# Angular velocity and rotation matrix

1. Aug 17, 2011

### srpvx

Hello. Sorry for my English

There are $R$ - rotation matrix (that performs transformation from associated coordinate system $IE$ to static coordinate system $OI$) and $\omega$ - angular velocity. The matrix $R$ depends on parameters $\xi$ (for example, Euler angles). I need to express $\omega$ as function of $\xi$.

Let $r^e$ - components of vector $r$ in the associated coordinate system: $r=Rr^{e}$ and $r^{e}=R^{T}r$. Than variation of vector $r$:
$\delta r=\sum_{i}\frac{\partial Rr^{e}}{\partial\xi_{i}}\delta\xi_{i}=$ $\sum_{i}\left(\frac{\partial R}{\partial\xi_{i}}\delta\xi_{i}\right)r^{e}=$ $\sum_{i}\left(\frac{\partial R}{\partial\xi_{i}}\delta\xi_{i}\right)R^{T}r$

On the other hand if I rotate $r$ about $l$ on angle $\Delta\varphi$ then variation of $r$ equals: $\delta r=\Delta\varphi\left[l\times r\right]=\delta\varphi\times r$, where $\delta\varphi = \Delta \varphi l$.

Comparing $\delta r=\delta\varphi\times r$ with $\delta r=\sum_{i}\left(\frac{\partial R}{\partial\xi_{i}}\delta\xi_{i}\right)R^{T}r$, we get the following expression for omega:
$\omega = \frac{\delta \varphi}{d t}$
$\left[\omega\times\right]=\left(\begin{array}{ccc}0 & -\omega_{3} & \omega_{2}\\ \omega_{3} & 0 & -\omega_{1}\\ -\omega_{2} & \omega_{1} & 0\end{array}\right)=\sum_{i}\left(\frac{\partial R}{\partial\xi_{i}}\dot{\xi_{i}}\right)R^{T}$

Is it right?

2. Aug 19, 2011

### A.T.

3. Aug 22, 2011

### srpvx

I found the answer on my question at http://en.wikipedia.org/wiki/Angular_velocity" [Broken] :
But $W$ is not a tensor, $W$ is a pseudotensor: $W_{ij} = e_{iwj} \omega_{w}$.

Last edited by a moderator: May 5, 2017