# Experimental Validation of a Model-Free High-Order Sliding Mode Controller with Finite-Time Convergence for Trajectory Tracking of Autonomous Underwater Vehicles

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## Abstract

**:**

## 1. Introduction

#### Related Work

## 2. Materials and Methods

#### 2.1. Autonomous Underwater Vehicles Kinematics and Hydrodynamics

- $M\u03f5{\mathbb{R}}^{6\times 6}$ is the inertial and added mass matrix,
- $C\u03f5{\mathbb{R}}^{6\times 6}$ is the rigid body and added mass centripetal and Coriolis matrix,
- $D\u03f5{\mathbb{R}}^{6\times 6}$ is the hydrodynamic damping matrix,
- $g\u03f5{\mathbb{R}}^{6\times 1}$ is the restitution forces vector,
- $\tau \u03f5{\mathbb{R}}^{6\times 1}$ is the control signal vector,
- ${B}_{t}\u03f5{\mathbb{R}}^{6\times 6}$ is the thruster allocation matrix,
- ${u}_{t}\u03f5{\mathbb{R}}^{6\times 1}$ is a vector containing the force generated by the thrusters, and
- $\omega \u03f5{\mathbb{R}}^{6\times 6}$ represents environmental disturbances.

#### 2.2. Finite-Time Controller with Convergence in a Predefined Time

**Remark**

**1.**

#### 2.3. BlueROV2 Robot

^{®}is a Remotely Operated Vehicle (ROV) with a 6-Thruster vectored configuration as shown in Figure 3. It has an open-source software and hardware configuration, which make it suitable for research purposes.

#### 2.4. Exact Differentiatior

## 3. Experimental Set-Up

#### 3.1. Hardware

^{®}3 (RPi) acts as the processor, and it is in charge of running the control algorithms and managing the different sensors. The Rpi runs Lubuntu as operative system, and it is connected to a control station through a tether. A BAR-30 high-resolution pressure sensor from BlueRobotics

^{®}is connected to the robot through an I${}^{2}$C port in the RPi to measure the depth z of the vehicle. A Smart sensor BNO-055 from Bosh

^{®}is connected to a serial port of the Rpi and is used to estimate the orientations $\varphi ,\theta ,\psi $. The thruster’s speeds are controlled by Pulse Wide Modulation (PWM) signals from the RPi, which goes through a set of 30 A Electronic Speed Controllers (ESC). Finally, a 14.8 V, 18A Ah battery supplies energy for the whole electronics. A diagram of the hardware configuration in the BlueROV2 used for the experiments is shown in Figure 5.

#### 3.2. Software

#### 3.3. Time-Parameterized Trajectories

#### 3.4. Control Algorithms

#### 3.4.1. PID Control

#### 3.4.2. Model-Free High-Order SMC (Asymptotic)

#### 3.4.3. Model-Free High-Order SMC (Finite-Time)

#### 3.5. Exact Differentiator Algorithm

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AUV | Autonomous Underwater Vehicles |

DoF | Degrees Of Freedom |

ESC | Electronic Speed Controllers |

PID | Proportional Integrate Derivative |

PWM | Pulse Wide Modulation |

RMS | Root Mean Square |

RMSE | Root Mean Square Error |

ROV | Remotely Operated Vehicle |

ROS | Robot Operating System |

RPi | Raspberry Pi |

SMC | Sliding Mode Control |

SNAME | Society of Naval Architects and Marine Engineers |

TBG | Time-Base Generator |

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**Figure 4.**Experimental set-up at Tecnologico de Monterrey, Campus Querétaro. (

**a**) Ground control station at a side of the semi-Olympic pool. (

**b**) BlueROV2 deployed into the water. (

**c**) Autonomous trajectory tracking mission.

**Figure 9.**Controlled trajectories with the asymptotic model-free 2nd-order SMC. (

**Left**) Depth. (

**Right**) Heading.

**Figure 11.**Controlled trajectory with the model-free high-order SMC with finite-time convergence in a 5 s predefined-time.

**Figure 15.**Controlled trajectories with the model-free high-order SMC with finite-time convergence in predefined-time. (

**Left**) Depth. (

**Right**) Heading.

**Figure 17.**RMSE comparison for the trajectory tracking with the different controllers. (

**Left**) Depth. (

**Right**) Heading.

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**MDPI and ACS Style**

González-García, J.; Gómez-Espinosa, A.; García-Valdovinos, L.G.; Salgado-Jiménez, T.; Cuan-Urquizo, E.; Escobedo Cabello, J.A.
Experimental Validation of a Model-Free High-Order Sliding Mode Controller with Finite-Time Convergence for Trajectory Tracking of Autonomous Underwater Vehicles. *Sensors* **2022**, *22*, 488.
https://doi.org/10.3390/s22020488

**AMA Style**

González-García J, Gómez-Espinosa A, García-Valdovinos LG, Salgado-Jiménez T, Cuan-Urquizo E, Escobedo Cabello JA.
Experimental Validation of a Model-Free High-Order Sliding Mode Controller with Finite-Time Convergence for Trajectory Tracking of Autonomous Underwater Vehicles. *Sensors*. 2022; 22(2):488.
https://doi.org/10.3390/s22020488

**Chicago/Turabian Style**

González-García, Josué, Alfonso Gómez-Espinosa, Luis Govinda García-Valdovinos, Tomás Salgado-Jiménez, Enrique Cuan-Urquizo, and Jesús Arturo Escobedo Cabello.
2022. "Experimental Validation of a Model-Free High-Order Sliding Mode Controller with Finite-Time Convergence for Trajectory Tracking of Autonomous Underwater Vehicles" *Sensors* 22, no. 2: 488.
https://doi.org/10.3390/s22020488