Robustness analysis of Passivity-based Controllers for Complementarity Lagrangian Systems

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(1)Robustness analysis of Passivity-based Controllers for Complementarity Lagrangian Systems Jean-Matthieu Bourgeot, Bernard Brogliato. To cite this version: Jean-Matthieu Bourgeot, Bernard Brogliato. Robustness analysis of Passivity-based Controllers for Complementarity Lagrangian Systems. [Research Report] RR-5385, INRIA. 2004, pp.19. �inria00070618�. HAL Id: inria-00070618 https://hal.inria.fr/inria-00070618 Submitted on 19 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. Robustness analysis of Passivity-based Controllers for Complementarity Lagrangian Systems Jean-Matthieu Bourgeot — Bernard Brogliato. N° 5385 Novembre 2004. ISSN 0249-6399. ISRN INRIA/RR--5385--FR+ENG. THÈME 4. apport de recherche.

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(56) £¯`k*gGs8dfnNkq{lm t gGw`¤kl]#`p!mNsvdfkqdrx` t ` Gn#dfkq`¢g#nGwkldfmon ixnGi~hruesvdjs¦*£`+k V t `n#mxkq`¨kl]#`{lÙsvkq{ldrw+kqdrmxn¬m~ V kqm Σ ¦ ± ; ; ¶ U ¶ & ³ UUe & ± ³ U ± < >& . V (·). kq]Gi~k*drhfh#svÙ{qx`¥dfnkl]#`,sqg#Gsq` Ng#Ùnok. )/ '6( #-. [Y @B >

(57) ! $ 2 8 9d8 @B )> @! 3 , ' > .

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(59) [&3b 5&b 43' #6 V > Ω Σ (

(60) !! ., ' ', -3 {t } ,,Ik P ,Σ! 3<3' /b/

(61) 6 3 ! t < +∞ ? k k∈IN ∞ n rC E *k¥m©w+hji~dr_s(ix{q`p#{lÙsq`nokl` t ]#drw ] i~{l`gGsv`¢g#hkqm¡p#{lmJx`®kq]G`sk i~#drhrd«ku6mxª£ixx{ i~n#xdji~n. ΣI. [. 7\Z]. ^]. I. a. ]. +-,. cyc. I. I. squeskl`_s,d«kl]{q`Hsvp!Ùw+k,kqm;kq]#` t ` Gn#dfkqdrmxnGsi~n t e¦ )8 + ¬7 ( )*+ ,) J)8(/< ') n;kq]Gdrs¤sqÙw+kqdrmxn;`#{ldr` Gu t ò`hrmxpQkq]G`kq{ ixw @dfn#w+mxnNkq{lmxhrhr`{|svkq{ i.kqÙxugGsv` t drn;kq]Gdrs|pBi~p!`{H¦ = _Qmo{q` Ùhrix!mx{ i.kq` t Ùslw+{ldfp#kqdrmxndrsªiH~i~drhrix#hr`¨drn Cf E

(62) dfkq]iw+mo_QpGhf`kq`svklix#drhfdfku¬ixnGi~hruesvdjsU¦ nkl]#drsªpGi~p!`{2¥` ¢mewgGs,mon¬kq]#`sqdf_g#hji~kqdrmxnGsi~n t {qmo#gGsvkqn#`Hsqsixsqp!Ùwk s¦ ZY. . XW. ZY. ^ L ` _%_ ` !Q`"!". G.F. \

(63) m_i~ x`kl]#`©w+monokq{lmxhrhr`{ t Ùsqdrxn Ùiosvdr`{kq]#` t u@nGix_;drwUi~h§` NgGi~kqdrmxnGs U Vi~{l`Ww+monGsqd t Ù{q` t drnkl]#` o`n#`{ i~hrd/?` t wm@mx{ t drnGi.klÙs,dfnNkq{lm t gGw` t drn C E ¦ = ìkl`{2kl{lixnGsv¢mx{l_;i~kqdrmxndfnkq]G`n#` w+m@mo{ t dfnGi~kq`Hs q = ekq]#` t u@nGix_;drwUi~h¯svuesvkqÙ_djsios¢mxhrhfm.2s 9 [q , q ] q = [q . . . q ] q = Q(X) ∈ IR 1. 2. T. 1 1. 1. n. m T 1.  M11 (q)q¨1 + M12 (q)q¨2 + C1 (q, q) ˙ q˙ + g1 (q) = T1 (q)U + λ      M21 (q)q¨1 + M22 (q)q¨2 + C2 (q, q) ˙ q˙ + g2 (q) = T2 (q)U i i   q1 ≥ 0, q1 λi = 0, λi ≥ 0, 1 ≤ i ≤ m   . U adV. ¥¹ mohfhrdjsvdrmxn¬{lg#hr` \,]#`ªx`Hwklmx{ t `nGm~kq`Hs|kl]#`¨w+monGskl{lixdfnNk¥w+memx{ t dfnBi.kq`ªixn t t Ùn#m~klÙs¤kq]#`ª¢{l``§wm@mx{ t drnGi.kl`x¦8\,]#` wmxnokl{qmohfhr`{ t `x`hrqmxp!` t dfn²kl]#drs§pGi~p!Ù{ªgGsv`Hskl]#{qÙ` t d87R`{l`nNk§hfm.q hfÙx`h

(64) w+monokq{lmxhhjiH2s¢mo{ª`Hixw ]²p#]Gixsq` Ω i~n t I 9 Ω 1. 2kcyc. 2. 2kcyc +1. kcyc.  Unc      Ut T (q)U =  Ut     Uc. = Unc (q, qd? , q, ˙ q˙d? , q¨, q¨d? ) = Unc (q, qd? , q, ˙ q˙d? , q¨, q¨d? ) =. Unc (q, qd? , q, ˙ 0, q¨, 0). ! `+¢mo{q`¨kl]#` G{lsvkdr_;pGixw+k i.ìkqÙ{,kq]G` G{lsvkdr_;pGixwk. U [V. ]#Ù{q` −P= + K (P − P ) djs|kl]#`2¢mo{lw`$#Jp!mNsvdfkqdrmxnQ¢`` t Giowl Q]Gdrw ]®djs¥i t#t ` t®t g#{qdrn#kq]#` Ω p#]Giosv`Hs¦ sqg#p!`{lNdjsqmx{svdfklw ]#`HsB`kÙ``n6kl]#Ùsq`kq]G{qÙ`®wmxnokl{qmoh|hriJ2sU¦\,]#`®m.x`{ i~hrh|skl{qgBwkqgG{q`mx kl]#`¡w+monokl{qmohªsvkl{li~kqòudjs¬sv]GmJnmon 8dfBU ¦¥aG¦ \,]G`kq{ i~nGsqd«kldfmon w+mxnNkq{lmxhªhjiHÛwmx{l{qÙsqp!mxn t s;klmkl]#` n#mo!n wmxnGsvkq{ i~drnok mon#`;dfkq] ¢{l[m ?U`n V+8i~n t kl]#`®w+monGsvkq{ i~drnokhriJ wmx{l{q`Hsvp!mxn t kqm kl]#`sqg#_Õm~n#mx!n Ùw+monGskl{lixdfnNqk w(t)mxnokl{qmohXi~n t qi=wmxnNq˙kq{lmx=h*dr0n¡¢mx{ w+`x¦;\,]#ìosvu@_;peklm~kqdjwQsvklix#dfhrdfkÙuWm~¥kq]#djs slwl]G`_;`;_i~ x`Hs¨kq]#`®squeskl`_Õhrixn t mxn©kq]#`®wmxnGsvkq{ i~drnoksvg#{qÆixwÙs¨kli~nGx`nNkqdji~hrhruWi.ìkqÙ{Ùn#mxg#o]©wuew+hrÙs d. f. q. = Unc − Pd + Kf (Pq − Pd ). 2kcyc +1. d. . ? d. ? d. ? d. Üìß e¯Ü. C.

(65)

(66) !"! #$%&' &(*)+ ,

(67) -.!/ -&. mxwmxnGsvkq{ i~drnok s #J¢{qÙ`_;mxkqdrmxnGs U mon#`¡w+uew+hr` V¦ = squN_;peklm~kldrwUi~hrhfu kl]#` kl{lixnGsvdfkqdrmxnBs*!`+k¥Ù`n;¢{qÙ`_;m~kldfmon;p#]GixsqÙs|ixn t Ωp!`{l_i~n#∪ÙnoIkqhruQw+∪mxnBΩskl{lixdfnNkXp#]Giosv`Hs|i~{l` t mon#`dfkq]#mogek ixnNuwmxhrhfdjsqdfmon¯¦ 2kcyc. kcyc. 2kcyc +1. Desired trajectories generator. Free motion trajectory. Transition trajectory. Constraint trajectory. Supervisor Force control. !#"$ %'& (! "$ )+* $ .

(68)

(69) . . Lagrangian System. Non linear controller. . 8dfog#{q` a 9|cNkq{lgGw+kqg#{l` mx

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(71) m®_i~ o`kq]#`QsvuesvkqÙ_ ]#dfkªkl]#`(wmxnGsvkq{ i~drnok§sqg#{vÆiow+` U i~n t kl]#`n t drslsvdrpGi~kq``n#`{lxu t g#{qdrn#df_ • pGiowk s Vd«8kq]#` kl{liowl @drn#Q`{l{lmx{djs,nGm~k ?`{lmG¦ \

(72) m_ix x`kq]G`§squ@svkqÙ_ i~p#pG{qmNixw ](kl]#`§wmxnGsvkq{ i~drnok¥sqg#{qÆixw`,k i~n#o`nNkqdji~hrhfu U dfkq]#mogek{qÙ!mxg#n t VXd« • kl]#` kq{ ixw @dfn#QdjspBÙ{v¢`HwkH¦ \,]#`Hsv`®k¥m©svdfkqgGi~kqdrmxnGs(ix{q`¬w+mx"n Gdrw+kqdrn#G¦ §nkq]G`mxkq]#Ù{(]Gi~n t kq]#`wmxg#pGhfdrn#W!`+k¥`Ùn i~n t drn U a VX_i~ xÙskq]#ìxsqu@_Qp#kqm~kldjw®sk i~#drhrd«ku Ng#dfkq` t 8d \®wg#h«kQkqmWmxek i~drndf2xÙhfmewd«kldf`Hsix{q`sqqg#e}Ù`Hwk(qkqm t djslw+mxnNkqdrnNg#dfkqdrÙTs 9i~n@u x`hrmewd«ku²}g#_;pi~k dr_QpGhfdrÙs ]#Ùn ¦ ¼2`nBw+`²d«¨kq]G` kl{lixnGsvdfkqdrmxnpG]Gixsq`djsªwmxnGsvkq{lgGwkl` t dfkq]dr_QtpBixwk sBmxnG`σ]Gios(tkqm ) Gn> t 0i_;ixn#n#VÙ{U ≡klm®0x`k V (t ) = 0 drn²mo{ t Ù{kqm®¢mx{ w+`kq]#`(squeskl`_kqm®{l`_ixdfn²mxnkq]G` t Ùsqdf{l` t kq{ i.}Ùw+kqmx{lu ]#`{l` V+¦2\,]#djsªdrs n#mxkmoN@dfmogGsdrnx`nG`{ i~h i~n tt ` GnGdfn# q (·) ixs t mon#` !Ùhfm.djs2iQiHu®kqXm;o(·)`+k,kl]#`{q`HqsvgG(·)h«kH¦ . $. 1. k. V. 2. k. kcyc f. d. . d. ? d. G

(73) ]. I. RO, L . - 0` /21 RO"31 ` M54 Q` bJ. g#{ldfn# kq]#`,kl{lixnGsvdfkqdrmxnQp#]Gixsq`kq]G`2w+mxnNkq{lmxh#svdrxnGixh drs t ` Bn#` t ixs8¢mohfhrm.2s U svÙ` 8drG¦oª¢mx{ q (·) ]#Ù{q` A, A , B , B i~n t C w+mx{l{lÙsqp!mxn t klm 8dfB¦# dqV¦|(t)£¯`+k2gGs t ` Bn#`[9 τ djs2wl]#mNsvÙn¬@u®kq]#` t `HsvdrxnG`{ioskq]#`svklix{vkm~

(74) kl]#` kq{ i~nBsvdfkqdrmxnp#]Gixsq`x 0. kcyc 0. efeÖg,iä k i. 0. . ? d. . ? 1d.

(75) . b-d & -.86 . . .

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(84) {li.}`Hwklmx{lu q (t) t drs,kq]#` kldf_;`wmx{l{qÙsqp!mxn t drn#klm q (t )¨G t w+mo{q{lÙsqp!mxn t s¥kqmQkl]#` G{ sk2df_;pGiowkU t w+mo{q{lÙsqp!mxn t s¤klmQkl]#` Gn#dfkqìoww+gG_g#hri~kqdrmxnp!mxdrnok2m~

(85) kl]#`sv`Ng#`nGw` {t } t drs,kq]#`Ùn t mx

(86) kq]#` kl{lixnGsqd«kldfmon¬p#]Giosvò τ djs2svgGw ]¬kl]Gi.k q (τ ) = −αV (τ ) i~n t q˙ (τ ) = 0 ¦ Ω = [t , t ] ¦ §n [τ , t ) !¥`df_;p!mosq`kq]Bi.k q (t) drs2kdrw` t 8d 7R`{l`nokldji~#hr`xRi~n t q (t) t Ùw{q`HixsqÙs,klmJix{ t s mxn ¦ nmx{ t `{¯kqm§w+mopB`¤d«kl]kq]#`w+mog#p#hrdfn#ªB`kÙ``n i~n t U V −αV (τ kl]#`,svdrxnGixh q) (t)[τ∈ C ,(IRτ )] djs

(87) ¢{qm ?`n t g#{ldfnG2kl]#`¥kq{ i~nGsqdfkqdrmxnp#]Bixsq`x.dL¦ `x/¦ 9 q q(t) = qq Mq˙ (t)6==0 0 mon [τ , t ] ¦ §n (t , t ] #` t ` Gn#` q i~n t q ixTs 9 q = (0, q ) q = (−αV (τ ), q ) ¦ §n ¥`¥sq`+k ¦8\,]#`¥p#g#{qp!mNsv`¤m~ djs¯klm§w+{lÙi~kq`¥i @dr{vklgGi~h Xp!mxkq`nNkqdji~h ¢mo{lw+`([t]#drw ]¡,svtkli~Gdfhr]Bd ?UÙsªkq]#`Qqsque=svkq`[0,_ qmon (t)]∂Φ `xÙndf|kq]#`Qp!mosqd«kldfqmonm~|kq]#`;w+monGskl{lixdfnNk§drs§g#nGw`{qklixdfn¯¦ ¹¥monGsv` Ng#`noklhrukl]#)` #ye` t p!mxdrnok mx!kq]#`ªw+mo_Qp#hr`_;Ùnoklix{qdfku(svuesvkq`_ drsXgBsv` t dfnQkl]#`2`+yep#{lÙslsvdrmxn mx¥kq]G`®£¯uoixp#g#n#m.²¢g#nGw+kqdrmxn (˜q(q=,qq˙ −) q )

(88) ]G`{lÙixs¨kq]G`g#n#{lÙiowl]Gix#hf` #ye` t p!modfnNk q djsgGsq` t dfn kl]#`(w+monokq{lmxhhjiH¦ nWsqg#_;_i~{luxBi~ìkqÙ{2kl]#` G{ sk§df_;pGiowkï.k t q (·) djs§sv`k2klm ?U`{lm]#drhr`drnWwiosv` djsQsv`kklm (τ ) U drnm~kl]#`{(mx{ t s U svdfklw ]#`Hs(iosdfn t drwUi.kq` t drn U [ V.V®¦ τ c@drnGw+` >q˙ t(t q) 6=(·)0 i~n t q (t −αV drn²o`n#Ù{lixhLekl]#`kl{li.}`Hwkqmo{qu !`]BiHx`Hshrdf x`drni®svmo{vk2mx p#hjixsvkqdjw;w+mxhrhrdrsqdfmon U e = 0V¦ )d«kl6=]©0{lÙsqp!Ùw+k¨kqm 8drG¦¯ #mxn#`;]Gixs τq (·)i.k A t i~k B t i.k A i.k C #i~n t B i.k t U kl]#`§kl`{l_ −P − K P t ` Gn#Ùs¥kl]#` sqdrxnGixh X (·) !`+k¥Ù`n B i~n t C t mon 8drG¦G V+¦ V (τ ) = 0 kl]#`n A wmx{l{q`Hsvp!mxn t s¥kqm;kq]#` kldf_;` τ ¦ . ? 1d. kcyc 0. kcyc 0. ? 1d. 0. k k≥0. ∞. kcyc f. kcyc 1. kcyc f. 2kcyc +1. $. kcyc 0. kcyc 0. $. $. kcyc 0. kcyc 0. kcyc ? 1d 1 kcyc d. kcyc 1. ? d. 0. kcyc kcyc 1 0 + 2. ? 2d. ? 1d. ? 1d. 1. 2 ? 2d. ? 2d. ∞. 0 f kcyc kcyc d f. ? d. d. ? T 2d. d T. 2d. d. d. kcyc 0. ? d. ? d. 10. ? T 2d. ? d. 0. kcyc 1. 0. 1d. − 0. ? 1d. 1d. 7@. − 0. kcyc 0. 1d kcyc 0. A. kcyc f. kcyc 0. d. 00. 1d. t. n. kcyc d. 32. d. d. . 12 ? 2d. f. ∞. 0. 0. 0. ? d. d. kcyc 1. Üìß e¯Ü. C.

(89)

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(92) . \,]#`wmxnokl{qmoh@hriJgGsq` t dfn(kl]#djsG{ sk|slwl]#Ù_;`djs*Gixsq` t mxnkl]#`wmxnokl{qmohfhr`{*p#{q`HsvÙnokl` t drn C TE xmx{ldfodfnGixhfhru t `Hsvdrxn#` t ¢mx{,¢{l` ` _;m~kldfmon¬p!mosqd«kldfmonixn t x`hrmewd«ku®xhrmxBi~h iosvu@_;pekqmxkqdjw§kq{ ixw @dfn#B¦*£¯`k2gGs,p#{lmxp!mNsv`9. U ZV. Unc = M (q)¨ qd? + C(q, q) ˙ q˙d? + g(q) − γ1 (q − qd? ) − γ2 (q˙ − q˙d? ). ¦|\,]#`£uoi~p#gGn#mJ¢g#nGw+kqdrmxn²iosqsqmew+dji.kl` t klmQkl]#drs2wmxnokl{qmohRhjiJdjs'9 U V edfkl] q˜(·) = q(·) − q (·) 1 1 V (t, q˜, q˜˙ ) = q˜˙ M (q)q˜˙ + γ q˜ q˜ 2 2 ³ 8 ; · & ¶ ; 8±x´ ; H ± ± Gmx{kl]#`sqÙwmxn t slwl]#Ù_;`x

(93) kq]G`®n#mon#hfdrn#`Hi~{w+monokq{lmxhrhr`{#hrmew U mxn 8dfB¦da V djsGixsq` t mon©kq]G`slwl]#Ù_;` p#{lÙsq`nNkq` t drn C E ¦|£¯`+k2gGsp#{lmxp!mosq`§kq]#` ¢mohfhrm.dfn# 9 U V U = M (q)¨ q + C(q, q) ˙ q˙ + g(q) − γ s ]#Ù{q` s = q˜˙ + γ q˜ s¯ = q¯˙ + γ q¯ q˙ = q˙ − γ q˜ i~n t q¯ = q − q γ > 0 ixn t γ > 0 i~{l`kÙ¥m slwixhrix{Ni~drnGs¦X£¯`kgBswmxnGsqd t `{kl]#` ¢mxhrhrmJdrn#;pBmNsvdfkldfx`¨¢g#nGw+kqdrmxnGTs 9 ]#Ù{q`. . γ1 > 0 γ 2 > 0. T. 1. T. d. ]. . . nc. 2. 2. V1 (t, s). r. r. =. d. 1. r. ? d. 2. 1 T 2 s(t) M (q)s(t) 1 T 2 s(t) M (q)s(t). 1. 2. U V. ¶ ¶ &*e. ± ± .o \,]#`wmxnokl{qmoh¤pBi~{ i~_;`+kl`{ s]Gdrw ] wi~n ! `klg#n#` t Nu©kl]#`¬`n t gGsq`{;i~{l` ¢`` t Giowl oixdfnGs γ γ i~n t K ekl]#òixdfn α i~n t kq]#` kldf_;` τ # ]#djwl] t ` Bn#` q ¦ . . V2 (t, s). =. T. + γ2 γ1 q˜(t) q˜(t). ^]. 1. 2. f. kcyc 0. ? d. ^ _ R 51"!_ $# R`bM%O%_.O `J MHL MH_ J RO%R ¬U &8 ; ¶ ] _ +-,, 53& #-. B '- Ut ! (' >& b-. ., ', -3 {tk}k≥0 % / 3 3 4

(94) / e& #;> 8H!. , lim k→+∞ tk = t∞ < +∞ ? ¶ ¶ ; ; ¶ ]`_

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(113) 1. 1. kcyc. 0.17.

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(115) −0.03 0.50. 0.54. 0.58. 0.62. #. 0.66. 0.70. 0.74. 0.78. 0.82.

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(150) Ua. b-d & -.86 . 0.9. 24e−3. Slotine−Li

(151) . 0.8. 0.09. 0.7. 20e−3. 0.07. 0.05.

(152) . 0.6. 16e−3. 0.03. 0.5 0.01.

(153) . .

(154) . −0.01 . 0.4. −0.03 15900. 0.3. 16300. 16700. 17100. 17500. 17900. 18300. 12e−3. 18700. 19100. 19500. ". 19900. #!. 8e−3.

(155) . 0.2. Slotine−Li. 0.11. 0.1 4e−3. 0.

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(159) H.

(160) !"! #$%&' &(*)+ ,

(161) -.!/ -& 0.9. Slotine−Li. 0.21. ')(+*,,.-/0(21.-34(5,6 1.798:(+*;=<. Slotine−Li. 0.8. . 0.7. . 0.17. "! #$. 0.6. . 0.13. ?. 0.5 0.4. 0.09. 0.3. % & ! #$. 0.05. . 0.2. % & ! #$. . 0.1 0. 0.01. . . −0.1. −0.03. 0. 4e3. 8e3.

(162) 20e3

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(164) kq]#`dr_;pGixw+k Slotine−Li. . 0.62. >. 0.66. 0.70. 0.74. $. 0.78. 0.82. Oxy. 18e−3. 0.13. @ L B CD. 0.8. 0.58. 8drxg#{l`QH 9 §{l#d«k s,dfnkq]G`p#hrixn#`. . 0.9. 0.54. Slotine−Li. 0.11. 16e−3 0.09. 0.7. @JL K B CD. 14e−3. @ A B CD. 0.07. 0.6. 0.05. 0.03. 0.5. 0.01. Q HMR. 0.4 −0.01. 0.3. 12e−3. HIEF GMEHNTS. HIEF GMEHNPU. 10e−3. HIEF GMEHNPO. @ JA K B CD. 8e−3. ^ [ _]. −0.03 8e3. 10e3. 12e3. 14e3. 16e3. 18e3. 20e3. 6e−3. @AB CD. 0.2. 4e−3. 0.1. 2e−3. @JA K B CD. 0 −0.1. 0. 0. 4e3. 8e3. EF G

(165) H 20e3 B G

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(167) [ X

(168) \]. 16e3. 24e3. 28e3. 32e3. 36e3. 40e3.

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(171) sqw ]#Ù_Q`d«kl]²{lÙsqp!Ùw+kklm®_;Ùiosvg#{l`_;Ùnok n#modrsq`monp!mosqdfkqdrmxnWixn t o`hrmew+dfkÙuo¦ 8drosU¦ e xsv]GmJ kl]#`Q£uoi~p#g#nGmJ¬¢gGnGwkldfmonWm~|kq]G`k¥mw+monokq{lmxh slwl]G`_;ÙsU8{q`Hsvp!Ùw+kqdrx`hrukl]#`z*i t `!n z*ixn.}visqw ]#`_;`®dfkq] ]#dfkq`n#mxdjsv`®m~ 5% i t#t ` t mon q ixn t q˙ . `. $. W. . +a. . +a. efeÖg,iä k i.

(172) "Z. b-d & -.86 . 28e−3. 3.6. Paden−Panja. . Slotine−Li. 3.2. 24e−3 2.8. 20e−3 2.4. 16e−3. 2.0. 12e−3. 1.6.

(173) . 1.2. 8e−3 0.8. 4e−3 0.4. 0. 0. 0. 4e3. 8e3. 12e3. 16e3. 20e3 . 24e3. 28e3. 8drxg#{lè9*mo#gGsvkqn#`Hsqs [¢g#nGw+kqdrmxn. 32e3. . 36e3. 40e3. 0. 4e3. 8e3. 12e3. . 16e3. . 20e3. 24e3. 28e3. 32e3. 8drxg#{l`xb9Xmo#gGsvkqn#`Hsqs |¢g#nGw+kqdrmxn . V (t). 0.21. Paden−Panja. 36e3. 40e3. V (t) Slotine−Li. 0.5 0.17. *40 132. 0.4 0.13. (. 0.3. 0.09. 0.2 0.05. !. 0.1. " #$ #%#& ' *,+ - ./0 132. 0.01. 0. −0.03. 0. 1e3. 2e3. 3e3. 5e3 . 4e3. 6e3. 7e3. 8e3. 9e3. 10e3. 0.52. 0.56. 0.60. 0.64. ). 0.68. 0.72. 0.76. 0.80. 0.84. 8drxg#{l`.a 9*mo#gGsvkqn#`Hsqs |kq{ i.}Ùw+kqmx{ldrÙs 8dfog#{l`x 9Xmx#gBskln#Ùsls X¢g#nBwkqdrmxn V (t) ixn t kl]#`¨cehfmxkqdrn# ` Ù£¯dBhjiH»dfkl]®n#modrsq`2hr`x`hBmx 40% ¦X\,]#Ùsq`2nGmxdjsv`ªhfÙx`hjs¤ix{q`kl]#`ª_i.yedr_g#_¢mx{¥Ùixw ] hjiH !`+¢mo{q` drnGsk i~#drhrd«kuo¦

(174) drG¦@~ sv]GmJ2sXkq]#`ª`xmohfgekldfmon;m~Rkq]G`ª£uoi~p#g#n#m.¢gGnGwkqdrmonmx isvdr_g#hji.kldfmon;m~Rkl]#`2zXi t Ùn![zXi~n.}vi slwl]G`_;`§i~k¤kl]#`§hrdf_;dfkm~¯kq]#`¨svklix#drhfdfkuxNdfkl]¬in#mxdjsq`ªhfÙx`h!m~ 10% ¦X\,]#`¤}g#_;pGs σ (τ ) dfnGw{q`Hixsq` m.x`{;kq]#`Ww+uew+hrÙU sU¦ \,]#djs`+y#ix_QpGhf`djsn#m~k®{lmxGgGsk!`HwixgGsv`dfk t meÙsqn k¢g#h Gh§ixhfh2{q` Ng#df{l`_;Ùnokls;mx p#{lmxp!mNsvdfkqdrmxn¡ sq``{qÙ_;ix{q mon 8dfB¦ HQsqÙw~¦#da V+¦ #{lmx_Ûkq]#`Hsv`®svdr_g#hji.kqdrmxnBs¯¥`wmxnGwhfg t `Qkq]Gi~k kq]G`;]Nu@#{ld t c@hrmxkqdrn# ` Ù£¯d|slwl]#Ù_;`;drs _;mx{l`Q{lmx#gGsvk kl]Gi~n(kl]#`]Nu@#{ld t zXi t Ù!n z*i~n~}vi¨mxnG`x¦8gek|mxn 8drG¦x~aG~¥`2svÙ`kq]GixnQ`xÙnQd«!kq]#`2whfmNsv` t hrm@mxp;svuesvkq`_ t meÙs

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(184) #@H C E{qmoxhrdji.kqmG¯¨¦r

(185) 2djw+g#hrÙslw+g8cR¦r

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(189) wmxnGsvkq{ i~drn#` t {lmx!mxklsU¦ W+W W \{ i~nGsU¦emxn = geklmx_i~kqdjw¹¥monokq{lmxh U V U xo V¤aG

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(192) h 9Wi wixsq`svkqg t uo¦ W+W W-\

(193) {lixnGsU¦Xmon = g#kqmx_i~kqdjw¹¥monokl{qmohL¦ U o" V U oxd V o

(194) !Ux~ @. 0. _. $. $. $. @. @. . efeÖg,iä k i.

(195) H. b-d & -.86 . %' Q

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