Anonymous leadership and stochastic resonance in collectives of self-propelled robots

TL;DR

This study addresses how a minimal, anonymous leader can influence a swarm of self-propelled robots. It couples Kilobot experiments with a stochastic majority-rule model and a detailed overdamped numerical framework to show that a periodically reversing leader does not significantly change global order but produces a resonance-like enhancement in leader–swarm synchronization at intermediate noise $p$. The optimal response occurs when the leader's reversal period $\tau$ matches the mean residence time of the leaderless system, a timescale-matching condition consistent with stochastic resonance. The results advance understanding of decision-making in active matter and suggest principles for steering robotic swarms with limited leadership input, with potential relevance to biological collective behavior as well as engineering applications.

Abstract

We investigate the influence of an anonymous leader on a collective of self-propelled robots using Kilobot experiments and numerical simulations. A single leader alternated deterministically between clockwise and counterclockwise motion, while the other robots followed a stochastic majority rule. Although the leader does not change global order, it induces correlations with the collective response that peak at intermediate perturbation levels, resembling stochastic resonance. Simulations confirm that this resonance occurs when the leader's reversal period matches the mean residence time of the unperturbed system. Our results contribute to understanding decision-making in active matter and suggesting principles for steering robotic swarms with minimal leadership input.

Figures (5)

• Figure 1: (a) Snapshot of the experimental setup. Nineteen Kilobots were placed inside a circular arena of radius 15 cm. (b)–(d) Trajectories of two robots over a period of 5 minutes for $p = 0.02$, $0.08$, and $0.20$, respectively.
• Figure 2: (a)–(d) Representative 20-minute windows of the temporal evolution of the instantaneous angular velocity of the center of mass and the order parameter of the system for $p = 0.02$, $0.08$, $0.14$, and $0.20$, respectively. (e)–(h) Joint distributions of the angular velocity and the order parameter for the same values of $p$ as in panels (a)–(d), considering the full 60 minutes of each experiment. (i)–(j) Temporal average of the order parameter and its standard deviation as a function of $p$.
• Figure 3: Representative 20-minute windows of the temporal evolution of the collective order parameter and the leader's state for $p = 0.02$, $0.08$, $0.14$, and $0.20$, respectively. Results correspond to a leader that reverses its state with $\tau = 100\ \mathrm{s}$. (e) Time-averaged order parameter: system with leader versus leader-free system. (f) Order parameter standard deviation: system with leader versus leader-free system. (g) Maximum cross-correlation between the system order parameter and the leader's state as a function of $p$. Results are shown for $\tau = 30$, $100$, and $300\ \mathrm{s}$.
• Figure 4: (a) Schematic of a two-particle collision illustrating the relevant quantities. (b)–(e) Simulated 20-minute windows of the temporal evolution of the instantaneous angular velocity of the center of mass and the order parameter of the system for $p = 0.02$, $0.08$, $0.14$, and $0.20$, respectively. (f) Mean residence time as a function of $p$.
• Figure 5: Simulated 20-minute windows of the temporal evolution of the collective order parameter and the leader's state for $p = 0.02$, $0.08$, $0.14$, and $0.20$, respectively. Results correspond to a leader that reverses its state with $\tau = 100\ \mathrm{s}$. (e) Maximum cross-correlation between the system order parameter and the leader's state as a function of $p$ and $\tau$. The black line indicates the MRT of the system in the absence of a leader revealing resonance-like behavior consistent with timescale matching.