Predictable Motion of Hyper-redundant Manipulators Using Constrained Optimization Control *

Size: px

Start display at page:

Download "Predictable Motion of Hyper-redundant Manipulators Using Constrained Optimization Control *"

Jordan Hunter
6 years ago
Views:

1 Prdctabl Moton of Hpr-rdundant Manpulators Usn Constrand Optmaton Control * Markus P.J. Fromhr, Warrn B. Jackson Xro PARC, 3333 Coot Hll Road, Palo Alto, CA 9434, U.S.A. {fromhr,wjackson}@parc.ro.com Jun 9, Abstract Hpr-rdundant robotc manpulators ar robots that hav man mor drs of frdom than rqurd for a tpcal task such as raspn an objct n 3D spac. Th lar numbr of jonts, whch ma ran from dons to thousands, offrs both opportunts and challns for th control of such robots. An opportunt s to us th tra drs of frdom to optmall control multpl objctvs. A challn s to dlvr prdctabl bhavor dspt th lar numbr of possbl confuratons. In ths papr, w propos to us a control approach basd on constrand optmaton. W dscuss a rprsntatv ampl for an undr-constrand task and sust and anal possbl solutons that nsur prdctabl bhavor.. Introducton Modular robots ar robots consstn of man ntrchanabl robotc moduls connctd nto a mchancal and functonal assmbl. Intrst n modular robots has ncrasd ovr rcnt ars du to thr potntal for robustnss, rducd costs, and wd ran of applcablt partcularl n hhl constrand nvronmnts [Chrkjan & Burdck 94, Ym 94]. Tpcall, thr ar onl a small numbr of modul tps, and ach modul tp onl has lmtd moton capablt,.., for th cas consdrd n ths papr onl on rotatonal jont. Th flblt and low cost of modular robots s achvd throuh th lar numbr of massfabrcatd moduls, pctd to ran from dons to thousands, and thr man possbl confuratons. Bcaus of th lar numbr of drs of frdom of th rdundant modular lmbs, such modular robots ar also calld hprrdundant [Chrkjan & Burdck 94]. Ths papr concrns th control of a subclass of such robots, naml hpr-rdundant manpulators that consst of a chan of moduls wth rotatonal jonts (cf. F. ). F. Modular robot prototp PolBot ( jonts) In ordr to ral th potntal of modular robots, robust, vrsatl and scalabl control alorthms must b dvlopd whl satsfn a numbr of chann jont and forc constrants. For ampl, a manpulator ma not onl b askd to achv tpcal oals such as follown a trajctor wth ts nd ffctor, but also to mnm vlocts, dstrbut torqus quall, and avod obstacls. At th sam tm, ach modul has to ob ts phscal lmts, such as lmts on jont anls and motor torqus. Fnall, n ordr to smplf prorammn dffrnt confuratons of modular robots, th softwar should also b nrc wth rspct to constrants and oals. In our work, w attmpt to robustl handl a vart of constrants for th man drs of frdom found n modular robots. W cast th control problm as a constrand optmaton problm. Th hard constrants rprsnt th (manl phscal) lmts of th robot, whl th objctv functon prsss th varous oals of ntrst (soft constrants). Furthrmor, w assum that ndffctor oals (.., poston and orntaton) ar vn b a hhr-lvl controllr such as a path plannr, and w focus on th problm of actuaton * Prsntd at IC-AI, Las Vas, NV, Jun.

2 allocaton, th problm to fndn optmal solutons for th jont anls at ach tm stp. W hav arud n favor of constrant-basd anl allocaton [Fromhr t al. 99] bcaus () t scals to larr numbrs of jonts than tradtonal, analtc control alorthms, () a vart of constrants can b asl addd thr as trms n th objctv functon or hard constrants, and (3) constrand optmaton radl handls a multtud of (somtms conflctn) objctvs. Bcaus of th complt of th objctv functon as wll as th constrants, constrand optmaton s snstv to ntal confuratons b fndn onl narb local mnma. Also, bcaus hprrdundant manpulators ar snfcantl undrconstrand, wdl dffrnt bhavors dvlop dpndn on mnor dffrncs n th startn condtons. For ampl, th foot of a spdr-robot l can b controlld to follow an arc trajctor, but th stat of th moduls n btwn foot and hp must b udd as wll to avod nffcnt or vn dstructv confuratons. In th rst of th papr, w show that that th flblt n objctv and constrant slcton n our constrand optmaton control framwork provds th ncssar tools to ovrcom ths local mnma rlatd problms. Aftr a rvw of rlatd work and our robot and task modls, w dscuss and valuat two aspcts of constrand optmaton control: () conflctn objctvs; () uncontrolld bhavors snstv to startn condtons.. Rlatd Work Prvous work on th control of rdundant manpulators has focusd on thr knmatcs, nvolvn th computaton and nvrson of th Jacoban mappn from jont spac to th ndffctor spac [Khatb 87, Ghosal & Roth 88, Chu 88]. Ths approachs tpcall nclud fw or no constrants and hav bn appld to robots wth a small numbr of drs of frdom (lss than ). Attmpts hav bn mad to nclud mor constrants b tndn Jacoban mthods wth projctd radnts of th constrants [Ballul 86] or numrcall computd radnts [Kln t al. 95]. Ths mthods can work rasonabl wll for sstms wth a small numbr of jonts and a fw wll-charactrd constrants. Howvr, modular robot control problms tnd to hav too man drs of frdom and md task spac, jont spac, and dnamc nqualt constrants to solv such sstms n ral tm. Also, bcaus ths mthods tnd to rqur a dtald constrant analss, t s dffcult to robustl adjust th prorts of dffrnt constrants ovr tm as robot nvronmnt and pos chan. Morovr, snc th solutons ar computd n vloct spac, poston rrors can buld up ovr tm. Control of hpr-rdundant robots usn contnuous backbon curvs has bn wll studd [Chrkjan & Burdck 94, Chrkjan & Burdck 95]. Ths approach scals vr wll to lar numbrs of moduls and s of partcular ntrst n hrarchcal control approachs. Aan, ths approach has onl bn studd wth knmatc constrants. Takn nto account constrants such as torqu lmts s partcularl rlvant for modular robots. Tradtonal robots ar dsnd such that thr jont torqu lmts ar nvr cdd (.., b makn th bas jonts stron and th nd jonts lhtwht). In a modular robot, all moduls ar smlar and oftn rlatvl wak. In summar, n contrast to tradtonal approachs to manpulator control, w attmpt to sparat constrants, oals, and control alorthms from ach othr, and to nclud a larr class of constrants. So far, w hav focusd on fndn ffctv constrants and objctv functons. W assum a nrc nonlnar constrant solvr and lav th dvlopmnt of a ddcatd solvr for ral-tm control to futur work. 3. Robot and Task Modls W hav prvousl prsntd a modl for our manpulator and shown how constrants and task objctvs can b prssd n trms of ths modl [Fromhr t al. 99]. W rvw ths dfntons n ths scton. Th planatons ar ncssarl brf, and w rfr th ntrstd radr to th arlr publcaton for dtals. 3. Manpulator Modl For th purpos of ths papr, w focus on a robotc manpulator that conssts of a chan of n lnks. Ths manpulator s attachd to a bas at on nd (lnk ), and th prmar task s to rach a oal poston, or follow a oal trajctor, wth th othr nd (lnk n). Such a manpulator ma, for ampl, b an arm (whch s attachd to a tabl or robot bod and whos hand nd rasps or pushs othr objcts) or a l (whch s attachd to a robot bod and whos foot nd s movn ovr th round durn locomoton). Not that for now w consdr onl statcall stabl confuratons, and, du to th rlatvl slow movmnt of th jonts, w ar not concrnd wth dnamcs.

3 Each manpulator lnk has ts own local coordnat sstm, or fram. Follown robotcs convntons [Cra 89], th knmatcs of ach lnk s dfnd b ts Dnavt-Hartnbr paramtrs α, a, θ, d, th tnsons and rotatons from on fram to th nt. Ths paramtrs dfn a homonous transform matr T, a rotaton plus a translaton, that maps a pont p dfnd n fram to pont p dfnd n fram b th multplcaton p = T p [Cra 89]. W wll also us th functonal dnotaton T = T ( α, a, θ, d ) to dscrb th matr. Transform matrcs can b composd b multpln thm. In partcular, th transformaton from bas fram to fram s vn b T = T TK T = T T In addton to th knmatcs paramtrs, ach lnk has th follown paramtrs: lnk mass m, cntr of mass poston c (n th lnk s fram), jont anl lmts θ mn, and θ ma,, mamum rotatonal vloct ω ma, achvabl b th jont, and mamum motor torqu τ ma,. Anls ar prssd n radans, tms n sconds, dstancs n mtrs, masss n klorams, and torqus n Nwton-mtrs. 3. Manpulator Constrants Th manpulator s onl actuator varabls ar th jont anls θ,,θ n, and thus all constrants ultmatl hav to b prssd as constrants on ths anls and/or thr drvatvs. Gvn th lnks paramtrs, w tak nto account at last thr tps of constrant: jont anl lmts, torqu lmts, and vloct lmts. Anl lmts. A jont s movmnt s rstrctd b th mchancs of th lnk n thr drcton of th rotaton and s dfnd trvall as θ mn, θ θ ma, for all =,,n Torqu lmts. For ach jont, th lnks from throuh n tothr rt a torqu onto jont, whch s lmtd b th strnth of th jont. As notd bfor, ths problm has tradtonall not bn addrssd n manpulator control, and rlatd work has bn rstrctd to D manpulators [Arawal & Garmlla 94, Gokc & Arawal 99]. A torqu s dtrmnd prmarl b th ravtatonal forc (appld at th cntr of ravt of lnks throuh n) and th momnt arm (th vctor from th jont as to whr th forc s appld). As a vctor, ths torqu s both prpndcular to th momnt arm and tanntal to th rotaton. Th torqu τ on jont can b computd as τ = ˆ ( R ) Z F whr Ẑ s th as of rotaton, R s th vctor from jont to th cntr of mass C of lnks throuh n, and F s th ravtatonal forc on lnks throuh n. Th nonlnar torqu constrant s dfnd as τ (θ) τ ma, for all =,,n Vloct lmts. For th movmnt from a prvous poston, w dfn as a constrant th mamum vloct achvabl b ach jont,.., ω ω ma, for all. W us a lnar appromaton for jont vloct,.., ω = (θ θ )/dt, whr dt s th tm stp for control and θ s th prvous anl valu for jont. Thus, ths constrant s dfnd as θ ω ma, dt θ θ + ω ma, dt for all =,,n 3.3 Task Modl For th purpos of ths papr, th man oal of th robot manpulator s to rach or follow a tart poston wth ts nd lnk. Thr ma b scondar oals, howvr, such as also achvn a tart orntaton wth th nd lnk, mnmn th dffrnc from a prvous poston, and dstrbutn actuaton quall. At th sam tm, th manpulator has to satsf ts constrants, and t ma not alwas b abl to achv ts oals actl. Thus, w dfn th actuaton allocaton task as th constrand optmaton problm mnm subjct to h(θ) c(θ) whr θ = {θ,,θ n } ar th fr varabls. Th hard constrants c(θ) ar thos prsntd abov. Th objctv functon s rvwd n th balanc of ths scton. Goals w hav mplmntd nclud a poston oal, an orntaton oal, mnmal actuaton nr, and actuaton dstrbuton. Th total objctv functon s a whtd, normald sum of th ndvdual oal functons h (θ): h( θ ) = w h ( θ ) w q Thus, both th ndvdual oal functons and th ntr sum ar normald to b n th ran [,]. Th whts w [,] dtrmn th rlatv prort of th oals, and th scaln factors q = ma θ (h (θ)) normal th oal functons (not dscussd n ths papr). 3

4 Poston oal. Gvn a oal poston p = [ ] T for th nd ffctor (th nd of lnk n), th oal functon s θ ) ( ) + ( ) + ( ) h ( = whr p s th poston of th nd ffctor. Our normalaton factor for ths oal s th sum of th dstanc from bas orn to oal poston and th mamum rach of th manpulator: q = a = Orntaton oal. W currntl allow on to dfn a oal twst α and rotaton θ for th nd ffctor. Ths anls dfn a fram T = T( α,, θ,). Usn Eulr s thorm on rotaton [Cra 89], w rlat th rotaton from ths fram to th fram T of th nd-ffctor wth a snl as and a snl rotaton anl Θ. Ths anl can b computd from th rotaton matr R btwn th two frams as an anl n th ntrval [,π]. W dfn th oal functon for an orntaton oal b ths rotaton anl: h (θ )= Θ Snc th mamum rotaton anl s b dfnton π, th normalaton factor for ths oal s q = π W hav oftn usd addtonal objctvs, such as mnmal actuaton nr and qual actuaton dstrbuton. S th arlr papr for dtals [Fromhr t al. 99]. 4. Achvn Prdctabl Bhavor Th problm s to us th objctv functons and constrants of th prvous sctons to obtan dsrabl trajctor-follown bhavor for a modular robot chan. Usn constrand optmaton, our approach to controlln a manpulator conssts of thr lvls: a sstm-lvl (fdback) controllr that dtrmns va ponts on th trajctor tothr wth othr objctvs and thr whts, th manpulator-lvl controllr whr a solvr dtrmns sutabl confuratons for th vn objctvs, and modul-lvl fdback controllrs that attmpt to achv th squnc of jont anls vn b ths confuratons. n Hr, w focus on th ntracton btwn th uppr two lvls. Gvn a squnc of objctvs ovr tm, th solvr fnds confuratons that optm ths objctvs and satsf th constrants. 4. Smulaton In ordr to plor th possblt of usn constrand optmaton n a control approach for modular robots, a tstbd was mplmntd n Matlab n whch a manpulator can b rprsntd accordn to th modl dscrbd n th prvous scton. In partcular, moduls ar dscrbd b th follown paramtrs: lnk twst α, lnk lnth a, rotaton θ, tnson d, lnk mass m, cntr of mass poston c, jont anl lmts θ mn, and θ ma,, mamum rotatonal vloct ω ma, achvabl b th jont, and mamum motor torqu τ ma,. In a tpcal manpulator confuraton, jonts ar attachd to ach othr at offsts of 9 drs to on anothr so that thr rotatons ar n orthoonal plans,.., α = ( ) π /. Th cntr of mass for ach jont, c, s st at th orn of fram, whr th motor would b locatd for a phscal manpulator. Lnk lnth a, lnk mass m, and mamum motor torqu τ ma, ar st such that a motor can lft appromatl 5 othr moduls wthout cdn th mamum motor torqu. Jont anl lmts θ mn, and θ ma, ar st to 8 drs and 8 drs rspctvl. Th rsults dscussd n ths papr wr achvd usn a manpulator composd of jonts (cf. F. ) foot -4-4 bas - trajctor -4-4 a) b) F. Rndrn of a) a tpcal robot l confuraton, b) snapshots from a l moton At prsnt w ar usn an actv st nonlnar constrand optmaton routn, fmncon, provdd as part of th Matlab optmaton toolbo. Ths nonlnar optmr allows us to solv th constrant optmaton problm drctl n jont spac wth a st of currnt jont anls as startn pont. In th follown valuatons, w wll contnu to us th ampl of a robot l whos foot s askd to mov on an arc trajctor (cf. F. b). Bsds 4

5 dffrnt objctvs, w also compar dffrnt startn confuratons; Fs. a and 3 show som of th varablt n possbl start confuratons for th sam bas and nd postons. Start F. 3 Two tst start confuratons 4. Conflctn Objctvs Start 3 4 On caus for undsrabl local mnma arss from multpl conflctn objctvs. In th l ampl, w tpcall hav multpl oals, such as puttn down and lftn th foot at approprat tms, puttn th foot down at th rht anl, lavn nouh claranc undr th l to avod common obstacls, tc. A soluton for ths tp of problm s to stablsh a rankn of th varous objctvs and turn off th lss mportant ons at frst whl optmn onl ovr th most mportant ons. Latr, th lss crucal objctvs ar raduall turnd on, thrb fndn a local soluton that prfrntall optms th mportant objctvs ovr th lss mportant ons. In valuatn th l ampl, th orntaton anl oal s lmnatd untl th l succssfull tracks th poston oal. Thn, th anl (orntaton) oal wht s adjustd to b nvrsl proportonal to th dstanc from th round to th foot,.., th closr th foot coms to th round, th mor mportant th anl oal bcoms. B th tm th foot arrvs at th oal, t s at th corrct orntaton as wll as th corrct poston. W compar ths dnamc adjustmnt of th anl porton of th objctv functon wth an anl oal that s fd from start to nd. Th rsults ar shown n F. 4, comparn th ffcts of fd and varabl anl oals on trajctor rror for fv dffrnt start confuratons. Varn th anl oal prforms snfcantl bttr, spcall on th hard start confuraton. In nral, constrand optmaton control wth conflctn objctvs can lad to jrk moton, whl adaptn objctv whts as ndd maks th moton smoothr and mor prdctabl avra f d anl var abl anl F. 4 Trajctor rror comparson of fd vs. varabl nd-ffctor anl oals for fv start confuratons 4.3 Uncontrolld Bhavor A scond problm arss whn th objctv functon dos not stronl dffrntat btwn a lar numbr of altrnatv local mnma. Dffrnt start confuratons can lad to qualtatvl vr dffrnt bhavors. Th snapshots n Fs. b and 5 from l motons for dffrnt objctvs ndcat that vn f th foot follows ts trajctor n th sam wa, th confuraton ovrall usuall rflcts th start confuraton (.., mddl moduls sta back, or low). - Obj. (p) from start Obj. (p,av) from start a) b) F. 5 Two l motons from start confuraton 3: a) wth just an nd-poston oal, b) wth nd-poston and varabl anl oals In such condtons, actl rproducbl bhavor s not rqurd, but qualtatvl smlar bhavor for ach run s prfrrd,.., an arc from bas to foot (to avod common objcts on th round) or a straht ln from bas to hand (to nsur mamum movmnt flblt n an drcton). In ths stuaton t s ncssar to add addtonal objctvs, whch s n contrast to th prvous scton whr w lmnatd objctvs. For th l ampl, on such soluton s to spcf oals for on or mor mddl modul postons. 5

6 Instad of spcfn oal postons, w dfn control ponts towards whch spcfc moduls ar pulld wth what w thnk of as vrtual, lastc strns. For ampl, a sutabl control pont for th l trajctor s a pont vrtcall abov th mdpont btwn bas and va pont, such that th mddl scton of th l s pulld upwards. To masur th dsrd bhavor, w us a bnchmark modl that dfns, for ach va pont on th foot trajctor, an llptcal arc from th bas to th va pont. Achvn th dsrd bhavor dscrbd abov s thn dfnd as conformn to ths arcs as closl as possbl, and w masur as arc rror th sum of th dstancs btwn l moduls and corrspondn ponts on th bnchmark arc. F. 6 shows rsults for runs wth dffrnt combnatons of objctvs from fv dffrnt ntal confuratons. Th objctvs ar foot poston (p), fd or varabl foot anl (af or av), and strns (s). W masur th total trajctor ( pos ) rror, arc rror, and tm. Th rsults show th postv mpact of strns wthout natv mpact on trajctor rror. akn to control ponts for a spln, hav provd to b a smpl and vrsatl tool n dffrnt knds of trajctor follown tasks. For ampl, spcfn strns for on or two ponts s usuall suffcnt, and th objctv can b dfnd ndpndnt of th numbr of moduls or thr partcular paramtrs. Furthrmor, as th rsults abov sust, such strn constrants not onl hlp nforc dsrabl robot confuratons, but also hlp rduc th trajctor follown rror. For ampl, w hav found stuatons whr th mddl part of th manpulator actuall ts n th wa of th ndffctor unlss t s pulld out of th wa n tm avra (p) (p,av) (p,af ) (p,av,s) (p,af,s ) (p,s) a) (p,s) (p,af,s) (p,av,s) (p,af) (p,av) (p) pos rror arc rror tm 5 F. 6 Masurmnts of trajctor (pos) rror, bhavor bnchmark (arc) rror, and tm for dffrnt combnatons of objctvs whn controlln l moton In F. 7, th masurmnts ar furthr brokn down b start confuratons. Ths rvals aan how som combnatons of objctvs dpnd on th start confuraton (and thus ar lss prdctabl), whl othrs ar mor ndpndnt of th ntal condtons. W hav found that th addton of objctvs for undr-constrand parts of a hpr-rdundant manpulator usuall maks ts controlld bhavor snfcantl mor robust. Th us of strn constrants n partcular, whch ud th robot avra (p) (p,av) (p,af ) (p,av,s) (p,af,s ) (p,s ) F. 7 Dtals of th masurmnts of F. 6, brokn down b start confuratons: a) trajctor rror, b) bhavor bnchmark (arc) rror b) 6

7 5. Conclusons In ths papr, w hav prsntd constrand optmaton basd control for hpr-rdundant, rconfurabl manpulators that can b tndd radl wth addtonal constrants on jonts and sts of jonts wthout dtald analss. In fact, constrants and objctvs can b addd dnamcall durn opraton of th robotc control sstm. Ths approach was appld to a problm of rachn varous objctvs such as oal poston and actuaton dstrbuton for a modular robot. In ths applcaton w found that, whl th constrand optmaton basd control was scalabl and flbl, occasonall t would fall nto undsrabl local mnma. Whn th problm was ovr-constrand wth conflctn objctvs ladn to a lar objctv rror, squntal adaptaton of prortd oals provd to rsult n ratl mprovd bhavor. In th scond tp of problm, th local mnma and start pont dpndnc aros from an undr-constrand spcfcaton. In ths cas prdctabl bhavor was obtand b addn trms to th objctv functon. Spcfn th poston of ntrnal moduls n addton to th nd moduls rsults n bttr bhavd solutons from th constrand optmaton. In both cass, th adjustmnt of th objctvs s prformd dnamcall as th trajctor s followd. Ths dnamc fdback btwn optmr prformanc and adaptaton of th objctv functons from th suprvsor lvl has th potntal to snfcantl mprov th robustnss of th constrand optmaton approach for hpr-rdundant modular robot control. In: IEEE Trans. On Robotcs and Automaton, vol., 995, pp [Chu 88] S. L. Chu, Task compatblt of manpulator posturs. In: Int. Journal of Robotcs Rsarch, vol. 7, no. 5, 988, pp. 3-. [Cra 89] J. J. Cra, Introducton to Robotcs Mchancs and Control. Addson Wsl, 989. [Fromhr t al. 99] M. P. J. Fromhr, M. Hobrchts, and W. B. Jackson, Towards Constrant-basd Actuaton Allocaton for Hpr-rdundant Manpulators. In: CP 99 Workshop on Constrants n Control (CC'99), Alandra, VA, Oct publcatons/sm-cc99-abstract.html [Ghosal & Roth 88] A. Ghosal and B. Roth, A nw approach for knmatc rsoluton of rdundanc. In: Int. Journal of Robotcs Rsarch, vol. 7, no., March/Aprl 988, pp [Gokc & Arawal 99] A. Gokc and S. K. Arawal, Mass cntr of planar mchansm usn aular paralllorams. In: Trans. of th ASME, vol., March 999, pp [Khatb 87] O. Khatb, A unfd approach to moton and forc control n robotc manpulators: Th opratonal spac formulaton. In: IEEE Journal on Robotcs and Automaton, vol. 3, no., 987, pp [Kln t al. 95] C. A. Kln, C. Chu-Jnq, and S. Ahmd, A nw formulaton of th tndd Jacoban mthod and ts us n mappn alorthmc snularts for knmatcall rdundant manpulators. In: IEEE Trans. On Robotcs and Automaton, vol., 995, pp [Ym 94] M. Ym, Locomoton wth a Unt-Modular Rconfurabl Robot. Ph.D. Thss, Dpt. of Mchancal Ennrn, Stanford Unvrst, Rfrncs [Arawal & Garmlla 94] S. K. Arawal and R. Garmlla, Workspac boundars of fr-floatn opn and closd chan planar manpulators. In: Journal of Mchancal Dsn, vol. 6, March 994, pp. 5-. [Ballul 86] J. Ballul, Avodn obstacls and rsolvn knmatc rdundanc. In: Proc. 986 IEEE Int. Conf. on Robotcs and Automaton, 986, pp [Chrkjan & Burdck 94] G. S. Chrkjan and J. W. Burdck, A modal approach to hpr-rdundant manpulator knmatcs. In: IEEE Trans. On Robotcs and Automaton,, 994, pp [Chrkjan & Burdck 95] G. S. Chrkjan and J. W. Burdck, Knmatcall optmal hpr-rdundant manpulator confuratons. 7

8-node quadrilateral element. Numerical integration

8-node quadrilateral element. Numerical integration Fnt Elmnt Mthod lctur nots _nod quadrlatral lmnt Pag of 0 -nod quadrlatral lmnt. Numrcal ntgraton h tchnqu usd for th formulaton of th lnar trangl can b formall tndd to construct quadrlatral lmnts as wll