Benefits of Positive Current Feedback Inner Loop for Robot Servos

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BENEFITS OF POSITIVE CURRENT FEEDBACK INNER LOOP FOR ROBOT SERVOS

J. D r/mr/lllm/

uI

O'Shea and A. B. Turgeon

Elre /riml EII!(ill f l'rill!(. Eru/t, P u/Y/fChlliqllf' . .\lulI/ rm /. COIIOt/O

Abstract. To remedy the limitations of the high-gain position servos used for contro" ing robots, a "load-equal izer" scheme (patented for gun fire systems) was investigated. Since that technique implies positive current feedback, it was decided to study the properties of PCF for DC-servomotor drives . Considered as a load-equal izer, PCF cannot completely null ified the effect of varying moment of inertias on robot control because the gain which is required makes the system unstable. However, it is shown in this study and it is confirmed experimentally that PCF with limited gain greatly improves the dynami c response and the sti ffness of posi ti on servos wi thout requi ri ng tachometer feedback. Control systems i ncorporati ng PCF are therefore seen as hi ghly suited for servoing robots and, specially, direct-drive arms. Keywords. motors.

Positi ve feedback;

load equal i za ti on; motor control; robots; D. C. Servo

INTRODUCTION The following study , like [4 ), borrows only on cl assi cal control theory to determi ne the characteri sti cs of pos iti on servos wi th PCF and compares them to those of position servos with The resul ts of 1aboratory tachometer feedback. experiments conclude this paper.

There are, in general, two types of feedback configurati ons which are actually used for DCmotor based robot servomechani sms. One resorts to negative current feedback to implement the "computed-torque method" whereas industrial robot designers generally take advantage of high gain classical servomechanisms to reduce the effects of varyi ng parameters, non-l i neari ti es and coup1 i ngs. For thi s second type of control, the addition of tachometer feedback is often required to attain the highest possible gain while insuring a satisfactory damping factor. But resonant modes constitute an upper limit to this techni que.

DC MOTORS WITH POSITIVE CURRENT FEEDBACK The obj ec ti ve of thi s sec ti on is to determi ne the fundamental properties of the system described in figure 4. The symbols used thereafter are defi ned below.

A means to obviate that limitation has been proposed [1], [2], [3] for gun fire systems and is patented . The strategy is call ed "loadequalizer" and, in its simpler fonn, constitutes posi tive current feedback (PCF). In search to improve robot controllers, the authors have studied more thoroughly the properties of PCF as no more refe r ences were found on the subject wh i le those mentioned did not supply the pertinent theoritical derivations. The lack of references can be explained by the widespread bel ief that positi ve feedback necessarily 1eads to instability . This presumably prevented the development and exploitation of the technique as a recent survey [4] on the desi gn of robot controllers indirectly confirms.

List of symbols Al = Pre-amplifier gain A2 = Power ampl ifier gain B = Coeffic i ent of viscous friction i = Armature current J = Moment of inertia (rotor + load) Kc = Motor torque constant K· = Shunt constant K~ = Potentiometer constant Kt = Tachometer constant Ko = Counter e.m.f. constant L = Armature inductance R = Annature resistance Ra = Shunt resistance s = Complex variable = CJ + jw t = Time T Torque perturbati on Va Voltage appl i ed to the motor Vi Voltage across the shunt Vs Set poi nt vol tage E Position error r Torque del ivered by the motor o Rotational velocity of the motor shaft a Angular position of the motor shaft

Although most industrial robots relay on PWM power drives, the experimental resul ts reported below were obtained from a servomechanism using a linear power amplifier. The aim was simply to Now, since the determine figures of merit. results obtained are sufficiently good, more efforts are justified. A previous design of a PWM drive [5] is being modified to accomodate PCF and it is hoped that it will be possible to communicate new results soon. 185

J.

186

O'Shea and A. B. Tuq,(con

STABILITY CONSTRAINTS Fi gure 1 represents by means of a block -di agram the important part of the system to be studied with the current loop opened. One easily obtains this transfert function: (R+sLl (Js+B) + (KcKQ) Because B is very small, both the rotor and the robotic joint having low viscous friction, the zero is very close to the left side of the origin on the real axis of the complex plane (figure 2). And, since armature inductance is also a small parameter, the poles are located on the real axis in the neighborhood of -KcKn/RJ for the slow mode and near -R/L for the fast mode. The pole located at -R/L is typically three to six decades away to the left of the pole located at -KcK /RJ. Therefore the current loop has a very wi de %andwidth which is effectively the property claimed But, as can be seen in figure 3 (the in [1]. critical point being located at + 1 due to positive feedback) there is a serious risk of instabi 1 i ty. Thus the loop gai n A2Ki must be set carefully. Another stability implication related to the increase of A,2Ki concerns the position loop. Refering to flgure 4, the open loop transfer function is readily obtained :

As ~ Ki tends toward R, the damping coefficient tends toward zero (si nce the product BL is very s mall ) . I nth e 1 i mit, ma kin g A2 Ki = R, th a t transfer function becomes : AtAfcKp ef{s) = Vs{s) s{JL s 2 + BLS+KCK Q)

In fact, one can benefit from positive current feedback without risking instability. The previous discussion simply aimed at showing the impossibility for this technique to make the control system completely insensitive to the varying moment of inertias in the case of robotic app 1 i cat ions. Good improvements of the dynami c response have been observed even with A2Ki as small as R /8 while higher experimental values of A2 Ki have been used and very good results are reported in the next section with ~Ki = 0.69 R. Another improvement that results from positive current feedback concerns the steady-state response and more specifically the stiffness. The di sturbance of the output due to a perturbating torque can be determined by this transfer function (obtained from figure 4) : e (s) =

fTST

R + SL - A2Ki

Using the final value theorem and assuming that T{s) = T/ s, it come s that: e {t)lt__ =

T{R-A2Ki ) At ~ K c K p

Hence, the stiffness factor AtA2KcKD/{R-A2Ki) can be greatly improved by choosing values tor ~Ki as high as stability permits. COMPARISON WITH TACHOMETRIC COMPENSATION The block-diagram of the conventional tachometric compensated position-servo is shown in figure 5. The overall transfer function is :

e

(s) =

Vs{s) Its root locus has two branches, leaving from the complex poles, which go toward the right half plane when increasing AtKp" Therefore the adjustment of A, cannot be made i ndependantly of that of A,. for ~table operations. LOAD EQUALIZATION From figure 4, the closed loop transfer function of the position servo with PCF is:

In order to get a better insight, let L = O. Then, by making A2Ki = R, it looks as if the effects of the load could be cancelled:

e

1

(s) =

Vs{s)

-----s~~--------

Kp (

+ 1)

At A2 Kp Hence, the expression "load-equalizer" was coined. But, due to stability considerations, as seen previously, A2Ki cannot be made as high as in practice the load effect will one desires: not be completely nullified. Nevertheless, the above simple reasoning points out that PCF can make robot servos much less sensitive to the wide variations of J.

The tachometer action, as is well known, increases the damping naturally provided by the counter-electromotive force. The stiffness can be determined as in the previous section: e {t)lt = . :.,.T:. :,:.R_ _ -At ~KcKp Therefore, the stiffne ss factor A A2KCK /R is independent of the tachometer gain Rt' Al~hough the same expres s i on wou 1d be obtained for the case of position servo without tachometer feedback, it must be remarked that the tachometer action permits to increase the gain At to values tha t wou 1d otherwi se crea te object i onnab 1e overshoot. Repl aci ng the parameters by thei r numeri cal val ues shows that the sti ffness factor can be made as high for either type of compensation, PCF or tachometri c, but the gai n At has to be hi gher for the later. Potentially, tachometric feedback can yield better performance s. However, due mostly to the power limitation of A., , higher gains result in eithe r current or voltage satu ration which practically limits the performances at levels comparable to tho se of PCF.

Be nefits of I'ositin' Currcnt Feedback Inn e r Loop For Robot S('n·os EXPERIMENTAL RESULTS The equipment used for evaluating the performances of the various feedback configurations discussed in this paper is described schematically by means of the block-diagram shown on figure 6. A list of components (see Table 1), gi ves the pertinent i nformat i on about each el ement of the di agram. A two-trace recorder and a torquemeter served as the basic measuring instruments. The numerical val ues for the parameters, due to measurement difficulties, are only 5% to 10% accurate (while it is probably 20% for the armature resistance) . Therefore one cannot expect that the responses derived from the previous equations would fit perfectly well with the recordings. The authors simply attempted to show the relative merits of both types of controller. Many experiments were carried to explore the numerous combinations of gain settings. An i nteresti ng "di scovery" was the fact that as R is increased so is the maximum of the produc~ ~Ki that allows stability. For instance, with Ro = . 053 ohm the highest possible value for AoKi was 2.7 ohms; beyond vii:>rations became noticf!able. But, with Ro = 1.35 ohm, I\K . can be made as high as 6.9 ohms before reaching the threshold of vibrations. This can be explained by the high noi se of V1 due to brush arci ng. The current feedback loop having a wide bandwith, all that no!se is amplify by ~ which needs to have a high ga 1 n to make the product Ki sign ifi cant when Ki is small. Therefore increasing Ki (or Ro) makes the system much less sensitive to noise. That is why the resu lts gi ven therea fter correspond to measurements taken with the 1.35 ohm shunt.

1\

Figure 8 is a recording of the position and armature current versus time for a step-input of 0.5 volt using the configuration of figure 6 with Rl = 40K ohms and ~ = lOOK ohms.

187

opposing the perturbation has the same value as the current recorded in Figure 9, one can see at once that the stiffness with PCF is approximately 8.1 times the stiffness without current feedback, all other parameters being kept constant. Figure 12 shows the position and armature current versus time for the servomechani sm wi th tachometri c compensation (fi gure 7) when R =50K ohms 1 and ~=30K ohms. The transfer function is then: e (s)

Vs{s)

40.19 2.14x10-S S3 + (l7.15x10- 4 )s2 + 0.17s +1

which corresponds to a highly damped k = 2.0) second .order system with wn = 23.9 rad/sec. One can notlce that the rise tlme is longer for this case then it was in the case of current feedback. Figure 13 shows that, for the tachometric confi~uration with 0.0 volt input the output moves an lncrement of 0.26 deg. when a perturbation is app 1 i ed by tu rni ng the flywheel with the hand unt i 1 the counter-acti ng current in the armature reaches the same value as in the case of the PCF configuration (figure 9). One can thus notice that the stiffness of the tachometer feedback configuration with these setting is slightly greater than the stiffness obtained for the curr~nt feedbac~ configuration with the settings prevlously mentloned but the transient was much shorter for the PCF configuration. The comparison of these two cases is based on the cri teri on of the bes t overa 11 performances achievable without saturating the amplifiers and ~ithout . producing vibrations. For example, lncreaslng further the gain A (R =100K) to get a better rise time for the fac~ometer feedback configuration, resulted in high current peaks (fig~r~ 14) that would have saturated the power ampllfler lf the over-current protection had been a?justed correctly at 5 amperes on both polaritles (a defect found while running these tests).

Then the transfer function is : e (s) _ Vs{s) -

40.19 2.14x10- S s 3 + 5.5x10- 3 s 2 + 0.1255s + 1

Roughly, the characteristic polynomial yields a frequency response closed to that of a second order with wn =13.5 rad/sec and C=0.85. Figure 9 shows the perturbation that results when manually turning the flywheel while keeping Vs=O.O volt. This gives a measure of the stiffness for the PCF configuration : the displacement due to the perturbation being 0.32 deg and the current opposing the perturbation being 2.94 amperes, a simple cal c ulation yields 0.33 deg/N-m x 192 refered to the output. Figure 10 is a recordi ng of the positi on and armature current versus time when the current feedback loop is opened in the configuration of figure 6, for a step-input of 0.5 vol t. The servomecanism behaved like an underdamped second- order system. The comparison of figure 10 with figure 8 demonstrates clearly the beneficial action of the PCF loop. The same conclusion is reached when compari ng fi gure 9 and fi gure 11. Thi s 1ast fi gure is a recordi ng showi ng the effects of the perturbation that resulted with no current feedback when turning the flywheel by hand while keeping Vs constant. Since the armature current

CONCLUSION It was shown in section 2 that using positive current feedback as a form of compensation can improve considerably the performance of position servomechani sms although perfect load equal i zation is not feasible. Comparing the tachometer feedback controller with the PCF controller, one can see that i:>oth are competitive forms of compensd ti on based on performance criteri d from an automatic control standpoint. Nevertheless, instrumentation wise, PCF offers a better solution as a shunt (an ordinary resistor will do) is potentially more reliable than a tachometer, adds no weight to the robot joint and is i nexpens i ve. The PCF method is also superi or to the fi nite difference method using the potentiometer signal as a replacement to the tachometer, since it is much less susceptible to noise. The benefits of positive current feedback become more striking in the case of direct drive arms where the motors move so slowly that enc?ders with gear multipliers are required (6) to lnsure proper tachometer feedback. Therefore, the PCF controller presents valuable improvements over existing designs and constitutes a substitute solution for the coming generation of robots.

J. O' Shea

188

and A. B. T urgeon

( 4]

Tak.ing the state-space approach and the po1elocation formulation as the authors did more recently (7], one also finds that positive state feedback. is required for the current loop in order to get the best performance.

[ 5]

REFERENCES [ 1]

[ 2] [ 3]

( 6]

Bi gl ey, W.J., Ri zzo, V. (1978) . Resonance Equalization in Feedback. Control Systems. A.S.M.E. Winter Meeting Proceedings, Paper No. /8-WA/DSC-24. Eng, B.P. (1979). Microprocessor Based Servo Control for Gun Fire Control. IECI, 79 Proceedings. Rizzo, V. J., Big1ey, W.J. (1980). Microprocessor Based Control System. Proceedin~s of the IEEE 1980 National Aerospace an Electronlc Conference, 211-218.

(7]

Lun, - J ~Y.-S-: - - (1983). Conventional Controller Design for Industrial Robots - A Turoria1 . IEEE Trans. on Systems, Man and CybernetlCS, Vol. SMC-13 No. 3, 298-316. C1lvi'O, P.M., O'Shea, J. (1984). A Triaxia1 Control Interface Using the AIM-65 Microcomputer, (in French). Proceedings of the 1984 Can. Conf. on Ind. Computer Systems, Paper No. 88. Asada, H., Knadet, Tak.eyama, I. (1982). Control of Di rect Dri ve ARM. uReport CMU-RI-TR-82-4, Carnegie-Me110n nlverSlty, Plttsburg. Turgeon, A. B., 0' Shea, J. (1986). Improvements of Robot Servos Performances via State Feedback. (in Frel1ch). Research Report to be published, Eco1e Polytechnlque, Montreal.

TABLE 1 LIST OF COMPONENTS WITH NUMERICAL CONSTANTS Component

~

DC-motor wi th tachometer

El ectrocraft Model 650

Flywheel

Steel Disc mounted on a shaft with ba11bearing supports

Gear reducer

Globe inc. Part #1A1172

Servo potentiometer

TIC, Model JC4788 Kp= Kp!192

Shunt

Power resi stor

Pre-amp1ifiers

~-741

Adjustable resistors

General-Radio decade resistor Type 1432-M

Recorder

Techni-write Model TR-722

Torquemeter

Lebow Mode 1 11 02-50

Figure

Parameters B=10- SN-m . sec/rad. J r =2.5x10- 4 Kg-m 2 R=10 ohms L=10- 2 henry Kc =0. 32N-m/amp KQ=0.32 v.sec/rad. Kt =0.3 v.sec/rad.

N=1/192 K =4.77v./rad. -0.024v./rad. KpE RO=1.35 ohm

1 ohm increment to 10K ohm increment

Bene fit s of Positi\'e Current Feedback Inne r Loop For Robot Ser\'os

189

jw

Js+B

~ (R+SL)(JS+B)+KcK~ 1 IT

:.< _ R

KiAZ

-----+~------+-----~~--------.Re

B

- J

[

Fi (Jure Z

~----------------------~ Kp ~----------------------~

Fiqure 4

L-..---------------l Kt

--------------1

.....

~------------------------~K

P

FIGURE 5 +15V

,, :

-15V

11 :1,92 1 /(2)F1Y-Wheel ~1otor

Figure 6

J.

190

O'Shea and A. B. Turgeon

+15V

I

51K

+15V

Kepco

:

~ ,0 -15V

-15V

Fly-wheel

Motor

Tachometer

Figure 7

L~i~:' C:u' :~~t[! L::L L ~_L _L.LLL

,e),_L-LL

2 5' .....",..,... / A.e.c.

-

I

FIGURE 8

FIGURE 11

FIGURE 9

FIGURE 10

FIGURE 12

FIGURE 1~

Benclits of Positiye Current Feedback Inner Loop For Robot Ser\'os

FIGURE 14

191