The approximate synthesis of a planar four-bar linkage for rigid-body guidance consists in ®nding 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 ®nding 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 de®ne 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 re®ne a solution to the desired accuracy. An example is included to illustrate the method.

A forward displacement analysis in closed form is performed for the decahedral variable geometry truss manipulator. Based on the simplified geometry model, a system of the constraint equations are formulated. With elimination, this system of equations can be reduced to a 48th degree polynomial equation in one output variable.

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