  This paper is concerned with the problem of continuous-time nonlinear
filtering for stochastic processes on a connected matrix Lie group. The main
contribution of this paper is to derive the feedback particle filter (FPF)
algorithm for this problem. In its general form, the FPF is shown to provide a
coordinate-free description of the filter that automatically satisfies the
geometric constraints of the manifold. The particle dynamics are encapsulated
in a Stratonovich stochastic differential equation that retains the feedback
structure of the original (Euclidean) FPF. The implementation of the filter
requires a solution of a Poisson equation on the Lie group, and two numerical
algorithms are described for this purpose. As an example, the FPF is applied to
the problem of attitude estimation - a nonlinear filtering problem on the Lie
group SO(3). The formulae of the filter are described using both the rotation
matrix and the quaternion coordinates. Comparisons are also provided between
the FPF and some popular algorithms for attitude estimation, namely the
multiplicative EKF, the unscented quaternion estimator, the left invariant EKF,
and the invariant ensemble Kalman filter. Numerical simulations are presented
to illustrate the comparisons.
