Homotopic approach and pseudospectral method applied jointly to low thrust trajectory optimization

Abstract

The homotopic approach and the pseudospectral method are two popular techniques for low thrust trajectory optimization. A hybrid scheme is proposed in this paper by combining the above two together to cope with various difficulties encountered when they are applied separately. Explicitly, a smooth energy-optimal problem is first discretized by the pseudospectral method, leading to a nonlinear programming problem (NLP). Costates, especially their initial values, are then estimated from Karush–Kuhn–Tucker (KKT) multipliers of this NLP. Based upon these estimated initial costates, homotopic procedures are initiated efficiently and the desirable non-smooth fuel-optimal results are finally obtained by continuing the smooth energy-optimal results through a homotopic algorithm. Two main difficulties, one due to absence of reasonable initial costates when the homotopic procedures are being initiated and the other due to discontinuous bang–bang controls when the pseudospectral method is applied to the fuel-optimal problem, are both resolved successfully. Numerical results of two scenarios are presented in the end, demonstrating feasibility and well performance of this hybrid technique.