# Robust Flow Field Signal Estimation Method for Flow Sensing by Underwater Robotics
## 4 Comparative Analysis Results  
### Analysis of Regular Undesired Signals  
In the natural underwater environment, there are interference factors, such as ocean currents, shock waves, and surrounding objects. An approximate mathematical model  
Figure 11: Predicted error responses in comparison with diffusion coefficient.  
Figure 12: Predicting results of RBF-NNM.  
describing these factors was built, and the KKF-VSM and RBF-NNM were compared when processing the signal time series.  
First, the interference of a regular wave on the flow field signal was analyzed. From the first to 400 th value of the original test samples, one regular undesired signal series \(S(n)\) with a sine wave was put in, as shown in Equation (21). The three-order KKF-VSM and RBF-NNM were used to filter the undesired signal and predict the original test samples. The variation curve of MSE with the amplitude of \(S(n)\) is shown in Figure 13.  
\[S(n)=A\sin(\frac{n\pi}{4}),n=1,2,\cdots,500 \tag{21}\]  
where \(S(n)\) is the interfering signal series with the sine wave, and \(A\) is the amplitude.  
As seen in Figure 13, when the strength of undesired signals was the same, the MSE of KKF-VSM was smaller than that of RBF-NNM. In addition, the gradient of the KKF-VSM curve was also smaller than that of the RBF-NNM one. These results imply that the robustness of RBF-NNM was poor when the undesired signals were strong. In comparison, the KKF-VSM showed a more robust performance because KKF-VSM contained both the linear and the nonlinear components at the same time.  
When the order of KKF-VSM changed, the contribution of each part to the method's robustness was analyzed, as shown in Table 2. From it, we can see that MSE decreased when KKF-VSM changed from one order to two orders. That implies the second component of KKF-VSM could express the most nonlinear characteristic of the test sample. Furthermore, when the KKF-VSM changed from two orders to three orders, the gradient of MSE obviously decreased, implying that the three-order component provided a clear contribution to the robustness.  
The sensitivity of KKF-VSM and RBF-NNM to regular undesired signals was compared. The results are shown in Figure 14. From it, it appears that when the amplitude of undesired signals was small, the \(\Delta MSE\) in KKF-VSM changed more obviously than that in RBF-NNM, but the result was the opposite when the amplitude became larger. This  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Amplitude/\(A\)** & **0.1** & **0.2** & **0.3** & **0.4** & **0.5** \\ \hline \(MSE_{\text{1-order}}\times 10^{-3}\) & 1.94 & 2.43 & 3.17 & 4.29 & 5.87 \\ \(MSE_{\text{2-orders}}\times 10^{-4}\) & 1.46 & 1.77 & 2.27 & 2.97 & 3.84 \\ \(MSE_{\text{3-orders}}\times 10^{-4}\) & 0.97 & 1.09 & 1.28 & 1.55 & 1.83 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Influence of Volterra orders on the robustness.  
Figure 13: Predicted error responses in comparison with disturbance amplitude.  
phenomenon implies that KKF-VSM can capture weak interference signals more easily than RBF-NNM; in other words, KKF-VSM has a greater ability to filter regular undesired signals than RBF-NNM.