Research Article

Trajectory tracking control of an underactuated unmanned underwater vehicle synchronously following mother submarine without velocity measurement

Advances in Mechanical Engineering 2015, Vol. 7(7) 1–11 Ó The Author(s) 2015 DOI: 10.1177/1687814015595340 aime.sagepub.com

Jian Xu, Man Wang and Gengshi Zhang

Abstract This article addresses the trajectory tracking control of underactuated unmanned underwater vehicles, where the control structure is designed to synchronously follow the mother submarine with only available position information. The controller is derived using the virtual guidance strategy and backstepping-based design techniques. The virtual guidance system is first constructed to eliminate the measurement of the mother submarine’s velocity and dynamics. The convergence and effectiveness of the virtual guidance system are proven using Lyapunov stability theory. In addition, to enhance the robustness of the underactuated vehicle against the unmodeled dynamics and unknown disturbances from the environment, radial basis function neural networks are finally used to approximate the nonlinear uncertainties. Under the proposed control, semiglobal uniform boundedness of the closed-loop system is guaranteed for adequate choices of the controller gains. A simulation example is included to demonstrate the effectiveness and robustness of the approaches suggested. Keywords Trajectory tracking, underwater robotics, underactuated system, backstepping, Lyapunov stability theory

Date received: 9 June 2015; accepted: 11 June 2015 Academic Editor: Fakher Chaari

Introduction The high-precision tracking control of unmanned underwater vehicles (UUVs) is extremely important, for it is a basic technique for successfully accomplishing underwater specified tasks such as deep sea inspections, long-distance surveys, oceanographic mapping and resource exploration.1 However, the control problem of underactuated underwater vehicles is still very challenging. The main difficulties can be summarized in the following issues: (1) the dynamics of UUVs is usually highly nonlinear and coupled, and the hydrodynamic coefficients are impossible to be determined accurately;2 (2) the ocean disturbances cannot be negligible in the

trajectory tracking control for UUVs;3 (3) reliable information exchange between UUV and submarine is quite difficult due to the weak underwater communication;4 and (4) motivated by cost and weight considerations, more and more UUVs are underactuated, which represent fewer independent actuators than the number of degrees of freedom and then dynamic models of College of Automation, Harbin Engineering University, Harbin, China Corresponding author: Man Wang, College of Automation, Harbin Engineering University, Harbin 150001, China. Email: [email protected]

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2 underactuated vehicles result in systems with secondorder nonholonomic constraints.5 Control of underactuated UUVs is an active topic of research.6 In Do et al.,7 a global output feedback controller was proposed for stabilization and tracking of underactuated omni-directional intelligent navigator (ODIN), a spherical underwater vehicle. Also, based on a linearized error space model, a static output feedback controller was designed and implemented using Serret– Frenet frame in Subudhi et al.8 In general, however, the velocities of the vehicles are very difficult to be accurately measured, which causes full-state feedback scheme to be not feasible. Hence, in Zhang et al.,9 an adaptive output feedback controller based on dynamic recurrent fuzzy neural network (DRFNN) was proposed, in which the locational information is only needed for the controller design. Based on the model, one can predict the motion of the vehicle using controller actuator inputs and available state measurements.10 Experimental results, reported in Whitcomb and Yoerger11 and Zhao and Yuh,12 demonstrated successful controller performance. In Refsnes et al.,13 a modelbased output feedback controller was proposed for slender-body underactuated autonomous underwater vehicles (AUVs) where sea trials of Minesniper MKII were performed. Despite the wide range of applications and the large number of control strategies developed, most of them highly rely on the assumption that both the velocity and position of the reference trajectory are known for navigation, guidance, and control. Nowadays, in numerous practical applications, there is an increasing demand that UUVs should synchronously track all states of a moving target, which can be another underwater vehicle, including the real-time position, attitude, and velocities rather than a predefined trajectory. In this article, we are interested in the problem when the only information available about the reference signals is the position of the moving target. This type of problem is motivated by an interesting and attractive scenario. The case is that a mother submarine is performing a maneuver along a predefined trajectory, while an UUV in a configuration master or slave is required to follow the mother submarine. Note that the UUV can only use the Doppler Velocity Log (DVL) to generate accurate velocity measurements when the distance to seafloor is within a certain boundary.14 This restriction in the instrumentation contributes to increase challenges for accurate tracking. Therefore, to improve the performance, the virtual guidance control scheme is proposed in this article to eliminate measurement requirement of velocity and dynamics of mother submarine. Since the presented virtual guidance system can work independently of these velocity measurements, it contributes to increase reliability of the control system by providing analytical redundancy to the measurements.

Advances in Mechanical Engineering In fact, the submarine or UUV application scenario considered in this article is more like one type of pursuing or target tracking problem. Different from leader– follower formation control in previous works,15–17 the dynamics and velocity of the maneuvering submarine are unknown and uncontrollable. To overcome this problem, one should obtain the reference trajectory of the UUV based on the only position information from the mother submarine. Fortunately, there are some schemes described in the literature to resolve the tracking control problem without velocity measurement. Particular examples can be found in the area of leader– follower formation control and the pursuing or target tracking problem. In the leader–follower formation control of multiple underactuated AUVs,18 the follower tracked a reference trajectory based on the leader’s position and predetermined formation without the need for its velocity. In the tracking control of robot manipulators,19 a pseudo-filtered tracking error signal was designed to eliminate the need for velocity measurement. A similar control strategy, in combination with backstepping and Lyapunov direct design technique, was applied for robust formation control of multiple nonholonomic mobile robots.20 In target tracking problem of autonomous robotic vehicles,21 a switched logic-based control strategy was proposed to solve the pursuing problem using range-only measurements. It is important to stress that most of the aforementioned tracking controllers would make the vehicle to move toward a maneuvering target at a constant speed, which results in unnatural trajectories. This significantly restricts the class of reference trajectories to be used for practical applications. However, uncertainties exist in any realistic system because of systematical modeling errors and the perturbations from external environment. Therefore, trajectory tracking controller designed an assumption that the model is accurate, and no disturbances may fail to work or significantly degrade performance when the systematical parameters are unknown, and the environmental disturbances cannot be neglected. This article presents successful results of an underactuated UUV synchronously tracking a maneuvering mother submarine without the measurement of its velocity and dynamics, even in the presence of possibly large systematical modeling uncertainty and unknown disturbances. The virtual guidance system is first constructed based on the only available position of the mother submarine. The convergence and effectiveness of the virtual guidance system are proven using Lyapunov stability theory, and then, trajectory tracking controller are designed to make the actual velocities follow these virtual guidance signals to achieve the position and attitude tracking. The main contributions of this article can be summarized as follows: (1) a virtual guidance system is constructed for an underactuated UUV using

Xu et al. only position information of maneuvering submarine; (2) with the aid of backstepping and Lyapunov direct method, the trajectory tracking controller for an underactuated UUV is derived, and semiglobal uniform boundedness of the closed-loop system can be guaranteed. Simulation results demonstrate the effectiveness and robustness of the proposed methods; and (3) successful results of radial basis function (RBF) neural networks against systematical uncertainty and unknown disturbances are presented in this article. The key feature of the technique can handle the dynamical uncertainties without the need for explicit knowledge of the model. The remainder of this article is organized as follows: the mathematical model and preliminaries are presented in section ‘‘Mathematical model and preliminaries.’’ The controller design and analyses are given in sections ‘‘Virtual guidance system design’’ and ‘‘Trajectory tracking control design,’’ respectively. Furthermore, a case study on an underactuated UUV is presented in section ‘‘Numerical simulations,’’ and, finally, the simulation results are shown. Some concluding remarks are given in section ‘‘Conclusion.’’

Mathematical model and preliminaries This section describes the kinematic and dynamic models of an underactuated UUV moving in the horizontal plane and then formulates the problem of trajectory tracking control.

Mathematical model In general, the dynamic behaviors of an underwater vehicle are commonly described in two-coordinate frames, namely, earth-fixed frame and body-fixed frame as shown in Figure 1. In the body-fixed frame, the mathematical model of an UUV presented in this article, with standard notation, can be described as follows3

3 h_ = JðhÞv M_v = t + td  CðvÞv  DðvÞv  gðhÞ

where h = ½x, y, cT 2 R3 and v = ½u, v, rT 2 R3 are the earth-fixed frame position or orientation and bodyfixed frame velocities in 3 degrees of freedom (DOF), respectively; M, J(h), C(v), and D(v) are the inertia matrix, the Jacobian transformation matrix, the centrifugal and Coriolis matrices, and the hydrodynamic damping matrix, respectively; g(h) is an unknown vector of restoring forces due to buoyancy and gravitational forces and moments; and t = ½t u , tv , t r T 2 R3 and td = ½td1 , td2 , t d3 T 2 R3 are the vectors of the input signals and the vectors of external disturbances, respectively. Without loss of generality, we consider an underactuated UUV synchronously tracking a maneuvering mother submarine in the horizontal plane for applications at a constant depth, such as autonomous underwater recovery, and assume that g(h) = 0. Due to the underactuation, t = ½t u , 0, tr T 2 R3 . Then, other parameters are given as follows 2

m11 6 M=40 0 2

0

0 m22

0 0

0

m33 0

3 7 5, m22 v

3

6 7 CðvÞ = 4 0 0 m11 u 5, 0 m22 v m11 u 2 3 0 d11 0 6 7 d22 0 5 and D ðv Þ = 4 0 0 0 d33 2 3 cos c  sin c 0 6 7 JðhÞ = 4 sin c cos c 05 0 0 1 also m11 = m  Xu_ , m22 = m  Yv_ , m33 = Iz  Nr_ , d11 = Xu + Xujuj juj, d22 = Yv + Yvjvj jvj, and d33 = Nr + Nrjrj jrj. The mathematical model of an underactuated UUV is described by 8 x_ = cos (c)u  sin (c)v > > > > > y_ = sinðcÞu + cosðcÞv > > > > > c_ = r > > > > < m22 d11 t u + td1 vr  u+ u_ = m11 m11 m11 > > > > m d td2 > 11 22 > ur  v+ v_ =  > > > m m m 22 22 22 > > > > m11  m22 d33 t r + td3 > : uv  r+ r_ = m33 m33 m33

Figure 1. Reference frames of unmanned underwater vehicles.

ð1Þ

ð2Þ

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Problem formulation In the trajectory tracking control considered in this article, an underactuated UUV should track a maneuvering mother submarine rather than a predefined trajectory. First, the work we should carry out is that how to implement the guidance dynamics when the UUV cannot accurately obtain the available information of mother submarine’s velocities and dynamics by inertial measurement units (IMU). Second, due to the underactuated property with second-order nonholonomic constraints, UUV cannot be transformed into a driftless-chained system as the mobile robot, so that the control approaches applied for the robots cannot be directly employed in underactuated underwater vehicles. Thus, a much simpler control structure, requiring much less computational effort along with the rigorous stability analysis, is required. Finally, when the dynamics is included, the system has a drift vector field that poses new demands on the controllability analysis and the control design. The issues mentioned above all will be resolved in this article.

Preliminaries In this subsection, we will recall some definitions and lemmas, which are necessary in the design of the proposed control. Lemma 1. For bounded initial conditions, if there exists a C 1 continuous and positive definite Lyapunov function V (j) satisfying k1 (jjjjj)  V (j)  k2 (jjjjj), such that V_ (jjjjj)   mV (jjjjj) + c, where k1 and k2 : Rn ! R are class K functions and c is a positive constant, then the solution j = 0 is uniformly bounded.18 In the following, RBF neural network is introduced to adaptively approximate the unknown continuous function f (j). Definition 1. The algorithm of RBF neural network can be presented as follows ! jjj  ci jj2 , ui = g b2i

i = 1, 2, . . . , n

Assumption 1. The output of RBF neural network is a continuous function ^f (j, u). Assumption 2. Under the condition that e0 is a positive constant to be arbitrarily small, the ideal approximation output of RBF neural network ^f (j, u ) exists and satisfies

where

max jj^f ðj, u Þ  f ðjÞjj  e0 ( )  u = arg min sup jjf ðjÞ  ^f ðj, uÞjj , u2Rn

ð3Þ

Y = u ϕðjÞ where the input vector is j = ½j1 , j2 , . . . , jq  2 Rq , Gaussian membership value is ϕ = ½u1 , u2 , . . . , un  2 Rn , weight vector is u 2 Rn , and output is Y 2 R1 . According to Ge and Wang,22 under the following assumptions, RBF neural network can approximate any continuous function to any desired accuracy over a compact set j 2 Oj to arbitrarily any degree of accuracy.

and

j2Oj

u 2 Rn is defined as the optimal value of u.

Virtual guidance system design The key idea in this subsection is how to design a virtual guidance system, which can ensure that the underactuated UUV could synchronously track the maneuvering mother submarine only by the available position measurement. To design the control input for the virtual guidance system, we generate a pseudofiltered tracking error signal as in previous works,18–20 to eliminate the need for velocity measurement and dynamics in the control design, and the details are given by the following implementable equations ef = ev + ~e ~e_ =  b ð~eÞ  Kef 1

ð5Þ

~eð0Þ = 0 where the virtual guidance tracking error is defined as ev = hv  hr 2 R3 ; hv and hr are the virtual and actual positions of the submarine, respectively; and ~e = ½~e1 , ~e2 , ~e3 T and b1 (~e) = diag½li tanh (~ei =li ) for i = 1, 2, 3. The gain matrices of the filter are assumed to have the form K = diag½ki , where ki and li , i = 1, 2, 3, are positive scalar constants. Then, the control input for the virtual guidance system can be designed as follows vv = J1 ðhv Þðb1 ð~eÞ + b2 ð~eÞ  aef Þ

T

ð4Þ

ð6Þ

where a = diag½a, a.0, is a feedback gain matrix and b2 (~e) = diag½ki tanh (~ei =ki ) for i = 1, 2, 3, which is similar to b1 (~e) as before. Taking the time derivate of ef and using the virtual control variable vv yields e_ f =  ðK + aÞef  Jðhr Þvr + b2 ð~eÞ

ð7Þ

Note that the induced norm of the matrix J(hr ) is always bounded; thus, clearly, the solutions of the equation (7) are bounded when vr is bounded. In the practical applications, it is a reasonable assumption that the velocity of the vehicle vr is bounded. The

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convergence and effectiveness of the virtual guidance system is stated in the following theorem. Theorem 1. Consider the filter dynamics described by equations (5) and (7) satisfying the assumption that the velocity of the vehicle is bounded. Applying the inputs (6) for the virtual guidance system, then for any bounded initial conditions, all the closed-loop signals are uniformly semiglobally practically asymptotically stable (USPAS). Proof. Similar to Cui et al.18 and Ghommam et al.,20 consider the following Lyapunov function candidate 1 Ve ðtÞ = eTf ef 2 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 3T 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 3 ~e1 ~e1 6 k1 ln cosh k1 7 6 k1 ln cosh k1 7 6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6  7 6  7 6 7 6 7 + 6 k2 ln cosh ~ek2 7 6 k2 ln cosh ~ek2 7 2 2 6 7 6 7 6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7  5 4  5 4 k3 ln cosh ~ek33 k3 ln cosh ~ek33

ð8Þ

1 _ T

+ K ~e b2 ð~eÞ

=  ðK + aÞeTf ef  eTf Jðhr ÞVr  K1 b1 ð~eÞT b2 ð~eÞ    3V jjef jj2  K1 b1 ð~eÞT b2 ð~eÞ   lmin ðK + aÞ  2jjef jj

ð9Þ

ð10Þ

 denotes the upper bound of the vehicle’s velocity. and V Then, choose the gain lmin ðK + aÞ 

3V 0 2jjejj

Trajectory tracking control design This section presents a Lyapunov-based control law using backstepping technique to design the trajectory tracking control of an underactuated UUV with the dynamic parameters that are uncertainty and unknown disturbances from the environment. Before stating the main result of the note, we first do the coordinate transformation in a form that is easily amenable for stabilization.

To facilitate the design of the controller, the position and orientation errors by coordinate transformation can be obtained easily as follows 32 3 2 3 x  xv cosðcÞ sinðcÞ 0 xe 4 ye 5 = 4  sinðcÞ cosðcÞ 0 54 y  yv 5 0 0 1 ce c  cv 2

ð12Þ

where the orientation and position of the virtual guidance system can satisfy   y_ v cv = arc tan x_ v

ð13Þ

Then, the derivatives of the position tracking error variables along equation (2) can be obtained

where we have used the fact that   d ðln coshðxðtÞÞÞ = x_ ðtÞ tanhðxðtÞÞ dt    3V jjef jj2  eTf Jðhr ÞVr  2jjef jj

Remark 1. At this stage, the convergence and effectiveness of the virtual guidance system is guaranteed using the measurements of the available signals such as the position and orientation. These signals are exploited later to track the trajectory of the virtual guidance with an underactuated UUV.

Coordinate transformation

Clearly, note that Ve (t) is positive definite and upper bounded by a positive increasing function. More precisely, using the inequality property ln cosh (jjxjj)  jjxjj2 , it can be derived that Ve  f (ef , ~e), where f (ef , ~e) is a class K-function such that f (ef , ~e) = 1=2(jjef jj2 + jj~ejj2 ). Then, taking the derivate of equation (8) along the solutions of equations (5)–(7) yields V_ e ðtÞ = eTf e_ f

dependency of 1=e in V_ e (t), the solutions of the closedloop system are USPAS according to the results in Purwar et al.19

ð11Þ

For jjef jj2 .e2 , since b1 (~e)T b2 (~e) is a positive definite function for all ~e, it is straightforward to allow for the conclusion that V_ e (t)  0. However, due to the

x_ e = u  vp cosðce Þ + rye y_ e = v + vp sinðce Þ  rxe pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where vp = x_ 2v + y_ 2v .

ð14Þ

Remark 2. Note that the derivatives of the orientation error in this subsection are not introduced. Because it may cause the constraints on initial error conditions of the vehicle according to the results previously investigated in literature7. However, fortunately, we can clearly see that the orientation error can also reflect on the position errors due to the coupled property of underwater vehicle from equation (14). Then, we will introduce an indirect method to work out the problems of the orientation and velocity tracking.

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Backstepping-based dynamic control

u_ e =

Based on equation (14) derived in the prior subsection, the trajectory tracking controller is designed using the backstepping method. Step 1: Consider the following Lyapunov function candidate V1 =

1

2

 2

x2e + ye

ð15Þ

Taking the derivate of equation (15) along the solutions of equation (14) yields



 V_ 1 = xe u  vp cosðce Þ + ye v + vp sinðce Þ

ð16Þ

The traditional design method is to choose ce to make ye converge to a neighborhood of the origin that can be made arbitrarily small. It is, however, a drawback that results in the constraints on initial errors of the vehicle and also make the expression of the controller more complex. Hence, to avoid the constraints, we utilize an indirect method to choose as follows av = vp sinðce Þ

ð17Þ

From the expression of the virtual variation, it makes the control of ce transform to the control of av , which is novel in the trajectory control design. Then, u and av are not the real control inputs. Hence, in order to make V_ 1 negative, their desired values ud and avd are chosen as ud = vp cosðce Þ  avd =  v 

k4 xe E

k5 ye E

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where E = 1 + x2e + y2e and k4 and k5 are positive constants needed to be chosen later. According to the analysis above, we have to introduce appropriate error variables as follows ue = u  ud ave = av  avd

ð19Þ

Then, computing the corresponding error dynamics by equations (18) and (19) yields

 k4 x2e + k5 y2e + ue xe + ave ye V_ 1 =  E

ð20Þ

Step 2: Consider the Lyapunov function candidate V2 = V1 +

1 1 m11 u2e + a2ve 2 2

ð21Þ

Now, the task is to stabilize the errors ue and ave . First, differentiating ue with respect to time and using equation (2) yields

ð22Þ

The control input t u is chosen as tu =  xe  k6 ue + m11 u_ d  m22 vr + d11 u  td1 ð23Þ where k6 is a positive constant. Accordingly, we have

V_ 2 = 

 k4 x2e + k5 y2e  k6 u2e + ave ðye + a_ ve Þ E

ð24Þ

Second, as a similar procedure, differentiating ave with respect to time and using equation (17) yields

 a_ ve = v_ p sinðce Þ + vp cosðce Þ r  c_ d  a_ vd

ð25Þ

Before proceeding to the next step of the design, some manipulations on the virtual error dynamics equation can be performed. In order to make V_ 2 negative, r is considered as a virtual control, and its desired value rd is chosen as _vp sinðce Þ + a_ vd  k7 ave  ye rd = c_ d + vp cosðce Þ

ð26Þ

where k7 is a positive constant. Then, we have to introduce appropriate error variables as follows re = r  rd

ð27Þ

Finally, computing the corresponding error dynamics by equations (25)–(27) yields

ð18Þ

m22 vr  d11 u + t d1  m11 u_ d + tu m11

V_ 2 = 

 k4 x2e + k5 y2e  k6 u2e  k7 a2ve + vp ave re cosðce Þ E ð28Þ

Step 3: Consider the following Lyapunov function candidate V3 = V2 +

1 m33 re2 2

ð29Þ

Then, differentiating re with respect to time and using equation (2) yields r_ e =

ðm11  m22 Þuv  d33 r + td3  m33 r_ d + tr m33

ð30Þ

The control input t r is chosen as tr =  k8 re  vp ave cosðce Þ + m33 r_ d  ðm11  m22 Þuv + d33 r  t d3

ð31Þ

where k8 is a positive constant needed to be chosen later. Accordingly, we have

 k4 x2e + k5 y2e  k6 u2e  k7 a2ve  k8 re2  0 ð32Þ V_ 3 =  E

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Remark 3. In the practical applications, since the parameters m11 , m22 , m33 , d11 , d22 , d33 , t d1 , t d2 , and t d3 are highly uncertain, the model-based controls (23) and (31) are not feasible. Motivated by the uncertain parameters and unknown disturbances from the environment, RBF neural network is presented to adaptively estimate and compensate the uncertainty in the following subsection.

For underwater vehicles, it is difficult to accurately measure the hydrodynamic coefficients and environmental disturbances. Hence, the system dynamics are not exactly known. Here, we define the uncertain terms of the systematical model as follows



 f1 ðxÞ =  m011  m11 u_ d + m022  m22 vr

0   d11 u  td1  d11



 f2 ðxÞ = m033  m33 r_ d + m011  m022  m11 + m22 uv

0   d33 r  t d3  d33 ð33Þ where m0ii and dii0 are the nominal models, and mii and dii are the actual models. Then, consider the following control laws for underactuated UUV 8 0 t u =  xe  k6 ue + m011 u_ d  m022 vr + d11 u + ^f ðx, uÞ > >

0 1 0 > > 0 0 ^ > _ t =  k r  v a cos ð c Þ + m  m r r 8 e p ve d > e 33 11  m22 uv + d33 r + f2 ðx, uÞ < ^fi ðx, uÞ = u ^T ϕðxÞ > i T   ! > > ue k ^ > ^_ > > ϕðxÞ + 9 u :u=  g re k10

ð34Þ T

^ = ½^ where u u1 , ^u2  is the estimation of the weight vector, g is a positive constant to be chosen later, and x = ½xe , ue , ave , re T are the input variables of the network. Consider the following Lyapunov function candidate 2 1 X ~u2 2g i = 1 i

 k4 x2e + k5 y2e  k6 u2e  k7 a2ve E  k8 re2  k9 ~u1 ^u1  k10 ~u2 ^u2

ð37Þ

Accordingly, it yields

 k4 x2e + k5 y2e  k6 u2e  k7 a2ve  k8 re2 E k9 k10 ~2 k9 k10  2 u2 + jju1 jj2 + jju2 jj  ~u21  2 2 2 2

ð38Þ

V_ 4   2rV4 + C 2k6 2k8 r = min 2k4 , 2k5 , , 2k7 , , k9 g, k10 g m11 m33 k9 k10  2 jju2 jj C = jju1 jj2 + 2 2

ð39Þ

V_ 4  

Theorem 2. Consider the underactuated UUV with dynamics (2) satisfying assumptions 1 and 2, under the actions of the control law (34). For each compact set O, where (h(0), V(0), ^u1 (0), ^u2 (0)) 2 O, the trajectory tracking control system is semiglobally uniformly bounded. And the tracking errors converge to a compffiffiffiffiffiffiffiffiffiffiffi pact set Oe : = fjjzjj  C=2rg, where r and C are pffiffiffiffiffiffiffiffi defined in equation (39) and z = ½xe , ye , m11 ue , ave , pffiffiffiffiffiffiffiffi ~ pffiffiffi ~ pffiffiffi m33 re , u1 = g, u2 = g . Proof. From equation (39), using the Comparison Lemma yields V4 (t) V4 (0)e2rt + C=2r for t 2 ½0, + ‘). Using the fact that 2V4 (t)= jjzjj2 , straightforward computations allow for the conclusions that sffiffiffiffi C t 2 ½0, + ‘Þ jjzðtÞjj  jjzð0Þjjert + r

ð40Þ

Therefore, the closed-loop system is semiglobally uniformly bounded. The convergence and effectiveness of the tracking errors can be guaranteed by properly increasing r.

ð35Þ

~=u ^  u . Then, differentiating V4 with respect where u to time along with the solutions of equations (34) and (35) yields

V_ 4 = 

kj ~ 2 kj jjui jj + jjui jj2 2 2

Then

Adaptive compensation and stability analysis

V4 = V3 +

 kj ~ui ^ui  

Remark 4. Although this article can guarantee semiglobally uniformly bounded of the system, global uniform asymptotic stabling (UGAS), which is preferred in the control design, is not given. Hence, the design techniques based should be further improved in the future. The control scheme of the UUV system is shown in Figure 2.

ð36Þ

Applying Young’s inequality to the term  kj ~ui ^ui yields

Numerical simulations In this section, a simulation example is included to illustrate the effectiveness and efficiency of the proposed control scheme. The simulation studies are performed

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Figure 2. Control scheme of an underactuated UUV.

Figure 3. MATLAB or Simulink framework for the trajectory tracking controller.

on an underactuated UUV without side thruster, which has only the control inputs on surge and yaw directions, implementing in MATLAB or Simulink, and the framework of the system is shown in Figure 3. The aim of the simulation tests is summarized as follows: (1) to verify the feasibility of the designed virtual guidance system, (2) whether the proposed trajectory tracking controller can satisfy the demands of underaction and synchronization in submarine or UUV application scenario, and (3) to show the robustness of RBF neural network against the systematical uncertainties and unknown disturbances. Simulation case is the trajectory tracking control of underactuated UUV in the horizontal plane. The mother submarine will move as a trajectory with a specified timing law, which looks like ‘‘N.’’ Specially, the trajectory is described by

xd ðtÞ = 100 sinð0:02tÞ yd ðtÞ = 50 sinð0:04tÞ

ð41Þ

In all simulations, the vehicle moves from the initial position h0 = ½20, 0, 0T and the velocity V0 = ½0:1, 0:1, 0:1T . The gains of the controller are chosen as follows:

Virtual guidance system: a = 0:01, l1 = 3, l2 = 3, l3 = 0:5, k1 = 5, k2 = 10, and k3 = 0:5 Backstepping-based dynamic controller: k4 = 0:1, k5 = 0:1, k6 = 300, k7 = 4, and k8 = 200 RBF neural network: k9 = 0:02, k10 = 0:1, and g = 50 And other design parameters are selected as 2 2 ϕ(x) = ejjxci jj =bi , ci = 1, bi = 10, i = 1, 2, ., 5. In order to more comprehensively illustrate the performance of the proposed scheme, without loss of generality, we define the disturbance as time-varying forces or moment in frame {E} fc ðtÞ = 2 3 10 + 1:8sinð0:07tÞ + 1:2sinð0:05tÞ + 1:2sinð0:09tÞ 6 7 5 + 0:4sinð0:01tÞ + 0:2cosð0:06tÞ 4 5 0 ð42Þ Then, in body-fixed frame {B}, the external disturbances acting on the UUV can be presented as follows: td ðtÞ = JT ðhÞfc ðtÞ

ð43Þ

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Table 1. UUV hydrodynamic parameters. Parameter

Symbol

Value

Unit

Mass Rotational inertia Added mass Added mass Added mass Surge linear drag Surge quadratic drag Sway linear drag Sway quadratic drag Yaw linear drag Yaw quadratic drag

m Iz Xu_ Yv_ Nr_ Xu Xujuj Yv Yvjvj Nr Nrjrj

185 50 230 280 230 70 100 100 200 50 100

kg kg m2 kg kg kg kg/s kg/m kg/s kg/m kg m2/s kg m2

Figure 5. Tracking errors.

submarine virtual guidance uuv

u(m/s)

10 5 0

0

50

100

150 t(s)

200

250

300

0

50

100

150 t(s)

200

250

300

0

50

100

150 t(s)

200

250

300

v(m/s)

5 0 -5

Figure 4. Tracking trajectory in the x–y plane.

In addition, the uncertainty of the systematical model is considered in the simulations. Specially, we assume that the system parameters will simultaneously increase 10% to the actual model. The hydrodynamic coefficients and inertial parameters of an UUV are shown in Table 1. Figure 4 shows the trajectories of the maneuvering mother submarine, underactuated UUV, and virtual guidance system. In order to more comprehensively illustrate the tracking performance, the synchronously tracking errors between the UUV and mother submarine are shown in Figure 5. It is clearly seen that the tracking errors converge to a compact set, which is bounded, by the analysis described in subsection ‘‘Adaptive compensation and stability analysis.’’ One of the features of the trajectory tracking is the requirement of time-varying velocities of the vehicles compared with the path following. Thus, the tracking performance of the system, when the velocity in surge, sway, and yaw changes with timing law, is duly analyzed in Figure 6. These results confirm that the

r(rad/s)

0.5 0 -0.5

Figure 6. Simulation results of surge, sway, and yaw velocities.

proposed control scheme is able to regulate and stabilize the dynamic behaviors of the underactuated UUV in the trajectory tracking in spite of the systematical uncertainty and additional disturbances. Furthermore, to verify the effectiveness and efficiency of RBF neural network in the control design, Figure 7 shows the results of the systematical uncertainty and self-adaptive estimation. It is clearly seen that the uncertainties are time varying and unknown, not constant as presented in many articles. This is the advantage of RBF neural network and the reason why we utilize the technique in this article. Finally, values taken by the control actions so that the underactuated UUV can synchronously track the maneuvering mother submarine are shown in Figure 8. It is noted that the magnitude of the control inputs is considerably influenced by the initial position errors. Fortunately, under the proposed control, the control inputs are restrained in the limited range even

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Advances in Mechanical Engineering

Figure 7. Simulation results of self-adaptive estimation.

Figure 8. Control actions of the underactuated UUV.

in the presence of large initial constraints. This can be verified in Figure 8. As shown in Figures 5 and 8, with the initial position error xe = 20, the control force tu is within 0–8000 N, and the moment t r is within 2100 to 300 N m, all of which are acceptable in the practical operations. Furthermore, the control procedure is always a finite-time process in practical tracking control of UUVs. Thus, the actual control actions can always be kept within a limited range. Even while the control inputs and velocity of the vehicle are constrained, there are still some available methods to solve this problem in the literature. Further work can study the global tracking control of underactuated UUVs with control inputs and velocity constraints.

Funding

Conclusion In this article, a trajectory tracking controller is proposed for an underactuated UUV, which can synchronously follow a maneuvering mother submarine without the need for the velocity measurement and dynamics of the submarine. To enhance the robustness of the vehicle against the systematical uncertainties caused by the unmodeled parameters and unknown disturbance from underwater environment, RBF neural network is utilized in this article and a simulation example is included to demonstrate the effectiveness and advantages of the technique. Furthermore, the complete analysis and proofs are given using Lyapunov stability theory, and semiglobal uniform boundedness of the overall system is guaranteed. Further work involves achieving more improved theory that the controller is UGAS in attempt to optimize the performance of the vehicle and underwater experiments. Declaration of conflicting interests The authors declare that there is no conflict of interest.

This work was supported by the National Nature Science Foundation of China under grant 51409055 and Harbin Engineering University Science and Technology on Underwater Vehicle Laboratory under grant 9140C270208140C27123 and partly supported by the Fundamental Research Funds for the Central Universities under grant HEUCFX041402.

References 1. Hai H, Tang Q, Li Y, et al. Dynamic control and disturbance estimation of 3D path following for the observation class underwater remotely operated vehicle. Adv Mech Eng 2013; 2013: 604393 (16 pp.). 2. Evans J and Nahon M. Dynamics modeling and performance evaluation of an autonomous underwater vehicle. Ocean Eng 2004; 31: 1835–1858. 3. Fossen T. Guidance and control of ocean vehicles. New York: Wiley Interscience, 1994. 4. Schoenwald D. AUVs: in space, air, water, and on the ground. IEEE Contr Syst Mag 2000; 20: 15–18. 5. Repoulias F and Papadopoulos E. Planar trajectory planning and tracking control design for underactuated AUVs. Ocean Eng 2007; 34: 1650–1667. 6. Meng D, Li A, Tao G, et al. High performance motion trajectory tracking control of pneumatic cylinders: a comparison of some nonlinear control algorithms. Adv Mech Eng 2014; 2014: 485704 (17 pp.). 7. Do KD, Jiang ZP and Nijmeijer H. A global outputfeedback controller for stabilization and tracking of underactuated ODIN: a spherical underwater vehicle. Automatica 2004; 40: 117–124. 8. Subudhi B, Mukherjee K and Ghosh S. A static output feedback control design for path following of autonomous underwater vehicle in vertical plane. Ocean Eng 2013; 63: 72–76. 9. Zhang LJ, Qi X and Pang YJ. Adaptive output feedback control based on DRFNN for AUV. Ocean Eng 2009; 36: 716–722.

Xu et al. 10. Smallwood DA and Whitcomb L. Model-based dynamic positioning of underwater robotic vehicles: theory and experiments. IEEE J Oceanic Eng 2004; 29: 169–186. 11. Whitcomb LL and Yoerger D. Development, comparison and preliminary experimental validation of nonlinear dynamic thruster models. IEEE J Oceanic Eng 1999; 24: 481–494. 12. Zhao S and Yuh J. Experimental study on advanced underwater robot control. IEEE T Robot 2005; 21: 695–703. 13. Refsnes JE, Sorensen AJ and Pettersen KY. Model-based output feedback control of slender-body underactuated AUVs: theory and experiments. IEEE T Contr Syst T 2008; 16: 930–946. 14. Aguiar AP and Hespanha JP. Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty. IEEE T Automat Contr 2007; 52: 1362–1379. 15. Edwards D, Bean T, Odell D, et al. A leader-follower algorithm for multiple AUV formations. In: Proceedings of IEEE/OES autonomous underwater vehicles, vol. 1, Sebasco, ME, 17–18 June 2004, pp.40–46. New York: IEEE.

11 16. Huang N, Duan Z and Zhao Y. Leader-following consensus of second-order nonlinear multi-agent systems with directed intermittent communication. IET Control Theory A 2014; 8: 782–795. 17. Tao Y. Leader-following coordination problem with an uncertain leader in a multi-agent system. IET Control Theory A 2014; 8: 773–781. 18. Cui RX, Ge SS, How BVE, et al. Leader-follower formation control of underactuated autonomous underwater vehicles. Ocean Eng 2010; 37: 1491–1502. 19. Purwar S, Kar I and Jha A. Adaptive output feedback tracking control of robot manipulators using position measurements only. Expert Syst Appl 2008; 34: 2789–2798. 20. Ghommam J, Mehrjerdi H and Saad M. Robust formation control without velocity measurement of the leader robot. Control Eng Pract 2013; 21: 1143–1156. 21. Shoushtari ON, Aguiar AP and Sedigh AK. Target tracking of autonomous robotic vehicles using range-only measurements: a switched logic-based control strategy. Int J Robust Nonlin 2012; 22: 1983–1998. 22. Ge SS and Wang C. Adaptive neural network control of uncertain MIMO nonlinear systems. IEEE T Neural Networ 2004; 15: 674–692.