Continuous-Discrete Filtering and Smoothing on Submanifolds of Euclidean Space

In this paper the issue of filtering and smoothing in continuous discrete time is studied when the state variable evolves in some submanifold of Euclidean space, which may not have the usual Lebesgue measure. Formal expressions for prediction and smoothing problems are reviewed, which agree with the classical results except that the formal adjoint of the generator is different in general. These results are used to generalise the projection approach to filtering and smoothing to the case when the state variable evolves in some submanifold that lacks a Lebesgue measure. The approach is used to develop projection filters and smoothers based on the von Mises-Fisher distribution, which are shown to be outperform Gaussian estimators both in terms of estimation accuracy and computational speed in simulation experiments involving the tracking of a gravity vector.