In this paper, a robust optimization method is developed to solve the Satellite Launch Vehicle (SLV) trajectory design problem in the presence of uncertainties using a powerful Particle Swarm Optimization (PSO) algorithm. Given the uncertainties such as uncertainties in the actual values ​​of aerodynamic coefficients, engine thrust, and mass in the ascent phase of a SLV, it is important to achieve an optimal trajectory that is robust to these uncertainties; because it improves the flight performance, reduces the workload of the guidance-control system, and increases the reliability of the satellite. For this purpose, first the optimization problem is considered by using the criterion of minimizing the flight time of the SLV as a cost function, and three-dimensional equations of motion as constraints governing the problem. Then, by adding the mean parameters and the standard deviation of uncertainties in the cost function, a robust optimizer model is developed and the algorithm is used to numerically optimize the model. Monte Carlo's perspective has also been used to analyze the results of uncertainties and their continuous feedback to the optimization model. Finally, the optimal trajectory is obtained that is robust to the uncertainties. The resulting simulation results show the accuracy of this claim.


1. Wazed, M. A., Ahmed, Sh., and Yusoff, N.,“Uncertainty Factors in Real Manufacturing Environment”, Australian Journal of Basic and Applied Sciences, Vol. 2, No. 3, pp. 342- 351, 2009.
2. Yao, W., Chen X., Luo, W., Tooren, M., and Guo J., “Review of Uncertainty-Based Multidisciplinary Design Optimization Methods for Aerospace Vehicles”, Journal of Progress in Aerospace Sciences, Vol. 47, No. 6, pp. 450-479, 2011.
3. Weck, O., Eckert C., and Clarkson, J., “A Classification of Uncertainty for Early Product and System Design”, International Conference on Engineering Design (ICED), pp. 159-160, 2007.
4. Koch, P. N., Wujek, B., and Golovidov, O., “A Multi-Stage, Parallel Implementation of Probabilistic Design Optimization in an MDO Framework”, 8th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, CA.,, 2000.
5. Marvis, D. N., and Delaurentis, D. A., “Uncertainty Modeling and Management in Multidisciplinary Analysis and Synthesis”, 38th Aerospace sciences meeting and exhibit, AIAA Journal,, 2000.
6. Sues, R., and Cesare, M., “An Innovative Framework for Reliability-Base MDO”, 41st Structures, Structural Dynamics, and Materials Conference and Exhibit,, 2000.
7. Akhtar, A., and Linshu, H., “An Efficient Evolutionary Multi-Objective Approach for Robust Design of Multi-Stage Space Launch Vehicle”, 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Portsmouth, Virginia,, 2006.
8. Zaman, K., McDonald, M., Mahadevan S., and Green L., “Robustness-Based Design Optimization Under Data Uncertainty”, Struct Multidisc Optim journal,, January 2011.
9. Bataleblu, A. A., Roshanian, J., and Ebrahimi, M., “Robust Design Optimization of a Launch Vehicle with Liquid Fuel”, MS.c Thesis, Tehran, 2011 (In Persian).
10. Liu, X., and Lu, P., “Robust Trajectory Optimization for Highly Constrained Rendezvous and Proximity Operations”, AIAA Guidance, Navigation, and Control (GNC) Conference, 2013.
11. Ricardo M. P., “Robust and Reliability-Based Design Optimization Framework for Wing Design”, AIAA Journal, Vol. 52, No. 4, 2014.
12. Okada M., “Robust Trajectory Design for Object Throwing based on Sensitivity for Model Uncertainties”, IEEE International Conference on Robotics and Automation (ICRA), DOI: 10.1109/ICRA.2015.7139623, 2015.
13. Su, Z., and Wang, H., “A Novel Robust Hybrid Gravitational Search Algorithm for Reusable Launch Vehicle Approach and Landing Trajectory Optimization”, Elsevier Neurocomputing, Vol. 162, No. 25, pp. 116-127, 2015.
14. Luo, Y., and Yang, Z., “A Review of Uncertainty Propagation in Orbital Mechanics”, Elsevier- J. Progress in Aerospace Sciences, Vol. 89 , pp. 23-39, 2017.
15. Michael J, G. and Michael J, B., Rapid, Robust Trajectory Design Using Indirect Optimization Methods, Elsevier 2016.
16. Xue, Q., and Haibin Duan, H., “Robust Attitude Control for Reusable Launch Vehicles Based on Fractional Calculus and Pigeon-inspired Optimization”, Ieee/Caa Journal of Automatica Sinica, Vol. 4, No. 1, 2017.
17. Bataleblu, A., and Roshanian, J.,“Robust trajectory optimization of space launch vehicle using computational intelligence”, IEEE Congress on Evolutionary Computation (CEC), Sendai, 2015,
18. Jiaye, Ch., Rongjun, Mu, Zhang, Xin, Zh.,Yanpeng, and D., “Reusable Launch Vehicle Model Uncertainties Impact Analysis”, In Young Scientists Forum 2017, Vol. 10710, pp. 393-400, 2018.
19. Roshanian, J., Bataleblu, A., and Ebrahimi, M., “Robust Ascent Trajectory Design and Optimization of a Typical Launch Vehicle”, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Vol. 232, No. 24, pp. 4601-4614, 2018.
20. Zipfel, P. H., Modeling and Simulation of Aerospace Vehicle Dynamics, Third edition, American Institute of Aeronautics and Astronautics, AIAA education series, 2007.
21. Vinh, N., Optimal Trajectories in Atmospheric Flight, Elsevier, New York, 1981.
22. Shaver, A., and Hull, D. G., “Advanced Launch System Trajectory Optimization Using Suboptimal Control”, AIAA GNC Conference, 1990.
23. Rostami, R. H., and Toloei, A., “Mid-course Trajectory Design of a Ground-to-Air Missile using GA and PSO”, M.Sc Thesis, Tehran, 2015 (In Persian).
24. Hosseini, S. M., Nosratolahi, M., Toloei, A., “Multi- Disciplinary Optimization Design of a Launch Vehicle”, M.Sc Thesis, Tehran, 2015 (In Persian).
25. Mohan. N S. “Robust design”, Ph.D. Thesis, Indian Institute of Technology, 2002.

تحت نظارت وف ایرانی