Magnetic field odometry method and device based on a magnetic tensor measurement array

Abstract

This invention discloses a magnetic field odometry method and apparatus based on a magnetic tensor measurement array. The method includes the following steps: Step S1, measuring the magnetic tensor at the current location using a magnetic tensor measurement array; Step S2, obtaining the position increment using both analytical positioning and magnetic odometry calculation methods, and updating the position of the magnetic tensor measurement array using an adaptive analytical positioning and magnetic odometry fusion algorithm. Specifically, the magnetic odometry calculation method combines the second-order forward Taylor expansion and the second-order backward Taylor expansion of the magnetic induction intensity at adjacent times, eliminates the second-order Hessian term using the even function property of the second-order Hessian term with respect to the position increment, and calculates the position increment based on the average tensor matrix of the magnetic gradient tensor at adjacent times. This invention employs an adaptive weighted fusion strategy of two algorithms to achieve high-precision autonomous navigation without requiring a global geomagnetic field model, relying entirely on inertial navigation systems and prior geomagnetic maps, thus improving the long-term stability of positioning in complex magnetic field environments.

Description

Technical Field

[0001] This application relates to the field of navigation technology, and in particular to a magnetic field odometry method and apparatus based on a magnetic tensor measurement array. Background Technology

[0002] Geomagnetic navigation is a method of navigation that utilizes information about the Earth's magnetic field. Leveraging the relatively stable nature of the Earth's magnetic field and its relatively significant spatial variations, geomagnetic navigation can operate globally, around the clock. Traditional geomagnetic navigation largely relies on prior geomagnetic maps, establishing a mapping between magnetometer measurements and the map to determine its location. However, acquiring geomagnetic maps is a time-consuming and labor-intensive process, making it difficult to obtain large-scale maps and limiting the application scope of geomagnetic navigation. Magnetic field navigation methods that do not rely on prior geomagnetic maps can be broadly categorized according to their principles as follows: magnetic field SLAM, heuristic geomagnetic navigation, data-driven geomagnetic navigation, and magnetic field gradient navigation.

[0003] Magnetic field SLAM is an autonomous navigation technology that, in the absence of a priori magnetic map, simultaneously achieves pose estimation and magnetic field distribution map construction by acquiring real-time characteristics of magnetic induction intensity changes generated by a carrier during its spatial motion. Taylor N. Lee et al. published their findings in the Journal of the Institute of Navigation in 2019. MagSLAM: Aerial Simultaneous Localization and Mapping Using Earth's Magnetic Anomaly Field This paper proposes a MagSLAM algorithm that effectively combines aircraft pose estimation with continuous geomagnetic anomaly mapping based on Gaussian process regression by improving the Rao-Blackwellized particle filter architecture. Furthermore, it introduces a selective resampling strategy to maintain particle diversity. In a 100-minute flight test over a 9×12km area, the algorithm constrained the cumulative drift of the inertial navigation system to within a root mean square error range of 10–20 meters. This method, along with other mainstream magnetic field SLAM algorithms, is primarily limited by its strong dependence on trajectory loops, making it unable to effectively suppress drift in unidirectional tasks lacking revisited paths. Simultaneously, the computational complexity introduced by Gaussian process modeling leads to a significant increase in computational overhead over time.

[0004] In the field of heuristic magnetic field navigation, Wang Qiong et al. published "A Parallel Approach Geomagnetic Gradient Bionic Navigation Method" in the Journal of Northwestern Polytechnical University in 2018. The article proposed a geomagnetic gradient bionic navigation algorithm based on the parallel approach method. This method simulates the sensitivity of homing pigeons to the spatial gradient of the magnetic field and utilizes the gradient information of the geomagnetic vector components to construct a nonlinear guidance law, achieving target approach and long-range orientation capabilities for the carrier without any prior magnetic map. However, this algorithm is highly dependent on the saliency of local magnetic field characteristics and is prone to failure in regions with stable magnetic gradients. Furthermore, this method is currently mainly limited to two-dimensional spatial guidance and has not yet fully addressed the perception uncertainty caused by the attenuation of the magnetic field with altitude in complex three-dimensional dynamic environments. In addition, although this type of method does not require a prior geomagnetic field map, it must obtain the magnetic feature fingerprint of the target area beforehand as a heuristic guidance target; otherwise, the system will be unable to converge to the predetermined target point through search.

[0005] In the field of data-driven navigation, Lee et al. published in the journal Sensors in 2018. AMID: Accurate Magnetic Indoor Localization Using Deep Learning This paper proposes a pure data-driven positioning system called AMID. This technique acquires three-dimensional geomagnetic vector sequences and transforms them into recursive graph features. Then, a deep convolutional neural network is used to perform pattern recognition and classification on these fingerprint images, achieving high-precision identification of geomagnetic features in indoor environments. Its two-dimensional positioning accuracy is less than 1 meter in 88% of the tests, significantly solving the identification difficulties caused by uneven magnetic field distribution in complex indoor layouts using traditional methods. The limitations of this system lie in its reliance on the classification of magnetic field feature points. The model performs poorly in open areas lacking significant geomagnetic disturbances, and its performance is highly dependent on the coverage of the training samples. Once the vehicle deviates from the pre-trained sampling path, its positioning robustness will decrease significantly.

[0006] Regarding magnetic field gradient navigation, in 2025, Liang Yuxiao et al. published "A Geomagnetic / Inertial Fusion Navigation Method Based on Magnetic Anomaly Gradient Measurement" in the *Journal of Space Science*. This article proposed a geomagnetic / inertial fusion navigation method based on magnetic anomaly gradient measurement. This method achieves real-time estimation and correction of the position and velocity errors of the inertial navigation system by extracting the spatial gradient characteristics of the magnetic anomaly field, without requiring a pre-built high-precision geomagnetic map. It significantly suppresses the cumulative drift of the inertial navigation system over time, giving the system stronger autonomous positioning capabilities and lower computational overhead. This technology imposes relatively strict requirements on the magnetic field environment: the environment must have significant magnetic anomaly characteristics. If the geomagnetic background is too smooth, causing the magnetic field gradient change to be less than the sensor's measurement noise, the system will be unable to make corrections due to a lack of effective observations. Furthermore, the environmental magnetic field should have good physical characteristics of a single magnetic dipole model. When the magnetic dipole characteristics of the field source are difficult to separate in a dense multi-magnetic dipole field, it will affect the robustness of positioning and produce large errors in specific boundary regions. Summary of the Invention

[0007] To address the reliance of traditional geomagnetic navigation on high-precision prior geomagnetic maps, and considering the computational demands and trajectory limitations of traditional magnetic field SLAM methods, the need for prior magnetic field information at target points in heuristic magnetic field navigation methods, the requirement for large amounts of prior magnetic field data for model training in data-driven magnetic field navigation methods, and the accuracy degradation of magnetic gradient navigation methods in dense multi-magnetic dipole navigation, this invention provides a magnetic field odometry method and device based on a magnetic tensor measurement array. It employs analytical positioning and / or magnetic odometry calculation methods, particularly an adaptive weighted fusion positioning method based on a cross-shaped sensor array, combining analytical positioning and magnetic odometry calculation. The analytical positioning method originates from the physical model of a single magnetic dipole field, inverting absolute displacement through the mathematical relationship between the magnetic gradient tensor and magnetic induction intensity. The magnetic odometry calculation method originates from the total differential of time and the second-order Taylor series expansion of the magnetic field, using forward and backward difference approximations to derive the position update equation. The adaptive fusion technique uses the absolute position constraints of the analytical positioning method to suppress trajectory drift in the near field of the magnetic source, and uses the magnetic odometry calculation method to handle the superimposed field of multiple magnetic dipole sources in the far field. This invention enables the estimation of the positional changes of a moving platform without relying on an inertial navigation system or prior geomagnetic maps.

[0008] To solve the above-mentioned technical problems, this application provides a magnetic field odometry method based on a magnetic tensor measurement array, characterized by comprising the following steps: Step S1: Measure the magnetic tensor at the current location using a magnetic tensor measurement array; Step S2, update the position of the magnetic tensor measurement array using either method one or method two: Method 1: Analytical positioning method, its position update equation is:

[0009] in, Located within the magnetic dipole field The magnetic gradient tensor matrix of the position; for The magnetic field strength at the location; Method 2: Magnetic odometry calculation method, whose position update equation is:

[0010] in .

[0011] As one method described above, step S2 also includes updating the position of the magnetic tensor measurement array using the following method three: Method 3: Adaptive analytical positioning and magnetometry fusion algorithm, with the following position update equation:

[0012] in and These are the position increments obtained using analytical positioning and magnetic odometry, respectively. To set the maximum reasonable single-step displacement, For adaptive weights, the expression is:

[0013] in The condition number of the average tensor matrix. This is the maximum condition number set.

[0014] As another embodiment of the above method, the magnetic tensor measurement array includes: four magnetic sensors respectively installed at the four ends of the cross array, the cross lines of the cross array being orthogonal, and the four magnetic sensors being equidistant from the intersection of the cross lines.

[0015] As a further embodiment of the above method, the maximum condition number is set based on the signal-to-noise ratio of the magnetic gradient tensor.

[0016] As a further embodiment of the above method, the maximum condition number is set as follows: in is the signal-to-noise ratio of the magnetic gradient tensor.

[0017] To achieve the above objectives, the present invention also provides a magnetic field odometry device based on a magnetic tensor measurement array, characterized in that it comprises: A magnetic tensor measurement array is used to measure the magnetic tensor data at a given location. A memory for storing the magnetic tensor data and the computer program; A processor for executing the computer program to implement the steps of the above-described magnetic field odometry method.

[0018] As one embodiment of the aforementioned device, the magnetic tensor measurement array comprises: four magnetic sensors respectively installed at the four ends of the cross array, the cross lines of the cross array being orthogonal, and the four magnetic sensors being equidistant from the intersection of the cross lines.

[0019] To achieve the above objectives, the present invention also provides a readable storage medium, characterized in that it stores a computer program thereon, the computer program being executable by a processor to implement the steps of the above-described magnetic field odometer method.

[0020] Compared with the prior art, the advantages of this application are: This invention employs a cross-array tensor measurement device consisting of four high-precision triaxial vector magnetometers. It uses the difference between magnetic field vectors at adjacent sampling points to approximate the magnetic anomaly gradient tensor. It then uses analytical positioning and / or magnetic odometry calculation methods, and further employs an adaptive weighted fusion strategy that combines analytical positioning and magnetic odometry calculation methods. This enables high-precision autonomous navigation without the need for a global geomagnetic field model, and without relying on an inertial navigation system or prior geomagnetic map assistance. This improves the long-term stability of the motion platform's positioning in complex magnetic field environments. Attached Figure Description

[0021] Figure 1 This is a schematic diagram of the magnetometer mounting cross array in a specific implementation. Detailed Implementation

[0022] The technical solutions provided in this application are further described below in conjunction with the embodiments.

[0023] like Figure 1 As shown, in a specific embodiment of the present invention, a magnetometer comprising four high-precision triaxial vector magnetometers is used to measure magnetic anomaly gradients. The four sensors form a magnetic tensor measurement array, which is installed at the four ends of a cross array with a diameter of, for example, 2 meters. Strict requirements govern the installation and numbering of the cross array. The intersection lines of the cross array must be strictly orthogonal, and the distance from each sensor to the central origin at the intersection point of the intersection lines must be strictly equal to the radius. The first sensor in the forward direction of the cross array is designated as sensor 1, and in clockwise order, the right end is sensor 2, the rear end is sensor 3, and the left end is sensor 4, and so on. Figure 1As shown. The diameters of the first sensor 1 and the third sensor 3 are aligned with the direction of motion, while the diameters of the second sensor 2 and the fourth sensor 4 are perpendicular to the direction of motion. A spatial rectangular coordinate system is established with the center of the cross array as the origin. The positive x-axis points to the first sensor, the positive y-axis points to the second sensor, and the positive z-axis points vertically upwards into the xy-plane.

[0024] The magnetic flux density of each sensor is defined as follows: ,in This indicates that the magnetic field strength value is determined by the first... The data was obtained from the sensor. ; This represents the measurement value of the magnetic sensor in a certain direction. .

[0025] 1. Analytical Positioning Method When the magnetic sensor is located at a distance greater than three times the size of the magnetic field source, the magnetic field source can be considered as a single magnetic dipole source. According to the magnetic dipole model, the distance vector to the center of the dipole... The magnetic field strength at that location is Similarly, within the magnetic dipole field magnetic induction intensity at the location for:

[0026] in , is the magnetic permeability of air. Let be the magnetic moment of the magnetic dipole, n represent the direction vector, and dr represent the distance differential. According to formula (1), the magnetic difference between two points can be expressed as:

[0027] The magnetic field strength at two points in space can be expressed in terms of three components. The difference between the magnetic field strengths at the two points is:

[0028] Where G is the magnetic gradient tensor matrix. Represented using sensor measurement information:

[0029] Combining formulas (2) and (3), we can obtain:

[0030] After sorting, we can get:

[0031] Since the magnetic source is set to be fixed when the position of the inversion magnetometer is determined, the distance vector from the sensor to the center of the dipole can be calculated at time k+1 and time k, respectively. and Therefore, the displacement change of the magnetometer is calculated as follows:

[0032] Therefore, when there is only one magnetic dipole source in space, the position update equation of the magnetometer can be obtained as follows:

[0033] This method is referred to as the Analytical Localization Method (ALM) in this invention.

[0034] The ALM method described above can be used for magnetic field odometers when there is only one magnetic dipole source in space (or can be considered as having only one magnetic dipole source) to locate the position of the magnetometer.

[0035] 2. Magnetic Oscillation Calculation Method When there is more than one magnetic field source in space and these multiple sources cannot be considered as a single magnetic dipole, the position update equation of the analytical positioning method will no longer be valid. In this case, a completely new formula is needed to characterize the relationship between spatial position and changes in magnetic intensity.

[0036] Assume the magnetic sensor moves in a non-single magnetic dipole field. The sensor measures the magnetic flux density. The rate of change over time depends not only on the change over time, but also on the motion of the magnetometer in space.

[0037] According to the total differential chain rule, the magnetic induction intensity changes with time. The derivative of the change can be expressed as:

[0038] When using magnetic anomaly fields for sensor localization, it is assumed that the magnitude and location of the magnetic field source are constant within a certain time range, i.e. Equation (9) can be simplified to obtain:

[0039] in It is the sensor in The speed of motion at any given moment is important, but since this invention lacks speed and inertial sensors, the speed value cannot be directly obtained. Therefore, it is necessary to... magnetic induction intensity at the location Perform a second-order Taylor expansion:

[0040]

[0041] Simplifying and rearranging, we get:

[0042] in Let be the Hessian matrix of the magnetic field, and be the second spatial derivative of the magnetic field.

[0043] Existing cross array hardware can only measure the first-order gradient tensor. If the second-order term is obtained through numerical differentiation, the sensor noise will be amplified exponentially, resulting in severe positioning distortion. Therefore, this invention attempts to avoid the numerical instability caused by the calculation of higher-order derivatives without increasing hardware complexity. Thus, formula (12) is expanded forward and backward.

[0044] (1) In Expand forward :

[0045] in This is the abbreviation for the second-order Hessian term vector.

[0046] (2) In Expand backward :

[0047] because It is about The even function, and when the magnetic field changes drastically in the region where the carrier moves within the sampling interval, it is considered that... Therefore, it is believed that under the above conditions Substituting into the formula for backward expansion, we get:

[0048] Combining the forward expansion formula (13) and the backward expansion formula (15), we get:

[0049] After simplification and organization:

[0050] in .

[0051] Therefore, by constructing a dual function, the complex Hessian matrix calculation process is avoided. Under the assumption that the magnetic field does not change drastically within the sampling interval, the position update equation of the magnetometer can be obtained as follows:

[0052] This method is referred to as magnetic odometry (MO) in this patent. This method can be used to locate the position of a magnetometer when there is more than one magnetic field source in space and the multiple magnetic sources cannot be regarded as a single magnetic dipole.

[0053] 3 Adaptive analytical positioning / magnetometry fusion algorithm Since magnetic odometry calculation methods have high-order truncation errors in displacement estimation and rely solely on the magnetic gradient tensor integral with cumulative errors, when the magnetic sensor is in a region close to a single magnetic dipole source, the position inversion error can be further reduced by fusing analytical positioning algorithms.

[0054] Both methods have their advantages and disadvantages. MO is sensitive to high-frequency motion, but due to the lack of absolute correction, there is a certain degree of position drift. The ALM method strictly follows the law of magnetic field decay, so there is no position drift, but it can only be used within the range where the magnetic source can be regarded as a single dipole model.

[0055] To combine the advantages of both algorithms, the weighted fusion equation is constructed as follows:

[0056] in and These are the position increments obtained using the ALM and MO algorithms, respectively. The maximum reasonable single-step displacement can be set, and the set range is related to the moving speed of the magnetometer, with the unit being meters. For adaptive weights, the expression is:

[0057] in The condition number of the average tensor matrix. The maximum condition number is determined by the signal-to-noise ratio (SNR) of the magnetic gradient tensor, which is defined as:

[0058] in The signal-to-noise ratio of the magnetic gradient tensor. This is a measurement of the magnetic gradient tensor. This represents the noise value of the magnetic gradient tensor.

[0059] The relationship between the maximum condition number and the magnetic gradient tensor signal-to-noise ratio can be set as follows:

[0060] Formula (22) demonstrates that the algorithm selects high-quality data to ensure the accuracy of analytical positioning when the signal-to-noise ratio is high; and relaxes the tolerance when the signal-to-noise ratio is low, extracting spatial geometric residual features to the maximum extent to compensate for integral drift under weak magnetic environment or strong interference conditions. The specific implementation gives the above specific settings for the maximum condition number, but it should be noted that the max_cond is an empirical value that can be set by those skilled in the art. This formula (22) is not a unique solution, but it is a better solution. Therefore, all possible settings of max_cond should be covered within the protection scope of this invention.

[0061] because ,therefore It changes in real time with position. When the magnetic sensor is in the near-field region of the magnetic dipole... The value is relatively small. At this point, the value is close to 1, exhibiting clear single magnetic dipole characteristics, allowing for accurate estimation of motion displacement using the ALM algorithm; when the sensor is in the far field... When the value is close to 0, the MO algorithm can be used to directly calculate the motion displacement of the sensor under the superposition of multiple magnetic dipole fields.

[0062] By constructing a fusion equation for the two methods, an architecture based on adaptive fusion and switching models using magnetic source characteristics is built. The MO method provides motion continuity under high dynamic conditions, while the ALM method provides structural consistency based on the laws of the single magnetic dipole model. Through adaptive weight allocation, the fusion algorithm can introduce absolute distance constraints to suppress drift when the local environment conforms to the dipole model, and degenerate into position estimation when the model mismatches, thus achieving higher robustness and long-term stability than the single algorithm overall.

[0063] Corresponding to the above method, a specific embodiment of the present invention also provides a magnetic field odometry device based on a magnetic tensor measurement array, characterized in that it includes: A magnetic tensor measurement array is used to measure the magnetic tensor data at a given location. Memory for storing the magnetic tensor data and computer program; A processor for executing the computer program to implement the steps of the above-described magnetic field odometry method.

[0064] As one embodiment of the aforementioned device, the magnetic tensor measurement array comprises: four magnetic sensors respectively installed at the four ends of a cross array, the cross lines of the cross array being orthogonal, and the four magnetic sensors being equidistant from the intersection of the cross lines.

[0065] Corresponding to the above method, a specific embodiment of the present invention also provides a readable storage medium, characterized in that it stores a computer program thereon, the computer program being executable by a processor to implement the steps of the above-described magnetic field odometer method.

[0066] Through the above detailed description of the embodiments, it can be seen that the innovation of the present invention lies in: (1) Displacement calculation model driven by pure magnetic field This invention breaks through the dependence of traditional geomagnetic navigation on external information sources and innovatively proposes two calculation methods that do not require prior maps or inertial sensors. The analytical positioning method (ALM) establishes a direct analytical mapping from the magnetic tensor to absolute displacement through a magnetic dipole model; the magnetic odometry (MO) method utilizes a mathematical construction combining second-order Taylor expansion and forward / backward expansions to eliminate the complex and difficult-to-measure second-order derivative terms in the calculation, thus achieving high-precision position increment calculation relying solely on the first-order gradient tensor.

[0067] (2) Adaptive fusion architecture driven by magnetic source characteristics This invention implements an architecture based on an adaptive fusion and switching model using magnetic source characteristics. By introducing the condition number of the magnetic gradient tensor matrix as a quality evaluation index, an adaptive weighted fusion equation is constructed. This enables automatic switching of navigation modes based on the physical characteristics of the environmental magnetic field: in regions with prominent single magnetic dipole characteristics, the analytical equation of the ALM is used to calibrate accumulated errors; in complex multi-magnetic-source superimposed field regions, the MO is used to maintain motion continuity.

[0068] Due to the aforementioned innovative features, the beneficial effects of this invention include: This invention employs a cross array tensor measurement device consisting of four high-precision triaxial vector magnetometers. It uses the difference between the magnetic field vectors of adjacent sampling points to approximate the magnetic anomaly gradient tensor. Combined with the analytical positioning method and the adaptive weighted fusion strategy of magnetic odometry, it achieves high-precision autonomous navigation without the need for a global geomagnetic field model, and without relying on an inertial navigation system or prior geomagnetic map assistance. This improves the long-term stability of the motion platform's positioning in complex magnetic field environments.

[0069] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to the embodiments, those skilled in the art should understand that common-sense modifications or equivalent substitutions to the technical solutions of the present invention do not depart from the spirit and scope of the technical solutions of the present invention, and all such modifications and substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A magnetic field odometry method based on a magnetic tensor measurement array, characterized in that, The steps include the following: Step S1: Measure the magnetic tensor at the current location using a magnetic tensor measurement array; Step S2: The position increments are obtained using both analytical positioning and magnetic odometry methods, and then the position increments obtained from the analytical positioning and magnetic odometry methods are fused to update the position of the magnetic tensor measurement array. The position update equation is: in and These are the position increments obtained using analytical positioning and magnetic odometry, respectively. To set the maximum reasonable single-step displacement, For adaptive weights, the expression is: in The condition number of the average tensor matrix. The maximum condition number is set; the maximum condition number is set according to the signal-to-noise ratio of the magnetic gradient tensor as follows: in The signal-to-noise ratio of the magnetic gradient tensor; The position update equation for the analytical positioning method is: in, Located within the magnetic dipole field The magnetic gradient tensor matrix of the position; for The magnetic field strength at the location; The position update equation for the magnetic odometry calculation method is: in In the magnetic odometry calculation method, a second-order forward Taylor expansion is performed on the magnetic flux density at time k+1 at position k, and a second-order backward Taylor expansion is performed on the magnetic flux density at time k+1 at position k. Utilizing the even function property of the second-order Hessian term with respect to the position increment, and when the magnetic field change in the region where the magnetic tensor measurement array moves within the sampling interval makes the Hessian matrix equal for two adjacent positions, the second-order forward Taylor expansion and the second-order backward Taylor expansion are combined to eliminate the second-order Hessian term, resulting in: ； The position update equation for the magnetic odometry calculation method is obtained from the above relationship.

2. The magnetic field odometer method according to claim 1, characterized in that, The magnetic tensor measurement array includes four magnetic sensors respectively installed at the four ends of the cross array, the cross lines of the cross array are orthogonal, and the four magnetic sensors are equidistant from the intersection of the cross lines.

3. A magnetic field odometer device based on a magnetic tensor measurement array, characterized in that, include: A magnetic tensor measurement array is used to measure the magnetic tensor data at a given location. Memory for storing the magnetic tensor data and computer program; A processor for executing the computer program to implement the steps of the magnetic field odometry method of claim 1.

4. The magnetic field odometer device according to claim 3, characterized in that, The magnetic tensor measurement array includes four magnetic sensors respectively installed at the four ends of the cross array, the cross lines of the cross array are orthogonal, and the four magnetic sensors are equidistant from the intersection of the cross lines.

5. A readable storage medium, characterized in that, It stores a computer program that can be executed by a processor to implement the steps of the magnetic field odometry method of claim 1.

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