Low frequency error analysis and calibration for multiple star sensors system of GaoFen7 satellite

ABSTRACT

The GaoFen7 (GF7) optical satellite is the first Chinese civilian sub-meter high-resolution stereo mapping satellite and is equipped with a double linear array camera and laser altimeter to achieve large-scale topographic mapping. To improve the accuracy of attitude determination, an attitude determination system comprised of four star sensors is loaded. According to the measurement accuracy and steady performance, the star sensors 1a and 1b is usually used together for satellite attitude calculation, which is called the conventional mode of attitude determination. Then, the combination of star sensors 2a and 2b is called the unconventional mode of attitude determination. Affected by variations in the incident angle of sunlight and solar radiation, thermal deformation occurs in the body and installation structure of the star sensor, which causes Attitude Low-Frequency Error (ALFE) and seriously influences the consistency of attitude determination results of different combination modes for multiple star sensors system. This study proposes an ALFE analysis and calibration approach for the multiple star sensors system of GF7 satellite to ensure the consistency of attitude determination results of different combination modes. Based on the statistical characteristics of the angles of the three axes, the installation parameters of the four star sensors are first calibrated. After analyzing the characteristics of the optical axis angles within 1420 orbit periods over 135 days, the segmented ALFE compensation model between the unconventional and conventional modes is proposed based on the Fourier series model and input parameter of latitude. Based on the on-orbit installation parameters and the ALFE model, the precise attitude determination results of the unconventional mode are calculated. Experimental results show that the attitude determination consistency after compensation is better than 2”. Moreover, the reliable application time range of the compensation model is 30 days to satisfy the requirements for high-precision attitude determination of GF7 satellite.

KEYWORDS:

• Attitude Low Frequency Error (ALFE)
• multiple star sensors system
• GaoFen7
• attitude determination consistency
• Fourier series

1. Introduction

The GaoFen7 (GF7) optical satellite, launched on 3 November 2019, is the Chinese civilian first sub-meter high-resolution stereo mapping satellite. GF7 is equipped with a double liner array camera, which consists of a forward optical linear sensor with a resolution of 0.79 m and an installation angle of +26° and a backward optical linear sensor with a resolution of 0.64 m and an installation angle of −5° to determine the 3-dimensional object coordinates (Li et al. 2020; Li, Wang, and Jiang 2021; Shan et al. 2020; Zhang et al. 2021). To further improve the altitude accuracy of stereo mapping of the double liner array camera, a dual-beam laser altimeter with a wavelength of 1064 nm has been applied. With the ground resolution of these sensors, GF7 can achieve 1:10,000 scale topographic mapping, which is widely used in land surveying and mapping, urban construction, and statistical investigation. The detailed satellite parameters of GF7 are listed in Table 1. The geometric positioning accuracies without Ground Control Points (GCPs) of the GF7 image and laser footprint are influenced by the orbit measurement error, attitude measurement error, time measurement error, and observation conditions (Guan et al. 2019; Pi, Li, and Yang 2019; Toutin 2004; Wang et al. 2019, 2020a; Xu et al. 2017; Yan et al. 2016). After the precise orbit determination, the geometric positioning error introduced by orbit error is about 2 cm. The time measurement error is better than 30 us, which causes geometric positioning error of 0.2 m. However, an attitude error of 1” introduces a geometric positioning error of 2.448 m, which is a key factor influencing the positioning accuracy of GF7 imagery orientation and laser footprint positioning without GCPs.

Table 1. Design and specification information for GF7 satellite.

type	parameter
orbit	altitude: 505 km; type: SunSync. Descending
life	8 years
double linear dynamic range	field of view	2.7°
	dynamic range	11-bits per pixel
	ground resolution	forward sensor: 0.79 m; backward sensor: 0.64 m
	installation angle	forward sensor: +26°; backward sensor: −5°
	swath width	>20 km
laser altimeter	number of beams	2
	wavelength	1064 nm
	ranging accuracy	≤0.3 m(slope less than 15°)
star senor	sensor	star sensors 1a, 1b, 2a, 2b
	frequency	8 Hz
	measurement accuracy	optical axis: 1”(3σ)

To improve the accuracy of attitude measurement, an Attitude Determination System (ADS) comprised of four star sensors is loaded. The measurement error of the optical axis (z-axis) of the four star sensors is 1”(3σ) and the frequency is 8 Hz, as listed in Table 1. Star sensors 1a and 1b are active pixel star sensors installed in forward sensor, and star sensors 2a and 2b are Hydra star sensors installed in backward sensor, as shown in Figure 1. These four star sensors employ a light detector to collect the lines of sight from two or more stars in space and calculate the attitude transformation matrices from their own star sensor coordinate system to the Julian Year 2000 (J2000) celestial coordinate system (Liebe 1995; Lu et al. 2017). Then, based on the installation parameters of these star sensors in the satellite body coordinate system, the precise orientation of the satellite body in the J2000 celestial coordinate system can be acquired with the requirement of at least two star sensors working simultaneously. With the high measurement accuracy and steady performance, the star sensors 1a and 1b is usually used together to determinate the attitude of the satellite body, named the conventional mode, as shown in Figure 1. Sometimes, influenced by exposure to the sun light or an insufficient number of stars, only star sensors 2a and 2b work simultaneously, called the unconventional mode of attitude determination. Variations in the incident angle of the sunlight and solar radiation may cause a thermal deformation in the installation structure and surface of each star sensor, which in turn causes fluctuations in the measurement data of the star sensor with satellite orbit. These fluctuations are termed the Attitude Low-Frequency Error (ALFE) (Lai et al. 2014; Schmidt, Elstner, and Michel 2013). Owing to the differences in the ALFE characteristics for each star sensor, there is a periodical fluctuation in the attitude determination results between the conventional and unconventional modes as shown in Figure 1. It causes the inconsistency of attitude results between conventional and unconventional modes for ADS. The geometric positioning accuracy of GF7 image and laser footprint seriously decreases when the unconventional mode is applied for attitude determination, which reaches more than 10 m and cannot satisfy the requirement for large scale topographic mapping.

Figure 1. The diagram of installation of star sensor of GF7.

Currently, several researchers are focusing on the calibration and compensation of ALFE. Blarre et al. optimized the thermodynamic structure design of a star sensor and enhanced the optical distortion calibration of the lens to reduce ALFE (Blarre et al. 2006). Based on the difference in the error characteristics, Wang et al. derived the analytical expression between the estimated gyro drift and the ALFE of the star sensor. Then, based on the power spectrum of the constant drift of the gyroscope, the ALFE of the star senor was calibrated using Kalman filtering (Wang, Xiong, and Zhou 2012). Similarly, Xiong et al. extracted ALFE according to the frequency spectrum of the drift estimate of the gyroscope sensor from a standard Kalman filter. A bank of filters were applied to compensate ALFE and improve the accuracy of attitude determination (Xiong, Zhang, and Liu 2012; Xiong, and Liu 2013). To eliminate the error caused by the spatial thermal deformation of geostationary optical satellites, Li et al. elaborates a stellar-based geometric positioning model and proposes a compensation method for the geostationary orbit camera payload. The distribution of positioning error caused by the spatial thermal deformation decreases from ±18 pixels to ± 3 pixels (Li et al. 2019). Based on the reference data, Fan et al. presented the ALFE compensation model for the YaoGan26 satellite, and the geometric accuracy of optical image positioning was improved by 40% (Fan et al. 2016). However, the amount of data for ALFE analysis is relatively small only within dozens of orbit periods. Little research has been conducted on the spatial and temporal characteristic analysis of ALFE and evaluated the reliability of a compensation model for a long time range. Moreover, the ALFE between the conventional and unconventional modes for multiple star sensors system cannot be solved using these approaches, causing an inconsistency in the attitude determination results.

Herein, an approach for ALFE analysis and calibration for multiple star sensors system of the GF7 satellite is presented. Based on the statistical characteristics of the angle of the three axes of the star sensor in the J2000 celestial coordinate system, the installation parameters of star sensors are first calibrated. By analyzing the characteristics of the optical axis angle over 135 days, the ALFE characteristics with respect to location and time are acquired. Then, an ALFE compensation model from the unconventional mode to the conventional mode is established with the input of latitude and the parameters of the Fourier series to achieve consistency in the results of attitude determination. The proposed approach was tested using the measurement data of four star sensors of GF7. The error of attitude determination consistency between star sensors 1a and 1b mode and star sensors 2a and 2b mode was found to be improved from more than 200” to less than 2”.

2. Methodology

The proposed approach for the ALFE analysis and calibration of multiple star sensor system includes three parts: installation parameter calibration, ALFE calibration, and ALFE compensation, as shown in Figure 2. Based on the on-ground installation parameters of star sensor 1a and statistical characteristics of the angle of the three axes, the on-orbit installation parameters of the other three star sensors are calibrated first. Then, the ALFE between the conventional and unconventional modes is extracted from the divergence of the attitude determination results and is calibrated using the Fourier series model with the input parameter of geographical latitude of the World Geodetic System 1984 (WGS84). With the on-orbit installation parameters, the initial attitude determination results of the unconventional mode are calculated. Using the model parameters and latitude information, the precise attitude determination results are acquired after ALFE compensation.

Figure 2. Flowchart of ALFE calibration and compensation for multiple star sensors system of GF7.

2.1. Installation parameter calibration

In this study, star sensor 1a is set as the fiducial star sensor to calibrate the on-orbit installation parameters of other star sensors. Therefore, the on-orbit installation parameters of star sensors 1a are set as the on-ground installation parameters. In the satellite body coordinate system, the on-ground installation parameters of x-axis, y-axis, and z-axis of star sensor 1a are $x_{1a}^{\mathrm{body}}$, $y_{1a}^{\mathrm{body}}$, $z_{1a}^{\mathrm{body}}$, respectively. At time t, the measurement data of the three axes of star sensor 1a in the J2000 celestial coordinate system are $x_{1a}^{J2000,t}$, $y_{1a}^{J2000,t}$, $z_{1a}^{J2000,t}$, respectively.

At time t, the measurement data of the z-axis of star sensor 1b is $z_{1b}^{J2000,t}$. The angles between $z_{1b}^{J2000,t}$ and the three axes of star sensor 1a in the J2000 celestial coordinate system are ${\alpha }_1^t$, ${\alpha }_2^t$, ${\alpha }_3^t$, as shown in Equation (1)(1) $\left\{\begin{array}{c}{\alpha }_1^t=arccos\left(x_{1a}^{J2000,t}\cdot z_{1b}^{J2000,t}\right)\\ {\alpha }_2^t=arccos\left(y_{1a}^{J2000,t}\cdot z_{1b}^{J2000,t}\right)\\ {\alpha }_3^t=arccos\left(z_{1a}^{J2000,t}\cdot z_{1b}^{J2000,t}\right)\end{array}\right.$(1) .

(1) $\left\{\begin{array}{c}{\alpha }_1^t=arccos\left(x_{1a}^{J2000,t}\cdot z_{1b}^{J2000,t}\right)\\ {\alpha }_2^t=arccos\left(y_{1a}^{J2000,t}\cdot z_{1b}^{J2000,t}\right)\\ {\alpha }_3^t=arccos\left(z_{1a}^{J2000,t}\cdot z_{1b}^{J2000,t}\right)\end{array}\right.$(1)

When there are no measurement errors, the angles between the on-orbit installation parameter of the z-axis of star sensor 1b ${\hat{z}}_{1\mathit{b}}^{\mathit{body}}$ and the three axes of star sensor 1a in the satellite body system are equal to those in the J2000 system. Therefore, ${\hat{z}}_{1\mathit{b}}^{\mathit{body}}$ can be calibrated using the following equation:

(2) $\left\{\begin{array}{c}{x}_{1a}^{J2000,t}\cdot z_{1b}^{J2000,t}=x_{1a}^{\mathrm{body}}\cdot {\hat{z}}_{1\mathit{b}}^{\mathit{body}}\\ y_{1a}^{J2000,t}\cdot z_{1b}^{J2000,t}=y_{1a}^{\mathrm{body}}\cdot {\hat{z}}_{1\mathit{b}}^{\mathit{body}}\\ z_{1a}^{J2000,t}\cdot z_{1b}^{J2000,t}=z_{1a}^{\mathrm{body}}\cdot {\hat{z}}_{1\mathit{b}}^{\mathit{body}}\end{array}\right.$(2)

Further, to eliminate the influences of the measurement noise and ALFE, star sensor data within an orbit period is adopted herein to obtain the optimal solution of ${\hat{z}}_{1\mathit{b}}^{\mathit{body}}$.

Let

(3) $\mathit{A}={\left[x_{1a}^{\mathrm{body}},y_{1a}^{\mathrm{body}},z_{1a}^{\mathrm{body}}\right]}^{\mathit{T}}$(3)

(4) $\mathit{B}={\left[\begin{array}{c}\mathit{mean}\left({\mathit{x}}_{1\mathit{a}}^{J2000,t_1}\cdot z_{1\mathit{b}}^{J2000,t_1},x_{1\mathit{a}}^{J2000,t_2}\cdot z_{1\mathit{b}}^{J2000,t_2},\dots ,x_{1\mathit{a}}^{J2000,t_n}\cdot z_{1\mathit{b}}^{J2000,t_n}\right)\\ \mathit{mean}\left(y_{1\mathit{a}}^{J2000,t_1}\cdot z_{1\mathit{b}}^{J2000,t_1},y_{1\mathit{a}}^{J2000,t_2}\cdot z_{1\mathit{b}}^{J2000,t_2},\dots ,y_{1\mathit{a}}^{J2000,t_n}\cdot z_{1\mathit{b}}^{J2000,t_n}\right)\\ \mathit{mean}\left(z_{1\mathit{a}}^{J2000,t_1}\cdot z_{1\mathit{b}}^{J2000,t_1},z_{1\mathit{a}}^{J2000,t_2}\cdot z_{1\mathit{b}}^{J2000,t_2},\dots ,z_{1\mathit{a}}^{J2000,t_n}\cdot z_{1\mathit{b}}^{J2000,t_n}\right)\end{array}\right]}^{\mathit{T}}$(4)

where $t_1,t_2,\dots ,t_n$ are the measurement times within an orbit period.

${\hat{z}}_{1\mathit{b}}^{\mathit{body}}$ can be calibrated by following Equation (5)(5) ${\hat{z}}_{1\mathit{b}}^{\mathit{body}}={\mathit{A}}^T\mathit{B}$(5) .

(5) ${\hat{z}}_{1\mathit{b}}^{\mathit{body}}={\mathit{A}}^T\mathit{B}$(5)

Similarly, the installation parameters of the x-axis and y-axis of star sensor 1b can be calibrated. Although the measurement noise of the x-axis and y-axis is larger than that of the z-axis, the number of calibration data within an orbit period can reach 45,400, which is sufficient to eliminate the influence of random errors. Therefore, the calibration accuracy of the x-axis or y-axis is the same as the calibration accuracy of the z-axis. Based on the statistical characteristics of the angles, the on-orbit installation parameters of star sensors 1b, 2a, and 2b can be calculated.

2.2. ALFE analysis

Due to the influence of thermal deformation, the installation of optical axis of the star sensor in satellite body coordinate system changes gradually. The angle of optical axes of two star sensors also fluctuates gradually. Here, the angle of optical axes is used for the analysis of the spatial and temporal characteristics of ALFE.

At time t, the measurement data of star sensors 1a and 1b are $q_{1a}^t={\left[q_{1a,0}^t,q_{1a,1}^t,q_{1a,2}^t,q_{1a,3}^t\right]}^{\mathit{T}}$ and $q_{1b}^t={\left[q_{1b,0}^t,q_{1b,1}^t,q_{1b,2}^t,q_{1b,3}^t\right]}^{\mathit{T}}$, respectively (Crassidis, Markley, and Cheng 2007; Tang et al. 2015; Wang et al. 2016, 2020b; Wu et al. 2018). The fluctuation of the optical axis angle ${\beta }_t$ in the J2000 celestial coordinate system can be calculated by the following equation:

(6) ${\beta }_t=arccos\left(z_{1a}^{J2000,t}\cdot z_{1b}^{J2000,t}\right)-{\beta }_0$(6)

(7) $z_{1a}^{J2000,t}=\left[\begin{array}{c}2\left(q_{1a,1}^t{q}_{1a,3}^t+q_{1a,0}^t{q}_{1a,2}^t\right)\\ 2\left(q_{1a,2}^t{q}_{1a,3}^t-q_{1a,0}^t{q}_{1a,1}^t\right)\\ 1-2\left(q_{1a,1}^t{q}_{1a,1}^t+q_{1a,2}^t{q}_{1a,2}^t\right)\end{array}\right]$(7)

(8) $z_{1b}^{J2000,t}=\left[\begin{array}{c}2\left(q_{1b,1}^t{q}_{1b,3}^t+q_{1b,0}^t{q}_{1b,2}^t\right)\\ 2\left(q_{1b,2}^t{q}_{1b,3}^t-q_{1b,0}^t{q}_{1b,1}^t\right)\\ 1-2\left(q_{1b,1}^t{q}_{1b,1}^t+q_{1b,2}^t{q}_{1b,2}^t\right)\end{array}\right]$(8)

where ${\beta }_0$ is the optical axis angle calculated by the on-orbit installation parameters.

For the optical axis angle of star sensors 1a and 1b and the optical axis angle of star sensors 2a and 2b of GF7, the fluctuations of LFE within 1420 orbit periods over 135 days are analyzed, as shown in Figure 3. The time range is from 15 February 2020 to 28 June 2020. The relative time refers to the time interval beginning with the start time of first track data of the first day and the unit is days. The range of LFE of the optical axis angle of star sensors 1a and 1b are from −5.734” to 5.793” with a Root Mean Square Error (RMSE) of 0.900”. The range of LFE of the optical axis angle of star sensors 2a and 2b are from −13.507” to 10.860” with the RMSE of 4.426”. These differences are caused by the discrepancy in the installation position and temperature control. The ALFE of the optical axis angle of star sensors 1a and 1b is significantly less than that of the star sensors 2a and 2b. Hence, to satisfy the requirements of the geometric positioning accuracy of GF7, the combination mode of star sensors 1a and 1b is usually used together for satellite attitude calculation, which is called the conventional mode. Then, to effectively decrease the influence of ALFE on the consistency of attitude determination, the conventional mode with star sensors 1a and 1b is used as the reference mode, and the ALFE of the unconventional mode with star sensors 2a and 2b is calibrated to maintain consistency with the reference mode.

Figure 3. The LFE of optical axis angle of GF7 within 135 days.

Based on the LFE of optical axis angle, the spatial characteristics of ALFE with latitude in WGS84 are analyzed here. In Figure 3(a), for the first data on 15 February 2020, the LFEs of the optical axis angles of star sensors 1a and 1b with latitudes of 0°, 30°N, and 60°N in daytime area are 0.407”, 1.790” and −0.881”, respectively. As shown in Figure 3(b), the LFEs of the optical axis angles of star sensors 2a and 2b with latitudes of 0°, 30°N, and 60°N in daytime area on 15 February are −0.251”, −0.027” and −4.983”, respectively. Owing to the space thermal environment discrepancy with latitude, the values of the optical axis vary considerably for different latitudes, as shown by the green lines in Figure 3(a,b). Furthermore, the temporal characteristics of ALFE with different time intervals are analyzed. As shown by red, yellow, and blue lines in Figure 3(a,b), for the same latitude, when the acquisition time varies, the temporal characteristics of ALFE change gradually. The differences in the optical axis angle between the first data acquired on 15 February and other data acquired over 135 days are calculated as listed in Table 2. The differences in the optical axis angle of star sensors 2a and 2b at the latitude of 0° within 5, 10, 30, 50, 70, and 120 days are 0.714”, 0.806”, 1.506”, 1.850”, 1.980”, and 3.296”, respectively; for the latitude of 60°N, the differences are 0.383”, 0.846”, 1.707”, 2.006”, 2.029”, and 3.214”, respectively. With the increase in time interval, the difference in the LFE increases gradually. When the time is more than 30 days, the difference is more than 2”, which introduces a geometric error of 4.896 m. Over 135 days, the direct point of sunlight gradually moves from the Tropic of Capricorn to the Tropic of Cancer, which results in changes in the solar radiation for the same latitude position. Moreover, the temperature control system of satellite is another important factor influencing the temperature and heating surface of star sensors and causing the changes of ALFE characteristics. Thus, the temporal characteristics of ALFE is more complicated than the spatial characteristics.

Table 2. The temporal characteristics of LFE in different latitudes.

star sensor	latitude	error(”)
		5 days	10 days	30 days	50 days	70 days	120 days
1a, 1b	0°	0.198	0.199	0.315	0.311	0.334	0.693
	30°N	0.509	0.892	1.051	1.078	1.101	1.158
	60°N	0.332	0.405	1.001	1.178	1.208	1.416
2a, 2b	0°	0.714	0.806	1.506	1.850	1.980	3.296
	30°N	1.267	2.061	2.164	2.810	2.902	2.653
	60°N	0.383	0.846	1.707	2.006	2.029	3.214

The characteristics of ALFE are influenced by the space thermal environment of multiple star sensors system, which is related to the solar radiation, the incident angle of sunlight, temperature control system of satellite, thermal radiation of correlated loads and so on. Thus, to compensate ALFE accurately, the temperature and heating area of star sensors should be acquired. However, these data are not measured by GF7 satellite. It is impossible to directly establish relationship between the ALFE characteristics and thermal environment of multiple star sensors system. For the same geographic latitude in WGS84, the solar radiation and the effect of the temperature control system is similar in a short time range, which causes the similar value of ALFE. As listed in Table 2, the LFE of star sensors 2a and 2b within 10 days is about 1.3”, which satisfies the requirement for large scale topographic mapping. With the time interval decreasing, the consistency of characteristic of ALFE in the same latitude increases. Therefore, due to the limitation of observation data and the stability of the spatial thermal environment over a period of time, the geographic latitude is adopted here as the alternative input parameter to establish the compensation model. Meanwhile, the time range of application of ALFE model is discussed to ensure the accuracy and reliability of the compensation.

2.3. ALFE calibration

Based on the TRIAD algorithm (Shuster 1981, 2004), the attitude determination result of the conventional mode with star sensors 1a and 1b is adopted as the reference, and the unconventional mode with star sensors 2a and 2b is calibrated and unified to the reference to eliminate ALFE. At time t, the difference in the attitude determination results between the conventional and unconventional modes caused by ALFE is expressed by the following equation:

(9) ${\mathbf{R}}_1^2{\left(t\right)}_{\mathit{\psi },\mathit{\varphi },\mathit{\theta }}={\left({\mathbf{R}}_2^{J2000}\left(\mathit{t}\right)\right)}^{-1}{\mathbf{R}}_1^{J2000}\left(\mathit{t}\right)$(9)

where ${\mathbf{R}}_1^{J2000}\left(\mathit{t}\right)$, ${\mathbf{R}}_2^{J2000}\left(\mathit{t}\right)$ are the transformation matrices form the satellite body coordinate system to the J2000 celestial coordinate system calculated by the conventional mode, and the unconventional mode, respectively; ${\mathbf{R}}_1^2\left(\mathit{t}\right)$ is the transformation matrix from the conventional mode to the unconventional mode; $\mathit{\psi },\mathit{\varphi },\mathit{\theta }$ are the pitch, roll, and yaw angles, respectively, that express the transformation matrix and reflect ALFE.

To eliminate the influence of measurement noise, a median filter with a width of 101 epoch is applied to $\mathit{\psi },\mathit{\varphi },\mathit{\theta }$ within an orbit period. Then, the smoothed transformation angles ${\psi }_{\mathit{ALFE}},{\varphi }_{\mathit{ALFE}},{\theta }_{\mathit{ALFE}}$ are extracted as ALFE. Based on the measured time and orbit data, the corresponding latitude of the satellite can be calculated, which is adopted as the input parameter in the ALFE calibration model.

Compared with polynomial model and rational model, Fourier series model has the advantages of high fitting accuracy and simple calculation (Fefferman 1973; Per 1971; Mackay, Keulen, and Smith 2011). However, owing to the complexity of ALFE, it is difficult to build a Fourier series model to calibrate the errors from 90°N to 90°S in daytime areas. Therefore, the latitudes from 90°N to 90°S in descending orbit are equally segmented into 20 sections to exactly calibrate errors. In each section, the Fourier series model is adopted as the compensation model, as shown by the following equation:

(10) $\left\{\begin{array}{c}{\psi }_{\mathit{ALFE}}\left(\mathit{lat}\right)=a_{\mathit{\psi },0}+\sum _{\mathit{j}=1}^8\left[a_{\mathit{\psi },\mathit{j}}\mathit{cos}\left(\mathit{j\omega lat}\right)+b_{\mathit{\psi },\mathit{j}}\mathit{sin}\left(\mathit{j\omega lat}\right)\right]\\ {\varphi }_{\mathit{ALFE}}\left(\mathit{lat}\right)=a_{\mathit{\varphi },0}+\sum _{\mathit{j}=1}^8\left[a_{\mathit{\varphi },\mathit{j}}\mathit{cos}\left(\mathit{j\omega lat}\right)+b_{\mathit{\varphi },\mathit{j}}\mathit{sin}\left(\mathit{j\omega lat}\right)\right]\\ {\theta }_{\mathit{ALFE}}\left(\mathit{lat}\right)=a_{\mathit{\theta },0}+\sum _{\mathit{j}=1}^8\left[a_{\mathit{\theta },\mathit{j}}\mathit{cos}\left(\mathit{j\omega lat}\right)+b_{\mathit{\theta },\mathit{j}}\mathit{sin}\left(\mathit{j\omega lat}\right)\right]\end{array}\right.$(10)

where lat is the latitude of the satellite in WGS84 as the input parameter; ${\psi }_{\mathit{ALFE}}\left(\mathit{lat}\right),{\varphi }_{\mathit{ALFE}}\left(\mathit{lat}\right),{\theta }_{\mathit{ALFE}}\left(\mathit{lat}\right)$ are the values of ALFE at lat; $\mathit{\omega }=2\mathit{\pi }/\left(max\left(\mathit{lat}\right)-min\left(\mathit{lat}\right)\right)$ can be seen as frequency parameter; $a_{\mathit{\psi },0}$, $a_{\mathit{\psi },\mathit{j}}$, $b_{\mathit{\psi },\mathit{j}}$, $a_{\mathit{\varphi },0}$, $a_{\mathit{\varphi },\mathit{j}}$, $b_{\mathit{\varphi },\mathit{j}}$, $a_{\mathit{\theta },0}$, $a_{\mathit{\theta },\mathit{j}}$ and $b_{\mathit{\theta },\mathit{j}}$ are the parameters of the Fourier series. To ensure the calibration accuracy of the proposed model, the number of terms is set to 8.

The parameters of the Fourier series is calculated by the least squares principle. For the calibration of ${\psi }_{\mathit{ALFE}}\left(\mathit{lat}\right)$, the parameter model is established as shown by the following equation:

(11) $\mathit{Z}=\mathbf{\Phi }\mathit{X}$(11)

(12) $\mathbf{\Phi }=\left[\begin{array}{cccccc}1& \mathit{cos}\left(\mathit{\omega la}{t}_1\right)& \mathit{sin}\left(\mathit{\omega la}{t}_1\right)& \dots & \mathit{cos}\left(8\mathit{\omega la}{t}_1\right)& \mathit{sin}\left(8\mathit{\omega la}{t}_1\right)\\ 1& \mathit{cos}\left(\mathit{\omega la}{t}_2\right)& \mathit{sin}\left(\mathit{\omega la}{t}_2\right)& \dots & \mathit{cos}\left(8\mathit{\omega la}{t}_2\right)& \mathit{sin}\left(8\mathit{\omega la}{t}_2\right)\\ \dots & \dots & \dots & \dots & \dots & \dots \\ 1& \mathit{cos}\left(\mathit{\omega la}{t}_n\right)& \mathit{sin}\left(\mathit{\omega la}{t}_n\right)& \dots & \mathit{cos}\left(8\mathit{\omega la}{t}_n\right)& \mathit{sin}\left(8\mathit{\omega la}{t}_n\right)\end{array}\right]$(12)

(13) $\mathit{X}={\left[{\mathit{\alpha }}_{\mathit{\psi },0},{\alpha }_{\mathit{\psi },1},{\beta }_{\mathit{\psi },1},\dots ,{\alpha }_{\mathit{\psi },8},{\beta }_{\mathit{\psi },8}\right]}^{\mathit{T}}$(13)

(14) $\mathit{Z}={\left[{\mathit{\psi }}_{\mathit{ALFE}}\left(\mathit{la}{t}_1\right),{\psi }_{\mathit{ALFE}}\left(\mathit{la}{t}_2\right),\dots ,{\psi }_{\mathit{ALFE}}\left(\mathit{la}{t}_n\right)\right]}^{\mathit{T}}$(14)

where $\mathit{la}{t}_i\left(\mathit{i}=1,2,\dots ,\mathit{n}\right)$ is the latitude of the measurement data with index i; n is the number of measurement data; and ${\psi }_{\mathit{ALFE}}\left(\mathit{la}{t}_i\right)$ is the ALFE along yaw angle for the measurement data with index i.

This equation is linear. Thus, the value of the Fourier series parameter with the least squares adjustment principle is as follows:

(15) $\mathit{X}={\left({\mathbf{\Phi }}^{\mathrm{T}}\mathbf{\Phi }\right)}^{-1}\left(\mathbf{\Phi }\mathit{Z}\right)$(15)

Similarly, the parameters of Fourier series for ${\varphi }_{\mathit{ALFE}}\left(\mathit{lat}\right)$and ${\theta }_{\mathit{ALFE}}\left(\mathit{lat}\right)$ can be calculated by the least square fitting. With the section ranges and corresponding Fourier series parameters, the ALFE model between the unconventional mode and the conventional mode is established.

2.4. ALFE compensation

For the ALFE compensation, based on the on-orbit installation parameters, the initial attitude determination results of the unconventional mode ${\mathit{R}}_{2\mathrm{\_}\mathrm{initial}}^{J2000}\left(\mathit{t}\right)$ are first calculated. Then, the latitude of the satellite in WGS84 can be calculated using the measured time and orbit data. According to the parameters of the segmented compensation model, the compensation parameters ${\psi }_{\mathit{ALFE}}\left(\mathit{lat}\left(\mathit{t}\right)\right),{\varphi }_{\mathit{ALFE}}\left(\mathit{lat}\left(\mathit{t}\right)\right),{\theta }_{\mathit{ALFE}}\left(\mathit{lat}\left(\mathit{t}\right)\right)$ of the star sensors 2a and 2b mode can be acquired. The transformation matrix for ALFE compensation $R_{ALFE}^{2\mathrm{\_}\mathrm{initial}}\left(\mathit{t}\right)$ can be obtained by the following equation:

(16) ${\mathit{R}}_{ALFE}^{2\mathrm{\_}\mathrm{initial}}\left(\mathit{t}\right)={\mathit{R}}_Y\left({\psi }_{\mathit{ALFE}}\left(\mathit{lat}\left(\mathit{t}\right)\right)\right){\mathit{R}}_X\left({\varphi }_{\mathit{ALFE}}\left(\mathit{lat}\left(\mathit{t}\right)\right)\right){\mathit{R}}_Z\left({\theta }_{\mathit{ALFE}}\left(\mathit{lat}\left(\mathit{t}\right)\right)\right)$(16)

where ${\mathit{R}}_Y,{\mathit{R}}_X,{\mathit{R}}_Z$ are the transformation matrices corresponding to the pitch, roll, and yaw angles, respectively.

With transformation matrix for ALFE compensation, the precision attitude determination results $R_{2\mathrm{\_}\mathrm{precise}}^{J2000}\left(\mathit{t}\right)$ can be calculated by the following equation:

(17) ${\mathit{R}}_{2\mathrm{\_}\mathrm{precise}}^{J2000}\left(\mathit{t}\right)={\mathit{R}}_{2\mathrm{\_}\mathrm{initial}}^{J2000}\left(\mathit{t}\right){\mathit{R}}_{ALFE}^{2\mathrm{\_}\mathrm{initial}}\left(\mathit{t}\right)$(17)

After ALFE compensation, the difference of attitude determination results between conventional and unconventional modes is eliminated. Moreover, the geometric positioning accuracies of the GF7 imagery and laser footprint can satisfy the requirement for large scale topographic mapping, regardless of the conventional mode or unconventional mode of the attitude determination.

3. Experimental results and discussion

3.1. Experimental data

To calibrate the ALFE of the multiple star sensors system of GF7, the original measurement data of four star sensors within an orbit period is acquired as the calibration data, collected on 3 April 2020; the number of calibration data is 45,462 as listed in Table 3. To evaluate the accuracy and reliability of the proposed model, the original measurement data within 346 orbit periods are applied as the verification data. The acquisition time is from 19 March to 29 April. The time range is 42 days and the number of data is 15,728,691. These verification data can be used to comprehensively analyze the compensation accuracy of the proposed model in different time ranges.

Table 3. Description of experimental data.

type	orbit period	acquisition time range	number of data
calibration	1	2020-04-03T20:14:36.733 2020-04-03T21:01:49.196	45,462
verification	346	2020-03-19T10:50:54.207 2020-04-29T19:47:11.694	15,728,691

To evaluate the accuracy of the proposed method, the attitude determination consistency between the star sensor 1a and 1b mode and the star sensor 2a and 2b mode is adopted here as shown in Equation (18)(18) $\mathrm{\Delta }\mathit{c}=\left[\begin{array}{c}{\theta }_1^{J2000}-{\theta }_2^{J2000}\\ {\varphi }_1^{J2000}-{\varphi }_2^{J2000}\\ {\psi }_1^{J2000}-{\psi }_2^{J2000}\end{array}\right]$(18) :

(18) $\mathrm{\Delta }\mathit{c}=\left[\begin{array}{c}{\theta }_1^{J2000}-{\theta }_2^{J2000}\\ {\varphi }_1^{J2000}-{\varphi }_2^{J2000}\\ {\psi }_1^{J2000}-{\psi }_2^{J2000}\end{array}\right]$(18)

where $\mathrm{\Delta }\mathit{c}$ is the attitude determination consistency; ${\theta }_1^{J2000},{\varphi }_1^{J2000},{\psi }_1^{J2000}$ are the yaw，roll, and pitch transformation angles, respectively, calculated from the attitude determination results of star sensor 1a and 1b; ${\theta }_2^{J2000},{\varphi }_2^{J2000},{\psi }_2^{J2000}$are the yaw，roll, and pitch transformation angles, respectively, calculated from the attitude determination results of star sensor 2a and 2b.

3.2. Calibration for multiple star sensors system

3.2.1. Calibration for installation parameter

To evaluate the calibration accuracy of the installation parameters, based on the measurement data, the angles of the three axes between the fiducial star sensor and the other sensors are calculated to compare the differences with the installation values before and after installation parameter calibration. Figure 4(a–c) show the x-axis, y-axis, and z-axis angles of star sensor 1a and 1b, respectively. Figure 4(d–f) show angle of three axes of star sensors 1a and 2a. Figure 4(g–i) show angle of three axes of star sensor 1a and 2b. The measured angles are represented by blue points. The installation angles before and after calibration are represented by red and yellow lines, respectively. Moreover, the Mean Error (MEAN) and RMSE are also applied to analyze the errors of the installation angles before and after the calibration, as listed in Table 4.

Figure 4. The angle of coordinate axes of two star sensors before and after installation parameter calibration.

Table 4. The angle error of coordinate axes before and after installation parameter calibration.

sensor		MEAN(”)			RMSE(”)
		x-axis	y-axis	z-axis	x-axis	y-axis	z-axis
1a, 1b	before	26.717	−145.695	−38.263	27.790	146.131	38.273
	after	−0.008	0.002	−0.017	7.647	11.279	0.871
1a, 2a	before	−186.578	−105.513	202.702	186.795	106.142	202.765
	after	−0.001	−0.003	−0.003	8.998	11.537	5.056
1a, 2b	before	−153.826	28.319	−224.661	154.177	30.716	224.670
	after	−0.006	−0.001	−0.009	10.402	11.896	2.034

As shown by blue points and red line in Figure 4, there is a large difference between the measured angle and on-ground angle owing to the vibrations and thermal shocks during the launch and orbit penetration processes. Before calibration, the mean errors of the x-axis, y-axis, and z-axis angles of star sensors 1a and 1b are 26.717”, −145.695”, and −38.263”, respectively. For star sensors 1a and 2a or star sensors 1a and 2b, the angle errors of the three axes reach hundreds of arc-seconds. After installation parameter calibration, the angles of the three axes approximate to the measured angles, as shown by blue points and yellow line in Figure 4. The MEAN is less than 0.02” for all three axes angles of the four star sensors, which means that the installation parameters are calibrated well.

Moreover, RMSE represents the influences of the measurement noise and ALFE. As shown in Figure 4, the measurement noise of the x-axis and y-axis is larger than that of the z-axis. Thus, the RMSE of the x-axis or y-axis angles is larger than that of the z-axis angle as listed in Table 4. The RMSE of the x-axis and y-axis angles of star sensors 1a and 1b are 7.647”, and 11.279”, respectively. It demonstrates that the measurement noise of the x-axis or y-axis of the star sensors of GF7 is approximately 30”(3σ). The RMSE of z-axis angle between star sensor 1a and star sensors 1b, 2a, and 2b are 0.871”, 5.056”, and 2.034”, respectively, which express the ALFE characteristics. It can be seen that the ALFE of star sensors 1a and 1b are less than 1”. The ALFE of star sensor 2a is the largest among these four star sensors.

3.2.2. Calibration for ALFE

As shown in Figure 5, for the latitude from 0° to 80°N in the daytime area, the errors of attitude consistency between conventional and unconventional modes are marked with blue points. The extracted ALFE and calibrated result of the segmented compensation model are represented by red and yellow curves, respectively. In Figure 5, the distribution ranges of the ALFE along the yaw, roll, and pitch angles are [−22.990”, −1.633”], [−7.924”, 4.260”], and [−0.909”, 14.042”], respectively. The geometric error introduced by these errors is approximately 15.737 m, which severely decreases the reliability of GF7 imagery with the unconventional attitude determination mode. For the ALFE along the yaw, roll, and pitch angles, red curves coincide with yellow curves, demonstrating that the parameters of the segmented compensation model calculated by the proposed method fit the ALFE well.

Figure 5. Calibration results of the segmented ALFE compensation model.

To evaluate the calibration accuracy of the ALFE model within different positions, the calibration errors are segmented into six sections from 90°N to 90°S with a latitude interval of 30° as listed in Table 5. In these six sections, the mean errors of calibration error along the yaw, roll, and pitch angles are less than 0.001”, and RMSE of calibration error along the yaw, roll, and pitch angles are less than 0.06”. The proposed model can be used to compensate for the ALFE with the consistent accuracy in any latitude. For the whole sections, the yaw MEAN, roll MEAN, and pitch MEAN are −6.854E-08”, −7.582E-07”, and 2.173E-07”, respectively; the yaw RMSE, roll RMSE, pitch RMSE are 0.049”, 0.040”, and 0.026”, respectively. Therefore, the parameters of the segmented compensation model obtain fine calibration results with an accuracy of 0.06”.

Table 5. Calibration accuracy of the segmented ALFE compensation model.

	MEAN(”)			RMSE(”)
	yaw	roll	pitch	yaw	roll	pitch
60°N-90°N	−1.828E-05	1.660E-04	−3.683E-05	0.057	0.046	0.032
30°N-60°N	−3.630E-05	−2.342E-04	4.008E-05	0.044	0.037	0.024
0°-30°N	1.148E-04	4.969E-05	8.014E-06	0.054	0.037	0.024
0°-30°S	−4.500E-04	−4.082E-04	−2.192E-05	0.049	0.039	0.021
30°S-60°S	5.127E-04	5.303E-04	9.427E-05	0.043	0.033	0.022
60°S-90°S	−1.380E-04	−1.142E-04	−8.940E-05	0.043	0.046	0.030
ALL	−6.854E-08	−7.582E-07	2.173E-07	0.049	0.040	0.026

3.2.3. Attitude determination consistency for calibration data

The attitude determination consistency between conventional and unconventional modes is applied to evaluate the calibration accuracy. Figure 6(a–c) show the attitude determination consistencies of the original data, attitude determination result after installation parameter calibration, and attitude determination result after ALFE compensation, respectively. Moreover, the Min Error (MIN), Max Error (MAX), MEAN, and RMSE are adopted here to analyze the errors, as listed in Table 6.

Figure 6. Attitude determination consistency for calibration data.

Table 6. Attitude determination consistency for calibration data.

		yaw(”)	roll(”)	pitch(”)
original	MIN	−269.000	−21.818	419.377
	MAX	−210.631	−1.266	451.405
	MEAN	−240.996	−9.424	434.063
	RMSE	241.616	10.182	434.144
after installation calibration	MIN	−27.993	−12.352	−14.686
	MAX	30.353	8.168	17.344
	MEAN	3.151E-04	−2.663E-04	6.259E-04
	RMSE	17.301	3.858	8.420
after ALFE compensation	MIN	−8.548	−4.162	−4.464
	MAX	5.482	5.201	4.400
	MEAN	1.695E-04	1.876E-05	4.438E-05
	RMSE	1.064	1.022	1.061

For the original data with the on-ground parameters, the yaw MEAN, roll MEAN, and pitch MEAN of attitude determination consistency are −240.996”, −9.424”, and 434.063”, respectively. After the calibration of the installation parameters, the yaw MEAN, roll MEAN, and pitch MEAN of attitude determination consistency are decreased to 3.151E-04”, −2.663E-04”, and 6.259E-04”, respectively. Moreover, the yaw RMS, roll RMS, and pitch RMS of the attitude determination consistency are 17.301”, 3.858”, and 8.420”, respectively. This reflects the influence of ALFE. With ALFE compensation, attitude determination consistency is improved significantly, with a yaw RMS of 1.064”, roll RMS of 1.022”, and pitch RMS of 1.061”, respectively. They are similar to the measurement noise of the optical axis of star sensors 2a and 2b. As shown in Figure 6(b), the errors of attitude determination consistency are systematic errors related to latitude. After ALFE compensation, the errors of attitude determination consistency are random, as shown in Figure 6(c). Therefore, based on the proposed method, the attitude determination consistency between the star sensors 2a and 2b mode and the star sensors 1a and 1b mode is improved significantly from more than 200” to less than 1.1” without systematic error.

3.3. Attitude determination consistency verification

3.3.1. Attitude determination consistency improvement

For the verification data within 346 orbit periods over 42 days, the attitude determination consistencies between conventional and unconventional modes for the original data, installation calibrated result, and ALFE compensation result are applied to evaluate the accuracy and reliability of the proposed method, as listed in Table 7.

Table 7. Attitude determination consistency for verification data within 346 orbit periods.

		yaw(”)	roll(”)	pitch(”)
original	MIN	−268.843	−20.889	420.322
	MAX	−214.270	−2.183	450.189
	MEAN	−240.342	−9.532	433.670
	RMSE	240.982	10.286	433.750
after installation calibration	MIN	−27.858	−11.421	−13.741
	MAX	26.718	7.232	16.128
	MEAN	0.654	−0.108	−0.392
	RMSE	17.570	3.869	8.329
after ALFE compensation	MIN	−8.555	−6.601	−7.102
	MAX	7.521	5.793	9.504
	MEAN	−0.442	0.072	−0.132
	RMSE	1.963	1.475	1.731

For the original data, owing to the large differences between the on-orbit and on-ground installation parameters, the attitude determination consistency is very low, with the yaw MEAN of −240.342”, roll MEAN of −9.532”, pitch MEAN of 433.670”, yaw RMS of 240.982”, roll RMS of 10.286”, and pitch RMS of 433.750”. After the calibration of the installation parameters, the yaw MEAN, roll MEAN, and pitch MEAN of the attitude determination consistency are decreased to 0.654”, −0.108”, and −0.392”, respectively, which are influenced by the measurement noise. Moreover, the yaw RMSE, roll RMSE, and pitch RMSE of the attitude determination consistency are 17.570”, 3.869”, and 8.329”, respectively. They reflect the value of ALFE. Based on the parameters of segmented compensation model, the yaw RMSE, roll RMSE, and pitch RMSE are decreased to 1.963”, 1.475”, and 1.731”, respectively. For the verification data set, it demonstrates that the proposed method can eliminate ALFE well and improve the attitude determination result consistency from more than 200” to less than 2”.

3.3.2. Analysis with different acquisition time

To further analyze the reliability of the proposed method over a long time range, the attitude determination consistencies of the verification data within 346 orbit periods before and after ALFE compensation are calculated as shown in Figure 7. Figure 7(a,c,e) show the errors of attitude determination consistency along the yaw, roll, and pitch angles before ALFE compensation, respectively. Figure 7(b,d,f) show the errors of attitude determination consistency after ALFE compensation.

Figure 7. Attitude determination consistencies of verification data within 346 orbit period.

As shown by the curves of the small images in Figure 7, the ALFE characteristics at latitudes of 0°, 30°N, 60°N are different before compensation. Using the proposed method, these systematic errors are compensated well, and the error of attitude determination consistency distributes randomly. It demonstrates that the proposed model can be applied to the ALFE compensation at various latitudes and eliminate the systematic error. Moreover, when the time interval between the acquisition time of the verification data and the acquisition time of the calibration data is longer, the error of attitude determination consistency after the ALFE compensation is larger. As shown by the curves of the small images in Figure 7(b,d,f), the errors in the attitude determination consistency of the verification data within the orbit numbers from 300 to 346 are larger than those within other orbit numbers. The acquisition time of the verification data within the orbit numbers from 300 to 346 is 20 April to 29 April and the time interval to the acquisition date of calibration data is more than 17 days. It is caused by changes in the spatial thermal environment for the same latitude.

Attitude determination consistencies of the conventional and unconventional modes before and after ALFE compensation are given in Tables 8 and 9, respectively. When the time interval is 3 days, i.e. the acquisition time of 31 March, the errors of attitude determination consistency along the yaw, roll, and pitch angles are 1.907”, 1.175”, and 1.336”, respectively. When the time interval is 7 days, i.e. the acquisition time of 10 April, the errors of attitude determination consistency along the yaw, roll, and pitch angles are 1.554”, 1.359”, and 1.458”, respectively. When the time interval is 15 days, i.e. the acquisition time of 19 March, the errors of attitude determination consistency of three angles are 2.055”, 1.421”, and 1.693”. When the time interval is 26 days, i.e. the acquisition time of 29 April, the errors of attitude determination consistency of three angles are 2.229”, 1.754”, and 2.339”. With the time interval increasing, the consistency of characteristic of ALFE decreases. It is caused by the changes of the solar radiation and the effect of the temperature control system for the same latitude. Thus, to ensure an attitude determination consistency better than 2”, the time interval between the calibration data and the test data should be smaller than 15 days, i.e. the time range of reliable compensation is about 30 days.

Table 8. The attitude determination consistencies of conventional mode and unconventional mode before ALFE compensation.

Date	MEAN(”)			RMSE(”)
	yaw	roll	pitch	yaw	roll	pitch
2020-03-19T15:35:03.241	−0.024	0.465	0.486	16.738	3.846	8.110
2020-03-23T19:02:13.828	−0.549	0.260	0.325	16.874	3.866	8.178
2020-03-27T19:19:57.155	−0.108	0.066	−0.191	16.964	3.792	8.311
2020-03-31T00:41:03.843	−0.966	−0.087	−0.049	17.044	3.775	8.361
2020-04-04T00:58:45.514	−0.333	−0.080	−0.186	17.429	3.859	8.373
2020-04-10T19:34:30.207	−0.119	−0.302	0.336	17.627	3.671	8.532
2020-04-13T18:36:42.508	0.087	−0.333	0.064	17.678	3.483	8.472
2020-04-16T19:13:37.103	0.304	−0.381	0.077	17.398	3.572	8.237
2020-04-20T14:47:06.174	0.078	−0.474	−0.101	17.247	3.733	7.894
2020-04-24T05:36:25.947	−0.228	−0.797	−0.550	17.227	3.579	7.764
2020-04-29T19:47:11.694	−0.163	−0.918	−0.147	17.611	3.631	7.677
MEAN	−0.138	−0.218	0.037	17.256	3.709	8.175

Table 9. The attitude determination consistencies of conventional mode and unconventional mode after ALFE compensation.

Date	MEAN(”)			RMSE(”)
	yaw	roll	pitch	yaw	roll	pitch
2020-03-19T15:35:03.241	−0.086	0.489	0.503	2.055	1.421	1.693
2020-03-23T19:02:13.828	−0.615	0.283	0.344	1.907	1.334	1.491
2020-03-27T19:19:57.155	0.325	0.271	0.170	1.927	1.249	1.553
2020-03-31T00:41:03.843	−1.037	−0.065	−0.027	1.907	1.175	1.336
2020-04-04T00:58:45.514	−0.404	−0.058	−0.165	1.210	1.045	1.107
2020-04-10T19:34:30.207	−0.186	−0.281	0.357	1.554	1.359	1.458
2020-04-13T18:36:42.508	0.022	−0.312	0.084	1.697	1.432	1.549
2020-04-16T19:13:37.103	0.243	−0.360	0.095	1.724	1.733	1.668
2020-04-20T14:47:06.174	0.021	−0.454	−0.085	2.266	2.266	2.006
2020-04-24T05:36:25.947	−0.280	−0.777	−0.536	2.283	1.777	2.307
2020-04-29T19:47:11.694	−0.208	−0.898	−0.136	2.229	1.754	2.339
MEAN	−0.201	−0.197	0.055	1.887	1.504	1.683

4. Conclusions

For the multiple star sensors system, owing to the variations in the incident angle of the sunlight and solar radiation, there is a thermal deformation in the body and installation structure of each star sensor, causing ALFE and seriously affecting the consistency of attitude determination results for different star sensor combination modes. This study was aimed at solving the ALFE analysis and calibration for multiple star sensors system of GF7 satellite. The installation parameters of star sensors are first calibrated using the statistical characteristics of the angle of three axes between the fiducial star sensor and other star sensors. Then, after analyzing the characteristics of the optical axis angles over 135 days, the ALFE between the conventional and unconventional modes for attitude determination is extracted from the divergence of the attitude calculation results and calibrated using the segmented Fourier series model with latitude as the input parameter. Based on the on-orbit installation parameters and the ALFE model parameters, the precise attitude determination results of unconventional mode are calculated.

The proposed method is tested with the on-orbit measurement data of the four star sensors of GF7. Experimental results show that the calibration accuracy of the ALFE model is better than 0.06”. The error of attitude determination consistency between the star sensors 2a and 2b mode and the star sensors 1a and 1b mode is improved from more than 200” to less than 2”, which satisfies the demand for high-precision geometric positioning for GF7. Moreover, to ensure the reliability of the compensation accuracy, the time range of the ALFE calibration model can be set to 30 days.

Acknowledgement

The authors would like to thank the Editor and the reviewers for their constructive comments and valuable suggestions to improve the quality of the letter. GF7 data are provided by CRESDA. These supports are valuable.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data that support the findings of this study are available on reasonable request from the corresponding author.

Additional information

Funding

This work was supported by the National Science Fund for Distinguished Young Scholars [grant number 61825103] and the Shanghai Aerospace Science and Technology Innovation Fund.

Notes on contributors

Yanli Wang

Yanli Wang is a lecturer in Shandong University of Science and Technology. She received the PhD degree from Wuhan University in 2021. Her current research interest is the attitude determination of remote sensing satellite.

Mi Wang

Mi Wang is a professor in Wuhan University. He received the PhD degree in photogrammetry and remote sensing from Wuhan University in 2001. His research interests include the geometric processing and intelligent service of remote sensing imagery of high resolution optical satellite.

Ying Zhu

Ying Zhu is an associate professor in Wuhan Institute of Technology. She received the PhD degree from Wuhan University in 2016. Her research interests include geometric accuracy improvement of remote sensing imagery and platform jitter analysis and processing of optical satellite.

Xiaoxiang Long

Xiaoxiang Long is senior engineer with China Center for Resources Satellite Data and Application. His research interest is the geometric processing of remote sensing imagery of high resolution optical satellite.