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In a paper I've been reading ("Non-linear complementary filters on the special orthogonal group", Robert Mahony et al. link: warning PDF) there is an operation:

$P_a(\tilde{R}) = \frac{1}{2} (\tilde{R} - \tilde{R}^T)$,

where $R \in SO(3)$ and $\tilde{R}$ is an error of $R$'s estimate. In the particular case $P_a(\tilde{R})$ seems to be 'transforming' the rotation-error matrix to a skew-symmetric matrix, maybe even its derivative. Or can it really be its derivative?

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    ...if you'd mention the paper you're reading, it'd be very helpful.2012-07-08
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    If $\tilde R\in SO(3)$, $\tilde R^T=\tilde R^{-1}$ so $P_a(\tilde R)=0$ if and only if $\tilde R^2=1$, which happens if and only if $\tilde R$ is the identity or the reflection about some axis.2012-07-08
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    @J.M. I added the paper's title.2012-07-08
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    Great. Please do that the next time you ask about some result you've found in a paper.2012-07-08
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    In the paper we do have $\tilde R\in SO(3)$, since it is the quotient of rotation matrices $\hat R^T R\ (=\hat R^{-1} R)$.2012-07-08
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    OK, I corrected my post again. Infact $\tilde{R} \in SO(3)$. I don't know how I could get confused so much not to realize that. So what can $P_a(\tilde{R})$ be? Can "anti-symmetric projection" mean a derivative in this case?2012-07-08

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