# A Trajectory Tracking Approach for Aerial Manipulators Using Nonsingular Global Fast Terminal Sliding Mode and an RBF Neural Network

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Kinematic Model

#### 2.2. Dynamics Model

#### 2.3. Model Simplification

## 3. Materials and Methods

#### 3.1. Position Controller Design

#### 3.2. Attitude Controller Design

#### 3.3. RBF Neural Network Design and Stability Judgement

## 4. Evaluation Criteria for UAM

- Robustness

- Convergence

- Accuracy

## 5. Simulations Results

#### 5.1. Aerial Hovering

#### 5.2. Square Trajectory Tracking

#### 5.3. Spiral Trajectory Tracking

## 6. Experimental Results

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

UAV | Unmanned aerial vehicle |

UAM | Unmanned aerial manipulation |

RBF | Radial basis function |

CNGFTSM | Composite nonsingular global fast terminal sliding mode |

NGFTSM | Nonsingular global fast terminal sliding mode |

SM | Sliding mode |

SMPID | Sliding Mode proportional-integral-derivative |

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**Figure 3.**The position error of the system. (

**a**) Error in the $x$ direction. (

**b**) Error in the $y$ direction. (

**c**) Error in the $z$ direction.

**Figure 4.**Euler angle tracking error of the system. (

**a**) Roll angle error. (

**b**) Pitch angle error. (

**c**) Yaw angle error.

**Figure 5.**Square trajectory tracking error of the system. (

**a**) Error in the $x$ direction. (

**b**) Error in the $y$ direction. (

**c**) Error in the $z$ direction. (

**d**) Actual and desired trajectory of the system with the designed controller.

**Figure 6.**Euler angle tracking error of the square trajectory. (

**a**) Roll angle error. (

**b**) Pitch angle error. (

**c**) Yaw angle error.

**Figure 7.**The tracking error between the system with external disturbance and the system without external disturbance. (

**a**) Error in the $x$ direction. (

**b**) Error in the $y$ direction. (

**c**) Error in the $z$ direction.

**Figure 8.**Spiral trajectory tracking error of the system. (

**a**) Error in the $x$ direction. (

**b**) Error in the $y$ direction. (

**c**) Error in the $z$ direction. (

**d**) Actual and desired trajectory of the system.

**Figure 9.**Euler angle tracking error of the spiral trajectory. (

**a**) Roll angle error. (

**b**) Pitch angle error. (

**c**) Yaw angle error.

**Figure 10.**Tracking error between the system with external disturbance and the system without external disturbance. (

**a**) Error in the $x$ direction. (

**b**) Error in the $y$ direction. (

**c**) Error in the $z$ direction.

**Figure 14.**Trajectory tracking experimental results. (

**a**) Results in the $x$ direction. (

**b**) Results in the $y$ direction. (

**c**) Results in the $z$ direction. (

**d**) Desired and actual trajectory of the system.

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

$m$ | 1.8 kg | ${m}_{1}$ | 0.5 kg | ${m}_{2}$ | 0.5 kg |

${I}_{xx}/(\mathrm{kg}\cdot {\mathrm{m}}^{2})$ | 1.24 | ${I}_{x1}/(\mathrm{kg}\cdot {\mathrm{m}}^{2})$ | 10^{−3} | ${I}_{x2}/(\mathrm{kg}\cdot {\mathrm{m}}^{2})$ | 10^{−3} |

${I}_{yy}/(\mathrm{kg}\cdot {\mathrm{m}}^{2})$ | 1.24 | ${I}_{y1}/(\mathrm{kg}\cdot {\mathrm{m}}^{2})$ | 10^{−3} | ${I}_{y2}/(\mathrm{kg}\cdot {\mathrm{m}}^{2})$ | 10^{−3} |

${I}_{zz}/(\mathrm{kg}\cdot {\mathrm{m}}^{2})$ | 2.48 | ${I}_{z1}/(\mathrm{kg}\cdot {\mathrm{m}}^{2})$ | 0 | ${I}_{z2}/(\mathrm{kg}\cdot {\mathrm{m}}^{2})$ | 0 |

${l}_{1}$ | 0.15 m | ${l}_{2}$ | 0.15 m |

NGFTSM Function | RBF Neural Network | ||
---|---|---|---|

Parameter | Value | Parameter | Value |

$\Gamma $ | $diag(50,50,50)$ | ${c}_{j}$ | $0.1\ast [\begin{array}{ccccc}-1& -0.5& 0& 0.5& 1\\ -1& -0.5& 0& 0.5& 1\end{array}]$ |

$\Lambda $ | $diag(\frac{1}{400},\frac{1}{400},\frac{1}{400})$ | ${b}_{j}$ | 5 |

$\lambda $ | $diag(20,\cdots ,20)$ | ${\delta}_{p}$ | 0.01 |

$\eta $ | $diag(0.01,\cdots ,0.01)$ | ${\delta}_{\Phi}$ | 15 |

Convergence/s | CNGFTSM | NGFTSM | SM | SMPID |
---|---|---|---|---|

Simulation 1 | 0.452 | 0.354 | 0.952 | 0.911 |

Simulation 2 | 0.455 | 0.365 | 1.031 | 0.942 |

Simulation 3 | 0.461 | 0.363 | 0.763 | 0.794 |

Accuracy/m | CNGFTSM | NGFTSM | SM | SMPID |
---|---|---|---|---|

Simulation 1 | 10^{−6} | 0.002 | 0.023 | 0.014 |

Simulation 2 | 10^{−6} | 0.002 | 0.024 | 0.015 |

Simulation 3 | 0.001 | 0.0032 | 0.112 | 0.106 |

Robustness/m | CNGFTSM | NGFTSM | SM | SMPID |
---|---|---|---|---|

Simulation 2 | 10^{−6} | 0.002 | 0.023 | 0.014 |

Simulation 3 | 10^{−6} | 0.002 | 0.023 | 0.025 |

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**MDPI and ACS Style**

Shen, L.; Mao, P.; Fang, Q.; Wang, J.
A Trajectory Tracking Approach for Aerial Manipulators Using Nonsingular Global Fast Terminal Sliding Mode and an RBF Neural Network. *Machines* **2022**, *10*, 1021.
https://doi.org/10.3390/machines10111021

**AMA Style**

Shen L, Mao P, Fang Q, Wang J.
A Trajectory Tracking Approach for Aerial Manipulators Using Nonsingular Global Fast Terminal Sliding Mode and an RBF Neural Network. *Machines*. 2022; 10(11):1021.
https://doi.org/10.3390/machines10111021

**Chicago/Turabian Style**

Shen, Lirui, Pengjun Mao, Qian Fang, and Jun Wang.
2022. "A Trajectory Tracking Approach for Aerial Manipulators Using Nonsingular Global Fast Terminal Sliding Mode and an RBF Neural Network" *Machines* 10, no. 11: 1021.
https://doi.org/10.3390/machines10111021