## 1 Introduction  
Autonomous underwater vehicles attract significant attention due to their great potential. These underwater vehicles could be applied to many underwater tasks, such as coastal defense. These coastal defense activities include cruising, reconnaissance, surveillance, etc., and they are key to safeguarding maritime rights and interests. However, when facing coastal defense tasks, large-scale underwater vehicles are limited in some applications due to their large size, high energy consumption, and poor maneuverability. In recent years, driven by the rapid development and integration of disciplines such as bio-science, mechanical manufacturing, materials science, control science, and micro-electronic-mechanical technology, small bio-mimetic underwater robots have gradually become important tools for defending coastal security due to their good concealment ability, strong mobility, and strong environmental adaptability. Compared with a single, small underwater robot, a coordinated group is more capable and flexible in performing complicated, large-scale, high-efficiency underwater cruising missions. The research on multiple-vehicle systems has been receiving increasing attention from engineers and scientists in recent decades [1; 2; 3]. One core problem concerned with the multi-small underwater robot system is formation tracking control, which requires the robots to keep a prescribed formation while simultaneously tracking a given reference trajectory [4]. Existing formation control strategies can be divided into the leader-follower method [5] and the virtual structure method [6]. The leader-follower formation strategy is a relatively common and practical method, where a single or multiple underwater robots in the formation are first determined as the leaders, and the other robots are determined as followers. The followers only need to track the leader and ensure the normal operation of the formation by keeping a certain distance and direction with respect to the leader. The key to the leader-follower method is whether the followers can acquire the information of the leader in real time and accurately track the leader behavior. It being relatively easy to design and implement are the advantages of this method. However, the followers rely on the leaders excessively, and they lack the ability to deliver feedback to the leaders, which leads to poor reliability. In addition, due to the lack of interaction between the followers, internal collision problems are prone to occur. The virtual structure-based formation strategy was proposed by [PERSON] [7], in which the entire formation is considered as a single rigid body. The virtual structure passes completely through the formation path, and then the corresponding trajectory of each structure point is obtained. Each robot tracks the trajectory provided by the virtual structure point to complete the formation. The virtual structure is able to reduce the difficulty and workload of task allocation, and, at the same time, it can guarantee high formation accuracy. For the above two strategies, it is important that an effective controller is designed for achieving a formation tracking effect. [PERSON] et al. combined consensus with backstepping to address the problem of formation tracking for multiple AUVs [8]. [PERSON] et al. studied the leader-follower formation control problem of AUVs under uncertain dynamics and ocean disturbances, and they proposed a combined tracking controller based on backstepping, adaptive neural networks, and dynamic sliding mode control techniques [9]. In reference [10], the researchers adopted the leader-follower method to obtain the reference trajectory of the followers and designed a tracking controller based on the adaptive neural network method for followers. Coordinated transportation tasks of distributed multiple AUVs were implemented based on the virtual structure strategy under weak underwater communication conditions and the nonlinear model predictive controller was designed for tracking [11; 12].