Homological invariants for classification of kinematic singularities

Abstract

This paper provides a classification of kinematic singularities for the mechanisms, serial manipulators and arrays of control moment gyros via homological invariants. It is shown here that rather than solely using local geometric methods to describe kinematic singularity type, analysis of subsets of the input space through their associated homological invariants provide the additional ability to classify and identify specific types of kinematic singularities with nonlocal behaviour such as those of the ”degenerate” type (path-connected impassable singularities) and those that are only locally passable. In addition, as a bi-product of the analysis, a partition of the input space by its fibres of a smooth map over points in the workspace is given that separates regions of singular from nonsingular points. Some proofs are provided utilizing singular and cellular homology on the existence of singularity types and classes within these fibres of the input space.