Document Type : Original Article


Mechanical Engineering; University of Guilan


In this paper, in order to simultaneously optimize the staging and trajectory of launch vehicles, changes are made in the structure of the trajectory optimization problem. In this approach, the flight times of all stages are considered freely and as optimization variables. During the solution and in each iteration, by using the values of the flight times in that iteration and the fuel consumption rate of each stage, the masses of the fuel and structure of the stages and the initial and instantaneous masses of the vehicle are calculated. By minimizing the initial mass as the objective function of the integrated optimization problem, the optimal flight trajectory is obtained in the form of the optimal state and control values, and the optimal budget of the fuel and structure masses between different stages is calculated. In this paper, to implement the dynamic equations, the direct collocation method is used, and to approximate the variables, the B-spline curves are used. By using the B-spline curves, despite the discreteness of the relevant parameters as optimization variables, a continuous concept for the optimal solution can be created. The presented approach in this paper for integrated optimization of staging and trajectory and the use of B-spline curves in the approximation of the multiphase problem with free final times can lead to reduce the initial weight of the Europa 2 launch vehicle by 30% to perform a specific mission.


Main Subjects

[1] E. Keith, The New Rocket Science, Lulu Enterprises Incorporated, 2010.
[2] G. Palaia, , M. Pallone, M. Pontani, “Ascent Trajectory Optimization and Neighboring Optimal Guidance of Multistage Launch Vehicles”, Springer Optimization and Its Applications, Vol. 144, 2019, pp. 343-371.
[3] A. Koch, “Optimal Staging of Serially Staged Rockets with Velocity Losses and Fairing Separation”, Aerospace Science and Technology, Vol. 88, 2019, pp. 65-72.
[4] B. Jo, J. Ahn, “Optimal Staging of Reusable Launch Vehicles Considering Velocity Losses”, Aerospace Science and Technology, Vol. 109, 2021.
[5] B. Jo, J. Ahn, “Optimal Staging of Reusable Launch Vehicles for Minimum Life Cycle Cost”, Aerospace Science and Technology, Vol. 127, 2022.
[6] Q. Zhang, Z. Xu, “Autonomous Ascent Guidance with Multiple Terminal Constraints for All-Solid Launch Vehicles”, Aerospace Science and Technology, Vol. 97, 2020.
[7] Y. Liu, X. Li, P. Wang, X. Zhang, “Multi-Objective, Multi-Disciplinary Design Optimization and Multi-Attribute Evaluation of Hybrid Rocket Motors Used for Manned Lunar Lander”, Aerospace, MDPI, Vol. 10, No. 3, 2023.
[8] K. Wang, B. Zhang, “Multiobjective Trajectory Optimization for A Suborbital Spaceplane Using Directed Search Domain Approach”, Aerospace Science and Technology, Vol. 77, 2018, pp. 713-724.
[9] J. Betts, “Survey of Numerical Methods for Trajectory Optimization”, Journal of the Guidance, Control and Dynamics, Vol. 21, No. 2, 1998, pp. 193-207.
[10] R. Jamilnia, “Multidisciplinary Design Optimization of Multistage Launch Vehicles”, M.Sc. Thesis, Amirkabir University of Technology, 2005 (in Persian).
[11] R. Jamilnia, “Optimal Guidance of a Reentry Vehicle Based on Online Trajectory Optimization”, Journal of Aerospace Science and Technology (JAST), Vol. 13, No. 1, 2020, pp. 98–105.
[12] R. Jamilnia, “Optimal Trajectory Design for Soft Landing on the Moon by Using Differential Flatness”, Modares Mechanical Engineering, Vol. 17, No. 10, 2018, pp. 9–19.
[13] J. Betts, Practical Methods for Optimal Control Using Nonlinear Programming, SIAM, 2010.
[14] C. De Boor, A Practical Guide to Splines, New York: Springer-Verlag, 2001.
[15] K. Höllig, J. Hörner, Approximation and Modeling with B-Splines, SIAM, 2013.
[16] A. Wächter, 2008, Introduction to IPOPT, Carnegie Mellon University, 2008.
[17] B. Harvey, 2003, Europe's Space Programme: To Ariane and Beyond, Springer, 2003.