Relative orbit determination algorithm of space targets with passive observation

doi:10.23919/JSEE.2024.000051

Journal of Systems Engineering and Electronics ›› 2024, Vol. 35 ›› Issue (3): 793-804.doi: 10.23919/JSEE.2024.000051

• CONTROL THEORY AND APPLICATION • Previous Articles

Relative orbit determination algorithm of space targets with passive observation

Chenchao DAI¹(), Hongfu QIANG¹(), Degang ZHANG²^,³(), Shaolei HU⁴(), Baichun GONG²^,*()

¹ Zhijian Laboratory, Rocket Force University of Engineering, Xi’an 710025, China
² College of Astronautics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
³ China Flight Test Research Institute, Xi’an 710089, China
⁴ Beijing Institute of Control and Electronic Technology, Beijing 100038, China

Received:2023-03-08 Online:2024-06-18 Published:2024-06-19
Contact: Baichun GONG E-mail:dai32917zr@sina.com;Qiang@263.net;zdg765049828@qq.com;hushaolei028@163.com;baichun.gong@nuaa.edu.cn
About author:
DAI Chenchao was born in 1993. He received his B.S. degree from Rocket Force University. Currently, he is studying for his Ph.D. degree in Rocket Force University of Engineering. His research interests are multi-agent game confrontation and solid attitude-orbit control motor. E-mail: dai32917zr@sina.com

QIANG Hongfu was born in 1965. He received his Ph.D. degree from Xi’an Jiaotong University. Currently he is a professor in Rocket Force University of Engineering. His research interests include agent game confrontation, structural integrity of solid rocket motor and advanced computational mechanics. E-mail: Qiang@263.net

ZHANG Degang was born in 1997. He received his B.S. and M.S. degrees from Nanjing University of Aeronautics and Astronautics. His research interest is aerospace engineering. E-mail: zdg765049828@qq.com

HU Shaolei was born in 1983. He received his M.S. degree from Xi’an Institute of Optics and Precision Mechanics. Currently, he is an engineer of Beijing Institute of Control and Electronic Technology. His research interest is navigation. E-mail: hushaolei028@163.com

GONG Baichun was born in 1987. He received his Ph.D. degree from Northwestern Polytechnical University. Currently, he is an associate research fellow of Nanjing University of Aeronautics and Astronautics. His current research interests are space situational awareness and relative navigation for flight vehicles formation. E-mail: baichun.gong@nuaa.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (12272168) and the Foundation of Science and Technology on Space Intelligent Control Laboratory (HTKJ2023KL502015).

Abstract

Abstract:

Angles-only relative orbit determination for space non-cooperative targets based on passive sensor is subject to weakly observable problem of the relative state between two spacecraft. Previously, the evidence for angles-only observability was found by using cylindrical dynamics, however, the solution of orbit determination is still not provided. This study develops a relative orbit determination algorithm with the cylindrical dynamics based on differential evolution. Firstly, the relative motion dynamics and line-of-sight measurement model for near-circular orbit are established in cylindrical coordinate system. Secondly, the observability is qualitatively analyzed by using the dynamics and measurement model where the unobservable geometry is found. Then, the angles-only relative orbit determination problem is modeled into an optimal searching frame and an improved differential evolution algorithm is introduced to solve the problem. Finally, the proposed algorithm is verified and tested by a set of numerical simulations in the context of high-Earth and low-Earth cases. The results show that initial relative orbit determination (IROD) solution with an appropriate accuracy in a relative short span is achieved, which can be used to initialize the navigation filter.

Key words: angles-only measurement, relative orbit determination, cylindrical coordinate, differential evolution

Chenchao DAI, Hongfu QIANG, Degang ZHANG, Shaolei HU, Baichun GONG. Relative orbit determination algorithm of space targets with passive observation[J]. Journal of Systems Engineering and Electronics, 2024, 35(3): 793-804.

Figures/Tables 12

Fig 1

Fig 2

Table 1

Table 2

Fig 3

Table 3

Table 4

Table 5

Table 6

Table 7

Table 8

Fig 4

References 55

1	GONG B C, WANG S, LI S, et al Review of space relative navigation based on angles-only measurements. Astrodynamics, 2023, 7 (2): 131- 152. doi: 10.1007/s42064-022-0152-2
2	GAIAS G, ARDAENS J S In-orbit experience and lessons learned from the AVANTI experiment. Acta Astronautica, 2018, 153, 383- 393. doi: 10.1016/j.actaastro.2018.01.042
3	D’AMICO S, ARDAENS J, GAIAS G, et al Noncooperative rendezvous using angles-only optical navigation: system design and flight results. Journal of Guidance, Control, and Dynamics, 2013, 36 (6): 1576- 1595. doi: 10.2514/1.59236
4	FLORES-ABAD A, MA O, PHAM K, et al A review of space robotics technologies for on-orbit servicing. Progress in Aerospace Sciences, 2014, 68, 1- 26.
5	DAVID B. The phoenix project. https://www.unoosa.org/pdf/pres/stsc2013/tech-11E.pdf.
6	CHARI R Autonomous orbital rendezvous using angles-only navigation. Cambridge: Massachusetts Institute of Technology, 2012.
7	CHEN T, XU S J Approach guidance with double-line-of-sight measuring navigation constraint for autonomous rendezvous. Journal of Guidance, Control, and Dynamics, 2011, 34 (3): 678- 687. doi: 10.2514/1.52963
8	HIPPELHEUSER J, ELGOHARY T A Inertial space-based orbit estimation: a new measurement model for multiple observers. Acta Astronautica, 2021, 181, 717- 732. doi: 10.1016/j.actaastro.2020.05.033
9	HIPPELHEUSER J, ELGOHARY T. A novel approach for initial orbit determination for space-based observation networks. Proc. of the 3rd IAA Conference on Space Situational Awareness, 2022: IAA-ICSSA-20-00-19.
10	TASIF T H, HIPPELHEUSER J E, ELGOHARY T A Analytic continuation extended Kalman filter framework for perturbed orbit estimation using a network of space-based observers with angles-only measurements. Astrodynamics, 2022, 6 (2): 161- 187. doi: 10.1007/s42064-022-0138-0
11	LEGRAND K, DEMARS K, PERNICKA H Bearings-only initial relative orbit determination. Journal of Guidance, Control, and Dynamics, 2015, 38 (9): 1699- 1713. doi: 10.2514/1.G001003
12	WANG Z W, GONG B C, YUAN Y H, et al Incomplete information pursuit-evasion game control for a space non-cooperative target. Aerospace, 2021, 8 (8): 211. doi: 10.3390/aerospace8080211
13	ANDREWS B A, GELLER D K Analysis of angles-only hybrid space-based/ground-based approach for geosynchronous orbit catalog maintenance. The Journal of the Astronautical Sciences, 2022, 69, 473- 510. doi: 10.1007/s40295-022-00313-3
14	WOFFINDEN D C, GELLER D K Observability criteria for angles-only navigation. IEEE Trans. on Aerospace and Electronic Systems, 2009, 45 (3): 1194- 1208. doi: 10.1109/TAES.2009.5259193
15	WOFFINDEN D C, GELLER D K Optimal orbital rendezvous maneuvering for angles-only navigation. Journal of Guidance, Control, and Dynamics, 2009, 32 (4): 1382- 1387. doi: 10.2514/1.45006
16	GRZYMISCH J, FICHTER W Observability criteria and unobservable maneuvers for in-orbit bearings-only navigation. Journal of Guidance, Control, and Dynamics, 2014, 37 (4): 1250- 1259. doi: 10.2514/1.62476
17	GRZYMISCH J, FICHTER W Analytic optimal observability maneuvers for in-orbit bearings-only rendezvous. Journal of Guidance, Control, and Dynamics, 2014, 37 (5): 1658- 1664. doi: 10.2514/1.G000612
18	GRZYMISCH J, FICHTER W Optimal rendezvous guidance with enhanced bearings-only observability. Journal of Guidance, Control, and Dynamics, 2015, 38 (6): 1131- 1140. doi: 10.2514/1.G000822
19	ANJALY P, RATNOO A Observability enhancement of maneuvering target with bearings-only information. Journal of Guidance, Control, and Dynamics, 2018, 41 (1): 184- 198. doi: 10.2514/1.G003003
20	PI J, BANG H Trajectory design for improving observability of angles-only relative navigation between two satellites. The Journal of the Astronautical Sciences, 2014, 61, 391- 412. doi: 10.1007/s40295-014-0016-y
21	HEBERT L. Angles-only initial relative-orbit determination via maneuver. Auburn: Auburn University, 2016.
22	HOU B, WANG D Y, WANG J Q, et al Optimal maneuvering for autonomous relative navigation using monocular camera sequential images. Journal of Guidance, Control, and Dynamics, 2021, 44 (11): 1947- 1960. doi: 10.2514/1.G005706
23	LUO J J, GONG B C, YUAN J P, et al Angles-only relative navigation and closed-loop guidance for spacecraft proximity operations. Acta Astronautica, 2016, 128, 91- 106. doi: 10.1016/j.actaastro.2016.06.032
24	ZHANG Y J, WANG J Q, HOU B W, et al Optimal maneuvering strategy of spacecraft evasion based on angles-only measurement and observability analysis. Journal of Systems Engineering and Electronics, 2023, 34 (1): 172- 184. doi: 10.23919/JSEE.2023.000026
25	KLEIN I, GELLER D K. Zero Δv solution to the angles-onlyrange observability problem during orbital proximity operations. Proc. of the Bar-Itzhack Memorial Symposium on Estimation, Navigation, and Spacecraft Control, 2015: 351−369
26	GELLER D K, PEREZ A Initial relative orbit determination for close-in proximity operations. Journal of Guidance, Control, and Dynamics, 2015, 38 (9): 1833- 1841. doi: 10.2514/1.G000933
27	GONG B C, GELLER D K, LUO J J Initial relative orbit determination analytical covariance and performance analysis for proximity operations. Journal of Spacecraft and Rockets, 2016, 53 (5): 822- 835. doi: 10.2514/1.A33444
28	GONG B C, LI W D, LI S, et al Angles-only initial relative orbit determination algorithm for non-cooperative spacecraft proximity operations. Astrodynamics, 2018, 2 (3): 217- 231. doi: 10.1007/s42064-018-0022-0
29	GONG B C, LUO J J, LI S, et al. Observability criterion of angles-only navigation for spacecraft proximity operations. Proceeding of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 2019, 233(12): 4302–4315.
30	CHRISTENSEN R, GELLER D. Spin-assisted angles-only navigation and control for SmallSats. Proc. of the 37th Annual AAS Guidance and Control Conference, 2014: AAS14-067.
31	KAUFMAN E, LOVELL T A, LEE T Nonlinear observability measure for relative orbit determination with angles-only measurements. The Journal of Astronautical Science, 2016, 63, 60- 80. doi: 10.1007/s40295-015-0082-9
32	LOVELL T A, LEE T. Nonlinear observability for relative satellite orbits with angles-only measurements. Proc. of the 24th International Symposium on Space Flight Dynamics, 2014. https://issfd.org/ISSFD_2014/ISSFD24_Paper_S16-3_Lovell-pdf.
33	NEWMAN B A, PRATT E, LOVELL A, et al. Quadratic hexa-dimensional solution for relative orbit determination. Proc. of AIAA/AAS Astrodynamics Specialist Conference, 2014. DOI: 10.2514/6.2014-4309.
34	NEWMAN B A, LOVELL T A, PRATT E, et al Hybrid linear-nonlinear initial determination with single iteration refinement for relative motion. Advances in the Astronautical Sciences, 2015, 155, 2149- 2168.
35	SHUBHAM G. Initial relative orbit determination using second order dynamics and line-of-sight measurements. Auburn: Auburn University, 2015.
36	SHUBHAM G. Initial relative-orbit determination using second-order dynamics and line-of-sight measurements. Auburn: Auburn University, 2015.
37	GONG B C, MA Y Q, ZHANG W F, et al Deep-neural-network-based angles-only relative orbit determination for space non-cooperative target. Acta Astronautica, 2023, 204, 552- 567. doi: 10.1016/j.actaastro.2022.09.024
38	TOMBASCO J, AXELRAD P Observability of relative hybrid elements, given space-based angles-only observations. Journal of Guidance, Control, and Dynamics, 2012, 35 (5): 1681- 1686. doi: 10.2514/1.54981
39	GAIAS G, D’AMICO S, ARDAENS J S Angles-only navigation to a noncooperative satellite using relative orbital elements. Journal of Guidance, Control, and Dynamics, 2014, 37 (3): 439- 451.
40	YIM J R, CRASSIDIS J L, JUNKINS J L. Autonomous orbit navigation of two spacecraft system using relative line of sight vector measurements. Proc. of the AAS/AIAA Spaceflight Mechanics Conference, 2004: AAS04-257.
41	SULLIVAN J, D’AMICO S Nonlinear Kalman filtering for improved angles-only navigation using relative orbital elements. Journal of Guidance, Control, and Dynamics, 2017, 40 (9): 2183- 2200. doi: 10.2514/1.G002719
42	SULLIVAN J, LOVELL T A, D’AMICO S. Angles-only navigation for autonomous on-orbit space situational awareness applications. Proc. of the AAS/AIAA Astrodynamics Specialist Conference, 2018: AAS 18-468.
43	ARDAENS J S, GAIAS G A numerical approach to the problem of angles-only initial relative orbit determination in low earth orbit. Advances in Space Research, 2019, 63 (12): 3884- 3899. doi: 10.1016/j.asr.2019.03.001
44	KOENIG A W, D’AMICO S. Observability-aware numerical algorithm for angles-only initial relative orbit determination. Proc. of the AAS/AIAA Astrodynamics Specialist Conference, 2020: AAS 20-594.
45	GRZYMISCH J, FICHTER W, CASASCO M, et al. A spherical coordinate parametrization for an in orbit bearings-only navigation filter. Proc. of the 2nd CEAS Specialist Conference on Guidance, Navigation and Control, 2013: 215−231.
46	GELLER D K, LOVELL T A Non-iterative approximate solution to the angles-only initial relative orbit determination problem in spherical coordinates. Proc. of the 26th AAS/AIAA Space Fight Mechanics Meetings, 2016, 158, 356.
47	PEREZ A C, GELLER D K, LOVELL T A Non-iterative angles-only initial relative orbit determination with J2 perturbations. Acta Astronautica, 2018, 151, 146- 159. doi: 10.1016/j.actaastro.2018.06.033
48	GELLER D, LOVELL T A. Initial relative orbit determination performance analysis in cylindrical coordinates using angles-only measurements. Proc. of the 24th AAS/AIAA Space Flight Mechanics Meeting, 2014: AAS 14-212.
49	GELLER D K, LOVELL T A Angles-only initial relative orbit determination performance analysis using cylindrical coordinates. The Journal of the Astronautical Sciences, 2017, 64, 72- 96. doi: 10.1007/s40295-016-0095-z
50	PEREZ A, LOVELL T A, GELLER D, et al. Relative satellite motion solutions using curvilinear coordinate frames. Proc. of the AAS/AIAA Astrodynamics Specialist Conference, 2015: AAS 15−437.
51	GONG B C, ZHANG D G, ZHANG W F, et al Angles-only relative navigation algorithm for space non-cooperative target in cylindrical frame. Journal of Chinese Inertial Technology, 2021, 29 (6): 752- 762.
52	BILAL, PANT M, ZAHEER H, et al Differential evolution: a review of more than two decades of research. Engineering Applications of Artificial Intelligence, 2020, 90, 103479. doi: 10.1016/j.engappai.2020.103479
53	OPARA K R, ARABAS J Differential evolution: a survey of theoretical analyses. Swarm and Evolutionary Computation, 2019, 44, 546- 558. doi: 10.1016/j.swevo.2018.06.010
54	SI X G, BAO J D Evaluate linearity error based on Hill-Climbing and differential evolutionary algorithm. Journal of Test and Measurement Technology, 2019, 33 (5): 418- 420.
55	DENG W, SHANG S F, CAI X, et al An improved differential evolution algorithm and its application in optimization problem. Soft Computing, 2021, 25 (7): 5277- 5298. doi: 10.1007/s00500-020-05527-x

Orbital element	Chief	Deputy
Orbital element	Chief	Case 1	Case 2	Case 3
Semimajor axis/km	42164.17	42264.17	42264.17	42364.17
Eccentricity	0	0.0001	0.0001	0.0001
Inclination/(°)	0	0.00002	0.00002	0.00002
Right ascension of ascending node/(°)	0	0	0	0
Argument of perigee/(°)	0	0	0	0
True anomaly/(°)	322.7645	324.1151	325.46	324.093

State	Minimum	Maximum
$ {\rho _{{\mathrm{rel}}}} $/m	100000	3000000
$ {\theta _{{\mathrm{rel}}}} $/rad	0	0. 1$\alpha $
$ {z_{{\mathrm{rel}}}} $/m	−2000	2000
$ {\dot \rho _{{\mathrm{rel}}}} $/(m/s)	−2	2
$ {\dot \theta _{{\mathrm{rel}}}} $/(rad/s)	−0.00001$\alpha $	0.00001$\alpha $
$ {\dot z_{{\mathrm{rel}}}} $/(m/s)	−20	20

State	Run #1	Run #2	Run #3	Run #4
Sel_best	0.000482	0.003928	0.001034	0.000814
$ {\rho _{{\mathrm{rel}}}} $/m	99342.4	183869.0	175715.5	130469.7
$ {\theta _{{\mathrm{rel}}}} $/rad	0.023193	0.040047	0.038522	0.029747
$ {z_{{\mathrm{rel}}}} $/m	0.007945	0.007754	−0.059337	−0.016427
$ {\dot \rho _{{\mathrm{rel}}}} $/(m/s)	−0.530838	−1.018791	−0.876775	−0.525139
$ {\dot \theta _{{\mathrm{rel}}}} $/(rad/s)	−2.483 7 e−07	−4.605 3 e−07	−4.391 6 e−07	−3.285 6 e−07
$ {\dot z_{{\mathrm{rel}}}} $/(m/s)	−3.743 0 e-06	4.148 0 e−06	1.547 0 e−05	5.170 7 e−06
Error/%	1.6019	70.1985	63.6927	26.2843

State	Run #1	Run #2	Run #3	Run #4
Sel_best	0.000297	0.000149	0.000193	0.000132
$ {\rho _{{\mathrm{rel}}}} $/m	129523.9	108668.7	128224.8	120088.4
$ {\theta _{{\mathrm{rel}}}} $/rad	0.055997	0.049735	0.055685	0.053446
$ {z_{{\mathrm{rel}}}} $/m	−0.041245	0.008858	−0.022159	0.006543
$ {\dot \rho _{{\mathrm{rel}}}} $/(m/s)	−0.109385	−0.368671	−0.230537	0.241181
$ {\dot \theta _{{\mathrm{rel}}}} $/(rad/s)	−3.455 5 e−07	−2.744 5 e−07	−3.313 3 e−07	−3.302 6 e−07
$ {\dot z_{{\mathrm{rel}}}} $/(m/s)	5.887 4 e−06	−1.634 8 e−05	1.297 9 e−05	−1.354 8 e−05
Error/%	19.0946	5.7362	18.4289	13.6489

State	Run #1	Run #2	Run #3	Run #4
Sel_best	0.000154	0.000251	0.000062	0.000290
$ {\rho _{{\mathrm{rel}}}} $/m	238250.4	187168.5	200431.8	212928.5
$ {\theta _{{\mathrm{rel}}}} $/rad	0.021730	0.020696	0.023536	0.024695
$ {z_{{\mathrm{rel}}}} $/m	0.004079	0.020357	−0.012792	−0.022222
$ {\dot \rho _{{\mathrm{rel}}}} $/(m/s)	−0.816903	−0.321888	−0.570304	0.201456
$ {\dot \theta _{{\mathrm{rel}}}} $/(rad/s)	−6.953 6 e−07	−4.822 3 e−07	−5.037 8 e−07	−5.603 5 e−07
$\dot {z}_{{\mathrm{rel}}} $/(m/s)	−1.557 9 e−06	−1.325 0 e−06	−4.368 8 e−06	−1.963 6 e−06
Error/%	4.1185	5.8605	1.4823	6.5278

Relative orbit determination algorithm of space targets with passive observation

RichHTML

PDF (PC)

Knowledge

Abstract

Cite this article

Share this article

Figures/Tables 12

References 55

Related Articles 9

Recommended Articles

Metrics

Comments

Relative state	Minimum	Maximum
$ {\rho _{{\mathrm{rel}}}} $/m	10000	180000
$ {\theta _{{\mathrm{rel}}}} $/rad	−0. 1$\alpha $	0. 1$\alpha $
$ {z_{{\mathrm{rel}}}} $/m	−2000	2000
$ {\dot \rho _{{\mathrm{rel}}}} $/(m/s)	−20	20
$ {\dot \theta _{{\mathrm{rel}}}} $/(rad/s)	−0.00001$\alpha $	0.00001$\alpha $
$ {\dot z_{{\mathrm{rel}}}} $/(m/s)	−20	20

Case	Run #1	Run #2	Run #3	Run #4	Run #5	Run #6	Run #7	Run #8	Run #9	Run #10
1	10.5	18.5	11.4	12.1	14.3	16.7	1.7	2.9	0.4	0.9
2	13.0	14.0	14.5	16.9	7.6	8.4	16.4	9.0	20.8	19.6
3	3.3	4.2	22.1	11.9	14.3	19.9	0.3	8.8	3.8	28.5
4	1.2	2.7	4.0	12.8	0.4	12.9	33.3	7.3	30.1	36.9
5	1.0	3.5	10.3	13.4	13.3	14.5	2.3	34.6	12.1	12.0
6	21.3	25.3	1.1	30.6	5.8	6.4	22.1	5.5	5.6	29.5

[1]	Yijie ZHANG, Jiongqi WANG, Bowen HOU, Dayi WANG, Yuyun CHEN. Optimal maneuvering strategy of spacecraft evasion based on angles-only measurement and observability analysis [J]. Journal of Systems Engineering and Electronics, 2023, 34(1): 172-184.
[2]	Kai Zhao and Yongcheng Sun. Genetic-optimization framework for SVC transmission based on partial cooperative communication [J]. Systems Engineering and Electronics, 2017, 28(5): 861-870.
[3]	Kangsheng Dong, Hanqiao Huang, Changqiang Huang, and Zhuoran Zhang. Trajectory online optimization for unmanned combat aerial vehicle using combined strategy [J]. Systems Engineering and Electronics, 2017, 28(5): 963-970.
[4]	Hong Li, Li Zhang, and Yongchang Jiao. Discrete differential evolution algorithm for integer linear bilevel programming problems [J]. Systems Engineering and Electronics, 2016, 27(4): 912-.
[5]	Yu Xue,Yi Zhuang, Tianquan Ni, Siru Ni, and Xuezhi Wen. Self-adaptive learning based discrete differential evolution algorithm for solving CJWTA problem [J]. Journal of Systems Engineering and Electronics, 2014, 25(1): 59-68.
[6]	Yong Liu and Liang Ma. Improved gravitational search algorithm based on free search differential evolution [J]. Journal of Systems Engineering and Electronics, 2013, 24(4): 690-698.
[7]	Ye Xu and Ling Wang. Differential evolution algorithm for hybrid flow-shop scheduling problems [J]. Journal of Systems Engineering and Electronics, 2011, 22(5): 794-798.
[8]	Lianghong Wu, Yaonan Wang, and Shaowu Zhou. Improved differential evolution algorithm for resource-constrained project scheduling problem [J]. Journal of Systems Engineering and Electronics, 2010, 21(5): 798-805.
[9]	Wu Lianghong, Wang Yaonan, Zhou Shaowu & Tan Wen. Design of PID controller with incomplete derivation based on differential evolution algorithm [J]. Journal of Systems Engineering and Electronics, 2008, 19(3): 578-583.

Orbital element	Chief	Deputy
Semi major axis/m	6731140	6771140
Eccentricity	0.000212	0.000212
Inclination/(°)	41.471	41.471
Right ascension of ascending node/(°)	281.9962	281.9962
Argument of perigee/(°)	349.1571	349.1571
True anomaly/(°)	269.3134	277.3134