## 1 Introduction  
In recent years, USV formation control has attracted large amounts of attention in the control science and engineering areas for different purposes such as mine clearance, patrol, investigation, transportation of strategic materials, coordinated operations, target rounding and marine science applications [1; 2; 3; 4]. The objectives of formation control are to achieve the desired distance and orientation among multiple agents. Meanwhile, to avoid collision and keep effective communication during the formation control, the relative distance among USV should be guaranteed to be within a reasonable interval. Therefore, it is necessary to deal with model uncertainties and constraints to design the control system. A motivating example is controlling a group of moving cars in the form of closely spaced vehicular platoons while ensuring safety [5]. Vehicular platoons could deal well with traffic flow capacity and reduce the consumption of fuel in the automated highway systems. Therefore, many researchers extend the platoon formation control problem to robots or autonomous vehicles [5; 6; 7]. Great progress has been made in related research [8; 9; 10; 11; 12]. A control scheme based on the backstepping method technique and the Lyapunov direct method for formation control of ships using the leader-follower strategy was proposed in [13]. Based on [13], a control strategy was developed in [14] based on the bioinspired method for the underactuated surface vehicles that solved the problem of computation exploration and realized the constraints on virtual velocity. However,the ship models they used were all simplified and the model uncertainties were not considered. The unmodeled dynamics and parameter uncertainties induced by hydrodynamic damping terms may lead to problems that make it difficult or impossible to build an accurate ship model and this affects the transient and steady-state performance which leads to collisions and gives rise to invalid connectivity in formation control. In [15], the NN-based DSC technique was employed to solve uncertain local dynamics of leader-follower formation control. The adaptive NNs were used in [16] to deal with model uncertainties. The extended dissipativity criterion for a discrete-time NN with time-varying delay was established in [17]. An NN formation control for multiple USV cooperated with adaptive filtering methods were employed to extract the low-frequency content of the model uncertainties and ocean disturbances in [18]. The input constraint, model uncertainties, and external disturbances were considered for spacecraft formation control in [19]. Though the uncertain parameters and unmodeled dynamics or input constraints were considered in the above research, they have not taken the output constraints into consideration. The output and error constraints of the formation are very critical factors affecting formation stability, for the reason that if the distance between the two vehicles is too large and exceeds the effective range of communication in the process of formation pattern establishing, the connectivity between each vehicle will be invalid which will affect the formation pattern. On the other hand, if the distance between them is too short, it will cause intervehicle collisions and damage the system. Therefore, it is necessary to consider the output constraint of formation control. To deal with the problem of constraint, some methods have also been proposed to achieve the desired performance [6; 20; 21; 22]. For example, the exponentially decaying functions of time and formation error transformation were used in [6]. An output feedback control was proposed in [23] for a certain class of systems that guaranteed the convergence of generic bounded solutions to the desired set-point. Under these methods, the formation errors are always within the predefined regions. To deal with the issue of connectivity preserving and collision avoidance problems between each other, a linear transformed error surface was introduced in [24].