Analytical solution to and error analysis of the quaternion based similarity transformation considering measurement errors in both frames

The similarity transformation between two coordinate frames, is widely adopted in science and engineering. The transformation parameters are estimated using coordinate determinations of a set of common points in both frames. The quaternion is employed to represent the rotation transformation; and a 3 × 1 error vector is defined to represent the quaternion estimation error. Coordinate determinations in both frames are assumed noisy. An analytical least-squares solution is derived in which the quaternion estimate is the eigenvector of a 4 × 4 symmetric matrix corresponding to its largest eigenvalue. It is found that as long as a practically meaningful quaternion estimate exists, the largest eigenvalue must be single. Error analysis of this solution is investigated in detail in which the error analysis of the largest eigenvalue-eigenvector pair plays a pivotal role. Monte Carlo experiments are conducted and the results validate the consistency of the developed error analysis.