Equivariant Filter Design for Discrete-time Systems

The kinematics of many nonlinear control systems, especially in the robotics field, admit a transitive Lie-group symmetry, which is useful in high performance observer design. The recently proposed equivariant filter (EqF) exploits equivariance to generate high performance filters for a wide range of real-world systems. However, existing work on the equivariant filter, and equivariance of control systems in general, is based on a continuous-time formulation. In this paper, we first present the equivariant structure of a discrete-time system. We then use this to propose a discrete-time version of the equivariant filter. A novelty of the approach is that the geometry of the symmetry group naturally appears as parallel transport in the reset step of the filter. Preliminary results for linear second order kinematics with separate bearing and range measurements indicate that the discrete EqF significantly outperforms both a discretized version of the continuous EqF and a classical discrete EKF.