Control of underactuated mechanical systems by the transverse function approach

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HAL Id: inria-00070482

https://hal.inria.fr/inria-00070482

Submitted on 19 May 2006

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Control of underactuated mechanical systems by the transverse function approach

Pascal Morin, Claude Samson

To cite this version:

Pascal Morin, Claude Samson. Control of underactuated mechanical systems by the transverse func-

tion approach. [Research Report] RR-5525, INRIA. 2005, pp.18. �inria-00070482�

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ISRN INRIA/RR--5525--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème NUM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Control of underactuated mechanical systems by the transverse function approach

Pascal Morin — Claude Samson

N° 5525

March 2005

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©Qx³l}h\;\;'vlZ\\;zgqpfQ \;oqpµ}\Dgokv[ iqWZpsg+\h|Zok\;gkgpvlQikW}zi+iqW}\ok\2szikpv~l ¦j=N~¨psg gkzikpgµ}\DpiqW

c 1 (θ) = ε 1 cos θ, c 2 (θ) = −ε 2 sin θ, c 3 = −(ε 1 ε 2 )/2

VXW}\½z|Z|Zpy;zikpv~l%Jikv ikWZ\gfhgiq\;[ ¦jF1¨£´v17iqWZ\z|Z|Zokv~zyWT|Zoqv~|Svtg\Dpl :DK<vo]ikWZ\y vltikoqv~!v

Zoqp§iq\;gkggfhgiq\;[]gXpl< z1okpzl~iv~l/p\okvm}|}g´iqWZ\;lftp\2sZg!ikv]Z\ µ}lZ\ikWZ\lZ\;A z1okpsz1¥Z\

z := y ◦ (f (θ)) ⁻ ¹

=





y 1 − f 1 (θ) y 2 − f 2 (θ)

y 3 − f 3 (θ) − af 1 (θ)(y 2 − f 2 (θ))





¦«hD¨

Jp¬ikWZ\2oJ¥<fz|Z|Zpy;zikpv~lvSikWZ\oq\;z1iqpvl½¦jD~¨ pl9:;K< v~o¥<fhpoq\Dy£iy2zy2mZsziqpvl/iqWZ\Xikp[]\2¢h\;oqp zikp ~\

v

z

psgp \;l¥<f

˙

z = ∆(f 1 (θ), z)Y (f (θ))





ϑ 1 − c 1 (θ) ˙ θ u 2 − c 2 (θ) ˙ θ

−c 3 θ ˙ + ¯ v 1 v 2





¦«¨

piqW

∆(f 1 , z) =





1 0 0

0 1 0

az 2 −af 1 1





v~o;h\DrtmZp z\;l~ikf~







˙

z 1 = ϑ 1 − c 1 (θ) ˙ θ

˙

z 2 = u 2 − c 2 (θ) ˙ θ

˙

z 3 = −ac 3 θ ˙ + a¯ v 1 v 2 − af 1 (θ)(u 2 − c 2 (θ) ˙ θ) + ay 2 (ϑ 1 − c 1 (θ) ˙ θ)

¦« ¨

VXW}\?p[]|Sv~oiz1lti´Fz~y£iDZzi!iqWZpsgX|Sv~pltiDhpg´iqWSzi;<ikW}z1l}¡<g´ikvikWZ\7iqoz1l}gq \;okgkz1p¬ijf|Zokv|9\2oqijf+v

f

^¦FWZpsyW

p[]|Zp\DgXiqW}z1i

c 3

psg(hp9\;oq\;ltioqv~[ ¯2\;oqv<¨£

ϑ 1

u 2

Sz1lS

θ ˙

y;z1l½¥S\mSg\Dikv&[&z1¡\

z

v~v ¥}z~gpsy2zf zl<fh\;gqpoq\Dikokz1wj\;y£ikvokf© Zv~opl}gikzl}y \~hvlZ\y;z1l½yWZv<v~gq\

ϑ 1 = c 1 (θ) ˙ θ − k 1 z 1 (k 1 > 0)

= c 1 (θ) ˙ θ − k 1 (p 1 − f 1 (θ))

¦«Dt¨

gqvz~g(ikv\h|9vlZ\;ltiqpsz1f½gjiz1¥Zpp¯;\

z 1

ikv¯2\;oqvzv~lZ&ikWZ\+gqvmhikpv~l}g(viqWZ\+y vltikoqv~\;gqfhgjik\2[½© A!f

m}oikWZ\2o(gq\ iqiqplZ

p 1 (0) = f 1 (θ(0))

^}iqWZ\;l

∀t,

p 1 (t) = f 1 (θ(t)) ϑ 1 (t) = c 1 (θ(t)) ˙ θ(t)

¦«¨

7 lSy \?iqWZpsg(yWZv~py2\?psgX[&zh\~~ikWZ\iqW}oq\;\7ok\2[&zplZplZ]y v~l~ikoqv~SplZ|ZmZikg¦Fl}z1[]\2f

u 1

u 2

}zl}

θ ˙

^{¨´y;z1l¥S\}

mSg\Dikv[+v~lZpiqvoNikWZ\ \Dy£ikvo

(¯ v 1 , v 2 , v 3 ) ^T

z1|}|S\Dz1okplZpl+iqWZ\okp~Wti¢W}z1l}gpsh\vN¦D¨ ©VXW}\Xy2vltiqokv

(13)

giqozik\2~f]iqW}z1i(´\|Zokv|9v~gq\?W}\2ok\7psgXikv+\ h|Sv~lZ\2ltiqpsz1fgjiz1¥Zpp¯;\

ξ ¯ :=





¯ v 1

z 2

z 3



 −

^X (¯ g ⁻ ¹ )v r − v ^? (¯ g)

^¦K8L~¨

ikv½¯2\2okvQ¦Fy v~[]|}z1ok\piqW ¦;~¨¨£/piqW

v ^?

p \;l¥<f ¦$N~¨£/iqWZ\]m}l}y£ikpv~l#y2vl}gqpZ\2ok\; pl iqWZ\y;zgq\v

m}zy£ikm}zikpv~l%© 7 lZ\\Dzgqpf ~\2okp¬µ}\Dg!iqW}z1i(iqW}\iqp[+\2¢Z\2okp zikp ~\v1

ξ ¯

zv~lZ+iqWZ\gfhgiq\2[ g(gqvmhiqpvl}g WSzg!iqW}\vvplZ]gjikoqm}y iqmZok\



 



 



ξ ˙¯ 1 = u 1 + ε 1 θ ˙ ² sin θ − ε 1 θ ¨ cos θ + r 1 + ¯ ξ 1 s 1

ξ ˙¯ 2

ξ ˙¯ 3

!

= M(θ) u 2

θ ˙

+ r 2

r 3

+ ¯ ξ 1

s 2

s 3

¦«F1¨

piqW

M(θ)

iqWZ\pl< \2oqiqp¥Z\[]z1iqokph\ µ}l}\;¥<f

M(θ) :=

1 ε 2 sin θ

−aε 1 sin θ ^aε ¹ ₂ ^ε ² cos 2θ

zl}

r i , s i (i = 1, 2, 3)

9gqv[]\mZl}y£ikpv~l}gh\2|9\2lShpl}mZ|9vl

¯

g, ξ ¯ 2 , ξ ¯ 3 , θ, v r

'z1lS

˙ v r

U¥ZmZi7lZvi7m}|Sv~l

ξ ¯ 1

©(VXWZ\;oq\zoq\v~¥< tpvmSgf[&z1l<f!z®fhgXikv\h|Zvpi(iqW}pg?gjikoqm}y iqmZok\plvoh\2o(iqvgikz¥Zpp¯2\

ξ ¯

iqv¯2\2okv}©

7

l}\7v ikWZ\2[ pgX|9vpltiq\Dv~mhiplikWZ\vvplZ]\2[][&zZ©

´· Q ± B ?4@q° ±&$ 8$1J±/ @ ²



 

  u 2

θ ˙

:= (M(θ)) ⁻ ¹

−k ξ ¯ 2

ξ ¯ 3

− r 2

r 3

u 1 := −ε 1 θ ˙ ² sin θ + ε 1 θ ⁽²⁾ cos θ − r 1 − ξ ¯ 1 s 1 − k ξ ¯ 1 − ξ ¯ 2 s 2 − ξ ¯ 3 s 3

¦«1~¨

+$

k > 0

θ(0)

^° ^°±S²%®° ^G ^°±/

θ ⁽²⁾

^B ^±/ ^±;8 ^O ^2±'§± ^KO ^±

¯ g, ξ, θ, v ¯ r , v ˙ r

°1±

¨ v r

>%®° G 0 §±8

+$9$)E+0#8+%®°'$+% B0 ± §± O

θ ˙

^° ^* ^±°±S² ^G ^±

B ±+ ²

)2±

B° O8O&k°'$ ± $ ±&$' B ² ² $) +

§±6 ° +«²

1 2

d

dt k ξk ¯ ² = −kk ξk ¯ ²

^¦«'N~¨

°1±

)

$)> 2 O

±/ ±&$F°

°4@

D°'$

±

ξ ¯ = 0

VXW}\7|}oqv<v1pg`gjikokzp~W~iqvok!zok%©

5#\lZv gqWZv ikW}ziikWZ\7\2\;Z¥}zy¡y2vltiqokv9sz®Ah\2µ}lZ\;plikWZ\|Zok\2 <pv~m}g!\;[][]z]zgqv\;l}gqmZoq\Dg2

m}l}h\2o!y \;oiz1ply v~l}hpiqpvl}g;1ikWZ\?m}¬ikp[&zik\`¥9vmZlSh\;hl}\;gkgv9ikWZ\hpgikzl}y \`¥S\2ij´\;\2l

g

z1l}]ikWZ\`ok\ \;o¢

\;l}y \7gpiqmSziqpvl

g r

y

z

^<z1l}

ξ ¯

ikWZ\`\Drtm}zikpv~l¦j®¨y;z1l¥9\`ok\2okp¬iqiq\;l

(14)

z~g

˙¯

g = X (¯ g)

^X (h 1 )



T (f 1 )(v ^? (¯ g) + ¯ ξ ) +



 0 f 2

f 3



 +

T (f 1 )

^X (¯ g ⁻ ¹ ) −

^X (˜ g ⁻ ¹ ) v r





¦ ~¨

piqW

T (f 1 ) :=





1 0 0

0 1 0

0 af 1 1





¦ D¨

A!fª¦j Dt¨ ¦«¨£

h 1 (t) = h 1 (0) • exp((f 1 (θ(t)) − f 1 (θ(0)))X 1 )

©`/\ i`m}g7zgkgqmZ[]\¦§v~o(iqW}\gkz1¡\v gqp[+|}pµSy2z1iqpvlS¨?ikW}zi

h 1 (0)

psgyWZv~gq\2l#\;rtm}zNiqv

e

©VXWZ\;l% W}\2l

max θ kf (θ)k

psgg[&z«d<fhgiq\;[

¦ t¨[&z®f¥9\gq\2\2l½zgzlqz1|}|Zoqv hp[&ziqpvl `v1

˙˜

g = X(˜ g)v ^? (˜ g)

^¦ ^¨

v~oJWZpsyW]iqWZ\?|Sv~plti

˜

g = e

^psgJ\ <|9vl}\2ltiqpsz1f]gikz1¥}\~© 7 lZ\`[]z®fiqW<m}gWZv|9\iqWSzi´piok\ iz1pl}ggqv[]\v1

ikWZ\gikz¥Zppijf|}oqv~|S\;oikp\Dg(v1`¦ ~¨£©`VXWZ\vvplZ|}oqv~|Svtgpiqpvl%SWZpsyW¤pg(iqWZ\+[&z1ploq\Dgm}¬i7v1iqW}pg

|Sz1|9\2oDhp \;gXy2vl}y2oq\2iq\7vok[ iqv]ikWZpsgXWZv|9\©

»J» § »

h 1 (0) := e

θ(0) = ±π/2

^°±

η

^± ^° ^° ^#

K

^±$ ^± ^h°

max θ (kf (θ)k +

G (h 1 , e) + kI 3 −

^X (h 1 )k) ≤ η(ε)

^+$

ε := k(ε 1 , ε 2 )k

^)2± ^¤°±}²

± °1±

K r

$) 2

ε 0 , γ g , γ v , β > 0

^0h° ^°1±}²E? ^2± ^$k° ^£²

g r

h°

kv r k ≤ K r

°±/

°±S²

ε ∈ (0, ε 0 ]

G (˜ g(0), e) ≤ γ g

k(v − v r )(0)k ≤ γ v

= ⇒

G (˜ g, e)

^psgXm%©^¥/©J¥<f

βη(ε)

^¦⁸ ^¨

B?Fm/©¥%© °± B+&° ² @ ' ±8 %K

k˙ v r (t)k

^°±/

k¨ v r (t)k

^°? ^@ ^±/8 ^{B ±}

kv(t)k

^°±/$)> ^±^' ^§± ^O

u 1 (t)

u 2 (t)

^°±/

θ(t) ˙

^°' ^@^' ^±?

VXWZ\]p[]|Sv~oiz1lti7|9vpltikgviqW}pg|Zokv|9v~gqpiqpvlzoq\ iqW}\&\hpgiq\;l}y \+v1Xz1l mZ¬ikp[&z1iq\]¥Sv~mZl}vo

ikWZ\y2vtg\D<¢«v<v| iqozy¡<plZª\2okoqv~o; + iqWZ\T¦iqWZ\;vok\ ikpy;z1§¨]|Svtgqgqp¥ZppijfTv1ok\;hmSy plZQiqWZpsg¥9vm}l} zg

[m}yWzg´h\;gqpoq\D¥<f&yWZv<v~gqplZ

ε 1

z1l}

ε 2

gq[&z1}\;lZvmZ~W%<z1l}9++ ikWZ\`|9v~gkgp¥Zppijf+v1%g|9\;y2p¬f<plZzl

kv r k

^{¨ }z1lS}

©oD©iD©

ε ∈ (0, ε 0 ]

^©

Zokv[-¦ t¨£

˙¯

g = L 1 + L 2 + L 3 + L 4

piqW



 



 



L 1 = X (¯ g)v ^? (¯ g)

L 2 = X (¯ g)(

^X (h 1 )T (f 1 ) − I 3 )v ^? (¯ g)

L 3 = X (¯ g)

^X (h 1 )(T (f 1 )

^X (¯ g ⁻ ¹ ) −

^X (˜ g ⁻ ¹ ))v r + X (¯ g)

^X (h 1 )(0, f 2 , f 3 ) ^T

L 4 = X (¯ g)

^X (h 1 )T (f 1 ) ¯ ξ

(15)

/\ i

γ

h\;lZv1ik\z1l<f¤y v~l}gjiz1ltigmSyWikW}ziiqW}\&|Zoqv~|S\;oikp\Dg

P 1

zl}

P 2

pl¦F1¨zoq\]gqz1iqpsgjµS\;vo

g ∈

A

G (γ)

©JVXW}\2l%<v~o

¯

g ∈

^A

G (γ)

<iqWZ\h\;oqp ziqp \Dg

dV (¯ g)L i

v

V (¯ g)

^zv~lZ

L i (i = 1, . . . , 4)

gkziqpsgf ikWZ\vvplZ+ok\2sziqpvlSg



 



 



dV (¯ g)L 1 ≤ −β m k m V (¯ g) dV (¯ g)L 2 ≤ α 2 (η + η ² )(ε)V (¯ g)

dV (¯ g)L 3 ≤ α 3 (η + η ² )(ε) (1 + K r ζ (¯ g)) V ¹ ² (¯ g) dV (¯ g)L 4 ≤ α 4 (1 + η) ² (ε)k ξ(0)k ¯ exp(−kt)V ¹ ² (¯ g)

¦ Dt¨

piqW

ζ

zgq[]v<v1iqWmZl}y£ikpv~l%hz1l}WZ\2ok\

α 1 , . . . , α 4

v

η

}zl}iqWZ\Z\ µ}lZpiqpvl¦ ®¨v

T (f 1 )

ikWZ\Fzy iXiqW}z1i

kv r k ≤ K r

}zl}iqWZ\ok\2szikpv~l

˜ g = ¯ gh 1

WZpsyWp[]|Zp\DgXiqW}z1i

X (˜ g ⁻ ¹ ) =

^X (h ⁻ ₁ ¹ )

^X (¯ g ⁻ ¹ )

θ(0) = ±π/2

zl}iqWZ\Fz~y£iiqWSzi

G (¯ g, e) ≤

G (˜ g, e) +

G (h 1 , e) ≤

G (˜ g, e) + η(ε)

^¦ ^~¨

v~lZ\gW}v(gokv[ ¦K8L~¨iqW}z1i;ZW}\2l

˜

g(0)

^z1l}

v(0)

gkziqpsgf&iqWZ\[&z1wjvoziqpvlSg´plª¦ ¨

k ξ(0)k ≤ ¯ α 5 ((1 + K r )γ g + (1 + η(ε))γ v + (1 + K r + η(ε))η(ε))

dhpl}y2\

η(ε)

zlZpgqWZ\Dgzi

ε = 0

%vl}\]Z\;hm}y2\;g7okv[ ikWZ\&z1¥9v \+pl}\;rtm}z1pijf/zl}plQ <p\;v1&¦ Dt¨ ikW}ziDhvo

ε 0 , γ g , γ v

g[&z1'\2l}vmZ~W%<vo(zltf

ε ∈ (0, ε 0 ]

{¯ g ∈

^A

G (γ)

^zl}

V (¯ g) = α m γ ² } = ⇒ V ˙ (¯ g) < 0

^¦ ^L~¨

piqW

V ˙ (¯ g) :=

4 X

i=1

dV (¯ g)L i

A!fok\;Zm}y plZ

ε 0

z1l}

γ g

mZoikWZ\2oDZpl}\;y \Dgqgkz1okf<v~lZ\h\;hmSy \;g!okv[-¦F1¨Xzl}Q¦ ¨iqW}z1i

G (˜ g(0), e) ≤ γ g = ⇒

G (¯ g(0), e) ≤ γ p α m /α M

= ⇒ V (¯ g(0)) ≤ α m γ ²

^zl}

¯ g(0) ∈

^A

G (γ)

gqvQiqWSzi;¥<f¦F¨zl} ¦ Lt¨£

¯

g(t) ∈

^A

G (γ)

^voz

t

^zl}

G (¯ g(t), e)

G (˜ g(t), e)

G (˜ g, e)

|9vpltiq\DvmZi+¥<f ¦ 8

¨pgiqW}\2l

v~¥hikzpl}\;#¥<fm}gqpl}½iqWZ\]Fz~y£iikW}zi

η(0) = 0

¦Fgqv½iqW}z1i

η ² (ε) η(ε)

^WZ\2l

ε

psggq[&z1s¨ zl}#¥tf mSgplZ¤¦F¨£/¦ D<¨}z1lSikWZ\plZ\;rtm}zpijf

G (˜ g, e) ≤

G (¯ g, e) +

G (h 1 , e) ≤

G (¯ g, e) + η(ε)

G (¯ g, e)

k ξk ¯

kv ^? k

^~z1l}

kv r k

f<p\2sZgiqW}\¥Sv~mZl}h\DhlZ\;gkgNv1

v 2

z1l}

v 3

©Nd<pl}y2\

k v ˙ r k

psgN¥Sv~mZl}h\D'1ikWZ\¥Sv~mZl}h\DhlZ\Dgqg v

u 2

˙ θ

^%z1lS

v 1

u 1

vv(g`okv[ z1iqWZ\Dg\¥Sv~mZl}Zgz1lS

ikWZ\¥9vmZl}Z\;hlZ\DgqgXv

k¨ v r k

^©

(16)

4½ "

) 4) $ #

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v z1okpvm}gXlZv~lZWZv~v~lZv[]psy`[]\;yWSz1lZpsg[&g;<iqWZ\v~vplZ+gq\;y2vl}h¢«v~okh\;o!yWSz1plZ\;gfhgiq\;[







¨

x 1 = u 1

¨

x 2 = u 2

¨

x 3 = u 1 x 2

¦ F1¨

y;z1l¥9\Jm}gq\;7iqv`[+vhh\;1ikWZ\hf<l}z[+psy2g/v1hz(y \;oiz1plltm}[¥9\2o/vtm}l}h\2ozy iqm}z1iq\D[+\DyW}z1l}py;z11gqf<giq\;[&g2

p¡\|}zl}z1ouJuJ{ z1l}{({({H[]zlZp|ZmZsziqv~okg;9ph\Dz1p¯;\;¤gqmZoqFzy2\ \Dgqgq\2sg7z1l}m}l}h\2okXziq\;o? ~\2WZpsy \;g

gq\ i v1}gqf<giq\;[&g¦j~D¨ DpiqW

G = R ³

g = x

X 1 (x) = (1, 0, x 2 ) ^T

X 2 = (0, 1, 0) ^T

X 3 = (0, 0, 1) ^T

zl}

a = −1

•

X 1

X 2

}z1l}

X 3

z1ok\\2§i¢pl< z1okpzlti

v~l

R ³

x • y =





x 1 + y 1

x 2 + y 2

x 3 + y 3 + y 1 x 2





VXW}\lZ\<i`\ hz[]|Z\DgXpm}gjikokz1iq\ikW}ziDSzgvo(lZv~lZWZv~v~lZv[]psy7gqfhgjik\2[&g;hiqWZ\iqoz1l}gvok[&ziqpvlv1N[]\ ¢

yWSz1lZpsy2z!\Drtm}zikpv~l}g]pltikv ikWZ\¤yW}zplZ\Dv~oq[½!z1iqWZv~mZW|Svtgqgqp¥Z\~´psg]lZvilZ\;y2\;gkgqzoqf#voy2vltiqokv

) -$&,-+ )

VXW}\gqfhgjik\2[ z1l}iqWZ\\Drtm}zikpv~l}gz1ok\`ikWZv~gq\h\;gky okp¥S\Dpl;:K<







m x x ¨ − m 3 l α ¨ sin α − m 3 l α ˙ ² cos α = τ 1

m y y ¨ + m 3 l α ¨ cos α − m 3 l α ˙ ² sin α = τ 2

I α ¨ − m 3 l x ¨ sin α + m 3 l y ¨ cos α = 0

¦ ~¨

piqW

m x > m y > m 3

z1l}

I = I 3 + m 3 l ²

\Drtm}zikpv~l}g!pg\DrtmZp z\;l~iXikv







¨ ¯

x = _m ^τ ¹ ₃ − ( ^m _m ^x ₃ − 1)¨ x

¨ ¯

y = _m ^τ ² ₃ − ( ^m _m ^y ₃ − 1)¨ y

¨

α = δ x ¨ ¯ sin α − δ y ¨ ¯ cos α

(17)

piqW

δ = ^m _I ₃ ³ ^l

x ¯ := x + l cos α

^%z1lS

y ¯ := y + l sin α

ikW}ziikWZpsggfhgiq\2[ y2zlzgqv¥9\okp¬iqiq\2lz~g





˙¯

x

˙¯

y

˙ α



 =

R(α) 0 2 × 1

0 1 × 2 1

¯ v

˙¯

v =





¯

u 1 + ¯ v 2 v ¯ 3

¯

u 2 − v ¯ 1 v ¯ 3

−δ¯ u 2





¦ N~¨

piqW

R(α)

iqW}\´okv1iziqpvl[&z1iqokppliqWZ\!|Zsz1lZ\´v1SzlZ\

α

^z1lS

(¯ u 1 , ¯ u 2 )

l}\2y v~ltiqokv zoqpsz1¥Z\;g gmSyW ikW}zi

(τ 1 , τ 2 )

psg\;rtm}zJikv¤gv~[+\mZl}y£ikpv~l ¦Fh\2µ}lZ\Dª\2 ~\2okftW}\2ok\D¨?v1

(¯ u 1 , u ¯ 2 , α, α) ˙

y;z1lplikmZoql¥9\oq\;oqpiik\2lzg



 



 



˙

g = X(g)v

˙

v 1 = u 1

˙

v 2 = u 2

˙

v 3 = −v 1 v 2

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u 1 := ¯ u 1 + ¯ v 2 ¯ v 3

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0 1 0

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α

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{e 1 , e 2 , e 3 }

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Control of underactuated mechanical systems by the transverse function approach

HAL Id: inria-00070482

https://hal.inria.fr/inria-00070482

Submitted on 19 May 2006

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