A High-Fidelity Star Map Simulation Method for Airborne All-Time Three-FOV Star Sensor Under Dynamic Conditions

Highlights

What are the main findings?

• A comprehensive star map simulation method for airborne All-Time Three-FOV star sensors is proposed, integrating coordinate transformation, energy transfer, and image degradation models.

What are the implications of the main findings?

• The method offers a reliable technical basis for optimizing the design and assessing the performance of airborne All-Time Three-FOV star sensors under dynamic conditions.
• It enables the validation of star centroid extraction and identification algorithms under controlled disturbance scenarios, reducing dependency on costly and time-consuming real-world stargazing experiments.

Abstract

To address the lack of reliable test data for evaluating star sensor performance in dynamic airborne environments, this paper presents a high-fidelity star map simulation method for all-time three-Field of View (FOV) star sensors. A comprehensive simulation framework integrating stellar radiation, atmospheric transmission, and detector noise models was developed to accurately model star trailing effects under dynamic conditions. First, a stellar position calculation model incorporating atmospheric refraction correction and platform motion parameters was established through coordinate transformations between the Geocentric Celestial Reference System (GCRS) and FOV coordinate system. Next, a complete energy transfer chain was constructed by combining star catalog data, atmospheric radiative properties, and detector noise characteristics. Finally, a quantitative evaluation system was introduced, employing metrics such as signal-to-noise ratio (SNR), total grayscale value (G_total), grayscale concentration index (GCI), and dynamic star displacement (DSD). Field experiments at 2388 m altitude (100.23°E, 26.86°N) demonstrated the average relative error of all evaluation metrics below 9% for static conditions and approximately 8% for dynamic scenarios between simulated and real star maps. The method effectively reproduces stellar radiation, atmospheric noise, and dynamic degradation, providing reliable simulation conditions for airborne star sensor testing and star trailing restoration algorithm development.

1. Introduction

With the continuous development of aerospace technology, navigation is becoming increasingly important in near-Earth space and atmospheric missions, and star sensors have become core components of navigation systems [1,2,3,4]. The star sensor is a high-precision optoelectronic attitude measurement device that utilizes stars as observation targets. By capturing and analyzing stellar distributions in real-time, it provides spacecraft with precise three-axis attitude information relative to an inertial reference frame [5,6,7]. As one of the most accurate attitude sensors for spacecraft, star sensors achieve measurement precision at the sub-arcsecond level, significantly outperforming other attitude determination instruments such as sun sensors [8,9]. Compared with traditional inertial navigation systems, star sensors offer multiple advantages, including (1) superior attitude measurement accuracy, (2) non-cumulative errors, (3) strong resistance to electromagnetic interference, (4) high concealment capability, and (5) independence from external attitude references. These advantages have led to their widespread adoption in extraterrestrial satellite platforms [10,11]. In recent years, with growing demands for high-precision, interference-resistant, and fully autonomous navigation systems among atmospheric platforms-including high-altitude vehicles (e.g., missiles, aircraft), marine mobile platforms (e.g., ships), and stratospheric aerostats (e.g., high-altitude balloons)-star sensor research has gradually expanded from traditional deep-space exploration to atmospheric platform applications [12,13,14,15].

However, compared to star sensors deployed on satellite platforms, all-time star sensors for airborne platforms face significant challenges from sky background radiation [16]. To effectively suppress sky background radiation, enhance detection capability, and increase detectable star counts, their optical systems typically adopt a combined strategy of long focal length with small FOV and multiple FOVs, while preferentially operating in the short-wave infrared (SWIR) band. This design is referred to as an all-time multi-FOV star sensor (typically configured with three FOVs) [17].

The core functionality of all-time three-FOV star sensors relies on stellar optical imaging and identification: First, the optical system images detected stars onto the image sensor located at the focal plane. Subsequently, star centroid extraction algorithms is employed to determine stellar centroid coordinates and match them with a star catalog stored in the system memory. Finally, attitude determination algorithms compute the spacecraft’s attitude relative to the inertial reference frame [18]. Among these processes, the design and optimization of high-precision star centroid extraction algorithms and robust star identification algorithms under strong sky background radiation and dynamic conditions have become critical technical challenges limiting the realization of all-time autonomous navigation capability for star sensors.

To thoroughly investigate the performance and verify the reliability of these algorithms, evaluation using real star maps is essential. However, obtaining star maps captured by star sensors on actual airborne platforms is considerably challenging due to experimental constraints and costs [19]. This scarcity of real airborne star map data creates a critical bottleneck: without sufficient test data, it is impossible to systematically validate the performance of star centroid extraction and identification algorithms under diverse dynamic airborne conditions (e.g., platform angular motion and random vibrations) or optimize the design of all-time three-FOV star sensors. In this context, high-fidelity star map simulation emerges as an indispensable solution. It can generate controllable, repeatable, and low-cost simulated star maps that replicate the key characteristics of real airborne scenarios (including stellar radiation, atmospheric interference, detector noise, and dynamic star trailing). These simulated star maps not only fill the gap regarding the scarcity of real test data but also enable quantitative evaluation of algorithm robustness and sensor performance under extreme or variable conditions. Thus, reliable star map simulation methods are of significant importance: they serve as a foundational prerequisite for advancing the development and practical application of airborne all-time three-FOV star sensors and provide effective validation means for the development of star spot extraction and star map recognition algorithms.

Zhao et al. proposed a star map simulation method for airborne star sensors, detailing the calculation of stellar projection positions on the detector focal plane [20]. Building upon this, Wu et al. further incorporated atmospheric refraction effects and derived formulas for calculating star position offsets [19]. Li et al. employed Moderate-Resolution Atmospheric Transmission (MODTRAN) software to construct a sky background radiation model that comprehensively considers parameters including viewing zenith angle (VZA), solar zenith angle (SZA), solar azimuth angle (SAA), observation band, and altitude, enabling daytime star map simulation for airborne star trackers across different bands [21]. Wang et al. developed atmospheric interference and detector noise models for the star map simulation process. The aforementioned studies focus on static stellar imaging models, but the influence of platform motion on such imaging has not been taken into account [22]. Zheng et al. established a stellar radiation transfer model based on blackbody radiation laws while considering the impact of airborne platform motion on star imaging. However, this study has several limitations: First, the relative spectral distribution of stars based on blackbody radiation deviates from actual spectra; Second, simplifying complex vibrational motion into one-dimensional rapid simple harmonic motion results in significant discrepancies from actual star map motion blur characteristics, failing to accurately represent motion properties and complex blur effects in real physical scenarios [23].

Currently, existing star map simulation methods primarily focus on coordinate transformation, stellar radiation transfer modeling, and sky background radiation transfer modeling in the simulation workflow. Relatively few studies have addressed the construction and validation of dynamic star imaging models for daytime airborne conditions.

Since the flight speed of the platform considered in this study is below 4 Ma, the influence of aero-optical effects on the star sensor is negligible [12], so such effects are not included in the current model scope. Building upon existing research, this study establishes a systematic star map simulation model for all-time three-FOV star sensors and generates corresponding simulated star maps. Specifically, the work conducts in-depth investigations into the dynamic star imaging mechanism under airborne conditions, developing corresponding mathematical models and constructing a multi-dimensional evaluation metric system to systematically analyze the impact patterns of platform motion on star imaging quality. Finally, through comparative analysis between real star maps and simulation results, the effectiveness of the proposed simulation method is validated. This research lays a solid theoretical foundation for subsequent studies on high-precision star extraction algorithms and robust star map identification algorithms under daytime dynamic conditions.

2. The Fundamental Workflow of Star Map Simulation

The system architecture of the all-time three-FOV star sensor employed in this study is illustrated in Figure 1. The system adopts a tri-optical configuration, featuring three independent optical systems with identical structural parameters arranged in a 120° symmetrical distribution azimuthally. Each optical axis has an elevation angle of 45°, with an inter-axis angle of 75.5° between adjacent optical axes. Figure 2 presents the star map simulation workflow for this star sensor: First, based on the geographic coordinates (longitude, latitude), attitude, and local time of the observation point, the right ascension (RA) and declination (Dec) parameters of the three optical axes in the GCRS are calculated. Second, according to the FOV parameters and optical axis orientation, observable star data within each FOV are retrieved and extracted from the star catalog. Subsequently, using a perspective projection transformation algorithm, the ideal projection coordinates of observable stars on the corresponding image plane are computed, with precise corrections applied to these coordinates through the introduction of optical system distortion and atmospheric refraction models. Finally, by comprehensively considering platform motion parameters, detector noise characteristics, stellar magnitudes, and sky background radiation intensity, the required simulated star maps are generated through the establishment of a grayscale calculation model.

2.1. Coordinate Transformation

2.1.1. Coordinate System Definition

In star simulation, the precise description of celestial body positions and airborne platform attitude relies on multiple coordinate systems and their transformation relationships. The primary coordinate systems used in this study are defined as follows:

I. Geocentric Celestial Reference System (GCRS, Ox_iy_iz_i): With its origin at the Earth’s center of mass, the Ox_i axis points toward the vernal equinox, the Oz_i axis aligns with the Earth’s rotation axis, and the Oy_i axis completes the right-handed orthogonal system. The GCRS is commonly used to describe celestial body positions in inertial space.

II. Earth-Centered, Earth-Fixed Coordinate System (ECEF, Ox_ey_ez_e): Also centered at the Earth’s mass center, the Ox_e axis points toward the intersection of the prime meridian and the equator, the Oz_e axis aligns with the geographic North Pole, and the Oy_e axis is determined by the right-hand rule. This system rotates synchronously with the Earth.

III. East-North-Up Coordinate System (ENU, Ox_gy_gz_g): With its origin at the platform’s center of mass, the Ox_g, Oy_g, and Oz_g axes point toward geographic east, north, and the local zenith, respectively. This system is typically used to characterize the airborne platform’s attitude relative to the Earth.

IV. Star Sensor Coordinate System (Ox_sy_sz_s): Serving as the measurement reference frame, this coordinate system maintains a fixed rotational transformation matrix $C_b^s$ relative to the body coordinate system (Ox_by_bz_b).

V. FOV Coordinate System (Ox_fy_fz_f): The origin of each FOV coordinate system lies at the optical axis center, with the Oz_f axis aligned along the optical axis. The Ox_f and Oy_f axes are parallel to the detector edges. For FOV1 Coordinate System, the Oz_f1 axis projects along the Ox_s axis in the star sensor coordinate system.

VI. Image Coordinate System (Ox_py_p): Taking the image plane center point as coordinate origin, the Ox_p and Oy_p axes are parallel to Ox_f and Oy_f, respectively, representing the two-dimensional star position on the image plane.

2.1.2. Extraction of Observable Stars

When calculating the projection positions and grayscale values of observable stars on the detector focal plane, it is essential to obtain key parameters such as stellar magnitudes, RA, and Dec, all of which are sourced from the star catalog. Considering that the observation bands of The Two Micron All-Sky Survey (2MASS) cover the J-band (1.235 µm, bandwidth: 0.162 µm), H-band (1.662 µm, bandwidth: 0.251 µm), and Ks-band (2.159 µm, bandwidth: 0.262 µm), which significantly overlap with the SWIR band adopted in this study, we selected 2MASS as the data source for the star catalog. The 2MASS catalog contains an extensive collection of stellar data. Therefore, preprocessing of the 2MASS catalog data is necessary to ensure the reliability of subsequent analysis results. The specific screening criteria are determined based on the magnitudes of the J, H, and K infrared bands provided by the catalog, combined with the magnitude measurement uncertainty flag (Rd_flg) and source confidence flag (Cc_flg): only stellar samples with an H-band magnitude less than or equal to 6; an Rd_flg value of “1”, “2”, or “3” (this range of values corresponds to the highest observation quality grade in the catalog); and a Cc_flg value of 0 (this value indicates that the target source is minimally affected by image artifacts or contamination from nearby sources of equal or higher luminosity) are retained.

To accurately identify observable stars within the three FOVs, a filtering process is performed based on the optical axis orientation, FOV parameters, and the RA/Dec coordinates of stars. Assuming the directional vector of the optical axis in the FOV coordinate system is denoted as Z_f, the corresponding RA and Dec in the GCRS can be calculated using Equation (1).

$$Z_i=C_f^i\cdot Z_f=\left[\begin{array}{c}\cos ({\alpha }_{\mathrm{i}})\cdot \cos ({\delta }_i)\\ \sin ({\alpha }_i)\cdot \cos ({\delta }_i)\\ \sin ({\delta }_i)\end{array}\right]$$

(1)

where α_i and δ_i represent the RA and Dec of the optical axis in the GCRS, respectively, while $C_{\mathrm{f}}^{\mathrm{i}}$ denotes the coordinate transformation matrix from the FOV coordinate system to GCRS. The detailed derivation of this transformation matrix is presented in Section 2.1.3 of this paper, and Z_i corresponds to the directional vector of the optical axis in GCRS.

Given that the optical axes of the three FOVs are all located at the geometric center of their respective FOVs, Equation (2) can be subsequently utilized to determine the coverage range of RA and Dec in the GCRS for the circular FOV with a half-angle R centered on each optical axis. More specifically, any stellar object whose directional vector Zₛ in GCRS satisfies the condition defined by Equation (2), and whose instrument magnitude is below the extreme detection magnitude, will be classified as an observable star.

$$\left\{\begin{array}{l}\arccos \left({\displaystyle \frac{Z_s\cdot Z_i}{{‖Z_s‖}_2\cdot {‖Z_i‖}_2}}\right)<R\\ R={\displaystyle \frac{FOV}{2}}\end{array}\right.$$

(2)

2.1.3. Transformation from Celestial to Image Coordinate Systems

In this study, the conversion from GCRS to ECEF is achieved using a vernal equinox-based transformation method [24], implemented via the rotation matrix $C_i^e$ (Equation (3)).

$$C_i^e=W(t)\cdot R(\mathrm{GAST})\cdot N(t)\cdot P(t)$$

(3)

where P(t) represents the precession matrix, N(t) denotes the nutation matrix, and R(GAST) is the rotation matrix transforming coordinates from the instantaneous true celestial reference system to the instantaneous true terrestrial reference system, computed based on the Greenwich Apparent Sidereal Time (GAST) referenced to the true equinox. Additionally, W(t) corresponds to the polar motion matrix, where t denotes the number of Julian centuries elapsed.

Based on the geographic longitude and latitude of the airborne platform, the rotation matrix from the ECEF coordinate system to the ENU coordinate system, denoted as C, can be derived, as shown in Equation (4).

$$C_e^g=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos ({\displaystyle \frac{\pi }{2}}-{\phi }_g)& \sin ({\displaystyle \frac{\pi }{2}}-{\phi }_g)\\ 0& -\sin ({\displaystyle \frac{\pi }{2}}-{\phi }_g)& \cos ({\displaystyle \frac{\pi }{2}}-{\phi }_g)\end{array}\right]\cdot \left[\begin{array}{ccc}\cos ({\displaystyle \frac{\pi }{2}}+{\lambda }_g)& \sin ({\displaystyle \frac{\pi }{2}}+{\lambda }_g)& 0\\ -\sin ({\displaystyle \frac{\pi }{2}}+{\lambda }_g)& \cos ({\displaystyle \frac{\pi }{2}}+{\lambda }_g)& 0\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}-\sin {\lambda }_g& \cos {\lambda }_g& 0\\ -\cos {\lambda }_g\sin {\phi }_g& -\sin {\lambda }_g\sin {\phi }_g& \cos {\phi }_g\\ \cos {\lambda }_g\cos {\phi }_g& \sin {\lambda }_g\cos {\phi }_g& \sin {\phi }_g\end{array}\right]$$

(4)

where λ_g and φ_g represent the geographic longitude and latitude of the airborne platform, respectively.

Using the airborne platform’s attitude, the rotation matrix from the ENU coordinate system to the body coordinate system can be derived, as expressed mathematically in Equation (5).

$$C_g^b=\left[\begin{array}{ccc}\cos ({\beta }_b)\cos ({\gamma }_b)& \cos ({\beta }_b)\sin ({\gamma }_b)& -\sin ({\beta }_b)\\ \cos ({\gamma }_b)\sin ({\alpha }_b)\sin ({\beta }_b)-\cos ({\alpha }_b)\sin ({\gamma }_b)& \cos ({\alpha }_b)\cos ({\gamma }_b)+\sin ({\alpha }_b)\sin ({\beta }_b)\sin ({\gamma }_b)& \cos ({\beta }_b)\sin ({\alpha }_b)\\ \sin ({\alpha }_b)\sin ({\gamma }_b)+\cos ({\alpha }_b)\cos ({\gamma }_b)\sin ({\beta }_b)& \cos ({\alpha }_b)\sin ({\beta }_b)\sin ({\gamma }_b)-\cos ({\gamma }_b)\sin ({\alpha }_b)& \cos ({\alpha }_b)\cos ({\beta }_b)\end{array}\right]$$

(5)

where α_b, β_b, and γ_b denote the attitude angles of the airborne platform.

Since the star sensor is equipped with three independent FOVs, the rotation matrices from the star sensor coordinate system to each individual FOV coordinate system must be derived separately. Let $C_s^{f_i}$ (i = 1,2,3) denote the rotation matrices from the star sensor coordinate system to the three FOV coordinate systems, which can be uniformly expressed as:

$$C_{\mathrm{s}}^{f_i}=\left[\begin{array}{ccc}\cos ({\displaystyle \frac{\pi }{4}})& 0& -\sin ({\displaystyle \frac{\pi }{4}})\\ 0& 1& 0\\ \sin ({\displaystyle \frac{\pi }{4}})& 0& \cos ({\displaystyle \frac{\pi }{4}})\end{array}\right]\cdot \left[\begin{array}{ccc}\cos ((i-1){\displaystyle \frac{2\pi }{3}})& \sin ((i-1){\displaystyle \frac{2\pi }{3}})& 0\\ -\sin ((i-1){\displaystyle \frac{2\pi }{3}})& \cos ((i-1){\displaystyle \frac{2\pi }{3}})& 0\\ 0& 0& 1\end{array}\right]$$

(6)

In the subsequent processes of star map identification and attitude determination, the system processor must accurately compute the positions of stars in the image coordinate system. To achieve this, the three-dimensional star vectors in the FOV coordinate system must be projected onto the two-dimensional image plane via perspective projection. FOV1 is used here as an example for illustration. The three-dimensional star vector S₁ in the FOV1 coordinate system can be calculated using Equation (7).

$$S_1=C_i^{f_1}\cdot \left[\begin{array}{c}\cos ({\alpha }_1)\cdot \cos ({\delta }_1)\\ \sin ({\alpha }_1)\cdot \cos ({\delta }_1)\\ \sin ({\delta }_1)\end{array}\right]=\left[\begin{array}{c}{X}_1\\ Y_1\\ Z_1\end{array}\right]$$

(7)

where $C_i^{f_1}$ denotes the rotation matrix from the GCRS to the FOV1 coordinate system, and X₁, Y₁, and Z₁ represent the components of the vector S₁ along the three coordinate axes of the FOV1 coordinate system, respectively.

The spatial geometric relationship between the FOV coordinate system and the image coordinate system is illustrated in Figure 3. Based on the principles of perspective projection, the coordinate position of the star in the image coordinate system can be derived. The corresponding transformation relationship is given in Equation (8).

$$\left\{\begin{array}{l}{x}_1=-{\displaystyle \frac{X_1}{Z_1}}\cdot {\displaystyle \frac{f}{d_{pixel}}}\\ {\mathrm{y}}_1=-{\displaystyle \frac{Y_1}{Z_1}}\cdot {\displaystyle \frac{f}{d_{pixel}}}\end{array}\right.$$

(8)

where x₁ and y₁ represent the horizontal and vertical coordinates of the star in the image coordinate system, respectively, f denotes the focal length of the optical system, and d_pixel is defined as the pixel size of the detector.

In addition, to improve imaging accuracy, the effect of optical system distortion on the star’s imaged position must be taken into account. In this study, a first-order radial distortion model is introduced for correction. The corresponding calculation is shown in Equation (9).

$$\left\{\begin{array}{l}{x}_1^{\prime }=x_1+x_1\cdot g_1{r}^2\\ y_1^{\prime }=y_1+y_1\cdot g_1{r}^2\\ r=\sqrt{{x_1}^2+{y_1}^2}\end{array}\right.$$

(9)

where $x_1^{’}$ and $y_1^{’}$ represent the star coordinates corrected by the first-order radial distortion model, g₁ denotes the radial distortion coefficient, and r is the Euclidean distance from the point (x₁, y₁) to the principal point O_p.

During the generation of the simulated star map, the computer screen uses an image coordinate system with its origin located at the top-left corner of the image. Therefore, the star coordinates must be transformed into this coordinate system using the transformation relationship presented in Equation (10).

$$\left\{\begin{array}{l}{x}_2={\displaystyle \frac{N_x}{2}}+x_1^{\prime }\\ {\mathrm{y}}_2={\displaystyle \frac{N_y}{2}}-y_1^{\prime }\end{array}\right.$$

(10)

where x₂ and y₂ represent the horizontal and vertical coordinates of the star in the image coordinate system with the origin located at the top-left corner of the image, while N_x and N_y denote the total number of pixels in each row and column of the detector, respectively.

After obtaining the coordinates of the stars, it is necessary to further verify the accuracy of the results. According to the principle of invariant stellar angular distance, the angular distance between stars in the GCRS should be consistent with that in the FOV coordinate system. Building on this, suppose there are two observable stars in the FOV1 coordinate system, whose right ascension and declination are (α_x, δ_x) and (α_y, δ_y), respectively. Their corresponding coordinates in the image coordinate system are (m_x, n_x) and (m_y, n_y), respectively. The direction vectors of these two stars in the GCRS can be expressed as Z_x = [cosα_xcosδ_x, sinα_xcosδ_x, sinδ_x]^T and Z_y = [cosα_ycosδ_y, sinα_ycosδ_y, sinδ_y]^T, while their direction vectors in the FOV1 coordinate system are S_x = [m_x, n_x, -f]^T and S_y = [m_y, n_y, -f]^T, respectively. Based on these direction vectors, the angular distances θ_αδ in the GCRS and θ_mn in the FOV 1 coordinate system can be calculated using Equation (2).

If the condition |θ_αδ-θ_mn| ≤ ε is satisfied, where ε denotes a predefined threshold for coordinate transformation accuracy, the coordinate transformation can be considered sufficiently accurate. This criterion provides a quantitative basis for verifying the correctness of the star positions and ensures the reliability of the coordinate transformation results.

2.1.4. Impact of Atmospheric Refraction on Stellar Imaging

Unlike star sensors deployed on satellite platforms, those designed for airborne platforms operating within the atmosphere must additionally consider the effects of atmospheric refraction on the imaging positions of stars. Atmospheric refraction reduces the VZA of the stars. Its magnitude primarily depends on the VZA, atmospheric pressure, and temperature, as expressed in Equation (11) [22,25].

$$\left\{\begin{array}{l}R=\left[{\displaystyle \frac{-0.000383T}{1+0.00367T}}+{\displaystyle \frac{H^{\prime }(1-0.00264\cos 2\phi -0.000163(T^{\prime }-T))}{760}}\right]\cdot R_0\cdot n_g\\ R_0=-0.006734+60.13008\cdot \tan (VZA)-0.025433\cdot {\tan }^2(VZA)-0.058699\cdot {\tan }^3(VZA)\\ n_g=0.965+{\displaystyle \frac{0.0164}{{\lambda }_m^2}}+{\displaystyle \frac{0.00028}{{\lambda }_m^4}}\end{array}\right.$$

(11)

where R denotes the atmospheric refraction, and R₀ represents the reference atmospheric refraction, both expressed in arcseconds (″). R₀ is defined as the atmospheric refraction under standard conditions at sea level with a temperature of 0 °C, pressure of 760 mmHg, and latitude of 45°, and its specific value is closely related to the VZA. The approximate calculation formula for R₀ is obtained by fitting data from the atmospheric refraction tables provided in the Chinese Astronomical Almanac, and it is valid for VZAs ranging from 0° to 76°. T is the ambient air temperature during observation, and T′ is the temperature of mercury in the barometer, both measured in degrees Celsius (°C). H′ denotes the atmospheric pressure during observation, measured in millimeters of mercury (mmHg). Additionally, n_g is the atmospheric refraction correction factor, which depends on the detector’s response band; λ_m is the central wavelength of the detector’s response band, expressed in micrometers (µm).

Based on the atmospheric refraction R, the stellar coordinates can be corrected. As illustrated in Figure 4, atmospheric refraction induces three typical imaging scenarios. Assuming the angle between the starlight and the optical axis before refraction is θ_r, its specific value can be calculated using Equation (12) according to the geometric relationships depicted in Figure 4.

$${\theta }_r=\arctan \left({\displaystyle \frac{\sqrt{x_r^2+y_r^2}\cdot d_{pixel}}{f}}\right)$$

(12)

where x_r and y_r represent the projected position of the star.

Atmospheric refraction alters the angle between incident starlight and the optical axis, resulting in three typical imaging scenarios, as illustrated in Figure 4. In Case 1, when the star’s altitude angle is greater than that of the optical axis, refraction increases the angle between starlight and the optical axis to θ_r + R; in Case 2, when the star’s altitude angle is below that of the optical axis and θ_r > R, refraction decreases this angle to θ_r-R; in Case 3, when the star’s altitude angle is below that of the optical axis and θ_r < R, refraction causes the angle between starlight and the optical axis to become R-θ_r. Based on the law of similar triangles, the projected positions of the refracted starlight in the image coordinate system under these three cases, denoted as (x_r1, y_r1), (x_r2, y_r2), and (x_r3, y_r3), can be derived. The corresponding formulas are provided in Equations (13)–(15).

$$\left\{\begin{array}{l}{x}_{r1}=x_r\cdot {\displaystyle \frac{f\cdot \tan ({\theta }_r+R)}{\sqrt{x_r^2+y_r^2}\cdot d_{pixel}}}\\ y_{r1}=y_r\cdot {\displaystyle \frac{f\cdot \tan ({\theta }_r+R)}{\sqrt{x_r^2+y_r^2}\cdot d_{pixel}}}\end{array}\right.$$

(13)

$$\left\{\begin{array}{l}{x}_{r2}=x_r\cdot {\displaystyle \frac{f\cdot \tan ({\theta }_r-R)}{\sqrt{x_r^2+y_r^2}\cdot d_{pixel}}}\\ y_{r2}=y_r\cdot {\displaystyle \frac{f\cdot \tan ({\theta }_r-R)}{\sqrt{x_r^2+y_r^2}\cdot d_{pixel}}}\end{array}\right.$$

(14)

$$\left\{\begin{array}{l}{x}_{r3}=-x_r\cdot {\displaystyle \frac{f\cdot \tan (R-{\theta }_r)}{\sqrt{x_r^2+y_r^2}\cdot d_{pixel}}}\\ y_{r3}=-y_r\cdot {\displaystyle \frac{f\cdot \tan (R-{\theta }_r)}{\sqrt{x_r^2+y_r^2}\cdot d_{pixel}}}\end{array}\right.$$

(15)

Through the aforementioned procedures, the imaging position of the star can be determined.

2.2. Star Imaging Model Under Dynamic Conditions

The above study focused on the star imaging model of the all-time three-FOV star sensor under static conditions. However, in practical applications, angular motion and angular vibration of the airborne platform induce trailing and distortion of star spots during exposure. Therefore, it is necessary to analyze the platform’s motion characteristics and develop a corresponding dynamic star imaging model.

2.2.1. Rotation of the Airborne Platform About an Arbitrary Axis

The angular motion of an airborne platform about an arbitrary axis can be decomposed into rotational components about the non-optical axes (X_f1, Y_f1) and the optical axis (Z_f1) (taking one FOV as an example, see Figure 5). To establish a mathematical model of stellar position offset, the platform’s angular velocity vector is defined as ω = [ω_x, ω_y, ω_z]^T. Given the initial stellar coordinates P(x,y) and the angle θ₀ between the starlight and the optical axis, the stellar image position during exposure time t_e is affected by composite rotation [26]:

• Non-optical axes component contribution: The angular velocities ω_x and ω_y induce linear displacement of the star spot, as illustrated in Figure 5a. At time t, the displacement components along the image coordinate system axes X_p and Y_p, denoted L_x1(t) and L_y1(t) respectively, are given by:

$$L_{\mathrm{i}}(t)={\displaystyle \frac{f\cdot \left(1+ta{n}^2{\theta }_0\right)\cdot {\omega }_j\cdot t}{d_{pixel}}},i\in \{x_1,y_1\},j\in \{y,x\}$$

(16)

where t ∈ [0, t_e]. Under the initial conditions, when θ₀ is sufficiently small, the approximation tan(θ₀) ≈ 0 can be adopted, allowing the nonlinear terms to be neglected and the model to be simplified as a linear offset model.

2.

Optical-axis component contribution: The angular velocity ω_z induces circumferential motion of the star spot about point O_p. At time t, the corresponding arc length L_z(t) is given by:

$$L_z(t)=r_t\cdot {\omega }_z\cdot t=\sqrt{x^2+y^2}\cdot {\omega }_z\cdot t$$

(17)

Here, r_t represents the radial distance between point P(x,y) and point O_p. Notably, star spots farther from the optical axis exhibit longer blur trajectories, resulting in more pronounced blurring effects in the peripheral regions of the star map. Let α_start denote the initial angle between the line connecting P(x,y) to O_p and the X_p axis. At time t, the star spot moves to position P₂(x,y) (Figure 5b), with the corresponding angle relative to the X_p axis being α(t). Based on geometric analysis, the displacement components L_x2(t) and L_y2(t) at this moment can be derived as:

$$\left\{\begin{array}{l}{L}_{x2}(t)=r_t(\cos \alpha (t)-\cos {\alpha }_{start})\\ L_{y2}(t)=r_t(\sin \alpha (t)-\sin {\alpha }_{start})\end{array}\right.$$

(18)

Therefore, by combining the displacement components L_x1(t), L_y1(t), L_x2(t), and L_y2(t), the stellar position offsets L_xr(t) and L_yr(t) induced by the platform’s angular motion can be expressed as:

$$\left\{\begin{array}{l}{L}_{x\mathrm{r}}(t)=L_{x1}(t)+L_{x2}(t)\\ L_{yr}(t)=L_{y2}(t)-L_{y1}(t)\end{array}\right.$$

(19)

2.2.2. Angular Vibration of the Airborne Platform About an Arbitrary Axis

The imaging quality of the all-time three-FOV star sensor is susceptible to disturbances caused by random angular vibrations of the airborne platform about arbitrary axes. Since starlight can be approximated as a parallel beam due to its long-distance propagation, the effect of vibration on the star spot imaging can be decomposed into angular motions about the platform’s three axes, while the influence of translational components on the star spot position is negligible, as illustrated in Figure 6. During the exposure time t_e, angular vibrations lead to dispersion in the energy distribution of the star spot on the imaging plane, accompanied by random shifts in the centroid position. The cumulative effect ultimately results in dynamic blurring phenomena.

To accurately model the blurring effects induced by random vibrations, it is essential to analyze the statistical characteristics of the angular vibration signal from the frequency-domain perspective. Assuming that the single-axis angular vibration signal X(t) is a stationary random process that satisfies the Dirichlet conditions, the time-domain signal can be decomposed into a linear superposition of discrete frequency components [27]:

$$X(t)={\displaystyle \sum _{n=1}^{\infty }{A}_n\cos ({\omega }_{n}t-{\theta }_n)}$$

(20)

where A_n denotes the amplitude of the nth frequency component, and θ_n is the corresponding random initial phase, which follows a uniform distribution over the interval [0, 2π].

To quantify the vibration energy, the mean square value of the signal E[X²(t)] is introduced, which is related to the angular vibration power spectral density function S(ν) as follows:

$$E[X^2(t)]={\displaystyle \frac{1}{2}}{\displaystyle \sum _{n=1}^{\infty }{A}_n^2}={\displaystyle {\int }_{-\infty }^{+\infty }S(\nu )d\nu }$$

(21)

With reference to the angular vibration power spectral data developed for the US Altair UAV and ESA’s OLYMPUS satellite, we employ Equation (22) to generate the corresponding angular vibration power spectral density data [27]. The angular vibration power spectral data from the Altair UAV was selected as a reference because it is representative of high-altitude, high-endurance unmanned aerial platforms. The Altair UAV operates at altitudes of 15.8–18.3 km, which is analogous to the operational environment considered for our platform. On this basis, the discrete formulation for the vibration amplitude A_n can be derived as follows:

$$S(\nu )={\displaystyle \frac{A}{1+{\nu }^2}}$$

(22)

$$A_n=\sqrt{2{\displaystyle {\int }_{{\nu }_{n-1}}^{{\nu }_n}{S}_n(\nu )d\nu }}$$

(23)

where A denotes the angular vibration amplitude coefficient, ν_n and ν_n−1 represent the upper and lower bound frequencies, respectively, of the nth sampling band in the vibration power spectrum, and ν is the central frequency of that band.

Considering that vibration energy is predominantly concentrated in the low-frequency band, a logarithmically uniform sampling strategy is adopted in this study to more accurately capture low-frequency-dominated vibration characteristics. By simultaneously solving Equations (16), (18)–(20), and (22)–(23), the random positional deviations L_xa(t) and L_xb(t) of the star spot induced by airborne platform angular vibrations at time t can be obtained. Further combining with the stellar position offsets L_xr(t) and L_yr(t) generated by platform angular motion, the complete expressions for the total stellar position offsets L_x(t) and L_y(t) at time t are derived as:

$$\begin{array}{l}{L}_{\mathrm{x}}(t)=L_{xr}(t)+L_{xa}(t)\\ L_y(t)=L_{yr}(t)+L_{ya}(t)\end{array}$$

(24)

2.3. Energy Modeling of Star Map

The grayscale values of star maps primarily consist of three key components: (1) the radiant energy from target stars, (2) sky background radiation, and (3) detector noise, as illustrated in Figure 7. To accurately characterize the energy distribution properties of star maps, each component requires individual modeling analysis. Specifically, stellar radiation modeling captures the brightness information of stars, sky background modeling characterizes interference from atmospheric or spatial backgrounds, while detector noise modeling quantifies random errors introduced by the sensor. By integrating these three modeling components, a comprehensive energy model of star maps can be established.

2.3.1. Stellar Radiation Model

Since the detector operates based on photon counting, it is essential to accurately calculate the number of photons radiated by the star. The photon count is closely related to the star’s instrument magnitude, color temperature, and absolute spectral distribution. To precisely determine the photon count, the first step is to establish the star’s relative spectral distribution. In traditional methods, the relative spectral distribution is typically derived from the star’s color temperature. By cross-validating the 2MASS and High-Precision Parallax-Collecting Satellite (HIPPARCOS) catalogs, the spectral type of the star can be determined, from which the color temperature is inferred. Based on the color temperature data, the relative spectral distribution of the star can be approximated using Planck’s law of blackbody radiation, expressed mathematically as follows:

$$I(\lambda ,T)={\displaystyle \frac{2\pi h{c}^2}{{\lambda }^5(e^{hc/\lambda k_{B}T}-1)}}$$

(25)

where h represents Planck’s constant, with a value of 6.626 × 10⁻³⁴ Js; c denotes the speed of light, given as 2.997 × 10⁸ m/s; λ is the wavelength, expressed in micrometers (μm); k_B stands for the Boltzmann constant, valued at 1.38 × 10⁻²³ J/K; and T corresponds to the color temperature, measured in Kelvin (K).

However, the relative spectral distribution of stars approximated using Planck’s law of blackbody radiation may exhibit certain deviations from the actual spectral distribution. To improve the accuracy of the spectral distribution, this study employs stellar spectral data from the INGS and Pickles stellar spectral catalogs to approximate the relative spectral distribution of stars. The INGS stellar spectral catalog integrates 143 stellar spectra from three catalogs (IUE, NGSL, and SpeX/IRTF), while the Pickles stellar spectral catalog contains 131 stellar spectra. By merging these two catalogs, we established a comprehensive spectral catalog comprising 161 stellar spectra [28,29]. As illustrated in Figure 8, the relative spectral distribution curves of two representative stellar types (e.g., K-type and B-type stars) are presented. These databases provide extensive spectral data, enabling a more accurate representation of the true spectral characteristics of stars, thereby offering reliable data support for the precise calculation of stellar photon counts.

Since the absolute spectral distribution of a star is closely related to its magnitude, and the detector’s response band is the SWIR band, this study calibrates the relative spectral distribution using the H-band magnitude to derive the absolute spectral distribution. Based on the absolute spectral distribution, the photon count S_m incident on the detector can be further calculated, as expressed in Equation (26) [30].

$$S_m={\displaystyle {\int }_{{\lambda }_1}^{{\lambda }_2}f(\lambda )}\cdot {\tau }_0(H,\mu ,\lambda )\cdot {\displaystyle \frac{\pi D^2}{4}}\cdot T_0(\lambda )\cdot t_e\cdot {\displaystyle \frac{\lambda }{hc}}\cdot Q_E(\lambda )\cdot d\lambda$$

(26)

where f(λ) represents the absolute spectral distribution of the target star in units of W·m⁻²·μm⁻¹, τ₀(H,μ,λ) denotes the atmospheric transmittance at the observation altitude H, zenith angle μ, and wavelength λ, D is the aperture diameter of the optical system, T₀(λ) stands for the transmittance of the optical system, and Q_E(λ) indicates the quantum efficiency of the detector.

For stars with unknown spectral types or those lacking spectral data in the Pickles and INGS stellar spectral catalogs, this study employs the method from Reference [17] to calculate photon counts. The specific procedure consists of the following steps: First, instrument magnitudes (M_ins) are determined for stars with known spectral distribution, adopting Vega (A0V spectral type) as the zero-point reference (M_ins = 0), as formulated in Equations (27) and (28).

$$M_{ins}=-2.5\lg {\displaystyle \underset{{\lambda }_1}{\overset{{\lambda }_2}{\int }}\frac{f(\lambda )}{hc/\lambda }{T}_0(\lambda )Q_E(\lambda )\lambda \mathrm{d}\lambda }-M_{ref}$$

(27)

$$M_{ref}=-2.5\lg {\displaystyle \underset{{\lambda }_1}{\overset{{\lambda }_2}{\int }}\frac{f_{ref}(\lambda )}{hc/\lambda }{T}_0(\lambda )Q_E(\lambda )\lambda \mathrm{d}\lambda }$$

(28)

where M_ref denotes the instrument magnitude of Vega, f_ref(λ) represents the absolute spectral distribution of Vega.

Subsequently, based on the target star’s J-band magnitude (M_J) and H-band magnitude (M_H), combined with the instrument magnitude calculation method proposed in Reference [17], the instrument magnitude is derived using the target star’s regression coefficients (β₀ and β₁). The specific calculation formula is given below:

$$M_{ins}-M_H={\beta }_0+{\beta }_1(M_J-M_H)$$

(29)

Finally, the photon count of the target star on the detector focal plane can be derived from its instrument magnitude and the photon count S_m0 of Vega received by the detector focal plane, as given in Equation (30).

$$S_{\mathrm{m}}=S_{m0}\cdot {2.512}^{-M_{ins}}$$

(30)

2.3.2. Sky Background Radiation Model

When observing stars during daytime, the all-time three-FOV star sensor is susceptible to interference from sky background radiation. Therefore, to accurately assess the impact of sky background radiation on stellar observations, it is necessary to calculate the sky background radiation and convert it into corresponding photon counts. In this study, MODTRAN software was employed for simulation—a widely used tool for atmospheric radiative transfer modeling. By inputting atmospheric parameters (e.g., aerosol distribution, visibility) and observational conditions (e.g., VZA, SZA, SAA, and observation altitude), MODTRAN generates corresponding atmospheric transmittance and sky background radiance data. The simulation results are presented in Figure 9. It is important to note that at a given location, atmospheric transmittance is highly dependent on the VZA. A smaller VZA results in a shorter atmospheric path length and consequently higher atmospheric transmittance. Conversely, the intensity of the sky background radiance is primarily governed by the angular distance between the optical axis and the direction of the sun. For nighttime stellar observation, the MODTRAN configuration for lunar-induced sky background radiation follows a similar logic to that of solar-induced daytime radiation: by inputting the VZA, lunar zenith angle (LZA), lunar azimuth angle (LAA), and lunar phase angle (LPA), we can determine the intensity of lunar-related sky background radiation.

With the sky background radiation data provided by the MODTRAN, the number of photons received by each detector pixel from the sky background can be calculated as follows:

$$S_{bgk}={\displaystyle {\int }_{{\lambda }_1}^{{\lambda }_2}{f}_b(\lambda )}\cdot {\tau }_0(H,\mu ,\lambda )\cdot {\displaystyle \frac{\pi D^2}{4}}\cdot T_0(\lambda )\cdot t_e\cdot {\displaystyle \frac{\lambda }{hc}}\cdot Q_E(\lambda )\cdot {\displaystyle \frac{\sigma }{f^2}}\cdot d\lambda$$

(31)

where f_b(λ) represents the spectral distribution of sky background radiance, λ₁ and λ₂ denote the lower and upper bounds of the detector’s response band respectively, and σ corresponds to the area of an individual pixel.

2.3.3. Detector Noise Model

The star maps acquired by the all-time three-FOV star sensor typically contain multiple noise sources, primarily including stellar photon shot noise, sky background radiation shot noise, dark current shot noise, photo-response non-uniformity (PRNU) noise, dark current non-uniformity (DCNU) noise, and readout noise. Under conditions of weak starlight and sky background radiation, the dominant noise source in the detector is readout noise, which generally remains constant. As the intensity of starlight and sky background radiation increases, the primary noise source transitions to shot noise, whose amplitude is proportional to the square root of the signal intensity. When the signal further intensifies, PRNU and DCNU noise become the predominant noise sources, manifesting as pixel response non-uniformity.

Following the theoretical analysis presented above, a systematic noise model can be established for simulated star maps of the all-time three-FOV star sensor. Let I_n denote the noise matrix. The influence of readout noise, stellar photon shot noise, sky background radiation shot noise, and dark current shot noise on the star maps is first considered, which can be calculated as follows:

$$I_n=\sqrt{I_s+I_b+I_d\cdot t_e+{\sigma }_{read}^2}$$

(32)

where I_s denotes the matrix containing only photon counts from stellar radiation, I_b represents the matrix containing only photon counts from sky background radiation, I_d corresponds to the dark current in units of electrons per second (e^-/s), and ${\sigma }_{read}^2$ denotes the readout noise.

Building upon this foundation, PRNU and DCNU noise are further incorporated. Assuming the pixel response non-uniformity matrix U_r follows a Gaussian distribution with zero mean and standard deviation std, the total star map energy I_m can be expressed as:

$$I_m=(I_s+I_b+I_n)\cdot (1+U_r)$$

(33)

2.4. Stellar Energy Distribution

To achieve subpixel-level centroid localization accuracy, a defocused optical design is typically employed to disperse stellar energy across a 3 × 3 pixel region on the detector, ensuring the energy distribution closely approximates a two-dimensional Gaussian profile. Consequently, under static conditions, the stellar energy distribution on the detector focal plane can be mathematically described by the following expression:

$$I_s(x,y)={\displaystyle {\int }_{-0.5+m_0}^{0.5+m_0}{\displaystyle {\int }_{-0.5+n_0}^{0.5+n_0}\frac{I_0}{2\pi {\sigma }_{psf}^2}\exp \left(-\frac{x^2+y^2}{2{\sigma }_{psf}^2}\right)dxdy}}$$

(34)

where m₀ and n₀ denote the distances from pixel (x,y) to the stellar projection center along the x and y axis directions, respectively. I₀ represents the total stellar energy accumulated on the detector focal plane during the exposure time t_e, while σ_psf stands for the Gaussian dispersion radius, which characterizes the defocusing level of the optical system. The Gaussian dispersion radius σ_psf adopted in this work was calculated from real star maps acquired in the laboratory using a collimator. The extraction method corresponded to the parameter estimation algorithm detailed in Section 3.1 of Reference [31], which computes σ_psf by solving equations based on the ratios of grayscale sums within a 3 × 3 pixel window surrounding the stellar image spot [31]. To enhance reliability, the final value of σ_psf was determined by averaging the results obtained from multiple star spots.

For dynamic conditions, Equation (34) requires modification to reflect platform motion effects on the stellar energy distribution, as shown in Equation (35):

$$I_s(x,y)={\displaystyle {\int }_0^{t_e}{\displaystyle {\int }_{-0.5+m_0-L_x(t)}^{0.5+m_0-L_x(t)}{\displaystyle {\int }_{-0.5+n_0-L_y(t)}^{0.5+n_0-L_y(t)}\frac{I_0}{2\pi {\sigma }_{psf}^2{t}_e}\exp \left[-\frac{x^2+y^2}{2{\sigma }_{psf}^2}\right]dxdydt}}}$$

(35)

To reduce computational complexity and improve real-time performance, the integral can be approximated in summation form. Through temporal discretization, the stellar energy distribution under dynamic conditions can be approximated as follows:

$$I_s(x,y)={\displaystyle \sum _{i=0}^{N-1}{\displaystyle {\int }_{-0.5+m_0-L_x(\Delta t\cdot i)}^{0.5+m_0-L_x(\Delta t\cdot i)}{\displaystyle {\int }_{-0.5+n_0-L_y(\Delta t\cdot i)}^{0.5+n_0-L_y(\Delta t\cdot i)}\frac{I_0}{2\pi {\sigma }_{psf}^{2}N}\exp \left(-\frac{x^2+y^2}{2{\sigma }_{psf}^2}\right)dxdy}}}$$

(36)

where N denotes the total number of time steps, and Δt represents the duration of each time step, satisfying Δt = t_e/N. To ensure the accuracy of temporal discretization, Δt must be maintained well below 1 millisecond, with N = 1000 selected in this study. While this choice of N = 1000 ensures high fidelity in modeling the dynamic trailing effect, it does increase the computational load proportionally. In practice, the simulation runtime for a single dynamic star map is approximately 13 s on a standard workstation (Intel i7-14700HX, 16 GB RAM) using MATLAB 2022b. For real-time applications or large-scale batch simulations, the value of N could be adaptively reduced based on the angular velocity or the required accuracy, trading off some precision for computational efficiency.

3. Star Map Simulation and Validation

3.1. Evaluation Metrics

This study establishes a multi-dimensional star map simulation quality assessment system, which systematically verifies simulation results through four evaluation metrics: the signal-to-noise ratio (SNR) characterizes the relative relationship between stellar signal intensity and background noise intensity, the total grayscale value (G_total) reflects the total energy of star points, the grayscale concentration index (GCI) describes the spatial distribution characteristics of energy, and the dynamic star displacement (DSD) quantifies the degree of geometric deformation. This comprehensive evaluation framework assesses the degradation characteristics of star maps under dynamic disturbance conditions from multiple dimensions, providing a scientific quantitative basis for validating the effectiveness and accuracy of simulation methods.

The SNR is defined as the ratio of the difference between the average grayscale value of the star spot (I_f) within a 3 × 3 pixel region and the average grayscale value of the background region (I_g) to the system noise. A higher SNR value indicates that the stellar signal is more distinguishable from the background noise. The calculation formula is as follows:

$$\mathrm{SNR}={\displaystyle \frac{I_f-I_g}{{\sigma }_b/3}}$$

(37)

where σ_b represents the standard deviation of the grayscale values in the background region.

The G_total is defined as the cumulative grayscale value of all pixels within the star’s core region (3 × 3 pixels) after subtracting the background grayscale value, expressed as follows:

$${\mathrm{G}}_{\mathrm{total}}={\displaystyle \sum _{i=x_m-1}^{x_m+1}{\displaystyle \sum _{j=y_m-1}^{y_m+1}(I_{ij}-I_g)}}$$

(38)

where I_ij represents the grayscale value of each pixel within the 3 × 3 pixel region, x_m and y_m denote the stellar centroid coordinates.

The GCI is defined as the ratio of the background-subtracted cumulative grayscale values between the star’s core region (3 × 3 pixels) and a larger surrounding area (7 × 7 pixels), serving to quantitatively characterize the spatial distribution characteristics of stellar energy, expressed mathematically as:

$$\mathrm{GCI}={\displaystyle \frac{{\displaystyle \sum _{i=x_m-1}^{x_m+1}{\displaystyle \sum _{j=y_m-1}^{y_m+1}(I_{ij}-I_g)}}}{{\displaystyle \sum _{i=x_m-3}^{x_m+3}{\displaystyle \sum _{j=y_m-3}^{y_m+3}(I_{ij}-I_g)}}}}$$

(39)

The DSD is defined as the major axis length of the star spot along the direction of motion in the image plane, which quantitatively reflects the star trailing effect induced by platform motion. Specifically, the original image undergoes Gaussian filtering for noise reduction, followed by Otsu threshold segmentation to extract the binarized star spot regions. Subsequently, morphological processing is applied to identify connected domains. Finally, the major axis length is determined through region property analysis of the measured image.

3.2. Simulation Results

The parameters of the all-time three-FOV star sensor employed in this study are presented in

Table 1. Parameters of the optical system.

Parameter	Value
FOV/°	1.96 × 1.96
Focal length/mm	298
Optics aperture/mm	100
Transmittance	90%
g1/mm−2	2.31 × 10−5

(optical system) and

Table 2. Parameters of the Bobcat-640.

Parameter	Value
Array format/pixels	640 × 512
Pixel pitch/μm	20
Wavelength range/μm	1.3–1.7
Quantum efficiency	80%
Dark current/fA	30

(detector), wherein the SWIR detector adopts the Bobcat-640 from Xenics. The simulation observation site was geographically positioned at 100.23°E longitude and 26.86°N latitude (altitude: 2388 m). Considering that the sky background radiation intensity encountered by airborne platforms is significantly lower than ground-based observations, the morning of 30 December 2024 (UTC + 8) was selected as the simulation period, during which the sky background radiation intensity better approximates airborne conditions. The MODTRAN simulation parameters were configured as follows: the 1976 U.S. Standard Atmosphere model was adopted with a CO₂ mixing ratio of 419.3 ppmv, rural aerosol model (visibility: 30 km), and no cloud or rain. Extreme weather events are not yet incorporated in the current model; relevant research on their impact will be conducted in subsequent tasks. The pixel response non-uniformity matrix U_r conforms to a normal distribution characterized by zero mean and a variance of 0.0053². The optical axis orientations of the three FOV coordinate systems in GCRS are specified as: FOV1 (169.2313°, −14.3330°), FOV2 (241.7068°, 8.8859°), and FOV3 (165.6247°, 61.1346°).

Given the rigid connection of the three FOVs to the same airborne platform, only FOV1 was subjected to motion disturbances in the simulation, while the motion states of the remaining FOVs were derived through coordinate system transformation relationships. All simulations employed a 5 ms exposure time, with results shown in Figure 10, Figure 11, Figure 12, Figure 13, Figure 14: Figure 10, presents the star map simulation results of the three FOVs under static conditions, containing four observable stars; Figure 11 and Figure 12 demonstrate the star map degradation characteristics under angular motion conditions around the non-optical axis (X_f1 axis angular velocity: −2° to −10°/s at 2°/s intervals, Y_f1 axis angular velocity: 2–10°/s at 2°/s intervals) and optical axis (Z_f1 axis angular velocity: 5°/s and 10°/s) of the FOV1 coordinate system, respectively; Figure 13 and Figure 14 display the imaging results under angular vibration conditions around the non-optical axis (X_f1/Y_f1 axis angular vibration amplitude coefficients: 5 × 10⁵ and 10⁶) and optical axis (Z_f1 axis angular vibration amplitude coefficients: 5 × 10⁶ and 10⁷), respectively.

Under non-optical axis rotation conditions (Figure 11), when the angular velocity increases from 2°/s to 10°/s, the star spots in FOV1 and FOV3 exhibit significant dynamic degradation characteristics, manifested as increased trailing length and diminished brightness. Quantitative analysis (

Table 3. Star map degradation characteristics of All-Time Three-FOV star sensor under various dynamic conditions.

Condition Type	Motion Parameters	FOV	SNR	Gtotal	GCI	DSD
Static Baseline	-	FOV1	34.5/ 39.9, 47.6/ 158.6	75,847/ 79,915, 95,281/ 310,800	68.1%/ 64.0%, 70.6%/ 60.4%	8.9/ 8.1, 8.2/ 5.8
		FOV2
		FOV3
Non- Optical Axis Rotation	Xf1: −2°/s Yf1: 2°/s	FOV1	27.0/ 38.0, 42.0/ 144.3	59,439/ 75,751, 83,599/ 283,100	51.8%/ 63.3%, 62.9%/ 48.0%	11.0/ 9.2, 9.5/ 7.0
		FOV2
		FOV3
	Xf1: −4°/s Yf1: 4°/s	FOV1	15.0/ 36.6, 39.7/ 95.1	33,262/ 73,369, 79,498/ 187,055	33.7%/ 55.3%, 57.1%/ 38.3%	13.2/ 10.1, 10.3/ 9.2
		FOV2
		FOV3
	Xf1: −6°/s Yf1: 6°/s	FOV1	10.3/ 34.3, 34.0/ 70.0	22,855/ 68,539, 68,000/ 137,232	32.0%/ 53.7%, 52.0%/ 32.5%	17.1/ 11.0, 11.4/ 13.2
		FOV2
		FOV3
	Xf1: −8°/s Yf1: 8°/s	FOV1	9.3/ 31.0, 31.2/ 53.8	20,460/ 62,090, 62,471/ 105,696	31.0%/ 52.6%, 50.1%/ 31.8%	20.3/ 11.9, 11.9/ 16.4
		FOV2
		FOV3
	Xf1: −10°/s Yf1: 10°/s	FOV1	7.3/ 25.9, 27.9/ 39.2	16,102/ 51,830, 55,710/ 77,154	27.7%/ 45.5%, 44.5%/ 29.0%	22.9/ 12.7, 13.1/ 19.7
		FOV2
		FOV3
Optical Axis Rotation	Zf1: 5°/s	FOV1	34.4/ 20.7, 24.1/ 121.7	75,562/ 41,574, 48,273/ 239,275	66.0%/ 38.7%, 40.0%/ 41.1%	10.1/ 13.1, 14.9/ 8.4
		FOV2
		FOV3
	Zf1: 10°/s	FOV1	33.4/ 12.5, 13.5/ 66.1	73,678/ 24,971, 26,978/ 130,187	61.6%/ 36.0%, 37.1%/ 37.9%	11.0/ 18.2, 18.5/ 11.6
		FOV2
		FOV3
Non- Optical Axis Angular Vibration	Xf1: 5 × 105 Yf1: 5 × 105	FOV1	24.3/ 34.0, 44.5/ 145.6	53,596/ 68,401, 89,398/ 285,175	46.2%/ 59.1%, 64.9%/ 49.4%	12.0/ 10.8, 11.1/ 7.1
		FOV2
		FOV3
	Xf1: 106 Yf1: 106	FOV1	7.5/ 9.8, 10.1/ 140.3	16,551/ 19,672, 20,638/ 275,581	36.1%/ 35.5%, 34.9%/ 47.6%	24.0/ 25.2, 24.9/ 8.0
		FOV2
		FOV3
Optical Axis Angular Vibration	Zf1: 5 × 106	FOV1	34.4/ 10.4, 8.7/ 44.5	76,062/ 20,827, 17,363/ 87,411	64.5%/ 41.4%, 34.9%/ 35.7%	10.1/ 31.1, 32.0/ 28.3
		FOV2
		FOV3
	Zf1: 107	FOV1	32.2/ 6.7, 5.3/ 29.9	70,925/ 13,542, 10,658/ 58,787	59.0%/ 33.0%, 26.4%/ 34.7%	11.3/ 36.9, 38.4/ 40.6
		FOV2
		FOV3

) reveals varying degrees of performance degradation across FOV1, FOV2, and FOV3 under this condition: SNR decreases by 78.1%, 35.1%, 41.4%, and 75.3%; G_total declines by 78.8%, 35.1%, 41.5%, and 75.2%; GCI attenuates by 59.3%, 28.9%, 37.0%, and 52.0%; while DSD increases by 157.3%, 56.8%, 59.8%, and 239.7%, respectively. Notably, the degradation amplitude of all evaluation metrics in FOV2 is significantly lower than in other FOVs. This phenomenon stems from its unique kinematic decomposition property—non-optical axis rotation primarily translates into a component rotating around its optical axis in FOV2. Theoretical analysis (Equations (16) and (17)) demonstrates that the trailing effect induced by non-optical axis rotation is positively correlated with the focal length, whereas optical axis rotation-induced trailing depends on the pixel distance between the star spot and the principal point. Given that the focal length (298 mm) is substantially larger than the pixel size (20 μm), star spot trailing in FOV2 consequently remains relatively mild.

Under optical axis rotation conditions (Figure 12), as the rotational speed increases from 5°/s to 10°/s, FOV2 and FOV3 exhibit pronounced trailing effects, while FOV1 is relatively less affected. The quantitative data (Table 3) reveal significant performance degradation in FOV2 and FOV3: SNR decreases by 68.7%, 71.6%, and 58.3%; G_total declines by 68.8%, 71.7%, and 58.1%; GCI attenuates by 43.8%, 47.5%, and 37.3%; and DSD increases by 124.7%, 125.6%, and 100.0%, respectively. In contrast, all evaluation metrics for FOV1 fluctuate within a 25% variation range. This discrepancy primarily originates from the alignment between the optical axis and rotation axis in FOV1, which significantly mitigates the impact of optical axis rotation on star imaging. Conversely, the angular deviation between optical axes and rotation axes in FOV2 and FOV3 leads to more pronounced dynamic degradation.

For non-optical axis angular vibration (Figure 13), when the angular vibration amplitude coefficient increased to 10⁶, FOV1 and FOV2 exhibited significant degradation (SNR decreased by 78.3%, 75.4%, and 78.8%, G_total declined by 78.2%, 75.4%, and 78.3%, GCI attenuates by 47.0%, 44.5%, and 50.6%, and DSD increased by 169.7%, 211.1%, and 203.7%, respectively). In contrast, FOV3 was less affected (average variation of evaluation metrics: 21.0%).

For optical axis angular vibration (Figure 14), when the angular vibration amplitude coefficient reached 10⁷, the imaging quality of FOV2 and FOV3 is primarily affected (SNR decreased by 83.2%, 88.9%, and 81.2%, G_total declined by 83.1%, 88.8%, and 81.1%, GCI attenuates by 48.4%, 62.6%, and 42.6%, and DSD increased by 355.6%, 368.3%, and 600.0%, respectively). However, the star characteristics in FOV1 remain relatively stable (average variation of evaluation metrics: 13.4%).

Simulation results indicate that the degree of star trailing is predominantly influenced by two key factors:

• Motion parameters of the airborne platform, including the angular velocity and angular vibration amplitude coefficient;
• FOV installation parameters, specifically the pointing deviation between each FOV’s optical axis and the rotation/vibration axis.

By optimizing the relative orientation of each FOV’s optical axis with respect to the platform’s primary motion axes, the degradation of imaging quality under dynamic conditions can be mitigated.

3.3. Experimental Verification

This study rigorously adhered to the experimental parameters (observation time, geographic coordinates, and altitude) established in Section 3.2, conducting systematic field validation experiments using an all-time three-FOV star sensor. Both experiments and simulations employed a 2.5 ms exposure time, with results presented in Figure 15, Figure 16 and Figure 17. The selected experimental site (2388 m altitude, 100.23°E, 26.86°N) was chosen because it presents convenient observation conditions and favorable meteorological conditions with abundant clear days—both factors collectively ensure the acquisition of high-quality stellar observation data. To ensure experimental reliability and avoid random errors, 20 sets of real star map data were collected under each experimental condition (static, non-optical axis rotation, and optical axis rotation). Note that the “real star map” (labeled as “R”) in

Table 4. Comparative analysis of multi-dimensional evaluation metrics between real and simulated star maps.

Condition Type	Motion Parameters	SNR[R/Sp/S1] 1	Gtotal[R/Sp/S1]	GCI[R/Sp/S1]	DSD[R/Sp/S1]
Static Baseline	-	24.0/27.8/29.1	74,680/74,746/82,123	78.1%/67.3%/66.1%	10.11/9.58/11.15
Non-Optical Axis Rotation	Yf1: 2°/s	24.0/26.6/28.6	71,623/71,233/80,764	68.2%/62.9%/61.3%	11.84/10.80/13.06
	Yf1: 4°/s	21.2/24.7/26.0	63,629/66,760/73,645	69.4%/63.5%/59.7%	13.32/12.24/13.92
	Yf1: 6°/s	18.6/21.0/23.6	55,726/56,217/66,604	54.5%/55.5%/56.7%	14.52/13.56/15.80
	Yf1: 8°/s	15.8/18.4/19.9	47,994/49,480/56,288	44.7%/45.9%/50.8%	16.39/15.10/17.84
	Yf1: 10°/s	13.5/15.7/16.0	41,564/42,429/45,280	42.9%/48.9%/49.2%	17.57/16.02/18.42
Optical Axis Rotation	Zf1: 5°/s	23.4/27.1/28.6	71,379/73,385/80,346	73.8%/65.4%/62.8%	10.85/10.13/11.90
	Zf1: 10°/s	23.7/26.6/28.1	71,335/71,583/79,134	71.9%/64.3%/63.9%	11.53/10.78/12.55

¹ R: real star map; S_p: simulated star map generated using the proposed method; S₁: simulated star map generated using Method 1.

corresponds to the first set of data among the 20 repetitions, and the standard deviation of each metric across 20 repetitions is supplemented to characterize data stability. We conduct a comparative analysis of the proposed method and the method in Reference [23] (hereafter referred to as Method 1): the actual observational data of the Polaris region in FOV1 is presented in Figure 15a, the simulated results for the same celestial area using the method proposed in this study are shown in Figure 15b, and the simulated results of the same celestial area using Method 1 are displayed in Figure 15c. Moreover, the standard deviations of the 20 sets of real star map metrics are SNR = 1.1, G_total = 2329, GCI = 2.4%, DSD = 0.32, indicating low random fluctuations in static observation. Quantitative evaluation (Figure 18) reveals that under static conditions, the average relative error of all evaluation metrics for the star maps generated by the method proposed in this study remains below 9%; in contrast, the average relative error of all evaluation metrics for the star maps generated using Method 1 is 14.2%, which is higher than that of the proposed method. This conclusively validates the accuracy of the proposed static simulation model.

Experimental results from dynamic condition testing under non-optical axis rotation (Y_f1: 2–10°/s), with the results presented in Figure 16 and Table 4, demonstrate that with the increase in rotation speed, the SNR, G_total, and GCI of real star maps exhibit a gradual decreasing trend, while the DSD shows a gradual increasing trend. For the simulated star maps generated using the proposed method and those generated using Method 1, the variation trends of these four evaluation metrics are consistent with those of the real star maps. Furthermore, the metric fluctuations across 20 repeated experiments under different operating conditions are as follows: at Y_f1 = 2°/s, the standard deviations are SNR = 1.4, G_total = 2782, GCI = 2.8%, DSD =0.41; at Y_f1 = 4°/s, the standard deviations are SNR = 1.7, G_total = 3328, GCI = 3.3%, DSD = 0.57; at Y_f1 = 6°/s, the standard deviations are SNR = 1.6, G_total = 3179, GCI = 3.1%, DSD = 0.49; at Y_f1 = 8°/s, the standard deviations are SNR = 1.8, G_total = 3572, GCI = 3.8%, DSD = 0.66; and at the maximum angular velocity of Y_f1 = 10°/s, the standard deviations are SNR = 1.9, G_total = 3896, GCI = 4.3%, DSD = 0.80, demonstrating favorable stability. Error analysis in Figure 18 further reveals that the average relative error of all evaluation metrics for the star maps generated using the proposed method is merely 8.0%; in contrast, the average relative error of all evaluation metrics for the star maps generated using Method 1 is 14.1%. This result confirms the excellent adaptability of the model proposed in this study to non-optical axis rotation conditions.

For optical axis rotation conditions (Z_f1: 5–10°/s), as illustrated in Figure 17, both simulated and real star maps maintain stable stellar morphology and energy distribution. The data presented in Table 4 demonstrate that the average fluctuation amplitude of the evaluation metrics for the simulation results generated using the proposed method is 4.80%, and that for the simulation results generated using Method 1 is 4.82%. The average fluctuation amplitude of the evaluation metrics for these two sets of simulation results is relatively close to that of the evaluation metrics for the measured data (5.93%). Notably, the metrics of the 20 sets of real star maps exhibit low volatility, similar to the static condition: at Z_f1 = 5°/s, the standard deviations are SNR = 1.2, G_total = 2526, GCI = 2.6%, and DSD = 0.35; at Z_f1 = 10°/s, the standard deviations are SNR = 1.3, G_total = 2686, GCI = 2.7%, and DSD = 0.39. In addition, Figure 18 reveals that under these conditions, the average relative error of all evaluation metrics for the simulated star maps generated via the proposed method is 8.3%, while that for the simulated star maps generated using Method 1 is 13.6%. This result further reaffirms the effectiveness of the model proposed in this study.

The experimental results verify that the simulated star maps generated using the method proposed in this study exhibit high consistency with actual measurement results under both static and various dynamic conditions and can accurately reproduce the imaging characteristics of real star sensors in diverse dynamic environments. Furthermore, these results also confirm the superiority of the proposed method over existing methods.

4. Conclusions

This study conducts systematic research on star map simulation under dynamic airborne conditions. A star map simulation model for an all-time three-FOV star sensor is established, which comprehensively integrates key factors including stellar radiation, sky background, detector noise, and platform motion. The model achieves precise modeling of star map generation in complex environments. For model validation, a multi-dimensional quality evaluation framework is developed. Comparative analysis between simulated and real star maps confirms the model’s validity. The proposed method and evaluation system provide significant value for assessing dynamic performance and optimizing algorithms of airborne star sensors. Future work may incorporate aero-optical effects at speeds above 4 Ma and more complex platform motion patterns to enhance the simulation system’s applicability under extreme operating conditions.