Celestial Mechanics and Dynamical Astronomy

, Volume 127, Issue 3, pp 301–330 | Cite as

The theory of asynchronous relative motion I: time transformations and nonlinear corrections

  • Javier RoaEmail author
  • Jesús Peláez
Original Article


Using alternative independent variables in lieu of time has important advantages when propagating the partial derivatives of the trajectory. This paper focuses on spacecraft relative motion, but the concepts presented here can be extended to any problem involving the variational equations of orbital motion. A usual approach for modeling the relative dynamics is to evaluate how the reference orbit changes when modifying the initial conditions slightly. But when the time is a mere dependent variable, changes in the initial conditions will result in changes in time as well: a time delay between the reference and the neighbor solution will appear. The theory of asynchronous relative motion shows how the time delay can be corrected to recover the physical sense of the solution and, more importantly, how this correction can be used to improve significantly the accuracy of the linear solutions to relative motion found in the literature. As an example, an improved version of the Clohessy-Wiltshire (CW) solution is presented explicitly. The correcting terms are extremely compact, and the solution proves more accurate than the second and even third order CW equations for long propagations. The application to the elliptic case is also discussed. The theory is not restricted to Keplerian orbits, as it holds under any perturbation. To prove this statement, two examples of realistic trajectories are presented: a pair of spacecraft orbiting the Earth and perturbed by a realistic force model; and two probes describing a quasi-periodic orbit in the Jupiter-Europa system subject to third-body perturbations. The numerical examples show that the new theory yields reductions in the propagation error of several orders of magnitude, both in position and velocity, when compared to the linear approach.


Relative motion Nonlinear effects Time transformations Regularization Stability Clohessy-Wiltshire solution 



This work was carried out within the framework of the research project entitled “Dynamical Analysis, Advanced Orbital Propagation, and Simulation of Complex Space Systems” (ESP2013-41634-P) supported by the Spanish Ministry of Economy and Competitiveness. Authors thank the Spanish Government for its support and H. Urrutxua for reading the manuscript carefully and making valuable suggestions. J. Roa especially thanks “La Caixa” for his doctoral fellowship. The comments from an anonymous reviewer greatly improved the quality of the paper.


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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Space Dynamics GroupTechnical University of Madrid (UPM)MadridSpain
  2. 2.Jet Propulsion LaboratoryCalifornia Institute of TechnologyPasadenaUSA

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