Aubry-Andr\'e Localization Transition for an Active Undulator

TL;DR

This study explores how a snake-like robot's ability to traverse a channel with aperiodic obstacles is analogous to quantum localization phenomena, revealing a transition from passable to trapped states influenced by landscape disorder.

Contribution

It demonstrates a novel analogy between Aubry-André localization in quantum systems and the locomotion of a self-propelling robot in a disordered environment, supported by experiments, simulations, and theory.

Findings

Robot passes in periodic landscapes

Robot becomes trapped in sufficiently aperiodic landscapes

Transition driven by fluctuations in propulsion torque

Abstract

The transport of deformable self-propelling objects like bacteria, worms, snakes, and robots through heterogeneous environments is poorly understood. In this paper, we use experiment, simulation, and theory to study a snake-like robot as it undulates without sensory feedback through a narrow channel containing a linear array of boulder-like hemispherical obstacles. The profile of the boulder landscape approximates a one-dimensional potential introduced by Aubry and Andr\'e (AA) to study wave function localization in aperiodic lattices. The AA model provides a deterministically disordered alternative to the better-known phenomenon of Anderson localization, which occurs in truly random disordered lattices. When the boulder landscape is strictly periodic, the robot can pass completely through the channel. But if the landscape is sufficiently aperiodic, the robot becomes trapped and fails to exit the channel. The metrics we use to quantify this transition -- including exponential distributions of robot position when localized -- agree well with earlier experimental and theoretical work on a localization transition that occurs when quantum waves interact with the AA potential. A theoretical treatment of the robot's motion using resistive force theory modified to include spatially varying drag forces reproduces the behavior we observe. Further, our results indicate that the transition is generated by large fluctuations in the driving torques required for self-propulsion. These results point to a potentially fundamental connection between classical and quantum wave mechanics and the locomotion of undulators. Our study illustrates how analogies with models from condensed matter physics and wave optics can lead to the discovery of principles of self-propulsion in non-periodic landscapes.