Orbit period modulation for relative motion using continuous low thrust in the two-body and restricted three-body problems

C. S. Arnot, C. R. McInnes, R. J. McKay, M. Macdonald, J. Biggs

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This paper presents rich new families of relative orbits for spacecraft formation
ight generated through the application of continuous thrust with only minimal intervention into the dynamics of the problem. Such simplicity facilitates implementation for small, low-cost spacecraft with only position state feedback, and yet permits interesting and novel relative orbits in both two- and three-body systems with potential future applications in space-based interferometry, hyperspectral sensing, and on-orbit inspection. Position feedback is used to modify the natural frequencies of the linearised relative dynamics through direct manipulation of the system eigenvalues, producing new families of stable relative orbits. Specifically, in the Hill-Clohessy-Wiltshire frame, simple adaptations of the linearised dynamics are used to produce a circular relative orbit, frequency-modulated out-of-plane motion, and a novel doubly-periodic cylindrical relative trajectory for the purposes of on-orbit inspection. Within the circular restricted three-body problem, a similar minimal approach with position feedback is used to generate new families of stable, frequency-modulated relative orbits in the vicinity of a Lagrange point, culminating in the derivation of the gain requirements for synchronisation of the in-plane and out-of-plane frequencies to yield a singly-periodic tilted elliptical relative orbit with potential use as a Lunar far-side communications relay. The deltav requirements for the cylindrical relative orbit and singly-periodic Lagrange point orbit are analysed, and it is shown that these requirements are modest and feasible for existing low-thrust propulsion technology.
Original languageEnglish
Number of pages23
JournalCelestial Mechanics and Dynamical Astronomy
Issue number12
Early online date18 Jan 2018
Publication statusPublished - 28 Feb 2018


  • non-Keplerian orbits
  • continuous low thrust
  • relative spacecraft motion
  • orbit period modulation
  • Hill-Clohessy-Wiltshire equations
  • restricted three-body problem


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