## 1 Introduction  
With the rapid development of scientific ocean exploration, underwater robots have become one of the most important tools for exploiting and utilizing marine resources [1; 2]. When compared with autonomous underwater vehicles (AUVs), the core feature of bionic small-scale robots refers to the capability of operating missions such as tracking ocean creatures and monitoring the marine environment in narrow underwater spaces. Therefore, these small-scale biomimetic robots have been receiving increasing interest from academia. Most researchers have focused on a prototype design [3; 4; 5] and the control methods of the driving system [6; 7; 8; 9]. However, the closed-loop motion control (such as trajectory tracking, etc.) has seldom been considered, which is of primary importance for most applications. Over the past few years, a large number of studies have been dedicated to the underwater trajectory tracking control for AUVs. Sliding mode control (SMC) is preferred for achieving underwater trajectory tracking, because it is insensitive to model uncertainties [10; 11; 12]. In [13], the authors combined SMC with the proportional-integral-derivative (PID) algorithm to enhance the trajectory tracking performance. In [14], an adaptive high order sliding mode controller was designed in order to achieve a AUV tracking trajectory on the yaw and depth. However, the \"chattering\" effect is the main drawback associated with SMC. In [15], a new adaptive term was developed to achieve a chattering-free sliding mode controller. The neural network algorithm is another commonly utilized algorithm in underwater trajectory tracking. In [16], the authors proposed an efficient neural network approach with a single-layer structure for tracking control. In [17], a novel adaptive neutral network tracking controller was constructed by combining dynamic surface control (DSC) and a minimal learning parameter(MLP), and a radial basis function neural network was employed to account for tracking error. The simulation results illustrate that the algorithm guarantees convergence of the tracking errors into a small neighborhood of the desired trajectory. Neutral-network based tracking control is insensitive to the dynamic model of the robot and has strong adaptability and learning capabilities [18]. However, because of the changeable underwater environment, the training and learning process is difficult, which leads to a poor real-time performance of the tracking algorithm. Backstepping control has been widely applied to the trajectory tracking of underwater robots. In [19], a bio-inspired model-based filter was integrated with the backstepping control for three-dimensional trajectory tracking of a manned submarine vehicle. The simulation results illustrate that this algorithm achieved satisfactory tracking results. In [20], a controller combined with the Lyapunov theory and a backstepping algorithm was designed for AUV motion control. A common drawback of backstepping control is that the speed sharply increases when there is a large tracking error, which leads to poor tracking results. Most of the above studies on trajectory tracking control have achieved a good performance, but they do not consider actual system constraints such as the saturation of the control input. To solve the practical constraints in the controller design, model predictive control (MPC), which is the ability to cope with hard system constraints through optimization procedures, is an ideal algorithm. MPC-based trajectory tracking of AUVs has been conducted in recent years [21]. Path planning and trajectory tracking problems were integrated, which was resolved using nonlinear MPC in [22]. To improve the efficiency in solving the optimization problem, in [23] the authors incorporated log barrier functions into the cost function and modified the C/GMRES algorithm for use with a traditional MPC algorithm. In [24], to solve the trajectory tracking control problem, a novel Lyapunov-based nonlinear model predictive control (LMPC) scheme was developed for AUVs.