Performance Analysis of Stellar Refraction Autonomous Navigation for Cross-Domain Vehicles

Abstract

Stellar refraction autonomous navigation provides a promising alternative for cross-domain vehicles, particularly in near-space environments where traditional inertial and satellite navigation methods face limitations. This study develops a stellar refraction navigation system that utilizes stellar refraction angle observations and the Implicit Unscented Kalman Filter (IUKF) for state estimation. A representative orbit with altitudes ranging from 60 km to 200 km is designed to simulate cross-domain flight conditions. The navigation performance is analyzed under varying conditions, including orbital altitude, as well as star sensor design parameters, such as limiting magnitude, field of view (FOV) value, and measurement error, along with different sampling intervals. The simulation results show that increasing the limiting magnitude from 5 to 8 reduced the position error from 705.19 m to below 1 m, with optimal accuracy reaching 0.89 m when using a 20° × 20° field of view and a 3 s sampling interval. In addition, shorter sampling intervals improved accuracy and filter stability, while longer intervals introduced greater integration drift. When the sampling interval reached 100 s, position error grew to the kilometer level. These findings validate the feasibility of using stellar refraction for autonomous navigation in cross-domain scenarios and provide design guidance for optimizing star sensor configurations and sampling strategies in future near-space navigation systems.

1. Introduction

Cross-domain vehicles, characterized by advanced features, such as morphing structures and adaptive propulsion systems, represent a new generation of aerospace platforms capable of wide-speed-range flight, multi-environment adaptation, and cross-medium penetration. With the increasing demand for air–space integration, such vehicles have achieved seamless transitions between atmospheric and orbital domains, emerging as a critical breakthrough in next-generation aerospace systems [1]. However, while offering revolutionary potential, their complex dynamics and extreme operating environments also pose unprecedented challenges to autonomous navigation systems [2].

Unlike traditional spacecraft, cross-domain vehicles operate in highly dynamic trajectories, often involving large-scale maneuvering across a wide altitude range and diverse atmospheric conditions [3]. These highly dynamic and complex environments impose stringent demands on onboard navigation systems, including the need to maintain continuous positioning accuracy, robustness to observability changes, and resilience to signal degradation. Conventional navigation techniques suffer from various limitations: satellite navigation, though accurate, is vulnerable to signal blockage and electromagnetic interference [4]; inertial navigation is fully autonomous but suffers from error accumulation over time [5,6]; celestial navigation, with its inherent immunity to jamming, strong stealth, and drift-free nature, has been increasingly regarded as a promising alternative [7,8]. Although some progress has been made in the development of navigation systems for cross-domain vehicles [9,10,11], detailed investigations into the applicability of celestial navigation in such environments remain limited.

Celestial navigation methods can be broadly categorized into direct and indirect horizon-sensing schemes [12]. Direct sensing typically uses optical or infrared sensors to measure celestial body–horizon angles [13], but the required hardware is often complex and costly. In contrast, stellar refraction navigation indirectly infers the horizon by detecting refracted starlight through a high-precision star sensor, combined with atmospheric refraction models [14]. This method offers practical advantages, including lower hardware dependency and compatibility with digital platforms.

Stellar refraction navigation emerged in the 1960s as a conceptual approach [15]. Its feasibility was later confirmed when refracted starlight was first observed by the Orbiting Astronomical Observatory (OAO) [16], demonstrating the physical observability of the phenomenon. Since then, it has rapidly evolved along two primary directions. The first is the refinement of observation models, evolving from the refraction apparent height measurements [17] to more precise formulations, such as stellar refraction angle [18], stellar image coordinates [19], and even time-differential pixel tracking [20]. The second is the development of innovative navigation algorithms tailored for atmospheric flight, including Earth-referenced positioning models and single star sensor strategies [14,21]. These advancements have significantly expanded the applicability of the technique.

Despite these promising developments, existing research still exhibits notable limitations. Most studies focus on theoretical modeling or initial algorithm verification, lacking comprehensive evaluations of system-level performance. Moreover, the impact of varying mission conditions—such as altitude changes, sensor design parameters, and sampling intervals—on navigation accuracy has not been systematically analyzed. To address these gaps, this paper constructs a representative cross-domain satellite flight trajectory and employs an IUKF framework to process stellar refraction angle observations. A quantitative assessment is then conducted to investigate the influence of mission parameters on navigation accuracy, thereby providing theoretical support and performance benchmarks for the practical deployment of stellar refraction navigation systems.

This study focuses on evaluating the performance of stellar refraction autonomous navigation under cross-domain flight trajectories. To isolate the effects of sensor parameters and measurement conditions on navigation accuracy, the simulations are conducted under idealized assumptions, without incorporating detailed propulsion or atmospheric interaction models. These simplifications allow for a clearer analysis of how system-level navigation performance varies across altitude, sensor design, and sampling configurations.

The remainder of this paper is organized as follows. Section 2 introduces the theoretical foundation of stellar refraction navigation. Section 3 presents the modeling framework. Section 4 investigates the effects of key mission parameters on navigation accuracy through a series of simulation experiments. Finally, Section 5 summarizes the conclusions and discusses potential directions for future research.

2. Principle of Stellar Refraction Navigation

Stellar refraction navigation leverages the phenomenon whereby starlight is bent as it traverses the Earth’s atmosphere. When light from a distant star passes through the atmospheric limb, it is refracted toward the Earth’s center. As a result, the observed position of the star appears higher than its true celestial position. This effect enables star sensors on high-altitude or near-space vehicles to detect stars even when they are geometrically below the horizon. The basic geometry of this principle is illustrated in Figure 1 [14].

The apparent altitude of a refracted star, as observed by the onboard star sensor, is denoted as $h_a$, while $h_g$ represents the effective altitude at which the refraction occurs. Due to atmospheric refraction, the apparent direction of a star deviates from its true position. This angular difference is defined as the stellar refraction angle, denoted as $R$. In addition, several other quantities are defined in the diagram. Among them, $R_e$ denotes the Earth’s radius, $r$ represents the spacecraft’s position vector in the Earth-centered inertial (ECI) frame, and $u$ is the unit direction vector of the stellar before refraction.

According to the geometric relationship in Figure 1, the following formula can be obtained [14]:

$$h_a=\sqrt{r^2-u^2}+u\tan R-R_e$$

(1)

where $r$ and $u$ can be obtained by the following formula:

$$\left\{\begin{array}{l}r=\left|r\right|=\sqrt{x^2+y^2+z^2}\hfill \\ u=|r\cdot u|=|x{s}_x+y{s}_y+z{s}_z|\hfill \end{array}\right.$$

(2)

where $r={[x\hspace{0.33em}y\hspace{0.33em}z]}^T$ represents the position vector coordinates of the aircraft in the ECI system; $u={[s_x\hspace{0.33em}{s}_y\hspace{0.33em}{s}_z]}^T$ represents the coordinates of the unit direction vector of the starlight before refraction in the ECI frame.

Furthermore, using empirical atmospheric models [22], the refraction apparent height can also be approximated as a function of the refraction angle $R$:

$$h_a=-21.74089877-6.441326\ln R+69.21177057R^{0.9805}$$

(3)

Combining Equations (1) and (3), we can get

$$\sqrt{r^2-u^2}+u\tan R-R_e=-21.74089877-6.441326\ln R+69.21177057R^{0.9805}$$

(4)

In the above formulation, the incident starlight direction vector $u$ and the refraction angle $R$ are derived through a process of star map identification and refracted star detection. The procedure is as follows: First, non-refracted stars in the observed star map are matched with stars in the onboard navigation star catalog. Based on the geometric imaging model of the star sensor, the pixel coordinates of each matched star point $(x_{si},y_{si})$ can be converted into a unit direction vector $(X_{si},Y_{si},Z_{si})$ in the star sensor coordinate frame:

$$\left[\begin{array}{c}{X}_{si}\\ Y_{si}\\ Z_{si}\end{array}\right]={\displaystyle \frac{1}{\sqrt{x_{si}^2+y_{si}^2+f^2}}}\left[\begin{array}{c}-x_{si}\\ -y_{si}\\ f\end{array}\right]$$

(5)

Meanwhile, the corresponding right ascension and declination values $({\alpha }_i,{\delta }_i)$ provided by the catalog yield the unit direction vector $(X_{ii},Y_{ii},Z_{ii})$ of the same star in the inertial frame:

$$\left[\begin{array}{c}{X}_{ii}\\ Y_{ii}\\ Z_{ii}\end{array}\right]=\left[\begin{array}{c}\cos {\alpha }_i\cos {\delta }_i\\ \sin {\alpha }_i\cos {\delta }_i\\ \sin {\delta }_i\end{array}\right]$$

(6)

By aligning these vectors, the coordinate transformation matrix $M_c^i$ from the sensor frame to the inertial frame can be determined. Next, using the transformation matrix $M_c^i$ and the navigation star catalog, a simulated non-refracted star map is generated for the current boresight orientation. For each star in the original observed star map, the Euclidean distance $d$ to the nearest neighbor in the simulated non-refracted star map is computed. If $d$ exceeds a threshold determined by the star map’s positional accuracy and identification precision, the star is classified as a refracted star. For each detected refracted star, its inertial unit direction vector $u$ is obtained from the navigation star catalog. The corresponding unit vector in the sensor coordinate frame is computed by combining the star’s pixel position $(x_{ri},y_{ri})$ in the observed map and its projected position $(x_{ni},y_{ni})$ in the non-refracted map. These two pixel positions are then converted into unit direction vectors $S_{c1},S_{c2}$ in the sensor coordinate system using the star sensor’s imaging geometry. The angle between these two unit vectors yields the stellar refraction angle $R$:

$$R=\arccos (S_{c1}\cdot S_{c2})$$

(7)

Then, Equation (4) becomes an equation with the spacecraft position vector as a variable, which can be expressed as

$$h(r,R)=0$$

(8)

3. Navigation System Model

3.1. System Structure

The overall structure of the stellar refraction navigation system is shown in Figure 2. It consists of three main components: initialization of spacecraft state, construction of the navigation system model, and filtering via Kalman-based estimation.

The initialization phase provides the system inputs, including the spacecraft’s initial position and velocity, as well as a time series of refraction angle measurements recorded by the onboard star sensor during flight. In this study, a satellite is selected as the navigation carrier, and a highly eccentric orbit is constructed to simulate cross-domain flight, serving as a representative scenario for subsequent analysis.

The navigation system model includes both a dynamic model and a measurement model. The dynamic model governs the orbital motion and serves as the state transition model for time propagation within the filtering framework. The measurement model, constructed based on stellar refraction observations, is used in the update step to correct the predicted state using current measurements. Together, these models enable real-time estimation of the spacecraft state through a variant of the IUKF capable of handling nonlinear and implicit observation functions.

3.2. State Model

This paper takes the cross-domain flying satellite as the research object, so the system state equation is established based on the orbital dynamics model. Due to the complex operating environment of cross-domain vehicles, their motion is influenced by various perturbative forces, including the gravitational model, the second zonal harmonic (J2) perturbation, atmospheric density model, third-body gravitational effects and solar radiation pressure [23,24]. However, as the primary focus of this work is on evaluating the effectiveness of stellar refraction navigation across dynamic orbital regimes, the dynamic model is simplified accordingly. Specifically, the gravitational perturbation is expanded to include the J2 term, which captures the dominant effect of Earth’s oblateness, while all remaining perturbations are aggregated into a generalized perturbation term $\Delta F_x$, $\Delta F_y$ and $\Delta F_z$. The corresponding equation of motion is expressed as

$$\left\{\begin{array}{l}{\displaystyle \frac{dx}{dt}}=v_x\hfill \\ {\displaystyle \frac{dy}{dt}}=v_y\hfill \\ {\displaystyle \frac{dz}{dt}}=v_z\hfill \\ {\displaystyle \frac{d{v}_x}{dt}}=-\mu {\displaystyle \frac{x}{r^3}}\left[1-J_2\left({\displaystyle \frac{R_e}{r}}\right)\left(7.5{\displaystyle \frac{z^2}{r^2}}-1.5\right)\right]+\Delta F_x\hfill \\ {\displaystyle \frac{d{v}_y}{dt}}=-\mu {\displaystyle \frac{y}{r^3}}\left[1-J_2\left({\displaystyle \frac{R_e}{r}}\right)\left(7.5{\displaystyle \frac{z^2}{r^2}}-1.5\right)\right]+\Delta F_y\hfill \\ {\displaystyle \frac{d{v}_z}{dt}}=-\mu {\displaystyle \frac{z}{r^3}}\left[1-J_2\left({\displaystyle \frac{R_e}{r}}\right)\left(7.5{\displaystyle \frac{z^2}{r^2}}-1.5\right)\right]+\Delta F_z\hfill \end{array}\right.$$

(9)

In this expression, the state vector is defined as $X=[x,y,z,v_x,v_y,v_z]$, where $[x,y,z]$ and $[v_x,v_y,v_z]$ represent the three-dimensional coordinate results of the position and velocity of the aircraft in the ECI frame, respectively, and $r=\sqrt{x^2+y^2+z^2}$ represents the distance from the satellite to the earth’s center. In addition, $\mu =3.986\times {10}^{14}{ \mathrm{m}}^3{/\mathrm{s}}^2$ represents the earth’s gravitational constant, and $J_2=1.08263\times {10}^{-3}$ is the earth’s flattening perturbation coefficient. Additionally, $R_e=6378.137\mathrm{km}$ is the radius of the earth.

In this study, $\Delta F_x$, $\Delta F_y$, and $\Delta F_z$ represent unmodeled perturbative accelerations beyond the J2 effect, including atmospheric drag, third-body gravitational forces, and solar radiation pressure. These perturbations are not explicitly computed or input into the filter. Instead, their influence is statistically modeled as process noise and incorporated into the process noise covariance matrix $Q$ within the IUKF framework. This simplification aligns with our focus on isolating the influence of observation-related parameters under idealized orbital conditions.

3.3. Measurement Model

By selecting the stellar refraction angle as the observation variable, the corresponding measurement model is derived from Equation (8). However, the refraction angle R in Equation (8) describes the relationship between a single refraction angle and the corresponding position vector. In practical stellar refraction navigation, multiple refraction angles can be extracted from the star map captured by the onboard star sensor. Therefore, these individual angles are combined into a one-dimensional observation vector Z, which is used in the measurement model. In the filtering framework, the system dynamics are represented by the nonlinear state transition function $f(X,t)$, corresponding to the orbital motion model described in Equation (9). The observation model is formulated using an implicit nonlinear measurement function $h(X,Z,t)$, derived from the stellar refraction angle model in Equation (8). Combining the system’s state and measurement models yields the following overall system formulation:

$$\left\{\begin{array}{l}\stackrel{.}{X}=f(X,t)+w(t)\hfill \\ 0=h(X,Z,t)+v(t)\hfill \end{array}\right.$$

(10)

where $X\in {ℝ}^n$ denotes the state vector of the satellite, and $w\in {ℝ}^n$ is the process noise vector. Similarly, $Z\in {ℝ}^m$ represents the m-dimensional observation vector, and $v\in {ℝ}^m$ is the n-dimensional measurement noise.

3.4. IUKF

In this work, IUKF was employed as the estimation algorithm for the autonomous navigation system. The choice of IUKF was motivated by the nature of the observation model: the starlight refraction angle, which serves as the system’s measurement input, is inherently an implicit nonlinear function of the spacecraft’s position. As such, it cannot be expressed in an explicit closed-form equation suitable for traditional Kalman filtering frameworks.

While alternative nonlinear filters, such as the Extended Kalman Filter (EKF), Iterative EKF, and standard UKF, have been applied in similar contexts, prior comparative studies [25] have demonstrated that IUKF achieves better robustness and computational efficiency in handling implicit measurement models. Accordingly, this work directly adopted IUKF as the estimation framework, building on those established findings rather than repeating comparative analysis. The implementation used in this paper is consistent with the general formulation of the IUKF presented in [25], particularly in its treatment of implicit measurement functions and sigma point propagation.

To enhance the clarity of the IUKF framework, the main computational procedure of the method adopted in this work is illustrated in Figure 3. The algorithm is structured into two main stages: the time update step and the measurement update step. In the time update phase, sigma points are generated from the current state estimate and propagated through the nonlinear system dynamics. The predicted mean and covariance of the state are then computed based on the propagated sigma points. In the measurement update phase, the observation function—formulated implicitly based on the starlight refraction angle model—is applied to the predicted sigma points. The measurement residual is used to correct the predicted state using a Kalman gain derived through unscented transformations. This structure allows the IUKF to handle implicit and highly nonlinear measurement models without requiring explicit inversion or linearization.

4. Simulations and Results

4.1. Cross-Domain Aircraft Trajectory Simulation Design

To enable the testing of astronomical navigation and positioning techniques for cross-domain vehicles, a satellite orbit was designed. The orbital parameters are as follows: semimajor axis $a$ = 6508.14 km, eccentricity $e$ = 0.0107558, inclination $i$ = 70°, argument of perigee $\omega$ = 0°, and right ascension of the ascending node (RAAN) $\mathsf{\Omega }$ = 0°. Other navigation system input parameters are listed in

Table 1. Navigation system input parameter design baseline.

Type	Name	Value
Orbit	Perigee altitude	60 km
	Apogee altitude	200 km
Star sensor	Stellar database	Tycho2
	Field of view	20° × 20°
	Resolution	1024 × 1024
	Limiting magnitude	7
Filter	Sampling frequency	3 s/frame

.

The reference orbit used in this study was generated using Systems Tool Kit (STK, v11) software, adopting the J2 perturbation model while neglecting atmospheric drag and other higher-order effects. This idealized setting is intended to isolate the impact of altitude variation on the visibility of refracted stars and the corresponding navigation performance. Figure 4 provides a schematic representation of the simulated cross-domain trajectory and the corresponding star sensor field of view. Specifically, Figure 4a shows the two-dimensional projection of the Earth and the orbital path, illustrating the eccentric trajectory used for the experiment. Figure 4b depicts the visible field of view of the onboard star sensor, aligned with its optical axis. In this figure, the red coordinate frame represents the spacecraft body frame, while the green coordinate frame denotes the star sensor coordinate system, with the z-axis pointing along the sensor’s line of sight (optical axis). The orientation of the star sensor is defined relative to the body frame, with an azimuth angle of 0° and an elevation angle of 5°. This configuration reflects a typical installation scenario and determines the visible region of refracted stars at each orbital position.

The satellite traverses altitudes between 60 km and 200 km in approximately 1.5 h cycles. Such an orbital configuration satisfies both the physical definition of cross-domain flight and the operational altitude requirements of stellar refraction navigation (typically above 40 km), thereby providing a suitable foundation for subsequent simulation experiments.

In addition, the specific model parameters of the Kalman filter used in the experiment were set as follows:

The initial state error: $\Delta X={[1000 \mathrm{m},1000 \mathrm{m},1000 \mathrm{m},10 \mathrm{m}/\mathrm{s},10 \mathrm{m}/\mathrm{s},10 \mathrm{m}/\mathrm{s}]}^{\mathrm{T}}.$

Accordingly, the initial state covariance matrix $P_0$ was defined as a diagonal matrix with the position variances set to ${(1000 \mathrm{m})}^2$ and the velocity variances set to ${(100 \mathrm{m}/\mathrm{s})}^2$. The velocity variances were deliberately set larger than the initial velocity error to account for potential uncertainty in the initial state and to enhance the robustness of the filter during the early stages:

$$P_0=\mathrm{diag}[{(1000 \mathrm{m})}^2,{(1000 \mathrm{m})}^2,{(1000 \mathrm{m})}^2,{(100 \mathrm{m}/\mathrm{s})}^2,{(100 \mathrm{m}/\mathrm{s})}^2,{(100 \mathrm{m}/\mathrm{s})}^2].$$

The process noise covariance matrix $Q$ was also defined as a diagonal matrix, with position-related noise variances set to ${(1 \mathrm{m})}^2$ and velocity-related noise variances set to ${(0.0001 \mathrm{m}/\mathrm{s})}^2$:

$$Q=\mathrm{diag}[{(1 \mathrm{m})}^2,{(1 \mathrm{m})}^2,{(1 \mathrm{m})}^2,{(0.0001 \mathrm{m}/\mathrm{s})}^2,{(0.0001 \mathrm{m}/\mathrm{s})}^2,{(0.0001 \mathrm{m}/\mathrm{s})}^2].$$

The measurement noise covariance matrix $N$ corresponds to the angular error in the observed refraction angles. It was defined as a diagonal matrix with variances set to ${(0.005″)}^2$, where the dimension of $N$ depends on the number of observed refraction angles in each frame.

All navigation errors presented in this study were computed in the ECI frame, consistent with the simulation and filtering processes. The position and velocity errors reported in this study were calculated as the root mean square errors (RMSEs), defined as

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{‖{\widehat{x}}_i-x_i^{\mathrm{ground} \mathrm{truth}}‖}^2}}$$

(11)

where ${\widehat{x}}_i$ denotes the estimated state, $x_i^{\mathrm{ground} \mathrm{truth}}$ represents the ground-truth value at time step $i$, and $N$ is the total number of time steps. This metric reflects the average value of estimation error across the simulation period.

4.2. Performance Analysis of Autonomous Navigation at Different Orbital Altitudes

To analyze the relationship between orbital altitude and the number of observable refracted stars, a simulation was performed based on the constructed highly eccentric trajectory (60–200 km). At each time step along the orbit, the refraction star map was generated according to the current spacecraft position, attitude, star sensor configuration (including FOV value and limiting magnitude), and the atmospheric refraction model. The number of refracted stars captured in each star map was then recorded together with the corresponding orbital altitude. By collecting these data throughout the orbital period, the statistical relationship shown in Figure 5 was obtained.

As shown in Figure 5, the number of observable refracted stars varied with orbital altitude, exhibiting a general trend of initial decrease, followed by a slight increase, and then another decline. Local fluctuations were also observed due to changes in orbital position. The blue dashed line in the figure represents the threshold of three observable refracted stars. Under the current orbital configuration and star sensor mounting angle, more than three refracted stars were visible at most altitudes, which meets the minimum requirement for stellar refraction-based autonomous navigation.

Figure 6 presents the IUKF filtering results under the system parameter settings listed in Table 1. The left subfigure shows the position error, which initially converges rapidly but exhibits significant fluctuations between 2000 s and 3000 s. To investigate this, the local variations in both orbital altitude and the number of refracted stars are shown in the right subfigure. During this time window, the spacecraft operated at altitudes between 180 km and 200 km, where the number of refracted stars visibly declined and stabilized at a relatively low level. This suggests that when the vehicle is at higher altitudes, the number of refracted stars within the star sensor’s field of view significantly decreases, thereby degrading the overall navigation performance.

In summary, starlight refracted navigation is more suitable for low-Earth orbit (LEO) missions, where the number of observable refracted stars is higher and the navigation accuracy is more stable. For cross-domain flight applications, special attention should be paid to the apogee altitude, as a reduced number of refracted stars at higher altitudes can lead to degraded or unstable positioning performance.

4.3. Performance Analysis Under Different Star Sensor Design Parameters

The design parameters of the star sensor directly influence the resulting star maps, particularly the number of detectable refracted stars, which in turn significantly affects the performance of starlight refracted navigation. This section investigates how variations in key sensor design parameters—specifically, limiting magnitude, field of view, and measurement accuracy—affect the autonomous navigation and positioning performance of the spacecraft.

These parameter settings were selected with reference to the performance ranges of existing spaceborne star sensors [26], ensuring that the simulation conditions remained within the bounds of practical feasibility.

4.3.1. Limiting Magnitude

In this experiment, the limiting magnitude of the star sensor varied from 5 to 8, while other simulation parameters were kept consistent with those listed in Table 1. A series of star maps was generated for each configuration, and stellar refraction navigation simulations were conducted accordingly. The variation in the number of refracted stars under different limiting magnitudes is shown in Figure 7. As the limiting magnitude increased, the sensitivity of the star sensor improved, allowing more refracted stars to be detected. When the limiting magnitude reached 8, at least one refracted star was observed in every star map, ensuring sufficient observability for navigation.

The impact of limiting magnitude on navigation accuracy is illustrated in Figure 8 and Figure 9, which present the position and velocity errors, respectively. As observed, both errors decreased significantly with increasing limiting magnitude, indicating improved positioning and velocity estimation performance. Although a reduced number of refracted stars does not prevent convergence of the IUKF, it does degrade the system’s observability. Consequently, when the number of refracted stars becomes too low, or none are available at a given time, position and velocity estimates may exhibit noticeable fluctuations.

A quantitative summary of navigation performance under different limiting magnitudes is provided in

Table 2. Navigation performance results under different limiting magnitudes.

Limiting Magnitude	Number of Refracted Stars	3D Position Error/m	3D Velocity Error (m·s−1)
5	0.88	705.19	4.45
6	3	62.73	1.99
7	10	3.49	1.74
8	32	0.89	1.73

, where the 3D position and velocity errors are expressed as RMSE values. The number of refracted stars reported represents the average per frame over the full simulation duration. As the limiting magnitude increases, the average number of refracted stars per frame increases substantially, and the overall positioning error decreases significantly. These results demonstrate that a higher limiting magnitude enables the detection of more refracted stars, thereby improving the robustness of navigation. For instance, when the limiting magnitude is set to 5, the position error reaches up to 705.19 m. Therefore, it is recommended that star sensors used for autonomous navigation based on stellar refraction have a limiting magnitude of at least 6vM.

4.3.2. FOV Value

Simulations were conducted using star sensors with three different FOV values: 10° × 10°, 14° × 14°, and 20° × 20°, while keeping other parameters consistent with those listed in Table 1. Since the FOV value directly affects the number of stars observable in a single frame, the number of refracted stars detected over time under each configuration was statistically compared, as shown in Figure 10. The results indicate that increasing the FOV value significantly improved the number of refracted stars observed by the sensor.

The influence of the FOV value on position and velocity estimation accuracy is illustrated in Figure 11. Since the initial error and the final convergence error in the position sub-graph differ greatly, the difference between the sub-graphs is not obvious, so the first value estimated by the filter is uniformly used as the starting point of the horizontal axis. As shown, the positioning error decreases progressively with the expansion of the FOV values. This trend is positively correlated with the increase in the number of refracted stars: more observable stars enhance system observability and improve the accuracy of the measurement model, enabling more effective correction of position estimates and yielding results closer to the true trajectory. In contrast to position accuracy, the effect of the FOV value on velocity estimation is less pronounced. From the perspective of the measurement model, this is expected, since the refracted angle mainly constrains the spacecraft’s position, while the velocity state is primarily propagated through the dynamic model. The velocity at each step depends on the position estimation from the previous step. Therefore, improving position accuracy indirectly contributes to better velocity estimation.

Table 3. Navigation performance results under different FOV values.

FOV	Number of Refracted Stars	3D Position Error/m	3D Velocity Error/(m·s−1)
10° × 10°	5.1033	12.04	1.71
14° × 14°	7.2733	5.80	1.69
20° × 20°	10	3.49	1.74

provides quantitative results for the effect of the FOV value on navigation performance, including the average number of refracted stars per frame computed over the entire simulation period, as well as the 3D position and velocity errors, which are presented in terms of the RMSE. As the FOV value increased, the average number of refracted stars rose, and both position and velocity errors decreased accordingly. These results reaffirm that the number of refracted stars plays a critical role in determining navigation accuracy: the greater the number of observable refracted stars, the better the system performance.

In summary, star sensors with larger FOV values offer better navigation performance by enabling the detection of more refracted stars. However, in practical sensor design, the FOV value should not be increased arbitrarily. Larger FOV values lead to higher data throughput and processing demands, which may affect the computation rate of the onboard navigation processor. Therefore, FOV value selection should be optimized by jointly considering both observation capability and processing efficiency.

4.3.3. Stellar Refraction Angle Measurement Accuracy

The measurement accuracy of the star sensor is the primary factor affecting the accuracy of stellar refraction angle measurements. In this section, it is assumed that the refraction angle error arises solely from the measurement precision of the star sensor.

To analyze the effect of measurement accuracy on autonomous navigation performance, simulations were conducted under star sensor measurement errors of 0″, 0.01″, 0.02″, 0.03″, 0.04″, and 0.05″, while keeping all other parameters unchanged. These values correspond to the standard deviation (1σ) of zero-mean Gaussian white noise applied to the starlight refraction angle measurements, simulating the inherent measurement uncertainty of the star sensor. The selected range of refraction angle measurement errors (0″–0.05″) reflects the achievable precision of current-generation star sensors. Advanced optical systems used in high-accuracy spacecraft attitude determination have demonstrated sub-arcsecond performance [27], which supports the practical feasibility of implementing stellar refraction navigation with similar accuracy requirements. The RMSE value of position and velocity under measurement accuracies ranging from 0.01″ to 0.05″ are shown in Figure 12. Selected detailed simulation results are also listed in

Table 4. The influence of measurement accuracy on navigation accuracy.

Measurement Error/″	3D Position Error/m	3D Velocity Error/(m·s−1)
0	3.4954	1.7423
0.01	239.4764	2.2005
0.02	455.6859	3.1258
0.03	675.232	4.2843
0.04	968.3264	5.6127
0.05	1161.6	6.9177

.

The results clearly indicate that the measurement accuracy of the star sensor had a significant impact on the overall navigation performance. A near-linear relationship was observed between sensor precision and navigation accuracy; higher angular measurement precision led to smaller position and velocity errors. This finding suggests that, in practical sensor design and system implementation, improving the measurement accuracy of the star sensor is essential for achieving high-precision autonomous navigation and positioning.

4.4. Performance Analysis Under Different Sampling Intervals

To evaluate the impact of sampling frequency on autonomous navigation performance, six different sampling intervals—1 s, 3 s, 6 s, 10 s, 20 s, and 100 s—were tested in cross-domain navigation experiments. The corresponding position errors are compared in Figure 13. As the sampling interval increased, both the overall positioning error and its fluctuation range became more pronounced.

Quantitative RMSE statistics of the position estimates, computed against ground-truth trajectories under different sampling intervals, are summarized in

Table 5. Quantitative comparison of position error at different sampling intervals.

Sampling Interval	3D Error/m	X Error/m	Y Error/m	Z Error/m
Δt = 1 s	0.98	0.77	0.33	0.52
Δt = 3 s	3.49	3.03	0.89	1.49
Δt = 6 s	10.74	9.39	2.66	4.48
Δt = 10 s	26.99	23.63	6.08	11.56
Δt = 20 s	104.29	91.36	22.52	44.98
Δt = 100 s	2092.46	1907.29	324.17	797.2

. A vertical comparison shows that the positioning error increases steadily as the sampling interval grows. The difference in accuracy between the 1 s and 3 s intervals is relatively small; however, when the interval reaches 100 s, the positioning error increases sharply to the kilometer level. This is attributed to the fact that the state prediction in the filter relies on numerical integration of the state equations, which inevitably accumulates error over time. When the interval is large, state updates occur less frequently, and error correction becomes insufficient, resulting in divergence of the position estimate.

A horizontal comparison of the three-axis position errors reveals that the error along the x-axis is the largest, followed by the z-axis, while the y-axis shows the smallest error. This anisotropic behavior can be explained by the nature of the stellar refraction observations: the star image plane primarily lies in the y–z plane, while the x-axis represents the sensor boresight direction. Consequently, the refraction angle measurements provide more direct constraints on the y and z positions of the spacecraft, leading to better estimation accuracy in those directions.

The above results indicate that the sampling interval has a substantial impact on navigation accuracy. When the interval is small (e.g., 1 s or 3 s), the positioning accuracy is high and relatively stable. As the interval increases, the accuracy gradually deteriorates. When the interval reaches 100 s, the positioning performance becomes unacceptable for most navigation applications. On the other hand, shorter intervals impose stricter requirements on system processing speed and computational resources. Specifically, high-frequency star maps must be processed in real time, and the navigation filter must complete prediction and update cycles within tight time constraints. These requirements place a considerable computational burden on the onboard system. Therefore, selecting an appropriate sampling interval requires a careful balance between positioning accuracy and onboard processing capability.

5. Conclusions

In this paper, we investigated the performance of autonomous navigation and positioning using stellar refraction for cross-domain vehicles. A nonlinear navigation system based on stellar refraction angle observations and IUKF estimation was constructed. Extensive simulations were conducted to evaluate the effects of orbital altitude, star sensor design parameters including limiting magnitude, field of view, and measurement accuracy, and sampling frequency on navigation performance.

The key findings are as follows:

(1)

Feasibility and precision: Starlight refraction navigation is feasible and effective for cross-domain spacecraft operating in ultra-low orbits. When a star sensor with a limiting magnitude of 8, a field of view of 20° × 20°, and a sampling interval of 3 s/frame is used, the average positioning error can reach as low as 0.89 m.

(2)

Altitude sensitivity: The number of observable refracted stars decreases as orbital altitude increases, leading to reduced system observability and degraded navigation performance. Therefore, the orbital design must be carefully considered when applying this method to cross-domain missions.

(3)

Sensor configuration: Increasing the sensor field of view, improving the limiting magnitude, and reducing measurement noise all contribute to enhanced positioning accuracy and robustness. Although high-performance star sensors may be challenging to implement, hardware configurations with a limiting magnitude of 7° × 7° and a 14° × 14° field of view can still achieve meter-level accuracy.

(4)

Sampling interval trade-off: Due to nonlinear propagation error, longer sampling intervals introduce greater uncertainty. A sampling interval of less than 6 s is recommended. When the interval exceeds 6 s, the positioning error increases to the decimeter level; above 20 s, it reaches the 100 m level; and above 100 s, it can grow to the kilometer level.

While this study provides a quantitative understanding of how system parameters affect navigation performance, the simulations were conducted under idealized conditions. Future work will integrate more comprehensive models, including atmospheric drag, propulsion dynamics, and thermal effects, to evaluate the practical feasibility of deploying the proposed navigation method in real cross-domain mission scenarios, and investigate robust refracted star map matching techniques under such operational conditions.