# What's rotation matrix minus its transpose?

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. –  Ｊ. Ｍ. Jul 8 '12 at 16:23
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. –  Generic Human Jul 8 '12 at 16:35
@J.M. I added the paper's title. –  mmm Jul 8 '12 at 16:56
Great. Please do that the next time you ask about some result you've found in a paper. –  Ｊ. Ｍ. Jul 8 '12 at 17:01
In the paper we do have $\tilde R\in SO(3)$, since it is the quotient of rotation matrices $\hat R^T R\ (=\hat R^{-1} R)$. –  Generic Human Jul 8 '12 at 17:07

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 $\hat R$).