# Configuration Optimization for Free-Floating Space Robot Capturing Tumbling Target

## Abstract

**:**

## 1. Introduction

## 2. Integrated Effective Mass

#### 2.1. Inertial Properties Perceived at End-Effector

#### 2.2. Analytical Expression of Integrated Effective Mass

## 3. Continuous Contact Model between Space Robot and Tumbling Target

## 4. Configuration Optimization for Capturing Tumbling Target

## 5. Numerical Simulations

#### 5.1. Simulation for a 3-Degree-of-Freedom Free-Floating Space Robot

#### 5.1.1. Accuracy of Proposed Maximum-Contact-Force Model for 3-DOF Space Robot

- I.
- Different capture configurations produce different integrated effective masses;
- II.
- The maximum contact force increases as the integrated effective mass increases;
- III.
- The optimization of maximum contact force is necessary as its value may vary widely.

#### 5.1.2. Configuration Optimization for 3-DOF Space Robot

#### 5.2. Simulation for a 7-Degree-of-Freedom Free-Floating Space Robot

#### 5.2.1. Accuracy of Proposed Maximum-Contact-Force Model for 7-DOF Space Robot

#### 5.2.2. Configuration Optimization for 7-DOF Space Robot

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Liu, H.; Liu, D.Y.; Jiang, Z.N. Review and Prospect of Space Manipulator Technology. Hangkong Xuebao
**2021**, 42, 524164. [Google Scholar] - Flores-Abad, A.; Ma, O.; Pham, K.; Ulrich, S. A Review of Space Robotics Technologies for On-orbit Servicing. Prog. Aerosp. Sci.
**2014**, 68, 1–26. [Google Scholar] [CrossRef] [Green Version] - Pan, G.; Jia, Q.; Chen, G.; Li, T.; Liu, C. A Control Method of Space Manipulator for Peg-in-Hole Assembly Task Considering Equivalent Stiffness Optimization. Aerospace
**2021**, 8, 310. [Google Scholar] [CrossRef] - Li, Z.; Wang, B.; Liu, H. Target Capturing Control for Space Robots with Unknown Mass Properties: A Self-Tuning Method Based on Gyros and Cameras. Sensors
**2016**, 16, 1383. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sun, Y.; Wang, Q.; Liu, Y.; Xie, Z.; Jin, M.; Liu, H. A Survey of Non-cooperative Target Capturing Methods. Guofang Keji Daxue Xuebao
**2020**, 42, 74–90. [Google Scholar] - Stolfi, A.; Gasbarri, P.; Sabatini, M. A Parametric Analysis of a Controlled Deployable Space Manipulator for Capturing a Non-cooperative Flexible Satellite. Acta Astronaut.
**2018**, 148, 317–326. [Google Scholar] [CrossRef] - Zhang, L.; Jia, Q.; Chen, G.; Sun, H. Pre-impact Trajectory Planning for Minimizing Base Attitude Disturbance in Space Manipulator Systems for a Capture Task. Chin. J. Aeronaut.
**2015**, 28, 1199–1208. [Google Scholar] [CrossRef] [Green Version] - Yoshida, K.; Sashida, N. Modeling of Impact Dynamics and Impulse Minimization for Space Robots. In Proceedings of the 1993 IEEE/RSJ International Conference on Intelligent Robots and Systems, Yokohama, Japan, 26–30 July 1993; pp. 2064–2069. [Google Scholar]
- Flores-Abad, A.; Wei, Z.; Ma, O.; Pham, K. Optimal Control of Space Robots for Capturing a Tumbling Object with Uncertainties. J. Guid. Control Dyn.
**2014**, 37, 2014–2017. [Google Scholar] [CrossRef] - Liu, Y.; Du, Z.; Wu, Z.; Liu, F.; Li, X. Multiobjective Preimpact Trajectory Planning of Space Manipulator for Self-assembling a Heavy Payload. Int. J. Adv. Rob. Syst.
**2021**, 18, 17298806. [Google Scholar] [CrossRef] - James, F.; Shah, S.; Singh, A.; Krishna, K.; Misra, A. Reactionless Maneuvering of a Space Robot in Precapture Phase. J. Guid. Control Dyn.
**2016**, 39, 2417–2423. [Google Scholar] [CrossRef] - Xu, W.; Hu, Z.; Yan, L.; Yuan, H.; Liang, B. Modeling and Planning of a Space Robot for Capturing Tumbling Target by Approaching the Dynamic Closest Point. Multibody Syst. Dyn.
**2019**, 74, 203–241. [Google Scholar] [CrossRef] - Huang, P.; Wang, M.; Meng, Z.; Zhang, F.; Liu, Z. Attitude Takeover Control for Post-capture of Target Spacecraft using Space Robot. Aerosp. Sci. Technol.
**2016**, 51, 171–180. [Google Scholar] [CrossRef] - Wang, M.; Luo, J.; Wang, J.; Yuan, J. Detumbling Strategy and Impedance Control for Space Robot after Capturing an Uncooperative Satellite. Jiqiren
**2018**, 40, 750–761. [Google Scholar] - Gangapersaud, R.; Liu, G.; Ruiter, A. Detumbling of a Non-cooperative Target with Unknown Inertial Parameters using a Space Robot. Adv. Space Res.
**2019**, 63, 3900–3915. [Google Scholar] [CrossRef] - Chen, G.; Wang, Y.; Wang, Y.; Liang, J.; Zhang, L.; Pan, G. Detumbling Strategy based on Friction Control of Dual-arm Space Robot for Capturing Tumbling Target. Chin. J. Aeronaut.
**2020**, 33, 1093–1106. [Google Scholar] [CrossRef] - Wee, L.; Walker, M. On the Dynamics of Contact between Space Robots and Configuration Control for Impact Minimization. IEEE Trans. Robot. Autom.
**1993**, 9, 581–591. [Google Scholar] - Yoshida, K.; Dragomir, N. Space Robot Impact Analysis and Satellite-Base Impulse Minimization Using Reaction Null-Space. In Proceedings of the IEEE International Conference on Robotics and Automation, Nagoya, Japan, 21–27 May 1995; pp. 1271–1277. [Google Scholar]
- Nenchev, D.; Yoshida, K. Impact Analysis and Post-Impact Motion Control Issues of a Free-Floating Space Robot Subject to a Force Impulse. IEEE Trans. Robot. Autom.
**1999**, 15, 548–557. [Google Scholar] [CrossRef] - Huang, P.; Xu, Y.; Liang, B. Contact and Impact Dynamics of Space Manipulator and Free-flying Target. In Proceedings of the 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems, Edmonton, AB, Canada, 2–6 August 2005; pp. 1935–1940. [Google Scholar]
- Zhang, L.; Jia, Q.; Chen, G.; Sun, H. The Precollision Trajectory Planning of Redundant Space Manipulator for Capture Task. Adv. Mech. Eng.
**2014**, 2014, 1–10. [Google Scholar] [CrossRef] - Xie, R.; Kong, X.; Shi, P.; Zhao, Y. Impact Analysis of Space Manipulators During Target Capture. Comput. Simul.
**2016**, 33, 438–441. [Google Scholar] - Luka, S.; Janko, S.; Miha, B. A Review of Continuous Contact-force Models in Multibody Dynamics. Int. J. Mech. Sci.
**2018**, 145, 171–187. [Google Scholar] - Machado, M.; Moreira, P.; Flores, P.; Lankarani, H. Compliant Contact Force Models in Multibody Dynamics: Evolution of the Hertz Contact Theory. Mech. Mach. Theory
**2012**, 53, 99–121. [Google Scholar] [CrossRef] - Zhang, J.; Li, W.; Zhao, L.; He, G. A Continuous Contact Force Model for Impact Analysis in Multibody Dynamics. Mech. Mach. Theory
**2020**, 153, 103946. [Google Scholar] [CrossRef] - Yoshikawa, S.; Katsuhiko, Y. Dynamics Evaluation for Capturing a Target by a Space Robot. J. Robot. Soc. Jpn.
**1997**, 15, 1034–1042. [Google Scholar] [CrossRef] - Ji, H.; Li, M.; Wang, X.; Li, Y.; Dou, L. Contact-impact Dynamics Simulation for Space Manipulator using Equivalent Spring-damping Model. In Proceedings of the World Congress on Intelligent Control and Automation, Shenyang, China, 29 June–4 July 2014; pp. 3186–3191. [Google Scholar]
- Wu, S.; Mou, F.; Liu, Q.; Cheng, J. Contact Dynamics and Control of a Space Robot Capturing a Tumbling Object. Acta Astronaut.
**2018**, 151, 532–542. [Google Scholar] [CrossRef] - She, Y.; Li, S.; Hu, J. Contact Dynamics and Relative Motion Estimation of Non-cooperative Target with Unilateral Contact Constraint. Aerosp. Sci. Technol.
**2020**, 98, 105705. [Google Scholar] [CrossRef] - Yoshida, K.; Sashida, N.; Kurazume, R.; Umetani, Y. Modeling of Collision Dynamics for Space Free-Floating Links with Extended Generalized Inertia Tensor. In Proceedings of the IEEE International Conference on Robotics and Automation, Nice, France, 12–14 May 1992; pp. 899–904. [Google Scholar]
- Jia, Q.; Zhang, L.; Chen, G.; Sun, H. Collision Analysis of Space Manipulator Multi-body System based on Equivalent Mass. Yuhang Xuebao
**2015**, 36, 1356–1362. [Google Scholar] - Zhang, L.; Jia, Q.; Chen, G.; Sun, H.; Cao, L. Impact Analysis of Space Manipulator Collision with Soft Environment. In Proceedings of the 9th IEEE Conference on Industrial Electronics and Applications, Hangzhou, China, 9–11 June 2014; pp. 1965–1970. [Google Scholar]
- Chen, G.; Liu, D.; Wang, Y.F.; Jia, Q.; Liu, X. Contact Force Minimization for Space Flexible Manipulators based on Effective Mass. J. Guid. Control Dyn.
**2019**, 42, 1870–1877. [Google Scholar] [CrossRef] - Gilardi, G.; Sharf, I. Literature Survey of Contact Dynamics Modelling. Mech. Mach. Theory
**2002**, 37, 1213–1239. [Google Scholar] [CrossRef] - Khatib, O. Inertial Properties in Robotic Manipulation. An Object-level Framework. Int. J. Rob. Res.
**1995**, 14, 19–36. [Google Scholar] [CrossRef] - Cong, P.; Sun, Z. Pre-impact Configuration Planning for Capture Object of Space Manipulator. In Proceedings of the 2nd International Symposium on Systems and Control in Aerospace and Astronautics, Shenzhen, China, 10–12 December 2008; p. 4776179. [Google Scholar]

**Figure 4.**Errors with different stiffness parameters (${\dot{\delta}}^{(-)}=0.2\mathrm{m}/\mathrm{s},{c}_{\mathrm{r}}=0.9$).

**Figure 5.**Errors with different initial relative contact velocities ($K={10}^{9}\mathrm{N}/{\mathrm{m}}^{1.5},{c}_{\mathrm{r}}=0.9$).

**Figure 6.**Errors with different restitution coefficients ($K={10}^{9}\mathrm{N}/{\mathrm{m}}^{1.5},{\dot{\delta}}^{(-)}=0.2\mathrm{m}/\mathrm{s}$).

**Figure 9.**Slice map at ${\theta}_{1}=0\mathrm{deg}$, ${\theta}_{2}=0\mathrm{deg}$ and ${\theta}_{3}=0\mathrm{deg}$.

**Figure 14.**Capture process of 7-dof space robot with maximum-contact-force optimization and the constraint of specified pose of the end-effector.

**Figure 15.**Capture process of 7-dof space robot with maximum-contact-force optimization and the constraint of specified position of the end-effector.

Model | Hysteresis Damping Factor | Model | Hysteresis Damping Factor |
---|---|---|---|

Herbert–McWhannell | $\lambda =\frac{6\left(1-{c}_{\mathrm{r}}\right)}{\left({\left(2{c}_{\mathrm{r}}-1\right)}^{2}+3\right)}\frac{K}{{\dot{\delta}}^{(-)}}$ | Hunt-Crossley | $\lambda =\frac{3\left(1-{c}_{\mathrm{r}}\right)}{2}\frac{K}{{\dot{\delta}}^{(-)}}$ |

Lankarain–Nikravesh | $\lambda =\frac{3\left(1-{c}_{\mathrm{r}}{}^{2}\right)}{4}\frac{K}{{\dot{\delta}}^{(-)}}$ | Lee-Wang | $\lambda =\frac{3\left(1-{c}_{\mathrm{r}}\right)}{4}\frac{K}{{\dot{\delta}}^{(-)}}$ |

Flores et al. | $\lambda =\frac{8\left(1-{c}_{\mathrm{r}}\right)}{5{c}_{\mathrm{r}}}\frac{K}{{\dot{\delta}}^{(-)}}$ | Gonthier et al. | $\lambda =\frac{1-{c}_{\mathrm{r}}{}^{2}}{{c}_{\mathrm{r}}}\frac{K}{{\dot{\delta}}^{(-)}}$ |

Zhiying–Qishao | $\lambda =\frac{3\left(1-{c}_{\mathrm{r}}{}^{2}\right){\mathrm{e}}^{2\left(1-{c}_{\mathrm{r}}\right)}}{4}\frac{K}{{\dot{\delta}}^{(-)}}$ | Hu-Guo | $\lambda =\frac{3\left(1-{c}_{\mathrm{r}}\right)}{2{c}_{\mathrm{r}}}\frac{K}{{\dot{\delta}}^{(-)}}$ |

Part | Mass (kg) | Inertia Matrix (kg•m^{2}) |
---|---|---|

Link 1 | 5 | diag([0.01, 0.82, 0.82]) |

Link 2 | 6 | diag([0.01, 1.28, 1.28]) |

Link 3 | 4 | diag([0.00, 0.09, 0.09]) |

Base | 500 | diag([200, 200, 200]) |

Target | 200 | diag([100, 100, 100]) |

Capture Configuration (Deg) | Integrated Effective Mass (kg) | Proposed Method (kN) | Numerical Integration Method (kN) | Model Accuracy |
---|---|---|---|---|

[36, 30, −4] | 1.555 | 0.834 | 0.844 | 98.83% |

[85, −41, 11] | 1.648 | 0.863 | 0.873 | 98.84% |

[−75, 30, 12] | 3.679 | 1.397 | 1.414 | 98.86% |

[10, 86, −76] | 4.299 | 1.534 | 1.552 | 98.85% |

[0, 90, −90] | 5.817 | 1.840 | 1.861 | 98.84% |

[30, 21, −56] | 5.926 | 1.860 | 1.882 | 98.84% |

[54, −30, −5] | 7.446 | 2.133 | 2.158 | 98.87% |

[−10, −18, 30] | 9.635 | 2.490 | 2.519 | 98.84% |

[46, −45, 0] | 13.74 | 3.081 | 3.119 | 98.78% |

[−6, 20, −7] | 21.093 | 3.985 | 4.036 | 98.73% |

Capture Configuration (Deg) | Integrate Effective Mass (kg) | The Maximum Contact Force (kN) | |
---|---|---|---|

Without optimization | [−2.29, 91.75, −79.77] | 5.43 | 1.71 |

With optimization | [−16.87, 88.06, 11.33] | 1.22 | 0.72 |

Part | Mass (kg) | Inertia Matrix (kg•m^{2}) |
---|---|---|

Link 1 | 5 | diag([0.01, 0.02, 0.02]) |

Link 2 | 5 | diag([0.02, 0.01, 0.02]) |

Link 3 | 10 | diag([0.84, 0.01, 0.84]) |

Link 4 | 10 | diag([0.01, 0.84, 0.84]) |

Link 5 | 5 | diag([0.02, 0.02, 0.01]) |

Link 6 | 5 | diag([0.02, 0.02, 0.01]) |

Link 7 | 8 | diag([0.03, 0.03, 0.01]) |

Base | 1000 | diag([500, 500, 500]) |

Target | 200 | diag([100, 100, 100]) |

Capture Configuration (Deg) | Integrated Effective Mass (kg) | Proposed Method (kN) | Numerical Integration Method (kN) | Model Accuracy |
---|---|---|---|---|

[0, 29, 54, 10, −10, 120, 32] | 4.875 | 1.664 | 1.659 | 99.74% |

[10, −2, 160, 0, 33, −5, 80] | 7.876 | 2.218 | 2.213 | 99.76% |

[55, −22, 45, 11, 36, −10, 160] | 2.848 | 1.205 | 1.202 | 99.74% |

[−10, 30, 60, −45, 0, 22, 4] | 3.061 | 1.258 | 1.255 | 99.76% |

[23, 33, −64, 145, 34, 88, 60] | 3.167 | 1.284 | 1.281 | 99.76% |

[74, −20, 2, 19, 112, 75, −10] | 5.003 | 1.690 | 1.686 | 99.76% |

[5, 100, 26, −69, 20, 108, −66] | 2.513 | 1.118 | 1.115 | 99.75% |

[100, 23, 126, −6, 34, 34, −1] | 1.658 | 0.871 | 0.869 | 99.76% |

[1, 110, −90, 90, 14, 150, 45] | 2.677 | 1.161 | 1.158 | 99.75% |

[−36, 88, 46, 123, −110, 15, 0] | 1.495 | 0.819 | 0.817 | 99.77% |

Capture Configuration (Deg) | Integrate Effective Mass (kg) | The Maximum Contact Force (kN) | |
---|---|---|---|

Without optimization | [−0.81, 90.80, 68.96, −47.37, 67.83, 91.01, −0.92] | 17.24 | 3.55 |

With optimization (Specified pose of the end-effector) | [−47.56, 44.57, 55.96, −24.88, 96.29, 121.85, 56.33] | 15.88 | 3.38 |

With optimization (Specified position of the end-effector) | [43.84, 127.47, 69.31, −22.02, 121.81, 86.01, 0] | 2.70 | 1.17 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the author. 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**

Zhang, L.
Configuration Optimization for Free-Floating Space Robot Capturing Tumbling Target. *Aerospace* **2022**, *9*, 69.
https://doi.org/10.3390/aerospace9020069

**AMA Style**

Zhang L.
Configuration Optimization for Free-Floating Space Robot Capturing Tumbling Target. *Aerospace*. 2022; 9(2):69.
https://doi.org/10.3390/aerospace9020069

**Chicago/Turabian Style**

Zhang, Long.
2022. "Configuration Optimization for Free-Floating Space Robot Capturing Tumbling Target" *Aerospace* 9, no. 2: 69.
https://doi.org/10.3390/aerospace9020069