Averaging quaternions

This paper presents an algorithm to average a set of quaternion observations. The average quaternion is determined by minimizing the weighted sum of the squared Probenius norms of the corresponding attitude matrix differences, subject to the unit-norm constraint in the determined solution. Two cases are presented: one that incorporates scalar weights and one that incorporates general weights on the quaternions. For both cases, the optimal quaternion is found from an eigenvalue/eigenvector decomposition of a matrix formed by the weights and quaternion observations. The relationship between the solution for the average quaternion and familiar q-method for attitude estimation is discussed. Also, the uniqueness of the solution is shown and the relationship of the matrix weighted case to maximum likelihood estimation is given.