Control of a robotic manipulator- A model following approach

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Control of a robotic manipulator- A model following approach

S Mukherjee & S N Sengupta

Department of Mechanical Engineering, Regional Engineering College, Durgapur 713209, India Received 21 July 1993; accepted 4 November 1993

A model following approach has been used to produce a consistently good dynamic performance in a manipulator, whose effectiveness is illustrated through the simulated responses of a two degree of freedom plant with variable inertia and fluctuating payload. The model is provided with the reference input and produces the "ideal response" that sets the standard to be followed by the plant which runs totally on the basis of the differences of its states from those of the model. More eruditely, the plant is constrained by the model to emulate its states. A higher order plant was intentionally made to follow a lower order model to test the effectiveness of the controller. Satisfactory simulation results assure the usefulness of the scheme even under considerable payload fluctuations (:t 10%), and a 1:8 range of inertia variation that follows a quadratic law. This system is found to have a noteworthy performance robustness despite its simple and cost effective hardware implementation.

Precision and accuracy of modem robotics de- from "unmodelled dynamics". Bar-Kanas in 1991 mand highly efficient manipulators and therefore, showed that 'lower order adaptive controllers can it is important that the performance of these de- also be used to control higher order plants if vices be ~nhanced. The conventional methods of proper caution is taken on the bounds of stability.

control are not always effective in producing a de- In the present study, a simple version of the sired performance as in many applications there Model Reference appr<!>ach by deleting the idea of are changes in system parameters during opera- adaptation has been investigated. This is the mod- tion. The changes may arise due to variation of in- el following control (MFC). In this process a mod- ertia as a result of changing configuration or due el is first designed, which is a clone of the original to fluctuations in the payload. It is necessary, system. The parameters of the model are so chos- therefore, to seek more appropriate methods of en that its response is considered ideal in terms of

control. the control objectives such as overshoot, settling

Often a robot joint drive consists of a smaller time, speed of response, etc. The model, running inertia connected flexibly to a configuration de- simultaneously with the plant, is given the refer- pendent larger inertia by means of either a belt, ence input R. States of the model are then com- chain or a gear drive. Configuration changes of the pared with the corresponding states of the plant arm is associated with variation of inertia. It is in1- and the errors are fed to the plant in the feed for- perative that in order to maintain a level perform- ward path, thus providing the control necessary to ance, it shall be necessary to adjust the feedback constrain the plant states to coincide with the gains in accordance with this inertia variation. corresponding ones of the model. Further, the This problem has been resolved by means of sof;. plant chosen is a two degree of freedom robot tware servo controll-3. One non-conventional whose shoulder joint is driven by a dc servomotor method4 receiving increased attention in recent through a voltage source amplifier. The plant dy- times-the Model Reference Adaptive Control namics is of 5th order, while the chosen model i~

(MRAC)-has also been tried with success to an implicit one of 4th. order.

prove equal to this task5-? Nevertheless, these

methods are not sin1ple and hardware in1plemen- Model Following Control

tation for them are both complex and expensive. The optin1al control theory has been very well Most adaptive control algorithms, developed be- developed and can meet the requirements for com- fore 1982 assumed adaptive controllers of the plex and high performance control systems. But same order as that of the plant. It was believed the theory cannot always be applied successfully that a lower order adaptive controller cannot con- since the (quadratic) performance index is often trol a higher order plant due to errors resulting difficult to construct in terms of the real control

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r '

I

I I

~ R I q

m

R Input

Fig:. I-Linear model following control system

objectives. One effective way to avoid this difficul- : : P

ty is to use the Linear Model Following Control ! i

system (IMFC)4. IMFC specifies the design ob- LPJ~~, :~ ~j

jectives through a model which may either be con- Ap. -.

ceptual (implicit), utilized only for the construction

of the control law or real (explicit), i.e., a part of Fig. 2-Block diagram of the MFC scheme the original system. Variation of plant parameters

during operation requires the application of Adap- which is proportiollal to the state errors. It is im- tive Control for consistent precision. LMFC sys- material whether the model is implicit or explicit.

.tem with an explicit model can be extended to The advantage in this scheme is that the model Adaptive Model Following Control (AMFC). can have constant parameters. It is noteworthy Here, the system containing a plant and a model that use of the model enables the determination of as shown in Fig. 1, has three matrices: KM-the errors in all the states of the plant w.hich is other- model feedback gain, Kp-~e pl~t feedback gain wise unavailable if the input R is used directly for and Ku-the feedforward gam. VIS the controlled comparison.

input to the plant. The elements of thes~ matrices When the plant and'model are in exact corre- are so ~h?~en that, ~erfect ~o~el folloWIng occurs spondence, it is expected that the plant should be at null IllItlal conditions as mdlcated by null state able to emulate the ideal states of the model. The errors (disl;>lacement & velocity), de~oted here ~y displacement and the velocity error vectors are e for any mput R. However, Adaptive Control IS given as:

always more complex and expensive, The authors

are of the opinion that satisfactory plan~ response e = qm -qp and e = «'1m -«'Ip '- can also be obtained by suitable choice of a model

-( in the simple IMFC scheme even under consider- These errors are provlde~ WIth suitable weigh- able amount of parameter variations and this is on tages by m,eans of e,rror gam vectors :4.P ~d AD the basis of studies made earlier with wide range for prod~cmg effective c?ntrol. For smgl~ m,put, of inertia variations (1:8), where disturbance is al- vector B m the feedback lmeof the model IS given so imposed on the plant in terms of fluctuating by:

pay load (:t 10%): Encouraging results were ob- B={O I}

tained to justify this contention. Here, the general

LMFC proposal is applied to a simple configura- Here, the control is simplified to the extent of tion. Though the requirement for the state errors selecting only the gain vectors Ap and AD in ac- to be null for any input may not be satisfied, espe- cordance with the performance expected from the cially at the earlier stages of response, still very manipulator,

good end results can be obtained by suitable de-

sign of the model, The block diagram representa- Model Fonnulation

tion of the scheme is given in Fig. 2. This scheme Fig, 3 shows the schematic diagram of the robot ., utilizes a rilodel, which can be considered ideal in manipulator being modelled after the shoulder~

terms of the control objectives mentioned earlier. joint. This is a two degree of freedom system with The model runs simultaneously with the plant and lumped parameters. A smaller inertia is connected exe~ the requisite control to force the plant states with a larger (variable) one by means of a flexible to coincide with those of the former, i.e., to pro- belt. This is a follow up of the studies of Seto et duce null state errors. Further, in this paper, the aLl,2 and Mukhopadhyay & Sengupta3. Schematic reference input is applied only to the model. diagram of the model is given in Fig. 4. Here a ro- Clearly, the plant is made to run under a control tational model with constant inertia is chosen. -The

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QP2

c ~

R Input

J Jp2

M~I dynami..

Fig. 4- The reference model P

.from which,

Fig. 3- The mechanIcal system

i={1/(R.+Ls)}(V-Eb)=£(V-Eb} ...(2)

model IS so desIgned that Its response IS slower

than that of the plant and its natural frequencies^...were h ^{s IS}

.

^thê ôperator ^did ^1,Ê ^{b IS}^. ^thê ^bâc^k ê^mf

are I made

.

close _high to

...

those of the ^Wivanable ^hall mertia^{. d} ^prod^uced

.

^m ^th^{e motor} ^-^-b'^K ^q

.

^and £

.

^{IS a} tin^ear

p ant at Its er mertla settmg. ..exponen e s COnsi er tiallag operator given y:

.

bPI that the state vecwrs qm and qm are Ideal at any

instant. £=1/(R +Ls)

Here a digital simulation is made with an impli- .

cit model providing ideal response norms for the voltage applied to the motor (from the voltage plant. Also the plant or the robot joint is driven source) is given by:

by a linear dc servomotor, which is powered by a V=KI(A ."pI eI+A p2 e +A e +A e2 dId 2 ⁾ ."/' ⁽³ )

voltage source. Such a voltage source could be a I 2

cost effective analog device like a power opera- Eqs (2) and (3) combine with others to form the tional amplifier. The model in this paper assumes plant equation as follows:

a motor driven by a current source. It is to be not- -'. ..

ed that for the given specifications, the plant is of Jpqp+Jpqp+cp,qp+ Kt£Kbqp+ Kpqp+Te

5th order and is being made to follow a 4th order d I = K £(A..n + A iIep rI m O'lm ) ... ( 4 )

mo e.

where, Stability Analysis

The model (current source driven motor) dy-

^...

^C ^-PI

[

c + C ~2 + K £,It ^{P PI} ^ep~, K £,It -C ~ ~^{ep ~2} ^{p PI P2}

]

namiCS are represented by the followmg equations: p -- C C ²

Jmq m ..+

c

mq m .+ Kmq m = Kem ^(R

-

Bq m ) ' .. (1) prp,rp2 prp2

where Kp~ [Kpr~, + Kep£Ap, Kep£~2 -~prp,1"P2

]

-Kprp,rp2 Kprp2

Jm = [Jm, 0

]

^; ^{J =}

[

^{Jpl 0}

]

^.j ⁼

[

⁰ ^?

]

^.K ⁼

^[

^{Kb 0}

]

0 Jm2 p 0 J' p 0 J' bOO

P2 P2

[

Cm+Cmr~ -Cmrmrm ] T

C =' , , 2 T ={O 7:}. q ={

q q }T

m -C r ~ C + C ~ 2 e e' P PI P2

m m, m, m, m m,

[ K 2 K ] Ap={Ap Ap}; Ao={Ad Ad}

mrm -mrm rm, I 2 I 2

Km = -Kmrm..rm2 K~r~~ -; From, Eq. (4) the characteristic equation for the plant IS derived as

Kem = {Kem O}T

b5S + 4.)'+b3.)~+b2.)-+bIs+bo=0 ⁵ b ^-A ^-~ ^_2 ... (5)

qm

=

^{qm, m2 ' ^q }T_{'B= {}01_} where

, For the model f<:>llowing pl~t, the equation for a b = J J L

voltage source dnven motor IS 5 PI P2

V

-

^Eb-

-

^Ld I ^'A'dt+R .b4=Jp .' 2 I Cp L+Jp Cpr~L+Jp Jp R.+Jp C ~2 I 12 I PP2L

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b3 =Jp2Kp~lL+Jp2CplRa+Jp2Cp~lRa+ KtKbJp2 rpl = 9.55 mm; rp2 = 38.2 mm; Jpl =0.0018 kgm2;

+ Ke Ad J + CpCp Lr2p +J p^RaCpr2^p Jp =0.0089 kgm2 to 0.071 kgm2;

~ p"'£ 1 P2 1 2 1 2 2

+J LK r2 _PI _{P P2} L=0.003H; Ra=0.5 Ohm;

b2=J K r2 R + K A Jp + CpCp r2p Ra Kb = Back emf constant ofmotor=0.15 Vs/rad.

P2 P PI a ep PI 2 1 2

+ Kt Kb C r2 + 1: Kpr2pRa + Ke~d Cpr~ Proper design of a reference model is the first

p P2 PI 2 1 2. . MFC h Thi '

d b

.lffiportant step m sc erne. s IS one y

+ KepAd2 Cprpl r P2 + Kp~2 CPl L ~djusting its parameters, such that its natural fre- b = K A C r2 + K r2 C R + KKK r2 q~enci~s cl?sely ~atch ~ose. ~f the pl~t at its i ep PI P P2 P P2 PI a t b P P2 higher mertia settmg. An lffiplicrt model IS chosen

+ K ~ K r2 + K A Cr. r and for the sake of convenience, it is built to 1: 1

e dl P P2 ep P2 p PI P2 scale with the plal)t. The larger inertia of the mod-

+ Ke~d Kpr p r p el is made equal to the maximum value of higher

2 1 2 ..

fth I

-2 mertla 0 e pant.

bo -KepApl Kpr P2 + K~P2Kpr PI r P2 The parameters of the model are:

The conditions for the system to be stable are ob- C = 2100 N s/m' C = 0.4 N ms/rad' 4 tained by the application of Routh's criteria as fol- m , ml '

lows: Cm = 3 N ms/rad;

2

b5 >0, b4>0, b3>0, b2>0, b1 >0, bo>O ...(6) Km=5 x 105N/m;Kt=0.65 Nm/A;.

b4b3> b5b2 ...(7) K1 = (Current source) Amplifier~ gain = 100 Nrad;

b2>{(b4b1-b5bo)ob4}/(b4b3-b5b2) ...(8) rm =9.55mm;rm =38.2mm;

1 2

b4b1> b5bo + [bo(b4b3 -b5b2f/{.b2(b4b3 -b5b2) 1m = 0.0018 kg m2; Jm2= 0.071 kg m2.

1 .

-b4(bi b4 -b5bo)}] ...(9) Main objective of the MFC scheme is to choose

, ...suitable gains for the computed errors to produce

Eqs. (6)-(9) proVide only the stability ranges for the requisite control action, irrespective of the the gams of the system: ApI' Ap2' Adl and AdZ But plant parameter variations and impart reasonable to visualize the .system response and its stability it performance robustness to the system. To achieve

~ is necess'i!.fY to study the root loci of the system by this, tlie root loci of the plant are obtained from v~gthe gain vectors Ap and AD its characteristic equation, by varying one gain at a time and keeping the others fixed at some nominal

P ti^° aIE I values, taken as A^p ^=A ^p ^=1; ^Ad ^=Ad ^{= 0.001.}

rac c xamp e ...1 2. 1 2 .

A pr~ctical e~ample, s~ar to the one ?f Seto Higher mertla of the 2plant!s kept. at the geom~c et aLl is considered to illustrate the different mean (0.0251 kg m ) of its ffilDlffium and maxI- aspects of the MFC scheme for a voltage source mum values for the purpose of this computation.

driven joint motor. The root loci are useful in selecting the starting The plant is assumed to have the following par- values of the gains on the basis of the margin of

ameters: stability. Based on these values, extensive search is

made for the optimum gains to ensure minimum Kp = 5 X 105 N/m; Cp.= 2100 N s/m; deviation between the model and plant responses

C = 0.1 N ms/rad' K = K oK = 125 NmV/rad A' for both. increas~g and d.ecr~ing plant iner.tia.

PI ' ep 1. t , The choice of optlffium gams is made by definmg...,.4t

where, a quadratic performance index Z as follows:

K1=(Voltagesource)Amplifiergain=100V/rad;

f

^T ² ^.2 ^.2

Z= [w1°e2+W2°e2+W3°1 ]dt Output swing of the voltage source amplifi- 0

er= ::t: 15V;

where e2 = q m2 -QP2 and i is motor current in the Kt = Motor torque constant = 1.25 N ml A; energy term. T is chosen to be 300 ms and the

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weight factors are taken as WI = 10000; W2 = 1 and variations during operation. This is given major W3 = 0.1, where maxirnt!m emphasis is given to the importance here while simulating the system equ- position error. ations. Inertia Jp of the output rotor of the plant The average of tWo sets of gains for both in- is made to vary ~ithin a range as wide as 0.0089 creasing and decreasing inertia is then forwarded kg m2 to 0.071 kg m2 (1:8 range). The output iner- to give a final set of gains as a compromise for tia of the plant is assumed to increase or decrease both cases. These are obtained as: according to the following law:

A p =0.08;A p =5.37;Ad =O.Oll;Ad =0.05. _I ₂ _I ₂ Jp.=J p [initial]+j p ot,wherej is given by₂ ₂ ₂ _P2 Final gains thus obtained are utilized to draw j = a + bt.

the root loci for the plant by varying the gains in P2

turn, keeping others fixed at the above calculated This law presumes a cylindrical robot with uni- values. These are shown in Figs 5-8 depicting form radial motion where the slewing at the clearly the stability ranges of the plant for all th,e shoulder joint is controlled by the proposed servo- gains. The operating points, marked with 6., system. Further, when the inertia J p increases, j p indicate that this choice produces the near-highest is assumed to rise uniformly from 6.12 kg m2/s t~

margin of stability as judged by their distance in 0.21 kg m2/s in 300 ms time. For decreasing J , the left half plane from the vertical (imaginary) j is assumed to be attenuated uniformly fr;tit

axis. -0.12 kg m2/s to -0.21 kg m2/s m 300 ms..Ad-p~.

Simulation Results ditionally a sinusoidally fluctuating disturbance of Control systems often find themselves in diffi- magnitude Te = 14 + 1.4 sinwt is imposed on the culty while handling the problem of parameter system at near resonant frequencies. For increasing

700 600

350 300 0

'" V'

~ ~

0 0 g 0

c -

.-'"₀₁ ₀

0 E

E -

-35 -30 0

-700 -600

-300 -200 -100 0 100 200 300 -500 -0 150 300

R~I S R.al 5

Fig. 5-Root1ocus (A^p varyin g) Fig. 7-Root locus (Ad varying)

.I I

800 700

/000 35

V' V'

~ 0 ,.. ~ 0.6

coO 0

-c

'" -

0 '"

E 0

-~

-/00 -350

-800 -1000 -1000 -700 -800 -800

R~I 5 R.aIS

Fig. 6-Root locus (Ap varying) Fig. 8-Root locus (Ap2 varying)

2

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Load:T.zI4+1.4 .in SOOt (0) SSC control (b) MFC control

10 '& 1.0 I-

~ ~~ 0053kg m2"8 ontinwusly

"0 a':' ~ ing from

: C f7 \!089 kg m2.1' 89kg m2 to

~ E 0 -~ 0.5 ~ 05 5 kg m 2

goo. 5 i is is

N

.-~

-.~

J0.5 .;: ~

0. .~ ~ 200 I 200

a ~ Tim~. ms Tim~.ms

Fig. ii-Comparison of step response for SSC and MFC

10

Fig. 10 gives the transient response of the sys- Tim. .ms tern with decreasing inertia, other conditions being similar to the previous one. Again the plant re- Fig. 9- Transient response of loaded plant with varying Jp (Jp sponse shows remarkable coincidence with the

uniformly increasing) 2 2 ideal states qf the model after tolerable departures within the first 120 ms and the residual oscill-

, Load: T.:14+1.4sinI50t 0 ations die down at around 200 ms. Current re-

~~ quired and energy expenditure are again in agre;e-

I. 0 ment with those noted for an increasing rate of in-

0 ertia.

-g E Fi gs 9 and 10 document quantitatively the level

~ .

~ ~ c performance provided by the present schem,e un-

ct. 0 -;. i der very wide parameter variations. Note that be-

J 05 t t tween 100 ms and 200 ms J

p increases from

15 0 ~ 2

.~ u 0.024 kg m2 to 0.045 kg m2 in Fig. 9 while the

c 0 same falls from 0.056 kg m2 to 0.035 kg m2 in

10 Fig. 10 and the shape of the response remains quite uniform upto 200 ms (truncated here). The

0 100 00 performance robustness is therefore noteworthy.

Ti~ ms Fig. 11 shows a comparison of the step re-

"- 0 0 ) o. sponse obtained by the present MFC control with

--:.I FIg. 10- TransIent response of loaded plant with varyIng Jp, (Jp,

th SSC t I d b S t et l In the

Of rml d . ) --e con ro as Propose y e 0 a .

urn 0 y ecreasmg ..

SSC the inertia is not varymg contInuously but is

...

⁵⁰⁰ ^dl ⁽² ^d ^d ^ed ^. ^t. ⁾ fixed at two representative values Jp = 0.0089 kg mertla W IS .ra. s .n ~o e pr ornma m& m2 and 1. =0.071 k m2. The res! nse of the and for decreasmg mertla W IS taken as 150 radl s d ^{PJ. '}FC ' gh

.

tpo

th t f th

...propose M IS muc supenor 0 a 0 e

(1st mode predoffilnatmg). This may be regarded SSC

.

^th ^t. J . t.

.' .even WI con muous vana Ion.

as a substantIal pumshment to which the control P2

system is subjected. The simulation has been carri- Conclusion

ed out in a high speed PC AT. Parameter variation of a system during opera- Fig. 9 shows the transient response at uniformly tion is considered to have the most damaging ef- increasing rate of inertia variation when a step in- fect on its dynamic characteristics. A number of put is applied to the model, starting with null in- methods. have been used with success to mitigate itial states. Plant and mod6l responses show a little this effect. One of these methods, the Model Fol- difference at the initial stages but after a small lowing Control, is experimented here with some time interval of around 100 ms they practically modification in that the reference input is applied 1)c;' merge together. Energy expenditure and current only to the model.

required are also within a very reasonable limit. As mentioned earlier, the output inertia Jp1 i~

However, the armature current of the servomotor, modelled after the shoulder joint of a cylindrical clipped at the extreme of ::!: 30A due to saturation robot with uniform radial motion at the end ef- of the voltage amplifier, shows considerable fecter where the inertia variation (over a 1:8 amount of initial transients that die down in about range) is a quadratic function of time. For increas- 100 ms. The speed of response is around JOO ms. ing inertia,

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J = 0.0089 + j p t, Kep = gain vector for the motor cum amplifier (plant)

P2 2 qp' lip = state vectors (plant)

where J p' Kp, C p = inertia, stiffness and damping matrices

j =0.12+0.29t. respectively (plant)

P2 J'p = rate of inertia variation matrix (plant)

While for decreasing inertia, Kb = back emf constant matrix of the servomotor .(plant)

Jp2=0.071 + Jp2t, Jm ,Jm = moment of inertia of the input and output rotors I 2 (model),kgm2

where, Cm = damping constant of the coupler (model), N s/m

j = -0.12-0.29t. Cml,Cm2 = damping constant of the dampers connected with

P2 the input and output rotors (model), respectively,

Further, the joint ha.~ been subjected to a sinusoid- N ms/ rad

.

^d

.

^b

.

h K = stiffness of the coupler (model) N/m

all^y fluctuatm g load torque lstur ance WIt a m d.. f

.

^t ^d ^t ^t ^t ^' ^U.

., rm ,rm = ra 11 0 InpU an ou pu ro ors respec ve yI

:t 10% pulsatIon at frequencIes close to resonance. I 2 (model), m

The plant in this simulation has no doubt been Kem = gain of the motor cum amplifier, N m/ A subjected to a very severe punishment by way of ~I' qm2 = system co-ordinates (model), rad above variations, but its response is quite unifonn qm ,lIm = system velocities (model), rad/s (Figs 9 and 10). The perfonnance robustness is R I 2 = reference input to the model

thus noteworthy. Additionally, there is a high en- ApI' ~2 = positional error feedback gains (plant) ough speed of response and negligible residual os- Ad)' Ad2 = velocity error feeqback gains (pfant) cillations. The authors' have chosen a voltage Kem = gain vector for motor cum amplifier (model) source amplifier (linear) which, as mentioned earli~ q m' qm = ~tate.vect.ors for the model. .

er, can be very simply realized by power opera- J m' K m' C m = Inertia, .sUffness and damping matnces

.

^al ^lifi ^h ^d

^.

^all ^d ^h ..r~specuvely (model)

tlon amp ers t at rastl.c y re uce t e cIrcuIt B = feedback row vector (model) manufacture cost. Further, m the absence of adap- e, e = error vectors for the system tation, the model can be implemented as an ana- Ap = positional error feedback gain vector log circuit with low cost linear micro electronic Ad = veclocity error feedback gain vector

devices. Te = payload vecto:r associated with the plant

.. ^highi

^.Z = performance lJ1dex

The results of sImulatIon are y ~ncourag1ng £ = exponential lag operator, 1/( R. + Ls), NV and the arrangement can well compete m perform- v = voltage supplied to the motor (plant), V ance with other more expensive and complex Eb = back emf produced in the motor (plant), V schemes using adaptation. This proposal is fully R. = ~rmature resistance of the motor (plant), Ohm practical also from the standpoint of simple hard~ 1:- : Inductance of the motor armature (plant), H

.

^I

.

^Of ^fu ^h

.

^t

.

th f t I -current, A

ware Imp ementatlon. rt er mteres IS e ac k = back emf clJnstant of the motor (plant) V s/rad .that a reduced ord~r model (4th order) is used to k: = gain of the amplifier '

make a higher order plant (5th order) follow the k, = motor torque constant, N m/ A same. It is thus observed that even if the dynam- t = time, s

ics of the plant be somewhat "neglected" it will Tp = plant torque input = k,.i still be possible to achieve successful MFC, as References

with a reduced order model chosen here. The 1 Seto K, Takita Y & Tominari N, Bull JSME, 28 (239) MFC arrangement developed here is perhaps wor- (19~5) 937-942. .

th y of inclusion in the family. of efficient and prac- 2 Takita Y, Seto K & NokamJzo T, Bull JSME, 29 (1986) .1274:'1279.

tIcal control methods. 3 Mukhopadhyay S & Sengupta S N, Mech Eng Bu/~ 18

Nomenclature (3&4)(1987)93-98.

J , 1 = moment 0 Inertia 0 f.. f the mpu an ou pu ro ors,. t d t t t 4 Landau I D, Adaptive control- The model reference ap- PI P2 ^U. ¹ ⁽ ^l ^t) ^k ² proach(MarcelDekker Inc,NewYorK),1979,203-210.

respec veypan gm

-.'. . ) f 5 Dubowsky S & DesForges D T, ASME J Dyn Sys Meas

CPI -damping constant (beanng and WIndage effect 0 Contr~ 101 (1979) 193-20Q.

motor(plant),Nms/rad ⁶ H . R, T . M An G & T .

uk M Pr

C _d ' fth bel ( 1 ) N s/m \ oroWitz Sal, war 10ffilZ a , oc

p -amplng constant 0 e t pant, IEEE I C ,f D- b ' & A . ₃ ₍₁₉₈₇₎ ₁₂₁₆ _-

K p = SUo .ffness 0 f th belt(e pan, .m 1 t) N/ 1222. nt on, on 1M' otlCS utornatlon,

Kep = gain of the motor cum amplifier (plant), N 7 Asare H R & Wilson D G, Proc IEEE 1m Con! on Roboties &

mV/radA .

( ) 2

... d . I Autorna:wn, 3 1987 1531-154 .

rpl,rp2 =radlloflnputan outputro.orsrespecuvey 8 Bar-Kana I, ASME J Dyn Sys Meas Contr, 113 (1991) (plant), m .721-728.

qp, q_I ^p₂ = system co-ordmates (plant), rad 9 M_oran

.

^{M &} ^Zafinou , 0 us process con I)

.

^{E R}^b ^t ^tr ^I^(Pren ce^U

.

li^p, lip = system velocities (plant), rad/:. HaD N J

) 1989 39-84

I 2 ' , , .

j = rate of inertia variatiC'n of the output rotor 10 Sugiuchi H & Yoshimoto K, JSME 1m J SeT /II, 32 (3)

P2 (plant), kg m2/s (1989)421-427.

Control of a robotic manipulator- A model following approach