## 3 Novel Robust Adaptive CKF
\tag{29}\]  
where \(\overline{p}_{t_{ii}}\) and \(\overline{p}_{t_{ij}}\) are diagonal elements and non-diagonal elements of \(\overline{\mathbf{P}}\), respectively; \(\sigma_{ii}\) and \(\sigma_{ij}\) are diagonal elements and non-diagonal elements of the original \(\mathbf{R}_{k}\) array; \(r_{i}\) is a residual component corresponding to the observation quantity \(Z_{i}\), \(r_{i}^{\prime}\) is a corresponding standard residual component, and \(\sigma_{r_{i}}\) is a mean square error of \(r_{i}\); and \(c\) is a given constant, usually taken from 1.3 to 2.0. The above operation involves that \(\sigma_{r_{i}}\) and \(r_{i}\) are deterministic. In practice, because the covariance matrix of the measurement residuals is obtained from Equation (21), that is, the variable quantity \(\mathbf{P}_{zz,k|k-1}\) before being modified, the expressions \(\sigma_{r_{i}}\) and \(r_{i}\) are:  
\[\sigma_{r_{i}}\ =\ \left(\mathbf{P}_{zz,k|k-1}\right)_{ii} \tag{30}\]  
\[r_{i}\ =\ \left(\mathbf{z}_{k}-\hat{\mathbf{z}}_{k|k-1}\right)_{i} \tag{31}\]  
Then, \(\overline{\mathbf{P}}_{zz,k|k-1}\) is substituted for \(\mathbf{P}_{zz,k|k-1}\) in Equation (15), and the gain matrix is modified. Furthermore, the subsequent filtering solution is continued. The robust CKF algorithm is actually based on the standard CKF filter and modifies the noise covariance matrix \(\mathbf{R}_{k}\), so as to adjust the filter gain matrix, and ultimately enhance the performance of algorithm against observation errors consisting in satellite signal to restrain the divergence of Kalman filter. Therefore, robust CKF algorithm is more suitable for the near space environment, in which satellite signals are unstable and measurement noise statistical characteristics deviate considerably from the prior information. The robust CKF algorithm solves the influence of the observation abnormal errors in the system. In other words, aiming at the inaccurate measurement noise, the CKF algorithm was improved, and then completing the improvement of the CKF algorithm when there are errors in the dynamic model. It is notable that the inaccuracy of system noise statistical characteristics is also an inducement of dynamic model errors. The dynamic equations established for the satellite signal link may deviate greatly from the real model of the system when a nearby space vehicle makes a highly dynamic maneuvering flight, that is to say, a large dynamic disturbance error may exist in the state equation of the system, and the CKF filter algorithm fixed by the model cannot estimate the state parameters of the system. Therefore, based on the above robust CKF algorithm, a robust adaptive CKF algorithm is proposed for dynamic model errors. It is pointed out in [13] that the error of the dynamic model usually destroys the effect of parameter estimation as a whole; in other words, the errors of the dynamic model will affect the estimation of all state parameter components. Therefore, we consider using an adaptive adjustment factor to modify the model as a whole, which also ensures the computational efficiency of the algorithm. The specific algorithm is that adaptive regulator \(\vartheta_{k}\) modify \(\mathbf{P}_{k|k-1}\), obtaining:  
\[\widehat{\mathbf{P}}_{k|k-1}\ =\ \vartheta_{k}^{-1}\mathbf{P}_{k|k-1}\ =\ \vartheta_{k}^{-1}\left[\frac{1}{2n}\sum_{j=1}^{2n}\mathbf{X}_{j,k-1|k-1}^{*} \mathbf{X}_{j,k-1|k-1}^{*\mathsf{T}}-\hat{\pi}_{k|k-1}\hat{\pi}_{k|k-1}^{\mathsf{ T}}+\mathbf{Q}_{k-1}\right] \tag{32}\]  
where the optimal value of \(\vartheta_{k}\) may be obtained from the residual covariance matrix after predicting and updating, where \(\vartheta_{k}\) is given by the following theorem. **Theorem 1**.