Revisiting the fuel-optimal four-impulse rendezvous problem near circular orbits

This paper takes a revisit of the fuel-optimal four-impulse rendezvous problem near circular orbits. For coplanar impulsive rendezvous based on the Hill-Clohessy-Wiltshire (HCW) equations, the primer vector hodograph for an optimal four-impulse rendezvous is symmetric about the rendezvous time halfway and can be expressed as an analytical function of the third impulse time. By utilizing the associated necessary and sufficient conditions of optimality, the third and fourth impulse times are numerically determined. For practical applications, relations between the third and fourth impulse times can be well approximated as polynomial functions, which enable analytical formulas to obtain fuel-optimal four-impulse solutions. It is shown that analysis and derivations based on the HCW equations can be directly extended to the J2-perturbed fuel-optimal four-impulse rendezvous. Finally, numerical examples are given to illustrate and validate the obtained results.