## 5 Conclusions and Discussion  
This paper discusses how to realize a complex, varied time-varying formation of multi-AUVs and the formation-control strategy. In the current ocean exploration and technology  
Figure 9: State trajectories of the five AUVs in the simulation and \(c(t)\). (**a**) The position trajectories of the five AUVs in the simulation; (**b**) the velocity trajectories of the five AUVs in the simulation.  
application, an AUV is an efficient and autonomous underwater exploration tool, and its formation control and cooperative work ability are critical in completing complex tasks. Through theoretical analysis and algorithm design, we provided a feasible solution for multi-AUVs to establish the desired time-varying formation.  
Consensus-based time-varying formation problems for multi-AUVs systems are investigated and simulation of the formation theories to multi-AUVs are presented. On the formation control level, based on the dynamic characteristics of the AUV, a consensus-based formation protocol for multi-AUV systems to achieve time-varying formations is proposed. Then, formation problems are transformed into consensus problems. Necessary and sufficient conditions for multi-AUV systems to achieve time-varying formation are presented, and an explicit expression of the time-varying formation center function is given. The formations can be any time-varying vectors satisfying condition (12). Different from most cases of collaboration, the velocities of the multi-AUVs studied in this paper are not equal when they reach the cooperative state, and the research content is more general. In addition, the interaction topology of each AUV only needs to have a spanning tree, without the information mentioned previously. Furthermore, a procedure to design the protocol for multi-AUV systems to achieve time-varying formations is proposed. The method was then verified by three typical simulations. These results not only enrich the theoretical system of formation control, but also provide an important theoretical basis for practical application.  
However, owing to the complexity and variability of the marine environment, the limitation of sensor accuracy, and the unreliability of underwater communication, it is often difficult to establish the expected effect with the traditional formation-control algorithm.  
Therefore, how to determine the practicability of AUV formation has become the focus of research.  
Future research needs to pay more attention to practical application scenarios and to develop robust formation-control algorithms that can adapt to complex marine environments. In this way, we can ensure that AUVs can perform tasks stably and reliably under various harsh conditions.  
In addition, the research content of this study was aimed at isomorphic AUVs, but the formation algorithm of heterogeneous AUVs is more valuable to study. For practicality, the heterogeneous AUV formation structure can make the formation system have stronger flexibility and adaptability to complete more complex tasks.  
Conceptualization, L.W.; methodology, G.S.; software, G.S.; validation, G.S. and H.X.; writing--original draft preparation, G.S.; writing--review and editing, H.X.; visualization, G.S.; supervision, H.X.; project administration, H.X.; funding acquisition, H.X. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Key R&D Program of China [grant no. 2021 YFC 2800100] and the Natural Science Foundation of Liaoning Province, China, [grant no. 2022-MS-035].  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The research was performed at Harbin Engineering University and Shenyang Institute of Automation (SIA), as well as the Chinese Academy of Sciences.  
The authors declare no conflicts of interest.