Computation of all optimum dyads in the approximate synthesis of planar linkages for rigid-body guidance

The approximate synthesis of a planar four-bar linkage for rigid-body guida nce consists in finding all the relevant parameters of the linkage that pro duces a set of poses of its coupler link that best approximate a large numb er of prescribed poses. By "large" we mean here a number larger than that a llowing for an exact matching of poses. Moreover, the approximation error i n the synthesis equations is measured in the least-square sense, the proble m thus giving rise to an optimization problem. Each solution of this proble m, producing a local minimum of the approximation error yields one dyad, th e combination of any pail. of these then yielding one linkage. While purely numerical methods yield only isolated local minima, we apply here the cont our method in an attempt to finding all the real stationary points of the p roblem at hand. First, symbolic computations are used to derive the underly ing normal equations of the optimization problem. The normal equations are then reduced to a set of two bivariate polynomial equations. These two equa tions are plotted in the plane of the two unknown variables, the two contou rs that they define in this plane being then overlapped. In principle, thei r intersections provide, visually, all the teal solutions of the problem un der study as well as the numerical conditioning of these solutions. Finally , numerical techniques are used to refine a solution to the desired accurac y. An example is included to illustrate the method. (C) 2000 Elsevier Scien ce Ltd. All rights reserved.