線性軸為各工具機中不可或缺的零組件，因此針對線性軸的誤差量測非常重要。有鑑於此，本論文提出一個光學量測系統，用來量測線性軸六自由度誤差，並結合干涉法及幾何光學法解析誤差。其中干涉法主要用於定位誤差解析，利用此方法可解決過往幾何光學受限於感測器大小的問題，並提高定位誤差量測精度；幾何光學法則用於其餘五自由度誤差解析，利用歪斜光線追跡法與齊次座標轉換建立數學模型與光學模擬軟體Zemax進行比較，此方法相比於干涉法，可大幅簡化數據擷取時間及運算量，透過上述兩種方法的同時應用，可以提高過往使用單一方法時所面臨的問題，進而提高量測精度及降低運算時間。
除此之外，本論文也提出補償雷射源擾動及系統安裝誤差的方法。首先，本論文利用兩個感測器量測雷射源的四自由度擾動，並將其加入至數學模型中，如此即解決量測時雷射源擾動造成的誤差；接著，本論文利用靈敏度分析，分析各元件安裝誤差對量測精度的影響，進而找出需考慮之安裝誤差項，接著將其設為未知數帶入數學模型中，透過市售干涉儀量出標準數據，並以最小平方法去解析各安裝誤差，如此即降低安裝誤差造成的影響。

This paper proposes an optical measurement system to measure the six-degree-of-freedom errors for linear stage, and combines the interferometry and geometrical optics to analyze these errors. Among them, the interferometric method is mainly used for position error analysis. This method can solve the problem which geometrical optics is limited by the size of the sensor in the past, and improve the measurement accuracy of position error. The geometrical optics is used for the other five degrees of freedom errors analysis, using skew-ray tracing to compare with the optical simulation software Zemax. In addition, this paper also proposes methods to compensate for laser beam drifts and installation errors. First, we use two sensors to measure the four-degree-of-freedom beam drifts of the laser source, and add them to the mathematical model, so as to solve the errors caused by the laser beam drifts during measurement. Then, we use the sensitivity analysis to analyze the influence of each installation error on the measurement accuracy, and find out the installation errors that need to be considered. Next, we set them as an unknown and bring them into the mathematical model. Finally, the standard data through a commercial interferometer are measured, and the least squares method is used to analyze the installation errors, so that the influence of installation errors can be reduced.

[1]行政院新聞傳播處，"協助工具機產業因應中美貿易戰衝擊"，2019。 [Online]. Available: https://www.ey.gov.tw/Page/5A8A0CB5B41DA11E/977ec967-d10d-46e4-8849-29c7a66c313f
[2]楊德華，"台灣工具機產業的回顧與展望"，機械資訊 592期: 臺灣機械工業同業公會，2006。 [Online]. Available: http://www.tami.org.tw/print/592/592_02.htm
[3]莊瀅芯，"工具機發展與趨勢"，機械工業雜誌418期:工業技術研究院機械所，2018。 [Online]. Available: https://www.airitilibrary.com/Publication/alDet ailedMesh?docid=P20171221002-201801-201801040017-201801040017-32-39
[4]R. J. Hocken, "Technology of Machine Tools," Lawrence Livermore National Laboratory in University of California., 1980.
[5]W. Knapp, and E. Matthias, "Test of the Three-Dimensional Uncertainty of Machine Tools and Measuring Machines and its Relation to the Machine Errors," CIRP Annals, vol. 32, no. 1, pp. 459-464, 1983.
[6]J. A. Soons, "Accuracy Analysis of Multi-axis Machines," Eindhoven University of Technology, 1993.
[7]H. A. M. Spaan, "Software error compensation of machine tools," Technische Universiteit Eindhoven, 1995.
[8]W. G. Weekers, and P. H. J. Schellekens, "Compensation for dynamic errors of coordinate measuring machines," Measurement, vol. 20, no. 3, pp. 197-205, 1996.
[9]B. Chen, B. Xu, E. Zhang, and Y. Liu, "Laser straightness interferometer system with rotational error compensation and simultaneous measurement of six degrees of freedom error parameters," Optics Express, vol. 23, no. 7, pp. 9052-9073, 2015.
[10]O. Borisov, S. Fletcher, A. Longstaff, and A. Myers, "Performance evaluation of a new taut wire system or straightness measurement of machine tools," Precision Engineering, vol. 38, no. 3, pp. 492–498, 2014.
[11]P. Yang, T. Takamura, S. Takahashi, K. Takamasu, O. Sato, S. Osawa, and T. Takatsuji, "Development of high-precision micro-coordinate measuring machine: Multi-probe measurement system for measuring yaw and straightness motion error of XY linear stage," Precision Engineering, vol. 35, no. 3, pp. 424–430, 2011.
[12]J. M. Linares, J. Chaves-Jacob, H. Schwenke, A. Longstaff, S. Fletcher, J. Flore, E. Uhlmann, and J. Wintering, "Impact of measurement procedure when error mapping and compensating a small CNC machine using a multilateration laser interferometer," Precision Engineering, vol. 38, no. 3, pp. 578–588, 2014.
[13]P. Huang, Y. Li, H. Wei, L. Ren, and S. Zhao, "Five-degrees-of-freedom measurement system based on a monolithic prism and phase-sensitive detection technique," Applied Optics, vol. 52, no. 26, pp. 6607-6615, 2013.
[14]R. Ramesh, M. Y. Mannan, and A. N. Poo, "Error compensation in machine tools — a review Part I: geometric, cutting-force induced and fixture-dependent errors," International Journal of Machine Tools & Manufacture, vol. 40, no. 9, pp. 1235-1256, 2000.
[15]X. Yu, S. R. Gillmer, S. C. Woody, and J. D. Ellis, "Development of a compact, fiber-coupled, six degree-of-freedom measurement system for precision linear stage metrology," Review of Scientific Instruments, vol. 87, no. 6, 2016.
[16]K. C. Fan, H. Y. Wang, H. W. Yang, and L. M. Chen , "Techniques of multi-degree-of-freedom measurement on the linear motion errors of precision machines," Advanced Optical Technologies, vol. 3, no. 4, pp. 375-386, 2014.
[17]Q. Feng, B. Zhang, C. Cui, C. Kuang, Y. Zhai, and F. You , "Development of a simple system for simultaneously measuring 6DOF geometric motion errors of a linear guide," Optical Society of America, vol. 21, no. 22, pp. 25805-25819, 2013.
[18]RENISHAW，"XL-80 雷射系統" [Online]. Available: https://www.renishaw.com.tw/tw/xl-80-laser-system--8268
[19]RENISHAW，"Renishaw 發表新款 XM-60 多光束校正儀" [Online]. Available: https://www.renishaw.com.tw/tw/renishaw-unveils-the-new-xm-60-multi-axis-calibrator--39275
[20]API," XD LASER" [Online]. Available: http://www.join-star.com.tw/joinstar2_3.asp?num=616
[21]Y. T. Chen, W. C. Lin, and C. S. Liu, "Design and experimental verification of novel six-degree-of freedom geometric error measurement system for linear stage,"Optics and Lasers in EngineeringVolume, vol. 92, pp. 94-104, 2017.
[22]戴暐哲，"基於幾何光學之線性軸六自由度誤差量測系統開發與驗證" 國立成功大學機械工程學系，碩士論文，2020。
[23]曾絜妤，"幾何光學長行程線性軸定位誤差量測系統之開發" 國立成功大學機械工程學系，碩士論文，2019。
[24]林威辰，"工具機單線性軸六自由度量測系統之設計" 國立中正大學機械工程學系暨研究所，碩士論文，2014。
[25]A. H. Slocum, "Precision machine design: macromachine design philosophy and its applicability to the design of micromachines," IEEE Micro Electro Mechanical Systems, pp. 37-42, 1992.
[26]V. S. B. Kiridena and P. M. Ferreira, "Kinematic modeling of quasistatic errors of three-axis machining centers," International Journal of Machine Tools and Manufacture, vol. 34, no. 1, pp. 85-100, 1994.
[27]V. Kiridena and P. M. Ferreira, "Computational approaches to compensating quasistatic errors of three-axis machining centers," International Journal of Machine Tools and Manufacture, vol. 34, no. 1, pp. 127-145, 1994.
[28]J. Ni, "CNC machine accuracy enhancement through real-time error compensation," ASME Trans Journal of Manufacturing Science and Engineering, vol. 119, no. 4B, pp. 717-724, 1997.
[29]陳昱達，"五軸工具機旋轉軸幾何誤差量測與驗證" 國立中正大學機械工程學系暨研究所，博士論文，2019。
[30]R. Ramesh, M. A. Mannan, and A. N. Poo, "Error compensation in machine tools — a review Part II: thermal errors," International Journal of Machine Tools & Manufacture, vol. 40, no. 9, pp. 1235-1256, 2000.
[31]J. Bryan, "International status of thermal error research," Annals of the CIRP, vol. 39, no. 2, pp. 645-656, 1990.
[32]C. Cui, Q. Feng, B. Zhang, and Y. Zhao, "System for simultaneously measuring 6DOF geometric motion errors using a polarization maintaining fiber-coupled dual-frequency laser," Optical Society of America, vol. 24, no. 6, pp. 6735-6748, 2016.
[33]X. Wei, Z. Su, X. Yang, Z. Lv, Z. Yang, H. Zhang, X. Li, and F. Fang, "A Novel Method for the Measurement of Geometric Errors in the Linear Motion of CNC Machine Tools," applied sciences, vol. 9, no. 16, pp. 3357, 2019.
[34]RENISHAW，"干涉儀系統的運作方式？" [Online]. Available: https://www.renishaw.com.tw/tw/how-do-interferometric-systems-work--38612
[35]T. B. Eom, J. Y. Kim, and K. Jeong, "The dynamic compensation of nonlinearity in a homodyne laser interferometer," Measurement Science and Technology, vol. 12, no. 10, pp. 1734-1738, 2001.
[36]P. Hu, J. Zhu, X. Zhai, and J. B. Tan, "DC-offset-free homodyne interferometer and its nonlinearity compensation," Optics Express, vol. 23, no. 7, pp. 8399-8408, 2015.
[37]P. Gregorcic, T. Pozar, and J. Mozina, "Quadrature phase-shift error analysis using a homodyne laser interferometer," Optics Express, vol. 17, no. 18, pp. 16322-16331, 2009.
[38]E. Zhang, B. Chen, H. Zheng, L. Yan, and X. Teng, "Laser heterodyne interferometer with rotational error compensation for precision displacement measurement," Optics Express, vol 23, no. 1, pp. 90-98, 2018.
[39]Q. Lv, Z. Liu, W. Wang, X. Li, S. Li, Y. Song, H. Yu, and W. Li, "Simple and compact grating-based heterodyne interferometer with the Littrow configuration for high-accuracy and long-range measurement of two-dimensional displacement," Applied Optics, vol. 57, no. 31, pp. 9455-9463, 2018.
[40]K. N. Joo, E. Clark, Y. Zhang, J. D. Ellis, and F. Guzman, "A compact high-precision periodic-error-free heterodyne interferometer," Journal of the Optical Society of America A, vol. 37, no. 9, pp. B11-18, 2020.
[41]S. Li., X. Jia, M. Chen, and Y. Yang, "Error analysis and correction for color in laser triangulation measurement," Optik, vol. 168, pp. 165-173, 2018.
[42]J. H. Wu, R. S. Chang, and J. A. Jiang, "A Novel Pulse Measurement System by Using Laser Triangulation and a CMOS Image Sensor," Sensors, vol. 7, no. 12, pp. 3366-3385, 2007.
[43]B. Sun, and B. Li, "A Rapid Method to Achieve Aero-Engine Blade Form Detection." Sensors, vol. 15, no. 6, pp3 12782-12801, 2015.
[44]L. Linder, O. Sergiyenko, J. C. Rodríguez-Quiñonez, V. Tyrsa, P. Mercorelli, W. F. Fuentes, F. N. Murrieta-Rico, and J. I. Nieto-Hipolito, "Continuous 3D scanning mode using servomotors instead of stepping motors in dynamic laser triangulation," IEEE 24th International Symposium on Industrial Electronics, pp. 944-949, 2015.
[45]Q. Yao, and M. Cao, "Design of optical emission system in 3D shape detection with oblique laser triangulation probe," Journal of Physics: Conference Series, 2021.
[46]F. Struckmeier, J. Zhao, and F. P. Leon, "Measuring the supporting slats of laser cutting machines using laser triangulation," The International Journal of Advanced Manufacturing Technology, vol. 108, pp. 3819–3833, 2020.
[47]P. Siekański, K. Magda, K. Malowany, J. Rutkiewicz, A. Styk, J. Krzesłowski, T. Kowaluk, and A. Zagórski, "On-Line Laser Triangulation Scanner for Wood Logs Surface Geometry Measurement," Sensors, vol. 19, no. 5, pp. 1074-1096, 2019
[48]C. Wu, B. Chen, and C. Ye, "Detecting defects on corrugated plate surfaces using a differential laser triangulation method," Optics and Lasers in Engineering, vol. 129, 2020.
[49]Y. Selami, W. Tao, Q. Gao, H. Yang, and H. Zhao, "A Scheme for Enhancing Precision in 3-Dimensional Positioning for Non-Contact Measurement Systems Based on Laser Triangulation," Sensors, vol. 18, no. 2, pp. 504-515, 2018.
[50]M. Pillarz, A. V. Freyberg, D. Stöbener, and A. Fischer, "Gear Shape Measurement Potential of Laser Triangulation and Confocal-Chromatic Distance Sensors," Sensors, vol. 21, no. 3, pp. 937-958, 2021.
[51]C. Kuang, E. Hong, Q. Feng, B. Zhang, and Z. Zhang, "A novel method to enhance the sensitivity for two-degrees-of-freedom straightness measurement," MEASUREMENT SCIENCE AND TECHNOLOGY, vol. 18, no. 12, pp. 3795-3800, 2007.
[52]C. H. Liu, W. Y. Jywe, H. L. Huang, T.H. Hsu, M.S. Wang, and S.Y. Deng, "Development of a straightness measuring system and compensation technique using multiple corner cubes for precision stages," Journal of Engineering Manufacture, vol. 224, no. 3, pp. 483-492, 2010.
[53]K. C. Fan, C. L. Chu, J. L. Liao, and J. I. Mou, "Development of a high-precision straightness measuring system with DVD pick-up head," Measurement Science and Technology, vol. 14, no. 1, pp. 47-54,2003.
[54]Q. Feng, B. Zhang, and C. Kuang, "A straightness measurement system using a single-mode fiber-coupled laser module," Optics & Laser Technology, vol. 36, no. 4, pp. 279-283, 2004.
[55]C. M. Wu, "Heterodyne interferometric system with subnanometer accuracy for measurement of straightness," Applied Optics, vol. 43, no 19, pp. 3812-3816, 2004.
[56]S. T. Lin, "A laser interferometer for measuring straightness," Optics & Laser Technology, vol. 33, no. 3, pp. 195-199, 2001.
[57]H. L. Hsieh, B. Y. Sun, G. Y. Fan, and J. Y. Lee, "2-DOF in-Plane Displacement Measurement by Wollaston Prism-based Interferometer," 14th IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications (MESA), 2018.
[58]Y. Lou, Z. Li, B. Chen, and J. Jian, "A phase differential heterodyne interferometer for simultaneous measurement of straightness error and displacement," Optics Communications, Optics Communications, vol. 497,2021.
[59]S. T. Lin, S. L. Yeh, C. S. Chiu, and M. S. Huang, "A calibrator based on the use of low-coherent light source straightness interferometer and compensation method," Optics Express, vol. 19, no 22, pp. 21929-21937, 2011.
[60]Q. Chen, D. Lin, J. Wu, J. Yan, and C. Yin, "Straightness/coaxiality measurement system with transverse Zeeman dual-frequency laser," Measurement Science and Technology, vol. 16, no. 10, pp.2030-2037, 2005.
[61]K. Li, C. Kuang, and X. Liu, "Small angular displacement measurement based on an autocollimator and a common path compensation principle," Review of Scientific Instruments, vol. 84, no. 1, pp. 015108-1-7, 2013.
[62]F. Zhu, J. Tan, and J. Cui, "Beam splitting target reflector based compensation for angular drift of laser beam in laser autocollimation of measuring small angle deviations," Review of Scientific Instruments, vol. 84, no. 6, pp. 065116-1-8, 2013.
[63]G. Kikuchi, and R. Furutani, "Interferometer for pitch and yaw measurement using LC-screen and four ball lenses," Measurement Science and Technology, vol. 31, no. 9, pp. 094016, 2020.
[64]Z. Ge, and M. Takeda, "High-resolution two-dimensional angle measurement technique based on fringe analysis," Applied Optics, vol. 42, no. 34, pp. 6859-6868, 2003.
[65]M. H. Chiu, S. F. Wang, and R. S. Chang, "Instrument for measuring small angles by use of multiple total internal reflections in heterodyne interferometry," Applied Optics, vol. 43, no, 29, pp. 5438-5442, 2004.
[66]S. Chatterjee, and Y. P. Kumar, "Measurement of two-dimensional small angle deviation with a prism interferometer," Applied Optics, vol. 47, no 27, pp. 4900-4906 2008.
[67]I. Hahn, M. Weilert, X. Wang, and R. Goullioud, "A heterodyne interferometer for angle metrology," Review of Scientific Instruments, vol. 81, no. 4, pp. 045103-1-6, 2010.
[68]Y. Zhai, Q. Feng, and B. Zhang, "A simple roll measurement method based on a rectangular-prism," Optics & Laser Technology, vol. 44, no. 4, pp. 839-843, 2012.
[69]Z. Yusheng, Z. Zhifeng, S. Yuling, W. Xinjie, and F. Qibo, "A high-precision roll angle measurement method," Optik, vol. 126, no. 24, pp. 4837-4840, 2015.
[70]Y. Cai, B. Yang, B. Yang, and K. C. Fan, "Robust roll angular error measurement system for precision machines," Optics Express, vol. 27, no. 6, pp. 8027-8036, 2019.
[71]S. Li, C. Yang, E. Zhang, and G. Jin, "Compact optical roll-angle sensor with large measurement range and high sensitivity," Optics Letters vol. 30, no. 3, pp. 242-244, 2005.
[72]Y. Le, W. Hou, K. Hu, and K. Shi, "High-sensitivity roll-angle interferometer," Optics Letters, vol. 38, no. 18, pp. 3600-3603, 2013.
[73]S. R. Gillmer, X. Yu, C. Wang, and J. D. Ellis, "Robust high-dynamic-range optical roll sensing," Optics Letters, vol. 40, no 11, pp. 2497-2500, 2015.
[74]C. Chen, H. Zhang, S. Chen, B. Liu, K. Zhang, and H. Ji, "Decoupling and detecting angular motion errors based on a line laser sensor for motion platforms," Applied Optics, vol. 59, no 2, pp. 500-507, 2020.
[75]S. Gao, B. Zhang, Q. Feng, C. Cui, S. Chen, and Y. Zhao , "Errors crosstalk analysis and compensation in the simultaneous measuring system for five-degree-of-freedom geometric error," Applied Optics, vol. 59, no. 2, pp. 500-507, 2020.
[76]C. Cui, Q. Feng, and B. Zhang, "Compensation for straightness measurement systematic errors in six degree-of-freedom motion error simultaneous measurement system," Applied Optics, vol. 54, no. 11, pp. 3122-3131, 2015.
[77]F. Zheng, Q. Feng, B. Zhang, J. Li, and Y. Zhao, "Effect of detector installation error on the measurement accuracy of multi-degree of freedom geometric errors of a linear axis," Measurement Science and Technology, vol. 31, no. 9, pp. 094018-1-7 2020.
[78]Y. Fan, Z. Lou, Y. Huang, and K. C. Fan, "Self-compensation method for dual-beam roll angle measurement of linear stages," Optics Express, vol. 29, no. 17, pp. 26340-26352, 2021.
[79]C. Kuang, Q. Feng, B. Zhang, B. Liu, S. Chen, and Z. Zhang, "A four-degree-of-freedom laser measurement system (FDMS) using a single-mode fiber-coupled laser module," Sensors & Actuators: A. Physical, vol. 125, no. 1, pp. 100-108, 2005.
[80]Y. Zhao, B. Zhang, and Q. Feng, "Measurement system and model for simultaneously measuring 6DOF geometric errors," Optics Express, vol. 25, no 18, pp. 20993-21007, 2017.
[81]J. Li, Q. Feng, C. Bao, and Y. Zhao, "Method for simultaneous measurement of five DOF motion errors of a rotary axis using a single-mode fiber-coupled laser," Optics Express, vol. 26, no 3, pp. 2535-2545 2018.
[82]C. Bao, J. Li, Q. Feng, and B. Zhang, "Error-compensation model for simultaneous measurement of five degrees of freedom motion errors of a rotary axis," Measurement Science and Technology, vol. 29, no. 7, pp. 075004-1-10, 2018.
[83]F. Zheng, Q. Feng, B. Zhang, and J. Li, "A Method for Simultaneously Measuring 6DOF Geometric Motion Errors of Linear and Rotary Axes Using Lasers," Sensors,2019
[84]P. Jia, B. Zhang, Q. Feng, and F. Zheng, "Simultaneous Measurement of 6DOF Motion Errors of Linear Guides of CNC Machine Tools Using Different Modes," sensors, vol. 19, no. 8, pp. 1764-1767, 2020.
[85]F. Zheng, Q. Feng, B. Zhang, J. Li, Y. Zhao, and Y. Zhao, "A high-precision laser method for directly and quickly measuring 21 geometric motion errors of three linear axes of computer numerical control machine tools," The International Journal of Advanced Manufacturing Technology, vol. 109, pp. 1285–1296, 2020.
[86]Y. Zhao, Q. Feng, B. Zhang, and C. Cui, "Influence of beam radii on a common-path compensation method for laser beam drifts in laser collimation systems," Measurement Science and Technology, vol. 27, no. 8, pp. 084013-1-9, 2016.
[87]F. You, B. Zhang, and Q. Feng, "A novel laser straightness measurement method with beam bend compensation," Optik, vol. 122, no. 17, pp. 1530-1534, 2011.
[88]H. Yang, W. Tao, Z. Zhang, S. Zhao, X. Yin, and H. Zhao, "Reduction of the Influence of Laser Beam Directional Dithering in a Laser Triangulation Displacement Probe," Sensors, vol. 17, no. 5, pp. 1126-1136, 2019.
[89]Q. Feng, B. Zhang, and C. Kuang, "Four Degree-of-Freedom Geometric Error Measurement System with Common-Path Compensation for Laser Beam Drift," International Journal of Precision Engineering and Manufacturing, vol. 9, no. 4, pp. 26-31, 2008.
[90]B. Chen, L. Cheng, L. Yan, E. Zhang, and Y. Lou, "A heterodyne straightness and displacement measuring interferometer with laser beam drift compensation for long-travel linear stage metrology," Review of Scientific Instruments, vol. 88, no. 3, pp. 035114 1-9, 2017.
[91]W. Zhao, J. Tan, L. Qiu, L. Zou, J. Cui, and Z. Shi, "Enhancing laser beam directional stability by single-mode optical fiber and feedback control of drifts," Review of Scientific Instruments, vol. 76, no. 3, pp. 36101-1-3, 2005.
[92]W. Zhao, L. Qiu, Z. Feng, and C. Li, "Laser beam alignment by fast fedback control of both linear and angular drifts," Optik, vol. 117, no. 11, pp. 505-510, 2006.
[93]S. Liu, S. Zhang, Y. Huang, Y. Wang and K. C. Fan, "The Method for Restraining Laser Drift Based on Controlling Mirror," Nanomanufacturing and Metrology, vol. 1, pp. 58–65, 2018.
[94]Y. Huang, K. C. Fan, W. Sun, and S. Liu, "Low cost, compact 4-DOF measurement system with active compensation of beam angular drift error," Optics Express, vol. 26, no. 13, pp. 17185-17198, 2018.
[95]S. Liu, S. Tan, Y. Huang, Y. Wang, and K. C. Fan, "Design of a compact four degree-of-freedom active compensation system to restrain laser’s angular drift and parallel drift," Review of Scientific Instruments, vol. 90, no. 11, pp. 115002-1-8, 2019.
[96]Y. Huang, K. C. Fan, and W. Sun, "Embedded sensor system for five-degree-of-freedom error detection on machine tools," Mechanical Engineering Science, vol. 1, no. 2, 2020.
[97]Y. Cai, Q. Sang, Z. Lou, and K. C. Fan, "Error Analysis and Compensation of a Laser Measurement System for Simultaneously Measuring Five-Degree-of-Freedom Error Motions of Linear Stages," Sensors, vol. 19, no. 18, pp. 3833-3846, 2019.
[98]R. Li, Y. Wang, P. Tao, R. Cheng, Z. Cheng, Y. Wei, and X. Dang, "Drift Reduction of a 4-DOF Measurement System Caused by Unstable Air Refractive Index," Sensors, vol. 20, no. 21, pp. 6329-6342, 2020.
[99]Y. Cai, Y. Gao, K. Yin, Q. Fu, and K. C. Fan, "Active methods to improve the accuracy of a laser diode-based geometric errors measurement system for machine tools," Applied Sciences, vol. 12, no. 7, pp. 3479-3491, 2022.
[100]S.W. Lee, R.M. Mayor, and J. Ni, "Development of a Six-Degree-of-Freedom Geometric Error Measurement System for a Meso-Scale Machine Tool," Journal of Manufacturing Science and Engineering, vol. 127, no. 4, pp. 857-865, 2005.
[101]F. You, Q. Feng, and B. Zhang, "A new five degree-of-freedom measurement method and system," International Conference on Optical Instruments and Technology:Optical Systems and Optoelectronic Instrument, vol. 7156, pp. 715634-1-11, 2008.
[102]S. Gao, B. Zhang, Q. Feng, C. Cui, S. Chen, and Y. Zhao, "Errors crosstalk analysis and compensation in the simultaneous measuring system for five-degree-of-freedom geometric error," Applied Optics, vol. 54, no. 3, pp. 458-466, 2015.
[103]C. S. Liu, J. J. Lai, and Y. T. Luo, "Design of a Measurement System for Six-Degree-of-Freedom Geometric Errors of a Linear Guide of a Machine Tool," Sensors, vol. 19, no. 1, pp. 5-17, 2018.
[104]Y. Cai, Q. Sang, B. Yang, B. Feng, and K. C. Fan, "Construction of a robust three-degree-of-freedom laser measurement system for measuring motion error of long-travel linear stages," International Symposium on Precision Mechanical Measurements, vol. 11343, pp. 1134320-1-6, 2019.
[105]C. Sun, S. Cai, Y. Liu, and Y. Qiao, "Compact Laser Collimation System for Simultaneous Measurement of Five-Degree-of-FreedomMotion Errors," Applied Sciences vol. 10, no. 15, pp. 5057-5069, 2020.
[106]R. Furutani, "Measurement of six degree motion error of linear stage," Materials Science and Engineering," International Conference on Aerospace, Mechanical and Mechatronic Engineering, vol. 211, 2017.
[107]C. H. Liu, W. Y. Jywe, C. C. Hsu, and T. H. Hsu, "Development of a laser-based high-precision six-degrees-of-freedom motion errors measuring system for linear stage," Review of Scientific Instruments, vol. 76, no. 5, pp. 055110-1-6, 2005.
[108]C. Kuang, E. Hong, and J. Ni, "A high-precision five-degree-of-freedom measurement system based on laser collimator and interferometry techniques," Review of Scientific Instruments, vol. 78, no. 9, pp. 095105-1-7, 2007.
[109]P. Hu, S. Mao, and J. B. Tan, "Compensation of errors due to incident beam drift in a 3 DOF measurement system for linear guide motion," Optics Express, vol. 23, no 22, pp. 28389-28401, 2015.
[110]Y. Lou, L. Yan, B. Chen, and S. Zhang, "Laser homodyne straightness interferometer with simultaneous measurement of six degrees of freedom motion errors for precision linear stage metrology," Optics Express, vol. 25, no 6, pp. 6805-6821, 2017.
[111]Y. Zhao, B. Zhang, and Q. Feng, "Measurement system and model for simultaneously measuring 6DOF geometric errors," Optics Express vol. 25, no 18, pp. 20993-21007, 2017.
[112]C. S. Liu, Y. F. Pu, Y. T. Chen, and Y. T. Luo, "Design of a Measurement System for Simultaneously Measuring Six-Degree-Of-Freedom Geometric Errors of a Long Linear Stage," Sensors, vol. 18 no. 11, pp. 3875-3887, 2018.
[113]S. J. A. G. Cosijns, "Displacement laser interferometry with sub-nanometer uncertainty," Eindhoven University of Technology, 2004.
[114]L. S. Pedrotti, "Basic geometrical optics, Fundamentals of photonics, pp. 73-116, 2008.
[115]P. Mouroulis, J. Macdonald, and J. Macdonald, "Geometrical optics and optical design," Oxford University Press New York, 1997.
[116]M. Katz, "Introduction to geometrical optics, "World Scientific Publishing Company," 2002.
[117]F. A. Jenkins, and H. E. White, "Fundamentals of optics, "ata McGraw-Hill Education," 1937.
[118]W. J. Smith, "Modern optical engineering," Tata McGraw-Hill Education, 2008.
[119]蔡忠佑，"稜鏡成像位姿變化之分析與設計" 國立成功大學機械研究所，博士論文，2007。
[120]陳俊仁，"使用歪斜光線追蹤法發展光電式多自由度量測系統" 國立成功大學機械研究所，博士論文，2007。
[121]P. D. Lin, and C. H. Lu, "Analysis and design of optical systems by use of sensitivity analysis of skew ray tracing," Applied optics, vol. 43, no. 4, pp. 796-807, 2004.
[122]P. D. Lin, and T. T. Liao, "Skew ray tracing and sensitivity analysis of geometrical optics," J. Manuf. Sci. Eng., vol. 122, no. 2, pp. 338-349, 2000.
[123]P. D. Lin, "New computation methods for geometrical optics," Springer, 2014.
[124]盧嘉鴻，"雷射追蹤器的光學建模與分析" 國立成功大學機械研究所，博士論文，2004。
[125]廖德潭，"雙照相機立體成像系統的建模與分析" 國立成功大學機械研究所，博士論文，2003。
[126]江適文，"幾何光學之光跡追蹤研究" 國立成功大學機械研究所，碩士論文，2001。
[127]K. C. Fan, F. Cheng, and Y. J. Chen, "Nanopositioning Control on a Commerical Linear Stage by Software Error Correction," Nanotechnology and Precision Engineering., vol. 4, no. 1, pp. 1-9, 2006.
[128]K. Diao, C. Chen, R. Leach, X. Liu, Wenlong Lu, and W. Yang, " Crosstalk decoupling measurement method to determine the six degrees of freedom of motion error of linear stages," Applied Optics., vol. 61, no. 6, pp. 1284-1291, 2022.
[129]Y. Cai, L. Wang, Y. Liu, C. Li, and K. C. Fan, " Accuracy improvement of linear stages using on-machine geometric error measurement system and error transformation model," Optics Express., vol. 30, no. 5, pp. 7539-7550, 2022.
[130]Duma Optronics，"SpotOn Analog & USB"，2019。 [Online]. Available: https://www.dumaoptronics.com/_files/ugd/a5b484_e0f424c88f5f49709dcc959fa57b26cd.pdf
[131]Thorlabs，" PDA100A-Manual"，2017。 [Online]. Available: https://www.thorlabs.com/drawings/525993e6820e6cc4-7F74178C-90D0-1B71-384BBCC3081FA3D9/PDA100A-Manual.pdf