Lin Chao（林超），Ji Jiuxiang（紀(jì)久祥），Tao Youtao（陶友淘），Huo Dehong（霍德鴻），Cai Lizhong（才立忠），Cheng Kai（程凱）

1.State Key Laboratory of Mechanical Transmission，Chongqing University，Chongqing，400030，P.R.China；2.School of Mechanical and Systems Engineering，Newcastle University，Newcastle Upon Tyne，NE1 7RU，UK；3.Advanced Manufacturing and Enterprise Engineering Department，Brunel University，Uxbridge，Middlesex UB8 3PH，UK

1 Introduction

Micro／nano transmission platform（MNTP）is used to produce small linear and angular displacement.With rapid development of microelectronics technology，aerospace and biological engineering，the research of MNTP with high accuracy and stability，multiple degrees of freedom and high controllability is increasingly gaining academic attention.

So far，MNTP has become a research focus and many MNTPs have been developed in the past two decades.Zhang，et al.［1］developed a micro positioning table with long stroke.The micro positioning mechanism driven by piezoelectric actuators（PZTs）has two degrees of freedom and the displacement in Xor Y directions is about 288 μm.Wang，et al.［2］proposed an MNTP based on 3－revolute－revolute－revolute（3－RRR）parallel mechanism，and successfully realized±70μm displacement in both Xand Ydirections，as well as±0.1°rotation about Zaxis.Zhang，et al.［3］designed a precision positioning table with three degrees of freedom，and acquired±300μm displacements in Zdirection and±0.25°rotation about both Xand Yaxes.Chu，et al.［4］developed a long－travel linear nano－positioning stage driven by PZTs，and 35μm displacement along Xaxis was thus realized with a precision of 10nm.Yuen，et al.［5］presented a micro positioning platform with rapid movement in Xand Ydirections，followed by 25μm×25μm displacement successfully.Deepkishore，et al.［6］developed a new linkage mechanism which could achieve 18μm displacement in Xor Ydirection and±1.72°ro－tation around Zaxis.Li，et al.［7］proposed a new type of decoupled large－displacement micro position platform with two degrees of freedom，on which the stroke of each two linear axes is 117 μm.Although the above－mentioned micro positioning platforms have high accuracy and repeatability，there still exist some disadvantages，such as small movement range.Hu，et al.［8］designed a long range nano－transmission platform based on the principle of inchworm motion.It featured a maximum linear speed of 13.9μm／s，a minimum step of 10nm and a movement distance of more than 20mm.Though it met the requirements of both long displacement and high precision，it can only move along a single direction.Therefore，the study of MNTP with large displacement，high precision，and multiple degrees of freedom is challenging researchers in the field.

In addition，the control strategies are also central to the development of micro positioning platforms.Since precision can be improved by using an effective control means，the research on control algorithms and control strategies has been carried out to improve the performance of micro positioning platforms.The digital proportion integration differentiation（PID）method is widely adopted due to its effective and simple setup.Moreover，some advanced control algorithms such as fuzzy algorithm，neural network algorithm and genetic algorithm were also applied to the precision control of MNTP［9－12］.Currently，several control strategies have been used，including feedforward open－loop control，PID closed－loop control，and hybrid control which combines feedforward，closed－loop control and adaptive inverse control［13－14］.

This paper presents a fully flexure MNTP with five degrees of freedom driven by PZTs.The platform adopts leaf－springs and bridge type amplification mechanism（BTAM）to realize the amplification of input displacements.Firstly，the theoretical output displacements and driving mechanism of MNTP are analyzed according to the kinematic theory and elastic beam theory.PZT actuator is calibrated.Meanwhile，the whole closed－loop control strategy of MNTP is established using the feedforward PID compound con－trol algorithm based on the Preisach model.Moreover，the total transfer function of the positioning system is derived，and the calculation of output signal in the positioning system is performed.Finally，the finite element analysis（FEA）simulation and positioning experiments are conducted to verify the effectiveness of the control strategies.

2 Principle of Structural Design

According to the principles of machinery and compliant mechanism，the large displacement MNTP with five degrees of freedom is designed（Fig.1），whose dimensions are 260mm×260mm×120mm.It consists of a upper platform，a lower platform and six PZTs.The upper platform is composed of four micro displacement amplification mechanisms driven by four PZTs.And the four displacement amplifiers，symmetrically arranged，can make the upper platform to move along Zaxis and rotate around both Xand Yaxes.The lower platform is composed of two micro displacement amplification mechanisms driven by two PZTs.It employs an asymmetric structure to realize the motion in Xand Ydirections.This configuration ensures the high accuracy and fast response of MNTP.MNTP is manufactured by drilling，milling，grinding，and wire electrical discharge machining（EDM），etc.，on one piece of material，therefore，without gap and mechanical friction.By controlling the input voltages of the six PZTs，an MNTP with high－precision，high dynamics，and multiple degrees of freedom are obtained.

MNTP consists of 56leaf springs and 42rigid rods（Fig.1），and it is fixed on an experiment table by bolts.To ensure that the piezoelectric actuator is fully engaged with the micro displacement amplification mechanism，pre－loads are applied by threaded screws.The force and motion are conveyed by the pure elastic deformation of leaf springs.

3 Performance Analysis of MNTP

Fig.1 MNTP with five degrees of freedom

Fig.2shows the micro－displacement amplification mechanism of MNTP.BTAM is a key com－ponent of MNTP.The input displacement of PZT is amplified by the amplification mechanism which also makes the platform move along a specified direction.Different rigid rods are connected with each other by leaf springs.In order to analyze the amplification performance of MNTP，the amplification ratio of BTAM should be first considered with the assumption that the material of different rigid rods and leaf springs is isotropic.

According to the kinematic theory and the elastic beam theory，the amplification ratio of BTAM can be expressed as

Fig.2 Drawing of displacement amplification mechanism

where lais the center distance of adjacent leaf springs，t the thickness of leaf springs，andαthe amplifying angle of BTAM（Fig.2）.

Considering the relationship of geometric parameters of MNTP，the theoretical output displacements in different directions can be derived.

The linear displacements along X，Y，and Z axes can be described by the equation

The rotation angle around Xor Yaxis can be expressed as

whereΔLis the elongation displacement of PZT under variable force，which can be obtained as

whereΔL0is the elongation displacement of PZT under free condition，kPZTthe stiffness of PZT，and ksthe stiffness of MNTP.

Based on the elastic beam theory，the stiffness of the leaf spring can be determined by the equation

According to Eq.（5）and the principle of virtual work，the stiffness of BTAM can be calculated

From Eqs.（5，6），the stiffness of MNTP can be expressed as

where t，α，l，Land L1are the structural parameters of the displacement amplification mechanism in Fig.2，b the thickness of BTAM，and Ethe Young′s modulus.

Substituting Eq.（7）into Eq.（4），ΔLis obtained.Then according to Eqs.（2，3），the displacements along X，Yand Zaxes can thus be determined.

4 Drive and Control Strategies

4.1 Drive mechanism of MNTP

PZT is usually adopted as the actuator of micro positioning system.It works by utilizing the inverse piezoelectric effect，which means when an electric field is applied on PZT，it will generate a small deformation，accompanied by the micro－displacement of MNTP.The inverse piezoelectric effect of PZT can be expressed as follows

where d33is the piezoelectric constant，S the strain，T′the stress，E the electric field strength，and xEthe elastic constant.

For accurate and reliable positioning of MNTP，PZT used in the experiments has been calibrated under a free loading condition.By varying input displacements for PZT（model 40VS15），the output displacement values are collected by a capacitance displacement sensor（model Micro－Epsilon CS5）.The calibration tests are repeated three times and the average values measured in the tests are shown in Fig.3.

Fig.3 Calibration test results

From Fig.3，it can be concluded that there is slight deviation between actual output displace－ments and target displacements.The overall er－rors are within 0.1μm range mainly due to the effect of the electromechanical coupling.The effect makes PZT exhibit hysteresis，creep，and nonlinear，etc.，which are the main factors that generate errors.Therefore，a well－calibrated PZT can also help to improve the control accuracy of MNTP.

4.2 Control strategies of MNTP

To effectively eliminate the displacement difference of PZT，the encapsulated ceramic with strain gauge sensor（SGS）is utilized.In addition，in order to reduce the stiffness of MNTP and obtain larger motion distance and swing angles，the lower platform adopts the aforementioned asymmetric structure.But the asymmetric structure will increase the coupled displacement and affect the platform precision.A full closedloop control system is thus designed to improve the positioning performance of the platform.Fig.4presents the schematic of the positioning system and Fig.5the block diagram of the basic control algorithm.The main control process can be expressed as follows：first，the driving control signal Ri（kT）is sent out by computer，and then the feedforward controller based on the Preisach model predicts the output values Ud（kT）more accurately by a certain control voltage sequence.Meanwhile，by a D／A converter，high voltages are generated to drive PZT，which will make the platform have an extremely small displacement C（s）.The micro displacement can be measured by capacitance displacement sensors.And the position voltage signal x（kT）is sent back to the computer by the A／D converter，then the signal x（kT）will compare it with the signal Ri（kT）to obtain the deviation signal e（kT）.By the PID controller，a certain offsetΔU（kT）will be generated to compensate the hysteresis and creep effects of the piezoelectric ceramic as well as external disturbances，so as to realize precise positioning.

Taking the movement in Xdirection as an example，the differential equation of the positioning system holds

Fig.4 Full closed－loop control principle of positioning system

Fig.5 Feedforward PID control algorithm based on Preisach model

where mis the mass of MNTP，μthe damping coefficient，kPZTthe stiffness of PZT，ksthe stiffness of MNTP，kgthe stiffness of BTAM，ΔLthe elongation displacement of PZT under variable force，and xthe output displacement along Xdirection of MNTP.

According to Eq.（9），the transfer function of the MNTP in Xdirection can be obtained as

whereωnis the natural frequency of MNTP，ζthe relative damping coefficient of MNTP.

Similarly，MNTP transfer functions in Yand Zdirections are introduced by the equations

Since the output of computer is a digital signal，D／A conversion and high voltage amplification are prerequisite for PZT actuator to generate a DC high voltage，which makes MNTP accom－plish precise positioning.The PZT controller here is equivalent to a scaling up link whose transfer function can be expressed by

whereγvis the amplification coefficient of the PZT controller.

Therefore，the transfer function of the feedback system is as follows

Based on Eqs.（10－12），the transfer function of the positioning system moving along Xdirection can be expressed as

Similarly，the transfer function of the positioning system along Y and Z directions can be written as

Based on Eqs.（13－15），the output of the positioning system in X，Yand Zdirections can be obtained as

where Tis the sampling period of the control system.kp，kiand kdrepresent the proportion，integral and differential coefficients of the PID controller，respectively.

5 Simulation and Experimental Analyses

The simulation is performed by the commer－cial FEA package ANSYS.The FEA model and displacement contours in different directions are illustrated in Fig.6.The platform is made of 65Si2Mn，and its mechanical properties are shown in Table 1.The structural parameters are shown in Fig.2and Table 2.

Table 1 Mechanical properties of 65Si2Mn

Table 2 Key structural parameters of MNTP

Fig.6 FEA model and movement simulation

In order to validate the effectiveness of the positioning control system，the feedforward PID rcontroller based on the Preisach model is integrated into the PZT controller.The device in Fig.7is to implement the control principle shown in Fig.5，and it includes PZTs（40VS15），XE－500／501modularized PZT controller，the capacitance displacement sensor（CS5with a measuring range of 5mm and the highest resolution of 3.75 nm），a capacitance displacement sensor controller（capaNCDT6500），an RS6500circuit controller and computers.Taking the movement in Xaxis direction as an example，the specific testing method is as follows：the PZT controller controls the output displacement of PZT，and then the capacitance displacement sensor collects the output displacement signals of MNTP，the signals are subsequently sent back to the computer and the PID controller so as to readjust the platform output.

For the better test results，the range of 20—40μm is chosen as the PZT′s testing output.The test results of MNTP in different directions are shown in Figs.8（a－f）.

Fig.7 Experimental apparatus for testing positioning performance

From Figs.8（a，c，e），it can be concluded that when PZT output displacement is within 20—40μm，the positioning system has a good repeated positioning performance，and the measured results in X，Yand Zaxes directions are basically identical.It can also be known that the minimum and maximum output displacement is 117.612μm and 355.863μm，respectively，and the repeated positioning errors are within 6—8μm range.In Figs.8（b，d，f），the experimental results agree well with the theoretical results and the FEA results，which further verifies the efficiency of the PID control algorithm based on the Preisach model.Besides，the positioning errors are mainly due to the change of material properties caused by local high temperature in the wire EDM processing.The change of material properties in turn results in the variation of stiffness and the increase of processing errors.Moreover，the asymmetric structure of the lower platform can also cause the nonlinear displacement coupling errors of the platform.All of these will affect the precision of the micro positioning system.Therefore，further optimization design about the control system，structural parameters and machining method of the platform are essential to MNTP for achieving the higher precision.

Fig.8 Positioning performance tests in X，Y，and Zaxis directions

6 Conclusions

The following conclusions can be drawn from this research：

（1）A long stroke MNTP，driven by PZT，with five degrees of freedom is designed.The amplification and guidance mechanism of micro displacement is achieved by flexure hinge.In addition，the structural design and displacement amplification performance of MNTP are analyzed.

（2）The driving mechanism of MNTP is analyzed，and the PZT actuator is calibrated.As a result，the overall positioning errors are within 0.1μm range.Meanwhile，the full closed－loop control strategy of MNTP is established using the feedforward PID compound control algorithm based on the Preisach model.Moreover，the total transfer functions of the positioning system are derived，and the calculation method of the output signal of the positioning system is obtained.

（3）The experimental apparatus is built to assess the local performance.The maximum output displacement of the positioning system is 355.863μm and the repetitive positioning errors are kept within 6—8μm.Finally，the experiment results，the FEA results and the theoretical values are in good agreement.The results also further verify the effectiveness of the PID control algorithm based on the Preisach model.

［1］ Zhang Y F，Gong J L.Structure and parameter design for two degrees of freedom micro－positioning mechanism with large travel［J］.Journal of Mechanical Engineering，2010，46（23）：30－35.（in Chinese）

［2］ Wang W，Hu P H，Zeng Q，et al.Theory design method of a flexure mechanism for parallel micromotion working table［J］.Metrology Test Technology＆Verification，2006，16（6）：4－7.（in Chinese）

［3］ Zhang J L，Chen W Y，Ke X R，et al.Design and simulation of a 3－DOF micro－positioning worktable［J］.Modern Manufacturing Engineering，2009（3）：39－42.（in Chinese）

［4］ Chu C L，F(xiàn)an S H.A novel long－travel piezoelectric－driven linear nano－positioning stage［J］.Precision Engineering，2005（30）：85－95.（in Chinese）

［5］ Yuen K Y，Sumeet S A，Aphale S O.Design，analysis and control of a fast nano positioning stage［C］／／Proceedings of the 2008IEEE／ASME International Conference on Advanced Intelligent Mechatronics.［S.l.］：IEEE，2008：451－456.

［6］ Deepkishore M，Dong J Y，Eakkachai P W，et al.A SOI－MEMS－based 3－DOF planar parallel－kinematics nano－positioning stage［J］.Sensors and Actuators，A Physical，2008，147：340－351.

［7］ Li Y M，Xu Q S.Structural design and control strategy analysis of micro／nano transmission platform［J］.IEEE／ASME Transactions on Mechatronics，2010，15（1）：125－135.

［8］ Hu C D，Zhao M R，Li Y Q，et al.Study on design and experimentation of piezoelectric micro－moving platform with nano－step and long－range［J］.Chinese Journal of Sensors and Actuators，2009，22（6）：803－807.（in Chinese）

［9］ Jie D G，Sun L N，Qu D S，et al.Fuzzy－reasoning based self－tuning PID control for piezoelectric microdisplacement system［J］.Journal of Harbin Institute of Technology，2005，37（2）：145－147.

［10］Yong Y K，Aphale S S，Moheimani S O R.Design，identification，and control of a flexure－based XY stage for fast nanoscale positioning［J］.IEEE Transactions on Nanotechnology，2009，8（1）：46－53.

［11］Li C，Tan Y.A hybrid neural network based modeling for hysteresis［C］／／Proceedings of the 2005IEEE International Symposium on，Mediterrean Conference on Control and Automation.Piscataway，USA：［s.n.］，2005：53－58.

［12］Wai R J，Lin C M，Peng Y F.Robust CMAC neural network control for LLCC resonant driving linear piezoelectric ceramic motor［J］.IEE Proceedings：Control Theory and Applications，2003，150（3）：221－232.

［13］Yang C，Zhao Q，Wang H R，et al.Study on intelligent control system of two－dimensional platform based on ultra－precision positioning and large range［J］.Precision Engineering，2010，34：627－633.

［14］Liaw H C，Shirinzadeh B，Smith J.Robust neural network motion tracking control of piezoelectric actuation systems for micro／nanomanipulation［J］.IEEE Transactions on Neural Networks，2009，20（2）：356－366.