## 5 Conclusions  
This study presents a comprehensive exploration of the control performance of the Gaussian-process-based model predictive control (GP-MPC) technique concerning trajectory tracking and obstacle avoidance in AUVs. The research commences with the establishment of the kinematic and dynamic equations governing AUV motion, providing a fundamental framework for subsequent developments. The state prediction equation, incorporating the GP-MPC methodology, is then formulated, accounting for both self-constraints and obstacle avoidance considerations while integrating an optimization function. Extensive simulations are conducted to comprehensively evaluate the performance of the proposed approach in terms of trajectory tracking and static as well as dynamic obstacle avoidance. The results validate the efficacy of the GP-MPC method in facilitating AUVs to navigate adeptly within their own constraints and effectively circumvent external obstacles. This facilitates precise and efficient obstacle avoidance control, which is crucial for ensuring AUV safety and mission success in complex underwater environments. Moreover, the study extends to the exploration of three-dimensional obstacle avoidance, providing valuable insights into the algorithm's resilience and performance within intricate underwater environments. The research outcomes not only contribute significantly to advancing AUV capabilities and safety, but also hold substantial potential for the evolution of autonomous underwater systems. Future research endeavors may concentrate on further refining the algorithm and delving into its practical applications, with the aim of enhancing the overall dependability and efficacy of AUV obstacle avoidance strategies in real-world scenarios.  
In future research, the in-depth integration of Gaussian Process Model Predictive Control (GP-MPC) with other advanced artificial intelligence techniques will become an important exploration direction, aiming to significantly enhance the intelligence level and overall performance of control systems. Such integration will not only help strengthen the adaptability and robustness of the system but also improve the accuracy and efficiency of decision-making. In addition, it is crucial to investigate how to better handle multi-objective optimization problems and constraints using the GP-MPC method. In practical applications, control systems often need to make decisions while satisfying multiple objectives and constraints, posing higher demands on optimization algorithms. Therefore, we need to delve deeper into multi-objective optimization theory, explore effective constraint handling methods, and integrate them with the GP-MPC algorithm to achieve a more intelligent and efficient control system design.  
T.L.: conceptualization, writing--original draft, methodology. J.Z.: data curation, visualization, writing--review and editing, modification. J.H.: preparation, software, modification. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Natural Science Foundation of China (grant numbers: 51279040), and Southern Marine Science and Engineering Guangdong Laboratory (Zhuhai) (SML2023 SP232).  
Not applicable.  
Data Availability Statement: Data is contained within the article.  
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.