Low-frequency Periodic Error Identification and Compensation for Star Tracker Attitude Measurement

Chinese Journal of Aeronautics 25 (2012) 615-621

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Chinese Journal of Aeronautics journal homepage: www.elsevier.com/locate/cja

Low-frequency Periodic Error Identification and Compensation for Star Tracker Attitude Measurement WANG Jiongqia,*, XIONG Kaib, ZHOU Haiyina,b a

Department of Mathematics and System Science, National University of Defense Technology, Changsha 410073, China b

National Laboratory of Space Intelligent Control, Beijing 100048, China

Received 16 May 2011; revised 27 June 2011; accepted 29 September 2011

Abstract The low-frequency periodic error of star tracker is one of the most critical problems for high-accuracy satellite attitude determination. In this paper an approach is proposed to identify and compensate the low-frequency periodic error for star tracker in attitude measurement. The analytical expression between the estimated gyro drift and the low-frequency periodic error of star tracker is derived firstly. And then the low-frequency periodic error, which can be expressed by Fourier series, is identified by the frequency spectrum of the estimated gyro drift according to the solution of the first step. Furthermore, the compensated model of the low-frequency periodic error is established based on the identified parameters to improve the attitude determination accuracy. Finally, promising simulated experimental results demonstrate the validity and effectiveness of the proposed method. The periodic error for attitude determination is eliminated basically and the estimation precision is improved greatly. Keywords: star trackers; attitude determination; low-frequency periodic error; gyro drift; precision analysis

1.

Introduction

1

High-accuracy attitude determination plays an important role in Earth-orientation and satellite control. Kalman filter has been widely used in the classical high-accuracy attitude determination system with star trackers and gyros [1-3]. Generally, the measurement error of star tracker is assumed to be Gaussian in the Kalman filter. However, from a more accurate point of view, the measurement error of star tracker can be concluded into three parts, which are constant deviation, random error and low-frequency periodic error [4]. Therefore, the assumption of Gaussian white noise for star tracker measurement is not appropriate [5-6]. A more *Corresponding author. Tel.: +86-731-84573260. E-mail address: [email protected] Foundation items: National Natural Science Foundation of China (61004081, 11126033); School Advanced Research Foundation of National University of Defense Technology (JC11-02-22) 1000-9361/$ - see front matter © 2012 Elsevier Ltd. All rights reserved. doi: 10.1016/S1000-9361(11)60426-3

accurate measurement error model of star tracker should be provided. Generally, the constant deviation can be considered as the equivalent installation error, which can be compensated by on-orbital calibration [7]. The random error can be weakened by filter algorithm. However, the low-frequency periodic error caused by the satellite orbital motion and the complicated space environment such as thermal effect [8], cannot be directly dealt with using the Kalman filter, and is also difficult to be eliminated by the general on-orbital calibration. Therefore, it is one of the most critical problems for high accuracy satellite attitude determination [9]. Over the past decade, two kinds of approaches have been investigated to overcome this problem. One mainly contributes to hardware, which includes structure optimization and on-orbital temperature control [10-11]. The other one depends on improving attitude determination algorithm, such as optimization design for filter parameters [12] and algorithm structure [13-14]. Since it is difficult to distinguish the low-frequency periodic error and the satellite attitude motion, the es-

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timated satellite attitude parameters cannot be used to identify low-frequency periodic error. Fortunately, the gyro drift is independent of the satellite attitude motion and the orbital period. Therefore, the periodic change of the estimated gyro drift is caused by the low-frequency periodic error of star tracker. In this paper, we analyze the effect of the low-frequency periodic error on the estimated gyro drift, and propose an approach for low-frequency periodic error identification and compensation. 2.

K

¦ [am, j cos( jZ k )  bm, j sin( jZ k )]

where am,j and bm,j are amplitudes of cosine and sine function, Z is periodic signal frequency, and K a positive integer. am,j, bm,j and Z are parameters to be identified. Kalman filter algorithm for attitude determination can be derived according to Eq. (1) and Eq. (3). Its optimal performance index function is [16] xˆ k

where J ( xk )

Gqk x4:6, k

ª x1:3, k º «x » is system state vector, x1:3,k ¬ 4:6, k ¼

[Gq1, k

Gq2, k

Gq3, k ]T error quaternion vector,

Gbk

[Gbx , k

Gby ,k

ª w1:3, k º «w » is system state error vector, w1:3,k and ¬ 4:6, k ¼ w4:6,k are gyro angle random walk and angular rate ranW ª º I  W [k u]  I 3u3 » dom walk respectively. Fk « 3u3 2 « » I 3u3 ¼» 03u3 ¬« is state transition matrix, W filter period, and k [Z x, k Z y , k Z z , k ]T the satellite angular velocity

in the body frame provided by gyros. The form of matrix [k u] is [4] [k u]

Z z , k 0

Zx,k

Z y ,k º

» Z x, k » 0 »¼

(2)

The linear observation equation for the attitude determination system is given as follows: yk

H k xk  'yk  vk

(3)

where vk is observation error vector, the observation vector and the observation matrix are yk

qk , H k

[ I 3u3

03u3 ]

'y2,k

'y3, k ]T

denotes the periodic

error caused by the low-frequency periodic error of star tracker. The periodic error signal 'ym,k (m =1, 2, 3) is assumed as Fourier series form:

xk  xˆ k |k 1

2 Pk|k11

 yk  H k xk

2 Rk1

,

xˆ k

measurement covariance matrix. Assume xˆ k  xˆ k |k 1  k , yk  H k xˆ k k ;  k and k are unknown vector. Usually, if the filter is properly designed, the value of  k and k are very small. The estimated state vector and predicted state vector can be written as follows: xˆ k

ª xˆ1:3, k º « xˆ » , xˆ k |k 1 ¬ 4:6, k ¼

ª xˆ1:3,k |k 1 º « » «¬ xˆ 4:6, k |k 1 »¼

(6)

And then,

yk  xˆ1:3, k

k , xˆ1:3,k  xˆ1:3, k |k 1

1:3,k

(7)

2.2. Effect of low-frequency periodic error on gyro drift Since it is difficult to distinguish the low-frequency periodic error of star tracker and the satellite attitude, the satellite attitude quaternion is not suitable to identify the low-frequency periodic error of star tracker. Nevertheless, the gyro drift is independent of the satellite attitude motion and the orbital period. Therefore, the periodic change of the estimated gyro drift is caused by the low-frequency periodic error of star tracker. In the following part, the effect of the low-frequency periodic error on the estimated gyro drift is analyzed before addressing the problem of low-frequency periodic error identification. The observation yk can be calculated from Eqs. (3)-(4): yk

x1:3, k  'yk  vk

(8)

According to Eqs. (7)-(8), we obtain

(4)

where yk is quaternion observation error vector, and ['y1, k

xk

Pk |k 1 predicted error covariance matrix, and Rk

Gbz ,k ]T gyro drift vector.

wk

ª 0 « « Zz ,k « Z ¬ y,k

arg min J ( xk )

is estimated state vector, xˆ k |k 1 predicted state vector,

Attitude quaternion is used to describe the satellite attitude for the gyro/star tracker integrated attitude determination system [15]. The motion equation for such a system can be represented by xk Fk xk 1  wk (1) where xk

(5)

j 1

Influence Analysis of Low-frequency Periodic Error

2.1. Attitude determination system equations

'yk

'ym, k

No.4

xˆ1:3, k

x1:3, k  'yk  vk  k

(9)

Substituting Eq. (1) and Eq. (6), we get xˆ1:3,k 1|k

( I3u3  W [k 1u]) xˆ1:3,k  W xˆ 4:6, k 2 (10)

Combining with Eq. (7), we can rewrite Eq. (10) as xˆ1:3, k 1 ( I 3u3  W [k 1u]) xˆ1:3,k  W xˆ 4:6, k 2  1:3, k 1 (11)

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No.4

Then, xˆ1:3, k  vk 1  k 1  1:3, k 1 ] W

(12)

From Eq. (1), we can see that the state vector can be expressed as x1:3,k 1

ª0 2 «« 0 «¬Z0

2[ x1:3,k 1  'yk 1  ( I 3u3  W [k 1u]) ˜

xˆ 4:6, k

'x4,k



(14)

2

W

x4:6, k  'x4:6, k  Ok (vk :k 1 , k :k 1 , w1:3, k 1 , 1:3, k 1 )

where  2('yk 1  'yk ) W  2[k 1u]'yk

(16)

Ok (vk:k 1, k:k 1, w1:3,k 1, 1:3,k 1 ) 2[( I3u3  [k 1u]) ˜ (vk  k )  (vk 1  k 1 )  w1:3,k 1  1:3,k 1 ] W

the periodic error 'yk . For the case that the satellite orbit is approximately circular, the movement rate roughly turns out to be [17] 0]

(18)

where Z0 is the orbital angular rate. Substituting Eq. (18) into Eq. (2), we can rewrite the matrix [k u] as [k u]

ª0 «0 « «¬Z0

0 Z0 º 0 »» 0 »¼

0 0

W

K

¦{a1, j [cos( jZ(k  1))  cos( jZk )]}  j 1

K

¦{b1, j [sin( jZ(k  1))  sin( jZk )]}  j 1

(22)

j 1

Since the orbital angular rate is small, the first and second items on the right side of Eq. (22) can be simplified as cos( jZ (k  1))  cos( jZ k )

 jZ sin( jZ k )

(23)

sin( jZ (k  1))  sin( jZ k )

jZ cos( jZ k )

(24)

And then Eq. (22) can be simplified as 'x4, k

K

¦ [a4,( xj) cos( jZ k )  b4,( xj) sin( jZ k )]

(25)

j 1

where a4,( x j)

The analytical form of estimated gyro drift error is presented in this section. It is described in Fourier series to indicate the relationship between the estimated gyro drift error 'x4:6,k and the Fourier coefficients of

[0 Z0

(21)



2

W

jZb1, j  2Z0 a3, j , b4,( xj)

2

W

jZ a1, j  2Z0 b3, j

Similarly, we have the expression for the second and the third components of 'x4:6,k respectively˖

2.3. Estimated error expression for gyro drift

k

2

K

(17)

From Eq. (14), we know that the estimated gyro drift will be affected by the low-frequency periodic error of star tracker. The way in which the estimated error of low-frequency periodic error 'yk has influence on the estimated gyro drift 'x4:6,k is shown in Eq. (16). The first item on the right side of Eq. (16) can be regarded as the change rate of periodic error and the second item is affected by the periodic error and the satellite attitude motion.

T

W

2Z0 ¦[a3, j cos( jZk )  b3, j sin( jZk )]

(15) 'x4:6, k

(20)

Substituting Eq. (5), we have

We simplify the definition as xˆ 4:6, k

2  ('y1,k 1  'y1,k )  2Z0 'y3, k

'x4, k

2[k 1u]'yk  2[( I 3u3  [k 1u])(vk  k )  (vk 1  k 1 )  w1:3, k 1  1:3, k 1 ] W

0 Z0 º ª 'y1,k º « » 0 0 »» « 'y2,k » 0 0 »¼ «¬ 'y3,k »¼

From Eq. (20), 'x4,k in 'x4:6,k is

( I3u3  W [k 1u]) x1:3, k  W x4:6,k 2  w1:3,k 1 (13)

Thus, Eq. (12) is changed to xˆ 4:6,k x4:6, k  2('yk 1  'yk ) W 

· 617 ·

(19)

Therefore, Eq. (16) can be further rewritten as § ª 'y1, k 1 º ª 'y1, k º · ª 'x4, k º 2¨« « » » « »¸ 'x4:6, k « 'x5, k »  ¨ « 'y2, k 1 »  « 'y2, k » ¸  W ¨ « 'y « 'x6, k » » « »¸ ¬ ¼ © ¬ 3, k 1 ¼ ¬ 'y3, k ¼ ¹

'x5, k

K

¦ [a5,( xj) cos( jZ k )  b5,( xj) sin( jZ k )]

(26)

j 1

'x6, k

K

¦ [a6,( xj) cos( jZ k )  b6,( xj) sin( jZ k )]

(27)

j 1

where 2

a5,( xj)



a6,( xj)



W 2

W

jZb2, j , b5,( xj)

2

W

jZ a2, j

jZb3, j  2Z0 a1, j , b6,( xj)

2

W

jZ a3, j  2Z0 b1, j

Equations (25)-(27) show the relationship between the frequency and amplitude of the estimated drift error 'xl , k (l = 4, 5, 6) and the low-frequency periodic error 'ym,k (m = 1, 2, 3) of star tracker. The frequency of 'xl , k is equal to that of 'ym, k . Since the period of the low-frequency periodic error and the satellite orbit are the same, the error with period approximating to jZ0 in the estimated gyro drift is brought by the

WANG Jiongqi et al. / Chinese Journal of Aeronautics 25(2012) 615-621

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where the periodic error 'xl , k is shown in Eq. (28),

low-frequency periodic error of star tracker. 3.

Low-frequency Periodic Error Identification

and el , k the random error. Assume that el , k is Gaussian white noise with

3.1. Identification principle

0, E (el , k 'xl , k )

E (el , k )

The proposed principle for low-frequency periodic error identification is as follows. The low-frequency periodic error identification can be transformed into the identification of Fourier coefficients of periodic error in the estimated gyro drift. From Eq. (15), one can see that the estimated gyro drift xˆl , k (l = 4, 5, 6) includes the periodic error

because the sine component sin( jZ k ) and cosine component cos( jZ k ) of zl,k are the peak value at jZ in the spectrum of 'xl , k . Meanwhile, when the parameters

bl(,xj)

and

culated by Eqs. (25)-(27). The periodic signal 'xl , k can be expressed by the Fourier series form: 'xl , k

­ (z) 2 N ¦ zl ,k cos( jZ k ) °al , j Nk 1 ° °° ( z ) 2 N ®bl , j ¦ zl ,k sin( jZ k ) Nk 1 ° ° 2S °Z N °¯

(28)

E (al(,zj) )

al(,xj) , Var(al(,zj) )

2V 2 N

(33)

E (bl(,zj) )

bl(,xj) , Var(bl(,zj) )

2V 2 N

(34)

Proof Substituting Eq. (30) into Eq. (32), we get the mean value of al(,zj) as follows: º 2 ªN E « ¦ ('xl , k  el , k ) cos( jZ k ) » N ¬k 1 ¼

E (al(,zj) )

In ideal scenario, the data sample 'xl , k

N; N=2K+1) can be used directly, and the Fourier coefficients can be calculated as ­ ( x) °al , j ° ® °b( x ) °¯ l , j

2 N 2 N

E (al(,zj) )

al(,xj)

(36)

The variance value of al(,zj) is Var(al(,zj) )

E ((al(,zj) ) 2 )  ( E (al(,zj) )) 2

(37)

N

¦ 'xl ,k cos( jZ k ) k 1 N

(29)

¦ 'xl ,k sin( jZ k )

The first item on the right side of Eq. (37) can be written as follows according to Eq. (30) and Eq. (32): E ((al(,zj) )2 )

k 1

However, the periodic signal 'xl , k cannot be directly used for analysis because of gyro’s constant deviation. Thus, the estimation of Fourier coefficients al(,xj) and bl(,xj) should be based on the data samples zl,k.

N 4 E{[¦ ('xl ,k  el ,k ) cos( jZk )]2 } 2 N k 1 N 4 {[ E 'xl ,k cos( jZk )]2  ¦ N2 k 1

N

N

2[¦ 'xl ,k cos( jZk )][¦ el ,k cos( jZk )]  k 1

The effect of random error in zl,k on the estimation of parameters al(,xj) and bl(,xj) is proved in Theorem 1. Theorem 1 zl,k (k=1, 2, …, N) is the sample data used for parameter identification and can be expressed by

zl , k

(35)

Combining with Eq. (29) and Eq. (31), we have

j 1

(k=1, 2,…,

(32)

And the mean and variance values of parameters satisfy

K

¦ [al(,xj) cos( jZ k )  bl(,xj) sin( jZ k )]

k j kz j

calculated by

(l = 4, 5, 6) are obtained, the

parameters am, j and bm, j (m = 1, 2, 3) can be cal-

­°V 2 ® °¯0

Then, similar to Eq. (29), the parameters to the sample data zl,k are defined as al(,zj) and bl(,zj) , which are

random error. Generally, the constant deviation can be removed by extracting the mean from the original data xˆl , k [18]. Denote the data sample after removing the mean value as zl,k (l = 4, 5, 6). Then, the task is to identify the coefficients of the low-frequency periodic error according to the data sample zl,k. The frequency Z of periodic signal 'xl , k can be directly identified

0, E (el , k el , j )

(31)

'xl , k . Gyro drift contains both constant deviation and

al(,xj)

No.4

'xl , k  el , k

(30)

k 1

N

[¦ el ,k cos( jZk )]2 }

(38)

k 1

Combining with Eq. (29) and Eq. (31), we have E[(al(,zj) ) 2 ] (al(,xj) ) 2  4V 2 N2

N

¦ (cos( jZ k ))2 k 1

(al(,xj) )2 

2V 2 N

(39)

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No.4

2V 2 N .

Then, Var(al(,zj) )

4

Equation (34) can be verified by the similar process. According to Theorem 1, the parameters al(,zj) and bl(,zj) can be regarded as the estimation of the parameters al(,zj) , bˆl(,xj)

al(,xj) and bl(,xj) , i.e., aˆl(,xj)

bl(,zj) . The mean

is equal to its actual value, and the variance value decreases with the increase of the sample number. It means that the effect of random error can be overcome by increasing the sample number. According to Eqs. (25)-(27), the estimated value of am , j and bm, j is therefore given by 4

aˆ1, j

(4Z02 

bˆ1, j

(4Z02 

aˆ2, j

W 2 jZ

W

j 2Z 2 ) 1 (

2

4

W

(4Z02 

bˆ3, j

(4Z02 

4

W

4

W

2

2



jZbˆ4,( xj)  2Z0 aˆ6,( xj) )

W

j 2Z 2 ) 1 (

2

bˆ5,( xj) , bˆ2, j

aˆ3, j

2

W

2

W

2 jZ

jZ aˆ4,( x j)  2Z0 bˆ6,( xj) )

aˆ5,( xj)

j 2Z 2 ) 1 (2Z0 aˆ4,( x j) 

j 2Z 2 ) 1 (2Z0 bˆ4,( xj) 

2

W

2

W

jZbˆ6,( xj) )

jZ aˆ6,( xj) )

3.2. Identification algorithm The previous sections show that the parameters for low-frequency periodic error of star tracker can be identified by extracting the Fourier parameters of estimated gyro drift. The identified parameters include the Fourier parameters am,j, bm,j and Z. The algorithm process is described in Fig. 1.

· 619 ·

Simulation and Results

4.1. Simulation condition Simulated experiments are conducted to validate the proposed identifying and compensating approach in this section. The attitude determination results obtained by Kalman filter and the proposed approach are also compared here. Initial orbit parameters of the satellite are as follows: semi-major axis a=7 087.45 km, eccentricity e=1.99×103, orbital inclination i=98.153°, ascending node longitude : =30.534°, perigee Z * =0.133°. The measurement data of gyro is generated by numerical simulation. The error model of gyro is the sum of angle random walk and angular rate random walk; the standard deviation of angular random walk and angular rate random walk is 4×104 (°)/h and 6.7×105 (°)/h respectively. Three star trackers are used, and the measurement error model of star tracker includes the random error (1), the constant deviation (0.1) and the low-frequency periodic error. The low-frequency periodic error is described by [19] ­ °G l , x ° ® °G ° l, y ¯

8

¦ [ax, j cos( jZ k )  bx, j sin( jZ k )] j 1 8

(40)

¦ [a y, j cos( jZ k )  by , j sin( jZ k )] j 1

where ax,j and bx,j are amplitudes of cosine and sine function in x axis, and ay,j and by,j that in y axis. Parameters in Eq. (40) are simulated and the real values are shown in Table 1. It is clear that the simulated Fourier series coefficients are much larger at 2Z and 4Z than at any other frequency ranges. 4.2. Simulation results When traditional Kalman filter is used, which does not identify and compensate the low-frequency periodic error, the three-axis attitude determination results are shown in Fig. 2. The solid line in Fig. 2 is the attitude determination error; the dashed line indicates the boundary of ±0.5. And the root mean square (RMS) of attitude determination error is about 1.67 (3V). As seen in Fig. 2, the satellite attitude determination error is changed periodically because of the effect of the low-frequency periodic error, and moreover the change period is basically equal to the orbit period. Figures 3-4 show the estimated gyro drift and its spectrum by the traditional Kalman filter respectively. As analyzed in Section 2.2, the estimated gyro drift xˆl , k in Fig. 3 (l = 4, 5, 6) includes the periodic error 'xl , k as well as the constant deviation and the random error el , k ( l = 4, 5, 6). As seen from Fig. 3, the constant deviation is the mean of xˆl , k , which is ap-

Fig. 1

Algorithm process.

proximately equal to zero. Besides, from Fig. 4, there

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Table 1

No.4

Star tracker low-frequency periodic error parameters ()

Star tracker

1

2

3

j

Error parameter

1

2

3

4

5

6

7

8

ax,j

0.008 5

0.400 7

0.031 3

0.200 0

0.001 9

0.007 9

0.003 9

0.004 0

bx,j ay,j

0.008 0 0.007 8

0.407 5 0.404 5

0.030 6 0.030 6

0.217 8 0.200 0

0.007 5 0.001 2

0.001 3 0.033 2

0.008 5 0.002 7

0.003 0 0.003 7

by,j

0.012 3

0.315 3

0.020 3

0.108 8

0.023 9

0.017 8

0.014 6

0.022 1

ax,j

0.013 5

0.309 4

0.021 2

0.100 0

0.005 3

0.010 2

0.011 9

0.007 7

bx,j ay,j

0.018 8 0.002 5

0.306 7 0.311 3

0.020 4 0.020 1

0.100 0 0.100 0

0.023 3 0.034 3

0.025 5 0.018 8

0.007 1 0.026 1

0.002 9 0.013 1

by,j

0.020 6

0.310 1

0.021 1

0.100 0

0.001 7

0.014 7

0.011 5

0.014 3

ax,j

0.003 9

0.313 6

0.020 1

0.110 4

0.014 0

0.002 6

0.007 6

0.001 3

bx,j ay,j by,j

0.010 4 0.009 3 0.014 0

0.302 0 0.311 7 0.301 4

0.020 3 0.020 4 0.021 6

0.100 0 0.113 1 0.100 0

0.000 2 0.010 8 0.003 5

0.011 5 0.001 6 0.023 8

0.008 1 0.030 2 0.003 5

0.001 5 0.012 0 0.002 5

are much more significant peak values in the spectrum at the frequency of 2Z and 4Z than any other frequency ranges. This is consistent with the simulated Fourier series coefficients of low-frequency periodic error in Table 2. It also indicates that the estimated gyro drift is greatly affected by the low-frequency periodic error of star tracker. The major frequencies of the low-frequency periodic error are at 2Z and 4Z.

mination error is significantly reduced. The attitude determination error is within 0.5. And the RMS of attitude determination error is about 0.98 (3V). To further illustrate the effect of compensating the low-frequency periodic error on the attitude determination accuracy, Monte Carlo simulation is conducted. Three-axis attitude estimation error curve are shown in Fig. 6. The solid and dotted lines indicate the attitude determination error after and before the calibration of the low-frequency periodic error, respectively. Obviously, the low-frequency periodic error is compensated effectively and the attitude determination accuracy is

Fig. 2 Attitude error with Kalman filter. Fig. 4

Fig. 3

Spectrum of estimated gyro drift.

Estimated gyro drift.

Figure 5 shows the attitude determination results with the proposed approach. As seen in Fig. 5, the period change is greatly eliminated. The attitude deter-

ügü Determination error üü r0.5s

Fig. 5

Attitude error with the proposed approach.

WANG Jiongqi et al. / Chinese Journal of Aeronautics 25(2012) 615-621

No.4

[6]

[7]

[8]

ügü Before calibration üü After calibration

Fig. 6

[9]

Comparison of attitude determination results.

improved greatly. This indicates that the reconstruction of low-frequency periodic error model is reasonable.

[10]

5.

[11]

Conclusions

The low-frequency periodic error, which is caused by the complicated space environment such as thermal effects and usually changes with the orbital period, cannot be directly dealt with the Kalman filter. It is also difficult to be eliminated by the general calibration. Therefore, it is one of the most critical problems for high-accuracy satellite attitude determination. An approach of low-frequency periodic error identification and compensation for star tracker measurement is presented in this paper. The derivation of relationship between low-frequency periodic error of star tracker and the estimated gyro drift shows that the parameters for low-frequency periodic error of star tracker can be identified by extracting the Fourier parameters of estimated gyro drift. Simulated experimental results demonstrate the validity of reconstruction for low-frequency periodic error model and effectiveness of the proposed method.

[12]

[13]

[14]

[15]

[16]

[17]

References [18] [1]

[2]

[3]

[4]

[5]

Iwata T, Hoshino H, Yoshizawa T, et al. Precision attitude determination for the advanced land observing satellite (ALOS): design, verification and on-orbit calibration. AIAA-2007-6817, 2007. Srinivasa Rao J, Pullaiah D, Padmasree S, et al. Star tracker alignment determination for resourcesat-I. AIAA-2004-5392, 2004. Holzapfel F, Theil S. Advances in aerospace guidance, navigation and control. Berlin: Springer, 2011; 373-384. Tu S C. Satellite attitude dynamics and control. Beijing: Chinese Astronautic Publishing House, 2003; 125-168. [in Chinese] Wang J Q, Jiao Y Y, Zhou H Y, et al. Star sensor attitude measuring data processing technique in condition of complex satellite dithering. Journal of Electronics &

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Biography: WANG Jiongqi received Ph. D. degree from National University of Defense Technology in 2008 and then became a teacher there. His main research interests are measurement data process and satellite attitude determination. E-mail: [email protected]