Position Coordination of a Linear Teleoperation System with Constant Time Delay

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1 Poo Coordao of a Lear eleoperao Syem wh Coa me Delay Kamra Raz, S. MJ. Yazdapaah, Saeed Shry Ghdary Abrac A lear me vara coroller deg for a blaeral eleoperao of a par of N-DO lear roboc yem uder coa me delay preeed. h framework ue poo velocy gal ogeher wh he force gal o compeae huma/evrome durbace acheve poo coordao boh free moo coac ak. I h approach, by ug Pareval dey pavy aaly doe he frequecy doma o reder eleoperaor pave. I pafyg poo velocy feedback erm o eed o coder he dyamc of he roboc yem a log a hey afy Euler-Lagraga dyamc yem equao[]. h fac mplfe he aaly deg roue. owever how ha whe force feedback appled, gag advaage of coroller robo dyamc decouplg o loger avalable. Smulao performed o llurae he propoed algorhm clarfy ome po he deg procedure. I. INRODUCION A eleroboc yem amed a raferrg a ee of remoe ak o he operaor a he feel f he performg ha ak drecly. o do h, he eleoperaor, cog of maer lave robo, her corol, he commucao bewee hem (ee g.), hould be rapare,.e. ha o rafer he mechacal mpedace o he operaor wh he mmum doro he frequecy of ere []. Alo, h goal may be erpreed a he equaly bewee wo force gal,, wo poo gal, q q. So, he mo mple cae, - DO maer lave robo, we deal wh a mulvarable corol problem. Moreover, here are wo ucera ubyem amely huma evrome erferg he dyamc propere of he overall yem. oruaely, huma operaor proved o be pave workg codo aumpo of couplg wh a pave evrome make he corolled yem applcable o a large cla of ak. A corol yem o be codered afe hould be robu o hee ubyem. I ome work [5 8], huma he evrome are modeled by a lear ma-damperprg mpedace wh ukow ga bu belog o a cera boud. Some oher work [6] preume evrome huma operaor pave make ue of Llewelly crera or fy orm of he caerg marx of K.Raz (emal: k.raz@ece.u.ac.r) SMJ. Yazdapaah (emal: yazda@u.ac.r) are wh he Corol Iellge Proceg Ceer of Excellece (CIPCE), Uvery of ehra, Ira. S. Sh. Ghdary (emal: hry@au.ac.r) wh he deparme of compuer egeerg formao echology, Amrkabr uvery of echology, ehra, Ira. he eleoperaor o guaraee he ably of he yem. hee approache are applcable he cae of lear yem. Poo coordao a mpora cocer degg uch yem. I caerg-baed approache exra corol loop hould be added o eure he poo rackg becaue of he lack of explc poo feedback []. Alhough a real world applcao of a eleoperao yem co of mul-do roboc yem, may corbug work make ue of a -DO lear model [, 6, ] o focu he dcuo o he mpora ue a eleoperao yem raher ha mulvarable olear corol. robu corol approache have a advaage ha hey are ealy applcable o mul-do cae [6,,, 5]. Ulke approache, ome oher work chooe a pre-pecfed corol rucure adu he parameer o oba pavy of he eleoperaor performace [,, ]. A a cae po, Lee e al [], frly, rafer he problem of he eleoperao of wo -DO roboc yem o corol of wo decoupled -DO yem amely??. he fr yem corolled wh a PD coroller o acheve poo coordao of he overall yem?? o hape he dyamc of he overall yem by ue of mpedace corol approache. her work geeralzed [] for a Euler-Lagraga olear yem. I [], hey apply h approach o he cae of me-delayed eleoperao wh olear dyamc. hey ue u poo formao he feedback loop. So, poo coordao ca o be acheved he preece of huma evrome durbace force. Meoed work are mlar ha pave mplemeao of he coroller doe for he corol erm dvdually. Clearly, h would mplfy he corol deg ak. h paper he le of hee work. We ag a pre-pecfed corol rucure e codo for he pavy for erm. I addo o PD corol, a force compeao erm added o acheve poo coordao boh cae of free moo coac ak. Mapulaor moo ca be dvded o hree dffere phae, free moo, coac moo, he rao bewee hem. here large umber of paper coderg he fr wo ue, bu he complexy whch he varey of he evrome add modelg make he rao dcuo dffcul. Impora ue coac rao lke mpac force umber amplude of bouce are

2 dcued [7]. N e al dcu hee ue [8] for a -DO lear eleoperao yem learly modeled huma evrome. Oher work prefer o bypa h problem by ome praccal oluo lke addg damper mpedace o he yem a repored o be he mo mpora facor ably of he coac rao [9]. I h work, he addoal erm added o pafy he corolled yem are maly dpave oe, o eble o leave he performace of coac ak o he mulao. II. MODELLING Coder a eleoperao yem whch co of wo N- DO roboc yem a he maer lave robo govered by wo lear yem a follow, M& () = () + () () M & () = () + () q, q R are he poo cofgurao, R, repree he huma evrome force repecvely,, R are corol gal, M, M R are he pove-defe (PD) era marce. Each of he robo ca be aumed a a Euler-Lagraga dyamc yem Corol marx zero. I alo kow ha he coroller pavy,.e. [ ( θ ) q( θ ) + ( θ ) q( θ )] d θ c & & () mple eergec pavy of he eleoperao yem (),.e. [ + ] d d () h propery o beefcal ha oe ca u eure he pavy of he eleoperaor whou cocerg he olear dyamc of he robo a log a Euler- Lagraga dyamc. owever, whe force feedback appled o compeae he effec of, =, o acheve poo compeao, h decouplg deg ca o be ued he dyamc of he eleoperaor hould alo be codered. III. CONROL DESIGN Poo error-baed corol ued [] eure poo coordao u free moo (whe =, =, ). ere, force feedback alo added o provde yem wh poo coordao boh cae of free moo coac ak. ece, he overall corol deged o be Dervave erm Proporoal erm K q q K q q () = v( & ( ) & ()) + p( ( ) ) + K () K ( ) K () K () + f f d d = Compeao erm Dpao erm Derv Proporoal erm 6 78 ave erm 678 K q q () = v (& ( ) & ()) + Kp ( q q ) + K f + Kf () Kd () Kd Compeao erm = Dpao erm K v, K p are he ymmerc pove-defe PD ga K d are four dpao ga for each robo. Oe of hem o pafy he effec of proporoal corol aco he oher hree o pafy he effec of he force compeao erm. τ, τ are he coa commucao delay. Deg of he corol co of wo ep. r, pafy he effec of he PD corol ga. I h ep decouplg of he coroller roboc mapulaor ulzed o mplfy he deg procedure. I he ecod ep force compeao erm become pave. A wll be how h requre corporao of he yem dyamc a well a PD corol aco. he power geeraed by he coroller gve by P = + c () () () = P () + P () pd whch p pd p f are he power aocaed wh he PDcorol force compeao erm repecvely are defed by P = (). v ( τ ) () (). ( τ ) () PD K ( ) + Kp( q q ) Kd () + Kv( ) + Kp( q q ) Kd () P() = (). K f() + K f( ) Kd () = + (). K f( ) + K f() Kd () = A. Pave mplemeao of he PD corol I [] uffce codo for pafyg h par propoed. he mehod maly baed o aalyzg he eergy produco of he coroller he frequecy doma. he codo ca be ummarzed o be a follow, τrd Kd = for =, (8) K p τ rd he roud rp delay he um of wo delay f () (5) (6) (7)

3 τ τ. B. Pave mplemeao of he force compeao erm I h eco we are gog o ae codo of pavy of he power aocaed wh he force compeao erm. he P f expreed erm of boh poo force gal. I order o aalyze pavy oe may frly expre h power erm by u poo gal. h ca be doe ug he equao of he moo defed (). Laplace raform appled o olve hee lear dffereal equao, ( ) M Q K e Q Q K e Q Q v p d d ( f) f = K Q K Q I K e K + + = + + ( ) M Q K e Q Q K e Q Q v p d d f ( f ) = + + = + + K Q K Q e K I K (9) () Q, Q are Laplace raform of he rucaed q% & θ, q% & θ,repecvely, whch are defed by, gal [ ] & q%& θ = u( θ ) q θ, u( λ) oe f λ oherwe zero. he, ug hee equao, calculaed a, = C A + C B = = = C A + C B = = A, B, C are defed a, ( ) ( ) A = M Q, A = K e Q Q, v A =K e Q Q, A = K Q + K Q p d d = B = M Q, B = K e Q Q, v B =K e Q Q, B = K Q + K Q () p d d = C I K C + f e K f C C = e K f I + K f () hu, huma evrome force gal co of four par whch are eparaed de he gma operaor by ubcrp. he erm = dcae he par added by he robo era marx, he erm =, how he par added by dervave proporoal corol aco, fally he erm = he par added by dpave aco. Ug Pareval heorem, we ca rewre P () (7) a + ω P () d = Q. K f e K f K Q π + d = ω + Q. e K f + K f Kd Q d = Clearly, he par aocaed wh dpave erm alway dpae eergy. hu, we wll drbue h par amog he oher oe o pafy hem. oal eergy aocaed wh he force feedback ca be eparaed coderg four par of a + P () d = Q DA DB Kd Q π + = = + Q ( DA + DB) Kd Q d = = + k = P ω dω π k = he erm ( ω f f ) ( ω f f ) ( f + ω f ) ( τ ω f + f ) D= K C+ e K C D= K C+ e K C D= K C e K C D= K C e K C k P, k=, are defed by [ ].[ + ] P = Q. D M Q + D M Q Q DMQ D MQ Q Kd Q Q Kd Q= Q Q d DM DM + Kd ω () (.a) D M + K D M = (.b) P = Q DKv( e Q Q ) + DKv( e Q Q ) Q DKv( e Q Q ) + DKv( e Q Q ) Q KdQ Q KdQ= Q Q D Kve D Kv+ Kd DKve DKv = D K ed K D K ed K + K v v v v d (5.a) (5.a) ω

4 [ ] p + p Q [ DKp( e Q Q ) DKp( e Q Q )] d d [ ] τ P τ = Q D K e Q Q D K e Q Q + QK Q QK Q= Q Q (6.a) ( D Kpe DKp) / K ( d )/ D Kpe D Kp + = ( DKpe DKv) / ( DKpe DKp) / + Kd (6.b) = Q. D K d Q + D K d Q = = P + Q. D Kd Q + D Kd Q = = Q K d Q Q Kd Q= Q Q D K d + K d D Kd = = = D K d D Kd K + = = x A = xax, =w. d (7.a) (7.b) Now we ca reder he compeao par pave by makg he marce aocaed wh each par power erm pove defe (PD). h ca be doe by a raformao of hee marce o he caerg marce eurg ha he reulg marce have a fy orm le ha oe. o clarfy h le coder oe of he power erm P, Q P =Q Q Q P he power aocaed wh he pu oupu of a lear fler. So, he eceary uffce codo for o be pave ha he raformed caerg marx have a fy orm le ha oe [],.e., S S ( )( = I + I) for =,,. C. Seg he force compeao ga he four force feedback ga K f,, =, have o eleced appropraely order o cacel ou he effec of he force durbace o he poo coordao. ollowg [], le u fr decompoe he yem o locked haped yem o ee he dyamc equao of he poo error. Coder followg ae raformao vl I-φ φ = ve I -I [ ] φ = M + M M Ug h raform, he dyamc of he eleoperaor yem ca be wre a M Lv& L = L + L M v& = + E E E E v E he velocy of he poo error, M E, M L are ymmerc PD marx gve by ML = M + M ME = φ Mφ + [ φ I] M[ φ I] E, E, L, L are raform for he, =, gve a L L =, = E E o mplfy he dcuo uppoe ha he eady ae, delayed gal q (-τ ) =, equal o q (). he, by choog E =- E he effec of exeral force o he poo error dyamc wll be compeaed. h wll add followg erm o he corol gal: E E = = =, E = φ + ( φ I) E E E o, K f,, =, are choe a, K f = φ K f = ( φ I) K = φ K = φ I f f IV. SIMULAION I h eco deg mulao of a -DO eleoperao yem preeed o llurae he deg procedure. I fr ep of deg PD corol ga are e aocaed dpao ga e o pafy. he he ecod ep afer choog approprae ga for force compeao, oher dpao ga are e o pafy hee erm. Coder a eleoperao yem decrbed by, & () = () + () & () = () + () commucao delay τ, τ are choe o be.5. orce meaureme poo velocy gal are avalable a boh e. hu, corol gal are e o be he form of Eq.. PD ga K p K v are e o.5 repecvely. Ug Eq.8 Dpao erm for PD ga are e o be.5 o pafy PD aco. Now coder he fr par of he power aocaed wh he force compeao erm.e. he erm P. K f, K f are e o be.5/(5+w) dpao erm K d, K d are e

5 o 5. A how g. he large gular value of he marx S le ha oe coequely he choe value reder h par pave r gular value Secod gular value gure. Sgular value of he marx he ecod erm co of he effec of he dervave corol he power erm. he dpao erm K d, K d are e o be. h erm relavely ealy pafed ha oher erm. I g. he gular value of he correpodg caerg operaor are how..... gure. Sgular value of he marx or he hrd erm P dpao erm K d, K d are choe o be. g. depc ha wh h eg he gular value of he marx are boh maller ha S gure. Sgular value of he marx Dpao erm K d, K d are ued o pafy he forh power erm P. hey are e o.5. g.5 how he pavy of h erm frequecy, Rada gure. Sgular value of he marx o demorae he performace of he coroller we coder he followg cearo. he huma operaor modeled by a PD poo rackg coroller by prg dampg ga a K ph, K vh repecvely. rly, he puhe he maer robo o he poo.8. he, he re o puh he maer robo o he poo.5. Whle he movg he robo o h arge, he realze he exece of a hard wall whle recevg ep-lke force feedback. A he eady ae moo of puhg h wall, we wll have poo coordao a well a force rackg h po he ma corbuo of h paper. ally, he puhe he maer robo back o al poo. Choe deged value for he mulao are ummarzed able.. Smulao reul are how g.5. able.. Parameer he mulao Choe Deged Value parameer parameer Value τ K d = K d 5 τ.5 K d = K d 5 M =M K d = K d K p =K p K d = K d K v =K v.5 K d = K d.5 K = K.5/(5+) V. CONCLUSION A corol framework for corol of a eleoperao yem he preece of large me delay wa preeed. h framework provde poo coordao boh cae of free coac moo whle preerve pavy of he eleoperaor. Pre-pecfed mple rucure of he coroller pree low co ealy mplemeed corol framework. Corol yem eparaed o poo force feedback par. he pafcao of he fr par uded [] by Lee e al. o compeae he evrome force, hece, achevg poo rackg coac ak force feedback are alo added o he corol yem. he effec of h par o oal eergy are aalyzed uffce codo for pavy obaed hrough udyg he effec of everal par h force feedback add. o 5

6 Poo orce, Newo me, Secod q q me, Secod gure.5 Reul of he mulao demorae how velocy feedback are appled o pafy he overall corol yem a llurave example are alo preeed. Nex reearch proec may co of exedg he reul o he cae of he eleoperao of olear robo. REERENCES [] D. Lee & M. W. Spog, Pave blaeral eleoperao wh coa me delay, IEEE raaco o Roboc, v,, p.69-8, 6. [] D. J. Lee P. Y. L, Pave blaeral corol ool dyamc rederg for olear mechacal eleoperaor, IEEE ra. Robo., vol., o. 5, pp.96-95, Oc. 5. [] D. J. Lee P. Y. L, Pave blaeral feedforward corol of lear dyamcally mlar eleoperaed mapulaor, IEEE ra. Robo. Auom., vol. 9, o., pp. -56, Ju [] D. Lawrece, Sably raparecy Blaeral eleoperao, IEEE ra. Robo. Auom., vol. 9, o., pp.6-67, 99. [5] Mc. Cavuoglu, A. Sherma,. edck, Deg of blaeral eleoperao coroller for hapc explorao elemapulao of of evrome, IEEE raaco o roboc auomao, Augu. [6] S. Sroupour, Modellg corol of cooperave eleoperao yem, IEEE raaco o Roboc, vol., o. 6, Dec. 5, pp. -5. [7].J. ar, Y.Y. Wu, N. X A. Idor, orce Regulao Coac rao Corol, IEEE Corol Syem Magaze, eb [8] Lya N Davd W. L. Wag, A Ga-Swchg Corol Scheme for Poo-Error-Baed Blaeral eleoperao: Coac Sably Aaly Coroller Deg, he Ieraoal Joural of Roboc Reearch, pp. 55-7,. [9] D.. Surdlovc, Coac rao Sably he Impedace Corol, IEEE, ICRA, Apr, 997. [] G. Nemeyer J. J.Sloe, Sable Adapve eleoperao, IEEE Joural of Oceac Egeerg, vol.6, o., pp.5-6, 99. [] R.J. Adero M.W. Spog, Blaeral corol of operaor wh me delay, IEEE ra. Auoma. Co., vol. AC-,pp. 9{5, May 989. [] N. oga, O he Sably of Mapulaor Performg Coac ak, J. of Roboc Auomao, -6, pp , 988. [] G.M.. Leug, B.A. rac, J. Apkara, Blaeral coroller for eleoperaor wh me delay va μ_yhe," Jauary 995. [] J.E. Colgae, Robu Impedace Shapg elemapulao," IEEE raaco o Roboc Auomao, vol. 9, pp. 7-8, Augu 99. [5] J. Ya S. Salcudea, eleoperao coroller deg ug - fy Opmzao wh applcao o moo-calg, IEEE ra.cor. Sy. echol., vol. 5, o., pp. 58, May 996. [6] K. ahrud-zaad S.E. Salcudea, Aaly Evaluao of Sably Performace Robue for eleoperao Corol Archeecure, IEEE I. Cof. o Roboc Auo., Sa racco, CA, Aprl. 6