## 3 Switched Variational Estimation Filtering in Underwater Navigation
### Proposed Optimal Variational Estimation in Navigation Information Acquisition
: _The initialization values of \(\alpha(t_{k})\) and \(\beta(t_{k})\) influence the estimation accuracy; experiments were employed to find the optimal solutions of \(\alpha(0)\) and \(\beta(0)\). When \(\alpha\in[1,10)\) and \(\beta\in[1,10)\), the algorithm was easily divergent, and when \(\alpha\in(25,+\infty)\) and \(\beta\in(25,+\infty)\), the algorithm also diverged. Therefore, the range of the parameters was set as \(\alpha\in[10,25]\) and \(\beta\in[10,25]\) in the following experiments._

When the observation equation is linear, the solutions are as expressed in Figure 3. When the observation equation is nonlinear, the solutions are as denoted in Figure 4. From Figures 3 and 4, it can be concluded that \(\alpha(0)\) and \(\beta(0)\) impact the RMSE. From the results of experiments, \((\alpha(0),\beta(0))=(15,15)\) was chosen as an optimal solution to guarantee an algorithm with better accuracy and robustness. **Remark 8**. : _For the robustness verification of the proposed SVEF, we compared the proposed SVEF algorithm and other conventional algorithms by using the Monte Carlo (MC) method. The definition of MC RMSE is:_

\[\text{MC RMSE}= \frac{1}{mc}\sum_{j=1}^{mc}\text{RMSE}(j) \tag{65}\]

_where \(\text{RMSE}(j)\) refers to the RMSE at the jth Monte Carlo operation; mc denotes the total number of Monte Carlo operations, which was set to 100._

### Comparison of State Estimation Algorithms

**Case 1: _The measurement model is linear_**

In Case 1, the measurement vector was \(\mathbf{Z}(t_{k})= \begin{bmatrix}x(t_{k})&y(t_{k})&z(t_{k})\end{bmatrix} ^{T}\). Figure 5 shows that the proposed SVEF achieved the best performance among all the comparison methods. Figure 6 shows that the robustness of SVEF was also the best among all compared algorithms. Aiming to more comprehensively evaluate the performance of these four methods, Table 2 also shows the calculated MC RMSE values. Table 2 shows that the MC RMSE value of SVEF was the smallest of the four algorithms. The improvement in the MC RMSE was about 7.79% over EKF, 5.17% over UKF, and 6.70% over CKF. Figure 4: RMSE of the localization (nonlinear observation equation). Figure 3: RMSE of the localization (linear observation equation). The traditional EKF uses the partial derivative to linearize the nonlinear process and measurement functions. However, truncation errors are introduced when calculating the Kalman gain, so that the errors of \(\mathbf{X}(t_{k})\) and \(\mathbf{P}(t_{k})\) become larger. The sample method for UKF and CKF is fundamentally the sample of a few points. When the dimensions of the state vector \(n_{x}\) are more than three, the weight function of the UKF central sampling point is negative, but, in reality, the UKF and CKF have similar accuracies when the dimensions of state vector \(n_{x}\) are less than seven. The proposed SVEF uses the VE method to update the \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithms** & **EKF** & **UKF** & **CKF** & **SVEF** \\ \hline MC RMSE (m) & 6.6054 & 6.4225 & 6.5284 & 6.0907 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: MC RMSE for different algorithms. Figure 5: Position errors for different algorithms. Figure 6: RMSEs for different algorithms. state vector in real time in each iteration process, and a predicted error covariance matrix with high accuracy is achieved. **Case 2:**_The measurement model is nonlinear_

In Case 2, the measurement vector was selected as \(\mathbf{Z}(t_{k})= \begin{bmatrix}r(t_{k})&d(t_{k})&\theta(t_{k})\end{bmatrix} ^{T}\). Figure 7 shows that the proposed SVEF achieved the best performance among the four algorithms. It is easy to see from Figure 7 MISSING that the robustness of SVEF was better than that of the other three algorithms. To more comprehensively evaluate the performance of these four methods, Figure 8 and Table 3 show the calculated MC RMSE values. The MC RMSE value of the SVEF was the smallest of the four algorithms. The improvement in the MC RMSE was about 62.84% over EKF, 59.90% over UKF, and 59.39% over CKF.