*Article* **Horizontal Trajectory Tracking Control for Underactuated Autonomous Underwater Vehicles Based on Contraction Theory**

**Caipeng Ma 1, Jinjun Jia 1, Tiedong Zhang 1,2, Shaoqun Wu <sup>3</sup> and Dapeng Jiang 1,2,\***


**Abstract:** In this paper, contraction theory is applied to design a control law to address the horizontal trajectory tracking problem of an underactuated autonomous underwater vehicle. Suppose that the vehicle faces challenges such as model uncertainties, external environmental disturbances, and actuator saturation. Firstly, a coordinate transformation is introduced to solve the problem of underactuation. Then, a disturbance observer is designed to estimate the total disturbances, which are composed of model uncertainties and external environmental disturbances. Next, a saturated controller is designed based on singular perturbation theory and contraction theory. Meanwhile, contraction theory is used to analyse the convergence properties of the observer and the full singular perturbation system, and make quantitative analysis of the estimation error and the tracking error. Finally, the results of numerical simulations prove that the method in this paper enables the vehicle to track the desired trajectory with relatively high accuracy, while the control inputs do not exceed the limitations of the actuators.

**Keywords:** underactuated autonomous underwater vehicle; trajectory tracking; actuator saturation; singular perturbation system; contraction theory

**Citation:** Ma, C.; Jia, J.; Zhang, T.; Wu, S.; Jiang, D. Horizontal Trajectory Tracking Control for Underactuated Autonomous Underwater Vehicles Based on Contraction Theory. *J. Mar. Sci. Eng.* **2023**, *11*, 805. https:// doi.org/10.3390/jmse11040805

Academic Editor: Sergei Chernyi

Received: 15 March 2023 Revised: 4 April 2023 Accepted: 5 April 2023 Published: 10 April 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).
