Evaluation of a Three Degree of Freedom Revolute-Spherical-Revolute Joint Configuration Parallel Manipulator A thesis presented to the faculty of the Russ College of Engineering and Technology of Ohio University In partial fulfillment of the requirements for the degree Master of Science Joseph J. Feck August 2013 © 2013 Joseph J. Feck. All Rights Reserved. 2 This thesis titled Evaluation of a Three Degree of Freedom Revolute-Spherical-Revolute Joint Configuration Parallel Manipulator by JOSEPH J. FECK has been approved for the Department of Mechanical Engineering and the Russ College of Engineering and Technology by Robert L. Williams II Professor of Mechanical Engineering Dennis Irwin Dean, the Russ College of Engineering and Technology 3 Abstract FECK, JOSEPH J., M.S., August 2013, Mechanical Engineering Evaluation of a Three Degree of Freedom Revolute-Spherical-Revolute Joint Configuration Parallel Manipulator (142 pp.) Director of Thesis: Robert L. Williams II. The purpose of this thesis is to evaluate a three degree of freedom (DOF) revolute-spherical-revolute (RSR) configured robotic manipulator for the use of controlling yaw, pitch, and roll of a model aircraft. Studies of past 3-DOF parallel manipulators were used to gain an understanding of how to design such a system without creating singularity conditions. Inverse orientation and forward orientation kinematics were derived from the vector loop closure equation of the system. This study investigated what effects would occur to the maximal rotational capabilities of the robotic system when the constant lengths of the parallel manipulator were changed. It was determined that the IOK and FOK solutions of the system were valid and that such a system would be able to attain requested angular rotations by the aircraft’s team. However, modeling of the system led to a case were another leg configuration wanted to be used, as it proved to reduce the torque applied to the servo motors within the system. A prototype of the new parallel manipulator was machined and future work on this project will require the derivation of the kinematics equations, such as was done with the 3-RSR orientation. 4 Acknowledgements First and foremost, I would like to give special thanks to God for bestowing many blessings upon me and helping me shape into the man I am today. I would like to thank my advisor, Dr. Robert L. Williams II. Without him I would have never gained such a passion for robotics, and it is that passion which fueled my pursuit of this thesis. He has helped me every step of the way with great enthusiasm. I would like to thank Dr. Zhu for providing me with a project and helping me obtain the materials needed to complete this project. He has been very helpful along the way. I want to thank Dr. Choi. The knowledge I obtained in his class has helped me in my work. I would also like to thank Dr. Chapin for his time and participation with my thesis. I would like to give a special thanks to Dean Irwin for funding me and making all of this possible. I would also like to thank Elvedin Klijuno, as his work jumpstarted this project. Finally, I would like to thank my family. It is their constant love and support that has encouraged me to always work hard and do my best in all aspects of life. 5 Table of Contents Page Abstract ............................................................................................................................... 3 Acknowledgements ............................................................................................................. 4 List of Figures ..................................................................................................................... 7 List of Tables .................................................................................................................... 10 1 Introduction ............................................................................................................... 11 1.1 Background ........................................................................................................ 11 1.2 Literature Review ............................................................................................... 14 1.2.1 3-PRUR Parallel System ............................................................................. 14 1.2.2 3-UPU Parallel System ............................................................................... 18 1.2.3 3-RRUR Parallel System ............................................................................ 20 1.2.4 3-RPR Parallel Manipulator ........................................................................ 23 1.3 Project Information ............................................................................................ 26 1.4 Thesis Objectives ............................................................................................... 29 2 Kinematics Analysis .................................................................................................. 31 2.1 Vector Loop Closure Equation ........................................................................... 31 2.2 Inverse Orientation Kinematics Solution ........................................................... 41 2.3 Forward Orientation Kinematics Solution ......................................................... 44 3 Kinematics Results .................................................................................................... 51 3.1 Inverse Orientation Kinematics .......................................................................... 51 3.1.1 Obtaining the IOK Solution ........................................................................ 51 3.1.2 Graphical Interpretation of the IOK Solution ............................................. 56 6 3.1.3 IOK Results for Windmobile Project .......................................................... 76 3.2 Forward Orientation Kinematics ........................................................................ 82 4 Discussion of Kinematics Results ............................................................................. 88 5 Modified Design 3-RSR to 3-USR ............................................................................ 94 6 Conclusion/Future Work ......................................................................................... 102 References ....................................................................................................................... 106 Appendix. MATLAB Programs..................................................................................... 108 7 List of Figures Page Figure 1: Stewart Platform (Shao, Tang, Chen, & Wang, 2012, p. 650) .......................... 11 Figure 2: 3-PRUR Parallel System (Zeng et al., 2008)..................................................... 16 Figure 3: 3-UPU Parallel System (Di Gregorio, 2003)..................................................... 19 Figure 4: 3-RRUR Parallel System (Deidda et al., 2009) ................................................. 21 Figure 5: Reference frames for analysis of 3-RRUR (Deidda et al., 2009) ...................... 22 Figure 6: 3-RPR parallel manipulator (Zhang, 2012) ....................................................... 24 Figure 7: Galah aircraft (Dibenedetto, 2012 a) ................................................................. 27 Figure 8: Truss system conceptual design (Dibenedetto, 2012 b) .................................... 28 Figure 9: Generalized model of 3-RSR system ................................................................ 32 Figure 10: Fixed Base Platform Kinematic Details .......................................................... 33 Figure 11: Moving Base Platform Kinematic Details ....................................................... 34 Figure 12: Model of RSR system with virtual middle leg ................................................ 36 Figure 13: {L} Kinematic Detail ...................................................................................... 38 Figure 14: {r} Kinematic Details ...................................................................................... 39 Figure 15: {r} Kinematic Details with Rotated Servomotor ............................................ 40 Figure 16: General Graphical Interpretation of IOK Solution with variable α ................. 57 Figure 17: Change in α vs θ and ϕ with L = 10in ............................................................ 59 Figure 18: Change in α vs θ and ϕ with L = 5in ............................................................... 60 Figure 19: Change in α vs θ and ϕ with R = P = 3.5in...................................................... 61 Figure 20: Change in α vs θ and ϕ with R = P = 1.5in...................................................... 62 Figure 21: Change in α vs θ and ϕ with r = 2in ................................................................ 63 8 Figure 22: Change in α vs θ and ϕ with r = 0.5in ............................................................. 64 Figure 23: General Graphical Interpretation of IOK Solution with variable β ................. 65 Figure 24: Change in β vs θ and ϕ with L = 10in ............................................................. 66 Figure 25: Change in β vs θ and ϕ with L = 5in ............................................................... 66 Figure 26: Change in β vs θ and ϕ with R = P = 3.5in ...................................................... 67 Figure 27: Change in β vs θ and ϕ with R = P = 1.5in ...................................................... 68 Figure