# Globally Attractive Hyperbolic Control for the Robust Flight of an Actively Tilting Quadrotor

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## Abstract

**:**

## 1. Introduction

#### 1.1. Literature Review about Control of Actively Tilting Quadrotors

#### 1.2. Problem Statement and Proposed Solution

#### 1.3. Contributions of the Work

- The proposed hyperbolic control is inspired by some related works that have been successfully applied to other types of robotic systems [3]. To the best of the authors’ knowledge, this is the first time that a hyperbolic controller has been applied to the field of UAVs.
- Different from some texts, this paper presents a theoretically sound controller. Moreover, the allocation technique is included, which places this paper within a reduced set of published manuscripts that include both aspects.
- To the best of the authors’ knowledge, this is the first time that a robust model-free controller is implemented in an actively tilting quadrotor whose behaviour is almost identical to an actual UAV, which is possible due to the utilization of a physics-engine-based simulation approach.

## 2. Mathematical Preliminaries

## 3. Dynamics of Actively Tilting Quadrotor

#### Properties and Assumptions of the Actively Tilting Quadrotor Dynamics

**Property 1.**

**Property 2.**

**Property 3.**

**Assumption A1.**

**ξ**is available.

**Assumption A2.**

## 4. Hyperbolic Control for Robust Flight

#### 4.1. Globally Attractive Ultimate Boundedness

**Theorem 1.**

**Proof.**

#### 4.2. Control Allocation

## 5. Physics-Engine-Based Simulation Results

#### 5.1. Implementation of the Controller

#### 5.2. Tracking

#### 5.3. Regulation towards Far Set Points

#### 5.4. Robustness against Disturbances

#### 5.5. Comparative Study

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Error norms while the actively tilting quadrotor tracks the given trajectory. Top: position error. Bottom: orientation error.

**Figure 4.**Performed trajectory in Cartesian space. Left: top view. Right: projected view.

**- - -**Desired trajectory.

**––**Performed trajectory.

**Figure 5.**Position and orientation error norms while achieving set points far away from the current position. Top: position error norm. Bottom: orientation error norm.

**Figure 6.**Performed trajectory in Cartesian space given a set of far set points. Left: top view. Right: projected view.

**- - -**Distances between the desired points.

**––**Performed trajectory.

**Figure 8.**Motion of the on board manipulator. Top: joint positions.

**––**${q}_{1}$.

**––**${q}_{2}$.

**––**${q}_{3}$. Bottom: end effector’s Cartesian motion norm.

**Figure 9.**Time evolution of the position and orientation error norms of the UAV while moving the onboard manipulator. Top: position error norm. Bottom: orientation error norm.

**Figure 10.**Position and orientation error norms while achieving the commanded set points. Top: position error norm. Bottom: orientation error norm.

**––**Hyperbolic controller.

**––**nonlinear PID controller.

**Figure 11.**Control signals. Top Left: angular speeds of the rotors using the hyperbolic controller. Top right: Tilting angle of the propellers using the hyperbolic controller.

**––**${\omega}_{1}$.

**––**${\omega}_{2}$.

**––**${\omega}_{3}$.

**––**${\omega}_{4}$. Bottom Left: angular speeds of the rotors using the nonlinear PID controller. Bottom right: Tilting angle of the propellers using the nonlinear PID controller.

**––**${\omega}_{1}$.

**––**${\omega}_{2}$.

**––**${\omega}_{3}$.

**––**${\omega}_{4}$.

Gains | Value |
---|---|

${\mathbf{K}}_{p}$ | diag$(5,5,200,10,10,10)$ |

${\mathbf{K}}_{i}$ | diag$(4,4,100,2,2,2)$ |

${\mathbf{K}}_{v}$ | diag$(0.1,0.1,50,0.3,0.3,0.3)$ |

Parameter | Value |
---|---|

l | 0.183847763 |

${k}_{f}$ | 8.54858$\times {10}^{-5}$ |

${k}_{\tau}$ | 1.75e$\times {10}^{-4}$ |

Parameter | Value |
---|---|

m | 5 kg |

${\mathbf{J}}_{b}$ | diag$(0.1522,0.1522,0.1841)$ kg$\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{2}$ |

g | 9.81 m/${\mathrm{s}}^{2}$ |

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**MDPI and ACS Style**

Orozco Soto, S.M.; Ruggiero, F.; Lippiello, V.
Globally Attractive Hyperbolic Control for the Robust Flight of an Actively Tilting Quadrotor. *Drones* **2022**, *6*, 373.
https://doi.org/10.3390/drones6120373

**AMA Style**

Orozco Soto SM, Ruggiero F, Lippiello V.
Globally Attractive Hyperbolic Control for the Robust Flight of an Actively Tilting Quadrotor. *Drones*. 2022; 6(12):373.
https://doi.org/10.3390/drones6120373

**Chicago/Turabian Style**

Orozco Soto, Santos Miguel, Fabio Ruggiero, and Vincenzo Lippiello.
2022. "Globally Attractive Hyperbolic Control for the Robust Flight of an Actively Tilting Quadrotor" *Drones* 6, no. 12: 373.
https://doi.org/10.3390/drones6120373