# Bézier Curves-Based Optimal Trajectory Design for Multirotor UAVs with Any-Angle Pathfinding Algorithms

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## Abstract

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## 1. Introduction

## 2. Path Planning

#### 2.1. Grid Representation

- E < 50. These nodes were highlighted in magenta color. These nodes represent the safest elevation for the UAV to fly above taking into consideration the UAV’s flight altitude.
- 50 < E < 59. These nodes were highlighted in red color. They are defined as a risky elevation, which is close to the flight altitude of the UAV and it is preferred to be avoided unless there is another path to be traversed, so it will be used to fly above.
- E > 60. These nodes were highlighted in black color. They can be avoided by the UAV while it is trying to arrive at its goal by turning around them.

#### 2.2. Any-Angle Pathfinding Algorithms

#### 2.2.1. Basic Theta* Algorithm

#### 2.2.2. Lazy Theta* Algorithm

#### 2.2.3. Phi* Algorithm

#### 2.3. Cost Function

## 3. Trajectory Design

#### 3.1. Bezier Curves

#### 3.2. Proposed Approach

## 4. Results and Discussion

#### 4.1. Implementation of the Modified Any-Angle Pathfinding Algorithms

#### 4.2. Performance Comparison of the Three Any-Angle Pathfinding Algorithms

#### 4.3. Implementation of Bézier Curves-Based Approaches on the Pathfinding Algorithms

- A single midpoint has been located at the center of each segment of the generated path excluding the first and the final segments. This approach was named the Bézier1 curve approach.
- Two midpoints have been located at [0.25, 0.75] positions of each segment of the generated path excluding the first and the final segments. This approach was named the Bézier2 curve approach.
- A single midpoint has been located at the center of each segment of the generated path including the first and the final segments. This approach was named the Bézier3 curve approach.
- Two midpoints have been located at [0.25, 0.75] positions of each segment of the generated path including the first and the final segments. This approach was named the Bézier4 curve approach.

#### 4.3.1. Basic Theta* Algorithm

#### 4.3.2. Lazy Theta* Algorithm

#### 4.3.3. Phi* Algorithm

#### 4.4. Evaluation Process of the Proposed Bézier Curves-Based Approaches with the Safety Feature

#### 4.5. Re-Evaluation Process of the Proposed Bézier Curves-Based Approaches without the Safety Feature

#### 4.6. A New Approach to Increasing the Efficiency of Bézier4 Curve-Based Approach to the Optimality

## 5. Conclusions

- Feasible paths were calculated by employing three modified algorithms of any-angle pathfinding algorithms over a conventional grid.
- The resulting paths were characterized by angular or sharp turns that are inappropriate for the flying vehicles; therefore, the properties of Bézier curves were exploited for smoothing the path by presenting four Bézier curves-based approaches.
- An innovative evaluation process was proposed to determine the most proper approach of the four Bézier curves-based approaches in constructing a flyable and collision-free trajectory.
- Based on the evaluation process results, a novel algorithm was proposed to push the chosen Bézier curve-based approach to be optimal by adding further midpoints. As a result, it has proven its effectiveness by generating an optimal trajectory with a success rate of up to 100%.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Grid Size | Basic Theta* | Lazy Theta* | Phi* | |
---|---|---|---|---|

Time [Sec] | 50 × 50 | 0.08818 | 0.03178 | 0.03403 |

Time [Sec] | 100 × 100 | 0.1939 | 0.0603 | 0.06112 |

Properties | Values |
---|---|

Computer | LEGION—Lenovo |

CPU | Intel(R) Core(TM) i7-8750h 2.20 GHz |

RAM | 8.00 GB |

Operating System | 64 bit Windows 10 Pro edition |

Programming Language | Python |

IDE | JetBrains PyCharm Community Edition 2019.1.1 x64 |

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**MDPI and ACS Style**

Satai, H.A.; Zahra, M.M.A.; Rasool, Z.I.; Abd-Ali, R.S.; Pruncu, C.I.
Bézier Curves-Based Optimal Trajectory Design for Multirotor UAVs with Any-Angle Pathfinding Algorithms. *Sensors* **2021**, *21*, 2460.
https://doi.org/10.3390/s21072460

**AMA Style**

Satai HA, Zahra MMA, Rasool ZI, Abd-Ali RS, Pruncu CI.
Bézier Curves-Based Optimal Trajectory Design for Multirotor UAVs with Any-Angle Pathfinding Algorithms. *Sensors*. 2021; 21(7):2460.
https://doi.org/10.3390/s21072460

**Chicago/Turabian Style**

Satai, Haitham AL, Musaddak M. Abdul Zahra, Zaid I. Rasool, Ridhab Sami Abd-Ali, and Catalin I. Pruncu.
2021. "Bézier Curves-Based Optimal Trajectory Design for Multirotor UAVs with Any-Angle Pathfinding Algorithms" *Sensors* 21, no. 7: 2460.
https://doi.org/10.3390/s21072460