# Robust Flow Field Signal Estimation Method for Flow Sensing by Underwater Robotics
## 3 Description of the Pretreatment Method  
### Description of the KKF-VSM  
The signal time series obtained in Section 2 was processed by phase space reconstruction, as shown in Equation (6). If the flow field can be seen as a nonlinear system with a single input parameter \(n\in Z^{+}\to x(n)\), where \(n\) is the sample time, and \(x(n)\) is the collected signal time series, the corresponding predicted signal \(n\in Z^{+}\to y(n)\) is single and can be expressed by the VSM, as shown in Equation (9). The output of the system can be divided into linear and nonlinear parts [50], as shown in Equation (10). If the length of the expanded series is too big, this method is not easy to converge. It is necessary to minimize the length to ensure prediction accuracy. Therefore, the Kautz function was used to expand the Volterra series, as shown in Equation (11). It can predict the response of a nonlinear system and monitor the structure of interest [51; 52]. \[y(n)=\sum\limits_{p=1}^{\infty}\sum\limits_{m_{1}}^{N_{1}-1}\ldots\sum \limits_{m_{p}=0}^{N_{p}-1}h_{p}(m_{1},\ldots,m_{p}){\prod\limits_{i=1}^{p}u( n-m_{i})} \tag{9}\]  
\[y(n)=\underbrace{y_{1}(n)}_{linear}+\underbrace{y_{2}(n)+y_{3}(n)+\ldots+y_{ p}(n)}_{nonlinear} \tag{10}\]  
\[h_{p}(m_{1},\ldots,m_{p})\approx\sum\limits_{i_{1}=1}^{I_{1}}\ldots\sum \limits_{i_{p}=1}^{I_{p}}\Re_{p}(i_{1},\ldots,i_{p}){\prod\limits_{j=1}^{p} \textbf{v}_{p;j_{j}}(m_{j})} \tag{11}\]  
Figure 6: The original signal spectrum. where \(h_{p}(m_{1},\ldots,m_{p})\) is the \(p\)-order Volterra kernel, \(J_{1,.....,J_{p}}\) is the number of Kautz functions used in each orthonormal projections of the Volterra kernels, \(\Re_{p}(i_{1},\ldots,i_{p})\) is the \(p\)-order Volterra kernel expanded in the orthonormal basis, \(\Psi_{p,j_{j}(m_{j})}\) is the \(i\)-th Kautz filter, which is shown in Equation (12). \[\left\{ \begin{array}{l}\Psi_{p,2j}(z)=\Psi_{p,2j-1}(z)\frac{z-b_{p}}{\sqrt{1- b_{p}^{2}}}\\ \\ \Psi_{p,2j-1}(z)=\frac{z\sqrt{(1-b_{p}^{2})(1-c_{p}^{2})}}{z^{2}+b_{p}(c_{p}-1 )z-c_{p}}[\frac{-c_{p}z^{2}+b_{p}(c_{p}-1)z+1}{z^{2}+b_{p}(c_{p}-1)z-c_{p}}]^{j -1}\\ \\ b_{p}=\frac{\Gamma_{p}+\widetilde{\Gamma}_{p}}{1+\Gamma_{p}\Gamma_{p}}\\ \\ c_{p}=-\Gamma_{p}\widetilde{\Gamma}_{p}\\ \\ \Gamma_{p}=\exp(\frac{g_{p}}{F_{s}})\\ \\ S_{p}=-\tilde{x}_{p}\omega_{p}\pm j\omega_{p}\sqrt{1-\tilde{x}_{p}^{2}}\end{array} \right. \tag{12}\]  
where \(\omega_{p}\) is the natural frequency of the system, \(\tilde{x}_{p}\) is the damping ratio of the flow field, \(F_{s}\) is the sampling frequency, \(S_{p}\) is the continuous poles of the input signal series, \(\Gamma\) and \(\widetilde{\Gamma}\) are the Kautz poles and its complex conjugate in the discrete domain. The steps for predicting the flow field signal based on the KKF-VSM were as follows:  
1. Establish a reference system: the known signal series was used to train the KKF-VSM, as the reference system;
2. Input the test series: the test signal series \(x_{n}=[x(1),x(2),\ldots,x(N)]\) was normalized by Equation (13), and Equation (14) was used to establish the input signal matrix; \[x^{*}(n)=\frac{x(n)-\frac{1}{N}\sum\limits_{i=1}^{N}x(i)}{\max(x(n))-\min(x(n))}\] (13) \[X(n)=\left[ \begin{array}{cccc}x^{*}(1)&x^{*}(2)&\cdots&x^{*}(N-(m-1))\\ x^{*}(2)&x^{*}(3)&\cdots&x^{*}(N-(m-2))\\ \vdots&\vdots&&\vdots\\ x^{*}(m)&x^{*}(m+1)&\cdots&&x^{*}(N)\end{array} \right]\] (14) where \(x^{*}(n)\) is the processed data, \(x(n)\) is the original data, \(\min(x(n))\) and \(\max(x(n))\) respects the minimum and maximum values of the original signal. 3. Phase space reconstruction: the input signal matrix was reconstructed in higher phase space, and the signal matrix \(U(n)\) to the input into the KKF-VSM was obtained;
4. Sample prediction test: the KKF-VSM was used to obtain the output of the sample series, consisting of the orthonormal kernels and the input signal, as shown in Equation (15). The prediction error was expressed as the mean-square error (MSE), as shown in Equation (16).