Instabilities of Robot Motion

TL;DR

This paper explores the topological reasons behind robot motion instabilities, linking the configuration space's algebraic topology to unavoidable motion planning challenges and introducing a new measure of instability.

Contribution

It establishes a relationship between the cohomology algebra of configuration spaces and the inherent instabilities in robot motion planning, introducing the concept of order of instability.

Findings

Minimal number of motion planning sets determined by topology

Introduction of order of instability as a new measure

Application to various robotic motion scenarios

Abstract

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 {\it motion planner} is given by a splitting $X\times X = F_1\cup F_2\cup ... \cup F_k$ (where $F_1, ..., F_k$ 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 {\it 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 $\R^3$, a robot arm, motion in $\R^3$ in the presence of obstacles, and others.