Orbiting Spacecraft Relative Motion in the Inertial Frame

Abstract While the description of orbiting spacecraft relative motion is usually done in the rotating Hill frame due to analytical first-order solutions and the intuitive shape of the relative orbits, it is disadvantageous for mission design requirements that are fixed in the inertial frame. This includes distributed space telescopes aligned with inertial targets as well as formations and servicing operations with inertially fixed keep-in/out zones, e.g. constraints imposed by the Sun direction. This paper studies the analytical first-order inertial frame solutions of the relative motion of orbiting spacecraft and derives geometrically meaningful relative orbit elements and invariants of motion for inertial frame relative orbits. It is found that the relative motion for a circular chief orbit corresponds to the motion of an epitrochoid. For elliptic chief orbits, the inertial frame relative orbits are stretched and distorted compared to the epitrochoid curve for circular chief orbits, but similar relative orbit elements are defined as well. Finally, the variational equations of the inertial relative orbit elements are developed and their use is demonstrated through an asymptotically stabilizing continuous feedback control law.