在本篇論文中，我們提出了一個三維人體運動處理系統，以解決許多現有的人體運動數據處理問題。我們開發了一種新型雙模光學三維定位器，整合了人體全身掃描和動作捕捉功能，解決了個人化的身體模型的建立、運動追蹤標誌點和人體模型間的註冊問題。網格化的數位人體模型的建立是來自從實際的人體掃描數據。身體連桿經過分割後，該連桿的運動學參數可被非常精確地計算出來。人體模型可以複製個人化的運動到人形機器人上，甚至可代替人類執行危險嚴峻的任務。
對於使用不同的人體運動學模型，我們提出了一個兩階段的最佳化逆向運動學方法來解運動關節角，解出的關節角為與人體運動資料相應的最低姿勢誤差，並提供準確的四肢末端位置軌跡。藉由運動分析可比較不同模型之間姿勢的誤差。解出的關節角度可以進一步下載到一個真正的人形機器人來展現複製的人體運動。
透過演算法可將三維人體運動，編碼成二維影像格式，並稱為人體運動譜。運動譜可以編碼成任何解析度也可利用現有的影像處理技術來過濾、編輯和插補人體運動。人體運動譜的作用就像音樂的樂譜一般，可以記錄、編修與重現運動。本篇論文所提出的這套系統可以成為一個很方便的工具來處理人體的三維運動。

An intelligent 3D body motion processing system is developed to tackle many existing human body motion data processing problems. A novel dual-mode optical 3D locater integrates body scanning and motion capturing functions into one that solves the personalized body model creation, marker labeling, and marker-model registration problems. A mesh-structured digital human model is created from the scanned data. After the body links are segmented, the kinematics parameters of that body can be computed very accurately. The body model may replicate personalized motions to the humanoid robot for dangerous tasks.
Using different body kinematic models, a two-phase optimization method is proposed to solve the joint angles that yield the minimum posture error to the corresponding motion captured data with accurate end-point trajectories. A motion analysis compares the errors of different postures between different models. The joint angles can further be downloaded to a real humanoid robot for replicating the body motions.
An algorithm encodes 3D body motions into an image format, called the body motion staff. The staff can be encoded into any resolution and manipulated by filtering, editing, and interpolating using available image processing techniques. The body motion staff performs the role to human motion just like staff does to the music. The 3D human body motions have never been manipulated with such a convenient tool before.

[1] J. Denavit and R. S. Hartenberg, "A kinematic notation for lower pair mechanisms based on matrices," ASME J. of Appl. Mech., vol. 22, pp. 215-221, 1955.
[2] T. W. Yang, "A Research on the Motion Editing System for General Humanoid Robots," Msc., Mechanical Engineering, National Cheng Kung University, 2011.
[3] T. P. Koninckx and L. V. Gool, "Efficient, Active 3D Acquisition, Based on a Pattern-Specific Snake," Pattern Recognition, vol. 2449, pp. 557-565, 2002.
[4] T. P. Koninckx and L. V. Gool, "High-Speed Active 3D Acquisition Based on a Pattern-Specific Mesh," in Proc. of SPIE's 15th Ann. Symp. on Electronic Imaging-Videometrics VII, 2003, pp. 26-37.
[5] T. P. Koninckx, I. Geys, T. Jaeggli, and L. V. Gool, "A Graph Cut Based Adaptive Structured Light Approach for Real-Time Range Acquisition," in Proc. of .2nd Int. Symp. on 3D Data Processing, Visualization, and Transmission, 2004, pp. 413-421.
[6] A. Griesser and L. V. Gool, "A High-Speed Adaptive Multi-Module Structured Light Scanner," in Proc. of the Workshop on Modeling and Motion Capture Techniques for Virtual Environments, 2004.
[7] A. Griesser, T. P. Koninckx, and L. V. Gool, "Adaptive Real-Time 3D Acquisition and Contour Tracking within a Multiple Structured Light System," in Proc. of the 12th Pacific Conf. on the Computer Graphics and Applications, 2004, pp. 361-370.
[8] C. Y. L. Liu, "Three-Dimensional Hand Tracking and Surface-Geometry Measurement for a Robot-Vision System," Master Dept. of Systems Design Eng., Waterloo Univ., Waterloo, Ontario, Canada, 2009.
[9] J. Lee and K. H. Lee, "Precomputing Avatar Behavior from Human Motion Data," Graphical Models, vol. 68, pp. 158-174, 2006.
[10] V. B. Zordan and J. K. Hodgins, "Motion capture-driven simulations that hit and react," in Proc. of 2002 ACM SIGGRAPH/EurographicsSymp. on Computer Animation, 2002, pp. 89-96.
[11] M. J. Tsai, H. W. Lee, and T. W. Yang, "Body motion staff, producing module, image processing module and motion replication module," U.S. Patent Pending Patent.
[12] M. Petrov, A. Talapov, T. Robertson, A. Lebedev, A. Zhilyaev, and L. Polonskiy, "Optical 3D digitizers: bring life to the virtual world," IEEE Computer Graphics and Application, pp. 28-37, May/June 1998.
[13] Cyberware. Available: http://www.cyberware.com, 2013
[14] TC2. Available: http://www.tc2.com, 2013
[15] Vitus. Available: http://www.vitronic.de, 2013
[16] Creaform. Available: http://www.creaform3d.com, 2013
[17] J. H. Nurre, J. Connor, E. A. Lewark, and J. S. Collier, "On segmenting the three-dimensional scan data of a human body," IEEE Trans. on Medical Imaging, vol. 19, pp. 787-797, 2000.
[18] X. Ju, N. Werghi, and J. P. Siebert, "Automatic segmentation of 3D human body scans," in Procs. of IASTED Int. Conf. on Comp. Graphics and Imaging, 2000.
[19] C. Lovato, U. Castellani, and A. Giachetti, "Automatic segmentation of scanned human body using curve skeleton analysis," in MIRAGE '09 Procs. 4th Int. Conf. on Computer Vision/Computer Graphics Collaboration Techniques, 2009, pp. 34-35.
[20] K. Robinette, H. Daanen, and E. Paquet, "The Caesar project: A 3-D surface anthropometry survey," in Procs. of IEEE 2nd Int. Conf. on 3-D Digital Imaging and Modeling, 2001, pp. 380-386.
[21] K. Robinette, M. Boehmer, and D. Burnsides, "3-D landmark detection and identification in the CAESAR project," in Procs. 3rd Int. Conf. on 3-D Digital Imaging and Modeling, 2001, pp. 393-398.
[22] R. P. Pargas, N. J. Staples, and J. S. Davis, "Automatic measurement extraction for apparel from a three-dimensional body scan," Optics and Lasers in Engineering, vol. 28, pp. 157-172, 1997.
[23] K. P. Simmons, "Body measurement techniques: a comparison of three-dimensional body scanning and physical anthropometric methods," North Carolina State University, Raleigh, N.C., 2001.
[24] M. Soucy and D. Laurendeau, "Multiresolution surface modeling based on hierarchical triangulation," Computer Vision and Image Understanding, vol. 63, pp. 1-14, 1996.
[25] Y. J. Liu and M. M. F. Yuen, "Optimized triangle mesh reconstruction from unstructured points," The Visual Computer, vol. 19, pp. 23-37, 2003.
[26] C. C. L. Wang, T. K. K. Chang, and M. M. F. Yuen, "From laser-scanned data to feature human model: A system based on fuzzy logical concept," Computer-Aided Design, vol. 35, pp. 241-253, 2003.
[27] M. J. Tsai, Z. P. Chen, and Y. S. Liu, "A study on the automatic feature search from 3D body scanners," in Procs. 20th National Conference, Society of Chinese Mechanical Engineering, Taipei, 2003.
[28] I. F. Leong, J. J. Fang, and M. J. Tsai, "Automatic body feature extraction from a marker-less scanned human body," Computer-Aided Design, vol. 39, pp. 568-582, 2007.
[29] M. J. Tsai and J. J. Fang, "Feature based data structure for computer manikin," US Patent No. 7,218,752 B2 Patent, 2007.
[30] Vicon Motion Systems. Available: http://www.vicon.com/, 2013
[31] OptiTrack System. Available: http://www.naturalpoint.com/optitrack/, 2013
[32] A. Jaumei-i-Capó, J. Varona, M. González-Hidalgo, and F. J. Perales, "Adding Image Constraints to Inverse Kinematics for Human Motion Capture," EURASIP J. Adv. Signal Process, vol. 2010, pp. 1-13, 2010.
[33] Microsoft XBOX 360 Kinect. Available: http://www.xbox.com/, 2013
[34] L. A. Schwarz, A. Mkhitaryan, D. Mateus, and N. Navab, "Human Skeleton Tracking from Depth Data Using Geodesic Distances and Optical flow," Image and Vision Computing, vol. 30, pp. 217-226, 2012.
[35] A. Aristidou and J. Lasenby, "Real-time Marker Prediction and CoR Estimation in Optical Motion Capture," Vis Comput, vol. 29, pp. 7-26, 2013.
[36] M. J. Tsai, H. W. Lee, and H. Y. Lung, "Dual-mode optical measurement apparatus and system," U.S. Patent Pending Patent.
[37] A. R. Klumpp, "Singularity-Free Extraction of a Quaternion from a Direction-Cosine Matrix," Journal of Spacecraft and Rockets, vol. 13, pp. 754-755, 1976.
[38] T. R. Kane, D. A. Levinson, and Likins, Spacecraft Dynamics. NewYork: McGraw-Hill, 1983.
[39] K. Shoemake, "Animating rotation with quaternion curves," in Proc. SIGGRAPH '85, 1985, pp. 245-254.
[40] R. Lake and M. Green, "Dynamic motion control of an articulated figure using quaternion curves," in Proc. of the 2nd Internat. Conf. on Computer-Aided Design and Computer Graphics, Hangzhou, China, 1991, pp. 37-44.
[41] B. Juttler, "Visualization of moving objects using dual quaternion curves," Comput. Graphics, vol. 18, pp. 315-326, 1994.
[42] M. J. Kim, M. S. Kim, and S. Y. Shin, "A C2-continuous B-spline quaternion curve interpolating a given sequence of solid orientations," in Proc. of Computer Animation '95, Geneva, Switzerland, 1995, pp. 72-81.
[43] G. Farin, G. Rein, N. Sapidis, and A. Worsey, "Fairing cubic B-spline curves," Comput. Aided Geom. Design, vol. 4, pp. 91-103, 1987.
[44] B. Choi, Geometric Modeling for CAD/CAM. New York: Elsevier Science, 1991.
[45] Z. Xue, S. Wan, and J. Zhou, "Motion Sequence Filtering using Geometric Algebra," in Proceedings of the Second Symposium International Computer Science and Computational Technology(ISCSCT '09), 2009, pp. 393-397.
[46] Y. C. Fang, C. C. Hsieh, M. J. Kim, J. J. Chang, and T. C. Woo, "Real-Time Motion Fairing with Unit Quaternions," Computer-Aided Design, vol. 30, pp. 191-198, 1998.
[47] C. C. Hsieh, "B-Spline Wavelet-Based Motion Smoothing," Computers & Industrial Engineering, vol. 41, pp. 59-76, 2001.
[48] N. I. Badler, C. B. Phillips, and B. L. Webber, "Simulating Humans: Computer Graphics, Animation, and Control," Oxford University Press, New York, 1993.
[49] J. K. Hodgins, W. L. Wooten, D. C. Brogan, and J. F. O'Brien, "Animating Human Athletics," in Proc. of the 22nd Ann. Conf. on Computer Graphics and Interactive Techniques (SIGGRAPH'95), 1995, pp. 71-78.
[50] K. Kreutz-Delgado, M. Long, and H. Seraji, "Kinematic Analysis of 7 DOF Anthropomorphic Arms," in Proc. of the IEEE International Conf. on Robotics and Automation, 1990, pp. 824-830.
[51] K. Kreutz-Delgado, M. Long, and H. Seraji, "Kinematic Analysis of 7-DOF Manipulators," Int. J. of Robotics Res., vol. 11, pp. 469-481, 1992.
[52] B. Tondu, "A Closed-Form Inverse Kinematic Modeling of a 7R Anthropomorphic Upper Limb Based on a Joint Parameterization," in Proc. of the 6th IEEE-RAS Int. Conf. on Humanoid Robots, 2006, pp. 390-397.
[53] T. Asfour, K. Berns, J. Schelling, and R. Dillmann, "Programming of Manipulation Tasks of the Humanoid Robot ARMAR," in Proc. of the IEEE International Conf. on Advanced Robotics, 1999, pp. 107-112.
[54] T. Asfour and R. Dillmann, "Human-Like Motion of a Humanoid Robot Arm Based on a Closed-Form Solution of the Inverse Kinematics Problem," in Proc. of the IEEE/RSJ International Conf. on Intelligent Robots and System, 2003, pp. 1407-1412.
[55] M. Shimizu, H. Kakuya, W. K. Yoon, K. Kitagaki, and K. Kosuge, "Analytical Inverse Kinematic Computation for 7-DOF Redundant Manipulators with Joint Limits and Its Application to Redundancy Resolution," IEEE Trans. on Robotics and Automation, vol. 24, pp. 1131-1142, 2008.
[56] J. K. Parker, A. R. Khoogar, and D. E. Goldberg, "Inverse Kinematics of Redundant Robots Using Genetic Algorithms," in Proc. of the IEEE International Conf. on Robotics and Automation, 1989, pp. 271-276.
[57] A. A. Ata and T. R. Myo, "Optimal Point-to-Point Trajectory Tracking of Redundant Manipulators Using Generalized Pattern Search," International J. of Advanced Robotic Systems, vol. 2, pp. 239-244, 2005.
[58] A. Guez and Z. Ahmad, "Solution to the Inverse Kinematics Problem in Robotics by Neural Networks," in Proc. of the IEEE International Conf. on Neural Networks, 1988, pp. 617-624.
[59] H. Moradi and S. Lee, "Joint Limit Analysis and Elbow Movement Minimization for Redundant Manipulators Using Closed Form Method," Advances in Intelligent Computing, vol. 3645, pp. 423-432, 2005.
[60] A. A. Goldenberg, J. A. Apkarian, and H. W. Smith, "A New Approach to Approach to Kinematic Control of Robot Manipulators," ASME J. Dynamic Syst. Measure., Control, vol. 109, pp. 97-103, 1987.
[61] Y. T. Tsai and E. D. Orin, "A Strictly Convergent Real-Time Solution for Inverse Kinematics of Robot Manipulators," J. Robotic System, vol. 4, pp. 477-501, 1987.
[62] M. Tucker and N. D. Perreira, "Generalized Inverse for Robotic Manipulators," Mech. and Mach. Theory, vol. 22, pp. 507-514, 1987.
[63] K. Kazerounian, "On the Numerical Inverse Kinematics of Robotics Manipulators," ASME J. Mechanisms, Transmissions, Automat. Design, vol. 109, pp. 8-13, 1987.
[64] L. C. T. Wang and C. C. Chen, "A Combined Optimization Method for Solving the Inverse Kinematics Problems of Mechanical Manipulators," IEEE Trans. on Robotics and Automation, vol. 7, pp. 489-499, 1991.
[65] J. A. Reinbolt, J. F. Schutte, B. J. Fregly, B. I. Koh, R. T. Haftka, A. D. George, et al., "Determination of Patient-Specific Multi-Joint Kinematic Models Through Two-Level Optimization," J. of Biomechanics, vol. 38, pp. 621-626, 2005.
[66] J. Zhao and N. I. Badler, "Inverse Kinematics Positioning Using Nonlinear Programming for Highly Articulated Figures," ACM Trans. on Graphics, vol. 13, pp. 313-336, 1994.
[67] P. Baerlocher and R. Boulic, "An Inverse Kinematics Architecture Enforcing an Arbitrary Number of Strict Priority Levels," The Visual Computer, vol. 20, pp. 402-417, 2004.
[68] P. Hoonsuwan, S. Sillapaphiromsuk, K. Sukvichai, and A. Fish, "Designing a stable humanoid robot trajectory using a real human motion," in Proc. of the 6th Int. Conf. on Electrical Engineering/Electronics, Computer, Telecommunications and Information Technology, Pattaya, Chonburi, 2009, pp. 336-339.
[69] Q. Huang, K. Yokoi, S. Kajita, K. Kaneko, H. Arai, N. Koyachi, et al., "Planning walking patterns for a biped robot," IEEE Trans. on Robotics and Automation, vol. 17, pp. 280-289, 2001.
[70] S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Harada, K. Yokoi, et al., "Biped walking pattern generation by using preview control of zero-moment point," in Proc. of the IEEE Int. Conf. on Robotics and Automation, 2003, pp. 1620-1626.
[71] K. Nishiwaki, S. Kagami, Y. Kuniyoshi, M. Inaba, and H. Inoue, "Online generation of humanoid walking motion based on a fast generation method of motion pattern that follows desired ZMP," in Proc. of theIEEE/RSJ Int.Conf. on Intelligent Robots and Systems, 2002, pp. 2684-2689.
[72] S. Nakaoka, A. Nakazawa, F. Kanehiro, K. Kaneko, M. Morisawa, and K. Ikeuchi, "Task model of lower body motion for a biped humanoid robot to imitate human dances," in 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2005. (IROS 2005), 2005, pp. 3157-3162.
[73] V. B. Zordan and J. K. Hodgins, "Tracking and modifying upper-body human motion data with dynamic simulation," in Proc. of Computer Animation and Simulation ’99 (Eurographic), 1999, pp. 13-22.
[74] A. Dasgupta and Y. Nakamura, "Making feasible walking motion of humanoid robots from human motion capture data," in Proc. of the IEEE Int. Conf. on Robotics and Automation, Detroit, MI, 1999, pp. 1044-1049.
[75] W. Suleiman, E. Yoshida, F. Kanehiro, J. P. Laumond, and A. Monin, "On human motion imitation by humanoid robot," in Proc. of the IEEE Int. Conf. on Robotics and Automation, Pasadena, CA, 2008, pp. 2697-2704.
[76] J. Yang, Q. Huang, Z. Peng, L. Zhang, Y. Shi, and X. Zhao, "Capturing and analyzing of human motion for designing humanoid motion," in Proc. of IEEE Int. Conf. Information Acquisition, 2005, pp. 332-337.
[77] E. Chaudhry, L. H. You, and J. J. Zhang, "Character skin deformation: A survey," presented at the 7th Int. Conf. on Computer Graphics, Imaging and Visualization, 2010.
[78] J. E. Chadwick, D. R. Haumann, and R. E. Parent, "Layered construction for deformation animated characters," ACM SIGGRAPH Computer Graphics, vol. 23, pp. 243-252, 1989.
[79] J. Wilhelms and A. V. Gelder, "Anatomically based modeling," in Procs. of the 24th Annual Conf. on Computer Graphics and Interactive Techniques, 1997, pp. 173-180.
[80] F. Scheepers, R. E. Parent, W. E. Carlson, and S. F. May, "Anatomy-based modeling of the human musculature," in Procs. of the 24th Annual Conf. on Computer Graphics and Interactive Techniques, 1997, pp. 163-172.
[81] L. Zuo, J. T. Li, and Z. Q. Wang, "Anatomical human musculature modeling for realtime deformation," in Procs. of Computer Graphics, Visualization and Computer Vision WSCG, 2003.
[82] N. M. Thalmann, R. Laperriere, and D. Thalmann, "Joint dependent local deformations for hand animation and object grasping," in Procs. of Graphics Interface, 1988, pp. 26-33.
[83] J. P. Lewis, M. Cordner, and N. Fong, "Pose space deformation: a unified approach to shape interpolation and skeleton-driven deformation," in Procs. of the ACM SIGGRAPH Conf. on Computer Graphics, 2000, pp. 165-172.
[84] L. Kavan and J. Zara, "Spherical blend skinning: a real-time deformation of articulated models," in Procs. of the 2005 Symposium on Interactive 3D Graphics and Games, 2005, pp. 9-15.
[85] X. S. Yang, A. Somasekharan, and J. J. Zhang, "Curve skeleton skinning for human and creature characters," Computer Animation and Virtual Worlds, vol. 17, pp. 281-292, 2006.
[86] S. Jianhua, N. Magnenat-Thalmann, and D. Thalmann, "Human skin deformation from cross-sections," in Procs. of Computer Graphics International CGI'94, 1994.
[87] L. Kavan, S. Collins, J. Zara, and C. O'Sullivan, "Geometric skinning with approximate dual quaternion blending," ACM Transactions on Graphics, vol. 25, pp. 1-23, 2008.
[88] J. Chang, X. Yang, and J. J. Zhang, "Continuous skeleton-driven skinning: a general approach for modeling skin deformation," Int. Journal of Image and Graphics, vol. 9, pp. 591-608, 2009.
[89] B. Allen, B. Curless, and Z. Popovic, "Articulated body deformation from range scan data," ACM SIGGRAPH Computer Graphics, pp. 612-619, 2002.
[90] S. I. Park and J. K. Hodgins, "Capturing and animating skin deformation in human motion," ACM Transactions on Graphics (TOG), vol. 25, pp. 881-889, 2006.
[91] J. F. Oliveira, D. Zhang, B. Spanlang, and B. F. Buxton, "Animating scanned human models," Journal of WSCG, vol. 11, 2003.
[92] Y. Y. Ma, H. Zhang, and S. W. Jiang, "Realistic modeling and animation of human body based on scanned data," Journal of Computer Science and Technology, vol. 19, pp. 529-537, 2004.
[93] P. Kalra, N. M. Thalmann, L. Moccozet, G. Sannier, A. Aubel, and D. Thalmann, "Real-time animation of realistic virtual humans," Computer Graphics Applications, vol. 18, pp. 42 -55, 1998.
[94] H. J. Shin, L. Kovar, and M. Gleicher, "Physical touch-up of human motions," in Proc. of IEEE Computer Society of Pacific Graphics, 2003, pp. 194-203.
[95] M. Riley, A. Ude, and C. G. Atkeson, "Methods for motion generation and interaction with a humanoid robot: case studies of dancing and catching," in Proc. of2000 Workshop on Interactive Robotics and Entertainment, Carnegie Mellon Univ., Pittsburgh, 2000, pp. 35-42.
[96] M. Inaba, T. Igarashi, S. Kagami, and H. Inoue, "A 35 DOF humanoid that can coordinate arms and legs in standing up, reaching and grasping an object," in Proc. of IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, Osaka, Japan, 1996, pp. 29-36.
[97] Labanotation. Available: http://user.uni-frankfurt.de/~griesbec/LABANE.HTML, 2013
[98] C. H. Huang, "Development of a Dual-Mode 3D Optical Measurement System," Msc., Mechanical engineering, Nation Cheng Kung University, 2009.
[99] Y. J. Huang, "A Study on an Automatic Joint Finding Process for Human Body Motion," Msc., Mechanical Engineering, National Cheng Kung Univeristy, 2010.
[100] J. Z. Liang, "Constructing an auto-registration process between coded markers and human model in a motion tracking system," Msc., Mechanical Engineering, National Cheng Kung University, 2010.
[101] L. N. Trefethen and D. B. III, Numerical linear algebra: Philadelphia: Society for Industrial and Applied Mathematics, 1997.
[102] F. Gray, "Pulse code communication," U.S. Patent 2,632,058. Patent, 1953
[103] M. J. Tsai and C. C. Hung, "A Fast Evaluation Approach of Geometrical Correspondence Uncertainty for 3-D Vision Measurement System," Journal of Japan Society of Mechanical Engineering, Series C, vol. 49, pp. 527-534, 2006.
[104] R. W. Webster and Y. Wei, ""ARINE P."-A Robot Golfing System Using Binocular Stereo Vision and a Heuristic Feedback Mechanism," in Proceeding of the 1992 IEEE/RSJ International Conference on Intelligent Robots and System, 1992, pp. 2027-2034.
[105] M. J. Tsai, H. W. Lee, and P. W. Hsu, "Patterned marker for coding," U.S. Patent Pending Patent.
[106] M. J. Tsai, H. W. Lee, and H. Y. Lung, "Feature-based data structure for digital manikin," U.S. Patent Pending Patent.
[107] M. J. Tsai, "人形機器人之人體運動擷取與編修系統研究-行政院國家科學委員會補助專題研究計畫NSC 96-2221-E-006 -275 -MY3," 國立成功大學機械工程學系2010.
[108] M. J. Tsai, H. W. Lee, and H. Y. Lung, "A Point-Structured Geometric Modeling Approach to Foot Shape Representation," in Procs. of the ASME 2012 Int. Design Engineering Technical Conf. & Computers and Information in Engineering Conf., Chicago, USA, 2012, pp. DETC 2012-70603.
[109] K. J. Waldron and J. Schmiedeler, Springer Handbook of Robotics, Chapter 1: Berlin: Springer-Verlag, 2008.
[110] A. T. Yang and F. Freudenstein, "Application of dual-number quaternion algebra to the analysis of spatial mechanisms," ASME J. of Appl. Mech., vol. 31, pp. 300-308, 1964.
[111] J. M. Selig, Geometrical Methods in Robotics. NJ, USA: Springer-Verlag New York, Inc. Secaucus, 1996.
[112] 鄭秀瑗, 現代運動生物力學, 2 ed. 國防工業出版社: 國防工業出版社, 2007.
[113] I. Aizenberg and C. Butakoff, "Effective impulse detector based on rank-order criteria," Signal Processing Letters, vol. 11, pp. 363-366, 2004.
[114] K. Soudan and P. Dierckx, "Calculation of Derivatives and Fourier Coefficients of Human Motion Data, While Using Spline Functions," J. Biomechanics, vol. 12, pp. 21-26, 1979.
[115] E. Pennestrì and R. Stefanelli, "Linear Algebra and Numerical Algorithms Using Dual Numbers," Multibody System Dynamics, vol. 18, pp. 323-344, 2007.
[116] C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis: Addison-Wesley Publishing Co., 1989.
[117] Y. H. Dai and Y. Yuan, "Convergence Properties of the Fletcher-Reeves Method," IMA J. of Numerical Analysis, vol. 16, pp. 155-164, 1996.
[118] R. Fletcher and C. M. Reeves, "Function Minimization by Conjugate Gradients," The Computer J., vol. 7, pp. 149-154, 1964.
[119] M. J. Tsai and K. J. Waldron, "Operating Barriers within the Workspace of Industrial Robots," IASTED Int. J. of Robotics and Auto., vol. 4, pp. 142-147, 1989.