  An unscented Kalman filter for matrix Lie groups is proposed where the time
propagation of the state is formulated on the Lie algebra. This is done with
the kinematic differential equation of the logarithm, where the inverse of the
right Jacobian is used. The sigma points can then be expressed as logarithms in
vector form, and time propagation of the sigma points and the computation of
the mean and the covariance can be done on the Lie algebra. The resulting
formulation is to a large extent based on logarithms in vector form, and is
therefore closer to the UKF for systems in $\mathbb{R}^n$. This gives an
elegant and well-structured formulation which provides additional insight into
the problem, and which is computationally efficient. The proposed method is in
particular formulated and investigated on the matrix Lie group $SE(3)$. A
discussion on right and left Jacobians is included, and a novel closed form
solution for the inverse of the right Jacobian on $SE(3)$ is derived, which
gives a compact representation involving fewer matrix operations. The proposed
method is validated in simulations.
