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Linearized impulsive rendezvous problem

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The solution of the problem of impulsive minimization of a weighted sum of characteristic velocities of a spacecraft subject to linear equations of motion is presented without the use of calculus of variations or optimal control theory. The geometric structure of the set of boundary points associated with an optimal primer vector is found to be a simplex composed of convex conical sets. Eachk-dimensional open face of the simplex consists of boundary points having nondegeneratek-impulse solutions. This geometric structure leads to a simple proof that at mostn-impulses are required to solve a problem inn-dimensional space. This work is applied to the problem of planar rendezvous of a spacecraft with a satellite in Keplerian orbit using the Tschauner-Hempel equations of motion, with special emphasis on four-impulse solutions. Primer vectors representing four-impulse solutions are sought out and found for elliptical orbits, but none were found for orbits of higher eccentricity. For highly eccentric elliptical orbits, degenerate fiveimpulse solutions were found. In this situation, computer simulations reveal vastly different optimal trajectories having identical boundary conditions and cost.

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  1. 1.

    Edelbaum, T. N.,Minimum-Impulse Transfers in the Near Vicinity of a Circular Orbit, Journal of the Astronautical Sciences, Vol. 14, pp. 66–73, 1967.

  2. 2.

    Jones, J. B.,Optimal Rendezvous in the Neighborhood of a Circular Orbit, Journal of the Astronautical Sciences, Vol. 24, pp. 53–90, 1976.

  3. 3.

    Prussing, J. E.,Illustration of the Primer Vector in Time-Fixed Orbit Transfers, AIAA Journal, Vol. 7, pp. 1167–1168, 1969.

  4. 4.

    Prussing, J. E.,Optimal Four-Impulse Fixed-Time Rendezvous in the Vicinity of a Circular Orbit, AIAA Journal, Vol. 7, pp. 928–935, 1969.

  5. 5.

    Prussing, J. E.,Optimal Two-and Three-Impulse Fixed-Time Rendezvous in the Vicinity of a Circular Orbit, AIAA Journal, Vol. 8, pp. 1211–1228, 1970.

  6. 6.

    Jezewski, D. J., andDonaldson, J. D.,An Analytic Approach to Optimal Rendezvous Using the Clohessy-Wiltshire Equations, Journal of the Astronautical Sciences, Vol. 27, pp. 293–310, 1979.

  7. 7.

    Jezewski, D.,Primer Vector Theory Applied to the Linear Relative-Motion Equations, Optimal Control Applications and Methods, Vol. 1, pp. 387–401, 1980.

  8. 8.

    Prussing, J. E., andClifton, R. S.,Optimal Multiple-Impulse Satellite Avoidance Maneuvers, Paper No. AAS 87-543, AAS/AIAA Astrodynamics Specialist Conference, Kalispell, Montana, 1987.

  9. 9.

    Carter, T. E.,Optimal Impulsive Space Trajectories Based on Linear Equations, Journal of Optimization Theory and Applications, Vol. 70, pp. 277–297, 1991.

  10. 10.

    Lembeck, C. A., andPrussing, J. E.,Optimal Impulsive Intercept with Low-Thrust Rendezvous Return, Journal of Guidance, Control, and Dynamics, Vol. 16, pp. 426–433, 1993.

  11. 11.

    Carter, T. E., andBrient, J.,Fuel-Optimal Rendezvous for Linearized Equations of Motion, Journal of Guidance, Control, and Dynamics, Vol. 15, pp. 1411–1416, 1992.

  12. 12.

    Neustadt, L. W.,Optimization, a Moment Problem, and Nonlinear Programming, SIAM Journal on Control, Vol. 2, pp. 33–53, 1964.

  13. 13.

    Neustadt, L. W.,A General Theory of Minimum-Fuel Space Trajectories, SIAM Journal on Control, Vol. 3, pp. 317–356, 1965.

  14. 14.

    Wheelon, A. D.,Midcourse and Terminal Guidance, Space Technology, Wiley, New York, New York, 1959.

  15. 15.

    Clohessy, W. H., andWiltshire, R. S.,Terminal Guidance System for Satellite Rendezvous, Journal of the Aerospace Sciences, Vol. 27, pp. 653–658, 674, 1960.

  16. 16.

    Geyling, F. T.,Satellite Perturbations from Extraterrestrial Gravitation and Radiation Pressure, Journal of the Franklin Institute, Vol. 269, pp. 375–407, 1960.

  17. 17.

    Spradlin, L. W.,The Long-Time Satellite Rendezvous Trajectory, Aerospace Engineering, Vol. 19, pp. 32–37, 1960.

  18. 18.

    De Vries, J. P.,Elliptic Elements in Terms of Small Increments of Position and Velocity Components, AIAA Journal, Vol. 1, pp. 2626–2629, 1963.

  19. 19.

    Tschauner, J., andHempel, P.,Rendezvous zu ein Min Elliptischer Bahn Umlaufenden Ziel, Astronautica Acta, Vol. 11, pp. 104–109, 1965.

  20. 20.

    Carter, T. E., andHumi, M.,Fuel-Optimal Rendezvous Near a Point in General Keplerian Orbit, Journal of Guidance, Control, and Dynamics, Vol. 10, pp. 567–573, 1987.

  21. 21.

    Carter, T. E.,New Form for the Optimal Rendezvous Equations Near a Keplerian Orbit, Journal of Guidance, Control, and Dynamics, Vol. 13, pp. 183–186, 1990.

  22. 22.

    Lion, P. M., andHandelsman, M.,Primer Vector on Fixed-Time Impulsive Trajectories, AIAA Journal, Vol. 6, pp. 127–132, 1968.

  23. 23.

    Jezewski, D. J., andRozendaal, H. L.,An Efficient Method for Calculating Optimal Free-Space N-Impulse Trajectories, AIAA Journal, Vol. 6, pp. 2160–2165, 1968.

  24. 24.

    Stern, R. G., andPotter, J. E.,Optimization of Midcourse Velocity Corrections, Report RE-17, Experimental Astronomy Laboratory, MIT, Cambridge, Massachusetts, 1965.

  25. 25.

    Klamka, J.,Controllability of Dynamical Systems, Kluwer, Dordrecht, Holland, 1991.

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Communicated by D. G. Hull

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Carter, T.E., Brient, J. Linearized impulsive rendezvous problem. J Optim Theory Appl 86, 553–584 (1995). https://doi.org/10.1007/BF02192159

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Key Words

  • Optimization
  • primer vector
  • linear problems
  • rendezvous problems
  • impulsive minimization
  • conical sets