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.

翻译：参考文献[1]提出了一种基于两个矢量观测值的新型闭式四元数估计器。该估计器的简洁性使其能够获得清晰的物理洞察，并得到以四元数误差协方差矩阵函数表示的偏差闭式表达式。后者可在假设任意概率分布的情况下，利用底层测量噪声逼近至二阶。本文放宽了二阶假设，在高斯分布条件下给出了精确至四阶的误差协方差表达式。这不仅提高了精度，还缓解了与奇异性相关的问题。本技术报告详细推导了四元数加法误差协方差矩阵的各个分量。