Learning Control of Robotic Manipulators Based on Neurological Model CMAC

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LEARNING CONTROL OF ROBOTIC MANIPULATORS BASED ON NEUROLOGICAL MODEL CMAC H. Kano and K. Takayama International Institute for Advanced Study of Social Information Science, Fujitsu Limited, 140 Miyamoto, Numazu, Shizuoka 410-03, Japan Abstract. A learning control method for robotic manipulators is proposed by using the CMAC (Cerebeller Model Arithmetic Computer), a neurological model of cerebellum. The control system consists of a feedforward controller with the CMAC and a fixed-gain linear feedback controller. This system is designed in such a way that desired hand trajectories which are specified in the operational space can be tracked directly without being converted into the ones in the joint space, thus so-called inverse kinematics problems need not be solved. Moreover this system requires no knowledge on manipula tor dynamics for which an internal model is to be built in the control system by learning. Feasibility of the method is studied through various kinds of computer simulation experiments by using a 2-DOF planar manipulator. Desired trajectories are generated from the data measured in experiments of human handwriting motion, and are fairly complex for the model manipulator. Such simulation experiments demonstrate high control system performance from the viewpoints of the convergence in repetitive motions as well as of the so-called generalization phenomenon in learning. Keywords . Robots; manipulators; learning systems; motor control; nonlinear control systems; tracking systems; simulation.

INTRODUCTION As is well-known, motion of robotic manipulators can be described by a system of coupled second order differential equations which are highly nonlinear( e.g., Brady et aL 1984). Such nonlinearities are due to centrifugal forces, Coriolis forces, gravitational forces, and these effects, which increase as manipulators are operated faster, present difficulties in designing the control systems. Various methods in the so-called modern control theory have thus been applied to problems of manipulator control, while on the other hand, techniques which are inherent in this field have also been developed. As typical methods, we may mention linear control method by Golla, Garg and Hughes (1981), resolved acceleration method by Luh, Walker and Paul (1980), non linear control method by Freund (1982), and sliding mode ' control by Young (1978). The so-called adaptive- and learning control methods (Dubowsky and DesForges 1979, Nicosia and Tomei 1984, Balestrino et aL 1983, Takegaki and Arimoto 1981 , Ambrosino et aL 1986) form a class where modeling errors of dynamical equations and/or payload changes are taken into account ( Narendra 1986). Most of these studies considered the trajectory tracking problems in which desired trajectories of manipulator hand are preplanned mostly in the joint space. Optimal control methods have also been studied extensively by employing cost functionals such as minimum-time control(Shin and McKay 1986, Ailon and Langholz 1985), minimum-energy control (Ohsaka and Ono 1989), and minimum-torque-rate motion (Uno et aL 1989). However, these methods usually require heavy computational load due to the nonlinearities in the dynamical modeL On the other hand , considerable attention has been paid to the control not only of the position of manipulator but also of its force (Paul and Shimano 1976, Raibert and Craig 1981). In particular. the concept of impedance control proposed by Hogan (1985) plays important roles in dealing with contact motions of manipulator with its environment.

A trajectory control problem for robotic manipulators is considered in this paper. In particular, we study a learning control method based on a neurological model CMAC (Cerebellar Model Arithmetic Computer) proposed by J.S.Albus (cf. Albus 1979, 1981 ). Applications of CMAC to the manipulator control problems have been reported in several literatures. Albus himself, although details are not described, demonstrated its use in the manipulator control (Albus 1975, 1981). Miller applied CMAC to a sensor-based control problem (1987a), and also studied various characteristics by computer simulations in detail (1987b) . It has been applied also to solving the so-called inverse kinematics problems of redundant manipulators (Tanaka et aL 1988). In this paper, we investigate a control system in which, unlike the case of Miller (1987b), the desired trajectories of hand specified in the operational space can be directly processed (Kano and Takayama 1988a,b). Thus , the so-called inverse kinematics problem which presents various difficulties needs not be solved a-priori. The remainder of this paper is organized as follows. First, the control problem of robotic manipulators is described . CMAC is then briefly reviewed , and based on it, a learning control system is constituted. Performances of such a control system are examined by computer simulation experiments from various aspects. Finally, concluding remarks are given. STATEMENT OF CONTROL PROBLEM As is well-known, the dynamics of robotic manipulators can be described by a system of second order nonlinear differential equations in each joint angle. R(8)8 + e(8 , 17 ) + £(8)

=

T

(1 )

Here, 8 is an n-dimensional joint angle vector, R(8) is an n x n inertia matrix, and T is an n-dimensional joint torque vector. The first , second, and third term in the left-handside of eq.( 1) respectively represent inertia force, centrifugal and Coriolis forces, and gravitational force .

On the other hand, denoting by x an m-dimensional position/orientation vector of hand, we obtain from the kinematics (2) x = f(B)

C(x) = rT(B)E(B)

and its differentiation with respect to time t yields x i

=

=

J(B)O

J(B)8

+ J(B)O

T

(3) ( 4)

The problem of trajectory tracking control is considered in this paper. That is, we find joint torques which drive a manipulator in such a way that the hand follows given (desired) trajectory as close as possible. Typical approach to this problem is as follows. A desired trajectory of hand is usually specified as the trajectory Xd of position/orientation vector of the hand in the operational space. Such a trajectory Xd is then transformed into the one in the joint space, say Bd, by the so-called inverse kinematics. This process is introduced since, as we see in eq.(I) , the dynamics of manipulators is given in terms of joint angles B and torques T. Finally, based on the dynamical equation, some kind of controller is designed. The solution to the inverse kinematics problem involves considerable amount of computation and some inherently difficult problems. Assuming non-redundant manipulators, i.e. n = rn, we formally obtain the following relations from eqs.(2), (3) and (4).

0=

= JT(B)p

(12)

Substituting Band 0 obtained from eqs.(5) and (6) into eqs.(9)-(1l) results in functions in x and X, and thus, in principle, equations for hand trajectories can be derived.

where J(B) is the Jacobian of f(B), i.e., J(B) = 8f(B)/8B.

B=rl(X)

(11)

Further, p in the right hand side of eq.(8) is an n-dimensional force/moment vector acting on the hand , and from the principle of virtual work , it holds that

It should, however, be noted that transformation from x to Bin eq.( 5) (i.e., inverse kinematics problem) is not unique in general, and nonsingulari ty of the J acobian must be assumed in eqs.(6) and (7). Besides, the resulting equation (8) will be very complicated and it is almost impossible to get an explicit expression. As noted earlier, we consider a control scheme for robotic manipulator such that its hand follows desired trajectories specified in the operational space. In particular, by employing a neurological model CMAC, we present a learning control method that will not require solving inverse kinematics problem for given trajectories. In fact, CMAC will be used to learn and approximate the complicated dynamic equation (8) in manipulator hand motion. CMAC Cerebellar Model Arithmetic Computer (CMAC) was proposed by J.S.Albus as a neurological model of cerebellum (see, e.g., Albus 1979). Here for the sake of later discussions, it will be briefly reviewed. CMAC possesses the function of computing, learning, and storing scalar-valued functions in several variables

(5)

q = hiS)

(13)

rl(B)x

(6)

rl(B)(i - J(B)O)

(7)

essentially by the so-called table look-up method. Here S is a v-dimensional input vector (SI , S2, ... , Sy) with each element

Besides the computational amount, eq.(5) has in general multiple solutions and the Jacobian matrix has to be inverted.

taking suitably quantized values, and q is a scalar output. An approximation and learning of function h(-) proceed as summarized below (see e.g. Albus 1979 for details).

Such a control problem as given above is treated in this paper and, in particular, a control system is examined which consists of CMACs and possesses the following properties: (a) The desired trajectories specified in the operational ~p~ce can be directly input to the control system, whereby aVOldmg the explicit solution of inverse kinematics problem, (b) the control system has the ability of learning the motions, and (c) it can construct an internal model of the manipulator by learning.

(PI) Given an input vector S, it is coded by using the socalled quantizing functions (let the number of the functions be C), and it specifies C addresses in the CMAC memory of table form.

8=

(P2) An approximate value of h( S) (let it be ij) is then obtained by summing the C current values (weights) Wj in memory specified by these addresses: (14)

LEARNING CONTROL SYSTEM Formulation in Operational Space In this section, we show how CMAC is used in learning control system. As was presented in the previous section, motions of robotic manipulators are governed by a set of differential equations in joint angles. Here, we consider their counterparts in the operational space, namely, equations of motion with respect to the position/orientation vector x of manipulator hand. Now we assume that the manipulator is non-redundant, namely, the vector x has the same dimension n as B. l'sing eqs.(5)-(7) in eq.(I), it is seen that the hand equations of motion in operational space can be expressed in the same form as in eq.(I) (see also Khatib 1981). ]t.f(x)i

+ B(x,x) + C(x) = p

(P3) For a true value (or supervisory signal) q of hiS), a quantity (15) is added to each of the weights w, that were used to compute ij in eq.(14) , and thus the C~IAC memory is updated. Here f3 (0 < f3 1) is a gain for the weight adjustment.

:s

The above is an outline of approximating and learning a function. The number of addresses C and a gain f3 are parameters that have to be fixed a priori. C is related to the degree of the so-called generalization, and greater C implies stronger generalization. On the other hand, f3 is related to learning speed and the stability of learning process. Greater fJ implies faster learning, but at the same time, often results in more unstable learning process.

(8)

Here Af(x) is an n x n symmetric matrix , B(x,x) and C(x) are n-dimensional vectors related to the terms in eq.(I) as follows.

The size of the memory table required is CQY where Q is the number of outputs that each quantizing function can assume. It is noted that the simplest table look-up method 2.'iO

requires RV memory locations, where R is the number of distinguishable values that each element of 5 can take. Although CQv is in general considerably smaller than RV, it still is too large to be applied to moderate size of control problems of manipulators. Thus a method of hash-coding is employed, as suggested by Albus, by using a uniform random mapping technique, and the memory size is reduced to a practical one. Although only functions with single output are considered in above, the case of multi-outputs can be easily devised by simply preparing the same number of CMACs as outputs. Learning Control System As has been shown above, CMAC has the ability of learning and storing an unknown function by repeating a process of computing an approximate value of the function for given input and then by updating the memory based on the approximation error. In the present study, the system of n differential equations (8) in x is regarded as a vector-valued function in x, x and X,

P = AI(x)x

+ B(x, x) + C(x) ~f h(5)

(16)

with 5 ~f (x, x, x), for the approximation of which 11 CMACs are used. Notice in eq.(13) that if two discrete input vectors 51 and 52 are close to each other, then the corresponding two sets of addresses share many overlapping elements, implying that the resulting two function values are close to each other. When CMAC is applied to the manipulator dynamical equation (16), this property of the so-called generalization leads to such an advantage that the information gained or the function learned for an input (X(tk), x( tk), X(tk)) can well be used to approximate an input (X(tk+I) , :i:(tHI),X(tk+d) at the next instant since desired trajectories are usually continuous functions of time. Moreover, we can also expect the same kind of property for neigh boring trajectories: information gained for a trajectory by learning can be effectively utilized for nearby trajectories. Now we suggest the learning control system shown in Fig.l for the problems of trajectory tracking control of manipulators (I
trol system, and the feed forward portion with CMACs and Jacobian can be regarded as an inverse system of manipulator. In a general case, it is expected that, as the learning proceeds, Pc(t) approaches p(t) and the errors decrease. The characteristics of this control system are as follows: ( a) As is seen from Fig.l, a priori know ledge concerning the kinematics is required, whereas no knowledge on manipulator dynamics is necessary and in effect inverse dynamics will be built within the CMACs by learning. (b) The system is a learning control system. By repeating the motions for a single desired trajectory, we expect that the tracking error decreases. In addition, learning for one trajectory helps tracking for other neighboring trajectories even in its first trial of control action. These are apparently due to the generalization property of CMAC. (c) Desired trajectories specified in the operational space can directly be input to the control system, and unlike conventional method, it is not necessary to solve the inverse kinematics problem explicitly. Thus the difficulties associated with multiple solutions and with heavy computational load can be circumvented.

SIMULATION EXPERIMENTS

For numerical simulation experiments, 2-link manipulator shown in Fig.2 is used with arm length I1 = 12 = 0.3m and the weight ml = m2 = 5kg. Then the dynamical equation (1) of the manipulator is obtained with

+ 0.45 cos 82 = 0.1505 + 0.225 cos 82

rll

= 0.751

r12

= r21

rn

= 0.1505

Cl

C2 el e2

= (-0.45B I B2 - 0.225B~) sin 82

(17)

= 0.225Bi sin 82 = 22.05 cos 81 + 7.35cos(8 1 + 82 ) = 7.35cos(8 1 + 82 )

where, rij is the ij-element of matrix R, and Ci and ei are the i-th elements of vectors C and E respectively. \-Ve, however, consider motions in the horizontal plane (let it be x-y plane with its origin located at the first joint), and the gravity term E above can be omitted. Notice that the work space of this model manipulator is a disc with its radius 0.6m and the center at the origin. y

(x ,y)

Fig.l Learning control system for manipulators x

Learning in the C!\L\C proceeds by updating the memory using p( = Pc + Pf) as the supervisory signal. In an ideal case where the learning is perfectly achieved , i.e., Pc(t) = p(t) for all t :::: 0, we have PI(t) = 0 and thus the position error X(t)-Xd(t) as well as the velocity error :i:(t)-Xd(t) are zero. The system then behaves as if it were the feedforward con-

Fig.2 2-link model manipulator used in the simulations Desired Trajectories Desired trajectories are obtained from hand-writing motions of a Japanese letter '
of size approximately 0.2m square and is written in the first quadrant of the x-y plane (see, e.g., dotted line in Fig.3). In this- paper,-we pay our attention only to horizontal motion (projection onto the x-y plane), and vertical motion will not be simulated. Moreover, portions of the trajectory where the pen is being lifted are not shown resulting in three seperated segments in the figures representing the operational space (xy-plane; see e.g. Fig.3), although the trajectory actually is continuous and is so processed in the control system.

Experimental Results and Discussions In the results to be presented below, the feedback gain K in the control system of Fig.1 is fixed as 70h for both the position and velocity terms. It is noted that, at least qualitatively, similar results were obtained for other choice of K. It has been observed that larger value of K yields faster learning, but too large K results in unstable control system. Smaller K, on the other hand, results in more steady but slower learning.

< Experiment 1 >.

The trajectory is given as a time series data which is obtained by sampling the motion with period 28msec, and it consists of 121 points. Thus the duration of motion is about 3.4sec., and we believe that the trajectory is quite complex for the model manipulator. As we shall see, other trajectories that are obtained by parallel displacement of above trajectory will also be used. CMACs CMAC is used to approximate the function in eq.(16), and hence there are six input variables, namely x, y, x, if, x and ij. Two such CMACs are prepared in response to the two outputs PI and P2 in eq.(16). Each input variable is quantized by dividing preassigned intervals Ix =[0.0,0.6], Iy =[0.0,0.6], Ix =[-0.5, 0.5], Iy =[-0.5, 0.5], I1: =[-5 .0, 5.0] and Iy =[-5.0, 5.0] into 500 equal intervals. These intervals Ix. Iy etc. are determined so that they are large enough to cover various motions to be considered. The parameter C for CMAC is set as 100 unless otherwise stated. In this case, the required memory size exceeds 4 million locations. Here, by employing the so-called hash-coding method based on uniform random mapping, the memory size is reduced. The size actually used is 10000 for each output. At the start of each experiment, the contents in the memory are all initialized to zeros. Finally, the gain f3 used to update the memory in step (P3) is set as 0.1 unless otherwize stated.

Here, the experimental results of repetitive motions for a single trajectory are presented. The controlled trajectories in the hand coordinate are plotted in Fig.3 in solid line with (a) for the first trial and (b) for the 50th trial. The desired trajectory is plotted in dotted lines, and we see that the hand followed the desired trajectory with sufficient accuracy after 50 repetitions. Fig.4 shows the root mean square (rms) tracking error for x against the number of trials. It can be seen that the error decays quite fast and smoothly with the increase of the number of trials. Similar results have been obtained for x, y, and if. The dotted straight line in the figure shows the resolution that resulted in the quantization, and in this case it is 1.2x 10- 3 = (0.6 - 0.0)/500. It should be noted that the rms tracking error became smaller than this value although it is the least value which the CMAC can distinguish. Fig.5 (a) and (b) show PI = Pc! + PII (solid line) and PII (dotted line) with respect to time t for the 1st trial and the 50th trial, respectively. We see that the output signal of the controller PI mostly consists of feedback signal PII in the first trial, and that P II is almost zero in the 50th trial hence the learning is performed successfully. In Fig.6, rms value of CMAC memory adjustment {j in eq.(15) for the output PI is plotted against the number of trials . VIe also see rapid and smooth convergence.

Y

Y

0.4

0.4

(m)

Cm)

0.3

0.3

Cm)

0")

I

0

.-<

" "

4

4-<

0

'0" I-<

0.2

0.2

2

I-<

0.2

0.3

0.2

0.4

0.3

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'"E

0.4

x Cm)

x

'"

Cm)

0

25

0

50

no. o f trials

(a) first trial

(b) 50th trial Fig.4 Root-mean-square tracking error of x.

Fig.3 Desired trajectory (dotted line) and controlled trajectory (solid line) in repetitive motions.

'-'

c:

0.01

'E"

'-'
:0

-g '-'

0.005

Oc

o

1

(a) first trial

3

o

1

2

3

'3" 'E"

t

l-

0.0

0

25

50

no. o f trials

(b) 50th trial

Fig.5 x-component PI of total force vector acting on the hand (solid line) , and its feedback component Pil (dotted line).

Fig.6 rms weight adjustment in CMAC memory.

From these experiments, we may observe that the learning is done in quite a steady manner and the learning control system is operating satisfactorily. Moreover, the computational amount required in the process seems not to be so large, since the substantial part consists of only simple arithmetics in (PI), (P2) and (P3), random mapping in the hash-coding, and the computation for the forward kinematics.

< Experiment 2 >. Some experimental results concerning with the inverse kinematics problem are presented. As has been discussed, the desired trajectories of hands are specified in the operational space. In the case of 2-D OF planar manipulator considered here, generally there correspond two points in the joint space to each of a point in the operational space, represented as elbow-down configuration and elbowup configuration. Thus if there are no constraints on the joint angles, then it is expected that the control system can yield any of the two configurations. Fig.7 (a) and (b) show the trajectories of 82 which resulted after 50 trials starting respectively from elbow-down and elbow-up initial configurations. For the sake of comparison, we also obtained two 'desired trajectories in the joint space' corresponding to elbow-down and elbow-up configurations respectively by solving the inverse kinematics problem for the given desired trajectory. These have been plotted in Fig.7 (a) and (b) in dotted lines, but almost completely agree with the controlled trajectories (solid lines). It is noted that the curves in (a) and (b) have opposite sign from each other as is expected. We may observe that the controller outputs such joint torques that maintain the same configuration as was initially specified; elbow-down or elbow-up. This probably is due to the property of generalization of CMAC: for given continuous trajectory, it has the tendency of producing a continuous torque and thus a control which involves the changes in configurations seems to be avoided automatically.

2 rad.

-1.0 8 2 rad.

2.0

-1.5

1.5

-2.0

8

< Experiment 3 >. In the above, the experiments were performed for repetitive motions for a single desired trajectory (let it be called the trajectory A). Here, we examine properties of thecbIftrol system for repeating motions for different trajectories, such as the effect of learning for one trajectory to the control for other trajectories. The desired trajectories used below are generated by parallel displacement of A on xy-plane by the amount of (-a, a) with -0.15 ::; a ::; 0.15. Thus the velocity and acceleration components are the same with all the trajectories. The trajectory for a = 0.02 is called B for simplicity. Fig.8 shows the result of 1st trial for trajectory A right after the 50 trials for trajectory B. Comparing with the case in Fig.3(a) where no prior trial for B is performed, considerable improvement can be admitted: In words, if one is trained to write a letter, then he can write the letter much better as long as it is to be written in the neighborhood of the original one. This implies that, by repeating the trials for B, the functional relation in eq.(16) has been approximately formed in the memory also for trajectory A which is regarded as close to B. Every 20 trials are repeated alternatingly for trajectories A and B, and the learning process is examined. An example is shown in Fig.9 (a) and (b). The ratio of feedback value in the total control value is plotted in (a), and the rms weight adjustment in the memory is plotted in (b), both against the number of trials. It can be seen that both of the quantities in (a) and (b) present peaks each time the desired trajectories are switched from one to the other. These peaks, however, become smaller as the number of trials increase, and as a whole, the process converges. Similar curves are obtained for trajectory tracking errors. Thus, the learning is being completed simultaneously for trajectories A and B.

Y

Cl.4 (m)

*

0.3

tJ0

0.2 1.0 0

1

2

3

-2.5

(a) elbow-down configuration

0

1

3

0.2

w

1.0

'"E"

a.

Fig.S 1st trial (solid line) for the trajectory A right after 50 trials for trajectory B.

0

"

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....

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0.5

'-'

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0.005


~ .....

a.

o


0.0

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100 200 no. of tri a ls

(a) normalized rms value of x-component Pil in feedback signal.

0.0

200 no. o f trials

(b) rms weight adjustment in CMAC memories.

Fig.9 Learning by switching the desired trajectories A and B alternatingly every

20 trials starting with A.

0.01

... E

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( m)

0 .02

...c ......

!I.

~

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E

---

0.010 , - - - - - - ----,

w


0.4 x (m)

Fig.7 Controlled trajectory rh (solid line) in the 50th trial with (a) elbow-down and (b) elbow-up initial configurations, and the corresponding inverse kinematics solutions of desired trajectory (dotted line).

-<

0.3

(b) elbow-up configuration

o

0.1 Cl

(m)

Fig.lO rms tracking errors in the 1st trial for the trajectory Q with and without the prior 50 trials for A (i.e. a=J).

Finally, the degree of generalization in this control system is examined. First, we set C = 100. Prior to each first trial for each trajectory specified by 0', all the memory contents are ini tialized to those right after the 50 trials performed for trajectory A (corresponding to 0' = 0). The rms tracking error for y in the first trial performed in such a manner for each 0' are shown in thick solid line in Fig.lO against 0', a measure of closeness of the trajectory to A . In these figures, dotted lines correspond to the case where the prior 50 trials for trajectory A are not carried out. Thus the learning effects for trajectory A to nearby trajectories with small 10'1 are apparent. On the other hand, for large 10'1. namely when the two trajectories are far apart, such effects are negligibly small. We may also observe that the errors decreases almost linearly with 0' in the neighborhood of 0' = O. Also, small C resulted narrower valley in Fig.lO (see thin solid line for the case of C = 50), or the region influenced by the trials at 0' = 0 is smaller. That is, the smaller the value C, so is the degree of generalization. CONCLUDING REMARKS We considered the trajectory tracking problem for robotic manipulators using a neurological model CMAC. A learning control system has been suggested and it possesses the following properties. (1) In general, the desired trajectories for the hand are specified in the operational space, and this control system can process such trajectories directly. That is, unlike most of the existing methods, it is not necessary to transform such trajectories into the ones in the joint space by solving the inverse kinematics problem. (2) The only information on manipulator needed to design the control system is the one related to kinematics, i.e., geometrical information of manipulator. Dynamical equation of the manipulator is being built in the CMAC memory by repeating tracking motions for various desired trajectories. (3) As the learning proceeds, the system gradually shifts from feedback control system to feedforward control system. At the extreme when the learning becomes perfect, it is in effect a feed forward control system and the controller consisting of CMAC plays a role as an inverse system of manipulator. Note that the control system still is robust to unexp ected disturbances by the existence of feedback block. Various properties of the learning contol system have been examined by numerical simulation experiments using a 2link planar manipulator, and we verified the followings. (4) By repeating tracking motions for a single trajectory or multiple trajectories, the tracking errors approached zero quite rapidly. Moreover, such a learning process was sufficiently stable. (5) The generalization property of Ct-.lAC plays an important role in the learning control system. (6) As for the inverse kinematics problems , such a solution seemed to have been chosen that yields a continuous transition of movement from an initially spec ified arm configuration. This probably is due to th(' property of C~IAC that it tends to approximate input/output pairs as a continuous function from the way it is constructed. As we see from the abO\'e observations on numerical experiments, the learning control sys tem proposed here is attractive and seems to be promising for futur e practical applications. Further st udies are needed from various aspects: Among them are the stability analysis of the control system, sensitivity analysis for the erroneous kinet ic parameters , etc. REFEREl\CES Ailon , A. and G. Langholz (1985). On the Existence of TimeOptimal Control of ~fechanicalll!anipulators. J. Optimiz. Theory Appl., 46, 1-2l. Albus, J.S. (1975). A New Approach to ~Ianipulator Control: The Cerebellar !>.!odel Articulation Controller (Cl\!AC). ASME J.

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Robot Motion : Planning and

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