CN114509090A - Error correction device and method for inclinometer for coal mine - Google Patents

Error correction device and method for inclinometer for coal mine Download PDF

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CN114509090A
CN114509090A CN202111580994.5A CN202111580994A CN114509090A CN 114509090 A CN114509090 A CN 114509090A CN 202111580994 A CN202111580994 A CN 202111580994A CN 114509090 A CN114509090 A CN 114509090A
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probe
bracket
calibrated
support rod
inclinometer
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CN114509090B (en
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宫浩
王信文
徐维泽
刘京科
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Xian Research Institute Co Ltd of CCTEG
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    • GPHYSICS
    • G01MEASURING; TESTING
    • G01CMEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C25/00Manufacturing, calibrating, cleaning, or repairing instruments or devices referred to in the other groups of this subclass
    • EFIXED CONSTRUCTIONS
    • E21EARTH OR ROCK DRILLING; MINING
    • E21BEARTH OR ROCK DRILLING; OBTAINING OIL, GAS, WATER, SOLUBLE OR MELTABLE MATERIALS OR A SLURRY OF MINERALS FROM WELLS
    • E21B47/00Survey of boreholes or wells
    • E21B47/02Determining slope or direction
    • E21B47/022Determining slope or direction of the borehole, e.g. using geomagnetism
    • GPHYSICS
    • G01MEASURING; TESTING
    • G01CMEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C1/00Measuring angles
    • GPHYSICS
    • G01MEASURING; TESTING
    • G01CMEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
    • G01C17/00Compasses; Devices for ascertaining true or magnetic north for navigation or surveying purposes
    • G01C17/02Magnetic compasses
    • G01C17/28Electromagnetic compasses
    • G01C17/32Electron compasses
    • YGENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02ATECHNOLOGIES FOR ADAPTATION TO CLIMATE CHANGE
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    • Y02A90/30Assessment of water resources

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Abstract

The invention discloses an error correction device of an inclinometer for a coal mine, which comprises a base and a non-magnetic rotary table arranged on the base, wherein the non-magnetic rotary table comprises a bracket and a coaxial plate connected with the bracket, and the coaxial plate is used for arranging a standard probe and a probe to be calibrated; the coaxial plates can pitch and roll on the bracket. After the probe tube to be calibrated is calibrated by adopting the method, the error of the azimuth angle can be controlled within +/-1.5 degrees, and the errors of the inclination angle and the facing angle can be controlled within 0.2 degrees. Therefore, the method can effectively correct the error of the probe tube to be calibrated, and avoids the problems of poor timeliness and increased cost caused by the fact that the calibration equipment is limited and cannot be recalibrated on a coal mine site.

Description

Error correction device and method for inclinometer for coal mine
Technical Field
The invention relates to the technical field of coal mine inclinometers while drilling, in particular to an error correction device and method for a coal mine inclinometer.
Background
In coal mine safety production, a drilling method is usually adopted to realize exploration of hidden disaster-causing factors such as underground old dead zones, gas-enriched zones and the like, and how to judge whether the actual track of a drill hole drills according to a designed track is a key in the drilling method and can greatly influence the exploration effect. The mining inclinometer is used as measuring equipment, is widely applied to the field of underground drilling measurement of coal mines, and achieves a good effect. Due to the special use environment of the underground coal mine, most of the existing mine borehole inclinometers adopt a three-axis geomagnetic field sensor and a three-axis acceleration sensor to determine the posture of the inclinometer in a borehole. However, when the inclinometer is installed and used, the inclinometer is affected by various errors such as soft and hard ferromagnetic interference errors, installation errors, sensitivity errors, orthogonal errors and the like, and the errors can cause large deviation between measured data and real data, so that the measurement effect is affected. Therefore, the inclinometer needs to perform compensation correction on the error before being used so as to reduce the influence of the error on the measurement result.
At present, an error compensation method for an inclinometer mainly compensates an electronic compass in the inclinometer, and the adopted methods comprise least square fitting, an ellipse hypothesis method and the like, but the methods need a non-magnetic turntable with high precision to provide a space standard position for calibration. Before the inclinometer leaves a factory, calibration work can be completed on a high-precision nonmagnetic rotary table in a laboratory, the 0-degree azimuth line of the nonmagnetic rotary table needs to be consistent with the local magnetic north azimuth line, and the local magnetic north azimuth line needs to be determined by means of a high-precision measuring geomagnetic field measuring device. However, after the inclinometer probe is used for a period of time in a coal mine, the measurement error is increased due to the fact that the internal sensor drifts and the external magnetic field generates residual magnetization intensity, and recalibration is needed. This situation usually requires that the probe be returned to the manufacturer and recalibrated on a laboratory magnetic-free turntable, but this approach can result in delayed coal mine field work, reduced efficiency and increased cost.
Disclosure of Invention
Aiming at the defects in the prior art, the invention aims to provide an error correction device and method for a coal mine inclinometer, and solve the problem of poor timeliness caused by the fact that a high-precision non-magnetic rotary table cannot be used in a coal mine field in the prior art.
In order to solve the technical problems, the invention adopts the following technical scheme: an error correction device for a coal mine inclinometer comprises a base and a non-magnetic rotary table arranged on the base, wherein the non-magnetic rotary table comprises a support and a coaxial plate connected with the support, and the coaxial plate is used for arranging a standard probe and a probe to be calibrated;
the coaxial plates can pitch, roll or pitch and roll on the support.
The invention also has the following technical characteristics:
the support is of a T-shaped structure and comprises a first support rod and a second support rod vertically connected with the first support rod, and one end of the first support rod is connected with the middle part of the second support rod;
the first bracket rod is connected with the base through a bracket connecting rod, and the coaxial plate is connected to one end of the second bracket rod;
the first support rod can rotate along the axis of the first support rod, and the coaxial plate can rotate along the axis of the second support rod.
The coaxial plate is provided with a plurality of grooves parallel to the second support rod, the coaxial plate is also provided with a probe tube fastening screw and a probe tube fixing clamp, and the grooves, the probe tube fastening screw and the probe tube fixing clamp are jointly used for fixing a standard probe tube and a probe tube to be calibrated;
the coaxial plate is sleeved with the second support rod, and the position of the coaxial plate is fixed through fastening a screw.
The first support rod is sleeved with a support sleeve, the support sleeve is fixedly connected with the support connecting rod, and the first support rod can rotate in the support sleeve.
One end of the bracket sleeve is provided with a bracket rotating handle, and the other end of the bracket sleeve is provided with an inclination angle dial;
the bracket sleeve is provided with a bracket fastening screw which is used for fixing the relative position of the first bracket rod;
the balance weight is installed at the other end of the first support rod connected with the second support rod, and the balance weight is also installed at the other end of the second support rod connected with the coaxial plate.
The coaxial plate is provided with a coaxial plate rotating handle, and the second support rod is also provided with a face angle dial.
And a base horizontal adjusting screw is arranged on the base.
An error correction method for a coal mine inclinometer is carried out by adopting the device.
The method comprises the following steps:
step 1, adjusting a base horizontal adjusting screw on a base by using a level ruler, adjusting the base to be horizontal, fixing a standard probe and a probe to be calibrated in grooves on a coaxial plate by using a probe fastening screw and a probe fixing clamp respectively, connecting the standard probe and the probe to be calibrated with a computer through a communication cable, opening calibration software, and reading probe data in a static state;
step 2, loosening probe fastening screws on the probe to be calibrated, and rotating the probe to be calibrated to enable the initial facing angle gamma of the probe to be calibrated to be gamma0Initial face angle gamma to standard probe1The same;
step 3, rotating the bracket according to the given position, and recording the components H of the three-axis magnetic sensor of the standard probe tube 3 and the probe tube 4 to be calibrated at each position1=[Hx1,Hy1,Hz1],H0=[Hx0,Hy0,Hz0]Component G of a triaxial acceleration sensor1=[Gx1,Gy1,Gz1],G0=[Gx0,Gy0,Gz0];
The rotation position of the nonmagnetic turntable is given as follows:
in that
Figure BDA0003427018710000041
In the range, the inclination angle alpha of the calibration table is rotated for M times, the orientation angle gamma is rotated for N times under each inclination angle position, N groups of three-axis data are collected, and M.N groups of data are collected;
step 4, performing data fitting on the acquired data of the magnetic sensors and the data of the acceleration sensors of the standard probe and the probe to be calibrated to obtain a calibration matrix of the magnetic sensors and a calibration matrix of the acceleration sensors;
taking the magnetic sensor as an example, the data H1 collected by the magnetic sensor of the standard probe 3 is set to [ H ═ Hx1,Hy1,Hz1]As a three-axis magnetic component theoretical value of the probe 4 to be calibrated, and modeling the probe error to be calibrated, the formula is:
H1=Kh·H0+Bh (1)
in the formula H1Data vectors, H, are collected for the three-axis magnetic sensor of the standard probe 30Data vectors, K, are collected for a three-axis magnetic sensor of the probe 4 to be calibratedhIs a 3 × 3 matrix of calibration coefficients, BhIs a constant matrix. The coefficient matrix K can be solved by the least square methodhSum constant matrix BhThe optimal solution of (1).
Similarly, the acceleration sensor coefficient matrix K can be obtainedgAnd Bg
Step 5, writing the calculated calibration matrix of the magnetic sensor and the calibration matrix of the acceleration sensor into a memory in the probe 4 to be calibrated for storage, and calculating the calibrated inclination angle, azimuth angle and facing angle according to the formulas (1) to (3);
Figure BDA0003427018710000042
Figure BDA0003427018710000043
Figure BDA0003427018710000044
compared with the prior art, the invention has the following technical effects:
the device effectively solves the problems of poor timeliness and cost increase caused by the fact that the existing inclinometer for the coal mine is incapable of being recalibrated on site due to limited calibration equipment after leaving factory.
The method of the invention can effectively correct the error of the probe tube to be calibrated, and has the advantages of simple method and manpower and material resource saving.
Drawings
FIG. 1 is a schematic diagram of the present invention.
Fig. 2 is a schematic structural diagram of the present invention in a specific application.
FIG. 3 is a flow chart of a method for inclinometer error correction.
Fig. 4 is a comparison diagram of data before and after correction of a certain axis of the three-axis magnetic sensor.
Fig. 5 is a corrected rear azimuth error curve.
Fig. 6 is a corrected facing angle error curve.
FIG. 7 is a corrected caster error curve.
The various reference numbers in the drawings have the meanings given below: 1-base, 2-non-magnetic turntable, 3-stand, 4-coaxial plate, 5-first stand bar, 6-second stand bar, 7-stand connecting rod, 8-groove, 9-probe fastening screw, 10-probe fixing clip, 11-coaxial plate fastening screw, 12-stand sleeve, 13-stand turning handle, 14-inclination dial, 15-stand fastening screw, 16-counterweight, 17-coaxial plate turning handle, 18-facing angle dial, 19-base horizontal adjusting screw, 20-communication cable, 21-computer, 22-standard probe, 23-probe to be calibrated,
the present invention will be explained in further detail with reference to examples.
Detailed Description
The following embodiments of the present invention are provided, and it should be noted that the present invention is not limited to the following embodiments, and all equivalent changes based on the technical solutions of the present invention are within the protection scope of the present invention.
As used herein, the terms "upper," "lower," "front," "back," "top," "bottom," and the like are used in an orientation or positional relationship that is indicated for convenience in describing the invention and to simplify the description, but does not indicate or imply that the referenced devices or elements must be in a particular orientation, constructed and operative in a particular orientation, "inner" and "outer" refer to the inner and outer of the contours of the corresponding parts and are not to be construed as limiting the invention.
In the present invention, the terms "mounted," "connected," "fixed," and the like are used broadly, and may be, for example, fixedly connected, detachably connected, or integrated without being described to the contrary; can be mechanically or electrically connected; either directly or indirectly through intervening media, either internally or in any other relationship. The specific meanings of the above terms in the present invention can be understood by those skilled in the art according to specific situations.
All components in the present invention, unless otherwise specified, are all those known in the art.
Example 1:
according to the technical scheme, as shown in fig. 1 to 7, the error correction device for the inclinometer for the coal mine comprises a base 1 and a non-magnetic rotary table 2 arranged on the base 1, wherein the non-magnetic rotary table 2 comprises a bracket 3 and a coaxial plate 4 connected with the bracket 3, and the coaxial plate 4 is used for arranging a standard probe and a probe to be calibrated;
the coaxial plates 4 can pitch, roll or pitch and roll on the support 3.
The pitching means that the coaxial plate 4 rotates by taking the first support rod 5 as an axis, and the rolling means that the coaxial plate 4 rotates by taking the second support rod 6 as an axis.
As a preference of this embodiment:
the support 3 is of a T-shaped structure and comprises a first support rod 5 and a second support rod 6 vertically connected with the first support rod 5, and one end of the first support rod 5 is connected with the middle part of the second support rod 6;
the first bracket rod 5 is connected with the base 1 through a bracket connecting rod 7, and the coaxial plate 4 is connected with one end of the second bracket rod 6;
the first support rod 5 can rotate along the axis of the first support rod, and the coaxial plate 4 can rotate along the axis of the second support rod 6.
A plurality of grooves 8 parallel to the second support rod 6 are arranged on the coaxial plate 4, probe fastening screws 9 and probe fixing clamps 10 are further arranged on the coaxial plate 4, and the grooves 8, the probe fastening screws 9 and the probe fixing clamps 10 are jointly used for fixing a standard probe and a probe to be calibrated;
the coaxial plate 4 is sleeved with the second support rod 6 and is fixed in position through a coaxial plate fastening screw 11.
The first support rod 5 is sleeved with a support sleeve 12, the support sleeve 12 is fixedly connected with the support connecting rod 7, and the first support rod 5 can rotate in the support sleeve 12.
A bracket rotating handle 13 is arranged at one end of the bracket sleeve 12, and an inclination angle dial 14 is arranged at the other end of the bracket sleeve 12; after the bracket fastening screw 15 is loosened, the bracket rotating handle 13 is rotated, so that the bracket 3 can axially rotate 360 degrees around the bracket sleeve 12, the inclination angle of the coaxial plate 4 is changed, and the inclination angle rotation angle can be read through the inclination angle dial 14.
The bracket sleeve 12 is provided with a bracket fastening screw 15, and the bracket fastening screw 15 is used for fixing the relative position of the first bracket rod 5;
the other end of the first support rod 5 connected with the second support rod 6 is provided with a counterweight 16, and the other end of the second support rod 6 connected with the coaxial plate 4 is also provided with a counterweight 16. The counterweight 16 can be hooked with a counterweight according to the weight of the symmetrical end, so that the two ends are balanced. The counterweight can be adjusted along with the weight change of the symmetrical end, so that the support is kept from toppling.
The coaxial plate 4 is provided with a coaxial plate rotating handle 17, and the second support rod 6 is also provided with a facing angle dial 18. After the coaxial plate fastening screw 11 is loosened, the coaxial plate rotating handle 17 is rotated, so that the coaxial plate 4 can axially rotate 360 degrees around the second support rod 6, the facing angle of the coaxial plate 4 is changed, and the facing angle rotating angle can be read through the facing angle dial 18.
The base 1 is provided with a base horizontal adjusting screw 19.
The flow of the inclinometer error correction method provided by the invention is shown in figure 2, and the working method comprises the following steps: and respectively installing the calibrated standard probe 22 and the probe 23 to be calibrated on the coaxial plate 4, randomly rotating the coaxial plate 4 at a plurality of positions, and simultaneously acquiring data of the three-axis magnetic sensor and the acceleration sensor of the two probes. After data acquisition is finished, error correction coefficients of the data of the three-axis magnetic sensor and the data of the three-axis acceleration sensor of the probe to be calibrated are calculated through a least square fitting algorithm, and the coefficients are downloaded to the probe to be calibrated to be stored.
An error correction method for a coal mine inclinometer is carried out by adopting the device.
The method comprises the following steps:
step 1, adjusting a base horizontal adjusting screw 19 on a base 1 by using a level ruler, adjusting the base 1 to be horizontal, fixing a standard probe and a probe to be calibrated in a groove 8 on a coaxial plate 4 by using a probe fastening screw 9 and a probe fixing clamp 10 respectively, connecting the standard probe and the probe to be calibrated with a computer through a communication cable, opening calibration software, and reading probe data in a static state;
step 2, loosening the probe fastening screw 9 on the probe to be calibrated, and rotating the probe to be calibrated to enable the initial facing angle gamma of the probe to be calibrated0Initial face angle gamma to standard probe1The same;
step 3, rotating the bracket 2 according to the given position, and recording the components H of the three-axis magnetic sensor of the standard probe 3 and the probe 4 to be calibrated at each position1=[Hx1,Hy1,Hz1],H0=[Hx0,Hy0,Hz0]Component G of a triaxial acceleration sensor1=[Gx1,Gy1,Gz1],G0=[Gx0,Gy0,Gz0];
The rotation position of the nonmagnetic turntable is given as follows:
in that
Figure BDA0003427018710000081
In the range, the inclination angle alpha of the calibration table is rotated for M times, the orientation angle gamma is rotated for N times under each inclination angle position, N groups of three-axis data are collected, and M.N groups of data are collected;
step 4, performing data fitting on the acquired data of the magnetic sensors and the data of the acceleration sensors of the standard probe and the probe to be calibrated to obtain a calibration matrix of the magnetic sensors and a calibration matrix of the acceleration sensors;
taking the magnetic sensor as an example, the magnetic sensor of the standard probe 3 collects data H1=[Hx1,Hy1,Hz1]As a three-axis magnetic component theoretical value of the probe 4 to be calibrated, and modeling the probe error to be calibrated, the formula is:
H1=Kh·H0+Bh (1)
in the formula H1Data vectors, H, are collected for the three-axis magnetic sensor of the standard probe 30Data vectors, K, are collected for a three-axis magnetic sensor of the probe 4 to be calibratedhIs a 3 × 3 matrix of calibration coefficients, BhIs a constant matrix. The coefficient matrix K can be solved by the least square methodhSum constant matrix BhThe optimal solution of (1).
Similarly, the acceleration sensor coefficient matrix K can be obtainedgAnd Bg
Step 5, writing the calculated calibration matrix of the magnetic sensor and the calibration matrix of the acceleration sensor into a memory in the probe to be calibrated for storage, and calculating the calibrated inclination angle, azimuth angle and facing angle according to the formulas (1) to (3);
Figure BDA0003427018710000091
Figure BDA0003427018710000092
Figure BDA0003427018710000093
fig. 3 is a comparison graph of data before and after correction of a certain axis of the three-axis magnetic sensor, and fig. 4 to 6 are error curves between the corrected inclination angle, the corrected facing angle and the corrected azimuth angle of the probe to be calibrated and a given theoretical angle. It can be seen that after the probe to be calibrated is calibrated by adopting the invention, the error of the azimuth angle can be controlled within +/-1.5 degrees, and the errors of the inclination angle and the facing angle can be controlled within 0.2 degrees. Therefore, the method can effectively correct the error of the probe tube to be calibrated, and avoids the problems of poor timeliness and increased cost caused by the fact that the calibration equipment is limited and cannot be recalibrated on a coal mine site.
The above description is only a preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be made by those skilled in the art without inventive work within the technical scope of the present invention are included in the scope of the present invention.

Claims (8)

1. An inclinometer error correction device for a coal mine comprises a base (1) and a non-magnetic rotary table (2) arranged on the base (1), and is characterized in that the non-magnetic rotary table (2) comprises a bracket (3) and a coaxial plate (4) connected with the bracket (3), wherein the coaxial plate (4) is used for arranging a standard probe and a probe to be calibrated;
the coaxial plate (4) is pitching, rolling or pitching and rolling on the bracket (3).
2. The inclinometer error correction device for the coal mine according to claim 1, wherein the support (3) is of a T-shaped structure and comprises a first support rod (5) and a second support rod (6) vertically connected with the first support rod (5), one end of the first support rod (5) is connected with the middle part of the second support rod (6);
the first bracket rod (5) is connected with the base (1) through a bracket connecting rod (7), and the coaxial plate (4) is connected to one end of the second bracket rod (6);
the first support rod (5) can rotate along the axis of the first support rod, and the coaxial plate (4) can rotate along the axis of the second support rod (6).
3. The coal mine inclinometer error correction device according to claim 2, characterized in that a plurality of grooves (8) parallel to the second support rod (6) are arranged on the coaxial plate (4), a probe fastening screw (9) and a probe fixing clamp (10) are also arranged on the coaxial plate (4), and the grooves (8), the probe fastening screw (9) and the probe fixing clamp (10) are jointly used for fixing a standard probe and a probe to be calibrated;
the coaxial plate (4) is sleeved with the second support rod (6) and is fixed in position through a coaxial plate fastening screw (11).
4. The coal mine inclinometer error correction device as defined in claim 2, characterized in that a bracket sleeve (12) is sleeved on the first bracket rod (5), the bracket sleeve (12) is fixedly connected with the bracket connecting rod (7), and the first bracket rod (5) can rotate in the bracket sleeve (12).
5. The error correction device of the inclinometer for the coal mine according to claim 4, characterized in that one end of the bracket sleeve (12) is provided with a bracket rotating handle (13), and the other end of the bracket sleeve (12) is provided with an inclination angle dial (14);
the bracket sleeve (12) is provided with a bracket fastening screw (15), and the bracket fastening screw (15) is used for fixing the relative position of the first bracket rod (5);
the other end of the first support rod (5) connected with the second support rod (6) is provided with a balance weight (16), and the other end of the second support rod (6) connected with the coaxial plate (4) is also provided with the balance weight (16).
6. An error correction device for a coal mine inclinometer as defined in claim 3, characterized in that the coaxial plate (4) is provided with a coaxial plate rotating handle (17), and the second support rod (6) is provided with a facing angle dial (18).
7. The error correction device of the coal mine inclinometer as defined in claim 1, characterized in that a base horizontal adjusting screw (19) is installed on the base (1).
8. An error correction method for a coal mine inclinometer is characterized by comprising the following steps:
step 1, adjusting a base horizontal adjusting screw (19) on a base (1) by using a level ruler, adjusting the base (1) to be horizontal, fixing a standard probe and a probe to be calibrated in a groove (8) on a coaxial plate (4) by using a probe fastening screw (9) and a probe fixing clamp (10) respectively, connecting the standard probe and the probe to be calibrated with a computer through a communication cable, opening calibration software, and reading probe data in a static state;
step 2, loosening a probe fastening screw (9) on the probe to be calibrated, and rotating the probe to be calibrated to enable the initial facing angle gamma of the probe to be calibrated to be gamma0Initial face angle gamma to standard probe1The same;
step 3, rotating the support (2) according to the given position, and recording the component H of the three-axis magnetic sensor of the standard probe tube 3 and the probe tube 4 to be calibrated at each position1=[Hx1,Hy1,Hz1],H0=[Hx0,Hy0,Hz0]Component G of a triaxial acceleration sensor1=[Gx1,Gy1,Gz1],G0=[Gx0,Gy0,Gz0];
The rotation position of the nonmagnetic turntable is given as follows:
in that
Figure FDA0003427018700000031
In the range, the inclination angle alpha of the calibration table is rotated for M times, the orientation angle gamma is rotated for N times under each inclination angle position, N groups of three-axis data are collected, and M.N groups of data are collected;
step 4, performing data fitting on the acquired data of the magnetic sensors and the data of the acceleration sensors of the standard probe and the probe to be calibrated to obtain a calibration matrix of the magnetic sensors and a calibration matrix of the acceleration sensors;
step 5, writing the calculated calibration matrix of the magnetic sensor and the calibration matrix of the acceleration sensor into a memory in the probe to be calibrated for storage, and calculating the calibrated inclination angle, azimuth angle and facing angle according to the formulas (1) to (3);
Figure FDA0003427018700000032
Figure FDA0003427018700000033
Figure FDA0003427018700000034
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