Hierarchically clustered navigation of distinct Euclidean particles

Arslan, Omur; Guralnik, Dan P.; Koditschek, Daniel E.

doi:10.1109/allerton.2012.6483320

Cited by 5 publications

(23 citation statements)

References 13 publications

(17 reference statements)

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“…Many of the concepts and some of the technical constructions we develop here were presented in preliminary form in the conference paper [1], building on the initial results of the conference paper [29]. This presentation gives a unified view of the detailed results (with some tutorial background) and contributes a major new extension by generalizing the construction of [1] from point particles to thickened disks of non-zero radius (necessitating a more involved version of the hierarchy invariant fields in Section V-B).…”

Section: Related Literaturementioning

confidence: 99%

“…For every cluster I ∈ C (τ ) we recall the standard notion of parent (cluster) Pr (I, τ ) and lists of children Ch (I, τ ), ancestors Anc (I, τ ) and descendants Des (I, τ ) of I in τ -see [29] for explicit definitions of cluster relations. Additionally, we find it useful to define the local complement (sibling) of cluster I ∈ C (τ ) as I −τ : = Pr (I, τ ) \ I.…”

Section: B Cluster Hierarchiesmentioning

confidence: 99%

“…Given such an HC, for any x ∈ Conf R d , J, r and τ ∈ BT J , we say x supports τ if and only if (x, τ ) ∈ HC. The stratum associated with a binary hierarchy τ ∈ BT J , denoted by S(τ ) ⊂ Conf R d , J, r , is the set of all configurations x ∈ Conf R d , J, r supporting the same tree τ [29],…”

Section: Configuration Hierarchiesmentioning

confidence: 99%

“…Denote by ϕ t the flow [79] on Conf R d , J, r induced by the vector field f . For a choice of hierarchical clustering HC, the class of hierarchy-invariant vector fields maintaining the robot group in a specified hierarchical arrangement of clusters, τ ∈ BT J , is defined as [29],…”

Section: A Generic Components Of Hierarchical Navigation 1) Hierarchmentioning

confidence: 99%

“…15 Hence, navigation in a hierarchical stratum is carried out by aligning separating hyperplane normals, illustrated in Fig. 8; and using this geometric intuition, we construct in [29] a family of hierarchy preserving control policies for point particle configurations, and in the following we extend that construction to thickened disk configurations. Proof It is well known that k-means clustering is a multifunction generally yielding different k-partitions of any given data, and so is HC 2-means (Property 1) [8], [72].…”

Section: Theoremmentioning

confidence: 99%

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Coordinated Robot Navigation via Hierarchical Clustering

2016

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Abstract-We introduce the use of hierarchical clustering for relaxed, deterministic coordination and control of multiple robots. Traditionally an unsupervised learning method, hierarchical clustering offers a formalism for identifying and representing spatially cohesive and segregated robot groups at different resolutions by relating the continuous space of configurations to the combinatorial space of trees. We formalize and exploit this relation, developing computationally effective reactive algorithms for navigating through the combinatorial space in concert with geometric realizations for a particular choice of hierarchical clustering method. These constructions yield computationally effective vector field planners for both hierarchically invariant as well as transitional navigation in the configuration space. We apply these methods to the centralized coordination and control of n perfectly sensed and actuated Euclidean spheres in a d-dimensional ambient space (for arbitrary n and d). Given a desired configuration supporting a desired hierarchy, we construct a hybrid controller which is quadratic in n and algebraic in d and prove that its execution brings all but a measure zero set of initial configurations to the desired goal with the guarantee of no collisions along the way.

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Section: Related Literaturementioning

confidence: 99%

Section: B Cluster Hierarchiesmentioning

confidence: 99%

Section: Configuration Hierarchiesmentioning

confidence: 99%

Section: A Generic Components Of Hierarchical Navigation 1) Hierarchmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

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Coordinated Robot Navigation via Hierarchical Clustering

2016

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Navigation of Distinct Euclidean Particles via Hierarchical Clustering

Arslan

Guralnik

Koditschek

2015

Springer Tracts in Advanced Robotics

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We present a centralized online (completely reactive) hybrid navigation algorithm for bringing a swarm of n perfectly sensed and actuated point particles in Euclidean d space (for arbitrary n and d) to an arbitrary goal configuration with the guarantee of no collisions along the way. Our construction entails a discrete abstraction of configurations using cluster hierarchies, and relies upon two prior recent constructions: (i) a family of hierarchy-preserving control policies and (ii) an abstract discrete dynamical system for navigating through the space of cluster hierarchies. Here, we relate the (combinatorial) topology of hierarchical clusters to the (continuous) topology of configurations by constructing "portals"-open sets of configurations supporting two adjacent hierarchies. The resulting online sequential composition of hierarchy-invariant swarming followed by discrete selection of a hierarchy "closer" to that of the destination along with its continuous instantiation via an appropriate portal configuration yields a computationally effective construction for the desired navigation policy.

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Discriminative measures for comparison of phylogenetic trees

Arslan

Guralnik

Koditschek

2017

Discrete Applied Mathematics

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In this paper we introduce and study three new measures for efficient discriminative comparison of phylogenetic trees. The NNI navigation dissimilarity d nav counts the steps along a "combing" of the Nearest Neighbor Interchange (NNI) graph of binary hierarchies, providing an efficient approximation to the (NP-hard) NNI distance in terms of "edit length". At the same time, a closed form formula for d nav presents it as a weighted count of pairwise incompatibilities between clusters, lending it the character of an edge dissimilarity measure as well. A relaxation of this formula to a simple count yields another measure on all trees -the crossing dissimilarity d CM . Both dissimilarities are symmetric and positive definite (vanish only between identical trees) on binary hierarchies but they fail to satisfy the triangle inequality. Nevertheless, both are bounded below by the widely used RobinsonFoulds metric and bounded above by a closely related true metric, the cluster-cardinality metric d CC . We show that each of the three proposed new dissimilarities is computable in time O(n 2 ) in the number of leaves n, and conclude the paper with a brief numerical exploration of the distribution over tree space of these dissimilarities in comparison with the Robinson-Foulds metric and the more recently introduced matching-split distance.

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Hierarchically clustered navigation of distinct Euclidean particles

Cited by 5 publications

References 13 publications

Coordinated Robot Navigation via Hierarchical Clustering

Coordinated Robot Navigation via Hierarchical Clustering

Navigation of Distinct Euclidean Particles via Hierarchical Clustering

Discriminative measures for comparison of phylogenetic trees

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