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Finding reference quaternion of a satellite in orbit

How can i find the reference quaternion of a satellite after it has detumbled? I need the quaternion to be able to successfully actuate my payload, which in this case will be a hyperspectral imaging ...
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What are quaternions and how are they used to represent spacecraft dynamics?

The title says it all. Quaternions are widely used to represent the orientation of a spacecraft. Why is that, and how do quaternions compare to other alternatives?