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Rik
on 23 Aug 2018

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

z =

0.5132 0.6911 0.2549 0.4405

>> a = 2*acosd(z(4)) % min angle rotation from BODY1 to BODY2

a =

127.7227

But, again, these calculations are dependent on how I have the quaternions defined. Your specific case may be different.

James Tursa
on 6 May 2021

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.

James Tursa
on 13 May 2020

What you are describing is just the math for the scalar part of the quaternion multiply I described. This still only recovers the half angle of the rotation since it is the same calculation. I.e., the scalar part of the quatmultiply(quatconj(x),y) is this:

xs*ys - dot(-xv,yv) = xs*ys + dot(xv,yv)

And this is the same as the dot(x,y) operation you describe.

So if the quaternions represent two coordinate system transformations (my assumption), this result contains cos(half rotation angle), so acos( ) of this will recover the half rotation angle, not the full rotation angle. Or maybe I am misunderstanding you?

To be more robust for numerical issues, I could have done this:

z = quatmultiply(quatconj(x),y)

angle = 2 * atan2(norm(zv),zs)

where zv is the vector part of z and zs is the scalar part of z (either first or last element depending on convention).

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