Trajectory Optimization Under Uncertainty with Nonlinear Programming and Forward–Backward Shooting

Uncertainties are an unavoidable concern for all real-world operations in space. When dealing with uncertainties, it is preferable to have mathematical guarantees on the effects of these uncertainties rather than heuristic approaches that have traditionally been used. This paper derives a linear error feedback policy whose feedback gains can be optimized simultaneously with the nominal trajectory, which allows explicit constraint on the final state distribution; these gains significantly reduce the number of design variables associated with each feedback matrix compared to existing methods. This paper develops a nonlinear-programming-based framework for uncertainty-aware trajectory optimization with forward–backward shooting, a technique that has been used in deterministic trajectory optimization to improve convergence properties. Nonlinear programming allows any numerically tractable objectives and constraints at the cost of computational complexity. Additionally, this paper formulates a framework for nonlinear path chance constraints and applies it to collision avoidance. This paper also develops a framework to avoid numerical issues due to singularities in numerical integration; this approach is beneficial to deterministic and stochastic trajectory optimization. A stochastic mass leak is also derived that is theoretically and numerically necessary to avoid singularities in the sensitivities of the objectives and constraints. This framework is applied to transfers in the circular restricted three-body problem, which provide additional insight into the continuous-time stochastic fuel problem in nonlinear trajectory design.