Second-order-optimal minimum-energy filters on Lie groups

Systems on Lie groups naturally appear as models for physical systems with full symmetry. We consider the state estimation problem for such systems where both input and output measurements are corrupted by unknown disturbances. We provide an explicit formula for the second-order-optimal nonlinear filter on a general Lie group where optimality is with respect to a deterministic cost measuring the cumulative energy in the unknown system disturbances (minimum-energy filtering). The resulting filter depends on the choice of affine connection which encodes the nonlinear geometry of the state space. As an example, we look at attitude estimation, where we are given a second order mechanical system on the tangent bundle of the special orthogonal group SO(3), namely the rigid body kinematics together with the Euler equation. When we choose the symmetric Cartan-Schouten (0)-connection, the resulting filter has the familiar form of a gradient observer combined with a perturbed matrix Riccati differential equation that updates the filter gain. This example demonstrates how to construct a matrix representation of the abstract general filter formula.