View source for PlanetPhysics/Euler Angle Velocity of 123 Sequence

The method of deriving the [[../EulerAngleVelocity/|Euler angle velocity]] for a given sequence is to transform each of the derivatives into the [[../CosmologicalConstant2/|reference frame]]. Remember that an [[../EulerAngleSequence/|Euler angle sequence]] is made up of three successive rotations. In other words, the angular [[../Velocity/|velocity]] <math>\dot{\phi}</math> needs one rotation, <math>\dot{\theta}</math> needs two and <math>\dot{\psi}</math> needs three.

<math> \vec{\omega} = R_3(\psi)  R_2(\theta) R_1(\phi) \left[  \begin{matrix}
\dot{\phi} \\
0 \\
0 \end{matrix} \right] +  R_3(\psi)  R_2(\theta)  \left[  \begin{matrix}
0 \\
\dot{\theta} \\
0 \end{matrix} \right] + R_3(\psi) \left[  \begin{matrix}
0 \\
0 \\
\dot{\psi} \end{matrix} \right] </math>

Carrying out the [[../Matrix/|matrix multiplication]] with <math> R_3(\psi)  R_2(\theta) R_1(\phi)</math> being the [[../Euler123Sequence/|Euler 123 sequence]]
<math> R_3(\psi)  R_2(\theta) =  \left[  \begin{matrix}
c_{\psi} c_{\theta} & s_{\psi} & -c_{\psi} s_{\theta} \\
-s_{\psi} c_{\theta} & c_{\psi} & s_{\theta} s_{\psi} \\
s_{\theta} & 0 & c_{\theta} \end{matrix} \right] </math>

and

<math> R_3(\psi) = \left[ \begin{matrix}
c_{\psi} & s_{\psi} & 0 \\
-s_{\psi} & c_{\psi} & 0 \\
0 & 0 & 1 \end{matrix} \right] </math>

gives us

<math> \left[ \begin{matrix}
\omega_x \\
\omega_y \\
\omega_z \end{matrix} \right] =  \left[ \begin{matrix}
c_{\theta} c_{\psi} \dot{\phi} \\
-c_{\theta} s_{\psi} \dot{\phi} \\
s_{\theta} \dot{\phi} \end{matrix} \right] + \left[ \begin{matrix}
s_{\psi} \dot{\theta} \\
c_{\psi} \dot{\theta} \\
0 \end{matrix}  \right] + \left[ \begin{matrix}
0 \\
0 \\
\dot{\psi} \end{matrix} \right] </math>

Adding the [[../Vectors/|vectors]] together yields

<math> \left[ \begin{matrix}
\omega_x \\
\omega_y \\
\omega_z \end{matrix} \right] =  \left[ \begin{matrix}
\dot{\phi} c_{\theta} c_{\psi} + \dot{\theta} s_{\psi}  \\
\dot{\theta} c_{\psi} - \dot{\phi} s_{\psi} c_{\theta} \\
\dot{\phi} s_{\theta} + \dot{\psi} \end{matrix} \right]  </math>

Of course, we also wish to have the Euler angle velocities in terms of the angular velocities which requires us to solve the linear equations for them. Using a [[../RecursiveFunction/|program]] like Matlab makes it easy for us to get

<math> \left[ \begin{matrix}
\dot{\phi} \\
\dot{\theta} \\
\dot{\psi} \end{matrix} \right] =  \left[ \begin{matrix}
(\omega_x c_{\psi} - \omega_y s_{\psi}) / c_{\theta} \\
\omega_x s_{\psi} + \omega_y c_{\psi} \\
(-\omega_x c_{\psi} + \omega_y s_{\psi}) s_{\theta} / c_{\theta}   + \omega_z \end{matrix} \right]  </math>

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