Talk:PlanetPhysics/Euler Angles

Original TeX Content from PlanetPhysics Archive
%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Euler angle sequences %%% Primary Category Code: 45.40.-f %%% Filename: EulerAngles.tex %%% Version: 5 %%% 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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\begin{document}

An Euler angle sequence is a rotation matrix that is completely determined by three parameters, called Euler angles. These Euler angles are represented by the $ \phi $, $ \theta $ and $ \psi $ variables with each corresponding to a rotation about an axis. There are several different conventions. Only one will be shown here, since it is more important to understand the underlying math thoroughly.

A list of the Euler angle rotation matrices for different sequences

\htmladdnormallink{Euler 123 sequence}{http://planetphysics.org/encyclopedia/Euler123Sequence.html} \\

$ R_3(\psi)R_2(\theta)R_1(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\theta} & c_{\psi} s_{\theta} s_{\phi} + s_{\psi} c_{\phi} & -c_{\psi} s_{\theta} c_{\phi} + s_{\psi} s_{\phi} \\ -s_{\psi} c_{\theta} & -s_{\psi} s_{\theta} s_{\phi} + c_{\psi} c_{\phi} & s_{\psi} s_{\theta} c_{\phi} + c_{\psi} s_{\phi} \\ s_{\theta} & -c_{\theta} s_{\phi} & c_{\theta} c_{\phi} \end{array} \right] $

\htmladdnormallink{Euler 132 sequence}{http://planetphysics.org/encyclopedia/Euler132Sequence.html} \\

$ R_2(\psi)R_3(\theta)R_1(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\theta} & c_{\psi} s_{\theta} c_{\phi} + s_{\psi} s_{\phi} & c_{\psi} s_{\theta} s_{\phi} - s_{\psi} c_{\phi} \\ -s_{\theta} & c_{\theta} c_{\phi} & c_{\theta} s_{\phi} \\ s_{\psi} c_{\theta} & s_{\psi} s_{\theta} c_{\phi} - c_{\psi} s_{\phi} & s_{\psi} s_{\theta} s_{\phi} + c_{\psi} c_{\phi} \end{array} \right] $

\htmladdnormallink{Euler 121 sequence}{http://planetphysics.org/encyclopedia/Euler121Sequence.html} \\

$ R_1(\psi)R_2(\theta)R_1(\phi) = \left[ \begin{array}{ccc} c_{\theta} & -s_{\theta} s_{\phi}  & s_{\theta} c_{\phi}  \\ -s_{\psi} s_{\theta} & c_{\psi} c_{\phi} - s_{\psi} c_{\theta} s_{\phi} & c_{\psi} s_{\phi} + s_{\psi} c_{\theta} c_{\phi} \\ -s_{\theta} c_{\psi} & -s_{\psi} c_{\phi} - c_{\psi} c_{\theta} s_{\phi} & -s_{\psi} s_{\phi} + c_{\psi} c_{\theta} c_{\phi} \end{array} \right] $

\htmladdnormallink{Euler 131 sequence}{http://planetphysics.org/encyclopedia/Euler131Sequence.html} \\

$ R_1(\psi)R_3(\theta)R_1(\phi) = \left[ \begin{array}{ccc} c_{\theta} & s_{\theta} c_{\phi}   & s_{\theta} s_{\phi}   \\ -c_{\psi} s_{\theta} & c_{\psi} c_{\theta} c_{\phi} - s_{\psi} s_{\phi} & c_{\psi} c_{\theta} s_{\phi} + s_{\psi} c_{\phi} \\ s_{\psi} s_{\theta} & -s_{\psi} c_{\theta} c_{\phi} - c_{\psi} s_{\phi} &  - s_{\psi} c_{\theta} s_{\phi} +  c_{\psi} c_{\phi}  \end{array} \right] $

\htmladdnormallink{Euler 213 sequence}{http://planetphysics.org/encyclopedia/Euler213Sequence.html} \\

$ R_3(\psi)R_1(\theta)R_2(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\phi} + s_{\psi} s_{\theta} s_{\phi} & s_{\psi} c_{\theta} & -c_{\psi} s_{\phi} + s_{\psi} s_{\theta} c_{\phi} \\ -s_{\psi} c_{\phi} + c_{\psi} s_{\theta} s_{\phi} & c_{\psi} c_{\theta} & s_{\psi} s_{\phi} + c_{\psi} s_{\theta} c_{\phi} \\ c_{\theta} s_{\phi} & -s_{\theta} & c_{\theta} c_{\phi} \end{array} \right] $

\htmladdnormallink{Euler 231 sequence}{http://planetphysics.org/encyclopedia/Euler231Sequence.html} \\

$ R_1(\psi)R_3(\theta)R_2(\phi) = \left[ \begin{array}{ccc} c_{\theta} c_{\phi} & s_{\theta} & -c_{\theta} s_{\phi}  \\ -c_{\psi} s_{\theta} c_{\phi} + s_{\psi} s_{\phi} & c_{\psi} c_{\theta} & c_{\psi} s_{\theta} s_{\phi} + s_{\psi} s_{\theta} s_{\phi} \\ s_{\psi} s_{\theta} c_{\phi} + c_{\psi} s_{\phi} & -s_{\psi} c_{\theta} & -s_{\psi} s_{\theta} s_{\phi} +c_{\psi} c_{\phi} \end{array} \right] $

\htmladdnormallink{Euler 212 sequence}{http://planetphysics.org/encyclopedia/Euler212Sequence.html} \\

$ R_2(\psi)R_1(\theta)R_2(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\phi} - s_{\psi} c_{\theta} s_{\phi} & s_{\psi} s_{\theta} & -c_{\psi} s_{\phi} - s_{\psi} c_{\theta} c_{\phi} \\ s_{\theta} s_{\phi} & c_{\theta} & s_{\theta} c_{\phi} \\ s_{\psi} c_{\phi} + c_{\psi} c_{\theta} s_{\phi} & -c_{\psi} s_{\theta} & -s_{\psi} s_{\phi} + c_{\psi} c_{\theta} c_{\phi} \end{array} \right] $

\htmladdnormallink{Euler 232 sequence}{http://planetphysics.org/encyclopedia/Euler232Sequence.html} \\

$ R_2(\psi)R_3(\theta)R_2(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\theta} c_{\phi} - s_{\psi} s_{\phi} & c_{\psi} s_{\theta}  & -c_{\psi} c_{\theta} s_{\phi} - s_{\psi} c_{\phi}   \\ -s_{\theta} c_{\phi} & c_{\theta} & s_{\theta} s_{\phi} \\ s_{\psi} c_{\theta} c_{\phi} + c_{\psi} s_{\phi} & s_{\psi} s_{\theta} &  - s_{\psi} c_{\theta} s_{\phi} +  c_{\psi} c_{\phi}  \end{array} \right] $

\htmladdnormallink{Euler 312 Sequence}{http://planetphysics.org/encyclopedia/Euler312Sequence.html} \\

$ R_2(\psi)R_1(\theta)R_3(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\phi} - s_{\psi} s_{\theta} s_{\phi} & c_{\psi} s_{\phi} + s_{\psi} s_{\theta} c_{\phi} & -s_{\psi} c_{\theta} \\ -s_{\phi} c_{\theta} & c_{\theta} c_{\phi} & s_{\theta} \\ s_{\psi} c_{\phi} + c_{\psi} s_{\theta} s_{\phi} & s_{\psi} s_{\phi} - c_{\psi} s_{\theta} c_{\phi} & c_{\psi} c_{\theta} \end{array} \right] $

\htmladdnormallink{Euler 321 sequence}{http://planetphysics.org/encyclopedia/Euler321Sequence.html} \\

$ R_1(\psi)R_2(\theta)R_3(\phi) = \left[ \begin{array}{ccc} c_{\theta} c_{\phi} & c_{\theta} s_{\phi} & -s_{\theta}  \\ - c_{\psi} s_{\phi} + s_{\psi} s_{\theta} c_{\phi} & c_{\psi} c_{\phi} + s_{\psi} s_{\theta} s_{\phi} & s_{\psi} c_{\theta} \\ s_{\psi} s_{\phi} - c_{\psi} s_{\theta} c_{\phi} & -s_{\psi} c_{\phi} + c_{\psi} s_{\theta} s_{\phi} & c_{\psi} c_{\theta}  \end{array} \right]  $

\htmladdnormallink{Euler 313 sequence}{http://planetphysics.org/encyclopedia/Euler313Sequence.html} \\

$ R_3(\psi)R_1(\theta)R_3(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\phi} - s_{\psi} s_{\phi} c_{\theta} & c_{\psi} s_{\phi} + s_{\psi} c_{\theta} c_{\phi} & s_{\psi} s_{\theta} \\ -s_{\psi} c_{\phi} - c_{\psi} s_{\phi} c_{\theta} & -s_{\psi} s_{\phi} + c_{\psi} c_{\theta} c_{\phi} & c_{\psi} s_{\theta} \\ s_{\theta} s_{\phi} & -s_{\theta} c_{\phi} & c_{\theta} \end{array} \right] $

\htmladdnormallink{Euler 323 sequence}{http://planetphysics.org/encyclopedia/Euler323Sequence.html} \\

$ R_3(\psi)R_2(\theta)R_3(\phi) = \left[ \begin{array}{ccc} c_{\psi} c_{\theta} c_{\phi} - s_{\psi} s_{\phi} & c_{\psi} c_{\theta} s_{\phi} + s_{\psi} c_{\phi} & -c_{\psi} s_{\theta} \\ - s_{\psi} c_{\theta} c_{\phi} - c_{\psi} s_{\phi} & -s_{\psi} c_{\theta} s_{\phi} + c_{\psi} c_{\phi} & s_{\psi} s_{\theta} \\ s_{\theta} c_{\phi} & s_{\theta} s_{\phi} & c_{\theta} \end{array} \right] $

\end{document}