# A Comparison of Closed-Loop Performance of Multirotor Configurations Using Non-Linear Dynamic Inversion Control

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## Abstract

**:**

## 1. Introduction

## 2. Review of the Literature

#### 2.1. Multirotor Platforms

#### 2.2. NDI Control of Multirotors

## 3. Multirotor Dynamics

#### 3.1. Shared Properties

#### 3.1.1. Rigid Body Model

#### 3.1.2. Rotor Model

#### 3.1.3. General Model Description and Control Allocation

#### 3.2. The Quadrotor

#### 3.3. The Hexarotor

#### 3.4. The Octorotor

## 4. Controller Design

#### 4.1. General Control Approach

#### 4.1.1. Non-Linear Dynamic Inversion

#### 4.1.2. State Feedback Control

#### 4.2. Control of the Quadrotor

#### 4.3. Control of the Hexarotor

#### 4.4. Control of the Octorotor

## 5. Operational Differences Between Configurations

Property | Symbol | Value | Unit |
---|---|---|---|

Quadrotor mass | ${m}_{Q}$ | $0.5$ | kg |

Hexarotor mass | ${m}_{H}$ | $0.7$ | kg |

Octorotor mass | ${m}_{O}$ | $1.0$ | kg |

Quadrotor inertia about ${x}^{\mathcal{B}}$ | ${I}_{x,Q}$ | $0.0054$ | kg m^{2} |

Quadrotor inertia about ${y}^{\mathcal{B}}$ | ${I}_{y,Q}$ | $0.0054$ | kg m^{2} |

Quadrotor inertia about ${z}^{\mathcal{B}}$ | ${I}_{z,Q}$ | $0.0107$ | kg m^{2} |

Hexarotor inertia about ${x}^{\mathcal{B}}$ | ${I}_{x,H}$ | $0.0085$ | kg m^{2} |

Hexarotor inertia about ${y}^{\mathcal{B}}$ | ${I}_{y,H}$ | $0.0085$ | kg m^{2} |

Hexarotor inertia about ${z}^{\mathcal{B}}$ | ${I}_{z,H}$ | $0.0149$ | kg m^{2} |

Octorotor inertia about ${x}^{\mathcal{B}}$ | ${I}_{x,O}$ | $0.0167$ | kg m^{2} |

Octorotor inertia about ${y}^{\mathcal{B}}$ | ${I}_{y,O}$ | $0.0734$ | kg m^{2} |

Octorotor inertia about ${z}^{\mathcal{B}}$ | ${I}_{z,O}$ | $0.0901$ | kg m^{2} |

Quadrotor/hexarotor rotor arm | l | $0.2$ | m |

Octorotor near rotor arm in x | ${l}_{x1}$ | $\sqrt{0.02}$ | m |

Octorotor far rotor arm in x | ${l}_{x2}$ | $\sqrt{0.02}+0.3$ | m |

Octorotor rotor arm in y | ${l}_{y}$ | $\sqrt{0.02}$ | m |

Hexarotor rotor inclination | α | 15 | ° |

Thrust gain | ${K}_{T}$ | 60 | N |

Torque gain | ${K}_{Q}$ | 1 | N m |

Lower input limit | ${u}_{\text{min}}$ | 0 | – |

Upper input limit | ${u}_{\text{max}}$ | $0.05$ | – |

Hexarotor max lateral acceleration | ${a}_{\text{max}}$ | $1.2$ | m s^{−2} |

#### 5.1. Controller Gains and Trajectory Definition

#### 5.2. Control Effort During Accurate Trajectory Tracking

**Figure 7.**Dynamic response in position and heading for each multirotor when accurately tracking a commanded path.

**Figure 8.**Motor input histories for each configuration while tracking a flight path with equivalent accuracy.

**Figure 9.**Dynamic response in roll and pitch for each multirotor when accurately tracking a commanded path.

Platform | Energy (J) |
---|---|

Quadrotor | 197.53 |

Hexarotor | 290.37 |

Octorotor | 396.22 |

#### 5.3. Tracking Performance During Specific Manoeuvres

#### 5.3.1. Fast Horizontal Translation

**Figure 11.**Dynamic response in longitudinal outputs for each multirotor when performing a fast horizontal translation in x.

#### 5.3.2. Fast Yaw Rotation

**Figure 12.**Rotor input histories for each configuration while performing a fast horizontal translation.

**Figure 13.**Dynamic response in yaw and height outputs for each multirotor when performing a fast yaw rotation.

#### 5.3.3. Fast 4D Manoeuvre

**Figure 15.**Dynamic response in position and heading for each multirotor when tracking a fast four-dimensional path.

**Figure 16.**Motor input histories for each configuration while tracking a fast four-dimensional path.

**Figure 17.**Dynamic response in roll and pitch for each multirotor when accurately tracking a commanded path.

## 6. Discussion and Conclusions

#### 6.1. Discussion

#### 6.2. Conclusions

#### 6.3. Further Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Ireland, M.L.; Vargas, A.; Anderson, D.
A Comparison of Closed-Loop Performance of Multirotor Configurations Using Non-Linear Dynamic Inversion Control. *Aerospace* **2015**, *2*, 325-352.
https://doi.org/10.3390/aerospace2020325

**AMA Style**

Ireland ML, Vargas A, Anderson D.
A Comparison of Closed-Loop Performance of Multirotor Configurations Using Non-Linear Dynamic Inversion Control. *Aerospace*. 2015; 2(2):325-352.
https://doi.org/10.3390/aerospace2020325

**Chicago/Turabian Style**

Ireland, Murray L., Aldo Vargas, and David Anderson.
2015. "A Comparison of Closed-Loop Performance of Multirotor Configurations Using Non-Linear Dynamic Inversion Control" *Aerospace* 2, no. 2: 325-352.
https://doi.org/10.3390/aerospace2020325