EDIT
Lets assume my transformation does the following mapping:
\begin{align*}
x = -y \\
y = -x \\
z = -z
\end{align*}
Which produces this transformation matrix $R_t$:
$$
\begin{pmatrix}
0 & -1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & -1
\end{pmatrix}
$$
A rotation around the $x$-axis, $R_{x}$ in one coordinate system should equal a rotation around the negative $y$-axis, $R_{-y}$, in the other coordinate system. Why is this relation not satisfied?
$I = R_x^{-1}*R_t*R_{-y}$
I'm a bit confused, so bear with me if I don't make total sense. I have two sets of rotations, $R_{1,k}$ and $R_{2,k}$, $k =1,\dots,N$ expressed in different coordinate systems. I want to find the transformation, $R_t$ between these coordinate systems, such that
$I = R_{2,k}^{-1} * R_t * R_{1,k}$
If $R_{1,k}$ represents the same rotation as $R_{2,k}$. I'm doing an optimization approach, using the above function as a cost function. I'm not getting the results I expecting. I know for a fact that $R_1$ and $R_2$ represent the same rotations and that the transformation between the coordinate systems is constant. Even then, I receive different $R_t$ depending on how $R_1$ and $R_2$ looks.
What is it Im doing wrong? Is it even possible to find such a transformation?
Help is greatly appreciated!
Best Answer
Solving the equation $I = R_{2,k}^{-1} * R_t * R_{1,k}$ for $R_t$, you would conclude that $R_{2,k}*R_{1,k}^{-1} = R_t$.