This is my first post here, so if I unknowingly vioated any rules, mods are welcome to edit my post accordingly.
Ok, so my problem is that following Craig's conventions, I can't seem to find the expected homogeneous transform after a series of transformations.
I have included the image for clarity.
We are given the initial frame {0} as usual and then:
-$\{A\}$ is the frame under rotating {0} $90^\circ$ around $z$ and translating, with $OA = \begin{bmatrix}
1
\\ 1
\\ 1
\end{bmatrix}$
-$\{B\}$ is obtained after translating $A$ by $AB = \begin{bmatrix}
-2
\\ -2
\\ 0
\end{bmatrix}$
What I found is:
$$ {}_A^OT = \left[ {\begin{array}{*{20}{c}}
0&-1&0&1\\
1&0&0&1\\
0&0&1&{1}\\
0&0&0&1
\end{array}} \right], \;\;{}_B^AT=\left[ {\begin{array}{*{20}{c}}
1&0&0&-2\\
0&1&0&-2\\
0&0&1&{0}\\
0&0&0&1
\end{array}} \right],\\ {}_B^OT = {}_A^OT {}_B^AT=\left[ {\begin{array}{*{20}{c}}
0&-1&0&3\\
1&0&0&-1\\
0&0&1&{1}\\
0&0&0&1
\end{array}} \right] $$
This is wrong, since the last column should obviously have the coordinates $-1,-1,1$, the origin of B
What am I missing?