於多軸循跡運動中，如何降低輪廓誤差為極重要的研究課題。有鑑於此，為達降低輪廓誤差之目的，本文提出三種不同型式的運動控制架構。首先，結合位置控制器、速度前饋控制器、即時輪廓誤差估測器、模糊邏輯進給率調節器及切線輪廓誤差控制器，發展出以切線輪廓誤差控制器為基礎之運動控制架構以提升循跡運動精度。其中所發展之模糊邏輯進給率調節器為基於動態循跡誤差資訊來加以設計，可提供適當的進給命令以達到降低輪廓誤差之目的。其次，為提升自由曲線循跡運動之精度，提出一包含速度前饋控制器、修正型交叉耦合控制器及模糊邏輯進給率調節器之運動控制架構。不同於前述之模糊邏輯進給率調節器，此架構所發展之模糊邏輯進給率調節器此外，透過所提出的修正型交叉耦合控制器，可將傳統交叉耦合控制器進一步推廣至可適用於任意命令軌跡形式之循跡運動。最後，於搭配交叉耦合控制器為發展架構下，提出位置誤差補償器，其特點為透過預先補償機制可同時降低輪廓誤差及追蹤誤差。而欲進ㄧ步達到兼具良好的加工精度及效率之目的，則透過結合模糊邏輯進給率調節器與位置誤差補償器而發展出一整合型運動控制架構，並藉由多種參數式自由曲線循跡運動實驗來驗證所探討方法之可行性，其中軌跡位置命令為經由NURBS即時插値器產生。經由雙軸循跡運動實驗結果可知，本文所提出之各方法皆能有效地降低輪廓誤差，同時也能滿足循跡精度及效率的要求。

Reducing contour error is an important issue in multi-axis contour following tasks. To reduce contour error, this dissertation presents three different motion control schemes. First, to improve contouring accuracy, a Tangential Contouring Controller (TCC) based motion control scheme was developed. The scheme consists of a position loop controller with velocity command feedforward, a TCC, a real time contour error estimator, and a fuzzy logic based feedrate regulator. The feedrate regulator is designed to reduce contour error based on dynamic machining error information, which adaptively adjusts the desired feedrate value. The second scheme focuses on improving the machining accuracy in free-form contour following tasks. This scheme consists of a velocity command feedforward, a modified Cross-Coupled Controller (CCC), and a fuzzy logic based feedrate regulator different from the one employed in the first scheme. The feedrate regulator, which can adaptively adjust the desired feedrate to a proper value, is designed based on the dynamic machining error and curvature information. In addition, the modified CCC extends the conventional CCC to arbitrary types of contour following tasks. Finally, the third scheme, using the CCC concept as a starting point, develops a Position Error Compensator (PEC) approach to reduce contour error. The main advantage of PEC is that it can simultaneously improve tracking and contouring performance by compensating for position errors in advance. To further meet contouring accuracy and efficiency requirements, the integrated motion control scheme equipped with a fuzzy logic based feedrate regulator and PEC is developed. To test the feasibility of the proposed approaches, several parametric free-form contour following experiments were conducted, in which the position commands were generated by a real-time NURBS (Non-Uniform Rational B-Spline) interpolator. Experimental results demonstrate that all of the proposed approaches can significantly reduce contour error for biaxial contour following tasks. Additionally, the results demonstrate that the contouring efficiency and accuracy of the proposed approaches meet machining requirements.

[1] AC servo unit maintenance manual. FANUC Ltd., 1987.
[2] Y. Altintas and K. Erkorkmaz, “Feedrate optimization for spline interpolation in high speed machine tools,” Ann. CIRP, vol. 52, pp. 297-302, 2003.
[3] S. Bedi, I. Ali, and N. Quan, “Advanced interpolation techniques for NC. machines,” Trans. ASME, J. Eng. Ind., vol. 115, pp. 329-336, 1993.
[4] H. Bin, K. Yamazaki, and M. De Vries, “A microprocessor-based control scheme for the improvement of contouring accuracy,” Ann. CIRP, vol. 32, pp. 275-279, 1983.
[5] R. H. Brown, S. C. Schneider, and M. G. Mulligan, “Analysis of algorithms for velocity estimation from discrete position versus time data,” IEEE Trans. Ind. Electron., vol. 39, pp. 11-19, 1992.
[6] C. S. Chen and A. C. Lee, “New direct velocity and acceleration feedforward tracking control in a retrofitted milling machine,” Int. J. Jpn. Soc. Precis. Eng., vol. 33, pp. 178-184, 1999.
[7] C. S. Chen, Y. H. Fan, and S. P. Tseng, “Position command shaping control in a retrofitted milling machine,” Int. J. Mach. Tools. Manuf., vol. 46, pp. 293-303, 2006.
[8] C. T. Chen, Analog and digital control system design: Transfer-function, state-space, and algebraic methods. New York: Oxford University Press, Inc., 1995.
[9] S. L. Chen, H. L. Liu, and S. C. Ting, “Contouring control of biaxial systems based on polar coordinates,” IEEE/ASME Trans. Mechatron., vol. 7, pp. 329-345, 2002.
[10] S. L. Chen and K. C. Wu, “Contouring control of smooth paths for multiaxis motion systems based on equivalent errors,” IEEE Trans. Contr. Syst. Technol., vol. 15, pp. 1151-1158, 2007.
[11] C. W. Cheng, “Design and implementation of real-time NURBS curve and surface interpolators for motion controllers,” Ph.D. Dissertation, Mechanical Engineering, National Cheng Kung University, 2003.
[12] M. Y. Cheng, M. C. Tsai, and J. C. Kuo, “Real-time NURBS command generators for CNC servo controllers,” Int. J. Mach. Tools Manuf., vol. 42, pp. 801-813, 2002.
[13] M. Y. Cheng and C. C. Lee, “Motion controller design for contour-following tasks based on real-time contour error estimation,” IEEE Trans. Ind. Electron., vol. 54, pp. 1686-1695, 2007.
[14] M. Y. Cheng, K. H. Su, and S. F. Wang, “Contour error reduction for free-form contour following tasks of biaxial motion control systems,” Robot. Comput. Integr. Manuf., 2008 (doi:10.1016/j.rcim.2008.01.003).
[15] Y. M. Cheng and J. H. Chin, “Machining contour errors as ensembles of cutting, feeding and machine structure effects,” Int. J. Mach. Tools Manuf., vol. 43, pp. 1001-1014, 2003.
[16] J. H. Chin and T. C. Lin, “Cross-coupled precompensation method for the contouring accuracy of computer numerically controlled machine tools,” Int. J. Mach. Tools Manuf., vol. 37, pp. 947-967, 1997.
[17] J. H. Chin and H. W. Lin, “The algorithms of the cross-coupled precompensation method for generating the involute-type scrolls,” Trans. ASME, J. Dyn. Syst., Meas., Control, vol. 121, pp. 96-104, 1999.
[18] J. H. Chin, Y. M. Cheng, and J. H. Lin, “Improving contour accuracy by Fuzzy-logic enhanced cross-coupled precompensation method,” Robot. Comput. Integr. Manuf., vol. 20, pp. 65-76, 2004.
[19] G. Tao, Adaptive control design and analysis. New Jersey: John Wiley & Sons, Inc., 2003.
[20] G. T. C. Chiu and M. Tomizuka, “Coordinated position control of multi-axis mechanical systems,” Trans. ASME, J. Dyn. Syst., Meas., Control, vol. 120, pp. 389-393, 1998.
[21] G. T. C. Chiu and M. Tomizuka, “Contouring control of machine tool feed drive systems: a task coordinate frame approach,” IEEE Trans. Contr. Syst. Technol., vol. 9, pp. 130-139, 2001.
[22] J. J. Chou and D. C. H. Yang, “On the generation of coordinated motion of five-axis CNC/CMM machines,” Trans. ASME, J. Eng. Indust., vol. 114, pp. 15-22, 1992.
[23] H. Y. Chuang and C. H. Liu, “Cross-coupled adaptive feedrate control for multiaxis machine tools,” Trans. ASME, J. Dyn. Syst., Meas., Control, vol. 113, pp. 451-457, 1991.
[24] D. Du, Y. Liu, C. Yan, and C. Li, “An accurate adaptive parametric curve interpolator for NURBS curve interpolation,” Int. J. Adv. Manuf. Technol., vol. 32, pp. 999-1008, 2007.
[25] K. Erkorkmaz and Y. Altintas, “High speed CNC system design. Part I: jerk limited trajectory generation and quintic spline interpolation,” Int. J. Mach. Tools. Manuf., vol. 41, pp. 1323-1345, 2001.
[26] K. Erkorkmaz and Y. Altintas, “High speed CNC system design. Part II: modeling and identification of feed drives,” Int. J. Mach. Tools. Manuf., vol. 41, pp. 1487-1509, 2001.
[27] K. Erkorkmaz and Y. Altintas, “High speed CNC system design. Part III: high speed tracking and contouring control of feed drives,” Int. J. Mach. Tools. Manuf., vol. 41, pp. 1637-1658, 2001.
[28] R. T. Farouki and S. Shah, “Real-time CNC interpolators for Pythagorean- hodograph curves,” Comput. Aided Geom. D. , vol. 13, pp. 583-600, 1996.
[29] R. T. Farouki and Y. F. Tsai, “Exact Taylor series coefficients for variable-feedrate CNC curve interpolators,” Comput. Aided Design, vol. 33, pp. 155-165, 2001.
[30] Y. Funahashi and M. Yamada, “Zero phase error tracking controllers with optimal gain characteristics,” Trans. ASME, J. Dyn. Syst., Meas., Control, vol. 115, pp. 311-318, 1993.
[31] L. Guo and M. Tomizuka, “High-speed and high-precision motion control with an optimal hybrid feedforward controller,” IEEE/ASME Trans. Mechatron., vol. 2, pp. 110-122, 1997.
[32] R. E. Haber, C. R. Peres, A. Alique, S. Ros, C. Gonzalez, and J. R. Alique, “Toward intelligent machining: Hierarchical fuzzy control for the end milling process,” IEEE Trans. Contr. Syst. Technol., vol. 6, pp. 188-199, 1998.
[33] H. C. Ho, J. Y. Yen, and S. S. Lu, “A decoupled path-following control algorithm based upon the decomposed trajectory error,” Int. J. Mach. Tools. Manuf., vol. 39, pp. 1619-1630, 1999.
[34] J. T. Huang and D. C. H. Yang, “Precision command generation for computer controlled machines,” ASME, Pred. Eng., vol. PED-58, pp. 89-104, 1992.
[35] S. J. Huang and Y. W. Lin, “Application of grey predictor and fuzzy speed regulator in controlling a retrofitted machining table,” Int. J. Mach. Tools Manuf., vol. 36, pp. 477-489, 1996.
[36] Y. Kakino, Y. Ihara, and Y. Nakatsu, “The measurement of motion errors of NC machine tools and diagnosis of their origins by using telescoping magnetic ball bar method,” Ann. CIRP, vol. 36, pp. 377-380, 1987.
[37] B. K. Kim, W. K. Chung, H. T. Choi, I. H. Suh, and Y. H. Chang, “Robust internal loop compensator design for motion control of precision linear motor,” in Proc. IEEE Int. Symposium Ind. Electron. Bled, Slovenia, July, 1999, pp. 1045-1050.
[38] Y. Koren, “Cross-coupled biaxial computer control for manufacturing systems,” Trans. ASME, J. Dyn. Syst., Meas., Control, vol. 102, pp. 265–272, 1980.
[39] Y. Koren and C. C. Lo, “Variable-gain cross-coupling controller for contouring,” Ann. CIRP, vol. 40, pp. 371-374, 1991.
[40] Y. Koren, C. C. Lo, and M. Shpitalni, “CNC interpolators: Algorithms and analysis,” ASME Manuf. Sci. Eng., vol. PED-64, pp. 83-92, 1993.
[41] U. Kunz and W. Oberschelp, “High performance control for machine tools using microprocessors,” in Proc. PCIM Conf. München, Germany, June,1989, pp. 192-198.
[42] J. C. Kuo, “Design and Implementation of real-time NURBS Interpolators for CNC servo controllers,” Master Thesis, Mechanical Engineering, National Cheng Kung University, 2000.
[43] Y. D. Landau, Adaptive control: The model reference approaches. New York: Marcel Dekker, 1979.
[44] C. C. Lee, “Fuzzy logic in control systems: Fuzzy logic controller. Part I,” IEEE Trans. Syst., Man., Cyb., vol. 20, pp. 404-418, 1990.
[45] C. C. Lee, “Fuzzy logic in control systems: Fuzzy logic controller, Part II,” IEEE Trans. Syst., Man., Cyb., vol. 20, pp. 419-435, 1990.
[46] H. S. Lee and M. Tomizuka, “Robust motion controller design for high-accuracy positioning systems,” IEEE Trans. Ind. Electron., vol. 43, pp. 48-55, 1996.
[47] C. T. Lin and C. S. G. Lee, Neural fuzzy systems: A neuro-fuzzy synergism to intelligent systems. Taiwan: Prentice Hall, Inc., 1996.
[48] C. T. Lin, I. F. Chung, and S. Y. Huang, “Improvement of machining accuracy by fuzzy logic at corner parts for wire-EDM,” Fuzzy Set. Syst., vol.122, pp. 499-511, 2001.
[49] M. T. Lin, M. S. Tsai, and H. T. Yau, “Development of a dynamics-based NURBS interpolator with real-time look-ahead algorithm,” Int. J. Mach. Tools Manuf., vol. 47, pp. 2246-2262, 2007.
[50] X. Liu, F. Ahmad, K. Yamazaki, and M. Mori, “Adaptive interpolation scheme for NURBS curves with the integration of machining dynamics,” Int. J. Mach. Tools Manuf., vol. 45, pp. 433-444, 2005.
[51] C. C. Lo and C. Y. Chung, “Tangential-contouring controller for biaxial motion control,” Trans. ASME, J. Dyn. Syst., Meas., Control, vol. 121, pp. 126-129, 1999.
[52] O. Masory, “Improving contouring accuracy of NC/CNC systems with additional velocity feed forward loop,” J. Eng. Ind., vol. 108, pp. 227-230, 1986.
[53] P. V. O'Neil, Advanced engineering mathematics. 5th ed., Calif.: Books/Cole Thomson Learning, 2003.
[54] S. L. Omirou and A. K. Barouni, “Integration of new programming capabilities into a CNC milling system,” Robot. Comput. Integr. Manuf., vol. 21, pp. 518-527, 2005.
[55] M. P. do Carmo, Differential geometry of curves and surfaces. Upper Saddle River, New Jersey: Prentice-Hall, Inc., 1976.
[56] C. C. Peng and C. L. Chen, “Biaxial contouring control with friction dynamics using a contour index approach,” Int. J. Mach. Tools Manuf., vol. 47, pp. 1542-1555, 2007.
[57] L. A. Piegl and W. Tiller, The NURBS book. 2nd ed., New York: Springer, 1997.
[58] A. N. Poo, J. G. Bollinger, and G. W. Younkin, “Dynamic errors in type 1 contouring systems,” IEEE Trans. Ind. Applicat. , pp. 477-484, 1972.
[59] R. Ramesh, M. A. Mannan, and A. N. Poo, “Tracking and contour error control in CNC servo systems,” Int. J. Mach. Tools Manuf., vol. 45, pp. 301-326, 2005.
[60] T. Sata, F. Kimura, N. Okada, and M. Hosaka, “A new method of NC interpolation for machining the sculptured surface,” Ann. CIRP, vol. 30, pp. 369-372, 1981.
[61] Y. T. Shih, C. S. Chen, and A. C. Lee, “A novel cross-coupling control design for bi-axis motion,” Int. J. Mach. Tools Manuf., vol. 42, pp. 1539-1548, 2002.
[62] D. J. Shin, H. S. Ryu, and U. Y. Huh, “Fuzzy logic control for the contouring accuracy of XY positioning system,” in Proc. IEEE Int. Symposium Ind. Electro., vol. 2, Pusan, South Korea, June, 2001, pp. 1248-1252.
[63] K. Srinivasan and P. K. Kulkarni, “Cross-coupled control of biaxial feed drive servomechanisms,” Trans. ASME, J. Dyn. Syst., Meas., Control, vol. 112, pp. 225-232, 1990.
[64] 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 Proc. IEEE Int. Conf. Mechatron., Budapest, Hungary, July, 2006, pp. 209-214.
[65] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man. Cyb., vol. 15, pp. 116-132, 1985.
[66] 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,” Int. J. Mach. Tools Manuf., vol. 39, pp. 1673-1692, 1999.
[67] M. Tomizuka, “Zero phase error tracking algorithm for digital control,” Trans. ASME, J. Dyn. Syst., Meas., Control, vol. 109, pp. 65-68, 1987.
[68] M. C. Tsai, S. J. Lau, and S. C. Lee, “DSP-based motion control by H∞ axis-controller and fuzzy adaptive feedrate,” Control Eng., vol. 3, pp. 1587-1597, 1995.
[69] M. C. Tsai and C. W. Cheng, “A real-time predictor-corrector interpolator for CNC machining,” Trans. ASME, J. Manuf., vol. 125, pp. 449-460, 2003.
[70] M. C. Tsai, C. W. Cheng, and M. Y. Cheng, “A real-time NURBS surface interpolator for precision three-axis CNC machining,” Int. J. Mach. Tools Manuf., vol. 43, pp. 1217-1227, 2003.
[71] M. S. Tsai, H. W. Nien, and H. T. Yau, “Development of an integrated look-ahead dynamics-based NURBS interpolator for high precision machinery,” Comput. Aided Design, vol.40, pp.554-566, 2008.
[72] A. G. Ulsoy and Y. Koren, “Control of machining processes,” Trans. ASME, J. Dyn. Syst., Meas., Control, vol. 115, pp. 301-308, 1993.
[73] L. X. Wang, A course in fuzzy systems and control. Upper Saddle River, New Jersey: Prentice-Hall, Inc., 1996.
[74] H. T. Yau and M. J. Kuo, “NURBS machining and feed rate adjustment for high-speed cutting of complex sculptured surfaces,” Int. J. Prod. Res., vol. 39, pp. 21-41, 2001.
[75] H. T. Yau, M. T. Lin, and M. S. Tsai, “Real-time NURBS interpolation using FPGA for high speed motion control,” Comput. Aided Design, vol. 38, pp. 1123-1133, 2006.
[76] S. S. Yeh and P. L. Hsu, “Theory and applications of the robust cross-coupled control design,” Trans. ASME, J. Dyn. Syst., Meas., Control, vol. 121, pp. 524-530, 1999.
[77] S. S. Yeh and P. L. Hsu, “Analysis and design of the integrated controller for precise motion systems,” IEEE Trans. Contr. Syst. Technol., vol. 7, pp. 706-717, 1999.
[78] S. S. Yeh and P. L. Hsu, “The speed-controlled interpolator for machining parametric curves,” Comput. Aided Design, vol. 31, pp. 349-357, 1999.
[79] S. S. Yeh and P. L. Hsu, “Adaptive-feedrate interpolation for parametric curves with a confined chord error,” Comput. Aided Design, vol. 34, pp. 229-237, 2002.
[80] S. S. Yeh and P. L. Hsu, “Estimation of the contouring error vector for the cross coup -led control design,” IEEE/ASME Trans. Mechatron., vol. 7, pp. 44-51, 2002.
[81] Z. M. Yeh, Y. S. Tarng, and C. Y. Nian, “A self-organizing neural fuzzy logic controller for turning operations,” Int. J. Mach. Tools Manuf., vol. 35, pp. 1363-1374, 1995.
[82] D. S. Yun and G. J. Jeon, “Feedrate control of CNC machines by inverse mapping method,” in Proc. IEEE International Symposium Intelligent Control, Rio, Patras, Greece, July, 2000 , TB-3, pp. 279-284.
[83] L. A. Zadeh, “Fuzzy sets,” Informat. Control, vol. 8, pp. 338-353, 1965.
[84] Q. G. Zhang and R. B. Greenway, “Development and implementation of a NURBS curve motion interpolator,” Robot. Comput. Integr. Manuf., vol. 14, pp. 27-36, 1998.
[85] X. Zhiming, C. Jincheng, and F. Zhengjin, “Performance evaluation of a real-time interpolation algorithm for NURBS curves,” Int. J. Adv. Manuf. Technol., vol. 20, pp. 270-276, 2002.