THE DUAL OF SU(2) IN THE ANALYSIS OF SPATIAL LINKAGES, SU(2) IN THE SYNTHESIS OF SPHERICAL LINKAGES, AND ISOTROPIC COORDINATES IN PLANAR LINKAGE SINGULARITY TRACE GENERATION Dissertation Submitted to The School of Engineering of the UNIVERSITY OF DAYTON In Partial Fulfillment of the Requirements for The Degree of Doctor of Philosophy in Engineering By Saleh Mohamed Almestiri Dayton, Ohio May, 2018 THE DUAL OF SU(2) IN THE ANALYSIS OF SPATIAL LINKAGES, SU(2) IN THE SYNTHESIS OF SPHERICAL LINKAGES, AND ISOTROPIC COORDINATES IN PLANAR LINKAGE SINGULARITY TRACE GENERATION Name: Almestiri, Saleh Mohamed APPROVED BY: Andrew P. Murray, Ph.D. David H. Myszka, Ph.D. Advisor Committee Chairman Committee Member Professor, Department of Mechanical Associate Professor, Department of and Aerospace Engineering Mechanical and Aerospace Engineering Vinod Jain, Ph.D. Muhammad Islam, Ph.D. Committee Member Committee Member Professor, Department of Mechanical Professor, Department of Mathematics and Aerospace Engineering Robert J. Wilkens, Ph.D., P.E. Eddy M. Rojas, Ph.D., M.A., P.E. Associate Dean for Research and Innovation Dean, School of Engineering Professor School of Engineering ii © Copyright by Saleh Mohamed Almestiri All rights reserved 2018 ABSTRACT THE DUAL OF SU(2) IN THE ANALYSIS OF SPATIAL LINKAGES, SU(2) IN THE SYNTHESIS OF SPHERICAL LINKAGES, AND ISOTROPIC COORDINATES IN PLANAR LINKAGE SINGULARITY TRACE GENERATION Name: Almestiri, Saleh Mohamed University of Dayton Advisor: Dr. Andrew P. Murray This research seeks to efficiently and systematically model and solve the equations associated with the class of design problems arising in the study of planar and spatial kinematics. Part of this work is an extension to the method to generate singularity traces for planar linkages. This extension allows the incorporation of prismatic joints. The generation of the singularity trace is based on equations that use isotropic coordinates to describe a planar linkage. In addition, methods to analyze and synthesize spherical and spatial linkages are presented. The formulation of the analysis and the synthesis problem is accomplish through the use of the special unitary matrices, SU(2). Special unitary matrices are written in algebraic form to express the governing equations as polynomials. These polynomials are readily solved using the tools of homotopy continuation, namely Bertini. The analysis process presented here include determining the displacement and singular configuration for spherical and spatial linkages. Formulations and numerical examples of the analysis problem are presented for spherical four-bar, spherical Watt I linkages, spherical eight-bar, the RCCC, and the RRRCC spatial linkages. Synthesis problem are formulated and solved for spherical linkages, and with lesser extent for spatial linkages. Synthesis formulations for the spherical linkages are done in iii two different methods. One approach used the loop closure and the other approach is derived from the dot product that recognizes physical constraints within the linkage. The methods are explained and supported with Numerical examples. Specifically, the five orientation synthesis of a spherical four-bar mechanism, the eight orientation task of the Watt I linkage, eleven orientation task of an eight-bar linkage are solved. In addition, the synthesis problem of a 4C mechanism is solved using the physical constraint of the linkage between two links. Finally, using SU(2) readily allows for the use of a homotopy-continuation-based solver, in this case Bertini. The use of Bertini is motivated by its capacity to calculate every possible solution to a system of polynomials . iv TABLE OF CONTENTS ABSTRACT . iii LIST OF FIGURES . vii LIST OF TABLES . ix CHAPTER I. INTRODUCTION . .1 1.1 Review of Related Work . .3 1.1.1 Singularity Trace . .3 1.1.2 Unitary Matrices and Spatial Linkages . .4 1.2 Contribution . .7 1.3 Organization . .9 CHAPTER II. MATHEMATICAL BACKGROUND . 11 2.1 Singularity Traces and Isotropic Cordinates . 11 2.1.1 Forward Kinematics . 13 2.1.2 Singularity Analysis . 14 2.1.3 Critical Points . 15 2.2 Special Unitary Matrices . 15 2.2.1 Point and Line Transformations . 17 2.2.2 Derivatives . 18 2.3 Homogeneous Transformation Review . 19 CHAPTER III. SINGULARITY TRACES FOR LINKAGES WITH PRISMATIC AND REVOLUTE JOINTS . 22 3.1 General Method . 22 3.2 Offset Slider-Crank Linkage . 23 3.2.1 Loop Closure . 24 3.2.2 Forward Kinematics . 24 3.2.3 Singularity Points . 25 3.2.4 Critical Points . 25 3.2.5 Motion Curve and Singularity Trace . 26 3.2.6 Validation . 27 3.3 Inverted Slider Crank Linkage . 28 3.3.1 Loop Closure . 29 3.3.2 Forward Kinematics . 29 3.3.3 Singularity Points . 29 3.3.4 Critical Points . 30 3.3.5 Motion Curve and Singularity Trace . 30 3.3.6 Validation . 32 v 3.4 Assur IV/3 with Two Prismatic Joints . 33 3.4.1 Loop Closure . 35 3.4.2 Forward Kinematics . 36 3.4.3 Singularity Points . 36 3.4.4 Critical Points . 37 3.4.5 Singularity Trace . 37 3.4.6 Motion Curve . 38 CHAPTER IV. SPHERICAL LINKAGE ANALYSIS . 40 4.1 The 3-Roll Wrist on SU(2) . 40 4.2 Spherical Four-Bar Linkage Analysis . 43 4.2.1 Loop Closure and Forward Kinematics . 44 4.2.2 Spherical Four-Bar Singularity Points . 44 4.3 Spherical Watt I Linkage Analysis . 46 4.3.1 Loop Closure and Forward Kinematics . 47 4.3.2 Singularity Points . 48 4.4 Spherical Eight-Bar Linkage Analysis . 50 4.4.1 Loop Closure and Forward Kinematics . 51 4.4.2 Singularity Points . 54 CHAPTER V. SPHERICAL LINKAGE SYNTHESIS . 59 5.1 Four-Bar Synthesis Using Dot Product Approach . 59 5.2 Watt I Synthesis Using Dot Product Approach . 61 5.3 Eight-Bar Synthesis Using Dot Product Approach . 67 5.4 Four-Bar Synthesis Using Loop Closure Approach . 69 5.5 Watt I Synthesis Using Loop Closure . 71 5.6 Discussion . 72 CHAPTER VI. SPATIAL LINKAGE ANALYSIS AND 4C SYNTHESIS . 74 6.1 RCCC Spatial Linkage Analysis . 74 6.1.1 RCCC Loop Closure and Forward Kinematics . 75 6.1.2 RCCC Singularity Points and Motion Curve . 75 6.2 RRRCC Linkage Analysis . 77 6.2.1 RRRCC Loop Closure and Forward Kinematics . 77 6.2.2 RRRCC Singularity Points and Motion Curve . 78 6.3 4C Spatial Linkage Synthesis . 80 CHAPTER VII. CONCLUSIONS AND FUTURE WORK . 83 7.1 Conclusion . 83 7.2 Future Work . 85 REFERENCES . 87 vi LIST OF FIGURES 1.1 Spherical four-bar linkage is used to place an object in three consecutive positions [1].2 2.1 Prismatic joint on a moving line of slide. 12 3.1 The position vector loop for an offset, slider-crank linkage. 24 3.2 The slider-crank singularity trace. Red markers represent the critical points. Regions of equal GI and circuits are identified. Singularities at different values of a1 are indicated. Both circuits within the gray zone exhibit a fully rotatable crank. 26 3.3 Inverted slider-crank linkage position vector loop. 28 3.4 The inverted slider-crank singularity trace. Red markers denote the critical points. Region of equal GIs and circuits are identified. Singularities at different values of a1 are indicated. Both circuits within the gray zones exhibit a fully rotatable crank. 31 3.5 Traces of the motion curve at various lengths of a1 = 0:2, 0:3, and 0:4, from the first zone on the singularity trace. 32 3.6 Traces of the motion curve at various lengths of a1 = 0:5, 1:0, and 1:5, from the second zone on the singularity trace. Singularity points are identified with red markers. 33 3.7 Traces of the motion curve at various lengths of a1 = 1:58, 1:65,and 2:00, from the third zone on the singularity trace. 34 3.8 Assur IV/3 linkage position vector loop. 35 3.9 The singularity trace for Assur IV/3 with respect to a1. Red markers denote the critical points. Regions of equal GIs and circuits are identified. The zone shaded in gray contains at least one circuit with a fully rotatable crank. 38 3.10 The singularity trace for Assur IV/3 with respect to a12. Red markers denote the critical points. The zone shaded in gray contains at least one circuit with a fully rotatable crank . 39 3.11 Motion curve for Assur IV/3 with a1 = 0:25 projected onto θ1-θ3.......... 39 vii 4.1 A 3-roll wrist in a robot assembly. Adopted from[2]. 41 4.2 A spherical four-bar linkage. 43 4.3 Motion curve for spherical Four-Bar linkage projected onto θ1 − θ4. Singularity points are identified with red circles. 46 4.4 Spherical Watt I six-bar linkage. 47 4.5 Motion curve for Watt I spherical linkage projected onto θ1-θ7. Singularity points are identified with red circles. 50 4.6 Spherical eight-bar linkage schematic diagram. 51 4.7 Spherical eight-bar linkage. 52 4.8 Motion curve for spherical eight-bar linkage projected onto θ1-θ5. Singularity points are identified with red circles. 57 4.9 Motion curve for spherical eight-bar linkage projected onto θ1-θ6. Singularity points are identified with red circles. ..