EulerAngleVelocityOf321Sequence 2007-02-14 21:19:03 2007-02-14 21:19:03 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 The method of deriving the Euler angle velocity for a given sequence is to transform each of the derivatives into the reference frame. Remember that an Euler angle sequence is made up of three successive rotations. In other words, the angular velocity $\dot{\psi}$ needs one rotation, $\dot{\theta}$ needs two and $\dot{\phi}$ needs three. $$ \vec{\omega} = R_1(\psi) R_2(\theta) R_3(\phi) \left[ \begin{array}{c} 0 \\ 0 \\ \dot{\phi} \end{array} \right] + R_1(\psi) R_2(\theta) \left[ \begin{array}{c} 0 \\ \dot{\theta} \\ 0 \end{array} \right] + R_1(\psi) \left[ \begin{array}{c} \dot{\psi} \\ 0 \\ 0 \end{array} \right] $$ Carrying out the matrix multiplication with $ R_1(\psi) R_2(\theta) R_3(\phi)$ being the Euler 321 sequence $$ R_1(\psi)R_2(\theta) = \left[ \begin{array}{ccc} c_{\theta} & 0 & -s_{\theta} \\ s_{\phi} s_{\theta} & c_{\psi} & s_{\psi} c_{\theta} \\ c_{\psi} s_{\theta} & -s_{\psi} & c_{\psi} c_{\theta} \end{array} \right] $$ and $$ R_1(\psi) = \left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & c_{\psi} & s_{\psi} \\ 0 & -s_{\psi} & c_{\psi}\end{array} \right] $$ gives us $$ \left[ \begin{array}{c} \omega_x \\ \omega_y \\ \omega_z \end{array} \right] = \left[ \begin{array}{c} -s_{\theta} \dot{\phi} \\ s_{\psi} c_{\theta} \dot{\phi} \\ c_{\psi} c_{\theta} \dot{\phi} \end{array} \right] + \left[ \begin{array}{c} 0 \\ c_{\psi} \dot{\theta} \\ -s_{\psi} \dot{\theta} \end{array} \right] + \left[ \begin{array}{c} \dot{\psi} \\ 0 \\ 0 \end{array} \right] $$ Adding the vectors together yields $$ \left[ \begin{array}{c} \omega_x \\ \omega_y \\ \omega_z \end{array} \right] = \left[ \begin{array}{c} -s_{\theta} \dot{\phi} + \dot{\psi} \\ s_{\psi} c_{\theta} \dot{\phi} + c_{\psi} \dot{\theta} \\ c_{\psi} c_{\theta} \dot{\phi} - s_{\psi} \dot{\theta} \end{array} \right] $$ 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 program like Matlab makes it easy for us to get $$ \left[ \begin{array}{c} \dot{\phi} \\ \dot{\theta} \\ \dot{\psi} \end{array} \right] = \left[ \begin{array}{c} (\omega_x c_{\psi} - \omega_y s_{\psi}) / c_{\theta} \\ \omega_x s_{\psi} + \omega_y c_{\psi} \\ (-\omega_x c_{\psi} + \omega_y s_{\psi}) s_{\theta} / c_{\theta} + \omega_z \end{array} \right] $$