Asymptotic behavior of the covariance of quaternion estimators

The asymptotic behavior of the estimation error covariance of quaternion estimators is mathematically examined. It is proved that the condition number of the asymptotic covariance matrix is of the order of the inverse of its largest eigenvalue, so that this matrix becomes asymptotically ill-conditioned as its trace tends to zero. Nevertheless, it is proved that the aforementioned asymptotic behavior cannot be captured by low-order Taylor approximations of the covariance, such as the one computed by the extended Kalman filter. Geometrical interpretation of the results is provided, using tools from differential geometry. The analytical results are demonstrated via a simulation study using the recently introduced quaternion particle filter and the additive quaternion extended Kalman filter.