Intern
ati
o
n
a
l Jo
urn
a
l
o
f
R
o
botics
a
nd Au
tom
a
tion
(I
JR
A)
V
o
l.
2, N
o
. 3
,
Sep
t
em
b
e
r
2013
, pp
. 10
4
~
11
1
I
S
SN
: 208
9-4
8
5
6
1
04
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
Modeling and Simulation of Wa
ve Gait of a Hexapod Walking
Robot: A CAD/
CAE Approach
Abhiji
t Mahapatra
1
,
Shibe
n
du S
h
ekh
a
r
Roy
2
, Dilip
Kuma
r Pratiha
r
3
1
Virtual Prototyping and
Imme
rsive Visua
liz
ation
Lab
.
,
CSIR-CMERI, Durg
apur
2
Departement of
Mechan
ical
Eng
i
neer
ing, Na
tion
a
l Institute of
Technolog
y
,
Durgapur
3
Departem
ent
of
Mechan
ic
al
Eng
i
neer
ing,
I
ndian
Institute
of Te
ch
nolog
y,
Kha
r
agp
u
r
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Dec 25, 2012
Rev
i
sed
May 19
, 20
13
Accepte
d
J
u
n 7, 2013
In the pres
ent pa
per, an a
ttem
p
t h
a
s
been m
a
de to
carr
y
out d
y
nam
i
c ana
l
y
s
is
of a hexapod r
obot using the concep
t of m
u
ltibod
y
d
y
n
a
m
i
cs. A CAD
(Computer Aided Design) model of a r
ealistic hexapod robot has been made
for dy
namic simulation of its lo
comotion using ADAM
S (Automatic
D
y
nam
i
c Anal
ys
is
of M
echanic
al
S
y
s
t
em
s
)
m
u
ltibod
y
d
y
nam
i
cs
s
o
lver. Th
e
kinem
a
ti
c m
odel
for each
leg of
three d
e
gre
e
s
of freedom
has
bee
n
des
i
gned
using CATIA (Computer Aided
Three Di
mensional Interactiv
e
Application)
and S
i
m
D
es
igner packag
e in o
r
der to
dev
e
lop
its overall model, when
it
follows a straigh
t
path
. Th
e var
i
ations of join
t tor
que and
aggregate center o
f
mass of the ro
bot were analyzed fo
r the wave tetr
apod gait. Simulation
results provide the basis for
develop
i
ng th
e contro
l algo
r
ithm and an
intelligen
t d
e
cision making s
y
s
t
em for the robot,
while
in motion
.
Keyword:
Aut
o
m
a
tic Dy
nam
i
c Analy
s
is
C
o
m
put
er Ai
d
e
d
Desi
g
n
Hexa
p
o
d
r
o
bot
Th
ree Dim
e
n
s
io
n
a
l
In
teractive
Wave
gai
t
Copyright ©
201
3 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
:
Shi
b
e
n
du
S
h
ek
har
R
o
y
,
Depa
rtem
ent of Mecha
n
ical E
ngi
neeri
n
g,
Natio
n
a
l
In
stitu
te of Techn
o
l
o
g
y
, Du
rg
apu
r
M
.
G.
Ave
n
ue,
Du
rg
ap
ur
, WB
,
7
1
3
2
0
9
, I
ndi
a
.
Em
a
il: ssro
y
9
9
@yaho
o
.co
m
1.
INTRODUCTION
Walk
ing
robo
t
s
are
v
e
ry co
mp
lex
m
ech
atron
i
cs sy
stem
s,
wh
ere th
ei
r mu
lti-d
e
g
r
ees of freedo
m
leg
s
ar
e co
nn
ected
to
on
e ano
t
h
e
r
th
ro
ugh
th
e trun
k bod
y. M
o
r
e
o
v
e
r
,
a
h
e
x
a
pod
r
obo
t in
m
o
tio
n, at an
y m
o
men
t
,
i
s
a com
p
l
e
x c
o
m
b
i
n
at
i
on
of
ope
n a
n
d cl
ose
d
l
o
o
p
ki
nem
a
ti
c chai
ns
. T
h
e
ro
b
o
t
’
s l
e
gs t
h
at
are i
n
c
ont
a
c
t
wi
t
h
th
e gro
und
for
m
clo
s
ed
loop
s
w
ith
th
e tru
n
k
bo
d
y
and g
r
ou
nd
, wh
ereas no
n-
co
n
t
actin
g
legs r
e
pr
esen
t
in
d
i
v
i
d
u
a
l,
b
r
an
ch
i
n
g
o
p
en
l
o
op
k
i
n
e
m
a
tic
ch
ain
s
. To
desig
n
algorithm
s
fo
r th
e con
t
ro
l o
f
m
u
ltil
eg
g
e
d
wal
k
i
n
g r
o
bot
s
,
i
t
i
s
im
port
a
nt
t
o
have g
o
o
d
m
odel
s
, w
h
i
c
h desc
ri
be t
h
e
dy
nam
i
c behavi
o
r
of t
h
e r
o
b
o
t
[1
]
.
Seve
ral m
e
thods exist
for open and closed
chain sim
u
lati
o
n
of
legg
ed
ro
bo
ts. Th
e algo
r
i
t
h
m
o
f
Ro
dr
iguez an
d
Kre
u
t
z
[
2
]
use
d
l
i
n
ear
o
p
erat
or m
e
t
hod
s t
o
deri
ve si
m
p
l
e
form
s of
dy
nam
i
c equat
i
o
ns.
Y
e
t
i
n
an
ot
he
r m
e
t
h
o
d
,
Lilly an
d
Orin
[3
] treated
a leg
g
e
d
ro
bo
t as a syste
m
o
f
mu
ltip
le
m
a
n
i
p
u
l
ato
r
s (i.e. leg
s
) co
n
t
actin
g
an
o
b
j
ect
(i
.e. t
r
u
n
k
b
o
d
y
)
,
wi
t
h
g
r
o
u
n
d
c
o
nt
act
m
odel
e
d as a
m
a
ni
pul
at
or
j
o
i
n
t
.
H
o
we
ve
r
,
b
o
t
h
t
h
e m
odel
s
ar
e
app
r
oxi
m
a
t
e
m
odel
of
t
h
e c
o
m
p
l
e
x l
e
gged
r
o
b
o
t
.
In
o
r
de
r t
o
d
e
si
g
n
m
o
re
effi
ci
ent
c
ont
r
o
l
al
g
o
ri
t
h
m
fo
r a si
x-
l
e
gge
d r
o
bot
, i
t
i
s
very
m
u
ch essent
i
a
l
t
o
de
vel
o
p m
o
re ac
curat
e
dy
nam
i
c
m
odel
o
f
t
h
e
real
l
e
gge
d
ro
bot
s
.
I
n
th
is conn
ectio
n, wor
k
of
Song and
W
a
ldr
o
n
[1
], Sh
ih et al
.
[4
], Pf
eif
f
e
r
et
al. [5
], Lin an
d
Son
g
[6
],
K
i
mu
r
a
et
al. [7], and Silva et al. [8-9] are im
port
a
nt
t
o
m
e
nt
i
on. B
a
rret
o
et
al
. [1
0
]
devel
o
ped t
h
e free b
ody
di
agram
m
e
t
hod f
o
r
ki
n
e
m
a
t
i
c
and dy
nam
i
c
m
odel
i
ng of a si
x
-
l
e
g
g
e
d m
achi
n
e. Erden [
1
1]
i
nvest
i
g
at
ed t
h
e dy
n
a
m
i
cs
of a
he
xa
po
d
wal
k
i
n
g
ro
b
o
t
i
n
a l
e
vel
wav
e
gai
t
base
d
o
n
Newt
on
-E
ul
er f
o
rm
ul
at
i
on.
K
oo a
n
d
Yo
o
n
[
1
2]
obt
ai
ne
d a m
a
them
ati
cal
m
odel
for
q
u
ad
ru
p
e
d wal
k
i
n
g r
o
b
o
t to
inv
e
stig
at
e th
e d
y
n
a
m
i
cs after con
s
id
eri
n
g
all
th
e in
ertial effects in
th
e syste
m
. Li et al. [1
3
]
d
e
v
e
lop
e
d
a d
y
n
a
m
i
c
m
o
d
e
l an
d ob
tained
a to
rqu
e
i
n
d
e
x to
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA I
S
SN
:
208
9-4
8
5
6
Mod
e
lin
g
and
S
i
mu
la
tion
o
f
Wa
ve Ga
it
o
f
a
H
e
xa
pod
Wa
l
k
in
g
Robo
t:
A CAD
/
CAE…
(Abh
ijit
Ma
hapa
tra
)
10
5
opt
i
m
i
ze t
h
e
con
f
i
g
urat
i
o
n
of t
h
e l
e
gs a
n
d o
p
er
at
i
on
fo
r co
ns
um
i
ng t
h
e m
i
nim
u
m
po
we
r. M
a
ny
m
o
re
anal
y
t
i
cal
dy
nam
i
c
m
odel
s
had bee
n
devel
ope
d
by
t
h
e p
r
evi
o
us res
earc
h
ers a
n
d m
o
st
of t
h
em
were deri
ve
d
base
d o
n
Ne
wt
on
-E
ul
er eq
uat
i
on.
Due t
o
t
h
e
com
p
l
e
xi
t
y
of
a realistic wal
k
ing
ro
bo
t, it i
s
n
o
t
an
easy task
to
in
clu
d
e
th
e i
n
ertial term
s in
t
h
e m
o
d
e
lin
g.
Mo
st of th
e
st
udi
es
o
n
wal
k
i
n
g
dy
nam
i
cs
were
co
n
duct
e
d
wi
t
h
a
sim
p
l
i
f
i
e
d m
o
del
o
f
l
e
gs a
n
d b
o
d
y
as wel
l
as fo
ot
/
g
r
o
un
d i
n
t
e
ract
i
o
n
[
14]
,
[1
5]
. T
h
e
r
ef
ore
,
t
h
e
dev
e
l
ope
d
dy
nam
i
c
m
odel
s
were
fa
r
f
r
om
act
ual
dy
nam
i
c behavi
o
r
o
f
t
h
e sy
st
e
m
. B
u
t
,
i
n
o
r
der
t
o
ha
ve a
bet
t
e
r
u
n
d
e
rstand
ing
o
f
wal
k
ing
,
d
y
n
a
m
i
cs an
d
o
t
her im
p
o
r
tan
t
issu
es
o
f
wal
k
ing
,
su
ch
as d
y
na
m
i
c stab
ilit
y, en
erg
y
effi
ci
ency
a
nd
on
-l
i
n
e c
ont
rol
,
ki
nem
a
t
i
c
and dy
nam
i
c
m
o
del
s
base
d
on a
real
i
s
t
i
c
wal
k
i
ng
r
o
b
o
t
desi
g
n
usi
n
g
the
conce
p
t of virtual pr
ot
otyping
a
r
e necess
a
ry.
Here
, an atte
mpt has
bee
n
m
a
de to
de
velop a m
o
re accurate dynam
i
c
model
of the
he
xapod robot
usi
n
g La
gra
n
g
e
-Eul
e
r
f
o
rm
ul
at
i
on an
d si
m
u
l
a
t
e
t
h
e C
AD
m
odel
of t
h
e re
al
ro
bot
i
n
AD
AM
S s
o
l
v
er
us
i
ng t
h
e
conce
p
t
o
f
ri
gi
d m
u
l
t
i
body
d
y
n
am
i
c
s and a
n
al
y
ze i
t
s
dy
n
a
m
i
c perfo
rm
ance. T
h
e
vari
at
i
ons
of t
h
e ag
g
r
egat
e
Cen
t
er
o
f
Mass (CM) at an
y
in
stan
t
o
f
ti
m
e
and
jo
in
t torqu
e
s
for tetrapod
g
a
it wal
k
ing on
flat terrai
n
h
a
v
e
also been
disc
ussed
.
2.
MODELING OF
THE HEX
A
POD
ROB
O
T
A CAD m
o
d
e
l o
f
t
h
e h
e
x
a
p
o
d
rob
o
t
is
b
u
ilt in
CA
TIA C
A
D/CAE so
ft
ware [1
6
]
as shown
i
n
Figure
1.
The
r
o
b
o
t
t
r
un
k
b
ody
i
s
4
9
5
m
m
i
n
l
e
n
g
t
h
, 2
0
5
m
m
i
n
wi
dt
h a
n
d
89m
m
i
n
hei
g
ht
a
n
d
m
a
de o
f
al
um
ini
u
m
allo
y (d
en
sity= 2
.
7
e
-6 Kg
/mm
3
). Th
e leg
s
are id
en
tical an
d symmetrica
l
l
y
di
st
ri
but
e
d
on
ei
t
h
er
si
de
of
t
h
e
trun
k
bo
d
y
. Each
leg con
s
ists o
f
three link
s
, n
a
m
e
ly l
i
n
k
i
1
(cox
a), link
i2
(fem
u
r) an
d lin
k
i
3
(tib
ia) with
effectiv
e leng
t
h
s 83
.5
mm
, 1
1
9
.34
m
m
,
9
8
.
7
9
m
m
resp
ectiv
ely (i=1
to
6). All th
e jo
in
ts are m
o
to
rized ro
tary
j
o
i
n
ts with
ro
t
a
tio
n
a
l
ax
is co
nfigu
r
ation Z-Y-Y (
i
1
-
i
2
-
i
3
)
for t
h
e t
h
ree
joi
n
ts
respectively as shown
i
n
Fi
gu
re
1.
T
o
t
a
l
n
u
m
b
er
of
D
O
F
o
f
t
h
e
sy
st
em
i
s
2
4
(6
D
O
F
o
f
t
h
e
t
r
un
k
bo
dy
a
n
d
18
DO
F
of
t
h
e
l
e
gs)
.
T
h
e
ro
b
o
t
consi
s
t
s
of 1
9
m
a
i
n
part
s al
on
g wi
t
h
1
8
ser
vom
ot
ors
.
A fl
o
w
-c
hart
sh
o
w
n i
n
Fi
gu
re 2 sh
o
w
s t
h
e
C
AD/
C
A
E a
p
p
r
oac
h
u
nde
rt
ak
en i
n
o
r
der t
o
m
odel
and
si
m
u
l
a
t
e
t
h
e
ro
b
o
t
.
Fi
gu
re 1.
3D
C
AD
m
odel
of
a
he
xa
po
d r
o
b
o
t
Fi
gu
r
e
2
.
Fl
owc
h
a
r
t
f
o
r m
ode
lin
g
a
n
d
s
im
u
l
at
i
o
n
o
f
t
h
e
ADAM
CAT
I
A
SimDesigner
CAT
I
A V5
Assem
b
l
i
ng of
t
h
e
Pa
rt
s
Ru
n Sim
u
latio
n
Defi
ni
n
g
of Joi
n
t
s
a
n
d
C
o
nt
act
s
Export file as
‘.cm
d’
Defi
ne I
n
put
s
Mo
delin
g of
t
h
e
d
i
f
f
e
r
e
n
t
p
a
r
t
s of
t
h
e Robot
Po
st Pro
cessing
o
f
th
e
Sim
u
la
ted
Data
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
56
I
J
RA
Vo
l. 2
,
N
o
. 3
,
Sep
t
emb
e
r
201
3
:
1
04
–
11
1
10
6
The
ki
nem
a
t
i
c
and
dy
nam
i
c param
e
t
e
rs of t
h
e r
o
b
o
t
obt
ai
ne
d
fr
om
C
AD m
odel
a
r
e l
i
s
t
e
d
i
n
Ta
bl
e
1.
Tabl
e
1.
Ki
ne
m
a
t
i
c
and
dy
na
m
i
c param
e
t
e
rs o
f
t
h
e
r
o
b
o
t
Par
a
m
e
ter
s
T
r
unk body
L
i
nk i1
(i=1
to
6
)
L
i
nki 2
(i=1
to
6
)
L
i
nk i3
(i=1
to
6
)
Mass (Kg)
0.
65
0.
15
0.
041
0.
11
Mass Mo
m
e
n
t
o
f
In
ertia (Kg
-
mm
2
)
J
0x
1665
2.
9
70.
827
20.
188
98.
446
J
0y
2518.
5
108.
37
7
86.
698
87.
515
J
0z
1689
7.
2
56.
745
100.
26
5
20.
777
L
e
ngth (m
m
)
495
83.
50
119.
34
98.
79
Total mass of the
six legged rob
o
t (
i
ncluding additional parts) = 3
.
08
7
46
Kg.
In order to de
velop
m
o
re a
c
c
urate dy
na
m
i
c
m
o
del, co
m
p
u
t
er aided
si
m
u
l
a
tion tool
s ba
sed on rigid
m
u
ltib
ody
dy
nam
i
cs called A
D
A
MS h
a
s
been
used. A virtu
a
l
prototy
p
e of the hexap
od robo
t has b
een d
e
veloped
and si
m
u
l
a
ted
in ADA
M
S/So
lver [
17]. The
following assu
m
p
tio
n
s are
made to si
m
p
lify
the rigid
m
u
lt
ibody
dy
na
m
i
c analy
s
is of robot.
(a)
The
robot
m
oves forward in a
straight
path
on
flat surface with
wa
ve tetrapod gait.
(b
) T
h
e t
r
u
n
k
b
ody
i
s
hel
d
at
a
co
nst
a
nt
hei
g
h
t
and
pa
ral
l
e
l
t
o
t
h
e
g
r
ou
n
d
pl
ane
du
ri
n
g
l
o
c
o
m
o
t
i
on.
(
c
)
Sw
i
n
g leg
s
ar
e co
n
s
i
d
er
ed
n
o
t
t
o
cr
o
s
s th
e sup
por
tin
g legs so th
at
f
o
r
t
h
c
o
m
in
g
supp
or
t
p
o
l
ygon
is conv
ex.
In
th
e
p
r
esen
t
work, fo
llowing
two
asp
ects
with
resp
ect to d
y
n
a
m
i
c stab
i
lity o
f
tetrap
od g
a
it o
f
th
e
robo
t ov
er p
e
rfectly flat terrain
have
bee
n
c
h
ecked
f
o
r
,
nam
e
l
y
,
a. Variatio
n of
to
rq
u
e
w
ith stroke
at eac
h ste
p
b. Va
ri
at
i
on of
t
o
r
que
wi
t
h
cycle tim
e
at eac
h ste
p
The i
n
put
m
o
t
i
ons
fo
r eac
h o
f
t
h
e rot
a
ry
joi
n
t
s
are defi
ned t
h
r
o
ug
h
St
e
p
M
a
t
h
F
unct
i
on
i
n
A
DAM
S.
As an
ex
am
p
l
e
,
th
e
v
e
lo
city in
pu
t m
o
tio
n
of leg
2 h
a
v
i
ng
strok
e
=0.14
m
a
n
d cycle ti
m
e
=2
.4
s is as shown
i
n
Fi
gu
re
3.
Fo
r a
l
l
t
y
pes of si
m
u
l
a
t
i
ons
, m
a
xi
m
u
m
vari
at
i
on of
j
o
i
n
t
a
n
gl
e
00
0
12
20
,
0
to
6
&
ii
jo
in
t ang
l
e
3
i
i
s
kept
co
nst
a
nt
at
0
90
. Im
pact
-base
d
co
nt
act
param
e
t
e
rs have bee
n
de
fi
ne
d bet
w
ee
n t
h
e
l
e
gs an
d t
h
e
g
r
ou
nd
t
o
m
a
k
e
th
e sim
u
latio
n
m
o
re realistic (Fi
g
ure
4
)
.
Th
e sim
u
latio
n is ev
al
u
a
ted for
1
0
s
and 600
t
i
m
e
step
s to st
udy at
m
o
st three com
p
lete cycles. In t
h
e
si
m
u
latio
n
,
each
ti
m
e
step
represen
ts an
in
teg
r
ation
st
ep
in wh
ich
th
e
n
e
w po
sition
s
,
o
r
ien
t
atio
n
s
,
v
e
l
o
cities
and accelerations
of the
robot’s
body
pa
rts are com
puted
ba
sed
on the
forc
es acting on t
h
e
m
.
(a)
(b
)
Fi
gu
re
3.
Vel
o
ci
t
y
St
ep Fu
nct
i
ons
f
o
r
st
r
oke
= 0.
1
4
m
,
Cycle ti
me=2
.4
s
for t
h
e tetrap
od
g
a
it (a)
21
L
(b
)
22
L
Tim
e
(
s
)
Angu
lar v
e
lo
ci
ty
(ra
d/s)
Angu
lar v
e
lo
ci
ty
(ra
d/s)
Tim
e
(
s
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA I
S
SN
:
208
9-4
8
5
6
Mod
e
lin
g
and
S
i
mu
la
tion
o
f
Wa
ve Ga
it
o
f
a
H
e
xa
pod
Wa
l
k
in
g
Robo
t:
A CAD
/
CAE…
(Abh
ijit
Ma
hapa
tra
)
10
7
Fi
gu
re
4.
C
o
nt
act
param
e
t
e
rs use
d
du
ri
n
g
si
m
u
l
a
t
i
o
n
3. RES
U
LTS AN
D DIS
C
US
SION
Sim
u
l
a
t
i
on o
f
t
h
e si
x
l
e
g
g
ed
ro
b
o
t
has
bee
n
do
ne t
o
inv
e
stig
ate th
e
g
e
neratio
n of statically stab
le
g
a
it p
a
ttern
s
fo
r
v
a
riou
s stro
k
e
and
cycle ti
m
e
o
f
th
e
walk
ing
robo
t. In th
is section
,
t
h
e resu
lts
o
f
t
h
e
agg
r
e
g
at
e C
M
and t
o
r
que
-di
s
t
r
i
b
ut
i
on i
n
t
h
e
di
ffe
re
nt
joi
n
ts o
f
th
e lin
k
s
wh
ile th
e rob
o
t
is in
m
o
tio
n
on
a flat
terrain a
r
e
disc
usse
d.
3.
1.
A
ggre
g
ate
Cen
t
er
of
M
a
ss o
f
the w
a
l
k
i
n
g r
o
b
o
t
To
determ
in
e t
h
e po
sition
of th
e aggreg
ate C
M
o
f
th
e
walk
i
n
g
robo
t at an
y in
stan
t of ti
m
e
in
o
r
d
e
r t
o
ch
eck its
d
y
n
a
mic stab
ility i
s
a
v
e
ry
essential p
a
rt
o
f
our inv
e
stig
ation
.
Th
e
v
a
riation of agg
r
eg
ate
CM is
in
v
e
stig
ated
in ADAMS u
s
ing
a u
s
er
written
sub
r
ou
tin
e.
Th
e u
s
er written
sub
r
ou
tin
e cen
tro
i
d
.cm
d
file is
im
port
e
d
t
o
t
h
e
A
DAM
S
w
o
r
kbe
nc
h a
n
d
t
h
e
si
m
u
l
a
ti
on i
s
r
u
n
.
Fo
r
v
i
su
alizatio
n
o
f
th
e
v
a
ri
atio
n
of th
e ag
greg
ate CM
at an
y in
stan
t o
f
ti
m
e
, th
e u
s
er
written
sub
r
outine A
G
G
_CM
_
ST
AT
E_
VARI
A
BL
E_IM
PORT.cm
d
file
is i
m
p
o
rted
. Fi
gure 5 shows that there is no
suc
h
off
bit pat
h
variation
of the CM a
n
d the
trace is m
o
re
or less a st
raight line.
Thus, t
h
e system
is stable.
3.2. Joint
Tor
ques for
Tetr
apod
Gait Wal
k
ing on
Flat
T
errain
Dyn
a
m
i
c si
m
u
latio
n
o
f
t
h
e h
e
x
a
pod
is do
n
e
to
stu
d
y
th
e torq
u
e
requ
irem
en
t in
th
e jo
in
ts
o
f
th
e leg
s
.
Th
e
o
b
j
ectiv
e
is to
m
i
n
i
mize
th
e torqu
e
i
n
th
e jo
in
ts so
as to
m
a
k
e
th
e
d
r
i
v
e m
o
re easily co
n
t
ro
llab
l
e. The
to
rq
u
e
s at all t
h
e jo
in
ts are calcu
l
ated
b
y
varyin
g
th
e
g
a
it p
a
ram
e
ters. Th
e to
rqu
e
req
u
ired
at th
e jo
ints, to
achi
e
ve sy
st
e
m
m
o
t
i
on are sim
u
l
a
t
e
d i
n
A
DAM
S e
n
vi
ro
nm
ent
based o
n
t
h
e vi
rt
ual
m
odel
i
n
Fi
gu
re 1 an
d
dy
nam
i
c t
o
rq
u
e
eq
uat
i
o
ns as
m
e
nt
i
oned i
n
APPE
N
D
I
X
A. Th
e tetrap
od g
a
it is co
m
p
osed
o
f
th
e fo
llo
wi
n
g
seq
u
ences:
i
)
l
e
gs 1
-
6 ret
r
act
i
n
g
,
l
e
gs 2
-
3
pr
ot
ract
i
n
g an
d l
e
gs 4
-
5 i
n
t
h
e
m
i
ddl
e;
i
i
)
l
e
gs 1-6
pr
ot
ract
i
n
g, l
e
g
s
2
-
3
i
n
th
e m
i
d
d
l
e and
legs 4
-
5
retracting
iii) leg
s
1-
6 in
th
e m
i
d
d
l
e, leg
s
2-3 retractin
g
and
l
e
g
s
4
-
5
p
r
o
t
racting
.
The walk
is rep
e
ated
for to
rqu
e
-d
istribu
tio
n
cal
cu
latio
n
s
. Figu
re
6
sho
w
s the to
rqu
e
d
i
stri
bu
tio
n
i
n
t
h
e joi
n
t
s
of
al
l t
h
e l
i
nks for t
w
o cy
cl
es (Dut
y
fact
o
r
= 2
/
3, St
ro
ke=
0
.
1
4m
, Cy
cl
e
t
i
me= 2.4s
). I
n
Fi
gu
re 6
,
o
n
e
can
easily o
b
s
erve th
at fo
r all th
e leg
s
, th
e j
o
i
n
t to
rqu
e
s in
ret
r
action
tak
e
sign
ifican
t v
a
lu
es, while th
e
j
o
i
n
t torqu
e
ses in
p
r
o
t
raction
are
v
e
ry less.
Ag
ai
n
,
t
h
e to
rqu
e
in jo
in
t
1 see
m
s to
b
e
h
i
g
h
er th
an
t
h
e torq
u
e
in
othe
r joi
n
ts. Si
nce these are consi
d
ere
d
to be propor
tion
a
l to
th
e av
erag
e d
i
ssip
a
ted
p
o
wer o
n
th
e m
o
to
rs, the
m
o
t
o
r
sel
ect
i
on has
t
o
be bas
e
d on
t
h
e
m
a
xi
m
u
m
t
o
rq
ue o
b
t
ai
ned fr
om
t
h
e
si
m
u
l
a
ti
on.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
56
I
J
RA
Vo
l. 2
,
N
o
. 3
,
Sep
t
emb
e
r
201
3
:
1
04
–
11
1
10
8
Fi
gu
re
5.
A
g
gr
egat
e C
M
va
ri
at
i
on
of
t
h
e
he
xap
o
d
ro
b
o
t
d
u
r
i
n
g si
m
u
l
a
t
i
on
Fi
gu
re
6.
Va
ri
at
i
on
of
j
o
i
n
t
t
o
r
que
s f
o
r t
e
t
r
ap
od
gai
t
wal
k
i
n
g
on
fl
at
t
e
r
r
ai
n
(D
ut
y
fact
o
r
= 2/
3
,
St
ro
ke=
0.
14m
,
Trace
m
a
rk of
aggregate CM
Aggregate CM
location
Z
X
Y
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA I
S
SN
:
208
9-4
8
5
6
Mod
e
lin
g
and
S
i
mu
la
tion
o
f
Wa
ve Ga
it
o
f
a
H
e
xa
pod
Wa
l
k
in
g
Robo
t:
A CAD
/
CAE…
(Abh
ijit
Ma
hapa
tra
)
10
9
Cycle time= 2.4s)
4. CO
N
C
L
U
S
I
ON
A 3
D
vi
rt
ual
pr
ot
ot
y
p
e
of t
h
e r
o
bot
sy
st
e
m
i
s
creat
ed i
n
C
A
T
I
A V
5
wo
rk
be
nc
h an
d ex
p
o
rt
e
d
t
o
ADAMS using
CATIA SimDesign
e
r an
d
si
m
u
lated
in
real
time. The variations of aggre
g
ate CM
of the
sy
st
em
and t
o
r
que
s at
di
ffe
re
nt
joi
n
t
s
ha
ve
been st
udi
e
d
.
The res
u
l
t
s
sh
ow t
h
at
t
h
e t
o
rq
ue re
qui
red
du
ri
n
g
retraction phas
e is m
u
ch higher tha
n
t
h
at during protrac
tion
p
h
a
se fo
r all th
e
j
o
i
n
ts.
It is ob
v
i
o
u
s th
at
d
u
ri
ng
the ret
r
action
phase, legs
are
c
a
rrying t
h
e
weight
of t
h
e
trunk
b
o
d
y
, p
a
y
load
. Fo
r m
o
st
of th
e cases, torq
u
e
in
j
o
i
n
t 1
is
f
ound
to
b
e
h
i
gh
er th
an
t
h
at o
f
th
e o
t
h
e
r
jo
in
ts. So, th
e m
o
tor
h
a
s t
o
b
e
selected
b
a
sed
on
th
e
max
i
m
u
m
to
rqu
e
. Th
is work
will allo
w fu
rt
h
e
r b
e
n
c
h
m
ark
i
n
g
of th
e m
e
c
h
an
ical ev
en
t si
m
u
latio
n
o
f
the six
legge
d robot s
y
ste
m
, such as
traject
ory
planni
ng,
ki
nem
a
tic workspace
constraints a
n
d c
o
ordi
nation issue
s
with
o
t
h
e
r syste
m
referen
ces etc. Fu
tu
re
wo
rk
will
fo
cu
s o
n
th
e
d
e
termin
atio
n
o
f
relatio
n
s
h
i
p
s
of en
erg
y
efficien
cy with
d
i
fferen
t g
a
it
p
a
tte
r
n
s fo
r d
i
ff
er
en
t w
a
lk
ing
sp
eed
s
.
Appe
ndix A:
Form
ul
at
i
on of
M
a
t
h
em
ati
cal
M
odel
of
t
h
e D
y
nam
i
c
Sy
st
em
The La
g
r
an
ge
’
s
eq
uat
i
o
n
of
m
o
ti
on
fo
r
unc
onst
r
ai
ne
d sy
st
em
i
n
a m
a
t
r
i
x
-vect
o
r
f
o
rm
i
s
:
()
(
,
)
Mq
q
h
q
q
Q
nc
,
(
1
)
Whe
r
e
24
24
MR
is th
e m
a
ss m
a
trix
o
f
t
h
e
robo
tic syste
m
,
24
hR
i
s
t
h
e f
o
rce
vec
t
or c
ont
ai
ni
ng
vel
o
ci
t
y
depe
n
d
ent
fo
rc
es
an
d gra
v
i
t
a
t
i
onal
f
o
rces,
24
nc
QR
i
s
t
h
e
no
n-c
o
ns
ervat
i
v
e
f
o
rce/
t
o
r
que
vect
or
appl
i
e
d
t
o
t
h
e r
o
bot
.
Due
t
o
fo
ot
c
ont
act
wi
t
h
t
h
e gr
o
u
n
d
fo
r
a co
nst
r
ai
ne
d
si
x-l
e
g
g
e
d
r
o
b
o
t
i
c
sy
st
em
, o
n
e m
a
y
use
Lag
r
ang
e
M
u
ltip
liers and
write in
th
e m
a
trix
-v
ector form
as fo
llows:
()
(
,
)
λ
T
nc
q
Mq
q
h
q
q
Φ
Q
,
(
2
)
Whe
r
e
λ
is th
e v
ector
o
f
Lag
r
ang
e
m
u
ltip
l
i
ers,
wh
ich
is
id
en
tical to
t
h
e terrain
reactio
n
forces
of th
e
su
ppo
r
ting
f
eet
, an
d
32
4
g
n
q
Φ
R
is th
e con
s
train
t
Jacob
i
an
m
a
trix
,
g
n
is th
e nu
m
b
er
of
f
e
et g
r
ou
nd
ed
.
The accele
r
ation c
o
nstraint e
q
uation can be
written as follows:
(,
)
q
Φ
q
γ
qq
0
,
(3
)
Whe
r
e
3
g
n
γ
R
is the
velocity
depende
nt acceleration
vector
c
ontaini
ng the
Centripetal a
n
d Corioli
s
accelerations.
The c
o
nstraine
d e
quations
of m
o
t
i
on of
the whole
system
can
be written in
vect
or-m
atri
x
form
as fo
llo
ws:
T
nc
q
q
M
Φ
q
Qh
Φ
0
λγ
(4)
Th
e co
llid
ing
p
h
e
no
m
e
n
a
will o
ccur,
wh
en th
e
foo
t
stri
kes th
e
g
r
ou
nd
. Assu
m
i
n
g
th
at th
e id
eally
p
l
astic i
m
p
act o
ccurs
b
e
t
w
een
th
e co
llid
in
g fo
o
t
and
t
h
e
g
r
ou
nd
surface in
stan
tan
e
ou
sl
y, th
e co
llision
h
a
s
been m
odel
e
d
as t
h
e occ
u
r
r
en
ce of i
n
st
a
n
t
a
n
e
ou
s vel
o
ci
t
y
change.
Here
, the ideally plastic im
pact
mea
n
s tha
t
th
e tip
v
e
l
o
city o
f
th
e co
llidin
g
foo
t
h
a
s been
v
a
n
i
sh
ed
rig
h
t
after th
e
foo
t
stri
k
e
. Then
fro
m
Lag
r
an
g
e
’s
im
pul
se equat
i
on f
o
r a ki
ne
m
a
t
i
call
y
cons
t
r
ai
ned sy
st
em
wi
t
h
t
h
e assu
m
p
ti
on of i
d
e
a
l
l
y
pl
ast
i
c
i
nput
, a
d
i
fferen
tial-algeb
r
aic equ
a
tio
n is ob
tain
ed
in
th
e fo
llowing
fo
rm
:
gT
c
T
qq
gg
q
cc
c
q
q
0
00
0
00
M
ΦΦ
Φ
v
Φ
,
(5
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
56
I
J
RA
Vo
l. 2
,
N
o
. 3
,
Sep
t
emb
e
r
201
3
:
1
04
–
11
1
11
0
Whe
r
e n
c
is t
h
e nu
m
b
er of collid
in
g
feet h
a
vin
g
v
e
lo
city
v
e
cto
r
c
3n
c
vR
,
g
3n
2
4
g
q
Φ
R
and
g
3n
g
R
are
the Jacobian
matrices and t
h
e im
pact forc
e vectors
of
t
h
e origi
n
ally contacted
feet respectively, wherea
s
c
3n
2
4
c
q
Φ
R
and
c
3n
c
R
are the Ja
cobia
n
m
a
trice
s
and the
im
p
act
force
vectors
of
th
e co
ll
id
in
g feet
,
resp
ectiv
ely. Th
e in
stan
tan
e
ou
s ch
ang
e
o
f
velo
city d
u
e
to
th
e co
llisio
n
o
f
feet is d
e
termi
n
ed
as
qq
q
,
wh
ere t
h
e su
p
e
rscri
p
ts – and
+ rep
r
esen
t th
e qu
an
tities
ri
g
h
t b
e
fo
re and
aft
e
r th
e co
llision, resp
ectiv
ely.
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y
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etric six-legged
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y
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y
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al
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BIOGRAP
HI
ES
OF AUTH
ORS
Abhijit Mah
a
pat
r
a re
ceiv
e
d his
B.E and
M.Te
ch
degrees
in Mec
h
anic
al Eng
i
ne
e
r
ing from
B.E.
College (presen
tly
, BESU), Shib
pur, Howrah, In
dia
and NIT Dur
g
apur, Ind
i
a
in 2
002 and 2008
,
respectively
.
Cu
rrently
, he is working as a Sc
ientist
in Virtu
a
l Prototy
p
ing an
d Immersive
Vis
u
aliz
ation
La
b., CS
IR- Cen
t
r
a
l M
ech
ani
cal
E
ngineer
ing Res
e
arch Ins
t
i
t
ut
e, D
u
rgapur, Ind
i
a
.
He has publish
e
d number of res
earch
pap
e
rs in
nation
a
l
and in
ternational journ
a
ls,
confer
ence
proceed
ings and
fil
e
d num
ber o
f
pat
e
nts in
ar
ea
of produc
t dev
e
lopm
ent. His c
u
rrent r
e
sear
ch
inter
e
sts include
design & anal
ysis, m
u
lti-bod
y
d
y
nam
i
c
s, m
odeling and sim
u
lat
i
on of legge
d
robots.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA I
S
SN
:
208
9-4
8
5
6
Mod
e
lin
g
and
S
i
mu
la
tion
o
f
Wa
ve Ga
it
o
f
a
H
e
xa
pod
Wa
l
k
in
g
Robo
t:
A CAD
/
CAE…
(Abh
ijit
Ma
hapa
tra
)
11
1
Shibendu Shekhar Ro
y
receiv
ed
his B.E and
M.T
ech d
e
grees in
Mechanical En
gineer
ing from
R.E. Co
lleg
e
(p
resently
, NIT D
u
rgapur), Durgapur, India in 19
99 and 2001, r
e
spectiv
ely
.
He
obtain
e
d his Ph.D from IIT Khar
agpur, India, in 2011.
Currently
,
He is an
Assista
n
t Professor in
the Depar
t
m
e
nt
of Mechani
cal
E
ngineer
ing, Nat
i
onal Institu
te of
Techno
log
y
, D
u
rgapur, Indi
a.
From
March 2001 to Decem
ber 2006 he was a Scie
ntist a
t
the CSIR-C
entr
al Mechan
ic
al
Engineering R
e
search Institute, Durgapur, Ind
i
a.
He has published a great
deal of r
e
sear
ch
papers at nation
al and internatio
nal journals
, co
nference proceedings, book chapters and filed
num
ber of pat
e
n
t
s in product dev
e
lopm
ent. He
has written
a book
on “Modeling and anal
y
s
is o
f
six-legged robots”, which has been published b
y
Lap Lambert
Academic Publishing Gmb
H
&
Co. KG, Germ
an
y. His
current
res
earch
inter
e
sts include modelling a
nd simulation of legged
robots and o
t
her
robotic s
y
stems.
Dilip Kum
a
r Pratihar (B
.E
. (Hon
s.), M.T
ech
., Ph
.D.) receiv
ed his
Ph.D. from
IIT
Kanpur, Indi
a,
in the
y
ear
2000. He receiv
ed th
e University
Go
ld
Medal
in 1988,
A.M.
Das Memorial Med
a
l
in
1987, Institu
tion
of Engin
eers’ (
I) Medal
in 200
2, and o
t
hers. H
e
com
p
let
e
d his
post-doctoral
studies in Japan (6 months)
and German
y
(
1
y
e
ar)
under the
Alexand
e
r von
Humbold
t
Fellowship Programme. He is
working at pr
esent as
a Professor in the Depar
t
ment of Mechan
ical
Engineering, II
T Kharagpur, I
ndia. His resear
ch
areas includ
e robotics
,
soft computing and
manufacturing
scien
c
e. He has
published
more th
an 150
pap
e
rs in d
i
ffer
e
nt journals and
conferen
ce pro
c
eedings. He h
a
s authored
a tex
t
b
ook on ‘‘Soft Computing’’, co-au
t
hored anoth
e
r
textbook
on “Analy
tical Eng
i
neer
ing Mech
anics”
and two o
t
her
reference books.
He has ed
ited
a
book on ‘‘Intelligent
and Auton
o
m
ous Sy
stems’’, which was
in
2010 published
b
y
Spring
er-
Verlag,
Germ
any.
He has
b
een
includ
ed as
a
m
e
m
b
er of the
program
com
m
i
tte
e for s
e
ver
a
l
intern
ation
a
l
co
nferenc
e
s. He h
a
s been se
lec
t
e
d
as the
Editor
i
al Bo
ard Mem
b
er of th
irte
en
intern
ation
a
l jou
r
nals. He has be
en el
ect
ed as
a
Fellow of the In
stitution of
Engi
neers (I) and
M
e
m
b
er of IE
E
E
.
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