Sequential Rotations and Error Analysis of a Simple Quaternion Estimator

Reference [1] introduces a novel closed-form quaternion estimator from two vector observations. The simplicity of the estimator sometimes yields singular expressions, the current annotation provides the simple rotation schemes for four singular cases. The estimator enables clear physical insights and a closed-form expression for the bias as a function of the quaternion error covariance matrix. The latter could be approximated up to second order with respect to the underlying measurement noise assuming arbitrary probability distribution. This note relaxes the second-order assumption, provides an expression for the error covariance that is exact to the fourth order, and a comprehensive derivation of the individual components of the quaternion additive error covariance matrix, under the assumption of Gaussian distribution. It not only provides increased accuracy but also alleviates issues related to singularity.