Optimal Multiple-Burn Injection of a Satellite into Geostationary orbit Using an Upper Stage

In this paper the problem of optimal multiple-burn injection of a satellite into geostationary orbit using an upper stage with a limited thrust and restart capability, and comparison with sub-optimal case is considered. The goal is finding thrust vector angle, times of the engine firings and optimal duration of active phases of the upper stage so as to minimize fuel consumption and to meet desired boundary conditions. The contribution of this research is developing an accurate and rapid convergence algorithm for solving multiple-burn trajectory for satellite injection into geostationary orbit. To solve the multipoint boundary value problem, an improved indirect shooting method with high performance and modified Newton’s method is presented and used for optimal solution. Moreover, the novel method presented for multi burn problem, not only has very good accuracy, but also, it converges very fast to the desired end conditions. Various flight sequences with multiple burns are considered and the optimal trajectory with minimum fuel consumption criteria, for each flight sequence is derived. The verification and validation of the proposed algorithm is made via comparison with references. Finally, the results of optimal solutions are compared with the results of sub-optimal solution which its thrust direction is aligned to the velocity vector direction.