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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.

To solve the navigation accuracy problems of multi-Unmanned Underwater Vehicles (multi-UUVs) in the polar region, a polar cooperative navigation algorithm for multi-UUVs considering communication delays is proposed in this paper. UUVs are important pieces of equipment in ocean engineering for marine development. For UUVs to complete missions, precise navigation is necessary. It is difficult for UUVs to establish true headings because of the rapid convergence of Earth meridians and the severe polar environment. Based on the polar grid navigation algorithm, UUV navigation in the polar region can be accomplished with the Strapdown Inertial Navigation System (SINS) in the grid frame. To save costs, a leader-follower type of system is introduced in this paper. The leader UUV helps the follower UUVs to achieve high navigation accuracy. Follower UUVs correct their own states based on the information sent by the leader UUV and the relative position measured by ultra-short baseline (USBL) acoustic positioning. The underwater acoustic communication delay is quantized by the model. In this paper, considering underwater acoustic communication delay, the conventional adaptive Kalman filter (AKF) is modified to adapt to polar cooperative navigation. The results demonstrate that the polar cooperative navigation algorithm for multi-UUVs that considers communication delays can effectively navigate the sailing of multi-UUVs in the polar region.

The conventional initial alignment algorithms are invalid in the polar region. This is caused by the rapid convergence of the Earth meridians in the high-latitude areas. However, the initial alignment algorithms are important for the accurate navigation of Unmanned Underwater Vehicles. The polar transversal initial alignment algorithm is proposed to overcome this problem. In the polar transversal initial alignment algorithm, the transversal geographic frame is chosen as the navigation frame. The polar region in the conventional frames is equivalent to the equatorial region in the transversal frames. Therefore, the polar transversal initial can be effectively applied in the polar region. According to the complex environment in the polar region, a large misalignment angle is considered in this paper. Based on the large misalignment angle condition, the non-linear dynamics models are established. In addition, the simplified unscented Kalman filter (UKF) is chosen to realize the data fusion. Two comparison simulations and an experiment are performed to verify the performance of the proposed algorithm. The simulation and experiment results indicate the validity of the proposed algorithm, especially when large misalignment angles occur.

Multilevel thresholding is an effective and indispensable technology for image segmentation that has attracted extensive attention in recent years. However, the multilevel thresholding method has some disadvantages, such as a large computational complexity and low segmentation accuracy. Therefore, this paper proposes a whale optimization algorithm (WOA) based on Kapur's entropy method to solve the image segmentation problem. The WOA can effectively balance exploration and exploitation to avoid falling into premature convergence and obtain the global optimal solution. To verify the segmentation performance of the WOA, a series of experiments on underwater images from the experimental pool of Harbin Engineering University are conducted, and the segmentation results are compared with those of the BA, the FPA, MFO, the MSA, PSO and WWO by maximizing the fitness value of Kapur's entropy method. The fitness value, peak signal-to-noise ratio (PSNR), structure similarity index (SSIM), execution time and Wilcoxon's rank-sum test are used to evaluate the overall performance of each algorithm. The experimental results reveal that the WOA is superior to the other comparison algorithms and has a higher segmentation accuracy, better segmentation effect and stronger robustness. In addition, the feasibility and efficiency of the WOA are verified.