# Robust Flow Field Signal Estimation Method for Flow Sensing by Underwater Robotics
## 1 Introduction
In nonlinear systems, time and frequency domains are commonly used to describe nonlinear behaviors [39; 40; 41]. The VSM and the radial basis function neural network model (RBF-NNM) are the two mainly used methods. The RBF-NNM is a three-layer neural network model. With the help of a basis function in the hidden layer, the output layer can produce a local response. As a result, it has a good local approximation capability. Furthermore, in each study cycle, only a small number of weights has to be adjusted; therefore, it has the ability of fast learning. It is mostly used for the prediction of underwater acoustic signals and the tracking control for robots. For example, [PERSON] et al. used the RBF-NNM to predict radiated noise data of a ship, and the prediction error was compared with that of the BP neural network model. The results showed that the RBF-NNM has higher predicting efficiency than the BP network model [42]. [PERSON] et al. used the RBF-NNM to provide a tracking control strategy for solving the uncertainties of robotic dynamics and improving system robustness. By choosing proper parameters, the tracking error can converge to a small neighborhood of zero [43]. The VSM is a generalization of the linear convolution concept, allowing the separation of the system response in linear and nonlinear components. Therefore, this methodology can describe the whole dynamic behavior. The few available research reports on nonlinear analysis methods of flow field signals mainly focused on the nonlinear behavior of waves. For example, [PERSON] provided a method that combines the strip method with the third-order VSM to create a practical tool to simulate the statistics of nonlinear pressure responses in irregular waves [44]. [PERSON] and [PERSON] et al. proposed an integrated methodology using a nonlinear autoregressive with exogenous input (NARX) technique and VSM to predict the top tension of a mooring line of the box-type floating production storage and offloading. They extracted the frequency response function of the system by applying the harmonic probing method to the dynamic response of slender marine structures using the NARX-based truncated Volterra kernel [45; 46]. However, these models have many limitations, and it is difficult to obtain a general model that describes all the structures of interest. No study has been carried out so far on the nonlinear characteristics of the flow field. Therefore, there is no effective method of signal processing for flow sensing. In this paper, the VSM is improved by optimizing its kernel function, and a new robust KKF-VSM is provided. Then, the performance of KKF-VSM is studied systematically. Moreover, its accuracy, robustness, and sensitivity are compared with those of RBF-NNM, one of the methods commonly used. The final goal is to obtain a more suitable pretreatment method for flow sensing. The paper is organized as follows. In Section 2, the method for obtaining the signal and the phase space reconstruction method is described. First, computational fluid mechanics (CFD) and particle imaging velocimetry (PIV) are used to study the flow field around different targets. The pressure signal series in the flow field is processed by the C-C method, and a chaotic signal time series is obtained. In Section 3, the building of the new KKF-VSM is described. The impact of Volterra kernels coefficients on the results is studied in this section. Furthermore, one common RBF-NNM is built and compared with the new model. Section 4 shows the results of the two methods obtained by processing regular and irregular undesired signals. Finally, the main conclusions and some future research directions are presented in Section 5.