This paper addresses the problem of square-rooting in the Cubature Kalman Filtering (CKF) originated from Arasaratnam and Haykin in 2009. Presently, this technique has been accommodated to various cubature rules, including high-degree ones. Since its discovery the CKF has become one of the most powerful state estimation methods because of outstanding performance and robustness in numerous engineering applications. Its high-degree versions are shown to be accurate and even comparable to particle filters, which are considered to be among the most effective algorithms for treating nonlinear stochastic systems. However, the lack of square-root implementations within high-degree CKFs makes them vulnerable to round-off and other errors committed because of a potential covariance matrix positivity loss, which may encounter in practice. This shortcoming affects severely and fails high-degree CKFs since the Cholesky factorization of predicted and filtering covariances underlying the filters in use may not be fulfilled for indefinite matrices. Here, we resolve it by means of hyperbolic $QR$ transforms applied for yielding $J$-orthogonal square roots. Our novel square-root algorithms are justified theoretically and examined and compared numerically to the existing non-square-root CKF and some other available filters in a simulated flight control scenario, including that with ill-conditioned measurements.

本文讨论了Cubature卡尔曼滤波中的平方根问题（CKF）起源于Arasaratnam和Haykin于2009年。目前，此技术已适应各种培养规则，包括高学历规则。自发现以来，由于在众多工程应用中出色的性能和鲁棒性，CKF已成为最强大的状态估计方法之一。它的高阶版本被证明是准确的，甚至可以与粒子滤波器相媲美，后者被认为是处理非线性随机系统的最有效算法之一。但是，由于在实践中可能会遇到潜在的协方差矩阵正性损失，因此高级CKF中缺少平方根实现方式会使它们很容易出现舍入和其他错误。由于不确定的矩阵可能无法满足预测和滤波协方差的Cholesky因式分解，因此该缺点会严重影响高级CKF并使之失败。在这里，我们通过双曲来解决它$问\mathrm{[R}$ 转换用于屈服 $Ĵ$-正交平方根。我们的新颖平方根算法在理论上是合理的，并已进行了检查，并在模拟飞行控制方案中将其与现有的非平方根CKF以及其他一些可用的滤波器进行了数值比较，包括条件不佳的测量。