# Effect of East Gyro Drift and Initial Azimuth Error on the Compass Azimuth Alignment Convergence Time

Authors: Dongxu He; Xinle Zang; Lei Ge
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

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


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## Body

## 1. Introduction

Compass alignment is a typical initial alignment approach for inertial navigation systems (INSs). It is based on the principle of the compass effect and adopts classic control theory in the frequency domain to design a compass aligning circuit. Compass alignment has the advantages of few parameters, low computational complexity, and easy implementation [1]. At present, studies on compass alignment have 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-alignment technique was investigated for the strapdown INS (SINS) under the swing state. The horizontal and azimuth alignment parameters were designed and optimized; they subsequently performed well under different sea conditions, different initial attitude errors, and omnidirectional conditions. Zhu et al. [3] introduced an intelligent optimization algorithm into compass alignment and optimized its parameters using a particle swarm optimization algorithm to improve the initial alignment performance of the strapdown INS. In [4], a mechanization scheme for gyrocompassing to an arbitrary attitude was proposed.In terms of error analysis, Xu and Xao [5] studied the initial alignment of a compass loop under a sailing turntable and analyzed the alignment error based on the equivalence of the device error. Zhang et al. [6] analyzed the effect of random noise on compass azimuth alignment and proposed an innovative method in the time domain. Moreover, a method based on the inverse attitude calculation proposed that the periodic oscillation error of the gyro output can cause additional oscillation errors in compass azimuth alignment [7]. Furthermore, the azimuth error is amplified with increasing switching frequency. Ben et al. [8] described the effect of an outer lever arm on in-motion gyrocompass alignment, and a method for outer lever arm correction was provided to counteract the outer lever arm effect on the performance of the alignment. In [9], a complete error covariance analysis for strapdown inertial navigation system was presented, and, from this paper, it can be found that the cross-coupling terms in gyrocompass alignment errors can significantly influence the system error propagation.A considerable number of studies have been carried out on compass alignment for a large azimuth misalignment angle. Abbas et al. [10] derived a nonlinear error model of the SINS with a large azimuth misalignment and proposed the static base alignment of the SINS employing simplified unscented Kalman filter (UKF) on the 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 model with a small misalignment angle; it also improved the alignment speed of a large azimuth misalignment angle. He et al. [12] proposed a time-varying parameter compass alignment algorithm based on an optimal model and used a genetic algorithm to optimize the parameters of compass alignment for a large azimuth misalignment angle. In [13], a general nonlinear psi-angle approach for large misalignment errors that does not require coarse alignment was presented.Owing to its rapid development in recent years, many researchers have also introduced rotation modulation technology [14] into the initial alignment of the SINS to eliminate the influence of the inertia device constant error on the initial alignment. To eliminate the influence of the east gyro drift on the azimuth alignment accuracy of the SINS, Yanling et al. [15] proposed a compass alignment method suitable for a rotation modulation SINS based on an analysis of the frequency characteristics of the compass. Liu et al. [16] introduced an azimuth rotating modulation method to classical compass alignment for SINS and designed an alignment method based on repeated data calculation to improve the alignment accuracy with certain accuracy sensors and eliminate the effect of the carrier’s attitude on alignment accuracy.Thus, although many studies have dealt with compass alignment, the convergence time has received minimal attention, despite being a relatively significant index for compass alignment. This is because the system is expected to (1) have a strong anti-interference ability to minimize the random environmental interference and (2) be able to converge within a limited initial alignment time. However, these two requirements are always conflicting [17]. Previous research designed a fourth-order compass azimuth alignment control system and indicated that the convergence time is related to the selected second-order damping oscillation period [17]. That study only analyzed the classic second-order system, which is directly applied to the fourth-order compass azimuth alignment system; however, the authors revealed clear similarities between the second-order and fourth-order systems. A three-order compass alignment system was designed in [18], which indicated that when selecting the corresponding parameters, the system can converge within 30–50 min. However, a concrete analysis method has not yet been provided.Therefore, this study analyzes the effect of east gyro drift and initial azimuth error on the convergence time based on the fourth-order compass azimuth alignment system. We propose a novel method that converts the azimuth error response from the frequency domain to the time domain and then analyzes the convergence time in the time domain. A theoretical reference is provided to set the corresponding parameters of compass azimuth alignment and to control the convergence time.


## 2. Gyrocompass Azimuth Alignment Principle

Compass azimuth alignment is a self-alignment method based on the compass effect and classic control theory. The initial alignment is divided into two stages: horizontal leveling and azimuth alignment, where horizontal leveling is the basis of azimuth alignment. In general, horizontal alignment is rapid, precise, and simple, whereas azimuth alignment is problematic during the alignment process. Here, we briefly describe the principle of compass azimuth alignment, shown in Figure1, where ωie is the angular velocity of earth’s rotation; L is the local geographic latitude; g is the acceleration of gravity; ∇N is the 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 the initial alignment precision; δVN is the north velocity error; ϕx is the pitch error; ϕz is the azimuth error; K1 and K2 are the designed parameters of the north horizontal loop; and Ks=K3/ωiecosLs+K4 is the control link of the compass loop, where the input is δVN and the output is KsδVN, which replaces the command angular velocity of vertical control. During the process of compass azimuth alignment, beginning from ϕz through each link of the compass effect to output δVN and then through the azimuth control link Ks, the output ϕz is adjusted.Figure 1
Principle of compass azimuth alignment (see text for explanation of symbols).According to the principle shown in Figure1, the fourth-order system response is(1)ϕzs=1ωiecosLsK3Δs∇Ns+1ωiecosLgK3Δs−εEs+ϕx0+ss+K1+ωs2K2+1s+K4ΔsεUs+ϕz0,where ϕx0 is the initial error of the east error angle, which is very small and has a minimal influence on compass azimuth alignment after compass horizontal leveling, and ϕz0 is the initial error of the azimuth error angle, which affects the convergence characteristics of compass azimuth alignment. Thus, this is the parameter that requires research. The east gyro drift εE also affects the azimuth alignment accuracy. Δs is the characteristic equation of the compass azimuth alignment system:(2)Δs=s4+K1+K4s3+ωs2K2+1+K1K4s2+ωs2K2+1K4s+gK3,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 two identical second-order systems. The characteristic root then has the following form [2]:(3)s1,2=s3,4=−σ±jωd,where σ=ξωn is the attenuation coefficient; ξ is the damping ratio; ωn is the undamped oscillation frequency of the designed second-order system; ωd=2π/Td is the damping oscillation frequency; and Td is the damping oscillation period of the second-order system. The damping ratio is generally set to ξ=2/2. Therefore, ωd=σ, with the corresponding parameters of K1=K4=2σ, K2=4σ2/ωs−1, and K3=4σ4/g. For compass azimuth alignment, the other parameters are subsequently determined only if Td is set.According to the response function ofϕz, the output is influenced by five parameters. However, according to previous research, ∇N, ϕx0, and εU have a smaller impact on compass azimuth alignment. Their orders of magnitude are also small, so these parameters are not considered here. The focus of this study is analyzing the effects of east gyro drift and initial azimuth error on the convergence time of compass azimuth alignment.


## 3. Convergence Time Analysis of Compass Azimuth Alignment

### 3.1. Determination of Compass Azimuth Alignment

Generally, automatic control theory regards the controlled parameter in a certain error band as entering a steady system process, which means that the system converges. Meanwhile, the error band is generally assumed as 2% or 5% of the steady value. However, this selection is not appropriate for the study of compass azimuth alignment because the steady value of the effect of initial azimuth error on compass azimuth alignment is zero; thus, the error band cannot be assumed to be a percentage of the steady value. Additionally, determination of the azimuth convergence should be comprehensively considered during initial alignment based on inertial device precision and azimuth angle accuracy. Hence, whether the azimuth angle enters the error band (the unit of this error band is angle) is used as a criterion for the convergence of the compass azimuth.In this study, our analysis is based on the fiber optic gyroscope, and the gyro drift stability is restricted to 0.05°/h. Based on the initial alignment error formula, the initial alignment precision is constrained to 0.35° for a latitude of 53° north. Then, an error band of 0.01° is used with a comprehensive consideration of the effect of random error on the initial alignment, which is considered to have converged for medium-accuracy inertial devices. Certainly, during practical applications, this convergence determination may be adjusted according to the requirements of the environment, inertial device precision, and alignment accuracy. If the gyro drift stability is in the order of 0.001°/h, the error band can be up to 0.005°; however, if the gyro drift stability is in the order of 0.1°/h, the error band can be reduced to 0.02° or 0.03°.


### 3.2. Effect of Gyro Constant Drift on Convergence Time

According to Section2, the system response term related to east gyro drift is(4)ϕz1s=gK3ωiecosLεEss+σ2+ωd22.In order to examine the time characteristics, the response of the frequency domain is converted into that of the time domain. Thus, we apply the inverse Laplace transform toϕz1s and obtain(5)ϕz1t=−gK3εEωiecosL1σ2+ωd22+2σcosωdt4ωd2σ2+ωd2−2ωdsinωdt4ωd2σ2+ωd2te−σt+2σ2−ωd2cosωdt4ωd2σ2−ωd22+4σ2ωd2−4σωdsinωdt4ωd2σ2−ωd22+4σ2ωd2−2σsinωdt4σ2+ωd2ωd3−2ωdcosωdt4σ2+ωd2ωd3e−σt.According to the values of the corresponding parameters in Section2, σ=ωd=2π/Td and gK3=4σ4. Therefore, equation (5) can be simplified as follows:(6)ϕz1t=−4σ4εEωiecosL14σ4+2σcosωdt4σ2σ2+σ2−2σsinωdt4σ2σ2+σ2te−σt+−4σ2sinωdt4σ24σ2σ2−2σsinωdt4σ2+σ2σ3−2σcosωdt4σ2+σ2σ3e−σt=−4σ4εEωiecosL14σ4+cosωdt4σ3−sinωdt4σ3te−σt+−sinωdt2σ4−cosωdt4σ4e−σt=−εEωiecosL1+σcosωdt−σsinωdtte−σt+−2sinωdt−cosωdte−σt.From equation (6), ϕz1t converges to −εE/ωiecosL, which is the same as the formulae related to the effect of east gyro drift on initial alignment. However, we need to consider when this convergence occurs.Because the gyro drift stability is constrained to less than 0.05°/h in this study, the gyro constant drift is generally less than 0.05°/h. The latitude is set to 53° and(7)εEωiecosL=0.32∘.Consider the four decay oscillation error terms in equation (7):(8)Δϕz1t=0.32°σcosωt−σsinωtte−σt+−2sinωt−cosωte−σt=0.32°2σtcosωt+φ1e−σt+5cosωt+φ2e−σt=0.32°2σt2+5cosωt+φ3e−σt≤0.32°2σt2+5e−σt.Then, taking the 0.01° error band to determine whether the azimuth alignment converges, we obtain(9)Δϕz1t≤0.32°2σt2+5e−σt=0.01∘.Then,σt=5.57 can be calculated, so t=5.57/σ=5.57Td/2π≈0.9Td, which indicates that the compass azimuth alignment converges to the 0.01° error band after approximately 0.9 damping oscillation periods.Therefore, we conclude that the effect of east gyro drift on compass azimuth alignment can converge within 0.9 damped oscillation periods. However, the analysis here is too conservative; as the gyro precision is improved, its constant drift will be lower and its convergence time will be reduced. Table1 lists the convergence time of several typical gyro constant drifts.Table 1
Convergence time of different gyro constant drifts.
Gyro drift (°/h)0.050.040.030.020.01Convergence time (Td)0.90.850.790.70.58According to Table1, the convergence time of compass azimuth alignment is related to the selected second-order damping oscillation period and the gyro constant drift. When the gyro constant drift is determined, there is a fixed-proportion relationship between the convergence time and the second-order damping oscillation period. When the second-order damping oscillation period is determined, a larger gyro constant drift results in a longer convergence time, and vice versa.It should be noted that, due to adoption of the inequality amplification in the theoretical calculation of convergence time, the actual convergence time is often less than the calculated theoretical time. In other words, the convergence time given here is more conservative and denotes the maximum time that the system takes to stabilize.


### 3.3. Effect of Initial Azimuth Error on Convergence Time

This section mainly analyzes the influence of initial azimuth error on convergence time. According to Section2, the system response term related to the initial azimuth error is(10)ϕz2s=ss+K1+ωs2K2+1s+K4Δsϕz0.According to the values of the corresponding parameters in Section2,(11)ϕz2s=s3+4σs2+8σs+8σs+σ2+ωd22ϕz0.By performing the inverse Laplace transform, the obtained function in the time domain is(12)ϕz2t=ϕz0t23σ2−ωd2ωd2+12σ3σ2+ωd2e−σtsinωdtωd3+ωd2−t2σ3σ2−ωd2e−σtcosωdtωd2.Becauseωd=σ,(13)ϕz2t=ϕz0σt+2e−σtsinωdt+−σt+1e−σtcosωdt.In equation (13), ϕz2t eventually converges to 0; however, the target of this research is determining when the convergence occurs.Thus, due to(14)σt+2e−σtsinωt+−σt+1e−σtcosωt=2σt2+2σt+5cosωt+φ1e−σt≤2σt2+2σt+5e−σt,the following is true:(15)ϕz2t≤ϕz02σt2+2σt+5e−σt.Then, using an error band of 0.01° to determine the initial alignment convergence,(16)ϕz2t≤ϕz02σt2+2σt+5e−σt=0.01∘.In equation (16), ϕz2t is proportionate to the initial azimuth error ϕz0, so its convergence time is also related to the initial azimuth error. Furthermore, the greater the value of ϕz0, the longer the convergence time.However, before compass alignment, coarse alignment is generally required to guarantee the linear characteristic of the compass azimuth alignment error model. When the initial azimuth error is within 5°, the inertial error model has better linearity; the initial alignment performs well for compass azimuth alignment. When the initial azimuth error is more than 5°, the inertial error model has inferior linearity; the performance of compass azimuth alignment gradually decreases. Therefore, in practical applications, the coarse alignment error is always controlled within 5°. In fact, this section only considers a convergence time of the initial azimuth error within 5°. Based on the typical initial errors listed above, the convergence time of compass azimuth alignment is shown in Table2.Table 2
Convergence time of different initial azimuth misalignment.
Initial azimuth misalignment (°)0.51234Convergence time (Td)11.11.251.31.35We conclude from Table2 that the convergence time of compass azimuth alignment is related to the selected second-order oscillation period and the initial azimuth error. When the initial azimuth error is determined, the convergence time is fixed in proportion to the second-order oscillation period. When the second-order oscillation period is determined, the greater the initial azimuth error, the longer the convergence time. As in Section 3, part B, the convergence times listed in Table 2 are relatively conservative, and the actual convergence time is generally less than the calculated value.Due to adoption of the inequality amplification in the theoretical calculation of convergence time, the actual convergence time is often less than the calculated theoretical time. In other words, the convergence time given here is more conservative and denotes the maximum time that the system takes to stabilize.


### 3.4. Combined Effect of Both Errors on Convergence Time

During the actual initial alignment of INS, constant drift and an initial azimuth error both exist. Therefore, it is necessary to analyze the influence of both errors on the convergence time to provide theoretical guidance for parameter setting in practical applications.The gyro constant drift and initial azimuth error are mutually independent. Based on automatic control theory, both responses obtained by the transfer function of compass azimuth alignment can exhibit linear superposition. So, under both errors, the response function of the compass azimuth alignment error is(17)ϕz3t=ϕz1t+ϕz2t=−εEωiecosL1+σcosωdt−σsinωdtte−σt+−2sinωdt−cosωdte−σt+ϕz0σt+2e−σtsinωdt+−σt+1e−σtcosωdt.In Equation17, when both errors exist, ϕz3t converges to −εE/ωiecosL, for which the error decay oscillation term is(18)Δϕz3t=−εEωiecosLσcosωdt−σsinωdtte−σt+−2sinωdt−cosωdte−σt+ϕz0σt+2e−σtsinωdt+−σt+1e−σtcosωdt=ϕz0+εEωiecosLσt+2sinωdt+−σt+1cosωdte−σt.Therefore,(19)Δϕz3t=ϕz0+εEωiecosLσt+2sinωdt+−σt+1cosωdte−σt≤ϕz0+εEωiecosL2σt2+2σt+5e−σt,Taking the 0.01° error band as the criteria of initial alignment convergence,(20)Δϕz3t≤ϕz0+εEωiecosL2σt2+2σt+5e−σt=0.01∘.From equation (20), Δϕz3t is proportional to ϕz0+εE/ωiecosL, so the convergence time is associated with the initial azimuth angle. When the absolute values of ϕz0 and εE are unchanged, Δϕz3t under opposite signs of ϕz0 and εE is less than that under the same signs. As we perform the most conservative estimation in our convergence time analysis, both signs are the same. Typical convergence times of the initial azimuth error and gyro constant drift are listed in Table 3.Table 3
Convergence time (Td) of initial azimuth error and gyro constant drift.
Initial azimuth error (°)0.5123450.01°/h1.001.121.251.311.351.400.02°/h1.011.131.251.311.361.400.03°/h1.031.141.261.321.371.400.04°/h1.051.151.261.321.371.410.05°/h1.071.161.261.331.381.41By comparing Tables2 and 3, we see that, relative to the initial azimuth error, the gyro constant drift has a minimal influence on the convergence time, whereas the main factor influencing compass azimuth alignment is the initial azimuth error. When this error is within 5°, compass azimuth alignment will converge within 1.41 Td.


## 3.1. Determination of Compass Azimuth Alignment

Generally, automatic control theory regards the controlled parameter in a certain error band as entering a steady system process, which means that the system converges. Meanwhile, the error band is generally assumed as 2% or 5% of the steady value. However, this selection is not appropriate for the study of compass azimuth alignment because the steady value of the effect of initial azimuth error on compass azimuth alignment is zero; thus, the error band cannot be assumed to be a percentage of the steady value. Additionally, determination of the azimuth convergence should be comprehensively considered during initial alignment based on inertial device precision and azimuth angle accuracy. Hence, whether the azimuth angle enters the error band (the unit of this error band is angle) is used as a criterion for the convergence of the compass azimuth.In this study, our analysis is based on the fiber optic gyroscope, and the gyro drift stability is restricted to 0.05°/h. Based on the initial alignment error formula, the initial alignment precision is constrained to 0.35° for a latitude of 53° north. Then, an error band of 0.01° is used with a comprehensive consideration of the effect of random error on the initial alignment, which is considered to have converged for medium-accuracy inertial devices. Certainly, during practical applications, this convergence determination may be adjusted according to the requirements of the environment, inertial device precision, and alignment accuracy. If the gyro drift stability is in the order of 0.001°/h, the error band can be up to 0.005°; however, if the gyro drift stability is in the order of 0.1°/h, the error band can be reduced to 0.02° or 0.03°.


## 3.2. Effect of Gyro Constant Drift on Convergence Time

According to Section2, the system response term related to east gyro drift is(4)ϕz1s=gK3ωiecosLεEss+σ2+ωd22.In order to examine the time characteristics, the response of the frequency domain is converted into that of the time domain. Thus, we apply the inverse Laplace transform toϕz1s and obtain(5)ϕz1t=−gK3εEωiecosL1σ2+ωd22+2σcosωdt4ωd2σ2+ωd2−2ωdsinωdt4ωd2σ2+ωd2te−σt+2σ2−ωd2cosωdt4ωd2σ2−ωd22+4σ2ωd2−4σωdsinωdt4ωd2σ2−ωd22+4σ2ωd2−2σsinωdt4σ2+ωd2ωd3−2ωdcosωdt4σ2+ωd2ωd3e−σt.According to the values of the corresponding parameters in Section2, σ=ωd=2π/Td and gK3=4σ4. Therefore, equation (5) can be simplified as follows:(6)ϕz1t=−4σ4εEωiecosL14σ4+2σcosωdt4σ2σ2+σ2−2σsinωdt4σ2σ2+σ2te−σt+−4σ2sinωdt4σ24σ2σ2−2σsinωdt4σ2+σ2σ3−2σcosωdt4σ2+σ2σ3e−σt=−4σ4εEωiecosL14σ4+cosωdt4σ3−sinωdt4σ3te−σt+−sinωdt2σ4−cosωdt4σ4e−σt=−εEωiecosL1+σcosωdt−σsinωdtte−σt+−2sinωdt−cosωdte−σt.From equation (6), ϕz1t converges to −εE/ωiecosL, which is the same as the formulae related to the effect of east gyro drift on initial alignment. However, we need to consider when this convergence occurs.Because the gyro drift stability is constrained to less than 0.05°/h in this study, the gyro constant drift is generally less than 0.05°/h. The latitude is set to 53° and(7)εEωiecosL=0.32∘.Consider the four decay oscillation error terms in equation (7):(8)Δϕz1t=0.32°σcosωt−σsinωtte−σt+−2sinωt−cosωte−σt=0.32°2σtcosωt+φ1e−σt+5cosωt+φ2e−σt=0.32°2σt2+5cosωt+φ3e−σt≤0.32°2σt2+5e−σt.Then, taking the 0.01° error band to determine whether the azimuth alignment converges, we obtain(9)Δϕz1t≤0.32°2σt2+5e−σt=0.01∘.Then,σt=5.57 can be calculated, so t=5.57/σ=5.57Td/2π≈0.9Td, which indicates that the compass azimuth alignment converges to the 0.01° error band after approximately 0.9 damping oscillation periods.Therefore, we conclude that the effect of east gyro drift on compass azimuth alignment can converge within 0.9 damped oscillation periods. However, the analysis here is too conservative; as the gyro precision is improved, its constant drift will be lower and its convergence time will be reduced. Table1 lists the convergence time of several typical gyro constant drifts.Table 1
Convergence time of different gyro constant drifts.
Gyro drift (°/h)0.050.040.030.020.01Convergence time (Td)0.90.850.790.70.58According to Table1, the convergence time of compass azimuth alignment is related to the selected second-order damping oscillation period and the gyro constant drift. When the gyro constant drift is determined, there is a fixed-proportion relationship between the convergence time and the second-order damping oscillation period. When the second-order damping oscillation period is determined, a larger gyro constant drift results in a longer convergence time, and vice versa.It should be noted that, due to adoption of the inequality amplification in the theoretical calculation of convergence time, the actual convergence time is often less than the calculated theoretical time. In other words, the convergence time given here is more conservative and denotes the maximum time that the system takes to stabilize.


## 3.3. Effect of Initial Azimuth Error on Convergence Time

This section mainly analyzes the influence of initial azimuth error on convergence time. According to Section2, the system response term related to the initial azimuth error is(10)ϕz2s=ss+K1+ωs2K2+1s+K4Δsϕz0.According to the values of the corresponding parameters in Section2,(11)ϕz2s=s3+4σs2+8σs+8σs+σ2+ωd22ϕz0.By performing the inverse Laplace transform, the obtained function in the time domain is(12)ϕz2t=ϕz0t23σ2−ωd2ωd2+12σ3σ2+ωd2e−σtsinωdtωd3+ωd2−t2σ3σ2−ωd2e−σtcosωdtωd2.Becauseωd=σ,(13)ϕz2t=ϕz0σt+2e−σtsinωdt+−σt+1e−σtcosωdt.In equation (13), ϕz2t eventually converges to 0; however, the target of this research is determining when the convergence occurs.Thus, due to(14)σt+2e−σtsinωt+−σt+1e−σtcosωt=2σt2+2σt+5cosωt+φ1e−σt≤2σt2+2σt+5e−σt,the following is true:(15)ϕz2t≤ϕz02σt2+2σt+5e−σt.Then, using an error band of 0.01° to determine the initial alignment convergence,(16)ϕz2t≤ϕz02σt2+2σt+5e−σt=0.01∘.In equation (16), ϕz2t is proportionate to the initial azimuth error ϕz0, so its convergence time is also related to the initial azimuth error. Furthermore, the greater the value of ϕz0, the longer the convergence time.However, before compass alignment, coarse alignment is generally required to guarantee the linear characteristic of the compass azimuth alignment error model. When the initial azimuth error is within 5°, the inertial error model has better linearity; the initial alignment performs well for compass azimuth alignment. When the initial azimuth error is more than 5°, the inertial error model has inferior linearity; the performance of compass azimuth alignment gradually decreases. Therefore, in practical applications, the coarse alignment error is always controlled within 5°. In fact, this section only considers a convergence time of the initial azimuth error within 5°. Based on the typical initial errors listed above, the convergence time of compass azimuth alignment is shown in Table2.Table 2
Convergence time of different initial azimuth misalignment.
Initial azimuth misalignment (°)0.51234Convergence time (Td)11.11.251.31.35We conclude from Table2 that the convergence time of compass azimuth alignment is related to the selected second-order oscillation period and the initial azimuth error. When the initial azimuth error is determined, the convergence time is fixed in proportion to the second-order oscillation period. When the second-order oscillation period is determined, the greater the initial azimuth error, the longer the convergence time. As in Section 3, part B, the convergence times listed in Table 2 are relatively conservative, and the actual convergence time is generally less than the calculated value.Due to adoption of the inequality amplification in the theoretical calculation of convergence time, the actual convergence time is often less than the calculated theoretical time. In other words, the convergence time given here is more conservative and denotes the maximum time that the system takes to stabilize.


## 3.4. Combined Effect of Both Errors on Convergence Time

During the actual initial alignment of INS, constant drift and an initial azimuth error both exist. Therefore, it is necessary to analyze the influence of both errors on the convergence time to provide theoretical guidance for parameter setting in practical applications.The gyro constant drift and initial azimuth error are mutually independent. Based on automatic control theory, both responses obtained by the transfer function of compass azimuth alignment can exhibit linear superposition. So, under both errors, the response function of the compass azimuth alignment error is(17)ϕz3t=ϕz1t+ϕz2t=−εEωiecosL1+σcosωdt−σsinωdtte−σt+−2sinωdt−cosωdte−σt+ϕz0σt+2e−σtsinωdt+−σt+1e−σtcosωdt.In Equation17, when both errors exist, ϕz3t converges to −εE/ωiecosL, for which the error decay oscillation term is(18)Δϕz3t=−εEωiecosLσcosωdt−σsinωdtte−σt+−2sinωdt−cosωdte−σt+ϕz0σt+2e−σtsinωdt+−σt+1e−σtcosωdt=ϕz0+εEωiecosLσt+2sinωdt+−σt+1cosωdte−σt.Therefore,(19)Δϕz3t=ϕz0+εEωiecosLσt+2sinωdt+−σt+1cosωdte−σt≤ϕz0+εEωiecosL2σt2+2σt+5e−σt,Taking the 0.01° error band as the criteria of initial alignment convergence,(20)Δϕz3t≤ϕz0+εEωiecosL2σt2+2σt+5e−σt=0.01∘.From equation (20), Δϕz3t is proportional to ϕz0+εE/ωiecosL, so the convergence time is associated with the initial azimuth angle. When the absolute values of ϕz0 and εE are unchanged, Δϕz3t under opposite signs of ϕz0 and εE is less than that under the same signs. As we perform the most conservative estimation in our convergence time analysis, both signs are the same. Typical convergence times of the initial azimuth error and gyro constant drift are listed in Table 3.Table 3
Convergence time (Td) of initial azimuth error and gyro constant drift.
Initial azimuth error (°)0.5123450.01°/h1.001.121.251.311.351.400.02°/h1.011.131.251.311.361.400.03°/h1.031.141.261.321.371.400.04°/h1.051.151.261.321.371.410.05°/h1.071.161.261.331.381.41By comparing Tables2 and 3, we see that, relative to the initial azimuth error, the gyro constant drift has a minimal influence on the convergence time, whereas the main factor influencing compass azimuth alignment is the initial azimuth error. When this error is within 5°, compass azimuth alignment will converge within 1.41 Td.


## 4. Simulation Verification

### 4.1. Simulation of the Effect of Gyro Constant on Convergence Time

We assume that the reference coordinate system is the east-north-up coordinate system and the local latitude is 53°. Only theX-axis gyro has a constant drift of 0.05°/h. The INS attitudes are 0°, 0°, and 0°. The initial attitude errors are all 0°. Td is set to 200 s, 300 s, 400 s, and 500 s, respectively. The simulation time is 600 s. The obtained convergence curve of the initial alignment error is shown in Figure 2 for different convergence times.Figure 2
Convergence curves of the initial alignment error for differentTd.According to the related theory of initial alignment (formula (7)), at this time, the initial alignment error limit is −0.3165°. Therefore, according to Figure 2 and the definition of the 0.01° error band used in this study, we assume that the compass azimuth alignment converges when the initial alignment error curve finally passes through −0.3165°. Thus, the convergence times for different Td are shown in Table 4.Table 4
Convergence times for differentTd with a 0.05° gyro constant drift.
Td (s)200300400500Convergence time (s)177265354443Ratio0.890.880.890.89According to Table4, although the convergence time is different for different Td, the ratio of convergence time to Td coincides well with the theoretical analysis, which verifies the validity of our proposed analytical method.Due to limited space, the convergence curves of other gyro drifts are not presented here. For gyro drifts of 0.01°/h, 0.02°/h, 0.03°/h, and 0.04°/h, theTd convergence times are 200 s, 300 s, 400 s, and 500 s, respectively. Therefore, the ratio of convergence time to Td is shown in Table 5.Table 5
Convergence times for differentTd and different gyro constant drifts.
Gyro drift (°/h)Td (s)2003004005000.04Convergence time (s)170255341426Ratio0.850.850.850.850.03Convergence time (s)158238317396Ratio0.790.790.790.790.02Convergence time (s)101152203254Ratio0.510.510.510.510.01Convergence time (s)93139186232Ratio0.470.460.470.46It can be seen from Table5 that when the gyro drift is over 0.02°/h, it exhibits good agreement with the theoretical calculation. Moreover, the actual convergence time is less than the theoretical convergence time with a gyro drift of 0.02°/h and 0.01°/h. This is because the amplification of inequality is adopted during the process of theoretical derivation, resulting in an overconservative convergence time in the theoretical calculation. Despite this, the analysis method of this study is generally valid.


### 4.2. Simulation of the Effect of Initial Azimuth Error on Convergence Time

We assume that there is no inertial device error, the initial azimuth error is 5°,Td is set to 200 s, 300 s, 400 s, and 500 s, respectively, the simulation time is 1000 s, and the attitude is 0°, 0°, and 0°. The resulting simulation results are shown in Figure 3 and Table 6.Figure 3
Convergence curves of the initial azimuth error for differentTd.Table 6
Convergence times for differentTd with a 5° initial azimuth misalignment.
Td (s)200300400500Convergence time (s)277416555694Ratio1.391.391.391.39According to Table6, although the convergence time is different for different Td, the ratio of convergence time to Td is consistent with the theoretical analysis, which verifies the validity of our proposed analytical method. Due to limited space, the convergence curves of other gyro drifts are not shown 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 of convergence time to Td.Table 7
Convergence times for differentTd and different initial azimuth misalignment.
Gyro drift (°)Td (s)2003004005004Convergence time (s)270405541676Ratio1.351.351.351.353Convergence time (s)257386515645Ratio1.291.291.291.292Convergence time (s)207311414518Ratio1.041.041.041.041Convergence time (s)199299399498Ratio1.001.001.001.000.5Convergence time (s)188281375469Ratio0.940.940.940.94It can be seen from Table7 that when the initial error is more than 2°, it agrees almost perfectly with the theoretical calculation results. When the initial error is less than 2°, the actual convergence time is less than the theoretical calculation time. This phenomenon, explained briefly in Section 4, part A, is due to the inequality amplification in the theoretical analysis.


### 4.3. Simulation of the Effect of Both Errors on Convergence Time

We assume that there is no error of inertial device, the initial azimuth error is 5°, thex-axis gyro constant drift is 0.05°/h, Td is equal to 200 s, 300 s, 400 s, and 500 s, the simulation time is 1000 s, and the attitude is 0°, 0°, and 0°. The resulting simulation results are shown in Figure 4 and Table 8.Figure 4
Convergence curves for both errors for differentTd.Table 8
Convergence times for differentTd with a 0.05°/h gyro drift and a 5° initial azimuth misalignment.
Td (s)200300400500Convergence time (s)279419559699Ratio1.41.41.41.4Due to limited space, other convergence curves are not shown here. If the initial azimuth errors are 4°, 3°, 2°, 1°, and 0.5° and the gyro constant drift is 0.05°/h,Td is 200 s, 300 s, 400 s, and 500 s, respectively. The convergence times and ratios of convergence time to Td are listed in Table 9. When the gyro drift is set to 0.05°/h, there is a small difference of convergence time for different initial errors. All convergence times are proportionally related to Td. This verifies the validity of our proposed analysis method.Table 9
Convergence times for differentTd and different initial azimuth misalignments.
Initial azimuth error (°)Td (s)2003004005004Convergence time (s)273410548685Ratio1.371.371.371.373Convergence time (s)263395526658Ratio1.321.321.321.322Convergence time (s)209314418523Ratio1.051.051.051.051Convergence time (s)203304406507Ratio1.021.021.021.020.5Convergence time (s)200300400500Ratio1111


## 4.1. Simulation of the Effect of Gyro Constant on Convergence Time

We assume that the reference coordinate system is the east-north-up coordinate system and the local latitude is 53°. Only theX-axis gyro has a constant drift of 0.05°/h. The INS attitudes are 0°, 0°, and 0°. The initial attitude errors are all 0°. Td is set to 200 s, 300 s, 400 s, and 500 s, respectively. The simulation time is 600 s. The obtained convergence curve of the initial alignment error is shown in Figure 2 for different convergence times.Figure 2
Convergence curves of the initial alignment error for differentTd.According to the related theory of initial alignment (formula (7)), at this time, the initial alignment error limit is −0.3165°. Therefore, according to Figure 2 and the definition of the 0.01° error band used in this study, we assume that the compass azimuth alignment converges when the initial alignment error curve finally passes through −0.3165°. Thus, the convergence times for different Td are shown in Table 4.Table 4
Convergence times for differentTd with a 0.05° gyro constant drift.
Td (s)200300400500Convergence time (s)177265354443Ratio0.890.880.890.89According to Table4, although the convergence time is different for different Td, the ratio of convergence time to Td coincides well with the theoretical analysis, which verifies the validity of our proposed analytical method.Due to limited space, the convergence curves of other gyro drifts are not presented here. For gyro drifts of 0.01°/h, 0.02°/h, 0.03°/h, and 0.04°/h, theTd convergence times are 200 s, 300 s, 400 s, and 500 s, respectively. Therefore, the ratio of convergence time to Td is shown in Table 5.Table 5
Convergence times for differentTd and different gyro constant drifts.
Gyro drift (°/h)Td (s)2003004005000.04Convergence time (s)170255341426Ratio0.850.850.850.850.03Convergence time (s)158238317396Ratio0.790.790.790.790.02Convergence time (s)101152203254Ratio0.510.510.510.510.01Convergence time (s)93139186232Ratio0.470.460.470.46It can be seen from Table5 that when the gyro drift is over 0.02°/h, it exhibits good agreement with the theoretical calculation. Moreover, the actual convergence time is less than the theoretical convergence time with a gyro drift of 0.02°/h and 0.01°/h. This is because the amplification of inequality is adopted during the process of theoretical derivation, resulting in an overconservative convergence time in the theoretical calculation. Despite this, the analysis method of this study is generally valid.


## 4.2. Simulation of the Effect of Initial Azimuth Error on Convergence Time

We assume that there is no inertial device error, the initial azimuth error is 5°,Td is set to 200 s, 300 s, 400 s, and 500 s, respectively, the simulation time is 1000 s, and the attitude is 0°, 0°, and 0°. The resulting simulation results are shown in Figure 3 and Table 6.Figure 3
Convergence curves of the initial azimuth error for differentTd.Table 6
Convergence times for differentTd with a 5° initial azimuth misalignment.
Td (s)200300400500Convergence time (s)277416555694Ratio1.391.391.391.39According to Table6, although the convergence time is different for different Td, the ratio of convergence time to Td is consistent with the theoretical analysis, which verifies the validity of our proposed analytical method. Due to limited space, the convergence curves of other gyro drifts are not shown 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 of convergence time to Td.Table 7
Convergence times for differentTd and different initial azimuth misalignment.
Gyro drift (°)Td (s)2003004005004Convergence time (s)270405541676Ratio1.351.351.351.353Convergence time (s)257386515645Ratio1.291.291.291.292Convergence time (s)207311414518Ratio1.041.041.041.041Convergence time (s)199299399498Ratio1.001.001.001.000.5Convergence time (s)188281375469Ratio0.940.940.940.94It can be seen from Table7 that when the initial error is more than 2°, it agrees almost perfectly with the theoretical calculation results. When the initial error is less than 2°, the actual convergence time is less than the theoretical calculation time. This phenomenon, explained briefly in Section 4, part A, is due to the inequality amplification in the theoretical analysis.


## 4.3. Simulation of the Effect of Both Errors on Convergence Time

We assume that there is no error of inertial device, the initial azimuth error is 5°, thex-axis gyro constant drift is 0.05°/h, Td is equal to 200 s, 300 s, 400 s, and 500 s, the simulation time is 1000 s, and the attitude is 0°, 0°, and 0°. The resulting simulation results are shown in Figure 4 and Table 8.Figure 4
Convergence curves for both errors for differentTd.Table 8
Convergence times for differentTd with a 0.05°/h gyro drift and a 5° initial azimuth misalignment.
Td (s)200300400500Convergence time (s)279419559699Ratio1.41.41.41.4Due to limited space, other convergence curves are not shown here. If the initial azimuth errors are 4°, 3°, 2°, 1°, and 0.5° and the gyro constant drift is 0.05°/h,Td is 200 s, 300 s, 400 s, and 500 s, respectively. The convergence times and ratios of convergence time to Td are listed in Table 9. When the gyro drift is set to 0.05°/h, there is a small difference of convergence time for different initial errors. All convergence times are proportionally related to Td. This verifies the validity of our proposed analysis method.Table 9
Convergence times for differentTd and different initial azimuth misalignments.
Initial azimuth error (°)Td (s)2003004005004Convergence time (s)273410548685Ratio1.371.371.371.373Convergence time (s)263395526658Ratio1.321.321.321.322Convergence time (s)209314418523Ratio1.051.051.051.051Convergence time (s)203304406507Ratio1.021.021.021.020.5Convergence time (s)200300400500Ratio1111


## 5. Experiment

In order to test the effect of east gyro drift and initial azimuth error on the compass azimuth alignment convergence time, we implemented three sets of actual ship experiments in Harbin, China. The main equipment includes the self-made strapdown INS and a high-precision inertial navigation system PHINS. We used the data output by PHINS as the reference value. In the experiment, the constant gyro drifts and the accelerometer biases were set to 0.05 deg/h and 0.0001 g, respectively.The experiment procedure was as follows. The initial azimuth errors are 5° and 1°, and two sets of experiments were performed based on different azimuth errors. In each set of experiments,Td is equal to 200 s, 300 s, 400 s, and 500 s. The experiment results are shown in Table 10 and Figure 5.Table 10
Experiment results.
Initial azimuth error (°)Td (s)2003004005005Convergence time (s)278420558699Ratio1.3901.4001.3951.3981Convergence time (s)201305409510Ratio1.0051.0171.0221.020Figure 5
Convergence curves of the initial azimuth error 1° for differentTd.Obviously, the experimental results and the simulation results are basically the same. When the east gyro drift and the initial azimuth error are considered, the initial azimuth error plays a major role in the convergence time, compared to the gyro constant drift.


## 6. Conclusion

In compass azimuth alignment, precision conflicts with rapidity. Within a limited initial alignment time, the expected random disturbance is filtered as much as possible, and compass azimuth alignment is required to converge. Thus, it is necessary to analyze the convergence time of compass azimuth alignment and determine optimum parameters based on the precision of inertial devices and a reasonable selection of corresponding parameters. In this article, by analyzing the system transfer function of compass azimuth alignment, we obtain the response function of east gyro drift and initial azimuth error in the frequency domain, which are then transformed to the time-domain response function by the inverse Laplace transform. Therefore, we analyze the effect of east gyro drift and initial azimuth error on the convergence time of compass azimuth alignment in the time domain. Our analytical results indicate that convergence time is related to gyro drift, initial azimuth error, and the second-order damping oscillation period. When an error band of 0.01° is used to determine the convergence and the gyro drift is less than 0.05°/h, the compass azimuth alignment will converge within 0.9 damping oscillation periods due to gyro drift. When the initial azimuth error is less than 5°, the compass azimuth alignment will converge within 1.4 damping oscillation periods due to the initial azimuth error. When both errors are considered, the initial azimuth error plays a major role in the convergence time, compared to the gyro constant drift. Our proposed method provides a theoretical basis for setting the corresponding parameters and controlling the convergence time during compass azimuth alignment.



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