Computational and Applied Mathematics

, Volume 37, Supplement 1, pp 27–54

# Interplanetary patched-conic approximation with an intermediary swing-by maneuver with the moon

• Luiz Arthur Gagg Filho
• Sandro da Silva Fernandes
Article

## Abstract

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.

## Keywords

Earth–Mars trajectory Earth–Venus trajectory Lunar swing-by maneuver Interplanetary patched-conic approximation

## Notes

### Acknowledgements

This research is supported by Grant 2012/25308-5, São Paulo Research Foundation (FAPESP), and by CNPq under contract 304913/2013-8.

## References

1. Arthur Gagg Filho L, Da Silva Fernandes S (2016) Optimal round trip lunar missions based on the patched-conic approximation. Comput Appl Math 35:753–787.
2. Bate RR, Mueller DD, White JE (1971) Fundamentals of astrodynamics. Courier Dover Publications, USAGoogle Scholar
3. Breakwell JV, Speyer JL, Bryson AE (1963) Optimization and control of nonlinear systems using the second variation. J Soc Ind Appl Math Ser A Control 1:193–223.
4. Broucke R (1988) The celestial mechanics of gravity assist, 1988 [C]. In: AIAA/AAS astrodynamics conference. American Institute of Aeronautics and Astronautics, Minneapolis pp 69–78Google Scholar
5. Bryson AE, Denham WF (1962) A Steepest-ascent method for solving optimum programming problems. J Appl Mech 29:247–257
6. Curtis HD (2005) Orbital Mechanics for Engineering Students. Elsevier Butterworth-Heinemann, OxfordGoogle Scholar
7. D’amario LA, Byrnes DV, Stanford RH (1982) Interplanetary trajectory optimization with application to Galileo. J Guid Control Dyn 5:465–471.
8. Da Silva Fernandes S, Golfetto WA (2005) Numerical computation of optimal low-thrust limited-power trajectories—transfers between coplanar circular orbits. J Braz Soc Mech Sci Eng 27:178–185
9. Da Silva Fernandes S, Silveira Filho CR, Golfetto WA (2012) A numerical study of low-thrust limited power trajectories between coplanar circular orbits in an inverse-square force field. Math Probl Eng 2012:168632Google Scholar
10. Flandro GA (1966) Fast reconnaissance missions to the outer solar system utilizing energy derived from the gravitational field of Jupiter1. Acta Astronaut 12:329–337Google Scholar
11. Fletcher R, Reeves CM (1964) Function minimization by conjugate gradients. Comput J 7:149–154
12. Jones DR (2016a) Trajectories for Europa flyby sample return. In: AIAA/AAS astrodynamics specialist conference. AIAA SPACE Forum. American Institute of Aeronautics and Astronautics.
13. Jones DR (2016b) Trajectories for Europa flyby sample return. American Institute of Aeronautics and AstronauticsGoogle Scholar
14. Jones DR (2016c) Trajectories for Flyby Sample Return at Saturn’s Moons. In: AIAA/AAS Astrodynamics Specialist Conference. AIAA SPACE Forum. American Institute of Aeronautics and Astronautics.
15. Kelley HJ (1960) Gradient theory of optimal flight paths. ARS J 30:947–954.
16. Kenneth P, Mcgill R (1964) Solution of variational problems by means of a generalized Newton–Raphson operator. AIAA J 2:1761–1766
17. Lasdon L, Mitter S, Waren A (1967) The conjugate gradient method for optimal control problems. IEEE Trans Autom Control 12:132–138
18. Longmuir AG, Bohn EV (1969) Second-variation methods in dynamic optimization. J Optim Theory Appl 3:164–173
19. Marec J-P (1979) Optimal space trajectories. Elsevier, Amsterdam
20. Miele A, Huang HY, Heideman JC (1969) Sequential gradient-restoration algorithm for the minimization of constrained functions–ordinary and conjugate gradient versions. J Optim Theory Appl 4:213–243
21. Miele A, Wang T (1999) Optimal transfers from an Earth orbit to a Mars orbit. Acta Astronaut 45:119–133
22. Prado AFBDA (2007) A comparison of the “patched-conics approach” and the restricted problem for swing-bys. Adv Space Res 40:113–117
23. Pu CL, Edelbaum TN (1975) Four-body trajectory optimization. AIAA J 13:333–336.
24. Tang S, Conway BA (1995) Optimization of low-thrust interplanetary trajectories using collocation and nonlinear programming. J Guid Control Dyn 18:599–604.