Inverse optimal stabilization of an underactuated spacecraft in a zero angular momentum mode

The problems of optimal stabilization of an underactuated spacecraft using two wheels in a zero angular momentum mode are investigated in this paper. Considering this kind of spacecraft cannot be stabilized by time-invariant smooth control laws, a nonlinear discontinuous control law is designed based on the Lyapunov direct method and the backstepping technique. Then an inverse optimal control law is presented which circumvents the task of solving the Hamilton-Jacobi equation and minimizes a meaningful cost function using control Lyapunov function (CLF). A consequence of the optimality is that the control law has a sector margin which guarantees robustness with respect to input uncertainties. The simulation results show the asymptotical stability and the optimality of the proposed control law. In addition, the sector margin guarantees robustness when the moments of inertia are of uncertainty.