Polynomial modeling method and device based on rotation-free and dispersion-free characteristics of static magnetic field

Abstract

The invention provides a polynomial modeling method and device based on static magnetic field rotation-free and dispersion-free characteristics, and belongs to the technical field of indoor localization, and the method comprises the steps: employing an array type magnetic sensor to collect the data of a triaxial magnetometer in a space range, and obtaining the spatial distribution information of the magnetic field intensity; analyzing the acquired magnetic field data, determining the change rule of the magnetic field intensity along with the spatial position, and performing modeling preparation on the magnetic field based on the non-spin and non-dispersion characteristic of a Maxwell equation set; establishing a magnetic field polynomial model without spinning and scattering constraints, fitting local change of the magnetic field intensity through polynomial, and embedding a differential constraint condition without spinning and scattering into the magnetic field polynomial model to ensure that the magnetic field polynomial model meets a physical law; and calculating parameters of the polynomial model of the magnetic field by using the collected magnetic field data to obtain the polynomial model of the indoor magnetic field. The invention provides a lightweight solution for real-time magnetic field positioning and navigation in a large-scale complex environment.

Description

Technical Field

[0001] This invention belongs to the field of indoor positioning technology, specifically involving a polynomial modeling method and device based on the irrotational and divergent characteristics of a static magnetic field. It analyzes the variation law of indoor magnetic field with spatial position and models the magnetic field as an irrotational and divergent polynomial model constrained by Maxwell's equations. Background Technology

[0002] The spatiotemporal stability and quasi-homogeneity of the Earth's magnetic field vector characteristics in outdoor environments have led to its widespread application in navigation, primarily as a heading reference. It can correct heading drift errors caused by the accumulation of gyroscope noise. Modern inertial navigation systems integrate data from three-axis magnetometers and gyroscopes, effectively suppressing related errors and reducing azimuth errors by 60%-80%. However, in indoor environments, building structures, electrical equipment, and metal components cause the magnetic field to become spatially heterogeneous, rendering traditional heading calculation methods based on the assumption of a uniform magnetic field ineffective and posing a challenge to conventional magnetic navigation technology. Nevertheless, indoor magnetic fields possess unique "magnetic fingerprint" characteristics, which can serve as a positioning information source for navigation satellite systems in denied environments. Therefore, magnetic field modeling has become a key research direction.

[0003] The current mainstream magnetic field model is the Gaussian process model. Based on the irrotational and non-scattering physical properties of the magnetic field, it meets the requirements of global physical consistency through covariance kernel function design, etc. However, it has problems such as the kernel function design relying on prior knowledge, the extrapolation prediction error increasing dramatically with distance, the large amount of computation in high-dimensional scenarios making it difficult to expand, and the isotropic kernel being unable to characterize the directional heterogeneous features of the magnetic field. Summary of the Invention

[0004] To address the problems of high computational complexity, difficulty in describing complex magnetic field details, poor high-dimensional scalability, and limitations in dynamic field modeling of existing Gaussian process models for magnetic fields, this invention provides a polynomial modeling method and apparatus based on the irrotational and divergent characteristics of static magnetic fields. Analysis of the variation characteristics of indoor magnetic field strength with spatial location reveals that the magnetic field strength changes continuously with spatial location and satisfies Maxwell's equations. In the absence of external power or magnetic sources, the magnetic field exhibits zero curl and zero divergence. Therefore, the magnetic field is modeled as an irrotational and divergent magnetic field polynomial model. By directly fitting the local variation of magnetic field strength with location using polynomials and explicitly embedding irrotational and divergent differential constraint equations into the model structure, this approach avoids the computational burden of complex kernel function and covariance matrix operations required by Gaussian process models. Furthermore, the magnetic field gradient characteristics can be analytically represented through polynomial coefficients, significantly improving computational efficiency and adaptability to dynamic scenarios. Simultaneously, it ensures that the model strictly satisfies physical laws, providing a lightweight solution for real-time magnetic field positioning and navigation in large-scale complex environments.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] A polynomial modeling method based on the irrotational and divergence-free properties of a static magnetic field includes the following steps:

[0007] Step 1: Use an array-type magnetic sensor to collect data from a triaxial magnetometer within a spatial range to obtain spatial distribution information of the magnetic field strength;

[0008] Step 2: Analyze the collected triaxial magnetometer data to determine the variation law of magnetic field strength with spatial position, and prepare a model of the magnetic field based on the irrotational and divergence-free characteristics of Maxwell's equations.

[0009] Step 3: Construct a magnetic field polynomial model with irrotation-free and divergence-free constraints. Fit the local changes in magnetic field strength using polynomials and embed irrotation-free and divergence-free differential constraints into the magnetic field polynomial model to ensure that the magnetic field polynomial model satisfies physical laws.

[0010] Step 4: Using the collected triaxial magnetometer data, calculate the parameters of the magnetic field polynomial model to obtain the indoor magnetic field polynomial model.

[0011] The present invention also provides a polynomial modeling device based on the irrotational and divergence-free properties of a static magnetic field, comprising the following modules:

[0012] The spatial distribution information acquisition module uses an array of magnetic sensors to collect triaxial magnetometer data within a spatial range to obtain spatial distribution information of magnetic field strength.

[0013] The modeling preparation module analyzes the collected triaxial magnetometer data, determines the variation law of magnetic field strength with spatial position, and prepares the modeling of the magnetic field based on the irrotational and divergence-free characteristics of Maxwell's equations.

[0014] The module for constructing irrotation-free and divergence-free constraint models builds irrotation-free and divergence-free magnetic field polynomial models. It fits the local changes in magnetic field strength through polynomial fitting and embeds irrotation-free and divergence-free differential constraint conditions into the magnetic field polynomial model to ensure that the magnetic field polynomial model satisfies physical laws.

[0015] The module for constructing a polynomial model of the indoor magnetic field uses the collected data from a triaxial magnetometer to calculate the parameters of the magnetic field polynomial model and obtain the indoor magnetic field polynomial model.

[0016] The present invention also provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the steps of the above-described polynomial modeling method based on the irrotational and divergent properties of a static magnetic field.

[0017] The present invention also provides a non-transient computer-readable storage medium storing a computer program thereon, characterized in that the computer program, when executed by a processor, implements the steps of the above-described polynomial modeling method based on the irrotational and divergent properties of a static magnetic field.

[0018] Beneficial effects:

[0019] This invention proposes an irrotational and divergence-free magnetic field polynomial model, addressing the problems of computational complexity, difficulty in describing complex magnetic field details, and poor high-dimensional scalability of existing Gaussian process models. This model analyzes the variation of indoor magnetic field strength with spatial location, directly fitting the local changes in magnetic field strength using polynomials. Furthermore, it embeds irrotational and divergence-free differential constraint equations into the model, avoiding the dependence of Gaussian process models on complex kernel functions and covariance matrix operations. Simultaneously, it analyzes the magnetic field gradient characteristics through polynomial coefficients, significantly improving computational efficiency and adaptability to dynamic scenarios while strictly satisfying physical laws. Attached Figure Description

[0020] Figure 1 This is a flowchart of a polynomial modeling method based on the irrotational and divergent properties of a static magnetic field according to the present invention.

[0021] Figure 2 This is a schematic diagram of a polynomial modeling device based on the irrotational and divergent properties of a static magnetic field according to the present invention.

[0022] Figure 3 Error plots for estimating the three-axis magnetic field using a polynomial model of the magnetic field; where a is the first subplot representing the error in estimating the magnetic field along the X-axis, b is the second subplot representing the error in estimating the magnetic field along the Y-axis, and c is the third subplot representing the error in estimating the magnetic field along the Z-axis. Detailed Implementation

[0023] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0024] This invention constructs a polynomial model of the magnetic field with zero curl and zero divergence constraints. By analyzing the variation of indoor magnetic field strength with spatial location and combining it with Maxwell's equations, the magnetic field is modeled as a polynomial model satisfying irrotational and divergence-free constraints. The polynomial model is used to directly fit the variation of magnetic field strength in a local region. Irrotational and divergence-free differential constraint equations are explicitly embedded in the structural design, and the gradient characteristics of the magnetic field are effectively characterized by the analytical expression of the polynomial coefficients. This method not only significantly improves computational efficiency and adaptability to dynamic scenarios but also ensures that the model strictly conforms to physical laws, thereby achieving high-precision magnetic field distribution inversion.

[0025] Specifically, such as Figure 1 As shown, the polynomial modeling method based on the irrotational and divergence-free properties of a static magnetic field according to the present invention includes the following steps:

[0026] Step 1: Use an array-type magnetic sensor to collect triaxial magnetometer data within the spatial range;

[0027] Step 2, Indoor Static Magnetic Field Characteristics Analysis: Analyze the collected magnetic field information to explore the variation of the magnetic field with spatial location;

[0028] Step 3: Construct magnetic field polynomial models without rotation or divergence constraints, including magnetic field polynomial models with zero curl, zero divergence, and magnetic field polynomial models with zero curl and zero divergence.

[0029] Step 4: Solve the parameters of the magnetic field polynomial model to obtain the indoor magnetic field polynomial model.

[0030] Specifically, step 1 includes:

[0031] The tester uses a handheld planar array of magnetic sensors to collect triaxial magnetic field data from each sensor and stores the collected data. The array of magnetic sensors is arranged according to a preset spatial distribution to cover the magnetic field measurement needs within the target area, ensuring the comprehensiveness and accuracy of the collected data.

[0032] Specifically, step 2 includes:

[0033] In indoor environments, magnetic fields primarily originate from the Earth's magnetic field, permanent magnets in building materials, electronic devices, and the magnetization of various potential ferromagnetic materials. Due to the highly random spatial distribution of the magnetic field vector, the variation of magnetic field strength with spatial location is unpredictable. Based on Maxwell's equations, under static field conditions, the following simplification is made:

[0034] Under static conditions, Ampere's circuital law simplifies to:

[0035] ;

[0036] in, Represents the Hamiltonian operator. It is the magnetic field strength. This refers to current density. If there are no obvious, stable current sources in the indoor environment (e.g., no DC wires running through the space, no continuously discharging electronic components), and assuming current density is ignored or minimized, the indoor static magnetic field can be considered, on a macroscopic scale, as... It approximately satisfies the irrotational assumption.

[0037] magnetic induction intensity non-dispersion: This is a fundamental law of nature, meaning that magnetic field lines are closed loops, without a beginning or end, and there are no so-called "magnetic monopoles." Even in various complex indoor environments, the non-dispersibility of the magnetic field remains unchanged. Ferromagnetic materials present in indoor environments (such as reinforced concrete) alter the magnetic field distribution, causing magnetic field lines to bend, converge, or sparse. This alteration effectively makes the magnetic field strength distribution uneven in local areas, appearing as if the magnetic field lines have "converged" or "divered" in some places. However, this is merely a visual illusion, because the magnetic field lines are still closed loops; they have simply undergone complex bending and redistribution within the material. Even if building materials have a magnetic shielding effect, blocking or absorbing external magnetic fields, the non-dispersibility of the magnetic field remains unchanged. Magnetic shielding only reduces the indoor magnetic field strength and alters the distribution of magnetic field lines, but the magnetic field lines remain closed loops.

[0038] Specifically, step 3 includes:

[0039] To model a magnetic field as a polynomial model with zero curl, the following conditions must be met:

[0040] ;

[0041] in, Indicates magnetic field strength The curl of.

[0042] Model the magnetic field as Order-inrotation polynomial model One approach is to represent it as polynomial model The derivative form of is expressed as follows:

[0043] ;

[0044] in, It is a vector whose elements are composed of Given, satisfying , Spatial location coordinates, Represents natural numbers, superscript , , Representing spatial position coordinates , , powers of, It is an arbitrary constant. These are the parameters of the polynomial model; the number of model parameters is... The superscript T indicates the transpose of the matrix.

[0045] The order-inrotational magnetic field polynomial model is as follows:

[0046] ,

[0047] ;

[0048] in, Indicates to Find the derivative, assuming that in the volume Within the array, centered at the origin, a magnetic field polynomial model satisfying zero curl can be used... express, For model parameters, It is a position-dependent function.

[0049] To model the magnetic field as an irrotational and divergence-free polynomial model based on the irrotational polynomial model, the condition of zero divergence must be met:

[0050] ;

[0051] in, Indicates magnetic flux density. express The divergence.

[0052] By designing the parameters of the irrotational magnetic field polynomial model, the irrotational magnetic field polynomial model can satisfy the zero divergence constraint:

[0053] ;

[0054] Where m represents the x, y, and z coordinate axes respectively, and when m = x, the above formula is expressed as: Differentiate the x-component with respect to x; when m = y, the above expression is: Differentiate the y-component with respect to y; when m=z, the above expression is: Differentiate the z-axis component with respect to z.

[0055] in, It's about variables. For a polynomial of order 1, to make the above expression in order 2 For the sum of the three polynomials to hold true, the coefficient of each power must be zero. Therefore, the constraint in the above equation can be equivalently written as... The system of equations, where, Indicates satisfaction The number of equations required.

[0056] set up The system of equations represents the irrotational and divergent magnetic field polynomial model as follows:

[0057]

[0058]

[0059] By introducing a matrix Its column spans Zero space, then set This model can be parameterized into a low-order model. Here, null{} represents the null space, null{B} represents the null space of matrix B, and st represents the constraints.

[0060] ,

[0061] ,

[0062] ;

[0063] in, It is a polynomial model of an irrotational and divergent magnetic field. For model parameters, To simplify the parameters of the low-order polynomial model, This refers to position-dependent functions in low-order models. (Superscript) It represents the orthogonal fill space.

[0064] The specific expression is as follows (taking the first order as an example):

[0065] .

[0066] Specifically, step 4 includes:

[0067] use The magnetic field data measured by the time-array magnetic sensor, combined with the magnetic field measurement model, is used to solve the model parameters using the least squares method. (Parameters of the magnetic field polynomial model) to obtain the indoor magnetic field polynomial model M.

[0068] The magnetic field measurement model is as follows: ;

[0069] in, These are observations from an array-type magnetic sensor. To measure noise, the mean is 0 and the variance is... White noise.

[0070] The model parameters are solved using the least squares method. The process is as follows:

[0071] ;

[0072] Based on the solved model parameters The polynomial model M of the indoor magnetic field (taking the first-order model as an example) is obtained as follows:

[0073] ;

[0074] in, .

[0075] In other words, step three involves the step-by-step derivation of the indoor magnetic field polynomial model M, and step four involves the solution of the model parameters to obtain the final indoor magnetic field polynomial model without unknowns. .

[0076] Example:

[0077] A planar array magnetic sensor, consisting of 42 magnetic sensors arranged in 6 rows and 7 columns with a spacing of 10 cm between each sensor, was used to collect magnetic field data. The collected magnetic field data is as follows:

[0078] ;

[0079] in, , , ..., These represent the triaxial magnetic field data collected by 42 magnetic sensors.

[0080] The method for generating the sensor positions in an array-type magnetic sensor is as follows: Using the first sensor at the top left corner as the origin, positions r of 6 rows and 7 columns of sensors are generated by simulating intervals of 10cm to the right and downwards. Substituting... The model parameters can be obtained. The obtained magnetic field model is .

[0081] like Figure 3 As shown, the horizontal axis represents the label of each magnetic sensor in the array magnetic sensor, and the vertical axis represents the error of the estimated magnetic field, in microtesla. Figure 3 The first subplot, 'a', represents the X-axis estimation magnetic field error. Figure 3 The second subplot, b, represents the Y-axis estimation error of the magnetic field. Figure 3 The 'c' in the third subplot represents the Z-axis estimated magnetic field error. Figure 3 It can be seen that the errors of the three-axis magnetic field estimated by the magnetic field polynomial model are all less than 0.3 uT, with the X-axis error less than 0.15 uT, the Y-axis error less than 0.3 uT, and the Z-axis error less than 0.08 uT. This demonstrates that the magnetic field polynomial model can accurately predict the local magnetic field strength.

[0082] like Figure 2 As shown, the present invention also provides a polynomial modeling device based on the irrotational and divergence-free characteristics of a static magnetic field, comprising the following modules:

[0083] The spatial distribution information acquisition module uses an array of magnetic sensors to collect triaxial magnetometer data within a spatial range to obtain spatial distribution information of magnetic field strength.

[0084] The modeling preparation module analyzes the collected triaxial magnetometer data, determines the variation law of magnetic field strength with spatial position, and prepares the modeling of the magnetic field based on the irrotational and divergence-free characteristics of Maxwell's equations.

[0085] The module for constructing irrotation-free and divergence-free constraint models builds irrotation-free and divergence-free magnetic field polynomial models. It fits the local changes in magnetic field strength through polynomial fitting and embeds irrotation-free and divergence-free differential constraint conditions into the magnetic field polynomial model to ensure that the magnetic field polynomial model satisfies physical laws.

[0086] The module for constructing a polynomial model of the indoor magnetic field uses the collected data from a triaxial magnetometer to calculate the parameters of the magnetic field polynomial model and obtain the indoor magnetic field polynomial model.

[0087] The present invention also provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the steps of the above-described polynomial modeling method based on the irrotational and divergent properties of a static magnetic field.

[0088] The present invention also provides a non-transient computer-readable storage medium storing a computer program thereon, characterized in that the computer program, when executed by a processor, implements the steps of the above-described polynomial modeling method based on the irrotational and divergent properties of a static magnetic field.

[0089] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code. The solutions in the embodiments of the present invention can be implemented using various computer languages, such as the object-oriented programming language Java and the interpreted scripting language JavaScript.

[0090] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0091] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0092] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0093] Although preferred embodiments of the invention have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including both the preferred embodiments and all changes and modifications falling within the scope of the invention.

[0094] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, this invention also intends to include these modifications and variations.

Claims

1. A polynomial modeling method based on the irrotational and divergence-free properties of a static magnetic field, characterized in that, Includes the following steps: Step 1: Use an array-type magnetic sensor to collect data from a triaxial magnetometer within a spatial range to obtain spatial distribution information of the magnetic field strength; Step 2: Analyze the collected triaxial magnetometer data to determine the variation law of magnetic field strength with spatial position, and prepare a model of the magnetic field based on the irrotational and divergence-free characteristics of Maxwell's equations. Step 3: Construct a magnetic field polynomial model with irrotation-free and divergence-free constraints. Fit the local changes in magnetic field strength using polynomials and embed irrotation-free and divergence-free differential constraints into the magnetic field polynomial model to ensure that the magnetic field polynomial model satisfies physical laws. Step 4: Using the collected triaxial magnetometer data, calculate the parameters of the magnetic field polynomial model to obtain the indoor magnetic field polynomial model.

2. The polynomial modeling method based on the irrotational and divergence-free properties of a static magnetic field according to claim 1, characterized in that, In step 1, the array of magnetic sensors is arranged according to a preset spatial distribution to cover the magnetic field measurement needs within the target area, ensuring the comprehensiveness and accuracy of the collected data.

3. The polynomial modeling method based on the irrotational and divergence-free properties of a static magnetic field according to claim 1, characterized in that, In step 2, by analyzing the spatial variation law of the magnetic field strength, the irrotational and divergent characteristics of the magnetic field are identified, providing basic constraints for subsequent modeling.

4. The polynomial modeling method based on the irrotational and divergence-free properties of a static magnetic field according to claim 1, characterized in that, In step 3, the magnetic field polynomial model with irrotation-free and divergence-free constraints analyzes the magnetic field gradient characteristics through polynomial coefficients, effectively describing the local changes in the magnetic field.

5. The polynomial modeling method based on the irrotational and divergence-free properties of a static magnetic field according to claim 1, characterized in that, In step 3, the magnetic field polynomial model without rotation or divergence constraints is parameterized using low-order polynomials.

6. The polynomial modeling method based on the irrotational and divergence-free properties of a static magnetic field according to claim 1, characterized in that, In step 4, the parameters of the magnetic field polynomial model are solved using the least squares method.

7. The polynomial modeling method based on the irrotational and divergence-free properties of a static magnetic field according to claim 1, characterized in that, In step 4, the obtained magnetic field polynomial model is used to achieve high-precision inversion of the indoor magnetic field distribution, which is suitable for real-time magnetic field positioning and navigation in complex local environments.

8. A polynomial modeling device based on the irrotational and divergence-free characteristics of a static magnetic field, characterized in that, Includes the following modules: The spatial distribution information acquisition module uses an array of magnetic sensors to collect triaxial magnetometer data within a spatial range to obtain spatial distribution information of magnetic field strength. The modeling preparation module analyzes the collected triaxial magnetometer data, determines the variation law of magnetic field strength with spatial position, and prepares the modeling of the magnetic field based on the irrotational and divergence-free characteristics of Maxwell's equations. The module for constructing irrotation-free and divergence-free constraint models builds irrotation-free and divergence-free magnetic field polynomial models. It fits the local changes in magnetic field strength through polynomial fitting and embeds irrotation-free and divergence-free differential constraint conditions into the magnetic field polynomial model to ensure that the magnetic field polynomial model satisfies physical laws. The module for constructing a polynomial model of the indoor magnetic field uses the collected data from a triaxial magnetometer to calculate the parameters of the magnetic field polynomial model and obtain the indoor magnetic field polynomial model.

9. An electronic device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the program, it implements the steps of a polynomial modeling method based on the irrotational and divergent properties of a static magnetic field as described in any one of claims 1 to 7.

10. A non-transitory computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the steps of a polynomial modeling method based on the irrotational and divergent properties of a static magnetic field as described in any one of claims 1 to 7.

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