Contribution to the controller design in tasks of robot deburring

~

Mech. Math. Theory Vol. 30, No. 3, pp. 363-382, 1995

Pergamon

0094-114X(94)00040-9

CONTRIBUTION

Copyright © 1995 ElsevierScienceLtd Printed in Great Britain. All rights reserved 0094-114X/95 $9.50 + 0.00

TO THE CONTROLLER

DESIGN

IN

TASKS OF ROBOT DEBURRING ALEKSANDAR D. RODI(2 and MIOMIR K. VUKOBRATOVI(; Mihajlo Pupin Institute, Robotics Department, Volgina 15, P.O.B. 150, 11000 Belgrade, Serbia

(Received 9 September 1993; received for publication 1 August 1994)

Abstract--An adaptive control system is proposed and successfully implemented into the robotic deburring. The developed system consists of a controller of the manipulation robot and of a supplementary local controller of its active end-effector. The first controller ensures satisfactory tracking of the nominal robot trajectory, i.e. the robot's gross motion, while the second one enables its fine motion in the direction normal to the processed surface. In this way, a good quality of the processed workpiece edge is ensured, as well as the prescribed chamfer depth. The manipulator end-effector has been designed as a special device, and possesses a supplementary degree of freedom for fine positioning. For its control a digital regulator of minimal variance was applied, based on the estimated model parameters of the deterministic process and on the parameters of the external, stochastic disturbance model. These parameters have been experimentally identified on the basis of the measurement data obtained on the system itself. The purpose of the designed regulator is to minimize the chamfer surface roughness and control signal variance to the end-effector servomotor. Simulation results have been verified on an experimental system in the course of an on-line deburring process, whereby the possibilities of different processing speeds of the object rough edges were investigated and the obtained results analyzed from the viewpoint of the quality of the processed surface. Based on the CAD-model of the part edge or on the basis of the direct measurement of the burr during the experiment, a supplementary modification of the synthesized robot controller was carried out, which yielded a still better adaptability to external, stochastic perturbations arising in the robotic deburring.

1. I N T R O D U C T I O N

Automation of industrial processes have become indispensable in advanced technology systems. Enhancement of productivity, standardization of the processing quality, planning of the production volume, the desired commercial effects--all these requirements do inevitably urge further work on the development and improvement of both the performances and efficiency of the existing manipulation robots controllers in a broad class of contact tasks. In that sense, during the last decade or so, much work has been done on the development of robotic controllers dedicated to implementation into the robotized processes of machining. One of the typical tasks of robotic process, that on industrial scale has been performed manually for a long time, is the task of deburring the uneven part edges. In that, the manipulator replaces the unpleasant and inefficient human work. This operation is most often performed in the final stage of the workpiece processing, and is quite expensive; in some cases it attains even 35% of the finished workpiece value. Hence, the research efforts of the engineers involved in the field of control of manipulation robots applied in this class of technological operations, are highly justified. The demands posed to the robot in the deburring process are very stringent. It is necessary to ensure the chamfer depth to be within certain tolerance limits, and the minimum surface roughness of the processed edge. To achieve this manually is rather difficult, so that these operations are performed by robots more and more equipped with efficient control systems. Many investigations in the field of robot control synthesis for the robotic deburring processes have been devoted to the search for the most efficient control of the depth of cut when chamfering by using industrial manipulators, whose end-effector is in the form of a "noncompliant robot tool" [1]-[8]. As known, the chamfering force is proportional to the "tool feedrate" (tool velocity along the tool path), and the cross sectional area of a chamfer and a burr. When the cross sectional area is projected in the direction normal to the part edge, burr size variations have little effect on 363

364

ALEKSANDARD. RODI(~and MIOMmK. VUKOBRATOVI(~

the total area. If the control requirement is to maintain constant force, the burr variations will be followed and the control of the chamfer depth will be poor. This is unsuitable from the viewpoint of the processed surface quality, because a constant chamfer depth is desirable. A constant chamfer depth can therefore be obtained by controlling the normal force. In the course of the last few years, this has become an important control strategy in the field of improving the existing and designing new control algorithms. Usually, the position and force of the end-effector can be controlled by controlling the robot mechanism only. However, many investigations carried out on several industrial robots have shown that the low bandwidth of the manipulator mechanism limits its capability to control the chamfer depth at higher feedrates [3, 4]. The use of special actuators by which the cutting tool is controlled in a much more direct way resulted in improved performances of the system [6-8]. Such actuators are called active robotic end-effector, because they possess two supplementary degrees of freedom (DOFs) for fine positioning of the deburring tool, i.e. for realization of the normal chamfering force as precisely as possible. In this way, the robotic mechanism (its n DOFs) is utilized for the realization of the so-called gross motion along the processed workpiece edge. The so-called fine motion is controlled by the active end-effector of the manipulator, performing fine positioning of the tool tip relative to the workpiece in the direction perpendicular to the processed surface. The robot trajectory can be determined either from the off-line CAD-model of the workpiece that is being processed and the tool geometry [9], or by on-line tracking of the workpiece edge [10]. Since the nominal trajectory is known, fine motion can be realized by using either a passive (not demanding an external power to operate) or active end-effeetor. In [5], Kramer used a passive end-effector, specially designed for that purpose, so that its impedance reduces the effects of the robot position errors and of the variable workpiece edge profile. The author concluded that the precision can be further improved by adding a supplementary active force control to the end-effector. Kazerooni [8] has developed an end-effector with 2 DOFs which enabled to overcome the limitations of passive end-effectors. The impedance in the direction normal to the processed part edge (surface) was used for obtaining precise chamfer depth. At the same time, the impedance in the direction tangential to the edge, was used to control the rate of material removal. Using the passive end-effector design, the impedances in the normal and tangential direction are coupled. On the other hand, the active impedance control enabled the impedances in the normal and tangential direction to be selected independently, so that the system's performances are improved. This paper represents a contribution of studies of the control problems in the field of robotized deburring by using the one DOF milling tool. The attention is focused on the synthesis of the control laws for a 6-DOF manipulation robot, based on control algorithms proposed in [11]. Different from traditional hybrid control concept[12], the proposed control approach Ill] is based on the new formulation of position/force control which ensures asymptotic stability of both, desired motion and interaction force of robotic mechanisms in the contact with dynamic environments. In this paper, the proposed control scheme is used for the solution of gross motion control. Fine motion is achieved by using a local servo-controller in the form of a Digital Regulator of Minimal Variance (DRMV) with respect to the contact force in the active robot end-effector. This regulator is used to minimize the oscillations of the system output, i.e. to minimize the chamfer surface roughness. Instead of the surface roughness which is rather difficult (if not impossible) to measure on-line, minimization of the normal force variance is used as the control objective. At the same time DRMV ensures the minimization of the control signal oscillations to the end-effector actuator. Performance evaluation has been made in this paper, based on the force and control signal variance achieved by proposed controller in simulation and real-time experiment. To improve the performances of the control system, further steps were taken in the direction of achieving additional adaptibility of the control system to the presence of measurable variations of the part edge. This improvement was achieved using the prediction of the disturbance signal acting on the system. On the basis of the prediction of the disturbance signal and its influence to the robot end-effector during deburring, a supplement control signal was determined in order to correct fine motion control signal on DRMV. In this way, depending on the precision of the

Robotic deburring

365

disturbance signal prediction, the resident variation of the system output can be practically to a large extent eliminated. For the reasons of security and better efficiency of the robotic system, the controller was tested first off-line by computer simulation. Then, these results have been verified in real-time experiments. The controller performances were evaluated in respect to the criterion of the minimum of the normal force variance and the chamfer depth with minimum surface roughness for various feedrates. 2. M O D E L OF T H E M I L L I N G P R O C E S S The tool used in robotic deburring is usually chosen in the form of a milling tool which is characterized by hardness and the defined geometry [13]. The model of milling process is presented in Fig. 1. This model was used for simulations. The model adequacy has been confirmed by comparing the simulation results to those obtained experimentally by on-line measurement on the real system. A linear model of the cutting force F c was adopted, similar to that proposed by Yellowley [14] and Altintas[15]. As shown in Fig. 1, the cutting force Fc has two components: Fx and F~. Also there are friction forces F~ and F~' (i.e. the friction torque T~') acting between the tool and the surface which is being processed. The cutting force model can be written in the following form: F c = KsAt A t = as t s, = v r A t = v f ~°t = vf

(Dt

(1)

d

6'3t r t

vf

feedrate

J;i

[mm/sl <5~

I i

L Ao

Z

|

l

b u r r stze

o:

varlatlons

"..~-------------~ I

_~_xX

\ \VI\\\ \ \ \\ \¢'\\\\\

~t

r~

~ g

tee~

tooth

Fig, I. Model of the milling process.

366

ALEKSANDARD. RODI~and MIOMIRK. VUKOBRATOVI(~

where Ks is the specific cutting pressure (in [N/m2]) and At is the cross sectional area of the burr ([mS]) removed by action of one tool tooth only. This surface is a function of the axial chamfer depth a and the thickness of the material layer st removed during one full revolution of one tool tooth around its proper axis. The thickness st is proportional to the linear feedrate vr in the x-direction of motion (Fig. 1) and the time interval At. It represents the duration of the tool tooth contact with the processed material. This time interval is proportional to the angle of material engagement (fit and inversely proportional to the tool angular rotation speed tot. The angle of material engagement ~ot can be approximated with sufficient accuracy by the ratio of the average width of the burr d and the radius of the tool rt, provided the tool centre point (the point " O " in Fig. 1) is close to the burr axis of symmetry and the average width of the burr is small compared to the tool radius. The parameters Ks and st are determined experimentally. The friction forces F~ and F~' are acting in the X - Y plane, as reaction forces to the transfer translation and the proper tool rotation. Taking into account one tool tooth only and the engagement angle ~0t corresponding to it, and the friction force components in the contact zone of the tool and the processed surface, the chamfering force components in the X and Y directions are: F~,(~ot) = F~ + F~ = - F c sin tpt + F~ F'r(q~,) = F,. + F}' = - F c cos (pt + F}'

(2)

The instantaneous amplitude of the resultant chamfering force is obtained by summing the force components in the plane, for the total number of milling tool teeth nt:

FR(t)~
(3)

During the milling process the robot tool is sliding over the processed surface and acts on it by a prescribed pressure force Fp. At that, the elastic effects both of the robot mechanism itself and of its environment come to effect [16]. It is well known that every material resists penetration of a strange body into its structure, by which the speed of that body is reduced. Also, the tool in motion experiences certain inertial forces acting in the direction normal to the chamfer surface which is being processed. In Fig. 1 is shown the pressure force Fp, representing the contact force acting on the manipulator end-effector. The model of contact force in the critical Z-direction is defined by the general impedance model in a scalar form [17]: Fa(t) = meg(t) + f~3(t) + Ce6(t)

(4)

In this model, the parameters me, fe and ce can be identified experimentally on the system. Successively, they are representing the virtual (equivalent) mass of robot mechanism [17] and the coefficients of the equivalent damping and stiffness of the robot mechanism and the environment[16]. The variable 3(t) represents micro-displacement of the tool tip in the Z-direction perpendicular to the chamfer surface which is being processed (Fig. 1). The described process of robotic deburring is a convenient example where the process control can be realized by using a microcomputer with autonomous hardware and software. This process possesses typical dynamics enabling the given requirements in respect of the quality of behaviour and work precision in stationary state, as well as the quality of the dynamic behaviour of the regulation contour in a transient state to be achieved by using the corresponding DRMV. The mentioned regulator is especially suitable for application in control of the process exposed to the external, stochastic perturbations. The system structure with DRMV, the considered process as the control object, and external perturbations are illustrated in Fig. 2. It is assumed that the digital controller possesses at its input an A/D converter generating samples of the controlled variable y t or the error signal e*. Besides, the controller possesses a microcomputer, which, on the basis of samples of the error signals e ( k T ) = y o ( k T ) - y ( k T ) (k = 1, 2 . . . . ), realizes the prescribed program (control algorithm), described by the function of t Controlled variable y represents the system output. Physically,it represents the cutting depth of robotic deburring, and according to the Fig. 1, it consists of the axial chamfer depth a, the burr magnitude and the penetration depth 6(0 of the milling tool tip into the workpiece material structure.

367

Robotic deburring DI,$'IIJRBANCE FILTER v I--"-! .....................

.................... i

n

Yo

!--1................................................................. P R O C E S S Fig. 2. Structure of robotic deburring system with DRMV. a discrete transfer Gr(z). Also, it is supposed that the DRMV possesses at its output a D/A converter and an output device (OD) which transforms the output signal of the D/A converter into a desired standard current signal, demanded by the end-effector actuator (EA). In the text below, the process, and the detector of the controlled variable (DET) will be treated together as the control object in a broader sense. The digital filter of the disturbance signal is given by its discrete transfer function G,(z). For the D R M V synthesis it is useful to express the theoretical dynamic model of the milling process in the following form:

y(t) = f[u(t), n(t), O]

(5)

Here, u(t) and y(t) represent the measurable input and output signal, respectively, 0 is the so-called vector of process parameters, and n(t) is the unmeasurable disturbance signal. The unknown components of the process vector 0 are determined either by off-line or on-line estimation of parameters using the well known Recursioe Least Squares Estimation Method (RLSEM)[18, 19]. The process and perturbation parameters, identified in this way, were used for the synthesis of the mentioned digital regulator. Let us assume that the dynamic process, whose parameters are to be identified, is stable, linear and stationary. The process behaviour in the vicinity of the observed stationary state can be described by the discrete model in the form of a discrete transfer function [18]: Gp(z) = 6(z) = B(z -j) z_a= bl z -I -~- b 2 z - 2 - q - " " " --[- b,,z '~z_ a u(z) A(z -1) l +a~z-l + . . . + a , , z -m

(6)

where u(k ) = U(k ) - Uo and y(k ) = Y(k ) - Yo are the differences of the absolute input U(k ) and output Y(k) values and the values of the signals in stationary state U0 and Y0; d is the positive integer constant designating the transport delay or the process "dead time". The order and the coefficients of the transfer function are determined experimentally. The output signal y(k) of the system used, illustrated in Fig. 2, is to a smaller or greater extent contaminated by the stochastic, unmeasurable signal n (k) due to perturbation action on the process and/or the presence of a noise, as shown in Fig. 3.

v(k)

D(z'I) A(z "1 )

u(k)

B(z -1 ) A(z "1 )

n(k)

Fig. 3. Model of the process and perturbations.

y(k)

368

ALEKSANDARD, RODICand MIOMIRK. VUKOBRATOVI~

For this purpose we have assumed that the model of the perturbation signal in a general form is described by the known [18] discrete model of the form:

n(k)+atn(k-1)+'.'+apn(k-p)=dov(k)+dlv(k-1)+...+dpv(k-p)

(7)

Further on, let us assume that the disturbance signal n (k) is generated at the output of an imaginary digital filter. At the input of it, an unmeasurable independent perturbation signal--the discrete white noise v(k), is acting, with a normal distribution of amplitudes having the average value = E{v(k)} = 0 and the variance a~ = 1. By the recursive relation (7) a stochastic signal n(k) is generated, which can be presented as the output of the imaginary filter of the discrete transfer function:

Gv(z) -

n(z) v(z)

-

2D(z-t) A(z -1)

=

,~(do+ dtz-I _ ~ . . . . ~ dpz-") l+alz-l+...+apZ "

(8)

in input of which are the samples of the Pseudo-Random Binary Signal (PRBS). In the discrete transfer function Gv(z), the parameter 2 represents the estimated proportionality coefficient. The parameters of the discrete model of the process and perturbations were estimated for the open-loop system conditions by assigning the manipulator actuator the task to track the PRBS function and measuring simultaneously the realized chamfering forces during the milling process. Namely, the depth of the chamfer y(k), obtained during the robotic deburring, represents the output variable of the control system illustrated in Fig. 2. As this quantity is practically unmeasurable or hardly measurable in a direct way, it has been determined in an indirect way by measuring the instantaneous chamfering forces. Using the combined discrete models of the deterministic process (6) and the corresponding discrete perturbation model (8), the discrete model of the system output signal is:

B(z -I) ).D(z -1) y(z) = A(z-I) z-du(z) + A (z-'-----5-v(z)

(9)

The importance of the presented models (6) and (8) for the design of a local controller for the robot end-effector will be emphasized below. 3. S Y N T H E S I S

OF ROBOT

CONTROLLER

AND SIMULATIONS

The robot controller implemented in the considered class of tasks should ensure the following system's performances: precise tracking of the nominal tool trajectory; realization of the manipulator end-effector motion with prescribed speed; maintenance of the constant tool orientation with respect to the workpiece. The main goal is to attain a good quality of the processed chamfer surface, a uniform load of the milling tool, obtaining of desired chamfer depth and surface roughness of the processed part edge. The designed controller must be robust to the action of external, stochastic, unmeasurable perturbations, as well as to enable attaining of the desired dynamic behaviour. These, quite severe criteria of behaviour quality, cannot be achieved by means of the robot mechanism only. Many previous studies in this area [3, 4] have shown that with the majority of industrial robot the low bandwidth of robot arm limits the possibility of controlling the chamfering depth at greater feedrates. For these reasons, we have developed a controller consisting of two functional parts: the robot mechanism controller, and a supplementary, local controller of the robot end-effector ensuring higher system's reaction speed as well as a better control of the chamfer depth. In Fig. 4 is presented a control scheme of the robotic control system applied in deburring. The output variables in this control scheme are the chamfer depth y(t) and the tool feedrate vr(t). They represent the measure of the process quality.

3.1. Synthesis of the robot arm controller Let us remind that the known dynamic robot model in contact with the environment has the following form [20]:

P(t) = H(q)i] + h(q, (t) - jT(q)F(t)

(10)

where P is the driving torques vector at the robot mechanism joints of dimensions (n × 1) where n is the number of the mechanism DOFs; q and 0 are the (n x 1) vector of the internal coordinates

Robotic deburring

369

~io(t) ~ ~.~:.:.:.:,:.:.:+:.:.:.x.:,:+:.:,:.:.:.:+:

Fp (tl

chamfer deptll

v,

qttl feed~te

Fig. 4. Control scheme for robotic deburring.

and vector of the corresponding velocities at the manipulator joints; H ( q ) is the (n x n) matrix of the mechanism inertia; h(q, (1) is the (n x 1) vector of the gravitational, centrifugal and Coriolis forces acting on the mechanism; J ( q ) is the (n x m) Jacobian matrix (m ~< n), and F i s the ( m x 1) vector of the external, generalized forces, acting on the manipulator end-effector. In [11] several different control laws have been proposed for controlling manipulation robots with respect to position and force. They ensure asymptotic stability of the system and the required quality of the transient response of tracking the manipulator trajectory, or, of the contact force on the manipulator end-effector. To attain a desired quality of tracking the robot nominal trajectory and for the sake of compensating for the dynamic effects which are very pronounced in the contact phase of the manipulator end-effector and the environment, the control law has been chosen in the following form [11]: t ( t ) = H(q) [~/0+ Q(q, it)] + rn(q, (1) - j V ( q ) F ( t )

(1 i)

where t is the (n x 1) vector of the control torques acting at the manipulator joints;/l(q),/~(q, q) and j r ( q ) are the estimated elements of the robot model (10);/10 is the (n x 1) vector of the nominal accelerations at the manipulator joints; Q (r/, it) is the n-dimensional vector function, continuous on the set of all its arguments [11]. Deviation of the real motion from the programmed motion represents the transient process which can be described as: rl(t) = q(t) - qo(t)

(12)

It is quite natural that the requirements to be satisfied by the transient process have to be defined in advance. First of all, it is necessary to ensure the basic control goal, the system's stability. This means the real process should converge the desired one. Certain initial perturbations are always present, i.e.: rlo = q(to) = q(to) - qo(to) ~ O.

(13)

These perturbations are related to a whole family of q(t) functions, which in the beginning can be initially perturbed. Thus the family of transient processes can always be represented by the vector differential function: /i(t) = a(q, it)

(14)

The solution of this differential equation must satisfy the requirements concerning the character of the transient process, and this is the task of the robot controller. The function Q( • ) has been adopted in the form [11]: ii(t) = F, it(t) + r 2 n ( t ) MMT

30/3--0

(I 5)

370

ALEKSANDAR D. RODIn; and MIOMIR K. VUKOBRATOVI(~

where F1 and F2 are the constant (n x n) matrices. Equation (15) can be written as:

Yc= Fx x = (rl, 0)~ F = [ O"L2F rI]]

(16)

where O, and I, are the zero and the unit matrix, respectively, both of the dimension (n x n). The matrices F~ and F2 can be chosen in such a way that the eigenvalues of the matrix F possess real, negative parts. In that way it is ensured that the process is asymptotically stable. It is easy to show that the matrices F~ i F 2 having the eigenvalues 2i (i = 1, 2 . . . . . n) can be chosen in the following form [11]:

FI=

0

)'3+24

"" "'"

0

0

01 ,,0 0

" F2=

0

'

" • " 22~_ 1+ 22~

0

-2324

"'" "'"

0

0

....

0] 0

(17)

0

22~_ 122,

The applied control law ensures stabilization simultaneously of both the programmed robot motion and the desired interaction force with the environment. In [11] it has been shown that (11) ensures a desired quality of robot's gross motion too. However, as the synthesized control law, in addition to the error signals of position and the velocity at the joints, contains the information on the real force measured only (but not on the force error), the quality of force information is not satisfactory although the global system stability is satisfied (see Ref. [11]). The goal has been, all the same, attained because the desired fine motion and higher bandwidth of the system are obtained by using a supplementary local controller with respect to force, as shown in Fig. 4. In the course of the last few years, various more or less effective, control algorithms have been used [21] for these purposes. To achieve fine motion control of the robot end-effector, the criterion of realization of the process quality as good as possible, and the criterion of simplicity for microprocessor implementation in the frame of robot controller, were decisive in choosing D R M V for the purpose of considered robot task.

3.2. Synthesis of the DRMV The D R M V was synthesized on the basis of the identified system parameters of the milling process and of the external, stochastic perturbation by applying RLSEM. In designing the regulator, use was made of the error minimization criterion of the output signal y(k) (Fig. 2), as well as the criterion of minimization of the control signal oscillations u(k) on the servomotor of the actuator. In the system with the one input and the one output variable, as is the control system considered, the D R M V should minimize the variance [18, 22]:

I(k) = var[y(k)] = E{y2(k)}

(18)

In the criterion function (18) appears only the controlled variable. However, with the aim of reducing the oscillation of the actuator input signal, the criterion function (18) contains more often the controlled variable introduced in the form:

I(k + 1) = E{y2(k + d + 1) + ru2(k)}

(19)

where d is the transport process delay and r is a positive weighting factor. This factor is chosen in such a way that each sample u(k) of the input signal can be carried out by the actuator. The considered process of cutting belongs to very fast processes, and hence, it can be assumed that the transport delay is negligibly small, i.e. d = 0. This process is represented by its discrete transfer function (6). The perturbation model is represented by the digital disturbance filter (8), the parameters of which are also being determined by off-line estimation, on the basis of measurements carried out on the real system. The perturbation signal v(k) is a statistically independent PRBS

Robotic deburring

371

signal having the average value v = E{v(k)} = 0. The structure of the considered control system with the feedback is illustrated in Fig. 2. Since the system considered is of regulator type, in which on a constant controlled variable in stationary state acts an external perturbation, we shall assume that the error signal can be written as e ( k ) = - y ( k ) . Then, the task reduces to the synthesis of the regulator G,(z ) = u(z )/e(z ) = Q ( z - l ) / P ( z - I ) , minimizing the variance: I(k + 1) = E{y2(k + 1) + ru2(k)} = E{e2(k + 1) + ru2(k)}

(20)

where e(k), i.e. y ( k ) represents the variation of the controlled variable around a stationary value Y0, due to the action of the stochastic perturbation v(k). The control variable u(k) influences the controlled variable y ( k + 1) in the subsequent time instant only, because the most recent coefficient b0 of the polynomial in the denominator of the discrete transfer function (6) is equal to zero. Hence, the sample y ( k + 1), and not y(k), appears in the criterion function (20). Since in the considered instant of sampling k, the value of the controlled variable is not known in advance of the subsequent time instant, y ( k + 1) must be estimated on the basis of the available values of u(k), u ( k - 1 ) . . . . . u ( k - m ) , y(k), y ( k - 1). . . . . y ( k - m ) , v(k), v(k - 1). . . . . v(k - m ) . For this purpose use was made of equation (9) supposing that time delay is negligible: B(z -I) , . 2D(z-]) zy(z) = atz')'-77"""-~-"zutz)-f ~ zv(z)

(21)

from which we obtain: (I + alz -I + . . • + amZ-")zy(z) = (blZ-l + . . . + bmz-m)zu(z) + 2(1 + d~z -~ + . . . + dmz-")zv(z)

(22)

or in the time domain: y ( k + 1) + a ~ y ( k ) + . . . + a , , y ( k - m + 1) = b ~ u ( k ) + . . . + bmu(k - m + 1) +2[v(k+l)+d~v(k)+".+d,,v(k-m+l)]

(23)

From the previous equation we have: y ( k + 1) = S + 2v(k + 1)

(24)

where the following substitution was introduced: S = ~ [-a~y(k - i + 1) + b~u(k - i + 1) + 2div(k - i + 1)].

(25)

i~l

By substituting y ( k + 1) from (24) in the criterion function (20) we obtain: I(k + I) = E{[S + 2v(k + 1)12+ ru2(k)}.

(26)

For further derivation it would be interesting to note that in the considered sampling instant k, all the samples of the process input and output appearing in the criterion function (26), should be known, except for the control variable sample u(k), which has to be generated in that instant and the value of the stochastic perturbation v(k + 1) in the next time instant. The control variable is independent of the values of other signals. Hence, on the right side of the criterion function (26) we take only the expected value of v(k + 1). Thus we obtain: I(k + 1) = S 2 + 22SE{v(k + 1)} + 22E{v2(k + 1) + ru2(k)}

(27)

The stochastic signal v(k) is presumably statistically independent of ~ = E{v (k)} = 0; hence the second term on the right hand side of equation (27) is equal zero. Then: I(k + 1) = S 2 + 22E{v2(k + 1) + ru2(k)}.

(28)

ALEKSANDAR D. RODI6 and MIOMIR K. VUKOBRATOVI(~

372

The value of optimal control u(k) in the instant k is obtained from: dl(k+l) 2~ dS -d~u(-k) = ~ - d - ~

+ 2ru(k ) = 2Sb~ + 2ru(k ) = 0

(29)

or after substituting S by (25), b~ ~ [ - a t y ( k

- i + 1) + b,u(k - i + 1) + 2div(k - i + 1)] + ru(k) = 0

(30)

i=1

It should be noted that on the basis of equation (23), the sum in the previous equation is in fact S = y ( k + 1) - 2v(k + 1), so that, after applying the Z-transformation to equation (30), we obtain: [zy(z ) - )~zv(z )]bt + ru(z ) = 0

(31)

After substituting expression 2zv(z) from (21) in (31): .,

A(z -~)

B(z -~)

zytz ) - D(z-~) zy(z ) + ~

-1

zu(z )|b, + ru(z ) = 0

(32)

we easily derive the discrete transfer function of the minimal variance regulator: u(z) [D(z-') - A ( z - l ) ] z Gr(z ) :-- e(z ) - z B ( z -i) + (r /bt )D (z -1)

(33)

After a slight rearranging of the coefficients, function (33) becomes: Q(z -I) G~(z) - -p ( -z _ l )

qo + qlz -I 1 + plz -I + p2z -2

=

(34)

where: = b~ + r/bj qo =

(dr

-

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q, = (dz - a2)/~ p, = [b2 + (r/b,)d, ]/~ P2 = [(r /b, )d2]/ ¢

(36)

The regulator (33) does not reduce either the zeros nor the poles of the process discrete transfer function, so that there are no limitations in its application. Stability conditions. The necessary condition for the existence of the minimum of the criterion function (20) is the stability of the closed-loop system. The control system shown in Fig. 2 is stable, if all the roots of its characteristic equation: 1 + Gr(z-~)Gp(z -t) = 0

(37)

lie inside the unit circle in the plane-z. In the case of the regulator obtained for the case r ~ 0, its characteristic equation (37) takes the form:

It can be concluded, that the system with this regulator will be stable if the disturbance digital filter is of the minimal phase [all zeros of the D ( z ) polynomial are within the unit circle in the z-plane], and if all the roots of the equation ( r / b ~ ) A ( z ) + z B ( z ) = 0 are also within that circle. It is known [18] that by increasing the weighting factor r the roots of this equation will approach the process poles, i.e. the zeroes of the polynomial A(z). 3.3. Simulation results

The designed regulator is very efficient and easily implementable into industrial controllers for controlling manipulation robots in machining processes. This statement will be supported by the obtained results in the simulation process, as well as in the experimental verification.

Robotic deburring

373

The first step in the synthesis of the robot controller to be applied in the deburring process, was to carry out several simulations aimed at adjusting the control system parameters. On the basis of the obtained results, the final readjustment o f the controller parameters was performed in the experimental work regime. The simulation was performed using the C O N M O T [23] package, dedicated to modelling, control synthesis, and manipulation robots simulation, for the case of constrained gripper motion. The parameter data for the industrial manipulation robot M A N U T E C r3 [24] were taken from the catalogue. The deburring parameters for removing the uneven rectilinear edge of 10 [cm] length from the workpiece in the form of a prism made of aluminium alloy were also defined. The process duration was 10 [s]. The milling tool progressed with a nominal speed of 11.5 [mm/s], and the tool linear speed changed according to a trapezoidal law, the beginning and the end of the process being characterized by the period ATac¢= 1 [s] for the tool acceleration and deceleration. Tool orientation was preset in such a way that it remained constant and in margins of the prescribed tolerances. The intensity of the pressure force was set at Fv = 5 [N], which ensured permanent contact between the end-effector and the workpiece. In simulation use was made of the parameters of the milling tool and its actuator as in the real-system experiment. In Table 1 are given the process parameters and the parameters of the tool used in the simulation phase. As mentioned above, the parameters of the process and disturbance model were determined for the open system state, for the requirement the robot end-effector is tracking the PRBS function (the amplitude of perturbation model a = 0.15 [mm]), while the forces realized in the course of milling process were measured simultaneously. The local controller was synthesized on the basis of the estimated process and disturbance parameters. The effect of model inaccuracies (i.e. differences between the model and the real process) were studied by comparing the simulated dynamics of the end-effector for variation of the roughness size of the part edge being processed in the margins of 0.164).20 [mm]. The intensity of these model inaccuracies is important because the controller reliability performance is estimated in the on-line work regime. Final adjustment of the controller parameters represents a compromise between the requirements for the system's robustness, the physical capabilities of realization and the minimization of the normal chamfering force variance and of tracking quality of the desired trajectory. As already stated, the D R M V was obtained by satisfying the criteria of function (19). The effects of adjustment of the weighting parameter r in the scope of the regulator (33), onto the variations of the contact force and control signal are very substantial. The choice of the parameter r was performed by the analysis of the obtained simulation results for a broader range of changing the value of the parameter mentioned. The obtained results for the variances of the controlled and control variable are shown in Table 2. For the case r = 0, or very small values of this weighting factor, an enormously high variance of the control variable is obtained, while the variance of the controlled variable is very small. Large oscillations of the control variable tend to excite the system's non-linearities. By this, they can damage the end-effector actuator, so that this situation is undesirable. In Fig. 5 are illustrated the simulation results for small values of the weighting parameter r. It can be noticed that the contact force oscillations are very small which would mean that by adopting the value r = 0.1, the oscillations of the normal chamfering force would be minimized. Table 1. Parameters of the deburring process 11.5 [mm/s] Tool feedrate 20000 [rpm] Maximal tool speed Maximal torque at tool 2 [N ml 1.5 [cm] Tool radius 2 Number of teeth of deburring-tool 0.5 [mm] Nominal burr width Nominal burr depth 0.3 lmm] 0.15 [mm] Maximal amplitude of surface variations 1.4 [KN/mm2] Specific cutting force resistance I0 s [N/m] Stiffness of workpiece 4500 [N/(m/s)] Damping of workpiece

ALEKSANDAR D . RODI~ and MIOMIR K . VUKOBRATOVI(~

374

Table 2. Force and fine motion control variance in process simulation for different values of the tunning parameter r

F o r c e v a r i a n c e [N 2] C o n t r o l v a r i a n c e [ m m 2]

r = 0.10

r = 1.00

r = 5.00

r = 10.00

r = 100.00

0.00374 111.000 x l 0 -4

0.00591 9.78 x 10 -4

0.05130 0.743 x 10 -4

0.07560 0.203 x 10 -4

0.09870 0.233 x 10 -6

However, to achieve this, the end-effector actuator should realize the fine motion control signal shown in Fig. 5. Since the oscillations of the signal are enormous (magnitude greater than 300 Lum]) and very frequent, the actuator should make a great effort in the realization and this would be accompanied by a large energy consumption. This could lead to severe motor damages. For this reason, some larger value of the adjustable parameter r (let us say r > 1) should be chosen. However, by analyzing the variance values, obtained for larger values of the parameter r and presented in Table 2, we arrive at the following conclusion, In the case when the parameter value r is too large, the consequence is reducing of the control signal oscillations (more even motor operation is achieved). However, the oscillations of the controlled variable show a significant increase. This means that the increase in r results in the deterioration of the quality of the processed surface. Therefore, the weighting factor r is determined as the result of a compromise between attaining a satisfactory quality of the processed surface (by reducing the value of factor r) and the uniformness of the end-effector actuator work (by increasing the value oft). Taking into account the above analysis, we have adopted the value of the parameter r = 3.4 which ensured the attaining of the following variances of the controlled and control value: ey-2 _ 0.0259 IN2] and e,2 = 1.47 × 10 - 4 [mmZ]. In Fig. 6 are shown the values of amplitudes of the chamfering forces Fp and Fc (see Fig. 1), as well as the form of the control signal u(t), ensuring a fine positioning of the milling tool tip. As can be seen (in Fig. 6) when a stationary regime of the robotic deburring is established (after 1 [s]), the maximal deviations of the normal chamfering force Fp amplitude are smaller than 9.6%. This means that the attained quality of the transient process is satisfactory, i.e. a good quality of the proposed surface is ensured. Also, the cutting force Fc exhibits an approximately trapezoidal

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profile, with very small amplitude variations. As for the oscillations of the control signal, they have been reduced to the margins of allowed values (__+40 #m). Such micro-displacements can be easily realized with the aid of the ball screw and thus a relatively even work of the actuator is ensured (with no large position jumps, which ensures lower energy consumption and longer motor service life). As is shown in Fig. 6 the proposed regulator enables attaining of a satisfactory behaviour of the force on the end-effector of the manipulation robot. This means that the synthesized DRMV ensures the required chamfer depth and minimum surface roughness• As for attaining the precision of the tool nominal trajectory tracking and of its realized constant orientation by applying the control scheme given in Fig. 4, the errors of tracking obtained by simulation of the desired trajectory are presented in Fig. 7. The elements of the matrices F~ i/'2 in expression (15) were preset in such a way that the control system is stable, and this is attained if the poles of the characteristic system equation are placed to lie in the left half of the space. In this way we account for the system's velocity (its dynamic performances), i.e. we take into account the low mechanism bandwidth, which for the MANUTEC r3 robot is 2-3 [Hz]. In conclusion, the control algorithm (11) in "cooperation" with the DRMV (34) ensure a sufficient precision and stability of the manipulator end-effector motion along a defined nominal

ALEKSANDARD. RODI(~and MIOMIRK. VUKOBRATOVI~

376

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trajectory, The obtained curves Ax(t) and Az(t) represent the position errors of the tool tip along the desired trajectory, expressed in the absolute coordinate system attached to the robot base. Simulation was carried out by prescribing the initial perturbations in the form of initial deviations of the joint positions. The convergence of these curves (Ax, Az) to zero with time, is evident. Also, the quality indices of keeping constant orientation (of the projection of the vector1" h onto the axes of the absolute coordinate system of the robot) indicate the convergence of the real value to the desired one. Thus it can be concluded, based on the previously given simulation results, that the proposed controller ensures good quality of robotic deburring for the process conditions given and an ideally determined model. However, it is indispensable to carry out investigations of the proposed controller robustness, before it was applied in the experimental system. Supposing that the process and perturbation models have been determined ideally, we have analyzed the effects of augmenting the linear tool feedrate onto the control system performances. Hereby we augmented the tool feedrate two times and compared the acquired results with the previous. It can be concluded that by increasing the tool feedrate two times, the realized process quality are not radically impaired as we perhaps expected (Fig. 8). On the contrary, the variance of the controlled variable Fp is diminished by about 41%, becoming now e~ = 0.01830 IN2]. The variance 1"Theindex of tool orientation was adopted in the form of a vector directed from the mass centre of the last manipulator link towards the point representing its tip. This vector frequentlycoincideswith the end-effectorsymmetryaxis, so it is very suitable for theoretical considerationsand simulation, and one can follow it more easily than the usually used Euler's angles.

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Fig. 8. Simulationresults via proposed robot controllerfor two times greater tool feedrate. of the fine motion control signal is now 1.62 x 10-4 [mm2], which contains increases of about 10.2% as compared with the case with two times smaller feedrate (that was 11.5 [mm/s]). The maximal amplitude of the contact force Fp was reduced by 0.6% and the maximal amplitude of the control variable was reduced by 10%. But, this does not mean that it is allowed to increase the tool feedrate in uncontrolled manner, for the reason that the tracking error of the tool trajectory is greater due to more expressed dynamical effects. Even more, the tool feedrate can be increased to a certain value, exclusively depending on the mechanic performances of the tool and the material feature characteristics of the workpiece which is being processed. The process and disturbance model cannot be determined ideally. Some error, done in scope of the process of system parameters identification, is a consequence of: inaccuracy of measuring the input/output variables, impossibility of measuring some values but their estimation, imperfectness of the identification method, etc. Robustness of the control system with respect to dynamic models parameters inaccuracy is an important characteristic of every system. The parameters of the deterministic process discrete model and of the stochastic disturbance model have been determined experimentally in conditions of open-loop system state. Hereby we commanded the manipulator end-effector to track the control PRBS signal, measuring simultaneously the input samples u(k) and output samples of the process y(k) (Fig. 2). On applying RLSEM [18, 19] we obtained the order and the polynomial coefficients A (z-i), B(z-i) and D(z- l) in models (6) and (8). Here, the PRBS

378

ALEKSANDARD. RODI~and MIOM1RK. VUKOBRATOVIC

signals with the following amplitudes were used: (I) 0.10 [mm], (II) 0.15 [mm], (III) 0.20 [mm]. Identified polynomial coefficients are: • (I) a = l . , al =-1.37930, a2=0.49092, b0=0., b1=0.88624, b2=-0.77458, dl = -0.33493, d2 = 0.54010, 2 = 0.28936 • (II) a0 =1., a t = - 1 . 3 7 9 3 0 , a2=0.49092, b0=0., b1=0.88624, b2=-0.77458, d l = --0.45395, d2 = 0.51267, 2 = 0.33076 (III) a0 = 1., aj =-1.37930, a2=0.49092, b0=0., bt=0.88624, b2=-0.77458, dl = --0.50957, d2 = 0.49246, 2 = 0.35703

d0=l., d0=l, do = 1.,

An average variation of the parameters d~, d2 i 2, with respect to case (II), is about 15% for case (I), and about 8% for case (III). Taking into account the variations of the input signals amplitudes u(k), of about 33%, the estimated parameters variations represent significant non-linearities of the device. Robustness of the control system with respect to the inaccuracies of the identified model parameters have been tested in using the synthesized DRMV and perturbation model with an amplitude of 0.15 [mm]. Such designed regulator was used for controlling robotic deburring, whereby the values of 0.10 and 0.20 [ram] were taken as the minimal values of the workpiece burr. The following conclusions were made. If in the synthesis of the end-effector controller the error was committed in that a PRBS type signal of maximal amplitude greater by 50% than the real model amplitude was used (the average variation of parameter values in that case was 15%), this regulator is sufficiently robust for ensuring the realization of good process quality. It means, that in the case when an error exists in the model parameters identification, so that the real perturbation is smaller than that one used in the regulator synthesis, the effects of inaccuracy will not be sensibly expressed. Also, if in the synthesis of the proposed regulator the error was committed in that the perturbation model with an amplitude 50% smaller than that one of the real model was used (the average variation was in that case 8%) a deterioration of the control system performances is happening. In that case we have noticed that the variances of the controlled and control variables increase by 62.22 and 77.30%, respectively. This represents a significant increase of the variances, because, due to the model inaccuracy, some non-linear system effects have been excited, influencing essentially the process quality. Thus, identification of the corresponding model parameters must be as accurate as possible. In the text to follow, we will propose a certain modification of the applied control scheme (Fig. 4) in order to show that the normal chamfering forcing and control signal variances can be even more decreased. Based on the DRMV, minimization of the chamfer depth variance has been ensured, as well as the minimization of the fine motion control signal oscillations. This does not mean that the controlled variable oscillations have been fully eliminated (Fig. 6). Theoretically regarded, they could be totally neutralized if the stochastic perturbation signal (the profile of the part edge) would be known with ideal accuracy. Since it cannot be identified with ideal precision, elimination of the output variable and control signal variations cannot be realized completely. However, it is possible to determine the profile of the workpiece edge [25] and to design its CAD-model with sufficient precision. On the basis of the knowledge of the disturbance model predicted a certain modification of the robot control scheme can be carried out. Let us note, that the justification of introducing a corrective signal based on the disturbance prediction into the control scheme is specially expressed in machining processes with highly significant variations of burr amplitudes. In that case, the disturbance signal predictor (DSP) is indispensable to prevent impulse forces, i.e. cutting force peaks. If the cutting force is increased near the tool breakage limit, the large overshoots would lead to breakage of slender end mills. A priori information about incoming dangerous changes in the workpiece geometry is therefore required by adaptive robot controllers. On-line geometric sensors such as vision cameras and capacitance probes are rather impractical due to the hostile cutting process environment. On the other hand, since contemporary practice is to generate numerical control tool paths using a CAD system, it is proposed that the tool/part edge intersection geometry be calculated by a solid modellar of subsequent on-line use by the adaptive robot controller. By examining the interaction of the cutter with the workpiece edge, the chamfer depth and width can be calculated at any point along the determined tool path. The local geometry of burr, which

Robotic deburring

379

would cause chamfering force variations, is examined and supplied to the adaptive control scheme earlier than the actual occurrence. Using a milling process model, CAD corrective forces AF(t) are added to the on-line measurement so that the feeding velocity will be safely changed by the adaptive robot controller before the cutter will be influenced by incoming burr geometry. Then it can be written:

F*(t) = F(t) + AF(t)

(39)

where F* is the corrected signal of the measured force F(t). Accordance to it, instead of the measured value F(t) the corrected value F*(t) is used in the control law (11). Since the proposed robot controller (Fig. 4) consists of two control parts (robot arm controller and local end-effector servocontroller), which are intercoupled, the influence of each other is rather expressed. In that way, the modification of the signal F(t) in control law (11) influences indirectly on diminishing the variations of the controlled variable. By using the described modified control scheme, significant improvement of the processing quality was achieved. The following variances of the controlled variable and the control signal variables were obtained: e~ = 0.00388 [N 2] and e2u= 0.232 x 10-~3. These are of order of magnitudes smaller than those obtained by the unmodified robot control scheme.

3.4. Experimental verification The experimental system used for the verification of the proposed controller is given by Fig. 9. The industrial manipulation robot M A N U T E C r3 [24] was used, having six DOFs, whereby the sixth DOF was passive. The active end-effector used a high-speed d.c. servomotor (see Table 1) coupled to a precision miniature ball screw to position the milling tool for achieving high tool positioning accuracy (up to 0.01 [mm]). An aluminium alloy casting was used as workpiece, the burr of which (20 [cm] length) was to be removed. Tool feedrate was 17 [mm/s]. The chamfering forces were measured by a small piezoelectric sensor mounted in the active end-effector. The sensor output was conducted via an amplifier to the low-pass noise filter. The control software was written in C language and implemented on a three MOTOROLA 68020 processor. The tasks were distributed in the following way. On the first processor the robot language and the interpreter were implemented, on the second one, tacted at every 64 [ms], the robot kinematics and dynamics, and on the third one, operating every 8 [ms], are the servos and on-line process and disturbance estimations.

MJ

"I

Fig. 9. Experimental system for robotic deburring--industrial robot MANUTEC r3 and its active end-effector.

380

ALEKSANDARD. R.ODI~ and MIOMIRK. VUKOBRATOVI(;

By measurements during the experiment, results of the robotic deburring were obtained (see Fig. 10). The contact force variance was measured as e~ = 0.03543 IN2] and the variance of the fine motion control signal e~= 3.02 x 10-4[mm2]. The obtained experimental results confirm the justification of the DRMV use in robotic metal processing. Good quality of the processed surface, as well as the corresponding chamfer depth, in desired tolerances have been achieved. Oscillations of the control variable were minimized, so that a relatively "smooth" work of the end-effector actuator was achieved. The desired precision of the trajectory tracking was achieved too. Next experiment was repeated on the workpiece with the same geometrical performances and process parameters (imposed tool velocity and referent contact force). By that we have also used the DRMV for the process control purposes and the DSP assistance. The results of this experiment are shown in Fig. 11. The enclosed results show that by using the proposed control method significant improvement of the control system performances can be realized. The normal chamfering force variance ey--2_ 0.00520 I N 2] and variance of the fine motion control signal e2u= 0.0346 x 10-4 [mm] were measured. These indices are by order of magnitude better than in the case of the described controller application but without the proposed DSP, notably advantageous when processing workpieces of greater burr variations. 4. C O N C L U S I O N S An active end-effector force based control system has been successfully implemented using the manipulation robot MANUTEC r3 in the deburring. By application of the control system shown, the basic goal of the corresponding controller design was achieved--minimization of the chamfer surface roughness. This has been enabled by on-line minimization of the chamfering force variance in the direction normal to the workpiece surface using DRMV and by supplementary modification of the algorithm by introducing the DSP into the control scheme. Based on combined discrete models of the deterministic process and of the stochastic perturbations, the corresponding DRMV was synthesized. Also, a dynamic control law was used for controlling the robot mechanism during its gross motion along the prescribed end-effector trajectory. In the simulation process the efficiency and robustness of the proposed control scheme was estimated. After that, its performances were successfully verified in the real-time experiment and they are acceptable for the majority of industrial applications from the standpoint of processing quality, simplicity of implementation and system integration price. 6.01

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Acknowledgement--We express special thanks to the research engineer Dragoljub ~urdilovi6 from the Fraunhofer-lnstitute IPK in Berlin (Germany) who significantly contributed to the experimental verification of the performances of the proposed position/force robot controller.

REFERENCES

1. H. Kazerooni, J. J. Bausch and B. M. Kramer, A S M E J. Dynamic Systems, Measurement and Control 108, 354-359 (Dec. 1986). 2. T. M. Stepien, L. M. Sweet, M. C. Good and M. Tomizuka, IEEE J. Robotics Autom. RA-3, 7-18 (1987). 3. K.B. Haefner, P. K. Houpt, T. E. Baker and M. E. Dausch, Robotics: Theory and Applications. DSC-Vol. 3, ASME, pp. 73-78 (1986). 4. F. M. Puls and M. M. Barash, J. Manufacturing Systems 4, 169-178 (1985). 5. B. M. Kramer, J. J. Bausch, R. L. Gott and D. M. Dombrowski, Robotics Comp.-Integrated Manufacturing 1, 365-374 (1984). 6. R. Hollowell and R. Guile, Modeling and Control o f Robotic Manipulators and Manufacturing Process. DSC-Vol. 6, ASME, 73-79 (1987). 7. R. Hollowell, Proc. A N S Third Topical Meeting on Robotics and Remote Systems, Charleston, S. C., pp. I ~ (March 1989). 8. H. Kazerooni, 1EEE Control Systems Magazine, pp. 21-25 (Feb. 1987). 9. A. Vishnu, M. Cutkosky and E. Erickson, Proc. U.S.A.-Japan Symposium on Flexible Automation, Minneapolis, pp. 325-332 (July 1988). 10. H. Kazerooni and M. G. Her, Proc. American Control Conf., Atlanta, GA, pp. 108-114 (June 1988).

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ALEKSANDARD. Rom~: and MIOMIRK. VUKOBRATOVI~

11. M. Vukobratovi6 and Y. Ekato, Proc. of the 1993 IEEE Int. Conf. on Robotics and Automation, Tutorial: Force and Contact Control in Robotic Systems, Atlanta, U.S.A., pp. 213-229. 12. M. H. Raibert and J. J. Craig, ASME J. Dynamic Systems, Measurement and Control 103, 126-133 (Jun 1981). 13. F. Koenigsberger and J. Tlusty, Structures of Machine Tools. Pergamon Press (1971). 14. I. Yellowley, Int. J. Machine Tool Design and Research 25, 337-346 (1985). 15. Y. Altintas, I. Yellowly and J. Tlusty, A S M E J. Engng Ind. 110, 271-277 (1988). 16. D. S. Eppinger and P. W. Seering, Proc. o f l E E E Int. Conf. on Robotics and Automation, pp. 29-34, San Francisco (April 1986). 17. H. Asada, A S M E J. of Dynam& Systems, Measurement and Control 105, 131-136 (1983). 18. K. J. A,strom and B. Wittenmark, Computer Controlled Systems, Theory and Design. Prentice-Hall, Englewood Cliffs, N.J. (1984). 19. R. lsermann, Digital Control Systems. Springer, Berlin (1977). 20. M. K. Vukobratovi6, Applied Dynamics of Manipulation Robots: Modeling, Analyses, Examples. Springer (1989). 21. G. M. Bone, M. A. Elbestaw, R. Lingakar and L. Liu, J. of Dynamic Systems, Measurement, and Control 113, 395-400 (September 1991). 22. H. R. Middleton and C. G. Goodwin, Digital Control and Estimation: A Unified Approach. Prentice-Hall International Editions (1990). 23. A. Rodi6 and M. Vukobratovi6, Mech. Mach. Theory 29, 455~,78 (1994). 24. S. Tfirk and M. Otter, Robotersysteme 3, pp. 101-106 (1987). 25. A. Spence and Y. Altintas, d. Dynamic Systems, Measurement, and Control-Transactions of the ASME 113, 444-~50 (Sept. 1991).