DirectionCosineMatrixToEuler123Sequence 2005-08-16 00:08:53 2005-08-16 00:08:53 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 Starting with a direction cosine matrix (DCM), we need to determine the three Euler angles. The connection is made by comparing the DCM elements with the combined Euler 123 sequence. The DCM matrix \begin{equation} DCM = \left[ \begin{array}{ccc} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{array} \right] \end{equation} The Euler 123 sequence is \begin{equation} 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] \end{equation} If we examine the coloum 1 row 3 element, then by inspection $A_{31} = sin(\theta)$ Solving for $\theta$ yields \begin{equation} \theta = sin^{-1}(A_{31}) \end{equation} Care must now be taken when evaluating the inverse sine. It is a multivalued function, which will will have values of $\theta$ and $\pi - \theta$. Analytically, the convention is to choose the principle value such that $-\pi/2 \le \theta \le \pi/2$ If a numerical program is used, a function asin() usually does this for us. The next step is to analyze the ratio $A_{32} \over A_{33}$. Using these values from the Euler sequence we get $\dfrac{A_{32}}{A_{33}}= \dfrac{-sin(\phi) cos(\theta)}{cos(\phi) cos(\theta)}$ Rearranging the minus sign and using the tangent yields $tan(\phi) = \dfrac{-A_{32}}{A_{33}}$ Solving the quadrant ambiquity caused by the inverse tangent is done by examining the signs of the numerator and denominator. Denoting y as the numerator and x as the denominator, the the quadrant is chosen by \begin{equation} \left[ \begin{array}{cc} \left[ \begin{array}{cc} y & x \\ + & - \\ \end{array} \right] & \left[ \begin{array}{cc} y & x \\ + & + \\ \end{array} \right] \end{array} \right] \end{equation}