Talk:PlanetPhysics/Euler Angle Velocity of 123 Sequence

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Euler angle velocity of 123 Sequence %%% Primary Category Code: 45.40.-f %%% Filename: EulerAngleVelocityOf123Sequence.tex %%% Version: 1 %%% Owner: bloftin %%% Author(s): bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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The method of deriving the \htmladdnormallink{Euler angle velocity}{http://planetphysics.us/encyclopedia/EulerAngleVelocity.html} for a given sequence is to transform each of the derivatives into the \htmladdnormallink{reference frame}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html}. Remember that an \htmladdnormallink{Euler angle sequence}{http://planetphysics.us/encyclopedia/EulerAngleSequence.html} is made up of three successive rotations. In other words, the angular \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} $\dot{\phi}$ needs one rotation, $\dot{\theta}$ needs two and $\dot{\psi}$ needs three.

$$ \vec{\omega} = R_3(\psi) R_2(\theta) R_1(\phi) \left[  \begin{array}{c} \dot{\phi} \\ 0 \\ 0 \end{array} \right] + R_3(\psi)  R_2(\theta)  \left[  \begin{array}{c} 0 \\ \dot{\theta} \\ 0 \end{array} \right] + R_3(\psi) \left[ \begin{array}{c} 0 \\ 0 \\ \dot{\psi} \end{array} \right] $$

Carrying out the \htmladdnormallink{matrix multiplication}{http://planetphysics.us/encyclopedia/Matrix.html} with $ R_3(\psi) R_2(\theta) R_1(\phi)$ being the \htmladdnormallink{Euler 123 sequence}{http://planetphysics.us/encyclopedia/Euler123Sequence.html} $$ R_3(\psi) R_2(\theta) =  \left[  \begin{array}{ccc} c_{\psi} c_{\theta} & s_{\psi} & -c_{\psi} s_{\theta} \\ -s_{\psi} c_{\theta} & c_{\psi} & s_{\theta} s_{\psi} \\ s_{\theta} & 0 & c_{\theta} \end{array} \right] $$

and

$$ R_3(\psi) = \left[ \begin{array}{ccc} c_{\psi} & s_{\psi} & 0 \\ -s_{\psi} & c_{\psi} & 0 \\ 0 & 0 & 1 \end{array} \right] $$

gives us

$$ \left[ \begin{array}{c} \omega_x \\ \omega_y \\ \omega_z \end{array} \right] = \left[ \begin{array}{c} c_{\theta} c_{\psi} \dot{\phi} \\ -c_{\theta} s_{\psi} \dot{\phi} \\ s_{\theta} \dot{\phi} \end{array} \right] + \left[ \begin{array}{c} s_{\psi} \dot{\theta} \\ c_{\psi} \dot{\theta} \\ 0 \end{array} \right] + \left[ \begin{array}{c} 0 \\ 0 \\ \dot{\psi} \end{array} \right] $$

Adding the \htmladdnormallink{vectors}{http://planetphysics.us/encyclopedia/Vectors.html} together yields

$$ \left[ \begin{array}{c} \omega_x \\ \omega_y \\ \omega_z \end{array} \right] = \left[ \begin{array}{c} \dot{\phi} c_{\theta} c_{\psi} + \dot{\theta} s_{\psi} \\ \dot{\theta} c_{\psi} - \dot{\phi} s_{\psi} c_{\theta} \\ \dot{\phi} s_{\theta} + \dot{\psi} \end{array} \right] $$

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 \htmladdnormallink{program}{http://planetphysics.us/encyclopedia/RecursiveFunction.html} like Matlab makes it easy for us to get

$$ \left[ \begin{array}{c} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{array} \right] = \left[ \begin{array}{c} (\omega_x c_{\psi} - \omega_y s_{\psi}) / c_{\theta} \\ \omega_x s_{\psi} + \omega_y c_{\psi} \\ (-\omega_x c_{\psi} + \omega_y s_{\psi}) s_{\theta} / c_{\theta}  + \omega_z \end{array} \right]  $$

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