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Algorithmic Foundations of Robotics XI pp 19–36Cite as

Navigation of Distinct Euclidean Particles via Hierarchical Clustering

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Part of the book series: Springer Tracts in Advanced Robotics ((STAR,volume 107))

Abstract

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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Notes

  1. 1.

    We will mention in the conclusion a few such extensions presently in progress.

  2. 2.

    Although clustering algorithms generating degenerate hierarchies are available, many standard hierarchical clustering methods return binary clustering trees as a default, thereby avoiding commitment to some “optimal” number of clusters [18, 37].

  3. 3.

    Here, \(\mathring{A}\) denotes the interior of set \(A\).

  4. 4.

    A long prior robotics literature motivates the utility of this fully actuated “generalized damper” dynamical model [24], and provides methods for “lifts” to controllers for second order plants [20, 21] as well.

  5. 5.

    Note that for all \(\tau \in \mathcal {BT}_{J}\), \(\mathfrak {S}_o \left( \tau \right) \subseteq \mathring{\mathfrak {S}}\left( \tau \right) \).

  6. 6.

    Here, \({\mathrm {\mathbf {A}}}^\mathrm {T}\) denotes the transpose of \(\mathrm {\mathbf {A}}\).

  7. 7.

    Here, we use \(\eta _{i, I, \tau } : \left( \mathbb {R}^d\right) ^{J} \rightarrow \mathbb {R}\) (8).

  8. 8.

    In a metric space \(\left( X, d\right) \), the open ball \(B\left( \mathrm {x},r\right) \) centered at \(\mathrm {x}\) with radius \(r \in \mathbb {R}_{\ge 0}\) is the set of points in \(X\) whose distance to \(\mathrm {x}\) is less than \(r\), i.e. \(B\left( \mathrm {x},r\right) = \left\{ \mathrm {y} \in X \; | \; d\left( \mathrm {x},\mathrm {y}\right) < r \right\} \).

  9. 9.

    Here, we set \(\frac{\mathrm {x}}{\left\| \mathrm {x}\right\| _2} = 0\) for \(\mathrm {x} = 0\).

  10. 10.

    Here, \(\mathrm {\mathbf {I}}_{d}\) is the \(d \times d\) identity matrix, and \(\mathrm {\mathbf {1}}_k\) is the \(\mathbb {R}^k\) column vector of all ones. Also, \(\otimes \) and \(\cdot \) denote the Kronecker product and the standard array product, respectively.

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Acknowledgments

This work was supported in part by AFOSR under the CHASE MURI FA9550-10-1-0567 and in part by ONR under the HUNT MURI N00014070829.

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Authors and Affiliations

  1. Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, PA, 19104, USA

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

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  1. Omur Arslan
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Correspondence to Omur Arslan .

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Editors and Affiliations

  1. Department of Computer Engineering, Bogazici University, Istanbul, Turkey

    H. Levent Akin

  2. Department of Computer Science and Engineering, Texas A&M University, College Station, Texas, USA

    Nancy M. Amato

  3. Department of Computer Science and Engineering, University of Minnesota, Minneapolis, Minnesota, USA

    Volkan Isler

  4. Department of Information and Computing Sciences, Utrecht University, Utrecht, Utrecht, The Netherlands

    A. Frank van der Stappen

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Arslan, O., Guralnik, D.P., Koditschek, D.E. (2015). Navigation of Distinct Euclidean Particles via Hierarchical Clustering. In: Akin, H., Amato, N., Isler, V., van der Stappen, A. (eds) Algorithmic Foundations of Robotics XI. Springer Tracts in Advanced Robotics, vol 107. Springer, Cham. https://doi.org/10.1007/978-3-319-16595-0_2

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