Equivariant Filters are Equivariant

TL;DR

This paper demonstrates that Equivariant Filters leverage system symmetries to provide intrinsic, coordinate-independent observer performance, validated through simulations in robot localization.

Contribution

It proves that the Equivariant Filter's performance is invariant to coordinate choices and origins, emphasizing its intrinsic nature tied to system symmetries.

Findings

EqF error dynamics are invariant to input transformations

Different coordinate choices yield identical performance

Origin selection can improve practical performance by reducing numerical errors

Abstract

Observers for systems with Lie group symmetries are an active area of research that is seeing significant impact in a number of practical domains, including aerospace, robotics, and mechatronics. This paper builds on the theory of the recently proposed Equivariant Filter (EqF), which is a general observer design for systems on homogeneous spaces that takes advantage of symmetries to yield significant performance advantages. It is shown that the EqF error dynamics are invariant to transformation of the input signal and equivariant as a parametrised vector field. The main theorem shows that two EqF's with different choices of local coordinates and origins and with equivalent noise modelling yield identical performance. In other words, the EqF is intrinsic to the system equations and symmetry. This is verified in a simulation of a 2D robot localisation problem, which also shows how the ability to choose an origin for the EqF can yield practical performance advantages by mitigating floating point precision errors.