applying differential geometry to kinematic modeling in mobile robotics

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Applying Differential Geometry to Kinematic Modeling in mobile Robotics

EBETIUC, SILVIA; STAAB, HARALD

Department of Robot Systems

Fraunhofer Insttute for !anufa"turn# En#neern# an$ Automaton I%A

&obe'stra(e )*, +-./ Stutt#art0ER!A&1

Abstract: - e e<amne the =nemat" mo$e' of a mob'e robot :th too's of $fferenta' #eometry7 These too's

a''o: "omprehens4e mo$e''n# of e4en "omp'e< mob'e art"u'ate$ me"han"a' systems7 Furthermore, they offer a

4ery ''ustrat4e stru"ture of the e>uatons of moton by pro4$n# a so "a''e$ tr4a' "onne"ton of pure moton an$

shape moton7 %ure moton s the systems e4o'uton n phys"a' spa"e, shape moton s the mo4ement of the

art"u'ate$ me"han"s su"h as :hee's, fns, f'aps, 'e#s, et"7 Dfferent from the most other pub'"atons, :e try to

#4e a #raph"a' nterpretaton of the "omp'e< mathemat"a' ob?e"ts as :e'' as a $eta'e$ mathemat"a' treatment

of "onf#uraton spa"es an$ ts tan#enta's7 Ths s mot4ate$ n the obser4aton that peop'e :th an en#neern# ba"=#roun$ often fn$ t $ff"u't to #et nto ths mathemat"a' $oman7

Key-Words: - manfo'$, #eometr" me"han"s, mob'e robot, Le #roups, fbre bun$'e, Ehresmann "onne"ton,

prn"pa' "onne"ton, "onstrants

1 Introduction

By e<p'orn# me"han"a' systems from a

$fferenta' #eometr" pont of 4e:, one :ants to

un$erstan$ the stru"ture of the e>uatons of moton n

a :ay, that he'ps to so'ate the mportant ob?e"ts :h"h

#o4ern the moton of the system7 The $e4e'opments n

the fe'$ of #eometr" me"han"s ha4e 'e$ to pro#ress

n the stu$y of #eometr"a' stru"ture an$ $ynam"s of

me"han"a' systems @,)+7 By mo$e''n# the

'o"omoton pro"ess, t s possb'e to fu''y un$erstan$

the beha4our of the system7 An ana'yss of "omp'e<

systems :as ma$e n @7 A re'ate$ mo$ern e<amp'e s

the sna=eboar$ ma$e n @)* usn# smu'atons an$

e<perments7 e''y an$ !urray @/ mo$e''e$ many

'o"omot4e systems usn# =nemat" "onstrants :th

resu'ts on "ontro''ab'ty an$ moton #eneraton, ?ustas n @) :here the "onf#uraton 4arab'es are

$4$e$ nto t:o "'asses shape an$ poston

a""or$n# to the bas" stru"ture of 'o"omoton7 In the

)/+s Bro"=ett put the theory of Le #roups an$ ther

Le a'#ebras n the "onte<t of non'near "ontro'7 For

moton "ontro' prob'ems n4o'4n# rotatn# an$

trans'atn# bo$es su"h as mob'e robots, spa"e

systems, un$er:ater 4e"h"'es, the natura' appearan"e

of "ertan Le #roups $er4es from the fa"t that they

$es"rbe the "onf#uraton spa"e of the system or a

part of t @)7 For an ntro$u"ton to robot"s n a

mathemat"a' sense @)- s re"ommen$e$ an$ for the

un$erstan$n# of mathemat"a' notons :e su##est

@)),),)/,*),G7 Furthermore n ths paper :e try to

#4e a #raph"a' an$ "on"rete $es"rpton of the

mathemat"a' notons use$ n mo$e''n# an$ re$u"n#

the e>uatons of moton of a mob'e robot7 e try to

"'arfy aspe"ts that are h$$en by the mathemat"a'

frame:or=, 7e7 the po:er of $fferenta' #eometr"forma'sm, from a $fferent pont of 4e:7 Ths

forma'sm may not seem 4ery fam'ar to en#neers

an$ robot" pra"ttoners, thus :e try to ma=e t

un$erstan$ab'e, #4n# a$ n the "ontnue$ stu$y of

robot" 'o"omoton7

The or#ansaton of ths paper s as fo''o:s8 after

the short o4er4e: n "hapter ), n "hapter * :e han$'e

some bas"s for :or=n# :th me"han"a' systems n a

#raph"a' :ay, fo''o:e$ by an ntro$u"ton of some

matr< #roups that ha4e #reat s#nf"an"e n robot"s,as :e'' as other usefu' too's an$ "on"epts7 e :''

app'y ths theory to a smp'e t:o :hee'e$ robot7 In

"hapter G, :e "ontnue to stu$y the robot n moton

a'so n a #raph"a' :ay7 e e<p'an some bas"s about

Le #roups an$ ther Le a'#ebras7

2 Basics of geometric mechanics

2.1 he configuration space is a manifold

For me"han"a' systems "ompose$ of r#$ bo$es,the configuration s the "o''e"ton of a'' poston

4arab'es7 The number of 4arab'es n s e>ua' to the

2005 WSEAS Int. Conf. on DYNAMICAL SYSTEMS and CONTROL, Venice, Italy, November 2-4, 2005 (pp106-112)



number of $e#rees of free$om DoF of the

me"han"a' system7 The set of a'' "onf#uratons s

"a''e$ configuration space Q 7 The 4e"tor

q =q 1,,q

n ∈Q $enotes a "onf#uraton :th

#enera'Je$ postons q i 7 F#7) ''ustrates some smp'e

me"han"s an$ ther "onf#uraton spa"es7 For a r#$ p'anar ro$ pen$u'um top 'eft the "onf#uraton spa"e

s S 1

={x , y ∈2

: x 2

y 2

=1} 7 The spot on the S 1

"r"'e represents the "urrent "onf#uraton7 Ho:e4er, a

"onf#uraton spa"e at frst s no mathemat"a' ob?e"t

one "an "a'"u'ate on7 Thus one nee$s an ob?e"t from

mathemat"s :h"h s "omparab'e to "onf#uraton

spa"es7 Cons$er a topo'o#"a' manfo'$ :h"h s a

topo'o#"a' spa"e an$ 'o"a''y 'oo=s '=e the eu"'$an

n )7 %ropertes of

n "an be transfere$ to

manfo'$s, n part"u'ar one "an use "a'"u'us on

manfo'$s7 &ote that a manfo'$ has no #'oba', but

on'y 'o"a' "oor$nates7 Su"h a manfo'$ s somorph"

to "onf#uraton spa"es of many me"han"a' systems7

0enera''y any q i s of un'mte$ ran#e*7 Thus one "an

spea= of the configuration manifold 7

F#7) K Smp'e me"han"s an$ ther "onf#uraton

manfo'$s7

F#7* K %art of "onf#uraton manfo'$ of mob'e

robot7

)more pre"se'y a manfo'$ s a spa"e :here for any pont

there e<sts an open ne#hbourhoo$ U an$ a

homeomorphsm :U U ⊂n

on any open set

U ⊂n

7 Refer to e7#7 @)),). for e<a"t $efntons7* the ran#e of q

i s often 'mte$ n pra"t"e but ths :ou'$

ma=e theory more "omp'e<7 It seems easer to omt 'mts n

theory an$ "ons$er them n pra"t"e K f nee$e$7

The art"u'ate$ me"han"s of the "ons$ere$ mob'e

robot "onsst of t:o n$epen$ent'y $r4en :hee's

:h"h form a S 1×S

1

part of the "onf#uraton

manfo'$ as sho:n n F#7*7 Ths s a'so "a''e$ shape

space an$ 1

, 2

are "a''e$ shape variables7 Sn"e

the robot s free to 'o"omote there s a se"on$ part:h"h represents the robots postonG on a p'ane an$

:'' be ntro$u"e$ n the ne<t subse"ton7

2.2 !pecial matri" groups are manifolds

If a bo$y on a horJonta' p'ane s free to rotate

about the 4ert"a' a<s, the orentaton ∈ "an be

represente$ as a matr<

R=

[

cos −sin

sin cos

]7

R s ortho#ona' 7e7 detR=1 n a r#ht3han$e$

"oor$nate frame an$ RT =R

−17 Hen"e R s a

member of the spe"a' ortho#ona' #roup SO 2 7

The mob'e robot s mo4ab'e on a p'ane an$ ts

poston "an a'so be :rtten n matr< form8

g =

[

cos −sin x

sin cos y

0 0 1

]

=[R p

0 1] 7 )

F#7G sho:s g of t:o e<amp'e postons8 g 0

at the

or#n of the p'ane "oor$nate frame an$ g 1

at another

poston7 A'' g form the spe"a' eu"'$an matr<

#roup SE 2 7 These matr"es ha4e the propertes

R∈SO 2 an$ p=x ,y ∈2

7 F#7G a'so #4es an

''ustraton of SE 2 7 Lo"a''y t 'oo=s '=e eu"'$an3

, but #'oba''y t s a stru"ture n4

7 ne may

$es"rbe t as an nfntesma''y 'on# tube :th an

nfntesma''y th"= :a''7

A beneft of usn# SE 2 s, that ts members not

on'y $es"rbe postons, but a'so trans'atons7 The

matr< n ) $es"rbes a trans'aton about x , y an$

a rotaton about -7 hen 'oo=n# at the propertes of

G There s sometmes "onfuson about the terms

MorentatonN, MpostonN, M'o"atonN, et"7 Throu#hout ths

paper, :e use them as fo''o:s8 location O :here a pont s,

orientation O :here an ob?e"t s $re"te$ to, position O

'o"aton an$ orentaton of an ob?e"t n spa"e7 the "o'umns of a rotaton matr< are orthonorma'7 Ths

ye'$s to RRT =R

T R=I ⇒R

T =R

−1

7-

e use the fo''o:n# terms for r#$ bo$y moton8 rotationO a"t of turnn# an ob?e"t about ts a<es, translation #shift$

O a unform mo4ement :thout rotaton or$&et, motion

O rotaton an$ trans'aton7 A'thou#h at frst s#ht, these terms




a $fferentab'e manfo'$, one fn$s that they a'so ho'$

for SE 2 , be"ause the #enera' matr< #roup

GL n ={R∈n ×n

:det R≠1} s the super#roup

of SE 2 an$ s a'so a $fferentab'e manfo'$7 An$

GL n , the #roup of 'near somorphsms from n

onto

n

, s a $fferentab'e manfo'$, be"ause t sopen n 4e"tor spa"e

n ,

n . t "an be seen as

the n4erse ma#e of ∖ {0} n re'aton to the

$etermnant map, 7e7 GL n =det −1

∖{0} 7 th a

f<e$ base n n

:e ha4e n

,n

≈n ×n

7

The propertes of SE 2 ye'$ to the "ontnuous an$

$fferentab'e transformaton

3∋x , y ,

[

cos −sin x

sin cos y

0 0 1

]∈SE 2 7

Thus one "an "ombne SE 2 :th the manfo'$ part

of the art"u'ate$ me"han"s n or$er to obtan the

entre "onf#uraton manfo'$ of the mob'e robot

:h"h s $one n the ne<t subse"ton7

F#7G K !atr< #roup SE 2 for $es"rbn#

postons

2.% Bundled manifolds

hen "ombnn# the t:o parts of the manfo'$ :e

ntro$u"e$ n the pre"e$n# subse"tons, one obtans

the "onf#uraton manfo'$ of the mob'e robot as

Q =SE 2×S 1

×S 1

7 *

The t:o S 1

be'on# to the an#'es of the t:o :hee's7

E>uaton * sho:s a natura' $e"omposton of Q n

the sense that SE 2 s the poston :here :e :ant to

may #4e the mpresson of obser4n# a "ontnuous moton,:e man'y use them :thout "ons$ern# the e4o'uton of

moton n tme7.the set of a'' 'near maps from

n nto

n 7

mo4e to, an$ S 1

×S 1

s :th :hat Ptoo'sP :e ha4e to

a"he4e that7 SE 2=G s "a''e$ fibre an$

S 1×S

1=M =Q /G s "a''e$ base7 At e4ery base

e'ement an ortho#ona' fbre s atta"he$7 A'' fbres

to#ether form a fbre bun$'e+, :h"h n turn s Q 7

F#7 ''ustrates the "onf#uraton manfo'$ of themob'e robot as a fbre bun$'e7 &ote that e4ery fbre

strn# s a 3$mensona' manfo'$ ob?e"t see F#7G7

Another ''ustraton s #4en n F#7-8 the fbres are

bun$'e$ n the sense that the he#ht of e4ery bun$'e3

rn# abo4e the base represents a un>ue poston of

the robot n the p'ane7 Qust as n F#7G t:o e<amp'e

postons are sho:n n the f#ure8 the Jero poston g 0

an$ another poston g 1

7 &ote that a bun$'e3rn# a#an

s the entre torus surfa"e sn"e at e4ery g the t:o

:hee's "an ha4e arbtrary an#'es 1

, 2

7

F#7 K Conf#uraton manfo'$ of the robot as a

fbre bun$'e

F#7- K Representatons of $fferent postons on

the bun$'e$ "onf#uraton manfo'$

+A fbre bun$'e s a smp'e #eometr" stru"ture "onsstn# of

a base spa"e an$ a fbre spa"e7 !ore pre"se'y t s a trp'etof the manfo'$s E tota' spa"e, M base spa"e an$ the

pro?e"ton : E M 7 For $eta'e$ $efntons refer to e7#7

@.7




2.& Mechanical constraints

!ost me"han"a' systems ha4e "onstrants :h"h

re$u"e the number of DoF7 ne :ay to "hara"terse

"onstrants s to $stn#ush bet:een ho'onom" an$

nonho'onom" "onstrants7 For a better un$erstan$n#

of $fferent "onstrants refer to @)G7 Any ro''n#

moton :thout s'$n# s nonho'onom" non

nte#rab'e :h"h s a'so the "ase for the mob'e robot7

In $eta' the "onstrants are

x =/2⋅cos ⋅ 1

2

y =/2⋅sin ⋅ 1

2

=/2w ⋅ 1−

2

G

:here s the ra$us of a :hee' an$ w s the a<a'

$stan"e bet:een the :hee's7 G sho:s no "onstrant

on the shape 4arab'es 7e7 from a startn#

1t

0,

2t

0 , the system s free to mo4e any:here

on the torus surfa"e F#7. 'eft7 hereas an

nterestn# >ueston s :hat mpa"t the "onstrants

ha4e on the poston n SE 2 7 Ths s ans:ere$ by

G an$ some phys"a' $e'beratons7 At =0 an$

= the robot may on'y mo4e a'on# x 3, an$ at

=/2 an$ =3 / 2 t may on'y mo4e n y 3

$re"ton7 Cuttn# ths out of the SE 2 manfo'$ as

sho:n n F#7G, one obtans a stru"ture :e "a'' thetwisted band manfo'$7 F#7. r#ht #4es an

''ustraton7 Ths manfo'$ stru"ture s true for a''

#roun$3mo4n# me"han"s :h"h ha4e "onstrants that

$o not a''o: s$e:ay mo4ements7

F#7. K Ima#es of the "onstrane$ "onf#uraton

manfo'$

% angent space and tangent bundle &o: :e are #on# to e<amne the mob'e robot n

moton, 7e7 there s a q ≠0 7 There are $fferent :ays

ho: to ntro$u"e tan#ent 4e"tors7 They "an ether be

$efne$ by "oor$nate representaton or as 4e'o"ty

4e"tors of a "ur4e on manfo'$s7 The set of a'' 4e'o"ty

4e"tors at q ∈Q be'on#s to a 4e"tor spa"e ass#ne$ to

q 7 Ths s "a''e$ the tangent space T q Q 7 &atura''y t

has the same $menson as Q 7 For an nter4a' I ⊂an$ a "ur4e c : I Q the tan#ent 4e"tor 4e'o"ty

4e"tor of c at tme t 0∈I s

c t 0:=T

t 0

c d

dt ∣

t 0

∈T c t

0Q 7

th a$ of the tan#ent spa"e one "an transfer the

$er4aton from rea' "a'"u'us to manfo'$s7 From the

unty of tan#ent spa"es at a'' ponts on a manfo'$, one

"an "reate the tangent bundle TQ =∪q ∈Q

T q

Q 7 The

tan#ent bun$'e TQ of a manfo'$ Q a'so s a fbre bun$'e :th fbres F =T

q Q , base Q , an$ the

pro?e"ton : TQ Q 7 E4ery base pont s tra4erse$

by a fbre7 F#7+ sho:s ho: TQ 'oo=s '=e for one of

the :hee's of the robot7 hen t rotates :th "onstant

spee$ 1

, :e are mo4n# on TQ at "onstant he#ht

abo4e Q 7

F#7+ K Tan#ent spa"e, tan#ent bun$'e an$ natura'

pro?e"ton of one :hee'

A 4e"tor fe'$ on a manfo'$ Q s a part "ross3

se"ton of the tan#ent bun$'e TQ 7 In other :or$s8 a

4e"tor fe'$ ass#ns a tan#ent 4e"tor to e4ery pont

q ∈Q 7 ne "an nterpret a 4e"tor fe'$ as the r#ht

s$e of a system of or$nary frst or$er $fferenta'e>uatons7

%.1 'ie groups and 'ie algebras

Le #roups are sho:n to be usefu' n robot

=nemat"s an$ "ontro', thus :e are #on# to ntro$u"e

the bas" "on"epts7 A Lie group s a manfo'$, :h"h

a$$tona''y has a #roup stru"ture :h"h s "ompatb'e

:th the stru"ture of the manfo'$7 Important e<amp'es

a spa"e G s "a''e$ Le #roup, f8 a G ,⋅ s a #roup; bG s a n 3$mensona' $fferentab'e manfo'$; " the

mappn#s G ×G G , g ,h gh an$ G G , g g −1

are smooth7




are the matr< #roups, :h"h are a'so Le #roups7 ne

may refer to @. for $eta'e$ proofs for the #enera'

'near #roup GL n a'' rea'34a'ue$ nonsn#u'ar

n ×n matr"es, an$ the spe"a' ortho#ona' an$

eu"'$an #roups SO 2,SO 3,SE 2 7 For the

mob'e robot :e :'' 'oo= "'oser at SE 2 n or$er to better un$erstan$ the stru"ture of a Le #roup7 ne

may :rte SE 2 as 2

×SO 2 7 Sn"e SO 2 an$

2

are Le #roups, SE 2 s as :e'' a Le #roup,

an$ dimSE 2=dimSO 2dim2=3 7 An e'ement

g =p ,R∈SE 2 has the "oor$nates g =x , y , :th x , y ∈

2 an$ ∈SO 2 7 g , :rtten as a

3×3 3matr<, s

g =

[cos −sin x

sin cos y 0 0 1

]7

SE 2 may be $entfe$ :th the spa"e of a'' 3×3 3

matr"es of the form

g =[R p

0 1] , R∈SO 2 , p ∈2

:h'e

g =p ,R g =

[R p

0 1]7

The #roup stru"ture s #4en by

g ⋅h =p1

,R1p

2,R

2=R

1p

2p

1,R

1R

2 an$

g −1

=p ,R −1

=−RTp ,R

T 7

rtten n "oor$nates SE 2 as a 4e"tor spa"e8

g ⋅h =(cos 1x

2−sin

1y

2x

1,

cos 1x

2sin

1y

2y

2,

1

2) an$

g −1

=−cos x −sin y , sin x −cos y ,− 7

As one "an see, the t:o operators are $fferentab'e

$fferentab'e n ther "omponents7

To a Le #roup G one "an ass#n a Lie algebra G .G s nothn# e'se but a "o''e"ton of 4e"tor fe'$s

somorph" to the tan#ent spa"e at the $entty T e

G 7

Stu$yn# the Le a'#ebra one "an $er4e propertes of

the stru"ture of the asso"ate$ Le #roup7 th these

propertes one "an $efne the exponential map of a Le

#roup, :h"h n turn s the 'n= bet:een Le a'#ebra G

an$ Le #roup G . Sn"e a Le #roup n #enera' s not

abe'an g ⋅h ≠h ⋅g , one may ntro$u"e a 'eft

trans'aton Lg

: G G as Lg h =g ⋅h an$ a r#ht

trans'aton Rg

: G G as Rg h =h ⋅g , :h"h

"ommute Lg R

g =R

g L

g 7 The 'eft or r#ht

trans'aton of a Le #roup "an be seen as an a"ton/,

:h"h s of "ertan nterest for the robot =nemat"s7e "an ma#ne a"tons as a moton of the robot n the

p'ane from one poston to another7

In the pre4ous se"ton :e ha4e ntro$u"e$ the

tan#ent bun$'e an$ the tan#ent spa"e of some smp'e

me"han"a' systems7 &o: :e are #on# to return to

our robot :here :e :ant to e<amne the Le #roup

part G ⊂Q 7 hene4er :e ha4e a"tua' or $esre$

motons of any non3statonary r#$ me"han"s, :e are

:or=n# on a Le #roup an$ ts tan#ent bun$'e7 e

ha4e seen that Le #roup e'ements $es"rbe both,

postons an$ trans'atons7 th the Le #roup

stru"ture :e "an e<press tan#ent 4e"tors n T g

G :th

e'ements of the asso"ate$ Le a'#ebra7 The

"orrespon$n# 4e"tor fe'$s are the set of a'' 'eft

n4arant 4e"tor fe'$s 7e7

g ∈G :T h Lg X h =X g ⋅h , for a'' h ∈G , so t $oes

not 4ary un$er any 'eft a"ton7 L=e 4e"tor spa"es the

set of a'' 'eft3n4arant 4e"tor fe'$s :h"h are a'so

$fferentab'e, for more $eta's see @*),* an$ T e

G

are somorph", :here e s the $entty e'ement7 Thus

f :e =no: X at the $entty e , :e =no: t on theentre G X g =T

e L

g X e 7 If :e :rte e'ements

of T e

G :th !, " , then the Le a'#ebra stru"ture s

$efne$ :th he'p of the Le bra"=et for the $efnton

refer to e7#7 @)/,)G8 [! ,"]=[X , Y ]e ,#!,"∈G 7

F#7 #4es an ''ustraton of G , ts $entty e , a 'eft

n4arant 4e"tor fe'$ X , an$ e'ements :e ha4e

$s"usse$ so far7

F#7 K SE 2 Le #roup :th a 'eft3n4arant

4e"tor fe'$ X

/an a"ton s a $fferentab'e map $ :G ×M M so that

$e , p =p , # p∈M , an$

$g ,$h , p =$g ⋅h , p , # p ∈M , # g ,h ∈G

:here G s a Le #roup an$ M s a manfo'$7




%.2 he e"ponential map

The e<ponenta' map e<p : G G 'n=s the Le

a'#ebra :th ts Le #roup7 It s a 'o"a' $ffeomorphsm

7e7 a smooth map :th ts n4erse a'so smooth7 In the

fo''o:n# :e :'' stu$y Le a'#ebra an$ e<ponenta'

map of the Le #roup SE 2 7 Further $eta'e$ proofs,

propertes, an$ app'"atons of e<ponenta' map,

a"tons, an$ other usefu' too's of #eometr" me"han"s

"an be foun$ n @., *),)G7 The Le a'#ebra of SE 2 ,

$enote$ as se 2 , s se 2=so 2×2

, an$ "an be

$entfe$ :th 3×3 matr"es

%!=

[

0 −!3 !

1

!3

0 !2

0 0 0

])7

E4ery e'ement of SE 2 one "an :rte as an e'ement

of the Le a'#ebra :th the e<ponenta' map8

exp!=(!

1

!3

sin !3

!2

!3

cos !3−1,

!2

!3

sin !3

!1

!3

1−cos !3,!

3)

7

th the 'eft trans'aton one "an $es"rbe the

4e'o"ty on a tra?e"tory g t ∈G :th a sn#'e Lea'#ebra e'ement7 The bo$y3 an$ spata' 4e'o"ty of a

r#$ bo$y are !=g

−1⋅g an$ !

s =g ⋅g

−1 respe"t4e'y7

The phys"a' nterpretatons are the trans'atory an$

rotatory 4e'o"ty of an ob?e"t :th respe"t to a bo$y3

an$ a spata' "oor$nate frame7 The re'aton bet:een

both 4e'o"tes s $es"rbe$ :th the lifted adjoint

action $eta's e7#7 n @*7 For SE 2 the 'fte$

a$?ont a"ton s

!d g =T

e

Rg

−1Lg

=

=T g Rg

−1T e Lg =[cos −sin y

sin cos x

0 0 1] 7

)as a 4e"tor spa"e se 2 s somorph" :th G

by

%!=

[

0 −& " 1

& 0 " 2

0 0 0

]!=& , " ∈

37

Ths somorphsm a'so s a Le a'#ebra somorphsm by

[! ,"]= %! %"− %" %! 7

%.% (onnections

As out'ne$ abo4e one "an $es"rbe postons of the

robot :th spe"a' Le #roups, :h"h n turn are part of

the "onf#uraton manfo'$7 For "ontro' prob'ems :e

are ntereste$ n the other part of the "onf#uraton, the

shape 4arab'es7 The shape spa"e s Q /G , a'so "a''e$

re$u"e$ spa"e7 Le #roup an$ shape spa"e "reate the

fbre bun$'e7 hen spea=n# of a trivial bundle one

"an span the "onf#uraton manfo'$ :th the base

Q /G an$ the fbres G atta"he$ to e4ery # ∈Q /G 7

In ths "onte<t one may ntro$u"e the so "a''e$

hresmann connection7 An Ehresmann "onne"ton !

s a 4e"tor34a'ue$ one3form on Q , so that

!q :T

q Q $e# T

q s 'near #q ∈Q an$ t s a

pro?e"ton7 The =erne' of !q

s "a''e$ horJonta'

spa"e an$ $e# T q =% q s "a''e$ 4ert"a' spa"e7Ths means that one "an sp't the tan#ent spa"e T

q Q

nto a horJonta' an$ a 4ert"a' part7 The tan#ent 4e"tor

s pro?e"te$ onto ts 4ert"a' part by a "onne"ton)) or

prn"pa' "onne"ton)* for $eta's on "onne"tons refer

to e7#7 @*7 ne fn$s that the 4ert"a' spa"e s tan#ent

to the 4ert"a' fbre abo4e q ∈Q , 7e7 a'' ponts :h"h

are mappe$ onto the same pont for $eta's refer to

@.7

%.& (onstrained distribution

The constrained distribution "onssts of a'' tan#ent

4e"tors :h"h meet the "onstrants7 A $strbuton s a

"o''e"ton of tan#ent spa"es at a pont on a manfo'$,

so t s a parta' bun$'e of ts tan#ent bun$'e7 In other

:or$s a $strbuton "an be represente$ :th 'near'y

n$epen$ent $fferentab'e 4e"tor fe'$s7 If one :rtes

the "onstrants as oneforms 1

, 2

, 3

, the

$strbuton s q =Ker {

1,

2,

3} 7 The tan#ent

spa"e T q Q "an be sp't un>ue'y nto a 4ert"a' part

% q =T

q O#

q ={!

Q

q ,!∈se 2} )G, an$ a horJonta'

part, :h"h n "ase of the mob'e robot s e>ua' to the

"onstrants G7 Then :e ha4e % q '

q ={0} :h"h s

not 4a'$ for other me"han"a' systems7 In other

:or$s, the 4ert"a' spa"e represents pure motons of

the robot on the p'ane :thout shape motons, an$ the

horJonta' spa"e represents shape motons, 7e7 turnn#

))a "onne"ton s an a''o"aton of horJonta' spa"es

& q ⊂T

q Q for e4ery q ∈Q , so that T

q Q =&

q Q (%

q Q ,

T q

$g &

q Q =&

gq Q for a'' q ∈Q an$ g ∈G an$ &

q Q

$epen$s $fferentab'y on q 7)*

a prn"pa' "onne"ton s a "onne"ton '=e a Le a'#ebra4a'ue$ oneform on Q 7)G !

Q

q =d /dt $

exp t !q ∣

t =0 s the nfntesma' #enerator

a""or$n# to an a"ton !∈G 7




:hee's, :thout pure motons7 The "onne"ton

$es"rbes the 'n= bet:een them7

& (onclusion and )utloo*

In ths paper :e e<p'ane$ the bas"s of the

mathemat"a' frame:or= ne"essary to mo$e' a

nonho'onom" me"han"a' system that n"'u$es

a'most a'' mob'e robots present'y n use n a :ay that

$4$es the "onf#uraton 4arab'es n shape an$

poston "'asses7 Here :e on'y treate$ the pure

=nemat" "ase @, :here the #roup 4arab'es $o not

ntera"t :th the shape 4arab'es the $ynam"s of the

system are "onstrane$ usn# on'y "onf#uraton

4e'o"tes7 Frst :e "hoose the poston 4arab'e

$es"rbn# the poston an$ orentaton of the robot

:th respe"t to an nerta' frame7 The remann#4arab'es bu'$ the shape of the system an$ ts

4araton n$u"es the 'o"omoton7 The re'atonshp

bet:een shape an$ poston "han#es s ma$e by a

mathemat"a' "onstru"ton name$ "onne"ton7

There are some systems :here the ntera"ton

bet:een "onstrants an$ the #roup a"ton s not tr4a',

pro$u"n# momentum "han#es :h"h resu't n

'o"omoton @),7 Ths ntera"ton p'ays an mportant

ro'e n $efnn# a "onne"ton7 The mathemat"a'

propertes of the "onne"ton a''o: to smp'fy resu'ts

for both $ynam"s an$ "ontro' of 'o"omoton systems7

@+ an$ @* ha4e 'ea$ to an un$erstan$n# of the

re$u"ton pro"ess @,) bu'$n# the me"han"a'

"onne"ton7

+ References

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Sprn#er &e: 1or=, &1, )///7

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!athemat"s, *)7

@)- !urray R7 !7, L 7, Sastry S7 S78 A

!athematical ntroduction to &obotic !anipulation, CRC %ress, Bo"a Raton, )//

@). '4a 7 !78 +eometric !echanics, Sprn#er3

Ver'a# Ber'n He$e'ber#, **7

@)+ stro:s= Q7 %78 "he !echanics and %ontrol of .ndulatory &obotic Locomotion, Te"hn"a'

Report for the Department of !e"han"a'

En#neern# an$ App'e$ S"en"e, Ca'forna

Insttute of Te"hno'o#y %asa$ena CA, )//.7@) stro:s= Q7 %78 "he !echanics and %ontrol of

.ndulatory &obotic Locomotion, %h7D7 thess,

Ca'forna Insttute of Te"hno'o#y, %asa$ena,

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%ress7, <for$, **7

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!ath, 4o'7), pp7 G-3GG), )/+.

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applying differential geometry to kinematic modeling in mobile robotics

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