Almost Global and Singularity-Free Fixed-Time Sliding Mode Satellite Attitude Control: A Geometric Control Framework

In this article, a robust singularity-free fixed-time satellite attitude control in a geometric control framework is proposed. The satellite attitude is represented by the rotation matrices defined on the special orthogonal matrix group <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text {SO}(3)$</tex-math></inline-formula> to achieve unique, global, and kinematic singularity-free attitude representation. A nonsingular fixed-time sliding manifold for the attitude control system evolving on the Lie group <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text {SO}(3)\times \mathbb {R}^{3}$</tex-math></inline-formula> is proposed. A complete mathematical analysis is carried out to prove and establish that the proposed sliding manifold is a Lie subgroup of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text {SO}(3)\times \mathbb {R}^{3}$</tex-math></inline-formula>. Along this manifold, the reduced-order dynamics is almost globally fixed-time attractive to a residual set containing the stable equilibrium point. Based on the proposed sliding manifold, a nonsingular geometric fixed-time sliding mode control (NGFTSMC) law is proposed for the satellite attitude control in the presence of external disturbance torques. It is proved through five propositions that the NGFTSMC facilitates an attitude maneuver on the phase space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\text {SO}(3)\times \mathbb {R}^{3}$</tex-math></inline-formula>, excluding a set of measure zero. This guarantees almost global, unwinding-free, and singularity-free closed-loop attitude dynamics. Simulation results are presented and compared with a quaternion-based fixed-time sliding mode control to show superiority of the NGFTSMC in terms of the time of convergence, unwinding, and control effort.