PlanetPhysics/Euler Angle Velocity

The velocity of the Euler angles of a rotating coordinate frame can be derived from the angular velocity vector of this frame with respect to a reference frame. The derivation is lengthy for each Euler angle sequence and is given in their respective entries. One useful result is acheived by combining the angular velocity vector, in terms of Euler angles, with Euler's moment equations yielding the differential equations of motion that characterizes rigid body motion for a given sequence.

Notation: $$cos(\theta) = c_{\theta}, sin(\theta) = s_{\theta} $$

Euler angle velocity of 123 Sequence:

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

Euler angle velocity of 321 Sequence:

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