Formation control of multiple autonomous underwater vehicles: a review

2.1. Basic knowledge on graph theory

To analyze the properties of a AUVs formation system, graph theory can be used as a useful tool. This subsection aims to introduce some fundamental concepts in graph theory. The graph, denoted by a triple $$ G = \{ {V,E,A} \} $$ , can be used to represent the communication topology among a AUVs fleet, including the vertex set $$ V = \{ { \nu _1, \nu _2, \ldots, \nu _N } \} $$ , edge set $$ E \subseteq V \times V $$ and weighted adjacency matrix $$ A = \left[ {{a_{ij}}} \right] \in \mathbb{R}^{\text{N} \times \text{N}} $$ . In particular, the element $$ \nu _i $$ in the vertex set $$ V $$ , termed a node, represents the AUV $$ i $$ in the group, where $$ i $$ belongs to an accountable index set $$ \Gamma = \left\{ {1, \ldots ,N} \right\} $$ . The element $$ \left( {{\nu _i},{\nu _j}} \right) $$ in the edge set $$ E $$ describes the interaction between AUVs $$ i $$ and $$ j $$ , and associated with weights $$ {a_{ij}} = {a_{ji}} > 0 $$ , which are the entries of adjacency matrix $$ A $$ . In such a case, we call AUV $$ j $$ a neighbor of AUV $$ i $$ , and all the neighbors of AUV $$ i $$ can be described by the set $$ {N_i} = \left\{ {j | {\left( {{\nu _i},{\nu _j}} \right) \in E} } \right\} $$ . If there is no connection between AUVs $$ i $$ and $$ j $$ , then let $$ a_{ij} = a_{ji} = 0 $$ and $$ \left( {{\nu _i},{\nu _j}} \right) $$ is not the element of $$ E $$ . We may further define $$ a_{ii} = 0 $$ for all $$ i \in \Gamma $$ , and out-degree of the node $$ i $$ as $$ d_{i} = \sum _{j \in N_i} {a_{ij}} $$ , after which the degree matrix and the Laplacian matrix of the graph $$ G $$ can then be defined as $$ D = \text{diag} \left\{ {d_1, \ldots, d_N} \right\} \in \mathbb{R} ^{\text{N} \times \text{N}} $$ and $$ L = D-A $$ , respectively.

In addition, a path in graph is defined by a sequence that contains a set of successive adjacent nodes, starting from the initial node and ending at the final node. If there exists at least one path between any two nodes in a graph $$ G $$ , then, say, graph $$ G $$ is connected. Furthermore, in order to make the AUVs fleet move along with a desired path as a whole, a reference trajectory must be defined ahead of time. Thus, the availability of the information of reference trajectory for $$ i $$ -th AUV is characterized by a parameter $$ b_i $$ ; that is, if AUV $$ i $$ is permitted to access this information, then $$ b_i >0 $$ ; otherwise, $$ b_i = 0 $$ , and define $$ B = \text{diag} \left\{ {b_1, \ldots, b_N} \right\} $$ . Based on that, we may have the following important lemma, which is useful to help analyze the stability of AUVs formation systems based on the graph theory.

Lemma 1 For the considered AUVs formation control network, described by graph $$ G $$ , if $$ G $$ is connected and there is at least one AUV able to access the information of the reference trajectory, i.e., the elements of $$ B $$ are not all equal to zero, then the matrix $$ L+B $$ is positive definite.

2.2. Mathematical models of AUVs

The kinematics of each AUV is described as ^[28]

where $$ (x_i, y_i, z_i) $$ and $$ (\theta _i, \psi _i ) $$ are the position and orientation of the $$ i $$ -th vehicle ($$ i \in \Gamma $$ ), respectively, expressed in the earth-fixed frame $$ E^{\rm I} = \left\{ {e_o^{\rm I}, e_x^{\rm I}, e_y^{\rm I}, e_z^{\rm I}} \right\} $$ , and $$ (u_i, v_i, w_i) $$ and $$ (q_i, r_i) $$ are the linear and angular velocities of the $$ i $$ -th vehicle, respectively, which is expressed in the body-fixed frame $$ E^{\rm B} = \left\{ { e_{o,i}^{\rm B}, e_{x,i}^{\rm B}, e_{y,i}^{\rm B}, e_{z,i}^{\rm B}} \right\} $$ , as shown in Figure 1.

Figure 1. Earth-fixed frame and body-fixed frame systems of $$ i $$ -th AUV

The dynamics of the $$ i $$ -th vehicle is modeled by

where $$ m_{i1} = m_i - \beta _{\dot u i} $$ , $$ m_{i2} = m_i - \beta _{\dot v i} $$ , $$ m_{i3} = m_i - \beta _{\dot w i} $$ , $$ m_{i4} = I_{yi} - \beta _{\dot q i} $$ and $$ m_{i5} = I_{zi} - \beta _{\dot r i} $$ ; $$ m_i $$ is the mass of the $$ i $$ -th vehicle; $$ I_{yi} $$ and $$ I_{zi} $$ are the moments of inertia around the axes of $$ e_{y,i}^{\rm B} $$ and $$ e_{z,i}^{\rm B} $$ , respectively; $$ \beta _{\left(\cdot \right)} $$ is set of hydrodynamic related terms associated with the $$ i $$ -th vehicle, $$ \tau _i = \left[{\tau _{i1},\tau _{i2},\tau _{i3} }\right]^{\rm T} \in \mathbb{R}^{3} $$ is the control input, and $$ d_i = \left[ {d_{i1},d_{i2},d_{i3},d_{i4},d_{i5} } \right]^{\rm T} \in \mathbb{R}^{5} $$ is the disturbance acting on the $$ i $$ -th vehicle.

Remark 1 It is worthwhile noting that the complete motion of the equation of an AUV is of 6 DOF, but, as we surveyed, almost all of the literature studying 3-dimensional (3D) formation applications employ the AUV model, as presented in Equation (1) and Equation (2), with 5 DOF. This relies on the fact that for an AUV formation fleet, the rotational motion around $$ e_{x,i}^{\rm B} $$ axis, i.e., roll motion, sometimes is not required in many practical AUV maneuvering, and hence the equation governing the roll motion is intentionally omitted, which does not cause loss of the practicality. Indeed, it is easy to check that the roll motion is passively bounded ^[28]. Particularly, its impacts on the other DOF can be treated as unmodeled dynamics and handled effectively by some disturbance rejection techniques.

Remark 2 It is clear that the dynamics of AUVs are highly nonlinear and underactuated as in Equation (2); in other words, the sway and heave velocities, i.e., $$ v_i $$ and $$ w_i $$ , are not fully actuated and there are no actual control actions allowed to be applied. In the sequel, this underactuation feature will be demonstrated as a major challenge to designing high-performance formation controllers for AUVs. In addition, besides the environmental disturbances described by $$ d_i $$ , as observed in Equation (2), there are many system parameters subject to the perturbation due to the effect of hydrodynamics, which is characterized by the time-varying parameters $$ \beta _{\left(\cdot \right)} $$ . Such uncertainties in the dynamic model of AUVs give rise to another significant difficulty for the formation control design. Furthermore, the angle of $$ \theta_i $$ is constrained and not allowed to take values at $$ \pm \pi/ 2 $$ in order to avoid the singularities, which should be guaranteed by the formation controller. Therefore, from the control point of view, addressing such a system continues to be challenging.

2.3. Problem statement

This subsection aims to formulate the considered formation control problem of a fleet of AUVs. As mentioned earlier, one of the requirements of formation control is to drive the AUVs to form a prescribed static or time-varying geometric shape and maintain it. Typically, the expected geometric pattern can be determined by the appropriate assignments of the relative positions between vehicles, denoting as $$ \Delta _{ij} = \left[ {\delta_{x,ij}, \delta_{y,ij}, \delta_{z,ij}, \delta_{\theta,ij}, \delta_{ \psi,ij} }\right]^{\rm T} $$ for AUV $$ i $$ $$ (i \in \Gamma) $$ and its neighboring nodes $$ j \in N_i $$ . To this end, the formation control is cast to the problem of controlling relative positions and orientations of vehicles with respect to their neighbors so that $$ \Delta _{ij} $$ can be achieved as time tends to infinity, and particularly these desired relative poses could even be set to be time-varying as necessary. Another practical requirement is that it is most desirable for an AUVs fleet in applications to track a reference trajectory as a group, in which case not only the positions, but also the velocities of the vehicles in the fleet are needed to be in consensus; that is, $$ (\dot x_i, \dot y_i, \dot z_i) $$ converges to a common reference speed $$ (\dot x ^d, \dot y^d, \dot z^d) $$ , as $$ t \to +\infty $$ . Such an objective is also referred to as the problem of formation tracking control in the literature.

In short, the formation control objective can be summarized as designing controllers for AUVs such that a set of desired relative positions and orientations, i.e., $$ \Delta _{ij} \ (i,j \in \Gamma) $$ , can be achieved and maintained while the AUVs fleet tracks a common reference trajectory, moving in the same speed together.

3.1. Leader-following structure

Leader-following structure is one of the most popular schemes used for the formation control of multiple agent systems because of its straightforward descriptions ^[11]. In such a scheme, one or several agents are selected as the leaders, and the rest of the agents are grouped into followers, as depicted in Figure 2. The desired reference signal is merely known to the leaders, and in conventional leader-following structure, the goal of the leaders is simply to track this prescribed reference and there is no explicit interaction between their following agents. The sole goal of the remainders is aimed to keep the desired relative pose (i.e., position and heading) with respect to their leading agents. As such, the formation control objective, as stated in section 2.3. can be achieved if each vehicle's goal is reached. The major advantages of such a method are that it is easy to be implemented and fairly flexible to add or remove vehicles in the fleet; besides, since there are no direct interactions between neighboring vehicles, the stability of the whole formation system can be easily analyzed based on the graph theory, as presented in section 2.1. Because of those nice features, there is a great amount of literature on AUV formation control adopting the leader-following structure ^[44-48].

Figure 2. A typical topology of leader-following structure.

Such an approach, however, suffers from a major defect that the performance of the overall formation system depends highly on the behavior of the leaders and the quality of the communication. In other words, once the lead agents or communication network fail to work as usual due to unpredictable faults, which is often the case in underwater environments, the entire formation system may be disabled. To overcome this issue and improve the robustness of the leader-following approach, a virtual leader based method is proposed, in which there are no physical vehicles employed to lead the group and, therefore, the above-mentioned issue can be appropriately addressed ^[49-52]. A typical virtual leader based formation structure can be referred to the Figure 3. Another critical consideration regarding this type of structure is that it is always assumed that every vehicle in the group is permitted to obtain the trajectory information of the virtual leaders, which is a strong assumption and may not be fulfilled in many realistic applications.

Figure 3. A typical topology of virtual leader structure.

3.2. Virtual structure approaches

Similar to the virtual leader approach, virtual structure coordination is another common method used to coordinate the multi-agent formation, which was first reported in ^[53,54] to address the cooperative control of multiple mobile robots. In this method, a set of virtual points are defined corresponding to each vehicle, which is determined by the desired formation configuration as well as the trajectory to be tracked. Since each vehicle is assigned its own reference point, the formation tracking problem is then converted into the tracking control problem associated, the goal of which is to drive the vehicles to minimize the errors between their actual positions and desired ones. A typical realization of such an approach is illustrated in Figure 4. Due to the fact that such a method is also straightforward and simple to analyze and realize, there have been many results reported to date based on this method to achieve the formation control requirements ^[54-59].

Figure 4. A typical topology of virtual structure approach.

The main drawbacks of the virtual structure approach may lie in several aspects: (1) Like the virtual leader approach, this method relies closely on the desired reference trajectories, which appears to be unrealistic in many practical scenarios; (2) It is not easy to expand the AUV formation, since the desired virtual reference points are designed in advance based on the prescribed formation pattern; (3) Due to the lack of information exchanges between the neighboring vehicles, there is no cooperation occurred in the formation system, which degrades the coordination performance.

3.3. Behavior-based approaches

Different from the above two methods, as shown in Figure 5, there exists an explicit mutual communication in formation systems synthesized using the behavior-based coordination approach. Instead of directly prescribing a priori reference trajectories, in behavior-based approach, each vehicle in the group makes its own decisions based on the local information (e.g., its own states, surroundings, and neighbors' states) and the goals predefined. The goals usually include target reaching, collision or obstacle avoidance, distance maintaining, etc. Particularly, the overall control actions of the vehicles are then generated by a weighted combination of achieving these different goals. Due to the multi-objective and distributed features, behavior-based approach has attracted extensive attention over the past few decades in the research areas of multi-agent cooperation and coordination ^[60-64].

Figure 5. A typical topology of behavior-based approach.

Although the behavior-based scheme is turned out to be able to achieve multiple objectives and is merely dependent on the limited local information to calculate control activities, it is hard to analyze the stability properties of the overall formation system based on such a method when more vehicles and behaviors are involved. This restricts its practical applications greatly.

3.4. Artificial potential field approaches

Artificial potential field (APF) approach is first invented by Khatib ^[65] in order to design an algorithm to generate an obstacle-free route for manipulator and mobile robot path planning. The main feature of the method is that a series of artificial potential functions are defined intentionally with the purpose of reaching the target and meanwhile avoiding obstacles. Analogous to the potential energy in physics, the APF functions defined can generate corresponding potential forces as well. Typically, there are two types of APF functions involved: One aims to generate attractive potential forces to bring the vehicles to the targets, and another attempts to yield the repulsive potential forces to make the vehicles keep away from the obstacles, which is shown in Figure 6. As a consequence, under both the attractive and repulsive forces, the prescribed objective can be attained. Inspired by such a formulation of clear physical significance, APF approach is also introduced in various multi-agent systems to help organize the cooperation and coordination ^[66-72].

Figure 6. A typical topology of APF-based approach.

Similar to the behavior-based approach, it is relatively easy for the APF approach to synthesize distributed controllers that achieve multiple goals depending only on the local information. However, one major drawback is that it has the chance to trap into points at which the resulting net force applied on vehicles is zero, which is also known as the issue of "local minima". Likewise, the stability analysis of APF based multi-agent systems is not easy as well, compared to the leader-follower structure and virtual structure approaches, when the scale of the group grows larger.

3.5. Other approaches

By a comprehensive literature review, there are some other commonly used approaches to achieving the multi-agent cooperation and coordination, while these kinds of schemes can be, in some sense, regarded as variants of those already presented. For example, the so-called formation reference point (FRP) method is actually a type of virtual leader method, in which a reference point is defined and parameterized with desired velocity profile, and then the control objective of vehicles in the group is simply to maintain a specific distance and bearing with respect to the reference point ^[73,74]. Like the conventional virtual leader method, there are no explicit interactions took place between neighboring vehicles.

In the field of multi-agent coordination, there exist a fundamental problem, termed consensus problem, which characterizes how the inter-agent cooperation can be emergent by merely using the local information (i.e., interacting with neighbors) ^[75]. Specifically speaking, it is possible that the state of the entire multi-agent system can be ultimately in consensus; that is, each agent's state converges to the same equilibrium point under a proper local control law that is designed only based on the neighboring information. Moreover, such a problem can be well formulated and tackled by the graph theory, as mentioned in section 2.1, including the basic issues such as the existence of solutions, stability and robustness properties of multi-agent systems. For this reason, a vital amount of related research results on networked multi-agent systems coordination based on consensus approach are reported both in theoretical and practical dimensions ^[31,76-81]. Such a problem is then extended to the case of formation control design, where a virtual leader is introduced to guide the vehicle group to move along with a reference trajectory together, and meanwhile, the formation is formed by the exchange of information between neighbors ^[82-84]. A typical communication topology of consensus-based formation control is illustrated in Figure 7.

Figure 7. A typical topology of consensus-based approach.

It should be noted that in stark contrast with the conventional virtual leader structure, there exists local information flow in the consensus based approach, and more importantly, it is not necessary that all members in the group have access to the information of reference trajectory ^[85-87]. It is, in effect, the sole condition that the graph associated has a spanning tree ^[11]. Those mentioned features seemingly make the consensus based method more practical and beneficial for real-word applications among various formation control methods. While such a method has attracted much attention in the studies of groups of unmanned ground vehicles ^[88-92], unmanned aerial vehicles ^[93-98], and spacecraft ^[99-102], there is still a lack of sufficient research efforts to apply consensus based approach to the multi-AUV formation where more practical challenges associated should be addressed further, most of which will be discussed in what follows.

4.1. Nonlinear constrained dynamics of AUVs

As mentioned earlier, different from the general multi-agent systems ^[11] where a point-mass model is typically used to describe the motion of agents, the governing equations of AUVs are much more complicated as explained in Remark2, including nonlinearity, underactuation and system constraints. As a result, such systems may suffer from some extra design complexities in respect of motion control. To deal with the nonlinearity, employing an approximated model is a likely choice by linearizing the nonlinear model at a specific operating point, and then linear control theory can be used to design the motion controller for AUVs. Based on this idea, the authors ^[103] proposed a nonlinear gain scheduling controller for heading control of AUVs in the horizontal plane. In this study, a finite number of linear static feedback controllers were derived at distinct speed conditions, after which the parameters of the controller were interpolated upon speed. The resulting scheduled controller was designed using the D-methodology ^[104] to guarantee the stability of the overall closed-loop system. To obtain an optimal control performance with respect to a quadratic-type objective, a linear-quadratic regulator (LQR) algorithm was implemented ^[105], and an optimal state feedback gain was figured out to regulate the AUVs' depth as well as stabilize the roll and pitch angles. In order to account for the approximating errors and the time-varying environmental conditions, a robust H$$ _\infty $$ based depth control was proposed ^[106], in which the aim of the controller is to minimize the cost function under the maximum effects of the parametric perturbations, and the resulting robust performance is verified by simulation.

Another class of optimization-based control methods, termed model predictive control (MPC), is inherently effective in handling internal uncertainties. Due to the preceding horizon formulation, an online optimization mechanism is enabled in MPC using real-time state feedback against the potential modeling errors. Furthermore, owing to a shortened horizon employed in optimization, such a method is also allowed to handle the state and input constraints, which is a very beneficial and useful feature in designing practical controllers as nearly all of the physical systems are subject to actuation saturation. By means of the MPC policy, the authors ^[107] studied three-dimension (3D) underwater tracking control of a fully-actuated AUV with the practical constraints in both state and input, and the simulation results verified the satisfactory performance in the presence of various disturbances. In underwater conditions, the measurement noises may be another non-negligible factor affecting the control performance. To this end, linear quadratic Gaussian (LQG) control, integrated with the Kalman filtering (KF) technique, serves as an efficient optimal control strategy to ensure that the controlled system operates at the optimum status in the sense of variance minimization ^[108]. Recently, a model predictive tracking controller was combined with an extended state based Kalman filter for a constrained remotely operated underwater vehicle in order to achieve a high-precision tracking performance, while the system nominal model is uncertain and subject to both sensor noises and external disturbances ^[109].

However, most of the aforementioned approaches are designed based on a linearized model at some specific operating points, which necessitate assumptions that the AUVs move at a relatively low speed and the impacts of disturbances, caused by the ocean waves and currents, are much limited such that the locally stable results can be obtained. Such a hypothesis may be quite conservative and, in effect, not always hold in practical situations. In view of that, nonlinear control techniques have been the research focal point over past decades to help develop high-performance motion controllers for AUVs with global stability guarantee. The researchers ^[110] proposed a nonlinear tracking controller for a remotely controlled AUV based on the feedback linearization technique, under which the resulting system becomes a decoupled linear system, and furthermore, an optimal error correcting term is added in the feedback loop, based on LQR approach, to compensate the uncertainty. To derive adaptive formation tracking controllers for a group of underactuated AUVs in a horizontal plane, the backstepping design procedure and neural network technique are used ^[46]. The resulting nonlinear formation controllers developed for each AUV are based on a virtual leader structure without knowledge of the leader's velocity and dynamics, and the overall AUVs formation is proved to be uniformly ultimately bounded stable using the Lyapunov stability theory. Due to the obvious benefits of MPC approach in handling systems constraints, a nonlinear MPC controller was presented for the tracking control of an AUV in 2-dimension (2D) case^[111,112], where an auxiliary nonlinear control law is proposed using the backstepping-based Lyapunov synthesis, and as a result, the linearization procedure can be avoided for standard MPC design, which improves the region of attraction of resulting system greatly. Furthermore, the resulting Lyapunov-based MPC is guaranteed with properties of iterative feasibility and stability. Following the same ideas, the authors ^[113] developed a leader-follower based receding horizon kinematic controller for formation control of a constrained underactuated AUV. Taking into consideration the uncertainties in modeling, the authors combined the extended state observer (ESO) with nonlinear MPC to deal with the formation tracking of AUVs ^[114].

Another class of important methods to address complex systems is intelligent control, including fuzzy logic control, neural network control, and data-driven control (sometimes in the literature also called machine learning-based control), etc. For instance, the authors ^[115]developed a nonlinear fuzzy logic based proportional-integral-derivative (PID) controller to regulate an AUV's heading and depth, and in particular, Mamdani fuzzy rules are used to tune the control gains of PID adaptively to address the nonlinearity and uncertainties in AUV modeling. As well, to address a similar issue, 3D path following of an underactuated AUV is investigated using fuzzy logic based backstepping controller ^[116]. It is worth noting that AUVs path following problem is slightly different from the tracking problem, while they are both expected to follow a desired route. The former does not impose a strict temporal constraint on the reference trajectory. Thus, an extra degree of freedom in the speed of the trajectory can be exploited to design controllers, which could be practical in some applications ^[6]. To handle the nonlinear and unknown dynamics of AUVs, a data-driven model-free predictive approach ^[117] was proposed for the accurate trajectory tracking control, in which the authors suppose that both parameters and order of the plant are unavailable to control design. To address a similar problem, an adaptive line of sight (LOS) guided reinforcement learning approach based on the long short-term memory (LSTM) neural network model was devised^[118]. Considering the unknown nonlinearity in AUV dynamic model, underactuation and input limitation, the author ^[119] designed a neural network based robust formation controller using virtual structure for a group of underactuated AUVs with actuation saturation, and furthermore, rigorous stability analysis was provided.

4.2. Adverse underwater environments

In addition to the nonlinear, uncertain and underactuated characteristics of the AUVs dynamics as discussed above, the complicated underwater conditions (e.g., marine disturbances, hydrodynamic effects, unpredictable static or dynamic obstacles) may also pose a great difficulty in AUVs formation design. In the preceding discussions, we have already mentioned numerous methods to address the uncertainties in AUV dynamic model, most of which adopt an adaptive idea, e.g., adaptive control methods, data-driven methods, neural network-based methods, etc., to estimate parameters (either control gains or model-related parameters) in a real-time manner based on the measured data. This class of methods, although superior, is basically suitable for the case when the structure of the uncertainties is known, but the associated parameters are required to be estimated. On the other hand, due to the integration with online parameter estimation, most existing adaptive approaches result in a nonlinear time-varying system whose robustness properties regarding the modeling errors are hard to be guaranteed. Nonetheless, as for the marine disturbances induced by the ocean waves, currents and winds, as well as the effects from the hydrodynamics, their interaction patterns with the AUVs are complicated and quite difficult to understand. In view of that, nonlinear robust control techniques, e.g., sliding mode control (SMC), have received much attention in various mechatronic control systems, for which a static switching-like feedback control law is used and the corresponding control gains are calculated to ensure that all types of disturbances, so long as within a given bound, can be handled^[120-122].

Based on this paradigm, there is a variety of literature employing SMC schemes to study the robust tracking control problem of AUVs subject to both unknown disturbances and modeling uncertainties ^[123-126]. In particular, considering the potential time delays between surface ships and vehicles, which happens quite often in practical situations, the authors proposed a discrete-time quasi-SMC method for AUVs depth control to guarantee the stability and robustness even in the presence of large sampling intervals, and concurrently, the chattering was addressed using the so-called equivalent control region ^[127]. Besides the robustness, in order to meet a fast converging requirement near the equilibrium point, terminal sliding mode control (TSMC) techniques are used by researchers to achieve the finite time control ^[128-130]. Due to the possibility of singularity appearing in TSMC schemes, some non-singular TSMC approaches were investigated to avoid this issue^[131-133]. While SMC based controllers exhibit a stunning robust performance in handling unknown environmental disturbances as well as inherent modeling uncertainties, a major drawback is that the chattering issue occurring in control activities should be carefully addressed, as it may inevitably deteriorate the performance in practical cases. Towards this end, higher-order SMC techniques were proposed to attenuate the high-frequency chattering to make SMC schemes more practical, and such methods were also applied to the control problems of AUVs tracking and formation ^[134-138]. To completely eliminate the chattering while obtaining good robustness, the authors developed a distributed neuro-dynamics-based sliding mode controller for the consensus formation tracking control of a group of AUVs ^[139]. In addition to the static robust control schemes that are considered more conservative, there are numerous disturbance observer-based controllers used for AUVs motion and formation control, which behave more actively in addressing the disturbances and hydrodynamic coefficients ^[140-145]. As well, observer-based schemes can also be used as an effective tool to tackle the output feedback control where the velocity sensing is unavailable, which is fairly beneficial in terms of reducing cost and meanwhile improving the performance of AUVs formation ^[146-149]. Furthermore, observer or estimator based techniques are also of critical importance and act as useful tools in designing systems of diagnosis and accommodation for sensor errors and faults. It is well known that failure to detect and isolate the errors and faults may lead to inaccurate control performance and even loss of the AUVs. To overcome it, the authors ^[150,151] employed certain observers and filters on the basis of vehicles' models to identify the errors in sensors and provide an estimate correspondingly, and thus a high-quality motion control objective can still be maintained.

Except for the handling of unknown disturbances and hydrodynamic uncertainties, avoiding obstacles and inter-vehicle collisions is another realistic aspect in synthesis of formation systems due to the poor knowledge of the obstacle distribution. To prevent unmanned aerial vehicles (UAVs) formation from any collisions, a collision avoidance mechanism was incorporated into the UAVs formation control strategy. In this approach, an additional velocity term is involved based on the mechanical impedance principle to avoid potential collisions, once the obstacles detected are within a certain range ^[152]. A new dual-mode strategy is designed to achieve cooperative UAV formation flying. In an obstacle-free environment, safe mode is activated to achieve global optimization. When faced with obstacles, danger mode enables a modified Grossberg neural network to plan an obstacle-free route, and a model predictive controller is also developed to achieve the route planned ^[153]. An obstacle avoidance strategy was proposed for UAV formation control design based on the artificial potential field (APF). An attractive potential field is designed for a leader to track the moving target, and the rest of the AUVs aims to follow the leader using the attractive potential force, while repulsive potential forces are also defined for UAVs to avoid obstacles ^[154]. Employing the same idea of APF, the authors proposed an obstacle-free formation controller for multi-agent systems based on virtual structure. To achieve the path planned by APF, a backstepping controller was developed based on the neural network and finite-time control technique ^[155]. Besides the APF-based obstacle avoiding strategies, as aforementioned, MPC-type controllers are capable of handling constraints effectively, which can be utilized to design collision-free formation controllers. This can be achieved by either extending the objective function or adding extra avoidance constraints into the receding horizon optimization procedure ^[156-159]. Other available approaches, such as ant colony algorithms^[160,161], particle swarm optimization^[162], machine learning-based algorithms^[163], etc., were also reported to optimize an obstacle-free path, yet such intelligent optimization-based algorithms quite often necessitate a great computational effort and take a relatively long time, which greatly restricts their applications in AUVs formation where computation resources are limited and rapid response is needed to handle the variable surroundings.

4.3. Communication constraints

Different from the multiple land robots, surface vehicles and unmanned aerial vehicles, the signals of the global position system (GPS) or base stations cannot be received by AUVs in underwater conditions, and therefore the localization and communication in AUVs fleet may become an unavoidable issue. It has been turned out that the traditional acoustic technologies still serve as the most efficient way to provide the wireless connecting services ^[27], in which case the communication capability, thereby, has major impact on the performance of AUVs formation. The effects of the communication constraints mainly result from the following several aspects.

● Propagation delay: The speed of acoustic waves is nearly 1.5 kilometers per second, which is far slower than the speed of electromagnetic waves in air, and thus the effects of propagation delays cannot be neglected. Moreover, since the speed of the acoustic propagation relies highly on environmental factors (e.g., pressure, temperature, salinity, disturbances), the calculation for propagation delays may be rather complicated and often time-varying.

● Path loss: As the acoustic waves propagate, their energy spreads and attenuates in the medium. This process is called path loss, especially the higher the frequency, the easier the path loss. More path loss means a shorter communication range.

● Limited bandwidth: For a specific transition channel, its bandwidth is always limited and related to the range of communication. Therefore, it is necessary to allocate the interactive data reasonably (e.g., sensor data, control data, localization data) in order to achieve an effective formation control.

● Multipath: During propagation, the acoustic signals could be reflected possibly by the sea surface, seabed and obstacles, which causes the multipath. Signals in multipath will result in distortion, consequently reducing the measurement or transmission accuracy.

● Doppler effect: Since the AUVs formation is always in motion, the frequency of the wave being transmitted by a moving base may be varied on its receiving side, which is called the Doppler effect.

There are a number of research efforts made to overcome the above communication constraints so that a better formation performance can be achieved. By assuming that the delays of propagation are bounded and the bounds on delays are less than the sampling period of AUVs, it was shown ^[164] that the leader-following based formation system can tolerate the delays and still achieve the formation stabilization, although it takes a longer time to stabilize than the case without delays. To provide a lower bound on control performance of an inspection ROV with sensor time delays, the authors employed an input-output controllability analysis to show that at which amount of delays the ROV system can tolerate irrespective of the forms of designed controllers, while the results are obtained based on an approximated linear model of the ROV^[165]. As well, the researchers demonstrated by experiments that there exists a dominant time delay in the underwater acoustic positioning system powered by the short baseline (SBL) with four transducers, and that the traditional Smith predictor failed to estimate due to the time-varying properties. To handle it, an online identification was proposed in their work to estimate the variation of the delay, thus minimizing its impacts as much as possible ^[166]. To be able to account for the package dropouts, the authors ^[167] treated it as an extra delay out of the total delays and a compensating law was designed in the H$$ _\infty $$ controller to ensure a robust formation performance against delays. To characterize the delays more precisely, researchers ^[168] represent the delays as a differentiable function and suppose that its time derivative is less than one, under which a leader-following controller is designed to reach the formation requirements. To overcome the bandwidth limitation, an observer-based formation controller is quite often used to reduce the amount of information exchange, and furthermore, a fault function is introduced to describe the effect of the multipath in communication ^[169]. Nonetheless, as indicated^[27], due to the fact that the information will be compressed before transmission, reducing control or sensor data do not contribute a lot to the relaxation of the bandwidth limitation. The authors provided a leader-following algorithm to optimize the formation configuration in terms of coverage efficacy and communication power consumption utilizing a calculation model for the path loss ^[170]. To reduce the Doppler effect, a Doppler effect compensation system is designed using an efficient multi-rate sampling technique ^[171]. Similarly, using diffident resampling strategies, reduction of the Doppler effect for the moving platform can be achieved ^[172,173]. Due to the unpredictable ambient disturbances and long communication distance, it is possible for the AUVs fleet to encounter a period of communication disconnection, in which case dynamic network topologies can be used to describe it. To achieve a dynamic topology, the authors predefine a set of topologies and put a random mechanism on it, and the objective is to design a formation controller such that the formation system can be stabilized under a jumped network topology ^[174,175]. Instead of random switching, a distance-based dynamic topology is defined among the group^[176], in which merely the vehicles within the communication range can create the connection.

Authors' contributions

Made substantial contributions to the research and investigation process, reviewed and summarized the literature, wrote and edited the original draft: Yan T, Xu Z

Performed oversight and leadership responsibility for the research activity planning and execution as well as developed ideas and provided critical review, commentary and revision: Yang SX, Gadsden SA

Availability of data and materials

Not applicable.

Financial support and sponsorship

This work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

Conflicts of interest

All authors declared that there are no conflicts of interest.

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Copyright

© The Author(s) 2023.