Time-varying exponential stabilization of nonholonomic systems in power form
Texte intégral
(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 De´cembre 1993. PROGRAMME 4. Robotique, image et vision. ISSN 0249-6399. apport de recherche. 1993.
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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 $ãpKeh¨|xzsft|ehri}dul qrehxzrVl1x@zt§¹ ª«eLdfk)¨|¦©pt|sf^dulrlbs6sfet|zt§sutwi}¨[beLrt|sft§eLrb¯ÀÅzt® t§^suxzdf t§k)z¨|t§lpsf i}s sf@leL¹ ¨|xzsft|ehr l»t®s eyl~d± : I@3 H@j¯\Åzl¦i}duli}¨®e.xzzt®vÍxzlt| Vt§l©`eªsfzldul~ghxz¨®i}dut§sK¦ Eehrt|sft|ehrt|k)reyl¡x@eLsuzl(ª«xzrbsut§eh i}r Á¯ Åzlqrehx@rl@lfeª$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}dutwisƦ@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¯.
(68) # : 4#7E47 8
(69) # D¦Ft§Vsfl~gLdji}sut§eL)e}ªM=Èh@j{ = J@\twght|yl~)qV¦ AJ ) =¶ 9@ = J@ bey- = 3 @ 3 + ©p@l~dfl(suzlÆbehrKsjis {){Vi} ^l~rl~ !eh =@\ir = : @Ç{ir ²t®lvyxri¨Bsfe ¼l~ dfe6t³ª2i} eLz¨§¦+t³ª = (@ :V {t¶¯ ly¯yt³ª2ir)ehz¨|¦!t ª t®t§sÇl~¨ ª>lvyxri¨Bsfe ¼l~dfe@¯ e©{ : twlvyxzt§Li¨§l~Vssue :V{zt¶¯ lï %) ! ( = @ 6 % : I =¶) F@ % ! 6 ©p@twj>{ͪ«duehk =È 9)@Ç{t|k)z¨|t§l ( bey- = ! 3
(70) @3 + ) % ) ! 6 = ! I888I (@ : =¶h @ % = @ % ! 6 ©p@l~dfl\i¨§¨ysuzl\beyl bt|l~sÇ 7 idfl[beLrsÇi}Vs~{~Kt§rElsuzl~¦Æl~rl~r(eLz¨§¦eh ! I888I (V¯ ½ª :V{suzldfl~¨®i}sft|ehr =¶L @ÈK¹ =È 9)@¸t§k)z¨|¦ sfris i} = @pidfllvyxri}¨ sue¼~l~dfei}rB{hª«duehk =Èh´ @j{ysfris H:V¯Él~rbly{~ª«dfeLk =Èh @Ç{ % :V{ = I888 I O @j¯ ½ª2l~t§suzl~d eLd t®! zehz¼l~dfe@{zsfzl()ª«xz rE%sf)t§eL! r <%(= J@ J> bey = 3
(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
(72) 1t®jzeyKl~!¨®i}dughl~d\sf i} eLzl =½ehd
(73) > N©pt§su qrl~t|zgsK©ptwbl.t Ál~dfl~Vsft®i}qz¨|ll~yldf¦V©pzl~dfl@j{\sfzl~ sf@l eLqzsjit§zl¡sut§k)l¹½Li}du¦t|zg EehVsfdfeL¨wpidfl6t Ál~dul~Vsftwiqz¨§lÆl~yl~df¦V©pzl~duly¯.
(74) L @ehsf@l~dpreyfKt§qzt|¨§t|s¦Ã{ÍkFehsft|Li}sfl¡!q¦Fc$dfeLeÃt|sft§eLh{@behrKtwKsjpt§jzeyeyKt§zg+sf@l ª«x@rbsft|ehr i}r e+sf i}s twzeLk)ehgLl~z l~eLxre}ªl~ghdul~l ! 3 =.O ^ @ I + ,,A=)@ = )@\©pt|sf K : I =¶) :@ ,,
(75) 3! 2 ! 2 8 . ¶)suztw\~iLly{yeh@l(lihKt§¨|¦ yl~dftBrl$sfrissuzl(EehVsfdfeL¨B¨®i¡© ; i} ; ! beLrt®l~dfl¡ @dfl~Vt§eLxr¨|¦ idflzehkFehghlzl~ehx e}ªÄl~ghdul~l i} 1 ! dulrlbsut§yl~¨|¦y{esuri}s\c$duehreyKt§¹ sut§eL(izz¨§t|lb¯ eLsfl(suri}sÁ{zi} sux ; {@~i}z@ehsqrlVt Ál~dflsutwi}q@¨§ll~yl~df¦V©pzl~dul Vxzl)sfesuzl)dÇi}sft|e ©pztwj^t® ki}¨§¨|l~dsfriehzl)t|^sf@tw ~iLly¯2Åzl)dfl¡xz¨|s tw Kxzk)kidft|¼~l t§suzl(ª«eh¨§¨|e©pt§zg zduehreyKt§sut§eh Ð \Ð Ô «Ð " 47 8 #45 6 "( "47 8 "/ 2 -7 2 M0 7& 4 %6&-7
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(81) 6 7 2 2-7 2 " 2 6 7 '46 / " . Å>e!t|¨§¨|xrsudji}sul(sf@twdfl¡xz¨|s~{zeh@l8ki¡¦Ã{ê«eLdl»zik)z¨§lÃ{y¾@eyeyl . = ! I888-I ( @ . (. % !. %5%. % 6 7 6 2 2 @. . ¶. = @. ©pt|sf %5% H: = I888IO>@[i}r ! 3 =.O ^ ¯ tw\sfzl~+zeLk)ehgLl~zl~eLxreª\lghdfll6i}r)EehVsft§VxzeLxr¨|¦+t Ál~dul~sutwiqz¨§ly¯ @l i¨wKe!riL;t|sfzt®p~ihKl . (. % ) ! 0 5O= %-@ % ) 6 7 6 2 2. ¶h $»Vz¨§t®bt§s[l~»@dflut§eLrª«ehdM; i}r<; ! idflsfzlÂeLqzsjit§zl¡ª«dfeLk =¶m@¸i}r =¶L @j¯ .l(ri~ylÆt§k+xz¨®i}sfl¡ sfzt®eL¨§xzsut§eL!ª«ehd¸i8ª«ehxzdu¹¶t|k)l~rKt§eLri}¨
(82) =.O m@[K¦zKsfl~kRt| re©ldª«ehduk`dul~zdull~Vsft|zgsfzlVt§zl~ki}sft®~eª2i ~idf¹«¨§t§ylyl~zt®b¨§lïhÅzt®Kt§k+xz¨wisft§eL sux Bt§¨§¨|xrsudjisflrsfzl\reyut§q@t§¨§t|sK¦eªzl~»Vrehzlsutwi}¨|¨§¦Ksji}q@t§¨§t|¼~t§@gpx ¾6iyl~zt®b¨§l2i}qreLxzs i Vlt|dfl)EehBrgLxzdjisft|ehqV¦+k)li} e}ªi beLsut§Vxzehxrsut§k)l¹½Li}du¦t|zg(ª«l~lq ihjB¯ .l jzeÃl ! I² I =¶h @ . . % ! !. 3H= . %5%. @ = . @ . = @.
(83) . 4#7E47 8
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(85) = " @ " ! 3 % $ $ !. t§gL. 3 ! 0 5O= . = @. = @K@ = @ 2 2. ¶}m @ =. ¶sfzl6Kt§k+xz¨wisft§eL>{ysfzlgÃi}t§ idfl + ,%&!' :&8 ´ : 8m % " : $ =¶h´@ , % $ 5 8 1 ´ !. : ! : 8§ . opl~i¨§¨Bsuri}ssf@l Ksjisfl6behkFeLzl~Vsj ? i}r ? ! eª\sf@l8l¡vÍxzt|Li}¨§ls ª«eLxzdf¹Èt|k)l~rKt§eh i}¨[¾ i}t§@lNK¦zKsfl~k:ki~¦zV¦zKtw~i¨§¨|¦qrl)t§Vsfldfzdul~sfl.ih suzl~idf¹ sult®i}.beyeLdjt|ri}sul(e}ª[sf@l yl~ztwE¨§l $ reht|s(¨|eV~i}suli}sk)t®¹ÈtwKsji} ble}ª[sfzl duli}d ©p@l~l~¨w =¶Kl~lF± :hµ¶{zehdÆt§i k)eLdflghl~@l~dji¨Äª«dÇi}k)l~©[ehdu.± L}µ¶{zª«ehdl~»@i}k)@¨§)l @ǯz ¶Vt§l~© eª[sfzl+jri}@ghl+e}ª¸beyehdÇ t§ i}! sfl¡ =«m @Ç{Bsfzl¡l+~idfsfl¡twi.beyehdÇt§ i}sfl¡8idfl+sfVxrgLt§yl~ qV¦ i}r $ " 3 ! ! dflKrlbsft|yl~¨§¦Ã¯yÅzlyl~zt®b¨§l $ ehdut§l~Vsjisft|ehi}zgL¨§l tw gLt§yl~qVL¦ %3J % O= ? " @= % J %O
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(87)
(88) 6ze©suzlpyl~zt®b¨§l $ã>~idfsfl¡twi k)ehsut§eL>{bª«dfeLksf@lpt§zt|sftwi¨zEeh BrgLxzdjisft|eh = ? :{ L{ : @\sfe+sf@l8gLeyi¨$beL BrgLxzdjisft§eL = ? : @ǯ Kze©sf@l EehVyl~dfgLl~rble}ª\suzlyl~zt®b¨§l $ $beyeLdjt|ri}sul;sue!¼l~dfe@¯ Kze©sf@l EehVsfdfeL¨>t§z@xzsj ; i}r ; ! yldjx sft§kFly¯ t|ri}¨|¨§¦y{ ze©sf@l(¨§t|zli}d¸lbdulihKt§zgeª = ) = @&@$©pzt®¾)t|¨§¨|xrsudji}sul suzll~»VeLzl~Vsft®i}¨>EehVyl~dfgLl~rbl(eª ) = @\sfe ¼l~dfe@¯ >!#" = =È»VtwKsfl~ bleªÀzehkFehghlzl~ehx B¦Vi}@xzze ª«x@rbsft|ehr$ª«ehd¸ih¦Vk)zsuehsutw~i¨§¨§¦ºsÇi}qz¨|l @ehk)eLghl~zlehxrsut§k)l~¹«Li}du¦t|zg K¦zKsfl~k@.
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(100) ('758 &8 " & BP = J-I?A@ M = J-I?A@43 ( M 0 (5= J-I?A@ H : J
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(106) &7
(107) {VqrihKtw~i¨§¨§¦behk)l¡2ª«duehk7sf i}spsf@l ª«x@rbsft|eh = =
(108) :{ :{
(109) !zeLst§ @Ç{B©pzt®¾t®zehkFehghlzl~ ehx eª Vl~ghdul~l
(110) ©pt§sf dflKl¡bs suesf@lt§¨®i}sut§eh
(111) = )@ y{>twehz¨|¦Neªb¨®ihu C ©pt§su <H
(112) y¯ 8*y
(113) [º./ 0 ±³Eµ D(¯2& $ãrd@l¡ i}¹ eyl~¨¶ { 6¯ DpiLsft|>
(114) { 6¯[ik)zt§eL>{$·eVVl~¨§t|zgir²\eLsudfeh¨peª ehzzeL¨§eLzehk)t® zl~l¨§l.o;eLqrehsjE{
(115) 7 " 7+
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