# Mothership-Cubesat Radioscience for Phobos Geodesy and Autonomous Navigation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mission Profile and Orbits

## 3. Budget Analyses and CubeSat Design

#### 3.1. Deployment $\Delta v$ Budget

#### 3.2. Doppler Measurement and Link Budget

#### 3.3. System Sizing

## 4. Models of Geodesy and Orbit Determination

#### 4.1. Baseline Model

`mar097`ephemeris developed at JPL [43] is adopted as the true ephemeris of Mars and Phobos. The Mars spherical harmonics gravity model is considered up to 10 degrees and orders, whose values are available from the JPL model

`jgmro_110b`. The used Phobos gravity model is up to 3 degrees and orders. The baseline gravity harmonics coefficients of Phobos are computed by the CNES Geodesy Group based on Gaskell’s Phobos shape model [44] under the assumption of homogeneous density and $GM$ of $7.1\times {10}^{-4}$ km${}^{3}$/s${}^{2}$.

`mar097`ephemeris, and true range-rates at measurement epochs are generated.

#### 4.2. Error Models

`mar097`and the

`NOE-4-2020`developed at IMCCE [47] are adopted as the known ephemerides to express the situations without and with ephemeris errors, respectively. The difference between

`mar097`and

`NOE-4-2020`during the period from 2025 to 2027 is around 2 km, mainly along the transverse direction (for more details, see Ref. [47]). The ephemeris error can be approximated by and regarded as a phase shift, $\Delta et$, of 0.9 s.

#### 4.3. Theoretical Covariance

## 5. Numerical Estimation Simulations

#### 5.1. Orbit Determination

`mar097`ephemeris is employed in the estimation routine) is displayed in Figure 7. The estimation process converges and starts to follow new orbit states in one day. Upon convergence, the average root-sum-squared (RSS) position uncertainty is around 0.2 m, and the RSS velocity uncertainty is 0.05 mm/s. Figure 8 shows the process of online estimation in the presence of ephemeris errors (i.e., the ephemeris

`NOE-4-2020`is used). The RSS position uncertainty is significantly increased to 81.6 m, and the RSS velocity uncertainty to 1.63 cm/s. Considering the influence of ephemeris error on the estimation, an ephemeris time error $\Delta et$ is taken as a consider parameter [49]. Figure 9 shows the estimation process considering the ephemeris time error. The RSS position uncertainty is substantially improved to 20.7 m, and the RSS velocity uncertainty to 0.4 cm/s.

#### Discussion on Orbiting Operations

#### 5.2. Parameter Identification

#### 5.3. Statistical Covariance

## 6. Effect of Inferring Interior Structure

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

`tycho`and

`cerfeuil`, financed and managed by Observatoire de Paris and IMCCE. The first author appreciates the constant support from the informatics department of IMCCE delivered by Stephane Vaillant and Mickaël Gastineau, and also wants to thank Zhuoxi Huo (QXSLab) for an invited visit, Jinsong Ping (NAOC/CAS) for his advice on the Doppler error analysis, and Long Long (CAST) for making the 3D model of the CubeSat.

## Conflicts of Interest

## Abbreviations

3D | Three-dimensional |

CR3BP | Circular-restricted three-body problem |

MoI | Moment of inertia |

QSO | Quasi-satellite orbit |

RSS | Root-sum-squared |

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**Figure 2.**Solutions of resonant 3D QSO (resonance ratio indicated by $i:j$) around Phobos and their effective stability, extracted from Ref. [34].

**Figure 5.**The 100 × 200 km and the 29 × 50 × 21 km spacecraft orbits propagated in the full-dynamic model for 7 days. Orbits are plotted in the Phobos-centered Mars-Phobos rotating frame with Mars on the −x axis.

**Figure 7.**Online determination of orbits of mothership and CubeSat when the ephemeris error is not present.

**Figure 8.**Online determination of orbits of mothership and CubeSat when the ephemeris error (i.e., around 2 km) exists and is not considered.

**Figure 9.**Online determination of orbits of mothership and CubeSat when the ephemeris error (i.e., around 2 km) exists and the ephemeris time error is considered in the estimation.

**Figure 10.**Errors of gravity coefficients estimated in the three situations; namely, the situation that the ephemeris error is not present, the situation that the ephemeris error exists and is not considered in the estimation, and the situation that the ephemeris error exists and the ephemeris time error is considered in the estimation.

**Figure 11.**Errors and approximated 1-$\sigma $ and 3-$\sigma $ certainty ellipses of estimated ${C}_{20}$ and ${C}_{22}$ under no ephemeris error (blue) and under the ephemeris error (red).

**Figure 12.**1-$\sigma $ uncertainty ellipsoids of inferred moments of inertia projected on the two-dimensional space.

**Figure 13.**Map from Ref. [9] on the distribution of structure families over ${C}_{20}$ and ${C}_{22}$ and the obtained 3-$\sigma $ uncertainty ellipse (arbitrarily centered on [0.028, −0.046]) under the ephemeris error.

Item | Symbol | Unit | Value |
---|---|---|---|

Frequency | f | MHz | 435 |

Carrier wave length | ${\mathsf{\lambda}}_{L}$ | m | 0.6892 |

Transmitter output power | ${P}_{t}$ | dBW | −3 ^{a} |

Transmit antenna gain | ${G}_{t}$ | dB | 0 ^{b} |

Equiv. isotropic radiated power | $EIRP$ | dBW | −3 |

Propagation path length | D | km | 250 ^{c} |

Space loss | ${L}_{s}$ | dB | −133.18 |

Receive antenna gain | ${G}_{r}$ | dBi | 0 |

Received power | C | dB | −136.18 |

System noise temperature | ${T}_{s}$ | K | 635 |

Carrier to noise density ratio | $C/{N}_{0}$ | MHz | 2.84 |

^{a}ISIS V/U tranceiver.

^{b}Endurosat UHF antenna.

^{c}Maximum mothership-CubeSat distance.

Item | Symbol | Unit | Value |
---|---|---|---|

Clock stability | ${\sigma}_{{f}_{\mathrm{time}}}/f$ | - | $5\times {10}^{-13}$^{a} |

Code loop noise bandwidth | ${B}_{n}$ | Hz | 20 |

Predetection integration time | ${t}_{p}$ | sec | 5 |

PLL thermal noise | ${\sigma}_{{f}_{\mathit{PLL}}}/f$ | - | $2.74\times {10}^{-13}$ |

Overall measurement noise ^{b} | ${\sigma}_{{v}_{r}}$ | mm/s | $8.55\times {10}^{-2}$ |

^{a}General Allen stability of a USO.

^{b}Root-sum-squared of clock and PLL noises.

Subsystem | Components | Weight, g | Power, W |
---|---|---|---|

Structure | 1 2U frame | 390 ^{a} | - |

Shielding and deployable panels | 523 ^{a} | - | |

Power | >10 1U solar arrays | 440 | +12.24 ^{b} |

1 battery | 258 ^{c} | +38.5 h ^{c} | |

ADCS | 4 reaction wheels | 220 ^{a} | −0.8 ^{a} |

1 star tracker | 170 ^{d} | −1.5 ^{d} | |

1 MEMS IMU | 20 | −0.6 | |

Sun sensors | - ^{e} | −0.33 | |

Communication | 1 UHF antenna set | 85 ^{f} | −1 ^{f} |

1 UHF transceiver | 75 ^{g} | −4 ^{g} | |

C&DH | 1 on-board computer | 100 | −0.4 |

Propulsion | 1 cold-gas thruster | 676 ^{h} | −0.25 ^{h} |

Margin | - | 444 ^{i} | −1.3 ^{i} |

Total | - | 3401 | 2.0 |

^{a}Based on a lunar CubeSat development experience [41,42].

^{b}During the sun-pointing phase with 51% of the efficiency obtained at 1 AU.

^{c}GOMSpace NanoPower BP4, https://gomspace.com/shop/subsystems/power/nanopower-bp4.aspx, accessed on 19 February 2019.

^{d}MAI-SS Space Sextant, https://www.cubesatshop.com/wp-content/uploads/2016/06/MAI-SS-Specification-10-11-17.pdf, accessed on 1 February 2021.

^{e}Attached to solar arrays.

^{f}EnduroSat UHF antenna, https://www.endurosat.com/cubesat-store/all-cubesat-modules/uhf-antenna/, accessed on 1 February 2021.

^{g}ISIS V/U tranceiver, https://www.isispace.nl/product/isis-uhf-downlink-vhf-uplink-full-duplex-transceiver/, accessed on 1 February 2021.

^{h}VACCO End-Mounted Standard MiPS (0.25U) https://www.cubesat-propulsion.com/wp-content/uploads/2015/10/End-mounted-standard-mips.pdf accessed on 1 February 2021.

^{i}Additional 15% to account for thermal control parts, resistors and coating, and margins of subsystem components.

Item | A Priori Uncertainty |
---|---|

Range-rate measurement | 0.1 mm/s |

Spacecraft position | (100, 100, 100) m |

Spacecraft velocity | (10, 10, 10) cm/s |

Phobos’ ${C}_{00}$ | 3% |

Non-spherical CS coefficients | 10% |

Libration amplitude $\theta $ | 0.11${}^{\circ}$ |

Ephemeris error (if present) | $\left(\right)$ |

Item | Truth | Initial Knowledge |
---|---|---|

${x}_{c}$ [km] | −16.361 | −16.387 |

${y}_{c}$ [km] | −10.789 | −10.745 |

${z}_{c}$ [km] | 30.088 | 30.128 |

${v}_{cx}$ [m/s] | 1.213 | 1.088 |

${v}_{cy}$ [m/s] | 7.249 | 7.154 |

${v}_{cz}$ [m/s] | 3.119 | 3.045 |

${x}_{m}$ [km] | −88.829 | −88.880 |

${y}_{m}$ [km] | −6.438 | −6.470 |

${z}_{m}$ [km] | 46.460 | 46.462 |

${v}_{mx}$ [m/s] | 3.592 | 3.289 |

${v}_{my}$ [m/s] | 20.561 | 20.515 |

${v}_{mz}$ [m/s] | 9.009 | 9.133 |

${\overline{C}}_{00}$ | 1.00000 | 0.995321 |

${\overline{C}}_{20}$ * | −0.04757 | −0.05173 |

${\overline{C}}_{21}$ | 0.00127 | 0.00123 |

${\overline{C}}_{22}$ | 0.02467 | 0.02863 |

${\overline{S}}_{21}$ | 0.00014 | 0.00014 |

${\overline{S}}_{22}$ | 0.00032 | 0.00029 |

${\overline{C}}_{30}$ | 0.00303 | 0.00288 |

${\overline{C}}_{31}$ | −0.00452 | −0.00464 |

${\overline{C}}_{32}$ | −0.00902 | −0.00923 |

${\overline{C}}_{33}$ | 0.00162 | 0.00154 |

${\overline{S}}_{31}$ | 0.00216 | 0.00249 |

${\overline{S}}_{32}$ | 0.00075 | 0.00084 |

${\overline{S}}_{33}$ | −0.01360 | −0.01565 |

$\theta $ [${}^{\circ}$] | −1.1000 | −1.1276 |

**Table 6.**Standard deviation of estimated gravity coefficients elucidated by the Monte-Carlo simulation.

Item | Absence of Ephemeris Errors | Presence of Ephemeris Errors |
---|---|---|

${C}_{00}$ | $2.5\times {10}^{-7}$ | 0.08‰ |

${C}_{20}$ | 0.04‰ | 0.65‰ |

${C}_{22}$ | 0.15‰ | 1.88‰ |

$\theta $ | 0.59‰ | 8.32% |

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**MDPI and ACS Style**

Chen, H.; Rambaux, N.; Lainey, V.; Hestroffer, D.
Mothership-Cubesat Radioscience for Phobos Geodesy and Autonomous Navigation. *Remote Sens.* **2022**, *14*, 1619.
https://doi.org/10.3390/rs14071619

**AMA Style**

Chen H, Rambaux N, Lainey V, Hestroffer D.
Mothership-Cubesat Radioscience for Phobos Geodesy and Autonomous Navigation. *Remote Sensing*. 2022; 14(7):1619.
https://doi.org/10.3390/rs14071619

**Chicago/Turabian Style**

Chen, Hongru, Nicolas Rambaux, Valéry Lainey, and Daniel Hestroffer.
2022. "Mothership-Cubesat Radioscience for Phobos Geodesy and Autonomous Navigation" *Remote Sensing* 14, no. 7: 1619.
https://doi.org/10.3390/rs14071619