Here, in this tutorial: http://ethaneade.com/lie.pdf, the author gives the composition of uncertain rotations for Gaussians in $SO(3)$ (Eqn. 47, Page 7). The author doesn't give the detailed derivation. However, I have some difficulty to derive this result. Here is what I'd tried,

Assume $\bf{\epsilon}_1$ and $\bf{\epsilon}_0$ are samples drawn from $\left({\bf{R}_1},\bf{\Sigma}_1\right)$ and $\left({\bf{R}_0},\bf{\Sigma}_0\right)$, respectively. Then the composition rotation by first transforming by sample_0 and then by sample_1 is given by

\begin{align} &\exp{\left({\bf{\epsilon}_1}_{\times}\right)}\bf{R}_1\exp{\left({\bf{\epsilon}_0}_{\times}\right)}\bf{R}_0 \\ =& \exp{\left({\bf{\epsilon}_1}_{\times}\right)}\exp{\left(\left({\rm{Adj_{\bf{R}_1}}\bf{\epsilon}_0}\right)_{\times}\right)}\bf{R}_1\bf{R}_0 \\ =& \exp{\left({\bf{\epsilon}_1}_{\times}\right)}\exp{\left(\left({\bf{R}_1\bf{\epsilon}_0}\right)_{\times}\right)}\bf{R}_1\bf{R}_0 \end{align}

If the the first two exponential items could be combined, then this looks pretty close to the final result. But I don't think I could combine those two items, since the matrices inside the exponential functions seem are not commutative. Am I wrong or am I missing something? I'd appreciate any help in getting me through this. Thank you very much.


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