Casually reading “orbital mechanics for engineering students” on rigid body attitude dynamics, i see the following passage:
$$M_{net}=\dot{H}_{rel} + \Omega\times H$$ Keep in mind that, whereas $\Omega$ (the angular velocity of the moving xyz coordinate system) and $\omega$ (the angular velocity of the rigid body itself) are both absolute kinematic quantities, ... If the comoving frame is rigidly attached to the body frame, then ... $\Omega=\omega$.
I can’t fathom any plausible reason why one would have $\Omega\ne\omega$ at all, much less for a satellite. Is there a situation that comes up where it would be useful to describe a rigid body satellite with a rotating coordinate axis that differs from the rotation of the body itself? It seems like the authors are extremely careful to avoid saying that $\Omega=\omega$ universally, which makes me suspect that there may be cases that come up where it is best to keep them separate. What are those situations in the context of a satellite’s attitude dynamics?