| Popis: |
Optimal space trajectories have been extensively investigated from both the analytical and the numerical perspective, using a variety of approaches [1-3]. In the last decades, several direct and indirect approaches have been developed for trajectory optimization. Their main limitation is related to the need of a suitable starting guess, which must be supplied in order for the algorithm to converge. The present research describes the indirect heuristic method (IHM) [4,5] and extends its application to minimum-fuel orbit transfers. The technique at hand is based upon the joint use of the necessary conditions for optimality and a heuristic algorithm. More specifically, the necessary conditions are employed to express the control variables (i.e., the thrust magnitude and direction) as functions of the adjoint variables, which are subject to the Euler-Lagrange equations. As a result, a reduced parameter set mainly composed of the unknown initial values of the adjoint variables suffices to transcribe the optimal control problem into a parameter optimization problem. Furthermore, the optimal control variables are determined without any restriction, because no particular representation is assumed. Lastly, satisfaction of all the necessary conditions guarantees the (local) optimality of the solution. The indirect heuristic technique is thus capable of circumventing the main disadvantages of using heuristic approaches, while retaining the main advantage, which is the absence of any starting guess. Minimum-fuel paths admit coast arcs and powered phases, whose sequence is unknown a priori and depends on the switching function [6]. This circumstance adds further complexity to the study of minimum-fuel transfers compared to minimum-time trajectories. In this research, minimum-fuel orbit transfers are sought with the use of modified equinoctial elements. These state variables were already proven to mitigate the hypersensitivity issues associated with the use of spherical coordinates, for minimum-time orbit transfers [7]. An illustrative example taken from the scientific literature [8] is considered. Minimum-fuel orbit transfers are identified for different values of the propulsion parameters. The numerical results point out the existence of a variety of structures for the optimal transfer. In fact, different sequences of powered phases and coast arcs are proven to exist, and their number reduces as the thrust magnitude increases. The minimum-fuel transfer found in this study can be regarded as globally optimal in a specified range of transfer times, and outperforms that reported in the scientific literature, found with identical propulsion parameters. Moreover, this research proves that further locally optimal solutions exist to the same minimum-fuel orbit transfer problem. All the optimal paths enjoy the analytical conditions to a great accuracy, and the numerical solution method did not encounter hypersensitivity issues. These circumstances unequivocally testify to the effectiveness and accuracy of the indirect heuristic methodology, with the use of modified equinoctial elements, in detecting minimum-fuel orbit transfers. References [1] Betts J.T., \"Survey of Numerical Methods for Trajectory Optimization,\" Journal of Guidance, Control, and Dynamics, Vol. 21, No. 2, 1988, pp. 193-207 [2] Rao A.V., \"A Survey of Numerical Methods for Optimal Control,\" Advances in the Astronautical Sciences, Vol. 135, 2010; paper AAS 09-334 [3] Conway B.A. \"A Survey of Methods Available for the Numerical Optimization of Continuous Dynamical Systems.\" Journal of Optimization Theory and Applications, Vol. 152, 2012, pp. 271-306 [4] Pontani, M. and Conway, B.A., \"Optimal Low-Thrust Orbital Maneuvers via Indirect Swarming Method,\" Journal of Optimization Theory and Applications, Vol. 162, No. 1, 2014, pp. 272-292 [5] Pontani, M. and Conway, B.A., \"Minimum-Fuel Finite-Thrust Relative Orbit Maneuvers via Indirect Heuristic Method,\" Journal of Guidance, Control, and Dynamics, Vol. 38, No. 5, 2015, pp. 913-924 [6] Prussing J.E., \"Primer Vector Theory and Applications,\" Spacecraft Trajectory Optimization, Ed. B. Conway, Cambridge University Press, New York, NY, 2010, pp. 16-36 [7] Pontani M., \"Optimal Low-Thrust Trajectories Using Nonsingular Equinoctial Orbit Elements,\" Advances in the Astronautical Sciences, Vol. 170, 2020, pp. 443-462 [8] Pan B., Lu P., and Chen Z., Three-dimensional closed-form costate solutions in optimal coast, Acta Astronautica, vol. 77, pp. 156-166, 2012. |
