DDP-Based Stochastic Trajectory Optimization With Safe Mode Uncertainty Modeled by an Alternative Renewal Process

Safe mode protects spacecraft from faults but in deep-space missions it can cause unplanned propulsion shutdowns and severe trajectory deviations. Conventional robust trajectory design relies on overly conservative margins, leading to inefficient fuel use. This study proposes a stochastic trajectory optimization framework that models safe mode activation probabilistically using Weibull-distributed intervals. Scenario-based formulations are integrated with Differential Dynamic Programming, whose internal computations are extended to account for discrete uncertainties. A numerical example with a double-integrator system shows that optimal margins emerge automatically without tuning.