Solving and analyzing relative Lambert's problem through differential orbital elements

A novel approach based on Lagrange's time equation and differential orbital elements is developed to solve the relative Lambert's problem on circular reference orbit. In comparison with the conventional Clohessy-Wiltshire equation, the proposed method can directly obtain the change of orbital elements. This advantage enables us to account for the singularities occurred in relative Lambert's problem. Results reveal that relative velocities depend on five differential orbital elements between the transfer orbit and the reference orbit. Accordingly, singularities can be attributed to any significant change of the semi-major axis, eccentricity or orbital plane. Furthermore, adjusting the initial and final relative positions properly can remove some of the singularities. A numerical simulation based on classic Lambert's formula for a rendezvous mission in closed range demonstrates all the analytical results.