## 3 Novel Robust Adaptive CKF
: _If \(\hat{\mathbf{P}}_{zz,k|k-1}\) is the residual covariance matrix estimated after the introduction of new measurement information, it is defined as the updating residual covariance matrix; \(\widetilde{\mathbf{P}}_{zz,k|k-1}\) is a theoretical residual covariance matrix obtained by adaptive filtering, and \(\mathbf{P}_{zz,k|k-1}\) is a residual covariance matrix obtained by the covariance propagation law. The selection of the optimal adaptive factor shall ensure that the following equation holds:_  
\[\widetilde{\mathbf{P}}_{zz,k|k-1}\ =\ \hat{\mathbf{P}}_{zz,k|k-1} \tag{33}\]  
_Then, the optimal adaptive factor is:_  
\[\vartheta_{k}\ =\ \frac{tr(\mathbf{P}_{zz,k|k-1}-\overline{\mathbf{R}}_{k})}{tr( \hat{\mathbf{P}}_{zz,k|k-1}-\overline{\mathbf{R}}_{k})} \tag{34}\]  
**Proof of Theorem 1. ** Let \(\widetilde{\mathbf{x}}_{k|k-1}\) a step-by-step prediction error, then:  
\[\hat{\pi}_{k|k-1}\ =\ x_{k}-\widetilde{\mathbf{x}}_{k|k-1} \tag{35}\]  
Thus, the filtered residual is:  
\[\mathbf{r}_{k}\ =\ z_{k}-h(\hat{\pi}_{k|k-1})\ =\ z_{k}-h(\mathbf{x}_{k}-\widetilde{\mathbf{ x}}_{k|k-1}) \tag{36}\]A first-order [PERSON] expansion of the second term on the right-hand side of the equation:  
\[h(\mathbf{x}_{k}-\widetilde{\mathbf{x}}_{k|k-1})\approx h(\mathbf{x}_{k})-\frac{\partial h}{ \partial\mathbf{x}}\Big{|}_{x=\hat{x}_{k|k-1}}\times\widetilde{\mathbf{x}}_{k|k-1}\ =\ h(\mathbf{x}_{k})-\mathbf{D}_{k}\times\widetilde{\mathbf{x}}_{k|k-1} \tag{37}\]  
Bringing the above equation into the following equation:  
\[\mathbf{r}_{k}\ =\ \mathbf{z}_{k}-h(\mathbf{x}_{k})+\mathbf{D}_{k}\widetilde{\mathbf{x}}_{k|k-1}\ =\ \mathbf{v}_{k}+\mathbf{D}_{k} \widetilde{\mathbf{x}}_{k|k-1} \tag{38}\]  
Then the residual covariance matrix A is:  
\[\mathbf{P}_{zz,k|k-1}\ =\ E\big{\{}\mathbf{r}_{k}\mathbf{r}_{k}^{\mathrm{T}}\big{\}}\ =\ E\Big{\{}\big{(}\mathbf{v}_{k}+\mathbf{D}_{k} \widetilde{\mathbf{x}}_{k|k-1}\big{)}\big{(}\mathbf{v}_{k}+\mathbf{D}_{k}\widetilde{\mathbf{x }}_{k|k-1}\big{)}^{\mathrm{T}}\Big{\}}\ =\ E\big{(}\mathbf{D}_{k}\widetilde{\mathbf{x}}_{k|k-1} \widetilde{\mathbf{x}}_{k|k-1}^{\mathrm{T}}\big{)}+\ E\big{(}\mathbf{v}_{k}\mathbf{v}_{k} ^{\mathrm{T}}\big{)}\ =\ \mathbf{D}_{k}\mathbf{P}_{k|k-1}\mathbf{D}_{k}^{\mathrm{T}}+ \overline{\mathbf{R}}_{k} \tag{39}\]  
Note the equation using the measurement noise covariance matrix after being modified by robust CKF. Furthermore, in adaptive filtering, \(\mathbf{P}_{k|k-1}\) is modified to \(\theta_{k}^{-1}\mathbf{P}_{k|k-1}\), and then the theoretical residual covariance matrix obtained by adaptive filtering is:  
\[r_{i}\ =\ \left(\mathbf{z}_{k}-\hat{\mathbf{z}}_{k|k-1}\right)_{i} \tag{31}\]0  
From the equation:  
\[r_{i}\ =\ \left(\mathbf{z}_{k}-\hat{\mathbf{z}}_{k|k-1}\right)_{i} \tag{31}\]1  
Multiply the two sides of the upper form by A and move the item to obtain  
\[r_{i}\ =\ \left(\mathbf{z}_{k}-\hat{\mathbf{z}}_{k|k-1}\right)_{i} \tag{31}\]2  
Further derivation:  
\[r_{i}\ =\ \left(\mathbf{z}_{k}-\hat{\mathbf{z}}_{k|k-1}\right)_{i} \tag{31}\]3  
Take the traces of the matrices on both sides of the expressions. Then, the expression of the optimal adaptive factor can be obtained by the migration transformation. \(\square\)  
In practical application, the regulatory factor in adaptive filtering algorithm is usually not greater than 1 [32; 33], so we further determine the optimal adaptive factor as follows:  
\[\theta_{k}\ =\ \left\{ \begin{array}{ll}1&tr(\mathbf{\hat{P}}_{zz,k|k-1})\leq tr( \mathbf{P}_{zz,k|k-1})\\ \frac{tr(\mathbf{P}_{zz,k|k-1}-\overline{\mathbf{R}}_{k})}{tr(\mathbf{P}_{zz,k|k-1}- \overline{\mathbf{R}}_{k})}&tr(\mathbf{\hat{P}}_{zz,k|k-1})>tr(\mathbf{P}_{zz,k|k-1})\end{array} \right. \tag{44}\]  
Taking into account the fact that the numerator and the denominator in the upper equation all contain the variance term of measurement noises. The approximate optimal adaptive factor expression can also be obtained by using the common polynomial. \[\theta_{k}\approx\left\{