# Three-Dimensional Flight Corridor: An Occupancy Checking Process for Unmanned Aerial Vehicle Motion Planning inside Confined Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Prior Work (Literature Review)

#### Proposed Methodology

## 3. Novel 3D Flight Corridor Generation

- (i)
- Given an environmental map, generate a hand-crafted flight path composed of multiple path segments.
- (ii)
- Generate suitable maximized ellipsoid geometries containing the hand-crafted path segments while avoiding obstacles.
- (iii)
- Define a set of hyperplanes for each path segment and their corresponding halfspaces based on the generated ellipsoids.
- (iv)
- Extract the intersection of the hyperplanes obtained from step (iii) as obstacle-free convex polyhedra surrounding each segment of the path.
- (v)
- Perform a cuboid bounding surrounding the UAV based on its dimensions to reduce the number of collision checks required during the polyhedra construction process (step (vi)).
- (vi)
- Add an augmenting occupancy checking process to reduce the search space by identifying regions free from obstacles and replace the previously constructed convex polyhedra with alternate reduced-in-volume convex polyhedra.
- (vii)
- Generate the 3D Collision-Free Safe Flight Corridor.

#### 3.1. Generation of Convex Polyhedra

#### 3.1.1. Generation of Ellipsoids

Algorithm 1: Pseudo-code of the proposed SFC${}^{+}$ flight corridor generation process. |

#### 3.1.2. Intersection of Hyperplanes

#### 3.1.3. Cuboid Bounding

#### 3.1.4. Occupancy Checking

#### 3.1.5. Final Flight Corridor

#### 3.2. Calculation of Corridor Volume

## 4. Simulation Results and Discussion

#### 4.1. Selected UAV

#### 4.2. Testing Scenarios

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CP | Convex Polyhedron |

IRIS | Iterative Regional Inflation by Semi-Definite Programming |

I-TRP | Improved “Teach-Repeat-Replan” |

SCM | Star Convex Method |

SFC${}^{+}$ | Enhanced Safe Flight Corridor |

SLAM | Simultaneous Localization and Mapping |

UAV | Unmanned Aerial Vehicle |

USAR | Urban Search and Rescue |

VTOL | Vertical Take-off and Landing |

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**Figure 5.**An example of generating a flight corridor $FC\left({P}_{hcr}\right)$ composed of connected polyhedra.

**Figure 6.**The process of finding an ellipsoid around $\overrightarrow{{P}_{i}{P}_{i+1}}$ by shrinking to form a collision-free one shown in 2D setting: (

**a**) an initial solid blue sphere is surrounding a path segment $\overrightarrow{{P}_{i}{P}_{i+1}}$; (

**b**) the initial sphere is gradually shrinking until it has no obstacle points by repeating the same shrinking process; (

**c**) the final phase of obtaining the maximal collision-free ellipse in size/area (dashed yellow).

**Figure 7.**Examples for finding ellipsoids around the provided ${P}_{hcr}$ segments in two environmental scenarios: (

**a**) $30\times 10\times 6$ m environment with 18 convex and concave obstacles, some with opening holes, (

**b**) $18\times 6\times 6$ m environment with 7 large and 5 small convex obstacles.

**Figure 8.**The process of finding a convex polyhedron $C{P}_{i}$ around $\overrightarrow{{P}_{i}{P}_{i+1}}$: (

**a**) an initial dashed yellow ellipse touching an obstacle point to generate the first hyperplane; (

**b**) a dashed blue dilated ellipse until touching the second obstacle point to generate the second hyperplane; (

**c**) a third dilated ellipse (solid red) to generate the third hyperplane; (

**d**) the final obstacle-free polyhedron (dashed green).

**Figure 11.**Example of generated polyhedra in an obstacle-free space using occupancy checking process.

**Figure 21.**Performance comparison (${t}_{c},{Vol}_{cor}$) for three different map scenarios against obstacle density ($\varsigma $):

**left**: collapsed building scenario,

**middle**: cave scenario,

**right**: obstructed mine corridor.

**Figure 22.**Performance comparison ($\Delta {t}_{c},\Delta {Vol}_{cor}$) for three different map scenarios against obstacle density ($\varsigma $):

**left**: computational time difference,

**right**: corridor volume difference.

**Figure 23.**Results from a 3D collapsed building environment ($\varsigma =0.1645$) based on a post-processed hand-crafted path.

Environment | Obstacle Density ($\mathit{\varsigma}$) | No. of Obstacles | Overall Size of the Space (w × h × d, m) | Coordinates of Entry Point (m) | Coordinates of Exit Points (m) |
---|---|---|---|---|---|

Collapsed building | $\varsigma =0.1645$ (confined space) | 20 | 30 × 10 × 6 | [0.5,4,2] | [30,4,2] |

$\varsigma =0.1233$ (confined space) | 10 | ||||

$\varsigma =0.081$ (cluttered space) | 9 | ||||

Steep cave | $\varsigma =0.18567$ (confined space) | 14 | 20 × 5 × 6 | [1,1,5] | [20,3,2] |

$\varsigma =0.13325$ (confined space) | 5 | ||||

$\varsigma =0.08125$ (cluttered space) | 3 | ||||

Obstructed mine corridor | $\varsigma =0.1455$ (confined space) | 21 | 20 × 5 × 6 | [0.5,4,2] | [19,4,2] |

$\varsigma =0.1188$ (confined space) | 19 | ||||

$\varsigma =0.077$ (cluttered space) | 17 |

**Table 2.**Performance comparison between the proposed SFC${}^{+}$ and SCM for 3D flight corridor generation based on a hand-crafted path.

Computational Time (${\mathit{t}}_{\mathit{c}}$, s) | Corridor Volume (m ^{3}) | No. of Generated Polyhedra | ||||||
---|---|---|---|---|---|---|---|---|

Obstacle Density ($\mathbf{\varsigma}$) | Map Resolution (Voxel Size, $\mathrm{cm}$) | Proposed SFC${}^{+}$ | SCM | Proposed SFC${}^{+}$ | SCM | Proposed SFC${}^{+}$ | SCM | |

Collapsed building map scenario | $\varsigma =0.1645$ | $20\times 20\times 20$ | 2.5818 | 3.5284 | 237.6235 | 500.5325 | 21 | 27 |

$10\times 10\times 10$ | 2.6651 | 3.712 | 237.6235 | 500.5325 | 21 | 27 | ||

$\varsigma =0.1233$ | $20\times 20\times 20$ | 1.8453 | 2.5013 | 257.3771 | 515.8238 | 21 | 25 | |

$10\times 10\times 10$ | 1.862 | 2.5227 | 257.3771 | 515.8238 | 21 | 25 | ||

$\varsigma =0.081$ | $20\times 20\times 20$ | 1.5007 | 2.0604 | 277.559 | 530.5415 | 21 | 24 | |

$10\times 10\times 10$ | 1.5351 | 1.9972 | 277.559 | 530.5415 | 21 | 24 | ||

Cave map scenario | $\varsigma =0.18567$ | $20\times 20\times 20$ | 1.3274 | 2.8754 | 221.0281 | 306.3768 | 21 | 33 |

$10\times 10\times 10$ | 1.3054 | 2.9787 | 221.0281 | 306.3768 | 21 | 33 | ||

$\varsigma =0.13325$ | $20\times 20\times 20$ | 0.90729 | 1.9332 | 265.5472 | 351.4842 | 21 | 28 | |

$10\times 10\times 10$ | 0.87448 | 1.9204 | 265.5472 | 351.4842 | 21 | 28 | ||

$\varsigma =0.08125$ | $20\times 20\times 20$ | 0.62353 | 1.0092 | 257.136 | 365.0643 | 21 | 27 | |

$10\times 10\times 10$ | 0.5778 | 1.0391 | 257.136 | 365.0643 | 21 | 27 | ||

Obstructed mining map scenario | $\varsigma =0.1455$ | $20\times 20\times 20$ | 1.196 | 2.0492 | 126.6911 | 211.6088 | 24 | 18 |

$10\times 10\times 10$ | 1.1476 | 2.0704 | 126.6911 | 211.6088 | 24 | 18 | ||

$\varsigma =0.1188$ | $20\times 20\times 20$ | 1.1009 | 1.9819 | 126.6911 | 206.3995 | 24 | 19 | |

$10\times 10\times 10$ | 1.0287 | 1.9481 | 126.6911 | 206.3995 | 24 | 19 | ||

$\varsigma =0.077$ | $20\times 20\times 20$ | 0.71836 | 1.5541 | 138.8408 | 240.288 | 24 | 19 | |

$10\times 10\times 10$ | 0.70418 | 1.6772 | 138.8408 | 240.288 | 24 | 19 |

**Table 3.**Performance comparison between the proposed SFC${}^{+}$ and SCM for 3D flight corridor generation based on a post-processed hand-crafted path.

Computational Time (${\mathit{t}}_{\mathit{c}}$, s) | Corridor Volume (m ^{3}) | No. of Generated Polyhedra | ||||||
---|---|---|---|---|---|---|---|---|

Obstacle Density ($\mathbf{\varsigma}$) | Map Resolution (Voxel Size, cm) | Proposed SFC${}^{+}$ | SCM | Proposed SFC${}^{+}$ | SCM | Proposed SFC${}^{+}$ | SCM | |

Collapsed building map scenario | $\varsigma =0.1645$ | $20\times 20\times 20$ | 2.1929 | 2.7444 | 231.7023 | 358.8962 | 7 | 24 |

$10\times 10\times 10$ | 1.9182 | 2.9326 | 309.0498 | 423.968 | 8 | 23 | ||

Cave map scenario | $\varsigma =0.18567$ | $20\times 20\times 20$ | 1.0666 | 2.7866 | 370.3862 | 307.041 | 13 | 32 |

$10\times 10\times 10$ | 1.0577 | 2.6848 | 362.4562 | 288.697 | 13 | 30 | ||

Obstructed mining environment | $\varsigma =0.1455$ | $20\times 20\times 20$ | 0.66375 | 1.806 | 113.088 | 177.5656 | 7 | 18 |

$10\times 10\times 10$ | 0.70554 | 1.6912 | 93.0473 | 172.254 | 6 | 19 |

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**MDPI and ACS Style**

Mostafa, S.; Ramirez-Serrano, A.
Three-Dimensional Flight Corridor: An Occupancy Checking Process for Unmanned Aerial Vehicle Motion Planning inside Confined Spaces. *Robotics* **2023**, *12*, 134.
https://doi.org/10.3390/robotics12050134

**AMA Style**

Mostafa S, Ramirez-Serrano A.
Three-Dimensional Flight Corridor: An Occupancy Checking Process for Unmanned Aerial Vehicle Motion Planning inside Confined Spaces. *Robotics*. 2023; 12(5):134.
https://doi.org/10.3390/robotics12050134

**Chicago/Turabian Style**

Mostafa, Sherif, and Alejandro Ramirez-Serrano.
2023. "Three-Dimensional Flight Corridor: An Occupancy Checking Process for Unmanned Aerial Vehicle Motion Planning inside Confined Spaces" *Robotics* 12, no. 5: 134.
https://doi.org/10.3390/robotics12050134