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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 Articles you may be interested in Green’s function and image system for the Laplace operator in the prolate spheroidal geometry AIP Advances 7, 015024015024 (2017); 10.1063/1.4974156 Control of Rayleigh-like waves in thick plate Willis metamaterials AIP Advances 6, 121707121707 (2016); 10.1063/1.4972280 Estimation of Curie temperature of manganite-based materials for magnetic refrigeration application using hybrid gravitational based support vector regression AIP Advances 6, 105009105009 (2016); 10.1063/1.4966043 Observation of high Tc one dimensional superconductivity in angstrom carbon nanotube arrays AIP Advances 7, 025305025305 (2017); 10.1063/1.4976847 AIP ADVANCES 7, 015026 (2017) Contour accuracy improvement of a flexure-based micro-motion stage for tracking repetitive trajectory Shi Jia,1 Yao Jiang,2,3 Tiemin Li,1,2,a and Yunsong Du1 Institute of Manufacturing Engineering, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, People’s Republic of China Beijing Key Lab of Precision/Ultra-precision Manufacturing Equipments and Control, Tsinghua University, Beijing 100084, People’s Republic of China Institute 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 precision industry due to their inherent merits, while model uncertainty, uncertain nonlinearity, and cross-coupling effect will obviously deteriorate their contour accuracy, especially in the high-speed application This paper aims at improving the contouring performance of a flexure-based micro-motion stage utilized for tracking 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 provides 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 applications, 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 flexurebased 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.7 Therefore, 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,2 and accelerometer transverse sensitivity testing.8 To meet the requirement of application, the optimal design of the mechanical structure and control system of the flexurebased 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,9–11 stiffness modeling and geometric parameter optimization,12 and dynamic modeling.13 Based on these researches, several flexure-based micro-motion stages with high-frequency6 have 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 guaranteeing 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 micromotion 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 piezoceramic actuator,16,17 the 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 combined with a PID controller to compensate the nonlinearities of a parallel micro-motion stage, and excellent positioning and tracking performances were achieved Additionally, several analytical 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 obtaining an accurate model of the micro-motion stage Instead of acquiring accurate knowledge of the dynamic model, iterative learning control (ILC) scheme22,23 takes full advantage of the characteristic 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 uncertainties 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) scheme24 and 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 calculate 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) scheme27,28 was proposed for improving the contour 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 II introduces the flexure-based micro-motion stage and its dynamic model is established and identified In Section III, 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 Section IV, 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 Section V 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 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 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 secondorder system according to the characteristics of its mechanical structure and control system.13 Thanks 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 me xă (t) + ce x˙ (t) + ke x(t) = Fp (t) (1) where me , ce , and k e are 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 p is 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 xă (t) + x˙ (t) + θ x(t) = θ u(t) (2) in where θ = mcee , θ = mke , θ = λk me , 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 realtime DFT algorithm An open-loop sweeping test is carried out on the stage and the swept sinusoidal signals range from 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 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 Gx (s) = 1.546 × 106 s2 + 9.463 × 103 s + 9.733 × 106 FIG Frequency response of the micro-motion stage in X- and Y -direction (3) 015026-5 Jia et al AIP Advances 7, 015026 (2017) 1.937 × 106 Gy (s) = (4) s2 + 1.254 × 104 s + 1.296 × 107 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 repetitive trajectories, a comprehensive control scheme is proposed, as shown in Fig 3, where P1 and 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 ui+1 (t) = Q(ui (t) + Lei (t)) (5) where ui (t) and ui+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,29 L is the learning function, and ei (t) is the tracking error of the system in the ith iterative process, which can be expressed as ei (t) = rd (t) − ri (t) FIG Control scheme for the micro-motion stage (6) 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.30 Many types of the learning functions have been developed for ILC,22 including 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 L(s) = Kip + Kid s (7) 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 eliminated 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,31 it 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 Section III A and the proposed nonlinear CCILC, can be written as ux (t) = Q(uix (t) + Le exi (t) + (1 + Φ(ε i ))Cx Lε ε i (t)) i+1 uy (t) = Q(uy (t) + L ey (t) + (1 + Φ(ε ))C L ε (t)) e i i y ε i i+1 i (8) where ui x and ui+1 x are the input signals of the X-axis in the ith and (i+1)th iterations, ui y and ui+1 y are the input signals of the Y -axis in the ith and (i+1)th iterations, ei x and ei y are the tracking errors of the X- and Y -axes in the ith iterations, L e and L are 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 ε = Kx ex + Ky ey (9) where the K x and K y are the coefficients related to the tracking error and the contour shape C x and C y are the cross-coupled gains along the X- and Y -axes Thus, let x and y be the components of contouring error along X- and Y -axes, respectively, C x and C y should 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 ( α(1 − |εδ | ), |ε| >δ (10) Φ(ε) = 0, |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 performed 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 ri (k) = P(q)ui (k) + d(k) (11) 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, ui 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 framework 22,33 is used by stacking the signals in vectors For N-sample sequences of input and output, Eq (11) can be written as ri (1) p1 · · · ui (0) di (1) ri (2) = p2 p1 · · · · ui (1) + di (2) (12) (N) · · · p1 ui (N − 1) di (N) ri {z pN pN−1 | } | {z } | {z } | {z } Rˆ i Pˆ ˆi U dˆ Based on Eq (12), the dynamics of the micro-motion stage using lifted system framework in Xand Y -axes can be expressed as x Xi+1 = Pˆ x Uˆ i + dˆ x (13) y Yi+1 = Pˆ y Uˆ i + dˆ y Then Eq (5) which is the iteration learning law of the ILC can be written in the lifted form as ˆ Uˆ i + Lˆ Eˆ i ) Uˆ i+1 = Q( (14) where Qˆ and Lˆ 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 x ˆ Uˆ xi + Lˆ e Eˆ xi + Nˆ Cˆ x Lˆ ε εˆi ) Uˆ = Q( i+1 (15) ˆy ˆ Uˆ yi + Lˆ e Eˆ yi + Nˆ Cˆ y Lˆ ε εˆi ) U i+1 = Q( where N = + Φ( i ) is the nonlinear gain and Nˆ is the lifted form of N According to Eq (13), the x y expression of Eˆ i and Eˆ i can be given as x x Eˆ i = Xˆ r − (Pˆ x Uˆ i + dˆ x ) y ˆy ˆ E i = Y r − (Pˆ y Uˆ i + dˆ y ) (16) where Xˆ r and Yˆ 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 " x# " # " # ˆx ˆ e Nˆ Cˆ x Lˆ ε E y + * Uˆ x Uˆ L Eˆ // = Qˆ ˆ y + (17) y Lˆ e Nˆ Cˆ y Lˆ ε Uˆ i+1 U ε ˆ , j 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 " x# " x #! " # U U Dˆ x ˆ ˆ (18) = Q M + y y U i+1 U Dˆ y i 015026-8 Jia et al AIP Advances 7, 015026 (2017) ˆ − Pˆ x (Lˆ e + Nˆ Cˆ x Lˆ ε Kˆ x )) Q(I −Qˆ Pˆ y Nˆ Cˆ x Lˆ ε Kˆ y where Mˆ = ˆ − Pˆ y (Lˆ e + Nˆ Cˆ y Lˆ ε Kˆ y )) , −Qˆ Pˆ x Nˆ Cˆ y Lˆ ε Kˆ x Q(I " # " # ˆ Lˆ e + Nˆ Cˆ x Lˆ ε Kˆ x )(Xˆ r − dˆ x ) + Nˆ Cˆ x Lˆ ε Kˆ y (Yˆ r − dˆ y )) Dˆ x Q(( ˆ ˆ = ˆ Lˆ e + Nˆ Cˆ y Lˆ ε Kˆ y )(Yˆ r − dˆ y ) + Nˆ Cˆ y Lˆ ε Kˆ x (Xˆ r − dˆ x )) , and K x and K x denote the lifted Dˆ y Q(( form of Kx and Ky , respectively The converged control in the iteration domain is defined as U∞ = lim Ui , and then a necessary " # i→∞ and sufficient condition for asymptotic convergence is sense of |U∞ − Ui + | < |U∞ − Ui | is presented for the system as22,27 ˆ ρ(M)