A method for correcting a magnetic gradient array based on a rotational invariant in a non-uniform magnetic field

Abstract

The application discloses a kind of non-uniform magnetic field based on rotation invariant magnetic gradient array correction method, belong to magnetic detection and correction technical field.The method includes the following steps: first, 8 triaxial magnetometers are placed according to hexahedral array, and their three coordinate axes are aligned respectively;Second, the measurement error model in non-uniform magnetic field is established;Then, the rotation invariant model in the rotation process of magnetic gradient array around center point is established;Finally, the objective function is designed and the error parameters of each sensor are calculated by solving constrained optimization problem.The application compared with the traditional non-uniform magnetic field correction method can obtain more accurate sensor measurement error parameters, can effectively reduce the error caused by non-uniform magnetic field, ensure that magnetic gradient array can still provide accurate data in complex environment, so as to improve the effect of positioning, detection and other related applications.

Description

Technical Field

[0001] This invention belongs to the field of magnetic detection and correction technology, specifically relating to a magnetic gradient array correction method based on rotational invariants in a non-uniform magnetic field, which is mainly applied to the correction before gradient array detection and positioning in harsh geomagnetic environments. Background Technology

[0002] Magnetic gradient arrays are an important structure in the fields of magnetic field detection and target localization, widely used in marine exploration, geological surveys, and national defense. Ideal magnetic gradient arrays assume a uniform magnetic field in the detection environment. However, in practical applications, the magnetic field of the detection environment is usually affected by various external factors, exhibiting non-uniform characteristics, such as interference from underground metal structures, power facilities, and complex geological structures. This non-uniformity leads to errors in magnetic gradient measurements, affecting positioning accuracy and detection results. Therefore, to achieve high-precision magnetic field measurements in non-uniform magnetic field environments, it is urgent to develop effective correction methods to address problems such as insufficient consideration of cost functions, long solution times, and susceptibility to local optima. Summary of the Invention

[0003] To address the aforementioned technical problems, this invention provides a magnetic gradient array correction method based on rotation invariant in non-uniform magnetic fields. By rotating the magnetic gradient array, gradient information can be collected from multiple directions, and the rotation invariant property can be used to eliminate errors caused by magnetic field non-uniformity, thereby completing array correction faster and improving correction accuracy. This method helps to significantly improve the application effect of magnetic gradient arrays in complex environments and promotes the development of high-precision detection and positioning technology.

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

[0005] A magnetic gradient array correction method based on rotational invariants in a non-uniform magnetic field includes the following steps:

[0006] Eight triaxial magnetometers were placed sequentially at the vertices of a regular hexahedral fixture structure, and the three coordinate axes of each magnetic sensor were aligned.

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

[0008] A rotation invariant model for the magnetic gradient array rotating around its center point is established based on the measurement error model.

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

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

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

[0012] (2) This invention fully considers the shortcomings of traditional methods for solving non-convex nonlinear problems, which tend to converge to local minima, and uses sequential quadratic programming to speed up the solution. Attached Figure Description

[0013] Figure 1 This is a flowchart of a magnetic gradient array correction method based on rotational invariants in a non-uniform magnetic field according to the present invention. Detailed Implementation

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

[0015] This invention discloses a magnetic gradient array correction method based on rotational invariants in a non-uniform magnetic field, comprising the following steps: First, eight triaxial magnetometers are sequentially placed at the vertices of a hexahedral fixture structure, and the three coordinate axes of each magnetic sensor are aligned. Second, a measurement error model in the non-uniform magnetic field is established. Then, a rotational invariant model is established during the rotation of the magnetic gradient array around its center point. 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 entire method is shown below. Figure 1 As shown, the specific implementation steps are as follows:

[0016] The first step is to model a ferromagnetic target at a distance from the measurement system as a magnetic dipole, represented as:

[0017] (1)

[0018] in, Represents the permeability of free space. This represents the position vector from the ferromagnetic anomaly to the sensor. Represents vector The model, Represents the magnetic dipole moment vector. Indicates the magnitude of the magnetic dipole moment. This indicates the transpose operation.

[0019] In most practical scenarios, due to the complex measurement environment and the inherent characteristics of magnetic field sensors, the true magnetic field of a magnetic dipole is... Typically, the measured disturbed magnetic field is subject to five sources of error. It can be represented as:

[0020] (2)

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

[0022] make and They 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, magnetic gradient arrays are typically composed of eight cubic magnetic field sensors, so there is a mismatch error between different sensors. By arbitrarily selecting a sensor (taking S1 as an example) as a reference, the coupling coefficient of S1 can be obtained using the traditional single-sensor calibration method. and So the first The magnetic field measured by the sensor can be expressed as:

[0027] (5)

[0028] in, and They are the first The coupling error coefficient matrix and offset vector of each magnetic sensor Indicates the first Mismatch error of each sensor Indicates the first The disturbed magnetic field of each sensor and They represent the first Intermediate parameters of a sensor.

[0029] The second step, according to (1), defines the magnetic gradient tensor as the rate of change of the three components of magnetic induction along the three coordinate axes, expressed as: ,in, , and They represent magnetic fields respectively. The three-axis components.

[0030] Similar to other third-order matrices, It also has eigenvalues. and eigenvectors satisfy Eigenvalues ​​can be obtained by solving... Received, among which It is the identity matrix. Expanding it to... cubic equation:

[0031] (6)

[0032] in:

[0033] (7)

[0034] In the above formula, Representing a tensor matrix The trace is always equal to zero. Representing a tensor matrix The determinant. In addition, two commonly used rotation invariants are:

[0035] (8)

[0036] in, For tensor shrinkage values, Normalized magnetic source strength, , and It is a tensor matrix The three eigenvalues, and have , , Therefore, due to the above five scalars , , , and It describes the physical properties of a point in a magnetic field. These are rotational invariants in a non-uniform magnetic field, which are related to the position and magnetic moment of the magnetic dipole, rather than to the choice of coordinate system.

[0037] The third step is to rotate the magnetic gradient array around the magnetic gradient array. , and When the axis rotates, the center point of the array remains unchanged, which means that the rotation invariants at the center point also remain 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] in, It is an unknown constant to be optimized, subscript Indicates circling Variables during axis rotation . , , Let x, y, and z represent the magnetic field components along the x, y, and z axes at the center point, and the corresponding unknown constants to be optimized. The parameters of the cost function and other cost functions described below have similar meanings and will not be repeated here. Furthermore, the magnetic field strength at the coplanar center of the sensor in both positive and negative directions along the coordinate axes should also remain constant; its cost function is:

[0041] (10)

[0042] Among them, subscript and They represent the corresponding coplanar centers. and .

[0043] (2) Next, consider rotation invariants. : It is a tensor matrix The trace can be represented by eigenvalues ​​as:

[0044] (11)

[0045] Similarly, regarding The cost function is:

[0046] (12)

[0047] (3) Then consider rotation invariants Invariants It is the coefficient of the quadratic term in (6), and can be rewritten using eigenvalues ​​as:

[0048] (13)

[0049] Therefore, regarding The cost function is:

[0050] (14)

[0051] (4) Next, consider rotation invariants. Using tensor matrices The eigenvalues ​​can be expressed as their determinants. Rewritten as:

[0052] (15)

[0053] Therefore, regarding The cost function is:

[0054] (16)

[0055] (5) Then consider tensor shrinking. Magnetic gradient tensor shrinkage Defined as a matrix The F-norm of can be further derived as:

[0056] (17)

[0057] in, Representing vectors and The angle between them. During rotation, the tensor at the array center point shrinks. Since it remains unchanged, the cost function can be established as:

[0058] (18)

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

[0060] (19)

[0061] With based The cost function is the same as, and The relevant cost function can be written as:

[0062] (20)

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

[0064] (twenty one)

[0065] in, and It measures the magnetic field and the parameters to be optimized. Is and The corresponding normalized cost function is:

[0066] (twenty two)

[0067] in, and These are rough estimates. The maximum and minimum values.

[0068] Therefore, the objective function is designed as follows:

[0069] (twenty three)

[0070] The error coefficient calibration problem of the magnetic gradient tensor can be obtained by solving the above optimization problem. For non-convex nonlinear optimization problems like (23), most existing studies use the Levenberg-Marquardt (LM) algorithm. However, the LM algorithm is a local optimization method and is prone to converge to a local minimum, which affects the time and accuracy of array calibration. Based on this, we chose sequential quadratic programming to solve this problem. This is an iterative optimization method that gradually approaches the optimal solution by solving a series of quadratic programming subproblems.

[0071] Contents not described in detail in this specification are prior art known to those skilled in the art. Although exemplary embodiments of the invention have been described for illustrative purposes, those skilled in the art will understand that various 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 modifications and substitutions should fall within the protection scope of the appended claims. Furthermore, the various parts of the product and the various steps of the method claimed in this invention can be combined in any combination. Therefore, the description of the embodiments disclosed in this invention is not intended to limit the scope of the invention, but rather to describe the invention. Accordingly, the scope of the invention is not limited by the above embodiments, but is defined by the claims or their equivalents.

Claims

1. A method for correcting a magnetic gradient array based on rotational invariants in a non-uniform magnetic field, characterized in that, Includes the following steps: Eight triaxial magnetometers were placed sequentially at the vertices of a regular hexahedral fixture structure, and the three coordinate axes of each magnetic sensor were aligned. Establish a measurement error model in a non-uniform magnetic field; A rotation invariant model for the magnetic gradient array rotating around its center point is established based on the measurement error model. The cost function corresponding to each parameter of the rotation invariant model is calculated, an objective function is designed to balance the cost function corresponding to each parameter, and the error parameters of each magnetic sensor are calculated by solving a constrained optimization problem. The parameters of the rotation invariant model include three rotation invariants in the non-uniform magnetic field, tensor shrinkage value, and normalized magnetic source intensity.

2. The magnetic gradient array correction method based on rotational 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 between a ferromagnetic object and the measurement system, including the 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 actual magnetic field of the magnetic dipole is... Represented as: （1） in, Represents the permeability of free space. This represents the position vector from the ferromagnetic object to the magnetic sensor. Represents position vector The model, Represents the magnetic dipole moment vector. Indicates the magnitude of the magnetic dipole moment. Indicates the transpose operation; Measured disturbed magnetic field Represented as: （2） in, It is a diagonal matrix representing the proportional error of each axis of the magnetic sensor. It is an upper triangular matrix representing the non-orthogonal error of the magnetic sensor. Represents the soft iron error matrix. It is a column vector representing the hard iron error. It is a column vector representing the drift error of the magnetic sensor; The corrected magnetic field is represented as: （4） superscript This indicates the inverse operation, with intermediate parameters. and They represent and ; No. The measured magnetic field of each sensor is represented as: （5） in, and They are the first The coupling error coefficient matrix and offset vector of each magnetic sensor Indicates the first Mismatch error of each sensor Indicates the first The disturbed magnetic field of each sensor and They represent the first Intermediate parameters of a sensor.

3. The magnetic gradient array correction method based on rotational invariants in a non-uniform magnetic field according to claim 1, characterized in that, The rotation invariant model established based on the measurement error model during the rotation of the magnetic gradient array around the center point includes: Define the magnetic gradient tensor matrix The rates of change of the three components of magnetic flux density along the three coordinate axes are expressed as: ,in, , and They represent magnetic fields respectively. The three-axis components, magnetic gradient tensor matrix Including eigenvalues and eigenvectors satisfy By solving The eigenvalues ​​are obtained, where Given the identity matrix, extend it to be about cubic equation: （6） in, （7） In the above formula, , , All are rotational invariants in non-uniform magnetic fields, among which Represents the magnetic gradient tensor matrix The trace is always equal to zero. Represents the magnetic gradient tensor matrix The determinant of; The other two rotation invariants included in the rotation invariant model are: （8） in, For tensor shrinkage values, Normalized magnetic source strength, , and It is the magnetic gradient tensor matrix The three eigenvalues, and have , , Rotation invariants , , , and It depends on the position and magnetic moment of the magnetic dipole, rather than on the choice of coordinate system.

4. The magnetic gradient array correction method based on rotational invariants in a non-uniform magnetic field according to claim 3, characterized in that, The design objective function balances the cost function corresponding to each parameter, and the error parameters of each sensor are calculated by solving a constrained optimization problem, including: Calculate the rotation invariants separately , , , and Cost function , To balance different cost functions, the objective function to be optimized is... The model is as follows: （21） in, and These are the coupling error coefficient matrix and offset vector between the measured magnetic field and the sensor to be optimized. Is and The corresponding normalized cost function: （22） in, and These are rough estimates. The maximum and minimum values; Therefore, the objective function to be optimized is designed as follows: （23） and They are the first The coupling error coefficient matrix and offset vector of each magnetic sensor , For the corresponding number Upper and lower limits of the coupling error coefficient matrix of a magnetic sensor , For the corresponding number The upper and lower limits of the offset vector of each magnetic sensor are determined; sequential quadratic programming is used to solve the objective function to be optimized, and the optimal solution is gradually approached by solving a series of quadratic programming subproblems.

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