Singularity analysis of a three-leg six-degree-of-freedom parallel platform mechanism based on grassmann line geometry

This paper addresses the determination of the singularity loci of a six-degree-of-freedom spatial parallel platform mechanism of a new type that can be statically balanced. The mechanism consists of a base and a mobile platform that are connected by three legs using five-bar linkages. A general formulation of the Jacobian matrix is first derived that allows one to determine the Plucker vectors associated with the six input angles of the architecture. The linear dependencies between the corresponding lines are studied using Grassmann line geometry, and the singular configurations are presented using simple geometric rules. It is shown that most of the singular configurations of the three-leg six-degree-of-freedom parallel manipulator can be reduced to the generation of a general linear complex. Expressions describing all the corresponding singularities are then obtained in closed form. Thus, it is shown that for a given orientation of the mobile platform, the singularity locus corresponding to the general complex is a quadratic surface (i.e., either a hyperbolic, a parabolic, or an elliptic cylinder) oriented along the z-axis. Finally, three-dimensional representations that show the intersection between the singularity loci and the constant-orientation workspace of the mechanism are given.