# Extracting angular velocity tensor from orthogonal matrices

Let us suppose we have two orthogonal rotation matrices representing a three-dimensional rotations $$\mathbf{R}(t)$$ and $$\mathbf{R}(t+\Delta t)$$

How is it possible to extract the angular velocity of the rotation $\boldsymbol \omega \in \mathbb{R}^3$ or equivalently the angular velocity tensor (represented by the skew-symmetric matrix) $$\boldsymbol \Omega = \begin{pmatrix} 0 & -\omega_z(t) & \omega_y(t) \\ \omega_z(t) & 0 & -\omega_x(t) \\ -\omega_y(t) & \omega_x(t) & 0 \\ \end{pmatrix}$$ from them?

I would ask you if the following approach make sense...

1) By mean of exponential map $\mathbf{R}(t+\Delta t) = e^{\boldsymbol \Omega \Delta t} \mathbf{R}(t)$

2) Solve for $\boldsymbol \Omega$ then $$\boldsymbol \Omega = \frac{ \log\left( \mathbf{R}(t)\mathbf{R}^{-1}(t+\Delta t) \right)}{\Delta t}$$

3) Call $\boldsymbol Y = \mathbf{R}(t) \mathbf{R}^{-1}(t+\Delta t) -\mathbf{I}$ and approximate the logarithm with its Taylor expansion

$$\boldsymbol \Omega(t) = \dfrac{ \log\left(\mathbf{R}(t) \mathbf{R}^{-1}(t+\Delta t) \right)}{\Delta t} \approx \frac{1}{\Delta t} \left( \mathbf{Y} - \frac{\mathbf{Y}^2}{2} + \frac{\mathbf{Y}^3}{3} - \frac{\mathbf{Y}^4}{4} + \ldots \right )$$

• I think you mean "between two consecutive points in time"? Though that raises the question what makes two points in time consecutive. I think it should simply say "between two points in time". – joriki Sep 7 '11 at 9:01
• Is $dt$ just a normal time interval? Then it might be preferable to denote it by $\Delta t$ to prevent confusion with the differential. – joriki Sep 7 '11 at 9:02
• Rotation matrices are evaluated on a rigid-body at two consecutive time intervals, so I would have used $\Delta t$, yes. – linello Sep 7 '11 at 9:07
• I pointed out that "at two consecutive time intervals" makes no sense, but you used it again -- do you disagree? – joriki Sep 7 '11 at 9:13
• excuse me corrected... – linello Sep 7 '11 at 9:21

You don't need any series expansions to do this. $\mathbf Y^{-1}=\mathbf R(t+\Delta t) \mathbf R^{-1}(t)$ is a rotation matrix that rotates the body from the position at time $t$ to the position at time $t+\Delta t$. This is a rotation around the axis along $\boldsymbol\omega$ through the angle $|\boldsymbol\omega|\Delta t$. The trace of a rotation matrix with angle $\phi$ is $1+2\cos\phi$, so you can calculate $|\boldsymbol\omega|$ directly from the trace. To get the direction, you can solve the homogoeneous linear system $\mathbf Yx=x$.
• Ok it works: first I compute the angle $\phi$ as $\phi = \arccos \left( \dfrac{\textrm{trace}(\mathbf{Y}(t)) - 1}{2}\right)$ and then the direction as the eigenvector of $\mathbf{Y}$ with the largest eigenvalue. Does this make sense? – linello Sep 7 '11 at 9:48
• @linello: The first part does; the second part is unnecessarily complicated -- you don't have to solve an eigenvalue problem; you already know that the eigenvalue for the axis is $1$, so you can solve the linear system $Yx=x$, which is easier. – joriki Sep 7 '11 at 9:55
• maybe even easier is to use the following? $$\boldsymbol \omega = \frac{1}{2 \sin(\phi)} \begin{bmatrix} \mathbf{Y}_{3,2}(t)-\mathbf{Y}_{2,3}(t) \\ \mathbf{Y}_{1,3}(t)-\mathbf{Y}_{3,1}(t) \\ \mathbf{Y}_{2,1}(t)-\mathbf{Y}_{1,2}(t) \end{bmatrix}$$ – linello Sep 7 '11 at 10:15
• @linello: Yes, that's a good idea. All methods (including these two) will become ill-conditioned as $\Delta t\to0$; I can't say at first sight which of these two will be worse numericaly in case you have to handle small $\Delta t$; you might want to try both and compare. – joriki Sep 7 '11 at 10:51