Given multiple quaternions representing orientations, and I want to average them. Each one has a different weight, and they all sum up to one.
How can I get the average of them? Simple multiplication by weights and addition won't work, since it doesn't take into account that (qw, qx, qy, qz) = (-qw, -qx, -qy, -qz)..
I assume you are thinking of unit quaternions and you are using them to represent rotations? If that is the case then here is a paper on the related subject of means and averages in the rotation group. It might not be a very easy read though if you don't understand the notation.
Barring that, Here's what I might try: Pick a canonical form for your quaternions. Then convert each to the canonical form and finally perform your weighted linear combination.