Accurate modeling and heuristic trajectory optimization of multistage launch vehicles

Multistage launch vehicles are commonly employed to place spacecraft in their operational orbits. Several characteristics, i.e. mass distribution and time variation, propulsion, and aerodynamics, affect the overall performance of the ascent vehicle of interest. Thus, it is apparent that accurate modeling is a central issue and an essential prerequisite for trajectory optimization. This research uses the Scout, a launch vehicle of reduced size used in the past, as the reference multistage rocket. Its ascending trajectory is assumed to be composed of five arcs: (i) first stage propulsion, (ii) second stage propulsion, (iii) third stage propulsion, (iv) coast arc (after release of the third stage), and (v) fourth stage propulsion. The Euler-Lagrange equations and the Pontryagin minimum principle, in conjunction with the Weierstrass-Erdmann corner conditions, are employed to express the thrust direction as a function of the adjoint variables conjugate to the dynamics equations. The use of these analytical conditions coming from the calculus of variations leads to obtaining the overall rocket dynamics as a function of a few parameters only, mainly represented by the unknown values of the initial costate components. Then, a heuristic algorithm, e.g. particle swarm, is used to determine the optimal parameter set. The path constraint related to the maximum dynamical pressure is taken into account. The numerical results unequivocally prove that the methodology at hand is rather effective and accurate, and definitely allows evaluating the performance attainable from multistage launch vehicles with accurate aerodynamics and propulsive modeling.