I have a 3D local xyz coordinate system placed in a world ENU (East-North-Up) coordinate system.
The current relationship between them is shown as follows:
As shown, the angle $\theta$ is known. z-axis is vertical to Up-axis and thus lies on the EN-plane.
Now, I rotate the xyz coordinate system clockwise (seen from above) around Up-axis by an angle $\alpha$. During the rotation, z-axis is always kept on the EN-plane. $\theta$ is also kept as constant.
After the rotation, the position is as follows:
My Question:
Given $\theta$, how can I decompose the rotation around Up-axis $\alpha$ into two rotations around x-axis and y-axis?
What are their relationships?