Observation angular distance error modeling and matching threshold optimization for terrestrial star tracker

Zhen Wang; Zhen Wang; Zhen Wang; Jie Jiang; Jie Jiang; Jie Jiang; Guangjun Zhang

doi:10.1364/OE.27.033518

1. Introduction

As a high-precision attitude measurement device, terrestrial star trackers are widely used in aircraft, ship, and other Earth’s surface working systems. Such device can determine the real-time three-axis attitude information, such as pitch, yaw, and roll angles, by matching the observation stars obtained from the focal plane of star tracker with the reference stars in star catalog. Among the attitude determination devices, the star tracker is deemed the most accurate [1–4].

Due to the effects of the Earth’s rotation and atmospheric refraction, the attitude measurement of the terrestrial star tracker is more complicated than the traditional star sensors that are used in satellites or space vehicles. Therefore, additional studies are needed to improve the working performance of terrestrial star trackers.

Li et al. [5] proposed an analytical method to estimate the stellar instrument magnitude, characterize associated errors, and make the guide star selection efficient. Yu et al. [6,7] divided the exposure time into segments to improve attitude update rate and estimate the complete motion parameters of the star tracker. In [8,9], the suppression of stray light is implemented to achieve accurate star spot extraction. Katake and Bruccoleri introduced the intensified image detector into the star-tracker field to solve the problem of low sensitivity [10,11]. Imaging performance of a star tracker was analyzed to obtain high positional accuracy in [12–14]. Fast centroid extraction methods have been proposed to break the application limitations of new image sensors and were beneficial to the improvement of attitude updating rate [15,16]. Additional efficient star identification algorithms have been presented to improve the success rate of star identification [17–20].

However, the difference between the actual observation angular distance and the reference angular distance, named observation angular distance error, remains to be unresolved by far. It is the criterion to determine whether stars are identified in star identification, and so that affects the identification success rate, attitude measurement accuracy, and real-time performance of terrestrial star trackers. Fewer studies can be found to determine the observation angular distance error, nor have studies determined the value of angular distance matching threshold in star identification. In [2,17,18,21,22], the size of angular distance matching threshold was set as different values directly.

The determination of observation angular distance error is complex and related not only to the properties of stars, which make up angular distances, but also to various factors, such as angular distance value, revolution and rotation of the Earth, observation time, observation location, observation environment, atmospheric refraction, and optical parameters of the star tracker. The demand to clarify the observation angular distance error of the terrestrial star tracker is substantial.

In this work, an observation angular error model is presented to resolve the issue. The optimal angular matching threshold expression for the terrestrial star tracker is obtained on the basis of the proposed model. Numerical simulation and night sky experiments support the conclusions.

2. Principle of terrestrial star tracker and observation angular distance error

2.1 Principle of terrestrial star tracker

As a high-precision attitude measurement device, terrestrial star trackers are widely used in aircraft, ship, and other Earth’s surface working systems to obtain the real-time attitude information, such as pitch, yaw, and roll angles.

Figure 1(a) shows the structure of the airborne terrestrial star tracker, where, XeYeZe, XnYnZn, XbYbZb, and XsYsZs represent the terrestrial, geographical, carrier, and star tracker coordinate systems, respectively. These coordinate systems are denoted as E-frame, N-frame, B-frame, and S-frame for simplicity.

Fig. 1. Airborne terrestrial star tracker: (a) coordinate systems; (b) imaging system.

Download Full Size | PDF

Assuming that the real-time attitude information of the aircraft (i.e., pitch, yaw, and roll angles) is denoted as $\left [ {\theta ,\psi ,\gamma } \right ]$, the real-time attitude transformation matrix between B- and N-frames can be expressed as follows [23]:

(1)$$\begin{aligned} C_n^b \! =\! \left(\!\!\! {\begin{array}{*{20}c} {c_{11} }\!\!\! & \!\!\!{c_{12} } \!\!\! & \!\!\! {c_{13} } \\ {c_{21} }\!\!\! & \!\!\!{c_{22} } \!\!\! & \!\! \!{c_{23} } \\ {c_{31} }\!\!\! & \!\!\!{c_{32} }\! \!\! & \! \!\!{c_{33} } \\ \end{array}}\!\!\! \right) \!=\! \left(\!\!\! {\begin{array}{*{20}c} {\cos \gamma \cos \psi - \sin \gamma \sin \theta \sin \psi } \!\! & \!\! { \cos \gamma \sin \psi + \sin \gamma \sin \theta \cos \psi } \!\! & \!\! { - \sin \gamma \cos \theta } \\ {-\cos \theta \sin \psi } \!\! & \!\! {\cos \theta \cos \psi } \!\! & \!\! {\sin \theta } \\ {\sin \gamma \cos \psi + \cos \gamma \sin \theta \sin \psi } \!\! & \!\! { \sin \gamma \sin \psi - \cos \gamma \sin \theta \cos \psi } \!\! & \!\! {\cos \gamma \cos \theta } \\ \end{array}}\!\!\! \right). \end{aligned}$$

If $C_n^b$ is known, according to Eq. (1), then the pitch, yaw, and roll angles of the aircraft are given by

(2)$$\theta = \sin ^{ - 1} c_{23} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,\gamma ={-} \tan ^{ - 1} \left( { \frac{{c_{13} }}{{c_{33} }}} \right){\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ,\psi ={-} \tan ^{ - 1} \left( { \frac{{c_{21} }}{{c_{22} }}} \right).$$

The terrestrial star tracker is used to obtain $C_n^b$. As shown in Fig. 1(b), stars S1, S2, and S3 are imaged on the focal plane of the star tracker in positions P1, P2, and P3, respectively. Each observation star corresponds to a reference star in the star catalog. If the corresponding relationship between the observation and the reference stars is known, then the real-time attitude transformation matrix between the S-frame and the coordinate system of star catalog (ref-frame) can be calculated as

(3)$$\textbf{W}_i = C_\textrm{ref}^s \textbf{V}_i ,$$

where $\textbf {W}_{\textbf {i}}$ is the observation vector of star $i$ in S-frame of the star tracker, ${\textbf {V}}_{i}$ is the reference vector of star $i$ in ref-frame of the star catalog. Here, the QUEST algorithm can be used to solve the optimal estimation of the attitude matrix $C_\textrm {ref}^s$ [20].

If the reference star vectors $\textbf {V}_i$ are obtained in the N-frame, the attitude transformation matrix between S- and N-frames (i.e., $C_{n}^s$ ) is obtained. Then, by using the installation matrix between B- and S-frames (i.e., $C_s^b$), the real-time attitude matrix $C_n^b$ is given by

(4)$$C_n^b = C_n^s \cdot C_s^b .$$

Finally, the real-time attitude information of the Earth’s surface working systems is obtained on the basis of Eqs. (1) and (2).

2.2 Observation angular distance error

As discussed in the principle of terrestrial star tracker, the corresponding relationship between the observation and the reference stars is the premise for the calculation of the real-time transformation matrix $C_\textrm {ref}^s$.

Suppose that the observation vectors of two stars $i$ and $j$ in the terrestrial star tracker are $\textbf {W}_i$ and $\textbf {W}_j$ , and their corresponding reference vectors in the star catalog are $\textbf {V}_i$ and $\textbf {V}_j$, respectively. Then, the observation angular distance $d_{i,j}^s$ and the reference angular distance $d_{i,j}^\textrm {ref}$ made up of stars $i$ and $j$ are shown respectively as follows:

(5)$$d_{i,j}^s = \textrm{acos}\left( {{\textbf{W}_i \bullet \textbf{W}_j} \mathord{\left/ {\vphantom {{} {}}} \right.} {{\left| {\textbf{W}_i } \right|}{\left| {\textbf{W}_j } \right|}}} \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} d_{i,j}^\textrm{ref} = \textrm{acos}\left( {{\textbf{V}_i \bullet \textbf{V}_j} \mathord{\left/ {\vphantom {{} {}}} \right.} {{\left| {\textbf{V}_i } \right|}{\left| {\textbf{V}_j } \right|}}} \right).$$

Theoretically, the real-time observation angular distance needs to be exactly equals to the reference angular distance (i.e., $d_{i,j}^s=d_{i,j}^\textrm {ref}$). However, since the S-frame and the ref-frame are different in most cases, various factors from the transmission between the ref-frame and the S-frame change the direction of the star vectors and result in a difference between $d_{i,j}^s$ and $d_{i,j}^\textrm {ref}$, named observation angular distance error, as shown in Fig. 2.

Fig. 2. Observation angular distance error between $d_{i,j}^s$ and $d_{i,j}^\textrm {ref}$.

Download Full Size | PDF

At observation time $t$, assuming that the real-time observed angular distance of stars $i$ and $j$ is $d_{i,j}^s \left ( t \right )$, the real-time observation angular distance error $\varepsilon _{i,j} \left ( t \right )$ is expressed as

(6)$$\varepsilon _{i,j} \left( t \right) = {\kern 1pt} {\kern 1pt} \left| {d_{i,j}^s \left( t \right) - d_{i,j}^{ref} } \right|.$$

$\varepsilon _{i,j} \left ( t \right )$ is the criterion to determine whether stars are identified in star identification algorithms [2,17,18,20–22,24]. Therefore, the observation angular distance error is an important parameter that affects the star identification success rate, attitude measurement accuracy, and real-time performance of the terrestrial star tracker. However, this issue is yet to be clarified to date.

3. Analysis and modeling of the observation angular distance error

In this section, we aim to contribute to the modeling of observation angular distance error. Given that the observation angular distance error results from the inconsistent change of star vectors, the factors influencing the observation star vectors from the ref-frame to the S-frame are analyzed in the first step.

3.1 Factors affecting the observation star vectors

The reference star catalog used by the star tracker is derived from astronomical catalogues [25]. Astronomical catalogue provide the position of a star at a specified epoch and referred to a selected reference system. In this work, the Hipparcos catalogue is selected.

The Hipparcos catalogue stores the position parameters of nearly 120,000 stars in the International Celestial Reference System (ICRS), and detailed positional parameters of the stars in the ICRS at epoch $J2000.0$ can be obtained in [26]. Evidently, a considerable difference exists between the star vectors provided in Hipparcos catalogue and observed from the terrestrial star tracker,as shonw in Fig. 3.

Fig. 3. Star vectors observed in the ICRS and the terrestrial star tracker.

Download Full Size | PDF

The chain of astrometric transformations linking the star vector from a catalog to the Earth’s surface observation discussed in the International Astronomical Union (IAU) Standards of Fundamental Astronomy (SOFA) [27] is shown in Fig. 4.

Fig. 4. Chain of astrometric transformations.

Download Full Size | PDF

Factors that affect the star vector in the chain of astrometric transformations include the space motion, annual parallax, solar gravitational light deflection, annual aberration, Earth nutation-precession, Earth rotation, polar motion, diurnal aberration and parallax, and atmospheric refraction. The coordinate systems based on the different observation positions include the ICRS, astronomical coordinate system (AT-frame), celestial inertial coordinate system (G-frame), and E-, N-, and S-frames.

On the basis of the Astronomical Almanac [28], the real-time star vector in the N-frame at observation time $t$ is transmitted as follows:

(7)$$\textbf{V}_{ref}^N \left( t \right) = \textbf{R}_2 \left( {90^\circ{-} \lambda } \right)\textbf{R}_3 \left( \phi \right)\textbf{W}\left( t \right)\textbf{R}_3 \left( { - \beta } \right)R_{\Sigma} f \left[g\left[\textbf{u}_B \left( {t_0 } \right) + \left( {t - t_0 } \right) \overline {\textbf{u}} _B \left({t_0}\right) - \pi \textbf{E}_B \left( t \right)\right]\right],$$

where $t$ is the epoch of observation in the TT timescale; $t_0$ is the reference epoch of catalog $t_0 = J2000.0$; $\textbf {u}_B ( {t_0 } )$ is the initial star vector in catalog; $\overline {\textbf{u}} _B ( {t_0 } )$ is the space motion vector (in au/day) of the star at reference epoch $t_0$; $\textbf {E}_B \left ( t \right )$ is the barycentric position of the Earth at the epoch of observation t referred to ICRS; $g\left [ {\cdots } \right ]$ is the function representing the gravitation deflection of light; $f\left [ {\cdots } \right ]$ is the function representing the aberration of light; $R_\Sigma$ is the transformation, including frame bias, precession and nutation; $\textbf {R}_3 \left ( { - \beta } \right )$ is the Earth’s rotation matrix; $\textbf {W}\left ( t \right )$ is the polar rotation matrix; and $\textbf {R}_2 \left ( {90^\circ - \lambda } \right )\textbf {R}_3 \left ( \phi \right )$ is the rotation matrix from the E-frame to the N-frame based on the longitude $\lambda$ and latitude $\phi$ of the observation position; $\textbf {R}_1$, $\textbf {R}_2$ and $\textbf {R}_3$ are the unit rotation matrixs around $x$, $y$, $z$ axis. The influence of diurnal parallax can be ignored because it is small.

In view of the influence of atmospheric refraction [29], the real-time observation reference vector of the star in the S-system (i.e., $\textbf {W}_S \left ( t \right )$) can be obtained as

(8)$$\textbf{W}_\textbf{s} = \left\langle {C_n^S } \right\rangle \textbf{Refr}\left[ {\textbf{V}_{ref}^N \left( t \right)} \right],$$

where $\textbf {Refr}\left [ \bullet \right ]$ is the function representing the atmospheric refraction, and $\left \langle {C_n^S } \right \rangle$ is the transformation matrix between the S- and N-frames that contain atmospheric refraction.

3.2 Analysis of observation angular distance error

The inconsistent changes of star vectors result in the observation angular distance error in real-time observation of terrestrial star trackers. In this section, we analyze the influence of the aforementioned astrometric transformation factors on the observation angular distance error.

First, the reference angular distance $d_{i,j}^\textrm {ref}$ is determined by setting the ref-frame and reference time ( $T_\textrm {ref}$) of the star catalog as ICRS and $t_0 = J2000.0$, i.e., $d_{i,j}^\textrm {ref} = d_{i,j}^\textrm {ICRS} (t_0 )$. The three-axis rotation transformations, such as Earth nutation-precession and rotation, are neglected because they are not influence the angular distance values.

Then, the factors, namely, space motion, annual parallax, solar gravitational light deflection, annual aberration, diurnal aberration, and atmospheric refraction, are analyzed as follows.

(1) Influences of space motion and annual parallax

The motion in space of stellar objects and the annual parallax are considered because they are functions of time and changes during the earth’s revolution.

Given that the corrections of space motion can be directly made to the right ascension and declination and the reduction for annual parallax refers to the barycentric coordinates of the Earth (Eqs.(7.27) and (7.37) in [28]), the observation angular distance error caused by space motion and the annual parallax should be

(9)$$\begin{aligned} \begin{array}{l} \left| {\varepsilon _{i,j}^\textrm{space - move} \left( {t - t_0 } \right)} \right| +\left| {\varepsilon _{i,j}^{\textrm{Annual}\_\textrm{par} } \left( t \right)} \right|\le \left| {d_{i,j}^\textrm{ref} \left( t \right) - d_{i,j}^\textrm{ref} \left( {t_0 } \right)} \right|+\left| {d_{i,j}^{AT} \left( t \right) - d_{i,j}^\textrm{ref} \left( t \right)} \right|\\ \le \left| {t - t_0 } \right|\left( {\sqrt {u^2 _{\alpha ,i} + u^2 _{\delta ,i} } + \sqrt {u^2 _{\alpha ,j} + u^2 _{\delta ,j} } } \right)+\left| {\pi _i } \right| + \left| {\pi _j } \right|, \end{array} \end{aligned}$$

where $u_{\alpha ,i}$ , $u_{\delta ,i}$ and $\pi _i$ are the proper motions and parallax of star $i$ in Hipparcos catalog, $d_{i,j}^\textrm {ref} \left ( t \right )$ is the real-time angular distance in the ref-frame at time $t$, and $d_{i,j}^{AT} \left ( t \right )$ is the real-time angular distance in the AT-frame after the space motion and annual parallax are corrected. Equation (9) equals only when the space motion and parallax of the two stars changes in the opposite directions.

(2) Influence of solar gravitational light deflection

When the stellar light passes through the gravitational field of Sun, the star direction will be changed by the Solar gravitational light deflection. According to Einstein’s general relativity theory, the solar gravitational light deflection is estimated as follows:

(10)$$\Delta \phi = \frac{{2\mu }}{{c^2 E}}\tan \left( {\frac{{180^\circ{-} \textrm{asin}\left( {r/E} \right)}}{2}} \right),$$

where $E$ is the distance between the barycenters of Earth and Sun, $\mu$ is the heliocentric gravitational constant, $c$ is the speed of light, and $r$ is the distance from the barycenter of Sun to the stellar light line (Fig. 5(a)).

Fig. 5. Solar gravitational light deflection: influence on the (a) star direction and (b) star angular distance.

Download Full Size | PDF

When two stars $i$ and $j$ make up an angular distance, the observation angular distance error caused by solar gravitational light deflection should be

(11)$$\left| {\varepsilon _{i,j}^{\textrm{Gravity}\_\textrm{Ref}} \left( t \right)} \right| \!\le\! \left| {\Delta \phi _i \pm \Delta \phi _j } \right| \!=\! \frac{{2\mu }}{{c^2 E}}\left[ {\tan \left( {\frac{{180^\circ \!\!-\! \textrm{asin}\left( {r_i \left( t \right)/E} \right)}}{2}} \right) \pm \tan \left( {\frac{{180^\circ \!\!- \textrm{asin}\left( {r_j \left( t \right)/E} \right)}}{2}} \right)} \right],$$

where $\Delta \phi _i$ and $\Delta \phi _j$ are the solar gravitational light deflection of stars $i$ and $j$; $r_i \left ( t \right )$ and $r_j \left ( t \right )$ are the distances from the barycenter of Sun to the stellar light lines of stars $i$ and $j$, respectively. The condition for the ’+’ established when the two stars are on either side of the sun (as star1 or star2 in Fig. 5(b)); the ’-’ established only when the two stars on the same side of the sun (as star3 and star4 in Fig. 5(b)).

Given that the barycentric coordinate of the Earth changes in annual periodic as shown in Fig. 6(a) (NASA Jet Propulsion Laboratory (JPL) DE405 [30] or [31]), $r_i \left ( t \right )$ will change with the Earth’s revolution from Sun’s radius ($R_\textrm {Sun}$) to $E$ in annual periodicin of each year. When $r_i \left ( t \right ) = R_\textrm {Sun}$, $\textrm {asin}\left ( {r_i \left ( t \right )/E} \right ) = 0.25^\circ$, and $\Delta \phi _i$ reaches its maximum value.

Fig. 6. (a) Barycentric coordinate of the Earth in 2017-2019; (b) $\varepsilon _{i,j}^{\textrm {Gravity}\_\textrm{Ref}} \left ( t \right )$ versus observation time $t$ in 2019 for different angular distances.

Download Full Size | PDF

If the star tracker is used in daytime, stars will satisfy the case $r_i \left ( t \right ) = R_\textrm {Sun}$ once in each year with the Earth’s revolution. Therefore, the observation angular distance error caused by solar gravitational light deflection should be

(12)$$\left| {\varepsilon _{i,j}^{\textrm {Gravity}\_\textrm{Ref}} \left( t \right)} \right| \le \frac{{2\mu }}{{c^2 E}}\left[ {\tan \left( {\frac{{180^\circ{-} 0.25^\circ }}{2}} \right) + \tan \left( {\frac{{\Delta d}}{2}} \right)} \right],$$

where $\Delta d = 180 - d_{i,j}^{AT} \left ( t \right ) + 0.25^\circ$, when $d_{i,j}^{AT} \left ( t \right ) \ge 0.5^\circ$; otherwise, $\Delta d = - d_{i,j}^{AT} \left ( t \right ) - 0.25^\circ$.

On the other hand, if terrestrial star trackers only work at night (as star3 and star4 in Fig. 5(b)), the observation angular distance error caused by solar gravitational light deflection for such trackers should be

(13)$$\left| {\varepsilon _{i,j}^{\textrm {Gravity}\_\textrm{Ref}} \left( t \right)} \right| \le \frac{{2\mu }}{{c^2 E}}\left[ {\tan \left( {\frac{{90^\circ }}{2}} \right) - \tan \left( {\frac{{90^\circ{-} d_{i,j}^{AT} \left( t \right)}}{2}} \right)} \right]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} used{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} in{\kern 1pt} {\kern 1pt} {\kern 1pt} night.$$

Therefore, the value of $\left | {\varepsilon _{i,j}^{\textrm {Gravity}\_\textrm{Ref}} \left ( t \right )} \right |$ changes with the Earth’s revolution. As for the star tracker used in daytime, the $\varepsilon _{i,j}^{\textrm {Gravity}\_\textrm{Ref}} \left ( t \right )$ of all the angular distances changed in the range of $[0,3.732'']$ in each year, as shown in Fig. 6(b).

(3) Influence of annual aberration

On the basis of special relativity, the apparent direction of a star from a moving observer is changed with the Earth’s motion around the Sun, named annual aberration. The annual aberration can be calculated as

(14)$$\Delta \theta = \textrm{asin}\left( {\frac{{V_E }}{c}\sin \theta - \frac{1}{4}\left( {\frac{{V_E }}{c}} \right)^2 \sin 2\theta + \cdots} \right),$$

where $V_E$ is the revolution speed of the Earth, and $\theta$ is the angle between the motion direction and the true direction of the star (Fig. 7(a)).

Fig. 7. Annual aberration: influence on (a) star direction and (b) star angular distance.

Download Full Size | PDF

Assuming that the star vectors of $i$ and $j$ are ${{\textbf {OS}}_i}$ and ${\textbf {OS}}_\textbf{j}$, respectively, and the vector resulting from annual aberration is $\textbf {A}_\textbf {E}$, the variation of the angular distance can be approximately expressed as:

(15)$$\Delta \textbf{W} = \left( {\textbf{OS}_\textbf{i} \textbf{+ A}_\textbf{E} } \right) \bullet \left( {\textbf{OS}_\textbf{j} \textbf{+ A}_\textbf{E} } \right)\textbf{- OS}_\textbf{i} \bullet \textbf{OS}_\textbf{j} = \left( {\textbf{OS}_\textbf{i} \textbf{+ OS}_\textbf{j}+\textbf{A}_\textbf{E} } \right) \bullet \textbf{A}_\textbf{E} .$$

When $\textbf {A}_\textbf {E}$ is collinear with $\left ( {\textbf {OS}_\textbf {i} \textbf {+ OS}_\textbf {j} } \right )$ (Fig. 7(b)), $\Delta \textbf {W}$ will obtain its maximum or minimum value. Therefore, the observation angular distance error caused by annual aberration should be

(16)$$\left| {\varepsilon _{i,j}^{\textrm{Annual}\_\textrm{abe}} \left( t \right)} \right| \!\le\! \alpha \!-\! \angle S^{\prime}_i OS^{\prime}_j \!=\! \angle S_i^{\prime\prime} OS_j^{\prime\prime} \!-\! \alpha = 2 \times \textrm{asin}\left( {\frac{{V_E }}{c}\sin \frac{\alpha }{2} \!-\! \frac{1}{4}\left( {\frac{{V_E }}{c}} \right)^2 \sin \alpha + \cdots} \right),$$

where $\alpha$ is the value of angular distance of stars $i$ and $j$ before annual aberration is corrected.

Since the annual aberration is influenced by the speed direction of Earth, the value of $\left | {\varepsilon _{i,j}^{\textrm {Annual}\_\textrm {abe}} \left ( t \right )} \right |$ changes with the Earth’s revolution. For instance, based on Eq. (16), the angular distance errors of values, $\alpha = 15.0089^\circ$, $14.9996^\circ$, $10.0059^\circ$, and $4.99942^\circ$ changed in the ranges $[0,5.3522'']$, $[0,5.3489'']$, $[0,3.5738'']$, and $[0,1.7855'']$, respectively, as shown in Fig. 8(a).

Fig. 8. Annual and diurnal aberrations for observation angular distance error in 2019: (a) $\varepsilon _{i,j}^{\textrm {Annual}\_ \textrm {abe}} \left ( t \right )$ versus observation time $t$; (b) $\varepsilon _{i,j}^{\textrm{Diurnal}\_ \textrm{abe}} (t)$ versus observation time $t$.

Download Full Size | PDF

(4) Influence of diurnal aberration

Given that the terrestrial star tracker is used in the Earth’s surface working systems, in addition to the speed of the carrier, the Earth’s rotation also provides a speed to the star tracker. These two speed components results in an aberration called diurnal aberration, as follows:

(17)$$\Delta \theta _{\textrm{Diurnal - Abe}} = \textrm{asin}\left( {\frac{{\left| {\textbf{v}_\textrm{Earth} + \textbf{v}_\textrm{Observe} } \right|}}{c}\sin \theta _E - \frac{1}{4}\left( {\frac{{\left| {\textbf{v}_\textrm{Earth} + \textbf{v}_\textrm{Observe} } \right|}}{c}} \right)^2 \sin 2\theta _E + \cdots} \right),$$

where $\textbf {v}_\textrm {Earth} = \left ( {R_E + H} \right )\cos \left ( {\textrm {L}_{\textrm {atitude}} } \right )2\pi /\left ( {24*3600} \right )$ is the speed of the Earth’s rotation at the observation place, and determined by the Earth’s radius $R_E$, height $H$ and latitude $\textrm {L}_\textrm {atitude}$ of the observation position; $\textbf {v}_\textrm {Observe}$ is the speed of the carrier relative to the ground; and $\theta _E$ is the angle between the direction of $\left ( {\textbf {v}_{\textrm {Earth}} + \textbf {v}_{\textrm {Observe}} } \right )$ and the star vector.

$\textbf {v}_\textrm {Observe}$ is usually considerably smaller than $\textbf {v}_\textrm {Earth}$; thus, regardless of $\textbf {v}_\textrm {Observe}$, the observation angular distance error caused by diurnal aberration should be

(18)$$\left| {\varepsilon _{i,j}^{\textrm{Diurnal}\_ \textrm{abe}} (t)} \right| \le 2 \times \textrm{asin}\left( {\frac{{\textbf{v}_\textrm{Earth} }}{c}\sin \frac{{\alpha '}}{2} - \frac{1}{4}\left( {\frac{{\textbf{v}_\textrm{Earth} }}{c}} \right)^2 \sin \alpha ' + \cdots} \right),$$

where $\alpha '$ is the value of angular distance before diurnal aberration is corrected.

The value of $\left | {\varepsilon _{i,j}^{\textrm{Diurnal}\_ \textrm{abe}} (t)} \right |$ changes with the Earth’s rotation. Based on Eq. (18), the variation range of $\left | {\varepsilon _{i,j}^{\textrm{Diurnal}\_ \textrm{abe}} (t)} \right |$ for the angular distance values can be obtained, as shown in Fig. 8(b).

(5) Influence of atmospheric refraction

When a stellar light passes through the atmosphere, its direction will be changed due to the influence of the Earth’s atmosphere (Fig. 9(a)). The value of refraction angle can be estimated on the basis of the atmospheric refraction model [29] as follows:

(19)$$R\left( {z_0 } \right) = \kappa \gamma _1 \left( {1 - \beta } \right)\tan z_0 - \kappa \gamma _1 \left( {\beta - \gamma _1 /2} \right)\tan ^3 z_0,$$

where ${z_0 }$ is the star zenith distance. In a spherical Earth, the value of parameter $\kappa$ is $\kappa = 1.0$; $\gamma _1 = n_0 - 1$, and $n_0$ is the refractive index of air at the observing site; $\beta = H_0 /r_0$, and $r_0$ is the geocentric distance of the observing site, $H_0$ is the height of an equivalent homogeneous atmosphere.

Fig. 9. Atmospheric refraction: influence on (a) star direction and (b) star angular distance.

Download Full Size | PDF

The atmospheric refraction model is influenced by pressure $P_a$, temperature $T_c$, relative humidity $r_h$, and wavelength $w_l$ [27,32], as shown as follows:

(20)$$\begin{aligned} \left\{ \begin{array}{l} ps = 10^{\left( {0.7859 + 0.03477 \cdot T_c } \right)/\left( {1 + 0.00412 \cdot T_c } \right)} \cdot \left( {1 + P_a \cdot \left( {4.5 \cdot 10^{ - 6} + 6 \cdot 10^{ - 10} T_c^2 } \right)} \right) \\ pw = r_h \cdot ps/\left( {1 - \left( {1 - r_h } \right)ps/P_a } \right) \\ \gamma _1 = \frac{{\left( {77.53484 \cdot 10^{ - 6} + \left( {4.39108 \cdot 10^{ - 7} + 3.666 \cdot 10^{ - 9} /w_l^2 } \right)/w_l^2 } \right)P_a - 11.2684 \cdot 10^{ - 6} pw}}{{T_c + 273.15}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \\ \beta = 4.4474 \cdot 10^{ - 6} \left( {T_c + 273.15} \right) \\ \end{array} \right.. \end{aligned}$$

With regard to the angular distance made up of two stars $i$ and $j$ (Fig. 9(b)), the observation angular distance error caused by atmospheric refraction should be

(21)$$\left| {\varepsilon _{i,j}^{\textrm{Atmo}\_ \textrm{ref }} \left( t \right)} \right| = \left| {\angle S_i O_E S_j - \angle S'_i O_E S'_j } \right| \le \left| {R\left( {z_{\max } } \right) - R\left( {z_{\max } - \alpha^{\prime\prime }} \right)} \right|,$$

where $z_{\max }$ is the maximum zenith distance of stars $i$ and $j$ (i.e., $\max \left [ {\angle S_i O_E U,\angle S_j O_E U} \right ]$), $\alpha ''$ is the value of angular distance before passing through the Earth’s atmosphere.

As the $\left | {\varepsilon _{i,j}^{\textrm {Atmo}\_ \textrm {ref }} \left ( t \right )} \right |$ increases with the increasing of $P_a$, and decreases with the increasing of $T_c$ and $r_h$, Eq. (21) is rewritten as follows:

(22)$$\left| {\varepsilon _{i,j}^{\textrm{Atmo}\_ \textrm{ref }} \left( t \right)} \right| \le R\left( {z_{\max }}, P_{a,\max },T_{c,\min} , r_{h,\min} \right)-R\left( {z_{\max }-\alpha^{\prime\prime}}, P_{a,\max },T_{c,\min} , r_{h,\min} \right),$$

where $P_a \in \left [ {P_{a,\min } ,P_{a,\max } } \right ]$, $T_c \in \left [ {T_{c,\min } ,T_{c,\max } } \right ]$ and $r_h \in \left [ {r_{h,\min } ,r_{h,\max } } \right ]$ are the observation environment rangs.

However, because of the Earth’s rotation, the value of a star zenith distance will change from $-90^\circ$ to $90^\circ$ every day and exceed the accuracy limitation of Eq. (19) (i.e., $\left [ {0^\circ ,75^\circ } \right ]$ ). It is difficult to obtain the value of $\left | {\varepsilon _{i,j}^{\textrm {Atmo}\_ \textrm {ref }} \left ( t \right )} \right |$ in the larger variation of stars’ zenith distance.

Actually, the star’s zenith distance that can be observed by the terrestrial star tracker is determined by its attitude performance (i.e., the range of pitch, yaw, and roll angles). If the pitch, yaw, and roll angles of the terrestrial star tracker are $\left | \theta \right | \in \left [ {0^\circ ,\theta _{\max } } \right ]$, $\left | \psi \right | \in \left [ {0^\circ ,\psi _{\max } } \right ]$ and $\left | \gamma \right | \in \left [ {0^\circ ,\gamma _{\max } } \right ]$, the zenith distances $z_0$ of a star observed by the terrestrial star tracker will be in the range of

(23)$$z_0 \in \left[ {z_{\min } ,z_{\max } } \right] = \left[ {0^\circ ,z_{s,\max } + \theta _\textrm{FOV} /2} \right],$$

where $z_{s,\max }$ is the maximum value of the pitch and roll angles, and $z_{s,\max } = \max \left [ {\theta _{\max } ,\gamma _{\max } } \right ]$.

By substituting Eq. (23) into Eq. (22), the observation angular distance error caused by the atmospheric refraction of the terrestrial star tracker can be obtained as follows:

(24)$$\left| {\varepsilon _{i,j}^{\textrm{Atmo}\_ \textrm{ref }} \left( t \right)} \right| \le R\left( z_{s,\max } + \theta _\textrm{FOV} /2 \right)-R\left( {z_{s,\max } + \theta _\textrm{FOV} /2-\alpha^{\prime\prime}} \right),$$

where, the pressure $P_a$, temperature $T_c$, relative humidity $r_h$ in Eq. (24) are equal to $P_{a,\max }$, $T_{c,\min }$ and $r_{h,\min }$ of the observation environment rangs.

3.3 Modeling of the observation angular distance error

In summary, the observation angular distance error is a complex issue, which is related not only to the properties of stars that make up the angular distances but also to the various factors, such as the angular value, revolution and rotation of the Earth, observation time, observation location, observation environment, atmospheric refraction, and optical parameters of the star tracker. The angular distance with different values or made up of distinct stars will have completely varying rules.

However, since the angular distance of a single pair of stars is identified only when they belong to the angular distance threshold range in star identification, the upper variation range of the angular distance errors is in demand. Therefore, the variation range of the observation angular distance error $\varepsilon _{i,j} \left ( t \right )$ for the terrestrial star tracker can be obtained by setting $d_{i,j}^{AT}=\alpha =\alpha '=\alpha ''=d_{i,j}^\textrm {ref}$, as follows:

(25)$$\begin{aligned} \begin{array}{l} \varepsilon _{i,j} \left( t \right) = \left| {\varepsilon _{i,j}^\textrm{space-move} \left( {t - t_0 } \right)} \right| + \left| {\varepsilon _{i,j}^{\textrm{Annual}\_ \textrm{par }} \left( t \right)} \right| + \left| {\varepsilon _{i,j}^{\textrm {Gravity}\_ \textrm{Ref}} \left( t \right)} \right| + \left| {\varepsilon _{i,j}^{\textrm{Annual}\_ \textrm{abe}} \left( t \right)} \right| \\+ \left| {\varepsilon _{i,j}^{\textrm{Diurnal}\_ \textrm{abe}} (t)} \right| + \left| {\varepsilon _{i,j}^{\textrm{Atmo}\_ \textrm{ref }} \left( t \right)} \right| \\ \le \left| {t - t_0 } \right|\left( {\sqrt {u^2 _{\alpha ,i} + u^2 _{\delta ,i} } + \sqrt {u^2 _{\alpha ,j} + u^2 _{\delta ,j} } } \right) + \left| {\pi _i } \right| + \left| {\pi _j } \right| + \dfrac{{2\mu }}{{c^2 E}}\left[ {\tan \left( {\dfrac{{\phi _1 }}{2}} \right) + \tan \left( {\dfrac{{\phi _2 }}{2}} \right)} \right] \\ + 2 \times \textrm{asin}\left( {\dfrac{{V_E }}{c}\sin \dfrac{{d_{i,j}^\textrm{ref} }}{2} - \dfrac{1}{4}\left( {\dfrac{{V_E }}{c}} \right)^2 \sin d_{i,j}^\textrm{ref} } \right) + 2 \times \textrm{asin}\left( {\dfrac{{\textbf{v}_\textrm{Earth} }}{c}\sin \dfrac{{d_{i,j}^\textrm{ref} }}{2}} \right.\left. { - \dfrac{1}{4}\left( {\dfrac{{\textbf{v}_\textrm{Earth} }}{c}} \right)^2 \sin d_{i,j}^\textrm{ref} } \right) \\ + \left| {R\left( z_{s,\max } + \theta _\textrm{FOV} /2 \right)-R\left( {z_{s,\max } + \theta _\textrm{FOV} /2-d_{i,j}^\textrm{ref}} \right)} \right| \\ \end{array}, \end{aligned}$$

where $d_{i,j}^\textrm {ref}$ is the value of reference angular distance, $\left [ {u_{\alpha ,i} ,u_{\delta ,i} ,\pi _i } \right ]$ represents the positional parameters of star $i$; $z_{s,\max } = \max \left [ {\theta _{\max } ,\gamma _{\max } } \right ]$ is the maximum value of pitch and roll angles; $\phi _1$ and $\phi _2$ are determined by Eqs. (12) and (13).

Evidently, the changes on $T_\textrm {ref}$ and ref-frame of the star catalog will influence $\varepsilon _{i,j} \left ( t \right )$. On the basis of the difference of ref-frame, the observation angular distance error can be divided into two items, that is,

(26)$$\varepsilon _{i,j} \left( t \right) = {\kern 1pt} \left| {d_{i,j}^\textrm{ref} (t) - d_{i,j}^\textrm{ref} (T_\textrm{ref} )} \right| + \left| {d_{i,j}^S \left( t \right) - d_{i,j}^\textrm{ref} (t)} \right| = \varepsilon _{i,j}^\textrm{ref} \left( t \right) + \varepsilon _{i,j}^\textrm{ref - S} \left( t \right),$$

where the first item $\varepsilon _{i,j}^\textrm {ref} \left ( t \right )$ is the observation angular distance error in the ref-frame, and the second one $\varepsilon _{i,j}^\textrm {ref - S} \left ( t \right )$ is the observation angular distance error caused by the transmission from the ref-frame to the S-frame. The ref-frame will be one of the five coordinate systems, namely, ICRS, AT-frame and G-, E-, and N-frames (Fig. 4).

For example, if the ref-frame of the star catalog is the AT-frame, $\varepsilon _{i,j}^\textrm {ref} \left ( t \right )$ will consist of two factars, i.e., ${\varepsilon _{i,j}^\textrm {space - move} \left ( {t - T_\textrm {ref} } \right )}$ and ${\varepsilon _{i,j}^{\textrm {Annual}\_ \textrm {par }} \left ( t \right )}$; $\varepsilon _{i,j}^\textrm {ref - S} \left ( t \right )$ will consist of four factors, i.e., ${\varepsilon _{i,j}^{\textrm {Gravity}\_ \textrm{Ref}} \left ( t \right )}$, ${\varepsilon _{i,j}^{\textrm {Annual}\_ \textrm {abe}} \left ( t \right )}$, ${\varepsilon _{i,j}^{\textrm{Diurnal}\_ \textrm{abe}} (t)}$, and ${\varepsilon _{i,j}^{\textrm {Atmo}\_ \textrm {ref }} \left ( t \right )}$. Then, the variation range of the observation angular distance error can be calculated on the basis of Eq. (25).

4. Angular distance matching threshold optimization

On the basis of the aforementioned analysis, the observation angular distance error is a complex issue related to various factors. It is unrealistic for a terrestrial star tracker to determine whether the observation angular distance error $\varepsilon _{i,j} \left ( t \right )$ of all possible observed angular distances is consistent with Eq. (6) all the time.

For convenience, the angular distance matching threshold ($TH$) is used to represent the observation angular distances error of all the possible observed angular distances in star tracker at any observation time $t$, as follows:

(27)$${\kern 1pt} TH \ge {\kern 1pt} \varepsilon _{i,j} \left( t \right) = \left| {d_{i,j}^s \left( t \right) - d_{i,j}^\textrm{ref} } \right|,$$

where ${d_{i,j}^s \left ( t \right )}$ and $d_{i,j}^\textrm {ref}$ are the observation and the reference angular distances may be observed in the working time of terrestrial star tracker.

Given that the observation angular distance error is divided into two items on the basis of the proposed observation angular distance error model Eq. (26), the angular distance matching threshold $TH$ is also rewritten as

(28)$${\kern 1pt} TH \ge \max \left| {d_{i,j}^\textrm{ref} (t) - d_{i,j}^\textrm{ref} (T_\textrm{ref} )} \right| + \max \left| {d_{i,j}^S \left( t \right) - d_{i,j}^\textrm{ref} (t)} \right| = TH_{\textrm{catalogue }} {\kern 1pt} + TH_{\textrm{ref - S}} ,$$

where $TH_{\textrm {catalogue }}$ represents the star catalog angular distance matching threshold, and $TH_{\textrm {ref - S}}$ represents the transmission angular distance matching threshold.

4.1 Star catalog angular distance matching threshold

The star catalog angular distance matching threshold $TH_{\textrm {catalogue }}$ is determined by the ref-frame and $T_\textrm {ref}$ of the star catalog. The ref-frame will be one of the five coordinate systems, namely, ICRS, AT-frame, and G-, E-, and N-frames (Fig. 4). $T_\textrm {ref}$ is determined by the stable time of the star catalog, which is usually larger than 1 year compared with the real-time observation time.

AT-frame is the most suitable coordinate system in practical applications, since the vector variation of stars in the G-frame ( larger than $\left [ {\textrm { - 20}\textrm {.4915'', + 20}\textrm {.4915''}} \right ]$ per year), E-frame, and N-frame ( approximately $\left [ {\textrm {- 180}^\circ \textrm {, + 180}^\circ } \right ]$ per day) are too large to obtain a high-accuracy star catalog over the stable time.

Therefore, on the basis of Eqs. (25) and (28), the star catalog angular distance matching threshold can be expressed as follows by setting the ref-frame of the star catalog as AT-frame:

(29)$$\begin{aligned} \begin{array}{l} TH_{\textrm{catalogue}} = \max \left| {d_{i,j}^{AT} (t) - d_{i,j}^{AT} (T_\textrm{ref} )} \right| = \max \left( {\left| {\varepsilon _{i,j}^\textrm{space - move} \left( {\Delta t} \right)} \right| + \left| {\varepsilon _{i,j}^{\textrm{Annual}\_ \textrm{par }} \left( t \right)} \right|} \right) \\ = \max \left( {\left| {t - T_\textrm{ref} } \right|\left( {\sqrt {u^2 _{\alpha ,i} + u^2 _{\delta ,i} } + \sqrt {u^2 _{\alpha ,j} + u^2 _{\delta ,j} } } \right) + \left| {\pi _i } \right| + \left| {\pi _j } \right|} \right) \\ \approx 2\max \left( {\left| {t - T_\textrm{ref} } \right|\sqrt {u^2 _{\alpha ,i} + u^2 _{\delta ,i} } + \left| {\pi _i } \right|} \right) \\ \end{array}, \end{aligned}$$

where, $\left [ {u_{\alpha ,i} ,u_{\delta ,i} ,\pi _i } \right ]$ represents the proper motions and parallax parameters of star $i$ in star catalog.

Suppose that the star catalog is stable in 2019, that is, the stable time of the star catalog is 1 year. Figure 10(a) shows the vector error of stars in the AT-frame versus observation time $t$ in 2019 by setting the $T_\textrm {ref}$ as $2019.07.01$. A total of 5,041 stars are selected from the Hipparcos catalogue, with their magnitudes brighter than 6.0. It is found that the vector variation of different stars keep near-linearly increasing with $\left | {t - T_\textrm {ref} } \right |$. The maximum value of star vector error reach to $3.2''$, and the minimum value equals to $0''$.

Fig. 10. Vector error of stars in the AT-frame (a) versus observation time t in 2019 when $T_\textrm {ref} = 2019.07.01$; (b)cumulative distribution of the number of stars versus star vector error.

Download Full Size | PDF

The cumulative distribution of the number of stars versus the star vector error is shown in Fig. 10(b). If using a value, denoted as $Err_\textrm {star}$, to represent the vector error range of stars during the available stable time, different value of $Err_\textrm {Star}$ will corresponding to different sets of stars in star catalog. For example, when $Err_\textrm {star} = \left [ {0,0.4''} \right ]$, 4,961 stars are present in the range of $Err_\textrm {star}$, and $TH_{\textrm {catalogue}} \approx 2Err_\textrm {star} = 0.8''$.

Therefore, The value of $TH_{\textrm {catalogue}}$ is determined by the selection of star in star catalog. According to the helix uniform distribution method [33,34], assume that there are $N_{\textrm {Axis}}$ uniformly directions of the FOV axis, denoted as $\textbf {b}^{AT}$ existed in the AT-frame. Then, on the basis of $Err_\textrm {star}$ and the FOV of star tracker $\theta _\textrm {FOV}$, the probability that $N_s$ stars exist in the FOV of the star tracker is

(30)$$P\left( {\theta _\textrm{FOV} ,Err_\textrm{star} ,N_s } \right) = \frac{1}{{N_{\textrm{Axis}} }}\sum\nolimits_1^{N_{\textrm{Axis}} } {\left[ {\left( {\sum {\left( {\theta \left( {{\textbf{V}_{AT}^i }, \textbf{b}^{AT} } \right) \le \dfrac{1}{2}\theta _\textrm{FOV} } \right)} \ge N_s } \right)} \right]} ,$$

where ${\textbf {V}_{AT}^i }$ is the vector of star $i$ in star catalog, $\theta ({\textbf {V}}_{{AT}}^{{i}},{\textbf {b}}^{{{AT}}} )$ is the angular distance between ${\textbf {V}}_{\textbf{AT}}^{\textbf{i}}$ and $\textbf {b}^{AT}$. Generally, $N_s$ is larger than 10.

If the FOV of a star tracker is increased from $14^\circ \times 14^\circ$ to $22^\circ \times 22^\circ$, the simulation results of $P\left ( {\theta _\textrm {FOV} ,Err_\textrm {star} ,N_s} \right )$ versus $Err_\textrm {star}$ when $N_\textrm {Axis}=100,000$ and $N_s=4$ is shown in Fig. 11(a); the results of $Err_\textrm {star}$ versus $N_s$ when $P\left ( {\theta _\textrm {FOV} ,Err_\textrm {star} ,N_s} \right )=100\%$ is shown in Fig. 11(b).

Fig. 11. Analysis results by helix uniform distribution method: (a) $P\left ( {\theta _\textrm {FOV} ,Err_\textrm {star} ,N_s} \right )$ vs. $Err_\textrm {star}$ when $N_\textrm {Axis}=100,000$ and $N_s=4$; (b) $Err_\textrm {star}$ vs. $N_s$ when $P\left ( {\theta _\textrm {FOV} ,Err_\textrm {star} ,N_s} \right )=100\%$.

Download Full Size | PDF

With the decreasing of $\theta _\textrm {FOV}$ and the increasing of $N_s$, the value of $Err_\textrm {star}$ increases for the case $P\left ( {\theta _\textrm {FOV} ,Err_\textrm {star} ,N_s} \right )=100\%$. Therefore, on the basis of $\theta _\textrm {FOV}$ and $N_s$, the star catalog angular distance matching threshold is determined by

(31)$$TH_{\textrm{catalogue}} = 2\max \left( {\left| {t - T_\textrm{ref} } \right|\sqrt {u^2 _{\alpha ,i} + u^2 _{\delta ,i} } + \left| {\pi _i } \right|} \right) \approx 2Err_\textrm{star} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {star}\_{i} \in \overline {\textrm{Star}_{\textrm{Catalogue}} } ,$$

where $\overline {\textrm {Star}_{\textrm {Catalogue}} }$ is the optimal star catalog containing the stars whose vector error is less than $Err_\textrm {star}$ when $P\left ( {\theta _\textrm {FOV} ,Err_\textrm {star} ,N_s} \right )=100\%$.

For instance, when $\theta _\textrm {FOV}= 20^\circ$ and $N_s=11$, the optimal $Err_\textrm {star} = \left [ {0,0.375''} \right ]$ (Fig. 11(b)), $TH_{\textrm {catalogue}} \approx 0.75''$, and a total of 4,950 stars are contained in $\overline {\textrm {Star}_{\textrm {Catalogue}} }$.

4.2 Transmission angular distance matching threshold

The transmission angular distance matching threshold is determined by the astrometric transmissions from the ref-frame to the S-frame. According to Eq. (25), when the ref-frame of the star catalog is the AT-frame, the transmission angular distance matching threshold is

(32)$$\begin{aligned} \begin{array}{l} TH_{\textrm{ref - S}} = \max \left| {d_{i,j}^s \left( t \right) - d_{i,j}^{AT} (t)} \right| \\ = \max \left( {\left| {\varepsilon _{i,j}^{\textrm {Gravity}\_ \textrm{Ref}} \left( t \right)} \right| + \left| {\varepsilon _{i,j}^{\textrm{Annual}\_ \textrm{abe}} \left( t \right)} \right| + \left| {\varepsilon _{i,j}^{\textrm{Diurnal}\_ \textrm{abe}} (t)} \right| + \left| {\varepsilon _{i,j}^{\textrm{Atmo}\_ \textrm{ref }} \left( t \right)} \right|} \right) \\ \end{array}. \end{aligned}$$

This function is related to the value of $d_{i,j}^\textrm {ref}$. When the terrestrial star tracker is used at night, the ${\varepsilon _{i,j}^{\textrm {Gravity}\_ \textrm{Ref}} \left ( t \right )}$, ${\varepsilon _{i,j}^{\textrm {Annual}\_ \textrm {abe}} \left ( t \right )}$, ${\varepsilon _{i,j}^{\textrm{Diurnal}\_ \textrm{abe}} (t)}$, and ${\varepsilon _{i,j}^{\textrm {Atmo}\_ \textrm {ref }} \left ( t \right )}$ increase with the increasing of $d_{i,j}^\textrm {ref}$; otherwise, ${\varepsilon _{i,j}^{\textrm {Gravity}\_ \textrm{Ref}} \left ( t \right )}$ obtains its maximum value when $d_{i,j}^\textrm {ref}\approx 0.5^\circ$.

Actually, the observed angular distance values of the terrestrial star tracker is determined by the value of $\theta _\textrm {FOV}$ (i.e., $d_{i,j}^\textrm {ref} \in \left ( {0,\theta _\textrm {FOV} } \right ]$). Therefore, on the basis of Eq. (25), when these four factors reach their maximum at the same time, the transmission angular distance matching threshold $TH_{\textrm {ref - S}}$ is obtained as

(33)$$\begin{aligned} \begin{array}{l} TH_{\textrm{ref - S}}\!=\! \dfrac{{2\mu }}{{c^2 E}}\left[ {\tan \left( {\dfrac{{\phi _1 }}{2}} \right) \!-\! \tan \left( {\dfrac{{\phi _2 }}{2}} \right)} \right] + 2 \!\times\! \textrm{asin}\left( {\dfrac{{V_E }}{c}\sin \dfrac{{\theta _\textrm{FOV} }}{2} \!-\! \dfrac{1}{4}\left( {\dfrac{{V_E }}{c}} \right)^2 \sin \theta _\textrm{FOV} } \right) \!+\!\\ 2 \!\times\!{\textrm{asin}}\!\left( {\dfrac{{\textbf{v}_\textrm{Earth} }}{c}\sin \dfrac{{\theta _\textrm{FOV} }}{2}} \right.\left. { \!- \dfrac{1}{4}\left( {\dfrac{{\textbf{v}_\textrm{Earth} }}{c}} \right)^2\!\! \sin \theta _\textrm{FOV} } \right) \!\!+\!\! R\left( {z_{s,\max } \!+\! \dfrac{\theta _\textrm{FOV}}{2}}, P_{a,\max },T_{c,\min} , r_{h,\min} \right)\\-R\left( {z_{s,\max } - \theta _\textrm{FOV} /2}, P_{a,\max },T_{c,\min} , r_{h,\min} \right)\!\! \\ \end{array}, \end{aligned}$$

where the observation environment $P_a \in \left [ {P_{a,\min } ,P_{a,\max } } \right ]$, $T_c \in \left [ {T_{c,\min } ,T_{c,\max } } \right ]$ and $r_h \in \left [ {r_{h,\min } ,r_{h,\max } } \right ]$; the pitch, yaw, and roll angles ranges of the terrestrial star tracker are $\left | \theta \right | \in \left [ {0^\circ ,\theta _{\max } } \right ]$, $\left | \psi \right | \in \left [ {0^\circ ,\psi _{\max } } \right ]$ and $\left | \gamma \right | \in \left [ {0^\circ ,\gamma _{\max } } \right ]$, $z_{s,\max } = \max \left [ {\theta _{\max } ,\gamma _{\max } } \right ]$; when the terrestrial star tracker is only used at night, $\phi _1 = 90^\circ$ and $\phi _2 = 90^\circ - \theta _\textrm {FOV}$; otherwise, $\phi _1 = 180^\circ - 0.25^\circ$ and $\phi _2 = - \phi _1$.

For instance, by setting the values of parameters as Tabel 1, the $TH_{\textrm {ref - S}}$ versus the value of $\theta _\textrm {FOV}$ when $z_{s,\max }=45^\circ$ is shown in Fig. 12(a), and the $TH_{\textrm {ref - S}}$ versus the value of $z_{s,\max }$ when $\theta _\textrm {FOV}=20^\circ$ is shown in Fig. 12(b). If $\theta _\textrm {FOV}=20^\circ$ and $z_{s,\max }=45^\circ$, the $TH_{\textrm {ref - S}}=58.1974''$ when the star tracker is only used in night; otherwise, $TH_{\textrm {ref - S}}=61.9281''$.

Fig. 12. Transmission angular distance matching threshold: (a) $TH_{\textrm {ref - S}}$ vs. $\theta _\textrm {FOV}$ when $z_{s,\max }=45^\circ$; (b) $TH_{\textrm {ref - S}}$ vs. $z_{s,\max }$ when $\theta _\textrm {FOV}=20^\circ$.

Download Full Size | PDF

Table 1. Values of parameters in angular distance matching threshold model.

View Table | View all tables in this article

4.3 Optimal angular distance matching threshold

In summary, the total angular distance matching threshold of the terrestrial star tracker is determined by accumulating the star catalog and transmission matching thresholds.

If the value of angular distance matching threshold $TH$ larger than these two matching thresholds resources, i.e., ($TH_{\textrm {catalogue}}$ +$TH_{\textrm {ref - S}}$), the observation angular distance error of the observation stars satisfies Eq. (27) and ensures the success of star identification. However, the value of $TH$ is not the larger the better. It is also related to the identification efficiency. A larger value of $TH$ means that more misrecognized pairs of stars satisfy Eq. (27), and make it more difficult and inefficient to identify the true stars form the misrecognized stars [22].

Therefore, the optimal angular matching threshold $\overline {TH}$ should be the minimum value of $TH$ that ensure the star identification success rate equals to 100%, as shown as follows:

(34)$$\begin{aligned} \begin{array}{l} \overline {TH} \! = \!\min \left( {TH} \right) \!= \!TH_{\textrm{catalogue}} {\kern 1pt} \! + \!TH_{\textrm{ref - S}} \\ \! =\! 2Err_\textrm{star} \! + \!\dfrac{{2\mu }}{{c^2 E}}\left[ {\tan \left( {\dfrac{{\phi _1 }}{2}} \right)\! -\! \tan \left( {\dfrac{{\phi _2 }}{2}} \right)} \right] \!+\! 2 \times \textrm{asin}\left( {\dfrac{{V_E }}{c}\sin \dfrac{{\theta _\textrm{FOV} }}{2} - \dfrac{1}{4}\left( {\dfrac{{V_E }}{c}} \right)^2 \!\!\sin \theta _\textrm{FOV} } \right)+ \\ 2 \!\times\!\textrm{asin}\!\left( {\dfrac{{\textbf{v}_\textrm{Earth} }}{c}\sin \dfrac{{\theta _\textrm{FOV} }}{2}} \right.\left. { \!- \dfrac{1}{4}\left( {\dfrac{{\textbf{v}_\textrm{Earth} }}{c}} \right)^2\!\! \sin \theta _\textrm{FOV} } \right) \!\!+\!\! R\left( {z_{s,\max } \!+\! \dfrac{\theta _\textrm{FOV}}{2}}, P_{a,\max },T_{c,\min} , r_{h,\min} \right)\\-R\left( {z_{s,\max } - \theta _\textrm{FOV} /2}, P_{a,\max },T_{c,\min} , r_{h,\min} \right)\\ \end{array}, \end{aligned}$$

where $Err_\textrm {star}$ is the angular distance matching threshold of the optimal star catalog $\overline {\textrm {Star}_{\textrm {Catalogue}} }$.

Additionally, in applications, the angular distance errors of the observation stars are also influenced by the measurement capabilities of the star tracker itself and the atmospheric turbulence. Therefore, the optimal angular distance matching threshold $\overline {TH}$ is rewritten as:

(35)$$\overline {TH} = TH_{\textrm{catalogue}} + TH_{\textrm{ref - S}} + 2\left( {\left| {\varepsilon _p } \right| + \left| {\varepsilon _{\textrm{turbulence}} } \right|} \right),$$

where ${\varepsilon _p }$ is the residual measurement error of the star tracker after laboratory calibrations [35–38], ${\varepsilon _{\textrm {turbulence}} }$ is the positioning error caused by atmospheric turbulence. Commonly, the limited resolution of the typical visible wavelengths is less than 1 arc-second, i.e., $\left | {\varepsilon _{\textrm {turbulence}} } \right |\le 1''$ [39].

5. Simulations and night sky experiment

In the previous sections, we present the observation angular distance error model and derive the optimal angular matching threshold expression for terrestrial star trackers. In this section, numerical simulations and night sky experiment are conducted to validate our conclusions.

5.1 Numerical simulations

A series of simulations is conducted to verify the optimization of angular distance matching threshold. Assuming that the values of simulations parameters are presented as Tabel 1, and the star tracker is only used at night.

Regardless of the positional accuracy ${\varepsilon _p }$ of the star tracker and the atmospheric turbulence error ${\varepsilon _{\textrm {turbulence}} }$, a total of 10,000 simulation vector star images are generated in the test as follows. (1) $\overline {\textrm {Star}_{\textrm {Catalogue}} }$, which consists of 4,950 stars, and $Err_\textrm {star} = 0.375''$ is used as the reference star catalog. (2) A total of 10,000 random unit vectors of the FOV axis in the N-frame $\textbf {b}^N$ is generated with random observed zenith distance $z_s \in \left [ {0^\circ ,45^\circ } \right ]$ and observed azimuth $\alpha _s \in \left [ {0,360^\circ } \right ]$ . (3) The unit vectors of FOV axis in the AT-frame $\textbf {b}^{AT}$ are obtained by transmitting the $\textbf {b}^N$ from the N-frame to the AT-frame on the basis of Eq. (7), where the observation longitude is random in $\left [ { - 180^\circ ,180^\circ } \right ]$, the latitude is random in $\left [ { - 90^\circ ,90^\circ } \right ]$, the height is 960 m, and the local time is 00:00:00 on any day in 2019. (4) All the stars $\textbf {V}_{AT} ^{i}$ from $\overline {\textrm {Star}_{\textrm {Catalogue}} }$, with their angular distances less than $10^\circ$ between $\textbf {V}_{AT} ^{i}$, are selected. (5) The stars $\textbf {V}_{AT} ^{i}$ from the AT-frame to the S-frame are transmitted on the basis of Eqs. (7) and (8) to generate the observation vector $\textbf {W}_{i}$ and formed the simulation vector star images, where $\left \langle {C_n^S } \right \rangle = {\textbf I}$. The examples of two typical simulation vector star images are shown in Fig. 13.

Fig. 13. Examples of the two typical simulation vector star images: (a) $z_s=2.5677^\circ$ and $\alpha _s=81.0825^\circ$; (b) $z_s=38.6821^\circ$ and $\alpha _s=339.1070^\circ$.

Download Full Size | PDF

Nine value of the tested angular distance matching threshold $TH$ (i.e., $30''$, $35''$, $40''$, $45''$, $50''$, $55''$, $58''$, $60''$, and $65''$) are used to implement the star identification for the 10,000 simulation vector star images. The star identification for the simulation vector star image is regarded successful only when all the stars are identified. The average number of angular distance pairs that satisfy Eq. (27) for different values of $TH$ is recoded.

Figure 14 shows the simulation results of the star identification success rate and the average number of angular distance pairs versus the angular distance matching threshold $TH$. Theoretically, on the basis of Eq. (35), the optimal angular distance matching threshold of the simulation terrestrial star tracker is $\overline {TH} = \textrm {58}\textrm {.9474''}$, as shown by the dashed line in Fig. 14.

Fig. 14. Simulation results of the star identification success rate and the average number of angular distances vs. the angular distance matching threshold $TH$.

Download Full Size | PDF

From the figure, the identification success rate has increased from 72% to 100% as $TH$ increased from $30''$ to approximate $59''$. Although the cases when $TH$ is larger than $59''$ also ensure the success of star identification, the average number of angular distance pairs is increased drastically. When the value of $TH$ increased from $59''$ to $65''$, the average number of angular distance pairs is increased by more than $0.71\times 10^4$. Thus, the optimal angular distance matching threshold for the simulation observation results $\overline {TH}$ is approximately equal to $59''$.

The value of theoretical optimal angular distance matching threshold $\overline {TH}$ is approximately equal to the simulation observation results, which means that all the angular distances in the simulation star images agree with the optimal angular distance matching threshold model.

5.2 Night sky experiment

A night sky experiment was conducted at the Xinglong Station (National Astronomical Observatories, China) to verify our theoretical conclusions. In Fig. 15, the star tracker is mounted on a two-axis turntable, the install zenith distance of the star tracker is approximately equals to $1.8^\circ$, and the parameters of the test star tracker is shown in Table 2.

Fig. 15. Setup of the night sky experiment: (a) turntable and star tracker; (2) FOV structure of the test terrestrial star tracker.

Download Full Size | PDF

Table 2. Design parameters of the star tracker

View Table | View all tables in this article

In the night sky experiment, a total of 1,000-frame centroid data were acquired by rotating the turntable within $-40^\circ$ to $40^\circ$. Twelve values of tested angular distance matching threshold (i.e., $25''$, $30''$, $35''$, $40''$, $45''$, $50''$, $55''$, $60''$, $65''$, $70''$, $75''$ and $80''$) are used to implement the star identification for the 1,000-frame centroid data. The $\overline {\textrm {Star}_{\textrm {Catalogue}} }$, which consists of 4,950 stars and $Err_\textrm {star} = 0.375''$, is used as the reference star catalog. The star identification is regarded successful only when the number of recognized stars is larger than six. The results of the identification success rate and the average number of angular distances versus the angular distance matching threshold $TH$ are shown in Fig. 16.

Fig. 16. Night sky experimental results of identification success rate and average number of angular distances vs. the angular distance matching threshold $TH$.

Download Full Size | PDF

On the basis of Eq. (35), the theoretical $\overline {TH} = \textrm {69}\textrm {.5430''}$, where, the observation longitude is $117.580176^\circ$, the latitude is $40.397073^\circ$, the height is 958 m, $z_{s,\max } = 40^\circ + 1.8^\circ$, $P_a \in \left [ {980,1020} \right ]$, $T_c \in \left [ {14^\circ C,20^\circ C} \right ]$, $r_h \in \left ( {0.2,0.4} \right )$, and $\max \theta _\textrm {FOV} = \sqrt 2 \times 20^\circ$, the residual measurement error of the star tracker after laboratory calibration $\varepsilon _p = 1''$, and $\left | {\varepsilon _{\textrm {turbulence}} } \right |\le 1''$.

As shown in Fig. 16, the identification success rate and the average number of angular distances increase with the distance matching threshold $TH$. As $TH$ increased from $25''$ to $70''$, the identification success rate has increased from 74.6% to 100%, and the average number of angular distances was increased from 0.5 to $2.339\times 10^6$. Therefore, the optimal angular distance matching threshold for the actual observation results is approximately equal to $70''$.

The difference between the theoretical value of $\overline {TH}$ ($\textrm {69}\textrm {.5430''}$) and the actual experiment results ($70''$) is less than $0.5''$. Therefore, the night sky experiment validates the reliability of our conclusions for the observation angular distance error model and angular distance matching threshold optimization.

6. Conclusion

This study focuses on the modeling of the observation angular distance error for terrestrial star trackers. This parameter is important and affects the identification success rate, attitude measurement accuracy, and real-time performance of the terrestrial star tracker but is yet to be clarified to date. To resolve this problem, a stellar observation angular distance error model is derived. The factors, namely, space motion, annual parallax, solar gravitational light deflection, annual aberration, diurnal aberration, and atmospheric refraction, are analyzed to determine the variation range of observation angular distance error. Then, the optimization solution of the angular distance matching threshold is obtained to achieve the optimal efficiency in the star identification. Numerical simulations and night sky experiment validate the reliability of our conclusions for error model and threshold optimization.

Funding

National Natural Science Foundation of China (61725501).

References

1. C. C. Liebe, “Accuracy performance of star trackers-a tutorial,” IEEE Trans. Aerosp. Electron. Syst. 38(2), 587–599 (2002). [CrossRef]

2. G. Zhang, STAR IDENTIFICATION: Methods, Techniques and Algorithms (SPRINGER, 2019).

3. C. C. Liebe, “Star trackers for attitude determination,” IEEE Aerosp. Electron. Syst. Mag. 10(6), 10–16 (1995). [CrossRef]

4. T. Sun, F. Xing, X. Wang, Z. You, and D. Chu, “An accuracy measurement method for star trackers based on direct astronomic observation,” Sci. Rep. 6(1), 22593 (2016). [CrossRef]

5. J. Li, G. Wang, and X. Wei, “Generation of guide star catalogue for star trackers,” IEEE Sens. J. 18(11), 4592–4601 (2018). [CrossRef]

6. W. Yu, J. Jiang, and G. Zhang, “Multi exposure imaging and parameter optimization for intensified star trackers,” Appl. Opt. 55(36), 10187–10197 (2016). [CrossRef]

7. W. Yu, J. Jiang, and G. Zhang, “Star tracking method based on multiexposure imaging for intensified star trackers,” Appl. Opt. 56(21), 5961–5971 (2017). [CrossRef]

8. G. Wang, F. Xing, M. Wei, and Z. You, “Rapid optimization method of the strong stray light elimination for extremely weak light signal detection,” Opt. Express 25(21), 26175–26185 (2017). [CrossRef]

9. T. Sun, F. Xing, J. Bao, S. Ji, and J. Li, “Suppression of stray light based on energy information mining,” Appl. Opt. 57(31), 9239–9245 (2018). [CrossRef]

10. A. Katake and C. Bruccoleri, “Starcam sg100: a high-update rate, high-sensitivity stellar gyroscope for spacecraft,” Proc. SPIE 7536, 753608 (2010). [CrossRef]

11. A. B. Katake, Modeling, image processing and attitude estimation of high speed star sensors Doctoral thesis (Texas A&M University, 2006).

12. T. Sun, F. Xing, Z. You, X. Wang, and B. Li, “Smearing model and restoration of star image under conditions of variable angular velocity and long exposure time,” Opt. Express 22(5), 6009–6024 (2014). [CrossRef]

13. G. Zhang, J. Jiang, and J. Yan, “Dynamic imaging model and parameter optimization for a star tracker,” Opt. Express 24(6), 5961–5983 (2016). [CrossRef]

14. Z. Wang, J. Jiang, and G. Zhang, “Global field-of-view imaging model and parameter optimization for high dynamic star tracker,” Opt. Express 26(25), 33314–33332 (2018). [CrossRef]

15. Z. Wang, J. Jiang, and G. Zhang, “Distributed parallel super-block-based star detection and centroid calculation,” IEEE Sens. J. 18(19), 8096–8107 (2018). [CrossRef]

16. M. S. Wei, F. Xing, and Z. You, “A real-time detection and positioning method for small and weak targets using a 1d morphology-based approach in 2d images,” Light: Sci. Appl. 7(5), 18006 (2018). [CrossRef]

17. G. Wang, J. Li, and X. Wei, “Star identification based on hash map,” IEEE Sens. J. 18(4), 1591–1599 (2018). [CrossRef]

18. D. Mortari, M. A. Samaan, C. Bruccoleri, and J. L. Junkins, “The pyramid star identification technique,” Navigation 51(3), 171–183 (2004). [CrossRef]

19. M. Aghaei and H. A. Moghaddam, “Grid star identification improvement using optimization approaches,” IEEE Trans. Aerosp. Electron. Syst. 52(5), 2080–2090 (2016). [CrossRef]

20. M. D. Shuster and S. D. Oh, “Three-axis attitude determination from vector observations,” J Guid. Control. Dynam 4(1), 70–77 (1981). [CrossRef]

21. Q. Fan and X. Zhong, “A triangle voting algorithm based on double feature constraints for star sensors,” Adv. Space Res. 61(4), 1132–1142 (2018). [CrossRef]

22. H. Zhu, B. Liang, and T. Zhang, “A robust and fast star identification algorithm based on an ordered set of points pattern,” Acta Astronaut. 148, 327–336 (2018). [CrossRef]

23. S. Huang, J. Huang, D. Tang, and F. Chen, Research on uav flight performance test method based on dual antenna gps/ins integrated system, in 2018 IEEE 3rd International Conference on Communication and Information Systems (ICCIS), (2018), pp. 106–116.

24. C. C. Liebe, “Pattern recognition of star constellations for spacecraft applications,” IEEE Aerosp. Electron. Syst. Mag. 8(1), 31–39 (1993). [CrossRef]

25. H. G. Walter and O. Sovers, Astrometry of fundamental catalogues: the evolution from optical to radio reference frames (Springer Science & Business Media, 2000).

26. VizieR, “Hipparcos catalogue,” http://vizier.u-strasbg.fr/.

27. “International astronomical union standards of fundamental astronomy,” http://www.iausofa.org/.

28. S. E. Urban and P. Kenneth Seidelmann, Explanatory supplement to the astronomical almanac (University Science Books, 2013).

29. R. C. Stone, “An accurate method for computing atmospheric refraction,” Publ. Astron. Soc. Pac. 108, 1051–1058 (1996). [CrossRef]

30. G. Sitarski, “Warsaw ephemeris of the solar system: De405/waw,” Acta Astronomica -Warsaw and Cracow- 52, 471–486 (2002).

31. X. Moisson and P. Bretagnon, “Analytical planetary solution vsop2000,” Celest. Mech. & Dyn. Astron. 80(3/4), 205–213 (2001). [CrossRef]

32. R. M. Green, Spherical astronomy (Cambridge University Press, 1985).

33. R. Bauer, “Distribution of points on a sphere with application to star catalogs,” J. Guid. Control. Dyn. 23(1), 130–137 (2000). [CrossRef]

34. W. Wang, X. Wei, J. Li, and G. Zhang, “Guide star catalog generation for short-wave infrared (swir) all-time star sensor,” Rev. Sci. Instrum. 89(7), 075003 (2018). [CrossRef]

35. J. Jiang, K. Xiong, W. Yu, J. Yan, and G. Zhang, “Star centroiding error compensation for intensified star sensors,” Opt. Express 24(26), 29830–29842 (2016). [CrossRef]

36. Y. Li, J. Zhang, W. Hu, and J. Tian, “Laboratory calibration of star sensor with installation error using a nonlinear distortion model,” Appl. Phys. B: Lasers Opt. 115(4), 561–570 (2014). [CrossRef]

37. T. Sun, F. Xing, and Z. You, “Optical system error analysis and calibration method of high-accuracy star trackers,” Sensors 13(4), 4598–4623 (2013). [CrossRef]

38. X. Wei, G. Zhang, Q. Fan, J. Jiang, and J. Li, “Star sensor calibration based on integrated modelling with intrinsic and extrinsic parameters,” Measurement 55, 117–125 (2014). [CrossRef]

39. T. S. McKechnie, “Atmospheric turbulence and the resolution limits of large ground-based telescopes,” J. Opt. Soc. Am. A 9(11), 1937–1954 (1992). [CrossRef]

Parameter	Value	Parameter	Value	Parameter	Value
Wavelength $ω_{l}$	$0.65 μ m$	Latitude $L_{atitude}$	$0^{\circ}$	Pressure $P_{a}$	$[900, 1100]$
Temperature $T_{c}$	${[- 20, 50]}^{\circ} C$	Humidity $r_{h}$	$(0, 1)$	Zenith distance $z_{s, max}$	$[0^{\circ}, 45^{\circ}]$
FOV angle $θ_{FOV}$	$20^{\circ}$	Catalog Error $E r r_{star}$	${0.375}^{″}$	Number of Stars	4950

Parameter	Value	Parameter	Value	Parameter	Value
Focal length $f$	31.44 mm	Lens diameter D	40 mm	Pixel length	5.5 $μ$ m
FOV angle $θ_{FOV}$	$20^{\circ}$	Central wavelength	650 nm	Detection sensitivity	6 magnitudes
Active pixels	2048 $\times$ 2048	Lens transmissivity $τ$	0.85	Principle point	(1008.2, 822.9)

Parameter	Value	Parameter	Value	Parameter	Value
Wavelength $ω_{l}$	$0.65 μ m$	Latitude $L_{atitude}$	$0^{\circ}$	Pressure $P_{a}$	$[900, 1100]$
Temperature $T_{c}$	${[- 20, 50]}^{\circ} C$	Humidity $r_{h}$	$(0, 1)$	Zenith distance $z_{s, max}$	$[0^{\circ}, 45^{\circ}]$
FOV angle $θ_{FOV}$	$20^{\circ}$	Catalog Error $E r r_{star}$	${0.375}^{″}$	Number of Stars	4950

Parameter	Value	Parameter	Value	Parameter	Value
Focal length $f$	31.44 mm	Lens diameter D	40 mm	Pixel length	5.5 $μ$ m
FOV angle $θ_{FOV}$	$20^{\circ}$	Central wavelength	650 nm	Detection sensitivity	6 magnitudes
Active pixels	2048 $\times$ 2048	Lens transmissivity $τ$	0.85	Principle point	(1008.2, 822.9)

Observation angular distance error modeling and matching threshold optimization for terrestrial star tracker

Abstract

1. Introduction

2. Principle of terrestrial star tracker and observation angular distance error

2.1 Principle of terrestrial star tracker

2.2 Observation angular distance error

3. Analysis and modeling of the observation angular distance error

3.1 Factors affecting the observation star vectors

3.2 Analysis of observation angular distance error

3.3 Modeling of the observation angular distance error

4. Angular distance matching threshold optimization

4.1 Star catalog angular distance matching threshold

4.2 Transmission angular distance matching threshold

4.3 Optimal angular distance matching threshold

5. Simulations and night sky experiment

5.1 Numerical simulations

5.2 Night sky experiment

6. Conclusion

Funding

References

Cited By

Figures (16)

Tables (2)

Equations (35)

Optics Express