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Direct-to-indirect map** for optimal low-thrust trajectories

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Abstract

Optimal, many-revolution spacecraft trajectories are challenging to solve. A connection is made for a class of models between optimal direct and indirect solutions. For transfers that minimize thrust-acceleration-squared, primer vector theory maps direct, many-impulsive-maneuver trajectories to the indirect, continuous-thrust-acceleration equivalent. The map** algorithm is independent of how the direct solution is obtained and requires only a solver for a boundary value problem and its partial derivatives. A Lambert solver is used for the two-body problem in this work. The map** is simple because the impulsive maneuvers and co-states share the same linear space around an optimal trajectory. For numerical results, the direct coast-impulse solutions are demonstrated to converge to the indirect continuous solutions as the number of impulses and segments increases. The two-body design space is explored with a set of three many-revolution, many-segment examples changing semimajor axis, eccentricity, and inclination. The first two examples involve a small change to either semimajor axis or eccentricity, and the third example is a transfer to geosynchronous orbit. Using a single processor, the optimization runtime is seconds to minutes for revolution counts of 10 to 100, and on the order of one hour for examples with up to 500 revolutions. Any of these thrust-acceleration-squared solutions are good candidates to start a homotopy to a higher-fidelity minimization problem with practical constraints.

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  1. Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, Texas, 78712, USA

    David Ottesen & Ryan P. Russell

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  1. David Ottesen
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Correspondence to David Ottesen.

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David Ottesen holds his bachelor, master, and Ph.D. degrees from The University of Texas at Austin, USA, in aerospace engineering. His Ph.D. thesis was under the guidance of Dr. Ryan P. Russell, working on and writing about many interesting spacecraft trajectory optimization problems. He has had the opportunity to work at Johns Hopkins University Applied Physics Laboratory, Emergent Space Technologies, and NASA Ames Research Center. Today, he is an astrodynamics engineer at LeoLabs in Menlo Park, CA, USA, hel** deliver the data needed to safely operate in low-Earth orbit. E-mail: davidottesen@utexas.edu

Ryan P. Russell is a professor in the Department of Aerospace Engineering and Engineering Mechanics at The University of Texas at Austin, USA. His research areas of interest include orbit mechanics, numerical optimization, and trajectory design. Russell began his professional career at NASA’s Jet Propulsion Laboratory as a member of the Guidance, Navigation and Control section where he worked as a mission designer and navigation analyst for a variety of space flight projects. He served on the faculty of Georgia Institute of Technology, USA, before joining the UT ASE/EM faculty in 2012. He is a fellow of the American Astronautical Society, an associate fellow of AIAA, and is the former chair of the AIAA Astrodynamics Technical Committee. He has authored or co-authored more than two hundred technical publications and is an associate editor for three esteemed journals. E-mail: ryan.russell@austin.utexas.edu

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Ottesen, D., Russell, R.P. Direct-to-indirect map** for optimal low-thrust trajectories. Astrodyn 8, 27–46 (2024). https://doi.org/10.1007/s42064-023-0164-6

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