Analytical solution to and error analysis of the quaternion based similarity transformation considering measurement errors in both frames. (November 2017)

• Record Type: Journal Article
• Authors: Chang, Guobin; Xu, Tianhe; Wang, Qianxin; Liu, Ming
• Is Part Of: Measurement. Volume 110(2017)
• Page Start: 1
• Page End: 10
• Publication Date: 2017-11
• Subjects: Similarity transformation -- Least-squares -- Quaternion -- Error analysis -- Eigen decomposition; Weights and measures -- Periodicals; Measurement -- Periodicals; Measurement; Weights and measures; Periodicals; 530.8
• DOI: 10.1016/j.measurement.2017.06.013
• Languages: English
• ISSNs: 0263-2241

Abstract

Highlights: A realistic error model is assumed considering error in both frames. Quaternion is employed to represent arbitrary rotation. An analytical solution is derived based on eigen decomposition. A rigorous error analysis is developed in detail. Monte Carlo experiments are conducted to validate the consistency. Abstract: 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.