Interacting particles with L\'{e}vy strategies: limits of transport equations for swarm robotic systems

TL;DR

This paper derives fractional PDE models from individual Lévy-based robot movement strategies, enabling analysis of swarm search efficiency and collective behavior over different time scales.

Contribution

It introduces a macroscopic fractional PDE framework for Lévy robotic systems, connecting microscopic movement rules to emergent collective dynamics.

Findings

Fractional PDEs effectively model long-term robot swarm behavior.

Optimal robot numbers depend on area coverage and time constraints.

Alignment dominates short-term dynamics regardless of long-range Lévy movements.

Abstract

L\'{e}vy 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 L\'{e}vy 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.