電腦數值控制工具機現在已經成為各種製造工業裡最主要的生產力提供者。面對目前的市場環境，產品製造商在維持高生產力的同時又得兼顧產品的品質下背負了龐大的壓力。為了維持一定的產品精度，機器所造成體積誤差必須校正與補償以避免製造出有缺陷的產品。為了迎合消費者喜好，產品的幾何形狀設計越來越複雜，設計人員也越來越習慣使用自由曲線與曲面來設計新的產品。NUBRS曲線與曲面定義近來被廣泛應用在CAD/CAM系統，NURBS格式定義對於工件幾何形狀的描述遠比傳統CAM系統用後處理器產生大量小直線線段來逼近的精度高。另外由於NURBS格式的檔案資料量遠比線段逼近小，因此傳輸上負擔比較輕，加上單一路徑長度足夠讓機器加速到較高的切削速度，使得複雜曲面的高速切削加工可行性大為提高，許多高階的CNC控制器製造商已經將NURBS插補功能視為標準模組。因此，開發一個具有NURBS插補功能並整合體積誤差補償的CNC控制器將會對產品製造商帶來莫大的幫助。

本論文提出一個快速的NURBS路徑插補機制。該插補機制將CNC控制器中NURBS路徑資料從預處理到即時插補模組進行了有效率的整合。NURBS路徑的長度計算採用了適應面積法(adaptive quadrature)對長度函數的一次微分進行積分，該方法根據設定的計算精度會自動將曲線分割成許多區段，數量多寡取決於曲線的複雜度。本論文利用細分區段的結果來建構曲線長度逆轉換函式(Inverse length function)，進行即時插補計算時可直接利用曲線長度逆轉換函式計算得到相關曲線位置參數，大幅地減輕了即時插補計算的負擔。本方法可快速且準確地提供NURBS即時插補運算，模擬與實際切削結果也驗證了該插補機制的效果。

此外，本論文也提出一個對於具NURBS插補功能CNC工具機進行幾何誤差補償的快速方法。該方法利用原始路徑的NURBS基底函數來估算實際機器運動時所造成的誤差函數，由於誤差函數使用與原始路徑同樣的NURBS基底函數，因此路徑修正時只需要修正原始路徑的控制點即可達到補償的目的。如果原始路徑的基底函數無法準確描述誤差函數，則可利用NURBS節點插入法來增加原始路徑的易曲性，也同時改善本方法的補償精度。本方法主要將補償功能實現在開放式架構CNC控制器中，雖然不是在即時位置控制模組直接進行定位補償，但在NURBS插補器之前進行補償處理，在適當控制資料區塊讀取條件下，機台的一些即時狀態可在補償過程中列入考慮，因此本方法可視為一種on-line補償機制。本方法不但具有on-line補償的優點，同時所有計算都在非即時模組中完成，完全不會增加即時控制模組的計算負擔。模擬與實驗量測結果均驗證了本方法可以有效的將機器幾何誤差有效的補償到要求精度範圍內。本方法同時也可以實現在CAM系統的後處理器中，但不具備相關on-line補償的特性。

Computer numerical control (CNC) machine tools have now become the primary production tool for all kinds of manufacturing industries. Manufacturers are under tremendous pressure to enhance product quality in terms of dimension accuracy while maintaining high productivity. To maintain product quality, the volumetric errors of machine tools should be calibrated or eliminated in advance to prevent the manufacture of defective parts. As modern products adopt more complex geometric shapes, it is becoming common for designers to employ free-form curves and surfaces. Non-uniform rational B-spline (NURBS) curves and surfaces are widely used in CAD/CAM systems. The NURBS formats have much more precise mold shape descriptions than conventional CAM approximations with tiny line segments. The data transfer burden is reduced due to the smaller NC data file with the NURBS format. The real cutting speed is increased and is stable for sufficient path length in the feedrate profile scheduler. All these benefits are important for high-speed cutting programming. The NURBS interpolator has recently become the standard module for the latest high-performance CNC controllers. Therefore, the designing and building of a CNC controller with NURBS interpolation and volumetric error compensation functions will be of great help to manufacturers.

In this thesis, a fast real-time NURBS path interpolation method is proposed. The proposed method efficiently integrates the data processing of a NURBS path in a CNC controller, from pre-processing to real-time interpolation. In the calculation of the total length of the NURBS path, the numerical adaptive quadrature method adapts to the integrand, i.e. the first derivative of the length function, automatically, dividing the parameter interval into subintervals with fine or coarse spacing according to the varying condition of the integrand. This new method takes full advantage of the subdivision scheme. The key point is to generate inverse length functions (ILF) for each resulting subinterval. In the real-time NURBS path interpolation, the new setting path parameter can be calculated directly using the inverse length function without any time-consuming computation for NURBS parameter approximation. The proposed method is extremely fast, accurate and suitable for real-time implementation, and simulations and a practical cutting test have proved its effectiveness.

Another important emphasis in this thesis is to present a new scheme for compensating the geometric errors of a CNC machine tool which is capable of NURBS interpolation. The proposed NURBS-based compensation method utilizes the same basis functions as the setting tool path to directly estimate the actual NURBS path. Due to the use of the same basis functions, the modification of control points is a fast and effective way to compensate the setting tool path. The knot insertion technique of NURBS is used to increase the flexibility and enhance the accuracy of compensation at the same time. This thesis proposes a fast method of on-line compensation of the total positioning errors of a setting NURBS tool path in advance without any additional real-time CPU loading. Both the simulation and experimental results show that the positioning deviations can be effectively enhanced using the fast geometric error compensation method. This proposed method can be also implemented in the post-processor of a CAM system for off-line compensation.

[1] M. Weck, Handbook of machine tools volume 2: Construction and mathematical analysis, John Wiley & Sons, 1984.
[2] Y. Kakino, Y. Ihara and A. Shinohara, Accuracy inspection of NC machine tools by Double ball bar method, Hanser publishers, Munich, Germany, 1993.
[3] L. Piegl, W. Tiller, The NURBS Book, Springer Verlag, Berlin, 1995.
[4] ISO 14649-10, Industrial automation systems and integration - Physical device control - Data model for computerized numerical controllers - Part 10: General process data, 2004.
[5] IMS Project 97006. 2003. STEP-Compliant data interface for numerical controls (STEP-NC), technical report 3.
[6] S. T. Newman, R. D. Allen, R. S. U. Rosso Jr., CAD/CAM solutions for STEP-compliant CNC manufacture, International Journal of Computer Integrated Manufacturing, 16 (7-8) (2003) 590-597.
[7] Y. Kakino, Y. Ihara, A. Shinohara, Accuracy Inspection of NC Machine Tools by Double Ball Bar Method, Hanser publishers, Munich, Germany (1993).
[8] N. A. Barakat, M. A. Elbestawi, A. D. Spence, “Kinematic and geometric error compensation of a coordinate measuring machine”, International Journal of Machine Tools and Manufacturing, 40 (6) (2000) 833-850.
[9] J. B. Bryan, “Simple method for testing measuring machines and machine tools”, Precision Engineering, 4 (2) (1982) 61-69.
[10] W. Knapp, “Measurement uncertainty and machine tool testing”, CIRP Annals – Manufacturing Technology, 51 (1) (2002) 459-462.
[11] J. S. Chen, J. X. Yuan, J. Ni and S. M. Wu, “Real-time compensation of time-variant volumetric error on a machining center”, Journal of Engineering for Industry, Transactions of the ASME, 115 (4) (1993) 472-479.
[12] M. H. Attia, L. Kops, “Thermometric design considerations for temperature monitoring in machine tools and CMM structures”, International Journal of Advanced Manufacturing Technology, 8 (5) (1993) 311-319.
[13] J. S. Chen, C. C. Ling, “Improving the machine accuracy through machine tool metrology and error correction”, International Journal of Advanced Manufacturing Technology, 11 (3) (1996) 198-205.
[14] J. Mou, “Method of using neural networks and inverse kinematics for machine tools error estimation and correction”, Journal of Manufacturing Science and Engineering, Transactions of the ASME, 119 (2) (1997) 247-254.
[15] M. C. Daniel, W. J. Sutherland, W. W. Olson, “Modeling the effects of component level geometric and form deviations on machine tool slideway errors”, Technical Paper – Society of Manufacturing Engineers, MR (1998) 1-6.
[16] A. H. Slocum, Precision Machine Design, Prentice Hall, Englewood Cliffs, 1992.
[17] M. Weck, Handbook of machine tools volume 4: Metrological Analysis and Performance Test, John Wiley & Sons, 1984.
[18] K. F. Eman, B. T. Wu, M. F. DeVries, “Generalized geometric error model for multi-axis machines”, CIRP Annals, 36 (1) (1987) 253-256.
[19] P. M. Ferreira, C. R. Liu, “Method for estimating and compensating quasistatic errors of machine tools”, Journal of Engineering for Industry, Transactions of the ASME, 115 (1) (1993) 149-159.
[20] V. S. B. Kiridena and P. M. Ferreira, Kinematic modeling of quasistatic errors of three-axis machining centers, International Journal of Machine Tools and Manufacturing 34 (1) (1994) 85-100.
[21] Y. Hatamura, T. Nagao, M. Mitsuishi, K. Kato, S. Taguchi, T. Okumura, G. Nakagawa, H. Suqishita, “Development of an intelligent machining center incorporating active compensation for thermal distortion”, Annals of the CIRP 42 (1) (1993) 549-552.
[22] M. Weck, P. McKeown, R. Bonse, U. Herbst, “Reduction and compensation of thermal errors in machine tools”, Annals of the CIRP – Manufacturing Technology 44 (2) (1995) 589-598.
[23] K. Lau, Q. Ma, X. Chu, Y. Liu and S. Olson, An advanced 6-degree-of-freedom laser system for quick CNC machine and CMM error mapping and compensation, Automated Precision Inc., Gaithersburg, MD 20879 U.S.A. (2000)
[24] W. Schroeder, A new 6D measuring device for rotary table calibration, Dr. Johannes Heidenhein GmbH, Germany, 2003.
[25] A. Koliskor, et. al., “Compensating for automatic-cycle machining errors”, Stanki Instrum, (5) (1970) 7-8.
[26] W. T. Lei, Y. Y. Hsu, “Accuracy enhancement of five-axis CNC machines through real-time error compensation”, International Journal of Advanced Manufacturing Technology 43 (9) (2003) 871-877.
[27] M. Rahman, J. Heikkala, K. Lappalainen, “Modelling, measurement and error compensation of multi-axis machine tools. Par I: theory”, International Journal of Machine Tools and Manufacture 40 (10) (2000) 1535-1546.
[28] K. C. Fan, “Generalized study of volumetric error analysis for NC machine tools and CMMs part one – mathematical model”, Journal of the Chinese Society of Mechanical Engineers, 10 (2) (1989) 135-144.
[29] W. Knapp, “Testing NC machine tools with the circular test”, Proceedings of the International Machine Tool Design and Research Conference 25th, (1985) 141-147.
[30] ISO 230-2, Acceptance Code for Machine Tools, Part 2 (1990) p 12.
[31] Sinumerik Extended Functions (Part 2), Sinumerik 840D/840Di/810D (CCU2) (2002) www.ad.siemens.de.
[32] N. Jun, “CNC machine accuracy enhancement through real-time error compensation”, Journal of Manufacturing Science and Engineering, 119 (4) (1997) 717-725.
[32] R. Md. Mahbubur et al, “Positioning accuracy improvement in five-axis milling by post processing”, International Journal of Machine Tool and Manufacture, 37 (2) (1997) 223-236.
[33] K. Morishiqe et al, “Collision-free tool path generation using 2-dimensional C-shape for 5-axis control machining”, International Journal of Advanced Manufacturing Technology, 13 (6) (1997) 393-400.
[34] M. Weck, J. Wolf, STEP-NC – The STEP compliant NC programming interface: evaluation and improvement of the modern interface, Proceeding of the IMS project forum (2001).
[35] J. Bulter, M. Tomizuka, Trajectory planning for high speed multiple axis contouring systems, Proceedings of the American Control Conference (1989) 87-94.
[36] P. H. Meckl, P. B. Arestides, M. C. Woods, Optimized s-curve motion profiles for minimum residual vibration, Proceedings of the American Control Conference (1998) 2627-2631.
[37] S. S. Yeh, P. L. Hsu, The speed-controlled interpolator for machining parametric curves, Computer-Aided Design 31 (5) (1999) 349-357.
[38] S. S. Yeh, P. L. Hsu, Adaptive feedrate interpolation for parametric curves with a confined chord error, Computer-Aided Design 34 (3) (2002) 229-237.
[39] X. Liu, F. Ahmad, K. Yamazaki, M. Mori, Adaptive interpolation scheme for NURBS curves with the integration of machining dynamics, International Journal of Machine Tools & Manufacture 45 (2005) 433–444.
[40] D. Kiritsis, “High precision interpolation algorithm for 3D parametric curve generation”, Computer Aided Design, 26 (11) (1994) 850-856.
[41] Y. Koren et al, “CNC interpolators: algorithm and analysis”, Manufacturing Science and Engineering, 64 (1993) 83-92.
[42] D.C.H. Yang et al, “Parametric interpolator versus linear interpolator for command generation in CNC machining”, Manufacturing Science and Engineering, 63 (1993) 59-67.
[43] C. McMahon et al, CAD/CAM from principle to practice, Addison Wesley (1993).
[44] Q.G. Zhang, R. B. Greenway, Development and implementation of a NURBS curve motion interpolator, Robotics and Computer-Integrated Manufacturing 14 (1) (1998) 27-36.
[45] X. Zhiming, C. Jincheng, F. Zhengjin, Performance evaluation of a real-time interpolation algorithm for NURBS curves, International Journal of Advanced Manufacturing Technology 20 (4) (2002) 270–276.
[46] M. Y. Cheng, M. C. Tsai, J. C. Kuo, Real-time NURBS command generators for CNC servo controllers, International Journal of Machine Tools & Manufacture 42 (7) (2002) 801–813.
[47] T. Otsuki et al, Fanuc Ltd., Free-Form Curve Interpolation Method and Apparatus, US patent 5,815,401, 1996.
[48] T. Otsuki et al, Fanuc Ltd., Free Curve Interpolation Apparatus and Interpolation Method, US patent 5,936,864, 1997.
[49] H. Wang et al, “Arc-length parameterized spline curves for real-time simulation”, Proceedings of the 5th International Conference on Curves and Surfaces, San Malo, France, (2002) 387-396.
[50] B. W. Mooring, Z. S. Roth, M. R. Driels, Fundamentals of manipulator calibration, John Wiley & Sons Inc. (1991).
[51] S. Ibaraki, Y. Kakino, K. Lee, Y. Ihara, J. Braasch, A. Eberherr, “Diagnosis and compensation of motion errors in NC machine tools by arbitrary shape contouring error measurement”, Laser Metrology and Machine Performance V (2001) 59-68.
[52] B. W. Mooring, Z. S. Roth, M. R. Driels, Fundamentals of manipulator calibration, John Wiley & Sons Inc. (1991).
[53] J. H. Mathews, K. K. Fink, Numerical Methods Using Matlab, 4th Edition, Prentice-Hall Inc. (2004).
[54] L.Y. Lin, Real-time NURBS interpolation in CNC, Master Dissertation, Department of Power Mechanical Engineering, National Tsing Hua University, R.O.C., 2004.
[55] M.L. Hung, Feedforward controller for high-speed and high precision CNC machine tools, Master Dissertation, Department of Power Mechanical Engineering, National Tsing Hua University, R.O.C., 2000.
[56] M. Rahman, Modeling and measurement of multi-axis machine tools to improve positioning accuracy in a software way, Doctoral Dissertation, Department of Mechanical Engineering, University of Oulu, Finland, 2004.
[57] J. H. Mathews, K. K. Fink, Numerical Methods Using Matlab, 4th Edition, Prentice-Hall Inc. (2004).