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.
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References
Polites, M.E.: An assessment of the technology of automated rendezvous and capture in space. Technical Report TP-208528, NASA MSFC (1970)
Zimpfer, D., Tuohy, S.: Autonomous rendezvous, capture, and in-space assembly: past, present, and future. AIAA Paper 05-2523, 1st Space Exploration Conference: Continuing the Voyage of Discovery, Orlando, FL, 2005
Ianni, J.D., Graves, J.D.: The human’s dismissing role in future on-orbit servicing missions. AIAA Paper 2001-4539, AIAA Space Conference and Exposition, Albuquerque, NM, 2001
Matsumoto, S., Oda, M., Kawano, I.: Attitude dynamics and control of space cargo for reusable orbital logistic supply servicing. AIAA Paper 2002-4785, AIAA Guidance, Navigation, and Control Conference, Monterey, CA, 2002
Defense Advanced Research Projects Agency (DARPA): Orbital Express space operation architecture. Website http://www.darpa.mil/tto/programs/oe.html
Air Force Research Laboratory (AFRL): XSS-11 Microsatellite. Website www.vs.afrl.af.mil/FactSheets/XSS11-MicroSatellite.pdf
Anonymous: Executive summary (introduction-CEV): NASA exploration system architecture study final report (DRAFT) (2005). Website http://www.spaceref.com/news/viewsr.html?pid=1967
Bailey, J.W.: NASA JSC Solicitation: commercial orbital transportation services (COTS) space flight demonstrations. NASA JSC Solicitation on spaceref.com. Website http://www.spaceref.com/news/viewsr.html?pid=18511
Dittmar, M.: Commercial avenues for space utilization. AIAA Paper 2003-6234, AIAA Space Conference and Exposition, Long Beach, CA, 2003
Long, A., Hastings, D.: Catching the wave: a unique opportunity for the development of an on-orbit satellite servicing infrastructure. AIAA Paper 04-6051, AIAA Space Conference and Exhibition, San Diego, CA, 2004
Pearson, D.J.: The glideslope approach. Adv. Astronaut. Sci. 69, 109–123 (1989), Paper AAS 89-162 (1989)
Fox, A.: Rendezvous Crew Training Handbook. Manual TD398B, Mission Operations Directorate, Training Division, NASA-MSFC (1998)
Jezewski, D.J., Brazzel, J.P., Jr., Prust, E.E., Brown, B.G., Mulder, T.A., Wissinger, D.B.: A survey of rendezvous trajectory planning. AAS Paper 91-505, AAS/AIAA Astrodynamics Conference, Durango, CO, pp. 1373–1396 (1991)
Pearson, D.J.: Baselining the Shuttle rendezvous technique. Informal Memorandum, Flight Design and Dynamics Division, NASA JSC (1990)
Chobotov, V.A. (ed.): Orbital Mechanics. AIAA Education Series, AIAA, Washington, DC (1991)
Prussing, J.A., Conway, B.A.: Orbital Mechanics. Oxford University Press, New York, NY (1993)
Thomson, W.T.: Introduction to Space Dynamics. Dover, New York, NY (1986)
Feshe, W.: Automated Rendezvous and Docking of Spacecraft. Cambridge University Press, Cambridge, MA (2003)
Bryson, A.E., Jr.: Control of Spacecraft and Aircraft. Princeton University Press, Princeton, NJ (1994)
Battin, R.H.: An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series, AIAA, New York (1987) (revised edn.)
Lion, P.M., Handelsman, M.: Primer vector on fixed-time impulsive trajectories. AIAA J. 6(1), 127–132 (1968)
Jezewski, D.J.: Primer vector theory applied to the linear relative-motion equations. Optim. Control Appl. Methods 1(4), 387–401 (1980)
Chiu, J.H.: Optimal multiple-impulse nonlinear orbital rendezvous. Ph.D. Thesis, University of Illinois at Urbana-Champaign (1984)
Prussing, J.E., Chiu, J.H.: Optimal multiple-impulse time-fixed rendezvous between circular orbits. J. Guid. Control Dyn. 9(1), 17–22 (1986)
Carter, T.E., Brient, J.: Linearized impulsive rendezvous problem. J. Optim. Theory Appl. 86(3), 553–584 (1995)
Guzman, J., Mailhe, L., Schiff, C., Hughes, S.: Primer vector optimization: survey of theory and some applications. Paper IAC-02-A. 6.09, 53rd International Astronautical Congress, Houston, TX, 2002
Shen, H., Tsiotras, P.: Optimal two-impulse rendezvous using multiple revolution Lambert solutions. J. Guid. Control Dyn. 26(1), 50–61 (2003)
Prussing, J.E.: Optimal two-impulse and three-impulse fixed-time rendezvous in the vicinity of a circular orbit. J. Spacecr. Rockets 40(6), 952–959 (2003)
Goldstein, A.A., Green, A.H., Johnson, A.T., Seidman, T.I.: Fuel optimization in orbital rendezvous. AIAA Paper 63-354, AIAA Guidance, Navigation, and Control Conference, Cambridge, MA, 1963
Paiewonsky, B., Woodrow, P.J.: Three-dimensional time-optimal rendezvous. J. Spacecr. Rockets 3(11), 1577–1584 (1966)
Carter, T.E., Humi, M.: Fuel-optimal rendezvous near a point in general Keplerian orbit. J. Guid. Control Dyn. 10(6), 567–573 (1987)
Van Der Ha, J.C.: Analytical formulation for finite-thrust rendezvous trajectories. Paper IAF-88-308, 39th Congress of the International Astronautical Federation, Bangalore, India, 1988
Carter, T.E., Brient, J.: Fuel-optimal rendezvous for linearized equations of motion. J. Guid. Control Dyn. 15(6), 1411–1416 (1992)
Carter, T.E., Pardis, C.J.: Optimal power-limited rendezvous with upper and lower bounds on thrust. J. Guid. Control Dyn. 19(5), 1124–1133 (1996)
Clohessy, W.H., Wiltshire, R.S.: Terminal guidance system for satellite rendezvous. J. Aerosp. Sci. 27(9), 653–658 (1960)
Miele, A., Pritchard, R.E., Damoulakis, J.N.: Sequential gradient-restoration algorithm for optimal control problems. J. Optim. Theory Appl. 5(4), 235–282 (1970)
Miele, A.: Method of particular solutions for linear two-point boundary-value problems. J. Optim. Theory Appl. 2(4), 260–273 (1968)
Rishikof, B.H., McCormick, B.R., Pritchard, R.E., Sponaugle, S.J.: SEGRAM: a practical and versatile tool for spacecraft trajectory optimization. Acta Astron. 26(8–10), 599–609 (1992)
Miele, A., Wang, T.: Multiple-subarc sequential gradient-restoration algorithm, part 1: algorithm structure. J. Optim. Theory Appl. 116(1), 1–17 (2003)
Miele, A., Wang, T.: Multiple-subarc sequential gradient-restoration algorithm, part 2: application to a multistage Launch vehicle design. J. Optim. Theory Appl. 116(1), 19–39 (2003)
Miele, A., Ciarcià, M., Weeks, M.W.: Guidance trajectories for spacecraft rendezvous. J. Optim. Theory Appl. 132(1) (2007)
Miele, A., Ciarcià, M., Weeks, M.W.: Rendezvous guidance trajectories via multiple-subarc sequential gradient-restoration algorithm. Paper IAC-06-C1.7.3, 57th International Astronautical Congress, Valencia, Spain, 2006
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This research has been supported by NSF under Grant CMS-0218878.
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Miele, A., Weeks, M.W. & Ciarcià, M. Optimal Trajectories for Spacecraft Rendezvous. J Optim Theory Appl 132, 353–376 (2007). https://doi.org/10.1007/s10957-007-9166-4
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DOI: https://doi.org/10.1007/s10957-007-9166-4