Solving the relative Lambert's problem and accounting for its singularities

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