Over the last two decades, the research and development of autonomous underwater vehicles (AUVs) and surface vessels have been important issues because of their usefulness in performing missions such as environmental surveying, undersea cable/pipeline inspection, monitoring of coastal shallow-water regions, and offshore oil installations [1-5].

In addition, since performing offshore missions is more efficiently accomplished by using multiple AUVs rather than a single AUV, a large number of studies have been published concerning the formation control of multiple AUVs. There are three popular strategies in designing the formation controller: the behavior-based strategy [6], the virtual structure strategy [7-8], and the leader-follower strategy [9-11]. Among these methods, the leaderfollower method is most widely used by many researchers due to advantages such as simplicity and scalability. In the leader-follower method, the leader tracks a predefined trajectory and the follower maintains a desired separation-bearing/separation-separation configuration with the leader. The follower can be designated as a leader for other vehicles because of the scalability of formation. In addition, the leader-follower method is simple to implement since the reference trajectory of the follower is clearly defined by the leader’s movement, and the internal formation stability is induced by the control laws of the individual vehicles. A leader-follower formation controller for underactuated AUVs was proposed in [11], in which the controller needs an ”exogenous” system and this exogenous system must know each vehicle position. Skejetne proposed a formation controller such that each individual vehicle has a position relative to a point called the formation reference point [12]. However, these papers did not consider the off-diagonal terms in the system matrix or the parametric uncertainties of the AUV.

In this paper, therefore, we propose a formation control algorithm for underactuated AUVs. First, we obtain the virtual leader in reference to the follower in the leader-follower strategy; the formation problem is then dealt with as a tracking problem. Second, in order to design the controller, we use the state transformation [13], which can avoid the difficulties caused by the off-diagonal terms in the system matrix, and we employ self-recurrent wavelet neural networks (SRWNNs) [14,15] to deal with the uncertainties in the hydrodynamic damping terms. Third, using the approach angle [16] and the formation error dynamics in the body-fixed frame, we solve the underactuated problem for AUVs. Fourth, the dynamic surface control (DSC) technique [17], which can solve the ”explosion of complexity” problem caused by the repeated differentiation of virtual controllers in the backstepping design procedure, is applied to design the formation controller for underactuated AUVs. Finally, we perform computer simulations to illustrate the performance of the proposed controller.

The asymmetric structured AUV has nonzero off-diagonal terms in the mass matrix, which are induced by the asymmetric shape of the bow and the stern of the vehicle. The kinematics and dynamics of the asymmetric AUV are described as follows [13]:

where η= [x y Ψ]^T denotes the position (x, y) and the yaw angle Ψ of the AUV in the earth-fixed frame, v= [u v r]^T is a vector denoting the surge, sway, and yaw velocities of the AUV in the body-fixed frame, respectively, and τ=[τ_u 0τ_r]^T is the control vector of the surge force τ_u and yaw moment τ_r. In the above equation, the matrices J (Ψ), D(v), C(v), and M are given as follows:

where

d22 (v, r) = ？(Yv + Yv│v│ + Yr│v│ │r│),

d23 (v, r) = ？(Yr + Yv│r│ + Yr│r│ │r│),

d32 (v, r) = ？(Nv + Nv│v│ + Nr│v│ │r│),

d33 (v, r) = ？(Nr + Nv│r│ │v│ + Nr│r│ │r│).

Here, X_u, X_u│u│, Y_v, Y_v│v│, Y_r│v│, Y_r, Y_v│r│, Y_r│r│, N_v, N_v│v│, N_r│v│, N_r, N_v│r│, and N_r│r│ are the linear and quadratic drag coefficients; m is the mass of the AUV;

Y_？, and N_？ are the added masses; x_g is the x-coordinate of the center of gravity (COG) of the AUV in the body-fixed frame; and I_z is the inertia with respect to the vertical axis.

Here the matrixMincludes the off-diagonal term m₂₃, which is present because the shape of bow is in general different from that of stern. Therefore, the sway dynamics is are affected by the yaw moment τ_r because of the parameter m₂₃. Furthermore, there are only two control inputs: the surge force τ_u and yaw moment τ_r. The number of control inputs is less than the number of degrees of freedom of the AUV in the horizontal plane. Moreover, the parameters of the AUV model (1) cannot be obtained exactly. Therefore, it is difficult to design the controller for the underactuated AUV, which has both off-diagonal terms and model uncertainties.

Since the yaw moment τ_r acts directly on the sway dynamics in (1), causing difficulty in designing the controller, we use the following state transformations [13]:

where ∈ = m₂₃=m₂₂. The transformed equation (2) indicates that the virtual center of mass is positioned by ？ in the longitudinal direction. Using (2), the AUV dynamics (1) can be rewritten as

where

Remark 1. Here, we assume that we do not know the exact values of φ_u, φ_v, and φ_r since the parameters of the system matrices cannot be obtained exactly by measurement or by calculation. Since the system matrices inevitably have uncertainties, we employ a SRWNN to compensate for the parametric uncertainties of φ_u, φ_v, and φ_r. The estimated parameters

and

of φ_u, φ_v, and φ_r comprise the SRWNN.

In this paper, we use a SRWNN to compensate for the parametric uncertainties of the AUV model. The SRWNN structure, which has N_i inputs, one output, and N_i × N_w mother wavelets, consists of four layers: an input layer, a mother wavelet layer, a product layer, and an output layer. The SRWNN output is composed of self-recurrent wavelets and parameters such that

where the subscript nk indicates the k-th input term of the n-th wavelet, N denotes the number of iterations, the output y is the estimate of the uncertainty, X_k denotes the k-th input of the SRWNN, a_k is the connection weight between the input nodes and the output node, ω_n is the connection weight between the product nodes and the output nodes, and g_nk(N) = (X_k(N) + ？_nk(N ？ 1) × ？_nk ？ ？_nk)/μ_nk. Here, ？_nk, μ_nk, and ？_nk are the translation factor, dilation factor, and weight of the selffeedback loop, respectively. The memory term ？_nk (N ？ 1) denotes the one step recurrent term of a wavelet. In addition, the mother wavelets are chosen as the first derivatives of a Gaussian function ？_nk (g_nk) =

which has the universal approximation property [18]. In this paper, the five weights a_k, ？_nk, μ_nk, ？_nk, and ω_n of the SRWNN will be trained online by the adaptation laws based on the Lyapunov theory.

The weighting vector W ∈ ？^{3N_iN_w+N_i+N_w} is defined as follows:

According to the powerful approximation ability [18], the SRWNN

can approximate the uncertainty term φ_j to a sufficient degree of accuracy as follows:

where j = u, v, r and X_j ∈ κ_Xj ⊂ ？¹⁰ is the input vector of the SRWNN

= diag(W_j), are the optimal and estimated matrices of the weighting vector of the SRWNN defined in (6), respectively, and

is the bounded reconstruction error. The optimal parameter vector

is given as

where α= 3N_iN_w+N_i+N_w.

Assumption 1. The optimal weight matrix is bounded such that

≤ │W_Mj, where ∥ㆍ∥_F denotes the Frobenius norm.

Taking the Taylor expansion of

we can obtain [19].

where

denotes the high-order terms, and

is the estimation error. Substituting (8) into (7), we have [20]

where

and ？_1j>0.

By the state transformation described in subsection 2.2, the new follower’s kinematics and dynamics are as follows:

Using (11), the follower’s dynamics (11) can be rewritten as

where

According to Remark 1, the hydrodynamic parameters φ₁, φ₂, and φ₃ are estimated by the SRWNN, respectively, as the estimated parameters

Further, to design the formation controller for underactuated AUVs, we obtain the position of the virtual reference vehicle of the followers as follows:

where η_r= [x_r, y_r, Ψ_r]^T is the reference position and yaw angle of the follower;

is the transformed position and yaw angle of the leader;

and

are the transformed positions of the x- and y-coordinates of the leader in the body-fixed frame; Ψ₁ is the yaw of the leader;

is the desired distance between the leader and the followers; and

is the desired angle between the x-coordinate of the leader in the body-fixed frame and the vector from the leader to the reference vehicle as shown in Figure 1. Using the form of (1), the dynamics of η_r can be described as

where

The reference vehicle’s dynamics (15) can be rewritten as

where u_r=u_l？d_yr_l and v_r=v_l+d_xr_l.

Step 1: Define the errors in the body-fixed frame as

where

and y_be are the x- and y-coordinates of the position error in the body-fixed frame, y_be

is the yaw angle error between the approach angle Ψ_a and the yaw angle Ψ_f of the follower, x_e and y_eare the position errors for the x- and y-axes and Ψ_e is the yaw angle error. Here, the approach angle Ψ_a is defined as follows [15]:

Using (16), we can obtain the following error dynamics in the body-fixed frame:

We select the virtual controls

of the follower’s surge velocity u_f and the yaw velocity r_f as follows:

where

and k₁ and k₂ are the control gains to be chosen in the stability analysis.

Step 2: Define the error surface as

where the filtered signals u_v and r_v are obtained by passing the virtual controls

through the first-order filter as follows:

Here, k₁ and k₂ are positive constants. The time derivatives of s₁ and s₂ are then obtained as follows:

We choose the actual controls τ_u and τ_r as follows:

where k₃ and k₄ are the control gains to be chosen in the stability analysis,

are, respectively, estimates of the unknown parameters φ₁ and φ₂, and are updated by

where ？₁ and ？₂ are positive definite matrices.

In this subsection, we show that all error signals of the closedloop control system are uniformly ultimately bounded. Define the boundary layer errors as

Then, their time derivatives are

Theorem 1. Consider the underactuated AUV (1) with parametric uncertainties controlled by (23). If the proposed control system satisfies Assumption 1, and unknown parameters φ₁ and φ₂ are trained by the adaptation laws (24) and (25), respectively, then for V (0) μ, where μ is a positive constant, all error signals are uniformly ultimately bounded.

Proof. We choose the Lyapunov function as follows:

where

are positive definite matrices, and

1, 2 are the estimation errors. The time derivative of (28) along with (18), (22), (26), and (27) yields

Substituting (19), (23), (24), and (25) into (29) yields

From Assumption 1, (30) can be written as

Consider a set A :=

Since the set A is compact in ？⁷, there exist positive constants p_i such

that

≤ p_i, i = 1, 2. Using Young’s inequality, we have

where ？_i, i = 1, 2, are positive constants. If we choose

where

= 1, 2, are positive constants, then we have

where

The constant ξ₁ satisfies

Multiplying (32) by e^ξ1t yields

Integrating (33) over [0, t] leads to

Since δ₁ is bounded, it follows that all error signals are uniformly ultimately bounded.

In this section, we report the results of some computer simulations that illustrate the performance of the proposed formation controller for underactuated AUVs with parametric uncertainties. The surge force and the yaw moment of the leader of the AUVs are chosen as follows:

0 ≤ t ≤ 50, τul= 30, τrl= 0,

50 < t ≤ 100, τul= 30, τrl= 3,

100 < t ≤150, τul= 30, τrl= ？3,

150 < t ≤ 200, τul= 30, τrl= 0.

For the simulations, we select the initial conditions of the leader and the followers of the AUV as follows:

(xl (0) , yl (0) , Ψl (0))=(0, 0, 0) ,

(xf1 (0) , yf1 (0) , Ψf1 (0))=(？1, 0, 0) ,

(xf2 (0) , yf2 (0) , Ψf2 (0))=(？1, 0, 0)

where the subscript l indicates a leader and the subscript f_i, i = 1, 2, indicates followers: x_l, y_l, and Ψ_l are the leader’s positions on the x -axis, y-axis, and the leader’s yaw angle, respectively. The desired formation parameters are as follows:

ldlf1= 2;; φdlf1= 3π/=4,

ldlf2= 2;; φdlf2= ？3π/=4.

The control gains are chosen as K₁ = 0.6, K₂ = 1, K₃ = 2, K₄ = 3, k₁=1, and k₂ = 1. The parameters of the SRWNN are N_i = 3, N_w = 10, and ？₁ = ？₂ = diag[0.1], σ₁ = σ₂ = 1. The initial values of the weights are randomly given in the range of [-1,1]. Various system parameters of a typical AUV for the simulations are given in Table 1.

Figure 2 shows the control result of formation control for the underactuated AUVs, where the trajectory of the leader (bold line) and the trajectories of the followers (dashed line and dash-dot line) are depicted. The proposed formation control algorithm requires the followers to maintain the desired distance and angle with respect to the leader. From the result of Figure 2, the followers move along the desired position with respect to the leader at the early stage of control. Figure 3 shows the actual surge velocity u, yaw velocity r, and control inputs, which are the yaw moment τ_r and surge force τ_u. The separation and bearing errors are shown in Figure 4. Each error stays in the neighborhood of zero at the early stage of control action.

We proposed a formation control algorithm for an underactuated AUV with parametric uncertainties. First, using the approach angle and formation error dynamics in the body-fixed frame, we solved the underactuated problem for AUVs. Second, the formation control algorithm was designed based on the leaderfollower strategy. Third, the state transformation was used to deal with off-diagonal terms resulting from the differences in shapes between the bow and stern. Next, the parametric uncertainties of the AUV were estimated by a SRWNN. Finally, the formation control algorithm was designed based on the DSC method. From the simulation results, we confirmed that the proposed control algorithm can maintain the predefined formation with good performance.

No conflict of interest relevant to this article was reported.