Geometric Fixed-Time Sliding Mode Control for Constrained Attitude Tracking on $\mathrm{SO}(3)$

Abstract

This paper studies constrained spacecraft attitude tracking on the Riemannian configuration manifold $\mathrm{SO}(3)$ in the presence of multiple attitude pointing constraints and matched external disturbances. To address this, an attitude potential function is proposed intrinsically on $\mathrm{SO}(3)$, and its key properties are established using intrinsic geometric analysis. Under mild conditions, the potential function is shown to admit a unique nondegenerate minimum at the desired attitude over the admissible subset of $\mathrm{SO}(3)$, defined by excluding the forbidden attitude regions as well as a measure-zero set, thereby ensuring a well-posed constrained attitude tracking problem. A Riemannian Hessian analysis shows that the Hessian of the potential function is locally uniform positive definite in an open neighborhood of the desired attitude, thereby establishing local strong convexity. A nonsingular fixed-time geometric sliding manifold is proposed using the Riemannian gradient of the potential function, leading to a geometric fixed-time sliding-mode-based constrained attitude control law. It is shown that, for every initial attitude in the admissible subset, the closed-loop state trajectory evolves on $\mathrm{SO}(3)\times\mathbb{R}^3$, with the attitude remaining in the admissible subset throughout the maneuver, while the state converges to a sufficiently small compact neighborhood of the desired equilibrium in a prescribed fixed time. Numerical simulations validate the proposed control approach and illustrate the theoretical results.