Singularities and Self-Motions of a Special Type of Platforms

Abstract

special class of parallel manipulators (Stewart-Gough platforms) with planar base and platform, where the plane of the platform is in a projective correspondence with the plane of the base. As a special case we obtain results for parallel manipulators for which the projective transformation simplifies to affine correspondence or a similarity or a congruence. We describe all singular positions of such manipulators. We show that such a manipulator is in a singular position if points of the platform lie on a conic section and then the platform is architecturally singular or the singularity of the position does not depend on the points of the platform and depends only on the projective correspondence. The condition for the projective correspondence, which leads to a singular position is geometrically interpreted. The paper is a substantial generalization of results in [Karger, 2001], where only the similarity case was treated. We also prove a surprizing result that manipulators of this type have practically no self-motions in spite of the fact that they have a large variety of singular positions.