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Presented in this paper is an optimization technique for solving problems with objective functions which are sums of squared quantities with linear constraints. The technique is based on Gauss’ unconstrained method, but is able to move along constraint surfaces in such a way that when the technique terminates, the Kuhn-Tucker conditions are satisfied. The resulting approach, called the Gauss constrained method, is shown to be very efficient and effective in solving problems with highly nonlinear objective functions often existing in mechanism design problems.

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The problem of synthesizing a four-bar linkage is presented as a mathematical programming problem. The objective is to synthesize a four-bar linkage whose coupler point will generate, as closely as possible, a given curve, and whose crank rotations will be as close as possible to desired values. Constraints are imposed on the design variables which force the result to be a four-bar linkage, limit the forces and torques within the linkage, restrict the location of the pivot points, limit the lengths of the links, and so on. The solution is found using an iterative technique with the aid of a digital computer. Several examples are presented which demonstrate the effectiveness of this approach. They include generation of a straight line, a figure eight, and a portion of a circular arc (previously investigated using a method developed by Freudenstein and Sandor). The work on this problem area is still in progress and there remain a number of open questions and unexplored alternatives.