Mathematical Models of Locomotion: Legged Crawling, Snake-like Motility, and Flagellar Swimming

Three different models of motile systems are studied: a vibrating legged robot, a snake-like locomotor, and two kinds of agellar microswimmers. The vibrating robot crawls by modulating the friction with the substrate. This also leads to the ability to switch direction of motion by varying the vibration frequency. A detailed account of this phenomenon is given through a fully analytical treatment of the model. The analysis delivers formulas for the average velocity of the robot and for the frequency at which the direction switch takes place. A quantitative description of the mechanism for the friction modulation underlying the motility of the robot is also provided. Snake-like locomotion is studied through a system consisting of a planar, internally actuated, elastic rod. The rod is constrained to slide longitudinally without slipping laterally. This setting is inspired by undulatory locomotion of snakes, where frictional resistance is typically larger in the lateral direction than in the longitudinal one. The presence of constraints leads to non-standard boundary conditions, that lead to the possibility to close and solve uniquely the equations of motion. Explicit formulas are derived, which highlight the connection between observed trajectories, internal actuation, and forces exchanged with the environment. The two swimmer models (one actuated externally and the other internally) provide an example of propulsion at low Reynolds number resulting from the periodical beating of a passive elastic filament. Motions produced by generic periodic actuations are studied within the regime of small compliance of the filament. The analysis shows that variations in the velocity of beating can generate different swimming trajectories. Motion control through modulations of the actuation velocity is discussed