I also really like arxiv.org/abs/1410.6443 for a discussion on how one defines a centre of mass for these spinning bodies in GR

Thanks to Lisa Drummond for showing me this cool result!

Deviations between the analytical and numerical solutions appear at second order in the spin. If you want a numerical solution, check out github.com/tomkimpson/R...

It turns out that, at linear order in the spin, you can shift the worldline of the spinning body to that of a geodesic. You can then deploy the usual closed-form analytic expressions for geodesics.

The motion of a spinning body in a curved spacetime is described by the Mathisson-Papapetrou-Dixon (MPD) equations. The spin drives the motion away from that of a geodesic. One typically solves for the motion numerically.