Covariance of an uncertain vector going through an uncertain rotation

Question

I have a vector $omega$ and it associated covariance, a rotation $mathbf R$ and its associated covariance. What is the covariance of $mathbf R cdot omega$ ?
More rigorously:
I have an estimation of a vector expressed in the $A$ frame: $^A mathbf omega in mathbb R^3$, and its associated covariance $Sigma_{^Aomega} in mathbb R^{3times3}$. I also have an estimate of the rotation between $A$ and $N$ expressed by the vector $mathbf theta in mathfrak{so}(3)$ such that the rotation matrix from $A$ to $N$ is: $^N mathbf R _{A} = text{expm}(theta) in SO(3)$. This estimate of the rotation vector $mathbf theta$ has the associated covariance $Sigma_{theta} in mathbb R^{3times3}$.
Given that $^{N}omega= ;^{N} mathbf{R}_{A} cdot ,^{A} mathbf omega ;$, what is $Sigma_{^Nomega}$ ?
Thank you.
[EDIT]: If the rotation is without covariance (fully known), then $Sigma_{^Nomega} = , ^N mathbf R _{A} cdot Sigma_{^Aomega} cdot ^N mathbf R _{A}^intercal  $

Maltergate · Accepted Answer

Turns out, it is not as complicated as it seems.
On a general basis, given a function $ f: mathbb{R}^mto mathbb{R}^n$ and a vector $mathbf x in mathbb R^m$ and its associated covariance $Sigma_x in mathbb R^{m times m}$, if $mathbf y = f(mathbf x)$ then:
$$Sigma_x simeq left . frac{partial f}{partial mathbf x}right |_{mathbf x} Sigma_x left . frac{partial f}{partial mathbf x}right |_{mathbf x} ^intercal$$
It requires to compute the jacobians, which can be tedious.

Covariance of an uncertain vector going through an uncertain rotation

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