Method and apparatus for measuring magnetic field intensity in high-temperature superconducting maglev transportation systems

Abstract

The present invention relates to the technical field of maglev transportation, specifically revealing a method and apparatus for measuring magnetic field intensity in high-temperature superconducting maglev transportation systems. The method includes: Establishing the top surface of the permanent magnet guideway (PMG) as the reference datum for magnetic field intensity measurements; developing a multi-point chord measurement system with parameters including system sampling interval, system order, and chord measurement configuration; computing intermediate chord values; constructing an inversion model incorporating the least squares method and employing this model to derive the vertical displacement of the reference chord baseline; determining gap sensor locations and deploying an array of Hall effect sensors along the direction of magnetic field intensity measurement; adjusting the reference datum position; and calculating the magnetic field intensity distribution above the reference datum using interpolation techniques based on the Hall effect sensor array measurements.

Claims

1. A magnetic field intensity measurement method for high-temperature superconducting maglev transportation systems, characterized by comprising the following steps: S1. Establishing the top surface of the permanent magnet guideway (PMG) as the reference datum for magnetic field intensity measurements; S2. Developing a chord-based multi-point measurement system to detect surface irregularities of the PMG, with system parameters including sampling interval, system order, and chord measurement configuration; S3. Computing chord measurement values for multiple intermediate points using data from the multi-point chord measurement system, and constructing an inversion model for the measurement system employing the least squares method; the inversion model is used to reconstruct the original track irregularity waveform from chord-based measurements, where the reconstructed track irregularity represents the vertical displacement of the chord baseline; S4. Determining gap sensor positions based on the optimal chord measurement configuration of the multi-point chord system, and deploying an array of Hall effect sensors along the direction of magnetic field intensity measurement; S5. Adjusting the reference datum position using the vertical displacement of the chord baseline, and calculating the magnetic field intensity distribution at any height above the reference datum using interpolation techniques based on Hall effect sensor readings; In step S3, the formula for calculating chord measurement values for multiple intermediate measurement points based on the multi-point chord measurement system data is expressed as:
G=H.Math.Z.sub.z In which G denotes the chord measurement value matrix, H denotes the measurement matrix, and Z.sub.z denotes the irregularity vector matrix of the PMG; The chord measurement value matrix G is represented as: $G=\left[\begin{array}{c}G\left(1\right)\\ G\left(2\right)\\ .Math.\\ G\left(n-1\right)\end{array}\right]={\left[\begin{array}{cccccc}{g}_{1,0}& g_{1,1}& .Math.& g_{1,i}& .Math.& g_{1,N-n-1}\\ g_{2,0}& g_{2,1}& .Math.& g_{2,i}& .Math.& g_{1,N-n-1}\\ .Math.& .Math.& & .Math.& .Math.& .Math.\\ g_{n-1,0}& g_{n-1,1}& .Math.& g_{n-1,i}& .Math.& g_{n-1,N-n-1}\end{array}\right]}_{\left(n-1\right)\left(N-n\right)}$ Where G(i) denotes all chord values of measurement points when the chord baseline begins at measurement point i, and g.sub.n1,i denotes the chord value of measurement point n1 when the chord baseline begins at measurement point i; The fully configured measurement matrix H for the chord baseline is presented as: $H=\left[\begin{array}{c}{h}_1\left[k\right]\\ h_2\left[k\right]\\ .Math.\\ h_{n-1}\left[k\right]\end{array}\right]={\left[\begin{array}{cccc}{}_1& 1& .Math.& \stackrel{\_}{{}_1}\\ {}_2& 1& .Math.& \stackrel{\_}{{}_2}\\ .Math.& & & .Math.\\ {}_{n-1}& .Math.& 1& \stackrel{\_}{{}_{n-1}}\end{array}\right]}_{\left(n-1\right)\left(n+1\right)}$ The measurement matrix H contains n1 rows and n+1 columns, with the number of rows corresponding to the number of measurement points in the full configuration, and the number of columns matching the dimension of the irregularity vector covered by the chord baseline, including endpoints; h.sub.n1[k] is the convolution kernel for measurement point n1, with elements in the 1st and n+1 columns corresponding to the split ratios .sub.i and .sub.i of measurement point i, while the middle columns from the 2nd to the n form an identity matrix of dimension n1; The PMG irregularity vector matrix Z.sub.z corresponding to each column of the chord measurement value matrix G is expressed as: $Z_Z=\left[\begin{array}{cccccc}{z}_0& z_1& .Math.& z_i& .Math.& z_{N-n-1}\\ z_1& z_2& .Math.& z_{i+1}& .Math.& Z_{N-n}\\ .Math.& .Math.& & .Math.& & .Math.\\ z_n& z_{n+1}& .Math.& z_{i+1}& .Math.& z_{N-1}\end{array}\right]$ Where Z.sub.i represents the irregularity vector corresponding to the chord baseline position i.

2. The magnetic field intensity measurement method for high-temperature superconducting maglev transportation systems as stated in claim 1, characterized in that: in step S2, the system sampling interval is s, and the formula for computing the system order is as follows:
=L/s Where denotes the system order, L denotes the system chord length, and s denotes the sampling step size.

3. The method for measuring magnetic field intensity in a high-temperature superconducting maglev transportation system as claimed in claim 1, characterized in that: an inversion model is established using the least squares method, which is subsequently employed to reconstruct the guideway irregularity waveform; during this process, the optimal solution is derived under the condition of best approximating the guideway track irregularity, thus ensuring the accuracy of the measurement results; the formula for deriving the optimal solution is as follows: $\underset{z}{\min }\frac{1}{2}{.Math.H.Math.Z^{*}-G.Math.}^2$ Where H denotes the measurement matrix, G denotes the chord measurement value matrix, and Z* denotes the optimal solution, which is the reconstructed waveform of the PMG (Permanent Magnet Guideway) irregularity.

4. The method for measuring magnetic field intensity in a high-temperature superconducting maglev transportation system as claimed in claim 1, characterized in that: an operator is incorporated into the inversion model of the measurement system to ensure solution stability and generate Z*, the reconstructed waveform of PMG irregularity; the operator expression is defined as: $={\left(\stackrel{k}{\underset{i=1}{.Math.}}{H}_i^T{H}_i+I\right)}^{-1}.Math.\underset{i=1}{\stackrel{k}{.Math.}}{H}_i^T$ Where H denotes the measurement matrix, T denotes the matrix transpose operation, denotes the regularization parameter coefficient ranging from 0.001 to 0.005, I denotes the identity matrix, and i denotes the position of the measurement point; The reconstructed waveform is expressed as:
Z*=.Math.G Where Z represents the reconstructed waveform of PMG irregularity, G represents the chord measurement value matrix, and represents the operator.

5. The method for measuring magnetic field intensity in a high-temperature superconducting maglev transportation system as claimed in claim 1, characterized in that: in step S4, gap sensor positions are determined based on the optimal chord measurement configuration of the multi-point chord system, and an array of Hall effect sensors is deployed along the direction of magnetic field intensity measurement; a Hall sensor measurement matrix B is obtained after placing j Hall sensors at each measurement point position; $B=\left[\begin{array}{cccc}{B}_{1,1}& B_{1,2}& .Math.& B_{1,j}\\ B_{2,1}& B_{2,2}& .Math.& B_{2,j}\\ .Math.& .Math.& .Math.& .Math.\\ B_{n-1,1}& B_{n-1,2}& .Math.& B_{n-1,j}\end{array}\right]$ Where B.sub.n1,j denotes the measurement value of the {dot over (j)}-th sensor in the measurement direction at measurement point n1.

6. The method for measuring magnetic field intensity in a high-temperature superconducting maglev transportation system as claimed in claim 1, characterized in that: in step S5, the adjusted reference datum position is determined by subtracting the vertical displacement from the initial position of the chord baseline, and the magnetic field intensity distribution at any arbitrary height above the reference datum is computed using interpolation techniques based on the Hall effect sensor readings; the mathematical expression is as follows: $R^{*}=R-Z^{*}$ $B^{}=i\mathrm{nterpl}\left(R^{},B,s\right)$ Where R* denotes the adjusted chord baseline position, R denotes the initial position, and B* denotes the interpolated magnetic field intensity at position S along the measurement direction.

7. A measurement apparatus fabricated according to the magnetic field intensity measurement method for high-temperature superconducting maglev transportation systems as described in claim 1, characterized in that: the said measurement apparatus includes: A trolley main frame (1), running wheels (2), sensor mounting bracket (3), measurement chord mounting bracket (4), measurement chord (5), Hall sensor mounting apertures (6), gap sensor mounting apertures (7), and an encoder (8); The running wheels (2) are attached to the trolley main frame (1) through bearings; both the sensor mounting bracket (3) and the measurement chord mounting bracket (4) are rigidly affixed to the trolley main frame (1); the sensor mounting bracket (3) features Hall sensor mounting apertures (6) for secure installation of Hall sensors (13) and gap sensor mounting apertures (7) for secure installation of gap sensors (14); the measurement chord (5) is rigidly attached to the measurement chord mounting bracket (4); the encoder (8) is linked to the bearing and rotates coaxially with it.

Description

DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 presents the procedural flow of the method described in this invention;

(2) FIG. 2 illustrates the magnetic field intensity measurement apparatus positioned above the PMG of the high-temperature superconducting maglev transportation system in the embodiment of this invention.

(3) FIG. 3 showcases the magnetic field intensity measurement apparatus situated above the PMG of the high-temperature superconducting maglev transportation system in the embodiment of this invention.

(4) FIG. 4 demonstrates the measurement process for magnetic field intensity at any location above the PMG of the high-temperature superconducting maglev transportation system in the embodiment of this invention;

(5) The numerical references in the figures are defined as follows: 1trolley main frame; 2running wheels; 3sensor mounting bracket; 4measurement chord mounting bracket; 5measurement chord; 6Hall sensor mounting apertures; 7gap sensor mounting apertures; 8encoder; 11magnetic field intensity irregularity; 12guideway irregularity; 13Hall sensor; 14gap sensor.

DETAILED DESCRIPTION OF THE EMBODIMENTS

(6) The following section will elucidate the technical solutions of the embodiments of this invention in a clear and comprehensive manner, in conjunction with the accompanying figures. Evidently, the embodiments described herein do not encompass all possible embodiments of this invention. Based on the embodiments presented in this invention, all other embodiments developed by those of ordinary skill in the pertinent field, which do not involve additional creative elements, fall within the protective scope of this invention.

Embodiment 1

(7) In high-temperature superconducting maglev transportation systems, the high degree of levelness of functional components is an essential prerequisite for ensuring the safe, stable, and comfortable operation of maglev vehicles throughout the entire speed domain. The combined effects of climatic variations such as rainfall, temperature extremes, guideway foundation settlement, concrete structure creep, and cyclic loading from maglev vehicles can result in non-uniform spatial distribution of the PMG magnetic field and alterations in the planarity of the traction DC motor. Should issues arise, such as magnetic field intensity irregularities with significant rates of change, the operational safety and stability of maglev vehicles will be directly impacted. As illustrated in FIG. 1, the method proposed in this invention enables the measurement of magnetic field intensity at any location above the PMG of the high-temperature superconducting maglev system. As a magnetic field intensity measurement method for high-temperature superconducting maglev transportation systems, it is characterized by comprising the following steps:

(8) S1. Establishing the top surface of the permanent magnet guideway (PMG) as the reference datum for magnetic field intensity measurements.

(9) In this embodiment, to assess the magnetic field intensity distribution irregularities above the PMG of the high-temperature superconducting maglev transportation system, the upper surface of the PMG is established as the measurement reference plane. The optimal operational height for high-temperature superconducting magnetic levitation is 10 mm; consequently, in this instance, the vertical magnetic field intensity is measured at a position 10 mm above the PMG's upper surface.

(10) S2. Developing a chord-based multi-point measurement system to detect surface irregularities of the PMG, with system parameters including sampling interval, system order, and chord measurement configuration (i.e., gap sensor installation positions);

(11) Said system sampling interval is s, and the formula for computing the system order is as follows:
=L/s

(12) Where denotes the system order, L denotes the system chord length, and s denotes the sampling step size.

(13) S3, Computing chord measurement values for multiple intermediate points using data from the multi-point chord measurement system, and constructing an inversion model for the measurement system employing the least squares method. The inversion model is used to reconstruct the original track irregularity waveform from chord-based measurements, where the reconstructed track irregularity represents the vertical displacement of the chord baseline;

(14) The formula for calculating chord measurement values for multiple intermediate measurement points based on the multi-point chord measurement system data is expressed as:
G=H.Math.Z.sub.z

(15) In which G denotes the chord measurement value matrix, H denotes the measurement matrix, and Z.sub.z denotes the irregularity vector matrix of the PMG;

(16) The chord measurement value matrix G is represented as:

(17) $G=\left[\begin{array}{c}G\left(1\right)\\ G\left(2\right)\\ .Math.\\ G\left(n-1\right)\end{array}\right]={\left[\begin{array}{cccc}{g}_{1,0}& g_{1,1}& .Math.g_{1,i}& .Math.g_{1,N-n-1}\\ g_{2,0}& g_{2,1}& .Math.g_{2,i}& .Math.g_{1,N-n-1}\\ .Math.& .Math.& \mathrm{.Math.}& .Math..Math.\\ g_{n-1,0}& g_{n-1,1}& .Math.g_{n-1,i}& .Math.g_{n-1,N-n-1}\end{array}\right]}_{\left(n-1\right)\left(N-n\right)}$

(18) Where G(i) denotes all chord values of measurement points when the chord baseline begins at measurement point i, and g.sub.n1,i denotes the chord value of measurement point n1 when the chord baseline begins at measurement point i;

(19) The fully configured measurement matrix H for the chord baseline is presented as:

(20) $H=\left[\begin{array}{c}{h}_1\left[k\right]\\ h_2\left[k\right]\\ .Math.\\ h_{n-1}\left[k\right]\end{array}\right]={\left[\begin{array}{cccc}{}_1& 1& .Math.& \stackrel{\_}{{}_1}\\ {}_2& 1& .Math.& \stackrel{\_}{{}_2}\\ .Math.& & & .Math.\\ {}_{n-1}& .Math.& 1& \stackrel{\_}{{}_{n-1}}\end{array}\right]}_{\left(n-1\right)\left(n+1\right)}$

(21) The measurement matrix H contains n1 rows and n+1 columns, with the number of rows corresponding to the number of measurement points in the full configuration, and the number of columns matching the dimension of the irregularity vector covered by the chord baseline, including endpoints. h.sub.n1[k] is the convolution kernel for measurement point n1, with elements in the 1st and n+1 columns corresponding to the split ratios .sub.i and .sub.i of measurement point i, while the middle columns from the 2nd to the n form an identity matrix of dimension n1;

(22) The PMG irregularity vector matrix Z.sub.z corresponding to each column of the chord measurement value matrix G is expressed as:

(23) 0$Z_Z=\left[\begin{array}{cccccc}{z}_0& z_1& .Math.& z_i& .Math.& z_{N-n-1}\\ z_1& z_2& .Math.& z_{i+1}& .Math.& Z_{N-n}\\ .Math.& .Math.& & .Math.& & .Math.\\ z_n& z_{n+1}& .Math.& z_{i+1}& .Math.& z_{N-1}\end{array}\right]$

(24) Where Z.sub.i represents the irregularity vector corresponding to the chord baseline position 1.

(25) An inversion model is established using the least squares method, which is subsequently employed to reconstruct the guideway irregularity waveform. During this process, the optimal solution is derived under the condition of best approximating the guideway track irregularity, thus ensuring the accuracy of the measurement results. The formula for deriving the optimal solution is as follows:

(26) $\underset{z}{\min }\frac{1}{2}{.Math.H.Math.Z^{*}-G.Math.}^2$

(27) Where H denotes the measurement matrix, G denotes the chord measurement value matrix, and Z* denotes the optimal solution, which is the reconstructed waveform of the PMG (Permanent Magnet Guideway) irregularity.

(28) When the condition number of the measurement matrix H is excessively high (termed an ill-conditioned matrix), it can result in instability of the equation solution Z*. To address this, an operator is incorporated into the inversion model of the measurement system to ensure solution stability and generate Z*, the reconstructed waveform of PMG irregularity. The operator expression is defined as:

(29) $={\left(\stackrel{k}{\underset{i=1}{.Math.}}{H}_i^T{H}_i+I\right)}^{-1}.Math.\underset{i=1}{\stackrel{k}{.Math.}}{H}_i^T$

(30) Where H denotes the measurement matrix, T denotes the matrix transpose operation, denotes the regularization parameter coefficient ranging from 0.001 to 0.005, I denotes the identity matrix, and i denotes the position of the measurement point; The reconstructed waveform is expressed as:
Z*=.Math.G

(31) Where Z* represents the reconstructed waveform of PMG irregularity, G represents the chord measurement value matrix, and represents the operator.

(32) S4, Determining gap sensor positions based on the optimal chord measurement configuration of the multi-point chord system, and deploying an array of Hall effect sensors along the direction of magnetic field intensity measurement;

(33) The optimal chord measurement configuration, which determines the optimal sensor installation positions, should satisfy the following criteria: (1) The greatest common divisor of the intervals between adjacent measurement points should be 1, ensuring the stability of the measurement system and the subsequent inversion model, thus guaranteeing the accuracy of chord values and reconstructed guideway irregularities; (2) Maximize the detectable wavelength while fulfilling measurement precision requirements, thereby ensuring the applicability of measurement results; (3) Satisfy measurement error requirements, ensuring the accuracy of measurement outcomes;

(34) The pertinent equations are presented as follows:

(35) $\{\begin{array}{c}d=GCD\left(o_1{p}_1,p_1{p}_2,p_2{p}_3,.Math.,p_{j-1}{p}_j.Math.,p_{n-1}{o}_2\right)=1\\ \min \left(z_e\right)\\ \underset{}{\min }-{}_k\end{array}$

(36) In which d denotes the greatest common divisor of measurement point spacings, GCD is the function for computing the greatest common divisor, min is the function for determining the minimum value, Z.sub.e signifies the error in the reconstructed guideway irregularity waveform, represents the standard deviation of the error waveform under the specified measurement precision, .sub.k denotes the standard deviation of the error waveform when the detectable wavelength is , and the formula for .sub.k is given by:

(37) ${}_k=\sqrt{\frac{\frac{1}{2}{.Math.}_{j=1}^{n-1}\left(\frac{z_j^2}{2.Math.s.Math.\left(n-1\right)}\right)}{N}}$

(38) In which Z.sub.j denotes the amplitude of the error waveform at location j, s represents the sampling step size, n1 signifies the number of wavelength domain partitions, and N is the length of the error waveform.

(39) For this embodiment, the measurement objective is the vertical magnetic field intensity at a height of 10 mm above the PMG's upper surface. Gap sensors are positioned in accordance with the optimal chord measurement configuration, while Hall sensors are vertically aligned at each measurement point.

(40) A Hall sensor measurement matrix B is obtained after placing j Hall sensors at each measurement point position;

(41) $B=\left[\begin{array}{cccc}{B}_{1,1}& B_{1,2}& .Math.& B_{1,j}\\ B_{2,1}& B_{2,2}& .Math.& B_{2,j}\\ .Math.& .Math.& .Math.& .Math.\\ B_{n-1,1}& B_{n-1,2}& .Math.& B_{n-1,j}\end{array}\right]$

(42) Where B.sub.n1,j denotes the measurement value of the j-th sensor in the measurement direction at measurement point n1.

(43) S5. Adjusting the reference datum position using the vertical displacement of the chord baseline, and calculating the magnetic field intensity distribution at any height above the reference datum using interpolation techniques based on Hall effect sensor readings.

(44) in step S5, the adjusted reference datum position is determined by subtracting the vertical displacement from the initial position of the chord baseline, and the magnetic field intensity distribution at any arbitrary height above the reference datum is computed using interpolation techniques based on the Hall effect sensor readings. The mathematical expression is as follows:

(45) $R^{*}=R-Z^{*}$$B^{}=i\mathrm{nterpl}\left(R^{},B,s\right)$

(46) Where R* denotes the adjusted chord baseline position, R denotes the initial position, and B* denotes the interpolated magnetic field intensity at position S along the measurement direction.

(47) For computing the vertical magnetic field intensity 10 mm above the reference baseline, B*=interpl(R*,B,10) denotes the interpolated vertical magnetic field intensity at a distance of 10 mm from the reference plane.

(48) In this embodiment, based on the proposed optimal chord measurement configuration, an apparatus for detecting the vertical magnetic field intensity above the PMG of high-temperature superconducting maglev transportation systems was devised, with the specific design illustrated in FIG. 2. Component 7 comprises gap sensor mounting apertures (aligned with the optimal chord measurement configuration) for installing gap sensors 14 (utilized to detect and collect the gap from the measurement surface); component 6 consists of Hall sensor mounting apertures, employed to vertically arrange Hall sensors 13 (for measuring vertical magnetic field intensity). Following the calculation of chord values and reconstruction of the PMG irregularity waveform through step S3, the reference plane is adjusted via step S5, and the vertical magnetic field intensity 10 mm above the PMG is computed. In the vertical magnetic field intensity measurement scheme above the PMG of the high-temperature superconducting maglev system depicted in FIG. 3, the measurement apparatus can traverse along the PMG surface. Throughout this process, the encoder can trigger the data acquisition card at equidistant pulse intervals to record gap sensor measurements, while concurrently uploading the acquired values to the host computer for subsequent offline analysis. The aforementioned measurement outcomes can be utilized for evaluating PMG magnetic field intensity irregularities and optimizing PMG positioning in high-temperature superconducting maglev systems, with the detailed procedure illustrated in FIG. 4.

(49) For the purpose of scientifically preserving the functional components of high-temperature superconducting maglev transportation systems and maintaining their levelness, thus fulfilling the requirements for safe, stable, and comfortable operation of high-speed maglev trains, and based on the same inventive concept as the aforementioned method embodiment, this invention additionally presents a magnetic field intensity measurement apparatus for high-temperature superconducting maglev transportation systems. This apparatus can be employed to execute the magnetic field intensity measurement method for high-temperature superconducting maglev transportation systems delineated in this invention.

(50) Additionally, this embodiment introduces an apparatus for assessing the smoothness of magnetic field intensity distribution above the PMG in high-temperature superconducting maglev transportation systems. As illustrated in FIG. 2, the apparatus comprises a trolley main frame 1, running wheels 2, sensor mounting bracket 3, measurement chord mounting bracket 4, measurement chord 5, Hall sensor mounting apertures 6, gap sensor mounting apertures 7, and an encoder 8. The apparatus's running wheels 2 are attached to the trolley main frame 1 through bearings, with the encoder 8 also connected to the bearings and rotating coaxially. The measurement chord 5 is affixed to the measurement chord mounting bracket 4, which is joined to the trolley main frame 1. The Hall sensor mounting apertures 6 and gap sensor mounting apertures 7 are situated on the sensor mounting bracket 3.

(51) Trolley main frame 1: This serves as the primary structure of the apparatus, to which all other hardware elements are attached, as detailed in FIG. 2.

(52) Running wheels 2: The apparatus features two running wheels, one at the front and one at the rear, which are attached to the trolley main frame through bearings. These enable the apparatus to traverse the PMG surface at velocities up to 3 m/s.

(53) Sensor mounting bracket 3: This component is chiefly employed for mounting gap sensors and Hall sensors, and is welded to the trolley main frame.

(54) Measurement chord mounting bracket 4: This component primarily serves to fix the measurement chord in place (it must provide adequate rigidity) and is welded to the trolley main frame.

(55) Measurement chord 5: A measurement chord with specified bending stiffness is manufactured according to the optimal chord measurement configuration. This chord is either welded or bolted into the measurement chord mounting bracket.

(56) Hall sensor mounting apertures 6: These are utilized to secure the Hall sensors 13 responsible for measuring magnetic field intensity. It is crucial to select suitable Hall sensors and ensure precise installation.

(57) Gap sensor mounting apertures 7: These are utilized to affix gap sensors 14 for measuring surface irregularities of the PMG's upper surface. It is crucial to choose suitable gap sensors and guarantee precision in their installation.

(58) Encoder 8: This component rotates coaxially with the trolley bearings and can activate the data acquisition card at equidistant intervals along the measurement apparatus's advancement direction based on encoder pulses. This records distance information, ensuring that sensor gap values correlate with mileage, which aids in subsequent data analysis.

(59) The apparatus conceived in this invention enables the measurement of magnetic field intensity at any location above the PMG in high-temperature superconducting maglev transportation systems. This apparatus is characterized by its light weight, ease of disassembly and transportation, controllable costs, and economic efficiency. This invention introduces, for the first time, the application of the multi-point chord measurement technique to high-temperature superconducting maglev systems. In comparison with conventional inertial reference methods, this approach provides superior measurement accuracy, enhanced stability, and improved resistance to external disturbances. It is particularly noteworthy that while the measurement apparatus experiences substantial impacts when traversing PMG joints, the measurement precision of the multi-point chord method remains unaffected by these vibrations.

(60) While the preceding description provides detailed explanations of certain embodiments of this invention, those of ordinary skill in the relevant field may modify the technical solutions described in the aforementioned embodiments or make equivalent substitutions for some technical features. Any such modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this invention should be considered as falling within the protective scope of this invention.