Robotic swimming in curved space via geometric phase

In flat space, locomotion requires momentum exchange with the surrounding environment. However, in curved space, the non-commutativity of translations can permit locomotion without momentum exchange, just as rotational non-commutivity allows falling cats and lizards to change their orientations. Here we illustrate this principle experimentally via a robot that locomotes via shape changes without any additional forces while confined to a spherical surface with a radius of 0.3 m. Permitting the robot to rotate about the vertical axis, we minimize external forces, particularly friction and gravity. We then observe an initial angular displacement per cycle matching the geometric (Berry) phase induced by the robot's shape changes for various patterns and magnitudes of such changes. A gait with each stroke displacing a robot's internal components 30 degrees advances the robot 0.6 degrees per cycle. In contrast with the ideal, force-free case, frictional dissipation and weak gravitational forces eventually arrest the robot, while also imbuing it with momentum in the opposite direction.  Our work demonstrates how the interaction between environmental curvature, active driving and geometric phases yields rich, exotic phenomena.