Model Time

Proceedingsof the TuB13.6

44th IEEE Conference on Decision andControl,and the European Control Conference 2005

Seville, Spain,December12-15,2005

Robust Trajectory Control of Robot Manipulators Using Time Delay Estimation and Internal Model Concept

Gun Rae

Cho,

Pyung-Hun

Chang, Member,

IEEE,

Sang

HyunPark

Abstract- This paper proposesanenhanced controller,using Time Delay Estimation(TDE) and Internal Model Control(IMC) concept for robot manipulators. The proposed controller has a compensator based on IMC to improve robustness of Time Delay Control(TDC) against friction; it is effective enough to handle bad effect of friction, moreover, simple andefficientas tomatchpositive attribute of TDC. The controllerwithTDE, doesnotneed whole modelofplants, thus, it is easily applicable. The analysis and experimental results show the effectiveness of the proposed controller against the friction effects of the robotmanipulators.

I. INTRODUCTION

IN

this paper, An enhanced controller to improve the robustness of TimeDelay Control(TDC) by usingInternal ModelControl(IMC)conceptisproposedandanalyzed.And it isapplied to control a robotmanipulator.

TDC is the control technique which compensates the unpredicted disturbances of the plant and/or the unknown dynamics of the plant with Time Delay Estimation (TDE) [ 1,4]. Although TDC has simpler structure than other advanced controlalgorithmsdevelopeduntil now, TDC hasa distinct robustness for the disturbances and the parameter variations. Its effectiveness to robot manipulators has been showedthroughmanysuccessfulapplications[3,4,9].

It has been observed, however, that in the presence of so-called hard nonlinearity such as coulomb friction/static friction, TDCreveals some problems commonly founded in other methods like PID control or disturbance observer[8].

More specially, in the system having coulomb friction, its tracking error increases at zero velocity. Moreover, in the systemhavingstatic friction and stribeckeffect,slow motions

Manuscript received October 9, 2001. (Write the date on whichyou submittedyour paperforreview.) This workwassupportedin partbytheU.S.

Departmentof Commerce under Grant BS123456 (sponsor and financial support acknowledgment goes here). Paper titles should be written in uppercase and lowercase letters, not all uppercase. Avoid writing long formulas withsubscriptsinthetitle; short formulas thatidentifythe elements arefine(e.g., "Nd-Fe-B").Do notwrite"(Invited)"inthe title. Fullnamesof authors^arepreferredinthe authorfield, butare notrequired. Putaspace between authors' initials.

F.A.Author is with the National Institute of Standards andTechnology, Boulder, CO 80305 USA (corresponding author to provide phone:

303-555-5555; fax: 303-555-5555; e-mail:authorw boulder.nist.gov).

S. B.Author,Jr.,waswith RiceUniversity,Houston,TX77005 USA. He is now with the Department ofPhysics, Colorado StateUniversity, Fort Collins,CO 80523 USA(e-mail:author@lamar. colostate.edu).

T.C.Author is with the ElectricalEngineering Department, University of Colorado, Boulder, CO 80309USA,onleave from the National Research Institute forMetals, Tsukuba, Japan (e-mail:author(nrim.go.jp).

often involvestick-slip phenomenaexpressedinthe form of limit cycle in the position control and oscillation intrajectory trackingcontrol.Therefore,the compensatortoremedythese problems is necessary in order to achieve better tracking performance.

There are a few researches to solve theseproblems. Chang and ParkproposedTDCSA[8].Itis TDC withacompensator of theswitchingaction basedonSidingModeControl(SMC).

Onthe otherhand,Kwon&Chung suggestedtheMPEC[11], which has multi-loop stages of the perturbation observer known as the similaralgorithm to TDC.And,Namadded the perturbationobserver as a compensatortoTDC[15].

IMC is the control algorithm which uses theplant model directly[5,6]. Ithasastraightforwarddesign method,and has good control performance as providing perfect control scheme theoretically. It has been widely used to control chemical processes, and usedto controlarobotmanipulator by addingtocomputedtorquecontrol[9].

In IMC, however, it is a weakpointthat itrequires plant model. Generally, many nonlinear plants, such as robot manipulators, are hard and need much time to be modeled.

Moreover, IMCisfundamentally applicabletostable system.

So,it has been seldom used in the mechanicalfield[9].

This paper proposes an enhanced controller for robot manipulators by adding two concepts, TDE and IMC, to make up for weakpoints of each algorithm. It doesn't need whole model of the plant and is better to control friction system than TDC. It is named Time Delay Control with Internal Model(TDCIM).

This paper structured asfollowing. InSectionII, TDCand IMCareintroducedtoandanalyzed.InSectionIII,enhanced controller named TDCIM isproposedand itssomeproperties areanalyzed.SectionIVprovidestheanalysis whyTDCIMis robustto frictionsystem,and SectionVshows its robustness throughtheexperimentalresults.Finally,wesummarized the results and draw the conclusions in SectionVI.

II. PRELIMINARIES

This section introduces to TDC and IMC, and points out theirpropertiesand weakpoints.

A. ReviewofTDC

We summarize TDC law for robot manipulators[4] and analyzeitsproblems concerned withTDE error.

0-7803-9568-9/05/$20.00 ©2005

IEEE

1) TDClaw

Dynamics ofrobot manipulators is generally described as follows.

M(0)O+ V(0,0)

⁺

G(0)

⁺

F(0, 0)

^{=T (1)}

Where M denotes inertia matrix, V the coriolis and the centrifugal forces, G gravity, F frictions and unmodeled disturbances and Tinputtorque.

Eq. (1) canbe rewritten as follows by importing constant matrix M which is selected based on M.

MO

+-H(0,OQ

^{)= T}

(2)

Where, H denotes nonlinear dynamics of robot manipulators and is described as follows.

H(0, , =)

-(M(O)

M)O+V(0, 0)+G(O)+F(0, 0) (3) Based on the computed torque control, T can be designed asfollows.

T=MU+H (4)

u=Od+

KD (°

+

KP

(5)

Where H denotes estimation values of H, KD andKPPD gain diagonalmatrices.

TDC uses TimeDelay Estimation(TDE) to get H. Under the assumption that the time delay L is sufficiently small, following approximation is valid dueto(2).

H(t) =H-(t) H(t(t- T(L(tL) MO(t L)

(6)

Finally, TDC can be summarized as follows.

T=T(tL-)-M (t L)

+d KD (°

⁺

KP

( (

If weselect M asdiagonal constant matrix, TDC canbe designedasseparatedjointcontrollerusing only M andPD gains, justlikeFig. 1.Thus,TDCis very efficient and has few burden ofcomputationby usingTDE,which doesnotneedto computewholedynamicsof robotmanipulators.

2) ProblemsofTDCconcerned withTDEerror

If we can use infinitesimal time delayL, it is possible to estimateHperfectly byTDE.However, it'simpossibleto set Las infinitesimal. Inmany cases, samplingtime is used for the time delay L, and there exists limitation of hardware.

Therefore, there exists TDE error due to finiteL. From(2), (4)and(6),thefollowing is induced.

Mi

^(t()

(=t) H(t) H1(t) H(t) H(t

^L)

(8)

The right term denotes TDE error. Here, we define TDE error asfollows.

£(t) -U

(t)-O(0

^=M

Ij(t) H(t-L) (9)

+ IMs

IsK1²KD _{PV+D 'u} Mm_' +

Od _T PLANT

The error dynamics of TDC is represented as follows; it shows the influence of TDE error to thetrackingerror.

e(;, +

KDe(t) +KPe(t)

^{(t) (10)}

Where, e ⁼ 09^-

If time delay L is sufficiently small compared to the changing rate of systemdynamics,TDEerror canbeignored.

In mechanical system, however, there exist fast dynamics such as coulomb friction/static friction, which can not be estimated wellbyTDEwith finiteL.Therefore,when TDC is appliedto such systems, TDE errorbecome large, and thus large tracking error is occurred.

B. Introduction to IMC

1) Structure ofIMC and its properties

IMC,proposed by Garcia &Morari[5], is showed in Fig.2.

Itis composed of IMC controller Q and internal modelGm.

The effect of the parallel path with the model(Gm) is to subtract the effect of themanipulatedvariables from theplant output. If we assume that the model represents the plant perfectly,then the feedbacksignalisequaltothe influence of disturbances and is not affected by the action of the manipulated variables. Thus, the system is effectively open-loop[6].

IMCcontroller Qdesigned based on

Gm-1

playsthe role of a feed-forward controller.Butth IMC controller doesnotsuffer from the disadvantages of feed-forward controllers; it can cancel the influence of disturbances because the feed-back signal is equal tothis influence and modifies the controller set-point accordingly[6].

Overall transfer function of IMC isgivenasfollows.

y(s) =I

^{GQ Yd 1 Q Gd}

^{(1 1)}

I±+)7(-

^{G 1}

^{Q ()} Gm

From the transfer function, we can easily find several propertiesof IMC, which can bearrangedas follows[5,7].

* PropertyI:DualStability Criterion. When the model is exact, stability of both controller and plant is sufficient for overall systemstability.

* PropertyII:Perfect Controller. Assume that

Q=Gm-'

is realizable and the IMC system is closed-loop-stable, then, perfect reference-tracking

control(y=yd)

^canbeachieved, for all t>O, despiteanydisturbance d.

* Property III: Zero offset. A controller Q which satisfies

Q(0)=Gm-'(0)

yieldszerooffset.

2) LimitationofIMC

IMC is an intuitive and prominent algorithm. However,

d Q

Yd Y V

+

+ G

+ ^y 0

KDS+KP

Fig.1. Blockdiagram ofTDC Fig.2. Blockdiagram ofIMC.Where, Q denotesIMCcontroller,Gplant,

G,.internal model.

3200

there exist some limitations of it to be applied solely to mechanical field. One thing is that it requires plantmodel, which is hard and takes much time to be obtained incomplex nonlinear plants such as robot manipulators. Moreover, its scheme, in its original design form, results in an open-loop control, which is applicable onlyto stable systems[7]. They aremaybe main reasonswhyIMChas been seldomappliedto the control of mechanical systems.

III. PROPOSITIONOFTDCIM

In this section, a robust controller named TDCIM(TDC with Internal Model), which is composed with TDC and compensatorbased on IMC, is proposed and its properties areanalyzed.

A. Derivationof TDCIM

1) Derivationoflinear dynamics from TDCsystem Plantlinearization usingTDE.

InTDC, TDEis used to cancel nonlineardynamics ofplant.

And the robotdynamics compensated byTDEcanbe written asfollowinglinearequation.

0 ^u H(t-L) ^{( (} (12)

This means that the relationship betweenjointvariable 0 and controlinputuis describedas alinear equation0=s 21u.

Where,TDEerror^Ecanbe treatedasdisturbance.

* ConsiderationofPDfeed-back

Controlinputuin(5)canbeseparatedreferenceinputand feed-backinput.

U =V

-0oD ^KPO

⁽¹³⁾

Where,vdenotes referenceinput; v- +KDd+KPd .

And then, (12) can berewritten as follows

O+KDO

⁺

KPO

=-v

(14)

Finally, we can find out that the dynamics of 0 can be described as the separated linear dynamics of v, and TDC system canbesimplified asFig. 3.

After all, TDC system canbe explained as a feed-forward controller with theplant linearizedbyTDEandPDfeedback.

Through the linearization process, however, TDE error is occurred, which is not compensated and affects on the manipulated variablesjust like as disturbance.

2) Addition Compensator basedonIMC

Now, let us add the compensator basedoninternal model concept to TDCsimplifiedinFig. 3.

According to design scheme of IMC[5,6], internal model Gm can beeasily selected fromFig.3asfollows.

Gm(s) =4)

^{I KDS}

^{KP (15)}

And, IMC controller can bedesigned as follows based on internal model.

Q(S)

=44 KDS

K1

(16)

Fig. 3 shows that the simplified TDC already has the controller such as (16). Therefore, TDCIM canbe achieved by adding only the internal model and IMC feed-back as described inFig. 4.

Fig. 5 represents overall block diagram of TDCIM. In TDCIM, afterall,

0Q

combined reference and IMC feed-back is used instead ofthe reference Od. The control law of TDCIM isgiven as follows.

0900

d + m

(17)

£ G

(S21±-KDS :K:)1

Fig.3.Simplified blockdiagram of TDC, with linearized plantbyTDEand PDfeed-back.

Q

s21+KDs K.

£ G

+ (s +_KDs Kp)

Fi(s 4+Ds Kb) ⁰

GM

Fig.4.Simplifiedblockdiagramof TDCIM.

e + T

- Ms'

PLANT

I ^{I 4}

m

KS D

Gm

...

:e

Fig. 5. Overall block diagram of TDCIM.

T T(t-L) -MO(t^L)

+

M4[# -+ KD

t +

KP

⁽¹⁸

TDCIMhas a simple structure as showed in Fig. 5, and does not need any extra gain. The control law described above shows that TDCIM canbedesigned by choosing only

M, KD

and

KP

^{same as} TDC case.

Therefore,

it is

easily

applicableand efficientas tomatchpositiveattribute of TDC.

Moreover, it provides the chance that IMC can be applied easily to robot manipulators, without whole computation of the robotdynamics.

* Remark

Feed-back timedelayin IMCfeed-back is included inFig.

5(dashed box).Itisnotincludedintentionally,but inevitable indiscreteimplementedcontrollers. If

Q=Gm-',

^IMCprovides

the perfect control scheme theoretically, but, it's possible onlyunder the continuous time condition. Inmany cases,the controllers are designedon the computer systems, andthey

1The controller needs thevelocity and the acceleration of the plants.In many cases,however, theyare notmeasurable, thus, obtained by numerical differentiation. Numerical differentiationcausesthedelayofresponsesand degrades trackingperformance,butitisnot acriticalproblemtoapply.Itis also well known thatitenlargethe influence ofsensing noise,butitcanbe overcomeby usinglowpassfilter[2,6].

Is21KDs Q

^{Kp v}

d

0 0a

+~/ ^{s ID}

Q

^{K + M}

d D " aIV- Tf^u-+

must work under the discrete time condition. And, it's impossible to prevent the time delay in feed-back due to sampling. Thus, IMC cannot work as theperfect controller.

So, it is necessary to include the timedelay dueto sampling forprecise analysis.

B. SomepropertiesofTDCIM

In this subsection, it is analyzed that what the IMC feed-back value means and how it affectstocontrol systems.

1) ThemeaningsofIMCfeed-backvalues

Atfirst, let usdefinethe controlinput of TDC asfollows, topreventconfusion with controlinputof TDCIM.

A .

UT()~O_{UTDC(t) - od(t)}+K_{D (t)} +K_{P (t)}e (1

9)

Then, controlinput of TDCIMincludingthe timedelayin IMCfeed-back can be described as follows.

u [40

~~...

u(t) TL) C(L-t)t- ^m(t L) D 9 L) m(t L) (20)

+Kp(0S9-L) ^m(t L)]

Theright terms exceptUTDC(t)aretheaddingpartsbyIMC feed-back. And, their meanings can be obtained by subtraction from theplant dynamicsGtotheinternal model dynamicsGm beforeonesampling time, i.e.^att-L.

09tm1--^{(t L) +}KD ^tL) m(t L) (21)

+KP(+9;)- m(t L) (t L)

This shows that the IMC feed-back value meansthe TDE error before onesamplingtime. Therefore, the control input and theinputtorqueof TDCIMcanbe rewrittenasfollows.

*(t) =I TDC(t) (t-L) (22)

*(t =MU(0 +4±(t- MuTDC H(t--L)

^{ME(t L)}

(23)

TDCIM canbe treatedasTDCwith the compensatorusing the TDE error before one sampling time. In addition, ^error dynamicsof TDCIM isgivenasfollows.

ei,+K_e(;) KD (t) +De +KKp(t)e =-,,t) (t-L) (4

(24)

Inotherwords,TDCIMcanberegardasthe controller with anestimator ofHworkingasfollows.

TDCIM(t) (f ⁷ (tL) 2H(tL) H(t 2L) (25)

Thismeansthat the compensator decreases the influence of TDEerror,andimprovesthe estimationperformanceofHby usingTDE errorbeforeonesampling time.

2) Propertyof TDCIMas the integrator

It is knownthat TDC has 1St orderintegrator[8]. Well, how isTDCIM case?

After takingLaplace transform, theinput torque of TDCIM include timedelay by sampling in the IMC feed-back can be rewritten as follows.

() e 2 ^(s)

^{20e s} (26)

If L is sufficiently small that holds the approximation e-Ls 1l-Ls, aboveequation can be approximatedtofollows.

T(S)=J OTDC(s)

202

^{e s s e (27)}

This means that TDCIM has2ndorderintegratorwithhigh gaininclude

L-2.

Thus,we canconjecturethat theinputtorque of TDCIM changes very fast to reduce the tracking error when it occurs duetodisturbancesorany otherreason.

C. Stabilityanalysis ofTDCIM

Thesufficient stability conditions have been derived based on the analysis in LN [14]. The notations used in stablity analysis arearrangedinTable I.

* Theorem.

Ifthe assumption and the stability condition presented belowaresatisfied, then therobotmanipulator controlledby TDCIM is

L,.

stable.

Assumption.* W,

0d, 0d,0dEdL00

Stability

condition

A(tA ||2c

^<

¹ &1| ^B' (|2c

^<

¹

Where, wdenotes disturbance, and the matricesA, B are definedasfollows.

J

is the diagonal positive definite NxN matrix, that satisfies ⁼M.

And Q1 =4

-MM-tl)

a isdiagonalizableinthe form described below.

(t) =

p(t) (t)P(t)

^{(2 )}

Where A(t) =

diagL2 lt),.t

^*,

AN(t)]

and P(t) is the orthogonalmatrix thatdiagonalizes

Q(t,.

Then

A'(t

and

B'(0

^{aredefinedas follows.}

TABLE I NOTATIONS

When * isa Nx1 vector When * isamatrix When * isa vectoror amatrix

istheinduced2norm

of..

^{* ,=}

liti2()(t)(t) (t)L

|@=@

^||

sup|

^{(t (Thismeans LN)}

^{II@()0 = x}

^(t)

^Ii21[t

^(-

0(0 (t ^c) (where, Ldenotesthesamplingtime)

(lt⁺V)

I,

^Sup₀ ^(t

J)|

^(where,Ldenotes thesampling time)

<: T

3202

A'(t) P

,1A

B'(t) =P B P

(29)

(t) N(t) (t)

and B( ) =

diag[b(t), ..,bN(t)

, b i(t (t) (t)

Note that A'(t) and B'(t) satisfythefollowings.

A' +B= 2Q A' B' B' A' Q

(30)

* Proof:

Including timedelay,(17)canbeexpressed asfollows.

09p

^{t (t) (t- ) (t L)}

⁽³¹⁾

From(18)and(31),inputtorqueof TDCIMcanbe derived asfollows.

t(t =-t-14

4i(t

^{L)] (t 2L) M(t 2L)] (32)}

+-M[Od (t)K KDe(O Kpe(t)]

And let F ofequation(1)as(33)where Fv denotes viscous friction, wis disturbanceincludingcoulomb and static frictions.

F =-F w

(33)

Then the closedlooperrordynamicsof TDCIM^canbe derivedas(34)from(1),

(32), (33).

£(t--1F

(t)M]E(t--L)

^[I

M(t)M]E(t

^2L)

=-w(t) (pt)

Where,

x(t) = l- M-

M()] [K

D (t)

Kpe(t) ]

I - .... ~

-4(t1,-Mt0tA M(t)o(t ^2L) V(t)

9p(t)

= (t)

w(t)

[- (t) d(t

Let e (t) and ^, (t) as(35).

eyt.0-:::=

M!e(t),

^{c (t) _I£ (t) (35)}

And(36)canbe derivedby

transforming (34)

which is of

e(t)

and

£(t)

^toequationwhich is of e (t) and ^f:7

£9£(t) (t--) (t) (t2L) =4

rp

-" ^(t) (36)

(36)canbeexpressedas

(37) by (30)

7#-A't

+B() )£

(t--L)

^B (t)^E (t2L) ⁽⁷

=

Mmfg -o/;

^(t)

If |B'I(t)|| < 1 is

satisfied,

(38)^canbe derived from

(37)

aftersome norm

operation.

(I-AB

It1 ² )(I IIAI(t)

Ili

²

J II

^{7(0 T (8}

<

-JMWT

^-BI(t)A (t)

7(t-2L)

^{|| || (t)r}

And if||A'(t)|| <1 is

satisfied,

(38)canbe derived from (39).

<

(i-B') 11

^7(t)

liToo-

^{(At) (t-2L)}

±Too I~

^{() TT (39(39)}

A1Bt (t)||i2o)(1 A^(t) zi2oo

Letting T ->oo, (40)is obtained.

-I~~J B'(+A()

⁽

I)

<|Xt-B

^{(t)A(t) 7(t-2L)}

||0

+ ||(t) (40)

/(t)co- (1-gB (t) ||i2oo( ^II1 (t) II i2Co

For sufficiently small L, | B(t)A' (t2L)| is very small2. And ifassumptionissatisfied,

||/iAi

(P <

c°

.

And || 7(t)|| <0° ^,

e(t)

^,

e(t)

arebounded.

As M is similar to M(t) ,

A'(t)

i2o and B'(t)

i2o

become smaller, so TDCIM has more

possibility

to

satisfy

condition 1 and 2.

Ithas been detected that stablegainof TDCIM is less then TDCfromanalysisand simulation.

IV. EFFECTOFTDCIMAGAINST FRICTION OF SYSTEM In this section, the effect of the

proposed

compensator is showed. And it is analyzed how it affects

against

fast dynamics of system,

especially including

coulomb friction and stribeck effectinciting

stick-slip phenomena.

Forquantitativeanalysis,weused the simulation results of TDCand TDCIMappliedto1DOF

manipulator

shown

Fig.

6.

We appliedTDC with M ⁼1.0 and TDCIM with M =0.5.

The PD gainsare set toKD=20,

KP=100

for both controllers that make natural frequencies

co,=1Orad/sec

and

damping

ratios$=1 of the^errordyanmicseq.

(10)

and

(24).

A. Effectofthecompensator againstslow

changes of

H TDCIM has a compensator using TDE error before one

sampling time. To show the role of the compensator more

visible,letusobserve its effect whenH

changes slowly.

Fig. 7 shows the estimation values of H in TDC and TDCIMcaseconceptually. InTDC

using

TDE H(t) H(t

L)'

TDE error occurs as much asthe

changes

ofH

during

one

samplingtime.InTDCIM,TDEerroris

compensated by

that before one

sampling time,

thus, more exact estimation is possible. It's a clue that control

performance

of TDCIM is better than that of TDC.

B. When H

changes rapidly

-Coulomb

friction

case When the

velocity

of

plant

passes

by 0,

coulomb friction changesits direction and the

plant

has

rapid nonlinearity.

In TDC, TDE cannot estimate it

exactly, thus, tacking

error become large. This subsection shows the effect of the compensator in TDCIM

against rapid nonlinearity

such ^as coulomb friction.

2Thisassumption,whichwasalso usedtoprovethestabilityofTDC in [2,4], isincomplete because Tf(t) isafunction ofmanipulatedvariables and affects thestabilitywhenL 0.Thisproblemhas been alsopresentedin[13].

Andwewill deal thisprobleminfurther works.

G_(t) F_V(t)

I

Generally, coulomb friction is modeled as follows.

F =

r-q sgn

() (41)

Where,

Tslip

denotes coulomb friction coefficient.

The simulation results of the effect of coulomb friction are showed in Fig. 8. (a) and (c) are conceptual figures of the estimation errors in TDC and TDCIM, (b) and (d) results of simulation of controlled system with the 1 DOF manipulator described inFig. 6.

InTDC, as described in Fig. 8 (a), TDE error jumps when coulomb friction changes its direction at zero velocity. And it incites tracking error to be large. On the other hand, in TDCIM, a reverseaction as described in (c) is appeared after occurring TDE error due to the compensator. And it eliminates the influence ofTDE errorand reduces tracking errorconsiderablyasshown in(d).

C.

Stick-Slip

case

When the controller with the integrator is applied to the systemwith stribeckeffect, stick-slip phenomenaoccurred.

TDC and TDCIM have the integrators, and can't avoid m

0

Fig.6. 1 DOFmanipulator. / 1.0(m), m =1.0(kg)

H(t)

M

'9-L

( L)

rH^DC

HDMC(:.

Fig.

to2L tDL t

Fig.7.Estimation of slow H inTDC,TDCIM

them. In this subsection, stick-slip phenomena of TDC and TDCIM are observed and analyzed.

* Friction model

There exist several friction models to represent stribeck effect[ 10]. Among them, we simulate and analyze the stick-slip phenomena based on Karnopp's model.

Karnopp's model, one ofthe friction model used widely, is represented in Fig. 9. The small zero velocity

zone(

+a) in Kornopp's model surrounding 0 =0 is necessary for numerical computation and digital implementation since an exactvalue of zero is hard to becomputed[10].

* Simulation results

Stick-slip phenomena of TDC and TDCIM are simulated with the plant in Fig. 6 under the regulation control with initial error of0.0057deg. Its results are showed in Fig. 10.

Through the results, it can be easily made out that stick-slip of TDCIM has smaller amplitude and chatters more frequently than that of TDC.

From now, the reasons are analyzed with the division of stick state and statechangefrom stick toslip.

* Analysisfor Stick state

Instick state with regulating error, input torque increases if the controller has anyintegrator.TDChas Ist orderintegrator, therefore,the controlinputincreaseslinearly.But, in TDCIM case with 2nd order integratorandhigh gain include L2, the controlinputincreases inproportiontotime square. It'seasily understood that the process of TDCIM is much faster and escapes stick state more rapidly than that of TDC. It's the

Vfriction

1lip

2cr

Fig.9.Karnopp's friction model. Where, Tmax denotesmax.static friction.

(a)Estimation error of TDC 2M -I

(c)Estimation error ofTDCIM 2M

rTslip I0.

-2M

Fig.8.Effect of Coulomb frictionin TDCandTDCIM.Where,rjj,=5Nm,

L=0.OOI sec.

Fig. 10. Simulation results ofStick-Slip in TDC and TDCIM. Where,

Tmax1=ONm,rjjv=5Nm,^a=10 rad/sec.

3204

reason why stick-slip of TDCIM chatters more frequently than that of TDC.

*Analysisfor thestatechangefrom SticktoSlip

Inchanging state from stick to slip, the friction value of the systemjumpsfrom static friction tomovingfriction. So,TDE errorbecomeslarge just like the case of coulomb friction and TDChas large tracking error. In TDCIM case, on the other hand, the tracking error reduces considerably by compensating TDE error. Therefore, the amplitude of Stick-Slipin TDCIMis smaller than that of TDC.

V. EXPERIMENTS

In this section, we confirm the robustness of TDCIM thoughtheexperimentswith 2 DOF SCARA typerobot, and comparewith that of TDC.

The SCARA type robot used in the experiments is showed inFig. 11. The first link length is 0.35m and the second is 0.2m.

=1 of the errordyanmics eq. (10) and (24).

Fig. 12 shows the reference trajectory, which let the end effecter make a circle with 10mm radius during 4 seconds.

Both the samplingtime and the time delay for TDE are set toL=0.00 I sec.

B. Experimentalresults

Experimental results are showed in Fig. 13 and Fig. 14. Fig.

13 representstracking errorand torque of each joint. And, Fig.

14 shows XY position responses of the end effecter.

Maximumtracking errors arearrangedin TABLEIII.

Fromthe results inFig. 13 (a), (c) compared to references inFig. 12, we canobserve that the plant controlled by TDC has large tracking error due to coulomb friction when the plant passes by zerovelocity. In TDCIM case, tracking error is reduced decently, and that confirms the compensator works well. Fig. 12 (b) and (d) show that TDC and TDCIM have simlier torque profiles, because input torque is decided to compensate theplant dynamics and to makeplanttrack the A. Experimentalsetup

M =

diag(aq

2) of TDC and TDCIM described in TABLEII arebest tuned to minimizetracking error. And,PD gains are set to

kDi=20, kpi=100

for both controllers to that make natural frequencies

wOn=

1Orad/sec and damping ratios

Fig.11.SCi _Fig.13.Experimentalresults(solid: TDCIM, dashed: TDC).

TABLE II

CONTROLGAINS(M=

diag(aP;

2))FORSCARAROBOT

TDC TDCIM

a1 0.1 0.03

a2 0.025 0.0058

Fig.12.Desiredtrajectory

TABLE III

MAXIMUM TRACKING ERRORSINTDCANDTDCIM

Symbol (1 TDC 2 TDCIM 211*100

Joint1 0.1257deg 0.0049deg 3.9%

Joint2 0.1883deg 0.0107deg 5.7%

time(sec)

Fig.14.Experimental results-XYplot

refereces. But, you can find that input torque of TDCIM changes faster than that of TDC to reduce tracking error,

when TDC has large trackingerrorduetoCoulomb friction.

Through the experimental results above, it is certificated that TDCIM is applicabletomulti DOF systemand improves the robustness of TDCagainst the friction.

VI. CONCLUSION

Thispaper proposes anenhanced controller usingTDEand internal modelconcept. Itisthe controller that improves the tracking performance of TDC by compensating TDE error

using IMC scheme. Ithasoutstanding robustness against the friction effects in the plants, and, moreover, has simple structure which is easily applicable without whole model computation of the plants. Through the analysis and experiments, its robustness is showed when TDE error

becomeslarge bycoulomb frictionorstribeck effect.

[12] M. W. Spong and M.Vidyasagar,"Robust linear compensatordesign for nonlinear robotic control," IEEE J. of Robotics andAutomation, Vol.RA-3,No.4, pp.345-351, 1987.

[13] J. H. Jung,P.H.Chang, O.S.Kown, "A new stability analysis of time delay control for input-output linearizable plants," in Proc. Of American Control Conference, pp.4972-4979, Boston, USA, June, 2004.

[14] Sang Hoon Kang, MaolinJin, Pyung H. Chang and Eunjeong Lee,

"Nonlinear Bang-Bang Impact Control for Free Space, Impact and Constrained Motion: Multi-DOFCase," American Control Conference, Oregon, USA, June, 2005.

[15] ByoungUk Nam, "Control of High Precision Dual-Stage," M.S. thesis, KAIST, 2005

ACKNOWLEDGMENT

This research is supported by Human-friendly Welfare Robot System Engineering Research Center (HWRS-ERC) of Korea Advanced Institute of Science and Technology

(KAIST).

This work ^was supported in part by the Ministry of information & Communications, Korea, under the Information Technology Research Center (ITRC) Support Program.

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