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的新型闭式四元数估计器。该估计器的简洁性使其能够提供清晰的物理洞察，并得到偏差关于四元数误差协方差矩阵的闭式表达式。在任意概率分布假设下，该表达式可近似至测量噪声的二阶项。本文放宽了二阶近似假设，并在高斯分布假设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。本技术报告全面推导了四元数加性误差协方差矩阵的各个分量。