# In-Plane Libration Suppression of a Two-Segment Tethered Towing System

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- With the elasticity of the tethers, the equilibrium configurations of a two-segment tethered towing system with constant thrust are obtained and the stabilities of equilibria are proved.
- (2)
- An in-plane sliding mode controller is designed to suppress the librations of the in-plane angles of the system.

## 2. Problem Formulation

## 3. Equilibria and Stability Analysis

#### 3.1. Equilibrium Configurations of the System

#### 3.2. Stability of Equilibrium Configuration

## 4. Libration Controller Design

^{T}, it is clear that $\dot{\mathit{V}}$ is negative. Hence, the system will converge to the sliding surface and reach the equilibrium asymptotically along the sliding surface. Therefore, the in-plane libration angles could converge to the desired state with ${u}_{1}$ and ${u}_{2}$.

## 5. Simulation Results

#### 5.1. Simulations for Equilibrium Configuration

#### 5.2. Simulations for Libration Controller

## 6. Conclusions

- (1)
- Seven sets of equilibrium configurations are given. The configuration in which the tug, sub-satellite, and debris remain in a straight line along the local horizontal is stable.
- (2)
- An in-plane libration controller is designed. According to the Lyapunov function, the system will converge to the sliding surface asymptotically. The system will reach the equilibrium asymptotically with the action of the designed libration controller. It can be seen from the Monte Carlo results that the control can converge within a short time.
- (3)
- It can be found from the simulation results that the librations in the direction of tether length can be effectively suppressed with the suppression of the oscillations of the in-plane angles. This is attributed to the coupling characteristics between in-plane angles and the tether length. As a result, the system under the action of the designed controller can reach the desired state while the librations of in-plane angles are effectively suppressed.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Schematic diagram of the solutions shown in Equation (23): (

**a**) ${E}_{2}$ and ${E}_{3}$; (

**b**) ${E}_{4}$ and ${E}_{5}$; (

**c**) ${E}_{6}$ and ${E}_{7}$.

**Figure 4.**Variations of the generalized coordinates: (

**a**) ${d}_{1}$; (

**b**) ${d}_{2}$; (

**c**) ${\theta}_{1}$; (

**d**) ${\theta}_{2}$.

**Figure 6.**Variations of the generalized coordinates with and without control: (

**a**) ${d}_{1}$; (

**b**) ${d}_{2}$; (

**c**) ${\theta}_{1}$; (

**d**) ${\theta}_{2}$.

**Figure 7.**Variations of the tether tensions with and without control: (

**a**) ${F}_{l1}$; (

**b**) ${F}_{l2}$.

Variables | Maximum Value without Control | Maximum Value with Control |
---|---|---|

${d}_{1}$ | 100.621 (m) | 100.482 (m) |

${d}_{2}$ | 9.121 (m) | 8.980 (m) |

${\theta}_{1}$ | 1.6519 (rad) | 1.5708 (rad) |

${\theta}_{2}$ | 1.6930 (rad) | 1.5708 (rad) |

${F}_{l1}$ | 130.146 (N) | 125.297 (N) |

${F}_{l2}$ | 129.745 (N) | 124.850 (N) |

Number | $\mathbf{Maximum}\mathbf{Control}\mathbf{Values}\left({\mathit{u}}_{1}\right)$ | $\mathbf{Maximum}\mathbf{Control}\mathbf{Values}\left({\mathit{u}}_{2}\right)$ | Settling Time | Number | $\mathbf{Maximum}\mathbf{Control}\mathbf{Values}\left({\mathit{u}}_{1}\right)$ | $\mathbf{Maximum}\mathbf{Control}\mathbf{Values}\left({\mathit{u}}_{2}\right)$ | Settling Time |
---|---|---|---|---|---|---|---|

1 | 5.3113 | 2.7763 | 0.85 | 26 | 5.0009 | 4.3501 | 0.77 |

2 | 4.8808 | 2.6520 | 0.87 | 27 | 9.1271 | 3.7823 | 0.89 |

3 | 3.9296 | 1.6403 | 0.82 | 28 | 1.0997 | 4.3210 | 0.79 |

4 | 1.1259 | 2.9257 | 0.78 | 29 | 5.9695 | 3.4041 | 0.86 |

5 | 17.5118 | 2.6065 | 0.97 | 30 | 13.4755 | 3.6302 | 0.98 |

6 | 4.5052 | 2.6899 | 0.84 | 31 | 0.9988 | 4.3829 | 0.72 |

7 | 3.9099 | 2.1041 | 0.87 | 32 | 18.5756 | 4.1907 | 1 |

8 | 23.4981 | 2.8793 | 1.03 | 33 | 1.2196 | 4.2315 | 0.71 |

9 | 9.7099 | 3.6143 | 0.93 | 34 | 2.1536 | 4.1356 | 0.87 |

10 | 2.9807 | 2.9669 | 0.84 | 35 | 2.8406 | 3.5581 | 0.74 |

11 | 9.0193 | 1.7830 | 0.91 | 36 | 5.4548 | 2.1753 | 0.85 |

12 | 22.0454 | 3.8974 | 1.03 | 37 | 20.7834 | 2.2356 | 1 |

13 | 10.2877 | 1.0020 | 0.91 | 38 | 21.7675 | 3.8868 | 1.03 |

14 | 3.0421 | 3.9762 | 0.67 | 39 | 6.7766 | 3.8235 | 0.83 |

15 | 12.9003 | 3.2324 | 0.94 | 40 | 3.2731 | 2.4333 | 0.69 |

16 | 18.3562 | 3.5349 | 1 | 41 | 15.2439 | 2.4094 | 0.97 |

17 | 4.3885 | 2.5658 | 0.84 | 42 | 4.5732 | 3.3041 | 0.83 |

18 | 23.1578 | 2.1886 | 1.04 | 43 | 3.2812 | 2.1940 | 0.78 |

19 | 2.0496 | 2.8360 | 0.73 | 44 | 3.6449 | 4.2722 | 0.80 |

20 | 12.4655 | 1.3433 | 0.93 | 45 | 13.9039 | 3.0007 | 0.97 |

21 | 4.7030 | 1.6657 | 0.84 | 46 | 5.8025 | 3.1464 | 0.81 |

22 | 17.4816 | 2.0840 | 0.98 | 47 | 5.2472 | 3.6166 | 0.84 |

23 | 12.4809 | 1.1169 | 0.94 | 48 | 5.3351 | 3.9363 | 0.79 |

24 | 5.6282 | 1.9968 | 0.84 | 49 | 17.3646 | 2.7448 | 0.99 |

25 | 2.4166 | 2.1073 | 0.83 | 50 | 14.0092 | 2.4622 | 0.96 |

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## Share and Cite

**MDPI and ACS Style**

Chen, S.; Chen, W.; Chen, T.; Kang, J.
In-Plane Libration Suppression of a Two-Segment Tethered Towing System. *Aerospace* **2023**, *10*, 286.
https://doi.org/10.3390/aerospace10030286

**AMA Style**

Chen S, Chen W, Chen T, Kang J.
In-Plane Libration Suppression of a Two-Segment Tethered Towing System. *Aerospace*. 2023; 10(3):286.
https://doi.org/10.3390/aerospace10030286

**Chicago/Turabian Style**

Chen, Shouxu, Weidong Chen, Ti Chen, and Junjie Kang.
2023. "In-Plane Libration Suppression of a Two-Segment Tethered Towing System" *Aerospace* 10, no. 3: 286.
https://doi.org/10.3390/aerospace10030286