Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 401794 10 pageshttpdxdoiorg1011552013401794

Research ArticleA Rapid Transfer Alignment Method for SINS Based on theAdded Backward-Forward SINS Resolution and Data Fusion

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 Ministry of Education Nanjing 210096 China

Correspondence should be addressed to Xixiang Liu scliuseu163com

Received 28 January 2013 Accepted 25 June 2013

Academic Editor Joao B R Do Val

Copyright copy 2013 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

Two viewpoints are given (1) initial alignment of strapdown inertial navigation system (SINS) can be fulfilled with a set of inertialsensor data (2) estimation time for sensor errors can be shortened by repeated data fusion on the added backward-forward SINSresolution results and the external reference data Based on the above viewpoints aiming to estimate gyro bias in a shortened timea rapid transfer alignment method without any changes for Kalman filter is introduced In this method inertial sensor data andreference data in one reference data update cycle are stored and one backward and one forward SINS resolutions are executedMeanwhile data fusion is executed when the corresponding resolution ends With the added backward-forward SINS resolutionin the above mentioned update cycle the estimating operations for gyro bias are added twice and the estimation time for it isshortened In the ship swinging condition with the ldquovelocity plus yawrdquo matching the effectiveness of this method is proved by thesimulation

1 Introduction

Transfer alignment is a rapid and effective initial alignmentmethod which is widely used for inertial navigation systems(INSs) in ships and planes For the quick response require-ment of weapon systems rapidity is always the main aimof the initial alignment In 1989 a fast transfer alignmentmethod was presented by Kain and Cloutier in whichalignment can be fulfilled in 10 s and 1 mrad accuracy canbe got with swinging movement and ldquovelocity plus attituderdquomatching [1] Based on reference 1 a prefilter was addedbefore the Kalman filter by Spalding [2] Together with theformer the calculation amount was reduced and alignmenttime was shortened And in this method 1 mrad accuracywas reached within 6 s when the update frequency of refer-ence data was 1Hz Aviation transfer alignment experimentson Apache helicopter and F-16 fighter were conducted byShortelle and Graham respectively [3ndash5] and the test resultsindicated that when the update frequency was 125Hz thealignment time could be reduced to 5 s and the accuracycould reach 1mradwith ldquovelocity plus attituderdquomatching andWing-Rock tactical action

In the above references the rapidity of initial alignmentwas studied while the estimation for inertial sensor errorswas not taken into account or not given sufficient attentionTo those strapdown INSs which are composed of low ormedium accuracy initial sensors sensor errors will decreasenavigation accuracy greatly due to the great drift of gyrosafter the long time storage [6] So it is certain that navigationaccuracywill be improved if the sensor errors are estimated inthe transfer alignment process But the observability degreeof each state variable in a Kalman filter for transfer alignmentrelies on the information matching method and the motionof carrier [7 8] With swinging movement and ldquovelocity plusattituderdquo matching an effective estimation for sensor errorscannot be completed within 10 s for it usually lasts for severalminutes [9]

In this paper after transfer alignment process in shipbetween the master INS (MINS) and slave INS has beenstudied a faster transfer alignment method is introducedin which the estimation for gyro bias can be fulfilled in ashorter time In ship the accuracy of platform INS (PINS)is usually several magnitudes higher than that of strapdownINS The PINS is used as MINS while the strapdown INS

2 Mathematical Problems in Engineering

is used as slave INS and the navigation parameters fromMINS are used as reference data In the paper the slave INSand strapdown INS are both abbreviated as SINS In theintegration of MS INS flexure deformation is a key factorwhichwill determine transfer alignment accuracy [10] In thispaper ldquovelocity plus yawrdquo matching method is chosen due tothe fact that the ldquoattituderdquo matching method is easily sufferedby flexure deformation [11 12] while ldquovelocityrdquo matchingmethod can ensure the rapidity and accuracy of estimationfor horizontal misalignment angles [8] According to theanalysis based on piece-wise constant systems (PWCS) lineartheory [7] with the excitation of the wave swinging thevelocity errors misalignment angles and gyro bias are theobservable parameters [13] but unfortunately the estimationfor gyro bias is a time-consuming process which usually lastsfor several minutes

To deal with the contradiction between the rapidity ofalignment and time consumption of gyro bias estimationtwo viewpoints are put forward by analyzing the alignmentprocess of SINS Firstly different from that in PINS the directcorresponding relationship between inertial senor data andmisalignment angles in SINS is severed by the mathematicalplatform Thus we can adjust the calculated mathematicalplatform 119899

1015840 constantly to speed up the initial alignment bythe repeated SINS resolution with the same set of sensordata Secondly analysis indicates that with a Kalman filter anew estimation for state vector will be provided With close-loop correction new initial navigation parameters and newcompensation parameters for gyro bias will be produced forthe next navigation period andwith the above new ones newnavigation results will be generated even built on the same setof sensor data So with the repeatedly data fusion of the addedbackward-forward SINS resolution results and the externalreference data the estimation time for sensor error will beshortened With the above two viewpoints a rapid transferalignment method based on the added backward-forwardSINS resolution without any changes to the Kalman filter isdesigned in detail In one reference data update cycle with

a normal resolution and data fusion an added backward andan added forward resolutions and their data fusions the timeconsumed for gyro bias estimation is effectively shortenedThe effectiveness of this method is proved by simulationresults

This thesis is constructed as follows In Section 2 thetransfer alignment model based on ldquovelocity plus yawrdquomatching is built and the reasons why the observabilitydegree of gyro bias is low are analyzed In Section 3 afterthe alignment process of INS is studied and compared apossible way to increase SINS alignment speed is introducedand the backward-forward SINS resolution is designed InSection 4 the transfer alignment model based on the addedbackward-forward resolution is also designed in detail Andin Section 5 the effectiveness of this method is proved bysimulation Finally some conclusions are given

2 Transfer Alignment Model Based onlsquolsquoVelocity Plus Yawrsquorsquo Matching Method

21 System Equation Choose velocity errors misalignmentangles and gyro bias as the state vector of the system

X = [120575119881119864120575119881119873

120601119864

120601119873

120601119880

120576119909

120576119910

120576119911]

119879

(1)

where 120575119881119864and 120575119881

119873are east and north velocity errors

respectively 120601119864 120601119873 and 120601

119880are misalignment angles of

pitch roll and yaw respectively 120576119909 120576119910 and 120576

119911are gyro bias

along 119909119910 and 119911 axes To ships which sail on the sea theheight and upward velocity can be set as zero

The system state equation can be constructed as

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

where A(119905) is the state matrix and W(119905) is the systemnoise matrix With the system vector velocity error andmisalignment angle equations the state matrix A(119905) can beexpressed as

A (119905) =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119881119873

119877

tan119871 2120596119894119890sin119871 + 119881119864

119877

tan119871 0 minus119891119880

119891119873

0 0 0

minus(2120596119894119890sin119871 + 119881119864

119877

tan119871) 0 119891119880

0 minus119891119864

0 0 0

0 minus1

119877

0 120596119894119890sin119871 + 119881119864

119877

tan119871 minus(120596119894119890cos119871 + 119881119864

119877

) minus11987911minus11987912minus11987913

1

119877

0 minus(120596119894119890sin119871 + 119881119864

119877

tan119871) 0 minus119881119873

119877

minus11987921minus11987922minus11987923

1

119877

tan119871 0 120596119894119890cos119871 + 119881119864

119877

119881119873

119877

0 minus11987931minus11987932minus11987933

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(3)

Mathematical Problems in Engineering 3

Inertial sensorAlignment process

Ideal navigation frame n

Physical platform frame n998400

(a) PINS

Body frame

Inertial sensor

Alignment process

Ideal navigation frame n

Mathematical platform frame n998400

(b) SINS

Figure 1 Alignment process for INS

where 119881119864and 119881

119873are the east and north velocity respectively

120596119894119890and 119877 are the rotational angular rate and radius of the

Earth 119871 is the latitude of ship position 119891119864 119891119873 and 119891

119880

denote the projection of the accelerometer measured data f119887in navigation frame 119899 and 119879

119894119895(119894 119895 = 1 2 3) are the elements

of the direct cosine matrix (DCM) of MINS

22 Measurement Equation The differences of velocity andyaw between MINS and SINS are used as measurement datafor data fusion Then the measurement vector is as

Z = [119881119864minus 119881119872119864

119881119873

minus 119881119872119873

119884119864minus 119884119872119864

]

119879

(4)

where 119881119864 119881119873and 119884

119864are the east north velocity and yaw

from SINS respectively 119881119872119864

119881119872119873

and 119884119872119864

are fromMINS respectively

The measurement equation can be constructed as

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

whereH(119905) is measurement matrix and V(119905) is measurementnoise matrix According to the relationship betweenmeasurement and state vectors the measurement matrixH(119905) can be expressed as

H =

[

[

[

[

[

[

[

[

1 0 0 0 0

0 1 0 0 0 03times3

0 0 minus

1198791211987932

1198792

12+ 1198792

22

minus

1198792211987932

1198792

12+ 1198792

22

minus1

]

]

]

]

]

]

]

]

(6)

23 Observability Degree of Each State Variable After theanalysis of the model combined with (2) and (5) by PWCStheory we can conclude that all of the eight selected variablesare observable but the observability degrees are differentfrom each other [13] The simulation results in Section 52indicate that the horizontal velocity errors and yaw mis-alignment converge immediately that the estimation forhorizontal misalignment angles lasts for 10sim20 s and thatthe estimation for gyro bias lasts for 3sim5min which meansthat the degrees of velocity errors and yaw misalignment

are the highest those of horizontal misalignment angles aremoderate and those of gyro bias are the lowest

The reasons which cause the different degrees of errorsare as follows The gyro bias needs integration to be reflectedon misalignment angles and the horizontal misalignmentangles need projection reflected on acceleration Furtherintegration of acceleration gyro bias can be reflected onvelocity errors In this process from gyro bias to velocityerrors integrating operations are needed twice and fromhorizontal misalignment angles to velocity errors only oneis needed When velocity errors and yaw misalignment areselected as components of a measurement vector there isno doubt that observability degrees of gyro bias are lowestwhile those of velocity errors and yaw misalignment arehighest which are determined by the mechanism of transferalignment To shorten the estimation time for gyro bias someother novel ways should be sought

3 A New Way to Speed UpSINS Alignment

31 Analysis of Alignment Process in INS An acceptedviewpoint is that in navigation sensor data is of real-timesignificance Based on initial navigation parameters INSobtains the attitude velocity and position by the integrationof sensor data After integration the sensor data can bediscarded

Take the alignment process of PINS as an example Asshown in Figure 1(a) 1198991015840 frame denotes the physical platformand 119899 frame denotes the ideal navigation frame In PINSinertial sensors are installed on the physical platform Inertialsensors measure the ship motion in 119899

1015840 frame and navigationresolution is also executed in this frame however referencedata are from 119899 frame Before aligning misalignment anglesbetween 119899

1015840 and 119899 frames will cause the differences betweennavigation resolution values in 119899

1015840 frame and reference val-ues in 119899 frame Because these differences can reflect themisalignments transfer alignment can be finished with themodel in Section 2 In the alignment process with theabove differences caused by misalignment angles physicalplatform can be adjusted to coincide with the ideal frameIn other words in PINS sensor data reflect the magnitudeof misalignment angle and other errors and there is direct

4 Mathematical Problems in Engineering

relationship between sensor data and misalignment anglesso the sensor data are of real-time significance And theadjustment for physical platform to get new sensor data is atime-consuming process

At the same time in SINS the alignment method isderived from that of PINS and so the alignment must bea time-consuming process for the need of real-time sensordata and the adjustment of platform But in INS as far as theprocess of alignment is concerned maybe sensor data is ofreal-time significance for PINS but not for SINS

In SINS as shown in Figure 1(b) amathematical platformreplaces the physical platform in PINS and inertial sensorsare directly installed on the ship Here the ship body isnamed 119887 frame and the mathematical platform is named 119899

1015840

frame In alignment the adjusting process of the 1198991015840 frame

is the same with that in PINS while the difference is thatthe calculated sensor data in 119899

1015840 frame is the projection ofmeasured sensor data from 119887 frame Only the projectiondata in 119899

1015840 frame reflect the misalignment angles between 1198991015840

and 119899 frames but sensor data cannot The calculated andmeasured sensor data are connected by calculated DCM C119899

1015840

119887

In comparison with that in PINS the direct relationshipbetween sensor data and misalignment angles is severedby the mathematical platform Then the 119899

1015840 frame can beadjusted constantly by the repeated calculation for DCM C119899

1015840

119887

with a same set of sensor data which means that alignmenttime can be shortened without too much time to be spent insampling real-time sensor data

The above set of data can be seen as that when the shipkeeps an ideal static state and there are no sensor errors allmeasured sensor data are equal and can be dealt with as aseries of single data In engineering because of the errorsfrom sensor and reference data the update frequency ofreference data and so forth a set of sensor data and referencedata should be used for repeated SINS resolution (addedbackward-forward resolution) and repeated data fusion toimprove the alignment accuracy

Also different from PINS SINS is a digital system inwhich there is only numerical data no mass spring andresistance So the adjustment for mathematical platform canbe as fast as lightning

32 Backward-Forward SINS Resolution The above analysisindicates that the alignment of SINS can be fulfilled with aset of same data which brings a new problemmdashthe way touse these data

In SINS with initial attitude velocity and positionthe navigation resolution is the real-time updating processfor navigation parameters with sensor data by integratedcalculation In this process ship moves from the origin tothe end In backward-forward SINS resolution as shown inFigure 2 backward SINS resolution is the process in whichship moves from the end to the originmdasha reverse process ofnormal navigation and forward resolution is that from theorigin to the endmdasha repeated process of normal navigationFrom the basic SINS resolution algorithm the deduction ofbackward-forward SINS algorithm is shown as follows

Backward

Forward

Normal

t0 tn tm

t01

t02

t0i

t0n

tn1

tn2

tni

tnn

middot middot middot

Figure 2 The process of backward-forward SINS resolution

321 Normal Resolution Algorithm in SINS The navigationresolution algorithm is [14]

C119899119887= C119899119887(120596119887

119899119887times)

V119899 = C119899119887f119887 minus (2120596

119899

119894119890+ 120596119899

119890119899) times V119899 + g119899

=

119881119899

119873

119877

120582 =

119881119899

119864sec 119871119877

(7)

where 120596119887119899119887

= 120596119887

119894119887minus (C119899

119887)119879

(120596119899

119894119890+ 120596119899

119890119899) 120596119899119890119899

=

[minus119881119899

119873119877 119881

119899

119864119877 (119881

119899

119864tan 119871)119877]

119879 A119863119861119862

(such as 120596119899119894119890) denotes

the projection of a motion vector A which means the relativemotion from 119862 frame to 119861 frame in119863 frame C119899

119887is the DCM

V119899 = [119881119899

119864119881119899

119873119881119899

119880] is the velocity vector 119871 and 120582 are the

latitude and longitude respectively and (sdottimes) denotes theantisymmetric matrix of the vector ldquosdotrdquo

For the recursive update with computer formulas (7)must be translated into discrete form in a certain samplingcycle and navigation resolution update cycle For a shipwhich always undergoes a low dynamical movement a singlesampling resolution algorithm can be adopted which meansonly one sample of sensor data per navigation cycle [15ndash17]Supposing the above cycle times are both equal to 119879

119904 then

the discrete form can be expressed as follows [14]

C119899119887119896

= C119899119887119896minus1

(I + 119879119904120596119887

119899119887119896times)

V119899119896= V119899119896minus1

+119879119904[C119899119887119896minus1

f119887119896minus1

minus (2120596119899

119894119890119896minus1+ 120596119899

119890119899119896minus1) times V119899119896minus1

+ g119899]

119871119896= 119871119896minus1

+

119879119904119881119873119896minus1

119877119872

+ ℎ119896minus1

120582119896= 120582119896minus1

+

119879119904119881119864119896minus1

sec119871119896minus1

119877119873

+ ℎ119896minus1

(8)

where 119896 denotes recursive number Without considerationof calculation error formulas (8) are composed of the basicupdate equations for navigation resolution

322 Backward Resolution Algorithm in SINS As shown inFigure 2 in a backward resolution process a ship needs toreturn from the end to the origin So in the normal process

Mathematical Problems in Engineering 5

all sensor data must be stored According to the formulas (8)the resolution process can be expressed as follows

C119899119887119896minus1

= C119899119887119896(I + 119879

119904120596119887

119899119887119896times)

minus1

asymp C119899119887119896

(I minus 119879119904120596119887

119899119887119896times) asymp C119899

119887119896(I minus 119879

119904120596119887

119899119887119896minus1times)

V119899119896minus1

= V119899119896minus 119879119904[C119899119887119896minus1

f119887119896minus1

minus (2120596119899

119894119890119896minus1+ 120596119899

119890119899119896minus1) times V119899119896minus1

+ g119899]

asymp V119899119896minus 119879119904[C119899119887119896f119887119896minus (2120596

119899

119894119890119896+ 120596119899

119890119899119896) times V119899119896+ g119899]

119871119896minus1

= 119871119896minus

119879119904119881119873119896minus1

119877

asymp 119871119896minus

119879119904119881119873119896

119877

120582119896minus1

= 120582119896minus

119879119904119881119864119896minus1

sec119871119896minus1

119877

asymp 120582119896minus

119879119904119881119864119896sec 119871119896

119877

(9)

where 119896 is reduced from 119899 to 0 As shown in Figure 2 1199051198991

isboth the starting point in the period 119905

1198991sim 11990501and the ending

point in the period 1199050

sim 119905119899 Take the attitude velocity and

position at 119905119899as the initial attitude velocity and position at

1199051198991 and the backward resolution can be fully realized In the

period of 1199050

sim 119905119899and 1199051198991

sim 11990501 with the same recursive

number 119896 the ship maintains the same attitude velocity andposition and maintains the same acceleration but oppositein direction Some errors are induced by the approximationin formulas (9) and all these errors can be ignored whenthe resolution cycle is short enough The simulation inSection 323 proves that the above approximation is effective

After the backward resolution a forward resolutionshould bemade in order for the ship to return from the originto the end In the forward resolution formulas (8) can beused

323 Simulation on Backward-Forward SINS Resolution Thesensor errors are listed inTable 1 Andwe assume that the shipis in a bad-moderate sea condition [18] and the ship swingingparameters are listed in Table 2 Ship initial attitudes are set as0 and the ship is assumed without linear motion and locatednorth latitude 32∘ and east longitude 118∘ Sensor samplingand navigation resolution cycle 119879

119904is set as 10ms

Simulation results are shown in Figure 3 in which (a)(b) and (c) show the resolution results of pitch east velocityand latitude The dot and solid lines denote the simulationwith no sensor error and with sensor error respectivelyThe simulation is divided into three stages firstly in the1199050sim 119905119899period sensor data are measured and stored normal

navigation is resolved and this period lasts for 1 s secondly inthe 1199051198991

sim 11990501period backward resolution is run and finally in

the 11990502

sim 1199051198992period forward resolution is run In the last two

stages the time consumed is determined by the performanceof computer

In Figure 3without consideration of calculating the errorthe ship can move from the end to the origin and move fromorigin to end with the backward and forward resolutionsrespectively

But the above processes also indicate that with onlybackward or forward or backward-forward SINS resolutionno new information will be generated

Table 1 Sensor errors

Gyro bias Acce biasConstant Random Constant Random

119909 05∘h 05∘h 500 120583g 500 120583g119910 05∘h 05∘h 500 120583g 500 120583g119911 05∘h 05∘h 500 120583g 500 120583g

Table 2 Swinging parameters

Pitch Roll YawAmplitude (∘) 9 12 14Cycle (s) 8 10 6

4 Rapid Transfer Alignment Based on theAdded Resolution and Data Fusion

The analysis in Section 31 indicates that alignment for SINScan be fulfilled with a set of data and in this section wecombine a Kalman filter with the added backward-forwardSINS resolution aiming to shorten the alignment time

As shown in Figure 4 transfer alignment model basedon Kalman filter introduced in Section 2 and a close-loopcorrection method are used in this rapid transfer alignmentalgorithm Close-loop correction means that after datafusion the new estimation for velocity errors and misalign-ment angles will be fed to SINS to revise those correspondingparameters and new estimation for gyro bias will be setas new compensation value to participate in the followingnavigation resolutions In other words after data fusion themathematical platform C119899

1015840

119887will be adjusted For a ship the

update frequency of the reference data from MINS is lowerthan that of SINS navigation resolution 119879

119904and 119879

119899are set

as update cycle of navigation resolution and reference datarespectively Also 119879

119899is set as the added backward-forward

SINS resolution cycleTwo transfer alignment methods are compared and the

estimation processes of gyro bias are schematically demon-strated in Figure 5 in which (a) and (b) indicate these twomethods respectively which are all based on the principleas shown in Figure 4 In the first one the added backward-forward SINS resolution and data fusion are not used whilethey are used in the second In Figure 5 the dot-dashed linesdenote the real gyro bias while the solid line denotes theestimation of gyro bias With the Kalman filter and matchingmethod introduced in Section 2 the estimation of gyro biasis a slow process which will be convergent towards real gyrobias after a long time

As shown in Figure 5(a) when the reference data isavailable such as at the point 119905

119899 data fusion is executed and

a new estimate for the state vector will be produced With thenew estimation for misalignment angles and velocity errorsinitial navigation parameters will be reset for the next periodsuch as 119905

119899sim 1199052119899 whichmeans that themathematical platform

1198991015840 will be adjusted and the new compensation value for gyro

bias will also be reset In this method along with time datafusion is run only once at every reference data update cycle

6 Mathematical Problems in Engineering

0

5

10

minus10

minus5

tn(tn1)t0 tn2t01(t02)

Ang

le (∘

)

(a) Pitch

Velo

city

(ms

)

0

005

minus005tn(tn1)t0 tn2t01(t02)

(b) East velocity

32

32

32

tn(tn1)t0 tn2t01(t02)

Ang

le (∘

)

(c) Latitude

Figure 3 Simulation of normal and backward-forward SINS resolution

Kalman filter

SINS

MINS Velocity + yaw

Velocity + yaw

State variables

Navigation parameters

Figure 4 Kalman filter

Estim

atio

n fo

rgy

ro b

ias

Data fusion

tt2n t3n t4nt0 tn

middot middot middot

(a) Without added backward-forward SINS resolution and datafusion

Estim

atio

n fo

rgy

ro b

ias

Data fusion

tt2n t3n t4nt0 tn

t01

t02

tn1

tn2

middot middot middot

(b) With added backward-forward SINS resolution and data fusion

Figure 5 Estimation process for gyro bias

But as shown in Figure 5(b) in the same reference dataupdate cycle such as 119905

0sim 119905119899 at the point 119905

119899 data fusion

will be run and new estimation will be produced andso will new initial navigation parameters And new initialparameters and new compensation value for gyro bias willproduce the new navigation parameters at the point 119905

01

which are different from those at 1199050 So a new estimate will

be generated at the point 11990501 which is different from that at

1199050 because the measurement vectors for the Kalman filter are

different at points 1199050and 11990501 which is caused by the same

reference data but different navigation parameters Similarlynew information will be got at the point 119905

1198992 which is different

from that at 119905119899 In Figure 5(b) with the added backward-

forward SINS resolution and data fusion the estimatingoperations for gyro bias and the adjustment for mathematicalplatform 119899

1015840 will be done with two more times In the second

method even with same observability degrees as in thefirst one the estimation time will be shortened because theestimation frequency is increased

There is no doubt that the added resolution and datafusion in Figure 5(b) will increase the burden of naviga-tion computer Though in the last decades the processingpowering of the employed microprocessors has dramaticallyincreased it is difficult in some way to complete a largecomputation in a relatively short period

As shown in Figure 6 in this rapid transfer alignmentmethod three tasks should be completed within one refer-ence data update cycle (1) Inertial sensor data needs to besampled and stored and navigation resolution should be run(2) Reference data needs to be sampled and stored and datafusion should be run (3) Backward and forward calculationsand data fusions should be executed The first task should be

Mathematical Problems in Engineering 7

executed at every navigation update cycle The last two tasksshould be run in the last navigation update cycle of everyreference data update cycle whichmeans that the above threetasks should be finished in 119879

119904 Otherwise the first task in the

first navigation cycle of the next reference data cycle will becompromised It is difficult to finish these three tasks in 119879

119904

with a computer of limited performanceHowever this problem can be resolved by the full use

of resources of high speed computers with the support ofreal-time multitasking operation system (RTOS) such asVxWorks In VxWorks the above three tasks can be set withdifferent priorities The first can be run preferentially whenTask 2 or Task 3 is being run whichmeans that the first task ofthe next reference data cycle can be run preferentially when

Tasks 2 and 3 of the previous reference data cycle are beingexecuted In this method the idle resources of CPU in thenext reference cycle can be used for the tasks 2 and 3 of theprevious cycle The paper is not involved in the programs inRTOS in detail

5 Simulation

51 Parameters for Simulation The ship moving parametersare set as in Section 323 The ideal velocity and yaw ofthe ship are used as reference data from MINS after whitenoise is added The variance of the white noise is set as[(04ms)2 (04ms)2 (03

∘)

2

] The sensor errors are listedin Table 1

The parameters for Kalman filter are

X0= [0 0 0 0 0 0 0 0]

119879

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

∘)

2

(15∘)

2

(15∘)

2

(15∘h)2 (15

∘h)2 (15

∘h)2]

Q = diag [(500 ug)2 (500 ug)2 (05∘)2

(05∘)2

(05∘)2

0 0 0]

R = diag [(04ms)2 (04ms)2 (03∘)

2

]

(10)

Two data fusion schemes were compared The secondscheme is a transfer alignment method with the addedbackward-forward SINS resolution and data fusion while thefirst scheme is not In these two schemes the same transferalignment model introduced in Section 2 is used

The update cycle of sensor data and navigation resolutionis set as 10ms and that of reference data from MINS is as1 s As analyzed in Section 4 in every reference data cyclenavigation resolution will be executed 300 times and datafusion 3 times in scheme 2 while in scheme 1 navigationresolution is executed 100 times and data fusion only once

In the ldquovelocity plus yawrdquo matching method accelerom-eter bias is unobservable so with the sensor errors assumedin Table 1 the limit alignment accuracy of pitch and roll areminus04990 mrad and 04990 mrad respectively and the limitalignment accuracy of yaw is 0

52 Simulation Results The simulation lasts for 500 s andthe simulation results are stored once per second The mis-alignment angle curves are shown in Figure 7The estimationof velocity error curves are in Figure 8 and the estimationcurves of gyros bias in Figure 9 In Figures 7ndash9 the dot-dashand solid lines denote the simulation results of scheme 1 andscheme 2 respectively The dotted lines in Figures 7 and 8are the limit alignment accuracy of misalignment angles andvelocity And the dotted lines in Figure 9 denote the settingvalue of constant gyro bias

The curves in Figure 7 show that either in scheme 1or scheme 2 misalignment angles can be estimated rapidlyand are oscillating in small amplitudes with the swinging

frequency of ship But the tendency of misalignment curvesespecially that of roll ones indicates that estimation speed isslightly higher in scheme 2 than in scheme 1 The statisticaldata about misalignment angles are shown in Table 3 and thestatistical results show that the alignment accuracy of thesetwo schemes is roughly equal after 60 sThe curves in Figure 8indicate that the estimation speed and accuracy for velocityerror are roughly equal in scheme 1 and scheme 2

The curves in Figure 9 show that in scheme 2 theestimation curves of gyro bias converge towards the settingvalues at about 50 s and the oscillating amplitudes are verysmall in 100 s while in scheme 1 200 s is needed for theconvergence of gyro bias and relatively large oscillations stillexist even after 300 s The statistical mean values shown inTable 4 indicate that in scheme 2 in the period from 50 sto 100 s about 96 and 93 and 72 of gyro bias can beestimated along 119909 119910 and 119911 axes respectively And about99 97 and 78 can be estimated in the period from 100 sto 150 s

6 Conclusion

Two viewpoints are given in this paper The first is that inSINS mathematical platform cuts off the direct relationshipbetween sensor data and misalignment angles which meansthat initial alignment can be fulfilled by the repeatedlyresolution on a same set of sensor data The second isthat with the added backward-forward SINS resolution andrepeated data fusion on the corresponding resolution resultsand the external data the alignment time can be greatlyreduced

8 Mathematical Problems in Engineering

Table 3 Statistical results for misalignment angles

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

1sim10 sScheme 1 minus08225 10610 11833 49672 minus09375 12756Scheme 2 04631 16724 02318 07429 minus03956 10700

11sim20 sScheme 1 minus02278 05136 minus00705 05156 minus11170 07199Scheme 2 minus04993 05865 04806 02647 minus04755 08203

31sim40 sScheme 1 minus04776 04653 09794 05194 00448 09782Scheme 2 minus06454 04553 02271 04726 minus00067 08820

41sim50 sScheme 1 minus08024 04560 10111 05086 minus03714 09385Scheme 2 minus08563 04557 04980 06148 minus03609 08828

51sim60 sScheme 1 minus05430 04296 05377 04614 minus02183 09682Scheme 2 minus07073 04854 05415 05531 02041 09117

61sim500 sScheme 1 minus05075 04498 04154 04852 minus00877 09168Scheme 2 minus06541 04505 04815 04831 00709 09011

Limited accuracy minus04990 04990 0

Table 4 Statistical results for gyro bias

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

51sim100 sScheme 1 minus04139 04563 minus07990 09143 07591 04807Scheme 2 05172 00757 05335 01329 06390 01588

101sim150 sScheme 1 03474 02102 minus03793 04709 19998 03182Scheme 2 05037 00505 04847 00499 06072 00628

151sim200 sScheme 1 05742 00632 02499 01254 08496 02778Scheme 2 04943 00382 04703 00284 05196 00256

201sim250 sScheme 1 04790 00283 02713 00744 05335 00648Scheme 2 04961 00224 04718 00231 05255 00224

251sim300 sScheme 1 05189 00344 04203 00630 05292 00530Scheme 2 05019 00217 04752 00258 05113 00236

301sim500 sScheme 1 05156 00272 04122 00410 05683 00412Scheme 2 04990 00116 04823 00169 05134 00129

Setting value 05 05 05

With the above two viewpoints (1) a backward-forwardSINS resolution algorithm is designed in detail and simu-lation results indicate that a ship can move from the endto the origin with the backward resolution and move fromthe origin to the end with the forward (2) A rapid transfer

alignment algorithm is also designed in detail in which thebackward-forward resolution and two more operations forthe estimation of gyro bias are added in one reference dataupdate cycle In addition the correctness of its algorithm isproved by simulation The simulation result produced with

Mathematical Problems in Engineering 9

middot middot middot

13

11 2

(1) Sensor data sampling storing and navigation resolution(2) Reference data sampling storing and data fusion(3) Reverse and forward calculationlowastCPU idle

TsTs Ts

Ts Ts

tnt2nlowast lowast

Tn Tn

t0

tn2

tn1t01

t02

middot middot middot

Figure 6 Calculation process for alignment program

50 100 150 200 250 300 350 400 450 500

Erro

r (m

rad)

012

Scheme 2Scheme 1

minus3minus2minus1

t (s)

(a) Pitch

Erro

r (m

rad)

50 100 150 200 250 300 350 400 450 500

3 Scheme 2Scheme 1

012

minus2minus1

t (s)

(b) Roll

50 100 150 200 250 300 350 400 450 500

024

Erro

r (m

rad) Scheme 2Scheme 1

minus2minus4

t (s)

(c) Yaw

Figure 7 Estimation curves for misalignment angles

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(a) East velocity

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(b) North velocity

Figure 8 Estimation curves for velocity errors

50 100 150 200 250 300 350 400 450 500

13 Scheme 1Scheme 2

minus3

minus1

Bias

(∘h

)

t (s)

(a) 119909 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 113

minus3

minus1

Bias

(∘h

)

t (s)

(b) 119910 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 1

13

minus3

minus1

Bias

(∘h

)

t (s)

(c) 119911 gyro

Figure 9 Estimation curves for gyro bias

the method in this paper indicates that the alignment timeis reduced from 300 s to 100 s compared with that of thetransfer alignment method without the added backward-forward SINS resolution and data fusion

One problem which may be encountered in engineeringapplications with this transfer alignment method is studiedand a possible solution is given in which RTOS is introducedto distribute computer resources coordinately

Acknowledgments

This work was supported in part by the National NaturalScience Foundation (61004125 61273056) and Chinese uni-versity research and operation expenses (10420525)

References

[1] J Kain and J Cloutier ldquoRapid transfer alignment for tacticalweapon applicationsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 1290ndash1300 BostonMass USA 1989

[2] K Spalding ldquoAn efficient rapid transfer alignment filterrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 1276ndash1286 Hilton Head Island SC USA 1992

[3] K Shortelle and W Graham ldquoAdvanced alignment conceptsfor precision-guided weaponsrdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 131ndash142Anaheim Calif USA January 1995

[4] W Graham and K Shortelle ldquoAdvanced transfer alignmentfor inertial navigators (A-train)rdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 113ndash124Anaheim Calif USA January 1995

[5] K J Shortelle W R Graham and C Rabourn ldquoF-16 flight testsof a rapid transfer alignment procedurerdquo in Proceedings of the

10 Mathematical Problems in Engineering

IEEEPosition Location andNavigation Symposium pp 379ndash386Palm Springs Calif USA April 1996

[6] P D Groves ldquoOptimising the transfer alignment of weaponINSrdquo Journal of Navigation vol 56 no 2 pp 323ndash335 2003

[7] DGoshen-Meskin and I Y Bar-Itzhack ldquoObservability analysisof piece-wise constant systems II Application to inertialnavigation in-flight alignment (military applications)rdquo IEEETransactions on Aerospace and Electronic Systems vol 28 no4 pp 1068ndash1075 1992

[8] X H Cheng D J Wan and X Zhong ldquoStudy on observabilityand its degree of strapdown inertial navigation systemrdquo Journalof Southeast University vol 37 no 6 pp 6ndash10 1997

[9] L-H Zhu X-H Cheng and Y-Y He ldquoFilter convergencecriterion in transfer alignmentrdquo Journal of Chinese InertialTechnology vol 19 no 3 pp 277ndash285 2011

[10] S Wang Z Wang Y Zhu and B Wang ldquoMonitoring onship hull deformation and correction for heading and attitudeinformationrdquo Journal of Chinese Inertial Technology vol 15 no6 pp 635ndash641 2007

[11] D R Tarrant C Jones and D Lin ldquoRapid and robust transferalignmentrdquo in Proceedings of the 1st IEEE Regional Conferenceon Aerospace Control Systems pp 758ndash762 1993

[12] J Lyou and Y C Lim ldquoTransfer alignment considering mea-surement time delay and ship body flexurerdquo Journal of Mechan-ical Science and Technology vol 23 no 1 pp 195ndash203 2009

[13] B-C Zhou X-H Cheng and D-J Liu ldquoAnalysis method ofobservable degree based on spectral decomposition in SINStransfer alignmentrdquo Journal of Chinese Inertial Technology vol18 no 5 pp 518ndash522 2010

[14] D H Titterton and J L Weston Strapdown Inertial NavigationTechnology Lavenham Press Ltd London UK 2nd edition2004

[15] R B Miller ldquoA new strapdown attitude algorithmrdquo Journal ofGuidance Control and Dynamics vol 6 no 4 pp 287ndash2911983

[16] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998

[17] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 2 velocity and position algorithmsrdquo Journalof Guidance Control and Dynamics vol 21 no 2 pp 208ndash2211998

[18] P J Gates and N M Lynn Ships Submarines and the SeaBrasseyrsquos London UK 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

is used as slave INS and the navigation parameters fromMINS are used as reference data In the paper the slave INSand strapdown INS are both abbreviated as SINS In theintegration of MS INS flexure deformation is a key factorwhichwill determine transfer alignment accuracy [10] In thispaper ldquovelocity plus yawrdquo matching method is chosen due tothe fact that the ldquoattituderdquo matching method is easily sufferedby flexure deformation [11 12] while ldquovelocityrdquo matchingmethod can ensure the rapidity and accuracy of estimationfor horizontal misalignment angles [8] According to theanalysis based on piece-wise constant systems (PWCS) lineartheory [7] with the excitation of the wave swinging thevelocity errors misalignment angles and gyro bias are theobservable parameters [13] but unfortunately the estimationfor gyro bias is a time-consuming process which usually lastsfor several minutes

To deal with the contradiction between the rapidity ofalignment and time consumption of gyro bias estimationtwo viewpoints are put forward by analyzing the alignmentprocess of SINS Firstly different from that in PINS the directcorresponding relationship between inertial senor data andmisalignment angles in SINS is severed by the mathematicalplatform Thus we can adjust the calculated mathematicalplatform 119899

1015840 constantly to speed up the initial alignment bythe repeated SINS resolution with the same set of sensordata Secondly analysis indicates that with a Kalman filter anew estimation for state vector will be provided With close-loop correction new initial navigation parameters and newcompensation parameters for gyro bias will be produced forthe next navigation period andwith the above new ones newnavigation results will be generated even built on the same setof sensor data So with the repeatedly data fusion of the addedbackward-forward SINS resolution results and the externalreference data the estimation time for sensor error will beshortened With the above two viewpoints a rapid transferalignment method based on the added backward-forwardSINS resolution without any changes to the Kalman filter isdesigned in detail In one reference data update cycle with

a normal resolution and data fusion an added backward andan added forward resolutions and their data fusions the timeconsumed for gyro bias estimation is effectively shortenedThe effectiveness of this method is proved by simulationresults

This thesis is constructed as follows In Section 2 thetransfer alignment model based on ldquovelocity plus yawrdquomatching is built and the reasons why the observabilitydegree of gyro bias is low are analyzed In Section 3 afterthe alignment process of INS is studied and compared apossible way to increase SINS alignment speed is introducedand the backward-forward SINS resolution is designed InSection 4 the transfer alignment model based on the addedbackward-forward resolution is also designed in detail Andin Section 5 the effectiveness of this method is proved bysimulation Finally some conclusions are given

2 Transfer Alignment Model Based onlsquolsquoVelocity Plus Yawrsquorsquo Matching Method

21 System Equation Choose velocity errors misalignmentangles and gyro bias as the state vector of the system

X = [120575119881119864120575119881119873

120601119864

120601119873

120601119880

120576119909

120576119910

120576119911]

119879

(1)

where 120575119881119864and 120575119881

119873are east and north velocity errors

respectively 120601119864 120601119873 and 120601

119880are misalignment angles of

pitch roll and yaw respectively 120576119909 120576119910 and 120576

119911are gyro bias

along 119909119910 and 119911 axes To ships which sail on the sea theheight and upward velocity can be set as zero

The system state equation can be constructed as

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

where A(119905) is the state matrix and W(119905) is the systemnoise matrix With the system vector velocity error andmisalignment angle equations the state matrix A(119905) can beexpressed as

A (119905) =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

119881119873

119877

tan119871 2120596119894119890sin119871 + 119881119864

119877

tan119871 0 minus119891119880

119891119873

0 0 0

minus(2120596119894119890sin119871 + 119881119864

119877

tan119871) 0 119891119880

0 minus119891119864

0 0 0

0 minus1

119877

0 120596119894119890sin119871 + 119881119864

119877

tan119871 minus(120596119894119890cos119871 + 119881119864

119877

) minus11987911minus11987912minus11987913

1

119877

0 minus(120596119894119890sin119871 + 119881119864

119877

tan119871) 0 minus119881119873

119877

minus11987921minus11987922minus11987923

1

119877

tan119871 0 120596119894119890cos119871 + 119881119864

119877

119881119873

119877

0 minus11987931minus11987932minus11987933

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(3)

Mathematical Problems in Engineering 3

Inertial sensorAlignment process

Ideal navigation frame n

Physical platform frame n998400

(a) PINS

Body frame

Inertial sensor

Alignment process

Ideal navigation frame n

Mathematical platform frame n998400

(b) SINS

Figure 1 Alignment process for INS

where 119881119864and 119881

119873are the east and north velocity respectively

120596119894119890and 119877 are the rotational angular rate and radius of the

Earth 119871 is the latitude of ship position 119891119864 119891119873 and 119891

119880

denote the projection of the accelerometer measured data f119887in navigation frame 119899 and 119879

119894119895(119894 119895 = 1 2 3) are the elements

of the direct cosine matrix (DCM) of MINS

22 Measurement Equation The differences of velocity andyaw between MINS and SINS are used as measurement datafor data fusion Then the measurement vector is as

Z = [119881119864minus 119881119872119864

119881119873

minus 119881119872119873

119884119864minus 119884119872119864

]

119879

(4)

where 119881119864 119881119873and 119884

119864are the east north velocity and yaw

from SINS respectively 119881119872119864

119881119872119873

and 119884119872119864

are fromMINS respectively

The measurement equation can be constructed as

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

whereH(119905) is measurement matrix and V(119905) is measurementnoise matrix According to the relationship betweenmeasurement and state vectors the measurement matrixH(119905) can be expressed as

H =

[

[

[

[

[

[

[

[

1 0 0 0 0

0 1 0 0 0 03times3

0 0 minus

1198791211987932

1198792

12+ 1198792

22

minus

1198792211987932

1198792

12+ 1198792

22

minus1

]

]

]

]

]

]

]

]

(6)

23 Observability Degree of Each State Variable After theanalysis of the model combined with (2) and (5) by PWCStheory we can conclude that all of the eight selected variablesare observable but the observability degrees are differentfrom each other [13] The simulation results in Section 52indicate that the horizontal velocity errors and yaw mis-alignment converge immediately that the estimation forhorizontal misalignment angles lasts for 10sim20 s and thatthe estimation for gyro bias lasts for 3sim5min which meansthat the degrees of velocity errors and yaw misalignment

are the highest those of horizontal misalignment angles aremoderate and those of gyro bias are the lowest

The reasons which cause the different degrees of errorsare as follows The gyro bias needs integration to be reflectedon misalignment angles and the horizontal misalignmentangles need projection reflected on acceleration Furtherintegration of acceleration gyro bias can be reflected onvelocity errors In this process from gyro bias to velocityerrors integrating operations are needed twice and fromhorizontal misalignment angles to velocity errors only oneis needed When velocity errors and yaw misalignment areselected as components of a measurement vector there isno doubt that observability degrees of gyro bias are lowestwhile those of velocity errors and yaw misalignment arehighest which are determined by the mechanism of transferalignment To shorten the estimation time for gyro bias someother novel ways should be sought

3 A New Way to Speed UpSINS Alignment

31 Analysis of Alignment Process in INS An acceptedviewpoint is that in navigation sensor data is of real-timesignificance Based on initial navigation parameters INSobtains the attitude velocity and position by the integrationof sensor data After integration the sensor data can bediscarded

Take the alignment process of PINS as an example Asshown in Figure 1(a) 1198991015840 frame denotes the physical platformand 119899 frame denotes the ideal navigation frame In PINSinertial sensors are installed on the physical platform Inertialsensors measure the ship motion in 119899

1015840 frame and navigationresolution is also executed in this frame however referencedata are from 119899 frame Before aligning misalignment anglesbetween 119899

1015840 and 119899 frames will cause the differences betweennavigation resolution values in 119899

1015840 frame and reference val-ues in 119899 frame Because these differences can reflect themisalignments transfer alignment can be finished with themodel in Section 2 In the alignment process with theabove differences caused by misalignment angles physicalplatform can be adjusted to coincide with the ideal frameIn other words in PINS sensor data reflect the magnitudeof misalignment angle and other errors and there is direct

4 Mathematical Problems in Engineering

relationship between sensor data and misalignment anglesso the sensor data are of real-time significance And theadjustment for physical platform to get new sensor data is atime-consuming process

At the same time in SINS the alignment method isderived from that of PINS and so the alignment must bea time-consuming process for the need of real-time sensordata and the adjustment of platform But in INS as far as theprocess of alignment is concerned maybe sensor data is ofreal-time significance for PINS but not for SINS

In SINS as shown in Figure 1(b) amathematical platformreplaces the physical platform in PINS and inertial sensorsare directly installed on the ship Here the ship body isnamed 119887 frame and the mathematical platform is named 119899

1015840

frame In alignment the adjusting process of the 1198991015840 frame

is the same with that in PINS while the difference is thatthe calculated sensor data in 119899

1015840 frame is the projection ofmeasured sensor data from 119887 frame Only the projectiondata in 119899

1015840 frame reflect the misalignment angles between 1198991015840

and 119899 frames but sensor data cannot The calculated andmeasured sensor data are connected by calculated DCM C119899

1015840

119887

In comparison with that in PINS the direct relationshipbetween sensor data and misalignment angles is severedby the mathematical platform Then the 119899

1015840 frame can beadjusted constantly by the repeated calculation for DCM C119899

1015840

119887

with a same set of sensor data which means that alignmenttime can be shortened without too much time to be spent insampling real-time sensor data

The above set of data can be seen as that when the shipkeeps an ideal static state and there are no sensor errors allmeasured sensor data are equal and can be dealt with as aseries of single data In engineering because of the errorsfrom sensor and reference data the update frequency ofreference data and so forth a set of sensor data and referencedata should be used for repeated SINS resolution (addedbackward-forward resolution) and repeated data fusion toimprove the alignment accuracy

Also different from PINS SINS is a digital system inwhich there is only numerical data no mass spring andresistance So the adjustment for mathematical platform canbe as fast as lightning

32 Backward-Forward SINS Resolution The above analysisindicates that the alignment of SINS can be fulfilled with aset of same data which brings a new problemmdashthe way touse these data

In SINS with initial attitude velocity and positionthe navigation resolution is the real-time updating processfor navigation parameters with sensor data by integratedcalculation In this process ship moves from the origin tothe end In backward-forward SINS resolution as shown inFigure 2 backward SINS resolution is the process in whichship moves from the end to the originmdasha reverse process ofnormal navigation and forward resolution is that from theorigin to the endmdasha repeated process of normal navigationFrom the basic SINS resolution algorithm the deduction ofbackward-forward SINS algorithm is shown as follows

Backward

Forward

Normal

t0 tn tm

t01

t02

t0i

t0n

tn1

tn2

tni

tnn

middot middot middot

Figure 2 The process of backward-forward SINS resolution

321 Normal Resolution Algorithm in SINS The navigationresolution algorithm is [14]

C119899119887= C119899119887(120596119887

119899119887times)

V119899 = C119899119887f119887 minus (2120596

119899

119894119890+ 120596119899

119890119899) times V119899 + g119899

=

119881119899

119873

119877

120582 =

119881119899

119864sec 119871119877

(7)

where 120596119887119899119887

= 120596119887

119894119887minus (C119899

119887)119879

(120596119899

119894119890+ 120596119899

119890119899) 120596119899119890119899

=

[minus119881119899

119873119877 119881

119899

119864119877 (119881

119899

119864tan 119871)119877]

119879 A119863119861119862

(such as 120596119899119894119890) denotes

the projection of a motion vector A which means the relativemotion from 119862 frame to 119861 frame in119863 frame C119899

119887is the DCM

V119899 = [119881119899

119864119881119899

119873119881119899

119880] is the velocity vector 119871 and 120582 are the

latitude and longitude respectively and (sdottimes) denotes theantisymmetric matrix of the vector ldquosdotrdquo

For the recursive update with computer formulas (7)must be translated into discrete form in a certain samplingcycle and navigation resolution update cycle For a shipwhich always undergoes a low dynamical movement a singlesampling resolution algorithm can be adopted which meansonly one sample of sensor data per navigation cycle [15ndash17]Supposing the above cycle times are both equal to 119879

119904 then

the discrete form can be expressed as follows [14]

C119899119887119896

= C119899119887119896minus1

(I + 119879119904120596119887

119899119887119896times)

V119899119896= V119899119896minus1

+119879119904[C119899119887119896minus1

f119887119896minus1

minus (2120596119899

119894119890119896minus1+ 120596119899

119890119899119896minus1) times V119899119896minus1

+ g119899]

119871119896= 119871119896minus1

+

119879119904119881119873119896minus1

119877119872

+ ℎ119896minus1

120582119896= 120582119896minus1

+

119879119904119881119864119896minus1

sec119871119896minus1

119877119873

+ ℎ119896minus1

(8)

where 119896 denotes recursive number Without considerationof calculation error formulas (8) are composed of the basicupdate equations for navigation resolution

322 Backward Resolution Algorithm in SINS As shown inFigure 2 in a backward resolution process a ship needs toreturn from the end to the origin So in the normal process

Mathematical Problems in Engineering 5

all sensor data must be stored According to the formulas (8)the resolution process can be expressed as follows

C119899119887119896minus1

= C119899119887119896(I + 119879

119904120596119887

119899119887119896times)

minus1

asymp C119899119887119896

(I minus 119879119904120596119887

119899119887119896times) asymp C119899

119887119896(I minus 119879

119904120596119887

119899119887119896minus1times)

V119899119896minus1

= V119899119896minus 119879119904[C119899119887119896minus1

f119887119896minus1

minus (2120596119899

119894119890119896minus1+ 120596119899

119890119899119896minus1) times V119899119896minus1

+ g119899]

asymp V119899119896minus 119879119904[C119899119887119896f119887119896minus (2120596

119899

119894119890119896+ 120596119899

119890119899119896) times V119899119896+ g119899]

119871119896minus1

= 119871119896minus

119879119904119881119873119896minus1

119877

asymp 119871119896minus

119879119904119881119873119896

119877

120582119896minus1

= 120582119896minus

119879119904119881119864119896minus1

sec119871119896minus1

119877

asymp 120582119896minus

119879119904119881119864119896sec 119871119896

119877

(9)

where 119896 is reduced from 119899 to 0 As shown in Figure 2 1199051198991

isboth the starting point in the period 119905

1198991sim 11990501and the ending

point in the period 1199050

sim 119905119899 Take the attitude velocity and

position at 119905119899as the initial attitude velocity and position at

1199051198991 and the backward resolution can be fully realized In the

period of 1199050

sim 119905119899and 1199051198991

sim 11990501 with the same recursive

number 119896 the ship maintains the same attitude velocity andposition and maintains the same acceleration but oppositein direction Some errors are induced by the approximationin formulas (9) and all these errors can be ignored whenthe resolution cycle is short enough The simulation inSection 323 proves that the above approximation is effective

After the backward resolution a forward resolutionshould bemade in order for the ship to return from the originto the end In the forward resolution formulas (8) can beused

323 Simulation on Backward-Forward SINS Resolution Thesensor errors are listed inTable 1 Andwe assume that the shipis in a bad-moderate sea condition [18] and the ship swingingparameters are listed in Table 2 Ship initial attitudes are set as0 and the ship is assumed without linear motion and locatednorth latitude 32∘ and east longitude 118∘ Sensor samplingand navigation resolution cycle 119879

119904is set as 10ms

Simulation results are shown in Figure 3 in which (a)(b) and (c) show the resolution results of pitch east velocityand latitude The dot and solid lines denote the simulationwith no sensor error and with sensor error respectivelyThe simulation is divided into three stages firstly in the1199050sim 119905119899period sensor data are measured and stored normal

navigation is resolved and this period lasts for 1 s secondly inthe 1199051198991

sim 11990501period backward resolution is run and finally in

the 11990502

sim 1199051198992period forward resolution is run In the last two

stages the time consumed is determined by the performanceof computer

In Figure 3without consideration of calculating the errorthe ship can move from the end to the origin and move fromorigin to end with the backward and forward resolutionsrespectively

But the above processes also indicate that with onlybackward or forward or backward-forward SINS resolutionno new information will be generated

Table 1 Sensor errors

Gyro bias Acce biasConstant Random Constant Random

119909 05∘h 05∘h 500 120583g 500 120583g119910 05∘h 05∘h 500 120583g 500 120583g119911 05∘h 05∘h 500 120583g 500 120583g

Table 2 Swinging parameters

Pitch Roll YawAmplitude (∘) 9 12 14Cycle (s) 8 10 6

4 Rapid Transfer Alignment Based on theAdded Resolution and Data Fusion

The analysis in Section 31 indicates that alignment for SINScan be fulfilled with a set of data and in this section wecombine a Kalman filter with the added backward-forwardSINS resolution aiming to shorten the alignment time

As shown in Figure 4 transfer alignment model basedon Kalman filter introduced in Section 2 and a close-loopcorrection method are used in this rapid transfer alignmentalgorithm Close-loop correction means that after datafusion the new estimation for velocity errors and misalign-ment angles will be fed to SINS to revise those correspondingparameters and new estimation for gyro bias will be setas new compensation value to participate in the followingnavigation resolutions In other words after data fusion themathematical platform C119899

1015840

119887will be adjusted For a ship the

update frequency of the reference data from MINS is lowerthan that of SINS navigation resolution 119879

119904and 119879

119899are set

as update cycle of navigation resolution and reference datarespectively Also 119879

119899is set as the added backward-forward

SINS resolution cycleTwo transfer alignment methods are compared and the

estimation processes of gyro bias are schematically demon-strated in Figure 5 in which (a) and (b) indicate these twomethods respectively which are all based on the principleas shown in Figure 4 In the first one the added backward-forward SINS resolution and data fusion are not used whilethey are used in the second In Figure 5 the dot-dashed linesdenote the real gyro bias while the solid line denotes theestimation of gyro bias With the Kalman filter and matchingmethod introduced in Section 2 the estimation of gyro biasis a slow process which will be convergent towards real gyrobias after a long time

As shown in Figure 5(a) when the reference data isavailable such as at the point 119905

119899 data fusion is executed and

a new estimate for the state vector will be produced With thenew estimation for misalignment angles and velocity errorsinitial navigation parameters will be reset for the next periodsuch as 119905

119899sim 1199052119899 whichmeans that themathematical platform

1198991015840 will be adjusted and the new compensation value for gyro

bias will also be reset In this method along with time datafusion is run only once at every reference data update cycle

6 Mathematical Problems in Engineering

0

5

10

minus10

minus5

tn(tn1)t0 tn2t01(t02)

Ang

le (∘

)

(a) Pitch

Velo

city

(ms

)

0

005

minus005tn(tn1)t0 tn2t01(t02)

(b) East velocity

32

32

32

tn(tn1)t0 tn2t01(t02)

Ang

le (∘

)

(c) Latitude

Figure 3 Simulation of normal and backward-forward SINS resolution

Kalman filter

SINS

MINS Velocity + yaw

Velocity + yaw

State variables

Navigation parameters

Figure 4 Kalman filter

Estim

atio

n fo

rgy

ro b

ias

Data fusion

tt2n t3n t4nt0 tn

middot middot middot

(a) Without added backward-forward SINS resolution and datafusion

Estim

atio

n fo

rgy

ro b

ias

Data fusion

tt2n t3n t4nt0 tn

t01

t02

tn1

tn2

middot middot middot

(b) With added backward-forward SINS resolution and data fusion

Figure 5 Estimation process for gyro bias

But as shown in Figure 5(b) in the same reference dataupdate cycle such as 119905

0sim 119905119899 at the point 119905

119899 data fusion

will be run and new estimation will be produced andso will new initial navigation parameters And new initialparameters and new compensation value for gyro bias willproduce the new navigation parameters at the point 119905

01

which are different from those at 1199050 So a new estimate will

be generated at the point 11990501 which is different from that at

1199050 because the measurement vectors for the Kalman filter are

different at points 1199050and 11990501 which is caused by the same

reference data but different navigation parameters Similarlynew information will be got at the point 119905

1198992 which is different

from that at 119905119899 In Figure 5(b) with the added backward-

forward SINS resolution and data fusion the estimatingoperations for gyro bias and the adjustment for mathematicalplatform 119899

1015840 will be done with two more times In the second

method even with same observability degrees as in thefirst one the estimation time will be shortened because theestimation frequency is increased

There is no doubt that the added resolution and datafusion in Figure 5(b) will increase the burden of naviga-tion computer Though in the last decades the processingpowering of the employed microprocessors has dramaticallyincreased it is difficult in some way to complete a largecomputation in a relatively short period

As shown in Figure 6 in this rapid transfer alignmentmethod three tasks should be completed within one refer-ence data update cycle (1) Inertial sensor data needs to besampled and stored and navigation resolution should be run(2) Reference data needs to be sampled and stored and datafusion should be run (3) Backward and forward calculationsand data fusions should be executed The first task should be

Mathematical Problems in Engineering 7

executed at every navigation update cycle The last two tasksshould be run in the last navigation update cycle of everyreference data update cycle whichmeans that the above threetasks should be finished in 119879

119904 Otherwise the first task in the

first navigation cycle of the next reference data cycle will becompromised It is difficult to finish these three tasks in 119879

119904

with a computer of limited performanceHowever this problem can be resolved by the full use

of resources of high speed computers with the support ofreal-time multitasking operation system (RTOS) such asVxWorks In VxWorks the above three tasks can be set withdifferent priorities The first can be run preferentially whenTask 2 or Task 3 is being run whichmeans that the first task ofthe next reference data cycle can be run preferentially when

Tasks 2 and 3 of the previous reference data cycle are beingexecuted In this method the idle resources of CPU in thenext reference cycle can be used for the tasks 2 and 3 of theprevious cycle The paper is not involved in the programs inRTOS in detail

5 Simulation

51 Parameters for Simulation The ship moving parametersare set as in Section 323 The ideal velocity and yaw ofthe ship are used as reference data from MINS after whitenoise is added The variance of the white noise is set as[(04ms)2 (04ms)2 (03

∘)

2

] The sensor errors are listedin Table 1

The parameters for Kalman filter are

X0= [0 0 0 0 0 0 0 0]

119879

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

∘)

2

(15∘)

2

(15∘)

2

(15∘h)2 (15

∘h)2 (15

∘h)2]

Q = diag [(500 ug)2 (500 ug)2 (05∘)2

(05∘)2

(05∘)2

0 0 0]

R = diag [(04ms)2 (04ms)2 (03∘)

2

]

(10)

Two data fusion schemes were compared The secondscheme is a transfer alignment method with the addedbackward-forward SINS resolution and data fusion while thefirst scheme is not In these two schemes the same transferalignment model introduced in Section 2 is used

The update cycle of sensor data and navigation resolutionis set as 10ms and that of reference data from MINS is as1 s As analyzed in Section 4 in every reference data cyclenavigation resolution will be executed 300 times and datafusion 3 times in scheme 2 while in scheme 1 navigationresolution is executed 100 times and data fusion only once

In the ldquovelocity plus yawrdquo matching method accelerom-eter bias is unobservable so with the sensor errors assumedin Table 1 the limit alignment accuracy of pitch and roll areminus04990 mrad and 04990 mrad respectively and the limitalignment accuracy of yaw is 0

52 Simulation Results The simulation lasts for 500 s andthe simulation results are stored once per second The mis-alignment angle curves are shown in Figure 7The estimationof velocity error curves are in Figure 8 and the estimationcurves of gyros bias in Figure 9 In Figures 7ndash9 the dot-dashand solid lines denote the simulation results of scheme 1 andscheme 2 respectively The dotted lines in Figures 7 and 8are the limit alignment accuracy of misalignment angles andvelocity And the dotted lines in Figure 9 denote the settingvalue of constant gyro bias

The curves in Figure 7 show that either in scheme 1or scheme 2 misalignment angles can be estimated rapidlyand are oscillating in small amplitudes with the swinging

frequency of ship But the tendency of misalignment curvesespecially that of roll ones indicates that estimation speed isslightly higher in scheme 2 than in scheme 1 The statisticaldata about misalignment angles are shown in Table 3 and thestatistical results show that the alignment accuracy of thesetwo schemes is roughly equal after 60 sThe curves in Figure 8indicate that the estimation speed and accuracy for velocityerror are roughly equal in scheme 1 and scheme 2

The curves in Figure 9 show that in scheme 2 theestimation curves of gyro bias converge towards the settingvalues at about 50 s and the oscillating amplitudes are verysmall in 100 s while in scheme 1 200 s is needed for theconvergence of gyro bias and relatively large oscillations stillexist even after 300 s The statistical mean values shown inTable 4 indicate that in scheme 2 in the period from 50 sto 100 s about 96 and 93 and 72 of gyro bias can beestimated along 119909 119910 and 119911 axes respectively And about99 97 and 78 can be estimated in the period from 100 sto 150 s

6 Conclusion

Two viewpoints are given in this paper The first is that inSINS mathematical platform cuts off the direct relationshipbetween sensor data and misalignment angles which meansthat initial alignment can be fulfilled by the repeatedlyresolution on a same set of sensor data The second isthat with the added backward-forward SINS resolution andrepeated data fusion on the corresponding resolution resultsand the external data the alignment time can be greatlyreduced

8 Mathematical Problems in Engineering

Table 3 Statistical results for misalignment angles

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

1sim10 sScheme 1 minus08225 10610 11833 49672 minus09375 12756Scheme 2 04631 16724 02318 07429 minus03956 10700

11sim20 sScheme 1 minus02278 05136 minus00705 05156 minus11170 07199Scheme 2 minus04993 05865 04806 02647 minus04755 08203

31sim40 sScheme 1 minus04776 04653 09794 05194 00448 09782Scheme 2 minus06454 04553 02271 04726 minus00067 08820

41sim50 sScheme 1 minus08024 04560 10111 05086 minus03714 09385Scheme 2 minus08563 04557 04980 06148 minus03609 08828

51sim60 sScheme 1 minus05430 04296 05377 04614 minus02183 09682Scheme 2 minus07073 04854 05415 05531 02041 09117

61sim500 sScheme 1 minus05075 04498 04154 04852 minus00877 09168Scheme 2 minus06541 04505 04815 04831 00709 09011

Limited accuracy minus04990 04990 0

Table 4 Statistical results for gyro bias

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

51sim100 sScheme 1 minus04139 04563 minus07990 09143 07591 04807Scheme 2 05172 00757 05335 01329 06390 01588

101sim150 sScheme 1 03474 02102 minus03793 04709 19998 03182Scheme 2 05037 00505 04847 00499 06072 00628

151sim200 sScheme 1 05742 00632 02499 01254 08496 02778Scheme 2 04943 00382 04703 00284 05196 00256

201sim250 sScheme 1 04790 00283 02713 00744 05335 00648Scheme 2 04961 00224 04718 00231 05255 00224

251sim300 sScheme 1 05189 00344 04203 00630 05292 00530Scheme 2 05019 00217 04752 00258 05113 00236

301sim500 sScheme 1 05156 00272 04122 00410 05683 00412Scheme 2 04990 00116 04823 00169 05134 00129

Setting value 05 05 05

With the above two viewpoints (1) a backward-forwardSINS resolution algorithm is designed in detail and simu-lation results indicate that a ship can move from the endto the origin with the backward resolution and move fromthe origin to the end with the forward (2) A rapid transfer

alignment algorithm is also designed in detail in which thebackward-forward resolution and two more operations forthe estimation of gyro bias are added in one reference dataupdate cycle In addition the correctness of its algorithm isproved by simulation The simulation result produced with

Mathematical Problems in Engineering 9

middot middot middot

13

11 2

(1) Sensor data sampling storing and navigation resolution(2) Reference data sampling storing and data fusion(3) Reverse and forward calculationlowastCPU idle

TsTs Ts

Ts Ts

tnt2nlowast lowast

Tn Tn

t0

tn2

tn1t01

t02

middot middot middot

Figure 6 Calculation process for alignment program

50 100 150 200 250 300 350 400 450 500

Erro

r (m

rad)

012

Scheme 2Scheme 1

minus3minus2minus1

t (s)

(a) Pitch

Erro

r (m

rad)

50 100 150 200 250 300 350 400 450 500

3 Scheme 2Scheme 1

012

minus2minus1

t (s)

(b) Roll

50 100 150 200 250 300 350 400 450 500

024

Erro

r (m

rad) Scheme 2Scheme 1

minus2minus4

t (s)

(c) Yaw

Figure 7 Estimation curves for misalignment angles

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(a) East velocity

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(b) North velocity

Figure 8 Estimation curves for velocity errors

50 100 150 200 250 300 350 400 450 500

13 Scheme 1Scheme 2

minus3

minus1

Bias

(∘h

)

t (s)

(a) 119909 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 113

minus3

minus1

Bias

(∘h

)

t (s)

(b) 119910 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 1

13

minus3

minus1

Bias

(∘h

)

t (s)

(c) 119911 gyro

Figure 9 Estimation curves for gyro bias

the method in this paper indicates that the alignment timeis reduced from 300 s to 100 s compared with that of thetransfer alignment method without the added backward-forward SINS resolution and data fusion

One problem which may be encountered in engineeringapplications with this transfer alignment method is studiedand a possible solution is given in which RTOS is introducedto distribute computer resources coordinately

Acknowledgments

This work was supported in part by the National NaturalScience Foundation (61004125 61273056) and Chinese uni-versity research and operation expenses (10420525)

References

[1] J Kain and J Cloutier ldquoRapid transfer alignment for tacticalweapon applicationsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 1290ndash1300 BostonMass USA 1989

[2] K Spalding ldquoAn efficient rapid transfer alignment filterrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 1276ndash1286 Hilton Head Island SC USA 1992

[3] K Shortelle and W Graham ldquoAdvanced alignment conceptsfor precision-guided weaponsrdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 131ndash142Anaheim Calif USA January 1995

[4] W Graham and K Shortelle ldquoAdvanced transfer alignmentfor inertial navigators (A-train)rdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 113ndash124Anaheim Calif USA January 1995

[5] K J Shortelle W R Graham and C Rabourn ldquoF-16 flight testsof a rapid transfer alignment procedurerdquo in Proceedings of the

10 Mathematical Problems in Engineering

IEEEPosition Location andNavigation Symposium pp 379ndash386Palm Springs Calif USA April 1996

[6] P D Groves ldquoOptimising the transfer alignment of weaponINSrdquo Journal of Navigation vol 56 no 2 pp 323ndash335 2003

[7] DGoshen-Meskin and I Y Bar-Itzhack ldquoObservability analysisof piece-wise constant systems II Application to inertialnavigation in-flight alignment (military applications)rdquo IEEETransactions on Aerospace and Electronic Systems vol 28 no4 pp 1068ndash1075 1992

[8] X H Cheng D J Wan and X Zhong ldquoStudy on observabilityand its degree of strapdown inertial navigation systemrdquo Journalof Southeast University vol 37 no 6 pp 6ndash10 1997

[9] L-H Zhu X-H Cheng and Y-Y He ldquoFilter convergencecriterion in transfer alignmentrdquo Journal of Chinese InertialTechnology vol 19 no 3 pp 277ndash285 2011

[10] S Wang Z Wang Y Zhu and B Wang ldquoMonitoring onship hull deformation and correction for heading and attitudeinformationrdquo Journal of Chinese Inertial Technology vol 15 no6 pp 635ndash641 2007

[11] D R Tarrant C Jones and D Lin ldquoRapid and robust transferalignmentrdquo in Proceedings of the 1st IEEE Regional Conferenceon Aerospace Control Systems pp 758ndash762 1993

[12] J Lyou and Y C Lim ldquoTransfer alignment considering mea-surement time delay and ship body flexurerdquo Journal of Mechan-ical Science and Technology vol 23 no 1 pp 195ndash203 2009

[13] B-C Zhou X-H Cheng and D-J Liu ldquoAnalysis method ofobservable degree based on spectral decomposition in SINStransfer alignmentrdquo Journal of Chinese Inertial Technology vol18 no 5 pp 518ndash522 2010

[14] D H Titterton and J L Weston Strapdown Inertial NavigationTechnology Lavenham Press Ltd London UK 2nd edition2004

[15] R B Miller ldquoA new strapdown attitude algorithmrdquo Journal ofGuidance Control and Dynamics vol 6 no 4 pp 287ndash2911983

[16] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998

[17] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 2 velocity and position algorithmsrdquo Journalof Guidance Control and Dynamics vol 21 no 2 pp 208ndash2211998

[18] P J Gates and N M Lynn Ships Submarines and the SeaBrasseyrsquos London UK 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Inertial sensorAlignment process

Ideal navigation frame n

Physical platform frame n998400

(a) PINS

Body frame

Inertial sensor

Alignment process

Ideal navigation frame n

Mathematical platform frame n998400

(b) SINS

Figure 1 Alignment process for INS

where 119881119864and 119881

119873are the east and north velocity respectively

120596119894119890and 119877 are the rotational angular rate and radius of the

Earth 119871 is the latitude of ship position 119891119864 119891119873 and 119891

119880

denote the projection of the accelerometer measured data f119887in navigation frame 119899 and 119879

119894119895(119894 119895 = 1 2 3) are the elements

of the direct cosine matrix (DCM) of MINS

22 Measurement Equation The differences of velocity andyaw between MINS and SINS are used as measurement datafor data fusion Then the measurement vector is as

Z = [119881119864minus 119881119872119864

119881119873

minus 119881119872119873

119884119864minus 119884119872119864

]

119879

(4)

where 119881119864 119881119873and 119884

119864are the east north velocity and yaw

from SINS respectively 119881119872119864

119881119872119873

and 119884119872119864

are fromMINS respectively

The measurement equation can be constructed as

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

whereH(119905) is measurement matrix and V(119905) is measurementnoise matrix According to the relationship betweenmeasurement and state vectors the measurement matrixH(119905) can be expressed as

H =

[

[

[

[

[

[

[

[

1 0 0 0 0

0 1 0 0 0 03times3

0 0 minus

1198791211987932

1198792

12+ 1198792

22

minus

1198792211987932

1198792

12+ 1198792

22

minus1

]

]

]

]

]

]

]

]

(6)

23 Observability Degree of Each State Variable After theanalysis of the model combined with (2) and (5) by PWCStheory we can conclude that all of the eight selected variablesare observable but the observability degrees are differentfrom each other [13] The simulation results in Section 52indicate that the horizontal velocity errors and yaw mis-alignment converge immediately that the estimation forhorizontal misalignment angles lasts for 10sim20 s and thatthe estimation for gyro bias lasts for 3sim5min which meansthat the degrees of velocity errors and yaw misalignment

are the highest those of horizontal misalignment angles aremoderate and those of gyro bias are the lowest

The reasons which cause the different degrees of errorsare as follows The gyro bias needs integration to be reflectedon misalignment angles and the horizontal misalignmentangles need projection reflected on acceleration Furtherintegration of acceleration gyro bias can be reflected onvelocity errors In this process from gyro bias to velocityerrors integrating operations are needed twice and fromhorizontal misalignment angles to velocity errors only oneis needed When velocity errors and yaw misalignment areselected as components of a measurement vector there isno doubt that observability degrees of gyro bias are lowestwhile those of velocity errors and yaw misalignment arehighest which are determined by the mechanism of transferalignment To shorten the estimation time for gyro bias someother novel ways should be sought

3 A New Way to Speed UpSINS Alignment

31 Analysis of Alignment Process in INS An acceptedviewpoint is that in navigation sensor data is of real-timesignificance Based on initial navigation parameters INSobtains the attitude velocity and position by the integrationof sensor data After integration the sensor data can bediscarded

Take the alignment process of PINS as an example Asshown in Figure 1(a) 1198991015840 frame denotes the physical platformand 119899 frame denotes the ideal navigation frame In PINSinertial sensors are installed on the physical platform Inertialsensors measure the ship motion in 119899

1015840 frame and navigationresolution is also executed in this frame however referencedata are from 119899 frame Before aligning misalignment anglesbetween 119899

1015840 and 119899 frames will cause the differences betweennavigation resolution values in 119899

1015840 frame and reference val-ues in 119899 frame Because these differences can reflect themisalignments transfer alignment can be finished with themodel in Section 2 In the alignment process with theabove differences caused by misalignment angles physicalplatform can be adjusted to coincide with the ideal frameIn other words in PINS sensor data reflect the magnitudeof misalignment angle and other errors and there is direct

4 Mathematical Problems in Engineering

relationship between sensor data and misalignment anglesso the sensor data are of real-time significance And theadjustment for physical platform to get new sensor data is atime-consuming process

At the same time in SINS the alignment method isderived from that of PINS and so the alignment must bea time-consuming process for the need of real-time sensordata and the adjustment of platform But in INS as far as theprocess of alignment is concerned maybe sensor data is ofreal-time significance for PINS but not for SINS

In SINS as shown in Figure 1(b) amathematical platformreplaces the physical platform in PINS and inertial sensorsare directly installed on the ship Here the ship body isnamed 119887 frame and the mathematical platform is named 119899

1015840

frame In alignment the adjusting process of the 1198991015840 frame

is the same with that in PINS while the difference is thatthe calculated sensor data in 119899

1015840 frame is the projection ofmeasured sensor data from 119887 frame Only the projectiondata in 119899

1015840 frame reflect the misalignment angles between 1198991015840

and 119899 frames but sensor data cannot The calculated andmeasured sensor data are connected by calculated DCM C119899

1015840

119887

In comparison with that in PINS the direct relationshipbetween sensor data and misalignment angles is severedby the mathematical platform Then the 119899

1015840 frame can beadjusted constantly by the repeated calculation for DCM C119899

1015840

119887

with a same set of sensor data which means that alignmenttime can be shortened without too much time to be spent insampling real-time sensor data

The above set of data can be seen as that when the shipkeeps an ideal static state and there are no sensor errors allmeasured sensor data are equal and can be dealt with as aseries of single data In engineering because of the errorsfrom sensor and reference data the update frequency ofreference data and so forth a set of sensor data and referencedata should be used for repeated SINS resolution (addedbackward-forward resolution) and repeated data fusion toimprove the alignment accuracy

Also different from PINS SINS is a digital system inwhich there is only numerical data no mass spring andresistance So the adjustment for mathematical platform canbe as fast as lightning

32 Backward-Forward SINS Resolution The above analysisindicates that the alignment of SINS can be fulfilled with aset of same data which brings a new problemmdashthe way touse these data

In SINS with initial attitude velocity and positionthe navigation resolution is the real-time updating processfor navigation parameters with sensor data by integratedcalculation In this process ship moves from the origin tothe end In backward-forward SINS resolution as shown inFigure 2 backward SINS resolution is the process in whichship moves from the end to the originmdasha reverse process ofnormal navigation and forward resolution is that from theorigin to the endmdasha repeated process of normal navigationFrom the basic SINS resolution algorithm the deduction ofbackward-forward SINS algorithm is shown as follows

Backward

Forward

Normal

t0 tn tm

t01

t02

t0i

t0n

tn1

tn2

tni

tnn

middot middot middot

Figure 2 The process of backward-forward SINS resolution

321 Normal Resolution Algorithm in SINS The navigationresolution algorithm is [14]

C119899119887= C119899119887(120596119887

119899119887times)

V119899 = C119899119887f119887 minus (2120596

119899

119894119890+ 120596119899

119890119899) times V119899 + g119899

=

119881119899

119873

119877

120582 =

119881119899

119864sec 119871119877

(7)

where 120596119887119899119887

= 120596119887

119894119887minus (C119899

119887)119879

(120596119899

119894119890+ 120596119899

119890119899) 120596119899119890119899

=

[minus119881119899

119873119877 119881

119899

119864119877 (119881

119899

119864tan 119871)119877]

119879 A119863119861119862

(such as 120596119899119894119890) denotes

the projection of a motion vector A which means the relativemotion from 119862 frame to 119861 frame in119863 frame C119899

119887is the DCM

V119899 = [119881119899

119864119881119899

119873119881119899

119880] is the velocity vector 119871 and 120582 are the

latitude and longitude respectively and (sdottimes) denotes theantisymmetric matrix of the vector ldquosdotrdquo

For the recursive update with computer formulas (7)must be translated into discrete form in a certain samplingcycle and navigation resolution update cycle For a shipwhich always undergoes a low dynamical movement a singlesampling resolution algorithm can be adopted which meansonly one sample of sensor data per navigation cycle [15ndash17]Supposing the above cycle times are both equal to 119879

119904 then

the discrete form can be expressed as follows [14]

C119899119887119896

= C119899119887119896minus1

(I + 119879119904120596119887

119899119887119896times)

V119899119896= V119899119896minus1

+119879119904[C119899119887119896minus1

f119887119896minus1

minus (2120596119899

119894119890119896minus1+ 120596119899

119890119899119896minus1) times V119899119896minus1

+ g119899]

119871119896= 119871119896minus1

+

119879119904119881119873119896minus1

119877119872

+ ℎ119896minus1

120582119896= 120582119896minus1

+

119879119904119881119864119896minus1

sec119871119896minus1

119877119873

+ ℎ119896minus1

(8)

where 119896 denotes recursive number Without considerationof calculation error formulas (8) are composed of the basicupdate equations for navigation resolution

322 Backward Resolution Algorithm in SINS As shown inFigure 2 in a backward resolution process a ship needs toreturn from the end to the origin So in the normal process

Mathematical Problems in Engineering 5

all sensor data must be stored According to the formulas (8)the resolution process can be expressed as follows

C119899119887119896minus1

= C119899119887119896(I + 119879

119904120596119887

119899119887119896times)

minus1

asymp C119899119887119896

(I minus 119879119904120596119887

119899119887119896times) asymp C119899

119887119896(I minus 119879

119904120596119887

119899119887119896minus1times)

V119899119896minus1

= V119899119896minus 119879119904[C119899119887119896minus1

f119887119896minus1

minus (2120596119899

119894119890119896minus1+ 120596119899

119890119899119896minus1) times V119899119896minus1

+ g119899]

asymp V119899119896minus 119879119904[C119899119887119896f119887119896minus (2120596

119899

119894119890119896+ 120596119899

119890119899119896) times V119899119896+ g119899]

119871119896minus1

= 119871119896minus

119879119904119881119873119896minus1

119877

asymp 119871119896minus

119879119904119881119873119896

119877

120582119896minus1

= 120582119896minus

119879119904119881119864119896minus1

sec119871119896minus1

119877

asymp 120582119896minus

119879119904119881119864119896sec 119871119896

119877

(9)

where 119896 is reduced from 119899 to 0 As shown in Figure 2 1199051198991

isboth the starting point in the period 119905

1198991sim 11990501and the ending

point in the period 1199050

sim 119905119899 Take the attitude velocity and

position at 119905119899as the initial attitude velocity and position at

1199051198991 and the backward resolution can be fully realized In the

period of 1199050

sim 119905119899and 1199051198991

sim 11990501 with the same recursive

number 119896 the ship maintains the same attitude velocity andposition and maintains the same acceleration but oppositein direction Some errors are induced by the approximationin formulas (9) and all these errors can be ignored whenthe resolution cycle is short enough The simulation inSection 323 proves that the above approximation is effective

After the backward resolution a forward resolutionshould bemade in order for the ship to return from the originto the end In the forward resolution formulas (8) can beused

323 Simulation on Backward-Forward SINS Resolution Thesensor errors are listed inTable 1 Andwe assume that the shipis in a bad-moderate sea condition [18] and the ship swingingparameters are listed in Table 2 Ship initial attitudes are set as0 and the ship is assumed without linear motion and locatednorth latitude 32∘ and east longitude 118∘ Sensor samplingand navigation resolution cycle 119879

119904is set as 10ms

Simulation results are shown in Figure 3 in which (a)(b) and (c) show the resolution results of pitch east velocityand latitude The dot and solid lines denote the simulationwith no sensor error and with sensor error respectivelyThe simulation is divided into three stages firstly in the1199050sim 119905119899period sensor data are measured and stored normal

navigation is resolved and this period lasts for 1 s secondly inthe 1199051198991

sim 11990501period backward resolution is run and finally in

the 11990502

sim 1199051198992period forward resolution is run In the last two

stages the time consumed is determined by the performanceof computer

In Figure 3without consideration of calculating the errorthe ship can move from the end to the origin and move fromorigin to end with the backward and forward resolutionsrespectively

But the above processes also indicate that with onlybackward or forward or backward-forward SINS resolutionno new information will be generated

Table 1 Sensor errors

Gyro bias Acce biasConstant Random Constant Random

119909 05∘h 05∘h 500 120583g 500 120583g119910 05∘h 05∘h 500 120583g 500 120583g119911 05∘h 05∘h 500 120583g 500 120583g

Table 2 Swinging parameters

Pitch Roll YawAmplitude (∘) 9 12 14Cycle (s) 8 10 6

4 Rapid Transfer Alignment Based on theAdded Resolution and Data Fusion

The analysis in Section 31 indicates that alignment for SINScan be fulfilled with a set of data and in this section wecombine a Kalman filter with the added backward-forwardSINS resolution aiming to shorten the alignment time

As shown in Figure 4 transfer alignment model basedon Kalman filter introduced in Section 2 and a close-loopcorrection method are used in this rapid transfer alignmentalgorithm Close-loop correction means that after datafusion the new estimation for velocity errors and misalign-ment angles will be fed to SINS to revise those correspondingparameters and new estimation for gyro bias will be setas new compensation value to participate in the followingnavigation resolutions In other words after data fusion themathematical platform C119899

1015840

119887will be adjusted For a ship the

update frequency of the reference data from MINS is lowerthan that of SINS navigation resolution 119879

119904and 119879

119899are set

as update cycle of navigation resolution and reference datarespectively Also 119879

119899is set as the added backward-forward

SINS resolution cycleTwo transfer alignment methods are compared and the

estimation processes of gyro bias are schematically demon-strated in Figure 5 in which (a) and (b) indicate these twomethods respectively which are all based on the principleas shown in Figure 4 In the first one the added backward-forward SINS resolution and data fusion are not used whilethey are used in the second In Figure 5 the dot-dashed linesdenote the real gyro bias while the solid line denotes theestimation of gyro bias With the Kalman filter and matchingmethod introduced in Section 2 the estimation of gyro biasis a slow process which will be convergent towards real gyrobias after a long time

As shown in Figure 5(a) when the reference data isavailable such as at the point 119905

119899 data fusion is executed and

a new estimate for the state vector will be produced With thenew estimation for misalignment angles and velocity errorsinitial navigation parameters will be reset for the next periodsuch as 119905

119899sim 1199052119899 whichmeans that themathematical platform

1198991015840 will be adjusted and the new compensation value for gyro

bias will also be reset In this method along with time datafusion is run only once at every reference data update cycle

6 Mathematical Problems in Engineering

0

5

10

minus10

minus5

tn(tn1)t0 tn2t01(t02)

Ang

le (∘

)

(a) Pitch

Velo

city

(ms

)

0

005

minus005tn(tn1)t0 tn2t01(t02)

(b) East velocity

32

32

32

tn(tn1)t0 tn2t01(t02)

Ang

le (∘

)

(c) Latitude

Figure 3 Simulation of normal and backward-forward SINS resolution

Kalman filter

SINS

MINS Velocity + yaw

Velocity + yaw

State variables

Navigation parameters

Figure 4 Kalman filter

Estim

atio

n fo

rgy

ro b

ias

Data fusion

tt2n t3n t4nt0 tn

middot middot middot

(a) Without added backward-forward SINS resolution and datafusion

Estim

atio

n fo

rgy

ro b

ias

Data fusion

tt2n t3n t4nt0 tn

t01

t02

tn1

tn2

middot middot middot

(b) With added backward-forward SINS resolution and data fusion

Figure 5 Estimation process for gyro bias

But as shown in Figure 5(b) in the same reference dataupdate cycle such as 119905

0sim 119905119899 at the point 119905

119899 data fusion

will be run and new estimation will be produced andso will new initial navigation parameters And new initialparameters and new compensation value for gyro bias willproduce the new navigation parameters at the point 119905

01

which are different from those at 1199050 So a new estimate will

be generated at the point 11990501 which is different from that at

1199050 because the measurement vectors for the Kalman filter are

different at points 1199050and 11990501 which is caused by the same

reference data but different navigation parameters Similarlynew information will be got at the point 119905

1198992 which is different

from that at 119905119899 In Figure 5(b) with the added backward-

forward SINS resolution and data fusion the estimatingoperations for gyro bias and the adjustment for mathematicalplatform 119899

1015840 will be done with two more times In the second

method even with same observability degrees as in thefirst one the estimation time will be shortened because theestimation frequency is increased

There is no doubt that the added resolution and datafusion in Figure 5(b) will increase the burden of naviga-tion computer Though in the last decades the processingpowering of the employed microprocessors has dramaticallyincreased it is difficult in some way to complete a largecomputation in a relatively short period

As shown in Figure 6 in this rapid transfer alignmentmethod three tasks should be completed within one refer-ence data update cycle (1) Inertial sensor data needs to besampled and stored and navigation resolution should be run(2) Reference data needs to be sampled and stored and datafusion should be run (3) Backward and forward calculationsand data fusions should be executed The first task should be

Mathematical Problems in Engineering 7

executed at every navigation update cycle The last two tasksshould be run in the last navigation update cycle of everyreference data update cycle whichmeans that the above threetasks should be finished in 119879

119904 Otherwise the first task in the

first navigation cycle of the next reference data cycle will becompromised It is difficult to finish these three tasks in 119879

119904

with a computer of limited performanceHowever this problem can be resolved by the full use

of resources of high speed computers with the support ofreal-time multitasking operation system (RTOS) such asVxWorks In VxWorks the above three tasks can be set withdifferent priorities The first can be run preferentially whenTask 2 or Task 3 is being run whichmeans that the first task ofthe next reference data cycle can be run preferentially when

Tasks 2 and 3 of the previous reference data cycle are beingexecuted In this method the idle resources of CPU in thenext reference cycle can be used for the tasks 2 and 3 of theprevious cycle The paper is not involved in the programs inRTOS in detail

5 Simulation

51 Parameters for Simulation The ship moving parametersare set as in Section 323 The ideal velocity and yaw ofthe ship are used as reference data from MINS after whitenoise is added The variance of the white noise is set as[(04ms)2 (04ms)2 (03

∘)

2

] The sensor errors are listedin Table 1

The parameters for Kalman filter are

X0= [0 0 0 0 0 0 0 0]

119879

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

∘)

2

(15∘)

2

(15∘)

2

(15∘h)2 (15

∘h)2 (15

∘h)2]

Q = diag [(500 ug)2 (500 ug)2 (05∘)2

(05∘)2

(05∘)2

0 0 0]

R = diag [(04ms)2 (04ms)2 (03∘)

2

]

(10)

Two data fusion schemes were compared The secondscheme is a transfer alignment method with the addedbackward-forward SINS resolution and data fusion while thefirst scheme is not In these two schemes the same transferalignment model introduced in Section 2 is used

The update cycle of sensor data and navigation resolutionis set as 10ms and that of reference data from MINS is as1 s As analyzed in Section 4 in every reference data cyclenavigation resolution will be executed 300 times and datafusion 3 times in scheme 2 while in scheme 1 navigationresolution is executed 100 times and data fusion only once

In the ldquovelocity plus yawrdquo matching method accelerom-eter bias is unobservable so with the sensor errors assumedin Table 1 the limit alignment accuracy of pitch and roll areminus04990 mrad and 04990 mrad respectively and the limitalignment accuracy of yaw is 0

52 Simulation Results The simulation lasts for 500 s andthe simulation results are stored once per second The mis-alignment angle curves are shown in Figure 7The estimationof velocity error curves are in Figure 8 and the estimationcurves of gyros bias in Figure 9 In Figures 7ndash9 the dot-dashand solid lines denote the simulation results of scheme 1 andscheme 2 respectively The dotted lines in Figures 7 and 8are the limit alignment accuracy of misalignment angles andvelocity And the dotted lines in Figure 9 denote the settingvalue of constant gyro bias

The curves in Figure 7 show that either in scheme 1or scheme 2 misalignment angles can be estimated rapidlyand are oscillating in small amplitudes with the swinging

frequency of ship But the tendency of misalignment curvesespecially that of roll ones indicates that estimation speed isslightly higher in scheme 2 than in scheme 1 The statisticaldata about misalignment angles are shown in Table 3 and thestatistical results show that the alignment accuracy of thesetwo schemes is roughly equal after 60 sThe curves in Figure 8indicate that the estimation speed and accuracy for velocityerror are roughly equal in scheme 1 and scheme 2

The curves in Figure 9 show that in scheme 2 theestimation curves of gyro bias converge towards the settingvalues at about 50 s and the oscillating amplitudes are verysmall in 100 s while in scheme 1 200 s is needed for theconvergence of gyro bias and relatively large oscillations stillexist even after 300 s The statistical mean values shown inTable 4 indicate that in scheme 2 in the period from 50 sto 100 s about 96 and 93 and 72 of gyro bias can beestimated along 119909 119910 and 119911 axes respectively And about99 97 and 78 can be estimated in the period from 100 sto 150 s

6 Conclusion

Two viewpoints are given in this paper The first is that inSINS mathematical platform cuts off the direct relationshipbetween sensor data and misalignment angles which meansthat initial alignment can be fulfilled by the repeatedlyresolution on a same set of sensor data The second isthat with the added backward-forward SINS resolution andrepeated data fusion on the corresponding resolution resultsand the external data the alignment time can be greatlyreduced

8 Mathematical Problems in Engineering

Table 3 Statistical results for misalignment angles

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

1sim10 sScheme 1 minus08225 10610 11833 49672 minus09375 12756Scheme 2 04631 16724 02318 07429 minus03956 10700

11sim20 sScheme 1 minus02278 05136 minus00705 05156 minus11170 07199Scheme 2 minus04993 05865 04806 02647 minus04755 08203

31sim40 sScheme 1 minus04776 04653 09794 05194 00448 09782Scheme 2 minus06454 04553 02271 04726 minus00067 08820

41sim50 sScheme 1 minus08024 04560 10111 05086 minus03714 09385Scheme 2 minus08563 04557 04980 06148 minus03609 08828

51sim60 sScheme 1 minus05430 04296 05377 04614 minus02183 09682Scheme 2 minus07073 04854 05415 05531 02041 09117

61sim500 sScheme 1 minus05075 04498 04154 04852 minus00877 09168Scheme 2 minus06541 04505 04815 04831 00709 09011

Limited accuracy minus04990 04990 0

Table 4 Statistical results for gyro bias

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

51sim100 sScheme 1 minus04139 04563 minus07990 09143 07591 04807Scheme 2 05172 00757 05335 01329 06390 01588

101sim150 sScheme 1 03474 02102 minus03793 04709 19998 03182Scheme 2 05037 00505 04847 00499 06072 00628

151sim200 sScheme 1 05742 00632 02499 01254 08496 02778Scheme 2 04943 00382 04703 00284 05196 00256

201sim250 sScheme 1 04790 00283 02713 00744 05335 00648Scheme 2 04961 00224 04718 00231 05255 00224

251sim300 sScheme 1 05189 00344 04203 00630 05292 00530Scheme 2 05019 00217 04752 00258 05113 00236

301sim500 sScheme 1 05156 00272 04122 00410 05683 00412Scheme 2 04990 00116 04823 00169 05134 00129

Setting value 05 05 05

With the above two viewpoints (1) a backward-forwardSINS resolution algorithm is designed in detail and simu-lation results indicate that a ship can move from the endto the origin with the backward resolution and move fromthe origin to the end with the forward (2) A rapid transfer

alignment algorithm is also designed in detail in which thebackward-forward resolution and two more operations forthe estimation of gyro bias are added in one reference dataupdate cycle In addition the correctness of its algorithm isproved by simulation The simulation result produced with

Mathematical Problems in Engineering 9

middot middot middot

13

11 2

(1) Sensor data sampling storing and navigation resolution(2) Reference data sampling storing and data fusion(3) Reverse and forward calculationlowastCPU idle

TsTs Ts

Ts Ts

tnt2nlowast lowast

Tn Tn

t0

tn2

tn1t01

t02

middot middot middot

Figure 6 Calculation process for alignment program

50 100 150 200 250 300 350 400 450 500

Erro

r (m

rad)

012

Scheme 2Scheme 1

minus3minus2minus1

t (s)

(a) Pitch

Erro

r (m

rad)

50 100 150 200 250 300 350 400 450 500

3 Scheme 2Scheme 1

012

minus2minus1

t (s)

(b) Roll

50 100 150 200 250 300 350 400 450 500

024

Erro

r (m

rad) Scheme 2Scheme 1

minus2minus4

t (s)

(c) Yaw

Figure 7 Estimation curves for misalignment angles

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(a) East velocity

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(b) North velocity

Figure 8 Estimation curves for velocity errors

50 100 150 200 250 300 350 400 450 500

13 Scheme 1Scheme 2

minus3

minus1

Bias

(∘h

)

t (s)

(a) 119909 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 113

minus3

minus1

Bias

(∘h

)

t (s)

(b) 119910 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 1

13

minus3

minus1

Bias

(∘h

)

t (s)

(c) 119911 gyro

Figure 9 Estimation curves for gyro bias

the method in this paper indicates that the alignment timeis reduced from 300 s to 100 s compared with that of thetransfer alignment method without the added backward-forward SINS resolution and data fusion

One problem which may be encountered in engineeringapplications with this transfer alignment method is studiedand a possible solution is given in which RTOS is introducedto distribute computer resources coordinately

Acknowledgments

This work was supported in part by the National NaturalScience Foundation (61004125 61273056) and Chinese uni-versity research and operation expenses (10420525)

References

[1] J Kain and J Cloutier ldquoRapid transfer alignment for tacticalweapon applicationsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 1290ndash1300 BostonMass USA 1989

[2] K Spalding ldquoAn efficient rapid transfer alignment filterrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 1276ndash1286 Hilton Head Island SC USA 1992

[3] K Shortelle and W Graham ldquoAdvanced alignment conceptsfor precision-guided weaponsrdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 131ndash142Anaheim Calif USA January 1995

[4] W Graham and K Shortelle ldquoAdvanced transfer alignmentfor inertial navigators (A-train)rdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 113ndash124Anaheim Calif USA January 1995

[5] K J Shortelle W R Graham and C Rabourn ldquoF-16 flight testsof a rapid transfer alignment procedurerdquo in Proceedings of the

10 Mathematical Problems in Engineering

IEEEPosition Location andNavigation Symposium pp 379ndash386Palm Springs Calif USA April 1996

[6] P D Groves ldquoOptimising the transfer alignment of weaponINSrdquo Journal of Navigation vol 56 no 2 pp 323ndash335 2003

[7] DGoshen-Meskin and I Y Bar-Itzhack ldquoObservability analysisof piece-wise constant systems II Application to inertialnavigation in-flight alignment (military applications)rdquo IEEETransactions on Aerospace and Electronic Systems vol 28 no4 pp 1068ndash1075 1992

[8] X H Cheng D J Wan and X Zhong ldquoStudy on observabilityand its degree of strapdown inertial navigation systemrdquo Journalof Southeast University vol 37 no 6 pp 6ndash10 1997

[9] L-H Zhu X-H Cheng and Y-Y He ldquoFilter convergencecriterion in transfer alignmentrdquo Journal of Chinese InertialTechnology vol 19 no 3 pp 277ndash285 2011

[10] S Wang Z Wang Y Zhu and B Wang ldquoMonitoring onship hull deformation and correction for heading and attitudeinformationrdquo Journal of Chinese Inertial Technology vol 15 no6 pp 635ndash641 2007

[11] D R Tarrant C Jones and D Lin ldquoRapid and robust transferalignmentrdquo in Proceedings of the 1st IEEE Regional Conferenceon Aerospace Control Systems pp 758ndash762 1993

[12] J Lyou and Y C Lim ldquoTransfer alignment considering mea-surement time delay and ship body flexurerdquo Journal of Mechan-ical Science and Technology vol 23 no 1 pp 195ndash203 2009

[13] B-C Zhou X-H Cheng and D-J Liu ldquoAnalysis method ofobservable degree based on spectral decomposition in SINStransfer alignmentrdquo Journal of Chinese Inertial Technology vol18 no 5 pp 518ndash522 2010

[14] D H Titterton and J L Weston Strapdown Inertial NavigationTechnology Lavenham Press Ltd London UK 2nd edition2004

[15] R B Miller ldquoA new strapdown attitude algorithmrdquo Journal ofGuidance Control and Dynamics vol 6 no 4 pp 287ndash2911983

[16] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998

[17] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 2 velocity and position algorithmsrdquo Journalof Guidance Control and Dynamics vol 21 no 2 pp 208ndash2211998

[18] P J Gates and N M Lynn Ships Submarines and the SeaBrasseyrsquos London UK 1990

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

relationship between sensor data and misalignment anglesso the sensor data are of real-time significance And theadjustment for physical platform to get new sensor data is atime-consuming process

At the same time in SINS the alignment method isderived from that of PINS and so the alignment must bea time-consuming process for the need of real-time sensordata and the adjustment of platform But in INS as far as theprocess of alignment is concerned maybe sensor data is ofreal-time significance for PINS but not for SINS

In SINS as shown in Figure 1(b) amathematical platformreplaces the physical platform in PINS and inertial sensorsare directly installed on the ship Here the ship body isnamed 119887 frame and the mathematical platform is named 119899

1015840

frame In alignment the adjusting process of the 1198991015840 frame

is the same with that in PINS while the difference is thatthe calculated sensor data in 119899

1015840 frame is the projection ofmeasured sensor data from 119887 frame Only the projectiondata in 119899

1015840 frame reflect the misalignment angles between 1198991015840

and 119899 frames but sensor data cannot The calculated andmeasured sensor data are connected by calculated DCM C119899

1015840

119887

In comparison with that in PINS the direct relationshipbetween sensor data and misalignment angles is severedby the mathematical platform Then the 119899

1015840 frame can beadjusted constantly by the repeated calculation for DCM C119899

1015840

119887

with a same set of sensor data which means that alignmenttime can be shortened without too much time to be spent insampling real-time sensor data

The above set of data can be seen as that when the shipkeeps an ideal static state and there are no sensor errors allmeasured sensor data are equal and can be dealt with as aseries of single data In engineering because of the errorsfrom sensor and reference data the update frequency ofreference data and so forth a set of sensor data and referencedata should be used for repeated SINS resolution (addedbackward-forward resolution) and repeated data fusion toimprove the alignment accuracy

Also different from PINS SINS is a digital system inwhich there is only numerical data no mass spring andresistance So the adjustment for mathematical platform canbe as fast as lightning

32 Backward-Forward SINS Resolution The above analysisindicates that the alignment of SINS can be fulfilled with aset of same data which brings a new problemmdashthe way touse these data

In SINS with initial attitude velocity and positionthe navigation resolution is the real-time updating processfor navigation parameters with sensor data by integratedcalculation In this process ship moves from the origin tothe end In backward-forward SINS resolution as shown inFigure 2 backward SINS resolution is the process in whichship moves from the end to the originmdasha reverse process ofnormal navigation and forward resolution is that from theorigin to the endmdasha repeated process of normal navigationFrom the basic SINS resolution algorithm the deduction ofbackward-forward SINS algorithm is shown as follows

Backward

Forward

Normal

t0 tn tm

t01

t02

t0i

t0n

tn1

tn2

tni

tnn

middot middot middot

Figure 2 The process of backward-forward SINS resolution

321 Normal Resolution Algorithm in SINS The navigationresolution algorithm is [14]

C119899119887= C119899119887(120596119887

119899119887times)

V119899 = C119899119887f119887 minus (2120596

119899

119894119890+ 120596119899

119890119899) times V119899 + g119899

=

119881119899

119873

119877

120582 =

119881119899

119864sec 119871119877

(7)

where 120596119887119899119887

= 120596119887

119894119887minus (C119899

119887)119879

(120596119899

119894119890+ 120596119899

119890119899) 120596119899119890119899

=

[minus119881119899

119873119877 119881

119899

119864119877 (119881

119899

119864tan 119871)119877]

119879 A119863119861119862

(such as 120596119899119894119890) denotes

the projection of a motion vector A which means the relativemotion from 119862 frame to 119861 frame in119863 frame C119899

119887is the DCM

V119899 = [119881119899

119864119881119899

119873119881119899

119880] is the velocity vector 119871 and 120582 are the

latitude and longitude respectively and (sdottimes) denotes theantisymmetric matrix of the vector ldquosdotrdquo

For the recursive update with computer formulas (7)must be translated into discrete form in a certain samplingcycle and navigation resolution update cycle For a shipwhich always undergoes a low dynamical movement a singlesampling resolution algorithm can be adopted which meansonly one sample of sensor data per navigation cycle [15ndash17]Supposing the above cycle times are both equal to 119879

119904 then

the discrete form can be expressed as follows [14]

C119899119887119896

= C119899119887119896minus1

(I + 119879119904120596119887

119899119887119896times)

V119899119896= V119899119896minus1

+119879119904[C119899119887119896minus1

f119887119896minus1

minus (2120596119899

119894119890119896minus1+ 120596119899

119890119899119896minus1) times V119899119896minus1

+ g119899]

119871119896= 119871119896minus1

+

119879119904119881119873119896minus1

119877119872

+ ℎ119896minus1

120582119896= 120582119896minus1

+

119879119904119881119864119896minus1

sec119871119896minus1

119877119873

+ ℎ119896minus1

(8)

where 119896 denotes recursive number Without considerationof calculation error formulas (8) are composed of the basicupdate equations for navigation resolution

322 Backward Resolution Algorithm in SINS As shown inFigure 2 in a backward resolution process a ship needs toreturn from the end to the origin So in the normal process

Mathematical Problems in Engineering 5

all sensor data must be stored According to the formulas (8)the resolution process can be expressed as follows

C119899119887119896minus1

= C119899119887119896(I + 119879

119904120596119887

119899119887119896times)

minus1

asymp C119899119887119896

(I minus 119879119904120596119887

119899119887119896times) asymp C119899

119887119896(I minus 119879

119904120596119887

119899119887119896minus1times)

V119899119896minus1

= V119899119896minus 119879119904[C119899119887119896minus1

f119887119896minus1

minus (2120596119899

119894119890119896minus1+ 120596119899

119890119899119896minus1) times V119899119896minus1

+ g119899]

asymp V119899119896minus 119879119904[C119899119887119896f119887119896minus (2120596

119899

119894119890119896+ 120596119899

119890119899119896) times V119899119896+ g119899]

119871119896minus1

= 119871119896minus

119879119904119881119873119896minus1

119877

asymp 119871119896minus

119879119904119881119873119896

119877

120582119896minus1

= 120582119896minus

119879119904119881119864119896minus1

sec119871119896minus1

119877

asymp 120582119896minus

119879119904119881119864119896sec 119871119896

119877

(9)

where 119896 is reduced from 119899 to 0 As shown in Figure 2 1199051198991

isboth the starting point in the period 119905

1198991sim 11990501and the ending

point in the period 1199050

sim 119905119899 Take the attitude velocity and

position at 119905119899as the initial attitude velocity and position at

1199051198991 and the backward resolution can be fully realized In the

period of 1199050

sim 119905119899and 1199051198991

sim 11990501 with the same recursive

number 119896 the ship maintains the same attitude velocity andposition and maintains the same acceleration but oppositein direction Some errors are induced by the approximationin formulas (9) and all these errors can be ignored whenthe resolution cycle is short enough The simulation inSection 323 proves that the above approximation is effective

After the backward resolution a forward resolutionshould bemade in order for the ship to return from the originto the end In the forward resolution formulas (8) can beused

323 Simulation on Backward-Forward SINS Resolution Thesensor errors are listed inTable 1 Andwe assume that the shipis in a bad-moderate sea condition [18] and the ship swingingparameters are listed in Table 2 Ship initial attitudes are set as0 and the ship is assumed without linear motion and locatednorth latitude 32∘ and east longitude 118∘ Sensor samplingand navigation resolution cycle 119879

119904is set as 10ms

Simulation results are shown in Figure 3 in which (a)(b) and (c) show the resolution results of pitch east velocityand latitude The dot and solid lines denote the simulationwith no sensor error and with sensor error respectivelyThe simulation is divided into three stages firstly in the1199050sim 119905119899period sensor data are measured and stored normal

navigation is resolved and this period lasts for 1 s secondly inthe 1199051198991

sim 11990501period backward resolution is run and finally in

the 11990502

sim 1199051198992period forward resolution is run In the last two

stages the time consumed is determined by the performanceof computer

In Figure 3without consideration of calculating the errorthe ship can move from the end to the origin and move fromorigin to end with the backward and forward resolutionsrespectively

But the above processes also indicate that with onlybackward or forward or backward-forward SINS resolutionno new information will be generated

Table 1 Sensor errors

Gyro bias Acce biasConstant Random Constant Random

119909 05∘h 05∘h 500 120583g 500 120583g119910 05∘h 05∘h 500 120583g 500 120583g119911 05∘h 05∘h 500 120583g 500 120583g

Table 2 Swinging parameters

Pitch Roll YawAmplitude (∘) 9 12 14Cycle (s) 8 10 6

4 Rapid Transfer Alignment Based on theAdded Resolution and Data Fusion

The analysis in Section 31 indicates that alignment for SINScan be fulfilled with a set of data and in this section wecombine a Kalman filter with the added backward-forwardSINS resolution aiming to shorten the alignment time

As shown in Figure 4 transfer alignment model basedon Kalman filter introduced in Section 2 and a close-loopcorrection method are used in this rapid transfer alignmentalgorithm Close-loop correction means that after datafusion the new estimation for velocity errors and misalign-ment angles will be fed to SINS to revise those correspondingparameters and new estimation for gyro bias will be setas new compensation value to participate in the followingnavigation resolutions In other words after data fusion themathematical platform C119899

1015840

119887will be adjusted For a ship the

update frequency of the reference data from MINS is lowerthan that of SINS navigation resolution 119879

119904and 119879

119899are set

as update cycle of navigation resolution and reference datarespectively Also 119879

119899is set as the added backward-forward

SINS resolution cycleTwo transfer alignment methods are compared and the

estimation processes of gyro bias are schematically demon-strated in Figure 5 in which (a) and (b) indicate these twomethods respectively which are all based on the principleas shown in Figure 4 In the first one the added backward-forward SINS resolution and data fusion are not used whilethey are used in the second In Figure 5 the dot-dashed linesdenote the real gyro bias while the solid line denotes theestimation of gyro bias With the Kalman filter and matchingmethod introduced in Section 2 the estimation of gyro biasis a slow process which will be convergent towards real gyrobias after a long time

As shown in Figure 5(a) when the reference data isavailable such as at the point 119905

119899 data fusion is executed and

a new estimate for the state vector will be produced With thenew estimation for misalignment angles and velocity errorsinitial navigation parameters will be reset for the next periodsuch as 119905

119899sim 1199052119899 whichmeans that themathematical platform

1198991015840 will be adjusted and the new compensation value for gyro

bias will also be reset In this method along with time datafusion is run only once at every reference data update cycle

6 Mathematical Problems in Engineering

0

5

10

minus10

minus5

tn(tn1)t0 tn2t01(t02)

Ang

le (∘

)

(a) Pitch

Velo

city

(ms

)

0

005

minus005tn(tn1)t0 tn2t01(t02)

(b) East velocity

32

32

32

tn(tn1)t0 tn2t01(t02)

Ang

le (∘

)

(c) Latitude

Figure 3 Simulation of normal and backward-forward SINS resolution

Kalman filter

SINS

MINS Velocity + yaw

Velocity + yaw

State variables

Navigation parameters

Figure 4 Kalman filter

Estim

atio

n fo

rgy

ro b

ias

Data fusion

tt2n t3n t4nt0 tn

middot middot middot

(a) Without added backward-forward SINS resolution and datafusion

Estim

atio

n fo

rgy

ro b

ias

Data fusion

tt2n t3n t4nt0 tn

t01

t02

tn1

tn2

middot middot middot

(b) With added backward-forward SINS resolution and data fusion

Figure 5 Estimation process for gyro bias

But as shown in Figure 5(b) in the same reference dataupdate cycle such as 119905

0sim 119905119899 at the point 119905

119899 data fusion

will be run and new estimation will be produced andso will new initial navigation parameters And new initialparameters and new compensation value for gyro bias willproduce the new navigation parameters at the point 119905

01

which are different from those at 1199050 So a new estimate will

be generated at the point 11990501 which is different from that at

1199050 because the measurement vectors for the Kalman filter are

different at points 1199050and 11990501 which is caused by the same

reference data but different navigation parameters Similarlynew information will be got at the point 119905

1198992 which is different

from that at 119905119899 In Figure 5(b) with the added backward-

forward SINS resolution and data fusion the estimatingoperations for gyro bias and the adjustment for mathematicalplatform 119899

1015840 will be done with two more times In the second

method even with same observability degrees as in thefirst one the estimation time will be shortened because theestimation frequency is increased

There is no doubt that the added resolution and datafusion in Figure 5(b) will increase the burden of naviga-tion computer Though in the last decades the processingpowering of the employed microprocessors has dramaticallyincreased it is difficult in some way to complete a largecomputation in a relatively short period

As shown in Figure 6 in this rapid transfer alignmentmethod three tasks should be completed within one refer-ence data update cycle (1) Inertial sensor data needs to besampled and stored and navigation resolution should be run(2) Reference data needs to be sampled and stored and datafusion should be run (3) Backward and forward calculationsand data fusions should be executed The first task should be

Mathematical Problems in Engineering 7

executed at every navigation update cycle The last two tasksshould be run in the last navigation update cycle of everyreference data update cycle whichmeans that the above threetasks should be finished in 119879

119904 Otherwise the first task in the

first navigation cycle of the next reference data cycle will becompromised It is difficult to finish these three tasks in 119879

119904

with a computer of limited performanceHowever this problem can be resolved by the full use

of resources of high speed computers with the support ofreal-time multitasking operation system (RTOS) such asVxWorks In VxWorks the above three tasks can be set withdifferent priorities The first can be run preferentially whenTask 2 or Task 3 is being run whichmeans that the first task ofthe next reference data cycle can be run preferentially when

Tasks 2 and 3 of the previous reference data cycle are beingexecuted In this method the idle resources of CPU in thenext reference cycle can be used for the tasks 2 and 3 of theprevious cycle The paper is not involved in the programs inRTOS in detail

5 Simulation

51 Parameters for Simulation The ship moving parametersare set as in Section 323 The ideal velocity and yaw ofthe ship are used as reference data from MINS after whitenoise is added The variance of the white noise is set as[(04ms)2 (04ms)2 (03

∘)

2

] The sensor errors are listedin Table 1

The parameters for Kalman filter are

X0= [0 0 0 0 0 0 0 0]

119879

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

∘)

2

(15∘)

2

(15∘)

2

(15∘h)2 (15

∘h)2 (15

∘h)2]

Q = diag [(500 ug)2 (500 ug)2 (05∘)2

(05∘)2

(05∘)2

0 0 0]

R = diag [(04ms)2 (04ms)2 (03∘)

2

]

(10)

Two data fusion schemes were compared The secondscheme is a transfer alignment method with the addedbackward-forward SINS resolution and data fusion while thefirst scheme is not In these two schemes the same transferalignment model introduced in Section 2 is used

The update cycle of sensor data and navigation resolutionis set as 10ms and that of reference data from MINS is as1 s As analyzed in Section 4 in every reference data cyclenavigation resolution will be executed 300 times and datafusion 3 times in scheme 2 while in scheme 1 navigationresolution is executed 100 times and data fusion only once

In the ldquovelocity plus yawrdquo matching method accelerom-eter bias is unobservable so with the sensor errors assumedin Table 1 the limit alignment accuracy of pitch and roll areminus04990 mrad and 04990 mrad respectively and the limitalignment accuracy of yaw is 0

52 Simulation Results The simulation lasts for 500 s andthe simulation results are stored once per second The mis-alignment angle curves are shown in Figure 7The estimationof velocity error curves are in Figure 8 and the estimationcurves of gyros bias in Figure 9 In Figures 7ndash9 the dot-dashand solid lines denote the simulation results of scheme 1 andscheme 2 respectively The dotted lines in Figures 7 and 8are the limit alignment accuracy of misalignment angles andvelocity And the dotted lines in Figure 9 denote the settingvalue of constant gyro bias

The curves in Figure 7 show that either in scheme 1or scheme 2 misalignment angles can be estimated rapidlyand are oscillating in small amplitudes with the swinging

frequency of ship But the tendency of misalignment curvesespecially that of roll ones indicates that estimation speed isslightly higher in scheme 2 than in scheme 1 The statisticaldata about misalignment angles are shown in Table 3 and thestatistical results show that the alignment accuracy of thesetwo schemes is roughly equal after 60 sThe curves in Figure 8indicate that the estimation speed and accuracy for velocityerror are roughly equal in scheme 1 and scheme 2

The curves in Figure 9 show that in scheme 2 theestimation curves of gyro bias converge towards the settingvalues at about 50 s and the oscillating amplitudes are verysmall in 100 s while in scheme 1 200 s is needed for theconvergence of gyro bias and relatively large oscillations stillexist even after 300 s The statistical mean values shown inTable 4 indicate that in scheme 2 in the period from 50 sto 100 s about 96 and 93 and 72 of gyro bias can beestimated along 119909 119910 and 119911 axes respectively And about99 97 and 78 can be estimated in the period from 100 sto 150 s

6 Conclusion

Two viewpoints are given in this paper The first is that inSINS mathematical platform cuts off the direct relationshipbetween sensor data and misalignment angles which meansthat initial alignment can be fulfilled by the repeatedlyresolution on a same set of sensor data The second isthat with the added backward-forward SINS resolution andrepeated data fusion on the corresponding resolution resultsand the external data the alignment time can be greatlyreduced

8 Mathematical Problems in Engineering

Table 3 Statistical results for misalignment angles

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

1sim10 sScheme 1 minus08225 10610 11833 49672 minus09375 12756Scheme 2 04631 16724 02318 07429 minus03956 10700

11sim20 sScheme 1 minus02278 05136 minus00705 05156 minus11170 07199Scheme 2 minus04993 05865 04806 02647 minus04755 08203

31sim40 sScheme 1 minus04776 04653 09794 05194 00448 09782Scheme 2 minus06454 04553 02271 04726 minus00067 08820

41sim50 sScheme 1 minus08024 04560 10111 05086 minus03714 09385Scheme 2 minus08563 04557 04980 06148 minus03609 08828

51sim60 sScheme 1 minus05430 04296 05377 04614 minus02183 09682Scheme 2 minus07073 04854 05415 05531 02041 09117

61sim500 sScheme 1 minus05075 04498 04154 04852 minus00877 09168Scheme 2 minus06541 04505 04815 04831 00709 09011

Limited accuracy minus04990 04990 0

Table 4 Statistical results for gyro bias

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

51sim100 sScheme 1 minus04139 04563 minus07990 09143 07591 04807Scheme 2 05172 00757 05335 01329 06390 01588

101sim150 sScheme 1 03474 02102 minus03793 04709 19998 03182Scheme 2 05037 00505 04847 00499 06072 00628

151sim200 sScheme 1 05742 00632 02499 01254 08496 02778Scheme 2 04943 00382 04703 00284 05196 00256

201sim250 sScheme 1 04790 00283 02713 00744 05335 00648Scheme 2 04961 00224 04718 00231 05255 00224

251sim300 sScheme 1 05189 00344 04203 00630 05292 00530Scheme 2 05019 00217 04752 00258 05113 00236

301sim500 sScheme 1 05156 00272 04122 00410 05683 00412Scheme 2 04990 00116 04823 00169 05134 00129

Setting value 05 05 05

With the above two viewpoints (1) a backward-forwardSINS resolution algorithm is designed in detail and simu-lation results indicate that a ship can move from the endto the origin with the backward resolution and move fromthe origin to the end with the forward (2) A rapid transfer

alignment algorithm is also designed in detail in which thebackward-forward resolution and two more operations forthe estimation of gyro bias are added in one reference dataupdate cycle In addition the correctness of its algorithm isproved by simulation The simulation result produced with

Mathematical Problems in Engineering 9

middot middot middot

13

11 2

(1) Sensor data sampling storing and navigation resolution(2) Reference data sampling storing and data fusion(3) Reverse and forward calculationlowastCPU idle

TsTs Ts

Ts Ts

tnt2nlowast lowast

Tn Tn

t0

tn2

tn1t01

t02

middot middot middot

Figure 6 Calculation process for alignment program

50 100 150 200 250 300 350 400 450 500

Erro

r (m

rad)

012

Scheme 2Scheme 1

minus3minus2minus1

t (s)

(a) Pitch

Erro

r (m

rad)

50 100 150 200 250 300 350 400 450 500

3 Scheme 2Scheme 1

012

minus2minus1

t (s)

(b) Roll

50 100 150 200 250 300 350 400 450 500

024

Erro

r (m

rad) Scheme 2Scheme 1

minus2minus4

t (s)

(c) Yaw

Figure 7 Estimation curves for misalignment angles

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(a) East velocity

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(b) North velocity

Figure 8 Estimation curves for velocity errors

50 100 150 200 250 300 350 400 450 500

13 Scheme 1Scheme 2

minus3

minus1

Bias

(∘h

)

t (s)

(a) 119909 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 113

minus3

minus1

Bias

(∘h

)

t (s)

(b) 119910 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 1

13

minus3

minus1

Bias

(∘h

)

t (s)

(c) 119911 gyro

Figure 9 Estimation curves for gyro bias

the method in this paper indicates that the alignment timeis reduced from 300 s to 100 s compared with that of thetransfer alignment method without the added backward-forward SINS resolution and data fusion

One problem which may be encountered in engineeringapplications with this transfer alignment method is studiedand a possible solution is given in which RTOS is introducedto distribute computer resources coordinately

Acknowledgments

This work was supported in part by the National NaturalScience Foundation (61004125 61273056) and Chinese uni-versity research and operation expenses (10420525)

References

[1] J Kain and J Cloutier ldquoRapid transfer alignment for tacticalweapon applicationsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 1290ndash1300 BostonMass USA 1989

[2] K Spalding ldquoAn efficient rapid transfer alignment filterrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 1276ndash1286 Hilton Head Island SC USA 1992

[3] K Shortelle and W Graham ldquoAdvanced alignment conceptsfor precision-guided weaponsrdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 131ndash142Anaheim Calif USA January 1995

[4] W Graham and K Shortelle ldquoAdvanced transfer alignmentfor inertial navigators (A-train)rdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 113ndash124Anaheim Calif USA January 1995

[5] K J Shortelle W R Graham and C Rabourn ldquoF-16 flight testsof a rapid transfer alignment procedurerdquo in Proceedings of the

10 Mathematical Problems in Engineering

IEEEPosition Location andNavigation Symposium pp 379ndash386Palm Springs Calif USA April 1996

[6] P D Groves ldquoOptimising the transfer alignment of weaponINSrdquo Journal of Navigation vol 56 no 2 pp 323ndash335 2003

[7] DGoshen-Meskin and I Y Bar-Itzhack ldquoObservability analysisof piece-wise constant systems II Application to inertialnavigation in-flight alignment (military applications)rdquo IEEETransactions on Aerospace and Electronic Systems vol 28 no4 pp 1068ndash1075 1992

[8] X H Cheng D J Wan and X Zhong ldquoStudy on observabilityand its degree of strapdown inertial navigation systemrdquo Journalof Southeast University vol 37 no 6 pp 6ndash10 1997

[9] L-H Zhu X-H Cheng and Y-Y He ldquoFilter convergencecriterion in transfer alignmentrdquo Journal of Chinese InertialTechnology vol 19 no 3 pp 277ndash285 2011

[10] S Wang Z Wang Y Zhu and B Wang ldquoMonitoring onship hull deformation and correction for heading and attitudeinformationrdquo Journal of Chinese Inertial Technology vol 15 no6 pp 635ndash641 2007

[11] D R Tarrant C Jones and D Lin ldquoRapid and robust transferalignmentrdquo in Proceedings of the 1st IEEE Regional Conferenceon Aerospace Control Systems pp 758ndash762 1993

[12] J Lyou and Y C Lim ldquoTransfer alignment considering mea-surement time delay and ship body flexurerdquo Journal of Mechan-ical Science and Technology vol 23 no 1 pp 195ndash203 2009

[13] B-C Zhou X-H Cheng and D-J Liu ldquoAnalysis method ofobservable degree based on spectral decomposition in SINStransfer alignmentrdquo Journal of Chinese Inertial Technology vol18 no 5 pp 518ndash522 2010

[14] D H Titterton and J L Weston Strapdown Inertial NavigationTechnology Lavenham Press Ltd London UK 2nd edition2004

[15] R B Miller ldquoA new strapdown attitude algorithmrdquo Journal ofGuidance Control and Dynamics vol 6 no 4 pp 287ndash2911983

[16] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998

[17] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 2 velocity and position algorithmsrdquo Journalof Guidance Control and Dynamics vol 21 no 2 pp 208ndash2211998

[18] P J Gates and N M Lynn Ships Submarines and the SeaBrasseyrsquos London UK 1990

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Mathematical PhysicsAdvances in

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

all sensor data must be stored According to the formulas (8)the resolution process can be expressed as follows

C119899119887119896minus1

= C119899119887119896(I + 119879

119904120596119887

119899119887119896times)

minus1

asymp C119899119887119896

(I minus 119879119904120596119887

119899119887119896times) asymp C119899

119887119896(I minus 119879

119904120596119887

119899119887119896minus1times)

V119899119896minus1

= V119899119896minus 119879119904[C119899119887119896minus1

f119887119896minus1

minus (2120596119899

119894119890119896minus1+ 120596119899

119890119899119896minus1) times V119899119896minus1

+ g119899]

asymp V119899119896minus 119879119904[C119899119887119896f119887119896minus (2120596

119899

119894119890119896+ 120596119899

119890119899119896) times V119899119896+ g119899]

119871119896minus1

= 119871119896minus

119879119904119881119873119896minus1

119877

asymp 119871119896minus

119879119904119881119873119896

119877

120582119896minus1

= 120582119896minus

119879119904119881119864119896minus1

sec119871119896minus1

119877

asymp 120582119896minus

119879119904119881119864119896sec 119871119896

119877

(9)

where 119896 is reduced from 119899 to 0 As shown in Figure 2 1199051198991

isboth the starting point in the period 119905

1198991sim 11990501and the ending

point in the period 1199050

sim 119905119899 Take the attitude velocity and

position at 119905119899as the initial attitude velocity and position at

1199051198991 and the backward resolution can be fully realized In the

period of 1199050

sim 119905119899and 1199051198991

sim 11990501 with the same recursive

number 119896 the ship maintains the same attitude velocity andposition and maintains the same acceleration but oppositein direction Some errors are induced by the approximationin formulas (9) and all these errors can be ignored whenthe resolution cycle is short enough The simulation inSection 323 proves that the above approximation is effective

After the backward resolution a forward resolutionshould bemade in order for the ship to return from the originto the end In the forward resolution formulas (8) can beused

323 Simulation on Backward-Forward SINS Resolution Thesensor errors are listed inTable 1 Andwe assume that the shipis in a bad-moderate sea condition [18] and the ship swingingparameters are listed in Table 2 Ship initial attitudes are set as0 and the ship is assumed without linear motion and locatednorth latitude 32∘ and east longitude 118∘ Sensor samplingand navigation resolution cycle 119879

119904is set as 10ms

Simulation results are shown in Figure 3 in which (a)(b) and (c) show the resolution results of pitch east velocityand latitude The dot and solid lines denote the simulationwith no sensor error and with sensor error respectivelyThe simulation is divided into three stages firstly in the1199050sim 119905119899period sensor data are measured and stored normal

navigation is resolved and this period lasts for 1 s secondly inthe 1199051198991

sim 11990501period backward resolution is run and finally in

the 11990502

sim 1199051198992period forward resolution is run In the last two

stages the time consumed is determined by the performanceof computer

In Figure 3without consideration of calculating the errorthe ship can move from the end to the origin and move fromorigin to end with the backward and forward resolutionsrespectively

But the above processes also indicate that with onlybackward or forward or backward-forward SINS resolutionno new information will be generated

Table 1 Sensor errors

Gyro bias Acce biasConstant Random Constant Random

119909 05∘h 05∘h 500 120583g 500 120583g119910 05∘h 05∘h 500 120583g 500 120583g119911 05∘h 05∘h 500 120583g 500 120583g

Table 2 Swinging parameters

Pitch Roll YawAmplitude (∘) 9 12 14Cycle (s) 8 10 6

4 Rapid Transfer Alignment Based on theAdded Resolution and Data Fusion

The analysis in Section 31 indicates that alignment for SINScan be fulfilled with a set of data and in this section wecombine a Kalman filter with the added backward-forwardSINS resolution aiming to shorten the alignment time

As shown in Figure 4 transfer alignment model basedon Kalman filter introduced in Section 2 and a close-loopcorrection method are used in this rapid transfer alignmentalgorithm Close-loop correction means that after datafusion the new estimation for velocity errors and misalign-ment angles will be fed to SINS to revise those correspondingparameters and new estimation for gyro bias will be setas new compensation value to participate in the followingnavigation resolutions In other words after data fusion themathematical platform C119899

1015840

119887will be adjusted For a ship the

update frequency of the reference data from MINS is lowerthan that of SINS navigation resolution 119879

119904and 119879

119899are set

as update cycle of navigation resolution and reference datarespectively Also 119879

119899is set as the added backward-forward

SINS resolution cycleTwo transfer alignment methods are compared and the

estimation processes of gyro bias are schematically demon-strated in Figure 5 in which (a) and (b) indicate these twomethods respectively which are all based on the principleas shown in Figure 4 In the first one the added backward-forward SINS resolution and data fusion are not used whilethey are used in the second In Figure 5 the dot-dashed linesdenote the real gyro bias while the solid line denotes theestimation of gyro bias With the Kalman filter and matchingmethod introduced in Section 2 the estimation of gyro biasis a slow process which will be convergent towards real gyrobias after a long time

As shown in Figure 5(a) when the reference data isavailable such as at the point 119905

119899 data fusion is executed and

a new estimate for the state vector will be produced With thenew estimation for misalignment angles and velocity errorsinitial navigation parameters will be reset for the next periodsuch as 119905

119899sim 1199052119899 whichmeans that themathematical platform

1198991015840 will be adjusted and the new compensation value for gyro

bias will also be reset In this method along with time datafusion is run only once at every reference data update cycle

6 Mathematical Problems in Engineering

0

5

10

minus10

minus5

tn(tn1)t0 tn2t01(t02)

Ang

le (∘

)

(a) Pitch

Velo

city

(ms

)

0

005

minus005tn(tn1)t0 tn2t01(t02)

(b) East velocity

32

32

32

tn(tn1)t0 tn2t01(t02)

Ang

le (∘

)

(c) Latitude

Figure 3 Simulation of normal and backward-forward SINS resolution

Kalman filter

SINS

MINS Velocity + yaw

Velocity + yaw

State variables

Navigation parameters

Figure 4 Kalman filter

Estim

atio

n fo

rgy

ro b

ias

Data fusion

tt2n t3n t4nt0 tn

middot middot middot

(a) Without added backward-forward SINS resolution and datafusion

Estim

atio

n fo

rgy

ro b

ias

Data fusion

tt2n t3n t4nt0 tn

t01

t02

tn1

tn2

middot middot middot

(b) With added backward-forward SINS resolution and data fusion

Figure 5 Estimation process for gyro bias

But as shown in Figure 5(b) in the same reference dataupdate cycle such as 119905

0sim 119905119899 at the point 119905

119899 data fusion

will be run and new estimation will be produced andso will new initial navigation parameters And new initialparameters and new compensation value for gyro bias willproduce the new navigation parameters at the point 119905

01

which are different from those at 1199050 So a new estimate will

be generated at the point 11990501 which is different from that at

1199050 because the measurement vectors for the Kalman filter are

different at points 1199050and 11990501 which is caused by the same

reference data but different navigation parameters Similarlynew information will be got at the point 119905

1198992 which is different

from that at 119905119899 In Figure 5(b) with the added backward-

forward SINS resolution and data fusion the estimatingoperations for gyro bias and the adjustment for mathematicalplatform 119899

1015840 will be done with two more times In the second

method even with same observability degrees as in thefirst one the estimation time will be shortened because theestimation frequency is increased

There is no doubt that the added resolution and datafusion in Figure 5(b) will increase the burden of naviga-tion computer Though in the last decades the processingpowering of the employed microprocessors has dramaticallyincreased it is difficult in some way to complete a largecomputation in a relatively short period

As shown in Figure 6 in this rapid transfer alignmentmethod three tasks should be completed within one refer-ence data update cycle (1) Inertial sensor data needs to besampled and stored and navigation resolution should be run(2) Reference data needs to be sampled and stored and datafusion should be run (3) Backward and forward calculationsand data fusions should be executed The first task should be

Mathematical Problems in Engineering 7

executed at every navigation update cycle The last two tasksshould be run in the last navigation update cycle of everyreference data update cycle whichmeans that the above threetasks should be finished in 119879

119904 Otherwise the first task in the

first navigation cycle of the next reference data cycle will becompromised It is difficult to finish these three tasks in 119879

119904

with a computer of limited performanceHowever this problem can be resolved by the full use

of resources of high speed computers with the support ofreal-time multitasking operation system (RTOS) such asVxWorks In VxWorks the above three tasks can be set withdifferent priorities The first can be run preferentially whenTask 2 or Task 3 is being run whichmeans that the first task ofthe next reference data cycle can be run preferentially when

Tasks 2 and 3 of the previous reference data cycle are beingexecuted In this method the idle resources of CPU in thenext reference cycle can be used for the tasks 2 and 3 of theprevious cycle The paper is not involved in the programs inRTOS in detail

5 Simulation

51 Parameters for Simulation The ship moving parametersare set as in Section 323 The ideal velocity and yaw ofthe ship are used as reference data from MINS after whitenoise is added The variance of the white noise is set as[(04ms)2 (04ms)2 (03

∘)

2

] The sensor errors are listedin Table 1

The parameters for Kalman filter are

X0= [0 0 0 0 0 0 0 0]

119879

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

∘)

2

(15∘)

2

(15∘)

2

(15∘h)2 (15

∘h)2 (15

∘h)2]

Q = diag [(500 ug)2 (500 ug)2 (05∘)2

(05∘)2

(05∘)2

0 0 0]

R = diag [(04ms)2 (04ms)2 (03∘)

2

]

(10)

Two data fusion schemes were compared The secondscheme is a transfer alignment method with the addedbackward-forward SINS resolution and data fusion while thefirst scheme is not In these two schemes the same transferalignment model introduced in Section 2 is used

The update cycle of sensor data and navigation resolutionis set as 10ms and that of reference data from MINS is as1 s As analyzed in Section 4 in every reference data cyclenavigation resolution will be executed 300 times and datafusion 3 times in scheme 2 while in scheme 1 navigationresolution is executed 100 times and data fusion only once

In the ldquovelocity plus yawrdquo matching method accelerom-eter bias is unobservable so with the sensor errors assumedin Table 1 the limit alignment accuracy of pitch and roll areminus04990 mrad and 04990 mrad respectively and the limitalignment accuracy of yaw is 0

52 Simulation Results The simulation lasts for 500 s andthe simulation results are stored once per second The mis-alignment angle curves are shown in Figure 7The estimationof velocity error curves are in Figure 8 and the estimationcurves of gyros bias in Figure 9 In Figures 7ndash9 the dot-dashand solid lines denote the simulation results of scheme 1 andscheme 2 respectively The dotted lines in Figures 7 and 8are the limit alignment accuracy of misalignment angles andvelocity And the dotted lines in Figure 9 denote the settingvalue of constant gyro bias

The curves in Figure 7 show that either in scheme 1or scheme 2 misalignment angles can be estimated rapidlyand are oscillating in small amplitudes with the swinging

frequency of ship But the tendency of misalignment curvesespecially that of roll ones indicates that estimation speed isslightly higher in scheme 2 than in scheme 1 The statisticaldata about misalignment angles are shown in Table 3 and thestatistical results show that the alignment accuracy of thesetwo schemes is roughly equal after 60 sThe curves in Figure 8indicate that the estimation speed and accuracy for velocityerror are roughly equal in scheme 1 and scheme 2

The curves in Figure 9 show that in scheme 2 theestimation curves of gyro bias converge towards the settingvalues at about 50 s and the oscillating amplitudes are verysmall in 100 s while in scheme 1 200 s is needed for theconvergence of gyro bias and relatively large oscillations stillexist even after 300 s The statistical mean values shown inTable 4 indicate that in scheme 2 in the period from 50 sto 100 s about 96 and 93 and 72 of gyro bias can beestimated along 119909 119910 and 119911 axes respectively And about99 97 and 78 can be estimated in the period from 100 sto 150 s

6 Conclusion

Two viewpoints are given in this paper The first is that inSINS mathematical platform cuts off the direct relationshipbetween sensor data and misalignment angles which meansthat initial alignment can be fulfilled by the repeatedlyresolution on a same set of sensor data The second isthat with the added backward-forward SINS resolution andrepeated data fusion on the corresponding resolution resultsand the external data the alignment time can be greatlyreduced

8 Mathematical Problems in Engineering

Table 3 Statistical results for misalignment angles

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

1sim10 sScheme 1 minus08225 10610 11833 49672 minus09375 12756Scheme 2 04631 16724 02318 07429 minus03956 10700

11sim20 sScheme 1 minus02278 05136 minus00705 05156 minus11170 07199Scheme 2 minus04993 05865 04806 02647 minus04755 08203

31sim40 sScheme 1 minus04776 04653 09794 05194 00448 09782Scheme 2 minus06454 04553 02271 04726 minus00067 08820

41sim50 sScheme 1 minus08024 04560 10111 05086 minus03714 09385Scheme 2 minus08563 04557 04980 06148 minus03609 08828

51sim60 sScheme 1 minus05430 04296 05377 04614 minus02183 09682Scheme 2 minus07073 04854 05415 05531 02041 09117

61sim500 sScheme 1 minus05075 04498 04154 04852 minus00877 09168Scheme 2 minus06541 04505 04815 04831 00709 09011

Limited accuracy minus04990 04990 0

Table 4 Statistical results for gyro bias

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

51sim100 sScheme 1 minus04139 04563 minus07990 09143 07591 04807Scheme 2 05172 00757 05335 01329 06390 01588

101sim150 sScheme 1 03474 02102 minus03793 04709 19998 03182Scheme 2 05037 00505 04847 00499 06072 00628

151sim200 sScheme 1 05742 00632 02499 01254 08496 02778Scheme 2 04943 00382 04703 00284 05196 00256

201sim250 sScheme 1 04790 00283 02713 00744 05335 00648Scheme 2 04961 00224 04718 00231 05255 00224

251sim300 sScheme 1 05189 00344 04203 00630 05292 00530Scheme 2 05019 00217 04752 00258 05113 00236

301sim500 sScheme 1 05156 00272 04122 00410 05683 00412Scheme 2 04990 00116 04823 00169 05134 00129

Setting value 05 05 05

With the above two viewpoints (1) a backward-forwardSINS resolution algorithm is designed in detail and simu-lation results indicate that a ship can move from the endto the origin with the backward resolution and move fromthe origin to the end with the forward (2) A rapid transfer

alignment algorithm is also designed in detail in which thebackward-forward resolution and two more operations forthe estimation of gyro bias are added in one reference dataupdate cycle In addition the correctness of its algorithm isproved by simulation The simulation result produced with

Mathematical Problems in Engineering 9

middot middot middot

13

11 2

(1) Sensor data sampling storing and navigation resolution(2) Reference data sampling storing and data fusion(3) Reverse and forward calculationlowastCPU idle

TsTs Ts

Ts Ts

tnt2nlowast lowast

Tn Tn

t0

tn2

tn1t01

t02

middot middot middot

Figure 6 Calculation process for alignment program

50 100 150 200 250 300 350 400 450 500

Erro

r (m

rad)

012

Scheme 2Scheme 1

minus3minus2minus1

t (s)

(a) Pitch

Erro

r (m

rad)

50 100 150 200 250 300 350 400 450 500

3 Scheme 2Scheme 1

012

minus2minus1

t (s)

(b) Roll

50 100 150 200 250 300 350 400 450 500

024

Erro

r (m

rad) Scheme 2Scheme 1

minus2minus4

t (s)

(c) Yaw

Figure 7 Estimation curves for misalignment angles

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(a) East velocity

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(b) North velocity

Figure 8 Estimation curves for velocity errors

50 100 150 200 250 300 350 400 450 500

13 Scheme 1Scheme 2

minus3

minus1

Bias

(∘h

)

t (s)

(a) 119909 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 113

minus3

minus1

Bias

(∘h

)

t (s)

(b) 119910 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 1

13

minus3

minus1

Bias

(∘h

)

t (s)

(c) 119911 gyro

Figure 9 Estimation curves for gyro bias

the method in this paper indicates that the alignment timeis reduced from 300 s to 100 s compared with that of thetransfer alignment method without the added backward-forward SINS resolution and data fusion

One problem which may be encountered in engineeringapplications with this transfer alignment method is studiedand a possible solution is given in which RTOS is introducedto distribute computer resources coordinately

Acknowledgments

This work was supported in part by the National NaturalScience Foundation (61004125 61273056) and Chinese uni-versity research and operation expenses (10420525)

References

[1] J Kain and J Cloutier ldquoRapid transfer alignment for tacticalweapon applicationsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 1290ndash1300 BostonMass USA 1989

[2] K Spalding ldquoAn efficient rapid transfer alignment filterrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 1276ndash1286 Hilton Head Island SC USA 1992

[3] K Shortelle and W Graham ldquoAdvanced alignment conceptsfor precision-guided weaponsrdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 131ndash142Anaheim Calif USA January 1995

[4] W Graham and K Shortelle ldquoAdvanced transfer alignmentfor inertial navigators (A-train)rdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 113ndash124Anaheim Calif USA January 1995

[5] K J Shortelle W R Graham and C Rabourn ldquoF-16 flight testsof a rapid transfer alignment procedurerdquo in Proceedings of the

10 Mathematical Problems in Engineering

IEEEPosition Location andNavigation Symposium pp 379ndash386Palm Springs Calif USA April 1996

[6] P D Groves ldquoOptimising the transfer alignment of weaponINSrdquo Journal of Navigation vol 56 no 2 pp 323ndash335 2003

[7] DGoshen-Meskin and I Y Bar-Itzhack ldquoObservability analysisof piece-wise constant systems II Application to inertialnavigation in-flight alignment (military applications)rdquo IEEETransactions on Aerospace and Electronic Systems vol 28 no4 pp 1068ndash1075 1992

[8] X H Cheng D J Wan and X Zhong ldquoStudy on observabilityand its degree of strapdown inertial navigation systemrdquo Journalof Southeast University vol 37 no 6 pp 6ndash10 1997

[9] L-H Zhu X-H Cheng and Y-Y He ldquoFilter convergencecriterion in transfer alignmentrdquo Journal of Chinese InertialTechnology vol 19 no 3 pp 277ndash285 2011

[10] S Wang Z Wang Y Zhu and B Wang ldquoMonitoring onship hull deformation and correction for heading and attitudeinformationrdquo Journal of Chinese Inertial Technology vol 15 no6 pp 635ndash641 2007

[11] D R Tarrant C Jones and D Lin ldquoRapid and robust transferalignmentrdquo in Proceedings of the 1st IEEE Regional Conferenceon Aerospace Control Systems pp 758ndash762 1993

[12] J Lyou and Y C Lim ldquoTransfer alignment considering mea-surement time delay and ship body flexurerdquo Journal of Mechan-ical Science and Technology vol 23 no 1 pp 195ndash203 2009

[13] B-C Zhou X-H Cheng and D-J Liu ldquoAnalysis method ofobservable degree based on spectral decomposition in SINStransfer alignmentrdquo Journal of Chinese Inertial Technology vol18 no 5 pp 518ndash522 2010

[14] D H Titterton and J L Weston Strapdown Inertial NavigationTechnology Lavenham Press Ltd London UK 2nd edition2004

[15] R B Miller ldquoA new strapdown attitude algorithmrdquo Journal ofGuidance Control and Dynamics vol 6 no 4 pp 287ndash2911983

[16] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998

[17] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 2 velocity and position algorithmsrdquo Journalof Guidance Control and Dynamics vol 21 no 2 pp 208ndash2211998

[18] P J Gates and N M Lynn Ships Submarines and the SeaBrasseyrsquos London UK 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

0

5

10

minus10

minus5

tn(tn1)t0 tn2t01(t02)

Ang

le (∘

)

(a) Pitch

Velo

city

(ms

)

0

005

minus005tn(tn1)t0 tn2t01(t02)

(b) East velocity

32

32

32

tn(tn1)t0 tn2t01(t02)

Ang

le (∘

)

(c) Latitude

Figure 3 Simulation of normal and backward-forward SINS resolution

Kalman filter

SINS

MINS Velocity + yaw

Velocity + yaw

State variables

Navigation parameters

Figure 4 Kalman filter

Estim

atio

n fo

rgy

ro b

ias

Data fusion

tt2n t3n t4nt0 tn

middot middot middot

(a) Without added backward-forward SINS resolution and datafusion

Estim

atio

n fo

rgy

ro b

ias

Data fusion

tt2n t3n t4nt0 tn

t01

t02

tn1

tn2

middot middot middot

(b) With added backward-forward SINS resolution and data fusion

Figure 5 Estimation process for gyro bias

But as shown in Figure 5(b) in the same reference dataupdate cycle such as 119905

0sim 119905119899 at the point 119905

119899 data fusion

will be run and new estimation will be produced andso will new initial navigation parameters And new initialparameters and new compensation value for gyro bias willproduce the new navigation parameters at the point 119905

01

which are different from those at 1199050 So a new estimate will

be generated at the point 11990501 which is different from that at

1199050 because the measurement vectors for the Kalman filter are

different at points 1199050and 11990501 which is caused by the same

reference data but different navigation parameters Similarlynew information will be got at the point 119905

1198992 which is different

from that at 119905119899 In Figure 5(b) with the added backward-

forward SINS resolution and data fusion the estimatingoperations for gyro bias and the adjustment for mathematicalplatform 119899

1015840 will be done with two more times In the second

method even with same observability degrees as in thefirst one the estimation time will be shortened because theestimation frequency is increased

There is no doubt that the added resolution and datafusion in Figure 5(b) will increase the burden of naviga-tion computer Though in the last decades the processingpowering of the employed microprocessors has dramaticallyincreased it is difficult in some way to complete a largecomputation in a relatively short period

As shown in Figure 6 in this rapid transfer alignmentmethod three tasks should be completed within one refer-ence data update cycle (1) Inertial sensor data needs to besampled and stored and navigation resolution should be run(2) Reference data needs to be sampled and stored and datafusion should be run (3) Backward and forward calculationsand data fusions should be executed The first task should be

Mathematical Problems in Engineering 7

executed at every navigation update cycle The last two tasksshould be run in the last navigation update cycle of everyreference data update cycle whichmeans that the above threetasks should be finished in 119879

119904 Otherwise the first task in the

first navigation cycle of the next reference data cycle will becompromised It is difficult to finish these three tasks in 119879

119904

with a computer of limited performanceHowever this problem can be resolved by the full use

of resources of high speed computers with the support ofreal-time multitasking operation system (RTOS) such asVxWorks In VxWorks the above three tasks can be set withdifferent priorities The first can be run preferentially whenTask 2 or Task 3 is being run whichmeans that the first task ofthe next reference data cycle can be run preferentially when

Tasks 2 and 3 of the previous reference data cycle are beingexecuted In this method the idle resources of CPU in thenext reference cycle can be used for the tasks 2 and 3 of theprevious cycle The paper is not involved in the programs inRTOS in detail

5 Simulation

51 Parameters for Simulation The ship moving parametersare set as in Section 323 The ideal velocity and yaw ofthe ship are used as reference data from MINS after whitenoise is added The variance of the white noise is set as[(04ms)2 (04ms)2 (03

∘)

2

] The sensor errors are listedin Table 1

The parameters for Kalman filter are

X0= [0 0 0 0 0 0 0 0]

119879

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

∘)

2

(15∘)

2

(15∘)

2

(15∘h)2 (15

∘h)2 (15

∘h)2]

Q = diag [(500 ug)2 (500 ug)2 (05∘)2

(05∘)2

(05∘)2

0 0 0]

R = diag [(04ms)2 (04ms)2 (03∘)

2

]

(10)

Two data fusion schemes were compared The secondscheme is a transfer alignment method with the addedbackward-forward SINS resolution and data fusion while thefirst scheme is not In these two schemes the same transferalignment model introduced in Section 2 is used

The update cycle of sensor data and navigation resolutionis set as 10ms and that of reference data from MINS is as1 s As analyzed in Section 4 in every reference data cyclenavigation resolution will be executed 300 times and datafusion 3 times in scheme 2 while in scheme 1 navigationresolution is executed 100 times and data fusion only once

In the ldquovelocity plus yawrdquo matching method accelerom-eter bias is unobservable so with the sensor errors assumedin Table 1 the limit alignment accuracy of pitch and roll areminus04990 mrad and 04990 mrad respectively and the limitalignment accuracy of yaw is 0

52 Simulation Results The simulation lasts for 500 s andthe simulation results are stored once per second The mis-alignment angle curves are shown in Figure 7The estimationof velocity error curves are in Figure 8 and the estimationcurves of gyros bias in Figure 9 In Figures 7ndash9 the dot-dashand solid lines denote the simulation results of scheme 1 andscheme 2 respectively The dotted lines in Figures 7 and 8are the limit alignment accuracy of misalignment angles andvelocity And the dotted lines in Figure 9 denote the settingvalue of constant gyro bias

The curves in Figure 7 show that either in scheme 1or scheme 2 misalignment angles can be estimated rapidlyand are oscillating in small amplitudes with the swinging

frequency of ship But the tendency of misalignment curvesespecially that of roll ones indicates that estimation speed isslightly higher in scheme 2 than in scheme 1 The statisticaldata about misalignment angles are shown in Table 3 and thestatistical results show that the alignment accuracy of thesetwo schemes is roughly equal after 60 sThe curves in Figure 8indicate that the estimation speed and accuracy for velocityerror are roughly equal in scheme 1 and scheme 2

The curves in Figure 9 show that in scheme 2 theestimation curves of gyro bias converge towards the settingvalues at about 50 s and the oscillating amplitudes are verysmall in 100 s while in scheme 1 200 s is needed for theconvergence of gyro bias and relatively large oscillations stillexist even after 300 s The statistical mean values shown inTable 4 indicate that in scheme 2 in the period from 50 sto 100 s about 96 and 93 and 72 of gyro bias can beestimated along 119909 119910 and 119911 axes respectively And about99 97 and 78 can be estimated in the period from 100 sto 150 s

6 Conclusion

Two viewpoints are given in this paper The first is that inSINS mathematical platform cuts off the direct relationshipbetween sensor data and misalignment angles which meansthat initial alignment can be fulfilled by the repeatedlyresolution on a same set of sensor data The second isthat with the added backward-forward SINS resolution andrepeated data fusion on the corresponding resolution resultsand the external data the alignment time can be greatlyreduced

8 Mathematical Problems in Engineering

Table 3 Statistical results for misalignment angles

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

1sim10 sScheme 1 minus08225 10610 11833 49672 minus09375 12756Scheme 2 04631 16724 02318 07429 minus03956 10700

11sim20 sScheme 1 minus02278 05136 minus00705 05156 minus11170 07199Scheme 2 minus04993 05865 04806 02647 minus04755 08203

31sim40 sScheme 1 minus04776 04653 09794 05194 00448 09782Scheme 2 minus06454 04553 02271 04726 minus00067 08820

41sim50 sScheme 1 minus08024 04560 10111 05086 minus03714 09385Scheme 2 minus08563 04557 04980 06148 minus03609 08828

51sim60 sScheme 1 minus05430 04296 05377 04614 minus02183 09682Scheme 2 minus07073 04854 05415 05531 02041 09117

61sim500 sScheme 1 minus05075 04498 04154 04852 minus00877 09168Scheme 2 minus06541 04505 04815 04831 00709 09011

Limited accuracy minus04990 04990 0

Table 4 Statistical results for gyro bias

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

51sim100 sScheme 1 minus04139 04563 minus07990 09143 07591 04807Scheme 2 05172 00757 05335 01329 06390 01588

101sim150 sScheme 1 03474 02102 minus03793 04709 19998 03182Scheme 2 05037 00505 04847 00499 06072 00628

151sim200 sScheme 1 05742 00632 02499 01254 08496 02778Scheme 2 04943 00382 04703 00284 05196 00256

201sim250 sScheme 1 04790 00283 02713 00744 05335 00648Scheme 2 04961 00224 04718 00231 05255 00224

251sim300 sScheme 1 05189 00344 04203 00630 05292 00530Scheme 2 05019 00217 04752 00258 05113 00236

301sim500 sScheme 1 05156 00272 04122 00410 05683 00412Scheme 2 04990 00116 04823 00169 05134 00129

Setting value 05 05 05

With the above two viewpoints (1) a backward-forwardSINS resolution algorithm is designed in detail and simu-lation results indicate that a ship can move from the endto the origin with the backward resolution and move fromthe origin to the end with the forward (2) A rapid transfer

alignment algorithm is also designed in detail in which thebackward-forward resolution and two more operations forthe estimation of gyro bias are added in one reference dataupdate cycle In addition the correctness of its algorithm isproved by simulation The simulation result produced with

Mathematical Problems in Engineering 9

middot middot middot

13

11 2

(1) Sensor data sampling storing and navigation resolution(2) Reference data sampling storing and data fusion(3) Reverse and forward calculationlowastCPU idle

TsTs Ts

Ts Ts

tnt2nlowast lowast

Tn Tn

t0

tn2

tn1t01

t02

middot middot middot

Figure 6 Calculation process for alignment program

50 100 150 200 250 300 350 400 450 500

Erro

r (m

rad)

012

Scheme 2Scheme 1

minus3minus2minus1

t (s)

(a) Pitch

Erro

r (m

rad)

50 100 150 200 250 300 350 400 450 500

3 Scheme 2Scheme 1

012

minus2minus1

t (s)

(b) Roll

50 100 150 200 250 300 350 400 450 500

024

Erro

r (m

rad) Scheme 2Scheme 1

minus2minus4

t (s)

(c) Yaw

Figure 7 Estimation curves for misalignment angles

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(a) East velocity

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(b) North velocity

Figure 8 Estimation curves for velocity errors

50 100 150 200 250 300 350 400 450 500

13 Scheme 1Scheme 2

minus3

minus1

Bias

(∘h

)

t (s)

(a) 119909 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 113

minus3

minus1

Bias

(∘h

)

t (s)

(b) 119910 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 1

13

minus3

minus1

Bias

(∘h

)

t (s)

(c) 119911 gyro

Figure 9 Estimation curves for gyro bias

the method in this paper indicates that the alignment timeis reduced from 300 s to 100 s compared with that of thetransfer alignment method without the added backward-forward SINS resolution and data fusion

One problem which may be encountered in engineeringapplications with this transfer alignment method is studiedand a possible solution is given in which RTOS is introducedto distribute computer resources coordinately

Acknowledgments

This work was supported in part by the National NaturalScience Foundation (61004125 61273056) and Chinese uni-versity research and operation expenses (10420525)

References

[1] J Kain and J Cloutier ldquoRapid transfer alignment for tacticalweapon applicationsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 1290ndash1300 BostonMass USA 1989

[2] K Spalding ldquoAn efficient rapid transfer alignment filterrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 1276ndash1286 Hilton Head Island SC USA 1992

[3] K Shortelle and W Graham ldquoAdvanced alignment conceptsfor precision-guided weaponsrdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 131ndash142Anaheim Calif USA January 1995

[4] W Graham and K Shortelle ldquoAdvanced transfer alignmentfor inertial navigators (A-train)rdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 113ndash124Anaheim Calif USA January 1995

[5] K J Shortelle W R Graham and C Rabourn ldquoF-16 flight testsof a rapid transfer alignment procedurerdquo in Proceedings of the

10 Mathematical Problems in Engineering

IEEEPosition Location andNavigation Symposium pp 379ndash386Palm Springs Calif USA April 1996

[6] P D Groves ldquoOptimising the transfer alignment of weaponINSrdquo Journal of Navigation vol 56 no 2 pp 323ndash335 2003

[7] DGoshen-Meskin and I Y Bar-Itzhack ldquoObservability analysisof piece-wise constant systems II Application to inertialnavigation in-flight alignment (military applications)rdquo IEEETransactions on Aerospace and Electronic Systems vol 28 no4 pp 1068ndash1075 1992

[8] X H Cheng D J Wan and X Zhong ldquoStudy on observabilityand its degree of strapdown inertial navigation systemrdquo Journalof Southeast University vol 37 no 6 pp 6ndash10 1997

[9] L-H Zhu X-H Cheng and Y-Y He ldquoFilter convergencecriterion in transfer alignmentrdquo Journal of Chinese InertialTechnology vol 19 no 3 pp 277ndash285 2011

[10] S Wang Z Wang Y Zhu and B Wang ldquoMonitoring onship hull deformation and correction for heading and attitudeinformationrdquo Journal of Chinese Inertial Technology vol 15 no6 pp 635ndash641 2007

[11] D R Tarrant C Jones and D Lin ldquoRapid and robust transferalignmentrdquo in Proceedings of the 1st IEEE Regional Conferenceon Aerospace Control Systems pp 758ndash762 1993

[12] J Lyou and Y C Lim ldquoTransfer alignment considering mea-surement time delay and ship body flexurerdquo Journal of Mechan-ical Science and Technology vol 23 no 1 pp 195ndash203 2009

[13] B-C Zhou X-H Cheng and D-J Liu ldquoAnalysis method ofobservable degree based on spectral decomposition in SINStransfer alignmentrdquo Journal of Chinese Inertial Technology vol18 no 5 pp 518ndash522 2010

[14] D H Titterton and J L Weston Strapdown Inertial NavigationTechnology Lavenham Press Ltd London UK 2nd edition2004

[15] R B Miller ldquoA new strapdown attitude algorithmrdquo Journal ofGuidance Control and Dynamics vol 6 no 4 pp 287ndash2911983

[16] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998

[17] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 2 velocity and position algorithmsrdquo Journalof Guidance Control and Dynamics vol 21 no 2 pp 208ndash2211998

[18] P J Gates and N M Lynn Ships Submarines and the SeaBrasseyrsquos London UK 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

executed at every navigation update cycle The last two tasksshould be run in the last navigation update cycle of everyreference data update cycle whichmeans that the above threetasks should be finished in 119879

119904 Otherwise the first task in the

first navigation cycle of the next reference data cycle will becompromised It is difficult to finish these three tasks in 119879

119904

with a computer of limited performanceHowever this problem can be resolved by the full use

of resources of high speed computers with the support ofreal-time multitasking operation system (RTOS) such asVxWorks In VxWorks the above three tasks can be set withdifferent priorities The first can be run preferentially whenTask 2 or Task 3 is being run whichmeans that the first task ofthe next reference data cycle can be run preferentially when

Tasks 2 and 3 of the previous reference data cycle are beingexecuted In this method the idle resources of CPU in thenext reference cycle can be used for the tasks 2 and 3 of theprevious cycle The paper is not involved in the programs inRTOS in detail

5 Simulation

51 Parameters for Simulation The ship moving parametersare set as in Section 323 The ideal velocity and yaw ofthe ship are used as reference data from MINS after whitenoise is added The variance of the white noise is set as[(04ms)2 (04ms)2 (03

∘)

2

] The sensor errors are listedin Table 1

The parameters for Kalman filter are

X0= [0 0 0 0 0 0 0 0]

119879

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

∘)

2

(15∘)

2

(15∘)

2

(15∘h)2 (15

∘h)2 (15

∘h)2]

Q = diag [(500 ug)2 (500 ug)2 (05∘)2

(05∘)2

(05∘)2

0 0 0]

R = diag [(04ms)2 (04ms)2 (03∘)

2

]

(10)

Two data fusion schemes were compared The secondscheme is a transfer alignment method with the addedbackward-forward SINS resolution and data fusion while thefirst scheme is not In these two schemes the same transferalignment model introduced in Section 2 is used

The update cycle of sensor data and navigation resolutionis set as 10ms and that of reference data from MINS is as1 s As analyzed in Section 4 in every reference data cyclenavigation resolution will be executed 300 times and datafusion 3 times in scheme 2 while in scheme 1 navigationresolution is executed 100 times and data fusion only once

In the ldquovelocity plus yawrdquo matching method accelerom-eter bias is unobservable so with the sensor errors assumedin Table 1 the limit alignment accuracy of pitch and roll areminus04990 mrad and 04990 mrad respectively and the limitalignment accuracy of yaw is 0

52 Simulation Results The simulation lasts for 500 s andthe simulation results are stored once per second The mis-alignment angle curves are shown in Figure 7The estimationof velocity error curves are in Figure 8 and the estimationcurves of gyros bias in Figure 9 In Figures 7ndash9 the dot-dashand solid lines denote the simulation results of scheme 1 andscheme 2 respectively The dotted lines in Figures 7 and 8are the limit alignment accuracy of misalignment angles andvelocity And the dotted lines in Figure 9 denote the settingvalue of constant gyro bias

The curves in Figure 7 show that either in scheme 1or scheme 2 misalignment angles can be estimated rapidlyand are oscillating in small amplitudes with the swinging

frequency of ship But the tendency of misalignment curvesespecially that of roll ones indicates that estimation speed isslightly higher in scheme 2 than in scheme 1 The statisticaldata about misalignment angles are shown in Table 3 and thestatistical results show that the alignment accuracy of thesetwo schemes is roughly equal after 60 sThe curves in Figure 8indicate that the estimation speed and accuracy for velocityerror are roughly equal in scheme 1 and scheme 2

The curves in Figure 9 show that in scheme 2 theestimation curves of gyro bias converge towards the settingvalues at about 50 s and the oscillating amplitudes are verysmall in 100 s while in scheme 1 200 s is needed for theconvergence of gyro bias and relatively large oscillations stillexist even after 300 s The statistical mean values shown inTable 4 indicate that in scheme 2 in the period from 50 sto 100 s about 96 and 93 and 72 of gyro bias can beestimated along 119909 119910 and 119911 axes respectively And about99 97 and 78 can be estimated in the period from 100 sto 150 s

6 Conclusion

Two viewpoints are given in this paper The first is that inSINS mathematical platform cuts off the direct relationshipbetween sensor data and misalignment angles which meansthat initial alignment can be fulfilled by the repeatedlyresolution on a same set of sensor data The second isthat with the added backward-forward SINS resolution andrepeated data fusion on the corresponding resolution resultsand the external data the alignment time can be greatlyreduced

8 Mathematical Problems in Engineering

Table 3 Statistical results for misalignment angles

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

1sim10 sScheme 1 minus08225 10610 11833 49672 minus09375 12756Scheme 2 04631 16724 02318 07429 minus03956 10700

11sim20 sScheme 1 minus02278 05136 minus00705 05156 minus11170 07199Scheme 2 minus04993 05865 04806 02647 minus04755 08203

31sim40 sScheme 1 minus04776 04653 09794 05194 00448 09782Scheme 2 minus06454 04553 02271 04726 minus00067 08820

41sim50 sScheme 1 minus08024 04560 10111 05086 minus03714 09385Scheme 2 minus08563 04557 04980 06148 minus03609 08828

51sim60 sScheme 1 minus05430 04296 05377 04614 minus02183 09682Scheme 2 minus07073 04854 05415 05531 02041 09117

61sim500 sScheme 1 minus05075 04498 04154 04852 minus00877 09168Scheme 2 minus06541 04505 04815 04831 00709 09011

Limited accuracy minus04990 04990 0

Table 4 Statistical results for gyro bias

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

51sim100 sScheme 1 minus04139 04563 minus07990 09143 07591 04807Scheme 2 05172 00757 05335 01329 06390 01588

101sim150 sScheme 1 03474 02102 minus03793 04709 19998 03182Scheme 2 05037 00505 04847 00499 06072 00628

151sim200 sScheme 1 05742 00632 02499 01254 08496 02778Scheme 2 04943 00382 04703 00284 05196 00256

201sim250 sScheme 1 04790 00283 02713 00744 05335 00648Scheme 2 04961 00224 04718 00231 05255 00224

251sim300 sScheme 1 05189 00344 04203 00630 05292 00530Scheme 2 05019 00217 04752 00258 05113 00236

301sim500 sScheme 1 05156 00272 04122 00410 05683 00412Scheme 2 04990 00116 04823 00169 05134 00129

Setting value 05 05 05

With the above two viewpoints (1) a backward-forwardSINS resolution algorithm is designed in detail and simu-lation results indicate that a ship can move from the endto the origin with the backward resolution and move fromthe origin to the end with the forward (2) A rapid transfer

alignment algorithm is also designed in detail in which thebackward-forward resolution and two more operations forthe estimation of gyro bias are added in one reference dataupdate cycle In addition the correctness of its algorithm isproved by simulation The simulation result produced with

Mathematical Problems in Engineering 9

middot middot middot

13

11 2

(1) Sensor data sampling storing and navigation resolution(2) Reference data sampling storing and data fusion(3) Reverse and forward calculationlowastCPU idle

TsTs Ts

Ts Ts

tnt2nlowast lowast

Tn Tn

t0

tn2

tn1t01

t02

middot middot middot

Figure 6 Calculation process for alignment program

50 100 150 200 250 300 350 400 450 500

Erro

r (m

rad)

012

Scheme 2Scheme 1

minus3minus2minus1

t (s)

(a) Pitch

Erro

r (m

rad)

50 100 150 200 250 300 350 400 450 500

3 Scheme 2Scheme 1

012

minus2minus1

t (s)

(b) Roll

50 100 150 200 250 300 350 400 450 500

024

Erro

r (m

rad) Scheme 2Scheme 1

minus2minus4

t (s)

(c) Yaw

Figure 7 Estimation curves for misalignment angles

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(a) East velocity

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(b) North velocity

Figure 8 Estimation curves for velocity errors

50 100 150 200 250 300 350 400 450 500

13 Scheme 1Scheme 2

minus3

minus1

Bias

(∘h

)

t (s)

(a) 119909 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 113

minus3

minus1

Bias

(∘h

)

t (s)

(b) 119910 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 1

13

minus3

minus1

Bias

(∘h

)

t (s)

(c) 119911 gyro

Figure 9 Estimation curves for gyro bias

the method in this paper indicates that the alignment timeis reduced from 300 s to 100 s compared with that of thetransfer alignment method without the added backward-forward SINS resolution and data fusion

One problem which may be encountered in engineeringapplications with this transfer alignment method is studiedand a possible solution is given in which RTOS is introducedto distribute computer resources coordinately

Acknowledgments

This work was supported in part by the National NaturalScience Foundation (61004125 61273056) and Chinese uni-versity research and operation expenses (10420525)

References

[1] J Kain and J Cloutier ldquoRapid transfer alignment for tacticalweapon applicationsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 1290ndash1300 BostonMass USA 1989

[2] K Spalding ldquoAn efficient rapid transfer alignment filterrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 1276ndash1286 Hilton Head Island SC USA 1992

[3] K Shortelle and W Graham ldquoAdvanced alignment conceptsfor precision-guided weaponsrdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 131ndash142Anaheim Calif USA January 1995

[4] W Graham and K Shortelle ldquoAdvanced transfer alignmentfor inertial navigators (A-train)rdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 113ndash124Anaheim Calif USA January 1995

[5] K J Shortelle W R Graham and C Rabourn ldquoF-16 flight testsof a rapid transfer alignment procedurerdquo in Proceedings of the

10 Mathematical Problems in Engineering

IEEEPosition Location andNavigation Symposium pp 379ndash386Palm Springs Calif USA April 1996

[6] P D Groves ldquoOptimising the transfer alignment of weaponINSrdquo Journal of Navigation vol 56 no 2 pp 323ndash335 2003

[7] DGoshen-Meskin and I Y Bar-Itzhack ldquoObservability analysisof piece-wise constant systems II Application to inertialnavigation in-flight alignment (military applications)rdquo IEEETransactions on Aerospace and Electronic Systems vol 28 no4 pp 1068ndash1075 1992

[8] X H Cheng D J Wan and X Zhong ldquoStudy on observabilityand its degree of strapdown inertial navigation systemrdquo Journalof Southeast University vol 37 no 6 pp 6ndash10 1997

[9] L-H Zhu X-H Cheng and Y-Y He ldquoFilter convergencecriterion in transfer alignmentrdquo Journal of Chinese InertialTechnology vol 19 no 3 pp 277ndash285 2011

[10] S Wang Z Wang Y Zhu and B Wang ldquoMonitoring onship hull deformation and correction for heading and attitudeinformationrdquo Journal of Chinese Inertial Technology vol 15 no6 pp 635ndash641 2007

[11] D R Tarrant C Jones and D Lin ldquoRapid and robust transferalignmentrdquo in Proceedings of the 1st IEEE Regional Conferenceon Aerospace Control Systems pp 758ndash762 1993

[12] J Lyou and Y C Lim ldquoTransfer alignment considering mea-surement time delay and ship body flexurerdquo Journal of Mechan-ical Science and Technology vol 23 no 1 pp 195ndash203 2009

[13] B-C Zhou X-H Cheng and D-J Liu ldquoAnalysis method ofobservable degree based on spectral decomposition in SINStransfer alignmentrdquo Journal of Chinese Inertial Technology vol18 no 5 pp 518ndash522 2010

[14] D H Titterton and J L Weston Strapdown Inertial NavigationTechnology Lavenham Press Ltd London UK 2nd edition2004

[15] R B Miller ldquoA new strapdown attitude algorithmrdquo Journal ofGuidance Control and Dynamics vol 6 no 4 pp 287ndash2911983

[16] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998

[17] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 2 velocity and position algorithmsrdquo Journalof Guidance Control and Dynamics vol 21 no 2 pp 208ndash2211998

[18] P J Gates and N M Lynn Ships Submarines and the SeaBrasseyrsquos London UK 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 3 Statistical results for misalignment angles

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

1sim10 sScheme 1 minus08225 10610 11833 49672 minus09375 12756Scheme 2 04631 16724 02318 07429 minus03956 10700

11sim20 sScheme 1 minus02278 05136 minus00705 05156 minus11170 07199Scheme 2 minus04993 05865 04806 02647 minus04755 08203

31sim40 sScheme 1 minus04776 04653 09794 05194 00448 09782Scheme 2 minus06454 04553 02271 04726 minus00067 08820

41sim50 sScheme 1 minus08024 04560 10111 05086 minus03714 09385Scheme 2 minus08563 04557 04980 06148 minus03609 08828

51sim60 sScheme 1 minus05430 04296 05377 04614 minus02183 09682Scheme 2 minus07073 04854 05415 05531 02041 09117

61sim500 sScheme 1 minus05075 04498 04154 04852 minus00877 09168Scheme 2 minus06541 04505 04815 04831 00709 09011

Limited accuracy minus04990 04990 0

Table 4 Statistical results for gyro bias

Pitch (mrad) Roll (mrad) Yaw (mrad)Mean Standard variance Mean Standard variance Mean Standard variance

51sim100 sScheme 1 minus04139 04563 minus07990 09143 07591 04807Scheme 2 05172 00757 05335 01329 06390 01588

101sim150 sScheme 1 03474 02102 minus03793 04709 19998 03182Scheme 2 05037 00505 04847 00499 06072 00628

151sim200 sScheme 1 05742 00632 02499 01254 08496 02778Scheme 2 04943 00382 04703 00284 05196 00256

201sim250 sScheme 1 04790 00283 02713 00744 05335 00648Scheme 2 04961 00224 04718 00231 05255 00224

251sim300 sScheme 1 05189 00344 04203 00630 05292 00530Scheme 2 05019 00217 04752 00258 05113 00236

301sim500 sScheme 1 05156 00272 04122 00410 05683 00412Scheme 2 04990 00116 04823 00169 05134 00129

Setting value 05 05 05

With the above two viewpoints (1) a backward-forwardSINS resolution algorithm is designed in detail and simu-lation results indicate that a ship can move from the endto the origin with the backward resolution and move fromthe origin to the end with the forward (2) A rapid transfer

alignment algorithm is also designed in detail in which thebackward-forward resolution and two more operations forthe estimation of gyro bias are added in one reference dataupdate cycle In addition the correctness of its algorithm isproved by simulation The simulation result produced with

Mathematical Problems in Engineering 9

middot middot middot

13

11 2

(1) Sensor data sampling storing and navigation resolution(2) Reference data sampling storing and data fusion(3) Reverse and forward calculationlowastCPU idle

TsTs Ts

Ts Ts

tnt2nlowast lowast

Tn Tn

t0

tn2

tn1t01

t02

middot middot middot

Figure 6 Calculation process for alignment program

50 100 150 200 250 300 350 400 450 500

Erro

r (m

rad)

012

Scheme 2Scheme 1

minus3minus2minus1

t (s)

(a) Pitch

Erro

r (m

rad)

50 100 150 200 250 300 350 400 450 500

3 Scheme 2Scheme 1

012

minus2minus1

t (s)

(b) Roll

50 100 150 200 250 300 350 400 450 500

024

Erro

r (m

rad) Scheme 2Scheme 1

minus2minus4

t (s)

(c) Yaw

Figure 7 Estimation curves for misalignment angles

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(a) East velocity

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(b) North velocity

Figure 8 Estimation curves for velocity errors

50 100 150 200 250 300 350 400 450 500

13 Scheme 1Scheme 2

minus3

minus1

Bias

(∘h

)

t (s)

(a) 119909 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 113

minus3

minus1

Bias

(∘h

)

t (s)

(b) 119910 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 1

13

minus3

minus1

Bias

(∘h

)

t (s)

(c) 119911 gyro

Figure 9 Estimation curves for gyro bias

the method in this paper indicates that the alignment timeis reduced from 300 s to 100 s compared with that of thetransfer alignment method without the added backward-forward SINS resolution and data fusion

One problem which may be encountered in engineeringapplications with this transfer alignment method is studiedand a possible solution is given in which RTOS is introducedto distribute computer resources coordinately

Acknowledgments

This work was supported in part by the National NaturalScience Foundation (61004125 61273056) and Chinese uni-versity research and operation expenses (10420525)

References

[1] J Kain and J Cloutier ldquoRapid transfer alignment for tacticalweapon applicationsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 1290ndash1300 BostonMass USA 1989

[2] K Spalding ldquoAn efficient rapid transfer alignment filterrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 1276ndash1286 Hilton Head Island SC USA 1992

[3] K Shortelle and W Graham ldquoAdvanced alignment conceptsfor precision-guided weaponsrdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 131ndash142Anaheim Calif USA January 1995

[4] W Graham and K Shortelle ldquoAdvanced transfer alignmentfor inertial navigators (A-train)rdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 113ndash124Anaheim Calif USA January 1995

[5] K J Shortelle W R Graham and C Rabourn ldquoF-16 flight testsof a rapid transfer alignment procedurerdquo in Proceedings of the

10 Mathematical Problems in Engineering

IEEEPosition Location andNavigation Symposium pp 379ndash386Palm Springs Calif USA April 1996

[6] P D Groves ldquoOptimising the transfer alignment of weaponINSrdquo Journal of Navigation vol 56 no 2 pp 323ndash335 2003

[7] DGoshen-Meskin and I Y Bar-Itzhack ldquoObservability analysisof piece-wise constant systems II Application to inertialnavigation in-flight alignment (military applications)rdquo IEEETransactions on Aerospace and Electronic Systems vol 28 no4 pp 1068ndash1075 1992

[8] X H Cheng D J Wan and X Zhong ldquoStudy on observabilityand its degree of strapdown inertial navigation systemrdquo Journalof Southeast University vol 37 no 6 pp 6ndash10 1997

[9] L-H Zhu X-H Cheng and Y-Y He ldquoFilter convergencecriterion in transfer alignmentrdquo Journal of Chinese InertialTechnology vol 19 no 3 pp 277ndash285 2011

[10] S Wang Z Wang Y Zhu and B Wang ldquoMonitoring onship hull deformation and correction for heading and attitudeinformationrdquo Journal of Chinese Inertial Technology vol 15 no6 pp 635ndash641 2007

[11] D R Tarrant C Jones and D Lin ldquoRapid and robust transferalignmentrdquo in Proceedings of the 1st IEEE Regional Conferenceon Aerospace Control Systems pp 758ndash762 1993

[12] J Lyou and Y C Lim ldquoTransfer alignment considering mea-surement time delay and ship body flexurerdquo Journal of Mechan-ical Science and Technology vol 23 no 1 pp 195ndash203 2009

[13] B-C Zhou X-H Cheng and D-J Liu ldquoAnalysis method ofobservable degree based on spectral decomposition in SINStransfer alignmentrdquo Journal of Chinese Inertial Technology vol18 no 5 pp 518ndash522 2010

[14] D H Titterton and J L Weston Strapdown Inertial NavigationTechnology Lavenham Press Ltd London UK 2nd edition2004

[15] R B Miller ldquoA new strapdown attitude algorithmrdquo Journal ofGuidance Control and Dynamics vol 6 no 4 pp 287ndash2911983

[16] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998

[17] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 2 velocity and position algorithmsrdquo Journalof Guidance Control and Dynamics vol 21 no 2 pp 208ndash2211998

[18] P J Gates and N M Lynn Ships Submarines and the SeaBrasseyrsquos London UK 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

middot middot middot

13

11 2

(1) Sensor data sampling storing and navigation resolution(2) Reference data sampling storing and data fusion(3) Reverse and forward calculationlowastCPU idle

TsTs Ts

Ts Ts

tnt2nlowast lowast

Tn Tn

t0

tn2

tn1t01

t02

middot middot middot

Figure 6 Calculation process for alignment program

50 100 150 200 250 300 350 400 450 500

Erro

r (m

rad)

012

Scheme 2Scheme 1

minus3minus2minus1

t (s)

(a) Pitch

Erro

r (m

rad)

50 100 150 200 250 300 350 400 450 500

3 Scheme 2Scheme 1

012

minus2minus1

t (s)

(b) Roll

50 100 150 200 250 300 350 400 450 500

024

Erro

r (m

rad) Scheme 2Scheme 1

minus2minus4

t (s)

(c) Yaw

Figure 7 Estimation curves for misalignment angles

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(a) East velocity

00204

50 100 150 200 250 300 350 400 450 500

Erro

r (m

s)

Scheme 2Scheme 1

minus04minus02

t (s)

(b) North velocity

Figure 8 Estimation curves for velocity errors

50 100 150 200 250 300 350 400 450 500

13 Scheme 1Scheme 2

minus3

minus1

Bias

(∘h

)

t (s)

(a) 119909 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 113

minus3

minus1

Bias

(∘h

)

t (s)

(b) 119910 gyro

50 100 150 200 250 300 350 400 450 500

Scheme 2Scheme 1

13

minus3

minus1

Bias

(∘h

)

t (s)

(c) 119911 gyro

Figure 9 Estimation curves for gyro bias

the method in this paper indicates that the alignment timeis reduced from 300 s to 100 s compared with that of thetransfer alignment method without the added backward-forward SINS resolution and data fusion

One problem which may be encountered in engineeringapplications with this transfer alignment method is studiedand a possible solution is given in which RTOS is introducedto distribute computer resources coordinately

Acknowledgments

This work was supported in part by the National NaturalScience Foundation (61004125 61273056) and Chinese uni-versity research and operation expenses (10420525)

References

[1] J Kain and J Cloutier ldquoRapid transfer alignment for tacticalweapon applicationsrdquo in Proceedings of the AIAA GuidanceNavigation and Control Conference pp 1290ndash1300 BostonMass USA 1989

[2] K Spalding ldquoAn efficient rapid transfer alignment filterrdquo inProceedings of the AIAA Guidance Navigation and ControlConference pp 1276ndash1286 Hilton Head Island SC USA 1992

[3] K Shortelle and W Graham ldquoAdvanced alignment conceptsfor precision-guided weaponsrdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 131ndash142Anaheim Calif USA January 1995

[4] W Graham and K Shortelle ldquoAdvanced transfer alignmentfor inertial navigators (A-train)rdquo in Proceedings of the NationalTechnical Meeting of the Institute of Navigation pp 113ndash124Anaheim Calif USA January 1995

[5] K J Shortelle W R Graham and C Rabourn ldquoF-16 flight testsof a rapid transfer alignment procedurerdquo in Proceedings of the

10 Mathematical Problems in Engineering

IEEEPosition Location andNavigation Symposium pp 379ndash386Palm Springs Calif USA April 1996

[6] P D Groves ldquoOptimising the transfer alignment of weaponINSrdquo Journal of Navigation vol 56 no 2 pp 323ndash335 2003

[7] DGoshen-Meskin and I Y Bar-Itzhack ldquoObservability analysisof piece-wise constant systems II Application to inertialnavigation in-flight alignment (military applications)rdquo IEEETransactions on Aerospace and Electronic Systems vol 28 no4 pp 1068ndash1075 1992

[8] X H Cheng D J Wan and X Zhong ldquoStudy on observabilityand its degree of strapdown inertial navigation systemrdquo Journalof Southeast University vol 37 no 6 pp 6ndash10 1997

[9] L-H Zhu X-H Cheng and Y-Y He ldquoFilter convergencecriterion in transfer alignmentrdquo Journal of Chinese InertialTechnology vol 19 no 3 pp 277ndash285 2011

[10] S Wang Z Wang Y Zhu and B Wang ldquoMonitoring onship hull deformation and correction for heading and attitudeinformationrdquo Journal of Chinese Inertial Technology vol 15 no6 pp 635ndash641 2007

[11] D R Tarrant C Jones and D Lin ldquoRapid and robust transferalignmentrdquo in Proceedings of the 1st IEEE Regional Conferenceon Aerospace Control Systems pp 758ndash762 1993

[12] J Lyou and Y C Lim ldquoTransfer alignment considering mea-surement time delay and ship body flexurerdquo Journal of Mechan-ical Science and Technology vol 23 no 1 pp 195ndash203 2009

[13] B-C Zhou X-H Cheng and D-J Liu ldquoAnalysis method ofobservable degree based on spectral decomposition in SINStransfer alignmentrdquo Journal of Chinese Inertial Technology vol18 no 5 pp 518ndash522 2010

[14] D H Titterton and J L Weston Strapdown Inertial NavigationTechnology Lavenham Press Ltd London UK 2nd edition2004

[15] R B Miller ldquoA new strapdown attitude algorithmrdquo Journal ofGuidance Control and Dynamics vol 6 no 4 pp 287ndash2911983

[16] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998

[17] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 2 velocity and position algorithmsrdquo Journalof Guidance Control and Dynamics vol 21 no 2 pp 208ndash2211998

[18] P J Gates and N M Lynn Ships Submarines and the SeaBrasseyrsquos London UK 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

IEEEPosition Location andNavigation Symposium pp 379ndash386Palm Springs Calif USA April 1996

[6] P D Groves ldquoOptimising the transfer alignment of weaponINSrdquo Journal of Navigation vol 56 no 2 pp 323ndash335 2003

[7] DGoshen-Meskin and I Y Bar-Itzhack ldquoObservability analysisof piece-wise constant systems II Application to inertialnavigation in-flight alignment (military applications)rdquo IEEETransactions on Aerospace and Electronic Systems vol 28 no4 pp 1068ndash1075 1992

[8] X H Cheng D J Wan and X Zhong ldquoStudy on observabilityand its degree of strapdown inertial navigation systemrdquo Journalof Southeast University vol 37 no 6 pp 6ndash10 1997

[9] L-H Zhu X-H Cheng and Y-Y He ldquoFilter convergencecriterion in transfer alignmentrdquo Journal of Chinese InertialTechnology vol 19 no 3 pp 277ndash285 2011

[10] S Wang Z Wang Y Zhu and B Wang ldquoMonitoring onship hull deformation and correction for heading and attitudeinformationrdquo Journal of Chinese Inertial Technology vol 15 no6 pp 635ndash641 2007

[11] D R Tarrant C Jones and D Lin ldquoRapid and robust transferalignmentrdquo in Proceedings of the 1st IEEE Regional Conferenceon Aerospace Control Systems pp 758ndash762 1993

[12] J Lyou and Y C Lim ldquoTransfer alignment considering mea-surement time delay and ship body flexurerdquo Journal of Mechan-ical Science and Technology vol 23 no 1 pp 195ndash203 2009

[13] B-C Zhou X-H Cheng and D-J Liu ldquoAnalysis method ofobservable degree based on spectral decomposition in SINStransfer alignmentrdquo Journal of Chinese Inertial Technology vol18 no 5 pp 518ndash522 2010

[14] D H Titterton and J L Weston Strapdown Inertial NavigationTechnology Lavenham Press Ltd London UK 2nd edition2004

[15] R B Miller ldquoA new strapdown attitude algorithmrdquo Journal ofGuidance Control and Dynamics vol 6 no 4 pp 287ndash2911983

[16] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 1 attitude algorithmsrdquo Journal of GuidanceControl and Dynamics vol 21 no 1 pp 19ndash28 1998

[17] P G Savage ldquoStrapdown inertial navigation integration algo-rithm designmdashpart 2 velocity and position algorithmsrdquo Journalof Guidance Control and Dynamics vol 21 no 2 pp 208ndash2211998

[18] P J Gates and N M Lynn Ships Submarines and the SeaBrasseyrsquos London UK 1990

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of