Modulation of orthogonal body waves enables versatile maneuverability in limbless locomotion

Limbless organisms can create different motions by modulating axial undulations that pass through their bodies. Sidewinding snakes generate horizontal and vertical waves, with a phase offset of $\pi/2$, resulting in posteriorly-propagating alternating regions of static contact with the substrate and elevated motion, resulting in a ``stepping'' motion of body segments. We have discovered that sidewinder rattlesnakes ({it Crotalus cerastes}) are quite maneuverable and possess at least two turning methods: ``differential turning'' and ``reversal turning.'' In differential turning, the amplitude of the horizontal wave changes along the body length, resulting in turns of average $25.6 \pm 12.9$, maximum $86.1^\circ$per cycle. In reversal turning, the vertical wave's phase rapidly changes by $\pi$, resulting in a sudden, large change in movement direction (average $77.8 \pm 27.4$, maximum $160.5^\circ$ per cycle) without body rotation. We applied these control mechanisms to a 16-link snake robot capable of sidewinding on sand. By modulation of horizontal wave amplitude gradient along the body, we replicated differential turning, and by producing a $\pi$ phase shift in the vertical wave, we replicated a reversal turn. More complex wave modulations lead to enhanced robot maneuverability.