## 3 Novel Robust Adaptive CKF  
When CKF is applied to a nonlinear system, it needs to know the mathematical model of the object to be studied and the prior knowledge of the noise statistics. However, if the filter is solved based on an inaccurate mathematical model and inaccurate noise statistics, it may result in a large state estimation error or even divergence. To solve this problem, it is necessary to study the adaptive filtering. For typical dynamic application systems, we focus on the study of adaptive filtering technology under the influence of two types of errors, that is, measuring abnormal errors and dynamic model errors. It is inevitable to produce the gross error in the observations for a dynamic system. For example, when tracking a satellite signal, it is shown that the occurrence of gross errors accounts for 1% to 10% of the total number of observations [30]. To a certain extent, the uncertainty factors are introduced into the statistical characteristics of the measurement noise. Therefore, it should be adjusted on line adaptively. In addition, in the condition of high dynamics and low SNR, the satellite signal tracking of the observed information is very easy to be influenced by a poor environment, and there is a large proportion of noise, so that the noise statistics deviate from the prior statistical characteristics. In serious cases, an abnormal perturbation error may exist. The above problem limits the application of CKF, so it is necessary to adjust the measurement noise in real-time to enhance the ability of the algorithm to resist gross and abnormal errors. In view of the gross error, robust M estimation in robust estimation theory is applied to the CKF algorithm, which can detect the influence degree of gross error, and then the model is adaptively adjusted and corrected, so as to eliminate the influence of the abnormal observation error on the algorithm. The improved CKF algorithm is called the robust CKF algorithm. In what follows, we will derive the algorithm. Since the measurement information only affects the updating process of the measurement, compared with the standard CKF algorithm, the robust CKF algorithm only adjusts and modifies the relevant expressions in the measurement updating equation as follows:  
\[\overline{\mathbf{P}}_{zz,k|k-1}\ =\ \sum_{j\,=\,1}^{2n}w_{j}\mathbf{Z}_{j,k|k-1}^{*} \mathbf{Z}_{j,k|k-1}^{*\mathrm{T}}-\hat{\mathbf{z}}_{k|k-1}\mathbf{\bar{z}}_{k|k-1}^{ \mathrm{T}}+\overline{\mathbf{R}}_{k} \tag{26}\]  
where \(\overline{\mathbf{R}}_{k}\) is an equivalent measurement noise variance matrix corresponding to \(\mathbf{R}_{k}\), and it can be obtained by obtaining an equivalent weight matrix \(\overline{\mathbf{P}}\) in an anti-difference M estimation method. That is:  
\[\overline{\mathbf{R}}_{k}\ =\ \overline{\mathbf{P}}^{-1} \tag{27}\]  
For the calculation of the equivalent weight matrix, the common methods are the IGGIII method, [PERSON] method, Tukey method, and Huber method [31]. Considering that the first three methods can make the diagonal elements of the p matrix 0, the Huber method can guarantee that the diagonal elements of \(\overline{\mathbf{P}}\) matrices are positive. The expression of equivalent is as follows:  
\[\overline{p}_{t_{ii}}\ =\ \left\{ \begin{array}{cc}1/\sigma_{ii}&|r_{i}/ \sigma_{r_{i}}|\ =\ |r_{i}^{\prime}|\leq c\\ c/\sigma_{ii}|r_{i}^{\prime}|&|r_{i}^{\prime}|>c\end{array} \right. \tag{28}\]  
\[\overline{p}_{t_{ij}}\ =\ \left\{ \begin{array}{cc}1/\sigma_{ij}&|r_{i}^{ \prime}|\leq c\text{ and }\Big{|}r_{j}^{\prime}\Big{|}\leq c\\ \frac{c}{\sigma_{ij}\max\left\{\big{|}r_{i}^{\prime}\big{|},\big{|}r_{j}^{ \prime}\big{|}\right\}}&|r_{i}^{\prime}|>c\text{ or }\Big{|}r_{j}^{\prime} \Big{|}>c\end{array} \right.