A Third-Order Analytical Solution for Relative Motion with a Circular Reference Orbit
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A Third-Order Analytical Solution for Relative Motion with a Circular Reference Orbit 1 D. L. Richardson' and J. W. Mitchel13 Abstract From a Lagrangian approach for the development of the relative motion equations viewed as a nonlinear Hill's problem, it is shown that the influence of a spherical primary mass takes the form of a third-body-like disturbing function expressed relative to the origin of Hill's rotating frame. The resulting Lagrangian and corresponding equations of motion are compact and provide an easily obtainable representation of the nonlinear contributions to the motion to an arbitrary order using recursion relations. The relative motion equations are expanded through third-order in the local Hill's coordinates and a correspondingly accurate successive approximations solution is developed to describe nonlinear periodic motions in Hill's frame.
Introduction Considerable attention has been given to the dynamics and control of formation flying satellites [1,7, 13]. In particular, much effort has been given to determining the relative effects of perturbations on satellite formations. Frequently, preliminary mission analyses begin with what are commonly known as the linearized Hill's [3] or Clohessy-Wiltshire [2] equations. These are the linearized equations describing the relative motion of two bodies under the gravitational influence of a point-mass central body. Traditionally, these equations have been used to describe rendezvous maneuvers and typically prove useful only for a few orbital revolutions. More indepth analyses have included the effects of central body oblateness, solar and electromagnetic perturbations, drag [14]; and eccentricity effects [5]. Consequently, it seems of interest to develop an analytical solution describing periodic motions in
'Originally presented as paper AAS 02-147 at the AAS/AIAA Space Flight Mechanics Meeting, San Antonio, Texas, January 27-30, 2002. 2professor, Aerospace Engineering and Engineering Mechanics, University of Cincinnati, P.O. Box 210070, Cincinnati OR 45221; member AAS. "Visiting Scientist, Air Vehicles Directorate, Control Theory Optimization Branch (VACA), 2210 Eighth St., Wright-Patterson AFB, OR 45433; member AAS.
Richardson and Mitchell
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Hill's frame containing the most significant nonlinear contributions that influence the relative motion as much as the effects of more well-known perturbations. In this paper, we describe a third -order analytical perturbation solution developed by successive approximations [8] that provides a very accurate periodic solution for relative motion viewed as a special case of Hill's equations.
Equations of Motion The typical geometry describing the relative motion of two satellites in orbit about a spherical central body can be seen in Fig. 1. In this configuration, a leader satellite moves in a circular reference orbit of radius R about the central body, while a follower satellite moves in close proximity to the leader at a relative distance r and a distance Rf from the central body. Further, a local dextral
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