Topological Complexity of Motion Planning |
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Authors: | Farber |
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Institution: | (1) School of Mathematical Sciences, Tel Aviv University, Ramat Aviv 69978, Israel mfarber@tau.ac.il, IL |
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Abstract: |
Abstract. In this paper we study a notion of topological complexity TC
(X) for the motion planning problem. TC
(X) is a number which measures discontinuity of the process of motion planning in the configuration space X . More precisely, TC
(X) is the minimal number k such that there are k different "motion planning rules," each defined on an open subset of X× X , 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 TC
(X) . We give an upper bound for TC
(X) (in terms of the dimension of the configuration space X ) and also a lower bound (in 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: spheres, two-dimensional
surfaces, 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. |
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Keywords: | |
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