Analytical Solution for Power-Limited Optimal Rendezvous near an Elliptic Orbit
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- 作者：P. Sengupta (1)
S. R. Vadali (1) - 关键词：Optimal rendezvous ; Power ; limited propulsion ; Eccentric orbits ; Tschauner ; Hempel equations ; Lambert W function
- 刊名：Journal of Optimization Theory and Applications
- 出版年：2008
- 出版时间：July 2008
- 年：2008
- 卷：138
- 期：1
- 页码：115-137
- 全文大小：537KB
- 参考文献：1. Hill, G.W.: Researches in the lunar theory. Am. J. Math. 1(1), 5鈥?6 (1878) CrossRef
2. Clohessy, W.H., Wiltshire, R.S.: Terminal guidance system for satellite rendezvous. J. Aerosp. Sci. 27, 653鈥?58, 674 (1960)
3. Tschauner, J.F.A., Hempel, P.R.: Rendezvous zu einemin Elliptischer Bahn umlaufenden ziel. Astronaut. Acta 11(2), 104鈥?09 (1965)
4. Carter, T.E., Humi, M.: Fuel-optimal rendezvous near a point in general Keplerian orbit. J. Guid. Control Dyn. 10(6), 567鈥?73 (1987) CrossRef
5. Carter, T.E.: New form for the optimal rendezvous equations near a Keplerian orbit. J. Guid. Control Dyn. 13(1), 183鈥?86 (1990)
6. Carter, T.E.: State transition matrices for terminal rendezvous studies: brief survey and new examples. J. Guid. Control Dyn. 21(1), 148鈥?55 (1998)
7. Yamanaka, K., Ankersen, F.: New state transition matrix for relative motion on an arbitrary elliptical orbit. J. Guid. Control Dyn. 25(1), 60鈥?6 (2002)
8. Lawden, D.F.: Optimal Trajectories for Space Navigation, 2nd edn. Butterworths, London (1963)
9. Billik, B.H.: Some optimal low-acceleration rendezvous maneuvers. AIAA J. 2(3), 510鈥?16 (1964)
10. Edelbaum, T.N.: Optimum low-thrust rendezvous and station keeping. AIAA J. 2(7), 1196鈥?201 (1964)
11. Gobetz, F.W.: Linear theory of optimum low-thrust rendezvous trajectories. J. Astronaut. Sci. 12(3), 69鈥?6 (1965)
12. Alfriend, K.T., Kashiwagi, Y.: Minimum time optimal rendezvous between neighboring elliptic orbits. J. Optim. Theory Appl. 4(4), 260鈥?76 (1969) CrossRef
13. Jezewski, D.J., Stoolz, J.M.: A closed-form solution for minimum-fuel, constant-thrust trajectories. AIAA J. 8(7), 1229鈥?234 (1970)
14. Carter, T.E., Brient, J.: Fuel-optimal rendezvous for linearized equations of motion. J. Guid. Control Dyn. 15(6), 1411鈥?416 (1992)
15. Humi, M.: Fuel-optimal rendezvous in a general central gravity field. J. Guid. Control Dyn. 16(1), 215鈥?17 (1993)
16. Carter, T.E., Brient, J.: Linearized impulsive rendezvous problem. J. Optim. Theory Appl. 86(3), 553鈥?84 (1995) CrossRef
17. Inalhan, G., Tillerson, M., How, J.P.: Relative dynamics and control of spacecraft formations in eccentric orbits. J. Guid. Control Dyn. 25(1), 48鈥?9 (2002)
18. Euler, E.A.: Optimal low-thrust rendezvous control. AIAA J. 7(6), 1140鈥?144 (1969)
19. Carter, T.E., Brient, J.: Optimal bounded-thrust space trajectories based on linearized equations. J.聽Optim. Theory Appl. 70(2), 299鈥?17 (1991) CrossRef
20. Carter, T.E.: Optimal power-limited rendezvous for linearized equations of motion. J. Guid. Control Dyn. 17(5), 1082鈥?086 (1994)
21. Coverstone-Carroll, V.L., Prussing, J.E.: Optimal cooperative power-limited rendezvous between neighboring circular orbits. J. Guid. Control Dyn. 16(6), 1045鈥?054 (1993)
22. Zanon, D.J., Campbell, M.E.: Optimal planner for spacecraft formations in elliptical orbits. J. Guid. Control Dyn. 29(1), 161鈥?71 (2006) CrossRef
23. Sengupta, P.: Comment on 鈥淥ptimal planner for spacecraft formations in elliptical orbits鈥? J. Guid. Control Dyn. 29(6), 1484鈥?485 (2006) CrossRef
24. Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series. AIAA, Reston (1999). (Revised edn.)
25. Broucke, R.A.: Solution of the elliptic rendezvous problem with the time as independent variable. J.聽Guid. Control Dyn. 26(4), 615鈥?21 (2003)
26. Melton, R.G.: Time explicit representation of relative motion between elliptical orbits. J. Guid. Control Dyn. 23(4), 604鈥?10 (2000)
27. Gim, D.-W., Alfriend, K.T.: State transition matrix of relative motion for the perturbed noncircular reference. J. Guid. Control Dyn. 26(6), 956鈥?71 (2003)
28. Marec, J.-P.: Optimal Spacecraft Trajectories. Studies in Astronautics, vol. 1. Elsevier, Amsterdam (1979)
29. Chang-D铆az, F.R., Hsu, M.M., Braden, E.M., Johnson, I.L., Yang, T.-F.: Rapid Mars transits with exhaust-modulated plasma propulsion. Tech. Rep. 3539, NASA (1995)
30. Starin, S.R., Yedavalli, R.K., Sparks, A.G.: Spacecraft formation flying maneuvers using linear quadratic regulation with no radial axis inputs. In: AIAA Guidance, Navigation and Control Conference and Exhibit, AIAA-2001-4029, AIAA, Montreal (2001)
31. Sengupta, P., Sharma, R., Vadali, S.R.: Periodic relative motion near a keplerian elliptic orbit with nonlinear differential gravity. J. Guid. Control Dyn. 29(5), 1110鈥?121 (2006) CrossRef
32. Lewis, F.L., Syrmos, V.L.: Optimal Control, 2nd edn. Wiley, New York (1995)
33. Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert / W function. Adv. Comput. Math. 5(1), 329鈥?59 (1996) CrossRef
34. Corless, R.M., Jeffrey, D.J., Knuth, D.E.: A sequence of series for the Lambert / W function. In: Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, pp. 197鈥?04. Assoc. Comput. Mech., New York (1997) CrossRef - 作者单位：P. Sengupta (1)
S. R. Vadali (1)
1. Department of Aerospace Engineering, MS 3141, Texas A&M University, College Station, TX, 77843-3141, USA - ISSN：1573-2878
文摘
This paper presents an analytical solution to the problem of the optimal rendezvous using power-limited propulsion for a spacecraft in an elliptic orbit in a gravitational field. The derivation of the result assumes small relative distances, but does not make any assumption on the eccentricity of the orbit and does not require numerical integration. The results are generalized to include the possibility of different weights on the control effort for each axis (radial, along-track, and out-of-plane). When the weights on the control efforts are unequal, several integrals are used whose solutions may be represented by infinite series in a small parameter dependent on the eccentricity. A methodology is introduced where the series can be extended trivially to as many terms as desired. Furthermore, for a given numerical tolerance, an upper bound on the number of terms required to represent the series is also obtained. When the weights are equal for all the three axes, the series representations are no longer necessary. The results can be used easily to design optimal feedback controls for rendezvous maneuvers, or for generating initial guesses for two-point boundary-value problems for numerical solutions to the nonlinear rendezvous problem.