Control of a Stewart Platform used in Biomechanical Systems

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1 Ffth LACCE nternatonal Latn Amercan an Carbbean Conference for Engneerng an Technology LACCE 7 Develong Entrereneural Engneers for the Sustanable Growth of Latn Amerca an the Carbbean: Eucaton, nnovaton, Technology an Practce 9 May June 7, Tamco, Méxco. Control of a Stewart Platform use n Bomechancal Systems Fernano Serrano, Flora nternaconal Unversty, Mam, Flora, USA, fserr@fu.eu Amaury A. Caballero Flora nternaconal Unversty, Mam, Flora, USA, caballer@fu.eu Kang K. Yen Flora nternaconal Unversty, Mam, Flora, USA, yenk@fu.eu Tomas Brezna Brno Unversty of Technology, Brno, Czech Reublc, brezna@fme.vutbr.cz ABSTRACT The aer s focuse on the analyss of the Stewart latform to be use n the eveloment of a evce for the etermnaton of mechancal roertes of materals use for substtutng snal segments of human boes. At the resent tme, the Stewart latform s wely use an ts oularty s ue manly to the followng facts, n comarson wth seral mechansms: hgher stffness, better ynamc roertes of the mechansm, hgher accuracy of the mechansm, recse an easy ostonng. Smulatons are conucte for the etermnaton of an accetable control system base on fuzzy logc. A controller escrton s nclue. Keywors: Stewart Platform, Bomechancs, Fuzzy Controllers. NTRODUCTON An mortant characterstc resente n the soluton of bomechancal roblems for clncal ractce s ther comlexty, whch requres mutual mxture of comutatonal an exermental moelng. Whereas n the comutatonal art, n most cases, commercal software can be use wth success, the exermental art requres at least the eveloment, esgn an manufacturng of fxators, an frequently, of the whole exermental evce, nclung the control system on whch secfc emans are often ut. The concet of arallel knematcs, whch s calle Stewart latform corresons to the evce of such esgn. n case that the lower late s frmly connecte to the base, the uer late s able to move wth sx egrees of freeom. At the resent tme, the Stewart latform s wely use an ts oularty s ue manly to the followng facts, n comarson wth seral mechansms: hgher stffness, better ynamc roertes of the mechansm, hgher accuracy of the mechansm, recse an easy ostonng, an we range of movements. The objectve of the aer s to analyze the Stewart latform an to conuct smulatons to evaluate the obtane results wth a fuzzy logc controller. n orer to conuct the analyss, a fuzzy logc controller has been mlemente. For the mlementaton, the followng stes have been followe: analyss of the cnematc moel of the latform Tamco, Méxco May 9-June, 7 5 th Latn Amercan an Carbbean Conference for Engneerng an Technology E.-

Ffth LACCE nternatonal Latn Amercan an Carbbean Conference for Engneerng an Technology LACCE 7 Develong Entrereneural Engneers for the Sustanable Growth of Latn Amerca an the Carbbean: Eucaton,

2 Ffth LACCE nternatonal Latn Amercan an Carbbean Conference for Engneerng an Technology LACCE 7 Develong Entrereneural Engneers for the Sustanable Growth of Latn Amerca an the Carbbean: Eucaton, nnovaton, Technology an Practce 9 May June 7, Tamco, Méxco. analyss of the ynamc moel efnton of the controller s structure esgn of the membersh functons esgn of the rules for the fuzzy controller system smulaton The followng restrctons were observe for the moel mlementaton: Each nut s relate only to ts resectve outut The nuctve reasonng metho wll be use for otmzng the membersh functons. CNEMATC MODEL The use of the Stewart Platform s unversal. n bomechancs t has been use n exermental backbone moellng an n exermental moellng of bomechancal roblems of large human jonts n Fgure, t s shown the 6 egrees of movement generate by sx extensble legs an the statc lower late consere nvually wth the uer late whch generate all the translatons an rotatons on the axs. n orer to erve the cnematc moel of the Stewart latform, t s use a smle moel consstng of a base an a latform whch are the center of the latform an the other sx coornate systems are locate n the base of each extensble leg. The cnematc moel for ths case has been resente n Brezna, T.; Efe, O.. Fgure : Scheme of Devce for Testng Snal Elements. 3. DYNAMC MODEL STATE SPACE MODEL To generate each movement t s necessary to use an actuator or motor connecte to the uer late of the latform, havng fxe the lower latform. Ths ynamc moel was eveloe frst usng the Newton-Euler Dynamcal Equatons an then converte nto a MMO state sace moel Shaowen Fu an Yu Yao. Ths moel s mlemente n MATLAB-SMULNK. The analyss has been base on the Newton-Euler Dynamc Equatons, usng the vector equatons of equlbrum of torques an forces nstea of matrx oeraton. The followng set of equatons for the Stewart latform s obtane: z z m Fzz t k y y m Fzy t k Tamco, Méxco May 9-June, 7 5 th Latn Amercan an Carbbean Conference for Engneerng an Technology E.-

3 Ffth LACCE nternatonal Latn Amercan an Carbbean Conference for Engneerng an Technology LACCE 7 Develong Entrereneural Engneers for the Sustanable Growth of Latn Amerca an the Carbbean: Eucaton, nnovaton, Technology an Practce 9 May June 7, Tamco, Méxco. Tamco, Méxco May 9-June, 7 5 th Latn Amercan an Carbbean Conference for Engneerng an Technology E.- 3 x k t F x m zx t M m x y zz θ φ ψ t M m z x zy ψ θ φ t M m y z zx ψ φ θ where F zx, F zy,f zz, M zx, M zy, Mzz are the forces an torques aroun each axs; m s the mass of the uer late; z, y, x are the resectve nerta values an θ,φ, ψ are the resectve angles to x, y, an z axs. State sace equatons were eveloe for the smulaton: The moel has been lnearze usng the Jacoban matrx, ue to the nonlnear characterstcs of the ynamc equatons t U M F V J A A + Y, 3 Where A 6 x 6 zero matrx entty matrx J nerta matrx that s shown next, ψ φ θ z y x ψ φ θ z y x

4 J Ffth LACCE nternatonal Latn Amercan an Carbbean Conference for Engneerng an Technology LACCE 7 Develong Entrereneural Engneers for the Sustanable Growth of Latn Amerca an the Carbbean: Eucaton, nnovaton, Technology an Practce 9 May June 7, Tamco, Méxco. -k/m k/m k/m S S 3 S S S S y Where V 3 zero vector F 3 3 nut forces vector M 3 3 nut torque vector S z y S x z S3 y x For the smulaton a snusoal nut Ut has been use. The followng control requrement have been establshe: Basc movements x,y,z oston n ±5 mm range accuracy, mm rotaton n x,y,z axes n range ± accuracy,5 loang Fx,y,z N accuracy N, Mx,y,z ± Nm accuracy,5 Nm 4. DESGN OF THE FUZZY CONTROLLER The selecte controller structure s shown n the block agram n Fgure [Bng Chen an aong Lu]. The esgn of the membersh functons has been accomlshe usng ata generate from the ynamc moel referre before. From revous ractcal stues, t has been foun that the ranom forces le n the range of - N an N an the torque n the range of -5 N.m an 5 N.m aroxmately, the obtane values were comare for the translatons an rotatons on each axs, usng the forces an torques ale at the nut of the system. Legs Trajectores + - Fuzzy Controller Devce Stewart Platform Rotatonal an ostonal sensors Fgure : Controller Block Dagram. Tamco, Méxco May 9-June, 7 5 th Latn Amercan an Carbbean Conference for Engneerng an Technology E.- 4

5 Ffth LACCE nternatonal Latn Amercan an Carbbean Conference for Engneerng an Technology LACCE 7 Develong Entrereneural Engneers for the Sustanable Growth of Latn Amerca an the Carbbean: Eucaton, nnovaton, Technology an Practce 9 May June 7, Tamco, Méxco. The range of each membersh functon was selecte as er the results gven by the software, usng the nuctve reasonng metho. Ross, T.J.; Coron O.. The selecte nuts for the fuzzy system are: Translaton errors : 3 varables n the, Y, an Z resectvely ex,ey,ez Rotaton errors 3 varables, veloctes 3varables an angular velocty errors 3 varables The 6 oututs of the fuzzy system are: Forces 3 varables Fx,Fy,Fz Torques 3 varables Tx, Ty, Tz For the esgn of the membersh functons stes were ale to etermne the range of the membersh functons: Clusterng nuctve reasonng Algorthm A clusterng algorthm s neee as a re rocessng ste before fnng the range of each membersh functon, because t gves an ea on how the ata s sort, then the nuctve reasonng metho s ale n orer to fn the overlang of each membersh functon. K-mean algorthm metho has been use for sortng the ata an formng the clusters. The metho s base on the arttonng of a gven set of ata onts nto a number of stnct grous, calle clusters. The artton s mae lookng for the maxmum smlarty of onts nto clusters, usng some global measure. The k-means algorthm attemts to fn the cluster centers c,...,c k n whch the norm D of each ata ont x s mnmze to ts nearest cluster center c k Chng-Hung Wang et al. D n mn x, c k k... k The algorthm conssts of the followng stes:. ntalze K center locatons c,...,c k.. Assgn each x to ts nearest cluster center. 3. Uate each cluster center c k as the mean of all x have been assgne as closest to t as ossble. 4. Calculate D 5. f the value of D converge, then return c,...,c k.; else go to ste. t s mortant to notce that fferent clusterng algorthms may be use: Har C means, Fuzzy Har C means etc. Ths s just a rerocessng ste that wll be use later n the arttonng rocess of the membersh functons. As an examle, the clusters formaton of the varable ex error n the translaton varable x, whch s one of the nut varables of the fuzzy controller, can be resente. Fgure 3, shows ths examle, usng the k mean algorthm for the oston error n the x coornate axs. Ths lot was mae from 8 samles out of Tamco, Méxco May 9-June, 7 5 th Latn Amercan an Carbbean Conference for Engneerng an Technology E.- 5

6 Ffth LACCE nternatonal Latn Amercan an Carbbean Conference for Engneerng an Technology LACCE 7 Develong Entrereneural Engneers for the Sustanable Growth of Latn Amerca an the Carbbean: Eucaton, nnovaton, Technology an Practce 9 May June 7, Tamco, Méxco. 3 taken from the ynamcal moel for a goo vsualzaton of the clusters. These clusters were use to esgn the three membersh functons for the nut varable ex. As er the ata locaton, there can be reresente three classes: negatve, zero, an ostve. 3 Clusters for the varable ex Poston error n x Cluster Cluster cluster Error mm Fgure 3. Clusters for the Varable ex n orer to fn the bounares of each membersh functon an the center of each cluster, the metho of nuctve reasonng s ale. The nuctve reasonng metho s use for the segmentaton of attrbutes for each fuzzy set an the scrmnant ower of each attrbute s use n a teratve metho calle chotomzer, the artton an scrmnant ower of an attrbute are etermne by a measurement calle entroy. The followng equatons are use n the nuctve reasonng metho Castro J. L. an J. M. Zurta nk x + k x ; n x + nk x + qk x ; q x x ; n x + x n x n S S q k k x ln x q xln q x k Proorton of samles of k class to the rght of the artton q k Proorton of samles of k class to the left of the artton Proorton of samles to the rght of the artton q Proorton of samles to the left of the artton. n k Number of samles of k. N Number of samle n the whole artton.. Tamco, Méxco, 7 5 th Latn Amercan an Carbbean Conference for Engneerng an Technology E.- 6 May 9-June

7 Ffth LACCE nternatonal Latn Amercan an Carbbean Conference for Engneerng an Technology LACCE 7 Develong Entrereneural Engneers for the Sustanable Growth of Latn Amerca an the Carbbean: Eucaton, nnovaton, Technology an Practce 9 May June 7, Tamco, Méxco. S Entroy to the rght of the artton. S q Entroy to the left of the artton. A comlete mang of the unverse of scourse an the ont where the mnmum entroy s obtane wll be consere as the frst artton r. Reeatng ths rocess to the left an rght of ths ont r k arttons are obtane, n whch the number of membersh functons s: n µ r k n µ number of membersh functons. The obtane membersh functons resultng from the artton of the varable error n ex are shown n Fgure 4. zero.8 Degree of membersh ex Fgure 4. Resultng Membersh Functon for the ex varable 6. RESULTS OBTANED WTH THE SMULATON The controllers smulatons were eveloe n SMULNK. As can be seen from Fgures 5 an 6, the errors of the tme resonse for the fuzzy controller reach the stablty n.5 sec aroxmately. A PD controller has been esgne n orer to comare ts resonse wth that of the fuzzy controller. The control sgnal nut for the ynamc system are the sx legs forces generate by the actuators, an the outut sgnals are the angular an lnear ostons. The gans for the PD controller K, K an K were selecte by tral an error. K 7; K ; K 4 Tamco, Méxco, 7 5 th Latn Amercan an Carbbean Conference for Engneerng an Technology E.- 7 May 9-June

8 Ffth LACCE nternatonal Latn Amercan an Carbbean Conference for Engneerng an Technology LACCE 7 Develong Entrereneural Engneers for the Sustanable Growth of Latn Amerca an the Carbbean: Eucaton, nnovaton, Technology an Practce 9 May June 7, Tamco, Méxco. The stablty wth the PD controller s obtane n.5 sec. aroxmately. For both t s esrable to lower ths tme. n a real alcaton, the fuzzy logc controller uses less harware than a PD controller, because t s not necessary to use amlfcaton to generate the feeback s gan n orer to reuce the steay state error. Fuzzy control systems are a goo oton n ths tye of alcaton, ue to the nonlnear characterstcs of the Stewart latform even f t s necessary to conser other kn of nonlnear characterstcs such as corols, gravty effects an centrfugal forces. The tme resonse of the PD controller can be mrove, but f the gans K, K an K are ncrease, the system coul be unstable or saturate. Sometmes the PD controller s not enough to get a accetable tme resonse of the system, ue to ntrnsc vbratons. Usually t becomes necessary to use a comensator such as a H controller n orer to reuce vbraton effects [Se-Han Lee, et al., 3]. Table shows the settlng tme for each controller. From ths table t can be seen that, uner the esgn contons, the fuzzy controller settlng tme s lower than the PD settlng tme. Table. Controllers Settlng tme. Fuzzy Controller PD Controller ttlng Tme s 73 5 ttlng Tme s 7 5 ttlng Tme s 7 5 Fgure 5: Results Obtane for the Fuzzy Contoller. Tamco, Méxco May 9-June, 7 5 th Latn Amercan an Carbbean Conference for Engneerng an Technology E.- 8

9 Ffth LACCE nternatonal Latn Amercan an Carbbean Conference for Engneerng an Technology LACCE 7 Develong Entrereneural Engneers for the Sustanable Growth of Latn Amerca an the Carbbean: Eucaton, nnovaton, Technology an Practce 9 May June 7, Tamco, Méxco. Fgure 6. Results Obtane wth the PD Controller. 7. CONCLUSONS An analyss of the Stewart latform has been eveloe. These latforms resent avantages n comarson to other mechancal structures use n the eveloment of bomechancal evces ue to ther flexblty, easy an recse ostonng, an we range of movement. A fuzzy an a PD controller are resente as goo alternatves for the control system ue to the nonlnear behavor of the controlle evce. The obtane results show the avantages of the fuzzy logc controller, because the PD controller yel a bgger settlng tme n comarson wth the fuzzy logc controller even wth hgh gan constants also, t s easer to mlement the fuzzy controller. REFERENCES Brezna, T. 5. Devce for Exermental Moelng of Proertes of Bomechancal Systems. Smulaton Moelng of Mechatronc Systems, Brno Unversty of Technology, Faculty of Mechancal Engneerng,, Efe, O., O. Kaynak.. A comaratve stuy of soft-comutng methoologes n entfcaton of robotc manulators. Robotcs an Autonomous Systems, Vol. 3, ssue 3,.-3 Shaowen Fu, Yu Yao. 6. Non-lnear robust control wth artal nverse ynamc comensaton for a Stewart latform manulator. nt. J. Moellng, entfcaton an Control, Vol., No.. Ross, T.J. 4. Fuzzy logc wth engneerng Alcatons.J. Wley an sons Lt. Corón O., F. Gome, F. Herrera, F. Hoffmann, L. Magalena. 4. Ten years of genetc fuzzy systems: current framework an new trens. Fuzzy Sets an Systems, Vol. 4, ssue,,.5-3. Chng-Hung Wang, et al. 999 A fuzzy nuctve learnng strategy for moular rules Fuzzy Sets an Systems, Volume 3, ssue, Arl 999, Pages 9-5. Tamco, Méxco May 9-June, 7 5 th Latn Amercan an Carbbean Conference for Engneerng an Technology E.- 9

10 Ffth LACCE nternatonal Latn Amercan an Carbbean Conference for Engneerng an Technology LACCE 7 Develong Entrereneural Engneers for the Sustanable Growth of Latn Amerca an the Carbbean: Eucaton, nnovaton, Technology an Practce 9 May June 7, Tamco, Méxco. Castro J. L. an J. M. Zurta. An nuctve learnng algorthm n fuzzy systems, Fuzzy Sets an Systems, Volume 89, ssue, 6 July 997, Pages 93-3 Authorzaton an Dsclamer Authors authorze LACCE to ublsh the aer n the conference roceengs. Nether LACCE nor the etors are resonsble ether for the content or for the mlcatons of what s exresse n the aer. Tamco, Méxco May 9-June, 7 5 th Latn Amercan an Carbbean Conference for Engneerng an Technology E.-

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