Control on the Sphere and Reduced Attitude Stabilization

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Francesco Bullo, Richard M. Murray, Augusto Sarti
1995 IFAC Symposium on Nonlinear Control Systems Design

This paper focuses on a new geometric approach to fully actuated control systems on the Riemannian manifold S^2. 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. Even for the stability analysis, the appropriate Lyapunov function relies upon the notion of distance and its properties. 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 attitude 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.