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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 multipledomain LegendreGaussRadau 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.more » « less

A structure detection method is developed for solving statevariable inequality path con strained optimal control problems. The method obtains estimates of activation and deactiva tion times of active statevariable 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 multipledomain formulation consisting of constrained and unconstrained domains. Within each domain, LegendreGaussRadau (LGR) orthogonal direct collocation is used to transcribe the infinite dimensional optimal control problem into a finitedimensional 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 wellknown optimal control problem where the analytical solution contains a state constrained arc.more » « less

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 resolved 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.more » « less

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 resolved 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.more » « less

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 userspecified 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 pathconstrained 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.more » « less

A numerical optimization study of a minimumfuel LEOtoMEO orbital trajectory trans fer is solved using a bangbang and singular optimal control (BBSOC) method with multi domain LegendreGaussRadau quadrature collocation. Modified equinoctial elements are used to avoid singularities that occur in orbital elements. The time, t, state components, (p, f,g,h,k,L,m), and control components, (ur,ut,un,T) are optimized in this one phase prob lem where seven cases of the initial thrust acceleration values are considered. The structure of the thrust was not assumed, therefore the optimizer determined the number of switch points. The solutions were categorized as partial and multiple revolution optimal trajec tories. The initial thrust accelerations considered for the partial revolution solutions are s0 = 1.0206 × 100, 5.1029 × 10−1, 1.0206 × 10−1, 5.1029 × 10−2 AU. Furthermore, as the ini tial thrust acceleration decreased, the final mass decreased while the total time thrusting increased. The initial thrust accelerations considered for the multiple revolution solutions are s0 = 1.0206 × 10−2, 5.1029 × 10−3, 1.0206 × 10−3 AU. Furthermore, as the initial thrust acceleration decreased, the final mass increased while the total time thrusting increased. An indepth study was completed for the cases of s0 = 1.0206 × 10−1, 1.0206 × 10−3 AU, where the final mass was [0.6683, 0.5991] MU and the total time thrusting was [4.0305, 487.3276] TU.more » « less