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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 $\Sigma_{^A\omega} \in \mathbb R^{3\times3}$. 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 $\Sigma_{\theta} \in \mathbb R^{3\times3}$.

Given that $^{N}\omega= \;^{N} \mathbf{R}_{A} \cdot \,^{A} \mathbf \omega \;$, what is $\Sigma_{^N\omega}$ ?

Thank you.

[EDIT]: If the rotation is without covariance (fully known), then $\Sigma_{^N\omega} = \, ^N \mathbf R _{A} \cdot \Sigma_{^A\omega} \cdot ^N \mathbf R _{A}^\intercal $

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Turns out, it is not as complicated as it seems.

On a general basis, given a function $ f: \mathbb{R}^m\to \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.

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