effectofeastgyrodriftandinitialazimutherroronthecompass ...on compass azimuth alignment can converge...

Research ArticleEffect of EastGyroDrift and Initial AzimuthError on theCompassAzimuth Alignment Convergence Time

Dongxu He,1 Xinle Zang ,1 and Lei Ge1,2

1College of Automation, Harbin Engineering University, Harbin 150001, China2Beijing Institute of Computer Technology and Application, Beijing 100854, China

Correspondence should be addressed to Xinle Zang; [email protected]

Received 10 September 2019; Revised 14 February 2020; Accepted 17 April 2020; Published 14 May 2020

Academic Editor: Saeed Eftekhar Azam

Copyright © 2020 Dongxu He et al. 0is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0e effect of gyro constant drift and initial azimuth error on the convergence time of compass azimuth is analyzed in this article.Using our designed compass azimuth alignment system, we obtain the responses of gyro constant drift and initial azimuth error inthe frequency domain. 0e corresponding response function in the time domain is derived using the inverse Laplace transform,and its convergence time is then analyzed. 0e analysis results demonstrate that the convergence time of compass azimuthalignment is related to the second-order damping oscillation period, the gyro constant drift, and the initial azimuth error. In thisstudy, the error band is set to 0.01° to determine convergence. When the gyro drift is less than 0.05°/h, compass azimuth alignmentcan converge within 0.9 damping oscillation periods. When the initial azimuth error is less than 5°, compass azimuth alignmentcan converge within 1.4 damping oscillation periods. When both conditions are met, the initial error plays a major role inconvergence, while gyro drift has a smaller effect on convergence time. Finally, the validity of our method is verifiedusing simulations.

1. Introduction

Compass alignment is a typical initial alignment approachfor inertial navigation systems (INSs). It is based on theprinciple of the compass effect and adopts classic controltheory in the frequency domain to design a compass aligningcircuit. Compass alignment has the advantages of few pa-rameters, low computational complexity, and easy imple-mentation [1]. At present, studies on compass alignmenthave mainly focused on parameter settings, error analysis,compass alignment of large azimuth misalignment angle,and rotation modulation compass alignment.

In terms of parameter setting, in [2], the self-alignmenttechnique was investigated for the strapdown INS (SINS)under the swing state.0e horizontal and azimuth alignmentparameters were designed and optimized; they subsequentlyperformed well under different sea conditions, differentinitial attitude errors, and omnidirectional conditions. Zhuet al. [3] introduced an intelligent optimization algorithminto compass alignment and optimized its parameters using

a particle swarm optimization algorithm to improve theinitial alignment performance of the strapdown INS. In [4], amechanization scheme for gyrocompassing to an arbitraryattitude was proposed.

In terms of error analysis, Xu and Xao [5] studied theinitial alignment of a compass loop under a sailingturntable and analyzed the alignment error based on theequivalence of the device error. Zhang et al. [6] analyzedthe effect of random noise on compass azimuth alignmentand proposed an innovative method in the time domain.Moreover, a method based on the inverse attitude cal-culation proposed that the periodic oscillation error of thegyro output can cause additional oscillation errors incompass azimuth alignment [7]. Furthermore, the azi-muth error is amplified with increasing switching fre-quency. Ben et al. [8] described the effect of an outer leverarm on in-motion gyrocompass alignment, and a methodfor outer lever arm correction was provided to counteractthe outer lever arm effect on the performance of thealignment. In [9], a complete error covariance analysis for

HindawiMathematical Problems in EngineeringVolume 2020, Article ID 9042197, 10 pageshttps://doi.org/10.1155/2020/9042197

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strapdown inertial navigation system was presented, and,from this paper, it can be found that the cross-couplingterms in gyrocompass alignment errors can significantlyinfluence the system error propagation.

A considerable number of studies have been carriedout on compass alignment for a large azimuth mis-alignment angle. Abbas et al. [10] derived a nonlinearerror model of the SINS with a large azimuth misalign-ment and proposed the static base alignment of the SINSemploying simplified unscented Kalman filter (UKF) onthe nonlinear error model. Sun et al. [11] proposed a time-varying parameter compass azimuth alignment method,which did not require the assumption of a linear modelwith a small misalignment angle; it also improved thealignment speed of a large azimuth misalignment angle.He et al. [12] proposed a time-varying parameter compassalignment algorithm based on an optimal model and useda genetic algorithm to optimize the parameters of compassalignment for a large azimuth misalignment angle. In [13],a general nonlinear psi-angle approach for large mis-alignment errors that does not require coarse alignmentwas presented.

Owing to its rapid development in recent years, manyresearchers have also introduced rotation modulationtechnology [14] into the initial alignment of the SINS toeliminate the influence of the inertia device constant erroron the initial alignment. To eliminate the influence of theeast gyro drift on the azimuth alignment accuracy of theSINS, Yanling et al. [15] proposed a compass alignmentmethod suitable for a rotation modulation SINS based on ananalysis of the frequency characteristics of the compass. Liuet al. [16] introduced an azimuth rotating modulationmethod to classical compass alignment for SINS anddesigned an alignment method based on repeated datacalculation to improve the alignment accuracy with certainaccuracy sensors and eliminate the effect of the carrier’sattitude on alignment accuracy.

0us, although many studies have dealt with compassalignment, the convergence time has received minimalattention, despite being a relatively significant index forcompass alignment. 0is is because the system is expectedto (1) have a strong anti-interference ability to minimizethe random environmental interference and (2) be able toconverge within a limited initial alignment time. How-ever, these two requirements are always conflicting [17].Previous research designed a fourth-order compass azi-muth alignment control system and indicated that theconvergence time is related to the selected second-orderdamping oscillation period [17]. 0at study only analyzedthe classic second-order system, which is directly appliedto the fourth-order compass azimuth alignment system;however, the authors revealed clear similarities betweenthe second-order and fourth-order systems. A three-ordercompass alignment system was designed in [18], whichindicated that when selecting the corresponding param-eters, the system can converge within 30–50min. How-ever, a concrete analysis method has not yet beenprovided.

0erefore, this study analyzes the effect of east gyro driftand initial azimuth error on the convergence time based onthe fourth-order compass azimuth alignment system. Wepropose a novel method that converts the azimuth errorresponse from the frequency domain to the time domain andthen analyzes the convergence time in the time domain. Atheoretical reference is provided to set the correspondingparameters of compass azimuth alignment and to controlthe convergence time.

2. Gyrocompass Azimuth Alignment Principle

Compass azimuth alignment is a self-alignment methodbased on the compass effect and classic control theory.0e initial alignment is divided into two stages: hori-zontal leveling and azimuth alignment, where horizontalleveling is the basis of azimuth alignment. In general,horizontal alignment is rapid, precise, and simple,whereas azimuth alignment is problematic during thealignment process. Here, we briefly describe the principleof compass azimuth alignment, shown in Figure 1, whereωie is the angular velocity of earth’s rotation; L is the localgeographic latitude; g is the acceleration of gravity; ∇N isthe north accelerometer bias; εE is the east gyro drift,which affects the azimuth alignment accuracy; εU is the z-axis gyro drift, which generally has a smaller effect on theinitial alignment precision; δVN is the north velocityerror; ϕx is the pitch error; ϕz is the azimuth error; K1 andK2 are the designed parameters of the north horizontalloop; and K(s) � K3/[ωie cos L(s + K4)] is the control linkof the compass loop, where the input is δVN and theoutput is K(s)δVN, which replaces the command angularvelocity of vertical control. During the process ofcompass azimuth alignment, beginning from ϕz througheach link of the compass effect to output δVN and thenthrough the azimuth control link K(s), the output ϕz isadjusted.

According to the principle shown in Figure 1, the fourth-order system response is

ϕz(s) �1

ωie cos LsK3

Δ(s)∇Ns

+1

ωie cosLgK3

Δ(s)−εEs

+ ϕx(0)

+s s + K1( + ω2s K2 + 1( ( s + K4(

Δ(s)εUs

+ ϕz(0) ,

(1)

where ϕx(0) is the initial error of the east error angle,which is very small and has a minimal influence oncompass azimuth alignment after compass horizontalleveling, and ϕz(0) is the initial error of the azimutherror angle, which affects the convergence characteris-tics of compass azimuth alignment. 0us, this is theparameter that requires research. 0e east gyro drift εEalso affects the azimuth alignment accuracy. Δ(s) is thecharacteristic equation of the compass azimuth align-ment system:

2 Mathematical Problems in Engineering

Δ(s) � s4 + K1 + K4( s3

+ ω2s K2 + 1( + K1K4 s2

+ ω2s K2 + 1( K4s + gK3,(2)

where ωs ��g/R

is the Schuler frequency and K1, K2, K3,

and K4 are the parameters to be set [19].In general, a relatively mature parameter setting method

separates a fourth-order system into a series formed of twoidentical second-order systems. 0e characteristic root thenhas the following form [2]:

s1,2 � s3,4 � − σ ± jωd, (3)

where σ � ξωn is the attenuation coefficient; ξ is thedamping ratio; ωn is the undamped oscillation frequencyof the designed second-order system; ωd � 2π/Td is thedamping oscillation frequency; and Td is the dampingoscillation period of the second-order system. 0edamping ratio is generally set to ξ �

�2

√/2. 0erefore,

ωd � σ, with the corresponding parameters of K1 � K4 �2σ, K2 � 4σ2/ωs − 1, and K3 � 4σ4/g. For compass azimuthalignment, the other parameters are subsequently deter-mined only if Td is set.

According to the response function of ϕz, the output isinfluenced by five parameters. However, according to pre-vious research, ∇N, ϕx(0), and εU have a smaller impact oncompass azimuth alignment. 0eir orders of magnitude arealso small, so these parameters are not considered here. 0efocus of this study is analyzing the effects of east gyro driftand initial azimuth error on the convergence time ofcompass azimuth alignment.

3. Convergence Time Analysis of CompassAzimuth Alignment

3.1. Determination of Compass Azimuth Alignment.Generally, automatic control theory regards the controlledparameter in a certain error band as entering a steadysystem process, which means that the system converges.Meanwhile, the error band is generally assumed as 2% or5% of the steady value. However, this selection is notappropriate for the study of compass azimuth alignmentbecause the steady value of the effect of initial azimutherror on compass azimuth alignment is zero; thus, theerror band cannot be assumed to be a percentage of thesteady value. Additionally, determination of the azimuthconvergence should be comprehensively considered dur-ing initial alignment based on inertial device precision andazimuth angle accuracy. Hence, whether the azimuth angleenters the error band (the unit of this error band is angle) isused as a criterion for the convergence of the compassazimuth.

In this study, our analysis is based on the fiber opticgyroscope, and the gyro drift stability is restricted to0.05°/h. Based on the initial alignment error formula, theinitial alignment precision is constrained to 0.35° for alatitude of 53° north. 0en, an error band of 0.01° is usedwith a comprehensive consideration of the effect ofrandom error on the initial alignment, which is con-sidered to have converged for medium-accuracy inertialdevices. Certainly, during practical applications, thisconvergence determination may be adjusted according tothe requirements of the environment, inertial deviceprecision, and alignment accuracy. If the gyro driftstability is in the order of 0.001°/h, the error band can beup to 0.005°; however, if the gyro drift stability is in theorder of 0.1°/h, the error band can be reduced to 0.02° or0.03°.

3.2. Effect of Gyro Constant Drift on Convergence Time.According to Section 2, the system response term related toeast gyro drift is

ϕz1(s) �gK3

ωie cos LεE

s (s + σ)2 + ω2d 2. (4)

In order to examine the time characteristics, the re-sponse of the frequency domain is converted into that of thetime domain. 0us, we apply the inverse Laplace transformto ϕz1(s) and obtain

(1/( s + K1)) (K2 + 1)/R)) (1/s)

(1/s)

g

∆N δVN ϕxδV•

N

εE

ϕzεU

K (s) ωie cos L

× ×

×

––

Figure 1: Principle of compass azimuth alignment (see text forexplanation of symbols).

Mathematical Problems in Engineering 3

ϕz1(t) � −gK3εEωie cos L

1

σ2 + ω2d 2 +

2σ cos ωdt( 4ω2d σ2 + ω

2d

⎛⎝ ⎞⎠ −2ωd sin ωdt( 4ω2d σ2 + ω

2d

⎛⎝ ⎞⎠⎛⎝ ⎞⎠te− σt

+

2 σ2 − ω2d( cos ωdt(

4ω2d σ2 − ω2d

2+ 4σ2ω2d

−4σωd sin ωdt(

4ω2d σ2 − ω2d

2+ 4σ2ω2d

−2σ sin ωdt( 4 σ2 + ω2d ω

3d

−2ωd cos ωdt( 4 σ2 + ω2d ω

3d

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

e− σt

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (5)

According to the values of the corresponding parametersin Section 2, σ � ωd � (2π/Td) and gK3 � 4σ4. 0erefore,equation (5) can be simplified as follows:

ϕz1(t) � −4σ4εE

ωie cosL

14σ4

+2σ cos ωdt( 4σ2 σ2 + σ2( )

−2σ sin ωdt( 4σ2 σ2 + σ2( )

te− σt

+ −4σ2 sin ωdt( 4σ2 4σ2σ2( )

−2σ sin ωdt( 4 σ2 + σ2( )σ3

−2σ cos ωdt( 4 σ2 + σ2( )σ3

e− σt

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

� −4σ4εE

ωie cosL

14σ4

+cos ωdt(

4σ3 −

sin ωdt( 4σ3

te− σt

+ −sin ωdt(

2σ4−cos ωdt(

4σ4 e− σt

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

� −εE

ωie cosL

1 + σ cos ωdt( − σ sin ωdt( ( te− σt

+ − 2 sin ωdt( − cos ωdt( ( e− σt⎛⎝ ⎞⎠.

(6)

From equation (6), ϕz1(t) converges to − (εE/ωie cos L),which is the same as the formulae related to the effect of eastgyro drift on initial alignment. However, we need to considerwhen this convergence occurs.

Because the gyro drift stability is constrained to less than0.05°/h in this study, the gyro constant drift is generally lessthan 0.05°/h. 0e latitude is set to 53° and

εEωie cos L

� 0.32∘. (7)

Consider the four decay oscillation error terms inequation (7):

Δϕz1(t)

� 0.32°(σ cos(ωt) − σ sin(ωt))te− σt

+(− 2 sin(ωt) − cos(ωt))e− σt

� 0.32°�2

√σt cos ωt + φ1( e

− σt

+�5

√cos ωt + φ2( e

− σt

� 0.32°��

2(σt)2 + 5

cos ωt + φ3( e− σt

≤ 0.32°��

2(σt)2 + 5

e− σt

.

(8)

0en, taking the 0.01° error band to determine whetherthe azimuth alignment converges, we obtain


Δϕz1(t)≤ 0.32°��

2(σt)2 + 5

e− σt

� 0.01∘. (9)

0en, σt � 5.57 can be calculated, so t � 5.57/σ �5.57Td/2π ≈ 0.9Td, which indicates that the compass azi-muth alignment converges to the 0.01° error band afterapproximately 0.9 damping oscillation periods.

0erefore, we conclude that the effect of east gyro drifton compass azimuth alignment can converge within 0.9damped oscillation periods. However, the analysis here is tooconservative; as the gyro precision is improved, its constantdrift will be lower and its convergence time will be reduced.Table 1 lists the convergence time of several typical gyroconstant drifts.

According to Table 1, the convergence time of compassazimuth alignment is related to the selected second-orderdamping oscillation period and the gyro constant drift.When the gyro constant drift is determined, there is a fixed-proportion relationship between the convergence time andthe second-order damping oscillation period. When thesecond-order damping oscillation period is determined, alarger gyro constant drift results in a longer convergencetime, and vice versa.

It should be noted that, due to adoption of the inequalityamplification in the theoretical calculation of convergencetime, the actual convergence time is often less than thecalculated theoretical time. In other words, the convergencetime given here is more conservative and denotes themaximum time that the system takes to stabilize.

3.3. Effect of Initial Azimuth Error on Convergence Time.0is section mainly analyzes the influence of initial azimutherror on convergence time. According to Section 2, thesystem response term related to the initial azimuth error is

ϕz2(s) �s s + K1( + ω2s K2 + 1( ( s + K4(

Δ(s)ϕz(0). (10)

According to the values of the corresponding parametersin Section 2,

ϕz2(s) �s3 + 4σs2 + 8σs + 8σ

(s + σ)2 + ωd( 2

2 ϕz(0). (11)

By performing the inverse Laplace transform, the ob-tained function in the time domain is

ϕz2(t) � ϕz(0)t

23σ2 − ω2d ω

2d +

12σ 3σ2 + ω2d

e− σt sin ωdt( ω3d

+ ω2d −t

2σ 3σ2 − ω2d

e− σt cos ωdt( ω2d

. (12)

Because ωd � σ,

ϕz2(t) � ϕz(0) (σt + 2)e− σt sin ωdt(

+(− σt + 1)e− σt cos ωdt( .(13)

In equation (13), ϕz2(t) eventually converges to 0;however, the target of this research is determining when theconvergence occurs.

0us, due to

(σt + 2)e− σt sin(ωt) +(− σt + 1)e− σt cos(ωt)

�

��

2(σt)2 + 2(σt) + 5

cos ωt + φ1( e− σt

≤��

2(σt)2 + 2(σt) + 5

e− σt

,

(14)

the following is true:

ϕz2(t)

≤ ϕz(0)��

2(σt)2 + 2(σt) + 5

e− σt

. (15)

0en, using an error band of 0.01° to determine the initialalignment convergence,

ϕz2(t)

≤ϕz(0)��

2(σt)2 + 2(σt) + 5

e− σt

� 0.01∘. (16)

In equation (16), ϕz2(t) is proportionate to the initialazimuth error ϕz(0), so its convergence time is also relatedto the initial azimuth error. Furthermore, the greater thevalue of ϕz(0), the longer the convergence time.

However, before compass alignment, coarse alignment isgenerally required to guarantee the linear characteristic of

the compass azimuth alignment error model. When theinitial azimuth error is within 5°, the inertial error model hasbetter linearity; the initial alignment performs well forcompass azimuth alignment. When the initial azimuth erroris more than 5°, the inertial error model has inferior linearity;the performance of compass azimuth alignment graduallydecreases. 0erefore, in practical applications, the coarsealignment error is always controlled within 5°. In fact, thissection only considers a convergence time of the initialazimuth error within 5°. Based on the typical initial errorslisted above, the convergence time of compass azimuthalignment is shown in Table 2.

We conclude from Table 2 that the convergence timeof compass azimuth alignment is related to the selectedsecond-order oscillation period and the initial azimutherror. When the initial azimuth error is determined, theconvergence time is fixed in proportion to the second-order oscillation period. When the second-order oscil-lation period is determined, the greater the initial azimutherror, the longer the convergence time. As in Section 3,part B, the convergence times listed in Table 2 are rela-tively conservative, and the actual convergence time isgenerally less than the calculated value.

Due to adoption of the inequality amplification in thetheoretical calculation of convergence time, the actualconvergence time is often less than the calculated theoreticaltime. In other words, the convergence time given here ismore conservative and denotes the maximum time that thesystem takes to stabilize.


3.4. Combined Effect of Both Errors on Convergence Time.During the actual initial alignment of INS, constant drift andan initial azimuth error both exist.0erefore, it is necessary toanalyze the influence of both errors on the convergence timeto provide theoretical guidance for parameter setting inpractical applications.

0e gyro constant drift and initial azimuth error aremutuallyindependent. Based on automatic control theory, both responsesobtained by the transfer function of compass azimuth alignmentcan exhibit linear superposition. So, under both errors, the re-sponse function of the compass azimuth alignment error is

ϕz3(t) � ϕz1(t) + ϕz2(t)

� −εE

ωie cos L

1 + σ cos ωdt( − σ sin ωdt( ( te− σt

+ − 2 sin ωdt( − cos ωdt( ( e− σt⎛⎝ ⎞⎠

+ ϕz(0) (σt + 2)e− σt sin ωdt(

+(− σt + 1)e− σt cos ωdt( .

(17)

In Equation 17, when both errors exist, ϕz3(t) converges to− (εE/ωie cos L), for which the error decay oscillation term is

Δϕz3(t) � −εE

ωie cos Lσ cos ωdt( − σ sin ωdt( ( te

− σt

+ − 2 sin ωdt( − cos ωdt( ( e− σt

+ ϕz(0) (σt + 2)e− σt sin ωdt(

+(− σt + 1)e− σt cos ωdt(

� ϕz(0) +εE

ωie cos L (σt + 2)sin ωdt( (

+(− σt + 1)cos ωdt( e− σt

.

(18)

0erefore,

Δϕz3(t)

� ϕz(0) +εE

ωie cos L

(σt + 2)sin ωdt(

+(− σt + 1)cos ωdt( e

− σt

≤ ϕz(0) +εE

ωie cosL

��

2(σt)2 + 2(σt) + 5

e− σt

,

(19)

Taking the 0.01° error band as the criteria of initialalignment convergence,

Δϕz3(t)

≤ ϕz(0) +εE

ωie cos L

��

2(σt)2 + 2σt + 5

e− σt

� 0.01∘.

(20)

From equation (20), |Δϕz3(t)| is proportional to|ϕz(0) + (εE/ωie cos L)|, so the convergence time is associ-ated with the initial azimuth angle. When the absolute valuesof ϕz(0) and εE are unchanged, |Δϕz3(t)| under oppositesigns of ϕz(0) and εE is less than that under the same signs.As we perform the most conservative estimation in ourconvergence time analysis, both signs are the same. Typicalconvergence times of the initial azimuth error and gyroconstant drift are listed in Table 3.

By comparing Tables 2 and 3, we see that, relative to theinitial azimuth error, the gyro constant drift has a minimalinfluence on the convergence time, whereas the main factorinfluencing compass azimuth alignment is the initial azi-muth error. When this error is within 5°, compass azimuthalignment will converge within 1.41 Td.

4. Simulation Verification

4.1. Simulation of the Effect of Gyro Constant on ConvergenceTime. We assume that the reference coordinate system is theeast-north-up coordinate system and the local latitude is 53°.Only the X-axis gyro has a constant drift of 0.05°/h. 0e INSattitudes are 0°, 0°, and 0°. 0e initial attitude errors are all 0°.Td is set to 200 s, 300 s, 400 s, and 500 s, respectively. 0esimulation time is 600 s. 0e obtained convergence curve ofthe initial alignment error is shown in Figure 2 for differentconvergence times.

According to the related theory of initial alignment(formula (7)), at this time, the initial alignment error limit is− 0.3165°. 0erefore, according to Figure 2 and the definitionof the 0.01° error band used in this study, we assume that thecompass azimuth alignment converges when the initialalignment error curve finally passes through − 0.3165°. 0us,the convergence times for different Td are shown in Table 4.

According to Table 4, although the convergence time isdifferent for different Td, the ratio of convergence time to Tdcoincides well with the theoretical analysis, which verifies thevalidity of our proposed analytical method.

Due to limited space, the convergence curves of othergyro drifts are not presented here. For gyro drifts of 0.01°/h,0.02°/h, 0.03°/h, and 0.04°/h, the Td convergence times are200 s, 300 s, 400 s, and 500 s, respectively.0erefore, the ratioof convergence time to Td is shown in Table 5.

It can be seen from Table 5 that when the gyro drift isover 0.02°/h, it exhibits good agreement with the theoreticalcalculation. Moreover, the actual convergence time is lessthan the theoretical convergence time with a gyro drift of0.02°/h and 0.01°/h. 0is is because the amplification ofinequality is adopted during the process of theoreticalderivation, resulting in an overconservative convergencetime in the theoretical calculation. Despite this, the analysismethod of this study is generally valid.

Table 1: Convergence time of different gyro constant drifts.

Gyro drift (°/h) 0.05 0.04 0.03 0.02 0.01Convergence time (Td) 0.9 0.85 0.79 0.7 0.58

Table 2: Convergence time of different initial azimuth misalignment.

Initial azimuth misalignment (°) 0.5 1 2 3 4Convergence time (Td) 1 1.1 1.25 1.3 1.35


4.2. Simulation of the Effect of Initial Azimuth Error onConvergence Time. We assume that there is no inertialdevice error, the initial azimuth error is 5°, Td is set to 200 s,300 s, 400 s, and 500 s, respectively, the simulation time is1000 s, and the attitude is 0°, 0°, and 0°. 0e resulting sim-ulation results are shown in Figure 3 and Table 6.

According to Table 6, although the convergence time isdifferent for different Td, the ratio of convergence time to Tdis consistent with the theoretical analysis, which verifies thevalidity of our proposed analytical method. Due to limitedspace, the convergence curves of other gyro drifts are notshown here. We assume that the initial azimuth errors are 4°,3°, 2°, 1°, and 0.5° and Td is 200 s, 300 s, 400 s, and 500 s,respectively. Table 7 lists the convergence times and ratios ofconvergence time to Td.

It can be seen from Table 7 that when the initial error ismore than 2°, it agrees almost perfectly with the theoreticalcalculation results. When the initial error is less than 2°, theactual convergence time is less than the theoretical calcu-lation time.0is phenomenon, explained briefly in Section 4,part A, is due to the inequality amplification in the theo-retical analysis.

4.3. Simulation of the Effect of Both Errors on ConvergenceTime. We assume that there is no error of inertial device, theinitial azimuth error is 5°, the x-axis gyro constant drift is0.05°/h, Td is equal to 200 s, 300 s, 400 s, and 500 s, thesimulation time is 1000 s, and the attitude is 0°, 0°, and 0°.0eresulting simulation results are shown in Figure 4 andTable 8.

Due to limited space, other convergence curves are notshown here. If the initial azimuth errors are 4°, 3°, 2°, 1°, and0.5° and the gyro constant drift is 0.05°/h, Td is 200 s, 300 s,400 s, and 500 s, respectively. 0e convergence times andratios of convergence time to Td are listed in Table 9. When

Table 3: Convergence time (Td) of initial azimuth error and gyroconstant drift.

Initial azimuth error (°) 0.5 1 2 3 4 50.01°/h 1.00 1.12 1.25 1.31 1.35 1.400.02°/h 1.01 1.13 1.25 1.31 1.36 1.400.03°/h 1.03 1.14 1.26 1.32 1.37 1.400.04°/h 1.05 1.15 1.26 1.32 1.37 1.410.05°/h 1.07 1.16 1.26 1.33 1.38 1.41

0 100 200 300 400 500 600–0.35

–0.3

–0.25

–0.2

–0.15

–0.1

–0.05

0

Second

Deg

ree

Td = 200Td = 300

Td = 400Td = 500

Figure 2: Convergence curves of the initial alignment error fordifferent Td.

Table 4: Convergence times for different Td with a 0.05° gyroconstant drift.

Td (s) 200 300 400 500Convergence time (s) 177 265 354 443Ratio 0.89 0.88 0.89 0.89

Table 5: Convergence times for different Td and different gyroconstant drifts.

Gyro drift (°/h) Td (s) 200 300 400 500

0.04 Convergence time (s) 170 255 341 426Ratio 0.85 0.85 0.85 0.85




Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

–0.01

0

0.01

0.02

0.03

0.04

0.05

Td = 200Td = 300

Td = 400Td = 500

Figure 3: Convergence curves of the initial azimuth error fordifferent Td.

Table 6: Convergence times for different Td with a 5° initial azi-muth misalignment.



the gyro drift is set to 0.05°/h, there is a small difference ofconvergence time for different initial errors. All convergencetimes are proportionally related to Td. 0is verifies thevalidity of our proposed analysis method.

5. Experiment

In order to test the effect of east gyro drift and initial azimutherror on the compass azimuth alignment convergence time,we implemented three sets of actual ship experiments inHarbin, China. 0e main equipment includes the self-madestrapdown INS and a high-precision inertial navigationsystem PHINS. We used the data output by PHINS as the

reference value. In the experiment, the constant gyro driftsand the accelerometer biases were set to 0.05 deg/h and0.0001 g, respectively.

0e experiment procedure was as follows. 0e initialazimuth errors are 5° and 1°, and two sets of experimentswere performed based on different azimuth errors. In eachset of experiments, Td is equal to 200 s, 300 s, 400 s, and 500 s.0e experiment results are shown in Table 10 and Figure 5.

Obviously, the experimental results and the simulationresults are basically the same. When the east gyro drift andthe initial azimuth error are considered, the initial azimutherror plays a major role in the convergence time, comparedto the gyro constant drift.

Table 7: Convergence times for different Td and different initial azimuth misalignment.

Gyro drift (°) Td (s) 200 300 400 500

4 Convergence time (s) 270 405 541 676Ratio 1.35 1.35 1.35 1.35





Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

Td = 200Td = 300

Td = 400Td = 500

–0.35

–0.34

–0.33

–0.32

–0.31

–0.3

–0.29

–0.28

Figure 4: Convergence curves for both errors for different Td.

Table 8: Convergence times for different Td with a 0.05°/h gyro drift and a 5° initial azimuth misalignment.



6. Conclusion

In compass azimuth alignment, precision conflicts withrapidity. Within a limited initial alignment time, the ex-pected random disturbance is filtered as much as possible,and compass azimuth alignment is required to converge.0us, it is necessary to analyze the convergence time ofcompass azimuth alignment and determine optimum pa-rameters based on the precision of inertial devices and areasonable selection of corresponding parameters. In thisarticle, by analyzing the system transfer function of compassazimuth alignment, we obtain the response function of eastgyro drift and initial azimuth error in the frequency domain,

which are then transformed to the time-domain responsefunction by the inverse Laplace transform. 0erefore, weanalyze the effect of east gyro drift and initial azimuth erroron the convergence time of compass azimuth alignment inthe time domain. Our analytical results indicate that con-vergence time is related to gyro drift, initial azimuth error,and the second-order damping oscillation period. When anerror band of 0.01° is used to determine the convergence andthe gyro drift is less than 0.05°/h, the compass azimuthalignment will converge within 0.9 damping oscillationperiods due to gyro drift. When the initial azimuth error isless than 5°, the compass azimuth alignment will convergewithin 1.4 damping oscillation periods due to the initial

Table 9: Convergence times for different Td and different initial azimuth misalignments.

Initial azimuth error (°) Td (s) 200 300 400 500





0.5 Convergence time (s) 200 300 400 500Ratio 1 1 1 1D

egre

e

–0.1

0

0.1

0.2

0.3

0.4

0.5

1000 2000 3000 4000 5000 60000Second

Td = 200Td = 300

Td = 400Td = 500

Figure 5: Convergence curves of the initial azimuth error 1° for different Td.

Table 10: Experiment results.

Initial azimuth error (°) Td (s) 200 300 400 500




azimuth error. When both errors are considered, the initialazimuth error plays a major role in the convergence time,compared to the gyro constant drift. Our proposed methodprovides a theoretical basis for setting the correspondingparameters and controlling the convergence time duringcompass azimuth alignment.

Data Availability

0e data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper.

Acknowledgments

0is work was supported by the Fundamental ResearchFunds for the Central University under GrantHEUCF180403, the Applied Technology Research andDevelopment Project of Harbin under Grant2017RAQXJ042, and the Autonomous Navigation 0eoryand Key Technology for Deep Sea Space Station in PolarRegion Project under Grant 61633008.

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