A
PARAMEIRIC FORMULATION
OF
‘I’KACKINC
ME‘IHODS:
APPLICATION
TO
CHAlNED
SYSTEMS
F.
DiAZ DEL
Rio,
G.
JIMkNE7-
J,
I,,
SEVILLANO, A.
CIVI‘T
BAI,CELI,S,
Escucla Tbcnica Supc io dc lngcnic ia In onnl ica. Uni c sidad dc Sc illa.
AV.
Kcin,
Mc ccdcs sin.
4
ID1
2
Sc illa. SPAIN,
L diaz.
pj!,
sei i.
[email p o ec ed]/.~.c.
ABS’IKAC’I
Whcn ollowing a pa h, hc c a c sc c al possibili ics a cnding
o
hc
way
in which hc ac ual
obo s a c can bc
cla cd
wi h
hc
wliolc pa h.
In
his wo k wc o mula c di c cn acking
mc hods
on
hc basc ha
a
mcmo izcd pa h can bc dcsc ibcd by
a
singlc dcsc ip o pa amc c ( hc
objcc i c poin is cadily gi cn by
his
pa amc c ). Wc classi y pa h acking acco ding
o
hc way
in which wc
impose
o
design
hc p og css o dcsc ip o pa amc c . Bcnc i s and disad an agcs
o
cach inc hod a c idcn i icd. Wc also sumnia izc how o cons uc
cach
mc hod. Dcspi c ha
his
classi ica ion
is
gcnc ic, applica ion
o
mobilc obo s and chaincd sys cms
is
c y aluablc duc
o
hc impo ancc o acking p oblc n in licm and bccausc hci dcsi cd pa i
is
usun ly mc no izcd.
KEY
WORDS:
pa amc lc cqualions, di c cn jal
gcomc y,
mobilc obo s, nonholonomic
cons ain s, pa h ollowing, ajcc o y acking, chaincd sys ems.
1.
INTROUUCTlON
i
is
wcll know ha hc con c gcncc o
a
non-omnidi cc ional mobi c obo o
a
ixcd pos u c
( hc s abiliza ion p oblcm), can no bc achic cd h ough
a
smoo h ccdback s abiliza ion con ol
law
duc
o
B ockc ’s hco em
[I].
On
hc
o hc hand,
in
mobilc obo s i
is
usual
ha1
hc
pa h
o
ajcc o y is mcmo izcd, and hc poin s abiliza ion p oblcm
is
c y di lic cn
o
hc pa h’s
con c gencc p oblcm.
Fo
mos
niobilc obo s imc dc c minism is no
a
cqui cmcn o hc
acking,
and
hc con ol objcc i c can bc hough
o
as
ollowing a pa h wi h a a c ha can bc
a iablc.
Thc c
is
no
doub
ha hc nunibc o mobilc obo s applica ions will bc la gc in hc ncx cw
yca s, cspccially in iclds such
as
in clligcnl anspo a ion
sys cms
(ITS),
cxplo c chiclcs, and
pc sonal
o
assis an obo s. Canscqucn ly, in hc las dccadc hc c has bccn a g ca in e cs
in
inding
con olle s
and acking mc hodologics
To
hcsc obo s.
As
a csul o hc wo k wc ha c
donc
du ing
hc
las
yca s
in
his icld,
wc
p cscn hc c
a
acking classi ica ion bascd on
a
pa amc ic o mula ion. Wc mus undc linc ha wc do no considc ncw con ol cchniqucs
in
his
wo k, nc c hclcss
wc
conccn a c
on
a p io subjcc : which mc hods
o
usc
whcn ollowing
a
pa h. In ac , con ol
law
applica ion can
bc
considc cd as
a
sccond s cp
in
hc acking p oblcm
dcsign. Mo co c , al hough his o mula ion can be applicd Eo c c y sys cm, wc cniphasizc same
aspcc s
e y
aluablc o chaincd sys cms.
In hc ncx scc ions wc will in oducc and analyzc hc ncw acking classi ica ion. In scc ion
2
and
3
wc o mula c hc di e cn acking mc hods. In scc ion
4
wc cxplain hc acking
classi ica ion bascd on
an
cxamp c and ina ly wc cxposc hc conclusions.
2.
PARAMETRIC FORMULATlON
A
memo ized,
e c cncc
o
dcsi cd pa h
(o
mc cly
pn h)
can bc dcsc ibcd by
a
singlc
dcsc ip o pa amc c
[13],
namcly
P,
and i can bc cxp csscd
as
a cc o
o
s a c coo dina cs
qacl( ).
As
a
csul , hc acking p ogcss can bc idcn i icd wi h hc p og css
o
.
Al hough hc
pa amc c
may
bc
imc, in cascs whc c imc dcpcndcncc is nob clc an , many o hc s a c possiblc.
Fo
cxamplc,
in
di c cn ial gcomchy hc na u al a c pa amc c
1131,
which
makcs
he linca
spced cqual o
I,
is
gcnc ally p c c cd. Fu hc mo c, whcn s udying con c gcncc o
a
pa h,
i
is
usual
o supposc ha
dcsi cd
ajcc o y has no cnd.
37
I
Whcn s udying hc acking p oblcm, wc can
hink
o
as a ncw s a c coo dina c, which mus
bc addcd
o
hc
o he
n
coo dina cs ha dc inc hc s a e
q( l
o hc cal sys cm. Thc c o c wc
inc casc hc nunibc
o
coo dinalcs in
1
i
wc
wan
lo
cxp css hc sia c
o
he
acking:
{ .
9).
In
addi ion, i
is
usual
lo
cxp css sys cm s alc h ough
c o
coo dina cs
e&),
which a c
somc
kind
o
ela ion bc wccn cal
nnd
dcsi cd s a cs. E u s will easily cxp css how a hc sys em
is
om
i s
con c gc icc objcc i c.
Thc c o c, wc can classi y hc acking
o
hc pa h acco ding
o
Ihe
way
in
which wc imposc
o
design (whcn possiblc) pa amc c
.
F om his poin o icw, wc can considc
a
ncw
sys cm
inpu :
ihc onc ha
go cms
pa amc c ’s
e olu ion,
which wc will call
a
Wc will
SCC
ha o
some
acking mc hods,
U
is
ob ious, bu o
somc
o he s, osclcc ion can in oducc in c cs ing ca u cs
on
hc acking.
To
sum up, wc
can
cxp css acking
o
whec cd mobilc
obo s
as:
wlic c
U
is he cc u
o
inpu a iables, which has a liiica da ion o chained sys cms.
Oncc hc acking mc hod has bccn sclcc cd, a sccond s cp is inding
a
con c gcncc.
Dcpcnding on hc sclcc cd acking mc hod, con ol
law
ex ac ion may bc di e en .
2.
TRACKING
METHODS’
CLASSIFICATION
Many csca chc s
ha c
s udicd a ious acking me hods whcn he dcsi cd pa h
is
mcmo izcd
o
p c iously gcnc a cd.
In
lic
icld
o
mobilc obo s
wo
main acking mc hods ha c bccn
s udicd.
In
a
i s
g oup
we
ind hose ha considc
hc
cxplici ly
in
hc acking
[2,3,4j
(usually
callcd
“/ ajecioq,
ading”
(TT)),
and
y
o
app oach he obo o a mo ing objcc i c poin .
A
usual
case
o
TT
is
acking
in
sc osys cms,
whc c
wc
ack
a
mobilc
sys cm
o
a ge
a
hc
imc
i
IIIOYC~.
The
dcsi cd
cuo dina cs
a c simply
qba(l),
bccause
= ,
and c o coo dina es may bc
dc incd
as
eq( )q(i)-qded ).
In
sc osys cms,
ime
is
c i ical
and
his
is
hc only possibili y
wc
ha c.
Nonc hclcss, when wc
I y
o
ack
a
mcmo izcd pa h, hc acking mc hodology can
bc
dcsigncd opcnly,
as
wc
know
a p io i he wholc ajec o y. O cou sc
classical
sc osys cm
acking can
bc
applicd
jus
by idcn i ying
hc
pa amc c associa cd
o
hc
pa h
wi h imc,
ha
is
( )= .
Fu hc mo c,
IT
can
bc
cx cndcd
in
a
mo c gcnc al assump ion han sc osys cms: Ic he
pa amc c
I’
bc
jhc ion
uj’ imc
= (l).
An
asynip olically s ablc con ol
law
may gua an cc ha
hc sys cm will con c gc
o
a poin
qdes( )
in
hc dcsi cd pa h in a dc c minis ic imc, cxccp o
hc inhe en pc u ba ions ha
il
may su e . Acco ding o
Eq.
(I),
U
will bc simply
I
o
se osys cms, and a unc ion o limc To
a
mo c gcnc al
TT:
a=u( ).
In
a
sccond
g oup,
wc
ind
lhosc
me hods ha
do
no considc iming cqui cmcn s and
y
o
con c gc
o
a pa h
[4,5,6,7,8],
which a c usually callcd
“p l h olk~wing”(PF)
o
mobilc obo s.
WE can
also
ind
sc c al cxccllcn compcndia
o
bo h
mc hods
in
[2,9].
PF
Is
based on
some
da ion
bc wccn ac ual sys cm’s s alc
q(l)
and hc wholc mcmo izcd pa h. This
cla ion
o
p ojeclic n
will gi c
us
hc dcsi cd
poin
qdJ ),
i.c. hc dcsc ip o pa amc c
Y
as
a unc ion
o
hc
ac ual posi ion and
pa h:
Nq,
)=O,
whc c
IT
is
somc
kind
o
p ojcc ion
o
hc ac ual posi ion
o
lhc pa h. Thcn hc eal sys cm
mus
y
o
ollow his poin
qdn( )
ins cad
o
hc
one
gi en by he
o hc app oach.
Fo
cxamplc,
he
dcsi cd poin is usually sc cc cd o bc hc ”closes poin
on
hc
pa h”
o
he
ac ual obo ’s posi ion
[13].
Thc
c m
“closes ”
mcans
hc pa h’s poin ha
makcs
ce ain dis ancc c ilc ion minimum.
Tlic
c o coo dina es may csul
also
as
eq(pq(+qdm( ).
O
cou sc using
his
app oach, i
is
no
gua an ccd
ha hc sys cm will each a poin
o
he dcsi cd
ajcc o y in
a
dc c minis ic imc. Bu hc main p oblcm wi h
PF
is
ha p ojec ion uniqucncss has
no
bccn gua an ied yc , hcncc
no
PF
ha c bccn dcsigncd ha can be applicable
o
all
possiblc
pa hs
( o
hc au ho s’ knowlcdgc). Wc mus no c ha
PF
p ojcc ion
can
bc considc cd
as
an
imposed holonomic cons ain on hc whole
s a e
{q,
},
and his mcans lha hc sys cm will
immcdia cly oosc
onc
o
i s
s a e
coo dina cs. Inpu
D
can bc ob aincd by di l’c cn ia ing hc
p ojcc ion wi h
cspcc
o
imc, subs i u ing s a c cqua ion
(I),
and hen sol ing o
IT=
i.
:
372
Thc c o c
PF
in oduces a dcpendencc likc
U
yu(q,
U)
=
/(q)
U.
No e ha i denomina o is
null in
Eq.
(2),
a ia ion o
is undc incd.
We
ha c shown in
141
ha his caw is cqi i alcn o hc
non-uniqucncss
o
hc choscn p ojcc ion.
In
o hc wo ds,
PF
is
no app icablc o a11 kind
o
pa hs.
Thc key ad an age
o
PF
is
ha i
is
mo c
sui ablc o
many
si ua ions
in
which imc is no
a
c i ical pa amc c
[5,6,10].
This
can bc unde s ood
iY
wc considc he ollowing cxamplc: i
pc u ba ions o cc hc mobile obo o bc a
cs ,
dcsi cd poin o
TT
will mo c una oidably,
This
mcans ha c o s will g ow
up
o
somc aluc ha
may
in oducc ins abili y.
On
hc o hc
hand, i
PF
wc c uscd, dcsi cd
poin
will
be
he samc
in
spi c
o
hcsc pc u ba ions, bccausc
pa h’s
shapc
and cal obo sh c cmain
hc
samc
(c is
linca
o
inpu s
U
To
chaincd sys ems).
A
di c cn acking mc hod was p oposcd in
a
p c ious papc o ou s
[I
I].
Wc namcd
i
e o
adop i e
ocking
(EAT),
bccausc acking adap s o sys cm c o s. Wc showcd ha
i
c ains hc
cxposcd ad an agcs
o
PF
and
i can
bc
applicd o
all
sc s o pa hs.
I s
dcsign is simila o ha o
TT
bu
i
is in cndcd o pa h cco c ing wi hou s ic iming cqui cmcn s. No e ha o mos
mobi c
obo s
(including ad anced whcclchai s
[
121)
imc dc cnninism
is
no
a
cqui cmcn o he
acking. Whcn imc
is
no c i ical,
wc
can “dcsign” he a ia ion
o
I’,
and
wc
can cga d obo
s a c, i.c. aking
c o s
in o accoun . Thus, con as ing hc
IT‘S
igid a ia ion o
,
ha is
o=o(/,),
wc
p oposcd
o=g(e,J,
whc c
g(eJ
is
a
“con cnicn ” unc ion
o
hc
c o s.
Hc c “con cnicn ” is
c c cd
o
hc
dcsignc objcc i cs, bu also
i
should mcan ha acking
is
done co ec ly, and
he
p c iously cxplaincd ad an agcs o
PF
a e p csc cd. Tha is, unc ion
g(eJ
should ul ill:
I
c o s
a c
small,
g{eJ
should
cnd
o
I
(o
nio c cxac ly o
lul/1udea( )l
i
wc
wan
o
mo c
along hc pa h a ano hc pacc),
so
he acking escmb cs
IT.
c o s a c la gc, hc c c cncc
obo
should
“wai
o ”
hc ac ual obo . Tha
is,
g(e.J
should
bc small un il a
good
can c gcncc
IS
cachcd
(D
should cnd o ze o). O cou sc in his
si ua ion,
no
dc c minis ic ollowing
is
cxpcc cd.
O cou sc many passible unc ions
g(eJ
can
bc
dcsigncd a cnding
o
hc
cha ac e is ics
and
pu poses
o
cach sys cm, bu
in
[I
I]
wc showcd ha
a
c y sui ablc unc ion is
U
=
g(eJ
=
ap(-
K#i)
;
whc c
K,>O
is a scalc ac o . Wi h hcsc condi ions hc acking canno bc dc c minis ic
and cminds ha
o
PF.
Mo co c , a sccond kind
o
EAT
can bc dcsigncd.
I
wc nccd somc aspcc s
o
imc
dc c minism, a ia ion
o
can bc cx cndcd o include
he
“inaccu acy
in
hc dc c minis ic
acking”, ha
is,
hc di c cncc bc wccn hc dcsc ip o pa amc c
( ha indica cs hc a gc
o
ou con ol dcsign
qdl ( ))
and imc
1
( ha
indica cs hc a gc
o
hc iming cqui cmcn s
qde,(i)).
Hcncc
awou d
bc
g( .e,J:
a
combined unc ion
o
c o s
eg
and hc di c cncc bc wccn pa amc c
and imc
.
Oncc again he e a c many possiblc unc ions
g(eJ,
bu an in c cs ing s a egy
[I
I]
is
in cndcd o gc a dc c minis ic acking
a
long
las
( ha
is,
a
“ claxcd” dc c minis ic acking).
Pa amc c
will cmain “a
es ”
whcn he obo is a om hc pa h (in spi c
o
hc inc casc
o
di c cncc
- ).
Whcn obo “ ccupc a cs” and app oachcs hc pa h,
wc
conccn a c
on
cducing
hc di c cncc
-/.
Thc c o c
g( ,e,J
can no bc boundcd by
I,
bccausc
in
hc sccond casc
I-
mus
app oach
/.
Mo co c a hc o igin
(/- =O.
e,+),
g
mus bc
I,
Thus wc showcd ha a con cnicn
choice o his
mo e
complcx
EAT
can bc:
g( ,eJ-
exp(-K,eqz)
(I+K,p c/on(/- ;)),
;
whc c
KpO,
K,,W
a c scalc ac o s ha indica cs how as hc con c gcncc
o
o
is. Hc c
g( ,eJ
is uppc
boundcd by
I
+
K,~
.and lowc boundcd by
I
-
K,~
2 2
o
0.
Thc o mula ion ad an age
o
EAT
is ha
i
can ob iously bc applicd
o
all
sc s o
pa hs,
and
implies a clnlion
To
hc inpu
c
Iikc
U
=c (q)
o
c
=u( ,
9).
Wc
can scc his cla ion
as
a
(in
gcnc al) non-in cg ablc da ion bc wccn lic whole s a c gi cn by
{q,
}.
To
sum
up,
wc can ind in hc
cu cn
li c a u c he ollowing p oposals o
,
which supposc
a
di cc cla ion cons ain bc wccn
{q,
}:
373
1.
7T: = ( ).
*
u=a( ).
2.
PF:
n(q,
)=O
u=o(q,u).
In
IT,
wc
“loose”
uoo dina c
,
and
in
PF
WE
usually loosc onc
o
hc
e o
coo dina cs
e,,
[4],
Mo co c hc
wo
p oposals
o
EAT
a c non-in cg ablc ela ions bc wccn coo dina cs
{q,
):
3.
EAT:
a--i=o(q)
4.
EAT:
= :=a( ,q).
5.
Q=
a(q,
U).
7.
u=cT( ,
U).
s.
u=a(u).
I
wc cambinc all possiblc dcpcndcnccs o
q
wc no icc ha hc c
a e
ou
cases
IcA:
6.
a=a( ,
u,q).
Wc
mus
poin
ou ha inc hod
5
is no hc samc as
PF,
bccausc wc suppose ha cla ion
5
is
no in cg ablc
(In
PF
i
is clca ly in cg ablc, bccausc
dq, )=Ocxis s).
Mo co c
we
will
no conside hc c
hc
las
wo
acking mc hods
(7
and
8).
Thcy
do
no sccm
o
bc
in c cs ing o acking pu poscs, bccausc a ia ion
o
pa amc c
docs
no
dcpcnd
on
hc
s a c
q.
On
hc con a y, cases
5
and
6
a c
a
mo c in ui i c
and
c cc i c
cx cnsion
o
hcm, and
wc will p cscn hci cha ac c islics and applica ions in his pape
o
he i s ime. Nc c hclcss
hcsc mc hods
7
and
8
will be s udicd in
ulu c
wo k
in o dc o dc c minc i hcy
can
bc
wo hwhile o
somc
pa icula applica ions.
In
o dc o
ix
idcas and cxplain he
wo
new
acking mc hods, wc will analyzc
a
simplc
sys cm
in
hc ncx sec ion. Dcspi e ha
his
examplc
is
c y simplc,
ou
classi ica ion
is
gcnc ic
o
c c y sys cm;
so
ex ac ion
o
con ol laws
o
chained
sys cins
can
akc ad anlagc
o
using somc
o
hcsc me hods.
Somc
cxamplcs
o
hcsc
bcnc i s
hs c
bccn
cxainincd
in
somc
o -ou
p c ious wo ks
[4,1
I].
3.
A
SIMPLE
EXAMPLE
Lc
us
conside
a
simplc sys cm wi h wo s a c coo dina cs
x={xl,
x },
whose
goal
is
o
ollow
a
c c cncc
pa h
xd,,=/. l,jJ i,
.k-ldw( )}
made
by
a
i ual
obo . S a c cqua ions a c simply:
. ,
=U*
(3)
. ,
=
11,
;
x:.,&
=
l ,,&(i-)
Xi,,&.”
=
u2,,A,
)
whc c
(‘)
holds o dc i a i c wi h cspcc o
,
Applying chain’s
law:
u,,k3(Q=
i
i+,*>( ),
As
wc
a c
in c cslcd
in
hc acking wc dclinc
hc
syslcm’s
c o s
as
hc
dij’c ence:
.j=1,2.
And hcn wc ha c
.ii,,&,
= lj,8h( )
,/=1,2.
ed j
=.VI(!)
-. lcle$i (i)l
;
L:,
(I)=
1 d )
-?141des( 1
;
(4)
.?(I)
=
a( )
-
. ~~~( ( ))
:
L;:
I )
=
u?( )
- uj&J ):
WC ha c cxp csscd
explici ly
hc depcndcncc
o
hcsc a iables bccausc hc dcsign
In
a
scwosys ciii
J.={,
a=l
and
a
c y siniplc con e gcn
con ol
law can
be:
I
TT
is
choscn, dcpcndencc
= ( )
will
only add
a
scalc
on
hc
clcc ioii
o -,
supposing ha
i>O
H.
NOIC
ha
in
bo h cascs, hc condi ion
iW
Vi
implics
ha
hc c c cncc poin
{xI&( )).
x2dc,( (g)/
will una oidably
ad ance
in
spi c
o
eal
obo mo ion.
Bu
i
wc could apply
PF
mc hod,
hc
si ua ion
would
bc c y di c cn . Fi s we mus
look
o
he mos sui ablc p ojcc jon
@x,
).
in
o dc
o
sclcc hc
desc ip o
pa amc c
and
hcncc, hc
c c cncc poin
/. lk,( ),
x?,kJ /]
a
any ins an .
The
mos ob ious choicc is hc p ojcc ion ha
sclcc s hc closcs poin
on
lic pa h o hc obo ’s posi ion, i.c.
ha makes
C:_,.,!
minimum. Fo
cxamplc, i hc dcsi cd pa h is hc [in,
{xJkS( )=s ,
xjb,( )=O,
s>O/
hcn
hc closcs poin wi 1 bc
xd6=j.xI.
O},
and bc p ojcc ion:
F
x,/s.
Di c cn ia Ion
o
p ojcc ion gi c
us
hc a c o
( J
D
=uI/s.
Thcn hc acking will p og ess
only
i
inc cascs,
ix.
i
U,>#
(o
uj<O
i hc acking
is
o
mclhodology
mus
now
choose
bc wccn di k cn
o ms
o
( )
o
a.
~#)
=
i?,,,J k&e,
i
~,~J )-K,c,,
j=1.2,
whc c
K/
>O.
374
bc donc
in
c c sc o dc ). Thcn wc
mus
impose some condi ion
o
“mo ion cxigcncy”[4]
o
cnsu c ha ( cal and hcncc i ual) obo s ad ancc, and hcncc o gua an cc ha hc acking
is
being donc.
The
simples mo ion cxigcncy
is
o cou sc
u,=com on >O,
bu o hc mo c
sophis ica cd can bc p oposcd.
I
hc i ial mo ion cxigcncy is p c c cd (and cnongh
o
ou
con ol
pu poses)
hcn hc p c ious con ol law s ill succccds
(o
cou sc only o inpu
2):
i 2
( )
=
u2
d&)
-
Kj
e?
=
-
K2
el.
This is ob iously a pa icula casc.
I
a
gcnc ic ajec o y had
o
bc
ackcd, hcn
a
mo c
gcnc al mo ion cxigcncy
is
p c c ablc, o cxamplc
U/’+
u?’=c in.~ioni>O
[4].
This has hc
addi ional ad an agc
o
main aining inpu s wi hin cc ain alucs, a oiding an cxccssi c inc casc
o inpu s (which may in oduce ins abili y).
Howc c , p oblcms
a ise
whcn
wc
canno
gua an cc
p ojcc ion uniqucncss. Fo cxamplc, i hc
pa h wc c a ci clc and hc p c ious p ojec ion wc c o bc applicd, hcn p ojec ion uniqucncss
is
b okcn whcn hc obo app oachcs hc ci clc ccn c .
So
i
is no
ully
applicablc
o
all
kind
o
pa hs, and p ojcc ion uniqucncss
should
bc
ca c ully analyzed (SCC
[4]).
In
EAT
wc can
design
he
mos
app op ia c acking alc, i.e. an cqua ion o
uas
a unc ion o
hc
c o s.
As
cxplaincd bc o c
a
c y
simp c and in c cs ing possibili y would bc:
CT
=g(e,J=exp
(-K
(e 2+
e:))
;
K,>O
is hc scalc ac o .
This
p oposal
ics
ha c o s
do
no inc casc g ca ly. Thc c o c,
i
consc cs hc
PF
ad au agcs, whilc a oiding hc p ojcc ion di licul ics.
Bcsidcs, o his i ial sys cm hc
p c ious
TT
con ol
law
s il wo ks:
ui( )=up(-KJe,’+
e )) u,,,jJ )-K,e,
:
j=1.2,
K,
>O.
Thc main di c cncc
is
ha whcn c o s a c Ia gc cnough, hc con c gcncc cscmblcs ha
o
hc
s abiliza ion p oblcm ( i ual obo
is
a
cs ).
Going u hc ,
we
p oposc he c hc wo ncw acking inc hods
5
and
6.
Mc hod
5
has hc
samc
kind o cla ion
c
=
u(q,
U)
han
PF,
which cll
us
ha
i
will bcha c
in
a
simila ashion
o
PF.
Bu
now
we
a c ce o “design” his cla ion, a oiding hc p oblcms
o
hc p ojcc ion di licul ics.
Fo cxamplc
a
o m
o
c
ha cminds
us
o
PF
can bc:
u.u,
n=-
2
(1
+
K.WF
a c an( ( ,x))
”
dcs
whc c
Kn F>O
is
a
scale ac o ha indica es haw as he con c gcncc o
o
hc p ojcc ing
poin
x,&pb)
will bc,
and
Nx,
)=O
is a sui ablc p ojcc ion. Thc p ojcc ing poin
xde,( p+j
is hc c
hc poin o hc pa h ha ul ils
n x,
,-)=O.
In
gcnc al o poin s
xaeS( )
ha a c nca
o
xdes( pp),
hc
sign
and magni udc o
Ns,
)
gi c
us
an idca o how a is
xdes( )
om
i
(in
121
some cxamplcs
can bc consul ed). Thcn hc sccond ac o
o
c
will bc
I
when
PF
objcc i c has bccn achic cd
(
cquals
pF),
and
will bc lcss
o
biggc o hc wisc, ying o app oxima c
I’
o
pF~
Func ion
a da
has bccn choscn
o
simila casons o ha o EAT mc hod. Thc
i s
ac o
has
bccn choscn
simila
o
hc dcpcndcncc ha can bc ound in hc
c7
o
PF.
O
cou sc
hc con ol law has
o
dccidc hc cxac o m o
U,
bu we mus calizc ha i obo app oachcs
sd,( },
U
will bc pa allcl
o
uaS.
In his casc pa ame e
will mo c
a
hc pace ha con ol law dccidcs. Finally no c
also
ha whcn obo is
s oppcd
(inpu s
U
a c
null),
Y
docs no p og csa. Likc in
PF
wi h his mc hod
i
is
no
p cdic ablc whcn hc obo will cach
a
pa h’s poin in hc gcnc al casc, and
i
may bc
mo c
s ablc han
TT
whcn c o s a c big.
On
ic o hc hand, p ojcc ion uniqucncss is no a p oblcm: i
i
wc c sa is icd o a wholc pa h, hc only conscqocncc
is
ha hc sccond ac o o
U
is
1.
In
addi ion, ano lw o m o c ha cminds
us
o
EAT
may bc:
whc c
KA+O
is
a
scalc Fac o ha indica cs how as hc con c gcncc
o
o
objcc i c poin
xd s( )
is.
The
p c ious analysis is alid again (considc ing
EAT
ca u cs).
As
his mc hod can bc
considc cd
as
a
PF
wi h adap a ion o c o s,
wc
can namc
i
“Adup ii~
Pa h
Fdollowing”
(APF).
375

Fo hc sainc dcsi cd pa h dcsc ibcd o
PF
(a
s aigh linc) wc
can
usc Lc samc p ojec ion:
n(x,
)=
-
x,Ls
-U.
Thcn
wc
can usc bc samc con ol
law
o
p c ious
PF:
imposc somc "mo ion
CXigC~lCy"[4]
o
cnsii c ha eal and i ual obo s ad ance, and imposc hc con o
law
o
inpu
2.
Hc c again
a
ino c sophis i a cd mo ion cxigcncy likc
U,'+
u2'=conSs an >0
would ha c
hc
addilional ad an agc
o
main aining
inpu s
wi hin ce ain
alues.
Mo co c hc c may
bc
o hc
possibili ics
o
dcsigning con ol
laws
ha can bc considc cd
in
u u c
wo ks
Finally wc can dcsign a acking wi h hc cla ion:
a
=U
(1,
U,
q).
Following hc samc
mc hodology p cscn cd
o
hc
EAT
wi h
U=
g( ,eJ,
and o hc wo
APF
a cla ions,
a
pai
o
in c cs ing cascs a c gi cn
by:
Thc
p c ious considc slions shown o
EAT
and APF will apply now combining bo h ac o s.
CONCLUSIONS
Wc
ha c o inula cd di 'c cn acking mc hods
o
mcmo izcd pa hs acco ding
o
hc way in
which
we
hpme
01-
design
hc p og ess
o
dcsc ip o pa amc c . Whilc hc so-called pa h
ollowing and ajcc o y acking ha c bccn classical mc hods,
wc
ha c in oduccd o hc mc hods,
cach ha ing sc c al ad an agcs:
somc
may
consc c mos
o
hc ad an agcs
o
pa h ollowing,
whilc a oiding
i s
p oblcins, o hc s a c alid
o
all
possiblc ajcc o ics,
somc
o hc s
may
p csc c
dc cm nis ic (non-s ic ) acking
a
hc
samc
mc
ha acili a c
ubus ncss
(i s
beha io
undc
la gc
c o b
o
dclaycd csponsc
is
much
bc c han ha
o
a ajcc o y acking),
c c.
Thc
gcnc ic
o muh ion
p cscn cd
hc c
can
opcn
a lo
o
possibili ies
u
many
applica ions.
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