CN112611310B - A magnetic dipole target ranging and direction finding method - Google Patents

Abstract

The invention discloses a magnetic dipole target ranging and direction finding method. A cross array of three-axis magnetometers arranged symmetrically with respect to the measurement axis is used to measure the magnetic dipole magnetic gradient tensor at the center of the array. The feature that the target attitude is independent can obtain the magnetic moment size range of the detected target. From the ratio of the magnetic gradient tensor at this point and the target magnetic moment size range, the distance between the detected target and the center of the cross array and the measurement coordinate system of the cross array can be inverted. The azimuth of the magnetic dipole target. The method proposed in the present invention does not require the magnetic gradient tensor measurement system to measure the magnetic field vector of the magnetic target, nor does the magnetic gradient tensor measurement system need to move to measure the magnetic gradient tensor values of multiple points in space; The cross array of triaxial magnetometers can be used as the magnetic gradient tensor measurement system, and the triaxial magnetometers used are relatively few, which reduces the complexity and weight of the measurement system, and is convenient for use on light platforms.

Description

A magnetic dipole target ranging and direction finding method

technical field

The invention relates to a magnetic dipole target ranging and direction finding method, in particular to a magnetic dipole target ranging and direction finding method based on the ratio of magnetic gradient tensor components, and belongs to the technical field of magnetic target detection and parameter inversion.

Background technique

The passive target detection technology based on the magnetic anomaly field has the advantages of good concealment, continuous detection, high efficiency, simple and reliable use, and rapid response. In addition, it has important application value in underwater target detection and tracking, mineral exploration, and non-invasive positioning of in vivo micro-diagnostic devices.

As a differential measurement method for the conjugate suppression of the background magnetic field, the measurement result of the magnetic gradient tensor is less affected by the inclination and declination of the geomagnetic field, and the formed tensor invariant can describe the magnetic field source well without special processing. Therefore, the magnetic target parameter inversion technology based on the magnetic gradient tensor has received attention and research.

Wiegert et al. proposed a localization method for tensor reduction STAR, which is an invariant of spherical isosurface (Roy Wiegert, J. Oeschger. Generalized magnetic gradient contraction based method for detection, localization and discrimination of underwater mines and unexploded ordnance [J]. Mine Warfare & Ship Self-Defence, 2010, 2:1325-1332). Clark uses the eigenvalues of the magnetic gradient tensor to calculate the normalized magnetic source strength, and combines the gradient tensor to determine the unique solution of the magnetic source position (DavidA Clark. New methods for interpretation of magnetic vector and gradient tensordata I: Eigenvector analysis and the normalized source strength [J]. Exploration geophysics, 2012, 43(4):267-282). Using the orthogonal relationship between the eigenvector corresponding to the minimum eigenvalue of the tensor matrix and the relative position vector, Teixeira et al. proposed a joint inversion method of tensor Euler deconvolution and magnetic field gradient tensor eigenanalysis, which enhanced the inversion method. Convergence of Algorithms (Francisco Curado Teixeira, António Pascoal. Magnetic navigation and tracking of underwater vehicles [J]. IFAC Proceedings Volumes, 2013, 46(33):239-244). Lee et al. used two scalar parameters derived from the magnetic gradient tensor to characterize the magnetic moment and position of the target, and then used the gradient method to solve the position and experimentally verified this method (Kok-Meng Lee, MinLi. Magnetic tensor sensor for gradient-based localization of ferrous object in geomagnetic field[J]. IEEE Transactions on Magnetics, 2016, 52(8):1-10).

The Naval University of Engineering proposed a positioning method based on the equivalent decay magnetic moment, and analyzed the effects of geomagnetic field fluctuations, three-axis magnetometer noise, and measurement baseline length on the positioning results (Jia Wenjiu, Lin Chunsheng, Sun Yuhui, Zhai Guojun. Equivalent decay magnetic moment positioning method of magnetic targets in the geomagnetic field [J]. Chinese Journal of Military Engineering, 2018, 39(2): 283-289). The Naval Academy of Aeronautical Engineering proposed an improved single-point positioning method based on the regular hexahedral magnetic gradient tensor system, which eliminated the systematic error of the tensor reduction STAR method (Lu Junwei, Chi Cheng, Yu Zhentao, Bi Bo, Song Qingshan. Research on elliptic error elimination method of magnetic gradient tensor invariant[J]. Acta Physica Sinica, 2015, 64(9): 52-59). Army Engineering University has studied a two-point magnetic gradient tensor localization algorithm and carried out ferromagnetic localization experiments (Gang Yin, Yingtang Zhang, Zhining Li, Hongbo Fan and Guoquan Ren. Detection of ferromagnetic target based on mobile magneticgradient tensor system [J] . Journal of Magnetism and Magnetic Material, 2016, 402(6):1-7).

Common magnetic gradient tensor localization methods need to measure the magnetic field vector of the magnetic target or measure the magnetic gradient tensors of the magnetic target at multiple points in space. On the one hand, it is not easy to measure the magnetic field vector of a magnetic target in the presence of the earth's background field; on the other hand, to measure the magnetic gradient tensor of multiple points in space, it is necessary to move the magnetic gradient tensor measurement system or use more The magnetometer constructs a suitable measurement array, which increases the complexity of the measurement system.

SUMMARY OF THE INVENTION

In view of the above-mentioned prior art, the technical problem to be solved by the present invention is to provide a magnetic dipole target ranging and direction finding method based on the ratio of magnetic gradient tensor components, which uses relatively few three-axis magnetometers and reduces the measurement The complexity and weight of the system make it easy to carry on a light platform.

In order to solve the above-mentioned technical problems, a magnetic dipole target ranging and direction finding method of the present invention comprises the following steps:

Step 1. Install four triaxial magnetometers of the same specification on a cross base made of non-magnetic material according to the symmetrical configuration of the measurement axis to form a triaxial magnetometer cross array; four triaxial magnetometers The measuring points are points A, B, C and D respectively, where points A and C are symmetrical about the y-axis, and points B and D are symmetrical about the x-axis;

Step 2. Read the measurement outputs of the four three-axis magnetometers, and calculate the independent component values a, b, d, e and f of the magnetic gradient tensor at the center of the cross array, specifically:

In the formula, δ _x and δ _y are the position errors of the x _s axis and y _s axis of the three-axis magnetometer deviating from the z _s axis, respectively, L _x is the length between point A and point C of the cross array, and L _y is The length between point B and point D, B _Ax , B _Ay and B _Az are the three components of the magnetic field of the magnetic body at point A, B _Bx , B _By and B _Bz are the three components of the magnetic body of the magnetic body at point B, B _Cx , B _Cy and B _Cz are the three components of the magnetic field of the magnetic body at point C, and B _Dx , B _Dy and B _Dz are the three components of the magnetic body of the magnetic body at point D;

Step 3. Set the position coordinate vector X=[x,y,z] ^T ∈R ³ , construct the equation about X, and establish the optimization objective function F(X), specifically:

F(X)=|f ₁ (X)|+|f ₂ (X)|+|f ₃ (X)|+|f ₄ (X)|

In the formula,

k _ax =(3r ² -5x ² )x, k _ay =(r ² -5x ² )y, k _az =(r ² -5x ² )z, k _bx =(r ² -5y ² )x, k _by =(3r ² -5y ² )y, k _bz =(r ² -5y ² )z, k _dx =ka y , k _dy =k _bx , k _dz = _-5xyz , k _ex =k _dz , _key =k _bz , k _ez =y(r ² -5z ² ), k _fx =k _az , k _fy =k _dz , k _fz =(r ² -5z ² )x, η _ab =a/b, η _ad =a/ d, η _ae = a/e and η _af = a/f, x, y and z are the spatial coordinates of the measurement point of the magnetic gradient tensor relative to the magnetic dipole, r is the measurement point of the magnetic gradient tensor and the magnetic the distance between the dipoles;

约束最优化问题具体为：Step 4. Determine the range of the magnetic moment [m _min , m _max ] according to the type of the detection target, and solve the constrained optimization problem to calculate the distance between the magnetic dipole target and the center of the triaxial magnetometer cross array

The constrained optimization problem is specifically:

In the formula, h ₁ (X)=xv ₃₁ +yv ₃₂ +zv ₃₃ , v ₃₁ , v ₃₂ and v ₃₃ are the eigenvector elements corresponding to the third eigenvalue of the magnetic gradient tensor matrix;

λ₁、λ₂和λ₃为磁梯度张量矩阵的三个特征值；where μ is the magnetic permeability,

λ ₁ , λ ₂ and λ ₃ are the three eigenvalues of the magnetic gradient tensor matrix;

Step 5. Calculate the three components m _x , m _y and m _z of the magnetic moment of the magnetic dipole, specifically:

In the formula, the elements of the matrix [K _ij ] are respectively K ₁₁ =(3r ² -5x ² )x, K ₁₂ =(r ² -5x ² )y, K ₁₃ =(r ² -5x ² )z, K ₂₁ =(r ² -5y ² )x, K ₂₂ =(3r ² -5y ² )y, K ₂₃ =(r ² -5y ² )z, K ₃₁ =K ₁₂ , K ₃₂ =K ₂₁ , K ₃₃ =- 5xyz, K ₄₁ =K ₃₃ , K ₄₂ =K ₂₃ , K ₄₃ =(r ² -5z ² )y, K ₅₁ =K ₁₃ , K ₅₂ =K ₃₃ and K ₅₃ =(r ² -5z ² )x;

Step 6. Solve the polynomial about k to obtain the six roots of k, which are as follows:

_A6k6 + ^A5k5 +A4k4 ₊ ^A3k3 _{+ A2k2 +} ^{A1k +} _{A0 =} ⁰

In the formula, A ₆ =d ² (a+2b)-e ² (ab)+2def, A ₅ =-2d[(ab)(a+2b)+(d ² +e ² +f ² )], A ₄ =(ab) ² (a+2b)+ ^d2 (4a-7b) ⁺ ( ^f2-2e2 )(ab)+6def, A3 ₌ -4d[(ab) ² +(- ^d2 +e ² +f ² )], A ₂ =(ab) ² (2a+b)+d ² (4b-7a)+(2f ² -e ² )(ab)+6def, A ₁ =2d[(ab)( 2a+b)-(d ² +e ² +f ² )], A ₀ =d ² (2a+b)+f ² (ab)+2def.

Step 7. Discard the complex roots of k, and calculate the q value according to the remaining real roots of k, specifically:

Step 8. Determine the positive and negative of z according to the position of the center point of the cross-symmetric array above or below the dipole magnetic target and the established measurement coordinate system, and calculate the z value according to the values of k, q and t, specifically:

where t is the magnetic field size of the magnetic dipole. When the center point of the cross-symmetric array is above the dipole magnetic target, z is negative. When the center point of the cross-symmetric array is below the dipole magnetic target, z is just.

Calculate x and y separately, specifically:

x=kqz

y=qz

Obtain the real number set of position coordinates {x _l , y _l , z _l |l=1,2,...,N _r }, where N _r is the number of real roots of k;

Step 9. Use {x _l , y _l , z _l |l=1,2,...,N _r } to calculate the error set {Δ _l }, specifically:

Step 10. Find the l corresponding to the minimum value from {Δ _l }, that is

Then the value of k is chosen as:

k= _kL

Then get the values of q and z;

Step 11. Determine the symbols of x and y, specifically:

In the formula, sign( ) represents the sign value, and the symbols of the position coordinates x _Q and y _Q of the magnetic dipole target equivalent to the array center point P are sign(x _Q )=-sign(x) and sign(y respectively. _Q )=-sign(y);

Calculate the azimuth angle ψ∈[0,2π) of the magnetic dipole target, specifically:

When sign(x _Q ) is "+" and sign(y _Q ) is "+", ψ=arctan(1/k);

When sign(x _Q ) is "+" and sign(y _Q ) is "-", ψ=2π+arctan(1/k);

When sign(x _Q ) is "-" and sign(y _Q ) is "+" or "-", ψ=π+arctan(1/k);

Among them, when |x|-0≤A and sign(y _Q ) is "+", ψ=π/2,; when |x|-0≤A and sign(y _Q ) is "-" , ψ=3π/2, A is a given value, A≤10 ^-4 ;

Step 12. Use the distance value r to calculate the size of the magnetic dipole magnetic field and magnetic moment, specifically:

In the formula,

Beneficial effects of the present invention: The present invention proposes a magnetic dipole target ranging and direction finding method based on the ratio of magnetic gradient tensor components, which uses a three-axis magnetometer cross array with a symmetrical configuration of the measurement axis to measure the magnetic dipole at the center of the array. Pole magnetic gradient tensor, according to the feature that the magnetic moment has nothing to do with the attitude of the magnetic target, the magnetic moment size range of the detected target can be obtained, and the ratio of the magnetic gradient tensor at this point and the target magnetic moment size range can be inverted. The distance to the center of the cross array and the azimuth of the magnetic dipole target in the cross array measurement coordinate system. The method proposed in the present invention does not require the magnetic gradient tensor measurement system to measure the magnetic field vector of the magnetic target, nor does the magnetic gradient tensor measurement system need to move to measure the magnetic gradient tensor values of multiple points in space; The cross array of triaxial magnetometers can be used as the magnetic gradient tensor measurement system, and the triaxial magnetometers used are relatively few, which reduces the complexity and weight of the measurement system, and is easy to carry on a light platform.

Compared with the existing magnetic dipole parameter inversion method based on magnetic gradient tensor, the present invention proposes a magnetic dipole target ranging and direction finding method based on the ratio of magnetic gradient tensor components, which does not require magnetic gradient tensor. The measurement system measures the magnetic field vector of the magnetic target, and the magnetic gradient tensor measurement system does not need to move to measure the magnetic gradient tensor values of multiple points in space, which expands the application object of the method; the method proposed in the present invention can use the three-axis magnetic intensity As a magnetic gradient tensor measurement system, the gage cross array uses relatively few three-axis magnetometers, which reduces the complexity and weight of the measurement system, and is convenient for light-weight platforms to carry and use, and has good practical application value; the proposed The method eliminates the influence of the medium permeability through the ratio operation between the magnetic gradient tensor components.

Description of drawings

Figure 1 is a cross array with a three-axis magnetometer sensitive axisymmetric configuration;

Fig. 2(a) is the variation curve of the ranging error with the noise standard deviation of the three-axis magnetometer;

Figure 2(b) is the variation curve of the direction finding error with the noise standard deviation of the triaxial magnetometer;

Figure 3(a) is the variation curve of the ranging error with the initial estimation error of the magnetic moment;

Figure 3(b) is the change curve of the direction finding error with the initial estimation error of the magnetic moment.

Detailed ways

The specific embodiments of the present invention will be further described below with reference to the accompanying drawings.

The realization principle of a magnetic dipole target ranging and direction finding method based on the ratio of magnetic gradient tensor components of the present invention is as follows:

和

其中B_x、B_y和B_z为磁性体磁场的三分量。The magnetic field of a magnetic body is a passive non-rotating field, so its magnetic gradient tensor matrix is a symmetric matrix with a trace of 0, that is, the magnetic gradient tensor matrix has only 5 independent components. make

and

Among them, B _x , _{By and B z are} the three components of the magnetic field of the magnetic body.

The three orthogonal measurement axes of the triaxial magnetometer usually do not intersect at one point, that is to say, there is a positional deviation of δ _x between the x _s axis of the triaxial magnetometer and its z _s axis, and the y of the triaxial magnetometer There is a positional deviation of δ _y between the _s axis and its z _s axis. In order to reduce the measurement error, the independent component value of the magnetic gradient tensor at the center of the array is measured by the cross array of the three-axis magnetometer as shown in Figure 1. The z- _s axis of the four three-axis magnetometers is the same as The z-axis of the cross-array coordinate system coincides, and they are placed at points A, B, C, and D, respectively.

The length between point A and point C is L _x , and the length between point B and point D is L _y , then the measured values of the five independent components a, b, d, e and f of the magnetic gradient tensor matrix are respectively for

In the formula, B _Ax , B _Ay and B _Az are the three components of the magnetic field at point A, B _Bx , B _By and B _Bz are the three components of the magnetic field at point B, B _Cx , B _Cy and B _Cz is the three components of the magnetic field of the magnetic body at point C, and B _Dx , B _Dy and B _Dz are the three components of the magnetic body of the magnetic body at point D .

For a magnetic dipole magnetic body, the expressions of the five independent components a, b, d, e and f of the magnetic gradient tensor matrix are respectively:

In the formula, μ is the magnetic permeability, x, y and z are the spatial coordinates of the measurement point of the magnetic gradient tensor relative to the magnetic dipole, and r is the distance between the measurement point of the magnetic gradient tensor and the magnetic dipole , m _x , m _y and m _z are the three components of the magnetic moment of the magnetic dipole.

According to formula (2), the ratios between the independent components are obtained as

In the formula, k _ax =(3r ² -5x ² )x, k _ay =(r ² -5x ² )y, k _az =(r ² -5x ² )z, k _bx =(r ² -5y ² )x , k _by =(3r ² -5y ² )y, k _bz =(r ² -5y ² )z, k _dx =ka _ay ,k _dy =k _bx ,k _dz =-5xyz,k _ex =k _dz ,k _ey = k _bz , k _ez =y(r ² -5z ² ), k _fx =k _az , k _fy =k _dz , k _fz =(r ² -5z ² )x.

Since m _x , m _y and m _z cannot all be zero, it can be obtained from formula (3)

Equation (4) can be rewritten as

In the formula, X=[x, y, z] ^T ∈ R ³ is the position coordinate vector of the center point of the array corresponding to the magnetic dipole.

Equation (5) can be rewritten as

Equation (6) can be rewritten as

Equation (7) can be rewritten as

The magnetic gradient tensor field of the magnetic dipole also has the relationship shown in equation (12).

h ₁ (X)=xv ₃₁ +yv ₃₂ +zv ₃₃ =0 (12)

In the formula, v ₃₁ , v ₃₂ and v ₃₃ are the eigenvector elements corresponding to the third eigenvalue of the magnetic gradient tensor matrix.

The range of the detection distance is determined according to the type of the detection target and the normalized intensity u, that is, there is

λ₁、λ₂和λ₃为磁梯度张量矩阵的三个特征值，m_max和m_min为磁偶极子磁矩大小m的最大值和最小值。In the formula,

λ ₁ , λ ₂ and λ ₃ are the three eigenvalues of the magnetic gradient tensor matrix, and m _max and m _min are the maximum and minimum values of the magnetic moment size m of the magnetic dipole.

The distance between the magnetic dipole target and the center of the triaxial magnetometer cross array is calculated by solving the constrained optimization problem shown in equation (14).

In the formula, F(X)=|f ₁ |+|f ₂ |+|f ₃ |+|f ₄ |.

The Sequential Quadratic Programming (SQP) algorithm is used to solve the constrained optimization problem shown in equation (14), and the distance value of the magnetic dipole target relative to the center point of the cross array is obtained.

Then, m _x , m _y and m _z are calculated from equation (15).

In the formula, the elements of the matrix [K _ij ] are respectively K ₁₁ =(3r ² -5x ² )x, K ₁₂ =(r ² -5x ² )y, K ₁₃ =(r ² -5x ² )z, K ₂₁ =(r ² -5y ² )x, K ₂₂ =(3r ² -5y ² )y, K ₂₃ =(r ² -5y ² )z, K ₃₁ =K ₁₂ , K ₃₂ =K ₂₁ , K ₃₃ =- 5xyz, K ₄₁ =K ₃₃ , K ₄₂ =K ₂₃ , K ₄₃ =(r ² -5z ² )y, K ₅₁ =K ₁₃ , K ₅₂ =K ₃₃ and K ₅₃ =(r ² -5z ² )x.

The polynomial with respect to k shown in Equation (16) is solved to obtain the six roots of k.

A ₆ k ⁶ +A ₅ k ⁵ +A ₄ k ⁴ +A ₃ k ³ +A ₂ k ² +A ₁ k+A ₀ =0 (16)

In the formula, A ₆ =d ² (a+2b)-e ² (ab)+2def, A ₅ =-2d[(ab)(a+2b)+(d ² +e ² +f ² )], A ₄ =(ab) ² (a+2b)+ ^d2 (4a-7b) ⁺ ( ^f2-2e2 )(ab)+6def, A3 ₌ -4d[(ab) ² +(- ^d2 +e ² +f ² )], A ₂ =(ab) ² (2a+b)+d ² (4b-7a)+(2f ² -e ² )(ab)+6def, A ₁ =2d[(ab)( 2a+b)-(d ² +e ² +f ² )], A ₀ =d ² (2a+b)+f ² (ab)+2def.

The complex roots of k are discarded, and the remaining real roots of k are substituted into equation (17) to calculate the value of q.

According to the position of the center point of the cross-symmetrical array above or below the dipole magnetic target and the established measurement coordinate system, the positive or negative of z is judged, so as to determine the positive and negative sign in formula (18), and k, q and The value of t is substituted into equation (18) to calculate the value of z.

Then according to formula (19) and formula (20), we can calculate

x=kqz (19)

y=qz (20)

Obtain the real number set of position coordinates {x _l , y _l , z _l |l=1,2,...,N _r }, where N _r is the number of real roots of k, and substitute the real solutions of these spatial coordinates into the formula (21 ) to calculate the error set {Δ _l }.

Find the l corresponding to the minimum value from {Δ _l }, that is,

Then the value of k is chosen as

k=k _L (23)

From this, the values of q and z are also determined.

Determine the signs of x and y according to equation (24), namely

In the formula, sign(·) means to evaluate the sign value. The symbols of the position coordinates x _Q and y _Q of the magnetic dipole target corresponding to the center point P of the array are sign(x _Q )=-sign(x) and sign(y _Q )=-sign(y), respectively.

The azimuth angle ψ∈[0,2π) of the magnetic dipole target is calculated by the calculation method described in Table 1.

Table 1 Calculation formula of azimuth angle ψ

Substituting the distance value r into equations (25) and (26) can also calculate the magnetic field size t and the magnetic moment size m of the magnetic dipole.

和

都为磁梯度张量不变量。In the formula,

and

Both are magnetic gradient tensor invariants.

1, the specific embodiment of the present invention includes the following steps:

Step 1. Install four triaxial magnetometers of the same specification on a cross base made of non-magnetic material according to the symmetrical configuration of the measurement axis to form a triaxial magnetometer cross array as shown in Figure 1.

Step 2: Read the measurement outputs of the four three-axis magnetometers, and calculate the independent component values a, b, d, e and f of the magnetic gradient tensor at the center of the cross array according to formula (1).

In the formula, δ _x and δ _y are the position errors of the x _s axis and y _s axis of the three-axis magnetometer deviating from the z _s axis, respectively, L _x is the length between point A and point C of the cross array, and L _y is The length between point B and point D, B _Ax , B _Ay and B _Az are the three components of the magnetic field of the magnetic body at point A, B _Bx , B _By and B _Bz are the three components of the magnetic body of the magnetic body at point B, B _Cx , B _Cy and B _Cz are the three components of the magnetic field of the magnetic body at point C, and B _Dx , B _Dy and B _Dz are the three components of the magnetic body of the magnetic body at point D.

Step 3. Set the position coordinate vector X=[x, y, z] ^T ∈ R ³ , construct an equation about X, and establish an optimization objective function F(X) according to formula (2).

F(X)=|f ₁ (X)|+|f ₂ (X)|+|f ₃ (X)|+|f ₄ (X)| (2)

In the formula,

k_ax＝(3r²-5x²)x，k_ay＝(r²-5x²)y，k_az＝(r²-5x²)z，k_bx＝(r²-5y²)x，k_by＝(3r²-5y²)y，k_bz＝(r²-5y²)z，k_dx＝k_ay，k_dy＝k_bx，k_dz＝-5xyz，k_ex＝k_dz，k_ey＝k_bz，k_ez＝y(r²-5z²)，k_fx＝k_az，k_fy＝k_dz，k_fz＝(r²-5z²)x，η_ab＝a/b，η_ad＝a/d，η_ae＝a/e和η_af＝a/f。

k _ax =(3r ² -5x ² )x, k _ay =(r ² -5x ² )y, k _az =(r ² -5x ² )z, k _bx =(r ² -5y ² )x, k _by =(3r ² -5y ² )y, k _bz =(r ² -5y ² )z, k _dx =ka y , k _dy =k _bx , k _dz = _-5xyz , k _ex =k _dz , _key =k _bz , k _ez =y(r ² -5z ² ), k _fx =k _az , k _fy =k _dz , k _fz =(r ² -5z ² )x, η _ab =a/b, η _ad =a/ d, η _ae =a/e and η _af =a/f.

Step 4. Determine the range of the magnetic moment size [m _min , m _max ] according to the type of the detection target, and solve the constrained optimization problem shown in equation (3) to calculate the relationship between the magnetic dipole target and the center of the triaxial magnetometer cross array. distance between

In the formula, h ₁ (X)=xv ₃₁ +yv ₃₂ +zv ₃₃ , and v ₃₁ , v ₃₂ and v ₃₃ are eigenvector elements corresponding to the third eigenvalue of the magnetic gradient tensor matrix.

λ₁、λ₂和λ₃为磁梯度张量矩阵的三个特征值。where μ is the magnetic permeability,

λ ₁ , λ ₂ and λ ₃ are three eigenvalues of the magnetic gradient tensor matrix.

Step 5. Calculate m _x , m _y and m _z according to formula (5).

In the formula, the elements of the matrix [K _ij ] are respectively K ₁₁ =(3r ² -5x ² )x, K ₁₂ =(r ² -5x ² )y, K ₁₃ =(r ² -5x ² )z, K ₂₁ =(r ² -5y ² )x, K ₂₂ =(3r ² -5y ² )y, K ₂₃ =(r ² -5y ² )z, K ₃₁ =K ₁₂ , K ₃₂ =K ₂₁ , K ₃₃ =- 5xyz, K ₄₁ =K ₃₃ , K ₄₂ =K ₂₃ , K ₄₃ =(r ² -5z ² )y, K ₅₁ =K ₁₃ , K ₅₂ =K ₃₃ and K ₅₃ =(r ² -5z ² )x.

Step 6: Solve the polynomial with respect to k shown in equation (6) to obtain six roots of k.

A ₆ k ⁶ +A ₅ k ⁵ +A ₄ k ⁴ +A ₃ k ³ +A ₂ k ² +A ₁ k+A ₀ =0 (6)

In the formula, A ₆ =d ² (a+2b)-e ² (ab)+2def, A ₅ =-2d[(ab)(a+2b)+(d ² +e ² +f ² )], A ₄ =(ab) ² (a+2b)+ ^d2 (4a-7b) ⁺ ( ^f2-2e2 )(ab)+6def, A3 ₌ -4d[(ab) ² +(- ^d2 +e ² +f ² )], A ₂ =(ab) ² (2a+b)+d ² (4b-7a)+(2f ² -e ² )(ab)+6def, A ₁ =2d[(ab)( 2a+b)-(d ² +e ² +f ² )], A ₀ =d ² (2a+b)+f ² (ab)+2def.

Step 7: Discard the complex root of k, and substitute the remaining real root of k into formula (7) to calculate the value of q.

Step 8. According to the position of the center point of the cross-symmetrical array above or below the dipole magnetic target and the established measurement coordinate system, determine the positive and negative of z, so as to determine the positive and negative sign in formula (8), and set k The values of , q and t are substituted into equation (8) to calculate the z value.

Then according to formula (9) and formula (10), we can calculate

x=kqz (9)

y=qz (10)

Obtain the real number set {x _l , y _l , z _l |l=1,2,...,N _r } of the position coordinates, where N _r is the number of real roots of k.

Step 9: Substitute {x _l , y _l , z _l |l=1,2,...,N _r } into formula (11) to calculate the error set {Δ _l }.

Step 10. Find the l corresponding to the minimum value from {Δ _l }, that is

Then the value of k is chosen as

k=k _L (13)

From this, the values of q and z are also determined.

Step 11. Determine the symbols of x and y according to formula (14), namely

In the formula, sign(·) means to evaluate the sign value. The symbols of the position coordinates x _Q and y _Q of the magnetic dipole target corresponding to the center point P of the array are sign(x _Q )=-sign(x) and sign(y _Q )=-sign(y), respectively.

The azimuth angle ψ∈[0,2π) of the magnetic dipole target is calculated by the calculation method described in Table 1.

Table 1 Calculation formula of azimuth angle ψ

Step 12: Substitute the distance value r into equations (15) and (16) to obtain the magnitude of the magnetic dipole magnetic field and magnetic moment.

和

In the formula,

and

In order to test the feasibility of the distance and direction finding method proposed by the present invention, the following numerical simulation experiments are carried out. The three components of the magnetic moment of the magnetic dipole are m _x = 8×10 ⁸ A·m, m _y = 6×10 ⁸ A·m and m _z = 3×10 ⁸ A·m, and the true position coordinates are x _Q = 180 m, y _Q = 220 m and z _Q = -150 m, the baseline lengths of the triaxial magnetometer cross array in the x and y directions are L _x = 0.8 m and L _y = 0.8 m, respectively, the position of the measurement axis The deviations are δ _x = 2 mm and δ _y = 2 mm. The noise of the three-axis magnetometer in each axis is an independent Gaussian process with a mean of 0 and a standard deviation of σ _m . The initial estimation error of the magnitude of the magnetic dipole moment is ρ.

和

在不同的三轴磁强计噪声标准差的情况下，进行磁偶极子目标测距与测向的50次蒙特卡洛仿真实验。得到三轴磁强计十字阵列对磁偶极子目标的测距与测向误差随三轴磁强计噪声标准差的变化曲线如图2(a)和2(b)所示，其中图2(a)为σ_m从0nT增加到0.8nT时的距离测量误差曲线，图2(b)为σ_m从0nT增加到0.8nT时的方向角测量误差曲线。由图2(a)和2(b)可知，距离测量误差随σ_m的增加而呈增大趋势，但增大的幅度不大；方向角测量误差随σ_m的增加而呈线性增大趋势。Simulation example 1: Let the initial estimation error of the magnetic moment of the magnetic dipole be 10%, that is

and

In the case of different noise standard deviations of the three-axis magnetometer, 50 Monte Carlo simulation experiments of ranging and direction finding of magnetic dipole targets were carried out. The variation curves of the ranging and direction finding errors of the three-axis magnetometer cross array to the magnetic dipole target with the noise standard deviation of the three-axis magnetometer are shown in Figures 2(a) and 2(b), in which Figure 2 (a) is the distance measurement error curve when σ _m increases from 0nT to 0.8nT, and Figure 2(b) is the direction angle measurement error curve when σ _m increases from 0nT to 0.8nT. It can be seen from Figures 2(a) and 2(b) that the distance measurement error increases with the increase of σ _m , but the increase is not large; the direction angle measurement error increases linearly with the increase of σ _m . .

Simulation example 2: Set the noise standard deviation σ _m of the three-axis magnetometer to 0.1nT, and in the case of different initial estimation errors of the magnetic moment size of the magnetic dipole, the distance and direction finding of the magnetic dipole target is 50%. A Monte Carlo simulation experiment. The variation curves of the ranging and direction finding errors of the three-axis magnetometer cross array to the magnetic dipole target with the initial estimation error of the magnetic moment of the magnetic dipole are shown in Figures 3(a) and 3(b), Figure 3(a) is the distance measurement error curve when ρ increases from 2% to 20%, and Figure 3(b) is the direction angle measurement error curve when ρ increases from 2% to 20%. It can be seen from Figures 3(a) and 3(b) that when ρ is large, the distance measurement error increases linearly with the increase of ρ, and when ρ is small, the distance measurement error is almost unchanged with the increase of ρ; the direction angle measurement The error does not change much with increasing ρ.

A Monte Carlo simulation experiment of distance and direction finding of a magnetic dipole magnetic target was carried out using the cross array of three-axis magnetometers arranged symmetrically to the measurement axis as the magnetic gradient tensor measurement device. The simulation experiment gives the variation of the range and direction finding errors of the triaxial magnetometer cross array to the magnetic dipole target with the noise standard deviation of the triaxial magnetometer under the condition of different noise standard deviations of the triaxial magnetometers The results show that the distance measurement error increases with the increase of the standard deviation of the measurement noise of the three-axis magnetometer, but the increase is not large; the measurement error of the direction angle increases with the increase of the standard deviation of the measurement noise of the axis magnetometer. Linear increase trend. The simulation experiments show that the range and direction finding errors of the three-axis magnetometer cross array to the magnetic dipole target vary with the magnetic moment of the magnetic dipole under the condition of different initial estimation errors of the magnetic moment of the magnetic dipole. The change curve of the initial estimation error of the magnetic moment size, the results show that the distance measurement error increases linearly with the increase of the initial estimation error when the initial estimation error of the magnetic moment size is large, and the distance measurement error when the initial estimation error of the magnetic moment size is small It is almost unchanged with the increase of the initial estimation error; the bearing angle measurement error does not change much with the increase of the initial estimation error. Simulation experiments prove the feasibility of the method of the present invention.

Compared with the general magnetic dipole parameter inversion method based on the magnetic gradient tensor, the method proposed in the present invention does not require the magnetic gradient tensor measurement system to measure the magnetic field vector of the magnetic target, nor does the magnetic gradient tensor measurement system need to move to measure the magnetic field vector. The magnetic gradient tensor values of multiple points in space expand the application object of the method; the magnetic gradient tensor measurement scheme proposed by the present invention requires relatively few three-axis magnetometers, the structure is simple, and the complexity of the measurement system is reduced The three-axis magnetometer measurement axis-symmetric configuration also reduces the error of the magnetic gradient tensor differential measurement, which has good practical application value; the ratio calculation between the magnetic gradient tensor components The influence of medium permeability is also reduced.

Claims (1)

1. a magnetic dipole target ranging and direction finding method, is characterized in that, comprises the following steps: Step 1. Install four triaxial magnetometers of the same specification on a cross base made of non-magnetic material according to the symmetrical configuration of the measurement axis to form a triaxial magnetometer cross array; four triaxial magnetometers The measuring points are points A, B, C and D respectively, where points A and C are symmetrical about the y-axis, and points B and D are symmetrical about the x-axis; Step 2. Read the measurement outputs of the four three-axis magnetometers, and calculate the independent component values a, b, d, e and f of the magnetic gradient tensor at the center of the cross array, specifically:

In the formula, δ _x and δ _y are the position errors of the x _s axis and y _s axis of the three-axis magnetometer deviating from the z _s axis, respectively, L _x is the length between point A and point C of the cross array, and L _y is The length between point B and point D, B _Ax , B _Ay and B _Az are the three components of the magnetic field of the magnetic body at point A, B _Bx , B _By and B _Bz are the three components of the magnetic body of the magnetic body at point B, B _Cx , B _Cy and B _Cz are the three components of the magnetic field of the magnetic body at point C, and B _Dx , B _Dy and B _Dz are the three components of the magnetic body of the magnetic body at point D; Step 3. Set the position coordinate vector X=[x,y,z] ^T ∈R ³ , construct the equation about X, and establish the optimization objective function F(X), specifically: F(X)=|f ₁ (X)|+|f ₂ (X)|+|f ₃ (X)|+|f ₄ (X)| In the formula,

k _ax =(3r ² -5x ² )x, k _ay =(r ² -5x ² )y, k _az =(r ² -5x ² )z, k _bx =(r ² -5y ² )x, k _by =(3r ² -5y ² )y, k _bz =(r ² -5y ² )z, k _dx =ka y , k _dy =k _bx , k _dz = _-5xyz , k _ex =k _dz , _key =k _bz , k _ez =y(r ² -5z ² ), k _fx =k _az , k _fy =k _dz , k _fz =(r ² -5z ² )x, η _ab =a/b, η _ad =a/ d, η _ae = a/e and η _af = a/f, x, y and z are the spatial coordinates of the measurement point of the magnetic gradient tensor relative to the magnetic dipole, r is the measurement point of the magnetic gradient tensor and the magnetic the distance between the dipoles; Step 4. Determine the range of the magnetic moment [m _min , m _max ] according to the type of the detection target, and solve the constrained optimization problem to calculate the distance between the magnetic dipole target and the center of the triaxial magnetometer cross array

The constrained optimization problem is specifically:

In the formula, h ₁ (X)=xv ₃₁ +yv ₃₂ +zv ₃₃ , v ₃₁ , v ₃₂ and v ₃₃ are the eigenvector elements corresponding to the third eigenvalue of the magnetic gradient tensor matrix;

where μ is the magnetic permeability,

λ ₁ , λ ₂ and λ ₃ are the three eigenvalues of the magnetic gradient tensor matrix; Step 5. Calculate the three components m _x , m _y and m _z of the magnetic moment of the magnetic dipole, specifically:

In the formula, the elements of the matrix [K _ij ] are respectively K ₁₁ =(3r ² -5x ² )x, K ₁₂ =(r ² -5x ² )y, K ₁₃ =(r ² -5x ² )z, K ₂₁ =(r ² -5y ² )x, K ₂₂ =(3r ² -5y ² )y, K ₂₃ =(r ² -5y ² )z, K ₃₁ =K ₁₂ , K ₃₂ =K ₂₁ , K ₃₃ =- 5xyz, K ₄₁ =K ₃₃ , K ₄₂ =K ₂₃ , K ₄₃ =(r ² -5z ² )y, K ₅₁ =K ₁₃ , K ₅₂ =K ₃₃ and K ₅₃ =(r ² -5z ² )x; Step 6. Solve the polynomial about k to obtain the six roots of k, which are as follows: _A6k6 + ^A5k5 +A4k4 ₊ ^A3k3 _{+ A2k2 +} ^{A1k +} _{A0 =} ⁰ In the formula, A ₆ =d ² (a+2b)-e ² (ab)+2def, A ₅ =-2d[(ab)(a+2b)+(d ² +e ² +f ² )], A ₄ =(ab) ² (a+2b)+ ^d2 (4a-7b) ⁺ ( ^f2-2e2 )(ab)+6def, A3 ₌ -4d[(ab) ² +(- ^d2 +e ² +f ² )], A ₂ =(ab) ² (2a+b)+d ² (4b-7a)+(2f ² -e ² )(ab)+6def, A ₁ =2d[(ab)( 2a+b)-(d ² +e ² +f ² )], A ₀ =d ² (2a+b)+f ² (ab)+2def; Step 7. Discard the complex roots of k, and calculate the q value according to the remaining real roots of k, specifically:

Step 8. Determine the positive and negative of z according to the position of the center point of the cross-symmetric array above or below the dipole magnetic target and the established measurement coordinate system, and calculate the z value according to the values of k, q and t, specifically:

where t is the magnetic field size of the magnetic dipole. When the center point of the cross-symmetric array is above the dipole magnetic target, z is negative. When the center point of the cross-symmetric array is below the dipole magnetic target, z is just; Calculate x and y respectively, specifically: x=kqz y=qz Obtain the real number set of position coordinates {x _l , y _l , z _l |l=1,2,...,N _r }, where N _r is the number of real roots of k; Step 9. Use {x _l , y _l , z _l |l=1,2,...,N _r } to calculate the error set {Δ _l }, specifically:

Step 10. Find the l corresponding to the minimum value from {Δ _l }, that is

Then the value of k is chosen as: k= _kL Then get the values of q and z; Step 11. Determine the symbols of x and y, specifically:

In the formula, sign() represents the sign value, and the symbols of the position coordinates x _Q and y _Q of the magnetic dipole target equivalent to the array center point P are sign(x _Q )=-sign(x) and sign(y _Q , respectively. ) = -sign(y); Calculate the azimuth angle ψ∈[0,2π) of the magnetic dipole target, specifically: When sign(x _Q ) is "+" and sign(y _Q ) is "+", ψ=arctan(1/k); When sign(x _Q ) is "+" and sign(y _Q ) is "-", ψ=2π+arctan(1/k); When sign(x _Q ) is "-" and sign(y _Q ) is "+" or "-", ψ=π+arctan(1/k); Among them, when |x|-0≤A and sign(y _Q ) is "+", ψ=π/2,; when |x|-0≤A and sign(y _Q ) is "-" , ψ=3π/2, A is a given value, A≤10 ^-4 ; Step 12. Use the distance value r to calculate the size of the magnetic dipole magnetic field and magnetic moment, specifically:

In the formula,

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