Simple method for performance evaluation of multistage rockets

Multistage rockets are commonly employed to place spacecraft and satellites in their operational orbits. Performance evaluation of multistage rockets is aimed at defining the maximum payload mass at orbit injection, for specified structural, propulsive, and aerodynamic data of the launch vehicle. This work proposes a simple method for a fast performance evaluation of multistage rockets. The technique at hand is based on three steps: (i) the flight-path angle at each stage separation is guessed, (ii) the spacecraft velocity is maximized at the first and second stage separation, and (iii) for the last stage the thrust direction is obtained through the particle swarm optimization technique, in conjunction with the use of the Euler–Lagrange equations and the Pontryagin minimum principle. The coast duration at the second stage separation is optimized as well. The method at hand is extremely simple and easy-to-implement, but nevertheless it proves to be capable of yielding near-optimal ascending trajectories for a multistage launch vehicle with realistic structural, propulsive, and aerodynamic characteristics. The solutions found with the technique under consideration can be employed either for a rapid evaluation of the multistage rocket performance or as guesses for more refined optimization algorithms.