Angle between 2 quaternions
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James Tursa on 23 Aug 2018
Edited: James Tursa on 23 Aug 2018
Assuming these represent attitude rotations from one coordinate frame to another, if you are simply asking what is the minimum rotation to take you from one quaternion to the other, you simply multiply one quaternion by the conjugate of the other and then pick off the rotation angle of the resulting quaternion.
But we really need to know what these quaternions represent, and what angle you are trying to recover, before we know what you want.
E.g., suppose x and y represent ECI->BODY rotation quaternions, and you want to know the minimum rotation angle that would take you from the x BODY position to the y BODY position. Then you could do this:
>> x = [ 0.968, 0.008, -0.008, 0.252]; x = x/norm(x); % ECI->BODY1
>> y = [ 0.382, 0.605, 0.413, 0.563]; y = y/norm(y); % ECI->BODY2
>> z = quatmultiply(quatconj(x),y) % BODY1->BODY2
0.5132 0.6911 0.2549 0.4405
>> a = 2*acosd(z(4)) % min angle rotation from BODY1 to BODY2
But, again, these calculations are dependent on how I have the quaternions defined. Your specific case may be different.
Erik Blake on 13 May 2020
Just as with vectors, the cosine of the rotation angle between two quaternions can be calculated as the dot product of the two quaternions divided by the 2-norm of the both quaternions. Normalization by the 2-norms is not required if the quaternions are unit quaternions (as is often the case when describing rotations).
As with vectors, the dot product is calculated by summing the products of the four elements of the quaternion.
Note that this calculation yields the full rotation angle, not the half-angle as when converting from quaternions to rotation vectors.