A PS (pseudo-spectral) method with improved parameters is proposed for the rapid footprint generation of reusable vehicles. Three aspects of the method were explored, including the consistency of the discrete NLP (Nonlinear Programming) problem with the original problem, the impact of the NLP problems' convergence accuracy on the results, and the final convergence accuracy factors. The reasonable ranges of the IPOPT solver’s convergence tolerance are redefined through these explorations. Furthermore, the solution process of the footprint is partially optimized. Finally, the results show that the method can fully improve the computational efficiency of solving the footprint using the PS method. The average time of each trajectory is minor than 0.9s, showing the parameter-improved PS method’s superiority.
LiuQ-GLiuX-XWuJ, et al.A fast computational method for the landing footprints of space-to-ground vehicles. Journal of Systems Engineering and Electronics2020; 31(5): 1062–1076, DOI: 10.23919/JSEE.2020.000080.
2.
HeR-ZZhangY-LLiuL-L, et al.Feasible footprint generation with uncertainty effects. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering2019; 233(1): 138–150, DOI: 10.1177/0954410017728971.
3.
ZhangY-LChenK-JLiuL-H, et al.Rapid generation of landing footprint based on geometry-predicted trajectory. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering2017; 231(10): 1851–1861, DOI: 10.1177/0954410016662066.
4.
WangTZhangHLiY, et al.An improved footprint generation method for entry vehicles. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering2017; 231(10): 1951–1956, DOI: 10.1177/0954410016662060.
5.
LiHZhangRLiZ, et al.Footprint problem with angle of attack optimization for high lifting reentry vehicle. Chinese Journal of Aeronautics2012; 25(2): 243–251, DOI: 10.1016/S1000-9361(11)60384-1.
6.
WangTZhangH-BLiY-Y, et al.Landing footprint generation of entry vehicle based on gauss PS method. Journal of National University of Defense Technology2016; 38(3): 75–80, DOI: 10.11887/j.cn.201603013.
7.
ZhaoJ-SZhangJ-HLiS. Rapid and high-accuracy approach for hypersonic glide vehicle reentry trajectory optimization. Journal of Astronautics2019; 40(9): 1034–1043, DOI: 10.3873/j.issn.1000-1328.2019.09.007.
8.
YangLChenW-CLiuX-M, et al.Steady glide dynamic modeling and trajectory optimization for high lift-to-drag ratio reentry vehicle. International Journal of Aerospace Engineering2016; 2016: 1–14, DOI: 10.1155/2016/3527460.
9.
YuC-MZhaoD-JYangY. Efficient convex optimization of reentry trajectory via the Chebyshev Pseudospectral method. International Journal of Aerospace Engineering2019; 2019: 1–9, DOI: 10.1155/2019/1414279.
10.
MaoYZhangDWangL. Reentry trajectory optimization for hypersonic vehicle based on improved Gauss PS method. Soft Computing2017; 21(16): 4583–4592, DOI: 10.1007/s00500-016-2201-3.
11.
XieYLiuLLiuJ, et al.Rapid generation of entry trajectories with waypoint and no-fly zone constraints. Acta Astronautica2012; 77: 167–181, DOI: 10.1016/j.actaastro.2012.04.006.
12.
RichieG. The Common Aero vehicle - Space delivery system of the future. Space Technology Conference and Exposition, 1999.
13.
SouzbanMSolaymani FardOBorzabadiAH. A rapid-based improvement on some mesh refinement strategies in solving optimal control problems. IMA Journal of Mathematical Control and Information2020; 37(2): 351–376, DOI: 10.1093/imamci/dnz004.
14.
MillerATHagerWWRaoAV. Mesh refinement method for solving optimal control problems with non-smooth solutions using Jump function approximations, 2020. ArXiv:2003.11676 Mathhttp://arxiv.org/abs/2003.11676.
15.
HuangJLiuZLiuZ, et al.A pk-Adaptive mesh refinement for Pseudospectral method to solve optimal control problem. IEEE Access2019; 7: 161666–161679, DOI: 10.1109/ACCESS.2019.2952139.
16.
WangLY. Pseudospectral mesh refinement algorithm based on density function and its application. Dynamical Systems and Control2017; 06: 35–42, DOI: 10.12677/DSC.2017.62005.
17.
KoeppenNRossIMWilcoxLC, et al.Fast mesh refinement in Pseudospectral optimal control. Journal of Guidance, Control, and Dynamics2019; 42(4): 711–722, DOI: 10.2514/1.G003904.
18.
FengZZhangQGeJ, et al. Mesh Adaptation method for optimal control with non-smooth control using second-generation Wavelet. IEEE Access2019; 7: 135076–135086, DOI: 10.1109/ACCESS.2019.2941539.
19.
RaoAVHagerWW. Mesh-generation method for real-time optimal control using Adaptive Gaussian Quadrature collocation. AIAA Guidance, Navigation, and Control Conference. FL: Kissimmee, 2018, DOI: 10.2514/6.2018-0848.
20.
Trefethen. Approximation theory and approximation practice. society for industrial & applied mathematics, 2013.
21.
WächterABieglerLT. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming2006; 106(1): 25–57, DOI: 10.1007/s10107-004-0559-y.
22.
BettsJTErbSO. Optimal Low thrust trajectories to the Moon. SIAM Journal on Applied Dynamical Systems2003; 2(2): 144–170, DOI: 10.1137/S1111111102409080.
23.
PattersonMARaoAV. GPOPS-II: a MATLAB Software for solving multiple-Phase optimal control problems using hp-Adaptive Gaussian Quadrature collocation methods and sparse nonlinear programming. ACM Transactions on Mathematical Software2014; 41(1): 1–37, DOI: 10.1145/2558904.
24.
PattersonMAHagerWWRaoAV. A ph mesh refinement method for optimal control. Optimal Control Applications and Methods2014; 36(4): 398–421, DOI: 10.1002/oca.2114.
