本文對非剛性機器手臂之連續分散式參數系統將輪軸（Hub）半徑及端點荷重列入考慮，作動態分析與控制系統設計，以達到非剛性機器手臂之精確軌跡控制。
首先，以漢米頓原理（Hamilton principle）推導得到此系統的非線性動態方程式（一非線性四階偏微分方程式）及相關邊界條件，經線性化及拉式轉換（Laplace transform）求得此系統開迴路轉移函數，利用Fortran for IMSL求出無限多組極零點位置分佈，然後利用根軌跡穩定分析法則使受控系統得以穩定並設計控制器，並比較當致動器與感測器考慮輪軸半徑及不考慮時對系統穩定度的差異，最後利用假設模態法（Assumed-mode method）推導出分離化後對時間相依函數的二階常微分方程式，再結合數值分析中的Runge-Kutta method 模擬線性及非線性運動方程式之暫態響應結果。
根據以上電腦程式模擬結果，我們可以看出此系統不論是線性或是非線性經由所設計的控制器，其響應結果皆可呈現出良好的穩定度及軌跡控制。

The stability and tracking control is discussed within a flexible robot arm system. The flexible robot arm is considered as a Bernoulli-Euler beam with tip-payload. At the same time, the position of the actuator and sensor are also taken into account.
In this thesis, first, the dynamic equation of the flexible robot arm is derived from Hamilton priciple. This equation is a fourth-order partial differential equation. Second, the open loop transfer function of the system is obtained from the dynamic equation and related boundary conditions. Third, the Fortran for IMSL subroutines is used to find all of the poles and zeros of the system in the s-plane. The root-locus method is applied to analyze the stablity of the accurately infinite poles and zeros’ locations. Then, in order to find the dynamic response, the assumed mode method is adopted and a very reasonable numer of vibration modes of the system is selected for computer simulations.
From the computer simulations, it can be seen that both the linear system and the nonlinear system have good stability property and tracking control response. The designed controller is quite effective to the flexible robot arm system.

誌謝
中文摘要…………………………………………..…………………Ⅰ
Abstract…………………………………………….………………..Ⅱ
目錄………………………………………………..…………………Ⅲ
圖索引……………………………………………….…………………Ⅴ
第一章 緒論……………….……………………………………………1
1.1 前言………………………………………………..……….….1
1.2 研究動機………………………………………..…….…...….2
1.3 文獻回顧………………………………………..………….….3
1.4 研究步驟………………………………………..….….…….6
第二章 非剛性機器手臂之數學模式推導…….…………………….…8
2.1 引言………………………………………………………..….8
2.2 非剛性機器手臂模型之建立…….……………………….….8
2.3 非剛性機器手臂動態方程式推導…..……………….…….….12
2.4 非剛性機器手臂自然頻率與形狀函數…………..……...…...17
第三章 轉移函數推導與控制器設計………………………..………..21
3.1 引言………………………………………………….………...21
3.2 非剛性機器手臂之開迴路系統之轉移函數………..………...21
3.3 對開迴路系統之轉移函數分析穩定度…………………….….27
3.4 致動器與感測器因兩者間輪軸之半徑不同對系統根軌跡之改變評估……………………………………………….…….…32
3.5 控制器設計、分析……..……………………..………………..38
3.6 非剛性機器手臂模擬之振動模態………..…..………………..47
第四章 結果與分析……………..….………………………………….54
4.1 引言………………………………………….…..….…………..54
4.2 系統運動過程中的輸出響應結果………………...….………..54
4.2.1 響應軌跡之追蹤……………………..………………..54
4.2.2 加入比例微分控制器輸出響應結果…………...…….63
4.2.3 加入比例積分微分控制器輸出響應結果………...….71
4.2.4 比較線性與非線性系統模型之回授控制模擬………78
第五章 結論與建議……………………..………………………… .…81
參考文獻……………………………………………..….…………… ..83

1. William Weaver and Paul R. Johnston “Structural dynamics by finite elements”, Prentice-Hall, 1987.
2. Singiresu S.Rao “Mechanical vibrations”, Addison-Wesley, pp.222-313,pp.468-510, 1984.
3. Roy R.CRAIG ,Jr “Structural Dynamic an introduction to computer methods”,Wiley,pp.207-233 pp.237-264 pp.297-319, 1981.
4. Lenard Meirovitch “Analytical Methods in vibration”, Macmillan,pp.30-51,pp. 328-381, 1967.
5. Sudhir Kumar Jain “Analytical models for the dynamics of building”, University Microfilms International, 1983.
6. F.Xi and R.G.Fenton “Point-To-Point Quasi-static Motion Planning for Flexible-Link Manipulators”,IEEE Transactions on Robotics and Automation ,Vo.11 ,No.5 ,pp.770-776 , 1995.
7. K. Liu and M.R. Kujath “Trajectory Optimization for a Two-Link Flexible Manipulator ”,International Journal of Robotics and Automation ,Vo.11 ,No.2 ,pp.56-61 , 1996.
8. Steven P. Eppinger and Warren P .Seering “Three Dynamic Problem in Robot Force Control Manipulator ”, IEEE Transactions on Robotics and Automation, Vo.8 ,No.6 , pp.751-758 , 1992.
9. Hirokazu Mayeda, Koji Yoshida and Koichi Osuka “Base Parameters of Manipulator Dynamic Models ”, IEEE Transactions on Robotics and Automation, Vo.6 ,No.3 , pp.313-321 , 1990.
10. Gang Feng and M. Palaniswami “An Adaptive Control Algorithm With Application to Single-Link Flexible Manipulator ”, International Journal of Robotics and Automation , Vo.7 ,No.3 ,pp.111-116 , 1992.
11. A. R. Mitchell “The Finite Element Method in Partial Differential Equations” Wiley, 1977.
12. Sydney H. Gould “Variational methods for eigenvalue problems: an introduction to the methods of Rayleigh, Ritz, Weinstein, and Aronszajn.”, University of Toronto Press,pp. 11-70, 1957.
13. O. Axelsson “Finite element solution of boundary value problems :theory and computation” Fla. :Academic Press, 1984.
14. Harvey Gould and Jan Tobochnik “An introduction to computer simulation methods :applications to physical systems”, Addison-Wesley, 1988.
15. Fernando Figueroa and Enrique Barbieri “An Ultrasonic Ranging System for Structural Vibration Measurements”, IEEE Transactions On Instrumentation And Measurement, Vo.40 ,No.4 , pp.764-769 , 1991.
16. A.T.Massoud and H.A.Elmaraghy “Design, Dynamics, and Identification of A Flexible Joint Robot Manipulator”, International Journal of Robotics and Automation , Vo.11 , No.1 ,pp.22-35 , 1996.
17. David Wang “Comparison of Optimal and Nonoptimal Control Strategies for The Single Flexible Link”, International Journal of Robotics and Automation , Vo.9 , No.3 ,pp.130-137 , 1994.
18. W. Yim and S.N. Singh “Trajectory Control of Flexible Manipulator Using Rigid Micromanipulator System”, International Journal of Robotics and Automation , Vo.12 , No.3, pp.93-97 , 1997.
19. W. Harmon Ray and Demetrios G. Lainiotis “Distributed parameter systems : identification, estimation, and control ”,M. Dekker, 1978.
20. F. Kappel, K. Kunisch and W. Schappacher “Control theory for distributed parameter systems and applications”, Springer -Verlag, 1983.
21. M.J.Balas “Feedback Control of Flexible Systems”,IEEE Trans. Auto,Control, Vol.AC-23.No.4, pp.673-679, 1978.
22. Ogata Katsuhiko “Modern Control Engineer ”, Prentice Hall, 1990.
23. Stanley M. Shinners “Modern Control System Theory and Design” , J. Wiley ,pp.185-505, 1992.
24. C. F. Beards “Vibrations and control systems”, E. Horwood,pp.21-35 ,1988.
25. John J. Craig “Introduction to Robotics: mechanics & control”, Addison-Wesley, 1986.
26. M.Hashimoto and Y.Kiyosawa and R.P. Paul “A Torque Sensing Technique for Robots with Harmonic Drives”,IEEE Transactions on Robotics and Automation. Vol.9 No.1,February, 1993.
27. Phillip John Mckerrow “Introduction to Robotics”, Addison-Wesley,pp.53-285,545-637, 1991.
28. IMSL MATH/Library User Manual,FORTRAN Subroutines for Mathematic Application,Visual Numerics ,pp.364-368,767-769, 1990.
29. G. J. Borse “FORTRAN 77 and numerical methods for engineers”, PWS Engineering, pp.320-436, 1985.
30. Larry R. Nyhoff “FORTRAN 77 and numerical methods for engineers and scientists”, Prentice Hall, 1995.
31. IMSL C/MATH/Library C Functions for Mathematic Application, Visual Numerics,pp.365-378, 1998.
32. Robert E. Skelton “Dynamic systems control :linear systems analysis and synthesis”, Wiley, 1988.
33. Wilson J. Rugh “Linear system theory”, Prentice Hall, 1996.
34. Benjamin C. Kuo “Automatic control systems”, Prentice Hall, 1991.
35. Stanley M. Shinners “Modern Control System Theory and Design” , J. Wiley ,pp.185-505, 1992.
36. Katsuhiko Ogata “Desiging Linear Control Systems With Matlab”,Prentice-Hall International editions,pp.145-185, 1994.
37. William F. Ames “Numerical methods for partial differential equations”, Academic Press,pp.41-105, 1977
38. Singiresu S.Rao “MECHANICAL VIBRATIONS second edition”, Addison-Wesley,pp.63-166 pp.488-513,pp.523-560, 1990
39. David Kahaner, Cleve Moler and Stephen Nash “Numerical methods and software”, Prentice Hall,pp51-105, 1989
40. Gene H. Hostetter, Mohammed S. Santina and Paul D'Carpio-Montalvo “Analytical, numerical, and computational methods for science and engineering”, Prentice Hall,pp.110-161, 1991
41. Richard L. Burden “Numerical Analysis second edition”, Weber & Schmidt, 1981.
42. Charles F. Van Loan “Introduction to scientific computing :a matrix-vector approach ”, Prentice Hall,pp.194-223 pp.258-302 pp.309-340, 1997
43. Shoichiro Nakamura “Numerical analysis and graphic visualization ”, Prentice Hall, 1996.
44. P.Lucibello and G. Ulivi “Design and Pealization of A Two-Link Direct-Drive Robot With A Very Flexible Forearm”, International Journal of Robotics and Automation , Vo.8 , No.3, pp.113-128 , 1993.
45. Jie Yang, Yangsheng Xu and C. S. Chen “Model Reduction of Flexible Manipulators”, International Journal of Robotics and Automation , Vo.9 , No.4, pp.199-204 , 1994.
46. M.C.Berg “Introduction to a SCB for multivariable linear system”,IEEE Proc-CONTROL THEORY AND APPLICATIONS,No 2,pp.204-209,March, 1998
47. Bernard Friedland “Control system design :an introduction to state-space methods”, McGraw-Hill, 1986
48. J.J.D’Azzo and C.H.Houpis “Feedback Control System Analysis & Synthsis”,Mcgraw-Hill, 1966.
49. John Vande Vegte “Feedback Control Systems third edition”,Prentice Hall, 1994.
50. Jan Marian Maciejowski “Multivariable feedback design”, Addison-Wesley, 1989
51. Frank L.Lewis and Vassilis L.Symos “OPTIMAL CONTROL Second Edition”,Wiley Interscience,pp.29-213, 1995
1. Schuler. Mcnamee原著,鄭慧玲編著 “Industrial Electronics And Robotics”,pp.73-102,187-208,全欣科技,1988.
53. Ward Cheney and David Kincaid 原著,劉睦雄、張任業編譯 “Numerical Mathematic And Computing Second Edition”,松岡電腦,pp.199-274,369-386,429-464, 1987.
54. 雨宮綾夫,田口武夫合著,黃德湧譯 “數值方法與FORTRAN”,儒林圖書,pp.455-544, 1986.
55. 陳世芳、陳昭綾編著 “數值方法入門-使用C語言”,全華科技, pp.6.1-6.53, 1986.
56. 劉賢淋 “結構動力入門”,天佑,cp2.3, 1992
57. Jaroslav Pachner原著,朱天賜編譯 “Handbook of Numerical analysis Applications with Programs for Engineers and Scientists”,全欣科技, pp.73-108, 1988.
58. 陳煌銘 “振動基礎分析與設計”,科技, 1986.