我國工具機產業近十年來發展迅速，加上工業自動化的潮流驅使，以致於數值工具機(CNC)的相關研究亟需進行。然而人們對品質要求不斷的提升，使得工具機須同時兼顧高速加工方式與高精密度的加工品質。運動控制的精度取決於三個因素：命令路徑的精確度、伺服控制架構及機械結構。其中機械結構的部分已臻成熟，因此為使CNC的加工架構精度與速度獲得提昇，控制系統的設計必須包含良好的運動控制器與正確的命令插值。
NURBS (non-uniform rational B-spline)插值器可精確、簡單的表示可解析曲線與自由型曲線，而交叉耦合控制器可有效的調節各軸動態響應的快慢，減少軌跡的輪廓誤差，期望結合兩者特點，發展出一套足夠讓工具機可由命令軌跡到控制器的設計，使得任意軌跡路徑的輪廓誤差有所改善的控制系統。
關鍵詞：工具機、NURBS、交叉耦合控制

During the latest ten years, development in the industry of machine tools has been growing rapidly in our country. Since industry automation is highly demanded, the study about CNC is therefore extensively proceeded. Owing to request for higher quality, the machine tool must achieve the high speed and high precision. There are three factors in precise motion control : the accuracy of reference command, the design of servo control structure, and machine structure. However the part of machine structure attains to maturity. The design of the control system need to include good motion control and correct reference command.
NURBS can represent analytic curves and free-form curves accurately and easily. Cross-coupled control is able to adjust the dynamic system of each axis to reduce the contour errors. It is expected to improve control performance in terms of contour errors by combine NURBS reference command and the cross-coupled control framework.
Keywords：machine tool, NURBS, cross-coupled control

目錄……………………………………………………………………….i
圖索引…………………………………………………………………...iii
表索引…………………………………………………………..…...…..vi
摘要………………………………………………………………..…....vii
Abstract………………………………………...………………...……viii
第一章 緒論……………………………………………………………..1
1-1 動機與目的……………………………………………………..1
1-2 文獻回顧………………………………………………………..2
1-3 論文架構………………………………………………………..6
第二章 NURBS特性與NURBS插值器之設計………………..……..7
2-1 NURBS的數學模型與實現……………...…………………….7
2-2 參數化曲線之即時插值器………………………….………...12
第三章 控制器之設計…………………………………………………18
3-1單軸的伺服控制………………………….……………………18
3-2 輪廓誤差………………………….…………………………...22
3-2-1 圓弧輪廓誤差………...……………..….………………22
3-2-2 任意軌跡輪廓誤差………...……………….…………..23
3-3 交叉耦合控制……………………..…………………………..26
3-3-1 交叉耦合控制器之架構……………………...……..….26
3-3-2 交叉耦合控制器之設計……………...………………...27
第四章 系統動態模式建立……………………………………………34
4-1 ARX(Auto-Regressive eXogeneous)模型………………….…34
4-2 馬達數學模式之建立………………………………………....38
第五章 模擬結果與分析………………………………………………41
5-1 軌跡設計………………………………………………………41
5-2 控制器參數選用………………………………………………44
5-3 模擬結果………………………………………………………47
第六章 輪廓誤差實驗…………………………………………………56
6-1 實驗系統架構…………………………………………………56
6-2 實驗結果………………………………………………………58
第七章 結論及未來展望………………………………………………72
參考文獻………………………………………………………………..73

[1] Bedi, S., Ali, I., and Quan, N., “Advanced Techniques for CNC Machine,” Journal of Engineering for Industry, Transactions of the ASME, Vol. 115, pp. 329-336, (1993).
[2] Blanc, C., and Schlick, C., “Accurate Parameterization of Conics by NURBS,” IEEE Computer Graphics and Applications, pp. 64-71, Nov. (1996).
[3] Chou, J.J., and Yang, D.C.H., “Command Generation for Three-Axis CNC Machining,” Journal of Engineering for Industry, Transactions of the ASME, Vol. 113, pp. 305-310, Aug. (1991).
[4] Gopi, M., and Manohar, S., “A Unified Architecture for the Computation of B-spline Curves and Surfaces,” IEEE Transactions on Parallel and Distributed Systems, Vol. 8, No. 12, Dec. (1997).
[5] Huang, J.T., and Yang, D.C.H., “Precision Command Generation for Computer Controlled Machines,” Precision Machining: Technology and Machine Development and Improvement, ASME, PED-Vol. 58, (1992).
[6] Koren, Y., and Lo, C.C., “Advanced Controller for Feed Drivers,” Annals of the CIRP, Vol. 41, pp.689-698, (1992).
[7] Koren, Y., “Cross-Coupled Biaxial Computer for Manufacturing Systems,” Journal of Dynamic System, Measurement and Control, Transactions of the ASME, Vol. 102, No.4, pp. 265-272, (1980).
[8] Koren, Y., and Lo, C.C., “Variable Gain Cross Coupling Controller for Contouring,” Annals of the CIRP, Vol. 40, pp. 371-374, (1991).
[9] Koren, Y., Lo, C.C. and Shpitalni, M., “CNC Interpolators: Algorithms and Analysis,” Manufacturing Science and Engineering ASME, PED-Vol. 64, (1993).
[10] Lo, C.C., and Chung, C.Y., “Curve Generation and Control for Biaxial Machine Tools,” Journal of CSME, Vol. 18, No. 2, pp. 175-182, (1997).
[11] Lo, C.C., “Feedback Interpolators for CNC Machine Tools,” Journal of Manufacturing Science and Engineering, Transactions of the ASME, Vol. 119, pp.587-592, (1997).
[12] Lo, C.C., “Real-Time Generation and Control of Cutter Path for 5-Axis CNC Machining,” International Journal of Machine Tools ＆ Manufacture, Vol. 39, No. 3, pp. 471-488, (1999).
[13] Piegl, L., and Tiller, W., “Cure and Surface Constructions Using Rational B-splines,” Computer-Aided Design, Vol. 19, No. 9, pp. 485-498, Nov. (1987).
[14] Piegl, L., “On NURBS: A Survey,” IEEE Computer Graphics ＆ Application, Vol. 11, No. 1, Jan. (1991).
[15] Piegl, L., and Tiller, W., “The NURBS Book,” (1995).
[16] Spivak M., “A Comprehensive Introduction to Differential Geometry,”(1975).
[17] Tiller, W., “Rational B-splines for Curve and Surface Representation,” IEEE Computer Graphics ＆ Application, pp. 61-69, Sep. (1983).
[18] Zhang, Q.G., and Greenway, R.B., “Development and Implementation of a NURBS Curve Motion Interpolator,” Robotics and Computer-Integrated Manufacturing, Vol. 14, pp. 27-36, (1998).
[19] The MathWorks, Inc., Real-Time Workshop － User’s Guide；Version 4, Natick, (2001).
[20] The MathWorks, Inc., Simulink － Writing S-Functions；Version 4, Natick, (2001).
[21] The MathWorks, Inc., System Identification Toolbox － User’s Guide；Version 4, Natick, (2001).
[22] DSPACE GmbH, Real-Time Interface to Simulink － User’s Guide；Version 3, (1997).
[23] 王仁傑，“CNC工具機加工路徑補間之研究，” 碩士論文，成功大學造船及船舶工程研究所，民國九十年。
[24] 足立修一，“控制之系統鑑別，” 全華科技圖書，民國九十年。
[25] 孫義欽，“MATLAB 與嵌入式控制器之整合控制應用，”碩士論文，中山大學機械與機電工程研究所，民國九十三年。
[26] 陳鳴吉，“交叉耦合控制在高速軌跡追蹤控制之設計與實作，” 碩士論文，中山大學機械與機電工程研究所，民國九十年。
[27] 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， 成功大學機械工程研究所，民國八十九年。
[28] 楊憲東，“精密機械控制原理與模擬，” 全華科技圖書，民國八十七年。
[29] 薛博文，“雙馬達之同步運動控制，” 碩士論文，中山大學機械與機電工程研究所，民國九十年。
[30] 程啟正，陳慶盈，“整合型設計在加工機上之應用，” 2004 第十三屆全國自動化科技研討會，民國九十三年，台北。