A compensation method for tensor calculation

Abstract

The present invention discloses a compensation method for tensor calculation, belonging to the technical field of target positioning based on magnetic fields. A magnetic sensor array structure that measures magnetic field gradients in three directions in the measurement space is called a spatial array structure, and a magnetic sensor array structure that measures magnetic field gradients in two directions in the measurement space is called a planar array structure, and compensation formulas for tensor calculation of all magnetic sensor array structures are given. The present invention proposes a compensation method for tensor calculation to compensate for the tensor calculation error caused by the non-coincidence of the spatial coordinate system and the magnetic field coordinate system, thereby reducing the errors of tensor measurement and magnetic target positioning, solving the problem of large current tensor calculation errors, effectively improving the accuracy of tensor measurement and magnetic target positioning, without correcting the position of the sensor, directly reducing the tensor calculation error by 77.38%, which is convenient and effective.

Description

Technical Field

[0001] The present invention relates to a compensation method for tensor calculation, belonging to the technical field of target positioning based on magnetic fields. Background Art

[0002] Magnetic positioning technology is a target positioning technology based on magnetic fields, which has the advantages of all-weather operation, high speed, high precision, etc., and shows its unique advantages and application prospects in many fields such as geophysics and biomedicine. When positioning and navigating a surgical robot, compared with optical tracking, magnetic positioning technology is not affected by occlusions and has a lower cost. When tracking a wireless capsule endoscope, tongue movement, and magnetic drug labeling, compared with CT with radiation and expensive MRI, magnetic positioning technology is safer, lower cost, and more efficient.

[0003] Magnetic positioning technology is mainly divided into scalar magnetic positioning technology, vector magnetic positioning technology, and tensor magnetic positioning technology. The magnetic gradient tensor is the gradient of the magnetic field vector in three directions in space, often simply referred to as the tensor. Compared with the magnetic field scalar and the magnetic field vector, the tensor has richer magnetic field information and higher spatial resolution. Therefore, the tensor magnetic positioning technology has higher positioning accuracy and faster positioning speed. In addition, since the gradient of the geomagnetic field is basically zero, the tensor magnetic positioning technology can be unaffected by the geomagnetic field and geomagnetic field fluctuations, and is considered to be the next breakthrough point of magnetic positioning technology.

[0004] The magnetic sensor array is the main instrument for measuring the tensor. The output, attitude, and position of the magnetic sensor will all affect the accuracy of tensor measurement and magnetic target positioning. Error parameter calibration and tensor calculation compensation must be performed before use. The current calibration and compensation methods for magnetic sensor arrays have the following problems:

[0005] Although various errors of the magnetic sensor array have been accurately calibrated, no compensation has been made for the tensor calculation errors caused by sensor position deviation.

[0006] The problem of calibrating various error parameters of the magnetic sensor array has existed for a long time, and generally recognized and effective calibration methods have been gradually formed. In a uniform magnetic field, the self-errors (scale factor error, non-orthogonal error, and zero bias) of the magnetic sensor can be accurately calibrated by using the scalar method to calibrate the sensor output; the misalignment error between the magnetic sensor coordinate systems can be accurately calibrated by using the vector method to calibrate the sensor attitude. In addition, there is a deviation between the actual position p i (i = 1, 2, 3, 4) of the magnetic sensor measurement point and the theoretical position o i (i = 1, 2, 3, 4), which is called the sensor position deviation, such as Figure 1As shown in the figure. The deviation between the magnetic induction center and the housing, as well as the installation deviation, will cause the position deviation of the sensor. The position deviation of the sensor makes the baseline distance between the magnetic sensors different from the designed value. More seriously, the spatial coordinate system o'-x'y'z' is non-orthogonal and does not coincide with the reference coordinate system o-xyz. These two types of errors will both cause tensor calculation errors. In the gradient magnetic field, the position coordinates of the sensor measurement point can be accurately calculated using the magnetic field gradient formula to correct the baseline distance error. However, the spatial coordinate system cannot be transformed into an orthogonal coordinate system like the magnetic field coordinate system, and the spatial coordinate system and the magnetic field coordinate system still do not coincide. No research has proposed a compensation method for such tensor calculation errors, and the tensor calculation errors have not been effectively compensated. Summary of the Invention

[0007] The present invention proposes a compensation method for tensor calculation to compensate for the tensor calculation errors caused by the non-coincidence of the spatial coordinate system and the magnetic field coordinate system, thereby reducing the errors in tensor measurement and magnetic target positioning. It solves the problem of large tensor calculation errors at present.

[0008] A compensation method for tensor calculation, the compensation method for tensor calculation includes the following steps:

[0009] The magnetic gradient tensor G is the gradient of the magnetic field vector B in three spatial directions. In Equation (1), B x , B y , B z are the three-axis components of B,

[0010]

[0011] In the measurement area without spatial current density, the divergence and curl of the magnetic field are both 0. The tensor G is symmetric and traceless, that is:

[0012]

[0013] At this time, G has only 5 independent components. Using this feature, the tensor can also be obtained by measuring the magnetic field gradients in two directions in space. The magnetic sensor array structure that measures the magnetic field gradients in three directions in space is called a spatial array structure, and the magnetic sensor array structure that measures the magnetic field gradients in two directions in space is called a planar array structure.

[0014] Define the equivalent measurement point on the positive axis of the magnetic sensor array as point P, and the equivalent measurement point on the negative axis as point N. The spatial vector from point N to point P is n, B P , B N are the magnetic field vectors at points P and N respectively. Then n X , n Y , n Z are the spatial vectors on the x, y, and z axes respectively, BPx , B Py , B Pz are the magnetic field vectors at the equivalent measurement points along the positive x, y, and z axes respectively, B Nx , B Ny , B Nz are the magnetic field vectors at the equivalent measurement points along the negative x, y, and z axes respectively,

[0015] For the spatial array structure, the tensor G 1 before compensation has the following calculation formula:

[0016]

[0017] Using the magnetic field gradient formula, for the compensated tensor G 2 we have:

[0018]

[0019] Then the calculation formula for G 2 is:

[0020] G 2 = [B Px - B Nx B Py - B Ny B Pz - B Nz [n X n Y n Z -1 (5)

[0021] According to the measurement direction, the planar array structure is divided into XY-plane array structure, XZ-plane array structure, and YZ-plane array structure.

[0022] For the XY-plane array structure, using the magnetic field gradient formula, for the compensated tensor G 2 we have:

[0023]

[0024] Combining Equation (2) and Equation (6), we get:

[0025]

[0026] where are the three-axis components of the spatial vector n X , are the three-axis components of the spatial vector n Y . Solving Equation (7) using the least squares method can obtain the compensated tensor G 2 , ​

[0027] For the XZ-plane array structure, using the magnetic field gradient formula for the compensated tensor G 2 There is:

[0028]

[0029] Combining Equation (2) and Equation (8), we can obtain:

[0030]

[0031] Where is the three-axis component of the spatial vector n Z Using the least squares method to solve Equation (9), the compensated tensor G can be obtained 2 ,

[0032] For the YZ-plane array structure, using the magnetic field gradient formula for the compensated tensor G 2 There is:

[0033]

[0034] Combining Equation (2) and Equation (10), we get:

[0035]

[0036] Using the least squares method to solve Equation (11), the compensated tensor G can be obtained 2 .

[0037] Advantages of the present invention:

[0038] (1) Although existing calibration methods can accurately calibrate various errors of magnetic sensors, the spatial coordinate system and the magnetic field coordinate system still do not coincide after calibrating the sensor position deviation, resulting in a large tensor calculation error. The present invention proposes a compensation method for tensor calculation, which does not require adjusting the position of the sensor and can directly compensate for the tensor calculation error caused by the non-coincidence of the spatial coordinate system and the magnetic field coordinate system, effectively improving the accuracy of tensor measurement and magnetic target positioning.

[0039] (2) A compensation method for tensor calculation error proposed by the present invention effectively compensates for the tensor calculation error. When the sensor position deviation is 2 mm, the tensor calculation error before compensation is 4.358%, and the tensor calculation error after compensation is reduced to 0.986%. The present invention does not require correcting the position of the sensor and directly reduces the tensor calculation error by 77.38%, which is convenient and effective. Description of the Drawings

[0040] Figure 1 is the sensor position deviation of the magnetic sensor array;

[0041] Figure 2 is the magnetic field gradient between two points;

[0042] Figure 3 is a magnetic sensor array with a square array structure;

[0043] Figure 4 is the tensor calculation error σ under different sensor position deviations. Detailed implementation manners

[0044] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0045] Due to the limitations of manufacturing and installation processes, various errors inevitably exist in the magnetic sensor array, which will affect the accuracy of tensor measurement and magnetic target positioning. Although existing calibration methods can accurately calibrate the self-error, misalignment error, and sensor position deviation of the magnetic sensor, the spatial coordinate system and the magnetic field coordinate system are still not coincident after the sensor position deviation is calibrated, and the tensor calculation error still exists. No scholar has proposed an effective compensation method for such tensor calculation errors. The purpose of the present invention is to propose a compensation method for tensor calculation to compensate for the tensor calculation error caused by the non-coincidence of the spatial coordinate system and the magnetic field coordinate system, thereby reducing the errors of tensor measurement and magnetic target positioning.

[0046] Referring to Figures 1 - 4 as shown, the present invention proposes that the magnetic gradient tensor G is the gradient of the magnetic field vector B in three directions in space. In Equation (1), B x , B y , B z are the three-axis components of B.

[0047]

[0048] In the measurement area without spatial current density, the divergence and curl of the magnetic field are both 0, and the tensor G is symmetric and traceless, that is:

[0049]

[0050] At this time, G has only 5 independent components. Utilizing this feature, the tensor can also be obtained by measuring the magnetic field gradients in two directions in space. The magnetic sensor array structure that measures the magnetic field gradients in three directions in space is called a spatial array structure, and the magnetic sensor array structure that measures the magnetic field gradients in two directions in space is called a planar array structure. The present invention proposes compensation methods for tensor calculation for the spatial array structure and the planar array structure respectively.

[0051] Define the equivalent measurement point on the positive axis of the magnetic sensor array as point P, and the equivalent measurement point on the negative axis as point N. The spatial vector pointing from point N to point P is n, and B P , B N are the magnetic field vectors at points P and N respectively, as Figure 2 shown. Then n X , n Y , n Z are the spatial vectors on the x, y, and z axes respectively, and B Px , B Py , B Pz are the magnetic field vectors at the equivalent measurement points on the positive axes of the x, y, and z axes respectively, and B Nx , B Ny , B Nz are the magnetic field vectors at the equivalent measurement points on the negative axes of the x, y, and z axes respectively.

[0052] For the spatial array structure, the calculation formula for the tensor G 1 before compensation is:

[0053]

[0054] Using the magnetic field gradient formula, for the compensated tensor G 2 there is:

[0055]

[0056] Then the calculation formula for G 2 is:

[0057] G 2 = [B Px - B Nx B Py - B Ny B Pz - B Nz [n X n Y n Z -1 (5)

[0058] Classify the planar array structure into XY-plane array structure, XZ-plane array structure, and YZ-plane array structure according to the measurement direction. For example, the XY-plane array structure means that the array structure measures the magnetic field gradients on the x-axis and y-axis.

[0059] For the XY-plane array structure, using the magnetic field gradient formula, for the compensated tensor G 2 there is:

[0060] ​

[0061] Combining Equation (2) and Equation (6), we can obtain:

[0062]

[0063] where are the three-axis components of the spatial vector n X , are the three-axis components of the spatial vector n Y . By using the least squares method to solve Equation (7), the compensated tensor G 2 can be obtained.

[0064] For the XZ plane array structure, using the magnetic field gradient formula for the compensated tensor G 2 , we have:

[0065]

[0066] Combining Equation (2) and Equation (8), we can obtain:

[0067]

[0068] where are the three-axis components of the spatial vector n Z . By using the least squares method to solve Equation (9), the compensated tensor G 2 can be obtained.

[0069] For the YZ plane array structure, using the magnetic field gradient formula for the compensated tensor G 2 , we have:

[0070]

[0071] Combining Equation (2) and Equation (10), we can obtain:

[0072]

[0073] By using the least squares method to solve Equation (11), the compensated tensor G 2 can be obtained. Thus far, the present invention provides the compensation formulas for calculating the tensors of all magnetic sensor array structures. When in use, first obtain the position coordinates and magnetic field vectors of the measurement points of each sensor by using the existing calibration method, select the corresponding tensor calculation compensation formula according to the array structure, and substitute the position coordinates and magnetic field vectors of the measurement points into the formula to calculate the compensated tensor G 2 .

[0074] The following is a specific embodiment of the present invention:

[0075] Taking the square array structure as an example, the implementation manner of the present invention is introduced. Four magnetic sensors are evenly distributed in the xoy plane, as shown in Figure 3As shown, the magnetic sensor array belongs to the XY plane array structure.

[0076] For the compensated tensor G 2 we have:

[0077]

[0078] where:

[0079]

[0080] In Equation (12), r 1 , r 2 , r 3 , r 4 are the position coordinates of the measurement points of sensors S1, S2, S3, and S4 respectively, and B 1 , B 2 , B 3 , B 4 are the magnetic field vectors at the positions of the measurement points of sensors S1, S2, S3, and S4 respectively. Solving Equation (13) using the least squares method can obtain the compensated tensor G 2 .

[0081]

[0082] The tensor G generated by the measurement magnetic dipole using the square array structure is used to illustrate the effect of the present invention. Table 1 shows the values of the magnetic moment vector m d , the position vector r d , the baseline distance D, and the sensor position deviation.

[0083] <![CDATA[m d (A·m 2 )]]> <![CDATA[r d (m)]]> Baseline distance D (m) Sensor position deviation (mm) <![CDATA[[1.768,0,1.768] T > <![CDATA[[0.5,0,0] T > 0.04 0-2

[0084] Table 1 Simulation conditions

[0085] Using (14) to measure the tensor calculation error σ t :

[0086]

[0087] where is the calculated value of the tensor component, and is the theoretical value of the tensor component. Taking the average of the calculation results repeated 50 times, Figure 4 the tensor calculation σ t is calculated for different sensor position deviations. The greater the sensor position deviation, the greater the tensor calculation error. If the sensor position deviation is 2 mm, σ t before compensation is 4.358%, and σ tIt drops to 0.986%. That is, without adjusting the position of the sensor, the present invention reduces the calculation error of the tensor by 77.38%.

Claims

1. A compensation method for tensor calculation, characterized in that, The compensation method for tensor calculation includes the following steps: The magnetic gradient tensor G is the gradient of the magnetic field vector B in three spatial directions. In Equation (1), B x , B y , B z are the three-axis components of B. In the measurement region without spatial current density, both the divergence and curl of the magnetic field are 0, and the tensor G is symmetric and traceless, that is: At this time, G has only 5 independent components. Using this feature, the tensor can also be obtained by measuring the magnetic field gradients in two directions in space. The magnetic sensor array structure for measuring the magnetic field gradients in three directions in space is called a spatial array structure, and the magnetic sensor array structure for measuring the magnetic field gradients in two directions in space is called a planar array structure. Define the equivalent measurement point on the positive axis of the magnetic sensor array as point P, and the equivalent measurement point on the negative axis as point N. The spatial vector pointing from point N to point P is n, B P and B N are the magnetic field vectors at points P and N respectively. Then n X , n Y , and n Z are the spatial vectors on the x, y, and z axes respectively. B Px , B Py , and B Pz are the magnetic field vectors at the equivalent measurement points on the positive axes of the x, y, and z axes respectively. B Nx , B Ny , and B Nz are the magnetic field vectors at the equivalent measurement points on the negative axes of the x, y, and z axes respectively. For a spatial array structure, the tensor G before compensation 1 is calculated by the formula: Using the magnetic field gradient formula for the compensated tensor G 2 There is: Then G 2 The calculation formula is as follows: G 2 = [B Px - B Nx B Py - B Ny B Pz - B Nz [n X n Y n Z -1 (5)​ According to the measurement directions, the planar array structure is divided into XY planar array structure, XZ planar array structure, and YZ planar array structure. For the XY-plane array structure, using the magnetic field gradient formula for the compensated tensor G 2 we have: Combining Equation (2) and Equation (6), we get: Among them are the three-axis components of the spatial vector n X , are the three-axis components of the spatial vector n Y . Solving Equation (7) using the least squares method yields the compensated tensor G 2 . For the XZ-plane array structure, using the magnetic field gradient formula for the compensated tensor G 2 There is: Combining Equation (2) and Equation (8), we get: where are the three-axis components of the spatial vector n Z and the compensated tensor G can be obtained by solving Equation (9) using the least squares method 2 , For the YZ-plane array structure, using the magnetic field gradient formula for the compensated tensor G 2 we have: Combining Equation (2) and Equation (10), we get: Solving Equation (11) using the least squares method gives the compensated tensor G 2 .

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