# A Novel Hybrid Approach to Deal with DVL Malfunctions for Underwater Integrated Navigation Systems
## 4 Performance Evaluation  
### Simulations and Results  
Simulations are carried out to validate the proposed approach. The PLSR-SVR hybrid predictor is employed to deal with DVL malfunctions for the SINS/DVL/MCP/PS integrated navigation system. The specifications of the instruments used are listed in Table 1. The time of simulation is 1500 s. Figure 6 shows the trajectory of the underwater vehicle. The initial latitude, longitude and height of the underwater vehicle are set as 32\({}^{\circ}\) N, 118\({}^{\circ}\) E and \(-\)20 m. The vehicle travels from point A to point B. The vehicle movements include acceleration, deceleration, turning, diving and surfacing. The detailed motion states of the vehicle are listed in Table 2. The proposed hybrid model is trained during the DVL normal operations and the trained model is further utilized for the predictions during the DVL malfunctions. Sample size for the training is 60, and the sample set updates online. When the DVL works well, the SINS calculating results and the DVL measurements are collected as the training  
Figure 4: The block diagram of modeling mode. Figure 5: The block diagram of prediction mode. samples. Then the PLSR-SVR hybrid model is updated online with constantly updated sample set when the DVL works well. If a DVL malfunction occurs, the trained PLSR-SVR hybrid model will be applied to predict the DVL measurements. According to Section 3.1, the principle of the PLSR predictor is using \(s+1\) SINS calculating velocities to predict the current DVL measurement. The predictor with \(s+1\) SINS calculating velocities is called PLSR predictor of \(s+1\) steps. In order to evaluate the impact of the step number on the prediction performance, comparative simulations on PLSR predictors of different steps are carried out. It is worth mentioning that the simulations are based on the PLSR-SVR hybrid predictor, and the only difference is the step number of PLSR. Specifically, the step number is assigned the value of 2, 3, 4 and 5 respectively. To simulate the DVL malfunction, the output of the DVL is frozen from 690 s to 750 s when the underwater vehicle is in uniform motion. Here the residual chi-squared test is applied to detect the fault. Figure 7 shows the velocity errors of the system with the PLSR predictors of different steps during the fault period. Table 3 shows the evaluation of PLSR predictor of different steps. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Instruments** & **Parameters** & **Accuracy** \\ \hline \multirow{4}{*}{IMU} & Gyroscope in run bias stability & 1\({}^{\circ}\)/hr (1\(\sigma\)) \\  & Gyroscope angular random walk & 0.0667\({}^{\circ}\)/vhr (1\(\sigma\)) \\  & Accelerometer in run bias variation & 0.25 mg (1\(\sigma\)) \\  & Accelerometer output noise & 55 \(\mu\)g/vHz (1\(\sigma\)) \\ \hline DVL & Velocity & \(\pm\)1\% \(\pm\) 2 mm/s \\ MCP & Heading & \(\pm\)0.5\({}^{\circ}\) \\ PS & Depth & \(\pm\)0.01\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Instrument specifications. \begin{table}
\begin{tabular}{c c} \hline \hline
**Motions States** & **Time (s)** \\ \hline Accelerated motion & 0–10 \\ Uniform motion & 10–110, 170–270, 330–430, 630–870, 1070–1170, 1230–1330, 1390–1490 \\ Diving motion & 110–170, 1330–1390 \\ Surfacing motion & 270–330, 1170–1230 \\ Left turning motion & 430–630 \\ Right turning motion & 870–1070 \\ Decelerated motion & 1490–1500 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Motion states of the underwater vehicle. Figure 6: The sailing trajectory of the underwater vehicle. According to Figure 7 and the maximum values of the velocity errors in Table 3, we can see that the velocity errors of the system with PLSR predictors of different steps are all within acceptable small range. Further from Table 3, the PLSR predictor of higher step performs better with smaller RMS in velocity error. However, PLSR predictor of high step means the dimension of the independent variable data matrix will be large, which will increase the computation effort.