Digitization Can Stall Swarm Transport: Commensurability Locking in Quantized-Sensing Chains

Abstract

We present a minimal model for autonomous robotic swarms in one- and higher-dimensional spaces, where identical, field-driven agents interact pairwise to self-organize spacing and independently follow local gradients sensed through quantized digital sensors. We show that the collective response of a multi-agent train amplifies sensitivity to weak gradients beyond what is achievable by a single agent. We discover a fractional transport phenomenon in which, under a uniform gradient, collective motion freezes abruptly whenever the ratio of intra-agent sensor separation to inter-agent spacing satisfies a number-theoretic commensurability condition. This commensurability locking persists even as the number of agents tends to infinity. We find that this condition is exactly solvable on the rationals -- a dense subset of real numbers -- providing analytic, testable predictions for when transport stalls. Our findings establish a surprising bridge between number theory and emergent transport in swarm robotics, informing design principles with implications for collective migration, analog computation, and even the exploration of number-theoretic structure via physical experimentation.

Figures (7)

• Figure 1: Field-driven robots on an environment with constant weak gradient. Our illustrations are for one-dimensional systems; denote $N$ for the number of robots in the swarm. (A) The perceived signal received by digital sensors, due to quantization, might be identical and thus the agent ($N=1$) has no motion bias. (B) Inter-agent interactions hold the group loosely cohesive while permitting internal motion; a small subset of agents that perceive the signal differences can then have motion bias, thus drive the entire group ($N=4$) forward. (C) The crawling mode of transport, in which a longitudinal traveling wave (green arrow) transmits information along the swarm. We show both the true signal and the perceived signal received be the quantized sensors. The snapshots are in chronological order, from top to bottom.
• Figure 2: Simulation results for the transport response of different robot swarm chains. For each specific choice for the number of robots $N$, the damping coefficient $\mu$, and the effective diffusivity $D$, we report the estimated transport response $\langle V \rangle /\alpha$ (the dark-to-light color gradient shows low-to-high values) for a range of constant slopes $\alpha \in [0,0.6]$ and spacing distances $L \in [2,20]$. (A) Baseline: $N=100$, $\mu=0.1$, $D=0.01$ (a long chain approximating the infinite-robot limit). Cyan lines mark loci in $(L,\alpha)$ where commensurability locking is predicted by theory (Sec. \ref{['sec:theo']}). (B1–B2) Vary $N$ only: $N=1$ and $N=2$ (with $\mu=0.1$, $D=0.01$). (C1–C2) Vary $\mu$ only: $\mu=0.3$ and $\mu=0.01$ (with $N=100$, $D=0.01$). (D1–D2) Vary $D$ only: $D=0.1$ and $D=0.001$ (with $N=100$, $\mu=0.1$). All other parameters are held at the baseline when not being varied.
• Figure 3: Simulation results for different formations of robot swarms in two-dimensional space. We use the same parameters, only change the number of robots and formation topologies: a solid triangle in (A), a Sierpinski fractal triangle in (B), and an empty triangle in (C). We report the time evolution of the center of mass $x$-position in (A–C1) and the corresponding swarm configurations in (A–C2). Cyan stars in (A–C1) indicate the times of the snapshots in (A–C2). All panels in (A–C2) use the same axis labels, so we show them only on the bottom-right panel.
• Figure 4: Experimental setups for a one-dimensional robotic swarm in chain formation. We are evaluating multiple methods for executing these experiments. (A) A chain of robots constrained by a guiding line passing through each unit to keep them collinear. (B) A robot placed on two parallel rail tracks. We show a preliminary experiment for robots running on tracks in the Supplementary Movie smovie06.mp4. Infrared sensors on each robot determine pairwise distances, which set the interaction strength.
• Figure 5: More simulation results for the transport response of different robot swarm chains. Here, every simulation's details are the same with the baseline in Fig. \ref{['fig02']}, except that we vary the number of robots in the chain. (A)$N=3$. (B)$N=4$. (C)$N=5$. (D)$N=6$. (E)$N=7$. (F)$N=8$. All panels use the same axis labels, so we show them only on the bottom-left panel.