# Minimum-Energy Transfer Optimization between Near-Circular Orbits Using an Approximate Closed-Form Solution

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The General Optimal Control Problem under Study

## 3. The Averaging Technique and Approximate Solutions

#### 3.1. Averaged Variational Equations in Modified Equinoctial Elements

#### 3.2. Closed-Form Averaged Solution of Orbital Motion Equations

## 4. Multirevolution Low-Thrust Trajectory Optimization

## 5. Results

#### 5.1. Maneuvering in the Vicinity of the Geostationary Orbit

#### 5.2. Spiral Orbit Raising Maneuver

#### 5.3. Comparison with Pontryagin Maximum Principle Control

## 6. Conclusions and Further Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a | semimajor axis |

E | eccentric anomaly |

e | eccentricity |

F | eccentric longitude |

$\mathbf{f}$ | vector of the low-thrust acceleration |

f | magnitude of the low-thrust acceleration |

${f}_{c}$ | circumferential component of the low-thrust acceleration |

${f}_{n}$ | normal component of the low-thrust acceleration |

${f}_{r}$ | radial component of the low-thrust acceleration |

i | inclination |

L | true longitude |

m | spacecraft mass |

P | jet propulsion power |

p, ${e}_{x}$, ${e}_{y}$, ${i}_{x}$, ${i}_{y}$ | modified equinoctial elements |

T | time of flight |

${v}_{\mathrm{ex}}$ | exhaust velocity |

$\mathbf{x}$ | vector of modified equinoctial elements |

$\overline{\mathbf{x}}$ | vector of averaged modified equinoctial elements |

${\alpha}_{k}^{r,c,n}$ | cosine coefficient, Fourier series of the low-thrust acceleration |

${\beta}_{k}^{r,c,n}$ | sine coefficient, Fourier series of the low-thrust acceleration |

$\lambda $ | mean longitude |

$\mu $ | gravitational parameter of the central body |

$\mathrm{\Omega}$ | longitude of the ascending node |

$\omega $ | argument of periapsis |

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Coefficient | Averaged Model | Nonaveraged Model |
---|---|---|

Radial, $\mathrm{mm}/{\mathrm{s}}^{2}$ | ||

${\alpha}_{0}^{r}$ | 0 | −0.0001 |

${\alpha}_{1}^{r}$ | 0.0005 | 0.0005 |

${\beta}_{1}^{r}$ | −0.0005 | −0.0005 |

Circumferential, $\mathrm{mm}/{\mathrm{s}}^{2}$ | ||

${\alpha}_{0}^{c}$ | −0.0070 | −0.0070 |

${\alpha}_{1}^{c}$ | −0.0008 | −0.0009 |

${\beta}_{1}^{c}$ | −0.0013 | −0.0013 |

${\alpha}_{2}^{c}$ | 0 | 0.0001 |

${\beta}_{2}^{c}$ | 0 | −0.0002 |

Normal, $\mathrm{mm}/{\mathrm{s}}^{2}$ | ||

${\alpha}_{0}^{n}$ | 0 | −0.0001 |

${\alpha}_{1}^{n}$ | 0.2129 | 0.2136 |

${\beta}_{1}^{n}$ | −0.1561 | −0.1559 |

${\alpha}_{2}^{n}$ | 0 | 0.0008 |

${\beta}_{2}^{n}$ | 0 | −0.0007 |

**Table 2.**Osculating modified equinoctial elements of the start, target, and final orbits for Case A.

Start Orbit | Target Orbit | Final Orbit | Final Orbit | |
---|---|---|---|---|

at First Stage | at Second Stage | |||

p, km | 42,500 | 42,164 | 42,163.9 | 42,163.9993 |

${e}_{x}$ | $7\xb7{10}^{-4}$ | $1\xb7{10}^{-4}$ | $1.2\xb7{10}^{-4}$ | $9.94\xb7{10}^{-5}$ |

${e}_{y}$ | $9\xb7{10}^{-4}$ | 0 | $-7\xb7{10}^{-6}$ | $2\xb7{10}^{-7}$ |

${i}_{x}$ | $1.4\xb7{10}^{-2}$ | $4.4\xb7{10}^{-2}$ | $4.38\xb7{10}^{-2}$ | $4.3998\xb7{10}^{-2}$ |

${i}_{y}$ | $2.2\xb7{10}^{-2}$ | 0 | $-2\xb7{10}^{-5}$ | $6\xb7{10}^{-7}$ |

Coefficient | Averaged Model | Nonaveraged Model |
---|---|---|

Radial, $\mathrm{mm}/{\mathrm{s}}^{2}$ | ||

${\alpha}_{0}^{r}$ | 0 | 0.0002 |

${\alpha}_{1}^{r}$ | 0 | 0.0017 |

${\beta}_{1}^{r}$ | 0 | 0.0011 |

Circumferential, $\mathrm{mm}/{\mathrm{s}}^{2}$ | ||

${\alpha}_{0}^{c}$ | 0.3783 | 0.3784 |

${\alpha}_{1}^{c}$ | 0 | 0.0019 |

${\beta}_{1}^{c}$ | 0 | −0.0033 |

${\alpha}_{2}^{c}$ | 0 | 0.0003 |

${\beta}_{2}^{c}$ | 0 | 0.0001 |

Normal, $\mathrm{mm}/{\mathrm{s}}^{2}$ | ||

${\alpha}_{0}^{n}$ | 0 | −0.0001 |

${\alpha}_{1}^{n}$ | 0 | 0 |

${\beta}_{1}^{n}$ | 0 | 0 |

${\alpha}_{2}^{n}$ | 0 | −0.0001 |

${\beta}_{2}^{n}$ | 0 | −0.0001 |

**Table 4.**Osculating modified equinoctial elements of the start, target, final, and corrected final orbits for Case B.

Start Orbit | Target Orbit | Final Orbit | Final Orbit | |
---|---|---|---|---|

at First Stage | at Second Stage | |||

p, km | 20,000 | 40,000 | 39,999 | 39,999.7 |

${e}_{x}$ | 0.0 | 0.0 | $-0.002$ | $-3\xb7{10}^{-6}$ |

${e}_{y}$ | 0.0 | 0.0 | $0.003$ | $-7\xb7{10}^{-6}$ |

${i}_{x}$ | 0.0 | 0.0 | $7\xb7{10}^{-8}$ | $-3\xb7{10}^{-8}$ |

${i}_{y}$ | 0.0 | 0.0 | $2\xb7{10}^{-8}$ | $-8\xb7{10}^{-8}$ |

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**MDPI and ACS Style**

Suslov, K.; Shirobokov, M.; Tselousova, A.
Minimum-Energy Transfer Optimization between Near-Circular Orbits Using an Approximate Closed-Form Solution. *Aerospace* **2023**, *10*, 1002.
https://doi.org/10.3390/aerospace10121002

**AMA Style**

Suslov K, Shirobokov M, Tselousova A.
Minimum-Energy Transfer Optimization between Near-Circular Orbits Using an Approximate Closed-Form Solution. *Aerospace*. 2023; 10(12):1002.
https://doi.org/10.3390/aerospace10121002

**Chicago/Turabian Style**

Suslov, Kirill, Maksim Shirobokov, and Anastasia Tselousova.
2023. "Minimum-Energy Transfer Optimization between Near-Circular Orbits Using an Approximate Closed-Form Solution" *Aerospace* 10, no. 12: 1002.
https://doi.org/10.3390/aerospace10121002