臺灣博碩士論文加值系統

本研究的重點是針對使用壓電致動器驅動的六自由度史都沃特型平台設計適當之控制方法，在受到外力影響時仍能保持很好的定位精度。首先藉由運動學分析模型與實驗測量終端位置之校正，以提高系統參數的精度。非線性的磁滯、潛變效應、漂移干擾、或溫升效應是使用壓電致動器常見之負面因素，會直接影響系統的精度以及穩定性，故設計控制方法須特別考量。本論文利用Preisach方法推導出壓電致動器的磁滯模型，以建立磁滯前饋控制器來解決磁滯問題。另利用已廣泛用於統計製程控制並能克服系統變化和漂移干擾之指數加權移動平均(Exponentially Weighted Moving Average, EWMA)方法，嘗試將EWMA方法轉移至兩次運轉對比(Run-to-Run)之數位模型參考適應系統(MRAS)，並將磁滯前饋控制器整合於內，完成一套定位控制系統。除了單級演算之EWMA方法，具有兩級演算之EWMA進階型預測校正控制器(Predictor Corrector Control, PCC)，亦將設計與驗證功能之優劣。
除了位移控制器，亦結合非線性PID控制器與常見之計算扭矩方法來設計力回饋控制器，以克服未知且隨環境變化之外力影響。配合選擇之表面研磨範例實驗，於系統中裝設力傳感器和電容式位移傳感器以即時量測工件之受力及形變。本論文進行了幾個加工案例研究，以評估所設計之複合式控制器的效能，亦與單獨使用前饋或力反饋控制器就無負載自由空間運動以及遭遇外力負載操作比較性能，證實其效果優異，無負載運行可以達到平移均方根誤差(x: 95.436 nm, y: 172.513 nm, z: 1111.581 nm)及旋轉均方根誤差(θx: 1.112 nrad, θy: 1.009 nrad, θz: 0.689 nrad)，遭遇外力負載可以達到平移均方根均方根誤差(x: 257.442 nm, y: 182.306 nm, z: 1187.987 nm)及旋轉誤差(θx: 3.35 nrad, θy: 7.015 nrad, θz: 0.687 nrad)的奈米級定位精度。

This study focuses on the development of appropriate control method for a 6DOF Stewart-type platform driven by piezoelectric actuators. It is aimed to preserve good positioning accuracy while encountered with external forces in particular. Kinematic calibration is firstly carried out by using pose measurement to improve the accuracy of kinematic parameters. Negative factors of using piezoelectric actuators such as nonlinear hysteresis, creep, drifting disturbance, and temperature rise that directly affect the accuracy and steadiness of the system are concerned. In this article, modeling of the hysteresis of a piezoelectric actuator is derived to build a hysteresis feedforward controller by means of Preisach method that is able to deal with the rate-independent nonlinear hysteresis. Exponentially Weighted Moving Average (EWMA) method has been widely used in statistical process control and verified its capability of overcoming systematic change and drift disturbance. An attempt is to map the EWMA method into a run-to-run (RtR) Model Reference Adaptive System (MRAS) and combine with the hysteresis feedforward controller for position control. Similarly, a Predictor Corrector Control (PCC) with two stages of EWMA formulas is also used and verified its capability of overcoming the drifting disturbance due to creep and temperature dependence of piezoelectric actuators.
Besides the position controller, an improved robust force feedback controller that is based on the idea of combining a nonlinear PID controller with the computed torque method is also investigated. The algorithm provides an essential way of dealing with unacknowledged interacting forces and variations of the environment characteristics. An example of surface grinding on different materials is investigated. Force sensor and capacitive displacement sensors are used to measure the interacting forces and the deformation respectively on the surface where the indentor tool is attached. Several case studies are performed to evaluate its effectiveness and robustness of the proposed controller. Comparison of the proposed controller with other controllers for free-space motion as well as for manipulation encountered with external load is carried out. The experiment results show that using the proposed composite controller is much better than using the feedforward or force feedback controller alone. The position accuracy can achieve root mean square error RMSE (x:95.436nm, y:172.513nm, z:1111.581nm) in translation and (θx:1.112nrad, θy:1.009 nrad, θz:0.689nrad) in orientation for free-space manipulation and RMSE (x:257.442nm, y:182.306nm, and z:1187.987nm) in translation and (θx:3.35nrad, θy:7.015 nrad, θz:0.687nrad) in orientation while encountered with external load.

摘要i
Abstract ii
Acknowledgements iv
Contents v
List of Figures viii
List of Tables xi
1 Introduction 1
1.1 Robotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Motivation and Objective of the Study . . . . . . . . . . . . . . . . 9
2 Literature Review 10
2.1 Mechanical Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Kinematic Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Error-model based calibration . . . . . . . . . . . . . . . . . 12
2.2.2 Self-calibration of parallel manipulators . . . . . . . . . . . . 13
2.2.3 Optimal calibration con gurations of parallel manipulators . 15
2.3 Control of Parallel Robots . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Model-based control . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Synchronized and adaptive control . . . . . . . . . . . . . . 17
2.3.3 Robust control . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Applications of Nanometer Positioning . . . . . . . . . . . . . . . . 19
3 Calibration and Sti ness Analysis 21
3.1 Mechanism Description . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Kinematics Model of the Platform . . . . . . . . . . . . . . . . . . . 24
3.2.1 Inverse kinematics . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Forward kinematics . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Jacobian Analysis of the Platform . . . . . . . . . . . . . . . . . . . 28
3.4 Kinematic Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4.1 Di erential error model . . . . . . . . . . . . . . . . . . . . . 32
3.4.2 Observability of kinematic error parameters . . . . . . . . . 33
3.4.3 Data acquisition . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.3.1 Measurement equipment . . . . . . . . . . . . . . . 34
3.4.3.2 Measurement method . . . . . . . . . . . . . . . . 35
3.4.3.3 Calibration algorithm . . . . . . . . . . . . . . . . 38
3.5 Sti ness of the Platform . . . . . . . . . . . . . . . . . . . . . . . . 41
3.5.1 Sti ness modeling of the platform . . . . . . . . . . . . . . . 42
3.5.2 Sti ness evaluation . . . . . . . . . . . . . . . . . . . . . . . 44
4 An Adaptive Control for Rate-Dependent Hysteresis 47
4.1 Fundamental of leg actuator . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Another look at limitations of PZT actuator . . . . . . . . . . . . . 50
4.2.1 Hysteresis e ect . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.2 Creep e ect . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.3 Temperature dependence . . . . . . . . . . . . . . . . . . . . 54
4.3 Adaptive controller design . . . . . . . . . . . . . . . . . . . . . . . 54
4.3.1 Modi ed model reference adaptive system . . . . . . . . . . 55
4.3.2 Rate-independent hysteresis compensation . . . . . . . . . . 56
4.3.2.1 Classical Preisach model . . . . . . . . . . . . . . . 56
4.3.2.2 Numerical Preisach model . . . . . . . . . . . . . . 58
4.3.2.3 Dynamics Preisach model . . . . . . . . . . . . . . 60
4.3.2.4 Inversion of Preisach model . . . . . . . . . . . . . 61
4.3.3 EWMA controller . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.4 PCC controller . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.4 Evaluation of controller performance . . . . . . . . . . . . . . . . . 70
4.4.1 Experiment setup . . . . . . . . . . . . . . . . . . . . . . . . 70
4.4.2 Evaluation on single piezoelectric actuator . . . . . . . . . . 71
4.4.3 Evaluation on the Stewart-type platform . . . . . . . . . . . 74
4.4.3.1 Case 1: Moving along X, Y, and Z axes without
rotation . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4.3.2 Position tracking error on X axis . . . . . . . . . . 83
4.4.3.3 Position tracking error on Y axis . . . . . . . . . . 83
4.4.3.4 Position tracking error on Z axis . . . . . . . . . . 83
4.4.3.5 Rotation tracking error around X axis . . . . . . . 83
4.4.3.6 Rotation tracking error around Y axis . . . . . . . 84
4.4.3.7 Rotation tracking error around Z axis . . . . . . . 84
4.4.3.8 Case 2: Moving along Z axis and rotating around
X and Y axes ..........................................................................84
5 Position/Force Feedback Control for Application 91
5.1 Dynamic Modeling ........................................................................................84
5.1.1 Velocity and acceleration of a leg .........................................................84
5.1.2 Dynamic formulation of the legs ..........................................................84
5.1.3 Dynamic formulation of the moving platform ......................................84
5.2 Dynamic Position/Force Control ...................................................................84
5.3 Gain Tuning of the Controller ........................................................................84
5.4 Case Study of Cutting Application ................................................................84
5.4.1 Experiment setup ..................................................................................84
5.4.2 Trajectory planning ...............................................................................84
5.4.3 Case 1 ....................................................................................................84
5.4.4 Case 2 ....................................................................................................84
5.4.5 Case 3 ....................................................................................................84
5.5 Experiment Results ........................................................................................84
5.5.1 Experiment results for case 1 ................................................................84
5.5.2 Experiment results for case 2 ................................................................84
5.5.3 Experiment results for case 3 ................................................................84
6 Conclusions 124
Publications 126
Bibliography 127



List of Figures
1.1 SCARA - Serial Robot Illustration . . . . . . . . . . . . . . . . . . 3
1.2 Examples of six degree-of-freedom serial manipulator . . . . . . . . 4
1.3 The original Gough platform at birth in 1954 . . . . . . . . . . . . 5
1.4 The rst
ight simulator based on an octahedral hexapod . . . . . . 6
1.5 The rst Delta robot, developed in 1985 . . . . . . . . . . . . . . . 6
1.6 Delta robot models o ered by ABB, Fanuc, Hiwin and Kawasaki . . 7
1.7 ALIO Hybrid Hexapod (Courtesy of ALIO Industries) . . . . . . . . 8
3.1 Structure (a) and Kinematic con guration (b) of the 6DOF Stewart
Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Schematic drawing of the 6DOF Stewart Platform . . . . . . . . . . 23
3.3 Flowchart of solving forward kinematics of the Stewart platform . . 27
3.4 Desired trajectory for inverse kinematic calculation . . . . . . . . . 28
3.5 Leg lengths - resulted from inverse kinematics - are used as inputs
for forward kinematics . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6 Error between the forward kinematic solution and the desired trajectory
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.7 The con guration of the employment of CMM . . . . . . . . . . . . 35
3.8 The entire con guration of the employment of the sensor . . . . . . 36
3.9 Geometric relation of measured parameters . . . . . . . . . . . . . . 37
3.10 A case study of computing angle z . . . . . . . . . . . . . . . . . 37
3.11 Angle z is a ected by orientation w.r.t axis y and x . . . . . . . 38
3.12 The
owchart of calibration algorithm . . . . . . . . . . . . . . . . 39
3.13 External force/torque acting on the moving platform and actuator
force on each leg of the platform . . . . . . . . . . . . . . . . . . . . 43
3.14 Reciprocal of a sti ness matrix of the Stewart-type nanoscale platform
over the constant orientation workspace: (a) x = 25 rad,
(b) y = = 25 rad, (c) z = 10 rad . . . . . . . . . . . . . . . . . 45
3.15 Maximum singular value of the Jacobian matrix over the constant
orientation workspace: (a) x = 25 rad, (b) y = 25 rad, (c)
z = 10 rad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1 Structure of a single PZT-based leg of the manipulator . . . . . . . 48
4.2 Hysteresis and its e ect on a PZT actuator . . . . . . . . . . . . . . 51
4.3 Rate-dependent hysteresis . . . . . . . . . . . . . . . . . . . . . . . 51
4.4 Slope change vs. Input signal frequency . . . . . . . . . . . . . . . . 52
4.5 Creep response on a PZT actuator . . . . . . . . . . . . . . . . . . 53
4.6 A slow creep response in output displacement after a short transient
response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.7 EWMA or PCC Controller embedded on an MRAS structure . . . . 55
4.8 Structure of model for a PZT actuator . . . . . . . . . . . . . . . . 57
4.9 Transfer characteristic of the elementary hysteresis operator (a) and
geometry representation of Preisach ( ; ) plane (b) . . . . . . . . . 58
4.10 Numerical implementation of Preisach model for a case study . . . . 59
4.11 Compensation of rate-independent hysteresis of a PZT actuator by
inverse control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.12 Performance of the identi ed weighting function . . . . . . . . . . 72
4.13 Tracking performance and tracking error (0.1 Hz) . . . . . . . . . . 73
4.14 Tracking performance and tracking error (1 Hz) . . . . . . . . . . . 73
4.15 Tracking performance and tracking error (10 Hz) . . . . . . . . . . 74
4.16 Contour of the desired trajectory - Case 1 . . . . . . . . . . . . . . 75
4.17 Desired displacement of legs -1, 2, 3 . . . . . . . . . . . . . . . . . . 76
4.18 Desired displacement of legs -4, 5, 6 . . . . . . . . . . . . . . . . . . 76
4.19 Tracking performance - Position (0.1 Hz) . . . . . . . . . . . . . . . 77
4.20 Tracking performance - Rotation (0.1 Hz) . . . . . . . . . . . . . . 77
4.21 Tracking error - Position (0.1 Hz) . . . . . . . . . . . . . . . . . . . 78
4.22 Tracking error - Rotation (0.1 Hz) . . . . . . . . . . . . . . . . . . . 78
4.23 Tracking performance - Position (1 Hz) . . . . . . . . . . . . . . . . 79
4.24 Tracking performance - Rotation (1 Hz) . . . . . . . . . . . . . . . 79
4.25 Tracking error - Position (1 Hz) . . . . . . . . . . . . . . . . . . . . 80
4.26 Tracking error - Rotation (1 Hz) . . . . . . . . . . . . . . . . . . . . 80
4.27 Tracking performance - Position (10 Hz) . . . . . . . . . . . . . . . 81
4.28 Tracking performance - Rotation (10 Hz) . . . . . . . . . . . . . . . 81
4.29 Tracking error - Position (10 Hz) . . . . . . . . . . . . . . . . . . . 82
4.30 Tracking error - Rotation (10 Hz) . . . . . . . . . . . . . . . . . . . 82
4.31 Tracking performance - Position (0.1 Hz) . . . . . . . . . . . . . . . 85
4.32 Tracking performance - Rotation (0.1 Hz) . . . . . . . . . . . . . . 86
4.33 Tracking error - Position (0.1 Hz) . . . . . . . . . . . . . . . . . . . 86
4.34 Tracking error - Rotation (0.1 Hz) . . . . . . . . . . . . . . . . . . . 87
4.35 Tracking performance - Position (1 Hz) . . . . . . . . . . . . . . . . 87
4.36 Tracking performance - Rotation (1 Hz) . . . . . . . . . . . . . . . 88
4.37 Tracking error - Position (1 Hz) . . . . . . . . . . . . . . . . . . . . 88
4.38 Tracking error - Rotation (1 Hz) . . . . . . . . . . . . . . . . . . . . 89
5.1 Schematic of the Stewart-type nanoscale platform with mechanical
parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2 Block diagram of the task-space position/force controller . . . . . . 100
5.3 Experiment setup for cutting application . . . . . . . . . . . . . . . 107
5.4 Prototype of the tool holder . . . . . . . . . . . . . . . . . . . . . . 108
5.5 Position trajectory of the moving platform in X, Y, and Z axes . . . 109
5.6 Rotation trajectory of the moving platform around X, Y, and Z axes 110
5.7 Trajectory of the moving platform in 3D . . . . . . . . . . . . . . . 110
5.8 Experiment setup for case 1 . . . . . . . . . . . . . . . . . . . . . . 111
5.9 Experiment setup for case 2 . . . . . . . . . . . . . . . . . . . . . . 111
5.10 Experiment setup for case 3 . . . . . . . . . . . . . . . . . . . . . . 112
5.11 External force on X, Y and Z axis - Case 1 . . . . . . . . . . . . . . 113
5.12 External torque on X, Y and Z axis - Case 1 . . . . . . . . . . . . . 113
5.13 Displacement tracking performance on X, Y and Z axis - Case 1 . . 114
5.14 Rotation tracking performance on x, y and z - Case 1 . . . . . . 114
5.15 Displacement error on X, Y and Z axis - Case 1 . . . . . . . . . . . 115
5.16 External force on X, Y and Z axis - Case 2 . . . . . . . . . . . . . . 116
5.17 External torque on X, Y and Z axis - Case 2 . . . . . . . . . . . . . 116
5.18 Rotation error on x, y and z - Case 2 . . . . . . . . . . . . . . . . 117
5.19 Displacement tracking performance on X, Y and Z axis - Case 2 . . 117
5.20 Rotation tracking performance on x, y and z - Case 2 . . . . . . 118
5.21 Displacement error on X, Y and Z axis - Case 2 . . . . . . . . . . . 118
5.22 Rotation error on x, y and z - Case 2 . . . . . . . . . . . . . . . . 119
5.23 External force on X, Y and Z axis - Case 2 . . . . . . . . . . . . . . 120
5.24 External torque on X, Y and Z axis - Case 2 . . . . . . . . . . . . . 120
5.25 Displacement tracking performance on X, Y and Z axis - Case 3 . . 121
5.26 Rotation tracking performance on x, y and z - Case 3 . . . . . . 121
5.27 Displacement error on X, Y and Z axis - Case 3 . . . . . . . . . . . 122
5.28 Rotation error on x, y and z - Case 3 . . . . . . . . . . . . . . . . 122
5.29 AFM images of micro-cutting on copper: (a) 2 m 2 m AFM topographic
in
at view (b) zoom in cutting edge of material (c) 3D
view of topographic of a cutting segment . . . . . . . . . . . . . . . 123


List of Tables
3.1 Measures of each measurement direction . . . . . . . . . . . . . . . 38
3.2 Calibration positions and errors after measurement (with respect to
frame fPg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Calibration orientations and errors after measurement (with respect
to frame fPg) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 The nal kinematic parameters . . . . . . . . . . . . . . . . . . . . 42
4.1 Slope vs. input signal frequency. . . . . . . . . . . . . . . . . . . . . 53
4.2 Position tracking error (nm) vs. input signal frequency. . . . . . . . 74
4.3 Position tracking error (nm) vs. input signal frequency - X-axis. . . 83
4.4 Position tracking error (nm) vs. input signal frequency - Y-axis. . . 83
4.5 Position tracking error (nm) vs. input signal frequency - Z-axis. . . 83
4.6 Position tracking error (nrad) vs. input signal frequency - x. . . . 84
4.7 Position tracking error (nrad) vs. input signal frequency - y. . . . 84
4.8 Position tracking error (nrad) vs. input signal frequency - z. . . . . 84
4.9 Position tracking error (nm) vs. input signal frequency - X-axis. . . 89
4.10 Position tracking error (nm) vs. input signal frequency - Y-axis. . . 89
4.11 Position tracking error (nm) vs. input signal frequency - Z-axis. . . 90
4.12 Position tracking error ( rad) vs. input signal frequency - x. . . . 90
4.13 Position tracking error ( rad) vs. input signal frequency - y. . . . 90
4.14 Position tracking error ( rad) vs. input signal frequency - z. . . . . 90
5.1 Parameters used in genetic algorithm . . . . . . . . . . . . . . . . . 106
5.2 Slope vs. input signal frequency. . . . . . . . . . . . . . . . . . . . . 107

[1] Y. Ting, C.-C. Li, T. V. Nguyen, Composite controller design for a 6DOF
Stewart nanoscale platform," Precision Engineering, vol. 37, pp. 671-683, 2013.
(SCI)
[2] Y. Ting, Y.-J. Shieh, T. V. Nguyen, and B.-K. Hou, Investigation and performance
evaluation of a d14 ceramic actuator," Journal of the European Ceramic
Society, vol. 34, pp. 2857-2864, 9/2014. (SCI)
[3] T. V. Nguyen, Y. Ting, and M. Leorna, Development of 6DOF nano-precision
Stewart platform for nano-milling application," in 2014 IEEE International Conference
on Robtics and Biomimetics (ROBIO 2014), 2014, pp. 1892-1897. (EI)
[4] T. V. Nguyen, Y. Ting, A Computed Feedforward Compensation and Robust
Dynamics Force Feedback Control for a 6DOF Stewart-type Platform," in
the 11th AIMS Conference on Dynamical Systems, Di erential Equations and Applications
(AIMS 2016), 2016. (EI)
[5] T. V. Nguyen, Y. Ting, Robust Nonlinear Force Control for a 6DOF Nano-
Precision Stewart Platform," in the International Conference on Advanced Technology
Innovation 2017 (ICATI 2017), 2017. (EI) (Accepted)
[6] Y. Ting, Suprapto, C. W. Peng, T. V. Nguyen, Design and Characterization
of a Composite Piezoelectric Ceramic Motor," in Twenty- fth International
Conference on Proceeding and Fabrication of Advanced Materials (PFAM-XXV),
22-25 January, 2017. (EI) (Accepted)
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