PlanetPhysics/Euler Angle Velocity of 321 Sequence

The method of deriving the [[../EulerAngleVelocity/|Euler angle velocity]] for a given sequence is to transform each of the derivatives into the [[../CosmologicalConstant/|reference frame]]. Remember that an [[../EulerAngleSequence/|Euler angle sequence]] is made up of three successive rotations. In other words, the angular [[../Velocity/|velocity]] $\dot{\psi }$ needs one rotation, $\dot{\theta }$ needs two and $\dot{\varphi }$ needs three.

$\overrightarrow{\omega }=R_1(\psi )R_2(\theta )R_3(\varphi )\left[\begin{array}{c}0\\ 0\\ \dot{\varphi }\end{array}\right]+R_1(\psi )R_2(\theta )\left[\begin{array}{c}0\\ \dot{\theta }\\ 0\end{array}\right]+R_1(\psi )\left[\begin{array}{c}\dot{\psi }\\ 0\\ 0\end{array}\right]$

Carrying out the [[../Matrix/|matrix multiplication]] with $R_1(\psi )R_2(\theta )R_3(\varphi )$ being the [[../Euler321Sequence/|Euler 321 sequence]] $R_1(\psi )R_2(\theta )=\left[\begin{array}{ccc}{c}_{\theta }& 0& -s_{\theta }\\ s_{\psi }{s}_{\theta }& c_{\psi }& s_{\psi }{c}_{\theta }\\ c_{\psi }{s}_{\theta }& -s_{\psi }& c_{\psi }{c}_{\theta }\end{array}\right]$

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

$\left[\begin{array}{c}{\omega }_x\\ {\omega }_y\\ {\omega }_z\end{array}\right]=\left[\begin{array}{c}-s_{\theta }\dot{\varphi }\\ s_{\psi }{c}_{\theta }\dot{\varphi }\\ c_{\psi }{c}_{\theta }\dot{\varphi }\end{array}\right]+\left[\begin{array}{c}0\\ c_{\psi }\dot{\theta }\\ -s_{\psi }\dot{\theta }\end{array}\right]+\left[\begin{array}{c}\dot{\psi }\\ 0\\ 0\end{array}\right]$

Adding the [[../Vectors/|vectors]] together yields

$\left[\begin{array}{c}{\omega }_x\\ {\omega }_y\\ {\omega }_z\end{array}\right]=\left[\begin{array}{c}-s_{\theta }\dot{\varphi }+\dot{\psi }\\ s_{\psi }{c}_{\theta }\dot{\varphi }+c_{\psi }\dot{\theta }\\ c_{\psi }{c}_{\theta }\dot{\varphi }-s_{\psi }\dot{\theta }\end{array}\right]$

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 [[../SupercomputerArchitercture/|program]] like Matlab makes it easy for us to get

$\left[\begin{array}{c}\dot{\varphi }\\ \dot{\theta }\\ \dot{\psi }\end{array}\right]=\left[\begin{array}{c}({\omega }_y{s}_{\psi }+{\omega }_z{c}_{\psi })sec(\theta )\\ {\omega }_y{c}_{\psi }-{\omega }_z{s}_{\psi }\\ {\omega }_x+{\omega }_y{s}_{\psi }{t}_{\theta }+{\omega }_z{c}_{\psi }{t}_{\theta }\end{array}\right]$

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 [[../LargeHadronCollider/|SPS]] 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

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

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