Gyrocompass Alignment

Journal of Computational Information Systems 11: 5 (2015) 1719–1727 Available at http://www.Jofcis.com

A Novel Gyrocompass Alignment Method for Strapdown Inertial Navigation System ⋆ Wei GAO, Baofeng LU ∗ College of Automation, Harbin Engineering University, Harbin 150001, China

Abstract In order to avoid the nonlinear problem of error model caused by the initial level misalignment, preliminary leveling operation is necessary for traditional gyrocompass alignment. However, due to the exist of preliminary leveling operation, the total alignment time will be lengthened. For the purpose of reducing alignment time, a novel gyrocompass alignment method is proposed in this paper. This method is accomplished by utilizing the different error propagation characteristics of two different level alignment algorithms, and the procedure of azimuth alignment is performed by opening the azimuth loop. So preliminary leveling operation can be removed and the duration of initial alignment can be shortened. Simulation results show that the alignment speed of novel gyrocompass alignment method is faster than that of the traditional one. Keywords: Style Strapdown Inertial Navigation System; Nonlinear Problem; Error Model; Gyrocompass Alignment

1

Introduction

The initial alignment of inertial navigation system (INS) is an important process performed prior to normal navigation [1]. It is well known that the initial alignment of the system is of fundamental importance to the resulting navigation accuracy [2]. Therefore, many researchers have investigated the topic [3-9], mainly concentrated on gyrocompass alignment and optimal estimation techniques. In contrast with optimal estimation techniques, the former method doesn’t need precise mathematical and noise model. And with many years’ development, gyrocompass alignment based on classical control theory is very mature. As early as in 1961, Cannon presented gyrocompass alignment method for platform INS [3], in which the initial level misalignment has a preponderant effect on the azimuth transient, so azimuth alignment is performed only after the system has been leveled using a preliminary leveling operation. Later, gyrocompass alignment is described extensively in the literatures [4, 5], including alignment technique and error analysis. In recent years, with the development of strapdown INS, gyrocompass alignment is applied to strapdown INS [1, 6]. In conclusion, the general process of azimuth alignment in gyrocompass ⋆ ∗

Project supported by the National Nature Science Foundation of China (No. 51379042). Corresponding author. Email address: lu bao [email protected] (Baofeng LU).

1553–9105 / Copyright © 2015 Binary Information Press DOI: 10.12733/jcis13342 March 1, 2015

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technique is accomplished by closing the azimuth loop. Therefore, preliminary leveling operation is necessary in order to avoid the nonlinear problem of error model caused by the initial level misalignment, and it will lengthen the total alignment time. In this paper, a novel gyrocompass alignment method for strapdown INS is provided, in which the procedure of azimuth alignment is performed by opening the azimuth loop. Then preliminary leveling operation can be removed, and the duration of initial alignment can be shortened. This method is realized by putting two different level alignment algorithms into operation simultaneously, and the estimate of azimuth misalignment can be obtained by the different characteristics of error propagation between them.

2

Strapdown INS Error Model

Before providing algorithms for level alignment, a strapdown INS error model is necessary to be derived. In the derivation process, we assume that the vehicle is at rest. Initial alignment is usually performed when the strapdown INS is resting at a location whose geographic coordinates are known almost perfectly [10]. So the position equation and the position error equation can be ruled out. Under the previously discussed hypothesis, the true equations of strapdown INS can be written as b n C˙ bn = Cbn [ωib − Cnb ωie ]× V˙ n = C n f b + g n = 0 b

(1) (2)

where Cbn is the direction cosine matrix (DCM), related the body frame b to the navigation frame b n n, ωib is the true angular velocity vector, ωie is the angular velocity vector of earth frame with n ˙ respect to inertial frame, V denotes the true velocity of strapdown INS, f b denotes the true specific force, g n is the gravity vector. And the computational equations of strapdown INS can be represented as b n C˙ bp = Cbp [e ωib − Cpb ωie ]×

(3)

˙n Vb = Cbp feb + g n

(4)

Cbp

where is the direction cosine matrix (DCM), related the body frame b to the platform frame b p, ω eib is the angular velocity vector measured by gyros, Vb n is the computed velocity of strapdown INS, feb is the specific force from the accelerometer output. Comparing Eq. (1) with Eq. (3), the attitude error equation is obtained, and can be represented by the psi-angle model [11, 12] n ψ˙n = −ωie × ψ n − Cbn εb (5) where ψ n = [ψE ψN ψU ]T is the misalignment vector, Cbn εb = [εE εN εU ]T , εb is the gyro error. Similarly, comparing Eq. (2) and Eq. (4), yields δ V˙ n = −g n × ψ n + Cbn ∇b

(6)

where δV n = [δVE δVN δVU ]T is the velocity error vector, Cbn ∇b = [∇E ∇N ∇U ]T , ∇b is the accelerometer error.

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3

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Two Kinds of Algorithms Designed for Level Alignment

In this section, we design two kinds of algorithms for level alignment from the control theoretical point of view, and look into the different characteristics between them. Obviously, system represented by Eq. (5) and Eq. (6) is an unstable system. In order to realize level alignment, the signals δ V˙ n need to be fed into controllers that analytically torque the strapdown INS to drive ψ˙n to zero.

3.1

Level alignment I

The first algorithm for level alignment is designed by adding control angular velocity ωIpI to the computational equations of strapdown INS. Then Eq. (3) and Eq. (4) can be rewritten as b n C˙ bpI = CbpI [e ωib − Cpb (ωie + ωIpI )]×

(7)

˙n Vb I = CbpI feb + g n

(8)

The attitude error model and the velocity error model are n ψ˙In = −ωie × ψIn − Cbn εb + ωIpI

δ V˙ In = −g n × ψIn + Cbn ∇b Where

ωIpI

(9) (10)

T

= [ωIE ωIN 0] , ωIE and ωIN are as shown in Fig. 1 and Fig. 2.

Fig. 1: East error model of level alignment I

Fig. 2: North error model of level alignment I Where kIa = 2ξωn , kIb = ωn2 /g. ξ is usually taken as 0.707, and ωωn is an adjustable variable. In our analysis, we assume that the inertial sensor errors are basically constant drifts; then the steady-state outputs of this algorithm due to constant error inputs are ψIN ss =

2ξ ∇E − · (εN + ψIEss Ω sin φ) g ωn

(11)

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∇N 2ξ − · (εE + ψIU ss Ω cos φ − ψIN ss Ω sin φ), g ωn Neglecting the small deviation, we have ψIEss = −

ψIN ss = ψIEss = −

3.2

∇E 2ξ ∇N − · (εN − Ω sin φ) g ωn g

∇E ∇N 2ξ · (εE + ψIU ss Ω cos φ − − Ω sin φ). g ωn g

(12)

(13) (14)

Level alignment II

Similarly, the second algorithm is designed by adding control angular velocity ωΠpΠ to the computational equations of strapdown INS. The form of ωΠpΠ is represented as ωΠpΠ = [ωΠE ωΠN 0]T , where ωΠE and ωΠN are as shown in Fig. 3 and Fig. 4.

Fig. 3: East error model of level alignment II

Fig. 4: North error model of level alignment II Where kΠa = 3ξωn , kΠb = ωn2 (1 + 2ξ 2 )/g, kΠc = ξωn3 /g. From Fig. 3 and Fig. 4, the steady-state outputs of this algorithm due to constant error inputs can be expressed as ∇E ψΠN ss = (15) g ∇N ψΠEss = − (16) g Comparing Eq. (14) with Eq. (16), the different characteristics of error propagation between these two algorithms can be found. That can be described as follows: different networks designed lead to the different steady-state outputs of strapdown INS, and the difference between them contains the azimuth misalignment information.

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4

1723

Open Azimuth Alignment Scheme

The characteristics of level misalignments have been investigated in the above section. Next, the error propagation of ψIU and ψΠU is necessary to be analyzed too. From Eq. (9), the error equation of ψIU is ψ˙ IU = ψIE Ω cos φ − εU (17) since ψIE Ω cos φ and εU are small values, they can be ignored. Then the error equation of ψIU becomes ψ˙ IU = 0, similarly, the error equation of ψΠU also can be written as ψ˙ ΠU = 0. Under the above analysis, we can obtain ψIU = ψIU (0) and ψΠU = ψΠU (0). The specific description of azimuth alignment scheme is provided in the following. Strapdown INS is the modern system that has removed most of the mechanical complexity of platform systems by having the sensors attached rigidly, or ‘strapped down’, to the body of the host vehicle [13]. So the measurements of specific force and angular rate provided by the inertial sensors can accurately represent the vehicle motions, without the effects of strapdown computational algorithms. In other words, no matter which algorithm is utilized in strapdown INS, the outputs of the inertial sensors are not changed. With the development of navigation computer, several strapdown computational algorithms can be utilized during its operation simultaneously, and these algorithms will be uncorrelated with each other. Here, after coarse alignment, the algorithms previously designed for level alignment are put into operation at the same time. Therefore, the error sources of these two algorithms are completely the same with each other. Then Eq. (14) becomes ψIEss = −

∇N 2ξ ∇E − · (εE + ψIU ss (0)Ω cos φ − Ω sin φ) g ωn g

(18)

From Eq. (7), we can get the attitude matrix of level alignment I, it can be written as CbpI = (I − ψIn ×)Cbn

(19)

Similarly, CbpΠ can be obtained and represented as CbpΠ = (I − ψΠn ×)Cbn

(20)

b = Cnb (I + ψΠn ×) CpΠ

(21)

Consequently, b Postmultiply Eq. (19) by CpΠ to obtain b = I − (ψIn − ψΠn )× CbpI CpΠ

(22)

On the other hand, comparing Eq. (16) with Eq. (18), yields ψΠU (0) = −

εE ∇E ωn · (ψIEss − ψΠEss ) − + tan φ 2ξΩ cos φ Ω cos φ g

(23)

From Eq. (22), we can get the value of ψIEss − ψΠEss , and the estimate of ψΠU (0) is provided by ψbΠU (0) = −

ωn · (ψIE − ψΠE ) 2ξΩ cos φ

(24)

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Thus, the steady error of this estimate is δψU = ψΠU − lim ψbΠU (0) = −

∇E εE + tan φ Ω cos φ g

An accurate attitude matrix Cbp can be obtained by   1 −ψbΠU (0) 0   pΠ C bΠU (0) Cbp =  ψ 1 0   b 0 0 1

(25)

(26)

With the attitude matrix CbpΠ compensated by ψbΠU (0), initial alignment could be realized. Due to ψbΠU (0) is not fed into the level alignment algorithms, the preliminary leveling operation can be removed. From Eq. (24), we note that the level and azimuth alignment can be accomplished at the same time. The steady-state alignment errors of this method affected by the sensor errors are as shown in the following ∇N δψE = ψΠEss = − (27) g ∇E δψN = ψΠN ss = (28) g ∇E εE + tan φ (29) δψU = ψΠU − lim ψbΠU (0) = − Ω cos φ g

5

Modified Open Azimuth Alignment Scheme

In many operational situations it is desired to align the inertial system with the disturbance of vehicle motions, and to do so on a ship or aircraft which is in disturbing motions caused by sea waves or gusty winds. The disturbing motions can be considered to be sinusoidal [3]. Each of the level alignment systems previously designed has a strong capability of restraining disturbance. On the other hand, the term appears in the denominator in Eq. (24), so that the disturbing motions have a large effect on the azimuth alignment. In order to reduce the influence of disturbance motions on azimuth alignment, a modified alignment method is designed, and expressed by Fig. 5.

Fig. 5: Modified open azimuth alignment scheme From Fig. 5, we would note that a low-pass filter is added to the alignment system. The reason for this design is to reduce the influence of disturbance motions on azimuth misalignment. The design law of this low-pass filter is provided in the following:

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1) The magnification of this low-pass filter in low frequency must be equivalent to 1, in order to assure the accuracy of azimuth misalignment information; 2) In high frequency, for the purpose of reducing the influence of disturbing motions, it needs to have a strong capability of restraining disturbance. Taking the last law into consideration, two kinds of typical minimum phase systems would meet our demands. They are given as follows G1 (s) = G2 (s) =

T 2 s2

1 Ts + 1

(30)

1 + 2ζT s + 1

(31)

Where T = 1/ωf . ζ is often set as 0.707, and ωf is an adjustable variable. If only one of them is applied to azimuth alignment, the value of ωf must be smaller for larger disturbances, and it would lengthen the duration of azimuth alignment. Therefore, the low-pass filter is provided as follows G(s) =

1 (T s +

1)(T 2 s2

+ 2ζT s + 1)

(32)

then the value of ωf can be greater, and the duration of azimuth alignment will be shortened. By adopting the alignment scheme shown by Fig. 5, the influence of disturbing motions can be controlled within allowable range.

6

Simulation Results

In this simulation, gyro and accelerometer outputs are generated by the strapdown INS simulator. We assume that the vehicle is in disturbing motions caused by sea waves, and modified open azimuth alignment is performed by strapdown INS. The sensor errors are set as: the gyro constant drift: 0.01◦ /h, the accelerometer bias: 1 × 10−4 g. Under the disturbing motions, the vehicle undertakes angular and lineal movements. In angular movement, the pitch θx , roll θy , and yaw θz are controlled as θx = 7◦ sin(2πt/5 + π/3), θy = 5◦ sin(2πt/7 + π/6), θz = 10◦ sin(2πt/6 + π/4)

(33)

In the lineal movement, the vehicle lineal movement velocities are taken as Vx = −0.3 sin(2πt/10) m/s2 , Vy = −0.2 sin(2πt/6) m/s2 , Vz = −0.4 sin(2πt/8) m/s2

(34)

We assume that the initial misalignments after coarse alignment are 0.2◦ , 0.2◦ and 0.5◦ . The local latitude is φ = 45◦ , and the control variables are set as ξ = ζ = 0.707, ωn = 0.02, ωf = 0.04.

(35)

The results of this simulation are shown in Fig. 6 and Fig. 7, and in order to make a comparison with traditional gyrocompass alignment scheme, the azimuth alignment result of traditional

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Fig. 6: The level misalignments of alignment for disturbing motions

Fig. 7: The azimuth misalignment of alignment for disturbing motions method is given in Fig. 7. Solid and dot lines are corresponding to the novel and traditional methods, respectively. The solid lines in Fig. 6 and Fig. 7 show the dynamic and steady characteristics of novel method, including dynamic amplitudes of the misalignments, steady-state alignment errors, and alignment time. In dynamic state, azimuth misalignment has a large peak mainly caused by the initial level misalignment. Then if the azimuth loop is closed, the nonlinear problem of error model will be produced. In traditional method, the nonlinear problem is resolved by using a preliminary leveling operation, the starting and ending time of this operation are corresponding to point A and point B in Fig. 7 respectively. In steady state, the mean values of these misalignments are about −0.0057◦ , 0.0057◦◦ and 0.048◦ , correspond to the Eqs. (27), (28) and (29). The influence of disturbance motions is controlled within allowable range, and comparing with traditional method, the duration of alignment is shortened.

7

Conclusions

A novel operational approach for gyrocompass alignment is proposed, and provides fresh insights into strapdown INS. This approach shows that different computational algorithms lead to different characteristics of error propagation, and azimuth alignment can be achieved by utilizing these

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different characteristics. The level and azimuth alignment can be accomplished simultaneously as the vehicle is at rest. In order to reduce the influence of disturbing motions, A modified alignment scheme is provided. From the scheme, we get the conclusions in the following: 1) The performance of level alignment is only determined by the choice of ωn ; 2) The performance of azimuth alignment is determined by both the choices of ωn and ωf . Then without changing the performance of level alignment, we can adjust the performance of azimuth alignment by altering the value of ωf . That is to say, navigation computer can estimate the azimuth misalignment more flexibly.

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