# A Divide and Conquer Strategy for Sweeping Coverage Path Planning

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(This article belongs to the Section F: Electrical Engineering)

## Abstract

**:**

## 1. Introduction

## 2. Related Work

## 3. Problem Modelling

## 4. Sweeping Coverage Path Planning

#### 4.1. Divide and Conquer Strategy

`RCPPforRoom`. The

`RCPPforRoom`receives as input the polygon of the room, the door points and the span of the cleaning area and returns the room cleaning pattern. The proposed adaptation is described separately in Section 4.1.1 because it relies only on computational geometry and keeps the property that it computes the shortest path in the room, as described in [26]. Once a path has been computed for each room, the paths are merged into one full path. Consequently, a complete graph, G, and its adjacency matrix, A, are constructed from the already computed door points and cleaning pattern distances. The path between the remaining door points is calculated by computing the medial axis of the map, converting the medial axis into a temporal graph and estimating the best path with the A* algorithm [42]. An alternative to the A* is the Dijkstra algorithm, as reported in [43]. Finally, the entire trajectory is obtained by finding the shortest travelled tour that satisfies the undirected rural postman problem. To determine the shortest tour, a genetic algorithm (GA) is proposed. The modelling and GA implementation are described in detail in Section 4.1.2.

Algorithm 1: Divide and conquer sweeping coverage path planner. |

#### 4.1.1. Single Room Path Planning

`getDoorPoint`function in Algorithm 1, the calculated room polygon ${\mathcal{P}}_{i}$ and the full map polygon $\mathcal{M}$ are used. First, the set difference [44] of the room with the intersection of both polygons provides a set of line segments, namely:

`CPPforRoom`function in Algorithm 1. To solve the problem, we adapt the RCPP planner [26], which estimates the optimal edge-vertex path that covers a convex polygon given starting and ending points. The proposed adaption is summarised in Algorithm 2. Given that such an algorithm requires a target convex polygon and two points as input, a new smaller polygon and two inner points are created (lines 1 to 3 of Algorithm 2). The reason is that the RCPP planner considers the edges of the polygon as valid positions. Though, for this use case, that implies that the robot will be in collision with the walls. The smaller polygon, ${\mathcal{P}}_{target}$, is computed by negatively offsetting the polygon ${\mathcal{P}}_{i}$ by a circle with a diameter equal to ${d}_{c}$, in other words, by computing the Minkowski difference [44,45] of ${\mathcal{P}}_{i}$ with the circle. Next, the starting and ending points, ${p}_{e}$ and ${p}_{x}$, are computed as the intersection with the polygon of the perpendicular line to the polygon that passes through the door points. Finally, the RCPP is called, and it returns a path, W, as a sequence of points that the robot has to follow, and the length of such path, l. To complete the path, the door points are appended to W, and the length is updated.

Algorithm 2:CPPforRoom$({\mathcal{P}}_{i},{p}_{i}^{a},{p}_{i}^{b},{d}_{c})$. Coverage path planning for a single room. |

Data: Room polygon ($\mathcal{P}$), door points (${p}_{i}^{a},{p}_{i}^{b},$), cleaning span (${d}_{c}$)Result: Path (W)1 ${\mathcal{P}}_{target}\leftarrow $ErodePolygon($\mathcal{P},{d}_{c}/2$)2 ${p}_{e}\leftarrow $NearestPoint(${\mathcal{P}}_{target},{p}_{i}^{a}$);3 ${p}_{x}\leftarrow $NearestPoint(${\mathcal{P}}_{target},{p}_{i}^{b},$);4 $W,l=$RCPP$({\mathcal{P}}_{target},{p}_{e},{p}_{x})$;5 $W\leftarrow \{{p}_{i}^{a},W,{p}_{i}^{b}\}$6 $l=l+{d}_{c}$;7 return $W,l$ |

#### 4.1.2. Shortest Cleaning Tour

`GetShortestTour`in Algorithm 1. Since we have identified that the problem can be addressed in different forms depending on the number of doors, we present two cases, when all the rooms have only one door and when they have a maximum of two doors.

#### 4.1.3. Rooms of One Door-Solving a TSP

#### 4.1.4. Rooms of Two Doors-Solving a RPP

`GetShortestTour`function of the general strategy when the rooms a maximum of two doors.

`GetShortestTour`function in Algorithm 1. Such order is specified as a list, $T=\{{e}_{1},\dots ,{e}_{\left|T\right|}\}$, whose elements are the edges to be travelled.

#### 4.2. General Case Insight

## 5. Experiments

#### 5.1. Synthetic Scenarios

#### 5.2. Case Study

^{®}LIDAR of an office building. The obtained probabilistic grid is displayed in Figure 5a. Such a probabilistic grid was first thresholded and manually cleaned to remove the laser misreadings. Then the contour was found using Suzuki’s algorithm [53]. Then, the contour was converted to a polygon by discarding collinear points. Then the rooms were segmented according to the cleaning needs specified by the final user.

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CPP | coverage path planning |

DnCS | divide and conquer strategy |

GA | genetic algorithm |

MDPI | multidisciplinary digital publishing institute |

RPP | rural postman problem |

RCPP | rotating callipers path planner |

S-CPP | sweeping coverage path planning problem |

TSP | travelling salesman problem |

UAV | unmanned aerial vehicle |

uRPP | undirected rural postman problem |

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**Figure 1.**Examples of room-based interior maps. In these maps, the rooms are described by convex polygons, and they are connected by corridors. (

**a**) Map A with three rooms and two corridors. (

**b**) Map B with four rooms and three corridors.

**Figure 2.**Example of a segmented map and its corresponding graph used for tour planning. We observe that each room is only accessible by the door points. In the corresponding graph, the door points are converted to vertices, while the movements through rooms and corridors are converted into edges. To cover all the rooms, the edges drawn with continuous lines must be travelled, while the edges drawn with dotted lines to be travelled in the formulation of the rural postman problem are optional. (

**a**) Segmented map and door points. (

**b**) Corresponding graph.

**Figure 4.**Comparison of computed paths for the proposed maps. The maps are delimited by a grey line. The computed path is drawn in blue. Starting point is drawn in green. Ending point is drawn in yellow. (

**a**) Map A; DnCS-RCPP. (

**b**) Map A; DnCS-Contour. (

**c**) Map A; TSPP [9] (

**d**) Map B; DnCS-RCPP. (

**e**) Map B; DnC-Contours. (

**f**) Map B; TSPP [9]. (

**g**) Map C; DnCS-RCPP. (

**h**) Map C; DnC-Contours. (

**i**) Map C; TSPP [9].

**Figure 5.**Case study. The map generated after SLAM and the computed coverage paths are displayed. Robot starting point is drawn in green. Ending point is drawn in yellow. (

**a**) The 2D initial map. Rooms (R1, R2 and R3), starting point (S) and ending point (E) are specified by the final user. (

**b**) Computed path by DnC-RCPP. The map polygon is displayed in grey. The planned route is drawn in blue. (

**c**) Computed path by DnC-Contour. The map polygon is displayed in grey. The planned route is drawn in blue. (

**d**) Computed path by TSPP [9]. The map polygon is displayed in grey. The planned route is drawn in blue.

DnCS-RCPP | DnCS-Contour | TSPP [9] | ||||
---|---|---|---|---|---|---|

Map | Trav. Dist. | C. Time (s) | Trav. Dist. | C. Time (s) | Trav. Dist. | C. Time (s) |

Map A | 6569 | 0.001 | 6663 | 0.001 | 6023 | 0.052 |

Map B | 12,315 | 0.036 | 13,249 | 0.041 | 11,881 | 0.078 |

Map C | 11,074 | 0.017 | 13,669 | 0.029 | 9919 | 0.024 |

DnCS-RCPP | DnCS-Contour | TSPP [9] | ||||
---|---|---|---|---|---|---|

Map | Trav. Dist. | C. Time (s) | Trav. Dist. | C. Time (s) | Trav. Dist. | C. Time (s) |

Building | 6284 | 0.012 | 7830 | 0.012 | 5972 | 0.031 |

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## Share and Cite

**MDPI and ACS Style**

Vasquez, J.I.; Merchán-Cruz, E.A.
A Divide and Conquer Strategy for Sweeping Coverage Path Planning. *Energies* **2022**, *15*, 7898.
https://doi.org/10.3390/en15217898

**AMA Style**

Vasquez JI, Merchán-Cruz EA.
A Divide and Conquer Strategy for Sweeping Coverage Path Planning. *Energies*. 2022; 15(21):7898.
https://doi.org/10.3390/en15217898

**Chicago/Turabian Style**

Vasquez, Juan Irving, and Emmanuel Alejandro Merchán-Cruz.
2022. "A Divide and Conquer Strategy for Sweeping Coverage Path Planning" *Energies* 15, no. 21: 7898.
https://doi.org/10.3390/en15217898