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Application of high order expansions of two-point boundary value problems to astrodynamics. (English) Zbl 1223.70081

Summary: Two-point boundary value problems appear frequently in space trajectory design. A remarkable example is represented by the Lambert’s problem, where the conic arc linking two fixed positions in space in a given time is to be characterized in the frame of the two-body problem. Classical methods to numerically solve these problems rely on iterative procedures, which turn out to be computationally intensive in case of lack of good first guesses for the solution. An algorithm to obtain the high order expansion of the solution of a two-point boundary value problem is presented in this paper. The classical iterative procedures are applied to identify a reference solution. Then, differential algebra is used to expand the solution of the problem around the achieved one. Consequently, the computation of new solutions in a relatively large neighborhood of the reference one is reduced to the simple evaluation of polynomials. The performances of the method are assessed by addressing typical applications in the field of spacecraft dynamics, such as the identification of halo orbits and the design of aerocapture maneuvers.

MSC:

70M20 Orbital mechanics
70F15 Celestial mechanics

Software:

Cosy
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[1] Armellin, R., Lavagna, M.: Multidisciplinary optimization of aerocapture maneuvers. J. Artif. Evol. Appl. Article ID 248798, 13 pp (2008) · Zbl 1223.70081
[2] Armellin R., Topputo F.: A sixth-order accurate scheme for solving two-point boundary value problems in astrodynamics. Celest. Mech. Dyn. Astron. 96, 289–309 (2006) · Zbl 1116.70005 · doi:10.1007/s10569-006-9047-4
[3] Battin R.H.: An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition. AIAA Education Series, Providence (1993)
[4] Bernelli Zazzera, F., Topputo, F., Massari, M.: Assessment of mission design including utilization of libration points and weak stability boundaries. Final Report, ESTEC Contract No. 18147/04/NL/MV (2004)
[5] Berz, M.: The new method of TPSA algebra for the description of beam dynamics to high orders. Technical Report AT-6:ATN-86-16, Los Alamos National Laboratory (1986)
[6] Berz M.: The method of power series tracking for the mathematical description of beam dynamics. Nucl. Instrum. Methods A258, 431 (1987) · doi:10.1016/0168-9002(87)90927-2
[7] Berz M.: High-order computation and normal form analysis of repetitive systems. Phys. Part. Accel. AIP 249, 456 (1991)
[8] Berz, M.: Differential algebraic techniques. In: Tigner, M., Chao, A. (eds.) Handbook of Accelerator Physics and Engineering. World Scientific (1999a)
[9] Berz, M.: Modern Map Methods in Particle Beam Physics. Academic Press (1999b)
[10] Berz, M., Makino, K.: COSY INFINITY version 9 reference manual, MSU Report MSUHEP-060803. Michigan State University, East Lansing, MI 48824 (2006)
[11] Bryson A.E., Ho Y.C.: Applied Optimal Control. Hemisphere Publishing Co., Washigton (1975)
[12] Cruz, M.I.: The aerocapture vehicle mission design concept. In: Joint Propulsion Conference, AIAA paper 79-0893 (1979)
[13] Di Lizia, P.: Robust space trajectory and space system design using differential algebra. Ph.D. Dissertation, Politecnico di Milano (2008a)
[14] Di Lizia P., Armellin R., Ercoli-Finzi A., Berz M.: High-order robust guidance of interplanetary trajectories based on differential algebra. J. Aerosp. Eng. Sci. Appl. 1, 43–57 (2008b)
[15] Gómez G., Mondelo J.M.: The dynamics around the collinear equilibrium points of the RTBP. Phys. D 157, 283–321 (2001) · Zbl 0990.70009 · doi:10.1016/S0167-2789(01)00312-8
[16] Gómez, G., Jorba, Á., Masdemont, J., Simó, C.: Dynamics and Mission Design Near Libration Points–Volume III: Advanced Methods for Collinear Points. World Scientic (2000) · Zbl 0971.70002
[17] Griffith, T.D., Turner, J.D., Vadali, S.R., Junkins, J.L.: Higher order sensitivities for solving nonlinear two-point boundary-value problems. In: AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Providence, Rhode Island, August 16–19, (2004)
[18] Guibout V., Scheeres D.J.: Solving relative two-point boundary value problems: spacecraft formulation flight transfers application. J. Guid. Control Dyn. 27, 693–704 (2004) · doi:10.2514/1.11164
[19] Guibout, V., Scheeres, D.J.: Solving two point boundary value problems using generating functions: theory and applications to astrodynamics. In: Gurfil, P. (ed.) Modern Astrodynamics. Elsevier Astrodynamics Series, Academic Press (2006a)
[20] Guibout V., Scheeres D.J.: Spacecraft formation dynamics and design. J. Guid. Control Dyn. 29, 121–133 (2006b) · doi:10.2514/1.13002
[21] Jorba Á., Masdemont J.: Dynamics in the center manifold of the collinear points of the restricted three body problem. Phys. D 132, 189–213 (1999) · Zbl 0942.70012 · doi:10.1016/S0167-2789(99)00042-1
[22] Masdemont J.J.: High order expansions of invariant manifolds of libration point orbits with applications to mission design. Dyn. Sys. Ann. Int. J. 20, 59–113 (2005) · Zbl 1093.70012
[23] Park R.S., Scheeres D.J.: Nonlinear mapping of Gaussian statistics: theory and application to spacecraft trajectory design. J. Guid. Control Dyn. 29, 1367–1375 (2006) · doi:10.2514/1.20177
[24] Park R.S., Scheeres D.J.: Nonlinear semi-analytic methods for trajectory estimation. J. Guid. Control Dyn. 30, 1668–1676 (2007) · Zbl 1183.85009 · doi:10.2514/1.29106
[25] Richardson D.L.: Analytic construction of periodic orbits about the collinear points. Celest. Mech. 22, 241–253 (1980) · Zbl 0465.34028 · doi:10.1007/BF01229511
[26] Stoer J., Bulirsch R.: Introduction to Numerical Analysis. Springer, New York (1993) · Zbl 0771.65002
[27] Thurman, R., Worfolk, P.A.: The geometry of halo orbits in the circular restricted three body problem. Technical Report GCG95, Geometry Center, University of Minnesota (1996)
[28] Vinh, N.X., Johnson, W.R., Longusky, J.M.: Mars aerocapture using bank modulation. In: Collection of Technical Papers AIAA/AAS Astrodynamics Specialist Conference, Denver, CO, August 14–17 (2000)
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