D Matrix for Keplerian State Error Propagation

Abstract

The D matrix is an error state transition matrix (STM) for Keplerian motion in cylindrical coordinates employing time as a state variable and reference true or eccentric anomaly as an independent variable. It has been only partially derived in previous articles and applied to different astrodynamics problems such as optimal collision avoidance and Lambert targeting. A rigorous and complete derivation of the full matrix in analytical form is presented. In addition, it is shown that the different terms of the proposed STM can lead to the definition of new variational equations and optimal steering laws applicable to impulsive and low-thrust orbit transfer optimization as well as reachability analysis. Finally, the effectiveness of the proposed STM is evaluated against the well-known Yamanaka–Ankersen STM showing more than one order of magnitude error reduction associated to the semimajor axis and eccentricity of the linearly propagated relative state.