  A novel single-frame quaternion estimator processing two vector observations
is introduced. The singular cases are examined, and appropriate rotational
solutions are provided. Additionally, an alternative method involving
sequential rotation is introduced to manage these singularities. 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
covariance 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. A comprehensive derivation of the individual components
of the quaternion additive error covariance matrix is presented. This not only
provides increased accuracy but also alleviates issues related to singularity.
