Luiz Arthur Gagg Filho

Optimal round trip lunar missions based on the patched-conic approximation

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.

Minimum fuel trajectories for round trip lunar missions

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.

A study of Earth–Moon trajectories based on analytical expressions for the velocity increments

A study of Earth–Moon bi-impulsive trajectories is presented in this paper. The motion of the space vehicle is described by the classic planar circular restricted three-body problem. The velocity increments are computed through analytical expressions, which are derived from the development of the Jacobi Integral expression. To determine the trajectories, a new two-point boundary value problem (TPBVP) with prescribed value of Jacobi Integral is formulated. Internal and external trajectories are determined through the solution of this new TPBVP for several times of flight. A relation between the Jacobi Integral and the Kepler’s energy at arrival is derived and several kinds of study are performed. Critical values of the Jacobi Integral, for which the Kepler’s energy of the space vehicle on the arrival trajectory becomes negative, are calculated for several configurations of arrival at the low Moon orbit in both directions: clockwise and counterclockwise. Results show that the proposed method allows the estimation of the fuel consumption before solving the TPBVP, and it facilitates the determination of trajectories with large time of flight. However, increasing values of the time of flight are not necessarily related with the increase of the Jacobi Integral value, which means that the obtaining of new trajectories becomes more difficult as the Jacobi Integral increases. Moreover, the proposed method provides results to be used as initial guess for more complex models and for optimization algorithms in order to minimize the total fuel consumption. For this case, this paper presents an example where an internal trajectory with large time of flight is optimized considering the Sun’s attraction.