Interacting particles with Lévy strategies: limits of transport equations for swarm robotic systems
This work provides a novel quantitative modeling framework for swarm robotic systems, addressing search and tracking problems by linking microscopic strategies to macroscopic transport equations.
The authors derived macroscopic fractional PDE descriptions from individual robot movement strategies combining Lévy-distributed long-distance runs with local alignment and collisions, enabling efficient parameter studies for search problems such as determining the optimal number of robots to cover an area in a given time.
Lévy robotic systems combine superdiffusive random movement with emergent collective behaviour 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 behaviour and anomalous diffusion, on the respective time scales.