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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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    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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Author of the paper clearly stated:

The two operations ${\mathbb P}_a(\tilde R)$ and $(RΩ)_×$ are maps from error space and velocity space into the tangent space of SO(3);

Yes, the “tangent space of SO(3)” (at the identity element) is the space of skew-symmetric matrices. But ${\mathbb P}_a$ is not a derivative; in combination with $\tilde R = \hat R^{\mathsf T} R$ expression (so-called “error space”) it is a (finite) difference operator on SO(3) with values in a convenient vector space (that has only 3 dimensions and doesn’t rotate together with $\hat R$).