## Abstract

The star tracker is widely used in attitude control systems of spacecraft for attitude measurement. The attitude update rate of a star tracker is important to guarantee the attitude control performance. In this paper, we propose a novel approach to improve the attitude update rate of a star tracker. The electronic Rolling Shutter (RS) imaging mode of the complementary metal-oxide semiconductor (CMOS) image sensor in the star tracker is applied to acquire star images in which the star spots are exposed with row-to-row time offsets, thereby reflecting the rotation of star tracker at different times. The attitude estimation method with a single star spot is developed to realize the multiple attitude updates by a star image, so as to reach a high update rate. The simulation and experiment are performed to verify the proposed approaches. The test results demonstrate that the proposed approach is effective and the attitude update rate of a star tracker is increased significantly.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The star tracker is a vision sensor for spacecraft attitude measurement in the missions such as earth observation and deep-space exploration [1–3]. For example, in order to acquire the high-resolution earth observation image with accurate geographical position, it is necessary to accurately control the attitude of the remote sensing satellite using the high-precision attitude information from a star tracker [4,5]. The attitude measurement by a star tracker can be regarded as the discrete sampling of spacecraft attitude motion and the attitude update rate represents the sampling frequency. According to the sampling theorem, the continuous attitude information is better recovered from the discrete attitude measurements with higher sampling frequency, and the accuracy of continuous attitude control is increased by using the more precise attitude information. Therefore, the attitude update rate of star tracker should be improved to meet the increasingly higher requirement on the performance of attitude control under dynamic conditions of spacecraft [6–8].

The attitude update rate of star tracker is mainly constrained by the exposure time and readout time in the tracking stage, and accordingly, some techniques are proposed to overcome these time constraints. In [9,10], the multiple pipelines are utilized to enhance the parallel processing capability such that the exposure time and readout time are overlapped and the attitude update rate can reach 8-10 Hz. Since the exposure time is determined by the sensitivity of image sensor, an image intensifier is placed in the optical path of star tracker for reducing the exposure time, and the high update rate is obtained [11,12]. In [13], a multiexposure imaging method is proposed based on the two-level exposure control structure in the image intensifier and sensor to further increase the attitude update rate of intensified star tracker. Although the intensifier star tracker has very short exposure time and higher than 100 Hz attitude update rate, the additional image intensifier inevitably leads to the increased weight, volume and power consumption of star tracker, especially limiting the application on micro- or nano-satellites. Therefore, in this paper, a more general approach based on the electronic Rolling Shutter (RS) mode of image sensor is proposed to improve the attitude update rate of star tracker without using the image intensifier.

The electronic Global Shutter (GS) and RS are common exposure modes for complementary metal-oxide semiconductor (CMOS) image sensors. In GS mode, the whole imaging plane of a CMOS image sensor is exposed simultaneously, while in RS mode, each row of the imaging plane is exposed from reset at a slightly different time [14]. In this paper, the RS mode is applied to improve the attitude update rate of a star tracker equipped with a CMOS image sensor. When the star tracker is rotating under RS mode, the star spots in a star image contain the motion information at different time due to the time offset of each row. Once a star spot is obtained, the attitude of the star tracker can be updated immediately based on the attitude estimation method with a single star spot. The multiple attitude updates by different star spots in a star image effectively improve the update rate of the star tracker. The main contributions of this paper are emphasized as follows: 1. A more general approach for improving the attitude update rate of a star tracker is proposed based on RS mode, without using the image intensifier. 2. A new attitude estimation method is developed for attitude update with a single star spot. 3. The simulation and experiment illustrate that the attitude update rate is increased significantly with the proposed approach.

## 2. High attitude update rate based on RS mode

In this section, the method for improving the attitude update rate is proposed by making full use of the advantage of RS mode. GS and RS modes are two exposure ways of the CMOS image sensor in the star tracker to acquire star images and the traditional star trackers usually work in GS mode. The operations of GS and RS modes are shown in Fig. 1.

It can be seen from Fig. 1 that the main difference between GS and RS modes is the exposure ways of the CMOS image sensor. In GS mode, the star image is acquired in the way that the whole imaging plane of the CMOS image sensor is exposed simultaneously, To be special, all rows of the imaging plane are exposed from $t=0$. After the exposure time ${t}_{e}$, the pixel data is read out row to row. Then the processes of centroid extraction, star identification and attitude determination are performed, and the attitude of the star tracker is yielded later than ${t}_{GS}$. Due to the simultaneously exposed imaging plane, all star spots in a star image contain the motion information at the same time and are used for attitude determination together. Therefore, the attitude update rate in GS mode is dependent on the exposure time and the readout time of the complete star image. Assuming that the imaging plane has $n$ rows, the attitude update rate of the star tracker in GS mode can be given by

where ${t}_{rd}$ is the readout time of pixel data for each row.In contrast to GS mode, RS mode is realized by the row-to-row sequential exposure of the imaging plane. Only the first row of the imaging plane is exposed at $t=0$, then the following rows are exposed in turn after a small row-to-row time offset ${t}_{row}$. It should be pointed out that although the row-to-row time offsets can be configured arbitrarily, the most effective way to obtain pixel data is that every row of the imaging plane is read out in sequence without additional time interval. Therefore, the row-to-row time offset is set as

The processing procedure of attitude update based on RS mode is shown in Fig. 2.

In Fig. 2, the star spots distributed in different rows of the star image can reflect the attitude of the star tracker at different times due to the different initial exposure moments caused by row-to-row time offset in RS mode. For example, there are $m$ star spots in a star image with initial exposure moments ${t}_{i},\text{\hspace{0.17em}}i=1,\dots ,m$. When the first star spot is exposed from ${t}_{1}$ and read out at time ${t}_{1}+{t}_{e}+{t}_{rd}$, the attitude of the star tracker can be updated after the processes of centroid extraction, star identification and attitude estimation. Similarly, the attitude can be updated with other star spots in the star image, meaning that $m$ updates are obtained by a star image. Moreover, noticing that the attitude update in RS mode may appears during the exposure period of GS mode in Fig. 1, the attitude update rate of the star tracker in RS mode satisfies

Therefore, the attitude update rate ${f}_{ERS}$ is much higher than ${f}_{GS}$.On the other hand, a CMOS image sensor has better signal to noise ratio (SNR) in RS mode than in GS mode because of the different pixel architectures between RS and GS. The pixel architectures of the commonly used CMOS image sensor in a star tracker are shown in Fig. 3. The function of transistor T1 is to reset the pixel photodiode D to a given level. When the photodiode is illuminated by light, the voltage of the photodiode drops with the electron accumulation. The transistor T2 amplifies the voltage signal of the photodiode and the transistor T3 is for reading out the amplified signal. The exposure time of a pixel is the period from reset to readout. Since the pixel signal can only be read out one by one, the initial readout moment is different for each row, essentially forming the RS mode as in Fig. 1. In order to implement the GS mode, a sample transistor T4 is added to lock the photodiode signal. The additional T4 causes more noises, lower SNR and smaller fill factor of a pixel. Usually, the readout noise in RS mode is about half of that in GS mode [15].

Since there is only one star spot used for an attitude update in RS mode, the existing attitude determination algorithms are unavailable, which brings some challenges for the realization of high update rate technique. It can be seen later that the attitude estimation method with a single star spot is developed to deal with this problem. On the other hand, the spatial distortion of a star image occurs in RS mode during spacecraft motion and affects the full-sky star identification in the initialization stage of the star tracker, because the full-sky star identification relies on the relative positions of star spots in a star image. The RS compensation method in [16] can be applied to correct the spatial distortion, and then the identification is completed with the traditional full-sky star identification technique [17].

## 3. Attitude estimation with a single star spot

Generally, the attitude of star tracker is determined by more than two noncollinear observation vectors [18,19]. However, in RS mode, only one observation vector obtained from a single star spot can be used for an attitude update. A new attitude estimation method based on a single star spot is proposed in this section. The process and measurement models for attitude estimation are established firstly.

#### 3.1 Kinematic equation of attitude error

The quaternion is introduced for convenient explanation of the system modeling, defined as $q={\left[\begin{array}{cc}{q}_{v}^{\text{T}}& {q}_{\text{4}}\end{array}\right]}^{\text{T}}$ with ${q}_{v}={\left[\begin{array}{ccc}{q}_{1}& {q}_{2}& {q}_{3}\end{array}\right]}^{\text{T}}$. In order to parameterize the attitude of star tracker, the quaternion is required to satisfy the norm constraint $\Vert q\Vert =1$. The corresponding attitude matrix is given by

Using attitude quaternion, the attitude kinematic equation of star tracker can be expressed by

withDefine attitude quaternion error as

where $\otimes $ indicates quaternion multiplication [21] and $\widehat{q}$ is the attitude quaternion estimate satisfyingwith the angular velocity estimate $\widehat{\omega}$.Then the differential equation for attitude quaternion error can be yielded as follow based on Eq. (6), Eq. (11) and Eq. (12).

Using small angle approximation [22], the attitude quaternion error $\delta q$ can be represented as

#### 3.2 Measurement model

The vector observation model of star tracker is established based on the pinhole imaging system shown in Fig. 4.

In Fig. 4, $r$ represents the reference vector in inertial frame ${o}_{i}-{x}_{i}{y}_{i}{z}_{i}$, which can be expressed as

$b$ is the observation vector in star tracker frame ${o}_{s}-{x}_{s}{y}_{s}{z}_{s}$ and satisfies

The optical design of the star tracker is shown in Fig. 5.

The coordinate $\left(\overline{x},\text{\hspace{0.17em}}\overline{y}\right)$ obtained through centroid extraction is affected by the shape of a star spot. The point spread function (PSF) of the optical system determines the shape of a star spot and is analyzed by ZEMAX software, seeing Fig. 5(b). In order to improve the precision of centroid extraction, the light path of the optical system should be designed to guarantee that more than 90% of the total energy of a star spot is distributed in a $5\times 5$ pixel region as shown in Fig. 5(c), satisfying the following PSF [23].

The precision of the coordinate $\left(\overline{x},\text{\hspace{0.17em}}\overline{y}\right)$ is also influenced by optical distortion, detector inclination and some optical and alignment errors. Therefore, a high-precision calibration is necessary to reduce the error of the coordinate $\left(\overline{x},\text{\hspace{0.17em}}\overline{y}\right)$. The residual error of $\left(\overline{x},\text{\hspace{0.17em}}\overline{y}\right)$ after the calibration is less than 0.02 pixels (about 100nm) [3].

The optical design involves stray light suppression as well. A big baffle as shown in Fig. 6 is employed to eliminate stay light effects from the sun, the moon, the earth albedo, etc [24]. The exclusive angle of the baffle is greater than 35° for the sun and 25° for the earth.

Even with the baffle, the stray light sometimes still gets into the field of view during spacecraft motion. The image processing technique that is robust to stray light can be applied to complete star extraction with stay light. See Fig. 7 that the moonlight moves into the field of view. The traditional global threshold method is hard to distinguish the star spots and the stay light from the moon, and morphology operations can be used to solve this problem. Define three flat square structuring elements ${B}_{s}$ ${B}_{i}$ and ${B}_{o}$ as shown in Fig. 8. $\text{\Delta B=}{\text{B}}_{o}-{B}_{i}$ is the margin region between ${B}_{i}$ and ${B}_{o}$. The star spots can be detected from the background with moonlight by the following operation [25]:

Based on the above morphology operations, the observation vector can be obtained using the detected star spot. Considering that the observation vector in star tracker frame is related to the reference vector in inertial frame with an attitude matrix, the vector observation model of star tracker is given by

It can be seen from Eq. (23) that the star identification with a single star spot is needed for obtaining the reference vector $r$. Firstly, the initial attitude and angular velocity of a star tracker are established through the full-sky star identification in the initialization stage. Then the star tracker enters the tracking process and works in RS mode. When a star spot is detected, the corresponding vector in inertial frame can be calculated through coordinate transformation using a predicted attitude quaternion yielded by Eq. (8) and is compared with the vectors in the guide star catalog. The single star identification is completed by finding the nearest reference vector in the catalog.

Based on the definition of attitude quaternion error and the small angle approximation, the vector observation model (23) can be rewritten as

From Eq. (16) and Eq. (25), the process and measurement models for attitude estimation in RS mode are established, where the angular velocity plays an important role in attitude propagation but has not been obtained. The unknown angular velocity can be calculated by observation vectors in successive frames of star images later.

#### 3.3 Angular velocity determination

The least-squares approach is applied to determine the angular velocity in the kinematic Eq. (16) based on observation vectors in sequential star images. If the star tracker rotates with angular velocity $\omega $, the positions of star spots changes in star images. Provided there are $N$, $N\ge 2$, pairs of star spots in two successive frames of star images (Image 1 and Image 2), the notations for the $i$-th pair of star spots related to same reference vector ${r}_{i}$ are summarized in Table 2. An effective method to find the pairs of star spots is present in [16].

Based on the vector observation model of star tracker, we have

Considering the angular velocity $\omega $ is almost constant in the small time interval ${t}_{i}^{\left(2\right)}-{t}_{i}^{\left(1\right)}$, the first-order approximation of attitude matrix propagation can be given by [22]

Substituting Eq. (29) into Eq. (28), gives

Generally, the approximation $\left({t}_{i}^{\left(2\right)}-{t}_{i}^{\left(1\right)}\right)\Vert \omega \Vert <<1$ is valid for star trackers, and thus have

According to Eq. (31), the least-squares approach can be applied to determine angular velocity, that is

Substituting Eq. (31) into Eq. (35), the angular velocity estimate error $\delta {\omega}_{k}$ can be expressed by

#### 3.4 Attitude estimation scheme under RS mode

With the results from previous discussion, the multiplicative extended Kalman filtering [26] is used for attitude estimation with a single star spot, outlined as follows:

- 1. Initialization
Set ${t}_{1}=0$. Giving initial values for attitude quaternion estimate, $\widehat{q}\left(0\right)$, attitude estimate error covariance, $P\left(0\right)$, angular velocity estimate, $\widehat{\omega}\left(0\right)$, and angular velocity estimate error covariance, $Q\left(0\right)$, the attitude is not updated until a star spot is detected under RS mode.

- 2. Time update
A star spot is obtained at time ${t}_{2}$. If it is the first star spot in a new frame of star image, the angular velocity estimate is updated dependent on the previous two frames of star images using the angular velocity determination method. Then the attitude kinematics Eq. (8) is propagated to yield a priori quaternion estimate.

$$\widehat{q}\left({t}_{2}|{t}_{1}\right)=\overline{\Omega}\left[{t}_{2},\widehat{\omega}\left({t}_{1}\right)\right]\widehat{q}\left({t}_{1}\right)$$Based on process Eq. (16), the prediction error covariance is obtained by

- 3. Measurement update
The filter gain is calculated by

$$K\left({t}_{2}\right)=P\left({t}_{2}|{t}_{1}\right)H{\left({t}_{2}\right)}^{\text{T}}{\left[H\left({t}_{2}\right)P\left({t}_{2}|{t}_{1}\right)H{\left({t}_{2}\right)}^{\text{T}}+R\right]}^{-1}$$and the attitude error estimate is obtained through

The estimate error covariance is obtained by

- 4. Attitude update
The posteriori attitude quaternion estimate $\widehat{q}\left({t}_{2}\right)$ is yielded from the attitude error estimate $\widehat{x}\left({t}_{2}\right)$ and the priori estimate $\widehat{q}\left({t}_{2}|{t}_{1}\right)$.

$$\widehat{q}\left({t}_{2}\right)=\left[\begin{array}{c}\frac{1}{2}\widehat{x}\left({t}_{2}\right)\\ 1\end{array}\right]\otimes \widehat{q}\left({t}_{2}|{t}_{1}\right)$$Then set ${t}_{1}={t}_{2}$ and the loop will start again from step 2.

## 4. Simulation and analysis

In this section, the simulation is performed in order to verify the effectiveness of the proposed approach to improve the attitude update rate.

The parameters of star tracker in Table 3 are used to obtain synthetic star images including the star spots from guide star projections in star tracker frame under RS mode, where the guide star catalog is selected from Tycho2n. A more detailed discussion for the simulation to acquire synthetic star images can be found in [27].

The angular velocity of star tracker is set as $\omega =\left[\begin{array}{ccc}0.0175& 0.0175& 0.0175\end{array}\right]{\text{\hspace{0.17em}}}^{\text{T}}\text{\hspace{0.17em}}rad/\mathrm{sec}$ and the initial attitude quaternion is taken as $q\left(0\right)={\left[\begin{array}{cccc}0& 0.3122& 0.95& 0\end{array}\right]}^{\text{T}}$.

The initialization for attitude estimation is done using the following values:

Using synthetic star images, the novel approach to improve the attitude update rate is simulated in MATLAB software.

In Fig. 10, the attitude estimate errors converge quickly and the predicted $3\text{\sigma}$-error boundaries do indeed bound the attitude estimate errors. Therefore, the attitude estimation method with a single star spot is effective for the star tracker under RS mode. This is due to the fact that we establish the mathematical model of attitude estimation under RS mode, determinate the angular rate from successive frames of star images and apply the multiplicative extended Kalman filtering algorithm to obtain the optimal attitude estimation of star tracker.

A frame of synthetic star image used in the simulation is shown in Fig. 11 and the positions of star spots in the star image are listed in Table 4. There are 15 star spots distributed in different rows of the star image, and the attitude quaternion updates by these star spots are plotted in Fig. 12. Since the star image is acquired in RS mode, the 15 star spots in the star image are exposed with row-to-row time offsets and can reflect the rotation of the star tracker at different times. Correspondingly, the attitude quaternion is updated 15 times based on the attitude estimation method with a single star spot.

The number of attitude updates varies with the simulation time is shown in Fig. 13, illustrating that the number of attitude updates in RS mode is much higher than that in GS mode. The coordinates of the last attitude updates in two modes are pointed out in the figure. For RS mode, the attitude is updated 880 times in 3 seconds, meaning that the attitude update rate is up to 293 Hz, compared with 9 Hz in GS mode. In conclusion, the novel approach based on RS mode can effectively improve the attitude update rate of a star tracker.

## 5. Experiment

The laboratory experiment is conducted to further show the applicability of the proposed approach. The experiment system is composed of three parts: the star tracker, the three-axis rotary table and the star simulator, as shown in Fig. 14.

The star tracker is fixed on the three-axis rotary table which can generate high-precision rotation angles and angular velocities. The star tracker used in the experiment is developed by Tsinghua University as shown in Fig. 15 and its parameters are listed in Table 5.

In the experiment, the angular velocity of the rotary table is set as ${\omega}_{com}=\left[\begin{array}{ccc}0.015& 0.015& 0\end{array}\right]{\text{\hspace{0.17em}}}^{\text{T}}\text{\hspace{0.17em}}rad/\mathrm{sec}$. The exposure time and the row-to-row time offset of the star tracker are configured as ${t}_{e}=50ms$ and ${t}_{row}=52\mu s$, respectively. The experiment results are shown in the following figures. Figure 16 demonstrates that the proposed approach is feasible for attitude determination of the star tracker in the real application. Moreover, it can be calculated from Fig. 17 that the attitude update rate under RS mode is about 360 Hz, indicating that the attitude update rate of the star tracker is remarkably improved.

## 6. Conclusions

In this paper, a novel approach to improve the attitude update rate of a star tracker has been proposed. The RS mode of a CMOS image sensor is used to acquire star images, where each row of the imaging plane is exposed with a time offset and the star spots in a star image reflect the rotation of the star tracker at different times. In order to reach a high attitude update rate, the new attitude estimation method has been developed and each star spot in a star image is able to update attitude once. Finally, the simulation and experiment have been given to illustrate that the proposed approach is effective for attitude update with a single star spot and the attitude update rate is increased significantly. In the future, we will intensively study the space radiation effects on a CMOS star tracker and use fault tolerant technique to enhance the radiation resistance of a star tracker.

## Funding

National Natural Science Foundation of China (NSFC) (No. 61377012 and No. 51522505); Key Research and Development Program of China (No. 2016YFB0501201); Postdoctoral Science Foundation of China (No. 2017M610882).

## Acknowledgments

The authors acknowledge the support from TY-Space Technology (Beijing) Ltd. for the cooperation in the experiment.

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