P oceedings
o
he 1999
IEEE
In e na ional Con e ence on Robo ics
&
Au oma ion
De oi , Michigan May 1999
A
Gene aliza ion o Pa h Following o Mobile Robo s
F.
Diaz del Rio,
G.
Jimknez,
J.
L.
Se illano,
S.
Vicen e,
A.
Ci i Balcells.
Facul ad de In o mh ica. Uni . Se illa.
T :
(+34) 54552
779.
E-mail: [email p o ec ed].
us.
es
Add ess: A da. Reina Me cedes
s/n.
41012 Se illa. SPAIN
Abs ac
Se e al au ho s ha e p oposed some me hoh o applying pa h
ollowing in
specijic
cases o mobile obo s
([3J
[9],
[Ill,
e c.).
When we
hy
o ex end he pa h ollowing app oach o he gene al
p oblem se e al dsy7cul ies a ise.
In
his pape we p esen a
gene alized echnique o apply pa h ollowing o a mobile obo wi h
nonholonomic cons ain s.
As
an applica ion example, we expose he
case o mobile obo s wi h a highe deg ee o maneu e abili y han
he ypical ca -like obo s.
In
pa icula we conside a obo ha can
u n a ound i sel making a ze o- adius u n; a case s ill no esol ed
as a as we know. Finally we p opose a sui able con ol law o his
example ha ensu es asymp o ical con e gence.
1
In oduc ion
The mos ex ended sys ems in au oma ic con ol heo y a e
se osys ems. He e we ack a mobile sys em a he ime i
mo es; i.e. posi ion, eloci y o , in gene al, any magni ude in
which we a e in e es ed, is he ins an aneous e e ence ha ou
sys em mus ollow. In igu e ig.
la
we show his case o he
s a e coo dina es
q( ).
he desi ed coo dina es
qdLF( ),
and he
e o coo dina es de ined as
e,,( )=q( )-qdes( ).
In se osys ems
his is he only possibili y we ha e, because he e e ence
ajec o y is collec ed as we do he acking.
On he o he hand, in mobile obo s i is usual ha he
ajec o y is memo ized o p e iously gene a ed by a pa h
gene a o module
[ diazl].
Fo ou pu poses bo h cases a e he
same, and he e m
memo izedpa h
o me ely
pa h
is used o
bo h o hem. A e e ence o desi ed pa h o be ollowed is
desc ibed by a single pa ame e , namely
,
and i can be
exp essed as a ec o o s a e coo dina es
qk( ).
Fu he mo e,
we mus emphasize he impo ance ha con e gence o a pa h
acqui es in mobile obo s, as con e gence o a ixed poin qo,
can no be achie ed h ough a smoo h eedback s abiliza ion
con ol law (a di ec esul o B ocke ’s heo em
[
11).
When we y o ack a memo ized pa h, he acking
me hodology can be e y di e en , as we know a p io i he
whole ajec o y. Thus we can ind se e al possibili ies o do
he acking. O cou se he classical se osys em acking can
be done jus by iden i ying he pa ame e
( )
associa ed o he
pa h wi h ime, ha is
( )= .
Ano he simila possibili y called
ajec oly acking
(U)
is based in a mo e gene al assump ion
han simple se osys ems (see ig.
Ib):
he pa ame e
( )
is a
gene ic unc ion o ime. The e o e he e o coo dina es a e
e,( )=q( )-qd-( ( )).
Then we can go h ough he s o ed
ajec o y wi h he mos app op ia e scale o
( ),
o example
=a .
Using an asymp o ically s able con ol law (e.g.
[SI),
i is
gua an eed ha he sys em will con e ge o he desi ed
ajec o y in a de e minis ic ime (excep o he inhe en
pe u ba ions ha i may su e ).
,*’
Ac uol desi ed
,’
mjec o y
,‘
Memo ized
mjec oy,
=a ;
a=scale
,‘
--
Fig. la: T acking
in
a se osys em Fig.
Ib:
T ajec o y T acking
Memo ized
ob ained h ough
some
ela ion
b
Fig.
IC:
Pa h Following
Al hough TT is s aigh o wa d, i is no he only me hod (no
he mos sui able) o ollowing memo ized pa hs. Du ing he
las yea s se e al al e na i es ha e been p oposed. The bes
es ablished in li e a u e and mos sui able o many si ua ions
in which ime is no a c i ical pa ame e ( his is he case o
mos cases in indus ial mobile obo s) is
pa h ollowing (PF).
This is based in some ela ion be ween ac ual sys em’s s a e
q( )
and he memo ized pa h. This ela ion will gi e us he
desi ed poin
qda( )
o memo ized pa h o be acked. The eal
sys em should y o ollow his poin ins ead o he one gi en
by he o he app oach (see ig.
IC).
The e o coo dina es a e
also
e,( )=q( )-qda( ).
Using his app oach, i is no gua an eed
ha he sys em will each a poin o he desi ed ajec o y in a
de e minis ic ime.
The i ues o PF can be unde s ood conside ing his example:
i big pe u ba ions o ce he sys em o be a es , o TT he
desi ed poin will mo e una oidably. This means ha e o s
will g ow up o some alue ha may in oduce ins abili y. On
he o he hand, i PF is used, he desi ed poin will be he same
in spi e o hese pe u ba ions. This allows he sys em o
o e come la ge pe u ba ions a oiding possible uns able s a es.
Mo eo e , he ex ac ion o an asymp o ically s able law using
PF is no mo e di icul han using TT, as we can see in he
0-7803-51 80-0-5/99 $10.00
0
1999
IEEE
7
men ioned li e a u e and also in he example o sec ion
IV.
Thus in e es in PF is g owing apidly.
The e ha e been se e al ials o apply PF (we desc ibe hem
below) in speci ic cases, bu when we y o gene alize PF
se e al p oblems a ise.
In
he nex sec ions we will y o sol e
hese p oblems, cla i ying hem wi h a comple e example.
2 De ini ions and Robo Model
Le 's conside a gene al mobile obo as shown in ig.
2
and le
q=(X,
Y,
4'
be i s s a e coo dina es, which ep esen he
Ca esian posi ion
(x,
VE@
o a ce ain poin PO ( ypically
he midpoin be ween he ea wheels) wi h espec o a ixed
ex insic coo dina e sys em
%(O,
i,
j)
and he o ien a ion
#E(-
z l
o he obo wi h espec o he
X
axis. We will choose
u=( ,
U)'€@
as he pai o con ol a iables o ou sys em
which ep esen he linea eloci y o poin
PO
and he angula
eloci y o he obo (o he pai o a iables such as o ques o
ol ages supplied o he mo o s, a e analogue o he acking
s udy, as showed by
[2]).
Fo hese ec o a iables he s a e equa ion o he mobile
obo a e he well-known equa ions ( ha a e non-linea in
q
and linea in
U):
q=B(q)u
;
B=
i4
sen4
0
11
;
U=
(I);
C ]
To s udy he acking o a memo ized e e ence pa h
qk( ) =(Xda( ), Yd,( ), 4,,,( ))'
le
us
de ine ano he in insic
coo dina e sys em
2?{qdes( ( )).
,
n]
linked o he pa h.
is he
uni a y ec o angen o he plana pa h in he desi ed poin
qks( ( ))
and
n
he no mal o i '. Le
(em e,,)€@
be he posi ion
e o s o poin
Po
ela i e o hese axis and
e&(-zn]
he
obo o ien a ion e o ,
so
e,( )=(e,
e,,,
e$'
will be ou ela i e
e o ec o 2. Le
udes=( des( ),
wdes( ))'
be he desi ed con ol
s a e exp essed as a unc ion o he desc ip o pa ame e
.
A his poin i is impo an o de ine exac ly which pa hs
qd,( )=(Xd,( ). Yk( ), q5d,( ))
a e alid. We can no choose
an a bi a y unc ion on
and assign i o he h ee desi ed
s a e coo dina es. These desi ed coo dina es mus ha e some
p ope ies o gua an ee ha he acking is possible. Fi s he
domain o
mus be in ini e ( o example he posi i e eal line
R')
in o de o gua an ee ha he e e ence ajec o y does no
end. This mus be done o ensu e he possibili y
o
con e gence, because, as i
is
well known o a mobile obo ,
B ocke 's heo em
[
11 p e en s eedback s abiliza ion3
o
he
obo o a ixed poin . Second, and o he same eason, he
e e ence ajec o y can no con ain singula poin s whe e he
inpu s a e null, i.e.
udes=O.
Finally, he pa h can be made by a
'
The ec o s
i
and
n
exis s only when he linea eloci y
hs
a
his
poin
o
he i ual obo ha wen h ough he e e ence pa h was no ze o; i i we e
null,
can be chosen pa allel o he i ual obo o ien a ion, as he
nonholonomic cons ain equi es he wo ec o s o be pa allel.
*
An
analogous coo dina e sys em was used in [Kanay90]. The e he sys em
was linked o he obo i sel .
We
mean smoo h con ol laws,
so
ha we elude non-smoo h laws o casons
o
con inui y on he con ol a iables.
8
obo like he s udied one, and his implies ha
(X&( ),
Yd,( ),
mus espec he nonholonomic cons ain s o ou
mobile obo .
e'
J
d=0.25m. R=0.16m
e*=
4-
4de,
X
W
jig.
2:
ex insic and in insic obo coo dina es.
Using he abo e ela i e a iables and coo dina es linked o
he pa h, and by simple calcula ions, he ollowing s a e
equa ions can be easily ound
[6]:
0
w,(~
0
e,
CO&,,)
In i s mo e gene al o m i we de ine e o s in a na u al
ashion, ha is e,
=
R(q,,)(q
-
qa,)
,
hen he ma ix o m o
he abo e equa ion can be exp essed as:
=
Bdcs(eq)Ude ( )+B(e,)u
(3)
3
The Gene al Me hod o Pa h Following
P e ious
s udies.
Du ing he las decade he e has been a g ea
esea ch e o o de elope a acking based in a
PF.
This has
lead o se e al good app oaches ha ha e made emphasis in
di e se aspec s o PF acco ding o he pa icula cha ac e is ics
o he analyzed sys em o he e e ence pa hs o be acked.
The mos impo an can be summa ized in he ollowing
ca ego ies:
1. In
[3]
and [9] he desi ed poin in he pa h is ob ained
h ough a no mal p ojec ion along he ec o ha we ha e
called
n.
The e o e his p ojec ion chooses he poin o he
e e ence pa h ha has null
e,
coo dina e (see
ig.
2).
They
ha e o p ohibi pa hs con aining ci cles wi h small adius
(we will call u ns wi h null adius and in ini e cu a u e
"ze o- adius u ns") o ensu e ha he no mal p ojec ion
exis s and is unique. A simila pa h ollowing was used in
Na lab [12]. As Na lab is a ca -like obo , i canno make
ze o- adius u ns,
so
hese pa hs we e no conside ed.
In
[ll]
he p ojec ion poin chosen by he au ho s is he
one ha minimizes he euclidian dis ance be ween he eal
and he e e ence poin s
PI
(see ig.
2).
Using poin
PI
hey
a oid pa hs wi h cu a u e ending o in ini e. Bu his
s a egy ails when he e e ence pa h is a u n a ound poin
Po.
In his case any ac ual con igu a ion (ha ing di e en
o ien a ions) whose poin
PI
is on he desi ed posi ion o
PI
will ha e ze o dis ance. Tha is, he couple
(XI,
YI)
does no
2.
ep esen he whole s a e o a mobile obo , al hough hese
coo dina es always change o e e y ajec o y. This
example shows
us
ha he whole s a e o he obo mus be
conside ed o cons uc a gene ic acking.
I is impo an o men ion ha p e ious s udies ha e no gi en
an exac de ini ion o cons uc ion o PF as a as we know;
hey ha e speci ied a pa icula me hod, sui able o hei
equi emen s, ha can be called PF. Con e sely, as ou goal in
his pape is o cons uc a gene al app oach o PF, and
conside ing he expe ience ex ac ed om p e ious wo ks, we
mus ha e in mind he ollowing wo s a emen s. Fi s we mus
conside he whole obo ’s s a e ( ep esen ed he e by
(X,
K
4)
o simplici y, and usually called he
obo
pos u e).
Second,
we mus con empla e all he possible e e ence pa hs ha can
be ollowed by he obo , including ze o- adius u ns.
Gene al Da h ollowinP cha ac e is ics.
As a i s s ep and in
o de o ge a gene al cons uc ion o PF, we a e going o
ex ac he gene ic cha ac e is ics o
pa h
ollowing.
Acco ding
o hese s udies and he in ui i e beha io men ioned a he
In oduc ion, he main cha ac e is ics ha mus ule PF and
ha di e en ia e i om
TT,
can be summa ized as ollows:
1.
We only mus conside he global shape o he pa h o do
he ollowing. The desi ed ajec o y e olu ion (go e ned by
)
mus no play any ole in he ack as i does in TT.
In opposi ion o TT (whe e he desi ed pos u e is exac ly
de e mined by a igid law like
= ( )),
in PF we mus choose
some ela ionship o de e mine he desi ed pos u e. We will
call his ela ionship “p ojec ing unc ion” as i p ojec s he
ac ual pos u e o he e e ence pa h. We will deno e i as
I he obo s ops, he e e ence o goal poin mus also
s ops, as he pa ame e
does no g ow by i sel . The
p og ess o
mus no be independen (as in TTj bu
dependen on he eal obo mo emen , ha is
i
equa ion
mus be d i less.
The exis ence o he igid law
= ( )
in TT implies he
e e ence e olu ion
o
be
qdes( ( )),
and consequen ly
“pulling” o “d agging” he obo o each he e e ence. On
he o he hand, in
PF
he e e ence pa h can no “pull” (o
“d ag”) he obo : he obo mus mo e independen ly by
some condi ion (o cou se, meanwhile a con ol law mus
ensu e con e gence o he pa h). We mus impose a mo ion
in he eal sys em o gua an ee i mo es o p og esses. We
will call his condi ion “mo ion exigency” and we will
deno e i as
mo eX g(u)=O.
A
di ec esul o wha is explained be o e
is
ha he e is
no ime exigency in he ollowing. This means ha we can
no ensu e ha he obo will each a e e ence poin in a
p edic able pe iod o ime.
Pa h ollowinsJ cons uc ion.
TT’s cons uc ion is elemen a y.
I equi es only choosing he mos sui able ela ion
= ([j,
o
a de e minis ic acking in he sys em. On he con a y PF
cons uc ion is no
so
ba e because i implies a special
ela ionship be ween he ac ual poin and he global pa h.
Pa adoxically, due o he mo e labo ious cha ac e o
PF,
i
pe mi s mo e lexibili y han TT.
2.
,*,O=O.
3.
4.
5.
9
[STEP
1:
p ojec ion une io?
.
p o,
(e,, )=o
...........................................................
..J
2 e o coo dina es
,
.............................................................
1
I
STEP
2:
mo ion exigency
I
1
................
*:!::!!.?!!:o
..................
.i
,
1
2 e o;cm&ina es
1
ig.
3.
Gene al pa h ollowing cons uc ion scheme.
Some me hods o PF cons uc ion ha e been de eloped in he
e e ences men ioned in his sec ion, bu hey only apply o
speci ic cases. Con e sely, and based on he p e ious
cha ac e is ics, we can s aigh o wa dly ind a PF
cons uc ion based in wo gene ic s eps (see
Jig.
3). In his
scheme we begin wi h a mobile obo ha has h ee s a e
coo dina es ( h ee e o coo dina es
eq
exp essed ela i e o he
e e ence pa h) and wo deg ees o eedom
U
(DOF).
Al hough we p esen he case o
a
mobile obo ; he case o a
gene ic nonholonomic sys em can be easily deduced.
Pi s s ep: “p ojec ing unc ion”.
This is de i ed as a
consequence o cha ac e is ics
1
and
2
o PF. Once a dis ance
c i e ion is chosen, a “p ojec ing unc ion” ,,i0
=O
ela es eal
pos u e wi h global pa h. This gi es us a p ojec ing poin on
he desi ed pa h: i is he desi ed pos u e
qh( ( ))
a his
ins an o ime. A he same ime he p ojec ion is an
holonomic cons ain be ween e o coo dina es;
so
i supposes
he elimina ion o one e o coo dina e. As he p ojec ion mus
be s a ed be ween e o coo dina es and he desi ed pa h, i
depends on ac ual e o s
eq
and on he memo ized pa h shape
(in gene al, pa ame e
);
ha is
p uj(eq,
)=O.
As
we a e
alking abou a geome ic p ojec ion, ec o
U
can no play a
ole in his s ep, because he eal obo eloci y does no
in luence on a geome ic p ojec ion. Consequen ly he
p ojec ion in oduces he ollowing coo dina e ans o ma ion:
Poin s
{eq}
ha obey
&,(eq,
)=O
de ine a su ace ( wo-
dimensional
in
ou
case) whe e he obo
is
placed. Hence
obo pos u e is now gi en by only wo e o coo dina es
ep=(el, e2)
ins ead o he h ee
eq=(ex, e e@)
on a
TT.
A
classical example o p ojec ing unc ion is he no mal
p ojec ion desc ibed in
[3]
and
[9],
equi alen o making
e,
null. Tha is, he i s e o coo dina e
e,
is elimina ed and he
obo pos u e is exp essed by only wo:
ep=(ep
e@)
(e o
coo dina es a e called
@,
@
in hese e e ences). The wo-
dimensional su ace is he
e,,
axis ex ended o all he possible
obo o ien a ions. This simple me hod o elimina ing one o
he e o coo dina es can no always be used, as we show in
he example o he nex sec ion.
I is impo an o ema k ha pa ame e
is a his ime he
hi d s a e coo dina e4, and we should men ion i o speci y he
/o,
91,
929
931
/qdes( ),
,
eh e2/
;
ep=(el,
‘
In
‘IT
gi es
us
no
s a e
because
is
de e mined only
by
ime
h ough
he
unc ion
= ( ).
whole obo pos u e, now gi en by
( , el, e2).
Bu
is no an
e o coo dina e,
ha is, i does no play any ole in he
con ol o in he s abiliza ion p oblem, ha is cen e ed only in
making e,( )-+O, ega dless o
.
A his s ep we can say ha
we ha e isola ed pa ame e
om he pa h con e gence
p oblem, and ha we ha e a sys em wi h wo deg ees o
eedom and wo s a e a iables ( o ge ing
),
whe e smoo h
s abiliza ion is possible. This asse ion does no con adic he
abo e men ioned impossibili y o eedback s abiliza ion o a
ixed pos u e in nonholonomic sys ems, because we only
s abilize wo coo dina es wi h his
PF
con e gence, neglec ing
he hi d coo dina e
.
In o he wo ds, we can s abilize e,,( )+O
bu no e,( )+O.
The addi ion o a p ojec ing unc ion ,,i(e,
)=O
gi es us he
way in which pa ame e
a ies. Di e en ia ion o his
unc ion lead
us
o5:
Now s a e equa ion
(3)
can be subs i u ed, and using he
“chain-law”
=
‘
,
we sol e o
i
and ha e inally:
i
L’
‘
p’
B(e,
, )u
d
e-
Now we ha e sol ed he p oblem o inding a closed
exp ession o he a ia ion o pa ame e
in an elegan way.
This is clea ly a
PF,
because a ia ion o
does no depend
implici ly on ime. In he equa ion
(4)
he e may be some
ope a ion es ic ions; o example i denomina o is null,
a ia ion o
is unde ined. This case mus be analyzed o
each applica ion and we will s udy i in dep h o he example
o sec ion IV.
The op imum p ojec ion depends on he mobile obo s uc u e
and e en on he applica ion, bu we can s a e some gene al
condi ions o a “good” p ojec ing unc ion o be cohe en :
1.
A p ojec ing poin can always be ound o any sys em’s
s a e
q
and o any alid e e ence pa h, ha is:
V(X,Y,Q))
E
@x(-z
Uniqueness o he goin on he pa h
q&( )
mus be
ensu ed (a leas locally
).
I he ac ual obo s a e is he same as a pos u e o he
pa h:
q( )=qdes( J,
hen he p ojec ed
mus be
,.
Tha is
p oy
has a ze o o eq=O: pmy(O,
)=O V .
I would be desi able ha he analy ic equa ions could
ha e a closed o m, o help he inding o a con ol law
whose s abili y is analy ically demons able.
Second s eD: “mo ion exieenc ”.
Finally, as we desc ibed in
PF
cha ac e is ic
4,
we need o imposed a “mo ion exigency”
mo eX,g(u)=O
o gua an ee ha he obo mo es. Al hough he
o m o his unc ion depends on he applica ion, we mus
ul ill he ollowing condi ions:
3
,~ 31/ p (q-qdes( J, J
=O
2.
3.
4.
1.
2.
No
solu ion a he o igin
u=O
so
U
is ne e null;
I is desi able ha
mo eXig(u)=O
is an e en unc ion on i s
componen s7. This ensu es ha he obo will app oach he
pa h h ough he mos sui able inpu s (nega i e o posi i e),
mindless o he di ec ion i mus ake on he pa h (inc easing
o dec easing
).
This is pa icula ly impo an when e o s
a e big.
To
help he con ol law o con e ge o he pa h, i would
be desi able ha co n onen s o U beha e symme ically, ha
is he o al mo ion [U
I
should spli iden ically be ween
and
w.
In he cu en mobile obo li e a u e mos mo ion exigencies
(no called wi h his e m) a e applied o ca -like obo s,
so
i is
usual o ha e =c e, which is in ui i e o ca s. Fo obo s wi h
highe maneu e abili y o he s au ho s ha e p e e ed he
exigency gi en by
I
F,,,,
I
=de,
ha is
I
w
I I
I
=C ee,
o a oid
slippage.
As
he las one has an inde e mina ion o null
w
o
,
and
ou
(2,O)
obo does no ha e mo ion es ic ions, we will
use he adequa e “mo ion exigency“ gi en by:
3.
(5)
i=l
whe e
Km
is he whole mo ion applied o he sys em and
bi
is
he scale ac o o each inpu .
4
An Example Applica ion: The Case
O
A
(2,O)
Mobile Robo .
One o he mos ex ended mobile obo con igu a ions is ha
wi h deg ee o maneu e abili y 2 and
0
s ee ing wheels (a
(2,0)- obo acco ding o he de ini ion o
[2]).
The ypical
opology o hese (2,O)- obo include wo d i e mo o s a each
ea wheel, ha can u n independen ly o wa d o backwa d.
Fu he mo e i
is
one o he obo s in which ajec o ies a e
mo e complica ed, as i can no ha e comple e maneu e abili y
( ha is, i is no omnidi ec ional) bu i can make ze o- adius
u ns.
So
i is a e y in e es ing p oblem o apply ou pa h
ollowing cons uc ion o hese obo s. As ou g oup has been
in e es ed du ing he las yea s in he imp o emen o elec ical
wheelchai s
([4][5][7])
ha inco po a e his con igu a ion, we
ha e s udied he complica ions ha his opology in oduces.
Fi s s eD: “woiec ing unc ion”.
Maneu e abili y in hese
obo s is e y high, and hey ha e no addi ional mo emen
es ic ions (excep o he inhe en nonholonomic cons ain ).
Thus we can choose he p ojec ing poin as ha in he pa h ha
is nea es o he obo , i.e. he one which dis ance is minimal.
As he h ee e o coo dina es mus play a ole in he dis ance,
a “good” elec ion o he dis ance
dq
can be:
i=l
whe e
Ki
a e he scale ac o s be ween he di e en e o s o
gua an ee dimensional homogenei y. In he case o ou obo
his leads o:
’
He e de i a ion espec o a ec o holds o a summa ion.
since i
can
each he same pos u e as
many
imes as i wishes.
Global uniqueness is impossible
o
a mobile obo ha is ully con ollable,
’
We
a e
assuming he e ha he obo mo o s ha e no a special s uc u e
o
he applica ion does no need o ma ch in
a
unique di ec ion.
A
ca would be
o example
his
case, because
i s
ea di ec ion is limi ed.
10
d:( , )
=
e: +e;
+
Kie
To choose he minimal dis ance poin we “ eeze” ac ual obo
pos u e ( ha is
u=O)
and “mo e” along he desi ed pa h ( ha is
we a y
)
looking o he poin wi h a local minimum:
lu=O
whe e we omi simple calcula ions (using
(2))
and use he
“chain-law”
j-
=
’i
. He e
(‘)
holds o di e en ia ion espec
o
and
(‘)
espec o 1.
So.
(7)
is ou
,,(e,( ), )=O,
whe e
des( ),
wdes( )
a e he inpu con ol p o iles o desi ed pa h.
This p ojec ion is ully in ui i e, because i always chooses he
nea es poin o he ac ual obo pos u e.
As we desc ibed in sec ion 111, di e en ia ion o his p ojec ing
unc ion gi es us he new equa ion o he a ia ion o
pa ame e
:
=
w&s( >cose+i
+
Kiwqgs( )
(8)
igs( )
-w&j< ) dgs< )ey
+
Ki&s( )
-
Kie#wies( )
-
ex ies( >
As his p ojec ion has been ob ained h ough a gene ic PF
cons uc ion, we can ge some o he classical p ojec ions as
pa icula cases. Fo example he no mal p ojec ion used in
[3][9],
is ound by doing
K,=O
(you should emind ha no mal
p ojec ion o a plane cu e coincides wi h minimal Euclidean
dis ance, gi en by he dis ance
dq
when
K,=O
[lo]). In e ec
he nulli y o
K,
and he use o he same pa ame e o hese
au ho s (cha ac e ized by des( )=I), leads o:
Once we ha e chosen his p ojec ion we should con i m i i
e i ies he condi ions o he abo e sec ion o be conside ed a
“good”
p ojec ing unc ion. All o hese condi ions excep
numbe
2
a e s aigh o wa dly sa is ied. Uniqueness is
equi alen in ou case o he non-nulli y o denomina o o
equa ion (4) [6]. A deep s udy is made in [6] and i s inal esul
is ha local uniqueness is eached unde ce ain non-se e e
condi ion?. These condi ions a e wo bounds o
K,
and
cu a u e de i a i e o he desi ed pa h ~;~~( ), ha can be
easily sa is ied i e o s a e no unbea able and desi ed pa hs
a e no ab up . In opposi ion o hese non-se e e condi ions,
no mal p ojec ion mus a oid pa hs con aining ci cles wi h
small adius o ensu e ha i exis s and is unique, ha is, i is
a mo e es ic i e o he easible e e ence pa hs.
Second s eD: “mo ion exieenc ”.
In he case o he
(2,
0)-
obo , we ha e only wo deg ees o eedom;
so
a e y sui able
mo ion exigency is he one men ioned be o e:
(9)
whe e K,,,,, is he whole eloci y applied o he sys em and b,
is he scale o he angula speed, ha gi e us how much he
sys em can u n.
In conclusion we ha e educed he sys em jus o wo s a e
a iables (no explici ly de ined bu ob ained h ough he
applica ion o a “p ojec ing” cons ain gi en by equa ion
(7)
2( )
+
bid( )
=
K:,,
>
0
o he h ee e o s
e,,),
and one deg ee o eedom ( esul ing
om he use o
(9)
o ec o inpu
U).
In hese a iables we
ha e condensed wha we need o con e ge o a gene ic pa h
h ough a pa h ollowing.
Con ol law.
Al hough con ol law selec ion is a om ou
objec i es, i is con enien o show he beha io o ou sys em
and he way o ge o an asymp o ically s able con ol law. We
will use Lyapuno ’s second me hod me hod wi h he quad a ic
e o unc ion as he Lyapuno unc ion (which ma ches wi h
he semidis ance):
(10)
Di e en ia ing wi h espec o ime and using s a e equa ions
(2)
we ha e:
V
=
(ex cos(e,)
+
ey sen(e,))+ Kie+,w
Now
we impose (as ou con ol law) his de i a i e
V
o be
nega i e semi-de ini e9 o ensu e con e gence, ha ing:
(1 1)
=
(ex cos(e,)
+
ey
sen(e+,)).I
+
(K,e,
)Kp
=
=
-k;(ex cos(e,)
+
ey sen(e,)p
+
:Kiei]
Equa ion (1
1)
ep esen s a line in he plane o he no malized
con ol inpu s ( ,
Kp).
I we choose, o con enience, b,=K,
in he mo ion exigency
(9),
his equa ion will ep esen a
ci cle. The in e sec ion o ci cle
(9)
and line (11) will gi e us
he eques ed alues o
and
w
(i i does no exis , ci cle’s
nea es poin o he line is chosen). Asymp o ic con e gence o
he p oposed law is demons a ed in [6] (i can be ob ained, as
usual o hese kind o laws in mobile obo s, h ough
Ba bala ’s lemma
[9]).
The in e sec ion
o
ci cle and line ( ha
is, he con ol law) educes o a i s o de di e en ial equa ion
when e,+O o ex4. In hese cases, he pa ame e s
Z,,
Z,
play
he ole o ime cons an .
Pa h ollowine e alua ion.
E en when asymp o ic
con e gence is ensu ed, simula ion is always a good way o
e i y and obse e he con ol beha io . The p oposed
e e ence pa hs whe e a smoo h PF con ol law mus be
e alua ed ha e o be alid (see sec ion
2).
Hence a ca -like
obo o ou
(2,O)
obo can no
go
h ough a piecewise pa h
including cu a u e discon inui ies (e.g.
a
s aigh line plus
a
ci cle); ne e heless he pieces should be linked by he pa h
planning o ensu e cu a u e con inui y a leas ( o example
h ough he addi ion o clo hoids o simila cu es). As we
ha e shown analy ically he gene ali y o ou p ojec ion
unc ion, hen e e y pa h complying wi h he cu a u e
con inui y is iden ical o e alua ing ou PF. We ha e selec ed
wo e y in e es ing examples:
1)
app oaching o a s aigh
line, whe e ypical con e gence is showed,
2)
con e ging o a
ze o- adius u n, whe e ou PF shows i s gene ali y. The alues
o cons an pa ame e s ha e been chosen o ensu e a smoo h
con e gence, as
5=0.5s,
2,=0.5s.
The whole mo ion is
K,,,,,=5Ocm/s,
K,,,,,=0.5
ds.
The cons an s
bo=K,
a e 0.25m.
1
1
2
2
.
V
=
-
di
( ,
)
=
-(ex2
+
e:
+
Kie:)
No e
ha Vcan ne e
be
nega i e de ini e because
he
exis ence
o
he
*
Mainly
he non-se e e condi ion
is
due
o
nonholomic na u e
o
he sys em. nonholonomic cons ain ,
see
[Diazgla]
o
a
demons a ion.
11

In he i s case we ha e selec ed big ini ial e o s o p o e he
good con e gence o he me hod in p esence o ex eme
condi ions (see
ig.
4)
and o compa e i wi h TT. In
PF
he
e o
e,
mus always be ze o acco ding o he p ojec ing
unc ion
(7)
and he solid line o he e o
ey
gi es us he eal
obo ajec o y. We ha e used o TT he con ol law o [8]
uning i s cons an s
so
i beha es in a simila ashion o s nall
e o s
(KX=20s-’; Ky=6.Ocm-‘; KB=3.2cm-’)
(i s eal ajec o y
is he dashed line). No e ha in PF he obo begins he
acking in e e se di ec ion in he i s ansien , in o de o
educe he e o s as e , and pa ame e
dec eases oo. In he
end his will imply ha PF me hod will each less dis ance in
he desi ed pa h. This case will ne e happen in a pu e TT,
whe e
= ,
and he e e ence obo s “pulls” he eal one and i
will ad ance he same as he e e ence ajec o y. This
is
a
well-known ad an age o
PF
equen ly commen ed in he
li e a u e [11][3], ha educes oscilla ions in he end.
Mo eo e inpu commands in
PF
a e limi ed by ou mo ion
exigency while in TT hey
a e
no . I hey we e in TT, i s
esponse would be e en poo e .
I
Ini ial
....
ig.
4:
PF
o
a line
(X
axis) unde big ini ial
e o s.
-*.....
-.
”.....
e-.
* qc ual
1.5
W ,(TT)
-0.1
--0.1
ig.
5:
PF
o a ze o- adius u n(e,(O)
=-0.03m,
e,(O)
=-0.
Im).
In he second example he desi ed ajec o y is he ‘ e ical
axis, because e e ence obo u ns a ound i s poin
Po
(see ig.
5).
The eal obo mo emen would be gi en by he p ojec ion
on plane
AY.
As
in he p e ious case, al hough ini ial e o s
we e big, PF chooses he nea es poin on he desi ed
ajec o y, i.e. ha wi h ze o
e)
(acco ding o p ojec ing
unc ion
(7)).
No e ha in
PF
inpu con ols
,
w
a e spli in he
mos con enien o m o ge a as con e gence. In TT
con e gence is slowe because cons an s we e uned o he
acking o a line. I cons an s we e uned o his las case hen
con e gence o
a
line would be slowe .
5
Conclusions.
We p esen a echnique o cons uc pa h ollowing in mobile
obo s ( ha has been shown by se e al s udies o be mo e
ad an ageous han ajec o y acking). I consis s o wo s eps:
choosing a “p ojec ing unc ion” o ela e ac ual pos u e o
desi ed pa h as a unc ion o e o s and in insic desc ip o
pa ame e , and imposing a “mo ion exigency” o ensu e obo
ad ances on he pa h. These s eps ha e been ob ained based on
he gene al pa h ollowing cha ac e is ics ha we ha e
p e iously ex ac ed. We also s a ed he condi ions ha bo h
s eps mus sa is y o be cohe en . The ea e we co obo a e
ou p oposi ion wi h a (2,0)- obo (acco ding o he de ini ion
o [2]) ha can make ze o- adius u ns, a case ha has ne e
been sol ed, as a as we know, using pa h ollowing. Finally
we p esen simula ion esul s unde big ini ial e o s o exhibi
he good and as con e gence o
ou
pa h ollowing app oach.
Acknowledgemen s
This wo k has been suppo ed by IASS ag eemen “Segundo
Con enio de Colab. IASS-Uni . de Se illa” and by CICYT
p ojec TER96-2056-C02-01. Au ho s will also hank P o . C.
Samson (INRIA, F ance) o his kind mailing o esea ch
epo s and he e iewe s o hei wo hwhile commen s.
Re e ences.
R.W. B ocke . “Asymp o ic S abili y and Feedback S abiliza ion”. in
Di e en ial Geom. Con . Theo y, Bi khause , pp. 181-208,1983.
G.
Campion,
G.
Bas in, d’ And ea-No el. “S uc u al P opie ies and
Classi ica ion o Kinema ic and Dynamic Models o Wheeled Mobile
Robo s”. In
IEEE
T ans. Rob.
And
Au om.,
V.
12, No 1, Feb. 1996.
C. Canudas de Wi ,
H.
Khennou , C.
Samson
and O.J. So dalen.
“Nonlinea Con ol design o Mobile Robo s”. In “Recen T ends in
Mobile Robo s”. Edi ed by Yuan
F.
Zheng. Ed. Wo ld Scien i ic Se ies
in Robo ics and Au oma ed Sys ems. 1993.
A. Ci i -Balcells, F. Diaz del Rio, J. Se illano,
G.
Jimknez
“SIRIUS:
A
Low
Cos High Pe o mance Compu e ized Wheelchai ”.
P oc.
o he
In . Wo kshop
on
Medical Robo s, pp. 23-30. Vienna. Oc obe 1996.
A. Ci i -Balcells, F. Diaz del
Rio,
G.
Jimhez, J.L. Se illano. “A
P oposal Fo A
Low
Cos Ad anced Wheelchai A chi ec u e”. The 4 h
Eu opean Con e ence o he Ad anc. Tech. AAATE Con e ence 1997.
ISBN 4274901831C3047 (Ohmsha). ISBN 9051993617
(10s
P ess). Oc
1997. Thessaloniki. G eece.
F. Diaz del Rio. “Analysis and E alua ion o Mobile Robo Con ol
:
Applica ion o Elec ic Wheelchai s”(In Spanish).
Ph.
D. Thesis.
Uni e si y o Se ille, (Spain), 1997.
F. Diaz del Rio, A. Ci i ,
G.
Jimhez,
J.L.
Se illano. “Pa h T acking
In The
SIRIUS
Wheelchai ”. Same as [Ci i 97].
Y.
Kanayama e al. “A S able T acking Con ol Me hod o an
Au onomous Mobile Robo ”.
Roc.
1990 IEEE In . Con . Robo ics and
Au om, Cinccina i,
Ohio,
pp. 384-389.
A. Micaelli, C. Samson. “T ajec o y T acking Fo Unicycle-Type And
Two-S ee ing-Wheels Mobile Robo s”. Rappo de Reche che N. 2097.
Ins i u Na ional de Reche che en In o ma iaue e en Au oma iaue. 1993.
[lo]
[l 11
Richa d
S.
Millman. “Elemen s o Di e en ial Geome y”. Edi .
P en ice Hall, Englewood Cli s, N.J. 1977.
N. Sa ka ,
X.
Yun,
V. Kuma . “Con ol o Mechanical Sys ems wi h
Rolling Cons ain s: Applica ion o Dynamic Con ol o Mobile Robo s”.
The In .
J.
Rob. Resea ch. V13.No.l Feb1994.
C. E. Tho pe. “Vision and Na iga ion
:
’be Camegie Mellon
Na lab.
Ed.
by Tho pe. Kluwe Academ. Publ., 1990.
[12]
12