#
A Tether System at the L_{1}, L_{2} Collinear Libration Points of the Mars–Phobos System: Analytical Solutions

^{*}

*Aerospace*: Advances in Aerospace Sciences and Technology)

## Abstract

**:**

_{1}or L

_{2}libration points of the Mars–Phobos system. The orbiting spacecraft deploying the tether is at the L

_{1}or L

_{2}libration point and is held at one of these unstable points by the low thrust of its engines. In this paper, the analysis is performed assuming that the tether length is constant. The equation of motion for the system in the polar reference frame is obtained. The stable equilibrium positions are found and the dependence of the tether angular oscillation period on the tether length is determined. An analytical solution in the vicinity of the stable equilibrium positions for small angles of deflection of the tether from the local vertical is obtained in Jacobi elliptic functions. The comparison of the numerical and analytical solutions for small angles of deflection is performed. The results show that the dependencies of the oscillation period on the length of the tether are fundamentally different for L

_{1}and L

_{2}points. Analytical expressions for the tether tension are derived, and the influence of system parameters on this force is investigated for static and dynamic cases.

## 1. Introduction

_{1}Mars–Phobos libration point and deployed toward Mars at a length slightly greater than the distance from Phobos. Paper [15] showed the maintenance of an L

_{1}-type artificial equilibrium point in the Sun (Earth + Moon) circular restricted three-body problem by means of an electric solar wind sail. The tether capture system is also a promising method for removing space debris [16,17,18,19,20,21]. The topic of dynamics and control of tether systems has received substantial attention [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36]. Huang et al. examined several new applications for the space tether during operation in orbit, focusing on the structure, dynamics and control [23]. Paper [24] discussed the diversity of tether modelling that has been undertaken recently, and showed that dynamics and control are the two fundamentally important aspects of all tether concepts, designs and mission architectures.

_{1}Operational Tether Experiment) to explore the surface of Phobos using a tether system ‘‘anchored’’ at the L

_{1}libration point of the Mars–Phobos system [10]. The tether release point was proposed to be an orbiting spacecraft hovering in the vicinity of the L

_{1}point. Once deployed, the small vehicle with a sensor package attached to the tether was expected to investigate Phobos. This mission concept is a synthesis of new technologies that would provide a unique platform for multiple sensors directed at Phobos as well as at Mars. The PHLOTE ConOps describes the PHLOTE mission and provides a key systems engineering document to support future mission development. However, such a complex innovative mission requires an additional theoretical justification and a variety of analytical models of the system motion. The work of [37] considers a mission similar to the PHLOTE mission, where a detailed study of the behavior of the tether system attached at the L

_{1}collinear libration point was performed using the classical Nehvil equations.

_{1}or L

_{2}collinear libration points of the Mars–Phobos system and to study the features of the tether motion near these positions. The system consists of the tether and the end mass attached to its end. The mathematical model is based on the differential equations of the classical circular restricted three-body problem [38,39,40,41,42]. The equation of motion for the tether system of constant length under the action of two gravitational fields (Mars–Phobos) and the centrifugal force associated with the rotation of the frame of the Mars–Phobos system are obtained in polar coordinates. The first integral of this differential equation is found and used to determine the phase trajectories and the stable equilibrium positions. The approximate analytical solutions of the equation of motion for the tether system are obtained using Jacobi elliptic functions. Next, the dependence of the oscillation period on the length of the tether is found. Finally, analytical expressions for the tether tension are derived, and the influence of system parameters on this force is investigated for static and dynamic cases. The results of this work can be used for PHLOTE-like mission design. It is worth noting that the obtained solutions for small tether deflection angles are of interest for the creation of the space elevator at the L

_{1}and L

_{2}libration points of the Mars–Phobos system in the future.

## 2. Mathematical Model: Finding Sustainable Positions

_{1}or L

_{2}libration points, either of which can be the attachment point for the tether.

#### 2.1. Tether Deflection Angle $\phi $

#### 2.2. Tether Deflection Angle $\psi $

## 3. Approximate Analytical Solutions

#### 3.1. Tether Deflection Angle $\phi $

#### 3.2. Tether Deflection Angle $\psi $

## 4. Oscillation Period of the Tether near the Stable Position

#### 4.1. Tether Deflection Angle $\phi $

#### 4.2. Tether Deflection Angle $\psi $

## 5. Tether Tension Force

#### 5.1. Static Tension

#### 5.2. Dynamic Tension

- The tether is stretched $(T>0)$ in all cases considered;
- The greater the amplitude of oscillation of the tether, the greater the period of oscillation.

## 6. Conclusions

_{1}or L

_{2}collinear libration points of the Mars–Phobos system, the equations of motion for the system for the case of massless and non-extensible tether with the end mass have been obtained. The first integrals of these differential equations have been found and used to determine the phase trajectories and the stable equilibrium positions. Simplified equations for small tether deflection angles in Jacobi elliptic functions have been obtained. The oscillation period of the system has been analytically found. It has been shown that the dependencies of the oscillation period on the tether length for L

_{1}and L

_{2}points are different. The obtained approximate analytical solutions and the results of the numerical integration of the original equations of motion for small angles of deflection of the tether are in good agreement. Analytical expressions have been obtained to determine the tether tension, and it has been shown that for the end mass of 50 kg, this force is small and does not exceed 1 N both for the static and dynamic states of the tether.

_{1}or L

_{2}libration point is a good stimulus for future research, which will also focus on the consideration of an elastic tether.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Olivieri, L.; Sansone, F.; Duzzi, M.; Francesconi, A. TED Project: Conjugating Technology Development and Educational Activities. Aerospace
**2019**, 6, 73. [Google Scholar] [CrossRef] [Green Version] - Shi, G.; Zhu, Z.H. Cooperative game-based multi-objective optimization of cargo transportation with floating partial space elevator. Acta Astronaut.
**2023**, 205, 110–118. [Google Scholar] [CrossRef] - Luo, S.; Cui, N.; Wang, X.; Fan, Y.; Shi, R. Model and Optimization of the Tether for a Segmented Space Elevator. Aerospace
**2022**, 9, 278. [Google Scholar] [CrossRef] - Aslanov, V.S.; Ledkov, A.S.; Misra, A.K.; Guerman, A.D. Dynamics of Space Elevator After Tether Rupture. J. Guid. Control. Dyn.
**2013**, 36, 986–992. [Google Scholar] [CrossRef] [Green Version] - Burov, A.A.; Guerman, A.D.; Kosenko, I.I. Tether orientation control for lunar elevator. Celest. Mech. Dyn. Astron.
**2014**, 120, 337–347. [Google Scholar] [CrossRef] - Weiwei, W.; Zhigang, W.; Jiafu, L. Conceptual Design and Mechanical Analysis of a Lunar Anchored Cislunar Tether. Cosm. Res.
**2023**, 61, 80–89. [Google Scholar] [CrossRef] - Ziegler, S.W.; Cartmell, M.P. Using Motorized Tethers for Payload Orbital Transfer. J. Spacecr. Rocket.
**2001**, 38, 904–913. [Google Scholar] [CrossRef] - Aslanov, V.S.; Ledkov, A.S. Swing Principle in Tether-Assisted Return Mission from an Elliptical Orbit. Aerosp. Sci. Technol.
**2017**, 71, 156–162. [Google Scholar] [CrossRef] - Luo, C.; Wen, H.; Jin, D.; Xu, S. Dynamics of a flexible multi-tethered satellite formation in a Halo orbit with uncertain parameters. Commun. Nonlinear Sci. Numer. Simul.
**2021**, 99, 105828. [Google Scholar] [CrossRef] - Kempton, K.; Pearson, J.; Levine, E.; Carroll, J.; Amzajerdian, F. Phase 1 Study for the Phobos L
_{1}Operational Tether Experiment (PHLOTE). End Report. NASA**2018**, 1, 1–91. Available online: https://ntrs.nasa.gov/citations/20190000916 (accessed on 1 September 2021). - Mashayekhi, M.J.; Misra, A.K. Optimization of tether-assisted asteroid deflection. J. Guid. Control. Dyn.
**2014**, 37, 898–906. [Google Scholar] [CrossRef] - Baião, M.F.; Stuchi, T.J. Dynamics of tethered satellites in the vicinity of the Lagrangian point L
_{2}of the Earth–Moon system. Astrophys. Space Sci.**2017**, 362, 134. [Google Scholar] [CrossRef] - Wong, B.; Patil, R.; Misra, A. Attitude dynamics of rigid bodies in the vicinity of the Lagrangian points. J. Guid. Control. Dyn.
**2008**, 31, 252–256. [Google Scholar] [CrossRef] - Aslanov, V.S. Dynamics of a Phobos-anchored tether near the L
_{1}libration point. Nonlinear Dyn.**2023**, 111, 1269–1283. [Google Scholar] [CrossRef] - Niccolai, L.; Caruso, A.; Quarta, A.A.; Mengali, G. Artificial Collinear Lagrangian Point Maintenance With Electric Solar Wind Sail. IEEE Trans. Aerosp. Electron. Syst.
**2020**, 56, 4467–4477. [Google Scholar] [CrossRef] - Shan, M.; Shi, L. Comparison of Tethered Post-Capture System Models for Space Debris Removal. Aerospace
**2022**, 9, 33. [Google Scholar] [CrossRef] - Lv, S.; Zhang, H.; Zhang, Y.; Ning, B.; Qi, R. Design of an Integrated Platform for Active Debris Removal. Aerospace
**2022**, 9, 339. [Google Scholar] [CrossRef] - Huang, P.; Zhang, F.; Cai, J.; Wang, D.; Meng, Z.; Guo, J. Dexterous tethered space robot: Design, measurement, control, and experiment. IEEE Trans. Aerosp. Electron. Syst.
**2017**, 53, 1452–1468. [Google Scholar] [CrossRef] - Feng, G.; Zhang, C.; Zhang, H.; Li, W. Theoretical and Experimental Investigation of Geomagnetic Energy Effect for LEO Debris Deorbiting. Aerospace
**2022**, 9, 511. [Google Scholar] [CrossRef] - Bourabah, D.; Field, L.; Botta, E.M. Estimation of uncooperative space debris inertial parameters after tether capture. Acta Astronaut.
**2023**, 202, 909–926. [Google Scholar] [CrossRef] - Zhang, F.; Sharf, I.; Misra, A.; Huang, P. On-line estimation of inertia parameters of space debris for its tether-assisted removal. Acta Astronaut.
**2015**, 107, 150–162. [Google Scholar] [CrossRef] - Aslanov, V.S.; Yudintsev, V.V. Chaos in Tethered Tug–Debris System Induced by Attitude Oscillations of Debris. J. Guid. Control. Dyn.
**2019**, 42, 1630–1637. [Google Scholar] [CrossRef] - Huang, P.; Zhang, F.; Chen, L.; Meng, Z.; Zhang, Y.; Liu, Z.; Hu, Y. A review of space tether in new applications. Nonlinear Dyn.
**2018**, 94, 1–19. [Google Scholar] [CrossRef] - Cartmell, M.P.; McKenzie, D.J. A Review of Space Tether Research. Prog. Aerosp. Sci.
**2008**, 44, 1–21. [Google Scholar] [CrossRef] - Shi, G.; Zhu, Z.H. Adaptive Anti-Saturation Prescribed-Time Control for Payload Retrieval of Tethered Space System. IEEE Trans. Aerosp. Electron. Syst.
**2023**, 99, 1–11. [Google Scholar] [CrossRef] - Salazar, F.J.T.; Prado, A.B.A. Deployment and Retrieval Missions from Quasi-Periodic and Chaotic States under a Non-Linear Control Law. Symmetry
**2022**, 14, 1381. [Google Scholar] [CrossRef] - Salazar, F.J.; Prado, A.F. Suppression of Chaotic Motion of Tethered Satellite Systems Using Tether Length Control. J. Guid. Control. Dyn.
**2022**, 45, 580–586. [Google Scholar] [CrossRef] - Misra, A.K.; Modi, V.J. Deployment and Retrieval of Shuttle Supported Tethered Satellites. J. Guid. Control. Dyn.
**1982**, 5, 278–285. [Google Scholar] [CrossRef] - Xu, S.; Sun, G.; Ma, Z.; Li, X. Fractional-order fuzzy sliding mode control for the deployment of tethered satellite system under input saturation. IEEE Trans. Aerosp. Electron. Syst.
**2018**, 55, 747–756. [Google Scholar] [CrossRef] - Ma, Z.; Huang, P. Discrete-Time Sliding Mode Control for Deployment of Tethered Space Robot With Only Length and Angle Measurement. IEEE Trans. Aerosp. Electron. Syst.
**2019**, 56, 585–596. [Google Scholar] [CrossRef] - Kang, J.; Zhu, Z.H.; Santaguida, L.F. Analytical and Experimental Investigation of Stabilizing Rotating Uncooperative Target by Tethered Space Tug. IEEE Trans. Aerosp. Electron. Syst.
**2021**, 57, 2426–2437. [Google Scholar] [CrossRef] - Kang, J.; Zhu, Z.H. Passivity-Based Model Predictive Control for Tethered Despin of Massive Space Objects by Small Space Tug. IEEE Trans. Aerosp. Electron. Syst.
**2022**, 59, 1239–1248. [Google Scholar] [CrossRef] - Ismail, N.A.; Cartmell, M.P. Three dimensional dynamics of a flexible Motorised Momentum Exchange Tether. Acta Astronaut.
**2016**, 120, 87–102. [Google Scholar] [CrossRef] [Green Version] - Chen, S.; Chen, W.; Chen, T.; Kang, J. In-Plane Libration Suppression of a Two-Segment Tethered Towing System. Aerospace
**2023**, 10, 286. [Google Scholar] [CrossRef] - Kim, M.; Hall, C.D. Control of a rotating variable-length tether system. J. Guid. Control. Dyn.
**2004**, 27, 849–858. [Google Scholar] [CrossRef] [Green Version] - Singh, S.; Junkins, J.; Anderson, B.; Taheri, E. Eclipse-conscious transfer to lunar gateway using ephemeris-driven terminal coast arcs. J. Guid. Control. Dyn.
**2021**, 44, 1972–1988. [Google Scholar] [CrossRef] - Aslanov, V.S. Prospects of a tether system deployed at the L
_{1}libration point. Nonlinear Dyn.**2021**, 106, 2021–2033. [Google Scholar] [CrossRef] - Szebehely, V. The Restricted Problem of Three Bodies; Academic Press: New York, NY, USA, 1967; 668p. [Google Scholar]
- Koon, W.S.; Lo, M.W.; Marsden, J.E.; Ross, S.D. Dynamical Systems, the Three-Body Problem and Space Mission Design; Springer: New York, NY, USA, 2011; 327p. [Google Scholar]
- Schaub, H.; Junkins, J.L. Analytical Mechanics of Space Systems; Aiaa: San Diego, CA, USA, 2003; 819p. [Google Scholar]
- Roy, A.E. Orbital Motion; CRC Press: Boca Raton, FL, USA, 2020; 505p. [Google Scholar]
- Kluever, C.A. Space Flight Dynamics; John Wiley & Sons: Hoboken, NJ, USA, 2018; 583p. [Google Scholar]
- Janke, E.; Emde, F.; Losch, F. Tafeln Hoherer Funktionen; BG Teubner Verlagsgeselschaft: Leipzig, Germany, 1960; 322p. [Google Scholar]

**Figure 2.**(

**a**) The potential energy $P\left(\phi \right)$ for the tether system attached at the ${L}_{1}$ libration point; (

**b**) the potential energy $P\left(\phi \right)$ for the tether system attached at the ${L}_{2}$ libration point; (

**c**) phase trajectories $\dot{\phi}\left(\phi \right)$ corresponding to different levels of the total energy ${E}_{j}\left(j=1,\hspace{0.17em}2,\hspace{0.17em}3,\hspace{0.17em}4\right)$ for the tether system attached at the ${L}_{1}$ libration point; (

**d**) phase trajectories $\dot{\phi}\left(\phi \right)$ corresponding to different levels of the total energy ${E}_{j}\left(j=1,\hspace{0.17em}2,\hspace{0.17em}3,\hspace{0.17em}4\right)$ for the tether system attached at the ${L}_{2}$ libration point.

**Figure 3.**(

**a**) The potential energy $P\left(\psi \right)$ for the tether system attached at the ${L}_{1}$ libration point; (

**b**) the potential energy $P\left(\psi \right)$ for the tether system attached at the ${L}_{2}$ libration point; (

**c**) the separatrices $\dot{\psi}\left(\psi \right)$ in the phase space corresponding to different levels of the total energy ${E}_{j}\left(j=1,\hspace{0.17em}2,\hspace{0.17em}3,\hspace{0.17em}4\right)$ for the tether system attached at the ${L}_{1}$ libration point; (

**d**) the separatrices $\dot{\psi}\left(\psi \right)$ in the phase space corresponding to different levels of the total energy ${E}_{j}\left(j=1,\hspace{0.17em}2,\hspace{0.17em}3,\hspace{0.17em}4\right)$ for the tether system attached at the ${L}_{2}$ libration point.

**Figure 4.**Time history of the tether deflection angle for the tether system attached in the ${L}_{1}$ libration point.

**Figure 5.**Time history of the tether deflection angle for the tether system attached in the ${L}_{2}$ libration point.

**Figure 6.**Time history of the tether deflection angle for the tether system attached in the ${L}_{1}$ libration point.

**Figure 7.**Time history of the tether deflection angle for the tether system attached in the ${L}_{2}$ libration point.

**Figure 8.**The oscillation periods of the tether systems attached at the ${L}_{1}$ and ${L}_{2}$ libration points.

**Figure 9.**The oscillation periods of the tether systems attached at the ${L}_{1}$ and ${L}_{2}$ libration points.

**Figure 10.**Tension force of the tether, as a function of its length for the end mass of 50 kg: (

**a**) tether attached at the ${L}_{1}$ libration point; (

**b**) tether attached at the ${L}_{2}$ libration point.

**Figure 11.**(

**a**) Tension force for the tether attached in the ${L}_{1}$ libration point; (

**b**) tension force for the tether attached in the ${L}_{2}$ libration point; (

**c**) tether deflection angle for the tether attached in the ${L}_{1}$ libration point; (

**d**) tether deflection angle for the tether attached in the ${L}_{2}$ libration point.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Aslanov, V.S.; Neryadovskaya, D.V.
A Tether System at the *L*_{1}, *L*_{2} Collinear Libration Points of the Mars–Phobos System: Analytical Solutions. *Aerospace* **2023**, *10*, 541.
https://doi.org/10.3390/aerospace10060541

**AMA Style**

Aslanov VS, Neryadovskaya DV.
A Tether System at the *L*_{1}, *L*_{2} Collinear Libration Points of the Mars–Phobos System: Analytical Solutions. *Aerospace*. 2023; 10(6):541.
https://doi.org/10.3390/aerospace10060541

**Chicago/Turabian Style**

Aslanov, Vladimir S., and Daria V. Neryadovskaya.
2023. "A Tether System at the *L*_{1}, *L*_{2} Collinear Libration Points of the Mars–Phobos System: Analytical Solutions" *Aerospace* 10, no. 6: 541.
https://doi.org/10.3390/aerospace10060541