Rocking, Rolling, and Hopping: Exploring the Multi-motion Capabilities of Rigid and Soft Ellipsoidal Actuators

The problem of a rigid disk rolling down a ramp is a classic problem given to students in introductory mechanics courses. In contrast, systematic studies on the rolling behavior of an ellipse have only recently emerged. Unlike a rolling disk, here the geometric center remains at a constant height from the floor, the center of a rotating ellipse changes nonlinearly due to its eccentric shape. This eccentricity introduces new modes of motion beyond rolling, including rocking and hopping. Leveraging this multi-motion behavior, we design an ellipsoidal actuator which exhibits both rolling and hopping behaviors in response to changes in the applied angular velocity. Using a simple geometric framework, we successfully capture the motion of the actuator as a force-driven rigid ellipsoid on a non-slip flat surface, and identify the critical angular velocity for the rolling-to-hopping transition. Furthermore, by adding deformability to the actuator, we unlock new functionalities, enabling soft actuators that can climb slopes and work together to collectively ascend stairs.