## Abstract

Star trackers, optical attitude sensors with high precision, are susceptible to space light from the Sun and the Earth albedo. Until now, research in this field has lacked systematic analysis. In this paper, we propose an installation orientation method for a star tracker onboard sun-synchronous-orbit spacecraft and analyze the space light distribution by transforming the complicated relative motion among the Sun, Earth, and the satellite to the body coordinate system of the satellite. Meanwhile, the boundary-curve equations of the areas exposed to the stray light from the Sun and the Earth albedo were calculated by the coordinate-transformation matrix under different maneuver attitudes, and the installation orientation of the star tracker was optimized based on the boundary equations instead of the traditional iterative simulation method. The simulation and verification experiment indicate that this installation orientation method is effective and precise and can provide a reference for the installation of sun-synchronous orbit star trackers free from the stray light.

© 2017 Optical Society of America

## 1. INTRODUCTION

In optical systems, stray light, which is the nonimaging light arriving at the image plane [1], is harmful to system performance and difficult to suppress or eliminate, especially for optical sensors with high precision [2–4]. Compared with other attitude sensors, the star tracker has an extremely high accuracy and has been widely used for attitude determination of the spacecraft [5–7]. However, star trackers suffer from the interference of stray light from the Sun and Earth, resulting in the loss of image quality, signal-to-noise ratio, and accuracy and even star tracker failure [8–10].

In order to suppress the interference of the stray light to star trackers, many solutions have been developed, such as employing a large baffle coated with black materials to block the light out of the field of view (FOV) [11–14], optimizing the star calibration method and extraction algorithm [15–18], and optimizing the installation directions of star trackers [19–21]. Furthermore, improper installation orientation will lead to star tracker failure when it suffers from the stray light. In such case, more than one star tracker with different orientations is required in one satellite to guarantee that at least one star tracker is functional during flight [13,22]. Therefore, the pointing orientation of a star tracker is a key parameter for effective attitude determination.

Regarding the aspect of installation orientation of the star tracker, the precision caused by the installation azimuth of the star tracker was presented in Ref. [19], but the influence factor—stray light—has not been considered. Given the stray light influence, Ref. [20] analyzed the geometric position of the satellite and different stray light sources (Sun, Earth, Moon) to determine the installation directions. However, this method failed to meet the needs of multimaneuver attitudes of the satellite. To solve this problem, Ref. [21] proposed an installation orientation method for satellites with multimaneuver attitudes; however, this method determined the installation orientations of star trackers by adjusting the positions based on repeat simulations, which is too complex to optimize the installation orientation under multimaneuver attitudes.

In this paper, we propose a novel method to optimize the installation of star trackers through an attitude sphere model, which is established to clarify the complicated relative motion among the Sun, the, Earth, and the satellite. Based on the initial boundary parameters and the mathematical model, we can calculate the boundary equations of the stray light from the Sun and Earth and determine the areas free from the stray light without any other simulations under multimaneuver attitudes for special orbits. We have utilized such method to optimize the installation directions of star trackers for sun-synchronous satellites.

## 2. VECTOR MODEL OF THE STRAY LIGHT BASED ON THE BODY COORDINATE SYSTEM OF A SATELLITE

For a sun-synchronous satellite, the relationship between the satellite and the stray light from the Sun and Earth is extraordinarily complex. To determine the relative direction between the stray light and the satellite, we assume that the satellite is fixed and the Sun and Earth rotate around the satellite. Given that the right ascension difference between the ascending node and the mean sun is constant for the sun-synchronous satellite, the relative direction between the satellite and the stray light is introduced by the mathematical model as follows.

#### A. Vector Model of the Stray Light from the Sun and the Earth

According to the space geometry, the space vector could be described by three cosine angles between the vector and the coordinate axis exclusively, and the cosine quadratic sum of the space vector angle is 1. Similarly, the relationship between the sunlight and the satellite body coordinate system could be described by the vector cosine. As shown in Fig. 1, the $X$ axis points to the direction of movement, the $Z$ axis points to Earth, and the included angles between the sun vector and $X$, $Y$, and $Z$ axes are $\alpha $, $\beta $, and $\gamma $ respectively. Therefore, the sun vector could be expressed by $[\mathrm{cos}\text{\hspace{0.17em}}\alpha ,\mathrm{cos}\text{\hspace{0.17em}}\beta ,\mathrm{cos}\text{\hspace{0.17em}}\gamma ]$. In this vector model, any single light ray will form a unit vector, as shown in Fig. 1(a). In this case, the light rays from all directions of the space will form a sphere with the radius of 1, as shown in Fig. 1(b).

Compared with the Sun, the satellite is quite close to Earth. Therefore, the stray light from the Earth reflection is in a cone area at a given time, as shown in Fig. 2. The stray light is in the scope of the cone, whose central line is ${R}_{s}$ and generatrix is ${V}_{\mathrm{atm}}$. Therefore, the half angle of the cone, ${\delta}_{\mathrm{atm}}$, is 1/2 of the earthlight radiation angle and can be approximately calculated with the equation below:

where ${R}_{s}$ denotes the distance between the satellite and the geocenter, ${R}_{e}$ is the radius of Earth, and $d$ is the thickness of the atmosphere.#### B. Angle Variation between the Orbit Plane and the Sunlight for a Sun-Synchronous Orbit

For a sun-synchronous orbit, the right ascension difference between the ascending node and the mean sun is constant; however, the angle between the sunlight and the orbit plane changes with small variance periodically. Moreover, because of the perturbation of the solar gravity and the atmospheric drag, the orbit inclination and the altitude will change slowly during a long period.

Figure 3 is the celestial coordinate system, where $N$ denotes the ascending node of the satellite orbit, ${S}_{T}$ denotes the true sun, ${S}_{M}$ denotes the mean sun, $\overrightarrow{OV}$ is perpendicular to the orbit plane, and ${N}_{P}$ denotes the north pole. Therefore, the angle between the sunlight and the orbit plane is $\xi (t)={\stackrel{\u2322}{VS}}_{T}$, which is related to the declination of the true sun, $p(t)$, and the right ascension difference, $\gamma $, between $V$ and ${S}_{T}$, where $\gamma $ can be derived by the following equation:

$A$ is the spring equinox, if the time is $t=0$ at this point, $\alpha (t)=2\pi t/365$ ($0\le t\le 365$). Based on the spherical right triangle $A{S}_{T}{S}_{P}$, $\mathrm{tan}\text{\hspace{0.17em}}p=\mathrm{sin}\text{\hspace{0.17em}}\mathrm{\alpha}/\mathrm{cot}\text{\hspace{0.17em}}\epsilon $. Therefore, we can obtain $p(t)$, as shown in the following equation:

Similarly, we can know that $\mathrm{cos}\text{\hspace{0.17em}}{\alpha}_{ST}=\mathrm{cos}\text{\hspace{0.17em}}\mathrm{\alpha}/\mathrm{cos}\text{\hspace{0.17em}}p$ through the inverse function transformation, where $\mathrm{\Delta}\alpha $ is derived by

According to the spherical right triangle ${N}_{P}V{S}_{T}$, ${\stackrel{\u2322}{VS}}_{T}$ can be derived by

By combining Eqs. (2), (3), (4), and (5), the angle between the sunlight and orbit plane, $\xi (t)$, can be described by

According to Eq. (6), $\xi (t)$ changes with a one-year period, and its maximum value can be obtained around the summer solstice. The orbit coordinate system ($O$) and the body coordinate system of the satellite ($B$) are coincident when the satellite does not rotate around the axes. Therefore, $\xi (t)$ is equivalent to the angle between the $Y$ axis of the body coordinate system of satellite ($B$) and the sunlight, and the angle variation between the $Y$ axis and the sunlight is similar to $\xi (t)$. Furthermore, the maximum angle between the $Y$ axis and the sunlight, $Y\_\text{\hspace{0.17em}}\mathrm{max}$, and the minimum angle between the $Y$ axis and the sunlight, $Y\_\text{\hspace{0.17em}}\mathrm{min}$, can be obtained by derivation from Eq. (6).

Under the perturbation of the solar gravity, the orbital inclination will change during a long period, and its rate of change, ${i}^{\prime}$, can be obtained by [23]

Based on aforementioned analysis, the angle between the sunlight and the orbit plane, $\xi (t)$, is mainly affected by two aspects: (1) the periodical variation, owing to the angle between the ecliptic and the equator, and (2) the secular variation, owing to the perturbation of the solar gravity and the atmospheric drag. Therefore, $\xi (t)$ contains periodical and secular variations simultaneously.

#### C. Vector Transformation Among Different Coordinate Systems

There are several coordinate systems among the coordinates transformation, such as the geocentric inertial coordinate system ($I$), the orbit coordinate system ($O$), the body coordinate system of the satellite ($B$), and the coordinate system of the star tracker ($S$). The $Z$ axis of $I$ points to the north pole of Earth, and the $X$ axis points to the spring equinox at 12:00 on 1 January 2000. The origin of $O$ is the spacecraft centroid, the $Z$ axis points to the geocenter, and the $X$ axis points to the flight direction. Similarly, when the attitude angle of $B$ is 0, $B$ and $O$ are coincident. As for $S$, the origin of which is the intersection of the optical axis and CCD, the $Z$ axis points to the optical axis, and the $X$ axis points to the direction of pixel increase. According to the orbital elements, the coordinate-transformation matrix ${M}_{OI}$ from $I$ to $O$ can be derived, as shown in the following equation:

Based on the $Z\u2013X$–Y rotation of the Euler angle, the coordinate-transformation matrix ${M}_{BO}$ from $O$ to $B$ can be derived, as shown in the following equation:

where $\theta $ denotes the pitch angle, $\mathrm{\varphi}$ denotes the roll angle, and $\lambda $ denotes the yaw angle. ${q}_{Y}(\phi )$ is the transformation matrix when the satellite rotates around the $Y$ axis.Similarly, the coordinate-transformation matrix ${M}_{SB}$ from $B$ to $S$ can also be derived if the Euler rotation angles are A, B, and C, as shown in the following equation:

The direction of the Sun in the geocentric inertial coordinate system ($I$) can be expressed by the right ascension and declination (${\alpha}_{\text{sun}}$ and ${\gamma}_{\text{sun}}$, respectively), and the sun vector can be described as

Therefore, based on aforementioned transformation matrix, the sun vector in the coordinate system of star tracker ($S$) can be expressed by

The view-axis vector of the star tracker in the orbit coordinate system ($O$), ${V}_{\text{vie}\_O}$, can be expressed by the azimuth angle $\alpha $ and the elevation angle $\delta $, and the elevation angle $\delta $ can be derived by

In this paper, the optimal method of installation directions for star trackers mainly includes: (1) based on the attitude sphere model and orbit elements, the boundary equations of the sunlight and the earthlight under normal attitude are achieved, (2) based on rotation matrices, the boundary equations of the earthlight and the sunlight under multimaneuver attitudes are calculated, (3) the vector areas free from the stray light in the attitude sphere are determined, (4) based on the special application and the exclusion angle of baffle, the installation directions of star trackers are optimized in aforementioned vector areas. A comparison of the new method and the general method is shown in Fig. 4.

It is very complex in the general method to determine the orientations of star trackers by ergodic simulations, especially under multimaneuver attitudes. As for the new method, the optimal orientation of star trackers can be determined in a more straightforward and accurate way, without any other simulations under multimaneuver attitudes and different periods for the special orbit. Based on the proposed model and the angle variation analysis in this section, we will extract the boundary equations of the stray light at normal attitude and calculate the boundary equations under different attitudes in the next section, which will be used to analyze the installation orientation of the star tracker.

## 3. BOUNDARY EQUATIONS OF THE STRAY LIGHT UNDER MULTIMANEUVER ATTITUDES

Remote sensing imaging is an important application for the sun-synchronous satellite, which requires that the satellite has maneuverability with a large angle to obtain 3D imaging or video, and the maneuver ability mainly includes left-swing (rotating around the $X$ axis clockwise), right-swing (rotating around the $X$ axis anticlockwise), front-swing (rotating around the $Y$ axis clockwise), and back-swing (rotating around the $Y$ axis anticlockwise). In this paper, the orbital altitude that we designed is 535 km, and the local time of the descending node is 10:30. According to Eq. (1), ${\delta}_{\mathrm{atm}}$ is 69.57°. The maximal swing angle is 45° under different maneuver attitudes. Additionally, since the satellite is equipped with orbit control devices, the orbital altitude is not affected by the perturbation force.

#### A. Sunlight General Boundary Equation under Normal Attitude

In this part, we simulated the vector position between the Sun and the body coordinate system of the satellite by satellite tool kit (STK) for a period of 3 years and extracted the general boundary equations under normal attitude in the special orbit. The simulation result indicated that the angle between the sunlight and the axis of the body coordinate system was related to the time and the latitude of subsatellite point, as shown in Fig. 5. Therefore, the angle between the sunlight and the coordinate axis, $\zeta $, can be expressed by

where $l$ denotes the latitude. The variation of the angle between the sunlight and the $Y$ axis has a minimum in one day; however, it also changed about 15° in three years, as shown in Table 1, which represented the maximum and minimum angles between the coordinate axis and the sunlight in three years. $X\_\text{\hspace{0.17em}}\mathrm{max}$ and $X\_\text{\hspace{0.17em}}\mathrm{min}$ represent the maximal and minimal angles between the sunlight and $X$ axis, respectively. Similarly, $Y\_\text{\hspace{0.17em}}\mathrm{max}$ and $Y\_\text{\hspace{0.17em}}\mathrm{min}$ represent the maximal and minimal angles between the sunlight and the $Y$ axis, respectively, and $Z\_\text{\hspace{0.17em}}\mathrm{max}$ and $Z\_\text{\hspace{0.17em}}\mathrm{min}$ represent the maximal and minimal angles between the sunlight and the $Z$ axis, respectively.The red area is the sunlight vector in 3 years, as shown in Fig. 6(a), where the sunlight represents periodic motion along the $Y$ axis, and the missing part of the red circle is attributed to the satellite moving into the earth’s shadow area.

According to the theoretical analysis in Section 2.B and simulation results, we can extract the boundary equations of the sunlight when the satellite is at normal attitude. In order to ensure that the star tracker is free from the sunlight during its lifetime, the included angle of the boundary should be expressed by the extreme value, ${\zeta}_{\mathrm{max}}$ or ${\zeta}_{\mathrm{min}}$, and then the equations of the boundary curves can be expressed by Eqs. (19) and (20), where $\mathrm{cos}(Z\_\mathrm{min})<z<\mathrm{cos}(Z\_\mathrm{max})$, and the boundary curves are shown in Fig. 6(b). We have

From the above, the sunlight represents periodic motion along the $Y$ axis, and the motion region is related to the orbit, time, and satellite body. Furthermore, for a satellite with orbit control devices, the motion region and extremum are constants $(Y\_\mathrm{min},Y\_\mathrm{max})$ in the special orbit during a long period, and it remains unchanged regardless of the launch time of the satellite. In contrast, for a satellite without orbit control devices, the motion region will drift along the $Y$ axis due to the drag, and the drift factor, ${\delta}_{\text{drift}}$, changes with time. Therefore, for satellites in special orbits with different maneuver requirements, we can directly calculate the optimal installation directions of star trackers without any other simulations based on the region and rotation matrices.

#### B. Sunlight Boundary Equations under Maneuver Attitudes

When the satellite is in the nadir-pointing mode, left- and right-swing are equivalent to rotation around the $X$ axis, and the body coordinates of the satellite after rotating around the $X$ axis are shown in the following equation:

In this paper, the maximal swing angle is 45°, and thus, based on Eq. (22), the boundary curves in Fig. 6(b) transform to those in Figs. 7(a) and 7(b) when the satellite swings leftward and rightward, respectively.

When the satellite is at front- or back-swing attitude, the stray light area rotates around the $Y$ axis compared with the normal attitude. Given that the area along the $+Z$ axis will be mostly exposed in the stray light from the Sun and the Earth reflection, it is not suitable for star tracker installation along such direction. Thus, it is not necessary to analyze the influence caused by the Sun when the satellite is at front- and back-swing attitudes.

#### C. Earthlight Boundary Equation

In Section 2, we analyzed the stray light from the Earth reflection, whose interference to a satellite is described by cone angle ${\delta}_{\mathrm{atm}}$. The boundary equation of the stray light from the Earth reflection at normal attitude is

The boundary curve of the earthlight is shown in Fig. 8, where the spherical cap area that is above the blue boundary curve is the vector area irradiated by the earthlight when the satellite is at normal attitude.

When the satellite is in the nadir-pointing mode, front- and back-swings are equivalent to rotation around the Y-axis, and the body coordinates of the satellite after rotating around the Y-axis are

In this paper, the maximal swing angle is 45°. Based on Eq. (25), the boundary curve in Fig. 8 transforms to those in Figs. 9(a) and 9(b) when the satellite swings frontward and backward, respectively.

When the satellite is at left- or right-swing attitude, the coordinate rotation matrix of the earthlight boundary equation is similar to the aforementioned sunlight situation. Therefore, based on Eq. (22), the earthlight boundary curves can be achieved when the satellite swings leftward and rightward, as shown in Figs. 10(a) and 10(b), respectively.

## 4. INSTALLATION ORIENTATION METHOD DESIGNED FOR STAR TRACKERS

Based on the boundary equations, the areas free from the stray light at any time will be determined in this section. Meanwhile, considering the baffle exclusion angle of the sunlight and the earthlight, the installation orientation could be optimized.

#### A. Area Not Affected by the Stray Light

For the star tracker, the installation directions should meet certain requirements: the incident angle of the sunlight and the earthlight should be greater than the corresponding exclusive angle of baffles, which are used to reject the sunlight and the earthlight. In Section 3, the vector areas and boundary curves of the stray light under different maneuver attitudes can be determined. The vector areas that are not affected by the stray light at any time can be also determined. Obviously, these vector areas are the perfect installation directions for star trackers, if they can meet the requirements. In this paper, the boundary curves of the stray light from the Sun and the Earth reflection under different maneuver attitudes are showed in Fig. 11 in the -X axis view. Only one vector area is not interfered by the stray light under all maneuver attitudes, as shown in Fig. 12, in the $Y$ axis view. This vector area is surrounded by the earthlight boundary curves when the satellite swings leftward, frontward, and backward and is surrounded by the sunlight boundary curve when the satellite swings rightward. Moreover, this vector area is equally divided by the $Y\u2013Z$ plane, and the angle between the right-swing boundary curve of the sunlight and the left-swing boundary curve of the earthlight will decrease while deviating from the $Y\u2013Z$ plane. Figure 11 shows that the angle region that is not affected by the stray light is 34.8° in the $Y\u2013Z$ plane, and there is an overlap area that is affected by the sunlight and the earthlight with 8.9° in the $Y\u2013Z$ plane. Meanwhile, the left- and right-swing angles are not so large, and the overlap area will change to a small nonstray light area.

#### B. Installation Method for Star Trackers

There are mainly two installation schemes for double star trackers. (1) When the vector area that is not affected by the stray light meets the requirement of the exclusive angles of the sunlight and the earthlight of star trackers, the installation directions of the double star trackers will be distributed on both sides of the $Y\u2013Z$ plane symmetrically, and thus, the double star trackers can operate free from the stray light under all maneuver attitudes. (2) When the vector area that is affected by the stray light cannot satisfy the aforementioned requirement, the double star trackers cannot work free from the stray light under all of the maneuver attitudes simultaneously. Therefore, we should ensure that there is at least one star tracker that can work under every attitude, and the installation direction should be in $Y\u2013Z$ plane $(Z<0)$, in which case, the front- and back-swing angles can reach maximization.

In this paper, the star tracker with baffle that we adopted is shown in Fig. 13, and the performance of the star tracker is shown in Table 2. Given that ${\eta}_{\text{sun}}+{\eta}_{\text{earth}}>34.8\xb0$, it is suitable for scheme 2, and the installation instruction is shown in Fig. 14.

As shown in Fig. 11, areas ① and ② do not suffer from the stray light when the satellite is at normal left-swing attitude. Similarly, areas ③ and ④ do not suffer from the stray light when the satellite is at normal right-swing attitude. To avoid the influence of the earthlight when the satellite is at front- or back-swing attitude, areas ② and ③ are not suitable for star tracker installation. To ensure that the star tracker does not suffer from the stray light when the satellite is at normal left-swing attitude, the installation direction of the first star tracker is in the range of 40°–69°. Because the installation angle range is 36°, which is less than ${\eta}_{\text{sun}}+{\eta}_{\text{earth}}$, the second star tracker pointed to area ② could not work when the satellite was at normal attitude. Based on the earthlight boundary curve in the right-swing attitude, the second star tracker points to 150° off the $Y$ axis in the $Y\u2013Z$ plane. Therefore, the second star tracker can work when the satellite is at 15°–45° right-swing attitude. To ensure that the first star tracker can work when the satellite is at right-swing attitude, which is less than 15°, the installation region of the first star tracker decreases to 40°–54°. Moreover, the precision of the double star trackers can be highest when they are installed perpendicularly, so the first star tracker is designed to point to 50° off the $Y$ axis in the $Y\u2013Z$ plane. The installation azimuth of the double star trackers in the body coordinate system of the satellite is shown in Table 3.

## 5. SIMULATION RESULTS AND DISCUSSION

In Section 4, based on the boundary equations of the stray light, we designed the orientations of the star trackers. In order to verify the aforementioned method, we simulated the relationship between the star tracker orientation and the stray light from the Sun and the Earth reflection by STK. Because the period of the stray light relative to the satellite body coordinate system is one year, the simulation duration is set to be one year as well.

Figure 15 shows the critical angles between the star tracker orientation and the stray light. In Fig. 15(a), the minimum angles between star tracker 2 orientation and the sunlight are all less than 35° at normal attitude, so star tracker 2 will suffer from the sunlight in every circle when the angle is less than 35°. When the satellite is at left-swing 45° attitude, the minimum angle between star tracker 1 and the earthlight is 25.43°, which corresponds with the analysis in Section 4, as shown in Fig. 15(b). Moreover, when the satellite is at right-swing 15° attitude, the included angles relative to the sunlight are all greater than 35°, as shown in Figs. 15(c) and 15(d), and the minimum angle corresponds with the analysis discussed above. Furthermore, when the satellite is at right-swing 45° attitude, star tracker 1 cannot work occasionally because of the sunlight, while star tracker 2 is still free from the earthlight, as shown in Figs. 15(e) and 15(f). The double star tracker does not suffer from the earthlight when the satellite is at front- or back-swing 45° attitude, as shown in Fig. 15(g). The simulation experiment indicates that the simulation results are consistent with the design, and the star tracker can work with a large-maneuver angle of 45° attitude.

In the design discussed in this paper, the star tracker does not suffer from the stray light at any time at the special attitude. However, when the minimum angle between the star tracker orientation and the sunlight is less than the sunlight exclusive angle, the star tracker can also work most of the time. For instance, when the satellite is at the normal attitude, the minimum angles between the star tracker 2 orientation and the sunlight in every circle are all less than 35°, while star tracker 2 can also work 78.5% of the time when the satellite is in the Sun area. The angle histogram is shown in Fig. 16, and the relationship between the sunlight exclusive angle and the available time ratio of the star tracker is represented in Fig. 17.

Star tracker 2 is available at any time when the latitude of the subsatellite point is less than southern latitude $-52\xb0$ and greater than northern latitude 52°. It is available at some time when the latitude of the subsatellite point is between $-52\xb0$ and 52°, and when star tracker 2 is disabled, star tracker 1 is available, as shown in Fig. 18. Based on aforementioned analysis, the star tracker orientation could be adjusted according to the actual requirement with at least one star tracker available at any time, and the available time is relevant to the orientation.

## 6. VERIFICATION EXPERIMENT AND RESULTS

The new method was verified through an on-orbit satellite with three star trackers. The orbital altitude of the satellite was 535 km, the local time of descending node was 10:30, and the maximal maneuver angle was 45°. The actual installation orientations of the three star trackers in the body coordinate system are shown in Table 4.

Based on the new method, the minimal angles between the installation orientations of star trackers and the sunlight/earthlight boundary under multimaneuver attitudes were calculated, and the obvious results are represented by valid or invalid, as shown in Table 5.

The exclusion angle of the sunlight and the earthlight of the baffle were 35° and 15°, respectively. Based on the data in Table 5, we can obtain the full-time-valid attitudes of the three star trackers, as shown in Table 6.

From Table 6, we know that it was valid full time for star tracker 1 under normal/front-swing 45°/back-swing 45° attitude during the operation period, and it was valid most of the time under other attitudes. Similarly, star tracker 2 was valid full time only under left-swing 45° attitude during the operation period, and star tracker 3 was affected by the stray light part of the time under all 45° maneuver attitudes. Compared with the actual operation modes of different star trackers under the maximal swing angle attitude, the verification results indicated that the calculated results using the new method were coincident with the actual on-orbit results.

## 7. CONCLUSIONS

In this paper, an installation orientation method for star trackers has been proposed to avoid the stray light from the Sun and the Earth reflection, which influences the image quality and star extraction of the star tracker. The vector areas of the sunlight and the earthlight in the attitude sphere model were determined based on orbit elements and a simple simulation when the satellite was at normal attitude, and the equations of the boundary curves were calculated by the coordinate-transformation matrix under different maneuver attitudes. Based on the vector area free from the stray light, the installation orientations of the double star trackers were optimized. Therefore, the orientation of the star tracker can be optimized without any other simulations under multimaneuver attitudes and different periods. The simulation and verification results indicated that the angles between the star tracker orientation and the stray light were consistent with the design, and this method can effectively solve the installation orientation problem of the star tracker when the satellite operates with large-angle maneuver attitudes. Additionally, based on the vector area affected by the stray light, the installation orientation of the sun sensor with high precision can be determined easily, which could be regarded as a vector of the multi-FOV star tracker [25,26].

## Funding

National Natural Science Foundation of China (NSFC) (51522505, 61377012, 61505094, 61605099).

## Acknowledgment

We gratefully acknowledge the support of the State Key Laboratory of Precision Measurement Technology and Instruments.

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