## Instabilities of robot motion.(English)Zbl 1106.68107

Summary: 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 $$X \times X=F_1\cup F_2 \cup \ldots \cup F_k$$ (where $$F_1, \ldots ,F_k$$ are pairwise disjoint ENRs, see below) and by continuous maps $$s_j :F_j \to PX$$, such that $$E \circ s_j=1_{F_j}$$. Here $$PX$$ denotes the space of all continuous paths in $$X$$ (admissible motions of the system) and $$E :PX \to X \times X$$ 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 $$\mathbf R^3$$, a robot arm, motion in $$\mathbf R^3$$ in the presence of obstacles, and others.

### MSC:

 68T40 Artificial intelligence for robotics 93C85 Automated systems (robots, etc.) in control theory
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### References:

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