# Online Trajectory Planning Method for Double-Pendulum Quadrotor Transportation Systems

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- The proposed method can be used for online trajectory planning without the need for offline design of the quadrotor speed and acceleration.
- The method can restrain the hook swing and payload swing without any adverse effects on the positioning performance.
- The method can ensure that the core indexes, e.g., the maximum acceleration and velocity of the quadrotor, are constrained.

## 2. Dynamical Model

**Assumption**

**1.**

## 3. Online Trajectory Generation

## 4. Convergence Analysis

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1**.

**Theorem**

**2.**

**Proof**

**of**

**Theorem**

**2**.

## 5. Simulation Analysis

#### 5.1. Comparison Test

- Final position of the quadrotor $(x,z)\left(m\right)$.
- Transportation time $t\left(s\right)$ (the time when the quadrotor reaches the target position).
- The maximum swing angles of the hook and payload during transportation ${\alpha}_{max}{(}^{\circ})$, ${\beta}_{max}{(}^{\circ})$.
- Consumption of driving energy in the entire transportation process $F\left(N\right)$.

#### 5.2. Robustness Test

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

PID | Proportional–integral–derivative |

PD | Proportional–derivative |

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Control Methods | $(\mathit{x},\mathit{z})\left(\mathit{m}\right)$ | $\mathit{t}\left(\mathit{s}\right)$ | ${\mathit{\alpha}}_{max}{(}^{\circ})$ | ${\mathit{\beta}}_{max}{(}^{\circ})$ | $\mathit{F}\left(\mathit{N}\right)$ |
---|---|---|---|---|---|

PD control | (5.0, 5.0) | 12.0 | 9.0 | 16.1 | 128.6 |

Nonlinear control | (5.0, 5.0) | 13.5 | 2.2 | 3.9 | 116.8 |

Online trajectory planning | (5.0, 5.0) | 17.4 | 0.7 | 0.7 | 114.0 |

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**MDPI and ACS Style**

Qi, J.; Ping, Y.; Wang, M.; Wu, C.
Online Trajectory Planning Method for Double-Pendulum Quadrotor Transportation Systems. *Electronics* **2022**, *11*, 50.
https://doi.org/10.3390/electronics11010050

**AMA Style**

Qi J, Ping Y, Wang M, Wu C.
Online Trajectory Planning Method for Double-Pendulum Quadrotor Transportation Systems. *Electronics*. 2022; 11(1):50.
https://doi.org/10.3390/electronics11010050

**Chicago/Turabian Style**

Qi, Juntong, Yuan Ping, Mingming Wang, and Chong Wu.
2022. "Online Trajectory Planning Method for Double-Pendulum Quadrotor Transportation Systems" *Electronics* 11, no. 1: 50.
https://doi.org/10.3390/electronics11010050