Minimum fuel trajectories for round trip lunar missions
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Minimum fuel trajectories for round trip lunar missions Luiz Arthur Gagg Filho1 · Sandro da Silva Fernandes1
Received: 27 April 2016 / Revised: 18 September 2017 / Accepted: 7 November 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
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 fourbody problem (PBR4BP). For computing the optimal trajectories, the Sequential GradientRestoration 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. Keywords Patched-conic approximation · Three-body problem · Four-body problem · Optimal Earth–Moon trajectories · Round trip lunar missions Mathematics Subject Classification 70M20
Communicated by Enrique Zuazua.
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Luiz Arthur Gagg Filho [email protected] Sandro da Silva Fernandes [email protected]
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Departamento de Matemática, Instituto Tecnológico de Aeronáutica, São José dos Campos, SP 12228-900, Brazil
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L. A. Gagg Filho, S. da. Silva Fernandes
1 Introduction Usually, in the study of optimal space trajectories the actuators of the space vehicle are assumed impulsive, that means that they produce an instantaneous velocity increment to put the vehicle in the desired trajectory. The determination of the velocity increments depends upon the mathematical model used to describe the motion of the space vehicle. These mathematical models can be divided in four classes according to the literature (Prado 1993): two-body model, perturbed two-body model, three-body model and N-body model. The two-body model is the simplest one and it is more studied among all because it provides a good approximation for more complex problems and it holds analytical solutions. The so-called Hohmann transfer appears from this model (Szebehely and Mark 1998). By the early 1960th decade, the determination of impulsive trajectories in more complex problems has become more s
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