# Finding and Comparing 2 Sensors Rotations, with same reference frame but different initial Orientation

Let's say we want to Compare two different Arm (Humerus) Rotations (series of quaternions) and we do not care about space translation but only for rotation. To measure each rotation we use the same single Inertial Measurement Unit that report orientation in Quaternions in the following Earth reference frame :

$$X$$-axis = North (roll) , $$Y$$-axis=West (pitch) , $$Z$$-axis=up (yaw) .(Right-handed coordinate system)

Sometimes in the following text I will refer to Euler Angles for better understanding of the problem.

Using Dynamic Time Warping or any similarity measuring algorithm, it's easy to compare these two rotations if both initial orientations are the same.

However, when the initial orientations differ, (i.e the people that are wearing the sensors are facing at different directions) even if from their perspective the rotation that they did is the same, the rotation sequence differs.

Is there a way to ''calibrate'' the sensors , so the initial orientation of each sensor does not concern us? If for example the sensor is placed on the humerus upside down or facing at different directions etc.

Things Ι tried: Assume that every Quartenion of the Quaternion series represents a frame of the rotation. I tried changing the reference frame of each sensor from the Earth frame I mentioned above, to the Initial Orientation Frame of the corresponding sensor. So I multiplied each Quaternion of the Series with the Conjugate Quaternion of the first sample (Initial Orientation Quaternion). It works pretty well provided that both sensors Roll and Pitch value are zero and they differ only in Yaw (Z-axis) but that is not a full solution.

The question is similar to this old unanswered : Determine similarity between two sequence of quaternions while allowing a degree of freedom around $$Z$$ -axis

Remarkable Information on Quaternions and Frame Changing: Springer Coordinate Systems and Transformations