Analysis and Synthesis of Parallel Robots for Medical Applications Nabil Simaan Analysis and Synthesis of Parallel Robots for Medical Applications RESEARCH THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MECHANICAL ENGINEERING by Nabil Simaan SUBMITTED TO THE SENATE OF THE TECHNION – ISRAEL INSTITUTE OF TECHNOLOGY HAIFA July 99 אב תשנ"ט This research was conducted under the supervision of Prof. Moshe Shoham in the Faculty of Mechanical Engineering , Technion – Israel Institute of Technology. Acknowledgments I am grateful for the dedicated supervision of Prof. Moshe Shoham. I spent about three busy years working with him in which I learned to appreciate his perseverance in research. I appreciate the time he spent with me on realizing this work both theoretically and especially practically in the stage of developing the prototype. I thank Mrs. Hanna Maller for her patience in answering all my endless questions regarding the bureaucratic procedures during my study period. I would like to thank the technical staff of the workshop in the robotics and manufacturing building of the Mechanical Engineering faculty for their work in preparing the prototype. Special thanks are dedicated to Yossi Barzily who expressed in stages of manufacturing his critical thoughts about my design and made me think about design issues repeatedly. I appreciate the work done by Avi Suliman who prepared the electrical system. Without his professional work, the electrical setup for the prototype would have been still missing. The generous financial support of the Technion, Mitchell-Sorf Foundation, and Marco and Louise Mitrani Memorial Fellowship is gratefully acknowledged. Finally, I would like to thank my family for their support and encouragement. Dedicated to my father, Butros, and my mother, Mariam, who spent endless efforts in bringing me up to be what I am today. Contents Abstract ...............................................................................................................................1 List of Symbols....................................................................................................................3 1 Introduction to Parallel Manipulators...........................................................................7 1.1...................................................................................................................Three robotic architectures ................................................................................................7 1.2 .................................................................................................................. Fully parallel and non-fully parallel manipulators............................................................11 2 Parallel Robots in Research and Industry: A Brief Review of the Literature ..............13 3 Robots for Medical Applications..................................................................................19 3.1 Medical robots and parallel robots in medical applications...........................19 3.2 The fundamental requirements from a medical robot ....................................20 3.3 Advantages of parallel robots in medical applications...................................21 4 Formulation of the Design Problem .............................................................................23 5 Type Synthesis..............................................................................................................26 5.1 Introduction....................................................................................................26 5.2 Synthesis of a class of parallel manipulators..................................................28 5.3 Selecting architectures based on design guidelines........................................30 5.4 A family of parallel manipulators ..................................................................36 5.5 The RSPR and the USR manipulators............................................................38 5.5.1 The USR manipulator.........................................................................38 5.52 The RSPR manipulator.......................................................................39 6 Kinematics of the USR and the RSPR Robots .............................................................41 6.1 Introduction....................................................................................................41 6.2 Inverse kinematics of the RSPR robot ...........................................................44 6.3 Inverse kinematics of the USR robot .............................................................49 7 Jacobian Formulation of A Class of Parallel Robots....................................................54 7.1 Introduction....................................................................................................54 7.2 The Jacobian matrix for parallel manipulators...............................................54 7.3 Jacobian formulation of a class of parallel Robots.........................................57 7.3.1 Jacobian of the tripod mechanism ......................................................57 7.3.2 Jacobian matrix of the RSPR robot ....................................................61 7.3.3 Jacobian matrix of the USR robot ......................................................62 7.4 Conclusions....................................................................................................64 Contents – (continued) 8 Dimensional Synthesis of the USR and the RSPR Robots and Performance-Based Comparison...................................................................................................................65 8.1 Introduction....................................................................................................65 8.2 Dimensional synthesis....................................................................................65 8.3 Selecting a prototype candidate......................................................................72 8.4 Work volume of the selected RSPR robot......................................................73 8.5 Conclusion......................................................................................................77 9 Introduction to Line Geometry.....................................................................................78 9.1 Introduction....................................................................................................78 9.2 Homogeneous coordinates of points, planes and lines in space.....................78 9.2.1 Homogeneous coordinates of a point in space ...................................78 9.2.2 Homogeneous coordinates of a plane in space...................................79 9.2.3 The principle of duality ......................................................................79 9.2.4 Homogeneous coordinates of a line....................................................80 9.3 The basic geometric forms .............................................................................82 9.4 Line families and linear dependence of line...................................................86 9.4.1 The linear complex .............................................................................86 9.4.2 The linear congruence ........................................................................88 9.4.3 The regulus.........................................................................................90 9.5 Line varieties..................................................................................................92 10 Singularity Analysis....................................................................................................96 10.1 Introduction...................................................................................................96 10.2 ..................................................................................................... A nalysis of Parallel singularities for a class of non-fully ................................................................................................................ ................................................................................................................parall el manipulators .......................................................................................100 10.2.1 The method of analysis ....................................................................100 10.2.2 Preliminary definitions.....................................................................100 10.2.3 Point singularities (1a).....................................................................103 10.2.4 Flat pencil singularities (2b).............................................................103 10.2.5 Planes singularities...........................................................................113 10.2.6 Linear Congruence singularities.......................................................119 10.2.7 Linear complex singularities ............................................................124 Contents – (continued) 10.2.8 Concluding remarks .........................................................................128 10.2.9 Design guidelines for minimizing the parallel singularities.............130 10.3 Analysis of serial singularities for the RSPR and USR robots ....................131 10.3.1 Serial singularities of the RSPR robot ...........................................131 10.3.2 Serial singularities of the USR robot .............................................133