# Minimum fuel trajectories for round trip lunar missions

## Abstract

In this work, a study about minimum fuel trajectories in a round trip journey to the Moon is presented. It is assumed that the velocity changes are instantaneous, that is, the propulsion system is capable of delivering impulses such that the fuel consumption is represented by the total velocity increment applied to the space vehicle. It is also assumed that the velocity increments are applied tangentially to the terminal orbits, and, the outgoing trip and the return trip are analyzed separately such that the whole mission is performed with four impulses (two impulses in each trip). The mathematical models used to describe the motion of the space vehicle are three: the lunar patched-conic approximation; the classic planar circular restricted three-body problem, and, the planar bi-circular restricted four-body problem (PBR4BP). For computing the optimal trajectories, the Sequential Gradient-Restoration Algorithm with constraints is used. The influence of the Sun on round trip lunar missions is analyzed through the PBR4BP model. For all models, the trajectories studied are direct ascent maneuvers, and, both the outgoing and return trips are considered. The results obtained through the different models are compared with each other. The optimal results for the PBR4BP model show that a smallmore »

- Authors:

- Instituto Tecnológico de Aeronáutica, Departamento de Matemática (Brazil)

- Publication Date:

- OSTI Identifier:
- 22769267

- Resource Type:
- Journal Article

- Journal Name:
- Computational and Applied Mathematics

- Additional Journal Information:
- Journal Volume: 37; Journal Issue: 3; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; 97 MATHEMATICAL METHODS AND COMPUTING; ALGORITHMS; APPROXIMATIONS; COMPARATIVE EVALUATIONS; FOUR-BODY PROBLEM; FUEL CONSUMPTION; MATHEMATICAL MODELS; PROPULSION SYSTEMS; SPACE VEHICLES; SUN; THREE-BODY PROBLEM

### Citation Formats

```
Gagg Filho, Luiz Arthur, E-mail: luizarthur.gagg@gmail.com, and Silva Fernandes, Sandro da, E-mail: sandro@ita.br.
```*Minimum fuel trajectories for round trip lunar missions*. United States: N. p., 2018.
Web. doi:10.1007/S40314-017-0533-Y.

```
Gagg Filho, Luiz Arthur, E-mail: luizarthur.gagg@gmail.com, & Silva Fernandes, Sandro da, E-mail: sandro@ita.br.
```*Minimum fuel trajectories for round trip lunar missions*. United States. doi:10.1007/S40314-017-0533-Y.

```
Gagg Filho, Luiz Arthur, E-mail: luizarthur.gagg@gmail.com, and Silva Fernandes, Sandro da, E-mail: sandro@ita.br. Sun .
"Minimum fuel trajectories for round trip lunar missions". United States. doi:10.1007/S40314-017-0533-Y.
```

```
@article{osti_22769267,
```

title = {Minimum fuel trajectories for round trip lunar missions},

author = {Gagg Filho, Luiz Arthur, E-mail: luizarthur.gagg@gmail.com and Silva Fernandes, Sandro da, E-mail: sandro@ita.br},

abstractNote = {In this work, a study about minimum fuel trajectories in a round trip journey to the Moon is presented. It is assumed that the velocity changes are instantaneous, that is, the propulsion system is capable of delivering impulses such that the fuel consumption is represented by the total velocity increment applied to the space vehicle. It is also assumed that the velocity increments are applied tangentially to the terminal orbits, and, the outgoing trip and the return trip are analyzed separately such that the whole mission is performed with four impulses (two impulses in each trip). The mathematical models used to describe the motion of the space vehicle are three: the lunar patched-conic approximation; the classic planar circular restricted three-body problem, and, the planar bi-circular restricted four-body problem (PBR4BP). For computing the optimal trajectories, the Sequential Gradient-Restoration Algorithm with constraints is used. The influence of the Sun on round trip lunar missions is analyzed through the PBR4BP model. For all models, the trajectories studied are direct ascent maneuvers, and, both the outgoing and return trips are considered. The results obtained through the different models are compared with each other. The optimal results for the PBR4BP model show that a small reduction of the fuel consumption can be achieved if the initial phase angle of the Sun is chosen properly.},

doi = {10.1007/S40314-017-0533-Y},

journal = {Computational and Applied Mathematics},

issn = {0101-8205},

number = 3,

volume = 37,

place = {United States},

year = {2018},

month = {7}

}