Skip to main content
SpringerLink
Log in

In-plane global dynamics and ground experiment of a linear tethered formation with three satellites

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

This study focuses on the global dynamics of a linear tethered satellite formation (LTSF). Specifically, the influence of the initial states of the system on the motion forms and their critical values are presented in detail. First, an approximate but useful model for a high-dimensional nonlinear system is established in a noninertial reference frame, where three satellites and two space tethers are deemed to be particles and massless springs, respectively. Three types of typical in-plane motions, including the motions with spin axes perpendicular to an orbital plane, tangent to the orbit and passing through the Earth center, are examined in conjunction with the suppression of possible out-of-plane motions via control. Then, a Poincaré map is utilized to analyze the stability of the motion. A dynamic parameter domain is proposed to reveal three forms of motions and their critical values numerically. Finally, an equivalent ground experimental system is structured by virtue of the dynamic similarity principle to reproduce the dynamic characteristics of the orbital system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
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
Fig. 18
Fig. 19

Similar content being viewed by others

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Abbreviations

\(\varvec{DP}\) :

Jacobian

\(d_{0i}\) :

Distance between adjacent satellites

\(d_{g0i}\) :

Distance between adjacent simulators

\(EA\) :

Stiffness of the tether

\({\mathbf{F}}_{gi}^{{\text{c}}}\) :

Equivalent control force acting on simulator \(s_{i}\)

\({\mathbf{F}}_{gi}^{{\text{G}}}\) :

Equivalent gravity acting on simulator \(s_{i}\)

\({\mathbf{F}}_{gi}^{{{\text{IC}}}}\) :

Equivalent Coriolis force acting on simulator \(s_{i}\)

\({\mathbf{F}}_{gi}^{{{\text{Ie}}}}\) :

Equivalent carrier inertial force acting on simulator \(s_{i}\)

\({\mathbf{F}}_{i}^{{\text{c}}}\) :

Control force acting on satellite \(S_{i}\)

\({\mathbf{F}}_{i}^{{{\text{IC}}}}\) :

Coriolis force of satellite \(S_{i}\)

\({\mathbf{F}}_{i}^{{{\text{Ie}}}}\) :

Carrier inertial force of satellite \(S_{i}\)

\({\overline{\mathbf{F}}}_{i}^{{\text{c}}}\) :

Dimensionless control force

\(\varvec{f}\) :

Vector field

\(\overline{G}_{i}\) :

Dimensionless gravity

\({\mathbf{g}}_{i}\) :

Gravitational acceleration of satellite \(S_{i}\)

\(\overline{K}_{0,i}\) :

Dimensionless stiffness of the tether

\(L_{0}\) :

Unstrained length of the space tether

\(L_{g}\) :

Current length of the experimental tether

\({\mathbf{T}}_{i,j}\) :

Tension force from satellite \(S_{j}\) acting on \(S_{i}\)

\(\overline{T}_{i,0}^{{}}\) :

Dimensionless tension force

\(t\) :

Orbital time

\(t_{g}\) :

Experimental time

\({\mathbf{v}}_{i}\) :

Relative velocity of satellite \(S_{i}\)

\(x_{i}\),\(y_{i}\),\(z_{i}\) :

Coordinate variable of satellite \(S_{i}\) in the noninertial orbital frame

\(x_{gi}\),\(y_{gi}\),\(z_{gi}\) :

Coordinate variable of simulator \(s_{i}\) in the ground reference frame

\(\overline{x}_{i}\),\(\overline{y}_{i}\),\(\overline{z}_{i}\) :

Dimensionless coordinate variable

\(\delta_{i,j}\),\(\delta_{gi,j}\) :

Sign function

\(\theta\) :

In-plane angle for Situation 1

\(\theta_{0}\) :

Initial in-plane angle for Situation 1

\(\lambda_{q}\) :

Characteristic root of the Jacobian

\(\mu_{gE}\) :

Equivalent gravitational parameter

\(\mu_{E}\) :

Earth gravitational parameter

\(\nu\) :

Orbital true anomaly

\(\nu_{g}\) :

Equivalent true anomaly

\(\varvec{\xi }\) :

Dimensionless vector variable

\(\varvec{\xi }^{(k)}\) :

The kth Poincaré point

\(L_{g0}\) :

Unstrained length of the experimental tether

\(\overline{L}_{0}\) :

Dimensionless unstrained length of the tether

\(l_{r}\) :

Reference length

\(m\) :

Satellite mass

\(m_{g}\) :

Simulator mass

N :

Ascending node

O :

Earth center

\(o{ - }xyz\) :

Noninertial orbital frame

\(o_{g} { - }x_{g} y_{g} z_{g}\) :

Ground reference frame

\(\varvec{P}\) :

Poincaré map

\({\mathbf{r}}_{gi}\) :

Position vector of simulator \(s_{i}\)

\({\mathbf{r}}_{gO}\) :

Corresponding vector to \({\mathbf{r}}_{O}\)

\({\mathbf{r}}_{i}\) :

Position vector of satellite \(S_{i}\)

\({\mathbf{r}}_{O}\) :

Position vector of the Earth center

\({\overline{\mathbf{r}}}\) :

Dimensionless vector to \({\mathbf{r}}_{O}\)

\(S_{i}\) :

Satellite i

\(s_{i}\) :

Simulator i

\({\mathbf{T}}_{gi,j}\) :

Equivalent tether tensile force from simulator \(s_{j}\) acting on \(s_{i}\)

\(\xi_{1}\),\(\xi_{2}\),\(\xi_{3}\),\(\xi_{4}\) :

Dimensionless variable

\(\varvec{\xi }_{p}\) :

Fixed point

\(\Sigma\) :

Poincaré section

\(\tau\) :

Dimensionless time

\(\varphi\) :

In-plane angle for Situation 2

\(\varphi_{0}\) :

Initial in-plane angle for Situation 2

\(\phi\) :

In-plane angle for Situation 3

\(\phi_{0}\) :

Initial in-plane angle for Situation 3

\(\varvec{\Omega }\) :

Orbital angular velocity

\(\Omega_{g}\) :

Equivalent orbital angular velocity

\(\overline{\Omega }\) :

Dimensionless orbital angular velocity

\(\omega_{gi0}\) :

Initial angular velocity of simulator \(s_{i}\)

\(\varvec{\omega }_{i}\) :

Angular velocity of the subsatellite \(S_{i}\)

\(\omega_{i0}\) :

Initial angular velocity of subsatellite \(S_{i}\)

\(\overline{\omega }_{i0}\) :

Initial dimensionless angular velocity

\({{\text{d}} \mathord{\left/ {\vphantom {{\text{d}} {{\text{d}}t}}} \right. \kern-\nulldelimiterspace} {{\text{d}}t}}\) :

Differential operation with respect to orbital time

\({{\text{d}} \mathord{\left/ {\vphantom {{\text{d}} {{\text{dt}}_{g} }}} \right. \kern-\nulldelimiterspace} {{\text{dt}}_{g} }}\) :

Differential operation with respect to experimental time

\( ( \cdot ) \) :

Differential operation with respect to dimensionless time \(\tau\)

References

  1. Kojima, H., Iwashima, H., Trivailo, P.M.: Libration synchronization control of clustered electrodynamic tether system using Kuramoto model. J. Guid. Control. Dyn. 34(3), 706–718 (2011)

    Article  Google Scholar 

  2. Iki, K., Kawamoto, S., Yoshiki, M.: Experiments and numerical simulations of an electrodynamical tether deployment from a spool-type reel using thrusters. Acta Astronaut. 94(1), 318–327 (2014)

    Article  Google Scholar 

  3. Steindl, A.: Optimal control of the deployment (and retrieval) of a tethered satellite under small initial disturbances. Meccanica 49(8), 1879–1885 (2014)

    Article  MathSciNet  Google Scholar 

  4. Pang, Z.J., Jin, D.P.: Experimental verification of chaotic control of an underactuated tethered satellite system. Acta Astronaut. 120(1), 287–294 (2016)

    Article  Google Scholar 

  5. Botta, E.M., Sharf, I., Misra, A.K.: Contact dynamics modeling and simulation of tether nets for space-debris capture. J. Guid. Control. Dyn. 40(1), 110–123 (2017)

    Article  Google Scholar 

  6. Xu, S.D., Sun, G.H., Ma, Z.Q., Li, X.L.: Fractional-order fuzzy sliding mode control for the deployment of tethered satellite system under input saturation. IEEE Trans. Aerosp. Electron. Syst. 55(2), 747–756 (2019)

    Article  Google Scholar 

  7. Ma, Z.Q., Huang, P.F.: Nonlinear analysis of discrete-time sliding mode prediction deployment of tethered space robot. IEEE Trans. Industr. Electron. 68(6), 5166–5175 (2021)

    Article  Google Scholar 

  8. Sanmartín, J.R., Lorenzini, E.C., Martinez-Sanchez, M.: Electrodynamic tether applications and constraints. J. Spacecr. Rocket. 47(3), 442–456 (2010)

    Article  Google Scholar 

  9. Peláez, J., Bombardelli, C., Scheeres, D.J.: Dynamics of a tethered observatory at Jupiter. J. Guid. Control. Dyn. 35(1), 195–207 (2012)

    Article  Google Scholar 

  10. Zhang, F., Huang, P.F.: Inertia parameter estimation for a noncooperative target captured by a space tethered system. J. Astronaut. 36(6), 630–639 (2015). ((in Chinese))

    Google Scholar 

  11. Aslanov, V.S., Ledkov, A.S.: Swing principle in tether-assisted return mission from an elliptical orbit. Aerosp. Sci. Technol. 71(1), 156–162 (2017)

    Article  Google Scholar 

  12. Qi, R., Yao, F.Z., Lu, S., Jiang, Z.H.: Stabilization of tethered tug-debris system with residual liquid fuel. J. Guid. Control. Dyn. 44(4), 880–888 (2021)

    Article  Google Scholar 

  13. Li, G.Q., Zhu, Z.H., Shi, G.F.: A novel looped space tether transportation system with multiple climbers for high efficiency. Acta Astronaut. 179(1), 253–265 (2021)

    Article  Google Scholar 

  14. Yu, B.S., Wen, H., Jin, D.P.: Advances in dynamics and control of tethered satellite formations. J. Dyn. Control 13(5), 321–328 (2015). ((in Chinese))

    Google Scholar 

  15. Tan, Z., Bainum P. M.: Tethered satellite constellations in auroral observation mission. In: Proceedings of AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Monterey, USA, 2002.

  16. Topal E., Daybege U.: Dynamics of a trianglar tethered satellite system on a low earth orbit. In: Proceedings of 2nd International Conference on Recent Advances in Space Technologies, Istanbul, Turkey, 2005.

  17. Chung S. -J., Nonlinear control and synchronization of multiple lagrangian systems with application to tethered formation flight spacecraft, Massachusetts Institute of Technology, Doctoral thesis, Cambridge, USA, 2007.

  18. Huang, P.F., Zhang, F., Chen, L., Meng, Z.J., Zhang, Y.Z., Liu, Z.X., Hu, Y.X.: A review of space tether in new applications. Nonlinear Dyn. 94(1), 1–19 (2018)

    Article  Google Scholar 

  19. Kumar, K.D., Yasaka, T.: Dynamics of rotating linear array tethered satellite system. J. Spacecr. Rocket. 42(2), 373–378 (2005)

    Article  Google Scholar 

  20. Williams, P.: Optimal deployment/retrieval of a tethered formation spinning in the orbital plane. J. Spacecr. Rocket. 43(3), 638–650 (2006)

    Article  Google Scholar 

  21. Larsen, M.B., Smith, R.S., Blanke, M.: Modeling of tethered satellite formations using graph theory. Acta Astronaut. 69(7–8), 470–479 (2011)

    Article  Google Scholar 

  22. Avanzini, G., Fedi, M.: Effects of eccentricity of the reference orbit on multi-tethered satellite formations. Acta Astronaut. 94(1), 338–350 (2014)

    Article  Google Scholar 

  23. Cai, Z.Q., Li, X.F., Zhou, H.: Nonlinear dynamics of a rotating triangular tethered satellite formation near libration points. Aerosp. Sci. Technol. 42(1), 384–391 (2015)

    Article  Google Scholar 

  24. Jung, W., Mazzoleni, A.P., Chung, J.: Nonlinear dynamic analysis of a three-body tethered satellite system with deployment/retrieval. Nonlinear Dyn. 82(1), 1127–1144 (2015)

    Article  MathSciNet  Google Scholar 

  25. Shi, G.F., Zhu, Z.X., Chen, S.Y., Yuan, J.P., Tang, B.W.: The motion and control of a complex three-body space tethered system. Adv. Space Res. 60(10), 2133–2145 (2017)

    Article  Google Scholar 

  26. Zhai, G., Bi, X.Z., Liang, B.: Optimal deployment of spin-stabilized tethered formations with continuous thrusters. Nonlinear Dyn. 95(1), 2143–2162 (2019)

    MATH  Google Scholar 

  27. Fang, G.T., Zhang, Y.Z., Huang, P.F., Zhang, F.: State estimation of double-pyramid tethered satellite formations using only two GPS sensors. Acta Astronaut. 180(1), 507–515 (2021)

    Article  Google Scholar 

  28. Chung, S.-J., Slotine, J.-J.E., Miller, D.W.: Nonlinear model reduction and decentralized control of tethered formation flight. J. Guid. Control. Dyn. 30(2), 390–400 (2007)

    Article  Google Scholar 

  29. Chung, S.-J., Miller, D.W.: Propellant-free control of tethered formation flight, Part 1: linear control and experiment. J. Guid. Control. Dyn. 31(3), 571–584 (2008)

    Article  Google Scholar 

  30. Iki, K., Kawamoto, S., Morino, Y.: Experiments and numerical simulations of an electrodynamic tether deployment from a spool-type reel using thrusters. Acta Astronaut. 94(1), 318–327 (2014)

    Article  Google Scholar 

  31. Mantellato, R., Lorenzini, E.C., Sternberg, D., Roascio, D., Saenz-Otero, A., Zachrau, H.J.: Simulation of a tethered microgravity robot pair and validation on a planar air bearing. Acta Astronaut. 138(1), 579–589 (2017)

    Article  Google Scholar 

  32. Yu, B.S., Huang, Z., Geng, L.L., Jin, D.P.: Stability and ground experiments of a spinning triangular tethered satellite formation on a low earth orbit. Aerosp. Sci. Technol. 92(1), 595–604 (2019)

    Article  Google Scholar 

  33. Forshaw, J.L., Aglietti, G.S., Fellowes, S., Salmon, T., Retat, I., Hall, A., Chabot, T., Pisseloup, A., Tye, D., Bernal, C., Chaumette, F., Pollini, A., Steyn, W.H.: The active space debris removal mission RemoveDebris. Part 1: From concept to launch. Acta Astronautica 168(1), 293–309 (2020)

    Article  Google Scholar 

  34. Yamagiwa, Y., Fujii, T., Nakashima, K., Oshimori, H., Okino, T., Komua, S., Arita, S., Nohmi, M., Ishikawa, Y.: Space experimental results of STARS-C CubeSat to verify tether deployment in orbit. Acta Astronaut. 177(1), 759–770 (2020)

    Article  Google Scholar 

  35. Chai, R.Q., Tsourdos, A., Savvaris, A., Chai, S.C., Xia, Y.Q., Chen, C.L.P.: Six-DOF spacecraft optimal trajectory planning and real-time attitude control: A deep neural network-based approach. IEEE Trans. Neural Netw. Learn. Syst. 31(11), 5005–5013 (2020)

    Article  Google Scholar 

  36. Chai, R.Q., Tsourdos, A., Savvaris, A., Xia, Y.Q., Chai, S.C.: Real-time reentry trajectory planning of hypersonic vehicles: A two-step strategy incorporating fuzzy multi-objective transcription and deep neural network. IEEE Trans. Industr. Electron. 67(8), 6904–6915 (2020)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the Natural Science Foundation of China (12072147, 11732006, and 11672125), the Natural Science Foundation of Jiangsu Province of China (BK20211177), the Aviation Science Foundation of China (2020Z063052001), and the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (MCMS-I-0120G03).

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People’s Republic of China

    B. S. Yu, K. Ji, Z. T. Wei & D. P. Jin

Authors
  1. B. S. Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Z. T. Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. D. P. Jin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. P. Jin.

Ethics declarations

Conflict of interest

The authors declare that they have no potential conflicts of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A: Cases of different motion forms for Situation 2

Figures 20, 21, 22, 23, 24, and 25.

Fig. 20
figure 20

Pendulum-like oscillation of the system (\(\omega_{i0} = - 1.{6}\Omega\))

Full size image
Fig. 21
figure 21

Pendulum-like oscillation of the experimental system (\(\omega_{gi0} = - 1.6\Omega_{g}\))

Full size image
Fig. 22
figure 22

Irregular motion of the system (\(\omega_{i0} = - {2}.{0}\Omega\))

Full size image
Fig. 23
figure 23

Irregular motion of the experimental system (\(\omega_{gi0} = - 2.0\Omega_{g}\))

Full size image
Fig. 24
figure 24

Spinning motion of the system (\(\omega_{i0} = - {2}.{6}\Omega\))

Full size image
Fig. 25
figure 25

Spinning motion of the experimental system (\(\omega_{gi0} = - 2.6\Omega_{g}\))

Full size image

Appendix B: Cases of different motion forms for Situation 3

Figures 26, 27, 28, 29.

Fig. 26
figure 26

Irregular motion of the system (\(\omega_{i0} = - 1.{3}\Omega\))

Full size image
Fig. 27
figure 27

Irregular motion of the experimental system (\(\omega_{gi0} = - 1.3\Omega_{g}\))

Full size image
Fig. 28
figure 28

Spinning motion of the system (\(\omega_{i0} = - 1.{6}\Omega\))

Full size image
Fig. 29
figure 29

Spinning motion of the experimental system (\(\omega_{gi0} = - 1.6\Omega_{g}\))

Full size image

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yu, B.S., Ji, K., Wei, Z.T. et al. In-plane global dynamics and ground experiment of a linear tethered formation with three satellites. Nonlinear Dyn 108, 3247–3278 (2022). https://doi.org/10.1007/s11071-022-07403-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07403-9

Keywords

Search

Navigation