# Numerical Solution for the Single-Impulse Flyby Co-Orbital Spacecraft Problem

^{*}

## Abstract

**:**

^{−6}.

## 1. Introduction

## 2. Description of the Problem

## 3. Mathematical Formulation of the Problem

#### 3.1. Nonlinear Equations for Terminal Constraints

#### 3.2. Nonlinear Equations for Coplanar Lambert Problem

## 4. Numerical Method without Derivation for the Two Equations

#### 4.1. Solution of Lambert Problem without Derivation

^{−6}, preventing the iterations from proceeding correctly. Simultaneously, the initial value-finding algorithm can significantly reduce the number of iterations. The second point that must be addressed is why quadratic functions are employed. Using quadratic functions reduces the amount of computing required, while quadratic functions fit the curve better than linear interpolation, with the mean degree-of-freedom-adjusted coefficient of the determination being 0.998 for 100,000 random tests.

#### 4.2. Solution of Single-Impulse Flyby Two Co-Orbital Spacecraft Problem without Derivation

## 5. Numerical Examples

^{®}Core™ i7-9750H CPU at 2.60 GHz with Windows 10 and run on MATLAB R2018b.

#### 5.1. The Lambert Problem

^{−7}as shown in Figure 7.

#### 5.2. The Single-Impulse Flyby Co-Orbital Spacecraft Problem

#### Random Initial Orbital Elements

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Glossary

$c$ | Length of the transfer chord |

${e}_{i}$ | Eccentricity vector of orbit $i$ |

${e}_{i}$ | Eccentricity of orbit $i$ |

${e}_{2T}$ | Perpendicular component of the transfer chord |

${e}_{2F}$ | Parallel component of the transfer chord |

${M}_{i,{t}_{j}}$ | Mean anomaly of spacecraft $i$ at time ${t}_{j}$ |

${n}_{i}$ | Mean motion of orbit $i$ |

${p}_{i}$ | Semi-latus rectum of orbit $i$ |

${r}_{i,{t}_{j}}$ | Position of spacecraft $i$ at time ${t}_{j}$ |

${r}_{i,{t}_{j}}$ | Distance between spacecraft $i$ and earth center at time ${t}_{j}$ |

${\theta}_{i,{t}_{j}}$ | Anomaly of spacecraft $i$ at time ${t}_{j}$ |

${\omega}_{{o}_{i}}$ | Perigee of orbit $i$ |

$\mathsf{\Delta}{\omega}_{{o}_{i},{o}_{j}}$ | Angle between perigees of two orbits (from $i$ to $j$) |

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Average Computation Time (ms) | 1000 Times | 10,000 Times | 50,000 Times | 100,000 Times | Efficiency Improvement |
---|---|---|---|---|---|

Universal variable method [35] | 1.3581 | 1.3593 | 1.3547 | 1.3543 | 92.09% |

Traversal search method | 0.2120 | 0.2039 | 0.1977 | 0.1972 | 45.69% |

Our method | 0.1118 | 0.1114 | 0.1075 | 0.1071 |

Spacecraft | ${\mathit{r}}_{\mathit{p}}\left(\mathbf{km}\right)$ | ${\mathit{r}}_{\mathit{a}}\left(\mathbf{km}\right)$ | $\mathit{i}$ (°) | $\mathbf{\Omega}$ (°) | $\mathit{\omega}$ (°) | $\mathit{f}$ (°) |
---|---|---|---|---|---|---|

$S{C}_{0}$ | 7134 | 7861 | 23 | 11 | 38 | 17 |

$S{T}_{1}$ | 9871 | 10,306 | 40 | 81 | ||

$S{T}_{2}$ | 82 |

Spacecraft | ${\mathit{r}}_{\mathit{p}}\left(\mathbf{km}\right)$ | ${\mathit{r}}_{\mathit{a}}\left(\mathbf{km}\right)$ | $\mathit{i}$ (°) | $\mathbf{\Omega}$ (°) | $\mathit{\omega}$ (°) | $\mathit{f}$ (°) |
---|---|---|---|---|---|---|

$S{C}_{0}$ | $\left[7000,7500\right]$ | $\left[7500,9000\right]$ | $\left[0,45\right]$ | $\left[0,90\right]$ | $\left[0,45\right]$ | $\left[0,45\right]$ |

$S{T}_{1}$ | $\left[9000,\mathrm{10,000}\right]$ | $\left[\mathrm{10,000},\mathrm{12,000}\right]$ | $\left[0,180\right]$ | $\left[0,90\right]$ | ||

$S{T}_{2}$ | $\left[0,270\right]$ |

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**MDPI and ACS Style**

Su, H.; Dong, Z.; Liu, L.; Xia, L.
Numerical Solution for the Single-Impulse Flyby Co-Orbital Spacecraft Problem. *Aerospace* **2022**, *9*, 374.
https://doi.org/10.3390/aerospace9070374

**AMA Style**

Su H, Dong Z, Liu L, Xia L.
Numerical Solution for the Single-Impulse Flyby Co-Orbital Spacecraft Problem. *Aerospace*. 2022; 9(7):374.
https://doi.org/10.3390/aerospace9070374

**Chicago/Turabian Style**

Su, Haoxiang, Zhenghong Dong, Lihao Liu, and Lurui Xia.
2022. "Numerical Solution for the Single-Impulse Flyby Co-Orbital Spacecraft Problem" *Aerospace* 9, no. 7: 374.
https://doi.org/10.3390/aerospace9070374