| This paper addresses the determination
of the singularity loci of a 6-DOF 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 Plücker 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 6-DOF 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. |
