Say I have a timeseries attitude of some body given in an inertial frame as axis-angles pairs $\bf{a}$$(t)$, $\phi(t)$ or alternatively just the 3D vector $\boldsymbol{\alpha}$$(t)$ where
$\boldsymbol{\alpha} = \bf{a}$$(t)\phi(t)$,
and $\bf{a}$ is always a unit vector. From this timeseries I extract the derivative $\dot{\boldsymbol{\alpha}}$, which is not equal to the body angular velocity ${\boldsymbol{\omega}}$. How do I then get the angular velocity as a function of $\dot{\boldsymbol{\alpha}}$?
The only mention of this I've seen is in this post: https://physics.stackexchange.com/questions/287830/is-there-a-formula-for-the-rotation-vector-in-terms-of-the-angular-velocity-vect?newreg=bd8f638066594d58b0070c6cd99e1e68, however it's not clear how this formula is arrived at, and I can't find sources of it elsewhere.