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Convex Hull Asymptotic Shape Evolution

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Algorithmic Foundations of Robotics X

Part of the book series: Springer Tracts in Advanced Robotics ((STAR,volume 86))

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Abstract

The asymptotic properties of Rapidly exploring Random Tree (RRT) growth in large spaces is studied both in simulation and analysis. The main phenomenon is that the convex hull of the RRT reliably evolves into an equilateral triangle when grown in a symmetric planar region (a disk). To characterize this and related phenomena from flocking and swarming, a family of dynamical systems based on incremental evolution in the space of shapes is introduced. Basins of attraction over the shape space explain why the number of hull vertices tends to reduce and the shape stabilizes to a regular polygon with no more than four vertices.

Supported by grants from AFOSR (FA9550-10-1-0567) and ONR (N00014-11-1-0178).

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Author information

Authors and Affiliations

  1. University of Illinois, Urbana, IL, 61801, USA

    Maxim Arnold, Yuliy Baryshnikov & Steven M. LaValle

  2. MA: Institute for Information Transmission Problems, Moscow, Russia

    Maxim Arnold, Yuliy Baryshnikov & Steven M. LaValle

Authors
  1. Maxim Arnold
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  2. Yuliy Baryshnikov
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  3. Steven M. LaValle
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Corresponding author

Correspondence to Maxim Arnold .

Editor information

Editors and Affiliations

  1. Associate Professor of Aeronautics and, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

    Emilio Frazzoli

  2. MIT Comp. Sci. & Artificial Intel. Lab., Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

    Tomas Lozano-Perez

  3. Robust Robotics Group, Massachusetts Institute of Technology CSAIL, Cambridge, Massachusetts, USA

    Nicholas Roy

  4. Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

    Daniela Rus

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Arnold, M., Baryshnikov, Y., LaValle, S.M. (2013). Convex Hull Asymptotic Shape Evolution. In: Frazzoli, E., Lozano-Perez, T., Roy, N., Rus, D. (eds) Algorithmic Foundations of Robotics X. Springer Tracts in Advanced Robotics, vol 86. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36279-8_21

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  • DOI: https://doi.org/10.1007/978-3-642-36279-8_21

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36278-1

  • Online ISBN: 978-3-642-36279-8

  • eBook Packages: EngineeringEngineering (R0)

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