On the Optimum Synthesis of Four-bar Linkages Using Differential Evolution and the Geometric Centroid of Precision Positions

Abstract

Mechanism synthesis, the identification of the parameters of a mechanism, has been extensively studied especially for four-bar linkages using graphical and numerical optimization approaches. Graphical techniques follow a number of predefined steps and rely heavily on the user. Numerical optimization techniques that require the user to provide "good initial guesses" or bounds for the design variables have also been applied. In general, a linkage is synthesized for function generation, motion generation, and path generation. This article studies four-bar mechanism synthesis by combining Differential Evolution, an evolutionary optimization scheme that can search outside the initial defined bounds for the design variables, and a newly developed novel technique called the Geometric Centroid of Precision Points (GCPP) and the distant precision point in defining the initial bounds for the design variables. The developed methodology has been applied to the synthesis of four-bar linkages for path generation with prescribed timing, where the coupler point is required to pass through a number of precision points within a prescribed accuracy level and in the correct order, and for the generation of families of coupler curves. Two penalty functions were used, one for constraint violation and one for relative accuracy. The results of the application of this approach could also be used as "good initial guesses" for improving the desired accuracy level. Examples demonstrating the successful application of the developed methodology are presented.