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Best Initial Conditions for the Rendezvous Maneuver

  • Angelo Miele
  • Marco Ciarcià
Conference paper
Part of the Springer Optimization and Its Applications book series (SOIA, volume 33)

Abstract

Abstract The majority of the papers dedicated to the rendezvous problem have employed the assumption of given initial position and given initial velocity of the chaser spacecraft vis-à-vis the target spacecraft. In this research, the initial separation velocity components are assumed free and the initial separation coordinates are subject to only the requirement that the chaser-to-target distance is given. Within this frame, two problems are studied: time-to-rendezvous free and time-to-rendezvous given. It is assumed that the target spacecraft moves along a circular orbit, that the chaser spacecraft has variable mass, and that its trajectory is governed by three controls, one determining the thrust magnitude and two determining the thrust direction. Analyses performed with the multiple-subarc sequential gradient-restoration algorithm for optimal control problems show that the fuel-optimal trajectory is zero-bang; namely, it includes a long coasting zero-thrust subarc followed by a short powered max-thrust braking subarc. While the thrust direction of the powered subarc is continuously variable for the optimal trajectory, its replacement with a constant (yet optimized) thrust direction produces a very efficient guidance trajectory.

The optimization of the initial separation coordinates and velocities as well as the time lengths of all the subarcs is performed for several values of the initial distance in the range 5 ≤ do ≤ 60 km with particular reference to the rendezvous between the Space Shuttle (SS) and the International Space Station (ISS). This study is of interest because, for a preselected initial distance SS-to-ISS, it supplies not only the best initial conditions for the rendezvous maneuver, but also the corresponding final conditions of the ascent trajectory.

Keywords

Optimal Control Problem International Space Station Initial Distance Initial Separation Thrust Direction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Clohessy, W.H., and Wiltshire, R.S., Terminal Guidance System for Satellite Rendezvous, Journal of the Aerospace Sciences, Vol. 27, No. 9, pp. 653–658, 1960.Google Scholar
  2. 2.
    Miele, A., Weeks, M.W., and Ciarcià, M., Optimal Trajectories for Spacecraft Rendezvous, Journal of Optimization Theory and Applications, Vol.132, No. 3, pp. 353–376, 2007.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Miele, A., Ciarcià, M., and Weeks, M.W., Guidance Trajectories for Spacecraft Rendezvous, Journal of Optimization Theory and Applications, Vol.132, No. 3, pp. 377–400, 2007.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Goldstein, A.A., Green, A.H., Johnson, A. T., Seidman, T.I., Fuel Optimization in Orbital Rendezvous, AIAA Paper 63–354, AIAA Guidance and Control Conference, Cambridge, Massachusetts, 1963.Google Scholar
  5. 5.
    Lion, P. M., and Handelsman, M., Primer Vector on Fixed-Time Impulsive Trajectories, AIAA Journal, Vol. 6, No. 1, pp. 127–132, 1968.MATHCrossRefGoogle Scholar
  6. 6.
    Jones, B. J., Optimal Rendezvous in the Neighborhood of a Circular Orbit, Journal of the Astronautical Sciences, Vol. 24, No. 1, pp. 55–90, 1976.Google Scholar
  7. 7.
    Jezewski, D. J., Primer Vector Theory Applied to the Linear Relative-Motion Equations, Optimal Control Applications and Methods, Vol. 1, No. 4, pp. 387–401, 1980.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Chiu, J. H., Optimal Multiple-Impulse Nonlinear Orbital Rendezvous, PhD Thesis, University of Illinois at Urbana-Champain, 1984.Google Scholar
  9. 9.
    Prussing, J. E., and Chiu, J. H., Optimal Multiple-Impulse Time-Fixed Rendezvous between Circular Orbits, Journal of Guidance, Control, and Dynamics, Vol. 9, No. 1, pp. 17–22, 1986.MATHCrossRefGoogle Scholar
  10. 10.
    Carter, T. E., and Brient, J., Linearized Impulsive Rendezvous Problem, Journal of Optimization Theory and Applications, Vol. 86, No. 3, pp. 553–584, 1995.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Guzman, J., Mailhe, L., Schiff, C., and Hughes, S., Primer Vector Optimization: Survey of Theory and Some Applications, Paper IAC-02-A. 6.09, 53rd International Astronautical Congress, Houston, Texas, 2002.Google Scholar
  12. 12.
    Shen, H., and Tsiotras, P., Optimal Two-Impulse Rendezvous Using Multiple Revolution Lambert Solutions, Journal of Guidance, Control, and Dynamics, Vol. 26, No. 1, pp. 50–61, 2003.CrossRefGoogle Scholar
  13. 13.
    Prussing, J.E., Optimal Two-Impulse and Three-Impulse Fixed-Time Rendezvous in the Vicinity of a Circular Orbit, Journal of Spacecraft and Rockets, Vol. 40, No. 6, pp. 952–959, 2003.CrossRefGoogle Scholar
  14. 14.
    Paiewonsky, B., and Woodrow, P.J., Three-Dimensional Time-Optimal Rendezvouz, Journal of Spacecraft and Rockets, Vol. 3, No. 11, pp. 1577–1584, 1966.CrossRefGoogle Scholar
  15. 15.
    Carter, T.E., and Humi, M., Fuel-Optimal Rendezvous Near a Point in General Keplerian Orbit, Journal of Guidance, Control, and Dynamics, Vol. 10, No. 6, pp. 567–573, 1987.MATHCrossRefGoogle Scholar
  16. 16.
    Van Der Ha, J. C., Analytical Formulation for Finite-Thrust Rendezvous Trajectories, Paper IAF-88-308, 39th Congress of the International Astronautical Federation, Bangalore, India, 1988.Google Scholar
  17. 17.
    Carter, T.E., and Brient, J., Fuel-Optimal Rendezvous for Linearized Equations of Motion, Journal of Guidance, Control, and Dynamics, Vol. 15, No. 6, pp. 1411–1416, 1992.CrossRefGoogle Scholar
  18. 18.
    Carter, T.E., Optimal Power-Limited Rendezvous of a Spacecraft with Bounded Thrust and General Linear Equation of Motion, Journal of Optimization Theory and Applications, Vol. 87, No. 3, pp. 487–515, 1995.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Carter, T.E., and Pardis, C. J., Optimal Power-Limited Rendezvous with Upper and Lower Bounds on Thrust, Journal of Guidance, Control, and Dynamics, Vol. 19, No. 5, pp. 1124–1133, 1996.MATHCrossRefGoogle Scholar
  20. 20.
    Park, C., Guibout, V., and Scheeres, D. J., Solving Optimal Continuous Thrust Rendezvous Problem with Generating Functions, Journal of Guidance, Control, and Dynamics, Vol. 29, No. 25, pp. 321–331, 2006.CrossRefGoogle Scholar
  21. 21.
    Miele, A., Method of Particular Solutions for Linear Two-Point Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 2, No. 4, pp. 260–273, 1968.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Miele, A., Pritchard, R.E., and Damoulakis, J.N., Sequential Gradient-Restoration Algorithm for Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 5, No. 4, pp. 235–282, 1970.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Miele, A, Tietze, J. L., and Levy, A. V., Summary and Comparison of Gradient-Restoration Algorithms for Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 10, No. 6, pp. 381–403, 1972.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Miele, A., Recent Advances in Gradient Algorithms for Optimal Control Problems, Journal of Optimization Theory and Applications, Vol. 17, Nos. 5–6, pp. 361–430, 1975.MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Gonzalez, S., and Miele, A., Sequential Gradient-Restoration Algorithm for Optimal Control Problems with General Boundary Conditions, Journal of Optimization Theory and Applications, Vol. 26, No. 3, pp. 395–425, 1978.MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Rishikof, B. H., McCormick, B. R., Prictchard, R. E., and Sponaugle, S. J., SEGRAM: A Practical and Versatile Tool for Spacecraft Trajectory Optimization, Acta Astronautica, Vol. 26, Nos. 8–10, pp. 599–609. 1992.CrossRefGoogle Scholar
  27. 27.
    Miele, A., and Wang, T., Multiple-Subarc Sequential Gradient-Restoration Algorithm, Part 1: Algorithm Structure, Journal of Optimization Theory and Applications, Vol. 116, No. 1, pp. 1–17, 2003.MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Miele, A., and Wang, T., Multiple-Subarc Sequential Gradient-Restoration Algorithm, Part 2: Application to a Multistage Launch Vehicle Design, Journal of Optimization Theory and Applications, Vol. 116, No. 1, pp. 19–39, 2003.MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Miele, A., and Ciarcià, M., Optimal Starting Conditions for the Rendezvous Maneuver: Analytical and Computational Approach, Aero-Astronautics Report 361, Rice University, 2007.Google Scholar

Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.Research Professor and Foyt Professor EmeritusAero-Astronautics Group, Rice UniversityHoustonUSA
  2. 2.PhD CandidateAero-Astronautics Group, Rice UniversityHoustonUSA

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