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.

翻译：参考文献[1]提出了一种基于两个矢量观测的新型封闭式四元数估计器。该估计器的简洁性有时会导致奇异性表达式，当前注解为四种奇异情况提供了简单的旋转方案。该估计器能够提供清晰的物理洞察，并给出偏差作为四元数误差协方差矩阵函数的封闭式表达式。后者可在任意概率分布假设下，相对于底层测量噪声近似至二阶精度。本注解放宽了二阶假设，给出了精确至四阶的误差协方差表达式，并在高斯分布假设下，对四元数加性误差协方差矩阵的各个分量进行了全面推导。这不仅提高了精度，还缓解了与奇异性相关的问题。