PlanetPhysics/Euler Angle Velocity of 123 Sequence

The method of deriving the [[../EulerAngleVelocity/|Euler angle velocity]] for a given sequence is to transform each of the derivatives into the [[../CosmologicalConstant2/|reference frame]]. Remember that an [[../EulerAngleSequence/|Euler angle sequence]] is made up of three successive rotations. In other words, the angular [[../Velocity/|velocity]] $\dot{\varphi }$ needs one rotation, $\dot{\theta }$ needs two and $\dot{\psi }$ needs three.

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

Carrying out the [[../Matrix/|matrix multiplication]] with $R_3(\psi )R_2(\theta )R_1(\varphi )$ being the [[../Euler123Sequence/|Euler 123 sequence]] $R_3(\psi )R_2(\theta )=\left[\begin{array}{ccc}{c}_{\psi }{c}_{\theta }& s_{\psi }& -c_{\psi }{s}_{\theta }\\ -s_{\psi }{c}_{\theta }& c_{\psi }& s_{\theta }{s}_{\psi }\\ s_{\theta }& 0& c_{\theta }\end{array}\right]$

and

$R_3(\psi )=\left[\begin{array}{ccc}{c}_{\psi }& s_{\psi }& 0\\ -s_{\psi }& c_{\psi }& 0\\ 0& 0& 1\end{array}\right]$

gives us

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

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

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

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

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