Topological complexity of motion planning

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dc.creatorFarber, Michael
dc.date2001-11-18
dc.date.accessioned2026-07-07T04:44:39Z
dc.date.available2026-07-07T04:44:39Z
dc.descriptionIn this paper we study a notion of topological complexity for the motion planning problem. The topological complexity is a number which measures discontinuity of the process of motion planning in the configuration space X. More precisely, it is the minimal number k such that there are k different motion planning rules, each defined on an open subset of XxX, so that each rule is continuous in the source and target configurations. We use methods of algebraic topology (the Lusternik - Schnirelman theory) to study the topological complexity . We give an upper bound (in terms of the dimension of the configuration space X) and also a lower bound (in the terms of the structure of the cohomology algebra of X). We explicitly compute the topological complexity of motion planning for a number of configuration spaces: for spheres, two-dimensional surfaces, for products of spheres. In particular, we completely calculate the topological complexity of the problem of motion planning for a robot arm in the absence of obstacles.
dc.description11 pages. 3 figures
dc.identifierhttps://arxiv.org/abs/math/0111197
dc.identifierhttp://arxiv.org/abs/math/0111197
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/62676
dc.subjectAlgebraic Topology
dc.subjectDifferential Geometry
dc.subjectFunctional Analysis
dc.subject37Cxx, 58Exx, 53Dxx
dc.titleTopological complexity of motion planning
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