Error Analysis of a Simple Quaternion Estimator: Proof of the Bias and Covariance Matrix

Reference [1] introduces a novel closed-form quaternion estimator from two vector observations. The simplicity of 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. The current note relaxes the second-order assumption and provides an expression for the error covariance that is exact to the fourth order, under the assumption of Gaussian distribution. This not only provides increased accuracy but also alleviates issues related to singularity. This technical note presents a comprehensive derivation of the individual components of the quaternion additive error covariance matrix.