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Let's have two vectors $\mathbf \omega \in \mathbb R^3$, $\mathbf \theta \in \mathbb R^3$ and their associated covariance, $\Sigma_{\omega} \in \mathbb R^{3\times3}$ and $\Sigma_{\theta} \in \mathbb R^{3\times3}$ respectively. A function $\mathbf{R}(\cdot): \mathbb{R}^3 \mapsto \mathbb{R}^{3\times3}$ defined as :

$$\mathbf{R}(\mathbf{x})=\: \mathbf{I}_{3} \:+ \: \sin \left \| \mathbf{x} \right \| \left [ \frac{\mathbf{x}}{\left \| \mathbf{x} \right \|} \right ]_{\times} \:+ \: (1-\cos \left \| \mathbf{x} \right \|) \left [ \frac{\mathbf{x}}{\left \| \mathbf{x} \right \|} \right ]_{\times}$$

with $\left [\cdot \right ]_{\times} $ the "cross product matrix". (Function R is known as Rodrigues' formulae for 3D rotations).

Given that $\mathbf{\alpha}= \mathbf{R}(\mathbf{\theta})\,*\,\mathbf \omega$, $\mathbf \;\alpha \in \mathbb R^3$, what is the covariance of $\mathbf{\alpha}$, $\:\Sigma_{\mathbf{\alpha}}$?

I'm a bit puzzled on how to handle this problem. Shoud I introduce little perturbations and come back to the very definition of the covariance? In that case, how do I compute $E[\mathbf{X}\mathbf{X}^{\intercal}]$?

Thank you very much for your help.

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  • $\begingroup$ Because$$\alpha:\mathbb{R}^6 \approx\mathbb{R}^3\otimes \mathbb{R}^3 \to \operatorname{Mat}(\mathbb{R}^3,\mathbb{R}^3)\approx\mathbb{R}^9,$$and--assuming your vectors are uncorrelated--the covariance is$$\operatorname{Cov}(\omega,\theta)=\Sigma=\pmatrix{\Sigma_\omega&\mathbf{0}\\\mathbf{0}&\Sigma_\theta},$$this is a standard instance of a differentiable map $\alpha:\mathbb{R}^n\to\mathbb{R}^m$ and, assuming $\left\|\mathbf{x}\right\|$ has little chance of being near $0$ and $\Sigma$ is small, apply the usual delta method machinery. $\endgroup$
    – whuber
    Nov 24 '20 at 13:15
  • $\begingroup$ Please add the self-study tag & read its wiki. $\endgroup$ Nov 24 '20 at 14:10
  • $\begingroup$ @gung This question doesn't look like the kind of routine exercise we would treat as self-study. $\endgroup$
    – whuber
    Nov 24 '20 at 15:11
  • $\begingroup$ Thanks a lot @whuber for you answer. I'm a bit puzzled because $\alpha$ is a vector of dimension 3x1 resulting of the matrix operation $\mathbf{R} * \omega$. Therefore, its covariance should be a 3x3 matrix. This problem is not self-study as I encounter it while propagating covariances of the angular speed of a robot joint to other frames. $\endgroup$
    – Maltergate
    Nov 24 '20 at 15:26
  • $\begingroup$ That's right. I was thinking of your function $R$ instead of $\alpha.$ The image of $\alpha$ is indeed in $\mathbb{R}^3$ and its covariance will be a $3\times 3$ matrix. But the general point still holds and it appears the delta method is what you are looking for. $\endgroup$
    – whuber
    Nov 24 '20 at 15:47
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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 $\Sigma_{\mathbf x} \in \mathbb R^{m \times m}$, if $\mathbf y = f(\mathbf x)$ then:

$$\Sigma_{\mathbf y} \simeq \left . \frac{\partial f}{\partial \mathbf x}\right |_{\mathbf x} \Sigma_{\mathbf x} \left . \frac{\partial f}{\partial \mathbf x}\right |_{\mathbf x} ^\intercal$$

Here our function $f$ is a vectorized version of the Rodrigues' formulae of 3D rotation. The Jacobians can be a bit tedious to compute by hand, hence I used a software to analytically derive the equation.

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