| The minimum time orbital transfer problem for spacecraft with steerable, continuous thrust of constant magnitude may be solved using Euler-Lagrange theory, which leads to the optimal control law in terms of Lagrange multipliers. However, the initial values of the Lagrange multipliers are not known from the orbital boundary conditions. Using analytical and empirical results, the optimal initial costates are modeled as functions of the problem parameters which are the initial thrust acceleration, A, and the final orbit radius, R, in canonical units. For circle-to-circle, coplanar orbit transfers, these approximate initial costate models lead to convergence in the shooting method for all practical values of A and R. The models also lead to convergence for a wide range of other problems, including circle-to-hyperbola transfers and non-coplanar transfers. To counter the extreme sensitivity to small changes in the initial costate conditions, a dynamic step limiter is introduced which improves the convergence properties. |
