Terminal sliding mode control for a type of special long-periodic orbit

doi:10.23919/JSEE.2026.000149

Abstract

Abstract:

A communication and navigation constellation which can cover the whole Cislunar space is useful for deep space exploration. In our previous work, we proposed a configuration of navigation constellations arranged in special long-period orbits of Cislunar space. In order to keep the navigation satellites strictly on the accurate nominal orbit, a terminal sliding mode control (TSMC) strategy based on the disturbance observer is proposed. Since the states of navigation satellites cannot be observed directly, a linear extended state observer (LESO) is designed to recover the state as well as estimate solar gravity to assist in spacecraft orbit determination. Then through the TSMC strategy, high-precision control of the spacecraft on special long-period orbits has been achieved. The stability conditions of the observer are deduced, and the stability of the system is proven using Lyapunov stability theory. The simulation results show that during the entire mission cycle of 76 days, the maximum position error is 259.95 m and the minimum is 1.41 m, and the maximum value of the velocity error is 0.08 m/s and the minimum velocity error is 6.15×10⁻⁶ m/s. The total velocity increment throughout the entire mission cycle is 30.96 m/s.

Key words: station-keeping control, special long-periodic orbit, linear extended state observer, terminal sliding mode control

Youtao Gao, Yinghui Xin, Chaoyong Hu, Chenliang Li. Terminal sliding mode control for a type of special long-periodic orbit[J]. Journal of Systems Engineering and Electronics, 2026, 37(4): 1383-1398.

Figures/Tables 24

Fig 1

Table 1

Table 2

Fig 2

Table 3

Fig 3

Fig 4

Table 4

Table 5

Fig 5

Fig 6

Fig 7

Fig 8

Fig 9

Fig 10

Fig 11

Fig 12

Fig 13

Fig 14

Fig 15

Fig 16

Fig 17

Table 6

Fig 18

References 30

1	Liu L, Wu W R, Liu Y Design and implementation of Chinese libration point missions. Science China Information Sciences, 2023, 66 (9): 191201. doi: 10.1007/s11432-022-3716-9
2	Farres A, Gao C, Masdemont J J, et al Geometrical analysis of station-keeping strategies about libration point orbits. Journal of Guidance, Control, and Dynamics, 2022, 45 (6): 1108. doi: 10.2514/1.G006014
3	Kayama Y, Howell K C, Bando M, et al Low-thrust trajectory design with successive convex optimization for libration point orbits. Journal of Guidance, Control, and Dynamics, 2022, 45 (4): 623. doi: 10.2514/1.G005916
4	Circi C, Romagnoli D, Fumenti F Halo orbit dynamics and properties for a lunar global positioning system design. Monthly Notices of the Royal Astronomical Society, 2014, 442 (4): 3511. doi: 10.1093/mnras/stu1085
5	Gao Y T, Xu B, Zhang L Feasibility study of autonomous orbit determination using only the crosslink range measurement for a combined navigation constellation. Chinese Journal of Aeronautics, 2014, 27 (5): 1199. doi: 10.1016/j.cja.2014.09.005
6	Gao Y T, Hu C Y, Xin Y H, et al Design of a Cislunar space navigation constellation based on special long-period orbits. Advances in Space Research, 2024, 74 (7): 3137. doi: 10.1016/j.asr.2024.06.020
7	Simo C, Gomez G, Llibre J, et al On the optimal station keeping control of halo orbits. Acta Astronautica, 1987, 15 (6-7): 391. doi: 10.1016/0094-5765(87)90175-5
8	Duan M, Jia J, Ito T Fast terminal sliding mode control based on speed and disturbance estimation for an active suspension gravity compensation system. Mechanism and Machine Theory, 2021, 155, 104073. doi: 10.1016/j.mechmachtheory.2020.104073
9	Howell K C, Pernicka H J Station-keeping method for libration point trajectories. Journal of Guidance, Control, and Dynamics, 1993, 16 (1): 151. doi: 10.2514/3.11440
10	Xu M Stationkeeping strategy of halo orbit in linear periodic control. Aerospace Control, 2008, 26 (3): 13.
11	Xin M, Dancer M W, Balakrishnan S N, et al. Stationkeeping of an L/sub 2/libration point satellite with/spl theta/-D technique [C]//American Control Conference, 2004: 1037.
12	Darvish K, Pourtakdoust S H, Assadian N Linear and nonlinear control strategies for formation and station keeping of spacecrafts within the context of the three body problem. Aerospace Science and Technology, 2015, 42, 12. doi: 10.1016/j.ast.2014.12.030
13	Lian Y, Gomez G, Masdemont J J, et al Station-keeping of real Earth–Moon libration point orbits using discrete-time sliding mode control. Communications in Nonlinear Science and Numerical Simulation, 2014, 19 (10): 3792. doi: 10.1016/j.cnsns.2014.03.026
14	Wu L, Liu J, Vazquez S, et al Sliding mode control in power converters and drives: a review. IEEE/CAA Journal of Automatica Sinica, 2021, 9 (3): 392.
15	Zhuang K Y, Su H Y, Zhang K Q, et al Adaptive terminal sliding mode control for high-order nonlinear dynamic systems. Journal of Zhejiang University. Science, 2003, 4 (1): 58. doi: 10.1631/BF02841080
16	Man Z H, Yu X H Terminal sliding mode control of MIMO linear systems. IEEE Trans. on Circuits and Systems I: Fundamental Theory and Applications, 1997, 44 (11): 1065. doi: 10.1109/81.641769
17	Narula A, Biggs J D Fault-tolerant station-keeping on libration point orbits. Journal of Guidance, Control, and Dynamics, 2018, 41 (4): 879. doi: 10.2514/1.G003115
18	Zhu M, Karimi H R, Zhang H, et al Active disturbance rejection station-keeping control of unstable orbits around collinear libration points. Mathematical Problems in Engineering, 2014, 2014 (1): 410989. doi: 10.1155/2014/410989
19	Sun G J, Zhou M Q, Jiang X Q Non-cooperative spacecraft proximity control considering target behavior uncertainty. Astrodynamics, 2022, 6 (4): 399. doi: 10.1007/s42064-022-0133-5
20	Jin H Y, Zhang R Q, Wang L, et al Root locus analysis on parameter tuning robustness of linear active disturbance rejection control. Control Theory & Applications, 2018, 35 (11): 1648.
21	Zheng Q, Gao Z Q Active disturbance rejection control: between the formulation in time and the understanding in frequency. Control Theory and Technology, 2016, 14, 250. doi: 10.1007/s11768-016-6059-9
22	Izzo D, Sprague C I, Tailor D V. Machine learning and evolutionary techniques in interplanetary trajectory design[M]//Modeling and Optimization in Space Engineering: State of the Art and New Challenges. Bethlehem, USA: Springer, 2019: 191.
23	Quartullo R, Garulli A, Kolmanovsky I Periodic model predictive control for tracking halo orbits in the elliptic restricted three-body problem. IEEE Trans. on Control Systems Technology, 2023, 31 (5): 1971. doi: 10.1109/TCST.2023.3291548
24	Li L C, Colombo C, Gkolias I, et al Low-thrust station-keeping strategy toward exploiting the resonances in the geostationary region. Journal of Aerospace Engineering, 2024, 37 (2): 04023123. doi: 10.1061/JAEEEZ.ASENG-5330
25	Lozano-Cuadra F, Soret B. Multi-agent deep reinforcement learning for distributed satellite routing [C]//IEEE International Conference on Machine Learning for Communication and Networking, 2024. doi: 10.1109/ICMLCN59089.2024.10624767.
26	Albidah A B Halo orbits under some perturbations in cr3bp. Symmetry, 2023, 15 (2): 481. doi: 10.3390/sym15020481
27	Xu M Overview of orbital dynamics and control for libration point orbits. Journal of Astronautics, 2009, 30, 1299.
28	Zhang R Y A review of periodic orbits in the circular restricted three-body problem. Journal of Systems Engineering and Electronics, 2022, 33 (3): 612. doi: 10.23919/JSEE.2022.000059
29	Wu X J, Zeng L C, Gong Y K DRO computation and its perturbative force in the Earth-Moon system. Journal of Beijing University of Aeronautics and Astronautics, 2020, 46 (5): 883.
30	Lou Z P, Wang Y Robust station-keeping control of sun-earth/moon libration point orbits using electric propulsion. Journal of Aerospace Engineering, 2019, 32 (2): 04018142. doi: 10.1061/(ASCE)AS.1943-5525.0000971

Source of acceleration	Average magnitude of acceleration /$ ({\mathrm{m}} \cdot {{\mathrm{s}}^{ - 2}}) $
Lunar-centered gravity	$ 1.00 \times {10^{ - 3}} $
Earth’s gravity	$ 1.33 \times {10^{ - 3}} $
Solar gravity	$ 5.60 \times {10^{ - 6}} $
Solar radiation pressure	$ 8.66 \times {10^{ - 8}} $
Asymmetric gravity of the Moon	$ 1.88 \times {10^{ - 10}} $
Venusian gravity	$ 6.63 \times {10^{ - 10}} $
Jovian gravity	$ 7.16 \times {10^{ - 11}} $

Gravity sources	Nominal orbit	Simulation environment
Earth’s gravity and Lunar gravity	√	√
Solar gravity		√

Characteristic	Traditional Halo orbits	Special long-period orbits
Orbit period/days	14−16	76
Nonlinearity (measured via phase portrait curvature)	Mild	Strong
Position error without control (Δr in 10 days)/m	<50	>200
Trajectory stability	Higher	Weaker
Maneuver cost/(m/s)	0.1−1.0	30−80
Navigation suitability	Limited spatial coverage	Wide coverage via extended drift regions

Parameter	Value
Insertion time	2035,11,1
Running time/days	76
Integration time step/s	10
Initial position error/m	(10,10,10)
Initial velocity error/m/s	(0.01, 0.01, 0.01)
Initial ranging error/m	0.17

Parameter	Value
$ {c_1},{c_2},{c_3} $	1000
$ {k_1},{k_2},{k_3} $	1000
$ {g_1},{g_2},{g_3} $	10
$ {w_1},{w_2},{w_3} $	10
$ {L_{x1}},{L_{y1}},{L_{z1}} $	200
$ {L_{x2}},{L_{y2}},{L_{z2}} $	$ 1 \times {10^6} $
$ {L_{x3}},{L_{y3}},{L_{z3}} $	$ 1 \times {10^8} $
$ \xi $/m	0.17