Geometric Reachability for Attitude Control Systems via Contraction Theory

TL;DR

This paper introduces a geometric approach using contraction theory and Riemannian metrics on manifolds to analyze the reachability of attitude control systems, enabling efficient over-approximation of their reachable sets.

Contribution

It develops a novel framework combining contraction theory and parametrized Riemannian metrics on manifolds for reachability analysis of attitude systems, with an efficient metric search via semidefinite programming.

Findings

Effective upper bounds on trajectory distances established

Simulation-based algorithms for over-approximating reachable sets

Numerical experiments confirm the approach's validity

Abstract

In this paper, we present a geometric framework for the reachability analysis of attitude control systems. We model the attitude dynamics on the product manifold $\mathrm{SO}(3) \times \mathbb{R}^3$ and introduce a novel parametrized family of Riemannian metrics on this space. Using contraction theory on manifolds, we establish reliable upper bounds on the Riemannian distance between nearby trajectories of the attitude control systems. By combining these trajectory bounds with numerical simulations, we provide a simulation-based algorithm to over-approximate the reachable sets of attitude systems. We show that the search for optimal metrics for distance bounds can be efficiently performed using semidefinite programming. Additionally, we introduce a practical and effective representation of these over-approximations on manifolds, enabling their integration with existing Euclidean tools and software. Numerical experiments validate the effectiveness of the proposed approach.