Neural-Network- and L2-Gain-Based Cascaded Control ... - IEEE Xplore

630

IEEE JOURNAL OF OCEANIC ENGINEERING, VOL. 39, NO. 4, OCTOBER 2014

Neural-Network- and L2-Gain-Based Cascaded Control of Underwater Robot Thrust Weilin Luo, Carlos Guedes Soares, and Zaojian Zou

Abstract—This paper presents a robust cascaded control strategy to underwater robot thrust. The dynamics of surge motion, of propeller axial flow, of propeller shaft, and of electrically driven circuit in the motor constitute a cascaded system with respect to propeller thrust. Instead of the usual parameter perturbation, generalized modeling errors are considered in the plant, which may be parametric errors, ignored high-order modes, or some unmodeled dynamics in the underwater thrust system. External disturbances are also taken into account, which may be the random noises from mechanical or electrical equipment, or the environmental forces possibly induced by nonuniform currents, ocean internal wave, or cable tension. Combined with state feedback control, an online neural network (NN) compensator is introduced to identify the modeling errors, while L2-gain design is used to suppress the externally continuous or instantaneous disturbances. The Lyapunov’s second method is applied to instruct the controller design, which guarantees the uniformly ultimately bounded (UUB) stability of the error system. By analyzing the tracking errors, it is recommended how to properly select the controller parameters. Good tracking performance and reasonable control inputs are illustrated by numerical simulations. Index Terms—L2-gain, neural networks (NNs), stability, uncertainties, underwater robots.

I. INTRODUCTION

U

NDERWATER vehicles play an important role in oceanic exploration and exploitation. Usually, three kinds of underwater vehicles are available, i.e., human occupied vehicles (HOVs), remotely operated vehicles (ROVs), and autonomous underwater vehicles (AUVs). Compared with HOVs, AUVs and ROVs are less expensive and capable of doing operations that are too dangerous or even impossible for humans. Usually, they are referred to as underwater robots or unmanned

Manuscript received January 05, 2013; revised June 23, 2013; accepted September 10, 2013. Date of publication November 25, 2013; date of current version October 09, 2014. The work of W. Luo and Z. Zou was supported by the China Scholarship Council, the National Natural Science Foundation of China under Grants 51079031 and 50979060, and the Natural Science Foundation of Fujian Province of China under Grant 2010J01004. Associate Editor: K. Takagi. W. Luo is with the Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, University of Lisbon, Lisbon 1049-001, Portugal and also with the College of Mechanical Engineering and Automation, Fuzhou University, Fujian 350108, China (e-mail: [email protected]). C. Guedes Soares is with the Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, University of Lisbon, Lisbon 1049-001, Portugal (e-mail: [email protected]). Z. Zou is with the School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China (e-mail: [email protected]). Digital Object Identifier 10.1109/JOE.2013.2282475

underwater vehicles (UUVs). During the last decades, underwater robots have been paid increasing attention both from the technological and methodological aspects, which is due to an increased number of commercial, military, and scientific applications. With the development of technologies, the use of underwater robots will be more intensive in the future. Underwater robotics refer to modeling; sensor systems; actuating systems; mission control system, guidance, and control; localization; underwater manipulation; fault detection/tolerance; and multiple underwater vehicles [1]. An operating underwater robot in a 3-D subsea space is a complicated nonlinear dynamic system. During navigation, the hull, rudder/fin, and thruster are affected by varying hydrodynamic forces and moments and by unpredictable oceanic environmental forces as well. The motion of an underwater robot can be characterized as time varying, highly nonlinear with multivariable coupling, and disturbed by uncertainties. To accomplish appointed deep-water missions in complicated oceanic environments, autonomy and safety are essential for an operating underwater robot. Such requirements can only be guaranteed provided the underwater robot has good controllability. As defined in [2], control is the development and application to a vehicle of appropriate forces and moments for operating point control, tracking, and stabilization, which involves the feedforward and feedback control laws. Point stabilization, path following, and trajectory tracking are usually concerned about in the guidance and control of underwater robots. During the last decades, various control techniques have been proposed for underwater robots, both in simulation environments and on the spot. Presented were proportional–integral–derivative (PID) control, linearization feedback control, sliding mode control, fuzzy control, parameter-adaptive control, control, model predictive control, linear-quadratic-Gaussian/loop-transfer-recovery (LQG/LTR) control, neural network (NN) control, and hybrid control. Among them, many efforts were devoted to robust control of underwater robots with uncertainties. For example, to deal with uncertain structural parameters, adaptive control was applied in [2] and [3], while sliding mode variable structure method was proposed in [4] and [5]. To suppress the environmental disturbances on AUVs, a quaternion-based adaptive controller was proposed in [6], while or robust controllers were proposed in [7], [8]. Mu synthesis [9] and linear matrix inequality (LMI) synthesis [10] were used to compensate for the uncertainties in UUVs. An robust controller was proposed for suppressing pitch and yaw coupling for a high-speed AUV [11]. For underactuated ROVs with uncertainties, an adaptive fuzzy sliding mode controller was proposed in [12].

0364-9059 © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

LUO et al.: NEURAL-NETWORK- AND L2-GAIN-BASED CASCADED CONTROL OF UNDERWATER ROBOT THRUST

Generally, most underwater vehicles are underactuated systems. Taking an AUV for example, usually actuators are only configured using three coordinates, i.e., surge, pitch, and yaw. The motion of the AUV is driven by the propeller (thruster), rudder, or fin at the stern. Since the main driving force (moment) is provided by the propeller, the design of thrust plays an important role for an operating underwater robot in accomplishing assigned missions, and was regarded as a critical system with respect to safety [13]. Many control strategies have been proposed to the thrusters of AUVs. However, much work cared little about the dynamics of actuators. In [14], the influence of thrust dynamics on underwater vehicles was discussed. As pointed out, if the dynamics of an actuator is ignored, the performance of the control system will degrade, and sometimes it is even a severe problem. In [15], an effective approach to modeling the thruster dynamics was proposed for the motion control of AUVs and ROVs. In [16], the problem of cascaded control of an ocean vehicle with significant actuator dynamics was discussed. In cascaded control of an underwater robot with uncertainties, parameter-adaptive control strategy is in common use. For example, in [17], this method was proposed to the speed control of cascaded AUVs; in [18] and [19], the state estimation method was applied; in [20], a hybrid power/torque thruster controller with loss estimation was designed; in [21], a synthesis method based on a decomposition principle was proposed for the motion control of cascaded AUVs. This paper presents an NN- and L2-gain-based approach for the cascaded control of an underwater robot with uncertainties. This hybrid control strategy makes use of the respective advantages of NN and methods. Uncertainties in the plant are dealt with separately according to the characteristics of uncertainties. For inherent uncertainties, referred to as the modeling errors, an online two-layer NN is introduced to be an approximation, while for stochastic uncertainties from external disturbances, control-based L2-gain design is used to suppress such uncertainties. In past work, these two uncertainties were both usually treated with the same control methodology, i.e., a parameter-adaptive scheme. By using this method, the uncertainties in the model are assumed to be described by unknown parameters. These parameters are estimated online, and the controller is adjusted adaptively according to those estimates. To a large degree, the control performance relies on accurate estimation of parameters. Sometimes, however, the convergence of the estimated parameters is hard to be guaranteed due to uncertain external disturbances or unmodeled dynamics, which are unable to be expressed by formulas. control is an alternative to deal with such uncertainties. However, this method is a conservative strategy because the controller is kept unchanged during the control process, hence, in a sense, it is not an optimal solver. To improve the control accuracy, NN is another alternative to deal with uncertainties. Owing to the nonlinear mapping and online learning abilities of NNs, theoretically all uncertainties can be identified and compensated for in an intelligent way. However, NN is not appropriate to deal with some instantaneous disturbance because it needs time to learn the unknown. Therefore, in this paper, a hybrid control strategy based on NN and theory is proposed to guarantee the con-

631

trol accuracy and to suppress unpredictable disturbances meanwhile. In this paper, a simple and effective technique in control, i.e., L2-gain design is adopted. II. PROBLEM FORMULATION The mathematical model for an underwater robot of six degrees of freedom (DOF) can be expressed in the general form [2] (1) where is the position and Euler angle vector with respect to the earth-fixed coordinate system; is the velocity vector with respect to the body-fixed coordinate system; is the inertia matrix; is the centripetal-Coriolis matrix; is the gravitational or restoring forces vector; is the hydrodynamic damping term; and is the external forces vector provided by rudders, fins, thrusters, or propellers. The motions of (1) include three translational motions and three rotations, in which surge motion can be described by (2) where is the surge velocity; is the mass of the underwater vehicle; is the added mass; and are the linear and nonlinear hydrodynamic derivatives, respectively; and is the thrust deduction coefficient. The thrust force is calculated by (3) where

and is water density; is the propeller diameter; and are constants in the linear regression formulas of thrust coefficient , respectively. The ratio determined by the advance velocity to the propeller and the propeller axial flow velocity is denoted as . In [17], to obtain a more accurate model, the surge dynamics is considered to be coupled to the propeller axial flow dynamics (4) where is the mass of the control volume; is the linear damping coefficient; and is the quadratic damping coefficient. The advance velocity to the propeller satisfies , where is the wake fraction. In [22], a practical modeling of the axial flow velocity was addressed. Assuming that the propeller is driven by a direct current (dc) motor, the actuator dynamics can be described as follows: (5) (6) where is the inertia moment of the motor and the propeller; is the propeller revolution; is the motor damping coefficient;

632


Fig. 1. Driven system of the propeller.

is the conversion coefficient from electrical current to torque; is the electrical current; is the armature inductance of motor; is the armature resistance; and is the applied armature voltage. The propeller torque has the form (7) where

and and are the constants in the linear regression formulas of torque coefficient . The relationship between thrust and torque can be easily found as (8) where

The propeller-driven system can be described as Fig. 1. Equations (2) and (4)–(6) constitute a cascaded thrust system. Taking uncertainties into account, one has

(9) where are the modeling errors and is an external disturbance signal. Note that instead of the usual parameter perturbation, denotes generalized modeling errors, which are referred to as parameter errors, high-order modes, ignored or unmodeled dynamics. For underwater robots, unmodeled dynamics in the thrust system may result from thrust and torque losses caused by viscous drag effects, crosscoupling drag, varying wake with turn or sway, air suction, and interaction between the thruster and the hull [23]. Modeling errors are inevitable and uncertainties caused by external disturbances are stochastic. They may refer to unknown random noises from mechanical or electrical equipment, or the environmental forces possibly induced by nonuniform currents, ocean internal waves,

or cable traction (for ROVs). Also note that modeling errors and disturbances are mainly concerned with the first subsystem in (9) because the focus in this paper is on the tracking control of surge speed . Though modeling errors and disturbances also exist in the other subsystems, they are translated into the first subsystem, except torque losses that are emphasized in the third subsystem, i.e., [18]. As seen, the cascaded system (9) is a complicated nonlinear coupling system, in which the terminated control input is , while control signals including and are interim state variables. The control objective is to design an appropriate , so that the actual surge speed can track well the desired speed . III. CONTROL STRATEGY To a complicated underwater robot system, systematic architecture is in popular use in controller design, which refers to the backstepping method [24], the gain-scheduled method [25], and the hierarchical architecture method [26]. The backstepping method is used in this paper so as to construct appropriate Lyapunov function candidates. The controller design observes two procedures. Step 1) By applying linearization feedback control to deal with the certainties in the plant, “ideal” or desired controllers are designed and a resultant error tracking system is obtained. The forthcoming control mission is to guarantee this system stable and convergent. Step 2) By using backstepping method, several positive Lyapunov function candidates are designed according to the error system in the first step. Then, calculate the derivative of the Lyapunov function candidates. Meanwhile, the NN approach and L2-gain design are used to deal with the uncertainties. To guarantee the derivative negative, appropriate controllers are constructed, so is the tuning algorithm of NN weight. IV. CONTROL DESIGN Without loss of generality, two assumptions are first given with respect to a continuous tracking control issue. Assumption 1: The desired signal is differentiable and the system is controllable. Assumption 2: The external disturbance is bounded as , , and . A. Linearization Feedback Control For certainties in the plant, state feedback control is preferable because of its simplicity and quick response. Two desired controllers and the terminated control are defined as

(10) (11) (12)


where is the desired thrust force; is the desired electrical current; and , , and are the auxiliary controllers. Denoting several tracking errors as

633

where and are positive constants. This inequality is equivalent to an L2-gain index

system (9) becomes the following error system: For a scalar, inequality (16) equals

(17) (13) which is linear with error signal. Appropriate designs of three auxiliary controllers answer for the performance of the tracking system. To obtain a stable controller, the Lyapunov stepping function method is used. This method has been widely applied to control in many engineering fields and is considered as an effective way to deal with a nonlinear system with uncertainties. By using this method, the controller design complies with a systematic and simple procedure. Meanwhile, the robustness and stability of the tracking error system can be guaranteed. With the help of Lyapunov function, it is not required to solve differential equations to judge whether the investigated system is stable. Additionally, the performance of the controller is paid attention to from the start, which is different from the conventional procedure in which the stability proof comes after a controller has been designed. Considering the nonlinear characteristics of the motion of an AUV as well as the effect of uncertainties, in this paper, the Lyapunov function method is employed to achieve a stable controller. By observing (13), a positive–definite Lyapunov function candidate is defined as (14) is not involved in the candidate funcNote that the error tion. Because the thrust force is a function of surge velocity and the propeller revolution , it can be inferred from the error definition that the stability of can be guaranteed provided the errors and converge. Calculating the derivative of yields (15) Because the mathematical expressions of uncertainties , , and are unknown, exact linearization feedback control is useless. NN and L2-gain design are the next ones used. B. L2-Gain Design To deal with an uncertain external disturbance , an evaluation signal is adopted, as recommended in [27] (16)

Let , then the term of disturbance in (15) can be treated by incorporating the L2-gain index

(18) Hence, inequality (15) becomes

(19) If the boundedness of modeling errors is known a priori, the robust control technique is an alternative to deal with it. However, it requires experienced or expert knowledge. Too large boundedness will degrade the tracking accuracy, while too small boundedness cannot guarantee the robust stability. In a sense, the performance of a controller depends on its designer’s experience and skills. To guarantee the tracking accuracy, it is proposed to use NN to approximate such uncertainties. Besides, the term is also dealt with by using NN. As seen from (19), to obtain , it is necessary to calculate the derivative of . According to (11), one has

(20) and . Moreover, It requires calculating the derivative of measurement of the acceleration is required. Hence, to obtain the explicit expression of is tedious and difficult. For simplicity, is also identified by NN. C. NN Identification Rather than the popular backpropagation neural networks (BPNNs), a simpler approach of functional link neural network (FLNN) is used in this paper. As pointed out in [28], this NN was proved to be a universal approximation of nonlinear functions with any accuracy, provided the activation function of the hidden layer is selected as basic or squashing function (such as sigmoid or Gaussian function, etc.) and an appropriate

634


number of the hidden layer nodes exist. For more details about this NN, see [28] or [29]. Let three nonlinear functions be approximated by (21) (22) (23) is the so-called “ideal” weight from the where hidden layer to the output layer and satisfies Fig. 2. Framework of the control system.

where

denotes the Frobenius norm, i.e., Note that no explicit expression of the desired propeller revolution is given. In [17], a lowpass filter was recommended to the desired propeller revolution in engineering practices

The input vector to the hidden layer is derived from preprocessing the original input variables. The activation function vector of the hidden layer is usually a basis or squeezed function. The reconstruction error will satisfy after the NN has successfully approximated the objective function. Note that the first NN does not involve the term in (19) because it relates to the ability of suppressing external disturbance. By applying NN and observing (19) and (21)–(23), three auxiliary controllers can be designed as

(33) where

is the cutoff frequency and

(34) The control system is shown in Fig. 2.

(24) (25)

is obtained from (3)

V. STABILITY ANALYSIS Substituting (21)–(26) into (19) yields

(26) is a positive control gain, is the updated weight where vector, and the iterative algorithms are constructed as (27)

(35)

(28) (29) are positive constants; and three general error vecwhere tors are introduced as

is the weight error vector. where To guarantee the convergence of the instant weight to the constant “ideal” weight , a stepping positive–definite Lyapunov function candidate is defined

Then, the three controllers (10), (11), and (12) become

(36) Its derivative is (30)

(31) (32)

(37)


Incorporating inequality (35) yields

635

, then Define a compact set two cases for starting from the initial state may be analyzed as follows. Case 1: If stays in at any time, one has

(46) (38) Introduce a constant

Let

and let (47) Assuming

in (46), then

then it holds

(48) By definition of

, one has

(49) (39) Further effort is concerned with how to make the right-hand side of the above inequality negative. It can be inferred that the following inequalities hold:

Letting

then

(40) (50)

(41) Multiplying both sides of (48) by

yields (51)

(42) which is equivalent to

(52) (43)

Integrating this expression from

to yields (53)

Note that inequality (41) holds true because of the property of boundedness of [17]. Based on (40)–(43), inequality (39) is reduced to

Taking account of inequality (50), one has

(54)

To guarantee that the right-hand side is negative, the following holds:

It is concluded that the error vector is uniformly ultimately bounded (UUB) stable [30]. According to the definition of and , obviously is also UUB stable. Back to (46), otherwise if , then

(45)

(55)

(44)

636


Integrating this expression from

to

yields (56)

which is equivalent to (57) where the initial Lyapunov function is

(58) By appropriate initialization and parameter selection, e.g., are large enough, a small positive constant exists and satisfies (59) where (60) which is consistent with the L2-gain index given in (17). That is to say, if the external disturbance exists, the controller can suppress this disturbance to guarantee the tracking performance. Case 2: If , then the following inequality holds:

(61) if parameters are properly selected. Inequality (44) is then reduced to (62) If

, then (63)

Thus, the stability condition is satisfied. It should be noted that it seems that the asymptotic stability can be obtained. In fact, when decreases and falls into the compact set, only UUB stability can be guaranteed, as discussed in Case 1. If , then (64) By analogical analysis as Case 1, the performance of suppressing the disturbance can be guaranteed. From the discussions in Cases 1 and 2, it is concluded that, no matter whether evolves within , any starting from will converge into under the proposed control. Hence, the general tracking errors are UUB stable, and so are the NN weight errors and the tracking error of the surge speed by the definition of

. Moreover, the tracking control system’s performance of suppressing disturbance is guaranteed. It is noted that no asymptotic stability but UUB stability is guaranteed, and the latter case is more reasonable in practical use. As pointed out in [31] and [32], underwater vehicles are underactuated systems and their asymptotic stabilities cannot be guaranteed by continuous time-invariant feedback laws. VI. PARAMETER SELECTION Without doubt, the selection of parameters affects the performance of the controller. As seen from controllers (30)–(32) and weight algorithms (27)–(29), 11 parameters are to be determined in the cascaded control system, i.e., , , , and . To improve the control performance, generally, the three controller gains should be large. Parameters and are related to the performance of suppressing disturbances. From the L2-gain inequality (16) and L2-gain index, it can be inferred that smaller and are preferable. Parameters are related to the NN. From inequality (54), the upper boundedness of error can be calculated as

(65) It can be inferred that larger , against smaller , will improve the tracking accuracy of the control system. The same conclusion can also be derived from (47), (58), and (61). VII. SIMULATIONS A simulation study is conducted to verify the validity of the proposed control methodology. An AUV driven by a dc motor is taken as the model to be simulated. The parameters are given in Tables I and II. More details about the AUV and the driven system can be seen in [17] and [33]. Note that the time constant in the electrical actuator is small compared to the mechanical time constant. Usually, the dynamics of an electrically driven circuit can be neglected and only its static gain is taken into account. Nevertheless, as a simulation study, this dynamics is still considered. In fact, to a small underwater vehicle in practice, the dynamics of the electrically driven circuit cannot be neglected because its time constant is not small compared to that of the mechanical system. The desired surge speed is assumed to be and an initial deviation exists, . It should be noted that generally an AUV travels in a wide frequency range from zero to several hertz. To study the effect of the frequency of AUV’s motion on the performance of the tracking system, several runs with different frequencies were carried out during simulations. It was found that the AUV’s dynamic response was affected by the frequency of AUV’s motion. Nevertheless, the accuracy and the stability of the tracking system did not degrade at all. The only difference was the response rate. The applicability of the proposed control method is verified. The controller gains are given as and the parameters of the weight are


637

TABLE I PARAMETERS OF AN AUV

TABLE II PARAMETERS OF A DC MOTOR

Fig. 4. Control inputs to the tracking system, in case of continuous random noise distributed normally.

Fig. 3. Surge velocity and axial flow velocity random noise distributed normally.

, in case of continuous

The initial weights are set to zero, which are different from conventional BPNN whose performance greatly relies on initial weights; the activation function of the hidden layer is selected as a sigmoid function, i.e., (66) The preprocessed inputs to the hidden layer are given as Fig. 5. Four-quadrant thrust force and torque, in case of continuous random noise distributed normally.

It is noted that the elements in the input vector can usually be selected according to the information contained in the approximated function. In this study, the selection of is verified through the learning results by NNs. The parameters of the L2-gain index are and ; the modeling error is assumed as ; the torque losses is assumed to be 5% of . The external disturbance is

first assumed as a random and normally distributed noise constrained in [ 150 N 150 N], and regarded as a continuous disturbance. The simulation results are shown in Figs. 3–7. In detail, Fig. 3 shows the tracking of the surge velocity and the axial flow velocity; Fig. 4 indicates the control inputs including propeller revolution, thrust force, and applied voltage; Fig. 5 illustrates the change of thrust and torque in four different quadrants; Fig. 6 shows the histories of the NN weights; and Fig. 7 demonstrates the NN approximation of uncertainties.

638


Fig. 8. Surge velocity disturbance.

and axial flow velocity

, in case of instantaneous

Fig. 6. Histories of the weights of NNs, in case of continuous random noise distributed normally.

Fig. 9. Control inputs to the tracking system, in case of instantaneous disturbance.

Fig. 7. Approximation abilities of NNs, in case of continuous random noise distributed normally.

Second, the external disturbance is assumed as an instantaneous impulse with the amplitude of 250 N when . The simulation results are shown in Figs. 8–12. Note that in Figs. 7 and 12, the third simulation results are obtained from the difference form of (20) because its explicit continuous expression is unknown, as aforementioned. As can be seen from the simulation results, not only the tracking errors but also the updated weights of the NNs are UUB, under the condition of external disturbance and initial deviation from the desired form, no matter whether the external disturbance is continuous or instantaneous. The robustness and stability are both guaranteed, tracking accuracy as well. Moreover, the control inputs are reasonable and applicable. VIII. CONCLUSION To deal with the uncertainties in the thrust system of an underwater robot, a robust cascaded control strategy is proposed.

Fig. 10. Four-quadrant thrust force and torque, in case of instantaneous disturbance.

The uncertainties refer to the dynamic modeling errors and external disturbances. The cascaded plant consists of the dynamics of surge motion of an underwater vehicle, that of the propeller axial flow, that of the propeller shaft, and that of the electrically driven circuit. Linearization feedback control, NN identification, and L2-gain design are employed to obtain a controller of good performance. State feedback control is used to track


639

In this paper, only numerical simulation verification is performed. Results in a real case would have been more convincing and, in the future work, efforts will be devoted to the validation of this method combined with experiments or sea trials. Also, efforts will be devoted to the application of this method to the other control issues of AUVs, including point stabilization and path following. REFERENCES

Fig. 11. Histories of the weights of NNs, in case of instantaneous disturbance.

Fig. 12. Approximation abilities of NNs, in case of instantaneous disturbance.

the certainties in the plant. An online two-layer NN is used to compensate for the dynamic modeling errors, while L2-gain design is used to suppress the external continuous or instantaneous disturbances. Lyapunov’s second method is applied to instruct the controller design, in which stepping Lyapunov functions are constructed according to the tracking error system. The UUB stability is guaranteed. Appropriate selection of parameters in the controller is recommended. Simulation results demonstrate that the controller is feasible for a complicated underwater robot cascaded system. The proposed methodology offers a systematic and effective way to the control of a class of nonlinear systems with uncertainties. NNs and L2-gain techniques are effectively incorporated into the Lyapunov-function-based controller design. The obtained controller is characterized by simple structure, real-time performance, robustness, and good accuracy. In addition, this method can be extended to other rigid bodies such as manipulators. It is known that the mathematical models of many rigid bodies are the same, in spite of the fact that their physical characteristics are different.

[1] G. Antonelli, T. I. Fossen, and D. R. Yoerger, “Underwater robotics,” Handbook on Robotics, vol. 43, pp. 987–1008, 2008. [2] T. I. Fossen, Guidance and Control of Ocean Vehicles. New York, NY, USA: Wiley, 1994, ch. 1. [3] J. Li and P. Lee, “Design of an adaptive nonlinear controller for depth control of an autonomous underwater vehicles,” Ocean Eng., vol. 32, pp. 2165–2181, 2005. [4] A. J. Healey and D. Lienard, “Multivariable sliding-mode control for autonomous diving and steering of unmanned underwater vehicles,” IEEE J. Ocean. Eng., vol. 18, no. 3, pp. 327–339, Jul. 1993. [5] D. B. Marco and A. J. Healey, “Command, control and navigation experimental results with the NPS ARIES AUV,” IEEE J. Ocean. Eng., vol. 26, no. 4, pp. 466–476, Oct. 2001. [6] G. Antonelli, F. Caccavale, S. Chiaverini, and G. Fusco, “A novel adaptive control law for underwater vehicles,” IEEE Trans. Control Syst. Technol., vol. 11, no. 2, pp. 221–232, Mar. 2003. and designs for diving [7] L. Moreira and C. Guedes Soares, “ and course control of an autonomous underwater vehicle in presence of waves,” IEEE J. Ocean. Eng., vol. 33, no. 2, pp. 69–88, Apr. 2008. [8] L. Moreira and C. Guedes Soares, “Autonomous underwater vehicle control in presence of waves,” presented at the 27th Int. Conf. Offshore Mech. Arctic Eng., Estoril, Portugal, 2008, ASME paper OMAE 200857927, unpublished. [9] G. Campa and M. Innocenti, “Robust control of underwater vehicles: Sliding mode control vs. mu synthesis,” in Proc. IEEE OCEANS Conf., Nice, France, 1998, pp. 1640–1644. [10] M. Innocenti and G. Campa, “Robust control of underwater vehicles: Sliding mode control vs. LMI synthesis,” in Proc. Amer. Control Conf., San Diego, CA, USA, 1999, pp. 3422–3426. [11] J. Petrich and D. Stilwell, “Robust control for an autonomous underwater vehicle that suppresses pitch and yaw coupling,” Ocean Eng., vol. 38, pp. 197–204, 2011. [12] W. M. Bessa, M. S. Dutra, and E. Kreuzerc, “An adaptive fuzzy sliding mode controller for remotely operated underwater vehicles,” Robot. Autonom. Syst., vol. 58, no. 1, pp. 16–26, 2010. [13] A. J. Sørensen, “Structural issues in the design and operation of marine control systems,” Annu. Rev. Control, vol. 29, pp. 125–149, 2005. [14] D. R. Yoerger, J. G. Cooke, and J. S. Slotine, “The influence of thruster dynamics on underwater vehicle behavior and their incorporation into control system design,” IEEE J. Ocean. Eng., vol. 15, no. 3, pp. 167–178, Jul. 1990. [15] A. J. Healey, S. M. Rock. S. Cody, D. Miles, and J. P. Brown, “Towards an improved understanding of thrust dynamics for underwater vehicles,” IEEE J. Ocean. Eng., vol. 20, no. 4, pp. 354–361, Oct. 1995. [16] T. I. Fossen and O. E. Fjellstad, “Cascaded adaptive control of ocean vehicle with significant actuator dynamics,” Model. Identif. Control, vol. 15, no. 2, pp. 81–92, 1994. [17] T. I. Fossen and M. Blanke, “Nonlinear output feedback control of underwater vehicle propeller using feedback form estimation axial flow velocity,” IEEE J. Ocean. Eng., vol. 25, no. 2, pp. 241–255, Apr. 2000. [18] L. Pivano, T. A. Johansen, Ø. N. Smogeli, and T. I. Fossen, “Nonlinear thrust controller for marine propellers in four-quadrant operations,” in Proc. Amer. Control Conf., New York, NY, USA, 2007, pp. 900–905, WeBo6.2. [19] L. Pivano, T. A. Johansen, and Ø. N. Smogeli, “A four-quadrant thrust estimation scheme for marine propellers: Theory and experiments,” IEEE Trans. Control Syst. Technol., vol. 17, no. 1, pp. 215–226, Jan. 2009. [20] Ø. N. Smogeli, A. J. Sørensen, and T. I. Fossen, “Design of a hybrid power/torque thruster controller with loss estimation,” in Proc. Conf. Control Appl. Marine Syst., Ancona, Italy, Jul. 2004, pp. 409–414. [21] V. Filaretov and D. Yukhimets, “Synthesis method of control system for spatial motion of autonomous underwater vehicle,” Int. J. Ind. Eng. Manage., vol. 3, no. 3, pp. 133–141, 2012.

640


[22] J. Kim and W. K. Chung, “Accurate and practical thruster modeling for underwater vehicles,” Ocean Eng., vol. 33, pp. 566–586, 2006. [23] M. Blank, K. P. Lindegaard, and T. I. Fossen, “Dynamic model for thrust generation of marine propellers,” in Proc. Conf. Man Control Marine Craft, Aalborg, Denmark, 2000, pp. 363–368. [24] L. Lapierre and D. Soetanto, “Nonlinear path-following control of an AUV,” Ocean Eng., vol. 34, pp. 1734–1744, 2007. varying sampling [25] E. Roche, O. Sename, and D. Simon, “ control for autonomous underwater vehicles,” in Proc. 4th IFAC Symp. Syst. Struct. Control, Ancona, Italy, 2010, pp. 17–24. [26] K. Valavanis, D. Gracanin, M. Matijasevic, R. Kolluru, and G. A. Demetriou, “Control architectures for autonomous underwater vehicles,” IEEE Control Syst. Mag., vol. 17, no. 6, pp. 48–64, Dec. 1997. [27] C. Ishii, T. L. Shen, and Z. H. Qu, “Lyapunov recursive design of robust adaptive tracking control with L2-gain performance for electricallydriven robot manipulators,” Int. J. Control, vol. 74, no. 8, pp. 811–828, 2001. [28] F. L. Lewis, K. Liu, and A. Yesildirek, “Neural net robot controller with guaranteed tracking performance,” IEEE Trans. Neural Netw., vol. 6, no. 3, pp. 703–715, May 1995. [29] C. Kwan, F. L. Lewis, and D. M. Dawson, “Robust neural-network control of rigid-link electrically driven robots,” IEEE Trans. Neural Netw., vol. 9, no. 4, pp. 581–588, Jul. 1998. [30] Z. H. Qu and D. M. Dawson, Robust Tracking Control of Robot Manipulators. Piscataway, NJ, USA: IEEE Press, 1996, ch. 5. [31] K. Y. Pettersen and O. Egeland, “Exponential stabilization of an underactuated surface vessel,” in Proc. 35th IEEE Conf. Decision Control, Kobe, Japan, 1996, vol. 1, pp. 967–972. [32] M. Reyhanoglu, A. van der Schaft, N. H. McClamroch, and I. Kolmanovsky, “Dynamics and control of a class of underactuated mechanical systems,” IEEE Trans. Autom. Control, vol. 44, no. 9, pp. 1663–1671, 1999. [33] G. Indiveri, S. M. Zanoli, and G. Parlangeli, “DC motor control issues for UUVs,” in Proc. 14th Mediterranean Conf. Control Autom., Ancona, Italy, 2006. Weilin Luo was born in Fujian, China, in 1973. He received the B.Sc. degree in mechanical manufacturing process and equipment from Jiaozuo Institute of Technology, Jiaozuo, Henan, China, in 1995, the M.Sc degree in solid mechanics from Fuzhou University, Fuzhou, China, in 2003, and the Ph.D. degree in design and manufacture of naval architecture and ocean structure from Shanghai Jiao Tong University (SJTU), Shanghai, China, in 2009. Since 2011, he has been an Associate Professor in the College of Mechanical Engineering and Automa-

tion, Fuzhou University. He is currently a Postdoctoral Fellow in the Centre for Marine Technology and Engineering (CENTEC), Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal. His current research interests include ship maneuvering and control, underwater robotics, and artificial intelligent techniques.

Carlos Guedes Soares received the Ocean Engineer degree from the Massachusetts Institute of Technology (MIT), Cambridge, MA, USA, in 1976, the Dr.Ing. degree in marine technology from the Norwegian Institute of Technology (presently Norwegian University of Science and Technology), Trondheim, Norway, in 1984, and the Doctor of Science (Habilitation) from the Technical University of Lisbon, Lisbon, Portugal, in 1991. Currently, he is a Professor at the Instituto Superior Técnico (IST), University of Lisbon, Lisbon, Portugal, where he heads the Centre for Marine Technology and Engineering (CENTEC). Prof. Guedes Soares is a Fellow of the Institute of Marine Engineering, Science and Technology (IMarEST), the Royal Institution of Naval Architects (RINA), the Society of Naval Architects and Marine Engineers (SNAME), the American Society of Mechanical Engineers (ASME), and the Portuguese Association of Engineers (Ordem dos Engenheiros).

Zaojian Zou was born in Jiangxi, China, in 1956. He graduated in naval architecture from Wuhan University of Technology, Wuhan, Hubei, China, in 1982, from where he also received the M.S. degree in ship hydrodynamics in 1984. He received the Dr.Ing. degree in ship theory from Hamburg University, Hamburg, Germany, in 1994. He came back to his alma mater in China in 1995 and worked as an Associate Professor (1995–1996) and Professor (1996–2003) of Ship Hydrodynamics. Since 2003 he has been with the School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China, as a Professor of Ship Hydrodynamics. His research interest includes maneuvering and control of ships and other marine vehicles, numerical ship hydrodynamics, artificial intelligence technology, and its application in marine hydrodynamics.