Minimally Actuated Serial Robot

Transcription

1 Mnmally Actuatd Sral Robot Mosh P. Mann, Lor Damt, Davd Zarrouk Abstract In ths papr, w propos a novl typ of sral robot wth mnmal actuaton. Th robot s a sral rgd structur consstng of multpl lnks connctd by passv jonts and of movabl actuators. Th novlty of ths robot s that th actuators travl ovr th lnks to a gvn jont and adjust th rlatv angl btwn th two adjacnt lnks. Th jonts passvly prsrv thr angls untl on of th actuators movs thm agan. Ths actuaton can b appld to any sral robot wth two or mor lnks. Ths unqu confguraton nabls th robot to undrgo th sam wd rang of motons typcally assocatd wth hypr-rdundant robots but wth much fwr actuators. Th robot s modular and ts sz and gomtry can b asly changd. W dscrb th robot s mchancal dsgn and knmatcs n dtal and dmonstrat ts capablts for obstacl avodanc wth som smulatd xampls. In addton, w show how an xprmntal robot fttd wth a sngl mobl actuator can manuvr through a confnd spac to rach ts targt. Indx Trms hypr-rdundant robot, mnmal actuaton, moton plannng, mobl actuator I. ITRODUCTIO Hypr rdundant robots ar robots wth srally connctd lnks that possss a larg knmatc rdundancy. Altrnatvly known as snak robots, thy ar th subjct of xtnsv rsarch ovr th past svral dcads [1] [] [3]. wth many dffrnt confguratons, mchansms, control stratgs, and moton plannng algorthms bng proposd ovr th yars. Th prncpl motvaton for dvlopng hypr rdundant robots s thr ablty to navgat around obstacls and n hghly confnd spacs. Algorthms for plannng th moton of hypr rdundant robot prsnt a formdabl challng [4] [5]. Early moton plannrs for hypr-rdundant robot moton plannng wr dvlopd by Grgory Chrkjan n [6] [7] [8] [9]. In thos works, th curvatur of th robotc snak was approxmatd as a contnuous modal functon wth th obstacls xprssd as boundary constrants on th robot s shap. Many rcnt works hav addrssd obstacl avodanc schms for hypr rdundant robots. Stat-of-th-art approachs ncludng gntc algorthms [10] [11], varatonal mthods [1], and probablstc roadmaps [13] ar usd to plan th motons of th robots. Thr s a contnuous progrss n rducng th plannng tm and mprovng thr capablty n ral lf scnaros such as robotc surgry, agrcultur and sarch and rscu. altrnatv. Also known as soft robots or contnuum robots, thy consst of a flxbl contnuous structur that possss, at last n thory, an nfnt numbr of dgrs of frdom. Th advantag of flxbl robots ovr hypr-rdundant robots s thr lghtwght and spd. Howvr, thr s stll ongong rsarch to mprov thr accuracy, control and poston and snsng capablts (s [14] and [15]). In ths work, w propos th Mnmally Actuatd Sral Robot (MASR) whch combns som charactrstcs and advantags from both hypr rdundant robots and complant robots. Th MASR s a sral robot consstng of multpl lnks connctd by passv jonts and of a small numbr of movabl actuators. Th actuators translat ovr th lnks to any gvn jont and adjust t to th dsrd angular dsplacmnt. Th jont passvly prsrvs ts angl untl t s actuatd agan. Th numbr of dgrs of rconfgurablty (DOR) s qual to th numbr of jonts. Ths nabls th MASR to achv smlar moblty (albt slowr) to rgular hypr rdundant robots. Th advantags of MASR ar ts smplcty, smallr wght, hghr nrgy dnsty (powr/mass), low cost and modularty, as th numbr of lnks and actuators can b asly and quckly changd. W dscrb th mchansm of th MASR n Scton II. In Scton III, th knmatcs of th robot ar outlnd. Scton IV provds som xampls of moton plannng around obstacls that th MASR achvs. In Scton V, w dmonstrat how th MASR can duplcat th moton of a fully actuatd hypr-rdundant robot to any dsrd dgr of accuracy. Svral xampls of ths ar gvn n Scton VI usng multpl lnks and sngl mobl actuator. Conclusons and drctons for furthr rsarch ar gvn n Scton VII. II. MECHAISM DESCRIPTIO AD KIEMATICS Our novl robot systm s composd of lnks connctd through passv jonts, M mobl actuators that travl ovr th lnks, and an nd ffctor as shown n Fgur 1. Th passvty of th jonts s dfnd by thr bng no motors n btwn thm, whl th angl btwn adjacnt lnks s prsrvd. Th numbr of lnks and mobl actuators can b asly vard dpndng on th proposd task. Whn a mobl actuator travls ovr th lnks, t can rotat th dsrd jont thrby changng th rlatv angl btwn th lnks by a dsrd angl. Th bas s whr th robot s connctd to a constant support or a mobl platform. In paralll, flxbl robots hav bn dvlopd as an

2 n,, 1 n n M passng through a pont, T p u = p1 p gvn by: 0 J A p (4) J U Th constrant on th st of jont angls θ = [θ 1, θ,, θ ] n c-spac s thus xprssd as: p k, k (, ) (5) u a na a aja Fgur 1. A D prototyp of th Mnmally Actuatd Robotc Snak. Th robot n ths fgur has 10 lnks, on mobl actuator, and an nd ffctor. Th mobl actuator can frly travl ovr th lnks and rotats thm upon command. For smplcty, w assum that ach lnk s of unform lngth L. Th angl btwn th -1th and th sral lnk s dnotd by θ. Th orntaton of ach lnk n world coordnats s α and ts poston s gvn by: l l1 (1) k k k 1 l1 k 1 l1 x, y L cos,l sn l l () Th coordnat of th jth actuator s gvn by th par ( n, ), whr n j s th lnk at whch actuator j s j j currntly locatd and θ j s th angl of th actuator and th jont that th actuator s currntly actuatng, bng that th lattr two must b qual. Th actuator angl has th sam rang as θ. W dnot th st of actuatd jonts as J A and th st of unactuatd jonts as J U, gvn formally by A J n n 1, M J 1, \ J U Th confguraton spac of th robot, assumng thr ar jont lmts, s an dmnsonal cub I, whr I s opn on dmnsonal ball. Howvr, th rducd actuaton of th sral robot rsults n a vry sgnfcant knmatc constrant. For any gvn st of actuator locatons n 1,n, n M, th moton of th robot s confnd to an M dmnsonal manfold mbddd n I. Ths manfold s an M dmnsonal plan n th coordnat spac spannd by th unt vctors A (3) Translatng th actuators of th robot thus corrsponds to movng th manfold to a dffrnt plan n coordnat spac. Ths has sgnfcant ramfcatons for moton plannng, as wll bcom apparnt n Scton III. Th trajctory of th robot through confguraton spac s gvn by th paramtrzd curv f :[0,1] that s not C 1 contnuous. Th total tm t total rqurd for th robot to rach a goal s thus comprsd of th tms rqurd to rotat ach jont plus th tms rqurd to travrs th actuator from on lnk to anothr plus a crtan ntrrupton dlay btwn th translaton and rotaton. Th lattr two ar a consumpton of tm unqu to th MASR robot, and t s th prc w pay for usng lss actuators than jonts thr must b a tmshar of th actuator btwn th lnks. If w assum constant translatonal spd V of th mobl actuator and constant rotatonal spd ω, and that th dlay s T dlay, thn th tm rqurd to prform a task s: T TOTAL L STEP n STEP STEP *T DELAY V whr STEP s th total numbr of stps, and Δn and Δθ ar th rspctv numbr of lnks and rotaton travrsd durng stp. Th total nrgy E consumd by th MASR, assumng a lnar modl of nrgy consumpton dpndnc on coordnat dsplacmnt, would b proportonal to th numbr of actuator translatons and jont dsplacmnts: TOTAL STEP STEP E k L n k n (6) (7) whr k n and k θ ar coffcnts that can b dtrmnd mprcally. Ths travrsals of th actuator ar th mor tm consumng acton, and t would thrfor b dsrabl to mnmz th numbr of travrsals. Ths would consttut a sutabl optmzaton goal of any moton plannng algorthm for th MASR robot, and s th subjct of ongong rsarch. III. FULLY ACTUATED MOTIO DUPLICATIO Th mnmal actuaton of th sral robot mans that ts

3 moton s mor lmtd than that of a fully actuatd sral robot. Th motons xcutd by a fully actuatd robot cannot b compltly mmckd by th MASR. Howvr, on may dsr to approxmat th motons of a fully actuatd robot wth an MASR to wthn a crtan dgr of accuracy. In approxmatng th moton of th robot, thr ar two possbl gnral objctvs: to approxmat th moton of th nd ffctor n th work spac, or to approxmat th moton of th robot n coordnat spac, or c-spac for short.. th jont angls. Th formr objctv sms to b th mor convnnt and usful goal, as th postonng of th nd-ffctor s what dfns th accuracy of th task for many robotc applcatons. Howvr, th constrants on moton, xprssd by Equaton (5), ar on th jont angls. Thrfor, t s mor straghtforward to xprss rror bounds on th jont angls of th robots than on th nd-ffctor. Ths rror bound s dnotd by δ, and s usd as a masur of th closnss of approxmaton of th MASR robot to a fully actuatd robot. To ths nd, w formulat th followng dfnton: Dfnton 1: A curv f(t) s a δ p approxmaton of a curv g(u) f for all t [0,1], thr xsts u [0,1] and for all u [0,1], thr xsts t [0,1] such that f( t) g( u) p. In othr words, f all ponts along th trajctory of th MASR n c-spac ar clos to at last som pont along th trajctory of th fully actuatd robot and vc vrsa.. wthn a sphr of radus δ, thn th moton of th robot s suffcntly approxmatd. Although any norm can b usd to dfn th sphr, w slct th -norm. Ths mans that f w dnot th th dmnson of a pont g(u) as g (u), thn: max f (t ) g (u ) 1,, (8) Dnotng f(t) = θ(t) as rprsntng th confguraton of th MASR and g(u) = θ 0(t) =[θ 10,θ 0,, θ 0] that of th fully actuatd robot, thn Equaton (8) s quvalnt to: (9) 0 1, Th rason for ths slcton s bcaus th constrant of Equaton confn th jont angls to an M-dmnsonal plan spannd by M vctors paralll to M out of axs. Ths plan s by dfnton concdnt wth th surfac of an - dmnsonal cub. Equaton (8), whch uss th -norm, dfns a cub n -dmnsonal spac, and thrfor th - norm s th most natural on to us. On mght ask f t s knmatcally possbl for th mnmally actuatd robot to approxmat th motons of a fully actuatd robot n any arbtrary confguraton spac and to any dgr of accuracy. Th answr s ys: Lmma 1: For all g(u), thr xsts a δ p approxmaton for all δ > 0. Such an approxmat curv can b constructd for th p= norm usng th procdur APPROXIMATIO- CURVE. A flowchart of APPROXIMATIO-CURVE s shown n Fgur. Procdur APPROXIMATIO-CURVE Inputs: a C 0 curv g n dmnsonal spac g : [ 01, ] curv paramtr u [ 01, ] numbr of actuators M rror norm δ > 0 Output: an array of -dmnsonal ponts [x(1) x() x(nd)] rprsntng th path of th MASR whch travrss n a straght ln n coordnat spac from x(j) to x(j+1) 1. Start at u 0 = 0 and x(1) = g(0).. Usng a nonlnar quaton solvr, fnd th lowst u > u 0 for whch 1 g( u) g ( u0). Labl ths u. 3. Construct an dmnsonal hyprcubod spannd by cornrs g(u 0) and g(u ). Th hyprcubod s dscrbd by th st of all ponts x such that max g (u ),g (u ) x mn g (u ),g (u ) Fnd th shortst path btwn g(u 0) and g(u ) along an M dmnsonal surfac of th hyprcubod. Such tchnqus for fndng th shortst path ar outlnd n [17]. Ths path wll consst of straght lns connctng -M vrtcs btwn g(u 0) and g(u ). 5. Appnd ths vrtcs [y(1) y() y(-m) g(u )] to th nd of [x]. 6. St u 0 = u. 7. Rturn to Stp and rpat th procss untl Proof: From Stp, th mtrc dstanc btwn g(u 0) and all g(u) from u 0 and u s lss than or qual to δ/. Stp 4 constructs th porton of f(t) that ls on a surfac of a hyprcubod btwn cornrs g(u 0) and g(u ). Labl th nds of th doman of ths porton t 0 and t, rspctvly. Bcaus all ponts n th hyprcubod hav a mtrc dstanc of lss than g(u )-g(u 0) from any of ts cornr s as Stp 3 dscrbs, w hav: 1 f ( t) g( u0) g( u) g ( u0), t t0, t (10) Usng th trangl nqualty for normd mtrc spacs and applyng th rsult of Equaton (10) ylds:

4 f ( t) g( u) f ( t) g( u ) g( u ) g( u ) , t t0, t, u u0, u (11) of th nd ffctor gvn δ? In othr words, f th absolut dvaton of ach angl of th MASR from th fully actuatd robot s lss than or qual to δ, x s th ndpont of th MASR, and x 0 s th ndpont of th corrspondng fully actuatd robot, thn what s t [ 01, ] 0 max x (t ) x (t ) f (1) whr f(δ) s an xplct formula rlatng th angular dvaton δ to th -norm of th ndpont dvaton? To calculat ths dpndnc, w rwrt th poston of th ndpont by combng Equatons (1) and () as:, cos sn x y L,L 1 1 x, y cos sn L,L (13) whr α s th orntaton of th th jont of th MASR and α of th fully actuatd robot. Usng Equaton (13) to xprss th rror norm btwn th two ndponts ylds: x x L cos cos, sn sn k1 k1 0 0 L cos cos sn sn k1 k1 (14) Fgur. Flowchart of APPROXIMATIO-CURVE. Ths algorthm constructs th trajctory of th MASR robot to track a fully actuatd robot whl adhrng to th moton constrants of M actuators. Stp 5-7 construct f(t) contnuously for th ntr doman of t. Th nqualty of Equaton (11) thus holds for all of f(t) and thrfor satsfs Dfnton 1. Th rsultng path s clarly C 1 dscontnuous. It gos wthout sayng that th closr th approxmaton s, th mor tms th actuator wll hav to translat btwn lnks, thrby lngthnng th tm consumd. By makng us of th trgonomtrc dntts A B A B sn A sn B cos sn A B A B cos A cos B sn sn th trms nsd th root symbol of Equaton (14) bcom 0 0 cos cos sn sn sn 1 sn cos 1 sn 0 0 (15) (16) Rarrangng Equaton (16), nsrtng th rsult nto Equaton (14), and squarng ylds: IV. ERROR AALYSIS Whl th aformntond approxmaton procdur prsrvs th moton of th MASR wthn th jont rror n c-spac, th rror of th robot ndpont n th robot workspac s n gnral th ultmat concrn. Ths lads to th quston: How can w dtrmn th bound on th rror

5 1 x x L sn *... (17) sn 0 cos sn 0 sn j j0* L sn 0 sn j j j 1, j 1 1 cos 0cos j j0 Usng th trgonomtrc dntts cos sn 1 cos A B cos Acos B sn Asn B Equaton (17) bcoms: 1 x L sn... x L 1 j 1, j 1 1 sn 0 sn j j0*... 1 cos 0 j j0 (18) (19) To dtrmn th bounds on ach lnk s orntaton rror α - α 0, nsrt Equaton (1) nto Equaton (9) and apply th trangl nqualty to obtan: 0 k k 0 k k 0 k1 k1 (0) Applyng th nqualty of Equaton (0) and th smpl nqualts cos 1 R (1) whl kpng n mnd that sn θ s a monotoncally ncrasng functon for -π/ < θ < +π/, Equaton (19) ylds j x x 0 4L sn sn sn 1 1 j 1, j () Rarrangng th rght hand sd of Equaton () nto polynomal form ylds: x x 0 4L sn 1 (3) Thus, for suffcntly small δ, takng th squar of Equaton (3) ylds th root man squar of th nd ffctor as a functon of δ: x x 0 L sn (4) 1 Corollary 1: For planar robots whr th th jont s usd to st th orntaton Θ of th ndpont and th frst -1 jonts ar usd to st ts poston, t follows drctly from th abov analyss that th rspctv rror bounds on poston and orntaton ar gvn by: 1 0 x x L sn (5) 0 1 V. EXAMPLES OF ROBOTS WITH THREE DEGREES OF RECOFIGURABILITY W dmonstrat th constructon of an approxmaton curv for two rlatvly smpl MASR wth thr rvolut jonts: on wth two actuators and on wth on actuator. Each lnk s 10cm long and 1 cm thck. Th MASR s taskd wth translatng from pont A to pont B, movng a cup uprght along a ln, drawng th lttr Z, and drawng a crcl. Th trajctory of th robot must b satsfd wthn th gvn rror radus of th coordnat spac of a fully actuatd robot. A. Robot movng tp from pont A to pont B Th smplst task possbl for a robot s to mov ts ndffctor from on pont to anothr. Th c-spac trajctory of th 3DOF robots prformng that task s shown n Fgur 3. Th trajctory n c-spac can tak any form that starts at th ntal coordnat θ A and nds at th fnal coordnat θ B. It must b mphaszd that angls n c-spac can affct both poston and orntaton of th ndpont. Assumng that all maxmum angular vlocts ar ω and ach angl rotats ndpndntly, th tm of travrsal s smply max [1, ] T (6) For a -actuator robot, th trajctory s confnd to a srs of two-dmnsonal plans dscrbd by Equaton (5). Ths plans consttut th surfac of th blu box shown n Fgur 3. Th axs that span th plan rprsnt th jonts that ar actuatd durng th travrsal. For xampl, th rght sd of th cubod n Fgur 3 s spannd by θ and θ 3; any c-spac trajctory on ths plan mans that th robot s actuatd at jonts and 3. Travrsng across thr dmnsonal c-spac ntals th trajctory travrsng at last two plans,.. t must cross at last on boundary btwn to plans. Dnotng k as th jont angl that rtans an actuator durng both phass of actuaton, th tm s thus gvn by k k j max, max, L T T V, j, k {1,,3} (7) snc thr s only on par of actuator translatons. DELAY

6 Th on-actuator robot s confnd to a srs of ondmnsonal plans,.. lns. Ths lns consttut th dgs of th blu box n Fgur 3. Th tm of travrsal would b gvn by Equaton (6). Fgur 4. Snapshot of MASR robot transportng a cup along th blu ln shown. Th actuator, rprsntd by th rd rctangl, translats from jont to jont. Th trac of th nd-ffctor s trajctory s shown by th black ln. For all fgurs n ths artcl, th unts ar normalzd by th lngth of a sngl lnk. Fgur 3. Th trajctory of th robots to travrs from ntal confguraton θ A to fnal confguraton θ B n thr dmnsonal c-spac. Th trajctory of th fully actuatd robot s rprsntd by th black ln. It has no spatal constrants. Th trajctory of th -actuator robot, shown by th grn ln sgmnts, s confnd to th surfacs of th hyprcubod shown n blu, and that of th 1-actuator robot, shown by th rd ln sgmnts, s confnd to ts dgs. B. Robot orntaton and poston Ths planar robot has thr dgrs of frdom: two for locaton and on for orntaton. Its task s to mov a glass of watr along a straght ln whl kpng t uprght. Th actuator translats along th robot lnks, altrnatng btwn th poston-sttng jonts (jonts 1&) and th orntatonsttng jont (jont 3). A tm-laps snapshot of th robot s shown n Fgur 4. Th trajctory of a fully-actuatd sral robot n c-spac that movs th cup s rprsntd by th blu dottd curv n Fgur 5. W st δ = 0.1 rad. Followng th aformntond procdur, th trajctory for th MASR wth two actuators s shown by th squnc of dagonal grn ln sgmnts on th surfac of th cubods, whl that of th MASR wth on actuator s shown by th squnc of straght rd ln sgmnts on th dgs of th cubods. Each sgmnt s confnd to th two dmnsonal surfac of ts rspctv cubod. Ths cubods ar constructd n Stp 3 of APPROXIMATIO-CURVE. Evry tm that th trajctory movs onto a dffrnt fac or cubod, on of th actuators commuts to a dffrnt jont. Th MASR ffctvly tracks th fully actuatd robot, nsurng that th maxmum dvaton of corrspondng jont angls btwn th two s nvr mor than δ. Fgur 5. Trajctory of th MASR robot movng a cup n confguraton spac. Th blu dottd ln s th trajctory of th fully actuatd robot. Th lmtd actuaton of th MASR robots rsults n th constrant on th MASR trajctory n c-spac; for any gvn st of actuator locatons, th trajctory s confnd to th surfac (two actuators-grn ln sgmnts) or dgs (on actuator- rd ln sgmnts) of th cubods shown n blu. Th nd-pont rror norm dfnd by Equaton (1) s only rlvant whn both th MASR ndpont x and th fully actuatd ndpont x 0 ar paramtrzd by th sam ndpndnt varabl yldng a on-to-on corrspondnc. Howvr, such a paramtrzaton s not ncssary; th ndpont rror norm may b dfnd n a smlar mannr to th c-spac rror of Dfnton 1. Usng th notaton of Dfnton 1, w dnot th paramtrzd rspctv ndponts as x (t) and x 0(u). Th ndpont rror norm Δ s dfnd as th largst of th dstancs btwn th closst dstancs btwn any two ponts on x (t) and x 0(u): 1 1 : C C R max mn (t ) 0(u) t u x x (8) Smlarly, th orntaton rror at th nd ffctor, bng by dfnton th sum of th angular dffrncs, s gvn by

7 0 0 1 (9) Ths rror norm, along wth th lmt on th rror norm of Equaton (5), ar shown n Fgur 6. Ths valdats th analyss of Scton IV th fgur clarly dmonstrats that th actual rror s always lss than th rror bound, although th gap btwn thm grows wth ncrasng δ. sgnfcant mplcatons for th slcton of actuators of th MASR. Th ndpont rror of th MASR robot compard wth ts thortcal lmt gvn by Equaton (4) s prsntd n Fgur 9. Fgur 7. Output of th nd-ffctors of th MASR robot attmptng to draw th lttr Z undr maxmum angular dvaton of δ = 0. rad. As th fgur dmonstrats, th nd-ffctor dvaton for th sngl actuator robot s gratr than for th two-actuator robot, vn though both ar boundd by δ. Fgur 6. Th robot nd-ffctor rror Δ (plus sgn & astrsks) and th calculatd rror lmt (crcls) as a functon of th jont angl lmt for both poston rror (top) and orntaton rror (bottom). As xpctd, th actual rror s blow ts maxmum possbl. Bcaus th lattr has thr rvolutonary jonts whl only two ndpont coordnats x,y, t has on rdundant DOF. Thr ar many dffrnt tchnqus for rsolvng jont rdundancy and dffrnt objctvs for thr rsoluton. Howvr, th mthod w slct to rsolv ths rdundancy s by slctng th jont angls so as to maxmz th dtrmnant of J T J whl constranng th ndponts to stay on targt, whr J s th Jacoban. Ths mthod s chosn bcaus t s a standard objctv n robotcs that ylds th maxmum manpulablty, or th ablty to xrt any dsrd moton at th manpulator s nd ffctor. Ths was accomplshd usng th fmncon functon n th MATLAB Optmzaton Toolbox. C. Robot drawng th lttr Z Th output of th robots nd ffctors n tracng th lttr Z s shown n Fgur 7. A snapshot of th MASR drawng s shown n Fgur 8. For robot applcatons whr th nd ffctors ar taskd wth tracng a path, ths rsult has Fgur 8. Snapshot of MASR robot drawng th lttr Z. Th actuator, rprsntd by th rd rctangl, translats from jont to jont. A planar robotc task can b achvd wth a mnmum of two lnks and two rvolutonary jonts. It thus may appar at frst glanc that havng two movabl actuators runnng along thr lnks s an unncssary complcaton. Howvr, th xtra dgr of rdundancy s ncssary for nablng th robot to navgat around obstacls. In addton, th thr DOF provd th robot wth xtra manuvrablty and dxtrty that cannot b achvd wth a DOF robot. Most mportantly, th thr lnk robot s manly a proof of concpt for largr hypr-rdundant robots wth many dgrs of frdom. Th ffct of lowrng th c-spac rror radus on th numbr of actuator travrsals n drawng th lttr Z s shown n Fgur 10. As xpctd, th tghtr th rror bound s, th mor th actuators must swtch btwn th jonts of th MASR. As thr ar two surfacs and thr dgs btwn

8 oppost dgs of a thr dmnsonal cubod, th numbr of travrsals for th sngl actuator MASR wll always b 50 prcnt mor than that of th doubl actuator MASR. Ths s bcaus ach cubod ntals two travrsals for th lattr, whl thr for th formr. D. Robot drawng a crcl. Th rsults of th sam MASR robot drawng a crcl s shown n Fgur 11. Th crcl has a radus of cm and ts orgn s locatd at (10cm, 0cm) from th robot bas. Bcaus th task workspac for th crcl s smallr than that of th Z, th rspctv MASR rror norm must b corrspondngly smallr. Th outln drawn n Fgur 1 s th rsult of δ = 0.01 radans. Th numbr of actuator travrsals and total tm rqurd to draw th crcl ar shown n Fgur 11. Onc agan, th smallr th rror bound s, th mor translatons ar rqurd. Fgur 9. Th robot nd-ffctor rror Δ (astrsks) and th calculatd rror lmt (crcls) as a functon of th jont angl lmt. As xpctd, th actual rror s blow ts maxmum possbl. Th actual rror for th sngl motor robot dcrass past a crtan rror norm bcaus th dvaton of th robot from ts trajctory placs t closr to othr ponts along th fully actuatd robot s trajctory. Thus, usng th dfnton of Equaton (8) to dscrb th ndpont rror may not b th most usful dfnton. If th total tm for travrsal s masurd, rathr than just th numbr of actuator shfts, thn a smlar pctur mrgs. Assumng a vry smpl knmatc modl whr th rotaton consums a constant tm pr radan t r and a constant tm pr actuator translaton t s, th total tm consumd s gvn by Equaton (6). Th total tm for th robot drawng th fgur Z s also shown n Fgur 10, whr t r s takn to b 1.0 sconds pr radan and t s s 1.0 sconds. Hr too, th tm consumpton sharply ncrass for ncrasngly small rror rad. Fgur 11. Output of th nd-ffctors of th MASR robot attmptng to draw a crcl undr maxmum angular dvaton of δ = 0.01 rad. Fgur 1. umbr of actuator travrsals and total tm rqurd for th MASR robot to trac a crcl whl rmanng wthn th c-spac norm. Fgur 10. umbr of actuator travrsals and total tm rqurd for th MASR robot to transport an uprght cup whl rmanng wthn th c-spac norm. VI. EXAMPLES WITH HIGHLY REDUDAT COFIGURATIOS To dmonstrat th capablts of th MASR, w smulat a moton plannng stuaton wth obstacls as summarzd n Fgur 13. In ths scton, th plannng was prformd by th human oprator. Th MASR n ths xampl conssts of a bas and tn lnks and jonts (10 DOF) actuatd by on mobl actuator. Th goal of th robot s to grab th blu

9 crcl and brng t back to th robot s orgnal confguraton. Th task s composd of two man challngs. Th frst s gong through th narrow pass of 15 mm, and th scond s rachng th targt wth th small scton of th robot that wnt through th opnng. Throughout th whol task, th robot must avod colldng wth th obstacls. Th robot accomplshs ths task by havng th motor translat and adjust th angls of th jonts on at a tm. Th robot frst passs through th narrow pass by transformng ts scond half nto an arc lk shap. Thn, th mobl actuator passs through th pass and thn rotats th lnks to rach th targt. Snc four jonts and lnks wnt through th pass, th robot had four dgrs of frdom to rach ts targt (only thr ar rqurd n a D spac to rach locaton and orntaton). In total, only ght translatonal stps for th motor ar rqurd n ach drcton, dmonstratng th dxtrty and manuvrablty of th MASR. TABLE I. MOTIO SUMMARY OF MASR. DURIG EACH ACTIO, THE MOBILE ACTUATOR ROTATES A SPECIFIC JOIT BY A AGLE Θ OR ADVACES FROM JOIT (START) TO AOTHER (ED). STEP Turnng [dgrs] (jont/angl) Translaton (start-nd) Rachng th targt (1-1) +45 (1-) 3-45 (-6) 4-45 (6-7) 5-45 (7-9) 6-45 (9-) (-9) (9-10) Rturnng aftr graspng 9-75 (10-10) (10-9) (9-) (-10) (10-9) (9-7) (7-6) (6-) (-1) total (absolut) 840 [dgrs] 48 L Fgur 13. Snapshots of th anmaton of MASR quppd wth a sngl mobl actuator rachs ts targt. Startng at (a), th mobl actuator advancs to th cntr aftr rotatng th bas lnk (b). At (c), th mobl actuator rotats th sx top lnks to mak an arc shap and thn rturns to th bas (d) to rotat th lnks and pntrat through th small cavty. Th actuator travls agan to th top lnks to rotat thm towards th targt (). Aftr rachng ts targt, th robot maks th nvrs plan of a-b-c-d- to rturn to ts orgnal confguraton (f). Th bottom row of Tabl I shows that th sum total of dgrs that th lnks rotat quals 840, and th actuator translats a total of 48 lnk-spans. Th total tm of th manuvr thus quals th tm rqurd to prform both mods of acton. Wth optmal moton plannng, howvr, th lattr should b rducd to ts mnmum possbl. Basd on Eq.(6), th tm rqurd for th locomoton s t TOTAL 48L tdelay (30) V As shown n Tabl I, ach stag of moton conssts of rotatng th gvn jont by th turnng angl, thn translatng th actuator to th dsrd jont, and rpatng th procss. Thr ar a total of ght actons rqurd to rach th objct, on acton to grasp t, and anothr ght actons rqurd to rturn to ts ntal stat wth th graspd objct n hand. A. Robot dsgn VII. EXPERIMETS To prov th fasblty of MASR, w dsgnd a manufacturd a mobl actuator, wth 10 lnks and a bas. Th robot parts ar 3D prntd usng Objct Connx 350 wth nomnal accuracy of narly 50 mcrons usng Vrogray matral. In ths vrson, th jont angl s passvly lockd by a sprng applyng a frcton forc. To ncras th frcton

10 forc w glud sand paprs to th lnks and nsrtd a mtal scrw to th clamp. At thr bottom, th lnks hav a track whch allows th mobl actuator to travl along thm to rach and actuat a dsrd jont. Each of th lnks s cm wd and 5 cm long, gvng th actv scton of th snak robot a total lngth of 50 cm. Th wght of th mobl actuator s 10 grams, whras th avrag wght of a lnks ncludng th clamp and jont s narly 5 grams. W attachd a magnt to th tp of th last lnk n ordr to grasp our targt. Howvr, othr graspng mchansms can b addd. locomoton s narly 3 cm/s and th rotatonal spd s narly 18 dgrs/s. Th robot s vry modular and th numbr of mobl actuators and lnks s asly changabl. W usd motors wth 1000:1 gar rato whch can produc 0.9 m of torqu at 3 rpm. Ths torqu s ncssary to ovrcom th frcton torqu btwn th dffrnt lnks and othr xtrnal forcs to produc moton. Durng all of th xprmnts, th mobl actuator was rmotly controlld by a human oprator. Th oprator had a two channl joystck. On channl s usd to drv th mobl actuator forward and backward along th lnks and th othr to rotat th lnks clockws or countr clockws. ot that n ths prlmnary prototyp, thr s no lockng/unlockng mchansm (whch w blv wll rsult n supror prformanc n trms of accuracy and loads). Rathr, th systm s passvly lockd wth frcton and th mobl actuator fttd wth a strong motor ovrcoms th frcton to rotat th lnks. Fgur 14. A top and bottom vw of two adjacnt lnks. Th rlatv orntaon btwn th lnks s passvly fxd by th clamps. Th mobl actuator (prsntd n Fgur 15) has two motors. On motor actuats th whls to drv th mobl actuator along th tracks of th lnks, and a scond motor to rotat th lnks. Th rotatonal motor s attachd to a lnar gar mchansm, allowng th tth to dsconnct from th lnks or push thm for rotaton. Th maxmum rlatv angl btwn th lnks s 45 dgrs. W usd a 4 Volts B. Exprmnts wth 5 lnks In ordr for th robot to oprat as plannd, t must b abl to prform th followng mchancal opratons: 1. Travl frly ovr th lnks forward and backward.. Travl ovr curvd jonts wthout changng thr orntaton. (th lnks ar passvly lockd) 3. Rotat th lnks. Th basc xprmnt s prsntd n Fgur 16. Th mobl actuator was tstd gong towards th nd of th lnks and rturnng back wth and wthout bndng th lnks. In both cass, th robot had no dffculty travllng ovr th lnks or rotat thm to thr drcton. Startng at (a), th robot advancs towards ts tp (b-c), thn rturns to th cntr (d). Th robot thn rotats th lnks clockws () and countr clockws (f). Th robot thn travls ovr th curvd jont (g) and rotats ts tp clockws (h) and countr clockws (). Th robot thn movs to th tp (j) (s mov). As th jonts can b rotatd by 45 dgrs to ach drcton, th robot can mak a c shap (half a crcl) by rotatng 4 lnks n th sam drcton (countr clockws). Ths xprmnt s llustratd n Fgur 17 (s mov). Fgur 15. Th mobl actuator that travls upon th lnks. Ths actuator has two motors, on motor to travl along th lnks and a scond motor to rotat th lnks. Lthum-on battry to actuat th motors. Th spd of th

11 Followng th sam algorthm, th robot succssfully rachd ts dsrd targt. Howvr, w found that snc th robot s mad of prntd matral, t slghtly curd downwards by narly 1 cm. Evn though th wght of th robot s largr and th torqu actng on th lnks substantally ncrasd, th lnks rmand lockd durng th xprmnt. Fgur 16. Th mobl actuators travl forward and backward ovr th lnks wthout changng thr orntaton and actvat thm to th dsrd locaton. Fgur 18. Th robot pntratng through a small pass to rach a targt bng th wall. Fgur 17. By rotatng th four lnks countr clockws, th robot gts a C shap. C. Exprmnts wth 10 lnks In th followng xprmnt, w addd 5 mor lnks to th robot (10 n total). Th robot s vry modular and addng th lnks rqurs narly mnuts. Wth th longr vrson, w prformd a task that s smlar to th xampl prsntd n Scton IV. Th rsults ar prsntd n VIII. SUMMARY AD COCLUSIOS Ths papr has ntroducd a mnmally actuatd robotc snak (MASR). Th MASR can xcut complx motons wth a small numbr of actuators. It conssts of a mobl actuator that shfts ts poston along th jonts of th robot. Ths nabls t to shap th robot to any dsrd poston by ncrmntally adjustng all of ts jonts. Ths was shown by an xampl of whr t succssfully manpulats an objct whl manuvrng around obstacls. W hav dscrbd th unqu knmatcs of th MASR and dmonstratd how t can duplcat th moton of a fully actuatd robot to wthn any dsrd dgr of accuracy.

12 Th robot s sutabl for applcatons n a complx and confnd nvronmnt wth low payload and that do not rqur rapd dploymnt. Whl th robot cannot hold larg wghts, t s a rgd mchansm (not complant) n th sns that t s not mant to dform du to prformanc of ts tasks. Th robot s also vry modular - th numbr of lnks and mobl actuators can b changd n a mattr of mnuts. W bult an xprmntal robot wth 10 lnks and on mobl actuator. W usd th robot to show how by usng a sngl mobl actuator, t s possbl to control th 10 jonts of our robot and pntrat through a confnd spac and rach th targt. W found that th control s smpl and ntutv, and only a fw mnuts ar rqurd for a human oprator to larn how to actuat th robot. W wr abl to prform th tasks that ncludd gong through a small pass and rachng a targt. Th robot can achv dffrnt confguratons as c shap or s shap. Furthr rsarch and dvlopmnt of th MASR s ongong. w mprovd dsgns ar bng dvlopd for th physcal actuatng mchansm that wll yld mor rgd structur (by producng mtal lnks), smoothr motons, and rduc rrors and malfunctons by fttng th mobl actuator wth a controllr and snsors. In our futur work w am at dvlopng a comprhnsv gnral moton plannng algorthm to yld optmal motons for th MASR n an obstacl nvronmnt for on or mor actuators. IX. BIBLIOGRAPHY [1] P. Lljback, K.Y. Pttrsn, O. Stavdahl, and J.T. Gravdahl, "A rvw on modllng, mplmntaton, and control of snak robots," Robotcs and Autonmous Systms, vol. 60, pp. 9-40, 01. [] P. Lljback, K.Y. Pttrsn, O. Stavdahl, and J.T. Gravdahl, Snak Robots - Modllng, Mchatroncs, and Control.: Sprngr, 013. [3] P.K Sngh and C.M. Krshna, "Contnuum Arm Robotc Manpulator: A Rvw," Unvrsal Journal of Mchancal Engnrngs, vol., no. 6, pp , 014. [4] H. Chost t al., Prncpls of Robot Moton Thory, Algorthms, and Implmntaton.: MIT Prss, 005. [5] S.M. Lavall, Plannng Algorthms.: Cambrdg Unvrsty Prss, 006. [6] G.S. Chrkjan, "Thory and Applcatons of Hypr- Rdundant Robotc Manpulators," Calforna Insttut Of Tchnology, Ph.D. dssrtaton 199. [7] G.S. Chrkjan and J.W. Burdck, "An obstacl avodanc algorthm for hypr-rdundant manpulators,", Cncnnatt, OH, May [8] G.S. Chrkjan and J.W. Burdck, "Dsgn and xprmnts wth a 30 DOF robot,", Atlanta, GA, May [9] G.S. Chrkjan, "Hypr-Rdundant Robot Mchansms and Thr Applcatons,", Osaka, Japan, ovmbr [10] M. d Graca and J.A. Tnrro Machado, "An volutonary approach for th moton plannng of rdundant and hypr-rdundant manpulators," onlnar Dynamcs, pp , 010. [11] E.K. Xdas and.a. Aspragathos, "Tm Sub- Optmal Path Plannng for Hypr Rdundant Manpulators Amdst arrow Passags n 3D Workspacs," n Advancs on Thory and Practc of Robots and Manpulators.: sprngr, 014, vol., pp [1] A. Shukla, E. Sngla, P. Wah, and B. Dasgupta, "A drct varatonal mthod for plannng monotoncally optmal paths for rdundant manpulators n constrand workspacs," Robotcs and Autonomous Systms, pp. 09-0, 013. [13]. Shvalb, B. Bn Mosh, and O. Mdna, "A raltm moton plannng algorthm for a hyprrdundant st of mchansms," Robotca, pp , 013. [14] D. Trvd, C.D. Rahn, W.M. Kr, and I.D. Walkr, "Soft robotcs: Bologcal nspraton, stat of th art, and futur rsarch," Appld Boncs and Bomchancs, vol. 5, no. 3, pp , 008. [15] I.D. Walkr, "Contnuous Backbon Contnuum Robot Manpulators," ISR Robotcs, vol. 013, 013. [16] D. Rollnson and H. Chost, "Pp twork Locomoton wth a Snak Robot," Journal of Fld Robotcs, pp. 1-15, 014. [17] E. Chng, S. Gao, K. Qu, and Z. Shn, "On Dsjont Shortst Paths Routng on th Hyprcub," n Combnatoral Optmzaton and Applcatons.: sprngr, 009, vol. 5573, pp