is measured with a laser interferometer sensor (SP-120 SIOS Mebtechnik GmbH).

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Nonlinear dynamic modeling

To dene the dynamic model of the complete microrobotic system, the dynamics of each axis must be well modeled. As such, the challenge is to dene an accurate dynamic model of an elementary stick-slip actuator.
The major impediment lies in the fact that the internal structure is no public information. As such, hypothesis must be made for the modeling. There are three main assumptions (Fig. 2.5):
(i) the Piezoelectric Element (PE) is attached to the base of the actuator.
(ii) the moving slider is guided by a linear crossed roller guideway and has only one translational DOF.
(iii) there is a friction material between the slider and the PE without lubricant.

Modeling of the Friction

In order to dene the dynamic transfer function between the displacement of the slider x2 and the input voltage U, the friction force Ff must be modeled. The single-state elasto-plastic friction model [Dupont 2002] is used in this work.
The relative motion x = x1 􀀀x2 between the PE and the slider can be generally decomposed into elastic (reversible) and plastic (irreversible) components, denoted by z and w respectively: x = z + w (2.9).
The physical representation is illustrated in the gure 2.7. The black bending bristle stands for the asperity between the slider and the friction material surface. In contrast to multiple random asperities representations in multi-states models, this only one connection stands for the average eect and thus gives rise to single state model.
The elasto-plastic friction force Ff is governed by the following equation: Ff = 0z + 1z_ + 2x_ (2.10). 0, 1 and 2 are respectively the contact stiness, the damping for the tangential compliance and the viscous friction constant. The viscous friction force 2x_ is considered negligible compared to dry friction. This is the case when the frictional phenomenon is presliding dominant with a small relative velocity x_ [Landolsi 2009]. The elastic component z is governed.

Step3 – identication of the break-away elastic strain and the steady-state elastic strain in stepping mode

The stick-slip actuator is driven by sawtooth signals. The velocity x_2 (slope of the displacement) can be measured given a sequence of sawtooth with varying frequencies and amplitudes in both drive directions. This measurement is used to identify the break-away elastic strain zba and the steady-state elastic strain jzss(z)j so that the velocity matches between the model and experimental data.
In the original work, the mixed phase of friction model with sinus implementation is proposed to ensure continuous transition. The physical motivation about the ratio between zba and jzss(z)j still needs further investigation. It is suggested by the authors that the mixed phase should be relatively narrow with respect to the elastic phase. In our study, the ratio has been kept constant as zba=jzss(z)j = 0:667.
Under this condition, it will be shown in the following part how the break-away elastic strain zba evolves with respect to various input conditions (amplitude and frequency of the sawtooth signal). The gure 2.13 illustrates the evolution of the identied zba versus the amplitude of an input sawtooth signal of 50 Hz frequency. In the gure, zba(􀀀) and zba(+) denote respectively the identied value for backward and forward motion directions of the slider (see Fig. 2.3) . The tted curves are shown for both backward and forward motion directions.
The tting result gives rise to a dynamic model for stepping mode with limited input conditions (xed frequency as shown in the example). At the time of writing, we are still working on the identication process to derive a complete tting, i.e. a ZbalforlForward+BackwardlMotionl6atl50Hz8 amplitude. The value of zba is identied for input conditions: 10, 20, 30, 40, 50, 60, 70, 80, 90 and 100 V, where the sawtooth frequency is xed at 50 Hz.

Table of contents :

1 State-of-the-Art 
1.1 Microrobotics
1.1.1 Microworld
1.1.2 Observation
1.1.3 Micro-Manipulation Systems
1.1.4 Chronogrip System
1.1.5 Flexible Micro-Manipulation Solutions
1.2 Actuators for Microrobotic Applications
1.2.1 Piezoelectric Actuators
1.2.2 Electrostatic Actuators
1.2.3 Thermal Actuators
1.2.4 Magneto-/Electrorheological Fluids
1.2.5 Actuators for Microrobotic Applications
1.2.6 Control Algorithm for Piezoelectric Stick-Slip Actuator
1.3 Haptic Interface
1.3.1 Series-Structure Interface
1.3.2 Parallel-Structure Interface
1.3.3 Haptic Interfaces Summary
1.3.4 Dedicated Haptic Interfaces for Micro-Manipulation
1.4 Challenges
2 Nonlinear modeling for a class of microrobotic systems using piezoelectric stick-slip actuators 
2.1 Introduction
2.2 State-of-the-Art
2.2.1 Microrobotic Actuators
2.2.2 Working Principle of Piezoelectric Stick-Slip Actuator
2.2.3 Friction Modeling
2.3 Presentation of the microrobotic system
2.4 Nonlinear dynamic modeling
2.4.1 Dynamic Modeling of the PE and the slider
2.4.2 Modeling of the Slider
2.4.3 Modeling of the Friction
2.5 Experimental analysis and identication
2.5.1 Description of the experimental setup
2.5.2 Parameters identication
2.5.3 Model Validation
2.6 Discussion
2.7 Velocity Control
2.8 Conclusions
3 Extending the Bandwidth of Dual Stage Haptic Interface by Signal Crossover 
3.1 Introduction
3.2 State of the art
3.3 System description
3.3.1 Material description
3.3.2 Crossover method
3.4 Model and control
3.4.1 Dynamic modeling
3.4.2 Identication
3.4.3 Compensation
3.5 Experimental evaluation
3.5.1 Performances of the haptic crossover method
3.5.2 Optical tweezers application
3.6 Conclusion
4 Design of a Haptic Interface for Watchmakers 
4.1 Introduction
4.2 State-of-the-Art
4.3 Technical Specications of the TéléTweez
4.3.1 Description of Tweezers in Watchmaking Industries
4.3.2 Description of Haptic Tweezers
4.3.3 Technical Specications of the TéléTweez
4.4 Mechatronics Design
4.4.1 Actuator System Design
4.4.2 Mechatronic Design of the TéléTweez
4.5 Dynamics Analysis and Virtual Environment Application
4.5.1 Dynamic Modeling
4.5.2 Frequency Bandwidth Identication
4.5.3 Virtual Environment Application
4.6 Conclusion
Conclusions and Future Works 
A Geometric Optimization of Mechanical Amplier 
B Static Analysis of Ball Screw 
C Publication List 
Bibliography 

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