A Randomized Greedy Algorithm for Piecewise Linear Motion Planning

We describe and implement a randomized algorithm that inputs a polyhedron, thought of as the space of states of some automated guided vehicle R, and outputs an explicit system of piecewise linear motion planners for R. The algorithm is designed in such a way that the cardinality of the output is pro...

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Bibliographic Details
Published inMathematics (Basel) Vol. 9; no. 19; p. 2358
Main Authors Ortiz, Carlos, Lara, Adriana, González, Jesús, Borat, Ayse
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.10.2021
Subjects
abstract simplicial complex
Algorithms
Approximation
Automated guided vehicles
barycentric subdivision
Complexity
contiguity of simplicial maps
Greedy algorithms
homotopic distance
Invariants
Mathematics
Motion planning
Planning
Polyhedra
Portable computers
randomized algorithm
Robot dynamics
Robustness (mathematics)
Subdivisions
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Summary:We describe and implement a randomized algorithm that inputs a polyhedron, thought of as the space of states of some automated guided vehicle R, and outputs an explicit system of piecewise linear motion planners for R. The algorithm is designed in such a way that the cardinality of the output is probabilistically close (with parameters chosen by the user) to the minimal possible cardinality.This yields the first automated solution for robust-to-noise robot motion planning in terms of simplicial complexity (SC) techniques, a discretization of Farber’s topological complexity TC. Besides its relevance toward technological applications, our work reveals that, unlike other discrete approaches to TC, the SC model can recast Farber’s invariant without having to introduce costly subdivisions. We develop and implement our algorithm by actually discretizing Macías-Virgós and Mosquera-Lois’ notion of homotopic distance, thus encompassing computer estimations of other sectional category invariants as well, such as the Lusternik-Schnirelmann category of polyhedra.
ISSN:2227-7390
2227-7390
DOI:10.3390/math9192358