The Constrained Least-Square Optimization of Spherical Four-Bar Path Generators

In this paper, the optimization of the spherical RRRR four-bar linkage for the problem of path generation is addressed. The problem is formulated as a two-loop minimization of the error between the path-generating point in the coupler curve and the prescribed position, while decoupling the linkage parameters from the configuration variables, namely, the input angles. The synthesis problem consists of evaluating a set of input angles {ψ_k}₁^m defining m linkage configurations and the linkage parameters independently. This leads to a constrained overdetermined system of nonlinear equations. The orthogonal decomposition algorithm, introduced elsewhere, is employed to solve the problem. Continuation and damping techniques are used in the numerical procedure to ensure convergence and speed up its rate. The optimization scheme is developed on a general basis and can handle the problem for any number of given path points. Three numerical examples are included.