The star tracker is a high-precision space attitude measurement device with the stars as the reference system and the starry sky as the working object. By detecting stars in different positions on the celestial sphere and performing calculations, the star tracker provide accurate spatial orientation and reference for satellites, intercontinental strategic missiles, spaceships and other aerospace vehicles. The same as the inertial gyroscope, the star tracker has autonomous navigation capability, which valued in important application.
Through more than half a century of the research, development and applications, and with the emergence of new materials and devices and the progress of process technology, the star trackers grow at the improved accuracy, reduced power consumption, and reduced cost. Therefore, timely collection, analysis and comparison of information of global star trackers is beneficial to the development of domestic attitude measurement and control technologies.
According to different calculation methods, there are two attitude calculation algorithms in common for star trackers: the static deterministic attitude calculation algorithms and the dynamic filter estimation algorithms.
Deterministic Attitude Resolving Algorithm
The Deterministic Attitude Resolving Algorithm refers to calculating the direction cosine matrix of the satellite body coordinate system relative to the inertial coordinate system based on a set of vector observations. The results obtained by the deterministic algorithm have clear geometric and physical meanings, and the instantaneous attitude of the satellite can be obtained by only one measurement. Therefore, the static deterministic algorithm has the advantages of high stability, fast calculation speed, and less memory usage, and it is also the main attitude determination algorithm used in star trackers.
However, it is difficult to solve the Wahba problem directly, and it is difficult to obtain the optimal solution. In 1968, Davenport proposed the q-method, using quaternions to parameterize the attitude matrix, transforming the Wahba problem into the problem of solving the largest eigenvalue of the K matrix, which greatly promoted the development of static deterministic attitude calculation algorithms. Later, researchers proposed TRI AD algorithm, Euler-q algorithm, QUEST algorithm and FORM algorithm. Shuster pointed out that when the TRIAD method is used to solve the problem, the optimal result cannot be obtained when the measurement accuracy of the two observation vectors is not equal . Baritzhack proposed a method to obtain a more accurate attitude matrix by using twice TRIAD method for weighting processing . Starting from the FOAM method, Markley deduced the closed solution form in the case of two-vector observation, which is the most concise form at present, and analyzed the variance of the algorithm.
The QUEST algorithm is the optimal quaternion estimation in the sense of least squares. This algorithm was first applied to the MAGSAT task in 1979, and it is also the most commonly used algorithm to solve the Wahba problem. Shuster proposed the QUEST measurement model and proved that it is relatively accurate for small field of view sensors, and used the QUEST measurement model to deduce the variance matrix of the TRIAD method and the QUEST method, which theoretically proved that the QUEST method is superior to the TRIAD method.
Dynamic Filter Estimation Algorithm
In practical applications, the measurement errors of satellite orbit parameters and the installation errors of star trackers will bring uncertainty errors to the measurement of observation vectors, and these errors are difficult to overcome. In order to meet the needs of high-precision attitude control, the dynamic filter estimation method can be used to calculate the attitude information of the spacecraft. The dynamic filtering estimation method uses the spacecraft to establish the state equation and observation equation according to the attitude dynamic equation, and obtains the real attitude of the spacecraft by the optimal estimation method under certain criteria according to the observation information. Compared with the static deterministic algorithm, the dynamic filter estimation algorithm utilizes more observation information, can provide the optimal solution in the statistical sense, can avoid the influence of uncertain factors on the attitude of the spacecraft, and improve the accuracy of attitude determination.
The Extended Kalman Filter (EKF) algorithm is the most commonly used real-time attitude determination algorithm for spacecraft. According to the selection of attitude parameters and different forms of observations, the common multiplicative extended Kalman filter (MEKF) and additive extended Kalman filter (AEKF), among which MEKF is widely used in various spacecraft attitude determination tasks and most developed. However, EKF is not robust and easy to diverge, and often cannot obtain the optimal solution for estimation problems with strong nonlinear characteristics. Julier and Uhlmann used UT transformation instead of local linearization and proposed the Unscented Kalman filter (UKF), which still has good convergence and obtains better results in the case of large initial errors. Both the EKF and UKF algorithms are based on the assumption that the random part of the system obeys the Gaussian distribution. In the case of uncertain moment model errors in the attitude dynamic model of the star tracker, the results will be Effectiveness cannot be guaranteed.
Based on the QUEST algorithm, Shuster proposed the filtering QUEST algorithm, which realizes the recursive processing function of the Kalman filter by using the propagation of the attitude distribution matrix B. Bar-Itzhack also extended the QUEST algorithm and proposed the RE-QUEST algorithm, which realizes the recursive function through the propagation of the K matrix. Essentially, the filtering QUEST algorithm and the RE-QUEST algorithm are equivalent in mathematics. However, because the accuracy of these two algorithms is relatively poor compared with the EKF algorithm, they are not widely used in engineering. In recent years, with the emergence of new filter estimation methods, more and more algorithms have been used in star tracker dynamic attitude estimation, such as particle filter algorithm, Gaussian filter algorithm and multi-mode adaptive estimation algorithm. The advantage of the dynamic filter estimation method is that it can use the prior knowledge to approach the optimal solution in the statistical sense, but because of the nonlinear method, the complexity of the algorithm is high, and there are still some difficulties in practical use.
Both the static deterministic attitude calculation method and the dynamic filtering algorithm have been practiced and applied in star tracker products. Table 5 compares and analyzes the characteristics of star tracker attitude calculation algorithm. For attitude determination systems with gyroscopes, the most practical and commonly used method is the multiplicative extended Kalman filter (MEKF) method. For attitude determination systems without gyroscopes, QUEST or predictive filter estimation methods can be used.