# Lunar Surface Fault-Tolerant Soft-Landing Performance and Experiment for a Six-Legged Movable Repetitive Lander

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Lander System

#### 2.1. HexaMRL

#### 2.2. Leg Mechanism

#### 2.3. Leg Residual Capacity

## 3. Landing Configuration Analysis

#### 3.1. Classification

#### 3.2. Stability

#### 3.3. Relationship between Fault Number and Configuration

## 4. Fault-Tolerant Landing Algorithms

#### 4.1. VI Configuration

#### 4.2. V Configuration

#### 4.3. IV and III Configurations

## 5. Quasi-Incentre Stability Optimization

#### 5.1. Quasi-Incentre Definition

#### 5.2. Quasi-Incentre Searching

#### 5.2.1. Searching in V Configuration

#### 5.2.2. Searching in IV Configuration

#### 5.2.3. Searching in III Configuration

## 6. Experiments

#### 6.1. Experiment Platform

#### 6.2. Five-Legged Landing

#### 6.3. Four-Legged Landing

#### 6.4. Three-Legged Landing

## 7. Discussion

## 8. Conclusions

- (1)
- To achieve the soft-landing as far as possible with a failed IDU, we systematically analyze the classification and stability of landing configurations. Then the relationship between fault number and landing configuration is concluded by equation and listed by table.
- (2)
- As for stable configuration, we have designed corresponding fault-tolerant landing algorithms to achieve buffer landing and further proposed a quasi-incentre stability optimization method to increase the stability margin during supported operations.
- (3)
- A series of experiments including five-legged, four-legged and three-legged soft-landing with a vertical landing velocity of −1.9 m/s and a payload of 140 kg were conducted to verify the fault-tolerant landing capacity on the constructed 5-DoF lunar gravity ground testing platform by means of counterweight.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 8.**Particle evolution process in V-2 configuration (the horizontal axis denotes the forward/backward direction illustrated in Figure 2 while the vertical axis represents the leftward/rightward direction, respectively).

**Figure 10.**Particle evolution process in IV-2 configuration (the definition of vertical or horizontal axis is same as the one in Figure 8).

**Figure 14.**5-Dof lunar gravity ground testing platform. (

**a**) components and Degrees of freedom; (

**b**) components of counterweight system.

Fault Number | Side | Thigh | Shank | Number | Symbol | G_{f} Set | Fault Tolerance |
---|---|---|---|---|---|---|---|

1 | LA | N | N | 4 | $\mathrm{E}{1}_{1}$ | ${G}_{F}^{II}\left(0,{R}_{\beta},0;{T}_{a},0,0\right)$ | 1 |

LB | N | N | $\mathrm{E}{1}_{2}$ | ${G}_{F}^{II}\left(0,{R}_{\beta},0;{T}_{a},0,0\right)$ | 0 | ||

N | L | N | ${G}_{F}^{II}\left({R}_{\alpha},{R}_{\beta},0;0,0,0\right)$ | 0 | |||

N | N | L | ${G}_{F}^{II}\left({R}_{\alpha},{R}_{\beta},0;0,0,0\right)$ | 0 | |||

2 | L | L | N | 3 | E2 | ${G}_{F}^{II}\left(0,{R}_{\beta},0;0,0,0\right)$ | 0 |

L | N | L | ${G}_{F}^{II}\left(0,{R}_{\beta},0;0,0,0\right)$ | 0 | |||

N | L | L | ${G}_{F}^{II}\left({R}_{\alpha},0,0;0,0,0\right)$ | 0 | |||

3 | L | L | L | 1 | E3 | ${G}_{F}^{I}\left(0,0,0;0,0,0\right)$ | 0 |

Type I | Number | Supporting Leg | Type II | Type I | Number | Supporting Leg | Type II |
---|---|---|---|---|---|---|---|

VI | ${\mathrm{C}}_{6}^{6}=1$ | 1-2-3-4-5-6 | VI-1 | III | ${\mathrm{C}}_{6}^{3}=20$ | 1-2-3 | III-1 |

V | ${\mathrm{C}}_{6}^{5}=6$ | 2-3-4-5-6 | V-1 | 1-2-6 | |||

1-3-4-5-6 | 3-4-5 | ||||||

1-2-3-5-6 | 4-5-6 | ||||||

1-2-3-4-6 | 1-2-4 | III-2 | |||||

1-2-4-5-6 | V-2 | 1-2-5 | |||||

1-2-3-4-5 | 1-4-5 | ||||||

IV | ${\mathrm{C}}_{6}^{4}=15$ | 3-4-5-6 | IV-1 | 2-4-5 | |||

1-2-3-6 | 1-3-4 | III-3 | |||||

2-4-5-6 | IV-2 | 1-4-6 | |||||

1-3-4-5 | 2-3-5 | ||||||

1-2-4-6 | 2-5-6 | ||||||

1-2-3-5 | 1-3-5 | III-4 | |||||

2-3-5-6 | IV-3 | 2-4-6 | |||||

1-3-4-6 | 1-3-6 | III-5 | |||||

2-3-4-6 | IV-4 | 2-3-6 | |||||

1-3-5-6 | 3-4-6 | ||||||

2-3-4-5 | IV-5 | 3-5-6 | |||||

1-4-5-6 | 2-3-4 | III-6 | |||||

1-2-5-6 | 1-5-6 | ||||||

1-2-3-4 | |||||||

1-2-4-5 | IV-6 |

Type I | Type II | Stability Radius d(m) | Supporting Area S(m^{2}) | SAI | SS |
---|---|---|---|---|---|

$\mathrm{VI}$ | $\mathrm{VI}-1$ | 0.5998 | 1.5233 | 1 | S |

$\mathrm{V}$ | $\mathrm{V}-1$ | 0.3577 | 1.2640 | 0.4948 | S |

$\mathrm{V}-2$ | 0.4324 | 1.2803 | 0.6058 | S | |

$\mathrm{IV}$ | $\mathrm{IV}-1$ | 0 | 0.7617 | 0 | CS |

$\mathrm{IV}-2$ | 0.3577 | 1.021 | 0.3997 | S | |

$\mathrm{IV}-3$ | 0.3577 | 1.0047 | 0.3933 | S | |

$\mathrm{IV}-4$ | 0.4324 | 1.0047 | 0.4754 | S | |

$\mathrm{IV}-5$ | 0 | 0.7617 | 0 | CS | |

$\mathrm{IV}-6$ | 0.4324 | 1.0373 | 0.4908 | S | |

$\mathrm{III}$ | $\mathrm{III}-1$ | −0.3577 | 0.2593 | −0.1015 | US |

$\mathrm{III}-2$ | 0 | 0.5187 | 0 | CS | |

$\mathrm{III}-3$ | 0 | 0.5023 | 0 | CS | |

$\mathrm{III}-4$ | 0.3577 | 0.7617 | 0.2982 | S | |

$\mathrm{III}-5$ | 0 | 0.5023 | 0 | CS | |

$\mathrm{III}-6$ | −0.4324 | 0.243 | −0.1150 | US |

N_{m} | Solve Sets | Fault Legs Group | Type I | N_{m} | Solve Sets | Fault Legs Group | Type I |
---|---|---|---|---|---|---|---|

1 | {1,0,0} | $\mathrm{E}{1}_{1}$ | $\mathrm{VI}$ | 4 | {1,0,1} | $\mathrm{E}{1}_{1}\mathrm{E}3$ | $\mathrm{V}$ |

$\mathrm{E}{1}_{2}$ | $\mathrm{V}$ | $\mathrm{E}{1}_{2}\mathrm{E}3$ | $\mathrm{IV}$ | ||||

2 | {0,1,0} | $\mathrm{E}2$ | $\mathrm{V}$ | {0,2,0} | $\mathrm{E}2\mathrm{E}2$ | $\mathrm{IV}$ | |

{2,0,0} | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}$ | $\mathrm{VI}$ | {2,1,0} | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}2$ | $\mathrm{V}$ | ||

$\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}$ | $\mathrm{V}$ | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}2$ | $\mathrm{IV}$ | ||||

$\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}$ | $\mathrm{IV}$ | $\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}\mathrm{E}2$ | $\mathrm{III}$ | ||||

3 | {0,0,1} | $\mathrm{E}3$ | $\mathrm{VI}$ | {4,0,0} | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}$ | $\mathrm{VI}$ | |

{1,1,0} | $\mathrm{E}{1}_{1}\mathrm{E}2$ | $\mathrm{V}$ | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}$ | $\mathrm{V}$ | |||

$\mathrm{E}{1}_{2}\mathrm{E}2$ | $\mathrm{IV}$ | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}$ | $\mathrm{IV}$ | ||||

{3,0,0} | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}$ | $\mathrm{VI}$ | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}$ | $\mathrm{III}$ | |||

$\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}$ | $\mathrm{V}$ | 6 | {0,0,2} | $\mathrm{E}3\mathrm{E}3$ | $\mathrm{IV}$ | ||

$\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}$ | $\mathrm{IV}$ | {1,1,1} | $\mathrm{E}{1}_{1}\mathrm{E}2\mathrm{E}3$ | $\mathrm{IV}$ | |||

$\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}$ | $\mathrm{III}$ | $\mathrm{E}{1}_{2}\mathrm{E}2\mathrm{E}3$ | $\mathrm{III}$ | ||||

5 | {0,1,1} | $\mathrm{E}2\mathrm{E}3$ | $\mathrm{IV}$ | {0,3,0} | $\mathrm{E}2\mathrm{E}2\mathrm{E}2$ | $\mathrm{III}$ | |

{2,0,1} | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}3$ | $\mathrm{V}$ | {3,0,1} | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}3$ | $\mathrm{V}$ | ||

$\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}3$ | $\mathrm{IV}$ | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}3$ | $\mathrm{IV}$ | ||||

$\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}\mathrm{E}3$ | $\mathrm{III}$ | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}\mathrm{E}3$ | $\mathrm{III}$ | ||||

{1,2,0} | $\mathrm{E}{1}_{1}\mathrm{E}2\mathrm{E}2$ | $\mathrm{IV}$ | {2,2,0} | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}2\mathrm{E}2$ | $\mathrm{IV}$ | ||

$\mathrm{E}{1}_{2}\mathrm{E}2\mathrm{E}2$ | $\mathrm{III}$ | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}2\mathrm{E}2$ | $\mathrm{III}$ | ||||

{3,1,0} | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}2$ | $\mathrm{V}$ | {4,1,0} | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}2$ | $\mathrm{V}$ | ||

$\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}2$ | $\mathrm{IV}$ | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}2$ | $\mathrm{IV}$ | ||||

$\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}\mathrm{E}2$ | $\mathrm{III}$ | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}\mathrm{E}2$ | $\mathrm{III}$ | ||||

{5,0,0} | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}$ | $\mathrm{VI}$ | {6,0,0} | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}$ | $\mathrm{VI}$ | ||

$\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}$ | $\mathrm{V}$ | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}$ | $\mathrm{V}$ | ||||

$\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}$ | $\mathrm{IV}$ | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}$ | $\mathrm{IV}$ | ||||

$\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}$ | $\mathrm{III}$ | $\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{1}\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}\mathrm{E}{1}_{2}$ | $\mathrm{III}$ |

**Table 5.**Classification of virtual supporting triangle of stable configuration in five-legged soft-landing.

Type II | Valid Virtual Supporting Triangle | N_{v} | Invalid Virtual Supporting Triangle | N_{i} |
---|---|---|---|---|

V-1 | 2-3-5, 2-3-6, 2-4-5, 2-4-6, 2-5-6, 3-4-6, 3-5-6 | 7 | 2-3-4, 3-4-5, 4-5-6 | 3 |

V-2 | 1-2-4, 1-2-5, 1-3-4, 1-3-5, 1-4-5, 2-3-5, 2-4-5 | 7 | 1-2-3, 2-3-4, 3-4-5 | 3 |

**Table 6.**Classification of virtual supporting triangle of stable configuration in four-legged soft-landing.

Type II | Valid Virtual Supporting Triangle | N_{v} | Invalid Virtual Supporting Triangle | N_{i} |
---|---|---|---|---|

IV-2 | 2-4-5, 2-4-6, 2-5-6 | 3 | 4-5-6 | 1 |

IV-3 | 2-3-5, 2-3-6, 2-5-6, 3-5-6 | 4 | 0 | |

IV-4 | 2-3-6, 2-4-6, 3-4-6 | 3 | 2-3-4 | 1 |

IV-6 | 1-2-4, 1-2-5, 1-4-5, 2-4-5 | 4 | 0 |

Index | Five-Legged Landing | Four-Legged Landing | Three-Legged Landing |
---|---|---|---|

Touch-ground velocity (m/s) | 1.9 | 1.9 | 1.9 |

System mass (kg) | 180 | 180 | 180 |

Thigh peak torque (Nm) | 203.7 | 175.2 | 184.7 |

Shank peak torque (Nm) | −84.14 | −98.39 | −78.57 |

Roll angle derivation (°) | −1.8~0° | −1.46~0.74° | −1.26~0.53° |

Pitch angle derivation (°) | −7.98~1.13° | −0.93~0.47° | −1.16~1.08° |

Damping vibration duration (s) | 1.8 | 1.6 | 2 |

Derivative x Velocity (m/s) | 0.355 | $\approx 0$ | $\approx 0$ |

Derivative y Velocity (m/s) | 0.04 | $\approx 0$ | $\approx 0$ |

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**MDPI and ACS Style**

Yin, K.; Zhou, S.; Sun, Q.; Gao, F.
Lunar Surface Fault-Tolerant Soft-Landing Performance and Experiment for a Six-Legged Movable Repetitive Lander. *Sensors* **2021**, *21*, 5680.
https://doi.org/10.3390/s21175680

**AMA Style**

Yin K, Zhou S, Sun Q, Gao F.
Lunar Surface Fault-Tolerant Soft-Landing Performance and Experiment for a Six-Legged Movable Repetitive Lander. *Sensors*. 2021; 21(17):5680.
https://doi.org/10.3390/s21175680

**Chicago/Turabian Style**

Yin, Ke, Songlin Zhou, Qiao Sun, and Feng Gao.
2021. "Lunar Surface Fault-Tolerant Soft-Landing Performance and Experiment for a Six-Legged Movable Repetitive Lander" *Sensors* 21, no. 17: 5680.
https://doi.org/10.3390/s21175680