CN105021211B - A kind of attitude test device and method based on autocollimator - Google Patents

A kind of attitude test device and method based on autocollimator Download PDF

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Publication number
CN105021211B
CN105021211B CN201510303942.1A CN201510303942A CN105021211B CN 105021211 B CN105021211 B CN 105021211B CN 201510303942 A CN201510303942 A CN 201510303942A CN 105021211 B CN105021211 B CN 105021211B
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mrow
angle
autocollimator
attitude
msubsup
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CN105021211A (en
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徐凯
孙晨
张群
王鑫
张世先
翟晓静
李莉
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707th Research Institute of CSIC
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707th Research Institute of CSIC
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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
    • G01C25/005Manufacturing, calibrating, cleaning, or repairing instruments or devices referred to in the other groups of this subclass initial alignment, calibration or starting-up of inertial devices

Abstract

The present invention relates to a kind of attitude test device and method based on autocollimator, the device mainly includes autocollimator, optics hexahedron, twin shaft electrolevel and the pedestal with levelling function.This method be by attitude test device be placed on it is firm on the basis of, autocollimator is sighted after target reflective surface, sights optics hexahedron using auto-collimation gyrotheodolite, and measure its angle with geographical north azimuth reference.Start the continuous data admission that autocollimator carries out target reflective surface posture, start the continuous data admission that twin shaft electrolevel carries out horizontal attitude.After data recording is finished, using attitude test device data processing method, data to autocollimator and twin shaft electrolevel are handled, and finally give the consecutive variations of target reflective surface posture in geographic coordinate system, so that the need for meeting continuous absolute measurement.

Description

A kind of attitude test device and method based on autocollimator
Technical field
The present invention relates to technical field of inertial, especially a kind of attitude test device and side based on autocollimator Method.
Background technology
, it is necessary to which the continuous optical measurement rotary part in real time of dynamic is relative to ground during inertial navigation set scheduling and planning The relativeness of coordinate system is managed, and precision prescribed is higher.Continuous measurement can not be realized using traditional transit survey.And use Autocollimator, which is measured, has the features such as continuous, real-time, precision is high, can continuously be measured.But autocollimator can only Relative angle is exported, its measurement result can not reflect target reflective surface normal and geographic coordinate system relativeness.
The content of the invention
It is an object of the invention to make up the deficiencies in the prior art part, there is provided a kind of attitude test based on autocollimator Device and method so that user can calculate target by the measurement result of the autocollimator on the attitude test device Reflective surface normal and the relativeness of Department of Geography, the need for meeting continuous absolute measurement.
The purpose of the present invention is realized by following technological means:
A kind of attitude test device based on autocollimator, it is characterised in that:Including autocollimator, optics hexahedron, double Axle electrolevel, the pedestal with levelling function, autocollimator are arranged on the pedestal with levelling function, on a left side for autocollimator Right both sides set optics hexahedron and twin shaft electrolevel respectively, and the optics hexahedron and twin shaft electrolevel are installed in On pedestal with levelling function.
Moreover, the pedestal with levelling function includes bottom plate and screw pair, two are symmetrically installed in the front end of bottom plate Screw pair, installs a screw pair in the middle part of the rear end of bottom plate;Autocollimator, optics hexahedron and twin shaft electrolevel are pacified On bottom plate.
A fluting, each support base are made moreover, being installed on described bottom plate on two support bases, each support base On hinge fill an arc jam plate, autocollimator is placed in the fluting of support base and positioned by arc jam plate.
A kind of attitude test method based on autocollimator, it is characterised in that:Comprise the following steps:
(1) places attitude test device:Attitude test device is placed on the basis of consolidating;
(2) autocollimators prepare:Autocollimator is started shooting, and sights target reflective surface;
(3) the measurement of horizontal angles:Start twin shaft electrolevel, measurement apparatus horizontal attitude and horizontal plane angle;
(4) the azimuthal measurements of:Azimuth can be measured the folder between optics hexahedron and outer benchmark by autocollimation theodolite Angle is obtained by auto-collimation gyrotheodolite direct measurement;
(5) measures targeted attitude:Start the measurement of absolute angle pattern of autocollimator, data record is carried out to target reflective surface Take, start the data recording that twin shaft electrolevel carries out horizontal attitude, record autocollimator output (x, y);
(6) data processings:After the completion of attitude test, with attitude test device data processing method, to autocollimator and The data of twin shaft electrolevel are handled, and obtain the consecutive variations of target reflective surface posture in geographic coordinate system.
Azimuthal it is measured as moreover, step is (4) described:Autocollimation theodolite is set up, makes its lens barrel pitch axis and optics Hexahedron is contour, sights optics hexahedron, then aims at reference mirror or acquisition benchmark is mutually taken aim at gyrotheodolite, so as to measure light Learn hexahedron and the angle in geographical north orientation;Because hexahedron has three reflectings surface to reflect the azimuth of device, depending on scene side Just the optional one measurement of degree, measurement result is designated as (ξ respectively1ξ2ξ3)。
Moreover, (6) described data handling procedure is step:
1. sets up device coordinate system:
Set up device coordinate system and set up the transition matrix of hexahedron coordinate system and Department of Geography, it is necessary to which to use hexahedron relative In three attitude angles of Department of Geography, attitude angle can be obtained by level meter output and azimuth injection as follows:
The angle of pitch:
The device vertical inclination angle measured by level meter is obtained plus error angle obtained by calibrating:
θ=S1+△1
Roll angle:
The device Angle of Heel measured by level meter is added after error angle divided by pitching cosine of an angle is obtained:
Azimuth:
Because hexahedron has three reflectings surface to inject azimuth, according to different notes when setting up coordinate system
Enter plane and be divided into situations below:
When injecting minute surface 1, azimuth is injector angle:
ξ=ξ1
When injecting minute surface 3, azimuth is that injector angle adds 180 degree:
ξ=ξ2+180;
When injecting minute surface 2, azimuth and injection angular dependence are as follows:
ξ=ξ3-tan(S1+△1)tan(S2+△2);
Above attitude angle can set up device and the transition matrix of Department of Geography after determining
2. sets up autocollimator coordinate system:
Autocollimator optical axis is considered as a vector, and its CCD plane can regard as the coordinate plane vertical with optical axis; Set up the coordinate system abbreviation light pipe coordinate system that is constituted with optical axis and CCD reference axis, by demarcation can obtain light pipe coordinate system and Transition matrix between device coordinate systemObtained by above stepCan obtain light pipe coordinate system and geographic coordinate system it Between transition matrix:
3. calculates target reflective surface normal attitude vectors and calculates normal with horizontal plane angle and in water by attitude vectors Projection and geographical north angle in plane:
Optical-autocollimator is auto-collimation light source and high-quality camera lens based on high-resolution CCD and generation directional light Group is to measure the instrument of objective plane mirror and local Coordinate System angulation;Its principle is:Objective lens focal length is f, in its focus There is a light source O position, and the light from light source O is reflected by objective lens, as directional light, is radiated at after objective plane mirror instead CCD planar imaging O ' are emitted back towards, if level crossing normal has an angle, θ with light pipe axle, according to the reflectivity properties of light, reflected light Line will be with the θ of incident ray angulation 2, and the picture O ' reflected to form will have certain distance d, high-resolution CCD to survey with its light source O D is measured, and angle theta is obtained by following formula:
When actual use, light source returns to light pipe and converged on CCD after reflection by graticle one cross light beam of formation For a cross hot spot, and coordinate (dx, dy) of the cross spot center point in CCD planes can be obtained;
It can thus be concluded that the angle between target reflecting surface normal and light pipe CCD reference axis:
Thus two angles can obtain direction unit vector of the target reflecting surface normal relative to light pipe coordinate system:
And then direction unit vector of the target reflecting surface normal relative to Department of Geography can be obtained:
Thus vector can be gone out target reflecting surface with inverse and project angle between geographical north in the horizontal plane, and target is anti- The angle penetrated between face and horizontal plane:
Advantages and positive effects of the present invention are:
1st, the present invention is a kind of attitude test device and method based on autocollimator of design science, this attitude test dress The autocollimator put can carry out continuous, real-time, high-precision attitude measurement after target reflective surface is sighted, and by certainly The measurement hexahedral to the optics on attitude test device of collimation theodolite obtains the attitude test device in geographic coordinate system Azimuth, obtains the horizontal attitude of the attitude test device in geographic coordinate system by the measurement of twin shaft electrolevel, enters And the consecutive variations of target reflective surface posture in geographic coordinate system can be obtained, meet measurement needs.
2nd, the optics hexahedron in the present invention is made of the K9 glass of excellent in optical properties, can be emitted onto at it and be pacified Incident ray outside dress face on five faces is reflected back, and is high-precision optical reference, and can realize multiangular measurement, is reduced The erection difficulty of autocollimation theodolite, improves service efficiency.
3rd, optics hexahedron of the invention is the orientation attitude reference of attitude test device, and twin shaft electrolevel is posture The horizontal attitude benchmark of test device, this design ensure that measurement accuracy.
4th, in the case of existing azimuth reference, 0.5 can be reached using autocollimation theodolite transmitting accuracy ", using certainly The measurement accuracy of collimation gyrotheodolite can reach 2 ".Compared to more traditional theodolite and gyrotheodolite side measured directly Method, the precision of attitude angle of the present invention can reach 0.1 ", it is obviously improved.
5th, the present invention can be realized continuously monitors to target level posture and azimuthal long-time, in for a long time The target of slowly varying (INS Platform that such as long-term posture is kept), traditional measuring method can only realize the point type survey to target Amount can not reflect the situation of change and changing rule of target, and the present invention can then realize continuous monitoring for a long time, so as to With detailed record object attitude angle and azimuthal change procedure so as to summarizing changing rule.Meanwhile, compared to biography The transit survey method of system, the present invention realizes unattended high-precision attitude measurement, is effectively reduced human cost raising Measurement efficiency.
6th, traditional measurement using theodolite or gyrotheodolite during being measured, due to manually-operated random Property so that final measurement error is unable to reach the theoretical precision of instrument, and the present invention is used as tester using autocollimator Automated imaging can be realized and record number, it is to avoid the error that artificial operation is brought, so as to improve measurement accuracy.
Brief description of the drawings
Fig. 1 is the schematic perspective view one of the present invention;
Fig. 2 is the schematic perspective view two of the present invention;
Fig. 3 is optical-autocollimator operation principle;
Fig. 4 is optical-autocollimator output schematic diagram.
Embodiment
Describe embodiments of the invention in detail below in conjunction with the accompanying drawings;It should be noted that the present embodiment is narrative, no It is limited, it is impossible to which protection scope of the present invention is limited with this.
A kind of attitude test device based on autocollimator, including autocollimator 2, optics hexahedron 4, twin shaft electronic horizon Instrument 1, the pedestal with levelling function, the pedestal with levelling function include bottom plate 3 and screw pair 5, are symmetrically installed in the front end of bottom plate Two screw pairs, install a screw pair in the middle part of the rear end of bottom plate.Pedestal with levelling function provides installation to autocollimator Benchmark, while providing reference for installation to optics hexahedron and twin shaft electrolevel, and is ensured in certain limit using screw pair The interior level to attitude test device is adjusted.
Autocollimator:It is to utilize optical autocollimating principle, coordinates speculum to measure small deflection angle, small angle measurement Or can be exchanged into a kind of conventional metrology and measurement instrument of small angle measurement.Auto-collimation technology is widely used in precision positioning, automatic In terms of angle locating ring collection, angular surveying, be optical manufacturing and detection, machine-building, shipbuilding, Aero-Space, metrology and measurement, The indispensable conventional measuring equipment of the departments such as scientific research.Particularly in terms of accurate, Ultra-precision positioning, more there is irreplaceable work With.Autocollimator is arranged on the pedestal bottom plate with levelling function, i.e.,:Two support bases 7, each support base are installed on bottom plate On make a fluting, the fluting is dovetail groove or deep-slotted chip breaker.Hinge fills an arc jam plate 6 on each support base, from Collimator is placed in the fluting of support base and positioned by arc jam plate.
In the left and right sides of autocollimator, optics hexahedron and twin shaft electrolevel are set respectively, the optics hexahedron and Twin shaft electrolevel is installed on the pedestal bottom plate with levelling function.Optics hexahedron:Using the K9 of excellent in optical properties The incident ray that the optics hexahedron that glass is made can be emitted onto on outer five faces of mounting surface at it is reflected back, and is high-precision The optical reference of degree, and multiangular measurement can be realized, the erection difficulty of autocollimation theodolite is reduced, service efficiency is improved, It is the orientation attitude reference of attitude test device.Twin shaft electrolevel:The high-precision of orthogonal two horizontal angles can be achieved Degree measurement in real time, is the horizontal attitude benchmark of attitude test device.Adjustment optics hexahedron makes the horizontal attitude in two face defeated Go out and differed with the output of twin shaft electrolevel as better than 5 ".Zero-bit is adjusted after accurate twin shaft electrolevel installation in place Measure the actual difference of the horizontal normal of the optical axis of its mounting surface and autocollimator.
A kind of attitude test method based on autocollimator, comprises the following steps:
(1) places attitude test device:
In laboratory, attitude test device is placed on the basis of consolidating;
(2) autocollimators prepare:
Autocollimator is started shooting, and sights target reflective surface;
(3) the measurement of horizontal angles:
Start twin shaft electrolevel, measurement apparatus horizontal attitude and horizontal plane angle;
(4) the azimuthal measurements of:
Azimuth can be by the angle between autocollimation theodolite measurement optics hexahedron and outer benchmark or by auto-collimation gyro Theodolite direct measurement is obtained:
Autocollimation theodolite is set up, makes its lens barrel pitch axis and optics hexahedron contour, sights optics hexahedron, then take aim at Quasi- reference mirror mutually takes aim at acquisition benchmark with auto-collimation gyrotheodolite, so as to measure the angle of optics hexahedron and geographical north orientation. Because hexahedron there are three reflectings surface to reflect the azimuth of device, depending on the optional one measurement of live comfort level, survey Amount result is designated as (ξ respectively1ξ2ξ3);
(5) measures targeted attitude:
Start the measurement of absolute angle pattern of autocollimator, data recording is carried out to target reflective surface, start twin shaft electronics Level meter carries out the data recording of horizontal attitude, record autocollimator output (x, y);
(6) data processings:
After the completion of attitude test, with attitude test device data processing method, to autocollimator and twin shaft electronic horizon The data of instrument are handled, and obtain the consecutive variations of target reflective surface posture in geographic coordinate system.The processing procedure is:
1. sets up device coordinate system:
Set up device coordinate system and set up the transition matrix of hexahedron coordinate system and Department of Geography, it is necessary to which to use hexahedron relative In three attitude angles of Department of Geography.Attitude angle can be obtained by level meter output and azimuth injection as follows:
The angle of pitch:
The device vertical inclination angle measured by level meter is obtained plus error angle obtained by calibrating:
θ=S1+△1
Roll angle:
The device Angle of Heel measured by level meter is added after error angle divided by pitching cosine of an angle is obtained:
Azimuth:
Because hexahedron has three reflectings surface to inject azimuth, according to different notes when setting up coordinate system
Enter plane and be divided into situations below:
When injecting minute surface 1, azimuth is injector angle:
ξ=ξ1
When injecting minute surface 3, azimuth is that injector angle adds 180 degree:
ξ=ξ2+180;
When injecting minute surface 2, azimuth and injection angular dependence are as follows:
ξ=ξ3-tan(S1+△1)tan(S2+△2);
Above attitude angle can set up device and the transition matrix of Department of Geography after determining
2. sets up autocollimator coordinate system:
Autocollimator optical axis is considered as a vector, and its CCD plane can regard as the coordinate plane vertical with optical axis. Set up the coordinate system abbreviation light pipe coordinate system that is constituted with optical axis and CCD reference axis, by demarcation can obtain light pipe coordinate system and Transition matrix between device coordinate systemObtained by above stepCan obtain light pipe coordinate system and geographic coordinate system it Between transition matrix:
3. calculates target reflective surface normal attitude vectors and calculates normal with horizontal plane angle and in water by attitude vectors Projection and geographical north angle in plane:
Optical-autocollimator is auto-collimation light source and high-quality camera lens based on high-resolution CCD and generation directional light Group is to measure the instrument of objective plane mirror and local Coordinate System angulation.Its principle as shown in figure 3, objective lens focal length be f, There is a light source O its focal position, and the light from light source O is reflected by objective lens, as directional light, is radiated at objective plane mirror CCD planar imaging O ' are reflected back afterwards, if there is an angle, θ with light pipe axle in level crossing normal, according to the reflectivity properties of light, Reflection light will be with the θ of incident ray angulation 2, and the picture O ' reflected to form will have certain distance d, high-resolution CCD with its light source O It can measure and obtain d, and angle theta is obtained by following formula:
When actual use, light source returns to light pipe and converged on CCD after reflection by graticle one cross light beam of formation For a cross hot spot, (as shown in Figure 4) and coordinate (dx, dy) of the cross spot center point in CCD planes can be obtained;
It can thus be concluded that the angle between target reflecting surface normal and light pipe CCD reference axis:
Thus two angles can obtain direction unit vector of the target reflecting surface normal relative to light pipe coordinate system:
And then direction unit vector of the target reflecting surface normal relative to Department of Geography can be obtained:
Thus vector can be gone out target reflecting surface with inverse and project angle between geographical north in the horizontal plane, and target is anti- The angle penetrated between face and horizontal plane:

Claims (6)

1. a kind of attitude test device based on autocollimator, it is characterised in that:Including autocollimator, optics hexahedron, twin shaft Electrolevel, the pedestal with levelling function, autocollimator are arranged on the pedestal with levelling function, in the left and right of autocollimator Both sides set optics hexahedron and twin shaft electrolevel respectively, and the optics hexahedron and twin shaft electrolevel are installed in band On the pedestal of levelling function.
2. a kind of attitude test device based on autocollimator according to claim 1, it is characterised in that:Described band is adjusted The pedestal of flat function includes bottom plate and screw pair, two screw pairs is symmetrically installed in the front end of bottom plate, in the middle part of the rear end of bottom plate One screw pair is installed;Autocollimator, optics hexahedron and twin shaft electrolevel are installed on bottom plate.
3. a kind of attitude test device based on autocollimator according to claim 2, it is characterised in that:Described bottom plate Make to cut with scissors on a fluting, each support base on two support bases of upper installation, each support base and fill an arc jam plate, from Collimator is placed in the fluting of support base and positioned by arc jam plate.
4. a kind of attitude test method using attitude test device as claimed in claim 1, it is characterised in that:Including following Step:
(1) places attitude test device:Attitude test device is placed on the basis of consolidating;
(2) autocollimators prepare:Autocollimator is started shooting, and sights target reflective surface;
(3) the measurement of horizontal angles:Start twin shaft electrolevel, measurement apparatus horizontal attitude and horizontal plane angle;
(4) the azimuthal measurements of:Azimuth can be measured by autocollimation theodolite angle between optics hexahedron and outer benchmark or Obtained by auto-collimation gyrotheodolite direct measurement;
(5) measures targeted attitude:Start the measurement of absolute angle pattern of autocollimator, data recording carried out to target reflective surface, Start the data recording that twin shaft electrolevel carries out horizontal attitude, record autocollimator output (x, y);
(6) data processings:After the completion of attitude test, with attitude test device data processing method, to autocollimator and twin shaft The data of electrolevel are handled, and obtain the consecutive variations of target reflective surface posture in geographic coordinate system.
5. a kind of attitude test method according to claim 4, it is characterised in that:Step (4) described azimuthal measurement For:Autocollimation theodolite is set up, makes its lens barrel pitch axis and optics hexahedron contour, sights optics hexahedron, then aim at base Quasi- mirror mutually takes aim at acquisition benchmark with gyrotheodolite, so as to measure the angle of optics hexahedron and geographical north orientation;Due to hexahedron There are three reflectings surface to reflect the azimuth of device, depending on the optional one measurement of live comfort level, measurement result difference It is designated as (ξ1ξ2ξ3)。
6. a kind of attitude test method according to claim 4, it is characterised in that:Step (6) described data handling procedure For:
1. sets up device coordinate system:
Set up device coordinate system and set up the transition matrix of hexahedron coordinate system and Department of Geography, it is necessary to use hexahedron relative to ground Three attitude angles of system are managed, it is as follows to obtain attitude angle by level meter output and azimuth injection:
The angle of pitch:
The device vertical inclination angle measured by level meter is obtained plus error angle obtained by calibrating:
θ=S11
Roll angle:
The device Angle of Heel measured by level meter is added after error angle divided by pitching cosine of an angle is obtained:
Azimuth:
Because hexahedron has three reflectings surface to inject azimuth, when setting up coordinate system according to different injection planes be divided into Lower situation:
When injecting minute surface 1, azimuth is injector angle:
ξ=ξ1
When injecting minute surface 3, azimuth is that injector angle adds 180 degree:
ξ=ξ2+180;
When injecting minute surface 2, azimuth and injection angular dependence are as follows:
ξ=ξ3-tan(S11)tan(S22);
Above attitude angle can set up device and the transition matrix of Department of Geography after determining
2. sets up autocollimator coordinate system:
Autocollimator optical axis is considered as a vector, and its CCD plane can regard as the coordinate plane vertical with optical axis;Set up The coordinate system abbreviation light pipe coordinate system constituted with optical axis and CCD reference axis, light pipe coordinate system and device can be obtained by demarcation Transition matrix between coordinate systemObtained by above stepIt can obtain between light pipe coordinate system and geographic coordinate system Transition matrix:
<mrow> <msubsup> <mi>C</mi> <mi>g</mi> <mi>N</mi> </msubsup> <mo>=</mo> <msubsup> <mi>C</mi> <mi>g</mi> <mi>b</mi> </msubsup> <mo>&amp;CenterDot;</mo> <msubsup> <mi>C</mi> <mi>b</mi> <mi>N</mi> </msubsup> <mo>;</mo> </mrow>
3. calculates target reflective surface normal attitude vectors and calculates normal with horizontal plane angle and in horizontal plane by attitude vectors Upper projection and geographical north angle:
Optical-autocollimator be based on high-resolution CCD and produce directional light auto-collimation light source and high-quality lens group with Measure the instrument of objective plane mirror and local Coordinate System angulation;Its principle is:Objective lens focal length is f, in its focal position There is a light source O, the light from light source O is reflected by objective lens, as directional light, be radiated at after objective plane mirror and be reflected back CCD planar imaging O ', if level crossing normal has an angle, θ with light pipe axle, according to the reflectivity properties of light, reflection light will With the θ of incident ray angulation 2, the picture O ' reflected to form will have certain distance d, high-resolution CCD to measure with its light source O Angle theta is obtained to d, and by following formula:
<mrow> <mn>2</mn> <mi>&amp;theta;</mi> <mo>=</mo> <mfrac> <mi>d</mi> <mi>f</mi> </mfrac> </mrow>
When actual use, light source returns to light pipe and convergence is one on CCD after reflection by graticle one cross light beam of formation Cross hot spot, and coordinate (dx, dy) of the cross spot center point in CCD planes can be obtained;
It can thus be concluded that the angle between target reflecting surface normal and light pipe CCD reference axis:
<mrow> <mi>&amp;alpha;</mi> <mo>=</mo> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>f</mi> </mrow> </mfrac> <mo>,</mo> <mi>&amp;beta;</mi> <mo>=</mo> <mfrac> <mrow> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mn>2</mn> <mi>f</mi> </mrow> </mfrac> </mrow>
Thus two angles can obtain direction unit vector of the target reflecting surface normal relative to light pipe coordinate system:
<mrow> <msubsup> <mover> <mi>X</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>j</mi> <mi>g</mi> </msubsup> <mo>=</mo> <mfenced open = "(" close = ")"> <mtable> <mtr> <mtd> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&amp;alpha;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;beta;</mi> </mtd> </mtr> <mtr> <mtd> <mo>-</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;alpha;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&amp;beta;</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mi>&amp;beta;</mi> </mtd> </mtr> </mtable> </mfenced> </mrow>
And then direction unit vector of the target reflecting surface normal relative to Department of Geography can be obtained:
<mrow> <msubsup> <mover> <mi>X</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>j</mi> <mi>N</mi> </msubsup> <mo>=</mo> <msubsup> <mi>C</mi> <mi>g</mi> <mi>N</mi> </msubsup> <mo>&amp;CenterDot;</mo> <msubsup> <mover> <mi>X</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>j</mi> <mi>g</mi> </msubsup> </mrow>
Thus vector can be gone out target reflecting surface with inverse and project angle between geographical north, and target reflecting surface in the horizontal plane Angle between horizontal plane:
<mrow> <mi>&amp;psi;</mi> <mo>=</mo> <mi>arctan</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msubsup> <mover> <mi>X</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>j</mi> <mi>N</mi> </msubsup> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mrow> <msubsup> <mover> <mi>X</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>j</mi> <mi>N</mi> </msubsup> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow>
<mrow> <mi>&amp;upsi;</mi> <mo>=</mo> <mi>arcsin</mi> <mrow> <mo>(</mo> <msubsup> <mover> <mi>X</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>j</mi> <mi>N</mi> </msubsup> <mo>(</mo> <mn>3</mn> <mo>)</mo> <mo>)</mo> </mrow> <mo>.</mo> </mrow> 3
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