Minimum-Time Low-Thrust Geocentric Transfer with Longitude Estimation and Costate Prediction

No AccessEngineering NotesMinimum-Time Low-Thrust Geocentric Transfer with Longitude Estimation and Costate PredictionZhao Li, Di Wu, Hengnian Li and Fanghua JiangZhao Li https://orcid.org/0000-0002-5118-540XTsinghua University, 100084 Beijing, People’s Republic of China*Ph.D. Candidate, School of Aerospace Engineering; also Assistant Research Fellow, State Key Laboratory of Astronautic Dynamics, Xi’an Satellite Control Center; .Search for more papers by this author, Di Wu https://orcid.org/0000-0002-5214-962XTsinghua University, 100084 Beijing, People’s Republic of China†Postdoctoral Researcher, School of Aerospace Engineering; .Search for more papers by this author, Hengnian LiXi’an Satellite Control Center, 710043 Xi’an, People’s Republic of China‡Professor, State Key Laboratory of Astronautic Dynamics; .Search for more papers by this author and Fanghua JiangTsinghua University, 100084 Beijing, People’s Republic of China§Associate Professor, School of Aerospace Engineering; . Senior Member AIAA (Corresponding Author).Search for more papers by this authorPublished Online:18 Sep 2023https://doi.org/10.2514/1.G007333SectionsRead Now ToolsAdd to favoritesDownload citationTrack citations ShareShare onFacebookTwitterLinked InRedditEmail About References [1] Poole M. and Ho M., “Boeing Low-Thrust Geosynchronous Transfer Mission Experience,” Proceedings of the 20th International Symposium on Space Flight Dynamics, NASA Goddard Space Flight Center, Annapolis, MD, Sept. 2007, Paper 23-1. Google Scholar[2] Boniface C., Charbonnier J.-M., Lefebvre L., Leroi V. and Lienart T., “An Overview of Electric Propulsion Activities at CNES,” 35th International Electric Propulsion Conference, Georgia Inst. of Technology, Atlanta, GA, Oct. 2017, Paper IEPC-2017-102. Google Scholar[3] Sreesawet S. and Dutta A., “Fast and Robust Computation of Low-Thrust Orbit-Raising Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 41, No. 9, 2018, pp. 1888–1905. https://doi.org/10.2514/1.G003319 LinkGoogle Scholar[4] Graham K. F. and Rao A. V., “Minimum-Time Trajectory Optimization of Multiple Revolution Low-Thrust Earth-Orbit Transfers,” Journal of Spacecraft and Rockets, Vol. 52, No. 3, 2015, pp. 711–727. https://doi.org/10.2514/1.A33187 LinkGoogle Scholar[5] Pan B., Pan X. and Lu P., “Finding Best Solution in Low-Thrust Trajectory Optimization by Two-Phase Homotopy,” Journal of Spacecraft and Rockets, Vol. 56, No. 1, 2019, pp. 283–291. https://doi.org/10.2514/1.A34144 LinkGoogle Scholar[6] Scheel W. A. and Conway B. A., “Optimization of Very-Low-Thrust, Many-Revolution Spacecraft Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 17, No. 6, 1994, pp. 1185–1192. https://doi.org/10.2514/3.21331 LinkGoogle Scholar[7] Caruso A., Quarta A. A. and Mengali G., “Comparison Between Direct and Indirect Approach to Solar Sail Circle-to-Circle Orbit Raising Optimization,” Astrodynamics, Vol. 3, No. 3, 2019, pp. 273–284. https://doi.org/10.1007/s42064-019-0040-x CrossrefGoogle Scholar[8] Leomanni M., Bianchini G., Garulli A., Quartullo R. and Scortecci F., “Optimal Low-Thrust Orbit Transfers Made Easy: A Direct Approach,” Journal of Spacecraft and Rockets, Vol. 58, No. 6, 2021, pp. 1904–1914. https://doi.org/10.2514/1.A34949 LinkGoogle Scholar[9] Graham K. F. and Rao A. V., “Minimum-Time Trajectory Optimization of Low-Thrust Earth-Orbit Transfers with Eclipsing,” Journal of Spacecraft and Rockets, Vol. 53, No. 2, 2016, pp. 289–303. https://doi.org/10.2514/1.A33416 LinkGoogle Scholar[10] Shannon J. L., Ozimek M. T., Atchison J. A. and Hartzell C. M., “Q-Law Aided Direct Trajectory Optimization of Many-Revolution Low-Thrust Transfers,” Journal of Spacecraft and Rockets, Vol. 57, No. 4, 2020, pp. 672–682. https://doi.org/10.2514/1.A34586 LinkGoogle Scholar[11] Ghosh P., “A Survey of the Methods Available for the Design of Many-Revolution Low-Thrust Planetocentric Trajectories,” 29th AAS/AIAA Space Flight Mechanics Meeting, AAS Paper 19-297, Jan. 2019. Google Scholar[12] Zuiani F. and Vasile M., “Extended Analytical Formulas for the Perturbed Keplerian Motion under a Constant Control Acceleration,” Celestial Mechanics and Dynamical Astronomy, Vol. 121, No. 3, 2015, pp. 275–300. https://doi.org/10.1007/s10569-014-9600-5 CrossrefGoogle Scholar[13] Petropoulos A., “Low-Thrust Orbit Transfers Using Candidate Lyapunov Functions with a Mechanism for Coasting,” AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Guidance, Navigation, and Control and Co-Located Conferences, AIAA Paper 2004-5089, 2004. https://doi.org/10.2514/6.2004-5089 Google Scholar[14] Petropoulos A. E., “Refinements to the Q-law for the Low-Thrust Orbit Transfers,” AAS/AIAA Space Flight Mechanics Conference, AAS Paper 05-162, Jan. 2005. Google Scholar[15] Prussing J. E., “Primer Vector Theory and Applications,” Spacecraft Trajectory Optimization, edited by Conway B. A., Cambridge Aerospace Series, Cambridge Univ. Press, Cambridge, England, U.K., 2010, pp. 16–36. https://doi.org/10.1017/CBO9780511778025.003 CrossrefGoogle Scholar[16] Quarta A. A., Izzo D. and Vasile M., “Time-Optimal Trajectories to Circumsolar Space Using Solar Electric Propulsion,” Advances in Space Research, Vol. 51, No. 3, 2013, pp. 411–422. https://doi.org/10.1016/j.asr.2012.09.012 CrossrefGoogle Scholar[17] Taheri E., Arya V. and Junkins J. L., “Costate Mapping for Indirect Trajectory Optimization,” Astrodynamics, Vol. 5, No. 4, 2021, pp. 359–371. https://doi.org/10.1007/s42064-021-0114-0 CrossrefGoogle Scholar[18] Pontani M., “Optimal Space Trajectories with Multiple Coast Arcs Using Modified Equinoctial Elements,” Journal of Optimization Theory and Applications, Vol. 191, No. 2, 2021, pp. 545–574. https://doi.org/10.1007/s10957-021-01867-2 Google Scholar[19] Wu D., Wang W., Jiang F. and Li J., “Minimum-Time Low-Thrust Many-Revolution Geocentric Trajectories with Analytical Costates Initialization,” Aerospace Science and Technology, Vol. 119, Dec. 2021, Paper 107146. https://doi.org/10.1016/j.ast.2021.107146 Google Scholar[20] Caillau J., Gergaud J. and Noailles J., “3D Geosynchronous Transfer of a Satellite: Continuation on the Thrust,” Journal of Optimization Theory and Applications, Vol. 118, No. 3, 2003, pp. 541–565. https://doi.org/10.1023/B:JOTA.0000004870.74778.ae CrossrefGoogle Scholar[21] Jiang F., Baoyin H. and Li J., “Practical Techniques for Low-Thrust Trajectory Optimization with Homotopic Approach,” Journal of Guidance, Control, and Dynamics, Vol. 35, No. 1, 2012, pp. 245–258. https://doi.org/10.2514/1.52476 LinkGoogle Scholar[22] Haberkorn T., Martinon P. and Gergaud J., “Low Thrust Minimum-Fuel Orbital Transfer: A Homotopic Approach,” Journal of Guidance, Control, and Dynamics, Vol. 27, No. 6, 2004, pp. 1046–1060. https://doi.org/10.2514/1.4022 LinkGoogle Scholar[23] Pan B., Lu P., Pan X. and Ma Y., “Double-Homotopy Method for Solving Optimal Control Problems,” Journal of Guidance, Control, and Dynamics, Vol. 39, No. 8, 2016, pp. 1706–1720. https://doi.org/10.2514/1.G001553 LinkGoogle Scholar[24] Kluever C. A. and Oleson S. R., “Direct Approach for Computing Near-Optimal Low-Thrust Earth-Orbit Transfers,” Journal of Spacecraft and Rockets, Vol. 35, No. 4, 1998, pp. 509–515. https://doi.org/10.2514/2.3360 LinkGoogle Scholar[25] Gao Y., “Near-Optimal Very Low-Thrust Earth-Orbit Transfers and Guidance Schemes,” Journal of Guidance, Control, and Dynamics, Vol. 30, No. 2, 2007, pp. 529–539. https://doi.org/10.2514/1.24836 LinkGoogle Scholar[26] Gao Y., “Direct Optimization of Low-Thrust Many-Revolution Earth-Orbit Transfers,” Chinese Journal of Aeronautics, Vol. 22, No. 4, 2009, pp. 426–433. https://doi.org/10.1016/S1000-9361(08)60121-1 Google Scholar[27] Kluever C. A., “Low-Thrust Trajectory Optimization Using Orbital Averaging and Control Parameterization,” Spacecraft Trajectory Optimization, edited by Conway B., Cambridge Univ. Press, Cambridge, England, U.K., 2010, pp. 112–138. https://doi.org/10.1017/CBO9780511778025.006 CrossrefGoogle Scholar[28] Falck R. and Dankanich J., “Optimization of Low-Thrust Spiral Trajectories by Collocation,” AIAA/AAS Astrodynamics Specialist Conference, AIAA Paper 2012-4423, 2012. https://doi.org/10.2514/6.2012-4423 Google Scholar[29] Sackett L. L., Malchow H. L. and Edelbaum T. N., “Solar Electric Geocentric Transfer with Attitude Constraints: Analysis,” NASA CR-134927, Aug. 1975. LinkGoogle Scholar[30] Geffroy S. and Epenoy R., “Optimal Low-Thrust Transfers with Constraints—Generalization of Averaging Techniques,” Acta Astronautica, Vol. 41, No. 3, 1997, pp. 133–149. https://doi.org/10.1016/S0094-5765(97)00208-7 CrossrefGoogle Scholar[31] Dargent T., “Averaging Technique in T-3D an Integrated Tool for Continuous Thrust Optimal Control in Orbit Transfers,” Advances in the Astronautical Sciences, Vol. 152, 2014, pp. 1599–1615. Google Scholar[32] Kelchner M. J. and Kluever C. A., “Rapid Evaluation of Low-Thrust Transfers from Elliptical Orbits to Geostationary Orbit,” Journal of Spacecraft and Rockets, Vol. 57, No. 5, 2020, pp. 898–906. https://doi.org/10.2514/1.A34630 LinkGoogle Scholar[33] Li Z., Li H., Zhou H. and Jiang F., “Learning-Based Polynomial Approximation of Minimum-Time Low-Thrust Transfers to Geostationary Orbit,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 59, No. 3, 2023, pp. 2388–2401. https://doi.org/10.1109/TAES.2022.3213533 CrossrefGoogle Scholar[34] Walker M. J. H., Ireland B. and Owens J., “A Set of Modified Equinoctial Orbit Elements,” Celestial Mechanics, Vol. 36, No. 4, 1985, pp. 409–419. https://doi.org/10.1007/BF01227493 CrossrefGoogle Scholar[35] Izzo D. and Öztürk E., “Real-Time Guidance for Low-Thrust Transfers Using Deep Neural Networks,” Journal of Guidance, Control, and Dynamics, Vol. 44, No. 2, 2021, pp. 315–327. https://doi.org/10.2514/1.G005254 LinkGoogle Scholar[36] Hubaux C., Lemaître A., Delsate N. and Carletti T., “Symplectic Integration of Space Debris Motion Considering Several Earth’s Shadowing Models,” Advances in Space Research, Vol. 49, No. 10, 2012, pp. 1472–1486. https://doi.org/10.1016/j.asr.2012.02.009 CrossrefGoogle Scholar[37] Topputo F. and Ceccherini S., “A Catalogue of Parametric Time-Optimal Transfers for All-Electric GEO Satellites,” Modeling and Optimization in Space Engineering: State of the Art and New Challenges, edited by Fasano G. and Pintér J. D., Springer International Publishing, Cham, Switzerland, 2019, pp. 459–478. https://doi.org/10.1007/978-3-030-10501-3_17 Google Scholar[38] Caillau J.-B. and Noailles J., “Sensitivity Analysis for Time Optimal Orbit Transfer,” Optimization, Vol. 49, No. 4, 2001, pp. 327–350. https://doi.org/10.1080/02331930108844536 Google Scholar[39] Ferrier CH. and Epenoy R., “Optimal Control for Engines with Electro-Ionic Propulsion Under Constraint of Eclipse,” Acta Astronautica, Vol. 48, No. 4, 2001, pp. 181–192. https://doi.org/10.1016/S0094-5765(00)00158-2 CrossrefGoogle Scholar[40] Standish E. M., “Keplerian Elements for Approximate Positions of the Major Planets,” JPL Solar System Dynamics, 2006, https://ssd.jpl.nasa.gov/. Google Scholar[41] Shampine L. F. and Gordon M. K., Computer Solution of Ordinary Differential Equations: The Initial Value Problem, W. H. Freeman, San Francisco, 1975, Chap. 10. Google Scholar[42] Moré J. J., Garbow B. S. and Hillstrom K. E., “User Guide for MINPACK-1,” Argonne National Lab. ANL-80-74, Argonne, IL, Aug. 1980. CrossrefGoogle Scholar Previous article Next article FiguresReferencesRelatedDetails What's Popular Articles in Advance CrossmarkInformationCopyright © 2023 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. All requests for copying and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-3884 to initiate your request. See also AIAA Rights and Permissions www.aiaa.org/randp. TopicsAerodynamicsAerospace SciencesAstrodynamicsAstronauticsAstronomical EventsAstronomyControl TheoryElectric PropulsionFluid DynamicsGuidance, Navigation, and Control SystemsOptimal Control TheoryPropulsion and PowerSpace OrbitSpace Science and TechnologySpacecraft Propulsion KeywordsElectric PropulsionTrajectory OptimizationPerturbed OrbitEclipsesLow Thrust TrajectoryEarthPontryagin's Maximum PrincipleThrustOrbital ElementsOrbital TransferAcknowledgmentsThis work was supported by the National Natural Science Foundation of China (Grant No. 12022214) and the National Key R&D Program of China (Grant No. 2020YFC2201200).PDF Received29 October 2022Accepted9 August 2023Published online18 September 2023