Efficient Reachable Sets on Lie Groups Using Lie Algebra Monotonicity and Tangent Intervals

In this paper, we efficiently compute overapproximated reachable sets for control systems evolving on Lie groups, building off results from monotone systems theory and geometric integration theory. We propose to consider intervals living in the Lie algebra, which through the exponential map, describe real sets on the Lie group. A local equivalence between the original system and a system evolving on the Lie algebra allows existing interval reachability techniques to apply in the tangent space. Using interval bounds of the Baker-Campbell-Hausdorff formula, a Runge-Kutta-Munthe-Kaas reachability algorithm is proposed, providing reachable set estimates for arbitrary time horizons at little computational cost. The algorithm is demonstrated on through consensus on a torus and attitude control on $SO(3)$.