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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}\\<br /> \omega_{3} & 0 & -\omega_{1}\\<br /> -\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?
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}\\<br /> \omega_{3} & 0 & -\omega_{1}\\<br /> -\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?