  The kinematics of many mechanical systems encountered in robotics and other
fields, such as single-bearing attitude estimation and SLAM, are naturally
posed on homogeneous spaces: That is, their state lies in a smooth manifold
equipped with a transitive Lie-group symmetry. This paper shows that any system
posed in a homogeneous space can be extended to a larger system that is
equivariant under a symmetry action. The equivariant structure of the system is
exploited to propose a novel new filter, the Equivariant Filter (EqF), based on
linearisation of global error dynamics derived from the symmetry action. The
EqF is applied to an example of estimating the positions of stationary
landmarks relative to a moving monocular camera that is intractable for
previously proposed symmetry based filter design methodologies.
