Talk:PlanetPhysics/Euler Angle Velocity

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Euler angle velocity %%% Primary Category Code: 45.40.-f %%% Filename: EulerAngleVelocity.tex %%% Version: 2 %%% 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 \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} of the Euler angles of a rotating coordinate frame can be derived from the angular velocity \htmladdnormallink{vector}{http://planetphysics.us/encyclopedia/Vectors.html} of this frame with respect to a \htmladdnormallink{reference frame}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}. The derivation is lengthy for each \htmladdnormallink{Euler angle sequence}{http://planetphysics.us/encyclopedia/EulerAngleSequence.html} and is given in their respective entries. One useful result is acheived by combining the angular velocity vector, in terms of Euler angles, with \htmladdnormallink{Euler's moment equations}{http://planetphysics.us/encyclopedia/EulersMomentEquations.html} yielding the \htmladdnormallink{differential equations}{http://planetphysics.us/encyclopedia/DifferentialEquations.html} of \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} that characterizes \htmladdnormallink{rigid body}{http://planetphysics.us/encyclopedia/RigidBody.html} motion for a given sequence.

Notation: $cos(\theta) = c_{\theta}, sin(\theta) = s_{\theta} $

\htmladdnormallink{Euler angle velocity of 123 Sequence}{http://planetphysics.us/encyclopedia/EulerAngleVelocityOf123Sequence.html}:

$ \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] $

\htmladdnormallink{Euler angle velocity of 321 Sequence}{http://planetphysics.us/encyclopedia/EulerAngleVelocityOf321Sequence.html}:

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

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