## 3 Novel Robust Adaptive CKF
\begin{array}{ll}1&tr(\mathbf{\hat{P}}_{zz,k|k-1})\leq tr (\mathbf{P}_{zz,k|k-1})\\ \frac{tr(\mathbf{P}_{zz,k|k-1})}{tr(\mathbf{P}_{zz,k|k-1})}&tr(\mathbf{\hat{P}}_{zz,k|k-1} )>tr(\mathbf{P}_{zz,k|k-1})\end{array} \right. \tag{45}\]  
\(\mathbf{\hat{P}}_{zz,k|k-1}\) can be obtained by the residual error estimation of the observations. The sliding window method is adopted, namely:  
\[tr(\mathbf{\hat{P}}_{zz,k|k-1})\ =\ tr(\mathbf{r}_{k}\mathbf{r}_{k}^{\mathrm{T}})\ =\ \mathbf{r}_{k}^{\mathrm{T}}\mathbf{r}_{k}\ =\ \sum_{i=1}^{m}\left(\mathbf{z}_{k}-\hat{\mathbf{z}}_{k|k-1}\right)_{i} \tag{46}\]In order to further clarify the robust adaptive CKF algorithm, the following steps are given:  
**Step 1. **: Set the initial condition. \[\hat{x}_{0}\ =\ E(x_{0})\qquad P_{0}\ =\ E[(x_{0}-\hat{x}_{0})(x_{0}-\hat{x}_{0})]^{ \text{T}} \tag{47}\]
**Step 2. **: Forecast the update. For a given A and B, according to Equations (12)-(26), the state prediction value C and its prediction covariance matrix D are obtained. **Step 3. **: Calculate the state volume points in the measurement updating. The matrix decomposition of the state prediction covariance matrix is completed by using Equation (17), and replacing Equation (18) with the following equation, adaptive adjustment of the model will be complete:  
\[\mathbf{X}_{j,k|k-1}^{h}\ =\ \sqrt{\theta_{k}^{-1}}\mathbf{S}_{k|k-1}\xi_{j}+\hat{x}_{ k|k-1} \tag{48}\]
**Step 4. **: Calculate the predicted values and the residual covariance matrix. Computational observational prediction \(\hat{\mathbf{z}}_{k|k-1}\) and predictive residual covariance matrix \(\mathbf{P}_{zz,k|k-1}\) are calculated by Equations (19)-(21). **Step 5. **: Robust correction. The robust correction of the algorithm is completed according to the Equations (26)-(29). **Step 6. **: Adaptive factor regulation model. Use the approximate expression Equation (45) to calculate the optimal adaptive factor, if the value is 1, then continue the following steps; if the value is less than 1, the value is substituted into Equation (48), and then by the Equations (19), (20) and (26) calculate the adaptive factor to adjust the revised prediction residual covariance matrix. **Step 7. **: Update the measurement. According to Equations (22)-(25), the update calculation of the state variable and its corresponding state covariance matrix is completed. **Remark 1**: _The proposed robust adaptive cubature Kalman filter (CKF) could deal with the problem of inaccurately known system model and noises statistics simultaneously. The proposed method includes two steps of improvement for the basic CKF. First, in order to overcome the kinematic model error, we introduce an adaptive factor to adjust the covariance matrix of state prediction, and process the influence introduced by the dynamic disturbance error. Second, aiming at overcoming the abnormality error, we adopt the robust estimation theory to adjust the CKF algorithm online. Our algorithm is not simply a cumulative combination of existing methods, but a derivation method considering the mathematical complexity and real-time, which ensures the computational efficiency of the algorithm and is more suitable for highly dynamical application scenarios._