臺灣博碩士論文加值系統

近年來並聯機器人機械手在工業應用和機器人學中備受關注，因為它們具有較高的精度、剛性、最大產量、負載和最高速度。目前常用的六個自由度的高夫·史都華並聯機器人機械手是最突出的並聯式機械手之一，包括一個固定底座、一個可動平台、六個線性致動器、六個直動關節、固定底座和致動器之間的六個萬向接頭，及在致動器和可動平台之間的六個萬向接頭。
本研究針對不同運動軌跡及其對高夫·史都華平台的動態影響進行比較。在許多工業應用中，機械手需要以平順、穩定和連續的方式在所需的軌跡上運動。高速運動可能會磨損機器人的结構，造成機械結構的共振頻率和增強振動，從而損壞致動器的結構。由於高夫·史都華平台具有高精度、剛性、負載和最高速度的特點，因此需要優化軌跡運動曲線以使平台連續平穩移動，從而提高運動精度。與傳統方法不同，本研究將軌跡運動曲線視為使用擺線運動曲線的優化問题，與三次和五次多項式之運動曲線進行比較。賈可比矩陣則用於識別沿移動平台路徑的任何地方是否出現奇異點 (det(J)=0) 。
最後，結合使用均方根誤差(RMSE)和平均差分誤差(ADE)的分析方法，研究和分析理論軌跡與實際軌跡之間的不準確性。此外，在 Recurdyn 和 MATLAB 上進行了模擬，結果證明了本研究提出的技術的有效性。

Parallel robotic manipulators have recently attracted a lot of interest in industrial applications and the robotic discipline due to their improved precision, maximum throughput, stiffness, heavy haul and high speed. The current model of general-purpose six degrees of freedom Gough-Stewart parallel robot manipulator is among the most prominent parallel manipulators comprises of a fixed base, a moving platform, six linear actuators, six prismatic joints, six universal joints between the fixed base and the actuators, and six universal joints between the actuators and the moving platform.
This work presents a comparative study among different trajectory motion profiles and their effect on dynamics of Gough-Stewart Platform. In many industrial applications, it is necessary for manipulator to move on a desired trajectory in a smooth, stable and continuous manner. High jerk in motion may wear out the robot structure, excite the body structures’ resonance frequencies and enhance vibrations which can damage the actuator's structure. In order to avoid unnecessary vibrations, trajectory generation profiles need to be optimized to move platform continuously smooth and hence enhance the movement accuracy. Unlike conventional methods, this work considered trajectory motion profiles as an optimization problem using Cycloidal motion curve in comparison with the cubic and quintic polynomial motion profiles with constraint conditions of kinematic parameters and execution time. The Jacobian matrix is employed to identify singularities (det(J)=0) that may occur anywhere along pathway of the moving platform.
Finally, the inaccuracies between the theoretical trajectory and actual model simulation trajectory will be investigated and analysed with considering the analytical methods using Root Mean Square Error (RMSE) and Average Difference Error (ADE). Moreover, Simulations on Recurdyn and MATLAB were carried out, and the results demonstrated the efficacy of the technique proposed in this work.

English abstract......i
Chinese abstract......iii
Acknowledgment......v
Table of Contents......vi
List of Tables......x
List of Figures......xi
Nomenclature......xiii
CHAPTER 1 INTRODUCTION......1
1.1 General View......1
1.1.1 Parallel robot Manipulators and their applications......2
1.1.2 Gough-Stewart robot manipulator......5
1.1.3 Degrees of mobility of Gough-Stewart Platform......6
1.1.4 Kinematics of Gough-Stewart manipulator robot......7
1.1.5 Trajectory Tracking of Gough-Stewart manipulator robot:......8
1.2 Research Objectives......10
1.3 Thesis Outline......11
CHAPTER 2 LITERATURE REVIEW......12
CHAPTER 3 THEORETICAL ANALYSIS......18
3.1 Introduction......18
3.2 Gough-Stewart robot manipulator kinematics analysis......18
3.2.1 homogeneous coordinates transformation of Gough-Stewart Platform......19
3.2.2 Inverse Kinematics Problem (IKP) of Gough-Stewart Platform......21
3.2.3 Forward Kinematics Problem (FKP) of Gough-Stewart Platform......22
3.3 Jacobian Matrix of Gough-Stewart Platform......26
3.4 Path planning of Gough-Stewart platform......30
3.5 Motion Profiles for Gough-Stewart Platform......31
3.5.1 Cubic Polynomial Motion Profile......32
3.5.2 Quintic Polynomial Motion Profile......32
3.5.3 Cycloid Motion Profile......33
3.6 Dynamic analysis of Gough-Stewart Platform......34
CHAPTER 4 MODELLING AND SIMULATION......37
4.1 Introduction......37
4.2 Mechanical Parts......37
4.3 Electronics Parts......40
4.4 Simulation Steps in Recurdyn......42
CHAPTER 5 RESULTS AND DISCUSSIONS......45
5.1 Introduction......44
5.2 Performance Analysis of Motion Profiles for Gough-Stewart Platform......45
5.2.1 Case 1: Cubic Polynomial Motion Profiles......46
5.2.2 Case 2: Quintic Polynomial Motion Profiles......48
5.2.3 Case 3: Cycloid Function Motion Profiles......50
5.3 Effect of Motion Profiles on Joints Reaction Forces and Torques......54
5.4 Root Mean Square Error (RMSE) and Average Difference Error (ADE) Analysis......58
5.5 Singularity detection by Jacobian matrix and dynamics analysis......61
CHAPTER 6 CONCLUSION......63
Reference......64
Appendices......71
Appendix A MATLAB Code for Kinematics and Jacobian Matrix
Appendix A.1 Inverse Kinematics
Appendix A.2 Forward Kinematics
Appendix A.3 Jacobian Matrix for singularity detection
Appendix B Reaction Forces and Torques in Joint of all six Actuators
Appendix B.1 Forces and Torques across Cubic Polynomial Function
Appendix B.2 Forces and Torques across Quintic Polynomial Function
Appendix B.1 Forces and Torques across Cycloid Function
Appendix C RMSE and ADE Values for Position, Velocity, Acceleration and Jerk
Appendix C.1 RMSE and ADE Value for Cubic Polynomial Function
Appendix C.2 RMSE and ADE Value for Quintic Polynomial Function
Appendix C.3 RMSE and ADE Value for Cycloid Function

[1]S. G. Tzafestas, "Fundamentals of Mechanics of Robotic Manipulation", volume 27. Italy: Springer, 2004.
[2]J. H. Ryu, "Parallel Manipulators", Austrian: I-Tech Education and Publishing, 2008.
[3]S. Briot, W. Khalil, "Dynamics of Parallel Robots", volume 35. Italy: Springer, 2015.
[4]D. Zhang, "Parallel Robotic Machine Tools", Canada: Springer, 2010.
[5]J. P. Merlet, "Parallel Robots", second edition, France: Springer, 2006.
[6]Y. D. Patel and P. M. George, "Parallel Manipulators Applications-A Survey", Modern Mechanical Engineering, vol. 2, pp. 57-64, April 2012.
[7]S. Küçük, "Serial and Parallel Robot Manipulators-Kinematics, Dynamics", Control and Optimization, Croatia: InTech, 2012.
[8]G. Gogu, "Structural Synthesis of Parallel Robots Part 1: Methodology", volume 149. France: Springer, 2008.
[9]E. Arnold, "Lufthansa Flight Simulator on 6-axis Platform," Wikipedia, 27 October 2004. [Online]. Available:
https://en.wikipedia.org/wiki/File:Simulator-flight-compartment.jpeg.
[10]Matthewbarry, "Universal Tyre-testing Machine," Wikipedia, 2 November 2007.[Online].Available:https://en.wikipedia.org/wiki/File:Gough-platform.jpg.
[11]G. Wu, "Error Modeling and Design Optimization of Parallel Manipulators", Ph.D. Thesis, Department of Mechanical and Manufacturing Engineering, Aalborg university, Denmark, 2013.
[12]W. Chen, "Static Force Capabilities and Dynamic Capabilities of Parallel Mechanisms Equipped With Safety Clutches", M.S. Thesis, Department of Science, Laval university, Canada, 2014.
[13]PageMaker, "A Taylor Spatial Frame on the Left Leg Consisting of Metal Rings, Pins and Struts," Wikipedia, 3 June 2013. [Online]. Available: https://commons.wikimedia.org/wiki/File:Taylor_Spatial_Frame.jpg.
[14]K. Lancaster, "Amiba, a Cosmic Microwave Background experiment located in Hawaii," Wikipedia, 8 June 2006. [Online]. Available: https://en.wikipedia.org/wiki/File:AMiBA_1.jpg.
[15]J. B. Sol, "Singularity-Invariant Transformations in Stewart-Gough Platforms: Theory and Applications", Ph.D. Thesis, Institute de robotica, Spain, 2009.
[16]R. Kumar, A. Chalanga, B. Bandyopahyay, "Smooth Integral Sliding-mode Controller for the Position Control of Stewart Platform" Elsevier, vol. 58, pp. 543-551, September 2015.
[17]H. D. Taghirad, "Parallel Robots Mechanics and Control", France: CRC Press is an Imprint of Taylor & Francis Group, an Informa business, 2013.
[18]E. F. Fichter, "A Stewart platform-based Manipulator: General Theory and Practical Construction", The International Journal of Robotics Research, Vol. 5, No. 2, pp. 157-182, Summer 1986.
[19]H. Wu, "Parallel Manipulators Towards New Applications", Vienna: Published by I-Tech Education and Publishing, 2008.
[20]F. C. Park and K. M. Lynch, "Introduction to Robotics Mechanics, Planning, and Control", UK: Cambridge University Press, July 7, 2017.
[21]Lin, Chun-Shin, Po-Rong Chang, and Js Ye So Luh. "Formulation and Optimization of Cubic Polynomial Joint Trajectories for Industrial Robots."Automatic Control, IEEE Transactions on 28.12 (1983): 1066- 1074.
[22]Piazzi, Aurelio, and Antonio Visioli, "Global Minimum-jerk Trajectory Planning of Robot Manipulators." Industrial Electronics, IEEE Transactions on 47.1, pp.140-149.2000
[23]Secco, Emanuele L., Antonio Visioli, and Giovanni Magenes. "Minimum Jerk Motion Planning for a Prosthetic Finger." Journal of Robotic Systems 21.7, pp.361-368.2004
[24]Gasparetto, A., and V. Zanotto. "A New Method for Smooth Trajectory Planning of Robot Manipulators." Mechanism and machine theory 42.4 (2007): 455-471.
[25]Huang, Panfeng, et al. "Motion Trajectory Planning of Space Manipulator for Joint Jerk Minimization." Mechatronics and Automation, 2007. ICMA 2007. International Conference on. IEEE, 2007.
[26]Lin, Hsien-I. "A Fast and Unified Method to Find a Minimum-Jerk Robot Joint Trajectory Using Particle Swarm Optimization." Journal of Intelligent & Robotic Systems 75.3-4 (2014): 379-392.
[27]Liu, Huashan, Xiaobo Lai, and Wenxiang Wu. "Time-optimal and Jerk Continuous Trajectory Planning for Robot Manipulators with Kinematic Constraints." Robotics and Computer-Integrated Manufacturing 29.2 (2013): 309-317.
[28]Choi, MinHee, Wheekuk Kim, and Byung-Ju Yi. "Trajectory Planning in 6-Degrees-of-freedom Operational Space for the 3-Degrees-of-freedom Mechanism Configured by Constraining the Stewart Platform Structure." Control, Automation and Systems, 2007. ICCAS'07. International Conference on. IEEE, 2007.
[29]Cong, Ming, et al. "Application of Triune Parallel-Serial Robot System for Full-mission Tank Training." Industrial Robot: An International Journal 38.5 (2011): 533-544.
[30]Afroun, M., et al. "Optimal Motions Planning for a GOUGH Parallel Robot."Control and Automation, 2008 16th Mediterranean Conference on. IEEE, 2008.
[31]Duan, Xuechao, et al. "Real-time Motion Planning-based Vibration Control of a Macro-Micro Parallel Manipulator System for Super Antenna." Journal of Vibroengineering 16.2 (2014): 694-703.
[32]Jiang, Huchao, et al. "Optimization Design of the Trajectory about Gunlaunched Missile Based on SQP." Control Conference (CCC), 2014 33rd Chinese. IEEE, 2014.
[33]B. Dasgupta and T. S. Mruthyunjaya, "A Canonical Formulation of the Direct Position Kinematics Problem for a General 6-6 Stewart Platform", Mech. Much. Theory, vol. 29, NO. 6, pp. 819-827, 1994.
[34]M. L. Husty, "An Algorithm for Solving the Direct Kinematics of General Stewart-Gough", Mech. Much. Theory, vol. 31, NO. 4, pp. 365-380, 1996.
[35]C. W. Wampler, "Forward Displacement Analysis of General Six-Inparallel SPS (Stewart) Platform Manipulators Using Some Coordinates", Mech. Much. Theory, vol. 31, NO. 3, pp. 331-337, 1996.
[36]D. Jakobovic and L. Jelenkovic, "The forward and Inverse Kinematics Problems for Stewart Parallel Mechanisms", CIM 2002 Computer Integrated Manufacturing and High-Speed Machining - 8th International Scientific Conference on Production Engineering, 2002.
[37]D. Gan, Q. Liao, J. S. Dai, S. Wei, L. D. Seneviratne, "Forward Displacement Analysis of the General 6–6 Stewart Mechanism Using Gröbner Bases", Mechanism and Machine Theory, vol. 44, pp. 1640- 1647, 2009.
[38]X. Huang, Q. Liao, S. Wei, "Closed-form Forward Kinematics for a Symmetrical 6-6 Stewart Platform Using Algebraic Elimination", Mechanism and Machine Theory, vol. 45, pp. 327-334, 2010.
[39]S. Pedrammehr, M. Mahboubkhah, S. Pakzad, "An Improved Solution to The Inverse Dynamics of the General Stewart Platform", Proceedings of IEEE International Conference on Mechatronics, April 13-15, 2011, Istanbul, Turkey.
[40]D. Karimi and M. J. Nategh, "A Statistical Approach to the Forward Kinematics Nonlinearity Analysis of Gough-Stewart Mechanism", Journal of Applied Mathematics, pp. 1-11, 2011.
[41]Y. Zhang, Q. Liao, H. J. Su, S. Wei, "A New Closed-form Solution to the Forward Displacement Analysis of a 5–5 in-Parallel Platform", Mechanism and Machine Theory, vol. 52, pp. 47-58, 2012.
[42]R. Kumar and B. Bandyopadhyay, "The Forward Kinematic Modeling of a Stewart Platform using NLARX Model with Wavelet Network ", 11th IEEE International Conference on Industrial Informatics, July 2013.
[43]J. G, Aivarado, "A Simple Method to Solve the Forward Displacement Analysis of the General Six-legged Parallel Manipulator", Robotics and Computer-Integrated Manufacturing, vol. 30, pp. 55-61, 2014.
[44]M. Cardona, "A new Approach for the Forward Kinematics of General Stewart-Gough Platforms", Proceeding of IEEE thirty fifth central American and Panama convention, 2015.
[45]W. Zhou, W. Chen, H. Liu, X. Li, "A New Forward Kinematic Algorithm for a General Stewart Platform", Mechanism and Machine Theory, vol. 87, pp. 177-190, 2015.
[46]E. S. Gebel, "Inverse Displacement Problem of Gough -Stewart Platform with Six Degrees of Freedom", Dynamics of Systems, Mechanisms and Machines, pp. 1-4, 2017.
[47]C.-T. Chen and H.-V. Pham, “Trajectory Planning in Parallel Kinematic Manipulators Using a Constrained Multi-objective Evolutionary Algorithm,” Nonlinear Dynamics, vol. 67, no. 2, pp. 1669–1681, 2012
[48]C.-T. Chen and T.-T. Liao, “On the Optimal Singularity-free Trajectory Planning of Parallel Robot Manipulators,” In Advances in Robot Manipulators. InTech, 2010.
[49]P. Wang, H. Yang, and K. Xue, “Jerk-optimal Trajectory Planning for Stewart Platform in Joint Space,” in Mechatronics and Automation (ICMA), 2015 IEEE International Conference on. IEEE, 2015, pp. 1932–1937
[50]S. Sen, B. Dasgupta, and A. K. Mallik, “Variational Approach for Singularity-free Path-planning of Parallel Manipulators,” Mechanism and Machine Theory, vol. 38, no. 11, pp. 1165–1183, 2003.
[51]S. Wu, C. Kang, X. Li et al., “Trajectory Planning of a Parallel Manipulator Based on Kinematic Transmission Property,” Intelligent Service Robotics, vol. 8, no. 3, pp. 129–139, 2015.
[52]A. Khoukhi, L. Baron, and M. Balazinski, “Constrained Multi-objective Trajectory Planning of Parallel Kinematic Machines,” Robotics and Computer-Integrated Manufacturing, vol. 25, no. 4, pp. 756–769, 2009.
[53]S. Kucuk, “Optimal Trajectory Generation Algorithm for Serial and Parallel Manipulators,” Robotics and Computer-Integrated Manufacturing, vol. 48, pp. 219–232, 2017.
[54]J. E. Marsden and M. West, “Discrete Mechanics and Variational Integrators,” Acta Numerica, vol. 10, pp. 357–514, 2001.
[55]O. Junge, J. E. Marsden, and S. Ober-Blobaum, “Discrete Mechanics ¨ and Optimal Control,” IFAC Proceedings Volumes, vol. 38, no. 1, pp. 538–543, 2005.
[56]A. Moore, S. Ober-blobaum, and J. E. Marsden, “Optimization of Space- ¨ Craft Trajectories: A Method Combining Invariant Manifold Techniques and Discrete Mechanics and Optimal Control,” 2009.
[57]O. Junge, J. E. Marsden, and S. Ober-Blobaum, “Optimal reconfiguration of formation flying spacecraft—a decentralized approach,” in Decision and Control, 2006 45th IEEE Conference on. IEEE, 2006, pp. 5210– 5215
[58]D. Wang, W. Zhang, and J. Shan, “A Multi-phase DMOC-based Optimal Trajectory Generation Methodology for a Quadrotor,” in Control Conference (CCC), 2014 33rd Chinese. IEEE, 2014, pp. 8984–8989.
[59]Z. Shareef and A. Trachtler, “Simultaneous Path Planning and Trajectory Optimization for Robotic Manipulators Using Discrete Mechanics and Optimal Control,” Robotica, vol. 34, no. 6, pp. 1322–1334, 2016.
[60]H. M. Alwan, "The Geometrical Analysis Accuracy for Parallel Robotic Mechanisms ", Eng. & Tech. Journal, vol. 29, pp. 3118-3129, Nov. 15, 2011.
[61]J. N. Pires, "Industrial Robots Programming: Building Applications for the Factories of the Future", University of Coimbra. Portugal: Springer, 2007.
[62]J. J. Craig, "Introduction to Robotics Mechanics and Control", third edition, United States of America: Pearson Education International, 2005.
[63]A. Sumnu, I. H. Guzelbey, M. V. Cakir, "Simulation and PID Control of a Stewart Platform with Linear Motor", Journal of Mechanical Science and Technology, vol. 31, pp. 345-356, 2017.