## 6 Conclusions  
This paper focuses on three sets of SRINSs and proposes a CKF method for multiple INSs based on topological distribution constraints. Considering the influence of the Earth's magnetic field and gravitational field, this paper optimizes the INS state equations and state transition matrix to establish a more accurate model. Based on a dynamic model of arm length variation under flexural deformation, the positional distance relationships between multiple SRINSs are used as measurement quantities to estimate the arm lengths. Simulation experiments indicate that the estimation results are good, with an error within  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Item**} & \multicolumn{3}{c}{**Positioning Error (RMS)**} \\  & **Longitude Error** & **Latitude Error** & **Total Error** \\ \hline \multirow{5}{*}{The First Trial} & SRINS1 & 0.2989 n m & 0.2799 n m & 0.4095 n m \\  & SRINS2 & 0.3212 n m & 0.2228 n m & 0.3909 n m \\  & SRINS3 & 0.3747 n m & 0.2497 n m & 0.4503 n m \\  & Data Fusion Result & 0.2801 n m & 0.2084 n m & 0.3491 n m \\  & Accuracy Increase & 6.3180\% & 6.4684\% & 10.6943\% \\ \hline \multirow{5}{*}{The Second Trial} & SRINS1 & 0.4854 n m & 0.4904 n m & 0.6900 n m \\  & SRINS2 & 0.4852 n m & 0.5935 n m & 0.7666 n m \\ \cline{1-1}  & SRINS3 & 0.4630 n m & 0.3933 n m & 0.6075 n m \\ \cline{1-1}  & Data Fusion Result & 0.4141 n m & 0.3643 n m & 0.5516 n m \\ \cline{1-1}  & Accuracy Increase & 10.5539\% & 7.3664\% & 9.2041\% \\ \hline \multicolumn{1}{c}{Average Accuracy Increase} & 8.4360\% & 6.9174\% & 9.9492\% \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of Fusion Accuracy.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Item**} & \multicolumn{2}{c}{**The First Experiment**} & \multicolumn{2}{c}{**The Second Experiment**} \\ \hline \multirow{3}{*}{SRINS1 and SRINS2} & x-axis & 0.7305 m & 0.6800 m \\  & y-axis & 0.8838 m & 0.9146 m \\  & z-axis & 0.0314 m & 0.0302 m \\ \hline \multirow{3}{*}{SRINS1 and SRINS3} & x-axis & \(-\)0.3975 m & \(-\)0.4069 m \\  & y-axis & 0.5823 m & 0.6060 m \\ \cline{1-1}  & z-axis & 0.0127 m & 0.0164 m \\ \hline \hline \end{tabular}
\end{table}
Table 5: Lever Arm Estimation Results.  
0.012 m. Additionally, this paper employs a data acquisition device to send 1 PPS signals, and simultaneously collects measurement data. Since the precision of the timestamp is 1 ms, the time difference between multiple SRINSs will be a constant value and less than 1 ms.  
The car test shows that our multi-INSs data fusion method can effectively improve the comprehensive positioning accuracy of SRINSs. Compared to before fusion, the overall positioning error RMS is reduced by about 9.9492 %. Under conditions without absolute positioning information, our data fusion method can be used with multiple low-cost, small-sized SRINSs to provide high-precision positional reference information for underwater unmanned submersibles, ensuring the reliability of navigation position.  
Conceptualization, G.Y. and Z.W.; methodology, Y.X.; software, Y.X.; validation, Y.X. and Z.W.; formal analysis, Y.X.; writing--original draft preparation, Y.X.; writing--review and editing, Z.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Foundation of National Key Laboratory of Underwater Information and Control in China (2023-JCJQ-LB-030-10).  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors declare no conflicts of interest.