Equivariant Filter Design for Kinematic Systems on Lie Groups

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 robotics applications such as inertial navigation or simultaneous localization and mapping (SLAM). 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. The EqF specializes to the Invariant Extended Kalman Filter (IEKF) in the case where the observed system is group affine or invariant.