Convex Hull Asymptotic Shape Evolution

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.

[1]  H. Kushner,et al.  Stochastic Approximation and Recursive Algorithms and Applications , 2003 .

[2]  Daniel Liberzon,et al.  Rendezvous Without Coordinates , 2012, IEEE Trans. Autom. Control..

[3]  David G. Kendall,et al.  Shape & Shape Theory , 1999 .

[4]  Steven M. LaValle,et al.  Planning algorithms , 2006 .

[5]  Dan Halperin,et al.  Sampling-Diagram Automata: A Tool for Analyzing Path Quality in Tree Planners , 2010, WAFR.

[6]  G. Parisi,et al.  Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study , 2007, Proceedings of the National Academy of Sciences.

[7]  Rajeev Motwani,et al.  Path Planning in Expansive Configuration Spaces , 1999, Int. J. Comput. Geom. Appl..

[8]  yuliy baryshnikov,et al.  SUB-RIEMANNIAN GEOMETRY AND PERIODIC ORBITS IN CLASSICAL BILLIARDS , 2006 .

[9]  A. Knutson.  The symplectic and algebraic geometry of Horn's problem , 1999, math/9911088.

[10]  Nancy M. Amato,et al.  Algorithmic Foundations of Robotics XIV, Proceedings of the Fourteenth Workshop on the Algorithmic Foundations of Robotics, WAFR 2021, Oulu, Finland, June 21-23, 2021 , 2021, WAFR.

[11]  S. LaValle.  Rapidly-exploring random trees : a new tool for path planning , 1998 .

[12]  Lydia E. Kavraki,et al.  On finding narrow passages with probabilistic roadmap planners , 1998 .

[13]  Nicholas Roy,et al.  Robust Models of Object Geometry , 2006 .

[14]  H. Robbins.  A Stochastic Approximation Method , 1951 .

[15]  Florent Lamiraux,et al.  On the expected complexity of random path planning , 1996, Proceedings of IEEE International Conference on Robotics and Automation.

[16]  T. K. Carne,et al.  Shape and Shape Theory , 1999 .

[17]  Emilio Frazzoli,et al.  Sampling-based algorithms for optimal motion planning , 2011, Int. J. Robotics Res..

[18]  Steven M. LaValle,et al.  Robot Motion Planning: A Game-Theoretic Foundation , 2000, Algorithmica.