This paper addresses the problem of coordinating a group of underactuated ships along given paths (path following) while holding a desired intership formation pattern. The solution to this problem unfolds into two basic subproblems. In the first step, a path-following controller is derived to force each underactuated ship to follow a reference path subject to constant disturbances induced by wave, wind, and ocean current. The controller is designed such that the ship moves on the path while its total velocity is maintained tangential to the path. In the second step, the speeds of the vehicles are adjusted so as to synchronize the positions of the corresponding virtual targets (or so-called coordination states), in the sense that the derivative of each path is left as a free input to synchronize the ships' motion. The proposed coordination controller uses a combination of Lyapunov direct method, backstepping, and concepts from graph theory. When dealing with the path-following coordination problem, it is considered that each ship transmits its coordination state to other ships with a varying time delay as determined by the communication topology. The coordination errors' convergence is achieved based on a proposed Lyapunov-Krasovskii function. Simulation results are provided to illustrate the effectiveness of the proposed approach.
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