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Topological complexity of a map
ID Pavešić, Petar (Author)

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Abstract
We study certain topological problems that are inspired by applications to autonomous robot manipulation. Consider a continuous map $f \colon X \to Y$, where $f$ can be a kinematic map from the configuration space $X$ to the working space $Y$ of a robot arm or a similar mechanism. Then one can associate to $f$ a number $\mathrm{TC}(f)$, which is, roughly speaking, the minimal number of continuous rules that are necessary to construct a complete manipulation algorithm for the device. Examples show that $\mathrm{TC}(f)$ is very sensitive to small perturbations of f and that its value depends heavily on the singularities of $f$. This fact considerably complicates the computations, so we focus here on estimates of $\mathrm{TC}(f)$ that can be expressed in terms of homotopy invariants of spaces ▫$X$▫ and ▫$Y$▫, or that are valid if f satisfies some additional assumptions like, for example, being a fibration. Some of the main results are the derivation of a general upper bound for $\mathrm{TC}(f)$, invariance of $\mathrm{TC}(f)$ with respect to deformations of the domain and codomain, proof that $\mathrm{TC}(f)$ is a FHE invariant, and the description of a cohomological lower bound for $\mathrm{TC}(f)$. Furthermore, if $f$ is a fibration we derive more precise estimates for $\mathrm{TC}(f)$ in terms of the Lusternik-Schnirelmann category and the topological complexity of $X$ and $Y$. We also obtain some results for the important special case of covering projections.

Language:English
Keywords:topological complexity, robotics, kinematic map, fibration, covering
Typology:1.01 - Original Scientific Article
Organization:FMF - Faculty of Mathematics and Physics
Year:2019
Number of pages:Str. 107-130
Numbering:Vol. 21, no. 2
PID:20.500.12556/RUL-115221 This link opens in a new window
UDC:515.14
ISSN on article:1532-0073
DOI:10.4310/HHA.2019.v21.n2.a7 This link opens in a new window
COBISS.SI-ID:18590297 This link opens in a new window
Publication date in RUL:18.04.2020
Views:1158
Downloads:362
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Record is a part of a journal

Title:Homology, homotopy, and applications
Publisher:International Press
ISSN:1532-0073
COBISS.SI-ID:513079065 This link opens in a new window

Secondary language

Language:Slovenian
Title:Topološka kompleksnost preslikave
Abstract:
V članku obravnavamo vprašanja, ki izvirajo iz primerov uporabe v krmiljenju robotov. Opazujemo preslikavo $f \colon X \to Y$, ki jo lahko razumemo kot kinematično preslikavo iz konfiguracijskega prostora $X$ v delovni prostor $Y$ robotske roke ali podobne naprave. Preslikavi $f$ lahko priredimo število $\mathrm{TC}(f)$, ki v grobem predstavlja minimalno število robustnih načrtov gibanja, ki so potrebni, da v celoti krmilimo dano napravo. Konkretni primeri kažejo, da je $\mathrm{TC}(f)$ precej občutljivo na majhne spremembe preslikave $f$, zlasti na njene singularnosti. Zato v članku največ časa posvetimo ocenam za $\mathrm{TC}(f)$, ki jih je mogoče izraziti na podlagi homotopskih invariant $X$ in $Y$ ter ocenam, ki jih dobimo, če je $f$ vlaknenje. Glavni rezultati obsegajo splošno veljavno zgornjo oceno za $\mathrm{TC}(f)$, invarianco glede na deformacije domene in kodomene ter kohomološke spodnje meje. Če je $f$ vlaknenje izpeljemo še natančnejše ocene z uporabo Lusternik-Schnirelmannove kategorije. Na koncu se še posvetimo pomembnem posebnem priimeru, ko je $f$ krovna projekcija.

Keywords:topološka komplesnost, robotika, kinematska preslikava, vlaknenje

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