Symmetries and Gaits for Purcell's Three-Link Microswimmer Model

Robotic locomotion typically involves using gaits-periodic changes of kinematic shape, which induce net motion of the body in a desired direction. An example is robotic microswimmers, which are inspired by motion of swimming microorganisms. One of the most famous theoretical models of a microswimmer is Purcell's planar three-link swimmer, whose structure possesses two axes of symmetry. Several works analyzed gaits for robotic three-link systems based on body-fixed velocity integrals. Using this approach, finite motion in desired directions can only be obtained approximately. In this study, we propose gaits which are based on analysis of the system's structural symmetries, and generate exact motion along principal directions without net rotation. Another gait that produces almost pure rotation is presented, and bounds on the small-amplitude residual translation are obtained by using perturbation expansion. Next, the theory is extended to more realistic swimmers which have only one symmetry axis. Gaits for such swimmers which generate net translation are proposed, and their small-amplitude motion is analyzed using perturbation expansion. The theoretical results are demonstrated by using numerical simulations and conducting controlled motion experiments with a robotic macroswimmer prototype in a highly viscous fluid.