## 3 Switched Variational Estimation Filtering in Underwater Navigation
### Proposed Optimal Variational Estimation in Navigation Information Acquisition
When the noises become complex and time-varying, there is a risk of losing measurement information. The above experimental results indicated that the SVEF performs better than the EKF, UKF, and CKF in dealing with the noise loss problem. Measurement information may be missing due to the interference of large noise with the sensor; therefore, precise noise information is difficult to obtain. In SVEF, \(\textbf{R}_{k}\), \(\textbf{P}\big{(}t_{k}^{+}\big{)}\), \(\xi(t_{k})\), and \(\tau(t_{k})\) can be adaptively adjusted to guarantee filtering stability. The problems of variable noise and measurement information loss lead to inaccuracy of the Kalman gain calculation and, thus, state variable divergence. The EKF cannot provide an accurate estimation of the state vector. The estimation effect of UKF relies on the weighting function. However, once the state dimension of the UKF is higher than three, the adjustment parameter becomes negative. In the presence of complex noise, the UKF cannot be able to obtain an accurate estimation. Therefore, the performance of UKF under complex noise conditions is also very poor. Although CKF overcomes the deficiencies of the EKF and UKF in some areas, CKF cannot obtain an accurate noise model, resulting in poor positioning performance. The loss of measurement information has a serious effect on \(\textbf{P}_{xz}^{(l+1)}\) and \(\textbf{P}_{zz}^{(l+1)}\), where the Kalman gain is also influenced. For the above problems, the proposed SVEF algorithm can achieve better results. We concluded that compared with other algorithms, the SVEF has a more accurate positioning effect for underwater vehicles. ## 5 Conclusions

The complex and variable ocean environment poses huge difficulties to the precise localization of MEMS-grade navigation systems mounted on AUVs. In addition, because the sampling frequency of the time update is 100 times higher than that of the measurement update, the ocean environment is not conducive to obtaining high-precision AUV positions. Obtaining accurate state estimates is difficult for conventional algorithms. The accuracy and robustness of the proposed SVEF were proved in theory and evaluated through a series of underwater experiments. The results demonstrated that SVEF achieves better AUV position estimation results than EKF, UKE, and CKF, even in the case of variations in measurement error and loss of measurement information. The SVEF can obtain more accurate estimations and error covariance matrices of states. Additionally, the state estimate is efficiently acquired by the SVEF and its robustness is excellent. Conceptualization, H.H. and X.S. ; methodology, H.H. ; software, C.R. ; validation, C.R. ; formal analysis, X.S. ; data curation, X.S. ; writing--original draft preparation, X.S. ; writing--review and editing, H.H. and X.S. ; supervision, H.H. ; project administration, H.H. ; funding acquisition, H.H. and X.S. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China, grant number 61703098; the Future Network Scientific Research Fund Project, grant numbers FNSRFP-2021-YB-29 and FNSRFP-2021-YB-28; the Natural Science Foundation of Jiangsu Province, grant number BK20160699; the Qing Lan Project of Jiangsu, grant number QLGC-2020; and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China, grant number 20 KJB130005. The research data used to support the findings of this study are included in the article. The data used to support the findings of this study are available from the corresponding author upon request. The authors declare that there is no conflict of interest regarding the publication of this paper. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithms** & **EKF** & **UKF** & **CKF** & **SVEF** \\ \hline MC RMSE (m) & 90.6664 & 84.0154 & 82.9580 & 33.6932 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: MC RMSE for different algorithms. * [PERSON] et al.