Robust Flow Field Signal Estimation Method for Flow Sensing by Underwater Robotics

: The ﬂow ﬁeld is difﬁcult to evaluate, and underwater robotics can only partly adapt to the submarine environment. However, ﬁsh can sense the complex underwater environment by their lateral line system. In order to reveal the ﬁsh ﬂow sensing mechanism, a robust nonlinear signal estimation method based on the Volterra series model with the Kautz kernel function is provided, which is named KKF-VSM. The ﬂow ﬁeld signal around a square target is used as the original signal. The sinusoidal noise and the signal around a triangular obstacle are considered undesired signals, and the predicting performance of KKF-VSM is analyzed after introducing them locally in the original signals. Compared to the radial basis function neural network model (RBF-NNM), the advantages of KKF-VSM are not only its robustness but also its higher sensitivity to weak signals and its predicting accuracy. It is conﬁrmed that even for strong nonlinear signals, such as pressure responses in the ﬂow ﬁeld, KKF-VSM is more efﬁcient than the commonly used RBF-NNM. It can provide a reference for the application of the artiﬁcial lateral line system on underwater robotics, improving its adaptability in complex environments based on ﬂow ﬁeld information. 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.


Introduction
The autonomous underwater vehicle (AUV) is an important tool for marine environment exploration that is widely used. Since the underwater environment is complex, the adaptability of AUV needs to be improved [1,2]. The main aim of the current study on AUV environmental adaptability is to improve its control [3][4][5][6][7]. In practical applications, it has been found that the poor environmental adaptability of AUV is due to its inability to sense the surrounding environment [8][9][10]. At present, AUVs sense the surrounding environment on the basis of a visual and an acoustic system. However, the sensing range of the underwater visual system is limited because visual signals decay rapidly in the water and are easily affected by the water quality. In the context of underwater target detection, the acoustic signal also has many issues, such as high cost and high-power consumption. In complex environments, the acoustic signal is affected and cannot be properly applied [11][12][13]. In addition, the visual and acoustic signals detect targets through reflection and are, thus, indirect signals. They are unable to capture information regarding the flow field, such as the eddy and ocean currents.
In a study of marine organisms, bionics researchers found that fish have a sensing organ that they called the lateral line system (LLS). The LLS can help fish acquire extraordinary environmental adaptability [14][15][16][17]. The LLS contains two kinds of sensory neuromasts: the superficial neuromast and the canal neuromast. The superficial neuromast can sense the magnitude and direction of flow velocity, and the canal neuromast is usually used to sense the acceleration of the flow field, such as fluctuations [18][19][20][21][22]. Fish can make adjustments to the surrounding environment based on the information from the LLS. This In this paper, the flow field signal around a square target was chosen as the original signal, as shown in Figure 1. The side length of the square target was named H. The computational domain was located in a Cartesian coordinate system, within a range of 30H × 10H. The virtual lateral line with the pressure sensors was under the center of the targets at a distance of D. CFD was used to obtain the signal series, according to a numerical method described in our previous work [47].
VSM is studied systematically. Moreover, its accuracy, robustness, compared with those of RBF-NNM, one of the methods commonly us to obtain a more suitable pretreatment method for flow sensing.
The paper is organized as follows. In Section 2, the method for o and the phase space reconstruction method is described. First, comp chanics (CFD) and particle imaging velocimetry (PIV) are used to s around different targets. The pressure signal series in the flow field is C method, and a chaotic signal time series is obtained. In Section 3, new KKF-VSM is described. The impact of Volterra kernels coefficien studied in this section. Furthermore, one common RBF-NNM is built the new model. Section 4 shows the results of the two methods obta regular and irregular undesired signals. Finally, the main conclusion research directions are presented in Section 5.

Numerical Scheme
In this paper, the flow field signal around a square target was ch signal, as shown in Figure 1. The side length of the square target w computational domain was located in a Cartesian coordinate system H H 10 30 × . The virtual lateral line with the pressure sensors was und targets at a distance of D . CFD was used to obtain the signal series, merical method described in our previous work [47]. In the numerical scheme, the two-step Taylor characteristic-base method was used to solve the flow governing equation, which was p al. [48]. The progress of the TCBG scheme for the momentum equation ing equations. The accuracy of the numerical scheme was validated p Strouhal numbers t S of the square target with different postures θ shown in Figure 2. The results were in accord with previous reports tional algorithm was adequate to solve the flow field around the unde In the numerical scheme, the two-step Taylor characteristic-based Galerkin (TCBG) method was used to solve the flow governing equation, which was presented by Bao et al. [48]. The progress of the TCBG scheme for the momentum equations is shown in flowing equations. The accuracy of the numerical scheme was validated previously [49]. The Strouhal numbers S t of the square target with different postures θ was computed, as shown in Figure 2. The results were in accord with previous reports, and the computational algorithm was adequate to solve the flow field around the underwater target.
where u i is the i-component velocity, ρ is the water density, p is the pressure, τ ij is the deviatoric stresses, Re is the Reynolds number, n, n + 1/2, and n + 1 denote the time points of t n , t n+1/2 and t n+1 , respectively.

Numerical Validation by a PIV Experiment
A two-dimension PIV experiment was carried out to validate the numerical scheme. PIV is a method of visualizing the flow field based on an optical principle. Tracer particles with reflective properties are placed in the flow field. A laser emits a light source, and a high-speed camera is used to capture the motion path of the tracer particles along with the fluid. The displacement of the particles is calculated by the Fourier algorithm based on two successive frames of images, and the flow field structure is visualized. The experimental set is shown in Figure 3; the length of the water tank was 5.0 m, and its width and height were 1.0 m and 0.8 m, respectively. The water tank was divided into three parts: the accelerating part, the experiment part, and the decelerating part. The length of the experiment part was 3.0 m.
A sliding rail was used to make the target body move at a uniform speed, and the maximum speed was set at 1 m/s. As shown in Figure 4, the diameter of the target was 0.1 m, and its length was 0.2 m. The support point was located in the middle, and the distance between the center of the target and the bottom was 0.4 m. To ensure the original flow field was at rest, the experiments were carried out one by one, and the time interval between two adjacent experiments was 20 min. on two successive frames of images, and the flow field structure is visualiz imental set is shown in Figure 3; the length of the water tank was 5.0 m, and height were 1.0 m and 0.8 m, respectively. The water tank was divided in the accelerating part, the experiment part, and the decelerating part. The experiment part was 3.0 m.  ppl. Sci. 2021, 11, x FOR PEER REVIEW A sliding rail was used to make the target body move at a uniform s maximum speed was set at 1 m/s. As shown in Figure 4, the diameter of the m, and its length was 0.2 m. The support point was located in the middle, an between the center of the target and the bottom was 0.4 m. To ensure the field was at rest, the experiments were carried out one by one, and the tim tween two adjacent experiments was 20 min. The two-dimension PIV system was provided by TSI. The model of YAG200-NWL, with maximum output energy of 200 mJ, and its pulse wid The 4MX-CCD camera was used to capture the image of the flow field at 20 els. The minimum span frame time of the CCD camera was 200 ns, and its f 15 frames per second. In this experiment, images were collected in double-fr ble-exposure mode and processed by the mutual algorithm of the system. rameters used in the experiment are shown in Table 1. The virtual LLS was set at a distance of 200 mm from the bottom. W moved to the middle of the experiment part, the axial velocity compone virtual LLS was captured. The comparison between the experimental and results is shown in Figure 5. From it, we can see that the results of calculati were in good agreement. The two-dimension PIV system was provided by TSI. The model of the layer was YAG200-NWL, with maximum output energy of 200 mJ, and its pulse width was 3-5 ns. The 4MX-CCD camera was used to capture the image of the flow field at 2048 × 2048 pixels. The minimum span frame time of the CCD camera was 200 ns, and its frame rate was 15 frames per second. In this experiment, images were collected in double-frame and double-exposure mode and processed by the mutual algorithm of the system. Some key parameters used in the experiment are shown in Table 1. The virtual LLS was set at a distance of 200 mm from the bottom. When the target moved to the middle of the experiment part, the axial velocity component v x,// of the virtual LLS was captured. The comparison between the experimental and the numerical results is shown in Figure 5. From it, we can see that the results of calculation and testing were in good agreement. moved to the middle of the experiment part, the axial velocity componen virtual LLS was captured. The comparison between the experimental and t results is shown in Figure 5. From it, we can see that the results of calculatio were in good agreement.

Phase Space Reconstruction
A virtual lateral line was set in the flow field, with 21 monitoring points. A weighting fusion algorithm was used to couple the 21-series signal, as described in our previous work [47] and shown in Equation (4).
where ε is the weight of position coefficient S i . A m (s i ) is the amplitude of signal collected by sensor i. c p (t) is the coupling signal series at the time of t.
In phase space reconstruction of the signal time series, the selection of delay time and embedding dimension was carried out by the C-C method, as shown in Equation (5). The first local minimum value times of ∆S(m, t) was considered the optimal delay time τ. The minimum value of S cor (t) can be seen as the delay time window Γ, which has a connection with the embedding dimension m, as shown in Equation (5).
∆S(m, t) = max S(m, r j , t) − min S(m, r j , t) ; The final signal time series at high phase space X n (t i ) and the correlation integral C(m, N, r, t) are shown in Equations (6) and (7). The pressure signal time series around a square target was defined as the original signal, as shown in Figure 6.
where N is the length of the signal series, r is the radius of the time series, σ is the time-series standard deviation, θ(x) is shown in Equation (8).

Description of the KKF-VSM
The signal time series obtained in Section 2 was processed by phase spac tion, as shown in Equation (6). If the flow field can be seen as a nonlinear s single input parameter where n is the sample time, and collected signal time series, the corresponding predicted signal Z n → ∈ + and can be expressed by the VSM, as shown in Equation (9). The output of th 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 t is necessary to minimize the length to ensure prediction accuracy. Therefo function was used to expand the Volterra series, as shown in Equation (11). the response of a nonlinear system and monitor the structure of interest [51,

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 ∈ Z + → x(n) , where n is the sample time, and x(n) is the collected signal time series, the corresponding predicted signal n ∈ Z + → 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].
where h p (m 1 , . . . , 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, p (i 1 , . . . , i p ) is the p-order Volterra kernel expanded in the orthonormal basis, Ψ p,i j (m j ) is the i-th Kautz filter, which is shown in Equation (12).
where ω p is the natural frequency of the system, ξ 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, Γ and Γ 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), . . . , x(N)] was normalized by Equation (13), and Equation (14) was used to establish the input signal matrix; 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).
The maximum number of samples was set as to. During the training process, KKF-VSM was trained with 2200 samples, and the other 800 samples were used in the test. When the delay time τ was 27 and the embedding dimension was 7, one-order, two-order, and three-order KKF-VSM was used to predict the samples. The predicting steps corresponded to the sample time n, and the results are shown in Figures 7-9.
The maximum number of samples was set as to. During the training process, KKF-VSM was trained with 2200 samples, and the other 800 samples were used in the test. When the delay timeτ was 27 and the embedding dimension was 7, one-order, two-order, and three-order KKF-VSM was used to predict the samples. The predicting steps corresponded to the sample time n , and the results are shown in Figures 7-9.  The MSE of the three models is shown in Figure 7. From it, it appears that the MSE obviously changed from one-order KKF-VSM to two-order KKF-VSM. The change was The maximum number of samples was set as to. During the training process, KKF-VSM was trained with 2200 samples, and the other 800 samples were used in the test. When the delay timeτ was 27 and the embedding dimension was 7, one-order, two-order, and three-order KKF-VSM was used to predict the samples. The predicting steps corresponded to the sample time n , and the results are shown in Figures 7-9. The MSE of the three models is shown in Figure 7. From it, it appears that the MSE obviously changed from one-order KKF-VSM to two-order KKF-VSM. The change was The maximum number of samples was set as to. During the training process, KKF-VSM was trained with 2200 samples, and the other 800 samples were used in the test. When the delay timeτ was 27 and the embedding dimension was 7, one-order, two-order, and three-order KKF-VSM was used to predict the samples. The predicting steps corresponded to the sample time n , and the results are shown in Figures 7-9.  The MSE of the three models is shown in Figure 7. From it, it appears that the MSE obviously changed from one-order KKF-VSM to two-order KKF-VSM. The change was The MSE of the three models is shown in Figure 7. From it, it appears that the MSE obviously changed from one-order KKF-VSM to two-order KKF-VSM. The change was slower when comparing two-order to three-order KKF-VSM. This implies that the contribution of the two-order component was the largest in this test sample. The third-order component could correct some specific values to fit the trend. As a result, three-order KKF-VSM was chosen in this paper to ensure predicting accuracy.

Description of the RBF-NNM
The RBF-NNM was a three-layer forward network, as shown in Figure 10. Through the radial basis function in each hidden layer, the input signal X(n) can find the optimal fitting plane in multidimensional space. Therefore, the RBF-NNM provides a good fit in the prediction of a chaotic signal time series.
slower when comparing two-order to three-order KKF-VSM. This implies t bution of the two-order component was the largest in this test sample. T component could correct some specific values to fit the trend. As a resu KKF-VSM was chosen in this paper to ensure predicting accuracy.

Description of the RBF-NNM
The RBF-NNM was a three-layer forward network, as shown in Figur the radial basis function in each hidden layer, the input signal ) (n X can fin fitting plane in multidimensional space. Therefore, the RBF-NNM provide the prediction of a chaotic signal time series. The Gaussian kernel function is normally used in the RBF-NNM, as sh tion (17); the output ) (n y is shown in Equation (18).
σ is the width of the i-th hidden node, m is the number of the perceivab the connection weight from the i-th hidden layer to the output. The above parameters can be obtained by using the K-means clustering follows: 1. Determine the cluster center: first, some clustering centers were de domly and defined as  The Gaussian kernel function is normally used in the RBF-NNM, as shown in Equation (17); the output y(n) is shown in Equation (18).
where X is the m-dimension input matrix, C i is the center of the i-th basis function, σ i is the width of the i-th hidden node, m is the number of the perceivable unit, ω i is the connection weight from the i-th hidden layer to the output. The above parameters can be obtained by using the K-means clustering algorithm, as follows:

1.
Determine the cluster center: first, some clustering centers were determined randomly and defined as c i (i = 1, 2, · · · , h). The training samples x j were allocated to each cluster set v j (j = 1, 2, · · · , P), based on the principle of minimizing the Euclidean distance between the training samples and the clustering center. The samples average in each set was iterated, and the result was defined as the new cluster center c i . Repeat the calculation cycle until the residual was less than the setting value and use the final c i as the cluster center of the basis function; 2.
Determine the width of the hidden node: the maximum c i,max was chosen from the cluster center set, and the width of the hidden node can be solved by Equation (19); 3.
Determine the connection weights: the least-square method was used to calculate the weight, as shown in Equation (20).
2 ), (i = 1, 2, · · · , h; j = 1, 2, · · · , P) (20) The training epoch N q and the testing epoch N p in Section 2.1 were used to build the RBF-NNM. The diffusion coefficient s of RBF was analyzed, and the optimum value was chosen. Commonly, the fitting radial basis function is smoother when s is larger, but if it is larger than a critical value, some numerical problem will appear. As seen in Figure 11, s was optimized in the range from 50 to 500, and when it was 400, the MSE was at its minimum. Therefore, in the following analysis, the diffusion coefficient was set to 400.
3. Determine the connection weights: the least-square method was used to weight, as shown in Equation (20). if it is larger than a critical value, some numerical problem will appear. As se 11, s was optimized in the range from 50 to 500, and when it was 400, the M minimum. Therefore, in the following analysis, the diffusion coefficient was  The RBF-NNM was used to predict the trend of the signal sample, as sho 12. From it, it appeared that the RBF-NNM can also predict the changing tren field signal.  The RBF-NNM was used to predict the trend of the signal sample, as shown in Figure 12. From it, it appeared that the RBF-NNM can also predict the changing trend of the flow field signal. The RBF-NNM was used to predict the trend of the signal sample, as sh 12. From it, it appeared that the RBF-NNM can also predict the changing tre field signal.

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 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 400th 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.
where S(n) is the interfering signal series with the sine wave, and A is the amplitude.
In the natural underwater environment, there are interference factors, currents, shock waves, and surrounding objects. An approximate mathemat scribing these factors was built, and the KKF-VSM and RBF-NNM were co processing the signal time series.
First, the interference of a regular wave on the flow field signal was an the first to 400th value of the original test samples, one regular undesired ) (n S with a sine wave was put in, as shown in Equation (21). The three-or and RBF-NNM were used to filter the undesired signal and predict the orig ples. The variation curve of MSE with the amplitude of is the interfering signal series with the sine wave, and A is th As seen in Figure 13, when the strength of undesired signals was the s of KKF-VSM was smaller than that of RBF-NNM. In addition, the gradien VSM curve was also smaller than that of the RBF-NNM one. These results robustness of RBF-NNM was poor when the undesired signals were stron son, the KKF-VSM showed a more robust performance because KKF-VSM c the linear and the nonlinear components at the same time.
When the order of KKF-VSM changed, the contribution of each part to robustness was analyzed, as shown in Table 2. From it, we can see that M when KKF-VSM changed from one order to two orders. That implies the s nent of KKF-VSM could express the most nonlinear characteristic of the tes thermore, when the KKF-VSM changed from two orders to three orders, t MSE obviously decreased, implying that the three-order component provid tribution to the robustness. 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 ∆MSE in KKF-VSM changed more obviously than that in RBF-NNM, but the result was the opposite when the amplitude became larger. This 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.
The sensitivity of KKF-VSM and RBF-NNM to regular undesired signa pared. The results are shown in Figure 14. From it, it appears that when the undesired signals was small, the MSE Δ in KKF-VSM changed more ob that in RBF-NNM, but the result was the opposite when the amplitude becam phenomenon implies that KKF-VSM can capture weak interference signals than RBF-NNM; in other words, KKF-VSM has a greater ability to filter regul signals than RBF-NNM.

Analysis of Irregular Undesired Signals
If there is an obstacle near the target in the flow field, it will have an im original signal, and the undesired signal is usually chaotic. In this paper, th around a triangle obstacle was used as an irregular undesired signal, as sho 15. Noise was added in correspondence of the regular undesired signal by position into the original test sample, as shown in Equation (22).

Analysis of Irregular Undesired Signals
If there is an obstacle near the target in the flow field, it will have an impact on the original signal, and the undesired signal is usually chaotic. In this paper, the flow field around a triangle obstacle was used as an irregular undesired signal, as shown in Figure 15. Noise was added in correspondence of the regular undesired signal by linear superposition into the original test sample, as shown in Equation (22).
x(n), 1 ≤ n ≤ 2200 x(n) + T(n), 2201 ≤ n ≤ 2400 x(n), 2401 ≤ n ≤ 3000 (22) where x(n) is the original signal, T(n) is the undesired signal, χ(n) is the final testing original signal series. 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 MSE Δ in KKF-VSM changed more obviously than that in RBF-NNM, but the result was the opposite when the amplitude became larger. This 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.

Analysis of Irregular Undesired Signals
If there is an obstacle near the target in the flow field, it will have an impact on the original signal, and the undesired signal is usually chaotic. In this paper, the flow field around a triangle obstacle was used as an irregular undesired signal, as shown in Figure  15. Noise was added in correspondence of the regular undesired signal by linear superposition into the original test sample, as shown in Equation (22).   The KKF-VSM and RBF-NNM were used to filter the irregular undesired signal and predict the original test sample. The results are shown in Figures 16 and 17. The results indicated that the irregular undesired signal could be filtered, and the change trend of the original samples could be predicted very well by both models. Furthermore, the MSE of RBF-NNM was 1.625 × 10 −4 , and that of three-order KKF-VSM was 1.374 × 10 −4 . That indicated that the robustness of the three-order KKF-VSM is higher than that of RBF-NNM.
The KKF-VSM and RBF-NNM were used to filter the irregular undesir predict the original test sample. The results are shown in Figures 16 and 17 indicated that the irregular undesired signal could be filtered, and the chang original samples could be predicted very well by both models. Furthermor

Conclusions
In order to estimate the flow field signal change based on the flow se nism of a lateral line system, so to improve the environment adaptability of A KKF-VSM was provided, using the Kautz function as the kernel function. Th compared to previous related works, which were mainly based on RBF-NN with the frequency response function as the kernel function. The flow fi square target was studied, and its pressure signal time series was obtained PIV. The predicting accuracy, sensitivity, and robustness of KKF-VSM wer this work. In addition, an RBF-NNM with a Gaussian kernel function was co KKF-VSM. Some interesting conclusions were made. The pressure signal of The KKF-VSM and RBF-NNM were used to filter the irregular undesi predict the original test sample. The results are shown in Figures 16 and indicated that the irregular undesired signal could be filtered, and the chan original samples could be predicted very well by both models. Furthermo RBF-NNM was , and that of three-order KKF-VSM was 374 . 1 dicated that the robustness of the three-order KKF-VSM is higher than that

Conclusions
In order to estimate the flow field signal change based on the flow s nism of a lateral line system, so to improve the environment adaptability of KKF-VSM was provided, using the Kautz function as the kernel function. T compared to previous related works, which were mainly based on RBF-N with the frequency response function as the kernel function. The flow square target was studied, and its pressure signal time series was obtaine PIV. The predicting accuracy, sensitivity, and robustness of KKF-VSM we this work. In addition, an RBF-NNM with a Gaussian kernel function was c KKF-VSM. Some interesting conclusions were made. The pressure signal o around the square target was chaotic, and KKF-VSM and RBF-NNM co

Conclusions
In order to estimate the flow field signal change based on the flow sensing mechanism of a lateral line system, so to improve the environment adaptability of AUVs, a novel KKF-VSM was provided, using the Kautz function as the kernel function. This is a novelty compared to previous related works, which were mainly based on RBF-NNM, and VSM with the frequency response function as the kernel function. The flow field around a square target was studied, and its pressure signal time series was obtained by CFD and PIV. The predicting accuracy, sensitivity, and robustness of KKF-VSM were analyzed in this work. In addition, an RBF-NNM with a Gaussian kernel function was compared with KKF-VSM. Some interesting conclusions were made. The pressure signal of the flow field around the square target was chaotic, and KKF-VSM and RBF-NNM could predict its change trend very well. However, in comparison with RBF-NNM, KKF-VSM showed higher accuracy in filtering regular and irregular noise. Furthermore, KKF-VSM appeared to capture weak undesired signals more easily, and its filter ability was more robust than that of RBF-NNM. In future work, the authors intend to apply this procedure considering an experimental setup in a natural underwater environment.