Indirect Optimization of Optimal Continuous Trajectory for Electric Sails with Refined Thrust Model in the Multi-target Exploration Scenario

Electric sail-based propulsion is an innovative propellant-less propulsion technology that generates continuous thrust through the interaction between an artificial electric field and the solar wind. In an electric sail, the propulsive acceleration is adjusted by controlling the attitude of its normal plane and the coefficient determining the maximum thrust. However, the attitude adjustment speed of the electric sail is relatively small and the direction of the normal vector is constrained. Consequently, the electric sail is required to maintain a continuous propulsive acceleration vector when flying by the intermediate targets in multi-target interplanetary exploration. Therefore, an indirect optimization of three-dimensional optimal continuous interplanetary trajectory for electric sails with refined thrust model is investigated in this study. First, the optimal propulsive angles and thrust adjustment coefficient of electric sails with a refined thrust model are derived using Pontryagin's minimum principle. Second, a homotopy function is introduced in the process of trajectory optimization with an indirect method to approximate the step of the thrust adjustment coefficient to improve the accuracy of numerical integration. Additionally, the initial costates of the electric sail are transformed from the numerical simulation results of the Bezier shaping approach (BSA) and integrative Bezier shaping approach (IBSA) using the Karush-Kuhn-Tucker (KKT) condition. The numerical simulation results for the Earth-Mars rendezvous mission and the multi-target flyby mission reveal that the initial values of the costates are effective to implement an indirect optimization process of the optimal continuous trajectory for the rapid convergence of the electric sail. According to the numerical simulation results for multi-target mission, the indirect method is on average 2.5% better than the Gauss pseudospectral method (GPM) in overall index, and the calculation time is only 1.06% of GPM.