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I was recently introduced to Euler Angles in a Dynamics course, but I am confused on the difference between precession and spin angles. Both precession and spin consist in rotating a coordinate system about the $z$-axis, which means that they have the same transformation matrices.

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Both precession and spin consist in rotating a coordinate system about the $z$-axis, which means that they have the same transformation matrices.

Yes, but in between the two there is a rotation about $x$. (Note: Some use $y$ for the middle rotation). This makes all the difference in the world for describing the orientation of a world with respect to the ecliptic, the original usage of Euler rotation sequences.

A bit overly simplified, the initial rotation about the original $z$-axis is the precession angle. Precession for a planet is very slow. The second rotation about the once-rotated $x$-axis is the nutation angle. For the Earth, this closely corresponds to the nearly constant obliquity. The final rotation about the twice-rotated $z$ axis is the daily rotation angle.

With two exceptions, given any orientation, the Euler sequence that rotates the initial $x$, $y$, and $z$ axes to the $X$, $Y$, and $Z$ axes that correspond to the desired orientation is unique with the constraint that the rotations about $z$ are between 0° (inclusive) and 360° degrees (exclusive) and the intermediate rotation about the once-rotated$x$-axis is between 0° and 180° (exclusive of both).

The two exceptions occur when the intermediate rotation about the once-rotated $x$-axis is 0° or 180°. These situations are called "gimbal lock." Here the distinction between precession and rotation is completely arbitrary; the standard approach is to arbitrarily set one of the two to zero.

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The difference is only obvious when visible. A good example is the precession of a gyroscope.

When a rotating body simply spins around an axle, we call it spin. Spin is on the third Euler angle. The first Euler angle defines the precession. Precession is the change in the orientation of the spin axis.

Please see here:

https://en.wikipedia.org/wiki/Precession

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