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Energy Efficient Control of Running Legged Robots. A Case Study : The Planar One-Legged Hopper

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(1)Energy Efficient Control of Running Legged Robots. A Case Study : The Planar One-Legged Hopper Charles François, Claude Samson. To cite this version: Charles François, Claude Samson. Energy Efficient Control of Running Legged Robots. A Case Study : The Planar One-Legged Hopper. RR-3027, INRIA. 1996. �inria-00073666�. HAL Id: inria-00073666 https://hal.inria.fr/inria-00073666 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. 

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(18) Ÿ. `A6A$- A) 

(19) \-A. ziy{jqrz¡uZt†xlk¢qrjmlk€69³uz  wgilC€#l}wyA;^ƒky{†x†xlk€~y{z  qrzi¢ lk†x†xy{†jmy  l}tr€(9ƒ¨uz œlC~¡u†ujml}w†ˆq@?kl  qrz wlk†xjm€&yA;uä€xqrzi¢trl|š{u†ˆqvuœetrl9wgilgiy{~i~lk†< €  lk€xqr†xl  ;Îy{†2Œu†  š•l}tryNƒ}q w¸‚9^;Qy{†ClU„¡ujm~etrl5 f gilz\sij6œlk†cyA;ƒky{zBw†xyt6uƒ}wxqry{zi€—wy œl u~i~et qrl   si†ˆqrzi¢;u €xqrzi¢trl€xwlk~ yA;6wgilgiy{~i~lk† ƒ¨uz œl  l  siƒkl  ;Άxy{j wgilC†xlNseqr†xlkjmlkz•wHyA;ƒky{z•w†xyt tvuœeq t q wµ‚ yA;wgil&u~i~i†xy¨„Nqrj­uZwl  lk†x†xy{† jmy  l}t 5 ‰µz;wgeqr€H†xlk€~lkƒ}(w 9Œq w+qr€H€gi(y 2zwg¡uZwCuրqrzi¢tr+l ;Îy{†xƒkl—qrjm~isetr€lcqrzwgilctrlk¢  si†ˆqrzi¢ wgil̀xw#uziƒklÌ~ig¡u€l 9eƒky{jm~etrlkjmlkzBwl  2 q wgCIw 2Œywy{† \sil5qrjm~isetr€lk€u~i~et qrl  uZw,wgilÌgeqr~  si†ˆqrzi¢ wgi_l Fiqr¢{g•w5~ig¡u€l 9u†xl€Ws J­ƒ}qrlkz•(w 92g¡uZwl}š•lk†wgil  lk€xqr†xl  ;Îy{† 2Œu†  š•l}try\ƒ}q wµ‚cAy ;wgil6giy{~i~lk† 5 ˜5y{zil}wgil}trlk€(€ 9Njmy{†xl5ƒky{z•w†xytÜuƒ}wxqry{zi€,ƒky{set  uZtr€#y6œlƒky{zi€xq  lk†xl  qrz+~i†uƒ}wxqrƒkl 9e'q ;Lwgeqr€Œ~i†xy¨š•l  wyœlÌsi€Cl ;QseDt ;Îy{†,†xy{œisi€xwzilk€€ lkzig¡uziƒklkjmlkzB(w 55ziƒklÌwgilԃky{z  q wxqry{zi€ ;Îy{†Œwgilԃky{z•w†xyt tvuœeq t q wµ‚ Ay ;Ôwgil|t qrzil¨u†ˆ@q ?kl  lk†x†xy{†Hjmy  l}tÌu†xlÍjml}(w 9uz•‚ ƒ}tvu€€xqrƒ¨uZtt qrzil¨u†&ƒky{zBw†xyt  lk€xqr¢{zßjml}wgiy  ƒ¨uz|œlsi€#l  wy  l}wlk†xjqrzilum€xw#uœetrlÔt qrzil¨u†^ƒky{z•w†xyt trlk†Ìuz  wgilƒky{†x†xlk€~y{z  qrzi¢Hj­uZw†ˆqʄcAy ; ƒky{zBw†xytA¢•uZqrzi(€ 5GUÖl6g¡ukš•lsi€l  uz—y{~ewxqrj­uZnt KLqrzil¨u† 5s¡u  †uZwxqrƒjml}wgiy  5 ‹^€Ì€gi6y 2zqrzÍwgil­pNlkƒ}wxqry{z%  9Lƒky{zBw†xytqrjm~isetr€lk€Ôƒ¨uz œl"†xlk~etvuƒkl  œ•‚Ïl \seq š{uZtrlkz•(w 9ǜisew jmy{†xl^l¨u€xq t ‚mqrjm~etrlkjmlkz•wl  9i~eqrlkƒkCl 2 qr€#lƒky{zi€xw#uzBwŒqrzi~isew6€ 5/‹^t wgiy{si¢{gCy{zet ‚mtry\ƒ¨uZt€xw#uœeq t q w¸‚­Ay ; wgilu~i~i†xy¨„Nqrj­uZwl  jmy  l}tAy ;wgil giy{~i~lk†qr€uz¡uZt ‚Nwxqrƒ¨uZt t ‚lk€w#uœet qr€gil  9B€xqrjsetvuZwxqry{zi€Û†xlk€#set w€ q t trsi€w†uZwl&wg¡uZw­wgil—~i†xy{~y{€#l  ƒky{z•w†xyt€xw†uZwlk¢‚ ~lk† ;Qy{†xjmi€ 2Œl}t ty{z wgil—lU„¡uƒ}w­jmy  l}tAy ; wgilCgiy{~i~lk† 5͋5€lU„i~lkƒ}wl  9wgilClkzilk†x¢‚ä€~lkzBw"œB‚ÖwgilCƒky{z•w†xyt,wỳw#uœeq t @q ?klHwgilCgiy{~i~lk† qr€š•lk†ˆ‚ €j­uZt tƒky{jm~¡u†xl  wywgil—¢try{œ¡uZtjmlkƒdg¡uzeqrƒ¨uZt^lkzilk†x¢‚ Ay ;wgilc€‚N€xwlk=j 5äf geqri€ 2Œu€ y{si†~i†ˆqrjml+jmywxq š{uZwxqry{> z 2gilkz €xw#u†ˆwxqrzi¢|wgeqr€€xws  ‚ 5—‰µwqr€gi(y 2Ûl}š•lk†ƒ}trl¨u†wg¡uZw6wgil+qrzi~isew lkzilk†x¢‚mƒ¨uziziyw,u€x‚Njm~ewywxqrƒ¨uZt t ‚mwlkz  wVy ?klk†xy6€xqrziƒklwgil€xw#uœeq t @q ?kl  giy{~i~lk† < €³¢•uZq w  y\lk€³ziyw lU„¡uƒ}wxt ‚Öƒkyqrziƒ}q  el 2 q wg uÍz¡uZwsi†uZt †xlk¢qrjml 5Íf gil  qr€ƒk†xlk~¡uziƒ}‚œl}Iw 2Œlklkz z¡uZwsi†uZt,†xlk¢qrjmlk€ uz  ziy{jqrz¡uZtԆxlk¢qrjmlk€Hqr€&uZtr€yä†xlk€~y{zi€qrœetr+l ;Îy{†+ƒ}try{€l  4ötryNy{~ €wl¨u  ‚ 4¸€xw#uZwlÏlk†x†xy{†x(€ 5)f geqr€ †xCl F¡lkƒ}w8€ ;Qy{†³lU„iujm~etrly{z­wgilƒky{z•w†xyt trl  ;Îy{† 2Œu†  š•l}tryNƒ}q w¸‚mAy ;Xwgilgiy{~i~lkL† 2geqrƒ#gHƒky{z•š•lk†x¢{lk€ wy­umšuZtrsilԀxt qr¢{gBwxt ‚  q lk†xlkzB^w ;Άxy{j wgil  lk€xqr†xl  y{zil 5 f gilÛ u ;Îy{†xlkjmlkz•wxqry{zizil  š•l}try\ƒ}q wµ‚ 4¸œeqvu€/ƒ¨uz6œl†xlkjmy›š•l  œB‚̃ky{jm~etrlkjmlkzBwxqrzi¢5wgil³ƒky{z•w†xybt 4 trlkr† 2 q wg&u6€xqrjm~etrl^qrz•wlk¢{†uZtÇuƒ}wxqry{Dz 9iu€Œ€gi(y 2zCqrz—pNlkƒ}wxqry{z—WŸ 5³p\qrj6setvuZwxqry{z—†xlk€set w€,uZtr€yqrz  bq 4 ƒ¨uZwlwg¡uZw5wgilƒky{zBw†xyt trlk†Ô€ y 4¸y{œew#uZqrzil  qr€5†xy{œisi€w^wyHšu†ˆqry{si€5€y{si†xƒklk€ÌAy ;³jmy  l}t t qrzi¢Hlk†x†xy{†x€ SÒsizi~i†xlkƒ}qr€#l 0Nzi(y 2 trl  ¢{lÏAy ;Ԁ~i†ˆqrzi¢{(€ <³€wxq zilk€€#lk(€ 9,y{†+sizijmy  l}t trl  ;Άˆqrƒ}wxqry{zqrzwgil|†xy{œy(w < € ‡xyqrz•w6€ 9:;Qy{† lU„¡ujm~etr6l T5 ŠŒy{ziƒ}trs  qrzi¢+ƒky{jmjmlkz•w€Ìu†xl M¡z¡uZt t ‚&¢q š•lkz—qrz—wgilÌtvu€xwÌpNlkƒ}wxqry{zÍ%– 5 Ô 'd#d6 "Ì1,#- 16' 6 1,#6f gilÏ~etvuz¡u†&y{zil4ötrlk¢{¢{l  giy{~i~lk†Cƒky{zi€xq  lk†xl  wgi†xy{si¢{giy{sew—wgilÍ~¡u~lk†Hqr€H†xlk~i†xlk€lkzBwl  qrz 

(20) §  5 KL@q 0{l wgil y{zil€xws  qrl  œ•‚ ‹5gij­u  quz  j,silkgetrlk†cqrz X nAZ95q wcqr€cœ¡u€#l  y{z 5uZqrœlk†ˆ(w < €,l¨u†ˆt qrlk€xw~i†xywywµ‚N~l X S” Z5‰µwÛqr€ ƒky{jm~y{€l  yA;/u"œy  ‚Cuz  u6wl}trlk€#ƒky{~eqrƒÌtrlk¢:5f gil ƒklkzBwlk†ÌAy ;j­u€€  Ay ;wgilœy  ‚—qr€ÌuZwwgilqrz•wlk†x€#lkƒ}wxqry{zÏAy ;³wgil6geqr~%4 ‡xyqrz•w(< €uS„etrluz  wgil . . Bm.

(21)

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