1. Introduction
The geomagnetic field is a passive signal and has the characteristics of one-to-one correspondence with the spatial position. However, the ferromagnetic material on the earth’s surface will generate an induced magnetic field due to the action of the geomagnetic field, which will cause the local geomagnetic field to be distorted. The geomagnetic field distortion also provides a theoretical basis for the detection of underwater magnetic targets, that is, the detection of magnetic anomalies [
1,
2]. The magnetic anomaly detection system based on the underwater unmanned vehicle (UUV) in the deep-sea environment can be close to the magnetic target and obtain anomalous magnetic signals with high signal-to-noise ratio. Therefore, as a highly maneuverable and flexible measurement method, UUV equipped with magnetometers began to be used for a series of tasks such as measuring the magnetic characteristics of magnetic targets on the water surface [
3,
4], locating and tracking underwater magnetic substances [
5], and detecting the magnetic field of the seabed [
6].
However, the actual measurement result of the magnetometer is superimposed by the earth’s magnetic field, the magnetic field of the magnetic target, and the interfering magnetic field of the UUV. Therefore, when the local standard geomagnetic field is known, the target magnetic field may also be submerged in the interference magnetic field of the UUV, resulting in the failure of the magnetic anomaly detection mission. Therefore, the UUV’s interference magnetic field compensation method has become one of the key factors restricting the development of underwater magnetic anomaly detection technology. In order to improve the measurement accuracy of the abnormal magnetic fields, it is necessary to conduct in-depth research on the removal of the UUV’s interference magnetic fields.
For the compensation research of the magnetometer, Tolles [
7] and Leliak [
8] decomposed the aircraft’s disturbance magnetic field into a constant magnetic field related to hard magnetic materials, an induced magnetic field caused by the change of carrier attitude, and an eddy current magnetic field. The corresponding mathematical model was established with the Tolles-Lawson equation (T-L equation), so that the compensation problem of the carrier interference magnetic field was transformed into the linear regression problem of the compensation parameters. Gebre-Egziabher et al. [
9] proposed a two-step calibration method based on the principle that the theoretical magnetic field strength after calibration is equal to the reference magnetic field strength. This method transformed the calibration compensation problem into a linear regression problem by introducing intermediate parameters. The linear least squares method (LSM) was first used to solve the intermediate parameters, and then the calibration compensation parameters were solved by the algebraic solution method.
Aiming at the problem of errors in the measurement data in the two-step method, Wu et al. [
10] and Zhang et al. [
11] proposed to use the overall least squares method and the truncated overall least squares method to solve the intermediate parameters, respectively. Based on the principle that the component information of the magnetic field after compensation is equal to the reference magnetic field, Zhang et al. [
12] introduced intermediate parameters to linearize the compensation parameters and obtained the compensation parameters by the least squares method. Pang et al. [
13] proposed a calibration compensation method based on a two-step LSM. After obtaining the intermediate parameters, the LSM was also used to solve the compensation parameters, which further increased the solution accuracy of the compensation parameters.
Yu et al. [
14] and Vasconcelos et al. [
15] fit the ellipsoid equation based on the magnetometer measurement value, and then used the LSM and the maximum likelihood estimation method to iteratively solve the compensation parameters. In order to fit the ellipsoid, this method requires the magnetometer to traverse various spatial attitude angles as much as possible, and the variation of the roll angle and pitch angle during the actual UUV navigation process is very limited, so this compensation method is not suitable for the compensation of UUV’s interference magnetic field.
There are also many scholars who have transformed the compensation problem of the carrier interference magnetic field into the nonlinear parameter optimization problem, and research has been made on the optimal estimation method of compensation parameters. Pang et al. [
16] established the calibration model of the fluxgate magnetometer, and the calibration parameters were iteratively solved using the “Fsolve” function in MATLAB. Li et al. [
17] and Li et al. [
18] used the UKF method and the trust region method to estimate the compensation parameters, respectively. These two methods can obtain better estimation results when the selection of the iteration initial value is appropriate. The above compensation parameter estimation results are greatly affected by the initial values of the parameters, and inappropriate initial values are also likely to cause a divergence of results.
Research using meta-heuristic algorithms has determined that this kind of algorithm has strong applicability to high-dimensional nonlinear optimization search problems, and has the advantages of insensitivity to the initial value, a simple structure and no gradient mechanism [
19]. Some scholars have begun to introduce heuristic algorithms into the estimation of compensation parameters, and have proved the superiority of such algorithms. The particle swarm optimization (PSO) algorithm has the advantages of insensitivity to the initial value and a fast convergence speed. Accordingly, many scholars have improved this method and applied it to the compensation of the carrier interference magnetic field. For example, Wu et al. [
20] and Wu et al. [
21] calibrated and compensated the measurement results of a fluxgate magnetometer based on a PSO and a stretching PSO algorithm. Li et al. [
22] improved the inertia weight in the PSO algorithm and proposed a damping PSO algorithm, and the results showed that the effect is better than the two-step method. Huang et al. [
23] proposed a magnetometer measurement error compensation method based on an immune adaptive particle swarm optimization algorithm. Zhang et al. [
24] used the differential evolution (DE) algorithm to estimate the compensation parameters and the local magnetic field. Gao et al. [
25] introduced the cuckoo optimization search (CS) algorithm into the compensation of the magnetometer. The experimental results showed that the compensation accuracy of this method is better than that of the UKF method.
The above compensation method is mainly based on the measurement results of the fluxgate magnetometer. Due to the limitations of production, processing and assembly, the fluxgate magnetometer has non-orthogonal errors and zero-scale factor errors [
26]. It needs to be specially calibrated, and the accumulation of errors will inevitably reduce its compensation accuracy. As a scalar magnetometer, the cesium optical pump probe has high geomagnetic measurement accuracy, no zero drift and is not affected by temperature. It does not need accurate orientation during operation, and is suitable for high-precision, rapid, and continuous measurement under moving conditions, such as airborne magnetic surveys and marine magnetic surveys.
At present, the compensation research for the cesium optical pump magnetometer’s measurement error mainly focuses on the aeromagnetic compensation method based on the T-L model. This compensation method assumes that the absolute value of the disturbance magnetic field at the scalar magnetometer is the projection of the vector disturbance magnetic field in the direction of the geomagnetic field [
27]. This assumption ignores the angle between the direction of the geomagnetic field and the direction of the interference magnetic field, so it is suitable for the case where the interference magnetic field is extremely small. In addition, the direction cosine is mostly provided by the fluxgate magnetometer in the aeromagnetic compensation, but the data collected by the fluxgate are not accurate, and the loading position of the fluxgate cannot be exactly the same as that of the optical pump magnetometer [
28]. Therefore, there is a certain deviation in the direction cosine value obtained by the fluxgate measurement result, which may lead to poor estimation accuracy of the compensation parameters. Wu et al. [
29] decomposed the total field data of the optical pump in the three-axis direction and used the genetic algorithm (GA) to estimate the compensation parameters, but this method used the three-axis magnetometer to calculate the three direction cosines containing the interference external magnetic field, which ignores the error of the fluxgate magnetometer itself, and thus may reduce the compensation accuracy.
In order to solve the problem of interference magnetic field when the remotely operated vehicle (ROV) is equipped with a cesium optical pump magnetometer for magnetic anomaly detection, this paper proposed an improved mayfly optimization algorithm to estimate the compensation parameters. The inertial navigation sensor was introduced to help decompose the total magnetic field measurement result of the optical pump magnetometer. Then, a compensation model based on the component value of the measurement results was constructed. The mayfly optimization algorithm (MOA) is a swarm intelligent optimization algorithm proposed by Zervoudakis and Tsafarakis [
30] in 2020. Compared with other intelligent optimization algorithms, this algorithm has certain advantages in local search performance and population diversity [
31] and is suitable for high-dimensional nonlinear parameter estimation. However, the algorithm also easily falls into local optimum, so this paper introduced the elite chaotic reverse learning strategy and Levy mutation strategy to improve the mayfly algorithm, and the improved mayfly optimization algorithm (IMOA) was used to estimate the compensation parameters. Finally, the practicability of the proposed compensation method was verified with a series of field experiments.
To summarize, the main contributions of this paper are as follows:
An ROV’s interference magnetic field removal method based on the measurement results of a cesium optical pump magnetometer was proposed in this paper, which used the inertial navigation sensor to help decompose the magnetic field measurement results;
This paper proposed an improved mayfly optimization algorithm, which improved the application ability of this algorithm in the field of estimation of the compensation parameters;
The proposed compensation method can achieve high compensation accuracy on actual magnetic field data and has feasibility and validity for different experiment data.
3. The Optimal Estimation Algorithm
In this section, we propose a compensation parameters estimation method based on the improved mayfly optimization algorithm (IMOA). The original mayfly optimization algorithm (MOA) is first described. Then, an IMOA based on an elite chaotic reverse learning strategy and Levy flight strategy is proposed. Finally, the compensation parameters estimation steps based on IMOA are given.
3.1. The Primitive Mayfly Optimization Algorithm
Like the classic PSO algorithm and GA, the MOA is also a heuristic search algorithm, which was inspired by the social behavior of mayfly groups, including behaviors such as aggregation and mating. The mayfly algorithm combines the advantages of PSO, GA and other algorithms in its structure. It not only has the ability of random search, but also enables the population to evolve according to the law of survival of the fittest. Compared with other algorithms, this algorithm possesses the advantages of strong local search performance, high population diversity, and fast convergence speed. Therefore, it is suitable for solving high-latitude nonlinear optimization problems.
Assuming that in the d-dimensional search space, the number of both the male and female mayflies in the population is m, the position of the male mayfly and female mayfly is expressed as x = (x1, x2, …, xd) and y = (y1, y2, …, yd), respectively. The speed of each mayfly is represented by v = (v1, v2, …, vd). The quality of each mayfly’s search result is evaluated by the objective function f(x) defined in the previous section.
After the
t-th iteration, the position, velocity, and historical optimal position of the
i-th male mayfly in the
j-th dimension search space are denoted as
,
, and
pij, respectively, and the population optimal position in the
j-th dimension search space is denoted as
gj. Since male mayflies generally gather and dance on the water’s surface to attract female mayflies, each male mayfly adjusts its flight speed according to its own flying experience and that of the group. Then each male mayfly updates its current speed and position according to Equations (11) and (12):
Among them, a1, a2 are the attraction coefficients of male mayflies, and β is the visibility coefficient. The Euclidean distance between the xi with the pij is represented as rp. The rg represents the Euclidean distance between the xi and the gj. d represents the dance coefficient, which is used to attract female mayflies, r is a random coefficient, and r ∈ [−1, 1]. The mayfly at the optimal position continuously updates its speed and position by introducing random element d to lead the mayfly group to fly to a better position.
Different from male mayflies, female mayflies will decide whether to fly close to the male mayfly according to the quality of its position. After the
t-th iteration, assuming that the position and velocity of the
i-th female mayfly in the
j-th dimensional search space are
and
, the female mayfly updates its current velocity and position according to Equations (13) and (14):
where
a3 is the attraction coefficient of the female mayfly,
rm represents the Euclidean distance between male and female mayflies. The
fl is the random walk coefficient, which means that the female mayfly will fly randomly to find a better male mayfly when not attracted.
The ultimate goal of the mayfly population is to mate to produce better offspring. In order to avoid falling into the local optimum, the mating process in the algorithm is mainly to randomly select a part of the samples in males and females, respectively. According to the mechanism of survival of the fittest, the male optimal individual and the female optimal individual mate, and the second optimal male individual mates with the female individual. The offspring produced after mating are generated by Equations (15) and (16).
Among them, off1 and off2 represent the two generated offspring, L is a random number and L ∈ [−1, 1], m represents a male mayfly, and f represents a female mayfly.
3.2. The Improved Mayfly Optimization Algorithm
Like most heuristic algorithms, the MOA also has the problem of easily falling into premature convergence in high-dimensional complex nonlinear problems. On the one hand, the MOA uses random initialization to generate the initial population which easily leads to a certain blindness in the iterative search process and may result in long search time, slow convergence speed, and the formation of local optimum. On the other hand, in the later stage of iteration, the inertia weight factor, dance coefficient and random walk coefficient will decrease gradually in order to meet the local search performance. If the algorithm falls into a local optimum in the early stage, it is difficult to jump out of the local optimum in the later stage.
In view of the above defects, we adopted the following measures to improve MOA: (1) we introduced the elite chaotic reverse learning strategy into the population initialization of the MOA. By combining the chaotic solution and the reverse solution, this strategy selects the mayfly population on a global scale to improve the initial population’s diversity and quality, avoiding the uncertainty caused by the random initial population, and improving the convergence efficiency; (2) the Levy mutation strategy was used to perturb some mayfly individuals that may fall into the local optimum, so as to improve the population’s diversity and anti-stagnation ability, and help the algorithm to jump out of the local optimum.
The elite chaotic reverse learning strategy mainly adopts two stages to initialize the mayfly population. Among the chaotic operators, the cubic chaotic mapping has better uniformity, so it can be considered to use cubic chaotic mapping to initialize the mayfly population. The initialization process is shown in Equations (17) and (18). First, the cubic mapping variable is generated by Equation (17), where
yi ∈ [−1, 1], and the initial value
y0 is randomly generated. Then, the cubic mapping variables are applied to the mayfly population using Equation (18), where
xi is the
i-th mayfly after the cubic chaotic mapping,
lb and
ub are the lower and upper bounds of the mayfly population position, respectively.
On the basis of the cubic chaotic mapping, the lens imaging reverse learning strategy [
33] is introduced to reversely solve the above mayfly population. The specific mathematical model is shown in Equation (19). Among them,
zi is the
i-th mayfly after the reverse learning, and
η is the lens’s zoom factor.
Two mayfly populations are obtained after the cubic chaotic mapping and the lens reverse learning strategy. In order to select the mayfly with better quality, the mayfly population is selected according to the fitness order. The mathematical model of screening is shown in Equation (20). The
i-th mayfly individual selected is represented by
Xi:
In the later stage of MOA iteration, mayfly individuals will rapidly assimilate, which may easily lead to the stagnation of the mayfly population at the local optimal position. In order to solve this problem, according to the roulette probability, some mayfly individuals are randomly selected for Levy mutation. Then the positions of the mutated mayfly before and after the mutation are compared according to the objective function. If the mutated position is better, the mutated mayfly individual is chosen to enter the next iteration, otherwise the mutation is invalid.
The Levy mutation strategy comes from Levy flight, which is a non-Gaussian random process proposed by French mathematician Levy. Levy flight is also an ubiquitous phenomenon in nature. Due to its large and small flight steps, both global optimization and local optimization can be taken into account in the optimization problem, which helps the population fly out of the local optimum. The Levy variation is shown below:
Among them, β ∈ [0, 2], μ obeys N (0, σ2) distribution, and ν obeys N (0, 1) distribution.
3.3. The Compensation Parameters Estimation Process Based on the IMOA
The main procedures for estimating compensation parameters using the IMOA are as follows:
Step 1: Setting the parameters in the algorithm, mainly including the population size, the maximum number of iterations, the upper and lower bounds of the compensation parameters, the attraction coefficients, the dance coefficient, and the random walk coefficient;
Step 2: Initializing the positions of male and female mayflies, respectively, using the elite chaotic reverse learning strategy, setting the initial speed value and calculating the initial fitness function value;
Step 3: Performing cyclic calculation on the male mayfly. Updating the speed, position and fitness function value of the male mayfly using the Equations (10)–(12) and then updating the global optimal fitness value;
Step 4: Performing cyclic calculation on the female mayfly. Updating the speed, position and fitness function value of the female mayfly using the Equations (13), (14) and (10) and then updating the global optimal fitness value;
Step 5: Selecting partial samples from the updated mayfly populations randomly. Obtaining the progeny mayfly according to the Equations (15) and (16) and then updating the global optimal fitness value;
Step 6: Deciding whether to perform Levy mutation according to the roulette probability and then updating the global optimal fitness value;
Step 7: Ending the iteration if the maximum number of iterations is reached, otherwise go to the Step 2 to reiterate.