This paper is concerned with the multi-UUVs formation control problem and proposes the control protocol with an additional function when considering environmental disturbances. Firstly, the leaderfollowing configuration is adopted to discuss the consensus problem for multi-UUVs system, the leader's behavior directs the follower's motion trajectory and the follower UUVs could, in turn, communicate with each other, respectively. Secondly, three types of coordination control protocols are proposed: the control protocol is designed without additional function and time delay; the control protocol with additional function and without time delay; the control protocol with additional function and time-varying delay. Sufficient consensus conditions are analyzed and derived by using the Lyapunov-Krasovskii functional theory, algebraic graph theory and matrix theory. Finally, two simulation experiments are given to illustrate the effectiveness of the proposed formation control methods. INDEX TERMS Multiple unmanned underwater vehicles, leader-following consensus, uncertain factors, time-varying delays. I. INTRODUCTION In the past decades, with the development of artificial intelligence and new engineering techniques, the cooperation and coordination problems of multi-Unmanned Underwater Vehicles (multi-UUVs) system have been rapidly developed and widely studied due to its broad applications in military and commercial fields, especially in ocean-graphic observation, ocean exploration, plane crash searches, trajectory tracking, etc. Compared with a single UUV, multi-UUVs formation equipped with more sensors can accomplish more complex and larger-scale deep-sea missions with efficient and stable communications. It is worth mentioning that the researches on motion control of a single UUV are getting deeper in recent years, there are several control methods such as robust control [1], adaptive control [2] and fuzzy control approaches [3] in ocean engineering applications, but the above algorithms cannot well applied to the formation problem of multi-UUVs system. The basic problem of formation The associate editor coordinating the review of this article and approving it for publication was Nagarajan Raghavan.

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In this paper, the formation coordination control of discrete-time distributed leaderless multiple autonomous underwater vehicle (AUV) system with double independent position–velocity communication topology and control inputs on a nonconvex set is studied. Firstly, the problem of formation coordination control of multi-AUV system is transformed into the problem of formation consensus of multi-AUV system, and the consistent state of leaderless multi-AUV system formation was defined. Secondly, considering the existence of bounded communication delay and nonconvex control input constraints for multi-AUV system formation under weak communication conditions, a formation consistent constraint controller algorithm for discrete-time leaderless multi-AUV system with double independent communication topology is proposed by introducing constraint operators. By using the properties of graph theory, random matrix and SIA matrix, and selecting appropriate controller parameters, the multi-AUV system formation can reach the defined consensus state. Furthermore, the unbounded communication delay of multi-AUV system formation is studied. Finally, the simulation results show that the proposed controller constraint algorithm is effective.

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