Unconstrained nonlinear least-square optimization of planar linkages for rigid-body guidance

Abstract

In this paper, unconstrained nonlinear least-square techniques are used in the optimization of planar linkages for rigid-body guidance. A variable-separation technique, consisting of decoupling the configuration variables from the linkage parameters, is applied here. In this way, the number of design parameters, namely eight, is constant, regardless of the number of prescribed configurations of the coupler link. Hence, the optimization problem consists of evaluating independently a set of values ψ k defining the kth linkage configuration, and eight design parameters which completely define the linkage sought. The problem is formulated so as to lead to an unconstrained overdetermined system of nonlinear algebraic equations whose least-square approximation is computed by the Newton-Gauss method. Continuation as well as damping techniques are introduced to ensure convergence and enhance its rate. Although the paper focuses only on the production of the linkage having a coupler link attaining a set of prescribed positions and orientations with the least-square error, disregarding further constraints, these can be readily incorporated, as illustrated in the examples.