Relative Orbital Elements Formulation Based upon the Clohessy-Wiltshire Equations
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Relative Orbital Elements Formulation Based upon the Clohessy-Wiltshire Equations Thomas A. Lovell · David A. Spencer
© American Astronautical Society 2015
Abstract For on-orbit applications where two or more spacecraft are flying in close proximity, it is often convenient to apply the Clohessy-Wiltshire differential relative motion equations in order to calculate the relative motion of a deputy spacecraft about a chief spacecraft that is assumed to be in a circular orbit. Under these assumptions, the solutions to the Clohessy-Wiltshire equations can be re-parameterized as a set of relative orbital elements that fully characterize the relative motion of the deputy about the chief. In contrast to the Cartesian relative position and velocity states, relative orbital elements provide a clear geometric interpretation of the relative motion and yield an intuitive understanding of how the unforced relative motion will evolve with time. In this paper, the derivation of relative orbital elements is given, and the transformation between relative orbital elements and Cartesian state elements expressed in the local-vertical, local-horizontal frame is provided. The evolution of relative orbital elements with time is evaluated, and characteristics of the unforced motion in terms of relative orbital elements are described. Keywords Orbital mechanics · Relative motion · Proximity operations
Thomas A. Lovell is a Research Aerospace Engineer, Space Vehicles Directorate, Air Force Research Laborabory; Associate Fellow, AIAA; Senior Member, AAS. David A. Spencer is a Professor of the Practice, Guggenheim School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150, Associate Fellow, AIAA, Member AAS. T. A. Lovell Air Force Research Laboratory, Kirtland AFB, Albuquerque, NM 87117, USA D. A. Spencer () Georgia Institute of Technology, Atlanta, GA 30332, USA e-mail: [email protected]
J of Astronaut Sci
Nomenclature ar = a Tx = a Ty = a Tz = Ax = Ay = Az = Er = ir = n= n¯ r = nˆ r = r= r¯ = r¯ C = r¯ 1 = r¯ 2 = t= x= x˙ = x¨ = xˆ = xr = xq = y= y˙ = y¨ = yˆ = yr = yq = ys = y˙s = z= z˙ =
semimajor axis of the instantaneous relative ellipse radial component of acceleration resulting from external forces along-track component of acceleration resulting from external forces cross-track component of acceleration resulting from external forces amplitude of the motion in the LVLH radial direction amplitude of the motion in the LVLH along-track direction amplitude of the motion in the LVLH cross-track direction relative eccentric anomaly relative inclination mean motion of the chief’s orbit vector normal to the instantaneous relative orbit unit vector normal to the instantaneous relative orbit magnitude of deputy position vector with respect to center of central body deputy position vector with respect to center of central body chief position vector with respect to center of central body position vector from the instantaneous center of motion to the point where Er = 0 position vector from the
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