  Stochastic inference on Lie groups plays a key role in state estimation
problems such as; inertial navigation, visual inertial odometry, pose
estimation in virtual reality, etc. A key problem is fusing independent
concentrated Gaussian distributions defined at different reference points on
the group. In this paper we approximate distributions at different points in
the group in a single set of exponential coordinates and then use classical
Gaussian fusion to obtain the fused posteriori in those coordinates. We
consider several approximations including the exact Jacobian of the change of
coordinate map, first and second order Taylor's expansions of the Jacobian, and
parallel transport with and without curvature correction associated with the
underlying geometry of the Lie group. Preliminary results on SO(3) demonstrate
that a novel approximation using parallel transport with curvature correction
achieves similar accuracy to the state-of-the-art optimisation based algorithms
at a fraction of the computational cost.
