Interacting particles with Lévy strategies: limits of transport equations for swarm robotic systems

Abstract

Lévy robotic systems combine superdiffusive random movement with emergent col-lective 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 theindividual robots. Starting from a kinetic equation which describes the movement of robots based onalignment, collisions, and occasional long distance runs according to a L\'evy distribution, we obtaina system of evolution equations for the fractional diffusion for long times. We show that the systemallows efficient parameter studies for a search problem, addressing basic questions like the optimalnumber of robots needed to cover an area in a certain time. For shorter times, in the hyperboliclimit of the kinetic equation, the PDE model is dominated by alignment, irrespective of the longrange 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 systemsprovide for the study of both emergent collective behavior and anomalous diffusion on the respectivetime scales

Peer Reviewed

Postprint (published version)