Optimal round trip lunar missions based on the patched-conic approximation
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Optimal round trip lunar missions based on the patched-conic approximation Luiz Arthur Gagg Filho1 · Sandro da Silva Fernandes1
Received: 23 February 2015 / Revised: 11 May 2015 / Accepted: 18 May 2015 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015
Abstract A study of optimal bi-impulsive trajectories of round trip lunar missions is presented in this paper. The optimization criterion is the total velocity increment. The dynamical model utilized to describe the motion of the space vehicle is a full lunar patched-conic approximation, which embraces the lunar patched-conic of the outgoing trip and the lunar patched-conic of the return mission. Each one of these parts is considered separately to solve an optimization problem of two degrees of freedom. The parameters to be optimized are two: the phase angle of the point at which the space vehicle reaches the edge of the Moon’s sphere of influence and the initial velocity at departure. The Sequential Gradient Restoration Algorithm is employed to achieve the optimal solutions. Analytical and numerical derivatives of expressions describing the lunar patched-conic approximations are utilized to ensure the results. The results based on the patched-conic approximation show a good agreement with the ones provided by literature, and the solution trajectories proved to be consistent with the image trajectories theorem. Keywords Optimal Earth–Moon trajectories · Patched-conic approximation · Round trip lunar missions Mathematics Subject Classification
70M20 Orbital mechanics
1 Introduction The problem of transferring a space vehicle from one orbit to another orbit has been growing in importance in last decades. The solution of this problem is of great concern to the space
Communicated by Eduardo Souza de Cursi.
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Luiz Arthur Gagg Filho [email protected] Departamento de Matemática, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP 12228-900, Brazil
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L. Arthur Gagg Filho, S. da Silva Fernandes
science. On May 2012, one variant of the Dragon spaceship belonging to the private company SpaceX became the first commercial ship to dock on the International Space Station defining a rendezvous problem that means that the final conditions are completely established for the trajectory calculus (Howell 2013). To send space probes to unveil the Solar System, it is vital to accomplish a lot of trajectories determination as, for instance, the Mars Science Laboratory mission, in which inventive stages were realized for the purpose of landing the Curiosity rover on Mars surface (Way et al. 2007). Other commercial applications can be found such as: the maintenance of telecommunications satellites, GPS constellations satellites, geostationary satellites and others (Prado and Rios-Neto 1993). However, the science development is the greatest benefit of this field because the trajectory design is one of many keys necessary to perform a space exploration and solve fundamentals issues of science, including the origin of life on Earth. In the
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