  The kinematics of many systems encountered in robotics, mechatronics, and
avionics are naturally posed on homogeneous spaces, that is, their state lies
in a smooth manifold equipped with a transitive Lie group symmetry. This paper
proposes a novel filter, the Equivariant Filter (EqF), by posing the observer
state on the symmetry group, linearising global error dynamics derived from the
equivariance of the system, and applying extended Kalman filter design
principles. We show that equivariance of the system output can be exploited to
reduce linearisation error and improve filter performance. Simulation
experiments of an example application show that the EqF significantly
outperforms the extended Kalman filter and that the reduced linearisation error
leads to a clear improvement in performance.
