Paper detail

Nearly Geodesic Riemannian Cubics in SO(3)

{\em Riemannian cubics} are curves in a manifold $M$ that satisfy a variational condition appropriate for interpolation problems. When $M$ is the rotation group SO(3), Riemannian cubics are track-summands of {\em Riemannian cubic splines}, used for motion planning of rigid bodies. Partial integrability results are known for Riemannian cubics, and the asymptotics of Riemannian cubics in SO(3) are reasonably well understood. The mathematical properties and medium-term behaviour of Riemannian cubics in SO(3) are known to be be extremely rich, but there are numerical methods for calculating Riemannian cubic splines in practice. What is missing is an understanding of the short-term behaviour of Riemannian cubics, and it is this that is important for applications. The present paper fills this gap by deriving approximations to nearly geodesic Riemannian cubics in terms of elementary functions. The high quality of these approximations depends on mathematical results that are specific to Riemannian cubics.