Solution of the Elliptic Rendezvous Problem with the Time as Independent Variable

Abstract

An explicit solution is given for the linearized motion of a chaser in a close neighborhood of a target in an elliptic orbit. The solution is a direct generalization of the Clohessy-Wiltshire equations that are widely used for circular orbits. In other words, when the eccentricity is set equal to zero in the new formulas, the well-known Clohessy-Wiltshire formulas are obtained. The solution is completely explicit in the time. As a starting point, a closed-form solution is found of the de Vries equations of 1963. These are the linearized equations of elliptic motion in a rotating coordinate system, rotating with a variable angular velocity. This solution is shown to be obtained simply by taking the partial derivatives, with respect to the orbit elements, of the two-body solution in polar coordinates. When four classical elements are used, four linearly independent solutions of the de Vries equations are obtained. However, the classical orbit elements turn out to be singular for circular orbits. The singularity is removed by taking appropriate linear combinations of the four solutions. This gives a 4 by 4 fundamental solution matrix R that is nonsingular and reduces to the Clohessy-Wiltshire solution matrix when the eccentricity is set equal to zero.