math - "Average" of multiple quaternions?

Question

I'm trying to make the switch from matrices to quaternions for skeletal animation in my OpenGL program, but I've encountered a problem:

Given a number of unit quaternions, I need to get a quaternion that when used to transform a vector will give a vector that is the average of the vector transformed by each quaternion individually. (with matrices I would simply add the matrices together and divide by the number of matrices)

Answer 1

Contrary to popular belief in the computer graphics industry, there is a straightforward algorithm to solve this problem which is robust, accurate, and simple that comes from the aerospace industry. It runs in time linear in the number of quaternions being averaged plus a (largish) constant factor.

Let Q = [a₁q₁, a₂q₂, ..., a_nq_n],

where a_i is the weight of the ith quaternion, and q_i is the ith quaternion being averaged, as a column vector. Q is therefore a 4×N matrix.

The normalized eigenvector corresponding to the largest eigenvalue of QQ^T is the weighted average. Since QQ^T is self-adjoint and at least positive semi-definite, fast and robust methods of solving that eigenproblem are available. Computing the matrix-matrix product is the only step that grows with the number of elements being averaged.

See this technical note in the Journal of Guidance, Control, and Dynamics from 2007, which is a summary paper of this and other methods. In the modern era, the method I quoted above makes a good tradeoff for implementation reliability and robustness, and was already published in textbooks in 1978!