# Path Planning in the Case of Swarm Unmanned Surface Vehicles for Visiting Multiple Targets

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Swarm Approach of USV Path Planning Problem

#### 2.2. Objective Terms of the USV Path Planning Problem

- -
- Term 1 for the minimization of traveled distance.$$minD={{\displaystyle \sum}}_{i\in \mathcal{N}}{{\displaystyle \sum}}_{\begin{array}{c}j\in N:\\ \left(i,j\right)\in \epsilon \end{array}}{d}_{ij}={{\displaystyle \sum}}_{i\in \mathcal{N}}{{\displaystyle \sum}}_{\begin{array}{c}j\in N:\\ \left(i,j\right)\in \epsilon \end{array}}\left(\sqrt{{\left({j}_{x}-{i}_{x}\right)}^{2}+{\left({j}_{y}-{i}_{y}\right)}^{2}}\right)$$
- -
- Term 2 for the minimization of brute changes along the path (Figure 2).$$min\theta ={{\displaystyle \sum}}_{i\in \mathcal{N}}{{\displaystyle \sum}}_{\begin{array}{c}j\in N:\\ \left(i,j\right)\in \epsilon \end{array}}{{\displaystyle \sum}}_{\begin{array}{c}k\in N:\\ \left(j,k\right)\in \epsilon \end{array}}{\theta}_{ijk}$$
- -
- Term 3 for the minimization of the fuel consumption of the USV.$$minFC={{\displaystyle \sum}}_{i\in \mathcal{N}}{{\displaystyle \sum}}_{\begin{array}{c}j\in N:\\ \left(i,j\right)\in \epsilon \end{array}}\frac{{d}_{ij}}{V+{v}_{c}}f$$

#### 2.3. Ant Colony Optimization Algorithm with Fuzzy Logic

Algorithm 1: ACO pseudoalgorithm |

Input: variables of ACO |

$InitializePheromoneValues\left(\tau \right)$ |

${\mathfrak{s}}^{*}\leftarrow NULL$ // current best solution does not exist |

while termination criteria are not met do |

${\mathcal{G}}_{iter}\leftarrow \varnothing $ // the set of the path at the current iteration is empty |

for $j=1,\dots ,{n}_{a}$ do |

$\mathfrak{s}\leftarrow ConstructSolution\left(\tau \right)$ |

if $\left(f\left(\mathfrak{s}\right)<f\left({\mathfrak{s}}^{*}\right)\right)$ or ${\mathfrak{s}}^{*}isNULL$ then ${\mathfrak{s}}^{*}\leftarrow \mathfrak{s}$ |

${\mathcal{G}}_{iter}\leftarrow {\mathcal{G}}_{iter}\cup \left\{{\mathfrak{s}}^{*}\right\}$ |

end for |

$ApplyPheromoneUpdate\left(\tau ,{\mathcal{G}}_{iter},{\mathfrak{s}}^{*}\right)$ |

end while |

Output: current best solution ${\mathfrak{s}}^{*}$ |

#### 2.3.1. FIS1 1: Mamdani Fuzzy Inference System (ACO-Mamdani)

#### 2.3.2. FIS 2: Takagi–Sugeno–Kang Fuzzy Inference System (ACO-TSK)

## 3. Evaluation Methodology

#### 3.1. Experimental Setup

#### 3.2. Comparative Evaluation of Clustering Algorithms

#### 3.3. Comparative Evaluation of Path Planning Algorithms

- The objective criteria: (i) distance; (ii) brute turns; and (iii) fuel consumption;
- Path quality based on the defuzzification value of Mamdani and TSK FISs;
- The computing time;

## 4. Results and Discussion

#### 4.1. Results

#### 4.2. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Membership functions of (

**a**) path distance; (

**b**) path turns; (

**c**) fuel consumption; and (

**d**) path optimality.

**Figure 7.**Clustering results of case study 2 with Mini Batch K-Means and Ward Clustering (

**a**) and Birch (

**b**).

**Figure 8.**Cumulative results of ACO-Mamdani and ACO-TSK over the objective criteria: (

**a**) distance; (

**b**) number of urns; and (

**c**) consumption for Case Study 1.

**Figure 9.**Cumulative results of ACO-Mamdani and ACO-TSK over the objective criteria: (

**a**) distance; (

**b**) number of urns; and (

**c**) consumption for Case Study 2.

Path Length | Path Deviations | Energy Consumption | Path Optimality |
---|---|---|---|

Short | Smooth | Low | Very High |

Short | Smooth | Medium | High |

Short | Moderate | Low | High |

Moderate | Smooth | Low | High |

Short | Moderate | Medium | Medium |

Moderate | Smooth | Medium | Medium |

Moderate | Moderate | Low or Medium | Medium |

Moderate | Moderate or Brut | Medium or High | Low |

Moderate or Long | Moderate | Medium or High | Low |

Moderate or Long | Moderate or Brut | Medium | Low |

Long | Brut | High | Very Low |

Clustering Algorithm | Silhouette Coefficient | Calinski–Harabasz Index | Davies–Bouldin Index | Cumulative Evaluation Score |
---|---|---|---|---|

Mini Batch K-Means | 0.82 | 1301.34 | 0.36 | 3 |

Ward | 0.82 | 1301.34 | 0.36 | 3 |

Birch | 0.77 | 1205.45 | 0.42 | 0 |

**Table 3.**Path planning mean results with standard deviation after 20 runs of the case studies for each ACO-FIS approach for the swarm of USVs. The number of turns are rounded. The best solutions are denoted in bold.

Case Study | ACO-FIS | Swarm USVs | Distance (km) | Number of Turns | Consumption (kg) |
---|---|---|---|---|---|

CS1 | ACO-Mamdani | USV1 (red) | 17.61 ± 1.02 | 8 ± 1.48 | 3.75 ± 0.25 |

USV2 (yellow) | 18.55 ± 0.98 | 9 ± 1.33 | 3.87 ± 0.13 | ||

USV3 (blue) | 18.43 ± 1.04 | 5 ± 0.87 | 3.73 ± 0.37 | ||

ACO-TSK | USV1 (red) | 17.63 ± 0.79 | 8 ± 1.08 | 3.78 ± 0.12 | |

USV2 (yellow) | 18.62 ± 1.14 | 8 ± 1.09 | 3.89 ± 0.24 | ||

USV3 (blue) | 18.43 ± 1.22 | 5 ± 0.88 | 3.72 ± 0.19 | ||

CS2 | ACO-Mamdani | USV1 (red) | 17.22 ± 2.24 | 7 ± 1.01 | 3.58 ± 0.45 |

USV2 (yellow) | 15.76 ± 1.95 | 6 ± 1.03 | 3.32 ± 0.54 | ||

USV3 (blue) | 19.04 ± 0.88 | 5 ± 0.86 | 3.64 ± 0.15 | ||

ACO-TSK | USV1 (red) | 17.37 ± 1.90 | 7 ± 1.03 | 3.65 ± 0.21 | |

USV2 (yellow) | 16.05 ± 1.46 | 6 ± 0.92 | 3.38 ± 0.17 | ||

USV3 (blue) | 19.18 ± 2.19 | 6 ± 0.88 | 3.79 ± 0.52 |

**Table 4.**Path planning optimality and computing time mean results with standard deviation after 20 runs of the case studies for each ACO-FIS approach for the swarm of USVs. The best solutions are denoted in bold.

Case Study | ACO-FIS | Optimality | Computing Time (ms) |
---|---|---|---|

CS1 | ACO-Mamdani | 0.82 ± 0.04 | 3.46 ± 0.03 |

ACO-TSK | 0.80 ± 0.05 | 3.39 ± 0.02 | |

CS2 | ACO-Mamdani | 0.75 ± 0.03 | 4.12 ± 0.02 |

ACO-TSK | 0.66 ± 0.04 | 4.01 ± 0.01 |

**Table 5.**Evaluation results of mean relative percentage deviation (RPD) and mean relative deviation index (RDI) for distance. The best solutions are denoted in bold.

Case Study | ACO-FIS | Swarm USVs | RPD | $\overline{\mathit{R}\mathit{P}\mathit{D}}$ | RDI | $\overline{\mathit{R}\mathit{D}\mathit{I}}$ |
---|---|---|---|---|---|---|

CS1 | ACO-Mamdani | USV1 (red) | 0.00% | 3.33% | 0.00% | 58.09% |

USV2 (yellow) | 5.34% | 93.07% | ||||

USV3 (blue) | 4.66% | 81.19% | ||||

ACO-TSK | USV1 (red) | 0.11% | 3.50% | 1.98% | 61.06% | |

USV2 (yellow) | 5.74% | 100.00% | ||||

USV3 (blue) | 4.66% | 81.19% | ||||

CS2 | ACO-Mamdani | USV1 (red) | 9.26% | 10.03% | 0.426900585 | 46.20% |

USV2 (yellow) | 0.00% | 0 | ||||

USV3 (blue) | 20.81% | 0.959064327 | ||||

ACO-TSK | USV1 (red) | 10.22% | 11.25% | 0.470760234 | 51.85% | |

USV2 (yellow) | 1.84% | 0.084795322 | ||||

USV3 (blue) | 21.70% | 1 |

**Table 6.**Evaluation results of mean relative percentage deviation (RPD) and mean relative deviation index (RDI) for brute turns. The best solutions are denoted in bold.

Case Study | ACO-FIS | Swarm USVs | RPD | $\overline{\mathit{R}\mathit{P}\mathit{D}}$ | RDI | $\overline{\mathit{R}\mathit{D}\mathit{I}}$ |
---|---|---|---|---|---|---|

CS1 | ACO-Mamdani | USV1 (red) | 60.00% | 46.67% | 75.00% | 58.33% |

USV2 (yellow) | 80.00% | 100.00% | ||||

USV3 (blue) | 0.00% | 0.00% | ||||

ACO-TSK | USV1 (red) | 60.00% | 40.00% | 75.00% | 50.00% | |

USV2 (yellow) | 60.00% | 75.00% | ||||

USV3 (blue) | 0.00% | 0.00% | ||||

CS2 | ACO-Mamdani | USV1 (red) | 40.00% | 20.00% | 100.00% | 50.00% |

USV2 (yellow) | 20.00% | 50.00% | ||||

USV3 (blue) | 0.00% | 0.00% | ||||

ACO-TSK | USV1 (red) | 40.00% | 26.67% | 100.00% | 66.67% | |

USV2 (yellow) | 20.00% | 50.00% | ||||

USV3 (blue) | 20.00% | 50.00% |

**Table 7.**Evaluation results of mean relative percentage deviation (RPD) and mean relative deviation index (RDI) for consumption. The best solutions are denoted in bold.

Case Study | ACO-FIS | Swarm USVs | RPD | $\overline{\mathit{R}\mathit{P}\mathit{D}}$ | RDI | $\overline{\mathit{R}\mathit{D}\mathit{I}}$ |
---|---|---|---|---|---|---|

CS1 | ACO-Mamdani | USV1 (red) | 0.81% | 1.70% | 17.65% | 37.25% |

USV2 (yellow) | 4.03% | 88.24% | ||||

USV3 (blue) | 0.27% | 5.88% | ||||

ACO-TSK | USV1 (red) | 1.61% | 2.06% | 35.29% | 45.10% | |

USV2 (yellow) | 4.57% | 100.00% | ||||

USV3 (blue) | 0.00% | 0.00% | ||||

CS2 | ACO-Mamdani | USV1 (red) | 7.83% | 5.82% | 55.32% | 41.13% |

USV2 (yellow) | 0.00% | 0.00% | ||||

USV3 (blue) | 9.64% | 68.09% | ||||

ACO-TSK | USV1 (red) | 9.94% | 8.63% | 70.21% | 60.99% | |

USV2 (yellow) | 1.81% | 12.77% | ||||

USV3 (blue) | 14.16% | 100.00% |

Case Studies | |||
---|---|---|---|

CS1 | CS2 | All | |

p-value | 1.05566 × 10^{−5} | 4.85828 × 10^{−122} | 1.05266 × 10^{−128} |

Chi-square | 305.97 | 544.35 | 603.97 |

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## Share and Cite

**MDPI and ACS Style**

Ntakolia, C.; Lyridis, D.V. Path Planning in the Case of Swarm Unmanned Surface Vehicles for Visiting Multiple Targets. *J. Mar. Sci. Eng.* **2023**, *11*, 719.
https://doi.org/10.3390/jmse11040719

**AMA Style**

Ntakolia C, Lyridis DV. Path Planning in the Case of Swarm Unmanned Surface Vehicles for Visiting Multiple Targets. *Journal of Marine Science and Engineering*. 2023; 11(4):719.
https://doi.org/10.3390/jmse11040719

**Chicago/Turabian Style**

Ntakolia, Charis, and Dimitrios V. Lyridis. 2023. "Path Planning in the Case of Swarm Unmanned Surface Vehicles for Visiting Multiple Targets" *Journal of Marine Science and Engineering* 11, no. 4: 719.
https://doi.org/10.3390/jmse11040719