# Compute the covariance of $R_2 R_1^T$ where $R_2$ and $R_1$ are rotation matrices with Gaussian uncertainty

I have estimates of two 3x3 rotation matrices $R_1$, $R_2$ expressed in terms of their expected values $R_{1\mu}$ and $R_{2\mu}$ and covariances $\Sigma_1$, $\Sigma_2$. The latter are expressed in the Lie algebra for SO(3). That is:

$$R_1 = R_{1\mu}\exp(\omega_1) ~~~~~~~ \omega_1 ~\tilde~ \ N(0,\Sigma_1)$$

$$R_2 = R_{2\mu}\exp(\omega_2) ~~~~~~~ \omega_2 ~\tilde~ \ N(0,\Sigma_2)$$

where $\omega_1$ and $\omega_2$ are independent random variables, exp is the exponential map from so(3) to SO(3), and $\Sigma_1$, $\Sigma_2$ are 3x3 covariance matrices.

I want to compute an expectation and covariance for $R_{\Delta} = R_2 R_1^T$, expressed again as an expected rotation matrix and 3x3 covariance in the Lie algebra for SO(3). I do not expect that this quantity is actually Gaussian-distributed so I am happy with an approximation based on an appropriate linearization.

Note that I am not guaranteed that $R_{\Delta}$ is close to the identity.

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Did you try a Monte-Carlo simulation, i.e. generate $\omega$, compute $R_\Delta(\omega)$ and then average out over many samples to get a feel what you are getting? – Sasha Feb 12 '13 at 1:33
As the question is currently stated, the expectation is $R_{2\mu}R_{1\mu}^T$ and the covariance is zero, since $R_\Delta$ has the constant value $R_{2\mu}R_{1\mu}^T$ independent of $\omega$. Did you intend the two instances of $\omega$ to be two different, independent random variables? – joriki Feb 12 '13 at 1:56
@joriki - yes, my bad, fixed now – Alex Flint Feb 12 '13 at 20:00
@Sasha - yes, thanks. I am convinced that the problem set up is reasonable but I'm still keen to find an analytic expression. – Alex Flint Feb 12 '13 at 20:03
@Alex: The information that $\omega_1$ and $\omega_2$ are independent is still missing in the question. As it's currently stated there's not enough information to determine the expectation and covariance of $R_\Delta$. – joriki Feb 12 '13 at 21:37