Research Article Research on the Trajectory Model for ZY-3downloads.hindawi.com/journals/tswj/2014/429041.pdfResearch Article Research on the Trajectory Model for ZY-3 YifuChenandZhongXie

Research ArticleResearch on the Trajectory Model for ZY-3

Yifu Chen and Zhong Xie

National Engineering Research Center for Geographic Information System China University of Geosciences 388 Lumo RoadWuhan 0086-430074 China

Correspondence should be addressed to Yifu Chen yifuchenyfgmailcom

Received 11 March 2014 Accepted 18 June 2014 Published 27 August 2014

Academic Editor Zhongmei Zhou

Copyright copy 2014 Y Chen and Z Xie This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The new generation Chinese high-resolution three-line stereo-mapping satellite Ziyuan 3 (ZY-3) is equipped with three sensors(nadir backward and forward views) Its objective is to manufacture the 1 50000 topographic map and revise and update the1 25000 topographic map For the push-broom satellite the interpolation accuracy of orbit and attitude determines directly thesatellitersquos stereo-mapping accuracy and the position accuracy without ground control point In this study a new trajectory model isproposed for ZY-3 in this paper according to researching and analyzing the orbit and attitude of ZY-3 Using the trajectory data setthe correction and accuracy of the new proposed trajectory are validated and compared with the other models polynomial model(LPM) piecewise polynomial model (PPM) and Lagrange cubic polynomial model (LCPM) Meanwhile the differential equationis derivate for the bundle block adjustment Finally the correction and practicability of piece-point with weight polynomial modelfor ZY-3 satellite are validated according to the experiment of geometric correction using the ZY-3 image and orbit and attitudedata

1 Introduction

Most high-resolution remote-sensing satellites are the nearpolar satellite these satellites generally run on their trajectorybelow 1000 km in order to acquire the higher resolutionfor earth observation [1] Ziyuan 3 (ZY-3) is the first Chi-nese civilian high-resolution stereo-mapping satellite that isequipped with three-line sensors (nadir backward and for-ward views) which have the separated optic system respec-tively and an additional multispectral sensorThe resolutionsof nadir backward and forward views are 25m 40m and40m respectively and the resolution of the multispectralsensor is 8m [2] The trajectory height of ZY-3 is relativelylower 505 km The satellite therefore is easily impacted byvarious disturbing forces from space and various flutters andjitters from the internal mechanical motion of the satellitesuch as high-frequency flutter from Gyro-Star and flywheeland the low-frequency jitter from solar panels All of thesefactors result in the high-frequency flutter and low-frequencyjitter of satellite when it is running on its trajectory [3]

For the linear push-broom satellite every acquired imageline has different data of orbit and attitude and the instru-ment just records the data at regular intervals but not

all The unrecorded data at a certain time therefore needsto be interpolated using exterior orientation model [4 5]For the traditional bundle block adjustment it is almostimpossible to acquire the solution using the orbit and attitudedata of every image line Thus the high-precision exteriororientation model is needed to be researched and proposedwhich is crucial for the geometry-data processing of linearpush-broom satellite In the geometry-data processing withblock bundle adjustment the difficult and key problem is howto reduce and eliminate the correlation between the interiorand exterior orientation model parameters and improve theinterpolationrsquos accuracy of orbit and attitude which avoidthe transmission of the interpolation error to the interiororientation model in order to improve the solved accuracy ofparameters [6 7]The interior and exteriormodel parameterswill affect each other and make the cross-correlations inthe calculation with bundle block adjustment When thecorrelation among the parameters is strong the systematicerror cannot be described completely and accurately withthese parameters which result in the unstable oscillationof solved parameters and the decreased solution accuracyIn this process the interpolation error of exterior elements

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 429041 9 pageshttpdxdoiorg1011552014429041

2 The Scientific World Journal

(orbit and attitude) with trajectory model will also be trans-mitted to interior orientation as a part of systematic errorincrease the systematic error of interior orientation anddecrease the solved accuracy and stability of geometric dataprocessing An ideal trajectory model not only can ensure ahigh interpolation accuracy for the attitude and orbit of everyimage line but also can decrease and eliminate the strongcorrelations among the solved parameters in bundle blockadjustment and reduce the transmission of systematic errorbetween the interior and exterior orientation models

Generally the satellite running trajectory is relativelystable in a short period so that the orbit and attitude in ashort interval trajectory can bemodeledwith the polynomialtherefore avoiding the complex stress analysis of the satellite[8] therefore avoiding the complex stress analysis of thesatellite Currently the trajectory models used for high-resolution remote-sensing satellite have polynomial model(LPM) piecewise polynomial model (PPM) and Lagrangecubic polynomial model proposed by Hofmann (LCPM) [9ndash11] By research and comparison with the trajectory modelshowever these models have some limitations The LPM andPPM can acquire the smooth fitted curves and the inter-polation accuracies are very low especially for the unstablyoscillated trajectory data The LCPM can acquire a higherinterpolation accuracy comparing with the LPM and PPMhowever the LCPM easily causes the data oscillation whenthe higher-order model is utilized in the block adjustment

For satellite ZY-3 a new trajectory model (piece-pointpolynomial with weight model) is proposed to acquire thehigher interpolationrsquos accuracy in this paper based on ana-lyzing the orbit and attitude data of ZY-3 In addition thedata set of ZY-3 is used to validate the correction of piece-point with weight polynomial model and compare it withthe other trajectory models LPM PPM and LCPM usedfor other satellites Meanwhile the differentiation equation ofthe proposed trajectory model is derivate and it is validatedwith the block bundle adjustment According to geometriccorrection experiment with the different ground controlpoints the correctness and applicability of piece-point withweight polynomialmodel are validated and assessed to ensureand improve the high accuracy of geometric correction forZY-3 satellite

2 ZY-3 Trajectory Model

21 Piece-Point with Weight Polynomial Model Trajectorymodel is a mathematic relationship elucidating the orbit andattitude of satellite vary with the different time in its trackFor the push-broom satellite every acquired image line has adifferent data of orbit and attitude and the instrument justrecords the data at regular intervals but not all the dataThe unrecorded data at a certain time thereby needs to beinterpolated using exterior orientation model

Piece-point with weight polynomial model (PWPM) isproposed according to the researching and analyzing ofthe data of ZY-3rsquos orbit and attitude in the long and shortperiod In comparison with the other models LPM PPMand LCPM the weight value is used in the new model to

pt119894+2pt119894minus1

timinus1 ti ti+2

pt119894 pt119894+1

ti+middotmiddotmiddot

pt119894+middotmiddotmiddot

Direction of flightti+1m

M

Figure 1 Diagram of piece-point with weight polynomial model

perform interpolationrsquos calculationThe newmodel thereforecan acquire a higher accuracy have a better flexibility andcan reduce the correlation among the exterior orientationelements The PWPM is an interpolation model with weightvalue Using the model to interpolate satellitersquos orbit andattitude the weight is calculated by the difference from anyinterpolationrsquos time to the time assumed as known Throughthe least square method the polynomial parameters aresolved and then the exterior orientation at any time on theorbit can be acquired with the polynomial parameters ThePWPMis represented by (1) and theweight value is describedwith 119875

119883119878119905= 119883119894

0+ 119883119894

1sdot 119905 + 119883

119894

2sdot 1199052

119884119878119905= 119884119894

0+ 119884119894

1sdot 119905 + 119884

119894

2sdot 1199052

119885119878119905= 119885119894

0+ 119885119894

1sdot 119905 + 119885

119894

2sdot 1199052

120601119905= 120601119894

0+ 120601119894

1sdot 119905 + 120601

119894

2sdot 1199052

120596119905= 120596119894

0+ 120596119894

1sdot 119905 + 120596

119894

2sdot 1199052

120581119905= 120581119894

0+ 120581119894

1sdot 119905 + 120581

119894

2sdot 1199052

119875 =1

1003817100381710038171003817119905 minus 1199051198941003817100381710038171003817

119875 =1

(119905 minus 119905119894)2

(1)

Figure 1 is the diagram of PWPM 119872 is a ground point Itsimaging point is shown by119898 119905 is the time that the point119898 isimaged On the orbit 119905

119894minus1 119905119894 119905119894+1

119905119894+2

and 119905119894+sdotsdotsdot

are the knowntimes that the exterior orientations are acquiredwithGPS andstar sensor which are represented by red square119901119894minus1 119901119894 119901119894+1119901119894+2 and 119901119894+sdotsdotsdot respectively represent the weight of the time119905to the known time 119905119894minus1 119905119894 119905119894+1 119905119894+2 and 119905119894+sdotsdotsdot The unrecordedorbit and attitude at every time like time 119905 can be interpolatedwith the exterior orientations at the time 119905119894minus1 119905119894 119905119894+1 119905119894+2 and119905119894+sdotsdotsdot using PWPM

Assuming that the four known times 1199051 1199052 1199053 and 1199054

for orbit and attitude are selected as known points thevalue of orbit and attitude at time t can be calculated andacquired with PWPM The same polynomial coefficients aresolved in the piece-wise polynomial model for the orbit andattitude respectively For the convenient expression the error

The Scientific World Journal 3

equation is represented only with the angle Kappa as (3)according to (1) and the general form of error

119881 = 119860 sdot 119883 minus 119871 (2)

V119894 = 120581119894

0+ 120581119894

1sdot 119905 + 120581

119894

2sdot 1199052minus 120581119905 (119894 = 1 2 3 4) (3)

Due to the four known times the four error equations canbe established and the coefficient matrix 119860 observed valuematrix 119871 and unknown vector matrix 119883 are constructed byerror equations Matrices 119860 119871 and119883 are represented by

119860 =

[[[[[[[

[

1 1199051 1199052

1

1 11990521199052

2

1 1199053 1199052

3

1 11990541199052

4

]]]]]]]

]

119871 =

[[[

[

1205811

1205812

1205813

1205814

]]]

]

119883 = [

[

1198960

1198961

1198962

]

]

(4)

According to the equation of weight the weights 1199011

1199012 1199013 and 119901

4for times 119905

1 1199052 1199053 and 119905

4 respectively are

calculated and shown in (5) and then the weight matrix 119875 isestablished The value of weight reflects the influence degreeof the orbit and attitude at times 119905

1 1199052 1199053 and 119905

4for the

unknown time 119905

1199011=

1

(119905 minus 1199051)2 119901

2=

1

(119905 minus 1199052)2

1199013 =

1

(119905 minus 1199053)2 1199014 =

1

(119905 minus 1199054)2

(5)

119875 =

[[[

[

1199011

1199012

1199013

1199014

]]]

]

(6)

According to the least square method normal equationcoefficient matrix 119873 and free vector 119880 are constructed andshown by

119873

= 119860119879119875119860

=

[[[[[

[

1199011+ sdot sdot sdot + 119901

411990111199051+ sdot sdot sdot + 119901

4119905411990111199052

1+ sdot sdot sdot + 119901

41199052

4

11990111199051+ sdot sdot sdot + 119901

4119905411990111199052


41199052

411990111199053


41199053

4

11990111199052


41199052

411990111199053


41199053

411990111199054


41199054

4

]]]]]

]

119880 =

[[[[

[

1205811sdot 1199011+ sdot sdot sdot + 120581

4sdot 1199014

1205811 sdot 1199051 sdot 1199011 + sdot sdot sdot + 1205814 sdot 1199054 sdot 1199014

1205811 sdot 1199052

1sdot 1199011 + sdot sdot sdot + 1205814 sdot 119905

2

4sdot 1199014

]]]]

]

(7)

For convenience the matrices119873minus1 and119880 are representedby the below formation shown in

119873minus1= [

[

119888111988821198883

119888411988851198886

119888711988881198889

]

]

119880 = [

[

1199061

1199062

1199063

]

]

(8)

Through least square adjustment the polynomial param-eters 119896

0 1198961 and 119896

2are solved and shown by

119883 = [

[

1198960

1198961

1198962

]

]

= [

[

119888111988821198883

119888411988851198886

1198887 1198888 1198889

]

]

sdot [

[

1199061

1199062

1199063

]

]

(9)

Afterwards the Kappa value at 119905 time can be calculatedusing (1) and the general formula of the Kappa value isrepresented by

120581 (119905) = (11988811199061+ 11988821199062+ 11988831199063)

+ (11988841199061+ 11988851199062+ 11988861199063) 119905

+ (11988871199061+ 11988881199062+ 11988891199063) 1199052

(10)

In the process of interpolation the PWPM can solve thedifferent parameters corresponding to the different attitudeand orbit at any time with the different weight value In thispaper the two weight equations are given out the reciprocalof the absolute value of time difference and the reciprocalof the square of the time difference The selection of weightequation has a large impact for the interpolationrsquos accuracyof orbit and attitude According to analysis and research thereciprocal of the square of the time difference is adoptedwhen the trajectory is relatively unstable On the contrary thereciprocal of the absolute value of time difference is utilized

22 Two Interpolation Methods of Trajectory Model For thePWPM the new trajectory has two kinds of interpolationmethods to acquire the data set of orbit and attitude at anytime One is using the several known times selected roundthe unknown time to interpolate the orbit and attitude atthe unknown time The other is using all the selected knowntimes to interpolate the orbit and attitude at the unknowntime The impaction from the selected known time for theorbit and attitude at the unknown time is measured andassessed according to the weight value In other words thetime difference between the unknown time and the knowntime is more far and the impaction from the unknown timeis more great The interpolation accuracies with the twomethods are different which is determined by the stability ofthe satellite trajectory the number of selected known timesand the location of the selected known time In the practicalapplication the two methods of PWPM are utilized togetheror respectively which is determined by the stability of orbitand attitude

23 The Differential Expression of PWPM Sensorrsquos imagingmodel describes the mathematic transformation relationshipbetween the coordinate of image point (119909 119910) and the coor-dinate of ground point (119883 119884 119885) It includes two kinds ofmodels rigorous imaging model and general imaging model[12 13] Based on the structure of CCD-array equipped on theZY-3 and the sight vector of every CCD scanning the groundthe rigorous imaging model for ZY-3 satellite is establishedrepresented by (11) and the one to one correspondencerelationship between image point and ground point is built


up with sensors coordinate systems satellitersquos trajectorycoordinate system and ground reference system

For ZY-3 satellite the data received from dual-frequencyGPS represents the location of the phase center of GPS andthe attitude data from star sensor is measured in the J2000coordinate [14 15] In the process the displacement matrixfrom the phase center ofGPS (GPS antenna) to the coordinateof satellite body [119863119909 119863119910 119863119911]

119879 the displacement matrixfrom CCD-array center to the coordinate of satellite body[119889119909 119889119910 119889119911]

119879 the rotation matrix from the coordinate ofstar sensor to the coordinate of satellite body 119877body

star andthe rotation matrix form imaging space coordinate to thecoordinate satellite body 119877body

camera are needed

[

[

119883

119884

119885

]

]WGS84

= 119898 sdot 119877 sdot [

[

[

[

119863119909

119863119910

119863119911

]

]

+ [

[

119889119909

119889119910

119889119911

]

]

+ 119877bodycamera sdot (

119909

119910

minus119891

)]

]

+ [

[

119883

119884

119885

]

]GPS

119877 = 119877WGS84J2000 sdot 119877

J2000orbit sdot 119877

orbitbody

(11)

where [119909 119910 minus119891]119879 are point coordinates in the image sys-

tem [119883 119884 119885]119879

GPS and [119883 119884 119885]119879

WGS84are perspective center

position and position coordinates in WGS84 coordinatesystem 119877orbit

body 119877J2000orbit and 119877

J2000orbit are rotation matrices respec-

tively from satellite body system to satellite orbit systemfrom satellite orbit system to J2000 coordinate system andfrom J2000 coordinate system to WGS84 coordinate system119898 represents the scale According to satellitersquos structuredesign and the result of laboratory calibration the three-line imaging model nadir forward and backward can beacquired with the different value of displacement matrix[119889119909 119889119910 119889119911]

119879 and rotation matrix 119877bodycamera

For the push-broom high-resolution satellite the objec-tive of the high-accuracy trajectory model is to acquire theaccurate elements of exterior orientation [119883GPS 119884GPS 119885GPS]

119879

and 119877orbitbody at any time

Assuming that the n known times are selected in a sceneimage of satellite and the orbit and attitude at time 119905 areneeded to be interpolated the four known times 119905

1 1199052 1199053 and

1199054round the unknown time 119905 are selected and the their weight

values 1199011 1199012 1199013 and 119901

4are correspondingly calculated with

(1) For the convenient expression the Pitch angle is picked upas a sample therefore the value of Pitch angle at 119905 time can becalculated and represented by (12) which has the same formexpression as (10)

Pitch (119905) = (11988811199061 + 11988821199062 + 11988831199063)

+ (11988841199061+ 11988851199062+ 11988861199063) 119905

+ (11988871199061+ 11988881199062+ 11988891199063) 1199052

(12)

In the calculation of bundle block adjustment the differ-ential expression of 119905 time for the four unknown times 119905

1 1199052

1199053 and 119905

4is derived and represented by

120597Pitch (119905)120597Pitch (119905

1)= (1198881+ 1198884119905 + 11988871199052) 1199011

+ (1198882+ 1198885119905 + 11988881199052) 11990111199051

+ (1198883 + 1198886119905 + 11988891199052) 11990111199052

1

120597Pitch (119905)120597Pitch (119905

2)= (1198881+ 1198884119905 + 11988871199052) 1199012

+ (1198882+ 1198885119905 + 11988881199052) 11990121199052

+ (1198883+ 1198886119905 + 11988891199052) 11990121199052

2

120597Pitch (119905)120597Pitch (1199053)

= (1198881 + 1198884119905 + 11988871199052) 1199013

+ (1198882 + 1198885119905 + 11988881199052) 11990131199053

+ (1198883+ 1198886119905 + 11988891199052) 11990131199052

3

120597Pitch (119905)120597Pitch (119905

4)= (1198881+ 1198884119905 + 11988871199052) 1199014

+ (1198882+ 1198885119905 + 11988881199052) 11990141199054

+ (1198883 + 1198886119905 + 11988891199052) 11990141199052

4

(13)

Similarly the differential expression of the other elementsof exterior orientation (Roll Yaw) and (119883

119904 119884119904 119885119904) can be

derived

3 Systematic Error Model

The systematic error model for the interior orientation isto describe the various distortions from satellitersquos sensorsuch as the CCD-array distortions the distortions of opticlenses and principal pointrsquos distortion In order to realizethe high-precision geometric correction for ZY-3 image it isvery necessary to establish the various error models basedon the analysis of satellitersquos structural parameters thus thesystem error coming from the interior orientation radialdirection and tangential direction distortion of optics lensand CCD-linersquos distortion and rotation will be modeled [1617] According to the analysis of satellitersquos structure and theimaging characteristics the systematic error model of theinterior orientation is established and represented by

Δ119909 = minusΔ119891

119891119909 + (119896

11199032+ 11989621199034) 119909

+ 1199011(1199032+ 21199092) + 2119901

2119909119910 + 119910 sin 120579


Δ119910 = minusΔ119891

119891119910 + (1198961119903

2+ 11989621199034) 119910

+ 1199012(1199032+ 21199102) + 2119901

1119909119910 + 119904

119910119910

119909 = (119909 minus 1199090) 119910 = (119910 minus 119910

0) 119903 = radic119909

2+ 1199102

(14)

where (minusΔ119891119891)119909 and (minusΔ119891119891)119910 represent the errors gen-erated by the image principal point and focal length Δ119891and 119891 mean the difference of focal length and the opticfocal length respectively (1198961119903

2+ 11989621199034)119909 and (1198961119903

2+ 11989621199034)119910

describe the optics lens distortion of radial direction inalong-track and cross-track directions respectively 1198961 1198962mean the distortionrsquos coefficient of radial direction 119903 meansthe distance from one point on the optic lens to the lensrsquoscenter 1199011(119903

2+ 21199092) + 21199012119909119910 and 1199012(119903

2+ 21199102) represent the

distortions of tangential direction in along-track and cross-track directions respectively 119901

1 1199012mean the distortionrsquos

coefficient of tangential direction 119910 sin 120579 represents theerror of CC-array rotation and 120579 is the rotation angle 119904

119910119910

represents the distortion of CCD-array in the cross-trackdirection generated by the temperature variation The CCD-array distortion in along-track direction is particle owingto only one CCD arranged in this direction the distortiontherefore can be ignored

According to the analysis of the correlations among themodelrsquos parameters in the block bundle adjustment thecorrelation between the principal point and focal length isvery strong so that the parameters are combined in order toreduce the parameters correlation and improve the stabilityand accuracy of the block bundle adjustment Equation (14)is represented as (15) after the parameters combination

Δ119909 = 1199090+ (11989611199032+ 11989621199034) 119909 + 119901

1(1199032+ 21199092)

+ 21199012119909119910 + 119910 sin 120579

Δ119910 = 1199100+ (11989611199032+ 11989621199034) 119910 + 119901

2(1199032+ 21199102)

+ 21199011119909119910 + 119904119910119910

119909 = (119909 minus 1199090) 119910 = (119910 minus 119910

0) 119903 = radic119909

2+ 1199102

(15)

4 Data and Method of Experiment

In this paper data set of orbit and attitude used to validatethe correction and accuracy of piece-point polynomialmodelis acquired from 609th track of ZY-3 In order to validatethe high accuracy of the new proposed trajectory model theLPM PPM LCPM piece-point with weight polynomial withfour known times model (PWP4M) and piece-point withweight polynomial model with all known times (PWPM)are utilized to interpolate and compare the interpolationaccuracy In the process the different numbers of the knowntimes 10 15 and 20 are selected from trajectory data set andare used to interpolate the other unknown timesrsquo orbit andattitude data with the different models respectively Finally

the result of interpolation is represented by the table andcurve and the advantage of PWPM is illuminated accordingto researching and analyzing the result

In order to validate the correction and accuracy ofPWPM ZY-3 orbit and attitude data ground control point(GCP) and systematic error model of interior orientation areused in the bundle block adjustment of geometric correctionBased on the nadir image of ZY-3 the 74 GCPs are pickedup from the image and 27 GCPs are selected as checkpoints (CPs) that do not take part in the block bundleadjustment For validating the correction and stability of theproposed models the 16 26 36 and 46 GCPs are performedrespectively in the geometric correction experiment Figure 2shows the distribution of GCPs and the errorrsquos distributionof image points corresponding to GCPs before geometriccorrection

5 Result and Validation of Experiment

51 Result and Analysis of Trajectory Model According toanalyzing the stability of the orbit and attitude of ZY-3 it canbe seen obviously that the orbit and attitude angles of Yawof ZY-3 are very stable but the attitude angles of Pitch andRoll are unstable relativelyThe curves of attitude angles in 10seconds are shown in Figures 3 4 and 5 From the diagramof curves it is obvious that the attitude angles of Pitch andRoll are unstable Hence the interpolation experiment isperformed using the angle Pitch and Roll

In Figure 6 the result of interpolation is represented usingthe different trajectory models with 15 selected known timesin a scene image The curves on the left of Figure 6 show theinterpolationrsquos result with angle Pitch and the curves on theright of Figure 6 show the result with angle YawThe red curveand red circle respectively mean the fitting curve and theselected known times and the green curvemeans the originalcurves From top to bottom Figures 6(a) 6(b) 6(c) 6(d) and6(e) represented respectively themodels LPM PPM LCPMPWP4M and PWPM

From Figure 6 it can be seen clearly that the interpola-tionrsquos accuracy with LCPM PWP4M and PWPM is higherthan LPM and PPM The curves of LCPM PWP4M andPWPM are relatively similar Furthermore Table 1 shows theinterpolation accuracy results with the different orbit andattitude models using the different numbers of known timesselected from609th track orbit in a scene imageThe accuracyof the interpolation for the angles Roll Pitch and Yaw withPWPM is the highest with 10 and 15 selected known timesUsing the 20 known times the interpolation accuracy of Yawis the highest with PWP4M 4539The accuracy of Pitch withPWP4M is 5727 lower than the accuracieswith PWPM5593In this case PWP4M and PWPM can be used together inorder to acquire the highest accuracy of interpolation for anyangle

52 Result and Analysis of Geometric Correction Based onPWPM The new proposed trajectory model (PWPM) hashigher interpolationrsquos accuracy and more flexibility than theothermodels according to the upper experiment and analysis


Y(p

ixel

)

X (pixel)

(a)

Y(p

ixel

)

X (pixel)

(b)

Figure 2 (a) Diagram of the distribution of GCPs (b) the errorrsquos distribution of image points corresponding to GCPs

4 1086 12 14 16

Roll

angl

e times10minus5

X (s)

Roll0

minus2

minus4

Y(∘

)

Figure 3 The attitude angle (Roll) curve in 10 seconds

4 6 8 10 12 14 16

Pitch

Pitc

h an

gle

X (s)

Y(∘

)

times10minus0

minus1

minus05

minus15

Figure 4 The attitude angle (Pitch) curve in 10 seconds

In order to validate the correction and accuracy of PWPMin the bundle block adjustment the geometric correctionexperiment is performed using the data set of ZY-3 Beforethe process which one interpolationrsquos method of PWPM isutilized according to the analysis of the orbit and attitudecorresponding to the used image range Thus geometriccorrection is performed and the result of correction isrepresented by Figure 7

In Figure 7(a) the residuals distribution of 46 GCPs afterthe geometric correction is represented and the residuals dis-tribution after checking with 27 CPs is shown in Figure 7(b)and the assessed accuracy is 00793 pixels From Figure 7 itis obvious that the accuracy of geometric correction is veryhigh based on the PWPM Furthermore Table 2 shows theassessment of accuracy for geometric correction using the 1016 26 36 and 46 GCPs respectively The accuracies with thedifferent number of GCPs are all high the highest accuracyreaches 05293 pixels with 26 GCPs and the lowest accuracyis 00841 pixels With the increasing number of GCPs the

Yaw

angl

e

Yaw

4 6 8 10 12 14 160034

0035

0036

X (s)

Y(∘

)

Figure 5 The attitude angle (Yaw) curve in 10 seconds

accuracy of geometric correction increased gradually untilthe 26 GCPs

53 Validation and Analysis Analyzing and comparingTable 1 and Figure 6 it is obvious that the interpolationaccuracy of PWPMis the highestWhen the orbit and attitudeare unstable the weight value is acquired with the reciprocalvalue of the absolute value of time differenceOn the contrarythe weight value is calculated with the reciprocal value ofthe square of time difference In comparison with the LPMPPM and LCPM PWPM can solve the different parameterscorresponding to the different attitude at any time with thedifferent weight value The proposed new trajectory modelcan therefore reach higher accuracy of interpolation thanothers especially when the orbit and attitude are unstableIn addition the PWPM has two interpolation methodsPWP4M and PWPM and the two methods can be usedtogether in order to acquire the higher interpolation accuracyOwing to the higher accuracy the new trajectory modelhas the ability that can improve the interpolation accuracyof orbit and attitude and avoid the interpolation error istransmitted into interior orientation as a part of systematicerror which will increase the systematic error of interiororientation Thus the solved accuracy of parameters in thesystematic model is improved and the accuracy of geometriccorrection is also increased correspondingly According tothe analysis and research the different form of PWPM inthe bundle block adjustment only relates to weight selected


4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

times10minus5

times10minus5

times10minus5

times10minus5

times10minus5

0

minus2

minus4

0

minus2

minus4

Roll

Roll

0

minus2

minus4

Roll

0

minus2

minus4

Roll

0

minus2

minus4

Roll

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

times10minus5

times10minus5

times10minus5

times10minus5

times10minus5

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

Time

X (s)

Time

X (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

Time

X (s)

Time

X (s)

Y(∘

)Y

(∘)

Y(∘

)Y

(∘)

Y(∘

)

Roll

Roll

Roll

Roll

Roll

Y(∘

)Y

(∘)

Y(∘

)Y

(∘)

Y(∘

)

(a)

(b)

(c)

(d)

(e)

Figure 6 The fitting curves with the different trajectory models

known time and needed interpolation time thus the cor-relation among the orbit and attitude is decreased and thecalculationrsquos accuracy and stability of the block adjustmentcan be improved

In the geometric correction experiment based on thePWPM the accuracies with the different number of GCPs

also reach a high level totally which is represented by Table 2and Figure 7 and validate the correctness and applicabilityof the PWPM For the different number of GCPs theaccuracy varies in the geometric correction mostly owingto two reasons the distribution of the different GCPs andthe correlations among the parameters of the systematic


Table 1 The fitting accuracy comparison of the different attitude and orbit modes selecting the different known times on the orbit (unitdegree)

120590 (119890 minus 007) 10 15 20Angle Roll Pitch Yaw Roll Pitch Yaw Roll Pitch Yaw1-LPM 64610 12772 15100 47859 9103 15297 47362 8569 151222-PPM 63136 12237 16210 47768 8737 15027 47281 8532 149483-LCPM 63523 11955 13919 51758 8275 10257 45901 5946 95544-PWP4M 63668 11732 13956 34760 8034 77186 15628 5727 45395-PWPM 60287 10387 12534 32837 7648 75958 22261 5593 5140

Y(p

ixel

)

X (pixel)

(a)

Y(p

ixel

)

X (pixel)

(b)

Figure 7 Diagram of geometric correction (a) the residuals of GCP (b) assessment with CP

Table 2The assessment of accuracy for geometric correction (unitpixel)

Number of GCPs 120590119909

120590119910

120590sum

46 00767 00215 0079736 00706 00217 0073926 00508 00146 0052916 00725 00161 0074310 00795 00276 00841

error model On the one hand the various distributions ofthe different GCPs will cause the accuracy to oscillate ina very small range on the other hand the bundle blockadjustment will generate correlations among the parametersThe correlations result in the following the solved results ofparameters vibrate in a range unstably and the systematicerror cannot be described completely and accurately withthese parameters Thus a better systematic error model isneeded to be proposed according to further researching andanalyzing of the satellite sensorrsquos overall structure design andthe imaging geometric characteristics

6 Conclusion

In this study the new trajectory model PWPM is proposedaccording to the researching and analyzing of the data ofZY-3rsquos orbit and attitude in the long and short period By

comparison with the other trajectory models the PWPMcan acquire a higher interpolationrsquos accuracy and has moreflexibility Meanwhile the differentiation equation of theproposed trajectory model is derivate and it is validatedthrough the bundle block adjustment In the geometriccorrection experiment based on the PWPM the accuraciesof geometric correction with the different number of GCPsalso reach a high level totally According to the analyzing andresearching of the assessment results with GCPs and CPsthe correctness and applicability of the PWPM are validatedand assessed to ensure and improve the high accuracy ofgeometric correction for ZY-3 satelliteThe further study willbe performed to experiment with the real image data of ZY-3 and GCP to research better systematic error model forinterior orientation in order to explore the potentials of usingZY-3 data for stereo mapping

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] F Long W Zhang and J Liu ldquoEffect of satellite attitude controlaccuracy on TDI CCD camerasrdquo Journal of Harbin Institute ofTechnology vol 34 no 3 pp 382ndash384 2002


[2] C Sun X Tang Z Qiu and X Wu ldquoIntroducing ZY-3 Chinarsquosfirst civilian high-res stereo mapping satelliterdquo in Proceedings ofthe Congress of the International Society for Photogrammetry andRemote Sensing (ISPRS rsquo12) Melbourne Australia September2012

[3] G Zhang Rectification for high resolution remote sensing imageunder lack of ground control points [PhD thesis] WuhanUniversity Wuhan China 2005

[4] A Bouillon E Breton F De Lussy and R Gachet ldquoSPOT5 geo-metric image qualityrdquo in Proceedings of the IEEE InternationalGeoscience and Remote Sensing Symposium (IGARSS rsquo03) vol 1pp 303ndash305 Toulouse France 2003

[5] T Toutin ldquoGeometric processing of remote sensing imagesmodels algorithms and methodsrdquo International Journal ofRemote Sensing vol 25 no 10 pp 1893ndash1924 2004

[6] K Jacobsen ldquoCalibration of optical satellite sensorsrdquo inProceed-ings of the International Calibration and Orientation WorkshopEuroCOW Casteldefels Spain 2006

[7] D Mulawa ldquoOn-orbit geometric calibration of the orb-view3high-resolution imaging satelliterdquo in Proceedings of the ISPRS20th Congress Commission 1 Remote Sensing and SpatialInformation Sciences Istanbul Turkey July 2004

[8] X Li L Zhang and W Xu ldquoPrecise acquisition of ZY-3 orbitand attitude parameters based on metadata filerdquo Journal ofAtmospheric and Environmental Optics vol 3 no 8 pp 166ndash173 2013

[9] F J Ponzoni J Zullo Jr R A C Lamparelli G Q Pellegrinoand Y Arnaud ldquoIn-flight absolute calibration of the Landsat-5 TM on the test site salar de uyunirdquo IEEE Transactions onGeoscience and Remote Sensing vol 42 no 12 pp 2761ndash27662004

[10] C Valorge ldquo40 years of experience with SPOT in-flight calibra-tionrdquo in Proceedings of the ISPRS Workshop on Radiometric andGeometric Calibration Gulfport Miss USA December 2003

[11] S Kocaman and A Gruen ldquoOrientation and self-calibration ofALOS PRISM imageryrdquo Photogrammetric Record vol 23 no123 pp 323ndash340 2008

[12] S Riazanoff SPOT 123-4-5 Geometry Handbook GAELConsul-tant 2004 httpwww-igmuniv-mlvfrsimriazanopublicationsGAEL-P135-DOC-001-01-04pdf

[13] X Zhu G Zhang X Tang and L Zhai ldquoResearch andapplication of CBRS02B image geometric exterior calibrationrdquoGeography andGeo-Information Science vol 25 no 3 pp 16ndash182009

[14] G Zhang Z Li H Pan Q Qiang and L Zhai ldquoOrientationof spaceborne SAR stereo pairs employing the RPC adjustmentmodelrdquo IEEE Transactions on Geoscience and Remote Sensingvol 49 no 7 pp 2782ndash2792 2011

[15] X Tang G Zhang X Zhu et al ldquoTriple linear-array imaginggeometry model of Ziyuan-3 surveying satelite and its valida-tionrdquo Acta Geodaetica et Cartographica Sinica vol 41 no 2 pp191ndash198 2012

[16] D PoliModelling of spaceborne linear array sensors [PhD the-sis] Swiss Federal Institute of Technology Zurich Switzerland2005

[17] D Poli ldquoIndirect georeferencing of airborne multiline arraysensors a simulated case studyrdquo in Proceedings of the ISPRSCommission Symposium International Archives of Photogram-metry and Remote Sensing vol 34 part B3 pp 246ndash251 GrazAustria September 2002 part B3

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Distributed Sensor Networks


Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014


ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014


Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications


Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia


Biomedical Imaging


ArtificialNeural Systems

Advances in


RoboticsJournal of



Computational Intelligence and Neuroscience

Industrial EngineeringJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in



(orbit and attitude) with trajectory model will also be trans-mitted to interior orientation as a part of systematic errorincrease the systematic error of interior orientation anddecrease the solved accuracy and stability of geometric dataprocessing An ideal trajectory model not only can ensure ahigh interpolation accuracy for the attitude and orbit of everyimage line but also can decrease and eliminate the strongcorrelations among the solved parameters in bundle blockadjustment and reduce the transmission of systematic errorbetween the interior and exterior orientation models

Generally the satellite running trajectory is relativelystable in a short period so that the orbit and attitude in ashort interval trajectory can bemodeledwith the polynomialtherefore avoiding the complex stress analysis of the satellite[8] therefore avoiding the complex stress analysis of thesatellite Currently the trajectory models used for high-resolution remote-sensing satellite have polynomial model(LPM) piecewise polynomial model (PPM) and Lagrangecubic polynomial model proposed by Hofmann (LCPM) [9ndash11] By research and comparison with the trajectory modelshowever these models have some limitations The LPM andPPM can acquire the smooth fitted curves and the inter-polation accuracies are very low especially for the unstablyoscillated trajectory data The LCPM can acquire a higherinterpolation accuracy comparing with the LPM and PPMhowever the LCPM easily causes the data oscillation whenthe higher-order model is utilized in the block adjustment

For satellite ZY-3 a new trajectory model (piece-pointpolynomial with weight model) is proposed to acquire thehigher interpolationrsquos accuracy in this paper based on ana-lyzing the orbit and attitude data of ZY-3 In addition thedata set of ZY-3 is used to validate the correction of piece-point with weight polynomial model and compare it withthe other trajectory models LPM PPM and LCPM usedfor other satellites Meanwhile the differentiation equation ofthe proposed trajectory model is derivate and it is validatedwith the block bundle adjustment According to geometriccorrection experiment with the different ground controlpoints the correctness and applicability of piece-point withweight polynomialmodel are validated and assessed to ensureand improve the high accuracy of geometric correction forZY-3 satellite

2 ZY-3 Trajectory Model

21 Piece-Point with Weight Polynomial Model Trajectorymodel is a mathematic relationship elucidating the orbit andattitude of satellite vary with the different time in its trackFor the push-broom satellite every acquired image line has adifferent data of orbit and attitude and the instrument justrecords the data at regular intervals but not all the dataThe unrecorded data at a certain time thereby needs to beinterpolated using exterior orientation model

Piece-point with weight polynomial model (PWPM) isproposed according to the researching and analyzing ofthe data of ZY-3rsquos orbit and attitude in the long and shortperiod In comparison with the other models LPM PPMand LCPM the weight value is used in the new model to

pt119894+2pt119894minus1

timinus1 ti ti+2

pt119894 pt119894+1

ti+middotmiddotmiddot

pt119894+middotmiddotmiddot

Direction of flightti+1m

M

Figure 1 Diagram of piece-point with weight polynomial model

perform interpolationrsquos calculationThe newmodel thereforecan acquire a higher accuracy have a better flexibility andcan reduce the correlation among the exterior orientationelements The PWPM is an interpolation model with weightvalue Using the model to interpolate satellitersquos orbit andattitude the weight is calculated by the difference from anyinterpolationrsquos time to the time assumed as known Throughthe least square method the polynomial parameters aresolved and then the exterior orientation at any time on theorbit can be acquired with the polynomial parameters ThePWPMis represented by (1) and theweight value is describedwith 119875

119883119878119905= 119883119894

0+ 119883119894

1sdot 119905 + 119883

119894

2sdot 1199052

119884119878119905= 119884119894

0+ 119884119894

1sdot 119905 + 119884

119894

2sdot 1199052

119885119878119905= 119885119894

0+ 119885119894

1sdot 119905 + 119885

119894

2sdot 1199052

120601119905= 120601119894

0+ 120601119894

1sdot 119905 + 120601

119894

2sdot 1199052

120596119905= 120596119894

0+ 120596119894

1sdot 119905 + 120596

119894

2sdot 1199052

120581119905= 120581119894

0+ 120581119894

1sdot 119905 + 120581

119894

2sdot 1199052

119875 =1

1003817100381710038171003817119905 minus 1199051198941003817100381710038171003817

119875 =1

(119905 minus 119905119894)2

(1)

Figure 1 is the diagram of PWPM 119872 is a ground point Itsimaging point is shown by119898 119905 is the time that the point119898 isimaged On the orbit 119905

119894minus1 119905119894 119905119894+1

119905119894+2

and 119905119894+sdotsdotsdot

are the knowntimes that the exterior orientations are acquiredwithGPS andstar sensor which are represented by red square119901119894minus1 119901119894 119901119894+1119901119894+2 and 119901119894+sdotsdotsdot respectively represent the weight of the time119905to the known time 119905119894minus1 119905119894 119905119894+1 119905119894+2 and 119905119894+sdotsdotsdot The unrecordedorbit and attitude at every time like time 119905 can be interpolatedwith the exterior orientations at the time 119905119894minus1 119905119894 119905119894+1 119905119894+2 and119905119894+sdotsdotsdot using PWPM

Assuming that the four known times 1199051 1199052 1199053 and 1199054

for orbit and attitude are selected as known points thevalue of orbit and attitude at time t can be calculated andacquired with PWPM The same polynomial coefficients aresolved in the piece-wise polynomial model for the orbit andattitude respectively For the convenient expression the error



119881 = 119860 sdot 119883 minus 119871 (2)

V119894 = 120581119894

0+ 120581119894

1sdot 119905 + 120581

119894

2sdot 1199052minus 120581119905 (119894 = 1 2 3 4) (3)


119860 =

[[[[[[[

[

1 1199051 1199052

1

1 11990521199052

2

1 1199053 1199052

3

1 11990541199052

4

]]]]]]]

]

119871 =

[[[

[

1205811

1205812

1205813

1205814

]]]

]

119883 = [

[

1198960

1198961

1198962

]

]

(4)


1199012 1199013 and 119901

4for times 119905

1 1199052 1199053 and 119905

4 respectively are


1 1199052 1199053 and 119905

4for the

unknown time 119905

1199011=

1

(119905 minus 1199051)2 119901

2=

1

(119905 minus 1199052)2

1199013 =

1

(119905 minus 1199053)2 1199014 =

1

(119905 minus 1199054)2

(5)

119875 =

[[[

[

1199011

1199012

1199013

1199014

]]]

]

(6)


119873

= 119860119879119875119860

=

[[[[[

[

1199011+ sdot sdot sdot + 119901

411990111199051+ sdot sdot sdot + 119901

4119905411990111199052


41199052

4

11990111199051+ sdot sdot sdot + 119901

4119905411990111199052


41199052

411990111199053


41199053

4

11990111199052


41199052

411990111199053


41199053

411990111199054


41199054

4

]]]]]

]

119880 =

[[[[

[


4sdot 1199014


1205811 sdot 1199052


2

4sdot 1199014

]]]]

]

(7)


119873minus1= [

[

119888111988821198883

119888411988851198886

119888711988881198889

]

]

119880 = [

[

1199061

1199062

1199063

]

]

(8)


0 1198961 and 119896


119883 = [

[

1198960

1198961

1198962

]

]

= [

[

119888111988821198883

119888411988851198886

1198887 1198888 1198889

]

]

sdot [

[

1199061

1199062

1199063

]

]

(9)


120581 (119905) = (11988811199061+ 11988821199062+ 11988831199063)

+ (11988841199061+ 11988851199062+ 11988861199063) 119905

+ (11988871199061+ 11988881199062+ 11988891199063) 1199052

(10)










camera are needed

[

[

119883

119884

119885

]

]WGS84

= 119898 sdot 119877 sdot [

[

[

[

119863119909

119863119910

119863119911

]

]

+ [

[

119889119909

119889119910

119889119911

]

]


119909

119910

minus119891

)]

]

+ [

[

119883

119884

119885

]

]GPS

119877 = 119877WGS84J2000 sdot 119877


orbitbody

(11)


tem [119883 119884 119885]119879

GPS and [119883 119884 119885]119879








119879



1 1199052 1199053 and


values 1199011 1199012 1199013 and 119901



Pitch (119905) = (11988811199061 + 11988821199062 + 11988831199063)

+ (11988841199061+ 11988851199062+ 11988861199063) 119905

+ (11988871199061+ 11988881199062+ 11988891199063) 1199052

(12)


1 1199052

1199053 and 119905


120597Pitch (119905)120597Pitch (119905

1)= (1198881+ 1198884119905 + 11988871199052) 1199011

+ (1198882+ 1198885119905 + 11988881199052) 11990111199051

+ (1198883 + 1198886119905 + 11988891199052) 11990111199052

1

120597Pitch (119905)120597Pitch (119905

2)= (1198881+ 1198884119905 + 11988871199052) 1199012

+ (1198882+ 1198885119905 + 11988881199052) 11990121199052

+ (1198883+ 1198886119905 + 11988891199052) 11990121199052

2

120597Pitch (119905)120597Pitch (1199053)

= (1198881 + 1198884119905 + 11988871199052) 1199013

+ (1198882 + 1198885119905 + 11988881199052) 11990131199053

+ (1198883+ 1198886119905 + 11988891199052) 11990131199052

3

120597Pitch (119905)120597Pitch (119905

4)= (1198881+ 1198884119905 + 11988871199052) 1199014

+ (1198882+ 1198885119905 + 11988881199052) 11990141199054

+ (1198883 + 1198886119905 + 11988891199052) 11990141199052

4

(13)


119904 119884119904 119885119904) can be

derived



Δ119909 = minusΔ119891

119891119909 + (119896

11199032+ 11989621199034) 119909

+ 1199011(1199032+ 21199092) + 2119901

2119909119910 + 119910 sin 120579


Δ119910 = minusΔ119891

119891119910 + (1198961119903

2+ 11989621199034) 119910

+ 1199012(1199032+ 21199102) + 2119901

1119909119910 + 119904

119910119910

119909 = (119909 minus 1199090) 119910 = (119910 minus 119910

0) 119903 = radic119909

2+ 1199102

(14)


2+ 11989621199034)119909 and (1198961119903

2+ 11989621199034)119910


2+ 21199092) + 21199012119909119910 and 1199012(119903





119910119910



Δ119909 = 1199090+ (11989611199032+ 11989621199034) 119909 + 119901

1(1199032+ 21199092)

+ 21199012119909119910 + 119910 sin 120579

Δ119910 = 1199100+ (11989611199032+ 11989621199034) 119910 + 119901

2(1199032+ 21199102)

+ 21199011119909119910 + 119904119910119910

119909 = (119909 minus 1199090) 119910 = (119910 minus 119910

0) 119903 = radic119909

2+ 1199102

(15)











Y(p

ixel

)

X (pixel)

(a)

Y(p

ixel

)

X (pixel)

(b)


4 1086 12 14 16

Roll

angl

e times10minus5

X (s)

Roll0

minus2

minus4

Y(∘

)


4 6 8 10 12 14 16

Pitch

Pitc

h an

gle

X (s)

Y(∘

)

times10minus0

minus1

minus05

minus15




Yaw

angl

e

Yaw

4 6 8 10 12 14 160034

0035

0036

X (s)

Y(∘

)





4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

times10minus5

times10minus5

times10minus5

times10minus5

times10minus5

0

minus2

minus4

0

minus2

minus4

Roll

Roll

0

minus2

minus4

Roll

0

minus2

minus4

Roll

0

minus2

minus4

Roll

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

times10minus5

times10minus5

times10minus5

times10minus5

times10minus5

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

Time

X (s)

Time

X (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

Time

X (s)

Time

X (s)

Y(∘

)Y

(∘)

Y(∘

)Y

(∘)

Y(∘

)

Roll

Roll

Roll

Roll

Roll

Y(∘

)Y

(∘)

Y(∘

)Y

(∘)

Y(∘

)

(a)

(b)

(c)

(d)

(e)








Y(p

ixel

)

X (pixel)

(a)

Y(p

ixel

)

X (pixel)

(b)




120590119910

120590sum

46 00767 00215 0079736 00706 00217 0073926 00508 00146 0052916 00725 00161 0074310 00795 00276 00841


6 Conclusion





References


























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





119881 = 119860 sdot 119883 minus 119871 (2)

V119894 = 120581119894

0+ 120581119894

1sdot 119905 + 120581

119894

2sdot 1199052minus 120581119905 (119894 = 1 2 3 4) (3)


119860 =

[[[[[[[

[

1 1199051 1199052

1

1 11990521199052

2

1 1199053 1199052

3

1 11990541199052

4

]]]]]]]

]

119871 =

[[[

[

1205811

1205812

1205813

1205814

]]]

]

119883 = [

[

1198960

1198961

1198962

]

]

(4)


1199012 1199013 and 119901

4for times 119905

1 1199052 1199053 and 119905

4 respectively are


1 1199052 1199053 and 119905

4for the

unknown time 119905

1199011=

1

(119905 minus 1199051)2 119901

2=

1

(119905 minus 1199052)2

1199013 =

1

(119905 minus 1199053)2 1199014 =

1

(119905 minus 1199054)2

(5)

119875 =

[[[

[

1199011

1199012

1199013

1199014

]]]

]

(6)


119873

= 119860119879119875119860

=

[[[[[

[

1199011+ sdot sdot sdot + 119901

411990111199051+ sdot sdot sdot + 119901

4119905411990111199052


41199052

4

11990111199051+ sdot sdot sdot + 119901

4119905411990111199052


41199052

411990111199053


41199053

4

11990111199052


41199052

411990111199053


41199053

411990111199054


41199054

4

]]]]]

]

119880 =

[[[[

[


4sdot 1199014


1205811 sdot 1199052


2

4sdot 1199014

]]]]

]

(7)


119873minus1= [

[

119888111988821198883

119888411988851198886

119888711988881198889

]

]

119880 = [

[

1199061

1199062

1199063

]

]

(8)


0 1198961 and 119896


119883 = [

[

1198960

1198961

1198962

]

]

= [

[

119888111988821198883

119888411988851198886

1198887 1198888 1198889

]

]

sdot [

[

1199061

1199062

1199063

]

]

(9)


120581 (119905) = (11988811199061+ 11988821199062+ 11988831199063)

+ (11988841199061+ 11988851199062+ 11988861199063) 119905

+ (11988871199061+ 11988881199062+ 11988891199063) 1199052

(10)










camera are needed

[

[

119883

119884

119885

]

]WGS84

= 119898 sdot 119877 sdot [

[

[

[

119863119909

119863119910

119863119911

]

]

+ [

[

119889119909

119889119910

119889119911

]

]


119909

119910

minus119891

)]

]

+ [

[

119883

119884

119885

]

]GPS

119877 = 119877WGS84J2000 sdot 119877


orbitbody

(11)


tem [119883 119884 119885]119879

GPS and [119883 119884 119885]119879








119879



1 1199052 1199053 and


values 1199011 1199012 1199013 and 119901



Pitch (119905) = (11988811199061 + 11988821199062 + 11988831199063)

+ (11988841199061+ 11988851199062+ 11988861199063) 119905

+ (11988871199061+ 11988881199062+ 11988891199063) 1199052

(12)


1 1199052

1199053 and 119905


120597Pitch (119905)120597Pitch (119905

1)= (1198881+ 1198884119905 + 11988871199052) 1199011

+ (1198882+ 1198885119905 + 11988881199052) 11990111199051

+ (1198883 + 1198886119905 + 11988891199052) 11990111199052

1

120597Pitch (119905)120597Pitch (119905

2)= (1198881+ 1198884119905 + 11988871199052) 1199012

+ (1198882+ 1198885119905 + 11988881199052) 11990121199052

+ (1198883+ 1198886119905 + 11988891199052) 11990121199052

2

120597Pitch (119905)120597Pitch (1199053)

= (1198881 + 1198884119905 + 11988871199052) 1199013

+ (1198882 + 1198885119905 + 11988881199052) 11990131199053

+ (1198883+ 1198886119905 + 11988891199052) 11990131199052

3

120597Pitch (119905)120597Pitch (119905

4)= (1198881+ 1198884119905 + 11988871199052) 1199014

+ (1198882+ 1198885119905 + 11988881199052) 11990141199054

+ (1198883 + 1198886119905 + 11988891199052) 11990141199052

4

(13)


119904 119884119904 119885119904) can be

derived



Δ119909 = minusΔ119891

119891119909 + (119896

11199032+ 11989621199034) 119909

+ 1199011(1199032+ 21199092) + 2119901

2119909119910 + 119910 sin 120579


Δ119910 = minusΔ119891

119891119910 + (1198961119903

2+ 11989621199034) 119910

+ 1199012(1199032+ 21199102) + 2119901

1119909119910 + 119904

119910119910

119909 = (119909 minus 1199090) 119910 = (119910 minus 119910

0) 119903 = radic119909

2+ 1199102

(14)


2+ 11989621199034)119909 and (1198961119903

2+ 11989621199034)119910


2+ 21199092) + 21199012119909119910 and 1199012(119903





119910119910



Δ119909 = 1199090+ (11989611199032+ 11989621199034) 119909 + 119901

1(1199032+ 21199092)

+ 21199012119909119910 + 119910 sin 120579

Δ119910 = 1199100+ (11989611199032+ 11989621199034) 119910 + 119901

2(1199032+ 21199102)

+ 21199011119909119910 + 119904119910119910

119909 = (119909 minus 1199090) 119910 = (119910 minus 119910

0) 119903 = radic119909

2+ 1199102

(15)











Y(p

ixel

)

X (pixel)

(a)

Y(p

ixel

)

X (pixel)

(b)


4 1086 12 14 16

Roll

angl

e times10minus5

X (s)

Roll0

minus2

minus4

Y(∘

)


4 6 8 10 12 14 16

Pitch

Pitc

h an

gle

X (s)

Y(∘

)

times10minus0

minus1

minus05

minus15




Yaw

angl

e

Yaw

4 6 8 10 12 14 160034

0035

0036

X (s)

Y(∘

)





4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

times10minus5

times10minus5

times10minus5

times10minus5

times10minus5

0

minus2

minus4

0

minus2

minus4

Roll

Roll

0

minus2

minus4

Roll

0

minus2

minus4

Roll

0

minus2

minus4

Roll

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

times10minus5

times10minus5

times10minus5

times10minus5

times10minus5

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

Time

X (s)

Time

X (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

Time

X (s)

Time

X (s)

Y(∘

)Y

(∘)

Y(∘

)Y

(∘)

Y(∘

)

Roll

Roll

Roll

Roll

Roll

Y(∘

)Y

(∘)

Y(∘

)Y

(∘)

Y(∘

)

(a)

(b)

(c)

(d)

(e)








Y(p

ixel

)

X (pixel)

(a)

Y(p

ixel

)

X (pixel)

(b)




120590119910

120590sum

46 00767 00215 0079736 00706 00217 0073926 00508 00146 0052916 00725 00161 0074310 00795 00276 00841


6 Conclusion





References


























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in









camera are needed

[

[

119883

119884

119885

]

]WGS84

= 119898 sdot 119877 sdot [

[

[

[

119863119909

119863119910

119863119911

]

]

+ [

[

119889119909

119889119910

119889119911

]

]


119909

119910

minus119891

)]

]

+ [

[

119883

119884

119885

]

]GPS

119877 = 119877WGS84J2000 sdot 119877


orbitbody

(11)


tem [119883 119884 119885]119879

GPS and [119883 119884 119885]119879








119879



1 1199052 1199053 and


values 1199011 1199012 1199013 and 119901



Pitch (119905) = (11988811199061 + 11988821199062 + 11988831199063)

+ (11988841199061+ 11988851199062+ 11988861199063) 119905

+ (11988871199061+ 11988881199062+ 11988891199063) 1199052

(12)


1 1199052

1199053 and 119905


120597Pitch (119905)120597Pitch (119905

1)= (1198881+ 1198884119905 + 11988871199052) 1199011

+ (1198882+ 1198885119905 + 11988881199052) 11990111199051

+ (1198883 + 1198886119905 + 11988891199052) 11990111199052

1

120597Pitch (119905)120597Pitch (119905

2)= (1198881+ 1198884119905 + 11988871199052) 1199012

+ (1198882+ 1198885119905 + 11988881199052) 11990121199052

+ (1198883+ 1198886119905 + 11988891199052) 11990121199052

2

120597Pitch (119905)120597Pitch (1199053)

= (1198881 + 1198884119905 + 11988871199052) 1199013

+ (1198882 + 1198885119905 + 11988881199052) 11990131199053

+ (1198883+ 1198886119905 + 11988891199052) 11990131199052

3

120597Pitch (119905)120597Pitch (119905

4)= (1198881+ 1198884119905 + 11988871199052) 1199014

+ (1198882+ 1198885119905 + 11988881199052) 11990141199054

+ (1198883 + 1198886119905 + 11988891199052) 11990141199052

4

(13)


119904 119884119904 119885119904) can be

derived



Δ119909 = minusΔ119891

119891119909 + (119896

11199032+ 11989621199034) 119909

+ 1199011(1199032+ 21199092) + 2119901

2119909119910 + 119910 sin 120579


Δ119910 = minusΔ119891

119891119910 + (1198961119903

2+ 11989621199034) 119910

+ 1199012(1199032+ 21199102) + 2119901

1119909119910 + 119904

119910119910

119909 = (119909 minus 1199090) 119910 = (119910 minus 119910

0) 119903 = radic119909

2+ 1199102

(14)


2+ 11989621199034)119909 and (1198961119903

2+ 11989621199034)119910


2+ 21199092) + 21199012119909119910 and 1199012(119903





119910119910



Δ119909 = 1199090+ (11989611199032+ 11989621199034) 119909 + 119901

1(1199032+ 21199092)

+ 21199012119909119910 + 119910 sin 120579

Δ119910 = 1199100+ (11989611199032+ 11989621199034) 119910 + 119901

2(1199032+ 21199102)

+ 21199011119909119910 + 119904119910119910

119909 = (119909 minus 1199090) 119910 = (119910 minus 119910

0) 119903 = radic119909

2+ 1199102

(15)











Y(p

ixel

)

X (pixel)

(a)

Y(p

ixel

)

X (pixel)

(b)


4 1086 12 14 16

Roll

angl

e times10minus5

X (s)

Roll0

minus2

minus4

Y(∘

)


4 6 8 10 12 14 16

Pitch

Pitc

h an

gle

X (s)

Y(∘

)

times10minus0

minus1

minus05

minus15




Yaw

angl

e

Yaw

4 6 8 10 12 14 160034

0035

0036

X (s)

Y(∘

)





4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

times10minus5

times10minus5

times10minus5

times10minus5

times10minus5

0

minus2

minus4

0

minus2

minus4

Roll

Roll

0

minus2

minus4

Roll

0

minus2

minus4

Roll

0

minus2

minus4

Roll

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

times10minus5

times10minus5

times10minus5

times10minus5

times10minus5

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

Time

X (s)

Time

X (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

Time

X (s)

Time

X (s)

Y(∘

)Y

(∘)

Y(∘

)Y

(∘)

Y(∘

)

Roll

Roll

Roll

Roll

Roll

Y(∘

)Y

(∘)

Y(∘

)Y

(∘)

Y(∘

)

(a)

(b)

(c)

(d)

(e)








Y(p

ixel

)

X (pixel)

(a)

Y(p

ixel

)

X (pixel)

(b)




120590119910

120590sum

46 00767 00215 0079736 00706 00217 0073926 00508 00146 0052916 00725 00161 0074310 00795 00276 00841


6 Conclusion





References


























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Δ119910 = minusΔ119891

119891119910 + (1198961119903

2+ 11989621199034) 119910

+ 1199012(1199032+ 21199102) + 2119901

1119909119910 + 119904

119910119910

119909 = (119909 minus 1199090) 119910 = (119910 minus 119910

0) 119903 = radic119909

2+ 1199102

(14)


2+ 11989621199034)119909 and (1198961119903

2+ 11989621199034)119910


2+ 21199092) + 21199012119909119910 and 1199012(119903





119910119910



Δ119909 = 1199090+ (11989611199032+ 11989621199034) 119909 + 119901

1(1199032+ 21199092)

+ 21199012119909119910 + 119910 sin 120579

Δ119910 = 1199100+ (11989611199032+ 11989621199034) 119910 + 119901

2(1199032+ 21199102)

+ 21199011119909119910 + 119904119910119910

119909 = (119909 minus 1199090) 119910 = (119910 minus 119910

0) 119903 = radic119909

2+ 1199102

(15)











Y(p

ixel

)

X (pixel)

(a)

Y(p

ixel

)

X (pixel)

(b)


4 1086 12 14 16

Roll

angl

e times10minus5

X (s)

Roll0

minus2

minus4

Y(∘

)


4 6 8 10 12 14 16

Pitch

Pitc

h an

gle

X (s)

Y(∘

)

times10minus0

minus1

minus05

minus15




Yaw

angl

e

Yaw

4 6 8 10 12 14 160034

0035

0036

X (s)

Y(∘

)





4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

times10minus5

times10minus5

times10minus5

times10minus5

times10minus5

0

minus2

minus4

0

minus2

minus4

Roll

Roll

0

minus2

minus4

Roll

0

minus2

minus4

Roll

0

minus2

minus4

Roll

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

times10minus5

times10minus5

times10minus5

times10minus5

times10minus5

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

Time

X (s)

Time

X (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

Time

X (s)

Time

X (s)

Y(∘

)Y

(∘)

Y(∘

)Y

(∘)

Y(∘

)

Roll

Roll

Roll

Roll

Roll

Y(∘

)Y

(∘)

Y(∘

)Y

(∘)

Y(∘

)

(a)

(b)

(c)

(d)

(e)








Y(p

ixel

)

X (pixel)

(a)

Y(p

ixel

)

X (pixel)

(b)




120590119910

120590sum

46 00767 00215 0079736 00706 00217 0073926 00508 00146 0052916 00725 00161 0074310 00795 00276 00841


6 Conclusion





References


























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




Y(p

ixel

)

X (pixel)

(a)

Y(p

ixel

)

X (pixel)

(b)


4 1086 12 14 16

Roll

angl

e times10minus5

X (s)

Roll0

minus2

minus4

Y(∘

)


4 6 8 10 12 14 16

Pitch

Pitc

h an

gle

X (s)

Y(∘

)

times10minus0

minus1

minus05

minus15




Yaw

angl

e

Yaw

4 6 8 10 12 14 160034

0035

0036

X (s)

Y(∘

)





4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

times10minus5

times10minus5

times10minus5

times10minus5

times10minus5

0

minus2

minus4

0

minus2

minus4

Roll

Roll

0

minus2

minus4

Roll

0

minus2

minus4

Roll

0

minus2

minus4

Roll

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

times10minus5

times10minus5

times10minus5

times10minus5

times10minus5

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

Time

X (s)

Time

X (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

Time

X (s)

Time

X (s)

Y(∘

)Y

(∘)

Y(∘

)Y

(∘)

Y(∘

)

Roll

Roll

Roll

Roll

Roll

Y(∘

)Y

(∘)

Y(∘

)Y

(∘)

Y(∘

)

(a)

(b)

(c)

(d)

(e)








Y(p

ixel

)

X (pixel)

(a)

Y(p

ixel

)

X (pixel)

(b)




120590119910

120590sum

46 00767 00215 0079736 00706 00217 0073926 00508 00146 0052916 00725 00161 0074310 00795 00276 00841


6 Conclusion





References


























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

times10minus5

times10minus5

times10minus5

times10minus5

times10minus5

0

minus2

minus4

0

minus2

minus4

Roll

Roll

0

minus2

minus4

Roll

0

minus2

minus4

Roll

0

minus2

minus4

Roll

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

4 1086 12 14 16

times10minus5

times10minus5

times10minus5

times10minus5

times10minus5

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

0

minus2

minus4

Time

X (s)

Time

X (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

TimeX (s)

Time

X (s)

Time

X (s)

Y(∘

)Y

(∘)

Y(∘

)Y

(∘)

Y(∘

)

Roll

Roll

Roll

Roll

Roll

Y(∘

)Y

(∘)

Y(∘

)Y

(∘)

Y(∘

)

(a)

(b)

(c)

(d)

(e)








Y(p

ixel

)

X (pixel)

(a)

Y(p

ixel

)

X (pixel)

(b)




120590119910

120590sum

46 00767 00215 0079736 00706 00217 0073926 00508 00146 0052916 00725 00161 0074310 00795 00276 00841


6 Conclusion





References


























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






Y(p

ixel

)

X (pixel)

(a)

Y(p

ixel

)

X (pixel)

(b)




120590119910

120590sum

46 00767 00215 0079736 00706 00217 0073926 00508 00146 0052916 00725 00161 0074310 00795 00276 00841


6 Conclusion





References


























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



Documents

Research Article Research on the Trajectory Model for ZY-3downloads.hindawi.com/journals/tswj/2014/429041.pdfResearch Article Research on the Trajectory Model for ZY-3 YifuChenandZhongXie