Footprint generation for reusable launch vehicles using a direct pseudospectral method

Footprint generation is a necessary element of mission design in the case of aborted flight condition. Mathematically, the problem is posed as finding the maximum cross-range for any down-range position for different heading angles. Since deriving and solving the necessary optimality conditions for the full-state equations prove to be a hard task, some simplifications have to be made. Traditional footprint generators either simplify the equations of motion using energy-state approximations or use approximate (suboptimal) control laws such as Vinh´s law. To remain faithful to the full-state physical model and use the optimal control law, we employ a new approach based on the direct Legendre pseudospectral method which avoids the pitfalls of the previous techniques. In this approach the optimal control problem is directly discretized and parameterized to a nonlinear programming problem. Both the reduced-state and full-state models for a vehicle example with tabular data are discretized using this method, and the results are compared.