How to find the angle between two quaternions?

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If quaternions represent an orientation in space, there is an axis between any two orientations and an angle between them. I'm looking for the procedure to find that angle.

Accepted Answer

James Tursa
James Tursa on 17 Aug 2019
Edited: James Tursa on 17 Aug 2019
For example purposes I am using the coordinate frames as ECI and BODY
Q1 = quaternion from ECI->BODY1
Q2 = quaternion from ECI->BODY2
Then perform the following calculation
Q12 = conj(Q1) * Q2 % <- quaternion conjugate and quaternion multiply
Q12 = quaternion from BODY1->BODY2
There may be MATLAB functions to do the conjugate and multiply, but I don't know at the moment. The conjugate of Q1 is simply [Q(1),-Q(2:4)] of course assuming the scalar is the first element.
If we assume the scalar is the first element of the quaternion, matching the MATLAB quaternion functions convention, then you have
Q12(1) = cos(angle/2)
and
Q12(2:4) = sin(angle/2) * e
where e is the unit axis of rotation
From these you can solve for the angle
angle = 2 * atan2(norm(Q12(2:4)),Q12(1))
See also this post:
and this post:
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More Answers (1)

Jim Riggs
Jim Riggs on 17 Aug 2019
Edited: Jim Riggs on 17 Aug 2019
I am more comfortable working with direction cosine matrices, so the way I would do this is to first convert the quaternions to DCM's;
Assume Quaternion A represents the orientation of body A in the I frame
Quaternion B represents the orientation of Body B in the I frame.
The direction cosine matrix, C, that transforms from I to A is defined as:
If the Quaternion is defined as [a, b, c, d], (where a is the scalar part and b, c, d is the vector part) then the direction cosine matrix in terms of the quaternion is
So, first compute the direction cosine matrix from quaternion A (DCMA) and from quaternion B (DCMB).
Now the direction cosine matrix for the transformation from A to B is
DCMAB = DCMB * transpose(DCMA).
Now that you have the transformation matrix from A to B, you can get the Euler angles or rotation vector from this DCM.
Matlab Answers 20190816c.JPG
Using DCMAB, this gives the rotation vector from A to B.
  2 Comments
Jim Riggs
Jim Riggs on 25 Mar 2020
Edited: Jim Riggs on 25 Mar 2020
It sounds like the problem you are working is exactly the same as the original question in this post.
I offered a solution based on converting quaterions to DCM's and then manipulating the DCM's to get the relative DCM from sensor 1 to sensor 2. Then, from this relative DCM, extract the Euler angles, or rotation vector. I suggested this approach because I am less comfortable working with quaternion math, as in James Tursa's answer. His approach is, no doubt, the most direct.
I would advise against working with Euler angles until the very last step, because they are fraught with potential problems, like the indeterminant angles (+-90 or +-180 situation). This is not a sign of gimbal lock, but of an indeterminant solution in the Euler angle. Quaternions and direction cosine matrices do not have these issues, so you should calculate the relative orientation between the two frames using quaternion math (per James Tursa's answer) or using DCM's (as I suggest) and then, in the final step, you can get the relative Euler angles from the relative quaternion or DCM.

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