The research on direct kinematic problem of the Stewart manipulators with two dissimilar semisymmetrical hexagons

This paper addresses the direct kinematics of a special class of the 6-6 SPS Stewart manipulators whose the mobile and base platforms are two dissimilar semisymmetrical hexagons. After proposing a mathematical model of the direct kinematic problem of this class of the Stewart manipulators, a multivariate polynomial equations set with respect to the mobile platform position parameters and orientation parameters is constructed. Based on this multivariate polynomial equations set, the analytical expression of the complete solutions to the direct kinematics of this special class of the Stewart manipulators can be successfully accomplished by utilizing a sophisticated commercial symbolic computation software, MATHEMATICA by which an arduous and complicated derivation and solving task can be avoided and a lot of computation time can be saved. It is shown that for any given set of design parameters and six given link-lengths of this special class of the 6-6 SPS Stewart manipulators, the direct kinematic problem of the manipulator considered in this paper have at most 28 solutions in the complex domain. Examples of the direct kinematic problem of a Stewart manipulator under investigation are given to demonstrate the aforementioned theoretical results. Direct kinematics analysis of this special class of the 6-6 SPS Stewart manipulators paves underlying theoretical grounds for the workspace analysis, path planning and control of this special class of the 6-6 SPS Stewart manipulators.