×

Numerical computation of optimal trajectories for coplanar, aeroassisted orbital transfer. (English) Zbl 1168.49310

Summary: This paper is concerned with the problem of the optimal coplanar aeroassisted orbital transfer of a spacecraft from a high Earth orbit to a low Earth orbit. It is assumed that the initial and final orbits are circular and that the gravitational field is central and is governed by the inverse square law. The whole trajectory is assumed to consist of two impulsive velocity changes at the begin and end of one interior atmospheric subarc, where the vehicle is controlled via the lift coefficient. The problem is reduced to the atmospheric part of the trajectory, thus arriving at an optimal control problem with free final time and lift coefficient as the only (bounded) control variable. For this problem, the necessary conditions of optimal control theory are derived. Applying multiple shooting techniques, two trajectories with different control structures are computed. The first trajectory is characterized by a lift coefficient at its minimum value during the whole atmospheric pass. For the second trajectory, an optimal control history with a boundary subarc followed by a free subarc is chosen. It turns out, that this second trajectory satisfies the minimum principle, whereas the first one fails to satisfy this necessary condition; nevertheless, the characteristic velocities of the two trajectories differ only in the sixth significant digit. In the second part of the paper, the assumption of impulsive velocity changes is dropped. Instead, a more realistic modeling with twofinite-thrust subarcs in the nonatmospheric part of the trajectory is considered. The resulting optimal control problem now describes the whole maneuver including the nonatmospheric parts. It contains as control variables the thrust, thrust angle, and lift coefficient. Further, the mass of the vehicle is treated as an additional state variable. For this optimal control problem, numerical solutions are presented. They are compared with the solutions of the impulsive model.

MSC:

49N90 Applications of optimal control and differential games
49K15 Optimality conditions for problems involving ordinary differential equations
93C95 Application models in control theory

Software:

BNDSCO
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] London, H. S., Change of Satellite Orbit Plane by Aerodynamic Maneuvering, Journal of the Aerospace Sciences, Vol. 29, pp. 323–332, 1962. · Zbl 0102.40203
[2] Walberg, G. D., A Survey of Aeroassisted Orbit Transfer, Journal of Spacecraft and Rockets, Vol. 22, pp. 3–17, 1985.
[3] Mease, K. D., Optimization of Aeroassisted Orbital Transfer: Current Status, Journal of the Astronautical Sciences, Vol. 36, pp. 7–33, 1988.
[4] Naidu, D. S., Aeroassisted Orbital Transfers: Guidance and Control Strategies, Lecture Notes in Control and Information Sciences, Springer, New York, NY, Vol. 188, 1994.
[5] Miele, A., Recent Advances in the Optimization and Guidance of Aeroassisted Orbital Transfer, The 1st John V. Breakwell Memorial Lecture, Acta Astronautica, Vol. 38, pp. 747–768, 1996.
[6] Mease, K. D., and Vinh, N. X., Minimum-Fuel Aeroassisted Coplanar Orbit Transfer Using Lift Modulations, Journal of Guidance, Control, and Dynamics, Vol. 8, pp. 134–141, 1985.
[7] Miele, A., and Venkataraman, P., Optimal Trajectories for Aeroassisted Orbital Transfer, Acta Astronautica, Vol. 11, pp. 423–433, 1984. · Zbl 0551.70019
[8] Miele, A., Wang, T., and Deaton, A. W., Properties of the Optimal Trajectories for Coplanar, Aeroassisted Orbital Transfer, Journal of Optimization Theory and Applications, Vol. 69, pp. 1–30, 1991. · Zbl 0724.49024
[9] Miele, A., and Wang, T., Gamma Guidance of Trajectories for Coplanar, Aeroassisted Orbital Transfer, Journal of Guidance, Control, and Dynamics, Vol. 15, pp. 255–262, 1992.
[10] Baumann, H., Complete Trajectories of an Aeroassisted Orbital Transfer, Optimal Control in Transatmospheric Flight, Workshop Greifswald, Greifswald, Germany, 1999 (to appear).
[11] Oberle, H. J., and Taubert, K., Existence and Multiple Solutions of the Minimum-Fuel Orbit Transfer Problem, Journal of Optimization Theory and Applications, Vol. 95, pp. 243–262, 1997. · Zbl 0896.49021
[12] Bryson, A. E., and Ho, Y. C., Applied Optimal Control, Ginn and Company, Waltham, Massachusetts, 1969.
[13] Bulirsch, R., The Multiple Shooting Method for the Numerical Solution of Nonlinear Boundary-Value Problems and of Optimal Control Problems, Report of the Carl–Cranz–Gesellschaft e.V., Oberpfaffenhofen, Germany, 1971 (in German).
[14] Stoer, J., and Bulirsch, R., Introduction to Numerical Analysis, 2nd Edition, Corrected 3rd Printing, Texts in Applied Mathematics, Springer, New York, NY, Vol. 12, 1996. · Zbl 0771.65002
[15] Oberle, H. J., and Grimm, W., BNDSCO–A Program for the Numerical Solution of Optimal Control Problems, Report 515, Institut for Flight Systems Dynamics, German Aerospace Research Establishment (DLR), Oberpfaffenhofen, Germany, 1989.
[16] Baumann, H., Numerical Computation of Optimal Aeroassisted Orbital Transfers, Diploma Thesis, University of Hamburg, 1998 (in German).
[17] Chudej, K., Generalized Necessary Conditions for State-Constrained Optimal Control Problems with Piecewise Model Functions, Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 75, pp. 587–588, 1995 (in German).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.