Considering the parametric uncertainties during the landing process of the vertical takeoff and vertical landing launch vehicle, an uncertainty optimization method of sequential optimization and reliability assessment based on nonintrusive polynomial chaos expansion for return trajectory is proposed. First, an optimization model of the return multi-flight-phase trajectory under the nominal conditions is established. Then, considering both robustness and reliability of the trajectory, an optimization model for the return trajectory under uncertainties is established, which consists of the robust optimization objective function, reliability-based path constraints and robust equality constraints. Finally, the robust optimization objective function and robust equality constraints are quantified based on the nonintrusive polynomial chaos expansion method, by which the original stochastic robust optimization problem is transformed into the equivalent deterministic optimization problem in the high dimensional state space. Meanwhile, to improve the evaluation efficiency of path constraints, the most probable point method is improved by using nonintrusive polynomial chaos expansion, and the sequence optimization and reliability assessment strategy are further developed. Numerical results show that the proposed method has good robustness and can meet the requirements of reliability in engineering, and also has high accuracy and efficiency.
[1] ZHANG L, WEI C Z, WU R, et al. Fixed-time extended state observer based non-singular fast terminal sliding mode control for a VTVL reusable launch vehicle[J]. Aerospace Science and Technology, 2018, 82-83:70-79.
[2] ZHANG L, WEI C Z, WU R, et al. Adaptive fault-tolerant control for a VTVL reusable launch vehicle[J]. Acta Astronautica, 2019, 159:362-370.
[3] LIU X F. Fuel-optimal rocket landing with aerodynamic controls[J]. Journal of Guidance, Control, and Dynamics, 2018, 42(1):65-77.
[4] RAGAB M, CHEATWOOD F M. Launch vehicle recovery and reuse[C]//AIAA SPACE 2015 Conference and Exposition. Reston:AIAA, 2015.
[5] OUNG R, D'ANDREA R. The distributed flight array:Design, implementation, and analysis of a modular vertical take-off and landing vehicle[J]. The International Journal of Robotics Research, 2014, 33(3):375-400.
[6] ZHAO J, LI H Y, HE X Y, et al. Uncertainty analysis for return trajectory of vertical takeoff and vertical landing reusable launch vehicle[J]. Mathematical Problems in Engineering, 2020, 2020:1-18.
[7] WANG J B, CUI N G, WEI C Z. Optimal rocket landing guidance using convex optimization and model predictive control[J]. Journal of Guidance, Control, and Dynamics, 2019, 42(5):1078-1092.
[8] SIMPLÍCIO P, MARCOS A, BENNANI S. Reusable launchers:development of a coupled flight mechanics, guidance, and control benchmark[J]. Journal of Spacecraft and Rockets, 2019, 57(1):74-89.
[9] LI S, JIANG X Q. Review and prospect of guidance and control for Mars atmospheric entry[J]. Progress in Aerospace Sciences, 2014, 69:40-57.
[10] MA L, WANG K X, SHAO Z J, et al. Direct trajectory optimization framework for vertical takeoff and vertical landing reusable rockets:case study of two-stage rockets[J]. Engineering Optimization, 2019, 51(4):627-645.
[11] MA L, WANG K X, XU Z H, et al. Trajectory optimization for powered descent and landing of reusable rockets with restartable engines[C]//69th International Astronautical Congress, 2018.
[12] WANG C, SONG Z Y. Trajectory optimization for reusable rocket landing[C]//2018 IEEE CSAA Guidance, Navigation and Control Conference (CGNCC). Piscataway:IEEE Press, 2018:1-6.
[13] 高朝辉, 张普卓, 刘宇, 等. 垂直返回重复使用运载火箭技术分析[J]. 宇航学报, 2016, 37(2):145-152. GAO Z H, ZHANG P Z, LIU Y, et al. Analysis of vertical landing technique in reusable launch vehicle[J]. Journal of Astronautics, 2016, 37(2):145-152(in Chinese).
[14] YAO W, CHEN X Q, LUO W C, et al. Review of uncertainty-based multidisciplinary design optimization methods for aerospace vehicles[J]. Progress in Aerospace Sciences, 2011, 47(6):450-479.
[15] DU X P, CHEN W. Sequential optimization and reliability assessment method for efficient probabilistic design[J]. Journal of Mechanical Design, 2004, 126(2):225-233.
[16] CHANDU S V L, GRANDHI R V. General purpose procedure for reliability based structural optimization under parametric uncertainties[J]. Advances in Engineering Software, 1995, 23(1):7-14.
[17] MíNGUEZ R, CASTILLO E. Reliability-based optimization in engineering using decomposition techniques and FORMS[J]. Structural Safety, 2009, 31(3):214-223.
[18] WANG F G, YANG S X, XIONG F F, et al. Robust trajectory optimization using polynomial chaos and convex optimization[J]. Aerospace Science and Technology, 2019, 92:314-325.
[19] 罗佳奇, 陈泽帅, 曾先. 考虑几何设计参数不确定性影响的涡轮叶栅稳健性气动设计优化[J]. 航空学报, 2020, 41(10):123826. LUO J Q, CHEN Z S, ZENG X. Robust aerodynamic design optimization of turbine cascades considering uncertainty of geometric design parameters[J]. Acta Aeronautica et Astronautica Sinica, 2020, 41(10):123826(in Chinese).
[20] JIANG X Q. Robust optimization of Mars entry trajectory under uncertainty[C]//2018 Space Flight Mechanics Meeting. Reston:AIAA, 2018.
[21] JIANG X Q, LI S. Mars entry trajectory planning using robust optimization and uncertainty quantification[J]. Acta Astronautica, 2019, 161:249-261.
[22] YADAV O P, BHAMARE S S, RATHORE A. Reliability-based robust design optimization:A multi-objective framework using hybrid quality loss function[J]. Quality and Reliability Engineering International, 2010, 26(1):27-41.
[23] LIM J, JANG Y S, CHANG H S, et al. Role of multi-response principal component analysis in reliability-based robust design optimization:An application to commercial vehicle design[J]. Structural and Multidisciplinary Optimization, 2018, 58(2):785-796.
[24] BOHLOURI V, JALALI-NAINI S H. Application of reliability-based robust optimization in spacecraft attitude control with PWPF modulator under uncertainties[J]. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2019, 41(10):1-15.
[25] SIMPLÍCIO P, MARCOS A, BENNANI S. Reusable launchers:development of a coupled flight mechanics, guidance, and control benchmark[J]. Journal of Spacecraft and Rockets, 2019, 57(1):74-89.
[26] STAPPERT S, WILKEN J, SIPPEL M. Evaluation of European reusable VTVL booster stages[C]//2018 AIAA SPACE and Astronautics Forum and Exposition. Reston:AIAA, 2018.
[27] RANGAVAJHALA S, MESSAC A. Designer's preferences regarding equality constraints in robust design optimization[J]. Structural and Multidisciplinary Optimization, 2010, 41(1):17-38.
[28] XIU D B, KARNIADAKIS G E. The Wiener:Askey polynomial chaos for stochastic differential equations[J]. SIAM Journal on Scientific Computing, 2002, 24(2):619-644.
[29] DU X P, CHEN W. A most probable point based method for uncertainty analysis[C]//Proceedings of ASME 2000 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2020:429-438.
[30] REN Y, SHAN J J. Reliability-based soft landing trajectory optimization near asteroid with uncertain gravitational field[J]. Journal of Guidance, Control, and Dynamics, 2015, 38(9):1810-1820.
[31] HUANG Y C, LI H Y. Reliability-based trajectory optimization using nonintrusive polynomial chaos for Mars entry mission[J]. Advances in Space Research, 2018, 61(11):2854-2869.
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