Research ArticleAn Improved Gyrocompass Alignment Method forLarge Azimuth Misalignment

Wei Gao1 Baofeng Lu1 Chunyang Yu1 and Haiyu Lan12

1College of Automation Harbin Engineering University Harbin 150001 China2Department of Geomatics Engineering University of Calgary Calgary AB Canada T2N 1N4

Correspondence should be addressed to Baofeng Lu lu bao feng163com

Received 3 August 2014 Revised 12 January 2015 Accepted 12 January 2015

Academic Editor Francesco Braghin

Copyright copy 2015 Wei Gao et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Due to the impact of the nonlinear factor caused by large azimuthmisalignment the conventional gyrocompass alignment methodis hard to favorablymeet the requirement of alignment speed under the condition of large azimuthmisalignment of INS In order tosolve this problem an improved gyrocompass alignmentmethod is presented in this paperThe improvedmethod is designed basedon the nonlinearmodel for large azimuthmisalignment and performed by opening the azimuth loopThe influence of the nonlinearfactor on gyrocompass alignment will be reduced when opening the azimuth loop Simulation and experimental results show thatthe initial alignment can be efficiently accomplished through using the improved method in the case of existing large azimuthmisalignment and in the same conditions the alignment speed of the improved method is faster than that of the conventional one

1 Introduction

The initial alignment of inertial navigation systems (INS) isan important process performed prior to normal navigationprocedure [1] It is well known that the initial alignment resultof the system is of fundamental importance to the followingnavigation accuracy [2] Therefore many researchers haveinvestigated this topic mainly concentrated on gyrocompassalignment and optimal estimation techniques In contrast tothe optimal estimation techniques the former method doesnot need precise mathematical and noise model [3] Withmany yearsrsquo development gyrocompass alignment methodbased on classical control theory is very mature now In 1961Cannon firstly presented gyrocompass alignment methodfor platform INS [4] After that gyrocompass alignmentis described extensively in the literatures [5 6] includingalignment technique and error analysis In recent years withthe development of strapdown INS gyrocompass alignmentis applied to strapdown INS [1 7ndash9] In all the previousworks gyrocompass alignment is usually designed basedon the small angle assumption (ie less than 5 degrees)and under this situation the alignment system can thenbe approximated as linear model in the case of small

azimuth misalignment However under the condition oflarge azimuth misalignment such approximation is invalidand the alignment system will then be influenced by the non-linear factorThen the conventionalmethod is not effective toproperly accomplish initial alignment in the case of existinglarge azimuth misalignment

Therefore in this paper a new improved gyrocompassalignment method which is applicable to the INS that causeslarge azimuth misalignment is established based on thenonlinear model So far many works are attempted to modellarge azimuth misalignment and several models have beenprovided such as the nonlinear psi-angle model [10 11]the rotation vector error and quaternion error models [12]and the nonlinear phi-angle model [13] In this work thenonlinear psi-angle model is adoptedThe improved methodproposed in this paper is performed by opening the azimuthloop and through using this scheme the nonlinear factorscan be regarded as constant inputs which contain azimuthmisalignment information The estimation of the azimuthmisalignment is implemented by using the horizontal veloc-ity signals to estimate those nonlinear factors At this timethe impact of the nonlinear factor caused by large azimuthmisalignment on gyrocompass alignment can be reduced

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 783640 12 pageshttpdxdoiorg1011552015783640

2 Mathematical Problems in Engineering

nablaN +

+

minus

minus

minus

minus

minus

g

1

s

1

s

1

skN

k1

120576U

120576E120595E

120595U

Ω cos 120593 Ω cos 120593 middotsin120595U

120595U

Large azimuthmisalignment

kUs + k2

Figure 1 Error diagram of conventional gyrocompass alignment

The remainder of this paper is organized as follows Section 2describes the conventional gyrocompass alignment methodand then analyzes the existing problem of this method indetail Section 3 details the improved gyrocompass alignmentmethod which includes the establishment operation andimplementation of the improved method Simulation andexperimental results that validate the proposed approach arepresented and discussed in Section 4 Finally conclusions aregiven in Section 5

2 Conventional GyrocompassAlignment Method

In literatures [8 9] a fourth-order gyrocompass alignmentsystem was used for INS alignment It is a higher-ordersystem which has a good performance in the case of smallazimuth misalignment In this paper the conventional gyro-compass alignmentmethod analyzed here is the fourth-ordersystem (as the representative of the similar methodologies)and the form of this system is simplified The error diagramof reduced conventional gyrocompass alignment is shown inFigure 1

In Figure 1nabla119873is the north accelerometer error 120576

119864and 120576119880

are the east and up gyro errors 120595119864and 120595

119880represent the east

level and azimuth misalignments respectively 119892 representsthe acceleration due to gravity Ω represents the earth rateand 120593 represents the local latitude 119896

1 1198962 119896119873 and 119896

119880are the

control gainsAccording to the Mason gain formula we can get the

characteristic equation of conventional gyrocompass align-ment from Figure 1

Δ (119904) = 1199042(119904 + 119896

1) (119904 + 119896

2) + 119904 (119904 + 119896

2) 119892119896119873

+ 119892Ω cos120593119896119880= 0

(1)

In order to ensure the stability of the alignment system foureigenvalues of (1) are set as 119904

12= minus120585120596

119899and 11990434

= minus120585120596119899plusmn

119895radic1 minus 1205852120596119899 where 120585 = 0707 and 120596

119899is the natural frequency

which is adjustable Then (1) can be rewritten as

(1199042+ 2120585120596

119899119904 + 1205962

119899) (119904 + 120585120596

119899)2= 0 (2)

Comparing (1) with (2) the values of the control gains underthe small angle can be obtained

1198961= 1198962= 2120585120596

119899 (3)

119896119873=

1205962

119899(1 + 120585

2)

119892 (4)

119896119880=

12058521205964

119899

119892Ω cos120593 (5)

In the knowledge of a classical control theory it is obvi-ous that the alignment speed of gyrocompass alignment isdetermined by the eigenvalues Worth noting is that thefour eigenvalues of the conventional gyrocompass alignmentsystem can be given any value desired by appropriate choiceof the control gains In other words the alignment speedof conventional gyrocompass alignment is determined bychoice of the control gains 119896

1 1198962 119896119873 and 119896

119880

However if the azimuth misalignment is large the scalefactor Ω cos120593 in Figure 1 will become the nonlinear factorΩ cos120593 sdot sin120595

119880120595119880 The characteristic equation of conven-

tional gyrocompass alignment becomes

Δ (119904) = 1199042(119904 + 119896

1) (119904 + 119896

2) + 119904 (119904 + 119896

2) 119892119896119873

+119892Ω cos120593119896

119880

119896 (120595119880)

= 0

(6)

where 119896(120595119880) denotes the nonlinear factor 120595

119880 sin120595

119880 Then

only if the values of the control gains are set as

1198961= 1198962= 2120585120596

119899 (7)

119896119873=

1205962

119899(1 + 120585

2)

119892 (8)

119896119880=119896 (120595119880) sdot 12058521205964

119899

119892Ω cos120593(9)

can the four eigenvalues of conventional gyrocompass align-ment system then be set as 119904

12= minus120585120596

119899and 11990434

= minus120585120596119899plusmn

119895radic1 minus 1205852120596119899(desired values) Since 120595

119880cannot be obtained

119896(120595119880) is an uncertain factor Thus the values of the control

Mathematical Problems in Engineering 3

0 45 90 135 180

102

101

k(120595

U)

|120595U| (deg)

Figure 2 Functional relationship between 119896(120595119880) and |120595

119880|

gains cannot be set as (7) (8) and (9) In fact under the largeazimuthmisalignment the values of the control gains are stillset as (3) (4) and (5) Then four eigenvalues of conventionalgyrocompass alignment cannot be set as the desired values Inthis case the conventional method is hard to favorably meetthe requirement of alignment speedThe particular reason ofthis problem also can be expressed by the following equation

119896119880=

1198961015840

119880

119896 (120595119880) (10)

where 1198961015840119880is the theoretical value calculated by (9) and 119896

119880is

the actual value obtained by (5) From (10) it is clear that theactual value 119896

119880adopted by the alignment system is smaller

than the theoretical value 1198961015840119880 due to the fact that 119896(120595

119880) ge

1 (domain 120595119880isin [minus180

∘ 180∘]) The functional relationship

between 119896(120595119880) and |120595

119880| is shown in Figure 2

It is obvious from Figure 2 that the value of 119896(120595119880) is

increasing along with the growth of |120595119880| then the difference

between 119896119880and 1198961015840

119880will increase together with the growth

of |120595119880| That means the difference between the actual and

desired eigenvalues increases along with the growth of |120595119880|

however the performance of the alignment system will bepoor

3 Improved Gyrocompass Alignment Method

Considering the poor performance of conventional gyro-compass alignment under the condition of large azimuthmisalignment an improved gyrocompass alignment methodis proposed in this paper to improve the performance ofgyrocompass alignment It is clear that the four eigenvaluesof conventional gyrocompass alignment cannot be set as thedesired values in the case of large azimuth misalignmentbecause the uncertain nonlinear factor is included in thecharacteristic equation So the key to design the improvedmethod is to reduce the impact of the nonlinear factor oncharacteristic equation This problem is solved by opening

the azimuth loop at this time the influence of the nonlin-ear factor on characteristic equation disappeared and theestimation of the azimuth misalignment is implemented byusing the horizontal velocity signals to estimate the nonlinearfactors

31TheEstablishment of the ImprovedGyrocompassAlignmentMethod In this paper 119894 119890 119887 119899 and 119901 denote the inertialframe the earth fixed coordinate frame the sensor bodyframe the navigation frame and the computed navigationframe respectively In this work we choose the local levelgeographic coordinate frame as the navigation frame Underthe large azimuth misalignment the direction cosine matrix(DCM) C119901

119899can be described as follows [13 14]

C119901119899

=[[

[

cos120595119880

sin120595119880

minus120595119873

minus sin120595119880

cos120595119880

120595119864

120595119873cos120595119880+ 120595119864sin120595119880120595119873sin120595119880minus 120595119864cos120595119880

1

]]

]

(11)

where 120595119873is the north level misalignment 120595

119864and 120595

119873(level

misalignments) are small and 120595119880(azimuth misalignment

the departure of computed north from actual north) is largeFor a quasistationary initial alignment the average value

for velocity will be zero [15] Then under quasistationary sit-uations the navigation equations for strapdown INS attitudeand velocity can be represented respectively as

C119901119887= C119901119887[120596119887

119901119887times] (12)

V119901 = C119901119887f119887 + g119899 (13)

where C119901119887represents the DCM relating the transformation

from 119887 frame to the 119901 frame 120596119887119901119887

is the angular velocityvector of 119887 frame with respect to 119901 frame resolved in 119887

frame [times] denotes a skew-symmetric matrix operator V119901 isthe computed velocity of strapdown INS f119887 is the specificforce from the accelerometer output g119899 is the gravity vectorresolved in 119899 frame The angular velocity 120596119887

119901119887is derived by

120596119887

119901119887= 119887

119894119887minus C119887119901120596119899

119894119890 (14)

where 119887119894119887is the angular velocity vector measured by gyros

120596119899

119894119890is the angular velocity vector of 119890 frame with respect to 119894

frame resolved in 119899 frameThese angular velocity vectors 119887

119894119887 120596119899119894119890and accelerometer

output f119887 can be described respectively as

119887

119894119887= 120596119887

119894119887+ 120576119887 (15a)

120596119899

119894119890= [0 Ω cos120593 Ω sin120593]119879 (15b)

f119887 = minusg119887 + 120575a119887 + nabla119887 (15c)

where 120596119887119894119887is the true angular velocity vector 120576119887 is the gyro

error vector g119887 is the gravity vector resolved in 119887 frame

4 Mathematical Problems in Engineering

120575a119887 is the disturbing motion vector under quasistationarysituations and nabla119887 is the accelerometer error vector

Substituting (15c) in (13) we obtain

V119901 = (I minus C119901119899) g119899 + C119901

119887120575a119887 + C119901

119887nabla119887 (16)

Then from (16) the horizontal components of V119901 can bedescribed respectively as

119864= minus119892120595

119873+ (C119901119887120575a119887)119864+ (C119901119887nabla119887)119864

119873= 119892120595119864+ (C119901119887120575a119887)119873+ (C119901119887nabla119887)119873

(17)

where 119881119864and 119881

119873represent the east and north horizontal

velocities respectively (sdot)119864and (sdot)

119873represent the 119864 and 119873

elements of column matrix (sdot) separately whose entries fromtop to bottom are the 119864119873 and 119880 elements

From (11) (12) and (14) the misalignment equation ofstrapdown INS under large azimuth misalignment can berepresented as [11 13]

= (I minus C119901119899)120596119899

119894119890minus C119901119887120576119887 (18)

where 120595 is the misalignment vector and 120595 = [120595119864 120595119873 120595119880]119879

The leveling alignment of the improved method is per-formed by introducing control angular velocity 120596119901

119888into the

calculation of angular velocity 120596119887119901119887 then we have 120596119887

119901119887= 119887

119894119887minus

C119887119901120596119899

119894119890minus C119887119901120596119901

119888 finally the misalignment equation could be

transformed into

= (I minus C119901119899)120596119899

119894119890minus C119901119887120576119887+ 120596119901

119888 (19)

where 120596119901119888= [120596119901

119888119864120596119901

1198881198730]119879

is the control angular velocityvector From (19) the horizontal components of 120595 can berespectively represented as

119864= minus sin120595

119880Ω cos120593 + 120595

119873Ω sin120593 minus (C119901

119887120576119887)119864+ 120596119901

119888119864 (20)

119873= (1 minus cos120595

119880)Ω cos120593 minus 120595

119864Ω sin120593 minus (C119901

119887120576119887)119873

+ 120596119901

119888119873

(21)

Because the azimuth misalignmentrsquos rate of change is small[10] the approximation

119880= 0 is admitted namely 120595

119880is a

constant value [15]The azimuth information is obtained by structuring

azimuth estimation functions The block diagram of 120596119901119888and

azimuth estimation functions is shown in Figure 3In Figure 3 119896

119886and 119896119887represent the control gains119866(119904) and

119867(119904) represent the control networks used for reducing theinfluence of disturbing motions 119878(120595

119880) and 119862(120595

119880) represent

the functions of 120595119880 From (17) (20) (21) and Figure 3 the

error diagrams of improved gyrocompass alignment can beobtained as in Figures 4 and 5

It can be seen from Figures 4 and 5 that the nonlinearfactors sin120595

119880Ω cos120593 and (1minuscos120595

119880)Ω cos120593 both are caused

by the large azimuth misalignment and become the constantinputs of the system through opening the azimuth loop So

minus

minus

ka

ka

G(s)

S(120595U)

P

N

E

minuskb

kb

H(s)

1

s

1

s

C(120595U)

120596pcE

120596pcN

120596pc

V

V

V

Figure 3 Block diagram of 120596119901119888and azimuth estimation functions

ka

1

s

1

s

+

+minus

minus

minus

minus

g

G(s)

kb

120595E

S(120595U)sin 120595UΩ cos 120593 120595NΩ sin 120593

(Cpb120576b)

E

(Cp b120575ab) N

+(C

p bnablab) N

Figure 4 Error diagram of improved gyrocompass alignment innorth loop

+

+

minus

minus

minus

minus

g

kb

ka

1

s

1

s

H(s)

120595N

C(120595U)(1 minus cos120595U)Ωcos 120593 120595EΩ sin 120593

(Cpb120576b)

N

(Cp b120575ab) E

+(C

p bnablab) E

Figure 5 Error diagram of improved gyrocompass alignment ineast loop

the nonlinear factors will not be included in the characteristicequation In addition these constant inputs contain azimuthmisalignment information and the estimation of the azimuthinformation can be obtained by 119878(120595

119880) and 119862(120595

119880)

32 The Operation of the Improved Gyrocompass AlignmentMethod In this section the control gains 119896

119886and 119896

119887and

control networks 119866(119904) and 119867(119904) are provided Then 119878(120595119880)

and 119862(120595119880) can be obtained and the azimuth estimation is

performed by these two functions

Mathematical Problems in Engineering 5

Firstly we provide the values of the control gains 119896119886and 119896119887

for improved gyrocompass alignment the detailed deductionfor these control gains can be found in the following

According to Mason gain formula we can get the charac-teristic equation of north loop from Figure 4

Δ119873 (119904) = 119904

2+ 119896119886119904 + 119892119896

119887= 0 (22)

In the same way from Figure 5 the characteristic equation ofeast loop can be obtained as follows

Δ119864 (119904) = 119904

2+ 119896119886119904 + 119892119896

119887= 0 (23)

It can be seen from (22) and (23) that the north and east loopshave the same characteristic equation In order to ensure thestability of alignment system we set two eigenvalues of (22)and (23) as

11990412= minus120585120596

119899plusmn 119895radic1 minus 1205852120596

119899 (24)

Then the characteristic equation can be rewritten as

Δ119873 (119904) = Δ119864 (119904) = 119904

2+ 2120585120596

119899119904 + 1205962

119899= 0 (25)

Comparing (22) with (25) we can get

119896119886= 2120585120596

119899

119896119887=1205962

119899

119892

(26)

Secondly in order to simplify the subsequent analysis weassume that the inertial sensor errors are basically constantdrifts and the vehicle in which the INS is mounted is totallystopped Under this assumption the disturbing motion vec-tor 120575a119887 is equal to zero At this time the control networks canbe set as 119866(119904) = 119867(119904) = 1 because no disturbing motion isintroduced into the alignment system The design procedureof azimuth estimation is described in the following and itcan be divided into two steps The first step the analysisof the relationships between functions 119878(120595

119880) and 119862(120595

119880)

and azimuth misalignment 120595119880is made with the aid of the

Laplace transformationThe second step based on the formeranalysis the azimuth estimation equations are provided

321 Relationships between Functions 119878(120595119880) and 119862(120595

119880) and

Azimuth Misalignment 120595119880 The response to east level mis-

alignment 120595119864and function 119878(120595

119880) can be written in Laplace

form directly from Figure 4 and can be seen as follows(according to the Mason gain formula)

120595119864 (119904)

= minus1205962

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot

(C119901119887nabla119887)119873 (119904)

119892

+119904 + 2120585120596

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot minus sin120595119880Ω cos120593 + 120595

119873Ω sin120593 minus (C119901

119887120576119887)119864 (119904)

119878 (120595119880) (119904)

=119904

1199042 + 2120585120596119899119904 + 1205962119899

sdot (C119901119887nabla119887)119873 (119904)

+119892

1199042 + 2120585120596119899119904 + 1205962119899

sdot minus sin120595119880Ω cos120593 + 120595

119873Ω sin120593 minus (C119901

119887120576119887)119864 (119904)

(27)

where sdot(119904) represents the Laplace transformation of sdotSimilarly from Figure 5 it yields

120595119873 (119904) =

1205962

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot

(C119901119887nabla119887)119864 (119904)

119892

+119904 + 2120585120596

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot (1 minus cos120595119880)Ω cos120593 minus 120595

119864Ω sin120593 minus (C119901

119887120576119887)119873

sdot (119904)

119862 (120595119880) (119904) =

119904

1199042 + 2120585120596119899119904 + 1205962119899

sdot (C119901119887nabla119887)119864 (119904)

minus119892

1199042 + 2120585120596119899119904 + 1205962119899

sdot (1 minus cos120595119880)Ω cos120593 minus 120595

119864Ω sin120593 minus (C119901

119887120576119887)119873

sdot (119904)

(28)

Equations (27)-(28) are solved for the steady-state errors byinvoking the final value theorem

lim119905rarrinfin

119891 (119905) = lim119904rarr0

119904119891 (119904) (29)

6 Mathematical Problems in Engineering

Then we have

120595119864ss = minus

(C119901119887nabla119887)119873

119892+2120585

120596119899

sdot [minus sin120595119880Ω cos120593

+ 120595119873ssΩ sin120593 minus (C119901

119887120576119887)119864]

119878 (120595119880)ss =

119892

1205962119899

sdot [minus sin120595119880Ω cos120593 + 120595

119873ssΩ sin120593

minus (C119901119887120576119887)119864]

120595119873ss =

(C119901119887nabla119887)119864

119892+2120585

120596119899

sdot [(1 minus cos120595119880)Ω cos120593

minus 120595119864ssΩ sin120593 minus (C119901

119887120576119887)119873]

119862 (120595119880)ss =

119892

1205962119899

sdot [(cos120595119880minus 1)Ω cos120593 + 120595

119864ssΩ sin120593

+ (C119901119887120576119887)119873]

(30)

where (sdot)ss represents the steady-state value of (sdot)We regard sin120595

119880 cos120595

119880 120595119864ss and 120595119873ss as the unknown

values (30) can then be easily solved Then the relationshipsbetween functions 119878(120595

119880) and 119862(120595

119880) and azimuth misalign-

ment 120595119880can be obtained

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593+

(C119901119887nabla119887)119873

119892tan120593

(31)

The detailed derivations of (31) can be found in Appendix A

322 The Azimuth Estimation Equations It is clear thatsin120595119880

and cos120595119880

can be supplied exactly by (31) if theaccelerometer errors and gyro errors (sensor errors) areknown Since the sensor errors are uncertain sin120595

119880and

cos120595119880are calculated by the following equations

sin 119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880) minus

2120585120596119899tan120593119892

sdot 119862 (120595119880) (32)

cos 119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880) minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)

(33)

where sin 119880and cos

119880are the computed values of sin120595

119880

and cos120595119880 respectively When the improved alignment

system is stable the errors between computed values (sin 119880

and cos 119880) and theoretical values (sin120595

119880and cos120595

119880) are

only determined by sensor errors and these errors areallowable Worth noting is that (32) and (33) do not preservethe unit-norm property of the trigonometric function thatis (sin

119880)2+ (cos

119880)2

= 1 So normalization of sin 119880

and cos 119880should be given and the simple normalization is

provided as follows

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(34)

The detailed derivations of (34) can be found in Appendix BAt this time the azimuth estimation

119880can be obtained

from (34) and is shown as follows

119880119898

= arcsin (sin 119880)

119880=

119880119898

cos 119880gt 0

180∘minus 119880119898

cos 119880lt 0 sin

119880gt 0

minus180∘minus 119880119898

cos 119880lt 0 sin

119880lt 0

(35)

It is obvious from the previous discussion that the azimuthestimation equations consist of (32)ndash(35)

Finally we take the disturbing motions into consid-eration because the alignment is often performed underquasistationary conditions The disturbing motions can begenerally considered to be sinusoidal [4 16] Then in orderto reduce the influence of the disturbing motions on azimuthestimation low pass filters need to be added to the systemnamely the control networks 119866(119904) and119867(119904) should have thecapability of restraining disturbance The design law of thecontrol networks 119866(119904) and119867(119904) is provided as follows

(a) The magnification of 119866(119904) and119867(119904) in low frequencymust be equivalent to 1 that is the following equationshould be satisfied

lim119904rarr0

119866 (119904) = lim119904rarr0

119867(119904) = 1 (36)

The reason for this requirement is that the rela-tionships between functions 119878(120595

119880) and 119862(120595

119880) and

azimuthmisalignment120595119880under quasistationary con-

ditions also can be represented by (31) as the require-ment is met

(b) In high frequency for the purpose of reducing theinfluence of disturbing motions they need be capableof restraining disturbance

Then under quasistationary conditions the azimuth estima-tion equations also consist of (32)ndash(35) In this work thecontrol networks 119866(119904) and119867(119904) are set as

119866 (119904) = 119867 (119904) =(120585120596119899)2

(119904 + 120585120596119899)2 (37)

Mathematical Problems in Engineering 7

bib

minus

minus

Gyros

AccelerometersCalculation of control angular

velocity and azimuth estimation functions

DSP

Final process

S(120595U) C(120595U)

120596bpb

E

N

U

120596n

ieCbp

pb =

pb[120596bpb times]CC

120596p

cCbp

fb

Vp= Cp

bfb+ gn

sin 120595U

cos 120595U

01

+

minus1205962n

gΩ cos 120593minus2120585120596n tan 120593

g

minus2120585120596n tan 120593

g

1205962n

gΩ cos 120593

S(120595U)C(120595U)lfloor

lceilrfloorrceillfloorlceillfloorlceil rfloorrceilrfloorrceil

V

V

Figure 6 Flow chart of improved gyrocompass alignment in real time operating device

At this time the denominators of 119878(120595119880)(119904) and 119862(120595

119880)(119904)

can be set as

(1199042+ 2120585120596

119899119904 + 1205962

119899) (119904 + 120585120596

119899)2 (38)

It is obvious from the azimuth estimation equations that theperformance of the azimuth alignment is determined by (32)and (33) Since both (32) and (33) consist of 119878(120595

119880) and119862(120595

119880)

the four eigenvalues of sin 119880(119904) and cos

119880(119904) could be

set as 11990412

= minus120585120596119899and 11990434

= minus120585120596119899plusmn 119895radic1 minus 1205852120596

119899 without

the influence of the nonlinear factors Therefore the impactof the nonlinear factor caused by large azimuthmisalignmenton gyrocompass alignment is reduced and the performanceof the improved gyrocompass alignment will be better thanthe conventional one

33 The Implementation of the Improved Gyrocompass Align-ment Method In this section the implementation of theproposed method in real time operating device is discussedand meanwhile a brief summary of this method is providedAn inertial navigation system implements the proposedalignment method using a cluster of accelerometers to sensethe specific force vector components f119887 a triad of gyrosto measure the angular velocity 119887

119894119887 and a digital signal

processor (DSP) to perform the alignment algorithm Thedirect expression of this implementation can be seen inFigure 6 According to Figure 6 the specific implementationsteps of the improved gyrocompass alignment in real timeoperation device are described as follows

(a) At the beginning a preliminary alignment oftencalled coarse alignment is performed and after thata rough DCM C119901

119887is obtained The specific coarse

alignment method can be found in [15] the levelmisalignments are small and azimuth misalignmentis usually large

(b) Secondly by utilizing the specific force f119887 and DCMC119901119887 the derivatives of the east and north horizontal

velocities 119864and 119873can be obtainedThen according

to the calculation method shown in Figure 3 thecontrol velocity 120596119901

119888and azimuth functions 119878(120595

119880) and

119862(120595119880) can be acquired

(c) On the one hand the control velocity 120596119901119888and angular

velocity 119887119894119887are used to update the DCM C119901

119887used for

the next calculation step(d) On the other hand the azimuth functions 119878(120595

119880) and

119862(120595119880) are used to compute sin

119880and cos

119880 And

after the final process which is achieved by utilizing(34)-(35) the azimuth estimation

119880is obtained

(e) Then steps (b) (c) and (d) are performed repeatedlyFinally by spending a period of time an accurateazimuth estimation

119880can be obtained and with the

compensation of azimuth misalignment using 119880 an

accurate DCM C119901119887can be obtained

4 Simulations and Experiments

In this section to evaluate the performance of the improvedgyrocompass alignment method under the condition of largeazimuth misalignment simulations and experiments arecarried out

41 Simulation Results and Analysis Both the conventionaland improved gyrocompass alignment methods are per-formed during the simulations simultaneously and the sim-ulations are conducted under the conditions of respec-tively choosing different azimuth misalignments namely120595119880= 50

∘ 100∘ 145∘ minus50∘ minus100

∘ minus145

∘ In these simu-

lations gyro and accelerometer outputs are generated bythe strapdown INS simulator we assume that the vehicleis in disturbing motions caused by sea waves The initialconditions are presented as follows

The constant gyro errors 120576119887 =

[[001∘ℎ 001

∘ℎ 001

∘ℎ]]119879

8 Mathematical Problems in Engineering

0 5 10 15 20

0 0

50

100

150

Time (min) Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0 5 10 15 20

Improved method

minus50

50

100

150

minus50

Figure 7 Gyrocompass alignment for 120595119880= 50

∘ 100∘ 145∘

The constant accelerometer errors nabla119887 =

[[10minus4119892 10minus4119892 10minus4119892]]119879

The acceleration due to gravity 119892 = 978049ms2The local longitude and latitude 120582 = 1266705∘ 120593 =457796

∘The control networks of improved method 119866(119904) =119867(119904) = (120585120596

119899)2(119904 + 120585120596

119899)2

The natural frequency of conventional and improvedmethods 120596

119899= 002 rads

Under the disturbingmotions the vehicle undertakes angularand linealmovements In angularmovement the pitch 120579

119909 the

roll 120579119910 and the yaw 120579

119911are controlled as

120579119909= 7∘ sin(2120587

5119905 +

120587

3)

120579119910= 5∘ sin(2120587

7119905 +

120587

6)

120579119911= 10∘ sin(2120587

6119905 +

120587

4)

(39)

In the lineal movement the vehicle lineal movement veloci-ties are taken as

119881119909= minus03 sin(2120587

10119905) ms2

119881119910= minus02 sin(2120587

6119905) ms2

119881119911= minus04 sin(2120587

8119905) ms2

(40)

Figures 7 and 8 provide the azimuth estimation errors ofconventional and improved methods under the conditions

of 120595119880= 50

∘ 100∘ 145∘ minus50∘ minus100

∘ minus145

∘ It can be seen

that azimuth alignment could be accomplished by both con-ventional and improved gyrocompass alignment methodsBut under the condition of large azimuth misalignment theperformance of conventional method will be poor along withthe growth of |120595

119880| The reason is that the difference between

the actual and desired eigenvalues is increasing alongwith thegrowth of |120595

119880|

Figure 9 shows convergence time for different azimuthmisalignments with a converged azimuth estimation errorof less than 1 degree It is clear that large azimuth misalign-ment needs more time to converge under the conventionalmethod However for the improved method the changeof the convergence time is not obvious That means theperformance of the improved method is not affected bythe large azimuth misalignment The reason is that theeigenvalues of improved method can be set as the desiredvalues without the influence of the nonlinear factors causedby the large azimuth misalignment Moreover it is obviousthat the alignment speed of the improved method is fasterthan that of the conventional one

42 Experimental Results and Analysis The experimentswere implemented in the lab as the true yaw angle ofthe strapdown INS was known so different 120595

119880could be

exactly set We fixed the strapdown INS on the SGT-3 three-axis turntable to implement the alignment experiments Thestrapdown INS and the turntable can be seen in Figure 10 Atthe start of each experiment the turntable turned to a staticposition for 20 minutes that is pitch angle 0∘ roll angle 0∘and yaw angle 135∘ Two alignmentmethods were performedunder the condition of respectively setting initial yaw angleas 185∘ and 235∘ namely 120595

119880= 50∘ 100∘

Mathematical Problems in Engineering 9

0 5 10 15 20

0

50

Time (min)0 5 10 15 20

Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0

50 Improved method

minus50

minus100

minus150

minus50

minus100

minus150

Figure 8 Gyrocompass alignment for 120595119880= minus50

∘ minus100

∘ minus145

∘

50 100 1450

5

10

15

20

Azimuth misalignment (deg)

Con

verg

ence

tim

e (m

in)

Conventional methodImproved method

minus145 minus50minus100

Figure 9 Convergence time comparison with different azimuthmisalignments

Figure 11 provides the comparison of the yaw angleusing the two alignment methods under the condition ofdifferent 120595

119880

As the results above show the yaw angle can convergeto 135∘ through using the two alignment methods How-ever for the conventional method the convergence time islengthened with the growth of the azimuth misalignmentCompared with the conventional method the convergence

Strapdown INS

SGT-3

Figure 10 The SGT-3 turntable and strapdown INS

time of improved method is hardly changed The exper-imental results are consistent with the simulation resultsThe experiments demonstrate that the performance of theimproved method is better than that of the conventional one

5 Conclusions

In this paper an improved gyrocompass alignment methodhas been proposed for large azimuth misalignment of INSThe improved new method is performed by opening theazimuth loop and the estimation of the azimuth misalign-ment is implemented by using the horizontal velocity signalsto estimate the nonlinear factorsThrough using thismethodthe impact of the nonlinear factor caused by large azimuthmisalignment on gyrocompass alignment is reduced andthe performance of gyrocompass alignment is improvedThealignment speed of the improved method is faster than theconventional one in the same conditions The performance

10 Mathematical Problems in Engineering

0 5 10 15 20100

135

150

185200

235250

300

Yaw

(deg

)

Time (min)

Conventional method 120595U = 50∘

Conventional method 120595U = 100∘

Improved method 120595U = 50∘

Improved method 120595U = 100∘

Figure 11 Comparison of the yaw angle using the two alignmentmethods

of the improved method is well validated by simulations andexperiments

Appendices

A The Derivations of (31)

In this section (31) are obtained in the followingAccording to (30) their matrix form can be written as

AX = B (A1)

where

A

=

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

[

2120585Ω cos120593120596119899

0 1 minus2120585Ω sin120593

120596119899

119892Ω cos1205931205962119899

0 0 minus119892Ω sin1205931205962119899

02120585Ω cos120593

120596119899

2120585Ω sin120593120596119899

1

0119892Ω cos1205931205962119899

119892Ω sin1205931205962119899

0

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

]

(A2)

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

(A3)

B =

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

[

minus

(C119901119887nabla119887)119873

119892minus2120585

120596119899

(C119901119887120576119887)119864

minus119878 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119864

2120585

120596119899

Ω cos120593 +(C119901119887nabla119887)119864

119892minus2120585

120596119899

(C119901119887120576119887)119873

119892

1205962119899

Ω cos120593 + 119862 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119873

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

]

(A4)

Multiply the first and third rows of (A1) by1205961198992120585Ω cos120593 and

multiply the second and fourth rows of (A1) by 1205962119899119892Ω cos120593

we can obtain

A1X = B

1 (A5)

where

A1=

[[[[[[[[

[

1 0120596119899

2120585Ω cos120593minus tan120593

1 0 0 minus tan120593

0 1 tan120593120596119899

2120585Ω cos1205930 1 tan120593 0

]]]]]]]]

]

X =[[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]]

]

B1=

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

[

minus120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

1 +120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892minus

(C119901119887120576119887)119873

Ω cos120593

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

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

]

(A6)

Furthermore by utilizing the elementary row transformation(A5) can be transformed into

A2X = B

2 (A7)

Mathematical Problems in Engineering 11

where

A1=

[[[[[[[[

[

1 0 0 minus tan120593

0 0 minus120596119899

2120585Ω cos1205930

0 1 tan120593 0

0 0 0 minus120596119899

2120585Ω cos120593

]]]]]]]]

]

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

B1

=

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

[

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos1205931205962

119899

119892Ω cos120593119862 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892

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

]

(A8)

Then according to matrix equation (A7) we have

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

+

(C119901119887nabla119887)119873

119892tan120593

(A9)

B The Derivations of (34)

The derivations of (34) are carried out based on the principleof least squares fitting setting the Euclidean minimum norm119869 between sin

119880 cos

119880and sin

119880 cos

119880(the normalized

data) as the normalization indexFirst define

119869 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]12

(B1)

It is subjected to the constrained condition namely

ℎ = (sin 119880)2+ (cos

119880)2minus 1 = 0 (B2)

Second define an auxiliary function119898 namely

119898 = 1198692+ 120582ℎ (B3)

where 120582 is an unknown parameterBy substituting (B1) and (B2) into (B3) we can get

119898 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]

minus 120582 ((sin 119880)2+ (cos

119880)2minus 1)

(B4)

Obviously the parameters sin 119880and cos

119880that made 119869

minimum are equal to the ones that make 119898 minimumRespectively the partial derivatives of the parameters sin

119880

and cos 119880are calculated and bymaking them equal to zero

we can get

sin 119880=sin 119880

1 + 120582

cos 119880=cos 119880

1 + 120582

(B5)

Then by substituting (B5) into (B2) we have

(sin 119880)2+ (cos

119880)2

(1 + 120582)2

= 1 (B6)

Finally combining (B5) and (B6) the normalization equa-tions can be obtained

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(B7)

Conflict of Interests

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

Acknowledgments

The work described in the paper was supported by theNational Natural Science Foundation of China (61203225)and the National Science Foundation for Post-DoctoralScientists of China (2012M510083)The authors would like tothank all members of the Inertial Navigation Research Groupat Harbin Engineering University for the technical assistancewith the navigation system

References

[1] H W Park J G Lee and C G Park ldquoCovariance analysis ofstrapdown INS considering gyrocompass characteristicsrdquo IEEETransactions on Aerospace and Electronic Systems vol 31 no 1pp 320ndash328 1995

12 Mathematical Problems in Engineering

[2] R A Nash Jr J A DrsquoAppolito and K J Roy ldquoError analysis ofhybrid aircraft inertial navigation systemsrdquo in Proceedings of theAIAA Guidance and Control Conference Stanford Calif USA1972

[3] F D Jurenka and C T Leondes ldquoOptimum alignment of aninertial autonavigatorrdquo IEEE Transactions on Aerospace andElectronic Systems vol AES-3 no 6 pp 880ndash888 1967

[4] R H Cannon ldquoAlignment of inertial guidance systems bygyrocompassing-linear theoryrdquo Journal of Aerospace Sciencevol 28 no 11 pp 885ndash895 912 1961

[5] G R Pitman Jr Inertial Guidance John Wiley amp Sons NewYork NY USA 1962

[6] K R Britting Inertial Navigation Systems Analysis John Wileyamp Sons New York NY USA 1971

[7] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation vol 28 no4 pp 311ndash341 1982

[8] X Liu X Xu Y Liu and L Wang ldquoA fast and high-accuracycompass alignmentmethod to SINSwith azimuth axis rotationrdquoMathematical Problems in Engineering vol 2013 Article ID524284 12 pages 2013

[9] F Sun Q Wang Z Qi and C Wang ldquoResearch on theestimation method of DVL velocity error based on doubleprogram in Fiber Optic Gyro SINSrdquo Optik vol 124 no 22 pp5344ndash5349 2013

[10] B M Scherzinger ldquoInertial navigator error models for largeheading uncertaintyrdquo in Proceedings of the IEEE Position Loca-tion and Navigation Symposium (PLANS rsquo96) pp 477ndash484Atlanta Ga USA April 1996

[11] X Y Kong EM Nebot andH Durrant-Whyte ldquoDevelopmentof a non-linear psi-angle model for large misalignment errorsand its application in INS alignment and calibrationrdquo in Pro-ceedings of the 1999 IEEE International Conference on Roboticsamp Automation (ICRA rsquo99) pp 1430ndash1435 Detroit Mich USAMay 1999

[12] M-J Yu H-W Park and C-B Jeon ldquoEquivalent nonlinearerror models of strapdown inertial navigation systemrdquo in Pro-ceedings of the Guidance Navigation and Control ConferenceAIAA-97-3563 pp 581ndash587 1997

[13] T Abbas Z Yunyan and L Yanjun ldquoSINS initial alignment forsmall tilt and large azimuth misalignment anglesrdquo in Proceed-ings of the IEEE 3rd International Conference on CommunicationSoftware and Networks (ICCSN rsquo11) pp 628ndash632 Xirsquoan ChinaMay 2011

[14] S L Han and J L Wang ldquoA novel initial alignment scheme forlow-cost INS aided by GPS for land vehicle applicationsrdquo TheJournal of Navigation vol 63 no 4 pp 663ndash680 2010

[15] P G Savage Strapdown Analytics Strapdown Associates 2ndedition 2007

[16] I Yaesh and B Priel ldquoDesign of leveling loop for marine nav-igation systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 29 no 2 pp 599ndash604 1993

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

nablaN +

+

minus

minus

minus

minus

minus

g

1

s

1

s

1

skN

k1

120576U

120576E120595E

120595U

Ω cos 120593 Ω cos 120593 middotsin120595U

120595U

Large azimuthmisalignment

kUs + k2

Figure 1 Error diagram of conventional gyrocompass alignment

The remainder of this paper is organized as follows Section 2describes the conventional gyrocompass alignment methodand then analyzes the existing problem of this method indetail Section 3 details the improved gyrocompass alignmentmethod which includes the establishment operation andimplementation of the improved method Simulation andexperimental results that validate the proposed approach arepresented and discussed in Section 4 Finally conclusions aregiven in Section 5

2 Conventional GyrocompassAlignment Method

In literatures [8 9] a fourth-order gyrocompass alignmentsystem was used for INS alignment It is a higher-ordersystem which has a good performance in the case of smallazimuth misalignment In this paper the conventional gyro-compass alignmentmethod analyzed here is the fourth-ordersystem (as the representative of the similar methodologies)and the form of this system is simplified The error diagramof reduced conventional gyrocompass alignment is shown inFigure 1

In Figure 1nabla119873is the north accelerometer error 120576

119864and 120576119880

are the east and up gyro errors 120595119864and 120595

119880represent the east

level and azimuth misalignments respectively 119892 representsthe acceleration due to gravity Ω represents the earth rateand 120593 represents the local latitude 119896

1 1198962 119896119873 and 119896

119880are the

control gainsAccording to the Mason gain formula we can get the

characteristic equation of conventional gyrocompass align-ment from Figure 1

Δ (119904) = 1199042(119904 + 119896

1) (119904 + 119896

2) + 119904 (119904 + 119896

2) 119892119896119873

+ 119892Ω cos120593119896119880= 0

(1)

In order to ensure the stability of the alignment system foureigenvalues of (1) are set as 119904

12= minus120585120596

119899and 11990434

= minus120585120596119899plusmn

119895radic1 minus 1205852120596119899 where 120585 = 0707 and 120596

119899is the natural frequency

which is adjustable Then (1) can be rewritten as

(1199042+ 2120585120596

119899119904 + 1205962

119899) (119904 + 120585120596

119899)2= 0 (2)

Comparing (1) with (2) the values of the control gains underthe small angle can be obtained

1198961= 1198962= 2120585120596

119899 (3)

119896119873=

1205962

119899(1 + 120585

2)

119892 (4)

119896119880=

12058521205964

119899

119892Ω cos120593 (5)

In the knowledge of a classical control theory it is obvi-ous that the alignment speed of gyrocompass alignment isdetermined by the eigenvalues Worth noting is that thefour eigenvalues of the conventional gyrocompass alignmentsystem can be given any value desired by appropriate choiceof the control gains In other words the alignment speedof conventional gyrocompass alignment is determined bychoice of the control gains 119896

1 1198962 119896119873 and 119896

119880

However if the azimuth misalignment is large the scalefactor Ω cos120593 in Figure 1 will become the nonlinear factorΩ cos120593 sdot sin120595

119880120595119880 The characteristic equation of conven-

tional gyrocompass alignment becomes

Δ (119904) = 1199042(119904 + 119896

1) (119904 + 119896

2) + 119904 (119904 + 119896

2) 119892119896119873

+119892Ω cos120593119896

119880

119896 (120595119880)

= 0

(6)

where 119896(120595119880) denotes the nonlinear factor 120595

119880 sin120595

119880 Then

only if the values of the control gains are set as

1198961= 1198962= 2120585120596

119899 (7)

119896119873=

1205962

119899(1 + 120585

2)

119892 (8)

119896119880=119896 (120595119880) sdot 12058521205964

119899

119892Ω cos120593(9)

can the four eigenvalues of conventional gyrocompass align-ment system then be set as 119904

12= minus120585120596

119899and 11990434

= minus120585120596119899plusmn

119895radic1 minus 1205852120596119899(desired values) Since 120595

119880cannot be obtained

119896(120595119880) is an uncertain factor Thus the values of the control

Mathematical Problems in Engineering 3

0 45 90 135 180

102

101

k(120595

U)

|120595U| (deg)

Figure 2 Functional relationship between 119896(120595119880) and |120595

119880|

gains cannot be set as (7) (8) and (9) In fact under the largeazimuthmisalignment the values of the control gains are stillset as (3) (4) and (5) Then four eigenvalues of conventionalgyrocompass alignment cannot be set as the desired values Inthis case the conventional method is hard to favorably meetthe requirement of alignment speedThe particular reason ofthis problem also can be expressed by the following equation

119896119880=

1198961015840

119880

119896 (120595119880) (10)

where 1198961015840119880is the theoretical value calculated by (9) and 119896

119880is

the actual value obtained by (5) From (10) it is clear that theactual value 119896

119880adopted by the alignment system is smaller

than the theoretical value 1198961015840119880 due to the fact that 119896(120595

119880) ge

1 (domain 120595119880isin [minus180

∘ 180∘]) The functional relationship

between 119896(120595119880) and |120595

119880| is shown in Figure 2

It is obvious from Figure 2 that the value of 119896(120595119880) is

increasing along with the growth of |120595119880| then the difference

between 119896119880and 1198961015840

119880will increase together with the growth

of |120595119880| That means the difference between the actual and

desired eigenvalues increases along with the growth of |120595119880|

however the performance of the alignment system will bepoor

3 Improved Gyrocompass Alignment Method

Considering the poor performance of conventional gyro-compass alignment under the condition of large azimuthmisalignment an improved gyrocompass alignment methodis proposed in this paper to improve the performance ofgyrocompass alignment It is clear that the four eigenvaluesof conventional gyrocompass alignment cannot be set as thedesired values in the case of large azimuth misalignmentbecause the uncertain nonlinear factor is included in thecharacteristic equation So the key to design the improvedmethod is to reduce the impact of the nonlinear factor oncharacteristic equation This problem is solved by opening

the azimuth loop at this time the influence of the nonlin-ear factor on characteristic equation disappeared and theestimation of the azimuth misalignment is implemented byusing the horizontal velocity signals to estimate the nonlinearfactors

31TheEstablishment of the ImprovedGyrocompassAlignmentMethod In this paper 119894 119890 119887 119899 and 119901 denote the inertialframe the earth fixed coordinate frame the sensor bodyframe the navigation frame and the computed navigationframe respectively In this work we choose the local levelgeographic coordinate frame as the navigation frame Underthe large azimuth misalignment the direction cosine matrix(DCM) C119901

119899can be described as follows [13 14]

C119901119899

=[[

[

cos120595119880

sin120595119880

minus120595119873

minus sin120595119880

cos120595119880

120595119864

120595119873cos120595119880+ 120595119864sin120595119880120595119873sin120595119880minus 120595119864cos120595119880

1

]]

]

(11)

where 120595119873is the north level misalignment 120595

119864and 120595

119873(level

misalignments) are small and 120595119880(azimuth misalignment

the departure of computed north from actual north) is largeFor a quasistationary initial alignment the average value

for velocity will be zero [15] Then under quasistationary sit-uations the navigation equations for strapdown INS attitudeand velocity can be represented respectively as

C119901119887= C119901119887[120596119887

119901119887times] (12)

V119901 = C119901119887f119887 + g119899 (13)

where C119901119887represents the DCM relating the transformation

from 119887 frame to the 119901 frame 120596119887119901119887

is the angular velocityvector of 119887 frame with respect to 119901 frame resolved in 119887

frame [times] denotes a skew-symmetric matrix operator V119901 isthe computed velocity of strapdown INS f119887 is the specificforce from the accelerometer output g119899 is the gravity vectorresolved in 119899 frame The angular velocity 120596119887

119901119887is derived by

120596119887

119901119887= 119887

119894119887minus C119887119901120596119899

119894119890 (14)

where 119887119894119887is the angular velocity vector measured by gyros

120596119899

119894119890is the angular velocity vector of 119890 frame with respect to 119894

frame resolved in 119899 frameThese angular velocity vectors 119887

119894119887 120596119899119894119890and accelerometer

output f119887 can be described respectively as

119887

119894119887= 120596119887

119894119887+ 120576119887 (15a)

120596119899

119894119890= [0 Ω cos120593 Ω sin120593]119879 (15b)

f119887 = minusg119887 + 120575a119887 + nabla119887 (15c)

where 120596119887119894119887is the true angular velocity vector 120576119887 is the gyro

error vector g119887 is the gravity vector resolved in 119887 frame

4 Mathematical Problems in Engineering

120575a119887 is the disturbing motion vector under quasistationarysituations and nabla119887 is the accelerometer error vector

Substituting (15c) in (13) we obtain

V119901 = (I minus C119901119899) g119899 + C119901

119887120575a119887 + C119901

119887nabla119887 (16)

Then from (16) the horizontal components of V119901 can bedescribed respectively as

119864= minus119892120595

119873+ (C119901119887120575a119887)119864+ (C119901119887nabla119887)119864

119873= 119892120595119864+ (C119901119887120575a119887)119873+ (C119901119887nabla119887)119873

(17)

where 119881119864and 119881

119873represent the east and north horizontal

velocities respectively (sdot)119864and (sdot)

119873represent the 119864 and 119873

elements of column matrix (sdot) separately whose entries fromtop to bottom are the 119864119873 and 119880 elements

From (11) (12) and (14) the misalignment equation ofstrapdown INS under large azimuth misalignment can berepresented as [11 13]

= (I minus C119901119899)120596119899

119894119890minus C119901119887120576119887 (18)

where 120595 is the misalignment vector and 120595 = [120595119864 120595119873 120595119880]119879

The leveling alignment of the improved method is per-formed by introducing control angular velocity 120596119901

119888into the

calculation of angular velocity 120596119887119901119887 then we have 120596119887

119901119887= 119887

119894119887minus

C119887119901120596119899

119894119890minus C119887119901120596119901

119888 finally the misalignment equation could be

transformed into

= (I minus C119901119899)120596119899

119894119890minus C119901119887120576119887+ 120596119901

119888 (19)

where 120596119901119888= [120596119901

119888119864120596119901

1198881198730]119879

is the control angular velocityvector From (19) the horizontal components of 120595 can berespectively represented as

119864= minus sin120595

119880Ω cos120593 + 120595

119873Ω sin120593 minus (C119901

119887120576119887)119864+ 120596119901

119888119864 (20)

119873= (1 minus cos120595

119880)Ω cos120593 minus 120595

119864Ω sin120593 minus (C119901

119887120576119887)119873

+ 120596119901

119888119873

(21)

Because the azimuth misalignmentrsquos rate of change is small[10] the approximation

119880= 0 is admitted namely 120595

119880is a

constant value [15]The azimuth information is obtained by structuring

azimuth estimation functions The block diagram of 120596119901119888and

azimuth estimation functions is shown in Figure 3In Figure 3 119896

119886and 119896119887represent the control gains119866(119904) and

119867(119904) represent the control networks used for reducing theinfluence of disturbing motions 119878(120595

119880) and 119862(120595

119880) represent

the functions of 120595119880 From (17) (20) (21) and Figure 3 the

error diagrams of improved gyrocompass alignment can beobtained as in Figures 4 and 5

It can be seen from Figures 4 and 5 that the nonlinearfactors sin120595

119880Ω cos120593 and (1minuscos120595

119880)Ω cos120593 both are caused

by the large azimuth misalignment and become the constantinputs of the system through opening the azimuth loop So

minus

minus

ka

ka

G(s)

S(120595U)

P

N

E

minuskb

kb

H(s)

1

s

1

s

C(120595U)

120596pcE

120596pcN

120596pc

V

V

V

Figure 3 Block diagram of 120596119901119888and azimuth estimation functions

ka

1

s

1

s

+

+minus

minus

minus

minus

g

G(s)

kb

120595E

S(120595U)sin 120595UΩ cos 120593 120595NΩ sin 120593

(Cpb120576b)

E

(Cp b120575ab) N

+(C

p bnablab) N

Figure 4 Error diagram of improved gyrocompass alignment innorth loop

+

+

minus

minus

minus

minus

g

kb

ka

1

s

1

s

H(s)

120595N

C(120595U)(1 minus cos120595U)Ωcos 120593 120595EΩ sin 120593

(Cpb120576b)

N

(Cp b120575ab) E

+(C

p bnablab) E

Figure 5 Error diagram of improved gyrocompass alignment ineast loop

the nonlinear factors will not be included in the characteristicequation In addition these constant inputs contain azimuthmisalignment information and the estimation of the azimuthinformation can be obtained by 119878(120595

119880) and 119862(120595

119880)

32 The Operation of the Improved Gyrocompass AlignmentMethod In this section the control gains 119896

119886and 119896

119887and

control networks 119866(119904) and 119867(119904) are provided Then 119878(120595119880)

and 119862(120595119880) can be obtained and the azimuth estimation is

performed by these two functions

Mathematical Problems in Engineering 5

Firstly we provide the values of the control gains 119896119886and 119896119887

for improved gyrocompass alignment the detailed deductionfor these control gains can be found in the following

According to Mason gain formula we can get the charac-teristic equation of north loop from Figure 4

Δ119873 (119904) = 119904

2+ 119896119886119904 + 119892119896

119887= 0 (22)

In the same way from Figure 5 the characteristic equation ofeast loop can be obtained as follows

Δ119864 (119904) = 119904

2+ 119896119886119904 + 119892119896

119887= 0 (23)

It can be seen from (22) and (23) that the north and east loopshave the same characteristic equation In order to ensure thestability of alignment system we set two eigenvalues of (22)and (23) as

11990412= minus120585120596

119899plusmn 119895radic1 minus 1205852120596

119899 (24)

Then the characteristic equation can be rewritten as

Δ119873 (119904) = Δ119864 (119904) = 119904

2+ 2120585120596

119899119904 + 1205962

119899= 0 (25)

Comparing (22) with (25) we can get

119896119886= 2120585120596

119899

119896119887=1205962

119899

119892

(26)

Secondly in order to simplify the subsequent analysis weassume that the inertial sensor errors are basically constantdrifts and the vehicle in which the INS is mounted is totallystopped Under this assumption the disturbing motion vec-tor 120575a119887 is equal to zero At this time the control networks canbe set as 119866(119904) = 119867(119904) = 1 because no disturbing motion isintroduced into the alignment system The design procedureof azimuth estimation is described in the following and itcan be divided into two steps The first step the analysisof the relationships between functions 119878(120595

119880) and 119862(120595

119880)

and azimuth misalignment 120595119880is made with the aid of the

Laplace transformationThe second step based on the formeranalysis the azimuth estimation equations are provided

321 Relationships between Functions 119878(120595119880) and 119862(120595

119880) and

Azimuth Misalignment 120595119880 The response to east level mis-

alignment 120595119864and function 119878(120595

119880) can be written in Laplace

form directly from Figure 4 and can be seen as follows(according to the Mason gain formula)

120595119864 (119904)

= minus1205962

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot

(C119901119887nabla119887)119873 (119904)

119892

+119904 + 2120585120596

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot minus sin120595119880Ω cos120593 + 120595

119873Ω sin120593 minus (C119901

119887120576119887)119864 (119904)

119878 (120595119880) (119904)

=119904

1199042 + 2120585120596119899119904 + 1205962119899

sdot (C119901119887nabla119887)119873 (119904)

+119892

1199042 + 2120585120596119899119904 + 1205962119899

sdot minus sin120595119880Ω cos120593 + 120595

119873Ω sin120593 minus (C119901

119887120576119887)119864 (119904)

(27)

where sdot(119904) represents the Laplace transformation of sdotSimilarly from Figure 5 it yields

120595119873 (119904) =

1205962

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot

(C119901119887nabla119887)119864 (119904)

119892

+119904 + 2120585120596

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot (1 minus cos120595119880)Ω cos120593 minus 120595

119864Ω sin120593 minus (C119901

119887120576119887)119873

sdot (119904)

119862 (120595119880) (119904) =

119904

1199042 + 2120585120596119899119904 + 1205962119899

sdot (C119901119887nabla119887)119864 (119904)

minus119892

1199042 + 2120585120596119899119904 + 1205962119899

sdot (1 minus cos120595119880)Ω cos120593 minus 120595

119864Ω sin120593 minus (C119901

119887120576119887)119873

sdot (119904)

(28)

Equations (27)-(28) are solved for the steady-state errors byinvoking the final value theorem

lim119905rarrinfin

119891 (119905) = lim119904rarr0

119904119891 (119904) (29)

6 Mathematical Problems in Engineering

Then we have

120595119864ss = minus

(C119901119887nabla119887)119873

119892+2120585

120596119899

sdot [minus sin120595119880Ω cos120593

+ 120595119873ssΩ sin120593 minus (C119901

119887120576119887)119864]

119878 (120595119880)ss =

119892

1205962119899

sdot [minus sin120595119880Ω cos120593 + 120595

119873ssΩ sin120593

minus (C119901119887120576119887)119864]

120595119873ss =

(C119901119887nabla119887)119864

119892+2120585

120596119899

sdot [(1 minus cos120595119880)Ω cos120593

minus 120595119864ssΩ sin120593 minus (C119901

119887120576119887)119873]

119862 (120595119880)ss =

119892

1205962119899

sdot [(cos120595119880minus 1)Ω cos120593 + 120595

119864ssΩ sin120593

+ (C119901119887120576119887)119873]

(30)

where (sdot)ss represents the steady-state value of (sdot)We regard sin120595

119880 cos120595

119880 120595119864ss and 120595119873ss as the unknown

values (30) can then be easily solved Then the relationshipsbetween functions 119878(120595

119880) and 119862(120595

119880) and azimuth misalign-

ment 120595119880can be obtained

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593+

(C119901119887nabla119887)119873

119892tan120593

(31)

The detailed derivations of (31) can be found in Appendix A

322 The Azimuth Estimation Equations It is clear thatsin120595119880

and cos120595119880

can be supplied exactly by (31) if theaccelerometer errors and gyro errors (sensor errors) areknown Since the sensor errors are uncertain sin120595

119880and

cos120595119880are calculated by the following equations

sin 119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880) minus

2120585120596119899tan120593119892

sdot 119862 (120595119880) (32)

cos 119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880) minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)

(33)

where sin 119880and cos

119880are the computed values of sin120595

119880

and cos120595119880 respectively When the improved alignment

system is stable the errors between computed values (sin 119880

and cos 119880) and theoretical values (sin120595

119880and cos120595

119880) are

only determined by sensor errors and these errors areallowable Worth noting is that (32) and (33) do not preservethe unit-norm property of the trigonometric function thatis (sin

119880)2+ (cos

119880)2

= 1 So normalization of sin 119880

and cos 119880should be given and the simple normalization is

provided as follows

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(34)

The detailed derivations of (34) can be found in Appendix BAt this time the azimuth estimation

119880can be obtained

from (34) and is shown as follows

119880119898

= arcsin (sin 119880)

119880=

119880119898

cos 119880gt 0

180∘minus 119880119898

cos 119880lt 0 sin

119880gt 0

minus180∘minus 119880119898

cos 119880lt 0 sin

119880lt 0

(35)

It is obvious from the previous discussion that the azimuthestimation equations consist of (32)ndash(35)

Finally we take the disturbing motions into consid-eration because the alignment is often performed underquasistationary conditions The disturbing motions can begenerally considered to be sinusoidal [4 16] Then in orderto reduce the influence of the disturbing motions on azimuthestimation low pass filters need to be added to the systemnamely the control networks 119866(119904) and119867(119904) should have thecapability of restraining disturbance The design law of thecontrol networks 119866(119904) and119867(119904) is provided as follows

(a) The magnification of 119866(119904) and119867(119904) in low frequencymust be equivalent to 1 that is the following equationshould be satisfied

lim119904rarr0

119866 (119904) = lim119904rarr0

119867(119904) = 1 (36)

The reason for this requirement is that the rela-tionships between functions 119878(120595

119880) and 119862(120595

119880) and

azimuthmisalignment120595119880under quasistationary con-

ditions also can be represented by (31) as the require-ment is met

(b) In high frequency for the purpose of reducing theinfluence of disturbing motions they need be capableof restraining disturbance

Then under quasistationary conditions the azimuth estima-tion equations also consist of (32)ndash(35) In this work thecontrol networks 119866(119904) and119867(119904) are set as

119866 (119904) = 119867 (119904) =(120585120596119899)2

(119904 + 120585120596119899)2 (37)

Mathematical Problems in Engineering 7

bib

minus

minus

Gyros

AccelerometersCalculation of control angular

velocity and azimuth estimation functions

DSP

Final process

S(120595U) C(120595U)

120596bpb

E

N

U

120596n

ieCbp

pb =

pb[120596bpb times]CC

120596p

cCbp

fb

Vp= Cp

bfb+ gn

sin 120595U

cos 120595U

01

+

minus1205962n

gΩ cos 120593minus2120585120596n tan 120593

g

minus2120585120596n tan 120593

g

1205962n

gΩ cos 120593

S(120595U)C(120595U)lfloor

lceilrfloorrceillfloorlceillfloorlceil rfloorrceilrfloorrceil

V

V

Figure 6 Flow chart of improved gyrocompass alignment in real time operating device

At this time the denominators of 119878(120595119880)(119904) and 119862(120595

119880)(119904)

can be set as

(1199042+ 2120585120596

119899119904 + 1205962

119899) (119904 + 120585120596

119899)2 (38)

It is obvious from the azimuth estimation equations that theperformance of the azimuth alignment is determined by (32)and (33) Since both (32) and (33) consist of 119878(120595

119880) and119862(120595

119880)

the four eigenvalues of sin 119880(119904) and cos

119880(119904) could be

set as 11990412

= minus120585120596119899and 11990434

= minus120585120596119899plusmn 119895radic1 minus 1205852120596

119899 without

the influence of the nonlinear factors Therefore the impactof the nonlinear factor caused by large azimuthmisalignmenton gyrocompass alignment is reduced and the performanceof the improved gyrocompass alignment will be better thanthe conventional one

33 The Implementation of the Improved Gyrocompass Align-ment Method In this section the implementation of theproposed method in real time operating device is discussedand meanwhile a brief summary of this method is providedAn inertial navigation system implements the proposedalignment method using a cluster of accelerometers to sensethe specific force vector components f119887 a triad of gyrosto measure the angular velocity 119887

119894119887 and a digital signal

processor (DSP) to perform the alignment algorithm Thedirect expression of this implementation can be seen inFigure 6 According to Figure 6 the specific implementationsteps of the improved gyrocompass alignment in real timeoperation device are described as follows

(a) At the beginning a preliminary alignment oftencalled coarse alignment is performed and after thata rough DCM C119901

119887is obtained The specific coarse

alignment method can be found in [15] the levelmisalignments are small and azimuth misalignmentis usually large

(b) Secondly by utilizing the specific force f119887 and DCMC119901119887 the derivatives of the east and north horizontal

velocities 119864and 119873can be obtainedThen according

to the calculation method shown in Figure 3 thecontrol velocity 120596119901

119888and azimuth functions 119878(120595

119880) and

119862(120595119880) can be acquired

(c) On the one hand the control velocity 120596119901119888and angular

velocity 119887119894119887are used to update the DCM C119901

119887used for

the next calculation step(d) On the other hand the azimuth functions 119878(120595

119880) and

119862(120595119880) are used to compute sin

119880and cos

119880 And

after the final process which is achieved by utilizing(34)-(35) the azimuth estimation

119880is obtained

(e) Then steps (b) (c) and (d) are performed repeatedlyFinally by spending a period of time an accurateazimuth estimation

119880can be obtained and with the

compensation of azimuth misalignment using 119880 an

accurate DCM C119901119887can be obtained

4 Simulations and Experiments

In this section to evaluate the performance of the improvedgyrocompass alignment method under the condition of largeazimuth misalignment simulations and experiments arecarried out

41 Simulation Results and Analysis Both the conventionaland improved gyrocompass alignment methods are per-formed during the simulations simultaneously and the sim-ulations are conducted under the conditions of respec-tively choosing different azimuth misalignments namely120595119880= 50

∘ 100∘ 145∘ minus50∘ minus100

∘ minus145

∘ In these simu-

lations gyro and accelerometer outputs are generated bythe strapdown INS simulator we assume that the vehicleis in disturbing motions caused by sea waves The initialconditions are presented as follows

The constant gyro errors 120576119887 =

[[001∘ℎ 001

∘ℎ 001

∘ℎ]]119879

8 Mathematical Problems in Engineering

0 5 10 15 20

0 0

50

100

150

Time (min) Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0 5 10 15 20

Improved method

minus50

50

100

150

minus50

Figure 7 Gyrocompass alignment for 120595119880= 50

∘ 100∘ 145∘

The constant accelerometer errors nabla119887 =

[[10minus4119892 10minus4119892 10minus4119892]]119879

The acceleration due to gravity 119892 = 978049ms2The local longitude and latitude 120582 = 1266705∘ 120593 =457796

∘The control networks of improved method 119866(119904) =119867(119904) = (120585120596

119899)2(119904 + 120585120596

119899)2

The natural frequency of conventional and improvedmethods 120596

119899= 002 rads

Under the disturbingmotions the vehicle undertakes angularand linealmovements In angularmovement the pitch 120579

119909 the

roll 120579119910 and the yaw 120579

119911are controlled as

120579119909= 7∘ sin(2120587

5119905 +

120587

3)

120579119910= 5∘ sin(2120587

7119905 +

120587

6)

120579119911= 10∘ sin(2120587

6119905 +

120587

4)

(39)

In the lineal movement the vehicle lineal movement veloci-ties are taken as

119881119909= minus03 sin(2120587

10119905) ms2

119881119910= minus02 sin(2120587

6119905) ms2

119881119911= minus04 sin(2120587

8119905) ms2

(40)

Figures 7 and 8 provide the azimuth estimation errors ofconventional and improved methods under the conditions

of 120595119880= 50

∘ 100∘ 145∘ minus50∘ minus100

∘ minus145

∘ It can be seen

that azimuth alignment could be accomplished by both con-ventional and improved gyrocompass alignment methodsBut under the condition of large azimuth misalignment theperformance of conventional method will be poor along withthe growth of |120595

119880| The reason is that the difference between

the actual and desired eigenvalues is increasing alongwith thegrowth of |120595

119880|

Figure 9 shows convergence time for different azimuthmisalignments with a converged azimuth estimation errorof less than 1 degree It is clear that large azimuth misalign-ment needs more time to converge under the conventionalmethod However for the improved method the changeof the convergence time is not obvious That means theperformance of the improved method is not affected bythe large azimuth misalignment The reason is that theeigenvalues of improved method can be set as the desiredvalues without the influence of the nonlinear factors causedby the large azimuth misalignment Moreover it is obviousthat the alignment speed of the improved method is fasterthan that of the conventional one

42 Experimental Results and Analysis The experimentswere implemented in the lab as the true yaw angle ofthe strapdown INS was known so different 120595

119880could be

exactly set We fixed the strapdown INS on the SGT-3 three-axis turntable to implement the alignment experiments Thestrapdown INS and the turntable can be seen in Figure 10 Atthe start of each experiment the turntable turned to a staticposition for 20 minutes that is pitch angle 0∘ roll angle 0∘and yaw angle 135∘ Two alignmentmethods were performedunder the condition of respectively setting initial yaw angleas 185∘ and 235∘ namely 120595

119880= 50∘ 100∘

Mathematical Problems in Engineering 9

0 5 10 15 20

0

50

Time (min)0 5 10 15 20

Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0

50 Improved method

minus50

minus100

minus150

minus50

minus100

minus150

Figure 8 Gyrocompass alignment for 120595119880= minus50

∘ minus100

∘ minus145

∘

50 100 1450

5

10

15

20

Azimuth misalignment (deg)

Con

verg

ence

tim

e (m

in)

Conventional methodImproved method

minus145 minus50minus100

Figure 9 Convergence time comparison with different azimuthmisalignments

Figure 11 provides the comparison of the yaw angleusing the two alignment methods under the condition ofdifferent 120595

119880

As the results above show the yaw angle can convergeto 135∘ through using the two alignment methods How-ever for the conventional method the convergence time islengthened with the growth of the azimuth misalignmentCompared with the conventional method the convergence

Strapdown INS

SGT-3

Figure 10 The SGT-3 turntable and strapdown INS

time of improved method is hardly changed The exper-imental results are consistent with the simulation resultsThe experiments demonstrate that the performance of theimproved method is better than that of the conventional one

5 Conclusions

In this paper an improved gyrocompass alignment methodhas been proposed for large azimuth misalignment of INSThe improved new method is performed by opening theazimuth loop and the estimation of the azimuth misalign-ment is implemented by using the horizontal velocity signalsto estimate the nonlinear factorsThrough using thismethodthe impact of the nonlinear factor caused by large azimuthmisalignment on gyrocompass alignment is reduced andthe performance of gyrocompass alignment is improvedThealignment speed of the improved method is faster than theconventional one in the same conditions The performance

10 Mathematical Problems in Engineering

0 5 10 15 20100

135

150

185200

235250

300

Yaw

(deg

)

Time (min)

Conventional method 120595U = 50∘

Conventional method 120595U = 100∘

Improved method 120595U = 50∘

Improved method 120595U = 100∘

Figure 11 Comparison of the yaw angle using the two alignmentmethods

of the improved method is well validated by simulations andexperiments

Appendices

A The Derivations of (31)

In this section (31) are obtained in the followingAccording to (30) their matrix form can be written as

AX = B (A1)

where

A

=

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

[

2120585Ω cos120593120596119899

0 1 minus2120585Ω sin120593

120596119899

119892Ω cos1205931205962119899

0 0 minus119892Ω sin1205931205962119899

02120585Ω cos120593

120596119899

2120585Ω sin120593120596119899

1

0119892Ω cos1205931205962119899

119892Ω sin1205931205962119899

0

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

]

(A2)

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

(A3)

B =

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

[

minus

(C119901119887nabla119887)119873

119892minus2120585

120596119899

(C119901119887120576119887)119864

minus119878 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119864

2120585

120596119899

Ω cos120593 +(C119901119887nabla119887)119864

119892minus2120585

120596119899

(C119901119887120576119887)119873

119892

1205962119899

Ω cos120593 + 119862 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119873

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

]

(A4)

Multiply the first and third rows of (A1) by1205961198992120585Ω cos120593 and

multiply the second and fourth rows of (A1) by 1205962119899119892Ω cos120593

we can obtain

A1X = B

1 (A5)

where

A1=

[[[[[[[[

[

1 0120596119899

2120585Ω cos120593minus tan120593

1 0 0 minus tan120593

0 1 tan120593120596119899

2120585Ω cos1205930 1 tan120593 0

]]]]]]]]

]

X =[[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]]

]

B1=

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

[

minus120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

1 +120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892minus

(C119901119887120576119887)119873

Ω cos120593

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

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

]

(A6)

Furthermore by utilizing the elementary row transformation(A5) can be transformed into

A2X = B

2 (A7)

Mathematical Problems in Engineering 11

where

A1=

[[[[[[[[

[

1 0 0 minus tan120593

0 0 minus120596119899

2120585Ω cos1205930

0 1 tan120593 0

0 0 0 minus120596119899

2120585Ω cos120593

]]]]]]]]

]

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

B1

=

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

[

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos1205931205962

119899

119892Ω cos120593119862 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892

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

]

(A8)

Then according to matrix equation (A7) we have

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

+

(C119901119887nabla119887)119873

119892tan120593

(A9)

B The Derivations of (34)

The derivations of (34) are carried out based on the principleof least squares fitting setting the Euclidean minimum norm119869 between sin

119880 cos

119880and sin

119880 cos

119880(the normalized

data) as the normalization indexFirst define

119869 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]12

(B1)

It is subjected to the constrained condition namely

ℎ = (sin 119880)2+ (cos

119880)2minus 1 = 0 (B2)

Second define an auxiliary function119898 namely

119898 = 1198692+ 120582ℎ (B3)

where 120582 is an unknown parameterBy substituting (B1) and (B2) into (B3) we can get

119898 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]

minus 120582 ((sin 119880)2+ (cos

119880)2minus 1)

(B4)

Obviously the parameters sin 119880and cos

119880that made 119869

minimum are equal to the ones that make 119898 minimumRespectively the partial derivatives of the parameters sin

119880

and cos 119880are calculated and bymaking them equal to zero

we can get

sin 119880=sin 119880

1 + 120582

cos 119880=cos 119880

1 + 120582

(B5)

Then by substituting (B5) into (B2) we have

(sin 119880)2+ (cos

119880)2

(1 + 120582)2

= 1 (B6)

Finally combining (B5) and (B6) the normalization equa-tions can be obtained

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(B7)

Conflict of Interests

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

Acknowledgments

The work described in the paper was supported by theNational Natural Science Foundation of China (61203225)and the National Science Foundation for Post-DoctoralScientists of China (2012M510083)The authors would like tothank all members of the Inertial Navigation Research Groupat Harbin Engineering University for the technical assistancewith the navigation system

References

[1] H W Park J G Lee and C G Park ldquoCovariance analysis ofstrapdown INS considering gyrocompass characteristicsrdquo IEEETransactions on Aerospace and Electronic Systems vol 31 no 1pp 320ndash328 1995

12 Mathematical Problems in Engineering

[2] R A Nash Jr J A DrsquoAppolito and K J Roy ldquoError analysis ofhybrid aircraft inertial navigation systemsrdquo in Proceedings of theAIAA Guidance and Control Conference Stanford Calif USA1972

[3] F D Jurenka and C T Leondes ldquoOptimum alignment of aninertial autonavigatorrdquo IEEE Transactions on Aerospace andElectronic Systems vol AES-3 no 6 pp 880ndash888 1967

[4] R H Cannon ldquoAlignment of inertial guidance systems bygyrocompassing-linear theoryrdquo Journal of Aerospace Sciencevol 28 no 11 pp 885ndash895 912 1961

[5] G R Pitman Jr Inertial Guidance John Wiley amp Sons NewYork NY USA 1962

[6] K R Britting Inertial Navigation Systems Analysis John Wileyamp Sons New York NY USA 1971

[7] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation vol 28 no4 pp 311ndash341 1982

[8] X Liu X Xu Y Liu and L Wang ldquoA fast and high-accuracycompass alignmentmethod to SINSwith azimuth axis rotationrdquoMathematical Problems in Engineering vol 2013 Article ID524284 12 pages 2013

[9] F Sun Q Wang Z Qi and C Wang ldquoResearch on theestimation method of DVL velocity error based on doubleprogram in Fiber Optic Gyro SINSrdquo Optik vol 124 no 22 pp5344ndash5349 2013

[10] B M Scherzinger ldquoInertial navigator error models for largeheading uncertaintyrdquo in Proceedings of the IEEE Position Loca-tion and Navigation Symposium (PLANS rsquo96) pp 477ndash484Atlanta Ga USA April 1996

[11] X Y Kong EM Nebot andH Durrant-Whyte ldquoDevelopmentof a non-linear psi-angle model for large misalignment errorsand its application in INS alignment and calibrationrdquo in Pro-ceedings of the 1999 IEEE International Conference on Roboticsamp Automation (ICRA rsquo99) pp 1430ndash1435 Detroit Mich USAMay 1999

[12] M-J Yu H-W Park and C-B Jeon ldquoEquivalent nonlinearerror models of strapdown inertial navigation systemrdquo in Pro-ceedings of the Guidance Navigation and Control ConferenceAIAA-97-3563 pp 581ndash587 1997

[13] T Abbas Z Yunyan and L Yanjun ldquoSINS initial alignment forsmall tilt and large azimuth misalignment anglesrdquo in Proceed-ings of the IEEE 3rd International Conference on CommunicationSoftware and Networks (ICCSN rsquo11) pp 628ndash632 Xirsquoan ChinaMay 2011

[14] S L Han and J L Wang ldquoA novel initial alignment scheme forlow-cost INS aided by GPS for land vehicle applicationsrdquo TheJournal of Navigation vol 63 no 4 pp 663ndash680 2010

[15] P G Savage Strapdown Analytics Strapdown Associates 2ndedition 2007

[16] I Yaesh and B Priel ldquoDesign of leveling loop for marine nav-igation systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 29 no 2 pp 599ndash604 1993

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

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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 3

0 45 90 135 180

102

101

k(120595

U)

|120595U| (deg)

Figure 2 Functional relationship between 119896(120595119880) and |120595

119880|

gains cannot be set as (7) (8) and (9) In fact under the largeazimuthmisalignment the values of the control gains are stillset as (3) (4) and (5) Then four eigenvalues of conventionalgyrocompass alignment cannot be set as the desired values Inthis case the conventional method is hard to favorably meetthe requirement of alignment speedThe particular reason ofthis problem also can be expressed by the following equation

119896119880=

1198961015840

119880

119896 (120595119880) (10)

where 1198961015840119880is the theoretical value calculated by (9) and 119896

119880is

the actual value obtained by (5) From (10) it is clear that theactual value 119896

119880adopted by the alignment system is smaller

than the theoretical value 1198961015840119880 due to the fact that 119896(120595

119880) ge

1 (domain 120595119880isin [minus180

∘ 180∘]) The functional relationship

between 119896(120595119880) and |120595

119880| is shown in Figure 2

It is obvious from Figure 2 that the value of 119896(120595119880) is

increasing along with the growth of |120595119880| then the difference

between 119896119880and 1198961015840

119880will increase together with the growth

of |120595119880| That means the difference between the actual and

desired eigenvalues increases along with the growth of |120595119880|

however the performance of the alignment system will bepoor

3 Improved Gyrocompass Alignment Method

Considering the poor performance of conventional gyro-compass alignment under the condition of large azimuthmisalignment an improved gyrocompass alignment methodis proposed in this paper to improve the performance ofgyrocompass alignment It is clear that the four eigenvaluesof conventional gyrocompass alignment cannot be set as thedesired values in the case of large azimuth misalignmentbecause the uncertain nonlinear factor is included in thecharacteristic equation So the key to design the improvedmethod is to reduce the impact of the nonlinear factor oncharacteristic equation This problem is solved by opening

the azimuth loop at this time the influence of the nonlin-ear factor on characteristic equation disappeared and theestimation of the azimuth misalignment is implemented byusing the horizontal velocity signals to estimate the nonlinearfactors

31TheEstablishment of the ImprovedGyrocompassAlignmentMethod In this paper 119894 119890 119887 119899 and 119901 denote the inertialframe the earth fixed coordinate frame the sensor bodyframe the navigation frame and the computed navigationframe respectively In this work we choose the local levelgeographic coordinate frame as the navigation frame Underthe large azimuth misalignment the direction cosine matrix(DCM) C119901

119899can be described as follows [13 14]

C119901119899

=[[

[

cos120595119880

sin120595119880

minus120595119873

minus sin120595119880

cos120595119880

120595119864

120595119873cos120595119880+ 120595119864sin120595119880120595119873sin120595119880minus 120595119864cos120595119880

1

]]

]

(11)

where 120595119873is the north level misalignment 120595

119864and 120595

119873(level

misalignments) are small and 120595119880(azimuth misalignment

the departure of computed north from actual north) is largeFor a quasistationary initial alignment the average value

for velocity will be zero [15] Then under quasistationary sit-uations the navigation equations for strapdown INS attitudeand velocity can be represented respectively as

C119901119887= C119901119887[120596119887

119901119887times] (12)

V119901 = C119901119887f119887 + g119899 (13)

where C119901119887represents the DCM relating the transformation

from 119887 frame to the 119901 frame 120596119887119901119887

is the angular velocityvector of 119887 frame with respect to 119901 frame resolved in 119887

frame [times] denotes a skew-symmetric matrix operator V119901 isthe computed velocity of strapdown INS f119887 is the specificforce from the accelerometer output g119899 is the gravity vectorresolved in 119899 frame The angular velocity 120596119887

119901119887is derived by

120596119887

119901119887= 119887

119894119887minus C119887119901120596119899

119894119890 (14)

where 119887119894119887is the angular velocity vector measured by gyros

120596119899

119894119890is the angular velocity vector of 119890 frame with respect to 119894

frame resolved in 119899 frameThese angular velocity vectors 119887

119894119887 120596119899119894119890and accelerometer

output f119887 can be described respectively as

119887

119894119887= 120596119887

119894119887+ 120576119887 (15a)

120596119899

119894119890= [0 Ω cos120593 Ω sin120593]119879 (15b)

f119887 = minusg119887 + 120575a119887 + nabla119887 (15c)

where 120596119887119894119887is the true angular velocity vector 120576119887 is the gyro

error vector g119887 is the gravity vector resolved in 119887 frame

4 Mathematical Problems in Engineering

120575a119887 is the disturbing motion vector under quasistationarysituations and nabla119887 is the accelerometer error vector

Substituting (15c) in (13) we obtain

V119901 = (I minus C119901119899) g119899 + C119901

119887120575a119887 + C119901

119887nabla119887 (16)

Then from (16) the horizontal components of V119901 can bedescribed respectively as

119864= minus119892120595

119873+ (C119901119887120575a119887)119864+ (C119901119887nabla119887)119864

119873= 119892120595119864+ (C119901119887120575a119887)119873+ (C119901119887nabla119887)119873

(17)

where 119881119864and 119881

119873represent the east and north horizontal

velocities respectively (sdot)119864and (sdot)

119873represent the 119864 and 119873

elements of column matrix (sdot) separately whose entries fromtop to bottom are the 119864119873 and 119880 elements

From (11) (12) and (14) the misalignment equation ofstrapdown INS under large azimuth misalignment can berepresented as [11 13]

= (I minus C119901119899)120596119899

119894119890minus C119901119887120576119887 (18)

where 120595 is the misalignment vector and 120595 = [120595119864 120595119873 120595119880]119879

The leveling alignment of the improved method is per-formed by introducing control angular velocity 120596119901

119888into the

calculation of angular velocity 120596119887119901119887 then we have 120596119887

119901119887= 119887

119894119887minus

C119887119901120596119899

119894119890minus C119887119901120596119901

119888 finally the misalignment equation could be

transformed into

= (I minus C119901119899)120596119899

119894119890minus C119901119887120576119887+ 120596119901

119888 (19)

where 120596119901119888= [120596119901

119888119864120596119901

1198881198730]119879

is the control angular velocityvector From (19) the horizontal components of 120595 can berespectively represented as

119864= minus sin120595

119880Ω cos120593 + 120595

119873Ω sin120593 minus (C119901

119887120576119887)119864+ 120596119901

119888119864 (20)

119873= (1 minus cos120595

119880)Ω cos120593 minus 120595

119864Ω sin120593 minus (C119901

119887120576119887)119873

+ 120596119901

119888119873

(21)

Because the azimuth misalignmentrsquos rate of change is small[10] the approximation

119880= 0 is admitted namely 120595

119880is a

constant value [15]The azimuth information is obtained by structuring

azimuth estimation functions The block diagram of 120596119901119888and

azimuth estimation functions is shown in Figure 3In Figure 3 119896

119886and 119896119887represent the control gains119866(119904) and

119867(119904) represent the control networks used for reducing theinfluence of disturbing motions 119878(120595

119880) and 119862(120595

119880) represent

the functions of 120595119880 From (17) (20) (21) and Figure 3 the

error diagrams of improved gyrocompass alignment can beobtained as in Figures 4 and 5

It can be seen from Figures 4 and 5 that the nonlinearfactors sin120595

119880Ω cos120593 and (1minuscos120595

119880)Ω cos120593 both are caused

by the large azimuth misalignment and become the constantinputs of the system through opening the azimuth loop So

minus

minus

ka

ka

G(s)

S(120595U)

P

N

E

minuskb

kb

H(s)

1

s

1

s

C(120595U)

120596pcE

120596pcN

120596pc

V

V

V

Figure 3 Block diagram of 120596119901119888and azimuth estimation functions

ka

1

s

1

s

+

+minus

minus

minus

minus

g

G(s)

kb

120595E

S(120595U)sin 120595UΩ cos 120593 120595NΩ sin 120593

(Cpb120576b)

E

(Cp b120575ab) N

+(C

p bnablab) N

Figure 4 Error diagram of improved gyrocompass alignment innorth loop

+

+

minus

minus

minus

minus

g

kb

ka

1

s

1

s

H(s)

120595N

C(120595U)(1 minus cos120595U)Ωcos 120593 120595EΩ sin 120593

(Cpb120576b)

N

(Cp b120575ab) E

+(C

p bnablab) E

Figure 5 Error diagram of improved gyrocompass alignment ineast loop

the nonlinear factors will not be included in the characteristicequation In addition these constant inputs contain azimuthmisalignment information and the estimation of the azimuthinformation can be obtained by 119878(120595

119880) and 119862(120595

119880)

32 The Operation of the Improved Gyrocompass AlignmentMethod In this section the control gains 119896

119886and 119896

119887and

control networks 119866(119904) and 119867(119904) are provided Then 119878(120595119880)

and 119862(120595119880) can be obtained and the azimuth estimation is

performed by these two functions

Mathematical Problems in Engineering 5

Firstly we provide the values of the control gains 119896119886and 119896119887

for improved gyrocompass alignment the detailed deductionfor these control gains can be found in the following

According to Mason gain formula we can get the charac-teristic equation of north loop from Figure 4

Δ119873 (119904) = 119904

2+ 119896119886119904 + 119892119896

119887= 0 (22)

In the same way from Figure 5 the characteristic equation ofeast loop can be obtained as follows

Δ119864 (119904) = 119904

2+ 119896119886119904 + 119892119896

119887= 0 (23)

It can be seen from (22) and (23) that the north and east loopshave the same characteristic equation In order to ensure thestability of alignment system we set two eigenvalues of (22)and (23) as

11990412= minus120585120596

119899plusmn 119895radic1 minus 1205852120596

119899 (24)

Then the characteristic equation can be rewritten as

Δ119873 (119904) = Δ119864 (119904) = 119904

2+ 2120585120596

119899119904 + 1205962

119899= 0 (25)

Comparing (22) with (25) we can get

119896119886= 2120585120596

119899

119896119887=1205962

119899

119892

(26)

Secondly in order to simplify the subsequent analysis weassume that the inertial sensor errors are basically constantdrifts and the vehicle in which the INS is mounted is totallystopped Under this assumption the disturbing motion vec-tor 120575a119887 is equal to zero At this time the control networks canbe set as 119866(119904) = 119867(119904) = 1 because no disturbing motion isintroduced into the alignment system The design procedureof azimuth estimation is described in the following and itcan be divided into two steps The first step the analysisof the relationships between functions 119878(120595

119880) and 119862(120595

119880)

and azimuth misalignment 120595119880is made with the aid of the

Laplace transformationThe second step based on the formeranalysis the azimuth estimation equations are provided

321 Relationships between Functions 119878(120595119880) and 119862(120595

119880) and

Azimuth Misalignment 120595119880 The response to east level mis-

alignment 120595119864and function 119878(120595

119880) can be written in Laplace

form directly from Figure 4 and can be seen as follows(according to the Mason gain formula)

120595119864 (119904)

= minus1205962

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot

(C119901119887nabla119887)119873 (119904)

119892

+119904 + 2120585120596

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot minus sin120595119880Ω cos120593 + 120595

119873Ω sin120593 minus (C119901

119887120576119887)119864 (119904)

119878 (120595119880) (119904)

=119904

1199042 + 2120585120596119899119904 + 1205962119899

sdot (C119901119887nabla119887)119873 (119904)

+119892

1199042 + 2120585120596119899119904 + 1205962119899

sdot minus sin120595119880Ω cos120593 + 120595

119873Ω sin120593 minus (C119901

119887120576119887)119864 (119904)

(27)

where sdot(119904) represents the Laplace transformation of sdotSimilarly from Figure 5 it yields

120595119873 (119904) =

1205962

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot

(C119901119887nabla119887)119864 (119904)

119892

+119904 + 2120585120596

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot (1 minus cos120595119880)Ω cos120593 minus 120595

119864Ω sin120593 minus (C119901

119887120576119887)119873

sdot (119904)

119862 (120595119880) (119904) =

119904

1199042 + 2120585120596119899119904 + 1205962119899

sdot (C119901119887nabla119887)119864 (119904)

minus119892

1199042 + 2120585120596119899119904 + 1205962119899

sdot (1 minus cos120595119880)Ω cos120593 minus 120595

119864Ω sin120593 minus (C119901

119887120576119887)119873

sdot (119904)

(28)

Equations (27)-(28) are solved for the steady-state errors byinvoking the final value theorem

lim119905rarrinfin

119891 (119905) = lim119904rarr0

119904119891 (119904) (29)

6 Mathematical Problems in Engineering

Then we have

120595119864ss = minus

(C119901119887nabla119887)119873

119892+2120585

120596119899

sdot [minus sin120595119880Ω cos120593

+ 120595119873ssΩ sin120593 minus (C119901

119887120576119887)119864]

119878 (120595119880)ss =

119892

1205962119899

sdot [minus sin120595119880Ω cos120593 + 120595

119873ssΩ sin120593

minus (C119901119887120576119887)119864]

120595119873ss =

(C119901119887nabla119887)119864

119892+2120585

120596119899

sdot [(1 minus cos120595119880)Ω cos120593

minus 120595119864ssΩ sin120593 minus (C119901

119887120576119887)119873]

119862 (120595119880)ss =

119892

1205962119899

sdot [(cos120595119880minus 1)Ω cos120593 + 120595

119864ssΩ sin120593

+ (C119901119887120576119887)119873]

(30)

where (sdot)ss represents the steady-state value of (sdot)We regard sin120595

119880 cos120595

119880 120595119864ss and 120595119873ss as the unknown

values (30) can then be easily solved Then the relationshipsbetween functions 119878(120595

119880) and 119862(120595

119880) and azimuth misalign-

ment 120595119880can be obtained

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593+

(C119901119887nabla119887)119873

119892tan120593

(31)

The detailed derivations of (31) can be found in Appendix A

322 The Azimuth Estimation Equations It is clear thatsin120595119880

and cos120595119880

can be supplied exactly by (31) if theaccelerometer errors and gyro errors (sensor errors) areknown Since the sensor errors are uncertain sin120595

119880and

cos120595119880are calculated by the following equations

sin 119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880) minus

2120585120596119899tan120593119892

sdot 119862 (120595119880) (32)

cos 119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880) minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)

(33)

where sin 119880and cos

119880are the computed values of sin120595

119880

and cos120595119880 respectively When the improved alignment

system is stable the errors between computed values (sin 119880

and cos 119880) and theoretical values (sin120595

119880and cos120595

119880) are

only determined by sensor errors and these errors areallowable Worth noting is that (32) and (33) do not preservethe unit-norm property of the trigonometric function thatis (sin

119880)2+ (cos

119880)2

= 1 So normalization of sin 119880

and cos 119880should be given and the simple normalization is

provided as follows

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(34)

The detailed derivations of (34) can be found in Appendix BAt this time the azimuth estimation

119880can be obtained

from (34) and is shown as follows

119880119898

= arcsin (sin 119880)

119880=

119880119898

cos 119880gt 0

180∘minus 119880119898

cos 119880lt 0 sin

119880gt 0

minus180∘minus 119880119898

cos 119880lt 0 sin

119880lt 0

(35)

It is obvious from the previous discussion that the azimuthestimation equations consist of (32)ndash(35)

Finally we take the disturbing motions into consid-eration because the alignment is often performed underquasistationary conditions The disturbing motions can begenerally considered to be sinusoidal [4 16] Then in orderto reduce the influence of the disturbing motions on azimuthestimation low pass filters need to be added to the systemnamely the control networks 119866(119904) and119867(119904) should have thecapability of restraining disturbance The design law of thecontrol networks 119866(119904) and119867(119904) is provided as follows

(a) The magnification of 119866(119904) and119867(119904) in low frequencymust be equivalent to 1 that is the following equationshould be satisfied

lim119904rarr0

119866 (119904) = lim119904rarr0

119867(119904) = 1 (36)

The reason for this requirement is that the rela-tionships between functions 119878(120595

119880) and 119862(120595

119880) and

azimuthmisalignment120595119880under quasistationary con-

ditions also can be represented by (31) as the require-ment is met

(b) In high frequency for the purpose of reducing theinfluence of disturbing motions they need be capableof restraining disturbance

Then under quasistationary conditions the azimuth estima-tion equations also consist of (32)ndash(35) In this work thecontrol networks 119866(119904) and119867(119904) are set as

119866 (119904) = 119867 (119904) =(120585120596119899)2

(119904 + 120585120596119899)2 (37)

Mathematical Problems in Engineering 7

bib

minus

minus

Gyros

AccelerometersCalculation of control angular

velocity and azimuth estimation functions

DSP

Final process

S(120595U) C(120595U)

120596bpb

E

N

U

120596n

ieCbp

pb =

pb[120596bpb times]CC

120596p

cCbp

fb

Vp= Cp

bfb+ gn

sin 120595U

cos 120595U

01

+

minus1205962n

gΩ cos 120593minus2120585120596n tan 120593

g

minus2120585120596n tan 120593

g

1205962n

gΩ cos 120593

S(120595U)C(120595U)lfloor

lceilrfloorrceillfloorlceillfloorlceil rfloorrceilrfloorrceil

V

V

Figure 6 Flow chart of improved gyrocompass alignment in real time operating device

At this time the denominators of 119878(120595119880)(119904) and 119862(120595

119880)(119904)

can be set as

(1199042+ 2120585120596

119899119904 + 1205962

119899) (119904 + 120585120596

119899)2 (38)

It is obvious from the azimuth estimation equations that theperformance of the azimuth alignment is determined by (32)and (33) Since both (32) and (33) consist of 119878(120595

119880) and119862(120595

119880)

the four eigenvalues of sin 119880(119904) and cos

119880(119904) could be

set as 11990412

= minus120585120596119899and 11990434

= minus120585120596119899plusmn 119895radic1 minus 1205852120596

119899 without

the influence of the nonlinear factors Therefore the impactof the nonlinear factor caused by large azimuthmisalignmenton gyrocompass alignment is reduced and the performanceof the improved gyrocompass alignment will be better thanthe conventional one

33 The Implementation of the Improved Gyrocompass Align-ment Method In this section the implementation of theproposed method in real time operating device is discussedand meanwhile a brief summary of this method is providedAn inertial navigation system implements the proposedalignment method using a cluster of accelerometers to sensethe specific force vector components f119887 a triad of gyrosto measure the angular velocity 119887

119894119887 and a digital signal

processor (DSP) to perform the alignment algorithm Thedirect expression of this implementation can be seen inFigure 6 According to Figure 6 the specific implementationsteps of the improved gyrocompass alignment in real timeoperation device are described as follows

(a) At the beginning a preliminary alignment oftencalled coarse alignment is performed and after thata rough DCM C119901

119887is obtained The specific coarse

alignment method can be found in [15] the levelmisalignments are small and azimuth misalignmentis usually large

(b) Secondly by utilizing the specific force f119887 and DCMC119901119887 the derivatives of the east and north horizontal

velocities 119864and 119873can be obtainedThen according

to the calculation method shown in Figure 3 thecontrol velocity 120596119901

119888and azimuth functions 119878(120595

119880) and

119862(120595119880) can be acquired

(c) On the one hand the control velocity 120596119901119888and angular

velocity 119887119894119887are used to update the DCM C119901

119887used for

the next calculation step(d) On the other hand the azimuth functions 119878(120595

119880) and

119862(120595119880) are used to compute sin

119880and cos

119880 And

after the final process which is achieved by utilizing(34)-(35) the azimuth estimation

119880is obtained

(e) Then steps (b) (c) and (d) are performed repeatedlyFinally by spending a period of time an accurateazimuth estimation

119880can be obtained and with the

compensation of azimuth misalignment using 119880 an

accurate DCM C119901119887can be obtained

4 Simulations and Experiments

In this section to evaluate the performance of the improvedgyrocompass alignment method under the condition of largeazimuth misalignment simulations and experiments arecarried out

41 Simulation Results and Analysis Both the conventionaland improved gyrocompass alignment methods are per-formed during the simulations simultaneously and the sim-ulations are conducted under the conditions of respec-tively choosing different azimuth misalignments namely120595119880= 50

∘ 100∘ 145∘ minus50∘ minus100

∘ minus145

∘ In these simu-

lations gyro and accelerometer outputs are generated bythe strapdown INS simulator we assume that the vehicleis in disturbing motions caused by sea waves The initialconditions are presented as follows

The constant gyro errors 120576119887 =

[[001∘ℎ 001

∘ℎ 001

∘ℎ]]119879

8 Mathematical Problems in Engineering

0 5 10 15 20

0 0

50

100

150

Time (min) Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0 5 10 15 20

Improved method

minus50

50

100

150

minus50

Figure 7 Gyrocompass alignment for 120595119880= 50

∘ 100∘ 145∘

The constant accelerometer errors nabla119887 =

[[10minus4119892 10minus4119892 10minus4119892]]119879

The acceleration due to gravity 119892 = 978049ms2The local longitude and latitude 120582 = 1266705∘ 120593 =457796

∘The control networks of improved method 119866(119904) =119867(119904) = (120585120596

119899)2(119904 + 120585120596

119899)2

The natural frequency of conventional and improvedmethods 120596

119899= 002 rads

Under the disturbingmotions the vehicle undertakes angularand linealmovements In angularmovement the pitch 120579

119909 the

roll 120579119910 and the yaw 120579

119911are controlled as

120579119909= 7∘ sin(2120587

5119905 +

120587

3)

120579119910= 5∘ sin(2120587

7119905 +

120587

6)

120579119911= 10∘ sin(2120587

6119905 +

120587

4)

(39)

In the lineal movement the vehicle lineal movement veloci-ties are taken as

119881119909= minus03 sin(2120587

10119905) ms2

119881119910= minus02 sin(2120587

6119905) ms2

119881119911= minus04 sin(2120587

8119905) ms2

(40)

Figures 7 and 8 provide the azimuth estimation errors ofconventional and improved methods under the conditions

of 120595119880= 50

∘ 100∘ 145∘ minus50∘ minus100

∘ minus145

∘ It can be seen

that azimuth alignment could be accomplished by both con-ventional and improved gyrocompass alignment methodsBut under the condition of large azimuth misalignment theperformance of conventional method will be poor along withthe growth of |120595

119880| The reason is that the difference between

the actual and desired eigenvalues is increasing alongwith thegrowth of |120595

119880|

Figure 9 shows convergence time for different azimuthmisalignments with a converged azimuth estimation errorof less than 1 degree It is clear that large azimuth misalign-ment needs more time to converge under the conventionalmethod However for the improved method the changeof the convergence time is not obvious That means theperformance of the improved method is not affected bythe large azimuth misalignment The reason is that theeigenvalues of improved method can be set as the desiredvalues without the influence of the nonlinear factors causedby the large azimuth misalignment Moreover it is obviousthat the alignment speed of the improved method is fasterthan that of the conventional one

42 Experimental Results and Analysis The experimentswere implemented in the lab as the true yaw angle ofthe strapdown INS was known so different 120595

119880could be

exactly set We fixed the strapdown INS on the SGT-3 three-axis turntable to implement the alignment experiments Thestrapdown INS and the turntable can be seen in Figure 10 Atthe start of each experiment the turntable turned to a staticposition for 20 minutes that is pitch angle 0∘ roll angle 0∘and yaw angle 135∘ Two alignmentmethods were performedunder the condition of respectively setting initial yaw angleas 185∘ and 235∘ namely 120595

119880= 50∘ 100∘

Mathematical Problems in Engineering 9

0 5 10 15 20

0

50

Time (min)0 5 10 15 20

Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0

50 Improved method

minus50

minus100

minus150

minus50

minus100

minus150

Figure 8 Gyrocompass alignment for 120595119880= minus50

∘ minus100

∘ minus145

∘

50 100 1450

5

10

15

20

Azimuth misalignment (deg)

Con

verg

ence

tim

e (m

in)

Conventional methodImproved method

minus145 minus50minus100

Figure 9 Convergence time comparison with different azimuthmisalignments

Figure 11 provides the comparison of the yaw angleusing the two alignment methods under the condition ofdifferent 120595

119880

As the results above show the yaw angle can convergeto 135∘ through using the two alignment methods How-ever for the conventional method the convergence time islengthened with the growth of the azimuth misalignmentCompared with the conventional method the convergence

Strapdown INS

SGT-3

Figure 10 The SGT-3 turntable and strapdown INS

time of improved method is hardly changed The exper-imental results are consistent with the simulation resultsThe experiments demonstrate that the performance of theimproved method is better than that of the conventional one

5 Conclusions

In this paper an improved gyrocompass alignment methodhas been proposed for large azimuth misalignment of INSThe improved new method is performed by opening theazimuth loop and the estimation of the azimuth misalign-ment is implemented by using the horizontal velocity signalsto estimate the nonlinear factorsThrough using thismethodthe impact of the nonlinear factor caused by large azimuthmisalignment on gyrocompass alignment is reduced andthe performance of gyrocompass alignment is improvedThealignment speed of the improved method is faster than theconventional one in the same conditions The performance

10 Mathematical Problems in Engineering

0 5 10 15 20100

135

150

185200

235250

300

Yaw

(deg

)

Time (min)

Conventional method 120595U = 50∘

Conventional method 120595U = 100∘

Improved method 120595U = 50∘

Improved method 120595U = 100∘

Figure 11 Comparison of the yaw angle using the two alignmentmethods

of the improved method is well validated by simulations andexperiments

Appendices

A The Derivations of (31)

In this section (31) are obtained in the followingAccording to (30) their matrix form can be written as

AX = B (A1)

where

A

=

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

[

2120585Ω cos120593120596119899

0 1 minus2120585Ω sin120593

120596119899

119892Ω cos1205931205962119899

0 0 minus119892Ω sin1205931205962119899

02120585Ω cos120593

120596119899

2120585Ω sin120593120596119899

1

0119892Ω cos1205931205962119899

119892Ω sin1205931205962119899

0

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

]

(A2)

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

(A3)

B =

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

[

minus

(C119901119887nabla119887)119873

119892minus2120585

120596119899

(C119901119887120576119887)119864

minus119878 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119864

2120585

120596119899

Ω cos120593 +(C119901119887nabla119887)119864

119892minus2120585

120596119899

(C119901119887120576119887)119873

119892

1205962119899

Ω cos120593 + 119862 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119873

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

]

(A4)

Multiply the first and third rows of (A1) by1205961198992120585Ω cos120593 and

multiply the second and fourth rows of (A1) by 1205962119899119892Ω cos120593

we can obtain

A1X = B

1 (A5)

where

A1=

[[[[[[[[

[

1 0120596119899

2120585Ω cos120593minus tan120593

1 0 0 minus tan120593

0 1 tan120593120596119899

2120585Ω cos1205930 1 tan120593 0

]]]]]]]]

]

X =[[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]]

]

B1=

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

[

minus120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

1 +120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892minus

(C119901119887120576119887)119873

Ω cos120593

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

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

]

(A6)

Furthermore by utilizing the elementary row transformation(A5) can be transformed into

A2X = B

2 (A7)

Mathematical Problems in Engineering 11

where

A1=

[[[[[[[[

[

1 0 0 minus tan120593

0 0 minus120596119899

2120585Ω cos1205930

0 1 tan120593 0

0 0 0 minus120596119899

2120585Ω cos120593

]]]]]]]]

]

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

B1

=

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

[

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos1205931205962

119899

119892Ω cos120593119862 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892

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

]

(A8)

Then according to matrix equation (A7) we have

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

+

(C119901119887nabla119887)119873

119892tan120593

(A9)

B The Derivations of (34)

The derivations of (34) are carried out based on the principleof least squares fitting setting the Euclidean minimum norm119869 between sin

119880 cos

119880and sin

119880 cos

119880(the normalized

data) as the normalization indexFirst define

119869 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]12

(B1)

It is subjected to the constrained condition namely

ℎ = (sin 119880)2+ (cos

119880)2minus 1 = 0 (B2)

Second define an auxiliary function119898 namely

119898 = 1198692+ 120582ℎ (B3)

where 120582 is an unknown parameterBy substituting (B1) and (B2) into (B3) we can get

119898 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]

minus 120582 ((sin 119880)2+ (cos

119880)2minus 1)

(B4)

Obviously the parameters sin 119880and cos

119880that made 119869

minimum are equal to the ones that make 119898 minimumRespectively the partial derivatives of the parameters sin

119880

and cos 119880are calculated and bymaking them equal to zero

we can get

sin 119880=sin 119880

1 + 120582

cos 119880=cos 119880

1 + 120582

(B5)

Then by substituting (B5) into (B2) we have

(sin 119880)2+ (cos

119880)2

(1 + 120582)2

= 1 (B6)

Finally combining (B5) and (B6) the normalization equa-tions can be obtained

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(B7)

Conflict of Interests

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

Acknowledgments

The work described in the paper was supported by theNational Natural Science Foundation of China (61203225)and the National Science Foundation for Post-DoctoralScientists of China (2012M510083)The authors would like tothank all members of the Inertial Navigation Research Groupat Harbin Engineering University for the technical assistancewith the navigation system

References

[1] H W Park J G Lee and C G Park ldquoCovariance analysis ofstrapdown INS considering gyrocompass characteristicsrdquo IEEETransactions on Aerospace and Electronic Systems vol 31 no 1pp 320ndash328 1995

12 Mathematical Problems in Engineering

[2] R A Nash Jr J A DrsquoAppolito and K J Roy ldquoError analysis ofhybrid aircraft inertial navigation systemsrdquo in Proceedings of theAIAA Guidance and Control Conference Stanford Calif USA1972

[3] F D Jurenka and C T Leondes ldquoOptimum alignment of aninertial autonavigatorrdquo IEEE Transactions on Aerospace andElectronic Systems vol AES-3 no 6 pp 880ndash888 1967

[4] R H Cannon ldquoAlignment of inertial guidance systems bygyrocompassing-linear theoryrdquo Journal of Aerospace Sciencevol 28 no 11 pp 885ndash895 912 1961

[5] G R Pitman Jr Inertial Guidance John Wiley amp Sons NewYork NY USA 1962

[6] K R Britting Inertial Navigation Systems Analysis John Wileyamp Sons New York NY USA 1971

[7] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation vol 28 no4 pp 311ndash341 1982

[8] X Liu X Xu Y Liu and L Wang ldquoA fast and high-accuracycompass alignmentmethod to SINSwith azimuth axis rotationrdquoMathematical Problems in Engineering vol 2013 Article ID524284 12 pages 2013

[9] F Sun Q Wang Z Qi and C Wang ldquoResearch on theestimation method of DVL velocity error based on doubleprogram in Fiber Optic Gyro SINSrdquo Optik vol 124 no 22 pp5344ndash5349 2013

[10] B M Scherzinger ldquoInertial navigator error models for largeheading uncertaintyrdquo in Proceedings of the IEEE Position Loca-tion and Navigation Symposium (PLANS rsquo96) pp 477ndash484Atlanta Ga USA April 1996

[11] X Y Kong EM Nebot andH Durrant-Whyte ldquoDevelopmentof a non-linear psi-angle model for large misalignment errorsand its application in INS alignment and calibrationrdquo in Pro-ceedings of the 1999 IEEE International Conference on Roboticsamp Automation (ICRA rsquo99) pp 1430ndash1435 Detroit Mich USAMay 1999

[12] M-J Yu H-W Park and C-B Jeon ldquoEquivalent nonlinearerror models of strapdown inertial navigation systemrdquo in Pro-ceedings of the Guidance Navigation and Control ConferenceAIAA-97-3563 pp 581ndash587 1997

[13] T Abbas Z Yunyan and L Yanjun ldquoSINS initial alignment forsmall tilt and large azimuth misalignment anglesrdquo in Proceed-ings of the IEEE 3rd International Conference on CommunicationSoftware and Networks (ICCSN rsquo11) pp 628ndash632 Xirsquoan ChinaMay 2011

[14] S L Han and J L Wang ldquoA novel initial alignment scheme forlow-cost INS aided by GPS for land vehicle applicationsrdquo TheJournal of Navigation vol 63 no 4 pp 663ndash680 2010

[15] P G Savage Strapdown Analytics Strapdown Associates 2ndedition 2007

[16] I Yaesh and B Priel ldquoDesign of leveling loop for marine nav-igation systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 29 no 2 pp 599ndash604 1993

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

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4 Mathematical Problems in Engineering

120575a119887 is the disturbing motion vector under quasistationarysituations and nabla119887 is the accelerometer error vector

Substituting (15c) in (13) we obtain

V119901 = (I minus C119901119899) g119899 + C119901

119887120575a119887 + C119901

119887nabla119887 (16)

Then from (16) the horizontal components of V119901 can bedescribed respectively as

119864= minus119892120595

119873+ (C119901119887120575a119887)119864+ (C119901119887nabla119887)119864

119873= 119892120595119864+ (C119901119887120575a119887)119873+ (C119901119887nabla119887)119873

(17)

where 119881119864and 119881

119873represent the east and north horizontal

velocities respectively (sdot)119864and (sdot)

119873represent the 119864 and 119873

elements of column matrix (sdot) separately whose entries fromtop to bottom are the 119864119873 and 119880 elements

From (11) (12) and (14) the misalignment equation ofstrapdown INS under large azimuth misalignment can berepresented as [11 13]

= (I minus C119901119899)120596119899

119894119890minus C119901119887120576119887 (18)

where 120595 is the misalignment vector and 120595 = [120595119864 120595119873 120595119880]119879

The leveling alignment of the improved method is per-formed by introducing control angular velocity 120596119901

119888into the

calculation of angular velocity 120596119887119901119887 then we have 120596119887

119901119887= 119887

119894119887minus

C119887119901120596119899

119894119890minus C119887119901120596119901

119888 finally the misalignment equation could be

transformed into

= (I minus C119901119899)120596119899

119894119890minus C119901119887120576119887+ 120596119901

119888 (19)

where 120596119901119888= [120596119901

119888119864120596119901

1198881198730]119879

is the control angular velocityvector From (19) the horizontal components of 120595 can berespectively represented as

119864= minus sin120595

119880Ω cos120593 + 120595

119873Ω sin120593 minus (C119901

119887120576119887)119864+ 120596119901

119888119864 (20)

119873= (1 minus cos120595

119880)Ω cos120593 minus 120595

119864Ω sin120593 minus (C119901

119887120576119887)119873

+ 120596119901

119888119873

(21)

Because the azimuth misalignmentrsquos rate of change is small[10] the approximation

119880= 0 is admitted namely 120595

119880is a

constant value [15]The azimuth information is obtained by structuring

azimuth estimation functions The block diagram of 120596119901119888and

azimuth estimation functions is shown in Figure 3In Figure 3 119896

119886and 119896119887represent the control gains119866(119904) and

119867(119904) represent the control networks used for reducing theinfluence of disturbing motions 119878(120595

119880) and 119862(120595

119880) represent

the functions of 120595119880 From (17) (20) (21) and Figure 3 the

error diagrams of improved gyrocompass alignment can beobtained as in Figures 4 and 5

It can be seen from Figures 4 and 5 that the nonlinearfactors sin120595

119880Ω cos120593 and (1minuscos120595

119880)Ω cos120593 both are caused

by the large azimuth misalignment and become the constantinputs of the system through opening the azimuth loop So

minus

minus

ka

ka

G(s)

S(120595U)

P

N

E

minuskb

kb

H(s)

1

s

1

s

C(120595U)

120596pcE

120596pcN

120596pc

V

V

V

Figure 3 Block diagram of 120596119901119888and azimuth estimation functions

ka

1

s

1

s

+

+minus

minus

minus

minus

g

G(s)

kb

120595E

S(120595U)sin 120595UΩ cos 120593 120595NΩ sin 120593

(Cpb120576b)

E

(Cp b120575ab) N

+(C

p bnablab) N

Figure 4 Error diagram of improved gyrocompass alignment innorth loop

+

+

minus

minus

minus

minus

g

kb

ka

1

s

1

s

H(s)

120595N

C(120595U)(1 minus cos120595U)Ωcos 120593 120595EΩ sin 120593

(Cpb120576b)

N

(Cp b120575ab) E

+(C

p bnablab) E

Figure 5 Error diagram of improved gyrocompass alignment ineast loop

the nonlinear factors will not be included in the characteristicequation In addition these constant inputs contain azimuthmisalignment information and the estimation of the azimuthinformation can be obtained by 119878(120595

119880) and 119862(120595

119880)

32 The Operation of the Improved Gyrocompass AlignmentMethod In this section the control gains 119896

119886and 119896

119887and

control networks 119866(119904) and 119867(119904) are provided Then 119878(120595119880)

and 119862(120595119880) can be obtained and the azimuth estimation is

performed by these two functions

Mathematical Problems in Engineering 5

Firstly we provide the values of the control gains 119896119886and 119896119887

for improved gyrocompass alignment the detailed deductionfor these control gains can be found in the following

According to Mason gain formula we can get the charac-teristic equation of north loop from Figure 4

Δ119873 (119904) = 119904

2+ 119896119886119904 + 119892119896

119887= 0 (22)

In the same way from Figure 5 the characteristic equation ofeast loop can be obtained as follows

Δ119864 (119904) = 119904

2+ 119896119886119904 + 119892119896

119887= 0 (23)

It can be seen from (22) and (23) that the north and east loopshave the same characteristic equation In order to ensure thestability of alignment system we set two eigenvalues of (22)and (23) as

11990412= minus120585120596

119899plusmn 119895radic1 minus 1205852120596

119899 (24)

Then the characteristic equation can be rewritten as

Δ119873 (119904) = Δ119864 (119904) = 119904

2+ 2120585120596

119899119904 + 1205962

119899= 0 (25)

Comparing (22) with (25) we can get

119896119886= 2120585120596

119899

119896119887=1205962

119899

119892

(26)

Secondly in order to simplify the subsequent analysis weassume that the inertial sensor errors are basically constantdrifts and the vehicle in which the INS is mounted is totallystopped Under this assumption the disturbing motion vec-tor 120575a119887 is equal to zero At this time the control networks canbe set as 119866(119904) = 119867(119904) = 1 because no disturbing motion isintroduced into the alignment system The design procedureof azimuth estimation is described in the following and itcan be divided into two steps The first step the analysisof the relationships between functions 119878(120595

119880) and 119862(120595

119880)

and azimuth misalignment 120595119880is made with the aid of the

Laplace transformationThe second step based on the formeranalysis the azimuth estimation equations are provided

321 Relationships between Functions 119878(120595119880) and 119862(120595

119880) and

Azimuth Misalignment 120595119880 The response to east level mis-

alignment 120595119864and function 119878(120595

119880) can be written in Laplace

form directly from Figure 4 and can be seen as follows(according to the Mason gain formula)

120595119864 (119904)

= minus1205962

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot

(C119901119887nabla119887)119873 (119904)

119892

+119904 + 2120585120596

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot minus sin120595119880Ω cos120593 + 120595

119873Ω sin120593 minus (C119901

119887120576119887)119864 (119904)

119878 (120595119880) (119904)

=119904

1199042 + 2120585120596119899119904 + 1205962119899

sdot (C119901119887nabla119887)119873 (119904)

+119892

1199042 + 2120585120596119899119904 + 1205962119899

sdot minus sin120595119880Ω cos120593 + 120595

119873Ω sin120593 minus (C119901

119887120576119887)119864 (119904)

(27)

where sdot(119904) represents the Laplace transformation of sdotSimilarly from Figure 5 it yields

120595119873 (119904) =

1205962

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot

(C119901119887nabla119887)119864 (119904)

119892

+119904 + 2120585120596

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot (1 minus cos120595119880)Ω cos120593 minus 120595

119864Ω sin120593 minus (C119901

119887120576119887)119873

sdot (119904)

119862 (120595119880) (119904) =

119904

1199042 + 2120585120596119899119904 + 1205962119899

sdot (C119901119887nabla119887)119864 (119904)

minus119892

1199042 + 2120585120596119899119904 + 1205962119899

sdot (1 minus cos120595119880)Ω cos120593 minus 120595

119864Ω sin120593 minus (C119901

119887120576119887)119873

sdot (119904)

(28)

Equations (27)-(28) are solved for the steady-state errors byinvoking the final value theorem

lim119905rarrinfin

119891 (119905) = lim119904rarr0

119904119891 (119904) (29)

6 Mathematical Problems in Engineering

Then we have

120595119864ss = minus

(C119901119887nabla119887)119873

119892+2120585

120596119899

sdot [minus sin120595119880Ω cos120593

+ 120595119873ssΩ sin120593 minus (C119901

119887120576119887)119864]

119878 (120595119880)ss =

119892

1205962119899

sdot [minus sin120595119880Ω cos120593 + 120595

119873ssΩ sin120593

minus (C119901119887120576119887)119864]

120595119873ss =

(C119901119887nabla119887)119864

119892+2120585

120596119899

sdot [(1 minus cos120595119880)Ω cos120593

minus 120595119864ssΩ sin120593 minus (C119901

119887120576119887)119873]

119862 (120595119880)ss =

119892

1205962119899

sdot [(cos120595119880minus 1)Ω cos120593 + 120595

119864ssΩ sin120593

+ (C119901119887120576119887)119873]

(30)

where (sdot)ss represents the steady-state value of (sdot)We regard sin120595

119880 cos120595

119880 120595119864ss and 120595119873ss as the unknown

values (30) can then be easily solved Then the relationshipsbetween functions 119878(120595

119880) and 119862(120595

119880) and azimuth misalign-

ment 120595119880can be obtained

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593+

(C119901119887nabla119887)119873

119892tan120593

(31)

The detailed derivations of (31) can be found in Appendix A

322 The Azimuth Estimation Equations It is clear thatsin120595119880

and cos120595119880

can be supplied exactly by (31) if theaccelerometer errors and gyro errors (sensor errors) areknown Since the sensor errors are uncertain sin120595

119880and

cos120595119880are calculated by the following equations

sin 119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880) minus

2120585120596119899tan120593119892

sdot 119862 (120595119880) (32)

cos 119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880) minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)

(33)

where sin 119880and cos

119880are the computed values of sin120595

119880

and cos120595119880 respectively When the improved alignment

system is stable the errors between computed values (sin 119880

and cos 119880) and theoretical values (sin120595

119880and cos120595

119880) are

only determined by sensor errors and these errors areallowable Worth noting is that (32) and (33) do not preservethe unit-norm property of the trigonometric function thatis (sin

119880)2+ (cos

119880)2

= 1 So normalization of sin 119880

and cos 119880should be given and the simple normalization is

provided as follows

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(34)

The detailed derivations of (34) can be found in Appendix BAt this time the azimuth estimation

119880can be obtained

from (34) and is shown as follows

119880119898

= arcsin (sin 119880)

119880=

119880119898

cos 119880gt 0

180∘minus 119880119898

cos 119880lt 0 sin

119880gt 0

minus180∘minus 119880119898

cos 119880lt 0 sin

119880lt 0

(35)

It is obvious from the previous discussion that the azimuthestimation equations consist of (32)ndash(35)

Finally we take the disturbing motions into consid-eration because the alignment is often performed underquasistationary conditions The disturbing motions can begenerally considered to be sinusoidal [4 16] Then in orderto reduce the influence of the disturbing motions on azimuthestimation low pass filters need to be added to the systemnamely the control networks 119866(119904) and119867(119904) should have thecapability of restraining disturbance The design law of thecontrol networks 119866(119904) and119867(119904) is provided as follows

(a) The magnification of 119866(119904) and119867(119904) in low frequencymust be equivalent to 1 that is the following equationshould be satisfied

lim119904rarr0

119866 (119904) = lim119904rarr0

119867(119904) = 1 (36)

The reason for this requirement is that the rela-tionships between functions 119878(120595

119880) and 119862(120595

119880) and

azimuthmisalignment120595119880under quasistationary con-

ditions also can be represented by (31) as the require-ment is met

(b) In high frequency for the purpose of reducing theinfluence of disturbing motions they need be capableof restraining disturbance

Then under quasistationary conditions the azimuth estima-tion equations also consist of (32)ndash(35) In this work thecontrol networks 119866(119904) and119867(119904) are set as

119866 (119904) = 119867 (119904) =(120585120596119899)2

(119904 + 120585120596119899)2 (37)

Mathematical Problems in Engineering 7

bib

minus

minus

Gyros

AccelerometersCalculation of control angular

velocity and azimuth estimation functions

DSP

Final process

S(120595U) C(120595U)

120596bpb

E

N

U

120596n

ieCbp

pb =

pb[120596bpb times]CC

120596p

cCbp

fb

Vp= Cp

bfb+ gn

sin 120595U

cos 120595U

01

+

minus1205962n

gΩ cos 120593minus2120585120596n tan 120593

g

minus2120585120596n tan 120593

g

1205962n

gΩ cos 120593

S(120595U)C(120595U)lfloor

lceilrfloorrceillfloorlceillfloorlceil rfloorrceilrfloorrceil

V

V

Figure 6 Flow chart of improved gyrocompass alignment in real time operating device

At this time the denominators of 119878(120595119880)(119904) and 119862(120595

119880)(119904)

can be set as

(1199042+ 2120585120596

119899119904 + 1205962

119899) (119904 + 120585120596

119899)2 (38)

It is obvious from the azimuth estimation equations that theperformance of the azimuth alignment is determined by (32)and (33) Since both (32) and (33) consist of 119878(120595

119880) and119862(120595

119880)

the four eigenvalues of sin 119880(119904) and cos

119880(119904) could be

set as 11990412

= minus120585120596119899and 11990434

= minus120585120596119899plusmn 119895radic1 minus 1205852120596

119899 without

the influence of the nonlinear factors Therefore the impactof the nonlinear factor caused by large azimuthmisalignmenton gyrocompass alignment is reduced and the performanceof the improved gyrocompass alignment will be better thanthe conventional one

33 The Implementation of the Improved Gyrocompass Align-ment Method In this section the implementation of theproposed method in real time operating device is discussedand meanwhile a brief summary of this method is providedAn inertial navigation system implements the proposedalignment method using a cluster of accelerometers to sensethe specific force vector components f119887 a triad of gyrosto measure the angular velocity 119887

119894119887 and a digital signal

processor (DSP) to perform the alignment algorithm Thedirect expression of this implementation can be seen inFigure 6 According to Figure 6 the specific implementationsteps of the improved gyrocompass alignment in real timeoperation device are described as follows

(a) At the beginning a preliminary alignment oftencalled coarse alignment is performed and after thata rough DCM C119901

119887is obtained The specific coarse

alignment method can be found in [15] the levelmisalignments are small and azimuth misalignmentis usually large

(b) Secondly by utilizing the specific force f119887 and DCMC119901119887 the derivatives of the east and north horizontal

velocities 119864and 119873can be obtainedThen according

to the calculation method shown in Figure 3 thecontrol velocity 120596119901

119888and azimuth functions 119878(120595

119880) and

119862(120595119880) can be acquired

(c) On the one hand the control velocity 120596119901119888and angular

velocity 119887119894119887are used to update the DCM C119901

119887used for

the next calculation step(d) On the other hand the azimuth functions 119878(120595

119880) and

119862(120595119880) are used to compute sin

119880and cos

119880 And

after the final process which is achieved by utilizing(34)-(35) the azimuth estimation

119880is obtained

(e) Then steps (b) (c) and (d) are performed repeatedlyFinally by spending a period of time an accurateazimuth estimation

119880can be obtained and with the

compensation of azimuth misalignment using 119880 an

accurate DCM C119901119887can be obtained

4 Simulations and Experiments

In this section to evaluate the performance of the improvedgyrocompass alignment method under the condition of largeazimuth misalignment simulations and experiments arecarried out

41 Simulation Results and Analysis Both the conventionaland improved gyrocompass alignment methods are per-formed during the simulations simultaneously and the sim-ulations are conducted under the conditions of respec-tively choosing different azimuth misalignments namely120595119880= 50

∘ 100∘ 145∘ minus50∘ minus100

∘ minus145

∘ In these simu-

lations gyro and accelerometer outputs are generated bythe strapdown INS simulator we assume that the vehicleis in disturbing motions caused by sea waves The initialconditions are presented as follows

The constant gyro errors 120576119887 =

[[001∘ℎ 001

∘ℎ 001

∘ℎ]]119879

8 Mathematical Problems in Engineering

0 5 10 15 20

0 0

50

100

150

Time (min) Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0 5 10 15 20

Improved method

minus50

50

100

150

minus50

Figure 7 Gyrocompass alignment for 120595119880= 50

∘ 100∘ 145∘

The constant accelerometer errors nabla119887 =

[[10minus4119892 10minus4119892 10minus4119892]]119879

The acceleration due to gravity 119892 = 978049ms2The local longitude and latitude 120582 = 1266705∘ 120593 =457796

∘The control networks of improved method 119866(119904) =119867(119904) = (120585120596

119899)2(119904 + 120585120596

119899)2

The natural frequency of conventional and improvedmethods 120596

119899= 002 rads

Under the disturbingmotions the vehicle undertakes angularand linealmovements In angularmovement the pitch 120579

119909 the

roll 120579119910 and the yaw 120579

119911are controlled as

120579119909= 7∘ sin(2120587

5119905 +

120587

3)

120579119910= 5∘ sin(2120587

7119905 +

120587

6)

120579119911= 10∘ sin(2120587

6119905 +

120587

4)

(39)

In the lineal movement the vehicle lineal movement veloci-ties are taken as

119881119909= minus03 sin(2120587

10119905) ms2

119881119910= minus02 sin(2120587

6119905) ms2

119881119911= minus04 sin(2120587

8119905) ms2

(40)

Figures 7 and 8 provide the azimuth estimation errors ofconventional and improved methods under the conditions

of 120595119880= 50

∘ 100∘ 145∘ minus50∘ minus100

∘ minus145

∘ It can be seen

that azimuth alignment could be accomplished by both con-ventional and improved gyrocompass alignment methodsBut under the condition of large azimuth misalignment theperformance of conventional method will be poor along withthe growth of |120595

119880| The reason is that the difference between

the actual and desired eigenvalues is increasing alongwith thegrowth of |120595

119880|

Figure 9 shows convergence time for different azimuthmisalignments with a converged azimuth estimation errorof less than 1 degree It is clear that large azimuth misalign-ment needs more time to converge under the conventionalmethod However for the improved method the changeof the convergence time is not obvious That means theperformance of the improved method is not affected bythe large azimuth misalignment The reason is that theeigenvalues of improved method can be set as the desiredvalues without the influence of the nonlinear factors causedby the large azimuth misalignment Moreover it is obviousthat the alignment speed of the improved method is fasterthan that of the conventional one

42 Experimental Results and Analysis The experimentswere implemented in the lab as the true yaw angle ofthe strapdown INS was known so different 120595

119880could be

exactly set We fixed the strapdown INS on the SGT-3 three-axis turntable to implement the alignment experiments Thestrapdown INS and the turntable can be seen in Figure 10 Atthe start of each experiment the turntable turned to a staticposition for 20 minutes that is pitch angle 0∘ roll angle 0∘and yaw angle 135∘ Two alignmentmethods were performedunder the condition of respectively setting initial yaw angleas 185∘ and 235∘ namely 120595

119880= 50∘ 100∘

Mathematical Problems in Engineering 9

0 5 10 15 20

0

50

Time (min)0 5 10 15 20

Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0

50 Improved method

minus50

minus100

minus150

minus50

minus100

minus150

Figure 8 Gyrocompass alignment for 120595119880= minus50

∘ minus100

∘ minus145

∘

50 100 1450

5

10

15

20

Azimuth misalignment (deg)

Con

verg

ence

tim

e (m

in)

Conventional methodImproved method

minus145 minus50minus100

Figure 9 Convergence time comparison with different azimuthmisalignments

Figure 11 provides the comparison of the yaw angleusing the two alignment methods under the condition ofdifferent 120595

119880

As the results above show the yaw angle can convergeto 135∘ through using the two alignment methods How-ever for the conventional method the convergence time islengthened with the growth of the azimuth misalignmentCompared with the conventional method the convergence

Strapdown INS

SGT-3

Figure 10 The SGT-3 turntable and strapdown INS

time of improved method is hardly changed The exper-imental results are consistent with the simulation resultsThe experiments demonstrate that the performance of theimproved method is better than that of the conventional one

5 Conclusions

In this paper an improved gyrocompass alignment methodhas been proposed for large azimuth misalignment of INSThe improved new method is performed by opening theazimuth loop and the estimation of the azimuth misalign-ment is implemented by using the horizontal velocity signalsto estimate the nonlinear factorsThrough using thismethodthe impact of the nonlinear factor caused by large azimuthmisalignment on gyrocompass alignment is reduced andthe performance of gyrocompass alignment is improvedThealignment speed of the improved method is faster than theconventional one in the same conditions The performance

10 Mathematical Problems in Engineering

0 5 10 15 20100

135

150

185200

235250

300

Yaw

(deg

)

Time (min)

Conventional method 120595U = 50∘

Conventional method 120595U = 100∘

Improved method 120595U = 50∘

Improved method 120595U = 100∘

Figure 11 Comparison of the yaw angle using the two alignmentmethods

of the improved method is well validated by simulations andexperiments

Appendices

A The Derivations of (31)

In this section (31) are obtained in the followingAccording to (30) their matrix form can be written as

AX = B (A1)

where

A

=

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

[

2120585Ω cos120593120596119899

0 1 minus2120585Ω sin120593

120596119899

119892Ω cos1205931205962119899

0 0 minus119892Ω sin1205931205962119899

02120585Ω cos120593

120596119899

2120585Ω sin120593120596119899

1

0119892Ω cos1205931205962119899

119892Ω sin1205931205962119899

0

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

]

(A2)

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

(A3)

B =

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

[

minus

(C119901119887nabla119887)119873

119892minus2120585

120596119899

(C119901119887120576119887)119864

minus119878 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119864

2120585

120596119899

Ω cos120593 +(C119901119887nabla119887)119864

119892minus2120585

120596119899

(C119901119887120576119887)119873

119892

1205962119899

Ω cos120593 + 119862 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119873

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

]

(A4)

Multiply the first and third rows of (A1) by1205961198992120585Ω cos120593 and

multiply the second and fourth rows of (A1) by 1205962119899119892Ω cos120593

we can obtain

A1X = B

1 (A5)

where

A1=

[[[[[[[[

[

1 0120596119899

2120585Ω cos120593minus tan120593

1 0 0 minus tan120593

0 1 tan120593120596119899

2120585Ω cos1205930 1 tan120593 0

]]]]]]]]

]

X =[[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]]

]

B1=

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

[

minus120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

1 +120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892minus

(C119901119887120576119887)119873

Ω cos120593

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

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

]

(A6)

Furthermore by utilizing the elementary row transformation(A5) can be transformed into

A2X = B

2 (A7)

Mathematical Problems in Engineering 11

where

A1=

[[[[[[[[

[

1 0 0 minus tan120593

0 0 minus120596119899

2120585Ω cos1205930

0 1 tan120593 0

0 0 0 minus120596119899

2120585Ω cos120593

]]]]]]]]

]

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

B1

=

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

[

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos1205931205962

119899

119892Ω cos120593119862 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892

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

]

(A8)

Then according to matrix equation (A7) we have

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

+

(C119901119887nabla119887)119873

119892tan120593

(A9)

B The Derivations of (34)

The derivations of (34) are carried out based on the principleof least squares fitting setting the Euclidean minimum norm119869 between sin

119880 cos

119880and sin

119880 cos

119880(the normalized

data) as the normalization indexFirst define

119869 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]12

(B1)

It is subjected to the constrained condition namely

ℎ = (sin 119880)2+ (cos

119880)2minus 1 = 0 (B2)

Second define an auxiliary function119898 namely

119898 = 1198692+ 120582ℎ (B3)

where 120582 is an unknown parameterBy substituting (B1) and (B2) into (B3) we can get

119898 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]

minus 120582 ((sin 119880)2+ (cos

119880)2minus 1)

(B4)

Obviously the parameters sin 119880and cos

119880that made 119869

minimum are equal to the ones that make 119898 minimumRespectively the partial derivatives of the parameters sin

119880

and cos 119880are calculated and bymaking them equal to zero

we can get

sin 119880=sin 119880

1 + 120582

cos 119880=cos 119880

1 + 120582

(B5)

Then by substituting (B5) into (B2) we have

(sin 119880)2+ (cos

119880)2

(1 + 120582)2

= 1 (B6)

Finally combining (B5) and (B6) the normalization equa-tions can be obtained

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(B7)

Conflict of Interests

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

Acknowledgments

The work described in the paper was supported by theNational Natural Science Foundation of China (61203225)and the National Science Foundation for Post-DoctoralScientists of China (2012M510083)The authors would like tothank all members of the Inertial Navigation Research Groupat Harbin Engineering University for the technical assistancewith the navigation system

References

[1] H W Park J G Lee and C G Park ldquoCovariance analysis ofstrapdown INS considering gyrocompass characteristicsrdquo IEEETransactions on Aerospace and Electronic Systems vol 31 no 1pp 320ndash328 1995

12 Mathematical Problems in Engineering

[2] R A Nash Jr J A DrsquoAppolito and K J Roy ldquoError analysis ofhybrid aircraft inertial navigation systemsrdquo in Proceedings of theAIAA Guidance and Control Conference Stanford Calif USA1972

[3] F D Jurenka and C T Leondes ldquoOptimum alignment of aninertial autonavigatorrdquo IEEE Transactions on Aerospace andElectronic Systems vol AES-3 no 6 pp 880ndash888 1967

[4] R H Cannon ldquoAlignment of inertial guidance systems bygyrocompassing-linear theoryrdquo Journal of Aerospace Sciencevol 28 no 11 pp 885ndash895 912 1961

[5] G R Pitman Jr Inertial Guidance John Wiley amp Sons NewYork NY USA 1962

[6] K R Britting Inertial Navigation Systems Analysis John Wileyamp Sons New York NY USA 1971

[7] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation vol 28 no4 pp 311ndash341 1982

[8] X Liu X Xu Y Liu and L Wang ldquoA fast and high-accuracycompass alignmentmethod to SINSwith azimuth axis rotationrdquoMathematical Problems in Engineering vol 2013 Article ID524284 12 pages 2013

[9] F Sun Q Wang Z Qi and C Wang ldquoResearch on theestimation method of DVL velocity error based on doubleprogram in Fiber Optic Gyro SINSrdquo Optik vol 124 no 22 pp5344ndash5349 2013

[10] B M Scherzinger ldquoInertial navigator error models for largeheading uncertaintyrdquo in Proceedings of the IEEE Position Loca-tion and Navigation Symposium (PLANS rsquo96) pp 477ndash484Atlanta Ga USA April 1996

[11] X Y Kong EM Nebot andH Durrant-Whyte ldquoDevelopmentof a non-linear psi-angle model for large misalignment errorsand its application in INS alignment and calibrationrdquo in Pro-ceedings of the 1999 IEEE International Conference on Roboticsamp Automation (ICRA rsquo99) pp 1430ndash1435 Detroit Mich USAMay 1999

[12] M-J Yu H-W Park and C-B Jeon ldquoEquivalent nonlinearerror models of strapdown inertial navigation systemrdquo in Pro-ceedings of the Guidance Navigation and Control ConferenceAIAA-97-3563 pp 581ndash587 1997

[13] T Abbas Z Yunyan and L Yanjun ldquoSINS initial alignment forsmall tilt and large azimuth misalignment anglesrdquo in Proceed-ings of the IEEE 3rd International Conference on CommunicationSoftware and Networks (ICCSN rsquo11) pp 628ndash632 Xirsquoan ChinaMay 2011

[14] S L Han and J L Wang ldquoA novel initial alignment scheme forlow-cost INS aided by GPS for land vehicle applicationsrdquo TheJournal of Navigation vol 63 no 4 pp 663ndash680 2010

[15] P G Savage Strapdown Analytics Strapdown Associates 2ndedition 2007

[16] I Yaesh and B Priel ldquoDesign of leveling loop for marine nav-igation systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 29 no 2 pp 599ndash604 1993

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 5

Firstly we provide the values of the control gains 119896119886and 119896119887

for improved gyrocompass alignment the detailed deductionfor these control gains can be found in the following

According to Mason gain formula we can get the charac-teristic equation of north loop from Figure 4

Δ119873 (119904) = 119904

2+ 119896119886119904 + 119892119896

119887= 0 (22)

In the same way from Figure 5 the characteristic equation ofeast loop can be obtained as follows

Δ119864 (119904) = 119904

2+ 119896119886119904 + 119892119896

119887= 0 (23)

It can be seen from (22) and (23) that the north and east loopshave the same characteristic equation In order to ensure thestability of alignment system we set two eigenvalues of (22)and (23) as

11990412= minus120585120596

119899plusmn 119895radic1 minus 1205852120596

119899 (24)

Then the characteristic equation can be rewritten as

Δ119873 (119904) = Δ119864 (119904) = 119904

2+ 2120585120596

119899119904 + 1205962

119899= 0 (25)

Comparing (22) with (25) we can get

119896119886= 2120585120596

119899

119896119887=1205962

119899

119892

(26)

Secondly in order to simplify the subsequent analysis weassume that the inertial sensor errors are basically constantdrifts and the vehicle in which the INS is mounted is totallystopped Under this assumption the disturbing motion vec-tor 120575a119887 is equal to zero At this time the control networks canbe set as 119866(119904) = 119867(119904) = 1 because no disturbing motion isintroduced into the alignment system The design procedureof azimuth estimation is described in the following and itcan be divided into two steps The first step the analysisof the relationships between functions 119878(120595

119880) and 119862(120595

119880)

and azimuth misalignment 120595119880is made with the aid of the

Laplace transformationThe second step based on the formeranalysis the azimuth estimation equations are provided

321 Relationships between Functions 119878(120595119880) and 119862(120595

119880) and

Azimuth Misalignment 120595119880 The response to east level mis-

alignment 120595119864and function 119878(120595

119880) can be written in Laplace

form directly from Figure 4 and can be seen as follows(according to the Mason gain formula)

120595119864 (119904)

= minus1205962

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot

(C119901119887nabla119887)119873 (119904)

119892

+119904 + 2120585120596

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot minus sin120595119880Ω cos120593 + 120595

119873Ω sin120593 minus (C119901

119887120576119887)119864 (119904)

119878 (120595119880) (119904)

=119904

1199042 + 2120585120596119899119904 + 1205962119899

sdot (C119901119887nabla119887)119873 (119904)

+119892

1199042 + 2120585120596119899119904 + 1205962119899

sdot minus sin120595119880Ω cos120593 + 120595

119873Ω sin120593 minus (C119901

119887120576119887)119864 (119904)

(27)

where sdot(119904) represents the Laplace transformation of sdotSimilarly from Figure 5 it yields

120595119873 (119904) =

1205962

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot

(C119901119887nabla119887)119864 (119904)

119892

+119904 + 2120585120596

119899

1199042 + 2120585120596119899119904 + 1205962119899

sdot (1 minus cos120595119880)Ω cos120593 minus 120595

119864Ω sin120593 minus (C119901

119887120576119887)119873

sdot (119904)

119862 (120595119880) (119904) =

119904

1199042 + 2120585120596119899119904 + 1205962119899

sdot (C119901119887nabla119887)119864 (119904)

minus119892

1199042 + 2120585120596119899119904 + 1205962119899

sdot (1 minus cos120595119880)Ω cos120593 minus 120595

119864Ω sin120593 minus (C119901

119887120576119887)119873

sdot (119904)

(28)

Equations (27)-(28) are solved for the steady-state errors byinvoking the final value theorem

lim119905rarrinfin

119891 (119905) = lim119904rarr0

119904119891 (119904) (29)

6 Mathematical Problems in Engineering

Then we have

120595119864ss = minus

(C119901119887nabla119887)119873

119892+2120585

120596119899

sdot [minus sin120595119880Ω cos120593

+ 120595119873ssΩ sin120593 minus (C119901

119887120576119887)119864]

119878 (120595119880)ss =

119892

1205962119899

sdot [minus sin120595119880Ω cos120593 + 120595

119873ssΩ sin120593

minus (C119901119887120576119887)119864]

120595119873ss =

(C119901119887nabla119887)119864

119892+2120585

120596119899

sdot [(1 minus cos120595119880)Ω cos120593

minus 120595119864ssΩ sin120593 minus (C119901

119887120576119887)119873]

119862 (120595119880)ss =

119892

1205962119899

sdot [(cos120595119880minus 1)Ω cos120593 + 120595

119864ssΩ sin120593

+ (C119901119887120576119887)119873]

(30)

where (sdot)ss represents the steady-state value of (sdot)We regard sin120595

119880 cos120595

119880 120595119864ss and 120595119873ss as the unknown

values (30) can then be easily solved Then the relationshipsbetween functions 119878(120595

119880) and 119862(120595

119880) and azimuth misalign-

ment 120595119880can be obtained

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593+

(C119901119887nabla119887)119873

119892tan120593

(31)

The detailed derivations of (31) can be found in Appendix A

322 The Azimuth Estimation Equations It is clear thatsin120595119880

and cos120595119880

can be supplied exactly by (31) if theaccelerometer errors and gyro errors (sensor errors) areknown Since the sensor errors are uncertain sin120595

119880and

cos120595119880are calculated by the following equations

sin 119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880) minus

2120585120596119899tan120593119892

sdot 119862 (120595119880) (32)

cos 119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880) minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)

(33)

where sin 119880and cos

119880are the computed values of sin120595

119880

and cos120595119880 respectively When the improved alignment

system is stable the errors between computed values (sin 119880

and cos 119880) and theoretical values (sin120595

119880and cos120595

119880) are

only determined by sensor errors and these errors areallowable Worth noting is that (32) and (33) do not preservethe unit-norm property of the trigonometric function thatis (sin

119880)2+ (cos

119880)2

= 1 So normalization of sin 119880

and cos 119880should be given and the simple normalization is

provided as follows

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(34)

The detailed derivations of (34) can be found in Appendix BAt this time the azimuth estimation

119880can be obtained

from (34) and is shown as follows

119880119898

= arcsin (sin 119880)

119880=

119880119898

cos 119880gt 0

180∘minus 119880119898

cos 119880lt 0 sin

119880gt 0

minus180∘minus 119880119898

cos 119880lt 0 sin

119880lt 0

(35)

It is obvious from the previous discussion that the azimuthestimation equations consist of (32)ndash(35)

Finally we take the disturbing motions into consid-eration because the alignment is often performed underquasistationary conditions The disturbing motions can begenerally considered to be sinusoidal [4 16] Then in orderto reduce the influence of the disturbing motions on azimuthestimation low pass filters need to be added to the systemnamely the control networks 119866(119904) and119867(119904) should have thecapability of restraining disturbance The design law of thecontrol networks 119866(119904) and119867(119904) is provided as follows

(a) The magnification of 119866(119904) and119867(119904) in low frequencymust be equivalent to 1 that is the following equationshould be satisfied

lim119904rarr0

119866 (119904) = lim119904rarr0

119867(119904) = 1 (36)

The reason for this requirement is that the rela-tionships between functions 119878(120595

119880) and 119862(120595

119880) and

azimuthmisalignment120595119880under quasistationary con-

ditions also can be represented by (31) as the require-ment is met

(b) In high frequency for the purpose of reducing theinfluence of disturbing motions they need be capableof restraining disturbance

Then under quasistationary conditions the azimuth estima-tion equations also consist of (32)ndash(35) In this work thecontrol networks 119866(119904) and119867(119904) are set as

119866 (119904) = 119867 (119904) =(120585120596119899)2

(119904 + 120585120596119899)2 (37)

Mathematical Problems in Engineering 7

bib

minus

minus

Gyros

AccelerometersCalculation of control angular

velocity and azimuth estimation functions

DSP

Final process

S(120595U) C(120595U)

120596bpb

E

N

U

120596n

ieCbp

pb =

pb[120596bpb times]CC

120596p

cCbp

fb

Vp= Cp

bfb+ gn

sin 120595U

cos 120595U

01

+

minus1205962n

gΩ cos 120593minus2120585120596n tan 120593

g

minus2120585120596n tan 120593

g

1205962n

gΩ cos 120593

S(120595U)C(120595U)lfloor

lceilrfloorrceillfloorlceillfloorlceil rfloorrceilrfloorrceil

V

V

Figure 6 Flow chart of improved gyrocompass alignment in real time operating device

At this time the denominators of 119878(120595119880)(119904) and 119862(120595

119880)(119904)

can be set as

(1199042+ 2120585120596

119899119904 + 1205962

119899) (119904 + 120585120596

119899)2 (38)

It is obvious from the azimuth estimation equations that theperformance of the azimuth alignment is determined by (32)and (33) Since both (32) and (33) consist of 119878(120595

119880) and119862(120595

119880)

the four eigenvalues of sin 119880(119904) and cos

119880(119904) could be

set as 11990412

= minus120585120596119899and 11990434

= minus120585120596119899plusmn 119895radic1 minus 1205852120596

119899 without

the influence of the nonlinear factors Therefore the impactof the nonlinear factor caused by large azimuthmisalignmenton gyrocompass alignment is reduced and the performanceof the improved gyrocompass alignment will be better thanthe conventional one

33 The Implementation of the Improved Gyrocompass Align-ment Method In this section the implementation of theproposed method in real time operating device is discussedand meanwhile a brief summary of this method is providedAn inertial navigation system implements the proposedalignment method using a cluster of accelerometers to sensethe specific force vector components f119887 a triad of gyrosto measure the angular velocity 119887

119894119887 and a digital signal

processor (DSP) to perform the alignment algorithm Thedirect expression of this implementation can be seen inFigure 6 According to Figure 6 the specific implementationsteps of the improved gyrocompass alignment in real timeoperation device are described as follows

(a) At the beginning a preliminary alignment oftencalled coarse alignment is performed and after thata rough DCM C119901

119887is obtained The specific coarse

alignment method can be found in [15] the levelmisalignments are small and azimuth misalignmentis usually large

(b) Secondly by utilizing the specific force f119887 and DCMC119901119887 the derivatives of the east and north horizontal

velocities 119864and 119873can be obtainedThen according

to the calculation method shown in Figure 3 thecontrol velocity 120596119901

119888and azimuth functions 119878(120595

119880) and

119862(120595119880) can be acquired

(c) On the one hand the control velocity 120596119901119888and angular

velocity 119887119894119887are used to update the DCM C119901

119887used for

the next calculation step(d) On the other hand the azimuth functions 119878(120595

119880) and

119862(120595119880) are used to compute sin

119880and cos

119880 And

after the final process which is achieved by utilizing(34)-(35) the azimuth estimation

119880is obtained

(e) Then steps (b) (c) and (d) are performed repeatedlyFinally by spending a period of time an accurateazimuth estimation

119880can be obtained and with the

compensation of azimuth misalignment using 119880 an

accurate DCM C119901119887can be obtained

4 Simulations and Experiments

In this section to evaluate the performance of the improvedgyrocompass alignment method under the condition of largeazimuth misalignment simulations and experiments arecarried out

41 Simulation Results and Analysis Both the conventionaland improved gyrocompass alignment methods are per-formed during the simulations simultaneously and the sim-ulations are conducted under the conditions of respec-tively choosing different azimuth misalignments namely120595119880= 50

∘ 100∘ 145∘ minus50∘ minus100

∘ minus145

∘ In these simu-

lations gyro and accelerometer outputs are generated bythe strapdown INS simulator we assume that the vehicleis in disturbing motions caused by sea waves The initialconditions are presented as follows

The constant gyro errors 120576119887 =

[[001∘ℎ 001

∘ℎ 001

∘ℎ]]119879

8 Mathematical Problems in Engineering

0 5 10 15 20

0 0

50

100

150

Time (min) Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0 5 10 15 20

Improved method

minus50

50

100

150

minus50

Figure 7 Gyrocompass alignment for 120595119880= 50

∘ 100∘ 145∘

The constant accelerometer errors nabla119887 =

[[10minus4119892 10minus4119892 10minus4119892]]119879

The acceleration due to gravity 119892 = 978049ms2The local longitude and latitude 120582 = 1266705∘ 120593 =457796

∘The control networks of improved method 119866(119904) =119867(119904) = (120585120596

119899)2(119904 + 120585120596

119899)2

The natural frequency of conventional and improvedmethods 120596

119899= 002 rads

Under the disturbingmotions the vehicle undertakes angularand linealmovements In angularmovement the pitch 120579

119909 the

roll 120579119910 and the yaw 120579

119911are controlled as

120579119909= 7∘ sin(2120587

5119905 +

120587

3)

120579119910= 5∘ sin(2120587

7119905 +

120587

6)

120579119911= 10∘ sin(2120587

6119905 +

120587

4)

(39)

In the lineal movement the vehicle lineal movement veloci-ties are taken as

119881119909= minus03 sin(2120587

10119905) ms2

119881119910= minus02 sin(2120587

6119905) ms2

119881119911= minus04 sin(2120587

8119905) ms2

(40)

Figures 7 and 8 provide the azimuth estimation errors ofconventional and improved methods under the conditions

of 120595119880= 50

∘ 100∘ 145∘ minus50∘ minus100

∘ minus145

∘ It can be seen

that azimuth alignment could be accomplished by both con-ventional and improved gyrocompass alignment methodsBut under the condition of large azimuth misalignment theperformance of conventional method will be poor along withthe growth of |120595

119880| The reason is that the difference between

the actual and desired eigenvalues is increasing alongwith thegrowth of |120595

119880|

Figure 9 shows convergence time for different azimuthmisalignments with a converged azimuth estimation errorof less than 1 degree It is clear that large azimuth misalign-ment needs more time to converge under the conventionalmethod However for the improved method the changeof the convergence time is not obvious That means theperformance of the improved method is not affected bythe large azimuth misalignment The reason is that theeigenvalues of improved method can be set as the desiredvalues without the influence of the nonlinear factors causedby the large azimuth misalignment Moreover it is obviousthat the alignment speed of the improved method is fasterthan that of the conventional one

42 Experimental Results and Analysis The experimentswere implemented in the lab as the true yaw angle ofthe strapdown INS was known so different 120595

119880could be

exactly set We fixed the strapdown INS on the SGT-3 three-axis turntable to implement the alignment experiments Thestrapdown INS and the turntable can be seen in Figure 10 Atthe start of each experiment the turntable turned to a staticposition for 20 minutes that is pitch angle 0∘ roll angle 0∘and yaw angle 135∘ Two alignmentmethods were performedunder the condition of respectively setting initial yaw angleas 185∘ and 235∘ namely 120595

119880= 50∘ 100∘

Mathematical Problems in Engineering 9

0 5 10 15 20

0

50

Time (min)0 5 10 15 20

Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0

50 Improved method

minus50

minus100

minus150

minus50

minus100

minus150

Figure 8 Gyrocompass alignment for 120595119880= minus50

∘ minus100

∘ minus145

∘

50 100 1450

5

10

15

20

Azimuth misalignment (deg)

Con

verg

ence

tim

e (m

in)

Conventional methodImproved method

minus145 minus50minus100

Figure 9 Convergence time comparison with different azimuthmisalignments

Figure 11 provides the comparison of the yaw angleusing the two alignment methods under the condition ofdifferent 120595

119880

As the results above show the yaw angle can convergeto 135∘ through using the two alignment methods How-ever for the conventional method the convergence time islengthened with the growth of the azimuth misalignmentCompared with the conventional method the convergence

Strapdown INS

SGT-3

Figure 10 The SGT-3 turntable and strapdown INS

time of improved method is hardly changed The exper-imental results are consistent with the simulation resultsThe experiments demonstrate that the performance of theimproved method is better than that of the conventional one

5 Conclusions

In this paper an improved gyrocompass alignment methodhas been proposed for large azimuth misalignment of INSThe improved new method is performed by opening theazimuth loop and the estimation of the azimuth misalign-ment is implemented by using the horizontal velocity signalsto estimate the nonlinear factorsThrough using thismethodthe impact of the nonlinear factor caused by large azimuthmisalignment on gyrocompass alignment is reduced andthe performance of gyrocompass alignment is improvedThealignment speed of the improved method is faster than theconventional one in the same conditions The performance

10 Mathematical Problems in Engineering

0 5 10 15 20100

135

150

185200

235250

300

Yaw

(deg

)

Time (min)

Conventional method 120595U = 50∘

Conventional method 120595U = 100∘

Improved method 120595U = 50∘

Improved method 120595U = 100∘

Figure 11 Comparison of the yaw angle using the two alignmentmethods

of the improved method is well validated by simulations andexperiments

Appendices

A The Derivations of (31)

In this section (31) are obtained in the followingAccording to (30) their matrix form can be written as

AX = B (A1)

where

A

=

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

[

2120585Ω cos120593120596119899

0 1 minus2120585Ω sin120593

120596119899

119892Ω cos1205931205962119899

0 0 minus119892Ω sin1205931205962119899

02120585Ω cos120593

120596119899

2120585Ω sin120593120596119899

1

0119892Ω cos1205931205962119899

119892Ω sin1205931205962119899

0

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

]

(A2)

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

(A3)

B =

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

[

minus

(C119901119887nabla119887)119873

119892minus2120585

120596119899

(C119901119887120576119887)119864

minus119878 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119864

2120585

120596119899

Ω cos120593 +(C119901119887nabla119887)119864

119892minus2120585

120596119899

(C119901119887120576119887)119873

119892

1205962119899

Ω cos120593 + 119862 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119873

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

]

(A4)

Multiply the first and third rows of (A1) by1205961198992120585Ω cos120593 and

multiply the second and fourth rows of (A1) by 1205962119899119892Ω cos120593

we can obtain

A1X = B

1 (A5)

where

A1=

[[[[[[[[

[

1 0120596119899

2120585Ω cos120593minus tan120593

1 0 0 minus tan120593

0 1 tan120593120596119899

2120585Ω cos1205930 1 tan120593 0

]]]]]]]]

]

X =[[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]]

]

B1=

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

[

minus120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

1 +120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892minus

(C119901119887120576119887)119873

Ω cos120593

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

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

]

(A6)

Furthermore by utilizing the elementary row transformation(A5) can be transformed into

A2X = B

2 (A7)

Mathematical Problems in Engineering 11

where

A1=

[[[[[[[[

[

1 0 0 minus tan120593

0 0 minus120596119899

2120585Ω cos1205930

0 1 tan120593 0

0 0 0 minus120596119899

2120585Ω cos120593

]]]]]]]]

]

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

B1

=

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

[

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos1205931205962

119899

119892Ω cos120593119862 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892

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

]

(A8)

Then according to matrix equation (A7) we have

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

+

(C119901119887nabla119887)119873

119892tan120593

(A9)

B The Derivations of (34)

The derivations of (34) are carried out based on the principleof least squares fitting setting the Euclidean minimum norm119869 between sin

119880 cos

119880and sin

119880 cos

119880(the normalized

data) as the normalization indexFirst define

119869 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]12

(B1)

It is subjected to the constrained condition namely

ℎ = (sin 119880)2+ (cos

119880)2minus 1 = 0 (B2)

Second define an auxiliary function119898 namely

119898 = 1198692+ 120582ℎ (B3)

where 120582 is an unknown parameterBy substituting (B1) and (B2) into (B3) we can get

119898 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]

minus 120582 ((sin 119880)2+ (cos

119880)2minus 1)

(B4)

Obviously the parameters sin 119880and cos

119880that made 119869

minimum are equal to the ones that make 119898 minimumRespectively the partial derivatives of the parameters sin

119880

and cos 119880are calculated and bymaking them equal to zero

we can get

sin 119880=sin 119880

1 + 120582

cos 119880=cos 119880

1 + 120582

(B5)

Then by substituting (B5) into (B2) we have

(sin 119880)2+ (cos

119880)2

(1 + 120582)2

= 1 (B6)

Finally combining (B5) and (B6) the normalization equa-tions can be obtained

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(B7)

Conflict of Interests

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

Acknowledgments

The work described in the paper was supported by theNational Natural Science Foundation of China (61203225)and the National Science Foundation for Post-DoctoralScientists of China (2012M510083)The authors would like tothank all members of the Inertial Navigation Research Groupat Harbin Engineering University for the technical assistancewith the navigation system

References

[1] H W Park J G Lee and C G Park ldquoCovariance analysis ofstrapdown INS considering gyrocompass characteristicsrdquo IEEETransactions on Aerospace and Electronic Systems vol 31 no 1pp 320ndash328 1995

12 Mathematical Problems in Engineering

[2] R A Nash Jr J A DrsquoAppolito and K J Roy ldquoError analysis ofhybrid aircraft inertial navigation systemsrdquo in Proceedings of theAIAA Guidance and Control Conference Stanford Calif USA1972

[3] F D Jurenka and C T Leondes ldquoOptimum alignment of aninertial autonavigatorrdquo IEEE Transactions on Aerospace andElectronic Systems vol AES-3 no 6 pp 880ndash888 1967

[4] R H Cannon ldquoAlignment of inertial guidance systems bygyrocompassing-linear theoryrdquo Journal of Aerospace Sciencevol 28 no 11 pp 885ndash895 912 1961

[5] G R Pitman Jr Inertial Guidance John Wiley amp Sons NewYork NY USA 1962

[6] K R Britting Inertial Navigation Systems Analysis John Wileyamp Sons New York NY USA 1971

[7] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation vol 28 no4 pp 311ndash341 1982

[8] X Liu X Xu Y Liu and L Wang ldquoA fast and high-accuracycompass alignmentmethod to SINSwith azimuth axis rotationrdquoMathematical Problems in Engineering vol 2013 Article ID524284 12 pages 2013

[9] F Sun Q Wang Z Qi and C Wang ldquoResearch on theestimation method of DVL velocity error based on doubleprogram in Fiber Optic Gyro SINSrdquo Optik vol 124 no 22 pp5344ndash5349 2013

[10] B M Scherzinger ldquoInertial navigator error models for largeheading uncertaintyrdquo in Proceedings of the IEEE Position Loca-tion and Navigation Symposium (PLANS rsquo96) pp 477ndash484Atlanta Ga USA April 1996

[11] X Y Kong EM Nebot andH Durrant-Whyte ldquoDevelopmentof a non-linear psi-angle model for large misalignment errorsand its application in INS alignment and calibrationrdquo in Pro-ceedings of the 1999 IEEE International Conference on Roboticsamp Automation (ICRA rsquo99) pp 1430ndash1435 Detroit Mich USAMay 1999

[12] M-J Yu H-W Park and C-B Jeon ldquoEquivalent nonlinearerror models of strapdown inertial navigation systemrdquo in Pro-ceedings of the Guidance Navigation and Control ConferenceAIAA-97-3563 pp 581ndash587 1997

[13] T Abbas Z Yunyan and L Yanjun ldquoSINS initial alignment forsmall tilt and large azimuth misalignment anglesrdquo in Proceed-ings of the IEEE 3rd International Conference on CommunicationSoftware and Networks (ICCSN rsquo11) pp 628ndash632 Xirsquoan ChinaMay 2011

[14] S L Han and J L Wang ldquoA novel initial alignment scheme forlow-cost INS aided by GPS for land vehicle applicationsrdquo TheJournal of Navigation vol 63 no 4 pp 663ndash680 2010

[15] P G Savage Strapdown Analytics Strapdown Associates 2ndedition 2007

[16] I Yaesh and B Priel ldquoDesign of leveling loop for marine nav-igation systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 29 no 2 pp 599ndash604 1993

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

6 Mathematical Problems in Engineering

Then we have

120595119864ss = minus

(C119901119887nabla119887)119873

119892+2120585

120596119899

sdot [minus sin120595119880Ω cos120593

+ 120595119873ssΩ sin120593 minus (C119901

119887120576119887)119864]

119878 (120595119880)ss =

119892

1205962119899

sdot [minus sin120595119880Ω cos120593 + 120595

119873ssΩ sin120593

minus (C119901119887120576119887)119864]

120595119873ss =

(C119901119887nabla119887)119864

119892+2120585

120596119899

sdot [(1 minus cos120595119880)Ω cos120593

minus 120595119864ssΩ sin120593 minus (C119901

119887120576119887)119873]

119862 (120595119880)ss =

119892

1205962119899

sdot [(cos120595119880minus 1)Ω cos120593 + 120595

119864ssΩ sin120593

+ (C119901119887120576119887)119873]

(30)

where (sdot)ss represents the steady-state value of (sdot)We regard sin120595

119880 cos120595

119880 120595119864ss and 120595119873ss as the unknown

values (30) can then be easily solved Then the relationshipsbetween functions 119878(120595

119880) and 119862(120595

119880) and azimuth misalign-

ment 120595119880can be obtained

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593+

(C119901119887nabla119887)119873

119892tan120593

(31)

The detailed derivations of (31) can be found in Appendix A

322 The Azimuth Estimation Equations It is clear thatsin120595119880

and cos120595119880

can be supplied exactly by (31) if theaccelerometer errors and gyro errors (sensor errors) areknown Since the sensor errors are uncertain sin120595

119880and

cos120595119880are calculated by the following equations

sin 119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880) minus

2120585120596119899tan120593119892

sdot 119862 (120595119880) (32)

cos 119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880) minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)

(33)

where sin 119880and cos

119880are the computed values of sin120595

119880

and cos120595119880 respectively When the improved alignment

system is stable the errors between computed values (sin 119880

and cos 119880) and theoretical values (sin120595

119880and cos120595

119880) are

only determined by sensor errors and these errors areallowable Worth noting is that (32) and (33) do not preservethe unit-norm property of the trigonometric function thatis (sin

119880)2+ (cos

119880)2

= 1 So normalization of sin 119880

and cos 119880should be given and the simple normalization is

provided as follows

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(34)

The detailed derivations of (34) can be found in Appendix BAt this time the azimuth estimation

119880can be obtained

from (34) and is shown as follows

119880119898

= arcsin (sin 119880)

119880=

119880119898

cos 119880gt 0

180∘minus 119880119898

cos 119880lt 0 sin

119880gt 0

minus180∘minus 119880119898

cos 119880lt 0 sin

119880lt 0

(35)

It is obvious from the previous discussion that the azimuthestimation equations consist of (32)ndash(35)

Finally we take the disturbing motions into consid-eration because the alignment is often performed underquasistationary conditions The disturbing motions can begenerally considered to be sinusoidal [4 16] Then in orderto reduce the influence of the disturbing motions on azimuthestimation low pass filters need to be added to the systemnamely the control networks 119866(119904) and119867(119904) should have thecapability of restraining disturbance The design law of thecontrol networks 119866(119904) and119867(119904) is provided as follows

(a) The magnification of 119866(119904) and119867(119904) in low frequencymust be equivalent to 1 that is the following equationshould be satisfied

lim119904rarr0

119866 (119904) = lim119904rarr0

119867(119904) = 1 (36)

The reason for this requirement is that the rela-tionships between functions 119878(120595

119880) and 119862(120595

119880) and

azimuthmisalignment120595119880under quasistationary con-

ditions also can be represented by (31) as the require-ment is met

(b) In high frequency for the purpose of reducing theinfluence of disturbing motions they need be capableof restraining disturbance

Then under quasistationary conditions the azimuth estima-tion equations also consist of (32)ndash(35) In this work thecontrol networks 119866(119904) and119867(119904) are set as

119866 (119904) = 119867 (119904) =(120585120596119899)2

(119904 + 120585120596119899)2 (37)

Mathematical Problems in Engineering 7

bib

minus

minus

Gyros

AccelerometersCalculation of control angular

velocity and azimuth estimation functions

DSP

Final process

S(120595U) C(120595U)

120596bpb

E

N

U

120596n

ieCbp

pb =

pb[120596bpb times]CC

120596p

cCbp

fb

Vp= Cp

bfb+ gn

sin 120595U

cos 120595U

01

+

minus1205962n

gΩ cos 120593minus2120585120596n tan 120593

g

minus2120585120596n tan 120593

g

1205962n

gΩ cos 120593

S(120595U)C(120595U)lfloor

lceilrfloorrceillfloorlceillfloorlceil rfloorrceilrfloorrceil

V

V

Figure 6 Flow chart of improved gyrocompass alignment in real time operating device

At this time the denominators of 119878(120595119880)(119904) and 119862(120595

119880)(119904)

can be set as

(1199042+ 2120585120596

119899119904 + 1205962

119899) (119904 + 120585120596

119899)2 (38)

It is obvious from the azimuth estimation equations that theperformance of the azimuth alignment is determined by (32)and (33) Since both (32) and (33) consist of 119878(120595

119880) and119862(120595

119880)

the four eigenvalues of sin 119880(119904) and cos

119880(119904) could be

set as 11990412

= minus120585120596119899and 11990434

= minus120585120596119899plusmn 119895radic1 minus 1205852120596

119899 without

the influence of the nonlinear factors Therefore the impactof the nonlinear factor caused by large azimuthmisalignmenton gyrocompass alignment is reduced and the performanceof the improved gyrocompass alignment will be better thanthe conventional one

33 The Implementation of the Improved Gyrocompass Align-ment Method In this section the implementation of theproposed method in real time operating device is discussedand meanwhile a brief summary of this method is providedAn inertial navigation system implements the proposedalignment method using a cluster of accelerometers to sensethe specific force vector components f119887 a triad of gyrosto measure the angular velocity 119887

119894119887 and a digital signal

processor (DSP) to perform the alignment algorithm Thedirect expression of this implementation can be seen inFigure 6 According to Figure 6 the specific implementationsteps of the improved gyrocompass alignment in real timeoperation device are described as follows

(a) At the beginning a preliminary alignment oftencalled coarse alignment is performed and after thata rough DCM C119901

119887is obtained The specific coarse

alignment method can be found in [15] the levelmisalignments are small and azimuth misalignmentis usually large

(b) Secondly by utilizing the specific force f119887 and DCMC119901119887 the derivatives of the east and north horizontal

velocities 119864and 119873can be obtainedThen according

to the calculation method shown in Figure 3 thecontrol velocity 120596119901

119888and azimuth functions 119878(120595

119880) and

119862(120595119880) can be acquired

(c) On the one hand the control velocity 120596119901119888and angular

velocity 119887119894119887are used to update the DCM C119901

119887used for

the next calculation step(d) On the other hand the azimuth functions 119878(120595

119880) and

119862(120595119880) are used to compute sin

119880and cos

119880 And

after the final process which is achieved by utilizing(34)-(35) the azimuth estimation

119880is obtained

(e) Then steps (b) (c) and (d) are performed repeatedlyFinally by spending a period of time an accurateazimuth estimation

119880can be obtained and with the

compensation of azimuth misalignment using 119880 an

accurate DCM C119901119887can be obtained

4 Simulations and Experiments

In this section to evaluate the performance of the improvedgyrocompass alignment method under the condition of largeazimuth misalignment simulations and experiments arecarried out

41 Simulation Results and Analysis Both the conventionaland improved gyrocompass alignment methods are per-formed during the simulations simultaneously and the sim-ulations are conducted under the conditions of respec-tively choosing different azimuth misalignments namely120595119880= 50

∘ 100∘ 145∘ minus50∘ minus100

∘ minus145

∘ In these simu-

lations gyro and accelerometer outputs are generated bythe strapdown INS simulator we assume that the vehicleis in disturbing motions caused by sea waves The initialconditions are presented as follows

The constant gyro errors 120576119887 =

[[001∘ℎ 001

∘ℎ 001

∘ℎ]]119879

8 Mathematical Problems in Engineering

0 5 10 15 20

0 0

50

100

150

Time (min) Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0 5 10 15 20

Improved method

minus50

50

100

150

minus50

Figure 7 Gyrocompass alignment for 120595119880= 50

∘ 100∘ 145∘

The constant accelerometer errors nabla119887 =

[[10minus4119892 10minus4119892 10minus4119892]]119879

The acceleration due to gravity 119892 = 978049ms2The local longitude and latitude 120582 = 1266705∘ 120593 =457796

∘The control networks of improved method 119866(119904) =119867(119904) = (120585120596

119899)2(119904 + 120585120596

119899)2

The natural frequency of conventional and improvedmethods 120596

119899= 002 rads

Under the disturbingmotions the vehicle undertakes angularand linealmovements In angularmovement the pitch 120579

119909 the

roll 120579119910 and the yaw 120579

119911are controlled as

120579119909= 7∘ sin(2120587

5119905 +

120587

3)

120579119910= 5∘ sin(2120587

7119905 +

120587

6)

120579119911= 10∘ sin(2120587

6119905 +

120587

4)

(39)

In the lineal movement the vehicle lineal movement veloci-ties are taken as

119881119909= minus03 sin(2120587

10119905) ms2

119881119910= minus02 sin(2120587

6119905) ms2

119881119911= minus04 sin(2120587

8119905) ms2

(40)

Figures 7 and 8 provide the azimuth estimation errors ofconventional and improved methods under the conditions

of 120595119880= 50

∘ 100∘ 145∘ minus50∘ minus100

∘ minus145

∘ It can be seen

that azimuth alignment could be accomplished by both con-ventional and improved gyrocompass alignment methodsBut under the condition of large azimuth misalignment theperformance of conventional method will be poor along withthe growth of |120595

119880| The reason is that the difference between

the actual and desired eigenvalues is increasing alongwith thegrowth of |120595

119880|

Figure 9 shows convergence time for different azimuthmisalignments with a converged azimuth estimation errorof less than 1 degree It is clear that large azimuth misalign-ment needs more time to converge under the conventionalmethod However for the improved method the changeof the convergence time is not obvious That means theperformance of the improved method is not affected bythe large azimuth misalignment The reason is that theeigenvalues of improved method can be set as the desiredvalues without the influence of the nonlinear factors causedby the large azimuth misalignment Moreover it is obviousthat the alignment speed of the improved method is fasterthan that of the conventional one

42 Experimental Results and Analysis The experimentswere implemented in the lab as the true yaw angle ofthe strapdown INS was known so different 120595

119880could be

exactly set We fixed the strapdown INS on the SGT-3 three-axis turntable to implement the alignment experiments Thestrapdown INS and the turntable can be seen in Figure 10 Atthe start of each experiment the turntable turned to a staticposition for 20 minutes that is pitch angle 0∘ roll angle 0∘and yaw angle 135∘ Two alignmentmethods were performedunder the condition of respectively setting initial yaw angleas 185∘ and 235∘ namely 120595

119880= 50∘ 100∘

Mathematical Problems in Engineering 9

0 5 10 15 20

0

50

Time (min)0 5 10 15 20

Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0

50 Improved method

minus50

minus100

minus150

minus50

minus100

minus150

Figure 8 Gyrocompass alignment for 120595119880= minus50

∘ minus100

∘ minus145

∘

50 100 1450

5

10

15

20

Azimuth misalignment (deg)

Con

verg

ence

tim

e (m

in)

Conventional methodImproved method

minus145 minus50minus100

Figure 9 Convergence time comparison with different azimuthmisalignments

Figure 11 provides the comparison of the yaw angleusing the two alignment methods under the condition ofdifferent 120595

119880

As the results above show the yaw angle can convergeto 135∘ through using the two alignment methods How-ever for the conventional method the convergence time islengthened with the growth of the azimuth misalignmentCompared with the conventional method the convergence

Strapdown INS

SGT-3

Figure 10 The SGT-3 turntable and strapdown INS

time of improved method is hardly changed The exper-imental results are consistent with the simulation resultsThe experiments demonstrate that the performance of theimproved method is better than that of the conventional one

5 Conclusions

In this paper an improved gyrocompass alignment methodhas been proposed for large azimuth misalignment of INSThe improved new method is performed by opening theazimuth loop and the estimation of the azimuth misalign-ment is implemented by using the horizontal velocity signalsto estimate the nonlinear factorsThrough using thismethodthe impact of the nonlinear factor caused by large azimuthmisalignment on gyrocompass alignment is reduced andthe performance of gyrocompass alignment is improvedThealignment speed of the improved method is faster than theconventional one in the same conditions The performance

10 Mathematical Problems in Engineering

0 5 10 15 20100

135

150

185200

235250

300

Yaw

(deg

)

Time (min)

Conventional method 120595U = 50∘

Conventional method 120595U = 100∘

Improved method 120595U = 50∘

Improved method 120595U = 100∘

Figure 11 Comparison of the yaw angle using the two alignmentmethods

of the improved method is well validated by simulations andexperiments

Appendices

A The Derivations of (31)

In this section (31) are obtained in the followingAccording to (30) their matrix form can be written as

AX = B (A1)

where

A

=

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

[

2120585Ω cos120593120596119899

0 1 minus2120585Ω sin120593

120596119899

119892Ω cos1205931205962119899

0 0 minus119892Ω sin1205931205962119899

02120585Ω cos120593

120596119899

2120585Ω sin120593120596119899

1

0119892Ω cos1205931205962119899

119892Ω sin1205931205962119899

0

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

]

(A2)

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

(A3)

B =

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

[

minus

(C119901119887nabla119887)119873

119892minus2120585

120596119899

(C119901119887120576119887)119864

minus119878 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119864

2120585

120596119899

Ω cos120593 +(C119901119887nabla119887)119864

119892minus2120585

120596119899

(C119901119887120576119887)119873

119892

1205962119899

Ω cos120593 + 119862 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119873

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

]

(A4)

Multiply the first and third rows of (A1) by1205961198992120585Ω cos120593 and

multiply the second and fourth rows of (A1) by 1205962119899119892Ω cos120593

we can obtain

A1X = B

1 (A5)

where

A1=

[[[[[[[[

[

1 0120596119899

2120585Ω cos120593minus tan120593

1 0 0 minus tan120593

0 1 tan120593120596119899

2120585Ω cos1205930 1 tan120593 0

]]]]]]]]

]

X =[[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]]

]

B1=

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

[

minus120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

1 +120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892minus

(C119901119887120576119887)119873

Ω cos120593

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

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

]

(A6)

Furthermore by utilizing the elementary row transformation(A5) can be transformed into

A2X = B

2 (A7)

Mathematical Problems in Engineering 11

where

A1=

[[[[[[[[

[

1 0 0 minus tan120593

0 0 minus120596119899

2120585Ω cos1205930

0 1 tan120593 0

0 0 0 minus120596119899

2120585Ω cos120593

]]]]]]]]

]

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

B1

=

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

[

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos1205931205962

119899

119892Ω cos120593119862 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892

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

]

(A8)

Then according to matrix equation (A7) we have

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

+

(C119901119887nabla119887)119873

119892tan120593

(A9)

B The Derivations of (34)

The derivations of (34) are carried out based on the principleof least squares fitting setting the Euclidean minimum norm119869 between sin

119880 cos

119880and sin

119880 cos

119880(the normalized

data) as the normalization indexFirst define

119869 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]12

(B1)

It is subjected to the constrained condition namely

ℎ = (sin 119880)2+ (cos

119880)2minus 1 = 0 (B2)

Second define an auxiliary function119898 namely

119898 = 1198692+ 120582ℎ (B3)

where 120582 is an unknown parameterBy substituting (B1) and (B2) into (B3) we can get

119898 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]

minus 120582 ((sin 119880)2+ (cos

119880)2minus 1)

(B4)

Obviously the parameters sin 119880and cos

119880that made 119869

minimum are equal to the ones that make 119898 minimumRespectively the partial derivatives of the parameters sin

119880

and cos 119880are calculated and bymaking them equal to zero

we can get

sin 119880=sin 119880

1 + 120582

cos 119880=cos 119880

1 + 120582

(B5)

Then by substituting (B5) into (B2) we have

(sin 119880)2+ (cos

119880)2

(1 + 120582)2

= 1 (B6)

Finally combining (B5) and (B6) the normalization equa-tions can be obtained

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(B7)

Conflict of Interests

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

Acknowledgments

The work described in the paper was supported by theNational Natural Science Foundation of China (61203225)and the National Science Foundation for Post-DoctoralScientists of China (2012M510083)The authors would like tothank all members of the Inertial Navigation Research Groupat Harbin Engineering University for the technical assistancewith the navigation system

References

[1] H W Park J G Lee and C G Park ldquoCovariance analysis ofstrapdown INS considering gyrocompass characteristicsrdquo IEEETransactions on Aerospace and Electronic Systems vol 31 no 1pp 320ndash328 1995

12 Mathematical Problems in Engineering

[2] R A Nash Jr J A DrsquoAppolito and K J Roy ldquoError analysis ofhybrid aircraft inertial navigation systemsrdquo in Proceedings of theAIAA Guidance and Control Conference Stanford Calif USA1972

[3] F D Jurenka and C T Leondes ldquoOptimum alignment of aninertial autonavigatorrdquo IEEE Transactions on Aerospace andElectronic Systems vol AES-3 no 6 pp 880ndash888 1967

[4] R H Cannon ldquoAlignment of inertial guidance systems bygyrocompassing-linear theoryrdquo Journal of Aerospace Sciencevol 28 no 11 pp 885ndash895 912 1961

[5] G R Pitman Jr Inertial Guidance John Wiley amp Sons NewYork NY USA 1962

[6] K R Britting Inertial Navigation Systems Analysis John Wileyamp Sons New York NY USA 1971

[7] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation vol 28 no4 pp 311ndash341 1982

[8] X Liu X Xu Y Liu and L Wang ldquoA fast and high-accuracycompass alignmentmethod to SINSwith azimuth axis rotationrdquoMathematical Problems in Engineering vol 2013 Article ID524284 12 pages 2013

[9] F Sun Q Wang Z Qi and C Wang ldquoResearch on theestimation method of DVL velocity error based on doubleprogram in Fiber Optic Gyro SINSrdquo Optik vol 124 no 22 pp5344ndash5349 2013

[10] B M Scherzinger ldquoInertial navigator error models for largeheading uncertaintyrdquo in Proceedings of the IEEE Position Loca-tion and Navigation Symposium (PLANS rsquo96) pp 477ndash484Atlanta Ga USA April 1996

[11] X Y Kong EM Nebot andH Durrant-Whyte ldquoDevelopmentof a non-linear psi-angle model for large misalignment errorsand its application in INS alignment and calibrationrdquo in Pro-ceedings of the 1999 IEEE International Conference on Roboticsamp Automation (ICRA rsquo99) pp 1430ndash1435 Detroit Mich USAMay 1999

[12] M-J Yu H-W Park and C-B Jeon ldquoEquivalent nonlinearerror models of strapdown inertial navigation systemrdquo in Pro-ceedings of the Guidance Navigation and Control ConferenceAIAA-97-3563 pp 581ndash587 1997

[13] T Abbas Z Yunyan and L Yanjun ldquoSINS initial alignment forsmall tilt and large azimuth misalignment anglesrdquo in Proceed-ings of the IEEE 3rd International Conference on CommunicationSoftware and Networks (ICCSN rsquo11) pp 628ndash632 Xirsquoan ChinaMay 2011

[14] S L Han and J L Wang ldquoA novel initial alignment scheme forlow-cost INS aided by GPS for land vehicle applicationsrdquo TheJournal of Navigation vol 63 no 4 pp 663ndash680 2010

[15] P G Savage Strapdown Analytics Strapdown Associates 2ndedition 2007

[16] I Yaesh and B Priel ldquoDesign of leveling loop for marine nav-igation systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 29 no 2 pp 599ndash604 1993

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

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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

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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

Mathematical Problems in Engineering 7

bib

minus

minus

Gyros

AccelerometersCalculation of control angular

velocity and azimuth estimation functions

DSP

Final process

S(120595U) C(120595U)

120596bpb

E

N

U

120596n

ieCbp

pb =

pb[120596bpb times]CC

120596p

cCbp

fb

Vp= Cp

bfb+ gn

sin 120595U

cos 120595U

01

+

minus1205962n

gΩ cos 120593minus2120585120596n tan 120593

g

minus2120585120596n tan 120593

g

1205962n

gΩ cos 120593

S(120595U)C(120595U)lfloor

lceilrfloorrceillfloorlceillfloorlceil rfloorrceilrfloorrceil

V

V

Figure 6 Flow chart of improved gyrocompass alignment in real time operating device

At this time the denominators of 119878(120595119880)(119904) and 119862(120595

119880)(119904)

can be set as

(1199042+ 2120585120596

119899119904 + 1205962

119899) (119904 + 120585120596

119899)2 (38)

It is obvious from the azimuth estimation equations that theperformance of the azimuth alignment is determined by (32)and (33) Since both (32) and (33) consist of 119878(120595

119880) and119862(120595

119880)

the four eigenvalues of sin 119880(119904) and cos

119880(119904) could be

set as 11990412

= minus120585120596119899and 11990434

= minus120585120596119899plusmn 119895radic1 minus 1205852120596

119899 without

the influence of the nonlinear factors Therefore the impactof the nonlinear factor caused by large azimuthmisalignmenton gyrocompass alignment is reduced and the performanceof the improved gyrocompass alignment will be better thanthe conventional one

33 The Implementation of the Improved Gyrocompass Align-ment Method In this section the implementation of theproposed method in real time operating device is discussedand meanwhile a brief summary of this method is providedAn inertial navigation system implements the proposedalignment method using a cluster of accelerometers to sensethe specific force vector components f119887 a triad of gyrosto measure the angular velocity 119887

119894119887 and a digital signal

processor (DSP) to perform the alignment algorithm Thedirect expression of this implementation can be seen inFigure 6 According to Figure 6 the specific implementationsteps of the improved gyrocompass alignment in real timeoperation device are described as follows

(a) At the beginning a preliminary alignment oftencalled coarse alignment is performed and after thata rough DCM C119901

119887is obtained The specific coarse

alignment method can be found in [15] the levelmisalignments are small and azimuth misalignmentis usually large

(b) Secondly by utilizing the specific force f119887 and DCMC119901119887 the derivatives of the east and north horizontal

velocities 119864and 119873can be obtainedThen according

to the calculation method shown in Figure 3 thecontrol velocity 120596119901

119888and azimuth functions 119878(120595

119880) and

119862(120595119880) can be acquired

(c) On the one hand the control velocity 120596119901119888and angular

velocity 119887119894119887are used to update the DCM C119901

119887used for

the next calculation step(d) On the other hand the azimuth functions 119878(120595

119880) and

119862(120595119880) are used to compute sin

119880and cos

119880 And

after the final process which is achieved by utilizing(34)-(35) the azimuth estimation

119880is obtained

(e) Then steps (b) (c) and (d) are performed repeatedlyFinally by spending a period of time an accurateazimuth estimation

119880can be obtained and with the

compensation of azimuth misalignment using 119880 an

accurate DCM C119901119887can be obtained

4 Simulations and Experiments

In this section to evaluate the performance of the improvedgyrocompass alignment method under the condition of largeazimuth misalignment simulations and experiments arecarried out

41 Simulation Results and Analysis Both the conventionaland improved gyrocompass alignment methods are per-formed during the simulations simultaneously and the sim-ulations are conducted under the conditions of respec-tively choosing different azimuth misalignments namely120595119880= 50

∘ 100∘ 145∘ minus50∘ minus100

∘ minus145

∘ In these simu-

lations gyro and accelerometer outputs are generated bythe strapdown INS simulator we assume that the vehicleis in disturbing motions caused by sea waves The initialconditions are presented as follows

The constant gyro errors 120576119887 =

[[001∘ℎ 001

∘ℎ 001

∘ℎ]]119879

8 Mathematical Problems in Engineering

0 5 10 15 20

0 0

50

100

150

Time (min) Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0 5 10 15 20

Improved method

minus50

50

100

150

minus50

Figure 7 Gyrocompass alignment for 120595119880= 50

∘ 100∘ 145∘

The constant accelerometer errors nabla119887 =

[[10minus4119892 10minus4119892 10minus4119892]]119879

The acceleration due to gravity 119892 = 978049ms2The local longitude and latitude 120582 = 1266705∘ 120593 =457796

∘The control networks of improved method 119866(119904) =119867(119904) = (120585120596

119899)2(119904 + 120585120596

119899)2

The natural frequency of conventional and improvedmethods 120596

119899= 002 rads

Under the disturbingmotions the vehicle undertakes angularand linealmovements In angularmovement the pitch 120579

119909 the

roll 120579119910 and the yaw 120579

119911are controlled as

120579119909= 7∘ sin(2120587

5119905 +

120587

3)

120579119910= 5∘ sin(2120587

7119905 +

120587

6)

120579119911= 10∘ sin(2120587

6119905 +

120587

4)

(39)

In the lineal movement the vehicle lineal movement veloci-ties are taken as

119881119909= minus03 sin(2120587

10119905) ms2

119881119910= minus02 sin(2120587

6119905) ms2

119881119911= minus04 sin(2120587

8119905) ms2

(40)

Figures 7 and 8 provide the azimuth estimation errors ofconventional and improved methods under the conditions

of 120595119880= 50

∘ 100∘ 145∘ minus50∘ minus100

∘ minus145

∘ It can be seen

that azimuth alignment could be accomplished by both con-ventional and improved gyrocompass alignment methodsBut under the condition of large azimuth misalignment theperformance of conventional method will be poor along withthe growth of |120595

119880| The reason is that the difference between

the actual and desired eigenvalues is increasing alongwith thegrowth of |120595

119880|

Figure 9 shows convergence time for different azimuthmisalignments with a converged azimuth estimation errorof less than 1 degree It is clear that large azimuth misalign-ment needs more time to converge under the conventionalmethod However for the improved method the changeof the convergence time is not obvious That means theperformance of the improved method is not affected bythe large azimuth misalignment The reason is that theeigenvalues of improved method can be set as the desiredvalues without the influence of the nonlinear factors causedby the large azimuth misalignment Moreover it is obviousthat the alignment speed of the improved method is fasterthan that of the conventional one

42 Experimental Results and Analysis The experimentswere implemented in the lab as the true yaw angle ofthe strapdown INS was known so different 120595

119880could be

exactly set We fixed the strapdown INS on the SGT-3 three-axis turntable to implement the alignment experiments Thestrapdown INS and the turntable can be seen in Figure 10 Atthe start of each experiment the turntable turned to a staticposition for 20 minutes that is pitch angle 0∘ roll angle 0∘and yaw angle 135∘ Two alignmentmethods were performedunder the condition of respectively setting initial yaw angleas 185∘ and 235∘ namely 120595

119880= 50∘ 100∘

Mathematical Problems in Engineering 9

0 5 10 15 20

0

50

Time (min)0 5 10 15 20

Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0

50 Improved method

minus50

minus100

minus150

minus50

minus100

minus150

Figure 8 Gyrocompass alignment for 120595119880= minus50

∘ minus100

∘ minus145

∘

50 100 1450

5

10

15

20

Azimuth misalignment (deg)

Con

verg

ence

tim

e (m

in)

Conventional methodImproved method

minus145 minus50minus100

Figure 9 Convergence time comparison with different azimuthmisalignments

Figure 11 provides the comparison of the yaw angleusing the two alignment methods under the condition ofdifferent 120595

119880

As the results above show the yaw angle can convergeto 135∘ through using the two alignment methods How-ever for the conventional method the convergence time islengthened with the growth of the azimuth misalignmentCompared with the conventional method the convergence

Strapdown INS

SGT-3

Figure 10 The SGT-3 turntable and strapdown INS

time of improved method is hardly changed The exper-imental results are consistent with the simulation resultsThe experiments demonstrate that the performance of theimproved method is better than that of the conventional one

5 Conclusions

In this paper an improved gyrocompass alignment methodhas been proposed for large azimuth misalignment of INSThe improved new method is performed by opening theazimuth loop and the estimation of the azimuth misalign-ment is implemented by using the horizontal velocity signalsto estimate the nonlinear factorsThrough using thismethodthe impact of the nonlinear factor caused by large azimuthmisalignment on gyrocompass alignment is reduced andthe performance of gyrocompass alignment is improvedThealignment speed of the improved method is faster than theconventional one in the same conditions The performance

10 Mathematical Problems in Engineering

0 5 10 15 20100

135

150

185200

235250

300

Yaw

(deg

)

Time (min)

Conventional method 120595U = 50∘

Conventional method 120595U = 100∘

Improved method 120595U = 50∘

Improved method 120595U = 100∘

Figure 11 Comparison of the yaw angle using the two alignmentmethods

of the improved method is well validated by simulations andexperiments

Appendices

A The Derivations of (31)

In this section (31) are obtained in the followingAccording to (30) their matrix form can be written as

AX = B (A1)

where

A

=

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

[

2120585Ω cos120593120596119899

0 1 minus2120585Ω sin120593

120596119899

119892Ω cos1205931205962119899

0 0 minus119892Ω sin1205931205962119899

02120585Ω cos120593

120596119899

2120585Ω sin120593120596119899

1

0119892Ω cos1205931205962119899

119892Ω sin1205931205962119899

0

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

]

(A2)

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

(A3)

B =

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

[

minus

(C119901119887nabla119887)119873

119892minus2120585

120596119899

(C119901119887120576119887)119864

minus119878 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119864

2120585

120596119899

Ω cos120593 +(C119901119887nabla119887)119864

119892minus2120585

120596119899

(C119901119887120576119887)119873

119892

1205962119899

Ω cos120593 + 119862 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119873

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

]

(A4)

Multiply the first and third rows of (A1) by1205961198992120585Ω cos120593 and

multiply the second and fourth rows of (A1) by 1205962119899119892Ω cos120593

we can obtain

A1X = B

1 (A5)

where

A1=

[[[[[[[[

[

1 0120596119899

2120585Ω cos120593minus tan120593

1 0 0 minus tan120593

0 1 tan120593120596119899

2120585Ω cos1205930 1 tan120593 0

]]]]]]]]

]

X =[[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]]

]

B1=

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

[

minus120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

1 +120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892minus

(C119901119887120576119887)119873

Ω cos120593

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

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

]

(A6)

Furthermore by utilizing the elementary row transformation(A5) can be transformed into

A2X = B

2 (A7)

Mathematical Problems in Engineering 11

where

A1=

[[[[[[[[

[

1 0 0 minus tan120593

0 0 minus120596119899

2120585Ω cos1205930

0 1 tan120593 0

0 0 0 minus120596119899

2120585Ω cos120593

]]]]]]]]

]

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

B1

=

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

[

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos1205931205962

119899

119892Ω cos120593119862 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892

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

]

(A8)

Then according to matrix equation (A7) we have

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

+

(C119901119887nabla119887)119873

119892tan120593

(A9)

B The Derivations of (34)

The derivations of (34) are carried out based on the principleof least squares fitting setting the Euclidean minimum norm119869 between sin

119880 cos

119880and sin

119880 cos

119880(the normalized

data) as the normalization indexFirst define

119869 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]12

(B1)

It is subjected to the constrained condition namely

ℎ = (sin 119880)2+ (cos

119880)2minus 1 = 0 (B2)

Second define an auxiliary function119898 namely

119898 = 1198692+ 120582ℎ (B3)

where 120582 is an unknown parameterBy substituting (B1) and (B2) into (B3) we can get

119898 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]

minus 120582 ((sin 119880)2+ (cos

119880)2minus 1)

(B4)

Obviously the parameters sin 119880and cos

119880that made 119869

minimum are equal to the ones that make 119898 minimumRespectively the partial derivatives of the parameters sin

119880

and cos 119880are calculated and bymaking them equal to zero

we can get

sin 119880=sin 119880

1 + 120582

cos 119880=cos 119880

1 + 120582

(B5)

Then by substituting (B5) into (B2) we have

(sin 119880)2+ (cos

119880)2

(1 + 120582)2

= 1 (B6)

Finally combining (B5) and (B6) the normalization equa-tions can be obtained

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(B7)

Conflict of Interests

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

Acknowledgments

The work described in the paper was supported by theNational Natural Science Foundation of China (61203225)and the National Science Foundation for Post-DoctoralScientists of China (2012M510083)The authors would like tothank all members of the Inertial Navigation Research Groupat Harbin Engineering University for the technical assistancewith the navigation system

References

[1] H W Park J G Lee and C G Park ldquoCovariance analysis ofstrapdown INS considering gyrocompass characteristicsrdquo IEEETransactions on Aerospace and Electronic Systems vol 31 no 1pp 320ndash328 1995

12 Mathematical Problems in Engineering

[2] R A Nash Jr J A DrsquoAppolito and K J Roy ldquoError analysis ofhybrid aircraft inertial navigation systemsrdquo in Proceedings of theAIAA Guidance and Control Conference Stanford Calif USA1972

[3] F D Jurenka and C T Leondes ldquoOptimum alignment of aninertial autonavigatorrdquo IEEE Transactions on Aerospace andElectronic Systems vol AES-3 no 6 pp 880ndash888 1967

[4] R H Cannon ldquoAlignment of inertial guidance systems bygyrocompassing-linear theoryrdquo Journal of Aerospace Sciencevol 28 no 11 pp 885ndash895 912 1961

[5] G R Pitman Jr Inertial Guidance John Wiley amp Sons NewYork NY USA 1962

[6] K R Britting Inertial Navigation Systems Analysis John Wileyamp Sons New York NY USA 1971

[7] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation vol 28 no4 pp 311ndash341 1982

[8] X Liu X Xu Y Liu and L Wang ldquoA fast and high-accuracycompass alignmentmethod to SINSwith azimuth axis rotationrdquoMathematical Problems in Engineering vol 2013 Article ID524284 12 pages 2013

[9] F Sun Q Wang Z Qi and C Wang ldquoResearch on theestimation method of DVL velocity error based on doubleprogram in Fiber Optic Gyro SINSrdquo Optik vol 124 no 22 pp5344ndash5349 2013

[10] B M Scherzinger ldquoInertial navigator error models for largeheading uncertaintyrdquo in Proceedings of the IEEE Position Loca-tion and Navigation Symposium (PLANS rsquo96) pp 477ndash484Atlanta Ga USA April 1996

[11] X Y Kong EM Nebot andH Durrant-Whyte ldquoDevelopmentof a non-linear psi-angle model for large misalignment errorsand its application in INS alignment and calibrationrdquo in Pro-ceedings of the 1999 IEEE International Conference on Roboticsamp Automation (ICRA rsquo99) pp 1430ndash1435 Detroit Mich USAMay 1999

[12] M-J Yu H-W Park and C-B Jeon ldquoEquivalent nonlinearerror models of strapdown inertial navigation systemrdquo in Pro-ceedings of the Guidance Navigation and Control ConferenceAIAA-97-3563 pp 581ndash587 1997

[13] T Abbas Z Yunyan and L Yanjun ldquoSINS initial alignment forsmall tilt and large azimuth misalignment anglesrdquo in Proceed-ings of the IEEE 3rd International Conference on CommunicationSoftware and Networks (ICCSN rsquo11) pp 628ndash632 Xirsquoan ChinaMay 2011

[14] S L Han and J L Wang ldquoA novel initial alignment scheme forlow-cost INS aided by GPS for land vehicle applicationsrdquo TheJournal of Navigation vol 63 no 4 pp 663ndash680 2010

[15] P G Savage Strapdown Analytics Strapdown Associates 2ndedition 2007

[16] I Yaesh and B Priel ldquoDesign of leveling loop for marine nav-igation systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 29 no 2 pp 599ndash604 1993

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

0 5 10 15 20

0 0

50

100

150

Time (min) Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0 5 10 15 20

Improved method

minus50

50

100

150

minus50

Figure 7 Gyrocompass alignment for 120595119880= 50

∘ 100∘ 145∘

The constant accelerometer errors nabla119887 =

[[10minus4119892 10minus4119892 10minus4119892]]119879

The acceleration due to gravity 119892 = 978049ms2The local longitude and latitude 120582 = 1266705∘ 120593 =457796

∘The control networks of improved method 119866(119904) =119867(119904) = (120585120596

119899)2(119904 + 120585120596

119899)2

The natural frequency of conventional and improvedmethods 120596

119899= 002 rads

Under the disturbingmotions the vehicle undertakes angularand linealmovements In angularmovement the pitch 120579

119909 the

roll 120579119910 and the yaw 120579

119911are controlled as

120579119909= 7∘ sin(2120587

5119905 +

120587

3)

120579119910= 5∘ sin(2120587

7119905 +

120587

6)

120579119911= 10∘ sin(2120587

6119905 +

120587

4)

(39)

In the lineal movement the vehicle lineal movement veloci-ties are taken as

119881119909= minus03 sin(2120587

10119905) ms2

119881119910= minus02 sin(2120587

6119905) ms2

119881119911= minus04 sin(2120587

8119905) ms2

(40)

Figures 7 and 8 provide the azimuth estimation errors ofconventional and improved methods under the conditions

of 120595119880= 50

∘ 100∘ 145∘ minus50∘ minus100

∘ minus145

∘ It can be seen

that azimuth alignment could be accomplished by both con-ventional and improved gyrocompass alignment methodsBut under the condition of large azimuth misalignment theperformance of conventional method will be poor along withthe growth of |120595

119880| The reason is that the difference between

the actual and desired eigenvalues is increasing alongwith thegrowth of |120595

119880|

Figure 9 shows convergence time for different azimuthmisalignments with a converged azimuth estimation errorof less than 1 degree It is clear that large azimuth misalign-ment needs more time to converge under the conventionalmethod However for the improved method the changeof the convergence time is not obvious That means theperformance of the improved method is not affected bythe large azimuth misalignment The reason is that theeigenvalues of improved method can be set as the desiredvalues without the influence of the nonlinear factors causedby the large azimuth misalignment Moreover it is obviousthat the alignment speed of the improved method is fasterthan that of the conventional one

42 Experimental Results and Analysis The experimentswere implemented in the lab as the true yaw angle ofthe strapdown INS was known so different 120595

119880could be

exactly set We fixed the strapdown INS on the SGT-3 three-axis turntable to implement the alignment experiments Thestrapdown INS and the turntable can be seen in Figure 10 Atthe start of each experiment the turntable turned to a staticposition for 20 minutes that is pitch angle 0∘ roll angle 0∘and yaw angle 135∘ Two alignmentmethods were performedunder the condition of respectively setting initial yaw angleas 185∘ and 235∘ namely 120595

119880= 50∘ 100∘

Mathematical Problems in Engineering 9

0 5 10 15 20

0

50

Time (min)0 5 10 15 20

Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0

50 Improved method

minus50

minus100

minus150

minus50

minus100

minus150

Figure 8 Gyrocompass alignment for 120595119880= minus50

∘ minus100

∘ minus145

∘

50 100 1450

5

10

15

20

Azimuth misalignment (deg)

Con

verg

ence

tim

e (m

in)

Conventional methodImproved method

minus145 minus50minus100

Figure 9 Convergence time comparison with different azimuthmisalignments

Figure 11 provides the comparison of the yaw angleusing the two alignment methods under the condition ofdifferent 120595

119880

As the results above show the yaw angle can convergeto 135∘ through using the two alignment methods How-ever for the conventional method the convergence time islengthened with the growth of the azimuth misalignmentCompared with the conventional method the convergence

Strapdown INS

SGT-3

Figure 10 The SGT-3 turntable and strapdown INS

time of improved method is hardly changed The exper-imental results are consistent with the simulation resultsThe experiments demonstrate that the performance of theimproved method is better than that of the conventional one

5 Conclusions

In this paper an improved gyrocompass alignment methodhas been proposed for large azimuth misalignment of INSThe improved new method is performed by opening theazimuth loop and the estimation of the azimuth misalign-ment is implemented by using the horizontal velocity signalsto estimate the nonlinear factorsThrough using thismethodthe impact of the nonlinear factor caused by large azimuthmisalignment on gyrocompass alignment is reduced andthe performance of gyrocompass alignment is improvedThealignment speed of the improved method is faster than theconventional one in the same conditions The performance

10 Mathematical Problems in Engineering

0 5 10 15 20100

135

150

185200

235250

300

Yaw

(deg

)

Time (min)

Conventional method 120595U = 50∘

Conventional method 120595U = 100∘

Improved method 120595U = 50∘

Improved method 120595U = 100∘

Figure 11 Comparison of the yaw angle using the two alignmentmethods

of the improved method is well validated by simulations andexperiments

Appendices

A The Derivations of (31)

In this section (31) are obtained in the followingAccording to (30) their matrix form can be written as

AX = B (A1)

where

A

=

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

[

2120585Ω cos120593120596119899

0 1 minus2120585Ω sin120593

120596119899

119892Ω cos1205931205962119899

0 0 minus119892Ω sin1205931205962119899

02120585Ω cos120593

120596119899

2120585Ω sin120593120596119899

1

0119892Ω cos1205931205962119899

119892Ω sin1205931205962119899

0

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

]

(A2)

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

(A3)

B =

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

[

minus

(C119901119887nabla119887)119873

119892minus2120585

120596119899

(C119901119887120576119887)119864

minus119878 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119864

2120585

120596119899

Ω cos120593 +(C119901119887nabla119887)119864

119892minus2120585

120596119899

(C119901119887120576119887)119873

119892

1205962119899

Ω cos120593 + 119862 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119873

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

]

(A4)

Multiply the first and third rows of (A1) by1205961198992120585Ω cos120593 and

multiply the second and fourth rows of (A1) by 1205962119899119892Ω cos120593

we can obtain

A1X = B

1 (A5)

where

A1=

[[[[[[[[

[

1 0120596119899

2120585Ω cos120593minus tan120593

1 0 0 minus tan120593

0 1 tan120593120596119899

2120585Ω cos1205930 1 tan120593 0

]]]]]]]]

]

X =[[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]]

]

B1=

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

[

minus120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

1 +120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892minus

(C119901119887120576119887)119873

Ω cos120593

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

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

]

(A6)

Furthermore by utilizing the elementary row transformation(A5) can be transformed into

A2X = B

2 (A7)

Mathematical Problems in Engineering 11

where

A1=

[[[[[[[[

[

1 0 0 minus tan120593

0 0 minus120596119899

2120585Ω cos1205930

0 1 tan120593 0

0 0 0 minus120596119899

2120585Ω cos120593

]]]]]]]]

]

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

B1

=

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

[

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos1205931205962

119899

119892Ω cos120593119862 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892

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

]

(A8)

Then according to matrix equation (A7) we have

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

+

(C119901119887nabla119887)119873

119892tan120593

(A9)

B The Derivations of (34)

The derivations of (34) are carried out based on the principleof least squares fitting setting the Euclidean minimum norm119869 between sin

119880 cos

119880and sin

119880 cos

119880(the normalized

data) as the normalization indexFirst define

119869 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]12

(B1)

It is subjected to the constrained condition namely

ℎ = (sin 119880)2+ (cos

119880)2minus 1 = 0 (B2)

Second define an auxiliary function119898 namely

119898 = 1198692+ 120582ℎ (B3)

where 120582 is an unknown parameterBy substituting (B1) and (B2) into (B3) we can get

119898 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]

minus 120582 ((sin 119880)2+ (cos

119880)2minus 1)

(B4)

Obviously the parameters sin 119880and cos

119880that made 119869

minimum are equal to the ones that make 119898 minimumRespectively the partial derivatives of the parameters sin

119880

and cos 119880are calculated and bymaking them equal to zero

we can get

sin 119880=sin 119880

1 + 120582

cos 119880=cos 119880

1 + 120582

(B5)

Then by substituting (B5) into (B2) we have

(sin 119880)2+ (cos

119880)2

(1 + 120582)2

= 1 (B6)

Finally combining (B5) and (B6) the normalization equa-tions can be obtained

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(B7)

Conflict of Interests

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

Acknowledgments

The work described in the paper was supported by theNational Natural Science Foundation of China (61203225)and the National Science Foundation for Post-DoctoralScientists of China (2012M510083)The authors would like tothank all members of the Inertial Navigation Research Groupat Harbin Engineering University for the technical assistancewith the navigation system

References

[1] H W Park J G Lee and C G Park ldquoCovariance analysis ofstrapdown INS considering gyrocompass characteristicsrdquo IEEETransactions on Aerospace and Electronic Systems vol 31 no 1pp 320ndash328 1995

12 Mathematical Problems in Engineering

[2] R A Nash Jr J A DrsquoAppolito and K J Roy ldquoError analysis ofhybrid aircraft inertial navigation systemsrdquo in Proceedings of theAIAA Guidance and Control Conference Stanford Calif USA1972

[3] F D Jurenka and C T Leondes ldquoOptimum alignment of aninertial autonavigatorrdquo IEEE Transactions on Aerospace andElectronic Systems vol AES-3 no 6 pp 880ndash888 1967

[4] R H Cannon ldquoAlignment of inertial guidance systems bygyrocompassing-linear theoryrdquo Journal of Aerospace Sciencevol 28 no 11 pp 885ndash895 912 1961

[5] G R Pitman Jr Inertial Guidance John Wiley amp Sons NewYork NY USA 1962

[6] K R Britting Inertial Navigation Systems Analysis John Wileyamp Sons New York NY USA 1971

[7] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation vol 28 no4 pp 311ndash341 1982

[8] X Liu X Xu Y Liu and L Wang ldquoA fast and high-accuracycompass alignmentmethod to SINSwith azimuth axis rotationrdquoMathematical Problems in Engineering vol 2013 Article ID524284 12 pages 2013

[9] F Sun Q Wang Z Qi and C Wang ldquoResearch on theestimation method of DVL velocity error based on doubleprogram in Fiber Optic Gyro SINSrdquo Optik vol 124 no 22 pp5344ndash5349 2013

[10] B M Scherzinger ldquoInertial navigator error models for largeheading uncertaintyrdquo in Proceedings of the IEEE Position Loca-tion and Navigation Symposium (PLANS rsquo96) pp 477ndash484Atlanta Ga USA April 1996

[11] X Y Kong EM Nebot andH Durrant-Whyte ldquoDevelopmentof a non-linear psi-angle model for large misalignment errorsand its application in INS alignment and calibrationrdquo in Pro-ceedings of the 1999 IEEE International Conference on Roboticsamp Automation (ICRA rsquo99) pp 1430ndash1435 Detroit Mich USAMay 1999

[12] M-J Yu H-W Park and C-B Jeon ldquoEquivalent nonlinearerror models of strapdown inertial navigation systemrdquo in Pro-ceedings of the Guidance Navigation and Control ConferenceAIAA-97-3563 pp 581ndash587 1997

[13] T Abbas Z Yunyan and L Yanjun ldquoSINS initial alignment forsmall tilt and large azimuth misalignment anglesrdquo in Proceed-ings of the IEEE 3rd International Conference on CommunicationSoftware and Networks (ICCSN rsquo11) pp 628ndash632 Xirsquoan ChinaMay 2011

[14] S L Han and J L Wang ldquoA novel initial alignment scheme forlow-cost INS aided by GPS for land vehicle applicationsrdquo TheJournal of Navigation vol 63 no 4 pp 663ndash680 2010

[15] P G Savage Strapdown Analytics Strapdown Associates 2ndedition 2007

[16] I Yaesh and B Priel ldquoDesign of leveling loop for marine nav-igation systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 29 no 2 pp 599ndash604 1993

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

0 5 10 15 20

0

50

Time (min)0 5 10 15 20

Time (min)

Azi

mut

h es

timat

ion

erro

r (de

g)

Conventional method

0

50 Improved method

minus50

minus100

minus150

minus50

minus100

minus150

Figure 8 Gyrocompass alignment for 120595119880= minus50

∘ minus100

∘ minus145

∘

50 100 1450

5

10

15

20

Azimuth misalignment (deg)

Con

verg

ence

tim

e (m

in)

Conventional methodImproved method

minus145 minus50minus100

Figure 9 Convergence time comparison with different azimuthmisalignments

Figure 11 provides the comparison of the yaw angleusing the two alignment methods under the condition ofdifferent 120595

119880

As the results above show the yaw angle can convergeto 135∘ through using the two alignment methods How-ever for the conventional method the convergence time islengthened with the growth of the azimuth misalignmentCompared with the conventional method the convergence

Strapdown INS

SGT-3

Figure 10 The SGT-3 turntable and strapdown INS

time of improved method is hardly changed The exper-imental results are consistent with the simulation resultsThe experiments demonstrate that the performance of theimproved method is better than that of the conventional one

5 Conclusions

In this paper an improved gyrocompass alignment methodhas been proposed for large azimuth misalignment of INSThe improved new method is performed by opening theazimuth loop and the estimation of the azimuth misalign-ment is implemented by using the horizontal velocity signalsto estimate the nonlinear factorsThrough using thismethodthe impact of the nonlinear factor caused by large azimuthmisalignment on gyrocompass alignment is reduced andthe performance of gyrocompass alignment is improvedThealignment speed of the improved method is faster than theconventional one in the same conditions The performance

10 Mathematical Problems in Engineering

0 5 10 15 20100

135

150

185200

235250

300

Yaw

(deg

)

Time (min)

Conventional method 120595U = 50∘

Conventional method 120595U = 100∘

Improved method 120595U = 50∘

Improved method 120595U = 100∘

Figure 11 Comparison of the yaw angle using the two alignmentmethods

of the improved method is well validated by simulations andexperiments

Appendices

A The Derivations of (31)

In this section (31) are obtained in the followingAccording to (30) their matrix form can be written as

AX = B (A1)

where

A

=

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

[

2120585Ω cos120593120596119899

0 1 minus2120585Ω sin120593

120596119899

119892Ω cos1205931205962119899

0 0 minus119892Ω sin1205931205962119899

02120585Ω cos120593

120596119899

2120585Ω sin120593120596119899

1

0119892Ω cos1205931205962119899

119892Ω sin1205931205962119899

0

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

]

(A2)

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

(A3)

B =

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

[

minus

(C119901119887nabla119887)119873

119892minus2120585

120596119899

(C119901119887120576119887)119864

minus119878 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119864

2120585

120596119899

Ω cos120593 +(C119901119887nabla119887)119864

119892minus2120585

120596119899

(C119901119887120576119887)119873

119892

1205962119899

Ω cos120593 + 119862 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119873

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

]

(A4)

Multiply the first and third rows of (A1) by1205961198992120585Ω cos120593 and

multiply the second and fourth rows of (A1) by 1205962119899119892Ω cos120593

we can obtain

A1X = B

1 (A5)

where

A1=

[[[[[[[[

[

1 0120596119899

2120585Ω cos120593minus tan120593

1 0 0 minus tan120593

0 1 tan120593120596119899

2120585Ω cos1205930 1 tan120593 0

]]]]]]]]

]

X =[[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]]

]

B1=

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

[

minus120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

1 +120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892minus

(C119901119887120576119887)119873

Ω cos120593

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

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

]

(A6)

Furthermore by utilizing the elementary row transformation(A5) can be transformed into

A2X = B

2 (A7)

Mathematical Problems in Engineering 11

where

A1=

[[[[[[[[

[

1 0 0 minus tan120593

0 0 minus120596119899

2120585Ω cos1205930

0 1 tan120593 0

0 0 0 minus120596119899

2120585Ω cos120593

]]]]]]]]

]

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

B1

=

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

[

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos1205931205962

119899

119892Ω cos120593119862 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892

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

]

(A8)

Then according to matrix equation (A7) we have

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

+

(C119901119887nabla119887)119873

119892tan120593

(A9)

B The Derivations of (34)

The derivations of (34) are carried out based on the principleof least squares fitting setting the Euclidean minimum norm119869 between sin

119880 cos

119880and sin

119880 cos

119880(the normalized

data) as the normalization indexFirst define

119869 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]12

(B1)

It is subjected to the constrained condition namely

ℎ = (sin 119880)2+ (cos

119880)2minus 1 = 0 (B2)

Second define an auxiliary function119898 namely

119898 = 1198692+ 120582ℎ (B3)

where 120582 is an unknown parameterBy substituting (B1) and (B2) into (B3) we can get

119898 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]

minus 120582 ((sin 119880)2+ (cos

119880)2minus 1)

(B4)

Obviously the parameters sin 119880and cos

119880that made 119869

minimum are equal to the ones that make 119898 minimumRespectively the partial derivatives of the parameters sin

119880

and cos 119880are calculated and bymaking them equal to zero

we can get

sin 119880=sin 119880

1 + 120582

cos 119880=cos 119880

1 + 120582

(B5)

Then by substituting (B5) into (B2) we have

(sin 119880)2+ (cos

119880)2

(1 + 120582)2

= 1 (B6)

Finally combining (B5) and (B6) the normalization equa-tions can be obtained

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(B7)

Conflict of Interests

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

Acknowledgments

The work described in the paper was supported by theNational Natural Science Foundation of China (61203225)and the National Science Foundation for Post-DoctoralScientists of China (2012M510083)The authors would like tothank all members of the Inertial Navigation Research Groupat Harbin Engineering University for the technical assistancewith the navigation system

References

[1] H W Park J G Lee and C G Park ldquoCovariance analysis ofstrapdown INS considering gyrocompass characteristicsrdquo IEEETransactions on Aerospace and Electronic Systems vol 31 no 1pp 320ndash328 1995

12 Mathematical Problems in Engineering

[2] R A Nash Jr J A DrsquoAppolito and K J Roy ldquoError analysis ofhybrid aircraft inertial navigation systemsrdquo in Proceedings of theAIAA Guidance and Control Conference Stanford Calif USA1972

[3] F D Jurenka and C T Leondes ldquoOptimum alignment of aninertial autonavigatorrdquo IEEE Transactions on Aerospace andElectronic Systems vol AES-3 no 6 pp 880ndash888 1967

[4] R H Cannon ldquoAlignment of inertial guidance systems bygyrocompassing-linear theoryrdquo Journal of Aerospace Sciencevol 28 no 11 pp 885ndash895 912 1961

[5] G R Pitman Jr Inertial Guidance John Wiley amp Sons NewYork NY USA 1962

[6] K R Britting Inertial Navigation Systems Analysis John Wileyamp Sons New York NY USA 1971

[7] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation vol 28 no4 pp 311ndash341 1982

[8] X Liu X Xu Y Liu and L Wang ldquoA fast and high-accuracycompass alignmentmethod to SINSwith azimuth axis rotationrdquoMathematical Problems in Engineering vol 2013 Article ID524284 12 pages 2013

[9] F Sun Q Wang Z Qi and C Wang ldquoResearch on theestimation method of DVL velocity error based on doubleprogram in Fiber Optic Gyro SINSrdquo Optik vol 124 no 22 pp5344ndash5349 2013

[10] B M Scherzinger ldquoInertial navigator error models for largeheading uncertaintyrdquo in Proceedings of the IEEE Position Loca-tion and Navigation Symposium (PLANS rsquo96) pp 477ndash484Atlanta Ga USA April 1996

[11] X Y Kong EM Nebot andH Durrant-Whyte ldquoDevelopmentof a non-linear psi-angle model for large misalignment errorsand its application in INS alignment and calibrationrdquo in Pro-ceedings of the 1999 IEEE International Conference on Roboticsamp Automation (ICRA rsquo99) pp 1430ndash1435 Detroit Mich USAMay 1999

[12] M-J Yu H-W Park and C-B Jeon ldquoEquivalent nonlinearerror models of strapdown inertial navigation systemrdquo in Pro-ceedings of the Guidance Navigation and Control ConferenceAIAA-97-3563 pp 581ndash587 1997

[13] T Abbas Z Yunyan and L Yanjun ldquoSINS initial alignment forsmall tilt and large azimuth misalignment anglesrdquo in Proceed-ings of the IEEE 3rd International Conference on CommunicationSoftware and Networks (ICCSN rsquo11) pp 628ndash632 Xirsquoan ChinaMay 2011

[14] S L Han and J L Wang ldquoA novel initial alignment scheme forlow-cost INS aided by GPS for land vehicle applicationsrdquo TheJournal of Navigation vol 63 no 4 pp 663ndash680 2010

[15] P G Savage Strapdown Analytics Strapdown Associates 2ndedition 2007

[16] I Yaesh and B Priel ldquoDesign of leveling loop for marine nav-igation systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 29 no 2 pp 599ndash604 1993

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

0 5 10 15 20100

135

150

185200

235250

300

Yaw

(deg

)

Time (min)

Conventional method 120595U = 50∘

Conventional method 120595U = 100∘

Improved method 120595U = 50∘

Improved method 120595U = 100∘

Figure 11 Comparison of the yaw angle using the two alignmentmethods

of the improved method is well validated by simulations andexperiments

Appendices

A The Derivations of (31)

In this section (31) are obtained in the followingAccording to (30) their matrix form can be written as

AX = B (A1)

where

A

=

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

[

2120585Ω cos120593120596119899

0 1 minus2120585Ω sin120593

120596119899

119892Ω cos1205931205962119899

0 0 minus119892Ω sin1205931205962119899

02120585Ω cos120593

120596119899

2120585Ω sin120593120596119899

1

0119892Ω cos1205931205962119899

119892Ω sin1205931205962119899

0

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

]

(A2)

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

(A3)

B =

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

[

minus

(C119901119887nabla119887)119873

119892minus2120585

120596119899

(C119901119887120576119887)119864

minus119878 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119864

2120585

120596119899

Ω cos120593 +(C119901119887nabla119887)119864

119892minus2120585

120596119899

(C119901119887120576119887)119873

119892

1205962119899

Ω cos120593 + 119862 (120595119880)ss minus

119892

1205962119899

(C119901119887120576119887)119873

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

]

(A4)

Multiply the first and third rows of (A1) by1205961198992120585Ω cos120593 and

multiply the second and fourth rows of (A1) by 1205962119899119892Ω cos120593

we can obtain

A1X = B

1 (A5)

where

A1=

[[[[[[[[

[

1 0120596119899

2120585Ω cos120593minus tan120593

1 0 0 minus tan120593

0 1 tan120593120596119899

2120585Ω cos1205930 1 tan120593 0

]]]]]]]]

]

X =[[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]]

]

B1=

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

[

minus120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

1 +120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892minus

(C119901119887120576119887)119873

Ω cos120593

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

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

]

(A6)

Furthermore by utilizing the elementary row transformation(A5) can be transformed into

A2X = B

2 (A7)

Mathematical Problems in Engineering 11

where

A1=

[[[[[[[[

[

1 0 0 minus tan120593

0 0 minus120596119899

2120585Ω cos1205930

0 1 tan120593 0

0 0 0 minus120596119899

2120585Ω cos120593

]]]]]]]]

]

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

B1

=

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

[

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos1205931205962

119899

119892Ω cos120593119862 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892

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

]

(A8)

Then according to matrix equation (A7) we have

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

+

(C119901119887nabla119887)119873

119892tan120593

(A9)

B The Derivations of (34)

The derivations of (34) are carried out based on the principleof least squares fitting setting the Euclidean minimum norm119869 between sin

119880 cos

119880and sin

119880 cos

119880(the normalized

data) as the normalization indexFirst define

119869 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]12

(B1)

It is subjected to the constrained condition namely

ℎ = (sin 119880)2+ (cos

119880)2minus 1 = 0 (B2)

Second define an auxiliary function119898 namely

119898 = 1198692+ 120582ℎ (B3)

where 120582 is an unknown parameterBy substituting (B1) and (B2) into (B3) we can get

119898 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]

minus 120582 ((sin 119880)2+ (cos

119880)2minus 1)

(B4)

Obviously the parameters sin 119880and cos

119880that made 119869

minimum are equal to the ones that make 119898 minimumRespectively the partial derivatives of the parameters sin

119880

and cos 119880are calculated and bymaking them equal to zero

we can get

sin 119880=sin 119880

1 + 120582

cos 119880=cos 119880

1 + 120582

(B5)

Then by substituting (B5) into (B2) we have

(sin 119880)2+ (cos

119880)2

(1 + 120582)2

= 1 (B6)

Finally combining (B5) and (B6) the normalization equa-tions can be obtained

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(B7)

Conflict of Interests

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

Acknowledgments

The work described in the paper was supported by theNational Natural Science Foundation of China (61203225)and the National Science Foundation for Post-DoctoralScientists of China (2012M510083)The authors would like tothank all members of the Inertial Navigation Research Groupat Harbin Engineering University for the technical assistancewith the navigation system

References

[1] H W Park J G Lee and C G Park ldquoCovariance analysis ofstrapdown INS considering gyrocompass characteristicsrdquo IEEETransactions on Aerospace and Electronic Systems vol 31 no 1pp 320ndash328 1995

12 Mathematical Problems in Engineering

[2] R A Nash Jr J A DrsquoAppolito and K J Roy ldquoError analysis ofhybrid aircraft inertial navigation systemsrdquo in Proceedings of theAIAA Guidance and Control Conference Stanford Calif USA1972

[3] F D Jurenka and C T Leondes ldquoOptimum alignment of aninertial autonavigatorrdquo IEEE Transactions on Aerospace andElectronic Systems vol AES-3 no 6 pp 880ndash888 1967

[4] R H Cannon ldquoAlignment of inertial guidance systems bygyrocompassing-linear theoryrdquo Journal of Aerospace Sciencevol 28 no 11 pp 885ndash895 912 1961

[5] G R Pitman Jr Inertial Guidance John Wiley amp Sons NewYork NY USA 1962

[6] K R Britting Inertial Navigation Systems Analysis John Wileyamp Sons New York NY USA 1971

[7] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation vol 28 no4 pp 311ndash341 1982

[8] X Liu X Xu Y Liu and L Wang ldquoA fast and high-accuracycompass alignmentmethod to SINSwith azimuth axis rotationrdquoMathematical Problems in Engineering vol 2013 Article ID524284 12 pages 2013

[9] F Sun Q Wang Z Qi and C Wang ldquoResearch on theestimation method of DVL velocity error based on doubleprogram in Fiber Optic Gyro SINSrdquo Optik vol 124 no 22 pp5344ndash5349 2013

[10] B M Scherzinger ldquoInertial navigator error models for largeheading uncertaintyrdquo in Proceedings of the IEEE Position Loca-tion and Navigation Symposium (PLANS rsquo96) pp 477ndash484Atlanta Ga USA April 1996

[11] X Y Kong EM Nebot andH Durrant-Whyte ldquoDevelopmentof a non-linear psi-angle model for large misalignment errorsand its application in INS alignment and calibrationrdquo in Pro-ceedings of the 1999 IEEE International Conference on Roboticsamp Automation (ICRA rsquo99) pp 1430ndash1435 Detroit Mich USAMay 1999

[12] M-J Yu H-W Park and C-B Jeon ldquoEquivalent nonlinearerror models of strapdown inertial navigation systemrdquo in Pro-ceedings of the Guidance Navigation and Control ConferenceAIAA-97-3563 pp 581ndash587 1997

[13] T Abbas Z Yunyan and L Yanjun ldquoSINS initial alignment forsmall tilt and large azimuth misalignment anglesrdquo in Proceed-ings of the IEEE 3rd International Conference on CommunicationSoftware and Networks (ICCSN rsquo11) pp 628ndash632 Xirsquoan ChinaMay 2011

[14] S L Han and J L Wang ldquoA novel initial alignment scheme forlow-cost INS aided by GPS for land vehicle applicationsrdquo TheJournal of Navigation vol 63 no 4 pp 663ndash680 2010

[15] P G Savage Strapdown Analytics Strapdown Associates 2ndedition 2007

[16] I Yaesh and B Priel ldquoDesign of leveling loop for marine nav-igation systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 29 no 2 pp 599ndash604 1993

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 11

where

A1=

[[[[[[[[

[

1 0 0 minus tan120593

0 0 minus120596119899

2120585Ω cos1205930

0 1 tan120593 0

0 0 0 minus120596119899

2120585Ω cos120593

]]]]]]]]

]

X =[[[[[

[

sin120595119880

cos120595119880

120595119864ss

120595119873ss

]]]]]

]

B1

=

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

[

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

minus1205962

119899

119892Ω cos120593119878 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119873

119892

1 +1205962

119899

119892Ω cos120593119862 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos1205931205962

119899

119892Ω cos120593119862 (120595119880)ss minus

120596119899

2120585Ω cos120593(C119901119887nabla119887)119864

119892

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

]

(A8)

Then according to matrix equation (A7) we have

sin120595119880= minus

1205962

119899

119892Ω cos120593sdot 119878 (120595119880)ss minus

2120585120596119899tan120593119892

sdot 119862 (120595119880)ss minus

(C119901119887120576119887)119864

Ω cos120593

+

(C119901119887nabla119887)119864

119892tan120593

cos120595119880= 1 +

1205962

119899

119892Ω cos120593sdot 119862 (120595

119880)ss minus

2120585120596119899tan120593119892

sdot 119878 (120595119880)ss minus

(C119901119887120576119887)119873

Ω cos120593

+

(C119901119887nabla119887)119873

119892tan120593

(A9)

B The Derivations of (34)

The derivations of (34) are carried out based on the principleof least squares fitting setting the Euclidean minimum norm119869 between sin

119880 cos

119880and sin

119880 cos

119880(the normalized

data) as the normalization indexFirst define

119869 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]12

(B1)

It is subjected to the constrained condition namely

ℎ = (sin 119880)2+ (cos

119880)2minus 1 = 0 (B2)

Second define an auxiliary function119898 namely

119898 = 1198692+ 120582ℎ (B3)

where 120582 is an unknown parameterBy substituting (B1) and (B2) into (B3) we can get

119898 = [(sin 119880minus sin

119880)2+ (cos

119880minus cos

119880)2]

minus 120582 ((sin 119880)2+ (cos

119880)2minus 1)

(B4)

Obviously the parameters sin 119880and cos

119880that made 119869

minimum are equal to the ones that make 119898 minimumRespectively the partial derivatives of the parameters sin

119880

and cos 119880are calculated and bymaking them equal to zero

we can get

sin 119880=sin 119880

1 + 120582

cos 119880=cos 119880

1 + 120582

(B5)

Then by substituting (B5) into (B2) we have

(sin 119880)2+ (cos

119880)2

(1 + 120582)2

= 1 (B6)

Finally combining (B5) and (B6) the normalization equa-tions can be obtained

sin 119880=

sin 119880

radicsin2119880+ cos2

119880

cos 119880=

cos 119880

radicsin2119880+ cos2

119880

(B7)

Conflict of Interests

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

Acknowledgments

The work described in the paper was supported by theNational Natural Science Foundation of China (61203225)and the National Science Foundation for Post-DoctoralScientists of China (2012M510083)The authors would like tothank all members of the Inertial Navigation Research Groupat Harbin Engineering University for the technical assistancewith the navigation system

References

[1] H W Park J G Lee and C G Park ldquoCovariance analysis ofstrapdown INS considering gyrocompass characteristicsrdquo IEEETransactions on Aerospace and Electronic Systems vol 31 no 1pp 320ndash328 1995

12 Mathematical Problems in Engineering

[2] R A Nash Jr J A DrsquoAppolito and K J Roy ldquoError analysis ofhybrid aircraft inertial navigation systemsrdquo in Proceedings of theAIAA Guidance and Control Conference Stanford Calif USA1972

[3] F D Jurenka and C T Leondes ldquoOptimum alignment of aninertial autonavigatorrdquo IEEE Transactions on Aerospace andElectronic Systems vol AES-3 no 6 pp 880ndash888 1967

[4] R H Cannon ldquoAlignment of inertial guidance systems bygyrocompassing-linear theoryrdquo Journal of Aerospace Sciencevol 28 no 11 pp 885ndash895 912 1961

[5] G R Pitman Jr Inertial Guidance John Wiley amp Sons NewYork NY USA 1962

[6] K R Britting Inertial Navigation Systems Analysis John Wileyamp Sons New York NY USA 1971

[7] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation vol 28 no4 pp 311ndash341 1982

[8] X Liu X Xu Y Liu and L Wang ldquoA fast and high-accuracycompass alignmentmethod to SINSwith azimuth axis rotationrdquoMathematical Problems in Engineering vol 2013 Article ID524284 12 pages 2013

[9] F Sun Q Wang Z Qi and C Wang ldquoResearch on theestimation method of DVL velocity error based on doubleprogram in Fiber Optic Gyro SINSrdquo Optik vol 124 no 22 pp5344ndash5349 2013

[10] B M Scherzinger ldquoInertial navigator error models for largeheading uncertaintyrdquo in Proceedings of the IEEE Position Loca-tion and Navigation Symposium (PLANS rsquo96) pp 477ndash484Atlanta Ga USA April 1996

[11] X Y Kong EM Nebot andH Durrant-Whyte ldquoDevelopmentof a non-linear psi-angle model for large misalignment errorsand its application in INS alignment and calibrationrdquo in Pro-ceedings of the 1999 IEEE International Conference on Roboticsamp Automation (ICRA rsquo99) pp 1430ndash1435 Detroit Mich USAMay 1999

[12] M-J Yu H-W Park and C-B Jeon ldquoEquivalent nonlinearerror models of strapdown inertial navigation systemrdquo in Pro-ceedings of the Guidance Navigation and Control ConferenceAIAA-97-3563 pp 581ndash587 1997

[13] T Abbas Z Yunyan and L Yanjun ldquoSINS initial alignment forsmall tilt and large azimuth misalignment anglesrdquo in Proceed-ings of the IEEE 3rd International Conference on CommunicationSoftware and Networks (ICCSN rsquo11) pp 628ndash632 Xirsquoan ChinaMay 2011

[14] S L Han and J L Wang ldquoA novel initial alignment scheme forlow-cost INS aided by GPS for land vehicle applicationsrdquo TheJournal of Navigation vol 63 no 4 pp 663ndash680 2010

[15] P G Savage Strapdown Analytics Strapdown Associates 2ndedition 2007

[16] I Yaesh and B Priel ldquoDesign of leveling loop for marine nav-igation systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 29 no 2 pp 599ndash604 1993

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

12 Mathematical Problems in Engineering

[2] R A Nash Jr J A DrsquoAppolito and K J Roy ldquoError analysis ofhybrid aircraft inertial navigation systemsrdquo in Proceedings of theAIAA Guidance and Control Conference Stanford Calif USA1972

[3] F D Jurenka and C T Leondes ldquoOptimum alignment of aninertial autonavigatorrdquo IEEE Transactions on Aerospace andElectronic Systems vol AES-3 no 6 pp 880ndash888 1967

[4] R H Cannon ldquoAlignment of inertial guidance systems bygyrocompassing-linear theoryrdquo Journal of Aerospace Sciencevol 28 no 11 pp 885ndash895 912 1961

[5] G R Pitman Jr Inertial Guidance John Wiley amp Sons NewYork NY USA 1962

[6] K R Britting Inertial Navigation Systems Analysis John Wileyamp Sons New York NY USA 1971

[7] C San Giovanni Jr and E Levinson ldquoPerformance of a ringlaser strapdown marine gyrocompassrdquo Navigation vol 28 no4 pp 311ndash341 1982

[8] X Liu X Xu Y Liu and L Wang ldquoA fast and high-accuracycompass alignmentmethod to SINSwith azimuth axis rotationrdquoMathematical Problems in Engineering vol 2013 Article ID524284 12 pages 2013

[9] F Sun Q Wang Z Qi and C Wang ldquoResearch on theestimation method of DVL velocity error based on doubleprogram in Fiber Optic Gyro SINSrdquo Optik vol 124 no 22 pp5344ndash5349 2013

[10] B M Scherzinger ldquoInertial navigator error models for largeheading uncertaintyrdquo in Proceedings of the IEEE Position Loca-tion and Navigation Symposium (PLANS rsquo96) pp 477ndash484Atlanta Ga USA April 1996

[11] X Y Kong EM Nebot andH Durrant-Whyte ldquoDevelopmentof a non-linear psi-angle model for large misalignment errorsand its application in INS alignment and calibrationrdquo in Pro-ceedings of the 1999 IEEE International Conference on Roboticsamp Automation (ICRA rsquo99) pp 1430ndash1435 Detroit Mich USAMay 1999

[12] M-J Yu H-W Park and C-B Jeon ldquoEquivalent nonlinearerror models of strapdown inertial navigation systemrdquo in Pro-ceedings of the Guidance Navigation and Control ConferenceAIAA-97-3563 pp 581ndash587 1997

[13] T Abbas Z Yunyan and L Yanjun ldquoSINS initial alignment forsmall tilt and large azimuth misalignment anglesrdquo in Proceed-ings of the IEEE 3rd International Conference on CommunicationSoftware and Networks (ICCSN rsquo11) pp 628ndash632 Xirsquoan ChinaMay 2011

[14] S L Han and J L Wang ldquoA novel initial alignment scheme forlow-cost INS aided by GPS for land vehicle applicationsrdquo TheJournal of Navigation vol 63 no 4 pp 663ndash680 2010

[15] P G Savage Strapdown Analytics Strapdown Associates 2ndedition 2007

[16] I Yaesh and B Priel ldquoDesign of leveling loop for marine nav-igation systemrdquo IEEE Transactions on Aerospace and ElectronicSystems vol 29 no 2 pp 599ndash604 1993

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