Graduate Why are Euler's angles picked exactly that way?

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Euler's angles are defined by a specific sequence of rotations around the axes: first around the z-axis, then the x-axis, and finally the z-axis again. This particular order is crucial because it leads to a unique representation of orientation in three-dimensional space. Alternative sequences, such as rotating around the x-axis first, would result in different transformation equations and potentially ambiguous orientations. The choice of convention in parametrizing rotations is essential for consistency in applications like robotics and aerospace. Understanding these conventions is key to accurately describing motion and orientation in 3D systems.
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I'm wondering why exactly those angles are picked to describe the orientation of the rotating body.
So the Euler's angles are described like this:
xyz-x'y'z' (first rotation around z axis)
x'y'z'-x''y''z'' (second rotation around x')
x''y''z''-XYZ (third rotation around z'')
So I've been thought it goes like this, now I'm wondering why? Why exactly these angles and why this order? Why can't it go like this for example:
xyz-x'y'z' (rotate around x)
x'y'z'-x''y''z'' (rotate around y')
x''y''z''-XYZ (rotate around z'')
Can the motion be described this way? The equations of transformation of xyz-XYZ would be different for sure.
 
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There are all kinds of conventions around to parametrize the rotations. The Euler angles are just the most often used ones.
 
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Thread 'Reference frames, center of rotation, etc'

Topic about reference frames, center of rotation, postion of origin etc Comoving ref. frame is frame that is attached to moving object, does that mean, in that frame translation and rotation of object is zero, because origin and axes(x,y,z) are fixed to object? Is it same if you place origin of frame at object center of mass or at object tail? What type of comoving frame exist? What is lab frame? If we talk about center of rotation do we always need to specified from what frame we observe?
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