# 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

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**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 |

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**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