In this paper, we propose a formation control algorithm for underactuated autonomous underwater vehicles (AUVs) with parametric uncertainties using the approach angle. The approach angle is used to solve the underactuated problem for AUVs, and the leaderfollower strategy is used for the formation control. The proposed controller considers the nonzero offdiagonal terms of the mass matrix of the AUV model and the associated parametric uncertainties. Using the state transformation, the mass matrix, which has nonzero offdiagonal terms, is transformed into a diagonal matrix to simplify designing the control. To deal with the parametric uncertainties of the AUV model, a selfrecurrent wavelet neural network is used. The proposed formation controller is designed based on the dynamic surface control technique. Some simulation results are presented to demonstrate the performance of the proposed control method.
1. Introduction
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 shallowwater 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 behaviorbased strategy
[6]
, the virtual structure strategy
[7

8]
, and the leaderfollower 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 leaderfollower method, the leader tracks a predefined trajectory and the follower maintains a desired separationbearing/separationseparation configuration with the leader. The follower can be designated as a leader for other vehicles because of the scalability of formation. In addition, the leaderfollower 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 leaderfollower 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 offdiagonal 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 leaderfollower 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 offdiagonal terms in the system matrix, and we employ selfrecurrent 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 bodyfixed 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.
 2. Preliminaries
 2.1 Asymmetric Structured AUV Dynamics
The asymmetric structured AUV has nonzero offdiagonal 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 earthfixed frame,
v
= [u v r]
^{T}
is a vector denoting the surge, sway, and yaw velocities of the AUV in the bodyfixed 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
_{ṙ}
, and N
_{ṙ}
are the added masses; x
_{g}
is the xcoordinate of the center of gravity (COG) of the AUV in the bodyfixed frame; and I
_{z}
is the inertia with respect to the vertical axis.
Here the matrixMincludes the offdiagonal term m
_{23}
, 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 m
_{23}
. 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 offdiagonal terms and model uncertainties.
 2.2 State Transformation
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
∈
= m
_{23}
=m
_{22}
. The transformed equation (2) indicates that the virtual center of mass is positioned by ϵ 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.
 2.3 SelfRecurrent Wavelet Neural Network
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 selfrecurrent wavelets and parameters such that
where the subscript
n
k 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) + Փ
_{nk}
(N – 1) × ϑ
_{nk}
– ϱ
_{nk}
)/
μ
_{nk}
. Here, ϱ
_{nk}
,
μ
_{nk}
, and ϑ
_{nk}
are the translation factor, dilation factor, and weight of the selffeedback loop, respectively. The memory term Փ
_{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 Փ
_{nk}
(g
_{nk}
) =
which has the universal approximation property
[18]
. In this paper, the five weights a
_{k}
, ϱ
_{nk}
,
μ
_{nk}
, ϑ
_{nk}
, and
ω
_{n}
of the SRWNN will be trained online by the adaptation laws based on the Lyapunov theory.
The weighting vector W ∈ ℝ
^{3NiNw+Ni+Nw}
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}
⊂ ℝ
^{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
_{i}
N
_{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 highorder terms, and
is the estimation error. Substituting (8) into (7), we have
[20]
where
and ϵ
_{1j}
>0.
3. Main Results
 3.1 Controller Design
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
φ
_{1}
,
φ
_{2}
, and
φ
_{3}
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 ycoordinates of the leader in the bodyfixed frame;
Ψ
_{1}
is the yaw of the leader;
is the desired distance between the leader and the followers; and
is the desired angle between the xcoordinate of the leader in the bodyfixed 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
_{y}
r
_{l}
and v
_{r}
=v
_{l}
+d
_{x}
r
_{l}
.
Step 1: Define the errors in the bodyfixed frame as
where
and y
_{be}
are the x and ycoordinates of the position error in the bodyfixed 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
_{e}
are the position errors for the x and yaxes 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 bodyfixed frame:
We select the virtual controls
of the follower’s surge velocity u
_{f}
and the yaw velocity r
_{f}
as follows:
where
and k
_{1}
and k
_{2}
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 firstorder filter as follows:
Here, k
_{1}
and k
_{2}
are positive constants. The time derivatives of s
_{1}
and s
_{2}
are then obtained as follows:
Leaderfollowermodel of autonomous underwater vehicles.
We choose the actual controls τ
_{u}
and τ
_{r}
as follows:
where k
_{3}
and k
_{4}
are the control gains to be chosen in the stability analysis,
are, respectively, estimates of the unknown parameters
φ
_{1}
and
φ
_{2}
, and are updated by
where Ґ
_{1}
and Ґ
_{2}
are positive definite matrices.
 3.2 Stability Analysis
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
φ
_{1}
and
φ
_{2}
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 ℝ
^{7}
, there exist positive constants p
_{i}
such
that
≤ p
_{i}
, i = 1, 2. Using Young’s inequality, we have
where ϵ
_{i}
, i = 1, 2, are positive constants. If we choose
where
= 1, 2, are positive constants, then we have
where
The constant
ξ
_{1}
satisfies
Multiplying (32) by e
^{ξ1t}
yields
Integrating (33) over [0, t] leads to
Since
δ
_{1}
is bounded, it follows that all error signals are uniformly ultimately bounded.
4. Simulation Results
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, yaxis, 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 K
_{1}
= 0.6, K
_{2}
= 1, K
_{3}
= 2, K
_{4}
= 3, k
_{1}
=1, and k
_{2}
= 1. The parameters of the SRWNN are N
_{i}
= 3, N
_{w}
= 10, and Ґ
_{1}
= Ґ
_{2}
= diag[0.1],
σ
_{1}
=
σ
_{2}
= 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 dashdot 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.
Parameters of the asymmetric AUVAUV, autonomous underwater vehicle; COG, center of gravity.
Parameters of the asymmetric AUV AUV, autonomous underwater vehicle; COG, center of gravity.
5. Conclusion
We proposed a formation control algorithm for an underactuated AUV with parametric uncertainties. First, using the approach angle and formation error dynamics in the bodyfixed 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 offdiagonal 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.
Control result for formation control (bold line, the trajectory of the leader; dashed line, the trajectory of the first follower; dashdot line, the trajectory of the second follower).
Control inputs for formation control: (a) surge force, (b) yawmoments, (c) surge velocities, (d) yaw velocities (dashed line, the control inputs of the first follower; dashdot line, the control inputs of the second follower).
 Conflict of Interest
No conflict of interest relevant to this article was reported.
Formation errors: (a) separation errors, (b) bearing errors.
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