Scales and Locomotion: Non-Reversible Longitudinal Drag

Locomotion requires that an animal or robot be able to move itself forward farther than it moves backward in each gait cycle (formally, that it be able to break the symmetry of its interactions with the world). Previous work has established that a difference between lateral and longitudinal drag provides sufficient conditions for locomotion to be possible. The geometric mechanics community has used this principle to build a geometric framework for describing the effectiveness and efficiency of undulatory locomotion. Researchers in biology and robotics have observed that structures such as snake scales additionally provide a difference between forward and backward longitudinal drag. As yet, however, the impact of scales on the geometric features relevant to locomotion effectiveness and efficiency have not yet been explored. We present a geometric model for a single-joint undulating system with scales and identify the features needed to understand its motion. Mathematically, the scales can be treated as inducing a "Finsler metric" on the configuration space, and this paper lays the groundwork for further research into application of such Finsler metrics to robotic locomotion.