row is 1.33 milliseconds per pattern (row) for the neural networks versus 50.0 milliseconds for the linear regression. The neural networks were, on average, 37.5 ...
row is 1.33 milliseconds per pattern (row) for the neural networks versus 50.0 milliseconds for the linear regression. The neural networks were, on average, 37.5 times faster than the regression method. Given the size of the error threshold with respect to some of the coefficients, the errors appear very acceptable for real-time applications. These error values are significantly lower than those reported by Smith and Chase (1992) for the same problem using an absolute error threshold of ±0.25 for all coefficients. It must be noted, however, that the training and testing shown here is based on a nondegrading K and C. Given that structural parameters often degrade during a major loading, additional testing using a degrading model for K and C would be useful prior to real-time implementation. The dynamic architecture of Cascor allows the network to be augmented to accomodate the identification of these degrading properties.
Neural-Network Based Learning Control of Flexible Mechanism With Application to a Single-Link Flexible Arm Kazuhiko Takahashi 1 and Ichiro Yamada 2
This paper shows the effectiveness of a neural-network controllerfor controlling a flexible mechanism such as a flexible robot arm. An adaptive-type direct neural controller is formulated using state-space representation of the dynamics of the target system. The characteristics of the controller are experimentally investigated by using it to control the tip angular position of a single-link flexible arm.
Summary and Conclusions A method of performing structural system identification using a modified Cascade-Correlation neural network is described. The method entails training a network to emulate linear regression in the frequency domain to identify the stiffness and damping matrix coefficients, one row at a time. Overall, the Cascade-Correlation neural network method presented here performs very well in its application to structural system identification. The variable error threshold mechanism used in the trained networks produced output error performance superior to that of absolute error thresholds. The speed of identification was small enough that realtime applications become more feasible. In addition, the ability of the Cascor network to augment its training provides an additional flexibility and adaptability not available with conventional neural networks.
Acknowledgments The research is funded in part by the National Science Foundation Presidential Young Investigator Award No. BCS-9058316 and by Obayashi Corporation.
References Chu, S. R., Shoureshi, R., and Tenorio, M., 1990, "Neural Networks for System Identification," Proceedings of the 1989 American Control Conference, American Automatic Control Council, Green Valley, AZ, Vol. 1, pp. 916-921, Chu, S.R. and Shoureshi, R., 1991, "Neural Network Approach for Identification of Continuous Time, Non-Linear, Dynamic Systems," Proceedings of the 1991 American Control Conference, American Automatic Control Council, Evanston, IL, Vol. 1, pp. 1-6. Fahlman, S. E. and Lebiere, C , 1990, "The Cascade-Correlation Learning Architecture," Carnegie-Mellon University Technical Report No. CMU-CS-90100, Carnegie-Mellon University, Pittsburgh, PA. Masri, S. F., Chassiakos, A. G., and Caughey, T. K., 1992, "Structure-Unknown Nonlinear Dynamic Systems: Identification Through Neural Networks," Smart Materials and Structures, Vol. 1, pp. 45-56. Smith, H. A. and Chase, J. G., 1992, "Use of a Cascade-Correlation Neural Network for System Identification," Proceedings of the Third International Conference on Adaptive Structures, Wada et al., ed., Technomic Publishing Co., Lancaster, PA, pp. 619-630. Wen, Y. K„ Ghaboussi, J., Venini, P., and Nikzad, K„ 1992, "Control of Structures Using Neural Networks," Proceedings of the US/Japan/Italy Workshop on Structural Control and Intelligent Systems, Housner et al., ed., USC Publication No. CE-9210, Los Angeles, CA, pp. 232-251.
1
Introduction
Neural networks offer flexibility, learning ability, and nonlinear mapping ability. Therefore, many applications of neural networks to the control of mechanical systems, such as robot manipulators, have been studied (Kawato et al., 1987; Jordan, 1989, etc.), and several types of neural-network-based controller (hereafter called a neural controller) have been proposed for nonlinear and/or uncertain systems (Narendra, 1990; Yamada etal., 1990,1991, etc.). An important function of the neural controller is to determine the dynamics of the target system within the controller through a learning process in order to compensate for the nonlinearity and /or uncertainties of the system. However, the general design methods for neural controllers have not yet been established, since the stability of the neural controller depends greatly on the dynamics of the target system. We have shown a compensation method for nonlinear disturbances, such as solid friction, in a rigid body mechanism using a direct neural controller (Takahashi et al., 1992). When controlling elastic body mechanisms, such as controlling the tip position of a flexible arm, however, it is hard to converge the learning of the neural network. This is due to the nonminimum phase characteristics of the flexible arm from the applied torque to the arm tip position. Consequently, the control system often becomes unstable. In recent years, numerous studies have been carried out on the control of flexible arms by various control methods (Cannon et al., 1984; Bayo, 1987; Yuan et al., 1989; Cetinkunt et al., 1992, etc.) in many fields such as industrial robots and large space structures. However, only a few simulations have been reported on the control of flexible arms using a neural controller (Apolloni et al., 1990) and no experimental work has yet been reported. Research Engineer, Interdisciplinary Research Laboratories, Nippon Telegraph and Telephone Corporation, 3-9-11 Midori-cho, Musashino-shi, Tokyo 180, Japan. Deputy Executive Manager, Technology Research Department, Nippon Telegraph and Telephone Corporation, 1-1-6 Uchisaiwai-cho, Chiyoda-ku, Tokyo 10019, Japan. Contributed by the Dynamic Systems and Control Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received by the DSCD March 1993. Associate Technical Editor: C. de Silva.
792/Vol. 116, DECEMBER 1994
Transactions of the ASME
Copyright © 1994 by ASME
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x(k+l)
= Ax(k) + bu(k),
(5)
y{k) = cx(k)
(6)
where A = exp(AcA), Ac = [acij\ (i,j = 1,,...,4) fl
= a
cl3
fl
c33 = J~rl
c 2 4 = l-0>ac32
R
~~a Kt
=
^f>Co>
^r
K
b> « c 42 = J~r'
J
^
*
ac43 = J;. R~ c~ K,Kb, A
b = j exp(AcT)bcdT, bc = [bci] (i = 1.....4), bc3 = j;lR-lKt,bc4 Fig. 1 Model of a single-link flexible arm
This paper shows experimentally the effectiveness of a neural controller for controlling a flexible mechanism. Section 2 describes a neural controller for the tip angular position control of a single-link flexible arm. Section 3 investigates experimentally the characteristics of the neural controller. 2
Neural Controller for a Flexible Arm
2.1 Model of a Flexible Arm. We consider the simple model of a flexible arm shown in Fig. 1 for designing the neural controller. We assume that the arm is massless and that the end-point payload is a point. The arm is driven by a motor torque Tr(f) applied to the joint. We denote the angle of the arm tip elastic deflection by ip(0, the overall moment of inertia of the joint and the rigid part of the arm with respect to the arm rotation axis by Jr, the moment of inertia of the endpoint payload with respect to the arm rotation axis by Jt, and the angular stiffness of the arm with respect to the tip deflection by k . Neglecting the internal damping forces and Coulomb friction of the joint, the equations of motion are
=
j;lR-lc-lK„
c=[lco00], and A is the sampling time, k is the number of samples, and for the other values of both i and j not shown here, the identity ac = bc = 0 holds. Simple calculation confirms that cb * 0. 2.2 Neural Controller Design Using State-Space Representation. In this section, a simple direct neural controller, which determines the inverse dynamics of the target system after learning, is designed using the discrete-time state-space equation of the flexible arm Eqs. (5)-(6). We assume that the sign of the target system's gain (dy/du) is known. A three-layer PDP-type neural network is used as the controller. There is no inner feedback loop or direct connection from the input layer to the output layer. All the thresholds are assumed to be zero. A nonlinear function/is added to the neural network as an activation function of the neurons. The output of the neural network, which is used as a control input u{k), can be calculated by u(k)=j\W°»f[WhiINN(k)]},
(7)
/!
where W ' is the weight matrix (q x s) from the input layer to the hidden layer, Wo/' is the weight matrix ( l x ^ ) from the hidden layer to the output layer, and lNN(k) is the neural(Jr + J,)Q(t)+JtlJp(t) = Tr(t) (1) network input vector (s x 1). The learning of the neuralnetwork is performed according to the generalized 8-rule 7 , 6 ( 0 + -/,
