相較於常見的串聯式機構工具機，並聯式工具機由多組連桿構成閉迴路系統，使得機構具有高剛性、低移動慣量之特性；然而其剛性在空間中具有各向異性，在不同姿態下加工時，因機構變形造成的刀具位置改變較難預估，影響加工成品的幾何精度及表面粗糙度，甚至增加機台損耗風險。因此，本文將連桿結構件以及接頭零件視為撓性體，建立3-UPU型並聯式工具機之剛性解析模型，並針對結果進行分析。
本文首先介紹3-UPU型並聯式機構，定義相關參數後進行運動學推導以利後續座標轉換；接著以矩陣結構分析的方法建立剛性模型，導入機構中各零件剛性矩陣以及零件間的拘束方程式後，即可求得機構在不同姿態下的6x6空間剛性矩陣，此矩陣可以直接得到機構在工作空間座標系上不同方向的剛性數值以及受負載後末端效應器的變形量值；接著透過性能指標可以了解剛性的各向異性程度以及零件剛性對整體剛性的影響；最後以特徵螺旋向量分解找出剛性主軸，進而分析剛性在空間中的表現。

Compared with the serial kinematic mechanism that is usually utilized in machine tools, the parallel kinematic mechanism (PKM), which is composed of multiple linkages, is a closed-loop system and has the advantages of high stiffness and low inertia.
However, the structural stiffness of PKM is anisotropic in space, which results in the displacement of the tool point difficult to predict under various configurations. In addition, it further deteriorates the accuracy and the surface finish of the work piece, and even damages the machine tool. Therefore, this study establishes the stiffness analytical model of 3-UPU PKM and analyzes the performance in the workspace by taking the structural components and the joints as elastic bodies.
This study first introduces a 3-UPU PKM, and defines the parameters related to the derivation of kinematics, so that the coordinate transformation can be further proceeded. Then, the stiffness matrix of the components and the constraint equations of the joints are applied to the Matrix Structural Analysis. With the model, the complete spatial stiffness matrix of the whole PKM is derived. In order to thoroughly understand the performance of the stiffness, the following parameters are calculated, i.e., the displacement of the end-effector, the performance index, the influence of stiffness of component, and the parameter of eigenscrew. Through the analysis in difference aspects, the stiffness properties can be clearly described and visualized.

[1] 工具機發展基金會、王正青, 2019 全球工具機主要出口國家市場報導, 2020.
[2] 台灣機械工業同業公會、楊德華, 台灣工具機產業的回顧與展望, 2005.
[3] V. Gough, Universal tyre test machine. 1962.
[4] D. Stewart, “A Platform with Six Degrees of Freedom,” Proceedings of the Institution of Mechanical Engineers, vol. 180, no. 1, pp. 371–386, Nov. 2016.
[5] L. W. Tsai, Systematic Enumeration of Parallel Manipulators. Springer, London, 1999.
[6] R. Clavel, “Device for the movement and positioning of an element in space,” Sep. 06, 1989.
[7] L. Rey and R. Clavel, “The Delta Parallel Robot,” pp. 401–417, 1999.
[8] Z. Pandilov and V. Dukovski, “Parallel kinematics machine tools: Overview-from history to the future,” Annals of the Faculty of Engineering Hunedoara, pp. 111–124, 2021.
[9] “Exechon XMini.” https://exechon.com/xmini/
[10] “Loxin.” http://www.loxin2002.com/general-description
[11] “ECOSPEED.” https://www.starrag.com/en-us/series/ecospeed-ecoliner-series/27/product-range/11
[12] C. Gosselin, “Stiffness Mapping for Parallel Manipulators,” IEEE Transactions on Robotics and Automation, vol. 6, pp. 377–382, 1990.
[13] C. Gosselin, “Stiffness analysis of parallel mechanisms using a lumped model,” Int. J. Robotics Automat., vol. 17, pp. 17–27, 2002, Accessed: Nov. 27, 2021.
[14] A. Pashkevich, D. Chablat, and P. Wenger, “Stiffness analysis of overconstrained parallel manipulators,” Mechanism and Machine Theory, vol. 44, no. 5, pp. 966–982, May 2009.
[15] A. G. L. Hoevenaars, P. Lambert, and J. L. Herder, “Jacobian-based stiffness analysis method for parallel manipulators with non-redundant legs:,” Journal of Mechanical Engineering Science, vol. 230, no. 3, pp. 341–352, Sep. 2015.
[16] J. L. Meek, Matrix structural analysis. McGraw-Hill, 1971. Accessed: Jun. 15, 2022.
[17] T. Huang, X. Zhao, and D. J. Whitehouse, “Stiffness estimation of a tripod-based parallel kinematic machine,” IEEE Transactions on Robotics and Automation, vol. 18, no. 1, pp. 50–58, Feb. 2002.
[18] A. Klimchik, A. Pashkevich, and D. Chablat, “Fundamentals of manipulator stiffness modeling using matrix structural analysis,” Mechanism and Machine Theory, vol. 133, pp. 365–394, Mar. 2019.
[19] Q. Xu and Y. Li, “An investigation on mobility and stiffness of a 3-DOF translational parallel manipulator via screw theory,” Robotics and Computer-Integrated Manufacturing, vol. 24, no. 3, pp. 402–414, Jun. 2008.
[20] Y. Li and Q. Xu, “Stiffness analysis for a 3-PUU parallel kinematic machine,” Mechanism and Machine Theory, vol. 43, no. 2, pp. 186–200, Feb. 2008.
[21] Y. Wang, H. Liu, T. Huang, and D. G. Chetwynd, “Stiffness modeling of the tricept robot using the overall jacobian matrix,” Journal of Mechanisms and Robotics, vol. 1, no. 2, pp. 1–8, May 2009.
[22] Y. G. Li, H. T. Liu, X. M. Zhao, T. Huang, and D. G. Chetwynd, “Design of a 3-DOF PKM module for large structural component machining,” Mechanism and Machine Theory, vol. 45, no. 6, pp. 941–954, Jun. 2010.
[23] J. Zhang, Y. Zhao, and Y. Jin, “Kinetostatic-model-based stiffness analysis of Exechon PKM,” Robotics and Computer-Integrated Manufacturing, vol. 37, pp. 208–220, Feb. 2016.
[24] G. Carbone and M. Ceccarelli, “Comparison of indices for stiffness performance evaluation,” Frontiers of Mechanical Engineering in China 2010 5:3, vol. 5, no. 3, pp. 270–278, May 2010.
[25] R. S. Ball, A treatise on the theory of screws . Cambridge University Press, 1900. Accessed: Jun. 24, 2022.
[26] T. Patterson and H. Lipkin, “Structure of Robot Compliance,” Journal of Mechanical Design, vol. 115, no. 3, pp. 576–580, Sep. 1993.
[27] T. Patterson and H. Lipkin, “A Classification of Robot Compliance,” Journal of Mechanical Design, vol. 115, no. 3, pp. 581–584.
[28] S. Huang and J. M. Schimmels, “The eigenscrew decomposition of spatial stiffness matrices,” IEEE Transactions on Robotics and Automation, vol. 16, no. 2, pp. 146–156, 2000.
[29] 陳晏菘, 國立清華大學碩士論文, 利用螺旋理論探討具變化連桿長度3-UPU型並聯式機構之剛性分析, 2020.
[30] 曾勇智, 國立清華大學碩士論文, 3-PUU型並聯式機構應用於五軸加工機之參數優化與剛性分析, 2019.
[31] 向志軒, 國立清華大學碩士論文, 具變化連桿長度3-UPU型並聯式機構之平衡設計與最佳化, 2021.
[32] 楊培弘, 國立清華大學碩士論文, 考量關節撓性於具變化連桿長度3-UPU型並聯式機構之動態響應, 2020.
[33] K. Serope and R. S. Steven, Eds., Manufacturing Engineering and Technology, 7th ed. Pearson, 2014.
[34] L. W. Tsai, G. C. Walsh, and R. E. Stamper, “Kinematics of a novel three DOF translation platform,” Proceedings - IEEE International Conference on Robotics and Automation, vol. 4, pp. 3446–3451, 1996.
[35] 顏鴻森、吳隆庸, 機構學, 第四版, 台灣東華書局出版社, 2014.
[36] 上銀科技股份有限公司, 滾珠螺桿技術手冊.
[37] H. Lee and D. Tesar, “An Analytical Stiffness Analysis Between Actuator Structure and Principal Bearings Used for Robot Actuators,” Proceedings of the ASME Design Engineering Technical Conference, vol. 6, no. PARTS A AND B, pp. 1167–1176, Jun. 2012.
[38] NSK Technical Report. Accessed: Nov. 28, 2021.
[39] M. Wan, W. H. Zhang, J. W. Dang, and Y. Yang, “A novel cutting force modelling method for cylindrical end mill,” Applied Mathematical Modelling, vol. 34, no. 3, pp. 823–836, Mar. 2010.
[40] H. Onozuka, F. Tayama, Y. Huang, and M. Inui, “Cutting force model for power skiving of internal gear,” Journal of Manufacturing Processes, vol. 56, pp. 1277–1285, Aug. 2020.
[41] C. Golgels, H. Schlattmeier, F. K.-G. technology, “Optimization of the gear profile grinding process utilizing an analogy process,” 2006
[42] A. Klimchik, A. Pashkevich, and D. Chablat, “CAD-based approach for identification of elasto-static parameters of robotic manipulators,” Finite Elements in Analysis and Design, vol. 75, pp. 19–30, Nov. 2013.