euro-adaptive hybrid controller for robot- ulator tracking control
L.Behera S.Chaudhury M.Gopal
Indexing terms: Neurul control, Robot truclcing, Adaptive control luw
Abstract: The paper is concerned with the design
of a hybrid controller structure, consisting of the adaptive contrul law and a neural-network-based learning scheme for adaptation of time-varying controller parameters. The target error vector for weight adaptation of the neural networks is derived using the Lyapunov-function approach.
The global stability of the closed-loop feedback system is guaranteed, provided the structure of the robot-manipulator dynamics model is exact.
Generalisation of the controller over the desired trajectory space has been established using an on- line weight-learning scheme. Model learning, using a priori knowledge of a robot arm model, has been shown to improve tracking accuracy.
The proposed control scheme has been implemented using both MLN and RBF networks. Faster convergence, better generalisation and superior tracking accuracy have been achieved in the case of the RBF network.
1 Introduction
In the literature of neural control, as applied to robot tracking problems, the primary focus has been given to various schemes of off-line and on-line inverse dynam- ics learning using neural networks [14]. In this paper we have proposed a hybrid controller scheme, combin- ing positive aspects of both the classical adaptive control [S-lo] and the neural modelling [ll-161.
Adaptive control algorithms make use of properties of robot dynamics such as linearisability, skew-symmetric- ity etc., on the basis of a priori knowledge of the robot model. On the other hand, inverse dynamics learning, using neural networks, does not make use of well-established knowledge of the rigid robot model.
Combination of the adaptive tracking principle, based on a valid model, and neural modelling of robot model components subjected to uncertainties, is expected to provide an efficient performance-oriented controller.
Ozaki et ah [17] have proposed a controller in an
IEE Pyoceedings online no. 1996012I
Paper first received 11th April 1995 and in revised form 5th September 1995
The authors are with the Department of Electrical Engineering, Indian Institute of Technology, Delhi, Haus Khas, New Delhi-110 016, India
adaptive computed torque control configuration, where a neural network has been used to learn the elements of the inertial and Coriolis force matrices and the gravity force vector. It has been shown that this technique has the capability to compensate unmodelled effects, such as friction etc. However, convergence of the weight- learning scheme in the control loop has not been estab- lished. Kuan, Whittle and Bavarian [IS] have also pro- posed an adaptive controller, which uses single layer adaline-like neurons for learning the structured uncer- tainties of a three-degrees-of-freedom robot model, using the Lyapunov-function approach for the weight adaptation.
Motivated by these results, we have presented, in this paper, a new hybrid controller consisting of adaptive control law and a neural estimator. The control law has been adopted from the adaptive algorithm pro- posed by Slotine and Li [SI. The robot model has been linearly parametrised in terms of inertial, Coriolis and gravity force matrices and/or vector elements. These time-varying parameters have been estimated using feedforward neural networks as functions of joint posi- tion andlor velocity. Convergence of the weight-learn- ing scheme has been established using the Lyapunov- function approach. The proposed learning algorithm estimates the manipulator parameters, while the learn- ing algorithm of Kuan et al. [IS] only compensates for the uncertainties found in the available robot model.
Also, the present scheme uses more powerful MLN and RBFN as neural estimators, instead of simple adalines as in [18].
Two approaches have been followed to train the neu- ral estimator. In one approach, the weights are tuned using an on-line adaptation algorithm only. In the sec- ond approach the a priori model knowledge has been utilised to train the network off-line. Subsequently, on- line adaptation has been carried out to compensate for parameter uncertainties and unmodelled dynamics.
2 Parameter estimation using feedforward networks
Neural networks can be used for estimating parameters of an adaptive control algorithm designed for a robot manipulator. Such parameters are nonlinear functions of time-varying joint position and velocity vectors.
Both MLN [19] and RBFN [20] are capable of approx- imating any nonlinear mapping and, hence, can be used as estimators.
In case of a multi-layered network, the standard backpropagation algorithm is very effective, in spite of its limitations such as slow convergence and local
IEE Proc.-Control Theory Appl.. Vol 143, No 3, May 1996
minima trap. Training of the RBF network includes training of centre as well as connection weights. Centre learning can be done using an unsupervised learning algorithm, such as the c-means clustering technique [21], or supervised learning algorithms, such as gradient descent [22] and recursive predictive error (RPE) [23]
algorithms, etc. For weight learning, we can adopt either gradient descent, or any of the recursive least squares (RLS) algorithms available, as the response of the network is linear with respect to its weights.
3 Neuro-adaptive hybrid controller
The adaptive control law is so designed that feedback control operates in a stable mode, provided the exact structure of the manipulator dynainics model is known a priori. The proposed control scheme has been designed based on this concept. The controller struc- ture is shown in Fig. 1. The neural estimator tracks the time-varying controller parameters subjected to uncer- tainties. Final convergence of the scheme can be estab- lished, by ensuring the convergence of the weight learning scheme and global convergence of the closed- loop adaptive system. We now present the control and parameter adaptation law.
linear
jr
parametrised adaptive controller
X robot arm
q.q
Fig. 1 Neuro-adaptive controller structure
Consider an N-link rigid robot model given by M(q)G + ( 3%4 4 + G ( s ) = 7 (1) where z is the N x 1 vector of joint torques, q is the N x 1 vector of generalised joint positions, M(q) is an Nx N inertial matrix, C(q, q) represents torques arising from centrifugal and Coriolis forces and G(q) repre- sents torques due to gravitational effects when the manipulator is moving in its workspace.
We have chosen the control law, given by Slotine and Lee [SI
T = M ( q ) & + C(q,il)i17 + G ( 9 ) - KUS (2) where q, = qd - Aq, qr = ijd - Aq, s = $ i Rq“ and A is the gain matrix. The control law in eqn. 2 is globally cpnvergent for the model in eqn. 1 when M(q) = M(q), C(q, q) = C(q, q), and G(q) = G(q), and both the position and the velocity trackjng errors converge to the sliding surface given by s = q + Ag = 0 . In a gara- metrised adaptation algorithm [SI, M, C and G are computed on the basis of estimates of unknown manip- ulator parameters such as link length, link mass, payload etc., by assuming them to be either constant or slowly time varying. These assumptions may not be always valid. As robot dynamics is complex in struc- ture, calculation of M, C and G on the basis of these
parameters is also computationally expensive. There- fore, instead of estimating manipulator parameters, if we can directly estimate elements of these matrices, the computational cost can be reduced. Use of neural net- works as estimators for these matrix elements provides suitable means for handling time-varying uncertainties associated with the robot model.
_ In our proposed hybrid controller, we estimate M, C and G directly using a feedforward neural network.
Hence no restrictive assumptions regarding the nature of the controller parameters are required. Since ele- ments of function matrices are being estimated directly, computational complexity is less. We now derive the adaptation law and corresponding weight-learning scheme.
3.I Adaptation law for a neural estimator For an N-link robot manipulator, we define the param- eter vector a to be consisting of all elements of inertial, Coriolis and centrifugal, and gravitational force matri- ces and/or vectors. This p-dimensional vector (p = N(2N + 1)) can be expressed as
I , G-i...., GJV] (3) Each element of this parameter vector a is time vary- ing. These time-varying elements can be estimated using three feedforward neural networks, as shown in Fig. 2. Inertial and gravitational elements can be mod- elled as functions of a joint position vector while Cori- olis and centrifugal terms can be modelled as functions of both the joint position and velocity vectors. During the on-line adaptation process, these network weight vectors are tuned so that tracking convergence is achieved. As outputs of these supervised networks, i.e.
the target vectors are unknown. we need to estimate either the actual target vector a or the error e = a - 3 = -a at the target. Now we will provide a globally con- vergent scheme to estimate the target error e on-line, based on output tracking error.
CNN
F i g . 2 Structure oj neural estimators (NN!, NN, and NN,)
Using the linear parametrised property, robot dynamics, eqn. 1, can be written as
where
&...fjN0 0 &...&0 0... 0 fjl...@N 0... 0 0 ... 0 41...&
0 10 0 0... 0 01 0 0
. 0 0 &...qN0 0 ...qN0 01 (5)
IEE Proc-Control Theory Appl.. Vol. 143, No. 3, May 1996
Now, the Lyapunov function can be defined as
V = -sTMs1 + i a
Tr a (6)
2 2 '
where F is strictly Hurwitz.The time derivative of the Lyapunov function can be given as
V = sT~ S + aTri + - sTM s
= ST( . - c ; l - G - ~ 4~) + a Tri ('7) sT [-H
Using the skew symmetric property and control law given in eqn. 2, we have
V = sT(MqT + CqT + G - Kos ) + 5 ' r a Using the linearity relation, eqn. 4, associated with the robot dynamics model, eqn. I, we can write
M ( q ) q , + C(q, 4)ST + G(q) = Y(q. s, &; SI-) (8) where U(.) has the same function form as that of Yl(.), given in eqn. 4. Thus we have
V = sT( Y 6 — K o s ) + iiTr6
— — -2~ ~ ~ + aT( r i i + Y s) by choosing the adaptation law as
4 = -r --lYT(q,4,47.7GT)S (9) we have
V = - sTK n s 5 0 (10) Thus the target-error vector, estimated using the adap- tation law, eqn. 9, will result in global convergence of the closed-loop adaptive system. The recursive form of the target-error (e) vector estimation can be written as
eneui = cold + pYTs (11)
where p is the update rate. Thus, for each desired tra- jectory pattern, the error at the target of the neural estimator can be computed using eqn. 11 in an itera- tive fashion. This error now can be backpropagated to update the weights of the neural estimator. using a suit- able weight-adaptation algorithm.
As in the case of the parametrised adaptive scheme, the parameter vector consists of constant but unknown parameters. The parametrised adaptation algorithm can be derived from eqn. 9 as follows:
Before the proposed hybrid controller can be used for on-line trajectory tracking, the neural estimator should be trained properly. This training can be done in two ways.
3.2 On-line adaptive learning
In this approach, the neural weights are initialised ran- domly. Then the robot arm is made to track various random trajectories in its workspace. For each desired trajectory pattern, the target error for the neural esti- mator is computed using eqn. 11. The target error is backpropagated through the network estimator to update its weights. This training process is continued for various desired trajectory patterns spanning the robot workspace. Once training is over, the neural estimator can be directly used to predict the controller parameters, without further update of its weights, provided there are no on-line dynamic uncertainties
and disturbances. In case that the robot environment is not free from on-line uncertainties, an on-line adapta- tion scheme has to be operated, while the manipulator is tracking new trajectories for performing various tasks.
3.3 Hybrid learning
In hybrid learning, the training process is carried out in two phases: model learning and an on-line adaptation algorithm. In the first phase, neural estimators are directly trained using the available model. For example MI1, an element of the inertial matrix, can be modelled as a function of the joint position vector, using the data generated from the known functional available in the model. Thus, through model learning, the controller can be neurally expressed in terms of available knowl- edge of the robot model before we can start with the on-line adaptation scheme. On-line learning is carried out to take care of various uncertainties. The objective of thc hybrid learning is to effectively reduce the dura- tion of on-line learning and to increase the accuracy.
4 Simulation
Since nonlinear intricacies of a robot arm are largely reflected in the 2nd and 3rd links, we have considered a two-link manipulator model of PUMA 560 [24], con- sisting of 2nd and 3rd arms for simulation. The robot dynamics can be explicitly expressed as:
/ 0,2 \
(aj + 0,1 cos 52jQx + ( 0,3 H c o s (j2 I tx — o,2QiQ2 sin q-2
a4 cos 41 + a5 cos(q1 + q2) = (13)
/ 2 N 2 .
! a3 + -—cos(ft j q\ +a3e72 - —gj smg2 + «5 cos(gi + q2)
= n (14) where al = 3.82, a2 = 2.12, a3 = 0.71, a4 = 81.82 and a5
- 24.06.
The objective of simulation is to evaluate the per- formance of the neuro-adaptive scheme in comparison to the classical parametrised adaptive algorithm. Simul- taneously, the performance of the MLN based neural estimator has been compared with a RBF network- based neural estimator.
While doing simulation for the classical parametrised adaptive control scheme, we have assumed that eqns.
13 and 14 represent a true model of the manipulator, and the parameter vector is assumed to be unknown.
While doing simulation for the neuro-adaptive con- trol scheme, though the parameter vector is selected based on eqns. 13 and 14, viscous and Coulomb fric- tion forces have been introduced into the true model.
Thus, the true model is assumed as follows:
M(q)q + C(q, 4)s + G(q) (15)
where
and
V = 0 2
F = h = k2 = 1
We evaluate the performance of the proposed control- ler, with respect to the true model, so that the capabil- ity of the controller to compensate for unmodelled dynamics can be assessed.
212 IEE Pioc-Control Theory Appl, Vol 143, No 3, May I996
The control law 2 = Y(q, q, q,., 4,) 4 - KDs primarily involves the computation of matrix Y. For classical parametrised adaptive control, with the choice of the parameter vector as a = [a,, a2, T
of matrix Y can be written as:
, a3, a4, the elements
1.
= 9ri cosg2 + 7>cl
~22 = +er, cosqz + L4:L sinqa, y:M = 0, Y2 5 = cos(y1 + 92)
. r. .
2 cosq2 - sm q2 \qrlqr2 + -q Y15 = cOS(y1 + q2).
~23 = eT1 + ~~2
= 0
(16)
4r2
0 0 qrl
0
q.ri 0 0
0 qT1
0
g,,2
1 0
0 1 For the proposed control structure, the parameter vec- tor is chosen as a = [MI,, M12, M21, M22, CI1, Cl2, CZl, C22, GI, G,]. With this choice of parameter vector, the matrix Y can be expressed as
(17) Comparing eqns. 16 and 17, we can observe the simple structure of matrix Y for the case of the hybrid scheme.
As both the controller structure and the parameter adaptation algorithm involve this matrix Y, computa- tional complexity has been reduced considerably for the case of the proposed hybrid scheme.
4.I Architecture of neural estimators
Three neural networks (NN,, NN2 and NN3) are used to estimate the parameter vector (Fig. 2). Based on a priori knowledge of the model, the elements of the iner- tial matrix are modelled as functions of q2 (NN,), joint 2 position only, the elements of the Coriolis and cen- trifugal force matrix are modelled as functions of all the joint states (NN2), while elements of the gravity force vector have been modelled as functions of joint 1 and 2 positions (NN,). In the case of MLN, the struc- tures of NN1, NN2 and NN3 are 2-6-4, 5-6-4 and 3-20- 2, and the output layer is chosen to be linear. Each of the input vectors for all these networks contains a bias input. For the RBF network, the corresponding struc- tures are 1-20-4, 4-50-4 and 2-50-2, where no bias input has been introduced.
4.2 Tracking performance using only on-line adaptation
We selected 20 random sinusoid trajectories of varied frequency and amplitudes in robot workspace. Each trajectory is described by the following equation:
q"
4(1 — coswt) (181
Each of them can be represented as (qmuS, CO). Using the on-line adaptation algorithm, we trained the neural estimator using both MLN and RBFN. For MLN, we trained the network in 600,000 learning steps and RBFN has been trained using the same set of data pairs in 300,000 learning steps. After training, we tested the neural estimator on both the training set and test trajectories. Performance of the hybrid scheme has been compared with the parametrised adaptive scheme (eqn. 12). In all the cases, the gain matrices are assumed to be A = 101 and KD = 201. Results have been compiled for four trained trajectories in Table 1 and for four test (new) trajectories in Table 2. The cor- responding tracking performance for selected trajecto- ries are shown in Figs. 3 and 4, respectively.
4.3 Tracking performance using hybrid learning
Here we trained the neural estimators, using a priori knowledge of the robot dynamics model. For simula- tion, we assumed that the available manipulator parameters are subjected to 40-60 %I uncertainties over the true robot manipulators given by eqns. 13 and 14.
Also we assumed that viscous and friction terms are not available. Training data were generated using the available model. After model learning was over, on-line adaptation was carried out to compensate for both the parameter uncertainties and unmodelled effects, such as viscous and Coulomb friction. For on-line adaptation, for MLN we performed 240 000 learning steps, while for RBFN we performed 120 000 learning steps. After training was over, we evaluated the controller perform- ance by tracking four trajectories from the training set and four test (new) trajectories, as done in the previous sub-section. The corresponding tracking performance in terms of rms error is summarised in Tables 3 and 4.
Table 1: Tracking performance on learned trajectories Trajectory
parameters for joints 1 and 2 (qr™*, to) (1.2,6), (2,4) (3,5), (23) (1.6,3), (3,5) (2.4,4),(1.6,6) Table 2: Tracking
Trajectory parameters for joints 1 and 2
\q™*, to) (2.5), (23) (2.4,3),(1.6,5) (3,4).(1.6,6) (1.6,6). (3.4)
RMS error in joint I Adaptive MLN 0.038
0.0066 0.019 0.006 Performance
0,0093 0.0085 0.017 0.01
RBFN 0.0044 0.004 0.008 0.004 on test trajectories
RMS error in joint 1 Adaptive MLN 0.0145
0.0127 0.0127 0.0163
0.036 0.03 0.0177 0.047
RBFN 0.004 0.0044 0.0028 0.0058
RMS error in joint 2 Adaptive MLN 0.026
0.0078 0.017 0.009
0.0038 0.004 0.0029 0.0036
RMS error in joint 2 Adaptive MLN 0.0107
0.0158 0.01 0.046
0.013 0.014 0.0067 0.013
RBFN 0.0016 0.002 0.0017 0.0009
RBFN 0.001 0.0011 0.0008 0.00145
1EE Proc.-Control Theory Appl., Vol. 143, No. 3, Muy I996 273
0-15r
0 1 0
* 0-05
-0,051
0-05
2 timetime, s
Fig. 3 Tracking error for learned trajectories in robot workspace, joint 1 (p" = 1.2, w = 6)
...parametrised adaptive controller neuro-adaptive controller using MLN neuroadaptivc controller using RBFN
-O05
-0,101 0 Fig.6
2 3 time,s
learned trajectories in robot workspace, q — _\ OJ-J/
..parametrised adaptive controller
— neuro-adaptive controller using 1\
— neuroadaptive controller using RB g . 6 Tracking error for lea
™* = 3, w = 5)
joini 2
s ing MLN sing RBFN
0-05
D
-0.051
0-05r
2 time,
joint I
time, s
F i g . 4 Tracking error fbr learned trajectories in robot workspace, joint ( p ' = 1.6, w=3)
... pdrametrised adaptive controller
— neuro-adaptive controller using MLN neuroadaptive controller using RBFN
-0.051
0 1
time,s
F i g . 7 Tracking error for test trajectories in robot workspace, joint 1 (P,= 3, w= 4)
..parametrised adaptive controller neuro-adaptive controller using MLN neuroadaptive controller using RBFN
0 . 1 -
T3 O
-0.1 -
-0.2
2 time, s time, s
Fig .5 Trucking error ,for learned trajectories in robot i+'or.kspace, joint 2 (px = 2, w= 4)
'... pardmetrised adaptive controller neuro-adaptive controller using ML -.neuroadaptive controller using RBF
4.4 Summary of simulation results
From Tables 1 and 2, and Figs. 3-10, we can see that the neuro-adaptive scheme has performed much better in terms of tracking accuracy, in comparison to the classical parametrised adaptive scheme, in spite of the presence of unmodelled dynamics, such as viscous and Coulomb friction forces in the former case. The control scheme, using an RBFN-based neural estimator, has achieved better tracking accuracy than the MLN-based estimation scheme. Generalisation performance of RBFN is much better compared to MLN, as far as
U" 1
0
-0.1
-0.2 ,'
I
': v"
;\
'•'. V
t _
1 \
*-. t V
/ /
i . i | t i
^ \
v/ \ V ' I ' t ' I ' V
I
0 1 2 3 4
time,s
Fig. 8 Tracking error for test trajectories in robot workspace, joint 1 lef" = 1.6, w = 6)
..'.. pardmetrised adaptive controller neuro-adaptive controller using MLN neuroadaptive controller using RBFN
tracking errors over test trajectories are concerned.
Also we can see, in some cases MLN performance is comparable with that of classical adaptive scheme. In terms of learning speed, RBFN outperforms MLN, as shown in Table 5.
In the case of hybrid training, it has been found that, after model learning, learning steps involved during on- line adaptation have been drastically reduced. Simulta- neously significant improvement in the tracking accu- racy has been also observed.
IEEPPvoc-Control Theory Appl, Vol. 143, No 3, May 1596
0-02 r
- 0 0 2
-0-04
0 1 2 3 4
time,s
F i g . 9 Tracking error for test trajectories in robot workspace, joint 2 (if" = 1.6, 03=6)
,14“... irametrised adaptive controller
~-_neuro-adaptive controller using MLN neuroadaptive controller using RBFN
0-3r
0-2
-0.1
'":*£-'"\ • •./'' "•'/.'•'> « ' " /
0 1 2 3 4
t i m e , s
Fig. 10 Tracking error ,for test trajectories in robot workspace, joint2
(a1™* = 3, m = 4)
parametrised adaptive controller neuro-adaptive controller using MLN neuroadaptive controller using RBFN
Table 3: Tracking performance on learned trajectories (hybrid learning)
Trajectory RMS error in joint 1 RMS error in joint 2 parameters for joints
1 and Z (qmaX ,o) MLN RBFN MLN RBFN (1.2.6), (2,4) 0.0014 0.0006 0.0005 0.0003 ( 3 5 , (23) 0.0014 0.00066 0.0008 0.00035 ( 1 . 6 3 , (3,5) 0.0026 0.0013 0.0005 0.00038 (2.4,4),(1.6.6) 0.0015 0.0008 0.00025 0.0002 Table 4: Tracking performance on test trajectories (hybrid learning)
Trajectory RMS error in joint 1 RMS error in joint 2 parameters for joints
1 and z (qmax, O ) M L N R B F N M L N R B F N
12,5), (23) 0.0015 0.00057 0.00047 0.00027 ( 2 . 4 3 , (1.6.5) 0.0016 0.00077 0.00037 0.00015 (3,4),(15.6) 0.001 0.00056 0.00025 0.00015 (1.6,6),(3,4) 0.002 0.0008 0.00047 0.00029 Table 5: Learning speed
Training scheme
Learning steps
MLN RBFN
Online training Hvbrid trainina
600 000 240 000
300 000 120 000
5 Conclusion
We have presented a neuro-adaptive hybrid control scheme, where knowledge of a robot dynamics model has been properly utilised to obtain a high-precision and computationally less intensive control scheme.
Simulation results show that the proposed scheme is better than the classical parametrised adaptive scheme in providing very accurate tracking. Through simula- tion, it has also been established that the RBFN-based neural estimator outperforms its MLN counterpart in achieving better accuracy and generalisation.
6 References
1 GOLDBERG, K.Y., and PEARLMUTTER, B.A.: 'Using back- propagation with temporal windows to learn the dynamics of the CMU direct drive arm 11’ in TOURETSKY, D.S. (Ed.):
'Advances in neural inf. processing systems' (Morgan Kaufmann, 1989)
2 MILLER, W.T., GLANZ, F.H., and KRAFT, L.G.: 'Applica- tion of a general learning algorithm to the control of robotic manipulators', Int. J. Robot. Res., 1987, 6, pp. 84-98
3 JORDAN, M.I.: 'Supervised learning and systems with excess degree of freedom', COINS Tech. Rep., No. 88-27, University of Massachusetts at Amherst
4 MIYAMOTO, H., KAWATO, M., SETOYAMA, T., and SUZUKI, R.: 'Feedback error learning neural networks for tra- jectory control of a robotic manipulator', Neural Netw., 1988, 1, pp. 251-265
5 SPONG, M.W., and VIDYASAGAR, M.: 'Robot dynamics and control' (John Wiley and Sons, 1989)
6 HSIA, T.C.: 'Adaptive control of robotic manipulators - a review', IEEE international conferencc on Robotics and automa- tion, San Francisco, California, 1986, pp. 183-189
7 CRAIG, J.J., HSU, P. and SASTRY, S.: 'Adaptive control of mechanical manipulators', Int. J. Robot. Res., 1987, 6-2, pp. 10-20 8 SLOTINE J.-J.E. and LI W.: 'On the adaptive control of robotic
manipulators', Int. .I. Robot. Res., 1987, 6-3, pp. 49--59
9 MIDDLETON, R.H., and GOODWIN, G.C.: 'Adaptive com- puted torque control for rigid-link manipulator', Syst. Control Lett., 1988, 10, pp. 9-16
10 SLOTINE, J.-J.E., and LI, W.: 'Composite adaptive control of robot manipulators', Automatica, 1989, 25, (4), pp. 509-519 11 PSALTIS, D., SIDERIS, A., and YAMAMURA, A.: 'Neural
controllers', IEEE Proc. 1st international conference on Neural networks, San Diego, 1987, pp. 551-558
12 NARENDRA, K.S., and PARTHASARATHY, K.: 'Identifica- tion and control of dvnamic svstems usine neural networks'.
IEEE Trans. Neural Nefw., 1990, ’1, (l), pp. 4 2 7
13 BARTO, A.: 'Connectionist learning for control: an overview' in MILLER, W.T., SUTTON., R., and WERBOS, P.: 'Neural net- works for control' (MIT Press, 1990)
14 HUNT, K.J., SBARBARO, D., ZBIKOWSKI, R., and GAW- THROP. P.J.: 'Neural networks for control systems — a survey', Automatica, 1992, 28, (6), pp. 1080-1112
15 SANNER, R.M., and SLOTINE, J.-J.E.: 'Gaussian networks for direct adaptive control', IEEE Trans. Neural Netw., 1992, 3, (6) 16 JIN, L., NIKIFORUK, P.N., and GUPTA, M.M.: 'Direct adap-
tive output tracking control using multilayered neural networks', IEE Proc. D, 1993, 140, (6), pp. 393-399
17 OZAKI, T., SUZUKI, T., FURUHASI, T., OKAMA, S., and UCHIKAWA, Y.: 'Trajectory control of robotic manipulators using neural networks', IEEE Trans. Ind. Electron., 1991, 38, (3), -nn 195-702
18 KUAN, A., WHITTLE, C.B., JR, and BAVARIAN, B.: 'Neuro- controller selective learning from man-in-the-loop feedback con- pol actions' m BEKEY, G.A., and GOLDBERG, K.Y. (Eds.):
Neural networks in robotics' (Kluwer Academic Publisher, 1993) 19 HORNIK. K.. STINCHCOMBE. M.. and WHITE. H.: 'Multi- layer feedforward networks are universal approximators', Neural Netw., 1989, 2, pp. 356-366
20 POWELL, M.J.D.: 'Radial basis functions for multivariable interpolation: a review' in MASON, J.C., and COX, M.G. (Eds.):
'Algorithms for approximations' (Oxford, 1987), pp. 143-167 21 MOODY, J., and DARKEN, C.: 'Fast learning in networks for
locally tuned processing units', Neural Comput., 1989, 1, pp. 281—
294
22 LEE, S.: 'Supervised learning with Gaussian potential' in KOSKO, B. (Ed.): 'Neural networks for signal processing' (Pren- tice Hall Inc., 1992)
23 CHEN, S., COWAN, C.F.N., BILLINGS, S.A., and GRANT, P.M.: 'Parallel recursive prediction error algorithm for training layered neural networks', Int. J. Control, 1990, 51, (6). pp. 1215- 1328
24 SERAJI, H.: 'An approach to multivariable control of manipula- tors', Trans. ASME J. Dyn. Syst. Meas. Control, 1987, 109, pp.
146-154
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