Instabilities of robot motion

Instabilities of robot motion are caused by topological reasons. In this paper we find a relation between the topological properties of a configuration space (the structure of its cohomology algebra) and the character of instabilities, which are unavoidable in any motion planning algorithm. More specifically, let X denote the space of all admissible configurations of a mechanical system. A motion planner is given by a splitting X×X=F₁F₂F_k (where F₁,…,F_k are pairwise disjoint ENRs, see below) and by continuous maps s_j :F_j→PX, such that Es_j=1_Fj. Here PX denotes the space of all continuous paths in X (admissible motions of the system) and E :PX→X×X denotes the map which assigns to a path the pair of its initial–end points. Any motion planner determines an algorithm of motion planning for the system. In this paper we apply methods of algebraic topology to study the minimal number of sets F_j in any motion planner in X. We also introduce a new notion of order of instability of a motion planner; it describes the number of essentially distinct motions which may occur as a result of small perturbations of the input data. We find the minimal order of instability, which may have motion planners on a given configuration space X. We study a number of specific problems: motion of a rigid body in R³, a robot arm, motion in R³ in the presence of obstacles, and others.