Magnetic gradient tensor array two-stage calibration method based on direction regularization

Abstract

The invention relates to the field of magnetic field measurement and sensor error compensation, and provides a magnetic gradient tensor array double-stage calibration method based on direction regularization, which comprises the following steps: acquiring magnetic vector observation data under multiple attitudes; constructing a twelve-parameter error model for explicitly distinguishing an internal error of the sensor from an array attitude error; based on the model, establishing an optimization objective function considering both module value consistency and direction consistency; a two-stage optimization strategy based on a differential evolution algorithm is adopted, in the first stage, all parameters are estimated to achieve coarse calibration, and in the second stage, estimated array attitude error parameters are optimized on the basis of internal error parameters of a fixed sensor; and finally, correcting a magnetic vector observation value according to the estimated calibration parameter, and calculating a magnetic gradient tensor through a symmetric difference method. Under the condition that no external magnetic field reference is needed, the measurement precision is improved by correcting the magnetic vector observation value, the accuracy of magnetic gradient tensor calculation is finally guaranteed, and the method has good engineering adaptability and popularization prospects.

Description

Technical Field

[0001] The present invention relates to the technical field of magnetic field measurement and sensor error compensation, and in particular to a two-stage calibration method for a magnetic gradient tensor array based on directional regularization. Background Art

[0002] The Magnetic Gradient Tensor (MGT) is widely used in geomagnetic navigation, underwater passive positioning, and magnetic anomaly detection due to its superior interference immunity and spatial resolution compared to scalar magnetic field measurements. Currently, common MGT measurement systems primarily utilize multi-channel arrays of superconducting quantum interferometers (SQUIDs) or three-axis magnetometers (such as fluxgates, magnetoresistive devices, and Hall effect devices) to acquire the spatial gradient of the target magnetic field. Three-axis magnetometers, particularly fluxgates, have become a key implementation in portable MGT systems due to their low cost and ease of deployment.

[0003] However, in practical applications, the measurement accuracy of MGT arrays is susceptible to multiple error factors, including sensor scale factor error, non-orthogonality error, bias drift, and array attitude deviation during installation. These errors can severely distort the acquired magnetic field data, affecting the accuracy of MGT construction and subsequent inversion positioning performance. Therefore, high-precision system calibration to address these error sources is a key prerequisite for ensuring the reliable operation of the MGT measurement system.

[0004] In the existing art, calibration methods can be roughly divided into two categories: one is the vector calibration method, which uses Helmholtz coils or global geomagnetic models to provide a reference magnetic vector field, thereby minimizing the error between the observed vector and the reference vector to estimate the error parameters; the other is the scalar calibration method, which relies primarily on the physical property that the geomagnetic field modulus remains consistent under different attitudes, and constructs an objective function based on the modulus error to optimize the parameters. Although the vector calibration method has high accuracy, it is highly dependent on laboratory conditions (such as high-precision magnetic field sources or magnetically shielded environments), resulting in high deployment costs and poor feasibility in field environments. In contrast, the scalar calibration method, which only requires a scalar magnetometer to provide reference values, has better portability and field adaptability.

[0005] However, scalar calibration methods construct error models based solely on the magnetic field modulus, failing to provide directional constraints and making it difficult to effectively correct for deviations caused by directional errors. In the calibration results of these methods, the magnetic vector direction may deviate from the true direction, leading to directional mismatches in the gradient tensor components during calculation, which in turn adversely affects target positioning accuracy.

[0006] Furthermore, existing methods typically combine all error parameters to construct a unified optimization objective function for solution, without categorizing and modeling sensor internal errors and array-level attitude errors. This results in a lack of physical interpretability in the estimated parameters, making it difficult to accurately distinguish between various error sources, thus affecting the accuracy of subsequent error compensation. In actual deployment, such methods struggle to meet the high-precision and high-stability error correction requirements of magnetic gradient tensor systems. Therefore, there is an urgent need for a high-precision MGT array calibration method with good physical interpretability, dual modulus and direction constraints, and the ability to operate independently in field environments. Summary of the Invention

[0007] In view of this, the present invention proposes a two-stage calibration method for a magnetic gradient tensor array based on directional regularization. By constructing a twelve-parameter error model with physical interpretability, the internal sensor error and the array attitude error are explicitly distinguished and modeled, and a directional regularization term is introduced to implement vector alignment constraints. Finally, a differential evolution algorithm is used to solve the problem in stages, achieving high-precision, transferable, and self-contained error compensation for the magnetic sensor array system, thereby solving the problems existing in the above-mentioned prior art.

[0008] To achieve the above objectives, the present invention proposes a two-stage calibration method for a magnetic gradient tensor array based on directional regularization, which is characterized by comprising:

[0009] Collecting magnetic vector observation data in multiple postures, and constructing a twelve-parameter error model of the sensor to be calibrated in the magnetic gradient tensor array based on the magnetic vector observation data;

[0010] An overall optimization objective function is constructed based on the twelve-parameter error model, and calibration parameters are obtained using a two-stage optimization strategy based on a differential evolution algorithm;

[0011] A corrected magnetic vector observation value is obtained based on the calibration parameters, and a symmetric difference method is used to calculate the magnetic gradient tensor.

[0012] Furthermore, the twelve-parameter error model is as follows:

[0013]

[0014] in, is the calibration parameter, k x 、k y and k z represents the scale factor error of the sensor along the x, y and z axes respectively; the angles α, β and γ represent the non-orthogonal angular errors inside the sensor; b x 、b y and b z Indicates the three-axis fixed offset error; angle ψ, and θ represent the array-level Euler angle errors of the sensor to be calibrated relative to the internal reference sensor coordinate system. The first nine elements correspond to the sensor internal error, and the last three elements correspond to the array attitude error.

[0015] Furthermore, during the construction of the twelve-parameter error model, a measurement model of the sensor to be calibrated is constructed according to the twelve-parameter error model:

[0016]

[0017] Among them, B m is the magnetic field vector observed by the sensor to be calibrated; B ref is the reference vector, representing the true magnetic field vector; K is a third-order diagonal matrix, representing the scale factor error of the sensor in the three axes; A is the direction transformation matrix describing the three-axis non-orthogonality error, which is used to model the non-orthogonality of the sensor's internal coordinate system; b is the three-dimensional constant bias error vector, representing the constant offset term of the sensor output in each axis direction; ε is the random measurement noise vector; R array is the three-dimensional rotation matrix representing the array-level attitude error.

[0018] Each matrix and vector in the measurement model is defined by the corresponding elements in the twelve-parameter error model, and the calibrated magnetic field vector is obtained by the following formula:

[0019]

[0020] in, represents the calibrated magnetic field vector, represents the transformation matrix and .

[0021] Furthermore, the overall optimization objective function is as follows:

[0022]

[0023] in, represents the magnetic field vector measured by the sensor to be calibrated (peripheral sensor) at the u-th posture, represents the corresponding reference magnetic field vector provided by the central sensor S0, R array represents the array-level attitude error rotation matrix, b is the three-dimensional constant bias error vector, T represents the transformation matrix composed of the sensor scale factor and non-orthogonal error, and λ v is the direction regularization coefficient.

[0024] Furthermore, the process of obtaining calibration parameters using a two-stage optimization strategy based on differential evolution algorithm includes:

[0025] In the first stage, the directional regularization coefficient λv in the overall optimization objective function is set to 0, and the first stage objective function with the magnetic vector modulus error as the optimization target is constructed. The differential evolution algorithm is used to jointly estimate the 12 calibration parameters to obtain the preliminary calibration parameters of the first stage.

[0026] In the second stage, the directional regularization coefficient λv in the total optimization objective function is set to > 0, the magnetic vector direction consistency constraint is introduced, and the second stage objective function that integrates the modulus error and the directional error is constructed. On the basis of keeping the sensor internal error parameters estimated in the first stage unchanged, only the array attitude Euler angle parameters in the error model are optimized to obtain the final calibration parameters.

[0027] Furthermore, the objective function of the first stage is as follows:

[0028]

[0029] in, represents the magnetic field vector measured by the sensor to be calibrated at the u-th posture, represents the corresponding reference magnetic field vector provided by the central sensor S0, R array represents the array-level attitude error rotation matrix, b is the three-dimensional constant bias error vector, and T represents the transformation matrix composed of the sensor scale factor and non-orthogonal error.

[0030] Furthermore, the second-stage objective function is as follows:

[0031]

[0032] in, represents the Euler angle sequence, represents the magnetic field vector measured by the sensor to be calibrated at the u-th posture, represents the corresponding reference magnetic field vector provided by the central sensor S0, λ v is the direction regularization coefficient, R array represents the array-level attitude error rotation matrix, represents the first-stage estimate , represents the calibration parameter b estimated in the first stage.

[0033] Furthermore, the two-stage optimization strategy based on the differential evolution algorithm adopts a mutation strategy to iteratively update the candidate solution vectors, and generates a test vector according to the scaling difference between random individuals. The test vector is generated as follows:

[0034]

[0035] in, represents the test vector, is the mutation vector, is the current solution, CR is the crossover probability, j rand Make sure at least one dimension comes from the mutation vector.

[0036] Furthermore, in the process of constructing the measurement error model of the sensor to be calibrated based on the magnetic vector observation data, a reference magnetic vector with direction consistency constraint is provided by a central reference magnetometer arranged at the geometric center of the magnetic gradient tensor array.

[0037] Furthermore, the method also includes a verification process of the magnetic gradient tensor array calibration method:

[0038] At the vector level, the calibration effect of the magnetic vector is evaluated using the modulus consistency, direction consistency and vector residual.

[0039] At the tensor level, the magnetic gradient tensor is used to participate in the target positioning task, and the calibration effect is verified by comparing the positioning accuracy.

[0040] Compared with the prior art, the present invention has the following beneficial effects:

[0041] By constructing a physically interpretable error model, introducing a directional regularization term, and adopting a two-stage differential evolution optimization strategy, this paper achieves high-precision joint compensation for sensor internal errors and array-level directional misalignment errors in a magnetic sensor array system. This significantly improves the measurement consistency and target positioning accuracy of the magnetic gradient tensor system. The main technical effects are reflected in the following two aspects:

[0042] (1) At the vector level, high-precision magnetic vector error correction is achieved:

[0043] Based on the results of multiple simulation experiments, the proposed two-stage calibration method with directional regularization demonstrates significant error correction capabilities under various conditions. For example, without directional regularization, the average angular error reached 2.25°, the relative error of the Euler angles exceeded 100%, and the root mean square error (RMSE) of the modulus reached 45.815 nT. However, after introducing directional regularization and adopting the proposed two-stage optimization strategy, the angular error dropped to within 0.001°, the RMSE dropped to 0.418 nT, and the relative error of parameter estimation was less than 1%. In a comparison of calibration strategies, the proposed method achieved an average RMSE of 0.484 nT across all four peripheral sensors, with angular errors below 0.001°, significantly outperforming both single-stage calibration and traditional modulus methods.

[0044] (2) At the tensor level, the accuracy of magnetic target positioning is significantly improved:

[0045] After constructing the magnetic gradient tensor (MGT), the proposed method was further applied to target positioning tasks to verify the improved system performance of the present invention. Taking the positioning task of a typical magnetic dipole moving along a straight trajectory as an example, the uncalibrated measurement data resulted in an average positioning error exceeding 10.7 m and a directional error of 90.04°. However, after implementing the calibration and compensation proposed in the present invention, the average positioning error and directional error were reduced to 0.138 m and 0.48°, respectively, achieving relative error suppression rates of 98.7% and 99.5%, respectively. These results demonstrate that the proposed method not only significantly improves tensor construction accuracy but also effectively prevents the coupled propagation of errors in the inversion task.

[0046] In summary, the method proposed in the present invention has significant technical advantages and engineering application value. It not only realizes the precise compensation of the magnetic gradient tensor system in terms of amplitude and direction, but also verifies its robustness and effectiveness under complex interference conditions through positioning tasks, and has good promotion prospects. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Various other advantages and benefits will become apparent to those skilled in the art by reading the detailed description of the preferred embodiment below. The accompanying drawings are only for the purpose of illustrating the preferred embodiment and are not to be considered as limiting the present invention. In the accompanying drawings:

[0048] Figure 1 Schematic diagram of the structure of a cross-type magnetic gradient tensor sensor array in an embodiment of the present invention;

[0049] Figure 2 Schematic diagram of a magnetic vector error model in an embodiment of the present invention, wherein (a) is a schematic diagram of sensor-level non-orthogonality error, and (b) is a schematic diagram of array-level misalignment error;

[0050] Figure 3 This is the flowchart of the two-stage directional regularization calibration proposed by the present invention;

[0051] Figure 4 Schematic diagram of the application of the method of the present invention in the task of magnetic target positioning. DETAILED DESCRIPTION

[0052] The exemplary embodiments of the present disclosure will be described in more detail below with reference to the accompanying drawings. Although exemplary embodiments of the present disclosure are shown in the accompanying drawings, it should be understood that the present disclosure can be implemented in various forms and should not be limited by the embodiments set forth herein. On the contrary, these embodiments are provided to enable a more thorough understanding of the present disclosure and to fully convey the scope of the present disclosure to those skilled in the art. It should be noted that, unless there is a conflict, the embodiments of the present disclosure and the features described in the embodiments can be combined with each other. The present invention will be described in detail below with reference to the accompanying drawings and in combination with the embodiments.

[0053] This embodiment proposes a two-stage magnetic gradient tensor array calibration method based on directional regularization, which accurately compensates for internal errors and array-level directional deviations in magnetic sensor array systems. This technical solution is applicable to magnetic sensor array systems with three-axis magnetic measurement capabilities, particularly fluxgate sensor arrays with a cross-type magnetic gradient tensor (MGT) configuration. The technical solution of this invention includes the following key technical measures:

[0054] (1) Error modeling method

[0055] The present invention constructs a twelve-dimensional error parameter model with physical significance:

[0056]

[0057] Where: k x 、k y and k z represents the scale factor error of the sensor along the x, y, and z axes respectively; the angles α, β, and γ represent the non-orthogonal angular errors within the sensor (unit: degree); b x 、b y and b z Indicates the three-axis fixed offset error (unit: nT); angle ψ, and θ represent the array-level attitude Euler angle error (unit: degree).

[0058] Through this model, the sensor internal error and array attitude error can be explicitly separated, improving the interpretability of calibration parameters and system migration capabilities.

[0059] (2) Introduction of direction regularization

[0060] This paper introduces a directional regularization term for the first time in the error compensation process for magnetic sensor arrays to minimize the angular error between the calibrated magnetic vector direction and the reference vector. This method effectively suppresses the directional inconsistency that can occur with traditional modulus constraint methods, significantly improving the accuracy and stability of magnetic gradient tensor calculations. Through this regularization strategy, the present invention effectively constrains magnetic vector directional errors without relying on an external directional reference.

[0061] (3) Two-stage optimization strategy

[0062] The present invention adopts a two-stage optimization framework based on the Differential Evolution (DE) algorithm, including:

[0063] ① Phase 1: Without introducing any directional constraints (i.e., setting the directional regularization weight factor to zero), a preliminary optimization is performed with only the consistency of the magnetic vector modulus as the goal, estimating all 12-dimensional parameters and achieving rough error compensation.

[0064] ② The second stage: Based on the fixed internal parameters of the sensor, only the array-level attitude angle is optimized to minimize the magnetic vector direction residual to achieve high-precision direction alignment.

[0065] This phased optimization strategy effectively reduces the convergence problems caused by high-dimensional coupling and significantly improves the error compensation accuracy and stability of the optimization algorithm.

[0066] (4) Self-contained calibration architecture

[0067] The present invention uses a high-precision fluxgate three-axis sensor (such as the BartingtonMag690) located at the center of the array as an internal reference magnetic vector source (internal vector reference). Without relying on external magnetic field excitation, magnetic shielding rooms, or auxiliary equipment such as scalar magnetometers, a self-contained error compensation mechanism can be implemented throughout the entire process, with good adaptability to field deployment.

[0068] (5) Tensor construction and verification mechanism

[0069] After the calibration is completed, the symmetric difference method is used to calculate the magnetic gradient tensor (MGT) based on the corrected magnetic vector observation value, and the performance is verified by the following two means: On the one hand, at the vector level, the three indicators of modulus consistency, direction consistency and vector residual are used to evaluate the calibration effect of the magnetic vector; on the other hand, at the tensor level, the constructed MGT is used to participate in the target positioning task, and the tensor construction quality and system performance improvement are indirectly verified by comparing the positioning accuracy. Figure 4 The MGT positioning device consisting of two sampling points is demonstrated, and the process of data acquisition and target inversion is demonstrated under the simulated target motion path.

[0070] Figure 3 This paper describes the complete process of this embodiment, from data acquisition, error modeling, first-stage modulus consistency optimization, second-stage directional regularization optimization, to output calibration parameters, reflecting the structural characteristics of the present invention's "phased + modular + regularized guidance" approach. λ_v represents the weight factor of the directional regularization term, which is used to adjust the influence of the magnetic vector direction error on the overall optimization objective. When λ_v = 0, only the consistency of the magnetic vector modulus is considered; when λ_v > 0, a vector direction consistency constraint is introduced to improve the accuracy and stability of magnetic field direction calibration.

[0071] Without departing from the spirit of the present invention, the following technical alternatives may also be used:

[0072] (1) Optimization algorithm replacement: The differential evolution algorithm can be replaced by particle swarm optimization (PSO), genetic algorithm (GA) or other meta-heuristic optimization methods to adapt to different computing resources and convergence speed requirements;

[0073] (2) Regularization expression replacement: The directional regularization term can be replaced by various forms such as cosine similarity, directional vector residual or directional difference function, which can flexibly adapt to different directional control requirements;

[0074] (3) Reference source replacement method: In addition to a single-point high-precision flux gate, the reference magnetic vector can also use a multi-point redundant sensor or a virtual reference source based on vector field interpolation to improve the robustness and fault tolerance of the calibration results.

[0075] The technical solution of this embodiment is described in detail based on the above content, including:

[0076] 1. Establish a cross-shaped magnetic gradient tensor array computational model

[0077] MGT describes the magnetic field vector B = [B x , B y , B z ] T The spatial rate of change of . Its formal definition is as follows:

[0078] (1)

[0079] here, represents the partial derivative of the i-th magnetic field component with respect to the j-th spatial coordinate, where i, j∈{x, y, z}.

[0080] In typical weak magnetic anomaly detection scenarios, such as those involving static or slow-moving underwater ferromagnetic targets, the ambient magnetic field can be reasonably approximated as passive and static. Therefore, the divergence and curl of the magnetic field vector B can be considered zero in the local observation area, which leads to the following structural constraints on the MGT elements:

[0081] (2)

[0082] Therefore, the MGT can be represented by an asymmetric, zero-trace matrix:

[0083] (3)

[0084] Thus, although the complete MGT contains nine components, it is entirely determined by five independent elements.

[0085] Constructing a magnetic gradient tensor matrix using a cross-shaped sensor array

[0086] In this embodiment, the MGT is constructed using a cross-shaped planar sensor array, such as Figure 1 As shown in Figure 1, four peripheral triaxial fluxgate magnetometers (S1-S4) are arranged symmetrically along the x- and y-axes, with each sensor spaced a distance d from the central reference sensor S0. The spatial derivatives required for the MGT calculation are approximated using the central finite difference method.

[0087]

[0088] (4)

[0089] Therefore, MGT can be expressed in the following symmetric matrix form:

[0090] (5)

[0091] Here, represents the i component of the magnetic field measured by sensor Sn, where n ∈ {1,2,3,4}. Given the cross-shaped structure of G, all tensor components are derived from the differences between the sensor outputs. Therefore, ensuring accurate vector measurements from each sensor is crucial, as even small deviations in magnitude or direction can significantly distort the constructed tensor.

[0092] Figure 2 (a) Schematic diagram of sensor-level non-orthogonality error, where there are small angular misalignments (α, β, γ) between the internal axes of the magnetometer. (b) Schematic diagram of array-level directional misalignment error, using the X–Y–Z Euler angle sequence (ψ, ,θ) to describe the rotation of the peripheral sensors relative to the central reference sensor.

[0093] In an ideal situation, the central sensor S0 is considered to be correct and serves as a reference, providing an approximate true field B t The magnetic field vector B ref . All peripheral sensors are expected to have the same orientation with respect to S0 and produce the same readings under a uniform background field. However, each sensor exhibits inherent measurement errors due to manufacturing imperfections and electronic drift. In addition, mechanical tolerances during assembly often lead to orientation deviations of the peripheral sensors relative to the ideal configuration. While displacement errors can be minimized through precise physical alignment, rotational errors are more difficult to eliminate and represent the main source of array-level distortion. These errors significantly reduce the accuracy of the MGT build and highlight the necessity of calibration that accounts for both sensor-internal and array-level distortions.

[0094] 2. Construct a physics-inspired error model for the sensor to be calibrated

[0095] Figure 1A cross-shaped array layout consisting of four peripheral three-axis magnetic sensors (S1–S4) and a central high-precision reference magnetic sensor (S0) is demonstrated, in which the sensors form an orthogonal structure through fixed brackets to collect multi-channel magnetic vector information.

[0096] In order to accurately compensate for the sensor internal error and array-level attitude error, this embodiment defines a physics-inspired error model that combines each peripheral sensor ( Figure 1 The magnetic field vectors measured by the central sensor S1-S4 are related to the reference field vector provided by the central sensor S0. The measurement error model of each peripheral sensor (sensor to be calibrated) is:

[0097] (6)

[0098] Among them B m is the magnetic field vector observed by the sensor to be calibrated; B ref is the reference vector, representing the true magnetic field vector; K is a third-order diagonal matrix, representing the scale factor error of the sensor in the three axes; A is the direction transformation matrix describing the three-axis non-orthogonality error, which is used to model the non-orthogonality of the sensor's internal coordinate system; b is the three-dimensional constant bias error vector, representing the constant offset term of the sensor output in each axis direction; ε is the random measurement noise vector; R array is the three-dimensional rotation matrix representing the array-level attitude error. The calibration matrix and vector are defined as:

[0099] (7)

[0100] (8)

[0101] (9)

[0102] where k x 、k y and k z The deviation term b represents the scale factor of the sensor along the x, y, and z axes, respectively, reflecting the change in directional sensitivity due to manufacturing differences. x 、b y and b z Represents a constant offset along the three axes. The angles α, β, and γ describe the non-orthogonality within the sensor: α represents the in-plane deviation between the x-axis and its projection on the XOY plane; β describes the tilt of the x-axis relative to the XOY plane; and γ quantifies the deviation between the actual y-axis and its ideal orientation on the YOZ plane, as Figure 2 (a). At the same time, the angle ψ, and θ describe the orientation misalignment at the array level, following the XYZ Euler angle sequence, as Figure 2 (b) shown.

[0103] For simplicity, the scale and non-orthogonality errors are combined into a single transformation matrix:

[0104] (10)

[0105] Considering that the noise term ε can be neglected compared with the system error, the measurement model is simplified to:

[0106] (11)

[0107] It defines the ref To distortion measurement B m Deterministic mapping.

[0108] All calibration parameters are compactly represented as a twelve-dimensional vector:

[0109] (12)

[0110] The first nine elements correspond to the sensor internal errors, and the last three elements represent the array-level attitude errors. Once ξ is estimated, the calibrated magnetic field vector is calculated as:

[0111] (13)

[0112] 3. Directional Regularization Optimization Objective Function

[0113] Traditional scalar calibration methods estimate the error parameter vector ξ by minimizing the error between the magnitude of the calibrated magnetic field vector and the magnitude of a reference magnetic field, where the reference magnitude can be obtained from a scalar magnetometer or a global geomagnetic field model. The optimization objective function is defined as follows:

[0114] (14)

[0115] here represents the reference geomagnetic field strength corresponding to the u-th sensor posture, is the calibrated magnetic field vector calculated according to formula (13). However, since this method only considers the consistency of the modulus value and lacks constraints on the vector direction, it is easy to cause the direction of the calibrated magnetic vector to deviate from the true direction, thereby introducing significant errors and affecting the accuracy of the magnetic gradient tensor construction.

[0116] To this end, this method introduces a directional regularization term and proposes an optimization objective function that jointly constrains the magnetic vector modulus and directional consistency. The specific form is as follows:

[0117] (15)

[0118] in Represents the magnetic vector observation value measured by the peripheral sensor (to be calibrated) at the u-th attitude, Represents the corresponding reference magnetic vector provided by the central sensor S0. array represents the array-level attitude error rotation matrix, b is the three-dimensional constant bias error vector, T represents the transformation matrix composed of the sensor scale factor and non-orthogonal error, and λ v is the directional regularization coefficient, which is used to balance the relative weights of the modulus consistency term and the directional consistency term.

[0119] In this optimization objective function, the first term constrains the consistency between the calibrated magnetic vector magnitude and the reference magnitude, while the second term enforces directional consistency by measuring the directional difference between the vectors using the Euclidean norm. This joint objective function explicitly introduces vector alignment constraints, thereby improving overall calibration accuracy and the quality of gradient tensor construction.

[0120] 4. Optimization strategy based on two-stage DE

[0121] In the calibration model defined in Equation (15), the orientation regularization term has an influence on the Euler angles (ψ, ,θ) is very sensitive to small perturbations, while the modulus error term is more sensitive to internal errors and less responsive to attitude deviations. Directly jointly optimizing all twelve calibration parameters can easily lead to unstable convergence and trapping in local optimality, which in turn affects the final reconstruction accuracy. To improve the stability and accuracy of the calibration process, this paper proposes a two-stage optimization strategy based on a differential evolution algorithm. This strategy divides the error estimation process into two consecutive stages: first, global parameter estimation is performed to obtain internal parameters, and then, based on the fixed internal parameters, the array attitude error is locally refined.

[0122] Phase 1: Global parameter estimation based on module consistency

[0123] In the first stage, the directional regularization coefficient λ in the overall optimization objective function is set v = 0, and construct an optimization objective function that only includes the modulus error term. All twelve calibration parameters are jointly optimized using the differential evolution algorithm. The objective function of stage 1 is as follows:

[0124] (16)

[0125] in, represents the magnetic field vector measured by the sensor to be calibrated at the u-th posture, represents the corresponding reference magnetic field vector provided by the central sensor S0, R array represents the array-level attitude error rotation matrix, b is the three-dimensional constant bias error vector, and T represents the transformation matrix composed of the sensor scale factor and non-orthogonal error.

[0126] This stage mainly estimates internal error parameters such as sensor scale factor, non-orthogonal angle and bias term, and provides a rough approximation of array-level Euler angle error parameters.

[0127] Stage 2: Pose parameter refinement based on orientation regularization

[0128] In the second stage, the sensor internal error parameters estimated in the first stage (denoted as T fixed and b fixed ), only for array-level Euler angles (ψ, ,θ) error parameters are further optimized. The objective function of stage 2 is as follows:

[0129] (17)

[0130] in, represents the Euler angle sequence, represents the magnetic field vector measured by the sensor to be calibrated at the u-th posture, represents the corresponding reference magnetic field vector provided by the central sensor S0, λ v is the direction regularization coefficient, R array represents the array-level attitude error rotation matrix, represents the first-stage estimate , represents the calibration parameter b estimated in the first stage.

[0131] This stage further reduces the direction error and achieves fine compensation for the attitude error.

[0132] Both phases employ a standard differential evolution algorithm for parameter solution. Specifically, each candidate solution vector is iteratively updated using a mutation strategy that generates trial vectors based on the scaling difference between two randomly selected individuals.

[0133] (18)

[0134] The scaling factor is usually between [0.5, 1.0], and q, r, s are different indices randomly selected from the population. With the current solution Perform binomial crossover to generate the trial vector :

[0135] (19)

[0136] Where CR is the crossover probability, j rand Ensure that at least one dimension comes from the mutant. If the trial vector yields a lower objective function value, the original vector is replaced. This selection mechanism ensures greedy optimization while maintaining population diversity.

[0137] In the first stage, this DE-based strategy is applied to the full 12-dimensional calibration vector ξ. In the second stage, the optimization is limited to the 3 array-level Euler angles (ψ, ,θ), while the sensor internal parameters are fixed. This two-stage decomposition improves convergence stability and estimation accuracy by separating the global parameter search from the local direction refinement.

[0138] To comprehensively evaluate the performance of different calibration strategies, we conducted comparative tests in a simulation environment against three representative methods: (i) a traditional scalar calibration method (corresponding to λ_v = 0, without directional constraints); (ii) a single-stage optimization method (which simultaneously estimates twelve-dimensional error parameters and introduces a directional regularization term); and (iii) our proposed dual-stage optimization method (which decouples the estimation of sensor- and array-level error parameters in stages). During the tests, we performed a joint parameter sweep over sensor bias error (in the range of ±3000 nT), array-level misalignment error angle (in the range of ±5°), and the directional regularization weight λ_v (varying from {0.01, 0.1, 1, 10}). We systematically evaluated the compensation effectiveness of each method under varying error strengths and regularization weights, and determined the optimal strategy parameter configuration accordingly. Table 1 shows selected simulation results, demonstrating the performance of uncalibrated, scalar calibration, single-stage calibration, and our dual-stage calibration method under typical error conditions.

[0139] Table 1

[0140]

[0141] Table 1 shows that the proposed method outperforms other strategies across all performance metrics. For four peripheral sensors (S1–S4), the root mean square error (RMSE) of the magnetic vector modulus in the uncalibrated state generally ranges from 400 to 780 nT, the orientation error is approximately 0.4° to 1°, and the vector residual exceeds 500 nT. After scalar calibration, although the RMSE decreases for some sensors, the orientation error increases significantly (maximum 19.75°), and the vector residual exceeds 18,000 nT, indicating that this method cannot effectively control vector alignment accuracy.

[0142] A single-stage calibration method that introduces directional regularization significantly improves these issues, reducing the RMSE to 47.11–160.18 nT, keeping the directional error within 0.56°, and reducing the vector residual to 300–560 nT. Furthermore, the proposed two-stage calibration method achieves optimal performance across all evaluation metrics: RMSE is compressed to below 6.12 nT, the directional error is below 0.12°, and the vector residual is reduced to 29.35–112.77 nT. Compared to the uncalibrated state, this method achieves over 98% reduction in modulus error, over 90% suppression of directional error, and over 95% improvement in vector consistency.

[0143] The simulation results show that the two-stage optimization strategy effectively solves the error diffusion problem caused by the coupling of modulus distortion and directional deviation by structuring the parameter estimation process, and has good robustness and adaptability.

[0144] To further verify the engineering applicability of this method, the calibration performance of the three methods described above was compared using a portable cross-shaped magnetic gradient tensor array system under field measurement conditions. The calibration results of the four peripheral sensors are shown in Table 2.

[0145] Table 2

[0146]

[0147] Table 2 shows that, in the uncalibrated state, the RMSE of the magnetic vector ranges from 412.89 to 779.23 nT, the directional error is approximately 0.4°–1°, and the vector residual exceeds 500 nT. While the scalar method improves on the RMSE (minimum 168.40 nT), the directional error reaches a maximum of 19.75°, and the residual exceeds 18,000 nT, highlighting its limitations in directional calibration.

[0148] In comparison, the single-stage calibration method significantly improves the shortcomings of the scalar method, controlling the RMSE to 47.11–160.18 nT, compressing the directional error to within 0.56°, and reducing the vector residual to 300.03–562.52 nT. However, under the same test conditions, the proposed method further improves calibration accuracy, reducing the RMSE to 2.74–6.12 nT, the directional error to within 0.04°, and the vector residual to 29.35–112.77 nT, fully demonstrating its stability and high precision in real-world measurement environments.

[0149] This result further verifies the adaptability and practicality of the present invention under different working conditions, and its suitability for field deployment and high-precision mission environments.

[0150] In summary, this paper proposes a two-stage calibration framework with directional regularization for magnetic gradient tensor arrays. Based on twelve-parameter error modeling, this method explicitly distinguishes and decouples internal sensor distortion from array-level attitude deviation. In combination with an internal reference magnetic sensor, it enables system self-calibration without relying on an external magnetic source or scalar magnetometer.

[0151] In terms of optimization strategy, the differential evolution algorithm is adopted. In the first stage, a rough estimation of twelve-dimensional parameters is achieved based on the consistency of module values. In the second stage, directional regularization is introduced to perform array-level directional fine compensation, ultimately achieving coordinated suppression of amplitude and directional errors.

[0152] Simulation results show that the RMSE can be reduced to below 0.6 nT and the directional error can be suppressed to 0.001°. Field measurements demonstrate that all metrics achieve compression rates exceeding 90%. This method exhibits excellent physical interpretability, stability, and adaptability to engineering deployment, and has significant potential for application in portable or resource-constrained magnetic gradient tensor measurement systems.

[0153] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, ordinary technicians in the field should understand that the specific implementation methods of the present invention can still be modified or replaced by equivalents. Any modification or equivalent replacement that does not depart from the spirit and scope of the present invention should be covered by the scope of protection of the claims of the present invention.

Claims

1. A two-stage calibration method for magnetic gradient tensor array based on directional regularization, characterized in that: include: Collecting magnetic vector observation data in multiple postures, and constructing a twelve-parameter error model of the sensor to be calibrated in the magnetic gradient tensor array based on the magnetic vector observation data; An overall optimization objective function is constructed based on the twelve-parameter error model, and calibration parameters are obtained using a two-stage optimization strategy based on a differential evolution algorithm; A corrected magnetic vector observation value is obtained based on the calibration parameters, and a symmetric difference method is used to calculate the magnetic gradient tensor.

2. The method according to claim 1, characterized in that The twelve-parameter error model is as follows: ， in, is the calibration parameter, k x 、k y and k z represents the scale factor error of the sensor along the x, y and z axes respectively; the angles α, β and γ represent the non-orthogonal angular errors inside the sensor; b x 、b y and b z Indicates the three-axis fixed offset error; angle ψ, and θ represent the array-level Euler angle errors of the sensor to be calibrated relative to the internal reference sensor coordinate system. The first nine elements correspond to the sensor internal error, and the last three elements correspond to the array attitude error.

3. The method according to claim 1, characterized in that During the construction of the twelve-parameter error model, a measurement model of the sensor to be calibrated is constructed according to the twelve-parameter error model: ， Among them, B m is the magnetic field vector observed by the sensor to be calibrated; B ref is the reference vector, representing the true magnetic field vector; K is a third-order diagonal matrix, representing the scale factor error of the sensor in the three axes; A is the direction transformation matrix describing the three-axis non-orthogonality error, which is used to model the non-orthogonality of the sensor's internal coordinate system; b is the three-dimensional constant bias error vector, representing the constant offset term of the sensor output in each axis direction; ε is the random measurement noise vector; R array is the three-dimensional rotation matrix representing the array-level attitude error; Each matrix and vector in the measurement model is defined by the corresponding elements in the twelve-parameter error model, and the calibrated magnetic field vector is obtained by the following formula: ， in, represents the calibrated magnetic field vector, represents the transformation matrix and .

4. The method according to claim 1, wherein The overall optimization objective function is as follows: ， in, represents the magnetic field vector measured by the sensor to be calibrated (peripheral sensor) at the u-th posture, represents the corresponding reference magnetic field vector provided by the central sensor S0, R array represents the array-level attitude error rotation matrix, b is the three-dimensional constant bias error vector, T represents the transformation matrix composed of the sensor scale factor and non-orthogonal error, and λ v is the direction regularization coefficient.

5. The method according to claim 1, wherein The process of obtaining calibration parameters using a two-stage optimization strategy based on differential evolution algorithm includes: In the first stage, the directional regularization coefficient λ in the overall optimization objective function is set v =0, construct the first-stage objective function with the magnetic vector modulus error as the optimization target, use the differential evolution algorithm to jointly estimate the 12 calibration parameters, and obtain the first-stage preliminary calibration parameters; In the second stage, the directional regularization coefficient λv in the total optimization objective function is set to > 0, the magnetic vector direction consistency constraint is introduced, and the second stage objective function that integrates the modulus error and the directional error is constructed. On the basis of keeping the sensor internal error parameters estimated in the first stage unchanged, only the array attitude Euler angle parameters in the error model are optimized to obtain the final calibration parameters.

6. The method according to claim 5, characterized in that The objective function of the first stage is as follows: ， in, represents the magnetic field vector measured by the sensor to be calibrated at the u-th posture, represents the corresponding reference magnetic field vector provided by the central sensor S0, R array represents the array-level attitude error rotation matrix, b is the three-dimensional constant bias error vector, and T represents the transformation matrix composed of the sensor scale factor and non-orthogonal error.

7. The method according to claim 5, characterized in that The second-stage objective function is as follows: ， in, represents the Euler angle sequence, represents the magnetic field vector measured by the sensor to be calibrated at the u-th posture, represents the corresponding reference magnetic field vector provided by the central sensor S0, λ v is the direction regularization coefficient, R array represents the array-level attitude error rotation matrix, represents the first-stage estimate , represents the calibration parameter b estimated in the first stage.

8. The method according to claim 1, characterized in that The two-stage optimization strategy based on the differential evolution algorithm adopts a mutation strategy to iteratively update the candidate solution vectors, and generates a test vector based on the scaling difference between random individuals. The test vector is generated as follows: ， in, represents the test vector, is the mutation vector, is the current solution, CR is the crossover probability, j rand Make sure at least one dimension comes from the mutation vector.

9. The method according to claim 1, characterized in that In the process of constructing the measurement error model of the sensor to be calibrated based on the magnetic vector observation data, a reference magnetic vector with direction consistency constraint is provided by a central reference magnetometer arranged at the geometric center of the magnetic gradient tensor array.

10. The method according to claim 1, characterized in that Also included is a verification process for the magnetic gradient tensor array calibration method: At the vector level, the calibration effect of the magnetic vector is evaluated using the modulus consistency, direction consistency and vector residual. At the tensor level, the magnetic gradient tensor is used to participate in the target positioning task, and the calibration effect is verified by comparing the positioning accuracy.

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