Topological complexity of motion planning and Massey products

Grant, Mark

doi:10.4064/bc85-0-14

Cited by 6 publications

(6 citation statements)

References 14 publications

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“…In the final example we consider the link complement of the Borromean rings. Here our lower bound improves on the zero-divisors cuplength, and recovers the bound of [20,Example 4.3] which was obtained using sectional category weight and Massey products.…”

Section: Examplessupporting

confidence: 83%

“…In the case of pure braid groups, the lower bounds for their topological complexity obtained by Farber and Yuzvinsky in [18] follow from our Theorem 1.1 using an appealing geometric argument. We also consider the Borromean rings, and recover the conclusion of [20,Example 4.3], thus showing that our bounds can improve on zero-divisors cup-length bounds with very little computational effort. In Section 4 we consider Higman's curious acyclic group, introduced in [23], and show that its topological complexity is 4.…”

Section: Introductionsupporting

confidence: 65%

“…On the other hand, it is known that zero-divisors in cohomology with untwisted coefficients are not always sufficient to determine topological complexity. In [20] the topological complexity of the link complement of the Borromean rings was studied, and sectional category weight and Massey products were applied to obtain lower bounds. This is, to the best of our knowledge, the only previously known example of an aspherical space X for which TC(X) is greater than the zero-divisors cup-length for any field of coefficients.…”

Section: Introductionmentioning

confidence: 99%

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New lower bounds for the topological complexity of aspherical spaces

Grant

Lupton

Oprea

2015

Topology and its Applications

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Abstract. We show that the topological complexity of an aspherical space X is bounded below by the cohomological dimension of the direct product A × B, whenever A and B are subgroups of π 1 (X) whose conjugates intersect trivially. For instance, this assumption is satisfied whenever A and B are complementary subgroups of π 1 (X). This gives computable lower bounds for the topological complexity of many groups of interest (including semidirect products, pure braid groups, certain link groups, and Higman's acyclic four-generator group), which in some cases improve upon the standard lower bounds in terms of zerodivisors cup-length. Our results illustrate an intimate relationship between the topological complexity of an aspherical space and the subgroup structure of its fundamental group.

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Section: Examplessupporting

confidence: 83%

Section: Introductionsupporting

confidence: 65%

Section: Introductionmentioning

confidence: 99%

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New lower bounds for the topological complexity of aspherical spaces

Grant

Lupton

Oprea

2015

Topology and its Applications

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“…We refer the reader to surveys [9], [11] for detailed treatment of the invariant TC(X). Computation of TC(X) in various practically interesting examples has received much recent interest, see for instance papers [1], [2], [10], [12], [13].…”

Section: Introductionmentioning

confidence: 99%

Topological complexity of configuration spaces

Farber

Grant

2008

Proc. Amer. Math. Soc.

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The topological complexity TC(X) is a homotopy invariant which reflects the complexity of the problem of constructing a motion planning algorithm in the space X, viewed as configuration space of a mechanical system. In this paper we complete the computation of the topological complexity of the configuration space of n distinct points in Euclidean m-space for all m ≥ 2 and n ≥ 2; the answer was previously known in the cases m = 2 and m odd. We also give several useful general results concerning sharpness of upper bounds for the topological complexity.

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“…In this study, the path planning of a drone in an uncertain environment is presented using the computational application of topology (Grant, 2009). The navigational complexity, path safety, path optimization, obstacle avoidance and goal tracing are defined in the topological space.…”

Section: Introductionmentioning

confidence: 99%

Topological drone navigation in uncertain environment

Patle

Chen

Patel

et al. 2021

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Purpose With the increasing demand for surveillance and smart transportation, drone technology has become the center of attraction for robotics researchers. This study aims to introduce a new path planning approach to drone navigation based on topology in an uncertain environment. The main objective of this study is to use the Ricci flow evolution equation of metric and curvature tensor over angular Riemannian metric, and manifold for achieving navigational goals such as path length optimization at the minimum required time, collision-free obstacle avoidance in static and dynamic environments and reaching to the static and dynamic goals. The proposed navigational controller performs linearly and nonlinearly both with reduced error-based objective function by Riemannian metric and scalar curvature, respectively. Design/methodology/approach Topology and manifolds application-based methodology establishes the resultant drone. The trajectory planning and its optimization are controlled by the system of evolution equation over Ricci flow entropy. The navigation follows the Riemannian metric-based optimal path with an angular trajectory in the range from 0° to 360°. The obstacle avoidance in static and dynamic environments is controlled by the metric tensor and curvature tensor, respectively. The in-house drone is developed and coded using C++. For comparison of the real-time results and simulation results in static and dynamic environments, the simulation study has been conducted using MATLAB software. The proposed controller follows the topological programming constituted with manifold-based objective function and Riemannian metric, and scalar curvature-based constraints for linear and nonlinear navigation, respectively. Findings This proposed study demonstrates the possibility to develop the new topology-based efficient path planning approach for navigation of drone and provides a unique way to develop an innovative system having characteristics of static and dynamic obstacle avoidance and moving goal chasing in an uncertain environment. From the results obtained in the simulation and real-time environments, satisfactory agreements have been seen in terms of navigational parameters with the minimum error that justifies the significant working of the proposed controller. Additionally, the comparison of the proposed navigational controller with the other artificial intelligent controllers reveals performance improvement. Originality/value In this study, a new topological controller has been proposed for drone navigation. The topological drone navigation comprises the effective speed control and collision-free decisions corresponding to the Ricci flow equation and Ricci curvature over the Riemannian metric, respectively.

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Topological complexity of motion planning and Massey products

Cited by 6 publications

References 14 publications

New lower bounds for the topological complexity of aspherical spaces

New lower bounds for the topological complexity of aspherical spaces

Topological complexity of configuration spaces

Topological drone navigation in uncertain environment

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