A BIOMIMETIC ELASTIC CABLE DRIVEN QUADRUPED ROBOT THE ROBOCAT

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1 Proceedings of te ASME Interntion Mecnic Engineering Congress & Exposition IMECE November -7 Denver Coordo USA IMECE-5 A BIOMIMETIC ELASTIC CABLE DIVEN QUADUPED OBOT THE OBOCAT Evedin Kjuno J. Jim Zu obert L. Wiims II Stepen M. eiy # Mecnic Engineering Deprtment Scoo of Eectric Engineering nd Computer Science # Bioogic Sciences Deprtment Oio University Atens Oio 57 USA ABSTACT Stte of te rt egged robots suc s te Hond s series of biped robots ending in te test dvnced wing robot ASIMO nd te series of biped robots of Wsed University incuding te test dvnced robot WABIAN empoy jointmount motors wic simpifies te nysis/design nd trces te route for n effective contro system but resuts in egs tt re evy nd buy. Cbe-driven robots overcome tis sortcoming by owing te motors to be mounted on or ner te torso tereby reducing te weigt nd inerti of te egs resuting in ower over weigt nd power consumption. To fciitte nysis nd design typic cbe-driven robots use non-stretcbe cbes wic require t est n+ motors for n n Degree-of-Freedom (DoF) joint. Terefore for robot wit N joints t est N ddition motors re needed compring to joint-mount motor drives. Moreover te drive trin of bot joint-mount nd cbe-driven designs re rigid wic cnnot effectivey bsorb ground impct socs nor trnsfer potenti energy to inetic energy nd vice vers wen te robot is in motion s bioogic nims do. In tis pper we present te design nd test of ct-size qudruped robot ced oboct wic empoys stretcbe estic cbe-driven joints s inspired by bioogic qudruped nims. Atoug it compictes inemtics nd dynmics nysis nd design te estic cbes ow n motors to be used for n n-dof joint tereby eiminting N motors for robot wit N joints compring to non-stretcbe cbes furter reizing te weigt nd power svings of te cbe driven design. Moreover te estic cbe driven joints not ony effectivey bsorb ground contct soc but so effectivey trnsfer potenti nd inetic energy during wing or running tereby improving te robot motion performnce nd energy efficiency. In te pper we wi discuss te inemtics nd dynmics nysis of estic cbe driven joints impementtion of estic cbe-driven joints on te Oio University oboct nd contro.. INTODUCTION Trdition direct drive ctution system of robotic mniputors is probby one of te esiest wys to ctute robotic wers due to its simpicity in mecnic impementtion nd te fct tt te rottion motion of motors is directy mpped into rottion motion of te joints. Consequenty if we just require n pproprite functionity of robotic wer te direct drives wit ger set woud be convenient soution. However bioogic wers tt use n inverted penduum ie mecnism [ ] re considered energy efficient retivey wit respect to te stte of te rt robotic wers [ ] using different ind of ctutors te musces wic cn be considered s estic (stretcbe) iner ctutors. Energy efficiency nd te eve of te w cyce precision nd smootness re mong importnt resons for mimicing bioogic wers. As te fundment ctutor unit te musce bevior nd structure ttrct speci ttention of reserc in robotics. Tere ve been number of ttempts to produce rtifici musces for use in robotics [ - 8] bsed on different principes suc s pneumtics piezoeectric effect mgnetostriction etc. One of te possibiities of wing robot musce-ie ctution is to use (estic) cbes. Appictions of te cbe Contct utor emi: zuj@oio.edu Tis pper is bsed on reserc sponsored by te Air Force eserc Lbortory under greements number FA85 7 nd number FA85 9. Te U.S. Government is utorized to reproduce nd distribute reprints for Government purposes notwitstnding ny copyrigt nottion tereon. Te views nd concusions contined erein re tose of te utors nd soud not be interpreted s necessriy representing te offici poicies or endorsements eiter expressed or impied of te Air Force eserc Lbortory or te U.S. Government. Copyrigt by ASME

2 ctution in gener robotics [9] sow tt te min feture mong oter interesting fetures of te cbe ctution is te possibiity to cieve retivey ig cceertions due to te reduced mss of te most ineticy-ctive segments of te robots. Since te wing robots usuy ve to crry n independent energy source (btteries) it is te most critic to reduce te energy consumed per distnce wed. Using te cbes te motors re moved to te sections of te robot tt re te est ineticy-ctive nd experience te owest cceertions. Te min benefits re: te bncing stbiity of te robot is improved nd te energy consumption is reduced due to te reduced mss of te fst moving segments of te wing robot. It wi so ed to significnty reduced over weigt of te robot. Tere re oter benefits too but not directy incuded in te scope of tis pper. Some wor s been done in te re of cbe ctution in te wing robotics. A prtiy cbe ctuted expod is nyzed in [].. KINEMATICS AND DYNAMICS ANALYSIS OF THE QUADUPED WALKING OBOT. Te wing robot rcitecture Te wing robot rcitecture under considertion is sown in Figure.. Te robot rcitecture s ctuted degrees of freedom (ip nee ne nd te puey) for ec eg corresponding to te ongitudin motion nd ddition degrees of freedom (ip nd ne) for ec eg corresponding to te ter motion. Te ctuted revoute joints of one eg re mred by in Figure.. Te min objective of te design ws to provide minimum DoF suc tt te qudruped cn perform retivey smoot w witout frequent requests for w direction cnge. Terefore te oboct design considered ere is te resut of prti biomimicing wit significnt reduction in DoF. Te trun segment does not contin intern fexibiity wie bioogic ct s significnt fexibiity in te trun enbing te ocomotion direction cnge. Te robot rcitecture sti enbes te ocomotion cnge troug te incusion of te two (per eg) revoute ntero-posterior oriented joints t te ip nd t te ne. Oter revoute joints (ip nee nd ne) connecting te eg segments provide te motion pre to te sgitt (symmetry) pne. Te cbes for te nee joint ctution re pued using speciy sped pueys situted on te trun. Tis prticur design deti devites from te bioogic rcitecture nd s significnt disdvntge in te fct tt te cbe forces cn produce significnt couping between te nee nd te ip joint since te cbes for te nee cn generte significnt torque for te ip if not propery trced. A bioogic musce for te nee does not cross te ip joint nd does not generte ny torque wit respect to te ip. Figure. Te oboct rcitecture Te engt nd te position of te brs wit te cbe ttcment points bout te nee joint directy infuence te required (vribe) cbe tension over te joint motion rnge s we s te (vribe) cbe puing speed. Furter te required cbe tension nd speed (troug ccution of power) re te input prmeters for te motor sizing wie te trnsmission rtio of te motor gerbox seection directy depends on te size nd te spe of te puey. Besides te rcitecture sown in Figure. sever oter options ve been considered nd/or pnned to be nyzed in te future wor. Considering te compctness of te rcitecture nd te impemented rdwre it is convenient to use speric joints rter ten combintion of revoute joints for te ip. A bioogic ip joint is speric joint ( DoF) but it is difficut to design direct drive bsed ctution system for robotic ip joint. However using te cbes te option of using speric joints is recommended since it reduces te number of cbes needed ( cbes for speric joint) compring wit seprte tree revoute joints ( cbes for tree seprte DoF joints). For te se of te simpicity we decided to ctute ony nee joint using cbes. So fr te over rcitecture is presented witout specifying detis. Te proper sizing nd te specifiction design detied need to be done itertivey using mtemtic mode of te system.. Mtemtic modeing prmeters nd ssumptions Te most significnt ssumptions for te mtemtic mode derivtion re te foowing: ) Te contct surfces between cbes nd te guiding oes re frictioness ) Te cbes re idey fexibe i.e. te bending moments re zero ) Te stretcbe cbes beve s iner springs Copyrigt by ASME

3 ) Te ter motion does not produce significnt inerti effects on te ongitudin motion 5) Te w is performed on orizont ft pne ) For te se of te dynmics nysis te trun cn be represented s concentrted mss t te center of grvity (CG) for te upper body 7) Te w of te robotic system is stticy stbe. ' F ty t F tx Te wing cyce for eg is consisted of two min pses: te support pse nd te swing pse. Since te dynmics of stnce eg wit te trun is significnty different from te dynmics of swing eg one we wi consider te two corresponding modes of one eg dynmics wit te concentrted mss of te trun segment. Te stnce eg wit te concentrted mss of te trun mode is sown in Figure.. Te infuence of te oter egs to te trun is represented troug te orizont nd te vertic component of te interction force reocted to te trun CG (C t ) ong wit te torque resuting from te reoction of te forces. Since te dynmics of te system is governed by sever prmeters of te system tt re subject of te optimiztion troug te dynmics simution nd te testing on te re rdwre we wi prticury point out te sensitivity of te system bevior wit respect to tose quntities nd nyze te wy to optimize tem. ' ' m g y c c t m t g c m g x Cbe stiffness. One of te most importnt prmeters is te cbe spring stiffness. Te pysic quntity in nims tt corresponds to te stiffness of te cbes is te stiffness of te corresponding nim s musces nd tendons. Tendons ccumute portion of energy tt woud be normy ost due to foot-ground coision. Te ccumuted energy is reesed during te next git cyce to decrese te mount of te inetic energy input vi te musces. Simir effect is expected using te esticy stretcbe cbes for te wing robot. Te contct forces re expected to smoot out s we s te joint torques wic is desirbe effect on te robot contro. However potenti probem using te stretcbe cbes is tt te system cn esiy experience n osciting bevior if tere is no sufficient dmping wic requires dynmic controer to stbiize te system. Points of te cbe ttcments. Te positions of te points were te cbes re ttced retive to te joints nd te geometry of te cbe guidnce re very importnt for coupe of resons. Te points were te cbe guiding oes re positioned points A nd B in Figure. determine te torque rm wit respect to te joint xis ong wit te rnge of ngur motion of te joint. Inpproprite position of te guiding oes cn resut in very ig cbe tension required to generte certin required torque t specific joint nges. Terefore optimized positions of te guiding rings nd te ttcment points wi be determined vi numeric soutions nd simutions. A' A B' B F grx F gry Figure. Leg-trun mode Pueys profie. Te puey is designed to ve vribe rdius profie sown in Figure.. r ( ) z Heic cbe guide Figure. Te puey profie Te objective of te vribe rdius is to compenste for te difference in te cbe engt increse on one side nd te cbe engt decrese on te oter side of te corresponding joint in tis cse te nee joint. Considering te rcitecture sown in Figure. te cbe engt increse between points B nd B is not te sme s te cbe engt decrese between points A nd A. Tis cuses probems if te vrition of te rr ( ) z r xis of rottion Copyrigt by ASME

4 sum AA' BB' f ( ) is significnt since puey wit constnt rdius woud reese te sme cbe engt s te cbe engt tt it stores for te sme nge of rottion. Tis difference cn be compensted to certin extent wit te preoded springs ttced in series ong te cbes. However te experiments sowed tt te spring stiffness must be significnty decresed to compenste rge vrition of AA' ( ) BB'( ) wic wi reduce te effective torques it cn provide to te joints. Te vribe rdius puey is n effective soution to enbe singe motor to drive estic cbes. Te design objective is tt if te cbe were not stretced te puey nge of rottion soud be proportion to te rottion of te nee joint tt is constnt. Te constnt is modeing prmeter tt s infuence on te controer sensitivity nd performnce nd wi be considered troug te simutions resuts. Cery tere is no need for te vribe rdius puey in te cse wen two motors re used to pu te two cbes seprtey; owever te intention ere is to use ony one motor to drive revoute joint. Considering te mode specifics te mtemtic mode derivtion requires detied inemtic nysis wic is discussed next.. Kinemtics Te objective of inemtics nysis of te robotic wer ctuted by te estic cbes is to find te retionsips between te joint nges nd te positions in n inerti Crtesin coordinte system s we s te corresponding veocities. Besides tis te inemtic nysis needs to provide te retionsip between te cbe speed nd te ngur speed of te corresponding joint. In tis prticur cse te forementioned retionsip represents te retionsip between te puey ngur speed nd te nee joint ngur speed for te cse wen tere is no cnge in te cbe engt. Te cbes re stretced nd re ssumed to beve s preoded iner springs deformtion nd inemtics re couped wit te dynmics troug te deformtion nd forces of te springs. Expressing te veocities of te CGs in te inerti (ground ttced) Crtesin coordinte frme requires expressions of te ortogon coordintes in terms of te joint nges wic re obtined s foows. r C C C s s s c r C C (.) c Cc s s r T c c p p s c p p c s were Ci ( i ) re te ower prts of te eg segments engts from te joints to te CGs p nd p re te y nd x coordintes of te trun CG wit respect to te ip joint i ( i ) re te joint nges denoted in Figure. ij nd ij ( i j ) re te combined nges s ij i j nd ij i j nd bbrevition for sine nd cosine functions is used s s sin c cos wit denoting ny of te forementioned (combined) nges. By differentiting (.) we obtin te veocities c c C c vc C vc s s s C c c p c p s vt. (.) s s s c p p Besides te retionsips (.) nd (.) we need n pproprite function tt indictes ow muc nd wit wic rte te cbe soud be pued to obtin desired joint nge nd n ngur speed. Te cbe engt cnge on te two sides of te puey in Figure. is due to te cnge in te joint nge nd due to te cnge in te cbe tension. Te cbe engt cnge rte due to te cnge in te nee nge is d c BBs BBc r ( Sr ) BB'( ) d BB'( ) d c s AAs AAc ( S ) AA'( ) d AA'( ) s (.) were r nd re te cbe engts on te rigt nd on te eft of te puey respectivey S r nd S re corresponding cbe tensions nd wit BBs r r AAs r BBc r r AAc r denoting te position of te cbes ttcment points retivey to te nee joint s sown in Figure.. Te tot cbe engt cnges on te two sides of te puey re r ( Sr ) Sr Sr BB'( ) BB'() sr Copyrigt by ASME

5 S ) S S AA'( ) AA'() (.) ( s r A' A Figure. Te cbe ttcment points prmeters were sr s re te cbe spring stiffness coefficients for te rigt nd eft cbe segments respectivey. Now we need to determine te puey profie function tt wi provide proportion rottions of te nee joint nd te puey for n pproximtey constnt cbe tension. Te gener profie of te puey is sown in Figure. wic indictes two rigidy joined segments corresponding to te rigt nd eft cbe. Te objective of tis design is to reduce normy rge spring s deformtions due to te joint nges cnges (geometric cnges). To compenste te necessry difference in te cbe stored on nd reesed from te puey te rdius functions r ( ) nd r ( ) must cnce noninerity in AA' ( ) nd BB' ( ). In tis wy te two functions re r dpp' ( ) min for min d des dpp' ( ) r ( ) for s mx mx (.) d des dpp' ( ) oterwise. d des were is te desired rtio of te two rottions des min is te minimum (positive) rdius of te puey nd te derivtives re given in (.) te pus sign is for s=r nd " P B". Te rtio cnnot be precisey constnt due to te fct tt te cbe is stretcbe nd we cnnot compenste te gener cbe tension force since it is not ony function of te nges but so depends on te inerti forces nd te pyod. B B' For te foowing prmeters: r. m r.m =.5 te rdii functions of te two puey segments re: r s cos(.5 ) for.5 min sin(.5 ) min cos(.5 ) ( ) mx for.5 mx cos(.5 ) sin(.5 ).5 oterwise. sin(.5 ) ( in meters) were pus sign is for s=r. Te profie pot for te set of te prmeters is sown in Figure.5. Te puey profie sown in Figure.5 is consisted of te two sections corresponding to te bc nd te front cbe of te nee drive unit. Te derivtion of te puey ssumes tt te nee joint design restricts yperextensions wic mens tt te nge tes ony positive vues for te bc egs nd ony negtive vues for te front egs of te oboct. We cn see tt two rdii ve te sme vue ony for wen te symmetry exists. rdius (m) r (z) Puey profie bc cbe puey section connecting - - front cbe - - r r (z) puey xis z (m) Figure.5 Te puey profie Finy depending on te metod tt is used for te dynmics mode derivtion we woud need te cceertion vector expressions for te centers of grvity of te mjor rcitectur prts (ins motors bttery). However te metod tt we wi use requires te expressions for te veocity of te centers of grvity. Now we wi use te inemtics expressions to derive te dynmics mode of te robot. 5 Copyrigt by ASME

6 . Dynmics of te oboct Using te Lgrnge energy metod set of noniner differenti equtions of second order is derived. Since te derivtion detis woud te significnt spce we wi incude te fin resuts for every prticur DOF. Te dynmics of te system cn be represented by te mtrix eqution (.7) M( ( C( ( t) ( ( t) P( ( were (t) (t) nd (t) re te joint nge veocity nd cceertion vectors respectivey M( ( is te inerti properties mtrix; C( ( t) ( is te ngur speed couping mtrix P( ( is te vector tt incudes te grvity terms nd cbe tension terms nd (t) is te vector of torques cting t te joints. Te product C( ( t) ( ( t) represents combined products of te joints ngur speed wic is consisted of te Coriois nd retive norm cceertions. For te se of conciseness te mtrices M( ) nd C( ( t) ( re given by te coumns in te Appendix. Te vector of te conservtive generized forces P( ( requires n expntion tt is reted to te furter nysis nd we present it beow. Te vector of te conservtive generized forces P ( ) is P g( ps pc) m gcs g( s ps pc) BBsc BBcs AAsc AAcs Sr S L BB'( ) '( ) AA m ( ) gcs m g s Cs g( s s ps pc) (.8) were Sr S re te cbe stiffness coefficients for te front nd te bc cbe respectivey nd S ( ) BB' ( ) BB' P d ( ) Sr (.9) SL L L ( ) AA' ( ) AA' PL d ( ) S re te deformtion of cbe springs. Te new quntities incuded in (.9) denote te foowing: S nd S L re te cbes pre-tensions P( ) nd PL( ) re te two puey vribe rdii given by (.) nd is te geometric nge of te cbe guide on te puey sown in Figure.. Te integrs in (.9) represent te stored cbe ong te puey tred. Te vector of te torques in (.7) is t T ( ) (.) were nd re te torques t te ip nd te ne joints respectivey. Te torque t te nee joint is considered troug te cbe tension expressions in te vector P ( ). If we consider te system sown in Figure. precisey we woud ve DOF nd consequenty mtrices in (.7) woud ve te size by nd te vectors woud be by wic woud increse te computtion efforts in te system controer goritm. By negecting te moment of inerti of te puey drive system te mtemtic mode is reduced to tree differenti equtions of te second order. However we need te moment bncing eqution of te puey cbes system. Te eqution is ) ( ) ( ) ( ) (.) Sr ( P S L PL were is te torque provided by te puey drive. Te equtions (.7) troug (.) ong wit te inemtic retions nd te mtrices M( ) nd C( ( t) ( sown in Appendix represent te mtemtic mode of te wing robot. For te purpose of te controer design te mtemtic mode of te form (.7) needs to be converted into stte spce mode wic is set of te first order differenti equtions..5 Stte spce mode Since te mss mtrix in (.7) is invertibe te eqution cn be expicity soved wit respect to te ngur cceertions s or were M ( ) C ( ) M ( ) P( ) M ( ) f ( ) f ( ) G( ) (.) M ( ) C ( ) M ( ) P( ) G ( ). M ( ) nd Copyrigt by ASME

7 By ssigning vribes: 5 nd 5 te stte spce mode s te form f( ) G( ) f ( ) G( ) 5 f( ) G( ) G ( ) G ( ) G ( ) G ( ) (.) G( ) G( ) were f ( i ) G ( ij ) (i=; j=) represents te entries of te functions in (.).. Interconnections wit oter supporting egs Te interconnections wit oter supporting egs cn be viewed s disturbnces in cse wen it is not convenient to expnd te mtemtic mode of te system nd te controer compexity. However if it is necessry to obtin better performnce of te controer te interconnections cn be modeed s foows. Te interconnections between te supporting eg modes cn be interpreted troug te interconnecting force nd torque s sown in Figure.. Te interconnecting forces nd torque cn be incuded in te existing mode troug te joint torques. Te ugmented torque vector given by (.) becomes F c s ) F ( s c ) t tx( P P ty P P t Ftx( c P c Ps) Fty ( c P s Pc) t Ftx( c c P c Ps) Fty ( s s P s Pc) (.) were te mening of te interconnecting forces nd te torque is expined erier nd sown in Figure.. Te next section wi sow ow te mode is used to design controer.. CONTOLLE DESIGN Te controer design for te robotic wer wi be bsed on te trjectory regution contro [].. Controer rcitecture Te contro system rcitecture sown in Figure. is consisted of [ ]: () Nomin trjectories genertor (b) Inverse dynmics for nomin contro ccution (c) Trcing error regution controer (d) Mesurement system nd (e) Pnt Nomin motion specifiction - motion error vector Figure. Te trjectory regution controer rcitecture Te nomin motion specifiction boc genertes te joint trjectories tt wi provide bnced w. Te informtion bout te nomin joint nges t every time-step is sent to te error dynmics controer nd te nomin torques genertor. Te nomin torques re generted bsed on te inverse dynmics mtemtic mode. Since te mtemtic mode of te robot is not n exct description of te dynmic bevior tere wi be errors in te resuting motion. Te mount of te resuting motion devition from te desired motion is ccuted bsed on te mesurements of te joint s nges wic is used by te error dynmics controer to generte te correction torques. Te inverse dynmics tt is used to generte te nomin torques is obtined directy from (.7) were te torques re expicity expressed in terms of te functions of nges nd teir first two derivtives. However te derivtives of te input signs must be obtined vi pseudodifferentitors of te form (in te Lpce domin) s ( s) s to obtin pysicy reizbe derivtives. Since te controer rcitecture is bsed on te error dynmics we need to obtin te pproprite error dynmics mode.. Error dynmics Nomin motor torques ccution nom + + Error egution + FL Controer corr mesurements obotic wer Te error dynmics mode is bsed on te stte spce mode (.) nd it s te foowing form 7 Copyrigt by ASME

8 f ( ) f ( ) G( ) G ( ) G( ) f ( ) f ( ) G( ) G ( ) G( ) 5 ( ) ( ) ( ) ( ) ( ) G G G f f (.) were i i i nd i (i= ) re respectivey te error nd te nomin vue of te i-t component of te stte vector nd re te corrective torque vues generted by te feedbc controer wit te objective to stisfy (exponentiy). Te wy te error vector is stbiized is discussed in te foowing section.. Contro w Te contro w soud provide te corrective torques suc tt te errors converge to zero wit n exponenti decy. To cieve tis go te contro inputs cnce te noninerity (FL tecnique) nd introduce te terms proportion to te errors of te sttes s foows. were G G G ) G ) G ) ( ) G( ) b (.) ( ) G ( ) b ( ) G( ) G( bg ( g ( g b g b g b g f( ) f( ) f( ) f( ) f ( ) f ( ) 5 5 nd ij (i=; j=i- i) re te constnts tt need to be determined suc tt te cosed contro oop error dynmics re exponentiy stbe nd ve desired trnsient bevior. Prticury we cn set up te constrint tt te system s ess tn 5% oversoot nd te setting time ess tn.5 seconds for ec joint rottion DOF wic resuts in te dmping coefficient. 9 nd te ntur frequency rd n.59. To obtin te two vues te coefficients ij s need to ve te foowing vues Finy te contro w is (.) G b r g were G r is te reduced input mtrix evuted t te nomin trjectory. Next we wi evute te performnce of te contro w vi simution resuts.. Simution resuts nd te performnce nysis Te boc digrm sown in Figure. is impemented in te Mtb/Simuin; te contro w is tested on combintion of rmp inputs for te joints nges nd te simution resuts re sown in te foowing figures. Te nomin joint nges nd te smoot pseudodifferentitor obtined ngur veocity trjectories re sown in Figure.. joint nges (rd) nd speed (rd/s) Te ip nee nd ne nges nd inverse system estimted speed ip nee ip speed ne (=ip) nee speed ne speed 5 Figure. Nomin trjectories Te nomin torques predicted by te inverse dynmics boc re sown in Figure.. Te resuts in te nomin torques pot gree wit te expected resuts since force rm is significnty greter for te nee joint tn for te oter two joints. Te ip joint s ow predicted torque due to te fct tt te CG of te trun for te prescribed motion is considered verticy bove te ip. Cbe tensions needed to provide te nee joint ngur trjectory re sown in Figure.. Te pot sows tt te cbes pre-tension ws 5 N nd coud ve been even reduced nd sti void sc cbe. rd rd.. s s 8 Copyrigt by ASME

9 nomin torques (Nm) Te ip nee nd ne nomin torques ip nee ne nges (rd) Te ip nee nd ne desired nd ctu nges ip nom nee nom ne nom ip nee ne Figure. Nomin joint torques 5 Figure.5 Te ctu versus desired trjectories Force [N] Cbe tensions Spring Force Bc Spring Force Front errors (rd) x Te ip nee nd ne nge errors ip nee ne 5 Figure. Te cbe tensions Te comprison between te ctu nges versus te desired nges is sown in Figure.5. Te pot sows very ow errors of te ctu trjectories (te ne nd te ip ve te sme nomin trjectory in te pot). Devition is noticebe t te instnts wen tere re srp cnges in te desired trjectory sope. Te devition of te ctu nges from te desired nges cn be seen in te errors pot sown in Figure.. Tot torques suppied to te joints of te robot mode re sown in Figure.7. If te nomin torques sown in Figure. re compred wit te tot torques significnt corrective vues of te torques cn be noticed t te instnts wen tere is significnt cnge in te desired speed of te joints due to te inerti effects nd te effect of te pseudo-differentition s we Figure. Te ip nee nd ne joint nge errors tot torques (Nm) Te ip nee nd ne tot joint torques ip nee ne -5 5 Figure.7 Tot joint torques 9 Copyrigt by ASME

10 . CONCLUSION AND FUTUE WOK Te pper presented nove ctution system of robotic wer. Te ctution system is bsed on combintion of te direct drive ctution nd novety in wing robotics te stretcbe cbe ctution. Sever benefits re introduced using te cbes to ctute joints. Te energy consumption is reduced troug te reduction of te inerti forces on te most motion ctive prts of te robot. Te need of using two different puey-motor pirs to ctute revoute joint in cse of using non-stretcbe cbes is compensted using te stretcbe cbes wit te design of te speci puey profie. Te treded puey profie wit te vribe rdius ensured tt ong wit retivey sm deformtions of te cbe spring te cbes do not become oose wic woud ed directy into compictions wit pure trnsport deys in te contro w. Te mtemtic mode is derived wit respect to te ssumptions tt re isted. Te eg-trun dynmics mode is presented in stte spce form nd te corresponding error dynmics is used to design te controer using te trjectory regution contro wit n open-oop nomin controer nd cosed oop trcing error regution controer. Te nomin controer is bsed on te inverse dynmics mode of te pnt. Te cosed-oop controer is bsed on te feedbc ineriztion contro were pnt noninerity is cnceed by stte feedbc nd desired iner dynmics re ssigned. It is sown ow te mode cn be combined troug te interconnection quntities wit noter eg-trun mode to consider te impct from te oter supporting egs. Te performnce of te joint trjectories trcing ws nyzed using simutions wic sowed stisfctory resuts of trcing te prescribed joint trjectories. Te possibe probemtic cses of te trcing woud be te cses wit te srp cnges nd/or ssocited noise in te desired trjectories due to te need of finding (pproximte) derivtives. Pnned future wor is to impement te robot contro goritm in re rdwre. Certin steps in tis direction ve redy been mde on qudruped robot oboct owever using n open oop contro goritm tt produced certin oscition in te joints motion. It is expected tt te impementtion of te presented goritm wi resut in better performnce besides te inerent benefici fetures of using te (stretcbe) cbes for te robot ctution wic were described in te pper. Currenty te oboct empoys stticy stbe wing git wit stndrd dio Controed (/C) servo motors s ctutors s sown in Figure.. Figure. Te oboct (see te video cip in beow) Te souder/ip joints ve one DOF direct ctution nd te nee/ebow joints ve one DOF estic cbe driven ctution. Te servo motors ve buit-in ngur position controers wic restricted te motion contro to be sequence of position commnds to te joints. Suc joint motion contro sceme is open-oop in nture nd te performnce is very imited wic cn be seen t ttp:// ttp:// We wi impement te cosed-oop motion contro sceme deveoped in tis pper in te future wic soud improve te performnce significnty. Due to imittion on te number of pges we ve not incuded noter importnt segment of nysis te swing eg dynmics nd contro. Tis wi be incuded in future pper. ACKNOWLEDGMENT We tn Mr. Justin Mmr PD student t Eectric Engineering nd Computer Science Deprtment t Oio University for prticipting in ssembing te rdwre. EFEENCES [] Giovnni A. Cvgn Normn C. Hegund nd C. icrd Tyor Mecnic wor in terrestri ocomotion: two bsic mecnisms for minimizing energy expenditure. Am J Pysio.; 977. [] Brin. Umberger nd Piip E. Mrtin "Mecnic power nd efficiency of eve wing wit different stride rtes" Te Journ of Experiment Bioogy pp [] uin A. Bertrm J. E. & Srinivsn M. A coision mode of te energetic cost of support wor quittivey expins eg sequencing in wing nd goping pseudoestic eg bevior in running nd te w-to-run trnsition. Journ of Teoretic Bioogy 7 pp [] Vnderborgt Brm Vn Hm ond Verrest Bjorn Vn Dmme Mice Lefeber Dir Overview of te Lucy Project: Dynmic Stbiiztion of Biped Powered by Copyrigt by ASME

11 Pneumtic Artifici Musces Advnced obotics pp [5] ttp:// [] Ai E. Aiev et. Gint-Stroe Superestic Crbon Nnotube Aeroge Musces Science pp [7] y H. Bugmn Pying Nture's Gme wit Artifici Musces Science 8 pp [8] Yosep Br-Coen nd Sen Lery Eectroctive Poymers s Artifici Musces cnging obotics Prdigms Ntion Spce nd Missie Mteris Symposium. [9] E. Kjuno nd.l. Wiims II Veice Simution System: Contros nd Virtu-eity-Bsed Dynmics Simution Journ of Inteigent nd obotic Systems 5 pp [] Ki H. K. "Noniner Systems" nd Edition Engewood Ciffs NJ Prentice-H 99. [] Jim J. Zu Lecture Notes on Noniner Systems Contro: Bue Boo Oio University 8. [] ttp://word.ond.com/asimo/ [] ttp:// wbin/index.tm [] An P. Bowing Mss Distribution Effects on Dynmic Performnce of Cbe-Driven Hexpod ASME Journ of Mecnic Design vo. 9 no. 8 pp APPENDIX THE INETIAL POPETIES AND THE COUPLING MATICES Te inerti properties mtrix M( ( foowing coumns. M(: ) ( p M(: ) is given by te m T( p p) IT ( ( pc ps) ( p p)) IT p ( pc ps) ( pc ps)) IT m T( ( pc ps) ( p p)) IT m C ( p p ( pc ps)) IT I m ( C C c ) ( p p ( pc ps) ( pc ps c )) IT I M(: ) m T( p p ( pc ps) ( pc ps)) IT m ( C C c ) ( p p ( pc ps) ( pc ps c )) IT I IT m ( C C c ) mc ( p p ( pc ps) ( pc ps c )) I I were I T I I re te moments of inerti of te trun upper eg ower eg respectivey nd m m m T re te msses of te upper eg ower eg nd te trun segment respectivey. Moments of inerti of te motors re negected in te stnce eg mode. Te joints ngur speed couping mtrix C( ( t) ( is given by coumns s foows C( :) m T m ( )( ps pc) ( ( )( ) ( ) ps pc ps pc C( :) m T ( )( ps pc) ( ) s( m C ) s ( ( ps pc) ( )( ps pc) C( :) m T( ( ( ( p p s pc) ( ps pc)) s pc) ( m C ) s) m were m nd m re te effective dmping coefficients for te ip nee nd ne joints respectivey. Copyrigt by ASME