Robust Fuzzy Control of Electrical Manipulators

Transcription

1 J Intell Robot Syst (21) 6: DOI 1.17/s y Robust Fuzzy Control of Electrcal Manpulators Mohammad Mehd Fateh Receved: 1 September 29 / Accepted: 19 Aprl 21 / Publshed onlne: 22 June 21 Sprnger Scence+Busness Meda B.V. 21 Abstract Stablty analyss for fuzzy control of robot manpulators has been a serous challengng problem n lterature. The theoretcal dffcultes are hghly ncreased due to the complexty of both manpulator dynamcs and fuzzy controller structure. Ths paper develops a novel robust fuzzy control approach for electrcal robot manpulators usng the drect method of Lyapunov. We pass analytcal dffcultes by the use of voltage control strategy n replace of torque control strategy. Then, a normalzed and decentralzed Takag Sugeno fuzzy controller s presented n a smple structure. A smple Lyapunov canddate s proposed to apply stablty analyss wthout knowng the explct dynamcs of system. Consequently, fuzzy control s analyzed and desgned as a robust nonlnear control. Roles of scalng factors, gans n consequent parts, and membershp functons n condton parts are consdered n the control desgn. The proposed control approach s appled on a Puma56 robot arm. Keywords Decentralzed control Electrcal robot manpulator Robust control Takag Sugeno fuzzy controller Stablty analyss 1 Introducton Industral robots exhbt hgh accuracy, good resoluton, and sutable repeatablty. Therefore, a smple control strategy such as ndependent jont control [1] usng an ordnary proportonal-dervatve-ntegral (PID) controller shows a good performance on ndustral robots. However, an ndustral robot s constructed wth a hgh expense to have hgh mechancal specfcatons. To save the cost, we may pay more attenton on mprovng the control of robot. In order to use a cheaper robot for performng M. M. Fateh (B) Department of Electrcal and Robotc Engneerng, Shahrood Unversty of Technology, Shahrood, Iran e-mal: mmfateh@shahroodut.ac.r

2 416 J Intell Robot Syst (21) 6: precse applcatons, the vsual feedback of end-effector n a vson control has been proposed to compensate the naccuracy nherent n a cheaply-constructed robot [2]. Snce robust control can guarantee the stablty and show a good performance [3], the robust control of cheaply-constructed robot wth uncertan knematcs and dynamcs wll provde a satsfactory performance. Many valuable robust control approaches have been developed to control robot manpulators n the jont-space [3, 4] and the task-space [5, 6]. However, robust control may nvolve the complexty of manpulator dynamcs. A great attenton has been attracted to remove ths shortage. A proper uncertanty bound parameter has been proposed to smplfy and mprove the robust control of robot manpulators [7]. The voltage control strategy [8] s superor to the torque control strategy n the robust control of electrcal manpulators [9] n terms of smplcty n desgn and performance of control. Alternatvely, fuzzy control as a model-free approach s smply desgned to control complcated systems [1]. To form fuzzy rules, an exact knowledge of model s not requred. Fuzzy controller s an ntellgent controller usng lngustc fuzzy rules to nclude nformaton from experts. Consequently, fuzzy control of robot manpulators has attracted a great deal of researches to overcome uncertanty, nonlnearty and couplng [11 14]. In many applcatons, fuzzy controllers were utlzed and treated as black-box controllers that when constructed properly by tral-and-error method, could produce satsfactory control results [15]. However, an analytcal proof should be gven to guarantee stablty and provde a desred performance. An analytcal well-defned form of fuzzy system wll mprove the analytcal aspect of fuzzy control n nonlnear control theory. The Takag Sugeno (TS) system performs well n an adaptve manner to cope wth nonlnear systems [16]. It performs lke a part-to-part lnear functon that changes smoothly from one part to another. Stablty analyss of fuzzy control to robot manpulators has been a serous challenge n lterature. The theoretcal dffcultes are hghly ncreased due to the complexty of manpulator dynamcs and fuzzy controller together [17]. To pass dffcultes n analyss and desgn, we use the voltage control strategy [8] n replace of the torque control strategy. As a result, the motor dynamcs whch s much smpler than manpulator dynamcs can be employed to decentralze and decouple the robotc system [9]. Ths paper presents stablty analyss wthout knowng the explct dynamcs of system, and smple structures are proposed to the fuzzy controller and the Lyapunov canddate. Then, a novel robust decentralzed TS fuzzy control approach s developed to electrcal robot manpulators based on the nonlnear control theory. Ths paper s organzed as follows. Secton 2 presents the decentralzed system. Secton 3 desgns the controller and presents the stablty analyss. Secton 4 llustrates smulaton results and Secton 5 concludes the paper. 2 Decentralzed System Decentralzed control s a domnant control scheme n the multvarable control because t has many advantages, such as flexblty n operaton, falure tolerance, smplfed desgn and tunng [18]. The decentralzed controller s stll adopted by majorty of modern robots n favor of ts computaton smplcty and low-cost

3 J Intell Robot Syst (21) 6: hardware setup [19]. As a result, mprovng the trackng performance of robot manpulators usng the decentralzed control s stll an attractve research. The decentralzed strategy employs a smple nput output form of fuzzy control for controllng a sngle output varable. Consequently, the number of fuzzy rules n the system s greatly reduced and thus, the computatonal complexty of algorthm s sgnfcantly smplfed. To desgn the decentralzed fuzzy controllers, the robotc system s decomposed. The torque vector s commonly used as the nput of robot manpulator for decentralzng the system [2, 21]. The torque vector of manpulator s gven by D (q) q + C (q, q) q + g (q) = T (1) where q R n s a vector of generalzed jont postons, D(q) R n n s a matrx of manpulator nerta, C (q, q) q R n s a vector of centrfugal and Corols generalzed forces, g(q) R n s a vector of gravtatonal generalzed forces and T R n s the nput generalzed forces. Dynamcs of manpulator as stated by (1) s hghly coupled such that moton of the k-th jont s not only dependent on the k-th nput [1]. Therefore, ths model s not properly suted for decentralzng robotc system. Moreover, the complexty of control desgn and analyss s hghly ncreased snce the robot dynamcs s hghly nonlnear, very large, and uncertan. Consder an electrcal robot drven by permanent magnet dc motors. The electrcal equaton of motor s n the form of u = R a I a + L a İ a + k b r 1 q k (2) where u R s the motor voltage, I a R s the motor current, and q k R s the velocty of k-th jont. Motor parameters, R a, L a k b and r are resstance, nductance, back-emf constant, and the reducton gear coeffcent, respectvely. Equaton 2 s more sutable than (1) for decentralzed control snce velocty of the k-th jont only depends on the voltage and current of the k-th motor that drves the jont. Moreover, as a man advantage, (2) s smple and free of manpulator dynamcs. 3 Control Desgn and Stablty Analyss The control approach s appled on a Puma56 robot arm, whch possesses three degrees of freedom drven by three revolute jonts. Each jont s drven by a dc motor usng the decentralzed fuzzy controller. The motor voltage s consdered as the output of fuzzy controller, and the jont poston error and ts dervatve are the two nputs of fuzzy controller. The consequent part of TS fuzzy rule s a lnear functon of nput varables [22]. The general TS fuzzy systems wth lnear rule consequent are unversal approxmators [23] and nonlnear controllers wth varable gans [24]. They can be nherently nonlnear gan schedulng controllers wth dfferent varable gans n dfferent regons of nput space [25]. Due to the generalty n the confguraton, the expressons of general fuzzy controllers cannot be mathematcally dervable. Thus, stablty analyss should be presented wthout knowng the explct structure of fuzzy controller. To make the dervaton mathematcally feasble, the components of fuzzy controllers must be specfc. All parameters n the rule consequent are adjustable and

4 418 J Intell Robot Syst (21) 6: unknown, and the number of parameters grows exponentally wth ncrease n the number of nputs. The parameters n theory provde a tunng of local control acton, possbly resultng n superor control performance. However, manual tunng of these parameters n practce could be neffectve, neffcent, napproprate, or sometmes even mpossble when the parameters are too many [15]. We desgn the decentralzed TS fuzzy controller by the use of two nputs namely trackng error and ts dervatve, respectvely. If we select three fuzzy sets for each nput, the whole control space wll be covered by nne fuzzy rules. The TS Fuzzy rules are n the form of: Ru l : If x 1 s A l 1 and x 2 s A l 2 Then yl = a l 1 x 1 + a l 2 x 2 + a l (3) where Ru l denotes the fuzzy rule for l = 1,..., 9, and the nputs of rule Ru l are x 1 and x 2. The nput vector s formed as x = [x 1 x 2 ] T U R 2 where U s the unverse of dscourse. In the l-th rule s denoted as Ru l, A l s the fuzzy set, a l s the gan n consequent part for = 1, 2, andy l s the crsp output. The output of TS fuzzy system f (x) V R by usng product nference engne, sngleton fuzzfer and center average defuzzfer s n the form of [1] f (x) = 9 2 μ A l (x ) y l l=1 =1 9 l=1 =1 2 μ A l (x ) (4) where μ A l (x ) [, 1] s the Membershp Functon (MF). An mportant contrbuton of fuzzy systems theory s to provde a systematc procedure for transformng a set of lngustc rules nto a nonlnear mappng. Consequently, the fuzzy controller defned by f (x) s a nonlnear functon. Substtutng y l from (3) nto(4) yelds a smple form for f (x) as where c j (x) for j =, 1, 2 s gven by f (x) = c 1 (x) x 1 + c 2 (x) x 2 + c (x) (5) c j (x) = 9 l=1 =1 9 2 μ A l (x ) a l j l=1 =1 2 μ A l (x ) (6) The obtaned analytcal structure of TS fuzzy controller mproves our study to develop the analyss and desgn. When x 1 s the error and x 2 s the frst tme dervatve of error, (6) presents a decentralzed PD lke TS fuzzy controller. However, we can generalze the proposed structure for other types of varable gan controllers f we select the relevant varable for x. As a result, we smply provde other types of varable gan controller, such as Proportonal (P), Proportonal Integral (PI), or Proportonal Integral Dervatve (PID) fuzzy controller. Moreover, a hgher order decentralzed controller can be obtaned f we use nputs x 1,..., x n. The varable gan c j (x) s a hgh mert of ths controller to make t adaptve n dfferent condtons. To

5 J Intell Robot Syst (21) 6: normalze the controller, we use the nput scalng factors k 1 > and k 2 >, andthe output scalng factor k o >. The nput vector s formed as where for the k-th jont z 1 and z 2 are defned as x = [ k 1 z 1 k 2 z 2 ] T (7) z 1 = q dk q k (8) z 2 = q dk q k (9) where q dk and q k are the desred and actual trajectores, respectvely. Substtutng (8)and(9) nto(7)forx 1 = k 1 z 1, x 2 = k 2 z 2,andż 1 = z 2 yelds where α = k 1 /k 2 >. Fuzzy controller by the use of scalng factors s formed as ẋ 1 = αx 2 (1) u (x) = k o (c 1 (x) x 1 + c 2 (x) x 2 + c (x)) (11) Ths general structure shows a nonlnear varable gan controller that fnds many applcatons n control. The nonlnear gan c j (x) covers the nonlnearty of controller by parameters n hand. The control purposes are smply descrbed by lngustc rules n fuzzy controller transformed to a nonlnear functon as stated by (11). c j (x) s comprsed by MFs μ A l (x ) n the rule condton parts and the gans a l j n the rule consequent parts as descrbed by (6). The MFs play a sgnfcant role n the performance of control system. The center, range and shape of MF are the most sgnfcant parameters. The operatng range of varables should be covered by the selected fuzzy sets characterzed by MFs. The MFs may be classfed accordng to have ether a lmted range or an unlmted range. For example, a trangular type covers a lmted range whle a sgmod type covers an unlmted range. In the trackng control, to cover a large unwanted error caused by dsturbances, ntal trackng error, or large scaled error, we should cover a wde range by MFs of nputs. For ths purpose, the MFs located n the left and rght sdes of nput can be sgmod type whle the MF n the center can be a Gaussan or trangular shape. When a fuzzy controller s desgned as a normalzed one, the scalng factors are used to scale the actual range of varables n the range of MFs. The archtecture of TS controller n (11) ncludes the scalng factors. Substtutng (11)nto(1) forms the closed loop system as follows: (c 1 (x) x 1 + c 2 (x) x 2 + c (x)) k o = R a I a + L a İ a + k b r 1 q k (12) Assume that the motor voltage u expressed by (2) s lmted such that R a I a + L a İ a + k b r 1 q k u max (13) where < u max s a maxmum permtted voltage for the motor. Ths assumpton s a techncal regard to protect motor aganst over voltages. The complexty of desgn and analyss has been changed to smplcty for usng the model of motor n place of model of manpulator. Here, we should know only the upper lmts for the motor voltages as nputs of robotc system. Because electrcal motors drve the electrcal manpulator, the motor voltages are the system nputs. The desred trajectory should

6 42 J Intell Robot Syst (21) 6: be planned wth regardng the maxmum permtted voltages for motors somehow each motor s so strong such that can track the desred trajectory under the permtted voltage. Moreover, the desred trajectory should be smooth such that ts dervatves up to the requred order are avalable and lmted. The drect method of Lyapunov s utlzed for asymptotc stablty analyss of TS fuzzy controller. To fnd a control law for a guaranteed stablty, a Lyapunov canddate s proposed as V (x) = x1 c 2 (x) x 1 dx 1, V () = (14) where V(x) s a postve defnte functon f c 2 (x) s postve. c 2 (x) s stated by (6)as c 2 (x) = 9 l=1 =1 9 2 μ A l (x ) a l 2 l=1 =1 2 μ A l (x ) To satsfy < c 2 (x) t s suffcent that a l 2 and l such that < 2 μ A l (x ) a l 2. Proof Assume that C 2 c 2 (x) where C 2 s a postve constant. Thus x1 C 2 x 1 dx 1 x1 =1 (15) c 2 (x) x 1 dx 1 (16) x1 We have C 2 x 1 dx 1 =.5C 2 x 2 1 and <.5C 2x 2 1 for x 1 =. Thus, (16) mples that < V(x) for x =. Snce c 2 (x) x 1 dx 1 = and c 2 (x)x 1 s lmted, V() =. Thus, V(x) s a postve defnte functon. For smplcty, the feature < c 2 (x) s used n replace of the explct form of c 2 (x). The tme dervatve of V(x) by the use of (1)scalculatedas V = c 2 (x) x 1 ẋ 1 = αc 2 (x) x 1 x 2 (17) From (12)wecanwrte c 2 (x) x 2 = c 1 (x) x 1 c (x) + ( R a I a + L a İ a + k b r 1 q k ) /ko (18) Substtutng (18)nto(17) yelds V = α ( c 1 (x) x 2 1 x ( 1c (x) + x 1 Ra I a + L a İ a + k b r 1 ) ) q k /ko (19) Snce c 1 (x) x 2 1 for c 1(x), tosatsfy V for stablty, t s requred that ( x 1 c (x) + x 1 Ra I a + L a İ a + k b r 1 ) q k /ko (2) It can be wrtten as (( x 1 Ra I a + L a İ a + k b r 1 ) q k ko c (x) ) (21) Assume that c (x) =. To establsh stablty, k o s selected as x 1 u max / (x 1 c (x)) k o (22)

7 J Intell Robot Syst (21) 6: Proof Equaton 21 can be wrtten as The use of (13) yelds x 1 ( Ra I a + L a İ a + k b r 1 q k ) ko x 1 c (x) (23) x 1 ( Ra I a + L a İ a + k b r 1 q k ) x1. R a I a + L a İ a + k b r 1 q k = x 1. u (24) To satsfy (23), t s suffcent that x 1. u k o x 1 c (x) (25) Snce < k o, to guarantee stablty < x 1 c (x). Thsmeansthatc (x) must be desgned wth the same sgn as x 1. Ths condton s smply satsfed f a l s selected wth the same sgn as x 1. Ths fact s understood by studyng the behavor of dc motor as stated by (2) where q k has a drect relaton wth u. Ths means that a postve u ncreases q k. Thus, when x 1 = k 1 (q dk q k ) >,.e. the actual poston q k s under the desred poston q kd then u should be postve to reduce x 1.Wth the same reasonng when x 1 < thenu should be negatve to reduce the sze of x 1. Therefore, the sgn u s selected the same as sgn of x 1 to reduce the sze of x 1. From (25)andx 1 c (x) >, we obtan That s x 1 u / (x 1 c (x)) k o (26) u / c (x) k o (27) To know c (x), we pay attenton on c (x) as gven by (6)as c (x) = 9 l=1 =1 9 2 μ A l (x ) a l l=1 =1 2 μ A l (x ) Snce c (x) s an average functon due to 2 μ A l (x ) 1, thus where a,mn = mn 9 { } a l and a,max = 9 l=1 =1 (28) a,mn c (x) a,max (29) max l=1 { a l }. To select a constant value, we should select a value for k o that satsfes (27) n all cases. On the other hand, a larger value for k o may provde an over voltage. Thus, to protect the motor from over voltages we should regard u u max. The maxmum permtted value for k o s then selected as k o = u max / a,max (3) where u max and a l are known n advance. If the selected value s not suffcently large to satsfy (27), we should select an stronger motor wth a larger u max that n the operatng range u / c (x) u max / a,max = k o (31)

8 422 J Intell Robot Syst (21) 6: Therefore, stablty s guaranteed under assumptons < c 1 (x),< c 2 (x),< x 1 c (x), and k o = u max / a,max. The output scalng factor plays an effectve role n stablty of closed loop system. We may nterest n the role of nput scalng factors. It can be wrtten that c (x) x = c 1 (x) x 1 + c 2 (x) x 2 (32) where c(x) = [ c 1 (x) c 2 (x) ]. Then, (5) srewrttenas c (x) x = ( ) R a I a + L a İ a + k b q k /ko c (x) (33) By the use of (7), we can wrte [ ][ ] k1 z1 x = = kz (34) k 2 z 2 [ ] k1 where k = and z = [ ] T.Hence(33)sformedas z k 1 z 2 2 c (x) kz = ( ) R a I a + L a İ a + k b q k /ko c (x) (35) Thus c (x). k. z R a I a + L a İ a + k b q k /k o + c (x) (36) Therefore, by the use of (13), the norm of error z s bounded as z < (u max /k o + c (x) ) / ( k. c (x) ) (37) Membershp functons of x 1 1 N Z P.8 Degree of membershp x 1 Fg. 1 The membershp functons for the nputs

9 J Intell Robot Syst (21) 6: Table 1 Fuzzy rules f (x) x 2 N Z P x 1 P Z.5 15x 1 + 1x N Ths mples that selectng larger values for k o, k 1 and k 2 provdes a smaller norm of error for z. Fuzzy rules n the 9 subsectons for l = 1,...,9 are desgned where the followng cases occur: Case 1 Assume that x 1 x 2 < resultng n asymptotc stablty by V < n (17). Thus, a small control effort s gven to u. Case 2 Assume that x 1 = or x 2 = resultng n Lyapunov stablty by V = n (17). Thus, a medum control effort s gven to u. Case 3 Assume that x 1 and x 2 both are postve or negatve resultng n nstablty by V > n (17). Thus, a very hgh effort s gven to u. Assumptons < c 1 (x), < c 2 (x), < x 1 c (x), andk o = u max / a,max must be satsfed, as well. In the l-th subsecton, a l s selected wth the same sgn as x 1 to satsfy < x 1 c (x). Wecanselecta l 1 and al 2 n all subsectons but l that a l 1 > and al 2 > to satsfy < c 2 (x), < x 1 c (x), respectvely. Fuzzy controller as a nonlnear functon 1.5 f(x) x x Fg. 2 The normalzed TS fuzzy controller as a nonlnear functon

10 424 J Intell Robot Syst (21) 6: The robust normalzed TS fuzzy controller s then smply desgned as follows: Choose the two nputs x 1 and x 2, for error and ts frst tme dervatve, and y or the output. a) Gve three MFs for each nput as shown n a range of [ 1 1]nFg.1. b) Defne nne fuzzy rules n the form of (3)asfollows: Rule 1 If x 1 s P and x 2 s P Then y = 1 Rule 2 If x 1 s P and x 2 s Z Then y =.75 Rule 3 If x 1 s P and x 2 s N Then y =.25 Rule 4 If x 1 s Z and x 2 s P Then y =.5 Rule 5 If x 1 s Z and x 2 s Z Then y = a 1 x 1 + a 2 x 2 Rule 6 If x 1 s Z and x 2 s N Then y =.5 Rule 7 If x 1 s N and x 2 s P Then y =.25 Rule 8 If x 1 s N and x 2 s Z Then y =.75 Rule 9 If x 1 s N and x 2 s N Then y = 1 Table 1 summarzed the fuzzy rules. In Rule 5, a 1 > and a 2 >. Rule 5 s specal snce t governs the equlbrum subsecton. The fuzzy system as a nonlnear functon of mappng from nput vector to output n the case of a 1 = 1 and a 2 = 1 s shown n Fg Smulaton The control system s smulated for trackng control of PUMA 56 Arm drven by brushed permanent magnet dc motors wth specfcatons [26] gven n Table 2. The nductances of motors are ncluded to consder a more complcated model n smulatons. Motors that drve the frst, second and thrd jonts are 4 V and 16 W [27]. Thus, the maxmum permtted voltage s gven 4 V. The smulaton model of PUMA 56 n MATLAB [28] s used n the smulatons. The desred trajectory s suffcently smooth such that all requred dervatves up to requred order s avalable as shown n Fg. 3. It starts from zero to reach a jont angle of 2rad n a perod of 4 S. The fuzzy controller desgned n Secton 3 s appled on the system whle the same motors, controllers and desred trajectores are used for the jonts. The dsturbance s nserted to the nput of each motor as a perodc pulse functon wth a perod of 2 S, ampltude of 2 V, tme delay of.7 S, and pulse wdth 3% of perod. Ths form of dsturbance s an example of any form that can be appled but t ncludes jumps to cover the complex cases. The output scalng factor k o s gven 4 as calculated from (31) to protect motors from over voltages Table 2 Parameters of permanent magnet dc motors Jont R L K b J B 1/r

11 J Intell Robot Syst (21) 6: Desred jont angle Jont angle (rad) tme (s) Fg. 3 Desred trajectory and overcome uncertantes n the operatng range of motors under the permtted voltages of motors. Smulaton 1 The control system s smulated for trackng control wth zero ntal errors. The nput scalng factors are set to k 1 = 1 and k 2 = 1. The gans of consequent part of Rule 5 are gven a 1 = 15 and a 2 = 1. The control system performs well as shownnfg.4 whle the maxmum values of errors for jonts 1 3 are , and rad, respectvely. The effects of dsturbances are represented n the left and rght sdes of Fg. 4 as small jumps on the curves of trackng errors. Snce the ntal errors are zero, the control system starts from equlbrum subsecton governed by Rule 5 and stays there as confrmed by the trackng errors. Therefore, the gans of a 1 and a 2 n ths rule, play a sgnfcant role n the performance of controller. They exhbt a behavor smlar as the proportonal and dervatve gans n an ordnary PD controller. The control system overcomes dsturbances very well as shownn Fg.4. The motor voltages behave well under the maxmum permtted value of 4V as shown n Fg. 5. The control efforts compensate the dsturbances such that the jumps on the curves respond to the jumps nserted by dsturbances otherwse the

12 426 J Intell Robot Syst (21) 6: x Performance of control system jont 1 jont 2 jont 3 5 Trackng erorr (rad) tme (s) Fg. 4 Performance of robust fuzzy control system curves are smooth. There are not any chatterng, oscllatons, and over voltages n the control efforts. Smulaton 2 Three dfferent values of 15, 3 and 6 are gven to a 1 n Rule 5 and the value of 1 s gven to a 2 for consderng the role of gan a 1 on trackng error as shown n Fg.6. Other condtons are the same as Smulaton 1. The control system behaves well n these cases, as well. The maxmum value of trackng error for jont 2 n the smulatons s reached to the value of rad, rad and rad n respondng to the values of 15, 3 and 6 gven to a 1, respectvely. Snce the control system s around the equlbrum pont, a larger a 1 provdes smaller trackng error. Smulaton 3 A very large dsturbance of 1 tmes as the value n Smulaton 1 s nserted n the control system whle other condtons are the same as Smulaton 1. The control system s robust aganst the large dsturbances as verfed by performance of trackng error and the voltages of motors shown n Fgs. 7 and 8, respectvely. It s very nterestng that the system does not go out of the equlbrum even n the case of a very large dsturbance. The control efforts behave well to compensate the

13 J Intell Robot Syst (21) 6: Control efforts n trackng control u 1 u 2 u 3 15 motor voltage (V) tme (S) Fg. 5 Voltages of motors to perform trackng control dsturbances. The maxmum value of trackng error for jont 2 s ncreased from rad n Smulaton 1 to a value of rad n Smulaton 3. Smulaton 4 The effect of ntal error s studed whle a value of 2 rad s gven as ntal errors to jonts. The nserted dsturbance s removed to pay attenton on the effect of ntal error whereas other condtons are the same as smulaton 1. Control system starts from a pont that s far away from the equlbrum; however t goes to the equlbrum well. The trackng errors approach the value of about rad whle they start from 2 rad as shown n Fg. 9. The responses behave smoothly to reduce the sze of trackng errors. The robustness of system n terms of stablty and trackng performance s presented by ths smulaton n drectng the trackng system to the equlbrum. The control efforts show jumps when startng to control the effect of nonzero ntal errors. They behave well under the maxmum permtted voltages as shown n Fg. 1. We can see the role of MFs of nputs namely P and N n the performance of control system. If we select the trangular shape n replace of the sgmod shape for the P and N MFs of nputs, then the startng pont wll not n the range of nputs. As a result, the control system wll not operate snce the control efforts wll be zero. Smulaton 5 The control system s smulated for set pont control where a desred value of 1 rad s gven to the jonts. Other condtons are the same as Smulaton 1. Set pont applcaton s used for pont-to-pont moton control of robot manpulators. In ndustry, the set pont control s a domnant approach n the process control, as well. The control system responses very well wthout over shoot and relatvely fast wth an gnorable steady state error and robust aganst dsturbances. The actual jont

14 428 J Intell Robot Syst (21) 6: x 1-3 Role of nput scalng factors a1 a2.8.7 Norm of trackng error (rad) tme (s) Fg. 6 Increasng nput scalng factors reduces the trackng error. a1 k 1 = 2, k 2 =.2, k o = 16, a2 k 1 = 1, k 2 = 1, k o = 16 poston reaches to a value of about 1 rad at tme of 4S as shown n Fg. 11. The effect of dsturbances s compensated such that the responses are not affected by them. The control efforts show jumps when startng to control the effect of hgh value of errors. They behave well to compensate the effects of dsturbances and reduce the errors under the maxmum permtted voltages as shown n Fg. 12. Smulaton 6 The role of nput scalng factors becomes more mportant for usng the trangular MFs for nputs. The nput scalng factor s employed to take the nput nto the operatng range covered by MFs otherwse the controller wll not respond to the nput. When the ntal error s very large ether n trackng control or n set pont control, the nput scalng factor s selected small. If the error s very small, then a large nput scalng factor can be selected to operate effcently. In ths smulaton, the trangular shape MFs of nputs are arranged n [ 1 1] as shown n Fg. 13. To control the system for a desred set pont of 2 rad, the gans of nput scalng factors are gven k 1 =.5 and k 2 = 1, and the gans of Rule 5 are gven a 1 = 1 and a 2 = 1. Other condtons are the same as Smulaton 5. The control system responses very

15 J Intell Robot Syst (21) 6: x Control performance under a very large dsturbance jont1 jont2 jont3 5 trackng error (rad) tme (S) Fg. 7 Control performance under a very large dsturbance 5 4 Control efforts under a very large dsturbance u 1 u 2 u 3 3 motor voltge (V) tme (S) Fg. 8 Control efforts under a very large dsturbance

16 43 J Intell Robot Syst (21) 6: jont1 jont2 jont3 trackng error (rad) tme (S) Fg. 9 Trackng performance under a large ntal error 3 u 1 2 u 2 u 3 1 voltage of motor (V) tme (S) Fg. 1 Control effort under a large ntal error

17 J Intell Robot Syst (21) 6: q d 1.8 q 1 q 2 q 3 jont angle (rad) tme (S) Fg. 11 Performance of set pont control u 1 u 2 u 3 25 motor voltge (V) tme (S) Fg. 12 Control efforts n set pont control

18 432 J Intell Robot Syst (21) 6: Trangular MFs for x 1 n Smulaton 6 N Z P 1.8 Degree of membershp x 1 Fg. 13 Trangular MFs for nputs n Smulaton Set pont control usng trangular MFs and scalng factors q d q 1 q 2 q jont angle (rad) tme (S) Fg. 14 Performance of set pont control usng trangular MFs and scalng factors

19 J Intell Robot Syst (21) 6: Control efforts n set pont control usng trangular MFs u 1 u 2 u 3 25 motor voltage (V) tme (S) Fg. 15 Control efforts n set pont control usng trangular MFs and scalng factors well wthout over shoot and relatvely fast wth an gnorable steady state error and robust aganst dsturbances. The actual jont poston reaches to a value of about 2 rad at tme of 4S as shown n Fg. 14. The control efforts behave well to compensate the effects of dsturbances and reduce the errors under the maxmum permtted voltages as shown n Fg Concluson A robust normalzed TS fuzzy controller was desgned for both trackng and set pont control of electrcal manpulators. The analyss and desgn of fuzzy control as a robust and nonlnear control approach was presented by nonlnear control theory usng the drect method of Lyapunov. The stablty was analyzed wthout knowng the explct dynamcs of system. Maxmum permtted voltages of motors are the only requred knowledge about the robotc system. The complextes n analyss and control desgn have been removed well usng the voltage-based control. Fnally, a smple algorthm was provded to desgn a robust normalzed TS fuzzy controller. References 1. Spong, M.W., Hutchnson, S., Vdyasagar, M.: Robot Modelng and Control. Wley, Hoboken (26)

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