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-deg ree-of-freedom spatial parallel platform mechanism of a new type that can b e statically balanced. The mechanism consists of a base and a mobile platfo rm that are connected by three legs using five-bar linkages. A general form ulation of the Jacobian matrix is first derived that allows one to determin e the Plucker vectors associated with the six input angles of the architect ure. The linear dependencies between the corresponding lines are studied us ing Grassmann line geometry, and the singular configurations are presented using simple geometric rules. It is shown that most of the singular configu rations of the three-leg six-degree-of-freedom parallel manipulator can be reduced to the generation of a general linear complex. Expressions describi ng all the corresponding singularities are then obtained in closed form. Th us, it is shown that for a given orientation of the mobile platform, the si ngularity 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 wor kspace of the mechanism are given.