本文探討3-PUU並聯式機構應用於五軸工具機之運動學參數與機構受力關係，並提出剛性分析及最佳化之方法。
理論方面，首先定義各零件之座標及機構參數，再由轉換矩陣得出各座標間之關係，進而計算出機構之逆向運動學，利用其推導出順向運動學公式，再推導出Jacobian矩陣，並利用程式分析運動學找出機構之運動空間。最後利用機構之幾何關係與限制，得出機構中連桿與接頭之力平衡以及力矩平衡方程式。
分析方面，首先定義工作空間內實際使用範圍，並從使用範圍中計算各機構參數對於驅動軸推力之影響，並在符合速度及加速度的限制內找出優化解。並在不同機構角度配置下，分析使用範圍內之連桿受最大力矩分佈。
對於工具機而言，剛性是重要的性能表現，而並聯式機構之剛性會隨著姿態不同而改變，本論文已推導出該機構之剛性計算方式，並利用剛性指標作為基因演算法的依據以提升工作區域中之最弱剛性。

This paper investigated mainly on the kinetostatic analysis of a 3-PUU type parallel manipulator (PKM). The kinematics, Jacobian matrix, and kinetostatic equation were derived as well.
For theoretical derivation, the coordinates of crucial mechanical parts as well as parameters of mechanism were first defined. Then, the homogeneous transformation matrices governing the relationship among each prescribed coordinate were derived. Next, the inverse and forward kinematics was solved. Finally, the kinetostatic analysis is conducted.
For analyzing perspective, the useful workspace within the entire workspace of PKM is first determined based on the velocity and acceleration requirement. Then, the relationship between each parameter and the actuation force is derived to find the optimized parameter. In addition, the moment analysis for joints and links was presented to minimize the moments among three links.
Stiffness analysis is one of the most crucial concerns when evaluating the performance of a PKM machine. This study derived the stiffness matrix of the 3-PUU PKM. Furthermore, the stiffness index was taken into account to describe the rigidity performance in various postures of PKM. Based on the stiffness matrix and the stiffness index, this thesis proposed a method to improve the weakest stiffness within prescribed working area by adjusting the major parameters of parallel mechanism using Genetic Algorithm (GA).

[1] Gough V E and Whitehall (1962). Universal Tyre Test Machine.
[2] D. Stewart (1965). A Platform with Six Degrees of Freedom. Proceedings of the Institution of Mechanical Engineers, 180: 371.
[3] L. W. Tsai (1999) Systematic Enumeration of Parallel Manipulators. Parallel Kinematic Machines, 33-49.
[4] Lung-Wen Tsai and Sameer Joshi (2002). Kinematic Analysis of 3-DOF Position Mechanisms for Use in Hybrid Kinematic Machines. Journal of Mechanical Design, Vol. 124.
[5] M. Bouri and R. Clavel (2010). The Linear Delta: Developments and Applications. ISR / ROBOTIK 2010.
[6] C. Gosselin and J. Angeles (1989). The Optimum Kinematic Design of a Spherical Three-Degree-of-Freedom Parallel Manipulator. J. Mech., Trans., and Automation 111(2), 202-207
[7] Karol Miller (2002). Maximization of Workspace Volume of 3-DOF Spatial Parallel Manipulators. J. Mech. Des 124(2), 347-350
[8] Michael Stock and Karol Miller (2003). Optimal Kinematic Design of Spatial Parallel Manipulators: Application to Linear Delta Robot. J. Mech. Des 125(2), 292-301
[9] Yunjiang Lou , Guanfeng Liu, Ni Chen, and Zexiang Li (2005). Optimal Design of Parallel Manipulators for Maximum Effective Regular Workspace. IEEE/RSJ International Conference on Intelligent Robots and Systems.
[10] Zhongfei Wang , Guan Wang, Shiming Ji, Yuehua Wan and Qiaoling Yuan (2007). Optimal Design of a Linear Delta Robot for the Prescribed Cuboid Dexterous Workspace. IEEE International Conference on Robotics and Biomimetics (ROBIO).
[11] Ridha Kelaiaia, Olivier Company, and Abdelouahab Zaatri (2012). Multiobjective optimization of a linear Delta parallel robot. Mechanism and Machine Theory 50 (2012) 159–178
[12] Jia-Feng Wu, Rui Zhang, Rui-He Wang, and Ying-Xue Yao (2014). A systematic optimization approach for the calibration of parallel kinematics machine tools by a laser tracker. International Journal of Machine Tools & Manufacture 86 (2014) 1–11
[13] Yimin Song, Hao Gao, Tao Sun, Gang Dong, Binbin Lian, and Yang Qi (2014). Kinematic analysis and optimal design of a novel 1T3R parallel manipulator with an articulated travelling plate. Robotics and Computer-Integrated Manufacturing, 30 (2014) 508–516.
[14] Liping Wang, Huayang Xu, and Liwen Guan (2017). Optimal design of a 3-PUU parallel mechanism with 2R1T DOFs. Mechanism and Machine Theory, 114 (2017), 190–203.
[15] Clement Gosselin (1990). Stiffness Mapping for Parallel Manipulators. IEEE Transactions on Robotics and Automation ( Volume: 6 , Issue: 3 , Jun 1990 )
[16] Placid M. Ferreira and Bashar S. El-Khasawneh (1999). Computation of stiffness and stiffness bounds for parallel link manipulators. International Journal of Machine Tools & Manufacture, 39 (1999), 321–342.
[17] Yangmin Li & Qingsong Xu (2008). Stiffness analysis for a 3-PUU parallel kinematic machine. Mechanism and Machine Theory, 43 (2008), 186–200.
[18] Dan Zhang (2010). Global Stiffness Optimization of Parallel Robots Using Kinetostatic Performance Indices. Robot Manipulators Trends and Development.
[19] V.T. Portman, V.S. Chapsky , and Y. Shneor (2012). Workspace of parallel kinematics machines with minimum stiffness limits:Collinear stiffness value based approach. Mechanism and Machine Theory, 49 (2012), 67–86.
[20] Hao Wang, Linsong Zhang, Genliang Chen, and Shunzhou Huang (2017). Parameter optimization of heavy-load parallel manipulator by introducing stiffness distribution evaluation index. Mechanism and Machine Theory, 108 (2017), 244–259.
[21] Ting-Nung Shiau, Yi-Jeng Tsai, and Meng-Shiun Tsai (2008). Nonlinear dynamic analysis of a parallel mechanism with consideration of joint effects. Mechanism and Machine Theory, 43 (2008), 491–505.
[22] Stefan Staicu (2009). Dynamics analysis of the Star parallel manipulator. Robotics and Autonomous Systems, 57 (2009), 1057–1064.
[23] Meng-Shiun Tsai & Wei-Hsiang Yuan (2010). Inverse dynamics analysis for a 3-PRS parallel mechanism based on a special decomposition of the reaction forces. Mechanism and Machine Theory, 45 (2010), 1491–1508.
[24] Miguel Díaz-Rodríguez, Vicente Mata, Ángel Valera, and Álvaro Page (2010). A methodology for dynamic parameters identification of 3-DOF parallel robots in terms of relevant parameters. Mechanism and Machine Theory, 45 (2010), 1337–1356.
[25] Yunjiang Lou, Zhibin Li, and Yingying Zhong (2011). Dynamics and contouring control of a 3-DoF parallel kinematics machine. Mechatronics, 21 (2011), 215–226.
[26] Behrouz Afzali-Far, Per Lidström, and Kristina Nilsson (2014). Parametric damped vibrations of Gough–Stewart platforms for symmetric configurations. Mechanism and Machine Theory, 80 (2014), 52–69.
[27] Peter Vischeer & Reymond Clavel (1998). Kinematic calibration of the parallel Delta robot. Robotica volume, 16, pp. 207-218.
[28] Kuang-Chao Fan, Hai Wang, Jun-Wei Zhao, and Tsan-Hwei Chang (2003). Sensitivity analysis of the 3-PRS parallel kinematic spindle platform of a serial-parallel machine tool. International Journal of Machine Tools & Manufacture, 43 (2003), 1561–1569.
[29] Hai Wang & Kuang-Chao Fan (2004). Identification of strut and assembly errors of a 3-PRS serial–parallel machine tool. International Journal of Machine Tools & Manufacture, 44 (2004), 1171–1178.
[30] Wenjie Tian, Fuwen Yin, Haitao Liu, Jinhe Li, Qing Li, Tian Huang, and Derek G. Chetwynd (2016). Kinematic calibration of a 3-DOF spindle head using a double ball bar. Mechanism and Machine Theory, 102 (2016), 167–178.
[31] Tiemin Li, Fengchun Li, Yao Jiang, Jinglei Zhang, and Haitong Wang (2017). Kinematic calibration of a 3-P(Pa)S parallel-type spindle head considering the thermal error. Mechatronics, 43 (2017), 86–98.