Supplemental Software for Finding Straight Line Generators through the Approximate Synthesis of Symmetric Four-bar Coupler Curves

Results from the approximate path synthesis of four-bar linkages with symmetric coupler curves to synthesize straight line generating linkages. The related work includes a formulation of the optimization problem as polynomial root finding, a characterization of the maximum number of critical points, a complete numerical solution of all critical points by homotopy continuation, and application to the design of straight line generators. Our approach specifies a desired trace curve, and finds a number of four-bar linkages with a coupler trace that approximates the desired curve. The minima of a posed objective represent the dimensions of a four-bar linkages. The objective posed simultaneously enforces kinematic accuracy, loop closure, and leads to polynomial first order necessary conditions with a polynomial structure that remains the same for any desired trace. 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 (1440 in total) was executed. These computations found the expected linkages (Watt, Evans, Roberts, Chebyshev) and other previously unreported linkages with sizable variations on their dimensions. The UMAP algorithm is employed to organize the computed straight line generators into a 2D visual atlas.