Design of Mechanisms Via Constrained Least-Squares Method and Its Variants

Abstract

Based on the design equation of mechanism and the least-squares techniques, a rapidly convergent iteration method and simple direct methods for the synthesis of mechanisms are presented. It is proved that the so-called linear superposition method is a special case of the direct methods whose effectiveness depends upon the singular nature of the normal equations of the least-squares method as well as the smallness of the Lagrange multipliers of the compatibility equations for the mechanism. While the significance of the latter has been recognized in the literature, that of the former has not been documented in the literature. By examining the correlation matrix and the condition number for the normal equations, we show that these are near-singular. This property provides a fundamental basis for the direct methods presented in this paper. The sensitivity of solutions to the design specifications and to the precision of floating point computations also is discussed. The theory and associated algorithms can be applied to the synthesis of any planar or spatial mechanism where the use of the least-squares technique is contemplated.