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In this technical note, we solve the control problem of rigid bodies with only quaternion measurements for all initial rotations and angular velocities. The proposed solution is based on the theory of cascades using any switching certainty equivalence controller satisfying certain assumptions along with an in the large hybrid observer. The equilibrium point of the proposed observer in closed loop with the rigid body dynamics is proven to be $\kappa$-exponentially stable in the large i.e., we prove that the equilibrium point is stable and that the error states converge exponentially fast towards the origin for all initial rotations and angular velocities. Until now, stability results for quaternion-based observers have typically only been valid for a bounded set of initial conditions. To overcome this issue, our observer design is based on dynamic scaling and switching logic. Furthermore, we show that the origin of the proposed switching certainty equivalence controller in closed loop with the hybrid observer is asymptotically stable in the large for all available initial conditions associated with the quaternion space. Simulation results for the proposed scheme are presented with the particular case of the PD $+$ controller, revealing that all states converge as expected from our theoretical findings. [ABSTRACT FROM AUTHOR]