Research Article A Method for SINS Alignment with Large

Research ArticleA Method for SINS Alignment with Large Initial MisalignmentAngles Based on Kalman Filter with Parameters Resetting

Xixiang Liu12 Xiaosu Xu12 Yiting Liu12 and Lihui Wang12

1 School of Instrument Science amp Engineering Southeast University Nanjing 210096 China2 Key Laboratory of Micro-Inertial Instrument and Advanced Navigation Technology of Ministry of Education Nanjing 210096 China

Correspondence should be addressed to Xixiang Liu scliuseu163com

Received 24 September 2013 Revised 18 February 2014 Accepted 3 March 2014 Published 1 April 2014

Academic Editor Shihua Li

Copyright copy 2014 Xixiang Liu 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

In the initial alignment process of strapdown inertial navigation system (SINS) large initial misalignment angles always bringnonlinear problem which causes alignment failure when the classical linear error model and standard Kalman filter are usedIn this paper the problem of large misalignment angles in SINS initial alignment is investigated and the key reason for alignmentfailure is given as the state covariance fromKalmanfilter cannot represent the true one during the steady filtering process Accordingto the analysis an alignment method for SINS based on multiresetting the state covariance matrix of Kalman filter is designed todeal with large initial misalignment angles in which classical linear error model and standard Kalman filter are used but the statecovariance matrix should be multireset before the steady process until large misalignment angles are decreased to small ones Theperformance of the proposedmethod is evaluated by simulation and car test and the results indicate that the proposedmethod canfulfill initial alignment with large misalignment angles effectively and the alignment accuracy of the proposed method is as preciseas that of alignment with small misalignment angles

1 Introduction

For its merits of complete independence strong anti-interference ability comprehensive and high update fre-quency of navigation message strapdown inertial navigationsystem (SINS) has been widely used for military application[1 2] As a navigationmethod based on integration operationinitial alignment should be done in SINS before navigationis started The purpose of initial alignment of SINS is todetermine the attitude matrix from body frame to navigationframe that is initial attitude because initial velocity andposition are easy to determine by external navigation systemssuch as GPS [1ndash4] In SINS alignment misalignment anglesare defined as the Euler angles which describe the rotationbetween calculated navigation frame and ideal navigationframe [1]

The compassing alignment method based on compasseffect and the optimal estimation methods based on Kalmanfilter (eg transfer alignment zero-velocity alignment) arethe mostly used techniques to deal with alignment problems

in SINS [2ndash8] Among these zero-velocity is widely used asexternal reference navigation information by virtue of thecharacteristic of no linear velocity for static vehicles in zero-velocity alignment due to its merits of allowing swingingmotion and no requirement for external navigation systemThe alignment process of zero-velocity can be divided intotwo stages in chronological order the first one is coarsealignment based on analytical method and the second is finealignment based on Kalman filter [2 7 8] Among thesestages alignment result of coarse alignment is the precondi-tion of fine one and the accuracy of coarse alignment affectsthe accuracy of fine alignment

Error propagation model and filter method are two keyissues to addresswhenKalmanfilter ormethods derived fromKalman filter are used to fulfill SINS alignment [2 9] Theequations describing SINS algorithm are a set of nonlineardifferential equations so error propagation model of SINSis also nonlinear essentially [10] The classical linear errorpropagation model either with 120601 or 120595 representations isobtained by the assumption of small misalignment angles

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 346291 10 pageshttpdxdoiorg1011552014346291

2 Mathematical Problems in Engineering

and the standard Kalman filter can fulfill initial alignmentonly with small misalignment angles when linear model isused [1ndash9] However because of environmental disturbanceand sensor errors large errors between initial attitude andthe true one that is large misalignment problems or non-linear problems are often generated in SINS when analyticalmethod is used for coarse alignment [11ndash15] The problemsaddressed in this paper are concentrated on the investigationof initial alignment for SINS with large initial misalignmentangles caused by low accuracy of coarse alignment

Recently SINS error propagation models and nonlinearestimation methods for large misalignment angles are widelyreported in the literatures Error propagation models canbe divided into two types one is nonlinear model forlarge misalignment angles among which the large azimuthmisalignment error model is the most representative [11]the other is linearized method for nonlinear problem whichgains linear model for large misalignment angles by approx-imation or state transformation [12 13] Among nonlinearfilters extended Kalman filter (EKF) is widely used andestimation methods based on approximate probability den-sity distribution such as particle filter (PF) and unscentedKalman filter (UKF) are also popular currently [14 15]Thesefilters can solve nonlinear function directly by using theevolution of probability density distributionwithout the needof detailed expansion equations of nonlinear function Largemisalignment problems can be solved with the above modelsand methods to some extent However there are still manyshortages needed to be concerned such as the complexityproblem in nonlinear error model for large misalignmentangles state dimension extending problem in the lineariza-tion process and heavy computational problem in PF andUKF In addition investigation on stability of the methods isstill blank field when PF and UKF are used in engineering

For the purpose of simplifying error model withoutadding computational load an initial alignment method forlarge initial misalignment angles is proposed based on clas-sical linear error propagation model and standard Kalmanfilter in which the state covariance matrix of Kalman filteris reset several times before the steady filtering process is

achieved With the parameters resetting the state covariancefrom Kalman filter can truly represent the true one andthe high utilization of measurement vector can be ensuredCoupled with the help of closed loop correction mode largemisalignment angles can be converted into small ones so thatalignment can be finished effectively

The rest of this paper is organized as followsThe classicallinear error propagation model and standard Kalman filterused in zero-velocity alignment are described in Section 2Then the reason for alignment failure with large misalign-ment angles is analyzed in Section 3 and a potential solutionmethod is proposed which is verified by simulation InSection 4 the validity of the proposed method is furtherverified by car test Finally some conclusions are given inSection 5

2 Error Propagation Model and Kalman Filterfor Zero-Velocity Alignment

21 System Equation Choose velocity errors and misalign-ment angles of SINS as the state variables and then the statevector of the system can be constructed as

X = [120575119881119864120575119881119873

120601119864

120601119873

120601119880]

T (1)

where 120575119881119864

and 120575119881119873

are east and north velocity errorsrespectively and 120601

119864 120601119873 and 120601

119880are misalignment angles

of pitch roll and yaw respectively To vehicles running onthe earth surface the height and upward velocity can be setas zero In this paper east-north-up (ENU) is defined asnavigation frame 119899

The system state equation can be constructed as [1]

X (119905) = A (119905)X (119905) +W (119905) (2)

where A(119905) is the state matrix and W(119905) is the white systemprocess noise with the power spectral density Q Based onthe classical linear error propagation model the state matrixA(119905) can be expressed as [1]

A (119905)

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119881119873

119877

tan 119871 2120596119894119890sin 119871 +

119881119864

119877

tan119871 0 minus119891119880

119891119873

minus(2120596119894119890sin 119871 +

119881119864

119877

tan119871) 0 119891119880

0 minus119891119864

0 minus

1

119877

0 120596119894119890sin 119871 +

119881119864

119877

tan119871 minus(120596119894119890cos 119871 +

119881119864

119877

)

1

119877

0 minus(120596119894119890sin 119871 +

119881119864

119877

tan 119871) 0 minus

119881119873

119877

1

119877

tan 119871 0 120596119894119890cos 119871 +

119881119864

119877

119881119873

119877

0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(3)

Mathematical Problems in Engineering 3

Kalman filter

SINS

Zero-velocity Velocity

VelocityState variables

(feedback)

Navigation parameters

Figure 1 Kalman filter with closed loop correction mode

where 119881119864and 119881

119873are the east and north velocity from SINS

respectively120596119894119890and119877 are the rotation angular rate and radius

of the Earth respectively 119871 is the latitude of vehicle positionand 119891

119864 119891119873 and 119891

119880denote the projection in 119899 frame of the

measured value f119887 by accelerometer in 119887 frame

22 Measurement Equation When the vehicle is static thezero-velocity information in navigation frame is used asexternal reference and the measurement vector can be con-structed as

Z = [119881119864minus 0 119881

119873minus 0]

T= [120575119881119864

120575119881119873]

T (4)

where Δ119881119864and Δ119881

119873are the measurement variables which

equal 119881119864and 119881

119873from SINS respectively

The system measurement equation can be constructed as[1]

Z (119905) = H (119905)X (119905) + V (119905) (5)

whereH(119905) is measurementmatrix andV(119905) is the whitemea-surement noise with the power spectral density R Accordingto the relationship between measured vector and state vectorH(119905) can be expressed as

H = [

1 0 0 0 0

0 1 0 0 0

] (6)

23 Closed Loop Correction Mode The system and measure-ment equations given in Sections 21 and 22 are derived by theassumption that initial misalignment angles are small With(2) and (5) SINS alignment with small initial misalignmentangles can be finished by standard Kalman filter and the fiveequations of standard Kalman filter will be demonstrated inSection 31

The ldquosystemrdquo in (2) and (5) refers to the ldquoerror combina-tionrdquo of SINS navigation parameters The state variables donot participate in SINS navigation solution whichmeans thatthe SINS solution and error estimation based on (2) and (5)are two independent processes Based on this closed loopcorrection mode shown in Figure 1 is used to feed the esti-mation of state variables back to SINS to correct navigationparameters and the sequent SINS navigation solution will becarried out according to the corrected navigation parametersonce the correction operation is run When misalignmentangles can be estimated by Kalman filter the attitude curvesfrom SINS solution will gradually approach the true valueswith the help of closed loop correctionmode in other wordsthe error attitude curves (misalignment angles) from SINSsolution will reduce to zero (or limit alignment values)

24 Simulation

241 Simulation Setting Inertial measurement unit (IMU)is composed of medium or low precision sensors The gyroconstant bias is set as 01∘h and random bias is whiteGaussian noise with zero mean and standard variance 01∘hThe accelerometer constant error is 500 ug and random erroris also white Gaussian noise with zero mean and standardvariance 500 ug With the above assumption about sensorprecision the limit alignment accuracy can be obtained byzero-velocity alignment method as follows pitchmdashminus171891015840(minute of degree) rollmdash171891015840 and yawmdashminus2702481015840

The vehicle is without linear movement but with swing-ing motion The swinging amplitudes of pitch roll andyaw are set as 18∘ 24∘ and 28∘ respectively and thecorresponding oscillation cycle is 6 8 and 10 s Based on theabove ideal motion values the ideal sensor output can begotten by back-stepping of navigation solution When sensorerrors are added into ideal sensor output the true sensordata can be obtained with which navigation solution andalignment operation can be runMeanwhile the idealmotionvalues can be used as references to judge the alignment resultswith different initial misalignment angles

The initial parameters for Kalman filter are set as

X0= [0 0 0 0 0]

T

P0= diag [(01ms)2 (01ms)2 (15

∘)

2

(15

∘)

2

(80

∘)

2]

Q = diag [(500 ug)2 (500 ug)2 (01

∘)

2

(01

∘)

2

(01

∘)

2]

R = diag [(01ms)2 (01ms)2] (7)

There is no unified definition about the magnitude oflarge misalignment angles in SINS until now To show howthe initialmisalignment angles with differentmagnitudes willaffect alignment results the following five sets of differentmisalignment angles are used in simulation Condition 1120601

0=

[0

∘0

∘0

∘]

T Condition 21206010= [5

∘5

∘5

∘]

T Condition 31206010=

[5

∘5

∘10

∘]

T Condition 4 1206010= [5

∘5

∘20

∘]


∘5

∘30

∘]

T Generally speaking the level alignmentaccuracy is much higher than that of azimuth one obtainedby coarse alignment with analytical method so the levelmisalignment angles from Condition 2 to Condition 5 are allset as 5∘

In simulation the update cycle of sensor data navigationparameter and alignment filtering are all set as 10ms and thestatistical period is set as 1 s


0 100 200 300 400 500 600 700 800 900

421 422 423Condition 5

Condition 4

Limit alignment accuracy

Condition 2

Condition 3

Condition 1

t (s)

0

minus2

minus16

minus17

minus18

minus19

minus4

Erro

r(998400)

(a) Pitch

0 100 200 300 400 500 600 700 800 900

0

2

4

4068 407 4072

17217

174

Condition 1

Condition 5

Condition 4 Condition 2 Condition 3


Erro

r(998400)

t (s)

(b) Roll


0 100 200 300 400 500 600 700 800 900

437 438 439 440 441Condition 5

Condition 4 Condition 3

Condition 1 Condition 2 Limit alignment accuracy

0

200

400

minus200

minus400

minus40

minus30

minus20

Erro

r(998400)

t (s)

(c) Yaw

Figure 2 Alignment results with different initial misalignment angles

242 Simulation Results The simulation lasts for 900 s andthe misalignment angle curves of different conditions areshown in Figure 2 It can be seen that all misalignment curvescan converge to the curves of limit alignment accuracy ina short time under Conditions 1 and 2 under Condition3 misalignment curves can converge but with a longertime than those of Conditions 1 and 2 under Condition 4misalignment curves can converge but with a longer time andlarger errors while under Condition 5 misalignment curvescannot converge

3 Analysis for Alignment Failure andan Improved Method

31 Analysis for Alignment Failure The simulation resultsin Section 242 indicate that there is a close relationshipbetween initial misalignment angles and alignment resultswhen the classical linear error propagation model and stan-dard Kalman filter are used for zero-velocity alignmentLarger initial misalignment angles will produce lower accu-racy even alignment failure so this alignmentmethod cannotdeal with nonlinear problems caused by large misalignmentangles

Nevertheless the simulation results in Figure 2 alsoshow that there is a convergence trend for all five differentconditions at the beginning process that is 0ndash100 s inFigure 2(c) Maybe this short convergence process impliesthat Kalman filter can estimate misalignment angles to some

extent though the estimation is not very accurate In viewof this the change of the state covariance from Kalmanfilter is further studied and compared with the true one Forthe sake of brevity Figure 3 only shows the covariance ofyaw under Conditions 1 and 5 where the true covariance ofyaw can be calculated according to the alignment error inFigure 2(c) with square operation The curves in Figure 3indicate that under the condition of small misalignmentangles the covariance of yaw fromKalman filter is larger thanthe true one throughout the alignment process while underthe condition of large misalignment angles the covarianceis larger than the true one only at the beginning and oneshort period during the alignment and smaller than thetrue one during the whole steady filtering process Herethe period of 300ndash900 s during which the change of Pis smooth is defined as the steady filtering process Thedifferences between Figures 3(a) and 3(b) indicate that thestate covariance fromKalman filter cannot truly represent thetrue covariance of state vector under the condition of largemisalignment angles

In order to evaluate the effect of P on alignment resultsthe equations composing standard Kalman filter are furtherstudiedThe five equations of standardKalman filter are listedas follows [2 9 11ndash15]

Prediction for state vector

X119896119896minus1

= Φ119896119896minus1

X119896minus1

(8)


0 100 200 300 400 500 600 700 800 9000

20

40

True P

t (s)

P from Kalman filter

Elem

ent o

fPm

atrix

(∘2)

(a) Yaw in Condition 1

00 100 200 300 400 500 600 700 800 900

200

400

200 400 600 800

0102030P from Kalman filter

True P

t (s)

Elem

ent o

fPm

atrix

(∘2)

minus10

(b) Yaw in Condition 5

Figure 3 Comparison of state covariance

Update for state vector

X119896=

X119896119896minus1

+ K119896(Z119896minusH119896X119896119896minus1

) (9)

Calculation for gain

K119896= P119896119896minus1

HT119896(H119896P119896119896minus1

HT119896+ R119896)

minus1

(10)

Prediction for state covariance

P119896119896minus1

= Φ119896119896minus1

P119896minus1Φ

T119896119896minus1

+Q119896minus1

(11)

Update for state covariance

P119896= (I minus K

119896H119896)P119896119896minus1

(12)

In the above equations the subscript 119896 denotes the 119896thtime-step X denotes the estimation of state vector Φ

119896119896minus1is

the translation matrix which is a discretization of A(119905) in (2)and K

119896is the gain matrix which represents the utilization

degree between the measurement vector and prediction forstate vector In Kalman filter a larger K

119896means a higher

utilization degree of measurement vectorFurthermore by analyzing the relationship between (10)

and (11) it can be drawn that with a fixed 119896 all parameters in(8)ndash(12) will be fixed with a larger P

119896minus1 a larger P

119896119896minus1in (11)

will be produced and a larger K119896in (10) will be generated

which leads to a higher utilization degree of measurementvector

The above analysis indicates that Kalman filter can usemeasurement vector with a higher utilization degree duringthe whole alignment process under the condition of smallmisalignment angles while under the condition of largemisalignment angles Kalman filter cannot because the P issmaller than true ones during the steady filtering processAt the same time the convergence trends in Figure 2 at thebeginning of alignment process can be attributed to large

initial parameters of Kalman filter that is the initial P0is

much larger than the true onesBased on the above analysis it can be concluded that

under large misalignment condition the change of statecovariance from Kalman filter cannot truly represent thatof state vector and Kalman filter cannot utilize externalreference navigation information effectively It should be oneof the key reasons which lead to the alignment failure whenthe classical linear error propagation model and standardKalman filter are used for zero-velocity alignment with largemisalignment angles

32 A Possible Way to Solve This Problem

321 Kalman Filter with the Parameters Resetting In align-ment process in order to get a higher convergence speed andmake full use of external reference information initial P

0is

usually set as a large value when Kalman filter is usedBased on the understanding about the effect of P and

initial setting of P0 an initial alignment method for SINS

based on multiresetting Kalman filter parameters as shownin Figure 4 is designed in this paper In this methodthe classical linear error propagation model and standardKalman filter are used but the parameters of Kalman filterwill be reset before the steady filtering process to ensurethe state covariance larger than the true one and improveutilization degree of measurement vector In Figure 4 Δ119905denotes the update cycle of sensor data navigation solutionand alignment filtering Δ119879 denotes the parameter resettingcycle Δ119879 denotes the resetting number

During the alignment process based on parameter reset-ting at themoment 119905

0 the coarse alignment results will be set

as initial parameter for SINS solution and initial parameterssuch as P

0will be set to Kalman filter at the moment

1199051 the state covariance will be set to initial value that is

P rArr P0 when the preset resetting times are finished no


(1) State estimation with Kalman filter

(2) Close loop correction for SINS parameters

SINS solution SINS solution SINS solution





(1) SINS solution(2) Alignment without resetting

(1) Initialize SINS withcoarse alignment results

(2) Initialize Kalman filter Kalman filter

Kalman filter

nΔT

ΔT ΔT

Δt Δt Δt

t

t

1

with X0 P0 Q and R

Reset P rArr P0 for

Reset P rArr P0 for

t0

Figure 4 Initial alignment process with the state covariance resetting

resetting but normal alignment will be done According tothe parameters defined in Kalman filter the matrixes Q andR that denote the process noise and measurement noiserespectively will remain unchanged when the instrumentprecision and operating condition remain unchanged thusthe resetting operation to Kalman filter parameters refers inparticular to P in the proposed method

As mentioned in Section 23 Kalman filter and SINSsolution are two independent processes so resetting Kalmanfilter parameters will only affect the parameters of the filterand the state estimation but have no effect on navigationparametersThat is to say the state estimation before resetting(the convergence result at the beginning of filtering) will bepreserved in SINS navigation solutionWith the help of reset-ting and closed loop correction mode large misalignmentangles can be gradually converged to small ones until theclassical linear error propagation model can be applied

322 Analysis of the Proposed Method

(1) Analysis of Advantages and Disadvantages Obviouslythe proposed method in Section 321 adopts the standardKalman filter rather than nonlinear filter such as UKF andPF thus it owns the advantage of less calculation At the sametime it adopts the classical linear error propagation modelrather nonlinear model thus it owns less calculation andsimplified model

But during alignment process the resetting operationshould be executed several times and accordingly the stableprocess of filtering should be done at the same times passivelyThus a disadvantage about this method is the prolongedalignment time

(2) Two Problems about the Proposed Method Two problemscaused by the resetting operation should be solved The firstone is how to select the resetting cycle and the second is howto select the resetting times Generally speaking there are twosolutions to these problems The first one is a fixed resetting

cycle and resetting times which can be preset according tothe instrument precision and operation condition and thesecond one is adaptive ones which can be set adaptivelyaccording to some criteria Obviously the second is an idealmethod but it is difficult to be fulfilled

The aim of this paper is to find an alignment method forSINS with large initial misalignment angles without addingmodel complexity and adding computational load thus nogeneral or adaptive rules about the selection of resetting cycleand times are studied In the rest of this paper fixed resettingcycle and times are selected

In engineering a certain safety factor should be added tothe above parameters to ensure a successful alignment thusthe alignment time will further be prolonged

33 Simulation

331 Simulation Setting The simulation is conducted underthe same conditions as those in Section 241 and Condition5 is used to verify the validity of the proposed method

332 Simulation Results The simulation lasts for 900 s andthe P of Kalman filter is reset at 50 s 100 s and 150 srespectively during alignment The curves of misalignmentangles are shown in Figure 5 where the curves from Sec-tion 242 under Condition 1 are also given for comparison InFigure 5(c) the dotted lines perpendicular to the horizontalcoordinate denote the resetting moment The comparison ofstate covariance fromKalman filter under Conditions 1 and 5with the true one underCondition 5 as the reference is shownin Figure 6

The curves in Figure 5 indicate that initial alignmentcan be fulfilled by the classical linear error propagation andstandard Kalman filter after resetting the covariance matrixthree times And the partial enlargement in Figure 5(c)indicates that the alignment accuracy processed for 600 sis as precise as that with small initial misalignment anglesThe curves also show that there are two added passive stable


Condition 1

Condition 5 0 100 200 300 400 500 600 700 800 900

t (s)

0

minus2

minus4

Erro

r(998400)

(a) Pitch

Condition 5

Condition 1

0 100 200 300 400 500 600 700 800 900t (s)

0

2

4

Erro

r(998400)

(b) Roll

Condition 5

Condition 1

0 100 200 300 400 500 600 700 800 900500 600 700 800 900


t (s)

Erro

r(998400)

500

0

minus500

minus1000

minus1500

minus25minus30minus35minus40

(c) Yaw

Figure 5 Alignment results with state covariance resetting

processes compared with those in Figure 3 which is causedby the operation of resetting

The curves in Figure 6 indicate that the covariance of yawfrom Kalman filter under Condition 5 is larger than that oftrue one during thewhole alignment process Comparedwiththat in Figure 3 the true covariance gradually converges to asmall value and when the alignment is processed for 600 sthe covariance from Kalman filter under Conditions 1 and5 are approximately equal which means that the alignmentaccuracy are approximately equal at that moment as shownin Figure 5(c)

From the above analysis it can be concluded that withthe resetting to the state covariance of Kalman filter initialalignment for SINSwith large initial misalignment angles canbe fulfilled with classical linear error propagation model andstandard Kalman filter

4 Car Test

41 Test Setup Considering the small angular motion with-out linear movement caused by engine vibration undershutdown condition the car test is executed to verify thevalidity of the proposed alignment method The fiber-opticgyro SINS as shown in Figure 7 is provided by SoutheastUniversity and the precision parameters of sensor are listed inTable 1 In this type of SINS the update frequencies of sensordata and navigation data are both 100Hz and PC104 with aCPU frequency of 1 GHz is used as the navigation computer

Table 1 Instrument precision of SINS

Gyro AccelerometerZero bias lt003∘h Zeros bias plusmn5 times 10minus5 gRandom walkcoefficient lt0005∘h Zero bias stability lt4 times 10minus4 g

Scale factor error lt20 ppm Scale factor error lt100 ppm

Table 2 Accuracy of PHINS

Azimuth Level attitude005∘ secant latitude (without GPS aid) Less than 001∘

The navigation parameters from PHINSmdasha high precisionSINS from IxSea France as shown in Figure 7mdashare used asreference The precision of PHINS is listed in Table 2

Three different alignment schemes as follows are com-pared Scheme 1 standard Kalman filter with small initialmisalignment angles Scheme 2 standard Kalman filter withlarge initialmisalignment angles Scheme 3 standardKalmanfilter with large initial misalignment angles but with resettingoperation For the three schemes the filtering frequencies areall set as 100Hz and the classical linear error propagationmodel is used for the calculation

The test is run as offline semiphysical simulation basedon the saved data Before fine alignment analytical coarsealignment is executedThe results of coarse alignment can be


0 100 200 300 400 500 600 700 800 9000

50

100

True P in Condition 5

400 500 600 700 800 9004648

55254 P from Kalman filter in Condition 5

P from Kalman filter in Condition 1

Elem

ent o

fPm

atrix

(∘2)

t (s)


FOG-SINS PHINS

Figure 7 SINS and PHINS in car

Table 3 Statistic results of Scheme 1 Scheme 3 and PHINS

Pitch Roll YawMean Std Mean Std Mean Std

PHINS (∘) minus08292 00011 minus02441 00046 855903 00024Scheme 1 (∘) minus08200 00092 minus02433 00056 846823 03092Scheme 3 (∘) minus08202 00098 minus02455 00061 850917 01906

considered as small misalignment angles because the sensorsused in this test are high and the vibration caused by engineis small To simulate large misalignment angles the angles[5

∘5

∘30

∘]

T are added to the coarse alignment resultsDuring the alignment process in Scheme 3 the P of

Kalman filter is reset at 50 s 100 s 150 s and 200 s respec-tively

42 Test Results The curves of alignment results from differ-ent schemes are shown in Figure 8The trends and oscillatingprocesses in Figure 8 are similar to those in Figure 5 Thealignment results in Figure 8 indicate that standard Kalmanfilter can fulfill initial alignment with small initial mis-alignment angles but cannot with large initial misalignmentangles while standard Kalman filter with multiresetting canfulfill alignment with large initial misalignment angles

Table 3 lists the statistic results in 300sim500 s of Scheme 1Scheme 3 and PHINS According to the attitude of PHINSalignment accuracy of Scheme 3 is as precise as that of Scheme1 after alignment for 300 s

From Figure 8 and Table 3 it can be concluded that withthe help of multiresetting zero-velocity alignment with largeinitial misalignment angles can be fulfilled when classical

linear error propagation model and standard Kalman filterare used

43 Further Comparison between Simulation and Car TestIn Section 33 three times of resetting are required to fulfillalignment while in Section 42 four times are requiredThere are two possible reasons for this difference the firstis that it is easy to find a set of Kalman parameters fit forsensors in simulation while it is difficult in engineering andthe second is that the initial misalignment angles are largerin car test than those in simulation because of the addedcoarse alignment errors Although no general rules about theselection of resetting times are proposed in this paper thedifference in Sections 33 and 42 implies that more resettingtimes are needed when larger initial misalignment anglesappear

5 Conclusions

The problem of large misalignment angles in SINS initialalignment is investigated in this paper and simulation resultsverified that nonlinear problem brought by large initialmisalignment angles will cause alignment failure when theclassical linear error model and standard Kalman filter areusedThe reasons for alignment failure are discussed in detailand the analysis indicates that the state covariance of Kalmanfilter is smaller than the true one and the measurementvector cannot be effectively utilized during the steady filteringprocess At the same time the simulation also shows thatthere is a convergence trend for misalignment angles at thebeginning of alignment process when the state covariance ismuch larger than the true one


100 200 300 400 500 600 700 800 900

500 520 540 560 580 PHINS

Scheme 2 Scheme 1

Scheme 3

t (s)

minus09

minus08

minus082minus08

minus07

Ang

le (∘)

(a) Pitch

100 200 300 400 500 600 700 800 900

500 520 540 560 580PHINS

Scheme 2 Scheme 3

Scheme 1

minus015

minus02

minus025

minus026minus025minus024

Ang

le (∘)

t (s)

(b) Roll

100 200 300 400 500 600 700 800 90060

70

80

90

520 530 540 550 560

85

855PHINS

Scheme 2 Scheme 1 Scheme 3

t (s)

Ang

le (∘)

(c) Yaw

Figure 8 Alignment results with resetting in car test

Based on the analysis of state covariance an initialalignmentmethod for SINS is designed in which the classicallinear error propagation model and standard Kalman filterwith multiresetting are used The state covariance of Kalmanfilter is reset several times before the steady filtering processand ensures the state covariance larger than the true onewhich improves the utilization degree ofmeasurement vectorThis proposed method does not change the classical errorpropagation model and filter structure and owns the meritsof simple model and less calculation

Simulation and car test results both indicate that theproposed method can fulfill initial alignment with largeinitial misalignment angles effectively and the alignmentaccuracy can be as precise as that with small ones

Conflict of Interests

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

Acknowledgments

This work was supported in part by the National NaturalScience Foundation (61273056) and the Chinese universityresearch and operation expenses (10420525)

References

[1] H T David and L W John Strapdown Inertial NavigationTechnology Institute of Electrical Engineers Hampshire UK2nd edition 2004

[2] J Ali and J C Fang ldquoAlignment of strapdown inertialnavigation system a literature survey spanned over the last14 yearsrdquo ftplabattmoteleitabrelealessandroLeitura0420-20PAPERSARTIGOSKALMANkalmanlibStrapdown-Alignmentpdf

[3] Groves and PD Principles of GNSS Inertial and Multi-SensorIntegrated Navigation Systems Artech House London UK2008

[4] J A Frarell and A Navigation With High Rates SenorsMcGraw-Hill New York NY USA 2008

[5] K J Shortelle W R Graham and C Rabourn ldquoF-16 flight testsof a rapid transfer alignment procedurerdquo in Proceedings of theIEEEPosition Location andNavigation Symposium pp 379ndash386Palm Springs Calif USA April 1996

[6] A Ramanandan A Chen and J A Farrell ldquoInertial navigationaiding by stationary updatesrdquo IEEE Transactions on IntelligentTransportation Systems vol 13 no 1 pp 235ndash248 2012

[7] P M G Silson ldquoCoarse alignment of a shiprsquos strapdown inertialattitude reference system using velocity locirdquo IEEE Transactionson Instrumentation and Measurement vol 60 no 6 pp 1930ndash1941 2011

[8] J L Li J C Fang and M Du ldquoError analysis and gyro-biascalibration of analytic coarse alignment for airborne POSrdquo IEEETransactions on Instrumentation and Measurement vol 61 no11 pp 3058ndash3064 2012


[9] G M Yan W S Yan and D M Xu ldquoApplication of simplifiedUKF in SINS initial alignment for large misalignment anglesrdquoJournal of Chinese Inertial Technology vol 16 no 3 pp 253ndash2642008 (Chinese)

[10] M D Shuster ldquoSurvey of attitude representationsrdquo Journal ofthe Astronautical Sciences vol 41 no 4 pp 439ndash517 1993

[11] H S Hong J G Lee and C G Park ldquoIn-flight alignment ofSDINS under large initial heading errorrdquo in AIAA GuidanceNavigation and Control Conference and Exhibit pp 1ndash6 2001

[12] M J Yu HW Park and C B Jeon ldquoEquivalent nonlinear errormodels of strapdown inertial navigation systemrdquo in GuidanceNavigation and Control Conference pp 581ndash587 1997

[13] S E Dmitriyev O A Stepanov and S V Shepel ldquoNonlinearfiltering methods application in INS alignmentrdquo IEEE Trans-actions on Aerospace and Electronic Systems vol 33 no 1 pp260ndash272 1997

[14] Z Zhanxin G Yanan and C Jiabin ldquoUnscented Kalmanfilter for SINS alignmentrdquo Journal of Systems Engineering andElectronics vol 18 no 2 pp 327ndash333 2007

[15] J Xiong J Liu J Lai and B Jiang ldquoAmarginalized particle filterin initial alignment for sinsrdquo International Journal of InnovativeComputing Information and Control vol 7 no 7 pp 3771ndash37782011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


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Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


and the standard Kalman filter can fulfill initial alignmentonly with small misalignment angles when linear model isused [1ndash9] However because of environmental disturbanceand sensor errors large errors between initial attitude andthe true one that is large misalignment problems or non-linear problems are often generated in SINS when analyticalmethod is used for coarse alignment [11ndash15] The problemsaddressed in this paper are concentrated on the investigationof initial alignment for SINS with large initial misalignmentangles caused by low accuracy of coarse alignment

Recently SINS error propagation models and nonlinearestimation methods for large misalignment angles are widelyreported in the literatures Error propagation models canbe divided into two types one is nonlinear model forlarge misalignment angles among which the large azimuthmisalignment error model is the most representative [11]the other is linearized method for nonlinear problem whichgains linear model for large misalignment angles by approx-imation or state transformation [12 13] Among nonlinearfilters extended Kalman filter (EKF) is widely used andestimation methods based on approximate probability den-sity distribution such as particle filter (PF) and unscentedKalman filter (UKF) are also popular currently [14 15]Thesefilters can solve nonlinear function directly by using theevolution of probability density distributionwithout the needof detailed expansion equations of nonlinear function Largemisalignment problems can be solved with the above modelsand methods to some extent However there are still manyshortages needed to be concerned such as the complexityproblem in nonlinear error model for large misalignmentangles state dimension extending problem in the lineariza-tion process and heavy computational problem in PF andUKF In addition investigation on stability of the methods isstill blank field when PF and UKF are used in engineering

For the purpose of simplifying error model withoutadding computational load an initial alignment method forlarge initial misalignment angles is proposed based on clas-sical linear error propagation model and standard Kalmanfilter in which the state covariance matrix of Kalman filteris reset several times before the steady filtering process is

achieved With the parameters resetting the state covariancefrom Kalman filter can truly represent the true one andthe high utilization of measurement vector can be ensuredCoupled with the help of closed loop correction mode largemisalignment angles can be converted into small ones so thatalignment can be finished effectively

The rest of this paper is organized as followsThe classicallinear error propagation model and standard Kalman filterused in zero-velocity alignment are described in Section 2Then the reason for alignment failure with large misalign-ment angles is analyzed in Section 3 and a potential solutionmethod is proposed which is verified by simulation InSection 4 the validity of the proposed method is furtherverified by car test Finally some conclusions are given inSection 5

2 Error Propagation Model and Kalman Filterfor Zero-Velocity Alignment

21 System Equation Choose velocity errors and misalign-ment angles of SINS as the state variables and then the statevector of the system can be constructed as

X = [120575119881119864120575119881119873

120601119864

120601119873

120601119880]

T (1)

where 120575119881119864

and 120575119881119873

are east and north velocity errorsrespectively and 120601

119864 120601119873 and 120601

119880are misalignment angles

of pitch roll and yaw respectively To vehicles running onthe earth surface the height and upward velocity can be setas zero In this paper east-north-up (ENU) is defined asnavigation frame 119899

The system state equation can be constructed as [1]

X (119905) = A (119905)X (119905) +W (119905) (2)

where A(119905) is the state matrix and W(119905) is the white systemprocess noise with the power spectral density Q Based onthe classical linear error propagation model the state matrixA(119905) can be expressed as [1]

A (119905)

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119881119873

119877

tan 119871 2120596119894119890sin 119871 +

119881119864

119877

tan119871 0 minus119891119880

119891119873

minus(2120596119894119890sin 119871 +

119881119864

119877

tan119871) 0 119891119880

0 minus119891119864

0 minus

1

119877

0 120596119894119890sin 119871 +

119881119864

119877

tan119871 minus(120596119894119890cos 119871 +

119881119864

119877

)

1

119877

0 minus(120596119894119890sin 119871 +

119881119864

119877

tan 119871) 0 minus

119881119873

119877

1

119877

tan 119871 0 120596119894119890cos 119871 +

119881119864

119877

119881119873

119877

0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(3)


Kalman filter

SINS



(feedback)



where 119881119864and 119881




119864 119891119873 and 119891




Z = [119881119864minus 0 119881

119873minus 0]

T= [120575119881119864

120575119881119873]

T (4)

where Δ119881119864and Δ119881


equal 119881119864and 119881



Z (119905) = H (119905)X (119905) + V (119905) (5)


H = [

1 0 0 0 0

0 1 0 0 0

] (6)



24 Simulation




X0= [0 0 0 0 0]

T

P0= diag [(01ms)2 (01ms)2 (15

∘)

2

(15

∘)

2

(80

∘)

2]

Q = diag [(500 ug)2 (500 ug)2 (01

∘)

2

(01

∘)

2

(01

∘)

2]

R = diag [(01ms)2 (01ms)2] (7)


0=

[0

∘0

∘0

∘]


∘5

∘5

∘]


[5

∘5

∘10

∘]


∘5

∘20

∘]


∘5

∘30

∘]




0 100 200 300 400 500 600 700 800 900


Condition 4


Condition 2

Condition 3

Condition 1

t (s)

0

minus2

minus16

minus17

minus18

minus19

minus4

Erro

r(998400)

(a) Pitch

0 100 200 300 400 500 600 700 800 900

0

2

4

4068 407 4072

17217

174

Condition 1

Condition 5



Erro

r(998400)

t (s)

(b) Roll


0 100 200 300 400 500 600 700 800 900

437 438 439 440 441Condition 5



0

200

400

minus200

minus400

minus40

minus30

minus20

Erro

r(998400)

t (s)

(c) Yaw









X119896119896minus1

= Φ119896119896minus1

X119896minus1

(8)


0 100 200 300 400 500 600 700 800 9000

20

40

True P

t (s)


Elem

ent o

fPm

atrix

(∘2)


00 100 200 300 400 500 600 700 800 900

200

400

200 400 600 800


True P

t (s)

Elem

ent o

fPm

atrix

(∘2)

minus10




X119896=

X119896119896minus1

+ K119896(Z119896minusH119896X119896119896minus1

) (9)


K119896= P119896119896minus1

HT119896(H119896P119896119896minus1

HT119896+ R119896)

minus1

(10)


P119896119896minus1

= Φ119896119896minus1

P119896minus1Φ

T119896119896minus1

+Q119896minus1

(11)


P119896= (I minus K

119896H119896)P119896119896minus1

(12)


119896119896minus1is








119896119896minus1in (11)









0is





















Kalman filter

nΔT

ΔT ΔT

Δt Δt Δt

t

t

1

with X0 P0 Q and R

Reset P rArr P0 for

Reset P rArr P0 for

t0











33 Simulation





Condition 1

Condition 5 0 100 200 300 400 500 600 700 800 900

t (s)

0

minus2

minus4

Erro

r(998400)

(a) Pitch

Condition 5

Condition 1

0 100 200 300 400 500 600 700 800 900t (s)

0

2

4

Erro

r(998400)

(b) Roll

Condition 5

Condition 1

0 100 200 300 400 500 600 700 800 900500 600 700 800 900


t (s)

Erro

r(998400)

500

0

minus500

minus1000

minus1500


(c) Yaw





4 Car Test











0 100 200 300 400 500 600 700 800 9000

50

100


400 500 600 700 800 9004648



Elem

ent o

fPm

atrix

(∘2)

t (s)


FOG-SINS PHINS






∘5

∘30

∘]








5 Conclusions



100 200 300 400 500 600 700 800 900

500 520 540 560 580 PHINS

Scheme 2 Scheme 1

Scheme 3

t (s)

minus09

minus08

minus082minus08

minus07

Ang

le (∘)

(a) Pitch

100 200 300 400 500 600 700 800 900

500 520 540 560 580PHINS

Scheme 2 Scheme 3

Scheme 1

minus015

minus02

minus025


Ang

le (∘)

t (s)

(b) Roll

100 200 300 400 500 600 700 800 90060

70

80

90

520 530 540 550 560

85

855PHINS


t (s)

Ang

le (∘)

(c) Yaw






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Kalman filter

SINS



(feedback)



where 119881119864and 119881




119864 119891119873 and 119891




Z = [119881119864minus 0 119881

119873minus 0]

T= [120575119881119864

120575119881119873]

T (4)

where Δ119881119864and Δ119881


equal 119881119864and 119881



Z (119905) = H (119905)X (119905) + V (119905) (5)


H = [

1 0 0 0 0

0 1 0 0 0

] (6)



24 Simulation




X0= [0 0 0 0 0]

T

P0= diag [(01ms)2 (01ms)2 (15

∘)

2

(15

∘)

2

(80

∘)

2]

Q = diag [(500 ug)2 (500 ug)2 (01

∘)

2

(01

∘)

2

(01

∘)

2]

R = diag [(01ms)2 (01ms)2] (7)


0=

[0

∘0

∘0

∘]


∘5

∘5

∘]


[5

∘5

∘10

∘]


∘5

∘20

∘]


∘5

∘30

∘]




0 100 200 300 400 500 600 700 800 900


Condition 4


Condition 2

Condition 3

Condition 1

t (s)

0

minus2

minus16

minus17

minus18

minus19

minus4

Erro

r(998400)

(a) Pitch

0 100 200 300 400 500 600 700 800 900

0

2

4

4068 407 4072

17217

174

Condition 1

Condition 5



Erro

r(998400)

t (s)

(b) Roll


0 100 200 300 400 500 600 700 800 900

437 438 439 440 441Condition 5



0

200

400

minus200

minus400

minus40

minus30

minus20

Erro

r(998400)

t (s)

(c) Yaw









X119896119896minus1

= Φ119896119896minus1

X119896minus1

(8)


0 100 200 300 400 500 600 700 800 9000

20

40

True P

t (s)


Elem

ent o

fPm

atrix

(∘2)


00 100 200 300 400 500 600 700 800 900

200

400

200 400 600 800


True P

t (s)

Elem

ent o

fPm

atrix

(∘2)

minus10




X119896=

X119896119896minus1

+ K119896(Z119896minusH119896X119896119896minus1

) (9)


K119896= P119896119896minus1

HT119896(H119896P119896119896minus1

HT119896+ R119896)

minus1

(10)


P119896119896minus1

= Φ119896119896minus1

P119896minus1Φ

T119896119896minus1

+Q119896minus1

(11)


P119896= (I minus K

119896H119896)P119896119896minus1

(12)


119896119896minus1is








119896119896minus1in (11)









0is





















Kalman filter

nΔT

ΔT ΔT

Δt Δt Δt

t

t

1

with X0 P0 Q and R

Reset P rArr P0 for

Reset P rArr P0 for

t0











33 Simulation





Condition 1

Condition 5 0 100 200 300 400 500 600 700 800 900

t (s)

0

minus2

minus4

Erro

r(998400)

(a) Pitch

Condition 5

Condition 1

0 100 200 300 400 500 600 700 800 900t (s)

0

2

4

Erro

r(998400)

(b) Roll

Condition 5

Condition 1

0 100 200 300 400 500 600 700 800 900500 600 700 800 900


t (s)

Erro

r(998400)

500

0

minus500

minus1000

minus1500


(c) Yaw





4 Car Test











0 100 200 300 400 500 600 700 800 9000

50

100


400 500 600 700 800 9004648



Elem

ent o

fPm

atrix

(∘2)

t (s)


FOG-SINS PHINS






∘5

∘30

∘]








5 Conclusions



100 200 300 400 500 600 700 800 900

500 520 540 560 580 PHINS

Scheme 2 Scheme 1

Scheme 3

t (s)

minus09

minus08

minus082minus08

minus07

Ang

le (∘)

(a) Pitch

100 200 300 400 500 600 700 800 900

500 520 540 560 580PHINS

Scheme 2 Scheme 3

Scheme 1

minus015

minus02

minus025


Ang

le (∘)

t (s)

(b) Roll

100 200 300 400 500 600 700 800 90060

70

80

90

520 530 540 550 560

85

855PHINS


t (s)

Ang

le (∘)

(c) Yaw






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600 700 800 900


Condition 4


Condition 2

Condition 3

Condition 1

t (s)

0

minus2

minus16

minus17

minus18

minus19

minus4

Erro

r(998400)

(a) Pitch

0 100 200 300 400 500 600 700 800 900

0

2

4

4068 407 4072

17217

174

Condition 1

Condition 5



Erro

r(998400)

t (s)

(b) Roll


0 100 200 300 400 500 600 700 800 900

437 438 439 440 441Condition 5



0

200

400

minus200

minus400

minus40

minus30

minus20

Erro

r(998400)

t (s)

(c) Yaw









X119896119896minus1

= Φ119896119896minus1

X119896minus1

(8)


0 100 200 300 400 500 600 700 800 9000

20

40

True P

t (s)


Elem

ent o

fPm

atrix

(∘2)


00 100 200 300 400 500 600 700 800 900

200

400

200 400 600 800


True P

t (s)

Elem

ent o

fPm

atrix

(∘2)

minus10




X119896=

X119896119896minus1

+ K119896(Z119896minusH119896X119896119896minus1

) (9)


K119896= P119896119896minus1

HT119896(H119896P119896119896minus1

HT119896+ R119896)

minus1

(10)


P119896119896minus1

= Φ119896119896minus1

P119896minus1Φ

T119896119896minus1

+Q119896minus1

(11)


P119896= (I minus K

119896H119896)P119896119896minus1

(12)


119896119896minus1is








119896119896minus1in (11)









0is





















Kalman filter

nΔT

ΔT ΔT

Δt Δt Δt

t

t

1

with X0 P0 Q and R

Reset P rArr P0 for

Reset P rArr P0 for

t0











33 Simulation





Condition 1

Condition 5 0 100 200 300 400 500 600 700 800 900

t (s)

0

minus2

minus4

Erro

r(998400)

(a) Pitch

Condition 5

Condition 1

0 100 200 300 400 500 600 700 800 900t (s)

0

2

4

Erro

r(998400)

(b) Roll

Condition 5

Condition 1

0 100 200 300 400 500 600 700 800 900500 600 700 800 900


t (s)

Erro

r(998400)

500

0

minus500

minus1000

minus1500


(c) Yaw





4 Car Test











0 100 200 300 400 500 600 700 800 9000

50

100


400 500 600 700 800 9004648



Elem

ent o

fPm

atrix

(∘2)

t (s)


FOG-SINS PHINS






∘5

∘30

∘]








5 Conclusions



100 200 300 400 500 600 700 800 900

500 520 540 560 580 PHINS

Scheme 2 Scheme 1

Scheme 3

t (s)

minus09

minus08

minus082minus08

minus07

Ang

le (∘)

(a) Pitch

100 200 300 400 500 600 700 800 900

500 520 540 560 580PHINS

Scheme 2 Scheme 3

Scheme 1

minus015

minus02

minus025


Ang

le (∘)

t (s)

(b) Roll

100 200 300 400 500 600 700 800 90060

70

80

90

520 530 540 550 560

85

855PHINS


t (s)

Ang

le (∘)

(c) Yaw






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600 700 800 9000

20

40

True P

t (s)


Elem

ent o

fPm

atrix

(∘2)


00 100 200 300 400 500 600 700 800 900

200

400

200 400 600 800


True P

t (s)

Elem

ent o

fPm

atrix

(∘2)

minus10




X119896=

X119896119896minus1

+ K119896(Z119896minusH119896X119896119896minus1

) (9)


K119896= P119896119896minus1

HT119896(H119896P119896119896minus1

HT119896+ R119896)

minus1

(10)


P119896119896minus1

= Φ119896119896minus1

P119896minus1Φ

T119896119896minus1

+Q119896minus1

(11)


P119896= (I minus K

119896H119896)P119896119896minus1

(12)


119896119896minus1is








119896119896minus1in (11)









0is





















Kalman filter

nΔT

ΔT ΔT

Δt Δt Δt

t

t

1

with X0 P0 Q and R

Reset P rArr P0 for

Reset P rArr P0 for

t0











33 Simulation





Condition 1

Condition 5 0 100 200 300 400 500 600 700 800 900

t (s)

0

minus2

minus4

Erro

r(998400)

(a) Pitch

Condition 5

Condition 1

0 100 200 300 400 500 600 700 800 900t (s)

0

2

4

Erro

r(998400)

(b) Roll

Condition 5

Condition 1

0 100 200 300 400 500 600 700 800 900500 600 700 800 900


t (s)

Erro

r(998400)

500

0

minus500

minus1000

minus1500


(c) Yaw





4 Car Test











0 100 200 300 400 500 600 700 800 9000

50

100


400 500 600 700 800 9004648



Elem

ent o

fPm

atrix

(∘2)

t (s)


FOG-SINS PHINS






∘5

∘30

∘]








5 Conclusions



100 200 300 400 500 600 700 800 900

500 520 540 560 580 PHINS

Scheme 2 Scheme 1

Scheme 3

t (s)

minus09

minus08

minus082minus08

minus07

Ang

le (∘)

(a) Pitch

100 200 300 400 500 600 700 800 900

500 520 540 560 580PHINS

Scheme 2 Scheme 3

Scheme 1

minus015

minus02

minus025


Ang

le (∘)

t (s)

(b) Roll

100 200 300 400 500 600 700 800 90060

70

80

90

520 530 540 550 560

85

855PHINS


t (s)

Ang

le (∘)

(c) Yaw






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Kalman filter

nΔT

ΔT ΔT

Δt Δt Δt

t

t

1

with X0 P0 Q and R

Reset P rArr P0 for

Reset P rArr P0 for

t0











33 Simulation





Condition 1

Condition 5 0 100 200 300 400 500 600 700 800 900

t (s)

0

minus2

minus4

Erro

r(998400)

(a) Pitch

Condition 5

Condition 1

0 100 200 300 400 500 600 700 800 900t (s)

0

2

4

Erro

r(998400)

(b) Roll

Condition 5

Condition 1

0 100 200 300 400 500 600 700 800 900500 600 700 800 900


t (s)

Erro

r(998400)

500

0

minus500

minus1000

minus1500


(c) Yaw





4 Car Test











0 100 200 300 400 500 600 700 800 9000

50

100


400 500 600 700 800 9004648



Elem

ent o

fPm

atrix

(∘2)

t (s)


FOG-SINS PHINS






∘5

∘30

∘]








5 Conclusions



100 200 300 400 500 600 700 800 900

500 520 540 560 580 PHINS

Scheme 2 Scheme 1

Scheme 3

t (s)

minus09

minus08

minus082minus08

minus07

Ang

le (∘)

(a) Pitch

100 200 300 400 500 600 700 800 900

500 520 540 560 580PHINS

Scheme 2 Scheme 3

Scheme 1

minus015

minus02

minus025


Ang

le (∘)

t (s)

(b) Roll

100 200 300 400 500 600 700 800 90060

70

80

90

520 530 540 550 560

85

855PHINS


t (s)

Ang

le (∘)

(c) Yaw






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Condition 1

Condition 5 0 100 200 300 400 500 600 700 800 900

t (s)

0

minus2

minus4

Erro

r(998400)

(a) Pitch

Condition 5

Condition 1

0 100 200 300 400 500 600 700 800 900t (s)

0

2

4

Erro

r(998400)

(b) Roll

Condition 5

Condition 1

0 100 200 300 400 500 600 700 800 900500 600 700 800 900


t (s)

Erro

r(998400)

500

0

minus500

minus1000

minus1500


(c) Yaw





4 Car Test











0 100 200 300 400 500 600 700 800 9000

50

100


400 500 600 700 800 9004648



Elem

ent o

fPm

atrix

(∘2)

t (s)


FOG-SINS PHINS






∘5

∘30

∘]








5 Conclusions



100 200 300 400 500 600 700 800 900

500 520 540 560 580 PHINS

Scheme 2 Scheme 1

Scheme 3

t (s)

minus09

minus08

minus082minus08

minus07

Ang

le (∘)

(a) Pitch

100 200 300 400 500 600 700 800 900

500 520 540 560 580PHINS

Scheme 2 Scheme 3

Scheme 1

minus015

minus02

minus025


Ang

le (∘)

t (s)

(b) Roll

100 200 300 400 500 600 700 800 90060

70

80

90

520 530 540 550 560

85

855PHINS


t (s)

Ang

le (∘)

(c) Yaw






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 100 200 300 400 500 600 700 800 9000

50

100


400 500 600 700 800 9004648



Elem

ent o

fPm

atrix

(∘2)

t (s)


FOG-SINS PHINS






∘5

∘30

∘]








5 Conclusions



100 200 300 400 500 600 700 800 900

500 520 540 560 580 PHINS

Scheme 2 Scheme 1

Scheme 3

t (s)

minus09

minus08

minus082minus08

minus07

Ang

le (∘)

(a) Pitch

100 200 300 400 500 600 700 800 900

500 520 540 560 580PHINS

Scheme 2 Scheme 3

Scheme 1

minus015

minus02

minus025


Ang

le (∘)

t (s)

(b) Roll

100 200 300 400 500 600 700 800 90060

70

80

90

520 530 540 550 560

85

855PHINS


t (s)

Ang

le (∘)

(c) Yaw






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










100 200 300 400 500 600 700 800 900

500 520 540 560 580 PHINS

Scheme 2 Scheme 1

Scheme 3

t (s)

minus09

minus08

minus082minus08

minus07

Ang

le (∘)

(a) Pitch

100 200 300 400 500 600 700 800 900

500 520 540 560 580PHINS

Scheme 2 Scheme 3

Scheme 1

minus015

minus02

minus025


Ang

le (∘)

t (s)

(b) Roll

100 200 300 400 500 600 700 800 90060

70

80

90

520 530 540 550 560

85

855PHINS


t (s)

Ang

le (∘)

(c) Yaw






Acknowledgments


References
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A Method for SINS Alignment with Large