Topology of Robot Motion Planning

Farber, Michael

doi:10.1007/1-4020-4266-3_05

Cited by 64 publications

(61 citation statements)

References 28 publications

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“…As mentioned in the Introduction, the number TC(X) provides a measure of the complexity of the motion planning problem for a system with configuration space homotopy equivalent to X. More details can be found in Farber [Far1], [Far2], [Far3].…”

Section: Grantmentioning

confidence: 99%

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Topological complexity of motion planning and Massey products

Grant

2009

Algebraic Topology - Old and New

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Abstract. We employ Massey products to find sharper lower bounds for the Schwarz genus of a fibration than those previously known. In particular we give examples of non-formal spaces X for which the topological complexity TC(X) (defined to be the genus of the free path fibration on X) is greater than the zero-divisors cup-length plus one.1. Introduction. Motion planning is a fundamental area of research in Robotics. A motion planning algorithm for a given mechanical system S is a function which assigns to each ordered pair (A, B) of physical states of S a continuous motion of S starting at A and ending at B. We may regard the admissible physical states of S as being parameterised by the points of a topological space X (the configuration space of the system) such that motions of the system correspond to continuous paths γ : [0, 1] = I → X. A motion planning algorithm for the system is then a section s : X × X → X I (not necessarily continuous) of the free path fibration

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Section: Grantmentioning

confidence: 99%

“…The invariant TC(X) is a close relative of the Lusternik-Schnirelmann category cat(X), and although independent the two satisfy the inequalities cat(X) ≤ TC(X) ≤ cat(X × X) ≤ 2 · cat(X) − 1. We refer the reader to [Far3] for an excellent survey of results in this area.…”

mentioning

confidence: 99%

Topological complexity of motion planning and Massey products

Grant

2009

Algebraic Topology - Old and New

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“…coverings by open or closed sets, or by Euclidean neighborhood retracts, etc., but most of them agree on simplicial polyhedra (cf. [5], Theorem 13.1). The topological complexity T C(X) of X is the minimal number n of domains of continuity, i.e.…”

Section: Farber's Topological Complexitymentioning

confidence: 97%

“…It is simple and has the same Ktheory as the usual torus T 2 [2], hence T C(A θ ) = ∞, while for a usual torus T 2 one has T C(C(T 2 )) = 3 (cf. [5], Example 16.4). Nevertheless, tensoring by matrices does not increase topological complexity.…”

Section: General Casementioning

confidence: 99%

A noncommutative version of Farber's topological complexity

Manuilov¹

2017

TMNA

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Abstract. Topological complexity for spaces was introduced by M. Farber as a minimal number of continuity domains for motion planning algorithms. It turns out that this notion can be extended to the case of not necessarily commutative C * -algebras. Topological complexity for spaces is closely related to the Lusternik-Schnirelmann category, for which we do not know any noncommutative extension, so there is no hope to generalize the known estimation methods, but we are able to evaluate the topological complexity for some very simple examples of noncommutative C * -algebras.

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“…The problem has been studied extensively from this viewpoint by the first author in [8], [9], and [10], where a new homotopy invariant was explored. For any space X, the Topological Complexity of X is a positive natural number TC(X), which may be defined in a number of equivalent ways.…”

Section: Introductionmentioning

confidence: 99%

Symmetric motion planning

Farber¹,

Grant²

2007

Topology and Robotics

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Abstract. In this paper we study symmetric motion planning algorithms, i.e. such that the motion from one state A to another B, prescribed by the algorithm, is the time reverse of the motion from B to A. We experiment with several different notions of topological complexity of such algorithms and compare them with each other and with the usual (non-symmetric) concept of topological complexity. Using equivariant cohomology and the theory of Schwarz genus we obtain cohomological lower bounds for symmetric topological complexity. One of our main results states that in the case of aspherical manifolds the complexity of symmetric motion planning algorithms with fixed midpoint map exceeds twice the cup-length.We introduce a new concept, the sectional category weight of a cohomology class, which generalises the notion of category weight developed earlier by E. Fadell and S. Husseini. We apply this notion to study the symmetric topological complexity of aspherical manifolds.

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Topology of Robot Motion Planning

Cited by 64 publications

References 28 publications

Topological complexity of motion planning and Massey products

Topological complexity of motion planning and Massey products

A noncommutative version of Farber's topological complexity

Symmetric motion planning

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