Verification of the trajectories of Stewart platform manipulators against singularities

This paper deals with the problem of determining conclusively if a given spatial trajectory of a semi-regular Stewart platform manipulator (SRSPM) passes through singularities. The algorithm presented here is restricted to trajectories defined in terms of polynomial functions of a continuous path parameter. As such it can complement existing motion-planning strategies which use piecewise continuous cubics or other polynomials to describe the trajectories. The algorithm, however, can be extended to functions such as rational or trigonometric, which can be reduced readily to polynomial forms. The advantage of the formulation presented here is that the computation is relatively simple and the results are guaranteed theoretically. Further, if a given trajectory does pass through a singularity, the algorithm returns the set of points where it does so. The theory presented in the paper is illustrated with numerical examples involving 3-D and 6-D trajectories defined in terms of cubic polynomials in a single path parameter.