Fully Adaptive Feedforward Feedback Synchronized Tracking Control for Stewart Platform Systems

Transcription

1 Ieraoal Fully Joural Adapve of orol, Feedforward Auomao, Feedback ad Sysems, Sychrozed vol 6, rackg o 5, pp orol , for Sewar Ocober Plaform 008 Sysems 689 Fully Adapve Feedforward Feedback Sychrozed rackg orol for Sewar Plaform Sysems Dogya Zhao, Shaoyua L*, ad Feg ao Absrac: I hs paper, a fully adapve feedforward feedback sychrozed rackg corol approach s developed for precso rackg corol of 6 degree of freedom (6DOF Sewar Plaform he proposed coroller s desged deceralzed form for mplemeao smplcy Iercoecos amog dffere subsysems ad gravy effec are elmaed by he feedforward corol aco Feedback corol aco guaraees he sably of he sysem he gas of he proposed coroller ca be updaed o le whou requrg ay pror kowledge of Sewar Plaform mapulaor hus he corol approach s clamed o be fully adapve By employg cross-couplg error echology, he proposed approach ca guaraee boh of poso error ad sychrozao error coverge o zero asympocally Because he acuaors work sychroous maer, he rackg performaces are mproved he correspodg sably aalyss s also preseed hs paper Fally, smulao s demosraed o verfy he effecveess of he proposed approach Keywords: ross-couplg error, deceralzed corol, fully adapve corol, Sewar plaform, sychrozed rackg corol INRODUION As a ypcal 6DOF parallel mapulaor, Sewar Plaform has bee exesvely suded [-4] hs parallel mapulaor ca provde beer accuracy, rgdy, load o wegh rao ad load dsrbuo ha seral mapulaor I has bee used he area of low speed ad large payload codos, such as moo base of flgh smulaor ad moo bed of a mache ool [,5] How o develop a effecve corol approach for precse rackg corol of Sewar Mauscrp receved May 8, 007; revsed March 0, 008; acceped Aprl 8, 008 Recommeded by Edoral Board member Dog Hwa Km uder he dreco of Edor Jae Weo ho hs work was suppored by Naoal Naural Scece Foudao ( , , , he Hgh echology Research ad Developme Program (006AA04Z73, he Specalzed Research Fud for he Docoral Program of Hgher Educao of ha ( , ad parly by Shagha Naural Scece Foudao (07J406 & 06ZR4044 he auhors wsh o hak Dr Dog Su ad Dr Lu Re, hey provded some leraures for us o sudy ad gave some good advce our sudyg Dogya Zhao ad Shaoyua L are wh Isue of Auomao, Shagha Jao og Uversgy, 800 Dogchua Road, Mhag, Shagha 0040, ha Ad Dogya Zhao s also wh ollege of Mechacal ad Elecroc Egeerg, ha Uversy of Peroleum, Shadog, ha (e-mals: zhaodogya@hdpueduc, syl@sjueduc Feg ao s wh School of Mechacal Egeerg, Shagha Jao og Uversgy, 800 Dogchua Road, Mhag, Shagha 0040, ha (e-mal: fegg@sjueduc * orrespodg auhor Plaform aracs much aeo from academe ad dusry he rackg corol of parallel mapulaor ca be classfed as wo kds of approaches [6] he frs oe s depede jo corol scheme (omodel-based, such as proporoal-egraldervave (PID corol [7,8], fuzzy corol [9] ad eural ework corol [0] Aoher oe s modelbased approach, such as robus corol [4,6], mpedace corol [] ad adapve corol [,3] he mos of exsed approach for rackg corol of parallel mapulaor dd o cosder he sychrozao amog corol loops he lack of sychrozao wll lead o large couplg errors whch degrade he performace of overall mechacal sysem or eve damage he mache ool [4-7] By employg cross-couplg error echology [8-], some sychrozed rackg corol approaches were developed for parallel mapulaors Nomodel-based approach cludes egraed sauraed PI sychroous corol plus PD feedback corol approach [] ad feedforward compesao plus cross-couplg error feedback corol approach [3] Model-based approach cludes adapve sychrozed corol [4] ad covex sychrozed corol [5] hough modebased sychrozed rackg corol approach ca mprove he performace, s coroller srucure s complex due o he complcaed dyamc model of parallel robo he PID sychrozed rackg corol srucure s smple whou requrg ay pror kowledge of pla However he deermao of he PID coroller s parameers s dffcul pracce

2 690 Dogya Zhao, Shaoyua L, ad Feg ao hough he approach proposed by leraure [3] s a excelle mehod for parallel robo sychrozed rackg corol whou requrg he explc use of he sysem dyamc model, some pror kowledge s requred coroller desgg I s dffcul o ge he pror kowledge of parallel mapulaor pracce hese sychrozed rackg corol approaches meoed above are all developed for a 3 degree of freedom plaar parallel robo, he gravy effec eed o o be cosdered coroller desgg Sewar Plaform s a geeral cofgurao of parallel mapulaor Wh cosderg he gravy effec, he corol approach developed for ca be exeded o oher ype parallel mapulaors Due o he complexy of dyamc model of Sewar Plaform, a omodel-based sychrozed rackg corol approach s developed hs paper Based o he achevemes of leraure [6-7,3,6], a fully adapve feedforward feedback sychrozed rackg corol approach s developed for Sewar Plaform he proposed coroller s desged deceralzed form for mplemeao smplcy Iercoecos amog subsysems ad gravy effec are elmaed by he feedforward corol aco Feedback corol aco guaraees he sably of he sysem he gas of he proposed coroller ca be updaed o le whou requrg ay pror kowledge of Sewar Plaform mapulaor Fully adapve corol meas ha he desred corol objecve ca be acheved wh adapve corol whou ay pror kowledge of plas [6] By employg cross-couplg error echology, he proposed approach ca guaraee boh of poso error ad sychrozao error coverge o zero asympocally Because he acuaors work sychroous maer, he rackg performaces are mproved he res of hs paper s orgazed as follows: dyamc model of he Sewar Plaform ad some of s properes are depced Seco, he defos of sychrozao error ad cross-couplg error are also preseed hs seco A fully adapve feedforward feedback sychrozed rackg corol approach ad he correspodg sably aalyss are preseed Seco 3 I Seco 4, a smulao for 6DOF Sewar Plaform sychrozed rackg corol s demosraed suppor of he proposed approach Fally, cocludg remarks are gve Seco 5 PROBLEM FORMULAION As show Fg, he Sewar Plaform s a 6DOF mechasm wh wo bodes coeced by he sx exesble legs Usually, he eral frame O-XYZ s fxed a he base plaform wh s org a he geomery ceer of he base plaform, he body-fxed frame (movg frame P-xyz s aached o he mass L5 B5 org of he movg plaform he sx degree of freedom of Sewar Plaform are raslaos alog he X, Y, Z axes ad he roaos abou he axs O-X, O-Y, O-Z he work space coordaes of he mass ceer of movg plaform ca be wre as [ ], P = X Y Z α β γ ( where X, Y, Z represe he raslaos ad α, β, γ represe he roaos q= q L q s legh of legs, whch represes geeralzed coordaes he relaoshp bewee P ad q s he [ ], ( = ( ( ( P& J q&, ( where J( s Jacoba marx By usg he aural orhogoal compleme mehod, he dyamc model of Sewar Plaform ca be derved jo space [,5,7]: = τ, M q q+ && q,q& q+ & q (3 where 5 L4 B4 B6 4, y 6 L6 Y Fg oordaes of 6 DOF Sewar Plaform Z q R s he geeralzed coordae, M ( q dag{ M ( q } ( q, q = dag{ ( q, q } R cerfugal force marx, ( q ( q L ( q = R s he era marx, & & s he orols ad =,, R s he gravy force vecor, = [ L ],, τ τ τ R s he acuag force vecor Remark : omparg wh dyamcs of plaar parallel robo preseed by leraure [-5], he dyamcs of Sewar Plaform has a gravy force z P O L X x 3 B L3 B3 L B

3 Fully Adapve Feedforward Feedback Sychrozed rackg orol for Sewar Plaform Sysems 69 vecor, whch mus be compesaed coroller desgg he compesaor for gravy effec wll be dealed laer he dyamc model (3 has he followg properes ha wll be used coroller desgg [,3,6,7] M q s a symmerc ad Propery : he marx posve-defe marx, whch sasfes M ( q ad ( a λmax M q b for some cosas ab>, 0, where λmax ( / λm ( represes maxmum /mmum egevalue of he marx q, q& sasfes q q Propery : he marx (, & c for some cosa c > 0 Propery 3: he vecor ( q sasfes ( q d for some cosa d > 0 & & s Propery 4: he marx M ( q ( q, q skew-symmerc marx o formulae he deceralzed corol sysem srucure, he h subsysem of dyamc equao (3 ca be wre as he followg deceralzed form M qq+ && q,qq+ & & q = τ =, L,, (4 where τ represes h corol pu for leg For parallel mapulaor, he sychrozao error ca provde each acuaed jo wh moo formao boh from self ad oher acuaed jos As a resul, hs error represes he degree of coordao amog he acuaed jos [6,7] By employg cross-couplg error echology, he sychrozed corol approach ca mprove rackg performace of Sewar Plaform Defe he poso error d Δ q = q q, (5 where Δ q s poso error, q d s he desred coordae jo space, =, L, he coordaed acuaors of Sewar Plaform are subjeced o he followg sychrozao fuco [6,3] f q, Lq : c q = c q = L = c q, (6 where c s he couplg coeffce of h acuaor ad s assumed o be ozero hs sychrozao fuco represes a ew ask requreme kemacs he fuco (6 s vald for all desred d coordaes q (, amely d d d d d = = = f q, Lq : c q c q L c q, (7 From (6 ad (7, he followg sychrozao goal s defed as [6,3] c Δ q = c Δ q = L = c Δq (8 he sychrozao goal (8 ca be dvded o sub-goals such as cδ q = c+ Δ q+, whe =, + = he poso sychrozao errors of Sewar Plaform ca be defed as follows [6,3]: ε = cδq cδq ε = cδq c3δq3 (9 M ε = cδq cδq, where ε represes he sychrozao error of h ε = for all =, L,, he acuaed jo Whe 0 jos wll work a sychroous maer For developg a sychrozed rackg coroller, he cross-couplg error s defed as Su e al [6,7, 3]: ( = Δ + 0 e c q μ ε ω ε ω dω, (0 where μ s a posve couplg parameer, ω s a varable from me zero o, whe =, = Remark : he poso error ad sychrozao error are cluded cross-couplg error expresso By desgg a approprae coroller, boh of hem ca coverge o zero asympocally Remark 3: he couplg parameer μ plays a mpora role he sychrozed rackg corol As μ creases, he sychrozao error decreases Dffereag e wh respec o me yelds = Δ + Δ + ( e& c& q c q& μ ε ε ( he commad vecor u [3] d = + Δ + μ ε ε +Λ u cq& c& q e ( he geeralzed error r [3] r = u cq& = e& +Λe, (3 where Λ R s a feedback ga marx Subsug (3 o (4, he dyamc equao ca be rewre as M ( q c ( u& c& q& M q c r& + q, q& c r = τ + ξ, (4 ξ = (5 + q, q& c u + q,

4 69 Dogya Zhao, Shaoyua L, ad Feg ao where =, L, for all of he expressos above Remark 4: ξ represes he ercoecos amog dffere subsysems he sychrozed rackg corol s he o fd a class of fully adapve feedforward law o compesae he effec of ξ ad fully adapve feedback law o coverge r 0 ad ε 0 as Remark 5: he corol law proposed by leraure [3] s M = K c u cq N + K c u+ sg( c r K + Krc r + c K ε ( ε ε, N M = ( + % τ & & & (6 K D c u& c& q& D c u, (7 M where K, K > 0 are posve feedforward corol gas, Kr, K ε > 0 are posve feedback gas, M D, D > 0 are scalars hese gas are requred o sasfy he followg codos: ( λm ( Kr λmax M ( q c& c M M ( D K M ( q, D K ( q q&, I s obvous ha he corol law proposed by [3] does o cosder gravy effec Some pror kowledge s requred by hs corol law he corol objecve s o desg a fully adapve feedforward feedback sychrozed rackg corol pu τ o coverge r 0 ad ε 0 as From expressos (9~(3, oe ca see Δq 0 as r 0 ad ε 0 Ulke he coveoal o-sychrozed rackg corol approach whch cosders he poso error oly, he sychrozed rackg corol ca guaraee boh poso error ad sychrozao error coverge o zero asympocally 3 ONROLLER DESIN I hs seco, a fully adapve feedforward feedback sychrozed rackg coroller s developed for 6DOF Sewar Plaform Whou loss of geeraly, several echcal assumpos are made o pose he problem a racable maer Assumpo : he feedback ga Λ s cosa, dagoal ad posve-defe Assumpo : Desred jo poso rajecory d d q ( s a real umber ad ozero q ( d d me dervaves q& (, q ( && are bouded ad he Based o he achevemes of leraure [3,6], he fully adapve feedforward feedback sychrozed rackg coroller s desged as τ (& & & M = K c u cq + K c u + K + sg c r K ˆ N ( ( ( ε ε + Kˆ r c r + c K ε, (8 for =, L,, where he K, K > 0 are he feedforward gas o compesae he effecs of M q c u& c& q& ad ( q, q& s u, respecvely K > 0 s he gravy compesaor Kˆ > 0 ad K ε > 0 are posve feedback r corol gas o compesae for he effec due o he errors bewee he feedforward corol gas ad he modelg parameers, a sauraed corol ulzg he sg fuco sg( c r K ˆ N ( ˆ N K ca be wre as M ( ˆM = ( (& & & ˆ N K D c u c q ( ( ˆ + D c u + Dˆ, s employed, (9 for,, ˆM, ˆ, ˆ D D D 0 Assume ha * M *,, * Kr D D, D * 0 are he desred value of ˆ ˆM ˆ,,, ˆ Kr D D D (, whch sasfy he followg expresso = L where * ( Kr λmax ( M ( q c& c λ m M* M D K M q * D K q q * D K ( q (, & (0 Defe he esmao errors of ˆ M K (, D ( Dˆ (, D ˆ as ( ˆ ( ( ˆ ( ( ˆ ( ( ( % = D = D D Kr % M D% Kr M = D * Kr M* * D ˆ * D% = D D r ˆ, ( o cope wh uceray of dyamcs of Sewar Plaform, he adapve laws are desged as follows:

5 Fully Adapve Feedforward Feedback Sychrozed rackg orol for Sewar Plaform Sysems 693 K &% & ( ˆ r = Kr ( = kr c r D &% & = D = d c r c u& c& q& ( ˆ D &% & = D ( = d c r c u ( ˆ D &% & = D ( = d c r, ( ˆ ( M M M for,,, ( = L where K ˆ ( 0, ˆ M D ( 0, D r ˆ 0, ˆ D 0 > 0 are he al value ad he M kr, d, d, d > 0 are adapve gas Remark 6: From comparg he corol law (8~( wh (6~(7, oe ca fd ha he gravy compesaor K ad ˆ D ( are employed by he proposed approach of hs paper o elmae he effec of gravy Remark 7: orol gas Kˆ (, ˆ M D (, D (, D ˆ ˆ of he proposed approach of hs paper ca be updaed ole whou requrg ay pror kowledge of he Sewar Plaform, whle he correspodg gas of he corol law proposed by leraure [3] are requred o sasfy some codos whch are requred o be deermed a pror heorem : Uder Assumpos ad, cosder he error dyamcs of Sewar Plaform (4~(5 subjec o he proposed corol law (8~( If he M gas K, K, K, K ε > 0, he sysem s asympoc sably, e, Δq ( 0 ad ε ( 0 as me Proof: Defe a Lyapuov fuco caddae as V = r c M ( q c r + K = ε εε + ( ε 0 w ε w dw = Λ K ( ( w ( w dw μ ε ε 0 ε = M M kr K% r d ( D% = + + ( % ( % + d D + d D r (3 Dffereag V wh respec o me alog he rajecores of he sysem defed by (4 ad (8 = V& = r c M q c r & & + r c M q c r + r c M& ( q c r + ε Kε & ε ( ε ε μkε = 0 = M M M kr K% rk %& r d D% D% & = d D% D% & d D% D% & + Λ w w dw ( ε ε Mulplyg boh sdes of (4 by ( c r subsug he resul o (4, oe ca ge ( ˆ & V& = c r Kr M q c c c r = ˆ N c r N c r K = r Kε( ε ε ε Kεε & = = ( ε ε μkε = ( ε 0 w ε w dw = M M M kr K% rk %& r d D% D% & = Λ d & D% D% + d D% D% & (4 ad (5 Subsue he adapve law ( o (5, oe ca ge = = ( & * V& c r Kr M q c c c r M M ( * c r D K M q c ( u& c& q& ( (, & * * ( D K ( q * * + D K q q c u + (5

6 694 Dogya Zhao, Shaoyua L, ad Feg ao r Kε( ε ε ε Kεε & = = ( ε ε μkε = ( ε 0 w ε w dw = + + Λ Accordg o (0, (6 ca be wre as ε( ε ε ε εε & = = ( ε ε μkε = ( ε 0 w ε w dw = V& r K + K + Λ From he expresso (9~(3, oe ca ge [3] r Kε( ε ε = = & ε Kεε + ε ΛμKεε = = + ( ε ε+ μkε = ( ε 0 w ε+ = = ( ε ε+ μkε( ε ε+ = = w dw + Subsug (8 o (7 yelds V& = ε ΛμK ε ε ( ε ε+ μkε( ε ε+ = = (6 (7 (8 (9 I s obvous ha V& 0 from expresso (9 From he Barbǎla s Lemma [8], r 0 adε 0 as me From (3, oe furher has e 0 as he resul meas ha he sychrozao goal s acheved From (9 ad (0, oe ca ge Δ q = 0 as ε = 0 ad e = 0 herefore, he closed loop sysem s asympocally sable hs s he ed of proof Remark 8: From Propery ~Propery 3 ad ( M q c c Assumpo, oe ca see ha λ &, max M, K ( q, q&, K ( q K M q are depede of K r * M * * *, D, D ad D, respecvely hus, equaly (7 ca be derved * M * * * from (6 whe K r, D, D ad D are large eough Remark 9: he approach heorem ca be mplemeed whou ay pror kowledge of Sewar Plaform mapulaor, such as al codos ad parameers expresso (3 Remark 0: Whe all kowledge of Sewar Plaform mapulaor ad desred rajecory are provded a pror, he bouds of λ M ( M q c& c max,, K ( q, q&, K ( q K M q ca be esmaed advace, ad he he desred K r *, M * * * D, D ad D ca be deermed by makg equaly (0 sasfed Subsug hese deermed desrable values o (8 ad (9 o replace ole updaed gas, he approach heorem becomes he oe proposed by leraure [3] plus gravy compesao, whch ca be guaraeed by he smlar procedure poof of heorem 4 SIMULAION RESULS AND DISUSSION Smulaos were performed Malab SmMechcs oolbox he parameers of 6DOF Sewar Plaform were gve as: he mass ad mass mome of era values of upper plaform are m = 69kg, X, Y ( Z 30448(60846kg m, upper/lower par of h leg are ( m / ( m I I I = he mass of = 58/ 9kg, he mass mome of era values of upper ad lower par of h leg are, ux, uy ( uz 47( 003 kg m ( I = 430(056kg m I I I = ad I, I dz For comparso purpose, hree corol algorhms cludg he proposed approach, adapve sychrozed corol (A-S corol [4] ad PID corol were used o corol he sysem respecvely he A-S corol [4] s expressed as follows ˆ, &&&, θ τ = Y qqq Kr Ke a r e he adapao law s ( = PY ( ( qqq &&& r( + Y( qqq &&& ˆ θ,,,, % θ, d P d P Y q, q, q Y q, q, q, λ = λ + ( &&& ( &&& ( = λ ( P k, 0 0 u dx d dy

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12 700 Dogya Zhao, Shaoyua L, ad Feg ao From hese comparsos, oe ca see ha, he performace of he proposed corol ad A-S corol are beer ha PID corol Especally, he sychrozao errors of he proposed corol ad A-S corol are much smaller ha PID corol he large sychrozao errors are udesred pracce he proposed corol s developed whou ay pror kowledge of plas bu he A-S corol s model based Due o he complexy of dyamcs of Sewar Plaform sysem, he srucure of A-S corol s more complex ha he proposed corol hough he performace of A-S corol s almos as good as he oe of he proposed corol, he proposed corol s more applcable ha A-S corol pracce 5 ONLUSIONS A fully adapve feedforward feedback sychrozed rackg corol approach s developed for precso corol of 6DOF Sewar Plaform By corporag cross-couplg error echology, he proposed approach ca guaraee boh of poso error ad sychrozao error coverge o zero asympocally he rackg performaces are mproved for he acuaors workg coordag maer he gas of he proposed coroller ca be updaed o le whou requrg ay pror kowledge of Sewar Plaform mapulaor he correspodg sably aalyss s also preseed hs paper he proposed coroller s desged deceralzed form for mplemeao smplcy I ca be mplemeed pracce easly Fally, performaces of he proposed approach are compared wh performaces of A-S corol ad PID corol hrough smulao he resuls verfy he effecveess of he proposed approach REFERENES [] J P Merle, Parallel Robos, Kluwer Academc Publshers, Dordrech, 000 [] B Dasgupa ad S Mruhyujaya, he Sewar Plaform mapulaor: A revew, Mechasm ad Mache heory, vol 35, o, pp 5-40, 000 [3] Y g, Y-S, he, ad H- Jar, Modelg ad corol for a ough-sewar Plaform N mache, Joural of Roboc Sysems, vol, o, pp , 004 [4] H S Km, Y M ho, ad K I Lee, Robus olear ask space corol for 6DOF parallel mapulaor, Auomaca, vol 4, o 9, pp , 005 [5] L W sa, ROBO ANALYSIS he Mechacs of Seral ad Parallel Mapulaors, Joh Wley, New York, 999 [6] S Fu, Y Yao, ad She, No-lear robus corol wh paral verse dyamc compesao for a Sewar Plaform mapulaor, Modellg, Idefcao ad orol, vol, o, pp 44-5, 006 [7] F H horbel, O heela, R uawardaa, ad R Logchamp, Modelg ad se po corol of close-cha mechasms: heory ad experme, IEEE ras o orol Sysems echology, vol 8, o 9, pp 80-85, 000 [8] P haccho, F Perro, L Scavcco, ad B Sclao, Robus desg of depede jo corollers wh expermeao o a hgh-speed parallel robo, IEEE ras o Idusral Elecrocs, vol 40, o 4, pp , 993 [9] P Bego, F Perro, ad P Dauchez, Fuzzy sldg mode corol of a fas parallel robo, Proc IEEE I of Robocs ad Auomao, pp 78-83, 995 [0] Perechele, F Boroleo, ad E ro, Neural ework algorhm corollg a hexapod plaform, Proc IEEE of Neural Neworks, pp , 000 [] E D Fasse ad M ossel, Spaogeomerc mpedace corol of ough-sewar plaforms, IEEE ras o Roboc ad Auomao, vol 5, o, pp 8-88, 999 [] M R Srouspour ad S E Salcudea, Nolear corol of hydraulc robos, IEEE ras o Roboc ad Auomao, vol 7, o, pp 73-8, 00 [3] H M L ad J E McIroy, Adapve susodal dsurbace cacellao for precse pog of Sewar Plaform, IEEE ras o orol Sysems echology, vol, o, pp 67-7, 003 [4] A Rodrguez-Ageles ad H Njmejer, Muual sychrozao of robos va esmaed sae feedback: A cooperave approach, IEEE ras o orol Sysems echology, vol, o 4, pp , 004 [5] H Njmejer ad A Rodrguez-Ageles, Sychrozao of Mechacal Sysem, World Scefc, Sgapore, 003 [6] D Su ad J K Mlls, Adapve sychrozed corol for coordao of mulrobo assembly asks, IEEE ras o Robocs ad Auomao, vol 8, o4, pp , 00 [7] D Su, Poso sychrozao of mulple moo axes wh adapve couplg corol, Auomaca, vol 39, o 6, pp , 003 [8] Y Kore, ross-coupled baxal compuer corols for maufacurg sysems, ASME Joural of Dyamc Sysems, Measureme ad orol, vol 0, o 4, pp 65-7, 980 [9] P K Kulkar ad K Srvasa, rosscoupled corol of baxal feed drve servomechasms, ASME Joural of Dyamc Sysems, Measureme ad orol, vol, o

13 Fully Adapve Feedforward Feedback Sychrozed rackg orol for Sewar Plaform Sysems 70, pp 5-3, 990 [0] M omzuka, J S Hu, ad hu, Sychrozao of wo moo corol axes uder adapve feedforward corol, ASME Joural of Dyamc Sysems, Measureme ad orol, vol 4, o 6, pp 96-03, 99 [] L Feg, Y Kore, ad J Borese, rosscouplg moo coroller for moble robos, IEEE orol Sysems Magaze, vol 3, o 6, pp 35-43, 993 [] Y Su, D Su, L Re, ad J K Mlls, Iegrao of sauraed PI sychroous corol ad PD feedback for corol of parallel mapulaors, IEEE ras o Robocs, vol, o, pp 0-07, 006 [3] D Su, L Re, J K Mlls, ad Wag, Sychroous rackg corol of parallel mapulaors usg cross-couplg approach, Ieraoal Joural of Robocs Research, vol 5, o, pp 37-47, 006 [4] L Re, J K Mlls, ad D Su, Adapve sychrozed corol of a plaar parallel mapulaor: heory ad expermes, ASME Joural of Dyamc Sysems, Measureme ad orol, vol 8, o 4, pp , 006 [5] L Re, J K Mlls, ad D Su, ovex sychrozed corol for a 3-DOF plaar parallel mapulaor, Proc of he IEEE Ieraoal oferece o Robocs ad Auomao, pp 9-34, Orlado, Florda, May 006 [6] S-H Hsu ad L- Fu, A fully adapve deceralzed corol of robo mapulaors, Auomaca, vol 4, o 0, pp , 006 [7] J Ageles, Fudameals of Roboc Mechacal Sysems, heory, Mehods, ad Algorhms, d edo, Sprger-Verlag, New York, 003 [8] J J Sloe ad W L, Appled Nolear orol, Prece Hall, New Jersey, 99 Dogya Zhao receved he BS degree from Shadog Uversy, Ja, ha, 998 ad he MS degree from ahua Isue of hemcal Machery & Auomao, Lazhou, ha, 00 He s currely a PhD caddae a Deparme of Auomao, Shagha Jao og Uversy Sce 00, he has bee wh ollege of Mechacal ad Elecroc Egeerg, ha Uversy of Peroleum, where he s currely a Lecure Hs research eress clude robo corol, sldg mode corol ad olear corol Shaoyua L receved he BS ad MS degrees from Hebe Uversy of echology, aj, ha, 987 ad 99, respecvely Ad he receved he PhD degree from he Deparme of ompuer ad Sysem Scece of Naka Uversy, aj, ha, 997 Now he s a Professor wh he Isue of Auomao of Shagha Jao og Uversy Hs research eress clude predcve corol, fuzzy sysem, robo corol, sldg mode corol, olear corol, ad so o Feg ao receved he BS ad MS degrees from Norheas Heavy Machery Isue, Qhuagdao, ha, 979 ad 98, respecvely He receved he PhD degree Mechacal Egeerg, Bejg Uversy of Aeroaucs ad Asroaucs, Bejg, ha, 99 He s currely a Professor wh School of Mechacal Egeerg, Shagha Jao og Uversy Hs research eress clude parallel roboc mechasm desg heory, mul-dof parallel mcromapulaors, N parallel mache ools