# How to define an axis based on multiple quaternions?

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Silas Waxter on 18 Aug 2019
Commented: James Tursa on 7 Feb 2020
I have two sensors (IMUs) that stream quaternions to a data set. During streaming, the sensors are rotated around an unknown axis. I'm looking for a way to get the axis using the set of quaternions.

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James Tursa on 7 Feb 2020
See this link for a discussion of the MATLAB quaternion convention:

James Tursa on 2 Dec 2019
Edited: James Tursa on 2 Dec 2019
Assume you have the following:
q1 = quaternion from ECI to BODY at time1 (i.e., BODY1 frame)
q2 = quaternion from ECI to BODY at time2 (i.e., BODY2 frame)
Then do a quaternion multiply as follows
q = conjugate(q1) * q2 = quaternion from BODY1 to BODY2
And extract the rotation axis as follows
v = vector part of q
u = v / norm(v) = unit axis of rotation in BODY coordinates
How you do the conjugation and the quaternion multiply and how you extract the vector part depends on the quaternion convention used. For MATLAB toolboxes the vector part is in the last three elements of the quaternion. But for your streaming quaternions it might be different. You would need to verify all this in order to write the multiply and extraction code correctly.

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Silas Waxter on 5 Dec 2019