# Optimal On-Orbit Inspection of Satellite Formation

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## Abstract

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## 1. Introduction

## 2. Mission Description and Mathematical Model

#### 2.1. IS Relative Dynamics

#### 2.2. Total Velocity Variation Evaluation

## 3. Trajectory Optimization and Numerical Simulations

#### 3.1. Two-Step Optimization Procedure

#### 3.2. Test Case and Model Validation

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Local–vertical–local–horizontal (or Hill’s) reference frame ${\mathcal{T}}_{H}(C;{\widehat{\mathit{i}}}_{x},{\widehat{\mathit{i}}}_{y},{\widehat{\mathit{i}}}_{z})$.

**Figure 9.**Optimal IS inspection trajectories in the four optimal sequences listed in Table 2.

**Figure 10.**Duration of the generic arc in percentages of the total flight time $\mathrm{\Delta}t\simeq 5.05\phantom{\rule{0.166667em}{0ex}}\mathrm{h}$.

SC Label | ${\mathit{\rho}}_{{\mathbf{SC}}_{\mathit{i}}}\xb7{\widehat{\mathit{i}}}_{\mathit{x}}$ [km] | ${\mathit{\rho}}_{{\mathbf{SC}}_{\mathit{i}}}\xb7{\widehat{\mathit{i}}}_{\mathit{y}}$ [km] | ${\mathit{\rho}}_{{\mathbf{SC}}_{\mathit{i}}}\xb7{\widehat{\mathit{i}}}_{\mathit{z}}$ [km] |
---|---|---|---|

$\mathrm{\u2460}$ | 10 | 0 | 0 |

$\mathrm{\u2461}$ | −10 | 0 | 0 |

$\mathrm{\u2462}$ | 0 | 10 | 0 |

$\mathrm{\u2463}$ | 0 | −10 | 0 |

$\mathrm{\u2464}$ | 0 | 0 | 10 |

$\mathrm{\u2465}$ | 0 | 0 | −10 |

**Table 2.**Inspection sequences that give a total velocity variation of about $70\phantom{\rule{0.166667em}{0ex}}\mathrm{m}/\mathrm{s}$.

Sequence | SC Label |
---|---|

first | $C\to \mathrm{\u2461}\to \mathrm{\u2460}\to \mathrm{\u2463}\to \mathrm{\u2464}\to \mathrm{\u2465}\to \mathrm{\u2462}$ |

second | $C\to \mathrm{\u2460}\to \mathrm{\u2461}\to \mathrm{\u2462}\to \mathrm{\u2465}\to \mathrm{\u2464}\to \mathrm{\u2463}$ |

third | $C\to \mathrm{\u2461}\to \mathrm{\u2460}\to \mathrm{\u2463}\to \mathrm{\u2465}\to \mathrm{\u2464}\to \mathrm{\u2462}$ |

fourth | $C\to \mathrm{\u2460}\to \mathrm{\u2461}\to \mathrm{\u2462}\to \mathrm{\u2464}\to \mathrm{\u2465}\to \mathrm{\u2463}$ |

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**MDPI and ACS Style**

Caruso, A.; Quarta, A.A.; Mengali, G.; Bassetto, M.
Optimal On-Orbit Inspection of Satellite Formation. *Remote Sens.* **2022**, *14*, 5192.
https://doi.org/10.3390/rs14205192

**AMA Style**

Caruso A, Quarta AA, Mengali G, Bassetto M.
Optimal On-Orbit Inspection of Satellite Formation. *Remote Sensing*. 2022; 14(20):5192.
https://doi.org/10.3390/rs14205192

**Chicago/Turabian Style**

Caruso, Andrea, Alessandro A. Quarta, Giovanni Mengali, and Marco Bassetto.
2022. "Optimal On-Orbit Inspection of Satellite Formation" *Remote Sensing* 14, no. 20: 5192.
https://doi.org/10.3390/rs14205192