Uncertainty quantification for the ascent phase of launch vehicles using Bayesian inference

doi:10.23919/JSEE.2026.000063

Journal of Systems Engineering and Electronics ›› 2026, Vol. 37 ›› Issue (2): 485-503.doi: 10.23919/JSEE.2026.000063

• SYSTEMS ENGINEERING • Previous Articles

Uncertainty quantification for the ascent phase of launch vehicles using Bayesian inference

Tao CHAO¹^,²^,*(), Xiaonan LI¹^,²(), Xiaobing SHANG³(), Ping MA¹^,²(), Ming YANG¹^,²()

¹School of Astronautics, Harbin Institute of Technology, Harbin 150001, China
²National Key Laboratory of Modeling and Simulation for Complex Systems, Harbin 150001, China
³College of Intelligent Systems Science and Engineering, Harbin Engineering University, Harbin 150001, China

Received:2024-08-26 Accepted:2026-03-19 Online:2026-04-18 Published:2026-04-30
Contact: Tao CHAO E-mail:chaotao2000@163.com;hfutlxn@163.com;shangxiaobing@163.com;pingma@hit.edu.cn;myang@hit.edu.cn
About author:
CHAO Tao was born in 1983. He received his B.Sc., M.Sc., and Ph.D. degrees from Harbin Institute of Technology, Harbin, China, in 2005, 2007, and 2011, respectively. He is currently an associate professor with the Control and Simulation Center, Harbin Institute of Technology. His current research interest includes intelligent unmanned system. E-mail: chaotao2000@163.com

LI Xiaonan was born in 1996. He received his M.Sc. degree in engineering mechanics from the School of Astronautics, Harbin Institute of Technology, Harbin, China, in 2020, where he is currently pursuing his Ph.D. degree in control and science engineering. His research interests include Bayesian estimation, state estimation, and signal processing. E-mail: hfutlxn@163.com

SHANG Xiaobing was born in 1992. He received his B.Sc. degree from Northwestern Polytechnical University, China, in 2013 and M.Sc. and Ph.D. degrees from Harbin Institute of Technology, Harbin, China, in 2015 and 2020, respectively. He is currently an associate professor with Harbin Engineering University. His current research interests include reliability analysis modeling, and simulation. E-mail: shangxiaobing@163.com

MA Ping was born in 1970. She received her B.Sc. degree in computer and application engineering from Harbin Institute of Technology, Harbin, China, in 1992, and M.S. degree in guidance control and simulation from Harbin Institute of Technology, Harbin, China, in 1998. She received her Ph.D. degree from Harbin Institute of Technology, Harbin, China, in 2003.Her research interests include modeling, simulation, evaluation of intelligent systems, and reliability evaluation of complex simulation. E-mail: pingma@hit.edu.cn

YANG Ming was born in 1963. He received his Ph.D. degree from Harbin Institute of Technology, Harbin, China, in 1997. He is currently a professor with the Control and Simulation Center, Harbin Institute of Technology. His research interests include system simulation theory and unmanned systems. E-mail: myang@hit.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (62273119;62173103).

Abstract

Abstract:

The launch process of a multi-stage launch vehicle is significantly influenced by uncertain parameters, including air density, aerodynamic parameters, and engine thrust, which often exhibit deviation. Predicting the trajectory range of the launch vehicle under the influence of uncertainty is essential before launch, and uncertainty quantification serves as a crucial method to address this challenge. In traditional uncertainty quantification for launch vehicles, unknown parameters are often assigned specific distributions based on prior knowledge. However, prior knowledge is sometimes subjective, and unknown parameters are often assigned conservative ranges to meet safety margins. In addition, the flight data of the past launch is precious, especially in quantifying the uncertainty of reusable or same-type launch vehicles. This paper utilizes flight data to estimate parameters base on Bayesian methods and integrates the estimation results with prior knowledge, which can more objectively set the distribution of uncertain parameters. Reasonable distribution has a positive impact on uncertainty quantification, which can avoid control strategies that are not robust enough or overly redundant. Therefore, the uncertainty quantification for launch vehicles is discussed under different information sources. In addition, the algorithm is accelerated based on Gaussian process regression and polynomial chaos expansions.

Key words: launch vehicle, uncertainty quantification, Bayesian inference, launch experience, Gaussian process regression, polynomial chaos expansions

Tao CHAO, Xiaonan LI, Xiaobing SHANG, Ping MA, Ming YANG. Uncertainty quantification for the ascent phase of launch vehicles using Bayesian inference[J]. Journal of Systems Engineering and Electronics, 2026, 37(2): 485-503.

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Parameter	Value
Initial position $ {x}_{0} $	0
Initial position $ {y}_{0} $	0
Initial velocity $ {v}_{{{x}_{0}}} $/(m/s)	0
Initial velocity $ {v}_{{{y}_{0}}} $/(m/s)	30
Initial mass/kg	4e+04
Initial thrust/N	5.5e+05

Uncertainty parameter	Mean/%	16th−84th quantiles/%	2.5th−97.5th quantiles/%
$ \delta Pe $	1.00	[0.972, 1.031]	[0.953,1.052]
$ \delta \rho \cdot \delta {C}_{x} $	3.02	[2.60, 3.39]	[2.14, 3.90]
$ \delta \rho \cdot \delta {C}_{ya} $	−1.96	[−3.89, −0.04]	[−5.43,1.51]

Index	Uncertainty parameter	Distribution	Value
$ {\theta }_{1} $	$ \delta Pe $/%	Uniform	$ \text{U}[-2,2] $
$ {\theta }_{2} $	$ \delta \rho \cdot \delta {C}_{x} $ /%	Uniform	$ \text{U}[-20,20] $
$ {\theta }_{3} $	$ \delta \rho \cdot \delta {C}_{ya} $ /%	Uniform	$ \text{U}[-20,20] $
$ {\theta }_{4} $	Random wind $ {W}_{x} $/(m/s)	Uniform	$ \text{U}[-15,15] $
$ {\theta }_{5} $	Random wind $ {W}_{y} $/(m/s)	Uniform	$ \text{U}[-15,15] $

Parameter	Traditional method	The proposed method
$ {\sigma }_{h} $	570.65	397.23
$ {\sigma }_{v} $	25.84	14.57
$ {\sigma }_{{\mathrm{angle}}} $	0.48	0.29
$ {h}_{68\mathrm{\% }} $	933.3	485.9
$ {v}_{68\mathrm{\% }} $	21.7	11.1
$ {{\mathrm{angle}}}_{68\mathrm{\% }} $	0.04	0.022

Number	Bayesian inference	Uncertainty propagation	Total time
1	GPR-based	PCE	$ 4{\mathrm{h}}2{1}^{\prime} $
2	GPR-based	MC	$ 8{\mathrm{h}}5{4}^{\prime} $
3	PMCMC	PCE	$ 33{\mathrm{h}}5{5}^{\prime} $
4	PMCMC	MC	$ 38{\mathrm{h}}2{7}^{\prime} $

Uncertainty quantification for the ascent phase of launch vehicles using Bayesian inference

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Method	Number of samples	Total time
PCE	200	$ 1{3}^{\prime} $
MCS	70000	$ 23{\mathrm{h}}4{8}^{\prime} $

Uncertainty parameter	Distribution	Interval
$ \delta Pe $/%	Norm	$ \text{N}(0,1) $
$ \delta \rho \cdot \delta {C}_{x} $/%	Norm	$ \text{N}(0,10) $
$ \delta \rho \cdot \delta {C}_{ya} $/%	Norm	$ \text{N}(0,10) $
$ {W}_{x} $/(m/s)	Norm	$ \text{N}(0,10) $
$ {W}_{y} $/(m/s)	Norm	$ \text{N}(0,10) $

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