A fast computational method for the landing footprints of space-to-ground vehicles

doi:10.23919/JSEE.2020.000080

Abstract

Abstract:

Fast computation of the landing footprint of a space-toground vehicle is a basic requirement for the deployment of parking orbits, as well as for enabling decision makers to develop real-time programs of transfer trajectories. In order to address the usually slow computational time for the determination of the landing footprint of a space-to-ground vehicle under finite thrust, this work proposes a method that uses polynomial equations to describe the boundaries of the landing footprint and uses back propagation (BP) neural networks to quickly determine the landing footprint of the space-to-ground vehicle. First, given orbital parameters and a manoeuvre moment, the solution model of the landing footprint of a space-to-ground vehicle under finite thrust is established. Second, given arbitrary orbital parameters and an arbitrary manoeuvre moment, a fast computational model for the landing footprint of a space-to-ground vehicle based on BP neural networks is provided. Finally, the simulation results demonstrate that under the premise of ensuring accuracy, the proposed method can quickly determine the landing footprint of a space-to-ground vehicle with arbitrary orbital parameters and arbitrary manoeuvre moments. The proposed fast computational method for determining a landing footprint lays a foundation for the parking-orbit configuration and supports the design of real-time transfer trajectories.

Key words: space-to-ground vehicle, landing footprint, back propagation (BP) neural network, fast computational method, Pontryagin's minimum principle

Qingguo LIU, Xinxue LIU, Jian WU, Yaxiong LI. A fast computational method for the landing footprints of space-to-ground vehicles[J]. Journal of Systems Engineering and Electronics, 2020, 31(5): 1062-1076.

Figures/Tables 27

Fig 1

Fig 2

Fig 3

Fig 4

Fig 5

Fig 6

Fig 7

Fig 8

Table 1

Fig 9

Fig 10

Fig 11

Fig 12

Fig 13

Fig 14

Fig 15

Fig 16

Fig 17

Fig 18

Table 2

Table 3

Table 4

Table 5

Table 6

Table 7

Table 8

Table 9

References 32

1	NAZARENKO A I, USOVIK I V. The effect of parameters of the initial data updating algorithm on the accuracy of spacecraft reentry time prediction. Journal of Space Safety Engineering, 2019, 6 (1): 24- 29. doi: 10.1016/j.jsse.2019.01.002
2	DESIKAN S L N, SRINIVASAN K. Longitudinal aerodynamics of a winged reentry vehicle at supersonic speed. Journal of Spacecraft and Rockets, 2018, 55 (5): 1144- 1153. doi: 10.2514/1.A34088
3	LOBBIA M A. Multidisciplinary design optimization of waverider-derived crew reentry vehicles. Journal of Spacecraft and Rockets, 2017, 54 (1): 233- 245. doi: 10.2514/1.A33253
4	SGUBINI S, PALMERINI G B. Evaluation of main parameters in re-entry trajectories. Aerotecnica Missili & Spazio, 2016, 95 (1): 42- 49.
5	GANGIREDDY S, ASHOK J. Re-entry trajectory optimization using pigeon inspired optimization based control profiles. Advances in Space Research, 2018, 62 (11): 3170- 3186. doi: 10.1016/j.asr.2018.08.009
6	JIANG X Q, LI S. Mars atmospheric entry trajectory optimization via particle swarm optimization and Gauss pseudo-spectral method. Proc. of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 2016: 2320-2329.
7	WANG Z B, GRANT M J. Constrained trajectory optimization for planetary entry via sequential convex programming. Journal of Guidance, Control, and Dynamics, 2017, 40 (10): 1- 13.
8	GUO C M, ZHANG J, LUO Y Z, et al. Phase-matching homotopic method for indirect optimization of long-duration low-thrust trajectories. Advances in Space Research, 2018, 62 (3): 568- 579. doi: 10.1016/j.asr.2018.05.007
9	SHEN H X. No-guess indirect optimization of asteroid mission using electric propulsion. Optimal Control Applications and Methods, 2018, 39 (2): 1061- 1070. doi: 10.1002/oca.2396
10	RASOTTO M, ARMELLIN R, DI L P. Multi-step optimization strategy for fuel-optimal orbital transfer of low-thrust spacecraft. Engineering Optimization, 2016, 48 (3): 519- 542. doi: 10.1080/0305215X.2015.1025773
11	GAO Y, WANG J, WU W, et al. A hybrid method for mobile agent moving trajectory scheduling using ACO and PSO in WSNs. Sensors, 2019, 19 (3): 3- 15.
12	LEOBARDO C M, DAVID G G, RODRIGO A L, et al. A hybrid method for online trajectory planning of mobile robots in cluttered environments. IEEE Robotics and Automation Letters, 2017, 2 (2): 935- 942. doi: 10.1109/LRA.2017.2655145
13	SARAF A, LEAVITT J A, MEASE K D. Landing footprint computation for entry vehicles. Proc. of the AIAA Guidance, Navigation, and Control Conference and Exhibit, 2004, 1- 14.
14	LI D W, JIANG R K, LIU B. Reentry landing boundary prediction. Applied Mechanics and Materials, 2013, 433-435, 1123- 1126. doi: 10.4028/www.scientific.net/AMM.433-435.1123
15	HU H L, NAN Y, WEN X. Research on dynamic reentry footprint for the spacecraft based on the genetic algorithm. Chinese Space Science and Technology, 2014, 34 (1): 35- 41.
16	NGUYEN B H, GERMAN R, TROVAO J, et al. Real-time energy management of battery/supercapacitor electric vehicles based on an adaptation of Pontryagin's minimum principle. IEEE Trans. on Vehicular Technology, 2019, 68 (1): 203- 212. doi: 10.1109/TVT.2018.2881057
17	SARI C, SUBCHAN S. Application of Pontryagin's minimum principle in optimum time of missile manoeuvring. Cauchy, 2016, 4 (3): 107- 111. doi: 10.18860/ca.v4i3.3534
18	NADIR O, LOUNIS A, RUSTEM A. From offline to adaptive online energy management strategy of hybrid vehicle using Pontryagin's minimum principle. International Journal of Automotive Technology, 2018, 19 (3): 571- 584. doi: 10.1007/s12239-018-0054-8
19	GWAK H S, BEA S H, LEE D E. Resource leveling using genetic algorithm. Journal of the Architectural Institute of Korea Structure & Construction, 2018, 34 (2): 67- 74.
20	KITA H. Genetic algorithms for noisy fitness functions—applications, requirements and algorithms. Proc. of the ISCIE International Symposium on Stochastic Systems Theory and Applications, 2001, 137- 142.
21	SHOJAEDINI E, MAJD M, SAFABAKHSH R. Novel adaptive genetic algorithm sample consensus. Applied Soft Computing, 2019, 77, 635- 642. doi: 10.1016/j.asoc.2019.01.052
22	NAEENI A A, ROSHANIAN J. Developing a hybrid algorithm to design the optimal trajectory of reentry vehicles. Modares Mechanical Engineering, 2015, 14 (13): 143- 149.
23	ROBERTO P, GIANFRANCO M, MARCO C. Reentry trajectory optimization for mission analysis. Journal of Spacecraft and Rockets, 2017, 54 (1): 331- 336. doi: 10.2514/1.A33465
24	EDWARDS M R. Deep mantle plumes and an increasing Earth radius. Geodesy and Geodynamics, 2019, 10 (3): 173- 178. doi: 10.1016/j.geog.2019.03.002
25	KOPECZ S, MEISTER A. On order conditions for modified Patankar-Runge-Kutta schemes. Applied Numerical Mathematics, 2018, 123, 159- 179. doi: 10.1016/j.apnum.2017.09.004
26	RUBIO A. Propagators for the time-dependent Kohn-Sham equations: multistep, Runge-Kutta, exponential Runge-Kutta, and commutator free Magnus methods. Journal of Chemical Theory and Computation, 2018, 14 (6): 3040- 3052. doi: 10.1021/acs.jctc.8b00197
27	AMINE E, HICHAM A, NOURA A. An intelligent model for enterprise resource planning selection based on BP neural network. Proc. of the 2nd Mediterranean Symposium on Smart City Applications, 2017, 212- 222.
28	MA L, LIN X, JIANG L H. Differential-weighted global optimum of BP neural network on image classification. Proc. of the International Conference on Information Science and Applications, 2017, 544- 552.
29	CHANG X L, LI W J. Evaluation of creative talents in cultural industry based on BP neural network. International Journal of Performability Engineering, 2018, 14 (11): 8- 20.
30	CSATO L. A characterization of the logarithmic least squares method. European Journal of Operational Research, 2019, 276 (1): 212- 216. doi: 10.1016/j.ejor.2018.12.046
31	MACIEJ K, MARCIN P. The least squares method for option pricing revisited. Applicationes Mathematicae, 2018, 45 (1): 5- 29. doi: 10.4064/am2354-2-2018
32	KHAN T, YADAV J S. Adaptive learning based improved performance of activation functions in hidden layer using artificial neural network. International Journal of Engineering and Technology, 2018, 7 (4): 3223- 3227.

No.	$a$/km	$e$	$i/(^{\circ})$	$\omega /(^{\circ})$	${\it\Omega} /(^{\circ})$	$f/(^{\circ})$	$t_m /{{{\rm{s}}}}$
1	9 406	0.234 450	29.321	224.038	307.947	135.862	4 605.5
2	7 159	0.000 325	98.084	76.415	185.381	283.332	4 486.3
3	9 337	0.000 265	52.006	171.286	350.025	188.701	4 509.2
4	7 838	0.001 419	101.618	87.162	293.441	282.681	4 466.1
5	7 717	0.054 058	99.013 1	56.362	347.918	308.882	4 486.0
6	7 654	0.002 862	90.045	245.644	85.205	113.968	4 524.4
7	8 328	0.201 675	82.382	262.455	29.931	44.341	4 438.5
8	9 511	0.213 250	28.126	34.617	152.421	325.392	4 459.4
9	9 431	0.016 526	64.422	302.102	337.357	56.304	4 483.6
10	10 024	0.321 619	56.927	231.699	218.468	128.371	4 518.3

Number	$m_1 ^1(n_1 ^1) $	$m_1 ^2(n_1 ^2) $	$m_1 ^3(n_1 ^3) $	$m_1^4(n_1 ^4) $	$m_1 ^5(n_1 ^5) $
1	–3.194 444 e–6	1.233 779 e–3	–1.919 443 e–1	1.514 211 e+1	–3.883 817 e+2
2	–8.256 123 e–5	2.024 390 e–2	–1.619 387	5.305 329 e+1	–6.151 001 e+2
3	–4.012 587 e–6	1.495 236 e–3	–2.784 367 e–1	2.249 619 e+1	–6.293 869 e+2
4	–5.772 980 e–5	1.255 224 e–2	–1.005 445	2.842 024 e+1	–1.540 246 e+2
5	–1.519 489 e–4	2.663 004 e–2	1.478 789	4.130 832 e+1	–3.109 007 e+2
6	–4.037 333 e–5	8.784 321 e–3	–7.144 400 e–1	2.549 818 e+1	–2.075 674 e+2
7	–1.397 543 e–4	3.408 221 e–2	–2.787 432	9.969 876 e+1	–1.309 384 e+3
8	–4.214 520 e–6	1.114 532 e–3	–1.340 298 e–1	8.091 355	–1.224 080 e+2
9	–2.331 988 e–4	3.725 921 e–2	–3.201 335	1.244 620 e+2	–1.789 788 e+3
10	–3.090 332 e–4	3.844 785 e–2	3.145 540 9	1.084 894 e+2	–1.384 916 e+3

Number	$m_2 ^1(n_2 ^1) $	$m_2 ^2(n_2 ^2) $	$m_2 ^3(n_2 ^3) $	$m_2^4(n_2 ^4) $	$m_2 ^5(n_2 ^5) $
1	6.441 445 e–7	–2.563 355 e–4	2.536 902 e–2	–1.817 822 e–1	–2.005 818 e+1
2	3.573 245 e–4	–3.856 786 e–2	3.328 224	–1.309 137 e+2	1.941 689 e+e3
3	8.439 678 e–7	–3.980 074 e–4	3.934 290 e–2	–8.826 228 e–1	–1.826 913 e+1
4	2.699 356 e–5	6.645 901 e–3	5.467 708 e–1	2.391 602 e+1	5.170 087 e+2
5	–3.102 356 e–5	4.248 810 e–3	2.511 124 e–1	6.101 356	5.977 024 e+1
6	–4.884 679 e–6	1.253 366 e–3	–9.820 098 e–2	3.268 925	8.756 833 e+1
7	–7.493 214 e–6	5.948 623 e–4	1.142 389 e–1	–1.444 100 e+1	3.714 782 e+2
8	2.332 990 e–7	–3.002 315 e–4	4.742 596 e–3	9.270 998 e–1	–1.701 552 e+1
9	–5.454 633 e–5	7.603 279 e–3	–6.165 638 e–1	1.584 644 e+1	–5.327 344 e+1
10	–3.295 180 e–5	1.211 239 e–2	–8.845 530 e–1	2.358 660 e+1	–1.759 743 e+2

Number	$m_1 ^1(n_1 ^1) $	$m_1 ^2(n_1 ^2) $	$m_1 ^3(n_1 ^3) $	$m_1^4(n_1 ^4) $	$m_1 ^5(n_1 ^5) $
1	–2.364 762 e–6	1.093 990 e–3	–1.919 619 e–1	1.514 202 e+1	–3.883 818 e+2
2	–9.386 719 e–5	2.053 956 e–2	–1.619 389	5.305 330 e+1	–6.151 000 e+2
3	–3.243 843 e–6	1.545 236 e–3	–2.784 475 e–1	2.249 620 e+1	–6.293 869 e+2
4	–7.052 381 e–5	1.415 389 e–2	–1.005 241	2.842 022 e+1	–1.540 246 e+2
5	–1.409 326 e–4	2.343 824 e–2	1.478 954	4.130 829 e+1	–3.109 008 e+2
6	–4.277 951 e–5	8.904 313 e–3	–7.144 316 e–1	2.549 826 e+1	–2.075 673 e+2
7	–1.492 619 e–4	3.358 251 e–2	–2.787 314	9.969 881 e+1	–1.309 385 e+3
8	–2.874 275 e–6	1.004 532 e–3	–1.340 367 e–1	8.091 336	–1.224 080 e+2
9	–1.491 477 e–4	3.585 920 e–2	–3.201 586	1.244 616 e+2	–1.789 787 e+3
10	–1.840 043 e–4	3.954 786 e–2	3.145 550 4	1.084 895 e+2	–1.384 915 e+3

Number	$m_2 ^1(n_2 ^1) $	$m_2 ^2(n_2 ^2) $	$m_2 ^3(n_2 ^3) $	$m_2^4(n_2 ^4) $	$m_2 ^5(n_2 ^5) $
1	7.261 776 e–7	–2.546 767 e–4	2.542 910 e–2	–1.817 815 e–1	–2.005 819 e+1
2	1.683 400 e–4	–3.843 956 e–2	3.328 331	–1.309 139 e+2	1.941 689 e+e3
3	9.789 098 e–7	–3.551 292 e–4	3.932 091 e–2	–8.826 921 e–1	–1.826 912 e+1
4	3.519 243 e–5	6.694 692 e–3	5.467 192 e–1	2.391 697 e+1	5.170 088 e+2
5	–2.993 518 e–5	4.384 012 e–3	2.510 960 e–1	6.101 730	5.977 056 e+1
6	–5.956 112 e–6	1.193 456 e–3	–9.824 865 e–2	3.268 925	8.756 860 e+1
7	–8.504 437 e–6	6.485 160 e–4	1.147 097 e–1	–1.444 156 e+1	3.714 789 e+2
8	7.390 693 e–7	–1.630 918 e–4	4.747 751 e–3	9.270 948 e–1	–1.701 510 e+1
9	–3.333 035 e–5	7.864 112 e–3	–6.164 476 e–1	1.584 621 e+1	–5.327 374 e+1
10	–5.666 975 e–5	1.208 691 e–2	–8.844 547 e–1	2.358 612 e+1	–1.759 747 e+2