ON THE TOPOLOGICAL-STRUCTURE OF CONFIGURATION-SPACES

We investigate the topological structure of configuration spaces of ki nematic scenes, i.e., mechanisms consisting of rigid, independently mo vable objects in a 2- or 3-dimensional environment. We demonstrate the practical importance of the considered questions by giving a motivati on from the viewpoint of qualitative reasoning. Especially, we investi gate topological invariants of the configuration space as a means for characterizing and classifying mechanisms. This paper focuses on the t opological invariants homeomorphy, homotopy equivalence and fundamenta l group. We describe a procedure for computing a finite representation of the fundamental group of a given kinematic scene, and investigate its possible structure and complexity for simple classes of scenes. We show that for any finitely represented group one can construct a kine matic scene in the plane such that the fundamental group of its config uration space is isomorphic to the given group. This construction show s the undecidability of a variety of problems concerning the topologic al structure of configuration spaces, and reveals that the considered invariants in their general form do not provide an algorithmic method for characterizing or classifying arbitrary mechanisms.