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I've got a skew-symmetric matrix representing gyroscope measurements, say $\Omega = [p,q,r]^T$, with $p$, $q$, $r$ being the angular velocities around $X$, $Y$ and $Z$ axes. I know my system's dynamics is:

$\dot{R} = R \Omega_\times$

with $\Omega_\times$ being a skew-symmetric matrix built on $\Omega$. How do I integrate $R$ to obtain new rotation matrix from its derivative? I already have the derivative, so just the integration process seems to overwhelm me. Simple addition of $R_{new} = R + \dot{R}$ violates $SO(3)$ group's constraints ($det(R) \neq 1$).

Thanks for any help.

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The general solution is $R(t)=R_0\mathrm e^{t\Omega_\times}$, where $R_0$ is any rotation matrix. You can rotate to coordinates in which $\Omega_\times$ takes the form

$$\Omega_\times=\pmatrix{0&\omega&0\\-\omega&0&0\\0&0&0}\;,$$

which leads to

$$\mathrm e^{t\Omega_\times}=\pmatrix{\cos\omega t&\sin\omega t&0\\-\sin\omega t&\cos\omega t&0\\0&0&1}\;.$$

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  • $\begingroup$ I believe that the last matrix should have a 1 in the lower corner... $\endgroup$
    – Fabian
    Jul 30 '12 at 20:42
  • $\begingroup$ @Fabian: Indeed, thanks, fixed. $\endgroup$
    – joriki
    Jul 31 '12 at 1:27
  • $\begingroup$ what's the relationship between [p,q,r]T and w ? @joriki $\endgroup$
    – user73336
    Apr 19 '13 at 8:14
  • $\begingroup$ @user73336: I don't see a $w$ anywhere. Perhaps you mean $\omega$? You can produce that letter using the command \omega. $\endgroup$
    – joriki
    Apr 23 '13 at 22:26
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    $\begingroup$ @joriki Maybe in general case only $\omega$ is not enough. We need components $\omega_x, \omega_y, \omega_z$ in the skew-symmetric matrix because axis can be changing every moment ...? $\endgroup$
    – Widawensen
    Jan 17 '17 at 20:07

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