A general automatic method for mechanism optimization based on kinematic constraints and analytical Jacobian matrix

This paper proposes a general formulation intended for dimensional synthesis of kinematic chains based on the derivation of kinematic constraint equations in their matrix form and their subsequent assembly in a system of linear equations. In this way, the proposed formulation allows to calculate in a simple way the partial derivatives of the nodal coordinates of a mechanism with respect to the dimensional or input parameters. Here, a complete mathematical development of the equations is presented, obtaining a formulation suitable for its implementation in computer programs. The main utility of this type of calculation is its application to gradient calculation within optimal dimensional synthesis, being valid for different types of objectives (function, path, and motion generation), and for any planar mechanism composed of any number of elements connected by joints of revolution (R), prismatic (P) or the combination R + P. The developments described in this paper have been implemented in general-purpose GUI software to solve path generation problems.