I have two quaternions, $q_1$ and $q_2$. I want to rotate the axis of rotation of $q_1$ by $q_2$. Is there any way of doing this directly, without extracting the axis of rotation from $q_1$, rotating it and re-inserting it again?
Please note that I am not trying to perform a combined rotation by quaternion multiplication ($q' = q_2q_1$). Instead I want to rotate the axis of rotation of $q_1$.
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If $q_1$ can be written as a real $c$ and a vector $a$ (so that $q_1 = c + a$), then using rschwieb's method, you get
$$q_1' = c + q_2 a q_2^{-1} = q_2 (c + a) q_2^{-1} = q_2 q_1 q_2^{-1}$$
where only normalization of a quaternion has been used to manipulate the expression. So what you end up doing is back-rotating the whole system by $q_2^{-1}$, rotating by $q_1$ as originally written, and then rotating again by $q_2$.
The whole thing can be done with quaternion arithmetic.
The axis of $q_1$ is $a=\frac{q_1-\overline{q_1}}{2}$, and by computing $q_2aq_2^{-1}$ you will be rotating the vector $a$ into a new direction.