2014_10_8_3255_3264

Journal of Computational Information Systems 10: 8 (2014) 3255{3264Available at http://www.Jofcis.com

Strapdown INS Initial Alignment in Inertial Frame using

H1 Filter ?

Fujun PEI, Xuan LIU, Li ZHUSchool of Electronic Information & Control Engineering, Beijing University of Technology, Beijing

100124, China

Abstract

Initial alignment is the key procedure in the strapdown inertial navigation system (INS), which directlyrelates to the working precision of strapdown INS. But the tradition alignment method can't nishalignment under the dynamic disturbance condition. In this paper, a novel alignment method using H1lter was investigated to solve this problem. In the coarse alignment procedure, the invariance propertyof gravity in the inertial frame was used to deal with the disturbances. In the ne alignment procedure,strapdown INS alignment error model under the dynamic condition was established. Moreover, becauseof the uncertainty observation noise in the velocity measurements, the H1 robust lter was designed toestimate the misalignment angles accurately. Finally, simulation results demonstrated that the proposedmethod can depress the random disturbances more eectively than the traditional alignment method.

Keywords: Strapdown INS; Initial Alignment; Inertial Frame; H1 Filter

1 Introduction

The purpose of strapdown INS initial alignment is to determine the coordinate transformationmatrix from body frame to navigation frame. Then misalignment angles can be compensatedbased on their estimated values [1]. Since the initial alignment directly relates to the workingprecision and start-up time of strapdown INS, it is one of the most important factors in theoperation of strapdown INS [2]. In recent years, a lot of research has been developed for strapdownINS alignment which contains two phases: coarse alignment and ne alignment.

For the coarse alignment step, the system attitude can be determined directly by using theknown gravity and Earth rate signals in the local level frame and the measurements obtainedby using accelerometers and gyros [3, 4]. However, when marine strapdown INS is under thedynamic disturbance environment, where the disturbed acceleration and rotation velocity exist,the static coarse alignment method cannot be used. Therefore, a new coarse alignment methodfor the marine strapdown INS using the gravity in the inertial frame as a reference has been

?Project supported by the National Science Foundation of China (No. 60975065).Corresponding author.Email address: [email protected] (Fujun PEI).

1553{9105 / Copyright 2014 Binary Information PressDOI: 10.12733/jcis9764April 15, 2014

3256 F. Pei et al. /Journal of Computational Information Systems 10: 8 (2014) 3255{3264

discussed. This coarse alignment considers projection of gravity in the inertial frame to calculatea coarse value of the initial attitude matrix between body initial inertial frame and inertial frame[5]. It could withstand random movements under the dynamic disturbance condition. And theresults can meet the requirement of coarse alignment for strapdown INS in accuracy [6]. But theinertial frame coarse alignment method can only provide a fairly good initial condition for thealignment phase, the ne alignment should be used to compensate the misalignment angles.

As for the ne alignment step, a lot of research has been done on the stationary base algorithms.It is well known that the Kalman lter (KF) is an eective method for linear systems to distinguishthe real states from noisy information and it is an optimal tracking method when the noise issubject to Gaussian distribution. In order to solve the strapdown INS alignment problem underthe marine mooring condition, Gao et al proposed the rapid strapdown INS ne alignment methodin [7]. However, when it comes to uncertain disturbances, the Kalman lter becomes disabled. Inrecent years, H1 lter was widely used in initial alignment for strapdown INS instead of Kalmanlter, and have made a lot of research and simulation [8, 9]. Studies show that the advantageof the H1 lter in comparison with the Kalman lter is that no statistical assumptions on thenoises are required, and the lter is more robust when the noises are uncertainties in a system[10, 11, 12].

In order to deal with the lineal and angular disturbances, a novel initial alignment method wasinvestigated in this paper. On one hand, using of the gravity projection along the inertial frame,could avoid the eects of disturbed accelerates and angular velocities in alignment process. Onthe other hand, compared with the traditional Kalman lter, H1 robust lter can depress therandom disturbances more eectively. Therefore, the proposed novel initial alignment methodcan meet the requirements of strapdown INS alignment under the dynamic disturbance condition.

The remainder of this paper is organized as follows. The algorithm approaches for the inertialframe coarse alignment is presented in Section 2. Section 3 provides the strapdown INS nealignment error model on inertial frame and the mathematical description ofH1 lter is presentedin Section 4. The simulation results are illustrated in Section 5. Finally, the conclusion makes upin Section 6.

2 Coarse Alignment in the Inertial Frame

As mentioned above, the traditional coarse alignment cannot provide accurate values for theinitial attitude angles under the dynamic disturbance condition. To tackle these limitations, aspecial inertial frame, the initial time strapdown INS body inertial frame (ib0 frame) is dened andselected as transition reference frames in coarse alignment. The specic denition of coordinateframes used in this paper are quoted from [13]. Therefore, the initial attitude matrix, whichrelates the body frame to the navigation frame, could be described as follows:

cbn (t) = cne c

ei (t) c

iib0cib0b (t) (1)

Where cne and cei (t) are as follows:

cne =

26640 1 0

sinL 0 cosLcosL 0 sinL

3775 (2)

F. Pei et al. /Journal of Computational Information Systems 10: 8 (2014) 3255{3264 3257

cei (t) =

2664cos!ie (t t0) sin!ie (t t0) 0 sin!ie (t t0) cos!ie (t t0) 0

0 0 1

3775 (3)Where L is latitude, !ie is the Earth rotation rate, t0 is the start time of the coarse alignment.

cib0b which represents the transform matrix from the body frame b to the base inertial frame ib0,can be computed recurrently with the gyro outputs through the quaternion attitude algorithm.The matrix initial value for the recurrent computation is unit matrix. Thus the main problem ofthe inertial frame alignment is to compute the matrix ciib0 . While the vehicle is mooring on thesea or aside the dock, liner movements such as surge, sway and heave are exist aroused by thesway. The accelerometer's measured specic force not only contains gravity. The integration ofthe accelerometer's measured specic force projected in the body inertial frame can be describedas follows:

bV ib0 = R tkt0C ib0b f^

b

= R tkt0Cib0i C

ibg

bdt+R tkt0Cib0b a

bLAdt+

R tkt0C ib0b a

bDdt

= C ib0iR tkt0gidt+ V ib0LA + V

ib0

(4)

Where V ib0LA is the induced velocity due to the lever-arm and can be calculated using the knownlever-arm as described:

V ib0LA=

Z tkt0

aib0LAdt = Cib0b

!bib rb

(5)

Where abD is the disturbing acceleration and Vib0 can be ignored because it is much smaller

than the velocity caused by the gravity. Consequently, Eq. (4) becomes:

gV ib0 = dV ib0 V ib0LA = C ib0i Z tkt0

gidt = Cib0i V i (6)

Considering,

gi = C ieCeng

n =

2664g cosL cos!ietg cosL sin!iet

g sinL

3775 : (7)

vi (tk) =

2664g cosL!ie

sin (!ietk)g cosL!ie

(1 cos (!ietk))tkg sinL

3775 (8)In term of Eq. (6), we calculate ~V ib0 and V i at tk1 and tk2 respectively. Then,(

ib0 (tk1) = cib0i

i (tk1)

ib0 (tk2) = cib0i

i (tk2)(9)


From (9), cib0i could be calculated by the following Eq. (10):

C iib0 =

2664[vi (tk1)]

T

[vi (tk2)]T

[vi (tk1) vi (tk2)]T

37751 2664

[vib0 (tk1)]T

[vib0 (tk2)]T

[vib0 (tk1) vib0 (tk2)]T

3775 (10)After deriving ciib0 , the attitude matrix c

bn can be calculated through Eq. (1).

3 Strapdown INS Alignment Error Model

The coarse alignment method investigated in the Section 2 is eective for strapdown INS indisturbance environment. But it only accomplished the coarse estimation of attitude matrix fromthe body frame to the navigation frame. In this section, the strapdown INS error model in inertialframe is provided for the ne alignment process.

Use (1) for reference, cbn(t) at the end of the ne alignment process can be represented as

cbn = cne c

ei cib (t) (11)

cbi(t) can be described asCib(t) = [C

i0i (t)]

1Ci0b (t) (12)

where,

C i0i (t) = I 'i(t) =

26641 'iz (t) 'iy (t)

'iz (t) 1 'ix (t)'iy (t) 'ix (t) 1

3775 (13)The attitude matrix C i

0b (0), which relates the body frame to the computed inertial frame at the

beginning of the ne alignment. And the misalignment angles between inertial frame i and thecomputed i0 are

'i (t) =h'ix (t) '

iy (t) '

iz (t)

iT(14)

Using the attitude quaternion algorithm C i0b (t) can be computed. Thus the main problem of

the ne alignment becomes the estimation of misalignment angles. The accelerometer's measuredspecic force projected in the frame i' can be described as follows:

Ci0b f^

b = C i0b [(I + KA)(I + A)(gb + abLA + abD) +r]

gi + 'i gi + Cibr+ C i0b abLA + Ci0b abD+C i

0b (KA + A)(gb + abLA + abD)

(15)

Considering, accelerometer's constant bias rb and observation white noise ab,r = rb + ab (16)

Then, (15) can be written as:

Ci0b f^

b + gi C i0b abLA = 'i gi + Cibrb + C ibab + ai (17)


ai = Ci0b (KA + A)(gb + abLA + abD) (18)

Integrating (17) from t1 to t yields,R tt1C i

0b f^

b +R tt1gi R t

t1Ci

0b a

bLAd

=R tt1'i gid + R t

t1C ibrbd +

R tt1Ciba

bd +R tt1aid

(19)

The measurement vector is:

Z=~V i = V if V ig V iLA = V i + VW + V iD (20)

Where, V if =R

tt0Ci

0b f^

bd , V ig =R

tt0gid , V iLA=

R tt1C i

0b a

bLAd = C

i0b (!

bib rb), V iD=

R tt1aid ,

V i =

Z tt1

'i gid +Z tt1

C ibrbd +Z tt1

Cibabd (21)

In order to estimate the misalignment angles by using a H1 lter, the velocity-error equationand the misalignment-angle equation on the inertial frame can be written as

_V i(t) = gi(t) 'i(t) + Cib(t)rb + Cib(t)ab_'i(t) = Cib(t)"b Cib(t)!b

(22)

Then the state equation can be represented as

_X(t) = A(t)X(t) +B(t)W (23)

where,

X(t) =

"V ix(t); V

iy (t); V

iz (t); '

ix(t); '

iy(t); '

iz(t);

"bx; "by; "

bz;rbx;rby;rbz

#T(24)

W =abx; a

by; a

bz; w

bx; w

by; w

bz; 0; 0; 0; 0; 0; 0

T(25)

A(t) =

266664033 [gi(t)] 033 Cib(t)033 033 Cib(t) 033033 033 033 033033 033 033 033

377775 (26)

B(t)=

266664Cib(t) 033 033 033033 C ib(t) 033 033033 033 033 033033 033 033 033

377775 (27)Furthermore, the measurement equation can be formulated as

Z(t)=HX(t) + VW + ViD (28)

H=hI33 039

i: (29)

Where V iD is uncertainty observation noises, consists of disturbing velocity caused by the ship'spitch and roll. Vw is observation white noise.


4 H1 Filter Design for Fine Alignment

For the strapdown INS error equation described as Eq. (28), the interference of acceleration isintroduced. V iD is uncertainty observation noises, consists of disturbing velocity caused by theship's pitch and roll. The Kalman lter is a common way for linear systems when the noise issubject to Gaussian distribution. However, when it comes to random disturbances, the Kalmanlter becomes not so eective. Because of the uncertainty observation noise, the H1 robust lteris designed in the ne alignment procedure. The error model which we discussed above can beconsidered as the following state-space system:

XK = K1XK1 + K1WK1YK = HKXK + VK

ZK = LKXK

(30)

LK makes the ratio of estimation error signal energy and disturbance energy less than thepre-specied number [14]. And ltering error is dened as:

ek = Z^K LKXK (31)

For a xed positive number > 0, the job is nding H1 suboptimal estimation Z^K = Ff (y0; y1;: : : ; yk) to make jjTk(Ff )jj1 < . It can be shown as:

infFf

supX0;W2h2;V 2h2

jjekjj22jjX0 X^0jj2P10 + jjWkjj

22 + jjVkjj22

< 2 (32)

The mathematical equations of H1 ltering is as follows:

S^KnK = LKX^KnKX^K+1nK+1 = K+1nKX^KnK +KK+1(ZK+1 HK+1K+1nKX^KnK)KK+1 = PK+1H

TK+1(HK+1PK+1H

TK+1 +RK+1)

1

PK+1 = K+1nKPKTK+1nK + KTK

K+1nKPKhHTK L

TK

iR1e;k

"HK

LK

#PK

TK+1nK

Re;k =

"RK 0

0 2I

#+

"HK

LK

#PK

hHTK L

TK

i(33)

By comparing, it can be seen that the structure of Kalman lter and H lter has a lot incommon. In fact, when ! 1, H1 lter will degenerate into Kalman lter. That is to say, ifthe norm index is removed, H1 lter is equivalent to Kalman lter. Therefore in the design ofH1 lter, is set as small as possible, under the condition that the Riccati equation solution Pis denite positive.


5 Simulation Results

In this section, the simulation was carried out to test the coarse alignment and the ne alignmentused in our proposed method. The simulation conditions are set as follows: the gyro constantdrift: 0.01/h, the gyro random noise: 0.001/h, the accelerator bias: 1104g, the acceleratormeasurement noise: 1105g. In these simulations, the ship is assumed to be on the berth andis rocked by the surf. The roll , pitch and heading , resulting from the ship's rocking aredescribed as follows:

= 30+5cos(2

7t+

3); = 7cos(

2

5t+

4); = 10cos(

2

6t+

7)

The velocity caused by heave, surge and sway is as follows:

VDi = ADi!Di cos(!Dit+ 'Di):

Where i = x; y; z, ADx = 0:02m, ADy = 0:03m, ADz = 0:3m, !Di =2TDi

, TDx = 7s, TDy = 6s,

TDz = 8s. 'Di obeys the uniform distribution on the interval [0, 2].

Simulation 5.1

The coarse alignment lasts for 120 seconds. The values of tk1 and tk2 in (9) are set to the50 second and the 120 second respectively. The simulation for the coarse alignment runs for 50times. The statistics for coarse alignment results are listed in Table 1.

Table 1: Statistics for coarse alignment results

Parameter item(deg) Max. Min. Mean

Pitch error 0.3012 -0.6135 0.0054

Roll error 0.1046 -0.0687 0.0322

Yaw error 1.6502 -0.2894 0.3973

From Table 1, it is clear that the level attitude errors of the coarse alignment are less than0.3 degrees and the yaw error is less than 1.7 degrees. The coarse alignment guarantees thesubsequent ne alignment. But it only accomplished the coarse estimation of attitude matrix.So 600 seconds ne alignment was followed. Next, the mean and the maximum of misalignmentangles in Table 1 are used respectively as input for ne alignment to validate the method. Themean of misalignment angles are used in simulation 5.2 and 5.3, whereas maximum misalignmentangles are used in simulation 5.4.

Simulation 5.2

Consider the measurement noises as white noise or random disturbances respectively, compar-isons are made on the H1 lter and the traditional Kalman lter for the ne alignment. Theresults of simulation under white noise condition are shown in Fig. 1. The simulation results showthat the accuracy of H1 lter is a 1ittle lower than that of Kalman lter under white noise. Andthe Kalman lter has better speed of convergence than that of H1 lter. It can be concludedfrom the comparison that Kalman lter is good in dealing system with white noise like staticbase. But taking into account the unknown outside interference and the complicated workingenvironment, the strength of H1 lter becomes apparent.


Fig. 1: The Results of Kalman lter and H1 lter with white noise

Fig. 2: The Results of Kalman lter and H1 lter with mean misalignment angles

Simulation 5.3

When it comes to the random disturbances, the results in Fig. 2 show that the horizon mis-alignment angles can still be accurately estimated with Kalman lter under random disturbances,but the speed and accuracy of azimuth angle are all worse. While H1 lter can still result in a


quick response and a relatively equal accuracy as white noise. According to Fig. 2, convergencetime of level misalignment angle is within 30s, precision for 1300, while azimuth angle convergencetime is 60s, precision about 40. Azimuth misalignment angle even was emanated using Kalmanlter. In a sense it could be said that H1 lter is a robust lter and has a better performancethan Kalman lter. Moreover, since the practical noises are usually uncertain, H1 lter can beapplied more widely than Kalman lter, proved to an available method for initial alignment.

The simulation results in Fig. 1 and Fig. 2 show that the accuracy of H1 lter is a littlelower than that of Kalman lter under white noise. The advantages of H1 lter mainly turn outunder random disturbances condition. Next the simulation 5.4 with the maximum initial guresis made.

Simulation 5.4

The simulation results show that the H1 lter is ecient with large initial attitude error. Theresults are similar to Fig. 2. Compared with the traditional Kalman lter, the H1 lter can eec-tively depress the random disturbances in the velocity measurements caused by the ship rocking.Using H1 lter can improve system sensitivity to noise and the divergence speed of azimuth.Taking into account the unknown outside interference and the complicated working environment,H1 ltering with more robustness should be applied to initial alignment of strapdown INS.

Fig. 3: The Results of Kalman lter and H1 lter with maximum misalignment angles

6 Conclusions

In this paper the gravity in the inertial frame as a reference is used, so as to counteract thedisturbed components. With the deduced strapdown INS model on disturbances base, the relevantH1 lter and Kalman lter are designed, and the performance of the lter is compared. The


simulation results show that: 1) the attitude determined by this novel method can meet theaccuracy requirement of coarse alignment and it guarantees the subsequent ne alignment. 2)compared with the traditional Kalman ltering, the H1 ltering can eectively deal with therandom lineal and angular disturbances in the velocity measurements caused by the ship rocking.

Acknowledgement

This work was supported by the National Science Foundation of China under Grants 60975065,and Supported by Program for the Top Young Innovative Talents of Beijing J2002013201301.

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