Optimizing Under Uncertainty: Robust Trajectory Design for Uranus

Abstract

Spacecraft trajectories represent the path that a satellite takes from one point to another. This path depends on the state and the control, and optimization allows us to find the best choice of trajectory by selecting the appropriate controls (e.g., the thrust vector with respect to time). Considering that every kilogram of propellant saved through optimization provides an additional kilogram of payload at the final destination, or reduces launch mass and cost significantly, we are often motivated to find trajectories that use the least amount of propellant possible. However, uncertainty in navigational knowledge and in the parameters of a system, whether dynamic or due to the spacecraft, are not typically accounted for in these deterministic optimal control solutions. Therefore, a framework that accounts for these uncertainties in order to define a robust optimal trajectory would need to (a) satisfy necessary and sufficient conditions for optimality, (b) optimize for fuel consumption with numerical efficiency, (c) propagate uncertainty to account for navigational uncertainties and incorporate probabilistic constraints, and (d) balance the robustness and deterministic optimality of the solution for different use cases.

This thesis proposes using a robust hybrid primer vector approach that combines direct methods, the indirect primer vector, and uncertainty propagation to find solutions to the robust optimal trajectory problem. The direct optimization finds an optimal trajectory to minimize fuel consumption while constrained by the desired end state. To propagate initial navigational uncertainty, an unscented transform is used. The likelihood of collision with a ring, a moon, or the planet itself is then determined by evaluating the propagated uncertainty and applying a probabilistic constraint. Finally, the optimality of the solution is checked by propagating the primer vector between specified impulses. As a motivating example, the main concepts of the proposed algorithm are demonstrated on an orbital maneuver between two elliptical orbits around Uranus with a probability constraint to prevent collision with the epsilon ring. This work will be extended and applied to specific trajectories in a Uranus moon tour mission.