Attitude control of a rigid spacecraft with one variable-speed control moment gyro

Nonlinear controllability and attitude stabilization are studied for the underactuated nonholonomic dynamics of a rigid spacecraft with one variable-speed control moment gyro (VSCMG), which supplies only two internal torques. Nonlinear controllability theory is used to show that the dynamics are locally controllable from the equilibrium point and thus can be asymptotically stabilized to the equilibrium point via time-invariant piecewise continuous feedback laws or time-periodic continuous feedback laws. Specifically, when the total angular momentum of the spacecraft-VSCMG system is zero, any orientation can be a controllable equilibrium attitude. In this case, the attitude stabilization problem is addressed by designing a kinematic stabilizing law, which is implemented through a nonlinear proportional and derivative controller, using the generalized dynamic inverse (GDI) method. The steady-state instability inherent in the GDI controller is elegantly avoided by appropriately choosing control gains. In order to obtain the command gimbal rate and wheel acceleration from control torques, a simple steering logic is constructed to accommodate the requirements of attitude stabilization and singularity avoidance of the VSCMG. Illustrative numerical examples verify the efficacy of the proposed control strategy.