\begin{align} \mathbf{R}_z(\psi)\mathbf{R}_y(\theta)\mathbf{R}_x(\phi) \begin{bmatrix} \omega_x \\ \omega_y \\ \omega_z\end{bmatrix} &= \begin{bmatrix} 0 \\ 0 \\ \dot{\psi} \end{bmatrix} + \mathbf{R}_z(\psi)\begin{bmatrix} 0 \\ \dot{\theta} \\ 0 \end{bmatrix} + \mathbf{R}_z(\psi)R_y(\theta)\begin{bmatrix} \dot{\phi} \\ 0 \\ 0 \end{bmatrix},\\ \begin{bmatrix} \omega_x \\ \omega_y \\ \omega_z\end{bmatrix} &=\begin{bmatrix} \dot{\phi} \\ 0 \\ 0 \end{bmatrix} +\mathbf{R}_x^T(\phi) \begin{bmatrix} 0 \\ \dot{\theta} \\ 0 \end{bmatrix}+\mathbf{R}_x^T(\phi)\mathbf{R}_y^T (\theta) \begin{bmatrix} 0 \\ 0 \\ \dot{\psi} \end{bmatrix} \end{align}
There are two equations. And I know that below equation is derived from above equation. But I don't know how that can be derived. Can someone explain details about process of deriving below equation?(I guess may below one be derived by multiplying reverse matrices of above rotation matrices. But I don't know why there are no yaw transposes matrix at right side of below equation. And I also can't know why there is no pitch transposes matrix at the side of [0 pitch dot 0].)
\begin{align} \ * \ \omega_x\ \omega_y\ \omega_z\ : Angular \ velocity \ of \ rigid \ body \ frame \end{align}
\begin{align}\ * \mathbf{R}_z(\psi)\mathbf{R}_y(\theta)\mathbf{R}_x(\phi) \ : \ Rotation \ matrices\ between\ reference\ frame\ and\ rigid \ body \ frame.\ \end{align} \begin{align}\ * \ dot{\psi} \ dot{\phi} \ dot{\theta} \ : \ Euler \ angle \ rotation \ speed \end{align}