Why doesn't converting a difference quaternion to euler angles give angular velocity?

Question

According to https://stackoverflow.com/questions/22157435/difference-between-the-two-quaternions

I can get the difference between two quaternions as

diff * q1 = q2  --->  diff = q2 * inverse(q1)

where:  inverse(q1) = conjugate(q1) / abs(q1)

and:  conjugate( quaternion(re, i, j, k) ) = quaternion(re, -i, -j, -k)

Intuitively, I can understand this diff as an "angular velocity" applied to q1 that brings it to q2 in unit time.

If I understood this section of Wikipedia correctly (which I probably don't)

Euler proved that the projections of the angular velocity pseudovector on each of these three axes is the derivative of its associated angle (which is equivalent to decomposing the instantaneous rotation into three instantaneous Euler rotations)

To actually get the angular velocity in the usual sense, I could just convert diff into euler angles.

However, this gives me different results from other sources which calculate the angular velocity more rigorously.

I would assume the more rigorous calculation is correct. My question is, what went wrong in my understanding?

Answer 1

The mistake is, as I correctly guessed, my misinterpretation of

Euler proved that the projections of the angular velocity pseudovector on each of these three axes is the derivative of its associated angle (which is equivalent to decomposing the instantaneous rotation into three instantaneous Euler rotations)

By casting diff as a set of euler angles, what I get is not an angular velocity, but the euler angle rate of change. This is different from angular velocity in that angular velocity is usually expressed as the axis of rotation times speed of rotation (i.e. more akin to axis angle form) and expressed in the inertial frame (euler angles are not expressed in inertial frame).

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Additional resources

• How to get angular velocity from difference orientation quaternion and time?