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=F₁∪F₂∪⋯∪ F_k (where F₁,...,F_k are pairwise disjoint ENRs, see below) and by continuous maps s_j:F_j→PX, such that E○s_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.