research article error analysis and compensation of ...transfer alignment, integrated alignment, and...

Research ArticleError Analysis and Compensation of GyrocompassAlignment for SINS on Moving Base

Bo Xu,1 Yang Liu,1 Wei Shan,1 Yi Zhang,2,3 and Guochen Wang1

1 Harbin Engineering University, 145 Nantong Road, Harbin 150001, China2 Beijing Aerospace Automatic Control Institute, No. 50 Yongding Road, Haidian District, Beijing 100039, China3National Key Laboratory of Science and Technology on Aerospace Intelligence Control, Beijing 100854, China

Correspondence should be addressed to Bo Xu; [email protected]

Received 29 January 2014; Accepted 24 May 2014; Published 25 June 2014

Academic Editor: Bin Jiang

Copyright © 2014 Bo Xu et al. This 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.

An improved method of gyrocompass alignment for strap-down inertial navigation system (SINS) on moving base assisted withDoppler velocity log (DVL) is proposed in this paper. After analyzing the classical gyrocompass alignment principle on static base,implementation of compass alignment on moving base is given in detail. Furthermore, based on analysis of velocity error, latitudeerror, and acceleration error on moving base, two improvements are introduced to ensure alignment accuracy and speed: (1) thesystem parameters are redesigned to decrease the acceleration interference and (2) a data repeated calculation algorithm is usedin order to shorten the prolonged alignment time caused by changes in parameters. Simulation and test results indicate that theimproved method can realize the alignment on moving base quickly and effectively.

1. Introduction

Initial alignment is the process of determining the axesorientation of strap-down inertial navigation system (SINS)with respect to the reference navigational frame. To meet thequick response of ships and enhance survivability, alignmenton moving base has become a key technique for SINS [1].Different from the alignment for SINS on static base, externalinformation should be brought in to assist alignment for SINSon moving base [2]. At present, the research of alignment onmoving base mainly focuses on the assist of GPS location.However, GPS systemmay have some restrictions in practicalapplication [3–5]. Compared with GPS, Doppler velocity log(DVL) is an underwater available, independent, and highaccuracy velocity measuring element commonly used onships, and the research of initial alignment for SINS onmoving base assisted with DVL has attracted much moreattention [6].

The initial alignment methods on moving base canbe commonly classified into three mainstream directions:transfer alignment, integrated alignment, and gyrocompassalignment. In transfer alignment, by means of velocity

matching and attitude matching, a misaligned slave inertialnavigation system can be aligned with the assistance of amaster inertial navigation system. It can accomplish theinitial alignment quickly and accurately, but the overallsystem is very complex [7]. Integrated alignment is an initialalignment method based on modern estimation theory andstate space description. It can accomplish alignment rapidlyand precisely by using modern filtering methods to estimatethemisalignment angle.However, it is difficult to establish theabsolutely accurate mathematical model and noise model ofthe system, and the large amount of calculation in alignmentprocess always leads to poor instantaneity [8]. Gyrocompassalignment is built on the basis of classical control theory, sothere is no need to establish accurate mathematical modeland noise model. Its algorithm is simple, and the calculationamount is greatly reduced. However, as its fundamentals areestablished on static base or quasi-static base, when appliedonmoving base, gyrocompass alignmentwill be inaccurate oreven impracticable with the effect of speed and acceleration.

In recent years, to improve the performance of initialalignment for SINS on moving base, gyrocompass alignmentmethods have already been analyzed by some researchers.

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 373575, 18 pageshttp://dx.doi.org/10.1155/2014/373575

2 Mathematical Problems in Engineering

In paper [9], azimuth axis rotating is used to improve theaccuracy of compass loop, but the paper does not expandit to moving base [9]. Based on the principle of strap-downgyrocompass alignment, Cheng Xianghong from Southeast-ern University points out that the carrier velocity can affectthe gyrocompass alignment and puts forward a calibrationmethod for the inertial sensors of the SINS in the processof alignment on moving base [10]. Yan Gongmin fromNorthwestern Polytechnical University proposes a calculatemethod applied to the strap-down gyrocompass alignmenton moving base [11]. As gyrocompass alignment needs lesscalculation amount but still remains reliable, it has a greatapplication prospect for marine SINS alignment. Hwanget al. conduct precalibration using dual-electric compasses tominimize the error of spreader pose control [12].

However, few papers analyzed the problems faced by thegyrocompass alignment on moving base in a systemic andcomprehensive way. Aiming at this problem, after analyzingthe principle of classical gyrocompass initial alignment, thispaper put forward a gyrocompass alignmentmethod for SINSon moving base aided with Doppler velocity log (DVL) anddeduces the process of algorithm realization in detail. Basedon the characteristics of DVL’s measuring error, we analyzethe influence of velocity error of gyrocompass alignmenton moving base and then establish a misalignment anglemodel. It can be found from the analysis that the mostsevere interference comes from the acceleration. The systemparameters can be redesigned to restrain this kind of error,but it can also cause time growth. To shorten the gyrocompassinitial alignment time, a data repeated calculation algorithmis also introduced [13].

The paper is organized as follows. In Section 2, theprinciple and realization of classical gyrocompass alignmentfor SINS on static base are introduced and the systemcharacteristic is analyzed and in the end it leads to gyrocom-pass alignment on moving base. In Section 3, a DVL aidedcompass alignment method on moving base is proposed.The effect of velocity error, latitude error, and accelera-tion interference are analyzed, respectively, and the systemparameters are redesigned to reduce the most serious effect.In Section 4, a fast compass alignment method based onreversed navigation algorithm is put forward, and the detailedcalculating equations are given. In Section 5, simulationsabout the alignment methods mentioned above are done. InSection 6, a lake test with a certain type of SINS is carried out.Finally, conclusions are drawn in Section 7.

2. Gyrocompass Alignmentfor SINS on Static Base

2.1. Gyrocompass Alignment Principle. Gyrocompass align-ment is commonly used in many kinds of inertial navigationsystems [13–15] based on compass effect principle. Thisazimuth alignment method is proceeded after horizontalleveling adjust in application. By using control theory andadding dampings, it can make the platform coordinateapproach the navigation coordinate gradually. In this section,the operating principle and implementation of gyrocompass

yt yp

𝜙z 𝜙z

xt

xp

Ωcos𝜑

−sin𝜙zΩcos𝜑

Figure 1: Schematic diagram of gyrocompass effect.

alignment for SINS on static base are described, and theaccuracy on static base is also analyzed.

Compared with eastern horizontal loop, northern hori-zontal loop has an extra coupling term𝜙

𝑧𝜔𝑖𝑒cos𝜑, which is in

proportional relationship with the azimuth error angle. Thistermhas the same functionwith eastern gyro drift, and it is anangular rate that comes from projection of the earth rotationangular rate in essence [16]. When there is an azimuth errorangle 𝜙

𝑧between the platform coordinate system and the

geographic coordinate system, the northern earth rotationangular rate Ω cos𝜑 will partly be projected to the platformcoordinate system in eastern axis, and its projection value is− sin𝜙

𝑧𝜔𝑖𝑒cos𝜑. After coarse alignment, the projection value

can approximately be simplified to −𝜙𝑧𝜔𝑖𝑒cos𝜑. Then the

azimuth error angle can be coupled to northern horizontalloop by term −𝜙

𝑧𝜔𝑖𝑒cos𝜑. This coupling relationship is

defined as gyrocompass effect, as shown in Figure 1 [17].Due to gyrocompass effect, the horizontal error angle 𝜙

𝑥

is influenced additionally by the effect of azimuth error angle𝜙𝑧, and the projection value of gravity acceleration along

northern axis in platform coordinate system will change. Itwill lead to the change of velocity error in the northern loop.Making use of this coupling relationship, the gyrocompassalignment method controls the up axis gyro with the velocityerror information and forms a new closed loop circuits calledthe gyrocompass loop [18]. Reasonable designed gyrocom-pass circuit parameters can make the system stable, fast, andmore accurate; thus the gyrocompass alignment process canbe accomplished.

2.2. Realization of Gyrocompass Alignmentfor SINS on Static Base

2.2.1. System Realization. The direction cosine matrix 𝐶𝑝𝑏

from carrier coordinate system to the platform coordinatesystem is an important matrix in the process of calculation.The angular velocity and acceleration information measuredby IMU must be transformed into platform coordinate

Mathematical Problems in Engineering 3

fb

fp = Cp

bfb

Cp

b

Cp

b

Cp

b

𝜔bc = Cp

b𝜔pc

𝜔pc

fp

𝜔bip

𝜔bib

𝜔pie

𝜔pie

𝜔pep

𝜔bip = Cbp( + 𝜔pep)

Attitude

Calculation of therevised angular rate

𝜔bc −

−

Cp

b = Cp

b× 𝜔bpb

Figure 2: Schematic diagram of gyrocompass alignment of SINS.

system via matrix 𝐶𝑝𝑏before participating in the navigation

process. While 𝐶𝑝𝑏has the same function with the physical

platform of SINS, it is also called the mathematical platform.As the mathematical platform is used instead of the phys-

ical platform in SINS,𝐶𝑝𝑏has become the control object of the

revised angular velocity in gyrocompass method principle.The updating algorithm of the mathematical platform 𝐶𝑝

𝑏is

as follows:

𝜔𝑏

𝑝𝑏= 𝜔𝑏

𝑖𝑏− 𝐶𝑏

𝑝(𝜔𝑝

𝑒𝑝+ 𝜔𝑝

𝑖𝑒) ,

�̇�𝑝

𝑏= 𝐶𝑝

𝑏× 𝜔𝑏

𝑝𝑏.

(1)

The angular velocity to controlmathematical platform𝐶𝑝𝑏

is𝜔𝑏𝑝𝑏. Considering the drift error of gyro and the error caused

by interference movement of carrier, the revised angularvelocity of SINS is added. After adding the control angularvelocity 𝜔𝑝

𝑐, the corresponding mathematical platform of

SINS control equation is as follows:

𝜔𝑏

𝑝𝑏= 𝜔𝑏


𝑝(𝜔𝑝

𝑒𝑝+ 𝜔𝑝

𝑖𝑒) − 𝐶𝑏

𝑝(𝜔𝑝

𝑐) . (2)

The schematic diagram of gyrocompass initial alignmentof SINS is given in Figure 2.

2.2.2. Calculation of the Revised Angular Rate. The revisedangular rate can be obtained as shown in Figure 2. Figures 3and 4 are the north channel and azimuth channel, respec-tively, in the compass alignment loop.

In Figure 3, as the dash line shows, 𝐾1is a damping term

used to decrease the oscillation amplitude of Schuler loop; asthe dash-dot line shows,𝐾

2is applied to shorten the system’s

natural period of oscillating period by √1 + 𝐾2times. After

term 𝜔𝑝𝑐𝑥there is a horizontal angle error caused by gyro drift

and azimuthmisalignment angle. As the double dash-dot lineshows, 𝐾

3is an energy storage term introduced to offset this

error. All the 𝐾 values above can be calculated by dampedcoefficient 𝜉 and time constant 𝜎:

𝐾1= 3𝜎, 𝐾

2= (2 +

1

𝜉2)𝜎2

𝜔2𝑠

− 1, 𝐾3=

𝜎3

𝜉2𝜔2𝑠

,

𝜔𝑠= √

𝑅

𝑔, 𝜎 = 𝜉𝜔

𝑛.

(3)

Compared with Figure 3, 𝐾3is replaced by 𝐾(𝑠) in

Figure 4 to reflect the compass effect term, 𝐾(𝑠) =𝐾3/𝑅𝜔𝑖𝑒cos𝜑⋅(𝑠+𝐾

4) and its purpose is to reduce the azimuth

angle 𝜙𝑧to an allowed range. All the𝐾 values above can also

be calculated by damped coefficient 𝜉 and time constant 𝜎:

𝐾1= 𝐾3= 2𝜎, 𝐾

2=𝜎2 + 𝜔2

𝑛

𝜉2 ⋅ 𝜔2𝑠

− 1, 𝐾4=

4𝜎4

𝜉2 ⋅ 𝜔2𝑠

,

𝜔𝑠= √

𝑅

𝑔.

(4)The alignment accuracy on static base is mainly decided

by eastern and northern accelerometer zero bias ∇𝐸, ∇𝑁and

eastern gyro drift 𝜀𝐸:

𝜙𝑠𝑥= −

1

𝑔∇𝑁, (5)

𝜙𝑠𝑦=1

𝑔∇𝐸, (6)

𝜙𝑠𝑧=

𝜀𝐸

𝜔𝑖𝑒cos𝜑

+𝐾4(1 + 𝐾

2) 𝜀𝑢

𝑅 ⋅ 𝐾3

. (7)

2.3. Static Base Gyrocompass Circuit Characteristic Analysis.Gyrocompass alignment on static base or quasi-static basehas the following characteristics.

2.3.1. No External Acceleration Effect. Gyrocompass align-ment changes the strap-down inertial navigation controlsystem into a stable system in principle. However, Schulerloop of the system is destroyed and external accelerationimpact is introduced into the system. According to Figures 3and 4, although both of the two gyrocompass alignmenthorizontal loopswill be infected by acceleration, the influencebrought by motion acceleration can be ignored as the carrieraccelerations 𝐴

𝑁and 𝐴E can be approximately regarded in

this status.

2.3.2. Dispense with Updating of 𝜔𝑝𝑒𝑝

and 𝜔𝑝𝑖𝑒. According to

Figure 2, besides the measured value of gyro and accelerator,there are inputs 𝜔𝑝

𝑒𝑝and 𝜔𝑝

𝑖𝑒in gyrocompass alignment

realization process. In gyrocompass alignment system, thereis only attitude calculation but no velocity and positioncalculation, so the value of 𝜔𝑝


𝑖𝑒cannot be got except

for bringing in external information. As the velocity of carrieris zero and the position of carrier remains the same on staticbase or quasi-static base, the value of 𝜔𝑝


𝑖𝑒can be got

directly without updating calculation.


K3s

K2R

1

R

1

s

K1

+

+ +

+

−

−𝜔pcx

𝜀n∇n

ΔAN

Figure 3: Schematic diagram of gyrocompass alignment in north channel.

K(s)

K2R

1

R

1

s

K1

+

−

+

+−

pcx

n∇n

pcz

ΔAN

Figure 4: Schematic diagram of gyrocompass alignment in azimuthchannel.

2.3.3. Fixed Instrument Error inGeographyCoordinate System.SINS is strapped to carrier coordinate system, so its instru-ment error is defined in carrier coordinate system. Becausethe inertial navigation error equation is established in geogra-phy coordinate system, the analysis of instrument error has tobe projected in geography coordinate system. As the carriercoordinate system remains relatively unchangeable with thegeography coordinate system on static base, the instrumenterror in geography coordinate system is still constant.

2.4. Gyrocompass Alignment on Moving Base. Motion ofcarrier will change the relative position inevitably betweengeographical coordinate and inertial space. One reason isthat the earth’s rotation angular velocity 𝜔

𝑖𝑒will change the

direction of the earth coordinate system in inertial space; theother is that the movement of carrier on surface of the earthwill cause relative rotation between geographical coordinatesystem and earth coordinate. Assuming the velocity of carrieris 𝑉 and the azimuth angle is 𝜓 in carrier coordinate,then their projections along north and east of geographicalcoordinate are𝑉

𝑁= 𝑉⋅ cos𝜓 and𝑉

𝐸= 𝑉⋅ sin𝜓, respectively.

As shown in Figure 5, the rotational angular velocity 𝜔𝑛𝑖𝑛

of the geographical coordinate system relative to the inertialspace can be regarded as sum of the earth’s rotational angularvelocity 𝜔

𝑖𝑒and the relative rotational angular velocity 𝜔𝑛

𝑒𝑛

yV

x

z

O

Oi

R

VE

VN

𝜑

Rcos𝜑

pN

Figure 5: The projection in north and east of geographical coordi-nate.

between the geographical coordinate and the earth coordi-nate:

[

[

𝜔𝑥

𝜔𝑦

𝜔𝑧

]

]

= [

[

0𝜔𝑖𝑒cos𝜑

𝜔𝑖𝑒sin𝜑

]

]

+

[[[[[[

[

−𝑉 cos𝜓𝑅

𝑉 sin𝜓𝑅

𝑉 sin𝜓 ⋅tan𝜑𝑅

]]]]]]

]

. (8)

The output of gyroscope projected in navigation coordi-nates is 𝜔𝑛

𝑖𝑏= 𝜔𝑛𝑖𝑒+ 𝜀𝑛 on static base, but on moving base it

becomes𝜔𝑛𝑖𝑏= 𝜔𝑛𝑖𝑒+𝜔𝑛𝑒𝑛+𝜔𝑛𝑛𝑏+𝜀𝑛.𝜔𝑛

𝑛𝑏can be regarded as zero

in uniform motion. The output of accelerator projected innavigation coordinates is the gravitational acceleration 𝑓𝑛 =−𝑔𝑛 + ∇𝑛 on static base. However it becomes 𝑓𝑛 = (2𝜔𝑛

𝑖𝑒+

𝜔𝑛𝑒𝑛) × 𝑉𝑛 − 𝑔𝑛 + ∇𝑛 in uniform motion [13].To make error analysis of misalignment caused by move-

ment directly is relatively difficult. Therefore, the angularmotion and the linear motion caused by movement areequivalent to gyro drift 𝜀𝑛

𝑑and zero bias of acceleration ∇𝑛

𝑑


Static base

𝜔nib = 𝜔nie + 𝜀

n

fn = −gn + ∇n

𝜔nib = 𝜔nie + 𝜔

nen + 𝜀

n = 𝜔nie + 𝜀nd + 𝜀

n

fn = −gn + (2𝜔nie + 𝜔nen) × n + ∇n = −gn + ∇nd + ∇n

Angularmotion

Constantvelocity

Linearmotion

Uniform motion

Figure 6: The equivalent error caused by uniform motion.

on static base correspondingly [19]. The equivalent error isshown in Figure 6.

The equivalent error can be calculated as follows:

𝜀𝑛

𝑑= [𝜀𝑑𝑒𝜀𝑑𝑛

𝜀𝑑𝑢]𝑇

= 𝜔𝑛

𝑒𝑛

= [−𝑉 cos𝜓𝑅

𝑉 sin𝜓𝑅

𝑉 sin𝜓 tan𝜑𝑅

]𝑇

,

∇𝑛

𝑑= [

[

∇𝑑𝑒

∇𝑑𝑛

∇𝑑𝑛

]

]

= (2𝜔𝑛

𝑖𝑒+ 𝜔𝑛

𝑒𝑛) × 𝑉𝑛

=

[[[[[[[

[

2𝜔𝑖𝑒sin𝜓 sin𝜑 ⋅ 𝑉 +

𝑉2sin2𝜓 tan𝜑𝑅

−2𝜔𝑖𝑒cos𝜓 sin𝜑 ⋅ 𝑉 −

𝑉2 sin𝜓 cos𝜓 tan𝜑𝑅

−𝑉2 sin𝜓 cos𝜓

𝑅+ 2𝜔𝑖𝑒cos𝜑 cos𝜓 ⋅ 𝑉 +

𝑉2sin2𝜓𝑅

]]]]]]]

]

.

(9)

The final accuracy of misalignment angle along eastern,northern, and up orientation directions can be got by addingthe instrument error and errors caused by carrier’s motioninto (5) to (7) that can be expressed as follows:

𝜙𝑠𝑥= −

1

𝑔(∇𝑁+ ∇𝑑𝑛) , (10)

𝜙𝑠𝑦=1

𝑔(∇𝐸+ ∇𝑑𝑒) , (11)

𝜙𝑠𝑧=𝜀𝐸+ 𝜀𝑑𝑒

𝜔𝑖𝑒cos𝜑

+𝐾4(1 + 𝐾

2) (𝜀𝑢+ 𝜀𝑑𝑢)

𝑅 ⋅ 𝐾3

. (12)

In (12), 𝜔𝑖𝑒is earth’s rotational velocity, 𝜑 is latitude of

carrier’s position, and ∇𝐸, ∇𝑁, 𝜀𝐸, and 𝜀

𝑈are the equivalent

gyro drift and equivalent accelerator bias in navigationcoordinate system.The corresponding solution inmotionwillbe introduced in the following sections.

3. DVL Aided Gyrocompass Alignment onMoving Base

3.1. DVL Aided Gyrocompass Alignment. The analysis inSection 2 gives conclusion that the influence factors ofgyrocompass alignment become more complicated when

the complexity of motion rises. From the perspective of sys-tem, the influencing form of acceleration in motion is similarto accelerometer bias, but its input value is much larger thanaccelerometer bias. What is more, from the perspective ofDVL aided velocity information, error becomesmore instablein motion.

From the analysis of (9), compared with gyrocompassalignment on static base, error compensations are needed infour parts, respectively, on moving base. They are angularvelocity 𝜔𝑝

𝑒𝑝, earth rotation angular velocity 𝜔𝑝

𝑖𝑒, harmful

acceleration𝐵𝑝, andmotion acceleration caused by seawaves.The value of acceleration is only affected by waves in

uniformmotion, so it can be treated as disturbance.The otherthree parts can be calculated by the following equations:

𝜔𝑝

𝑖𝑒= [0 𝜔

𝑖𝑒cos𝜑 𝜔

𝑖𝑒sin𝜑]𝑇, (13)

𝜔𝑝

𝑒𝑝= [−

𝑉𝑁

𝑅

𝑉𝐸

𝑅

𝑉𝐸

𝑅 cos𝜑]𝑇

, (14)

𝐵𝑝

= (𝜔𝑝

𝑒𝑝+ 2𝜔𝑝

𝑖𝑒) × 𝑉𝑝

. (15)

It can be found that the precise information of carrier’svelocity and position is needed in compensation calculation.As the information cannot be obtained in gyrocompassalignment process, external information is essential to com-plete the calculation. If the initial position is known, withthe assistance of DVL velocity information, the dynamiccompensation can be calculated by the following methodsafter coarse alignment.

3.1.1. Velocity Projection Calculation. The velocity measuredby DVL is 𝑉𝑏 in carrier coordinate system, but the velocityin navigation coordinate system has to be calculated. Aftercoarse alignment, mathematical platform has been estab-lished and misalignment angle is controlled within a certainrange, so the velocity in platform coordinate system can begot by projection calculation of mathematical platform. SetDVL measurement velocity as 𝑉𝑏dvl, and its expression is

𝑉𝑝

dvl = 𝐶𝑝

𝑏𝑉𝑏

dvl. (16)

3.1.2. Latitude Calculation. After coarse alignment, the mis-alignment angle is controlled within a smaller range, so


fb

Cp

b

Cp

b

𝜔pc

𝜔bip

𝜔bib

𝜔pie

R

𝜔pep

𝜔bip = Cbp

𝜔bc = 𝜔pcC

bp

(𝜔pie + 𝜔pep)

𝜔bc

−

−

−

−

Cp

b

Cp

b

fp = Cp

bfb

𝜑

𝜃

𝛾

𝜓

Ap

Bp

Compensationalgorithm

Modificationcalculation

Cp

b = Cp

b× 𝜔bpb

Vp

dvl

Vp

dvlVb

dvlVp

dvl = Cp

bVbdvl

𝜑 = 𝜑0 − ∫VPdvlN

Figure 7: DVL aided gyrocompass alignment on moving base.

the carrier position can be got by integral calculation of theDVL velocity projection value 𝑉𝑝dvl:

𝜑 = 𝜑0− ∫

𝑉𝑝

dvl𝑁𝑅

. (17)

𝑉𝑝

dvl𝑁 is the projection of carrier velocity 𝑉𝑝

dvl along northin platform coordinate system.

3.1.3. Compensation Value Calculation. We can use latitudeand velocity information to calculate compensation value𝜔𝑝

𝑖𝑒, 𝜔𝑝𝑒𝑝, and 𝐵𝑝 in (13)–(15). The implementation principle

scheme of gyrocompass alignment on moving base is shownin Figure 7.

3.2. Error Analysis of DVL Aided Compass Alignment onMoving Base. There are still some error factors existing in thecompensation calculation method mentioned in Section 3.1.On one hand, as there are errors in compensation calculationprocess, the calculation of DVL velocity 𝑉𝑝dvl and latitude 𝜑will be effected accordingly. On the other hand, the errorcaused by sea waves is regarded as interference and ignoredin compensation calculation. So we need to analyze the errorof gyrocompass alignment from three aspects: velocity error,latitude error, and acceleration error.

3.2.1. The Effect of Velocity Error. As 𝑉𝑝dvl can be calculatedby (16), error factors mainly come from the error of attitudematrix 𝐶𝑝

𝑏and the error of DVL measured velocity 𝑉𝑏dvl.

In alignment process, misalignment angle becomes smallergradually, so it is unnecessary to make further analysis of itsinfluence.

The velocity 𝑉𝑏dvl measured by DVL with constant errorcan be written as

𝑉𝑏

dvl = 𝑉𝑏

+ Δ𝑉𝑏

𝑐. (18)

Equation (18) can be converted to the platform coordi-nate:

𝑉𝑝

dvl = 𝑉𝑝

+ 𝐶𝑝

𝑏Δ𝑉𝑏

𝑐. (19)

Due to swing of carrier and convergence of misalignmentangle, there are some tiny variations in 𝐶𝑝

𝑏. The swing with

small amplitude canmake𝐶𝑝𝑏Δ𝑉𝑏𝑐shake around a constant in

limited range. So the error of 𝑉𝑝dvl can be regarded as the sumof a constant error and a small high frequency oscillation.Thecalculation related to velocity is the angular velocity 𝜔𝑝

𝑒𝑝and

the harmful acceleration 𝐵𝑝.


(1) The Influence 𝜔𝑝𝑒𝑝

Calculation Errors. As shown in (14),velocity is linear to the angular rate, so the error in 𝜔𝑝

𝑒𝑝by

the effect of speed error 𝛿𝑉𝐸and 𝛿𝑉

𝑁can be expressed as

𝛿𝜔𝑝v𝑒𝑝= [−

𝛿𝑉𝑁

𝑅

𝛿𝑉𝐸

𝑅

𝛿𝑉𝐸

𝑅 cos𝜑]𝑇

. (20)

Under the influence of this error, (2) can be rewritten as(21) on moving base:

𝜔𝑏

𝑝𝑏= 𝜔𝑏


𝑝(𝜔𝑝

𝑒𝑝+ 𝛿𝜔𝑝

𝑒𝑝+ 𝜔𝑝

𝑖𝑒+ 𝛿𝜔𝑝


𝑝(𝜔𝑝

𝑐)

= 𝜔𝑏


𝑝(𝛿𝜔𝑝V𝑒𝑝) − 𝐶𝑏

𝑝(𝜔𝑝

𝑒𝑝+ 𝜔𝑝


𝑝(𝜔𝑝

𝑐) .

(21)

After adding gyroscopic drift, (21) can be written as

𝜔𝑏

𝑝𝑏= 𝜔𝑏

𝑖𝑏+ 𝜀𝑏− 𝐶𝑏

𝑝(𝜔𝑝

𝑒𝑝+ 𝜔𝑝


𝑝(𝜔𝑝

𝑐)

= 𝜔𝑏

𝑖𝑏+ 𝐶𝑏

𝑝𝜀𝑝− 𝐶𝑏

𝑝(𝜔𝑝

𝑒𝑝+ 𝜔𝑝


𝑝(𝜔𝑝

𝑐) .

(22)

The comparison of (21) and (22) gives conclusion that theerror of 𝜔𝑝

𝑒𝑝and gyro drift in carrier coordinate system have

the same influence form to attitude updating calculation. If itis considered as equivalent gyro drift 𝜀

𝑉𝑑𝑒and 𝜀𝑉𝑑𝑢

, the errorwith small high-frequency oscillationwill be restrained by thesystem and constant error is the only thing to be considered.Based on (7), with the effect of constant error, the alignmenterror caused by equivalent gyro drift in form (20) is as follows:

𝜙𝛿𝑉1

𝑧=𝜀𝑉𝑑𝑒

𝜔𝑛𝑖𝑒

+𝐾𝑧(1 + 𝐾

2) 𝜀𝑉𝑑𝑢

𝑅 ⋅ 𝐾3

=1

𝜔𝑖𝑒cos𝜑

⋅ (−𝛿𝑉𝑁

𝑅) +

𝐾𝑧(1 + 𝐾

2)

𝐾3

⋅𝛿𝑉𝐸

𝑅2 cos𝜑.

(23)

As analyzed above, east gyro drift −𝛿𝑉𝑁/𝑅 has greater

influence on azimuth angle than azimuth gyro drift𝛿𝑉𝐸/𝑅 cos𝜑, so the second part in (23) can be neglected and

(23) can be simplified as

𝜙𝛿𝑉1

𝑧=

1

𝜔𝑖𝑒cos𝜑


𝑅) . (24)

(2) The Influence of Harmful Acceleration Error. As shownin Figure 7, the influence form of harmful acceleration 𝐵𝑝 issimilar to𝐴𝑝.Therefore the influence of harmful accelerationand the accelerometer bias can be written in the same form,and the acceleration bias error analysis method can also beused to analyze the influence of harmful acceleration.

The projection of errors caused by harmful accelerationalong east and north of the platform is as follows:

𝐵𝑝

𝐸= (2𝜔

𝑖𝑒sin𝜑 + 𝑉𝐸

𝑅 cos𝜑) ⋅ 𝑉𝑁,

𝐵𝑝

𝑁= −(2𝜔

𝑖𝑒sin𝜑 + 𝑉𝐸

𝑅 cos𝜑) ⋅ 𝑉𝐸.

(25)

With the effect of velocity errors 𝛿𝑉𝐸and 𝛿𝑉

𝑁, (25) can

be converted as

𝐵𝑝

𝐸= (2𝜔

𝑖𝑒sin𝜑 + 𝑉𝐸 + 𝛿𝑉𝐸

𝑅 cos𝜑) ⋅ (𝑉𝑁+ 𝛿𝑉𝑁) ,

𝐵𝑝

𝑁= −(2𝜔

𝑖𝑒sin𝜑 + 𝑉𝐸 + 𝛿𝑉𝐸

𝑅 cos𝜑) ⋅ (𝑉𝐸+ 𝛿𝑉𝐸) .

(26)

The approximate value of harmful acceleration error canbe got by subtracting (26) from (25), and the result is

𝛿𝐵𝑝

𝐸= 2𝜔𝑖𝑒sin𝜑 ⋅ 𝛿𝑉

𝑁+𝛿𝑉𝑁⋅ 𝑉𝐸+ 𝛿𝑉𝐸⋅ 𝑉𝑁+ 𝛿𝑉𝐸⋅ 𝛿𝑉𝑁

𝑅 cos𝜑,

𝛿𝐵𝑝

𝑁= −(2𝜔

𝑖𝑒sin𝜑 ⋅ 𝛿𝑉

𝐸+2𝛿𝑉𝐸⋅ 𝑉𝐸+ 𝛿𝑉2𝐸

𝑅 cos𝜑) .

(27)

There is little change in latitude, so 𝜑 can be consideredas a constant in alignment process. The velocity and its errorcan be considered as sum of constant and high frequencyoscillation. As the swing frequency is high, the system has aninhibition to this oscillation error, so its influence is relativelyweak, so (26) can be analyzed as the constant gyro drift. Theharmful acceleration errors 𝛿𝐵𝑝

𝐸and 𝛿𝐵𝑝

𝑁and the constant

accelerometer bias ∇𝐵𝑑𝑒

, ∇𝐵𝑑𝑛

are equivalent, so accordingto (6) and (7), error angle can be obtained in (27) with theinfluence of the equivalent accelerometer bias:

𝜙𝛿𝑉2

𝑥= −

∇𝐵𝑑𝑛

𝑔

= −1

𝑔⋅ ( − 2𝜔

𝑖𝑒sin𝜑 ⋅ 𝛿𝑉

𝑁

−𝛿𝑉𝑁⋅ 𝑉𝐸+ 𝛿𝑉𝐸⋅ 𝑉𝑁+ 𝛿𝑉𝐸⋅ 𝛿𝑉𝑁

𝑅 cos𝜑) ,

𝜙𝛿𝑉2

𝑦=∇𝐵𝑑𝑒

𝑔=1

𝑔⋅ (2𝜔𝑖𝑒sin𝜑 ⋅ 𝛿𝑉

𝐸

+2𝛿𝑉𝐸⋅ 𝑉𝐸+ 𝛿𝑉2𝐸

𝑅 cos𝜑) .

(28)

Synthesizing themisalignment angles caused by two partsof the velocity error, the error equation can be rewritten as

𝜙𝛿𝑉

𝑥=1


𝑁

+𝛿𝑉𝑁⋅ 𝑉𝐸+ 𝛿𝑉𝐸⋅ 𝑉𝑁+ 𝛿𝑉𝐸⋅ 𝛿𝑉𝑁

𝑅 cos𝜑) ,

𝜙𝛿𝑉

𝑦=1


𝐸+2𝛿𝑉𝐸⋅ 𝑉𝐸+ 𝛿𝑉2𝐸

𝑅 cos𝜑) ,

𝜙𝛿𝑉

𝑧=

1

𝜔𝑖𝑒cos𝜑


𝑅) .

(29)

3.2.2.The Effect of Latitude Error. Latitude calculation can bemainly divided into two parts: one is the calculation of 𝜔𝑝

𝑖𝑒


and 𝜔𝑝𝑒𝑝

and the other is the calculation of the parametersin feedback loop. They are analyzed, respectively, in thefollowing sections.

(1) The Influence Caused by Calculation Error of 𝜔𝑝𝑖𝑒and 𝜔𝑝

𝑒𝑝.

From (13) and (14), we can know that the calculation of 𝜔𝑝𝑖𝑒

and 𝜔𝑝𝑒𝑝

is related to the latitude error. Set 𝜑 as inaccuratelatitude, and 𝛿𝜑 = 𝜑 − 𝜑 as latitude error.

While there exits error in latitude, the value of 𝜔𝑝𝑖𝑒can be

calculated by

𝜔𝑝

𝑖𝑒= [0 𝜔

𝑖𝑒cos𝜑 𝜔

𝑖𝑒sin𝜑]𝑇. (30)

Therefore, the miscalculation of 𝜔𝑝𝑖𝑒can be got by sub-

tracting (13) from (30):

𝛿𝜔𝑝𝜑

𝑖𝑒=[[

[

0

𝜔𝑖𝑒(cos𝜑 − cos𝜑)

𝜔𝑖𝑒(sin𝜑 − sin𝜑)

]]

]

=

[[[[[

[

0

𝜔𝑖𝑒⋅ (−2 sin(𝜑 +

𝛿𝜑

2) sin 𝛿𝜑)

𝜔𝑖𝑒⋅ (2 cos(𝜑 +

𝛿𝜑

2) sin 𝛿𝜑)

]]]]]

]

.

(31)

While there exists an error, the calculation value of 𝜔𝑝𝑒𝑝is

𝜔𝑝

𝑒𝑝= [−

𝑉𝑁

𝑅

𝑉𝐸

𝑅

𝑉𝐸

𝑅 cos𝜑 ]𝑇

. (32)

The calculation error of 𝜔𝑝𝑒𝑝can be got by subtracting (14)

from (32):

𝛿𝜔𝑝𝜑

𝑒𝑝= [0 0

𝑉𝐸

𝑅 cos𝜑−

𝑉𝐸

𝑅 cos𝜑 ]𝑇

= [0 0𝑉𝐸

𝑅⋅ [−

2 sin (𝜑 + 𝛿𝜑/2) sin 𝛿𝜑cos𝜑 cos𝜑

]]

𝑇

.

(33)

Similar to the analysis of 𝛿𝜔𝑝V𝑒𝑝, 𝛿𝜔𝑝𝜑𝑒𝑝

and 𝛿𝜔𝑝𝜑𝑖𝑒

can betreated as equivalent gyro drift; then combining with (31) and(33), the gyro drift error can bewritten as the sumof 𝛿𝜔𝑝𝜑

𝑒𝑝and

𝛿𝜔𝑝𝜑

𝑖𝑒:

𝛿𝜔𝑝

𝑒𝑝+ 𝛿𝜔𝑝

𝑖𝑒=

[[[[[

[

0

𝜔𝑖𝑒sin 𝛿𝜑 ⋅ (−2 sin(𝜑 +

𝛿𝜑

2))

𝜔𝑖𝑒sin 𝛿𝜑 ⋅ (2 cos(𝜑 +

𝛿𝜑

2)) −

sin 𝛿𝜑𝑅

⋅ [2𝑉𝐸sin (𝜑 + 𝛿𝜑/2)cos𝜑 cos𝜑

]

]]]]]

]

. (34)

From the analysis of (23), as cos𝜑 is in the denominator,the error will be infinite in theory when the carrier is sailingin high latitudes. However, the gyrocompass alignment isapplied in mid or low latitudes, so this situation is out ofconsideration [20]. As the radius of the earth is very large,the carrier’s change in position can only lead to tiny changein latitude, and the latitude error 𝛿𝜑 is even fainter. Deadreckoning latitude error terms 𝜔

𝑖𝑒sin 𝛿𝜑 and sin 𝛿𝜑/𝑅 in (34)

can be neglected compared with the relatively larger error ofvelocity.

(2) The Influence of Corrected Angular Velocity. The correctedangle rate of gyrocompass alignment needs to be calculatedthrough the form of Figures 3 and 4. Therefore, the values ofparameters 𝐾

1, 𝐾2, 𝐾3, and 𝐾

4are needed to be determined.

Equation (4) in Section 2.2 shows that the azimuth loopparameters are calculated as follows:

𝐾1= 𝐾3= 2𝜉𝜔

𝑛,

𝐾2=𝑅𝜔2𝑛(1 + 𝜉2)

𝑔− 1,

𝐾 (𝑠) =𝑅𝜉2𝜔4𝑛

𝑔.

(35)

𝜉 and 𝜔𝑛are adjustable variables in system; 𝜔

𝑖𝑒and 𝑔 are

known values. When errors occurred in 𝜑, the accurate valueof parameter𝐾(𝑠) cannot be obtained, and the incorrect𝐾(𝑠)will affect the convergence speed of the system. However, asthe value of 𝛿𝜑 is small, this influence on the convergencespeed is weak.

3.2.3. The Effect of Acceleration Interference. The main accel-eration of carrier is caused by wind and waves in uniformstraight line motion. Assuming that acceleration caused bywaves is a sine periodic oscillation, and its value is 𝐴

𝑝=

𝐴 sin(𝜔𝑡+𝜑), then the velocity error can be got by integratingthe acceleration. It is a cosine periodic oscillation and itsvalue is 𝑉

𝑝= (𝐴/𝜔) cos𝜑 − (𝐴/𝜔) cos(𝜔𝑡 + 𝜑). For

the acceleration interference, there is harmful accelerationinterference caused by the velocity error except for 𝐴𝑝, andall the above factors can be equivalent to acceleration zerobiases ∇

𝐴𝑑𝑛and ∇

𝐴𝑑𝑒as described in Section 3.2.1:

∇𝐴𝑑= [

[

∇𝐴𝑑𝑒

∇𝐴𝑑𝑛

∇𝐴𝑑𝑢

]

]

= (2𝜔𝑛

𝑖𝑒+ 𝜔𝑛

𝑃) × 𝑉𝑛

𝑃


=

[[[[[[[[

[

𝐴𝑁+ 2𝜔𝑖𝑒sin𝜓 sin𝜑 ⋅ 𝑉

𝑃+𝑉2𝑝sin2𝜓 tan𝜑𝑅

𝐴𝐸− 2𝜔𝑖𝑒cos𝜓 sin𝜑 ⋅ 𝑉

𝑃−𝑉2𝑝sin𝜓 cos𝜓 tan𝜑

𝑅−𝑉2𝑝sin𝜓 cos𝜓𝑅

+ 2𝜔𝑖𝑒cos𝜑 cos𝜓 ⋅ 𝑉

𝑝+𝑉2𝑝sin2𝜓𝑅

]]]]]]]]

]

.

(36)

There are also equivalent gyro drifts in three directionsproduced by velocity error: 𝜀

𝐴𝑑𝑛, 𝜀𝐴𝑑𝑒

, and 𝜀𝐴𝑑𝑢

:

𝜀𝐴𝑑= [𝜀𝐴𝑑𝑒

𝜀𝐴𝑑𝑛

𝜀𝐴𝑑𝑢]𝑇

= 𝜔𝑛

𝑝

= [−𝑉𝑃cos𝜓𝑅

𝑉𝑃sin𝜓𝑅

𝑉𝑃sin𝜓 tan𝜑𝑅

]𝑇

.

(37)

The horizontal alignment is mainly affected by acceler-ation bias, and the azimuth alignment is mainly affected byeast gyro drift. At this time, the misalignment angle equationin the frequency domain can be written as

𝜙𝐴

𝑥(𝑠) =

− ((1 + 2𝜉2) 𝜔2𝑛/𝑔) (𝑠 + 𝜉𝜔

𝑛/ (1 + 2𝜉2))

(𝑠2 + 2𝜉𝜔𝑛𝑠 + 𝜔2𝑛) (𝑠 + 𝜉𝜔

𝑛)

⋅ ∇𝐴𝑑𝑛

(𝑠) ,

𝜙𝐴

𝑦(𝑠) =

((1 + 2𝜉2) 𝜔2𝑛/𝑔) (𝑠 + 𝜉𝜔

𝑛/ (1 + 2𝜉2))

(𝑠2 + 2𝜉𝜔𝑛𝑠 + 𝜔2𝑛) (𝑠 + 𝜉𝜔

𝑛)

⋅ ∇𝐴𝑑𝑒

(𝑠) ,

𝜙𝐴

𝑧(𝑠) =

𝜉2𝜔4𝑛/Ω cos𝜑

(𝑠2 + 2𝜉𝜔𝑛𝑠 + 𝜔2𝑛) (𝑠 + 𝜉𝜔

𝑛)2⋅ 𝜀𝐴𝑑𝑛

(𝑠) .

(38)

The acceleration 𝐴𝑃produced by waves is in form of

periodic oscillationwith small amplitude and high frequency.Its input frequency is generally limited in [𝜔

𝑛, +∞), so the

effect of 𝐴𝑃on misalignment angle can be greatly reduced

by lowering the value of 𝜔𝑛. The effect of uniform motion

interference acceleration can be suppressed by changing theparameters in the system, but the alignment time will also beincreased accordingly.

In this section the error of gyrocompass alignment inuniform straight line motion is analyzed from three aspects:latitude error, velocity error, and acceleration error.The influ-ence of velocity error and latitude error on the misalignmentangle is too weak to be considered, but the interferenceacceleration brought by swing and waves in motion has agreat influence on the misalignment angles. We can reducethis influence by changing the system parameters, but thealignment time will be increased accordingly.

4. A Rapid Implementation of DVL AidedGyrocompass Method in Alignment

In platform inertial navigation system (PINS), it is difficultfor the platform to revert to former states and adding anew control method again, whereas for SINS, assuming thatthe storage capacity of navigation computer is large andcomputing power is strong enough, it is feasible for thenavigation computer to make a storage of the sampling data

of SINS and calculate the data repeatedly with different kindsof algorithms. By using this kind of repeated calculationmethod, the increased alignment time caused by changes inparameters can be solved to some extent.

There exists a certain convergence in the gyrocompassalignment process, and it is themain factor to affect the align-ment time, so alignment time can be shortened by reducingthe convergent time or accomplishing the convergence inother processes. By calculating the data repeatedly, with theconvergence completed in this repeated calculation process,the original alignment process is shortened, which in turnreduces the alignment time although the overall convergenceprocess did not change.

From the analysis above, the conception and structure ofan improved rapid alignment algorithm is given as follows:the gyro and acceleration sampling data of SINS can beregarded as a group of time series. The traditional navigationprocess calculates this data series according to time order,and real-time navigation results can be got without the storedprocedure. For the same reason, if this data series is storedby navigation computer, they can be calculated backward toconduct the processing and analyzing procedure as well. Itis called data repeated calculation algorithm in this paper,and, by analyzing the sampling data forward and backwardrepeatedly, the accuracy is increased and the actual length ofthe analyzed data series is shortened, in return reduced thealignment time. The schematic diagram of the data repeatedcalculation algorithm is shown in Figure 8, in whichΔ𝑇 is thesampling period.

Attitude, velocity, and position calculation of the compassalignment algorithm for SINS are expressed in the followingdifferential equation:

�̇�𝑛

𝑏= 𝐶𝑛

𝑏Ω𝑛

𝑛𝑏, (39a)

V̇𝑛 = 𝐶𝑛𝑏𝑓𝑏

𝑖𝑏− (2𝜔

𝑛

𝑖𝑒+ 𝜔𝑛

𝑒𝑛) × V𝑛 + 𝑔𝑛, (39b)

�̇� =V𝑛𝑁

𝑅, �̇� =

V𝑛𝐸sec𝜑𝑅

. (39c)

Among them

Ω𝑏

𝑛𝑏= (𝜔𝑏

𝑛𝑏×) , 𝜔

𝑏

𝑛𝑏= 𝜔𝑏

𝑖𝑏− (𝐶𝑛

𝑏)𝑇

(𝜔𝑛

𝑖𝑒+ 𝜔𝑛

𝑒𝑛) ,

𝑔𝑛

= [0, 0, −𝑔]𝑇

,

(40a)

𝜔𝑛

𝑖𝑒= [0, 𝜔

𝑖𝑒cos𝜑, 𝜔

𝑖𝑒sin𝜑]𝑇,

𝜔𝑛

𝑒𝑛= [−

V𝑛𝑁

𝑅,V𝑛𝐸

𝑅,V𝑛𝐸tan𝜑𝑅

]

𝑇

.

(40b)

𝛿V𝑛 is obtained by compass circuit and 𝐶𝑛𝑏, V𝑛 =

[V𝑛𝐸, V𝑛𝑁, V𝑛𝑈]𝑇, 𝜑, and 𝜆 are inertial attitude matrix, speed,

latitude, and longitude, respectively.𝜔𝑏𝑖𝑏and𝑓𝑏𝑖𝑏aremeasuring

gyro angular velocity and measuring acceleration, respec-tively. 𝜔

𝑖𝑒and 𝑔 are the angle rate of the earth and the

local acceleration of gravity, respectively. 𝑅 is the radius ofearth. Operator 𝑖(∙×) is the antisymmetric matrix composedby ∙ vector. Assuming the sampling period of gyroscope


Normal compass alignment

Reversed compassalignment

Forward compassalignment

Save the sampled data

...

...

...

......

...

tk

tk1

tk2

tkn

tmt0

tk1 + ΔT

tk2 + ΔT

tkn + ΔT

tk + ΔT

Figure 8: Data repeated calculation alignment process diagram.

and accelerometer in SINS are both Δ𝑇, the differentialequations ((39a), (39b), and (39c)) are discrete recursionmethod suitable for computer calculating:

𝐶𝑛

𝑏𝑘= 𝐶𝑛

𝑏𝑘−1(𝐼 + Δ𝑇 ⋅ Ω

𝑏

𝑛𝑏𝑘) , (41a)

V𝑛𝑘= V𝑛𝑘−1

+ Δ𝑇 ⋅ [𝐶𝑛

𝑏𝑘−1𝑓𝑏

𝑖𝑏𝑘− (2𝜔

𝑛

𝑖𝑒𝑘−1+ 𝜔𝑛

𝑒𝑛𝑘−1) × V𝑛𝑘−1+𝑔𝑛

] ,

(41b)

𝜑𝑘= 𝜑𝑘−1

+Δ𝑇 ⋅ V𝑛

𝑁𝑘−1

𝑅,

𝜆𝑘= 𝜆𝑘−1

+Δ𝑇 ⋅ V𝑛

𝐸𝑘−1sec𝜑𝑘−1

𝑅.

(41c)

Among them

Ω𝑏

𝑛𝑏𝑘= (𝜔𝑏

𝑛𝑏𝑘×) ,

𝜔𝑏

𝑛𝑏𝑘= 𝜔𝑏

𝑖𝑏𝑘− (𝐶𝑛

𝑏𝑘−1)𝑇

(𝜔𝑛


𝑒𝑛𝑘−1+ 𝜔𝑛

𝑐𝑘−1) ,

𝜔𝑛

𝑖𝑒𝑘= [0, 𝜔

𝑖𝑒cos𝜑, 𝜔

𝑖𝑒sin𝜑𝑘]𝑇

,

(42a)

𝜔𝑛

𝑒𝑛𝑘= [−

V𝑛𝑁𝑘

𝑅,V𝑛𝐸𝑘

𝑅,V𝑛𝐸𝑘

tan𝜑𝑘

𝑅]

𝑇

(𝑘 = 1, 2, 3, . . .) .

(42b)

𝜔𝑛𝑐= 𝛿V𝑛/𝑅, 𝛿V𝑛 can be obtained by compass circuit.From the equations above, if we take the opposite value

of gyro output and the earth rotation angle rate of theforward navigation algorithm, set the initial value of thealgorithm as 𝐶𝑛

𝑏0= 𝐶𝑛𝑏𝑚, V̂𝑛0= −V𝑛

𝑚, 𝜑0= 𝜑𝑚, and

�̂�0= 𝜆𝑚, and calculate the sampling data repeatedly, the

repeated calculation algorithm can be achieved. It has thesame expression with the forward navigation calculation, andthe reversed navigation calculating process from 𝑡

𝑚(point B)

to 𝑡0(point A) can simply be got by using this algorithm.

Regardless of calculating error, attitude matrix and positioncoordinates are both equal at the same time of the data serieswhile the velocity has the same absolute value with oppositesign in forward and reversed calculation.

The reversed navigation algorithm of SINS is as follows:

𝐶𝑛

𝑏𝑘−1= 𝐶𝑛

𝑏𝑘(𝐼 + Δ𝑇 ⋅ Ω

𝑏

𝑛𝑏𝑘)−1

≈ 𝐶𝑛

𝑏𝑘(𝐼 − Δ𝑇 ⋅ Ω

𝑏

𝑛𝑏𝑘) ≈ 𝐶

𝑛

𝑏𝑘(𝐼 + Δ𝑇 ⋅ Ω̃

𝑏

𝑛𝑏𝑘−1) ,

(43a)

V𝑛𝑘−1

= V𝑛𝑘− Δ𝑇

⋅ [𝐶𝑛

𝑏𝑘−1𝑓𝑏

𝑖𝑏𝑘− (2𝜔

𝑛


𝑒𝑛𝑘−1) × V𝑛𝑘−1

+ 𝑔𝑛

]

≈ V𝑛𝑘− Δ𝑇 ⋅ [𝐶

𝑛

𝑏𝑘𝑓𝑏

𝑖𝑏𝑘−1− (2𝜔

𝑛

𝑖𝑒𝑘+ 𝜔𝑛

𝑒𝑛𝑘) × V𝑛𝑘+ 𝑔𝑛

] ,

(43b)

𝜑𝑘−1

= 𝜑𝑘−Δ𝑇 ⋅ V𝑛

𝑁𝑘−1

𝑅≈ 𝜑𝑘−Δ𝑇 ⋅ V𝑛

𝑁𝑘

𝑅, (43c)

𝜆𝑘−1

= 𝜆𝑘−Δ𝑇 ⋅ V𝑛

𝐸𝑘−1sec𝜑𝑘−1

𝑅≈ 𝜆𝑘−Δ𝑇 ⋅ V𝑛

𝐸𝑘sec𝜑𝑘

𝑅.

(43d)

Among them,

Ω̃𝑏

𝑛𝑏𝑘−1= (�̃�𝑏

𝑛𝑏𝑘−1×) ,

�̃�𝑏

𝑛𝑏𝑘−1= − [𝜔

𝑏

𝑖𝑏𝑘−1− (𝐶𝑛

𝑏𝑘)𝑇

(𝜔𝑛

𝑖𝑒𝑘+ 𝜔𝑛

𝑒𝑛𝑘+ 𝜔𝑛

𝑐𝑘)] .

(44)

5. Simulation

5.1. Simulation Experiment of Traditional Gyrocompass Align-mentMethod. Thecomparison of gyrocompass alignment onstatic base and moving base is given, respectively, as follows.

5.1.1. The Simulation Conditions. Simulation is proceeded atlatitude 𝜑 = 45.7796∘ and longitude 𝜆 = 126.6705∘ (Harbinarea); in order to make a better observation of effect inmotion, the triaxial gyro drift of SINS is set as 0.01∘/h andthe bias of accelerator is set as 0.0001 g. The parameters of


Table 1: Misalignments of gyrocompass alignment in differentconditions.

Eastern errorangle (∘)

Northern errorangle (∘)

Azimuth errorangle (∘)

In motion −4.16 × 10−3 −4.34 × 10−3 1.211Static base 0.22 × 10−3 0.21 × 10−3 −0.059

Time (min)0 10 20 30 40 50 60

−0.05

0

0.05

0.1

Easte

rn m

isalig

nmen

tan

gle (

∘ )

Figure 9: Comparison of the misalignment in east axis.

gyrocompass alignment are set as 𝜉 = 0.707 and 𝜔𝑛= 0.008;

it means that the alignment parameters configuration is

𝑘1= 𝑘2= 0.0113,

𝑘𝐸= 𝑘𝑁= 9.81 × 10

−6

,

𝑘𝑈= 4.1 × 10

−6

.

(45)

Assuming that the carrier’s speed is 10m/s and heading isalong 315∘, the swing and sway of sailing are set as sinusoidaloscillation form.The extent of pitch, roll, and yaw axis swingis set as 6∘, 8∘, and 5∘, and the periods are set as 8 s, 6 s, and10 s; the extent of surge, sway, and heave is set as 0.1m/s2, andthe periods are 5 s. Set the axis misalignment angles of coarsealignment as 0.1∘, 0.1∘, and 1∘, respectively.

5.1.2. The Simulation Results. The gyrocompass alignmentmethod is used both on static base and in uniform motion.After maintaining one hour of alignment process, the align-ment results in both conditions are compared and shownin Figures 6–8. The thick dash line represents gyrocompassalignment on static base and the thin solid line representsgyrocompass alignment in uniform motion.

The error curves in Figures 9, 10, and 11 indicate that thegyrocompass alignment has good performance on static base,but there is constant error caused by velocity which existsduring the alignment process while the ship is in motion.By choosing the mean value of alignment errors in twominutes before the alignment process ends, the results of bothconditions are recorded in Table 1.

In theory, three misalignment angles on static base canbe obtained by substituting the gyro drift and acceleratorzero bias into (5)–(7). The values are 0.209 × 10−3(∘), 0.223 ×10−3(∘), and 0.062(∘), respectively, which is nearly the same tothe simulation results. Then a conclusion can be drawn thatvelocity will cause a sharp alignment error. Substitute velocityinto (10)–(12); three misalignment angles can be obtained as

Time (min)

0 10 20 30 40 50 60−0.05

0

0.05

0.1

Nor

ther

n m

isalig

nmen

tan

gle (

∘ )

Figure 10: Comparison of the misalignment in north axis.

Time (min)0 10 20 30 40 50 60

Hea

ding

misa

lignm

ent

angl

e (∘ )

3

4

2

1

0

−1

Figure 11: Comparison of the heading misalignment.

0

−50

−100Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

0

−50

−100

−150

−20010−3 10−2 10−1 100

Frequency (rad/s)

wn = 0.02

wn = 0.01

X: 0.314

X: 0.314

X: 0.314

X: 0.314

Y: −60.5

Y: −72.53

Y: −176.3

Y: −172.5

Figure 12: BODE plot of north acceleration to east misalignment.

−4.2 × 10−3(∘), −4.5 × 10−3(∘), and 1.248(∘), that is nearly thesame with the simulation results as well.

On one hand, the simulation result proves the perfor-mance of gyrocompass alignment on static base; on theother hand, it also proves the validity of error analysisfor gyrocompass alignment on moving base discussed inSection 2.4.Therefore the error caused by carrier’s movementhas to be amended on moving base.

5.2. Simulation Experiment of DVL Aided GyrocompassAlignment on Moving Base

5.2.1.The Simulation Conditions. Simulation experiments areproceeded in Harbin area, where the latitude 𝜑 = 45.7796∘


−100

−200

100

50

0

0

−300

−40010−3 10−2 10−1 100

Mag

nitu

de (d

B)Ph

ase (

deg)

Bode diagram

Frequency (rad/s)

−50

−100

wn = 0.02

wn = 0.01

X: 0.314

X: 0.314

X: 0.314

X: 0.314

Y: −13.68

Y: −37.74

Y: −354.2

Y: −348.3

Figure 13: BODE plot of north acceleration to heading misalign-ment.

0 500 1000 1500 2000 2500 3000−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Misa

lignm

ent a

ngle

(deg

)

2050 2060 2070 2080

0

0.005

0.01

Time (s)

Time (s)

wn = 0.02

wn = 0.01

Misa

lignm

ent

angl

e (de

g)

−0.015

−0.01

−0.005

Figure 14: Curve of eastmisalignment caused by north acceleration.

and the longitude 𝜆 = 126.6705∘. Ignoring all the otherfactors, the ship is assumed to sail along northeast direction.Set the initial velocity and gyro drift as zero and there existsacceleration on the gyrocompass alignment when the shipis moving. The period of acceleration oscillation is 20 s andits value is 𝐴

𝑁= 𝐴𝐸= 0.2 sin(2𝜋 ⋅ 𝑡/20). Set the axis

misalignment angles of coarse alignment as 0.5∘, 0.5∘, and0.5∘, respectively.

5.2.2. The Simulation Results. We change 𝐾 value of thehorizontal loop and the azimuth loop in which the dampingratio is still 𝜉 = 0.8 while oscillation frequency is adjustedfrom 𝜔

𝑛= 0.02 to 𝜔

𝑛= 0.01. In horizontal loop, 𝐾

1=

0.0240, 𝐾2= 147.69, and 𝐾

3= 0.5217. In azimuth loop,

𝐾1= 𝐾3= 0.016, 𝐾

2= 105.9534, and 𝐾

4= 0.0042. The

whole simulation time is 3000 s and the simulation diagramsare drawn in Figures 12, 13, 14, and 15.

0

10

20

30

40

50

2140 2150 2160 2170

−0.05

0

0.05

0.1

0 500 1000 1500 2000 2500 3000

Time (s)

Time (s)

wn = 0.02

wn = 0.01

Misa

lignm

ent a

ngle

(deg

)

Misa

lignm

ent

angl

e (de

g)

−10

Figure 15: Curve of heading misalignment caused by north acceler-ation.

Figures 12–15 show the BODE figure and correspondingmisalignment curve before and after adjusting system param-eters. At frequency of 0.314 rad/sec (the corresponding periodis 20 s), the magnitude is reduced from −60.5 dB to −72.5 dBin horizontal loop (as shown in Figure 12). Correspondinglythe steady-state oscillation of misalignment is reduced from0.6 to around 0.18 (as shown in Figure 14). In azimuthloop, the magnification is reduced from −13.7 dB to −37.7 dB(as shown in Figure 13), and the steady-state oscillation ofmisalignment is reduced from 4.5 to around 0.5 (as shownin Figure 15). It can be seen that the influence of accelerationcan be effectively reduced by changing the parameters ofsystem appropriately.

However, as shown in Figures 12–15, alignment time willalso be prolonged accordingly. Therefore the data repeatedcalculation algorithm introduced in Section 4 is necessary,and it will efficiently shorten the alignment process.

6. Test Verification

6.1. Test Equipment Set-Up. To evaluate the performance ofgyrocompass alignment deeply, a sailing test was conductedin testing field on Tai Lake (Wuxi, China). The test wasconducted on a high-speed yacht platform equipped withseveral devices. It consists of anAHRS based on a fiber opticalgyro (FOG) produced by our own research center (similarto [12], we conducted precalibration process of AHRS tominimize the error), a high-precise FOG-INS system calledPHINS combined with GPS used as a reference system, anda DVL used to assist the AHRS system. Based on the headingand attitude information supplied by PHINS, the accuracy ofgyrocompass initial alignment onmoving basewas evaluated.The characteristics of AHRS are shown in Table 2, and theperformances of PHINS are shown in Table 3.

The high-speed yacht platform used in the test and theset-up of the equipment it carried are shown in Figure 16.


AHRS PHINSData collection

computer

UPS power

DVL

GPS

Figure 16: Yacht platform and the test equipment.

120.14

120.16

120.18

120.2

120.22

120.14

120.16

120.18

120.2

120.22

120.24

120.1231.15 31.2

31.25 31.3 31.35 31.4

31.25 31.3 31.35 31.4 31.45 31.5

Trajectory

Starting point3800 s to 5800 s

Finishing point

Long

itude

Figure 17: Test trajectory.

Table 2: The characteristics of AHRS.

Gyroscope AccelerometersBias-error 0.01∘/h Threshold ±5 × 10−5 gRandom walkcoefficient


−1

−0.5

−1.5

0.5

00.5

1

Pitc

h er

ror(

∘ )Ro

ll er

ror(∘)

1960 1970 1980 1990

−0.2

0

0.2

0.4

Method 1Method 2

Method 1Method 2

Method 1Method 2

−5

−4

−3

− 2

2

−1

1

0

0.2

0

−0.2

1990 1995 2000

Hea

ding

erro

r(∘ )

−10

−5

0

5

10

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

0 200 400 600 800 1000 1200 1400 1600 1800 2000

t (s)

Figure 19: The restrain curves of heading and attitude error.

to the gyrocompass alignment algorithm on static base,while Method 2 corresponds to the gyrocompass alignmentalgorithm on moving base).

As shown in Figure 19, the pitch and roll error of gyro-compass alignment algorithm on moving base are of slight

difference compared with algorithm on static base. But thereis a significant difference in performance of heading angleon moving base. The error of algorithm on moving base isreduced to 1.5∘ compared with that on static base. However, itstill can be improved. As introduced in Section 3.2.3, while


Table 3: The performance of PHINS.

Position accuracy (CEP50%3) Heading accuracy (1𝜎 value) Attitude accuracy (1𝜎 value)With stand-alone GPS aiding 5–15m With GPS aiding 0.01∘ secant latitude Roll and pitch error Less than 0.01∘

With differential GPS aiding 0.5–3mWith RTK differential GPS aiding 2–5m No aiding 0.05∘ secant latitudeNo aiding for 5min 20mPure internal mode 0.6NM/h

−0.2

−0.10

0.1

0.2

Time (s)3800 4000 4200 4400 4600 4800 5000 5200 5400 5600 5800

3800 4000 4200 4400 4600 4800 5000 5200 5400 5600 5800

024

6

8

Acce

lera

tion

(m)

Velo

city

(m)

Time (s2)

DVL velocity

DVL acceleration

Figure 20: The velocity and acceleration curves under sailingcondition.

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Frequency spectrum×10−4

Frequency (Hz)

Am

plitu

deF(j𝜔)

Figure 21: Frequency spectrogram of acceleration.

there exists a periodic interference acceleration in sailingcondition, it will produce a periodic oscillation to heading.Inhibition of acceleration with this periodic oscillation canenhance the alignment performance further.

The velocity and acceleration curves of the yacht (from3800 s to 5800 s) are shown in Figure 20.

In order to give a clear expression about how the acceler-ation affects the alignment system, a fast Fourier transform(FFT) is presented to the acceleration, and its frequencyspectrogram is given in Figure 21.

Method 2Method 3

Hea

ding

erro

r(∘ )

200 400 600 800 1000 1200 1400 1600 1800 2000

−4

−2

0

2

4

6

8

t (s)

Figure 22: Heading error restrain curves before and after reset.

Figure 21 provides a factor that there exists an oscillationperiod of 0.05Hz in acceleration, and it will be equivalentto instrument error, which will seriously affect the result ofheading alignment, so it is necessary to reset the alignmentparameter of the gyrocompass loop.

The convergence curves are compared in Figure 22(Method 2 corresponds to 𝜔

𝑛= 0.05, while Method 3

corresponds to 𝜔𝑛= 0.007).

Curves in Figure 22 prove that, after reset of parameters,initial alignment results are much better than the former,but the alignment time is significantly increased from about700 s to 1400 s. To shorten the prolonged time, data repeatedcalculation algorithm is used, and the result is shown inFigure 23. In the first 220 s, the coarse alignment process iscarried out which can decrease the error angle to certainrange quickly, but the precision cannot be guaranteed and theerror vibration that causes a valley at nearly 200 s in Figure 23is obvious. After 220 s, the proposed alignment method isimplemented after coarse alignment.

The chart in the middle represents the data repeated cal-culating process, which avoids the sampling step and almosttakes less than 1 s, so the alignment time is shortened to about650 s. After comparing the alignment curve in Figure 23 withthat in Figure 22, it can be known that the alignment timeis much shorter than before when the accuracy remainsunchanged.


0 200 400 600

0

2

4

6

8

10

12

0

2

4

6

8

10

12

0

2

4

6

8

10

12

600 400 200 0 200 400 600

Hea

ding

erro

r(∘ )

t (s) t (s)t (s)

Figure 23: Time of heading alignment after using the data repeated calculation method.

Table 4: Statistics of 4 methods.

Method 1 Method 2 Method 3 Method 4Roll error()

Mean 0.0316 0.0314 0.0313 0.0313Variance −0.1623 −0.1618 −0.1636 −0.1636

Pitch error()

Mean 0.0646 0.0646 0.0645 0.0645Variance 0.1107 0.1109 0.1105 0.1105

Yaw error(∘)

Mean −1.4517 −0.3507 −0.1327 −0.1327Variance 0.3149 0.1791 0.1079 0.1079

Alignment time (s) 700 700 1400 650

The experimental results of all four methods are puttogether in Table 4 in order to make a comparison. Method1 is the gyrocompass alignment method using the algorithmon static base. Method 2 is the gyrocompass alignmentmethod using the algorithm on moving base. Method 3is the gyrocompass alignment method on moving base inwhich the control parameters are reset and Method 4 isthe improvement of Method 3 after using the data repeatedcalculation algorithm. The mean and variance values inTable 4 are the statistics of error angle in the last 20 s ofalignment process (from 1980 s to 2000 s) comparing thealignment results with the standard value collected fromPHINS.

From data in Table 4, it can be drawn that Method 4has much better alignment results compared with the otherthree methods. That is to say, after using the gyrocompassalignment algorithm on moving base with resetting thesystem control parameters, the accuracy of initial alignmentis guaranteed and the alignment time is also in acceptablerange with the use of data repeated calculation algorithm.

7. Conclusion

Based on the principle analysis of classic platform initial gyro-compass alignment, a DVL aided gyrocompass alignmentmethod for SINS on moving base is proposed in this paper.The implementation of algorithm is given and the influenceof external velocity error is also analyzed. More specifically,two methods are adopted to cope with the gyrocompassalignment on moving base: first, an improved algorithm ofgyrocompass alignment for SINS on moving base aided withDVL is introduced; then, after the error analysis, the systemparameters are reset to decrease the acceleration interference.However, from results it turns out that alignment time istoo long to be accepted. Aiming at this problem, a datarepeated calculation algorithm is put forward to shorten theprolonged time. The simulation and experimental resultsverify the performance of the proposed alignment methodboth in accuracy and convergence time.

Abbreviations

𝑅 : Earth radius𝑔: Gravitational acceleration𝜔𝑖𝑒: Rotational angular velocity of the earth

𝜙𝑥, 𝜙𝑦, 𝜙𝑧: The east, north, and azimuthmisalignment angle of platform

𝜑: Latitude𝜆: Longitude𝑡: The geographical coordinates𝑛: The navigation coordinates𝑝: The platform coordinates𝑏: The body coordinates𝑖: The geocentric inertial coordinates𝑒: The earth coordinates


𝐶𝑝

𝑏: The transform matrix from the platform

frame 𝑝 to the SINS’s body frame 𝑏𝜔𝑏𝑝𝑏: Angular rate of the body frame with

respect to the platform frame𝜔𝑝𝑐: The control angular rate

𝜔𝑏𝑖𝑏: The angular rate of the body frame with

respect to the inertial frame𝜀: Gyroscopic drift∇: Accelerometer bias𝜉: The damping coefficient of the system𝜎: The system constant time𝜔𝑛: The system oscillation frequency

𝜙𝑠𝑥, 𝜙𝑠𝑦, 𝜙𝑠𝑧: The error angle caused by device error in

three directions𝜓: Heading angle𝑓𝑏: Specific force directly measured by the

IMU in the body frame𝑓𝑛: Specific force directly measured by the

IMU in the navigation frame𝐵𝑝: Bad acceleration𝜀𝑑𝑒, 𝜀𝑑𝑛, 𝜀𝑑𝑢: The equivalent gyro drift on uniform

motion in three directions∇𝑑𝑒, ∇𝑑𝑛, ∇𝑑𝑢: The equivalent accelerometer bias on

uniform motion in three directions𝑉𝑏dvl: The speed of body measured by DVL𝜙𝛿𝑉𝑥, 𝜙𝛿𝑉𝑦, 𝜙𝛿𝑉𝑧: The error angle caused in three direc-

tions by uniform motion𝜀𝐴𝑑𝑒

, 𝜀𝐴𝑑𝑛

, 𝜀𝐴𝑑𝑢

: The equivalent gyro drift in three direc-tions on accelerated movement

∇𝐴𝑑𝑛

, ∇𝐴𝑑𝑒

, ∇𝐴𝑑𝑢

: The equivalent accelerometer biason accelerated movement in threedirections

𝜙𝐴𝑥, 𝜙𝐴𝑦, 𝜙𝐴𝑧: The error angle caused by accelerated

movement in three directions.

Conflict of Interests

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

Acknowledgments

This study is supported in part by the National NaturalScience Foundation of China (Grant no. 61203225), theState Postdoctoral Science Foundation (2012M510083), andthe Central college Fundamental Research Special Fund(no. HEUCF110427). The authors would like to thank theanonymous reviewers for their constructive suggestions andinsightful comments.

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