Locomotion without force: exploiting curved spaces

Conventional locomotion requires momentum exchange with the surrounding environment. As noted by Wisdom (2003), this relies on the commutativity of generators of translations and consequently locomotion can occur in curved spaces without the application of external forces, just as a falling cat can reorient itself. Here, we report the first experimental realization of this effect via a T-shaped robotic model with servos driving 34 gm masses via pinion-gears along three curved rack-type arms (lengths 59.5cm, 25cm, total mass 33 gm). The robot is confined to a spherical surface (radius 59.5 cm) via a carbon rod that rotates about a central pivot to allow low-friction swimming. The effect can be described via a gauge potential in which translations accumulate as a geometric (Berry) phase that depends on the path through shape space. This long-neglected effect is significant for locomotors that cover areas comparable to the inverse Gaussian curvature of the underlying surface--from macroscale robots to biological cells rearranging during embryogenesis.