20.500.12556/RUL-115221 Topological complexity of a map Topološka kompleksnost preslikave 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. 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. topological complexity robotics kinematic map fibration covering topološka komplesnost robotika kinematska preslikava vlaknenje true false false Angleški jezik Slovenski jezik Neznano 2020-04-18 11:10:07 2020-04-18 11:10:10 2022-08-27 04:00:13 0000-00-00 00:00:00 2019 0 0 Str. 107-130 no. 2 Vol. 21 Jan. 2019 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 515.14 1532-0073 10.4310/HHA.2019.v21.n2.a7 18590297 513079065 RAZ_Pavesic_Petar_2019.pdf RAZ_Pavesic_Petar_2019.pdf 1 E826FE5813320DE272A47D3625415E2D 755e64c158829d4960747b31aec28d3149e2272d6840884781ea0aa4c128e5d9 6ebe3867-a1b8-11eb-a523-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=127919 Fakulteta za matematiko in fiziko 0 0 0