Control on the Sphere and Reduced Attitude Stabilization

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Francesco Bullo, Richard M. Murray and Augusto Sarti
CDS Technical Report 95-005

This paper focuses on a new geometric approach to (fully actuated) control systems on the sphere. Our control laws exploit the basic and intuitive notions of geodesic direction and of 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 then applies 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 in fact defined on the sphere. For this reduced stabilization problem our approach allows us not only to deal optimally with the inevitable singularity, but also to achieve simplicity, versatility and (coordinate independent) adaptive capabilities.