We aim to better evaluate the accuracy of star sensors and provide accurate guarantees for satellite measurement accuracy. When remote sensing satellites are in orbit, the requirements for imaging quality are increasingly high. In addition to high-resolution imaging of the payload, the satellite platform attitude must also have high accuracy. Star sensors are high-precision satellite attitude measurement sensors, and their measurement accuracy directly determines the accuracy of satellite attitude determination. Their measurement accuracy is mainly influenced by factors such as measurement noise of star sensors, calibration errors of installation matrices, and thermal deformation of satellite structures. Usually, when calculating attitude measurement errors, a reference attitude, also known as the true attitude, is required. The measurement error is obtained by statistically comparing the measured attitude with the reference attitude. After the satellite enters orbit, it is usually difficult to establish this true reference attitude, so it cannot be compared with the reference attitude. The current method for calculating the measurement error of star sensors in orbit is to statistically analyze the change in attitude quaternion output by the star sensor or to quantitatively analyze the representation of the star sensor’s optical axis in inertial space. The implementation of determining the measurement error of star sensors through camera payload or using landmarks to measure the low-frequency error of star sensors is complex and difficult to generalize. At present, there is little systematic analysis in the literature on the low-frequency errors, noise equivalent angles, optical aberration, and precession of star sensors, especially the influence of satellite thermal environment changes on the accuracy of star sensors, including thermal deformation of the optical mechanical structure and support convex platform of the star sensor itself. Therefore, to better evaluate the accuracy of star sensors and provide accurate guarantees for satellite measurement accuracy, it is necessary to conduct a detailed analysis of multiple aspects such as random measurement errors, thermal deformation errors, optical aberration, and precession based on the in-orbit measurement data of star sensors.

First, we classify the in-orbit errors of star sensors. Second, we present three methods for analyzing in-orbit errors: epoch difference method, polynomial fitting method, and optical axis angle method. To eliminate the shaking of the satellite platform, we use the optical axis angle method to analyze the four star sensors of the four-dimensional Gaojing-3 satellite. We obtain the orbital period errors (including the thermal stability error of the star sensors and the thermal deformation error of the support convex platform) and the random errors of the star sensors such as low-frequency error and noise equivalent angle (Figs. 1?4). Based on the analysis of in-orbit data, we identify the reason for the large orbital period error of star sensor 2 and propose a method for correcting optical aberration. Using UTC, satellite orbital velocity, and star sensor quaternion, we calculate the attitude matrix after optical aberration correction and the quaternion after optical aberration compensation. We upgrade the optical aberration function of star sensor 2 in application software through in-orbit programming method and analyze the corrected data of optical aberration, resulting in a significant reduction in in-orbit error. Through the analysis of the angle between the optical axes of star sensors 1 and 2, we find that there is an error with an orbital time period (Fig. 8), and provide a precession correction method (Eqs. 10?14). We verify the necessity of precession correction by analyzing the optical axis angles of different star sensors before and after precession correction (Fig. 9).

The orbital period error of single star sensors 1 and 2 is 1.1112″, while the orbital period error of single star sensors 3 and 4 is 11.2637″. The difference in orbital period error between star sensors 1 and 2 is significant (Table 1). Considering that the materials, installation positions, and angles of the two types of star sensor bracket protrusions are basically same in the design of the entire star, and the thermal deformation of the bracket protrusions caused by the alternation of cold and hot orbits is basically same, the main reasons for the differences are the measurement errors and thermal deformation of the star sensors themselves. Optical aberration correction can significantly reduce orbital period errors, with little effect on noise equivalent angle and low-frequency errors. After optical aberration correction, the orbital period error of individual star sensors 3 and 4 is reduced from 11.2637″ to 2.5689″, and the optical aberration is effectively eliminated (Table 2). The main component of the orbital period error of star sensors 3 and 4 after optical aberration correction is the thermal stability error caused by the thermal deformation of the support and the star sensor. Meanwhile, the noise equivalent angle and low-frequency error of the star sensor are not affected before and after optical aberration correction. From the in-orbit data, it can be proven that the thermal stability error of star sensor 1 is about 4.73% of the thermal stability error of star sensor 2 [0.1203 (″)/℃)/(2.5414 (″)/℃]. By analyzing the angle between the optical axes of star sensors 2 and 4, it can be found that there is an error with a period of one orbital time, and the difference between the maximum and minimum values reaches 144″. One of the two star sensors has not enabled precession correction, which may result in a deviation in orbital period and requires precession correction (Fig. 8). Before and after correcting the precession of the star sensor, the average range of change in the optical axis angle between star sensor 1 and star sensor 2 decreases by 79″, accounting for 65.24%, which can effectively reduce the orbital period error of the star sensor (Table 3).

We provide a detailed classification of the in-orbit errors of star sensors, analyze the sources of errors, and make corrections to the errors. Star sensors are high-precision satellite attitude measurement sensors, and their measurement accuracy directly determines the accuracy of satellite attitude determination. We analyze the in-orbit data of four star sensors 1, 2, 3, and 4, based on the requirements of the Four Dimensional Gaojing-3 satellite for the allocation of measurement link accuracy indicators such as star sensors, GNSS, satellite platforms, and time benchmarks. Firstly, we introduce the in-orbit errors of star sensors, including random measurement errors of star sensors, thermal deformation errors of star sensor brackets, thermal deformation of bracket mounting surfaces relative to camera reference, thermal drift of star sensor visual axis pointing, aberration, and precession. The methods for analyzing in-orbit errors of star sensors, such as epoch difference method, polynomial fitting method, and optical axis angle method, are provided. We provide a detailed analysis of the low-frequency error, noise equivalent angle, and thermal stability error of the four star sensors 1, 2, 3, and 4. The random noise of the star sensor is 1.3525″, with a noise equivalent angle of 0.9940″ and a low-frequency error of 0.9197″, which meets the accuracy index allocation requirements of the sensing satellite for the star sensor. The orbital period error (X or Y axis of a single star sensor) of star sensor 1 (including thermal deformation of the bracket) is 0.7857″@±1 ℃, and the calculated thermal deformation of the star sensor bracket boss is 0.5458″. The orbital period error of star sensor 2 is 1.8165″@±0.25 ℃. The thermal stability error of star sensor 1 is 0.1203 (″)/℃, and the thermal stability error of star sensor 2 is 2.5414 (″)/℃. Star sensor 1 can effectively improve the thermal stability of in-orbit star sensors by independently isolating and installing a light shield, optimizing the design of the main frame material and structure, and improving the optical and mechanical assembly process. This method reduces the thermal stability error by 95.27% compared to star sensor 2 and can provide a reference for the subsequent thermal stability design of star sensors. Finally, through in-orbit data analysis, the aberration and precession of star sensors are discovered, and corresponding in-orbit correction methods are proposed. After aberration correction, the orbital period error of individual star sensors 3 and 4 is reduced from 11.2637″ to 2.5689″, effectively eliminating aberration. By using precession correction, the error between star sensor 1 and star sensor 3 with one orbital period is reduced from 144″ to 25″, which can reduce the orbital period error by 85.63%. The correction method can improve the in-orbit measurement accuracy of star sensors, assure satellite measurement accuracy, and also provide a reference for future high-precision star sensor design.