Abstract
The nonlinear initial value problem that models the relative orbital Keplerian motion is expressed using the quaternions algebra. Then, this problem is solved and closed-form expressions for the relative position and relative velocity are obtained. The procedure that allows solving completely the relative orbital motion is purely analytic, without any geometrical considerations. The closed-form expressions hold for any chief and deputy inertial trajectories and have no singularities. The solution offered in this paper is presented in a coordinate-free form that allows a variety of expressions, depending on the coordinate system that is chosen and on the orbit elements that one wants to use as independent variables (time, eccentric anomaly, true anomaly).
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