Rendezvous Equations in a Central-Force Field with Linear Drag

Abstract

The terminal phase of a rendezvous between a satellite and a spacecraft in a central-force e eld with a drag force that is linear in the velocity is considered. To simplify the work, we assume identical drag coefe cients on both. In this general setting, we linearize the equations of motion of the spacecraft and show that they can be transformed into a reasonably simple form. Under certain simplifying assumptions, they can be reduced to one second-order linear differential equation. We then specialize to the case of inverse square laws. If we assume that the drag accelerations are roughly identical on the satellite and spacecraft, we obtain a set of linear differential equations that can be solved in terms of integrals. This enables us to represent the solution of this version of the problem in terms of a state-transition matrix. The work is then placed in the context of various control models that have been developed in previous work. The kind of transformations presented may be useful in the analysis of problems having more realistic drag models.