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 guidance consists in finding all the relevant parameters of the linkage that produces a set of poses of its coupler link that best approximate a large number of prescribed poses. By “large” we mean here a number larger than that allowing for an exact matching of poses. Moreover, the approximation error in the synthesis equations is measured in the least-square sense, the problem thus giving rise to an optimization problem. Each solution of this problem, producing a local minimum of the approximation error yields one dyad, the combination of any pair of these then yielding one linkage. While purely numerical methods yield only isolated local minima, we apply here the contour method in an attempt to finding all the real stationary points of the problem at hand. First, symbolic computations are used to derive the underlying normal equations of the optimization problem. The normal equations are then reduced to a set of two bivariate polynomial equations. These two equations are plotted in the plane of the two unknown variables, the two contours that they define in this plane being then overlapped. In principle, their intersections provide, visually, all the real solutions of the problem under study as well as the numerical conditioning of these solutions. Finally, numerical techniques are used to refine a solution to the desired accuracy. An example is included to illustrate the method.