Topological complexity of a map

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;topological complexity;robotics;kinematic map;fibration;covering;

Data

Language:	English
Year of publishing:	2019
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	515.14
COBISS:	18590297
ISSN:	1532-0073
Views:	434
Downloads:	210
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	Topološka kompleksnost preslikave
Secondary 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.
Secondary keywords:	topološka komplesnost;robotika;kinematska preslikava;vlaknenje;
Pages:	str. 107-130
Volume:	ǂVol. ǂ21
Issue:	ǂno. ǂ2
Chronology:	Jan. 2019
DOI:	10.4310/HHA.2019.v21.n2.a7
ID:	11551950