Finite‐time coordination controls for multiple autonomous underwater vehicle systems

Coordination control of multiple autonomous underwater vehicles (AUVs) has attracted considerable attention owing to its widespread applications in deep‐sea exploration, marine environment monitoring, regional searches, and detector recovery. In addition, finite‐time control theory is applied to ensure that the system achieves its control goal as soon as possible. To the authors' knowledge, few studies have systematically summarized the finite‐time coordination control of multiple AUV systems. Thus this review aims to systematically and completely introduce the research progress into the coordination control methods for multi‐AUV systems and their combination with finite‐time theory. The primary areas of focus are the research significance, present state, and prospects of coordination control and finite‐time control theory in multiple AUV systems, as well as their applications.

The inertial coordinate system is shown in Figure 1, which mainly describes the motion task and navigation position of the AUV.One of its coordinate planes o e x e y e is a tangent plane with a tangent point of o e near the AUV system's moving region.Axis o e x e can be seen as the x-axis pointing north, axis o e y e as the y-axis pointing east, and axis o e z e as the z-axis pointing perpendicular to the tangent plane to the center of the sphere.
The body coordinate system is a coordinate system based on the AUV itself, which moves with the AUV, and takes a fixed point on the AUV body as the coordinate origin o b .The body coordinate system is shown in Figure 2. Its axis o b x b can be regarded as the x-axis pointing to the bow of the ship in the usual sense, the axis o b y b can be regarded as the y-axis perpendicular to the x-axis and point to the starboard side of the ship, and the axis o b z b can be regarded as the z-axis perpendicular to the plane o b x b y b and point down.

AUV space motion model
Studies on AUV system usually consider that it have six degrees of freedom, as shown in Table 2, namely Surge, Sway, Heave, Roll, Pitch and Yaw.The AUV system's states (position and attitude) defined in the inertial coordinate system, its velocities (linear and angular velocities) defined in the body coordinate system, and its controls (forces and moments) defined in the body coordinate system correspond to these six degrees of freedom respectively.In view of the fact that AUV system moves in a complex underwater environment, in order to facilitate modeling and theoretical analysis, the mathematical model is usually established in the body coordinate system.Kinematic model of AUV system.Define F  ] T , then the translation motion of the AUV system in the inertial coordinate system and the body coordinate system can be expressed as: and the rotation motion in the inertial coordinate system and the body coordinate system can be expressed as: where and s = sin(⋅), c = cos(⋅), t = tan(⋅).It is worth noting that the transformation matrix J 1 (p 2 ) is orthogonal matrix and satisfies J −1 1 (p 2 ) = J T 1 (p 2 ).
T , the kinematic model of the AUV system can be obtained as follows: where ] .
Dynamic model of AUV system.AUV system is affected by both rigid body dynamics and fluid dynamics during underwater motion.Therefore, the dynamic model of AUV system will be given based on rigid body dynamics and fluid dynamics.All external forces acting on the AUV system are equivalent to a combined external force (moment), and based on Newton's law of motion and Euler's law of motion, the 6-DOF rigid body dynamics model of the AUV system is obtained as follows: where M RB is the inertial mass matrix and satisfies M RB = M T RB > 0, C RB is the Coriolis centrifugal force matrix and satisfies C RB = −C T RB ,  RB = [ T 1 ,  T 2 ] T is the sum vector of all external forces.The concrete form of M RB and C RB is 0 −I yz q − I xz p + I z r I yz r + I xy p − I y q I yz q + I xz p − I z r 0 −I xz r − I xy q + I x p − I y r − I xy p + I y q I xz r + I xy q − I x p 0 where m is the mass of AUV, I 3×3 is the identity matrix, S 0 is the skew symmetric matrix, I 0 is the inertial tensor matrix, and (x G , y G , z G ) is the barycentric coordinate of AUV.AUV system will be subjected to various external forces during underwater motion, and based on the principle of linear superposition, the external forces and vectors acting on AUV rigid body dynamics model are obtained.These external forces mainly include: (1) The driving force  generated by the designed controller and acted on the AUV by the thruster; (2) The unknown interference force  E caused by ocean currents and waves in the complex working environment; (3)  The fluid force  H generated by the fluid's additional mass, damping, restoring force, and so forth.The driving force  can be obtained by artificially designing the controller, and the fluid force  H can be obtained by experimental modeling.Therefore, the external forces on the AUV system can be described as: where M A is the additional inertial mass matrix, C A (v) is the Coriolis centrifugal force matrix, D(v) is the damping matrix, and g(p) is the restoring force vector.The inertial additional mass M A is formed by the hydrodynamic force generated by the AUV's accelerated motion in the water.Like the inertial mass, the additional mass of the AUV also produces the Coriolis centrifugal force matrix C A (v).The damping matrix D(v) is generally generated by the combined action of potential flow damping, friction damping, wave damping and vortex shedding damping when AUV system moves underwater.The restoring force vector g(p) is generated by the gravity and buoyancy of the AUV system in underwater motion.In the research, it is usually assumed that the AUV system has a low velocity and a symmetric plane, so the inertial additional mass matrix M A , the Coriolis centrifugal force matrix C A (v), the damping matrix D(v) and the restoring force vector g(p) of the AUV system can be simplified into the following forms where D 1 (v) is the linear damping term, D 2 (v) is the nonlinear damping term, W is the gravity of AUV, and B is the buoyancy of AUV.Based on the above analysis, this review substituted the above formulas ( 5) and ( 6) into the rigid body dynamics Equation ( 4) under the body coordinate system, and combined with the kinematic equation, the mathematical model of the AUV system can be obtained as follows: where More details are also available in reference. 1oordination control research, controller design, and simulations of multi-AUV systems are typically based on the mathematical model of AUV system.Additionally, the researchers have also studied the problem of coordination control in multi-AUV systems when the AUVs move horizontally or are underactuated.When considering an AUV system moving in the horizontal plane, its position and velocity vectors are represented as p = (x, y, ) T and v = (u, v, r) T , and the model's parameter matrix in (11) is simplified accordingly.In these cases, specific forms of kinematic and dynamic models of AUV systems can be found in the literature. 2 When considering an AUV system is underactuated, the number of variables in its controller is less than the number of degrees of freedom of the system, that is, some control variables are zero.

APPLICATIONS OF MULTI-AUV SYSTEMS
This section provides an overview of the state of application research on multi-AUV systems, covering domestic and international research history, developments, and trends.

Research status abroad
Research on multi-AUV systems originated in the United States in the 1980s.In 1987, Albus and Blidberg proposed a layered robot control architecture and a real-time decision-making, perception, and communication system based on world modeling. 3In October of that year, in collaboration with the Defense Advanced Research Projects Agency (DARPA), they conducted formation, convergence, piloting, following, collaborative search, and area-scanning confirmatory experiments with other partner units based on dual AUV systems.In June 1996, to understand the dynamics of seawater in a channel, Schmidt and Bellingham used an underwater acoustic profile imaging network to obtain approximate oceanographic data.These were then used to assist multiple AUV systems in conducting collaborative detection to obtain more accurate data. 4The above were small-scale and short-term preliminary validation experiments based on a multi-AUV system, which had limited significance in practical underwater applications.Over the past two decades, the United States has conducted underwater experiments on numerous large-scale practical ocean application scenarios, including scanning the ocean environment and regional searches for multi-AUV systems, to demonstrate their use in ocean exploration.
The experimental results have entered the practical application stage as a feasible technology.
In terms of environmental perception, Princeton and Harvard Universities launched an adaptive sampling and prediction (ASAP) project in 2006 to improve the efficiency of ocean environmental data collection. 5This experimental project primarily focused on the collaborative control technology of multi-AUV systems.A sampling model based on system feedback and human-machine collaborative loop decisions was used to verify a novel ocean data-sampling tool based on integrated collaborative motion control and adaptive sampling.Since 2010, the Consortium for Ocean Leadership (COL) and the National Science Foundation (NSF) have constructed and implemented the massive Ocean Observatories Initiative (OOI) to provide novel perspectives of the world's oceans. 6The OOI project will place sensors offshore, in the open ocean, and beneath the surface to monitor complex changes in the oceanic environment, such as acidification, circulation, and climate variability.It integrates hundreds of continuous monitoring data streams into a complex computer network based on sensor networks and makes them publicly available.By 2016, the program had established an extensive sensor network encompassing all significant US coastal areas and the high seas, enabling the estimation of complex ocean processes and the transmission, processing, and integration of ocean detection data.To enhance the capability of multi-AUV systems for acquiring ocean monitoring data, MIT's Lincoln Laboratory and Ocean Engineering Center studied the time and energy optimal path planning problem of the system using Multidisciplinary Simulation, Estimation, and Assimilation systems (MSEASs) from 2014 to 2016, and a sea trial was conducted in 2016 to verify the practical capability of the system. 7To observe and sample natural phenomena in mid-ocean, the Woods Hole Oceanographic Institution (WHOI) uses multi-AUV systems in collaboration to decode the "twilight zone" of the ocean and track a range of mid-water organisms, such as jellyfish and juvenile fish. 8The system can accommodate numerous auxiliary payloads, such as samplers, sonars, and cameras, and can perform preprogrammed missions to collect images as well as oceanographic flora and fauna data.
Regarding regional search, based on the AOSN project, the MIT Joint NATO Underwater Research Center launched the Generic Oceanographic Array Technology System (GOATS) research project in 1998, which uses multiple AUVs to form an underwater mobile detection network and uses the underwater acoustic equipment on the AUVs to scan the coastal environment and search for and detect minefields. 9In 2008, the Center for Intelligent Systems Research (CISR) at the University of Idaho launched a research project called "Cooperative Autonomous Underwater Vehicles Used to Search Large Ocean Areas for Mines". 10This project aims to investigate a communication protocol for multi-AUV systems based on specific underwater acoustic communication modems and to conduct theoretical research and practical field experiments on the formation control algorithm of the system during the coordinated detection of minefields.In the same year, MIT launched the Generic Littoral Interoperable Network Technology (GLINT) based on the GOATS project.It uses numerous AUVs with sensors to collaborate to search, locate, and track underwater targets and conducted numerous subsea tests between 2008 and 2011 to verify the capability of the system to complete practical missions. 11dditionally, the GREX research project for the "coordination and control of cooperating heterogeneous unmanned systems in uncertain environments" was launched by several European Union research institutions from 2006 to 2009.It promotes the development of multi-AUV-system cooperation theory and method through practical engineering application to narrow the gap between concepts and practice. 12The project mainly focuses on designing and developing the command-and-control system architectures of heterogeneous multi-AUV systems.This team designs individual communication structures based on underwater acoustic communication, radio, and wireless networks and researches cooperative navigation and formation control technology in unknown environments.In 2009, the project successfully completed a mapping task using multi-AUV systems coordinated in the ocean environment in a test, which demonstrated that the cooperative navigation and control of multiple AUVs can be realized by underwater acoustic communication.To map, diagnose, clear, and protect sub-marine coastal archaeological sites, the European Union undertook the "ARchaeological RObot systems for the World's Seas (ARROWS)" research project from 2012 to 2015. 13The project investigated underwater communication and task-allocation decision-making when multiple AUVs collaborated, such that the system could more efficiently coordinate positioning, navigation, formation control, seabed mapping, target classification, and virtual scene construction.To investigate effective methods and tools to complete seabed exploration and mapping tasks in complex ocean environments, the European Union conducted a research project named "Marine robotic system of self-organizing, logically linked physical nodes" from 2012 to 2016, aiming at integrating multiple AUVs' structural and communication systems.Cooperative positioning, navigation, and formation control were studied. 14Based on the Horizon 2020 framework plan of the EU, since 2014, relevant scientific research institutions in the UK, France, Germany, and other major European countries have conducted the "Widely scalable Mobile Underwater Sonar Technology (WiMUST)" research project, which aimed to build a multi-AUV expandable underwater acoustic network system that can conduct large-scale seabed mapping, seismic wave detection, hydrological sampling, search and rescue, archaeology, enemy reconnaissance, minefield search, and other tasks. 15

Research status in China
Research on AUV systems in China began relatively late and was primarily concentrated at a few universities and institutes.These include the Institute of Ocean Engineering at Tsinghua University, the Shenyang Institute of Automation at the Chinese Academy of Sciences, Shanghai Jiao Tong University, Northwestern Polytechnical University, and Harbin Engineering University.In the mid-1990s, the Shenyang Institute of Automation jointly developed two types of deep-sea AUVs, "CR-01" and "CR-02", which were successfully applied to investigating Pacific polymetallic nuclei.Subsequently, to investigate deep-sea resources and conduct oceanic research, the Institute developed two series of deep-sea AUV technologies, "Qianlong" and "Tansuo", with the support of the State Oceanic Administration during the "12th Five-Year Plan" period.During the same period, Tianjin University developed a 2000-meter-level submarine survey AUV, and Harbin Engineering University developed a 1000-meter-level Wise Water series. 16The successful development of AUV systems demonstrates the country's advanced international standing; however, a single-system verification experiment in engineering applications is still required.For multi-AUV systems, Harbin Engineering University conducted collaborative underwater experiments at Weihai between 2014 and 2015. 17This experiment was based on radio and underwater acoustic communication systems and included a series of ocean missions, such as path and target tracking and behavior-based cooperative formation.In December 2019, the Shenyang Institute of Automation and the Tianjin Shenzhilan Marine Equipment Co., Ltd.jointly conducted clustered collaborative observation in the East Indian Ocean in a 300 by 300 nautical mile area supported by the National Key Research and Development Program and the Strategic Pilot Special Program of the Chinese Academy of Sciences. 18The expedition deployed twelve "Sea Wing" underwater gliders that operated for 550 days, completed over 3400 profile observations, and collected various hydrological data, such as oxygen content, turbidity, salinity, and temperature.In August 2017, the Qingdao National Laboratory of Marine Science and Technology and Tianjin University launched the "Mesoscale Eddy" ocean network observation project.By 2020, the project had conducted collaborative observation experiments on a heterogeneous multi-AUV system formation and undertook environmental monitoring of multiscale dynamic ocean processes, typhoons, and other extreme environments, thereby verifying the system's observation capability in extreme environments. 19

COORDINATION CONTROL OF MULTI-AUV SYSTEMS
This section analyzes and expounds on various coordination control problems of multi-AUV systems and their combination with finite-time stability theory based on current research, both domestically and abroad.First, the state of research into the coordination control of multi-AUV systems is introduced, including the research status of consensus, trajectory tracking, formation, and circumnavigation.Second, the development and state of research into finite-time theory are introduced.Finally, the application status of finite-time theory to the coordination control of multi-AUV systems is introduced.
As is known, completing the control task for multiple AUV systems requires exchanging information between individual AUVs.A general two-way communication pattern is shown in Figure 3, and the corresponding figure is an undirected graph.When the system communicates in only one direction, the graph corresponding to multiple AUVs becomes a directed graph.Additionally, AUV systems that perform tasks in complex ocean environments typically have complex nonlinear dynamic characteristics, such as the strong coupling of model variables, parameter uncertainty, and various unknown external disturbances, which makes research on AUV systems challenging.In addition, in a complex task environment, the coordination control of multi-AUV systems has a series of advantages, including greater efficiency, higher robustness, and a wider task range compared to a single AUV system.Therefore, with the development of science and technology and an increasing focus on ocean resources, distributed coordination control of multi-AUV systems has recently become a research hotspot in both the theoretical and application fields.Distributed coordination control in multi-AUV systems includes consensus, trajectory tracking, formation, and circumnavigation.This section elaborates on the current research status of all types of problems.

Consensus control of multi-AUV systems
The classic and fundamental problem in the coordination control of multi-agent systems is the consensus problem, which has significance in both theory and practical applications.This requires individuals in the system to achieve the same The reference frames of an AUV.
state (such as position, speed, acceleration, etc.) of all individuals through local information exchange and cooperation under a distributed control algorithm.In the multi-AUV systems' consensus problem, we must typically develop a distributed controller for each AUV to realize a state of consensus for all AUVs.Based on the AUV system model ( 11), Li et al. overlooked the Coriolis matrix and external disturbances and considered that all AUVs have the following special kinematic and dynamic models: where i denotes the ith AUV in the system. 20A distributed finite-time control law was designed to achieve a state of consensus for a system without a leader.It can be seen that the controller relies on complete state information.In order to facilitate the controller to adjust the control input online in time, Chen et al. further designed a distributed adaptive control protocol based on an AUV system model to achieve positional consensus for multi-AUV systems in a finite time. 21t can be found that the control protocols designed in these studies depend on the real-time communication between individual AUVs in the system and the state information of neighboring AUVs.In fact, an AUV system typically operates in areas with complex electromagnetic effects, which makes it difficult for the system to maintain real-time communication and obtain all relevant status information from its neighbors.In addition, due to the influence of the physical environment such as the information transmission distance and the medium, the delay in the communication process is inevitable.
When considering that the system has communication delay, Zhang et al. proposed a consensus control strategy for discrete multi-AUV systems. 22It can be seen that the controller is designed based on fixed communication topology.In order to cope with the situation that the communication topology may change, Zhang et al. designed a consensus control protocol based on switching communication topology. 23In view of the communication faults of the system, Chen et al. investigated the state consensus control problem under the condition of incomplete neighbor information. 24The designed distributed controller enables multiple AUV systems can reach a consensus when the neighbor information is incomplete.Although the controller does not require complete information on neighboring AUVs, it relies on continuous communication between individual AUVs, without which the convergence of the system cannot be realized.In practical applications, individual AUVs in a system may fail to achieve continuous communication because of energy, equipment, and environmental limitations.Moreover, in practical applications, there may be situations where the status information is not measurable due to equipment faults.Thus, Chen et al. studied the rendezvous control problem of multi-AUV systems when the velocity information of each AUV is unavailable and system communication is intermittent. 25Without loss of generality, the intermittent communication mode is shown in Figure 4, where the time lengths  k > 0 and  k = t k+1 − t k −  k > 0 in the time interval [t k , t k+1 ), k ∈ N indicate whether the system is communicating.To solve this problem, a distributed finite-time rendezvous control protocol was designed.The rendezvous control protocol designed in this study does not depend on continuous communication between each AUV in the system.However, through the intermittent communication between AUVs, a distributed hybrid control strategy is developed using the limited state information of neighbor AUVs.In addition, a finite-time distributed observer independent of the velocity information was

F I G U R E 5
The trajectories of all AUVs. 25signed to estimate the AUV's state information and address situations in which the velocity information of individual AUVs in the system is not measurable.More importantly, the controller designed in this study ensures that multi-AUV systems can achieve a state consensus in a finite time, even when system communication is intermittent and velocity information is unavailable.Figure 5 depicts the motion trajectories of all AUVs under the designed controller.It can be seen that all AUVs achieve consensus of their positions.Additionally, when the system velocity information is not measurable, Yan et al. also studied the consensus control problem under the conditions of environment disturbance and switching topology. 26

Trajectory tracking control of multi-AUV systems
Trajectory tracking is a key problem in the practical applications of multi-AUV systems and in the coordination control of the system.This requires all follower AUVs to track the desired or leader trajectory from any initial state.In recent years, scholars have proposed numerous coordination control strategies for multi-AUV systems, including sliding mode control, neural networks, adaptive control, and backstepping control.Sliding-mode control is commonly used for nonlinear systems and exhibits excellent control performance.The convergence of a system based on sliding mode control can be divided into the approaching and sliding mode stages.The approaching stage refers to the process by which the system moves from its initial state to the sliding mode surface, whereas the sliding mode stage refers to the process by which the system moves along the sliding mode surface to a stable state after reaching the sliding mode surface.The dynamic surface of the sliding mode control can be designed artificially, and the state of the system is almost unaffected by model parameter uncertainties and external disturbances when the system enters the sliding mode stage.Therefore, sliding mode control is widely used in the tracking control of multi-AUV system trajectories.Elmokadem et al. studied the control problem of under-actuated autonomous underwater vehicles, and proposed a trajectory tracking control strategy based on sliding mode control technology. 27Considering the influence of AUV system state and control input quantization, Yan et al. designed trajectory tracking control law based on sliding mode control technology by introducing quantization error. 28It can be seen that the traditional linear sliding mode surface can achieve exponential convergence of the system state, but can not achieve finite-time convergence.In order to achieve the finite-time convergence of the system, terminal sliding mode control is the most widely used method at present.For example, Elmokadem et al. proposed a robust trajectory-tracking control strategy based on terminal sliding mode control technology for an underactuated AUV system. 2 The AUV system tracks the preset trajectory using the designed underactuated tracking controller.For the AUV system affected by ocean currents and model uncertainties, Liu et al. proposed a quaterny-based terminal sliding mode trajectory tracking control method. 29It should be noted that this technique has two drawbacks: (1) When the system state is far from the equilibrium point, the system convergence speed will slow down; (2) There exists singular problem within this technology.For this reason, scholars propose fast terminal sliding mode control technology to solve the first problem, and non-singular terminal sliding mode control technology to solve the second problem.For the autonomous recovery of AUV system, Yang et al. use the Dubins path planning method to transform the recovery problem into the trajectory tracking problem. 30Then, a trajectory tracking control method based on non-singular terminal sliding mode control technology is designed.Considering the underactuated AUV system with external disturbance, Luo et al. proposed a non-singular fast terminal sliding mode control scheme based on disturbance observer to solve the trajectory tracking control problem of the system. 31In addition, Liu et al. proposed a trajectory tracking control scheme based on improved non-singular terminal sliding mode control technology to solve the finite time tracking problem of AUV systems with external interference and modeling uncertainty. 32daptive control is a type of control method with online parameter identification.This method can automatically adjust the control input according to the operating conditions of the system to adapt to changes in the operating characteristics of the agent.Considering that the parameters in the AUV model are uncertain and have unmodeled parts and external disturbances, Qi designed a collaborative tracking-control strategy for multi-AUV systems using an adaptive control method. 33A robust adaptive distributed controller that relies only on the information of its two neighboring AUVs is proposed for each AUV so that all AUVs can finally synchronize to the expected path.Through the literature investigation, it can be found that the adaptive method is usually combined with the sliding mode control method in the coordination control research of multi-AUV systems.For the trajectory tracking control problem of AUV system, Qiao et al proposed an adaptive fast non-singular integral terminal sliding mode control method.Compared with the existing non-singular integral terminal sliding mode control strategy, this method can effectively improve the convergence rate of the system. 34In fact, in the application of AUV system, the environment is constantly changing, which leads to the design of controllers based on the determinate model parameters may not be the most practical.Thus, Chen et al. proposed a distributed finite-time tracking control law combining adaptive method and sliding mode control technology to address the finite-time trajectory tracking control problem of multi-AUV systems with heterogeneous uncertain dynamics. 35Moreover, for the trajectory tracking control problem of AUV system with unknown thruster faults and uncertain dynamics, a fault-tolerant control method based on adaptive non-singular integral terminal sliding mode is proposed.Compared with the existing sliding mode fault-tolerant control, the proposed control strategy has a higher convergence rate and can effectively overcome the singularity problem of the system. 36The control law depends on the velocity information of the AUV system, which may be unobtainable during practical missions because of equipment and energy limitations.Therefore, Chen et al. 37 designed a distributed finite-time tracking control law independent of velocity information to address the finite-time trajectory tracking control problem of multi-AUV systems with heterogeneous uncertain dynamics and without velocity information.Aimed at the unmeasurable velocity information of individual AUVs in the system, this study proposes a distributed finite-time observer to estimate the AUV's state information and designs an observer-based trajectory tracking control strategy to ensure that all follower AUVs can track the leader's trajectory for a finite time without velocity information.Figure 6 depicts the position coordinate evolution of all AUVs under the designed adaptive velocity-free controller.Clearly, all AUVs can track the movement trajectory of the leader even in the absence of velocity information.
A neural network is an approximation technology that comprises numerous nodes (neurons) and their interconnections.Each node represents a specific excitation function, and the weighted value of the connection signal between two nodes is termed the weight.Neural networks have a nearly universal model approximation capability and are used by researchers to describe, amongst others, the nonlinear dynamic characteristics of multi-AUV systems.This approach is highly effective for handling systems with model uncertainties and unknown disturbances.For the target tracking control of underactuated AUV system, Elhaki et al. transformed the bearing and range angles of the AUV system and target into a second-order open-loop error system, and used a multi-layer neural network to approximate the unmodeled dynamics and external interference, and then designed a new tracking control strategy. 38It can be seen that the control strategy is designed based on the determined target model, without considering the search problem of the target.Thus, Cao et al. further used this method to study the multi-target search and tracking control problems of multi-AUV systems in unknown underwater environments. 39They first used a Glasius bioinspired-neural network to aid multiple Position coordinate evolution under the adaptive velocity-free controller. 37lti-AUV systems in searching for their targets.Then, a bioinspired cascaded tracking control method tracks the target and prevents its escape.The proposed integrated tracking control algorithm combines a Glasius bioinspired-neural network and a bioinspired cascaded tracking control method to reduce tracking errors and improve search efficiency.
The control algorithm can realize static or dynamic target searches and track different trajectories around obstacles in an underwater environment.Usually, the neural network method is combined with the adaptive method to further improve the control performance of the controller.Zhang et al. designed a trajectory tracking control strategy based on neural network compensation and adaptive estimation technology for underactuated AUVs with unknown asymmetrical actuator saturation and unknown dynamics.In this study, a new bounded saturation function is used to deal with the unknown saturation of asymmetrical actuator, the neural network is used to approximate the complex hydrodynamics and expected tracking velocity differentials, and the adaptive estimation technique is used to approximate the neural network approximation errors and ocean disturbances. 40Additionally, Li et al. further combined the command filtering method to propose a new robust adaptive neural network control strategy with predetermined performance to solve the three-dimensional trajectory tracking problem of underactuated AUVs with uncertain dynamics and unknown disturbances. 41ackstepping control essentially uses a series of simple systems back pushed onto the original complex system.First, the complex nonlinear system is decomposed into numerous lower-order subsystems, and then the intermediate virtual control law is designed for each subsystem according to the Lyapunov stability theorem.To design the entire control law for the system based on the subsystem, "back push" is introduced.Considering that the matching conditions and external disturbances of a horizontally moving AUV system are unsatisfactory, Cho et al. investigated the trajectory tracking control problem of a torpedo-shaped AUV system based on the backstepping control method. 42A time-delay estimation method was developed to represent the nonlinear dynamics created by external disturbances and ocean currents.Subsequently, a robust control law using backstepping technology and time-delay estimation was proposed to improve the robustness of the control system and realize accurate tracking of the AUV system's desired trajectory.In addition, by combining adaptive, sliding mode control, and backstepping methods, An et al. proposed an adaptive backstepping sliding mode control strategy for an AUV system with external disturbances, dynamic uncertainties, and quantization errors. 43he above control strategies have good control effect, but it is worth noting that in the practical application tasks of AUV system, due to the limitation of system equipment and energy, the actuator saturation is a problem that cannot be ignored.Thus, considering that the AUV system has constraints such as ocean current disturbance, model uncertainty, actuator failure, and propeller amplitude and velocity saturation, Zhang et al. proposed a fault-tolerant control method for area tracking based on backstepping technology. 44Moreover, Liu et al. used the command filter to deal with the unknown model dynamics and disturbances for such AUV system, and designed a novel command filter backstepping control strategy. 45

Formation control of multi-AUV systems
The formation-control problem is also a major issue in the coordinated control of multi-AUV systems.This requires all the AUVs in the system to track the desired reference trajectory and maintain a relatively fixed position between them to proceed in the desired formation.Presently, leader-follower, virtual structure, artificial potential field function, behavior-based, and path-following are the main methods used to realize the coordinated formation movement of multi-AUV systems.The leader-follower method: This method designates one AUV in the system as the leader, which leads the entire formation's movement, and the other AUVs as followers.The leader navigates a predetermined or set path, and the followers control their movements using its position, speed, or bearing information to execute the formation movement by maintaining relative angles and distances from the leader.Recently, this method has become among the most commonly used methods for solving formation control problems in multi-AUV systems.For multiple underactuated AUV systems traveling horizontally, Cui et al. designed a formation control protocol based on the leader-follower method to solve the formation motion problem of a system with external disturbances and parameter uncertainties. 46In this study, neural network technology was used to approximate the uncertain parameters and unknown disturbances of an AUV system, and Lyapunov and backstepping synthesis methods were used to design the formation-tracking controller, so that the residual trajectory errors of all AUVs converge to a bounded compact set.It can be found that the control strategy is designed for AUV system with planar motion.Considering that multi-AUV systems move in three-dimensional space, Yan et al. designed formation-tracking control strategies for systems with and without communication delays. 47In this study, a two-layer distributed formation-tracking control strategy was designed by combining an observer and a lower controller.Additionally, due to the complex underwater environment in which the AUV system is located, weak communication and actuator saturation are also problems that cannot be ignored in practical applications.Thus, Yan et al. further studied the formation control problem of multi-AUV systems under weak communication and actuator saturation conditions, and designed formation control strategies that can cope with different situations based on leader-follower method. 48Additionally, for the situation that the velocity information of underactuated AUV system is unmeasurable, Gao et al. designed a formation control strategy under line-of-sight and angle constraints based on leader-follower method. 49t should be noted that although the leader-follower method is the most widely used, it also has the disadvantage of over-reliance on the leader.If the leader fails, it is difficult for the system to achieve the desired control tasks.
The virtual structure method: In this method, the system formation pattern is regarded as a virtual rigid-body structure in which each vertex has a predesigned motion trajectory and each AUV in the system corresponds to a vertex in the virtual structure.During the formation's movement, the distributed control law for each AUV is designed to ensure that each overlaps its corresponding vertex.Considering multiple AUV systems moving along the horizontal plane, Yan et al. studied the formation-tracking control problem of the system for the target. 50In this study, the virtual structure method is combined with graph theory to propose a formation control strategy based on the fixed triangle structure formed by the leader and the neighboring AUVs.Moreover, considering the underactuated AUV system moving in three-dimensional space, Li et al. used the relative displacement and angle between the leader and follower to generate a virtual structure, and then design a robust formation control strategy based on the structure. 51When considering that the mathematical model of AUV system is discrete-time, Yan et al. examined the formation trajectory tracking control problem of multiple AUV systems with time-varying communication delays in environments with weak communication. 52The discrete kinematic and dynamic models of an AUV system with time-varying delays are expressed as follows: where  v ij (k) and  p ij (k) denote the communication delays in the velocity and position layers, respectively.u v ij (k) denotes the control input.Subsequently, the formation-tracking control law is designed using the virtual structure method for bounded and unbounded communication delays, such that a fixed formation pattern of discrete-time multiple-AUV systems can be formed.In addition, combining this method with leader-follower method, the author also do a preliminary research on formation control of multi-AUV systems under intermittent communication conditions.Figure 7 shows the the numerical simulation results of the motion trajectories of all AUVs under the designed controller.It is found that the controller can make the multi-AUV systems realize formation motion even under intermittent communication condition.
The artificial potential field function method: This method considers obstacles and goals as objects that, respectively, exert repulsive and gravitational forces on an AUV.A virtual artificial potential field is formed in the operating environment of the AUV by the interaction between the repulsive field generated by the obstacle and the gravitational field generated by the target.Formation control is achieved by determining the direction of the descending potential function.For the formation and safe obstacle avoidance control problems of multiple AUV systems, Chen et al. developed a formation control strategy using an artificial potential field function and a self-organizing mapping neural network. 53It can be seen that the attitude of AUV is not considered in the above study.Then, for attitude adjustment and complex seabed terrain detection problems in multiple AUV systems deployed underwater, Zhen et al. combined the virtual structure and artificial potential field methods to propose a novel finite-time attitude tracking and position formation control strategy. 54n this study, based on special Euclidean group theory, the AUV system moving in a 3D space is divided into position and attitude subsystems, and the AUV system model is described as follows: where x ∈ R 3 denotes position, v ∈ R 3 denotes velocity, R ∈ SO(3) denotes the attitude, and Ω ∈ SO(3) denotes the angular velocity.In this study, the separately designed controllers ensure that the AUV system can track the desired trajectory with the desired direction angle in a finite time.This control method ensures that all AUVs form up and change formations after reaching the set depth, thereby avoiding collisions.It should be noted that although the artificial potential field function can solve the problem of collision avoidance in formation, it is easy to fall into the problem of local optimal solution in optimization and expected path planning.For this problem, Yu et al. introduced the auxiliary potential field perpendicular to the direction of AUV motion, and proposed a coordination formation obstacle avoidance control algorithm based on improved artificial potential field method and consensus protocol to solve the formation avoidance problem of multi-AUV systems. 55It should be pointed out that the problem of local optimal solution for artificial potential field function has not been completely solved so far.The behavioral method: In this method, the system tasks are divided into a series of behaviors, such as running toward the target, keeping formation, avoiding obstacles, and keeping formation, and the formation control of multi-AUV systems is realized through behavior fusion.Kang et al. used fuzzy logic to coordinate multiple response behaviors and proposed a behavioral fusion method to ensure that multi-AUV systems can maintain space formation for detecting large-scale unknown regions and have real-time obstacle avoidance capabilities. 56Wen et al. combined a behavioral and an artificial potential field method to study the formation control of multi-AUV systems with power constraints, delays, and communication faults. 57In this study, an improved artificial potential field strategy based on the maritime Internet of Things was established to control AUVs, and a behavior-based path optimization method was used to realize the local optima.The experimental results demonstrate that this strategy can effectively realize the coordination control of multiple AUV systems.Based on this strategy, maritime communication and AUV physical engine control models have been established.
This following is based on the path-following method, which aims to decompose tasks into the coordinated synchronization of time and path following in space to realize coordinated formation control in multi-agent systems.Based on this method, Filaretov et al. designed a novel path-planning strategy for multi-AUV systems to navigate an unknown environment with obstacles in a desired formation. 58This strategy enabled the leader AUV to define a movement trajectory based on a given task at a safe distance from the detected obstacle.Subsequently, the follower AUVs move along a preset trajectory in the formation to guarantee a safe distance between them.Matouvs et al. designed a novel formation-path tracking control strategy for multiple underactuated AUVs based on line-of-sight guidance and null-space behavior control techniques. 59This strategy enables all the AUVs to move along a curved path while maintaining the desired formation.In addition, Ouyang et al. used an improved fast-search random tree algorithm to design a formation path planning technology for unmanned craft to address the global path planning and local autonomous collision avoidance problems of unmanned craft in formation operations. 60

Circumnavigation control of multi-AUV systems
The realization of multiple AUVs proceeding in formation around a target is a common task in the cooperative control of multiple AUVs.This requires that all AUVs in the system reach a specific position on a circular orbit with the moving target as the center, the desired distance as the radius, and the desired formation and orbit in a fixed relative position.Through the investigation of literatures, it is found that there have been a lot of good research results in the field of UAV or unmanned vehicle for the circumnavigation control of stationary or moving targets.For example, Chen et al. designed a distributed discrete control algorithm to solve the circular formation control and collision avoidance problems of a group of non-holonomic robots in the absence of a leader and global beacon. 61It can be seen that the control strategy is designed based on individual state information.In practical applications, information transmission is usually subject to communication conditions, while bearing information measurement depends on optics or radar, which is not affected by communication conditions.When considering that only bearing information can be obtained between system individuals, Li et al. established an estimator based only on target bearing information, and then designed the circumnavigation controller of multi-robot system, which solved the circling motion problem of the system around unknown stationary targets in three-dimensional space. 62And Chun et al. designed a geometric center estimator and a two-dimensional elliptical circumnavigation controller to solve the elliptic circumnavigation control problem of multiple targets in a complex environment for a multi-robot system. 63It can be seen that these control strategies are almost designed for linear systems, while AUV system is usually described by second-order nonlinear model.For the circumnavigation control problem of multi-AUV systems, some satisfactory results have been also obtained in recent years.Seuret et al. used a collaborative control scheme containing the Laplacian matrix of a communication graph to solve the communication constraint problem and designed a novel distributed control algorithm to solve the circumnavigation control problem of multi-AUV systems with a limited communication distance. 64Shi et al. combined the kinematic and dynamic models of AUVs and proposed a cascade-based distributed control law to address the circumnavigation control problem of multiple AUVs for targets without a common reference direction or global coordinate system. 65It should be pointed out that these results only consider the kinematic model of the AUV system and design the control strategy based on it.Considering that AUV system is a second-order nonlinear model composed of kinematics and dynamics, Xia et al. designed a distributed control protocol based on a leader-follower framework by combining the artificial potential field function and consensus theory to address the problem of uniform circumnavigation control of the system for communication delay or data loss. 66This control strategy uses the state-feedback linearization technique to deal with nonlinear coupling terms in the AUV model, and a distributed circumnavigation controller using an artificial potential field function is designed to realize leader-centered circumnavigation by multiple AUV systems.In addition, the authors also do a preliminary research on circumnavigation control of multi-AUV systems under intermittent communication conditions.Figure 8 shows the the numerical simulation results of the motion trajectories of all AUVs under the designed controller.It is found that the controller can make the multi-AUV systems realize circumnavigation motion even under intermittent communication condition.

F I G U R E 8
Motion trajectories of and all AUVs under the circumnavigation controller.

Finite time control of multi-AUV systems
The finite-time coordination control of a multi-AUV system mainly combines finite-time control theory with the coordination control of a multi-AUV system to aid it in achieving its control objectives in a finite time.This section summarizes the state of research into finite-time control and its applications in multiple AUV systems.

Finite time control
Most of the aforementioned control strategies designed for the coordinated control problem of multi-AUV systems can achieve asymptotic or exponential stability.However, in practical applications, people prefer the system can achieve its control mission in a finite time.Therefore, scholars proposed the concept of finite time.However, it was not until the late 1990s that the Lyapunov finite-time stability theory 67 and finite-time homogeneous theory 68 were put forward, so that the concept was gradually applied to the study of multi-agent systems.Bhat indicated that if a system is asymptotically stable and has negative homogeneous degree, then the system is finite time stable. 68However, this result can only be used to analyze the stability of simple homogeneous systems.Thus, Hong et al. further provided a finite-time stability criterion for a nonhomogeneous complex system, that is, if the homogeneous subsystem is finite-time stable, the nonhomogeneous term satisfies certain constraints, and the nonhomogeneous system is asymptotically stable, then the nonhomogeneous system is finite-time stable. 69Unfortunately, this criterion cannot give an upper bound on the settling time for finite-time convergence.Therefore, Bhat further provided a finite-time stability Lyapunov criterion that can estimate the upper bound on the convergence time, that is, if there is a continuously differentiated Lyapunov function V such that where  and p ∈ (0, 1) are normal numbers, then the system is finite time stable, and the settling time (1−p) . 67It can be seen that this conclusion has a requirement for the differential upper bound of the Lyapunov function In order to reduce this requirement, Shen et al. proposed a local finite-time stability theorem. 70In this study, when the continuous differentiated Lyapunov function V satisfies V(x) ≤ −cV  (x) + kV(x) (17)   where c, k > 0,  ∈ (0, 1), then the system is finite time stable, and the settling time satisfies T ≤ ln . Unfortunately, this conclusion is local finite time stable, which requires the initial value of the system to be within a certain range.Based on this theorem, finite-time stability theory has been rapidly developed.Subsequently, Yu et al. proposed a fast finite time stability theorem. 71In this study, when the continuous differentiated Lyapunov function V satisfies where c, b > 0, 0 <  1 < 1,  2 ≥ 1, then the system is fast finite time stable, and the settling time . Lu et al. proposed the real finite time stability theorem. 72In this study, when the continuous differentiated Lyapunov function V satisfies V(x) ≤ −cV  (x) +  (19)   where c,  > 0, 0 <  < 1, then the system is real finite time stable, and the settling time c (1−) .Up to now, the finite time control theory has been widely used in the research of multi-agent systems.For example, for linear heterogeneous multi-agent systems, Wu et al. used the finite-time control theory to study the output regulation problem. 73In addition, for high-order uncertain nonlinear systems, Guo et al. studied the finite-time stability problem of the system with external disturbances. 74At present, the commonly used finite-time control methods can be roughly divided into continuous and discontinuous finite-time controls.
When the control signal is continuous, the finite time control problem of the system is mainly solved based on the adding power integral method and the homogeneous system theory method.The adding power integral method provides better anti-interference properties and can achieve global convergence.Recently, this method has been widely applied by scholars, and various results have been reported.Based on this method, Fu et al. designed an adaptive finite-time control strategy to address the global stability problem for a class of uncertain nonlinear systems with positive odd rational number powers. 75It can be found that this method is conservative in the estimation of convergence time.Thus, Sun et al. proposed a new fast finite time controller design method to solve the global finite-time adaptive stability problem for a class of higher-order uncertain nonlinear systems. 76In fact, in practical applications, the motion direction of the dynamic system is unknown, that is, the control gain/direction is unknown.To solve this problem, wu et al. designed a logic switching rule based on Lyapunov function to overcome the design difficulties caused by unknown gains, and then solved the global finite-time stabilization problem of the system. 77It is worth noting that most of the above research results assume that the state of the system is completely measurable, but some states of the system are often unmeasurable in engineering practice.In view of the situation that the velocity of the agent is unmeasurable, du et al. used dynamic output feedback method to solve the finite time formation control problem of multi-agent system. 78Compared with the adding power integral method, the homogeneous system method has some advantages, such as a simple control process and controller form.Yin et al. studied the finite-time stability of a class of stochastic nonlinear systems based on the homogeneous theory. 79Yu et al. used the homogeneous theory to design a distributed tracking control protocol to ensure that a multi-agent system with an undirected communication topology could be stabilized in finite time. 80It can be seen that these control strategies are designed for theoretical systems.For the multi-arm system, Zhang et al. designed a controller based on the homogeneous theory, which makes all the robot arms converge to the target position in a finite time. 81Similarly, the control strategy is designed based on state feedback when the system state can be measured.For frictionless mechanical systems, Zamora-Gomez et al. designed a finite time control strategy based on output feedback technology using homogeneous theory. 82t present, the terminal sliding mode control is the most important finite time control method when the system control signal is discontinuous.Early terminal sliding mode control technology experienced two problems: slow convergence far from the equilibrium point and unbounded input.Yu et al. designed a fast terminal sliding mode control technology to address the problem of slow convergence when the system is far from the equilibrium point. 83Feng et al. designed a non-singular terminal sliding-mode control technique to address the problem in which the system may produce unbounded inputs. 84These methods are discontinuous controls that may cause the system to chatter during motion.Therefore, the boundary layer function method 85 and high-order sliding mode control techniques 86 have been proposed to reduce or eliminate possible chattering in the system.In recent years, the use of terminal sliding mode control technology has also appeared some new research results.For the time-varying uncertain second-order nonlinear system, Wu et al. designed a new robust control strategy based on the terminal sliding mode control is technology. 87For the nonlinear non-affine systems, Zhang et al. used the non-singular fast terminal sliding mode control technology to solve the finite time stability problem of the system. 88

5.5.2
Finite time coordination control for multi-AUV systems In the practical applications of multi-AUV systems, control missions should be completed rapidly.Therefore, it is necessary to study the finite-time cooperative control problem of multi-AUV systems with the finite-time theory.This section provides an overview of current research results.
Regarding the finite-time coordination control of multi-AUV systems, Li et al. designed a series of finite-time control strategies based on the leader-follower framework to solve the problem of position consistency control whether the system has a leader or not. 20It can be seen that the results do not consider the system disturbance.Subsequently, li et al. proposed a finite-time observer to estimate the system disturbances, and designed a finite-time output feedback trajectory tracking control strategy for multi-AUV systems based on sliding mode control technology. 89To solve the formation-tracking control problem of multi-AUV systems with external interference and parameter uncertainty, Gao et al. designed a novel distributed fixed-time control strategy based on a sliding-mode control technique and an interference observer. 90he practical application of multi-AUV systems is affected by complex ocean environments, such as temperature, salinity, and viscous resistance, as well as by the limitations of their energy and sensor equipment.Consequently, in practical applications, multi-AUV systems may encounter a series of problems, such as state constraint, input saturation, and unmeasurable parameters and states.For multi-AUV systems with state constraints, Fan et al. designed a finite-time consensus tracking control strategy based on barrier Lyapunov function and neural network. 91Considering that the AUV system have error constraints, Qin et al. proposed a distributed finite-time enveloping control strategy to solve the enveloping control problem of multi-AUV systems with fault-tolerant constraints. 92These control strategies need to be implemented based on the ideal control output.In fact, limited by the capability of the actuator, input saturation is a problem that needs to be considered in the study.Considering the input saturation constraints and actuator faults in the AUV system, Zhu et al. designed a finite-time control strategy based on rotation matrix by using the sliding mode control method to realize global trajectory tracking. 93For the intervention-AUV system with input saturation and output constraints, Hou et al. proposed a finite-time trajectory tracking control scheme combining high-order control barrier function and quadratic program. 94It should be noted that these control strategies can achieve good control effect, which is inseparable from real-time information transmission.Unfortunately, real-time communication is difficult to achieve in a practical application environment.Considering that the variable-weighted network topology of multi-AUV systems has time-varying communication delay, Zeng et al. studied the finite-time coordinated formation control problem of discrete multi-AUV systems by using the virtual structure method. 95he above control strategy can solve the cooperative control problem under various constraints.However, it should be noted that almost all of these control strategies are designed based on a definite AUV system model.In complex and variable underwater environments, it may not be practical to design controllers using prior AUV system parameters and models.Thus, Chen et al. studied the finite-time trajectory tracking control problem of multi-AUV systems with heterogeneous uncertain dynamics, 35 and the adaptive controller is designed as follows Additionally, for multiple underactuated AUV systems with disturbances and unknown dynamic parameters, Thuyen et al. designed a finite time double-loop formation control strategy using leader-follower method and integral terminal sliding mode control technology. 96It can be found that these control strategies all depend on the velocity information of the AUV system, which may be difficult to obtain in some extreme application scenarios.Therefore, considering the unavailability of velocity information, Chen et al. further designed a finite-time trajectory tracking control strategy independent of velocity information for multi-AUV systems with heterogeneous uncertain dynamics, 37 and the adaptive controller is designed as follows In addition, when considering that the multiple underactuated AUV systems have switching directed communication topologies, uncertain dynamics, and unknown time-varying external disturbances, Wang et al. proposed a distributed finite-time velocity-free robust formation control scheme. 97It should be emphasized that the above control strategies are almost based on continuous communication.However, in practical application scenarios, limited by equipment, energy and environment, it is difficult for multiple AUV systems to communicate in real time.Therefore, in recent years, scholars have studied the cooperative control of multi-AUV systems under intermittent communication condition.Gao et al. proposed an event-triggered intermittent communication strategy to control the communication between the leader AUV and the follower AUVs, and designed a finite-time formation control strategy based on the leader-follower framework. 98Additionally, the author also carried out the corresponding research.In Reference 25, the homogeneous theory was used to study the finite-time rendezvous control problem of multi-AUV systems when the velocity information of each AUV was unavailable and the system communication was intermittent, and the hybrid controller is designed as follows The simulation results shown in Figure 5, which depicts that the controller can make multi-AUV systems realize rendezvous motion.

CONCLUSIONS
In this study, the state of development in multi-AUV systems was systematically introduced from the perspectives of experimental research, collaborative control theory research, and finite-time theory.Although a large number of relevant research results have emerged in recent years, there are still many related control problems that have not been deeply studied, mainly including the following aspects: 1.The strong nonlinearity and coupling of the AUV system model, unknown external disturbances, and uncertain model parameters inevitably caused by complex ocean environments such as ocean currents and tides lead to difficulty in obtaining an accurate AUV system model.In addition, the special working environment of the AUV system and its characteristics lead to numerous problems such as actuator saturation and communication delay.Given the multitude of external factors, a distributed control strategy was designed for an AUV system such that all AUV systems can realize a variety of control behaviors, such as tracking and formation, while maintaining high precision and robustness.2. In practical applications of AUV system, the system is often required to achieve rapid convergence while completing the tasks.However, the current coordination control strategies for multi-AUV systems mainly focus on asymptotic or exponential stability and do not apply to systems that perform special tasks.Therefore, introducing the finite-time control theory into various cooperative control tasks of multi-AUV systems and designing a controller to achieve fast convergence while ensuring the task completion ability of the system is worthy of further study.3. Presently, control algorithms designed for resolving various coordination control problems of multiple AUV systems rely on position, attitude, velocity, acceleration, and other state information as well as continuous communication between individual AUVs.However, owing to the influence of subjective and objective factors, such as the complex ocean environment, payload, energy, and equipment limitations, the system cannot measure all status information and communicate continuously in practical applications.Therefore, the design of a controller that can ensure the stability of the system in a finite time to complete the specified tasks under the condition of partial information and intermittent communication is worthy of further study.4. Formation and circumnavigation are two important tasks in the coordinated control of multi-AUV systems.Most current research results can only control the system to achieve independent formation or circumnavigation motion.However, in some special application scenarios, these two motion patterns need to be switched quickly and smoothly to complete the task.Therefore, further study is warranted to ensure that the system can complete tasks in a finite time and realize transient formation-circumnavigation switching when the information and working environments of multi-AUV systems impose limitations.validation (lead); writing -original draft (supporting); writing -review and editing (lead).Bijoy Kumar Ghosh: conceptualization (supporting); investigation (supporting); supervision (supporting); writing -review and editing (supporting).

F I G U R E 4
The intermittent communication pattern.

F I G U R E 7
Motion trajectories of all AUVs under the designed controller.
The inertial coordinate system.Each state of AUV system.
F I G U R E 2 The body coordinate system.TA B L E 2