Comprehensive approach for the dimensional synthesis of a four-bar linkage based on path assessment and reformulating the error function

Abstract

The optimal dimensional synthesis methodology proposed in this article does not follow the more common approaches of recent years, such as heuristic methods, but aims to perfect one of the earlier and efficient approaches—gradient methods. The four-bar linkage in its general form is applied, analyzing crank and rocker input, ensuring the nonexistence of circuit and order defects. The procedure for obtaining the optimal mechanism is based on a characterization of the circuits and branches of the path traced by the coupler point, establishing the corresponding actuation mode based on a series of indices that avoid the use of penalty functions. The proposed optimization methodology defines a new error function that includes an additional characteristic, the slope of the tangent to each point of the path. Including the tangent enables better fitting to the desired curve without sacrificing the possibility of calculating the gradient analytically, and obtaining, in general, better results than by defining additional points. Although this paper focuses primarily on the dimensional synthesis of path generation, the bases of the procedure have been also applied to rigid body motion, as shown in the illustrative examples.