SIAM Journal on Applied Mathematics, Volume 80, Issue 1, Page 476-498, January 2020.
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.

中文翻译：

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与Lévy策略相互作用的粒子：群机器人系统的运输方程的极限

SIAM应用数学杂志，第80卷，第1期，第476-498页，2020年1月。
Lévy机器人系统将超扩散随机运动与本地通信和对准中出现的集体行为相结合，以找到稀有目标或跟踪物体。在本文中，我们从单个机器人的运动策略中得出宏观分数PDE描述。从动力学方程式开始，该动力学方程式根据对齐，碰撞和偶尔的长距离行驶根据Lévy分布描述了机器人的运动，我们获得了分数扩散的长时间演化方程组。我们展示了该系统允许对搜索问题进行有效的参数研究，解决基本问题，例如在一定时间内覆盖某个区域所需的最佳机器人数量。对于较短的时间，在动力学方程的双曲极限中，PDE模型由对齐控制，不论远距离运动。这与以前的自推进粒子成簇结果一致。文章指出了群体机器人系统为研究在相应时间尺度上出现的集体行为和异常扩散提供的新颖且定量的建模机会。