Interacting Particles with Lévy Strategies: Limits of Transport Equations for Swarm Robotic Systems

Gissell Estrada-Rodriguez, Heiko Gimperlein

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Abstract

Lévy robotic systems combine superdiffusive random movement with emergent collective behavior from local communication and alignment in order to find rare targets or track objects. In this article we derive macroscopic fractional PDE descriptions from the movement strategies of the individual robots. Starting from a kinetic equation which describes the movement of robots based on alignment, collisions, and occasional long distance runs according to a Lévy distribution, we obtain a system of evolution equations for the fractional diffusion for long times. We show that the system allows efficient parameter studies for a search problem, addressing basic questions like the optimal number of robots needed to cover an area in a certain time. For shorter times, in the hyperbolic limit of the kinetic equation, the PDE model is dominated by alignment, irrespective of the long range movement. This is in agreement with previous results in swarming of self-propelled particles. The article indicates the novel and quantitative modeling opportunities which swarm robotic systems provide for the study of both emergent collective behavior and anomalous diffusion on the respective time scales.
Original languageEnglish
Pages (from-to)476-498
Number of pages23
JournalSIAM Journal on Applied Mathematics
Volume80
Issue number1
Early online date18 Feb 2020
DOIs
Publication statusPublished - 2020

Keywords

  • Anomalous diffusion
  • Fractional Laplacian
  • Lévy walk
  • Swarm robotics
  • Transport equation
  • Velocity jump model

ASJC Scopus subject areas

  • Applied Mathematics

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