臺灣博碩士論文加值系統

本研究為發展以壓電陶瓷致動器驅動之具六自由度史都渥特型式奈米平台，內容包括了精密機構之研製、壓電陶瓷致動器之功能分析、磁滯模型及其前饋控制器之設計、平台運動數模之建立、系統量測校正及誤差補償方法之開發，以及系統之整合並驗證其功能。
精密機構研製之項目包括撓性接頭、壓電致動器、固定下平台及可動上平台，並利用有限元素法ANSYS軟體，分析接頭之剛性、平台之工作空間及其最大應力，找出適合之最佳設計。
利用所設計之動態Preisach模型進行磁滯現象之研究，不僅考慮其靜態行為，並同時分析因輸入電壓變化率所造成之動態行為。以所推導之動態Preisach模型，設計前饋控制器以降低磁滯之非線性影響。
設計工作空間之軌跡規畫，而期望之軸空間腳長值可經由逆向運動學求得。經由所提出之三點三軸量測法可獲得終端夾爪之工作空間變化量。由於製造誤差及組裝誤差的因素，期望夾爪變化量會與真實變化量不同，藉此建立誤差補償模型以校正機構所產生的誤差。
最後以螺旋軌跡進行平台追蹤性能之測試，透過不同行程之實驗，驗證所發展之平台系統能夠達到奈米等級定位之目標。故此設計不需要昂貴之感測器作為回授控制之用，符合經濟有效之實用目標。

A 6DOF Stewart-type nanoscale platform driven by piezoelectric actuator is developed. In this dissertation, several topics including the precision mechanism design, the piezoelectric actuator function analysis, the hysteresis model as well as its feed-forward controller design, the kinematics modeling, the system measurement and calibration as well as the error compensation method, and the system integration are investigated in this dissertation.
The design and manufacture of precision mechanisms includes the flexure joints, the piezoelectric actuators, and the lower fixed-base platform and the upper movable platform. By means of ANSYS simulation, the stiffness of the joint, and the workspace as well as the maximum stress of the platform are studied in order to attain better design purpose.
The hysteresis phenomenon of the piezoelectric actuator is analyzed by means of proposed dynamic Preisach model, which concerns not only the static analysis, but also the dynamic behavior of the actuator corresponding to the change rate of input voltage. A feed-forward controller is designed based on the proposed dynamic Preisach model to deal with the nonlinear effect.
Path planning in the task-space is designed for practical implementation. The desired joint-space leg length is calculated by inverse kinematics. Variation of the end-effector is measured by the developed 3-points-3-axes method. As a matter of fact, the desired variation of the end-effector is different from the actual variation due to manufacture error and assembly error. Therefore, error compensation model is established for calibration.
The tracking performance of the platform is experimented following a spiral trajectory. From experimental data, it indicates that the developed platform system is able to achieve the target of nanoscale positioning for numerous ranges of manipulating stroke. Therefore, it verifies a cost-effective design with no need of sensor for feedback control for a complex 6DOF platform is practicable.

Contents

Abstract (In English) I
Abstract (In Chinese) III
Acknowledgement IV
Contents V
List of Figures VII
List of Tables XI

Chapter 1 Introduction 1
1.1 Background and Motivation 1
1.2 Literature Review 2
1.3 Organization 9

Chapter 2 Mechanism Design and Analysis 11
2.1 Fundamentals of Piezoelectric Actuators 12
2.2 Design of the Mechanism 17

Chapter 3 Hysteresis Modeling 28
3.1 Classical Preisach Model 28
3.1.1 Definition of the Classical Preisach Model 28
3.1.2 Geometric of the Classical Preisach Model 30
3.1.3 Numerical Method of the Preisach Model 32
3.1.4 Database of the Preisach Model 35
3.1.5 Wipe-out Property 39
3.2 Extended Preisach Model 46
3.2.1 Problem of the Classical Preisach Model 47
3.2.2 Time-saving of the Interpolation Method 50
3.2.3 Interval of the Interpolation Method 53
3.3 Proposed Dynamic Preisach Model 55
3.4 Controller Design for Hysteresis Model 64
3.5 Simplified Rules 67

Chapter 4 Configuration and Calibration of the Stewart-type Nanoscale Platform 73
4.1 Kinematics of the Stewart-type Platform 73
4.2 Configuration of the Stewart-type Platform 77
4.3 Measurement Method of the Stewart-type Platform 82
4.4 Error Compensation 89

Chapter 5 Experiment Results 92
5.1 Experiment Results of Single Piezoelectric Actuator 93
5.2 Experiment Results of Leg 96
5.3 Experiment Results of Stewart-type Platform 100

Chapter 6 Conclusion and Future Work 107
6.1 Conclusion 107
6.2 Future work 108

References 109
Publications 115
Curriculum Vitae 117

List of Figures

Figure 2.1 Diagram of a leg actuator 11
Figure 2.2 Zero-point offset of piezoelectric actuator with constant force 15
Figure 2.3 Effective displacement of piezoelectric actuator acting against a spring load 16
Figure 2.4 Diagram of flexure joint design 18
Figure 2.5 Deformation of several cases of flexure joints 19
Figure 2.6 Design of flexure joint with a stainless steel rod 20
Figure 2.7 Flexure joint with a stainless steel rod 21
Figure 2.8 Casing design for piezoelectric actuators 22
Figure 2.9 Extension of the leg simulated by ANSYS 22
Figure 2.10 Experiment results of the six legs 23
Figure 2.11 Taper design for the whole platform 23
Figure 2.12 Workspace of the nanoscale platform 24
Figure 2.13 Maximum deformation of the Stewart platform 26
Figure 2.14 Force output of the platform 27
Figure 3.1 Hysteresis operator of Preisach model 28
Figure 3.2 Illustration of the hysteresis operator of Preisach model 29
Figure 3.3 Diagram of the Preisach model 30
Figure 3.4 Region S+ referring to the input voltage 31
Figure 3.5 Region R1 referring to Preisach function 33
Figure 3.6 Hyeteresis loop and integrated region for increasing voltage 34
Figure 3.7 Hyeteresis loop and integrated region for decreasing voltage 35
Figure 3.8 Mesh of discrete outputs of Preisach model 36
Figure 3.9 First-order reversal curve of Preisach model 36
Figure 3.10 Simulation of the Preisach model (ascending) 37
Figure 3.11 Simulation of the Preisach model (descending) 38
Figure 3.12 Simulation of the Preisach model (with problem) 38
Figure 3.13 Experiment results of the wipe-out property 40
Figure 3.14 Illustration of the wipe-out property 41
Figure 3.15 Diagram of the wiped-out property 42
Figure 3.16 Input voltage of two different sequences 43
Figure 3.17 Hysteresis loop without concern of the wipe-out property 43
Figure 3.18 Hysteresis loop with concern of the wipe-out property 44
Figure 3.19 Output displacement without concern of the wipe-out
property 44
Figure 3.20 Output displacement with concern of the wipe-out property
45
Figure 3.21 Example of the classical Preisach model 46
Figure 3.22 Limiting triangle with a “virtual” column of input voltage
48
Figure 3.23 Example of the extended Preisach model 49
Figure 3.24 Bilinear interpolation method 51
Figure 3.25 Calculation for triangle by use of square method 52
Figure 3.26 Limiting triangle with a “virtual” column of input voltage
and modified square cell 52
Figure 3.27 Model error for different interval of input voltage 54
Figure 3.28 Profile of arbitrary input voltage 56
Figure 3.29 Hysteresis model for various applied input voltage 60
Figure 3.30 Dynamic Preisach model for different cases 62
Figure 3.31 Tracking control under external force 63
Figure 3.32 Block diagram of the feed-forward controller 64
Figure 3.33 Flow chart of the inverse Preisach model 66
Figure 3.34 Diagram of searching the input voltage 67
Figure 3.35 Hysteresis curve for different ascending voltage 68
Figure 3.36 Hysteresis curve for different descending voltage 70
Figure 3.37 Desired trajectory and the distance between the desired
points 72
Figure 4.1 Definition of Stewart platform coordinates 74
Figure 4.2 Geometry of the nanoscale platform 75
Figure 4.3 Schematic drawing of Stewart-type platform 77
Figure 4.4 Leg configuration while the platform acting with linear motion 79
Figure 4.5 Geometric relation of measurement direction 83
Figure 4.6 Geometric relation of measurement 84
Figure 4.7 Motion of points S1 manipulating in the z direction 85
Figure 4.8 Motion of points S3 manipulating in the z direction 86
Figure 4.9 Output displacement computed by S1 and S3 87
Figure 4.10 Output orientation computed by S1 and S3 87
Figure 4.11 Output orientation computed by S1 and S2 88
Figure 4.12 Output orientation computed by (4.23) 88
Figure 4.13 Diagram of error compensation process 89
Figure 4.14 Relationship between h and hd 90
Figure 4.15 Position error after compensation in the z direction 91
Figure 5.1 Experiment Setup of Stewart-type nanoscale platform 92
Figure 5.2 Tracking control with long stroke about 12μm (case (a)) 94
Figure 5.3 Tracking control with long stroke about 12μm (case (b)) 94
Figure 5.4 Tracking control with short stroke about 1.2μm (case (c)) 95
Figure 5.5 Tracking control with short stroke about 1.2μm (case (d)) 95
Figure 5.6 Tracking control of the leg with stroke of 60μm (case (a1))
97
Figure 5.7 Tracking control of the leg with stroke of 60μm (case (b1))
97
Figure 5.8 Tracking control of the leg with stroke of 15μm (case (c1))
98
Figure 5.9 Tracking control of the leg with stroke of 15μm (case (d1))
98
Figure 5.10 Tracking control of the leg with stroke of 5μm (case (e1)) 99
Figure 5.11 Tracking control of the leg with stroke of 5μm (case (f1)) 99
Figure 5.12 Spiral trajectory for tracking in the X-Y plane (case (a2))
before compensation 101
Figure 5.13 Spiral trajectory for tracking in the X-Y plane (case (a2))
after compensation 101
Figure 5.14 Tracking error of the spiral trajectory (case (a2)) 102
Figure 5.15 Orientation error of the spiral trajectory (case (a2)) 102
Figure 5.16 Tracking control of the spiral trajectory (case (a2)) 103
Figure 5.17 Spiral trajectory for tracking in the X-Y plane (case (b2))
before compensation 104
Figure 5.18 Spiral trajectory for tracking in the X-Y plane (case (b2))
after compensation 104
Figure 5.19 Tracking error of the spiral trajectory (case (b2)) 105
Figure 5.20 Orientation error of the spiral trajectory (case (b2)) 105
Figure 5.21 Tracking control of the spiral trajectory (case (b2)) 106

List of Tables

Table 2.1 Deformation of the flexure joint under applied force 10N
(unit: μm) 19
Table 2.2 Stiffness of the flexure joint (unit: N/μm) 20
Table 3.1 Model error for different interval of input voltage 54
Table 3.2 Computation efficiency without/with simplified rule 72
Table 4.1 Maximum pure translations and rotations of the platform and the corresponding leg configuration 81
Table 4.2 Maximum pure translations and rotations of the platform and the corresponding leg configuration (modified) 81
Table 4.3 Measures of each measurement direction 84

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