於現代電腦數值控制之工具機中，多軸伺服系統的循跡運動為一重要應用，且廣泛使用於各種類型加工。因此，如何降低循跡運動過程之誤差即為重點發展技術。而現今主要用於循跡精度之指標為追蹤誤差與輪廓誤差，其中追蹤誤差定義為命令點至刀具實際位置點之距離；而輪廓誤差定義為實際位置點至整個命令軌跡上最短之距離，因此輪廓誤差即為判斷刀具是否偏離加工命令軌跡之重要依據。然而，對於外型多變之自由曲線而言，即時計算其精確輪廓誤差相當困難，也相對增加輪廓誤差控制難度。有鑑於此，本論文即針對參數式曲線提出數種即時輪廓誤差估測法。這些估測法主要基於參數式曲線插值器之運算，經由追蹤誤差向量與輪廓誤差之關係，求出命令軌跡上之一參考點做為更精確估算輪廓誤差之用。本論文藉由NURBS參數式曲線之循跡運動實驗，驗證所提出方法之可行性，並與其他現有之即時輪廓誤差估測法進行估測效能之比較。經由雙軸循跡運動實驗之結果可知，本論文提出之方法對於不同曲率變化之自由曲線均可有效提昇輪廓誤差估測之準確度，進而降低循跡運動中產生之輪廓誤差。

In modern Computer Numerical Control machine tools, the contour following task of a multi-axis servo system is one of the most important applications, and is widely used in many types of machining. Therefore, the reduction of error occurring during the contouring process is a key technology deserves more study and development. The main indices used to assess contouring accuracy are tracking error and contour error. The definition of tracking error is the distance from the current command position to the current cutting point. The definition of contour error is the shortest distance between the current cutting point and the entire command trajectory. Consequently, the contour error is an important indicator for judging whether the cutting trajectory deviates from the desired command path. However, it is difficult to calculate the accurate contour error for a variable shaped free-form curve thus increasing the difficulty with which of contour error control. To deal with such problems, this thesis proposes several real-time contour error estimation methods for parametric curves. These estimation methods are mainly based on the parametric curve interpolator. It approaches the actual point of which contour error occurs on the command path through the relationship between the tracking error vector and contour error so as to more accurately derive the approximated contour error. In this thesis, several NURBS parametric curve contour following experiments are performed to verify the feasibility of the proposed method and compare their performance with that of other existing real-time contour error estimation methods. The result of the biaxial contouring experiment reveals that the proposed methods can efficiently enhance estimating accuracy for different free-form curves with varying curvature, thereafter reducing the contour error during contour following tasks.

[1] 台灣區機器工業同業公會, "台灣機械進出口統計分析表," 2014.
[2] 台灣區工具機暨零組件工業同業公會, "2012-台灣工具機與零組件進出口報告," 2013.
[3] H. Y. Chuang and C. H. Liu, "Cross-Coupled Adaptive Feedrate Control for Multiaxis Machine Tools," Journal of Dynamic Systems, Measurement, and Control, vol. 113, pp. 451-457, 1991.
[4] Y. S. Tarng, H. Y. Chuang, and W. T. Hsu, "An optimisation approach to the contour error control of CNC machine tools using genetic algorithms," The International Journal of Advanced Manufacturing Technology, vol. 13, pp. 359-366, 1997.
[5] H. C. Lee and G. J. Jeon, "Real-time compensation of two-dimensional contour error in CNC machine tools," in Proceedings of 1999 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 1999, pp. 623-628.
[6] S. S. Yeh and P. L. Hsu, "A new approach to bi-axial cross-coupled control," in Proceedings of the 2000 IEEE International Conference on Control Applications, 2000, pp. 168-173.
[7] S. L. Chen, H. L. Liu, and S. C. Ting, "Contouring control of biaxial systems based on polar coordinates," IEEE/ASME Transactions on Mechatronics, vol. 7, pp. 329-345, 2002.
[8] M. Y. Cheng and C. C. Lee, "On real-time contour error estimation for contour following tasks," in Proceedings of 2005 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 2005, pp. 1047-1052.
[9] K. Erkorkmaz, C. H. Yeung, and Y. Altintas, "Virtual CNC system. Part II. High speed contouring application," International Journal of Machine Tools and Manufacture, vol. 46, pp. 1124-1138, 2006.
[10] S. L. Chen and K. C. Wu, "Contouring control of smooth paths for multiaxis motion systems based on equivalent errors," IEEE Transactions on Control Systems Technology, vol. 15, pp. 1151-1158, 2007.
[11] J. Z. Yang and Z. X. Li, "A Novel Contour Error Estimation for Position Loop-Based Cross-Coupled Control," IEEE/ASME Transactions on Mechatronics, vol. 16, pp. 643-655, 2011.
[12] F. Huo and A. N. Poo, "Free-form two-dimensional contour error estimation based on NURBS interpolation," Applied Mechanics and Materials, vol. 157, pp. 236-240, 2012.
[13] G. Y. Zhao, H. J. An, and Q. Z. Zhao, "Contour Error Coupled-Control Strategy based on Line Interpolation and Curve Interpolation," Journal of Computers, vol. 8, 2013.
[14] L. M. Zhu, H. Zhao, and H. Ding, "Real-time contouring error estimation for multi-axis motion systems using the second-order approximation," International Journal of Machine Tools and Manufacture, vol. 68, pp. 75-80, 2013.
[15] M. Tomizuka, "Zero Phase Error Tracking Algorithm for Digital Control," Journal of Dynamic Systems, Measurement, and Control, vol. 109, pp. 65-68, 1987.
[16] Y. Koren, "Cross-Coupled Biaxial Computer Control for Manufacturing Systems," Journal of Dynamic Systems, Measurement, and Control, vol. 102, pp. 265-272, 1980.
[17] K. Srinivasan and P. K. Kulkarni, "Cross-Coupled Control of Biaxial Feed Drive Servomechanisms," Journal of Dynamic Systems, Measurement, and Control, vol. 112, pp. 225-232, 1990.
[18] Y. Koren and C. C. Lo, "Variable-Gain Cross-Coupling Controller for Contouring," CIRP Annals - Manufacturing Technology, vol. 40, pp. 371-374, 1991.
[19] Y. Koren and S. Jee, "Fuzzy logic cross-coupling control," in Proceedings of 27th CIRP International Seminar on Manufacturing Systems, 1995, pp. 104-108.
[20] J. H. Chin and T. C. Lin, "Cross-coupled precompensation method for the contouring accuracy of computer numerically controlled machine tools," International Journal of Machine Tools and Manufacture, vol. 37, pp. 947-967, 1997.
[21] S. S. Yeh and P. L. Hsu, "Theory and applications of the robust cross-coupled control design," in Proceedings of the 1997 American Control Conference, 1997, pp. 791-795 vol.1.
[22] S. S. Yeh and P. L. Hsu, "Estimation of the contouring error vector for the cross-coupled control design," IEEE/ASME Transactions on Mechatronics, vol. 7, pp. 44-51, 2002.
[23] J. H. Chin, Y. M. Cheng, and J. H. Lin, "Improving contour accuracy by Fuzzy-logic enhanced cross-coupled precompensation method," Robotics and Computer-Integrated Manufacturing, vol. 20, pp. 65-76, 2004.
[24] K. L. Barton and A. G. Alleyne, "A Cross-Coupled Iterative Learning Control Design for Precision Motion Control," IEEE Transactions on Control Systems Technology, vol. 16, pp. 1218-1231, 2008.
[25] F. Huo and A. N. Poo, "Improving contouring accuracy by using generalized cross-coupled control," International Journal of Machine Tools and Manufacture, vol. 63, pp. 49-57, 2012.
[26] H. Y. Chuang and C. H. Liu, "A model-referenced adaptive control strategy for improving contour accuracy of multiaxis machine tools," IEEE Transactions on Industry Applications, vol. 28, pp. 221-227, 1992.
[27] Y. S. Tarng, H. Y. Chuang, and W. T. Hsu, "Intelligent cross-coupled fuzzy feedrate controller design for CNC machine tools based on genetic algorithms," International Journal of Machine Tools and Manufacture, vol. 39, pp. 1673-1692, 1999.
[28] H. T. Yau and M. J. Kuo, "NURBS machining and feed rate adjustment for high-speed cutting of complex sculptured surfaces," International Journal of Production Research, vol. 39, pp. 21-41, 2001.
[29] K. H. Su, C. Y. Hsieh, and M. Y. Cheng, "Design and Implementation of Biaxial Motion Control System Using Fuzzy Logic Based Adjustable Feedrate," in Proceedings of 2006 IEEE International Conference on Mechatronics, 2006, pp. 209-214.
[30] M. Y. Cheng, K. H. Su, and S. F. Wang, "Contour error reduction for free-form contour following tasks of biaxial motion control systems," Robotics and Computer-Integrated Manufacturing, vol. 25, pp. 323-333, 2009.
[31] L. Tang and R. G. Landers, "Predictive contour control with adaptive feed rate," IEEE/ASME Transactions on Mechatronics, vol. 17, pp. 669-679, 2012.
[32] G. T. C. Chiu, "Contour tracking of machine tool feed drive systems," in Proceedings of the 1998 American Control Conference, 1998, pp. 3833-3837 vol.6.
[33] C. X. Hu, B. Yao, and Q. F. Wang, "Global task coordinate frame-based contouring control of linear-motor-driven biaxial systems with accurate parameter estimations," IEEE Transactions on Industrial Electronics, vol. 58, pp. 5195-5205, 2011.
[34] B. Yao, C. X. Hu, and Q. F. Wang, "An Orthogonal Global Task Coordinate Frame for Contouring Control of Biaxial Systems," IEEE/ASME Transactions on Mechatronics, vol. 17, pp. 622-634, 2012.
[35] G. T. C. Chiu and M. Tomizuka, "Coordinated Position Control of Multi-Axis Mechanical Systems," Journal of Dynamic Systems, Measurement, and Control, vol. 120, pp. 389-393, 1998.
[36] L. Piegl, "On NURBS: a survey," IEEE Computer Graphics and Applications, vol. 11, pp. 55-71, 1991.
[37] Q. G. Zhang and R. B. Greenway, "Development and implementation of a NURBS curve motion interpolator," Robotics and Computer-Integrated Manufacturing, vol. 14, pp. 27-36, 1998.
[38] M. Y. Cheng, M. C. Tsai, and J. C. Kuo, "Real-time NURBS command generators for CNC servo controllers," International Journal of Machine Tools and Manufacture, vol. 42, pp. 801-813, 2002.
[39] L. Piegl and W. Tiller, "The NURBS book," ed: Springer, 1997.
[40] 郭洲成, "CNC伺服控制器之NURBS即時插值器設計與實現," 碩士論文, 機械工程學系, 國立成功大學, 2000.
[41] 汪曙峰, "循跡運動精度改善之探討," 碩士論文, 電機工程學系碩博士班, 國立成功大學, 2007.
[42] R. H. Brown, S. C. Schneider, and M. G. Mulligan, "Analysis of algorithms for velocity estimation from discrete position versus time data," IEEE Transactions on Industrial Electronics, vol. 39, pp. 11-19, 1992.
[43] K. H. Su, M. Y. Cheng, H. R. Chen, and Y. C. Chang, "Iterative Learning Control-based Tracking Error and Contour Error Compensations for Biaxial Contouring Control Systems," in Proceedings of the 5th International Conference on Positioning Technology Kaohsiung, Taiwan, 2012, pp. 107-110.