Finding Straight Line Generators Through the Approximate Synthesis of Symmetric Four-Bar Coupler Curves

Abstract

In this paper, equations for the approximate synthesis of symmetric four-bar coupler curves are formulated. Our approach specifies a number of desired trace points, and finds a number of four-bar linkages with a coupler trace that approximately passes through these points. The computed linkages correspond to all the minima of the posed objective. The objective posed simultaneously enforces kinematic accuracy, loop closure, and leads to polynomial first order necessary conditions with a monomial structure that remains the same for any number of specified desired trace points. This last characteristic makes our result more general. To simplify computations, ground pivot locations are set as chosen parameters, and a root count analysis is conducted that shows our objective has a maximum of 73 critical points. The theoretical work is applied to the computational design of straight line coupler paths. To perform this exercise, the choice of ground pivots was varied, and a parameter homotopy for each choice (504 in total) was executed. These computations found the expected linkages (Watt, Evans, Roberts, Chebyshev) and other linkages resembling them but with sizable variations on their dimensions. The t-SNE algorithm was employed to organize the computed straight line generators into a visual atlas.