Geometric Tracking on $\mathcal{S}^{3}$ Based on Sliding Mode Control

Attitude tracking on the unit sphere of dimension $3$ based on sliding mode is considered in this paper. The tangent bundle of Lagrangian dynamics that describes the rotational motion of a rigid body is first shown to be a Lie group, and then a sliding surface that emerged on it is defined. Next, a sliding-mode controller is designed for attitude tracking that relies on an intrinsic error defined on the Lie group. Almost global asymptotic stability of the closed loop is demonstrated using the Lyapunov analysis. Numerical simulations are included to compare the performance of the sliding mode controller designed on the Lie group with that designed in the embedding Euclidean space.