MATLAB: How to find the angle between two quaternions

Question

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.

Answer 1

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:

https://www.mathworks.com/matlabcentral/answers/283830-why-do-we-use-q-1-r-q-when-convert-quaternion-to-dcm?s_tid=answers_rc1-1_p1_MLT

and this post:

https://www.mathworks.com/matlabcentral/answers/415936-angle-between-2-quaternions?s_tid=answers_rc1-2_p2_MLT