  Equivariance is a common and natural property of many nonlinear control
systems, especially those associated with models of mechatronic and navigation
systems. Such systems admit a symmetry, associated with the equivariance, that
provides structure enabling the design of robust and high performance
observers. A key insight is to pose the observer state to lie in the symmetry
group rather than on the system state space. This allows one to define a
globally defined intrinsic equivariant error but poses a challenge in defining
internal dynamics for the observer. By choosing an equivariant lift of the
system dynamics for the observer internal model we show that the error dynamics
have a particularly nice form. Applying the methodology of Extended Kalman
Filtering (EKF) to the equivariant error state yields the Equivariant Filter
(EqF). The geometry of the state-space manifold appears naturally as a
curvature modification to the classical EKF Riccati equation. The equivariant
filter exploits the symmetry and respects the geometry of an equivariant system
model and yields high performance robust filters for a wide range of
mechatronic and navigation systems.
