# Optimizing Cycle Time of Industrial Robotic Tasks with Multiple Feasible Configurations at the Working Points

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## Abstract

**:**

## 1. Introduction

- to find the optimal task order in the case of multiple working points with multiple configurations and no fixed sequence;
- to find the optimal configurations in the working points with multiple configurations and a fixed sequence.

## 2. Industrial Robotic Tasks with Multiple Feasible Configurations at the Working Points

## 3. Optimizing Cycle Time through the Travelling Salesman Problem

- The path must enter a cluster through one configuration and exit through its mirror copy.
- To avoid discrepancies in the final cost, the connections within a cluster must have null cost.

- Define the working positions as Cartesian points (${P}_{i}$).
- Identify the feasible redundant configurations for each point (${P}_{i,1\dots m}$), excluding the configurations that are outside of robot mechanical limits or that result in collisions with the workcell equipment.
- Solve the modified TSP using the feasible configurations and the cost matrix as inputs.
- Finally, the optimal task sequence is provided.

#### Optimizing Cycle Time in the Case of a Fixed Point Sequence

## 4. Validation

#### 4.1. Simulation Results

#### 4.2. Experimental Testing

- Each point has a single robot configuration;
- Each point has two robot configurations (flip/no flip);
- Each point has two robot configurations (flip/no flip) and the rotation takes place around the camera axis (considered case study).

#### 4.3. Applicability of the Modified TSP

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**In this case, the same image of the workpiece (yellow) can acquired by the camera (red), rotated by 180 degrees, with the robot in both configurations shown.

**Figure 2.**Number of possible paths for a robotic tasks with multiple feasible configurations at the working points. (

**a**) Free sequence (Equation (2), ${N}_{g}=1$, ${k}_{1}=k$, ${m}_{1}=N$). The line with $k=1$ is coincident with ${n}_{N}$ (Equation (1)); (

**b**) number of possible paths with a fixed sequence (Equation (3), ${N}_{g}=1$, ${k}_{1}=k$, ${m}_{1}=N$).

**Figure 3.**Ratio between the cost of the minimum solution with multiple configurations (${t}_{k}$) and the corresponding solution with $k=1$ (${t}_{1}$): (

**a**) no fixed point sequence; (

**b**) fixed point sequence.

**Figure 4.**Example of a cluster. The TSP solution enters the cluster and visits all the positions within the cluster before moving to another cluster.

**Figure 5.**Example of connections within and between clusters. Each cluster comprises the feasible configurations of a working point, together with their mirror copies (identified with an asterisk ${}^{*}$).

**Figure 6.**Example of clusters for a working cycle made of a set of Cartesian paths. Each cluster represents a path, and comprises all possible configurations of the starting point (e.g., ${P}_{1,i}$) and of the ending point (e.g., ${P}_{2,i}$). The points are connected in such a way that the TSP will exit the cluster (i.e., the robot will exit the path) with the same configuration used while entering the cluster (i.e., with the same configuration held by the robot while entering the path).

**Figure 8.**Simulation results with $k=2$: (

**a**) normalized solution cost and (

**b**) computational time (log scale).

**Figure 9.**Experimental setup including one Adept s650 anthropomorphic manipulator, one AVT Pike F-505 industrial camera, and one piece to be inspected (the pyramid trunk, to the left).

**Figure 12.**Joint displacements (

**a**) and speeds (

**b**) for the Test 1, Scenarios 1 (blue) and 3 (red) of Table 2.

${\mathit{N}}_{\mathit{g}}$ ($\mathit{N}=5$) | ${\mathit{m}}_{\mathit{i}}$ | ${\mathit{k}}_{\mathit{i}}$ |
---|---|---|

1 | 4 | 4 |

2 | 1 | 1 |

Scenario | Movement Time [s] | Mean Time Reduction | ||||
---|---|---|---|---|---|---|

Test 1 | Test 2 | Test 3 | Test 4 | Test 5 | ||

1 | 3.1755 | 3.2651 | 3.3701 | 3.376 | 3.216 | - |

2 | 3.0278 | 3.0972 | 3.1748 | 3.131 | 2.9833 | 6.02% |

3 | 2.969 | 3.0183 | 3.1519 | 3.131 | 2.9226 | 7.38% |

Cost | Movement Time [s] | ||||
---|---|---|---|---|---|

Test 1 | Test 2 | Test 3 | Test 4 | Test 5 | |

Minimum (TSP) | 2.969 | 3.0183 | 3.1519 | 3.131 | 2.9226 |

Mean | 6.5620 | 6.5752 | 6.6562 | 6.7475 | 6.5519 |

Maximum | 10.8018 | 10.7906 | 10.7883 | 10.9330 | 10.8422 |

**Table 4.**Movement times of the solutions provided by the TSP for different scenarios with a fixed sequence.

Scenario | Movement Time [s] | Mean Time Reduction |
---|---|---|

1 | 3.766 | - |

2 | 3.758 | <1% |

3 | 3.486 | 7.43% |

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**MDPI and ACS Style**

Bottin, M.; Boschetti, G.; Rosati, G.
Optimizing Cycle Time of Industrial Robotic Tasks with Multiple Feasible Configurations at the Working Points. *Robotics* **2022**, *11*, 16.
https://doi.org/10.3390/robotics11010016

**AMA Style**

Bottin M, Boschetti G, Rosati G.
Optimizing Cycle Time of Industrial Robotic Tasks with Multiple Feasible Configurations at the Working Points. *Robotics*. 2022; 11(1):16.
https://doi.org/10.3390/robotics11010016

**Chicago/Turabian Style**

Bottin, Matteo, Giovanni Boschetti, and Giulio Rosati.
2022. "Optimizing Cycle Time of Industrial Robotic Tasks with Multiple Feasible Configurations at the Working Points" *Robotics* 11, no. 1: 16.
https://doi.org/10.3390/robotics11010016