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

Abstract

Primer vector theory using averaged dynamics is well suited for optimizing low-thrust, many-revolution spacecraft trajectories, 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. The new model is illustrated on a GTO to GEO transfer, including up to 486 revolutions.