A geometric perspective of state estimation using Kalman filters

TL;DR

This paper introduces a unified geometric framework for Kalman filters using affine connections, bridging Lie group and Riemannian approaches, and demonstrates its advantages in robotics and motion tracking applications.

Contribution

It develops a novel affine connection-based description of Kalman filters that generalizes existing geometric methods and enhances flexibility in state space modeling.

Findings

Unified geometric framework for Kalman filters.

Successful application to robotics and motion tracking.

Enhanced flexibility in state space design.

Abstract

Geometry of the state space is known to play a crucial role in many applications of Kalman filters, especially robotics and motion tracking. The Lie group-centric approach is currently very common, although a Riemannian approach has also been developed. In this work we explore the relationship between these two approaches and develop a novel description of Kalman filters based on affine connections that generalizes both commonly encountered descriptions. We illustrate the results on two test problems involving the special Euclidean group and the tangent bundle of a sphere in which the state is tracked by geometric variants of the extended Kalman filter and the unscented Kalman filter. The examples use a newly developed library GeometricKalman.jl. The new approach provides a greater freedom in selecting the structure of the state space for state estimation and can be easily integrated with standard techniques such as parameter estimation or covariance matrix estimation.