Fast, robust and near-optimal approximation of GTO trajectories and payload capacities of multistage rockets

Abstract

A method for automatically generating approximate trajectories for multistage rockets, launching into a geostationary transfer orbit, is presented. It can either be used to generate an initial guess or to determine the payload capacity of a given launcher. Only the apogee is directly optimized. When the maximum payload of the launcher is used, the perigee will gravitate towards its target value. The method is applicable to configurations consisting of three stages or two stages plus boosters. The trajectory is divided into three steps, one for each stage plus one for all boosters. Five control parameters are used: The first is the constant pitch rate, used during the pitch over maneuver. The other four define the angle of attack rate functions. A declining exponential function is used for $$\dot{\alpha }_2(t)$$ , whereas $$\dot{\alpha }_3(t)$$ is chosen so that $$\alpha _3(t)$$ becomes an inverted parabola. The trajectory algorithm consists of an outer and an inner loop. The outer loop varies the pitch rate to attain the correct apogee. It calls the inner loop, which adjusts the two parameters that define $$\dot{\alpha }_2(t)$$ so that the flight path angle rate at the end of the second step becomes zero. Tests were performed for a model of Ariane 40. Its payload capacity was determined in less than $${30}\hbox { s}$$ and the result matched the one produced by a conventional approach. Moreover, a Monte Carlo simulation, based on the Ariane 40 model, was performed for both applications. The success rate was 94% for the first and 93% for the second case.