Second-order-optimal filters on Lie groups

We provide an explicit formula for the second-order-optimal nonlinear filter for state estimation of systems on general Lie groups with disturbed measurements of inputs and outputs. Optimality is with respect to a deterministic cost measuring the cumulative energy in the unknown system disturbances (minimum-energy filtering). We show that the resulting filter will depend on the choice of affine connection, thus encoding the nonlinear geometry of the state space. For the case of attitude estimation, where we are given a second order (dynamic) mechanical system on the tangent bundle of the special orthogonal group SO(3), and where we choose the symmetric Cartan-Schouten (0)-connection, the resulting filter has the familiar form of a gradient observer combined with a matrix Riccati differential equation that updates the filter gain.