Invariant Kalman Filter for Relative Dynamics

TL;DR

This paper introduces a geometric invariant filtering framework for relative dynamics on Lie groups, ensuring autonomous error evolution and consistent covariance propagation, with theoretical foundations and numerical demonstrations.

Contribution

It develops a new geometric framework for invariant filtering of relative dynamics on Lie groups, linking relative trajectory independence to error autonomy and enabling structure-preserving filters.

Findings

Invariant filters preserve Lie group structure.

Error dynamics are exactly linearized.

Numerical examples demonstrate effectiveness.

Abstract

This paper develops a geometric framework for invariant filtering of relative dynamics on Lie groups. We first revisit the notion of state trajectory independence, under which the estimation error evolves autonomously, and derive new equivalent conditions by decomposing the system vector field into left-invariant, intrinsic, and right-invariant components. Building on this result, we introduce the concept of relative trajectory independence to characterize when the relative motion between two dynamical systems is autonomous. A key theoretical finding is that relative trajectory independence automatically ensures state trajectory independence for the corresponding estimation error. This connection provides the foundation for constructing invariant filters that preserve the Lie group structure, maintain exact linearization of the error dynamics, and enable consistent covariance propagation. These are illustrated with numerical examples.