Optimal low-thrust earth orbit transfers with eclipses using indirect heuristic approaches

Low-thrust propulsion allows substantial propellant saving if compared to high-thrust systems. However, multi-revolution orbit transfers are affected by unavailability of electrical power along eclipse arcs. This research addresses the identification of minimum-time low-thrust Earth orbit transfers with spacecraft eclipsing. Two different indirect heuristic approaches, based on the use of the analytical conditions for optimality, in conjunction with stochastic fractal search, are applied: (a) regularization through the hyperbolic tangent function and (b) multi-arc formulation of the problem. Both of them require no averaging, and the respective advantages and shortcomings are identified. Approach (b) leads to an extended set of multipoint conditions for optimality, which are shown to be solvable sequentially in the numerical solution process. Implicit costate transformation is proven to be a key ingredient for the purpose of obtaining closed-form solutions of the jump relations that hold for the adjoint variables at light/eclipse transitions. Some illustrative examples prove effectiveness of the two indirect heuristic approaches (a) and (b) in finding minimum-time low-thrust orbit transfers between either two coplanar or two noncoplanar Earth orbits. Approach (b) proves to be superior in terms of both computational efficiency and accuracy of the numerical results.