Frozen-anomaly transformation for the elliptic rendezvous problem

Abstract

A new solution to relative motion on elliptical orbits is presented, based on a novel transformation from the reference state vector to the relative state vector. The relative orbit is constructed assuming that the anomaly, and not the time, is the invariant element throughout the transformation. This concept arises naturally from the variational form of anomaly-explicit formulations. In particular, this paper explores the capabilities of a formulation developed by Pelaez et al. (Celest Mech Dyn Astron. 97, 131–150, 2007), called Dromo. This formulation exploits the advantages of the ideal reference frames and quaternionic descriptions of the orbital plane. The linear variational form of the equations of motion in Dromo is developed herein, and the resulting transformation matrix is presented. When applied to the reference state vector, this linear transformation provides the relative state vector at any step. The invariance in the anomaly implies a certain time delay in the results. Physical times for leader and follower do not coincide after the transformation. To recover the sense of the solution an additional correction is applied a posteriori to cancel this intrinsic time delay. The performance of the new transformation is compared against previous solutions to the problem through a set of numerical examples. Important error reductions in determining the relative orbit are observed in these tests.