Abstract
This work considers the development of a numerical-analytical procedure for computing optimal time-fixed low-thrust limited-power transfers between arbitrary orbits. It is assumed that Earth’s gravitational field is described by the main three zonal harmonics J2, J3 and J4. The optimization problem is formulated as a Mayer problem of optimal control with the state variables defined by the Cartesian elements—components of the position vector and the velocity vector—and a consumption variable that describes the fuel spent during the maneuver. Pontryagin Maximum Principle is applied to determine the optimal thrust acceleration. A set of classical orbital elements is introduced as a new set of state variables by means of an intrinsic canonical transformation defined by the general solution of the canonical system described by the undisturbed part of the maximum Hamiltonian. The proposed procedure involves the development of a two-stage algorithm to solve the two-point boundary value problem that defines the transfer problem. In the first stage of the algorithm, a neighboring extremals method is applied to solve the “mean” two-point boundary value problem of going from an initial orbit to a final orbit at a prescribed final time. This boundary value problem is described by the mean canonical system that governs the secular behavior of the optimal trajectories. The maximum Hamiltonian function that governs the mean canonical system is computed by applying the classic concept of “mean Hamiltonian”. In the second stage, the well-known Newton–Raphson method is applied to adjust the initial values of adjoint variables when periodic terms of the first order are included. These periodic terms are recovered by computing the Poisson brackets in the transformation equations, which are defined between the original set of canonical variables and the new set of average canonical variables, as described in Hori method. Numerical results show the main effects on the optimal trajectories due to the zonal harmonics considered in this study.
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References
Rayman MD, Varghese P, Lehman DH, Livesay LL (2000) Results from the Deep Space 1 technology validation mission. Acta Astronaut 47(2–9):475–487. https://doi.org/10.1016/s0094-5765(00)00087-4
Racca GD, Marini A, Stagnaro L, van Dooren J, di Napoli L, Foing BH, Lumb R, Volp J, Brinkmann J, Grünagel R et al (2002) SMART-1 mission description and developments status. Planet Space Sci 50:1323–1336. https://doi.org/10.1016/S0032-0633(02)00123-X
Camino O, Alonso M, Blake R, Milligan D, Bruin JD, Ricken S (2005) SMART-1: Europe's lunar mission paving the way for new cost effective ground operations (RCSGSO). In: Sixth international symposium reducing the costs of spacecraft ground systems and operations (RCSGSO), European Space Agency ESA SP-601; Darmstadt, Germany
Kawaguchi J, Fujiwara A, Uesugi T (2008) Hayabusa—its technology and science accomplishment summary and Hayabusa-2. Acta Astronaut 62:639–647. https://doi.org/10.1016/j.actaastro.2008.01.028
Morante D, Rivo MS, Soler M (2021) A survey on low-thrust trajectory optimization approaches. Aerospace 8(88):2–39
Funase R, Koizumi H, Nakasuka S, Kawakatsu Y, Fukushima Y, Tomiki A et al (2014) 50kg-class deep space exploration technology demonstration micro-spacecraft PROCYON. In: 28th annual AIAA/USU conference on small satellites, SSC14-VI-3
Folta DC, Bosanac N, Cox A, Howell KC (2016) The Lunar IceCube mission design: construction of feasible transfer trajectories with a constrained departure. AAS/AIAA Space Flight Mechanics Meeting, AAS 16–285, Napa
Gobetz FW (1965) A linear theory of optimum low-thrust rendezvous trajectories. J Astronaut Sci 12(3):69–74
Edelbaum TN (1965) Optimum power-limited orbit transfer in strong gravity fields. AIAA J 3(5):921–925. https://doi.org/10.2514/3.3016
Edelbaum TN (1966) An asymptotic solution for optimum power limited orbit transfer. AIAA J 4(8):1491–1494. https://doi.org/10.2514/3.3725
Marec JP, Vinh NX (1980) Étude generale des transferts optimaux a poussee faible et puissance limitee entre orbites elliptiques quelconques. ONERA Publication 1980–1982
Haissig CM, Mease KD, Vinh NX (1993) Minimum-fuel, power-limited transfers between coplanar elliptical orbits. Acta Astronaut 29(1):1–15. https://doi.org/10.1016/0094-5765(93)90064-4
Geffroy S, Epenoy R (1997) Optimal low-thrust transfers with constraints-generalization of averaging techniques. Acta Astronaut 41(3):133–149. https://doi.org/10.1016/s0094-5765(97)00208-7
Bonnard B, Caillau JB, Dujol R (2006) Averaging and optimal control of elliptic Keplerian orbits with low propulsion. Syst Control Lett 55(9):755–760. https://doi.org/10.1016/j.sysconle.2006.03.004
Huang W (2012) Solving coplanar power-limited orbit transfer problem by primer vector approximation method. Int J Aerosp Eng. https://doi.org/10.1155/2012/480320
Da Silva Fernandes S, Das Chagas Carvalho F, Romão Bateli JV (2018) A numerical-analytical approach based on canonical transformations for computing optimal low-thrust transfers. Revista Mexicana de Astronomía y Astrofísica 54(1):111–128
Li H, Chen S, Baoyin H (2018) J2-Perturbed multitarget rendezvous optimization with low thrust. J Guid Control Dyn 41(3):802–808. https://doi.org/10.2514/1.g002889
Kelchner MJ, Kluever CA (2020) Rapid evaluation of low-thrust transfers from elliptical orbits to geostationary orbit. J Spacecr Rocket. https://doi.org/10.2514/1.a34630
Di Carlo M, Romero Martin JM, Vasile M (2017) CAMELOT: computational-analytical multi-fidElity low-thrust optimisation toolbox. CEAS Sp J 10(1):25–36. https://doi.org/10.1007/s12567-017-0172-6
Hori GI (1966) Theory of general perturbation with unspecified canonical variable. Publ Astron Soc Jpn 18(4):287–296
Chobotov VA (2002) Orbital mechanics, 3rd edn. AIAA, Reston, p 447
Kaula WM (1966) Theory of satellite geodesy. Blaisdell Publishing Company, Waltham, Toronto and London, p 124
Osório, JP (1973) Perturbações de órbitas de satélites no estudo do campo gravitacional terrestre. Publicações do Observatório Astronômico Prof. Manuel de Barros, Universidade do Porto, Porto, Imprensa Portuguesa, p 127
Marec JP (1979) Optimal space trajectories. Elsevier, New York
Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF (1962) Mathematical theory of optimal processes. John Wiley, New York, p 360
Battin RH (1987) An introduction to the mathematics and methods of astrodynamics. American Institute of Aeronautics and Astronautics, New York, p 796
Bate RR, Mueller DD, White JE (2013) Fundamentals of astrodynamics. Dover Publications Inc, New York, p 455
Da Silva FS (1994) Generalized canonical systems—I. General properties. Acta Astronaut 32:331–338. https://doi.org/10.1016/0094-5765(94)90154-6
Vallado DA (2007) Fundamentals of astrodynamics and applications, 3rd edn. Springer, New York, p 1055
Levallois JJ, Kovalevsky J (1971) Géodésie Générale, Tome IV, Géodésie Spatiale, Eyrolles, Paris
Longmuir AG, Bohn EV (1969) Second-variation methods in dynamic optimization. J Optim Theory Appl 3(3):164–173. https://doi.org/10.1007/bf00929441
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(2):193–223. https://doi.org/10.1137/0301011
Stoer J, Bulirsch R (2002) Introduction to numerical analysis, 3rd edn. Springer, New York, p 744. https://doi.org/10.1007/978-0-387-21738-3
Da Silva Fernandes S, Das Chagas Carvalho F (2019) Effects of the zonal harmonics J2, J3 and J4 on optimal low-thrust trajectories. In: 25th International Congress of Mechanical Engineering—COBEM 2019. doi: https://doi.org/10.26678/ABCM.COBEM2019.COB2019-0356
Brouwer D (1959) Solution of the problem of artificial satellite theory without drag. Astron J 64:378–397
Kozai Y (1959) The motion of a close earth satellite. Astron J 64:367–377
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This research has been supported by CNPq under contract 301875/2017-0.
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da Silva Fernandes, S., das Chagas Carvalho, F. Effects of the main zonal harmonics on optimal low-thrust limited-power transfers. J Braz. Soc. Mech. Sci. Eng. 43, 523 (2021). https://doi.org/10.1007/s40430-021-03229-5
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DOI: https://doi.org/10.1007/s40430-021-03229-5