## 3 Estimation Algorithm  
### Square Root Cubature Kalman Filter  
The SVSF is a model-based filter which efficiently manages structural or parametric uncertainty and takes into account the effects of uncertainties and noise in measurements on filter stability and convergence. The purpose of this study is to improve filter precision and resilient performance against parametric uncertainties by combining the SVSF and SR-CKF. According to the following system, which incorporates a nonlinear process function and a linear measurement equation as:  
\[x_{k}=f(x_{k-1},u_{k-1})+w_{k-1} \tag{6}\]  
\[z_{k}=H_{k}x_{k}+v_{k} \tag{7}\]where \(x_{k}\in R^{n}\) denotes the state variable, \(z_{k}\in R^{m}\) means the measurement vector, and \(u_{k-1}\in R^{n}\) represents the control vector. The \(f(\cdot)\) signifies the nonlinear function which is assumed to be differentiable to the states. The term \(w_{k-1}\) refers to the process noise with a covariance of \(Q_{k}\) and \(v_{k}\) denotes the measurement noise with a covariance of \(R_{k}\). The \(w_{k-1}\) and \(v_{k}\) are zero-mean Gaussian processes and independent of each other. The CKF is derived based on the third-order spherical-radial rule to compute the Gaussian weighted integrals. The cubature rule provides an approximation for m-dimensional integrals with Gaussian weights as detailed below [10]:  
\[\int\limits_{R^{n}}f(x)N(x,\mu,P_{1})dx\approx\frac{1}{2m}\sum\limits_{i=1}^{2 m}f(\mu+\sqrt{P_{1}\overline{\varepsilon}_{i}}) \tag{8}\]  
where the covariance \(P_{1}=\sqrt{P_{1}}{\sqrt{P_{1}}^{T}}\) and the \(\overline{\varepsilon}_{i}\) is the \(i\)th element of \(2m\) cubature points as follows:  
\[\xi_{i}=\left\{ \begin{array}{l}\sqrt{m}\overline{e}_{i},i=1,2,\ldots,m\\ -\sqrt{m}\overline{e}_{i-n},i=n+1,n+2,\ldots,2m\end{array} \right. \tag{9}\]  
\[\overline{e}=\left\{ \begin{array}{l}\left(\begin{array}{l}1\\ \vdots\\ 0\end{array}\right),\quad\left(\begin{array}{l}0\\ \vdots\\ 0\end{array}\right),\quad\ldots,\quad\left(\begin{array}{l}0\\ \vdots\\ 0\end{array}\right),\quad\left(\begin{array}{l}-1\\ \vdots\\ 0\end{array}\right),\quad\ldots\quad\left(\begin{array}{l}0\\ \vdots\\ 0\end{array}\right)\\ -1\end{array} \right\} \tag{10}\]  
where \(\overline{e}_{i}\) means the \(i\)-th element of \(\overline{e}\). It represents the \(i\)-th elementary column vector. Given the benefits of using the square-root form of the CKF, which includes assured covariance properties and a prevention of estimation error covariance divergence, the SR-CKF is employed in this study. Firstly, the state and error covariance are initialized. The Cholesky decomposition is used to calculate the error covariance matrix and measurement noise:  
\[P_{k-1|k-1}=S_{k-1|k-1}S^{T}{}_{k-1|k-1} \tag{11}\]  
\[S_{Qk-1}=Chol(Q_{k}) \tag{12}\]  
The SR-CKF methodology encompasses a time update phase and a measurement update phase. The process of implementing the SR-CKF is outlined. Step 1: The time update stage is designed to predict the state at the next time step based on the dynamic model of the system. Determining the cubature points and advancing them by the state function:  
\[X_{i,k-1}=S_{k-1}\xi_{i}+\hat{x}_{k-1} \tag{13}\]  
\[X^{*}{}_{i,k}=f(X_{i,k-1}) \tag{14}\]  
Then, calculating the state prediction and the square-root of the predicted state error covariance:  
\[\hat{x}_{k|k-1}=\sum\limits_{i=1}^{2n}\omega_{i}X^{*}{}_{i,k} \tag{15}\]  
\[S_{k|k-1}=Tia([\chi^{*}{}_{k|k-1}\quad S_{Qk-1}]) \tag{16}\]  
\[\chi^{*}{}_{k|k-1}=\frac{1}{\sqrt{2m}}[X^{*}{}_{1,k|k-1}-\hat{x}_{k|k-1}\quad X ^{*}{}_{2,k|k-1}-\hat{x}_{k|k-1}\quad X^{*}{}_{3,k|k-1}-\hat{x}_{k|k-1}\quad \cdots\quad X^{*}{}_{2n,k|k-1}-\hat{x}_{k|k-1}] \tag{17}\]  
Step 2: The measurement update stage utilizes new measurement data to correct the predicted state, thereby obtaining more precise estimations.