Semi-global exponential convergence nonlinear control of the quadrotor unmanned aerial vehicle based on Lie algebra so(3)

Abstract

This paper addresses the position-attitude tracking control problem of quadrotor unmanned aerial vehicles (QUAVs). Over the past decade, the attitude control of QUAVs has achieved global exponential stability (GES) by leveraging Lie group theory to address the singularities in the attitude error representation. However, the position control of QUAVs has not yet reached GES. Based on the geometric position control law and the Lie-algebra-based attitude control law, this paper proposes an improved robust position-attitude control law. For attitude control, a method for computing attitude error represented in Lie algebra $\mathrm{so}\left(3\right)$ is proposed to address the singularity of the logarithmic map. For position control, a novel hybrid computational scheme of the desired rotation matrix is proposed to address the body flipping problem when the desired yaw direction is parallel to the desired thrust direction. Both attitude and position errors are globally defined on their respective manifolds, ensuring singularity-free tracking performance over the entire configuration space. For stability analysis, some Lyapunov-based techniques are employed to bound the high-order coupling terms between position and attitude. In the presence of unknown disturbances, the position-attitude control system is proven to be semi-globally exponentially stable. The proposed control law is validated through numerical simulations and indoor flight experiments under wind disturbances, in comparison with the classical geometric control method.