Uncertainty Propagation on Unimodular Matrix Lie Groups

In this article, we propose a general method for uncertainty propagation on unimodular matrix Lie groups that have a surjective exponential map when the initial probability density function is concentrated. We derive the exact formula for the propagation of mean and covariance expressed in the form of expectation 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. For the latter one, a closed-form second-order propagation formula is derived. We apply the general method to the joint attitude and angular momentum uncertainty propagation problem and numerical experiments demonstrate the two approximation methods. These results show that our new methods have high accuracy while being computationally efficient.