近年來，工業4.0、機器學習與人工智慧等議題十分熱門，面對機器人時代來臨，機械手臂的重要性不可同日而語。而現行機械手臂為了提升運動精度，常見的作法是利用高強度、硬度的材料，提高系統剛性以減少末端點振動。然而其笨重的設計會導致較高之成本、不必要的能量消耗、有限的移動速度等缺點。相較於剛性機械手臂，若能以適當方法克服因剛性不夠導致的末端點振動問題，則使用較為輕便且剛性較低之材料而製成的撓性桿件機械手臂，將具有更低的成本、更低的能量消耗、較快的移動速度與更安全的操作等優點，必會廣泛應用至工業及其他相關場合中。故本論文之主要目的為針對撓性機械手臂的數學模型、運動學與動力學進行深入探討，建立一完整的且即時計算量較低之泛用撓性機械手臂模型，並以模擬與部份實驗方式驗證之。期望本論文所推導之模型，可提供後續研究者於前饋控制及其它先進控制架構相關探討之用，進而解決撓性機械手臂振動問題並提升末端點精度，並以較低計算量的優勢協助實現適合用於工業4.0之先進控制架構，最大化其使用價值。

In recent years, Industry 4.0, machine learning, and artificial intelligence have become very popular. As the robot era is coming, robotic manipulators have taken on increasing importance in research and practice. In order to improve the motion accuracy of the robotic arms, it is common to use high-strength materials to increase the rigidity of the system and reduce the vibration of the end point. However, such bulky designs lead to disadvantages such as higher cost, unnecessary energy consumption, and limited speed of movement. Compared to rigid-body manipulators, if the vibration of the end point caused by insufficient rigidity can be overcome, the flexible robot manipulator made of a more flexible and less rigid material will result in lower cost, lower energy consumption, faster speed of movement and safer operation for wider applications in industry and other aspects. Therefore, the main purpose of this thesis is to discuss the mathematical model, kinematics and dynamics of flexible manipulators to establish a complete universal flexible manipulator model with low real-time calculation—a model which is verified by simulations and experiments. It is expected that the model derived in this thesis can be used in feedforward control and other advanced control architectures in the future so as to solve the vibration problem of flexible manipulators. In addition, with its low computation load advantage, the model developed in this thesis can be exploited to help implement advanced control schemes that are suitable for Industry 4.0 so as to maximize its value.

[1] 台達電子工業股份有限公司。產品-水平關節機器人-DRS40L系列-台達集團。檢索日期2018年4月22日，取自http://www.deltaww.com/Products/CategoryListT1.aspx?CID=060601&PID=2182&hl=zh-TW&Name=DRS40L%E7%B3%BB%E5%88%97
[2] 台達電子工業股份有限公司。產品-垂直多關節機器人-DRV90L系列-台達集團。檢索日期2018年4月22日，取自http://www.deltaww.com/Products/CategoryListT1.aspx?CID=060603&PID=3815&hl=zh-TW&Name=DRV90L%E7%B3%BB%E5%88%97
[3] Direct Industry. Articulated robot/7-axis/inspection/packaging-Weight:23.9kg | LBR IIWA 7 R800-KUKA Roboter GmbH. Retrieved Apr. 22, 2018, from http://www.directindustry.cn.com/prod/kuka-roboter-gmbh/product-17587-1650349.html
[4] 聯鋐自動化工業有限公司。產品介紹-聯鋐自動化工業有限公司。檢索日期2018年4月22日，取自http://www.lhrobot.com/products_ii.html?gid=11
[5] L. Meirovitch, Analytical Methods in Vibrations, New York, U.S.A.: Macmillan, 1967.
[6] L. Meirovitch, Principle and Techniques of Vibrations, New Jersey, U.S.A.: Prentice-Hall, 1997.
[7] Y. Gao and F.Y. Wang, Flexible Manipulators Modeling, Analysis and Optimum Design, Portland, U.S.A.: Academic Press, 2012.
[8] M.O. Tokhi and A.K.M. Azad, Flexible Robot Manipulators Modelling, Simulation and Control, London, U.K.: The Institution of Engineering and Technology, 2008.
[9] M. Stachowsky, “Dynamic and Control of a Two-Link Flexible Manipulator,” Master Thesis, Department of Earth and Science and Engineering, York University, Toronto, Canada, 2010.
[10] W.J. Book, “Modeling, Design, and Control of Flexible Manipulator Arms: A tutorial Review,” in Proceedings of the 29th IEEE Conference on Decision and Control, Honolulu, U.S.A., 1990, pp. 500-506.
[11] S.K. Dwivedy and P. Eberhard, “Dynamic Analysis of Flexible Manipulators, a Literature Review,” Mechanism and Machine Theory, vol. 41, issue 7, pp. 749-777, 2006.
[12] K. Lochan, B.K. Roy, and B. Subudhi, “A Review on Two-Link Flexible Manipulators,” International Federation of Automatic Control, vol.42, pp. 346-367, 2016.
[13] R.J. Theodore and A. Ghosal, “Comparison of the Assumed Modes and Finite Element Models for Flexible Multi-link Manipulators,” The International Journal of Robotics Research, vol. 14, issue 2, pp. 91-111, 1995.
[14] Wikipedia. Timoshenko beam theory-Wikipedia. Retrieved Apr. 22, 2018, form https://upload.wikimedia.org/wikipedia/commons/5/57/TimoshenkoBeam.svg
[15] G.G. Hastings and W.J. Book, “A Linear Dynamic Model for Flexible Robotic Manipulators,” IEEE Control Systems Magazine, vol. 7, issue 1, pp. 61-64, 1987.
[16] J.M. Martins, Z. Mohamed, M.O. Tokhi, J. Sa da Costa, and M.A. Botto, “Approaches for Dynamic Modelling of Flexible Manipulator Systems”, IEEE Proceedings on Control Theory and Applications, vol. 150, issue 4, pp. 401-411, 2003.
[17] B.V. Chapnik, G.R. Heppler, and J.D. Aplevich, “Modeling Impact on a One-Link Flexible Robotic Arm,” IEEE Transactions on Robtics and Automation, vol. 7, issue 4, pp. 479-488, 1991.
[18] G.G. Hastings and W.J. Book, “Verification of a Linear Dynamic Model for Flexible Robotic Manipulators,” in Proceedings of the IEEE International Conference on Robotics and Automation, San Francisco, U.S.A., 1986, pp. 1024-1029.
[19] W.F. Faris, A.A. Ata, and M.Y. Sa’adeh, “Energy Minimization Approach for a Two-link Flexible Manipulator,” Journal of Vibration and Control, vol. 15, issue 4, pp. 497-526, 2009.
[20] F. Matsuno, T. Asano, and Y. Sakawa, “Modeling and Quasi-Static Hybrid Position/Force Control of Constrained Planar Two-Link Flexible Manipulators,” IEEE Transactions on Robotics and Automation, vol. 10, issue 3, pp. 287-297, 1994.
[21] P. Tomei and A. Tornambe, “Approximate Modeling of Robots Having Elastic Links,” IEEE Transactions on Systems, Man and Cybernetics, vol.18, issue 5, pp. 831-840, 1988.
[22] A.D. Luca and B. Siciliano, “Closed-Form Dynamic Model of Planar Multilink Lightweight Robots,” IEEE Transactions on System, Man, and Cybernetics, vol. 21, issue 4, pp. 826-839, 1991.
[23] B. Subudhi and A.S. Morris, “Dynamic Modelling, Simulation and Control of a Manipulator with Flexible Links and Joints,” Robotics and Autonomous System, vol. 41, issue 4, pp. 257-270, 2002.
[24] M. Benati and A. Morro, “Dynamics of Chain of Flexible Links,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 110, issue 4, pp. 410-415, 1988.
[25] 魏進，空間柔性機械臂的動力學建模和分析，碩士論文，哈爾濱工業大學航天學院，哈爾濱，中國，2013。
[26] R.J. Theodore and A. Ghosal, “Modeling of Flexible-Link Manipulators with Prismatic Joints,” IEEE Transcactions on Systems, Man, and Cybernetics — Part B: Cybernetics, vol. 27, issue 2, pp. 296-305, 1997.
[27] F. Bellezza, L. Lanari, and G. Ulivi, “Exact Modeling of the Flexible Slewing Link,” in Proceeding of the IEEE international Conference on Robotics and Automation, Cincinnati, U.S.A., 1990, pp. 734-739.
[28] E. Barbieri and U. Ozguner, “Unconstrained and Constrained Mode Expansions for a Flexible Slewing Link,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 110, issue 4, pp. 416-421, 1988.
[29] S. Cetinkunt and W.L. Yu, “Closed-Loop Behavior of a Feedback-Controlled Flexible Arm A Comparative Study,” The International Journal of Robotics Research, vol. 10, issue 3, pp. 263-275, 1991.
[30] R.H. Cannon Jr. and E. Schmitz, “Initial Experiments on the End-Point Control of a Flexible One-Link Robot,” The International Journal of Robotics Research, vol. 3, issue 3, pp. 62-75, 1984.
[31] Y. Sakawa, F. Matsuno, and S. Fukushima, “Modeling and Feedback Control of a Flexible Arm,” Journal of Robotic Systems, vol. 2, issue 4, pp. 452-472, 1985.
[32] S.K. Tso, T.W. Yang, W.L. Xu, and Z.Q. Sun, “Vibration Control for a Flexible Link Robot Arm with Deflection Feedback,” International Journal of Non-Linear Mechanics, vol. 38, issue 1, pp.51-62, 2003.
[33] J. Chung and H.H. Yoo, “Dynamic Analysis of a Rotating Cantilever Beam by Using the Finite Element Method,” Journal of Sound and Vibration, vol. 249, issue 1, pp.147-164, 2002.
[34] M.O. Tokhi, Z. Mohamed, and M.H. Shaheed, “Dynamic Characterisation of a Flexible Manipulator System,” Robotica, vol. 19, issue 5, pp. 571-580, 2001.
[35] S. Nagarajan and D.A. Turcic, “Lagrangian Formulation of the Equations of Motion for Elastic Mechanisms With Mutual Dependence Between Rigid Body and Elastic Motions. Part I: Element Level Equations,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 112, issue 2, pp. 203-214, 1990.
[36] S. Nagarajan and D.A. Turcic, “Lagrangian Formulation of the Equations of Motion for Elastic Mechanisms With Mutual Dependence Between Rigid Body and Elastic Motions Part II: System Equations,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 112, issue 2, pp. 215-224, 1990.
[37] M.O. Tokhi and A.K.M. Azad, “Real-Time Finite Difference Simulation of a Single-Link Flexible Manipulator System Incorporating Hub Inertia and Payload,” The Institution of Mechanical Engineers. Part I, Journal of Systems and Control Engineering, vol. 209, issue 1, pp.21-23, 1995.
[38] E. Bayo, “A Finite-Element Approach to Control the End-Point Motion of a Single-Link Flexible Robot,” Journal of Robotic systems, vol. 4, issue 1, pp. 63-75, 1987.
[39] S.S. Ge, T.H. Lee, and G. Zhu, “A Nonlinear Feedback Controller for a Single-Link Flexible Manipulator Based on a Finite Element Model,” Journal of Robotic Systems, vol. 14, issue 3, pp. 165-178, 1997.
[40] Z. Mohamed and M.O. Tokhi, “Command Shaping Techniques for Vibration Control of a Flexible Robot Manipulator,” Mechatronics, vol. 14, issue 1, pp. 69-90, 2004.
[41] H. Moulin and E. Bayo, “On the Accuracy of End-Point Trajectory Tracking for Flexible Arms by Non-Causal Inverse Dynamic Solutions,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 113, issue 2, pp. 320-324, 1991.
[42] W. Chen, “Dynamic Modeling of Multi-Link Flexible Robotic Manipulators,” Computers and Structures, vol. 79, issue 2, pp. 183-195, 2001.
[43] A.D. Luca and B. Siciliano, “Explicit Dynamic Modeling of a Planar Two-Link Flexible Manipulator,” in Proceedings of the 29th IEEE Conference on Decision and Control, Honolulu, U.S.A., 1990, pp. 528-530.
[44] S.S. Ge, T.H. Lee, and G. Zhu, “A New Lumping Method of a Flexible Manipulator,” in Proceedings of the American Control Conference, Albuquerque, U.S.A., 1997, pp.1412-1416.
[45] V. Feliu, K.S. Rattan, and H.B. Brown, Jr., “Modeling and Control of Single Link Flexible Arms with Lumped Masses,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 114, issue 1, pp. 59-69, 1992.
[46] C. Sun, W. He, and J. Hong, “Neural Network Control of a Flexible Robotic Manipulator Using the Lumped Spring-Mass Model,” IEEE Transactions on Systems, Man, and Cybernetics: System, vol. 47, issue 8, pp. 1863-1874, 2017.
[47] G. Zhu, S.S. Ge, and T.H. Lee, “Simulation Studies of Tip Tracking Control of a Single-link Flexible Robot Based on a Lumped Model,” Robotica, vol. 17, issue 1, pp. 71-78, 1999.
[48] J. Hong, W. He, Z. Li, and S. Zhang, “Vibration Control and Angular Tracking of a Flexible Link via Neural Networks,” in Proceedings of the 27th Chinese Control and Decision Conference, Qingdao, China, 2017, pp. 2951-2956.
[49] D.S. Kwon and W.J. Book, “A Time-Domain Inverse Dynamic Tracking Control of a Single-Link Flexible Manipulator,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 116, issue 2, pp. 193-200, 1994.
[50] J.C. Ower and J. Van De Vegte, “Classical Control Design for a Flexible Manipulator Modeling and Control System Design,” IEEE Journal of Robotics and Automation, vol. 3, issue 5, pp. 485-489, 1987.
[51] D. Wang, Y. Lu, Y. Liu, and X. Li, “Dynamic Model and Tip Trajectory Tracking Control for a Two Link Flexible Robotic Manipulators,” in Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics, Beijing, China, 1996, pp. 1020-1024.
[52] S.S. Ge, T.H. Lee, and G. Zhu, “Tip Tracking Control of a Flexible Manipulator Using PD Type Controller,” in Proceedings of the IEEE International Conference on Control Applications, Dearborn, U.S.A., 1996, pp. 309-313.
[53] Y.Q. Dai, A.A. Loukianov, and M. Uchiyama, “A Hybrid Numerical Method for Solving the Inverse Kinematics of a Class of Spatial Flexible Manipulators,” in Proceedings of the IEEE International Conference on Robotics and Automation, Albuquerque, U.S.A., 1997, pp. 3449-3454.
[54] A.P. Tzes, S. Yurkovich, and F.D.L. Siemens, “A Symbolic Manipulation Package for Modeling of Rigid or Flexible Manipulators,” in Proceedings of the IEEE International Conference on Robotics and Automation, Philadelphia, U.S.A., 1988, pp. 1526-1531.
[55] C.L. Damaren, “Approximate Inverse Dynamics and Passive Feedback for Flexible Manipulators with Large Payloads,” IEEE Transactions on Robotics and Automation, vol. 12, issue 1, pp. 131-138, 1996.
[56] A. Green and J.Z. Sasiadek, “Dynamics and Trajectory Tracking Control of a Two-Link Robot Manipulator,” Journal of Vibration and Control, vol. 10, issue 10, pp. 1415-1440, 2004.
[57] H. Asada, Z.D. Ma, and H. Tokumaru, “Inverse Dynamics of Flexible Robot Arms Modeling and Computation for Trajectory Control,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 112, issue 2, pp. 177-185, 1990.
[58] T.A. Trautt and E. Bayo, “Inverse Dynamics of Flexible Manipulators with Coulomb Friction or Backlash and Non-Zero Initial Conditions,” Dynamics and Control, vol. 9, issue 2, pp. 173-195, 1999.
[59] H.C. Moulin and E. Bayo, “Accuracy of Discrete Models for the Inverse Dynamics of Flexible Arms, Feasible Trajectories,” in Proceedings of the 29th IEEE Conference on Decision and Control, Honolulu, U.S.A., 1990, pp. 531-532.
[60] H.C. Moulin and E. Bayo, “Accuracy of Discrete Models for the Solution of the Inverse Dynamics Problem for Flexible Arms, Feasible Trajectories,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 119, issue 3, pp. 396-404, 1997.
[61] M.H. Shaheed and M.O. Tokhi, “Dynamic Modelling of a Single-Link Flexible Manipulator: Parametric and Non-Parametric Approaches,” Robotica, vol. 20, issue 1, pp. 93-109, 2002.
[62] D.S. Kwon and W.J. Book, “An Inverse Dynamic Method Yielding Flexible Manipulators State Trajectory,” in Proceedings of the American Control Conference, San Diego, U.S.A., 1990, pp. 186-193.
[63] M. Moallem, R.V. Patel, and K. Khorasani, “An Inverse Dynamics Control Strategy for Tip Position Tracking of Flexible Multi-Link Manipulators,” Journal of Robotic Systems, vol. 14, issue 9, pp. 649-658, 1997.
[64] E. Bayo, P. Papadopoulos, J. Stubbe, and M.A. Serna, “Inverse Dynamics and Kinematics of Multi-link Elastic Robot An Iterative Frequency Domain Approach,” The International Journal of Robotics Research, vol. 8, issue 6, pp. 49-62, 1989.
[65] A.D. Luca and B. Siciliano, “Inversion Based Nonlinear Control of Robot Arms with Flexible Links,” Journal of Guidance, Control and Dynamics, vol. 16, issue 6, pp. 1169-1176, 1993.
[66] A.D. Luca, P. Lucibello, and G. Ulivi, “Inversion Technique for Trajectory Control of Flexible Arms,” Journal of Robotic Systems, vol. 6, issue 4, pp. 325-344, 1989.
[67] J. Denavit and R.S. Hartenberg, “A Kinematic Notation for Lower-Pair Mechanisms Based on Matrices,” ASME Journal of Applied Mechanics, vol. 23, pp. 215-221, 1955.
[68] K.S. Fu, R.C. Gonzalez, and C.S.G. Lee, Robotics: Control, Sensing, Vision, and Intelligence, New York, U.S.A.: McGraw-Hill Book Company, 1987.
[69] S. Cetinkunt and B. Ittoop, “Computer-Automated Symbolic Modeling of Dynamics of Robotic Manipulators with Flexible Links,” IEEE Transactions on Robotics and Automation, vol. 8, issue 1, pp. 94-105, 1992.
[70] W.J. Book, “Recursive Lagrangian Dynamics of Flexible Manipulator Arms,” The International Journal of Robotics Research, vol. 3, issue 3, pp. 87-100, 1984.
[71] S. Cetinkunt and W.J. Book, “Symbolic Modeling and Dynamic Simulation of Robotic Manipulators with Compliant Links and Joints,” Robotics and Computer-Integrated Manufacturing, vol. 5, issue 4, pp.301-310, 1989.
[72] W.J. Book, “Analysis of Massless Elastic Chains with Servo Controlled Joints,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 101, issue 3, pp. 187-192, 1979.
[73] T.W. Yang, W.L. Xu, and S.K. Tso, “Dynamic Modeling Based on Real-Time Deflection Measurement and Compensation Control for Flexible Multi-Link Manipulators,” Dynamics and Control, vol. 11, issue 1, pp. 5-24, 2001.
[74] P ChedMail, Y. Aoustin, and C. Chevallereau, “Modelling and Control of Flexible Robots,” International Journal for Numerical Methods in Engineering, vol. 32, issue 8, pp. 1595-1619, 1991.
[75] M. Makarov, M. Grossard, P. Rodriguez-Ayerbe, and D. Dumur, “A Frequency Domain Approach for Flexible Joint Robot Modeling and Identification,” in Proceedings of the 16th IFAC Symposium on System Identification The International Federation of Automatic Control, Brussels, Belgium, 2012, pp. 583-588.
[76] W.J. Book and M. Majette, “Controller Design for Flexible Distributed Parameter Mechanical Arms via Combined State Space and Frequency Domain Techniques,” Journal of Dynamic Systems, Measurement, and Control, vol. 105, issue 4, pp. 245-554, 1983.
[77] A.A. Ata, W.F. Fares, and M.Y.S. Adeh, “Dynamic Analysis of a Two-Link Flexible Manipulator Subject to Different Sets of Conditions,” International Symposium on Robotics and Intelligent Sensors, vol. 41, pp. 1253-1260, 2012.
[78] J.L. Chen, H. Xu, and A.Z. Yan, “Frequency Analysis of a Rotating Cantilever Beam Using Assumed Mode Method with Coupling Effect,” Mechanics Based Design of Structures and Machines, vol. 34, issue 1, pp. 25-47, 2006.
[79] H.H. Yoo and S.H. Shin, “Vibration Analysis of Rotating Cantilever Beam,” Journal of Sound and Vibration, vol. 212, issue 5, pp. 807-828, 1998.
[80] V. Feliu, K.S. Rattan, and H.B. Brown Jr., “Control of a Two Degree of Freedom Light Weight Flexible Arm with Friction in the Joints,” Journal of Robotic Systems, vol. 12, issue 1, pp. 1-17, 1995.
[81] Z.H. Luo, “Direct Strain Feedback Control of Flexible Robot Arms New Theoretical and Experimental Results,” IEEE Transactions on Automatic Control, vol. 38, issue 11, pp. 1610-1622, 1993.
[82] M.W. Vandegrift, F.L. Lewis, and S.Q. Zhu, “Flexible-Link Robot Arm Control by a Feedback Linearization/Singular Perturbation Approach,” Journal of Robotic Systems, vol. 11, issue 7, pp. 591-603, 1994.
[83] S. Nicosia and P. Tomei, “Robot Control by Using Only Joint Position Measurements,” IEEE Transactions on Automatic Control, vol. 35, issue 9, pp. 1058-1061, 1990.
[84] B. Siciliano and W.J. Book, “A Singular Perturbation Approach to Control of Lightweight Flexible Manipulators,” International Journal of Robotics Research, vol. 7, issue 4, pp. 79-90, 1986.
[85] N.G. Chalhoub and A.G. Ulsoy, “Control of a Flexible Robot Arm: Experimental and Theoretical Results,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 109, issue 4, pp. 299-309, 1987.
[86] M. Hermle and W. Schiehlen, “Hierarchical Control of Flexible Robots,” European Control Conference, Karlsruhe, Germany, pp. 2363-2368, 1999.
[87] A. Ailon and R.Ortega, “An Observer-Based Set-Point Controller for Robot Manipulators with Flexible Joints,” Systems and Control Letters, vol. 21, issue 4, pp. 329-335, 1993.
[88] J. Cheong, W.K. Chung, Y. Youm, and S. Oh, “Control of Two-Link Flexible Manipulator Using Disturbance Observer with Reaction Torque Feedback,” in Proceedings of the 8th International Conference on Advanced Robotics, Monterey, U.S.A., 1997, pp. 227-232.
[89] J.D. Lee and B.L. Wang, “Optimal Control of a Flexible Robot Arm,” Computers and Structures, vol. 29, issue 3, pp. 459-467, 1988.
[90] A.K. Banerjee and W.E. Singhose, “Command Shaping in Tracking Control of a Two-Link Flexible Robot,” Journal of Guidance, Control, and Dynamics, vol. 21, issue 6, pp. 1012-1015, 1998.
[91] K.L. Hillsley and S. Yurkovich, “Vibration Control of a Two-Link Flexible Robot Arm,” Dynamics and Control, vol.3, issue 3, pp. 261-280, 1991.
[92] K.J. Park, “Flexible Robot Manipulator Path Design to Reduce the Endpoint Residual Vibration under Torque Constraints,” Journal of Sound and Vibration, vol. 275, issues 3-5, pp. 1051-1068, 2004.
[93] M. Mirshekaran, F. Piltan, Z. Esmaeili, T. Khajeaian, and M. Kazeminasab, “Design Sliding Mode Modified Fuzzy Linear Controller with Application to Flexible Robot Manipulator,” International Journal of Modern Education and Computer Science, vol. 10, pp. 53-63, 2013.
[94] B. Subudhi and A.S. Morris, “Soft Computing Methods Applied to the Control of a Flexible Robot Manipulator,” Applied Soft Computing, vol. 9, issue 1, pp. 149-158, 2009.
[95] M. Bodur and M.E. Sezer, “Adaptive Control of Flexible Multilink Manipulators,” International Journal of Control, vol. 58, issue 3, pp. 519-536, 1993.
[96] Y. Li, S. Tong, and T. Li, “Adaptive Fuzzy Output Feedback Control for a Single-link Flexible Robot Manipulator Driven Dc Motor via Backstepping,” Nonlinear Analysis: Real World Applications, vol. 14, issue 1, pp. 483-494, 2013.
[97] B.S. Yuan, W.J. Book, and B. Sicilliano “Direct Adaptive Control of a One-Link Flexible Arm with Tracking,” Journal of Robotic Systems, vol. 6, issue 6, pp. 663-680, 1989.
[98] J.H. Yang, F.L. Lian, and L.C. Fu, “Nonlinear Adaptive Control for Flexible-Link Manipulators,” IEEE Transactions on Robotics and Automation, vol. 13, issue 1, pp. 140-148, 1997.
[99] J.S. Kim, K. Suzuki, A. Konno, and M. Uchiyama, “Force Control of Constrained Flexible Manipulators,” in Proceedings of the IEEE International Conference on Robotics and Automation, Minneapolis, U.S.A., 1996, pp. 635-640.
[100] R.J. Wai and M.C. Lee, “Intelligent Optimal Control of Single-Link Flexible Robot Arm,” IEEE Transactions on Industrial Electronics, vol. 51, issue 1, pp. 201-220, 2004.
[101] Y.C. Ouyang, W. He, and X.J. Li, “Reinforcement Learning Control of a Single Link Flexible Robotic Manipulator,” IET Control Theory Applications, vol. 11, issue 9, pp. 1426-1433, 2017.
[102] F. Ghorbel, J.Y. Hung, and M.W. Spong, “Adaptive Control of Flexible-Joint Manipulators,” IEEE Control System Magazine, vol. 9, issue 7, pp. 9-13, 1989.
[103] R. Lozano and B. Brogliato, “Adaptive Control of Robot Manipulators with Flexible Joints,” IEEE Transactions on Automatic Control, vol. 37, issue 2, pp.174-181, 1992.
[104] M.W. Spong, K. Khorasani, and P.V. Kokotovic, “An Integral Manifold Approach to the Feedback Control of Flexible Joint Robots,” IEEE Journal of Robotics and Automation, vol. RA-3, issue 4, pp. 291-300, 1987.
[105] H. Asada, J.H. Park, and S. Rai, “A Control Configured Flexible Arm Integrated Structure/Control Design,” in Proceedings of the IEEE International Conference on Robotics and Automation, Sacramento, U.S.A., 1991, pp. 2356-2362.
[106] F.Y. Wang, “On the External Fundamental Frequencies of One-Link Flexible Manipulators,” The International Journal of Robotics Research, vol. 13, issue 2, pp. 162-170, 1994.
[107] F.Y. Wang and J.L. Russell, “Optimum Shape Construction of Flexible Manipulators with Total Weight Constraint,” IEEE Transactions on System, Man, and Cybernetics, vol. 25, issue 4, pp. 605-614, 1995.
[108] L.L. Cui and Z.Q. Xiao, “Optimum Structure Design of Flexible Manipulators Based on GA,” in Proceedings of the IEEE Intelligent Transportation Systems, Shanghai, China, 2003, pp. 1622-1626.
[109] U.S. Dixit, R. Kumar, and S.K. Dwivedy, “Shape Optimization of Flexible Robotic Manipulators,” ASME Journal of Mechanical Design, vol. 128, issue 3, pp. 559-565, 2005.
[110] O. Turhan and G. Bulut, “On nonlinear vibrations of a rotating beam,” Journal of Sound and Vibration, vol. 322, issues 1-2, pp. 314-335, 2009.
[111] M. Gautier, A. Janot, and P.O. Vandanjon, “A New Closed-Loop Output Error Method for Parameter Identification of Robot Dynamics,” IEEE Transactions on Control Systems Technology, vol. 21, issue 2, pp. 428-444, 2013.
[112] A. Janot, P.O. Vandanjon, and M. Gautier, “A Generic Instrumental Variable Approach for Industrial Robot Identification,” IEEE Transactions on Control Systems Technology, vol. 22, issue 1, pp. 132-145, 2014.
[113] A. Green and J.Z. Sasiadek, “Robot Manipulator Control for Rigid and Assumed Mode Flexible Dynamic Models,” in Proceedings of the American Institute of Aeronautics and Astronautics Guidance, Navigation, and Control Conference and Exhibit, Austin, U.S.A., 2003, pp. 1-11.
[114] M.W. Spong, S. Jutchinson, and M. Vidyasagar, Robot Modeling and Control, New York, U.S.A.: John Wiley & Sons, 2006
[115] 莊閔皓，六軸工業用機械手臂之系統鑑別與順應控制研究，碩士論文，國立成功大學電機工程學系，中華民國，2016。
[116] 維基百科。轉動慣量-維基百科，自由的百科全書。檢索日期2017年12月28日，取自https://zh.wikipedia.org/wiki/%E8%BD%89%E5%8B%95%E6%85%A3%E9%87%8F
[117] L. Meirovitch, Methods of Analytical Dynamics, New York, U.S.A.: McGraw-Hill Book Company, 1988.
[118] Wikipedia. König's theorem (kinetics)-Wikipedia. Retrieved Dec. 28, 2017, from https://en.wikipedia.org/wiki/K%C3%B6nig%27s_theorem_(kinetics)
[119] 方同，多自由度系統中的反共振，力學學報，第4期，pp. 360-366，1979。
[120] K. Erkorkmaz and Y. Altintas, “High Speed CNC System Design. Part I: Jerk Limited Trajectory Generation and Quintic Spline Interpolation,” International Journal of Machine Tools & Manufacture, vol. 41, issue 9, pp. 1323-1345, 2001.