Calculating new rotation matrix with its derivative given

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.

• There is an elaborate answer for this in a paper by Oliver J. Woodman, "An introduction to inertial navigation" see link below: cl.cam.ac.uk/techreports/UCAM-CL-TR-696.pdf Jan 17 '17 at 8:50

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

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

$$\mathrm e^{t\Omega_\times}=\pmatrix{\cos\omega t&\sin\omega t&0\\-\sin\omega t&\cos\omega t&0\\0&0&1}\;.$$
• @user73336: I don't see a $w$ anywhere. Perhaps you mean $\omega$? You can produce that letter using the command \omega. Apr 23 '13 at 22:26
• @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 ...? Jan 17 '17 at 20:07