# Stochastic Trajectory Generation Using Particle Swarm Optimization for Quadrotor Unmanned Aerial Vehicles (UAVs)

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## Abstract

**:**

## 1. Introduction

- Flight level (FL) trajectory;
- Take-off → Mission → Landing (TML);
- Complex maneuvering in 3D space (CMS).

## 2. Dynamic Model

#### 2.1. The Moment of Inertia

#### 2.2. Thrust and Torques

#### 2.3. Dynamic Model

## 3. Modeling of Three Representative Scenarios

#### 3.1. Scenario One (FL)

#### 3.2. Scenario Two (TML)

#### 3.3. Scenario Three (CMS)

## 4. Trajectory Realization under Practical Constraints

#### 4.1. Design of Trajectory with Minimum Length

#### 4.2. Basics of PSO

## 5. Numerical Generator and Results

#### 5.1. Scenario 1 (FL)

#### 5.2. Scenario 2 (TML)

#### 5.3. Scenario 3 (CMS)

#### 5.4. PSO vs. A*, RRT* and GA

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 23.**Optimal path by rapidly-exploring random tree (RRT*), standard (A*) algorithm, genetic algorithm (GA) and the PSO.

Symbol | Definition |
---|---|

$\stackrel{\xb4}{o}({x}_{e},{y}_{e},{z}_{e})$ | Earth axes |

NED | North east down |

$o({x}_{b},{y}_{b},{z}_{b})$ | Body axes |

$\varphi $ | Roll angle |

$\theta $ | Pitch angle |

$\psi $ | Yaw angle |

Euler angles | $[\varphi ,\theta ,\psi ]$ |

${[\dot{\varphi},\dot{\theta},\dot{\psi}]}^{T}$ | Rate of change of Euler angles |

${[p,q,r]}^{T}$ | Body angular rates |

${M}_{T}=M+{M}_{r}$ | Overall mass of the UAV |

${I}_{xx},{I}_{yy},{I}_{zz}$ | Moment of inertia |

R | Radius of central sphere |

g | Gravitational acceleration |

${F}_{i}$ | Force generated by ${i}_{th}$ rotor |

${T}_{i}$ | Torque generated by ${i}_{th}$ rotor |

${\Omega}_{i}$ | Angular velocity of ${i}_{th}$ rotor |

${U}_{1},{U}_{2},{U}_{3},{U}_{4}$ | Control inputs |

Symbol | Unit |
---|---|

${M}_{T}$ | 0.65 Kg |

l | 0.232 m |

g | 9.806 m/s${}^{2}$ |

R | $0.15$ m |

${I}_{xx}$ | $0.07582$ Kg m${}^{2}$ |

${I}_{yy}$ | $0.07582$ Kg m${}^{2}$ |

${I}_{zz}$ | $0.1457924$ Kg m${}^{2}$ |

b | $3.13\times {10}^{-5}$ Ns${}^{2}$ |

d | $7.5\times {10}^{-7}$ Nms${}^{2}$ |

${\Lambda}^{max}=-{\Lambda}^{min}$ | $0.04$ Nm |

${\varphi}^{max}=-{\varphi}^{min}$ | 0.1 Rad |

${\theta}^{max}=-{\theta}^{min}$ | 0.1 Rad |

${\psi}^{max}=-{\psi}^{min}$ | 0.1 Rad |

Inertia Weight ($\omega $) | 1 |

${c}_{1}$ | 2 |

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

Scenarios | Control Points | Swarm Size | Iterations | Execution Time |
---|---|---|---|---|

Scenario 1 | 4 | 50 | 250 | 5.090549 (s) |

Scenario 2 | 5 | 100 | 400 | 6.572763 (s) |

Scenario 3 | 5 | 100 | 450 | 7.464875 (s) |

Parameters | RRT* | A* | GA | PSO + B-Spline |
---|---|---|---|---|

Length (m) | 18.1265 | 16.5563 | 17.2111 | 14.2095 |

Execution time (s) | 17.343599 | 0.02315 | 9.230695 | 1.544244 |

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**MDPI and ACS Style**

Salamat, B.; Tonello, A.M.
Stochastic Trajectory Generation Using Particle Swarm Optimization for Quadrotor Unmanned Aerial Vehicles (UAVs). *Aerospace* **2017**, *4*, 27.
https://doi.org/10.3390/aerospace4020027

**AMA Style**

Salamat B, Tonello AM.
Stochastic Trajectory Generation Using Particle Swarm Optimization for Quadrotor Unmanned Aerial Vehicles (UAVs). *Aerospace*. 2017; 4(2):27.
https://doi.org/10.3390/aerospace4020027

**Chicago/Turabian Style**

Salamat, Babak, and Andrea M. Tonello.
2017. "Stochastic Trajectory Generation Using Particle Swarm Optimization for Quadrotor Unmanned Aerial Vehicles (UAVs)" *Aerospace* 4, no. 2: 27.
https://doi.org/10.3390/aerospace4020027