choi oh oh chung - auto-tuning pid controller for robotic manipulators

8/8/2019 Choi Oh Oh Chung - Auto-Tuning Pid Controller for Robotic Manipulators

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Auto-tuning PID Controller for Robotic ManipulatorsYoungjin Choi,' Yonghwan Oh,' Sang Rok Oh,3 and Wan Kyun Chung4

1,2,3 Intelligent System Control Research Center,Korea Institute of Science and Technology (K IST), KOR EADepartment of Mechanical E ngineering,Pohang University of Science and Technology (POS TEC H), KOR EA

A bs t r ac tThis paper suggests an auto-tuning method of PIDtrajectory tracking controller for robotic manipulators.In general, the PID trajectory tracking controller formechanical systems shows the performance limitation.Since the control system including performance limita-tion can not have equilibrium points, we define newlythe quasi-equilibrium region as an alternative for equi-librium point. Also, the quasi-equilibrium region isused as the targe t performance of the au to-tuning PIDtrajec tory tracking controller. The auto-tuning law isderived from the direct adaptive control scheme, basedon the extended disturbance input-to-state stability.Finally, experimental resu lts show th at the control per-formance is enhanced by an auto-tuning method assist-ing the achievement of target performance.

1 In t roduct ionAlthough many anto-tuning methods for PID con-troller were proposed in [l ] ,hey were for the chemicalprocess control systems. Since most process systemsshow very slow responses with time-delay effect, theauto-tuning algorithms developed for process controlsystems can not be directly applied to mechanical sys-tems. Though the performance tuning by gain changeshas brought one's interes t with wide acceptance, therestill exist no generally applicable auto-tuning laws formechanical systems.The PID controller for mechanical systems, especiallyrobotic manipulators, has been widely used with vari-ous usages. As a matter of fact, the importance of PIDcontrol comes from the easy applicability and clear ef-fects of each proportional, integral and derivative con-trol. Inspired by the extended disturbance input-to-state stability of PID control for Lagrangian systemsincluding robotic manipulator, the inverse optimalityof PID controller was proved in [Z ] with some condi-

'Post Doctoral Fellow, KIST, yjchoiQpostech.edu'Research Engineer, KIST, oyh~amadeus. kist.re.k r3Principal Research Engineer, KIST, sroh~amadeus.kist.re.kr4Professor, POSTECH, wkchungQpostech.edu

0-7803-7896-2/03/$17.002003 EEE

tions for gains. Recently, the noticeable 'square tuning'and 'linear tuning' rules were proposed using an inverseoptimal PID controller. They were derived from theperformance prediction equation suggested in [Z]. Ac-tually, the performance tuning by the name of squarelaw was suggested for the first times in 131. In this pa,per, an auto-tuning method of an inverse optimal PIDtrajectory tracking controller will be proposed by mak-ing use of the direct adaptive control scheme based onthe extended disturbance input-to-state stability.Recently, the direct adapt ive control scheme for nonlin-ear systems was developed in [ 4 , 5 ] . The direct adaptivecontrol is different from the indirect adaptive controlin that the control parameters are estimated directlywithout intermediate calculations involving plant pa-rameter estimates. Strictly speaking, the conventionaladaptive motion (trajectory tracking) control methodsgiven in [SI can be classified as the indirect adaptivecontrol for robotic manipulator systems because theplant parameters are estimated to construct a dynami-cal model compensator. In this paper, the auto-tuningmethod of an inverse optimal PID trajectory track-ing controller will be proposed according to the directadaptive control scheme.

2 Automat i c Per formance TuningSince the PID controller has t he performance limitationfor trajectory tracking of robotic manipulator as provedin [Z], the auto-tuning method is devised so that itcan accomplish the ta rget performance of PID controlsystem. To begin with, let us obtain the state-spacedescription for trajectory tracking system model in thefollowing section.2.1 Traje ctory Tracking System ModelTh e robotic manipulator system is described by usingLagrangian equation of motion with configuration co-ordinates q = [ q l , z, . , nlT E $3" as follows:

M (q )9+c(q,q)q S(Q)+ d ( t ) = 7, (1)where M ( q ) = M T ( q ) E Pnx"s Inertia matrix,C(q , )q E L" Coriolis and centrifugal torque vector,

350 Proceedings of the American Control ConferenceDenver, Colorado June 4-6. 2003
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g(q) E P" gravitational torque vector, T E P" th econtrol input torque vector and d ( t ) unknown externaldisturbances. In fac t, the system model (1) obtainedby Lagrangian equation of motion is for the set-pointregulation control. Here, let us define the extendeddisturbance as following form:

w ( t , ,e , Se) = M ( q ) i i d +K p e +K f e )+C(q,0) Q d +K p e +K IJ e ) +g(n) +W , (2)where K p ,K I are diagonal positive constant matri-

ces, e = qd - q is the configuration error vector anddesired configurations (qdr d , d )are the function oftime , hence, the extended disturbance zu is the functionof time, configuration error, its derivative and integral,since q(= qd - e ) an d q ( = qd - e ) are the functionof time, configuration error and its derivative. If th eextended disturbance defined above is applied to theset-point regulation system model (l), then the trajec-tory tracking system model can be obtained as

M ( q ) i + C ( q , q ) s = w t , e , e , e + U , (3)where the control input a nd composite error vector aredefined by

( J >JU = -Ts = e + K p e f K ~ edt

Th e difference an d common point between the set-pointregulation system model (1) and trajectory trackingsystem model (3) were explained by Remark 1-2 in [Z ] .By using the composite error vector, the inverse opti-mal P ID controller suggested in [Z ] can be expressed asfollowing compact form:

T = ( K+ 7-21) s, (4 )with three conditions:

( C l ) K ,K p ,K I > 0 , constant diagonal matrices(CZ) K $ > ~ K I ,(C3) y > 0 .

First, let us obtain the state space description for tra-jectory tracking system model (3). If we define 3n-dimensional st ate vector as follows:edt[ 21 = [ ] EP3n,then the state space representation of trajectory track-ing system model (3) can be obtained by

X =A ( z , ) x+ B ( z , )w +B ( I , ) u ( 5 )

whereA ( I , )=

0 I0-M- ' C K I -M- ' C K p -K I -M-

and0

1r-'C K pSecond, let us investigate the characteristics of ex-tended disturbance defined a s (2). The extended dis-turbance w can be divided into th e linear parameteriza-tion part and external disturbances as following form:

w ( x , t )= Y(s , t )O d( t ) , ( 6 )where the regressor matrix Y ( m , ) can be found byseparating a real constant parameter vector O from

Y ( z ,) O = M(q)(i jd K p e + K l e )+C(q,@ ) ( q d +K pe + K I e ) + g ( 9 ) ,

and t he real constant parameters consist of masses andmoments of Inert ia of each link. Using the extendeddisturbance of ( 6 ) , the state-space representation of ( 5 )can be modified toX =A ( z , t ) z+ B ( z ,)Y(z , t )O B ( z , ) d + B ( z ,) u .Here, the regressor Y ( z , t ) s not a zero matrix atx = 0 except the case of set-point regulation controland gravity free motion. In othe r words, if g ( q ) # 0or qd # 0, d # 0 , then Y ( x , ) # 0 at I = 0. Hence,z = 0 is not an equilibrium point of (7 ) even whend ( t ) = 0 , because one term B ( 0 , )Y(O, ) O has thetime-varying characteristics according to desired tra-jectories. Actually, since the equilibrium point can notbe found for the system (7), we will introduce the con-cept of quasi-equilibrium region in the following sec-tion.2.2 Quasi-Equilibrium RegionTo begin with, we ss u m e tha t there exist no externaldisturbances for system model (7), namely d = 0 . Ifthe inverse optimal PID controller (4 ) described by asta te vector as following form:

(7)

U = ( K + y - 2 1 ) [ K ~ , K p , I ] ~is applied to (7), hen the closed-loop control system isobtained as follows:

X = A,($, )a :+ B ( m , t ) Y ( x , ) O (8)where

A, = A - B K + Y21) K I ,p , ] .Proceedingsof lhe American Control ConferenceDenver, Colorado June 4-6,2w351


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Since above closed-loop system has no equilibriumpoints as explained in previous section, we will de-fine the quasi-equilibrium region and find it for aboveclosed-loop system in following Theorem.Tlieorem 1 If the quasi-equilibrium region is defined asthe int erio r region of ball with the largest radius am ongstate vectors satisfying x = 0 , then it is obtained asfol lowing form :

where x.(t) means the state vector satisfying x = 0 i n(8 ) and i ts Euclidian norm is as follows:Iz.(t)l = IK;' [K+ ~ ~ 1 ] - ~ Y , ( t ) O l(9)

an d Ye ( t ) e= M(qd)@d+ C ( q d ,Qd )Qd f g ( q d ) .

xel is determined as follows:

Third, the quasi-equilibrium region is obtained by itsdefinition as follows:

where Iz.(t)i = IK,' [ K + y - 2 1 ] - 1 Y , ( t ) O I . 0The size of quasi-equilibrium regioii is inversely propor-tional to the integral gain K I as shown in an equation(9) in Theorem 1. Also, lsrgc K and small y make thequasi-equilibrium region small. If we are to approachthe quasi-equilibrium region to the point x. = 0 usingan inverse optimal P ID controller, irrespective of theconstant parameter vector 0 and desired configurations(qd,Q d , q d ) , hen one of three conditions should be sat-isfied: the one is that K I gain matrix goes to infinity,another is that &gain y to zero and the other is thatthe gain K to infinity. This explains indirectly why aninverse optimal PID controller for robotic manipulatorscan not bring the global asymptotic stability(GAS).In fa ct, the quasi-equilibrium region of Theorem 1 hasvery close relation with performance limitation of PIDcontroller. Ailother expression for quasi-equilibrium re-gion will be suggested using n-dimensional compositeerror in following Rcmark.

R em a rk 1 L e t s = e + K p e + K I J e d t . I fw em ul ti p ly[ K I , p , I ] b y I, in Theorem 1:

As.(t) = [ K I , K P , ~ I G ( ~ )= [K+ y - 2 1 ] - ' Y , ( t ) t l , ( 1 1 )

then the quasi-equilibrium region expressed b y the com-posite erro r vector i s obtained as follows:

In above Remark, t he quasi-equilibrium region was d ofined using the composite error as shown in Figure 1 .Its size is dependent upon the gain K rndtrix, &gainy and the inverse dynamics Y , ( t ) O according to desiredconfigurations. Actually, the quasi-equilibrium regioncan be used as a criterion for target performance cho-sen by user. Also, it indirectly proves the existenceof gains which can achieve the targe t performance. Infollowing section, we will propose an automatic per-formance tuning m ethod of PI D controller assisting toaccomplish th e target performance.Proceedings of the American Control ConferenceDenver. Colorado June 4-6,200352


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Quasi-Eouiiibrium Ts 2

Figure 1: Quasi-equilibrium region

2.3 Auto- tuning LawSince the quasi-equilibrium region is determined bythe size of gains of PID controller, we shouldknow the inverse dynamics Y.(t)t9depending on de-sired configurations(qd, qdr d) and dynamic parame-ter vecto r(@) o calculate the quasi-equilibrium region.However, it is difficult to exactly identify the dynamicparameters of a general robotic manipulator. Actually,if they are known exactly, then the model-based con-troller using inverse dynamics will show better perfor-mance than a PID controller. Hence, we use the con-cept of target performance instead of quasi-equilibriumregion. If we determine the target performance, thenthe size of quasi-equilibrium region should be adjustedto achieve the target performance by using the auto-tuning law for gains. Here, we choose the gain matrixK in (11) as an auto-tuning parameter. The auto-tuning law is derived from the direct adaptive controlscheme and ISS characteristics of trajectory trackingsystem in following Theorem.

Theorem 2 Assume that there exists the smallest con-stant diagonal gain matrix Kn > 0 of PID controller(4 ) guaranteeing the target performance (Cl) as follows:

If the auto-tuning inverse optimal PID controller:

using the auto-tuning law as following f o rm:

is applied to the trajectory tracking system (3 ) whenK n > K ( t ) , hen the closed-loop control system is ex-tended disturbance in put-to-state stable(lSS), where s iis i - th element of composite error vector s,2; and r,are i-t h diagonal elements of the diagonal time-uaryinggain matr ix g(t)> 0 and the update gain matrixr > 0, respectively.

Proof.ing form:

First, we take Lyapunov function as follow-

A 12( s ,K , )= -sTMs +1- tT [(??(t)-K n ) r-' (??(t) - K O ) ]

where tr[.]means the trace of given matrix. If the autotuning inverse optimal PID controller (12) is applied tothe trajectory tracking system model (3) , then we canget the time derivative of Lyapunov function along thesolution trajec tory of closed-loop system as follows:

Here, if the following matrix trace property is appliedto above equation:

then above time derivative of Lyapunov function is ar-ranged as follows:

v = -2 (Kn+ ? I ) s - r'Jis -m (22 Y2-lW12Z + T [ ( g ( t )K O ) r - ' k ( t ) SS')] , (14)2Also, if the diagonal elements of ( I - ' k ( t ) - s arezeros, then the t race term of (14) becomes zero because(g(t) K n ) is a diagonal matrix. In other words,the auto-tuning law (13) is derived from the followingrelation:

d k i-t = r&),tr [( g ( t ) K n ) ( I- ' f?(t) - ss')] = 0

ifthen

for i = 1 , . . , n

Therefore, if the auto-tuning inverse optimal PID con-troller (12) is applied to trajectory tracking system,then we can get the following relation from (14):

Since the right hand side of above inequality (15) isunbounded functions for s and w , respectively, hence,

Proceedings of the AmericanControlConference353 Denver, Colorado June 4-6.2003


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the trajectory tracking system with an auto-tuning in-verse optimal PID controller is extended disturbanceinput-to-state stable(1SS). 0Actually, when PI D controller does not achieve the tar-get performance, namely Kn >g(t) ,he auto-tuninglaw suggested in Theorem 2 will help to achieve the tar -get performance. On the contrary, when K(t) > Kn,the target performance can he accomplished without anauto-tuning law, namely K = 0 for K 2 Kn. Also,since the time derivative of Lyapunov function (14) can :he arranged as follows:

1

th e ISS is also assured for 2 K n . As a matterof fac t, t,he controller (12) without an auto-tuning law(13) is equal to the conventional inverse optimal PIDcontroller. In following section, we will discuss aboutthe criterion on whether the auto-tuning law is neces-sary to achieve the target performance or not.2.4 Cri te r ion for Auto-tuningTo apply an auto-tuning law, we should exactly knowthe gain matrix Kn guaranteeing target performance(n).However, we can not know the matrix Kn till theexperimental result satisfies the target performance asfollows:

hut we can calculate the size of composite error Is(r)Iaccording to time progress. Moreover, since the auto-tuning law was composed of the decentralized type inTheorem 2 , we suggest the decentralized criterion forauto-tuning as follows:

where n is the number of configuration coordinates.As soon as the error arrives at the tuning region of(16), the anto-tuning law should he implemented toassist the achievement of tar get performance. Hence,the target performance is somewhat different from thenon-tuning region as shown in Figure 2.Though we do not use the constant gain matrix K nwhich offers the target performance, an auto-tuning in-verse optimal PID controller (12) results in the effect ofgain K n for lsil > LL as shown in an equation (15).On the contrary, if the composite error stays in non-tuning region of Figure 2 , namely l s i l 5 &, hen theauto-tuning process is stopped. For this case, we expectth at th e gain updat ed by an auto-tuning law (13)will be larger than the matr ix K n which brings the ta r-get performance. As a matter of fact, the'auto-tuning

Jz;;

Figure 2: Target performance and non-tuning region

F igure 3: 3-DOF Robotic Manipulator

law has the property of a nonlinear damping. Strictlyspeaking, the first term of anta-tuning PID controller(12) means the nonlinear damping which helps to sta-bilize the control system against extended disturbanceand the second term is a linear controller.

3 Exper imen ta l resultsTo show th e effectiveness of an auto-tuning inverse op-timal PID controller, we utilized three link robot ma-nipulator as shown in Figure 3. The desired configura-tion profiles of Figure 4.(a) are obtained hy solving in-verse kinematics for 6 line trajectories. Also, the giventrajectories require the fast motion (maximum velocity% 3 r a d / s ) of robotic manipulator as shown in Figure4.(b). First, we determine the static gains of auto-tuning inverse optimal PID controller as K p = 201,Kr = 1001 and y = 0.5 satisfying the design guide-lines ( Cl), (C2), (C3) in section 2. Then the controllerhas the following form: for i = 1,2,3,

7, = (f?,(t) + 4 ) S ,s, = e, +20e , + l o 0 e,dtJ

where ri is i-th element of input torque vector r ,K;(O)= 0.1 and n = 3. Second, since the composite er-ror is approximately proportional to the configurationProceedingsof the American Control Conference354 Denver. Colorado June 4-6. 2003


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(a ) Desired configuration (h) Desired configurationvelocity : Qd

TimeIsl Iimelsl

(a) Composite error: s (h) Auto-tuned gains:

(c) Configuration error : (d) Configuration velocitye error : eFigure 4: Performance of auto-tuning PID controller

error with proportional constant Kp. the target per-formance can be approximately determined as follows:

where lsijt and le,Jt are the target composite error andconfiguration error, respectively. For instance, if we areto obtain the performance of le,/t < 0.02 rad for eachdriving axis, then the target performance should bedetermined as R = 1.0 by (17). Also, the update gainr = 1000 was used for the experiment. Figure 4.(c )and (d) show experimental results such as the config-uration error and its velocity error. In figures, we cansee that the errors are large at initial time and errorsare reduced till target performance can be achieved byan automatic gain tuning. As a matter of fact, theautomatic gain tunings are executed at the exterior oftwo dotted lines of Figure 5.(a). After the auto-tuningic finished, the tuned gains arrived at 1?1 = 136.98,K z = 65.60 and K 3 = 6.83 as we can see in Figure5.(b). To see the tuning process in detail, the horizon-tal ranges of 0 - 1 second of Figure 5.(a) and (b ) areenlarged as shown in that figure (e) and (d). The auto-tuning of first axis is started at 0.11 second and endedat 0.26 second because the error goes over the dottedline ( l /& = 0.408) for the first time as shown in Fig-ure 5.(c). Also, the error of second axis goes over thedotted linc from 0.13 to 0.24 second. Finally, since theerror of third axis goes over the dot ted line downwardtwice, the auto-tuning is implemented twice as shownin Figure 5.(d). The experimental result of Figure 5. (a)shows tha t the targe t performance is achieved after 0.6second when the auto-tuning is finished.

(c) Composite error (s) (d) Auto-tuned gains (E)for 0 - s for 0 - 1sFigure 5 : Composite error and auto-tuned gains

4 Concluding RemarksIn this paper, the quasi-equilibrium region was definedto guarantee the existence of controller gain achievingtarget performance. Also, we proposed the auto-tuninginverse optimal PID controller assisting the achieve-ment of target performance. Finally, we showed thevalidity of auto-tuning law through the experiment.

References[ l] c. C. Yu, Autotuning of PID COntTolkTS: RelayFeedback Approach, Springer, 1999.[2] Y . Choi, W . K. Chung, and I. H. Suh, Perfor-mance and H m ptimality of PID trajectory trackingcontroller for Lagrangian systems, I E E E Trans. onRobotics and Automation, vol. 17, no. 6, pp. 857-869,Dec. 2001.131 J. Park and W . K . Chung, Design of a robustH, PID control for industrial manipulators, naris.ASME J . of Dyn. Syst., Meas. and Contr., pp. 803-812, 2000.[4 ] W . M. Haddad and T. Hayakawa, Direct adap-tive control for nonlinear systems with bounded energyC2 isturbances disturbances, I E E E Conf. on Decz-sion and Control, pp. 2419-2423, 2000.[5] V. Chellaboina, W. M. Haddad, andT. Hayakawa, Direct adaptive control for non-linear matrix second-order dynamical systems withstate-dependent uncertainty, PTOC.of the AmencanControl Conference, pp. 4247-4252, 2001.(61 R. Ortega and M . W . Spang, Adaptive motioncontrol of rigid robots: A tutorial, I E E E Conf. onDecision and Control, pp. 1575-1584, 1988.

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choi oh oh chung - auto-tuning pid controller for robotic manipulators

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