Theory of undulator localization in resistive force dominated disordered environments

Many wave-like systems, including electrons in a crystal, ions in an optical lattice, and classical acoustic waves, exhibit a sudden lack of diffusion in finitely disordered media. Experimental studies reveal that undulators (e.g. worms, snakes, and limbless robots) that move through their environment by propagating wave-like body deformations, also display both ballistic motion (crawling or swimming forward) and localization (being 'stuck' via jamming or an inability to gain thrust from obstacles) in heterogeneous and disordered terrain. Undulatory locomotion in a variety of homogeneous overdamped environments (e.g. viscous fluids, sand) has been well-described using resistive force theory (RFT), which treats the net force from the medium acting on a body as the sum of local forces acting on infinitesimal elements. To model locomotion in heterogeneous disordered environments, we modify RFT by allowing the drag coefficients that scale the forces from the medium to vary spatially. We observe a transition from ballistic transport to localization as a function of aperiodicity, using a spatial drag function inspired by the Aubry-André Model, a bichromatic potential that produces an aperiodicity-induced metal-insulator transition in a 1-dimensional crystal for incommensurate wavelengths. By applying this form of aperiodicity to undulation, we establish theoretical limits on the performance and robustness of open-loop undulatory gaits in complex terrain.