Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 1 down vote accepted

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


which leads to

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

share|cite|improve this answer
Thank you joriki. – mmm Jul 11 '12 at 14:50
I believe that the last matrix should have a 1 in the lower corner... – Fabian Jul 30 '12 at 20:42
@Fabian: Indeed, thanks, fixed. – joriki Jul 31 '12 at 1:27
what's the relationship between [p,q,r]T and w ? @joriki – user73336 Apr 19 '13 at 8:14
@user73336: I don't see a $w$ anywhere. Perhaps you mean $\omega$? You can produce that letter using the command \omega. – joriki Apr 23 '13 at 22:26

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.