On the Manifold Property of the Set of Singularities of Kinematic Mappings: Genericity Conditions

Abstract

Whether the singularities of a kinematic mapping constitute smoothmanifolds is an important question with significance to mechanism design and robot control. It is thus obvious to ask if this is generically so. In a preceding paper, two kinematically meaningful genericity concepts have been introduced. In this paper, geometric conditions for the manifold property of families of kinematic mappings are addressed, and a sufficient condition is presented. This condition involves the joint screws and screws representing feasible link geometries specific to a class of kinematic mappings. It admits to establish genericity of the manifold property for given classes of kinematic mappings. This is a step forward to prove that singularities of kinematic mappings form generically smooth manifolds. As an example it is shown that the singularities of 3-DOF forward kinematic mappings form generically smooth manifolds. Restricting this condition to a particular geometry allows to check whether singularities of a given kinematic mapping form smooth manifolds.