Control on the sphere and reduced attitude stabilization

This paper focuses on a new geometric approach to (fully actuated) control systems on the sphere. The control laws exploit the basic and intuitive notions of geodesic direction and distance between points, and generalize the classical proportional plus derivative feedback (PD) without the need of arbitrary local coordinate charts. The stability analysis relies on an appropriate Lyapunov function, where the notion of distance and its properties are exploited. This methodology is applied to spin-axis stabilization of a spacecraft actuated by only two control torques: discarding the rotation about the unactuated axis, a reduced system is considered whose state is defined on the sphere. For this reduced stabilization problem, the approach allows not only an optimal treatment of the inevitable singularity, but also simplicity, versatility and (coordinate independent) adaptive capabilities.