# Method for Robot Manipulator Joint Wear Reduction by Finding the Optimal Robot Placement in a Robotic Cell

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## Abstract

**:**

## 1. Introduction

#### The Concept of Robot Wear

## 2. Materials and Methods

#### 2.1. Optimization Criterion

- to minimize the overall wear of all joints, and
- to balance the wear of all joints.

#### 2.2. Optimization Process

#### 2.3. Experiment Setup

## 3. Results

## 4. Verification on a Real Robot

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

API | Application programming interface |

CSO | Chicken swarm optimization |

GCS | Global coordinate system |

IK | Inverse kinematics |

PSO | Particle swarm optimization |

RTDE | Real-time data exchange |

TCP/IP | Transmission control protocol/Internet protocol |

UR3 | Universal Robots 3 |

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**Figure 1.**Example of the time progression of joint torque $\tau $ and angular velocity $\omega $; the dotted area represents the meaning of the value W calculated as the integral of absolute value $\left|\tau \omega \right|$.

**Figure 2.**The UR3 robot. (

**a**) The simulation model in the CoppeliaSim environment with a path representing the end-point trajectory and a coordinate system in the center of the robot base. (

**b**) The real robot used in the experiments.

**Figure 3.**Visualization of the five testing robot end-point paths; the UR3 robot is shown as a scale reference.

**Figure 4.**Example of a grid of valid robot locations for a given trajectory (A1). Each blue point represents a possible position of the robot base center point; the robot is displayed in four distinct sample locations. GCS represents the global coordinate system.

**Figure 5.**Statistical distribution of the ${f}_{f}$ values in all possible robot locations for the ten testing trajectories (standard box-plot diagram—each box is bounded by the first and third quartile, the horizontal line represents the median, the × represents the arithmetic mean, and circles represent outliers).

**Figure 6.**All valid robot locations in the grid; the color-coding indicates the ${f}_{f}$ values using the gradient red–yellow–green, where green is the best location (B) and red is the worst (W). Locations numbered 1, 2, and 3 are the three other locations used in the experiment with a real robot.

**Figure 7.**Detailed comparison of the components forming the fitness function value in the best (B, green) and worst (W, red) robot locations for two selected trajectories; displayed are the relative wear factors of each joint (${w}_{j}$) and the total fitness function value (${f}_{f}$).

**Figure 8.**Comparison of the angular velocity ${\omega}_{3}$, torque ${\tau}_{3}$, and immediate power ${\omega}_{3}{\tau}_{3}$ of the third joint during the whole trajectories A1 and B1 for the best (${}^{B}$) and worst (${}^{W}$) robot locations.

**Figure 9.**Comparison of the fitness function values acquired from simulation and experiments on the real robot; the black lines represent the ratio between the values.

**Figure 10.**Comparison of the real and simulated values of angular velocity ${\omega}_{3}$ and torque ${\tau}_{3}$ of the third joint during the whole trajectory B1. (

**a**) Robot placed in the best location. (

**b**) Robot placed in the worst location.

**Table 1.**The joint parameters for the UR3 robot; ${\tau}_{{m}_{j}}$ is the maximal permissible torque, and ${\omega}_{{m}_{j}}$ is the maximal permissible angular velocity of the j-th joint.

j | ${\mathit{\tau}}_{{\mathit{m}}_{\mathit{j}}}$ [Nm] | ${\mathit{\omega}}_{{\mathit{m}}_{\mathit{j}}}$ [s^{−1}] |
---|---|---|

1 | 56 | $\pi $ |

2 | 56 | $\pi $ |

3 | 28 | $\pi $ |

4 | 12 | $2\pi $ |

5 | 12 | $2\pi $ |

6 | 12 | $2\pi $ |

**Table 2.**Results for the 10 testing trajectories showing the number of valid robot locations n, the fitness function value in the best location ${{f}_{f}}^{B}$, in the worst location ${{f}_{f}}^{W}$, the average value ${{f}_{f}}^{A}$; and improvement of the best location against the worst $im{p}_{W}^{B}$ (9) and the average $im{p}_{A}^{B}$ location (10).

Traj. | n | ${{\mathit{f}}_{\mathit{f}}}^{\mathit{B}}$ | ${{\mathit{f}}_{\mathit{f}}}^{\mathit{W}}$ | ${{\mathit{f}}_{\mathit{f}}}^{\mathit{A}}$ | ${\mathit{i}\mathit{m}\mathit{p}}_{\mathit{W}}^{\mathit{B}}$ | ${\mathit{i}\mathit{m}\mathit{p}}_{\mathit{A}}^{\mathit{B}}$ |
---|---|---|---|---|---|---|

A1 | 800 | 0.0730 | 0.2053 | 0.1239 | 64.4% | 41.1% |

A2 | 564 | 0.0773 | 0.1577 | 0.1046 | 51.0% | 26.1% |

B1 | 2265 | 0.0217 | 0.1090 | 0.0461 | 80.1% | 52.9% |

B2 | 2236 | 0.0210 | 0.1067 | 0.0448 | 80.3% | 53.1% |

C1 | 2365 | 0.0769 | 0.1998 | 0.1239 | 61.5% | 37.9% |

C2 | 2214 | 0.0757 | 0.1828 | 0.1170 | 58.6% | 35.2% |

D1 | 866 | 0.0907 | 0.1994 | 0.1163 | 54.5% | 22.0% |

D2 | 828 | 0.0841 | 0.1829 | 0.1097 | 54.0% | 23.3% |

E1 | 2224 | 0.0279 | 0.1028 | 0.0532 | 72.8% | 47.4% |

E2 | 2151 | 0.0247 | 0.0984 | 0.0502 | 74.9% | 50.7% |

**Table 3.**Fitness function values in the five robot locations for individual trajectories—results from the simulation. Color gradient red–yellow–green is used for better visual representation of the value (green is the best, red is the worst).

B | 1 | 2 | 3 | W | |
---|---|---|---|---|---|

A1 | 0.0730 | 0.1256 | 0.1281 | 0.1241 | 0.2054 |

A2 | 0.0774 | 0.1025 | 0.1156 | 0.1434 | 0.1578 |

B1 | 0.0217 | 0.0357 | 0.0389 | 0.0484 | 0.1090 |

B2 | 0.021 | 0.0289 | 0.0310 | 0.0588 | 0.1068 |

C1 | 0.0770 | 0.1074 | 0.1195 | 0.1421 | 0.1998 |

C2 | 0.0758 | 0.0939 | 0.1040 | 0.1160 | 0.1828 |

D1 | 0.0907 | 0.1045 | 0.1077 | 0.1187 | 0.1995 |

D2 | 0.0841 | 0.0967 | 0.1157 | 0.1203 | 0.1830 |

E1 | 0.028 | 0.0358 | 0.0388 | 0.0642 | 0.1028 |

E2 | 0.0247 | 0.0371 | 0.0585 | 0.0606 | 0.0985 |

**Table 4.**Fitness function values in the five robot locations for individual trajectories–results from the real robot. Color gradient red–yellow–green is used for better visual representation of the value (green is the best, red is the worst).

B | 1 | 2 | 3 | W | |
---|---|---|---|---|---|

A1 | 0.0506 | 0.0799 | 0.0815 | 0.0848 | 0.1343 |

A2 | 0.054 | 0.0752 | 0.0799 | 0.0900 | 0.1155 |

B1 | 0.019 | 0.0312 | 0.0317 | 0.0278 | 0.0681 |

B2 | 0.0191 | 0.0263 | 0.0280 | 0.0348 | 0.0679 |

C1 | 0.0635 | 0.0760 | 0.0828 | 0.0874 | 0.1176 |

C2 | 0.0621 | 0.0625 | 0.0810 | 0.0808 | 0.1124 |

D1 | 0.078 | 0.0877 | 0.0940 | 0.0783 | 0.1015 |

D2 | 0.0777 | 0.0869 | 0.1038 | 0.0857 | 0.0958 |

E1 | 0.0239 | 0.0310 | 0.0348 | 0.0457 | 0.0599 |

E2 | 0.025 | 0.0355 | 0.0481 | 0.0446 | 0.0621 |

**Table 5.**Ratios between the real and simulated values of fitness function in the five robot locations for individual trajectories. Color gradient red–white is used for better visual representation of the value (white is the ideal ratio of 1.0, red is the ratio 0.5).

B | 1 | 2 | 3 | W | |
---|---|---|---|---|---|

A1 | 0.6932 | 0.6359 | 0.6362 | 0.6833 | 0.6541 |

A2 | 0.6985 | 0.7334 | 0.6912 | 0.6278 | 0.732 |

B1 | 0.8743 | 0.8722 | 0.8168 | 0.5735 | 0.6244 |

B2 | 0.9056 | 0.9096 | 0.9025 | 0.5915 | 0.6359 |

C1 | 0.8246 | 0.7075 | 0.6933 | 0.6151 | 0.5887 |

C2 | 0.8199 | 0.6657 | 0.7792 | 0.6965 | 0.6149 |

D1 | 0.8594 | 0.8393 | 0.8728 | 0.6591 | 0.5087 |

D2 | 0.9243 | 0.8984 | 0.8968 | 0.7126 | 0.5238 |

E1 | 0.8541 | 0.8658 | 0.8960 | 0.7118 | 0.5825 |

E2 | 1.0100 | 0.9590 | 0.8224 | 0.7360 | 0.6303 |

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**MDPI and ACS Style**

Kot, T.; Bobovský, Z.; Vysocký, A.; Krys, V.; Šafařík, J.; Ružarovský, R. Method for Robot Manipulator Joint Wear Reduction by Finding the Optimal Robot Placement in a Robotic Cell. *Appl. Sci.* **2021**, *11*, 5398.
https://doi.org/10.3390/app11125398

**AMA Style**

Kot T, Bobovský Z, Vysocký A, Krys V, Šafařík J, Ružarovský R. Method for Robot Manipulator Joint Wear Reduction by Finding the Optimal Robot Placement in a Robotic Cell. *Applied Sciences*. 2021; 11(12):5398.
https://doi.org/10.3390/app11125398

**Chicago/Turabian Style**

Kot, Tomáš, Zdenko Bobovský, Aleš Vysocký, Václav Krys, Jakub Šafařík, and Roman Ružarovský. 2021. "Method for Robot Manipulator Joint Wear Reduction by Finding the Optimal Robot Placement in a Robotic Cell" *Applied Sciences* 11, no. 12: 5398.
https://doi.org/10.3390/app11125398