# A Novel Hybrid Approach to Deal with DVL Malfunctions for Underwater Integrated Navigation Systems
## 4 Performance Evaluation
### Simulations and Results
The computation consumptions of PLSR predictors of different steps are listed in the last column of Table 3. Apparently the computation consumption increases with higher step number. Considering both the prediction performance and the computation consumption, we choose PLSR predictor of 4 steps to constitute the hybrid predictor in the following simulations. In order to further validate the performance of the proposed hybrid approach, comparative simulations with different approaches are implemented. During DVL malfunctions, the SINS/DVL/MCP/PS integrated navigation system is processed by four different solutions: (1) insulate the faulted DVL measurements without predictor (i.e., the local filter 1 introduced in Section 2.1 only executes the time update process); (2) use only PLSR predictor; (3) use radial basis function (RBF) neural network predictor [34]; (4) use PLSR-SVR hybrid predictor. It is worth noting that the DVL malfunctions listed in the introduction are all short-term, lasting for a few seconds or minutes. To simulate the sudden malfunction of the DVL, the velocity outputs of the DVL are added by 3 m/s along three directions during the period of 400-460 s when the underwater vehicle is in uniform motion and left turning motion according to Table 2. Figure 8 shows the navigation errors of the four solutions when there is a DVL malfunction for a length of 60 s. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Step Number**} & \multicolumn{2}{c}{**East Velocity Error (m/s)**} & \multicolumn{2}{c}{**North Velocity Error (m/s)**} & \multirow{2}{*}{**Computation Consumption (s)**} \\ \cline{2-2} \cline{5-6}  & **Maximum** & **RMS** & **Maximum** & **RMS** & \\ \hline
2 steps & 0.1081 & 0.0545 & 0.1249 & 0.0663 & 0.005945 \\
3 steps & 0.0514 & 0.0364 & 0.1055 & 0.0647 & 0.006277 \\
4 steps & 0.0479 & 0.0346 & 0.1196 & 0.0639 & 0.006755 \\
5 steps & 0.0561 & 0.0326 & 0.0911 & 0.0630 & 0.007651 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Evaluation on PLSR predictors of different steps. Figure 7: Velocity errors: (**a**) East velocity error; (**b**) North velocity error. Figure 8a shows that, the east and north velocity errors of the system with the first solution increase significantly during the malfunction period, which is due to the lack of external velocity information. By comparison, the systems with the other three predictors work better. Since the predictors provide the predictions of the DVL measurements, the accumulated SINS velocity errors are corrected by the prediction information. The partial enlargement drawings in Figure 8a indicate that the horizontal velocity errors of the system with the hybrid predictor are smaller than that with the PLSR predictors. This is because that the PLSR solution only establishes the linear model between the SINS calculating results and the DVL measurements. However there also exists nonlinear relationship between them (especially, for the maneuver situations). The PLSR-SVR hybrid predictor shows better accuracy by using SVR to predict the residual components of PLSR. Besides, the hybrid predictor is proved to be more effective than the RBF predictor. When the DVL failure is eliminated, the horizontal velocity errors are reduced gradually because of the available DVL velocity information. However, the latitude and longitude errors of the system have nonzero stable values as shown in Figure 8b because the horizontal position errors accumulate with the horizontal velocity errors, and the calculation for the horizontal position is an open loop. The data and the partial enlargement drawing in Figure 8b shows that the latitude errors of the systems with four different solutions stabilize at around 7.3 m, 6.9 m, 5.8 m and 3.5 m respectively, and the longitude errors stabilize around 7.6 m, 2.9 m, 2.0 m and 1 m respectively. Thus, by applying the hybrid predictor, the latitude and longitude precisions are increased by 49.3% and 65.5% respectively, in comparison with the PLSR predictor.