The Bingham-Gauss Mixture Filter for Pose Estimation

The Bingham-Gauss density quantifies the uncertainty of a state vector comprised of an attitude quaternion and other Euclidean states on its natural manifold, the unit hypercylinder, without the need to assume that the attitude error is small. The Bingham-Gauss density is developed, which facilitates a tractable implementation of an approximate Bayesian filter in which the true state densities, as quantified by the Chapman-Kolmogorov equation and Bayes\u27 rule, are approximated by a Bingham-Gauss density through moment matching, which is the Kullback-Leibler optimal approximation. The Bingham-Gauss density is then used to develop the Bingham-Gauss mixture (BGM) density. Methods to approximate a Bingham-Gauss density by a BGM density are presented. The BGM filter is then developed, which is an approximate Bayesian filter quantifying the approximate temporal and measurement evolution of a BGM density. The BGM filter is then applied to estimate the relative dynamic pose of an inspector spacecraft with respect to a target spacecraft given nonlinear measurements in order to show its efficacy