RU2456642C2 - Method of localising magnetic dipole - Google Patents

Abstract

FIELD: physics.

SUBSTANCE: six sensors at one end of a bar are used to measure increments in magnetic field induction relative a reference sensor at the other end of the bar. These increments are used to determine the gradient of the induction vector at the centre of the coordinate system of the bar and the vector of induction increment between the centre and the reference sensor. The combination of the gradient and increment are then used to locate the magnetic source and find the distance to the source.

EFFECT: high stability and reliability of location, which enables to localise a magnetic source during motion.

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Description

The invention relates to the field of determining the location coordinates of hidden ferromagnetic objects, from submarines and unexploded ordnance (bombs, shells, ground mines), to small arms buried in the ground (rifle, machine gun, pistol), etc.

Many concentrated sources of the magnetic field can be approximated by a (point) dipole with a magnetic moment M = Mm, where M is the magnitude of the magnetic moment, m is the unit direction vector of the magnetic moment. There are distinguished tasks of detection, direction finding and localization (of a fixed or moving) dipole by the parameters of its magnetic field, measured at one or more observation points. In localization, the radius vector X = Xn is determined, where X is the distance modulus, n is the unit direction vector. In other words, the localization determines the unknown coordinates of the vector X in the coordinate system of the observation. During direction finding, the unit direction vector n or the direction cosines of its angles in the observation coordinate system are determined. Upon detection, the parameters of an inhomogeneous magnetic field are determined against the background of the Earth’s field, by which it is judged that “somewhere there is something”. Some specialists understand detection as direction finding and localization. However, for definiteness, we will distinguish between these tasks.

The problem of direction finding of a magnetic dipole, apparently, was first posed by O. P. Khvostov [Magnetic direction finder. // Geophysical instrumentation. - L .: Gostekhizdat. 1963. P.3-11], who also proposed an approximate solution to it according to the gradient of the dipole induction module.

The first analytical (explicit) solution to the problem of direction finding of a dipole by the gradient of the vector of dipole induction was obtained by C. Frahm [W.M. Wynn a.o. Advanced Superconducting Gradiometer. / Magnetometer Arrays and a Novel Signal Processing Technique. // IEEE Transactions on Magnetics, vol. MAG-11, no.2, March 1975 P.701-707]. Frahm′a's decision was mixed. It included 4 directions, one of which was true, the other three - false. However, the Frahm′a decision remained unsolved, since no convenient form for presenting this solution was found.

Then T. Kolbenhayer [Prispevok krieseniu obratenej magnetometrickey ulohy predipol. // Banickey listy. 1976. # 4 31-35 (Review Journal of Geology. 1980. No. 12. C.20)] found an explicit solution to the localization problem with respect to a given vector and gradient of the dipole induction vector, based on Euler's theorem for homogeneous functions. However, the Kolbenhayer′a solution turned out to be unstable to the orientation of the desired dipole.

Later, both of the mentioned solutions were radically improved in the work of [V.G.Semenov. The solution of the inverse problem for the source of the physical field of the dipole or quadrupole model. // Methods and means of accurate magnetic measurements. / Sat scientific works of NPO VNIIM named after D.I. Mendeleev. 1980. S.3-19]. As a result, the localization problem gained stability, and for the direction finding problem, a simple and successful form of expression of this solution was found. In subsequent years, the work of Frahm′a [1975] and Semenov [1980] was unfairly criticized for the ambiguity of the solution in the article [V.V. Tikhonov and B.N. Shalaev Features of solving inverse problems in the electromagnetic study of underwater environments. News of LETI named after Ulyanov (Lenin). Issue 327 L-D 1983, Page 28] and monographs [edited by laureate of the State. Prizes of the USSR EG Pashchenko "Optical-physical means of ocean research." L-d Shipbuilding 1984. P.207]. The authors of these publications proposed a supposedly unique solution to the problem instead of the solution [Semenov 1980]. Response article [V.G.Semenov and A.Ya. Zayonchkovsky. Comparative analysis of gradient methods of localization of the magnetic field sources of the dipole model. Sat scientific works “Creation of measuring instruments for metrological support of precision nanoteslameters”. VNIIM them. Mendeleev L-d. 1988, pp. 56-67] showed the bankruptcy of claims for the unambiguous decision of Tikhonov and Shalaev. At the same time, Semenov and Zayonchkovsky [1988] received an analytical expression of the error of the Frahm′a method [1975] / Semenov [1980], which revealed some instability of this method, which was previously considered completely stable. Many years later, this analytical expression was cross-checked by computer modeling [Yu.M. Ivanov and V.G.Semenov. Estimates of errors of direction finding and localization of the magnetic field source of the dipole model. // Measuring technique №5. 2008. S. 30-35 and No. 7. 2008. P.72]. Computer simulation results confirmed the existence of an instability region.

So, there are two known today the best ways to explicitly solve, and therefore, high speed:

a) the method of localization [Semenov, 1980]. This method is stable and unambiguous, but it does not allow localization in the movement of the carrier of measuring instruments;

b) the method of direction finding [Semenov and Zayonchkovsky, 1988]. This method allows you to take measurements when moving the measuring instruments, but it is ambiguous, since it gives four solutions (which of them is true is unknown), moreover, this method is unstable in some narrow region. But most importantly, this is a method of direction finding, not localization.

Finally, the following methods are known to increase the accuracy of measuring the spatial increment of magnetic induction, as well as the increment base:

- RF patent №2256930 A method of minimizing errors in a portable differential magnetometer. Priority of the invention of 07/07/2003;

- RF patent №2257593 A method of minimizing errors in a differential magnetometer. Priority of the invention of 07/07/2003;

- RF patent No. 2290657 A method for determining the location and orientation of a three-component magnetometer sensor relative to a measure of magnetic moment. The priority of the invention of 09/09/2005;

- RF patent No. 2313801 A method for measuring the increment of the magnetic field induction by a three-component differential magnetometer. Priority of the invention of 05/04/2006

As a prototype of the proposed method, the method of direction finding of the source of the magnetic field of the dipole model [Semenov and Zayonchkovsky, 1988] (hereinafter, simply [1988]) is selected according to the set of essential features closest to the claimed method.

The known method [1988] includes preliminary measurements of the distances between the sensors of three-component magnetometers, which measure the gradient of the magnetic field induction of the source, determining the gradient of its eigenvalues and vectors, through which they express the unit direction vector of the radius vector between the gradient measurement point and the desired source, and the unit direction of the magnetic moment of this source, and both directions are determined with an accuracy of one of four possible solutions lying in one plane and (one of the solutions is true, the other three are false), in addition, the scalar intensity of the gradient and the cosine of the angle between the directions found are determined through the eigenvalues found.

The reason that impedes the achievement of the technical result indicated below for the known method is that the known method does not have enough input measured information to solve the localization problem, namely, it uses five measured numbers, or five independent gradient components, which after processing determine five output parameters: four numbers in two unit directions and the fifth number is the scalar intensity of the gradient.

The problem to which the invention is directed is to refine the direction finding method to a localization method, as well as to eliminate its ambiguity and to reduce its instability.

The technical result obtained by carrying out the invention is to enable localization of the source of the magnetic field in motion.

, а также определение соответствующих собственных векторов u₁, u₂, u₃, по которым находят четыре единичных направления на диполь n_i и направления его магнитного момента m_i по формуламThe specified technical result is achieved by the fact that the claimed method of localizing a magnetic dipole, including preliminary measurements of the distances between the sensors of three-component magnetometers, then measuring the gradient of the magnetic field induction with these sensors, determining the eigenvalues of the gradient λ ₂ ≥λ ₁ ≥λ ₃ , which determine the scalar intensity him

, as well as the determination of the corresponding eigenvectors u ₁ , u ₂ , u ₃ , by which four unit directions to the dipole n _i and the directions of its magnetic moment m _{i are found} by the formulas

where i = 1, 2, 3, 4 is the index of the pair of solutions (one true, three false), characterized in that using sensors of three-component magnetometers located at one end of the portable rod on the semiaxes of its coordinate system, coaxially and at the same distance from its center, the spatial increments of the magnetic field induction are measured relative to the reference sensor at the other end of the rod, by which the average increment over all half-axes is calculated, as well as the increment differences for each pair of half-axes of the same name, a symmetric difference matrix is constructed, n and that of each diagonal element of the matrix is subtracted third of its track, and the single direction are according to the formulas depends on the size of φ = λ ₁ / N:

at -0.95≤φ <0.95:

at -1≤φ <-0.95:

at 0.95 <φ≤1:

then, normalizing the vector of average increment, we obtain a vector equation with one unknown k = a / r:

where B ₀₇ is the vector of average increment;

a is the distance between the center of the reference sensor and the center of the coordinate system;

ρ = (1 + k ² -2kn _a · n) ^1/2 ;

n _a is the unit vector of the direction of the rod,

in turn, the normalized vector equation is reduced to the modular | B ₀₇ | / N a = f (k), solving which for each of the four pairs of found vectors n _i and m _i , we obtain the corresponding value k _i , then each pair of values of n _i , m _i , if it corresponds to a physically acceptable value of k _i , it is substituted into a normalized vector equation, it is calculated and the three true solutions n, m and k are selected from the minimum modulus of the difference between the calculated and measured versions of this equation, after which the sought distance to the source r = a / k.

Figure 1 shows a diagram of the implementation of the proposed method.

The circuit depicted in FIG. 1 contains sensors of three-component magnetometers mounted on a portable rod. Sensors 1-6 are located at one end of the rod, a reference sensor 7 is located at the other end of it. Sensors 1-2 are located on the axles x ₊ and x _- the coordinate system (SC) of the rod, at the same distance from the center 0 SC, sensors 3-4 - on the axes y ₊ and y _- , sensors 5-6 - on the axles z ₊ and z _- . All sensors 1-7 are oriented by their axes along the axes of the SK rod.

The inventive method operates as follows. Sensors 1-6 are included in the scheme for measuring increments of magnetic induction relative to the reference sensor 7. This means that when measuring the vector of increments of magnetic induction, for example, by a pair of 1-7, the recording devices count three projections of the difference B ₁₇ = B ₁ -B ₇ , but not the vectors B ₁ and B ₇ themselves. Preliminary measure the distance between the sensors, as well as their relative orientation using an external three-component measure of magnetic moment (MM). To do this, use the method of determining the location and orientation of the three-component magnetometer sensor relative to the MM measures [RF Patent No. 2290657]. During operation, the following measurement results are operated on: increments of magnetic induction B ₁₇ , B ₂₇ , B ₃₇ , B ₄₇ , B ₅₇ , B ₆₇ and predetermined distances between the respective sensors a ₁₂ , a ₃₄ , a ₅₆ , as well as the distance between the centers of SC 0 and sensor 7 - a .

If the rod is in the Earth's magnetic field, which can be considered homogeneous, that is, independent of the coordinate, then the readings of the rod magnetometers will be zero. If at some distance from the rod there is a source of the magnetic field of the dipole model, that is, a source of an inhomogeneous field, or depending on the coordinate, then the readings of the magnetometers differ from zeros. The task of direction finding is to determine the direction of the dipole from these indications, and the task of localization is to determine the distance to it along this direction.

First, using the measured increments, a 3 × 3 matrix is built:

with which the gradient matrix is found

where A ′ is the transposed matrix A,

A ₁₁ + A ₂₂ + A ₃₃ - trace of matrix A,

I is the identity 3 × 3 matrix.

Operations (2) impart symmetry to the gradient matrix and zero out its trace, since it is known that the gradient of the magnetic induction vector is symmetrical and traceless. Then, using standard techniques on the gradient matrix, find its eigenvalues λ ₂ ≥λ ₁ ≥λ ₃ and vectors u ₁ , u ₂ , u ₃ , which determine the scalar intensity of the gradient N, the number φ, which is the scalar product of unit direction vectors to the source from the center SK n and magnetic moment m, as well as the directions n and m themselves according to formulas depending on the size φ:

where µ ₀ = 4π × 10 ^-7 GN / m is the magnetic constant;

M is the size of the magnetic moment of the source;

r is the distance between the source and the center of the SC;

(We draw attention to the fact that the quantities M and r themselves are not determined by the gradient);

at -0.95≤φ <0.95:

at -1≤φ <-0.95:

at 0.95 <φ≤1:

where i = 1, 2, 3, 4.

Formulas (5) - (7) give an ambiguous solution, because they determine 4 pairs of solutions, one pair - a true solution, three - false. Which pair is true is unknown, since all solutions satisfy the gradient. This is manifested in the fact that the eigenvectors are determined up to a sign.

Note that formulas (5) - (7) are used depending on the size of φ. An analysis of the errors showed that the first two lines (5), each of the two terms, are methodologically perfect, but if there are errors, one term diverges at φ≈-1, and the other at φ≈ + 1. Therefore, it is advantageous, for | φ | ≈1 ÷ 0.95, to use one of these terms, assuming a slight methodological error, but increasing the resistance to errors in the input data. Thus, the breakdown of (5) into subexpressions depending on the size φ eliminates the instability that occurs in the known method [1988].

Next, determine the average increment vector

Equality (8) is based on the well-known Gauss mean value theorem. For the source of the dipole model, the average increment vector can be written as

In the right-hand side of (9), in addition to the unknown k = a / r, two more unknowns appear: M and r. In order to get rid of these two unknowns, use the relation

Expression (10) is a vector nonlinear equation for only one scalar unknown k, given that the quantities φ, n _i , m _{i are} known from the above solutions, and the vector n _a is known as the SC unit vector along the longitudinal axis of the rod. Further, from (10), a modular equation of the form

where φ _a = mn _a ,

deciding which for each pair n _i , m _i , get the corresponding k _i . Sometimes the obtained k _{i is} not a physically acceptable solution, for example, too large, (NaN is the term MatLab′a), such solutions are discarded. Each triple of solutions n _i , m _i , k _i , where k _{i is a} physically acceptable solution, is substituted into equation (10) and calculated. The true three solutions are determined by the smallest modulus of the difference Δ _i measured and calculated versions of equation (10).

Thus, in the inventive method, the ambiguity of solutions inherent in the known method [1988] is eliminated.

Then, based on the found true k, the desired distance to the source r = a / k is determined, which completes the transition from (stable and unambiguous) direction finding to (stable and unambiguous) localization of the source.

, и осуществляют пеленгацию по известному способу [1988] 500 раз для каждого из 21 значения φ=-1, -0.9, -0.8 …, 1. Всю совокупность обрабатывают для определения СКО пеленгации σ_n.The inventive method is checked using computer simulation in the MatLab package. According to MatLab, the gradient components are calculated, random errors with zero mean and a given level of standard deviation of the component are added to them

, and carry out direction finding according to the known method [1988] 500 times for each of 21 values φ = -1, -0.9, -0.8 ..., 1. The whole set is processed to determine the standard deviation of direction finding σ _n .

только одно из слагаемых (5) содержит почти всю полезную информацию, а другое слагаемое (5) - только «шумит», поэтому его отбрасывают. Отбрасывание второго слагаемого (5) создает методическую погрешность пеленгации, которая в диапазоне 0.95≤|φ|≤1 еще не велика и ею можно пренебречь в этой части диапазона. Кривая 2 представляет СКО первого члена (5), кривая 3 - СКО второго члена (5), кривая 4 - сумму методической погрешности плюс удвоенное СКО первого члена (5), кривая 5 - сумму методической погрешности плюс удвоенное СКО второго члена (5). Как показано на Фиг.2, при -1≤φ≤-0.95 пеленгация по методически неточному уравнению (6) выгоднее, чем по точному уравнению (5), а при 0.95≤φ≤1 по уравнению (7) выгоднее, чем по (5). Значение φ определяют в начале пеленгации по уравнению (4).In figure 2, curve 1 corresponds to the standard deviation of direction finding according to equation (5), curve 2 - according to (6), 3 - according to (7). Thus, when

only one of the terms (5) contains almost all the useful information, and the other term (5) only “makes noise”, therefore it is discarded. Discarding the second term (5) creates a methodological error of direction finding, which is still not large in the range 0.95≤ | φ | ≤1 and can be neglected in this part of the range. Curve 2 represents the standard deviation of the first term (5), curve 3 represents the standard deviation of the second term (5), curve 4 is the sum of the methodical error plus twice the standard deviation of the first term (5), curve 5 is the sum of the methodical error plus twice the standard deviation of the second term (5). As shown in FIG. 2, at -1≤φ≤-0.95, direction finding by methodically inaccurate equation (6) is more advantageous than by exact equation (5), and when 0.95≤φ≤1, it is more advantageous by equation (7) than by ( 5). The value of φ is determined at the beginning of direction finding by equation (4).

The example presented in Fig. 2 corresponds to the errors of direction finding of a dipole with a magnetic moment M = 0.8 Am ² at a distance of r = 2.5 m from the center 0 of the SC rod at a distance between the sensors 1-6 1 = 0.75 m. The standard deviation of the noise of the sensor components is σ _B = 10 ^-10 T.

The analytical expression of the standard deviation of direction finding (determining the direction to the desired source) according to equation (5) has the following form:

As can be seen from curve 6 (smooth line) in figure 2, the analytical expression (12) almost completely coincides with the results of computer modeling. This coincidence proves that expression (12) is deduced correctly, and also that the simulation is carried out correctly.

Thus, the breakdown of the claimed method of equation (5) into two additional (6) and (7) reduces the direction finding instability inherent in the known method [1988] at 0.95 ≤ | φ | ≤1.

The following Table 1 presents examples of the elimination of the claimed method for the ambiguity of direction finding inherent in the known method [1988]. These examples were also prepared by computer simulation of direction finding / localization of a source remote at a distance of r = 3.5 m from the center 0 of the SC rod, in which the reference sensor 7 is located at a distance of a = 1.5 m from the center 0.

Table 1 Determination of the parameter k _i (11) and the corresponding Δ - the modulus of the difference between the measured and calculated parts of equation (10) for four pairs of solutions (5) defined by the signs of the eigenvectors u ₂ , u ₃ φ + u2, + u3 -u2, + u3 -u2, -u3 + u2, -u3 k ₁ Δ ₁ , nT k ₂ Δ ₂ , nT k ₃ Δ ₃ , nT k ₄ Δ ₄ , nT -0.9 0.4286 0.000 ... 0.2573 21.4 0.2619 24.8 0.4574 15.2 -0.7 0.4286 0.000 ... 0.2328 11.5 0.2499 21.1 0.0411 13.4 -0.5 0.4286 0.000 ... 0.1927 7.0 0.2286 17.8 0.0678 17.4 -0.3 0.4286 0.000 ... 0.1285 12.4 0.1885 15.3 0.0751 20.4 -0.1 0.4286 0.000 ... 0.0415 18.5 0.1066 13.8 0.0403 21.2 0 0.4286 0.000 ... NaN * - NaN - NaN - +0.1 0.4286 0.000 ... 0.0403 21.2 0.1066 13.8 0.0415 18.5 +0.3 0.4286 0.000 ... 0.0751 20.4 0.1885 15.3 0.1285 12.4 +0.5 0.4286 0.000 ... 0.0678 17.4 0.2286 17.8 0.1927 7.0 +0.7 0.4286 0.000 ... 0.0411 13.4 0.2499 21.1 0.2328 11.5 +0.9 0.4286 0.000 ... 0.4574 15.2 0.2619 24.8 0.2573 21.4 * NaN - not a number (term MatLab′a).

As can be seen from each row of Table 1, the value of the modulus of the difference Δ ₁ is the smallest, therefore, the three solutions n ₁ , m ₁ , k ₁ in this example are taken as the true directions n, m and the true ratio k = a / r. The remaining triples are discarded as false.

Thus, the claimed method eliminates the ambiguity of solutions inherent in the known method [1988].

Finally, from the found and selected true value, the ratios k = a / r determine the desired distance to the source r = a / k.

For example, in Table 1, r = 1.5 / 0.4286 = 3.4998 m. Deviation from the estimated distance of 3.5 m occurred due to rounding of the number 0.4285714286 to 0.4286.

Thus, the claimed method performs not only unambiguous (and stable) direction finding, but unambiguous (and stable) localization of the source, while the known method [1988] performs only (ambiguous and unstable) direction finding of the source.

Moreover, since the inventive method is based on measurements of increments of magnetic induction by sensors on a portable rod, included in a differential circuit, the inventive method allows you to localize the source and in motion, since the differential inclusion itself protects against the harmful effects of a uniform field of the Earth when moving the rod after zero correction its sensors.

Claims (1)

A method for localizing a magnetic dipole, including preliminary measurements of the distances between the sensors of three-component magnetometers, then measuring the gradient of the magnetic field induction with these sensors, determining the eigenvalues of the gradient λ ₂ ≥λ ₁ ≥λ ₃ , which determine its scalar intensity

, as well as the determination of the corresponding eigenvectors u ₁ , u ₂ , u ₃ , by which four unit directions to the dipole n _i and the directions of its magnetic moment m _{i are found} , where i = 1, 2, 3, 4 is the index of the pair of solutions (one true, three false), characterized in that the spatial increments of the magnetic field induction relative to the reference sensor at the other end are measured coaxially and at the same distance from its center using sensors of three-component magnetometers located at one end of the portable rod on the axles of its coordinate system rods, which calculate the average increment over all semiaxes, as well as the differences of increments for each pair of semiaxes of the same name, build a symmetric difference matrix, while a third of its trace is subtracted from each diagonal element of the matrix, and unit directions are found by formulas depending on the size of the number φ = λ ₁ / N:
at -0.95≤φ <0.95:

at -1≤φ <-0.95:

at 0.95 <φ≤1:

then, normalizing the vector of average increment, we obtain a vector equation with one unknown k = a / r:

where B ₀₇ is the vector of average increment;
a - the distance between the center of the reference sensor and the center of the coordinate system;
ρ = (1 + k ² -2kn _a · n) ^1/2 ;
n _a is the unit vector of the direction of the rod,
in turn, the normalized vector equation is reduced to the modular | B ₀₇ | / N _a = f (k), solving for which for each of the four pairs of found vectors n _i and m _i get the corresponding value k _i , then each pair of quantities n _i , m _i , if it corresponds to a physically acceptable value of k _i , it is substituted into a normalized vector equation, it is calculated and the three true solutions n, m and k are selected from the minimum modulus of the difference between the calculated and measured versions of this equation, after which the sought distance to the source r = a / k.

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