Research ArticleBased on Grid Reference Frame for SINSCNS IntegratedNavigation System in the Polar Regions

Song Lijun 1 Zhao Wanliang2 Cheng Yuxiang2 and Chen Xiaozhen3

1School of Information amp Control Engineering Xirsquoan University of Architecture and Technology Xirsquoan 710055 China2Shanghai Aerospace Control Technology Institute Shanghai 201109 China3Beijing Aerospace Control Instrument Research Institute Beijing 100039 China

Correspondence should be addressed to Song Lijun songlijun9071sinacom

Received 6 August 2018 Accepted 25 October 2018 Published 1 January 2019

Guest Editor Junpei Zhong

Copyright copy 2019 Song Lijun et al 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

As the inertial navigation system cannot meet the precision requirements of global navigation in the special geographicalenvironment of the Polar Regions this paper presents Strapdown Inertial Navigation System (SINS)Celestial Navigation System(CNS) integrated navigation system of airborne based on Grid Reference Frame (GRF) and the simulation is carried out Theresult of simulation shows that the SINSCNS integrated navigation system is superior to the single subsystem in precision andperformance which not only effectively inhibits the error caused by gyro drift but also corrects the navigation parameters of systemwithout delay Comparing the simulation in the middle and low latitudes and in the Polar Regions the precision of SINSCNSintegrated navigation system is the same in the middle and low latitudes and in the Polar Regions

1 Introduction

With the development of science and technology the changeof economy military situation energy shipping scientificresearch and military value of Polar Regions are becomingmore and more prominent and the activities of countriesin the Polar Regions are also becoming more and morefrequently However the problems of existing navigationdevices are low reliability and poor security in the PolarRegions because there are few inhabitants around the placeunique topographic and geomorphic conditions

At present the arrangement of algorithm and the analysisof error are focused on the research of airborne navigationin the Polar Regions There is not established completetheoretical system which can support the actual navigationsystem of the Polar Regions there are many problems whichurgently need to be solved SINS is considered as the firstchoice of self-contained navigational aids of polar navigationbut SINS has the limitation of accumulating error with timeIt is difficult to only rely on SINS to complete the highprecision and long navigation of the airborne Therefore itis urgent to introduce the combination of external reference

information and inertial navigation system for data fusion inpolar navigation [1 2]

2 The SINSCNS Integrated NavigationSystem in the Middle and Low Latitudes

The CNS is a navigation which based on indestructiblenatural celestial bodiesThe star sensor is often used to detectthe stellar map to determine the attitude of the aircraftrelative to the inertial coordinate system and the result ofattitude is very high precision However the update rateof CNS is very low and it cannot provide real-time infor-mation about velocity and position of aircraft By the highprecision attitude of CNS SINSCNS integrated navigationsystem revises the error of SINS with time to improve themeasurement precision of integrated navigation system [3 4]

21 The Principle of SINSCNS Integrated Navigation SystemAccording to the different installation methods of star sensorand inertial device the working mode of SINSCNS inte-grated navigation systemcan be used as strapdownmodeThe

HindawiComplexityVolume 2019 Article ID 2164053 8 pageshttpsdoiorg10115520192164053

2 Complexity

Strap down Inertial Navigation System

Celestial Navigation System

navigation solution

Kalman filter

the transformation matrix

the transformation matrix

estimating error

minus

Attitude Velocity position

(SINS)

(CNS)

Figure 1 The structure of SINSCNS integrated navigation system

three-axis attitude of aircraft is given by SINS in SINSCNSintegrated navigation system and the star sensor outputthe transformation matrix of inertial coordinate systemrelative to the star sensor coordinate system The process ofcombinatorial is as follows

The first the transformationmatrix of inertial coordinatesystem relative to the star sensor coordinate system which iscalculated by the position and attitude of SINS The secondthe transformation matrix of SINS and the transformationmatrix of star sensor output which is subtracted as themeasurement sent to Kalman filter for information fusion toobtain the optimal estimation value of integrated navigationsystem The last the optimal estimation value of integratednavigation system which is used to adjust the error of SINSit is the result of integrated navigation system [5 6] Thestructure chart of SINSCNS integrated navigation system isshown in Figure 1

22 The State Space Mode of SINSCNS Integrated NavigationSystem in the Middle and Low Latitudes It is S coordinatesystem as themeasuring coordinate system of star sensor theoutput of star sensor is attitude matrix C119904119894 which is the starsensor coordinate system relative to the inertial coordinatesystem Due to the high precision of star sensor the error ofstar sensor can be ignored as long as the installation errorof star sensor is strictly calibrated [7 8] The transformationmatrix of star sensor can be obtained as follows

C119878119862119873119878 = C119904119894 + C119908 (1)

The attitude error C119908 can be considered as Gauss whitenoise

C119908 = [[[

11986211990811 11986211990812 1198621199081311986211990821 11986211990822 1198621199082311986211990831 11986211990832 11986211990833

]]]

(2)

The various C119908 are satisfied

119864 [119862119908119894119895 (119905)] = 0119864 [119862119908119894119895 (119905) 119862119908119894119895 (120591)] = 119902119908119894119895120575 (119905 minus 120591)

119894 119895 = 1 2 3(3)

The transformation matrix from the measurement coor-dinate system of star sensor S to body coordinate system bis C119904119887 the attitude matrix of SINS is C119887119899119888 the position matrix

of SINS is C119899119890119888 and the transformation matrix with inertialcoordinate system relative to terrestrial coordinate system isC119890119894 then the transformation matrix calculated by the SINS is

C119878119878119868119873119878 = C119904119887C119887119899119888C119899119890119888C119890119894

(4)

where C119887119899119888 = C119887119899[I + 120593119899times] C119899119890119888 = C119899119890 [I minus (120575119875times)]So

C119878119878119868119873119878 = C119904119887C119887119899119888C119899119890119888C119890119894

= C119904119887C119887119899 [I + 120601times]C119899119890 [I minus (120575119875times)]C119890119894

= C119904119887C119887119899C119899119890C119890119894 + C119904119887C119887119899 (120601times)C119899119890C119890119894

minus C119904119887C119887119899 (120575119875times)C119899119890C119890119894

(5)

where

[120601times] = [[[

0 minus120575120601119880 120575120601119873120575120601119880 0 minus120575120601119864minus120575120601119873 120575120601119864 0

]]]

[120575119875times] = [[[

0 minus120575120582 sin 119871 120575120582 cos 119871120575120582 sin 119871 0 120575119871minus120575120582 cos 119871 minus120575119871 0

]]]

(6)

When the difference of attitude angles measured bySINS and CNS is very small the cross-coupled term oftransformation matrix can be approximated to zero Themeasurement of SINSCNS integrated navigation system isthat the transformation matrix of star sensor subtracts fromthe transformation matrix calculated by SINS it is

Z119878119868119873119878119862119873119878 = C119878119878119868119873119878 minus C119878119862119873119878= C119904119887C

119887119899 (120601times)C119899119890C119890119894 minus C119904119887C119887119899 (120575119875times)C119899119890C119890119894

minus C119908(7)

120575120582 and 120575119871 are ascensional difference and declinationdifference it is a small angle of arc-second so it can beignored the influence by the longitude and latitude [120575119875times] =03times3 it can be simplified as follows

Z119878119868119873119878119862119873119878 = C119878119878119868119873119878 minus C119878119862119873119878 = C119904119887C119887119899 (120601times)C119899119890C119890119894 minus C119908 (8)

Complexity 3

Appointment

C119904119887C119887119899 = [[

[

11990411 11990412 1199041311990421 11990422 1199042311990431 11990432 11990433

]]]

C119899119890C119890119894 = [[[

11989911 11989912 1198991311989921 11989922 1198992311989931 11989932 11989933

]]]

(9)

It can be obtained

Z119878119868119873119878119862119873119878 =

[[[[[[[[[[[[[[[[[[[[

119885111198851211988513119885211198852211988523119885311198853211988533

]]]]]]]]]]]]]]]]]]]]

=

[[[[[[[[[[[[[[[[[[[[

1199041311989921 minus 1199041211989931 1199041111989931 minus 1199041311989911 1199041211989911 minus 11990411119899211199041311989922 minus 1199041211989932 1199041111989932 minus 1199041311989912 1199041211989912 minus 11990411119899221199041311989923 minus 1199041211989933 1199041111989933 minus 1199041311989913 1199041211989913 minus 11990411119899231199042311989921 minus 1199042211989931 1199042111989931 minus 1199042311989911 1199042211989911 minus 11990421119899211199042311989922 minus 1199042211989932 1199042111989932 minus 1199042311989912 1199042211989912 minus 11990421119899221199042311989923 minus 1199042211989933 1199042111989933 minus 1199042311989913 1199042211989913 minus 11990421119899231199043311989921 minus 1199043211989931 1199043111989931 minus 1199043311989911 1199043211989911 minus 11990431119899211199043311989922 minus 1199043211989932 1199043111989932 minus 1199043311989912 1199043211989912 minus 11990431119899221199043311989923 minus 1199043211989933 1199043111989933 minus 1199043311989913 1199043211989913 minus 1199043111989923

]]]]]]]]]]]]]]]]]]]]

[[[

120575120601119864120575120601119873120575120601119880

]]]

minus

[[[[[[[[[[[[[[[[[[[[

119862119908111198621199081211986211990813119862119908211198621199082211986211990823119862119908311198621199083211986211990833

]]]]]]]]]]]]]]]]]]]]

= 119867119878119868119873119878119862119873119878119883119868119873119878 minus 119881119862119873119878

(10)

23 The Simulation and Analysis of SINSCNS IntegratedNavigation System in the Middle and Low Latitudes Thesimulation time of SINSCNS integrated navigation systemin the middle and low latitudes is 600s the time of attitudeupdating velocity updating and position updating are 20msand the sampling period of inertial sensor is 10ms [9] In thesimulation the parameters are set as follows

The initial error of SINS is as followsThe initial error of attitude is [051015840 051015840 201015840] the initial

error of velocity is [001 001 001] ms and the initial errorof position is [20 20 20]119898

The parameters of inertial sensor are as follows

(1) The random constant drift of gyroscope is 001∘ℎ therandom walk coefficient of gyroscope is 0001∘radicℎand the error of scale factor is 30ppm

(2) The random constant drift of accelerometer is 40ugand the random walk coefficient of accelerometer is5ugradic119904

The parameters of star sensor are as followsThe horizontal measuring precision of angle is 1010158401015840 the

azimuth measuring precision of angle is 2010158401015840 and the error ofinstallation which between CNS and SINS is [31015840 31015840 31015840]T

These parameters are the same as in the Polar RegionsThe simulation results are showed in Figure 2

(1) The attitude error of SINSCNS in the middle andlow latitudes is showed as Figure 2(a) The attitude of CNSis considered as observation information in the SINSCNSintegrated navigation system the convergence rate of attitudeis quickened the measurement accuracy of SINSCNS inte-grated navigation system is improved effectively and after afew seconds the attitude error is up to 0151015840

(2)The velocity error and the position error of SINSCNSin the middle and low latitudes are showed as Figures 2(b)and 2(c) They are divergent in the SINSCNS integratednavigation system because CNS was unable to provide real-time information of the velocity and position of aircraft

(3) The constant drift of gyroscope and accelerometerof SINSCNS in the middle and low latitude are showed asFigures 2(d) and 2(e) The horizontal attitude error is themain error of integrated navigation system the influence ofazimuth attitude error is relatively weak The constant driftof gyroscope directly affects attitude error in the attitudeinformation which is given by the inertial navigation systemIt is better of the constant drift of gyroscope and it is diver-gent of the constant drift of accelerometer in the SINSCNSintegrated navigation system

3 The Algorithm of SINSCNSIntegrated Navigation System Based onGrid Reference Frame

In order to avoid the convergence of azimuth reference line atthe poles all the longitudes are redefined which are parallelto the Greenwich meridian [10 11] In this way the azimuthattitude is relative to the Greenwich meridian and its parallelline which is grid navigation The grid navigation and thegrid coordinate system have been described in the authorrsquosliterature [12] so that the author don not bore you withunnecessary details

The celestial navigation is an old method of navigationand position and the irreplaceable of celestial navigation isdetermined by the autonomy of celestial navigation Evenin the present age the advanced development of radionavigation system and the accuracy and timeliness of shippositioning have been well solved the celestial navigationis still unwavering in the navigation As early as 1989Guo Honggui has proposed the celestial navigation of PolarRegions in the literature [13]

4 Complexity

0 100 200 300 400 500 600minus05

0

05

0 100 200 300 400 500 600minus05

0

05

0 100 200 300 400 500 600minus20

0

20

e(

)

n()

Ｏ()

(a) The attitude error of SINSCNS in themiddle and low latitudes

0 100 200 300 400 500 6000

02

04

Ve

(ms

)

0 100 200 300 400 500 600minus02

0

02

Vn

(ms

)

0 100 200 300 400 500 6000

1

2

Vu

(ms

)

(b) The velocity error of SINSCNS in themiddle and low latitudes

0 100 200 300 400 500 600minus50

0

50

L

(m)

0 100 200 300 400 500 6000

100

200

(m)

0 100 200 300 400 500 6000

200

400

h

(m)

(c) The position error of SINSCNS in themiddle and low latitude

0 100 200 300 400 500 6000

001

002

0 100 200 300 400 500 600minus002

0

002

0 100 200 300 400 500 6000

001

002

＜Ｒ

(∘ Ｂ)

＜Ｓ

(∘ Ｂ)

＜Ｔ (

∘ Ｂ)

(d) The constant drift of gyroscope ofSINSCNS in the middle and low latitude

0 100 200 300 400 500 600399995

40

400005

nablax

( g)

0 100 200 300 400 500 600399995

40

400005

nablay

( g)

0 100 200 300 400 500 600399999

40

400001

nablaz (

g)

(e) The constant drift of accelerometer ofSINSCNS in the middle and low latitude

Figure 2

31 The Relationship of the Grid Orientation and the TrueNorth It is very difficult to navigate from the true northbecause of the convergence of polar meridians The northazimuth inertial navigation system is adopted in the middleand low latitude so the true north is thought as azimuthreference in the celestial navigation system To overcomethe difficult of convergence of polar meridians the gridcoordinate system is adopted in the inertial navigate systemand the grid north is thought as azimuth reference in thecelestial navigation system [13 14]

Furthermore the position of grid inertial navigationsystem can be directlymatched with polar chart and it can berealized by superposing grid lines on chart of Polar RegionsBecause the grid line is parallel to the Greenwich meridianthe angle between the grid north (GN) and the true north(TN) is determined by the longitude of location and theconvergence factor of chart [15ndash17]

The Northern Hemisphere is as follows

GN = TN + longitude (W)lowast the convergence factor ofchartGN = TN ndash longitude (E)lowast the convergence factor ofchart

The Southern Hemisphere is as follows

GN = TN ndashlongitude (W)lowast the convergence factor ofchartGN = TN +longitude (E)lowast the convergence factor ofchart

32 The State Space Mode of SINSCNS Integrated NavigationSystem in the Polar Regions Because the star sensor opticalaxis is fixedly connected with aircraft the state variablecontained the installation deviation angle between the SINSand the star sensor It selects the state variable of SINSCNS asfollowsThemisalignment angles of120601 = [120601119866119864 120601119866119873 120601119866119880]119879 arethe misalignment angle of grid eastward the misalignmentangle of grid northward and the misalignment angle of gridazimuth The velocity errors of 120575119907119866 = [120575V119866119864 120575V119866119873 120575V119866119880]119879are the velocity error of grid eastward the velocity error ofgrid northward and the velocity error of grid vertical Theposition errors of 120575119877119890 = [120575119909 120575119910 120575119911]119879 are the position errorof x-axis the position error of y-axis and the position error ofz-axis The constant drift of gyroscope 120576 = [120576119909 120576119910 120576119911]119879 is theconstant drift of x-axis the constant drift of y-axis and theconstant drift of z-axis The constant drift of accelerometernabla = [nabla119909 nabla119910 nabla119911]119879 is the constant drift of x-axis the constantdrift of y-axis and the constant drift of z-axisThe installationdeviation angles of 120583 = [120583119909 120583119910 120583119911]119879 are the installationdeviation angle of x-axis the installation deviation angle of y-axis and the installation deviation angle of z-axisThe systemstate vector of SINSCNS is

119883119862 = [120601119866119864 120601119866119873 120601119866119880 120575V119866119864 120575V119866119873 120575V119866119880 120575119909 120575119910 120575119911 120576119909 120576119910120576119911 nabla119909 nabla119910 nabla119911 120583119909 120583119910 120583119911]119879

(11)

Based on the error state equation of SINS and the systemstate vector of SINSCNS integrated navigation system the

Complexity 5

090

80minus200

70minus100

5000

60 0Start

50

h (m

)

10040 200

End

10000

15000

80

70

60 SSStarSSSSSSSSSSSSS

EndEEEEEEEEEEEEEEEEndE

(∘) (∘)

(a) The three-dimensional trajectory of the SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

0

50

100

0 1 2 3 4 5 6t (h)

minus200

0

200

0 1 2 3 4 5 6t (h)

0

1

2

h (m

)

times104

L(∘)

(∘)

(b) The position trajectory of the SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

0

20

40

0 1 2 3 4 5 6t (h)

minus50

0

50

0 1 2 3 4 5 6t (h)

0

100

200

(∘)

(∘)

(∘)

(c) The attitude trajectory of the SINSCNS in the PolarRegions

Figure 3

state equation of SINSCNS integrated navigation system canbe received as follows

[[[[[[[[[[[[[

120575119866120575119890

nabla

]]]]]]]]]]]]]

=

[[[[[[[[[[[[[[

minus (120596119866119894119866times) 119862119908119866V 11986213 minus119862119866119887 03

(119891119866times) 11986222 11986223 03 11986211986611988703 119862119890119866 11986233 03 0303 03 03 03 030303

0303

0303

0303

0303

]]]]]]]]]]]]]]

[[[[[[[[[[[[

120601

120575119907119866120575119877119890120576

nabla

120583

]]]]]]]]]]]]

+

[[[[[[[[[[[[

minus119862119866119887 120576119887119908119862119866119887 nabla119887119908

031031031031

]]]]]]]]]]]]

(12)

where

11986213 = 119862119908119890119877 + 11986211990811986611987711986222 = 119907119866 times 119862119908119866V minus [(2120596119866119894119890 + 120596119866119890119866) times]11986223 = 119907119866 times (2119862119908119890119877 + 119862119908119866119877)11986233 = minus119862119890119866 (119907119866times)119862119877

(13)

The measurements of SINSCNS integrated navigationsystem in the Polar Regions are the difference between

6 Complexity

0 1 2 3 4 5 6t (h)

minus5

0

5

0 1 2 3 4 5 6t (h)

minus5

0

5

0 1 2 3 4 5 6t (h)

minus5

0

5

U

()

()

()

(a) The misalignment angle of SINSCNS in the PolarRegions

0 1 2 3 4 5 6

t (h)

minus5

0

5

0 1 2 3 4 5 6

t (h)

minus5

0

5

0 1 2 3 4 5 6

t (h)

minus01

0

01

Ｐ(Ｇ

Ｍ)

Ｐ(Ｇ

Ｍ)

Ｐ５(Ｇ

Ｍ)

(b) The velocity error of SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

minus2000

0

2000

L

(m)

0 1 2 3 4 5 6t (h)

0

5

10

(m)

0 1 2 3 4 5 6t (h)

minus50

0

50

H

(m)

times105

(c) The position error of SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

0

0005

001

0 1 2 3 4 5 6t (h)

0

0005

001

0 1 2 3 4 5 6t (h)

0

0005

001 Ｒ

(∘Ｂ

) Ｓ

(∘Ｂ

) Ｔ

(∘Ｂ

)

(d) The constant drift of gyroscope of SINSCNS in thePolar Regions

0 1 2 3 4 5 6

t (h)

minus20

0

20

0 1 2 3 4 5 6

t (h)

minus100

0

100

0 1 2 3 4 5 6

t (h)

minus100

0

100

nablaＴ(

Ａ)nabla

Ｒ(

Ａ)nabla

Ｓ(

Ａ)

(e) The constant drift of accelerometer of SINSCNS inthe Polar Regions

0 1 2 3 4 5 6

t (h)

minus5

0

5

0 1 2 3 4 5 6

t (h)

0

2

4

0 1 2 3 4 5 6

t (h)

minus5

0

5

Ｒ

Ｓ

Ｔ

()

()

()

(f) The error of installation deviation angle ofSINSCNS in the Polar Regions

Figure 4

the transformation matrix of SINS and the transformationmatrix of star sensor so the measurement of the SINSCNSintegrated navigation system in the Polar Regions is shown as

119885119862119873119878 = 119901 minus 119888

= [(119906119888times) 03 119906119888 times119872119901 03 03 119862119875119887 (119906119888times)]

[[[[[[[[[[[[

120601120575119907119866120575119877119890120576

nabla

120583

]]]]]]]]]]]]

+ 120575119906119901(14)

where

119872119901 = [[[

minus1 0 00 cos 119871 00 sin 119871 0

]]]

(15)

33 The Trajectory of SINSCNS Integrated Navigation Systemin the Polar Regions It is the 6h in the simulation ofSINSCNS integrated navigation system when the aircraft isin the Polar Regions the trajectory is shown in Figure 3

Complexity 7

The parameters of trajectory are the starting point whichis [45∘119873 108∘119864 500119898] and culmination of latitude which is8926∘119873

The parameters of velocity are the initial velocity which is0119898119904 and the top velocity which is 31011989811990434The Simulation of SINSCNS IntegratedNavigation Systemin the Polar Regions In the simulation the parameters ofdevice are the same as the middle and low latitudes Theresults of simulation are shown in Figure 4

(1) The misalignment angle of SINSCNS integrated nav-igation system in the Polar Regions is showed in Figure 4(a)they are convergent to 051015840 in 100s and basically the same tothe attitude error of SINSCNS integrated navigation systemin the middle and low latitude

(2)The velocity error and the position error of SINSCNSintegrated navigation system in the Polar Regions are showedin Figures 4(b) and 4(c) the error of velocity is Schuler periodoscillation themaximum amplitude is not more than 05msThe error of position is the same as the error of velocity it isalso Schuler period oscillation and the maximum amplitudeis not more than 30m

(3) The constant drift of gyroscope and accelerometer ofSINSCNS integrated navigation system in the Polar Regionsis showed in Figures 4(d) and 4(e) the amplitude error ofvelocity and position also decreases when the constant driftof gyroscope and accelerometer is estimated The amplitudeof velocity and position error also decreases

(4)The error of installation deviation angle of SINSCNSintegrated navigation system in the Polar Regions is showedin Figure 4(f) the error of installation deviation angle is inaccordance with the misalignment angle and the error ofinstallation deviation angle of SINSCNS integrated naviga-tion system in the Polar Regions converges to 0310158404 Summary

The SINSCNS integrated navigation system in the PolarRegions based on the Grid Reference Frame are the chore-ography of inertial mechanics and overcome the problemwhich is the difficulties in positioning and orientation ofairborne caused by meridian convergence In the papercombining with the error of SINS and the state variable ofSINSCNS integrated navigation system in the Polar Regionswhich based on the Grid Reference Frame the state spacemode of SINSCNS integrated navigation system in the PolarRegions is established The simulation results showed thatthe SINSCNS integrated navigation system in the PolarRegions could repress the error caused by the constantdrift of gyroscope and correct the navigation parametersof SINSCNS integrated navigation system With the sameprecision of inertial components the precision of SINSCNSintegrated navigation system in the Polar Regions and in themiddle and low latitudes is consistent

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

The research area is detection and information technology

Conflicts of Interest

Wedeclare that we do not have any commercial or associativeinterest that represents conflicts of interest in connectionwiththe work submitted

Acknowledgments

This work is partially supported by the School FoundationResearch Fund of 2018 and the National Natural ScienceFoundation of China Project no 51678470

References

[1] L Brigham ldquoPolar Ocean Navigationrdquo Encyclopedia of Earthsciences pp 512ndash515 2014

[2] Z Zhao Y-B Liang and J Hu ldquoArtic oil and natural gaspotential and exploration and development trendrdquo Earth Sci-ence Frontiers vol 21 no 3 pp 47ndash55 2014

[3] C Yang XWang Z Li Y Li and C Su ldquoTeleoperation controlbased on combination of wave variable and neural networksrdquoIEEE Transactions on Systems Man and Cybernetics Systems2017

[4] G-L Yang L-F Wang and S-J Yang ldquoOnline calibrationmethod of missile-borne SINSCNS integrated navigation sys-temrdquo Journal of Projectiles Rockets Missiles and Guidance vol34 no 5 pp 29ndash32 2014

[5] B Zhou W-L Zhao Y-J Rong X-M Hu and J-Y MaldquoRobust Adaptive Kalman Filtering Algorithm for IntegratedNavigation Based on MEMS-INSGNSSrdquo navigation and con-trol vol 17 no 4 pp 14ndash20 2018

[6] C Yang K Huang H Cheng Y Li and C Su ldquoHapticidentification by ELM-controlled uncertain manipulatorrdquo IEEETransactions on Systems Man and Cybernetics Systems vol 47no 8 pp 2398ndash2409 2017

[7] L-F Zhou X-M Zhao Z-N Hou and C-L Ding ldquoCNSSINSintegrated calibration technique based on level dampingrdquoZhongguo Guanxing Jishu XuebaoJournal of Chinese InertialTechnology vol 25 no 5 pp 561ndash565 2017

[8] C Yang J Luo Y Pan Z Liu and C Su ldquoPersonalized variablegain control with tremor attenuation for robot teleoperationrdquoIEEE Transactions on Systems Man and Cybernetics Systemspp 1ndash12 2017

[9] X-NHuang J-R Liu andW Yang ldquoAn EvaluationMethod forthe Head Accuracy of Inertial Navigation System in Flight Testrdquonavigation and control vol 4 pp 7ndash12 2017

[10] Q Zhou Y Qin Q Fu and Y Yue ldquoGrid mechanization ininertial navigation systems for transpolar aircraftrdquoXibei GongyeDaxue XuebaoJournal of Northwestern Polytechnical Universityvol 31 no 2 pp 210ndash217 2013

[11] Y Gao M Liu G Li and X Guang ldquoInitial alignment for SINSbased on pseudo-earth frame in polar regionsrdquo Sensors vol 17no 6 2017

[12] L-J Song Z Li Q-W Fu and B He ldquoResearched of SINSGPSIntegrated Navigation System based on Grid Reference Framefor Transpolar Aircraftrdquo Journal of System Simulation 2018

8 Complexity

[13] H-G Guo and W-S Wang ldquoCelestial navigation in polarRegionsrdquo Journal of Dalian Maritime University vol 15 no 3pp 43ndash48 1989

[14] V B Larin andAA Tunik ldquoOn inertial navigation system errorcorrectionrdquo International Applied Mechanics vol 48 no 2 pp213ndash223 2012

[15] H Zhen YWang andH-BWnag ldquoSimulation of geomagneticaided submarine navigation based on EMAG2rdquo Progress inGeophysics vol 27 no 4 pp 1795ndash1803 2012

[16] Q Li Y Ben F Yu and J Tan ldquoTransversal strapdown INSbased on reference ellipsoid for vehicle in the Polar RegionrdquoIEEE Transactions on Vehicular Technology vol 65 no 9 pp7791ndash7795 2016

[17] Y-Q Yao X-S Xu Y Li Y-T Liu J Sun and J-W TongldquoTransverse Navigation under the Ellipsoidal Earth Model andits Performance in both Polar and Non-polar areasrdquo Journal ofNavigation vol 69 no 2 pp 335ndash352 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 3

Appointment

C119904119887C119887119899 = [[

[

11990411 11990412 1199041311990421 11990422 1199042311990431 11990432 11990433

]]]

C119899119890C119890119894 = [[[

11989911 11989912 1198991311989921 11989922 1198992311989931 11989932 11989933

]]]

(9)

It can be obtained

Z119878119868119873119878119862119873119878 =

[[[[[[[[[[[[[[[[[[[[

119885111198851211988513119885211198852211988523119885311198853211988533

]]]]]]]]]]]]]]]]]]]]

=

[[[[[[[[[[[[[[[[[[[[

1199041311989921 minus 1199041211989931 1199041111989931 minus 1199041311989911 1199041211989911 minus 11990411119899211199041311989922 minus 1199041211989932 1199041111989932 minus 1199041311989912 1199041211989912 minus 11990411119899221199041311989923 minus 1199041211989933 1199041111989933 minus 1199041311989913 1199041211989913 minus 11990411119899231199042311989921 minus 1199042211989931 1199042111989931 minus 1199042311989911 1199042211989911 minus 11990421119899211199042311989922 minus 1199042211989932 1199042111989932 minus 1199042311989912 1199042211989912 minus 11990421119899221199042311989923 minus 1199042211989933 1199042111989933 minus 1199042311989913 1199042211989913 minus 11990421119899231199043311989921 minus 1199043211989931 1199043111989931 minus 1199043311989911 1199043211989911 minus 11990431119899211199043311989922 minus 1199043211989932 1199043111989932 minus 1199043311989912 1199043211989912 minus 11990431119899221199043311989923 minus 1199043211989933 1199043111989933 minus 1199043311989913 1199043211989913 minus 1199043111989923

]]]]]]]]]]]]]]]]]]]]

[[[

120575120601119864120575120601119873120575120601119880

]]]

minus

[[[[[[[[[[[[[[[[[[[[

119862119908111198621199081211986211990813119862119908211198621199082211986211990823119862119908311198621199083211986211990833

]]]]]]]]]]]]]]]]]]]]

= 119867119878119868119873119878119862119873119878119883119868119873119878 minus 119881119862119873119878

(10)

23 The Simulation and Analysis of SINSCNS IntegratedNavigation System in the Middle and Low Latitudes Thesimulation time of SINSCNS integrated navigation systemin the middle and low latitudes is 600s the time of attitudeupdating velocity updating and position updating are 20msand the sampling period of inertial sensor is 10ms [9] In thesimulation the parameters are set as follows

The initial error of SINS is as followsThe initial error of attitude is [051015840 051015840 201015840] the initial

error of velocity is [001 001 001] ms and the initial errorof position is [20 20 20]119898

The parameters of inertial sensor are as follows

(1) The random constant drift of gyroscope is 001∘ℎ therandom walk coefficient of gyroscope is 0001∘radicℎand the error of scale factor is 30ppm

(2) The random constant drift of accelerometer is 40ugand the random walk coefficient of accelerometer is5ugradic119904

The parameters of star sensor are as followsThe horizontal measuring precision of angle is 1010158401015840 the

azimuth measuring precision of angle is 2010158401015840 and the error ofinstallation which between CNS and SINS is [31015840 31015840 31015840]T

These parameters are the same as in the Polar RegionsThe simulation results are showed in Figure 2

(1) The attitude error of SINSCNS in the middle andlow latitudes is showed as Figure 2(a) The attitude of CNSis considered as observation information in the SINSCNSintegrated navigation system the convergence rate of attitudeis quickened the measurement accuracy of SINSCNS inte-grated navigation system is improved effectively and after afew seconds the attitude error is up to 0151015840

(2)The velocity error and the position error of SINSCNSin the middle and low latitudes are showed as Figures 2(b)and 2(c) They are divergent in the SINSCNS integratednavigation system because CNS was unable to provide real-time information of the velocity and position of aircraft

(3) The constant drift of gyroscope and accelerometerof SINSCNS in the middle and low latitude are showed asFigures 2(d) and 2(e) The horizontal attitude error is themain error of integrated navigation system the influence ofazimuth attitude error is relatively weak The constant driftof gyroscope directly affects attitude error in the attitudeinformation which is given by the inertial navigation systemIt is better of the constant drift of gyroscope and it is diver-gent of the constant drift of accelerometer in the SINSCNSintegrated navigation system

3 The Algorithm of SINSCNSIntegrated Navigation System Based onGrid Reference Frame

In order to avoid the convergence of azimuth reference line atthe poles all the longitudes are redefined which are parallelto the Greenwich meridian [10 11] In this way the azimuthattitude is relative to the Greenwich meridian and its parallelline which is grid navigation The grid navigation and thegrid coordinate system have been described in the authorrsquosliterature [12] so that the author don not bore you withunnecessary details

The celestial navigation is an old method of navigationand position and the irreplaceable of celestial navigation isdetermined by the autonomy of celestial navigation Evenin the present age the advanced development of radionavigation system and the accuracy and timeliness of shippositioning have been well solved the celestial navigationis still unwavering in the navigation As early as 1989Guo Honggui has proposed the celestial navigation of PolarRegions in the literature [13]

4 Complexity

0 100 200 300 400 500 600minus05

0

05

0 100 200 300 400 500 600minus05

0

05

0 100 200 300 400 500 600minus20

0

20

e(

)

n()

Ｏ()

(a) The attitude error of SINSCNS in themiddle and low latitudes

0 100 200 300 400 500 6000

02

04

Ve

(ms

)

0 100 200 300 400 500 600minus02

0

02

Vn

(ms

)

0 100 200 300 400 500 6000

1

2

Vu

(ms

)

(b) The velocity error of SINSCNS in themiddle and low latitudes

0 100 200 300 400 500 600minus50

0

50

L

(m)

0 100 200 300 400 500 6000

100

200

(m)

0 100 200 300 400 500 6000

200

400

h

(m)

(c) The position error of SINSCNS in themiddle and low latitude

0 100 200 300 400 500 6000

001

002

0 100 200 300 400 500 600minus002

0

002

0 100 200 300 400 500 6000

001

002

＜Ｒ

(∘ Ｂ)

＜Ｓ

(∘ Ｂ)

＜Ｔ (

∘ Ｂ)

(d) The constant drift of gyroscope ofSINSCNS in the middle and low latitude

0 100 200 300 400 500 600399995

40

400005

nablax

( g)

0 100 200 300 400 500 600399995

40

400005

nablay

( g)

0 100 200 300 400 500 600399999

40

400001

nablaz (

g)

(e) The constant drift of accelerometer ofSINSCNS in the middle and low latitude

Figure 2

31 The Relationship of the Grid Orientation and the TrueNorth It is very difficult to navigate from the true northbecause of the convergence of polar meridians The northazimuth inertial navigation system is adopted in the middleand low latitude so the true north is thought as azimuthreference in the celestial navigation system To overcomethe difficult of convergence of polar meridians the gridcoordinate system is adopted in the inertial navigate systemand the grid north is thought as azimuth reference in thecelestial navigation system [13 14]

Furthermore the position of grid inertial navigationsystem can be directlymatched with polar chart and it can berealized by superposing grid lines on chart of Polar RegionsBecause the grid line is parallel to the Greenwich meridianthe angle between the grid north (GN) and the true north(TN) is determined by the longitude of location and theconvergence factor of chart [15ndash17]

The Northern Hemisphere is as follows

GN = TN + longitude (W)lowast the convergence factor ofchartGN = TN ndash longitude (E)lowast the convergence factor ofchart

The Southern Hemisphere is as follows

GN = TN ndashlongitude (W)lowast the convergence factor ofchartGN = TN +longitude (E)lowast the convergence factor ofchart

32 The State Space Mode of SINSCNS Integrated NavigationSystem in the Polar Regions Because the star sensor opticalaxis is fixedly connected with aircraft the state variablecontained the installation deviation angle between the SINSand the star sensor It selects the state variable of SINSCNS asfollowsThemisalignment angles of120601 = [120601119866119864 120601119866119873 120601119866119880]119879 arethe misalignment angle of grid eastward the misalignmentangle of grid northward and the misalignment angle of gridazimuth The velocity errors of 120575119907119866 = [120575V119866119864 120575V119866119873 120575V119866119880]119879are the velocity error of grid eastward the velocity error ofgrid northward and the velocity error of grid vertical Theposition errors of 120575119877119890 = [120575119909 120575119910 120575119911]119879 are the position errorof x-axis the position error of y-axis and the position error ofz-axis The constant drift of gyroscope 120576 = [120576119909 120576119910 120576119911]119879 is theconstant drift of x-axis the constant drift of y-axis and theconstant drift of z-axis The constant drift of accelerometernabla = [nabla119909 nabla119910 nabla119911]119879 is the constant drift of x-axis the constantdrift of y-axis and the constant drift of z-axisThe installationdeviation angles of 120583 = [120583119909 120583119910 120583119911]119879 are the installationdeviation angle of x-axis the installation deviation angle of y-axis and the installation deviation angle of z-axisThe systemstate vector of SINSCNS is

119883119862 = [120601119866119864 120601119866119873 120601119866119880 120575V119866119864 120575V119866119873 120575V119866119880 120575119909 120575119910 120575119911 120576119909 120576119910120576119911 nabla119909 nabla119910 nabla119911 120583119909 120583119910 120583119911]119879

(11)

Based on the error state equation of SINS and the systemstate vector of SINSCNS integrated navigation system the

Complexity 5

090

80minus200

70minus100

5000

60 0Start

50

h (m

)

10040 200

End

10000

15000

80

70

60 SSStarSSSSSSSSSSSSS

EndEEEEEEEEEEEEEEEEndE

(∘) (∘)

(a) The three-dimensional trajectory of the SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

0

50

100

0 1 2 3 4 5 6t (h)

minus200

0

200

0 1 2 3 4 5 6t (h)

0

1

2

h (m

)

times104

L(∘)

(∘)

(b) The position trajectory of the SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

0

20

40

0 1 2 3 4 5 6t (h)

minus50

0

50

0 1 2 3 4 5 6t (h)

0

100

200

(∘)

(∘)

(∘)

(c) The attitude trajectory of the SINSCNS in the PolarRegions

Figure 3

state equation of SINSCNS integrated navigation system canbe received as follows

[[[[[[[[[[[[[

120575119866120575119890

nabla

]]]]]]]]]]]]]

=

[[[[[[[[[[[[[[

minus (120596119866119894119866times) 119862119908119866V 11986213 minus119862119866119887 03

(119891119866times) 11986222 11986223 03 11986211986611988703 119862119890119866 11986233 03 0303 03 03 03 030303

0303

0303

0303

0303

]]]]]]]]]]]]]]

[[[[[[[[[[[[

120601

120575119907119866120575119877119890120576

nabla

120583

]]]]]]]]]]]]

+

[[[[[[[[[[[[

minus119862119866119887 120576119887119908119862119866119887 nabla119887119908

031031031031

]]]]]]]]]]]]

(12)

where

11986213 = 119862119908119890119877 + 11986211990811986611987711986222 = 119907119866 times 119862119908119866V minus [(2120596119866119894119890 + 120596119866119890119866) times]11986223 = 119907119866 times (2119862119908119890119877 + 119862119908119866119877)11986233 = minus119862119890119866 (119907119866times)119862119877

(13)

The measurements of SINSCNS integrated navigationsystem in the Polar Regions are the difference between

6 Complexity

0 1 2 3 4 5 6t (h)

minus5

0

5

0 1 2 3 4 5 6t (h)

minus5

0

5

0 1 2 3 4 5 6t (h)

minus5

0

5

U

()

()

()

(a) The misalignment angle of SINSCNS in the PolarRegions

0 1 2 3 4 5 6

t (h)

minus5

0

5

0 1 2 3 4 5 6

t (h)

minus5

0

5

0 1 2 3 4 5 6

t (h)

minus01

0

01

Ｐ(Ｇ

Ｍ)

Ｐ(Ｇ

Ｍ)

Ｐ５(Ｇ

Ｍ)

(b) The velocity error of SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

minus2000

0

2000

L

(m)

0 1 2 3 4 5 6t (h)

0

5

10

(m)

0 1 2 3 4 5 6t (h)

minus50

0

50

H

(m)

times105

(c) The position error of SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

0

0005

001

0 1 2 3 4 5 6t (h)

0

0005

001

0 1 2 3 4 5 6t (h)

0

0005

001 Ｒ

(∘Ｂ

) Ｓ

(∘Ｂ

) Ｔ

(∘Ｂ

)

(d) The constant drift of gyroscope of SINSCNS in thePolar Regions

0 1 2 3 4 5 6

t (h)

minus20

0

20

0 1 2 3 4 5 6

t (h)

minus100

0

100

0 1 2 3 4 5 6

t (h)

minus100

0

100

nablaＴ(

Ａ)nabla

Ｒ(

Ａ)nabla

Ｓ(

Ａ)

(e) The constant drift of accelerometer of SINSCNS inthe Polar Regions

0 1 2 3 4 5 6

t (h)

minus5

0

5

0 1 2 3 4 5 6

t (h)

0

2

4

0 1 2 3 4 5 6

t (h)

minus5

0

5

Ｒ

Ｓ

Ｔ

()

()

()

(f) The error of installation deviation angle ofSINSCNS in the Polar Regions

Figure 4

the transformation matrix of SINS and the transformationmatrix of star sensor so the measurement of the SINSCNSintegrated navigation system in the Polar Regions is shown as

119885119862119873119878 = 119901 minus 119888

= [(119906119888times) 03 119906119888 times119872119901 03 03 119862119875119887 (119906119888times)]

[[[[[[[[[[[[

120601120575119907119866120575119877119890120576

nabla

120583

]]]]]]]]]]]]

+ 120575119906119901(14)

where

119872119901 = [[[

minus1 0 00 cos 119871 00 sin 119871 0

]]]

(15)

33 The Trajectory of SINSCNS Integrated Navigation Systemin the Polar Regions It is the 6h in the simulation ofSINSCNS integrated navigation system when the aircraft isin the Polar Regions the trajectory is shown in Figure 3

Complexity 7

The parameters of trajectory are the starting point whichis [45∘119873 108∘119864 500119898] and culmination of latitude which is8926∘119873

The parameters of velocity are the initial velocity which is0119898119904 and the top velocity which is 31011989811990434The Simulation of SINSCNS IntegratedNavigation Systemin the Polar Regions In the simulation the parameters ofdevice are the same as the middle and low latitudes Theresults of simulation are shown in Figure 4

(1) The misalignment angle of SINSCNS integrated nav-igation system in the Polar Regions is showed in Figure 4(a)they are convergent to 051015840 in 100s and basically the same tothe attitude error of SINSCNS integrated navigation systemin the middle and low latitude

(2)The velocity error and the position error of SINSCNSintegrated navigation system in the Polar Regions are showedin Figures 4(b) and 4(c) the error of velocity is Schuler periodoscillation themaximum amplitude is not more than 05msThe error of position is the same as the error of velocity it isalso Schuler period oscillation and the maximum amplitudeis not more than 30m

(3) The constant drift of gyroscope and accelerometer ofSINSCNS integrated navigation system in the Polar Regionsis showed in Figures 4(d) and 4(e) the amplitude error ofvelocity and position also decreases when the constant driftof gyroscope and accelerometer is estimated The amplitudeof velocity and position error also decreases

(4)The error of installation deviation angle of SINSCNSintegrated navigation system in the Polar Regions is showedin Figure 4(f) the error of installation deviation angle is inaccordance with the misalignment angle and the error ofinstallation deviation angle of SINSCNS integrated naviga-tion system in the Polar Regions converges to 0310158404 Summary

The SINSCNS integrated navigation system in the PolarRegions based on the Grid Reference Frame are the chore-ography of inertial mechanics and overcome the problemwhich is the difficulties in positioning and orientation ofairborne caused by meridian convergence In the papercombining with the error of SINS and the state variable ofSINSCNS integrated navigation system in the Polar Regionswhich based on the Grid Reference Frame the state spacemode of SINSCNS integrated navigation system in the PolarRegions is established The simulation results showed thatthe SINSCNS integrated navigation system in the PolarRegions could repress the error caused by the constantdrift of gyroscope and correct the navigation parametersof SINSCNS integrated navigation system With the sameprecision of inertial components the precision of SINSCNSintegrated navigation system in the Polar Regions and in themiddle and low latitudes is consistent

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

The research area is detection and information technology

Conflicts of Interest

Wedeclare that we do not have any commercial or associativeinterest that represents conflicts of interest in connectionwiththe work submitted

Acknowledgments

This work is partially supported by the School FoundationResearch Fund of 2018 and the National Natural ScienceFoundation of China Project no 51678470

References

[1] L Brigham ldquoPolar Ocean Navigationrdquo Encyclopedia of Earthsciences pp 512ndash515 2014

[2] Z Zhao Y-B Liang and J Hu ldquoArtic oil and natural gaspotential and exploration and development trendrdquo Earth Sci-ence Frontiers vol 21 no 3 pp 47ndash55 2014

[3] C Yang XWang Z Li Y Li and C Su ldquoTeleoperation controlbased on combination of wave variable and neural networksrdquoIEEE Transactions on Systems Man and Cybernetics Systems2017

[4] G-L Yang L-F Wang and S-J Yang ldquoOnline calibrationmethod of missile-borne SINSCNS integrated navigation sys-temrdquo Journal of Projectiles Rockets Missiles and Guidance vol34 no 5 pp 29ndash32 2014

[5] B Zhou W-L Zhao Y-J Rong X-M Hu and J-Y MaldquoRobust Adaptive Kalman Filtering Algorithm for IntegratedNavigation Based on MEMS-INSGNSSrdquo navigation and con-trol vol 17 no 4 pp 14ndash20 2018

[6] C Yang K Huang H Cheng Y Li and C Su ldquoHapticidentification by ELM-controlled uncertain manipulatorrdquo IEEETransactions on Systems Man and Cybernetics Systems vol 47no 8 pp 2398ndash2409 2017

[7] L-F Zhou X-M Zhao Z-N Hou and C-L Ding ldquoCNSSINSintegrated calibration technique based on level dampingrdquoZhongguo Guanxing Jishu XuebaoJournal of Chinese InertialTechnology vol 25 no 5 pp 561ndash565 2017

[8] C Yang J Luo Y Pan Z Liu and C Su ldquoPersonalized variablegain control with tremor attenuation for robot teleoperationrdquoIEEE Transactions on Systems Man and Cybernetics Systemspp 1ndash12 2017

[9] X-NHuang J-R Liu andW Yang ldquoAn EvaluationMethod forthe Head Accuracy of Inertial Navigation System in Flight Testrdquonavigation and control vol 4 pp 7ndash12 2017

[10] Q Zhou Y Qin Q Fu and Y Yue ldquoGrid mechanization ininertial navigation systems for transpolar aircraftrdquoXibei GongyeDaxue XuebaoJournal of Northwestern Polytechnical Universityvol 31 no 2 pp 210ndash217 2013

[11] Y Gao M Liu G Li and X Guang ldquoInitial alignment for SINSbased on pseudo-earth frame in polar regionsrdquo Sensors vol 17no 6 2017

[12] L-J Song Z Li Q-W Fu and B He ldquoResearched of SINSGPSIntegrated Navigation System based on Grid Reference Framefor Transpolar Aircraftrdquo Journal of System Simulation 2018

8 Complexity

[13] H-G Guo and W-S Wang ldquoCelestial navigation in polarRegionsrdquo Journal of Dalian Maritime University vol 15 no 3pp 43ndash48 1989

[14] V B Larin andAA Tunik ldquoOn inertial navigation system errorcorrectionrdquo International Applied Mechanics vol 48 no 2 pp213ndash223 2012

[15] H Zhen YWang andH-BWnag ldquoSimulation of geomagneticaided submarine navigation based on EMAG2rdquo Progress inGeophysics vol 27 no 4 pp 1795ndash1803 2012

[16] Q Li Y Ben F Yu and J Tan ldquoTransversal strapdown INSbased on reference ellipsoid for vehicle in the Polar RegionrdquoIEEE Transactions on Vehicular Technology vol 65 no 9 pp7791ndash7795 2016

[17] Y-Q Yao X-S Xu Y Li Y-T Liu J Sun and J-W TongldquoTransverse Navigation under the Ellipsoidal Earth Model andits Performance in both Polar and Non-polar areasrdquo Journal ofNavigation vol 69 no 2 pp 335ndash352 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

4 Complexity

0 100 200 300 400 500 600minus05

0

05

0 100 200 300 400 500 600minus05

0

05

0 100 200 300 400 500 600minus20

0

20

e(

)

n()

Ｏ()

(a) The attitude error of SINSCNS in themiddle and low latitudes

0 100 200 300 400 500 6000

02

04

Ve

(ms

)

0 100 200 300 400 500 600minus02

0

02

Vn

(ms

)

0 100 200 300 400 500 6000

1

2

Vu

(ms

)

(b) The velocity error of SINSCNS in themiddle and low latitudes

0 100 200 300 400 500 600minus50

0

50

L

(m)

0 100 200 300 400 500 6000

100

200

(m)

0 100 200 300 400 500 6000

200

400

h

(m)

(c) The position error of SINSCNS in themiddle and low latitude

0 100 200 300 400 500 6000

001

002

0 100 200 300 400 500 600minus002

0

002

0 100 200 300 400 500 6000

001

002

＜Ｒ

(∘ Ｂ)

＜Ｓ

(∘ Ｂ)

＜Ｔ (

∘ Ｂ)

(d) The constant drift of gyroscope ofSINSCNS in the middle and low latitude

0 100 200 300 400 500 600399995

40

400005

nablax

( g)

0 100 200 300 400 500 600399995

40

400005

nablay

( g)

0 100 200 300 400 500 600399999

40

400001

nablaz (

g)

(e) The constant drift of accelerometer ofSINSCNS in the middle and low latitude

Figure 2

31 The Relationship of the Grid Orientation and the TrueNorth It is very difficult to navigate from the true northbecause of the convergence of polar meridians The northazimuth inertial navigation system is adopted in the middleand low latitude so the true north is thought as azimuthreference in the celestial navigation system To overcomethe difficult of convergence of polar meridians the gridcoordinate system is adopted in the inertial navigate systemand the grid north is thought as azimuth reference in thecelestial navigation system [13 14]

Furthermore the position of grid inertial navigationsystem can be directlymatched with polar chart and it can berealized by superposing grid lines on chart of Polar RegionsBecause the grid line is parallel to the Greenwich meridianthe angle between the grid north (GN) and the true north(TN) is determined by the longitude of location and theconvergence factor of chart [15ndash17]

The Northern Hemisphere is as follows

GN = TN + longitude (W)lowast the convergence factor ofchartGN = TN ndash longitude (E)lowast the convergence factor ofchart

The Southern Hemisphere is as follows

GN = TN ndashlongitude (W)lowast the convergence factor ofchartGN = TN +longitude (E)lowast the convergence factor ofchart

32 The State Space Mode of SINSCNS Integrated NavigationSystem in the Polar Regions Because the star sensor opticalaxis is fixedly connected with aircraft the state variablecontained the installation deviation angle between the SINSand the star sensor It selects the state variable of SINSCNS asfollowsThemisalignment angles of120601 = [120601119866119864 120601119866119873 120601119866119880]119879 arethe misalignment angle of grid eastward the misalignmentangle of grid northward and the misalignment angle of gridazimuth The velocity errors of 120575119907119866 = [120575V119866119864 120575V119866119873 120575V119866119880]119879are the velocity error of grid eastward the velocity error ofgrid northward and the velocity error of grid vertical Theposition errors of 120575119877119890 = [120575119909 120575119910 120575119911]119879 are the position errorof x-axis the position error of y-axis and the position error ofz-axis The constant drift of gyroscope 120576 = [120576119909 120576119910 120576119911]119879 is theconstant drift of x-axis the constant drift of y-axis and theconstant drift of z-axis The constant drift of accelerometernabla = [nabla119909 nabla119910 nabla119911]119879 is the constant drift of x-axis the constantdrift of y-axis and the constant drift of z-axisThe installationdeviation angles of 120583 = [120583119909 120583119910 120583119911]119879 are the installationdeviation angle of x-axis the installation deviation angle of y-axis and the installation deviation angle of z-axisThe systemstate vector of SINSCNS is

119883119862 = [120601119866119864 120601119866119873 120601119866119880 120575V119866119864 120575V119866119873 120575V119866119880 120575119909 120575119910 120575119911 120576119909 120576119910120576119911 nabla119909 nabla119910 nabla119911 120583119909 120583119910 120583119911]119879

(11)

Based on the error state equation of SINS and the systemstate vector of SINSCNS integrated navigation system the

Complexity 5

090

80minus200

70minus100

5000

60 0Start

50

h (m

)

10040 200

End

10000

15000

80

70

60 SSStarSSSSSSSSSSSSS

EndEEEEEEEEEEEEEEEEndE

(∘) (∘)

(a) The three-dimensional trajectory of the SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

0

50

100

0 1 2 3 4 5 6t (h)

minus200

0

200

0 1 2 3 4 5 6t (h)

0

1

2

h (m

)

times104

L(∘)

(∘)

(b) The position trajectory of the SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

0

20

40

0 1 2 3 4 5 6t (h)

minus50

0

50

0 1 2 3 4 5 6t (h)

0

100

200

(∘)

(∘)

(∘)

(c) The attitude trajectory of the SINSCNS in the PolarRegions

Figure 3

state equation of SINSCNS integrated navigation system canbe received as follows

[[[[[[[[[[[[[

120575119866120575119890

nabla

]]]]]]]]]]]]]

=

[[[[[[[[[[[[[[

minus (120596119866119894119866times) 119862119908119866V 11986213 minus119862119866119887 03

(119891119866times) 11986222 11986223 03 11986211986611988703 119862119890119866 11986233 03 0303 03 03 03 030303

0303

0303

0303

0303

]]]]]]]]]]]]]]

[[[[[[[[[[[[

120601

120575119907119866120575119877119890120576

nabla

120583

]]]]]]]]]]]]

+

[[[[[[[[[[[[

minus119862119866119887 120576119887119908119862119866119887 nabla119887119908

031031031031

]]]]]]]]]]]]

(12)

where

11986213 = 119862119908119890119877 + 11986211990811986611987711986222 = 119907119866 times 119862119908119866V minus [(2120596119866119894119890 + 120596119866119890119866) times]11986223 = 119907119866 times (2119862119908119890119877 + 119862119908119866119877)11986233 = minus119862119890119866 (119907119866times)119862119877

(13)

The measurements of SINSCNS integrated navigationsystem in the Polar Regions are the difference between

6 Complexity

0 1 2 3 4 5 6t (h)

minus5

0

5

0 1 2 3 4 5 6t (h)

minus5

0

5

0 1 2 3 4 5 6t (h)

minus5

0

5

U

()

()

()

(a) The misalignment angle of SINSCNS in the PolarRegions

0 1 2 3 4 5 6

t (h)

minus5

0

5

0 1 2 3 4 5 6

t (h)

minus5

0

5

0 1 2 3 4 5 6

t (h)

minus01

0

01

Ｐ(Ｇ

Ｍ)

Ｐ(Ｇ

Ｍ)

Ｐ５(Ｇ

Ｍ)

(b) The velocity error of SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

minus2000

0

2000

L

(m)

0 1 2 3 4 5 6t (h)

0

5

10

(m)

0 1 2 3 4 5 6t (h)

minus50

0

50

H

(m)

times105

(c) The position error of SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

0

0005

001

0 1 2 3 4 5 6t (h)

0

0005

001

0 1 2 3 4 5 6t (h)

0

0005

001 Ｒ

(∘Ｂ

) Ｓ

(∘Ｂ

) Ｔ

(∘Ｂ

)

(d) The constant drift of gyroscope of SINSCNS in thePolar Regions

0 1 2 3 4 5 6

t (h)

minus20

0

20

0 1 2 3 4 5 6

t (h)

minus100

0

100

0 1 2 3 4 5 6

t (h)

minus100

0

100

nablaＴ(

Ａ)nabla

Ｒ(

Ａ)nabla

Ｓ(

Ａ)

(e) The constant drift of accelerometer of SINSCNS inthe Polar Regions

0 1 2 3 4 5 6

t (h)

minus5

0

5

0 1 2 3 4 5 6

t (h)

0

2

4

0 1 2 3 4 5 6

t (h)

minus5

0

5

Ｒ

Ｓ

Ｔ

()

()

()

(f) The error of installation deviation angle ofSINSCNS in the Polar Regions

Figure 4

the transformation matrix of SINS and the transformationmatrix of star sensor so the measurement of the SINSCNSintegrated navigation system in the Polar Regions is shown as

119885119862119873119878 = 119901 minus 119888

= [(119906119888times) 03 119906119888 times119872119901 03 03 119862119875119887 (119906119888times)]

[[[[[[[[[[[[

120601120575119907119866120575119877119890120576

nabla

120583

]]]]]]]]]]]]

+ 120575119906119901(14)

where

119872119901 = [[[

minus1 0 00 cos 119871 00 sin 119871 0

]]]

(15)

33 The Trajectory of SINSCNS Integrated Navigation Systemin the Polar Regions It is the 6h in the simulation ofSINSCNS integrated navigation system when the aircraft isin the Polar Regions the trajectory is shown in Figure 3

Complexity 7

The parameters of trajectory are the starting point whichis [45∘119873 108∘119864 500119898] and culmination of latitude which is8926∘119873

The parameters of velocity are the initial velocity which is0119898119904 and the top velocity which is 31011989811990434The Simulation of SINSCNS IntegratedNavigation Systemin the Polar Regions In the simulation the parameters ofdevice are the same as the middle and low latitudes Theresults of simulation are shown in Figure 4

(1) The misalignment angle of SINSCNS integrated nav-igation system in the Polar Regions is showed in Figure 4(a)they are convergent to 051015840 in 100s and basically the same tothe attitude error of SINSCNS integrated navigation systemin the middle and low latitude

(2)The velocity error and the position error of SINSCNSintegrated navigation system in the Polar Regions are showedin Figures 4(b) and 4(c) the error of velocity is Schuler periodoscillation themaximum amplitude is not more than 05msThe error of position is the same as the error of velocity it isalso Schuler period oscillation and the maximum amplitudeis not more than 30m

(3) The constant drift of gyroscope and accelerometer ofSINSCNS integrated navigation system in the Polar Regionsis showed in Figures 4(d) and 4(e) the amplitude error ofvelocity and position also decreases when the constant driftof gyroscope and accelerometer is estimated The amplitudeof velocity and position error also decreases

(4)The error of installation deviation angle of SINSCNSintegrated navigation system in the Polar Regions is showedin Figure 4(f) the error of installation deviation angle is inaccordance with the misalignment angle and the error ofinstallation deviation angle of SINSCNS integrated naviga-tion system in the Polar Regions converges to 0310158404 Summary

The SINSCNS integrated navigation system in the PolarRegions based on the Grid Reference Frame are the chore-ography of inertial mechanics and overcome the problemwhich is the difficulties in positioning and orientation ofairborne caused by meridian convergence In the papercombining with the error of SINS and the state variable ofSINSCNS integrated navigation system in the Polar Regionswhich based on the Grid Reference Frame the state spacemode of SINSCNS integrated navigation system in the PolarRegions is established The simulation results showed thatthe SINSCNS integrated navigation system in the PolarRegions could repress the error caused by the constantdrift of gyroscope and correct the navigation parametersof SINSCNS integrated navigation system With the sameprecision of inertial components the precision of SINSCNSintegrated navigation system in the Polar Regions and in themiddle and low latitudes is consistent

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

The research area is detection and information technology

Conflicts of Interest

Wedeclare that we do not have any commercial or associativeinterest that represents conflicts of interest in connectionwiththe work submitted

Acknowledgments

This work is partially supported by the School FoundationResearch Fund of 2018 and the National Natural ScienceFoundation of China Project no 51678470

References

[1] L Brigham ldquoPolar Ocean Navigationrdquo Encyclopedia of Earthsciences pp 512ndash515 2014

[2] Z Zhao Y-B Liang and J Hu ldquoArtic oil and natural gaspotential and exploration and development trendrdquo Earth Sci-ence Frontiers vol 21 no 3 pp 47ndash55 2014

[3] C Yang XWang Z Li Y Li and C Su ldquoTeleoperation controlbased on combination of wave variable and neural networksrdquoIEEE Transactions on Systems Man and Cybernetics Systems2017

[4] G-L Yang L-F Wang and S-J Yang ldquoOnline calibrationmethod of missile-borne SINSCNS integrated navigation sys-temrdquo Journal of Projectiles Rockets Missiles and Guidance vol34 no 5 pp 29ndash32 2014

[5] B Zhou W-L Zhao Y-J Rong X-M Hu and J-Y MaldquoRobust Adaptive Kalman Filtering Algorithm for IntegratedNavigation Based on MEMS-INSGNSSrdquo navigation and con-trol vol 17 no 4 pp 14ndash20 2018

[6] C Yang K Huang H Cheng Y Li and C Su ldquoHapticidentification by ELM-controlled uncertain manipulatorrdquo IEEETransactions on Systems Man and Cybernetics Systems vol 47no 8 pp 2398ndash2409 2017

[7] L-F Zhou X-M Zhao Z-N Hou and C-L Ding ldquoCNSSINSintegrated calibration technique based on level dampingrdquoZhongguo Guanxing Jishu XuebaoJournal of Chinese InertialTechnology vol 25 no 5 pp 561ndash565 2017

[8] C Yang J Luo Y Pan Z Liu and C Su ldquoPersonalized variablegain control with tremor attenuation for robot teleoperationrdquoIEEE Transactions on Systems Man and Cybernetics Systemspp 1ndash12 2017

[9] X-NHuang J-R Liu andW Yang ldquoAn EvaluationMethod forthe Head Accuracy of Inertial Navigation System in Flight Testrdquonavigation and control vol 4 pp 7ndash12 2017

[10] Q Zhou Y Qin Q Fu and Y Yue ldquoGrid mechanization ininertial navigation systems for transpolar aircraftrdquoXibei GongyeDaxue XuebaoJournal of Northwestern Polytechnical Universityvol 31 no 2 pp 210ndash217 2013

[11] Y Gao M Liu G Li and X Guang ldquoInitial alignment for SINSbased on pseudo-earth frame in polar regionsrdquo Sensors vol 17no 6 2017

[12] L-J Song Z Li Q-W Fu and B He ldquoResearched of SINSGPSIntegrated Navigation System based on Grid Reference Framefor Transpolar Aircraftrdquo Journal of System Simulation 2018

8 Complexity

[13] H-G Guo and W-S Wang ldquoCelestial navigation in polarRegionsrdquo Journal of Dalian Maritime University vol 15 no 3pp 43ndash48 1989

[14] V B Larin andAA Tunik ldquoOn inertial navigation system errorcorrectionrdquo International Applied Mechanics vol 48 no 2 pp213ndash223 2012

[15] H Zhen YWang andH-BWnag ldquoSimulation of geomagneticaided submarine navigation based on EMAG2rdquo Progress inGeophysics vol 27 no 4 pp 1795ndash1803 2012

[16] Q Li Y Ben F Yu and J Tan ldquoTransversal strapdown INSbased on reference ellipsoid for vehicle in the Polar RegionrdquoIEEE Transactions on Vehicular Technology vol 65 no 9 pp7791ndash7795 2016

[17] Y-Q Yao X-S Xu Y Li Y-T Liu J Sun and J-W TongldquoTransverse Navigation under the Ellipsoidal Earth Model andits Performance in both Polar and Non-polar areasrdquo Journal ofNavigation vol 69 no 2 pp 335ndash352 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 5

090

80minus200

70minus100

5000

60 0Start

50

h (m

)

10040 200

End

10000

15000

80

70

60 SSStarSSSSSSSSSSSSS

EndEEEEEEEEEEEEEEEEndE

(∘) (∘)

(a) The three-dimensional trajectory of the SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

0

50

100

0 1 2 3 4 5 6t (h)

minus200

0

200

0 1 2 3 4 5 6t (h)

0

1

2

h (m

)

times104

L(∘)

(∘)

(b) The position trajectory of the SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

0

20

40

0 1 2 3 4 5 6t (h)

minus50

0

50

0 1 2 3 4 5 6t (h)

0

100

200

(∘)

(∘)

(∘)

(c) The attitude trajectory of the SINSCNS in the PolarRegions

Figure 3

state equation of SINSCNS integrated navigation system canbe received as follows

[[[[[[[[[[[[[

120575119866120575119890

nabla

]]]]]]]]]]]]]

=

[[[[[[[[[[[[[[

minus (120596119866119894119866times) 119862119908119866V 11986213 minus119862119866119887 03

(119891119866times) 11986222 11986223 03 11986211986611988703 119862119890119866 11986233 03 0303 03 03 03 030303

0303

0303

0303

0303

]]]]]]]]]]]]]]

[[[[[[[[[[[[

120601

120575119907119866120575119877119890120576

nabla

120583

]]]]]]]]]]]]

+

[[[[[[[[[[[[

minus119862119866119887 120576119887119908119862119866119887 nabla119887119908

031031031031

]]]]]]]]]]]]

(12)

where

11986213 = 119862119908119890119877 + 11986211990811986611987711986222 = 119907119866 times 119862119908119866V minus [(2120596119866119894119890 + 120596119866119890119866) times]11986223 = 119907119866 times (2119862119908119890119877 + 119862119908119866119877)11986233 = minus119862119890119866 (119907119866times)119862119877

(13)

The measurements of SINSCNS integrated navigationsystem in the Polar Regions are the difference between

6 Complexity

0 1 2 3 4 5 6t (h)

minus5

0

5

0 1 2 3 4 5 6t (h)

minus5

0

5

0 1 2 3 4 5 6t (h)

minus5

0

5

U

()

()

()

(a) The misalignment angle of SINSCNS in the PolarRegions

0 1 2 3 4 5 6

t (h)

minus5

0

5

0 1 2 3 4 5 6

t (h)

minus5

0

5

0 1 2 3 4 5 6

t (h)

minus01

0

01

Ｐ(Ｇ

Ｍ)

Ｐ(Ｇ

Ｍ)

Ｐ５(Ｇ

Ｍ)

(b) The velocity error of SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

minus2000

0

2000

L

(m)

0 1 2 3 4 5 6t (h)

0

5

10

(m)

0 1 2 3 4 5 6t (h)

minus50

0

50

H

(m)

times105

(c) The position error of SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

0

0005

001

0 1 2 3 4 5 6t (h)

0

0005

001

0 1 2 3 4 5 6t (h)

0

0005

001 Ｒ

(∘Ｂ

) Ｓ

(∘Ｂ

) Ｔ

(∘Ｂ

)

(d) The constant drift of gyroscope of SINSCNS in thePolar Regions

0 1 2 3 4 5 6

t (h)

minus20

0

20

0 1 2 3 4 5 6

t (h)

minus100

0

100

0 1 2 3 4 5 6

t (h)

minus100

0

100

nablaＴ(

Ａ)nabla

Ｒ(

Ａ)nabla

Ｓ(

Ａ)

(e) The constant drift of accelerometer of SINSCNS inthe Polar Regions

0 1 2 3 4 5 6

t (h)

minus5

0

5

0 1 2 3 4 5 6

t (h)

0

2

4

0 1 2 3 4 5 6

t (h)

minus5

0

5

Ｒ

Ｓ

Ｔ

()

()

()

(f) The error of installation deviation angle ofSINSCNS in the Polar Regions

Figure 4

the transformation matrix of SINS and the transformationmatrix of star sensor so the measurement of the SINSCNSintegrated navigation system in the Polar Regions is shown as

119885119862119873119878 = 119901 minus 119888

= [(119906119888times) 03 119906119888 times119872119901 03 03 119862119875119887 (119906119888times)]

[[[[[[[[[[[[

120601120575119907119866120575119877119890120576

nabla

120583

]]]]]]]]]]]]

+ 120575119906119901(14)

where

119872119901 = [[[

minus1 0 00 cos 119871 00 sin 119871 0

]]]

(15)

33 The Trajectory of SINSCNS Integrated Navigation Systemin the Polar Regions It is the 6h in the simulation ofSINSCNS integrated navigation system when the aircraft isin the Polar Regions the trajectory is shown in Figure 3

Complexity 7

The parameters of trajectory are the starting point whichis [45∘119873 108∘119864 500119898] and culmination of latitude which is8926∘119873

The parameters of velocity are the initial velocity which is0119898119904 and the top velocity which is 31011989811990434The Simulation of SINSCNS IntegratedNavigation Systemin the Polar Regions In the simulation the parameters ofdevice are the same as the middle and low latitudes Theresults of simulation are shown in Figure 4

(1) The misalignment angle of SINSCNS integrated nav-igation system in the Polar Regions is showed in Figure 4(a)they are convergent to 051015840 in 100s and basically the same tothe attitude error of SINSCNS integrated navigation systemin the middle and low latitude

(2)The velocity error and the position error of SINSCNSintegrated navigation system in the Polar Regions are showedin Figures 4(b) and 4(c) the error of velocity is Schuler periodoscillation themaximum amplitude is not more than 05msThe error of position is the same as the error of velocity it isalso Schuler period oscillation and the maximum amplitudeis not more than 30m

(3) The constant drift of gyroscope and accelerometer ofSINSCNS integrated navigation system in the Polar Regionsis showed in Figures 4(d) and 4(e) the amplitude error ofvelocity and position also decreases when the constant driftof gyroscope and accelerometer is estimated The amplitudeof velocity and position error also decreases

(4)The error of installation deviation angle of SINSCNSintegrated navigation system in the Polar Regions is showedin Figure 4(f) the error of installation deviation angle is inaccordance with the misalignment angle and the error ofinstallation deviation angle of SINSCNS integrated naviga-tion system in the Polar Regions converges to 0310158404 Summary

The SINSCNS integrated navigation system in the PolarRegions based on the Grid Reference Frame are the chore-ography of inertial mechanics and overcome the problemwhich is the difficulties in positioning and orientation ofairborne caused by meridian convergence In the papercombining with the error of SINS and the state variable ofSINSCNS integrated navigation system in the Polar Regionswhich based on the Grid Reference Frame the state spacemode of SINSCNS integrated navigation system in the PolarRegions is established The simulation results showed thatthe SINSCNS integrated navigation system in the PolarRegions could repress the error caused by the constantdrift of gyroscope and correct the navigation parametersof SINSCNS integrated navigation system With the sameprecision of inertial components the precision of SINSCNSintegrated navigation system in the Polar Regions and in themiddle and low latitudes is consistent

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

The research area is detection and information technology

Conflicts of Interest

Wedeclare that we do not have any commercial or associativeinterest that represents conflicts of interest in connectionwiththe work submitted

Acknowledgments

This work is partially supported by the School FoundationResearch Fund of 2018 and the National Natural ScienceFoundation of China Project no 51678470

References

[1] L Brigham ldquoPolar Ocean Navigationrdquo Encyclopedia of Earthsciences pp 512ndash515 2014

[2] Z Zhao Y-B Liang and J Hu ldquoArtic oil and natural gaspotential and exploration and development trendrdquo Earth Sci-ence Frontiers vol 21 no 3 pp 47ndash55 2014

[3] C Yang XWang Z Li Y Li and C Su ldquoTeleoperation controlbased on combination of wave variable and neural networksrdquoIEEE Transactions on Systems Man and Cybernetics Systems2017

[4] G-L Yang L-F Wang and S-J Yang ldquoOnline calibrationmethod of missile-borne SINSCNS integrated navigation sys-temrdquo Journal of Projectiles Rockets Missiles and Guidance vol34 no 5 pp 29ndash32 2014

[5] B Zhou W-L Zhao Y-J Rong X-M Hu and J-Y MaldquoRobust Adaptive Kalman Filtering Algorithm for IntegratedNavigation Based on MEMS-INSGNSSrdquo navigation and con-trol vol 17 no 4 pp 14ndash20 2018

[6] C Yang K Huang H Cheng Y Li and C Su ldquoHapticidentification by ELM-controlled uncertain manipulatorrdquo IEEETransactions on Systems Man and Cybernetics Systems vol 47no 8 pp 2398ndash2409 2017

[7] L-F Zhou X-M Zhao Z-N Hou and C-L Ding ldquoCNSSINSintegrated calibration technique based on level dampingrdquoZhongguo Guanxing Jishu XuebaoJournal of Chinese InertialTechnology vol 25 no 5 pp 561ndash565 2017

[8] C Yang J Luo Y Pan Z Liu and C Su ldquoPersonalized variablegain control with tremor attenuation for robot teleoperationrdquoIEEE Transactions on Systems Man and Cybernetics Systemspp 1ndash12 2017

[9] X-NHuang J-R Liu andW Yang ldquoAn EvaluationMethod forthe Head Accuracy of Inertial Navigation System in Flight Testrdquonavigation and control vol 4 pp 7ndash12 2017

[10] Q Zhou Y Qin Q Fu and Y Yue ldquoGrid mechanization ininertial navigation systems for transpolar aircraftrdquoXibei GongyeDaxue XuebaoJournal of Northwestern Polytechnical Universityvol 31 no 2 pp 210ndash217 2013

[11] Y Gao M Liu G Li and X Guang ldquoInitial alignment for SINSbased on pseudo-earth frame in polar regionsrdquo Sensors vol 17no 6 2017

[12] L-J Song Z Li Q-W Fu and B He ldquoResearched of SINSGPSIntegrated Navigation System based on Grid Reference Framefor Transpolar Aircraftrdquo Journal of System Simulation 2018

8 Complexity

[13] H-G Guo and W-S Wang ldquoCelestial navigation in polarRegionsrdquo Journal of Dalian Maritime University vol 15 no 3pp 43ndash48 1989

[14] V B Larin andAA Tunik ldquoOn inertial navigation system errorcorrectionrdquo International Applied Mechanics vol 48 no 2 pp213ndash223 2012

[15] H Zhen YWang andH-BWnag ldquoSimulation of geomagneticaided submarine navigation based on EMAG2rdquo Progress inGeophysics vol 27 no 4 pp 1795ndash1803 2012

[16] Q Li Y Ben F Yu and J Tan ldquoTransversal strapdown INSbased on reference ellipsoid for vehicle in the Polar RegionrdquoIEEE Transactions on Vehicular Technology vol 65 no 9 pp7791ndash7795 2016

[17] Y-Q Yao X-S Xu Y Li Y-T Liu J Sun and J-W TongldquoTransverse Navigation under the Ellipsoidal Earth Model andits Performance in both Polar and Non-polar areasrdquo Journal ofNavigation vol 69 no 2 pp 335ndash352 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

6 Complexity

0 1 2 3 4 5 6t (h)

minus5

0

5

0 1 2 3 4 5 6t (h)

minus5

0

5

0 1 2 3 4 5 6t (h)

minus5

0

5

U

()

()

()

(a) The misalignment angle of SINSCNS in the PolarRegions

0 1 2 3 4 5 6

t (h)

minus5

0

5

0 1 2 3 4 5 6

t (h)

minus5

0

5

0 1 2 3 4 5 6

t (h)

minus01

0

01

Ｐ(Ｇ

Ｍ)

Ｐ(Ｇ

Ｍ)

Ｐ５(Ｇ

Ｍ)

(b) The velocity error of SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

minus2000

0

2000

L

(m)

0 1 2 3 4 5 6t (h)

0

5

10

(m)

0 1 2 3 4 5 6t (h)

minus50

0

50

H

(m)

times105

(c) The position error of SINSCNS in the PolarRegions

0 1 2 3 4 5 6t (h)

0

0005

001

0 1 2 3 4 5 6t (h)

0

0005

001

0 1 2 3 4 5 6t (h)

0

0005

001 Ｒ

(∘Ｂ

) Ｓ

(∘Ｂ

) Ｔ

(∘Ｂ

)

(d) The constant drift of gyroscope of SINSCNS in thePolar Regions

0 1 2 3 4 5 6

t (h)

minus20

0

20

0 1 2 3 4 5 6

t (h)

minus100

0

100

0 1 2 3 4 5 6

t (h)

minus100

0

100

nablaＴ(

Ａ)nabla

Ｒ(

Ａ)nabla

Ｓ(

Ａ)

(e) The constant drift of accelerometer of SINSCNS inthe Polar Regions

0 1 2 3 4 5 6

t (h)

minus5

0

5

0 1 2 3 4 5 6

t (h)

0

2

4

0 1 2 3 4 5 6

t (h)

minus5

0

5

Ｒ

Ｓ

Ｔ

()

()

()

(f) The error of installation deviation angle ofSINSCNS in the Polar Regions

Figure 4

the transformation matrix of SINS and the transformationmatrix of star sensor so the measurement of the SINSCNSintegrated navigation system in the Polar Regions is shown as

119885119862119873119878 = 119901 minus 119888

= [(119906119888times) 03 119906119888 times119872119901 03 03 119862119875119887 (119906119888times)]

[[[[[[[[[[[[

120601120575119907119866120575119877119890120576

nabla

120583

]]]]]]]]]]]]

+ 120575119906119901(14)

where

119872119901 = [[[

minus1 0 00 cos 119871 00 sin 119871 0

]]]

(15)

33 The Trajectory of SINSCNS Integrated Navigation Systemin the Polar Regions It is the 6h in the simulation ofSINSCNS integrated navigation system when the aircraft isin the Polar Regions the trajectory is shown in Figure 3

Complexity 7

The parameters of trajectory are the starting point whichis [45∘119873 108∘119864 500119898] and culmination of latitude which is8926∘119873

The parameters of velocity are the initial velocity which is0119898119904 and the top velocity which is 31011989811990434The Simulation of SINSCNS IntegratedNavigation Systemin the Polar Regions In the simulation the parameters ofdevice are the same as the middle and low latitudes Theresults of simulation are shown in Figure 4

(1) The misalignment angle of SINSCNS integrated nav-igation system in the Polar Regions is showed in Figure 4(a)they are convergent to 051015840 in 100s and basically the same tothe attitude error of SINSCNS integrated navigation systemin the middle and low latitude

(2)The velocity error and the position error of SINSCNSintegrated navigation system in the Polar Regions are showedin Figures 4(b) and 4(c) the error of velocity is Schuler periodoscillation themaximum amplitude is not more than 05msThe error of position is the same as the error of velocity it isalso Schuler period oscillation and the maximum amplitudeis not more than 30m

(3) The constant drift of gyroscope and accelerometer ofSINSCNS integrated navigation system in the Polar Regionsis showed in Figures 4(d) and 4(e) the amplitude error ofvelocity and position also decreases when the constant driftof gyroscope and accelerometer is estimated The amplitudeof velocity and position error also decreases

(4)The error of installation deviation angle of SINSCNSintegrated navigation system in the Polar Regions is showedin Figure 4(f) the error of installation deviation angle is inaccordance with the misalignment angle and the error ofinstallation deviation angle of SINSCNS integrated naviga-tion system in the Polar Regions converges to 0310158404 Summary

The SINSCNS integrated navigation system in the PolarRegions based on the Grid Reference Frame are the chore-ography of inertial mechanics and overcome the problemwhich is the difficulties in positioning and orientation ofairborne caused by meridian convergence In the papercombining with the error of SINS and the state variable ofSINSCNS integrated navigation system in the Polar Regionswhich based on the Grid Reference Frame the state spacemode of SINSCNS integrated navigation system in the PolarRegions is established The simulation results showed thatthe SINSCNS integrated navigation system in the PolarRegions could repress the error caused by the constantdrift of gyroscope and correct the navigation parametersof SINSCNS integrated navigation system With the sameprecision of inertial components the precision of SINSCNSintegrated navigation system in the Polar Regions and in themiddle and low latitudes is consistent

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

The research area is detection and information technology

Conflicts of Interest

Wedeclare that we do not have any commercial or associativeinterest that represents conflicts of interest in connectionwiththe work submitted

Acknowledgments

This work is partially supported by the School FoundationResearch Fund of 2018 and the National Natural ScienceFoundation of China Project no 51678470

References

[1] L Brigham ldquoPolar Ocean Navigationrdquo Encyclopedia of Earthsciences pp 512ndash515 2014

[2] Z Zhao Y-B Liang and J Hu ldquoArtic oil and natural gaspotential and exploration and development trendrdquo Earth Sci-ence Frontiers vol 21 no 3 pp 47ndash55 2014

[3] C Yang XWang Z Li Y Li and C Su ldquoTeleoperation controlbased on combination of wave variable and neural networksrdquoIEEE Transactions on Systems Man and Cybernetics Systems2017

[4] G-L Yang L-F Wang and S-J Yang ldquoOnline calibrationmethod of missile-borne SINSCNS integrated navigation sys-temrdquo Journal of Projectiles Rockets Missiles and Guidance vol34 no 5 pp 29ndash32 2014

[5] B Zhou W-L Zhao Y-J Rong X-M Hu and J-Y MaldquoRobust Adaptive Kalman Filtering Algorithm for IntegratedNavigation Based on MEMS-INSGNSSrdquo navigation and con-trol vol 17 no 4 pp 14ndash20 2018

[6] C Yang K Huang H Cheng Y Li and C Su ldquoHapticidentification by ELM-controlled uncertain manipulatorrdquo IEEETransactions on Systems Man and Cybernetics Systems vol 47no 8 pp 2398ndash2409 2017

[7] L-F Zhou X-M Zhao Z-N Hou and C-L Ding ldquoCNSSINSintegrated calibration technique based on level dampingrdquoZhongguo Guanxing Jishu XuebaoJournal of Chinese InertialTechnology vol 25 no 5 pp 561ndash565 2017

[8] C Yang J Luo Y Pan Z Liu and C Su ldquoPersonalized variablegain control with tremor attenuation for robot teleoperationrdquoIEEE Transactions on Systems Man and Cybernetics Systemspp 1ndash12 2017

[9] X-NHuang J-R Liu andW Yang ldquoAn EvaluationMethod forthe Head Accuracy of Inertial Navigation System in Flight Testrdquonavigation and control vol 4 pp 7ndash12 2017

[10] Q Zhou Y Qin Q Fu and Y Yue ldquoGrid mechanization ininertial navigation systems for transpolar aircraftrdquoXibei GongyeDaxue XuebaoJournal of Northwestern Polytechnical Universityvol 31 no 2 pp 210ndash217 2013

[11] Y Gao M Liu G Li and X Guang ldquoInitial alignment for SINSbased on pseudo-earth frame in polar regionsrdquo Sensors vol 17no 6 2017

[12] L-J Song Z Li Q-W Fu and B He ldquoResearched of SINSGPSIntegrated Navigation System based on Grid Reference Framefor Transpolar Aircraftrdquo Journal of System Simulation 2018

8 Complexity

[13] H-G Guo and W-S Wang ldquoCelestial navigation in polarRegionsrdquo Journal of Dalian Maritime University vol 15 no 3pp 43ndash48 1989

[14] V B Larin andAA Tunik ldquoOn inertial navigation system errorcorrectionrdquo International Applied Mechanics vol 48 no 2 pp213ndash223 2012

[15] H Zhen YWang andH-BWnag ldquoSimulation of geomagneticaided submarine navigation based on EMAG2rdquo Progress inGeophysics vol 27 no 4 pp 1795ndash1803 2012

[16] Q Li Y Ben F Yu and J Tan ldquoTransversal strapdown INSbased on reference ellipsoid for vehicle in the Polar RegionrdquoIEEE Transactions on Vehicular Technology vol 65 no 9 pp7791ndash7795 2016

[17] Y-Q Yao X-S Xu Y Li Y-T Liu J Sun and J-W TongldquoTransverse Navigation under the Ellipsoidal Earth Model andits Performance in both Polar and Non-polar areasrdquo Journal ofNavigation vol 69 no 2 pp 335ndash352 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Complexity 7

The parameters of trajectory are the starting point whichis [45∘119873 108∘119864 500119898] and culmination of latitude which is8926∘119873

The parameters of velocity are the initial velocity which is0119898119904 and the top velocity which is 31011989811990434The Simulation of SINSCNS IntegratedNavigation Systemin the Polar Regions In the simulation the parameters ofdevice are the same as the middle and low latitudes Theresults of simulation are shown in Figure 4

(1) The misalignment angle of SINSCNS integrated nav-igation system in the Polar Regions is showed in Figure 4(a)they are convergent to 051015840 in 100s and basically the same tothe attitude error of SINSCNS integrated navigation systemin the middle and low latitude

(2)The velocity error and the position error of SINSCNSintegrated navigation system in the Polar Regions are showedin Figures 4(b) and 4(c) the error of velocity is Schuler periodoscillation themaximum amplitude is not more than 05msThe error of position is the same as the error of velocity it isalso Schuler period oscillation and the maximum amplitudeis not more than 30m

(3) The constant drift of gyroscope and accelerometer ofSINSCNS integrated navigation system in the Polar Regionsis showed in Figures 4(d) and 4(e) the amplitude error ofvelocity and position also decreases when the constant driftof gyroscope and accelerometer is estimated The amplitudeof velocity and position error also decreases

(4)The error of installation deviation angle of SINSCNSintegrated navigation system in the Polar Regions is showedin Figure 4(f) the error of installation deviation angle is inaccordance with the misalignment angle and the error ofinstallation deviation angle of SINSCNS integrated naviga-tion system in the Polar Regions converges to 0310158404 Summary

The SINSCNS integrated navigation system in the PolarRegions based on the Grid Reference Frame are the chore-ography of inertial mechanics and overcome the problemwhich is the difficulties in positioning and orientation ofairborne caused by meridian convergence In the papercombining with the error of SINS and the state variable ofSINSCNS integrated navigation system in the Polar Regionswhich based on the Grid Reference Frame the state spacemode of SINSCNS integrated navigation system in the PolarRegions is established The simulation results showed thatthe SINSCNS integrated navigation system in the PolarRegions could repress the error caused by the constantdrift of gyroscope and correct the navigation parametersof SINSCNS integrated navigation system With the sameprecision of inertial components the precision of SINSCNSintegrated navigation system in the Polar Regions and in themiddle and low latitudes is consistent

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

The research area is detection and information technology

Conflicts of Interest

Wedeclare that we do not have any commercial or associativeinterest that represents conflicts of interest in connectionwiththe work submitted

Acknowledgments

This work is partially supported by the School FoundationResearch Fund of 2018 and the National Natural ScienceFoundation of China Project no 51678470

References

[1] L Brigham ldquoPolar Ocean Navigationrdquo Encyclopedia of Earthsciences pp 512ndash515 2014

[2] Z Zhao Y-B Liang and J Hu ldquoArtic oil and natural gaspotential and exploration and development trendrdquo Earth Sci-ence Frontiers vol 21 no 3 pp 47ndash55 2014

[3] C Yang XWang Z Li Y Li and C Su ldquoTeleoperation controlbased on combination of wave variable and neural networksrdquoIEEE Transactions on Systems Man and Cybernetics Systems2017

[4] G-L Yang L-F Wang and S-J Yang ldquoOnline calibrationmethod of missile-borne SINSCNS integrated navigation sys-temrdquo Journal of Projectiles Rockets Missiles and Guidance vol34 no 5 pp 29ndash32 2014

[5] B Zhou W-L Zhao Y-J Rong X-M Hu and J-Y MaldquoRobust Adaptive Kalman Filtering Algorithm for IntegratedNavigation Based on MEMS-INSGNSSrdquo navigation and con-trol vol 17 no 4 pp 14ndash20 2018

[6] C Yang K Huang H Cheng Y Li and C Su ldquoHapticidentification by ELM-controlled uncertain manipulatorrdquo IEEETransactions on Systems Man and Cybernetics Systems vol 47no 8 pp 2398ndash2409 2017

[7] L-F Zhou X-M Zhao Z-N Hou and C-L Ding ldquoCNSSINSintegrated calibration technique based on level dampingrdquoZhongguo Guanxing Jishu XuebaoJournal of Chinese InertialTechnology vol 25 no 5 pp 561ndash565 2017

[8] C Yang J Luo Y Pan Z Liu and C Su ldquoPersonalized variablegain control with tremor attenuation for robot teleoperationrdquoIEEE Transactions on Systems Man and Cybernetics Systemspp 1ndash12 2017

[9] X-NHuang J-R Liu andW Yang ldquoAn EvaluationMethod forthe Head Accuracy of Inertial Navigation System in Flight Testrdquonavigation and control vol 4 pp 7ndash12 2017

[10] Q Zhou Y Qin Q Fu and Y Yue ldquoGrid mechanization ininertial navigation systems for transpolar aircraftrdquoXibei GongyeDaxue XuebaoJournal of Northwestern Polytechnical Universityvol 31 no 2 pp 210ndash217 2013

[11] Y Gao M Liu G Li and X Guang ldquoInitial alignment for SINSbased on pseudo-earth frame in polar regionsrdquo Sensors vol 17no 6 2017

[12] L-J Song Z Li Q-W Fu and B He ldquoResearched of SINSGPSIntegrated Navigation System based on Grid Reference Framefor Transpolar Aircraftrdquo Journal of System Simulation 2018

8 Complexity

[13] H-G Guo and W-S Wang ldquoCelestial navigation in polarRegionsrdquo Journal of Dalian Maritime University vol 15 no 3pp 43ndash48 1989

[14] V B Larin andAA Tunik ldquoOn inertial navigation system errorcorrectionrdquo International Applied Mechanics vol 48 no 2 pp213ndash223 2012

[15] H Zhen YWang andH-BWnag ldquoSimulation of geomagneticaided submarine navigation based on EMAG2rdquo Progress inGeophysics vol 27 no 4 pp 1795ndash1803 2012

[16] Q Li Y Ben F Yu and J Tan ldquoTransversal strapdown INSbased on reference ellipsoid for vehicle in the Polar RegionrdquoIEEE Transactions on Vehicular Technology vol 65 no 9 pp7791ndash7795 2016

[17] Y-Q Yao X-S Xu Y Li Y-T Liu J Sun and J-W TongldquoTransverse Navigation under the Ellipsoidal Earth Model andits Performance in both Polar and Non-polar areasrdquo Journal ofNavigation vol 69 no 2 pp 335ndash352 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Complexity

[13] H-G Guo and W-S Wang ldquoCelestial navigation in polarRegionsrdquo Journal of Dalian Maritime University vol 15 no 3pp 43ndash48 1989

[14] V B Larin andAA Tunik ldquoOn inertial navigation system errorcorrectionrdquo International Applied Mechanics vol 48 no 2 pp213ndash223 2012

[15] H Zhen YWang andH-BWnag ldquoSimulation of geomagneticaided submarine navigation based on EMAG2rdquo Progress inGeophysics vol 27 no 4 pp 1795ndash1803 2012

[16] Q Li Y Ben F Yu and J Tan ldquoTransversal strapdown INSbased on reference ellipsoid for vehicle in the Polar RegionrdquoIEEE Transactions on Vehicular Technology vol 65 no 9 pp7791ndash7795 2016

[17] Y-Q Yao X-S Xu Y Li Y-T Liu J Sun and J-W TongldquoTransverse Navigation under the Ellipsoidal Earth Model andits Performance in both Polar and Non-polar areasrdquo Journal ofNavigation vol 69 no 2 pp 335ndash352 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom