EffectofEastGyroDriftandInitialAzimuthErrorontheCompass ...downloads.hindawi.com/journals/mpe/2020/9042197.pdf · Δ(s) s4 +K1 +K 4 s 3 +ω2 h s K 2 +1 +K 1K 4is 2 +ω2 s K 2 +1 K

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

Dongxu He1 Xinle Zang 1 and Lei Ge12

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 zangxinle_heu163com

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

Academic Editor Saeed Eftekhar Azam

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

0e effect of gyro constant drift and initial azimuth error on the convergence time of compass azimuth is analyzed in this articleUsing 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 transformand 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 001deg to determine convergence When the gyro drift is less than 005degh compass azimuth alignmentcan converge within 09 damping oscillation periods When the initial azimuth error is less than 5deg compass azimuth alignmentcan converge within 14 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 analysiscompass alignment of large azimuth misalignment angleand rotation modulation compass alignment

In terms of parameter setting in [2] the self-alignmenttechnique was investigated for the strapdown INS (SINS)under the swing state0e 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 domainMoreover 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

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strapdown inertial navigation system was presented andfrom 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 methodwhich did not require the assumption of a linear modelwith a small misalignment angle it also improved thealignment speed of a large azimuth misalignment angleHe 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 carrierrsquosattitude 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 systemhowever 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 30ndash50min 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 theory0e initial alignment is divided into two stages hori-zontal leveling and azimuth alignment where horizontalleveling is the basis of azimuth alignment In generalhorizontal alignment is rapid precise and simplewhereas 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 earthrsquos rotation L is the localgeographic latitude g is the acceleration of gravity nablaN isthe north accelerometer bias εE is the east gyro driftwhich 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 L

sK3

Δ(s)

nablaN

s1113876 1113877 +

1ωie cosL

gK3

Δ(s)minusεE

s+ ϕx(0)1113876 1113877

+s s + K1( 1113857 + ω2

s K2 + 1( 1113857( 1113857 s + K4( 1113857

Δ(s)

εU

s+ ϕz(0)1113876 1113877

(1)

where ϕx(0) is the initial error of the east error anglewhich 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 εE

also 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( 1113857s3

+ ω2s K2 + 1( 1113857 + K1K41113960 1113961s

2

+ ω2s K2 + 1( 1113857K4s + gK3

(2)

where ωs gR

1113968is 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]

s12 s34 minus σ plusmn 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 2πTd is thedamping oscillation frequency and Td is the dampingoscillation period of the second-order system 0edamping ratio is generally set to ξ

2

radic2 0erefore

ωd σ with the corresponding parameters of K1 K4

2σ K2 4σ2ωs minus 1 and K3 4σ4g 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 nablaN 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

31 Determination of Compass Azimuth AlignmentGenerally automatic control theory regards the controlledparameter in a certain error band as entering a steadysystem process which means that the system convergesMeanwhile 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 to005degh Based on the initial alignment error formula theinitial alignment precision is constrained to 035deg for alatitude of 53deg north 0en an error band of 001deg 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 0001degh the error band can beup to 0005deg however if the gyro drift stability is in theorder of 01degh the error band can be reduced to 002deg or003deg

32 Effect of Gyro Constant Drift on Convergence TimeAccording to Section 2 the system response term related toeast gyro drift is

ϕz1(s) gK3

ωie cos L

εE

s (s + σ)2 + ω2d1113872 1113873

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)) (1s)

(1s)

g

∆N δVN ϕxδVbull

N

εE

ϕzεU

K (s) ωie cos L

times times

times

ndashndash

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

Mathematical Problems in Engineering 3

ϕz1(t) minusgK3εE

ωie cos L

1

σ2 + ω2d1113872 1113873

2 +2σ cos ωdt( 1113857

4ω2d σ2 + ω2

d1113872 1113873⎛⎝ ⎞⎠ minus

2ωd sin ωdt( 1113857

4ω2d σ2 + ω2

d1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠te

minus σt

+

2 σ2 minus ω2d( 1113857cos ωdt( 1113857

4ω2d σ2 minus ω2

d1113872 11138732

+ 4σ2ω2d1113874 1113875

minus4σωd sin ωdt( 1113857

4ω2d σ2 minus ω2

d1113872 11138732

+ 4σ2ω2d1113874 1113875

minus2σ sin ωdt( 1113857

4 σ2 + ω2d1113872 1113873ω3

d

minus2ωd cos ωdt( 1113857

4 σ2 + ω2d1113872 1113873ω3

d

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

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

eminus σt

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

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

(5)

According to the values of the corresponding parametersin Section 2 σ ωd (2πTd) and gK3 4σ4 0ereforeequation (5) can be simplified as follows

ϕz1(t) minus4σ4εE

ωie cosL

14σ4

+2σ cos ωdt( 1113857

4σ2 σ2 + σ2( )1113888 1113889 minus

2σ sin ωdt( 1113857

4σ2 σ2 + σ2( )1113888 11138891113888 1113889te

minus σt

+ minus4σ2 sin ωdt( 1113857

4σ2 4σ2σ2( )minus

2σ sin ωdt( 1113857

4 σ2 + σ2( )σ3minus2σ cos ωdt( 1113857

4 σ2 + σ2( )σ31113888 1113889eminus σt

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus4σ4εE

ωie cosL

14σ4

+cos ωdt( 1113857

4σ31113888 1113889 minus

sin ωdt( 1113857

4σ31113888 11138891113888 1113889te

minus σt

+ minussin ωdt( 1113857

2σ4minuscos ωdt( 1113857

4σ41113888 1113889eminus σt

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minusεE

ωie cosL

1 + σ cos ωdt( 1113857 minus σ sin ωdt( 1113857( 1113857teminus σt

+ minus 2 sin ωdt( 1113857 minus cos ωdt( 1113857( 1113857eminus σt

⎛⎝ ⎞⎠

(6)

From equation (6) ϕz1(t) converges to minus (εEωie 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 than005degh in this study the gyro constant drift is generally lessthan 005degh 0e latitude is set to 53deg and

εE

ωie cos L 032∘ (7)

Consider the four decay oscillation error terms inequation (7)

Δϕz1(t)1113868111386811138681113868

1113868111386811138681113868 032deg(σ cos(ωt) minus σ sin(ωt))teminus σt

1113868111386811138681113868

+(minus 2 sin(ωt) minus cos(ωt))eminus σt

1113868111386811138681113868

032deg2

radicσt cos ωt + φ1( 1113857e

minus σt1113868111386811138681113868

+5

radiccos ωt + φ2( 1113857e

minus σt1113868111386811138681113868

032deg

2(σt)2 + 51113969

cos ωt + φ3( 11138571113874 1113875eminus σt

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le 032deg

2(σt)2 + 51113969

eminus σt

(8)

0en taking the 001deg error band to determine whetherthe azimuth alignment converges we obtain


Δϕz1(t)le 032deg

2(σt)2 + 51113969

1113874 1113875eminus σt

001∘ (9)

0en σt 557 can be calculated so t 557σ

557Td2π asymp 09Td which indicates that the compass azi-muth alignment converges to the 001deg error band afterapproximately 09 damping oscillation periods

0erefore we conclude that the effect of east gyro drifton compass azimuth alignment can converge within 09damped 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 reducedTable 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 driftWhen 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

33 Effect of Initial Azimuth Error on Convergence Time0is 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( 1113857 + ω2

s K2 + 1( 1113857( 1113857 s + K4( 1113857

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( 11138572

1113872 11138732 ϕz(0) (11)

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

ϕz2(t) ϕz(0)t

23σ2 minus ω2

d1113872 1113873ω2d +

12σ 3σ2 + ω2

d1113872 11138731113874 1113875eminus σt sin ωdt( 1113857

ω3d

1113888 1113889 + ω2d minus

t

2σ 3σ2 minus ω2

d1113872 11138731113874 1113875eminus σt cos ωdt( 1113857

ω2d

1113888 11138891113888 1113889 (12)

Because ωd σ

ϕz2(t) ϕz(0) (σt + 2)eminus σt sin ωdt( 11138571113872

+(minus σt + 1)eminus σt cos ωdt( 11138571113873

(13)

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

0us due to

(σt + 2)eminus σt sin(ωt) +(minus σt + 1)e

minus σt cos(ωt)1113868111386811138681113868

1113868111386811138681113868

2(σt)2 + 2(σt) + 51113969

cos ωt + φ1( 1113857eminus σt

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le

2(σt)2 + 2(σt) + 51113969

eminus σt

(14)

the following is true

ϕz2(t)1113868111386811138681113868

1113868111386811138681113868le ϕz(0)

2(σt)2 + 2(σt) + 51113969

eminus σt

(15)

0en using an error band of 001deg to determine the initialalignment convergence

ϕz2(t)1113868111386811138681113868

1113868111386811138681113868leϕz(0)

2(σt)2 + 2(σt) + 51113969

eminus σt

001∘ (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 5deg the inertial error model hasbetter linearity the initial alignment performs well forcompass azimuth alignment When the initial azimuth erroris more than 5deg the inertial error model has inferior linearitythe performance of compass azimuth alignment graduallydecreases 0erefore in practical applications the coarsealignment error is always controlled within 5deg In fact thissection only considers a convergence time of the initialazimuth error within 5deg 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 3part 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


34 Combined Effect of Both Errors on Convergence TimeDuring the actual initial alignment of INS constant drift andan initial azimuth error both exist0erefore 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)

minusεE

ωie cos L



⎛⎝ ⎞⎠

+ ϕz(0) (σt + 2)eminus σt sin ωdt( 11138571113872


(17)

In Equation 17 when both errors exist ϕz3(t) converges tominus (εEωie cos L) for which the error decay oscillation term is

Δϕz3(t) minusεE

ωie cos Lσ cos ωdt( 1113857 minus σ sin ωdt( 1113857( 1113857te

minus σt1113872


1113873



ϕz(0) +εE

ωie cos L1113888 1113889 (σt + 2)sin ωdt( 1113857(

+(minus σt + 1)cos ωdt( 11138571113857eminus σt

(18)

0erefore

Δϕz3(t)1113868111386811138681113868

1113868111386811138681113868 ϕz(0) +εE

ωie cos L

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868(σt + 2)sin ωdt( 11138571113868111386811138681113868

+(minus σt + 1)cos ωdt( 11138571113868111386811138681113868e

minus σt

le ϕz(0) +εE

ωie cosL

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

2(σt)2 + 2(σt) + 51113969

eminus σt

(19)

Taking the 001deg error band as the criteria of initialalignment convergence

Δϕz3(t)1113868111386811138681113868

1113868111386811138681113868le ϕz(0) +εE

ωie cos L

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

2(σt)2 + 2σt + 51113969

eminus σt

001∘

(20)

From equation (20) |Δϕz3(t)| is proportional to|ϕz(0) + (εEωie 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 signsAs 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 5deg compass azimuthalignment will converge within 141 Td

4 Simulation Verification

41 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 53degOnly the X-axis gyro has a constant drift of 005degh 0e INSattitudes are 0deg 0deg and 0deg 0e initial attitude errors are all 0degTd 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 isminus 03165deg 0erefore according to Figure 2 and the definitionof the 001deg error band used in this study we assume that thecompass azimuth alignment converges when the initialalignment error curve finally passes through minus 03165deg 0usthe 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 001degh002degh 003degh and 004degh the Td convergence times are200 s 300 s 400 s and 500 s respectively0erefore the ratioof convergence time to Td is shown in Table 5

It can be seen from Table 5 that when the gyro drift isover 002degh it exhibits good agreement with the theoreticalcalculation Moreover the actual convergence time is lessthan the theoretical convergence time with a gyro drift of002degh and 001degh 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 (degh) 005 004 003 002 001Convergence time (Td) 09 085 079 07 058

Table 2 Convergence time of different initial azimuth misalignment

Initial azimuth misalignment (deg) 05 1 2 3 4Convergence time (Td) 1 11 125 13 135


42 Simulation of the Effect of Initial Azimuth Error onConvergence Time We assume that there is no inertialdevice error the initial azimuth error is 5deg Td is set to 200 s300 s 400 s and 500 s respectively the simulation time is1000 s and the attitude is 0deg 0deg and 0deg 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 4deg3deg 2deg 1deg and 05deg and Td is 200 s 300 s 400 s and 500 srespectively 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 2deg it agrees almost perfectly with the theoreticalcalculation results When the initial error is less than 2deg theactual convergence time is less than the theoretical calcu-lation time0is phenomenon explained briefly in Section 4part A is due to the inequality amplification in the theo-retical analysis

43 Simulation of the Effect of Both Errors on ConvergenceTime We assume that there is no error of inertial device theinitial azimuth error is 5deg the x-axis gyro constant drift is005degh Td is equal to 200 s 300 s 400 s and 500 s thesimulation time is 1000 s and the attitude is 0deg 0deg and 0deg0eresulting 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 4deg 3deg 2deg 1deg and05deg and the gyro constant drift is 005degh Td is 200 s 300 s400 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 (deg) 05 1 2 3 4 5001degh 100 112 125 131 135 140002degh 101 113 125 131 136 140003degh 103 114 126 132 137 140004degh 105 115 126 132 137 141005degh 107 116 126 133 138 141

0 100 200 300 400 500 600ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

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 005deg gyroconstant drift

Td (s) 200 300 400 500Convergence time (s) 177 265 354 443Ratio 089 088 089 089

Table 5 Convergence times for different Td and different gyroconstant drifts

Gyro drift (degh) Td (s) 200 300 400 500

004 Convergence time (s) 170 255 341 426Ratio 085 085 085 085




Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

ndash001

0

001

002

003

004

005

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 5deg initial azi-muth misalignment



the gyro drift is set to 005degh 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 timewe 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 005 degh and00001 g respectively

0e experiment procedure was as follows 0e initialazimuth errors are 5deg and 1deg 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 s0e 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 (deg) Td (s) 200 300 400 500






Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

Td = 200Td = 300

Td = 400Td = 500

ndash035

ndash034

ndash033

ndash032

ndash031

ndash03

ndash029

ndash028

Figure 4 Convergence curves for both errors for different Td

Table 8 Convergence times for different Td with a 005degh gyro drift and a 5deg 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 possibleand compass azimuth alignment is required to converge0us 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 errorand the second-order damping oscillation period When anerror band of 001deg is used to determine the convergence andthe gyro drift is less than 005degh the compass azimuthalignment will converge within 09 damping oscillationperiods due to gyro drift When the initial azimuth error isless than 5deg the compass azimuth alignment will convergewithin 14 damping oscillation periods due to the initial

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

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






Deg

ree

ndash01

0

01

02

03

04

05

1000 2000 3000 4000 5000 60000Second

Td = 200Td = 300

Td = 400Td = 500

Figure 5 Convergence curves of the initial azimuth error 1deg for different Td

Table 10 Experiment results





azimuth error When both errors are considered the initialazimuth error plays a major role in the convergence timecompared 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

References

[1] X Liu X Xu L Wang and Y Liu ldquoA fast compass alignmentmethod for SINS based on saved data and repeated navigationsolutionrdquo Measurement vol 46 no 10 pp 3836ndash3846 2013

[2] Li Yao X-S Xu and B-X Wu ldquoGyrocompass self-alignmentof SINSrdquo Journal of Chinese Inertial Technology vol 16 no 4pp 386ndash389 2008

[3] B Zhu J Xu H He et al ldquoInitial alignment method ofstrapdown gyrocompass based on particle swarm optimiza-tion algorithmrdquo Journal of Chinese Inertial Technology vol 25no 1 pp 47ndash51 2017

[4] J T Kouba and L W Mason ldquoGyrocompass alignment of aninertial platform to arbitrary attitudesrdquo ARS Journal vol 32no 7 pp 1029ndash1033 1962

[5] B Xu and Y Hao ldquoError analysis of compass-circuit align-ment method in navigation staterdquo Journal of LiaoningTechnical University (Natural Science Edition) vol 31 no 1pp 46ndash49 2012

[6] J Zhang L Ge and Y Wang ldquoEffect of angle random walk oncompass azimuth alignmentrdquo Journal of Chinese InertialTechnology vol 25 no 1 pp 28ndash32 2017

[7] W Gao B Lu and C Yu ldquoForward and backward processesfor INS compass alignmentrdquo Ocean Engineering vol 98pp 1ndash9 2015

[8] Y Ben Q Zhang X Zang Q Li and G Wang ldquoEffect of theouter lever arm on in-motion gyrocompass alignment forfiber-optic gyro strapdown inertial navigation systemrdquo Op-tical Engineering vol 56 no 4 Article ID 044106 2017

[9] H W Park and J G Lee C G Park Covariance analysis ofstrapdown INS considering gyrocompass characteristicsrdquoIEEE Transactions on Aerospace and Electronic Systemsvol 31 no 1 pp 320ndash328 1995

[10] T Abbas Y Zhang and Y Li ldquoSINS initial alignment forsmall tilt and large azimuth misalignment anglesrdquo in Pro-ceedings of the 2011 IEEE 3rd International Conference on

Communication Software and Networks pp 628ndash632 IEEEXirsquoan China May 2011

[11] F Sun J Xia Y Ben and H Lan ldquoTime-varying parametersbased gyrocompass Alignment for marine SINS with largeheading misalignmentrdquo in Proceedings of the 2014 IEEEIONPosition Location and Navigation SymposiummdashPLANS 2014IEEE Monterey CA USA May 2014

[12] H He J Xu F Qin and F Li ldquoGenetic algorithm based fastalignment method for strap-down inertial navigation systemwith large azimuth misalignmentrdquo Review of Scientific In-struments vol 86 no 11 pp 1930ndash1941 2015

[13] X Kong E Mario Nebot and H Durrant-whyte ldquoDevel-opment of a non-linear psi-angle model for large misalign-ment errors and its application in INS alignment andcalibrationrdquo in Proceedings of the IEEE International Con-ference on Robotics and Automation IEEE Detroit MI USAAugust 1999

[14] G I Emelrsquoyantsev A P Stepanov and B A BlazhnovldquoCalibration of in-run drifts of strapdown inertial navigationsystem with uniaxial modulation rotation of measurementunitrdquo Gyroscopy and Navigation vol 8 no 4 pp 241ndash2472017

[15] H Yanling Y Zhang F Sun andW Gao ldquoAnalysis of single-axial rotation SINS azimuth alignmentrdquo Chinese Journal ofScientific Instrument vol 32 no 2 pp 309ndash315 2011

[16] X Liu X Xu Y Liu and L Wang ldquoA fast and high-accuracycompass alignment method to SINS with azimuth axis ro-tationrdquo Mathematical Problems in Engineering vol 2013Article ID 524284 12 pages 2013

[17] G Yan On SINS In-Movement Initial Alignment and SomeOther Problems Northwestern Polytechnical UniversityPostdoctoral Research Report Xirsquoan China 2008

[18] Z Gao Inertial Navigation System Technology TsinghuaUniversity Press Beijing China 2012

[19] H Wang ldquoResearch of compass initial alignment for FOGSINSrdquo Transducer and Microsystem Technologies vol 30no 10 pp 53ndash55 2011


strapdown inertial navigation system was presented andfrom 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 methodwhich did not require the assumption of a linear modelwith a small misalignment angle it also improved thealignment speed of a large azimuth misalignment angleHe 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 carrierrsquosattitude 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 systemhowever 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 30ndash50min 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 theory0e initial alignment is divided into two stages hori-zontal leveling and azimuth alignment where horizontalleveling is the basis of azimuth alignment In generalhorizontal alignment is rapid precise and simplewhereas 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 earthrsquos rotation L is the localgeographic latitude g is the acceleration of gravity nablaN isthe north accelerometer bias εE is the east gyro driftwhich 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 L

sK3

Δ(s)

nablaN

s1113876 1113877 +

1ωie cosL

gK3

Δ(s)minusεE

s+ ϕx(0)1113876 1113877

+s s + K1( 1113857 + ω2

s K2 + 1( 1113857( 1113857 s + K4( 1113857

Δ(s)

εU

s+ ϕz(0)1113876 1113877

(1)

where ϕx(0) is the initial error of the east error anglewhich 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 εE

also affects the azimuth alignment accuracy Δ(s) is thecharacteristic equation of the compass azimuth align-ment system


Δ(s) s4

+ K1 + K4( 1113857s3

+ ω2s K2 + 1( 1113857 + K1K41113960 1113961s

2

+ ω2s K2 + 1( 1113857K4s + gK3

(2)

where ωs gR






2

radic2 0erefore








ϕz1(s) gK3

ωie cos L

εE

s (s + σ)2 + ω2d1113872 1113873

2 (4)


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

(1s)

g


N

εE

ϕzεU

K (s) ωie cos L

times times

times

ndashndash



ϕz1(t) minusgK3εE

ωie cos L

1

σ2 + ω2d1113872 1113873

2 +2σ cos ωdt( 1113857

4ω2d σ2 + ω2

d1113872 1113873⎛⎝ ⎞⎠ minus


4ω2d σ2 + ω2

d1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠te

minus σt

+

2 σ2 minus ω2d( 1113857cos ωdt( 1113857

4ω2d σ2 minus ω2

d1113872 11138732

+ 4σ2ω2d1113874 1113875


4ω2d σ2 minus ω2

d1113872 11138732

+ 4σ2ω2d1113874 1113875


4 σ2 + ω2d1113872 1113873ω3

d


4 σ2 + ω2d1113872 1113873ω3

d

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

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

eminus σt

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

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

(5)



ωie cosL

14σ4

+2σ cos ωdt( 1113857

4σ2 σ2 + σ2( )1113888 1113889 minus

2σ sin ωdt( 1113857

4σ2 σ2 + σ2( )1113888 11138891113888 1113889te

minus σt



2σ sin ωdt( 1113857


4 σ2 + σ2( )σ31113888 1113889eminus σt

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus4σ4εE

ωie cosL

14σ4

+cos ωdt( 1113857

4σ31113888 1113889 minus

sin ωdt( 1113857

4σ31113888 11138891113888 1113889te

minus σt



4σ41113888 1113889eminus σt

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minusεE

ωie cosL



⎛⎝ ⎞⎠

(6)



εE

ωie cos L 032∘ (7)


Δϕz1(t)1113868111386811138681113868


1113868111386811138681113868


1113868111386811138681113868

032deg2


minus σt1113868111386811138681113868

+5


minus σt1113868111386811138681113868

032deg

2(σt)2 + 51113969

cos ωt + φ3( 11138571113874 1113875eminus σt

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le 032deg

2(σt)2 + 51113969

eminus σt

(8)



Δϕz1(t)le 032deg

2(σt)2 + 51113969

1113874 1113875eminus σt

001∘ (9)







ϕz2(s) s s + K1( 1113857 + ω2

s K2 + 1( 1113857( 1113857 s + K4( 1113857

Δ(s)ϕz(0) (10)


ϕz2(s) s3 + 4σs2 + 8σs + 8σ

(s + σ)2 + ωd( 11138572

1113872 11138732 ϕz(0) (11)


ϕz2(t) ϕz(0)t

23σ2 minus ω2

d1113872 1113873ω2d +

12σ 3σ2 + ω2


ω3d

1113888 1113889 + ω2d minus

t

2σ 3σ2 minus ω2


ω2d

1113888 11138891113888 1113889 (12)

Because ωd σ



(13)


0us due to


minus σt cos(ωt)1113868111386811138681113868

1113868111386811138681113868

2(σt)2 + 2(σt) + 51113969


1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le

2(σt)2 + 2(σt) + 51113969

eminus σt

(14)


ϕz2(t)1113868111386811138681113868

1113868111386811138681113868le ϕz(0)

2(σt)2 + 2(σt) + 51113969

eminus σt

(15)


ϕz2(t)1113868111386811138681113868

1113868111386811138681113868leϕz(0)

2(σt)2 + 2(σt) + 51113969

eminus σt

001∘ (16)










minusεE

ωie cos L



⎛⎝ ⎞⎠



(17)


Δϕz3(t) minusεE


minus σt1113872


1113873



ϕz(0) +εE



(18)

0erefore

Δϕz3(t)1113868111386811138681113868

1113868111386811138681113868 ϕz(0) +εE

ωie cos L

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868(σt + 2)sin ωdt( 11138571113868111386811138681113868

+(minus σt + 1)cos ωdt( 11138571113868111386811138681113868e

minus σt

le ϕz(0) +εE

ωie cosL

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

2(σt)2 + 2(σt) + 51113969

eminus σt

(19)


Δϕz3(t)1113868111386811138681113868

1113868111386811138681113868le ϕz(0) +εE

ωie cos L

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

2(σt)2 + 2σt + 51113969

eminus σt

001∘

(20)





















0 100 200 300 400 500 600ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

Second

Deg

ree

Td = 200Td = 300

Td = 400Td = 500










Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

ndash001

0

001

002

003

004

005

Td = 200Td = 300

Td = 400Td = 500






5 Experiment












Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

Td = 200Td = 300

Td = 400Td = 500

ndash035

ndash034

ndash033

ndash032

ndash031

ndash03

ndash029

ndash028





6 Conclusion










Deg

ree

ndash01

0

01

02

03

04

05

1000 2000 3000 4000 5000 60000Second

Td = 200Td = 300

Td = 400Td = 500








Data Availability




Acknowledgments


References






















Δ(s) s4

+ K1 + K4( 1113857s3

+ ω2s K2 + 1( 1113857 + K1K41113960 1113961s

2

+ ω2s K2 + 1( 1113857K4s + gK3

(2)

where ωs gR






2

radic2 0erefore








ϕz1(s) gK3

ωie cos L

εE

s (s + σ)2 + ω2d1113872 1113873

2 (4)


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

(1s)

g


N

εE

ϕzεU

K (s) ωie cos L

times times

times

ndashndash



ϕz1(t) minusgK3εE

ωie cos L

1

σ2 + ω2d1113872 1113873

2 +2σ cos ωdt( 1113857

4ω2d σ2 + ω2

d1113872 1113873⎛⎝ ⎞⎠ minus


4ω2d σ2 + ω2

d1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠te

minus σt

+

2 σ2 minus ω2d( 1113857cos ωdt( 1113857

4ω2d σ2 minus ω2

d1113872 11138732

+ 4σ2ω2d1113874 1113875


4ω2d σ2 minus ω2

d1113872 11138732

+ 4σ2ω2d1113874 1113875


4 σ2 + ω2d1113872 1113873ω3

d


4 σ2 + ω2d1113872 1113873ω3

d

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

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

eminus σt

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

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

(5)



ωie cosL

14σ4

+2σ cos ωdt( 1113857

4σ2 σ2 + σ2( )1113888 1113889 minus

2σ sin ωdt( 1113857

4σ2 σ2 + σ2( )1113888 11138891113888 1113889te

minus σt



2σ sin ωdt( 1113857


4 σ2 + σ2( )σ31113888 1113889eminus σt

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus4σ4εE

ωie cosL

14σ4

+cos ωdt( 1113857

4σ31113888 1113889 minus

sin ωdt( 1113857

4σ31113888 11138891113888 1113889te

minus σt



4σ41113888 1113889eminus σt

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minusεE

ωie cosL



⎛⎝ ⎞⎠

(6)



εE

ωie cos L 032∘ (7)


Δϕz1(t)1113868111386811138681113868


1113868111386811138681113868


1113868111386811138681113868

032deg2


minus σt1113868111386811138681113868

+5


minus σt1113868111386811138681113868

032deg

2(σt)2 + 51113969

cos ωt + φ3( 11138571113874 1113875eminus σt

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le 032deg

2(σt)2 + 51113969

eminus σt

(8)



Δϕz1(t)le 032deg

2(σt)2 + 51113969

1113874 1113875eminus σt

001∘ (9)







ϕz2(s) s s + K1( 1113857 + ω2

s K2 + 1( 1113857( 1113857 s + K4( 1113857

Δ(s)ϕz(0) (10)


ϕz2(s) s3 + 4σs2 + 8σs + 8σ

(s + σ)2 + ωd( 11138572

1113872 11138732 ϕz(0) (11)


ϕz2(t) ϕz(0)t

23σ2 minus ω2

d1113872 1113873ω2d +

12σ 3σ2 + ω2


ω3d

1113888 1113889 + ω2d minus

t

2σ 3σ2 minus ω2


ω2d

1113888 11138891113888 1113889 (12)

Because ωd σ



(13)


0us due to


minus σt cos(ωt)1113868111386811138681113868

1113868111386811138681113868

2(σt)2 + 2(σt) + 51113969


1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le

2(σt)2 + 2(σt) + 51113969

eminus σt

(14)


ϕz2(t)1113868111386811138681113868

1113868111386811138681113868le ϕz(0)

2(σt)2 + 2(σt) + 51113969

eminus σt

(15)


ϕz2(t)1113868111386811138681113868

1113868111386811138681113868leϕz(0)

2(σt)2 + 2(σt) + 51113969

eminus σt

001∘ (16)










minusεE

ωie cos L



⎛⎝ ⎞⎠



(17)


Δϕz3(t) minusεE


minus σt1113872


1113873



ϕz(0) +εE



(18)

0erefore

Δϕz3(t)1113868111386811138681113868

1113868111386811138681113868 ϕz(0) +εE

ωie cos L

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868(σt + 2)sin ωdt( 11138571113868111386811138681113868

+(minus σt + 1)cos ωdt( 11138571113868111386811138681113868e

minus σt

le ϕz(0) +εE

ωie cosL

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

2(σt)2 + 2(σt) + 51113969

eminus σt

(19)


Δϕz3(t)1113868111386811138681113868

1113868111386811138681113868le ϕz(0) +εE

ωie cos L

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

2(σt)2 + 2σt + 51113969

eminus σt

001∘

(20)





















0 100 200 300 400 500 600ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

Second

Deg

ree

Td = 200Td = 300

Td = 400Td = 500










Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

ndash001

0

001

002

003

004

005

Td = 200Td = 300

Td = 400Td = 500






5 Experiment












Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

Td = 200Td = 300

Td = 400Td = 500

ndash035

ndash034

ndash033

ndash032

ndash031

ndash03

ndash029

ndash028





6 Conclusion










Deg

ree

ndash01

0

01

02

03

04

05

1000 2000 3000 4000 5000 60000Second

Td = 200Td = 300

Td = 400Td = 500








Data Availability




Acknowledgments


References






















ϕz1(t) minusgK3εE

ωie cos L

1

σ2 + ω2d1113872 1113873

2 +2σ cos ωdt( 1113857

4ω2d σ2 + ω2

d1113872 1113873⎛⎝ ⎞⎠ minus


4ω2d σ2 + ω2

d1113872 1113873⎛⎝ ⎞⎠⎛⎝ ⎞⎠te

minus σt

+

2 σ2 minus ω2d( 1113857cos ωdt( 1113857

4ω2d σ2 minus ω2

d1113872 11138732

+ 4σ2ω2d1113874 1113875


4ω2d σ2 minus ω2

d1113872 11138732

+ 4σ2ω2d1113874 1113875


4 σ2 + ω2d1113872 1113873ω3

d


4 σ2 + ω2d1113872 1113873ω3

d

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

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

eminus σt

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

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

(5)



ωie cosL

14σ4

+2σ cos ωdt( 1113857

4σ2 σ2 + σ2( )1113888 1113889 minus

2σ sin ωdt( 1113857

4σ2 σ2 + σ2( )1113888 11138891113888 1113889te

minus σt



2σ sin ωdt( 1113857


4 σ2 + σ2( )σ31113888 1113889eminus σt

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minus4σ4εE

ωie cosL

14σ4

+cos ωdt( 1113857

4σ31113888 1113889 minus

sin ωdt( 1113857

4σ31113888 11138891113888 1113889te

minus σt



4σ41113888 1113889eminus σt

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

minusεE

ωie cosL



⎛⎝ ⎞⎠

(6)



εE

ωie cos L 032∘ (7)


Δϕz1(t)1113868111386811138681113868


1113868111386811138681113868


1113868111386811138681113868

032deg2


minus σt1113868111386811138681113868

+5


minus σt1113868111386811138681113868

032deg

2(σt)2 + 51113969

cos ωt + φ3( 11138571113874 1113875eminus σt

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le 032deg

2(σt)2 + 51113969

eminus σt

(8)



Δϕz1(t)le 032deg

2(σt)2 + 51113969

1113874 1113875eminus σt

001∘ (9)







ϕz2(s) s s + K1( 1113857 + ω2

s K2 + 1( 1113857( 1113857 s + K4( 1113857

Δ(s)ϕz(0) (10)


ϕz2(s) s3 + 4σs2 + 8σs + 8σ

(s + σ)2 + ωd( 11138572

1113872 11138732 ϕz(0) (11)


ϕz2(t) ϕz(0)t

23σ2 minus ω2

d1113872 1113873ω2d +

12σ 3σ2 + ω2


ω3d

1113888 1113889 + ω2d minus

t

2σ 3σ2 minus ω2


ω2d

1113888 11138891113888 1113889 (12)

Because ωd σ



(13)


0us due to


minus σt cos(ωt)1113868111386811138681113868

1113868111386811138681113868

2(σt)2 + 2(σt) + 51113969


1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le

2(σt)2 + 2(σt) + 51113969

eminus σt

(14)


ϕz2(t)1113868111386811138681113868

1113868111386811138681113868le ϕz(0)

2(σt)2 + 2(σt) + 51113969

eminus σt

(15)


ϕz2(t)1113868111386811138681113868

1113868111386811138681113868leϕz(0)

2(σt)2 + 2(σt) + 51113969

eminus σt

001∘ (16)










minusεE

ωie cos L



⎛⎝ ⎞⎠



(17)


Δϕz3(t) minusεE


minus σt1113872


1113873



ϕz(0) +εE



(18)

0erefore

Δϕz3(t)1113868111386811138681113868

1113868111386811138681113868 ϕz(0) +εE

ωie cos L

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868(σt + 2)sin ωdt( 11138571113868111386811138681113868

+(minus σt + 1)cos ωdt( 11138571113868111386811138681113868e

minus σt

le ϕz(0) +εE

ωie cosL

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

2(σt)2 + 2(σt) + 51113969

eminus σt

(19)


Δϕz3(t)1113868111386811138681113868

1113868111386811138681113868le ϕz(0) +εE

ωie cos L

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

2(σt)2 + 2σt + 51113969

eminus σt

001∘

(20)





















0 100 200 300 400 500 600ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

Second

Deg

ree

Td = 200Td = 300

Td = 400Td = 500










Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

ndash001

0

001

002

003

004

005

Td = 200Td = 300

Td = 400Td = 500






5 Experiment












Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

Td = 200Td = 300

Td = 400Td = 500

ndash035

ndash034

ndash033

ndash032

ndash031

ndash03

ndash029

ndash028





6 Conclusion










Deg

ree

ndash01

0

01

02

03

04

05

1000 2000 3000 4000 5000 60000Second

Td = 200Td = 300

Td = 400Td = 500








Data Availability




Acknowledgments


References






















Δϕz1(t)le 032deg

2(σt)2 + 51113969

1113874 1113875eminus σt

001∘ (9)







ϕz2(s) s s + K1( 1113857 + ω2

s K2 + 1( 1113857( 1113857 s + K4( 1113857

Δ(s)ϕz(0) (10)


ϕz2(s) s3 + 4σs2 + 8σs + 8σ

(s + σ)2 + ωd( 11138572

1113872 11138732 ϕz(0) (11)


ϕz2(t) ϕz(0)t

23σ2 minus ω2

d1113872 1113873ω2d +

12σ 3σ2 + ω2


ω3d

1113888 1113889 + ω2d minus

t

2σ 3σ2 minus ω2


ω2d

1113888 11138891113888 1113889 (12)

Because ωd σ



(13)


0us due to


minus σt cos(ωt)1113868111386811138681113868

1113868111386811138681113868

2(σt)2 + 2(σt) + 51113969


1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868

le

2(σt)2 + 2(σt) + 51113969

eminus σt

(14)


ϕz2(t)1113868111386811138681113868

1113868111386811138681113868le ϕz(0)

2(σt)2 + 2(σt) + 51113969

eminus σt

(15)


ϕz2(t)1113868111386811138681113868

1113868111386811138681113868leϕz(0)

2(σt)2 + 2(σt) + 51113969

eminus σt

001∘ (16)










minusεE

ωie cos L



⎛⎝ ⎞⎠



(17)


Δϕz3(t) minusεE


minus σt1113872


1113873



ϕz(0) +εE



(18)

0erefore

Δϕz3(t)1113868111386811138681113868

1113868111386811138681113868 ϕz(0) +εE

ωie cos L

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868(σt + 2)sin ωdt( 11138571113868111386811138681113868

+(minus σt + 1)cos ωdt( 11138571113868111386811138681113868e

minus σt

le ϕz(0) +εE

ωie cosL

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

2(σt)2 + 2(σt) + 51113969

eminus σt

(19)


Δϕz3(t)1113868111386811138681113868

1113868111386811138681113868le ϕz(0) +εE

ωie cos L

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

2(σt)2 + 2σt + 51113969

eminus σt

001∘

(20)





















0 100 200 300 400 500 600ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

Second

Deg

ree

Td = 200Td = 300

Td = 400Td = 500










Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

ndash001

0

001

002

003

004

005

Td = 200Td = 300

Td = 400Td = 500






5 Experiment












Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

Td = 200Td = 300

Td = 400Td = 500

ndash035

ndash034

ndash033

ndash032

ndash031

ndash03

ndash029

ndash028





6 Conclusion










Deg

ree

ndash01

0

01

02

03

04

05

1000 2000 3000 4000 5000 60000Second

Td = 200Td = 300

Td = 400Td = 500








Data Availability




Acknowledgments


References

























minusεE

ωie cos L



⎛⎝ ⎞⎠



(17)


Δϕz3(t) minusεE


minus σt1113872


1113873



ϕz(0) +εE



(18)

0erefore

Δϕz3(t)1113868111386811138681113868

1113868111386811138681113868 ϕz(0) +εE

ωie cos L

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868(σt + 2)sin ωdt( 11138571113868111386811138681113868

+(minus σt + 1)cos ωdt( 11138571113868111386811138681113868e

minus σt

le ϕz(0) +εE

ωie cosL

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

2(σt)2 + 2(σt) + 51113969

eminus σt

(19)


Δϕz3(t)1113868111386811138681113868

1113868111386811138681113868le ϕz(0) +εE

ωie cos L

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868

2(σt)2 + 2σt + 51113969

eminus σt

001∘

(20)





















0 100 200 300 400 500 600ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

Second

Deg

ree

Td = 200Td = 300

Td = 400Td = 500










Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

ndash001

0

001

002

003

004

005

Td = 200Td = 300

Td = 400Td = 500






5 Experiment












Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

Td = 200Td = 300

Td = 400Td = 500

ndash035

ndash034

ndash033

ndash032

ndash031

ndash03

ndash029

ndash028





6 Conclusion










Deg

ree

ndash01

0

01

02

03

04

05

1000 2000 3000 4000 5000 60000Second

Td = 200Td = 300

Td = 400Td = 500








Data Availability




Acknowledgments


References





























0 100 200 300 400 500 600ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

Second

Deg

ree

Td = 200Td = 300

Td = 400Td = 500










Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

ndash001

0

001

002

003

004

005

Td = 200Td = 300

Td = 400Td = 500






5 Experiment












Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

Td = 200Td = 300

Td = 400Td = 500

ndash035

ndash034

ndash033

ndash032

ndash031

ndash03

ndash029

ndash028





6 Conclusion










Deg

ree

ndash01

0

01

02

03

04

05

1000 2000 3000 4000 5000 60000Second

Td = 200Td = 300

Td = 400Td = 500








Data Availability




Acknowledgments


References























5 Experiment












Deg

ree

100 200 300 400 500 600 700 800 900 10000Second

Td = 200Td = 300

Td = 400Td = 500

ndash035

ndash034

ndash033

ndash032

ndash031

ndash03

ndash029

ndash028





6 Conclusion










Deg

ree

ndash01

0

01

02

03

04

05

1000 2000 3000 4000 5000 60000Second

Td = 200Td = 300

Td = 400Td = 500








Data Availability




Acknowledgments


References






















6 Conclusion










Deg

ree

ndash01

0

01

02

03

04

05

1000 2000 3000 4000 5000 60000Second

Td = 200Td = 300

Td = 400Td = 500








Data Availability




Acknowledgments


References























Data Availability




Acknowledgments


References






















EffectofEastGyroDriftandInitialAzimuthErrorontheCompass ...downloads.hindawi.com/journals/mpe/2020/9042197.pdf · Δ(s) s4 +K1 +K 4 s 3 +ω2 h s K 2 +1 +K 1K 4is 2 +ω2 s K 2 +1 K

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