A closed-form solution of $\textbf{R}\textbf{R}_1=\textbf{R}_2\textbf{R}$ w.r.t $\textbf{R}$
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Is there a closed-form solution of $\textbf{R}\textbf{R}_1=\textbf{R}_2\textbf{R}$ with respect to $\textbf{R}\in SO(3)$ given $\textbf{R}_1$, $\textbf{R}_2 \in SO(3)$?
After long hours of search, I realized that it is not a simple problem. There are many related papers exists published back in the 1980s 1, 2. The short summary is as follows
\[^L\textbf{R} \textbf{R}=\textbf{R} ^C\textbf{R}\]$$^L\textbf{R}_{i}=\boldsymbol{e}^{[^L\textbf{r}_i]_\times}, ^C\textbf{R}_i=\boldsymbol{e}^{[^C\textbf{r}_i]_\times}$$
\[\textbf{M} = \sum_{i=1}^{I}{^L\textbf{r}_i {^C\textbf{r}_i^\top}}\] \[\textbf{R} = (\textbf{M}^\top\textbf{M})^{-1/2}\textbf{M}^\top\]Single pair of rotation is not enough to decide the unique solution.
If scale translation should be considered as well, Eq 8.27 of this book has the answer.
Here is matlab implementation of this problem.here