Control-Constrained Indirect Optimization for Low-Thrust Trajectories with Duty Cycles

A new paradigm for space exploration is expected in the next years: deep-space missions will be performed in the future by several electric-propulsion-based missions. Fuel-optimal low-thrust trajectories show very long thrusting periods due to the solution of the necessary conditions for optimality. These are not the best options when operational compliance is taken into account in mission design: coasting periods allow for orbit determination, science, telecommunications, and other space operations. This is particularly true for CubeSats, having limited thrusting, power, and pointing budgets. For this reason, electric-based spacecraft will be required to follow operational-compliant trajectories: transfers with an alternation of coasting and thrusting periods imposed at predefined time instants, called duty cycles. Traditional trajectory optimization algorithms exhibit convergence problems when handling this kind of discontinuous constraint. In this work, an efficient and robust indirect method to compute operational-compliant trajectories is formulated through the imposition of coasting arcs as control constraints and a triple continuation scheme. The algorithm is applied to over 50,000 trajectories to demonstrate its consistency and robustness and to study the effects of duty cycle imposition on low-thrust trajectories. Results show that the operational-compliant solutions can be computed in a fast and accurate way and that they are similar to the fuel-optimal ones with no operational constraint, in terms of both thrusting profile and propellant mass.