Least-Square Optimization of Planar and Spherical Four-Bar Function Generator Under Mobility Constraints

An optimization scheme for four-bar function generators under mobility constraints, which can be applied to both planar and spherical four bar linkages, is presented in this paper. The design error, defined as the residual in the input-output equation, is minimized over the vector of linkage parameters. The mobility constraints, given as a set of inequalities, are converted into equalities by introducing slack variables. The problem is thus formulated as an equality-constrained minimization problem, which is then solved using the orthogonal-decomposition algorithm, an iterative numerical method introduced elsewhere. To reduce the dimensional unbalance, which often occurs in solving a synthesis problem, a penalty function is combined with the original objective function, whose minimization leads to dimensionally balanced linkages. A numerical example is included.