  It is known that invariance and equivariance properties for systems on Lie
groups can be exploited in the design of high performance and robust observers
and filters for real-world robotic systems. This paper proposes an analysis
framework that allows any kinematic system on a Lie group to be embedded in a
natural manner into an equivariant kinematic system. This framework allows us
to characterise the properties of, and relationships between, invariant
systems, group affine systems, and equivariant systems. We propose a new filter
design, the Equivariant Filter (EqF), that exploits the equivariance properties
of the system embedding and can be applied to any kinematic system on a Lie
group.
