# An Informative Path Planner for a Swarm of ASVs Based on an Enhanced PSO with Gaussian Surrogate Model Components Intended for Water Monitoring Applications

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## Abstract

**:**

## 1. Introduction

- The development of an informative path planner for a swarm of ASVs for the monitoring system of a water resource based on an improved meta-heuristic algorithm, PSO, with a GP as the underlying surrogate model.
- The application of the proposed path planning using as the simulated scenario Ypacarai Lake, showing the superiority of the proposed approach with respect to other techniques.

## 2. Related Works

## 3. Statement of the Problem and Assumptions

#### 3.1. Monitoring Problem

#### 3.2. Assumptions

- Ypacarai Lake: The model defined for the monitoring space is modelled as a matrix $\mathcal{N}$, where each element ${\mathcal{N}}_{i,j}$ has a value indicating the state of the grids. The matrix is composed by $m\times n$ squares of side d. If a square space is a natural obstacle, prohibited zone, land, among others, the square is painted white and has a value of 0, otherwise, the square is black and has a value of 1. Figure 2 shows the occupancy grid model of Ypacarai Lake used in the proposed approach. The ASVs are unable to move on land where the color of the square is white. For the simulations, the distribution map is scaled, each element ${\mathcal{N}}_{ij}$ is 100 m × 100 m.
- Coordinator: The system used for the fleet of ASVs is centralized. Therefore, the ASVs are linked to a global coordinator through the cloud, via 4 G or 5 G. In addition, it is considered a safety zone. For safety reasons, the ASVs are not allowed to travel near the shore of the lake. The distance considered is 2 squares of side d.
- Sensors: The sensors that take the samples of the water quality parameters are considered ideal. As a result, the collected data by the ASVs are noiseless. As well, the GPS equipped in the ASVs are considered ideal, so there are no errors in the positions of the vehicles. The maximum speed of the ASVs is 2 m/s.
- Navigation: The motions of the ASVs are error-free, and as a result, the traveled trajectories are faultless. Furthermore, obstructions and collisions are not taken into account.
- Vehicle autonomy: Battery usage is deemed adequate for the duration of the test. The maximum distance traveled by the ASVs is approximately 20,000 m.

## 4. Proposed Approach

#### 4.1. Particle Swarm Optimization

#### 4.2. Gaussian Process Regression (GPR)

#### 4.3. Monitoring System Based on PSO and GP

Algorithm 1: Enhanced GP-based PSO pseudo-code. |

#### 4.4. Hyper-Parameter Optimization

## 5. Performance Evaluation

#### 5.1. Simulation Setup

#### 5.1.1. Ground Truth

#### 5.1.2. Simulation Parameters

#### 5.2. Hyper-Parameter Optimization

#### 5.3. Comparison with Other Algorithms

#### Discussion of the Results

- The proposed enhanced GP-based PSO considers not only the area with the highest uncertainty (unexplored areas) of the lake surface like in [9], but it also considers the area with the highest contamination to carry out the monitoring task.
- The enhanced GP-based PSO is an algorithm capable to be used in any water resources. The proposed approach based the path generation in the samples that is collected from the surface of the water. As a consequence, the algorithm does not need to have more information about the environment than the data collected by the sensors. In addition, it is not necessary to perform previous calculations to generate an optimal path.
- To improve the simulation time of our previous work, data is collected from the sensors and the surrogate model is updated only after the ASVs have traveled a distance l. With this improvement, the algorithm is capable of performing 6000 iterations in approximately 120 s.
- To obtain the most suitable values for the enhanced GP-based PSO, a tuning has been carried out. The algorithm used for the hyper-parameter optimization is the BO and as acquisition function the EI is used. For kernel selection, three functions were tested, the Matérn $(nu=2.5,\phantom{\rule{4pt}{0ex}}\ell =1.0)$, the RBF $(\ell =1.0)$, and RBF $(\ell =0.4)$. The function that obtained the best MSE was the RBF $(\ell =1.0)$, therefore, it is used for the tuning task.
- With the hyper-parameter values obtained from the BO, the performance of three algorithms, the original PSO, the GP-based PSO and the enhanced GP-based PSO, were compared. The results showed that the proposed approach has the best MSE, being approximately $7\%$ lower than the MSE of the original PSO and $15\%$ lower than that of the GP-based PSO. Moreover, the enhanced GP-based PSO has the lowest variability of the three algorithms.

## 6. Conclusions and Future Works

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Hyper-parameter optimization using Matérn $(nu=2.5,\phantom{\rule{4pt}{0ex}}\ell =1.0)$ as kernel.

Case | ${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ${\mathit{c}}_{4}$ | ℓ | $\mathit{\lambda}$ |
---|---|---|---|---|---|---|

1 | 2.6971 | 0.6936 | 2.6694 | 2.9681 | 0.5589 | 0.1 |

2 | 2.5966 | 1.4761 | 2.9241 | 2.4693 | 1 | 0.1 |

3 | 1.2540 | 0.5788 | 0.1010 | 2.6108 | 0.4826 | 0.1109 |

4 | 3.0962 | 2.6647 | 2.3832 | 0.5391 | 0.6368 | 0.1 |

5 | 3.6672 | 0.3784 | 3.9466 | 3.7880 | 0.6724 | 0.1053 |

6 | 1.8240 | 2.7981 | 0 | 1.0480 | 0.4202 | 0.1 |

7 | 2.0687 | 1.7396 | 2.5661 | 1.2766 | 1 | 0.1 |

8 | 2.1811 | 1.2099 | 1.6372 | 1.7350 | 1 | 0.1 |

9 | 1.1649 | 0.7747 | 0.0306 | 2.7362 | 0.4 | 0.1 |

10 | 2.3295 | 1.4352 | 2.1793 | 1.8342 | 0.8791 | 0.1 |

All | 3.4643 | 1.6279 | 0 | 0.9880 | 0.4 | 0.1 |

**Table A2.**Comparison of the MSE and the average distance using the values of Table A1.

Case | Each Ground Truth | All Ground Truths | ||
---|---|---|---|---|

MSE | Average Distance [m] | MSE | Average Distance [m] | |

1 | 0.0062 | 17,048 | 0.0087 | 16,860 |

2 | 0.0075 | 17,919 | 0.0076 | 17,720 |

3 | 0.0123 | 17,934 | 0.0107 | 17,862 |

4 | 0.0111 | 17,063 | 0.0145 | 16,896 |

5 | 0.0176 | 17,647 | 0.0139 | 16,861 |

6 | 0.0037 | 16,382 | 0.0037 | 16,796 |

7 | 0.0071 | 17,042 | 0.0086 | 16,661 |

8 | 0.0080 | 17,274 | 0.0090 | 16,610 |

9 | 0.0042 | 16,942 | 0.0045 | 16,705 |

10 | 0.0025 | 17,281 | 0.0025 | 16,875 |

Case | ${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ${\mathit{c}}_{4}$ | ℓ | $\mathit{\lambda}$ |
---|---|---|---|---|---|---|

1 | 2.5810 | 0.8581 | 3.3424 | 3.3720 | 0.4 | 0.1 |

2 | 1.9555 | 1.0462 | 2.6077 | 3.3304 | 1 | 0.1 |

3 | 2.2666 | 3.3180 | 0 | 1.0986 | 0.4 | 0.1 |

4 | 3.4534 | 2.8453 | 3.6083 | 1.7296 | 0.4 | 0.1 |

5 | 1.8751 | 2.7764 | 0 | 0.9860 | 0.4089 | 0.1 |

6 | 1.5753 | 2.7054 | 0 | 1.2312 | 0.6546 | 0.1 |

7 | 2.2777 | 0.4624 | 2.2728 | 3.3438 | 1 | 0.1 |

8 | 2.8131 | 0.8359 | 2.1828 | 2.4807 | 0.6501 | 0.1 |

9 | 0.2728 | 3.8949 | 0 | 3.5305 | 0.8892 | 0.1 |

10 | 1.9160 | 1.3901 | 1.5319 | 2.0815 | 0.5870 | 0.1 |

All | 2.7575 | 1.9945 | 1.9899 | 2.6570 | 0.9813 | 0.1 |

**Table A4.**Comparison of the MSE and the average distance using the values of Table A3.

Case | Each Ground Truth | All Ground Truths | ||
---|---|---|---|---|

MSE | Average Distance [m] | MSE | Average Distance [m] | |

1 | 0.0062 | 17,053 | 0.0068 | 17,047 |

2 | 0.0074 | 17,920 | 0.0077 | 17,917 |

3 | 0.0107 | 17,730 | 0.0135 | 17,920 |

4 | 0.0108 | 17,157 | 0.0120 | 17,040 |

5 | 0.0138 | 16,860 | 0.0173 | 17,416 |

6 | 0.0038 | 16,081 | 0.0051 | 17,918 |

7 | 0.0074 | 17,048 | 0.0079 | 17,044 |

8 | 0.0080 | 17,348 | 0.0081 | 17,108 |

9 | 0.0042 | 16,905 | 0.0064 | 17,115 |

10 | 0.0027 | 17,306 | 0.0027 | 17,273 |

Kernel | MSE (Each Ground Truth) | MSE (All Ground Truths) |
---|---|---|

Matérn $(nu=2.5,\phantom{\rule{4pt}{0ex}}\ell =1.0)$ | 0.0080 ± 0.0046 | 0.0084 ± 0.0040 |

RBF $(\ell =1.0)$ | 0.0071 ± 0.0034 | 0.0081 ± 0.0040 |

RBF $(\ell =0.4)$ | 0.0075 ± 0.0035 | 0.0088 ± 0.0043 |

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**Figure 5.**Example of 3 ground truths of the 10 simulated ground truths obtained with the Shekel function using the Ypacarai Lake as simulated scenario.

**Figure 6.**Representation of movements of the ASVs and uncertainty in the exploration of the lake surface (top) and mean of the surrogate model (bottom) in two different ground truths.

**Figure 7.**Representation of movements of the ASVs and uncertainty in the exploration of the lake surface (top) and mean of the surrogate model (bottom) using three different algorithms.

Algorithm | Biological Motivation | Advantages | Disadvantages |
---|---|---|---|

ACO | Ant colonies | - Effective in discovering good solutions. | - Theoretical analysis is difficult. |

- Adaptation to changes. | - Research is more experimental than theoretical. | ||

- Converge. | - The convergence time is uncertain. | ||

BA | Echolocation of bats | - Few parameters. | - Can get trapped in multi-dimensional functions. |

- Simple to implement. | - If the dimensions of the problem increase, the chances of converging towards a global optimal solution decrease. | ||

- Low accuracy. | |||

FA | Fireflies attraction | - Simple to implement. | - Slow convergence. |

- Parallel implementation. | - Get easily stuck in local optimum for multi-modal problems. | ||

- Do not save the best solutions. | |||

PSO | Bird flocks | - Simple to implement. | - Get stuck in local optimum. |

- Few parameters to be tuned. | - Dispersion problems cannot be solved. | ||

- Do not overlap or mutate | - Initial design parameters can be difficult to define. |

Hyper-Parameter | Range |
---|---|

${c}_{1}$ | [0, 4] |

${c}_{2}$ | [0, 4] |

${c}_{3}$ | [0, 4] |

${c}_{4}$ | [0, 4] |

ℓ | [0.4, 1.0] |

$\lambda $ | [0.1, 0.5] |

Case | ${\mathit{c}}_{1}$ | ${\mathit{c}}_{2}$ | ${\mathit{c}}_{3}$ | ${\mathit{c}}_{4}$ | ℓ | $\mathit{\lambda}$ |
---|---|---|---|---|---|---|

1 | 1.9857 | 0.8604 | 2.8352 | 4 | 1 | 0.1 |

2 | 0.6741 | 0 | 3.4434 | 3.0503 | 0.6892 | 0.1 |

3 | 4 | 0 | 0.0464 | 1.4928 | 0.5178 | 0.1 |

4 | 3.7648 | 2.2512 | 1.9676 | 0.1526 | 0.4 | 0.1 |

5 | 2.1225 | 2.7236 | 0 | 0.7353 | 0.4 | 0.1 |

6 | 1.8154 | 3.1523 | 0 | 0.6271 | 0.4 | 0.1 |

7 | 1.6320 | 2.6733 | 2.3820 | 0 | 1 | 0.1 |

8 | 2.4401 | 3.5659 | 2.0532 | 0.1041 | 0.4 | 0.1 |

9 | 0.3339 | 3.8719 | 0 | 3.5302 | 0.8668 | 0.1 |

10 | 3.1286 | 2.568 | 0.7900 | 0 | 1 | 0.1 |

All | 4 | 1.0830 | 1.1316 | 0 | 0.4 | 0.1 |

Case | Each Ground Truth | All Ground Truths | ||
---|---|---|---|---|

MSE | Average Distance [m] | MSE | Average Distance [m] | |

1 | 0.0063 | 17,042 | 0.0067 | 17,041 |

2 | 0.0067 | 17,915 | 0.0073 | 18,094 |

3 | 0.0086 | 17,033 | 0.0115 | 18,879 |

4 | 0.0109 | 17,059 | 0.0110 | 17,193 |

5 | 0.0138 | 16,574 | 0.0166 | 17,765 |

6 | 0.0038 | 15,992 | 0.0049 | 17,956 |

7 | 0.0067 | 17,042 | 0.0070 | 17,038 |

8 | 0.0080 | 16,755 | 0.0081 | 17,542 |

9 | 0.0042 | 16,901 | 0.0057 | 17,478 |

10 | 0.0022 | 17,038 | 0.0025 | 17,563 |

Parameter / Hyper-Parameter | Original PSO | GP-Based PSO | Enhanced GP-Based PSO |
---|---|---|---|

Number of ASVs | 4 | 4 | 4 |

Maximum speed of the ASVs (m/s) | 2 | 2 | 2 |

Surface limits (m) | x [0, 10,000] | x [0, 10,000] | x [0, 10,000] |

y [0, 15,000] | y [0, 15,000] | y [0, 15,000] | |

Iterations | 6000 | 6000 | 6000 |

Ground truth | Case 0 | Case 0 | Case 0 |

w | 1 | - | - |

${w}_{min}$ | - | 0.4 | 0.4 |

${w}_{max}$ | - | 0.9 | 0.9 |

${c}_{1}$ | 1.9857 | 1.9857 | 1.9857 |

${c}_{2}$ | 0.8604 | 0.8604 | 0.8604 |

${c}_{3}$ | - | 2.8352 | 2.8352 |

${c}_{4}$ | - | - | 4 |

ℓ | 1 | 1 | 1 |

$\lambda $ | 0.1 | 0.1 | 0.1 |

Algorithm | MSE | Average Distance [m] |
---|---|---|

Original PSO | 0.0074 ± 0.0063 | 12,190 |

GP-based PSO | 0.0087 ± 0.0045 | 17,769 |

Enhanced GP-based PSO (Case 0) | 0.0069 ± 0.0033 | 17,937 |

Enhanced GP-based PSO (Case All) | 0.0074 ± 0.0045 | 17,829 |

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## Share and Cite

**MDPI and ACS Style**

Kathen, M.J.T.; Flores, I.J.; Reina, D.G.
An Informative Path Planner for a Swarm of ASVs Based on an Enhanced PSO with Gaussian Surrogate Model Components Intended for Water Monitoring Applications. *Electronics* **2021**, *10*, 1605.
https://doi.org/10.3390/electronics10131605

**AMA Style**

Kathen MJT, Flores IJ, Reina DG.
An Informative Path Planner for a Swarm of ASVs Based on an Enhanced PSO with Gaussian Surrogate Model Components Intended for Water Monitoring Applications. *Electronics*. 2021; 10(13):1605.
https://doi.org/10.3390/electronics10131605

**Chicago/Turabian Style**

Kathen, Micaela Jara Ten, Isabel Jurado Flores, and Daniel Gutiérrez Reina.
2021. "An Informative Path Planner for a Swarm of ASVs Based on an Enhanced PSO with Gaussian Surrogate Model Components Intended for Water Monitoring Applications" *Electronics* 10, no. 13: 1605.
https://doi.org/10.3390/electronics10131605