Formation Control for Underactuated Autonomous Underwater Vehicles Using the Approach Angle

  • cc icon
  • 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.


  • KEYWORD

    Formation control , Leader-follower strategy , Autonomous underwater vehicle , Underactuated systems , Dynamic surface control

  • 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]:

    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=[τur]T is the control vector of the surge force τu and yaw moment τr. In the above equation, the matrices J (Ψ), D(v), C(v), and M are given as follows:

    image

    where

    image

    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, Xu, Xu│u│, Yv, Yv│v│, Yr│v│, Yr, Yv│r│, Yr│r│, Nv, Nv│v│, Nr│v│, Nr, Nv│r│, and Nr│r│ are the linear and quadratic drag coefficients; m is the mass of the AUV;

    image

    Y, and N are the added masses; xg is the x-coordinate of the center of gravity (COG) of the AUV in the body-fixed frame; and Iz 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 τr because of the parameter m23. Furthermore, there are only two control inputs: the surge force τu and yaw moment τr. 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 τr acts directly on the sway dynamics in (1), causing difficulty in designing the controller, we use the following state transformations [13]:

    image

    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

    image

    where

    image

    Remark 1. Here, we assume that we do not know the exact values of φu, φv, and φr 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 φu, φv, and φr. The estimated parameters

    image

    and

    image

    of φu, φv, and φr 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 Ni inputs, one output, and Ni × Nw 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

    image

    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, Xk denotes the k-th input of the SRWNN, ak is the connection weight between the input nodes and the output node, ωn is the connection weight between the product nodes and the output nodes, and gnk(N) = (Xk(N) + ?nk(N ? 1) × ?nk ? ?nk)/μnk. Here, ?nk, μnk, 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 (gnk) =

    image

    which has the universal approximation property [18]. In this paper, the five weights ak, ?nk, μnk, ?nk, and ωn of the SRWNN will be trained online by the adaptation laws based on the Lyapunov theory.

    The weighting vector W ∈ ?3NiNw+Ni+Nw is defined as follows:

    image

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

    image

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

    image

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

    image

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

    image

    is the bounded reconstruction error. The optimal parameter vector

    image

    is given as

    image

    where α= 3NiNw+Ni+Nw.

    Assumption 1. The optimal weight matrix is bounded such that

    image

    ≤ │WMj, where ∥ㆍ∥F denotes the Frobenius norm.

    Taking the Taylor expansion of

    image

    we can obtain [19].

    image

    where

    image

    denotes the high-order terms, and

    image

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

    image
    image

    where

    image

    and ?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:

    image

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

    image

    where

    image

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

    image

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

    image

    where ηr= [xr, yr, Ψr]T is the reference position and yaw angle of the follower;

    image

    is the transformed position and yaw angle of the leader;

    image

    and

    image

    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;

    image

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

    image

    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 ηr can be described as

    image

    where

    image

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

    image

    where ur=ul?dyrl and vr=vl+dxrl.

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

    image

    where

    image

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

    image

    is the yaw angle error between the approach angle Ψa and the yaw angle Ψf of the follower, xe and yeare the position errors for the x- and y-axes and Ψe is the yaw angle error. Here, the approach angle Ψa is defined as follows [15]:

    image

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

    image

    We select the virtual controls

    image

    of the follower’s surge velocity uf and the yaw velocity rf as follows:

    image

    where

    image

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

    Step 2: Define the error surface as

    image

    where the filtered signals uv and rv are obtained by passing the virtual controls

    image

    through the first-order filter as follows:

    image

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

    image
    image

    We choose the actual controls τu and τr as follows:

    image

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

    image

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

    image
    image

    where ?1 and ?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

    image

    Then, their time derivatives are

    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 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:

    image

    where

    image

    are positive definite matrices, and

    image

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

    image

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

    image

    From Assumption 1, (30) can be written as

    image

    Consider a set A :=

    image

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

    that

    image

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

    image

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

    image

    where

    image

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

    image

    where

    image

    The constant ξ1 satisfies

    image

    Multiplying (32) by eξ1t yields

    image

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

    image

    Since δ1 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 fi, i = 1, 2, indicates followers: xl, yl, and Ψl 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 Ni = 3, Nw = 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 τr and surge force τu. 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.

    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.

      >  Conflict of Interest

    No conflict of interest relevant to this article was reported.

  • 1. Papoulias F. A., Oral Z. O. 1995 “Hopf bifurcation and nonlinear studies of gain margins in path control of marine vehicles” [Applied Ocean Research] Vol.17 P.21-32 google doi
  • 2. Papoulias F. A. 1994 “Cross track error and proportional turning rate guidance of marine vehicles” [Journal of Ship Research] Vol.38 P.123-132 google
  • 3. Pettersen K. Y., Nijmeijer H. 2001 “Underactuated ship tracking control: theory and experiments” [International Journal of Control] Vol.74 P.1435-1446 google doi
  • 4. Lefeber E., Pettersen K. Y., Nijmeijer H. 2003 “Tracking control of an underactuated ship” [IEEE Transactions on Control Systems Technology] Vol.11 P.52-61 google doi
  • 5. Jiang Z. P. 2002 “Global tracking control of underactuated ships by Lyapunov’s direct method” [Automatica] Vol.38 P.301-309 google doi
  • 6. Balch T., Arkin R. C. 1998 “Behavior-based formation control for multirobot teams” [IEEE Transactions on Robotics and Automation] Vol.14 P.926-939 google doi
  • 7. Do K. D. 2011 “Practical formation control of multiple underactuated ships with limited sensing ranges” [Robotics and Autonomous Systems] Vol.59 P.457-471 google doi
  • 8. Lewis M. A., Tan K. H. 1997 “High precision formation control of mobile robots using virtual structures” [Autonomous Robots] Vol.4 P.387-4-3 google doi
  • 9. Yang E., Gu D. 2007 “Nonlinear formation-keeping and mooring control of multiple autonomous underwater vehicles” [IEEE Transactions on Mechatronics] Vol.12 P.164-178 google doi
  • 10. Cui R., Ge S. S., How B. V. E., Choo Y. S. 2010 “Leader follower formation control of underactuated autonomous underwater vehicles” [Ocean Engineering] Vol.37 P.1491-1502 google doi
  • 11. Edward D. B., Beam T. A., Odell D. L., Anderson M. L. 2004 “A leader-follower algorithm for multiple AUV formations” [in Proceedings of IEEE/OES Autonomous Underwater Vehicles] P.40-46 google doi
  • 12. Skjetne R., Moi S., Fossen T. 2002 “Nonlinear formation control of marine craft” [in Proceedings of the 41st IEEE Conference on Decision and Control] P.1699-1704 google
  • 13. Do K. D. 2010 “Practical control of underactuated ships” [Ocean Engineering] Vol.37 P.1111-1119 google doi
  • 14. Yoo S. J., Choi Y. H., Park J. B. 2006 “Generalized predictive control based on self-recurrent wavelet neural network for stable path tracking of mobile robots: adaptive learning rate approach” [IEEE Transactions on Circuits and Systems I: Regular Papers] Vol.53 P.1381-1394 google doi
  • 15. Park B. S., Yoo S. J., Park J. B., Choi Y. H. 2010 “A simple adaptive control approach for trajectory tracking of electrically driven nonholonomic mobile robots” [IEEE Transactions on Control Systems Technology] Vol.18 P.1199-1206 google doi
  • 16. Kim K. J., Park J. B., Choi Y. H. 2012 “Path tracking control for underactuated autonomous underwater vehicles using approach angle” [IEICE Transactions on Fundamentals of Electronics, Communications and Computer Science] Vol.E95-A P.760-766 google doi
  • 17. Swaroop D., Hedrick J. K., Yip P. P., Gerdes J. C. 2000 “Dynamic surface control for a class of nonlinear systems” [IEEE Transactions on Automatic Control] Vol.45 P.1893-1899 google doi
  • 18. Oussar Y., Rivals I., Personnaz L., Dreyfus G. 1998 “Training wavelet networks for nonlinear dynamic inputoutput modeling” [Neurocomputing] Vol.20 P.173-188 google doi
  • 19. Lin F. J., Lee T. S., Lin C. H. 2003 “Robust H1 controller design with recurrent neural network for linear synchronous motor drive” [IEEE Transactions on Industrial Electronics] Vol.50 P.456-470 google doi
  • 20. Polycarpou M. M., Mears M. J. 1998 “Stable adaptive tracking of uncertain systems using nonlinearly parameterized on-line approximators” [International Journal of Control] Vol.70 P.363-384 google doi
  • [Figure 1.] Leader-followermodel of autonomous underwater vehicles.
    Leader-followermodel of autonomous underwater vehicles.
  • [Table 1.] Parameters of the asymmetric AUV
    Parameters of the asymmetric AUV
  • [Figure 2.] 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).
    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).
  • [Figure 3.] 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).
    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).
  • [Figure 4.] Formation errors: (a) separation errors, (b) bearing errors.
    Formation errors: (a) separation errors, (b) bearing errors.