## Abstract

We have developed a method for solving the minimum-time, constant, continuous low thrust, near circular-to-circular transfers using the method of particular solutions and collocation. The first step is to compute a sub-optimal solution by iteratively solving for the coefficients of the Chebyshev polynomials that parameterize the steering angle of the control vector. This unique implementation of minimum norm direct optimization is attractive in that it does not require partial derivatives, yet we have shown that we can accommodate a relatively high dimensional parameterization of the control variables. Once the sub-optimal solution has been obtained it is used as an initial guess to "warm start" a collocation algorithm. The collocation algorithm is simply used to generate a low fidelity solution for the costates (about 4 digit accuracy) that is fed as an initial guess to start the method of particular solutions shooting method. This low fidelity collocation solution is computed using finite difference derivatives and only a few iterations (and expensive matrix inversions) are required to produce an adequate initial guess. The method of particular solutions, which does not require partial derivatives to be computed or propagated, is extremely efficient and is used to solve the state/costate two-point boundary value problem by iterating on the initial costates that converge to a solution that satisfies the final boundary conditions in the minimum time-of-flight. One approach for solving a minimum-time problem is to map time onto the fixed domain from 0 to 1, and append a trivial differential equation to the set of state equations that integrates to give an unknown free constant (final time). Since our basis functions are the orthogonal Chebyshev polynomials, we map time onto the domain from -1 to 1 where Chebyshev polynomials exist, and thus our algorithm minimizes time-of-flight while converging to the optimal minimum-time solution. We demonstrate the capability of our algorithm by computing an example orbit transfer from a medium Earth orbit (a = 26,000 km) to rendezvous with a piece of debris in a geostationary orbit.

Original language | English (US) |
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Title of host publication | Spaceflight Mechanics 2017 |

Editors | Jon A. Sims, Frederick A. Leve, Jay W. McMahon, Yanping Guo |

Publisher | Univelt Inc. |

Pages | 675-686 |

Number of pages | 12 |

ISBN (Print) | 9780877036371 |

State | Published - 2017 |

Externally published | Yes |

Event | 27th AAS/AIAA Space Flight Mechanics Meeting, 2017 - San Antonio, United States Duration: Feb 5 2017 → Feb 9 2017 |

### Publication series

Name | Advances in the Astronautical Sciences |
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Volume | 160 |

ISSN (Print) | 0065-3438 |

### Other

Other | 27th AAS/AIAA Space Flight Mechanics Meeting, 2017 |
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Country/Territory | United States |

City | San Antonio |

Period | 2/5/17 → 2/9/17 |

## ASJC Scopus subject areas

- Aerospace Engineering
- Space and Planetary Science