Mechanism Analysis and Synthesis Using Numerical Algebraic Geometry

Problems in kinematics often yield large, structured polynomial systems. Homotopy continuation is a standard tool from numerical algebraic geometry that can be used to solve systems of polynomial equations and analyze the resulting solution sets arising from problems in mechanism analysis and synthesis. First, we analyze the motion of Stewart-Gough platforms and define a family of exceptional Stewart-Gough platforms called Segre-dependent Stewart-Gough platforms which arise from a linear dependency of point-pairs under the Segre embedding and compute an irreducible decomposition of this family. We also consider Stewart-Gough platforms which move with two degrees of freedom. Since the Segre embedding arises from a representation of the special Euclidean group in three dimensions which has degree 40, we consider the special Euclidean group in other dimensions and compute spatial Stewart-Gough platforms that move in 4-dimensional space. Next, we investigate coupling methods from numerical algebraic geometry with statistical models to estimate the number of solutions to polynomial systems arising from synthesis problems. These systems generically have far fewer solutions than traditional upper bounds would suggest. The new approach extends previous work on success ratios of parameter homotopies to using monodromy loops as well as the addition of a trace test for validating that all solutions have been found. Several examples are presented demonstrating the method including Watt I six-bar motion generation problems. Lastly, we consider computing curve cognates of six-bar planar linkages, that is, planar mechanisms with rotational joints whose tracing point draws the same curve. Cognate linkages are mechanisms that share the same motion, a property that can be useful in mechanical design. While Roberts cognates for planar four-bars are relatively simple to understand from a geometric drawing, the same cannot be said for planar six-bar cognates. We provide a simple method to generate these cognates as alternatives in a mechanical design. The simplicity of the approach enables the derivation of cognates for eight-bars, ten-bars, and possibly beyond.