Time-varying exponential stabilization of nonholonomic systems in power form

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(1)Time-varying exponential stabilization of nonholonomic systems in power form Jean-Baptiste Pomet, Claude Samson. To cite this version: Jean-Baptiste Pomet, Claude Samson. Time-varying exponential stabilization of nonholonomic systems in power form. [Research Report] RR-2126, INRIA. 1993. �inria-00074546v2�. HAL Id: inria-00074546 https://hal.inria.fr/inria-00074546v2 Submitted on 7 Aug 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. Time-varying Exponential Stabilization of Nonholonomic Systems in Power Form Jean-Baptiste POMET , Claude SAMSON. N˚ 2126 Dećembre 1993. PROGRAMME 4. Robotique, image et vision. ISSN 0249-6399. apport de recherche. 1993.

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(62) H:+i}r ! :¯ ¶dfdulrlEsft§yl[e}ª¿sfzlpqrl~ri~Vt§eLdeª =Èbeh twVl~dfl+ih\iª«xzrEsft§eL!eª>sut§k)ly{~gLeyl~df¹ @l(q¦ ; @Ç{~sfzt®>behVsfdueh¨Vghxridji}VsfllÄqreLxzrlVzlfÀe}ª = ! I888-I (@¿i}rl~yl~Vsfxri¨§¨§¦ EehVyl~dfgLl~rblsfe+¼~l~due ©pzl~ = J@\tw[xzt|sji}q@¨§¦jzeyKl~>¯@ÅÄe+sfzt®zxzdueÃly{zeLzlki~¦ sudf¦)sue)Vl~sfl~duk)t§@l(i beLsut§Vxzehx ¸ª«l~l¡qrihj ; =½©p@twjÂ ihpsfe+qlsut§k)lÆl~rl~rVl~s Kt§rEl EehVsft§VxzeLxrzxzdul~¹¶Ksjisfl8ª«llqriL¾@p~i}@zehsÆsÇi}qzt|¨§t§¼l(sfzl6beLk)z¨|l~sflÆ¦@sul~k @ ©p@twj yll~r qrehxz lB{\ki}yl¡ t§s+beLyl~dughlsfe.¼~l~duez{\ir zdueVxrbl¡)i = J3@ ©p@twji}xrKl = ! I888 I (@sue!gLe!sue!¼l~dfe@¯rt|k)z¨|l(jzeht®bly{@i}k)eLzg!eLsfzldreyut|¹ q@t§¨§t|sft|lb{ytw AJ 8 ; 3 = = ! I888#I (@ @¸t| =¶}m @ ©pt|sf H:V¯ Ð \Ð Ô «Ð ( 47# "' 7 0@ 6 7 #45.6 "6 7% " 4758 47#0 7& 6 7 ' (' &4 "' 7 0@ 6 7 '032 2545 <6 "04*1C 7! µ :I3H@ I. =¶h´@ A=)@ :. . : I.

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(66) 6 6 2 2 6 %6 7 " 2 # " . Ð>AÐ @+¤ Àt§djKs~{rt§s(t®zdue}yl¡Âsf i}s6i}¨|¨suzl+¦@sul~k $ãpKeh¨|xzsft|ehri}dul qrehxzrVl1x@zt§¹ ª«eLdfk)¨|¦©pt|sf^dulrlbs6sfet|zt§sutwi}¨[beLrt|sft§eLrb¯ÀÅzt® t§^suxzdf t§k)z¨|t§lpsf i}s sf@leL¹ ¨|xzsft|ehr l»t®s eyl~d± : I@3 H@j¯\Åzl¦i}duli}¨®e.xzzt®vÍxzlt| Vt§l©èªsfzldul~ghxz¨®i}dut§sK¦ Eehrt|sft|ehrt|k)reyl¡x@eLsuzl(ª«xzrbsut§eh i}r Á¯ Åzlqrehx@rl@lfeª$suzl6¦@sul~k $ã[eL¨§xzsut§eh beLk)l$ª«dfehk sfris = ! = J3@ I888-I (= J@K@6tw zehz@l~gyisft§yli}r @ehzt|rbdfl¡iht|zgz¯[ ¶s tw sfVxr qeLxzrl¡B¯ Å@tw!t|k)z¨§t|lb{ª«duehk =¶h @j{suri}s = ! = J@I888 I ( = J@K@ t®!xzzt ª«ehduk)¨§¦qrehxz lB{pir Kt§rEl =¶h@ 3 = = ! = J@ I888-I ( = J3@&@Ft§ AJ t®$sfzllvyxri}sut§eh+e}ª2iÉx@df©pt§su¼ =ÈsÇi}qz¨|)l @À¨§t§@li}d\K¦zKsfl~kRr!l~ dusfxzduqrl qV¦ sf@l(iLzt§sut§yl x@zt³ª«eLdfk)¨|¦pqeLxzrl¡8sfldfk A= = ! = J3@ I888-I (= J@K@K@Ít§ { = J@ztw>i¨wKexz@t³ª«ehduk)¨§¦ qreL xzrlÁ¯$·ehdul~eyl~d = ! = J@I888I (5= J@K@ÆbeLyldfghl¡sfeehkFl!¨|t§k)t|sphi¨§xzlÃ{ir = = J3@ I888-I ( = J@K@j{Áqlt§zg)x@zt³ª«eLdfk)¨|¦behVsft|x@ehxr©pt|sfdulrlbsÆsfe JE{sul~rz(sfe ¼l~dfe@¯ .l!suzl~izz¨|¦ >iÍ¥Vi¨§¨§l $ã8t|Lidftwirbl zdut§rEt§z¨|l ª«ehd sut§k)l¹½t|Li}dutwisÆ¦@sul~k ± ~V{zÅzl~eL dfl~k µ¶¯@Åztw[~i})qrlVehzl(qV¦ t§ behdueLdjisft§@g sft§kFlpt§)suzl(K¦zKsfl~k sÇi}sul yl¡bsfeLPd = J @j¯> ¶ eht|zge@{is¨§liLsehzl+e}ª¸sfzl+KsjisflFbehk)reLzl~VsjP =½suzl sft|k)l EeyehdjVt§risf)l @rt®BxzqreLxzrl6esfris>t§sÁki~¦(Kl~l~k sfris >ihfi¨§¨|l $ãBzdut§rEt§z¨|l\eylBzeLs izz¨|¦ t§6sfzt®$~iLly¯yÅ@tw$t Exz¨§sK¦(t®zdfl¡l~Vsf¨|¦(eyl~djbeLk)l(t§6sfzlª«eL¨§¨|e}©pt|zg(©¸i¡¦ Kt§rEl(sfzlÆsft§kFl~¹¶Vl~rl~rl~ blptw\ldft§eVVtwh{Vt§st®reyfKt§qz¨|lsfe dflz¨wiLbl(i¨§¨ÁeV~bxzdul~rbl eª JBt| suzl[behVsfdueh¨V¨wi~©b{ir sfVxrBt§6sfzl[b¨|eyKl¹«¨§eyeh yl¡bsfeLAd Brl~¨wÁ{bqV¦8suzl /jizgh¨|2l 1 ©pzt®¾¨§t§yl¡t§ =È @ i}rt®;sux ;risfxzdÇi}¨|¨§¦qrehxzrVlB¯>Åzt®twpt§ªÈihbs l¡vÍxzt|Li}¨§lspsue!t|rbehdurehdjisft|zg 0.O ! i}r !

(67) t§sfzlºsÇi}sfl yl¡bsfeLd~{t|rsuliheª JE¯ e©{qV¦ >ihfi¨§¨|l $ã\zdft|rbt§@¨§ly{yKt§rEl(i¨§¨Bsuzl(sudji}KlEsfehdut§l = = J@ I = J@I888 I ( = J@K@ idflqrehx@rlB{ir¬Kt§rElsfzlª«xz bsft|eh = ! I888-I (@FlbdulihKl)i¨§eh@g suzlKl sudji}Kl¡bsfeLdft§l¡b{Vi}¨|¨Àeh¨|xzsft|ehr pEehVyl~dfgLl(sfe suzllsbehrKtwKsft§@g e}ª2sfzl(dul~xzzt|ehe}ª2sf@l sudji}Kl¡bsfeLdft§l¡ª«ehd©pzt®¾ = I888-I ( @\t®twlsutw~i¨§¨§¦ ¼l~dfe@¯ \eLrt®l~d KxrjiÂsudji}KlEsfehdu¦ = = J@I = J3@ I888-I (= J@K@j¯$¥yt|rbl t® ; ! { ir ; ! t®$ght§yl q¦ =Èh @Ç{ ; ! t®$twl~Vsft®~i}¨|¨§¦¼~l~dueeh+sfzt®sfdÇi}Klbsuehdu¦y¯yÅzl~dulbª«ehduly{ht| Vt§l© eªpsuzl)K¦zKsfl~k $ãlvyxrisft§eLrP =¶´ @j{ ! = J@I888 I ( = J@Æi}dul)beLrsÇi}Vs~¯ twÆi}¨wKe EehrKsji}Vs(eLsfzlºbehrKtwldflsfdjiKlbsfeLdf¦y{rKt§rEl :'¯ Bl~ s lzehsul t§sÇLi}¨|xzly¯.

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(71) @ 3 + i}dfl+¨§t|zlidf¨§¦t|rl~rl~rVl~sesfris suzlbeLrsÇi}Vs beyl bt|l~Vsj 7 t§H =Èh @(k+xrsi¨§¨[qlFlvÍx i}¨\sfe¼~l~dfe@¯ duehk =¶L @j{ suztwpigyit§t|k)z¨|t§l % H:V{ = I888# IO @[ir :¯ Élrbly{ri¨§¨\Keh¨§x@sft§eLr;ª«eLd8©p@twj twt®l~Vsft®~i}¨|¨§¦)¼~ldfeidfl+beLsÇi}t§@lÂt| suzl[l~s2l B zl(q¦ ! :I888I ( :V¯ duehk >iÍ¥Íi}¨§¨|l $ãBt|Lidftwirbl\zdut§rbt|z¨§lÃ{¾suztw t|k)z¨|t§lrsfris\i}¨|¨sudji}KlEsfehdut§l>EehVyl~dfgLlsue(sf@tw$Kl~s~¯Åzldflbª«eLdfly{~ª«ehd\i¦sudji}KlEsfehdu¦y{ % = J@ = I888#IO @i}r = J@Àsfl~r+sfe ¼~l~duez{Vi} ÂKe+eyl+ A= = J@K@j¯V ¶)Vt§l~© eª =Èh @j{zsf@twpt§sfxzdut§k)z¨|t§lÀsfris = J@$sul~rzpsfe ¼~ldfez¯. Å zt®$sÇi}qzt|¨§t|¼i}sut§ehdulxz¨|st®Äsudfxzl\ª«ehd\i¦ i} 1 k)ll~sft|zgsuzlpiLfx@k)zsft|ehr =Èh@pir =Èh´ @[dfl¡rlbsut§yl~¨|¦y¯ ¶+ri}dusftwExz¨wid~{yt³ª !t®\t Ál~dul~sutwiqz¨§lpl~yldf¦V©pzl~dfl¸i}r

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