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

  • Aaron D. KochEmail author
Original Paper


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.


Multistage rockets Trajectory optimization Initial guess generation Performance evaluation 



German Aerospace Center


Geostationary transfer orbit


Monte Carlo simulation


Main engine cutoff


Not a number


Trajectory approximation method



Data were provided by the European Space Agency to DLR for the atmospheric model of Kourou (communication from W. Flury to D. Wolf, 21 Dec. 1989) and Ariane 40 (communication from J. F. Lieberherr to D. Wolf, 17 Jan. 1990). The flowcharts were designed with Microsoft Visio. All other plots were made with the Python package Matplotlib [15].


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Copyright information

© CEAS 2019

Authors and Affiliations

  1. 1.German Aerospace Center (DLR), Institute of Space SystemsBremenGermany

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