Attitude tracking of underactuated spacecraft based on transverse function

The attitude tracking of an underactuated spacecraft with two independent actuators of the momentum exchange type is considered, and an arbitrary attitude trajectory tracking control law is proposed based on the transverse function. The kinematic equations of the rigid spacecraft attitude are established using the three-dimensional special orthogonal group $\mathrm{S}\mathrm{O}(\mathrm{3})$. Based on the momentum conservation condition, the kinematic equations are unified with the kinematics as a reduced-order system with the output angular momentum of the actuator as the control input. The transverse condition is used to construct the transverse function based on the Lie algebraic rank condition, which is essentially an embedded submanifold that can approach the equilibrium point arbitrarily. The adjoint system of the reduced-order tracking control system is established based on the transverse function and the attitude error. For the accompanying system, a smooth static feedback control law is proposed, and the ultimately bounded stability of the closed-loop system for arbitrary trajectories is proved by using the Morse function. Furthermore, for the feasible trajectory, the parameter adjustment law of the transverse function is designed based on the zero dynamic system, so that the transverse function converges to the equilibrium point of the tracking system at zero dynamic, and the closed-loop system is proved to have exponential stability. Finally, the effectiveness of the proposed controller is verified by numerical simulation.