Exo-atmospheric launch trajectory definition using direct optimization and slack variables

This article introduces an optimization problem to define an exo-atmospheric launch ascent trajectory. The thrust orientation is discretized converting the optimal two-point boundary value problem into a direct optimization method. The proposed criterion complexifies the classical minimization of fuel consumption by adding three terms. The first is an input smoothing term to limit drastic changes between two consecutive steps. The second added term, coming from an analytical solution of a simplified formulation of the two-point boundary value problem, helps the convergence of the optimization. The third term, the slack variables, relaxes the optimization problem by implementing soft constraints. The equations of motion for the prediction of the final orbit are the pseudo-five degrees of freedom equations. This model results from the addition of the closed control loop dynamics to the equations of translation. The defined optimization problem is solved using a classic optimization algorithm and proved to be accurate and reliable even for a nonvalidated launcher configuration.