Question about recovering XZ rotation of root from the motion representation

[pangwong]

Hi,
in appendix, I see the motion representation is defined below:

Specifically, the i-th frame pose is defined by a tuple of root angular velocity ( ̇ra ∈ R) along Y-axis, root linear velocities ( ̇rx, ̇rz ∈ R) on XZ-plane, root height ry ∈ R, local joints positions (jp ∈ R3N −1), and velocities (jv ∈ R3N ), where N denotes the number of joints.

Obviously, we can recover 3D global position of root from XZ-plane velocities and root height. We can also recover the Y rotation of root by using Y-rotation velocity . My question is how the X-rotation and Z-rotation of root can be recovered from the representation defined above.