Minimum-Time Low-Thrust Rendezvous and Transfer Using Epoch Mean Longitude Formulation

Themathematicsofminimum-timerendezvous and transfer usinga set ofnonsingularorbital elements involving the mean longitude at epoch is presented and shown to represent a basis from which the current time formulations are derived. The state and adjoint differential equations are explicit functions of time in this formulation that involves natural orbital elements that stay constant if no perturbation is applied. The optimal Hamiltonian also is varying in time; however, the function that deŽ nes the transversality condition at the end time in minimum-time problems is shown to remain constant during the optimal transfer, effectively replacing the numerical check for the constantHamiltonian in accepting a converged trajectory as being truly optimal.Themappingmatrices for the Lagrangemultipliers of this as well as all of the other formulations previously developed also are shown explicitly. A numerical example is presented that uses the shooting method coupled with a quasi-Newtonian scheme to solve the two-point boundary-valueproblem and generate the numerically integrated trajectory.