Lie-Algebraic Approach to Geometric Nonlinear $\mathcal {H}_\infty$ Inverse Optimal Attitude Tracking on $SO(3)$

Abstract

This letter presents a geometric nonlinear $\mathcal {H}_\infty$ inverse optimal attitude tracking framework formulated on $SO(3)$ using its Lie algebra $\mathfrak {so}(3)$. By adopting exponential coordinates $\bm {\xi }\in \mathfrak {so}(3)$, attitude errors are mapped directly onto the Lie algebra. This preserves a bijective, distortion-free relationship with the geodesic distance within the injectivity radius ($\Vert \bm {\xi }\Vert < \pi$), while eliminating singularities and unwinding issues inherent in Euler angles and quaternions. A sliding-mode-inspired nonlinear $\mathcal {H}_\infty$ controller is developed, with robust stability established using the exponential coordinates and a systematic tuning methodology provided to facilitate practical implementation. Quadrotor simulations demonstrate faster convergence in large-angle maneuvers by mitigating the vanishing error and sluggish response of conventional methods. Experiments on a robotic manipulator validate the tuning rule for the prescribed disturbance attenuation level $\gamma$, showing that the peak tracking error scales as $\mathcal {O}(\gamma ^{2})$ for small gains and $\mathcal {O}(\gamma)$ for large gains. These results demonstrate the robustness and predictability of the proposed framework and the effectiveness of Lie algebra-based attitude control on $SO(3)$.