Optimization and guidance of very low-thrust transfers to geostationary orbit

A new hybrid direct-indirect optimization algorithm is presented to compute the minimum-time transfer between two orbits, including the phasing with a desired spacecraft. Very-low thrust means several hundred revolutions to perform the large change in orbital elements. The optimal control solution of the fast-evolution problem combined with a direct method for the secular trajectory avoids the numerical instability arising in very long propagations, decreases the computational time, reduces the sensitivity to the initial guess and provides a feasible transfer at every optimization step. Optimization of transfers from GTO to GEO is presented and two types of trajectories are analysed, sub-synchronous (apogee constrained below GEO altitude) and super-synchronous (free apogee altitude). The optimization of a transfer from LEO to a very high orbit (11 ' 23 R E) is presented, showing the applicability of the method to different problems. A guidance algorithm is presented to compensate the deviations of the real trajectory from the optimal one due to off-nominal conditions. The results in closed-loop simulation of the guidance scheme to compensate detenninistic perturbations not considered in the optimization show good performances in both analysed missions.