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Neuro-fuzzy control of underwater vehicle-manipulator systems
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Journal of the Franklin Institute 349 (2012) 1125–1138 www.elsevier.com/locate/jfranklin
Neuro-fuzzy control of underwater vehicle-manipulator systems Bin Xua, Shunmugham R. Pandianb, Norimitsu Sakagamic,n, Fred Petryd,1 a
Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA b Department of Computer Science & Industrial Technology, Southeastern Louisiana University, Hammond, LA, USA c Department of Navigation and Ocean Engineering, Tokai University, Orido, Japan d Geospatial Sciences and Technology Branch, Naval Research Laboratory, Stennis Space Center, MS 39529, USA Received 8 December 2007; received in revised form 31 December 2011; accepted 4 January 2012 Available online 20 January 2012
Abstract This paper presents an intelligent controller for underwater vehicle-manipulator systems (UVMS) based on the neuro-fuzzy approach. The controller is composed of fuzzy PD control with membership function tuning by linguistic hedge. A neural network compensator approximates the dynamics of the UVMS in decentralized form. The new controller has the advantages of simplicity of implementation due to decentralized design, precision, and robustness to payload variations and hydrodynamic disturbances. It has significantly low energy consumption compared to both the conventional PD and conventional fuzzy control methods. The effectiveness of the proposed controller is illustrated by results of simulations for a six degrees of freedom autonomous underwater vehicle with a three degrees of freedom on-board manipulator. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction The precise and robust operation of autonomous underwater vehicles (AUVs) and their on-board manipulators is a challenging task. In fact, most current generation AUVs are n
Corresponding author. E-mail addresses: [email protected] (B. Xu), [email protected] (S.R. Pandian), [email protected] (N. Sakagami), [email protected] (F. Petry). 1 His work was supported by the Naval Research Laboratory’s Base Program, Program Element No. 0602435N. 0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2012.01.003
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not fitted with manipulators and hence are mainly limited to underwater surveying and surveillance tasks because of this difficulty. Therefore, research focused on robust, intelligent control of autonomous underwater vehicle-manipulator systems (UVMS) will contribute significantly to enhance the state-of-art in underwater robotics technology. This will enable autonomous underwater intervention by UVMS for tasks such as handling and manipulation of payloads, interaction with the environment, and underwater maintenance and repair. It has been pointed out that ‘‘Manipulator use on AUVs is embryonic, and much R&D is needed before AUVs are able to perform more than simple tasks’’ [1]. When a multiple degrees-of-freedom manipulator is placed on an underwater vehicle, the system becomes a multi-body dynamic system of different bandwidths. The dynamics of the vehicle and the manipulator strongly influence each other making precise coordinated control quite difficult [2]. Moreover, robustness of performance in the presence of strong hydrodynamic forces and disturbances such as tidal waves and currents, particularly in shallow and very shallow water environments, is essential for effective deployment of UVMS. Research on coordinated control of UVMS has so far been largely limited to conventional control methods which require estimation of the system dynamics and hydrodynamic disturbances. Some of these methods include feedback linearization [3], non-adaptive [4] and adaptive [5] sliding mode control, nonlinear feedback [6,7], and model-based control [8]. Coordinated vehicle-arm control with model-based compensation of hydrodynamic disturbances has been proposed in [9]. In contrast to the difficulties encountered in the reliable estimation of system dynamics, non-model-based control methods offer significant advantages for complex, nonlinear, uncertain systems such as UVMS, e.g., ease of implementation and potential for learning of hydrodynamic disturbances and parametric variations. Computational intelligence techniques based on fuzzy logic, neural networks, and genetic algorithms have been extensively studied for process control systems, manipulators in free space, and for mobile robots, e.g., [10]. Several authors have recently applied them for the planning, navigation, and control of AUVs or underwater manipulators, e.g., [11–13]. However, these methods have not so far been studied in the case of UVMS. An experimental study of conventional error-based iterative learning control for a prototype underwater vehicle with an one-link manipulator has been recently presented in [14]. In this paper, we propose an intelligent controller based on the neuro-fuzzy approach for precise, robust, and energy-efficient control of UVMS. The method is not model-based, and the design is decentralized. Therefore, the proposed controller is easy to implement in practice. The controller is of PD type, with fuzzy tuning of the feedback gains using modified fuzzy membership functions. Further, the magnitude of the gains, and hence the energy expenditure, is reduced by neural network-based identification of the system dynamics as well as hydrodynamic disturbances. The effectiveness of the new approach is illustrated with simulation studies of a six degrees of freedom (DOF) AUV with a three DOF on-board manipulator.
2. Dynamics of underwater vehicle-manipulator systems Several researchers have addressed the modeling of underwater vehicle-manipulator systems, e.g., [3,15–17]. In this paper, based on one of the models [17], the dynamics of an
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Fig. 1. An underwater vehicle-manipulator system.
underwater vehicle-manipulator system as shown in Fig. 1 is expressed as _ qðtÞ _ € þ HA ðqðtÞÞ_v ðtÞ þ SðqðtÞ, qðtÞÞ HðqðtÞÞqðtÞ _ þ CsgnðqðtÞÞ _ þ DðqðtÞ,vðtÞÞvðtÞ þ BqðtÞ þ gðqðtÞÞ þ bðqðtÞÞ ¼ uðtÞ
ð1Þ
The matrices and vectors in Eq. (1) are as follows: T qðtÞ ¼ qTv ðtÞ,qTm ðtÞ is the (6þn)x1 vector, where qv ðtÞ 2 R61 is the vector of vehicle positions and orientations, qm ðtÞ 2 Rn1 is the vector of manipulator joint angles in the bodyfixed reference frame, and vðtÞ 2 Rð6þnÞm is the flow velocity on a segment of the UVMS. A strip theory [16] is applied to the above modeling using the flow velocity vðtÞ. H(q) is the _ is a matrix positive-definite UVMS inertia matrix, and HA(q) the added inertia matrix. Sðq, qÞ of Coriolis and centripetal forces, and D(q,v) is a hydrodynamic damping tensor. B and C are viscous friction and Coulomb friction coefficient matrices, g(q) and b(q) are vectors of gravity and buoyancy forces, respectively, and u is the (6þn) input torque vector. To simplify the controller design, we transform the composite system into a set of interconnected second-order systems. Let ! ! qi 0 0 1 xi ¼ , bi ¼ , Ai ¼ ,and q_ i mii 0 0 0
0
1
6þn C B X B mij uj þ fi C C B Fi ¼ B C C B j¼1 A @ jai
i ¼ 1,2,:::ð6 þ nÞ
where mij are elements of the inverse of the inertia matrix. The dynamics in Eq. (1) can then be written compactly as x_ i ¼ Ai xi þ bi ui þ Fi
ð2Þ
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where Fi is treated as a disturbance term, accounting for uncertainties in vehicle-manipulator dynamics as well as hydrodynamic forces. fi is the i-th component of the vector _ þ g þ bÞ H 1 ðHA v_ þ S q_ þ Dv þ Bq_ þ CsgnðqÞ 3. Fuzzy logic controller design 3.1. Conventional PD controller In the trajectory tracking problem, the state trajectory is required to track the nominal state trajectory xr ðtÞ,t 2 ½0,T, where T is the terminal time. We assume that the desired velocity and acceleration trajectories x_ r ðtÞ, x€ r ðtÞ, respectively, exist and are continuous over t 2 ½0,T: Let the i-th subsystem trajectory tracking error be ! ei Ei ¼ ¼ xi xri e_ i The PD law for the i-th subsystem is set as ui ¼ Kpi ei Kdi e_ i
ð3Þ
where Kpi and Kdi are proportional and derivative gains, respectively. 3.2. Fuzzy gain-tuning Considering the decentralized, non-model-based PD control law (Eq. (3)), the values of feedback gains need to be chosen large enough to ensure high performance in the presence of dynamic uncertainties and disturbances. However, high feedback gains are preferably avoided due to the effects of actuator saturation and from the view point of the energy consumption particularly since AUVs have limited on-board power supply. In order to avoid the problems of gain tuning with conventional PD controller and introduce an element of adaptation, in this section we propose the use of a Mamdani-type fuzzy logic controller (FLC) to adaptively tune these gains based on several sets of fuzzy rules. The structure of the proposed fuzzy PD controller is shown in Fig. 2. The general form for the j-th rule of a single-input-single-output Mamdani-type FLC is as follows: rule Rj : IF xi is Ai,j THEN yi is Bi,j j ¼ 1,2,. . .,m ð4Þ
Fig. 2. Fuzzy PD controller for UVMS.
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where xi is the input variable (tracking error), yi is the output variable (feedback gain), and Ai,j and Bi,j are the fuzzy sets for xi and yi, respectively. Next, we fuzzify the normalized xi by means of predefined membership function mAi,j ðxi Þ of the elements xi as in Fig. 3. For simplicity of implementation, we use triangular membership functions. Finally, we use the correlation-minimum inference method to geometrically sum the consequences of the active associations and apply centroid defuzzification to find the crisp values of normalized yi. Then, we denormalize it to obtain the actual output value (gain). The details of this procedure are omitted here due to limitations of space. The proptional gain Kp has the effect of reducing the rise time and the steady-state error. The larger it is the faster and more robust will be the system response. However, usually lower gains are preferred in practice in order to avoid an oscillatory response. The fuzzy rule base as shown in Table 1 is based on the heuristics that if the tracking error is large, the proportional gain is also expected to be large in order to force the system to track the desired trajectory as accurately and as speedily as possible. The fuzzy logic tuning for the derivative gain Kdi is set up similarly based on the error derivative. 3.3. Fuzzy gain-tuning with modified membership functions In order to further reduce the energy consumption and improve the accuracy of the control, a so-called linguistic hedge (also known as ‘‘linguistic modifier’’) is adopted next. The linguistic hedge is a function that tunes the membership function of the fuzzy set associated with the linguistic label, obtaining a definition with a higher or lower precision depending on the case. Two of the most well known modifiers are the concentration linguistic hedge ‘‘very’’ and the
Fig. 3. Fuzzification of xi. Table 1 Fuzzy rule base for gain Kpi. Fuzzy rules
Kpi
jei j Zero
Small
Medium
Large
Zero
Small
Medium
Large
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Fig. 4. Effects of linguistic hedges on membership functions.
dilation linguistic hedge ‘‘more-or-less’’ [18]. Fig. 4 shows the differences among the standard triangular membership function (T), the ‘‘very’’ one, and the ‘‘more-or-less’’ one. The expressions for the two modifiers are given by a mvery T ¼ ðmT Þ
ð5Þ
¼ ðmT Þ1=a mmoreorless T
ð6Þ
where a41. Using the linguistic membership functions, the symbolic forms of the fuzzy rules are altered as, e.g. rule Rj : IF xi is more or less Ai,j THEN yi is more or less Bi,j j ¼ 1,2,:::,m. Or rule Rj : IF xi is very Ai,j THEN yi is very Bi,j j ¼ 1,2,:::,m. By changing the fuzzy membership function’s surface structure, we find that the dilation linguistic hedge ‘‘more-or-less’’ is able to provide the most accurate performance with the least energy consumption. The reason for this is that the ‘‘very T’’ is more likely to be a crispy value which means it is more like the conventional PD controller. In contrast, the ‘‘more-or-less T’’ is obviously more flexible which means that it behaves more like an adaptive controller [18]. Therefore, the ‘‘more-or-less’’ surface has advantages including high performance and low energy consumption. The triangular and modified triangular membership functions are simple and computationally simple to implement in real-time. Nonlinear membership functions like Gaussian and generalized Bell provide smooth variation and can also be used in conjunction with neural networks, though computational complexity in implementation must be taken into account. The computational requirements involving fuzzy rule sets might pose a constraint in situations with limited on-line computation resources, e.g., in microcontroller-based implementation. In such cases, a simple weighting function can be used to reduce computations. Fuzzy look-up tables could also be used, though defuzzification will still take up some computing time. It must be remarked, however, that computational resources are becoming increasingly cheaper and more powerful, so fuzzy control algorithms may still be implemented in real-time with little modifications. 4. Neural network compensation Neural networks have the characteristic of approximating highly nonlinear functions. In order to optimize the performance of the controllers for UVMS, the neural network is adopted as a compensator for PD and fuzzy PD controllers. The idea is to use the wellknown backpropagation method to roughly approximate the dynamics of each degree of freedom in the form of the decentralized neural network [10]. The controller is of the form ui ¼ Kpi ei Kdi e_ i þ uNN i
ð7Þ
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where Kpi and Kdi can be the propotional and derivative gains of conventional PD controller, standard fuzzy PD, or fuzzy PD with modified membership functions. uNNi is the output of the neural network compensator. By comparison with Eq. (2), we see that the neural net-based feedforward compensation of the system dynamics means that the magnitudes of the feedback gains in Eq. (7) can be much smaller, resulting in the same advantages provided by the model-based control methods which are more difficult to implement. The robustness of neural network control schemes to hydrodynamic disturbances has been demonstrated in the case of autonomous underwater vehicles as well as large ships, e.g., [13,19]. € q, _ and q. From Eq. (1), we note that the input u of the UVMS is a function of q, € q, _ and q. Therefore, the neural network is required to approximate the relations of u to q, However, due to the high number of degrees of freedom of the UVMS and its high nonlinearity as well as uncertainty, it is usually computationally expensive to regress the overall dynamics. Therefore, the decentralized form of neural network compensator is adopted here. Decentralized implementations of neural network-based controllers have been shown to have bounded tracking errors for interconnected nonlinear systems with strong interactions [20], and precise tracking performance in uncertain systems such as robot manipulators [21]. Instead of learning the overall dynamics, we approximate the dynamics link by link. In this case, the expense of the computation will be largely decreased. Another advantage to use the decentralized form is that it is easy to implement in practice. However, because UVMS is a highly coupled system with high degrees of freedom, the neural network regressor provides approximate compensation for the conventional PD or Fuzzy PD controller which is the main controller here. As the neural network is mainly used to reduce the magnitudes of required gains using feedforward compensation, it is sufficient for the decentralized estimator to have an acceptable degree of accuracy. The training method is based on conventional off-line backpropagation. To train the neural network, we first apply the conventional PD controller on the UVMS system, and record the data of q€ di , q_ i , qi, and the output ui of the PD controller for each link. In order to generalize the neural network dynamics for different trajectories, we use different
Fig. 5. Structure of the neural network of each link.
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desired trajectories including high and low frequency sine waves and cycloid waves when we generate the training data. The neural network has input, hidden, and output layers. We assign six neurons in the hidden layer with sigmoid output funtions. The output layer uses pure linear function. The three-layer neural network is theoretically sufficient to approximate any arbitrary nonlinear funtions. The reason we choose q€ di instead of q€ i is that q€ i is usually quite noisy and often not available in practice. The structure of the proposed neural network is as shown in Fig. 5. 5. Simulation results Numerical simulations have been conducted in order to illustrate the effectiveness of the proposed neuro-fuzzy controller. The simulator was developed in Visual Cþþ. 5.1. Simulation model In this paper, a 9-DOF AUV-type UVMS has been considered. The vehicle has six degrees of freedom comprising translational motions x, y, z and rotational motions pitch, roll, yaw. The manipulator with three rotational degrees of freedom is mounted on the bottom of the vehicle with the joint axes parallel to the vehicle z-axis initially. We assume the AUV to be a cylinder-shaped tube deployed in water. The parameters of the assumed model are shown in Table 2. For details, see [22]. In practice, PD control gains are selected by a systematic method such as one of various tuning techniques [23]. However, in our work the PD gains are mainly used as a base for performance comparison with neuro-fuzzy control with similar gains. Therefore, the PD controller gains were chosen sufficiently large to provide good performance with nominal payload. The neuro-fuzzy controller’s error-dependent PD gains were specified to be mainly lower than the fixed PD gain values, though for some degrees of freedom they were found to occasionally exceed the fixed gain values. In the case of fuzzy rule implementation, based on simulation runs for position control using nominal gain values, errors greater than 60 degrees were treated as large, so the errors are normalized by dividing by 60. For trajectory gains, the peak errors were typically much smaller in magnitude, and correspondingly large error values were selected as maximum. The neural net training trajectories were specified to be sinusoidal with peak amplitudes of 1.0 and frequency of 0.2 Hz. Extensive simulations have been conducted to analyze and compare the performance of the conventional PD and fuzzy logic (triangular membership function) controllers with the modified fuzzy controller (very T and more or less T membership functions) and the proposed neurofuzzy controller. Due to limitations of space, only selected results are presented in this paper. Table 2 Dynamic parameters of 9-DOF UVMS. Description
Vehicle
Link 1
Link 2
Link 3
Length (m) Diameter (m) Mass (kg)
0.80 0.10 15.0
0.20 0.025 1.0
0.20 0.025 1.0
0.20 0.025 1.0
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Figs. 6 and 7 show, respectively, the total tracking error norm and the power consumption for the AUVMS, for fuzzy PD controllers with the three types of membership functions in Fig. 4. It is evident that with respect to both the measures, the ‘‘more or less’’ fuzzy PD control method outperforms both the conventional fuzzy PD control and the ‘‘very T’’ fuzzy PD control. This is because the dilation hedge ‘‘more or less’’ has greater adaptability with changes in tracking errors. So, in the rest of the paper we consider the performance of the neuro-fuzzy PD controller with the ‘‘more or less’’ type membership function.
Fig. 6. Combined tracking error norm for fuzzy PD controllers.
Fig. 7. Combined expended energy norm for fuzzy PD controllers.
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Similarly, to consider the effectiveness of the proposed decentralized neural networkbased compensation method, we have compared the conventional PD control method with the PDþ neural net compensation method. In terms of individual tracking errors, robustness, as well as combined tracking error norm and expended energy norm, the PD controller with compensation was found to outperform the regular PD controller. Fig. 8 shows the comparison of error norms. To consider the robustness of the neuro-fuzzy controller, the payload mass is increased from 0 to 2 kg at t¼ 4 s. The resulting responses of the conventional PD controller and the neuro-fuzzy controller are compared in Figs. 9–11. For desired constant velociy motions along the AUV x, y, and z axes, the pitch, roll, and yaw angles are required to be constant, while the manipulator joint angles are specified to move along cycloidal trajectories. It is clear that the neuro-fuzzy control method is quite precise and robust to payload variations. The slight delay in response of the system at t¼ 0 is attributable to the effects of friction. The results shown in Figs. 6–11 are obtained for the case of static water conditions. When the current velocities are increased for the above cases, it is again found that the neuro-fuzzy controller significantly outperforms the conventional PD controller as well as the fuzzy PD controller with various membership function shapes. For the case when the water flow velocity along the reverse direction of the x-axis is changed from 0 to 2 knots, the resulting performances of the PD and neuro-fuzzy PD controller are summarized in Figs. 12 and 13. Fig. 12 shows the RMS tracking error along the AUV’s x-axis. It is again obvious that the neuro-fuzzy controller is significantly superior to the conventional PD control method. The simulation studies presented here have not explicitly considered system uncertainties. However, the controller implementation is not model-based and the neural network training is based on results of implementation of a PD-type controller for typical test trajectories and so would include in practice the effects system or environmental uncertainties. Effects of sensor noise will be considered in future studies on experimental implementation.
Fig. 8. Comparision of PD and PDþneural net compensation.
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Fig. 9. Position tracking: PD (
), neuro-fuzzy PD controllers (- - -); desired trajectory:
Fig. 10. Pitch, roll, and yaw tracking: PD (
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.
), neuro-fuzzy PD controllers (- - -); desired trajectory:
.
The PD control used in the paper is a subsystem/joint-based independent control, and the NN control is used mainly to add feedforward compensation to the PD/fuzzy controller. This separates the control action between fuzzy logic and NN methods for
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Fig. 11. Arm trajectory tracking: PD (
), neuro-fuzzy PD controllers (- - -); desired trajectory:
.
Fig. 12. RMS error of AUV x-axis motion.
feedback gain tuning and feedforward control, respectively, making implementation direct and straight-forward. Depending on additional performance improvements desired and the affordability of computational complexity, the proposed method can be more sophisticated, for example neural net-based selection or tuning of fuzzy rules, layers, weights, and similarly selection of neural net layers and parameters using fuzzy rules, e.g., [13]. On the other hand, depending on additional performance desired further improvements can also be made to the basic backpropagation neural net employed in this paper, e.g., adding additional
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Fig. 13. Comparison of total energy consumption.
hidden layer(s), tuning of neuronal weights to ensure passivity, and stability of tracking error dynamics, guaranteed tracking, and so on [10]. The global stability of the proposed method is not addressed in this paper due to limitations of space. For a complex uncertain system such as an AUVMS, it is necessary to incorporate into the system model significant practical effects such as actuator saturation and other nonlinear effects. Given the major advantages of intelligent controllers over fixed gain PD controllers, several researchers have addressed the stability of neuro-fuzzy controllers for critical systems such as hazrdous environments in nuclear reactors [24] and underwater vehicles [25]. 6. Conclusions A neuro-fuzzy controller for underwater vehicle-manipulator systems has been proposed in this paper. The performance of the fuzzy PD control method has been improved by the tuning of triangular membership functions using linguistic hedges. To further improve the precision and robustness with reduced feedback gains, a decentralized neural network compensator for estimation of the system dynamics has been developed. The controller design is decentralized, making it easy to implement. Simulation results for a six DOF AUV with a three DOF on-board manipulator show that the proposed neuro-fuzzy controller significantly outperforms the conventional PD controller in terms of robustness to payload changes and hydrodynamic disturbanses, and has superior performance in terms of tracking error norms and expended energy norms. References [1] U.S. Department of Defense, Developing Science And Technologies List, Section 13 Marine Systems Technology, July 2002. [2] S.R. Pandian, N. Sakagami, System integration aspects of underwater vehicle-manipulator systems for oceanic exploration, Journal of the Society of Instrument and Control Engineers (Japan) 47 (2008) 830–836. [3] I. Scholberg, T. Fossen, Modeling and control of underwater vehicle-manipulator systems, in: Proceedings of the Conference Marine Craft Maneuvering and Control, Southampton, UK, 1994, pp. 45–57.
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[4] B. Xu, S. Abe, N. Sakagami, S.R. Pandian, A robust nonlinear controller for underwater vehiclemanipulator systems, in: Proceedings of ASME/IEEE International Conference on Advanced Intelligent Mechatronics, Monterey Bay, CA, 2005, pp. 711–716. [5] Y. Taira, M. Oya, S. Sagara, An adaptive controller for underwater vehicle manipulator systems including thruster dynamics, in: Proceedings of the International Conference on Modeling, Identification, and Control, Okayama, Japan, 2010, pp. 185–190. [6] S. Soylu, B.J. Buckham, R.P. Podhorodeski, Development of a coordinated controller for underwater vehicle-manipulator systems, in: Proceedings of the IEEE International Conference on Oceans, Quebec, Canada, 2008, pp. 1–9. [7] J. Han, W.K. Chung, Coordinated motion control of underwater vehicle-manipulator system with minimizing restoring moments, in: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Nice, France, 2008, pp. 3158–3163. [8] N. Sarkar, T.K. Podder, Coordinated motion planning and control of autonomous underwater vehiclemanipulator systems subject to drag optimization, IEEE Journal of Oceanic Engineering 26 (2001) 228–239. [9] H.H. Wang, S.M. Rock, OTTER: The design and development of an intelligent underwater robot, Autonomous Robots 3 (1996) 297–320. [10] R. Yusof, R.Z.A. Rahman, M. Khalid, M.F. Ibrahim, Optimization of fuzzy model using genetic algorithm for process control application, Journal of The Franklin Institute 348 (2011) 1717–1737. [11] S.R. Pandian, N. Sakagami, Neuro-fuzzy control of an underwater robot manipulator, in: Proceedings of the International Conference on Advanced Robotics, Control, and Computer Vision, Singapore, 2010, pp. 2135–2140. [12] F. Song, S.M. Smith, Design of sliding mode fuzzy controllers for an autonomous underwater vehicle without system model, IEEE/MTS Oceans Conference Record, Providence, RI, 2000, pp. 835–840. [13] J.S. Wang, C.S.G. Lee, Self-adaptive recurrent neuro-fuzzy control of an autonomous underwater vehicle, IEEE Transactions on Robotics and Automation 19 (2003) 283–295. [14] N. Sakagami, Precise control of underwater vehicle manipulator systems using iterative learning control, in: Proceedings of the ICROS/SICE International Joint Conference, Fukuoka, Japan, pp. 3089–3093. [15] K. Ioi, K. Itoh, Modeling and simulation of an underwater manipulator, Advanced Robotics 4 (1990) 303–317. [16] T.W. McLain, S.M. Rock, Experiments in the hydrodynamic modeling of an underwater manipulator, IEEE Conference on the Proceedings of the Symposium on Autonomous Underwater Vehicle Technology, 1996, pp. 463–469. [17] G. Antonelli, Underwater Robots: Motion and Force Control of Vehicle-Manipulator Systems, Springer, 2006. [18] J. Casillas, O. Cordon, M.J. del Jesus, F. Herrera, Genetic tuning of fuzzy rule deep structures preserving interpretability and its interaction with fuzzy rule set reduction, IEEE Transactions on Fuzzy Systems 13 (2005) 13–29. [19] Y. Zhang, G.E. Hearn, P. Sen, A neural network approach to ship track-keeping, IEEE Journal of Oceanic Engineering 21 (1996) 513–527. [20] S.N. Huang, K.K. Tan, T.H. Lee, Decentralized control of a class of large-scale nonlinear systems using neural networks, Automatica 41 (2005) 1645–1649. [21] L. Acosta, G.N. Marichal, L. Moreno, J.J. Rodrigo, A. Hamilton, J.A. Mendez, A robotic system based on neural network controllers, Artificial Intelligence in Engineering 13 (1999) 393–398. [22] B. Xu, Fuzzy Control Underwater Vehicle-Manipulator Systems, M.S. Thesis, Electrical Engineering and Computer Science Department, Tulane University, 2007. [23] K.J. Astrom, T. Hagglund, PID Controllers: Theory, Design and Tuning, International Society of Measurement and Control Press, 1995. [24] Y. Hacioglu, Y.Z. Arslan, N. Yagiz, MIMO fuzzy sliding mode controlled dual arm robot in load transportation, Journal of the Franklin Institute 348 (2011) 1886–1902. [25] A. Piegat, Stability of the PD-neuro-fuzzy control system of the underwater vehicle Krab II, in: T. Graczyk, T. Jastrzebski, C.A. Brebbia (Eds.), Marine Technology II, WIT Press, 1997.
Journal of the Franklin Institute 349 (2012) 1125–1138 www.elsevier.com/locate/jfranklin
Neuro-fuzzy control of underwater vehicle-manipulator systems Bin Xua, Shunmugham R. Pandianb, Norimitsu Sakagamic,n, Fred Petryd,1 a
Department of Electrical and Computer Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA b Department of Computer Science & Industrial Technology, Southeastern Louisiana University, Hammond, LA, USA c Department of Navigation and Ocean Engineering, Tokai University, Orido, Japan d Geospatial Sciences and Technology Branch, Naval Research Laboratory, Stennis Space Center, MS 39529, USA Received 8 December 2007; received in revised form 31 December 2011; accepted 4 January 2012 Available online 20 January 2012
Abstract This paper presents an intelligent controller for underwater vehicle-manipulator systems (UVMS) based on the neuro-fuzzy approach. The controller is composed of fuzzy PD control with membership function tuning by linguistic hedge. A neural network compensator approximates the dynamics of the UVMS in decentralized form. The new controller has the advantages of simplicity of implementation due to decentralized design, precision, and robustness to payload variations and hydrodynamic disturbances. It has significantly low energy consumption compared to both the conventional PD and conventional fuzzy control methods. The effectiveness of the proposed controller is illustrated by results of simulations for a six degrees of freedom autonomous underwater vehicle with a three degrees of freedom on-board manipulator. & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction The precise and robust operation of autonomous underwater vehicles (AUVs) and their on-board manipulators is a challenging task. In fact, most current generation AUVs are n
Corresponding author. E-mail addresses: [email protected] (B. Xu), [email protected] (S.R. Pandian), [email protected] (N. Sakagami), [email protected] (F. Petry). 1 His work was supported by the Naval Research Laboratory’s Base Program, Program Element No. 0602435N. 0016-0032/$32.00 & 2012 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2012.01.003
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not fitted with manipulators and hence are mainly limited to underwater surveying and surveillance tasks because of this difficulty. Therefore, research focused on robust, intelligent control of autonomous underwater vehicle-manipulator systems (UVMS) will contribute significantly to enhance the state-of-art in underwater robotics technology. This will enable autonomous underwater intervention by UVMS for tasks such as handling and manipulation of payloads, interaction with the environment, and underwater maintenance and repair. It has been pointed out that ‘‘Manipulator use on AUVs is embryonic, and much R&D is needed before AUVs are able to perform more than simple tasks’’ [1]. When a multiple degrees-of-freedom manipulator is placed on an underwater vehicle, the system becomes a multi-body dynamic system of different bandwidths. The dynamics of the vehicle and the manipulator strongly influence each other making precise coordinated control quite difficult [2]. Moreover, robustness of performance in the presence of strong hydrodynamic forces and disturbances such as tidal waves and currents, particularly in shallow and very shallow water environments, is essential for effective deployment of UVMS. Research on coordinated control of UVMS has so far been largely limited to conventional control methods which require estimation of the system dynamics and hydrodynamic disturbances. Some of these methods include feedback linearization [3], non-adaptive [4] and adaptive [5] sliding mode control, nonlinear feedback [6,7], and model-based control [8]. Coordinated vehicle-arm control with model-based compensation of hydrodynamic disturbances has been proposed in [9]. In contrast to the difficulties encountered in the reliable estimation of system dynamics, non-model-based control methods offer significant advantages for complex, nonlinear, uncertain systems such as UVMS, e.g., ease of implementation and potential for learning of hydrodynamic disturbances and parametric variations. Computational intelligence techniques based on fuzzy logic, neural networks, and genetic algorithms have been extensively studied for process control systems, manipulators in free space, and for mobile robots, e.g., [10]. Several authors have recently applied them for the planning, navigation, and control of AUVs or underwater manipulators, e.g., [11–13]. However, these methods have not so far been studied in the case of UVMS. An experimental study of conventional error-based iterative learning control for a prototype underwater vehicle with an one-link manipulator has been recently presented in [14]. In this paper, we propose an intelligent controller based on the neuro-fuzzy approach for precise, robust, and energy-efficient control of UVMS. The method is not model-based, and the design is decentralized. Therefore, the proposed controller is easy to implement in practice. The controller is of PD type, with fuzzy tuning of the feedback gains using modified fuzzy membership functions. Further, the magnitude of the gains, and hence the energy expenditure, is reduced by neural network-based identification of the system dynamics as well as hydrodynamic disturbances. The effectiveness of the new approach is illustrated with simulation studies of a six degrees of freedom (DOF) AUV with a three DOF on-board manipulator.
2. Dynamics of underwater vehicle-manipulator systems Several researchers have addressed the modeling of underwater vehicle-manipulator systems, e.g., [3,15–17]. In this paper, based on one of the models [17], the dynamics of an
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Fig. 1. An underwater vehicle-manipulator system.
underwater vehicle-manipulator system as shown in Fig. 1 is expressed as _ qðtÞ _ € þ HA ðqðtÞÞ_v ðtÞ þ SðqðtÞ, qðtÞÞ HðqðtÞÞqðtÞ _ þ CsgnðqðtÞÞ _ þ DðqðtÞ,vðtÞÞvðtÞ þ BqðtÞ þ gðqðtÞÞ þ bðqðtÞÞ ¼ uðtÞ
ð1Þ
The matrices and vectors in Eq. (1) are as follows: T qðtÞ ¼ qTv ðtÞ,qTm ðtÞ is the (6þn)x1 vector, where qv ðtÞ 2 R61 is the vector of vehicle positions and orientations, qm ðtÞ 2 Rn1 is the vector of manipulator joint angles in the bodyfixed reference frame, and vðtÞ 2 Rð6þnÞm is the flow velocity on a segment of the UVMS. A strip theory [16] is applied to the above modeling using the flow velocity vðtÞ. H(q) is the _ is a matrix positive-definite UVMS inertia matrix, and HA(q) the added inertia matrix. Sðq, qÞ of Coriolis and centripetal forces, and D(q,v) is a hydrodynamic damping tensor. B and C are viscous friction and Coulomb friction coefficient matrices, g(q) and b(q) are vectors of gravity and buoyancy forces, respectively, and u is the (6þn) input torque vector. To simplify the controller design, we transform the composite system into a set of interconnected second-order systems. Let ! ! qi 0 0 1 xi ¼ , bi ¼ , Ai ¼ ,and q_ i mii 0 0 0
0
1
6þn C B X B mij uj þ fi C C B Fi ¼ B C C B j¼1 A @ jai
i ¼ 1,2,:::ð6 þ nÞ
where mij are elements of the inverse of the inertia matrix. The dynamics in Eq. (1) can then be written compactly as x_ i ¼ Ai xi þ bi ui þ Fi
ð2Þ
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where Fi is treated as a disturbance term, accounting for uncertainties in vehicle-manipulator dynamics as well as hydrodynamic forces. fi is the i-th component of the vector _ þ g þ bÞ H 1 ðHA v_ þ S q_ þ Dv þ Bq_ þ CsgnðqÞ 3. Fuzzy logic controller design 3.1. Conventional PD controller In the trajectory tracking problem, the state trajectory is required to track the nominal state trajectory xr ðtÞ,t 2 ½0,T, where T is the terminal time. We assume that the desired velocity and acceleration trajectories x_ r ðtÞ, x€ r ðtÞ, respectively, exist and are continuous over t 2 ½0,T: Let the i-th subsystem trajectory tracking error be ! ei Ei ¼ ¼ xi xri e_ i The PD law for the i-th subsystem is set as ui ¼ Kpi ei Kdi e_ i
ð3Þ
where Kpi and Kdi are proportional and derivative gains, respectively. 3.2. Fuzzy gain-tuning Considering the decentralized, non-model-based PD control law (Eq. (3)), the values of feedback gains need to be chosen large enough to ensure high performance in the presence of dynamic uncertainties and disturbances. However, high feedback gains are preferably avoided due to the effects of actuator saturation and from the view point of the energy consumption particularly since AUVs have limited on-board power supply. In order to avoid the problems of gain tuning with conventional PD controller and introduce an element of adaptation, in this section we propose the use of a Mamdani-type fuzzy logic controller (FLC) to adaptively tune these gains based on several sets of fuzzy rules. The structure of the proposed fuzzy PD controller is shown in Fig. 2. The general form for the j-th rule of a single-input-single-output Mamdani-type FLC is as follows: rule Rj : IF xi is Ai,j THEN yi is Bi,j j ¼ 1,2,. . .,m ð4Þ
Fig. 2. Fuzzy PD controller for UVMS.
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where xi is the input variable (tracking error), yi is the output variable (feedback gain), and Ai,j and Bi,j are the fuzzy sets for xi and yi, respectively. Next, we fuzzify the normalized xi by means of predefined membership function mAi,j ðxi Þ of the elements xi as in Fig. 3. For simplicity of implementation, we use triangular membership functions. Finally, we use the correlation-minimum inference method to geometrically sum the consequences of the active associations and apply centroid defuzzification to find the crisp values of normalized yi. Then, we denormalize it to obtain the actual output value (gain). The details of this procedure are omitted here due to limitations of space. The proptional gain Kp has the effect of reducing the rise time and the steady-state error. The larger it is the faster and more robust will be the system response. However, usually lower gains are preferred in practice in order to avoid an oscillatory response. The fuzzy rule base as shown in Table 1 is based on the heuristics that if the tracking error is large, the proportional gain is also expected to be large in order to force the system to track the desired trajectory as accurately and as speedily as possible. The fuzzy logic tuning for the derivative gain Kdi is set up similarly based on the error derivative. 3.3. Fuzzy gain-tuning with modified membership functions In order to further reduce the energy consumption and improve the accuracy of the control, a so-called linguistic hedge (also known as ‘‘linguistic modifier’’) is adopted next. The linguistic hedge is a function that tunes the membership function of the fuzzy set associated with the linguistic label, obtaining a definition with a higher or lower precision depending on the case. Two of the most well known modifiers are the concentration linguistic hedge ‘‘very’’ and the
Fig. 3. Fuzzification of xi. Table 1 Fuzzy rule base for gain Kpi. Fuzzy rules
Kpi
jei j Zero
Small
Medium
Large
Zero
Small
Medium
Large
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Fig. 4. Effects of linguistic hedges on membership functions.
dilation linguistic hedge ‘‘more-or-less’’ [18]. Fig. 4 shows the differences among the standard triangular membership function (T), the ‘‘very’’ one, and the ‘‘more-or-less’’ one. The expressions for the two modifiers are given by a mvery T ¼ ðmT Þ
ð5Þ
¼ ðmT Þ1=a mmoreorless T
ð6Þ
where a41. Using the linguistic membership functions, the symbolic forms of the fuzzy rules are altered as, e.g. rule Rj : IF xi is more or less Ai,j THEN yi is more or less Bi,j j ¼ 1,2,:::,m. Or rule Rj : IF xi is very Ai,j THEN yi is very Bi,j j ¼ 1,2,:::,m. By changing the fuzzy membership function’s surface structure, we find that the dilation linguistic hedge ‘‘more-or-less’’ is able to provide the most accurate performance with the least energy consumption. The reason for this is that the ‘‘very T’’ is more likely to be a crispy value which means it is more like the conventional PD controller. In contrast, the ‘‘more-or-less T’’ is obviously more flexible which means that it behaves more like an adaptive controller [18]. Therefore, the ‘‘more-or-less’’ surface has advantages including high performance and low energy consumption. The triangular and modified triangular membership functions are simple and computationally simple to implement in real-time. Nonlinear membership functions like Gaussian and generalized Bell provide smooth variation and can also be used in conjunction with neural networks, though computational complexity in implementation must be taken into account. The computational requirements involving fuzzy rule sets might pose a constraint in situations with limited on-line computation resources, e.g., in microcontroller-based implementation. In such cases, a simple weighting function can be used to reduce computations. Fuzzy look-up tables could also be used, though defuzzification will still take up some computing time. It must be remarked, however, that computational resources are becoming increasingly cheaper and more powerful, so fuzzy control algorithms may still be implemented in real-time with little modifications. 4. Neural network compensation Neural networks have the characteristic of approximating highly nonlinear functions. In order to optimize the performance of the controllers for UVMS, the neural network is adopted as a compensator for PD and fuzzy PD controllers. The idea is to use the wellknown backpropagation method to roughly approximate the dynamics of each degree of freedom in the form of the decentralized neural network [10]. The controller is of the form ui ¼ Kpi ei Kdi e_ i þ uNN i
ð7Þ
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where Kpi and Kdi can be the propotional and derivative gains of conventional PD controller, standard fuzzy PD, or fuzzy PD with modified membership functions. uNNi is the output of the neural network compensator. By comparison with Eq. (2), we see that the neural net-based feedforward compensation of the system dynamics means that the magnitudes of the feedback gains in Eq. (7) can be much smaller, resulting in the same advantages provided by the model-based control methods which are more difficult to implement. The robustness of neural network control schemes to hydrodynamic disturbances has been demonstrated in the case of autonomous underwater vehicles as well as large ships, e.g., [13,19]. € q, _ and q. From Eq. (1), we note that the input u of the UVMS is a function of q, € q, _ and q. Therefore, the neural network is required to approximate the relations of u to q, However, due to the high number of degrees of freedom of the UVMS and its high nonlinearity as well as uncertainty, it is usually computationally expensive to regress the overall dynamics. Therefore, the decentralized form of neural network compensator is adopted here. Decentralized implementations of neural network-based controllers have been shown to have bounded tracking errors for interconnected nonlinear systems with strong interactions [20], and precise tracking performance in uncertain systems such as robot manipulators [21]. Instead of learning the overall dynamics, we approximate the dynamics link by link. In this case, the expense of the computation will be largely decreased. Another advantage to use the decentralized form is that it is easy to implement in practice. However, because UVMS is a highly coupled system with high degrees of freedom, the neural network regressor provides approximate compensation for the conventional PD or Fuzzy PD controller which is the main controller here. As the neural network is mainly used to reduce the magnitudes of required gains using feedforward compensation, it is sufficient for the decentralized estimator to have an acceptable degree of accuracy. The training method is based on conventional off-line backpropagation. To train the neural network, we first apply the conventional PD controller on the UVMS system, and record the data of q€ di , q_ i , qi, and the output ui of the PD controller for each link. In order to generalize the neural network dynamics for different trajectories, we use different
Fig. 5. Structure of the neural network of each link.
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desired trajectories including high and low frequency sine waves and cycloid waves when we generate the training data. The neural network has input, hidden, and output layers. We assign six neurons in the hidden layer with sigmoid output funtions. The output layer uses pure linear function. The three-layer neural network is theoretically sufficient to approximate any arbitrary nonlinear funtions. The reason we choose q€ di instead of q€ i is that q€ i is usually quite noisy and often not available in practice. The structure of the proposed neural network is as shown in Fig. 5. 5. Simulation results Numerical simulations have been conducted in order to illustrate the effectiveness of the proposed neuro-fuzzy controller. The simulator was developed in Visual Cþþ. 5.1. Simulation model In this paper, a 9-DOF AUV-type UVMS has been considered. The vehicle has six degrees of freedom comprising translational motions x, y, z and rotational motions pitch, roll, yaw. The manipulator with three rotational degrees of freedom is mounted on the bottom of the vehicle with the joint axes parallel to the vehicle z-axis initially. We assume the AUV to be a cylinder-shaped tube deployed in water. The parameters of the assumed model are shown in Table 2. For details, see [22]. In practice, PD control gains are selected by a systematic method such as one of various tuning techniques [23]. However, in our work the PD gains are mainly used as a base for performance comparison with neuro-fuzzy control with similar gains. Therefore, the PD controller gains were chosen sufficiently large to provide good performance with nominal payload. The neuro-fuzzy controller’s error-dependent PD gains were specified to be mainly lower than the fixed PD gain values, though for some degrees of freedom they were found to occasionally exceed the fixed gain values. In the case of fuzzy rule implementation, based on simulation runs for position control using nominal gain values, errors greater than 60 degrees were treated as large, so the errors are normalized by dividing by 60. For trajectory gains, the peak errors were typically much smaller in magnitude, and correspondingly large error values were selected as maximum. The neural net training trajectories were specified to be sinusoidal with peak amplitudes of 1.0 and frequency of 0.2 Hz. Extensive simulations have been conducted to analyze and compare the performance of the conventional PD and fuzzy logic (triangular membership function) controllers with the modified fuzzy controller (very T and more or less T membership functions) and the proposed neurofuzzy controller. Due to limitations of space, only selected results are presented in this paper. Table 2 Dynamic parameters of 9-DOF UVMS. Description
Vehicle
Link 1
Link 2
Link 3
Length (m) Diameter (m) Mass (kg)
0.80 0.10 15.0
0.20 0.025 1.0
0.20 0.025 1.0
0.20 0.025 1.0
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Figs. 6 and 7 show, respectively, the total tracking error norm and the power consumption for the AUVMS, for fuzzy PD controllers with the three types of membership functions in Fig. 4. It is evident that with respect to both the measures, the ‘‘more or less’’ fuzzy PD control method outperforms both the conventional fuzzy PD control and the ‘‘very T’’ fuzzy PD control. This is because the dilation hedge ‘‘more or less’’ has greater adaptability with changes in tracking errors. So, in the rest of the paper we consider the performance of the neuro-fuzzy PD controller with the ‘‘more or less’’ type membership function.
Fig. 6. Combined tracking error norm for fuzzy PD controllers.
Fig. 7. Combined expended energy norm for fuzzy PD controllers.
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Similarly, to consider the effectiveness of the proposed decentralized neural networkbased compensation method, we have compared the conventional PD control method with the PDþ neural net compensation method. In terms of individual tracking errors, robustness, as well as combined tracking error norm and expended energy norm, the PD controller with compensation was found to outperform the regular PD controller. Fig. 8 shows the comparison of error norms. To consider the robustness of the neuro-fuzzy controller, the payload mass is increased from 0 to 2 kg at t¼ 4 s. The resulting responses of the conventional PD controller and the neuro-fuzzy controller are compared in Figs. 9–11. For desired constant velociy motions along the AUV x, y, and z axes, the pitch, roll, and yaw angles are required to be constant, while the manipulator joint angles are specified to move along cycloidal trajectories. It is clear that the neuro-fuzzy control method is quite precise and robust to payload variations. The slight delay in response of the system at t¼ 0 is attributable to the effects of friction. The results shown in Figs. 6–11 are obtained for the case of static water conditions. When the current velocities are increased for the above cases, it is again found that the neuro-fuzzy controller significantly outperforms the conventional PD controller as well as the fuzzy PD controller with various membership function shapes. For the case when the water flow velocity along the reverse direction of the x-axis is changed from 0 to 2 knots, the resulting performances of the PD and neuro-fuzzy PD controller are summarized in Figs. 12 and 13. Fig. 12 shows the RMS tracking error along the AUV’s x-axis. It is again obvious that the neuro-fuzzy controller is significantly superior to the conventional PD control method. The simulation studies presented here have not explicitly considered system uncertainties. However, the controller implementation is not model-based and the neural network training is based on results of implementation of a PD-type controller for typical test trajectories and so would include in practice the effects system or environmental uncertainties. Effects of sensor noise will be considered in future studies on experimental implementation.
Fig. 8. Comparision of PD and PDþneural net compensation.
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Fig. 9. Position tracking: PD (
), neuro-fuzzy PD controllers (- - -); desired trajectory:
Fig. 10. Pitch, roll, and yaw tracking: PD (
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.
), neuro-fuzzy PD controllers (- - -); desired trajectory:
.
The PD control used in the paper is a subsystem/joint-based independent control, and the NN control is used mainly to add feedforward compensation to the PD/fuzzy controller. This separates the control action between fuzzy logic and NN methods for
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Fig. 11. Arm trajectory tracking: PD (
), neuro-fuzzy PD controllers (- - -); desired trajectory:
.
Fig. 12. RMS error of AUV x-axis motion.
feedback gain tuning and feedforward control, respectively, making implementation direct and straight-forward. Depending on additional performance improvements desired and the affordability of computational complexity, the proposed method can be more sophisticated, for example neural net-based selection or tuning of fuzzy rules, layers, weights, and similarly selection of neural net layers and parameters using fuzzy rules, e.g., [13]. On the other hand, depending on additional performance desired further improvements can also be made to the basic backpropagation neural net employed in this paper, e.g., adding additional
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Fig. 13. Comparison of total energy consumption.
hidden layer(s), tuning of neuronal weights to ensure passivity, and stability of tracking error dynamics, guaranteed tracking, and so on [10]. The global stability of the proposed method is not addressed in this paper due to limitations of space. For a complex uncertain system such as an AUVMS, it is necessary to incorporate into the system model significant practical effects such as actuator saturation and other nonlinear effects. Given the major advantages of intelligent controllers over fixed gain PD controllers, several researchers have addressed the stability of neuro-fuzzy controllers for critical systems such as hazrdous environments in nuclear reactors [24] and underwater vehicles [25]. 6. Conclusions A neuro-fuzzy controller for underwater vehicle-manipulator systems has been proposed in this paper. The performance of the fuzzy PD control method has been improved by the tuning of triangular membership functions using linguistic hedges. To further improve the precision and robustness with reduced feedback gains, a decentralized neural network compensator for estimation of the system dynamics has been developed. The controller design is decentralized, making it easy to implement. Simulation results for a six DOF AUV with a three DOF on-board manipulator show that the proposed neuro-fuzzy controller significantly outperforms the conventional PD controller in terms of robustness to payload changes and hydrodynamic disturbanses, and has superior performance in terms of tracking error norms and expended energy norms. References [1] U.S. Department of Defense, Developing Science And Technologies List, Section 13 Marine Systems Technology, July 2002. [2] S.R. Pandian, N. Sakagami, System integration aspects of underwater vehicle-manipulator systems for oceanic exploration, Journal of the Society of Instrument and Control Engineers (Japan) 47 (2008) 830–836. [3] I. Scholberg, T. Fossen, Modeling and control of underwater vehicle-manipulator systems, in: Proceedings of the Conference Marine Craft Maneuvering and Control, Southampton, UK, 1994, pp. 45–57.
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[4] B. Xu, S. Abe, N. Sakagami, S.R. Pandian, A robust nonlinear controller for underwater vehiclemanipulator systems, in: Proceedings of ASME/IEEE International Conference on Advanced Intelligent Mechatronics, Monterey Bay, CA, 2005, pp. 711–716. [5] Y. Taira, M. Oya, S. Sagara, An adaptive controller for underwater vehicle manipulator systems including thruster dynamics, in: Proceedings of the International Conference on Modeling, Identification, and Control, Okayama, Japan, 2010, pp. 185–190. [6] S. Soylu, B.J. Buckham, R.P. Podhorodeski, Development of a coordinated controller for underwater vehicle-manipulator systems, in: Proceedings of the IEEE International Conference on Oceans, Quebec, Canada, 2008, pp. 1–9. [7] J. Han, W.K. Chung, Coordinated motion control of underwater vehicle-manipulator system with minimizing restoring moments, in: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Nice, France, 2008, pp. 3158–3163. [8] N. Sarkar, T.K. Podder, Coordinated motion planning and control of autonomous underwater vehiclemanipulator systems subject to drag optimization, IEEE Journal of Oceanic Engineering 26 (2001) 228–239. [9] H.H. Wang, S.M. Rock, OTTER: The design and development of an intelligent underwater robot, Autonomous Robots 3 (1996) 297–320. [10] R. Yusof, R.Z.A. Rahman, M. Khalid, M.F. Ibrahim, Optimization of fuzzy model using genetic algorithm for process control application, Journal of The Franklin Institute 348 (2011) 1717–1737. [11] S.R. Pandian, N. Sakagami, Neuro-fuzzy control of an underwater robot manipulator, in: Proceedings of the International Conference on Advanced Robotics, Control, and Computer Vision, Singapore, 2010, pp. 2135–2140. [12] F. Song, S.M. Smith, Design of sliding mode fuzzy controllers for an autonomous underwater vehicle without system model, IEEE/MTS Oceans Conference Record, Providence, RI, 2000, pp. 835–840. [13] J.S. Wang, C.S.G. Lee, Self-adaptive recurrent neuro-fuzzy control of an autonomous underwater vehicle, IEEE Transactions on Robotics and Automation 19 (2003) 283–295. [14] N. Sakagami, Precise control of underwater vehicle manipulator systems using iterative learning control, in: Proceedings of the ICROS/SICE International Joint Conference, Fukuoka, Japan, pp. 3089–3093. [15] K. Ioi, K. Itoh, Modeling and simulation of an underwater manipulator, Advanced Robotics 4 (1990) 303–317. [16] T.W. McLain, S.M. Rock, Experiments in the hydrodynamic modeling of an underwater manipulator, IEEE Conference on the Proceedings of the Symposium on Autonomous Underwater Vehicle Technology, 1996, pp. 463–469. [17] G. Antonelli, Underwater Robots: Motion and Force Control of Vehicle-Manipulator Systems, Springer, 2006. [18] J. Casillas, O. Cordon, M.J. del Jesus, F. Herrera, Genetic tuning of fuzzy rule deep structures preserving interpretability and its interaction with fuzzy rule set reduction, IEEE Transactions on Fuzzy Systems 13 (2005) 13–29. [19] Y. Zhang, G.E. Hearn, P. Sen, A neural network approach to ship track-keeping, IEEE Journal of Oceanic Engineering 21 (1996) 513–527. [20] S.N. Huang, K.K. Tan, T.H. Lee, Decentralized control of a class of large-scale nonlinear systems using neural networks, Automatica 41 (2005) 1645–1649. [21] L. Acosta, G.N. Marichal, L. Moreno, J.J. Rodrigo, A. Hamilton, J.A. Mendez, A robotic system based on neural network controllers, Artificial Intelligence in Engineering 13 (1999) 393–398. [22] B. Xu, Fuzzy Control Underwater Vehicle-Manipulator Systems, M.S. Thesis, Electrical Engineering and Computer Science Department, Tulane University, 2007. [23] K.J. Astrom, T. Hagglund, PID Controllers: Theory, Design and Tuning, International Society of Measurement and Control Press, 1995. [24] Y. Hacioglu, Y.Z. Arslan, N. Yagiz, MIMO fuzzy sliding mode controlled dual arm robot in load transportation, Journal of the Franklin Institute 348 (2011) 1886–1902. [25] A. Piegat, Stability of the PD-neuro-fuzzy control system of the underwater vehicle Krab II, in: T. Graczyk, T. Jastrzebski, C.A. Brebbia (Eds.), Marine Technology II, WIT Press, 1997.