## 5 Simulations
### Simulation Setting
\right]^{\text{T}},\mathbf{Q}_{by0}=\left[ \begin{array}{cc}0.0005&0.0 05\\ \end{array} \right]^{\text{T}}\text{and }\mathbf{R}_{by0}=\left(0.5\sqrt{3}\right)^{2}.\) \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Axes** & \multicolumn{2}{c}{**Gyro Bias (\({}^{\circ}\)/h)**} & \multicolumn{2}{c}{**Acce Bias (ug)**} \\ \cline{2-4}  & **Constant** & **Random (White Noise)** & **Constant** & **Random (White Noise)** \\ \hline \(x\)-axis & 0.01 & 0.01 & 50 & 50 \\ \(y\)-axis & 0.01 & 0.01 & 50 & 50 \\ \(z\)-axis & 0.01 & 0.01 & 50 & 50 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Sensor errors. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Pitch** & **Roll** & **Yaw** \\ \hline Amplitude (\({}^{\circ}\)) & 1.5 & 1.5 & 1.0 \\ Cycle (s) & 8 & 7.5 & 6 \\ Initial Phase (\({}^{\circ}\)) & 0 & 0 & 0 \\ Swinging center (\({}^{\circ}\)) & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Swinging parameters. The curves in Figure 4 and the statistic results in Table 3 indicate that before and after tracing, the means of both velocities are similar, but the standard variances are significantly different, which means the random noise in the DVL velocity can be significantly restrained by the velocity tracing algorithm. The curves in Figure 5 indicate that when the traced velocity in the body frame is projected into the navigation frame, the random noise can be restrained. The partial enlarged drawing in Figures 4 and 5 indicate that the velocity tracing algorithm is more robust than the standard variance. Figure 8: Curves of the position errors: (**a**) Latitude; (**b**) Longitude; (**c**) Height. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Statistical**} & \multicolumn{3}{c}{**Velocity along the**} & \multicolumn{3}{c}{**East Velocity**} & \multicolumn{3}{c}{**North Velocity**} & \multicolumn{3}{c}{**Upward Velocity**} \\ \cline{2-13}  & **St.1** & **Sch.2** & **Ideal** & **Sch.1** & **Sch.2** & **Ideal** & **Sch.1** & **Sch.2** & **Ideal** & **Sch.1** & **Sch.2** & **Ideal** \\ \hline Mean & 5.0002 & 5.0033 & 5 & 0.0005 & 0 & 0 & 5.0009 & 5.0005 & 5.0008 & 0.0004 & 0 & 0 \\ Std & 0.4975 & 0.0220 & 0 & 0.4996 & 0.0617 & 0.0618 & 0.5001 & 0.0020 & 0.0220 & 0.5153 & 0.0926 & 0.0925 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Statistical results of the Doppler Velocity Log (DVL) velocity (m/s). Figure 7: Curves of the velocity errors: (**a**) East velocity; (**b**) North velocity; (**c**) Up velocity. indicates that the traced velocities almost coincide with the theoretical ones. When a sudden change occurs, the fault diagnosis algorithm based on \(\chi^{2}\) rules can judge this sudden change immediately, and the time update of the tracing algorithm can be used as a substitute output of the DVL. The navigation results of the integrated SINS/DVL/MCP/PS are shown in Figures 6-8 where the attitude, velocity and position errors are identified, and solid lines and dashed lines denote Schemes 1 and 2, respectively. To acquire a direct display, the unit of the position error in the level position is transformed from degrees to meters. The curves in Figure 6c coincide with each other because the same yaw from MCP is used in both schemes. The curves in Figures 6-8 indicate that when there is a sudden change in the DVL velocity, with Scheme 1, sudden changes in attitude, velocity, and position are introduced. When the sudden change in velocity is eliminated, the sudden change in attitude and velocity can be restrained gradually, whereas the sudden change in the level position cannot be restrained because it is an open loop. The height error is restrained because the calculation for height is a closed loop. With the height from the PS, the sudden change in height caused by a sudden change in DVL velocity is corrected.