## 3 Cubature-Principle-Assisted IMM-Adaptive UKF Algorithm (CPIMM-NAUKF)
### Cpiim
#### 3.1.3 Accuracy Analysis of the CPIMM  
In the literature [22], the approximate precision of the mean of the Gaussian random variable after nonlinear propagation, by using the cubature principle, has been analyzed. In this section, the accuracy of the covariance matrix of the cubature principle will be analyzed. Assume that \(f(\mathbf{\chi})\in R^{n}\) is the nonlinear mapping from \(\mathrm{R}^{l}\) to \(\mathrm{R}^{n}\); \(\mathbf{\chi}\sim N(\mathbf{\chi}:\hat{\mathbf{\chi}},\mathbf{P})\); \(\Delta\mathbf{\chi}\sim N(\Delta\mathbf{\chi}:0,\mathbf{P})\). If the Taylor series expansion of \(\mathbf{y}=f(\mathbf{\chi})\) is carried out at \(\hat{\mathbf{\chi}}\) with two derivatives, we have  
\[\mathbf{y}=f(\hat{\mathbf{\chi}}+\Delta\mathbf{\chi})=f(\hat{\mathbf{\chi}})+f^{\prime}(\hat{ \mathbf{\chi}})\Delta\mathbf{\chi}+\frac{1}{2}\Big{[}\Delta\mathbf{\chi}^{T}f_{i}^{\prime \prime}(\hat{\mathbf{\chi}})\Delta\mathbf{\chi}\Big{]}_{i} \tag{20}\]where \(f^{\prime}(\hat{\mathbf{\chi}})\) is the Jacobian matrix with a \(n\times l\) dimension; \(f^{\prime\prime}\left(\hat{\mathbf{\chi}}\right)\) represents \(n\) matrixes with \(n\times l\) dimensions, which are called Hessian matrixes; \(f_{i}^{\prime\prime}\left(\hat{\mathbf{\chi}}\right)\) is the \(i^{\text{th}}\) Hessian matrix. Thus, we have  
\[f_{i}^{\prime}(\hat{\mathbf{\chi}}) =\left[\frac{\partial f_{i}}{\partial x_{1}\partial x_{1}},\frac{ \partial f_{i}}{\partial x_{2}},\frac{\partial f_{i}}{\partial x_{1}\partial x _{2}}\right.\left.\right.\left.\right.\left.\right.\left.\right.\] \[f_{i}^{\prime\prime}\left(\hat{\mathbf{\chi}}\right) =\left[ \begin{array}{cccc}\frac{\partial^{2}f_{i}}{\partial x _{1}\partial x_{1}}&\frac{\partial^{2}f_{i}}{\partial x_{1}\partial x_{2}}& \cdots&\frac{\partial^{2}f_{i}}{\partial x_{1}\partial x_{l}}\\ \frac{\partial^{2}f_{i}}{\partial x_{2}\partial x_{1}}&\frac{\partial^{2}f_{i }}{\partial x_{2}\partial x_{2}}&\cdots&\frac{\partial^{2}f_{i}}{\partial x_{ 2}\partial x_{l}}\\ \ldots&\cdots&\cdots&\cdots\\ \frac{\partial^{2}f_{i}}{\partial x_{l}\partial x_{1}}&\frac{\partial^{2}f_{i }}{\partial x_{l}\partial x_{2}}&\cdots&\frac{\partial^{2}f_{i}}{\partial x_{ l}\partial x_{l}}\\ \end{array} \right]_{\chi=\hat{\mathbf{\chi}}}. \tag{21}\]  
Calculating the first and second moments of Equation (20), we obtain  
\[ \begin{split} E(\mathbf{y})&=f(\hat{\mathbf{\chi}})+\tfrac{1}{2}\big{[} tr\big{(}f_{i}^{\prime\prime}\left(\hat{\mathbf{\chi}}\right)\mathbf{P}\big{]}_{i}\\ D(\mathbf{y})&=f^{\prime}(\hat{\mathbf{\chi}})\mathbf{P}\big{(}f^{\prime}(\hat{\mathbf{ \chi}})\big{)}^{T}+\tfrac{1}{2}\Big{[}\text{tr}\big{(}f_{i}^{\prime\prime} \left(\hat{\mathbf{\chi}}\right)\mathbf{P}f_{j}^{\prime\prime}\left(\hat{\mathbf{\chi}} \right)\mathbf{P}\big{)}\Big{]}_{i,j}\end{split}  \tag{22}\]  
where \(\text{tr}(\bullet)\) represents the trace of a matrix. Equation (18) can be rewritten as  
\[\hat{\mathbf{y}}=f(\hat{\mathbf{\chi}})+\frac{1}{2l}\sum_{r=1}^{l}(f(\chi_{r})-2f(\hat {\mathbf{\chi}})+f(\chi_{-r})). \tag{23}\]  
Substituting Equation (23) to (19), and assuming \(\mathbf{D}=\sum\limits_{r=1}^{l}(f(\chi_{r})-2f(\hat{\mathbf{\chi}})+f(\chi_{-r}))\), we have  
\[ \begin{split}\mathbf{P}_{y}&=\tfrac{1}{2l}\sum\limits_{ 1}^{2l}(f(\mathbf{\chi}^{r})-f(\hat{\mathbf{\chi}}))(\cdot)^{T}-\tfrac{1}{4l^{2}}\sum \limits_{1}^{2l}(f(\chi^{r})-f(\hat{\mathbf{\chi}}))\mathbf{D}^{T}\\ &-\tfrac{1}{4l^{2}}\sum\limits_{1}^{2l}(f(\mathbf{\chi}^{r})-f(\hat{ \mathbf{\chi}}))^{T}\mathbf{D}+\tfrac{1}{4l^{2}}\mathbf{D}\mathbf{D}^{T}\\ &=\tfrac{1}{2l}\sum\limits_{r=-l}^{l}(f(\mathbf{\chi}^{r})-f(\hat{ \mathbf{\chi}}))(\cdot)^{T}-\tfrac{1}{4l^{2}}\mathbf{D}\mathbf{D}^{T}.\end{split}  \tag{24}\]  
When \(\Delta\chi_{r}=\sqrt{l}\sigma_{r}\mathbf{v}_{r}\to 0\) is correct by considering the selection process of cubature points shown in Equation (17), we have  
\[