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srpvx
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Hello. Sorry for my English
There are [itex]R[/itex] - rotation matrix (that performs transformation from associated coordinate system [itex]IE[/itex] to static coordinate system [itex]OI[/itex]) and [itex]\omega[/itex] - angular velocity. The matrix [itex]R[/itex] depends on parameters [itex]\xi[/itex] (for example, Euler angles). I need to express [itex]\omega[/itex] as function of [itex]\xi[/itex].
Let [itex]r^e[/itex] - components of vector [itex]r[/itex] in the associated coordinate system: [itex]r=Rr^{e}[/itex] and [itex]r^{e}=R^{T}r[/itex]. Than variation of vector [itex]r[/itex]:
[itex]\delta r=\sum_{i}\frac{\partial Rr^{e}}{\partial\xi_{i}}\delta\xi_{i}=[/itex] [itex]\sum_{i}\left(\frac{\partial R}{\partial\xi_{i}}\delta\xi_{i}\right)r^{e}=[/itex] [itex]\sum_{i}\left(\frac{\partial R}{\partial\xi_{i}}\delta\xi_{i}\right)R^{T}r[/itex]
On the other hand if I rotate [itex]r[/itex] about [itex]l[/itex] on angle [itex]\Delta\varphi[/itex] then variation of [itex]r[/itex] equals: [itex]\delta r=\Delta\varphi\left[l\times r\right]=\delta\varphi\times r[/itex], where [itex]\delta\varphi = \Delta \varphi l[/itex].
Comparing [itex]\delta r=\delta\varphi\times r[/itex] with [itex]\delta r=\sum_{i}\left(\frac{\partial R}{\partial\xi_{i}}\delta\xi_{i}\right)R^{T}r[/itex], we get the following expression for omega:
[itex]\omega = \frac{\delta \varphi}{d t}[/itex]
[itex]\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}[/itex]
Is it right?
There are [itex]R[/itex] - rotation matrix (that performs transformation from associated coordinate system [itex]IE[/itex] to static coordinate system [itex]OI[/itex]) and [itex]\omega[/itex] - angular velocity. The matrix [itex]R[/itex] depends on parameters [itex]\xi[/itex] (for example, Euler angles). I need to express [itex]\omega[/itex] as function of [itex]\xi[/itex].
Let [itex]r^e[/itex] - components of vector [itex]r[/itex] in the associated coordinate system: [itex]r=Rr^{e}[/itex] and [itex]r^{e}=R^{T}r[/itex]. Than variation of vector [itex]r[/itex]:
[itex]\delta r=\sum_{i}\frac{\partial Rr^{e}}{\partial\xi_{i}}\delta\xi_{i}=[/itex] [itex]\sum_{i}\left(\frac{\partial R}{\partial\xi_{i}}\delta\xi_{i}\right)r^{e}=[/itex] [itex]\sum_{i}\left(\frac{\partial R}{\partial\xi_{i}}\delta\xi_{i}\right)R^{T}r[/itex]
On the other hand if I rotate [itex]r[/itex] about [itex]l[/itex] on angle [itex]\Delta\varphi[/itex] then variation of [itex]r[/itex] equals: [itex]\delta r=\Delta\varphi\left[l\times r\right]=\delta\varphi\times r[/itex], where [itex]\delta\varphi = \Delta \varphi l[/itex].
Comparing [itex]\delta r=\delta\varphi\times r[/itex] with [itex]\delta r=\sum_{i}\left(\frac{\partial R}{\partial\xi_{i}}\delta\xi_{i}\right)R^{T}r[/itex], we get the following expression for omega:
[itex]\omega = \frac{\delta \varphi}{d t}[/itex]
[itex]\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}[/itex]
Is it right?