| Improved collocation methods for the solution of a set of ordinary differential equations are developed and used to solve minimum-time, Earth-moon orbit transfers. These collocation methods are based on a family of modified Gaussian quadrature rules. This family of rules includes the basic trapezoid and Simpson quadrature rules. The collocation schemes derived from the trapezoid and Simpson rules are in use today for solving a variety of optimal control problems. These methods are used in direct optimization methods where the original optimal control problem formulation is converted to a nonlinear programming problem. This conversion is performed by first discretizing the solution time history using parameters to represent discrete values of the problem variables. These parameters are then related through constraints that represent an integration of the governing differential equations between discrete times called nodes. A method for automating the process of distributing nodes is developed and shown to be more efficient in using a given number of nodes than the usual practice of evenly distributing the nodes. However, even with a more efficient nodal distribution, these basic collocation schemes are found to be inadequate for solving many low-thrust orbit transfer problems. Therefore, new collocation schemes based on higher-order quadrature rules are derived and applied to a low-thrust orbit transfer problem, showing the advantages of using the new methods. These methods are considerably more complex than those based on the trapezoid or Simpson rules, and care must be taken to avoid developing deficient methods. A new collocation scheme based on a 12th-order quadrature rule is used to solve low-thrust, three-dimensional, Earth-moon orbit transfer problems that include third-body gravity as a perturbation. |
