contour accuracy improvement of a flexure based micro motion stage for tracking repetitive trajectory

Contour accuracy improvement of a flexure based micro motion stage for tracking repetitive trajectory Shi Jia, Yao Jiang, Tiemin Li, and Yunsong Du Citation AIP Advances 7, 015026 (2017); doi 10 1063/[.]

Contour accuracy improvement of a flexure-based micro-motion stage for tracking repetitive trajectory

Shi Jia, Yao Jiang, Tiemin Li, and Yunsong Du

Citation: AIP Advances 7, 015026 (2017); doi: 10.1063/1.4973873

View online: http://dx.doi.org/10.1063/1.4973873

View Table of Contents: http://aip.scitation.org/toc/adv/7/1

Published by the American Institute of Physics

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AIP ADVANCES 7, 015026 (2017)

Contour accuracy improvement of a flexure-based

micro-motion stage for tracking repetitive trajectory

Shi Jia,1Yao Jiang,2,3Tiemin Li,1,2,aand Yunsong Du1

1Institute of Manufacturing Engineering, Department of Mechanical Engineering,

Tsinghua University, Beijing 100084, People’s Republic of China

2Beijing Key Lab of Precision/Ultra-precision Manufacturing Equipments and Control,

Tsinghua University, Beijing 100084, People’s Republic of China

3Institute of Instrument Science and Technology, Department of Precision Instrument,

Tsinghua University, Beijing 100084, China

(Received 10 November 2016; accepted 28 December 2016; published online 17 January 2017)

Flexure-based micro-motion mechanisms have been widely utilized in modern pre-cision industry due to their inherent merits, while model uncertainty, uncertain nonlinearity, and cross-coupling effect will obviously deteriorate their contour accu-racy, especially in the high-speed application This paper aims at improving the contouring performance of a flexure-based micro-motion stage utilized for track-ing repetitive trajectories The dynamic characteristic of the micro-motion stage is first studied and modeled as a second-order system, which is identified through an open-loop sinusoidal sweeping test Then the iterative learning control (ILC) scheme

is utilized to improve the tracking performance of individual axis of the stage A nonlinear cross-coupled iterative learning control (CCILC) scheme is proposed to reduce the coupling effect among each axis, and thus improves contour accuracy

of the stage The nonlinear gain function incorporated into the CCILC controller can effectively avoid amplifying the non-recurring disturbances and noises in the iterations, which can further improve the stage’s contour accuracy in high-speed motion Comparative experiments between traditional PID, ILC, ILC & CCILC, and the proposed ILC & nonlinear CCILC are carried out on the micro-motion stage to track circular and square trajectories The results demonstrate that the proposed control scheme outperforms other control schemes much in improving the stage’s contour accuracy in high-speed motion The study in this paper pro-vides a practically effective technique for the flexure-based micro-motion stage in

high-speed contouring motion © 2017 Author(s) All article content, except where

otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/ ) [http://dx.doi.org/10.1063/1.4973873]

I INTRODUCTION

High-precision micro-motion system has a wide range of need in high-tech industrial applica-tions, such as the scanning probe microscopy, lithography, atomic force microscopy, biological cell manipulator, and precision manufacturing.1 5 As a typical micro-motion mechanism, the flexure-based stage transmits the motions entirely through the deformations of materials, which has the inherent merits over the conventional mechanisms in terms of no friction, backlash, and wear, high frequency, and easy of fabrication.6 Additionally, the piezoceramic actuator is usually adopted as the actuation element in the flexure-based stage due to its merits of high-precision, fast response, high stiffness, and large force.7Therefore, the flexure-based micro-motion stage has the potential to provide high-precision and high-speed motions, and it has already been widely used

Among the applications of the micro-motion stage, there is an increasing demand in the repetitive trajectory tracking with high speed and accuracy, such as the scanning probe microscopy imaging,1

a Email: thu11js@163.com

2158-3226/2017/7(1)/015026/15 7, 015026-1 © Author(s) 2017

015026-2 Jia et al. AIP Advances 7, 015026 (2017)

lithography equipment,2and accelerometer transverse sensitivity testing.8 To meet the requirement

of application, the optimal design of the mechanical structure and control system of the flexure-based stage are necessary to be carried out Numerous studies have been done on the design and modeling of the flexure-based mechanism, including the type synthesis,9 11stiffness modeling and geometric parameter optimization,12 and dynamic modeling.13Based on these researches, several flexure-based micro-motion stages with high-frequency6have already been developed, which provide

a solid foundation for achieving high-speed motion

Except for the mechanical structure, the control system also plays an important role in guar-anteeing the high speed and accuracy of the micro-motion stage Though the conventional PID controller has the advantages of simplicity and strong robustness, it will cause a large phase lag

of individual axis in high-speed motion, which limits the closed-loop bandwidth of the micro-motion stage.14,15 Therefore, another effective control schemes are necessary to be investigated When the micro-motion stage moves with a high speed, the nonlinearity of the adopted piezoce-ramic actuator,16,17the model uncertainty existed in the system dynamics, and the coupling effect among each axis will have an obvious negative influence on the stage’s performance To deal with the actuator nonlinearity, feedforward controller is introduced into the control system by inverting the nonlinear effects For example, Lai18 utilized an inversion-based feedforward controller com-bined with a PID controller to compensate the nonlinearities of a parallel micro-motion stage, and excellent positioning and tracking performances were achieved Additionally, several analyt-ical models, such as the Prandtl-Ishlinskii model,19 Preisach model,20 and Bouc-Wen model,21

have been proposed and applied in the feedforward controller to compensate the hysteresis of the piezoceramic actuator The result of the model-based feedforward controller depends on the model accuracy However, the plant uncertainty and complexity lead to the difficulty in obtain-ing an accurate model of the micro-motion stage Instead of acquirobtain-ing accurate knowledge of the dynamic model, iterative learning control (ILC) scheme22,23takes full advantage of the character-istic of the repetitive motion and extract information from previous iteration trial to modify the control input in the next iteration trail to compensate for the repetitive nonlinearities and uncer-tainties Therefore, the ILC scheme can be adopted as the feedforward controller in the stage to effectively improve the tracking performance of individual axis for tracking repetitive trajectory For a multi-axis system, however, the reduction of the tracking error of individual axis cannot fully guarantee its contour performance The coupling effect and incoordination among each axis have great influence on the contour accuracy of the micro-motion stage To deal with this issue, plenty of approaches have been developed, where cross-coupled control (CCC) scheme24and global task coordinate frame (GTCF)25,26 are two main approaches CCC is much simpler than GTCF

in improving the machine’s contouring performance while it is still difficult for CCC to calcu-late the contouring error in real-time, especially for large-curvature contouring tasks However, the calculation of the contouring error can be done off-line by integrating ILC into CCC Thus, the cross-coupled iterative learning control (CCILC) scheme27,28was proposed for improving the con-tour tracking performance of the machine in repetitive tasks, which combines the advantages of the two control schemes The CCILC controller has not been explored in the flexure-based micro-motion stage yet Additionally, the non-recurring noises and disturbances are likely to be amplified in the CCILC, which will deteriorate the accuracy of the micro-motion stage, especially in high-speed motion

This paper aims to improve the contour accuracy of a flexure-based micro-motion stage for tracking repetitive trajectory with high speed The dynamics of the micro-motion stage is first modeled and identified To fully improve the contouring performance of the stage, the tracking errors of individual axis are reduced by adopting the ILC controller, and the contour accuracy of the stage is further improved by a proposed nonlinear CCILC controller

The remainder of this paper is organized as follows Section IIintroduces the flexure-based micro-motion stage and its dynamic model is established and identified In SectionIII, an integrated control scheme is proposed to improve the tracking accuracy of individual axis and the contour accuracy of the stage The stability of the proposed control scheme is also analyzed In SectionIV, comparative experiments are carried out on the micro-motion stage for tracking circular and square trajectories Finally, the conclusions of this paper are given in SectionV

015026-3 Jia et al. AIP Advances 7, 015026 (2017)

II MODELING AND IDENTIFICATION OF THE FLEXURE-BASED MICRO-MOTION STAGE

A Description of the flexure-based micro-motion stage

The detailed 3-D model and prototype of a flexure-based micro-motion stage are shown in Fig.1 To realize high stiffness, high natural frequency, and output decoupling characteristic of the micro-motion stage, a symmetric parallel structure composed of four identical limbs is adopted Each limb is composed of a parallelogram flexure and a fixed-fixed beam, which are serially connected and acted as prismatic joints Therefore, the micro-motion stage has two translational degrees of freedom

along the X- and Y -axes.

The prototype of the micro-motion stage is monolithically fabricated from a block of Al-7075 where its features are machined by using the milling process to guarantee accuracy Two 40-µm-stroke piezoceramic actuators (PSt 150/7/40, VS 12, XMT Harbin) are adopted to drive the micro-motion

stage along the X- and Y -axes The two actuators are controlled by a modular piezo servo controller

(XE-501, XMT Harbin) The output displacements of the terminal platform along the two axes are tested by two high precision length gauges (MT-1281, Heidenhain) A dSPACE processor board DS1007 equipped with a 16-bit ADC card (DS2102) and a 6-channel high resolution incremental

FIG 1 Flexure-based micro-motion stage (a) 3-D model of the micro-motion stage (b) Prototype and control system of the micro-motion stage.

015026-4 Jia et al. AIP Advances 7, 015026 (2017)

encoder interface card (DS3002) is utilized to output the excitation voltage of the piezo servo controller and capture the real-time data for the length gauges Since the piezoceramic actuator is made up of multiple piezoelectric layers glued together, it is sensitive to the pulling force which may bring damage to the actuator Therefore, preloading forces are applied on the two piezoceramic actuators

by tightening the bolts to guarantee their operational safety

B System dynamic modeling and identification

The dynamics of the micro-motion stage is first investigated and it can be treated as a second-order system according to the characteristics of its mechanical structure and control system.13Thanks

to the symmetric structure of the micro-motion stage, only the dynamics of the stage along the X- or

Y -axis is necessary to be studied The dynamic model of the stage’s mechanical structure along the X-axis can be expressed as

m e ¨x(t) + c e ˙x(t) + k e x(t) = F p (t) (1)

where m e , c e , and k eare the equivalent mass, damping coefficient, and stiffness of the stage along

the X-axis, respectively, x is the output displacement of the terminal platform, and F pis the driving force of the piezoceramic actuator

Then the dynamics of the stage’s control system is studied It is difficult to obtain a precise dynamic model of the control system due to the nonlinearity, hysteresis, and creeping phenomenon

of the piezoceramic actuator To simplify this issue, the transfer function from the input voltage to the output displacement of the piezoceramic actuator is assumed as a constant gain λ when a power amplifier with high bandwidth is adopted Therefore, the dynamic model of the whole stage can be derived as

¨x(t) + θ2˙x(t) + θ1x(t)= θ0u(t) (2) where θ2=c e

m e, θ1= k

m e, θ0=λkin m e , k in is the input axial stiffness of the micro-motion stage along the

X-axis, and u is the control input voltage.

According to Eq (2), the dynamics of the micro-motion stage can be identified through a real-time DFT algorithm An open-loop sweeping test is carried out on the stage and the swept sinusoidal signals range from 1 Hz to 1500 Hz with a sampling frequency of 20 kHz The frequency response

of the stage is conducted in the two axes and the DFT based frequency responses of the experimental data are shown in Fig.2 With the help of the system identification tool in Matlab and the captured

response data, the second order transfer functions of the micro-motion stage along the X- and Y -axes

can be estimated as

G x (s)= 1.546 × 106

s2+ 9.463 × 103

FIG 2 Frequency response of the micro-motion stage in X- and Y -direction.

015026-5 Jia et al. AIP Advances 7, 015026 (2017)

G y (s)= 1.937 × 106

s2+ 1.254 × 104s + 1.296 × 107 (4)

It can be seen that the estimated frequency responses of the stage can be well fitted with the experimental results within 100Hz However, the flexible modes and high order dynamics appear with the increase of frequency and it is difficult to be modeled precisely

III CONTROL SYSTEM

The control system of the flexure-based parallel micro-motion stage will be presented in this section To achieve excellent contouring performance of the micro-motion system for tracking repet-itive trajectories, a comprehensive control scheme is proposed, as shown in Fig.3, where P1and P2

represent the dynamics of the stage along the X- and Y -axes, respectively, and the details of the control

system will be described later In general, the individual axis ILC term is the fundamental part of the control system, which can basically guarantee the tracking performance of individual axis Then the nonlinear CCILC term, which combines the advantages of ILC and CCC, can further improve the contour accuracy of the stage iteratively by reducing the coupling effect and incoordination among the two axes without amplifying the non-recurring noises and disturbances

A ILC term of the control system

The improvement of the tracking accuracy of individual axis is first discussed ILC is a kind

of feedforward control algorithm, which tries to find the actual inverse process of the system by incorporating the past repetitive control information into the present control input signal As shown

in Fig.3, the past control information is first stored and then is incorporated into the present control period as the compensation component to suppress the repetitive nonlinearities, uncertainties and disturbances, and therefore improves the system’s tracking accuracy in a repetitive trajectory The iterative learning law of the ILC adopted in this paper is given as

u i+1(t) = Q(u i (t) + Le i (t)) (5)

where u i (t) and u i+1 (t) are the input signals in the ith and (i+1)th iterative process, respectively, Q is

a low-pass Q-filter which is utilized to enhance the system robustness and suppress the noise in the

iterative process,29L is the learning function, and e i (t) is the tracking error of the system in the ith

iterative process, which can be expressed as

FIG 3 Control scheme for the micro-motion stage.

015026-6 Jia et al. AIP Advances 7, 015026 (2017)

where r d (t) and r i (t) are the reference and actual positions of the system, respectively.

It is noted that low-pass filter may cause a phase shift while as the iterative learning input signals are calculated off-line, the phase shift can be eliminated by filtering the signal back and forth.30Many types of the learning functions have been developed for ILC,22including the PD type, H∞ method, and plant inversion Among them, PD-type learning function is a typical, simple, and tunable ILC learning function, which is adopted in this paper and it is written as

B Nonlinear CCILC term of the control system

Though the ILC method can effectively reduce the tracking error of individual axis in tracking repetitive trajectories, it is not sufficient for guaranteeing the contour accuracy of the system The contour error caused by the coupling effects between each axis should be also considered and elimi-nated The cross-coupled control (CCC) is an effective approach in considering the coupling effects and compensating the corresponding contour error directly The result of the CCC is mainly depended

on the accuracy of the real-time contour error estimation model, which is determined according to the tracking error of each axis and the contour shape Though several contour error estimation approaches have been proposed to calculate the contour error related to arbitrary contours,31it is still a challenge for CCC to estimate and compensate the contour error in real-time For a repetitive tracking task,

by adopting the ILC method, the calculation of the contour error of the previous iteration trial can

be done off-line by searching the nearest point on the trajectory to the actual point Therefore, the cross-coupled iterative learning control (CCILC) scheme is proposed to further improve the contour accuracy of the micro-motion stage iteratively In case that the contour error caused by non-recurring noises and disturbances is amplified in the iterative process, a nonlinear function is also incorporated into the CCILC scheme

The update law of the whole control scheme, which combines the individual axis ILC discussed

in SectionIII Aand the proposed nonlinear CCILC, can be written as









u x i+1(t) = Q(u x

i (t) + L e e x i (t)+ (1 + Φ(εi ))C x Lεεi (t))

u y

i+1(t) = Q(u y

i (t) + L e e y i (t)+ (1 + Φ(εi ))C y Lεεi (t)) (8) where u i x and u i+1x are the input signals of the X-axis in the ith and (i+1)th iterations, u i y and u i+1y

are the input signals of the Y -axis in the ith and (i+1)th iterations, e i x and e i yare the tracking errors of

the X- and Y -axes in the ith iterations, L e and Lare the learning functions of the individual axis ILC and the nonlinear CCILC, and i is the contour error of the micro-motion stage in the ith iteration.

The contour error for arbitrary contours can be expressed as24

where the K x and K y are the coefficients related to the tracking error and the contour shape C xand

C y are the cross-coupled gains along the X- and Y -axes Thus, let  xand y be the components of

contouring error along X- and Y -axes, respectively, C x and C yshould be x/ and y/ respectively

to ensure that the compensation for each axis is along the direction of the contouring error

The nonlinear function Φ(i) is incorporated into the CCILC scheme to avoid amplifying the non-recurring noises and disturbances in the iterative process and it is given as32

Φ(ε)= (α(1 − δ

|ε |), |ε| >δ

where α is a coefficient and δ is a given threshold, and it is shown that for the error below the threshold the additional gain is zero while for the error beyond the threshold, larger error corresponds to larger additional learning gain Using the nonlinear gain function, the contour error below the threshold caused by noises is considered to be non-recurring and will not be amplified in the iterative process of CCILC while the contour error beyond the threshold is basically caused by repeatable disturbances and nonlinearities will be attenuated

015026-7 Jia et al. AIP Advances 7, 015026 (2017)

C Stability analysis

The stability analysis of the proposed control scheme for the micro-motion stage will be per-formed The stability of the ILC is generally analyzed in the frequency domain based on the system response analysis to sinusoidal inputs For the nonlinear CCILC, the cross-coupled gain

is time-varying, and therefore the time domain stability analysis is required

Considering a class of discrete-time time-invariant (LTI) system given by22

where k is the time index, i is the trial number of the ILC, and q is a forward time shift operator defined

as qx(k) + x(k+1), r i is the output signal, u i is the control input signal, and d is the free response of

the system to the initial condition and periodic disturbance effects

As for the time domain analysis, lifted system framework22,33is used by stacking the signals in

vectors For N-sample sequences of input and output, Eq (11) can be written as













r i(1)

r i(2)

r i (N)













| {z }

ˆ

R i

=













p1 0 · · · 0

p2 p1 · · · 0

.

p N p N−1· · ·p1













| {z }

ˆ

P

·













u i(0)

u i(1)

u i (N − 1)













| {z }

ˆ

U i

+













d i(1)

d i(2)

d i (N)













| {z }

ˆd

(12)

Based on Eq (12), the dynamics of the micro-motion stage using lifted system framework in X-and Y -axes can be expressed as









X i+1= ˆP x Uˆx

i + ˆd x

Y i+1= ˆP y Uˆy

i + ˆd y

(13)

Then Eq (5) which is the iteration learning law of the ILC can be written in the lifted form as

ˆ

U i+1= ˆQ( ˆU i + ˆL ˆE i) (14) where ˆQ and ˆ L are the lifted form of Q and L, which are shifted to N×N matrices.

Using the lifted system framework, the proposed control system update of the micro-motion stage, combining the ILC for individual axis and the nonlinear CCILC for the two axes, can be written as









ˆ

U x i+1= ˆQ( ˆU x

i + ˆL e Eˆx i + ˆN ˆC x Lˆεεˆi) ˆ

U y i+1= ˆQ( ˆU y

i + ˆL e Eˆy

i + ˆN ˆC y Lˆεεˆi) (15)

where N= 1 + Φ(i) is the nonlinear gain and ˆN is the lifted form of N According to Eq (13), the expression of ˆE i xand ˆE i ycan be given as









ˆ

E i x= ˆX r−( ˆP x Uˆx

i + ˆd x) ˆ

E i y= ˆY r−( ˆP y Uˆy

where ˆX rand ˆY r denote the desired system output vectors of lifted form in X- and Y -axes, respectively

and they are assumed to be iteration invariant

Equation (17) can be written in matrix format as

"Uˆx ˆ

U y

#

i+1

= ˆQ *.

,

"Uˆx ˆ

U y

# +

"Lˆe 0 N ˆˆC x Lˆε

0 Lˆe N ˆˆC y Lˆε

#







ˆ

E x

ˆ

E y

ˆ ε









+ /

-j

(17)

Substituting the lifted form of Eq (9) into Eq (17) for the contouring error, then substituting

Eq (16) for the lifted form of the tracking errors in X- and Y -axes into Eq (17) yields the relationship between the update control inputs and the previous control inputs as

" U x

U y

#

i+1= ˆQ Mˆ " U x

U y

# !

i

+

"Dˆx ˆ

D y

#

(18)

015026-8 Jia et al. AIP Advances 7, 015026 (2017)

where ˆM=

"Q(I − ˆˆ P x( ˆL e + ˆN ˆC x LˆεKˆx)) − ˆQ ˆ P y N ˆˆC x LˆεKˆy

− ˆQ ˆ P x N ˆˆC y LˆεKˆx Q(I − ˆˆ P y( ˆL e + ˆN ˆC y LˆεKˆy))

# ,

"

ˆ

D x

ˆ

D y

#

=

"

ˆ

Q(( ˆ L e + ˆN ˆC x LˆεKˆx)( ˆX r− ˆd x)+ ˆN ˆC x LˆεKˆy( ˆY r− ˆd y))

ˆ

Q(( ˆ L e + ˆN ˆC y LˆεKˆy)( ˆY r− ˆd y)+ ˆN ˆC y LˆεKˆx( ˆX r− ˆd x))

# , and ˆK x and ˆK x denote the lifted

form of K x and K y, respectively

The converged control in the iteration domain is defined as U∞= lim

i→∞ U i, and then a necessary

and sufficient condition for asymptotic convergence is sense of |U∞−U i + 1|< |U∞−U i|is presented for the system as22,27

where ρ(·) is the spectral radius of a n × n matrix defined as ρ(·)= max λj(·), j = 1, 2, , n, and λ j(·)

is the ith eigenvalue of the matrix.

IV EXPERIMENTAL VALIDATION

The effectiveness of the proposed control scheme in improving the contour accuracy of the micro-motion stage for tracking the repetitive trajectories will be validated by experimental tests For comparison, the PI controller, ILC controller, ILC & CCILC controller, and the proposed ILC & nonlinear CCILC controller are all implemented on the micro-motion stage The controller parameters

of the four controllers are tuned to guarantee the system stability and excellent contouring performance with the help of the identified dynamic model of the micro-motion stage

A Tracking high-speed circular contouring trajectory

The contour accuracy of the micro-motion stage for tracking a circular trajectory is first tested

The origin of the circular trajectory is located at O(0, 0), and its radius is 5 µm.

FIG 4 Contouring performances of circular trajectory with 5 Hz using (a) PI controller, (b) ILC controller, (c) ILC & CCILC controller and (d) ILC & Nonlinear CCILC controller.

015026-9 Jia et al. AIP Advances 7, 015026 (2017)

FIG 5 Contour errors of circular trajectory with 5 Hz.

TABLE I Contouring performance indexes of circular trajectory with 5 Hz.

The four control strategies are carried out on the micro-motion stage to improve its contour accu-racy The contouring performance of the stage for tracking the circular trajectory at a low frequency

of 5Hz is first tested The contour results and the corresponding contour errors of the stage are given

in Figs.4 and5, respectively It should be noted that as the tracking or contouring errors of each motion period of the stead-state motion are same, only the tracking or contouring errors in one motion period are plotted in this paper The maximum and root mean square (RMS) values of the contour errors are also compared in TableI

The results show that the stage has already achieved a high contour accuracy even when the PI controller is adopted Though all the other three learning controllers can reduce the contour error of the stage, they do not show obvious advantages in improving the stage’s contour accuracy at a low frequency Therefore, the PI controller is sufficient for guaranteeing the contour performance of the micro-motion stage at a low frequency Additionally, the nonlinear function incorporated into the CCILC method does not improve the stage’s contour accuracy The reason may result from that the non-recurring noises are very small at a low frequency and their negative effect on the contouring performance of the stage can be neglected

FIG 6 Axis tracking errors of circular trajectory with 200 Hz (a) in X-axis and (b) in Y -direction.

The contour accuracy of the micro- motion stage for tracking a circular trajectory is first tested

The origin of the circular trajectory is located at O(0, 0), and its radius is... strategies are carried out on the micro- motion stage to improve its contour accu-racy The contouring performance of the stage for tracking the circular trajectory at a low frequency

of 5Hz...