Analysis of LNSS satellite occlusion in the southern polar region of the Moon based on DEM

Abstract

The Moon’s Southern Pole region is crucial due to the unique terrain features. This study suggests using Digital Elevation Model (DEM) data to assess the positioning performance of the Lunar Navigation Satellite System (LNSS) in this area. Two LNSS constellation types, Elliptical Lunar Frozen Orbit (ELFO) and Elliptical Repeat Ground-Track Orbit, (ERGO), are evaluated based on lunar DEM data. The focus is on Number of Visible Satellites (NVS) and Position Dilution of Precision (PDOP) variations with satellite numbers and elevation mask angles. ELFO (8 satellites) shows a 69% probability of 4 + visible satellites with a 52% PDOP within 10, while ERGO (8 satellites) has 64% and 60% probabilities, respectively. Hybrid constellations involving an additional near-rectilinear halo orbit (NRHO) satellite are also explored. ELFO-NRHO (9 satellites) has a 96% probability of 4 + visible satellites and a 73% PDOP within 5, while ERGO-NRHO (9 satellites) presents probabilities of 83% and 78%, respectively. The positioning accuracy performance of hybrid constellations with NRHO is approximately 30 m.

Introduction

Lunar Navigation Satellite System (LNSS)^1,2,3 is an envisioned navigation system designed to provide positioning, navigation, and timing (PNT) services for exploration and research on the Moon. Humans have not yet established LNSS on the Moon, and the LNSS navigation constellation is still in the stage of design, simulation, analysis, and validation.

Some progress has been made in the simulation and analysis of LNSS constellations. Filipe et al.⁴ verified the accuracy of pseudo-range measurements in two 24-satellite navigation constellations and assessed their economic cost. Sriramya et al.⁵ analyzed the trade-off between different design considerations related to the onboard clock and the lunar orbit type for designing an LNSS with Earth-GPS time transmission. Guan⁶ designed a constellation comprising six near rectilinear halo orbits (NRHOs) and analyzed its navigation performance. Wang et al.⁷ proposed combining the advantages of Distant Retrograde Orbit (DRO) and halo orbits to provide precise PNT services with a smaller number of satellites. Renwick et al.⁸ utilized a multi-objective optimization method to design a constellation, identifying the best 10 constellations from a pool of 1864 iterations.

The Moon’s Southern Pole is of interest, both as a scientific treasure trove for understanding the evolution of the solar system and as a potential exploration site^9,10. Masaya et al.¹¹ introduced a navigation constellation comprising eight satellites in two elliptical frozen orbits (ELFOs) and simulated the horizontal positioning accuracy at the southern pole.

The structure of the lunar surface is generally complex and variable^12,13. Huber et al.¹⁴ analyzed the terrain of the lunar landing area and simulated the communication between the Earth and the landing area. Additionally, Mazarico et al.¹⁵ and Gläser et al.¹⁶ have combined elevation data to investigate the lighting conditions in the Southern Pole region of the Moon.

To our knowledge, there are no studies that combine terrain analysis and navigation performance. The simulated and analyzed LNSS in this paper is primarily focused on the Southern Polar region of the Moon. Using the existing lunar Digital Elevation Model (DEM) data, the simulation analyzes the occurrence of LNSS satellite occlusion leading to non-visibility at various locations. The performance of LNSS constellations was investigated by calculating the number of visible satellites (NVSs) and Positioning Dilution of Precision (PDOP) for various elevation mask angles or different numbers of LNSS satellites. In the Southern Polar region of the Moon, the performances of availability and accuracy of LNSS constellations are calculated. Furthermore, the performance of a hybrid constellation incorporating an additional NRHO satellite was validated by comparing relevant performance metrics.

Navigation constellation simulation

There are multiple orbital options available for LNSS options. Sibu et al. evaluated the performance of two currently most promising constellation candidates (Halo and ELFO constellations) for LNSS implementation¹⁷. Reference⁴ presented a repeat ground-track orbit (hereafter uniformly abbreviated as the Elliptical Repeat Ground-Track Orbit, ERGO for illustrative convenience). ERGO maintains long-term stability (no orbital maintenance required) even when Earth-Solar perturbations are considered within realistic ephemeris force models⁴. No positional station-keeping maneuvers are required for at least a decade⁴. The orbital stability of a satellite allows for infrequent orbital adjustments, thus conserving propellant and reducing maintenance expenses¹⁸.

The research in reference 4 demonstrated the stability of the ERGO constellation, while reference 17 pointed out that the ELFO constellation is currently the most promising constellation candidate. The ELFO constellation design in reference 11 is consistent with the study area of this paper. Since this study focuses on using the DEM model to assess the satellite occlusion caused by terrain, the LNSS scheme in this paper is based on the previous constellations (references 4 and 11) without making adjustments to the constellation design and system architecture (e.g., number of satellites, orbital parameters, signal characteristics). Keplerian orbital parameters are represented as follows: a for the semi-major axis; e for the eccentricity; i for the orbit inclination; Ω for the right ascension of the ascending node (RAAN); ω for the perigee angle amplitude; and ν for the true periapsis angle. Using Keplerian parameters, LNSS satellites can be created in various lunar orbits within the High Precision Orbit Propagator (HPOP) of Systems Tool Kit (STK) to simulate LNSS orbits and obtain the coordinates of LNSS satellites at different times.

The ELFO constellation is comprised of two ELFOs, each orbiting with 4 satellites. Similarly, the ERGO constellation consists of two ERGOs, each orbiting with 4 satellites. The orbital parameters provided in Table 1 are defined using classical orbital elements in an Inertial Coordinate System frame. These parameters are used to configure the ELFO and ERGO constellations in STK, as illustrated in Fig. 1.

Table 1 Orbital parameters of simulated constellations.

Fig. 1

Structures of ELFO constellation (a) and ERGO constellation (b).

Table 2 The parameters of lunar simulation.

The lunar simulation parameters are shown in Table 2, the simulation time is selected as a random day in the future.

Lunar surface terrain analysis

Due to the absence of an atmosphere, the surface of the moon features numerous impact craters^20,21, which contribute to the complexity and variability of lunar surface geography. Hence, it is imperative to conduct satellite visibility calculations to assess whether the signal of a satellite at a given moment is obstructed by the terrain and whether it can be observed by the grids.

The Digital Elevation Model (DEM) is a mathematical model that represents terrain elevation information in digital form. DEM data is a type of raster data, also known as grid data in Geographic Information Systems. The raster is the basic unit of data, and each raster represents a fixed-size area (related to the data resolution), containing an elevation value that indicates the elevation (altitude) of that area. This paper utilized high-precision lunar elevation data provided by the Lunar Orbiter Laser Altimeter (LOLA) onboard the Lunar Reconnaissance Orbiter (LRO)²². The vertical accuracy (radius error) of the LOLA DEM is better than ~ 1 m after crossover correction²³, and the horizontal resolution is 118 m × 118 m²⁴.

This paper proposes a method for calculating the visibility of static users (receivers) on the lunar surface with respect to LNSS satellites based on DEM data. The resolution of DEM has a high precision, but it also increases the computational load for terrain analysis. Therefore, the original DEM was resampled to reduce the computational load to verify the feasibility of the proposed method. The resolution after resampling is 0.25° × 0.25°. The proposed method can also be used for terrain analysis based on high-precision DEM.

Fig. 2

Schematic diagram of the process of determining whether a satellite is blocked by terrain.

As shown in Fig. 2, we calculate the elevation $\:{\phi\:}_{i}$ and azimuth $\:{\lambda\:}_{i}$ between each satellite in the LNSS constellation (denoted as satellite i, where i ranges from 1 to the total number of satellites) and the current DEM grid cell at a specific time.

From the literature^25,26, at a certain position on the lunar surface, the maximum visible distance (MVD) to the horizon is 300 km. Using $\:{\lambda\:}_{i}$, we identify terrain DEM grid cells lying along the direction of $\:{\lambda\:}_{i}$ within the MVD range. We then calculate elevations from the current DEM grid cell to each identified terrain grid cell and determine the maximum elevation angle ($\:{\theta\:}_{max}$) among them.

• If the following condition is satisfied:

$${\theta _{max}} \geqslant {\varphi _i} \geqslant 5$$

(1)

satellite i is considered “NOT visible” due to terrain occlusion;

• If the elevation mask angle (5°) > $\:{\phi\:}_{i}$, the satellite i is also considered “NOT visible”;

• Only when $\:{\phi\:}_{i}$ and $\:{\theta\:}_{max}$ satisfies:

$${\varphi _i}>{\theta _{max}} \geqslant 5$$

(2)

the satellite i is considered “visible” at the current gird.

This process is repeated for each satellite i until all DEM grid cells in the lunar south polar region (70°S–90°S, 180°W–180°E) are processed across all time steps within the simulation period.

Taking static users (grid points) both outside and inside the lunar impact crater as an example. In the lunar geodetic coordinate system, the coordinate of point A outside the lunar impact crater is (47°E, 83.5°S, 3570 m), and the coordinate of point B inside the lunar impact crater is (47°E, 83.25°S, −1026 m). In the DEM with a resolution of 0.25 × 0.25° (resampled from the original LOLA DEM), the selected points A and B are adjacent to each other. Figure 3 depicts the positions of point A and point B, the zenith diagrams of satellites at points A and B, as well as the grid locations within the MVD where the azimuth angles of the visible satellites belonging to the ELFO constellation pass through at points A and B.

Fig. 3

(a) Location of point A and B. (b) Azimuthal direction between visible satellites and A. (c) Azimuthal direction between visible satellites and B (d)-(e): Radian of dots: azimuths $\:{\lambda\:}_{i}$, radius of dots: elevations $\:{\phi\:}_{i}$. Satellite status at a certain location, blue: visible, gray: obscured. Numbers indicate satellite index. (d) The status of satellites at grid A location. (e) The status of satellites at grid B location. (elevation mask angle: 5°, 20:00 p.m., August 13, 2028, UTC).

The availability and precision of LNSS localization are generally assessed using metrics such as NVS and PDOP^27,28. The calculation of PDOP is given by Eq. 3 to 5.

$${\varvec{G}}=\left( {\begin{array}{*{20}{c}} { - \cos {\theta ^{(1)}}\sin {\alpha ^{(1)}}}&{ - \cos {\theta ^{(1)}}\cos {\alpha ^{(1)}}}&{ - \sin {\alpha ^{(1)}}}&1 \\ { - \cos {\theta ^{(2)}}\sin {\alpha ^{(2)}}}&{ - \cos {\theta ^{(2)}}\cos {\alpha ^{(2)}}}&{ - \sin {\alpha ^{(2)}}}&1 \\ \cdots & \cdots & \cdots & \cdots \\ { - \cos {\theta ^{(n)}}\sin {\alpha ^{(n)}}}&{ - \cos {\theta ^{(n)}}\cos {\alpha ^{(n)}}}&{ - \sin {\alpha ^{(n)}}}&1 \end{array}} \right)$$

(3)

$${\varvec{H}}={({{\varvec{G}}^\mathbf{T}}{\varvec{G}})^{ - 1}}$$

(4)

$$HDOP=\sqrt {{{\varvec{h}}_{11}}+{{\varvec{h}}_{22}}}$$

(5)

In Eq. 3, n represents the NVS, θ is the elevation angle, and α is the azimuth angle. In Eq. 5, h_ij represents the element in the i-th row and j-th column of matrix H.

Considering that positioning is not possible with NVS < 4 or the results are unreliable with PDOP > 10^29,30. Therefore, PDOP is only calculated when NVS ≥ 4. The available positioning condition is set to NVS ≥ 4 and PDOP ≤ 10.0 in this paper. The moments that meet the positioning criteria are referred to as available epochs.

Figure 4 shows the variation of NVS over time for the ELFO constellation at grid points A and B (with NVS calculated at time intervals of 5 min and 4 h, respectively), considering the terrain.

Table 3 compares the Ave-NVS and Ave-PDOP of the ELFO constellation at grid points A and B for time intervals of 5 min and 4 h, respectively, considering the terrain. The simulation time is from 04:00 UTC on August 13, 2028, to 04:00 UTC on August 14, 2028. In Table 3, the Ave-NVS is calculated based on all moments (All moments are obtained by dividing the simulation time by the time interval.), while the Ave-PDOP is calculated only based on the available moments.

Fig. 4

The variation of NVS at points A and B with simulation time under different time granularities.

Table 3 Comparison of Ave-NVS and Ave-PDOP at point A and B under different time granularities.

As can be seen from Fig. 4, for NVS, the 5-minute granularity can better reflect the variation of NVS over time. The 4-hour granularity lacks details (such as the maximum values), but it can reflect the overall trend of NVS changes. From Table 3, it can be seen that for Ave-NVS and Ave-PDOP, there is not much difference between the 5-minute granularity and the 4-hour granularity.

Given that the resampled DEM consists of 115,200 grids and each LNSS satellite has 289 position coordinates within the simulation time using a 5-minute granularity, the visibility analysis requires calculating the elevation angle and azimuth for each DEM grid, as well as identifying the terrain DEM grids and the maximum elevation angle. The number of calculations for these results is approximately 266 million (115,200 × 289 × 8), with the total computational load exceeding 1 billion. Such a large computational load entails a significant time cost, posing difficulties for the validation of the method. Since the main indicators used subsequently are Ave-NVS and Ave-PDOP, this paper employs a 4-hour granularity for the visibility calculation of LNSS satellites.

Results of simulation analysis

Terrain analysis vs. no terrain analysis

Figure 5 shows the grayscale image of the resampled DEM overlaid with transparent colors, where the colors represent the values of Ave-NVS (in Fig. 5a, b and e, and 5f) or Ave-PDOP (in Fig. 5c, d and g, and 5h). Figures 6, 7, 8, 9 and 10, and 11 are similar to Fig. 5, with the DEM grayscale images being resampled from the original LOLA DEM introduced in Section Ⅲ. When the calculated Ave-PDOP per day is greater than 10 or PDOPs at every epoch are all NULL, Ave-PDOP per day is set to NULL, and black indicates that the Ave-PDOP per day is NULL.

Table 4 includes navigation performance metrics (NVS and PDOP). In Table 4, the average of NVS per day for the entire region is calculated using all grids, and the average of PDOP per day for the entire region is calculated by excluding the grids with PDOP is NULL. As shown in Fig. 5; Table 4, after taking terrain occlusion into account, the Ave-PDOP per day increases while the Ave-NVS per day decreases for both the ELFO constellation and the ERGO constellation. Considering the topography of the area will lead to a decrease in the number of observable satellites employed for positioning, so developed LNSS constellation simulation with the existing lunar DEM data can evaluate lunar positioning performance more effectively.

As can be seen from Table 4; Fig. 5, ELFO Constellation has a better Ave-NVS than ERGO Constellation. ELFO Constellation performs better than ERGO Constellation for Ave-PDOP. However, ELFO Constellation has also a higher percentage of poor PDOP (with terrain: 17.20%, without terrain: 16.67%) than the ERGO Constellation (with terrain: 3.13%, without terrain: 0.81%). The overall performance of the ELFO constellation’s NVS is superior to that of the ERGO constellation, the availability of the ELFO constellation is notably inferior compared to the ERGO constellation.

Table 4 Constellation navigation performance metrics. (elevation mask angle: 5°; number of satellites: 8).

Fig. 5

Positioning capability of the two constellations without and with terrain, elevation mask angle: 5°, number of satellites: 8, 2028.08.13. Ave-NVS or Ave-PDOP per day at each grid in the southern polar region of the Moon, black: Ave-PDOP value of the grid is NULL.

For the ELFO constellation, Table 5 provides the regional metric values without considering terrain from Reference¹¹, as well as the results with and without considering terrain in this paper. The calculation methods for average NVS and average PDOP in Table 5 are the same as those in Table 4. Since Reference¹¹ does not provide the value of the elevation mask angle, the elevation mask angle used in this section is 5° (both with and without terrain). Therefore, there are differences between the average NVS and average PDOP values without considering terrain in this paper and those in Reference¹¹, although the differences are small. The results without considering terrain are closer to those in Reference¹¹, which indicates that considering terrain is necessary.

Table 5 Comparison with literature on selected constellations.

Consider terrain under different conditions with terrain analysis

Increasing the number of satellites in orbit will change the value of true periapsis angle (the other five Kepler parameters remain unchanged). Two satellites are added to the two orbits of ELFO constellation and ERGO constellation respectively. After adding four satellites, the true periapsis angle (ν) of ELFO constellation becomes: 0, 60, 120, 180, 240, 300; the true periapsis angle (ν) of ERGO constellation becomes 30, 90, 150, 210, 270, 330. The results of varying the elevation mask angle and the number of satellites in the constellation with the terrain occlusion are shown in Fig. 6.

Figure 6a and d depict the results of overlaying the DEM with Ave-NVS (Fig. 6a and b) and Ave-PDOP (Fig. 6c and d) under the condition that the number of satellites in the constellation is 8 and the elevation mask angle is 10°.

Figure 6e and h display the results of overlaying the DEM with Ave-NVS (Fig. 6e and f) and Ave-PDOP (Fig. 6g and h) under the condition that the number of satellites in the constellation is 12 and the elevation mask angle is 5°.

Figure 7 shows the daily available positioning numbers per day for the entire region for both the ELFO and ERGO constellations, considering various factors such as the number of satellites, and elevation mask angles.

Fig. 6

Positioning capability of the two constellations with terrain under different elevation mask angles and varying numbers of satellites. (a-d): elevation mask angle: 10°, number of satellites: 8, (e-h): elevation mask angle: 5°, number of satellites: 12, 2028.08.13. Black indicates that the Ave-PDOP value of the grid is NULL.

Table 6 compares the values of Ave-PDOP and Ave-NVS under different conditions for the ELFO and ERGO constellations, and the proportions of NVS > 4 and Ave-PDOP < 10. In Table 6, the average of NVS per day for the entire region is calculated by using all grids, and the average of PDOP per day for the entire region is calculated by removing the grids of PDOP is NULL.

According to Fig. 6a and d; Table 6, increasing the elevation mask angle filters out more visible satellites, resulting in poorer performance of NVS and PDOP, and black regions have been increased. Because the ERGO constellation contains fewer visible satellites, its average of PDOP is more affected. Regardless of mask angle conditions, the percentages of Ave-NVS ≥ 4 and the available number are worse in ERGO constellation than in the ELFO constellation.

Improving the constellation’s number of satellites will greatly enhance navigation performance. Figure 6e and h; Table 6 show that the Ave-NVS has been significantly improved overall, and black regions have been significantly reduced, regardless of ELFO constellation or ERGO constellation. The Ave-PDOP of both constellations also has been improved significantly overall. After the addition of 4 satellites, the percentage of NVS ≥ 4 increased by 27.48% in the ELFO constellation and 24.05% in ERGO constellation; the percentage of PDOP ≤ 10 increased by 44.22% in the ELFO constellation and 25.27% in the ERGO constellation.

According to Fig. 7; Table 6, the increase in the elevation mask angle will lead to a decrease in epochs of available positioning. Meanwhile, with the incorporation of an additional 4 satellites, the ELFO and ERGO constellations have experienced a substantial rise in the daily number of available positioning across most regions.

Fig. 7

The available number (NVS ≥ 4 & PDOP ≤ 10) during a specific day (2028.08.13) across various conditions.

Table 6 Comparison of NVS and PDOP related indicators of simulated constellations with terrain under different conditions. (elevation mask angle of 5° and 10°; number of satellites: 8 and 12) on 2028.08.13 (UTC).

Hybrid constellation results analysis

Introduction to NRHO

The Near-Rectilinear Halo Orbit (NRHO) is a type of periodic orbit within the Earth-Moon three-body system, characterized by its extremely high eccentricity (e ≈ 0.9). It orbits with a period of about 7 days. Although it is not as stable as popular circumlunar orbits³¹, its visibility to the Moon’s polar regions is practically constant³², making it appropriate as a navigation satellite orbit targeting the polar regions.

The Southern L2 NRHO orbit family is a nearly stable member of the halo orbit family³³, it offers good coverage of the southern polar region of the Moon and has low maintenance costs^34,35.

This paper simulates the motion trajectory of a satellite in the given parameter NRHO for 7 days. The simulation parameters of NRHO are as follows:

$\:{r}_{x}$: 1.6062080E + 4 (km), $\:{r}_{y}$: −1.36955848E + 3 (km).

$\:{r}_{z}$: −7.13184099E + 4 (km), $\:{v}_{x}$: −2.53190657E-1 (km/s).

$\:{v}_{y}$: −1.18595078E + 0 (km/s), $\:{v}_{z}$: −1.06887542E-2 (km/s).

Starting time: 2028.08.13 04:00 UTC.

The reference frame used for the NRHO simulation parameters is the Earth-Moon rotating frame.

One NRHO satellite performance of the southern polar region of the Moon is investigated here. Record the satellite’s position every 4 h during the simulation time, and a total of 42 observations of every grid can be collected for 7 days. Then one NRHO’s visible epochs at each grid with or without terrain for 7 days are calculated and analyzed. Figure 8 shows the visible epochs’ number to one NRHO satellite at each grid within 7 days (2028.08.13–2028.08.20 UTC).

Fig. 8

One NRHO’s visible epochs’ number at each grid within 7 days (2028.08.13–2028.08.20 UTC. Elevation mask angle: 5°).

Figure 8 shows that the NHRO satellite has good visibility in most of the target area. After considering terrain, the number of observations in most locations can still be maintained at 40, with the maximum number of 41 and the lowest of 34, demonstrating the advantage of the designed NRHO constellation in the lunar southern pole region.

The results of the hybrid constellations

The performance of a hybrid constellation with an additional one NRHO on the southern polar region of the Moon is also investigated. ELFO-NRHO constellation consists of one NRHO satellite and 8 ELFO constellation satellites, while the ERGO-NRHO constellation consists of one NRHO satellite and 8 ERGO constellation satellites.

Figure 9a, b and e, and 9f display the Ave-NVS while Fig. 9c, d and g, and 9h show the Ave-PDOP for each grid of ELFO-NRHO constellation and ERGO-NRHO constellation with the addition of an NRHO satellite on 2028.08.13 (UTC).

The analysis presented in Fig. 9 reveals that hybrid orbital constellations exhibit enhanced characteristics when contrasted with singular orbital configurations. Specifically, there is an augmentation in the proportions of orange and red segments, a significant reduction in the black area, and an increased presence of blue and green segments. These findings underscore the heightened visibility and precision achieved through the utilization of hybrid constellations.

The comparative analysis in Fig. 10 assesses the available number in which the ELFO-NRHO and ERGO-NRHO constellations can be effectively positioned within a single day, with or without the influence of terrain. From Fig. 10, it can be observed that the availability of the ELFO-NRHO constellation is superior to that of the ERGO-NRHO constellation.

Fig. 9

Positioning capability of the two hybrid constellations without and with terrain. Elevation mask angle: 5°, number of satellites: 9 (8 + 1NRHO), 2028.08.13. Black indicates that the Ave-PDOP value of the grid is NULL.

To obtain the positioning accuracy of hybrid constellations, it is essential to consider error sources and establish an error model. The standard error model used in this study is derived from the research conducted by³⁶, where errors related to the ionosphere and troposphere are not considered due to the absence of a lunar atmosphere. The bias in satellite clock errors is based on the study by¹¹.

Table 7 provides an overview of various error sources and biases^11,36,37. UERE is a user equivalent range error.

In Table 8, the standard deviation of positioning accuracy performance (σ_p) is obtained by the formula 6³⁶, In the formula, σ_UERE represents the standard deviation of UERE.

$${\sigma _p}=\text{P}\text{D}\text{O}P \times {\sigma _{\text{U}\text{R}\text{U}\text{E}}}$$

(6)

Fig. 10

The available number of NVS ≥ 4 & PDOP ≤ 10 during a specific day (2028.08.13). Elevation mask angle: 5°, number of satellites: 9 (8 + 1 NRHO).

Table 7 Standard error model (UERE is defined as the Root-Sum-Square (RSS) of all error components, error components are satellite orbit error, multipath, receiver noise and satellite clock.).

From Table 8, it is evident that the terrain significantly influences the positioning accuracy performance at points A and B. This observation indicates the importance of considering terrain effects in regions characterized by substantial changes in elevation.

Figure 11 shows the average positioning accuracy performance (Ave-σ_p) in the lunar south pole region over a full day for two hybrid constellations that include NRHO.

Table 8 NVS, PDOP and positioning error of point A and B at different times of the day (ELFO-NRHO constellation, 2028.08.13 UTC).

Fig. 11

Average positioning accuracy performance of the two hybrid constellations without (a–b) and with (c–d) terrain. Elevation mask angle: 5°, number of satellites: 9 (8 + 1NRHO), 2028.08.13. Black indicates that NVS are less than 4 and cannot be positioned.

Table 9 Performance indicators for ELFO-NRHO and ERGO-NRHO hybrid orbit constellation navigation.

Table 9 compares the values with and without terrain obstruction calculations. In Table 9, average NVS per day for the entire region is calculated by using all grid, and average PDOP and average σ_p per day for the entire region is calculated by removing the grids of PDOP is NULL.

Table 9 shows that with the addition of an NRHO orbital satellite and taking terrain into account, the Ave-PDOP of the ELFO-NRHO constellation reduced from  5.536 (Table6) to 3.52, a 36.42% decrease. The Ave-NVS increased by 16.74%, the percentage of PDOP ≤ 10 climbed by 46.88%, and the percentage of NVS ≥ 4 increased by 29.69%. The ERGO-NRHO constellation, on the other hand, witnessed a 28.84% decline in Ave-PDOP, a 20.33% increase in Ave-NVS, a 27.97% increase in the proportion of PDOP ≤ 10, and a 24.84% increase in the proportion of NVS ≥ 4. In terms of positioning precision, the average σ_p of the ELFO-NRHO constellation in the entire southern polar region of the Moon is very close to that of the ERGO-NRHO constellation.

As shown in Figs. 9, 10 and 11, and Table 9, the visibility and positioning precision of the hybrid orbit constellations incorporating NRHO have been considerably enhanced. With only one NRHO orbit satellite added, the indicators (Ave-PDOP, proportion of PDOP ≤ 10, proportion of NVS ≥ 4, number of times can be positioned) of the ELFO-NRHO and ERGO-NRHO constellations are all close to or better than the values of the ELFO or ERGO constellations with 12 satellites (in Table 6), and the Ave-PDOP values in hybrid constellations containing NRHO are less affected by terrain. As a result, NRHO is suitable for a lunar navigation satellite system, it is more cost-effective and can provide excellent navigation and positioning performance. Regardless of terrain, the ELFO-NRHO constellation has a higher proportion of PDOP ≤ 10 than the ERGO-NRHO constellation. However, the proportion of PDOP ≤ 5 is not as good as the latter, its regional Ave-PDOP performance is inferior to that of the ERGO-NRHO constellation. Moreover, the average positional accuracy of the ERGO-NRHO constellation is significantly superior to that of the ELFO-NRHO constellation across the majority of the lunar southern polar regions.

Discussion

The objective of this paper is to analyze the impact of terrain on the performance of the Lunar Navigation Satellite System (LNSS). This study builds on previous constellation schemes, with the two selected LNSS constellations derived from References^4,11. Since Earth-based GNSS satellites are not utilized, errors resulting from the ionosphere and troposphere are not taken into account. Based on lunar DEM (Digital Elevation Model) data, the visibility of satellites is assessed, and the influence of the lunar south pole terrain on the selected LNSS performance is analyzed.

The limitations of this study include the lack of signal propagation analysis in combination with DEM data and the absence of estimation filters. The positioning performance and feasibility for static users in the simulated constellations are evaluated, but no in-depth quantitative research on high-precision positioning effects is conducted. Consequently, no further simulation of signals is performed. For positioning errors, only rough assessments are made using formulas.

If a more comprehensive quantitative analysis is required, the following factors need to be considered in the signal simulation domain:

• Receiver noise assumptions and code/carrier phase error models.

• Multipath and non-line-of-sight (NLOS) effects.

• Clock biases and stability assumptions (for both user and satellites).

• Doppler effects, line-of-sight visibility, and error propagation through filters.

While our work provides valuable insights into the improvements achieved through DEM-based methods, it is important to contrast our findings with recent studies in the field:

Bhamidipati et al. present a novel concept for a SmallSat-based lunar navigation and communication satellite system (LNCSS) near the lunar south pole, utilizing Earth-GPS time transfer³⁸.

Iess et al. assess the potential of low-mass, low-power MEMS accelerometers for directly measuring non-gravitational accelerations (NGA) and propose methods to enhance ephemeris validity by integrating accelerometer data, radio-metric measurements, and advanced modeling techniques³⁹.

Wang, Dixing, et al. investigate the challenges of spacecraft autonomous navigation in Earth–Moon space using Global Navigation Satellite Systems (GNSS), proposing an adaptive Kalman filter based on Carrier-to-Noise ratio (C/N0) and innovation vector to mitigate GNSS accuracy attenuation⁴⁰.

Fienga et al. introduce TCL, a lunar time scale based on IAU resolutions, assess its implementation for orbiting/ground-based clocks, and evaluate enhancements to the lunar reference frame through retro-reflector deployment and orbiter altimetry⁴¹.

In our future study, we will refer to the above research to conduct signal simulations and perform more precise orbit determination, in order to achieve more accurate simulation results, while also implementing positioning simulations for dynamic users.

Conclusion

The Moon’s Southern Pole is of interest, both as a scientific treasure trove for understanding the evolution of the solar system and as a potential exploration site. The architecture of two LNSS constellations is examined in this article. The first is ELFO, which is a popular choice for LNSS, and the second is ERGO, which has long-term stability (no orbital maintenance required within 10 years). Given the complexity of the terrain in the lunar southern pole region, the impact of terrain blocking satellite signal coverage was assessed using available lunar DEM data. Simultaneously, an experiment with changing the elevation mask angle and the number of satellites in the constellation is carried out to examine the effects on the NVS and PDOP indicators. NVS and PDOP are used to analyze and evaluate the positioning performance of the two constellations. Simultaneously, because NRHO has good polar coverage, the positioning performance analysis and evaluation of hybrid orbit constellations with NRHO are carried out.

• Performance comparison of the ELFO and ERGO constellations: increasing the number of satellites improves the visibility and accuracy of the navigation satellite constellations, while increasing the elevation mask angle degrades the constellations’ performance. When considering terrain or not considering terrain, the visibility of the ELFO constellation surpasses that of the ERGO constellation. However, in terms of positioning accuracy, the ERGO constellation’s Ave-PDOP and the proportion of PDOP ≤ 10 outperform the ELFO constellation, indicating that the ERGO constellation is expected to be more accurate in positioning.

• Performance comparison of ELFO-NRHO and ERGO-NRHO hybrid constellations: The verification of the NRHO over, for 7 days has demonstrated its suitability for satellite navigation orbits in the lunar southern pole region. Despite the requirement for mobile maintenance, which incurs additional economic costs, the visibility of NRHO remains consistently stable in polar areas. At the same time, NRHO has a significant effect on improving the overall performance of the constellation. The ELFO-NRHO constellation still has a higher average NVS than the ERGO-NRHO constellation. Meanwhile, The Ave-PDOP and the proportion of PDOP ≤ 5 of the ERGO-NRHO constellation are also better than those of the EFLO-NRHO constellation. With the addition of one NRHO, the navigation performance indicators are significantly improved compared to the single type orbital constellation of 8 satellites, and are close to or better than the values of single type orbital constellation of 12 satellites. The average positioning precisions are almost from 34 to 44 m. ERGO-NRHO constellation demonstrates significantly superior positioning precision, making it more suitable for the establishment of LNSS. One additional NRHO satellite can improve visibility and the accuracy significantly. For lunar polar research, NRHO contributes to a reduction in the total number of satellites in the constellation, lowering satellite production and launch costs.

The simulation results demonstrate that the LNSS simulator developed in this study, incorporating terrain data, is capable of providing a more effective assessment of lunar surface positioning performance. Furthermore, the study validates the positioning performance in the southern polar region of the Moon for different single and hybrid orbit constellations. The research on LNSS is still in the stage of simulation analysis and verification. This paper verifies the navigation performance of the selected satellite orbit and constellation, it provides a reference for the simulation analysis of LNSS. Adding the positioning error model, coverage, signal duration, and other indicators to verify and analyze the performance of LNSS at the southern pole of the Moon will be considered in the future.