Uncertainty Propagation of correlated quaternion and Euclidean states using partially-conditioned Gaussian mixtures

The partially-conditioned Gaussian (PCG) density, a variant of the Gauss-Bingham density, quantifies the uncertainty of a state vector comprised of an attitude quaternion and other Euclidean states on their natural manifold, the unit hypercylinder. The conditioned Gaussian density is first developed by conditioning a Gaussian density on the unit hypersphere, and is an alternate representation of the Bingham density. The PCG density is then developed, which conditions only the quaternion portion of the aforementioned state vector on the unit hypersphere. The PCG density is then extended to the PCG mixture density, which can be used to approximate an arbitrary density on the unit hypercylinder. A method to construct a PCG mixture density approximating the PCG density on the two-dimensional cylinder is then developed. The temporal evolution of the PCG mixture density given system dynamics is then quantified and is compared to a Monte Carlo approach in order to verify its performance.