Intern
ati
o
n
a
l Jo
urn
a
l
o
f
R
o
botics
a
nd Au
tom
a
tion
(I
JR
A)
Vol
.
3
,
No
. 2,
J
une
20
1
4
, p
p
. 13
9~
15
0
I
S
SN
: 208
9-4
8
5
6
1
39
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJRA
Optimizing Hexapod Robot Reco
nfiguration using Hexa-Quad
Trans
f
ormation
Addie Ir
aw
a
n
, Yee
Yin
Ta
n
Robotics
& Un
manned Research (RUS) group,
Faculty
Electrical
&
Electronics Engineering,
Un
iversiti Malay
s
ia Pahang
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Ma
r 8, 2014
Rev
i
sed
May
3, 201
4
Accepted
May 19, 2014
This paper presents a leg reconfigurab
le techn
i
qu
e to optimize th
e hexapod
robot reconfigu
r
ation flex
iblity. A hexapod-to-quadruped (
H
exa-Quad)
transformation techniqu
e is proposed
to optimize hexapod legs on certain
situation
that n
e
ed som
e
legs to be
disabled
as a leg to do o
t
her tasks and
operations. Th
is proposed meth
od used the factor of cen
ter
of
bod
y
(CoB)
stability
in th
e support poly
gon
and its bod
y
shape.
The r
e
initialized leg’s
shoulder method is proposed to
ensure
the support poly
gon is b
a
lan
ced an
d
confirmed
th
e CoM
nearly
or at the cen
ter
.
This
method
is modeled
an
d
simulated in a
real-
time based
m
odel of hexapod robot with 4-DOF/leg
control
arch
it
ect
ure.
The
m
odel i
s
veri
fi
ed in
nu
m
e
rical
model and
presented
using separated
3D simulators.
Keyword:
C
e
nt
er of
M
a
ss
Sup
p
o
r
t
Po
lygo
n
Trave
r
se
-trot g
a
it
Tri
p
od
gai
t
Copyright ©
201
4 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Ad
die Ira
wan
,
Faculty Electrical & Elect
r
oni
cs En
gi
nee
r
i
n
g
,
Un
i
v
ersiti Malaysia Pah
a
n
g
,
2
660
0 Pek
a
n,
Pah
a
n
g
, Malaysia.
Em
a
il: ad
d
i
eirawan@u
m
p
.
ed
u.my
1.
INTRODUCTION
Mu
lti-leg
g
e
d
ro
bo
t
o
r
so
called
activ
e susp
en
sion
v
e
h
i
cle (ASV)
h
a
s sign
ifican
t
ad
v
a
n
t
ag
es if
com
p
are to the
wheel type
robot es
peci
al
l
y
on
faci
n
g
i
r
reg
u
l
a
r a
nd m
o
u
n
t
ai
nou
s t
e
rrai
n
.
The a
dva
nt
ag
es o
f
m
u
lt
i
-
l
e
gged
o
r
l
e
gge
d r
o
b
o
t
can be see
n
ob
vi
o
u
sl
y
on
i
n
sp
ired
life liv
ing
fo
rm
; leg
g
e
d creatu
r
es. Raib
ert in
his book has mentioned
that
only
ab
out
ha
lf of the eart
h
‘s landm
a
ss is
accessible to e
x
isting
wheel
ed a
nd
tracke
d
vehicle
s
,whe
reas a m
u
ch larger
fract
ion ca
n
be
reac
hed
by
a
n
i
m
al
s o
n
f
o
ot
[
1
]
.
In
m
u
l
t
i
-
l
e
gge
d r
o
b
o
t
research
and
d
e
v
e
l
o
p
m
en
t, sev
e
ral stud
ies h
a
v
e
b
een
do
n
e
to
ach
i
eve g
o
o
d
ad
ap
tab
ility, fu
n
c
tion
,
h
i
gh
flex
ib
ility an
d
ex
ten
s
i
b
ility with
ex
trem
e an
d
unk
nown
te
rrain
.
Th
e
p
r
ogress em
p
h
a
sized
in
all exp
ect
s and
hi
erarc
h
y
of
m
u
lt
i
-
l
e
gged s
y
st
em
such as sy
st
em
m
e
chani
s
m
,
st
ruct
ure de
si
g
n
/
c
o
n
fi
gu
rat
i
o
n
,
soft
war
e
devel
opm
ent
/
c
ont
rol
t
e
c
hni
q
u
e a
n
d
el
ect
r
oni
cs
u
n
i
t
de
s
i
gn.
I
n
c
o
nt
r
o
l
t
echni
que
l
e
vel
,
reco
n
f
i
g
u
r
at
i
on
t
echni
q
u
e i
s
o
n
e
of t
h
e i
m
port
a
nt
part
s i
n
l
e
gge
d
r
o
b
o
t
c
o
nt
r
o
l
,
w
h
i
c
h i
s
em
phasi
zed
o
n
re
co
very
act
i
o
n
[
2
]
an
d
m
u
lti-task
in
g
.
Th
erefo
r
e stab
ility
b
ecome a
main
p
o
i
n
t
in
th
is research
th
at in
vo
l
v
ing
cen
ter of
m
a
ss
(C
oM
)
o
f
t
h
e l
e
gge
d
r
o
b
o
t
a
n
d i
t
s
s
u
pp
o
r
t
p
o
l
y
go
n.
T
h
e l
a
rge
r
t
h
e
sup
port po
lygo
n
dev
e
l
ope
d by
t
h
e ro
b
o
t
s
th
e b
i
gg
er t
h
e
p
r
ob
ab
ility fo
r th
e robo
t to
re
m
a
in
u
p
r
i
g
h
t
with
ou
t ov
ert
u
rn
i
n
g
wh
en
it sto
p
s
walk
i
n
g
at an
y
m
o
men
t
d
u
ring
th
e
walk
i
n
g p
e
ri
o
d
, an
d
t
h
is is called
staticall
y
stab
le walk
i
n
g
or st
atic stab
ility.
Static
stab
ility o
ccu
rs w
h
en
C
o
M lies co
m
p
letely w
ith
in
th
e
su
pport po
lyg
o
n
an
d th
e po
lygo
n’s
area is greater
th
an
zero, and he
nc
e static stability requi
re
s at least th
ree po
in
ts o
f
g
r
o
und
c
ontact [
3
]. Ro
bo
t
’
s Co
M r
e
p
r
esen
ted
a
sig
n
i
fican
t
aid
in
m
a
in
tain
in
g
th
e stab
ility[4
]
an
d as add
itio
nal sou
r
ce
of info
rm
atio
n
in
i
d
en
tified
p
r
o
cess
an
d
stability
indicator. Moreove
r
, CoM is
calculated to provide critical
t
o
access reha
bilitation succ
ess in
p
a
tho
l
og
y d
e
tectio
n
and
in
d
e
scrib
i
ng
g
a
its[5
]. In
reco
n
f
i
g
u
r
ation
asp
e
ct, th
e Co
M’s of leg
g
e
d
rob
o
t
is will b
e
reallocated
since the c
h
a
ngi
ng
of
in t
h
e structure
or leg
confi
g
ur
atio
n of
t
h
e
r
obo
t.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
56
IJR
A
V
o
l
.
3,
No
. 2,
Ju
ne
2
0
1
4
:
13
9 – 1
5
0
14
0
The
r
ef
ore
i
n
t
h
i
s
st
u
d
y
,
det
e
rm
i
n
at
i
on on
h
e
xap
o
d
c
o
n
f
i
g
urat
i
o
n t
o
q
u
a
d
r
u
ped c
o
nfi
g
urat
i
o
n
fo
r a
hexa
p
od
ro
b
o
t
(He
x
a-
Qu
ad
)
i
s
pro
p
o
se
d. H
e
xap
o
d
i
s
o
n
e
o
f
th
e statically stab
le co
n
f
i
g
uration
s
of
m
u
l
ti-
l
e
gge
d r
o
b
o
t
t
h
at
has p
o
t
e
nt
i
a
l
t
o
be reco
nfi
g
u
r
ed i
n
t
o
l
e
ss t
h
an si
x l
e
gs suc
h
as q
u
a
dr
u
p
ed a
nd
b
i
pedal
con
f
i
g
urat
i
o
n
.
Tran
sf
orm
i
ng
hexa
p
od t
o
bi
pedal
c
o
n
f
i
g
u
r
at
i
on i
s
co
nsi
d
ere
d
as c
r
i
t
i
cal
con
f
i
g
urat
i
o
n f
o
r
hexa
p
od
u
n
l
e
ss t
h
ere ha
ve a
speci
al
desi
g
n
o
n
l
e
g c
o
nfig
uration
and
ro
bo
t bod
y’s sh
ap
e itself (o
t
h
er than
com
m
on hexa
po
d
’
s b
ody
sha
p
es;
sq
uare, t
r
a
p
ezi
um
, rou
n
d
or he
xa
go
n b
o
d
y
)
. T
h
e qu
ad
r
upe
d co
nfi
g
u
r
a
t
i
on i
s
selected
sin
ce th
is con
f
i
g
uratio
n
is in
b
e
tween
stati
cally and dynam
i
cally s
t
able and s
u
itable for any c
o
mmon
sh
ap
e of h
e
x
a
p
o
d
rob
o
t
’s bod
y. Static stab
ilit
y assu
m
e
s
t
h
e v
e
rtical p
r
oj
ectio
n
of th
e
Co
M always re
m
a
in
in
sid
e
th
e suppo
rt
p
o
l
ygon
w
i
th
an
ad
eq
u
a
te
stab
ility
marg
in
during
all phase o
f
m
o
v
e
m
e
n
t
s [6
]. On
t
h
e
o
t
h
e
r
h
a
nd
,
d
y
n
a
m
i
c
a
lly stab
le d
e
p
e
nd
s on
th
e
stab
ility d
u
r
ing
th
e robo
t is
m
o
v
i
n
g
wh
ich
d
e
m
a
n
d
s
on activ
e
act
uat
i
on t
o
m
a
i
n
t
a
i
n
t
h
e
bal
a
nce an
d
per
f
o
r
m
i
ng fast
er
m
o
t
i
on[
7]
. As part
o
f
dy
nam
i
cal
l
y
st
abl
e
con
f
i
g
urat
i
o
n
,
qua
d
r
u
p
e
d
l
e
g
g
ed
r
o
bot
c
o
nf
i
g
u
r
at
i
o
n
al
so
pract
i
cal
o
n
p
e
rf
orm
i
ng l
o
c
o
m
o
ti
on
fo
r c
o
m
p
l
e
x
terrain
accord
i
n
g to
t
h
e sev
e
ral p
r
actical
ach
i
e
vem
e
nt
repo
r
t
ed i
n
[
8
,
9]
.
Reco
nfigu
r
ation
issu
e b
e
co
m
e
o
n
e
of th
e small
sectio
n
s
in
ro
bo
tic issue th
at h
a
s p
o
t
en
tial to
b
e
ex
p
l
o
r
ed
i
n
o
r
d
e
r to
op
timize
th
e
u
s
e of the
d
e
fau
lt m
ech
an
ism
o
f
th
e robo
t itself an
d increased its flexib
ility.
CONR
O fro
m Po
lym
o
rph
i
c Rob
o
tics Lab
o
ratory of
US
C In
fo
rm
atio
n
Scien
ce
Institu
te is o
n
e
o
f
th
e
exam
pl
es of
he
xap
o
d
ro
b
o
t
t
h
at
per
f
o
rm
i
ng
pr
o
pose
d
h
o
rm
one
-
b
ased
di
st
r
i
but
ed
c
ont
r
o
l
t
o
i
m
pl
em
ent
i
t
s gai
t
recon
f
i
g
uration
b
e
tween
cat
erp
illar
an
d sp
id
er g
a
it m
o
d
e
[2
]. Sh
en
et. al.
m
e
nt
i
one
d t
h
at
t
h
e n
u
m
ber of
supporte
d
leg
m
u
st
meet the stability
criteri
a according t
o
the num
b
er of
le
g that a
v
ailable for
walki
n
g
used.
It
i
s
di
ffe
rent
t
o
t
h
e
hy
bri
d
w
h
eel
-l
eg
ge
d
ro
bot
,
nam
e
l
y
Hy
l
o
s i
s
de
si
g
n
e
d
a
n
d
d
e
vel
o
p
e
d
by
La
bo
rat
o
i
r
e
de
Rob
o
tiq
ue de Paris (LRP
), U
n
ive
r
sit´
e de Pierre et
Marie Curie, France where
b
y
t
o
opt
i
m
i
ze bot
h t
h
e bal
a
nc
e
o
f
traction
forces an
d
t
h
e tip
ov
er stab
ility. A sp
ecific tr
aj
ecto
ry and
po
sture con
t
ro
l is d
e
sig
n
e
d
to
o
v
e
rco
m
e
bot
h
ro
b
o
t
’
s l
o
com
o
t
i
on i
t
s
el
f
an
d
ori
e
nt
at
i
on
of
t
h
e m
a
i
n
bo
dy
a
n
d
si
de
way
w
h
eel
ba
se
s [
10]
.
On
t
h
e ot
he
r
han
d
,
OSC
A
R
from
Uni
v
e
r
si
t
y
Lübeck
has
pr
op
ose
d
t
h
e
or
ga
ni
c sel
f-c
o
n
fi
gu
rabl
e i
n
h
e
xap
o
d
r
o
b
o
t
as i
t
s
nam
e
im
pli
e
d. The ai
m
of t
h
e
devel
opm
ent
i
s
t
o
o
v
erc
o
m
e
t
h
e m
a
l
f
unct
i
on l
e
g
(
s) a
n
d o
p
t
im
i
z
i
ng t
h
e
o
v
eral
l
ener
gy
du
ri
n
g
l
o
com
o
t
i
on
by
per
f
o
r
m
i
ng sel
f-am
put
at
i
o
n
[
11]
.
Acco
r
d
i
n
g t
o
t
h
e st
u
d
y
g
o
al
,
bot
h he
xa
po
d
and
q
u
ad
ru
pe
d
ro
bot
st
a
b
l
e
d
wal
k
i
n
g
pat
t
e
r
n
i
s
cr
uci
a
l
.
This is a fundam
ental proble
m
need
t
o
be sol
v
ed
fo
r
every
wal
k
i
n
g r
o
b
o
t
i
n
m
ovi
ng
o
p
erat
i
o
n. T
h
e
devel
opm
ent
o
f
w
a
l
k
i
n
g
pat
t
e
rn
o
f
a
wal
k
i
n
g
r
o
b
o
t
i
s
a c
h
al
l
e
ngi
n
g
t
a
s
k
be
cause t
h
e c
onsi
d
erat
i
o
n t
h
e
de
gre
e
o
f
freed
o
m
(D
o
F
)
w
ith
th
e sup
port p
o
l
ygon
is i
m
p
o
r
tan
t
for th
e stab
ility o
f
th
e robo
t [12
]
. Y
a
ng
J.M.
et. al
in
th
eir stud
ies
has con
s
id
ered
th
e an
alysis
o
f
th
e
j
o
i
n
t failure b
a
sed on
t
h
e m
a
n
i
p
u
l
ato
r
k
i
n
e
m
a
tics an
d
g
a
it
pat
e
r
n
. Th
us
p
r
o
p
o
sed t
h
e
pe
ri
o
d
i
c
qua
dr
u
p
e
d an
d he
xap
o
d
gai
t
t
o
ove
rc
om
e any
faul
t
t
o
l
e
rant
caus
e
d by
j
o
i
n
t failure and
to
m
a
in
tain
th
e stab
ility o
f
t
h
e rob
o
t
[13
]
.
On
t
h
e o
t
h
e
r
han
d
s, Tsuj
ita K.
et. al
has
o
v
e
r
com
e
t
h
e t
i
m
i
ng pr
o
b
l
e
m
bet
w
een t
r
ans
v
erse
, r
o
t
a
ry
, pace
, b
o
u
n
ce and t
r
ot
gai
t
pat
t
e
rn
fo
r q
u
a
dr
u
p
ed
ro
b
o
t
st
udi
es
co
nsid
ered
th
e an
alysis o
n
the su
itab
l
e g
a
it p
a
ttern
for th
e
q
u
a
drup
ed
robo
t b
y
p
r
op
o
s
ed th
e ad
ap
tiv
e co
n
t
ro
l
[1
4]
. Ot
her e
f
f
o
rt
ha
s bee
n
d
one
by
p
r
o
p
o
s
e
d t
h
e
Gai
t
regul
at
i
o
n t
ech
ni
que t
o
i
n
cre
a
s
e
t
h
e ro
b
u
st
ne
ss i
n
m
u
l
ti-leg
g
e
d
ro
bo
t wal
k
ing
p
a
ttern. For a sin
g
l
e du
ty of a d
e
v
e
lop
i
ng g
a
it p
a
ttern
,
n
eed
ju
st ig
nore th
e
kinem
a
tic
mapping and the c
onsi
d
eration
of keep m
o
re
legs contact with the s
u
rface.
Due t
o
the limitation
reci
rcul
at
i
o
n s
p
eed
, t
h
e t
r
ot
and t
r
i
p
o
d
gai
t
pat
t
e
rn ca
n per
f
o
r
m
si
gni
fi
cat
i
on fast
e
r
t
h
an
ot
he
r[
15]
. Ac
c
o
r
d
i
n
g
to
th
e lift an
d
release p
r
ob
ab
il
istic ev
en
ts [3
]
fo
r each
le
g o
f
l
e
g
g
ed
r
o
b
o
t
,
t
r
i
p
o
d
pat
t
e
r
n
fo
r
hexa
p
od
r
o
bot
i
s
l
e
ss and
pr
o
d
u
c
i
ng fa
st
er m
ovem
e
nt
. Qua
d
r
upe
d r
o
bot
on
t
h
e ot
he
r ha
n
d
havi
ng
bet
w
ee
n dy
nam
i
c and
st
at
i
c
stab
ility ran
g
e
wh
ich
is requ
ired
g
ood
co
m
b
i
n
atio
n of su
itab
l
e wal
k
ing
p
a
ttern
. Th
erefo
r
e, in
t
h
is article, the
com
b
i
n
at
i
on
of
t
r
ave
r
se a
n
d t
r
ot
wal
k
i
n
g
pat
t
e
rn
has
be
e
n
p
r
op
ose
d
fo
r
t
h
e ro
b
o
t
m
odel
i
n
qua
d
r
u
p
e
d
m
ode.
The
pr
o
pose
d
Hexa
-
Q
ua
d t
r
a
n
sf
orm
a
t
i
on f
o
r he
xa
po
d
ro
b
o
t
i
s
desi
gne
d
wi
t
h
t
w
o
di
f
f
e
rent
fo
rm
s
nam
e
ly
cent
e
r
l
e
gs di
sa
bl
e (
C
LD) a
n
d si
d
e
l
e
gs
di
sabl
e
(SL
D
)
.
T
h
e f
o
rm
i
s
deci
de
d b
a
sed
o
n
c
o
m
m
on
ap
p
lication
for th
e
h
e
x
a
pod
ro
bo
t su
ch
as co
nv
ertin
g leg
s
to
t
h
e
free man
i
pu
lato
rs
o
r
d
i
sab
ling
t
h
e l
e
g
for
ener
gy
savi
n
g
.
The pr
o
pose
d
t
r
ans
f
o
r
m
a
t
i
on t
ech
ni
q
u
e is created by inspi
r
ed
fro
m
t
h
e Co
M in
sup
por
t
p
o
l
ygon
and leg
sh
ou
ld
er
ang
l
e sy
mmetr
ical co
n
cep
t p
r
op
o
s
ed p
r
ev
iou
s
in [
1
6
]
. Th
e
p
r
op
o
s
ed
tran
sform
a
t
i
o
n
,
trip
od
p
a
ttern
an
d
tr
av
erse
-t
r
o
t
pat
t
e
r
n
a
r
e
m
odel
e
d i
n
a
h
e
xap
o
d
r
o
bot
r
eal
-t
im
e
m
odel
wi
t
h
4
DoF leg configu
r
ation
s
.
2.
HEX
A
-
Q
UAD
TRAN
SFOR
MA
TION
TEC
H
NIQU
ES M
ETHOD
Most of the proposed
tran
sformatio
n
techn
i
q
u
es for m
u
lti-legged walking r
obot are du
e to
the specifi
c
configuration of
the robot itself. In this stud
y
,
the tran
sformatio
n is proposed for general hexap
od robot config
uration
with an
y
number
of DOF legs. The proposed Hex
a
-Quad tr
ansf
ormation techniqu
e is design
ed b
y
considering
the s
upport
poly
gon or stability
ar
ea of
the r
obot as shown in
Figu
re 1
and
Figu
re 2
. The
larger the support
poly
gon d
e
velop
e
d
b
y
the robots the bigger the probability
for th
e robo
t to remain upright without overt
urning when it stops walking at an
y
m
o
m
e
nt during
walking p
e
riod
,
and th
is is
cal
led
stat
ica
l
l
y
st
abl
e
walking or
sta
tic
stabi
lit
y
[3]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Opt
i
m
i
z
i
n
g
He
xap
o
d
R
o
bot
R
econf
i
g
ur
at
i
o
n
usi
n
g
Hex
a
-
Q
u
a
d
Tr
an
sf
or
ma
t
i
on (
A
d
d
i
e
Ira
w
an)
14
1
(a)
(b)
Fi
gu
re
1.
The
pr
o
pose
d
f
o
rm
s o
f
Hexa
-Q
ua
d t
r
a
n
s
f
o
r
m
a
t
i
o
n;
(a)
C
L
D
f
o
rm
, (b)
SL
D
fo
rm
.
There
f
ore i
n
pr
o
pose
d
He
x
a
-Q
ua
d t
r
ans
f
orm
a
t
i
on t
echni
q
u
e, t
w
o f
o
rm
s of t
r
ansf
orm
a
t
i
on are
p
r
op
o
s
ed
b
y
co
n
s
i
d
er
i
n
g
t
h
e su
ppo
r
t
po
lygon
and
Co
M as
sh
own
in
Figur
e 1
.
CLD
is realized
b
y
lif
ti
n
g
u
p
two ce
nter legs as in
si
t
dow
n m
o
de
. Th
is fo
rm
is n
o
t
critical to
co
n
t
ro
l i
f
com
p
are to t
h
e SL
D (Figure 1(b))
th
at requ
ired
a p
r
o
p
e
r in
itial
stan
d
i
n
g
p
o
s
itio
n
for o
t
h
e
r le
g
s
. Th
erefore,
th
is p
r
o
p
o
s
ed
tech
n
i
q
u
e
i
n
trod
uced
separat
e
d cal
c
u
l
a
t
i
on f
o
r
C
L
D
an
d S
L
D
as s
h
ow
n i
n
Fi
gu
re
2 a
n
d
Fi
g
u
r
e
3
respect
i
v
el
y
.
A
s
show
n
in
Figur
e
2
,
the Co
M is at the cen
ter
o
f
t
h
e bo
d
y
(
C
o
B
)
of
th
e rob
o
t
and th
e sup
por
t
p
o
l
ygon
is
f
o
l
l
o
w
ed
b
y
th
e
sh
ap
e of
t
h
e stan
d
i
ng
leg
s
.
Th
e sh
ap
e
o
f
su
ppo
r
t
po
lygo
n is d
e
p
e
n
d
s on
th
e
num
ber
of t
o
u
c
hi
n
g
l
e
g
o
n
t
h
e g
r
ou
n
d
(
r
ed
dot
t
e
d l
i
n
e) a
s
sh
ow
n i
n
Fi
gu
re
2 an
d Fi
gu
re
3. T
h
us t
h
e n
e
w
m
a
xim
u
m
ext
e
nde
d a
n
gl
e o
f
sho
u
l
d
er
fo
r e
ach s
u
p
p
o
rt
i
n
g
l
e
gs
(ena
bl
ed
l
e
gs)
(
a
) after tran
sform
a
t
i
o
n
can
be
det
e
rm
i
n
ed
by
usi
n
g i
s
t
h
e
l
e
ngt
h
(
l
) a
n
d
wi
dt
h
(
w
)
o
f
th
e
robo
t bod
y as
fo
llo
ws;
11
0.5
t
a
n
|
|
0.5
t
a
n
|
|
oo
o
an
n
o
x
l
kk
wy
(1)
whe
r
e
0
x
is th
e v
e
rtical len
g
t
h
fro
m
th
e cen
ter
o
f
th
e
ro
bo
t bod
y wh
ile
0
y
i
s
t
h
e ho
ri
zo
nt
al
l
e
ngt
h f
r
om
t
h
e
cent
e
r
of
t
h
e
b
ody
a
n
d
o
n
is an
in
itial v
a
lu
e for each
sh
ou
lder.
In add
ition
,
k
i
s
t
uni
ng
pa
ra
m
e
t
e
rs i
n
or
der
to
ach
iev
e
s
s
lw
to
en
su
r
e
C
o
M
n
e
ar
th
e cen
ter
o
f
sup
por
t po
lygo
n.
0
0
z
x
y
Bo
dy
in
sta
b
l
e
ra
n
g
e
0
0
z
x
y
Bo
dy
in
sta
b
l
e
ra
n
g
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
56
IJR
A
V
o
l
.
3,
No
. 2,
Ju
ne
2
0
1
4
:
13
9 – 1
5
0
14
2
Fi
gu
re
2.
S
h
o
u
l
der a
n
gl
e det
e
rm
i
n
at
i
on f
o
r
C
L
D t
r
a
n
s
f
o
r
m
a
t
i
on m
ode.
Fi
gu
re
3.
S
h
o
u
l
der a
n
gl
e det
e
rm
i
n
at
i
on f
o
r
S
L
D t
r
a
n
s
f
o
r
m
a
t
i
on m
ode.
Th
is
ru
le is ap
p
lied with referen
ce to th
e sho
u
l
d
e
r-b
ased
co
ord
i
n
a
tion syste
m
(
S
CS)
an
d Co
B
-
based
sym
m
etrical app
r
oach
[1
6]
.
M
o
re
ove
r, t
h
e
rule is ve
ry
i
m
po
rt
ant
f
o
r t
h
e
pr
o
pose
d
S
L
D
fo
rm
m
ode whi
c
h i
s
sid
e
legs are
disab
l
ed
fro
m
walk
ing u
s
ed
.
Th
e
o
t
h
e
r le
g
s
n
e
ed
t
o
b
e
rei
n
itialized
its sh
ou
ld
er’s an
g
l
e u
s
i
n
g
Eq.
1
. A
s
sh
ow
n i
n
Fi
gu
re 3,
exam
pl
e si
t
u
ati
on o
f
t
w
o si
d
e
l
e
gs (l
eg 1 and l
e
g 4
)
i
s
di
sabl
ed an
d ot
h
e
r fo
u
r
l
e
gs (l
eg
2,
3,
5 and
6) i
s
rei
n
i
t
i
a
l
i
zed. The f
u
l
l
t
r
ansfo
r
m
a
t
i
on se
q
u
ence
of
pr
op
ose
d
He
x
a
-Q
ua
d i
s
pres
ent
e
d
as finite state
machine (FSM
) as s
h
ow
n i
n
F
i
gu
re
4.
R
o
b
o
t
b
ody
s
h
ape
al
so
t
h
e
i
m
port
a
nt
fact
o
r
t
h
at
nee
d
t
o
be
co
nsi
d
e
r
e
d
on
sel
ect
i
n
g
p
r
op
ose
d
He
xa
-
Qua
d
t
r
a
n
sf
o
r
m
a
t
i
on fo
rm
.
C
o
m
m
onl
y
,
for
defa
ul
t
hexa
p
od
ro
b
o
t
,
t
h
e
b
ody
desi
g
n
wi
l
l
consi
d
e
r
e
d
t
h
e st
abl
e
p
o
s
ition
fo
r t
h
e leg
to
m
o
v
e
an
d
stan
d
i
n
g
to
en
su
re t
h
e Co
M always at cen
ter of its su
ppo
rt
p
o
l
ygo
n.
As
sh
own
i
n
Figur
e
5
,
th
er
e ar
e thr
ee
d
i
f
f
er
ent co
mm
o
n
sh
ap
e
of
h
e
x
a
pod rob
o
t
’
s
bo
d
y
th
at po
ssi
b
l
e t
o
b
e
Le
g2
Leg1
Leg3
Leg4
Leg5
Leg6
S
upp
ort
Po
l
y
g
o
n
CoM
a
b
c
d
Le
g2
Leg
1
Le
g3
Leg4
Le
g5
Le
g6
Supp
ort
Poly
gon
Co
M
a
b
c
d
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
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:
208
8-8
7
0
8
Opt
i
m
i
z
i
n
g
He
xap
o
d
R
o
bot
R
econf
i
g
ur
at
i
o
n
usi
n
g
Hex
a
-
Q
u
a
d
Tr
an
sf
or
ma
t
i
on (
A
d
d
i
e
Ira
w
an)
14
3
designe
d
. Moreove
r the
figure also shows t
h
at
each s
h
a
p
e
has
dif
f
e
r
ent
support
polygon size
,
s
l
and
s
w
,
wi
t
h
di
f
f
ere
n
t
bo
dy
si
ze,
l
and
w
. Fi
gu
re
5
(
a
)
a
n
d
5(
b)
s
h
ow
s
t
h
e
lw
an
d
wl
m
a
kes
SL
D
m
e
t
h
od
al
m
o
st in
stab
le to b
e
ap
p
lied
u
n
l
ess th
e supp
ort
po
lyg
on size is tun
e
d
s
s
lw
t
o
b
e
tter
a
v
a
lu
e s
u
ch a
s
Fig
u
re 3. It is sam
e
to
th
e rou
n
d
b
o
d
y
sh
ape with
th
e size
lw
.
The
r
ef
ore i
t
m
a
kes C
L
D
m
e
t
hod m
o
st
l
i
k
el
y
sui
t
a
bl
e t
r
ans
f
o
r
m
a
ti
on
fo
rm
for com
m
on shape
of
hexa
p
od
r
o
b
o
t
suc
h
as exi
s
t
e
d
est
a
bl
i
s
hed
he
xap
o
d
ro
b
o
ts re
p
o
rted
in
[1
7,
1
8
]
.
Fi
gu
re
4
.
F
S
M
o
f
pr
op
ose
d
H
e
xa-
Q
uad
t
r
an
s
f
o
r
m
a
ti
on fo
r h
e
xap
o
d
ro
b
o
t
m
odel
.
(a)
(b)
(c
)
Fi
gu
re
5.
F
u
n
d
a
m
e
nt
al
shape
fo
r
hexa
p
o
d
r
o
bot
,
(a
)he
x
a
g
o
n
bo
dy
s
h
a
p
e,
(
b
)
R
ect
ang
u
l
a
r
b
ody
s
h
a
p
e,
(c
)
R
o
u
n
d
bo
dy
s
h
ape.
3.
WALKING PATTER
N AND SHOULDE
R
-B
ASED CO
ORDINATION SYSTE
M
The se
que
nces
of t
h
e l
e
gs
fo
r
qua
dr
u
p
ed a
n
d he
xa
po
d wal
k
i
n
g are p
r
ese
n
t
e
d i
n
fi
ni
t
e
st
at
e
m
achi
n
e
(
FSM)
as show
n
in
Figur
e 6
.
On
h
e
x
a
pod co
n
f
i
g
ur
ation
o
r
h
e
x
a
p
o
d
mo
d
e
as show
n
in
Fig
u
r
e
6
(
a)
, tr
ip
od
walk
ing
g
a
it pattern
is u
s
ed
sin
ce it p
e
rform
s
fastest walk
ing
with
m
i
n
i
m
u
m
area o
f
su
ppo
rt
p
o
l
ygo
n
i
n
h
e
x
a
po
d
rob
o
t
stab
ility. On
th
e o
t
h
e
r h
a
nd
, trav
erse-tro
t
g
a
it p
a
ttern
is selected
for q
u
a
drup
ed
m
o
d
e
as sh
own
R
e
i
n
itia
l
i
z
e
d
s
i
de legs
(L
eg 1
,
3,
4,
6)
S
i
de
L
e
gs
(
l
e
g
1
& 4 or
le
g
3
& 6)
disa
ble
d
T
r
ansf
or
m
a
ti
on
M
ode
Ce
nte
r
L
e
gs
(
l
e
g
2 &
5)
disa
bled
Q
u
a
d
r
uped
W
a
lking Mode
R
e
ini
tia
l
i
z
e
d
si
de legs
(Leg
2,
5,
1/3,
4
/
6
)
Fr
o
m
He
x
a
p
o
d
Mod
e
Ce
nte
r
L
e
gs
(
l
e
g
2
&
5)
Rel
e
as
ed
Side
L
e
gs
(le
g
1
&
4 or
le
g
3
& 6
)
re
l
eas
e
d
To
Hexap
o
d
Mod
e
C
o
B\
Co
M
Co
B\C
o
M
/
Su
ppo
rt
Po
ly
gon
C
o
B\
Co
M
s
l
s
w
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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:
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56
IJR
A
V
o
l
.
3,
No
. 2,
Ju
ne
2
0
1
4
:
13
9 – 1
5
0
14
4
i
n
Fi
g
u
r
e
6
(
b).
Thi
s
pr
o
p
o
s
ed
q
u
ad
ru
pe
d m
ode
gai
t
pat
t
e
r
n
per
f
o
r
m
i
ng m
a
xim
u
m
t
w
o l
e
gs at
a t
i
m
e d
u
r
i
n
g
lo
co
m
o
tio
n
which
is th
e fastest for th
is co
nfig
uratio
n.
Fu
rt
h
e
rm
o
r
e tran
sverse-tro
t gait pattern
s is used an
d
desi
g
n
e
d
wi
t
h
SC
S ki
nem
a
t
i
c
s refere
nce
as sh
ow
n i
n
Fi
gu
re
7 si
n
ce bot
h
hexa
po
d a
n
d
q
u
a
d
ru
pe
d
co
nfigu
r
ation
m
o
d
e
s are applied
in
th
e sam
e
h
e
xapod
ro
bo
t m
o
d
e
l. In ad
d
ition
th
e
fo
rce effectiv
e t
r
aj
ectory
m
o
t
i
o
n
as sh
ow
n in Figu
r
e
8 [1
9
]
is app
lied fo
r bo
th w
a
l
k
in
g m
o
d
e
s, t
h
us th
e
s
u
pp
ort
p
h
a
se
and
sw
i
n
g ph
as
e
eq
u
a
tion
s
are
g
e
n
e
ralized
as ex
pressed
as Eq
. 2
an
d
Eq
.3
.
Bo
th
po
sitio
ns in
clu
d
i
ng
v
e
rt
ical leg
p
o
s
itio
n
(
z
)
i
s
det
e
rm
i
n
ed
di
ffe
re
nt
l
y
i
n
e
ach s
u
pp
ort
a
n
d s
w
i
n
g
p
h
ase
by
usi
n
g t
h
ose
equat
i
o
ns
res
p
ect
i
v
el
y
.
(a)
(b
)
Fig
u
re
6
.
FSM
for
(a) t
r
ipo
d
gait p
a
ttern
and
(b) trav
erse-trot g
a
it p
a
ttern
in h
e
x
a
pod
robo
t m
o
d
e
l with
Hex
a
-
Qua
d
t
r
ans
f
or
m
a
t
i
on.
Le
g 2
,
4,
6 :
S
uppor
t
Ph
as
e
Le
g 1
,
3,
5:
S
w
i
ng
Ph
as
e
A
l
l
le
gs:
Tr
a
n
s
i
e
n
t
Ph
ase
L
e
g 2,4
,
6
:
S
u
p
port
Phase
L
e
g 1,
3,
5:
S
w
ing
Phase
ST
OP:
A
ll l
e
gs on
gr
ound
No
Ye
s
Tran
sfo
r
m
a
t
i
o
n
state
(Hex
a
Qu
a
d
)
Shou
l
der
a
n
g
l
e
s
r
e
i
n
i
t
i
a
liz
e
d
To
q
u
a
d
ruped
wa
lk
in
g
s
e
quenc
es
No
t
T
r
ans
f
or
med
Tra
n
sf
orme
d
T
r
ans
f
orm
a
tio
n
St
a
t
e
(Hex
a
→
Quad
)
Sh
o
u
ld
er an
gl
e
s
reini
tialized
L
e
g 3
,
4
,
6
:
S
u
ppor
t
Ph
ase
L
e
g
1:
S
w
i
ng P
h
a
s
e
All Legs
:
T
r
ans
i
en
t Phas
e
All L
e
gs
:
T
r
an
s
i
en
t P
h
as
e
L
e
g
1,
3,
4:
S
uppo
r
t
P
h
a
s
e
L
e
g
6:
S
w
i
n
g Ph
a
s
e
Le
g
3,4: Su
pp
o
r
t Ph
a
s
e
L
e
g
1,6: Swin
g Ph
a
s
e
L
e
g 3,
4,
6:
S
u
p
por
t
Ph
ase
L
e
g 1:
S
w
i
n
g
Ph
ase
All Legs
:
T
r
ansie
n
t Phase
A
ll Leg
s
:
T
r
an
sien
t Ph
ase
L
e
g
1,
3,
4:
Su
ppo
r
t
Ph
a
s
e
L
e
g
6:
Sw
i
n
g Ph
a
s
e
All
Legs
:
T
r
ans
i
en
t Ph
ase
T
o
H
e
xapod
M
ode
Ye
s
No
All L
e
g
s
:
O
n
t
h
e
gr
o
und
Fr
om
H
e
xa
pod
Mode
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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7
0
8
Opt
i
m
i
z
i
n
g
He
xap
o
d
R
o
bot
R
econf
i
g
ur
at
i
o
n
usi
n
g
Hex
a
-
Q
u
a
d
Tr
an
sf
or
ma
t
i
on (
A
d
d
i
e
Ira
w
an)
14
5
(
S
upp
or
t Ph
ase –
Step
and
p
u
sh
o
n
th
e
gr
ound
)
0
2
c
T
t
0
0
0
21
4
()
s
i
n
c
o
s
42
21
4
()
s
i
n
s
i
n
42
()
nn
n
nn
n
nn
o
s
a
cc
o
s
a
cc
s
S
tt
xt
x
TT
S
tt
yt
y
TT
zt
z
(
2
)
(Swi
ng
P
h
ase
)
0
2
c
T
t
0
0
00
2
()
1
c
o
s
c
o
s
2
2
()
1
c
o
s
s
i
n
2
2
()
s
i
n
nn
n
nn
n
nn
o
s
a
c
o
s
a
c
s
c
S
xt
x
t
T
S
yt
y
t
T
zt
z
H
t
T
(
3
)
whe
r
e,
c
T
= walki
n
g cycle tim
e
(s),
t
= up
d
a
te tim
e (real-tim
e) (s),
ex
t
= ad
di
t
i
onal
pe
ri
o
d
f
o
r a
ppl
y
i
ng
ext
r
a f
o
rce (
s
),
0
S
= distance
of
foot
placem
en
t for
one
cycle (m
), and
0
H
= h
e
igh
t
o
f
leg
lift fro
m
th
e in
itial p
o
s
itio
n (m
).
Fi
gu
re 7.
SC
S t
r
aject
o
r
y
ki
ne
m
a
t
i
c
s
m
o
t
i
on
fo
r
a 4
-
D
O
F
l
e
g of
he
xa
po
d r
o
b
o
t
m
odel
wi
t
h
pr
o
pose
d
He
xa-
Qua
d
t
r
ans
f
or
m
a
t
i
on
1
3
L3
L4
L1
2
L2
4
X
Y
Z
Evaluation Warning : The document was created with Spire.PDF for Python.
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56
IJR
A
V
o
l
.
3,
No
. 2,
Ju
ne
2
0
1
4
:
13
9 – 1
5
0
14
6
Fi
gu
re
8.
A
l
e
g
m
o
t
i
on sha
p
e
use
d
i
n
he
xa
po
d
ro
b
o
t
m
odel
wi
t
h
pr
o
pose
d
Hexa
-
Q
ua
d t
r
a
n
sf
orm
a
t
i
o
n
4.
R
E
SU
LTS AN
D ANA
LY
SIS
Sev
e
r
a
l sim
u
latio
n
s
run
n
i
n
g
h
a
v
e
b
een setu
p an
d run to
an
alyze t
h
e po
ten
tial of
t
h
e
pr
opo
sed
m
e
t
hod t
o
be i
m
pl
em
ent
e
d i
n
t
h
e real
-t
i
m
e sy
st
em
.
The fi
rs
t
sim
u
l
a
t
i
on i
n
do
ne
on t
h
e
pr
op
ose
d
C
L
D m
e
t
h
o
d
by
si
m
u
l
a
t
i
ng t
h
e
real
-t
i
m
e hexap
o
d
ro
b
o
t
m
odel
wi
t
h
t
h
e
3D
m
odel
t
h
at
desi
gne
d
sepa
rat
e
l
y
[2
0]
as
s
h
o
w
n
i
n
Fi
g
u
re
9
.
Fi
gu
re
9 s
h
o
w
s t
h
e ce
nt
er l
e
g
s
(Leg
2 a
n
d
5)
are di
sa
bl
ed
af
t
e
r r
o
b
o
t
st
o
p
wal
k
i
n
g i
n
he
x
a
po
d
m
ode. In t
h
i
s
c
a
se of
t
r
an
sf
or
m
a
t
i
on, si
de l
e
gs
becom
e
m
a
in l
e
gs a
n
d rea
d
y
for
q
u
a
d
r
upe
d m
ode wal
k
i
n
g.
T
h
e
initial angle
of
each m
a
in leg for qua
d
rupe
d
m
ode doe
sn
’
t
c
h
ange m
u
ch
due to the
calcula
tion
using Eq.1.
It is
d
i
f
f
eren
t t
o
t
h
e
SLD t
r
ansform
a
tio
n
whereb
y
certain
step
s o
f
in
itializatio
n
n
e
ed
s
to b
e
do
n
e
on
th
e rem
a
in
ed
l
e
g
s
that will be u
s
ed
in qu
adrup
e
d
m
o
d
e
walk
in
g.
As
shown
in
Figu
re 10
,
cen
t
er
leg
s
(Leg
s 2
an
d
5) and
si
d
e
leg
s
are
rein
itialized
(Figure
1
0
(b
) and
(c
)) t
o
appropriate an
g
l
e
b
e
fore ano
t
h
e
r sid
e
leg
s
(Leg
1
an
d 4)
f
lip
p
e
d
to th
e
f
r
on
t an
d d
i
sab
l
ed (
F
i
g
ur
e
10
(d
))
.
(a)
(
b
)
Fi
gu
re 9.
3
D
m
odel
si
m
u
l
a
ti
on resul
t
fo
r
C
L
D
t
r
a
n
s
f
o
r
m
a
tion
,
(a) he
xap
o
d
wal
k
i
n
g st
o
p
,
(
b
)
ce
nt
er
l
e
gs
di
sabl
e
d
S
e
nsing P
o
int
(1
)
(2
)
(3
)
(4)
Z-Axis (
m
)
Y
-
Axis (m)
Shou
lde
r
point
h
S
F
I
RS
T
-
phase
MOV
E
-pha
se (k)
Evaluation Warning : The document was created with Spire.PDF for Python.
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7
0
8
Opt
i
m
i
z
i
n
g
He
xap
o
d
R
o
bot
R
econf
i
g
ur
at
i
o
n
usi
n
g
Hex
a
-
Q
u
a
d
Tr
an
sf
or
ma
t
i
on (
A
d
d
i
e
Ira
w
an)
14
7
(a)
(
b
)
(c)
(d
)
Fi
gu
re 1
0
. 3
D
m
odel
1
sim
u
lati
o
n
resu
lt fo
r SLD tran
sform
a
tio
n
,
(a) h
e
x
a
po
d wal
k
ing
st
op
,
(b
) cen
t
er leg
s
sh
ou
l
d
er ang
l
e
rein
itialized
, (c) sid
e
leg
s
shou
ld
er an
g
l
e
rei
n
itialized
, (d
) tar
g
et leg
s
d
i
sab
l
ed
.
Th
is step is im
p
o
r
tan
t
to
mak
e
su
re ro
bo
t
is in
st
abl
e
ra
nge
an
d
o
v
ert
u
r
n
i
n
g i
s
a
v
oi
ded
.
Si
nce t
h
e
hexa
p
o
d
m
odel
wi
t
h
lw
, C
L
D
i
s
use
d
t
o
si
m
u
l
a
t
e
hexa
p
o
d
m
ode
t
o
q
u
ad
r
u
p
e
d m
ode t
r
a
n
s
f
orm
a
t
i
on.
As
sh
own in Figu
re
1
1
,
fu
ll w
a
l
k
in
g
f
r
o
m
h
e
x
a
po
d m
o
d
e
to quad
r
up
ed m
o
d
e
is pr
esen
ted
.
Th
e t
r
ipo
d
w
a
lk
i
n
g i
s
prese
n
ted
f
r
om
Fig
u
re
1
1
(a
) t
o
1
1
(
b
) a
n
d it s
t
op
fo
r CL
D tr
ansf
o
r
m
a
tion a
s
sh
o
w
n i
n
Fi
g
u
re
1
1
(c
).
The
ro
b
o
t
cont
i
n
ue
d wal
k
i
n
g
i
n
q
u
ad
ru
ped
m
ode usi
n
g pr
o
p
o
s
ed
t
r
a
v
erse
-t
r
o
t
gai
t
pat
t
e
rn
f
r
o
m
Fi
gu
re 1
1
(
d
)
t
o
Fi
gu
re
1
1
(
f
)
i
n
re
vers
e pat
h
.
As
s
h
ow
n i
n
Fi
g
u
re
1
1
(c)
cent
e
r l
e
gs a
r
e
di
sa
bl
ed a
n
d al
l
re
m
a
i
n
i
ng l
e
g
d
one
t
h
e
trav
erse-tro
t walk
in
g g
a
it
p
a
ttern
as
shown
d
e
tail in
Fi
g
u
r
e
1
2
vi
a
fo
ot
m
o
ti
on sam
p
l
e
res
u
l
t
s
(z-a
xi
s
)
.
A
s
sho
w
n i
n
Fi
g
u
r
e 1
2
(
a)
, t
h
e
f
oot
m
o
t
i
on
st
art
e
d
di
f
f
ere
n
t
support phase
l
e
ngt
h a
f
t
e
r c
h
an
gi
n
g
m
ode
fr
om
hexa
p
o
d
m
ode
t
o
q
u
a
d
r
upe
d
m
ode. M
o
reo
v
e
r
fo
r ce
nt
er
re
prese
n
t
e
d
by
L
e
g
5
sam
p
l
e
re
sul
t
s
i
n
Fi
gu
re
12
(
b
)
sh
ows th
at
foot m
o
tio
n
is i
d
en
tically retain
i
n
i
n
itial po
sitio
n
(sit
d
o
wn
m
o
d
e
).
On
t
h
e
o
t
her
h
a
n
d
,
bo
d
y
mass
co
ord
i
n
a
tion
(BMC)
in
Figu
re 13
show
s stab
le lin
e fo
r bo
t
h
w
a
lk
i
n
g m
o
des alth
oug
h in
q
u
a
dr
up
ed
m
o
d
e
t
h
e
p
a
th
of walk
ing
is rev
e
rsing
hex
a
pod
robo
t.
1
3D
m
odel si
m
u
lator
is cour
tesy of
Nonam
i
L
a
b, Chiba Univer
sity
, Japan
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
56
IJR
A
V
o
l
.
3,
No
. 2,
Ju
ne
2
0
1
4
:
13
9 – 1
5
0
14
8
Fig
u
re
1
1
.
3D
m
o
d
e
l si
m
u
lati
o
n
resu
lts for fu
ll wa
l
k
i
n
g
f
r
o
m
hexapo
d m
ode t
o
q
u
a
d
r
u
pe
d m
ode
wi
t
h
pr
o
pose
d
C
L
D
He
xa-
Q
uad
t
r
ansf
o
r
m
a
ti
on,
(
a
) t
r
i
p
o
d
cy
cl
e
1,
(
b
) t
r
i
p
od
cy
cl
e 2,
(c)
C
L
D
t
r
ans
f
o
r
m
a
ti
on,
(
d
)
traverse
cycle 1, (e)
tra
v
erse
cycle 2 and
(f
) trot
cycle.
(a)
(b
)
Fig
u
re
12
.
Po
si
tio
n
o
f
th
e
foo
t
po
in
t
o
n
th
e
z
ax
is: (a) sam
p
le of leg 1, (b) sa
m
p
le o
f
leg
5
.
(a)
(b)
(c)
(d)
(e
)
(f
)
0
50
10
0
15
0
200
250
30
0
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.
4
-1
.
2
-1
-0
.
8
-0
.
6
-0
.
4
-0
.
2
0
Ti
m
e
[s
]
Z-
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f
[
m
]
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is
O
ut[
m
]
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b
r
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e
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u
adr
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d
Mode
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i
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Pha
s
e
S
uppor
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Pha
s
e
0
50
100
150
200
250
300
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.
4
-1
.
2
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-0
.
8
-0
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6
-0
.
4
-0
.
2
0
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m
e
[
s
]
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[
m
]
Z-
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u
t
[
m
]
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b
ra
n
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e
Le
g
di
s
a
b
l
e
d
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x
a
po
d
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e
Quadr
u
ped
Mod
e
Evaluation Warning : The document was created with Spire.PDF for Python.