Optimal Low-Thrust Interplanetary Trajectories by Direct Method Techniques
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Optimal Low-Thrust Interplanetary Trajectories by Direct Method Technlquee' Craig A. Kluever/ Abstract A direct optimization method has been developed and utilized to compute a wide range of optimal low-thrust interplanetary trajectories. This method replaces the optimal control problem with a nonlinear programming problem which in tum is solved by using sequential quadratic programming. The direct approach is capable of modeling multiple powered and coast arcs, planetary gravity assists, and constant-thrust and variable-thrust electric propulsion systems. The advantages of the direct approach include reduction in the design space, ease of establishing good initial guesses for the design parameters, and improved flexibility for handling "mixed" optimal control problems with continuous control functions and discrete control parameters. Numerical results are presented for three interplanetary mission examples and the results from the direct method show an excellent match with published optimal trajectories. In addition, optimal trajectories are obtained for a new low-thrust interstellar transfer problem.
Introduction Low-thrust electric propulsion (EP) spacecraft demonstrate a greater payload capability compared to conventional chemical propulsion spacecraft. It has been established that the first New Millennium mission (Deep Space 1) will utilize an ion engine for primary propulsion [1]. Previous studies focusing on interplanetary applications include a manned Mars mission [2] and scientific missions to Jupiter, Uranus, Neptune and Pluto [3]. Low-thrust trajectory optimization techniques fall into two categories: indirect methods and direct methods. Indirect methods solve the optimal control problem by obtaining the solution to the corresponding two-point boundary value problem (TPBVP) which results from the calculus of variations. A well-known example is the low-thrust trajectory optimization program VARITOP [2-4]. A drawback to indirect methods is that the TPBVP is usually very sensitive and extremely difficult to solve unless a good initial guess 1Presented at the AAS/AIAA Space Flight Mechanics Meeting, Austin, Texas, February 1996. 2Assistant Professor, Department of Mechanical and Aerospace Engineering, University of MissouriColumbia/Kansas City, Kansas City, MO 64110.
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for the unknown initial costate variables is available. An advantage of indirect methods is that if the solution to the TPBVP is obtained, then the resulting trajectory is (in most cases) optimal. In contrast, direct methods solve the optimal control problem by adjusting the control variables at each iteration in an attempt to continually reduce the performance index. Continuous control functions and the state differential equations are often parameterized and the associated nonlinear programming (NLP) problem may be solved with a gradient-based parameter optimization method. It is usually easier to produce a good initial control guess for direct methods since the control variables are explicitly parameterized. Another adva
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