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Optimal power-limited rendezvous of a spacecraft with bounded thrust and general linear equations of motion

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Abstract

The solution of the fixed-time optimal power-limited rendezvous with a general linear system of ordinary differential equations and a bound on the magnitude of the applied thrust is presented. Necessary and sufficient conditions for thrust saturation in an optimal solution are included.

Because of the generality of the linear system of equations of motion, controllability considerations are required for a complete solution of this problem. It is shown that the condition of controllability can be defined completely in terms of a class of primer vectors associated with this problem. Moreover, it is shown that two distinct versions of the primer vector appear in this problem. Therefore, there is not a unique primer vector associated with every rendezvous problem.

The work is applied to the problem of the rendezvous of a spacecraft near a satellite in circular orbit. The optimal rendezvous trajectory is determined by the interaction of a primer vector and the bound on the thrust magnitude. The results of computer simulations are presented graphically.

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References

  1. Irving, J. H.,Low Thrust Flight: Variable Exhaust Velocity in Gravitational Fields, Space Technology, Edited by H. S. Seifert, Wiley, New York, New York, pp. 10–01 to 10-54, 1959.

    Google Scholar 

  2. Ross, S., andLeitmann, G.,Low Acceleration Trajectory Optimization in a Strong Central Force Field, Proceedings of the IAS Symposium on Vehicle Systems Optimization, New York, New York, pp. 127–131, 1961.

  3. Edelbaum, T. N.,The Use of High and Low-Thrust Propulsion in Combination for Space Missions, Journal of the Astronautical Sciences, Vol. 9, pp. 49–60, 1961.

    Google Scholar 

  4. Saltzer, C., andFetheroff, C. W.,A Direct Variational Method for Calculation of Optimum Thrust Programs for Power-Limited Interplanetary Flight, Astronautica Acta, Vol. 7, pp. 8–20, 1961.

    Google Scholar 

  5. Melbourne, W. G.,Three-Dimensional Optimum Thrust Trajectories for Power-Limited Propulsion Systems, ARS Journal, Vol. 31, pp. 1723–1728, 1961.

    Google Scholar 

  6. Melbourne, W. G., andSauer, C. G., Jr.,Optimum Thrust Programs for Power-Limited Propulsion Systems, Astronautica Acta, Vol. 8, pp. 205–227, 1962.

    Google Scholar 

  7. Melbourne, W. G., andSauer, C. G., Jr.,Optimum Interplanetary Rendezvous with Power-Limited Vehicles, AIAA Journal, Vol. 1, pp. 54–60, 1963.

    Google Scholar 

  8. Billik, B. H.,Some Low-Acceleration Rendezvous Maneuvers, AIAA Journal, Vol. 2, pp. 510–516, 1964.

    Google Scholar 

  9. Gobetz, F. W.,Optimal Variable-Thrust Transfer of a Power-Limited Rocket between Neighboring Circular Orbits, AIAA Journal, Vol. 2, pp. 339–343, 1964.

    Google Scholar 

  10. Edelbaum, T. N.,Optimum Low-Thrust Rendezvous and Station Keeping, AIAA Journal, Vol. 2, pp. 1196–1201, 1964.

    Google Scholar 

  11. Gobetz, F. W.,A Linear Theory of Optimum Low-Thrust Rendezvous Trajectories, Journal of the Astronautical Sciences, Vol. 12, pp. 69–76, 1965.

    Google Scholar 

  12. Euler, E. A.,Optimal Low-Thrust Rendezvous Control, AIAA Journal, Vol. 7, pp. 1140–1144, 1969.

    Google Scholar 

  13. 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 

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

    Google Scholar 

  15. Edelbaum, T. N.,Optimal Space Trajectories, Report 69-4, analytical Mechanics Associates, 1969.

  16. Marec, J. P.,Optimal Space Trajectories, Elsevier Scientific Publishing Company, New York, New York, 1979.

    Google Scholar 

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

    Google Scholar 

  18. Coverstone-Carroll, V., andPrussing, J.,Optimal Cooperative Power-Limited Rendezvous between Neighboring Circular Orbits, Journal of Guidance, Control, and Dynamics, Vol. 16, pp. 1045–1054, 1993.

    Google Scholar 

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

    Google Scholar 

  20. Kechichian, J. A.,Optimal Low-Thrust Transfer Using Variable Bounded Thrust, Paper IAF-93-A.2.10, 44th Congress of the International Astronautical Federation, Graz, Austria, 1993.

  21. Letov, A. M.,The Analytical Design of Control Systems, Part 2, Automation and Remote Control, Vol. 21, pp. 561–568, 1960.

    Google Scholar 

  22. Letov, A. M.,The Analytical Design of Control Systems, Part 3, Automation and Remote Control, Vol. 21, pp. 389–395, 1960.

    Google Scholar 

  23. Letov, A. M.,The Analytical Design of Control Systems, Part 4, Automation and Remote Control, Vol. 23, pp. 363–372, 1961.

    Google Scholar 

  24. Letov, A. M.,The Analytical Design of Control Systems, Part 5, Automation and Remote Control, Vol. 23, pp. 1319–1327, 1962.

    Google Scholar 

  25. Chang, J. W.,A Problem in the Synthesis of Optimal Systems using the Maximum Principle, Automation and Remote Control, Vol. 22, pp. 1170–1176, 1961.

    Google Scholar 

  26. Krasovski, N. N., andLetov, A. M.,The Theory of Analytic Design of Controllers, Automation and Remote Control, Vol. 23, pp. 649–656, 1962.

    Google Scholar 

  27. Rekasius, Z. V., andHsia, T. C.,On an Inverse Problem in Optimal Control, IEEE Transactions on Automatic Control, Vol. 9, pp. 370–375, 1964.

    Google Scholar 

  28. Johnson, C. D., andWonham, W. M.,On a Problem of Letov in Optimal Control, Transactions of the ASME, Series D, Journal of Basic Engineering, Vol. 87, pp. 81–89, 1965.

    Google Scholar 

  29. Neustadt, L. W.,Minimum Effort Control Systems, SIAM Journal on Control, Vol. 1, pp. 16–31, 1962.

    Google Scholar 

  30. Chang, A.,An Optimal Regulator Problem, SIAM Journal on Control, Vol. 2, pp. 220–233, 1965.

    Google Scholar 

  31. Sakawa, Y.,On a Solution of an Optimization Problem in Linear Systems with Quadratic Performance Index, SIAM Journal on Control, Vol. 4, pp. 382–393, 1965.

    Google Scholar 

  32. Anvari, M., andDatko, R. F.,The Existence of Optimal Controls for a Performance Index with a Positive Integrand, SIAM Journal on Control, Vol. 4, pp. 372–381, 1966.

    Google Scholar 

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

    Google Scholar 

  34. Lawden, D. F.,Optimal Trajectories for Space Navigation, Butterworths, London, England, 1963.

    Google Scholar 

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

    Google Scholar 

  36. Carter, T. E.,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 

  37. 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.

    Google Scholar 

  38. Carter, T., andBrient, J.,Optimal Bounded-Thrust Space Trajectories Based on Linear Equations, Journal of Optimization Theory and Applications, Vol. 7, pp. 299–317, 1991.

    Google Scholar 

  39. Kalman, R. E.,Contributions to the Theory of Optimal Control, Boletin de la Sociedad Matematica Mexicana, Vol. 5, pp. 102–119, 1960.

    Google Scholar 

  40. Kalman, R. E., Falb, P. L., andArbib, M. A.,Topics in Mathematical System Theory, McGraw-Hill, New York, New York, 1969.

    Google Scholar 

  41. Lee, E. B., andMarkus, L.,Foundations of Optimal Control Theory, Wiley, New York, New York, 1967.

    Google Scholar 

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Author notes

  1. T. E. Carter (Professor)

    Present address: Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, Massachusetts

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Eastern Connecticut State University, willimantic, Connecticut

    T. E. Carter (Professor)

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Communicated by D. G. Hull

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Carter, T.E. Optimal power-limited rendezvous of a spacecraft with bounded thrust and general linear equations of motion. J Optim Theory Appl 87, 487–515 (1995). https://doi.org/10.1007/BF02192130

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