本研究目的是尋找3PPSR六自由度並聯式機械手臂之正向奇異位置。此機構的運動可以分解為以下兩部份1.整個機構的剛體位移，機構形狀並未改變。2.旋轉接頭旋轉，造成機構形狀改變。本文分別探討這兩種運動形態下的正向奇異位置。針對機構形狀會改變的運動，我們以機構運動倒置的方法，推導出旋轉接頭旋轉角度與球窩接頭間距離的關係式，再將這些距離對時間微分後可得到3×3 Jacobian矩陣，若矩陣行列式值為零即會發生正向奇異位置。針對整個機構剛體位移的運動，我們發現正向奇異位置發生在三個球窩接頭位在同一直線上的情況。本研究使用上述條件找出3PPSR並聯式機械手臂正向奇異位置在工作空間的分佈，並發現此機構在工作空間內任何區域皆有正向奇異位置。

The purpose of this study is to locate direct singular positions of the 3PPSR 6dof parallel manipulator. The motion of this manipulator can be categorized into the following two types: 1. rigid body displacement without changing its configuration, and 2. rotation of at least one revolute joint-causing a change in configuration. Direct kinematic singularities for these two types of motion are discussed. For the first type of motion, we found that direct kinematic singularity occurs when all three spherical joints lie along a straight line. As for the second type of motion, namely motion with a configuration change, we may obtain the 3×3 Jacobian matrix by kinematic inversion. Direct singular positions for the second type of motion may be determined by solving a quartic polynomial equation. The technique discussed above helps us to determine the distribution of direct singular positions in the manipulators workspace.

目錄
中文摘要 .................................................................................Ⅰ
英文摘要 .................................................................................Ⅱ
目錄 .........................................................................................Ⅲ
圖目錄 .....................................................................................Ⅳ
第一章  緒論 .......................................................................... 1
1.1前言 ........................................................................... 1
1.2文獻回顧 ................................................................... 1
1.3研究動機 ................................................................... 2
第二章  6×6 Jacobian矩陣....................................................... 3
第三章  3×3 Jacobian矩陣....................................................... 6
3.1 機構形狀改變的正向奇異位置................................7
3.2 機構剛體位移情況的正向奇異位置..................... 13
第四章  結果與討論 ............................................................ 15
第五章  結論.......................................................................... 18
參考文獻 ................................................................................ 19

圖目錄
圖一  3PPSR並聯式機械手臂示意圖 ........................21
圖二  3PPSR機構的位移示意圖  ................................22
圖三  3PPSR機構起始狀態圖  ..................................23
圖四  實根的分佈圖；”+”，”o”，”×”，”△”分別表示第一個，第二個，第三個及第四個實根..................................................24
圖五  實根的分佈上視圖；”+”，”o”，”×”，”△”分別表示第一個，第二個，第三個及第四個實根............................................25
圖六  當 時，所代表的正向奇異位置............26
圖七  當 時，所代表的正向奇異位置..............27
圖八  當 時，活動平台與基座重疊......................28
圖九  實根數目為4和實根數目為2的分佈圖；×代表實根數目4，○代表實根數目2.............................................................29
圖十   、 時四個實根所代表的正向奇異位置......30
圖十一  、 時的二個實根所代表的正向奇異位置..............................................................................................31
圖十二  實根數目為3和實根數目為1的分佈圖，×代表實根數目3，○代表實根數目1....................................................32
圖十三   、 時三個實根所代表的正向奇異位置…….....33
圖十四   、 時一個實根所代表的正向奇異位置…………………………………………………………...34
圖十五   時，Tsai[7]的第二種正向奇異位置...35

參 考 文 獻
[1] Tsai, L.W., and Tahmasebi, F., “Synthesis and Analysis of a New Class of Six-Degree-of-Freedom Parallel Minimanipulators”, Journal of Robotic systems, Vol. 10, pp. 561-580, 1993.
[2] Tsai, L.W., and Tahmasebi, F., “Design and Analysis of a New Six-Degree-of-Freedom Parallel Minimanipulator”, Proc. of the 6th  International Conference on CAD/CAM, Robotics, and Factories of the Future, Springer-Verlag, Berlin.1991.
[3] Tsai, L.W., and Tahmasebi, F., “Jacobin and Stiffness anslysis of a novel class of six-dof parallel Minimanipulators”, the 1992 ASME Design Technieal and 22nd biennial mechanisms conference Scottsdale, Arizona, sep.13~16 , pp. 95-102, 1992
[4] Tsai, L.W., and Tahmasebi, F., “Closed-form direct kinematics solution of a new parallel minimanipulator”, Proceedings of the 1992 Japan-USA symposium on Flexible Automation, Part 1, pp. 117-124, 1992.
[5] Tsai, L.W., and Tahmasebi, F., “Closed-form direct kinematics solution of a new parallel minimanipulator”, Journal of Mechanical Design, Vol. 116, pp. 1141-1147, 1994.
[6] Tsai, L.W., and Tahmasebi, F., “On the stiffness of a Novel six-degree-of-freedom parallel minimanipulator”, Journal of Robotic systems, Vol. 12, pp. 845-856, 1995
[7] Tsai, L-W., Robot Analysis : the Mechanics of Serial and Parallel Manipulators, John Wiley & Sons, Inc., New York, 1999.
[8] 蕭凱澤，“並聯式機械手臂3PPSR的正向奇異位置”，淡江大學機械與機電工程學系碩士論文，2005年6月。
[9] Gosselin, C., and Angeles, J., “Singularity Analysis of Closed-Loop Kinematic Chains”, IEEE Transactions of Robotics and Automation, Vol. 6, pp. 281-290, 1990.
[10] Liu, C.H. and Cheng, S. “Direct singular position of 3RPS parallel manipulators”. Journal of Mechanical Design, 126, pp.1006-1016, 2004.
[11] Liu, J., Mathematical Handbook of Formulas and Tables, McGraw-Hill, New York,1999.