EulerAngleSequences 2005-08-14 16:36:29 2005-08-19 14:20:49 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here An Euler 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 Euler 123 sequence \\ $ 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] $ Euler 132 sequence \\ $ 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] $ Euler 121 sequence \\ $ 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] $ Euler 131 sequence \\ $ 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] $ Euler 213 sequence \\ $ 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] $ Euler 231 sequence \\ $ 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] $ Euler 212 sequence \\ $ 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] $ Euler 232 sequence \\ $ 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} & -s_{\psi} s_{\theta} & - s_{\psi} c_{\theta} s_{\phi} + c_{\psi} c_{\phi} \end{array} \right] $ Euler 312 sequence \\ $ 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] $ Euler 321 sequence \\ $ 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} c_{\phi} - c_{\psi} s_{\theta} s_{\phi} & -s_{\psi} c_{\phi} - c_{\psi} s_{\theta} s_{\phi} & c_{\psi} c_{\theta} \end{array} \right] $ Euler 313 sequence \\ $ 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] $ Euler 323 sequence \\ $ 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_{\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] $