A topological obstruction to global asymptotic stabilization of rotational motion and the unwinding phenomenon

We show that a continuous dynamical system on a state space that has the structure of a vector bundle on a compact manifold possesses no globally asymptotically stable equilibrium. This result is directly applicable to mechanical systems having rotational degrees of freedom. In particular, the result applies to the attitude motion of a rigid body. In light of this result, we explain how attitude stabilizing controllers appearing in the literature lead to unwinding instead of global asymptotic stability.