# Neural Network-Based Approximation Model for Perturbed Orbit Rendezvous

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Description of Orbit Rendezvous

## 3. Methodology

#### 3.1. Approximation Method of the Perturbed Orbit Rendezvous Problem

_{2}perturbation were used and the changes in the orbit elements by maneuvers were set as unknown parameters. We assumed that $\Delta {a}_{0},\Delta {i}_{0}\text{}\mathrm{and}\text{}\Delta {\mathsf{\Omega}}_{0}$ were the differences of the semi-major axis, inclination, and RAAN between the initial and target orbits. $\Delta {a}_{1}$, $\Delta {i}_{1}$, and $\Delta {\mathsf{\Omega}}_{1}$ then denoted the changes in the semi-major axis, inclination, and RAAN caused by the impulses at the beginning of the transfer. $\Delta {a}_{2}$, $\Delta {i}_{2}$, and $\Delta {\mathsf{\Omega}}_{2}$ denoted the changes in semi-major axis, inclination, and RAAN caused by the impulses at the end of the transfer. Thus, the equality constraint optimization model was obtained as:

#### 3.2. Features Analysis of the Perturbed Orbit Rendezvous Problem

#### 3.3. Neural Network and Training

## 4. Experiments

#### 4.1. Dataset Generalization and Training Result

^{−6}s using an AMD 4.2 GHz CPU, which demonstrated a higher efficiency than previous approximation methods [8,9,10,11].

#### 4.2. Application in Global Optimization

## 5. Conclusions

^{−6}s using an ordinary desktop processor and could be directly applied to the global optimization of multi-target rendezvous missions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a_{0} | Semi-major axis of initial orbit |

i_{0} | Inclination of initial orbit |

Δa_{0} | Difference of semi-major axis between initial and target orbits |

Δi_{0} | Difference of inclination axis between initial and target orbits |

ΔΩ_{0} | Difference of RAAN axis between initial and target orbits |

Δu_{0} | Difference of phase axis between initial and target orbits |

Δe_{x0} | Difference of e cosω between initial and target orbits |

Δe_{y0} | Difference of e sinω between initial and target orbits |

${\dot{\mathsf{\Omega}}}_{0}$ | Initial drift rate of RAAN |

${e}_{\mathrm{max}}$ | Upper limit of eccentricity |

$\Delta {a}_{\mathrm{max}}$ | Upper limit of change in semi-major axis |

$\Delta {i}_{\mathrm{max}}$ | Upper limit of change in inclination |

$\Delta {\mathsf{\Omega}}_{\mathrm{max}}$ | Upper limit of change in RAAN |

$\Delta {t}_{\mathrm{max}}$ | Upper limit of flight time |

$\overline{a}$ | Mean value of semi-major axis |

$\overline{i}$ | Mean value of inclination |

Δv | Velocity increment of orbit rendezvous |

OTV | Orbit transfer vehicle |

${m}_{0}$ | Launch mass of OTV |

${m}_{N}$ | Dry mass of OTV |

$\Delta {m}_{kit}$ | Mass of de-orbit package released at each debris point |

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**Figure 5.**Correlation between the proposed neural network and approximate method in Petropoulos 2018.

Number of Hidden Layers | Number of Nodes in Each Hidden Layer | MRE (%) | MAE (m/s) | Time of Each Training Epoch (s) | Training Time (s) | Time of Δv Evaluation (s) |
---|---|---|---|---|---|---|

2 | 30 | 1.34 | 5.3 | 4.6 | 1380 | 1.2 × 10^{−6} |

2 | 60 | 0.96 | 3.8 | 5.0 | 1500 | 4.8 × 10^{−6} |

2 | 90 | 0.89 | 3.7 | 5.2 | 1560 | 1.1 × 10^{−5} |

3 | 60 | 0.81 | 3.3 | 6.0 | 1800 | 8.9 × 10^{−6} |

4 | 60 | 0.79 | 3.2 | 7.0 | 2100 | 1.3 × 10^{−5} |

Model of Velocity Increment | Optimal Order of Debris | Total Δv (m/s) | J (MEUR) | Computational Time (s) |
---|---|---|---|---|

Method in [20] | 72, 107, 61, 10, 28, 3, 64, 66, 31, 90, 73, 87, 57, 35, 69, 65, 8, 43, 71, 4, 29 | 3409.5 | 97.1 | >3600 |

Method in [21] | 72, 107, 61, 73, 3, 69, 64, 66, 31, 10, 90, 87, 57, 35, 28, 65, 8, 43, 71, 4, 29 | 3357.0 | 95.6 | 600 |

Neural network model in this paper | 72, 61, 107, 73, 3, 69, 64, 66, 31, 10, 90, 87, 57, 35, 28, 65, 8, 43, 71, 4, 29 | 3407.5 | 97.1 | 120 |

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**MDPI and ACS Style**

Huang, A.; Wu, S.
Neural Network-Based Approximation Model for Perturbed Orbit Rendezvous. *Mathematics* **2022**, *10*, 2489.
https://doi.org/10.3390/math10142489

**AMA Style**

Huang A, Wu S.
Neural Network-Based Approximation Model for Perturbed Orbit Rendezvous. *Mathematics*. 2022; 10(14):2489.
https://doi.org/10.3390/math10142489

**Chicago/Turabian Style**

Huang, Anyi, and Shenggang Wu.
2022. "Neural Network-Based Approximation Model for Perturbed Orbit Rendezvous" *Mathematics* 10, no. 14: 2489.
https://doi.org/10.3390/math10142489