A new numerical optimization method based on Taylor series for challenging trajectory optimization problems

New methods to solve trajectory optimization problems are devised. The methods use direct transcription to convert the continuous optimal control problem into a parameter optimization problem that can be solved with non-linear programming. In direct transcription the system equations must be converted into algebraic constraints. Existing methods use only zeroth and first order derivatives of the system equations to formulate these constraints. The techniques of automatic differentiation allow the computation of the derivatives of the state equations to arbitrary order in reasonable time. The new methods devised use these higher-order derivatives to form the constraints. To investigate the performance of the new methods they are tested on a series of progressively more challenging optimal control problems, culminating in the capstone problem. This capstone problem is a low-thrust Earth-Moon transfer that uses the interesting dynamics of the circular restricted three body problem (CR3BP), in particular the stable and unstable manifolds of a halo orbit about the interior L1 Lagrange point in the Earth-Moon system