On the topological behaviour of motion planners

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MyIsmail Mamouni
Younes Derfoufi

Abstract

Let \(X\) be a contractible, path connected and compact manifold. On the field of topological robotics, motion planning algorithms (MPA) are some kind of algorithms which need (as an input) a pair of point \((A,B)\in X\times X\) and produce (as an output) a path in \(X\) from \(A\) to \(B\) without collision. We focus here on the topological study of the set \(\mathcal{M}(X)\) of that  algorithms. We first topologize \(\mathcal{M}(X)\) with the open compact topology and show that it is also contractible. Secondly, we  equip it with a metric, that induces the same topology than the open-comact one, and for which \(\mathcal{M}(X)\) is complete. This leads us to a contravariant functor \(\mathcal{M}: X\mapsto \mathcal{M}(X)\) between the category of contractible, path connected and compact manifolds \(X\) and that of their associated sets of motion planning algorithms \(\mathcal{M}(X)\). This enable us to classify, up to isometry, all motion planning algorithms. Many kind of topological and category theoretic interpretations, but also open questions, will arise for each established result. The awesome one is that the motion planning algorithms of a robot may inherit the topological behaviour of the configuration space on which the robot moves.

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How to Cite
Mamouni, M., & Derfoufi, Y. (2016). On the topological behaviour of motion planners. Journal of Advanced Studies in Topology, 7(3), 125-131. https://doi.org/10.20454/jast.2016.1035
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Original Articles