Degeneracy, Singularity, and Multiplicity in Least-Squares Design of a Function-Generating Mechanism

Abstract

Error Linearization (EL), an iterative curve-fitting procedure recently proposed for designing minimum squared error four-bar function generating mechanisms, suffers from frequent instability. The cause seems to be the near singularity of a certain 3×3 matrix, which produces artificially large steps, usually toward designs with unrealistically short driver and follower. This degenerate case proves unfortunately to be the true global minimum. To bring this behavior under control, the coupler length, formerly regarded as an independent design variable, is made to depend on the driver and follower lengths. They are determined by solving a now well-conditioned 2×2 set of error linearization equations. In an example this Stabilized EL procedure (SEL) located five reasonable locally minimal designs which would have been missed by the unstabilized version.