### Abstract

The restricted four-body problem is used to derive linearized equations of relative motion that take into account the perturbing effects of a larger, secondary gravitational source. The result is a system of coupled, first-order, linear differential equations that has a complete analytical solution. In this paper, we numerically integrate these linearized equations and compare the results to the outcomes of the restricted four-body problem and the well known Hill-Clohessy-Wiltshire equations. Various cases pertaining to two scenarios are analyzed: the relative motion of a chase spacecraft with respect to a target satellite orbiting an asteroid while both are perturbed by the Sun, and the relative motion of a chase spacecraft with respect to a target satellite orbiting the Moon while both are perturbed by the Earth. The results demonstrate that the Benavides-Spencer formulation is far more accurate than the results given by the Hill-Clohessy-Wiltshire equations when compared to the real-life outcomes returned by the numerical integration of the restricted four-body problem. Future work will unveil the complete analytical solution of the Benavides-Spencer formulation both as an initial value problem and a boundary value problem.

Original language | English (US) |
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Title of host publication | International Astronautical Federation - 59th International Astronautical Congress 2008, IAC 2008 |

Pages | 4690-4699 |

Number of pages | 10 |

State | Published - Dec 1 2008 |

Event | 59th International Astronautical Congress 2008, IAC 2008 - Glasgow, United Kingdom Duration: Sep 29 2008 → Oct 3 2008 |

### Publication series

Name | International Astronautical Federation - 59th International Astronautical Congress 2008, IAC 2008 |
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Volume | 7 |

### Other

Other | 59th International Astronautical Congress 2008, IAC 2008 |
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Country | United Kingdom |

City | Glasgow |

Period | 9/29/08 → 10/3/08 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Space and Planetary Science
- Aerospace Engineering
- Astronomy and Astrophysics

### Cite this

*International Astronautical Federation - 59th International Astronautical Congress 2008, IAC 2008*(pp. 4690-4699). (International Astronautical Federation - 59th International Astronautical Congress 2008, IAC 2008; Vol. 7).

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*International Astronautical Federation - 59th International Astronautical Congress 2008, IAC 2008.*International Astronautical Federation - 59th International Astronautical Congress 2008, IAC 2008, vol. 7, pp. 4690-4699, 59th International Astronautical Congress 2008, IAC 2008, Glasgow, United Kingdom, 9/29/08.

**Preliminary assessment of the next generation equations of relative motion.** / Benavides, Julio C.; Spencer, David Bradley.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Preliminary assessment of the next generation equations of relative motion

AU - Benavides, Julio C.

AU - Spencer, David Bradley

PY - 2008/12/1

Y1 - 2008/12/1

N2 - The restricted four-body problem is used to derive linearized equations of relative motion that take into account the perturbing effects of a larger, secondary gravitational source. The result is a system of coupled, first-order, linear differential equations that has a complete analytical solution. In this paper, we numerically integrate these linearized equations and compare the results to the outcomes of the restricted four-body problem and the well known Hill-Clohessy-Wiltshire equations. Various cases pertaining to two scenarios are analyzed: the relative motion of a chase spacecraft with respect to a target satellite orbiting an asteroid while both are perturbed by the Sun, and the relative motion of a chase spacecraft with respect to a target satellite orbiting the Moon while both are perturbed by the Earth. The results demonstrate that the Benavides-Spencer formulation is far more accurate than the results given by the Hill-Clohessy-Wiltshire equations when compared to the real-life outcomes returned by the numerical integration of the restricted four-body problem. Future work will unveil the complete analytical solution of the Benavides-Spencer formulation both as an initial value problem and a boundary value problem.

AB - The restricted four-body problem is used to derive linearized equations of relative motion that take into account the perturbing effects of a larger, secondary gravitational source. The result is a system of coupled, first-order, linear differential equations that has a complete analytical solution. In this paper, we numerically integrate these linearized equations and compare the results to the outcomes of the restricted four-body problem and the well known Hill-Clohessy-Wiltshire equations. Various cases pertaining to two scenarios are analyzed: the relative motion of a chase spacecraft with respect to a target satellite orbiting an asteroid while both are perturbed by the Sun, and the relative motion of a chase spacecraft with respect to a target satellite orbiting the Moon while both are perturbed by the Earth. The results demonstrate that the Benavides-Spencer formulation is far more accurate than the results given by the Hill-Clohessy-Wiltshire equations when compared to the real-life outcomes returned by the numerical integration of the restricted four-body problem. Future work will unveil the complete analytical solution of the Benavides-Spencer formulation both as an initial value problem and a boundary value problem.

UR - http://www.scopus.com/inward/record.url?scp=77950510345&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950510345&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:77950510345

SN - 9781615671601

T3 - International Astronautical Federation - 59th International Astronautical Congress 2008, IAC 2008

SP - 4690

EP - 4699

BT - International Astronautical Federation - 59th International Astronautical Congress 2008, IAC 2008

ER -