## 1 Introduction
### Related Works  
Over the past decade, from 2002 to 2010, most studies on formation control of ASVs are within the framework of leader-follower strategy [12; 13]. Nonlinear formation control of surface vehicles was first considered on paper [12] that used backstopping to keep formation. In 2003 and 2004, some works presented for underactuated surface vessel (i.e., the surface vessels with three degrees of freedom) so that the proposed controllers were adjustable with realistic models of vessels [14; 15]. The formation control problem with and without communication constraints and also considering the time delay for marine vessels and submarines with full actuated [16; 17] and underactuated [16; 17; 18] has been studied from 2003 to 2007. Sliding mode techniques for controlling leader-follower underactuated surface vehicles have been considered in [19]. By using the theory of pioneers and parallel integrals [20], a directed leader-follower controller for full actuated surface vessels has been proposed. In practice, most of the existing control methods that have been presented for dynamical systems, are not implementable. A considered model may not be accurate and a system dynamic may be exposed to bounded disturbances or unmolded dynamics. In addition, external disturbances always affect the behavior of a dynamic system. Based on the distance techniques in formation control, adaptive controllers are proposed to keep formation among vessels [21; 22]. Using neural networks, an adaptive controller is designed for the first time for surface vehicles to deal with unknown dynamics in formation control in [23; 24]; however, the results obtained for full actuated surface vessels were not appropriate for underactuated surface vehicles. Formation control of ASVs with unknown dynamics of leader and local dynamic of followers has been considered in [25]. However, the controller design was very complicated due to the numerical calculating of partial derivatives of control signals. In addition of what was studied in  
Figure 1: Leader-follower formation with triangle pattern in two dimensions. previous papers, the control inputs in surface vessels (e.g., torques that apply to move a vessel) cannot adopt any value. The torque that is applied by a motor on blades and arms of a vessel is exposed to actuator saturation. In the other words, input constraints are not considered in previous works and as a result, most of controllers which have been designed in previous studies for trajectory tracking or to keep formation, are not implementable to real surface vessels. In this paper, the problem of leader-follower formation control for a group of ASVs has been considered. To this end and after modeling of leader-follower formation, first considering uncertainty, disturbances, and unmolded dynamics in the formation model, an adaptive robust controller is designed to keep formation. Then, this controller is modified to keep formation in the presence of input saturation constraints in addition of model uncertainties. It is related to the maneuver multiple autonomous surface vehicles that guided by a virtual leader moving in a parametric path in [26]. This paper addressed the maneuver of the various distributions of autonomous surface vehicles (ASVs) that currently have an unknown and state-restricted movement of motion. A design method for maneuvering the distribution of multiple ASVs based on neurodynamic optimization and fuzzy approximation is presented. The neuro-optimization method is able to obtain an optimal guidance signal for proper speed constraints and reducing control effort. A predictor is designed so that a fuzzy system is used to approximate unknown kinetic energy based on input and output data. The Stability of the closed-loop system has used the virtue of cascade theory. The continuous path problem using the error-constrained line-of-sight (ECLOS) guidance method is discussed in [27]. ECLOS guidance is applied to a Surface Vessel to control the path under the uncertainty, actuator saturation, and faults.