Robust trajectory optimization of constrained re-entry flight via stochastic collocation based ensemble pseudospectral optimal control

Abstract

A new computational framework for constrained robust trajectory optimization problems in hypersonic flight and re-entry has been developed, called Stochastic Collocation based Ensemble Pseudospectral Optimal Control (SC-EPOC). Uncertainty space has been simplified by utilizing Sparse Grid methods, one of which is the Conjugate Unscented Transformation. Then, the uncertainty-aware optimal control problem (OCP) is rewritten as a computationally tractable deterministic OCP by utilizing Ensemble Optimal Control. The resulting problem is solved via Pseudospectral Optimal Control Methodology and Nonlinear Programming. A tailored Pseudospectral Optimal Control Software has been developed, and validated against benchmark cases taken from literature with greater results, thanks to the in-house developed hyper-dual differentiation library and sparse calculation of Jacobian and Hessian matrices of the resulting optimization problem for Nonlinear Programming. A mesh-refinement algorithm has been developed for singular OCPs where wild oscillations occur over the boundary arc, which is unacceptable for safety-critical mission design. The algorithm is tested and validated against theoretically derived optimality conditions on a simple thrust programming landing rocket problem. To make SC-EPOC computationally tractable, the software has been upgraded by leveraging vectorization and parallelization technologies for sparse calculation of 3D Jacobian and Hessian matrices. As a result, the software has been shown convergent under high uncertainties with more than a million of optimization variables and constraints. Furthermore, SC-EPOC is applied to a strictly constrained re-entry problem where, a re-entry vehicle is being commanded to steer from an Entry Interface with uncertainties towards a TAEM area under angle-of-attack limits, dynamic load and heat flux constraints while maximizing the cross-range. Problem has been solved for different combinations of uncertainties including model uncertainties and state uncertainties and the results, including the variations of optimized states, control, costates derived by utilizing Covector Mapping Theorem, their boundary values and Hamiltonian have been given. Their optimality conditions have been derived by utilizing ensemble optimal control theory. Computational results showed excellent agreement between theory. In the end, future research direction has been depicted, and a theoretical investigation has been conducted for what is called as the Integrated Ensemble Pseudospectral Guidance and Recovery Control (IEPG&RC) to optimize the robust trajectories by incorporating the control term, that makes it possible to incorporate higher uncertainties and steer the initial distribution towards the same endpoint.