# Design and Analysis of the Cis-Lunar Navigation for the ArgoMoon CubeSat Mission

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## Abstract

**:**

## 1. Introduction

## 2. The ArgoMoon Mission

#### 2.1. Mission Overview

_{D}is considered the beginning of the timeline of the operations. The mission is subdivided into three main phases of different durations. Phase 1, which covers the first day of flight, is the most critical since ArgoMoon will perform the automatic proximity flight operations (ProxOps) close to the ICPS. The S/C will operate autonomously for 30 min after the deployment without s direct ground link. The relative positioning and pointing with respect to ICPS is performed by ArgoMoon using pictures acquired on board and processed by an image recognition algorithm based on machine learning [4]. Then, the S/C will turn on its onboard radio and the link with the antennas of the DSN will be established, allowing to perform ground-based orbit determination and navigation through radiometric observables [13]. After Phase 1 is completed, an orbital maneuver will be executed to target the first fly-by of the Moon and shape the geocentric trajectory. Phase 2 covers up to 20 days after the S/C deployment, and it starts once the first orbital maneuver is completed. During Phase 2, ArgoMoon will then perform the first fly-by of the Moon and its first revolution around the Earth. Phase 3 starts 20 days after the deployment, and it covers the remainder of the mission. During the last week of Phase 3, about 180 days after the deployment, ArgoMoon will perform a final fly-by with the Moon to be injected into a heliocentric orbit for its final disposal. The End of Mission (EOM) will occur when the S/C reaches the heliocentric orbit and the tracking activities will be terminated.

#### 2.2. The Spacecraft

#### 2.3. Trajectory

_{D,}where the real initial state vector after the deployment cannot be fully characterized a priori because the ICPS will rotate around its primary inertia axis before initiating the dispensing activities. However, since the uncertainty introduced by the deployment is negligible if compared to the ICPS injection covariance, the S/C initial state has been assumed to be equal to the one of ICPS at ${T}_{D}$. After the deployment, ArgoMoon will follow its release path for 15 s and then it will perform a rotation of 180 degrees to point the PL2 towards ICPS for the autonomous target tracking. Then, during the first 30 min of flight, the S/C will perform two orbital maneuvers to maintain a close and stable distance to the ICPS. However, the uncertainties introduced by the automatic orbital maneuvers are negligible with respect to the ICPS injection covariance. Therefore, the ProxOps phase around the ICPS during the first 30 min has not been modeled. Seventy-five minutes after the deployment, ArgoMoon will perform the Keep Out Zone (KOZ) maneuver required to drift away from the ICPS and then start Phase 2 of the mission. The KOZ is a one-second impulse executed aligned with the ArgoMoon-ICPS line-of-sight but in the opposite direction with respect to the ICPS. At 20 h after the deployment, an Orbit Trim Maneuver (OTM), named OTM1, with a ΔV of 10.95 m/s, is executed to target the fly-by M0 and shape the later geocentric trajectory. Nominally, no other deterministic orbital maneuvers are planned for ArgoMoon’s reference orbit. The trajectory foresees eight revolutions (REVs) around the Earth where the perigees and apogees are identified using the capital letters P and A followed by an incremental number starting from zero (i.e., P0 is the first perigee, A8 is the last apogee). A REV is defined between two successive perigees (i.e., REVi is between Pi and Pi + 1), except for REV0 and REV9 at the beginning and at the end of the mission, where M0 and M3 take place and there is no initial or final perigee. Before M3, the maximum distance from the Earth that ArgoMoon will reach in its path is 830,000 km, where the closest one is 37,400 km.

#### 2.4. Navigation Requirements

**Impact avoidance**: the S/C shall not fly below the threshold altitudes of 1000 km with respect to the Earth and 100 km with respect to the Moon. The requirement applies to the whole mission and can become significant at the perigees and fly-bys of the Moon.**Heliocentric disposal**: the S/C shall reach the heliocentric disposal orbit after the last fly-by of the Moon. The ranges of tolerance for the disposal conditions have been determined through a Monte Carlo analysis with the requirement of having a low probability of crossing the Earth’s sphere of influence in successive years. The disposal requirement is displayed in Figure 3, where the green dots are the samples with a correct disposal and the red crosses are the ones that do not satisfy the requirement.**DSN pointing uncertainty**: to ensure the link with the DSN 34 m antennas, the pointing uncertainty due to S/C orbit determination shall be lower than 0.031 deg, which corresponds to the Half Power Beamwidth (HPB) of the antenna at X-band [15]. However, during the first day of the mission, the threshold value of the pointing uncertainty is relaxed to 1.05 deg, which corresponds to half of the HPB of the 34 m dishes equipped with the 1.2 m aided acquisition antenna above the sub-reflector [15].

#### 2.5. Navigation Concept

## 3. Flight Path Control Analysis

#### 3.1. Uncontrolled Trajectory

^{5}km and 10

^{6}km up to the end of the mission. The uncontrolled dispersion causes a probability of about 1.8% to fly below 1000 km of altitude with respect to the Earth, violating the impact risk requirement. Moreover, due to the chaotic behavior of the propagated samples, the last fly-by with the Moon (M3) is never achieved.

#### 3.2. Optimal Control Strategy

**V**. The use of a linear method has given the possibility to rapidly test many kinds of aimpoints and coordinates to be targeted, as well as the maneuver’s location. Based on previous studies, the optimal aimpoints were selected to be the fly-bys with the Moon (M0, M3) and apocenters (A1, A2, …, A8) of the orbit around the Earth [26]. Moreover, to reduce the trajectory dispersion by improving the control on the S/C orbital velocity, the pericenters (P1, P2, …, P7) of the orbit around the Earth have been selected as further aimpoints.

## 4. Orbit Determination Analysis

#### 4.1. Processing Assumptions

#### 4.2. Dynamical Model

^{−11}km/s

^{2}per axis, constant on 8 h batches, and uncorrelated in time.

#### 4.3. Tracking Schedule

#### 4.4. Filter Configuration

#### 4.5. Baseline Results

## 5. Sensitivity Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**ArgoMoon CubeSat external view render adapted with permission from Ref. [4]. Copyright year, IEEE and the assumed body-fixed frame.

**Figure 2.**ArgoMoon trajectory in the Earth Mean Orbit frame (at J2000 epoch) and the S/C distance with respect to the Earth and the Moon.

**Figure 3.**B-Plane [16] admissible region for the last fly-by of the Moon (M3).

**Figure 4.**Statistics of the uncontrolled trajectory error with respect to the nominal trajectory from the deployment up to the first apogee A1.

**Figure 5.**The 99th percentile of the trajectory dispersion on the B-Plane [16] of M0.

**Figure 6.**K-inverse norm evolution for the targeting of B.R, B.T and LTOF coordinates of the B-Plane for the fly-bys M0 (

**left**) and M3 (

**right**).

**Figure 7.**K-inverse norm evolution for the targeting of the cartesian position at the apogee A1 (

**left**) and the cartesian velocity at the perigee P1 (

**right**).

**Figure 8.**Controlled trajectory 99th percentile dispersion through the whole mission using the nominal control strategy. The error is computed with respect to the reference trajectory using the geocentric RTN frame.

**Figure 10.**DSN to ArgoMoon pointing uncertainty evolution (3-sigma), during REV0. The gray line is the uncertainty evolution using the designed delivery schedule, while the blue dots identify the tracking passes.

**Figure 11.**DSN to ArgoMoon pointing uncertainty evolution (3-sigma) from P0 to P2 (encompasses REV1 and REV2). The gray line is the uncertainty evolution with the nominal delivery schedule, while the blue dots identify the tracking passes.

**Figure 12.**B-Plane uncertainties (3-sigma) for the fly-by M0 with DCO at the end of each tracking (TRK) pass up to the DCO of the STM1 (last maneuver before the fly-by). “A-priori” refers to the results obtained propagating the ICPS dispersion at the deployment, without processing any data.

**Figure 13.**B-Plane uncertainties (3-sigma) for the fly-by M3 with DCO at the end of each tracking (TRK) pass up to the DCO of the STM19 (last maneuver before the fly-by). “A-priori” refers to the results obtained propagating the expected ephemeris uncertainty at the end of the previous arc (REV7).

**Figure 17.**Full mission DSN pointing requirement ratio between the expected uncertainty and the requirement, for each tracking (TRK) pass, with respect to each relevant sensitivity case.

Event | Event Epoch | Details |
---|---|---|

Bus Stop 1 (BS1) | Launch + 3 h 54 min | First CubeSats dispensing phase |

Bus Stop 2 (BS2) | Launch + 6 h 59 min | Last ArgoMoon observed deployment phase |

Deployment (T_{D}) | BS1 + 6 min | Release of ArgoMoon from the ICPS (close to BS1) |

Transponder ON | T_{D} + 30 min | ArgoMoon starts to communicate with DSN |

KOZ | T_{D} + 75 min | Keep Out Zone maneuver to drift away from the ICPS |

OTM1 | ~T_{D} + 20 h | Maneuver to trim the first fly-by of the Moon (M0) |

M0 | ~T_{D} + 5.23 days | First fly-by of the Moon: C/A at 7773 km |

M1 | ~T_{D} + 82.08 days | Mid-course fly-by of the Moon: C/A at 86,051 km |

M2 | ~T_{D} + 104.61 days | Mid-course fly-by of the Moon: C/A at 84,594 km |

M3 | ~T_{D} + 191.51 days | Last fly-by of the Moon: C/A at 5261 km |

EOM | ${\mathrm{T}}_{\mathrm{D}}+200\mathrm{days}$ | End of the mission |

Pi (i = 0,1…8) | Perigees | Total number of perigees: 9 |

Ai (i = 0,1…8) | Apogees | Total number of apogees: 9 |

REV0 | ${\mathrm{T}}_{\mathrm{D}}$ to P0 | First revolution that encompasses the fly-by M0 |

REVi (i = 1,8) | Pi to Pi + 1 | Revolutions around the Earth (i.e., REV3: from P2 to P3) |

REV9 | P8 to EOM | Last revolution that encompasses the fly-by M3 |

Injection covariance | ICPS state (Earth-RTN) uncertainty (3-sigma) at BS1 epoch: | |||||

X (km) | Y (km) | Z (km) | VX (km/s) | VY (km/s) | VZ (km/s) | |

30.0 | 60.0 | 15.0 | 0.0021 | 0.0027 | 0.0042 | |

Maneuvers execution error | Gates Model applied to both OTMs and STMs. | |||||

Mis-modeling and OD error | OD covariance mapped from the maneuver’s DCO to the aimpoint. |

Maneuvers execution error | Error Component (Per Axis) | ArgoMoon PS | |

Magnitude | Fixed (m/s) Proportional (%) | 0.011 3.5 | |

Pointing | Fixed (m/s) Proportional (deg) | 0.011 1.1 |

**Table 4.**Statistical ΔV report of the entire mission generated through a Monte-Carlo simulation with 10,000 samples using the optimal trajectory control strategy described in the text.

Maneuver | Epoch | Aimpoint | Coordinates (EME2000) | ΔV Mean (m/s) | ΔV 99% (m/s) |
---|---|---|---|---|---|

OTM1 | ${\mathrm{T}}_{\mathrm{D}}+22\mathrm{h}$ | N/A: deterministic open-loop burn | 11.031 | 11.921 | |

STM1 | OTM1 + 48 h | M0 | B.R, B.T, TCA | 5.706 | 17.306 |

STM2 | P0−48 h | A1 | X, Y, Z | 4.527 | 18.315 |

STM3 | P0 + 48 h | A1 | X, Y, Z | 0.405 | 2.195 |

STM4 | A1 | P1 | VX, VY, VZ | 0.398 | 1.174 |

STM5 | P1 + 48 h | A2 | X, Y, Z | 0.088 | 0.381 |

STM6 | A2 | P2 | VX, VY, VZ | 0.106 | 0.312 |

STM7 | P2 + 48 h | A3 | X, Y, Z | 0.067 | 0.191 |

STM8 | A3 | P3 | VX, VY, VZ | 0.096 | 0.269 |

STM9 | P3 + 48 h | A4 | X, Y, Z | 0.061 | 0.153 |

STM10 | A4 | P4 | VX, VY, VZ | 0.088 | 0.238 |

STM11 | P4 + 48 h | A5 | X, Y, Z | 0.053 | 0.141 |

STM12 | A5 | P5 | VX, VY, VZ | 0.086 | 0.245 |

STM13 | P5 + 48 h | A6 | X, Y, Z | 0.047 | 0.122 |

STM14 | A6 | P6 | VX, VY, VZ | 0.082 | 0.231 |

STM15 | P6 + 48 h | A7 | X, Y, Z | 0.073 | 0.216 |

STM16 | A7 | P7 | VX, VY, VZ | 0.093 | 0.251 |

STM17 | P7 + 48 h | A8 | X, Y, Z | 0.057 | 0.152 |

STM18 | A8 | M3 | B.R, B.T, TCA | 0.074 | 0.206 |

STM19 | P8 + 12 h | M3 | B.R, B.T, TCA | 0.146 | 0.425 |

Total cumulated statistical ΔV: | 23.287 | 49.443 |

Arc data | Tracking data of a single REV (between two perigees): ${P}_{i}\to {P}_{i+1}$ | |

Tracking data X/X band | Doppler | 2-way, 60 s of integration time |

Range | 2-way, 1 observable every 300 s | |

Data noise and weights | Doppler | 0.1 mm/s at 60 s of integration time (2.81 mHz at X-band) |

Range | 2 m | |

Stochastic accelerations | $1.0\times {10}^{-11}\mathrm{km}/{\mathrm{s}}^{2}$ per axis, uncorrelated white noise, 8 h of batch time | |

Orbital Maneuvers | DCO | 96 h before the maneuver’s epoch (nominal) 24 h before the maneuver’s epoch (minimum) |

Tracking | No tracking data during the maneuver execution | |

REV0 epoch state covariance | ICPS state (Earth-RTN) uncertainty (3-sigma) at BS1 epoch (Table 2) | |

REV1 to REV9 epoch state covariance | Previous arc’s mapped state covariance scaled by a safety factor of 4 |

Component | Specular Reflectivity (ρ) | Diffusive Reflectivity (δ) |
---|---|---|

Bus faces | 0.0 | 0.25 |

Solar arrays | 0.115 | 0.25 |

Parameter | Unit | A priori Uncertainty | Estimated/Considered | |
---|---|---|---|---|

S/C epoch state (REV0) | - | ICPS state covariance at BS1 (Table 5:) | Estimated | |

S/C epoch state (REV1-REV9) | - | Estimated covariance mapped from previous arc, multiplied by 4 | Estimated | |

Solar Radiation Pressure Scale Factor | - | 50% | Estimated | |

Deterministic impulse burns (OTM) | ΔV | m/s | 10% of nominal | Estimated |

Ra | deg | 1.1 | Estimated | |

Dec | deg | 1.1 | Estimated | |

Time | s | 3.0 | Estimated | |

Statistical impulse burns (STM) | ΔV(X) | m/s | 0.011 | Estimated |

ΔV(Y) | m/s | 0.011 | Estimated | |

ΔV(Z) | m/s | 0.011 | Estimated | |

Time | s | 3.0 | Estimated | |

Stochastic accelerations | X/Y/Z | km/s^{2} | 10^{−11}, 8-h batches | Estimated |

Range Bias (per pass) | m | 2 | Estimated | |

Earth GM | km^{3}/s^{2} | 5.0 × 10^{−4} | Considered | |

Moon GM | km^{3}/s^{2} | 1.4 × 10^{−4} | Considered | |

DSN station locations (per axis) | cm | 3 | Considered | |

Troposphere path delay (wet/dry) | cm | 1/1 | Considered | |

Ionosphere path delay (day/night) | cm | 5/1 | Considered | |

Earth Polar Motion X/Y | deg | 8.6 × 10^{−7} | Considered | |

UT1 bias | s | 2.5 × 10^{−4} | Considered |

Case | Mean (m/s) | Sigma (m/s) | ΔV 99% (m/s) |
---|---|---|---|

Baseline | 23.3 | 7.6 | 49.5 |

0.5 × Injection Covariance | 18.2 | 3.9 | 31.6 |

0.5 × Maneuvers Execution Error | 21.4 | 6.2 | 41.3 |

No LTOF targeting | 22.1 | 7.2 | 46.6 |

5 × OTM1 sigmas | 23.3 | 7.7 | 49.5 |

Maximum 1 pass per day | 23.4 | 7.6 | 49.6 |

No Doppler data | 23.6 | 7.6 | 49.6 |

No Range data | 23.5 | 7.6 | 49.8 |

10 × Stochastic sigmas | 24.4 | 7.6 | 50.3 |

5 × STM sigmas | 26.6 | 7.6 | 52.3 |

2 × Maneuvers Execution Error | 30.1 | 12.9 | 77.7 |

2 × Injection Covariance | 34.0 | 15.2 | 86.8 |

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**MDPI and ACS Style**

Lombardo, M.; Zannoni, M.; Gai, I.; Gomez Casajus, L.; Gramigna, E.; Manghi, R.L.; Tortora, P.; Di Tana, V.; Cotugno, B.; Simonetti, S.; Patruno, S.; Pirrotta, S. Design and Analysis of the Cis-Lunar Navigation for the ArgoMoon CubeSat Mission. *Aerospace* **2022**, *9*, 659.
https://doi.org/10.3390/aerospace9110659

**AMA Style**

Lombardo M, Zannoni M, Gai I, Gomez Casajus L, Gramigna E, Manghi RL, Tortora P, Di Tana V, Cotugno B, Simonetti S, Patruno S, Pirrotta S. Design and Analysis of the Cis-Lunar Navigation for the ArgoMoon CubeSat Mission. *Aerospace*. 2022; 9(11):659.
https://doi.org/10.3390/aerospace9110659

**Chicago/Turabian Style**

Lombardo, Marco, Marco Zannoni, Igor Gai, Luis Gomez Casajus, Edoardo Gramigna, Riccardo Lasagni Manghi, Paolo Tortora, Valerio Di Tana, Biagio Cotugno, Simone Simonetti, Silvio Patruno, and Simone Pirrotta. 2022. "Design and Analysis of the Cis-Lunar Navigation for the ArgoMoon CubeSat Mission" *Aerospace* 9, no. 11: 659.
https://doi.org/10.3390/aerospace9110659