## 3 Switched Variational Estimation Filtering in Underwater Navigation
### Proposed Optimal Variational Estimation in Navigation Information Acquisition
For the accuracy and robustness validation of the proposed SVEF method, realistic underwater tests were carried out. The underwater navigation system was based on MEMS technology and was mainly made up of inertial sensors aided by magnetic sensors. The position and speed solutions were provided by the above sensors mounted on the AUV as it worked in the water. Once the underwater vehicle surfaced, GPS was used to update the navigation information. The camera was used for under water image acquisition. Moreover, to establish a reliable experimental platform, the underwater vehicle was also equipped with some navigation devices. The performance comparison between the proposed SVEF and the conventional filtering algorithms was carried out through a series of underwater experiments. The time interval was 0.01 s for the process model, and the time interval was 1 s for the observation model. **Remark 6**. : _The normalized process noise covariance matrix and measurement noise covariance matrix are defined as \(\overline{\mathbf{Q}}(t)\) and \(\overline{\mathbf{R}}(t_{k})\), respectively. Limited by experimental conditions, the real experiment was performed in a lake. However, to simulate complicated and variable marine environments, the complex process noise and the measurement noise were defined as:_

\[\mathbf{Q}(t)= \begin{cases}\overline{\mathbf{Q}}(t)&\mathrm{P}_{S}(0\leq k_{1 }<0.3)\\ 10\overline{\mathbf{Q}}(t)&\mathrm{P}_{\mathrm{m}}(0.3\leq k_{1}<0.7)\\ 20\overline{\mathbf{Q}}(t)&\mathrm{P}_{I}(0.7\leq k_{1}\leq 1)\end{cases}  \tag{62}\]

\[\mathbf{R}(t)= \begin{cases}\overline{\mathbf{R}}(t)&\mathrm{P}_{S}(0\leq k_{2 }<0.4)\\ 10\overline{\mathbf{R}}(t)&\mathrm{P}_{\mathrm{m}}(0.4\leq k_{2}<0.6)\\ 20\overline{\mathbf{R}}(t)&\mathrm{P}_{I}(0.6\leq k_{2}\leq 1)\end{cases}  \tag{63}\]

_where \(k_{1}\) and \(k_{2}\) represent preset uncorrelated random numbers. The probability interval [0, 1] is roughly divided into three segments, namely [0, 0.3], [0.3, 0.7], and [0.7, 1], which indicate low, moderate, and high impacts on noise, respectively. \(P_{S}(\bullet)\), \(P_{m}(\bullet)\), and \(P_{I}(\bullet)\) refer to the different probabilities under low, moderate, and high noises, respectively. The definition of the observation noise matrix is similar. The probability interval [0, 1] is roughly divided into three segments, namely [0, 0.4], [0.4, 0.6], and [0.6, 1], which indicate low, moderate, and high impacts on the noise, respectively._

The root mean square error (RMSE) of the position was applied to evaluate the performance of these algorithms. The definition of RMSE is:

\[\mathrm{RMSE}= \frac{1}{\mathrm{M}}\sum_{k=1}^{\mathrm{M}}\sqrt{\left[x(t_{k})-x _{k}^{r}\right]^{2}+\left[y(t_{k})-y_{k}^{r}\right]^{2}} \tag{64}\]

where \(x(t_{k})\) and \(y(t_{k})\) are the estimated east and north positions, respectively; \(x_{k}^{r}\) and \(y_{k}^{r}\) are the reference east and north positions, respectively; \(\mathrm{M}\) is the number of the time sequence. The initial parameter settings are shown in Table 1. \begin{table}
\begin{tabular}{c c} \hline \hline
**Initialization Parameter** & **Value** \\ \hline \(\mathbf{P}(t_{0})\) & \(10^{2}\times diag[10(\mathrm{m})^{2},1(\mathrm{m}/\mathrm{s})^{2},10(\mathrm{m })^{2},1(\mathrm{m}/\mathrm{s})^{2},10(\mathrm{m})^{2},1(\mathrm{m}/\mathrm{s} )^{2}]\) \\ \(\overline{\mathbf{Q}}(t_{0})\) & \(10^{-7}\times diag[100(\mathrm{m})^{2},1(\mathrm{m}/\mathrm{s})^{2},100( \mathrm{m})^{2},1(\mathrm{m}/\mathrm{s})^{2},100(\mathrm{m})^{2},1(\mathrm{m}/ \mathrm{s})^{2}]\) \\ \(\overline{\mathbf{R}}_{k}(\mathrm{nonlinear})\) & \(10^{-5}\times diag[1(\mathrm{m})^{2},1(\mathrm{m})^{2},1(\mathrm{deg})]\) \\ \(\overline{\mathbf{R}}_{k}(\mathrm{linear})\) & \(10^{-5}\times diag[1(\mathrm{m})^{2},1(\mathrm{m})^{2},1(\mathrm{m})^{2}]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter settings. ### Selection of Initialization Values

**Remark 7**.