# A Method of Path Planning on Safe Depth for Unmanned Surface Vehicles Based on Hydrodynamic Analysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling

#### 2.1. Hydrodynamic Modeling

**u**,

**v**,

**w**are respectively the components of velocity on the directions of x, y, z (see Figure 1).

**U**and

**W**are velocity profiles on x and z direction. H is the wave amplitude, k is wave-number (2π/m). x-axis is the direction of wave propagation, and the z-axis is the direction of wave fluctuation. The wave propagation direction is parallel to the heading direction of USV.

#### 2.2. Path Planning Environment Modeling

## 3. Depth Safety Path Planning Method

#### 3.1. Analysis of Hydrodynamic Properties of Unmanned Surface Vehicle

#### 3.2. Path Planning Algorithms Considering Depth Hazard

## 4. Discussions

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Liu, Z.X.; Zhang, Y.M.; Yu, X.; Yuan, C. Unmanned surface vehicles.: An overview of developments and challenges. Annu. Rev. Control
**2016**, 41, 71–93. [Google Scholar] [CrossRef] - Zheng, H.; Negenborn, R.R.; Lodewijks, G. Survey of approaches for improving the intelligence of marine Surface Vehicles. In Proceedings of the 16th International IEEE Conference on Intelligent Transportation Systems (ITSC 2013), The Hague, The Netherlands, 6–9 October 2013; pp. 1217–1223. [Google Scholar]
- Mohanan, M.G.; Salgoankar, A. A survey of robotic motion planning in dynamic environments. Robot. Auton. Syst.
**2018**, 100, 171–185. [Google Scholar] [CrossRef] - Macharet, D.G.; Campos, M.F.M. A survey on routing problems and robotic systems. Robotica
**2018**, 36, 1781–1803. [Google Scholar] [CrossRef] - Campbell, S.; Naeem, W.; Irwin, G.W. A review on improving the autonomy of unmanned surface vehicles through intelligent collision avoidance manoeuvres. Annu. Rev. Control
**2012**, 36, 267–283. [Google Scholar] [CrossRef] [Green Version] - Liu, Y.C.; Bucknall, R. Path planning algorithm for unmanned surface vehicle formations in a practical maritime environment. Ocean Eng.
**2015**, 97, 126–144. [Google Scholar] [CrossRef] - Wu, P.P.; Campbell, D.; Merz, T. Multi-objective four-dimensional vehicle motion planning in large dynamic environments. IEEE Trans. Syst. Man Cybern. B Cybern.
**2011**, 41, 621–634. [Google Scholar] [CrossRef] [PubMed] - Ma, Y.; Hu, M.Q.; Yan, X.P. Multi-objective path planning for unmanned surface vehicle with currents effects. ISA Trans.
**2018**, 75, 137–156. [Google Scholar] [CrossRef] [PubMed] - Hart, P.E.; Nilsson, N.J.; Raphael, B. A formal basis for the heuristic determination of minimum cost paths. IEEE Trans. Syst. Sci. Cybern.
**1968**, 4, 100–107. [Google Scholar] [CrossRef] - Song, R.; Liu, Y.; Bucknall, R. Smoothed A* algorithm for practical unmanned surface vehicle path planning. Appl. Ocean Res.
**2019**, 83, 9–20. [Google Scholar] [CrossRef] - Nash, A.; Koenig, S. Any-Angle Path Planning. AI Mag.
**2013**, 34, 85–107. [Google Scholar] [CrossRef] [Green Version] - Choi, S.; Yu, W. Any-angle path planning on non-uniform costmaps. In Proceedings of the 2011 IEEE International Conference on Robotics and Automation ICRA, Shanghai, China, 9–13 May 2011; pp. 5615–5621. [Google Scholar]
- Kim, H.; Lee, T.; Chung, H.; Son, N.; Myung, H. Any-angle Path Planning with Limit-Cycle Circle Set for Marine Surface Vehicle. In Proceedings of the 2012 IEEE International Conference on Robotics and Automation ICRA, Saint Paul, MN, USA, 14–18 May 2012; pp. 2275–2280. [Google Scholar]
- Nash, A.; Koenig, S.; Likhachev, M. Incremental Phi*: Incremental Any-Angle Path Planning on Grids. In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI), Pasadena, CA, USA, 11–17 July 2009; pp. 1824–1830. [Google Scholar]
- Nash, A.; Koenig, S.; Tovey, C. Lazy Theta*: Any-Angle Path Planning and Path Length Analysis in 3D. In Proceedings of the Third Annual Symposium on Combinatorial Search, SOCS 2010, Stone Mountain, Atlanta, GA, USA, 8–10 July 2010; pp. 147–154. [Google Scholar]
- Daniel, K.; Nash, A.; Koenig, S.; Felner, A. Theta*: Any-Angle Path Planning on Grids. J. Artif. Intell. Res.
**2010**, 39, 533–579. [Google Scholar] [CrossRef] [Green Version] - Yang, J.M.; Tseng, C.M.; Tseng, P.S. Path planning on satellite images for unmanned surface vehicles. Int. J. Nav. Arch. Ocean
**2015**, 7, 87–99. [Google Scholar] [CrossRef] [Green Version] - Liu, C.G.; Mao, Q.Z.; Chu, X.M.; Xie, S. An Improved A-Star Algorithm Considering Water Current, Traffic Separation and Berthing for Vessel Path Planning. Appl. Sci.
**2019**, 9, 1057. [Google Scholar] [CrossRef] - Singh, Y.; Sharma, S.; Sutton, R.; Hatton, D.; Khan, A. A constrained A* approach towards optimal path planning for an unmanned surface vehicle in a maritime environment containing dynamic obstacles and ocean currents. Ocean Eng.
**2018**, 169, 187–201. [Google Scholar] [CrossRef] [Green Version] - Larson, J.; Bruch, M.; Ebken, J. Autonomous navigation and obstacle avoidance for unmanned surface vehicles. Proc. SPIE
**2006**, 6230. [Google Scholar] [CrossRef] - Svec, P.; Schwartz, M.; Thakur, A.; Gupta, S.K. Trajectory Planning with Look-Ahead for Unmanned Sea Surface Vehicles to Handle Environmental Disturbances. In Proceedings of the 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems, San Francisco, CA, USA, 25–30 September 2011; pp. 1154–1159. [Google Scholar]
- Naeem, W.; Irwin, G.W.; Yang, A.L. COLREGs-based collision avoidance strategies for unmanned surface vehicles. Mechatronics
**2012**, 22, 669–678. [Google Scholar] [CrossRef] - Svec, P.; Gupta, S.K. Automated synthesis of action selection policies for unmanned vehicles operating in adverse environments. Auton. Robot.
**2012**, 32, 149–164. [Google Scholar] [CrossRef] - Soulignac, M. Feasible and Optimal Path Planning in Strong Current Fields. IEEE Trans. Robot.
**2011**, 27, 89–98. [Google Scholar] [CrossRef] - Naus, K.; Wąż, M. The idea of using the A* algorithm for route planning an unmanned vehicle “Edredon”. Zesz. Nauk. Akad. Morska W Szczec.
**2013**, 36, 143–147. [Google Scholar] - Kim, H.; Park, B.; Myung, H. Curvature Path Planning with High Resolution Graph for Unmanned Surface Vehicle. In Robot Intelligence Technology and Applications 2012; Springer: Berlin/Heidelberg, Germany, 2013; pp. 147–154. [Google Scholar]
- Lee, T.; Kim, H.; Chung, H.; Bang, Y.; Myung, H. Energy efficient path planning for a marine surface vehicle considering heading angle. Ocean Eng.
**2015**, 107, 118–131. [Google Scholar] [CrossRef] - Fossen, T.I. Handbook of Marine Craft Hydrodynamics and Motion Controli, 1st ed.; John Wiley & Sons Ltd.: Chichester, UK, 2011. [Google Scholar]
- Dhanak, M.R.; Xiros, N.I. Springer Handbook of Ocean Engineering; Springer: Berlin/Heidelberg, Germany, 2016. [Google Scholar]
- Kim, H.; Kim, D.; Shin, J.U.; Kim, H.; Myung, H. Angular rate-constrained path planning algorithm for unmanned surface vehicles. Ocean Eng.
**2014**, 84, 37–44. [Google Scholar] [CrossRef]

**Figure 3.**Mesh Generation and Distribution. (

**a**) Three-dimensional Mesh Generation; (

**b**) Grid Distribution Around the Wall.

**Figure 4.**S57 Environmental Modeling Map (

**a**) Study Area of S57 Electronic Chart. (

**b**) Regional Grid Marine Environment Map with a resolution of 25 m × 25 m. In the picture, the white meshes represent the feasible region, while the black meshes are a barrier.

**Figure 5.**Electronic navigation Chart and Grid Depth Distribution Map. (

**a**) S57 Vector Electronic navigation Chart displayed by S52 Standard in the study area; (

**b**) Isobath Area Map; (

**c**) Depth Distribution Map obtained by spline function interpolation approach with obstacles, in which the blank area is an obstacle; (

**d**) Overlay with discrete depth points, contours, islands and reefs and other obstacles. In the depth distribution map, the magenta area is an obstacle.

**Figure 10.**A* path planning algorithm schematic diagram. (

**a**) Schematic of adjacent node expansion; (

**b**) The shortest path of A* algorithm. The green star represents the starting point, and the yellow diamond represents the goal point. The shortest path is the black line between the starting point and the end point. The color grids are expanded nodes. The deeper the blue is, the closer the starting point is, and the deeper the red is, the closer the end point is. A set of white grids represents a feasible space, and black grids are obstacles.

**Figure 12.**Comparison of three kinds path. (

**a**) Shortest distance path (SDP) that ignores the depth risk; (

**b**) Shorter and safer path (SSP) that considers the depth risk level and distance. (

**c**) Safest path (SFP) that considers only the depth risk. The colorful grids in figure (

**a**–

**c**) are extended nodes. The space of white grids represents the feasible space, and the black grids are obstacles. (

**d**) Three paths, shown on the depth distribution map. (

**e**) Three paths, shown on the depth risk level distribution map. (

**f**) Three paths, shown on S57 ENC. The black line is the SDP, the blue one is SSP, and the purple one is SFP.

**Figure 13.**Comparison of three kinds path. (

**a**) SDP that ignores the depth risk; (

**b**) SSP that considers the depth risk level and distance. (

**c**) SFP that considers only the depth risk. The colorful grids in figure (

**a**–

**c**) are extended nodes. The space of white grids represents the feasible space, and the black grids are obstacles. (

**d**) Three paths, shown on the depth distribution map. (

**e**) Three paths, shown on the depth risk level distribution map. (

**f**) Three paths, shown on S57 ENC. The black line is the SDP, the blue one is SSP, and the purple one is SFP.

**Figure 14.**The comparation of key parameters of three different sorts of paths. (

**a**) The total distance of the path. The unit of distance is m. (

**b**) Water depth risk level of the path. (

**c**) The number of expanded nodes of three sorts of cost functions.

Parameters | Value |
---|---|

Length | 3.2 m |

Molded breadth | 2.2 m |

Displacement | 92 kg (full load 140 kg) |

Location of gravity center | (1.285, 0.000, 0.600) |

Moment of inertia of Y axis | 400 kg/${\mathrm{m}}^{2}$ |

Draft | 0.5 m |

Case Number | Speed (kn) | Wave Height (m) | ${\mathit{F}}_{\mathit{r}\nabla}{\text{}}^{1}$ |
---|---|---|---|

1 | 5 | 0.5 | 0.42 |

2 | 5 | 1.0 | 0.42 |

3 | 5 | 1.5 | 0.42 |

4 | 8 | 0.5 | 0.63 |

5 | 8 | 1.0 | 0.63 |

6 | 8 | 1.5 | 0.63 |

7 | 11 | 0.5 | 0.84 |

8 | 11 | 1.0 | 0.84 |

9 | 11 | 1.5 | 0.84 |

^{1}${F}_{r\nabla}$: Volume Froude number, a non-dimensional parameter which presents the shape and the displacement of the ship.

Discretization Method | Finite Volume Method |
---|---|

Scheme | SIMPLE |

Solver | Implicit Unsteady |

Gradient | Green-Guass Cell Based |

Pressure | Second Order |

Momentum | Second Order Upwind |

Transient Formulation | Second Order Implicit |

Mesh motion | Overset motion |

Mesh | Mesh 1 | Mesh 2 | Mesh 3 |
---|---|---|---|

Number of meshs | 774,921 | 1,373,495 | 1,925,630 |

$E=\left|1-{\overline{C}}_{di}/{\overline{C}}_{d3}\right|$ | $1.13\times {10}^{-3}$ | $4.35\times {10}^{-3}$ | -- |

Plotting Scale | Depth Contour Interval |
---|---|

≥1:150,000 | 0/2/5/10/20/30/50/100/200/500 |

≥1:1,000,000 and ≤1:150,000 | 0/5/10/20/30/50/100/200/500/1000 |

≤1:1,000,000 | 0/10/50/200/500/1000/2000/4000/6000 |

Parameters | Value |
---|---|

Cell size | 25 m × 25 m |

L | 3.2 m |

$\overline{T}$ | 0.5 m |

${z}_{\mathrm{max}}$ | 0.3 m |

${\theta}_{\mathrm{max}}$ | $11\xb0$ |

${e}_{enc}$ | 0.2 m |

$\alpha $ | 10 |

Test | Start Point | Goal Point | Total Distance (m) | Risk Level | Nodes of Expanded | |||
---|---|---|---|---|---|---|---|---|

A* | WDLRA* | A* | WDLRA* | A* | WDLRA* | |||

1 | (710, 902) | (990, 1408) | 15,667.5 | 17,319.5 | 38.86142 | 23.46830 | 37,682 | 102,283 |

2 | (343, 975) | (417, 1066) | 3063 | 3063.0 | 5.68736 | 5.47936 | 1363 | 4469 |

3 | (408, 560) | (520, 680) | 4457.75 | 4472.5 | 9.95472 | 7.03769 | 5851 | 7679 |

4 | (200, 250) | (410, 530) | 9270.5 | 9255.75 | 18.68029 | 17.92211 | 21,407 | 41,840 |

5 | (500, 889) | (650, 1039) | 5641.5 | 5405.5 | 14.43746 | 9.53375 | 2971 | 19,301 |

6 | (500, 500) | (650, 650) | 5567.75 | 5567.75 | 7.39605 | 7.31804 | 7761 | 15,533 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, S.; Wang, C.; Zhang, A.
A Method of Path Planning on Safe Depth for Unmanned Surface Vehicles Based on Hydrodynamic Analysis. *Appl. Sci.* **2019**, *9*, 3228.
https://doi.org/10.3390/app9163228

**AMA Style**

Liu S, Wang C, Zhang A.
A Method of Path Planning on Safe Depth for Unmanned Surface Vehicles Based on Hydrodynamic Analysis. *Applied Sciences*. 2019; 9(16):3228.
https://doi.org/10.3390/app9163228

**Chicago/Turabian Style**

Liu, Shuai, Chenxu Wang, and Anmin Zhang.
2019. "A Method of Path Planning on Safe Depth for Unmanned Surface Vehicles Based on Hydrodynamic Analysis" *Applied Sciences* 9, no. 16: 3228.
https://doi.org/10.3390/app9163228