On the Direct Kinematics of Spherical Three-Degree-of-Freedom Parallel Manipulators with a Coplanar Platform

Abstract

This paper presents a polynomial solution to the direct kinematic problem of a class of spherical three-degree-of-freedom parallel manipulators. This class is defined as the set of manipulators for which the axes of the three revolute joints attached to the gripper link are coplanar and symmetrically arranged. It is shown that, for these manipulators, the direct kinematic problem admits a maximum of 8 real solutions. A polynomial of degree 8 is obtained here to support this result and cases for which all the roots of the polynomial lead to real configurations are presented. Finally, the spherical parallel manipulator with collinear actuators, which received some attention in the literature, is also treated and is shown to lead to a minimal polynomial of the same degree. Examples of the application of the method to manipulators of each category are given and solved.