Instabilities of Robot Motion

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dc.creatorFarber, Michael
dc.date2002-05-12
dc.date.accessioned2026-07-26T00:01:20Z
dc.descriptionInstabilities 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.
dc.description26 pages, 5 figures
dc.identifierhttps://arxiv.org/abs/cs/0205015
dc.identifierhttp://arxiv.org/abs/cs/0205015
dc.identifier.urihttps://dspace.dare.co.zw/handle/123456789/103405
dc.subjectRobotics
dc.subjectComputational Geometry
dc.subjectAlgebraic Topology
dc.subjectI.2.9; I.3.5
dc.titleInstabilities of Robot Motion
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