Extended Data Fig. 7: Energetic considerations for intermittent uphill rolling.

Energy landscapes illustrating design considerations for traversal of uphill excursions (as in the path in the main-text Fig. 4n, o). The gravitational acceleration vector \({\bf{g}}\) (vertical arrow) points down. The trajectoid’s total energy (blue surface) decreases with descent due to losses such as friction. The potential energy surface (green) is defined by trajectoid’s shape and the angle of the slope. Parts of the potential surface that are below the total energy surface are accessible to the rolling trajectoid. Proper balance between losses, shape, and slope are shown in a: the total energy suffices to overcome the potential barrier along the target path (uphill excursion, short arrows), yet is insufficient to escape walls of the potential trench that follow the target path. b, When losses are too high, the net energy is insufficient for an uphill excursion. When losses are too low, as in c, net energy decreases more slowly than the potential energy and becomes sufficient to escape the potential trench as indicated by the dashed arrow. Note that strictly speaking, the net energy is not a function of only the 2D location on the plane and, furthermore, the potential energy depends on the trajectoid’s 3D orientation, which is not an unambiguous function of the 2D planar location even for slipless rolling. Still, the concept of potential landscape is useful as a first approximation.