Unscented Kalman Filters for Riemannian State-Space Systems

TL;DR

This paper develops a comprehensive Riemannian extension of Unscented Kalman Filters, filling theoretical gaps and providing a unified, principled framework for systems on Riemannian manifolds, demonstrated through satellite attitude tracking.

Contribution

It introduces new Riemannian UKF concepts, proves key steps, and relates these to existing Euclidean UKFs, enhancing theoretical consistency and generality.

Findings

Proposed Riemannian UKFs are more consistent and principled.

Framework relates Riemannian UKFs to Euclidean UKFs.

Satellite attitude tracking example validates the approach.

Abstract

Unscented Kalman Filters (UKFs) have become popular in the research community. Most UKFs work only with Euclidean systems, but in many scenarios it is advantageous to consider systems with state-variables taking values on Riemannian manifolds. However, we can still find some gaps in the literature's theory of UKFs for Riemannian systems: for instance, the literature has not yet i) developed Riemannian extensions of some fundamental concepts of the UKF theory (e.g., extensions of $\sigma$-representation, Unscented Transformation, Additive UKF, Augmented UKF, additive-noise system), ii) proofs of some steps in their UKFs for Riemannian systems (e.g., proof of sigma points parameterization by vectors, state correction equations, noise statistics inclusion), and iii) relations between their UKFs for Riemannian systems. In this work, we attempt to develop a theory capable of filling these gaps. Among other results, we propose Riemannian extensions of the main concepts in the UKF theory (including closed forms), justify all steps of the proposed UKFs, and provide a framework able to relate UKFs for particular manifolds among themselves and with UKFs for Euclidean spaces. Compared with UKFs for Riemannian manifolds of the literature, the proposed filters are more consistent, formally-principled, and general. An example of satellite attitude tracking illustrates the proposed theory.