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Formation Control for Underactuated Autonomous Underwater Vehicles Using the Approach Angle
Formation Control for Underactuated Autonomous Underwater Vehicles Using the Approach Angle
International Journal of Fuzzy Logic and Intelligent Systems. 2013. Sep, 13(3): 154-163
Copyright ©2013, Korean Institute of Intelligent Systems
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/ by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • Received : July 26, 2013
  • Accepted : September 14, 2013
  • Published : September 25, 2013
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About the Authors
Kyoung Joo Kim
Department of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea
Jin Bae Park
Department of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea
Yoon Ho Choi
Department of Electronic Engineering, Kyonggi University, Suwon, Korea

Abstract
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 leader-follower strategy is used for the formation control. The proposed controller considers the nonzero off-diagonal terms of the mass matrix of the AUV model and the associated parametric uncertainties. Using the state transformation, the mass matrix, which has nonzero off-diagonal terms, is transformed into a diagonal matrix to simplify designing the control. To deal with the parametric uncertainties of the AUV model, a self-recurrent 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.
Keywords
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 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.
- 2. Preliminaries
- 2.1 Asymmetric Structured AUV Dynamics
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] :
Lager Image
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 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:
Lager Image
where
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  • 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;
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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 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 off-diagonal 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] :
Lager Image
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
Lager Image
where
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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
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and
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of φ u , φ v , and φ r comprise the SRWNN.
- 2.3 Self-Recurrent 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 self-recurrent wavelets and parameters such that
Lager Image
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 ) =
Lager Image
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:
Lager Image
According to the powerful approximation ability [18] , the SRWNN
Lager Image
can approximate the uncertainty term φ j to a sufficient degree of accuracy as follows:
Lager Image
where j = u, v, r and X j ∈ κ Xj ⊂ ℝ 10 is the input vector of the SRWNN
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= diag(W j ), are the optimal and estimated matrices of the weighting vector of the SRWNN defined in (6), respectively, and
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is the bounded reconstruction error. The optimal parameter vector
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is given as
Lager Image
where α= 3N i N w +N i +N w .
Assumption 1. The optimal weight matrix is bounded such that
Lager Image
≤ │W Mj , where ∥ㆍ∥ F denotes the Frobenius norm.
Taking the Taylor expansion of
Lager Image
we can obtain [19] .
Lager Image
where
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denotes the high-order terms, and
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is the estimation error. Substituting (8) into (7), we have [20]
Lager Image
Lager Image
where
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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:
Lager Image
Using (11), the follower’s dynamics (11) can be rewritten as
Lager Image
where
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According to Remark 1, the hydrodynamic parameters φ 1 , φ 2 , and φ 3 are estimated by the SRWNN, respectively, as the estimated parameters
Lager Image
Further, to design the formation controller for underactuated AUVs, we obtain the position of the virtual reference vehicle of the followers as follows:
Lager Image
where η r = [x r , y r , Ψ r ] T is the reference position and yaw angle of the follower;
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is the transformed position and yaw angle of the leader;
Lager Image
and
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are the transformed positions of the x- and y-coordinates of the leader in the body-fixed frame; Ψ 1 is the yaw of the leader;
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is the desired distance between the leader and the followers; and
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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
Lager Image
where
Lager Image
The reference vehicle’s dynamics (15) can be rewritten as
Lager Image
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 body-fixed frame as
Lager Image
where
Lager Image
and y be are the x- and y-coordinates of the position error in the body-fixed frame, y be
Lager Image
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 y-axes and Ψ e is the yaw angle error. Here, the approach angle Ψ a is defined as follows [15] :
Lager Image
Using (16), we can obtain the following error dynamics in the body-fixed frame:
Lager Image
We select the virtual controls
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of the follower’s surge velocity u f and the yaw velocity r f as follows:
Lager Image
where
Lager Image
and k 1 and k 2 are the control gains to be chosen in the stability analysis.
Step 2: Define the error surface as
Lager Image
where the filtered signals u v and r v are obtained by passing the virtual controls
Lager Image
through the first-order filter as follows:
Lager Image
Here, k 1 and k 2 are positive constants. The time derivatives of s 1 and s 2 are then obtained as follows:
Lager Image
Lager Image
Leader-followermodel of autonomous underwater vehicles.
Lager Image
We choose the actual controls τ u and τ r as follows:
Lager Image
where k 3 and k 4 are the control gains to be chosen in the stability analysis,
Lager Image
are, respectively, estimates of the unknown parameters φ 1 and φ 2 , and are updated by
Lager Image
Lager Image
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
Lager Image
Then, their time derivatives are
Lager Image
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:
Lager Image
where
Lager Image
are positive definite matrices, and
Lager Image
1, 2 are the estimation errors. The time derivative of (28) along with (18), (22), (26), and (27) yields
Lager Image
Substituting (19), (23), (24), and (25) into (29) yields
Lager Image
From Assumption 1, (30) can be written as
Lager Image
Consider a set A :=
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Since the set A is compact in ℝ 7 , there exist positive constants p i such
that
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≤ p i , i = 1, 2. Using Young’s inequality, we have
Lager Image
where ϵ i , i = 1, 2, are positive constants. If we choose
Lager Image
where
Lager Image
= 1, 2, are positive constants, then we have
Lager Image
where
Lager Image
The constant ξ 1 satisfies
Lager Image
Multiplying (32) by e ξ1t yields
Lager Image
Integrating (33) over [0, t] leads to
Lager Image
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, 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 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 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.
Parameters of the asymmetric AUVAUV, autonomous underwater vehicle; COG, center of gravity.
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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 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.
Lager Image
Control result for formation control (bold line, the trajectory of the leader; dashed line, the trajectory of the first follower; dash-dot line, the trajectory of the second follower).
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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; dash-dot line, the control inputs of the second follower).
- Conflict of Interest
No conflict of interest relevant to this article was reported.
Lager Image
Formation errors: (a) separation errors, (b) bearing errors.
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