Conceptual Development of Differential Lunar Navigation Satellite System on the Moon Using a Swarm of Lunar Rovers

Euiho Kim; Danim Jung,; Donguk Kim

doi:10.33012/navi.753

Abstract

This study introduces strategies for the establishment of a differential lunar navigation satellite system (DLNSS) on the Moon using a swarm of lunar rovers and provides a performance analysis. The proposed DLNSS can enhance user positioning accuracy by broadcasting differential correction messages generated using raw measurements collected at multiple reference receivers with accurately known antenna coordinates. However, accurately determining the coordinates of reference receiver antennas presents a challenge, particularly in the absence of precise positioning methods on the Moon. In our proposed scheme for establishing a DLNSS on the Moon, we address this challenge by employing cooperative positioning techniques using a swarm of lunar rovers with a lunar navigation satellite system (LNSS) and ultra-wideband technology. Our simulations show that the proposed LNSS differential corrections achieve a three-dimensional user positioning accuracy that is better than a few meters with four or eight LNSS satellites.

Keywords

1 INTRODUCTION

The United States National Aeronautics and Space Agency (NASA), European Space Agency (ESA), Japan Aerospace Exploration Agency (JAXA), and other international space agencies have recently undertaken lunar navigation satellite system (LNSS) development programs to support various manned and unmanned missions for lunar explorations (Israel et al., 2020; Murata et al., 2022). As system design considerations, elliptical lunar frozen orbits (ELFOs), near-circular polar orbits, and near-rectilinear halo orbits (NRHOs) have been proposed with four to eight satellites (Bhamidipati et al., 2023; Schönfeldt et al., 2020). An ELFO provides stable coverage of the lunar polar regions and is inherently insensitive to external perturbations, thereby eliminating the need for station-keeping under the influence of gravitational forces (Ely & Lieb, 2006). NRHOs have orbital stability properties that can maintain NRHO-like motion over a long duration without frequently requiring propellant resources (Zimovan et al., 2017). In addition to their orbit stability characteristics, near-circular polar orbits can allow Moon satellites to perform multiple functions, supporting a number of tasks such as navigation, communications, data relay, and observation of deep space objects (Ivashkin & Gordienko, 2023).

Lunar navigation systems proposed by NASA, ESA, and JAXA under the LunaNet interoperability specification initially aim to provide navigation capabilities in the Moon’s south pole region using sparse satellite constellations (Giordano et al., 2023; Ryden & Volle, 2025). These systems are designed to emulate terrestrial global navigation satellite system (GNSS) architectures, enabling rapid deployment and facilitating lunar exploration by both governmental and commercial stakeholders. Typically comprising four to eight satellites, each constellation employs tailored orbital parameters to achieve high temporal coverage and enhanced horizontal positioning accuracy in the targeted region. However, without a dedicated LNSS monitoring network on the surface of the Moon, the orbit determination and time synchronization of the LNSS would depend on terrestrial GNSS signals, which constrains overall system performance (Delépaut et al., 2020; Winternitz et al., 2019). As a result, early LNSS operations are expected to suffer from substantial ranging errors and poor dilution of precision (DOP), leading to positioning errors on the order of several tens of meters, whether used independently or in conjunction with a terrestrial GNSS (Murata et al., 2022). This level of positioning accuracy is inadequate for many missions involving autonomous vehicles and robotic platforms. Therefore, improved positioning performance is critical during the early stages of lunar exploration, until a more comprehensive and dedicated lunar constellation can be established (Stallo et al., 2023).

Precise positioning with a terrestrial GNSS on Earth is typically achieved through differential positioning techniques, such as ground-based augmentation systems, satellite-based augmentation systems, real-time kinematic positioning, and precise point positioning, all of which require accurately surveyed coordinates of reference receivers (Blanch et al., 2012; Hou et al., 2023; Kim & Kim, 2022; Walter, 2017). Similarly, to enable differential positioning techniques on the Moon, it is necessary to first obtain accurately surveyed coordinates of reference receivers. As discussed previously, an LNSS alone or an LNSS integrated with a terrestrial GNSS cannot determine reference receiver coordinates with sufficient accuracy, owing to poor positioning performance. To achieve a higher positioning accuracy in the absence of a well-developed positioning infrastructure, cooperative positioning (CP) has been studied as an effective positioning method that circumvents the need for a dense infrastructure or high-power transmission (Bulusu et al., 2000; Capkun et al., 2001; Niculescu & Nath, 2001; Savarese et al., 2001; Savvides et al., 2001; Zhang et al., 2020). In CP, the positioning infrastructure and inter-range measurements between agents are exploited to obtain better positioning accuracy. CP positioning algorithms are often classified as Bayesian or non-Bayesian estimators, depending on whether the position states are treated as deterministic or random parameters. Non-Bayesian estimators include the range least-squares (LS), squared-range LS, range weighted LS (WLS), and squared-range WLS (Beck et al., 2008; Chen & Ho, 2013; Nguyen et al., 2015; Wymeersch et al., 2009). The squared-range LS and WLS have been solved using generalized trust region subproblem techniques. The performance of CP based on these non-Bayesian estimators or using slightly different approaches has been evaluated in prior studies (Alsindi et al., 2006; Blatt & Hero, 2006; Cheng et al., 2005; Wymeersch et al., 2009).

Well-known Bayesian approaches for localization include sum-product algorithms and particle filtering (Kschischang et al., 2001; Loeliger, 2004; Yang, 2012). Although Bayesian approaches are more complex and require a relatively heavier computational load, they can result in better positioning accuracy than LS- or Kalman-filter-based CP methods. Early studies on particle-filter-based CP for multi-agent robots have been conducted by many research groups (Fox et al., 1999; Howard et al., 2003). Recently, Zhang et al. (2020) introduced a direct particle filter-based distributed network localization method that has low complexity and is suitable for a real-time dense-network localization. Zhang et al. (2022) introduced a framework and simulation approach for a cooperative pose estimation. Sottile et al. (2011) proposed a hybrid GNSS–terrestrial CP method that fuses ranging measurements from the GNSS with a local terrestrial network.

In this study, a range-based WLS CP method was applied to enable a differential LNSS (DLNSS) on the Moon. The proposed DLNSS architecture, shown in Figure 1, comprises multiple lunar DLNSS rovers equipped with onboard LNSS reference receivers and two-way ultra-wideband (UWB) ranging sensors. One of these rovers acts as a central processing unit receiving raw LNSS measurements from the DLNSS rovers, computes differential corrections, and monitors measurement integrity. We envision dispatching these mobile rovers from a mother-ship spacecraft, after which they autonomously navigate to their designated destinations using LNSS signals.

FIGURE 1

Illustration of the DLNSS set-up scheme

This image was generated using DALL-E, an AI model created by OpenAI.

CP techniques with LNSS and UWB measurements were employed to accurately estimate the position of the reference receiver onboard each rover. These techniques are capable of yielding decimeter- to meter-level positioning accuracy depending on the number of rovers and their layout. However, CP requires extensive message exchanges and meticulous measurement monitoring to avoid degradation in the state estimation of multiple rovers caused by a single measurement fault. Once the reference receiver antenna coordinates converged sufficiently, LNSS differential corrections were generated in the central processing unit using code and carrier measurements from the reference receivers. Although integrity monitoring and related parameter generations are important components of DLNSS, this study focuses on the establishment of DLNSS reference receivers and differential correction generations. The contributions of this study are summarized as follows:

A WLS-based CP approach using LNSS and UWB measurements is introduced.
The achievable positioning accuracy of the reference receiver antenna coordinates through CP is investigated.
A scheme for generating LNSS differential corrections is presented.
The effectiveness of differential corrections is analyzed for users near the south pole region of the Moon.

The above contributions shed light on strategies for developing ground reference stations on the Moon and the feasibility of DLNSS services, which will facilitate exploration of the Moon by providing more precise navigation capabilities.

This study begins with CP algorithms using LNSS and UWB measurements for estimating the reference receiver antenna coordinates of the DLNSS and an analysis of its asymptotic convergence. Section 3 discusses the scheme for generating differential corrections and analyzes the error characteristics of the differential corrections. Section 4 presents simulation results of the proposed schemes for a set of DLNSS rovers and ELFO LNSS constellations.

2 STRATEGY FOR DETERMINING DLNSS ROVER COORDINATES

In this study, it is assumed that the DLNSS rovers can drive to their destination using the LNSS position solution, such that they can be spread within a circle with a radius of a few hundred meters. Once the rovers are positioned, their positions are accurately estimated through CP by using LNSS and UWB inter-range measurements between the DLNSS rovers and post-process filtering techniques.

2.1 CP Algorithms

In the proposed CP method, the positions of the DLNSS rovers were determined in two steps. In the first step, the position of a set of DLNSS rovers was estimated using only the LNSS, which provides the initial position and position uncertainties of all DLNSS rovers for the following CP procedures. In the second step, the CP for all DLNSS rovers was implemented using the LNSS and UWB inter-rover range measurements. It was assumed that all DLNSS rover positions and corresponding covariances, as well as the UWB inter-rover range measurements in the DLNSS network, were available for each rover through dedicated communication channels throughout the entire CP procedure.

We assumed the presence of N_r DLNSS rovers and N_s LNSS satellites. Based on the CP framework reported by Nguyen et al. (2015), the index sets of the DLNSS rovers and LNSS satellites are $\begin{matrix} V_{r} = {1, 2, \dots, N_{r}} \end{matrix}$ and $V_{s} = {N_{r} + 1, N_{r} + 2, \dots, N_{r} + N_{s}}$ , respectively. The position vector of the DLNSS rovers is denoted as $p_{r} = {[\begin{matrix} p_{1}^{T}, \dots, p_{N_{r}}^{T} \end{matrix}]}^{T}$ , Where p_i includes the three-dimensional (3D) position and receiver clock errors. The first three elements of p_i are the position vectors, and the fourth element is the receiver clock error. The LNSS satellite position vector is $\begin{matrix} \begin{matrix} p_{s} = {[\begin{matrix} p_{N_{r} + 1}^{T}, \dots, p_{N_{r} + N_{s}}^{T} \end{matrix}]}^{T} \end{matrix} \end{matrix}$ . Denoting $\begin{matrix} p = [p_{r}^{T}, p_{s}^{T}] \end{matrix}$ , the CP problem can be written as follows:

${\hat{p}}_{r} = \underset{p_{r}}{\arg \min} f (p)$ 1

with:

$\begin{matrix} f (p) = \sum_{i \in ν_{r}} \sum_{j \in ν_{r} \cup ν_{s}, j > i} \frac{c_{i j}}{ξ_{i j}^{2}} {(d_{i j} - ‖ p_{i} (1 : 3) - p_{j} (1 : 3) ‖ - p_{i} (4))}^{2} \end{matrix}$

where d_ij is an inter-rover or rover-to-satellite range measurement. ξ_ij is a weight factor between rovers i and j or between rover i and LNSS satellite j. c_ij is a line-of-sight indicator between the two rovers or between the rover and satellite and is set to 1 if there is a line of sight. Otherwise, c_ij = 0.

To solve Equation (1), the initial ${\hat{p}}_{r}^{(0)}$ and its covariance matrix Q⁽⁰⁾ are obtained from the LNSS-only position solution. The superscript (·) indicates an iteration number. Because each rover estimates its own position using the LNSS only at the initial stage, it is assumed that $Q^{(0)} = diag (Q_{1}^{(0)}, \dots, Q_{r}^{(0)})$ where Q_i ∈ ℝ^4×4. Using ${\hat{p}}_{r}^{(0)}$ and Q⁽⁰⁾ as well as LNSS and inter-rover range measurements, the CP procedures in this paper estimate the rover positions sequentially from rover 1 to rover N_r. The position correction of rover 1, denoted as $δ {\hat{p}}_{1}$ , is computed via linearization:

$\begin{matrix} δ {\hat{p}}_{1}^{(1)} & = {(H_{1}^{⊤} W_{1} H_{1})}^{- 1} H_{1}^{⊤} W_{1} y_{1} \end{matrix}$ 2

where:

$\begin{matrix} H_{1} & = [\begin{matrix} H_{1, r} \\ H_{1, LNSS} \end{matrix}], W_{1} = [\begin{matrix} Σ_{1, r}^{(0) - 1} & 0 \\ 0 & Σ_{1, LNSS}^{- 1} \end{matrix}], y_{1} = [\begin{matrix} y_{1, r} \\ y_{1, LNSS} \end{matrix}] \end{matrix}$ 3

H_1,r is a geometry matrix that comprises the line-of-sight vectors from rover 1 to the other rovers, and H_1,LNSS is that from rover 1 to the LNSS satellites. Because H₁ changes slowly over time, it is considered to be time-invariant between the iterative CP procedures. y₁ is a linearized measurement vector, with UWB inter-rover measurements denoted as y_1,r and LNSS range measurements denoted as y_{1, LNSS}. The UWB and LNSS range measurement errors were assumed to be white noise, with standard deviations of σ_UWB and σ_L. The LNSS range measurement error model is discussed further in later sections. W₁ is a weighting matrix for the measurement y₁, based on the following:

$\begin{array}{l} Σ_{1, r}^{(0)} & = σ_{UWB}^{2} I_{N_{r} - 1, N_{r} - 1} + [\begin{matrix} 1_{1}^{2} {\tilde{Q}}_{2}^{(0)} 1_{1}^{2 ⊤} & 0 & 0 \\ 0 & ⋱ & 0 \\ 0 & 0 & 1_{1}^{N_{r}} {\tilde{Q}}_{N_{r}}^{(0)} 1_{1}^{N_{r} ⊤} \end{matrix}], \\ Σ_{1, LNSS} & = [\begin{matrix} σ_{L, N_{r} + 1}^{2} & \dots & 0 \\ ⋮ & ⋱ & ⋮ \\ 0 & \dots & σ_{L, N_{r} + N_{s}}^{2} \end{matrix}] \end{array}$ 4

where $\tilde{Q}$ is a 3×3 submatrix of Q having only position uncertainties. $1_{1}^{(\cdot)}$ is a line-of-sight row vector to a ranging source of (·) from the rover $1 . σ_{L, (\cdot)}^{2} = \frac{σ_{L}^{2}}{| \sin (e l (\cdot)) |}$ , and el(·) is an elevation angle of (·) satellite.

The covariance of ${\hat{p}}_{1}^{(1)}$ is as follows:

$\begin{matrix} Q_{1}^{(1)} & = {(H_{1}^{⊤} W_{1} H_{1})}^{- 1} \\ = {([\begin{matrix} H_{1, r} \\ H_{1, LNSS} \end{matrix}]}^{⊤} [\begin{matrix} Σ_{1, r}^{(0) - 1} & 0 \\ 0 & Σ_{1, LNSS}^{- 1} \end{matrix}] {[\begin{matrix} H_{1, r} \\ H_{1, LNSS} \end{matrix}])}^{- 1} \\ = {(\underset{B^{(1)}}{\underset{︸}{H_{1, r}^{⊤} Σ_{1, r}^{(0) - 1} H_{1, r}}} + \underset{A}{\underset{︸}{H_{1, LNSS}^{⊤} Σ_{1, LNSS}^{- 1} H_{1, LNSS}}})}^{- 1} \end{matrix}$ 5

Once the rover 1 state has been estimated, ${\hat{p}}_{1}^{(1)}$ and $Q_{1}^{(1)}$ are updated and sent to other rovers. Subsequently, the state estimation from rover 2 to rover N_r follows similarly, using Equation (2). When the state estimation process for rover N_r is completed, the first iteration is complete and the second iteration begins. The CP process is initiated whenever new LNSS measurements become available and may continue iterating until all rover positions stabilize with minimal change between LNSS measurement intervals, as shown in Figure 2. However, under conditions of poor ranging accuracy or significant noise, deviations in the position solution may persist beyond predefined thresholds for durations exceeding the typical convergence time. To mitigate this issue, fault detection and exclusion techniques based on measurement innovations or residuals can be employed (Wang et al., 2016; Xiong et al., 2021). Additionally, online noise estimation processes or fault-tolerant estimators may be utilized to enhance robustness (Bo et al., 2019; Xiong et al., 2023).

FIGURE 2

CP process for DLNSS rovers

The correlated errors resulting from the use of the LNSS within a small rover network introduce similar position biases across all CP solutions. Owing to the similarity of these biases, these errors cannot be effectively mitigated through inter-ranging measurements. The characteristics of these CP position errors will be validated through simulation results presented in later sections. Moreover, because of the iterative nature of this process, a single measurement fault in a single rover can adversely affect the state estimation of all rovers. Therefore, it is essential to implement reliable techniques for CP and monitoring. However, this topic is beyond the scope of this study and will be explored in detail in future work.

2.1.1 Asymptotic Convergence of CP Using LNSS and UWB Measurements

In the first step of the CP, the position of rover 1 was estimated through WLS using LNSS and UWB measurements from the other rovers. A and B in Equation (5) are covariance matrices whose characteristics are real symmetric and non-negative definite (NND). It should be noted that $\begin{matrix} Q_{1}^{(0)} = A . \end{matrix}$ Because $\sum_{1, r}^{(0)}$ is NND, B⁽¹⁾ is also NND (Horn & Johnson, 2012). We can follow similar steps to prove a further reduction in the positioning uncertainty during CP. Once all rover positions have been updated via UWB and LNSS measurements, the weighting matrix of the UWB measurements for rover 1 is formulated as follows:

$\begin{array}{r} \begin{matrix} Σ_{1, r}^{(1)} = σ_{U W B}^{2} I_{n - 1, n - 1} + [\begin{matrix} 1_{1}^{2} {\tilde{Q}}_{2}^{(1)} 1_{1}^{2 ⊤} & 0 & 0 \\ 0 & ⋱ & 0 \\ 0 & 0 & 1_{1}^{N_{r}} {\tilde{Q}}_{N_{r}}^{(1)} 1_{1}^{N_{r} ⊤} \end{matrix}] \end{matrix} \end{array}$ 6

Using $\sum_{1, r}^{(1)}$ the position uncertainty of rover 1 at the second epoch is similarly estimated with Equation (5):

$\begin{matrix} Q_{1}^{(2)} = {(\underset{B^{(2)}}{\underset{︸}{H_{1, r}^{⊤} Σ_{1, r}^{(1) - 1} H_{1, r}}} + \underset{A}{\underset{︸}{H_{1, LNSS}^{⊤} Σ_{1, LNSS}^{- 1} H_{1, LNSS}}})}^{- 1} \end{matrix}$ 7

In Equation (7), A is treated as shown in Equation (5), because it can be assumed to be constant for several seconds. In addition, H_1,r is similar in value for two consecutive epochs. If these terms are treated as identical in Equations (5) and (7), it is evident that trace (B⁽¹⁾) ≤ trace (B⁽²⁾) because trace $\begin{matrix} (Q_{i}^{(1)}) \leq trace (Q_{i}^{(0)}) \end{matrix}$ for i=2 through N_S. A and B are NND, and the sum of these NND matrices is also NND. Using the linear mapping property of a trace, we obtain the following:

$\begin{matrix} trace (A + B^{(1)}) = trace (A) + trace (B^{(1)}) \end{matrix}$ 8

and:

$\begin{matrix} trace (A + B^{(2)}) = trace (A) + trace (B^{(2)}) \end{matrix}$ 9

Therefore, trace $(A + B^{(1)}) \leq trace (A + B^{(2)})$ , and it also holds that trace ${\begin{matrix} {(A + B^{(2)})}^{- 1} \leq trace (A + B^{(1)}) \end{matrix}}^{- 1}$ . Then, trace $(Q_{1}^{(1)}) \leq trace (Q_{1}^{(2)})$ . A similar process can be applied to other rovers and iterations, which reduces the covariance of the rover position estimates during CP. It should be noted that H_1,r can differ significantly in two consecutive epochs, particularly at the beginning of the CP. However, the reduction in trace (Q) is also large in the early iteration stage, preserving the property of trace ${(A + B^{(k + 1)})}^{- 1} \leq trace {(A + B^{(k)})}^{- 1}$ , where k is the iteration number.

2.2 Post-Process Filtering of the CP Position Solutions

Unlike a terrestrial GNSS, LNSS constellations experience high DOP values that can result in large positioning errors exceeding 100 m. Consequently, the CP position solutions are inevitably affected by the ill-conditioned LNSS constellation geometry. To further refine the position of the stationary rover, it is beneficial to average the CP position solutions over an extended period. Because the positioning error is a function of the DOP, it is logical to use a weighted arithmetic mean to improve the accuracy. The proposed weighted arithmetic mean is as follows:

$\begin{matrix} {\hat{p}}_{i, a v g} & = \sum_{k} f_{i} (k) \cdot {\hat{p}}_{i} (k) \\ f_{i} (k) & = \frac{1 / D O P {(i, k)}^{α}}{\sum_{k} 1 / D O P {(i, k)}^{α}} \end{matrix}$ 10

where ${\hat{p}}_{i} (k)$ is the CP position estimate of the i-th rover at the k-th epoch. DOP(i,k) is the DOP value of the i-th rover at the k-th epoch, and α values of approximately 8–10 were found to be effective based on our simulations. The effectiveness of the post-process filtering is discussed in later sections.

3 LNSS DIFFERENTIAL CORRECTION AND ERROR ANALYSIS

This section first discusses methods for generating differential corrections by using the estimated coordinates of the lunar DLNSS rovers and their LNSS measurements. The uncertainty in the differential correction is analyzed as follows.

3.1 Differential Correction Generation

For an LNSS system, a carrier-smoothed pseudorange of satellite i received by rover j can be modeled as follows:

$\begin{array}{r} ρ_{j}^{i} = r_{j}^{i} + δ t_{j} - δ t^{i} + e_{j}^{i} \end{array}$ 11

where $r_{j}^{i}$ is the geometric range from rover j to LNSS satellite i, δt_j is the receiver clock error of rover j, δtⁱ is the clock error of satellite i, and $e_{j}^{i}$ represents carrier-smoothed multipath and noise. To generate a pseudorange correction (PRC) for the i-th LNSS satellite, the individual correction value for the j-th rover is calculated as follows:

$\begin{array}{r} {PRC}_{j}^{i} = R_{j}^{i} - ρ_{j}^{i} - (c \cdot Δ t_{SV}^{i}) \end{array}$ 12

where $R_{j}^{i}$ is the presumed geometric range from the j-th rover to the i-th satellite, based on the estimated rover position and broadcast LNSS ephemeris. $\begin{array}{r} Δ t_{S V}^{i} \end{array}$ is a satellite clock error correction, and c is the speed of light.

Similar to the terrestrial GNSS differential correction generation, the receiver clock error is estimated by averaging the PRC of all satellites visible at rover j with weighting factors k_i, such that we have the following:

$\begin{matrix} δ {\hat{t}}_{j} = \sum_{i \in S_{j}} k_{i} \cdot {PRC}_{j}^{i} \end{matrix}$ 13

where S_j is a set of LNSS satellites that are visible to the j-th rover. k_i is given by the following expression:

$\begin{matrix} k_{i} = \frac{l_{j}^{i^{⊤}} {\tilde{Q}}_{j} l_{j}^{i} + m_{j}^{i} (e l)}{\sum_{i \in S_{j}} l_{j}^{i^{⊤}} {\tilde{Q}}_{j} l_{j}^{i} + m_{j}^{i} (e l)} \end{matrix}$ 14

where m(el) is the variance of the multipath and noise as a function of the elevation angle el.

Now, subtracting $\begin{matrix} δ {\hat{t}}_{j} \end{matrix}$ from $\begin{matrix} {PRC}_{j}^{i} \end{matrix}$ yields the following:

$\begin{matrix} {PRC}_{c, j}^{i} = {PRC}_{j}^{i} - δ {\hat{t}}_{j} \end{matrix}$ 15

Further, a baseline differential correction for satellite i is calculated as follows:

$\begin{matrix} {PRC}_{TX}^{i} = \sum_{j \in R R_{i}} w_{j}^{i} \cdot {PRC}_{c, j}^{i} \end{matrix}$ 16

and:

$\begin{matrix} w_{j}^{i} = \frac{l_{j}^{i} {\tilde{Q}}_{j} l_{j}^{i ⊤}}{\sum_{j \in R R_{i}} l_{j}^{i} {\tilde{Q}}_{j} l_{j}^{i ⊤}} \end{matrix}$ 17

where RR_i is the set of DLNSS reference receivers that can view satellite i.

The overall CP process and differential correction generation are illustrated in Figure 3. Steps 2 and 3 are sufficiently long to refine the stationary DLNSS rover position. Methods for monitoring the integrity of the rover position are necessary and will be investigated in future work.

FIGURE 3

Overall process of the CP and differential correction generation using multiple rovers (RV) in the proposed DLNSS system

3.2 Error Analysis of Differential Corrections

In a terrestrial differential GNSS, the reference receiver coordinate error is typically ignored because it is relatively small compared with other errors. However, in this proposed DLNSS, the reference receiver coordinate errors, denoted as Δp_j, are large, and their impact on differential corrections can be significant. To address this, Equation (12) can be expanded as follows:

$\begin{matrix} {PRC}_{j}^{i} & = R_{j}^{i} - ρ_{j}^{i} - (c \cdot Δ t_{SV}^{i}) \\ = b^{i} - δ t_{j} - l_{j}^{i ⊤} \cdot Δ p_{j} - e_{j}^{i} \end{matrix}$ 18

where bⁱ includes the LNSS ephemeris error and the remaining satellite clock error. The receiver clock error, estimated via Equation (13), is impacted by Δp_j and bⁱ as follows:

$\begin{matrix} δ {\hat{t}}_{j} & = \sum_{i \in S_{j}} k_{i} \cdot {PRC}_{j}^{i} \\ = δ t_{j} + \sum_{i \in S_{j}} k_{i} \cdot (b^{i} - l_{j}^{i ⊤} \cdot Δ p_{j} - e_{j}^{i}) \\ = δ t_{j} + \underset{Δ δ t_{j}}{\underset{︸}{Δ δ t_{b, j} + Δ δ t_{p, j}}} + ϵ_{j} \end{matrix}$ 19

where:

$\begin{matrix} Δ δ t_{b, j} & = \sum_{i \in S_{j}} k_{i} \cdot b^{i}, Δ δ t_{p, j} = \sum_{i \in S_{j}} k_{i} \cdot (- l_{j}^{i^{⊤}} \cdot Δ p_{j}) \end{matrix}$ 20

∆δt_j can be considered as a reference receiver clock estimate error. ∆p_j further propagates to the differential corrections for rover j and the broadcast differential correction for satellite i as follows:

$\begin{matrix} {PRC}_{c, j}^{i} & = {PRC}_{j}^{i} - δ {\hat{t}}_{j} \\ = b^{i} - l_{j}^{i^{𝖳}} \cdot Δ p_{j} - e_{j}^{i} - Δ δ t_{j} \\ {PRC}_{T X}^{i} & = \sum_{j \in R R_{i}} w_{j}^{i} \cdot {PRC}_{c, j}^{i} \\ = b^{i} + \sum_{j \in R R_{j}} w_{j}^{i} \cdot (- l_{j}^{i^{𝖳}} \cdot Δ p_{j} - e_{j}^{i} - Δ δ t_{j}) \\ = b^{i} + \underset{Δ {PRC}^{i}}{\underset{︸}{μ^{i} + {\tilde{e}}^{i}}} \end{matrix}$ 21

and:

$\begin{matrix} μ^{i} = \sum_{j \in R R_{j}} w_{j}^{i} \cdot (- 1_{j}^{i ⊤} \cdot Δ p_{j} - Δ δ t_{j}) \end{matrix}$ 22

Therefore, the quality of the differential correction depends on the magnitude of Δ PRCⁱ, which is essentially an averaged value of rover position errors, receiver clock estimate error, and multipath-induced range errors. The rover position errors projected in the direction of satellite i can be considered as a bias in the short-term because the line-of-sight vectors change slowly. Then, the uncertainty of PRC_TX, i.e., Δ PRC, can be modeled as a Gaussian random vector as follows:

$ΔPRC \sim N (μ, Σ_{d c})$ 23

where:

$\begin{matrix} μ & = [\begin{matrix} \sum_{j \in R R_{j}} w_{j}^{i} \cdot (- 1_{j}^{1^{𝖳}} \cdot Δ p_{j} - Δ δ t_{j}) \\ \sum_{j \in R R_{j}} w_{j}^{2} \cdot (- 1_{j}^{2^{𝖳}} \cdot Δ p_{j} - Δ δ t_{j}) \\ ⋮ \\ \sum_{j \in R R_{j}} w_{j}^{N_{s}} \cdot (- 1_{j}^{N_{s}^{𝖳}} \cdot Δ p_{j} - Δ δ t_{j}) \end{matrix}], Σ_{d c} = [\begin{matrix} σ_{d c, 1}^{2} & 0 & \dots & 0 \\ 0 & σ_{d c, 2}^{2} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & σ_{d c, N_{s}}^{2} \end{matrix}] \end{matrix}$ 24

Here, $σ_{d c, i} = \sqrt{\sum_{j \in R R_{i}} w_{j}^{i^{2}} \cdot m_{j}^{i} (e l)}$ . The elements of µ are very similar because the weighting factors $w_{j}^{i}$ are also similar for all rovers. In addition, ∆δt_j is much larger than ∆p_j. Because µ is very similar for all satellites, the bias in µ will be removed in the user positioning process with a receiver clock. The characteristics of µ and ∑_dc will be presented via simulation results in later sections.

3.3 User Positioning Algorithms and Performance Evaluation

Similar to Equation (11), the code-phase measurement of LNSS satellite i for a user is as follows:

$\begin{matrix} ρ_{u}^{i} = r_{u}^{i} + δ t_{u} - δ t^{i} + e_{u}^{i} \end{matrix}$ 25

By applying a differential correction of Equation (16) to Equation (25) and subtracting the presumed geometric range $R_{u}^{i}$ from the user, we obtain the following:

$\begin{matrix} ρ_{c, u}^{i} & = ρ_{u}^{i} - R_{u}^{i} + {PRC}_{TX}^{i} - (c \cdot Δ t_{SV}^{i}) \\ = r_{u}^{i} - R_{u}^{i} + b^{i} + δ t_{u} + e_{u}^{i} + {ΔPRC}^{i} \\ = - l_{u}^{i ⊤} \cdot δ x_{u} - δ l_{u}^{i ⊤} \cdot δ x_{u} + δ t_{u} + e_{u}^{i} + {ΔPRC}^{i} \end{matrix}$ 26

where $δ l_{u}^{i}$ is an error in the value of $l_{u}^{i}$ . δx_u is a relative position vector from the true position to an approximated user position. $δ l_{u}^{i ⊤} \cdot δ x_{u}$ is an effective differential ranging error caused by ephemeris errors (Pullen et al., 2001). This linearized equation is used to evaluate the DLNSS user positioning accuracy in later sections.

The user position displacement and receiver clock error vector from presumed states, $δ {\hat{x}}_{u t}$ , can be estimated using a conventional LS formulation as follows:

$\begin{matrix} Y_{u} & = [\begin{matrix} ρ_{c, u}^{1} \\ ρ_{c, u}^{2} \\ ⋮ \\ ρ_{c, u}^{n} \end{matrix}], G_{u} = [\begin{matrix} - 1_{u}^{1^{⊤}} & 1 \\ - 1_{u}^{2^{⊤}} & 1 \\ ⋮ & ⋮ \\ - 1_{u}^{n^{⊤}} & 1 \end{matrix}], δ {\hat{x}}_{u t} = {(G_{u}^{⊤} Σ_{u}^{- 1} G_{u})}^{- 1} G_{u}^{⊤} Σ_{u}^{- 1} Y_{u} \end{matrix}$ 27

where $Σ_{u}^{- 1}$ is a weighting matrix. The user positioning accuracy within a coverage area is primarily affected by the differential correction errors in Equation (23), ephemeris decorrelation errors, user multipath, and noise. Thus, the total user differential ranging measurement uncertainty is modeled as $\begin{matrix} σ_{t o t}^{2} = σ_{e p}^{2} + σ_{d c}^{2} + σ_{u m}^{2} \end{matrix}$ , where σ_ep is the standard deviation of the ephemeris decorrelation error, σ_dc is the standard deviation of the differential corrections without ephemeris correction errors, and σ_um is the standard deviation of user multipath and noise errors.

4 SIMULATION RESULTS

This section presents the characteristics of the differential corrections generated by the DLNSS central processing unit through simulations. In this study, we considered the proposed constellations of four and eight LNSS satellites, referred to as ELFO4 and ELFO8, respectively (Akiyama et al., 2022; Melman et al., 2022). The ephemeris parameters are listed in Tables 1 and 2, and the LNSS ranging error characteristics are listed in Table 3. UWB ranging measurements can exhibit significant bias errors, reaching several tens of centimeters when the received signal strength is low. These biases have been modeled by UWB sensor manufacturers and researchers and can be effectively compensated for by using established models (Liu et al., 2023; Malajner et al., 2015). After bias correction, the UWB ranging accuracy exhibited a Gaussian distribution with a zero mean and a standard deviation of 10 cm. In this study, we modeled the UWB ranging accuracy with a zero-mean Gaussian distribution and a 10-cm standard deviation, assuming proper bias compensation.

View this table:

TABLE 1 Keplerian Orbital Parameters of the ELFO4 LNSS Constellation

View this table:

TABLE 2 Keplerian Orbital Parameters of the ELFO8 LNSS Constellation

View this table:

TABLE 3 Simulation Parameters Used for LNSS Ranging Error Characteristics

4.1 CP Results

This subsection provides CP test results for the south pole region of the Moon, obtained by using simulated LNSS constellations and UWB networks for 52.8 h. First, 10 DLNSS rovers in two different layouts, rover10₁ and rover10₂, were randomly spread within a circle with a 300-m radius to facilitate inter-ranging among the rovers, as shown in Figure 4. To inspect the positioning performance of the CP and its sensitivity to the rover layout, the DOPs and positioning errors of the standalone LNSS and CP with the networks of rover10₁ and rover10₂, respectively, were studied, with results shown in Figure 5. As shown in Figure 5, it is evident that the CP using the UWB inter-rover measurements significantly reduces the DOP and improves the positioning accuracy. Additionally, the two networks with rover10₁ and rover10₂ have almost identical DOPs and positioning errors; thus, the two layouts of the 10 rovers in Figure 4 exhibit little difference in CP positioning performance. To determine the sensitivity of the CP to a smaller number of rovers, the CP algorithm was implemented with three rovers, as shown in Figure 4. Figure 6 shows the DOP and positioning errors for the three rovers. Although the maximum values of the DOP and position errors are larger than those in the case of 10 rovers, the CP provides a better positioning performance than the standalone LNSS. Figure 7 shows the 3D ellipsoidal uncertainty of the position estimates of the rover10₁ network through CP with the ELFO4 and ELFO8 LNSS satellites. As expected, the rovers at the center of the rover network had the smallest ellipsoids, whereas the outer rovers had the largest ellipsoids. The major axes of the ellipsoid were aligned in the vertical direction because of the limited ranging source geometry in this direction. In general, a larger network size is advantageous, as it improves the network’s DOP and CP accuracy. However, caution must be exercised when increasing the transmission power of the UWB signals, as their wideband characteristics may cause interference with other communication systems.

DLNSS rover distribution on the lunar surface (specifically at CR1 [89.4555°S, 222.6192°E]) using ten rovers in two different layouts, rover101 and rover102, or three rovers

Figure 4

DLNSS rover distribution on the lunar surface (specifically at CR1 [89.4555°S, 222.6192°E]) using ten rovers in two different layouts, rover10₁ and rover10₂, or three rovers

DOPs and position errors of the (left) standalone LNSS, (middle) CP with the rover101 network, and (right) CP with the rover102 network For the standalone LNSS case, one rover in the rover101 network is used.

Figure 5

DOPs and position errors of the (left) standalone LNSS, (middle) CP with the rover10₁ network, and (right) CP with the rover10₂ network

For the standalone LNSS case, one rover in the rover10₁ network is used.

FIGURE 6

DOPs and position errors of the three rovers when using the CP

3D ellipsoid for the rover101 network using (left) ELFO4 and (right) ELFO8 LNSS satellites, based on the CP process results

FIGURE 7

3D ellipsoid for the rover10₁ network using (left) ELFO4 and (right) ELFO8 LNSS satellites, based on the CP process results

Figure 8(a) shows the relationship between the DOP and the root mean square (RMS) of the 3D position errors of the CP when the rover10₁ network is used. The color represents the number of occurrences in the cells. As shown in the figure, a higher number of occurrences of larger 3D RMS tend to be allocated in the cells with larger DOPs, as expected. Owing to the poor ranging accuracy of the LNSS, it can be noted that a small difference in DOP leads to position changes of tens of meters. As shown in Figure 5, similar DOPs were repeated, and the positioning errors tended to have a perturbed cyclic pattern approximated over 12 h. The weighting factors shown in Figure 8(b) based on Equation (10) were used to obtain the stationary DLNSS rover position. For example, Equation (10) assigns 18-fold-larger weights to the position solutions with a DOP of 1.8 compared to those with a DOP of 2.6, which is reasonable given that the position difference of the brightest cells at these DOP values is 15 m. Using the proposed weighting factor, we conducted 40 Monte Carlo simulations of the CP, each lasting 52.8 h. Each set of position solutions was averaged using Equation (10), and the averaged position errors of the rovers were 1.45, 0,48, and –1.32 m along the x, y, and z axes, respectively. Figure 9(a) shows the true and estimated rover positions with respect to the center of the rover10₁ network. Figure 9(b) shows the norms of the averaged 3D positioning errors of the 10 rovers from the 40 simulation cases. Figure 9 clearly illustrates the positioning error characteristics of the CP, showing that the position error vectors of all rovers generally align in the same direction with similar magnitudes. This behavior reflects the effect of correlated LNSS ranging errors that remain uncorrected during the CP process.

FIGURE 8

(a) Relationship between the DOP and 3D RMS positioning errors, as depicted through the 2D intensity of sample occurrence; (b) proposed weighting factor for average position solutions

FIGURE 9

(a) True and estimated position of all DLNSS rovers; (b) norm of averaged positioning errors of the 10 rovers over 40 cases of Monte Carlo simulations Each simulation ran for 52.8 h.

4.2 Differential Correction Errors and User Positioning Accuracy

The differential corrections, PRC_TX’ in Equation (21) were generated for the ELFO4 LNSS satellites using the rover10₁ network and their estimated antenna positions, as shown in Figure 9(a). Figure 10 shows the PRC_TX generated for each satellite over 24 h. The corresponding ∆PRC values are shown in Figure 11, and it can be seen that their values are very similar at one specific time. The reason for this result is that ∆δt_j, the receiver clock error for each rover, and the weighted sum have similar values in each ∆PRC, as shown in Figure 12, which displays the the contributions of the ephemeris and rover position errors to the differential correction errors. Here, the ∆δt_bj values for all rovers are almost identical because the DLNSS rovers are closely located and subject to virtually the same ephemeris errors, as expected from Equation (13). Overall, ∆δt_pj is significantly smaller than ∆δt_bj , ranging from –0.5 m to 2.5 m.

FIGURE 10

Simulation result of PRC_TX for the ELFO4 LNSS constellation and 10 DLNSS rovers over 24 h

FIGURE 11

∆PRC generated in the simulated differential corrections for each LNSS satellite over 24 h

FIGURE 12

(Left) ∆δt_bj and (right) ∆δt_pj terms included in the ∆PRC displayed in Figure 11

Figure 13 shows the distribution of ∆PRC for the four LNSS satellites over 24 h. In the figures, the mean value of ∆PRC for each time was removed because, as discussed earlier, the mean values will be eliminated by the receiver clock error. The mean and standard deviations of the histograms were 0 and 0.07 m for all satellites. It should be noted that all histograms appear similar because a pair of satellites traveled in the same orbits, and the effects of the two orbits on the ranging error characteristics were also very similar. Notably, the differential correction and its error characteristics for the ELFO8 LNSS constellations were similar and are not repeated here for brevity.

FIGURE 13

Histograms of mean-subtracted ∆PRC at each epoch for four LNSS satellites

Figures 14 and 15 show histograms of the user positioning error along the x, y, and z axes for the ELFO4 and ELFO8 satellites, respectively, when the horizontal DOP (HDOP) is less than 10. For the ELFO4 LNSS, the DLNSS user positioning accuracy using the generated differential corrections had means of 1.45, 0.48, and –1.32 m along the x, y, and, z axes, respectively. The corresponding standard deviations were 0.54, 0.19, and 1.75 m on the x, y, and, z axes, respectively. For the ELFO8 LNSS, the averaged user positioning errors were very similar to those observed for the ELFO4 LNSS, whereas the corresponding standard deviations were reduced to 0.28, 0.15, and 0.63 m, respectively. In fact, the averaged user position errors were derived from the averaged DLNSS rover position errors remaining in the differential corrections, which were 1.45, 0,48, and –1.32 m along the x, y, and z axes, respectively.

FIGURE 14

Histograms of user positioning errors (PE) in meters along the x, y, and, z axes at the center of the DLNSS network over 24 h for the ELFO4 LNSS satellites when the HDOP is less than 10

FIGURE 15

Histograms of user positioning errors (PE) in meters along the x, y, and, z axes at the center of the DLNSS network over 24 h for the ELFO8 LNSS satellites when the HDOP is less than 10

A user located within several tens of kilometers from the center of the DLNSS network is expected to experience positioning performance comparable to that of users within the network coverage, in contrast to terrestrial differential correction services. This difference is primarily due to the Moon’s extremely thin atmosphere, which minimizes atmospheric effects whereas the variations of DOPs and orbit errors remain small at those user locations. As illustrated in Figure 16, the 3D DOP values for users positioned 50 km east, north, west, and south of the DLNSS network center, when utilizing the ELFO4 constellation, show negligible variation. Additionally, the differential correction errors due to ephemeris decorrelation for these users are shown in Figure 17, demonstrating minimal degradation in correction performance over the covered area.

FIGURE 16

The 3D DOP values for users located 50 km east, north, west, and south of the DLNSS network center were nearly identical. As a result, the HDOPs of the users over a 24-h period overlapped, indicating minimal variation in horizontal positioning precision across these locations.

FIGURE 17

Differential correction error values due to ephemeris decorrelation for users located 50 km east, north, west, and south of the DLNSS network center over a 24-h period

5 CONCLUSIONS

This study presented strategies for establishing a DLNSS network using a swarm of rovers. The coordinates of 10 rover antennas were estimated using WLS-based CP techniques. The asymptotic convergence of the CP process was demonstrated using covariance matrix analysis. The paper then discussed the generation of differential corrections and examined the error characteristics caused by inherent DLNSS rover position errors. The proposed DLNSS scheme was simulated using LNSS ELFO constellations with four (ELFO4) or eight (ELFO8) satellites, as studied in previous works. Simulation results showed that the differential correction errors exhibited a bias derived from the DLNSS rover position errors and a standard deviation of 0.07 m. Owing to the differential correction bias, the DLNSS user positions also had biases of less than a few meters for both the ELFO4 and ELFO8 constellations when the HDOP was less than 10. The horizontal and vertical standard deviations of user positioning errors were 0.57 m and 1.74 m, respectively, for ELFO4 and 0.32 m and 0.63 m for ELFO8.

We believe that this study provides valuable insights for achieving precise positioning services on the Moon. The proposed DLNSS scheme can provide accurate guidance for spacecrafts and rovers during various lunar exploration missions. Additionally, the CP process can be applied to establish monitoring stations for LNSS constellations, which will be the focus of future research. The coverage area of a DLNSS network is expected to extend to several tens of coverage span or more owing to the extremely thin and weak atmosphere of the Moon, even though the network itself spans only a few hundred meters. A key advantage of the DLNSS is its ability to deliver a positioning accuracy of a few meters to an unlimited number of users. However, a more detailed understanding of the future LNSS system is required to better assess the coverage areas.

CONFLICT OF INTEREST

The authors declare that there are no conflicts of interest.

HOW TO CITE THIS ARTICLE:

Kim, E., Jung, D., & Kim, D. (2026). Conceptual development of differential lunar navigation satellite system on the Moon using a swarm of lunar rovers. NAVIGATION, 73. https://doi.org/10.33012/navi.753

ACKNOWLEDGMENTS

This research was supported by the Unmanned Vehicles Core Technology Research and Development Program through the National Research Foundation of Korea and the Unmanned Vehicle Advanced Research Center funded by the Ministry of Science and ICT, Republic of Korea (No. 2020M3C1C1A01086407). This work was also supported by the Future Space Navigation & Satellite Research Center through the National Research Foundation funded by the Ministry of Science and ICT, Republic of Korea (2022M1A3C2074404).

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

REFERENCES

↵
Akiyama, K., Murata, M., & Kogure, S. (2022). Differential positioning performance on lunar south pole region using lunar navigation satellite system. In Proc. of the 35th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+ 2022), Denver, Colorado, 628–638. https://doi.org/10.33012/2022.18370
Google Scholar
↵
Alsindi, N. A., Pahlavan, K., Alavi, B., & Li, X. (2006). A novel cooperative localization algorithm for indoor sensor networks. In 2006 IEEE 17th International Symposium on Personal, Indoor and Mobile Radio Communications, 1–6. https://doi.org/10.1109/PIMRC.2006.254056
Google Scholar
↵
Beck, A., Stoica, P., Li, J. (2008). Exact and approximate solutions of source localization problems. IEEE Transactions on Signal Processing, 56(5), 1770–1778. https://doi.org/10.1109/TSP.2007.909342
Google Scholar
↵
Bhamidipati, S., Mina, T., & Gao, G. (2023). A case study analysis for designing a lunar navigation satellite system with time transfer from the earth GPS. NAVIGATION, 70(4). https://doi.org/10.33012/navi.599
Google Scholar
↵
Blanch, J., Walter, T., Enge, P. (2012). Satellite navigation for aviation in 2025. Proc. of the IEEE, 100(Special Centennial Issue), 1821–1830. https://doi.org/10.1109/JPROC.2012.2190154
Google Scholar
↵
Blatt, D., & Hero, A. O. (2006). Energy-based sensor network source localization via projection onto convex sets. IEEE Transactions on Signal Processing, 54(9), 3614–3619. https://doi.org/10.1109/TSP.2006.879312
Google Scholar
↵
Bo, X., Asghar, R., & Yalong, L. (2019). Cooperative localisation of AUVs based on Huber-based robust algorithm and adaptive noise estimation. Journal of Navigation, 72(4), 875–893. https://doi.org/10.1017/S0373463319000018
Google Scholar
↵
Bulusu, N., Heidemann, J., & Estrin, D. (2000). GPS-less low-cost outdoor localization for very small devices. IEEE Personal Communications, 7(5), 28–34. https://doi.org/10.1109/98.878533
Google Scholar CrossRef
↵
Capkun, S., Hamdi, M., Hubaux, J.-P. (2001). GPS-free positioning in mobile ad-hoc networks. Proc. of the 34th Annual Hawaii International Conference on System Sciences (HICSS-34), 3481–3490. https://doi.org/10.1109/HICSS.2001.927202
Google Scholar
↵
Chen, S., & Ho, K. C. (2013). Achieving asymptotic efficient performance for squared range and squared range difference localizations. IEEE Transactions on Signal Processing, 61(11), 2836–2849. https://doi.org/10.1109/TSP.2013.2254479
Google Scholar
↵
Cheng, B. H., Hudson, R. E., Lorenzelli, F., Vandenberghe, L., & Yao, K. (2005). Distributed Gauss–Newton method for node localization in wireless sensor networks. In IEEE 6th Workshop on Signal Processing Advances in Wireless Communications, 915–919. https://doi.org/10.1109/SPAWC.2005.1506273
Google Scholar
↵
Delépaut, A., Giordano, P., Ventura-Traveset, J., Blonski, D., Schönfeldt, M., Schoonejans, P., Aziz, S., & Walker, R. (2020). Use of GNSS for lunar missions and plans for lunar in-orbit development. Advances in Space Research, 66(12), 2739–2756. https://doi.org/10.1016/j.asr.2020.05.018
Google Scholar
↵
Ely, T. A., & Lieb, E. (2006). Constellations of elliptical inclined lunar orbits providing polar and global coverage. Journal of the Astronautical Sciences, 54(1), 53–67. https://doi.org/10.1007/BF03256476
Google Scholar
↵
Fox, D., Burgard, W., Kruppa, H., & Thrun, S. (1999). A Monte Carlo algorithm for multi-robot localization (Technical Report No. CMU-CS-99-120). School of Computer Science, Carnegie Mellon University. https://doi.org/10.21236/ADA363774
Google Scholar
↵
Giordano, P., Swinden, R., Gramling, C., Crenshaw, J., & Ventura-Traveset, J. (2023). LunaNet position, navigation, and timing services and signals, enabling the future of lunar exploration. Proc. of the 36th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+ 2023), Denver, Colorado, 3577–3588. https://doi.org/10.33012/2023.19345
Google Scholar
↵
Horn, R. A., & Johnson, C. R. (2012). Matrix analysis. 2nd ed.Cambridge University Press.
Google Scholar
↵
Hou, P., Zha, J., Liu, T., & Zhang, B. (2023). Recent advances and perspectives in GNSS PPP-RTK. Measurement Science and Technology, 34(5), 051002. https://doi.org/10.1088/1361-6501/acb78c
Google Scholar
↵
Howard, A., Mataric, M. J., & Sukhatme, G. S. (2003). Putting the ‘I’ in ‘team’: An ego-centric approach to cooperative localization. Proc. of the 2003 IEEE International Conference on Robotics and Automation (ICRA 2003), 868–874. https://doi.org/10.1109/ROBOT.2003.1241702
Google Scholar
↵
Israel, D. J., Mauldin, K. D., Roberts, C. J., Mitchell, J. W., Pulkkinen, A. A., Cooper, L. V. D., Johnson, M. A., Christe, S. D., & Gramling, C. J. (2020). LunaNet: A flexible and extensible lunar exploration communications and navigation infrastructure. Proc. of the 2020 IEEE Aerospace Conference, 1–14. https://doi.org/10.1109/AERO47225.2020.9172509
Google Scholar
↵
Ivashkin, V. V., & Gordienko, E. S. (2023). Study of the possibility to develop a lunar navigation satellite system and a lunar orbital spaceport using the high circular orbits of the moon artificial satellite. Gyroscopy and Navigation, 14(1), 56–65. https://doi.org/10.1134/S2075108723010030
Google Scholar
↵
Kim, E., & Kim, S.-K. (2022). Antenna array–aided real-time kinematic using moving base stations. IEEE Access, 10, 116554–116563. https://doi.org/10.1109/ACCESS.2022.3219823
Google Scholar
↵
Kschischang, F. R., Frey, B. J., Loeliger, H.-A. (2001). Factor graphs and the sum–product algorithm. IEEE Transactions on Information Theory, 47(2), 498–519. https://doi.org/10.1109/18.910572
Google Scholar
↵
Liu, S., Yang, H., Mei, Z., Xu, X., He, Q. (2023). Ultra-wideband high-accuracy distance measurement based on hybrid compensation of temperature and distance error. Measurement, 206, 112276. https://doi.org/10.1016/j.measurement.2022.112276
Google Scholar
↵
Loeliger, H.-A. (2004). An introduction to factor graphs. IEEE Signal Processing Magazine, 21(1), 28–41. https://doi.org/10.1109/MSP.2004.1267047
Google Scholar
↵
Malajner, M., Planinšič, P., Gleich, D. (2015). UWB ranging accuracy. In Proc. of the 2015 International Conference on Systems, Signals and Image Processing (IWSSIP 2015), 61–64. https://doi.org/10.1109/IWSSIP.2015.7314177
Google Scholar
↵
Melman, F. T., Zoccarato, P., Orgel, C., Swinden, R., Giordano, P., & Ventura-Traveset, J. (2022). LCNS positioning of a lunar surface rover using a DEM-based altitude constraint. Remote Sensing, 14(16), 3942. https://doi.org/10.3390/rs14163942
Google Scholar
↵
Murata, M., Kawano, I., & Kogure, S. (2022). Lunar navigation satellite system and positioning accuracy evaluation. Proc. of the 2022 International Technical Meeting of the Institute of Navigation (ION ITM 2022), Long Beach, California, 582–586. https://doi.org/10.33012/2022.18220
Google Scholar
↵
Murata, M., Koga, M., Nakajima, Y., Yasumitsu, R., Araki, T., Makino, K., Akiyama, K., Yamamoto, T., Tanabe, K., Kogure, S., Sato, N., Toyama, D., Kitamura, K., Miyasaka, K., Kawaguchi, T., Sato, Y., Kakihara, K., Shibukawa, T., Iiyama, K., & Tanaka, T. (2022). Lunar navigation satellite system: Mission, system overview, and demonstration. Proc. of the 39th International Communications Satellite Systems Conference (ICSSC 2022), 12–15. https://doi.org/10.1049/icp.2023.1355
Google Scholar
↵
Nguyen, T. V., Jeong, Y., Shin, H., & Win, M. Z. (2015). Least square cooperative localization. IEEE Transactions on Vehicular Technology, 64(4), 1318–1330. https://doi.org/10.1109/TVT.2015.2398874
Google Scholar
↵
Niculescu, D., & Nath, B. (2001). Ad hoc positioning system. Proc. of the 2001 IEEE Global Telecommunications Conference (GLOBECOM 2001), 2926–2931. https://doi.org/10.1109/GLOCOM.2001.965964
Google Scholar
↵
Pullen, S., Lee, J., Luo, M., Pervan, B., Chan, F.-C., & Gratton, L. (2001). Ephemeris protection level equations and monitor algorithms for GBAS. Proc. of the 14th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GPS 2001), Salt Lake City, UT, 1738–1749. https://www.ion.org/publications/abstract.cfm?articleID=1852
Google Scholar
↵
Ryden, G., & Volle, M. (2025). NASA lunar communications relay and navigation systems (LCRNS) reference constellation 3.1 (tech. rep.). National Aeronautics and Space Administration. https://esc.gsfc.nasa.gov/static-files/LCRNS_Reference_Constellation_White_Paper_03_2025.pdf
Google Scholar
↵
Savarese, C., Rabaey, J. M., & Beutel, J. (2001). Location in distributed ad-hoc wireless sensor networks. Proc. of the 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2001), 2037–2040. https://doi.org/10.1109/ICASSP.2001.940391
Google Scholar
↵
Savvides, A., Han, C.-C., & Srivastava, M. B. (2001). Dynamic fine-grained localization in ad-hoc networks of sensors. Proc. of the 7th Annual International Conference on Mobile Computing and Networking (MobiCom ‘01), 166–179. https://doi.org/10.1145/381677.381693
Google Scholar
↵
Schönfeldt, M., Grenier, A., Delépaut, A., Giordano, P., Swinden, R., Ventura-Traveset, J., Blonski, D., & Hahn, J. (2020). A system study about a lunar navigation satellite transmitter system. Proc. of the 2020 European Navigation Conference (ENC 2020), 1–10. https://doi.org/10.23919/ENC48637.2020.9317521
Google Scholar
↵
Sottile, F., Wymeersch, H., Caceres, M. A., & Spirito, M. A. (2011). Hybrid GNSS–terrestrial cooperative positioning based on particle filter. Proc. of the 2011 IEEE Global Telecommunications Conference (GLOBECOM 2011), 1–5. https://doi.org/10.1109/GLOCOM.2011.6134002
Google Scholar
↵
Stallo, C., Carosi, M., De Leo, L., Musacchio, D., Cappa, M., & Boomkamp, H. (2023). Performance analysis of an extended lunar radio navigation system. Proc. of the 36th International Technical Meeting of the Satellite Division of the Institute of Navigation (ION GNSS+ 2023), Denver, Colorado, 3698–3708. https://doi.org/10.33012/2023.19290
Google Scholar
↵
Walter, T. (2017). Satellite-based augmentation systems. In Teunissen, P. J. G., & Montenbruck, O.(Eds.), Springer handbook of global navigation satellite systems, 339–361. https://doi.org/10.1007/978-3-319-42928-1_12
Google Scholar
↵
Wang, R., Xiong, Z., Liu, J., Xu, J., & Shi, L. (2016). Chi-square and SPRT combined fault detection for multisensor navigation. IEEE Transactions on Aerospace and Electronic Systems, 52(3), 1352–1365. https://ieeexplore.ieee.org/document/7511863
Google Scholar
↵
Winternitz, L. B., Bamford, W. A., Long, A. C., Hassouneh, M. (2019). GPS based autonomous navigation study for the lunar gateway. Annual American Astronautical Society (AAS) Guidance, Navigation, and Control Conference. https://ntrs.nasa.gov/citations/20190002311
Google Scholar
↵
Wymeersch, H., Lien, J., & Win, M. Z. (2009). Cooperative localization in wireless networks. Proc. of the IEEE, 97(2), 427–450. https://doi.org/10.1109/JPROC.2008.2008853
Google Scholar
↵
Xiong, J., Cheong, J. W., Xiong, Z., Dempster, A. G., Tian, S., & Wang, R. (2021). Integrity for multi-sensor cooperative positioning. IEEE Transactions on Intelligent Transportation Systems, 22(2), 792–807. https://ieeexplore.ieee.org/document/8926518
Google Scholar
↵
Xiong, J., Xiong, Z., Zhuang, Y., Cheong, J. W., & Dempster, A. G. (2023). Fault-tolerant cooperative positioning based on hybrid robust Gaussian belief propagation. IEEE Transactions on Intelligent Transportation Systems, 24(6), 6425–6431. https://doi.org/10.1109/TITS.2023.3249251
Google Scholar
↵
Yang, P. (2012). Efficient particle filter algorithm for ultrasonic sensor-based 2D range-only simultaneous localisation and mapping application. IET Wireless Sensor Systems, 2(4), 394–401. https://doi.org/10.1049/iet-wss.2011.0129
Google Scholar
↵
Zhang, S., Cokona, K., Pöhlmann, R., Staudinger, E., Wiedemann, T., & Dammann, A. (2022). Cooperative pose estimation in a robotic swarm: Framework, simulation and experimental results. Proc. of the 2022 30th European Signal Processing Conference (EUSIPCO 2022), 987–991. https://doi.org/10.23919/EUSIPCO55093.2022.9909666
Google Scholar
↵
Zhang, S., Staudinger, E., Jost, T., Wang, W., Gentner, C., Dammann, A., Wymeersch, H., & Hoeher, P. A. (2020). Distributed direct localization suitable for dense networks. IEEE Transactions on Aerospace and Electronic Systems, 56(2), 1209–1227. https://doi.org/10.1109/TAES.2019.2928606
Google Scholar
↵
Zimovan, E. M., Howell, K. C., & Davis, D. C. (2017). Near rectilinear halo orbits and their application in cis-lunar space. Proc. of the 3rd IAA Conference on Dynamics and Control of Space Systems (DyCoSS 2017), 161 1–20. https://engineering.purdue.edu/people/kathleen.howell.1/Publications/Conferences/2017_IAA_ZimHowDav.pdf
Google Scholar

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Conceptual Development of Differential Lunar Navigation Satellite System on the Moon Using a Swarm of Lunar Rovers

Abstract

1 INTRODUCTION