Stochastic Primer Vector for Robust Low-Thrust Trajectory Design Under Uncertainty

Any space trajectories are subject to state uncertainty due to imperfect state knowledge, random disturbances, and partially known dynamical environments. Ideally, such uncertainty and associated risks must be properly quantified and taken into account in the process of trajectory design, ensuring a sufficiently low risk of causing hazardous events. To bridge the gap between the ideal goal and current practice in mission design, this paper extends Lawden's primer vector theory and develops a solution method to solve the problem of low-thrust trajectory optimization under state uncertainty. The new primer vector, termed stochastic primer vector, provides an analytical open-loop optimal control law that respects a probabilistic path constraint with a user-specified confidence level (chance constraint). A numerical aspect of the indirect method is also extended by introducing a smoothing approach across constraint corner discontinuities, enabling efficient solution methods for constrained optimal control problems. The validity and effectiveness of the theoretical development are demonstrated with two numerical examples, which clarify the behavior of chance-constrained optimal low-thrust trajectories and confirm through Monte Carlo simulations that the designed trajectories indeed satisfy the imposed constraints under uncertainty with the prescribed confidence level.