22 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART C: APPLICATIONS AND REVIEWS, VOL. 30, NO. 1, FEBUARY 2000
Optimal Design of CMAC Neural-Network Controller
for Robot Manipulators
Young H. Kim and Frank L. Lewis, Fellow, IEEE
Abstract—This paper is concerned with the application
of quadratic optimization for motion control to feedback
control of robotic systems using cerebellar model arithmetic
computer (CMAC) neural networks. Explicit solutions to the
Hamilton–Jacobi–Bellman (H–J–B) equation for optimal control
of robotic systems are found by solving an algebraic Riccati equa-
tion. It is shown how the CMAC’s can cope with nonlinearities
through optimization with no preliminary off-line learning phase
required. The adaptive-learning algorithm is derived from Lya-
punov stability analysis, so that both system-tracking stability and
error convergence can be guaranteed in the closed-loop system.
The filtered-tracking error or critic gain and the Lyapunov
function for the nonlinear analysis are derived from the user input
in terms of a specified quadratic-performance index. Simulation
results from a two-link robot manipulator show the satisfactory
performance of the proposed control schemes even in the presence
of large modeling uncertainties and external disturbances.
Index Terms—CMAC neural network, optimal control, robotic
control.
I. INTRODUCTION
T
HERE has been some work related to applying optimal-
control techniques to the nonlinear robotic manipulator.
These approaches often combine feedback linearization and op-
timal-control techniques. Johansson [6] showed explicit solu-
tions to the Hamilton–Jacobi–Bellman (H–J–B) equation for
optimal control of robot motion and how optimal control and
adaptive control may act in concert in the case of unknown
or uncertain system parameters. Dawson et al. [5] used a gen-
eral-control law known as modified computed-torque control
(MCTC) and quadratic optimal-control theory to derive a pa-
rameterized proportional-derivative (PD) form for an auxiliary
input to the controller. However, in actual situations, the robot
dynamics is rarely known completely, and thus, it is difficult to
express real robot dynamics in exact mathematical equations or
to linearize the dynamics with respect to the operating point.
Neural networks have been used for approximation of non-
linear systems, for classification of signals, and for associative
memory. For control engineers, the approximation capability of
neural networksisusuallyusedforsystemidentification or iden-
tification-based control. More work is now appearing on the
use of neural networks in direct, closed-loop controllers that
yield guaranteed performance [13]. The robotic application of
Manuscript received June 2, 1997; revised June 23, 1999. This research was
supported by NSF Grant ECS-9521673.
The authors are with the Automation and Robotics Research Institute,
University of Texas at Arlington, Fort Worth, TX 76118-7115 USA (e-mail:
ykim50@hotmail.com; flewis@arri.uta.edu).
Publisher Item Identifier S 1094-6977(00)00364-3.
neural-network based, closed-loop control can be found [12].
For indirect or identification-based, robotic-system control, sev-
eral neural network and learning schemes can be found in the lit-
erature. Most of these approaches consider neural networks as
very general computational models. Although a pure neural-net-
work approach without a knowledge of robot dynamics may be
promising, it is important to note that this approach will not be
very practical due to high dimensionality of input–output space.
In this way, the training or off-line learning process by pure con-
nectionist models would require a neural network of impractical
size and unreasonable number of repetition cycles. The pure
connectionist approach has poor generalization properties.
In this paper, we propose a nonlinear optimal-design method
that integrates linear optimal-control techniques and CMAC
neural-network learning methods. The linear optimal control
has an inherent robustness against a certain range of model
uncertainties [9]. However, nonlinear dynamics cannot be taken
into consideration in linear optimal-control design. We use
the CMAC neural networks to adaptively estimate nonlinear
uncertainties, yielding a controller that can tolerate a wider
range of uncertainties. The salient feature of this H–J–B control
design is that we can use a priori knowledge of the plant
dynamics as the system equation in the corresponding linear
optimal-control design. The neural network is used to improve
performance in the face of unknown nonlinearities by adding
nonlinear effects to the linear optimal controller.
The paper is organized as follows. In Section II, we will re-
view some fundamentals of the CMAC neural networks. In Sec-
tion III, we give a new control design for rigid robot systems
using the H–J–B equation. In Section IV, a CMAC controller
combined with the optimal-control signal is proposed. In Sec-
tion V, a two-link robot controller is designed and simulated in
the face of large uncertainties and external disturbances.
II. B
ACKGROUND
Let denote the real numbers, the real -vectors, and
the real matrices. We define the norm of a vector
as and the norm of a matrix
as where and
are the largestandsmallesteigenvalues of a matrix. The absolute
value is denoted as
.
Given
and , the Frobenius norm is
defined by
with as the trace
operator. The associated inner product is
.
The Frobenius norm is compatible with the two-norm so that
with and .
1094–6977/00$10.00 © 2000 IEEE
