طراحی مسیر بهینه انتقال از مدار حول زمین به مدار هاله‌ای در سیستم زمین-ماه با رویکرد هموتوپی

نوع مقاله : مقالة‌ تحقیقی‌ (پژوهشی‌)

نویسندگان

1 دانشکده مهندسی هوافضا، دانشگاه صنعتی شریف، تهران، ایران

2 دانشکده مهندسی هوافضاT دانشگاه صنعتی شریف، تهران، ایران

3 پژوهشکده سامانه های ماهواره، پژوهشگاه فضایی ایران، تهران، ایران

10.30699/jsst.2020.109626

چکیده

مدارهای هاله‌ای به واسطه ویژگی‌هایی همچون حرکت‌های تناوبی حول نقطه تعادلی برای رصد و مطالعه فضا اهمیت ویژه‌ای دارند. از این رو، مقاله حاضر به طراحی مسیر انتقال بهینه به مدار هاله‌ای پرداخته است. برای حل این مسئله کنترل بهینه از رویکرد هموتوپی در کنار روش پرتابی استفاده شده‌است. این روش باعث بهبود سرعت همگرایی و رسیدن به حدس اولیه بهتری می‌شود. مسیر بهینه طراحی‌شده، فضاپیمایی در مدار حول زمین را به یک مدار هاله‌ای حول نقطه لاگرانژی L1 سیستم سه جسم زمین-ماه منتقل می‌کند. سیستم پیشران فضاپیما از نوع تراست-کم با ضربه ویژه ثابت در نظر گرفته شده است. رویکرد هموتوپی دامنه گسترده‌ای دارد که از آن میان، روش ادامه‌ی هموتوپی در این پژوهش مدنظر قرار گرفته‌است. هدف از مسیر طراحی شده، انتقال با کمینه مصرف سوخت است که به کمک رویکرد هموتوپی از تغییر شکل حل مساله کمینه مصرف انرژی به دست آمده است. در واقع رویکرد هموتوپی سبب تسهیل حل مساله دشوار کمینه مصرف سوخت شده است.

کلیدواژه‌ها


عنوان مقاله [English]

Homotopy-based optimal trajectory design to transfer from Earth orbit to halo orbits

نویسندگان [English]

  • Maryam Kiani 1
  • Ghasem Heydari 2
  • سید حسین پورتاکدوست 1
  • Mohammad Sayanjali 3
1 Department of Aerospace Engineering, Sharif university of technology, Tehran, IRAN
2 Department of Aerospace engineering, Sharif University of Technology, Tehran IRAN
3 Space System Research center, Iran Space Center, Tehran, Iran
چکیده [English]

Halo orbits are of importance for observation and study of the space due to their specific characteristics including the orbital position and the periodic motion. In this regards, present paper has focused on optimal trajectory planning to transfer to halo orbits. To this aim, homotopy approach has been adopted for optimal trajectory design. This approach has improved the convergence rate and insensitivity of the problem to initial guess. The designed trajectory transfers a spacecraft orbiting the Earth to a Halo orbit around Lagrangian point L1 of the Earth-moon restricted three-body system. The propulsion system has been assumed to be low thrust with constant specific impulse. Homotopy approach has a broad domain of applicability and methods in which continuation method has been employed here among them. The optimal designed trajectory minimizes the fuel consumption via transforming solution of the minimum energy problem utilizing the homotopy approach. This approach simplifies solution of the complex problem of minimum fuel indeed.

کلیدواژه‌ها [English]

  • Halo Orbit
  • Homotopy Approach
  • Homotopy Continuation Method
  • Shooting Method
  • Optimal Trajectory Design
  • Orbit Transfer
[1]         Sayanjali, S.P.M., "Optimal Trajectory Design to Halo Orbits via Pseudo-invariant Manifolds Using a Nonlinear Four Body Formulation," Acta Astronautica, Vol. 110, No. 1, pp. 115-128, 2015.
[2]           He, J., " Homotopy Perturbation Technique," Computer Methods in Applied Mechanics and Engineering, Vol. 178, No. 3-4, pp. 257-262, 1999.
[3]           Nayfeh, A.H., Perturbation Methods, New York: John Wiley & Sons, 2008.
[4]           Liao, S., Introduction to the Homotopy Analysis Method, Boca Raton: Chapman & HALL/CRC, 2003.
[5]           Allgower, K.G.E.L., Introduction to Numerical Continuation Methods, New York: SIAM, 2003.
[6]           Doedel, E., Keller, H.B. and Kernevez, J.P., "Numerical Analysis and Control of Bifurcation Problems," International Journal of Bifurcation and Chaos, Vol. 1, No. 3, pp. 493-520, 1991.
[7] Doedel,                         E.J., Champneys, A.R., Fairgrieve, T.E. and et.al., AUTO 97: Continuation And Bifurcation Software for Ordinary Differential Equations, Canada: Concordia University, 1998.
[8]           Richardson, D., "Halo Orbit Formulation for the ISEE-3 Mission," Journal of Guidance, Control, and Dynamics, Vol. 3, No. 6, pp. 543-548, 1980.
[9]           Xu, M., Liang,Y., Ren, K., "Survey on Advances in Orbital Dynamics and Control for Libration Point Orbits," Progress in Aerospace Sciences, Vol. 82, No. 1, pp. 24-35, 2016.
[10]         Stuhlinger, E., "The Flight Path of an Electrically Propelled Space Ship," Journal of Jet Propulsion, Vol. 27, No. 4, pp. 410-414, 1957.
[11]         Rodriguez, E., "Method for Determining Steering Programs for Low Thrust Interplanetary Vehicles," ARS, Vol. 29, No. 10, pp. 783-788, 1959.
[12]         Kelley, H., "Gradient Theory of Optimal Flight Paths," ARS, Vol. 30, No. 10, pp. 947-954, 1960.
[13]         Conley, C., "Low Energy Transit Orbits in the Restricted Three-body Problems," SIAM Journal on Applied Mathematics, Vol. 16, No. 4, pp. 732-746, 1968.
[14]         Kluever, C.A. and Pierson, B.L., "Optimal Low-thrust Three-dimensional Earth-moon Trajectories," Guidance, Control, and Dynamics, Vol. 18, No. 4, pp. 830-837, 1995.
[15]         Senent, J., Ocampo, C., and Capella, A., "Low-Thrust Variable-specific-impulse Transfers and Guidance to Unstable Periodic Orbits," Guidance, Control, and Dynamics, Vol. 28, No. 2, pp. 280-290, 2005.
[16]         Watson, L., "Globally Convergent Homotopy Algorithms for Nonlinear Systems of Equations," Nonlinear Dynamics, Vol. 1, No. 2, pp. 143-191, 1990.
[17]   Chen, L.W.Y., "Optimal Trajectory Planning for a Space Robot Docking with a Moving Target via Homotopy Algorithms," Robotic Systems, Vol. 12, No. 8, pp. 531-540, 1995.
[18]         Haberkorn, T., Martinon, P., and Gergaud, J., "Low Thrust Minimum-fuel Orbital Transfer: a Homotopic Approach," Guidance, Control, and Dynamics, Vol. 27, No. 6, pp. 1046-1060, 2004.
[19]         Jiang, F., Li, J., and Guo, T. "Homotopic Approach and Pseudospectral Method Applied Jointly to Low Thrust Trajectory Optimization," Acta Astronautica, Vol. 71, No. 1, pp. 38-50, 2012.
[20]         Chupin, M., Haberkorn, T., and Trélat, E., "Low-thrust Lyapunov to Lyapunov and Halo to Halo Missions with L2-minimization," ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 51, No. 3, pp. 965-996, 2017.
[21]         Chi, Z., Yang, H., Chen, S., and Li, J., "Homotopy Method for Optimization of Variable-specific-impulse Low-thrust Trajectories," Astrophysics and Space Science, Vol. 362, No. 11, p. 216, 2017.
[22]Kayama, Y., Bando, M., and Hokamoto, Sh., "Minimum Fuel Trajectory Design Using Sparse Optimal Control in Three-Body Problem", AIAA Scitech, Published Online, 5 Jan. 2020.
[23]Vallado, D.A., Fundamentals of Astrodynamics and Applications, United States: Microcosm Press, 2013.
[24]         Yang, H. and Li, Sh., "Fast Homotopy Method for Binary Astroid Landing Trajectory Optimization", EUCASS, 2019
 [25]        Kirk, D., Optimal Control Theory: an Introduction, Richardson: Springer, 1970.
[26]         Navabi, M. and Meshkinfam, E., "Space Low-thrust Trajectory Optimization Utilizing Numerical Techniques, a Comparative Study," In Recent Advances in Space Technologies (RAST), Istanbul, Turkey , 2013.
[27]         Beale, G., "Minimum Fuel Optimal Control Example for A Scalar System," 19 August 2013. [Online]. Available: https://ece. gmu. edu/~gbeale/ece_620/xmpl-620-min-fuel-01. pdf.
[28]         Russell, R., "Primer Vector Theory Applied to Global Low-thrust Trade Studies," Guidance, Control, and Dynamics, Vol. 30, No. 2, pp. 460-472, 2007.