  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.
