Optimization and sensitvity analysis for a launch trajectory

Abstract

Using modern algorithms, an ideal launch vehicle trajectory can be calculated based on the principles of optimal control theory. Conventional approaches, such as shooting, seek to find the solution to a Hamiltonian boundary value problem. Finding solutions to a boundary value problem can be time consuming and difficult due to the twin curses of sensitivity and dimensionality. In an effort to alleviate these problems, pseduospectral optimal control theory can be used to reduce the time and effort required to design optimal launch trajectories. Problem formulation is shown to be a key step in this process. To illustrate the idea, a launch vehicle trajectory optimization problem is solved for maximizing the final velocity of the first stage of a multi-stage rocket assuming that all fuel will be expended. The sensitivity of the solution to uncertainties is examined by modeling environmental uncertainties as Gaussian processes in a Monte Carlo simulation. Combining optimal control and Monte Carlo analysis improves the planning process by allowing for worst case scenarios to be identified and mitigated.