© 2025 COSPARThis work describes the development of a semi-analytic theory for a preliminary orbit analysis of the GARATÉA-L Brazilian lunar probe. The dynamical model includes the effects of the zonal harmonics J2 up to J12, the effects of second- and third-degree tesserals and sectorials, and the third-body perturbation due to the attraction of the Earth. The Hamiltonian describing the dynamics is implicitly expressed in Delaunay variables, and, Hori's method is applied to derive a semi-analytic solution which is expressed in closed form with respect to the eccentricity. Expressions for Keplerian orbital elements are obtained including short-period and medium-period terms. In order to avoid singularities in eccentricity, non-singular orbital elements are introduced to compute frozen orbit conditions considering several values of inclinations and semi-major axes. A preliminary analysis of the orbit of the GARATÉA-L Brazilian probe is conducted, and the results are compared to those provided by several models using Cowell's method. A realistic model based on ephemeris data is also used for comparison. The findings reveal that the probe's nominal orbit does not exhibit a frozen condition in terms of eccentricity. A new inclination is proposed to freeze the orbit without altering the pericenter and apocenter altitudes. However, orbital evolution results in a collision with the Moon, as revealed by the 50 × 50 models. A polar frozen orbit is then suggested, offering the advantage of gradually shifting the sub-pericenter point from the South Pole toward the center of the Aitken Basin region.

© 2023 COSPARThis work studies transfer between non-coplanar circular orbits around Earth with the space vehicle performing a powered lunar flyby maneuver. The complete transfer trajectory is accomplished by an application of two or three impulsive velocity increments. First and final velocity increments are applied tangentially, respectively, to the departing and the arrival orbits around Earth. An optional second velocity increment is applied at the perilune in order to increase the effects of the flyby maneuver. Despite many works consider the powered lunar flyby instead of a natural lunar flyby, it is important to compare both maneuvers in the context of the complete trajectory. In this direction, the present work formulates and solves multiple point boundary value problems that determine the transfer trajectories considering three models: a three-dimensional patched-conic approximation, a model based on the spatial restricted three-body problem, and, a model based on the spatial bi-circular restricted four-body in which the influence of the Sun is included. The transfer trajectory solutions are compared with classical maneuvers and with transfers that perform a natural flyby maneuver. An interesting result shows that a decelerating propulsion during the flyby maneuver can provide a transfer trajectory with a fuel consumption smaller than the one of bi-parabolic maneuver even if the Sun's attraction is considered. Moreover, the influence of the Sun can decrease the time of flight and the apogee of the trajectory and it can save fuel consumption if the Sun's initial phase angle is properly chosen.

© 2023: Instituto de Astronomía, Universidad Nacional Autónoma de México.This work describes several models to design optimal interplanetary trajectories. The transfer problem consists in transferring a space vehicle from a circular low Earth orbit (LEO) to a circular low orbit around a destiny planet (Venus or Mars). Models based on the two-body, four-body, and five-body problems are considered. Also, several versions of the patched-conic approximation are utilized including a detailed version that designs a lunar swing-by maneuver. The results show that the optimal trajectories for Earth-Mars and Earth-Venus missions collide with the Moon if a lunar swing-by maneuver with an unspecified altitude of the closest approach is included in the trajectory design; however, sub-optimal trajectories that do not collide with the Moon exist, presenting a smaller fuel consumption than the trajectories without lunar swing-by and with no greater changes in the time of flight.

© 2022, The Author(s), under exclusive licence to The Brazilian Society of Mechanical Sciences and Engineering.This work extends the classic lunar patched-conic approximation model for Earth–Moon transfers by adding two complexities: the eccentricity of the Moon’s orbit around Earth and the eccentricity of the terminal orbits. In this way, the initial low Earth orbit (LEO) and the final low Moon orbit (LMO) are assumed elliptic. The transfer trajectory is performed by application of two impulses at the terminal orbits; however, they are not necessarily applied at the pericenter of the terminal orbits (LEO and LMO). The positions of application of the impulses are specified by the values of the true anomalies that define the point of departure in the LEO and the point of arrival in the LMO. The transfer problem is also formulated in the context of the planar elliptic restricted three-body problem with the same complexities: eccentricity of the primaries Earth and Moon, and the eccentricity of the terminal orbits. However, an additional final constraint is added relating the flight path angle of the transfer trajectory and the one of the LMO at the arrival time. In the proposed patched-conic approximation, this constraint does not appear as it is solved geometrically. In both models, a two-point boundary value problem solves the Earth–Moon trajectory. A one-degree-of-freedom problem, which uses the Moon’s position as a parameters, and a two-degree-of- freedom optimization problem, which sets the Moon’s position as an unknown to be solved, are also formulated in both models and solved by the sequential-gradient restoration algorithm. The results show some impossible configurations of arrival at LMO, as well as the agreements between the models. Also, a huge importance in the orientation of the LEO, determined by its argument of pericenter, is observed in the fuel consumption. So, a study of penalty on the fuel consumption due to the use of non-optimal values of argument of pericenter of the LEO is performed.

© 2021, The Author(s), under exclusive licence to The Brazilian Society of Mechanical Sciences and Engineering.In this paper, a preliminary study of optimal round-trip trajectories for Earth–Moon–Earth missions is presented. The outgoing mission consists in transferring a space vehicle from a circular low Earth orbit (LEO) to a circular low Moon orbit (LMO) with minimum fuel consumption. The class of two-impulse trajectories is considered: A first accelerating velocity impulse is applied to insert the space vehicle into an Earth–Moon transfer trajectory, and a second braking velocity impulse is applied to insert the space vehicle into the terminal LMO. It is assumed that the velocity increments are applied tangentially to the terminal orbits. The fuel consumption is defined by the arithmetic sum of the velocity increments. The return trip is similarly described with the initial orbit corresponding to LMO and the final orbit corresponding to LEO. Two dynamical models are considered: an extended version of the patched-conic approximation which includes the eccentricity of the Moon’s orbit and the planar elliptic restricted three-body problem. The optimization problem is solved by means of two gradient techniques: Newton–Raphson–gradient algorithm and sequential gradient–restoration algorithm. Clockwise and counterclockwise arrivals at LMO are considered for outgoing trips, and clockwise and counterclockwise departures from Moon are considered for return trips. The time of flight varies from 4.5 to 5.3 days for outgoing trips or for return trips. Numerical results show that the fuel can be saved if the initial position of the Moon is appropriately determined.

© 2019 IAAThis work proposes an alternative method for solving the two-point boundary value problem concerning to Earth-Moon bi-impulsive trajectories in the dynamics of the planar bi-circular restricted four-body problem, which describes the motion of a space vehicle subjected to the gravitational attraction of Earth, Moon and Sun. Initially, the space vehicle is at a circular low Earth orbit (LEO) with prescribed altitude. After applying the first impulsive velocity increment, the space vehicle is inserted into a transfer trajectory. The second velocity increment is applied to decelerate and circularize the movement of the space vehicle at a circular low Moon orbit (LMO) with prescribed altitude. To solve this problem, a new two-point boundary value problem (TPBVP) is formulated, which includes an unknown value of the Jacobi integral at the departure time, and, a prescribed value at the arrival time. Since the Jacobi integral is not a first integral for the four-body problem, it is taken as additional state variable, and, its variational equation is added to the system of differential equation in the description of the dynamics of the space vehicle. Taking into account the boundary conditions, expressions for the velocity increments are deduced from the Jacobi integral computed at the initial and final times. Based on this new TPBVP, a numerical procedure is proposed to obtain different families of Earth-Moon trajectories with decreasingly fuel consumption.

© 2017, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.The present work quantifies the fuel consumption of a space vehicle in bi-impulsive interplanetary trajectories with an intermediary swing-by maneuver with the Moon. In this way, an interplanetary patched-conic approximation with a lunar swing-by maneuver is formulated with an important characteristic: the swing-by maneuver is designed before the determination of the trajectory by specifying its geometry. The transfer problem is then solved by a multi-point boundary value problem (MPBVP) with two constraints. The intermediary constraint is related to the geometry of the swing-by maneuver with the Moon, and the terminal constraint is related to the altitude of the arrival at the low orbit around the target planet. The proposed algorithm is built in such way that the MPBVP is split into two-point boundary value problems (TPBVPs): the first one is solved to ensure the satisfying of the intermediary constraint, and the second TPBVP is solved next to satisfy the final constraint. Both TPBVPs are solved by means of Newton–Raphson algorithm. The proposed algorithm is then utilized to determine the Earth–Mars and Earth–Venus trajectories with several geometric configurations. The geometric configuration with the smallest fuel consumption is obtained for both missions and compared to an interplanetary patched-conic approximation without swing-by maneuver with Moon. The results show advantages in performing swing-by maneuver with the Moon for interplanetary missions by saving fuel consumption without much increase of the time of flight.

© 2017, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.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.

© 2018 IAAThe present work formulates an orbital transfer for an Earth-to-Earth mission between non coplanar orbits with different altitudes with a special feature: the occurrence of a lunar flyby during the transfer orbit. This lunar flyby is intended to help change the plane of motion of the spacecraft without fuel consumption. Only two-impulsive trajectories are considered with the velocity increments applied at the initial and final orbits. In order to solve this problem, a 3D patched-conic approximation associated with a two-point boundary value problem is proposed. The same transfer problem is formulated considering the spatial circular restricted three-body problem (SCR3BP). The results of the patched-conic approximation is compared with the results of the SCR3BP showing a good agreement between the models. This work also determines several trajectories in order to perform a study of the fuel consumption considering several inclinations and altitudes of both initial and final orbits around the Earth. The longitude of the ascending node of the initial orbit, and, the altitude of close approach with the Moon during the flyby are also analyzed. According to the total velocity increment analysis, the changing plane assisted by a lunar flyby can be very favorable. Despite the increase of the time of flight, the saving of fuel is considerable. Indeed, the total velocity increment of this kind of maneuver is in some cases better than the velocity increment provided by the bi-parabolic transfer.

© 2017, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.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.