Relaxation Dynamics in Oblate Spherical Rolling Robots

Abstract

Spherical robots rolling on flat ground often exhibit a wobbling motion that, at higher speeds, can escalate into end-over-end flipping. This paper proposes a fundamental dynamic cause of this instability: a relaxation effect analogous to the Intermediate Axis Theorem. Rotating bodies with oblate inertial profiles under dissipative loads tend to reorient toward spinning about their major moment of inertia, leading to the observed wobbling in spherical robots. While relaxation dynamics are well-studied in satellites and asteroids, this effect has not been previously applied to rolling systems. We extend these methods to constrained spherical robots, derive the governing dynamics, and conduct experiments with an empty shell on a slope and a reduced pendulum on flat ground and in water to aid in the discussion. Results suggest that translational rolling constraints act as a pseudo-dissipative load to drive the relaxation effect. This work bridges the fields of satellite dynamics theory and ground robotics, providing new insights into the stability of high-speed rolling robots to influence future hardware and control design choices.