Doubt about axis\angle representation

Question

I am trying to understand the axis/angle representation.

So far, I have understood that the main idea is that every rotation can be represented as a rotation around an arbitrary axis of an angle $\theta $.

So, for example, if I have a rotation matrix $R$ I can find an axis $r$ and an angle $\theta$.

what I have not clear is the following:

Suppose I have a rotation matrix which represents the orientation of a frame, $R ^{i}_0$ for where $i$ is used to express the fact that this is an initial frame.

Now, suppose I want to change reference frame, and so I can obtain a final reference frame, for example applying Roll Pitch Yaw sequence, and so I obtain $R_{f}^{i}$, where $f$ stands for final frame. So I end up with the orientation of a final frame with respect to the initial frame.

What I don't understand is: In which frame is the unit vector around which the rotation happens expressed?

Answer 1

Depending on which matrix you convert to an axis angle representation.

If you do it successively, you end up with 3 axis with 3 angles in 3 different frames ($R_0$, $R_1$ and $R_2$). It is trivial to notice, that in this the axes will coincide with axes of the successive frames.

If you choose to take the result of the multiplication and write it in axis angle form you will end up with 1 axis and 1 angle in the $R_0$ Frame and the axis-angles expresses the effect of all 3 rotations.