# Tether Space Mobility Device Attitude Control during Tether Extension and Winding

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. TSMD Modeling and Formulation

#### 2.1. Analytical Model

#### 2.2. Flexible Body Formulation

**e**:

_{1},e

_{2}) and (e

_{5},e

_{6}) represent the XY coordinates of the node at the element end. Similarly, (e

_{3},e

_{4}) and (e

_{7},e

_{8}) represent the spatial derivative in the XY coordinate system of the node at each element end. Furthermore, the shape function $S$ is determined based on the assumption of a Bernoulli-Eulerian beam, as follows:

#### 2.3. Rigid Body Formulation

#### 2.4. Rigid Body Formulation

#### 2.5. Extending and Attaching the Tether

## 3. Validity of the Analytical Model

#### 3.1. Experimental Setup

#### 3.2. Analytical Model

^{®}Core

^{TM}i7-4790K CPU 4.00 GHz) is used.

#### 3.3. Winding the Tether at Constant Speed

^{2}from the motor characteristics obtained in advance.

#### 3.4. Comparison of Analytical and Experimental Results

## 4. Attitude Control When Extending and Winding the Tether

#### 4.1. Winding Control of the Tether Focusing on the Change in Kinetic Energy

- $T$: Tension of the tether
- $\phi $: Angle formed by the tether and the $X$ axis
- $I$: Moment of inertia of the rigid body
- $G$-${X}^{\prime}{Y}^{\prime}$: Object coordinate system with the center of gravity $G$ of the rigid body as the origin
- $w$: Distance from $G$ to the tip of the inlet
- $\alpha $: Angle of the tip of the inlet in the object coordinate system
- $\theta $: Rigid body rotation angle

#### 4.2. Prevention of Chattering by Hysteresis

#### 4.3. Numerical Simulation Conditions

#### 4.4. Numerical Simulation Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Chen, Y.; Huang, R.; He, L.; Ren, X.; Zheng, B. Dynamical modelling and control of space tethers: A review of space tether research. Nonlinear Dyn.
**2014**, 77, 1077–1099. [Google Scholar] [CrossRef] - Nohmi, M. Space verification experimental analysis for attitude control of a tethered space robot. Trans. JSME
**2014**, 80. [Google Scholar] [CrossRef] - Watanabe, T.; Fujii, H.; Kojima, H.; Singhose, W. Design of electric current profile for electrodynamic tether systems by input shaping method. J. Jpn. Soc. Aeronaut. Space Sci.
**2005**, 53, 569–576. [Google Scholar] - Aslanov, S.V.; Ledkov, S.A. Dynamics of towed large debris taking into account atmospheric disturbance. Acta Mech.
**2014**, 225, 2685–2697. [Google Scholar] [CrossRef] - Sabatini, M.; Gasbarri, P.; Palmerini, B.G. Elastic issues and vibration reduction in a tethered deorbiting mission. Adv. Space Res.
**2016**, 57, 1951–1964. [Google Scholar] [CrossRef] - Huang, P.; Hu, Z.; Zhang, F. Dynamic modelling and coordinated controller designing for the maneuverable tether-net space robot system. Multibody Syst. Dyn.
**2016**, 36, 115–141. [Google Scholar] [CrossRef] - Takehara, S.; Nishizawa, T.; Kawarada, M.; Hase, H.; Terumichi, H. Development of tether space mobility device. Comput. Methods Appl. Sci.
**2014**, 35, 255–274. [Google Scholar] - Shabana, A.A. Dynamics of Multibody Systems, 3rd ed.; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Berzeri, M.; Shabana, A.A. Development of simple models for the elastic forces in the absolute nodal co-ordinate formulation. J. Sound Vib.
**2000**, 235, 539–565. [Google Scholar] [CrossRef] - Wago, T.; Kobayashi, K.; Sugawara, Y. Improvement on evaluating axial elastic force in beam element based on the absolute nodal coordinate formulation by accurate mean axial strain measure. Trans. Jpn. Soc. Mech. Eng.
**2013**, 79, 2704–2713. (In Japanese) [Google Scholar] [CrossRef] - Komatsu, T.; Uenohara, M.; Iikura, S.; Murata, H.; Shimoyama, I. The development of an autonomous space robot operation testbed. J. Robot. Soc. Jpn.
**1990**, 8, 712–720. [Google Scholar] [CrossRef] - Murakami, W.; Shimachi, S.; Hagihara, Y.; Hashimoto, A.; Hakozaki, Y. Cargo transfer by tethered satellite system. Trans. JSME
**2006**, 41, 169–170. (In Japanese) [Google Scholar]

**Figure 12.**Relationship between the center of gravity (COG) of the TSMD and the point of action of the tension force at 0.5 s and 1.5 s.

Total length of the tether $l$ (m) | 2.355 |

Diameter of the tether $d$ (m) | 5.2 × 10^{−4} |

Density of the tether $\rho $ (kg/m^{3}) | 1.140 |

Transverse elastic modulus ${E}_{t}$ (GPa) | 1.93 |

Longitudinal elastic modulus ${E}_{l}$ (GPa) | 1.93 |

Element number of the tether | 70 |

Mass of the arm ${M}_{r1}$ (kg) | 0.0367 |

Mass of the TSMD ${M}_{r2}$ (kg) | 0.817 |

Mass of the human analog ${M}_{r3}$ (kg) | 7.78 |

Moment of inertia of the arm ${I}_{r1}$ (kg·m^{2}) | 9.4 × 10^{−6} |

Moment of inertia of the TSMD ${I}_{r2}$ (kg·m^{2}) | 2.6 × 10^{−3} |

Moment of inertia of the human analog ${I}_{r3}$ (kg·m^{2}) | 0.0548 |

Length of the arm ${l}_{r1}$ (m) | 0.055 |

Length of the TSMD (m) | 0.17 |

Length of the human analog (m) | 0.19 |

Width of the arm (m) | 0.006 |

Width of the TSMD (m) | 0.095 |

Width of the human analog (m) | 0.22 |

Gap of the arm ${G}_{p}$ (m) | 3.0 × 10^{−3} |

Spring constant of the inside wall of the arm ${k}_{1}$ (N/m) | 230 |

Spring constant of the edge of the arm ${k}_{2}$ (N/m) | 230 |

Damping coefficient of the inside wall of the arm ${c}_{1}$ (N/(m/s)) | 0.0866 |

Damping coefficient of the inside wall of the arm ${c}_{2}$ (N/(m/s)) | 0.0866 |

Coefficient of friction of the inside wall of the arm ${\mu}_{1}$ | 0.1 |

Coefficient of friction of the edge of the arm ${\mu}_{2}$ | 0.1 |

Condition | ${\mathit{\tau}}_{\mathit{l}}$ | ${\mathit{u}}_{+}$ | ${\mathit{u}}_{-}$ | Setting Value |
---|---|---|---|---|

Without control | - | - | - | - |

Control 1 | 0.1 | 0.02 | −0.03 | $\begin{array}{c}{a}_{1}={1.5310}^{-6}\\ {a}_{2}={1.0310}^{-3}\\ {\omega}_{c}=0.001\end{array}$ |

Control 2 | 0.1 | 0.02 | −0.03 | a = 1.5 × 10^{−6} |

Control 3 | 0.1 | 0.02 | −0.03 | - |

Distance to the attachment position (m) | 2.0 |

Mass of the tip of the tether (kg) | 0.1 |

Distance from natural length to the point mass (m) | 0.3 |

Spring constant for shooting the tip k (N/m) | 100 |

Spring constant for attaching the tip ${k}_{a}$ (N/m) | 1000 |

Target deflection during tether extension ${l}_{et}$ (m) | 0.01 |

Gain of drive constraint during tether extension ${K}_{e}$ | 10 |

© 2018 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Takehara, S.; Uematsu, Y.; Miyaji, W.
Tether Space Mobility Device Attitude Control during Tether Extension and Winding. *Machines* **2018**, *6*, 61.
https://doi.org/10.3390/machines6040061

**AMA Style**

Takehara S, Uematsu Y, Miyaji W.
Tether Space Mobility Device Attitude Control during Tether Extension and Winding. *Machines*. 2018; 6(4):61.
https://doi.org/10.3390/machines6040061

**Chicago/Turabian Style**

Takehara, Shoichiro, Yu Uematsu, and Wataru Miyaji.
2018. "Tether Space Mobility Device Attitude Control during Tether Extension and Winding" *Machines* 6, no. 4: 61.
https://doi.org/10.3390/machines6040061