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Symmetric minimum-impulse rendezvous between certain non-coplanar orbits

Part of the Lecture Notes in Mathematics book series (LNM,volume 132)

Abstract

A three-parameter family of optimal rendezvous maneuvers is investigated, describable as follows: rendezvous is to be accomplished in a specified time-interval between two equi-energy near-circular slightly non-coplanar orbits whose line-of-nodes is traversed exactly at mid-time; the required total displacement of the "vacant" focus during the rendezvous is perpendicular to the line-of-nodes; lastly, the required in-track position change is, like the eccentricities and the plane change, a small quantity. The three essential parameters for this family are the two ratios of the three small quantities Δe, i, and Δ*ϑ, together with the total angular travel (not small) by either vehicle, where Δe is the change of the eccentricity component perpendicular to the line of nodes, i is the plane rotation, and Δ*ϑ is a suitable in-track phase change.

The optimal rendezvous maneuvers for this family consists of pairs of equal impulses symmetrically spaced about the mid-position (including possibly an impulse at the mid-position), the forward component of the impulse being reversed within each pair.

The optimal number of impulses varies from one to six. For cases in which Δe/i exceeds 1/√3 there is, as in the time-free near-circular problem, a degeneracy, at least if Δ*ϑ/Δe lies within a certain range, the location of the impulses being no longer unique. In such cases the number of impulses need never exceed six, however.

A one-parameter family of charts is presented, indicating regions requiring different numbers of impulses, in some cases with an initial and final coasting period, and indicating the degenerate region, if any. A separate chart permits the selection of a variety of optimal impulsive maneuvers in the degenerate cases.

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References

  1. Marec, J. P., "Contribution a L'Etude des Rendez-vous, Multi-Impulsionnels Optimaux, de Duree Moyenne, Entre Orbites Quasi-Circulaires, Proches, Non-Coplanaires," presented at 2e Colloque International sur les Methodes d'Optimisation. Akademgorodek, U.S.S.R., 20–26 June 1968.

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  4. Marec, J. P., "Transfers Infinitesimaux Impulsionnels Economiques Entre Orbites Quasi-Circulaires non Coplanaires," Proc. 17th IAF Congress, Madrid, Oct. 1966.

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  5. Edelbaum, T. N., "Minimum Impulse Transfers in the Near Vicinity of a Circular Orbit," J. of Astronaut. Sci. XIV, 2, March–April 1967.

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© 1970 Springer-Verlag

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Heine, W., Breakwell, J.V. (1970). Symmetric minimum-impulse rendezvous between certain non-coplanar orbits. In: Balakrishnan, A.V., Contensou, M., de Veubeke, B.F., Krée, P., Lions, J.L., Moiseev, N.N. (eds) Symposium on Optimization. Lecture Notes in Mathematics, vol 132. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066679

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  • DOI: https://doi.org/10.1007/BFb0066679

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-04921-0

  • Online ISBN: 978-3-540-36275-3

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