Invitation to Topological Robotics

Farber, Michael

doi:10.4171/054

Cited by 191 publications

(239 citation statements)

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“…The upper bounds for the complexity of F that we obtained in the last two subsections are actually constructive, being derived from the complexities of C and W for which suitable roadmaps can be explicitly described. On the other side, the lower bound in Theorem 3.1 depends indirectly on the lower bound for cx(W), which is in turn based on some cohomological estimates as in [9,Section 4.5]. In this subsection we will obtain better estimates by considering the homomorphism in cohomology induced by the forward kinematic map.…”

Section: Estimates Of Cx(f )mentioning

confidence: 99%

“…Let c 0 be any configuration in C . Then cat(C) = cx(C|C × {c 0 }) by [9,Lemma 4.29] so we obtain the following chain of (in)equalities…”

Section: Estimates Of Cx(f )mentioning

confidence: 99%

“…On the other side, the complexity of the standard working space W = R 3 × SO(3) is 4 (see [9,Theorem 4.61]), so in the presence of a global inverse kinematic map the motion planning requires at least 4 robust partial roadmaps. This surprising result explains many of the practical difficulties that arise when we try to construct explicit motion planning algorithms.…”

Section: Estimates Of Cx(f )mentioning

confidence: 99%

“…It is interesting to see how this very classical and independently developed theory found its application in the study of problems in robotics. For an overview of principal results on topological complexity see [9], for a less technical and very readable account, see also [14,Chapter 8].…”

Section: Prior Workmentioning

confidence: 99%

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Complexity of the forward kinematic map

Pavešić

2017

Mechanism and Machine Theory

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Abstract. The main objective of this paper is to introduce a new method for qualitative analysis of various designs of robot arms. To this end we define the complexity of a map, examine its main properties and develop some methods of computation. In particular, when applied to a forward kinematic map associated to some robot arm structure, the complexity measures the inherent discontinuities that arise when one attempts to solve the motion planning problem for any set of input data. In the second part of the paper, we consider instabilities of motion planning in the proximity of singular points, and present explicit computations for several common robot arm configurations.

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Section: Estimates Of Cx(f )mentioning

confidence: 99%

“…Let c 0 be any configuration in C . Then cat(C) = cx(C|C × {c 0 }) by [9,Lemma 4.29] so we obtain the following chain of (in)equalities…”

Section: Estimates Of Cx(f )mentioning

confidence: 99%

Section: Estimates Of Cx(f )mentioning

confidence: 99%

Section: Prior Workmentioning

confidence: 99%

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Complexity of the forward kinematic map

Pavešić

2017

Mechanism and Machine Theory

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“…Thinking of X as the space of configurations of a given mechanical system, TC.X / gives a measure of the topological instabilities in a motion planning algorithm for X -a perhaps discontinuous (but global) section of the map ev. We refer the reader to Farber [10] for a very useful survey of results in this area, and to the Farber's book [11] for a thorough introduction to the new mathematical discipline of topological robotics.…”

Section: Definition 11mentioning

confidence: 99%

Symmetric topological complexity of projective and lens spaces

González

Landweber

2009

Algebr. Geom. Topol.

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For real projective spaces, (a) the Euclidean immersion dimension, (b) the existence of axial maps and (c) the topological complexity are known to be three facets of the same problem. But when it comes to embedding dimension, the classical work of Berrick, Feder and Gitler leaves a small indeterminacy when trying to identify the existence of Euclidean embeddings of these manifolds with the existence of symmetric axial maps. As an alternative we show that the symmetrized version of (c) captures, in a sharp way, the embedding problem. Extensions to the case of even-torsion lens spaces and complex projective spaces are discussed. 55M30, 57R40Dedicated to the memory of Bob Stong Main resultThe Euclidean immersion and embedding questions for projective spaces were topics of intense research during the beginning of the second half of the last century. In the case of real projective spaces, the immersion problem has recently received a fresh push, partly in view of a surprising reformulation in terms of a basic concept arising in robotics, namely, the motion planning problem of mechanical systems. In more detail, Farber, Tabachnikov and Yuzvinsky [14] showed that for r ¤ 1; 3; 7, the immersion dimension of P r -the r -dimensional real projective space-agrees with TC.P r / 1, one unit less than the topological complexity of P r (see Definition 1.1 and Theorem 4.2 below). In this paper we accomplish a completely analogous goal by connecting the Euclidean embedding dimension of P r with Farber-Grant's notion of symmetric motion planning. Before stating our main results, we recall the relevant definitions. Definition 1.1 The topological complexity of a space X , TC.X /, is defined as the genus of the endpoints evaluation map evW P .X / ! X X , where P .X / is the free path space X OE0;1 with the compact-open topology.TC.X / is a homotopy invariant of X . Thinking of X as the space of configurations of a given mechanical system, TC.X / gives a measure of the topological instabilities in a motion planning algorithm for X -a perhaps discontinuous (but global) section of the map ev. We refer the reader to Farber [10] for a very useful survey of results in this area, and to the Farber's book [11] for a thorough introduction to the new mathematical discipline of topological robotics.We now come to the main definition (introduced and explored by Farber and Grant [12]). For a topological space X , let X be the diagonal in X X and ev 1 W P 1 .X / ! X X X be the restriction of the fibration ev in Definition 1.1. Thus P 1 .X / is the subspace of P .X / consisting of paths W OE0; 1 ! X with .0/ ¤ .1/. Note that ev 1 is a Z=2-equivariant map, where Z=2 acts freely on both P 1 .X / and X X X , by running a path backwards in the former, and by switching coordinates in the latter. Let P 2 .X / and B.X; 2/ denote the corresponding orbit spaces, and let ev 2 W P 2 .X / ! B.X; 2/ denote the resulting fibration.Definition 1.2 With the above conditions, the symmetric topological complexity of X , TC S .X /, is defined by TC S .X / D genus.ev 2...

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Parametrized Motion Planning and Topological Complexity

Farber

Weinberger

2022

Algorithmic Foundations of Robotics XV

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Invitation to Topological Robotics

Cited by 191 publications

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Complexity of the forward kinematic map

Complexity of the forward kinematic map

Symmetric topological complexity of projective and lens spaces

Parametrized Motion Planning and Topological Complexity

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