136 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 15, NO. 1, FEBRUARY2010 Inverse Double NARX Fuzzy Modeling for System Identification Kyoung Kwan Ahn, Member, IEEE, and Ho Pham Huy Anh Abstract—In this paper, a novel inverse double nonlinear au- toregressive with exogenous input (NARX) fuzzy model is applied to simultaneously model and identify both joints of the prototype two-axis pneumatic artificial muscle (PAM) robot arm’s inverse dynamic model. Highly nonlinear features of both joints of the nonlinear manipulator system are identified by the proposed in- verse double NARX fuzzy (IDNF) model based on experimental input–output training data. The modified genetic algorithm (GA) optimally generates the appropriate fuzzy if–then rules to perfectly characterize the dynamic features of the two-axis PAM manipula- tor system. The evaluation of different IDNF models with various ARX model structures will be discussed. For the first time, the nonlinear IDNF model of the two-axis PAM robot arm is inves- tigated. The results show that the nonlinear IDNF model that is trained by GA performs better and has a higher accuracy than the conventional inverse fuzzy model. Index Terms—Dynamic system, genetic algorithm (GA), inverse double nonlinear autoregressive with exogenous input (NARX) fuzzy (IDNF) model, modeling and identification, two-axis pneu- matic artificial muscle (PAM) robot arm. I. INTRODUCTION T HE FIELD of intelligent models can be divided into two major classes, namely, the artificial neural networks (ANNs) and fuzzy models. Because of their approximation ca- pabilities, ANNs have been popularly employed for modeling, identification, and control [1], [2]. The output of a feedforward ANN depends on its current inputs, and thus, it can model only memoryless representations. However, for identification of dy- namical systems, recurrent neural networks should be employed that exhibit some kind of memory. The recurrent models can be regarded as closed-loop systems, with the feedback paths intro- ducing dynamics to the model; they can learn the system dy- namics without assuming much knowledge about the structure of the system under consideration. Abdollahi et al. [3] success- fully constructed a stable identification method for a nonlinear system using a recurrent neural model. Wei et al. presented a new neural model that was designed to produce a robust and smooth trajectory control of the servo manipulator system [4]. Gueaieb et al. [5] applied a hybrid neural model to force con- trol of the highly nonlinear cooperative manipulators. The main Manuscript received June 3, 2008; revised January 17, 2009. First published August 21, 2009; current version published November 18, 2009. Recommended by Technical Editor J. Ueda. K. K. Ahn is with the School of Mechanical and Automotive Engineering, University of Ulsan, Ulsan 680-749, Korea (e-mail: kkahn@ulsan.ac.kr). H. P. H. Anh is with the Graduate School of Mechanical and Auto- motive Engineering, University of Ulsan, Ulsan 680-749, Korea (e-mail: hphanh@hcmut.edu.vn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2009.2020737 drawback of most neural models is that neural structure is un- clear, and thus, it is hard to know how to modify neural model’s parameters. The combination of neural ANN model and fuzzy model contributes various neuro-fuzzy models that provide another in- teresting approach to solving the challenging nonlinear model problem that the nonlinear dynamic system requires. The neuro- fuzzy models are often hybrid systems in which the fuzzy techniques are used to create or enhance neural networks. Sun et al. [6] tried to install an inverse neuro-fuzzy model to rep- resent the dynamic characteristics of the dynamic manipulator system. Wai and Chen applied a robust neural fuzzy model to identify the robot manipulator’s dynamics [7]. Tsagarakis and Darwin developed an improved neuro-fuzzy model for the pneumatic artificial muscle (PAM) system [8]. Hui and Woo combined a sliding mode control with the neuro-fuzzy model to obtain a robotic manipulator control [9]. Additionally, some authors in [10]–[12] applied different sophisticated types of neuro-fuzzy models to identify and model the nonlinear dynamic system. Furthermore, based on neural and neuro-fuzzy models, many authors have designed various intelligent models to con- trol nonlinear robotic systems [13]–[16]. The disadvantage of all these neuro-fuzzy models is that this approach, which is mostly based on expert knowledge, seems too complex for industrial applications. The potentially approximating capability of the fuzzy model and the transparence of the fuzzy system’s structure make fuzzy model as another interesting approach to modeling and identify- ing highly nonlinear dynamic systems. Chan et al. developed a fuzzy model reference learning controller that was designed for a single PAM actuator [17]. Lilly and Chang [18] introduced a fuzzy proportional–integral–derivative (PID) controller in 2003 for the PAM manipulator system that was studied by Chan et al. [17]. Unfortunately, these fuzzy models seem rather clumsy and do not provide adequate precision. Balasubramanian and Rattan applied a feedforward + PID fuzzy controller to the one-link PAM system [19], [20]. Unfortunately, these models seem to be too specific to apply in practice. The authors in [21] and [22] tried to apply some learning algorithm in practical system to derive hybrid fuzzy models in order to control the nonlinear dynamic system. The disadvantage of these designed fuzzy models is that they seem too complex to be efficiently applied because of the lack of necessary expert knowledge. Recently, researchers have begun to pay attention to the in- tegration of the nonlinear autoregressive with exogenous in- put (NARX) model into the intelligent model [23], [24]. The strongly predictive potential of the NARX model, when inte- grated into the intelligent model, gives us a promising approach 1083-4435/$26.00 © 2009 IEEE