## 1 Introduction  
In recent years, the burgeoning demand for ocean exploration has driven the widespread adoption of AUVs, establishing them as pivotal instruments in marine resource exploration, seabed mapping, and oceanographic research. AUVs have substantially broadened the horizons of oceanic investigations while concurrently mitigating the risks associated with human-operated missions. In practical applications, AUVs are frequently required to accurately and swiftly adhere to designated target trajectories [1, 2]. However, within engineering frameworks, AUVs typically operate as underactuated systems, characterized by a limited number of control inputs relative to their degrees of freedom. Moreover, fluctuations in the marine environment introduce significant uncertainties in the parameters of the AUVs' nonlinear motion models during navigation. Numerous marine operations demand precise maneuvering and positioning of AUVs within three-dimensional space. Consequently, the development of trajectory tracking control techniques for underactuated AUVs in three dimensions has emerged as a central area of research interest among scholars both domestically and internationally in recent years. To address the performance and stability issues of AUVs in complex marine environments, various control algorithms have been proposed by researchers. In the case of known mathematical models, [PERSON] et al. [3] proposed a feedback linearization control method to handle external time-varying ocean current disturbances; [PERSON] et al. [4] combined backstepping and radial basis function neural networks (RBFNNs) to approximate external disturbances; [PERSON] et al. [5] employed backstepping control to solve the 3D trajectory tracking problem; and [PERSON] et al. [6] introduced a continuous-time disturbance observer to avoid the \"complexity explosion\" of backstepping. These methods show good performance in compensating for external disturbances, but most focus on planar trajectories and do not consider the effects of unmodeled system components. In cases where the AUV's mathematical model and ocean environmental disturbances are unknown, [PERSON] et al. [7] designed an adaptive neural network controller based on sliding mode control, achieving finite-time convergence; [PERSON] et al. [8] proposed a global finite-time control strategy; [PERSON] et al. [9] integrated RBFNNs with sliding mode control to ensure the stability of 3D trajectory tracking; and [PERSON] et al. [10] designed a fixed-time controller for a 2D multi-agent system. Although these methods improve the system's response speed, the convergence time of finite-time control depends on initial conditions, and the convergence time of fixed-time control is difficult to determine. As a result, predefined-time control methods have gained attention. [PERSON] et al. [11] studied the attitude control problem of rigid spacecraft under external disturbances, and [PERSON] et al. [12] investigated the predefined-time consensus tracking problem for multi-agent systems. However, research on predefined-time trajectory tracking for underactuated AUVs in 3D remains relatively scarce. In recent years, radial basis function neural networks (RBFNNs) [13; 14; 15] and predefined-time control [16; 17; 18] have gained significant attention. In the presence of external disturbances, RBFNNs exhibit superior approximation capabilities compared to traditional adaptive methods. Moreover, in contrast to fixed-time control methods, predefined-time control offers distinct advantages in determining convergence time. Based on these discussions, this paper proposes a predefined-time control strategy for 3D trajectory tracking of underactuated AUVs with unmodeled dynamics and external disturbances. Specifically, the aim of this research is to design an efficient and stable 3D trajectory tracking control method that ensures the AUV can achieve precise trajectory tracking within a predefined time, even in the presence of unmodeled dynamics and external disturbances.