Heuristic Optimization Algorithms for Initializing Indirect Minimum-Fuel Trajectory Optimization

Indirect optimal control methods rely on the solution of two-point boundary-value problems (TPBVP) and reduce unknowns to a small set of co-state variables which are known to exhibit extreme sensitivity. In this paper, heuristic optimization algorithms are exploited, together with recently proposed homotopy continuation techniques, to produce a quality guess for the unknown co-state variables in low-thrust minimum-fuel trajectory optimization problems. A heuristic optimizer performs a search of the solution space by attempting to minimize an auxiliary cost functional formulated as the weighted sum of squares of the TPBVP final boundary condition residuals and the optimal control cost function. Once a guess for the co-state variables is determined, an implementation of single shooting with continuation is used to converge to the fuel-optimal solution. The proposed methodology is applied to determine a low-thrust trajectory from a geostationary transfer orbit (GTO) to a L1 halo orbit in the Earth-Moon system and case studies are performed to draw conclusions which point towards the best choice of heuristic optimization algorithm, homotopy continuation method, and auxiliary cost. The proposed approach is of simple implementation and is applicable to a range of trajectory optimization problems, which is an advantageous feature for the preliminary phase of mission design.