Topology of Robot Motion Planning

In the course we shall describe a new application of the algebraic topology in engineering. This theory allows to use the structure of the cohomology algebra of the configuration space to estimate numerically the character of instabilities appearing in the robot motion planning algorithms. We shall apply this approach to a number of specific problems which are interesting from both topological and the computer science points of view: collision free motion planning, control of a rigid body in R, simultaneous control of many objects, and some others. Algorithmic motion planning in robotics is a well established discipline; we refer to [9] for a recent survey and to [7] for a comprehensive textbook. In general, one is given a moving system R with k degrees of freedom and a two or three-dimensional workspace V . The geometry of R and of V is given in advance which determines the configuration space of the system, X. The latter is a subset of R consisting of all placements (or configurations) of the system R, each represented by a tuple of k real parameters, such that in this placement R lies fully in V . For simplicity we may restrict our attention to a single connected component of R, the one containing a prescribed initial placement of R. Being a subset of the Euclidean space R, the configuration space X naturally inherits its topology. Many questions of control theory depend solely on the configuration space X viewed as a topological space. One of the advantages of this approach is that different control problems could be treated simultaneously for all systems having homeomorphic configuration spaces. It is known that any real analytic manifold can be realized as the configuration space of a simple mechanical system (linkage). Therefore topological questions of robotics lead to interesting new topological invariants of abstract manifolds.