## 1 Introduction
(2016 [22]) presented a nonlinear MPC-based adaptive dynamic positioning control for fully driven underwater vehicles, enhancing system robustness and velocity tracking accuracy through neural network integration, with the efficacy demonstrated using the horizontal plane model of a Remotely Operated Vehicle (ROV). However, the intensive computational demand of MPC can potentially affect the real-time performance of control systems. To address this, [PERSON] et al. (2022) proposed a self-adaptive predictive control methodology utilizing least squares and Lagrange functions to decrease computational strain while improving system robustness and interference mitigation. [PERSON] et al. (2019 [13]) streamlined the MPC optimization process into a convex quadratic problem, facilitating online resolution and proving its trajectory tracking capability under external disturbances. However, the computational intensity of MPC can affect real-time control system performance, a challenge addressed by [PERSON] et al. (2022 [23]) through a self-adaptive predictive control method using Lagrange functions and least squares algorithms, improving system robustness and reducing computational demands. Further innovations include that of [PERSON] et al. (2017 [24]), who combined probabilistic policy search with MPC for enhanced decision-making in control systems, demonstrating success in agile drone navigation. [PERSON] et al. (2011 [25]) designed a ship course-keeping algorithm based on a knowledge base, which utilizes a set of input and output signals to obtain ship dynamics equations, thus avoiding the problems that arise when designing classical control algorithms for complex nonlinear ship models. [PERSON] et al. (2018 [26]) improved MPC efficiency by adapting the weighting matrix based on error magnitude, enhancing control precision as verified in pool experiments. [PERSON] et al. (2021 [27]) proposed an MPC-based AUV docking control system, employing a linear parameter variation (LPV) model and a time-varying Kalman filter to manage power flow disturbances. Additionally, [PERSON] et al. (2013 [28]) introduced a fault-tolerant control method for Unmanned Aerial Vehicles (UAVs) combining the Gaussian process and MPC, with successful simulation outcomes. [PERSON] et al. (2017 [29]) developed an adaptive control approach for autonomous miniature race cars, leveraging the Gaussian process and chance-constrained formulations to account for model uncertainties. Collectively, these studies contribute significant insights and methodological advancements to the field of MPC, particularly in enhancing the control systems of AUVs and related autonomous platforms. Appropriate modeling plays a crucial role in achieving high-precision control ([PERSON] et al., 2021 [30]). Gaussian process regression effectively addresses model uncertainty in nonlinear data fitting by estimating probability distributions, mitigating the influence of system noise. Leveraging this ability, this study employs a Gaussian-Process-based Model Predictive Control (GP-MPC) approach to control AUVs, aiming to guide the AUVs in achieving precise trajectory tracking within a three-dimensional spatial domain. The proposed method can be applied to Level 2 autonomous vehicles, which are partially automated ([PERSON] et al., 2020 [31]). In addition to trajectory tracking tasks, we evaluate the AUV's planar model for its proficiency in evading obstacles in static and dynamic environments. This method integrates a non-parametric Gaussian process model with MPC, constructing a robust controller with predictive capabilities. It is designed to address various challenges posed by the complex marine environment, such as model uncertainties, external disturbances, and constraints, thereby providing a theoretical foundation and technical support for achieving the high-precision motion control of AUVs.