Optimal impulsive space trajectories based on linear equations

  • T. E. Carter
Contributed Papers


The problem of minimizing the total characteristic velocity of a spacecraft having linear equations of motion and finitely many instantaneous impulses that result in jump discontinuities in velocity is considered. Fixed time and fixed end conditions are assumed. This formulation is flexible enough to allow some of the impulses to be specifieda priori by the mission planner. Necessary and sufficient conditions for solution of this problem are found without using specialized results from control theory or optimization theory. Solution of the two-point boundary-value problem is reduced to a problem of solving a specific set of equations. If the times of the impulses are specified, these equations are at most quadratic. Although this work is restricted to linear equations, there are situations where it has potential application. Some examples are the computation of the velocity increments of a spacecraft near a real or fictitious satellite or space station in a circular or more general Keplerian orbit. Another example is the computation of maneuvers of a spacecraft near a libration point in the restricted three-body problem.

Key Words

Optimization of linear systems optimal space trajectories impulsive maneuvers fuel optimal trajectories minimum characteristic velocity orbital rendezvous linear equations of motion velocity increments 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Neustadt, L. W.,Optimization, a Moment Problem, and Nonlinear Programming, SIAM Journal on Control, Vol. 2, pp. 33–53, 1964.Google Scholar
  2. 2.
    Neustadt, L. W.,A General Theory of Minimum-Fuel Space Trajectories, SIAM Journal on Control, Vol. 3, pp. 317–356, 1965.Google Scholar
  3. 3.
    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.Google Scholar
  4. 4.
    Jones, J. B.,Optimal Rendezvous in the Neighborhood of a Circular Orbit, Journal of the Astronautical Sciences, Vol. 24, pp. 53–90, 1976.Google Scholar
  5. 5.
    Prussing, J. E.,Illustration of the Primer Vector in Time-Fixed Orbit Transfers, AIAA Journal, Vol. 7, pp. 1167–1168, 1969.Google Scholar
  6. 6.
    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.Google Scholar
  7. 7.
    Prussing, J. E.,Optimal Four-Impulse Fixed-Time Rendezvous in the Vicinity of a Circular Orbit, AIAA Journal, Vol. 7, pp. 928–935, 1969.Google Scholar
  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.Google Scholar
  9. 9.
    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.Google Scholar
  10. 10.
    Jezewski, D.,Primer Vector Theory Applied to the Linear Relative-Motion Equations, Optimal Control Applications and Methods, Vol. 1, pp. 387–401, 1980.Google Scholar
  11. 11.
    Wheelon, A. D.,Midcourse and Terminal Guidance, Space Technology, Wiley, New York, New York, 1959.Google Scholar
  12. 12.
    Clohessy, W. H., andWiltshire, R. S.,Terminal Guidance System for Satellite Rendezvous, Journal of the Aerospace Sciences, Vol. 27, pp. 653–658, 674, 1960.Google Scholar
  13. 13.
    Geyling, F. T.,Satellite Perturbations from Extra-Terresrial Gravitation and Radiation Pressure, Journal of the Franklin Institute, Vol. 269, pp. 375–407, 1960.Google Scholar
  14. 14.
    Spradlin, L. W.,The Long-Time Satellite Rendezvous Trajectory, Aerospace Engineering, Vol. 19, pp. 32–37, 1960.Google Scholar
  15. 15.
    De Vries, J. P.,Elliptic Elements in Terms of Small Increments of Position and Velocity Components, AIAA Journal, Vol. 1, pp. 2626–2629, 1963.Google Scholar
  16. 16.
    Tschauner, J., andHempel, P.,Rendezvous zu ein Min Elliptischer Bahn Umlaufenden Ziel, Astronautica Acta, Vol. 11, pp. 104–109, 1965.Google Scholar
  17. 17.
    Shulman, Y., andScott, J.,Terminal Rendezvous for Elliptical Orbits, AIAA Paper No. 66-533, AIAA 4th Aerospace Sciences Meeting, Los Angeles, California, 1966.Google Scholar
  18. 18.
    Euler, E. A.,Optimal Low-Thrust Rendezvous Control, AIAA Journal, Vol. 7, pp. 1140–1144, 1969.Google Scholar
  19. 19.
    Weiss, J.,Solution of the Equation of Motion for High Elliptic Orbits, Technical Note PRV-5, No. 7/81, ERNO Reumfahrttechnik, Bremen, Germany, 1981.Google Scholar
  20. 20.
    Carter, T., andHumi, M.,Fuel-Optimal Rendezvous Near a Point in General Keplerian Orbit, Journal of Guidance, Control, and Dynamics, Vol. 10, pp. 567–573, 1987.Google Scholar
  21. 21.
    Carter, T.,Effects of Propellant Mass Loss on Fuel-Optimal Rendezvous Near Keplerian Orbit, Journal of Guidance, Control, and Dynamics, Vol. 12, pp. 19–26, 1989.Google Scholar
  22. 22.
    Carter, T.,New Form for the Optimal Rendezvous Equations Near a Keplerian Orbit, Journal of Guidance, Control, and Dynamics, Vol. 13, pp. 183–186, 1990.Google Scholar
  23. 23.
    Szebehely, V.,Theory of Orbits, Academic Press, New York, New York, 1967.Google Scholar
  24. 24.
    Lawden, D. F.,Optimal Trajectories for Space Navigation, Butterworths, London, England, 1963.Google Scholar
  25. 25.
    Lion, P. M.,A Primer on the Primer, STAR Memo No. 1, Department of Aerospace and Mechanical Engineering, Princeton University, Princeton, New Jersey, 1967.Google Scholar
  26. 26.
    Lion, P. M., andHandelsman, M.,Primer Vector on Fixed-Time Impulsive Trajectories, AIAA Journal, Vol. 6, pp. 127–132, 1968.Google Scholar
  27. 27.
    Carter, T.,How Many Intersections Can a Helical Curve Have with the Unit Sphere During One Period?, American Mathematical Monthly, Vol. 93, pp. 41–44, 1986.Google Scholar
  28. 28.
    Carter, T.,Fuel-Optimal Maneuvers of a Spacecraft Relative to a Point in Circular Orbit, Journal of Guidance, Control, and Dynamics, Vol. 7, pp. 710–716, 1984.Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • T. E. Carter
    • 1
  1. 1.Department of Mathematics and Computer ScienceEastern Connecticut State UniversityWillimantic

Personalised recommendations