On the Manifold Property of the Set of Singularities of Kinematic Mappings: Modeling, Classification, and Genericity

Abstract

It is commonly assumed that the singularities of kinematic mappings constitute generically smooth manifolds but this has not yet been proven. Moreover, before this assumption can be verified, the concept of genericity needs to be clarified. In this paper, two different notions of generic properties of kinematic mappings are discussed. One accounts for the stability of the manifold property with respect to small changes in the geometry of a mechanism while the other concerns the likelihood that a mechanism possesses smooth manifolds of singularities. Singularities forming smooth manifolds is the condition for singularity-free motion of overconstrained mechanisms but also has consequences for reliable control of serial manipulators. As basis for establishing genericity, a formulation of the kinematic mapping is presented that takes into account the type of joints and feasible link geometries. The continuous transition between link geometries defines a deformation of a kinematic mapping. All mappings obtained in this way constitute a class of kinematic mappings. A basic characteristic of a kinematic chain is its motion space. An explicit expression for the motion spaces of individual as well as classes of kinematic mappings is given. The actual conditions for genericity will be addressed in a forthcoming publication.