Finding straight line generators through the approximate synthesis of symmetric four-bar coupler curves

Abstract

The approximate path synthesis of four-bar linkages with symmetric coupler curves is presented. This includes the formulation of a polynomial optimization problem, a characterization of the maximum number of critical points, a complete numerical solution by homotopy continuation, and application to the design of straight line generators. Our approach specifies a desired curve and finds several optimal four-bar linkages with a coupler trace that approximates it. The objective posed simultaneously enforces kinematic accuracy, loop closure, and leads to polynomial first order necessary conditions with a structure that remains the same for any desired trace leading to a generalized result. Ground pivot locations are set as chosen parameters, and it is shown that the objective has a maximum of 73 critical points. The theoretical work is applied to the design of straight line paths. Parameter homotopy runs are executed for 1440 different choices of ground pivots for a thorough exploration. These computations found the expected linkages, namely, Watt, Evans, Roberts, Chebyshev, and other previously unreported linkages which are organized into a 2D atlas using the UMAP algorithm.