Locomotion without force, and impulse via dissipation: Robotic swimming in curved space via geometric phase

TL;DR

This paper demonstrates that a robotic system on a curved surface can achieve self-propulsion without exchanging momentum with the environment, leveraging geometric phases and environmental curvature effects.

Contribution

It introduces a robophysical model showing propulsion via shape changes on curved surfaces, exploiting geometric phases in curved space without environmental momentum exchange.

Findings

Robotic system achieves 0.1 cm per gait cycle movement.

Shape changes generate gauge potentials causing translation.

Dissipation stores momentum that can be released later.

Abstract

Locomotion by shape changes (spermatozoon swimming, snake slithering, bird flapping) or gas expulsion (rocket firing) is assumed to require environmental interaction, due to conservation of momentum. As first noted in (Wisdom, 2003) and later in (Gu\'eron, 2009) and (Avron et al, 2006), in curved space or spacetime the non-commutativity of translations permits translation without momentum exchange, just as falling cats and lizards can self-deform to reorient in flat space without environmental interaction. Translation in curved space can occur not only in gravitationally induced curved spacetime (where translation is predicted to be on the order of $10^{-23}$ m per gait cycle) but also in the curved surfaces encountered by locomotors in real-world environments. Here we show that a precision robophysical apparatus consisting of motors driven on curved tracks (and thereby confined to a spherical surface without a solid substrate) can self-propel without environmental momentum exchange (impulse) via shape changes that can generate gauge potentials that manifest as translations. Our system produces shape changes comparable to the environment's inverse curvatures and generates from zero momentum forward movement of $10^{-1}$ cm per gait cycle even while resisted by weak gravitational and frictional forces. Dissipation via friction eventually arrests the robot but also imbues it with momentum which can be released upon a cessation of shape changes. This work demonstrates how the interaction between environmental curvature, active driving and geometric phases yields rich, exotic phenomena.