An Analysis of the Quaternion Attitude Determination Filter
- PDF / 2,896,695 Bytes
- 18 Pages / 496.063 x 722.835 pts Page_size
- 16 Downloads / 255 Views
An Analysis of the Quaternion Attitude Determination Filter1 Mark E. Pittelkau' Abstract The full-quatemion attitude determination filter is analyzed to address questions of covariance singularity and quatemion normalization. It is shown how nonsingularity of the covariance in the extended Kalman filter depends on the initial covariance, the process noise matrix, and implementation details of the filter. The covariance of a normalized quatemion estimate and the various means to achieve normalization are examined. The effect of a quaternion measurement update on the covariance and on the norm of the estimated quatemion is analyzed. It is also shown that the multiplicative and additive quatemion updates are equivalent. These are distinguished from an update called "rotational," which was proven elsewhere to be the constrained maximum-likelihood optimal update. It is demonstrated that the reduced-order body-referenced attitude determination filter is embedded in the fullquatemion filter.
Introduction Attitude determination filters that include the full attitude quaternion in their state vector and that model the quaternion error covariance potentially have a singular covariance matrix when subject to the quaternion norm constraint or otherwise may exhibit problems related to the quaternion norm and its covariance. Singularity of the quaternion covariance matrix was mentioned in the landmark paper on attitude determination [1] and is often quoted as fact [2], although some maintain that the covariance is nonsingular [3]-[5]. Given that there is disagreement in the community, it would appear that the nature of the covariance matrix has not been fully analyzed in the literature and is not well understood. The following analysis should give greater insight into the nature of the covariance matrix and should clarify and resolve issues regarding singularity and quaternion normalization. This analysis is in the context of linearization in the Extended Kalman Filter (EKF): the true quatemion covariance is not necessarily singular (although it can be ill conditioned-see Appendix A); however, the EKF doesn't know anything about that. The EKF is simply a first-order approximation to the true probability 'Presented as paper AAS 03-194 at the AAS/AIAA Space Flight Mechanics Meeting, Ponce, Puerto Rico, February 9-12, 2003. 2The Johns Hopkins University Applied Physics Laboratory, Space Department, Laurel, MD, 20723-6099. e-mail: [email protected] @ieee.org
103
Pittelkau
104
density, and the computed EKF covariance may become singular even when the true covariance is nonsingular. The purpose of this paper is to show under what conditions the EKF covariance is singular, or nonsingular, or nonsingular but ill conditioned. Another problem with the EKF in attitude determination is that the estimated quaternion may not maintain unity norm, depending on the implementation of the EKF. Various means of normalizing the estimated quatemion were addressed in [6, 7]. We reexamine the various means of normalizing the es
Data Loading...
