1. Introduction
Underwater submersibles have been widely used in defense and scientific fields, such as underwater operations, marine research and mineral exploration, due to their ability to be concealed and high mobility [
1,
2]. For the smooth implementation of military and civilian technology, it is very necessary to ensure the safety of the underwater navigation of submersibles [
3]. However, since the beginning of the 20th century, there have been nearly 500 accidents involving submersibles, of which about 20% were collision–sinking accidents, and these caused 84 submersibles to sink to the bottom of the sea [
4]. Due to the complexity of the seabed environment, the imperfection of existing detection methods is one of the important reasons for the occurrence of many underwater accidents. In order to maintain concealment, submarines usually do not use active detection methods such as active sonar. As a result, it is difficult for the submarine to discover unexpected obstacles, causing collision accidents. Therefore, developing real-time, accurate, and passive underwater obstacle detection technology is crucial to ensure the safety of the underwater navigation of submersibles [
5].
Underwater obstacle detection is an important part of underwater safety navigation; underwater obstacle detection technology uses acoustic signals, optical signals, electromagnetic signals, and gravity gradient signals for underwater obstacle detection [
6]. The acoustic signal method mainly records the propagation time and phase difference of acoustic signals through sonar detection to estimate the azimuth and distance of underwater obstacles, which has the advantages of a large working range and is not affected by water turbidity. However, the acoustic signal method actively emits signals outward, resulting in the exposure of the submersible’s own position, which limits the practical application of the acoustic signal method [
7]. The optical signal method obtains environmental information by receiving light from the surrounding environment with a high resolution and refresh rate. However, underwater light conditions (such as light absorption and scattering) are complex, and the detection range of optical signal methods is limited [
8]. The electromagnetic sensor method can be applied in the underwater environment to achieve the estimation of underwater distance [
9]. However, ambient electromagnetic fields can interfere with measurement accuracy [
10]. The abovementioned underwater obstacle detection technologies have their own advantages and disadvantages, and researchers need to combine a variety of underwater detection technologies to complete various underwater obstacle detection tasks in practice. Based on this, as a passive detection method, the gravity gradient signal method has the characteristics of high sensitivity, can be concealed and can used in all weather types and in real time, which meets the requirements of the safe navigation of submersibles under concealed conditions and has an important application value [
11,
12,
13].
Gravitational gradients reflect subtle changes in the gravitational field, and for high-density objects, accurate mass estimates can be made by measuring the gravitational gradient they cause [
14]. Wu et al. [
15,
16,
17] proposed an automatic full-tensor gravity gradient algorithm to estimate the mass, direction, and distance of underwater obstacles. Cheng et al. [
18] combined gravity and gravity gradient information to detect the position and density of obstacles and concluded that the relative error of obstacle detection is within 5% under the condition that the accuracy of the gravimeter is 10
−2 mGal and the accuracy of the gravimeter is 10
−4 E. Wu and Cheng proposed that the method is feasible if the gravity gradient reference map in the detection area is known. Aiming at the obstacle detection problem in unknown sea areas (such as missing gravity gradient reference map or insufficient resolution). Yan et al. [
19,
20] proposed a gravity gradient difference ratio method without a gravity gradient reference map, which established a gravity gradient difference ratio (GGDR) equation only related to the obstacle’s position and used the Newton–Raphson method (NRM) to solve GGDR equation to obtain the obstacle’s position. However, if the initial value selected is not suitable when one is using NRM to solve the GGDR equation, the calculation result will not converge, and the detection accuracy will not be high. Nowadays, many articles focus on developing new submersibles and applying the regression surrogate model to solve some submersible problems. Chen et al. [
21,
22] were the first to apply machine learning algorithm to submersible fluid mechanics calculation and proposed the integration of multiple surrogate models, which improved the robustness of the model.
Different from previous research, this paper combines the characteristics of the gravity gradient difference ratio method and the characteristics of the SVR algorithm to propose a novel SVR–gravity gradient joint method (SGJM). The novel SGJM uses the gravity gradient difference ratio method to convert the obstacle detection problem into a higher order nonlinear equation solving problem, and then solves the higher order nonlinear equation using an SVR algorithm to determine the location of underwater obstacles. Firstly, the gravity gradient differential ratio data generated by simulated obstacles are calculated by the gravity gradient difference ratio method, and the differential ratio dataset is constructed by using the gravity gradient difference ratio data and the corresponding obstacle locations as the input and output, respectively. Then, based on the differential ratio dataset, the SVR obstacle localization model is trained, tested, and verified for reliability. Finally, the underwater obstacle’s positioning results of SGJM and NRM are compared in the same experimental environment, which verifies the improvement of obstacle’s positioning accuracy of SGJM in this paper. The novel SGJM ingeniously combines the gravity gradient with machine learning, which simplifies the conditions required for obstacle detection and improves the detection accuracy, and thus, it is a simple and practical method.
2. Methods
Earth is an irregularly shaped ellipsoid with uneven density distribution of its surface areas, resulting in differences in gravitational gradients throughout [
23]. For example, the gravitational vertical gradient can be divided into two main parts: one part is the normal gravitational vertical gradient, assuming that it is caused by a rotating ellipsoid with uniform density distribution; the other part is the gravitational vertical gradient anomaly, which is caused by the difference between the gravitational vertical gradient and the normal gravitational vertical gradient caused by Earth. Similarly, the density of underwater obstacles and the surrounding environment will cause differences in gravity gradients, which include information such as the location and mass of underwater obstacles [
24]. Therefore, under the condition that the accuracy of the gravity gradiometer is high enough, it is feasible to detect underwater obstacles by analyzing the gravity gradient caused by underwater obstacles.
2.1. Gravity Gradient Difference Ratio Method
The gravity gradient difference ratio method transforms obstacle detection problems into higher order nonlinear equation solving problems [
20]. Suppose there is an obstacle with a uniform density distribution and a submersible carrying a gravity gradiometer in a seawater environment. The Cartesian coordinate system (right-handed system) is used to establish the obstacle at the origin of the center, and the
x-axis is parallel to the submersible motion route and takes the submersible motion direction as the positive direction. Then, the center of mass coordinate of the obstacle in the Cartesian coordinate system is
, and the coordinate of the submersible position is
(the coordinates of the gravity gradiometer measurement are relative to the obstruction material center, reflecting the position of the obstacle). The
time gravitational gradient recorded by the submersible at point
can be described as [
24]:
where
is the gravity gradient caused by the obstacle,
is the component of the gravity gradient caused by the obstacle in the
direction,
is the underwater environmental noise,
is the value of underwater environmental noise in the
direction,
is the universal gravitational constant,
is the mass of the obstacle,
is the coordinates at point
of the submersible position at the time of the
time measurement, and
is the distance from the submersible position
point to the
point of the obstacle’s center at the
time measurement.
If two gravity gradient data are recorded in a short period of time during underwater navigation of the submersible, it can be considered that the difference between the two gravity gradient data is mainly caused by the relative change in position that occurs between the submersible and the underwater obstacle [
24]. Therefore, the difference between the two adjacent gravity gradient data can eliminate underwater environmental noise
, thereby isolating the submersible and obstacles from the surrounding environment. The gravity gradient difference
is calculated as follows:
where
is the component of the gravity gradient difference in the
direction.
From Equation (2), the gravity gradient difference
still contains the mass of the obstacle after eliminating the ambient noise. Because the quality of the obstacle cannot be determined in advance, the influence of the quality of the obstacle can be eliminated by ratio, and the two gravity gradient difference data are divided to obtain the gravity gradient difference ratio function [
20], which is only related to the position of the obstacle:
where
is gravity gradient difference ratio, and
is the component of the gravity gradient difference ratio in the
direction.
Equation (3) is further expanded into the integral form, and the gravity gradient difference ratio integral function, which is only related to the position of the obstacle, is obtained:
where the integral area of the integral function is the space surrounded by the outline of the obstacle;
is the product element coordinate of the obstacle object;
; if
is the gravity gradient recording distance interval, then
,
, and
.
Equation (4) is converted to the following:
Equation (5) is a high-order nonlinear equation. Traditional NRM is generally used to solve higher order nonlinear equations. However, as NRM is an iterative algorithm, the selection of initial values is very important, and inappropriate initial values will lead to nonconvergence of the calculation results and a low detection accuracy. Therefore, in order to improve the accuracy of obstacle detection, the SVR algorithm is used to solve the GGDR equation.
2.2. Support Vector Regression
The SVR algorithm is a machine learning algorithm that was proposed by Vapnik et al. [
25] in the 1990s, which is used to solve regression problems. The SVR algorithm takes structural risk minimization as the basic idea of machine learning, and compared with the traditional statistical theory, this algorithm specializes in the statistical law of machine learning in the case of small samples [
26]. In addition, the SVR algorithm is able to approximate the solution of higher order nonlinear equations. Thus, SVR was chosen as a surrogate model in this study [
27]. As shown in
Figure 1, the idea of the SVR algorithm is to find a hyperplane so that as many points as possible are concentrated in a space that is as small as possible on both sides of the hyperplane.
Given a dataset
(where
represents the 6-dimensional input vector,
,
is the length of the dataset), the nonlinear kernel function is mapped to the high-dimensional feature space to complete linear regression. In the high-dimensional feature space, the SVR algorithm uses the following approximation function [
28]:
where
represents the high-dimensional feature space of the input vector,
, for nonlinear mapping. The weight factor
and constant
are solved with the optimization model of the optimal regression function [
28]:
where
is the penalty factor for balancing empirical risk and model flatness,
and
are relaxation variables that constrain the output of the system, and
is the tube size constant [
29].
Equation (7) is a convex optimization problem, which can be solved by the Lagrange multiplier method. The Lagrange multiplier is introduced, and the regression function of SVR algorithm is calculated as follows [
30]:
where
and
are Lagrange multipliers,
.
is a kernel function,
.
Kernel functions help researchers to deal with feature spaces of any dimension without explicitly calculating mapping,
. Any function that satisfies the Mercer condition can be used as a kernel function [
31].
Table 1 describes the categories and expressions of several commonly used kernel functions [
32]:
The kernel parameters are manually set, and when the parameters are determined, it means that the kernel function is determined. The parameter selection of the kernel function directly affects the prediction accuracy of SVR [
33]. The selection of kernel parameters should be cautiously performed because it implicitly defines the structure of high-dimensional feature space, thus controlling the complexity of the final solution [
34]. In this paper, RBF is used as the kernel function of the SVR algorithm.
As a machine learning algorithm, the SVR algorithm is a supervised learning algorithm that regresses the target samples according to the current sample information. SVR implicitly expresses the mathematical relationship between the input vector and output value.
2.3. SVR–Gravity Gradient Joint Method
From the analysis in
Section 2.1, the mathematical relationship between the gravity gradient difference ratio and the position of the obstacle is established by the GGDR equation, and the obstacle detection problem is transformed into the problem of solving higher order nonlinear equations [
20]. From the analysis in
Section 2.2, it can be seen that the SVR algorithm simplifies the mathematical relationship between the gravity gradient difference ratio and the position of the obstacle to some extent [
35], and it can better approach the solution of the equation for higher order nonlinear equations [
27]. Therefore, this paper combines the characteristics of the gravity gradient difference ratio method and the SVR algorithm and proposes the novel SVR–gravity gradient joint method (SGJM).
In the composition of SGJM, the simulated gravity gradient data are transformed into a gravity gradient difference ratio by Equation (3). Then, the gravity gradient difference ratio and the position of obstacles are used as the input and output of the SVR algorithm, respectively, to construct the difference ratio dataset, and the SVR obstacle location model is obtained by training. Finally, the location model is used to solve the GGDR equation and evaluate the positioning accuracy. As shown in
Figure 2, the novel SGJM consists of three parts: data simulation, positioning model training, and accuracy evaluation.
In SVR obstacle location model training, this paper uses the grid search method [
36] and the 5-fold cross-validation method [
37] to optimize the SVR regression model, and the super parameters are penalty parameter,
, kernel function coefficient,
, and tube size,
.
The result of obstacle location is compared with the real position of the obstacle, and the reliability of underwater obstacle location based on SGJM is evaluated by mean absolute error (MAE), root-mean-square error (RMSE), coefficient of determination (R
2), relative error (RE), and signal-to-noise ratio (SNR). The smaller the MAE and RMSE are, the closer the result is to the real value. The closer R
2 is to 1, the higher the fitting degree of the SVR regression model is. The relative error reflects the credibility of the measurement, and the smaller the RE is, the higher the credibility is. The signal-to-noise ratio reflects the influence of noise in the signal, and the larger the SNR is, the smaller the noise is. The corresponding definition is [
38,
39,
40,
41]:
where
is the true value of the obstacle coordinates,
is the calculated value of the obstacle coordinates,
,
is the length of the test set,
is the square sum of the residuals,
is the total square sum,
is the gravity gradient signal power, and
is the noise power.
4. Application
The novel SGJM is applied to an underwater obstacle detection task, and the traditional NRM [
20] is used as a comparison to prove that SGJM has better detection accuracy. The simulation experiment is established, and the underwater obstacle is a cuboid prism of 80 m × 40 m × 20 m with a density of 2.7 t/m
3. Taking the center of the obstacle matter as the coordinate origin, the Cartesian coordinate system (right-handed system) is established. The motion direction of the submersible is parallel to the positive direction of the
x-axis, the coordinates of the starting point are
, and the velocity is 10 m/s. The gravity gradiometer records data once a second. The environmental noise with RMSE of 10
−5 E is added to the simulated gravity gradient data, and a total of 23 sets of gravity gradient differential ratio data are obtained.
In this experiment, 23 simulated localizations were performed, and after statistical analysis, the MAE, RMSE, and standard deviation (STD) of the localization results were obtained, and the results are shown in
Table 5.
t-test was carried out based on the normal distribution, and the difference level of positioning error between SGJM and NRM was obtained, as shown in
Table 6. The positioning error of the novel SGJM is shown in
Figure 10. The positioning error of the traditional NRM is shown in
Figure 11. The SNR of SGJM and NRM is shown in
Figure 12a, and the RE comparison is shown in
Figure 12b and
Figure 13.
According to
Table 5, both MAE and RMSE of SGJM are smaller than NRM is, the MAE and RMSE of SGJM in the
y direction are slightly better than NRM is, and the MAE and RMSE of SGJM in the
x direction and
z direction are obviously better than those of NRM. Among them, the precision in the
x direction has been improved by 88%, the precision in the
y direction has been improved by 6%, and the precision in the
z direction has been improved by 85%. The improvement of positioning accuracy in the
y direction is lower because the positioning accuracy of NRM in the
y direction is higher.
According to
Table 6, the positioning accuracy of SGJM is better than that of NRM in
x and
z directions. There is no significant difference in the positioning results in the y direction.
According to
Figure 10 and
Figure 11, the positioning error of SGJM is obviously better than that of NRM. The positioning error of SGJM increases with the increase in distance, but it is still within the range of −4~4 m. According to
Figure 12a, with the increase in distance, the SNR decreases, and the influence of environmental noise increases. According to
Figure 12b and
Figure 13, the RE of NRM is less than 6% when the distance is less than 360 m; the RE is less than 4% when the distance of SGJM is less than 410 m, and the RE is less than 2% when the distance is less than 290 m. The obstacle location accuracy of SGJM is higher than that of NRM.
In summary, the average MEA of the obstacle localization results obtained by the novel SGJM is only about 1/5 of that of NRM. The relative error of SGJM is less than 4%, which is better than 6% of NRM. Therefore, the novel SGJM effectively improves the positioning accuracy of the traditional NRM.