# Non-Probabilistic Reliability Analysis of Robot Accuracy under Uncertain Joint Clearance

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## Abstract

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## 1. Introduction

## 2. Mathematical Modeling of Robotic Systems

#### 2.1. Robot System Kinematics Positive Solution

#### 2.2. Reliability Model for Robot Motion Based on Non-Probability Interval Theory

## 3. Parametric Modeling of 6-Degree-of-Freedom Robots

#### 3.1. Joint Clearance Modeling Ideas

#### 3.2. Parametric Modeling of Joint Clearance

#### 3.3. Setting of Model Drive Parameters

## 4. Simulation Analysis of Positioning Accuracy of Industrial Robots

#### 4.1. Analysis of End-Effector Coordinate Points

#### 4.2. Analysis of End-Effector Trajectories

#### 4.3. Analysis of Different Time Periods of Robot End-Effectors

## 5. Conclusions

- (1)
- Non-probabilistic reliability is introduced to analyze the robot position accuracy under the influence of uncertainty considering the joint clearance, and it is found that the method in this paper is consistent with the traditional probabilistic method in evaluating the reliability of the robot end position, but the method in this paper has a lower degree of data requirement and has obvious advantages for the reliability analysis of small sample and information-poor structured systems.
- (2)
- The simulation analysis of the robot under the influence of the ideal state and uncertain joint clearance parameters is carried out to obtain the displacement error range in each spatial direction, and the reliability analysis based on the non-probability interval theory is carried out. The reliability of the end-effector of the robot is related to the displacement curve. The reliability decreases at the inflection point of the displacement curve and is higher at the smooth displacement curve, and the phenomenon of uneven transition of the displacement curve should be avoided when planning the robot motion trajectory; the reliability of the end-effector is related to its position in the workspace, and the further away from the center of the workspace, the worse the reliability.
- (3)
- The motion path is divided into small segments in time, and the reliability of each segment’s motion is analyzed to compare the reliability of the end-effector in different workspace areas corresponding to different time segments, and the conclusion is drawn that the reliability of the end-effector changes in different working regions under the influence of uncertain joint clearance parameters. Based on this conclusion, the research direction of dividing the robot space and partitioning it to establish a non-probability-based reliability calibration model is proposed, which can realize the prediction of the robot end-effector position range and the interval identification of parameters and improve the positioning accuracy and calibration reliability of the robot in the full workspace region.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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$\mathit{i}$ | ${\mathit{a}}_{\mathit{i}}/(\mathit{m}\mathit{m})$ | ${\mathit{d}}_{\mathit{i}}/(\mathit{m}\mathit{m})$ | ${\mathit{\alpha}}_{\mathit{i}}/(\xb0)$ | ${\mathit{\theta}}_{\mathit{i}}/(\xb0)$ | Joint Range$/(\xb0)$ |
---|---|---|---|---|---|

1 | 180 | 400 | −90 | ${\theta}_{1}$ | (−170~170) |

2 | 600 | 0 | 0 | ${\theta}_{2}$ | (−180~65) |

3 | 40 | 0 | −90 | ${\theta}_{3}$ | (−120~165) |

4 | 0 | 620 | 90 | ${\theta}_{4}$ | (−180~180) |

5 | 0 | 0 | −90 | ${\theta}_{5}$ | (−120~120) |

6 | 0 | 80 | 0 | ${\theta}_{6}$ | (−360~360) |

Variable | Nominal Size /(mm) | Tolerance /(mm) | Upper and Lower Limit Dimensions /(mm) | Maximum Clearance /(mm) | Minimum Clearance /(mm) | Notes |
---|---|---|---|---|---|---|

DV 1 | 40 | 0.011 | 39.989,40 | 0.027 | 0 | Radius of base shaft |

DV 2 | 40 | 0.016 | 40,40.016 | Radius of swivel base hole | ||

DV 3 | 30 | 0.013 | 30,30.013 | 0.022 | 0 | Radius of swivel base hole |

DV 4 | 30 | 0.009 | 29.991,30 | Radius of large arm shaft (front end) | ||

DV 5 | 22.5 | 0.009 | 22.491,22.5 | 0.022 | 0 | Radius of large arm shaft (end) |

DV 6 | 22.5 | 0.013 | 22.5,22.513 | Radius of small arm rod hole | ||

DV 7 | 10 | 0.006 | 9.994,10 | 0.015 | 0 | Radius of small arm rod shaft |

DV 8 | 10 | 0.009 | 10,10.009 | Radius of small armhole (front end) | ||

DV 9 | 6 | 0.008 | 6,6.008 | 0.013 | 0 | Radius of small armhole (end) |

DV 10 | 6 | 0.005 | 5.995,6 | Radius of wrist shaft | ||

DV 11 | 5 | 0.008 | 5,5.008 | 0.013 | 0 | Radius of wrist hole |

DV 12 | 5 | 0.005 | 4.995,5 | Radius of flange shaft |

Joint | Connected Components | Constraints (Plane Pairs) | Contact Force | Driving Mode (Point Driving) | Drive Function |
---|---|---|---|---|---|

Joint 1 | Base, swivel base | joint_1 | contact_1 | general_motion_1 | STEP(time, 0, 0 d,1, −45 d) + STEP(time, 5, 0 d, 6, 45 d) |

Joint 2 | Swivel base, large arm | joint_2 | contact_2 | general_motion_2 | STEP(time, 1, 0 d,2 ,60 d) + STEP(time, 6, 0 d, 7, −60 d) |

Joint 3 | Large arm, small arm rod | joint_3 | contact_3 | general_motion_3 | STEP(time, 2, 0 d, 3, 60 d) + STEP(time, 7, 0 d, 8, −60 d) |

Joint 4 | Small arm rod, small arm | joint_4 | contact_4 | general_motion_4 | STEP(time, 3, 0 d, 4, 90 d) + STEP(time, 8, 0 d, 9, −90 d) |

Joint 5 | Small arm, wrist | joint_5 | contact_5 | general_motion_5 | STEP(time, 4, 0 d, 5, 45 d) + STEP(time, 9, 0 d, 10, −45 d) |

Joint 6 | Wrist, flange | joint_6 | contact_6 | general_motion_6 | STEP(time, 4, 0 d, 5, 180 d) + STEP(time, 9, 0 d, 10, −180 d) |

$\mathbf{Joint}\mathbf{Angle}/(\xb0)$ | Theoretical Coordinates | Range of Error /(mm) | Non-Probability Reliability | Traditional Probabilistic Reliability | ||||||
---|---|---|---|---|---|---|---|---|---|---|

$\Delta \mathit{x}$ | $\Delta \mathit{y}$ | $\Delta \mathit{z}$ | x | y | z | x | y | z | ||

(−45°, −30°, −60°, −90°, −45°, 180°) | 1013.1, −933.109, 735.25 | −1.1424 | −1.077 | −1.0096 | 0.909 | 0.9435 | 0.925 | 0.903 | 0.924 | 0.919 |

1.0588 | 1.0428 | 1.1521 |

Time Period | X | Y | Z |
---|---|---|---|

0–2 s | [−0.1667,0.1726] | [−0.1056,0.1183] | [−0.7244,0] |

2–4 s | [−0.0255,0.1218] | [−0.2035,0.0717] | [−0.5962,−0.3404] |

4–6 s | [−0.0826,0.6975] | [−0.0893,1.8808] | [−0.4825,3.0768] |

6–8 s | [−3.0183,0.1457] | [−0.0601,0.1421] | [2.121,2.9081] |

8–10 s | [−0.4716,0.0345] | [−3.2367,0.0064] | [−0.4892,1.8582] |

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**MDPI and ACS Style**

Tang, Z.; Peng, J.; Sun, J.; Meng, X.
Non-Probabilistic Reliability Analysis of Robot Accuracy under Uncertain Joint Clearance. *Machines* **2022**, *10*, 917.
https://doi.org/10.3390/machines10100917

**AMA Style**

Tang Z, Peng J, Sun J, Meng X.
Non-Probabilistic Reliability Analysis of Robot Accuracy under Uncertain Joint Clearance. *Machines*. 2022; 10(10):917.
https://doi.org/10.3390/machines10100917

**Chicago/Turabian Style**

Tang, Zhaoping, Jun Peng, Jianping Sun, and Xin Meng.
2022. "Non-Probabilistic Reliability Analysis of Robot Accuracy under Uncertain Joint Clearance" *Machines* 10, no. 10: 917.
https://doi.org/10.3390/machines10100917