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An adaptive mesh refinement method for numerically solving optimal control problems is developed using Legendre-Gauss-Radau direct collocation. In regions of the solution where the desired accuracy tolerance has not been met, the mesh is refined by either increasing the degree of the approximating polynomial in a mesh interval or dividing a mesh interval into subintervals. In regions of the solution where the desired accuracy tolerance has been met, the mesh size may be reduced by either merging adjacent mesh intervals or decreasing the degree of the approximating polynomial in a mesh interval. Coupled with the mesh refinement method described in this paper is a newly developed relative error estimate that is based on the differences between solutions obtained from the collocation method and those obtained by solving initial-value and terminal-value problems in each mesh interval using an interpolated control obtained from the collocation method. Because the error estimate is based on explicit simulation, the solution obtained via collocation is in close agreement with the solution obtained via explicit simulation using the control on the final mesh, which ensures that the control is an accurate approximation of the true optimal control. The method is demonstrated on three examples from the open literature, and the results obtained show an improvement in final mesh size when compared against previously developed mesh refinement methods. 

The trajectory optimization of a reusable launch vehicle entry is studied. The objective is to maximize the crossrange during atmospheric entry subject to a constraint on the stagnation point heating rate. The problem is solved by partitioning the domain of the independent variable into multiple subdomains such that each subdomain consists of a segment where the heating rate constraint is either active or inactive. Additional necessary conditions for optimality are enforced in segments where the heating rate constraint is active. A multiple-domain Legendre-Gauss-Radau direct collocation method is then used to solve the partitioned problem. Key features of the stagnation point heating rate constraint are presented, and it is observed that the aforementioned approach is capable of solving the problem under consideration more accurately than traditional direct collocation methods. 

A Monte Carlo analysis of a contingency optimal guidance strategy is conducted. The guidance strategy is applied to a Mars Entry problem in which it is assumed that the surface level atmospheric density is a random variable. First, a nominal guidance strategy is employed such that the optimal control problem is re-solved at constant guidance cycles. When the trajectory lies within a particular distance from a path constraint boundary, the nominal guidance strategy is replaced with a contingency guidance strategy, where the contingency guidance strategy attempts to prevent a violation in the the relevant path constraint. The contingency guidance strategy utilizes the reference optimal control problem formulation, but modifies the objective functional to maximize the margin between the path constraint limit and path constraint function value. The ability of the contingency guidance strat- egy to prevent violations in the path constraints is assessed via a Monte Carlo simulation. 

A structure detection method is developed for solving state-variable inequality path con- strained optimal control problems. The method obtains estimates of activation and deactiva- tion times of active state-variable inequality path constraints (SVICs), and subsequently al- lows for the times to be included as decision variables in the optimization process. Once the identification step is completed, the method partitions the problem into a multiple-domain formulation consisting of constrained and unconstrained domains. Within each domain, Legendre-Gauss-Radau (LGR) orthogonal direct collocation is used to transcribe the infinite- dimensional optimal control problem into a finite-dimensional nonlinear programming (NLP) problem. Within constrained domains, the corresponding time derivative of the active SVICs that are explicit in the control are enforced as equality path constraints, and at the beginning of the constrained domains, the necessary tangency conditions are enforced. The accuracy of the proposed method is demonstrated on a well-known optimal control problem where the analytical solution contains a state constrained arc. 

A Monte Carlo analysis of a contingency optimal guidance strategy is conducted. The guidance strategy is applied to a Mars Entry problem in which it is assumed that the surface level atmospheric density is a random variable. First, a nominal guidance strategy is employed such that the optimal control problem is re-solved at constant guidance cycles. When the trajectory lies within a particular distance from a path constraint boundary, the nominal guidance strategy is replaced with a contingency guidance strategy, where the contingency guidance strategy attempts to prevent a violation in the the relevant path constraint. The contingency guidance strategy utilizes the reference optimal control problem formulation, but modifies the objective functional to maximize the margin between the path constraint limit and path constraint function value. The ability of the contingency guidance strategy to prevent violations in the path constraints is assessed via a Monte Carlo simulation. 

A robust optimal guidance strategy is proposed. The guidance strategy is designed to reduce the possibility of violations in inequality path constraints in the presence of modeling errors and perturbations. The guidance strategy solves a constrained nonlinear optimal control problem at the start of every guidance cycle. In order to reduce the possibility of path constraint violations, the objective functional for the optimal control problem is modified at the start of a guidance cycle if it is found that the solution lies within a user-specified threshold of a path constraint limit. The modified objective functional is designed such that it maximizes the margin in the solution relative to the path constraint limit that could potentially be violated in the future. The method is validated on a path-constrained Mars entry problem where the reference model and the perturbed model differ in their atmospheric density. It is found for the example studied that the approach significantly improves the path constraint margin and maintains feasibility relative to a guidance approach that maintains the original objective functional for each guidance update.