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Optimal Trajectories for Spacecraft Rendezvous

Author

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  • A. Miele

    (Rice University)

  • M. W. Weeks

    (NASA-Johnson Space Center)

  • M. Ciarcià

    (Rice University)

Abstract

The efficient execution of a rendezvous maneuver is an essential component of various types of space missions. This work describes the formulation and numerical investigation of the thrust function required to minimize the time or fuel required for the terminal phase of the rendezvous of two spacecraft. The particular rendezvous studied concerns a target spacecraft in a circular orbit and a chaser spacecraft with an initial separation distance and separation velocity in all three dimensions. First, the time-optimal rendezvous is investigated followed by the fuel-optimal rendezvous for three values of the max-thrust acceleration via the sequential gradient-restoration algorithm. Then, the time-optimal rendezvous for given fuel and the fuel-optimal rendezvous for given time are investigated. There are three controls, one determining the thrust magnitude and two determining the thrust direction in space. The time-optimal case results in a two-subarc solution: a max-thrust accelerating subarc followed by a max-thrust braking subarc. The fuel-optimal case results in a four-subarc solution: an initial coasting subarc, followed by a max-thrust braking subarc, followed by another coasting subarc, followed by another max-thrust braking subarc. The time-optimal case with fuel given and the fuel-optimal case with time given result in two, three, or four-subarc solutions depending on the performance index and the constraints. Regardless of the number of subarcs, the optimal thrust distribution requires the thrust magnitude to be at either the maximum value or zero. The coasting periods are finite in duration and their length increases as the time to rendezvous increases and/or as the max allowable thrust increases. Another finding is that, for the fuel-optimal rendezvous with the time unconstrained, the minimum fuel required is nearly constant and independent of the max available thrust. Yet another finding is that, depending on the performance index, constraints, and initial conditions, sometime the initial application of thrust must be delayed, resulting in an optimal rendezvous trajectory which starts with a coasting subarc.

Suggested Citation

  • A. Miele & M. W. Weeks & M. Ciarcià, 2007. "Optimal Trajectories for Spacecraft Rendezvous," Journal of Optimization Theory and Applications, Springer, vol. 132(3), pages 353-376, March.
    • Handle: RePEc:spr:joptap:v:132:y:2007:i:3:d:10.1007_s10957-007-9166-4
    DOI: 10.1007/s10957-007-9166-4
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    References listed on IDEAS

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    1. A. Miele & M. Ciarcià & M. W. Weeks, 2007. "Guidance Trajectories for Spacecraft Rendezvous," Journal of Optimization Theory and Applications, Springer, vol. 132(3), pages 377-400, March.
    2. A. Miele & T. Wang, 2003. "Multiple-Subarc Gradient-Restoration Algorithm, Part 2: Application to a Multistage Launch Vehicle Design," Journal of Optimization Theory and Applications, Springer, vol. 116(1), pages 19-39, January.
    3. A. Miele & T. Wang, 2003. "Multiple-Subarc Gradient-Restoration Algorithm, Part 1: Algorithm Structure," Journal of Optimization Theory and Applications, Springer, vol. 116(1), pages 1-17, January.
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    Cited by:

    1. Sandeep K. Singh & Brian D. Anderson & Ehsan Taheri & John L. Junkins, 2021. "Low-Thrust Transfers to Southern $$L_2$$ L 2 Near-Rectilinear Halo Orbits Facilitated by Invariant Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 191(2), pages 517-544, December.
    2. A. Miele & M. Ciarcià, 2008. "Optimal Starting Conditions for the Rendezvous Maneuver, Part 2: Mathematical Programming Approach," Journal of Optimization Theory and Applications, Springer, vol. 137(3), pages 625-639, June.
    3. Y.-K. Chang & J. J. Nieto & W.-S. Li, 2009. "On Impulsive Hyperbolic Differential Inclusions with Nonlocal Initial Conditions," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 431-442, March.
    4. A. Miele & M. Ciarcià & M. W. Weeks, 2007. "Guidance Trajectories for Spacecraft Rendezvous," Journal of Optimization Theory and Applications, Springer, vol. 132(3), pages 377-400, March.
    5. Fliege, Jörg & Kaparis, Konstantinos & Khosravi, Banafsheh, 2012. "Operations research in the space industry," European Journal of Operational Research, Elsevier, vol. 217(2), pages 233-240.
    6. A. Miele & M. Ciarcià, 2008. "Optimal Starting Conditions for the Rendezvous Maneuver, Part 1: Optimal Control Approach," Journal of Optimization Theory and Applications, Springer, vol. 137(3), pages 593-624, June.

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