Rotating angles

Recently I saw a video of the T-bar rotating in space. Trying to work out how that happens, I needed to refresh the stuff about Euler / Tait-Bryan angles for corotating reference frame. But the amazing bit was that you can rotate angles.

So angular velocity in 3-d, ω, is also related to the derivatives of Euler angles, rotated using rotation matrices:

For 3-1-3 system of Euler angles it becomes:

ω = r3 * r2 * r1 * [0; 0; ψ'] + r3 * r2 * [θ'; 0; 0] + r3 * [0; 0; φ'] -- that is, matrices multiplied by column vectors

where r1, r2, r3 are rotation matrices to revert the rotation by ψ around z first, then θ around x', then φ around z" - translate the derivatives of the corotation basis into the "absolute" reference frame.

There's javascript somewhere for this...