# Simplified Maneuvering Strategies for Rendezvous in Near-Circular Earth Orbits

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## Abstract

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## 1. Introduction

## 2. Problem Statement

## 3. Reconfiguration Strategies

## 4. Results

#### 4.1. Test Case 1—In-Plane Control

#### 4.2. Test Case 2—3D Control

## 5. Trajectory Constraints

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Relative Orbital Element changes. (

**a**) Transfer in Relative Eccentricity Vector Plane. (

**b**) Transfer in Relative Semi-Major Axis—Relative Mean Longitude Plane.

**Figure 10.**Reconfiguration using Scheme 3. (

**a**) Relative 3D Trajectory in Hill’s Coordinates. (

**b**) Trajectory in the Relative Inclination Vector Plane.

**Figure 11.**Reconfiguration using Scheme 4. (

**a**) Relative 3D Trajectory in Hill’s Coordinates. (

**b**) Trajectory in the Relative Inclination Vector Plane.

**Figure 12.**Relative 2D Trajectory in Hill’s Coordinates. (

**a**) Unconstrained Trajectory using Scheme 1. (

**b**) Constrained Trajectory using Scheme 1.

**Figure 13.**Evolution of Relative Distance. (

**a**) Unconstrained Distance using Scheme 1. (

**b**) Constrained Distance using Scheme 1.

Scheme Identifier | Dimensionality | Number of Impulses | Maneuver Type |
---|---|---|---|

1 | 2D | 3 | RT-RT-RT |

2 | 3D | 4 | RT-RT-RT-N |

3 | 3D | 3 | RTN-RTN-RTN |

4 | 3D | 3 | RTN-RTN-RTN |

Step, ${}^{\circ}$ | Time, s | Samples | Best Points, m/s | Optimized Value, m/s | Deviation, % |
---|---|---|---|---|---|

20 | 0.01 | 350 | 0.3106 | 0.3075 | 0.9992 |

1 | 0.03 | 130139 | 0.3105 | 0.3075 | 0.9799 |

0.05 | 2.53 | 51850799 | 0.3105 | 0.3075 | 0.9799 |

- | Scheme 1 | AVANTI | Numerical |
---|---|---|---|

${u}_{1},\mathrm{rad}$ | 0 | 2.5830 | 0 |

${u}_{2},\mathrm{rad}$ | 8.8550 | 5.7246 | 9.4540 |

${u}_{3},\mathrm{rad}$ | 12.5573 | 8.8662 | 12.5664 |

$\delta {v}_{1T},\mathrm{m}/\mathrm{s}$ | −0.1654 | −0.2964 | −0.1645 |

$\delta {v}_{2T},\mathrm{m}/\mathrm{s}$ | 0.0084 | −0.0379 | 0.0079 |

$\delta {v}_{3T},\mathrm{m}/\mathrm{s}$ | 0.1308 | 0.3080 | 0.1304 |

$\delta {v}_{1R},\mathrm{m}/\mathrm{s}$ | −0.0264 | 0 | −0.0296 |

$\delta {v}_{2R},\mathrm{m}/\mathrm{s}$ | −0.0012 | 0 | −0.0002 |

$\delta {v}_{3R},\mathrm{m}/\mathrm{s}$ | −0.0204 | 0 | −0.0235 |

$\sqrt{J},\mathrm{m}/\mathrm{s}$ | 0.3083 | 0.6422 | 0.3075 |

% | 0.2716 | 108.8497 | - |

Mean Relative Orbital Elements, m | ||||
---|---|---|---|---|

${\mathit{a}}_{\mathit{c}}\mathit{\delta}\mathit{a}$ | ${\mathit{a}}_{\mathit{c}}\mathit{\delta}\mathit{\lambda}$ | ${\mathit{a}}_{\mathit{c}}\mathit{\delta}{\mathit{e}}_{\mathit{x}}$ | ${\mathit{a}}_{\mathit{c}}\mathit{\delta}{\mathit{e}}_{\mathit{y}}$ | |

Desired | 0 | −5000 | 150 | 0 |

Achieved | −0.2386 | −5000.5059 | 150.1075 | −2.7300 |

Mean Relative Orbital Elements, m | |||||||
---|---|---|---|---|---|---|---|

Algorithm | Metric | ${\mathit{a}}_{\mathit{c}}\mathit{\delta}\mathit{a}$ | ${\mathit{a}}_{\mathit{c}}\mathit{\delta}\mathit{\lambda}$ | ${\mathit{a}}_{\mathit{c}}\mathit{\delta}{\mathit{e}}_{\mathit{x}}$ | ${\mathit{a}}_{\mathit{c}}\mathit{\delta}{\mathit{e}}_{\mathit{y}}$ | ${\mathit{a}}_{\mathit{c}}\mathit{\delta}{\mathit{i}}_{\mathit{x}}$ | ${\mathit{a}}_{\mathit{c}}\mathit{\delta}{\mathit{i}}_{\mathit{y}}$ |

Scheme 2 | Desired | 0 | −5000 | 150 | 0 | 89.9863 | 1.5707 |

Achieved | −0.2774 | −5001.5310 | 150.1351 | −2.7309 | 89.9618 | 0.2859 | |

Scheme 3 | Desired | 0 | −5000 | 150 | 0 | 89.9863 | 1.5707 |

Achieved | −0.2781 | −5001.0765 | 150.1301 | −2.7313 | 89.9647 | −0.7769 | |

Scheme 4 | Desired | 0 | −5000 | 150 | 0 | 89.9863 | 1.5707 |

Achieved | −0.2897 | −4992.6705 | 150.6405 | 2.7705 | 89.9661 | −1.7231 |

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**MDPI and ACS Style**

Costigliola, D.; Casalino, L.
Simplified Maneuvering Strategies for Rendezvous in Near-Circular Earth Orbits. *Aerospace* **2023**, *10*, 1027.
https://doi.org/10.3390/aerospace10121027

**AMA Style**

Costigliola D, Casalino L.
Simplified Maneuvering Strategies for Rendezvous in Near-Circular Earth Orbits. *Aerospace*. 2023; 10(12):1027.
https://doi.org/10.3390/aerospace10121027

**Chicago/Turabian Style**

Costigliola, Davide, and Lorenzo Casalino.
2023. "Simplified Maneuvering Strategies for Rendezvous in Near-Circular Earth Orbits" *Aerospace* 10, no. 12: 1027.
https://doi.org/10.3390/aerospace10121027