Optimal trajectory design under uncertainty

Abstract

Reference trajectory design for atmospheric reentry vehicles can be accomplished through trajectory optimization using optimal control techniques. However, this method generally focuses on nominal vehicle performance and does not include robustness considerations during trajectory design. This thesis explores the use of linear covariance analysis to directly include trajectory robustness in the design process. The covariance matrix can be propagated along a trajectory to provide the expected errors about the nominal trajectory in the presence of uncertainties. During the optimization process, the covariance matrix is used as a performance metric to be minimized, directly penalizing expected errors so that the trajectory is shaped to reduce its sensitivity to uncertainties. This technique can penalize the open-loop covariance of the trajectory or the closed-loop covariance with the inclusion of a feedback guidance law. This covariance shaping technique is applied to reference trajectory design for a generic small reentry vehicle. A baseline trajectory is generated without any robustness considerations, along with an open-loop covariance shaped trajectory and a closed-loop covariance shaped trajectory, which uses a feedback guidance law based on a linear quadratic regulator scheme. Uncertainties in initial conditions, atmospheric density, aerodynamic coefficients, and unmodeled dynamics are applied to each trajectory and performance is analyzed using linear covariance analysis and Monte Carlo simulations. The results show that when the vehicle is flown closed-loop with feedback, shaping using the open-loop covariance produces a trajectory that is less robust than the baseline trajectory, while shaping using the closed-loop covariance generates a trajectory with reduced sensitivity to uncertainty for more robust performance.