Relative Motion in the Velocity Frame for Atmospheric Entry Trajectories

Relative motion models provide a method of directly describing the position and velocity of a deputy spacecraft with respect to a chief spacecraft. Common approaches such as the Clohessy–Wiltshire equations describe relative motion in a rotating orbit frame aligned with the radial position vector of the chief, and intuitive solutions exist in this frame for circular or near-circular chief orbits. However, as eccentricity of the chief orbit increases, the along-track and velocity directions become less aligned and the orbit frame becomes less intuitive. This work revisits several key relative motion descriptions in the orbit frame and reformulates them to describe motion in the velocity frame, which provides an intuitive description of motion with respect to the flight path. Highly elliptic and hyperbolic chief motions are considered, which are common for atmospheric entry trajectory scenarios. These models are combined with the extended Allen–Eggers equations into a procedure for analytically estimating the offset in landing location for formation flying on an atmospheric entry trajectory. Three representative examples are given and compared with simulation, and range offset predictions are within 6% of total chief range in all cases.