## 1 Introduction  
Due to the rapid development of communication, navigation, and computer technology related to ship motion control, cooperative ship control has an extensive range of application prospects in military and production fields, including fleet cooperative combat, ocean-going replenishment, environmental monitoring, oil and gas detection, etc. Because of higher operational security, lower cost, and greater autonomy and flexibility, unmanned surface vehicles (USVs) are applied to perform extensive missions in hazardous maritime environments instead of manned vehicles [1]. Compared with a single USV, multiple USVs' cooperation has the advantages of strong adaptability and fault tolerance. The fleet can form a dynamic network during the navigation. Through division and cooperation, each USV can perceive the environmental information about the area quickly and accurately to accelerate the completion of missions and improve the efficiency of the system. Formation control is the most fundamental problem of multiple USV cooperative control. Therefore, formation control of USVs has become one of the hot issues in the research of USV motion control. A collective scheme is necessary to ensure that the USVs work together to complete a common task and coordinate in time and space. As of today, many scholars have studied the formation control problem of a multi-USV system. The formation control problems are summarized into two fundamental problems: (1) formation generation, which refers to how to form a designated formation [2; 3], and (2) formation maintenance, which refers to how to keep formation unchanged in the process of movement [4]. USVs' formation originated from the study of biological cluster dynamics, which can be traced back to the Boid model proposed by [PERSON] [5]. Based on this model, [PERSON] [6] extended the multi-agent consistency work to the usual swarm formation control field, introduced obstacle avoidance and tracking agents, and designed a distributed control framework including gradient-based term, velocity consensus term, and navigational feedback. [PERSON] et al. [7] further developed the collective formation control strategy based on the work of [PERSON], using a virtual leader to replace the actual leader. [PERSON] et al. [8] proposed a consistency control method based on a predictive mechanism to accelerate the convergence speed of multi-agent consistency. [PERSON] et al. [9] proposed the collective circular motion behavior control of heterogeneous multi-agents under arbitrarily closed curves. In the research of multi-agent formation control, most researchers treat the model as a linear system of first-order or multi-orders [6; 7; 8]. Taking into consideration the dynamic characteristics (nonlinearity, coupling, underactuation, etc.) of the robots in a multi-robot formation, it is often difficult to directly consider it as a system for analysis. To better achieve stability and efficiency, and at the same time be useful to the theoretical analysis of the multi-robot formation, researchers have proposed the following formation methods: the virtual structure method, the behavior-based method, and the leader-follower method [10]. The virtual structure method [11] is not flexible, and it is difficult to achieve obstacle avoidance with that method. It is hard to express the entire system in mathematical form and difficult to prove and guarantee the stability of the system using the behavior-based method [12]. In contrast, the leader-follower method [13] is easy to design and implement, and easy to ensure stability. The advantages of the leader-follower method are that the moving goal is only assigned to the leader to navigate the movement of the agent swarm, and each member only needs to collect information about its immediate leader instead of the whole swarm. For example, when performing seafloor terrain scanning, hydrological sampling, target search, and resource detection, a team of robots performs tasks in formation, and the path trajectory is assigned to the leader of the formation.