The graph theory is a powerful tool to deal with the consensus problem of multi-agent systems (Pratap et al., 2020; Zhang and Han, 2020). It is very effective to use graphs to represent the communication topology of information exchange among AUVs. Let us assume that every node in the graph corresponds to an AUV in our group, and the edges in the graph represent the information connection among AUVs. Therefore, the multi-AUV system can be referred to as a graph. The basic theory of graphs can be found in Ţurlea et al. (2019), which is omitted for simplicity.

A graph can be represented by an adjacency matrix A(G). This matrix is always square and has zero on its diagonal unless it is a loop. A(G) can be used to characterize the information interaction topology of the AUV group. The element’s value in A(G) is described as follows: a k n = 1 v n , v k ∈ E G 0 , o t h e r s , (1)

where G is an undirected graph, and A(G) is a symmetric matrix with all zeros on the main diagonal. Generally, the weighted adjacency matrix A _w (G) is defined as follows: A w G = a k n = w k n v n , v k ∈ E G 0 , o t h e r s . (2)

The Laplacian matrix L(G) is another matrix that describes the relationship between nodes and edges in graph. The elements of L(G) are given by the following expression: l k n = − a k n , k ≠ n , l k k = ∑ n = 1 N a k n , k = n . (3)

The adjacency matrix and Laplacian matrix have the following remarkable properties:

Lemma 1. Given a directed graph G and its adjacence matrix A(G), if A(G) is irreducible, then G is a strongly connected graph.

Lemma 2. The rank of a strongly connected directed graph G with N nodes is rank(L(G)) = N − 1.

Lemma 3. A symmetric graph G is connected if and only if rank(L(G)) = N − 1.

Lemma 4. L(G) is positive semi-definite.

Lemma 5. If zero is the eigenvalue of L(G), the graph is connected. 1 _N ∈ R ^N is its corresponding eigenvector, where 1 _N = [1 … 1] ^T .

Lemma 6. The eigenvalues of the Laplacian matrix are always non-negative. Moreover, they can always be ordered as follows: 0 =   λ 1 L G < λ 2 L G < … , < λ n L G .

Usually, the AUV dynamic model is divided into two parts: kinematics that only examines the geometric dimension of motion and kinetics that analyzes the forces that generate motion (Zhang et al., 2015; Cole et al., 2020; Franchi et al., 2020). Figure 1 presents an example of the AUV model, which includes dynamic variables in the body-fixed coordinate frame and its position relative to the inertial coordinate frame (Ma et al., 2020; Gao et al., 2021). Table 1 indicates different model variables defined as AUV’s motion behaviors in accordance with the Society of Naval Architects and Marine Engineers (SNAME) (Duan et al., 2020).

The relationship between velocity and acceleration is mainly considered by the AUV dynamic model. The shape of AUV is symmetrical from left to right and approximately symmetrical from top to bottom. The six-degree-of-freedom motion model of AUV can be expressed as follows (Sun et al., 2020, 2021): η ̇ =   J η ν M ν ̇ + C ν ν + D ν ν + g η = τ , (4) where η ∈ R 6 is the spatial position and posture of AUV in the fixed coordinate system, ν ∈ R 6 is the linear velocity and angular velocity of the AUV in the motion coordinate system, J(η) is a rotation transformation matrix from the motion coordinate system to the fixed coordinate system, M is the inertia matrix of the system (including additional mass), C(ν) is the Coriolis force matrix (including additional mass), D(ν) is the damping matrix, g(η) is the gravity/buoyancy and moment vector, and τ is the thrust and moment vector. The specific meanings of the vectors and matrices mentioned previously are as follows: η = η 1 , η 2 T η 1 = x , y , z η 2 = φ , θ , ψ , (5) v = v 1 , v 2 T v 1 = u , v , w , v 2 = p , q , r , (6) τ = τ 1 , τ 2 T τ 1 = X , Y , Z , τ 2 = K , M , N , (7) η ̇ 1 η ̇ 2 = J 1 η 2 0 0 J 2 η 2 v 1 v 2 , (8) J 1 = c ⁡ cos ⁡ ψ ⁡ cos ⁡ θ cos ⁡ ψ ⁡ sin ⁡ θ ⁡ sin ⁡ φ − sin ⁡ ψ ⁡ cos ⁡ φ cos ⁡ ψ ⁡ sin ⁡ θ ⁡ cos ⁡ φ + sin ⁡ ψ ⁡ cos ⁡ φ sin ⁡ ψ ⁡ cos ⁡ θ sin ⁡ ψ ⁡ sin ⁡ θ ⁡ sin ⁡ φ + cos ⁡ ψ ⁡ cos ⁡ φ sin ⁡ ψ ⁡ sin ⁡ θ ⁡ cos ⁡ φ − cos ⁡ ψ ⁡ sin ⁡ φ − sin ⁡ θ cos ⁡ θ ⁡ sin ⁡ φ cos ⁡ θ ⁡ cos ⁡ φ J 2 = c 1 sin ⁡ φ t θ cos ⁡ φ ⁡ tan ⁡ θ 0 cos ⁡ φ − sin ⁡ φ 0 sin ⁡ φ cos ⁡ θ cos ⁡ φ cos ⁡ θ , θ ≠ ± π 2 . (9) The six-degree-of-freedom motion equation of AUV can be expressed as follows: X = m u ̇ − v r + w q − x G q 2 + r 2 + y G p q + r ̇ + z G p r + q ̇ , Y = m v ̇ − w p + u r + x G p q + r ̇ − y G r 2 + p 2 + z G q r + p ̇ , Z = m w ̇ − u q + v p + x G r p + q ̇ + y G r q + p ̇ − z G p 2 + q 2 , K = I x x p ̇ + I z z − I y y q r + m y G w ̇ − u q + v p − z G v ̇ − w p + u r , M = I y y q ̇ + I x x − I z z r p + m z G u ̇ − v r + w q − x G w ̇ − u q + v p , N = I z z r ̇ + I y y − I z z p q + m x G v ̇ − w p + u r − y G u ̇ − v r + w p . (10)

At present, only one leader is defined in a group in most cases. The consistency process of the system is controlled by controlling the information interaction between the leader and other AUVs. In this study, multiple leaders and followers in the topological structure are adopted. The AUV group is divided into multiple small groups to form a swarm network. Each group that includes multiple AUVs has its own leader AUV. Moreover, there is one and only one AUV in the first group as virtual AUV, which is named virtual leader. The dynamic and kinematic characteristics of the virtual leader and the real AUV are the same. The desired speed and desired position of the follower are given as control inputs of the virtual leader. In other words, a mixed-mode topology jointly is designed to solve the consistency problem of the AUV swarm system. The schematic diagram of the topology is shown in Figure 2.

The network topology of AUV clusters is described by a directed graph G = (V, E). The set of vertices V = v 1 , v 2 ⋅ ⋅ ⋅ , v n represents the AUV clusters. The set of edges E ⊂ V × V represents their interaction relationships. All AUVs in the system are divided into m + 1 clusters, which are denoted by β ₀, β ₁, … β _m . There is a special AUV in each cluster called the pilot vessel, denoted by l i ∈ β i , ∀ i ∈ 0,1 , … , m . The other AUVs in the cluster are denoted by u _j . Among them, there is only one AUV in cluster β ₀, and the node is the virtual leader specifically denoted as β i = l i , u i , 1 , … u i , n , ∀ i ∈ 1 , … , m .

The set of pilot boats is denoted by L = l 0 , l 1 , … l m . In the whole communication topology, only l ₀ has information transfer with the pilot boats of other clusters, and the pilot boats complete the information exchange between groups. l 1 , … , l m not only has inter-group information transfer and interaction with other pilot boats but each pilot boat also has intra-group information transfer and interaction with other nodes in the cluster to which it belongs. When the pilot boat communicates and interacts with the nodes in the cluster, the locally consistent equilibrium point of the cluster is periodically reset.

It is important to note that data transmission between AUVs is best done digitally so that there is no interference due to communication. Some data will be lost or miscoded during transmission, but no new interference will be introduced as in the case of analog transmission. In addition, each machine should add a GPS clock when sending data. When other AUVs receive the data, they unpack the data and decode the GPS clock when the data are sent, and compare it with the current GPS clock to know the size of the transmission delay. After avoiding the generation of transmission interference and accurately knowing the communication delay, the formation control accuracy and the control effect can be improved.

In the actual engineering environment, the communication between AUVs underwater relies on hydroacoustic communication, the unreliability of communication channels, packet loss of communication data, failure of sensors and sonar equipment, the existence of communication barriers, and other factors, thus leading to intermittent communication interactions between AUVs, and there is often a certain time delay when AUVs receive position and velocity information. Therefore, considering that the information transmission between AUV individuals through sensors or communication devices will inevitably generate a time delay problem, the time delay in the process of information transmission from individual i to individual j is denoted by τ _ij . If the time delay of the AUV’s own state is the same size as the communication time delay of the received information from its neighbors, a symmetric consistency algorithm is usually used as u i = − ∑ j ∈ N a i j t x i t − τ i j − x j t − τ i j .

The control process of AUV swarm often needs to end in a finite time when it completes its assigned mission task, and the existence of communication time delay τ _ij affects the stability of the system and the convergence time of the system. Therefore, in order to ensure that the AUV swarm achieves asymptotic stability in finite time and realize the practical application of the system, it is necessary to design the upper limit of time delay τ _ij ≤ τ ₀ in order to study the consistency of the AUV swarm under the condition of time-varying communication time delay with fixed upper limit.

Furthermore, the dynamic of the tracking signal is described as follows:

In general, the motion of the AUV in water can be regarded as the spatial motion of a rigid body in a fluid. When the heading of the AUV in motion is constant and only the depth is changed, the center of gravity of the AUV is always kept in the same plumb plane within its changing heading and not changing depth. The motion model in 2.3.1 is adopted to ignore the motion of the cross-rolling surface, and the spatial motion in the water is approximated and decomposed into a horizontal plane motion and a vertical plane motion. Usually, the plane motion can reflect the basic characteristics of motion–depth and heading control, and the hydrodynamic characteristics of space motion are also developed on the basis of plane motion.

Suppose the AUV group is finally kept at the same depth, all the AUVs follow the leader AUV to keep a fixed formation, and only horizontal motion is considered. At this time, the position information of the first AUV is η i = x i , y i , z i , and the velocity information is v _i . Let the final horizontal surface speed of the AUV group be v ₀, the angular velocity of rotary motion be r ₀, and the vertical velocity be v z 0 . Then, according to the AUV motion model established previously, the i AUV motion model can be simplified as follows: x ̇ i = v i ⁡ cos ψ i y ̇ i = v i ⁡ sin ψ i ψ ̇ i = r i v ̇ i = 1 λ v i v 0 − v i r ̇ i = 1 λ r i r 0 − r i z ̇ i = v z i v ̇ z i = 1 λ z i v z 0 − v z i , (11) where λ is a parameter greater than zero. Let x f i , y f i denote the position of the point f _i on the i AUV at a distance d _i from the center of gravity x i , y i . Then, the simplified control algorithm using x f i , y f i instead of x i , y i in the coordinated control of the AUV is expressed as follows: x f i y f i = x i y i + d i ⁡ cos ψ i d i ⁡ sin ψ i , (12) where v 0 , r 0 , v z 0 satisfies v 0 r 0 = v i r i + λ v i λ r i cos ψ i − d i ⁡ sin ψ i sin ψ i d i ⁡ cos ψ i − 1 u x i + v i r i ⁡ sin ψ i + d i r i 2 ⁡ cos ψ i u y i − v i r i ⁡ cos ψ i + d i r i 2 ⁡ sin ψ i , (13) v z 0 = v z i + λ z i u z i . (14) We can get x ̇ f i = v x i y ̇ f i = v y i z ̇ i = v z i v ̇ x i = u x i v ̇ y i = u y i v ̇ z i = u z i . (15) Therefore, coordinated consistency control of the AUV cluster can be achieved by designing the control inputs u x i , u y i and u z i , which means the tracking signal of the leader is u x i , u y i , and u z i . It is assumed that each AUV has a unique number and that each AUV knows its own, as well as the numbers in its neighborhood. In addition, all interactions among AUVs are synchronized, that is, all update their state parameters at the same time.

Since there is a hydroacoustic communication delay among AUVs in the underwater environment, the consistency control algorithm for the AUV under the time delay condition is considered later. Similarly, assuming that the AUV group maintains a fixed depth motion when there is a time delay in the inter-AUV communication, the controller is designed as follows: u x i = f ̇ x t − γ ( v x i t − τ i j t − f x t − τ i j t − ∑ j = 1 m a i j ( x f i t − x f j t − τ i j t − δ i x − δ j x − ∑ j = 1 m a i j γ v x i t − τ i j t − v x j t − τ i j t , (16) u y i = f ̇ y t − γ ( v y i t − τ i j t − f y t − τ i j t − ∑ j = 1 m a i j y f i t − y f j t − τ i j t − δ i y − δ j y − ∑ j = 1 m a i j γ ( v y i t − τ i j t − v y j t − τ i j t . (17) where control gain γ > 0, τ _ij (t) denotes communication time delay between AUVi and AUVj, f ^x (t) and f ^y (t) are continuous differentiable functions that denote the velocity characteristics of the AUV motion, and δ i x , δ i y denote the desired position. When the upper limit of time delay is τ ₀, 0 < τ _ij (t) < τ ₀. Suppose that the communication between any two different AUVs allows a common upper limit of time delay τ ₀, that is, 0 < τ _ij ≤ τ ₀.

Theorem 1. For AUV group with a directionless connected network topology, the control inputs are designed to be Eq. 16 and Eq. 17, respectively, and the communication time delay between its individuals τ _ij (t) satisfies 0 ≤ τ i j t ≤ 1 ω 0 arctan 1 + λ i ω 0 γ λ i , (18) where ω 0 = ( 1 + λ i ) 2 γ 2 ± ( 1 + λ i ) 4 γ 4 + 4 λ i 2 2 , and λ _i is the characteristic root of the Laplacian matrix L of the graph G .

Proof. For control input u x i , suppose x ̃ f i = x f i − ∫ 0 1 f x s d t − δ i x v ̃ x i = v x i − f x t . (19) According to Eq. 16, there is x ̃ ̇ f i t = v ̃ x i t v ̃ ̇ x i t = − γ v ̃ x i t − τ i j t − ∑ j = 1 m a i j x ̃ f i t − x ̃ f j t − τ i j t − ∑ j = 1 m a i j γ v ̃ x i t − τ i j t − v ̃ x j t − τ i j t . (20)

Assuming that all AUVs have the same upper limit of delay τ ₀, we have x ̃ ̇ f t v ̃ ̇ x t = 0 I n 0 0 x ̃ f t v ̃ x t + 0 0 − L − γ I n + L x ̃ f t − τ 0 v ̃ x t − τ 0 , (21) where x ̃ f ( t ) = x ̃ f 1 ( t ) , … , x ̃ f n ( t ) T , v ̃ x ( t ) = v ̃ x 1 ( t ) , … , v ̃ x n ( t ) T . Suppose x ( t ) = x ̃ f ( t ) , v ̃ x ( t ) T , we can get x ̇ t = 0 I n 0 0 x t + 0 0 − L − γ I m + L x t − τ 0 . (22)

A Laplace variation of Eq. 22 is s x s = 0 I m 0 0 x s + 0 0 − L − γ I m + L e − τ 0 s x s . (23) The characteristic equation satisfies det s I 2 n − 0 I n 0 0 − e − τ 0 s 0 0 − L − γ I n + L = 0 , (24) det s I n − I n e − τ 0 s L s I n + e − τ 0 s γ I n + L = 0 , (25) det s 2 I m + e − τ 0 s γ I m + L s + e − τ 0 s L = 0 . (26) It follows from Section 2.1 that for an undirected connected graph G, the rank of L is rank(L) = m − 1; the eigenvalue of L is 0 = λ ₁ < λ ₂ ≤ …, ≤ λ _m = λ _max, then s ∏ i = 2 m s 2 + e − τ 0 s γ 1 + λ i s + λ i = 0 . (27)

When i = 2, … , m, Eq. 27 can be organized as 1 + e − τ 0 s 1 s γ 1 + λ i + 1 s 2 λ i = 0 . (28) Suppose G i ( s ) = e − τ 0 s [ 1 s γ ( 1 + λ i ) + 1 s 2 λ i ] , the number of unstable poles p = 0 is consistent with the minimum phase system characteristics. Based on the Nyquist stability criterion (Chou et al., 2020; Wang et al., 2021), the roots of Eq. 28 are in the left half-open plane of the s-plane when the Nyquist curve G _i (s) does not enclose the critical point (−1, j0). Then, the system reaches asymptotic consistency. Consequently, when Eq. 18 is satisfied, the roots of Eq. 28 are in the left half-open plane of the s-plane, and the system can reach agreement. At this point, x f i − x f j → δ i x − δ i x , v x i → v x j → f x ( t ) . Similarly, when control inputs u y i satisfy Eq. 18, y f i − y f j → δ i y − δ i y , v y i → v y j → f y ( t ) .

Similarly, time delay exists for heterogeneous AUV groups in a natural environment for underwater communication. The following will investigate the consistency control algorithm for the AUV group with both heterogeneity and time delay conditions. Suppose that the state of each AUV in the group is measurable.

In Zheng et al. (2011), the heterogeneous multi-agents are studied. Considering heterogeneous AUV group with N linear dynamics, the dynamic model of AUVi can be given by Eq. 29. x ̇ i t = − C x i t + A f i x i t + B f i x i t − τ i t + u i t , (29) where i = 1, 2, … , N denotes follower AUV; x _i (t) ∈ R ⁿ and u _i (t) ∈ R ⁿ are the state vector and control input vector of the system, respectively; f _i (x _i (t)) ∈ R ⁿ is the unknown continuous non-linear function of the system; and τ _i (t) > 0 denotes the underwater communication delay. C = diag c 1 , c 2 , … , c n ∈ R n × n , c _i > 0, A ∈ R ^n×n , and B ∈ R ^n×n are the known system matrices. Each AUV has a different non-linear function f _i (x _i (t)) and time delay τ _i (t). Different types of AUVs are included by Eq. 29 through selecting different system matrices A, B, C.

Suppose global state vector x ( t ) = x 1 T ( t ) , … , x N T ( t ) T ∈ R n N , Eq. 29 can be written as Eq. (30). x ̇ t = − I N ⊗ C x t + I N ⊗ A f x t + I N ⊗ B f x t − τ t + u t , (30) where f x t = f 1 T x 1 t , … , f N T x N t T ∈ R n N , (31) f x t − τ t = f 1 T x 1 t − τ 1 t , … , f N T x N t − τ N t T ∈ R n N , (32) u t = u 1 T t , … , u N T t T ∈ R n N . (33) The dynamic system of the leader (AUV0) is as follows: x ̇ 0 t = − C x 0 t + A f 0 x 0 t + B f 0 x 0 t − τ 0 t , (34) where x ₀(t), f ₀(x ₀(t)), and τ ₀(t) are the leader’s state vector, non-linear function vector, and time-varying system delay, respectively. Similarly, the leader node 0 can be used as an external reference system or command controller to generate the required dynamic trajectory. Suppose that the leader node could not be affected by n follower nodes.

Suppose that only information on neighboring nodes is available, local consistency error for AUVi is shown as follows: e i t = ∑ j = 1 N a i j x j t − x i t + g i x 0 t − x i t . (35) If AUVi can get the state information of AUVj, the connection weight a _ij > 0 and the traction gain g _i > 0. Otherwise, a _ij = 0 and g _i = 0. Combined with the consistency control strategy, the following control law is designed to achieve the control objective of the AUV group. u i t = c K e i t − A φ i x i t − B t − τ i m φ i x t − τ i m , (36) where c > 0, K ∈ R ^n×n are the pending control gain and feedback gain matrices, respectively, and τ _im denotes the upper bound of the unknown time delay τ _i (t). Consistent stability proofs refer to Section 3.1.

In this section, a simulation experiment is conducted for the consistency control algorithm with communication delay proposed in Section 3.1. The difference is that the communication time delay is set to τ = 0.5.

In Figure 5, it can be seen that the AUV group has a lag in the state of the follower in the condition of the presence of communication delay. The followers cannot keep a horizontal line with the leader, which means that the parallel formation cannot be maintained. However, the AUV group can still maintain a steady-state moving forward. In Figure 6, it can be seen that under the condition of communication time delay, the velocity of each follower AUV in x, y, and z directions is jittered and then converges rapidly. Simultaneously, the acceleration of each follower finally converges to zero.

A simulation experiment is conducted for the consistency control algorithm in Section 3.2. Different types of AUVs are included in Eq. 29 by selecting different system matrices C. Figure 7 and Figure 8 show the formation process and the state of formation keeping navigation of the heterogeneous AUV group through the horizontal plane and three-dimensional views, respectively. Compared with Section 3.1, the specified formation is also formed at around t = 20s, which indicates that the formation process remains constant when the control gain remains the same. Moreover, the latency of the simulation is much smaller than that of Section 4.1. Therefore, the effectiveness of the control algorithm of the heterogeneous AUV group under communication delay is proved.

The distributed controller designed in this study can be applied to the formation control of heterogeneous AUV groups. Under the condition of time delay existence, the control protocol does not need to know the specific form of complex non-linear functions in each AUV motion model and also does not need to know the unknown time-varying time delay existing in the system; it needs to know only the neighbor information of local AUVs within the heterogeneous AUV network. In addition, the corresponding control gain can be derived off-line for a determined heterogeneous AUV group. Therefore, the proposed control scheme can be easily implemented in terms of design and implementation. The simulation results show that under the conditions of unknown time-varying time delay, variable network topology, and intermittent communication, all the followers can still converge to the leader’s trajectory quickly and exhibit their heterogeneous characteristics as well as the leader.