Localization of Alternating Magnetic Dipole in the Near-Field Zone with Single-Component Magnetometers

Tri-axis magnetometers are widely used to measure magnetic field in engineering of the magnetic localization technology. However, the magnetic field measurement precision is influenced by the nonorthogonal error of tri-axis magnetometers. A locating model of the alternating magnetic dipole in the near-field zone with single-component magnetometers was proposed in this paper. Using the vertical component of the low-frequency magnetic field acquired by at least six single-component magnetometers, the localization of an alternating magnetic dipole could be attributed to the solution for a class of nonlinear unconstrained optimization problem. In order to calculate the locating information of alternating magnetic dipole, a hybrid algorithm combining the Gauss–Newton algorithm and genetic algorithm was applied. The theoretical simulation and field experiment for the localization of alternating magnetic dipole source were carried out, respectively. The positioning result is stable and reliable, indicating that the locating model has better performance and could meet the requirements of actual positioning.


Introduction
Magnetic positioning technology, with characteristics of allweather, low power consumption, and simple signal processing, has gradually drawn people's attention. Due to the lower noise level of the magnetic sensor and the higher measurement accuracy, it would be easier to detect weak magnetic signals. Using the magnetic field signal of the target detected by a magnetic sensor or magnetic sensor array, the position information and motion state of the target were obtained by data inversion, which could be widely used in identification of vehicles [1], monitoring of magnetic field [2,3], prediction of earthquake [4], diagnosis of pipeline failure [5], and exploration of crude oil [6].
Because the positioning algorithm of the magnetic target based on the static magnetic field is greatly influenced by the interference of the geomagnetic environment and other magnetic sources, some researchers have studied the localization of alternating magnetic dipole sources. In 2001, Paperno et al. proposed a method for magnetic position and orientation tracking. Based on two-axis generation of a quasi-static rotating magnetic field and three-axis sensing, two mutually orthogonal coils fed with phase-quadrature currents comprise the excitation source could be equivalent to a mechanically rotating magnetic dipole [7]. In 2006, Nara et al. presented a simple reconstruction formula for localization of a magnetic dipole. In order to calculate the locating information, the dipole position is expressed in terms of the magnetic field and its spatial gradients at a single place [8]. In 2010, Plotkin et al. developed a new scleral search coil (SSC) to track the target. e theoretical deduction and numerous simulations have shown that the proposed method could obtain the orientation and location information of SSC [9]. In 2013, Sheinker et al. proposed a locating method in 3D using beacons of low-frequency magnetic field. e method could be used in many applications, such as the navigation of indoor robot and the mapping of underground cavity [10]. Using beacons of lowfrequency magnetic field, the authors proposed a method of remote tracking a year later [11]. In 2016, Pasku et al. described a positioning system based on low-frequency magnetic field. e system could accommodate an arbitrary number of users without any additional infrastructure [12]. In 2015, Li et al. proposed an approach based on the genetic algorithm to search the location of the dipole. Only an electric field sensor in seawater is needed to measure the modulus of electric field intensity at the corresponding positions. en, the position of the dipole could be determined accurately [13]. In 2017, the author proposed a positioning method for moving objectives with alternating magnetic fields using coherent demodulation. However, the magnetic fields were measured by using a tri-axis magnetometer.
e magnetic field measurement precision is influenced by the nonorthogonal error of tri-axis magnetometers [14,15]. In 2018, Dai et al. proposed a new 6D tracking method using the 3D linear motion, 2D rotational motion, and 3D orientation tracking. e hybrid method of magnetic tracking and inertial sensing verified that the full 6D pose could be used to track the target accurately [16]. In 2020, Song et al. proposed a positioning method of lowfrequency magnetic beacons based on the genetic algorithm. In a wide-range measurement, the theoretical simulation and field experiment had been tested to show the accuracy of localization for the target [17].
A positioning method of alternating magnetic dipole in the near-field zone with single-component magnetometers was introduced in this paper. A measuring array consisting of at least six single-component magnetic sensors was used to collect the magnetic field emitted by the alternating magnetic dipole. rough the process of coherent demodulation, the varying curve of alternating magnetic field could be obtained. A hybrid algorithm combining the Gauss-Newton algorithm and genetic algorithm was applied to obtain the track of a moving target, which showed a good agreement with the actual motion information [18][19][20][21][22].

e Vertical Component of Alternating Magnetic Dipole.
e alternating magnetic dipole source is a transmitting coil that radiates a low-frequency sinusoidal electromagnetic signal, and the working frequency of the signal is set as a fixed frequency. For example, the working frequency of the signal ranges from 100 Hz to 1000 Hz, and the corresponding wave length is between 3 × 10 5 meters and 3 × 10 6 meters correspondingly. e geometry of the radiation coil is much smaller than its working wavelength so that the radiation coil could be equivalent to a magnetic dipole. e schematic diagram of magnetic dipole in the cylindrical coordinate system is shown in Figure 1.
e radiation magnetic moment of the magnetic dipole is expressed by the formula as follows: In formula (1), μ 0 is the magnetic permeability of the medium I is the current intensity in the coil S is the cross-sectional area of the coil, whose direction is the normal direction of the right-handed spiral Using Maxwell's equations and boundary conditions, the electromagnetic field expression of magnetic dipole radiation could be expressed as In formula (2), ω is the angular frequency of the alternating electromagnetic field R is the distance from the magnetic dipole source to the observation point K is the number of complex waves, which is plural in the conductive medium Formula (2) contains the item labelled as KR. According to the distance labelled as R between the magnetic dipole source and the receiving point, the wavelength of the radiated electromagnetic wave in the propagation medium is labeled as λ. e electromagnetic field magnetic transmitted by the magnetic dipole could be divided into three regions.
(1) When KR ≪ 1, it is called the near zone, also known as the quasi-stationary zone or the zone of stability (2) When KR ≫ 1, it is called the far zone (3) e region between the near zone and the far zone is called intermediate zone Usually, R ≪ 0.1λ. Considering of the target's working frequency, the positioning region of the target is the nearfield zone of the magnetic dipole source. e distribution of the electromagnetic field in the near zone of the alternating magnetic dipole approximates that of the static magnetic dipole (ω � 0, promptly, K � 0), which is similar to the constant stability field. It is assumed that the magnetic moment of an alternating magnetic dipole source located at the point labelled as P 0 (x 0 , y 0 , z 0 ) could be recorded as Mathematical Problems in Engineering e magnetic vector potential and magnetic fields at the receiving point labelled as P(x, y, z) are as follows: e three component magnetic fields acquired by a triaxis magnetometer could be described as follows: From a strictly mathematical point of view, at least six single-component sensors are required since there are six unknown quantities: the three position coordinates labelled as P 0 (x 0 , y 0 , z 0 ) and the three moment components labelled as M 0 (M x0 , M y0 , M z0 ), and each sensor provides only one equation.

e Static Locating Method Based on Single-Component of
Magnetic Field. Assume that the measuring array consisting of six single-component magnetic field sensors is shown in Figure 2, and their coordinates are labelled as P n (x n , y n , z n ) where 1 ≤ n ≤ 6. e alternating magnetic dipole source is at the point labelled as P 0 (x 0 , y 0 , z 0 ). e vertical component of the magnetic field generated by the alternating magnetic dipole at the point labelled as P 0 (x 0 , y 0 , z 0 ) was recorded as Using the coherent demodulation, the alternating magnetic field labelled as H n could be transformed to the varying curve signed as H zn [14]. en, the formula could be described as follows: Mathematical Problems in Engineering where E 0 is the objective function of the nonlinear unconstrained optimization problem.
which is called the coefficient matrix of magnetic moment parameters.
which is the varying curve of the vertical component for the alternating magnetic dipole in the near-field using the coherent demodulation [14,15].
which is the coefficient matrix of positions for the target.

Simulations
e measuring array consisting of six single-component magnetic field sensors is in the plane labelled as xOy of the Cartesian coordinate system and is shown in Figure 3, and the origin is signed as O. e magnetic target at the point P moves along a straight line from the point P(−20, −20, 2) to the point Q (20,20,2). e velocity is a constant of 10 m/s in the x-axis and 10 m/s in the y-axis. e coordinates of the six sensors are labelled as P 1 (−2, 1, 0), P 2 (0, 1, 0), P 3 (2, 1, 0), P 4 (−2, −1, 0), P 5 (0, −1, 0), and P 6 (2, −1, 0). e vertical component of the alternating magnetic fields is acquired by a measuring array consisting of six single-component magnetic field sensors.
As shown in Figure 6, the locating results show a good agreement with the actual values as predetermined in the simulation. It could also be seen that the target moved from −20 m to 20 m in the X-axis, and the average velocity is 10 m/ s. e result in the Y-axis is the same as that in the X-axis. e result in the Z-axis is a constant value of 2 m at the time from 1 s to 5 s.
From the results of above simulation, the position information calculated by the model is completely consistent with the predetermined position information of the moving target. ese verify the feasibility of localization for alternating magnetic dipole source using the single-component magnetic field.  Figure 7(a)). e magnetic field is acquired by a measuring array consisting of eight single-component magnetometers (see Figure 7(b)). e frequency of the sinusoidal signal emitted by the solenoid is set as 500 Hz. e measuring array consisting of eight single-component magnetometers collected the vertical component of the magnetic field, which is transferred to a PC via a data acquisition card. e sampling rate is set as 5000 Hz. Because the experimental environment is not ideal, there is strong interference of power frequency and other frequencies. It is impossible to directly use the signals collected by the single-component inductive magnetic field sensors. e signals collected by the single-component inductive magnetic field sensors were passed through a band-pass filter with a cut-off frequency from 480 Hz to 520 Hz. Taking the signal collected by sensor #2 as an example, Figure 9 shows the time domain signals before and after filtering. It also showed that the signals emitted by the signal source collected by sensor #2 were well extracted. Figure 10 shows the magnetic signals collected by the induction magnetic field sensor #1 to #4 processed after filtering. Figure 11 shows the magnetic signals collected by the induction magnetic field sensor #5 to #8 processed after filtering.

Experiment
As shown in Figure 12, the varying curves of alternating magnetic fields collected by the induction magnetic field sensor labelled from #1 to #4 were obtained by coherent demodulation [14,15].
As shown in Figure 13, the varying curves of alternating magnetic fields collected by the induction magnetic field sensor labelled from #5 to #8 were obtained by coherent demodulation [14,15].
Since the radiating rod inevitably has the problem of swaying during the movement, the curves obtained by Mathematical Problems in Engineering coherent demodulation have a certain amount of shaking compared with the smooth curves in the simulation. e peak values of sensor #1 in Figure 12 are significantly greater than those of sensor #2, sensor #3, and sensor #4 located in the same line. At the same time, it could be found that the peak values of sensor #7 in Figure 13 are significantly smaller than those of sensor #5, sensor #6, and sensor #8 located in the same line. ese were caused by the different sensitivities of each sensor. In order to reduce the impact of different sensitivities of the sensor, the signals collected by sensor #7 and sensor #1 were excluded in the final positioning solution.
Using the hybrid algorithm combining the Gauss-Newton algorithm and genetic algorithm, the positioning results of the target in the X direction are shown in Figure 14. e location result is about −3 m in the X direction from the time of 20 s to 55 s. ese show a good agreement with the actual value. is is the reason that the magnetic field signal gradually increases as the distance between the target and the sensor becomes close.  Figure 15. It could be found that when the target passed through the array, the positioning error was very small and the positioning effect was very good. e average error of positioning is 0.17 m from the time of 20 s to 55 s. However, as the target moved away from the array, the positioning error became larger and the positioning effect became poorer.
As shown in Figure 16, the average velocity was about 0.2 m/s in the Y direction from the time of 20 s to 55 s. ese also showed a good agreement with the actual value and a disagreement in the other times. e error curve between the positioning result and the actual movement trajectory of the target in the Y direction is shown in Figure 17. It could be found that when the target passed through the array, the positioning error was very small and the positioning effect was very good. e average           As the simulation result in the Z direction was the same as that in the X direction, the analysis would not be repeated in this paper.

Conclusions
Most of the traditional research studies on magnetic positioning technology are based on the magnetic target location of static magnetic anomalies, and the positioning effect is easily affected by geomagnetic anomalies and other magnetic interference noise. e magnetic field positioning methods of the alternating magnetic dipole model are studied, which have strong anti-interference ability. e methods could overcome the interference of geomagnetic environmental interference and reduce the influence of other frequency interference signals on the positioning through some signal processing methods. Using single-component magnetometers could reduce costs and avoid the steering differential calibration of tri-axis magnetometers. e theoretical analysis of simulation and experimental results showed that the position information agreed well with the actual moving state of the target, which verified the feasibility and practicability of the localization algorithm. It is of great significance in the application of engineering.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.