## 4 Establishing the Data Fusion Kalman Filter Equations
### Calculation Process
\right]\)
8: \(\mathbf{\hat{X}}_{k/k-1}=\mathbf{F}_{k/k-1}\mathbf{\hat{X}}_{k-1}\)
9: \(\mathbf{P}_{k/k-1}=\mathbf{F}_{k/k-1}\mathbf{P}_{k-1}\mathbf{F}_{k/k-1}^{T}+\Gamma_{k-1}\mathbf{Q} _{k-1}\mathbf{\Gamma}_{k-1}^{T}\)
10: \(\mathbf{K}_{k}=\mathbf{P}_{k/k-1}\mathbf{H}_{k}^{T}(\mathbf{H}_{k}\mathbf{P}_{k/k-1}\mathbf{H}_{k}^{T}+ \mathbf{R}_{k})^{-1}\)
11: \(\mathbf{P}_{k}=(\mathbf{I}-\mathbf{K}_{k}\mathbf{H}_{k})\mathbf{P}_{k/k-1}\)
12: \(\mathbf{\hat{X}}_{k}=\mathbf{\hat{X}}_{k/k  
Figure 7: Position Errors Under Simulation Conditions. The estimation results of the lever arm are shown in Table 2. It can be seen that the arm length estimation error is small, and the estimation result is quite accurate. From Figure 6b, it can be seen that the estimation of the lever arm errors in the \(x\) and \(y\) directions is convergent. The lever arm between \(SRINS_{1}\) and \(SRINS_{2}\), and between \(SRINS_{1}\) and \(SRINS_{3}\) can be estimated, thereby compensating for the lever arm errors during the data fusion process. Due to the divergence in the vertical direction of the INS, the observability of the arm length in the \(z\) direction is weak, resulting in poorer estimation results. However, for positional information such as longitude and latitude, the arm lengths in the \(x\) and \(y\) directions alone are sufficient to constrain the positions of multiple SRINSs. External auxiliary equipment, such as depth sensors, can be used to measure altitude and constrain the altitude channel of inertial navigation calculations, thereby improving the observability of the z-axis lever arm. However, this is not the focus of the current study. Based on the position provided by simulation trajectory, the position errors of three SRINSs during pure inertial navigation independently are compared with the position errors after applying the multi-inertial navigation data fusion method proposed in this paper, as shown in Figure 7. The RMS value of the position error can be calculated using the following formula, and the calculated values are shown in Table 3:  
\[X_{RMS}=\sqrt{\frac{\sum\limits_{i=1}^{N}X_{i}^{2}}{N}} \tag{34}\]  
In the formula above, \(X_{RMS}\) represents the RMS calculation result of the position error, \(X_{i}\) represents the position error, and \(N\) represents the total data volume. The units of the RMS calculation results of the position error in in Table 3 are all nautical miles (n m), and this length unit satisfies 1 n m = 1852 m. In addition, the accuracy improvement results are calculated using the following formula:  
\[\eta=\frac{\left|Err_{Data\_Fusion}-\min\{Err_{SRINS1},Err_{SRINS2},Err_{SRINS3 }\}\right|}{\min\{Err_{SRINS1},Err_{SRINS2},Err_{SRINS3}\}} \tag{35}\]  
where \(Err_{Data\_Fusion}\) represents the RMS of the position error after data fusion, while \(Err_{SRINS1}\), \(Err_{SRINS2}\), and \(Err_{SRINS3}\), respectively, represent the RMS of the position er \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Item**} & & **Simulation** & \multirow{2}{*}{**Estimation Error**} \\  & & **Experiment** & \\ \hline \multirow{3}{*}{SRINS1 and SRINS2} & x-axis & 0.2596 m & 0.0046 m \\  & y-axis & 0.3749 m & \(-\)0.0001 m \\  & z-axis & 0.0117 m & 0.0117 m \\ \hline \multirow{3}{*}{SRINS1 and SRINS3} & x-axis & 0.3990 m & \(-\)0.0100 m \\  & y-axis & 0.0192 m & 0.0108 m \\ \cline{1-1}  & z-axis & 0.7925 m & 0.0075 m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Lever Arm Estimation Results. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Item**} & & \multicolumn{3}{c}{**Positioning Error (RMS)**} \\  & & **Longitude Error** & **Latitude Error** & **Total Error** \\ \hline \multirow{4}{*}{Simulation Experiment} & SRINS1 & 0.1058 n m & 0.2283 n m & 0.2516 n m \\  & SRINS2 & 0.1272 n m & 0.2514 n m & 0.2817 n m \\ \cline{1-1}  & SRINS3 & 0.1196 n m & 0.2609 n m & 0.2870 n m \\ \cline{1-1}  & Data Fusion Result & 0.09740 n m & 0.2145 n m & 0.2356 n m \\ \cline{1-1}  & Accuracy Increase & 7.9785\% & 6.0212\% & 6.3645\% \\ \hline \hline \end{tabular}
\end{table}