Single-Satellite Lunar Navigation via Doppler Shift Observables for the NASA Endurance Mission

2.1 Generation of Doppler Shift Measurements

Constrained to a single communication satellite, the Lunar Pathfinder, we utilize Doppler shift measurements as the observable for state estimation. Doppler shift measurements D are obtained by calculating the difference between the frequency of the signal emitted at the source f_source and the frequency of the signal received by the rover f_received due to relative motion between the satellite and the rover. Given that the Lunar Pathfinder satellite has not yet been launched into lunar orbit, in this work, we must simulate the Doppler shift measurements that are observed by the rover. In particular, the Doppler shift can be modeled as follows:

$D^{\left(t\right)}=f_{\mathrm{received}}\left(t\right)-f_{\mathrm{source}}=-\frac{{\stackrel{\cdot }{\rho }}_{\text{obs}}^{\left(t\right)}{f}_{\mathrm{source}}}{c}$1

where c = 299 792 458 m/s is the speed of light and ${\stackrel{.}{\rho }}_{\text{obs}}$ is the observed pseudorange rate, or the apparent rate of change in distance between the satellite and the rover.

The rover’s observed pseudorange rate measurements are a function of the true range rate ${\stackrel{.}{\rho }}_{\mathrm{true}}$, the relative clock drift between the satellite and the rover, and a measurement error term. The observed, or noisy, pseudorange rate model is as follows:

${\stackrel{\cdot }{\rho }}_{\text{obs}}^{\left(t\right)}={\stackrel{\cdot }{\rho }}_{\mathrm{true}}^{\left(t\right)}+c\left(\stackrel{\cdot }{\delta }{t}_{\mathrm{sat}}-\stackrel{\cdot }{\delta }{t}_{\mathrm{rov}}\right)+{ϵ}_{\stackrel{\cdot }{\rho }}^{\left(t\right)},{ϵ}_{\stackrel{\cdot }{\rho }}\sim N\left(0,{\sigma }_{\stackrel{\cdot }{\rho }}^2\right)$2

where $\stackrel{\cdot }{\delta }{t}_{\mathrm{sat}}$ and $\stackrel{\cdot }{\delta }{t}_{\mathrm{rov}}$ are the clock drifts of the satellite and rover, respectively. The pseudorange rate measurement error term ${ϵ}_{\stackrel{\cdot }{\rho }}$ is modeled as zero-mean white Gaussian noise with variance ${\sigma }_{\stackrel{\cdot }{\rho }}^2$, which is defined later in Section 2.5.3. The true range rate ${\stackrel{\cdot }{\rho }}_{\mathrm{true}}$ is defined as the projection of the satellite velocity onto the line of sight between the rover and the satellite (Misra & Enge, 2010):

${\stackrel{\cdot }{\rho }}_{\mathrm{true}}^{\left(t\right)}=v_{\mathrm{sat}}^{\left(t\right)}.\frac{x_{\mathrm{sat}}^{\left(t\right)}-x_{\mathrm{rov}}}{\left(\left(x_{\mathrm{sat}}^{\left(t\right)}-x_{\mathrm{rov}}\right)\right)},x^{\left(t\right)}={\left(\begin{array}{ccc}{x}^{\left(t\right)}& y^{\left(t\right)}& z^{\left(t\right)}\end{array}\right)}^T,v^{\left(t\right)}={\left(\begin{array}{ccc}{\stackrel{\cdot }{x}}^{\left(t\right)}& {\stackrel{\cdot }{y}}^{\left(t\right)}& {\stackrel{\cdot }{z}}^{\left(t\right)}\end{array}\right)}^T$3

where $x_{\mathrm{sat}}^{\left(t\right)}$ and $v_{\mathrm{sat}}^{\left(t\right)}$ represent the 3D satellite position and velocity, respectively, at time t and x_rov represents the stationary rover position in the Moon principal axis frame (Folta et al., 2022).

In this study, we assume that the rover clock drift remains constant throughout the measurement window and that the Lunar Pathfinder carries a stable atomic clock with negligible drift over this period. Although these assumptions simplify the formulation, they also reflect the lack of publicly available data on the expected performance of the clocks onboard the Lunar Pathfinder and the Endurance rover. In reality, the satellite clock may exhibit drift over extended observation periods, and the rover clock may experience higher-order drift effects. Although these effects are not modeled in this study, future work could incorporate higher-fidelity clock models to assess their impact on the rover’s state estimation as more information about the satellite and rover becomes available.

2.2 Satellite Orbit Model

The European Space Agency (ESA) and Surrey Satellite Technology Ltd. (SSTL) have designed the Lunar Pathfinder to orbit the Moon in an ELFO with orbital elements defined in the Moon orbital plane frame of reference, as shown in Table 1 (SSTL, 2022). The orbital period is provided, as well as the periselene and aposelene altitudes. This orbital path was chosen to ensure a long duration of coverage of the Moon’s southern hemisphere as well as long-term stability.

2.3 Communication Signal Model

In this section, we discuss the structure of the Lunar Pathfinder’s communication signal, the parameters of the satellite’s transmitter and the rover’s receiver, and the model for the carrier-to-noise density ratio C/N₀.

2.4 Satellite Ephemeris Error Model

ESA and SSTL have not yet released the expected ephemeris errors of the Lunar Pathfinder. We assume that the rover will be able to downlink ephemeris information through the communication signal, but the frequency at which the rover will have access to the updated information and the preciseness of the ephemeris are unknown. Given that a communication satellite relies less heavily on precise ephemeris knowledge than a navigation satellite would, we can infer that the ephemeris errors cited in the Lunar Relay Services Requirements Document for lunar navigation satellites will present a lower bound, or best-case scenario (NASA, 2022a). According to the signal-in-space error (SISE) requirements for a lunar relay positioning, navigation, and timing satellite, the SISE position error is 13.43 m and the SISE velocity error is 1.2 mm/s, both at 3σ (NASA, 2022a).

In this study, we model the satellite ephemeris errors as zero-mean white Gaussian noise. While time-correlated random walks may more accurately represent the accumulation of ephemeris errors, a white noise model avoids assumptions about the satellite’s ephemeris update rate and ensures that the ephemeris errors early in the measurement accumulation window are not underestimated, which is crucial for evaluating worst-case positioning performance. Because of the uncertainty in the Lunar Pathfinder’s ephemeris knowledge, we conduct a sensitivity analysis on the rover’s positioning error with varying degrees of satellite ephemeris error in Section 5.4. Thus, we provide the rover with erroneous satellite position and velocity states:

$x_{\mathrm{sat},\mathrm{noise}}=x_{\mathrm{sat},\mathrm{true}}+{ϵ}_{\mathrm{eph},\mathrm{pos}}$8

$v_{\mathrm{sat},\mathrm{noise}}=v_{\mathrm{sat},\mathrm{true}}+{ϵ}_{\mathrm{eph},\mathrm{vel}}$9

where the satellite position and velocity errors are sampled as ${ϵ}_{\mathrm{eph,pos}}\sim N\left(0,{\sigma }_{{\text{eph,pos}}_{\mathrm{xyz}}}^2{I}_3\right)$ and ${ϵ}_{\mathrm{eph,vel}}\sim N(0,{\sigma }_{\mathrm{eph},\mathrm{ve}{l}_{\mathrm{xyz}}}^2{I}_3)$ and where I₃ represents a 3 × 3 identity matrix.

2.5 Pseudorange Rate Measurement Error Model

Outside of ephemeris errors, we must account for uncertainty in the pseudorange rate measurements. In this section, we discuss how we model the noise due to carrier tracking and clock frequency errors.

3.1 Weighted Batch Filter Formulation

The weighted batch filter aims to refine the rover’s state estimate $\hat{x}$ over time. In this formulation, we define the first three elements of the state estimate $\hat{x}$ as the rover’s position estimate, denoted as ${\hat{x}}_p$, as follows:

${\hat{x}}_p={\left(\begin{array}{ccc}x& y& z\end{array}\right)}^T$15

$\hat{x}={\left(\begin{array}{cc}{\hat{x}}_p^T& c\cdot \stackrel{\cdot }{\delta }\end{array}t\right)}^T$16

The weighted batch filter first accumulates observed pseudorange rate measurements over time to diversify the measurement geometry. With each iteration k of the batch filter, the batch of measurements processed by the filter increases in size because we retain all previous measurements in this framework. Let us define y as the stack of N measurements in a given batch of measurements that are accumulated over time:

$y={\left[\begin{array}{cccc}{\stackrel{\cdot }{\rho }}_{\text{obs},1}& {\dot{\rho }}_{\text{obs,}2}& \cdots & {\dot{\rho }}_{\text{obs,}N}\end{array}\right]}^T$17

For each batch of measurements, we predict the pseudorange rate observables that the rover expects to obtain at time t, given that the rover will have noisy satellite knowledge. The expected pseudorange rate observables ${\widehat{\stackrel{\cdot }{\rho }}}_{\text{obs}}$ are defined in Equation (18):

${\widehat{\stackrel{\cdot }{\rho }}}_{\text{obs}}^{\left(t\right)}=v_{\mathrm{s\; a\; t},\mathrm{n\; o\; i\; s\; e}}^{\left(t\right)}\cdot \frac{x_{\mathrm{s\; a\; t\; n\; o\; i\; s\; e}}^{\left(t\right)}-{\hat{x}}_p}{x_{\mathrm{s\; a\; t\; n\; o\; i\; s\; e}}^{\left(t\right)}-{\hat{x}}_p}+c(\stackrel{\cdot }{\delta }{t}_{\text{sat}}-\stackrel{\cdot }{\delta }{t}_{\text{rov}})$18

Note that the expected pseudorange rate has a form similar to that of the observed pseudorange rate shown in Equation (2). The expected measurements incorporate the rover’s state estimate and the noisy satellite’s state, whereas the observed pseudorange rate accounts for the true rover and satellite states along with measurement errors. We set the satellite clock drift $\stackrel{\cdot }{\delta }{t}_{\mathrm{sat}}$ to be zero because we assume that the clock onboard the Lunar Pathfinder is stable, as discussed in Section 2.1. The stack of expected measurements ${\widehat{\stackrel{\cdot }{\rho }}}_{\text{obs}}$ for each filter iteration k is defined as ${\hat{y}}_{k^{\cdot }}$.

To refine the rover’s state estimate to ${\hat{x}}_{k+1}$, we must also refine the batch of expected pseudorange rate measurements to ${\hat{y}}_{k+1}$. The expected pseudorange rate model as defined in Equation (18) is nonlinear; thus, we obtain ${\hat{y}}_{k+1}$ by applying a first-order Taylor series approximation about the rover’s current state estimate ${\hat{x}}_k$:

${\hat{y}}_{k+1}={\hat{y}}_k+\frac{\partial {\hat{y}}_k}{\partial {\hat{x}}_k}\left({\hat{x}}_{k+1}-{\hat{x}}_k\right)$19

$={\hat{y}}_k+H_k\delta x$20

Above, we define δx as the difference in the current and updated rover state estimate. The first derivative of the expected pseudorange rate measurement stack ${\hat{y}}_k$ with respect to the current rover state estimate ${\hat{x}}_k$ is defined as the measurement Jacobian matrix H_k:

$H_k={\left(\begin{array}{cccc}\frac{\partial {\widehat{\stackrel{.}{\rho }}}_{\text{obs},1}}{\partial {\hat{x}}_k}& \frac{\partial {\widehat{\stackrel{.}{\rho }}}_{\text{obs},2}}{\partial {\hat{x}}_k}& \cdots & \frac{\partial {\widehat{\stackrel{.}{\rho }}}_{\text{obs},N}}{\partial {\hat{x}}_k}\end{array}\right)}^T∊{ℝ}^{N\times 4}$21

$\frac{\partial {\hat{\dot{\rho }}}_{\text{obs},t}}{\partial {\hat{x}}_k}=\left[\begin{array}{cccc}\frac{\partial {\hat{\dot{\rho }}}_{\text{obs},t}}{\partial x_k}& \frac{\partial {\hat{\dot{\rho }}}_{\text{obs},t}}{\partial y_k}& \frac{\partial {\hat{\dot{\rho }}}_{\text{obs},t}}{\partial z_k}& 1\end{array}\right]$22

When refining the rover’s state estimate, we must take into account the rover clock drift, as it contributes to the rover’s expected pseudorange rate measurements. Therefore, in the measurement Jacobian, we consider the rover clock drift (multiplied by the speed of light) as the fourth element of the rover state. Recall that we assume that the rover clock drift is constant throughout the measurement window; thus, the fourth column of the Jacobian is 1_{N × 1}.

Ultimately, the rover’s updated state estimate ${\hat{x}}_{k+1}$ is obtained by minimizing the difference between the observed pseudorange rate measurements and the expected pseudorange rate measurements that we have refined, as shown in Equation (23). We define this difference as the measurement residual δ y. With the Taylor series approximation from Equation (20), the cost function to minimize is shown in Equation (24), which has a closed-form solution, as shown in Equation (25):

$J={\left(y-{\hat{y}}_{k+1}\right)}_w^2$23

$={\left(\delta y-H_k\delta x\right)}_w^2$24

$\partial x={\left(H_k^{T}W{H}_k\right)}^{-1}{H}_k^{T}W\delta y$25

We use a weighting matrix W in the measurement filtering to prioritize the measurements with less variance. The weighting matrix W is a function of the total measurement error, as defined in Equation (14):

$W=\text{diag}\left({\sigma }_{\text{tot},1}^{-2},\cdots ,{\sigma }_{\text{tot},N}^{-2}\right)$26

For each batch of measurements, we iteratively update δx (and thus $\hat{x}$) until δx is negligibly small. This iterative process, the Gauss-Newton method, is outlined in Algorithm 1. The predict_pseudorange_rate function in Algorithm 1 outputs the expected measurement stack and the Jacobian, as defined in Equations (18) and (21), respectively. The updated state estimate is stored after each batch is processed. With each new batch of measurements, the rover’s state estimate is re-initialized and refined once again through the batch filter. When the satellite is not visible to the rover, measurements cannot be collected. During this occultation period, the rover’s current state estimate is not updated.

3.2 Filter Initialization

We modify the initialization of the Gauss–Newton method by embedding prior knowledge about the rover’s initial position estimate ${\hat{x}}_{p,0}$ in the algorithm. In other words, we augment H_k and δy by treating the residual between the initial position estimate ${\hat{x}}_{p,0}$ and the current position estimate ${\hat{x}}_{p,k}$ as a measurement. The uncertainty of the initial position estimate is also embedded in the weighting matrix W. The augmentations to H_k, δy, and W can be found in lines 5, 6, and 7 of Algorithm 1, respectively.

This modification ensures that there is no significant divergence from the rover’s true state early in the simulation. When the initial position estimate is not embedded into the algorithm, we observe a spike in the rover’s positioning error. This augmentation helps maintain a stable estimate until there are enough measurements to resolve the rover’s state at a finer accuracy.

4.1 Initialized Rover Position

According to the Endurance Mission Concept Study Report (Keane et al., 2022), two implementation options are being considered for the Endurance mission: (1) Endurance-R, where the rover will rendezvous with an Earth return vehicle to return the samples and (2) Endurance-A, where the rover will meet with the Artemis astronauts, who will then bring the samples back to Earth. Endurance-R will begin its path at the Poincaré Q basin and end at the Apollo peak ring, whereas Endurance-A will start in the central SPA basin and reunite with the astronauts at the Artemis Basecamp. For this study, we have selected three key locations–Poincaré Q, the Apollo peak ring, and the Artemis Basecamp–as shown in Figure 2, because these locations are sufficiently spatially distributed with respect to each other in the SPA basin. Cortinovis et al. (2024) also conducted their analyses at these locations. Table 5 details the latitude and longitude coordinates of the selected waypoints.

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FIGURE 2

Map of Endurance’s proposed path with chosen waypoints circled in yellow; figure adapted from Keane et al. (2022)

Upon landing on the surface of the Moon, the rover will have good initial state estimation knowledge via manual human-in-the-loop map comparison. Therefore, the initial positioning error is on the order of 100 m for this study. The initial state estimate is randomly sampled from a Gaussian normal distribution such that ${\hat{x}}_0$ ̴ N(0, diag[100², 100², 100²]) m.

4.2 Simulation Parameters

The starting epoch to determine the Lunar Pathfinder’s orbit with respect to the Moon is set to 2030 October 1, 00:00 UTC. We consider two orbital periods, or approximately 22 h, for our simulation time unless otherwise specified. Measurements are sampled at 1 Hz, and the batch filter provides a new update of the rover state every 180 s. Note that no measurements are acquired when the Lunar Pathfinder satellite is not visible to the rover. Metrics that define signal availability are described in Section 5.1. Because we perform random sampling of the initial position estimate, satellite ephemeris errors, and Doppler measurement errors, we conduct 100 Monte Carlo realizations of the batch filter per location of interest.

5.1 Signal Availability

The design of the Lunar Pathfinder’s ELFO allows for favorable coverage of the SPA region. Figure 3 presents the received C/N₀ and the elevation mask over two orbital periods. Contrary to intuition, the received C/N₀ is highest when the satellite is at its maximum elevation (i.e., when the satellite is farthest from the rover). We would expect C/N₀ to reach a minimum at the highest elevation point owing to the increased free-space path loss. However, this assumption only holds if the satellite antenna is actively tracking the rover. In our simulations, we assume that the Lunar Pathfinder provides services to multiple lunar assets; thus, its transmitting antenna is always pointing in the nadir direction rather than adjusting to individual users. As a result, the directionality of the transmitting antenna significantly influences the received C/N₀ When the satellite is directly overhead, it points most accurately toward the rover, leading to a higher C/N₀ despite the larger distance.

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FIGURE 3

[Left] Elevation angle of the Lunar Pathfinder and [right] received carrier-to-noise density ratio C/N₀ (given that the satellite is visible) over two orbital periods for each of the three lunar sites considered

We consider an elevation mask of 5° for satellite visibility, and we define the signal as available when the received signal strength is larger than 30 dB Hz (as defined by Nardin et al. (2023) and Melman et al. (2022)). Based on the respective thresholds, the 5° elevation mask is the limiting factor for satellite visibility in comparison to the signal strength threshold of 30 dB Hz. The regions in which the satellite is not visible to the rover are shown as gray occultation zones in later figures.

Table 6 presents the total time duration for which the satellite signal is unavailable, either because of an insufficient received C/N₀ or a lack of satellite visibility.

The total occultation period for each location is the union of the two metrics (i.e., when C/N₀ is less than 30 dB Hz or when the elevation mask is less than 5°). The Artemis Basecamp location has the shortest occultation period while the Apollo peak ring has the longest occultation period, owing to the elevation mask. Because the Artemis Basecamp is located exactly at the South Pole, the rover position is symmetric relative to the Lunar Pathfinder, as shown in the elevation plot in Figure 3. Owing to this symmetry, there is no change in the occultation period between the first and second orbits for the Artemis Basecamp location. In contrast, for the Poincaré Q and Apollo peak ring locations, the occultation periods increase slightly between the first and second orbits because of the satellite’s relative motion.

Cortinovis et al. (2024) also used the same metrics for signal availability. However, because those authors assumed that the satellite is equipped with a navigation payload, they modeled their transmitter antenna according to the Lunar Communications Relay and Navigation Systems (LCRNS) requirements, resulting in smaller C/N₀ values than we observe with the transmitter parameters used in this study. Thus, our modeling of a lunar satellite with a communication antenna results in longer periods of signal availability in comparison to a lunar satellite with a navigation antenna (Cortinovis et al., 2024).

Figure 4 illustrates the behavior of the Doppler shift measurements. Positive Doppler measurements signify that the Lunar Pathfinder is approaching its peak elevation over the stationary Endurance rover, and negative measurements indicate that the satellite is receding. In the right panel, the Lunar Pathfinder is traversing its path in the clockwise direction. The blue point in the figure marks the point in the satellite’s path at which the mean anomaly is 180°. At time t = 0 in the left panel, the satellite is at the blue point in the right panel. When the satellite recedes away from the blue point, we observe a negative Doppler shift until the satellite enters the first occultation zone. Once the satellite re-enters the rover’s line of sight, the satellite approaches its peak elevation, resulting in a positive Doppler shift until a zero Doppler shift is reached at the blue point.

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FIGURE 4

[Left] Doppler shift and elevation and [right] trajectory of the Lunar Pathfinder over two orbital periods

The rover location is set to be at the Artemis Basecamp. Time t = 0 in the left panel corresponds to the time when the satellite is located at the blue point in the right panel. Measurement collection for state estimation begins once the satellite is visible to the rover, as shown by the black arrow.

For our state estimation experiments, we begin measurement collection once the satellite is visible to the rover (as indicated by the black arrow in Figures 4 and 5) to maximize the time for the rover to refine its position estimate before hitting an occultation zone. The visible and non-visible parts of the Lunar Pathfinder orbit with respect to the rover are shown in the right panel of Figure 4 for when the rover is located at the Artemis Basecamp. Corresponding results for when the rover is located at Poincaré Q and the Apollo peak ring are shown in Figure 5. We observe that the satellite enters longer occultation periods when the rover is located in higher-latitude regions such as the Apollo peak ring.

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FIGURE 5

Visible and non-visible parts of the Lunar Pathfinder orbit when the rover is located at [left] Poincaré Q and [right] the Apollo peak ring

The black arrow denotes the start of measurement collection for filtering. We observe longer occultation periods for higher-latitude locations.

5.2 State Estimation Performance

Using the weighted batch filter framework, we refine the rover’s position estimate over time for each of the three locations. The mean µ, range of 1σ values, and 99% error of the position estimation for 100 Monte Carlo realizations are shown in Figure 6. The figure also shows the occultation zones and the 10-m threshold for each location. All three plots are simulated under the assumption that the rover is equipped with an SRS PRS 10 clock and that the ephemeris errors are 4.48 m in position and 0.40 mm/s in velocity (the justification for this ephemeris error is described in Section 5.4). Table 7 summarizes the time required for the mean µ and the 99% error of 100 Monte Carlo realizations to achieve sub-10-m accuracy.

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FIGURE 6

Positioning error over two orbital periods at three different locations These plots are simulated with satellite ephemeris errors based on the LCRNS’s SISE specification and with the assumption that the rover is equipped with an SRS PRS 10 clock.

From Figure 6, we observe that the mean position error μ reaches sub-10-m-level accuracy within the first 12 h for the Poincaré Q and Apollo peak ring locations and within 8 h for the Artemis Basecamp location. The 99% error converges to 10 m within the first two orbital periods for all three locations. The mean positioning error for the Artemis Basecamp location is the only metric that is able to converge to below 10 m before the first occultation period. For an autonomous rover such as Endurance, the mean contact time per day during which the Lunar Pathfinder will be providing communication services is 529 min, or 8.82 h (SSTL, 2022). While the exact periods of time during which the Lunar Pathfinder will provide communication services have not yet been specified, according to our analysis, we find that the rover, on average, will be able to localize itself within the desired accuracy within 8.82 h only when the rover is located in the more southern regions of the rover’s path. Otherwise, the rover will require a larger measurement window to successfully localize.

Despite having the shortest occultation period, the Artemis Basecamp location resulted in the largest 99% positioning error convergence time in comparison to the other locations, as shown in Table 7. Because the Artemis Basecamp is located exactly at the South Pole, the diversity of measurements between the first visibility zone to the second visibility zone is minimal. Therefore, the rover’s state estimate is slow to improve after the first occultation zone when the rover is located at the Artemis Basecamp.

Overall, we observe a plateau in positioning error in the first 3–4 h for all three locations. This plateauing behavior occurs because we embed prior knowledge about the rover’s initial position estimate, as described in Section 3.2. During the first few hours, the rover does not have sufficient measurement diversity to refine the rover’s state estimate to a finer accuracy. We find that the 99% error in this study takes approximately 10.7–13.6 h longer to converge than in a previous study using two-way ranging measurements with a weighted batch filter (Cortinovis et al., 2024). The convergence window of the 99% error in this study includes an entire occultation period, which adds an additional 2.92–5.68 h during which the estimate is not being refined. However, it is essential to note that a comprehensive comparison of this study and that of Cortinovis et al. (2024) is limited by the differences in modeling the measurement and ephemeris errors. Furthermore, we take into account noise due to the rover’s onboard clock, which is not assessed in the study of Cortinovis et al. (2024), as their model utilized two-way measurements.

5.3 Sensitivity Study on Clock Stability

NASA has not yet specified the type of clock that the Endurance rover will carry onboard. To help inform this decision, we conducted a sensitivity study to evaluate various clock candidates for the specific use case of rover localization. Table 4 details the SWaPs, TDEVs per day, and PSD coefficients for the four rover clock candidates we consider in this study. We evaluated the time to achieve sub-10-m accuracy for each of the rover clocks and have summarized the performance metrics in Table 8.

We prioritize minimizing weight and power consumption of the rover’s onboard clock, owing to the high cost of deploying heavy equipment to the Moon and the significant power demands of the rover’s 2000-km traverse. Therefore, we conduct a trade-off analysis between clock stability and these two metrics. Figure 7 displays a plot of the mean convergence time to achieve sub-10-m accuracy for the clock candidates, with the weights and power consumptions plotted on the y-axis. As expected, the heavier and higher-power-consuming clocks (which correlate to less clock drift) have a shorter time to convergence in comparison to the lower-SWaP clocks. The smallest clock that we consider–the Microchip chip-scale atomic clock (CSAC)–requires five orbital periods for the mean time to converge. The 99% error is unable to converge within six orbital periods for the Microchip CSAC. Therefore, we do not recommend the Microchip CSAC for single-satellite localization.

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FIGURE 7

Mean time to reach sub-10-m accuracy for the four clock candidates, with their respective [left] clock weights and [right] clock power consumptions plotted on the y-axis

This study was performed under the assumption that the rover is located at the Artemis Basecamp and is using the LCRNS’s SISE values to describe its ephemeris knowledge.

The most stable clock–the Excelitas rubidium atomic frequency standard (RAFS)–is able to converge in 4.9 h. If clock weight (or power consumption) and performance time both need to be minimized with equal magnitude, we recommend the SRS PRS 10 and the Microchip miniature atomic clock (MAC) for the Endurance rover, as they have a mass of less than 1 kg, require less than 15 W for operation, and converge within 1–2 orbital periods. However, if rapid localization is more critical than additional weight on the rover, we recommend the Excelitas RAFS.

5.4 Sensitivity Study on Satellite Ephemeris Errors

Because of the uncertainty of the Lunar Pathfinder’s ephemeris knowledge, we examine the effects of varying degrees of satellite ephemeris errors on achieving the desired position estimation accuracy. We consider the lowest satellite ephemeris errors to be those cited in the Lunar Relay Services Requirements Document for lunar navigation satellites, which are σ_eph,pos= 4.48 m and σ_eph,vel= 0.40 mm/s (NASA, 2022a). For our sensitivity analysis, we linearly inflate ephemeris errors ranging from 10.00 to 50.00 m in position and from 1.00 to 5.00 mm/s in velocity, as shown in Figure 8.

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FIGURE 8

Time for the mean and 99th percentile of Monte Carlo runs to converge to below 10 m for different ephemeris errors in position [m] and velocity [mm/s]

This study assumes that the rover is located at the Artemis Basecamp and is equipped with an SRS PRS 10 clock.

We observe that the positioning error converges to below the 10-m threshold in roughly the same amount of time for the lowest ephemeris errors and for σ_eph,pos = 10.00 m and σ_eph,vel = 1.00 mm/s. However, for larger ephemeris errors, the 99% positioning error is unable to converge within two orbital periods, resulting in an additional 2.92 h for the time to convergence. Based on this sensitivity study, we recommend that the Lunar Pathfinder’s ephemeris errors remain below 20.00 m in position and 2.00 mm/s in velocity.