Robust Adaptive Control of Voltage Saturated Flexible Joint Robots with Experimental Evaluations

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1 AU Joural of Modl ad Smulao AU J. Modl. Smul., 5((83-38 OI:.6/mscj Rous Adapv orol of Vola Saurad Flxl Jo Roos wh Exprmal Evaluaos A. Izadakhsh * parm of Elcrcal Er, Garmsar Brach, Islamc Azad Uvrsy, Garmsar, Ira ABSRA: hs papr s cocrd wh h prolm of ds ad mplm a rous adapv corol sray for h flxl jo lcrcally drv roos (FJER whl cosdr h cosras o h acuaor vola pu. h corol ds procdur s asd o h fuco approxmao chqu, o avod saurao sds rous aas oh srucurd ad usrucurd ucras assocad wh xral dsuracs ad u-modld dyamcs. Saly proof of h ovrall closd-loop sysm s v va h Lyapuov drc mhod. h aalycal suds as wll as xprmal rsuls oad us MALAB/SIMULINK xral mod corol o a sl-lk flxl jo lcrcally drv roo, dmosra a hh prformac of h proposd corol schms. Rvw Hsory: Rcvd: 7 Novmr 6 Rvsd: 5 July 7 Accpd: 3 July 7 Avalal Ol: 7 Spmr 7 Kywords: Rous adapv corol Ral-m mplmao Acuaor saurao Fuco approxmao chqu - Iroduco h acuaor pu cosras ar o of h major prolms ha ars whl coroll a acuad dyamc sysm. hs cosras ar du o hr physcal lmaos of h dvcs or praccal rasos ha rsrc h commad sal com from h corollr o h acuaors [-]. Wh a acuaor has rachd such a pu lm, furhr ffors o cras h acuaor pu would o rsul ay varao h oupu [3]. o dal wh hs prolms, may valual orqu-asd corol sras hav proposd y rsarchrs, am o prv saly ad omal prformac dradaos of h rooc sysms cosdr pu cosras [4-9]. h cosdral po s ha alhouh hs approachs ar sasfacory prcpl, hy ar of crczd for fw rasos, as mod []. o ackl hs prolms, som rlad works h fld of adapv/rous corol hav proposd [-6]. Morovr, svral approachs o mmz h prformac loss du o pu cosras hav rpord [7]. Howvr, hr ar y ohr prolms. h covoal adapv corol schm rqurs h compuao of h rrssor marx, prss xcao (PE codo of h rfrc pu sal du o h covrc of h paramr s vcor, ad slow havor of h dyamc sysm. hs prolm coms hyprssv spcally for hhr dr of frdom (OF roo mapulaors. Furhrmor, hy ar ual o hadl usrucurd ucray ad xral dsuracs adqualy, whch s a mss lk almos all h addrssd approachs [8]. o cop wh hs prolms, a rous adapv corol has proposd. Rous adapv corol hacs h rousss of adapv corol. h d for rous adapv corol s asd o h osrvao ha rooc mapulaors may hav uparamrzal dyamcs such as frco, orrspod auhor; Emal: zadakhsh_alrza@homal.com xral dsuracs, ad umodld dyamcs. Ay of hs dyamcs ca poally dsalz h sysm sc h m drvav of h Lyapuov fuco s oly av sm-df udr adapv corol. hr ar wo ways o ra rousfd adapv corols, calld rous adapv corol, amly, I h frs mhod s o add m- corol o h xs adapv corol. h rous corol par compsas for hos uparamrzal dyamcs, ad hrfor oly rqurs hr oud fuco [9]. II h scod mhod of ds rous adapv corol s o cha h adapo law y us h so-calld lakalk adapo law []. ompard wh h sadard adapv corol law, h laka-lk adapv corol law achvs rous saly h prsc of dsuracs ad ucray wh comproms rack prcso. Rcly, rrssor-fr corol of roo mapulaors has proposd whch s asd o fuco approxmao chqus (FA [-3]. [] shows ha ucras ca approxmad y a smpl p- ordr lar dffral quao. hus, ca hadld y mas of a smpl wllkow modl rfrc adapv corol chqu whch faclas h aalyss ad h ds ask as wll. [3] prsd a ack-spp lk corollr ds asd o slo-l schm. Howvr, h umr of OF ad h wh marcs dmso ar h mpora ssus ha mpos a xra compuaoal load, whch ur affc h corollr prformac [4]. h coruo of hs arcl ls h ds of a FA- Basd rous adapv corol schm for FJER, whch paramr ucras ad v acuaor saurao olary ar cosdrd. h corol ds sray s asd o a hrd ordr sad of ffh ordr dyamc modl. ompard o ohr prvous FA-asd adapv corol sras proposd for FJER, h proposd approach has a lss compuaoal load ha s sual for praccal 3

2 mplmao. I also cosdrs h xral dsuracs ffcs, whch s h ma cocr covoal Modl Rfrc Adapv orol (MRA []. hs papr s orazd as follows. Sco rfly prss h modl of h FJER. Sco 3 s dvod o h dscrpo of h proposd corol schm. Saly aalyss ad prformac valuao ar prsd sco 4. h xprmal sup ad ral m rsuls ar dscrd ad prsd sco 5. Fally, coclud rmarks ar draw sco 6. - Modl wh osdr Saurao h dyamcs jo spac of a sral-cha -lk FJER cosdr acuaor vola pu cosra ca wr as [] ( + (, + ( = K( r J + B + rk ( r = K I m m m m a L I + R I + K + ϕ( = sa( u( a a a a m whr R,, ar lk poso, vlocy, ad acclrao, rspcvly. ( R s a symmrc, posv-df fuco calld ra marx, (, R s a marx fuco calld crfual ad orols forcs marx, ad ( R s ravy rms. m, m, m R ar h jo poso, vlocy, ad acclrao, rspcvly; h cosa, posv daoal marcs K, J, B, Km, La, R a, ad K R rprs flxly, ra, damp, orqu cosas, lcrcal ducac, lcrcal rssac, ad ackmf, rspcvly, of h acuaors. h cosa rasmsso marx r R s daoal, I a R s h armaur curr vcor, ϕ( rprss h xral dsurac, sa( R dos h saurao fuco, ad u ( dos h vola corol pu. Bfor prs h formulao of h corol prolm, w rcall a usful dfo: fo : h hard saurao fuco sa ( u (, u ca dvdd o a lar fuco u ( ad a dad-zo fuco, dz( u (, u []. hus, h corol pu appld o h sysm hrouh h acuaor s xprssd as follows: sau ( ( = u ( dzu ( (, u (4 whr sa ( u( sa ( u ( =, (5 sa ( u ( dz( u(, u dz( u (, u = dz( u (, u whr dz( s h dad-zo fuco, ad u s h mum oud of h corol pu vcor. 3- Proposd orollr h prsd modl v y quaos ( o (3 s hhly olar ad dyamcally coupld mulvaral sysms ha maks h corol prolm xrmly dffcul. o ackl hs prolm, w ds a rous adapv corollr for FJER y mploy vola as corol pu sal. h procss m ( ( (3 s y ds h dsrd moor poso md for (, calld fcous corol sal, so ha h roo dyamc ca v propr prformac. h, h corol sal u ( s cosrucd (3 o sur h covrc of o m md whch rsuls a covrc of o h dsrd rajcory. d 3- - orol Law for Roo Susysm Suppos ha, Equao ( ca rwr as r ( + (, + ( + = m whr ( = r K (, (, = r K (,, ad ( = r K (. f a rror vcor as S = +Λ = ν whr = d s h lk poso rror, d R dos a dsrd rajcory h jo spac, ad Λ= da ( λ, λ, λ3,, λ wh λ > for all =,...,. h corol prolm s ow o ds h dsrd moor poso md so ha ca covrd o d. Bcaus m s o a corol varal, w ca rwr (6 y add ad surac h sam rm md as follows: ( (, ( ( r = md + whr = m md rprss h moor poso rack rror. Now, w df as md = ˆ ( v + ˆ (, v + ˆ ( + r K S md whr ˆ (, ˆ (, ad ˆ ( ar smas of (, (, ad (, rspcvly, ad K s a posv daoal a marx. For oaoal smplcy, h squl, w drop h arum ad from h marcs (, (,, ad from h vcor (. Nx, from (9 ad (8, afr som mapulao holds ha S + S+ K S= ν ν (6 (7 (9 ( whch ( = ( ( ˆ. If a corollr u ( ad som propr upda laws for, ˆ ad ˆ ar dsd so ha ˆ ˆ, ˆ, ˆ ad, ( ca v h dsrd prformac. oward hs d, w us fuco approxmao chqu o rprs, ad wh h assumpo ha propr umrs of ass fucos ar mployd ( = W, (, + ε = W + ε ( ( = W + ε β β β whr W R, W R ad W R ar β wh marcs ad β R, R ad β R ar marcs of ass fucos. h umr β( rprss h umr of ass fucos usd. Us h sam s of ass fucos, h corrspod smas ca also rprsd as ˆ ˆ (, ˆ (, ˆ = W = W ˆ ( ˆ ( = W hrfor, h rh-had sd of (9 ca wr as 3

3 ˆ ˆ ˆ = W ν + W ν + W md + r K S (3 Now, com quaos (8 ad (3, w hav a rror quao of h form S + S + K S= W ν W ν (4 W + ε whr ε (,,, = ε ε ε ε ν s h lumpd approxmao rror vcor orol Law for Moor Susysm Hr, h corol ojcv s o ds a corol pu u ( o ralz h prfc moor poso vcor (3, such ha ca hr covr o zro or a las oudd. I rfrs o h fac ha a cosa-oudd dsurac wll o dsroy h saly rsul udr rous corol md whch s a rsul of uform ulma ouddss of rack rror us Lyapuov-asd hory of uarad saly of ucra sysms []. Wh hs md, h corol pu s roducd as u ( = Kˆ ( α + ˆ ( md (5 whr K s h posv daoal cosa marx rprs ˆ a smao of K, α s a posv cosa a marx, ad ˆ( s h smao of ( calld rsdual ucra dod y ( = LI ( ( ( ˆ a + RIa K K m( + ϕ( + dz( u (, u (6 Wh Isr (5 o (3 ad from (4, afr som mapulao holds ha Kˆ ( + α = ˆ ( ( (7 If a appropra upda law for ( ca dsd, w may sur as m os o fy. oward hs d, w apply h fuco approxmao rprsao for ( as ( = W + ε (8 β β whr W R s wh marx, R s h marx of ass fuco, ad ε s h vcor of lumpd approxmao rror. I addo, h corrspod sma of h las quao s rprsd y ˆ ˆ( = W (9 hus, quao (7 ca rwr as ˆ ( K + α = W + ε ( 4- Saly Aalyss ad Prformac Evaluao o aalyz h saly of h ovrall sysm ha has saurao lms h acuaors, w d h wo follow Assumpos. Assumpo. h dsrd rfrc rajcory s assumd o oudd ad uformly couous, ad has oudd ad uformly couous drvavs up o a cssary ordr. Assumpo. m, I a, ad I a ar oudd sc h corol pu vcor s oudd [5] Saly aalyss o carry ou h saly aalyss of h closd-loop sysm formd y h roo dyamc modls ( - (3 ohr wh h corollrs (3 ad (5, a Lyapuov-lk fuco s dvsd as: V( S,, W, W, W, W = ˆ S S + K + r( W Q W + W Q W + W Q W + W QW ( β β whr r( s h rac opraor; Q R, β β β β β β Q R, Q R ad Q R ar posv df wh marcs rlad o h adapo laws. h las fuco has h follow uppr ad lowr oud whch s crucal wh h aalycal s hs work: [ ( ( ( V λ A + λ Q r W W ( + λ ( Q r( W ( ( W + λ Q r W W + λ ( Q r( W W ] whr m m [ m ( m ( ( V λ A + λ Q r W W + λ ( Q r( W W + λ ( Q r( W W + λ m ( Q r( W W ] (3 A = Kˆ. Us h propry ( (, =, h m drvav of ( alo h rajcors of sysms (4 ad ( s ˆ V = S K S Kα + S + S ε [ ( ˆ + ε r W Q ] W + νs [ ( ˆ r W Q ] W + νs (4 [ ( ˆ r W Q ] W + S [ ( ˆ r W Q W + ] L us slc h upda laws wh σ -modfcao as: ˆ W ( ˆ = Q νs W ˆ W ( ˆ = Q νs W ˆ W ( ˆ = Q S W ˆ W = Q ( Wˆ (5 33

4 whr σ ( ar posv umrs. hus, quao (4 ca rwr as: S ε V = S P S + ε (6 ( ˆ ( ˆ r W W r W W ( ˆ r W W r( W Wˆ Whr K.5I P =.5I Kˆ α (7 h wo follow codos uara ha h marx P s posv df, K >, K Kˆ α >.5I (8 Rmark. Suppos ha a suffc umr of ass fucos ar usd ad h approxmao rror ca ord, h s o cssary o clud h σ -modfcao rms (5. Hc, (6 ca rducd o V = S P S (9 ad covrc of S ad ca furhr provd y Barala s Lmma. Rmark. Ow o h xsc of ε ad ε (6, h av dfss of V cao drmd. I h follow, w wll vsa h closd-loop saly h prsc of hs approxmao rrors. I s vry asy o prov ha ε S P + ε ε λm ( P λm ( P ε ˆ r( W ( W ( r( W ( W ( r( W ( W ( (3 ohr wh h rlaoshp ad us (, w may rwr quao (6 as S V δv + [ δλ ( A λm ( P ] ε + λm ( P ε + δλ ( Q σ ( r W W + δλ ( Q σ ( r W W + δλ ( Q σ ( r W W (3 + δλ ( Q σ r( W W + r( W ( W r W W σ r( W ( W r W W whr δ s a cosa o slcd as δ λ ( P σ σ λ ( A λ ( Q λ ( Q m m,, σ σ,, λ ( Q λ ( Q h, (3 coms V ε δv + m ( P ε λ [ ( ( r W W r WW r( W ( ] W r W W hs mpls V < whvr V ε ( τ sup > δλm ( P τ ε ( τ [ ( ( r W W r W W δ r( W W r( W W ] Hc, w hav provd ha S,, W, W, W ar uformly ulmaly oudd prformac valuao From (33, w compu h uppr oud for V ( as ε ( τ δ ( ( ( + sup δλm ( P < τ < ε ( τ V V [ ( ( r W W r W W δ r( W W r( W W ] (3 (33 (34 ad W (35 Us h qualy (3, w fd h uppr oud for S as m λ ( A m δ ( V ( ε( τ + sup m ( A m ( P τ ε ( τ δλ λ < < + [ σr( W ( W r W W δλ ( A r( W W r( W W ] (36 34

5 hrfor, w compu h oud as m δ ( + δλm ( A λm ( P < τ < ε( τ m V ( λ ( A sup ε ( τ + [ σr( W ( W r W W δλ ( A r( W W r( W W ] (37 hs mpls ha h maud of h S ad moor poso rrors ar oudd y a xpoal fuco plus som cosas. hs also mpls ha y adjus corollr paramrs, w ca mprov oupu rror covrc ra. As a cosquc, ε( τ lm sup δλ ( A λ ( P ε( τ m m < τ < + [ σr( W ( W r W W δλ ( A m r( W W r( W W ] (38 h ras prformac aalyss s hus compld. Us Assumpo ( ad ouddss of, ca cocludd from h quao (9 ha s oudd sc md,, ˆ (, ˆ (, ad ˆ ( ar all oudd. Morovr, s oudd sc m ad ar oudd. hs rsuls md addo o Assumpo, yld ouddss of all h sysm s sas. h valdy of h proposd approach wll vrfd wh h xprmal rsuls o a sl-lk FJER. rack (rad Fur.h flxl lm Fur 3. Block daram of h sysm.4. srd rajcory Lk al m (s Fur 4. Oupu rack prformac Fur. Exprmal sup 35

6 S( m (s m (s Fur 5. Flrd rack rror Fur 8. Approxmao of orol sal (vol m (s Fur 6. orol sal m (s Fur 9.Approxmao of ( m (s Fur 7. Approxmao of m (s Fur. Approxmao of 36

7 5- Exprmal Sudy I hs sco, xprms ar coducd o s h prformac of h proposd corol sray. A phooraph of h xprmal sup, ha s a sl-lk flxl jo mapulaor, s llusrad F.. h flxl lm ulzd for powr rasmsso sysm s show F.. I has mad from polyurha ad s dsd so ha has a vry hh flxly. O d of h flxl lm s drcly coupld o a ard prma ma moor (characrzd y Barr-olma ompay opra wh ± vol pu ha s drv y a puls-wdh modulao (PWM drvr. h ohr d s cocd o a sl arm. wo poomrs ar salld o provd h fdacks from h moor ad h arm als. h masurd pu-oupu daa ar rasfrrd o h compur (Pum II 366 MHz y a daa acquso card wh h radmark AVANEH PL-88L. I ca sampl h aalo daa wh h mum sampl ra of khz. Also, hs card has a ul- - hh-spd A/ covrr wh h mum covrso ra of 4 khz. h daa acquso card allows us o corol h praccal mapulaor hrouh usr-dfd prorams MALAB/SIMULINK vrom. h proposd corollr s mplmd a mr-rrup srvc rou wh h -ms sampl ra. A lock daram of h sysm s show F. 3. h rfrc rajcory s a susod wav dfd as q ( d = -.5s(.4 p. h corollr paramrs wr slcd as K =, Kˆ =.6, α =, Λ=. (39 W assum ha h sysm paramrs ad hr varao ouds ar o kow. L us slc h fv frs rms of Fourr srs as h ass fuco for h approxmao. hrfor, W ˆ, W ˆ, Wˆ ad Wˆ 5 ar R. h al wh vcors for h rs ar also assd o zro. h ad marcs h upda laws ar slcd as Q = Q = Q = I, Q = I 5 5 (4 I hs sp, w assum ha h approxmao rror ca lcd, ad hc h σ -modfcao paramrs ar chos as σ ( =. Udr hs ss, h lk rajcory s h show F.4. Accord o hs furs, h lk al covrs o s dsrd valu wh a fas ras rspos, sp of lar al rack rror. h flrd rack rror (7 s also plod F. 5, whch s ll. h appld vola o h moor s v F. 6. Fs. 7 o llusra h fucos approxmaos, whch ar oudd as dsrd. hus, h proposd corollr ca ovrcom h sysm olars ad shows accpal rousss aas varous ucras. 6- ocluso hs papr prsd a rous adapv corollr for FJER cosdr ucras h oh acuaor ad mapulaor dyamcs. h corollr ds s o dpd o h mchacal dyamcs of h acuaors, hus, s fr from prolms assocad wh orqu corol sray h ds ad mplmao. I was show ha h closd-loop sysm has BIBO saly whl oas uformly ulmaly oudd (UUB saly of lk/acuaor poso rack rror asd o h drc mhod of Lyapuov. Exprmal rsuls vrfd h succssful praccal mplmao of h proposd corol sray. Exprmal rsuls showd ha rack prformac s sasfacory such ha h ffcs of jo flxly ar wll udr corol. h prformac of h corol sysm dcad ha h corol sysm s rous aas all ucras h mapulaor dyamcs ad s moors. Morovr, moor volas s prmd udr h mum valu. Rfrcs [] A. Izadakhsh, A. Akarzadh Kala, M.M. Fah, S.M.R. Raf, A rous A-Wdup corol ds for lcrcally drv roos-hory ad Exprm, Iraoal Joural of orol. Auomao, ad Sysms, 9(5 ( 5-. [] A. Izadakhsh. Rous corol ds for rd-lk flxl-jo lcrcally drv roo sujcd o cosra: hory ad xprmal vrfcao, Nolar yamcs, 85( ( [3] W. Gao, R.R. Slmc, Nural Nwork orol of a lass of Nolar Sysms wh Acuaor Saurao, IEEE rasacos o Nural Nworks, 7( ( [4] W. P,. L, J. Su, ompud orqu corol-asd compos olar fdack corollr for roo mapulaors wh oudd orqus, IE orol hory ad Applcaos, 3(6 ( [5] A. -Ro, V. Saaz, Smpl xsos of h P wh ravy compsao corol law for roo mapulaors wh oudd pus, IEEE rasacos o orol Sysms choloy, 4(5 ( [6] A. -Ro, V. Saaz, A aural saura xso of h P wh dsrd ravy compsao corol law for roo mapulaors wh oudd pus, IEEE rasacos o Roocs, 3( ( [7] E. Auaa-Ruz, A. avala-ro, V. Saaz, F. Rys, Gloal rajcory rack hrouh sac fdack for roo mapulaors wh oudd Ipus, IEEE rasacos o orol Sysms choloy, 7(4 ( [8] J. A. Ramrz, V. Saaz, R. ampa, Saly of roo mapulaors udr Saurad PI compsao, IEEE rasacos o orol Sysms choloy, 6(6 ( [9] V. Saaz, K. amarllo, J. M. Valzula, R. ampa, A praccal PI rulaor wh oudd orqus for roo mapulaors, Iraoal Joural of orol, Auomao, ad Sysms, 8(3 ( [] A. Izadakhsh, M. M. Fah, Rous Lyapuov-asd corol of flxl-jo roos us vola corol sray, Araa joural for scc ad Er, 39(4 ( [] W. P. L, B. Luo, H. Hua, Acv vrao corol of Flxl Jo Mapulaor us Ipu Shap ad Adapv Paramr Auo surac Rjco orollr, Joural of Soud ad Vrao, 363(7 ( [] A. M. Aaswamy, J. E. Wo, Adapv corol h prsc of saurao olary, Iraoal Joural of Adapv orol ad Sal Procss, ( (

8 [3] S. Purwar, I. N. Kar, A. N.Jha, Adapv corol of roo mapulaors us fuzzy loc sysms udr acuaor cosras, Fuzzy Ss ad Sysms, 5(3 ( [4] R. J. avrly,. E. lok, L. J. Brdma, J. R. Fors, Saurad proporoal drvav corol of flxl-jo mapulaors, Roocs ad ompur- Irad Maufacur, 3(6 ( [5] R. J. avrly,. E. lok, J. R. Fors, Saurad corol of flxl-jo mapulaors us a Hammrs srcly posv ral compsaor, Rooca, 34(6 ( [6] W. E. xo, Adapv rulao of amplud lmd for roo mapulaors wh ucra kmacs ad dyamcs, IEEE rasacos o Auomac orol, 5(3 ( [7]. Lu, J. Lu, W. H, Paral dffral quao oudary corol of a flxl mapulaor wh pu saurao, Iraoal Joural of Sysms Scc, 48( ( [8] A. Izadakhsh, M. Masoum, FA-asd rous adapv corol of flxl-jo roos: sular prurao approach, IEEE Idusral Socy s 8h Iraoal ofrc o Idusral choloy (II, 7, pp [9]. Qu,. M. awso, Rous rack corol of roo mapulaors, IEEE Prss, Ic., Nw York, 996. [] K. S. Nardra, A. M. Aaswamy, Sal adapv sysms, Prc Hall, El wood clffs, NJ, 989. [] W. Gao. R-R. Slmc, Adapv Nural Nwork oupu fdack orol of Nolar Sysms wh Acuaor Saurao, 44h IEEE ofrc o cso ad orol, 5, pp [] A. Izadakhsh, M. M. Fah, Ral-m rous adapv corol of roos sujcd o acuaor vola cosra, Nolar yamcs, 78(3 ( [3] A-ch. Hua, M-h. h, Adapv corol of roo mapulaors-a ufd rrssor fr approach, World scfc,. [4] A. Izadakhsh, losd-form dyamc modl of Puma56 roo arm, Procds of h 4h Iraoal of. o Auoomous Roos ad As, 9, pp [5] A. Izadakhsh, A o o h olar corol of lcrcal flxl-jo roos, Nolar yamcs, 89(4 ( Plas c hs arcl us: A. Izadakhsh, Rous Adapv orol of Vola Saurad Flxl Jo Roos wh Exprmal Evaluaos, AU J. Modl. Smul. E., 5(( OI:.6/mscj