Parametrized Motion Planning and Topological Complexity

Farber, Michael; Weinberger, Shmuel

doi:10.1007/978-3-031-21090-7_1

Cited by 2 publications

(3 citation statements)

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“…The total space E is the union of the fibres E = ⊔ b∈B X b and its topology reflects natural connectivity of the situation. We refer the reader to [12], [10], [11] where the motion planning problem of many autonomously moving objects in the presence of multiple obstacles was studied.…”

Section: Sequential Parametrized Topological Complexitymentioning

confidence: 99%

“…Compared to Theorem 20 this result has two advantages: firstly, it involves characteristic classes of the original vector bundle ξ : E → B and, secondly, it does not require the bundle ξ to be orientable. The case r = 2 appears in the paper [12].…”

Section: Lower Bounds Using Stiefel -Whitney Characteristic Classesmentioning

confidence: 99%

“…The authors of [2], [3] analysed in full detail parametrised topological complexity of the Fadell -Neuwirth bundle [5] which is equivalent to the motion planning problem for many robots and obstacles in the Euclidean space. Further progress was achieved in [12] where explicit parametrized motion planning algorithms of minimal complexity were constructed.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Parametrized topological complexity of sphere bundles

Farber¹,

Weinberger²

2023

TMNA

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Parametrized motion planning algorithms \cite{CFW} have high degree of flexibility and universality, they can work under a variety of external conditions, which are viewed as parameters and form part of the input of the algorithm. In this paper we analyse the parameterized motion planning problem in the case of sphere bundles. Our main results provide upper and lower bounds for the parametrized topological complexity; the upper bounds typically involve sectional categories of the associated fibrations and the lower bounds are given in terms of characteristic classes and their properties. We explicitly compute the parametrized topological complexity in many examples and show that it may assume arbitrarily large values.

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Section: Sequential Parametrized Topological Complexitymentioning

confidence: 99%

Section: Lower Bounds Using Stiefel -Whitney Characteristic Classesmentioning

confidence: 99%