CN107588715A - A kind of locus detection means and measuring method based on magnetic effect - Google Patents

A kind of locus detection means and measuring method based on magnetic effect Download PDF

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CN107588715A
CN107588715A CN201711034255.XA CN201711034255A CN107588715A CN 107588715 A CN107588715 A CN 107588715A CN 201711034255 A CN201711034255 A CN 201711034255A CN 107588715 A CN107588715 A CN 107588715A
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CN107588715B (en
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路翼畅
胡鹏浩
陈时毅
廖普
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Hefei University of Technology
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Hefei University of Technology
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    • Y—GENERAL 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
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Abstract

The invention discloses a kind of locus detection means and measuring method based on magnetic effect, tested part is placed in horizontal operation platform top surface, each magnetic effect sensor is arranged in the side of workbench two using horizontal stand, measurement probe pen is that in the permanent magnet of ball-type, the front end of probe is probe for fixed setting between penholder and probe；Measured position Dian Chu of the probe touching in tested part of probe pen will be measured, the magnetic induction intensity for obtaining the magnetic field that permanent magnet is formed in space is detected by magnetic effect sensor, the magnetic induction intensity detected value exported by multiple magnetic effect sensors being arranged on diverse location calculates the angles and positions relation obtained between permanent magnet and each magnetic effect sensor, so as to the position relationship popped one's head between each magnetic effect sensor, realize that locus is detected.The present invention can be achieved without guide rail, without tape measure, require low to measuring environment, easy to operate.

Description

A kind of locus detection means and measuring method based on magnetic effect
Technical field
The present invention relates to a kind of magnetic effect device, more specifically a kind of achievable locus inspection based on magnetic effect The measuring method of survey.Size and Form and position error detection suitable for non-magnetic component of machine.
Background technology
Three-dimensional coordinate measurement be to the surface basic size of part and the measurement of Form and position error, be widely used in automobile, The fields such as machinery, Aero-Space, military project, medical science.Existing method for three-dimensional measurement is summarized as three classes, first, tested part is placed In 3 d space coordinate system, if doing on detection piece surface, obtains its coordinate value, traditional right angle three coordinate measuring machine And the flexible joint coordinate measuring machine to reach its maturity at present is based on this method, its measuring speed is slow, requires high to measuring environment； Two are built upon the laser scanner technique in indoor GPS technical foundation, its high efficiency, high accuracy, but hardware cost requires high；Three It is the detection method based on machine vision, same point is detected from different angles, determines each point data on part, it is identical The stereo matching problem of point is the key technology in stereoscopic vision, and this method is low to hardware requirement, but operating personnel's technology is contained Amount requires higher.
The light pen measuring instrument based on laser scanner technique that Mai Zhuonuo companies propose accurately is inlayed on hand-held light pen Relative position between several LED, each LED is, it is known that determining light pen according to change in location of the LED on camera sensor Pose, so as to accurately calculate the coordinate of light pen bottom measured point.Its a whole set of measurement apparatus is light and handy, in being applied to, large scale The accurate measurement of part, measurement accuracy is medium, but in actual use, if the light that LED is sent on light pen by barrier or by Survey part to block in itself, complete measurement can not be realized, while cost is higher.
The iGPS e measurement technologies of Boeing manufacturing company of U.S. research are applied to the measurement problem of aircraft large-size components. Four infrared transmitters are fixed on each angle of measured zone in the measuring system, and receiver is placed on part, and receiver calculates The vertically and horizontally angle for going out relative transmission device can determine that the position coordinates of measured point.The device is parallel measurement, can be simultaneously Multiple targets are detected, each receiver can independently read vertically and horizontally angle, but its measurement point is influenceed by receiver, it is impossible to Detect at part defect, the measurement of deep hole, groove class part can not be realized.
The research team of Hebei University of Science and Technology proposes the method for being combined traditional three coordinate measuring machine with machine vision To detect the size of part and Form and position error, the X of three coordinate measuring machine, Y-axis drive video camera movement, distinguished at diverse location Tested part is shot, each characteristic point is obtained on tested part in camera coordinate system using corresponding image points matching method Three-dimensional coordinate, and then by the scale data of traditional three coordinates, three-dimensional appearance is carried out to surveyed part and rebuilds simultaneously evaluating parts Pattern error.Because traditional three coordinate measuring machine has the guide rail and scale structure of complexity, a whole set of measurement apparatus volume is big, causes Measurement period is grown, and is unfavorable for assembling measurement in real time.
The content of the invention
The present invention is to avoid the weak point present in above-mentioned prior art, there is provided a kind of intelligence based on magnetic effect is sat Method for three-dimensional measurement in mapping amount, can be achieved without guide rail, without tape measure, it is easy to operate, to measuring environment require it is low.
The present invention adopts the following technical scheme that to solve technical problem：
Locus detection means of the invention based on magnetic effect is structurally characterized in that：
Be provided for placement tested part is in horizontal workbench, using horizontal stand in two sides of workbench Arrange each magnetic effect sensor；
Measurement probe pen is set, is that the permanent magnet in ball-type, the front end of the probe are fixedly installed between penholder and probe For probe；
Tested part is placed on the top surface of the workbench, measures measured position of the probe touching in tested part of probe pen At putting, the magnetic induction intensity detected value work in the magnetic field that the permanent magnet obtained is formed in space is detected with the magnetic effect sensor For the detection output signal of detection means.
The design feature of locus detection means of the invention based on magnetic effect is lain also in：The permanent magnet is in axis side Uniformly magnetize upwards, magnetizing direction is along the axis direction of the penholder, and the penholder is coaxial with probe, and probe is in probe Axial location on.
The design feature of locus detection means of the invention based on magnetic effect is lain also in：The magnetic effect sensor is total to There are six, it is symmetrical and be distributed on two sides of workbench.
The measuring method of the locus detection based on magnetic effect is realized using spatial location detection means of the present invention Feature is：The magnetic induction intensity detected value exported using the multiple magnetic effect sensors being arranged on diverse location, which is calculated, obtains institute The angles and positions relation between permanent magnet and each magnetic effect sensor is stated, so as to be popped one's head between each magnetic effect sensor Position relationship, realize measured position point locus detection.
The characteristics of measuring method of the present invention, lies also in：
The axial distance of the centre of sphere of the permanent magnet and the probe is known；It is and known：Adjacent two magnetic in same side The spacing of effect sensor is a, and the spacing of two magnetic effect sensors on two side correspondence positions is b；
To be in the magnetic effect sensor S at the endpoint location of a horizontal stand1Inspection center be origin, with parallel In horizontal stand direction be XωAxle, using the in-plane perpendicular to workbench as ZωAxle, Y is determined by the right-hand ruleωAxle, establish Fixed coordinate system Oω-XωYωZω；
When the axis for measuring the probe in probe pen is perpendicular to the plane of workbench, established by origin of the centre of sphere of permanent magnet Basis coordinates system O-XYZ, the magnetizing direction of permanent magnet are basis coordinates Z axis, X of the basis coordinates X-axis parallel to fixed coordinate systemωAxle, by The right-hand rule determines basis coordinates Y-axis；The sphere center position of holding permanent magnet is constant, and changing the spatial attitude of probe makes the axis of probe Plane with workbench is in the angle of a setting, is considered as permanent magnet and α angles are first turned about the X axis under basis coordinates system, further around Y-axis Rotate β angles；With the change of the spatial attitude of probe, it is rotating coordinate system that corresponding postrotational coordinate system, which occurs, for basis coordinates system O-X′Y′Z′；
The measuring method is to carry out as follows：
Step 1：Set magnetic effect sensor S1Coordinate under basis coordinates system is S1(x0,y0,z0), under rotating coordinate system Coordinate be S1(u, v, w), coordinate transformation relation such as formula (1)：
Step 2：Establish Equivalent Magnetic Charge model
Permanent magnet was utilized the horizontal plane of the centre of sphere of permanent magnet be divided into episphere and lower semisphere, be in above horizontal plane Permanent magnet portion be episphere, it is lower semisphere to be in the permanent magnet portion below horizontal plane；
Permanent magnet upper half ball surface optional position set P points, under rotating coordinate system the coordinate of P points be P (x ', y ', z+'), cross P points and one straight line PQ of work parallel with the magnetizing direction of permanent magnet, straight line PQ and the lower semisphere of permanent magnet surface phase Q points are met at, then coordinate of the Q points under rotating coordinate system is Q (x ', y ', z-'), magnetic effect sensor S1Magnetic strength at position Answer intensity B ' expression formula such as formula (2)：
In formula (2), ds is area element, S+And S-To be accumulated the source area of curved surface, the respectively positive and negative magnetic charge of permanent magnet, It is respectively the episphere of permanent magnet and the surface area of lower semisphere；
μrFor the relative permeability of the Magnetized Material of permanent magnet；BrFor the remanent magnetism parameter of permanent magnet；
Correspond as along X ' axles, Y ' axles, the unit vector of Z ' axles；
Step 3：Obtained using formula (2) under rotating coordinate system, magnetic effect sensor S1Along X ' axles, Y ' axles at position With one-to-one magnetic induction intensity component B in Z ' axle all directionsx', By' and Bz' expression formula be respectively：
R is permanent magnet radius；By product curved surface S+And S-The expression formula of projection in X ' OY ' planes is identical, is designated as integrating Region Dx’y’, its expression formula is (x')2+(y')2=r2；
Step 4：Obtained using formula (1) under basis coordinates system along the magnetic induction intensity component B of each reference axis of X, Y, Zx、By、BzPoint It is not：
Step 5：Magnetic effect sensor S is set1Detection direction be vertical direction, then magnetic effect sensor S1Detection obtains Magnetic induction intensityAs：Thus magnetic effect sensor S is obtained1Detect the magnetic induction intensity obtainedRotating On u, v, w, α, β expression formula under coordinate system；
Magnetic effect sensor S is obtained by formula (1)1On x under basis coordinates system0、y0、z0, α and β expression formula such as formula (3)：
Step 6：By step 1- steps 5, each expression formula of magnetic effect sensor under basis coordinates system is obtained, and establish Equation group as shown in formula (4)：
In formula (4),Correspond defeated for the magnetic induction of each magnetic effect sensor Go out signal, the equation group for solving formula (4) obtains x0、y0、z0, α and β value, with the x0、y0、z0, α and β value characterize and obtain magnetic Effect sensor S1Relative to the relative position and relative angle of permanent magnet；
Step 7：Probe and the distance at the center of permanent magnet are L, and the coordinate popped one's head under rotating coordinate system is (0,0 ,-L), It is (- L sin β, L sin α sin β ,-Lcos α cos β) to obtain coordinate value of the probe under basis coordinates system according to formula (3), then pops one's head in In fixed coordinate system Oω-XωYωZωUnder coordinate value be (- Lsin β-x0,Lsinαsinβ-y0,-Lcosαcos-z0), complete inspection Survey.
Compared with the prior art, the present invention has the beneficial effect that：
1st, the present invention realizes no guide rail, the three-dimensional inspection in the locus without scale compared to traditional three coordinate measuring machine Survey, the hand-holdable probe pen of survey crew causes probe contacts with measured point to measure in any way, and operation requires low, measurement side Just；
2nd, the present invention is sensed using magnetic effect, compared to optical means, when effectively overcoming part and actually using Interference and influence of the outside environmental elements on measurement accuracy, and avoid barrier and part in itself block caused inconvenience；
3rd, due to using multiple magnetic effect sensors measurement in a closed series obtain redundancy technical characteristic, measurement accuracy and Resolution ratio has been lifted, while can be according to the size and remanent magnetism parameter for being actually needed configuration permanent magnet with range extension；
4th, due to probe pen probe can design length and shape according to the actual requirements, therefore zanjon, hole slot can be realized The measurement of class part；
5th, the present invention it is simple in construction, cost is low, can be achieved in real time measurement, be adapted to non-magnetic component of machine size and Form and position error detects, and such as detection of automobile industry mould, has certain promotional value.
Brief description of the drawings
Fig. 1 is dimensional structure diagram of the present invention；
Fig. 2 is Coordinate Conversion schematic diagram；
Fig. 3 is sensing station relation schematic diagram；
Label in figure：1 horizontal stand；2 magnetic effect sensors；3 workbenches；4 penholders；5 permanent magnets；6 probes；7 probes； 8 tested parts.
Embodiment
Referring to Fig. 1 and Fig. 3, the structure type of the locus detection means based on magnetic effect is in the present embodiment：
Be provided for placement tested part 8 is in horizontal workbench 3, using horizontal stand 1 the two of workbench 3 Each magnetic effect sensor 2 is arranged in side；Measurement probe pen is set, is the permanent magnetism that fixed setting is in ball-type between penholder 4 and probe 6 Body 5, the front end of probe 6 is probe 7；Tested part 8 is placed on the top surface of workbench 3, and the probe 7 for measuring probe pen is touched in quilt The measured position Dian Chu of part 8 is surveyed, the magnetic strength in the magnetic field that the permanent magnet 5 obtained is formed in space is detected with magnetic effect sensor 2 Answer detection output signal of the intensity detection value as detection means.
In specific implementation, permanent magnet 5 uniformly magnetizes in the axial direction, and magnetizing direction is along the axis direction of penholder 4, pen Bar 4 is coaxial with probe 6, and probe 7 is in the axial location of probe 6；Magnetic effect sensor 2 shares six, symmetrical and uniform In two sides of workbench 3.
The measuring method of locus detection is realized using the locus detection means based on magnetic effect in the present embodiment It is：The magnetic induction intensity detected value exported using the multiple magnetic effect sensors 2 being arranged on diverse location, which is calculated, obtains permanent magnetism Angles and positions relation between body 5 and each magnetic effect sensor 2, so as to obtain between probe 7 and each magnetic effect sensor 2 Position relationship, realize the locus detection of measured position point.
The measuring method of the locus detection means based on magnetic effect is to carry out as follows in the present embodiment：
The axial distance of the centre of sphere of permanent magnet 5 and probe 7 is known；It is and known：Adjacent two magnetic effects in same side pass The spacing of sensor 2 is a, and the spacing of two magnetic effect sensors 2 on two side correspondence positions is b；
To be in the magnetic effect sensor S at the endpoint location of a horizontal stand1Inspection center be origin, with parallel In horizontal stand direction be XωAxle, using the in-plane perpendicular to workbench 3 as ZωAxle, Y is determined by the right-hand ruleωAxle, build Vertical fixed coordinate system Oω-XωYωZω。
As shown in Fig. 2 work as plane of the axis perpendicular to workbench 3 of the probe 6 in measurement probe pen, with permanent magnet 5 The centre of sphere is that origin establishes basis coordinates system O-XYZ, and the magnetizing direction of permanent magnet 5 is basis coordinates Z axis, and basis coordinates X-axis is parallel to fixation The X of coordinate systemωAxle, basis coordinates Y-axis is determined by the right-hand rule；Keep the sphere center position of permanent magnet 5 constant, change the sky of probe 6 Between posture make the angle that the plane of the axis of probe 6 and workbench 3 is in a setting, be considered as permanent magnet 5 under basis coordinates system first α angles are turned about the X axis, β angles are rotated further around Y-axis；With the change of the spatial attitude of probe 6, corresponding rotation occurs for basis coordinates system Coordinate system after turning is rotating coordinate system O-X ' Y ' Z '.
Detection method is carried out as follows：
Step 1：Set magnetic effect sensor S1Coordinate under basis coordinates system is S1(x0,y0,z0), under rotating coordinate system Coordinate be S1(u, v, w), coordinate transformation relation such as formula (1)：
Step 2：Establish Equivalent Magnetic Charge model
Permanent magnet 5 was utilized the horizontal plane of the centre of sphere of permanent magnet 5 be divided into episphere and lower semisphere, place is in the horizontal plane The permanent magnet portion of side is episphere, and it is lower semisphere to be in the permanent magnet portion below horizontal plane.
Permanent magnet 5 upper half ball surface optional position set P points, under rotating coordinate system the coordinate of P points be P (x ', y′,z+'), mistake P points and work one straight line PQ, straight line PQ parallel with the magnetizing direction of permanent magnet 5 and the lower semisphere of permanent magnet 5 table Face intersects at Q points, then coordinate of the Q points under rotating coordinate system is Q (x ', y ', z-'), magnetic effect sensor S1At position Magnetic induction density B ' expression formula such as formula (2)：
In formula (2), ds is area element, S+And S-To be accumulated the source area of curved surface, the respectively positive and negative magnetic charge of permanent magnet, It is respectively the episphere of permanent magnet (5) and the surface area of lower semisphere；
μrFor the relative permeability of the Magnetized Material of permanent magnet 5；BrFor the remanent magnetism parameter of permanent magnet 5；
Correspond as along X ' axles, Y ' axles, the unit vector of Z ' axles.
Step 3：Obtained using formula (2) under rotating coordinate system, magnetic effect sensor S1Along X ' axles, Y ' axles at position With one-to-one magnetic induction intensity component B in Z ' axle all directionsx', By' and Bz' expression formula be respectively：
R is permanent magnet radius；By product curved surface S+And S-The expression formula of projection in X ' OY ' planes is identical, is designated as integrating Region Dx’y’, its expression formula is (x')2+(y')2=r2。
Magnetic induction intensity component Bx', By' and Bz' by magnetic induction density B ' to the calculation of curved surface integral of area, calculate Method is as follows：
The sphere of permanent magnet is smooth surface, and its expression formula is：(x')2+(y')2+(z')2=r2
By product curved surface S+It is expressed as：
By product curved surface S-It is expressed as：
z’+And z '-The respectively ball surface coordinate expressions of ball-type permanent magnet episphere and lower semisphere；
By product curved surface S+And S-The expression formula of projection in X ' OY ' planes is that integral domain is identical, is designated as Dx’y’,
Dx’y’Expression formula be：(x')2+(y')2=r2；Therefore area element ds is expressed as：
z’+And z '-Derivation to x ' square expression formula it is identical, be [(z 'x’)’]2；
z’+And z '-Derivation to y ' square expression formula it is identical, be [(z 'y’)’]2；
Area element ds is substituted into formula (2) and obtains Bx', By' and Bz' expression formula.
Step 4：Obtained using formula (1) under basis coordinates system along the magnetic induction intensity component B of each reference axis of X, Y, Zx、By、BzPoint It is not：
Step 5：Because magnetic effect sensor can only detect the magnetic induction intensity in a direction, and with the sensing element of sensor Placement it is relevant, magnetic effect sensor S is set in the present embodiment1Detection direction be vertical direction, then magnetic effect sensor S1Inspection Survey the magnetic induction intensity obtainedAs：Thus magnetic effect sensor S is obtained1Detect the magnetic induction intensity obtained On u, v, w, α, β expression formula under rotating coordinate system；
Magnetic effect sensor S is obtained by formula (1)1On x under basis coordinates system0、y0、z0, α and β expression formula such as formula (3)：
Step 6：By step 1- steps 5, each expression formula of magnetic effect sensor under basis coordinates system is obtained, and establish Equation group as shown in formula (4)：
In formula (4),Correspond defeated for the magnetic induction of each magnetic effect sensor Go out signal, the equation group for solving formula (4) obtains x0、y0、z0, α and β value, with the x0、y0、z0, α and β value characterize and obtain magnetic Effect sensor S1Relative to the relative position and relative angle of permanent magnet.
Only five sensors are needed to obtain five unknown quantity x in the present invention0、y0、z0, α and β value, and actually place biography Sensor is six, the number placed more than theoretical needs, reading existence information redundancy, and the number of equation is individual more than unknown quantity Number, error can be reduced, improve precision.
Step 7：Probe 7 and the distance at the center of permanent magnet 5 be L, 7 coordinate under rotating coordinate system of probe be (0,0 ,- L), it is (- L sin β, L sin α sin β ,-Lcos α cos β) to obtain 7 coordinate value under basis coordinates system of probe according to formula (3), then Probe 7 is in fixed coordinate system Oω-XωYωZωUnder coordinate value be (- Lsin β-x0,Lsinαsinβ-y0,-Lcosαcos-z0), it is complete Into detection.

Claims (5)

1. a kind of locus detection means based on magnetic effect, it is characterized in that：
Be provided for placement tested part (8) is in horizontal workbench (3), using horizontal stand (1) in workbench (3) Two sides arrange each magnetic effect sensor (2)；
Measurement probe pen is set, is that the permanent magnet (5) in ball-type, the probe are fixedly installed between penholder (4) and probe (6) (6) front end is probe (7)；
Tested part (8) is placed on the top surface of the workbench (3), and the probe (7) for measuring probe pen is touched in tested part (8) Measured position Dian Chu, the magnetic strength in magnetic field that the permanent magnet (5) obtained formed in space is detected with the magnetic effect sensor (2) Answer detection output signal of the intensity detection value as detection means.
2. the locus detection means according to claim 1 based on magnetic effect, it is characterized in that：The permanent magnet (5) Uniformly magnetize in the axial direction, magnetizing direction is to be with probe (6) along the axis direction of the penholder (4), the penholder (4) Coaxial, probe (7) is in the axial location of probe (6).
3. the locus detection means according to claim 1 or 2 based on magnetic effect, it is characterized in that：The magnetic effect Sensor (2) shares six, symmetrical and be distributed on two sides of workbench (3).
4. the measurement side of locus detection is realized using the locus detection means based on magnetic effect described in claim 1 Method, it is characterized in that：The magnetic induction intensity detected value meter exported using the multiple magnetic effect sensors (2) being arranged on diverse location Calculate the angles and positions relation between the permanent magnet (5) and each magnetic effect sensor (2) of obtaining, so as to popped one's head in (7) and Position relationship between each magnetic effect sensor (2), realize the locus detection of measured position point.
5. measuring method according to claim 4, it is characterized in that：
The centre of sphere of the permanent magnet (5) and the axial distance of the probe (7) are known；It is and known：Same side is adjacent two The spacing of magnetic effect sensor (2) is a, and the spacing of two magnetic effect sensors (2) on two side correspondence positions is b；
To be in the magnetic effect sensor S at the endpoint location of a horizontal stand1Inspection center be origin, with parallel to water Flat holder orientation is XωAxle, using the in-plane perpendicular to workbench (3) as ZωAxle, Y is determined by the right-hand ruleωAxle, establish Fixed coordinate system Oω-XωYωZω；
When the axis for measuring the probe (6) in probe pen is perpendicular to the plane of workbench (3), using the centre of sphere of permanent magnet (5) as original Point establishes basis coordinates system O-XYZ, and the magnetizing direction of permanent magnet (5) is basis coordinates Z axis, and basis coordinates X-axis is parallel to fixed coordinate system XωAxle, basis coordinates Y-axis is determined by the right-hand rule；Keep the sphere center position of permanent magnet (5) constant, change the space of probe (6) Posture makes angle of the axis of probe (6) with the plane of workbench (3) in a setting, is considered as permanent magnet (5) in basis coordinates system It is lower first to turn about the X axis α angles, rotate β angles further around Y-axis；With the change of the spatial attitude of probe (6), basis coordinates system occurs Corresponding postrotational coordinate system is rotating coordinate system O-X ' Y ' Z '；
The detection method is carried out as follows：
Step 1：Set magnetic effect sensor S1Coordinate under basis coordinates system is S1(x0,y0,z0), the seat under rotating coordinate system It is designated as S1(u, v, w), coordinate transformation relation such as formula (1)：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>u</mi> <mo>=</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;beta;</mi> <mo>+</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;alpha;</mi> <mo>&amp;CenterDot;</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;beta;</mi> <mo>-</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;alpha;</mi> <mo>&amp;CenterDot;</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;beta;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>v</mi> <mo>=</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;alpha;</mi> <mo>+</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;alpha;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>w</mi> <mo>=</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;beta;</mi> <mo>-</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mi>sin</mi> <mi>&amp;alpha;</mi> <mo>&amp;CenterDot;</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;beta;</mi> <mo>+</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;alpha;</mi> <mo>&amp;CenterDot;</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;beta;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
Step 2：Establish Equivalent Magnetic Charge model
Permanent magnet (5) was utilized the horizontal plane of the centre of sphere of permanent magnet (5) be divided into episphere and lower semisphere, place is in the horizontal plane The permanent magnet portion of side is episphere, and it is lower semisphere to be in the permanent magnet portion below horizontal plane；
Permanent magnet (5) upper half ball surface optional position set P points, under rotating coordinate system the coordinate of P points be P (x ', y ', z+'), mistake P points and work one straight line PQ, straight line PQ parallel with the magnetizing direction of permanent magnet (5) and the lower semisphere of permanent magnet (5) Surface intersects at Q points, then coordinate of the Q points under rotating coordinate system is Q (x ', y ', z-'), magnetic effect sensor S1At position Magnetic induction density B ' expression formula such as formula (2)：
<mrow> <mtable> <mtr> <mtd> <mrow> <msup> <mi>B</mi> <mo>&amp;prime;</mo> </msup> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;mu;</mi> <mi>r</mi> </msub> <msub> <mi>B</mi> <mi>r</mi> </msub> </mrow> <mrow> <mn>4</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <munder> <mrow> <mo>&amp;Integral;</mo> <mo>&amp;Integral;</mo> </mrow> <msub> <mi>S</mi> <mo>+</mo> </msub> </munder> <mfrac> <mrow> <mo>(</mo> <mi>u</mi> <mo>-</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> <msup> <mover> <mi>i</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>+</mo> <mo>(</mo> <mi>v</mi> <mo>-</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> <msup> <mover> <mi>j</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>+</mo> <mo>(</mo> <mi>w</mi> <mo>-</mo> <msup> <msub> <mi>z</mi> <mo>+</mo> </msub> <mo>&amp;prime;</mo> </msup> <mo>)</mo> <msup> <mover> <mi>k</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>&amp;prime;</mo> </msup> </mrow> <msup> <mrow> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>u</mi> <mo>-</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>v</mi> <mo>-</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>w</mi> <mo>-</mo> <msup> <msub> <mi>z</mi> <mo>+</mo> </msub> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </msup> </mfrac> <mi>d</mi> <mi>s</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mrow> <msub> <mi>&amp;mu;</mi> <mi>r</mi> </msub> <msub> <mi>B</mi> <mi>r</mi> </msub> </mrow> <mrow> <mn>4</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <munder> <mrow> <mo>&amp;Integral;</mo> <mo>&amp;Integral;</mo> </mrow> <msub> <mi>S</mi> <mo>-</mo> </msub> </munder> <mfrac> <mrow> <mo>(</mo> <mi>u</mi> <mo>-</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> <msup> <mover> <mi>i</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>+</mo> <mo>(</mo> <mi>v</mi> <mo>-</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> <msup> <mover> <mi>j</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>&amp;prime;</mo> </msup> <mo>+</mo> <mo>(</mo> <mi>w</mi> <mo>-</mo> <msup> <msub> <mi>z</mi> <mo>-</mo> </msub> <mo>&amp;prime;</mo> </msup> <mo>)</mo> <msup> <mover> <mi>k</mi> <mo>&amp;RightArrow;</mo> </mover> <mo>&amp;prime;</mo> </msup> </mrow> <msup> <mrow> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>u</mi> <mo>-</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>v</mi> <mo>-</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>w</mi> <mo>-</mo> <msup> <msub> <mi>z</mi> <mo>-</mo> </msub> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </msup> </mfrac> <mi>d</mi> <mi>s</mi> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
In formula (2), ds is area element, S+And S-To be accumulated the source area of curved surface, the respectively positive and negative magnetic charge of permanent magnet, that is, divide Wei not the episphere of permanent magnet (5) and the surface area of lower semisphere；
μrFor the relative permeability of the Magnetized Material of permanent magnet (5)；BrFor the remanent magnetism parameter of permanent magnet (5)；
Correspond as along X ' axles, Y ' axles, the unit vector of Z ' axles；
Step 3：Obtained using formula (2) under rotating coordinate system, magnetic effect sensor S1Along X ' axles, Y ' axles and Z ' at position One-to-one magnetic induction intensity component B in axle all directionsx', By' and Bz' expression formula be respectively：
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msup> <msub> <mi>B</mi> <mi>x</mi> </msub> <mo>&amp;prime;</mo> </msup> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;mu;</mi> <mi>r</mi> </msub> <msub> <mi>B</mi> <mi>r</mi> </msub> </mrow> <mrow> <mn>4</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <munder> <mrow> <mo>&amp;Integral;</mo> <mo>&amp;Integral;</mo> </mrow> <msub> <mi>D</mi> <mrow> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> </mrow> </msub> </munder> <mfrac> <mrow> <mi>u</mi> <mo>-</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> </mrow> <msup> <mrow> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>u</mi> <mo>-</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>v</mi> <mo>-</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>w</mi> <mo>-</mo> <msqrt> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </msup> </mfrac> <mo>&amp;CenterDot;</mo> <mfrac> <mi>r</mi> <msqrt> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <msup> <mi>dx</mi> <mo>&amp;prime;</mo> </msup> <msup> <mi>dy</mi> <mo>&amp;prime;</mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mrow> <msub> <mi>&amp;mu;</mi> <mi>r</mi> </msub> <msub> <mi>B</mi> <mi>r</mi> </msub> </mrow> <mrow> <mn>4</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <munder> <mrow> <mo>&amp;Integral;</mo> <mo>&amp;Integral;</mo> </mrow> <msub> <mi>D</mi> <mrow> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> </mrow> </msub> </munder> <mfrac> <mrow> <mi>u</mi> <mo>-</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> </mrow> <msup> <mrow> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>u</mi> <mo>-</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>v</mi> <mo>-</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>w</mi> <mo>+</mo> <msqrt> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </msup> </mfrac> <mo>&amp;CenterDot;</mo> <mfrac> <mi>r</mi> <msqrt> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <msup> <mi>dx</mi> <mo>&amp;prime;</mo> </msup> <msup> <mi>dy</mi> <mo>&amp;prime;</mo> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msup> <msub> <mi>B</mi> <mi>y</mi> </msub> <mo>&amp;prime;</mo> </msup> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;mu;</mi> <mi>r</mi> </msub> <msub> <mi>B</mi> <mi>r</mi> </msub> </mrow> <mrow> <mn>4</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <munder> <mrow> <mo>&amp;Integral;</mo> <mo>&amp;Integral;</mo> </mrow> <msub> <mi>D</mi> <mrow> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> </mrow> </msub> </munder> <mfrac> <mrow> <mi>v</mi> <mo>-</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> </mrow> <msup> <mrow> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>u</mi> <mo>-</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>v</mi> <mo>-</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>w</mi> <mo>-</mo> <msqrt> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </msup> </mfrac> <mo>&amp;CenterDot;</mo> <mfrac> <mi>r</mi> <msqrt> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <msup> <mi>dx</mi> <mo>&amp;prime;</mo> </msup> <msup> <mi>dy</mi> <mo>&amp;prime;</mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mrow> <msub> <mi>&amp;mu;</mi> <mi>r</mi> </msub> <msub> <mi>B</mi> <mi>r</mi> </msub> </mrow> <mrow> <mn>4</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <munder> <mrow> <mo>&amp;Integral;</mo> <mo>&amp;Integral;</mo> </mrow> <msub> <mi>D</mi> <mrow> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> </mrow> </msub> </munder> <mfrac> <mrow> <mi>v</mi> <mo>-</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> </mrow> <msup> <mrow> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>u</mi> <mo>-</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>v</mi> <mo>-</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>w</mi> <mo>+</mo> <msqrt> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </msup> </mfrac> <mo>&amp;CenterDot;</mo> <mfrac> <mi>r</mi> <msqrt> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <msup> <mi>dx</mi> <mo>&amp;prime;</mo> </msup> <msup> <mi>dy</mi> <mo>&amp;prime;</mo> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced>
<mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msup> <msub> <mi>B</mi> <mi>z</mi> </msub> <mo>&amp;prime;</mo> </msup> <mo>=</mo> <mfrac> <mrow> <msub> <mi>&amp;mu;</mi> <mi>r</mi> </msub> <msub> <mi>B</mi> <mi>r</mi> </msub> </mrow> <mrow> <mn>4</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <munder> <mrow> <mo>&amp;Integral;</mo> <mo>&amp;Integral;</mo> </mrow> <msub> <mi>D</mi> <mrow> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> </mrow> </msub> </munder> <mfrac> <mrow> <mi>w</mi> <mo>-</mo> <msqrt> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <msup> <mrow> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>u</mi> <mo>-</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>v</mi> <mo>-</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>w</mi> <mo>-</mo> <msqrt> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </msup> </mfrac> <mo>&amp;CenterDot;</mo> <mfrac> <mi>r</mi> <msqrt> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <msup> <mi>dx</mi> <mo>&amp;prime;</mo> </msup> <msup> <mi>dy</mi> <mo>&amp;prime;</mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mrow> <msub> <mi>&amp;mu;</mi> <mi>r</mi> </msub> <msub> <mi>B</mi> <mi>r</mi> </msub> </mrow> <mrow> <mn>4</mn> <mi>&amp;pi;</mi> </mrow> </mfrac> <munder> <mrow> <mo>&amp;Integral;</mo> <mo>&amp;Integral;</mo> </mrow> <msub> <mi>D</mi> <mrow> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> </mrow> </msub> </munder> <mfrac> <mrow> <mi>w</mi> <mo>+</mo> <msqrt> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> <msup> <mrow> <mo>&amp;lsqb;</mo> <msup> <mrow> <mo>(</mo> <mi>u</mi> <mo>-</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>v</mi> <mo>-</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mi>w</mi> <mo>+</mo> <msqrt> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>&amp;rsqb;</mo> </mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </msup> </mfrac> <mo>&amp;CenterDot;</mo> <mfrac> <mi>r</mi> <msqrt> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>x</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>y</mi> <mo>&amp;prime;</mo> </msup> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> </mfrac> <msup> <mi>dx</mi> <mo>&amp;prime;</mo> </msup> <msup> <mi>dy</mi> <mo>&amp;prime;</mo> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced>
R is permanent magnet radius；By product curved surface S+And S-The expression formula of projection in X ' OY ' planes is identical, is designated as integral domain Dx’y’, its expression formula is (x')2+(y')2=r2；
Step 4：Obtained using formula (1) under basis coordinates system along the magnetic induction intensity component B of each reference axis of X, Y, Zx、By、BzRespectively For：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>B</mi> <mi>x</mi> </msub> <mo>=</mo> <msup> <msub> <mi>B</mi> <mi>x</mi> </msub> <mo>&amp;prime;</mo> </msup> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;beta;</mi> <mo>+</mo> <msup> <msub> <mi>B</mi> <mi>z</mi> </msub> <mo>&amp;prime;</mo> </msup> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;beta;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>B</mi> <mi>y</mi> </msub> <mo>=</mo> <msup> <msub> <mi>B</mi> <mi>x</mi> </msub> <mo>&amp;prime;</mo> </msup> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;alpha;</mi> <mo>&amp;CenterDot;</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;beta;</mi> <mo>+</mo> <msup> <msub> <mi>B</mi> <mi>y</mi> </msub> <mo>&amp;prime;</mo> </msup> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;alpha;</mi> <mo>-</mo> <msup> <msub> <mi>B</mi> <mi>z</mi> </msub> <mo>&amp;prime;</mo> </msup> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;alpha;</mi> <mo>&amp;CenterDot;</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;beta;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>B</mi> <mi>z</mi> </msub> <mo>=</mo> <mo>-</mo> <msup> <msub> <mi>B</mi> <mi>x</mi> </msub> <mo>&amp;prime;</mo> </msup> <mi>cos</mi> <mi>&amp;alpha;</mi> <mo>&amp;CenterDot;</mo> <mi>sin</mi> <mi>&amp;beta;</mi> <mo>+</mo> <msup> <msub> <mi>B</mi> <mi>y</mi> </msub> <mo>&amp;prime;</mo> </msup> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;alpha;</mi> <mo>+</mo> <msup> <msub> <mi>B</mi> <mi>z</mi> </msub> <mo>&amp;prime;</mo> </msup> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;alpha;</mi> <mo>&amp;CenterDot;</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;beta;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>
Step 5：Magnetic effect sensor S is set1Detection direction be vertical direction, then magnetic effect sensor S1Detect the magnetic obtained InductionAs：Thus magnetic effect sensor S is obtained1Detect the magnetic induction intensity obtainedIn rotational coordinates On u, v, w, α, β expression formula under system；
Magnetic effect sensor S is obtained by formula (1)1On x under basis coordinates system0、y0、z0, α and β expression formula such as formula (3)：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <mi>u</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;beta;</mi> <mo>+</mo> <mi>w</mi> <mi> </mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;beta;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>=</mo> <mi>u</mi> <mi> </mi> <mi>sin</mi> <mi>&amp;alpha;</mi> <mo>&amp;CenterDot;</mo> <mi>sin</mi> <mi>&amp;beta;</mi> <mo>+</mo> <mi>v</mi> <mi> </mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;alpha;</mi> <mo>-</mo> <mi>w</mi> <mi> </mi> <mi>sin</mi> <mi>&amp;alpha;</mi> <mo>&amp;CenterDot;</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;beta;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>-</mo> <mi>u</mi> <mi> </mi> <mi>cos</mi> <mi>&amp;alpha;</mi> <mo>&amp;CenterDot;</mo> <mi>sin</mi> <mi>&amp;beta;</mi> <mo>+</mo> <mi>v</mi> <mi> </mi> <mi>sin</mi> <mi>&amp;alpha;</mi> <mo>+</mo> <mi>w</mi> <mi> </mi> <mi>cos</mi> <mi>&amp;alpha;</mi> <mo>&amp;CenterDot;</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;beta;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow>
Step 6：By step 1- steps 5, each expression formula of magnetic effect sensor under basis coordinates system is obtained, and establish such as formula (4) equation group shown in：
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>B</mi> <msub> <mi>S</mi> <mn>1</mn> </msub> </msub> <mo>=</mo> <msub> <mi>f</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>,</mo> <mi>&amp;alpha;</mi> <mo>,</mo> <mi>&amp;beta;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>B</mi> <msub> <mi>S</mi> <mn>2</mn> </msub> </msub> <mo>=</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>a</mi> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>,</mo> <mi>&amp;alpha;</mi> <mo>,</mo> <mi>&amp;beta;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>B</mi> <msub> <mi>S</mi> <mn>3</mn> </msub> </msub> <mo>=</mo> <msub> <mi>f</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>,</mo> <mi>&amp;alpha;</mi> <mo>,</mo> <mi>&amp;beta;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>B</mi> <msub> <mi>S</mi> <mi>4</mi> </msub> </msub> <mo>=</mo> <msub> <mi>f</mi> <mi>4</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>2</mi> <mi>a</mi> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>b</mi> <mo>,</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>,</mo> <mi>&amp;alpha;</mi> <mo>,</mo> <mi>&amp;beta;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>B</mi> <msub> <mi>S</mi> <mn>5</mn> </msub> </msub> <mo>=</mo> <msub> <mi>f</mi> <mn>5</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>a</mi> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>b</mi> <mo>,</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>,</mo> <mi>&amp;alpha;</mi> <mo>,</mo> <mi>&amp;beta;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>B</mi> <msub> <mi>S</mi> <mn>6</mn> </msub> </msub> <mo>=</mo> <msub> <mi>f</mi> <mn>6</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>y</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>b</mi> <mo>,</mo> <msub> <mi>z</mi> <mn>0</mn> </msub> <mo>,</mo> <mi>&amp;alpha;</mi> <mo>,</mo> <mi>&amp;beta;</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
In formula (4),Correspond and export letter for the magnetic induction of each magnetic effect sensor Number, the equation group for solving formula (4) obtains x0、y0、z0, α and β value, with the x0、y0、z0, α and β value characterize and obtain magnetic effect Sensor S1Relative to the relative position and relative angle of permanent magnet；
Step 7：The distance at center of (7) and permanent magnet (5) of popping one's head in be L, the coordinate of probe (7) under rotating coordinate system be (0, 0 ,-L), it is (- L sin β, L sin α sin β ,-Lcos α cos according to formula (3) coordinate value of (7) under basis coordinates system of being popped one's head in β), then (7) are popped one's head in fixed coordinate system Oω-XωYωZωUnder coordinate value be (- Lsin β-x0,Lsinαsinβ-y0,-Lcosα cos-z0), complete detection.
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