The following question is from Kinematic Analysis of Robot Manipulators by Carl D. Crane, III, Joseph Duffy:
"The transformation that relates the A and B coordinate systems is given as That is the rotation of frame B with respect to frame A.
$${}^{A}_{B}T= \left[ \begin{matrix} 0.866025 & 0 & 0.5 & 0.26795 \\ 0 & 1 & 0 & 0 \\ -0.5 & 0 & 0.866025 & 1\\ 0&0&0&1 \\ \end{matrix}\right] $$
Coordinate system B can be obtained from coordinate system A by initially aligning it with A and then rotating coordinate system B about an axis $\mathbf{m}$ by an angle $\gamma$ where the rotation axis passes through a point $\mathbf{p}$. Determine $\mathbf{m}$, $\gamma$ and $\mathbf{p}$"[1]
My first question is about the question itself. As far as I know a final pose of a body in 3D space can be achieved by a rotation about an axis and a translation about that same axis (The screw theory). If a body in 3D space has only experienced rotation and if we know the rotation matrix, then we can use equivalent angle-axis formulation to find the unit vector parallel to the rotation axis and the angle of rotation. In 2D a single rotation suffices to attain a final pose of a planar body, no matter how many translations and rotations that planar body has performed. Checking the transformation matrix ${}^{A}_{B}T$, there is a translational difference between the origins of frames A and B which means some translation occured during the transformation. How is it possible that the frame B can be obtained by a single rotation about an axis as the question suggests?
My second question is how do you solve the question?
[1]Carl D. Crane III, Joseph Duffy (1998) Kinematic Analysis of Robot Manipulators.Cambridge University Press.