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

摘要

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 a method to generate singularity traces for planar linkages. This method 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 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 .