In-flight quality evaluation of attitude measurements from STECE APS-01 star tracker

Acta Astronautica 102 (2014) 207–216

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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro

In-flight quality evaluation of attitude measurements from STECE APS-01 star tracker Yuwang Lai a,n, Junhong Liu a, Yonghe Ding b, Defeng Gu a, Dongyun Yi a a b

Department of Mathematics and Systems Science, National University of Defense Technology, Changsha 410073, China Beijing Institute of Tracking and Telecommunication Technology, Beijing 100094, China

a r t i c l e i n f o

abstract

Article history: Received 25 August 2013 Received in revised form 25 February 2014 Accepted 5 June 2014 Available online 12 June 2014

The STECE satellite is a small satellite mission of China for space technology experiment and climate exploration. To test the in-flight performance of the CMOS Active Pixel Sensor (APS) based star trackers, the APS-01 star tracker has been loaded on the STECE satellite. The APS-01 star tracker payload delivers 1 Hz real time telemetry attitude data during its in-orbit experiment mission. These attitude data have been analyzed and evaluated down to the specification requirements like random noise, systematic bias and low frequency error in this paper. The systematic bias of APS-01, including precession–nutation, aberration and relative installation error, has been calibrated. An approach is proposed to analyze the attitude low frequency errors of APS-01. The results of the evaluation can contribute to a better understanding as well as further refinements of the CMOS APS based star trackers. & 2014 IAA. Published by Elsevier Ltd. All rights reserved.

Keywords: CMOS APS star tracker Attitude determination Precession–nutation Aberration Low frequency error Quality evaluation

1. Introduction With the development of Earth-observing satellites and deep-space exploration satellites, requirements for attitude measurement accuracy are increasing. Star trackers are the high precision optical sensor used on-board of spacecrafts for providing the absolute 3-axis attitude of a spacecraft utilizing star observations. Currently, star trackers are the most accurate attitude sensors with accuracies down to arc seconds and become the primary optical instrument in the field of the attitude and orbit control system (AOCS) sensors [1]. Now most commercial available star trackers are based on charge coupled device (CCD) detectors. The CCD star tracker can achieve an accuracy of less than 3 arcsecs [2]. But CCD star trackers also have some disadvantages, such as small FOV, low

n

Corresponding author. E-mail address: [email protected] (Y. Lai).

http://dx.doi.org/10.1016/j.actaastro.2014.06.009 0094-5765/& 2014 IAA. Published by Elsevier Ltd. All rights reserved.

working angular rate limit, low sample rate, high power consumption and large size, which prevent the use of CCD star tracker in low-cost small satellites [3]. The CMOS active pixel sensor (APS) technology, introduced in the early 90ties, has improved significantly in terms of the electrooptical parameters during the last years, allows now the beneficial replacement of the CCD detectors with respect to larger FOV, higher dynamic range, lower power consumption and smaller size [4,5]. This new kind of star tracker will be more applicable on modern low-cost agile small satellites. Thus the in-flight performance evaluation of the APS star trackers has become particularly important. Space Technology Experiment and Climate Exploration (STECE) is a small satellite mission of China for space technology experiment and climate exploration. STECE satellite was launched into a sun-synchronous orbit, and altitude is about 791 km. The payloads equipped on STECE satellite mainly contain three star trackers, two fiber optic gyros and a dual-frequency GPS receiver, etc. The three star trackers include an ASTRO 10 star tracker [6] from

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Jena-Optronik GmbH as the AOCS sensor, an CMOS APS based star tracker (named as APS-01) and a CCD star tracker as the test payloads. The APS-01 star tracker is a new star tracker based on CMOS APS detector. The main technical data of APS-01 are: Field of view: 20 deg
20 deg square. Sun exclusion: 36 deg. Sensitivity: 5.5 mi Update rate: 8 Hz.

qðt i þ 1 Þ ¼ qðt i Þ  Δqðt i Þ

The output attitude data of APS-01 are quaternions (4 element vector with q0 as scalar part and q1 , q2 , q3 as vector part, which satisfy q20 þq21 þ q22 þ q23 ¼ 1). During its in-orbit experiment mission, APS-01 delivers 1 Hz real time telemetry attitude data for telemetry channel constraints. To evaluate the in-flight performance of APS-01, this paper analyzed and evaluated the telemetry data of APS-01 down to the specification requirements like attitude random noise, systematic bias and low frequency errors [7,8]. Section 2 analyzes the attitude random noise. Section 3 focuses on systematic bias calibration. Section 4 focuses on the analysis of the low frequency error. The final section of this paper contains the summary and conclusions. 2. Random noise

y-axis noise[arcsec]

x-axis noise[arcsec]

The random noise include photon count noise, shot noise caused by the photon to electron conversion process, the detector readout noise, analogue processing chain noise and analogue to digital conversion quantization noise etc. The random noise is usually filtered with gyroscopes data by the AOCS [7]. In order to evaluate the 3-axis attitude measurement random noise of the APS-01 star tracker, we use the uncorrelated property of random noise between successive

z-axisnoise[arcsec]

data samples. By building up the so called delta quaternion ðΔqÞ we can derive the angular rate as the mean rotational difference between the data stamps and the uncorrelated error, which is just the random noise. Given a group of successive attitude quaternion data samples qðt i Þ ði ¼ 1; 2; …; NÞ, the quaternion measured at t i þ 1 ðqðt i þ 1 ÞÞ can be described by a quaternion multiplication of the quaternion measured at t i ðqðt i ÞÞ and the searched delta quaternion ðΔqðt i ÞÞ ð1Þ

In quaternion mathematics, the  operator is typically defined as [9] 2 32 3 q1 p0 p3  p2 p1 6 76 7 p0 p1 p 2 76 q 2 7 6  p3 76 7 q  p¼6 ð2Þ 6 p2 6 7 p1 p0 p3 7 4 54 q 3 5  p1 p2  p3 p0 q0 with q ¼ ½q0 q1 q2 q3 T , p ¼ ½p0 p1 p2 p3 T . Therefore, the delta quaternion can be computed by

Δqðt i Þ ¼ qðt i Þ  1  qðt i þ 1 Þ

ð3Þ

1

where qðt i Þ ¼ ½q0  q1 q2 q3 T is the inverse quaternion of qðt i Þ. The delta quaternions can be converted to the 3-axis Euler angles [9]. Fig. 1 shows the 3-axis delta Euler angle streams converted from the calculated delta quaternions of APS-01 over a time period of an orbit. The delta Euler angle streams are plotted around the 0 arcs line with the removed mean value (the mean rotational angular rate) per axis. The standard deviation of these Euler angle streams is [1.90, 2.14, 20.13] arcsec 3σ. Considering the delta quaternion is the multiplication of two statistically independent data samples, the 3-axis noise can be computed from the standard deviation per axis divided by sqrt (2). So the 3-axis random noise of APS-01 is [1.34, 1.51, 14.23] arcsec 3σ. Noise about the boresight (APS-01þZ

6 4

x-axis noise: 1.34arcsec(3sigma)

x-axis mean: 1.04arcsec/sec

y -ax is nois e: 1.51arc s ec (3s igma)

y -ax is mean: 171.12arc s ec /s ec

2 0 -2 -4 -6 6 4 2 0 -2 -4 -6 60 z-axis noise: 14.23arcsec(3sigma)

40

z-axis mean: 130.38arcsec/sec

20 0 -20 -40 -60 0

1000

2000

3000

4000

5000

6000

Time[sec] Fig. 1. The 3-axis delta Euler angle streams converted from the calculated delta quaternions of APS-01 over an orbit.

Y. Lai et al. / Acta Astronautica 102 (2014) 207–216

0.10

0.10 kurtosis = 3.4276

209

0.10 kurtosis = 4.3015

kurtosis = 3.6810

0.09

0.09

0.09

0.08

0.08

0.08

0.07

0.07

0.07

0.06

0.06

0.06

0.05

0.05

0.05

0.04

0.04

0.04

0.03

0.03

0.03

0.02

0.02

0.02

0.01

0.01

0.01

-3

-2

-1

0

1

x-axis noise[arcsec]

2

3

-4

-2

0

2

4

-40

y-axis noise[arcsec]

-20

0

20

40

z-axisnoise[arcsec]

Fig. 2. Histograms of the 3-axis delta Euler angle streams.

axis) is roughly 10times larger than noise normal to the boresight, which is typical for many models of star trackers. It should be noted that the noise distributions of the 3axis error streams appear to be normal, but non-normal. Fig. 2 presents the histograms of these data. We can see that, the kurtosis of x, y and z axis error streams are 3.4276, 3.6810 and 4.3015, respectively, higher than the kurtosis of a normal sample which is close to 3. So the noise distributions of the 3-axis error streams are nonnormal. 3. Systematic bias The systematic bias terms encompass the precession and nutation, the aberration and the relative installation error. The in-flight precession–nutation correction for the star catalog stored in star tracker in the AttitudeDetermination Process is a fundamental option for most star trackers. The aberration is caused by relativistic effects and it mainly includes the orbital and annual aberration. In-flight correction for the aberration is possible and is also an option for most star trackers [10]. The relative installation error is caused by the effects of launch shock, dithering and vibrating while complex satellite running and thermal deformation. The magnitude of the relative installation error may be arc minutes, which is much larger than the intrinsic high accuracy of the star tracker. Due to its uncertainty, the relative installation error is difficult to be reduced by the general filter calibration. Shuster, Pittelkau and Wang et al. have presented some calibration methods to overcome this problem [11–13]. Unfortunately, due to the model uncertainty error of the filter, the relative installation error of star tracker cannot be removed thoroughly after calibration. Besides, the filters introduced by them are complicated that they are not so suitable for in-flight application. In this section we will present a non-filter method to calibrate the relative installation error of APS-01 relative to ASTRO 10. The ASTRO 10 star tracker has corrected the precession– nutation and the aberration in its Attitude-Determination

Process. But the in-flight precession–nutation correction and the in-flight aberration correction of the APS-01 star tracker are off during its experiment mission. Thus the measured quaternion data of APS-01 need to be corrected for the precession–nutation and the aberration before estimate the relative installation error. In the following we are using a very simple measure, namely the so-called Inter-Boresight Angle (IBA), to verify the precession-nutation and aberration correction of APS01 attitude data. The IBA is found of the dot-product of the measured pointing direction from simultaneous measurements of two star trackers. Ideally, the IBA is almost constant as time series if there are no deflections.

3.1. The precession–nutation and aberration correction Due to the precession and nutation, the discrepancy between The Instantaneous True Equatorial Coordinate System (ITECS) and the J2000.0 Mean Equatorial Coordinate System (J2000.0 MECS) will increase about 50″ per year as time goes on. The transformation algorithm between the J2000.0 MECS and the ITECS at epoch t can be expressed as [14] rðtÞ ¼ NðtÞPðt; t 0 Þrðt 0 Þ

ð4Þ

where t 0 is epoch J2000.0, rðt 0 Þ is three dimensional position vector in the J2000.0 MECS, rðtÞ is three dimensional position vector in the ITECS, Pðt; t 0 Þ is the precession matrix from t 0 to t, NðtÞ is the nutation matrix at epoch t. Set ΔqPN as the corresponding attitude quaternion of the precession–nutation matrix ðNðtÞPðt; t 0 ÞÞ, then the corrected attitude quaternion q is obtained by multiplying the inverse correction quaternion ðΔqPN Þ  1 with the initial one q0 q ¼ ðΔqPN Þ  1  q0

ð5Þ

Fig. 3 depicts the IBA streams between APS-01 and ASTRO 10 as time series before correcting the precession and nutation of APS-01 attitude data. The IBA streams have been removed the prelaunch inter-boresight angle

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Inter-Boresight Angle [arcsec]

600 RMS = 315.91arcsec

500

Prelaunch IBA = 82.7627°

400 300 200 100 0 -100 -200 -300 -400 -500 -600 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

6.5

7

Time[hour] Fig. 3. Inter-boresight angle as time series (precession–nutation not corrected). 120 RMS = 15.48arcsec

Inter-Boresight Angle [arcsec]

110

Prelaunch IBA = 82.7627°

100 90 80 70 60 50 40 30 20 10 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Time[hour] Fig. 4. Inter-boresight angle as time series (precession–nutation corrected).

Inter-Boresight Angle [arcsec]

100 Prelaunch IBA = 82.7627°

RMS = 6.48arcsec

90 80 70 60 50 40 30 20 10 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

Time[hour] Fig. 5. Inter-boresight angle as time series (aberration corrected).

82.76271 (calculated from the prelaunch installation, the same below). It can be seen that, the missing precession– nutation correction of the APS-01 attitude data causes an apparent periodic deflection in the IBA between the two star trackers with respect to each other of up to 7 450″. As a consequence the IBA varies like sinusoidal oscillations over an orbit. The same kind of plot is given in Fig. 4 but the precession and nutation of APS-01 attitude data were corrected. As seen in Fig. 4, the 7450″ periodic deflection in the IBA is greatly eliminated. Now the deflection of the IBA is reduced to 735″.

Note that in Fig. 4, there is still an evident periodic deflection in the IBA between APS-01 and ASTRO 10 from orbit to orbit. As mention before, it is caused by the astronomical aberration effect. The aberration correction algorithm can be seen in Ref. [10]. Here we just use the correction algorithm to correct the aberration of APS-01 attitude quaternion data, the results are showed in Fig. 5. It can be seen that, after the aberration correction, the periodic deflection in the IBA has been greatly eliminated and the error of the IBA is reduced to 720″. Now the IBA is flat much more and the consistency between the two star trackers has improved.

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211

3.2. The relative installation calibration

Note that in Fig. 5, there are still excursions (drift and periodic variations) in the range of about 40 arcsec in the IBA for each orbit. Comparing the IBA of the four orbits, it becomes evident that these excursions are repeated from orbit to orbit. Fig. 6 shows the same IBA versus the satellite latitude. It can be seen that, the four consecutive orbits superpose. Features reappear at the same position along the orbit. This predicts that, the residual errors of the IBA include not only random noise but also some lowfrequency errors. In Section 4 we will analyze the lowfrequency errors of APS-01 particularly. From Fig. 5 we can also see that, though the IBA streams have been removed the prelaunch inter-boresight angle (82.76271), the IBA streams still have an offset for about 60″. This is caused by the inaccurate prelaunch relative installation. In the following, we will discuss the estimation of the relative installation of APS-01 under in-orbit condition.

The prelaunch installation Euler angle of the APS-01 star tracker measurement reference frame (STMRF) relative to the ASTRO 10 STMRF is [31.4963, 81.5037, 43.7636]T deg. As mention above, this prelaunch relative installation Euler angle is inaccurate and needs to be calibrated. Base on the in-flight attitude data of ASTRO 10 and APS-01, the relative installation can be estimated. Let qðt i Þ ði ¼ 1; 2; …; NÞ be the successive attitude quaternion data of APS-01, pðt i Þ be the successive attitude quaternion data of ASTRO-10. The APS-01 attitude quaternion qðt i Þ can be described by a quaternion multiplication of the ASTRO-10 attitude quaternion pðt i Þ and a searched delta quaternion ðΔqðt i ÞÞ at the same time stamps ðt i Þ qðt i Þ ¼ pðt i Þ  Δqðt i Þ

ð6Þ

100

Inter-Boresight Angle [arcsec]

Track 1

Track 2

Track 3

Track 4

90 80 70 60 50 40 30 20 10 0 -90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

40

50

60

70

80

90

6.5

7

Latitude/deg Fig. 6. Inter-boresight angle vs. satellite latitude (aberration corrected).

31.31

Roll[deg]

x-mean: 31.2848deg

31.3 31.29 31.28 31.27

Pitch[deg]

31.26 81.57 y-mean: 81.5432deg

81.56 81.55 81.54 81.53 81.52 43.94

Yaw[deg]

z-mean: 43.8915deg

43.92 43.9 43.88 43.86 43.84

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Time[sec] Fig. 7. The 3-axis rotation Euler angles from ASTRO 10 to APS-01.

6

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Then the delta quaternion Δqðt i Þ ð ¼ ½dq0 ðt i Þdq1 ðt i Þdq2 ðt i Þdq3 ðt i ÞT Þ can be computed by

time (actually two times is enough) unless the periodic signal of quaternion data is weak.

Δqðt i Þ ¼ pðt i Þ  1  qðt i Þ

4.1. Vondrak filter

ð7Þ

Actually, the delta quaternions Δqðt i Þ represent the 3-axis attitude as rotation of the ASTRO 10 STMRF into the APS-01 STMRF. That is to say, all of the delta quaternions Δqðt i Þ can be treated as an estimation of the relative installation quaternion. Naturally, the mean value of the delta quaternions is the best estimation of the relative installation quaternion. Considering the 4 elements of Δqðt i Þ are not independent, we convert the delta quaternions to the independent 3-axis Euler angles

Δqðt i Þ ¼ ½dq0 ðt i Þdq1 ðt i Þdq2 ðt i Þdq3 ðt i ÞT ) ½Δφðt i Þ Δθðt i Þ Δψ ðt i ÞT

ð8Þ

Let ½Δφ Δθ Δψ  be the mean value of ½Δφðt i Þ Δθðt i Þ Δψ ðt i ÞT " #T 1 N 1 N 1 N T ½Δφ Δθ Δψ  ¼ ∑ Δφðt i Þ ∑ Δθðt i Þ ∑ Δψ ðt i Þ Ni¼1 Ni¼1 Ni¼1 T

ð9Þ Then the mean Euler angle ½Δφ Δθ Δψ  is the best estimation of the installation Euler angle as rotation of the ASTRO 10 STMRF into the APS-01 STMRF. Fig. 7 shows the 3-axis rotation Euler angles from ASTRO 10 STMRF to APS-01 STMRF over a time period of 4 orbits. The estimation relative installation Euler angle is [31.2848, 81.5432, 43.8915]T deg. T

4. Low frequency error The low frequency error is of systematic nature (correlated with time) [7,8]. Typical error contributors may be: the satellite orbital motion, the complicated space environment such as thermal effect and the field of view errors which depend on the position of tracked stars in the field of view. The low frequency error cannot be directly dealt with using the Kalman filter, and is also difficult to be eliminated by the general on-orbital calibration [15]. Therefore, it is one of the most critical problems for high accuracy satellite attitude determination. In Section 3.1 we have found low frequency errors in the IBA between APS-01 and ASTRO-10. But the IBA is calculated by the two star tracker’s attitude data, the lowfrequency errors in the IBA are caused by the two star trackers. In order to obtain a better understanding the low frequency errors of APS-01, we have to assess its own 3-axis attitude measurement low-frequency errors. So we need to create a reference quaternion which represents as good as possible the orbit track under investigation. For this purpose we use Vondrak Filter combined with Fourier method (For convenience of reference, we will name Vondrak filter that combined with Fourier method as FVF) to fit the APS-01 measured quaternion data to acquire the reference quaternions. The Fourier method is used to extract the big periodic signals of the quaternion data before using Vondrak Filter to make Vondrak Filter fit the quaternion data better, and may be used more than one

Vondrak Filter is a method based on the observed data, which can smooth equidistance or non-equidistance data serials reasonably through selecting the smoothing factor to control its degree even in the condition of unknown fit function [16,17]. For a series of observational data (xi,yi), i ¼ 1; 2; …; N, where xi and yi are the measurement epochs and the measurements, respectively. The basic concept of the Vondrak Filter is to derive filter values under the following condition [17] Q ¼ F þ λ S- min 2

ð10Þ

where F denotes the degree of filtering and S is the smoothness of the graduated curve. F and S are defined as: N

F ¼ ðN  3Þ  1 ∑ pi ðyi  y0i Þ2 ; i¼1

S ¼ ðs  rÞ  1

Z

s r

½φ‴ðxÞ2 dx

ð11Þ

ð12Þ

where yi ¼ φðxi Þ is the filter value corresponding to measurement y0i ; pi is the weight of y0i ; φ is the smoothed curve expressed in terms of x and φ‴ the third-difference of filter values calculated based on a cubic Lagrange polynomial; 2 s ¼ xN  1 , r ¼ x2 . The coefficient λ is a unitless positive coefficient that controls the degree of filtering or the smoothness of the filtered series. When the coefficients λ2 -1, S-0 and F- min, a smooth parabola will be derived, and the operation is called absolute smoothing. 2 When λ -0, F-0, the filtered values approach the measurements, a rough curve will result and the operation 2 is called absolute fitting. Let ε ¼ 1=λ , ε is named the smoothing factor. The main advantages of the Vondrak filter are that no predefined fitting function is required, and that the filter values at the two ends of the data series can be calculated. In addition, the method is applicable to data of equal and unequal intervals. It can also be used as a numerical filter for separating signals of different frequencies [18]. 4.2. Extract low-frequency errors of attitude measurements with FVF method Set qj;F1 ðt i Þ; qj;F2 ðt i Þ; ⋯, ðj ¼ 0; 1; 2; 3Þ be the big periodic signals extracted by the Fourier method and qj;v ðt i Þ be the Vondrak Filter fit value of j elements of the measured quaternion qðt i Þð ¼ ½q0 ðt i Þq1 ðt i Þq2 ðt i Þq3 ðt i ÞT Þ, then the final fit value of the j elements is qj;r ðt i Þ ¼ qj;F1 ðt i Þ þ qj;F2 ðt i Þ þ ⋯ þ qj;v ðt i Þ; j ¼ 0; 1; 2; 3

ð13Þ

And the reference quaternion qr ðt i Þ can be calculated as: qr ðt i Þ ¼ ½q0;r ðt i Þq1;r ðt i Þq2;r ðt i Þq3;r ðt i ÞT =sqrtðq0;r ðt i Þ2 þ q1;r ðt i Þ2 þq2;r ðt i Þ2 þq3;r ðt i Þ2 Þ

ð14Þ

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-3

2

x 10

213

Residual error = 0.0009196

1 0 -1 -2 4000

0

8000

12000

16000

20000

24000

1 measurement quaternion data periodic signal

0.5 0 -0.5 -1 4000

0

8000

12000

16000

20000

24000

Time (s) Fig. 8. The first time periodic signal extraction results of q1 element of APS-01 quaternion data (a periodic signal with period 12,096 s (two orbits) was extracted and the standard deviation of the residual error is 0.0009196).

-3

Residual error = 0.0003413

x 10

1 0.5 0 -0.5 -1

4000

0

8000

12000

16000

20000

24000

-3

x 10

Residual error periodic signal

2 1 0 -1 -2 0

4000

8000

12000

16000

20000

24000

Time (s) Fig. 9. The second time periodic signal extraction results of q1 element of APS-01 quaternion data (a periodic signal with period 4032 s (2/3 orbits) was extracted and the standard deviation of the residual error is 0.0003413).

Residual error = 1.45e-005

-4

1

x 10

0.5 0 -0.5 -1 0

4000

8000

12000

16000

20000

24000

-3

1

x 10

Residual error the Vondrak Filter value

0.5 0 -0.5 -1 0

4000

8000

12000

16000

20000

24000

Time (s) Fig. 10. The Vondrak Filter results of q1 element of APS-01 quaternion data (the standard deviation of the residual error is 1.45e  5).

Figs. 8 and 9 show the periodic signal extraction results of q1 element of APS-01 quaternion data qðt i Þ for 4 orbits. We can see that, a periodic signal q1;F1 ðt i Þ with period 12,096 s and a periodic signal q1;F2 ðt i Þ with period 4032 s

were extracted from the q1 ðt i Þ element data, and there is no evident periodic signal in the residual errors any more. The residual error will be then fitted by Vondrak Filter and Fig. 10 is the result. The smoothing factor of the Vondrak

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z-axis noise[arcsec]

y-axis noise[arcsec]

x-axis noise[arcsec]

30 x-axis error: 9.74arcsec(3sigma)

20 10 0 -10 -20 30

y-axis error:9.70arcsec(3sigma)

20 10 0 -10 -20 150

z-axis error: 66.30arcsec(3sigma)

100 50 0 -50 -100 0

1000

2000

3000

4000

5000

6000

5000

6000

Time[sec] Fig. 11. The 3-axis total error of APS-01 for a single orbit.

160 140

x-noise[arcsec]

120 100 80 60 40 20 0 -20 0

1000

2000

3000

4000

Time[sec] Fig. 12. The x-axis total error of APS-01 for four consecutive orbits (the error streams have been offset by 40″ for each orbit for clarity).

Filter is ε ¼ 1:0e  12 and the weight of the measurement data is pi ¼ 1. When the reference quaternion qr ðt i Þ is acquired, we can calculate the delta quaternion between the measured quaternion qðt i Þ and qr ðt i Þ at the same epochs ðt i Þ

Δqðt i Þ ¼ qðt i Þ  1  qr ðt i Þ

ð15Þ

Convert the delta quaternion to the 3-axis Euler angle, and then we can obtain the 3-axis attitude measurement errors. Fig. 11 shows the delta quaternions represented in

3-axis Euler angles over an orbit. The reference quaternions are smooth enough due to FVF method. So the axis error plots contain the total error which includes the lowfrequency errors and the random noise. The low-frequency errors (the excursions) are in the range of 20 arcsec in x-, y-axis and 100 arcsec in z-axis, larger than the magnitude of random noise. Further, we compare the APS-01 attitude data from several orbits. It becomes evident that the low-frequency errors are repeated from orbit to orbit. Figs. 12–14 show

Y. Lai et al. / Acta Astronautica 102 (2014) 207–216

215

140

120

y-noise[arcsec]

100

80

60

40

20

0

-20

0

1000

2000

3000

4000

5000

6000

Time[sec] Fig. 13. The y-axis total error of APS-01 for four consecutive orbits (the error streams have been offset by 40″ for each orbit for clarity).

800 700

z-noise[arcsec]

600 500 400 300 200 100 0 -100 0

1000

2000

3000

4000

5000

6000

Time[sec] Fig. 14. The z-axis total error of APS-01 for four consecutive orbits (the error streams have been offset by 200″ for each orbit for clarity).

the x-, y-, z-axis error streams for four consecutive orbits, respectively. The orbits all show the same characteristic variation. 5. Summary and conclusions This paper contains a thorough discussion and the results of an in-flight attitude data quality evaluation of the STACE APS-01 star tracker. The random noise, the systematic bias and the low frequency errors of APS-01 attitude measurement data have been characterized. A non-filter method is presented to estimate the relative installation error of APS-01 relative to ASTRO 10. The FVF method is proposed to analyze the attitude low-frequency errors of APS-01. It is expected that the results of the inflight attitude data quality evaluation can contribute to a

better understanding as well as further refinements of the CMOS Active Pixel Sensor (APS) based star trackers hardand software. From the attitude data quality evaluation results of the APS-01 star tracker, we know that, the systematic bias usually can be calibrated or estimated; the random noise is small and is usually filtered with gyroscopes data by the AOCS. However, the low-frequency errors are much larger than the random noise and can’t be directly dealt with using the Kalman filter or be eliminated by the general onorbital calibration. So the most critical problem for high accuracy satellite attitude determination is the lowfrequency errors. In this paper we have extracted the low-frequency errors from APS-01 measured data and analyze the characteristic of them. But the generation reason of the low frequency errors is still not very clear.

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The relationship between the environmental effects and the low-frequency errors needs further investigation. So the calibration of the low-frequency errors is difficult by now. More work need to be done to analyze the features of the low-frequency errors and finally find an approach to overcome this problem.

Acknowledgments The authors are grateful to Beijing Institute of Tracking and Telecommunication Technology for providing attitude observation data of the APS-01 star tracker and the ASTRO 10 star tracker, and GPS observation data of STECE satellite. This study is supported by the National Natural Science Foundation of China (Grant nos. 61370013 and 61304119). References [1] G.D. Rogers, M.R. Schwinger, J.T. Kaidy, et al., Autonomous star tracker performance, Acta Astronaut. 65 (2009) 61–74. [2] Sun Ting, Fei Xing, Zheng You, Optical system error analysis and calibration method of high-accuracy star trackers, Sensors 13.4 (2013) 4598–4623. [3] D. Lu, J. Guang, W. Shaoju, et al., Application of CMOS APS star tracker with large FOV in attitude and angular velocity determination, Proc. SPIE Int. Soc. Opt. Eng. Soc. Photo-Opt. Instrum. Eng. (2008) 683026.1–683026.5. [4] Schmidt, U.w.e., ASTRO APS–The next generation hi-rel star tracker based on active pixel sensor technology, in: AIAA Guidance, Navigation, and Control Conference, AIAA 2005-5925, San Francisco, CA, United States, 2005, pp. 743–752.

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