Position-singularity analysis of a special class of the Stewart parallel mechanisms with two dissimilar semi-symmetrical hexagons

Abstract

SUMMARYIn this paper, for a special class of the Stewart parallel mechanism, whose moving platform and base one are two dissimilar semi-symmetrical hexagons, the position-singularity of the mechanism for a constant-orientation is analyzed systematically. The force Jacobian matrix [J]T is constructed based on the principle of static equilibrium and the screw theory. After expanding the determinant of the simplified matrix [D], whose rank is the same as the rank of the matrix [J]T, a cubic symbolic expression that represents the 3D position-singularity locus of the mechanism for a constant-orientation is derived and graphically represented. Further research shows that the 3D position-singularity surface is extremely complicated, and the geometric characteristics of the position-singularity locus lying in a general oblique plane are very difficult to be identified. However, the position-singularity locus lying in the series of characteristic planes, where the moving platform coincides, are all quadratic curves compromised of infinite many sets of hyperbolas, four pairs of intersecting lines and a parabola. For some special orientations, the quadratic curve can degenerate into two lines or even one line, all of which are parallel to the ridgeline. Two theorems are presented and proved for the first time when the geometric characteristics of the position-singularity curves in the characteristic plane are analyzed. Moreover, the kinematic property of the position-singularity curves is obtained using the Grassmann line geometry and the screw theory. The theoretical results are demonstrated with several numeric examples.