Advancing Averaged Primer Vector Theory with Bang-Bang Control and Eclipsing

Abstract: Low-thrust, many-revolution spacecraft trajectories are increasingly required for mission design due to the efficiency and reliability of electric propulsion technology. Primer vector theory using averaged dynamics is well suited for such applications, but is difficult to implement in a way that maintains both optimality and computational efficiency. An improved model is presented that combines advances from several past works into a general and practical formulation for minimum-fuel, perturbed Keplerian dynamics. The model maintains computational efficiency of dynamics averaging with optimal handling of the eclipsing constraint and bang-bang control through the use of the Leibniz integral rule for multi-arc averaging. A subtle, but important singularity arising from the averaged eclipsing constraint is identified and fixed. A maximum number of six switching function roots per revolution is established within the averaged dynamics. This new theoretical insight provides a practical upper-bound on the number of thrusting arcs required for any low-thrust optimization problem. Variational equations are provided for fast and accurate calculation of the state transition matrix for use in targeting and optimization. The dynamics include generic two-body perturbations and an expanded state to allow for sensitivity calculations with respect to launch date and flight time. A 48-revolution GTO to GEO transfer is used to directly compare optimal averaged and unaveraged trajectories. The capabilities of averaged dynamics are then demonstrated with an optimal 486-revolution GTO to GEO minimum fuel transfer.