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

Abstract

In this paper, we efficiently compute overapproximating reachable sets for control systems evolving on Lie groups, building off results from monotone systems theory and geometric integration theory. We consider intervals in the tangent space, which describe real sets on the Lie group through the exponential map. 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, these reachable set estimates are extended to arbitrary time horizons in an efficient Runge-Kutta-Munthe-Kaas integration algorithm. The algorithm is demonstrated through consensus on a torus and attitude control on $SO(3)$.