A Closed-Form Solution for the Direct Kinematics of a Special Class of Spherical Three-Degree-of-Freedom Parallel Manipulators

Abstract

It has been shown elsewhere that the solution of the direct kinematic problem of spherical three-degree-of-freedom parallel manipulators leads to a maximum of 8 solutions. Moreover, a polynomial of degree 8 can be obtained, whose roots will lead to all the solutions of the problem. In this paper, a particular geometry of spherical parallel manipulator is studied. This geometry arises from kinematic optimization which has been performed in previous work. The direct kinematic problem associated with this special architecture is studied here and it is shown that a simple closed-form solution can be obtained for this manipulator, which contrasts with the very complex polynomial solution obtained for the general case. This work is mainly motivated by the real-time trajectory planning and control of a prototype of parallel manipulator which is based on the simplified geometry studied here.