J OURNAL OF G UIDANCE , C ONTROL , AND D YNAMICS Vol. 30, No. 4, July–August 2007 Low-Thrust, High-Accuracy Trajectory Optimization I. Michael Ross, ∗ Qi Gong, † and Pooya Sekhavat ‡ Naval Postgraduate School, Monterey, California 93943 DOI: 10.2514/1.23181 Multirevolution, very low-thrust trajectory optimization problems have long been considered difficult problems due to their large time scales and high-frequency responses. By relating this difficulty to the well-known problem of aliasing in information theory, an antialiasing trajectory optimization method is developed. The method is based on Bellman’s principle of optimality and is extremely simple to implement. Appropriate technical conditions are derived for generating candidate optimal solutions to a high accuracy. The proposed method is capable of detecting suboptimality by way of three simple tests. These tests are used for verifying the optimality of a candidate solution without the need for computing costates or other covectors that are necessary in the Pontryagin framework. The tests are universal in the sense that they can be used in conjunction with any numerical method whether or not antialiasing is sought. Several low-thrust example problems are solved to illustrate the proposed ideas. It is shown that the antialiased solutions are, in fact, closed-loop solutions; hence, optimal feedback controls are obtained without recourse to the complexities of the Hamilton–Jacobi theory. Because the proposed method is easy to implement, it can be coded on an onboard computer for practical space guidance. the field to exchange ideas over several workshops. These workshops, held over 2003–2006, further clarified the scope of the problems, and ongoing efforts to address them are described in [12]. From a practical point of view, the goal is to quickly obtain verifiably optimal or near-optimal solutions to finite- and low-thrust problems so that alternative mission concepts can be analyzed I. Introduction C ONTINUOUS-THRUST trajectory optimization problems have served as one of the motivating problems for optimal control theory since its inception [1–4]. The classic problem posed by Moyer and Pinkham [2] is widely discussed in textbooks [1,3,4] and research articles [5–7]. When the continuity of thrust is removed from such problems, the results can be quite dramatic as illustrated in Fig. 1. This trajectory was obtained using recent advances in optimal control techniques and is extensively discussed in [8]. In canonical units, the problem illustrated in Fig. 1 corresponds to doubling the semimajor axis (a 0 1, a f 2), doubling the eccentricity (e 0 0:1, e f 0:2), and rotating the line of apsides by 1 rad. Note that the extremal thrust steering program for minimizing fuel is not tangential over a significant portion of the trajectory. Furthermore, the last burn is a singular control as demonstrated in Fig. 2 by the vanishing of the switching function. Although such finite-thrust problems can be solved quite readily nowadays, it has long been recognized [9–11] that as the thrust authority is reduced, new problems emerge. These well-known challenges chiefly arise as a result of a long flight time measured in terms of the number of orbital revolutions. Consequently, such problems are distinguished from finite-thrust problems as low-thrust problems although the boundary between finite thrust and low thrust is not altogether sharp. Although ad hoc techniques may circumvent some of the low- thrust challenges, it is not quite clear if the solutions generated from such methods are verifiably optimal. As detailed in [8], the engineering feasibility of a space mission is not dictated by trajectory generation, but by optimality. This is because fuel in space is extraordinarily expensive as the cost of a propellant is driven by the routine of space operations, or the lack of it, and not the chemical composition of the fuel. In an effort to circumvent ad hoc techniques to efficiently solve emerging problems in finite- and low-thrust trajectory optimization, NASA brought together leading experts in al Or iti bit In it Fin rb al O Transfer Trajectory Fig. 1 A benchmark minimum-fuel finite-thrust orbit transfer problem. Thrust Acceleration, u s = 0 Switching Function, s Received 13 February 2006; accepted for publication 21 August 2006. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per- copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/07 $10.00 in correspondence with the CCC. Professor, Department of Mechanical and Astronautical Engineering; imross@nps.edu. Associate Fellow AIAA. Research Associate, Department of Mechanical and Astronautical Engineering; qgong@nps.edu. Research Scientist, Department of Mechanical and Astronautical Engineering; psekhava@nps.edu. Singular Control s u time (canonical units) Fig. 2 Extremal thrust acceleration (control) program t7 !u and the corresponding switching function t7 !s for the trajectory shown in Fig. 1.
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