Aubry–André localization transition for an active undulator

The transport of deformable self-propelling objects like bacteria, worms, snakes, and robots through heterogeneous environments is poorly understood. Here we use experiment, simulation, and theory to study a snake-like robot as it undulates without sensory feedback through a linear array of boulder-like hemispherical obstacles. The profile of the boulder landscape approximates a one-dimensional potential for quantum waves introduced by Aubry and André (AA), who found that wave function localization occurs when this potential is sufficiently aperiodic. This behavior is related to the phenomenon of Anderson localization, where waves localize in the presence of a sufficiently random potential. When the boulder landscape is periodic, the robot passes through the array. But if the landscape is sufficiently aperiodic, the robot becomes trapped. 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 an unexpected connection between the Hamiltonian mechanics of quantum and classical wave systems and the highly dissipative, continuously driven, and far-from-equilibrium dynamics of active embodied waves in nonperiodic landscapes.