Low-thrust trajectory optimization based on epoch eccentric longitude formulation

The mathematics of trajectory optimization based on the use of nonsingular orbit elements involving the eccentric longitude at epoch as the sixth elem ent are fully derived. The epoch eccentric longitude and epoch mean longitu de are fundamental nonsingular orbit elements, which stay constant in the a bsence of perturbations. These formulations constitute the basis from which the current time formulations are derived and, therefore, are important fr om a theoretical point of view. They are also beneficial in minimum-fuel pr oblems during the natural coasting parts of the trajectory, where the adjoi nt equations need not be propagated by either analytical or numerical integ ration. The state and adjoint differential equations are explicit functions of time in this formulation that involves natural orbit elements with the optimal Hamiltonian, also varying in time. The mathematics of this epoch fo rmulation provides added insight into the problem of trajectory optimizatio n by relating the various assumptions used in generating the differential e quations for the adjoint variables that correspond to various sets of orbit al elements. Furthermore, the function that defines the transversality cond ition at the end time in minimum-time problems is shown to remain constant during the optimal transfer providing a further check in accepting a conver ged trajectory as truly optimal. This formulation is also related to the on e that uses the current eccentric longitude as the sixth state variable, an d the mathematical relationship between the Hamiltonian and the Lagrange mu ltipliers of these two formulations is also shown. A pair of continuous con stant acceleration minimum time transfer examples are duplicated using this new formulation to validate the mathematical derivations.