即時-NURBS-運動插值器於六軸正三角構型並聯機器人之控制分析

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研究生：	簡國華 Kuo-Hua Chien
論文名稱：	即時-NURBS-運動插值器於六軸正三角構型並聯機器人之控制分析 Control of a Slide Equilateral Triangle Parallel Manipulator by Real-Time NURBS Motion Interpolator
指導教授：	莊華益 Hua-Yi Chuang
口試委員:	成維華 Wei-Hua Chieng 劉祖華 Thu-Hua Liu 楊勝明 Sheng-Ming Yang 劉昌煥 Chang-Huan Liu 唐永新 Yeong-Shin Tarng 黃安橋 An-Chyau Huang
學位類別：	博士 Doctor
系所名稱：	工程學院 - 機械工程系 Department of Mechanical Engineering
論文出版年：	2008
畢業學年度：	96
語文別：	英文
論文頁數：	128
中文關鍵詞：	並聯機器人、工作空間、奇異點、即時NURBS運動插值器、適應性控制
外文關鍵詞：	Workspace, Singularity, SETPM, Real-Time NURBS Motion Interpolator, Adaptive control
相關次數：	點閱：220 下載：3
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即時NURBS運動插值器於六軸正三角構型並聯機器人之控制分析
研究生：簡國華
指導教授：莊華益博士
日期：96 年 12 月
論文摘要
近來，隨著半導體、光電、資訊、生物科技與奈米技術的蓬勃發展，產品外觀朝向輕薄、短小與多曲面的自由造型，其在製造上需要提供一響應速度快、定位精度高與多自由度運動的操作平台，方能符合生產上的需求。並聯機構所構成的並聯機器人具有上述的特性與優於傳統的串聯機構所構成的機器人，故並聯機器人被廣泛的應用於各種需求響應速度快、定位精度高與多自由度運動的場合。因此，本論文的研究目的是設計一新的六軸正三角構型的並聯機器人，並應用即時NURB運動插值器，產生曲線的速度及加速度命令，設計控制器控制此並聯機器人，以符合上述的問題。
本論文包含理論分析與實驗兩部份。首先，推導此並聯機器人的正(反)向運動學與正(反)向動力學，並由求得之運動學探討其運動特性，包括工作空間及奇異點。經由數值模擬結果，可知此機器人在工作空間內並不存在奇異點及比傳統的並聯機器人具有體積小與較高的載重比等優點。其次，一般傳統是使用近似的方法，求得曲線的速度及加速度，雖然其方法簡單，但因其並不是真實的曲線速度及加速度，故在實際應用上會損失其精確度。有別於傳統的方法，本論文使用即時NURB運動插值器方法，獲得曲線的真實速度及加速度，並利用此速度與加速度設計控制器，構成一即時PC-based控制系統。
此外，經由所獲得之系統動態方程式及即時NURB運動插值器所產生的速度與加速度命令，本論文提出兩種不同的控制器，控制此六軸正三角構型的並聯機器人。第一個方法，是設計一順向(feedforward)補償控制與比例-微分控制器(PD)，此方法可減少計算反向動力學的時間與軌跡的追蹤誤差，進而實現即時控制系統。第二個方法，是藉由參數適應性律(adaptive law)方法，估測系統的動態參數如平台質量、慣性與摩擦等參數，並結合回授控制器，以減少其軌跡的追蹤誤差，提高其精確度。另外，此控制器滿足李亞譜諾(Lyapunov)的不確定系統的穩定性理論。
依據理論分析與實驗結果，本論文所提出的藉由即時NURB運動插值器所構成的PC-based控制系統，成功的應用於此新設計的六軸正三角構型的並聯機器人。本研究的結果，提供一響應速度快、定位精度高與多自由度運動的操作平台，可實際應用於各領域的工程上。

Abstract
Title of Dissertation: Control of a Slide Equilateral Triangle Parallel Manipulator by Real-Time NURBS Motion Interpolator
Kuo-Hua Chien , Doctor of Philosophy, 2007
Dissertation directed by: Professor Hua-Yi Chuang
Department of Mechanical Engineering

There are an increasing number of biotechnology, photoelectric, information, semiconductor and nanotechnology industrial applications where the profile of products is light and thin, short and small with freeform surface, it needs to offer a fast response and a very complex motion with many degrees of freedom system to fabricate. The parallel manipulators have high rigidity and accuracy, and high load capacity. The advantage of parallel manipulators over serial manipulators mean that the parallel manipulators have been used in any filed. The advantages of parallel manipulators over serial manipulators that parallel manipulators can be overcome the problem. This study was motivated by the problem of the trajectory generation method for control of a novel slide equilateral triangle parallel manipulator (SETPM) to provide a fast response and a very complex motion with many degrees of freedom system.
This dissertation consists of the theoretical development and experiments. First of all, the kinematics and dynamics solutions consist of inverse and forward are derived and its singularity and workspace of the manipulator are analyzed. The parallel manipulator has compact configuration and no singularities in the workspace. Secondary, the conventional method utilizes the approximate method to obtain the characteristics of curves of the velocity and acceleration for control of manipulator in the joint space. The conventional method is simple. However, it cannot get the true velocity and true acceleration of the curve, but create a loss in accuracy. This study utilizes the differential geometry to obtain the true velocity and true acceleration of the curve, to develop the real-time NURBS motion interpolator command generation, to the motion dynamics of the SETPM including position, velocity and acceleration for control of the manipulator on a PC-based system.
Also, this study presents two different types of controllers for controlling the parallel manipulator. The first method is a feedforward compensation scheme with proportional- derivate control based on the real-time NURBS motion interpolator command generation to replace inverse dynamic control, it reduces both the time to compute the inverse dynamic and the tracking error. The second method is an adaptive feedforward part with the inverse dynamic and nonlinear feedback loop based on the real-time NURBS motion interpolator command generation. The adaptive feedforward control algorithm ensures a parameter adaptation law that satisfies the Lyapunov-based stability theory of uncertain systems.
On the basis of the experimental results, we can conclude that the proposed real-time NURBS motion interpolator command generation for the two types of feedforward compensation with feedback PD controllers has also been successfully applied to a novel slide equilateral triangle parallel manipulator on a personal computer to achieve fast and precise motion. The results of this research provide a fast response and a very complex motion with many degrees of freedom system which can apply to various kinds of engineering applications.

Contents
中文摘要 _________________________________________________I
Abstract _________________________________________________III
Acknowledgements ________________________________________ V
Nomenclature ___________________________________________IX
List of Figures ____________________________________________XI
List of Tables ___________________________________________XIV
Chaper 1 Introduction	1
1. 1. Background………..	1
1. 2. Research issues	3
1. 3. Organization of dissertation	5
Chaper 2 Kinematic analysis of SETPM  _______________________7
2. 1. Introduction	7
2. 2. Description of the manipulator	8
2. 2. 1.	Coordinate system of the maipulator…………………………….………10
2. 2. 2.	Position analysis of the manipulator……………………………………..11
2. 3.Kinematic of  the manipulator	12
2. 3. 1.	Inverse kinematic ……………….…………………………….…………12
2. 3. 2.	Direct kinematic …………………..……………………………………..12
2. 4.Simulation results	14
2. 5.Summary	18
Chaper 3 Singularity and workspace analysis of SETPM	19
3. 1. Introduction	19
3. 2  Jacobin matrices	20
3. 3. Workspace of the manipulator	23
3. 4. Singularity analysis	26
3. 5. Simulation results	27
3. 6. Summary	35
Chaper 4 Dynamic analysis of SETPM	37
4. 1. Introduction	36
4. 2. Velocity and acceleration analysis	37
4. 3. Dynamics of the moving platform	38
4. 4  Dynamics of the actuator joint	40
4. 5  Principle of virtual work	41
4. 6  Simulation results	43
4. 7  Summary	46
Chaper 5 Trajectory planning _______________________________47
5. 1. Introduction	47
5. 2. Curve representations	48
5. 3. NURBS and real-time interpolator	49
5. 3. 1. Properties of NURBS curve  .…..……………………….…………….…49
5. 3. 2.	Derivatives of a NURBS curve ….……………………….…………..51
5. 3. 3. Real-time interpolator for NURBS………….…………………………..52
5. 4. Existing interpolator	55
5. 4. 1. Desired velocity   ………………..……………………….…………...…55
5. 4. 2.	Desired acceleration  ……………...……………………………………..55
5. 5. Proposed interpolator	56
5. 5. 1.Desired velocity..…..……………..……………………….…………...…56
5. 5. 2.	Desired acceleration ……………...………………………………….…..57
5. 6. Simulation results	58
5. 7. Summary	62
Chaper 6 Controller design for SETPM _______________________63
6. 1. Introduction……………………………………………………………………	63
6. 2. Dynamic equation	64
6. 3. Feedforward compensation controller	65
6. 4. Adaptive feedforward controller ………………………………………….….	68
6. 4.1. Stability analysis of the controller ….…………………….…………....…72
6. 5. Simulation results	74
6. 6. Summary	79
Chaper 7 Experiment	80
7. 1. Real-time operation system………………………………..	80
7. 2. Experiment system	82
7. 3. Parameter estimation	83
7. 4. Experimental results 	86
7. 4.1. Feedforward compensation control…...………………….…………....…86
7. 4.2. Adaptive feedforward compensation control….………….…………....…91
7.5.Summary………………………………………………………………………..94
Chaper 8 Conclusion	95
8. 1. List of contributions	95
8. 2. Future work	98
References _______________________________________________99
Appendix A______________________________________________107
Appendix B______________________________________________109
Appendix C  _____________________________________________111
Vita __________________________________________________113




Nomenclature
e	tracking error
 	force exerted at the center of mass of the input joint
 	X component of the curvature of curve
ml	input joint mass
mp	moving platform mass
 	moment exerted at the center of mass of the moving platform
 	position of joint
rb	base platform radius
rp	moving platform radius
ui	arbitrary parameter
wi    	weight vector of NURBS
A (X, Y, Z)	fixed base reference frame
B (x, y, z)	moving platform frame
ARB( α,β,γ)	rotation matrix
F	sum of applied and inertia wrenches about the center of mass of the moving platform
 	gravity
 	nonlinear terms including Coriolis and centrifugal force
 	inertia matrix of the moving platform taken about its center of mass and expressed in the fixed frame A
J	Jacobian matrix of manipulator
L	limb length
M(p)  	inertia matrix
 	blending function of NURBS
Pi   	ordinal control points of NURBS
T	sampling time in interpolation
U	knot vector of NURBS

V	feedrate
 	velocity of the mass center of the moving platform
Xi	x components of the control points
Yi	y components of the control points
Zi	z components of the control points
 	estimated constant matrix of the gravity
 	estimated constant matrix of the nonlinear term
 	estimated constant matrix of the inertia
 	corresponding velocity command
 	corresponding acceleration command
 	first order derivatives of the NURBS
 	second order derivatives of the NURBS
 	X-component of the velocity
 	X-component of the acceleration
 	regression matrix
 	dynamic parameters
 	parameter estimation error
 	regression matrix error
 	learning factors
 	X-component of the curve tangent vector
ε	dead-zone function
 	forces of actuate
 	angular velocity of the mass center of the moving platform
 	angular velocity of the limb i
 	virtual displacement of the actuated joints
 	virtual displacement of the moving platform

List of Figures
Figure 2.1	Schematic diagram of a slide equilateral triangle parallel………….……8
Figure 2.2	Joint and loop graph of a slide equilateral triangle parallel manipulator...9
Figure 2.3	Diagram of coordinate system……………………………………….….10
Figure 2.4	The solutions of the inverse kinematic of the constant orientation motion...……………..……….…………………………………………..15
Figure 2.5	The solutions of the direct kinematic of the constant orientation motion(a)position(b)orientation…………….……………………………………15
Figure 2.6	The solutions of the inverse   kinematic of the variable orientation motion.16
Figure 2.7	The solutions of the direct kinematic of the variable orientation motion (a) position(b)orientation…...……………....…………………….…....16
Figure 2.8	Configurations of the direct kinematic solutions….………………………..17
Figure 3.1	Flow chart of workspace search…………………………………………...25
Figure 3.2	The singular configuration of the SETPM: (a) one example of the first kind of     singularity (b) one example of the second kind of singularity (c )one example of the third kind of singularity……………………………..……..29
Figure 3.3	The constant orientation workspace of the SETPM (a) 3D view (b) x-y plane (c) x-z plane (d) y-z plane….…...………….…………………………….30
Figure 3.4	The 3D constant orientation workspace of the SETPM (a)   (b)   (c)  .31
Figure 3.5	Configurations of the constant orientation workspace and singularity of the SETPM……………………………………………………………………...33
Figure 3.6	Configurations of the workspace and singularity of the SETPM in z=152mm  (a)  (b)  (c)  (d) .……….........……………………………….34
Figure 4.1	The free-body-diagram of the moving platform……….…………………...38
Figure 4.2	The free-body-diagram of the actuator joint….…………………………….40
Figure 4.3	The force follows the constant orientation trajectory path (a) joint force(b) actuate force…………………………………………………………….44
Figure 4.4	The corresponding determent value of the Jacobian of the constant orientation trajectory path………………………..…………………………44
Figure 4.5	The force follows the variable orientation trajectory path (a) joint force (b) actuate force…………………..……………………………………….45
Figure 4.6	The corresponding determent value of the Jacobian of the variable orientation trajectory path………..…………………………………………45
Figure 5.1	Parametric representation of a path in space…..……………………………48
Figure 5.2	The NURBS relationship between the B-splines and the B&eacute;zier……..…….49
Figure 5.3	(a) A circle NURBS; (b) A two petal NURBS; (c) corresponding rational basis functions……………………………..………..………………………59
Figure 5.4	(a) A three-dimensional NURBS; (b) corresponding rational basis functions…………………………….……………………………………...60
Figure 5.5	Velocity and acceleration components for feedrate 100mm/s (simulation): (a) proposed method (b) conventional method……….…..……………………61
Figure 6.1	Schematic diagram of the feedforward compensation with feedback control based on the real-time NURBS motion interpolator command generation: (a) conventional method (b) proposed method…………...……………………66
Figure 6.2	Schematic diagram of the real-time NURBS motion interpolator for adaptive feedforward compensation with feedback control: (a) conventional method (b) proposed method……………………………..…………………………71
Figure 6.3	A circle path tracking for feedrate 100mm/s: (a) trajectory (b) tracking error of the x-axis (c) tracking error of the y-axis…………..…………………..75
Figure 6.4	A circle path tracking for feedrate 300mm/s: (a) trajectory (b) tracking error of the x-axis (c) tracking error of the y-axis………..……………………..76
Figure 6.5	A circle path tracking for feedrate 500mm/s: (a) trajectory (b) tracking error of the x axis (c) tracking error of the y axis…..……………….…………..77
Figure 6.6	A two-petals path tracking for feedrate 300mm/s: (a) trajectory (b) tracking error…………………………….…………………………………………78
Figure 6.7	A two-petals path tracking for feedrate 500mm/s: (a) trajectory (b) tracking error…………………………………………………………………………78
Figure 7.1	Block diagram of the real-time PC-based control system……………...…81
Figure 7.2	The experimental setup…….………..…………………………………….82
Figure 7.3	Parameter estimates for the AFFC: (a) mass estimates (b) friction estimates (c) moment estimates (d) inertia estimates……….………………………..85
Figure 7.4	Tracking error for feedrate 300mm/s: (a) conventional method (b) proposed method………………………………………………………………………87
Figure 7.5	Tracking error for feedrate 500mm/s: (a) conventional method (b) proposed method………………………………………………………………………88
Figure 7.6	3D position tracking for feedrate 100mm/s: (a) proposed method (b) conventional method………………………………………………………..89
Figure 7.7	3D position tracking for feedrate 300mm/s: (a) proposed method (b) conventional method………………….…………………………………….90
Figure 7.8	Tracking error for feedrate 300mm/s: (a) conventional method (b) proposed method………………………………………………………………92
Figure 7.9	Tracking error for feedrate 500mm/s: (a) conventional method (b) proposed method………………………………………………………………………93













List of Tables
Table 3.1	Parameter of the simulation…...…………………..……………………28
Table 3.2	The simulation results of the workspace………………..……………….34
Table 5.1	A comparison between conventional and proposed interpolators for adaptive feedforward control of the SETPM……….………..…………57
Table 5.2	Parameter setting of NURBS for 2D and 3D path…….….…………….58
Table 7.1	The comparison of estimated with actual parameters……….….………84

                                

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