Magnetic gradient array correction method based on rotation invariant in non-uniform magnetic field

Abstract

The invention discloses a magnetic gradient array correction method based on rotation invariants in a non-uniform magnetic field, and belongs to the technical field of magnetic detection and correction. The method comprises the following steps: firstly, eight three-axis magnetometers are placed according to a hexahedron array, and three coordinate axes of the eight three-axis magnetometers are respectively aligned; secondly, establishing a measurement error model in the non-uniform magnetic field; then, establishing a rotation invariant model in the process that the magnetic gradient array rotates around a central point; and finally, designing a target function and calculating an error parameter of each sensor by solving a constrained optimization problem. Compared with a traditional non-uniform magnetic field correction method, the non-uniform magnetic field correction method has the advantages that more accurate sensor error measurement parameters can be obtained, errors caused by a non-uniform magnetic field can be effectively reduced, it is ensured that the magnetic gradient array can still provide accurate data in a complex environment, and therefore the effects of related applications such as positioning and detection are improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of magnetic detection and correction, and specifically relates to a correction method for a magnetic gradient array based on rotation invariants in a non-uniform magnetic field, which is mainly applied to the correction before the detection and positioning of a gradient array in a harsh geomagnetic environment. Background Art

[0002] A magnetic gradient array is an important structure in the fields of magnetic field detection and target positioning, and is widely used in fields such as marine exploration, geological exploration, and national defense security. The ideal magnetic gradient array assumes that the magnetic field in the detection environment is a uniform field. However, in actual applications, the magnetic field in the detection environment is usually affected by various external factors and exhibits non-uniform characteristics, such as the interference of underground metal structures, power facilities, and complex geological structures. This non-uniformity will cause errors in magnetic gradient measurement, affecting the positioning accuracy and detection effect. Therefore, in order to achieve high-precision magnetic field measurement in a non-uniform magnetic field environment, it is urgent to develop an effective correction method to solve problems such as insufficient consideration of the cost function, long solution time, and easy entrapment in local optima. Summary of the Invention

[0003] To solve the above technical problems, the present invention provides a correction method for a magnetic gradient array based on rotation invariants in a non-uniform magnetic field. By rotating the magnetic gradient array, gradient information can be collected from multiple directions, and the errors caused by the non-uniformity of the magnetic field can be eliminated by using the rotation-invariant characteristics, so as to complete the correction of the array faster and improve the correction accuracy, which helps to significantly improve the application effect of the magnetic gradient array in a complex environment and promote the development of high-precision detection and positioning technologies.

[0004] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0005] A correction method for a magnetic gradient array based on rotation invariants in a non-uniform magnetic field, comprising the following steps:

[0006] Place eight three-axis magnetometers at the vertices of a regular hexahedron tooling structure in sequence, and align the three coordinate axes of each magnetic sensor;

[0007] Establish a measurement error model in a non-uniform magnetic field;

[0008] Based on the measurement error model, establish a rotation invariant model during the rotation of the magnetic gradient array around the center point;

[0009] Calculate the cost function corresponding to each parameter of the rotation invariant model, design an objective function to balance the cost function corresponding to each parameter, and calculate the error parameters of each magnetic sensor by solving a constrained optimization problem. The parameters of the rotation invariant model include three rotation invariants in a non-uniform magnetic field, a tensor contraction value, and a normalized magnetic source intensity.

[0010] The beneficial effects of the present invention are as follows:

[0011] (1) The present invention fully considers various rotation invariants in the rotation of the magnetic gradient array and establishes a more comprehensive objective function.

[0012] (2) The present invention fully considers the deficiency that the traditional method for solving non-convex non-linear problems is prone to converge to local minimum values, and uses sequential quadratic programming to accelerate the solution speed. Description of the Drawings

[0013] Figure 1 It is a flowchart of a method for calibrating a magnetic gradient array based on rotation invariants in a non-uniform magnetic field according to the present invention. Detailed Embodiment

[0014] The present invention will be further described below with reference to the drawings and embodiments.

[0015] A method for calibrating a magnetic gradient array based on rotation invariants in a non-uniform magnetic field according to the present invention includes the following steps: First, eight three-axis magnetometers are sequentially placed at the vertices of a regular hexahedron tooling structure, and the three coordinate axes of each magnetic sensor are aligned, and their three coordinate axes are respectively aligned; Second, a measurement error model in the non-uniform magnetic field is established; Then, a rotation invariant model during the rotation of the magnetic gradient array around the center point is established; Finally, an objective function is designed and the error parameters of each sensor are calculated by solving a constrained optimization problem. The principle block diagram of the whole method is as Figure 1 shown, and the specific implementation steps are as follows:

[0016] In the first step, a ferromagnetic target far from the distance measurement system can be modeled as a magnetic dipole, expressed as:

[0017] (1)

[0018] Among them, represents the vacuum permeability, represents the position vector from the ferromagnetic anomaly to the sensor, represents the vector modulus, represents the magnetic dipole moment vector, represents the magnitude of the magnetic dipole moment, represents the transpose operation.

[0019] For most actual scenarios, due to the complex measurement environment and the inherent characteristics of the magnetic field sensor, the true magnetic field of the magnetic dipole is usually disturbed by five error sources, and the measured disturbed magnetic field can be expressed as:

[0020] (2)

[0021] Wherein, is a diagonal matrix representing the proportional error of each axis of the magnetic sensor, is an upper triangular matrix representing the non-orthogonal error of the magnetic sensor, is a matrix representing the soft iron error, is a column vector representing the hard iron error, is a column vector representing the drift error of the magnetic sensor.

[0022] Let and respectively represent and Then, (2) can be rewritten as:

[0023] (3)

[0024] The corrected magnetic field can be expressed as:

[0025] (4)

[0026] On the other hand, the magnetic gradient array is usually composed of 8 magnetic field sensors with a cube structure, so there is also a misalignment error between different sensors. Arbitrarily select a sensor (take S1 as an example) as a reference, and the coupling coefficients and of S1 can be obtained by using the traditional single-sensor calibration method. Then, the measured magnetic field of the th sensor can be expressed as:

[0027] (5)

[0028] Wherein, and are respectively the coupling error coefficient matrix and the offset vector of the th magnetic sensor, represents the misalignment error of the th sensor, represents the perturbed magnetic field of the th sensor, and respectively represent the intermediate parameters of the th sensor.

[0029] In the second step, according to (1), the magnetic gradient tensor is defined as the change rate of the three components of the magnetic induction intensity along the three coordinate axes, expressed as , wherein, , and respectively represent the three-axis components of the magnetic field .

[0030] Similar to other third-order matrices, it also has eigenvalues and eigenvectors satisfying . The eigenvalues can be obtained by solving , where is the identity matrix. Expand it into a cubic equation about :

[0031] (6)

[0032] where:

[0033] (7)

[0034] In the above formula, represents the trace of the tensor matrix and is always equal to zero, represents the determinant of the tensor matrix . In addition, there are two commonly used rotation invariants:

[0035] (8)

[0036] where, is the tensor contraction value, is the normalized magnetic source strength, , and are the three eigenvalues of the tensor matrix , and there are , , . Therefore, since the above five scalars , , , and describe the physical properties of a point in the magnetic field, they are all rotation invariants in the non-uniform magnetic field, related to the position and magnetic moment of the magnetic dipole, rather than related to the choice of coordinate system.

[0037] In the third step, when the magnetic gradient array rotates around the , and axes respectively, the center point of the array remains unchanged, which means that the rotation invariants at the center point are also unchanged. Based on this, a series of cost functions can be designed.

[0038] (1) First, consider the magnetic field strength: In a non-uniform magnetic field, the magnetic field strength at a given measurement point is constant, and the corresponding cost function is:

[0039] (9)

[0040] where is the unknown constant to be optimized, and the subscript represents the variable when rotating around the axis, . , , respectively represent the magnetic field components of the x, y, and z axes at the center point and the corresponding unknown constants to be optimized. The meanings of the parameters in the cost function and the cost function below are similar and will not be elaborated further. In addition, the magnetic field strengths at the center of the sensor coplanar in the positive and negative directions of the coordinate axes should also be constant, and its cost function is:

[0041] (10)

[0042] where the subscripts and respectively represent the corresponding coplanar centers and .

[0043] (2) Secondly, consider the rotation invariant : is the trace of the tensor matrix and can be expressed in terms of eigenvalues as:

[0044] (11)

[0045] Similarly, the cost function for is:

[0046] (12)

[0047] (3) Then, consider the rotation invariant : The invariant is the quadratic term coefficient in (6) and can be rewritten using eigenvalues as:

[0048] (13)

[0049] Therefore, the cost function for is:

[0050] (14)

[0051] (4) Next, consider the rotation invariant : By using the eigenvalues of the tensor matrix , its determinant can be rewritten as:

[0052] (15)

[0053] Therefore, the cost function with respect to is:

[0054] (16)

[0055] (5) Then, consider the tensor contraction : The magnetic gradient tensor contraction , defined as the Frobenius norm of the matrix , can be further derived as:

[0056] (17)

[0057] where represents the angle between the vectors and . During the rotation process, the tensor contraction at the center point of the array remains unchanged. Therefore, a cost function can be established as:

[0058] (18)

[0059] (6) Finally, consider the normalized magnetic source strength : The normalized source strength of the magnetic gradient tensor is a scalar, independent of the magnetization direction, and only affected by the magnitude of magnetization, that is:

[0060] (19)

[0061] Similar to the cost function based on , the cost function related to can be written as:

[0062] (20)

[0063] There are obvious differences in the order of magnitude among the cost functions (9)-(20). If their sum is directly used as the objective function, the final optimization result will focus on minimizing the part with a higher order of magnitude. Therefore, in order to balance different cost functions, the objective function to be optimized It can be modeled as:

[0064] (21)

[0065] Wherein, and are the measured magnetic field and the parameters to be optimized, is the corresponding normalized cost function, which is:

[0066] (22)

[0067] Wherein, and are respectively the maximum and minimum values of the roughly estimated.

[0068] Therefore, the design objective function is:

[0069] (23)

[0070] The problem of calibrating the error coefficients of the magnetic gradient tensor can be obtained by solving the above optimization problem. For non-convex and non-linear optimization problems like (23), most existing studies use the Levenberg-Marquardt (LM) algorithm to solve them. However, the LM algorithm is a local optimization method and is prone to converging to a local minimum, thus affecting the time and accuracy of array calibration. On this basis, we chose sequential quadratic programming to solve this problem, which is an iterative optimization method that gradually approaches the optimal solution by solving a series of quadratic programming sub-problems.

[0071] The content not described in detail in the specification of the present invention belongs to the prior art well-known to those skilled in the art. Although the exemplary embodiments of the present invention have been described for the purpose of illustration, those skilled in the art will understand that various changes such as modifications, additions, and substitutions in form and detail can be made without departing from the scope and spirit of the invention disclosed in the appended claims, and all such changes should fall within the protection scope of the appended claims of the present invention. Moreover, each part of the product and each step of the method claimed by the present invention can be combined in any form. Therefore, the description of the embodiments disclosed in the present invention is not intended to limit the scope of the present invention, but to describe the present invention. Accordingly, the scope of the present invention is not limited by the above embodiments, but is defined by the claims or their equivalents.

Claims

1. A method for calibrating a magnetic gradient array based on rotation invariants in a non-uniform magnetic field, characterized in that Including the following steps: Place eight three-axis magnetometers at the vertices of a regular hexahedron tooling structure in sequence, and align the three coordinate axes of each magnetic sensor; Establish a measurement error model in a non-uniform magnetic field; Based on the measurement error model, establish a rotation invariant model during the rotation of the magnetic gradient array around the center point; Calculate the cost function corresponding to each parameter of the rotation invariant model, design an objective function to balance the cost functions corresponding to each parameter, and calculate the error parameters of each magnetic sensor by solving a constrained optimization problem. The parameters of the rotation invariant model include three rotation invariants in a non-uniform magnetic field, a tensor contraction value, and a normalized magnetic source strength.

2. A magnetic gradient array calibration method based on rotation invariants in a non-uniform magnetic field according to claim 1, characterized in that The establishment of the measurement error model in a non-uniform magnetic field includes: When the distance from a ferromagnetic object to a measurement system including a magnetic gradient array to be calibrated is greater than three times the size of the ferromagnetic object itself, the ferromagnetic object is modeled as a magnetic dipole, and the true magnetic field of the magnetic dipole is expressed as: （1） Among them, represents the vacuum permeability, represents the position vector from the ferromagnetic object to the magnetic sensor, represents the position vector modulus of, represents the magnetic dipole moment vector, represents the magnitude of the magnetic dipole moment, represents the transpose operation; Measured disturbed magnetic field Expressed as: （2） Among them, is a diagonal matrix representing the scale error of each axis of the magnetic sensor, is an upper triangular matrix representing the non-orthogonal error of the magnetic sensor, represents the soft iron error matrix, is a column vector representing the hard iron error, is a column vector representing the drift error of the magnetic sensor; The corrected magnetic field is expressed as: （4） Superscript represents the inverse operation, intermediate parameter and respectively represent and ; The measured magnetic field of the first sensor is expressed as: （5） Among them, and are the coupling error coefficient matrix and the offset vector of the th magnetic sensor respectively, represents the mismatch error of the th sensor, represents the perturbed magnetic field of the th sensor, and represent the intermediate parameters of the th sensor respectively.

3. A magnetic gradient array calibration method based on rotation invariants in a non-uniform magnetic field according to claim 1, characterized in that The establishment of the rotation invariant model during the rotation of the magnetic gradient array around the center point based on the measurement error model includes: Define the magnetic gradient tensor matrix as the rate of change of the three components of the magnetic induction intensity along the three coordinate axes, expressed as: , where , and respectively represent the three-axis components of the magnetic field . The magnetic gradient tensor matrix includes eigenvalues and eigenvectors satisfying . By solving , the eigenvalues are obtained, where is the identity matrix, and it is extended to a cubic equation about : （6） Where, （7） In the above formula, , , are all rotation invariants in the non-uniform magnetic field, where represents the trace of the magnetic gradient tensor matrix and is always equal to zero, represents the determinant of the magnetic gradient tensor matrix ; The other two rotation invariants included in the rotation invariant model are: （8） Among them, is the tensor contraction value, is the normalized magnetic source strength, , and are the three eigenvalues of the magnetic gradient tensor matrix , and there is , , ; The rotation invariants , , , and are related to the position and magnetic moment of the magnetic dipole, rather than related to the choice of coordinate system.

4. A magnetic gradient array calibration method based on rotation invariants in a non-uniform magnetic field according to claim 1, characterized in that The design of the objective function to balance the cost functions corresponding to each parameter and the calculation of the error parameters of each sensor by solving a constrained optimization problem include: Calculate the rotation invariants separately , , , and of the cost function , , in order to balance different cost functions, the objective function to be optimized is modeled as: （21） Among them, and are the measured magnetic field and the parameter to be optimized, is the corresponding normalized cost function: （22） Among them, and are respectively the maximum value and the minimum value of the roughly estimated. Therefore, the objective function to be optimized is designed as: （23） Select sequential quadratic programming to solve the above-mentioned objective function to be optimized, and gradually approximate the optimal solution by solving a series of quadratic programming sub-problems.

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