On the Forward Kinematics of Parallel Manipulators

In this article we present a novel procedure for the system atic analysis of the forward kinematics of a class of parallel manipulators that generalize the well-known Stewart plat form. The designs comprise a movable platform connected to a fixed base by a set of legs, the lengths of which can be con trolled. The legs are connected to the base and the platform by unactuated spherical joints. The forward kinematics prob lem is to determine all possible manipulator configurations (i.e., all possible positions and orientations of the platform) when a set of leg lengths is prescribed. The new formulation of the forward kinematics problem is general in the sense that it can, in principle, be applied to any manipulator design (i.e., to any fixed geometry of base and platform joints, with associated leg connectivities). One interesting aspect of our procedure is that it splits the forward kinematics problem into two parts, the first being a simple, design-dependent, inversion of a linear system of equations, and the second being a matrix- completion problem involving the solution of certain nonlinear, but design-independent, closure equations. The general formu lation provides some insight into which manipulator designs have simple forward kinematics. We analyze four specific ma nipulator designs from the new point of view, one of which is a design with previously unrecognized closed-form forward kinematics.