Enforcement of Practical Path Constraints With the Indirect Method For Aerospace Applications

Abstract

Many aerospace-related optimal control problems (OCPs) are subject to various path equality and inequality constraints on states, control, or a combination of both. Enforcement of these constraints has been studied for decades, most commonly with variational calculus, from which first-order necessary conditions (NCs) of optimality have been derived. The indirect formalism of optimal control theory (i.e., the indirect method, also known as the ``optimize-then-discretize'' approach) aims to explicitly satisfy the NCs, whereas direct approaches (direct methods, also known as the ``discretize-then-optimize'' approach) circumvent these NCs by discretizing the OCP and directly optimizing the cost over parameters representing the states and control.

Direct methods allow for straightforward enforcement of path constraints and represent the industry standard, with many general-purpose solvers available commercially. Moreover, triggered path constraints, often referred to as conjunctive, disjunctive, or implicative constraints (depending on the framing) in the general optimization community, have recently garnered attention in aerospace OCPs, and direct methods have been demonstrated to accommodate a wide range of these constraints. While indirect methods are notoriously difficult in application, they offer certain invaluable benefits over direct methods, namely, guaranteed satisfaction of the NCs. Furthermore, progress has already been made in enforcing triggered control path constraints with the indirect method.

The goal of this dissertation is to build on this progress, leveraging recent advances in the indirect approach, specifically, regularization functions which accommodate nonsmooth control variables, and penalty functions which avoid directly enforcing, yet still satisfy, the notoriously burdensome NCs for state path inequality constraints. We expand the class of OCPs involving practical path constraints of various types that can be solved with the indirect method. Herein, ``practical'' describes any standard or triggered constraints directly arising from the physical limitations of a controllable system. Two applications within aerospace engineering are considered: solar electric propulsion (SEP) spacecraft trajectory optimization, involving time- and state-triggered control equality constraints, and six degree-of-freedom (6DOF) powered descent landings, involving state-triggered state inequality constraints along with standard state inequality constraints.

The major novel contributions towards the indirect method include enforcement of an SEP spacecraft thruster duty cycle constraint (i.e., forced coasting arcs), the solution of the 6DOF powered descent landing OCP itself, and enforcement of a state-triggered state path constraint within it. Several other minor contributions are also highlighted throughout. Because these OCPs are being solved for the first time with the indirect method, various existing direct methods are used to further validate the results from the indirect method.