This animation is from a simulation of a uniform density torus rolling on a smooth flat surface without slip. The model parameters are: g = 9.81 m/s^2 r1 = 0.594 m r2 = 0.055 m
Initial conditions can be specified with eight quantities: x0 = [yaw, lean, spin, x, y, lean rate, yaw rate (about axis in plane of torus, perpendicular to direction of heading), spin rate]. All angles are in radians, distances in meters.
An interesting thing about this system is that the motion only depends upon gravity, and the two radii of the torus -- mass and inertia terms drop out.
The simulation was performed using the GNU Scientific Library, rendered with OpenGL, frames captured to png using libpng, and encoded to an Ogg Vorbis file using ffmpeg. The original equations of motion were derived using PyDy and Sympy.