Talk:PlanetPhysics/Euler Angle Velocity of 321 Sequence

Original TeX Content from PlanetPhysics Archive

%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: Euler angle velocity of 321 Sequence %%% Primary Category Code: 45.40.-f %%% Filename: EulerAngleVelocityOf321Sequence.tex %%% Version: 3 %%% 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}

The method of deriving the \htmladdnormallink{Euler angle velocity}{http://planetphysics.us/encyclopedia/EulerAngleVelocity.html} for a given sequence is to transform each of the derivatives into the \htmladdnormallink{reference frame}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}. Remember that an \htmladdnormallink{Euler angle sequence}{http://planetphysics.us/encyclopedia/EulerAngleSequence.html} is made up of three successive rotations. In other words, the angular \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} $\dot{\psi}$ 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 \htmladdnormallink{matrix multiplication}{http://planetphysics.us/encyclopedia/Matrix.html} with $ R_1(\psi) R_2(\theta) R_3(\phi)$ being the \htmladdnormallink{Euler 321 sequence}{http://planetphysics.us/encyclopedia/Euler321Sequence.html} $$ R_1(\psi)R_2(\theta) = \left[ \begin{array}{ccc} c_{\theta} & 0 & -s_{\theta} \\ s_{\psi} s_{\theta} & c_{\psi} & s_{\psi} c_{\theta} \\ c_{\psi} s_{\theta} & -s_{\psi} & c_{\psi} c_{\theta} \end{array} \right] $$

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 \htmladdnormallink{vectors}{http://planetphysics.us/encyclopedia/Vectors.html} 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 \htmladdnormallink{program}{http://planetphysics.us/encyclopedia/SupercomputerArchitercture.html} like Matlab makes it easy for us to get

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

In matlab solving for the Euler angle velocites can be done with the following commands. Using the notation $Ax = b$, we want to solve for $x$, such that $x = A^{-1}b$. For our problem then

{\emph syms wx wy wz phd thd psd \htmladdnormallink{SPS}{http://planetphysics.us/encyclopedia/LargeHadronCollider.html} cth cps b x A;

b = [wx wy wz]';

x = [phd thd psd]';

A = [ -sth 0 1; sps*cth cps 0; cps*cth -sps 0];}

and solve for the angle velocites with the command

\emph{ x = inv(A)*b}

Note that matlab spits out extra sine and cosine terms that just equal 1 through

$$ s_{\psi}^2 + c_{\psi}^2 = 1$$

The shorthand notation used in this article is

$$ s_{\psi} = sin(\psi) $$ $$ c_{\psi} = cos(\psi) $$ $$ t_{\psi} = tan(\psi) $$

\end{document}