Adaptive Learning with Large Variability of Teaching Signals for Neural Networks and Its Application to Motion Control of an Industrial Robot

International Journal of Automation and Computing 8(1), February 2011, 54-61 DOI: 10.1007/s11633-010-0554-0 Adaptive Learning with Large Variability of Teaching Signals for Neural Networks and Its Application to Motion Control of an Industrial Robot Fusaomi Nagata 1 Keigo Watanabe 2 1 Department of Mechanical Engineering, Faculty of Engineering, Tokyo University of Science, Sanyo-Onoda 756-0884, Japan 2 Department of Intelligent Mechanical Systems, Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan Abstract: Recently, various control methods represented by proportional-integral-derivative (PID) control are used for robotic control. To cope with the requirements for high response and precision, advanced feedforward controllers such as gravity compensator, Coriolis/centrifugal force compensator and friction compensators have been built in the controller. Generally, it causes heavy computational load when calculating the compensating value within a short sampling period. In this paper, integrated recurrent neural networks are applied as a feedforward controller for PUMA560 manipulator. The feedforward controller works instead of gravity and Coriolis/centrifugal force compensators. In the learning process of the neural network by using back propagation algorithm, the learning coefficient and gain of sigmoid function are tuned intuitively and empirically according to teaching signals. The tuning is complicated because it is being conducted by trial and error. Especially, when the scale of teaching signal is large, the problem becomes crucial. To cope with the problem which concerns the learning performance, a simple and adaptive learning technique for large scale teaching signals is proposed. The learning techniques and control effectiveness are evaluated through simulations using the dynamic model of PUMA560 manipulator. Keywords: Neural networks, large-scale teaching signal, sigmoid function, adaptive learning, servo system, PUMA560 manipulator, trajectory following control, nonlinear control. 1 Introduction In this decade, open architecture industrial robots have been produced from several industrial robot makers such as KAWASAKI Heavy Industries, Ltd., MITSUBISHI Heavy Industries, Ltd. and YASKAWA Electric Corp., and so on. Open architecture described in this article means that the servo system and kinematics of the robot are technically opened, so that various applications required in industrial fields are allowed to be planned and developed at the user side. For example, non-taught operation by using a CAD/CAM system can be considered due to the opened accurate kinematics. Also, force control strategy that uses a force sensor can be easily implemented on the technically open discrete-time servo system [1]. It is now possible to model and simulate many types of robots. For example, Chen et al. [2] presented a new design of an environment for simulation, animation, and visualization of sensor-driven robots. Although conventional computer-graphics-based robot simulation and animation software packages lack the capabilities for robot sensing simulation, the system was designed to overcome the deficiency. Also, Benimeli et al. [3] addressed the implementation and comparison of an indirect identification procedure and a direct identification procedure on an industrial robot provided with an open control architecture. The estimation of dynamic parameters in mechanical systems constituted an issue of crucial importance for dynamic simulations where high accuracy was required. Manuscript received March 2, 2010; revised June 17, 2010 This work was supported by Grant-in-Aid for Scientific Research (C) (No. 20560248) of Japan. We have introduced a simulation technique of velocitybased discrete-time control system for open architectural industrial robots [4]. In order to develop a novel velocitybased control system, which is represented in discrete-time domain for an open architecture industrial robot, it is required from the viewpoint of safety, cost and easiness to preliminarily examine and evaluate the characteristics and performance. In such a case, the proposed simulation techniques are useful. Neural network, which is one of the representative intelligent control approaches, was paid attention to improve the control performance of robotic systems. Han and Moraga [5] reported a fundamental and important result in which a sigmoid function was employed in each unit. A variant sigmoid function with three parameters that denoted the dynamic range, symmetry and slope of the function was discussed. How these parameters influence the speed of back propagation learning was illustrated, and a hybrid sigmoidal network with different parameter configuration in different layers was introduced. The error signal problem, oscillation problem and asymmetrical input problem could be handled by regulating and modifying the sigmoid function parameter configuration in different layers. As for the application of neural networks to robotic control, many control systems and their learning techniques have been developed. For example, Torras [6] described that to carry out their tasks autonomously in unknown environments, it is essential for robots to adapt to their environments. The most effective method to endow robots with this capability is to the use of neural networks.

F. Nagata and K. Watanabe / Adaptive Learning with Large Variability of Teaching Signals for 55 The related techniques were explained with mobile robots and manipulator arms. Hasan et al. [7] proposed an adaptive learning strategy using an artificial neural network to control the motion of a manipulator robot with six degreeof-freedom and to overcome the inverse kinematics problem, which are mainly singularities and uncertainties in arm configurations. Horng [8] showed that Levenberg-Marquardt back-propagation (LMBP) has faster convergence if compared with other three back-propagation modified algorithms, and it is suitable for system identification and controller design. Matlab/Simulink and LabVIEW with neural networks were successfully integrated to develop a supervisory control and data acquisition (SCADA) system of AC servo motor. Also, Huang et al. [9] developed an adaptive neural network algorithm for a class of interconnected nonlinear systems. Neural networks were applied to approximate the unknown nonlinear functions and interconnections in the subsystems. A systematic approach was established to synthesize the adaptive learning control scheme. The effectiveness was demonstrated by simulations. Furthermore, [10, 11] Wu et al. conducted the stability analysis of neural networks with time-varying delay. Up to now, however, it seems that the consideration about the scale of teaching signal in the output layer has not been sufficiently taken yet. Recently, various control methods represented by proportional-integral-derivative (PID) control are used for robotic control. To cope with the requirements for high response and precision, advanced feedforward controllers such as gravity compensator, Coriolis/centrifugal force compensator and friction compensators have been built in the controller. However, it generally requires a serious computational effort to calculate the compensating value within a short sampling period. In this paper, a recurrent neural network is applied for a feedforward controller of PUMA560 manipulator, which is a representative of industrial robots with six degree-offreedom. Hereafter, the recurrent neural network is called RNN. The feedforward controller includes gravity compensator and Coriolis/centrifugal force compensator. Also, a simple and adaptive learning technique is proposed for a robotic servo system to deal with large-scale joint driving torques. The learning technique and control effectiveness are evaluated through simulations using the dynamic model of PUMA560 manipulator as shown in Fig. 1. The remainder of this paper is organized as follows: Section 2 explains the computed torque control method used as a model-based servo system. Then, teaching signals for RNNs are calculated. In Section 3, integrated RNNs are designed to compensate nonlinear terms such as gravity term and Coriolis/centrifugal force term. Furthermore, a simple and adaptive learning technique by using a scaler is proposed for large-scale teaching signals processed in back propagation algorithm, then the effectiveness is discussed. In Section 4, a servo system using the integrated RNNs is applied to a trajectory following control problem. The result demonstrates that the integrated RNNs can work well instead of gravity and Coriolis/centrifugal force terms. Finally, the conclusions are presented. 2 Model-based robotic servo system In order to simulate the motion of an industrial robot, first of all, a servo system is considered and designed. Here, the computed torque control method [12, 13] is applied in a servo system. The computed torque control method is used for nonlinear control of an industrial manipulator, which is composed of a model based portion and a servo portion. The servo portion is a closed loop with respect to the position and velocity. On the other hand, the model based portion has the inertia term, gravity term and Coriolis/centrifugal force term, which work to cancel the nonlinearity of the manipulator. In order to realize high control stability, the position and velocity feedback gains used in the servo portion should be tuned suitably. In this section, a simple desired trajectory as shown in Fig. 1 is calculated for the computed torque control method. The computed torque control method generates joint driving torque to follow the trajectory. The desired trajectory and joint driving torque are used as teaching signals for neural networks. 2.1 Computed torque control method The dynamic model of a manipulator with friction term is generally given by M(θ) θ + H(θ, θ) + G(θ) + F r(θ, θ) = τ (1) where M(θ) R 6 6 is the inertia term in joint space. H(θ, θ) R 6 1 and G(θ) R 6 1 are the Coriolis/centrifugal force term and gravity term, respectively. F r(θ, θ) R 6 1 is the friction term consisting of viscous friction and Coulomb friction. θ R 6 1, θ R 6 1 and θ R 6 1 are the angle, angle velocity, and acceleration vectors in joint coordinate system, respectively. τ R 6 1 is the joint driving torque vector. In the case that the computed torque control law is employed in the servo system of a manipulator, the desired angle, angle velocity and acceleration vectors in joint coordinate system are respectively given to the references of the servo system, so that the joint driving torque can be calculated from τ = ˆM(θ) [ θr + K v{ θ r θ} ] + K p{θ r θ} + Fig. 1 Image of a desired trajectory composed of a sine curve Ĥ(θ, θ) + Ĝ(θ) (2) where, ˆM, Ĥ(θ, θ), and Ĝ(θ) denote the modeled terms. θ r R 6 1, θr R 6 1, and θ r R 6 1 are the desired

56 International Journal of Automation and Computing 8(1), February 2011 angle, angle velocity and acceleration vectors, respectively. K v = diag{k v1,, K v6} and K p = diag{k p1,, K p6} are the feedback gains of velocity and position, respectively. Note that θ, θ in (2) are actual values, i.e., controlled variables. The nonlinear compensation terms ˆM(θ), Ĥ(θ, θ) and Ĝ(θ) are calculated by using the recursive Newton- Euler formulation [14] to cancel the nonlinearity and are effective to achieve stable trajectory control. The block diagram of the computed torque control method is shown in Fig. 2. Although the gravity term and Coriolis/centrifugal force term are ordinarily obtained by computing the inverse dynamics through recursive Newton-Euler formulation, the computational load within a sampling period is considerably large. Fig. 3 Desired joint angle vector θ r R 6 1 realizing the trajectory shown in Fig. 1 Fig. 2 Block diagram of the computed torque control method, in which θ r, θ r, and θ r are the desired angle, velocity and acceleration vectors in joint coordinate system 2.2 Teaching signal for RNN As an example, the desired trajectory in Cartesian space has been designed as shown in Fig. 1. The tip of the robot arm draws a sine curve in x-y plane. The trajectory in Cartesian space is resolved into the trajectories in joint coordinate system by using the inverse kinematics of PUMA560, which are shown in Fig. 3. Also, Fig. 4 shows the joint driving torques needed for realizing the trajectory shown in Fig. 1, which are obtained through a trajectory following simulation by using (2). The simulation was carried out by using the dynamic model of PUMA560 manipulator on Matlab system [14 16]. The robotic dynamic model with the friction torque term given by (1) was applied. The friction torque term F r(θ, θ) consists of the viscous friction torque and Coulomb friction torque, which is represented by F r(θ, θ) = BG 2 r θ + G rτ c{sign( θ)} (3) where B is the coefficient matrix of viscous friction at each motor, G r is the reduction gear ratio matrix which represents the motor speed to joint speed, and τ c{sign( θ)} is the Coulomb friction torque, which appeared at each motor. If sign( θ i) > 0, then τ ci = τ + ci ; if sign( θ i) < 0, then τ ci = τ ci. Also, τ ci becomes 0 in case of sign( θ i) = 0. In simulations, B and G r are set to diag{0.0015, 0.0008, 0.0014, 0.0001, 0.0001, 0.0000} and diag{ 62.6, 107.8, 53.7, 76.0, 71.9, 76.7}, respectively; τ + c and τ c are set to [0.395 0.126 0.132 0.011 0.009 0.004] T and [ 0.435 0.071 0.105 0.017 0.015 0.011] T, respectively [14]. In the next section, six RNNs are respectively tried to acquire the teaching signals composed of desired joint angle and joint driving torque. Fig. 4 Desired joint driving torque vector τ R 6 1 performing the trajectory shown in Fig. 1 3 Independent recurrent neural networks for an industrial robot with six joints 3.1 Adaptive learning of RNNs In order to learn the input/output relation shown in Figs. 3 and 4, an RNN is designed as shown in Fig. 5, in which the output signal with one sampling time delay is used as the feedback signal. Each unit has a standard sigmoid function as shown in Fig. 6. The RNN consists of an input layer, two hidden layers, and an output layer. Each layer has seven units, thirty units, and one unit, respectively. Note that the output layer does not have an activation function, i.e., sigmoid function. Therefore, the RNN directly yields a joint driving torque from the calculation of weighted sum. The RNN acquires the relation between the desired trajectory in joint space and joint driving torque. Although the improvement of learning performance in employing such a sigmoid function is significant, it seems that the adaptability for large-scale teaching signals has not been discussed clearly in the earlier studies. In this subsection, the learning process of RNN is explained in detail. The input to the hidden layers and output

F. Nagata and K. Watanabe / Adaptive Learning with Large Variability of Teaching Signals for 57 layer is a weighted sum of the previous layer. The sum is squashed into ±0.5 by the sigmoid function, which is given by f(x) = 1 0.5. (4) 1 + exp( X) The input/output relation of each unit is given by X i,l (k) = m j=1 w i,l j,l 1 (k) o j,l 1(k) (5) where X i,l (k) is the state of i-th unit in l-th layer when k-th teaching signal is applied, w i,l j,l 1 (k) is the interconnection weight between the i-th unit in l-th layer and the j-th unit in (l 1)-th layer, o j,l 1 (k) is the output of the j-th unit in (l 1)-th layer, m is the number of units in (l 1)-th layer. Back propagation algorithm is employed in the learning process. The learning was iterated until error function E became small sufficiently. For example, in the case of first joint, the error function E is calculated by E = 1 2 1000 { S1τ1 (k) τ } 2 1(k) k=1 where τ 1 (k) and τ 1(k) are the teaching signal and output of RNN at the discrete time k (1 k 1000) with respect to the first joint, respectively. That means the pattern number of teaching signals is 1000. Here, S 1 is an adaptive element called the down scaler, which is automatically extracted from the teaching signals in output layer as S 1 = (6) 1, if max( τ 1 (k) ) λ λ max( τ 1 (k) ), otherwise (7) where max( τ 1 (k) ) is the maximum absolute value in teaching signals. λ is determined by considering the activation area in which f(x) is steeply varying from about 0.5 to 0.5 as shown in Fig. 6. For example, when the gain of sigmoid function is 1 as given by (4), λ can be set to about 5. Fig. 5 Recurrent type neural network to learn the input/output relation between six joint angles θ r1,, θ r6 and driving torque τ i (i = 1,, 6) Fig. 6 Sigmoid function used in each unit allocated in input and hidden layers w i,l j,l 1 (k) is updated by the well-known steepest descent method given by w i,l j,l 1 (k + 1) = wi,l j,l 1 (k) η E w i,l j,l 1 (k) (8) where η is the learning coefficient. If l-th layer is the output one, i.e., l = 4, the update quantity for a weight is calculated by In the above equation, E w 1,4 = δ1,4 oj,3(k). (9) j,3 (k) δ 1,4 = { S 1τ 1 (k) τ 1(k) } f { X 1,4(k) }. (10) Also, in case of the third layer, the update quantity is computed by In the above equation, E = w i,3 δi,3 oj,2(k). (11) j,2 (k) δ i,3 = δ 1,4 w 1,4 i,3 (k) f { X i,3(k) }. (12) In the same manner, w i,2 j,1 (k) is updated through where δ i,2 = E = w i,2 δi,2 oj,1(k) (13) j,1 (k) 30 n=1 δ n,3 w n,3 i,2 (k) f { X i,2(k) }. (14) After sufficient learning, it is expected that the RNN will give an output similar to the torque curve of the first joint as shown in Fig. 4. Other five RNNs designed for the joints from the second to the sixth can be also learned in the same manner. If an unknown trajectory is given and the dynamic torque for teaching signal is out of the nonlinear range, then the torque is adaptively squashed into the range of about ±5 by using (6) and (7) in the learning process.

58 International Journal of Automation and Computing 8(1), February 2011 Fig. 7 Learning results of joint driving torques for six joints 3.2 Learning results of RNNs In this subsection, the learning process of RNN is described. If orders of six torques in Fig. 4 were almost the same, then the output layer could have six units. However, for example, the torque within the range from 30 N m to 5 N m is required for the second joint control. The range is much wider compared with the other five joints. In such case, it is difficult to construct a sufficiently learned RNN with six units in the output layer. When the scales of the teaching signals for each unit in the output layer are quite different, the learning does not go well as a whole. That is the reason why six RNNs are independently designed. Six RNNs are learned for the trajectory following control, respectively. At first, the down scaler vector S = diag{s 1,, S 6} is set to diag{1,, 1}. The learnings of six joints were respectively repeated 20 000 times in order that the values of the error functions converge to almost zero. However, the learning of the second joint could not be finished satisfactorily, i.e., the error did not decrease from the initial value. To overcome the problem concerning the learning, the second joint s teaching signals in the output layer were adaptively scaled down with S 2 = λ max( τ2 (k) ) = 5 0.17 (15) 30 i.e., by giving S = diag{1, 0.17, 1, 1, 1, 1}, so that the error of the second joint could be reduced to almost zero as shown in Fig. 7 (b). The errors of other joints are also shown in Fig. 7. Note that when the learned RNN for the second joint is used for a feedforward controller which can compensate the nonlinear terms such as gravity term Ĝ(θ) and Coriolis/centrifugal term Ĥ(θ, θ) in (2), the output of the second joint must be scaled up with S 1 2 by multiplying S 1 = diag{1, 6, 1, 1, 1, 1}. As it can be seen, the learning is well performed as shown in Fig. 7. Strictly speaking, however, the dynamic BP algorithm is more suitable for the training algorithm of the RNN. 3.3 Discussion It is known that the form and position of sigmoid function can be changed by using a gain α and threshold β, respectively. The function and its differential are respectively represented by f(x) = 1 0.5 (16) 1 + exp{ α(x β)} f (X) = αf(x){1 f(x)}. (17) If the gain α is set to 10, then the slope becomes steeper and the nonlinear range on horizontal axis becomes narrow as shown in Fig. 8. Because the learning algorithm written by (8) includes the differential f (X), and then learning effectiveness can be expected only in the narrow range, i.e., activation area. Accordingly, when the absolute value of input to the sigmoid function is large, i.e., out of range, the updates of weights do not progress well. On the other hand, if α is set to 0.1, the slope becomes linear and gentle. The sigmoid function loses the flexible characteristics of nonlinearity. Fig. 9 illustrates the differentials with several α. The most important point is the adjustment of the input value to a sigmoid function by using the down scaler S. For example, if α is set to 1, it is desirable that the input values move within about ±5 from the center as related in the painted area of Fig. 9. In the learning process of the output layer of the second joint, the absolute value of the error between the teaching signal and output of the corresponding RNN 2 became much larger than 5, so that the updates of weights and thresholds could not progress satisfactorily. In this section, we have proposed a simple and adaptive learning method for the second joint, in which the teaching signals are scaled down and to be mapped within about ±5, i.e., into activation area, on the horizontal axis shown in Fig. 6. The effectiveness has been confirmed from the learning result of the second joint. Of course, a similar effectiveness test will be performed by setting the gain α smaller and the learning coefficient η larger. However, the syner-

F. Nagata and K. Watanabe / Adaptive Learning with Large Variability of Teaching Signals for 59 gistic tuning according to the error signals in the output layer is not so easy, and time-consuming as well. It is common that antecedent membership functions are manually or automatically placed on support set according to actual fuzzy input signals when a fuzzy approach is incorporated in a control system. In some way, the proposed adaptive scaling down technique for error signals in output layer is similar to the design of such membership functions. In this section, a servo control with the learned RNNs is introduced. The six RNNs independently learned are integrated for a feedforward controller as shown in Fig. 10. The RNNs act instead of the model-base portion, i.e., Coriolis/centrifugal force term and gravity term, in the computed torque control law given by (2). The PUMA560 manipulator has six degree-of-freedoms, i.e., and six joints. The integrated RNNs are composed of six networks to independently produce the driving torque for each joint. Therefore, the array of the networks represents a dynamic model to compensate the nonlinear terms consisting of Coriolis/centrifugal force term and gravity term, which can be seen by comparing Fig. 2 with Fig. 11. Fig. 11 shows the block diagram of the control system including the integrated RNNs, in which the torque vector τ NN from the integrated RNNs is multiplied by S 1 to call back the original scales of teaching signals. Fig. 8 β Sigmoid functions with different gains α and a threshold Fig. 10 Integrated RNNs for a feedforward controller Fig. 11 Block diagram of servo system comprising the integrated RNNs illustrated in Fig. 10 Fig. 9 Differentials f (X) of sigmoid functions with different gains α 4 Advanced servo system using integrated RNNs The control system is similarly applied to the trajectory following control problem as shown in Fig. 1. Fig. 12 (a) (f) show the control results of six joints, respectively. It is observed that each joint can be successfully controlled along the desired trajectory although there exist small delays between the desired joint angles and controlled ones. In this case, the elements of the output τ F B from the servo system without gravity term and Coriolis/centrifugal force term are shown in Fig. 13 (a) (f), respectively. Note that τ F B is the output only from the feedback control with K p and K v, which does not include the compensation τ NN using the RNN. As it can be seen from the small values of six joints driving torques, the integrated RNNs work well as a feedforward controller which can compensate both the gravity term and Coriolis/centrifugal force term.

60 International Journal of Automation and Computing 8(1), February 2011 Fig. 12 Control results of joint angles θ 1,, θ 6 Fig. 13 Joint driving torques τ F B generated from the servo system without gravity term and Coriolis/centrifugal force term as shown in Fig. 11 5 Conclusions and future work In this paper, RNNs have been proposed for PUMA560 manipulator to feedforwardly control a trajectory. A basic sigmoid function is used in each unit. The feedforward controller using the integrated RNNs approximately includes gravity compensator and Coriolis/centrifugal force compensator. In order to adaptively learn such large-scale teaching signals that are out of the activation area of sigmoid function used in back propagation algorithm, the proposed RNNs scale down the teaching signals into the activation area. When the integrated RNNs are used, the output from the RNNs have only to be scaled up inversely. The adaptive learning technique by using a scale down factor and the control effectiveness of integrated RNNs has been evaluated through simulations using the dynamic model of PUMA560 manipulator. Consequently, a promising result has been confirmed compared with the conventional intuitive learning technique, in which the gain of sigmoid function and learning coefficient is synergistically selected by trial and error. As for future work, it is being planned to apply the proposed adaptive learning technique to the feedback error learning [17, 18], which is expected to be effective so as to realize online learning of the integrated RNNs. References [1] F. Nagata, K. Watanabe, T. Hase, Z. Haga, M. Omoto, K. Tsuda, O. Tsukamoto, M. Komino, Y. Kusumoto. Application of open architectural industrial robots and its simulation technique. International Journal of Computer Research, vol. 17, no. 1 2, pp. 1 40, 2008. [2] C. Chen, M. M. Trivedi, C. R. Bidlack. Simulation and animation of sensor-driven robots. IEEE Transactions on Robotics and Automation, vol. 10, no. 5, pp. 684 704, 1994.

F. Nagata and K. Watanabe / Adaptive Learning with Large Variability of Teaching Signals for 61 [3] F. Benimeli, V. Mata, F. Valero. A comparison between direct and indirect dynamic parameter identification methods in industrial robots. Robotica, vol. 24, no. 5, pp. 579 590, 2006. [4] F. Nagata. Simulation technique of velocity-based discretetime control system with intelligent control concepts for open architectural industrial robots. The Open Automation and Control Systems Journal, vol. 1, pp. 31 43, 2008. [5] J. Han, C. Moraga. The Influence of the sigmoid function parameters on the speed of backpropagation learning. Lecture Notes in Computer Science, Springer, vol. 930, pp. 195 201, 1995. [6] C. Torras. Robot adaptivity. Robotics and Autonomous Systems, vol. 15, no. 1 2, pp. 11 23, 1995. [7] A. T. Hasan, A. M. S. Hamouda, N. Ismail, H. M. A. A. Al- Assadi. An adaptive-learning algorithm to solve the inverse kinematics problem of a 6 D.O.F serial robot manipulator. Advances in Engineering Software, vol. 37, no. 7, pp. 432 438, 2006. [8] J. H. Horng. Hybrid MATLAB and LabVIEW with neural network to implement a SCADA system of AC servo motor. Advances in Engineering Software, vol. 39, no. 3, pp. 149 155, 2008. [9] S. N. Huang, K. K. Tana, T. H. Lee. Neural network learning algorithm for a class of interconnected nonlinear systems. Neurocomputing, vol. 72, no 4 6, pp. 1071 1077, 2009. [10] Y. Y. Wu, Y. Q. Wu. Stability analysis for recurrent neural networks with time-varying delay. International Journal of Automation and Computing, vol. 6, no. 3, pp. 223 227, 2009. [11] Y. Y. Wu, T. Li, Y. Q. Wu. Improved exponential stability criteria for recurrent neural networks with time-varying discrete and distributed delays. International Journal of Automation and Computing, vol. 7, no. 2, pp. 199 204, 2010. [12] R. P. Paul. Modeling, Trajectory Calculation and Servoing of a Computer Controlled Arm, Technical Report AIM- 177, Artificial Intelligence Laboratory, Stanford University, USA, 1972. [13] J. J. Craig. Introduction to Robitics Mechanics and Control, 2nd ed., Reading MA, USA: Addison Wesley Publishing Company, 1989. [14] P. Corke. A robotics toolbox for MATLAB. IEEE Robotics & Automation Magazine, vol. 3, no. 1, pp. 24 32, 1996. [15] P. Corke. MATLAB toolboxes: Robotics and vision for students and teachers. IEEE Robotics & Automation Magazine, vol. 14, no. 4, pp. 16 17, 2007. [16] F. Nagata, K. Kuribayashi, K. Kiguchi, K. Watanabe. Simulation of fine gain tuning using genetic algorithms for model-based robotic servo controllers. In Proceedings of the IEEE International Symposium on Computational Intelligence in Robotics and Automation, IEEE, Jacksonville, USA, pp. 196 201, 2007. [17] M. Kawato. The feedback-error-learning neural network for supervised motor learning. Advanced Neural Computers, R. Eckmiller Ed., North-Holland, Holland: Elsevier, pp. 365 373, 1990. [18] J. Nakanishi, S. Schaal. Feedback error learning and nonlinear adaptive control. Neural Networks, vol. 17, no. 10, pp. 1453 1465, 2004. Fusaomi Nagata received the B. Eng. degree from the Department of Electronic Engineering at Kyushu Institute of Technology, Japan in 1985, and the D. Eng. degree from the Faculty of Engineering Systems and Technology at Saga University, Japan in 1999. He was a research engineer with Kyushu Matsushita Electric Co., Japan from 1985 to 1988, and a special researcher with Fukuoka Industrial Technology Center, Japan from 1988 to 2006. He is currently an associate professor at the Department of Mechanical Engineering, Faculty of Engineering, Tokyo University of Science, Yamaguchi, Japan. His research interests include intelligent control of industrial robot and its applications. E-mail: nagata@ed.yama.tus.ac.jp (Corresponding author) Keigo Watanabe received the B. Eng. and M. Eng. degrees in mechanical engineering from the University of Tokushima, Japan in 1976 and 1978, respectively, and the D. Eng. degree in aeronautical engineering from Kyushu University, Japan in 1984. From 1980 to 1985, he was a research associate at Kyushu University. From 1985 to 1990, he was an associate professor at the College of Engineering, Shizuoka University, Japan. From April 1990 to March 1993, he was an associate professor, and from April 1993 to March 1998, he was a full professor in the Department of Mechanical Engineering at Saga University, Japan. From April 1998, he was with the Department of Advanced Systems Control Engineering, Graduate School of Science and Engineering, Saga University. Currently, he is with the Department of Intelligent Mechanical Systems, Graduate School of Natural Science and Technology, Okayama University, Japan. His research interests include stochastic adaptive estimation and control, robust control, neural network control, fuzzy control, genetic algorithms and their applications to the robotic control. E-mail: watanabe@sys.okayama-u.ac.jp