  This paper addresses uncertainty propagation on unimodular matrix Lie groups
that have a surjective exponential map. We derive the exact formula for the
propagation of mean and covariance in a continuous-time setting from the
governing Fokker-Planck equation. Two approximate propagation methods are
discussed based on the exact formula. One uses numerical quadrature and another
utilizes the expansion of moments. A closed-form second-order propagation
formula is derived. We apply the general theory to the joint attitude and
angular momentum uncertainty propagation problem and numerical experiments
demonstrate two approximation methods. These results show that our new methods
have high accuracy while being computationally efficient.
