Invariant Kalman Filter for Relative Dynamics

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.