How to understand the convention of Euler angle?

Question

We have 3 DOF for rotation in 3D space. So to describe an arbitrary rotation, we need to describe its 3 DOF. Euler angle does this by dividing a rotation in 3 steps, first rotate along the Z axis of the world frame ($z_0$), then rotate along Y axis of current frame ($y_1$), and finally rotate along the Z axis of the current frame($z_2$).

And of course we have other conventions like $z-x-z$, $x-y-x$, $y-z-y$, $x-z-x$, $y-x-y$.

And my question is, why the euler angle is defined in this "ZYZ" manner? Why the first and third rotation are about the same axis (of course the first Z and the last Z are in different frames) ? Why not to use ZYX?

Answer 1

One of the advantages 'Z-Y-Z' convention is that the inverse operation is defined also with this convention.

Generally for matrices we have

$(ABC)^{-1}=C^{-1}B^{-1}A^{-1}$

For rotations 'ZYZ' we have

$(\ Rot_z(\alpha)\ Rot_y(\beta)\ Rot_z(\gamma)\ )\ ^{-1}=Rot_z(-\gamma)\ Rot_y(-\beta)\ Rot_z(-\alpha) $.

So the convention is preserved, only angles are taken with minuses and in reverse order.

If we had chosen 'ZYX' the reverse order, based on the general form for the inverse of matrices product, would have been 'XYZ' and to have the operation coded in 'ZYX' order it would require more additional calculations.

Answer 2

It depends entirely on the context, but one advantage of using ZYZ is that the pitch angle (2nd Euler angle) is the angle between the original and final Z axis. Most often, this is thought of as the tilt angle.