Sum-of-Squares Stability Verification on Manifolds with Applications in Spacecraft Attitude Control

TL;DR

This paper introduces a sum-of-squares based framework for verifying almost global stability of spacecraft attitude control systems on non-contractible manifolds, addressing the complexity of stability analysis with attitude parametrizations.

Contribution

It develops a novel stability verification method using LaSalle's invariance principle and sum-of-squares programming tailored for manifold-constrained attitude systems.

Findings

Successfully verified stability in low Earth orbit attitude acquisition scenarios.

Applied the framework to gravity gradient torque-based three-axis attitude control.

Simplified Lyapunov function search for systems on non-contractible manifolds.

Abstract

In the context of spacecraft attitude control, parametrizations such as direction vectors or quaternions are often used to avoid singularities in the attitude representation. This, however, complicates the stability analysis of the system since, given the additional unit constraints, the resulting dynamics evolve on non-contractible manifolds. In this paper, we present a framework to verify almost global asymptotic stability of such systems using LaSalle's invariance principle and sum-of-squares programming, simplifying the search for Lyapunov functions. The framework is then applied to two examples: two-axis attitude acquisition utilizing aerodynamics in very low Earth orbits, and three-axis attitude acquisition for a satellite subject to gravity gradient torques in a circular orbit.