I
n
t
e
r
n
at
i
on
al
Jou
r
n
al
of
R
o
bo
t
i
c
s
a
nd A
ut
o
m
a
t
i
o
n
(
I
J
RA
)
V
o
l.
4
,
No
.
1
,
M
ar
ch
2
01
5
,
p
p.
19
~
30
I
SSN
:
2089
-
4856
19
Jou
r
n
al
h
om
e
p
age
:
ht
t
p:
/
/
i
ae
s
j
our
nal
.
c
om
/
onl
in
e
/in
d
e
x
.
p
h
p
/I
J
R
A
O
nline
B
iped
W
a
l
k
ing
P
a
t
t
er
n
G
ene
ra
t
io
n w
it
h
C
o
nt
a
ct
C
o
ns
is
t
ency
H
o
u
Wenqi
*
,
J
ia
n W
a
ng
*
,
J
i
a
nw
e
n W
a
ng
*
a
nd H
o
ng
x
u M
a
*
∗
C
ol
l
e
g
e
of
M
e
c
ha
t
r
oni
c
E
ng
i
ne
e
r
i
ng
a
nd
A
ut
om
a
t
i
on,
N
a
t
i
ona
l
U
ni
v
e
r
s
i
t
y
o
f
D
e
f
e
ns
e
T
e
c
hnol
og
y
A
rt
i
cl
e I
n
f
o
AB
S
T
RAC
T
A
r
tic
le
h
is
to
r
y
:
R
ecei
v
ed
Apr 24
,
201
4
R
e
v
i
s
e
d
Oct
20,
201
4
A
ccep
t
ed
Nov
1
6
,
201
4
I
n
t
hi
s
pa
pe
r
,
a
n
ov
e
l
on
l
i
ne
bi
p
e
d
w
a
l
k
i
ng
g
a
i
t
pa
t
t
e
r
n
g
e
ne
r
a
t
i
ng
m
e
t
hod
w
i
t
h
co
n
t
act
co
n
s
i
s
t
en
cy
i
s
p
r
o
p
o
s
ed
.
G
en
er
al
l
y
,
i
t
’
s
d
es
i
r
ab
l
e
t
h
at
t
h
er
e
i
s
n
o
f
oot
-
g
r
oun
d
s
l
i
ppi
ng
dur
i
ng
bi
pe
d
w
a
l
k
i
ng
.
B
y
t
r
e
a
t
i
ng
t
he
hi
p
of
t
he
bi
pe
d
r
o
b
o
t
as
a
l
i
n
ear
i
n
v
er
t
e
d
pe
nd
ul
um
(
L
I
P
)
,
a
f
oot
pl
a
c
e
m
e
nt
c
ont
r
ol
l
e
r
t
ha
t
t
a
k
e
s
t
he
c
ont
a
c
t
c
ons
i
s
t
e
nc
y
i
nt
o
a
c
c
ount
i
s
pr
o
pos
e
d
t
o
t
r
a
c
k
i
ng
t
he
de
s
i
r
e
d
o
r
b
i
t
en
er
g
y
.
B
y
s
el
ect
i
n
g
t
h
e
h
i
p
’
s
h
o
r
i
zo
n
t
al
l
o
co
m
o
t
i
o
n
as
t
h
e
p
ar
am
et
er
,
t
h
e
t
r
aj
ect
o
r
i
es
i
n
t
as
k
s
p
ace
f
o
r
w
al
k
i
n
g
ar
e
pl
a
nne
d.
A
t
a
s
k
s
pa
c
e
c
ont
r
ol
l
e
r
w
ith
o
u
t
c
a
lc
u
la
ti
n
g
t
h
e
i
n
v
e
r
s
io
n
o
f
in
e
r
tia
l
m
a
tr
ix
is
p
r
e
s
e
n
te
d
.
S
im
u
la
tio
n
ex
p
er
i
m
en
t
s
ar
e
i
m
p
l
e
m
en
t
ed
o
n
a v
i
r
t
u
al
5
-
l
i
nk
p
oi
nt
f
oot
b
i
pe
d
r
obo
t
.
T
he
r
es
u
l
t
s
s
h
o
w
t
h
e
ef
f
ect
i
v
en
es
s
o
f
t
h
e
w
al
k
i
n
g
p
at
t
er
n
g
en
er
at
i
n
g
m
et
h
od
w
h
i
ch
can
r
eal
i
ze
a s
t
ab
l
e
p
er
i
o
d
i
c
g
ai
t
cy
cl
e
w
i
t
h
o
u
t
s
l
i
p
p
i
n
g
a
n
d
f
al
l
i
n
g
e
v
e
n
s
uf
f
e
r
i
ng
a
s
udde
n
d
i
s
t
ur
ba
n
c
e
.
Ke
y
wo
rd
:
G
a
it
p
a
tte
r
n
g
e
n
e
r
a
tio
n
C
o
n
ta
c
t
c
o
n
s
is
te
n
c
y
B
i
pe
d
r
obot
P
oi
n
t
f
oot
O
r
b
it
e
n
e
r
g
y
T
as
k
s
p
ace
co
n
t
r
o
l
l
er
C
opy
r
i
g
ht
©
201
5
I
ns
t
i
t
ut
e
of
A
dv
anc
e
d E
ngi
ne
e
r
i
ng
a
nd
Sc
i
e
nc
e
.
A
l
l
ri
g
h
t
s re
se
rv
e
d
.
Co
rre
sp
o
n
d
i
n
g
Au
t
h
o
r
:
W
e
n
qi
H
ou
C
o
l
l
e
ge
o
f
M
e
c
ha
t
r
o
ni
c
E
n
gi
n
e
e
r
i
ng
a
nd
A
ut
o
m
a
t
i
o
n,
N
at
i
o
n
al
U
n
i
v
er
s
i
t
y
o
f
D
e
f
en
s
e
T
ech
n
o
l
o
g
y
D
e
ya
Ro
a
d
109#
,
C
ha
ng
s
ha
,
C
hi
na
.
E
m
a
il:
h
ouw
nq@
126.
c
o
m
1.
IN
TR
O
D
UCT
I
O
N
B
i
pe
d
r
o
bot
s
h
a
v
e
a
r
ou
s
e
d
m
u
c
h
a
t
t
e
n
t
i
on
ov
e
r
t
h
e
pa
s
t
de
c
a
de
s
.
B
u
t
u
p
t
o
n
o
w
,
a
bi
pe
d
w
a
l
k
i
ng
r
o
b
o
t
t
h
at
p
er
f
o
r
m
s
as
w
el
l
as
h
u
m
an
h
as
n
o
t
y
et
b
ee
n
m
ad
e.
T
o
s
o
m
e
ex
t
e
n
t
,
i
t
i
n
d
i
cat
es
t
h
at
t
h
e
f
u
n
d
a
m
e
n
ta
l
p
r
in
c
ip
le
s
o
f
b
ip
e
d
w
a
lk
i
n
g
a
r
e
s
till
n
o
t
f
ul
l
y
und
e
r
s
t
o
o
d
[
1
]
.
G
e
ne
r
a
l
l
y
,
i
n
o
r
d
e
r
t
o
f
o
c
us
o
n
t
he
d
y
n
a
m
i
cal
c
h
ar
act
er
s
o
f
b
i
p
ed
w
al
k
i
n
g
,
i
t
i
s
as
s
u
m
ed
t
h
at
t
h
e
l
eg
s
ar
e
t
er
m
i
n
at
ed
i
n
p
o
i
n
t
s
a
nd
no
a
c
t
ua
t
i
o
n
i
s
co
n
s
i
d
er
ed
at
t
h
e
s
t
an
ce
f
o
o
t
[
2
-
4
]
.
A
s
a r
es
u
l
t
,
t
h
e
Z
er
o
-
M
o
m
e
n
t
P
o
in
t(
Z
M
P
)
c
r
ite
r
io
n
c
a
n
’
t
b
e
u
s
e
d
in
t
h
e
s
ynt
he
s
i
s
o
f
w
a
l
ki
n
g
p
a
t
t
e
r
n
[
5
]
.
A
ddi
t
i
on
a
l
l
y
,
s
l
i
ppi
n
g
be
t
w
e
e
n
t
h
e
s
t
a
n
c
e
f
oot
a
n
d
g
r
ou
n
d
i
s
n
ot
e
x
pe
c
t
e
d,
w
h
i
c
h
m
e
a
n
s
t
ha
t
t
he
c
o
nt
a
c
t
c
o
ns
i
s
t
e
nc
y
s
ho
ul
d
b
e
p
r
e
s
e
r
ve
d
.
T
he
c
o
up
l
i
ng
b
e
t
w
e
e
n
t
he
a
c
t
ua
t
e
d
t
o
r
q
ue
s
an
d
t
h
e
co
n
t
act
f
o
r
ces
m
ad
e
t
h
e
co
n
t
r
o
l
p
r
o
b
l
em
a
g
r
eat
ch
al
l
en
g
e
f
o
r
r
es
ear
ch
er
s
.
D
i
f
f
er
en
t
ap
p
r
o
ach
es
ar
e
p
r
o
p
o
s
ed
i
n
l
i
t
er
at
u
r
es
t
o
h
an
d
l
e
b
i
p
ed
w
al
k
i
n
g
co
n
t
r
o
l
.
B
y
a
s
s
u
m
i
n
g
t
h
e
f
oot
-
gr
o
und
i
m
p
a
c
t
i
s
r
i
gi
d
a
nd
no
s
l
i
p
p
i
ng
o
r
r
e
b
o
und
s
o
c
c
ur
s
d
ur
i
ng
w
a
l
k
i
n
g
,
J
.
W
.
G
r
i
zzl
e
an
d
E
.
R
.
W
e
s
te
r
v
e
lt
e
t
a
l.
p
r
o
v
e
d
th
e
e
x
is
te
n
c
e
a
n
d
s
ta
b
ilit
y
o
f
p
e
r
io
d
ic
o
r
b
it
f
o
r
b
ip
e
d
w
a
lk
i
n
g
o
n
a
f
iv
e
-
l
i
n
k,
p
l
a
na
r
pr
ot
ot
y
pe
R
A
B
B
I
T
[
6
,
7
]
.
S
e
n
t
i
s
e
t
a
l
.
pr
opos
e
d
a
n
e
x
t
e
n
s
i
on
of
ope
r
a
t
i
on
a
l
s
pa
c
e
c
ont
r
ol
l
e
r
s
f
or
f
l
oa
t
i
ng
-
ba
s
e
d
r
obot
s
,
by
pr
oj
e
c
t
i
n
g
t
h
e
ope
r
a
t
i
on
a
l
t
a
s
k
s
in
to
t
h
e
c
o
n
s
tr
a
i
n
t
n
u
ll
s
p
a
c
e
,
th
e
v
io
la
tio
n
s
b
e
t
w
e
e
n
d
if
f
e
r
e
n
t
ta
s
k
s
a
r
e
a
v
o
id
e
d
[
8
]
.
H
o
w
e
ve
r
,
t
hi
s
a
p
p
r
o
a
c
h
r
e
q
ui
r
e
s
t
he
i
nve
r
s
i
o
n
o
f
t
he
s
ys
t
e
m
i
n
e
r
tia
m
a
tr
i
x
i
n
th
e
c
o
n
tr
o
lle
r
,
w
h
ic
h
m
a
k
e
s
it s
e
n
s
iti
v
e
to
m
o
d
e
lin
g
a
n
d
p
a
r
a
m
e
te
r
e
s
ti
m
a
tio
n
e
r
r
o
r
s
[
9
]
.
B
as
ed
o
n
t
h
e
or
t
h
og
on
a
l
de
c
o
m
pos
i
t
i
on
of
t
h
e
c
ons
t
r
a
i
nt
J
a
c
obi
a
n
,
M
i
ch
ael
M
i
s
t
r
y
et
al
.
p
r
o
p
o
s
ed
an
i
n
v
er
s
e
d
y
n
a
m
i
c
s
al
g
o
r
i
t
h
m
f
o
r
l
eg
g
ed
r
o
b
o
t
s
,
w
h
i
ch
e
x
p
r
es
s
e
s
t
h
e
i
n
v
er
s
e
d
y
n
a
m
i
cs
co
n
t
r
o
l
l
er
i
n
d
ep
en
d
en
t
l
y
o
f
co
n
t
act
f
o
r
ce
s
an
d
o
n
l
y
r
eq
u
i
r
es
t
h
e
k
i
n
e
m
at
i
c
t
er
m
s
[
10
]
.
R
ecen
t
l
y
,
S
a
l
m
a
n
F
ar
aj
i
et
al
.
u
s
ed
s
u
ch
a
n
ap
p
r
o
ach
t
o
co
n
t
r
o
l
a
p
l
an
ar
m
on
opod
h
oppe
r
i
n
r
ou
g
h
t
e
r
r
a
i
n
[
11
]
.
Evaluation Warning : The document was created with Spire.PDF for Python.
20
I
SSN
:
20
89
-
4856
IJ
RA
V
o
l.
4
, N
o
.
1
,
March
201
5
:
19
–
30
L
u
d
o
v
i
c
R
i
g
h
et
t
i
et
al
.
d
e
m
o
n
s
t
r
at
ed
t
h
at
t
h
e
ap
p
r
o
ach
es
o
f
S
e
n
t
i
s
et
al
.
a
n
d
M
ic
h
a
e
l
M
is
tr
y
e
t
a
l.
ar
e
eq
u
i
v
al
en
t
i
n
es
s
en
ce
[
12
]
,
s
i
nc
e
t
he
y
(
i
nc
l
ud
i
ng
J
.
W
.
G
r
i
z
z
l
e
a
n
d
E
.
R
.
W
es
t
er
v
el
t
et
al
.
)
ar
e
al
l
b
as
ed
o
n
t
h
e
as
s
u
m
p
t
i
o
n
t
h
at
t
h
e
co
n
t
ac
t
co
n
s
t
r
ai
n
t
s
,
b
y
d
ef
i
n
i
t
i
o
n
,
ar
e
ab
l
e
t
o
ap
p
l
y
n
ece
s
s
ar
y
f
o
r
ces
t
o
m
ai
n
t
ai
n
t
h
ei
r
c
o
ns
i
s
t
e
nc
y
.
H
o
w
e
ve
r
,
t
he
c
o
ns
t
r
a
i
nt
s
m
a
y
n
ot
be
h
ol
d
un
d
e
r
c
e
r
t
a
i
n
s
i
t
u
a
t
i
on
s
e
s
pe
c
i
a
l
l
y
f
or
t
h
e
r
obot
w
i
t
h
poi
n
t
f
oot
.
B
y
a
ddi
ng
a
ddi
t
i
on
a
l
t
or
qu
e
s
t
o
c
on
t
r
ol
t
h
e
c
o
nt
a
c
t
f
or
c
e
s
,
J
a
e
h
e
u
ng
P
a
r
k
a
nd
O
u
s
s
s
a
m
s
K
a
h
t
i
b
pr
op
os
e
d
a
c
on
t
a
c
t
c
ons
i
s
t
e
n
t
c
on
t
r
ol
f
r
a
m
e
w
or
k
f
or
h
um
a
n
oi
d
r
obot
s
[
14
]
.
B
u
t
t
h
i
s
ap
p
r
o
ach
al
s
o
n
eed
s
t
o
c
a
lc
u
la
te
t
h
e
in
v
e
r
s
io
n
o
f
t
h
e
s
y
s
te
m
i
n
e
r
tia
m
a
tr
ix
.
I
n
o
r
d
er
t
o
g
en
er
at
e
w
al
k
i
n
g
g
ai
t
t
h
at
can
p
r
es
er
v
e
t
h
e
co
n
t
act
co
n
s
i
s
t
e
n
c
y
,
b
y
u
t
i
l
i
zi
n
g
t
h
e
L
i
n
ear
I
nv
er
t
ed
P
en
d
u
l
u
m
Mo
d
el
(
L
I
P
M
)
[
15
]
,
a
no
ve
l
o
nl
i
ne
w
a
l
ki
ng
ga
i
t
p
a
t
t
e
r
n
ge
ne
r
a
t
i
ng
m
e
t
ho
d
i
s
p
r
o
po
s
e
d
i
n
t
h
i
s
p
ap
er
,
an
d
t
h
e
co
m
p
l
et
e
c
o
n
t
r
o
l
ar
ch
i
t
ect
u
r
e
i
s
s
h
o
w
n
i
n
F
i
g
.
1
.
F
i
gu
r
e
1.
B
l
oc
k
di
a
g
r
a
m
f
or
t
he
w
a
l
k
i
n
g
c
ont
r
ol
I
n
s
e
c
tio
n
2
,
th
e
r
e
la
tio
n
s
h
i
p
b
e
tw
e
e
n
th
e
o
r
b
it
e
n
e
r
g
y
a
n
d
th
e
f
o
o
t
lo
c
a
tio
n
is
d
is
c
u
s
s
e
d
b
y
a
na
l
y
z
i
n
g
t
he
d
yna
m
i
c
s
o
f
a
2
D
-
L
I
P
M
.
W
i
t
h
acco
u
n
t
i
n
g
f
o
r
t
h
e
r
es
t
r
i
ct
i
o
n
o
f
t
h
e
co
n
t
act
f
o
r
ces
,
a f
o
o
t
p
l
ace
m
en
t
co
n
t
r
o
l
l
er
i
s
p
r
o
p
o
s
ed
t
o
t
r
ack
t
h
e
d
es
i
r
ed
o
r
b
i
t
en
er
g
y
.
I
n
s
ect
i
o
n
3
,
w
e
d
i
s
cu
s
s
t
h
e
g
en
er
at
i
o
n
o
f
th
e
d
e
s
ir
e
d
tr
a
j
e
c
to
r
ie
s
in
ta
s
k
s
p
a
c
e
f
o
r
b
ip
e
d
w
a
lk
in
g
b
y
s
e
le
c
tin
g
t
h
e
h
ip
’
s
h
o
r
iz
o
n
ta
l
lo
c
o
m
o
tio
n
a
s
th
e
p
ar
am
et
er
.
I
n
s
ect
i
o
n
4
,
a t
as
k
s
p
ace
co
n
t
r
o
l
l
er
w
i
t
h
o
u
t
cal
cu
l
at
i
n
g
t
h
e
i
n
v
er
s
i
o
n
o
f
i
n
er
t
i
al
m
a
t
r
i
x
i
s
p
r
es
en
t
ed
.
I
n
s
ect
i
o
n
5
,
w
e
t
es
t
t
h
e
o
n
l
i
n
e
w
al
k
i
n
g
p
at
t
er
n
g
en
er
at
i
n
g
m
et
h
o
d
an
d
t
h
e
t
as
k
s
p
ace
c
o
n
t
r
o
l
l
er
o
n
a
v
i
r
t
u
a
l
bi
pe
d
r
obot
.
S
e
c
t
i
on
6 c
on
c
l
u
de
s
t
h
e
pa
pe
r
a
n
d
pr
o
v
i
de
s
t
h
e
di
r
e
c
t
i
on
s
f
or
f
u
t
u
r
e
w
or
k
.
2.
F
O
O
T P
LA
C
EM
EN
T C
O
N
TR
O
L
LE
R
W
I
TH
C
O
N
TA
C
T C
O
N
S
IS
TEN
C
Y
A
p
l
an
ar
5
-
l
i
n
k
poi
n
t
f
oot
bi
pe
d
r
o
bo
t
,
a
s
s
ho
w
n
i
n
F
i
g.
2
.
(
a
)
,
c
o
m
p
r
i
s
e
s
t
w
o
s
ym
m
e
t
r
i
c
l
e
gs
a
nd
a
to
r
s
o
.
F
r
o
m
th
e
v
ie
w
p
o
i
n
t
o
f
n
a
tu
r
a
l
h
u
m
a
n
w
a
l
k
i
n
g
,
it is
d
e
s
ir
a
b
le
th
a
t
th
e
to
r
s
o
is
a
l
w
a
y
s
u
p
r
i
g
h
t
w
i
th
m
i
n
i
m
u
m
h
ip
’
s
v
e
r
tic
a
l
o
s
c
il
la
tio
n
d
u
r
i
n
g
w
a
l
k
i
n
g
.
T
h
a
t
i
s
s
i
m
ila
r
to
a
2
D
-
L
I
P
M
,
wh
i
ch
i
s
d
es
cr
i
b
ed
b
y
K
a
j
ita
e
t
a
l.
[
15
-
17
]
.
A 2
D
-
L
I
P
M
c
o
m
p
r
is
e
s
a
p
o
in
t
m
a
s
s
a
n
d
a
te
le
s
c
o
p
in
g
m
a
s
s
le
s
s
le
g
w
it
h
a
p
o
in
t
f
o
o
t,
w
h
ic
h
is
in
c
o
n
ta
c
t
w
it
h
t
h
e
f
l
a
t
g
r
o
u
n
d,
a
s
s
h
o
w
n
i
n
F
i
g
.
2.
(
c
)
.
I
n
t
h
i
s
pa
pe
r
w
e
m
ode
l
t
h
e
m
o
t
i
on
o
f
t
h
e
bi
pe
d
r
obot
’
s
h
i
p
a
s
a
2D
-
L
I
P
M
.
F
i
gu
r
e
.
2.
A
p
l
an
ar
5
-
l
i
nk
bi
pe
d
r
ob
ot
w
i
t
h
o
n
e
f
oot
s
t
a
n
di
ng
on
t
h
e
g
r
oun
d:
(
a
)
phy
s
i
c
a
l
pa
r
a
m
e
t
e
r
s
of
t
h
e
r
obot
;
(
b)
g
e
n
e
r
a
l
i
zed
co
o
r
d
i
n
at
es
;
(
c)
a 2
D
-
L
I
P
M
a
ppr
ox
i
m
a
t
e
s
t
h
e
m
ot
i
on
of
t
h
e
bi
pe
d
r
o
bot
’
s
h
i
p.
T
h
e
L
I
P
M
co
m
p
r
i
s
ed
a p
o
i
n
t
m
as
s
w
i
t
h
m
as
s
a
n
d
a
m
a
s
s
le
s
s
te
l
e
s
c
o
p
in
g
le
g
w
it
h
p
o
in
t
f
o
o
t.
T
h
e
p
o
in
t
m
a
s
s
i
s
k
e
p
t
a
t
c
o
n
s
ta
n
t
h
e
ig
h
t
b
y
a
n
act
u
at
o
r
t
h
at
e
x
er
t
s
a f
o
r
ce
o
n
it.
a
nd
ar
e
t
h
e
t
an
g
en
t
i
al
an
d
n
o
r
m
al
co
m
p
o
n
e
n
t
o
f
t
h
e
co
n
t
act
f
o
r
ce
b
et
w
ee
n
t
h
e
p
o
i
n
t
f
o
o
t
an
d
t
h
e
g
r
o
u
n
d
,
r
es
p
ect
i
v
el
y
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
RA
I
S
S
N
:
2088
-
8708
Online Biped Walking
Pattern Generation with Contact Consistency (Hou Wenqi)
21
2
.1
2
D
L
i
n
er I
n
v
ert
ed
P
en
d
u
l
u
m
T
h
e
e
qu
a
t
i
on
s
of
m
o
t
i
on
f
or
t
he
2D
-
L
I
P
M
ar
e
[
16
,
17
]
:
(
1
)
(
2
)
w
h
er
e
is
th
e
g
r
a
v
ita
tio
n
a
l
a
c
c
e
le
r
a
tio
n
c
o
n
s
ta
n
t,
a
nd
a
r
e
t
h
e
pos
i
t
i
on
of
t
h
e
poi
n
t
m
a
s
s
,
e
x
pr
e
s
s
e
d
i
n
a
lo
c
a
l
f
r
a
m
e
w
h
ic
h
i
s
lo
c
a
t
e
d
a
t
th
e
p
o
in
t
f
o
o
t,
is
a
c
o
n
s
ta
n
t.
T
h
e
o
r
b
it
e
n
e
r
g
y
o
f
t
h
e
p
o
in
t
m
a
s
s
is
co
n
s
er
v
ed
d
u
r
i
n
g
eac
h
s
t
a
n
ce
p
h
as
e,
an
d
i
t
i
s
d
e
f
i
n
ed
as
[
15
-
17
]
:
(
3
)
T
he
o
r
b
i
t
e
ne
r
g
y
d
e
t
e
r
m
i
ne
s
t
he
b
e
ha
vi
o
r
o
f
t
he
L
I
P
w
he
n
t
he
p
o
i
nt
m
a
s
s
i
s
m
ovi
ng
t
o
w
a
r
d
t
h
e
f
oot
.
T
h
er
e
ar
e
t
h
r
ee
cas
es
[
1
]
:
a.
.
T
h
e
p
o
in
t
m
a
s
s
w
ill
s
to
p
a
n
d
r
e
v
e
r
s
e
d
ir
e
c
tio
n
b
e
f
o
r
e
g
e
tti
n
g
o
ve
r
t
he
f
o
o
t
.
b.
.
T
h
e
p
o
i
n
t
m
as
s
w
i
l
l
co
m
e
t
o
a r
es
t
ex
act
l
y
o
n
t
h
e
f
o
o
t
.
c.
.
T
he
p
o
i
nt
m
a
s
s
w
i
l
l
go
o
ve
r
t
he
f
o
o
t
a
nd
c
o
nt
i
nue
o
n
i
t
s
w
a
y.
E
q
ua
t
i
o
n
(
3
)
in
d
ic
a
te
s
th
a
t
th
e
o
r
b
it
e
n
e
r
g
y
is
e
q
u
iv
a
le
n
t
to
th
e
v
e
lo
c
it
y
w
h
e
n
t
h
e
p
o
in
t
m
a
s
s
p
a
s
s
e
s
th
r
o
u
g
h
t
h
e
s
ta
n
c
e
p
o
in
t.
I
t
m
e
a
n
s
t
h
a
t
w
it
h
m
o
r
e
o
r
b
it
e
n
e
r
g
y
t
h
e
p
o
in
t
m
a
s
s
w
ill
m
o
v
e
f
a
s
te
r
.
S
o
th
e
b
ip
e
d
w
a
l
ki
ng
c
on
t
r
ol
i
s
a
n
or
bi
t
e
n
e
r
gy
t
r
a
c
k
i
ng
pr
obl
e
m
i
n
e
s
s
e
nc
e
.
A
ddi
t
i
on
a
l
l
y
,
E
qu
a
t
i
on
(
3
)
s
h
o
w
s
th
a
t
th
e
o
r
b
it
e
ne
r
g
y
f
ul
l
y
d
e
p
e
nd
s
o
n
t
he
l
o
c
a
t
i
o
n
o
f
th
e
f
o
o
t
r
e
la
tiv
e
to
th
e
p
o
in
t
m
a
s
s
g
i
v
e
n
a
c
e
r
ta
in
in
itia
l
v
e
lo
c
it
y
.
S
o
t
he
o
nl
y
w
a
y
t
o
c
ha
nge
t
he
p
oi
nt
m
a
s
s
’
s
o
r
b
i
t
e
ne
r
g
y
i
s
c
ha
n
gi
n
g
t
he
f
o
o
t
l
o
c
a
t
i
o
n
b
y
t
a
ki
n
g
a
s
t
e
p
.
T
o
c
l
a
r
i
f
y
t
h
e
an
al
y
s
i
s
b
et
t
er
,
w
e
as
s
u
m
e
each
s
t
ep
i
s
t
ak
en
i
n
s
t
a
n
t
a
n
eo
u
s
l
y
a
n
d
t
h
e
v
el
o
ci
t
y
o
f
t
h
e
p
o
in
t
m
a
s
s
w
il
l
n
o
t
b
e
af
f
ect
ed
as
s
h
o
w
n
i
n
F
i
g
.
3
.
F
i
g
ur
e
3
.
T
he
m
o
m
e
nt
w
h
e
n
t
he
2
D
-
L
I
P
M
i
s
t
ak
i
n
g
a s
t
ep
.
is
th
e
p
e
r
m
i
tte
d
m
a
x
i
m
u
m
h
o
r
iz
o
n
ta
l
d
is
ta
n
c
e
be
t
w
e
e
n
t
h
e
poi
n
t
m
a
s
s
a
n
d
t
he
f
oot
i
n
e
a
ch
s
t
an
ce
p
h
a
s
e,
w
h
i
ch
m
ea
n
s
o
n
ce
t
h
e
p
o
i
n
t
m
a
s
s
ar
r
i
v
es
at
a
s
t
ep
m
u
s
t
b
e
t
ak
e
n
.
i
s
t
h
e
h
o
r
i
zo
n
t
al
d
i
s
t
an
ce
b
et
w
ee
n
t
h
e
d
es
i
r
ed
f
o
o
t
l
o
cat
i
o
n
an
d
t
h
e
p
o
i
n
t
m
a
s
s
w
h
e
n
a
s
t
e
p
i
s
b
e
i
n
g
t
a
ke
n.
is
th
e
v
e
lo
c
it
y
o
f
t
h
e
p
o
in
t
m
a
s
s
a
t
th
a
t
m
o
m
e
n
t.
L
e
t
a
nd
be
t
h
e
or
bi
t
e
n
e
r
gy
of
t
h
e
poi
n
t
m
a
s
s
be
f
or
e
a
n
d
a
f
t
e
r
t
h
e
2D
-
L
I
P
M
t
ak
es
a
s
t
ep
,
r
es
p
ect
i
v
el
y
,
i
.
e.
(
4
)
(
5
)
E
q
ua
t
i
o
n
(
5
)
mi
n
u
s
(
4
)
is
(
6
)
E
q
u
a
tio
n
(
6
)
c
l
ear
l
y
i
n
d
i
cat
es
t
h
at
,
can
ch
a
n
g
e
t
h
e
o
r
b
i
t
en
er
g
y
o
f
t
h
e
p
o
i
n
t
m
as
s
fo
r
a
gi
ve
n
,
an
d
t
h
r
ee
t
y
p
i
cal
ca
s
es
ar
e
i
n
cl
u
d
ed
:
a.
.
T
h
e
o
r
b
i
t
en
er
g
y
w
i
l
l
i
n
cr
e
as
e.
b.
.
T
h
e
o
r
b
i
t
en
er
g
y
w
i
l
l
n
o
t
ch
an
g
e,
an
d
t
h
e
p
o
i
n
t
m
as
s
w
i
l
l
m
o
v
e
ex
act
l
y
as
i
n
t
h
e
p
r
e
vi
o
us
s
t
a
nc
e
p
ha
s
e
.
c.
.
T
h
e
o
r
b
i
t
en
er
g
y
w
i
l
l
d
ecr
e
as
e.
S
o,
i
n
or
de
r
t
o
r
e
a
l
i
z
e
t
h
e
m
ot
i
on
of
t
h
e
poi
n
t
m
a
s
s
w
i
t
h
de
s
i
r
e
d
or
bi
t
e
n
e
r
gy
,
t
h
e
f
oot
l
oc
a
t
i
on
s
h
oul
d
be
pr
o
pe
r
l
y
c
on
t
r
ol
l
e
d
f
or
e
v
e
r
y
s
t
e
p.
Evaluation Warning : The document was created with Spire.PDF for Python.
22
I
SSN
:
20
89
-
4856
IJRA Vol. 4, No. 1, March 2015 : 19 – 30
2
.2
F
o
o
t
P
l
a
cem
en
t
C
o
n
t
ro
l
l
er
F
or
bi
pe
d
w
a
l
ki
ng
c
on
t
r
ol
,
t
he
r
e
a
r
e
t
w
o
qu
e
s
t
i
ons
t
o
be
a
n
s
w
e
r
e
d:
w
h
e
n
a
s
t
e
p
s
h
ou
l
d
be
t
a
k
e
n
?
W
h
er
e
t
h
e
f
o
o
t
s
h
o
u
l
d
b
e
p
l
aced
?
H
er
e
w
e
w
i
l
l
a
n
s
w
er
t
h
es
e
q
u
es
t
io
n
s
b
y
ta
k
in
g
t
w
o
p
r
a
c
tic
a
l
r
e
s
tr
ic
tio
n
s
acco
u
n
t
.
F
i
r
s
t
,
i
n
or
de
r
t
o
m
a
i
n
t
a
i
n
t
h
e
f
oot
-
g
r
oun
d
c
on
t
a
c
t
w
i
t
h
ou
t
s
l
i
ppi
ng
,
t
h
e
m
a
gn
i
t
u
de
of
t
h
e
t
a
ng
e
nt
i
a
l
f
o
r
ce
,
cx
F
m
u
st
b
e
l
e
ss
t
h
a
n
,
cy
F
µ
, i
.e
.
:
,,
cx
c
y
FF
µ
<
(
7
)
wh
e
r
e
µ
is
th
e
c
o
e
f
f
ic
ie
n
t
o
f
f
r
ic
tio
n
.
F
r
o
m
F
i
g
.
2
(
c
)
,
w
e
c
a
n
g
e
t
,
si
n
cx
FF
θ
=
(
8
)
,
c
o
s
cy
F
F
θ
=
(
9
)
S
ub
s
t
i
t
ut
i
n
g
e
q
ua
t
i
o
n
(
8
)
a
nd
(
9
)
in
to
(
7
)
,
w
e
ha
ve
0
t
a
n
x
y
α
µ
=
<
(
10
)
T
h
at
m
ea
n
s
a s
t
ep
m
u
s
t
b
een
t
ak
en
b
e
f
o
r
e
t
h
e
p
o
i
n
t
m
as
s
d
ev
i
at
es
f
r
o
m
t
h
e
f
o
o
t
w
i
t
h
a
m
a
g
n
i
t
u
d
e
o
f
0
y
µ
.
I
n
o
r
d
e
r
t
o
c
ha
nge
t
he
o
r
b
i
t
e
ne
r
ge
w
i
t
h
a
l
a
r
ge
r
va
l
ue
.
w
e
l
e
t
t
he
f
o
o
t
c
ha
nge
i
t
s
l
o
c
a
t
i
o
n
w
h
e
n
t
he
p
o
i
nt
ma
s
s
a
r
r
i
v
es
at
X
,
an
d
s
et
0
0.5
X
y
µ
=
(
11
)
S
eco
n
d
,
d
u
e
t
o
t
h
e
r
es
t
r
i
ct
i
o
n
o
f
t
h
e
act
u
at
o
r
,
t
h
er
e
is
a
lo
w
e
r
li
m
it
to
t
h
e
ti
m
e
b
e
t
w
e
e
n
s
te
p
s
(
f
o
o
t
l
o
c
a
t
i
o
n
c
ha
nge
s
)
,
mi
n
T
,
w
h
ic
h
m
o
d
e
ls
th
e
p
r
a
c
tic
a
l
a
c
tu
a
t
io
n
a
b
i
lit
y
o
f
s
w
i
n
g
le
g
.
F
r
o
m
X
a
nd
mi
n
T
,
w
e
c
a
n
g
e
t
t
h
e
u
p
p
e
r
lim
it
o
f
t
h
e
o
r
b
i
t
en
er
g
y
:
2
2
0
2
1
mi
n
c
mi
n
c
T
T
ma
x
T
T
gX
e
E
y
e
=
−
(
12
)
F
r
o
m
e
q
ua
t
i
o
n
(3
)
a
nd
(4
)
,
w
e
ha
ve
(
)
2
22
0
0
11
22
2
c
x
g
Ex
X
x
y
−
=
=
−=
(
13
)
I
n
e
q
ua
t
i
o
n
(5
)
,
w
e
l
et
22
0
1
22
d
e
s
c
d
e
s
g
EE
x
L
y
+
=
=
−
(
14
)
F
r
o
m
e
q
ua
t
i
o
n
(
13
)
a
nd
(
14
)
,
w
e
can
h
a
v
e:
(
)
2
2
0
0
0
2
1
22
d
e
s
d
e
s
a
x
y
g
L
x
XE
E
gy
=
=
±
+
−=
±
(
15
)
N
o
te
th
a
t,
if
(
)
2
2
0
0
d
e
s
x
g
Ex
X
y
=
≥+
(
16
)
th
e
r
e
w
il
l
n
o
t
b
e
a
p
o
s
itiv
e
s
o
lu
tio
n
f
o
r
d
e
s
L
,
t
ha
t
m
e
a
ns
t
he
d
e
s
i
r
e
d
o
r
b
i
t
e
ne
r
g
y
i
s
m
uc
h
l
a
r
gge
r
t
ha
n
t
he
c
ur
r
e
nt
va
l
ue
,
s
o
t
h
a
t
j
u
s
t
ta
k
e
o
n
l
y
o
n
e
s
te
p
c
a
n
n
’
t
le
t
t
h
e
p
o
in
t
m
a
s
s
m
o
v
e
w
it
h
t
h
e
d
e
s
ir
e
d
o
r
b
it
e
n
e
r
g
y
.
I
n
th
is
s
it
u
a
tio
n
o
u
r
s
tr
a
te
g
y
is
i
n
c
r
e
a
s
in
g
th
e
o
r
b
it
e
n
e
r
g
y
s
te
p
b
y
s
te
p
.
If
d
e
s
L
,
cal
cu
l
at
ed
f
r
o
m
eq
u
at
i
o
n
(
15
)
,
is
la
r
g
e
r
th
a
n
0
y
µ
, i
.e
.
0
d
e
s
Ly
µ
≥
(
17
)
W
hi
c
h
m
e
a
ns
t
he
d
e
s
i
r
e
d
o
r
b
i
t
e
ne
r
g
y
i
s
m
u
c
h
l
e
s
s
t
ha
n
t
he
c
ur
r
e
nt
va
l
ue
,
s
o
t
ha
t
a
l
o
n
g
s
t
e
p
m
us
t
b
e
ta
k
e
n
,
w
h
ic
h
m
a
y
v
io
la
te
s
th
e
r
e
s
tr
ic
tio
n
o
n
c
o
n
ta
c
t
f
o
r
c
e
.
I
n
th
is
s
i
tu
a
t
io
n
o
u
r
s
tr
a
te
g
y
is
d
e
c
r
e
a
s
in
g
t
h
e
or
bi
t
e
n
e
r
gy
s
t
e
p
b
y
s
t
e
p.
F
r
o
m
th
e
v
ie
w
p
o
i
n
t
o
f
n
a
tu
r
a
l
h
u
m
a
n
w
a
lk
i
n
g
,
it’
s
d
e
s
ir
a
b
le
th
a
t
th
e
s
w
i
n
g
f
o
o
t
a
l
w
a
y
s
la
n
d
s
i
n
f
r
o
n
t
o
f
t
h
e
b
o
d
y
.
C
o
n
s
ie
r
in
g
a
ll
th
e
c
o
n
d
itio
n
s
a
s
(
11
)
,
(
12
)
,
(
16
)
a
nd
(
17
)
,
w
e
d
es
i
g
n
d
e
s
L
as
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
RA
I
S
S
N
:
2088
-
8708
Online Biped Walking
Pattern Generation with Contact Consistency (Hou Wenqi)
2
3
0.6
,
0
0.6
1.6
,
1.6
,
0.6
1.6
aa
d
e
s
a
aa
X
E
or
E
X
L
XE
X
E
X
E
X
≤≤
=
≥
<
<
(
18
)
T
h
eo
re
m
1
:
F
or
t
h
e
2D
L
I
P
M
w
i
t
h
poi
n
t
f
oot
,
a
f
oot
pl
a
c
e
m
e
nt
c
on
t
r
ol
l
e
r
a
s
s
h
o
w
n
i
n
e
qu
a
t
i
on
s
(
11
)
a
nd
(
18
)
c
a
n
c
o
n
tr
o
l
th
e
p
o
in
t
m
a
s
s
m
o
v
e
f
r
o
m
a
n
y
p
e
r
m
itte
d
s
ta
te
(
i.
e
.
th
e
o
r
b
it
e
n
e
r
g
y
is
le
s
s
t
h
a
n
th
e
u
p
p
e
r
li
m
it)
i
n
t
o
a d
es
i
r
ed
p
er
i
o
d
i
c
g
ai
t
(
i
.
e.
t
h
e
o
r
b
i
t
en
er
g
y
i
s
eq
u
al
t
o
t
h
e
d
es
i
r
ed
o
n
e
w
i
t
h
a p
er
m
i
t
t
ed
v
al
u
e)
w
i
t
h
co
n
t
act
co
n
s
i
s
t
e
n
c
y
.
T
he
o
r
e
m
1
m
e
a
ns
w
h
e
n
t
he
p
o
i
nt
m
a
s
s
e
ve
n
s
u
f
f
e
r
s
a
s
u
d
d
e
n
d
i
s
t
ur
b
a
nc
e
(
e
.
g.
a
s
ud
d
e
n
p
us
h)
du
r
i
ng
t
h
e
m
ot
i
on
,
i
f
t
h
e
or
bi
t
e
n
e
r
gy
doe
s
n’
t
e
x
c
e
e
d
ma
x
E
,
t
h
e
d
es
i
g
n
ed
f
o
o
t
p
l
ace
m
e
n
t
co
n
t
r
o
l
l
er
can
s
ta
b
iliz
e
th
e
m
o
t
i
o
n.
E
q
ua
t
i
o
n
(
11
)
an
s
w
er
s
t
h
e
q
u
es
t
i
o
n
:
w
h
en
a s
t
ep
s
h
u
o
l
d
b
e
t
ak
en
?
W
h
i
ch
p
r
ev
en
t
s
t
h
e
co
n
t
act
c
o
ns
i
s
t
e
nc
y
d
ur
i
n
g
w
a
l
ki
ng.
A
nd
e
q
ua
t
i
o
n
(
18
)
a
ns
w
e
r
s
t
he
q
ue
s
t
i
o
n
:
w
h
er
e
t
h
e
f
o
o
t
s
h
u
o
l
d
b
e
p
l
aced
?
W
h
i
ch
ac
h
i
ev
e
s
a d
es
i
r
ed
w
al
k
i
n
g
c
y
cl
e.
3.
O
NL
I
N
E
T
R
AJ
E
C
T
O
RI
E
S
P
L
ANNI
NG
I
N T
AS
K
S
P
A
CE
I
n
th
is
s
e
c
tio
n
w
e
w
ill
d
is
c
u
s
s
t
h
e
p
l
an
n
i
n
g
o
f
t
h
e
d
es
i
r
ed
t
r
aj
ect
o
r
i
es
i
n
t
as
k
s
p
ace
d
u
r
i
n
g
b
i
p
ed
w
a
l
k
i
n
g
ba
s
e
d
on
t
h
e
f
oot
pl
a
c
e
m
e
nt
c
on
t
r
ol
l
e
r
i
n
s
e
c
t
i
on
2.
A
s
m
e
n
t
i
on
e
d
a
bov
e
,
t
h
e
t
or
s
o’
s
pi
t
c
h
a
n
g
l
e
a
n
d
t
h
e
h
i
p’
s
h
e
i
gh
t
s
h
ou
l
d
be
s
t
a
bl
i
z
e
d
t
o
c
on
s
t
a
n
t
s
d
ur
i
n
g
w
a
l
ki
n
g.
A
d
d
t
i
o
na
l
l
y,
t
he
s
w
i
ng
l
e
g
p
l
a
ys
a
n
i
nd
i
s
p
e
ns
a
b
l
e
r
o
l
e
,
a
nd
i
t
s
m
a
i
n
t
a
s
k
i
nc
l
ud
e
s
t
a
ki
n
g
o
f
f
f
r
o
m
t
h
e
g
r
oun
d,
m
ov
i
ng
f
or
w
a
r
d
a
n
d
l
a
n
di
ng
t
o
a
de
l
i
be
r
a
t
e
l
oc
a
t
i
on
,
a
n
d
t
h
e
n
v
i
a
c
h
a
n
g
i
ng
i
t
s
r
ol
e
a
s
a
s
t
an
ce
l
e
g
t
o
co
n
t
i
n
u
e
t
h
e
w
al
k
i
n
g
.
S
o
t
h
e
t
as
k
s
p
ace
d
u
r
i
n
g
b
i
p
ed
w
al
k
i
n
g
ca
n
b
e
s
el
ect
ed
as
:
;
;;
ts
k
h
ip
s
wt
s
wt
P
yx
y
θ
=
(
19
)
wh
e
r
e
θ
is
th
e
p
itc
h
a
n
g
le
o
f
th
e
to
r
s
o
,
hi
p
y
i
s
t
he
he
i
g
ht
o
f
t
he
hi
p
,
sw
t
x
a
nd
sw
t
y
ar
e
t
h
e
h
o
r
i
zo
n
t
al
a
n
d
v
ir
t
ic
a
l
c
o
m
p
o
n
e
n
t
o
f
t
h
e
p
o
s
itio
n
o
f
th
e
s
w
i
n
g
f
o
o
t.
I
t’
s
c
le
a
r
th
a
t
t
h
e
ta
s
k
s
i
n
(
19
)
ar
e
l
i
n
ear
l
y
i
nd
e
p
e
nd
e
nt
.
T
h
e
d
es
i
r
ed
p
i
t
ch
an
g
l
e
o
f
t
h
e
t
o
r
s
o
an
d
t
h
e
h
ei
g
h
t
o
f
t
h
e
h
i
p
ar
e
s
et
as
:
=
=
0
,0
d
e
s
d
e
s
d
e
s
d
e
s
d
e
s
d
e
s
hi
p
hi
p
hi
p
y
H
c
ons
t
ant
y
y
θ
θθ
=
=
=
=
=
(
20
)
F
i
g
u
r
e
4
i
l
l
u
s
t
r
at
es
a d
es
i
r
ed
s
t
an
ce
p
h
as
e
d
u
r
i
n
g
w
al
k
i
n
g
.
T
h
e
h
i
p
m
o
v
e
s
as
a
2D
L
I
P
,
a
nd
P
is
th
e
d
es
i
r
ed
l
o
cat
i
o
n
w
h
er
e
t
h
e
s
t
a
n
ce
f
o
o
t
s
h
o
u
l
d
b
e
p
l
aced
i
n
n
ex
t
s
t
a
n
ce
p
h
as
e
cacu
l
a
t
ed
b
y
eq
u
a
t
i
o
n
(
18
)
.T
h
e
d
a
sh
-
d
o
t
t
ed
l
i
n
e
d
en
o
t
es
t
h
e
d
es
i
r
ed
t
r
a
j
ect
o
r
y
o
f
t
h
e
s
w
i
n
g
f
o
o
t
.
T
h
e
t
r
aj
ect
o
r
i
es
can
b
e
p
l
an
n
ed
as
a f
u
n
ct
i
o
n
o
f
t
i
me
[
11
,
18
]
or
t
he
ge
o
m
e
t
r
i
c
e
vo
l
ut
i
o
n
o
f
t
he
b
i
p
e
d
[
19
,
20
]
.
I
n
or
de
r
t
o
w
i
t
hs
t
a
n
d
a
di
s
t
u
r
be
n
c
e
r
obu
s
t
l
y
th
e
la
t
te
r
o
p
tio
n
i
s
a
d
o
p
te
d
in
th
is
p
a
p
e
r
.
D
u
r
in
g
w
a
lk
in
g
,
t
h
e
h
o
r
iz
o
n
ta
l
lo
c
a
tio
n
o
f
h
ip
hi
p
x
is
m
o
n
o
to
n
ic
a
ll
y
i
n
cr
eas
i
n
g
,
s
o
i
t
can
b
e
s
el
ect
e
d
as
t
h
e
p
ar
a
m
et
er
o
f
d
es
i
r
ed
t
r
aj
ect
o
r
i
es
.
x
y
o
h
H
a
b
c
d
es
L
′
d
es
L
P
X
X
F
i
gu
r
e
4.
A
de
s
i
r
e
d
s
t
a
n
ce
p
h
a
s
e
co
n
t
ai
n
t
h
r
ee
cr
u
ci
al
m
o
m
e
n
t
s
:
(
a)
,
t
h
e
b
eg
i
n
n
i
n
g
o
f
a s
t
a
n
ce
p
h
as
e,
i
.
e.
t
h
e
m
o
m
e
nt
j
us
t
a
f
t
e
r
t
he
s
w
i
ng
l
e
g
b
e
c
a
m
e
t
he
s
t
a
nc
e
l
e
g;
(
b
)
,
t
he
m
o
m
e
n
t
w
he
n
t
he
hi
p
p
a
s
s
t
ho
u
g
h
t
he
s
t
a
nc
e
p
o
i
nt
;
(
c
)
,
t
he
m
o
m
e
nt
w
h
e
n
t
he
hi
p
a
r
r
i
ve
s
a
t
X
.
T
he
d
a
s
he
d
l
i
ne
d
e
no
t
e
s
s
w
i
ng
l
e
g
a
nd
t
he
s
o
l
i
d
l
i
ne
d
e
no
t
e
s
t
an
ce
l
e
g
.
L
e
t
(
)
(
)
;
d
e
s
d
e
s
d
e
s
s
wt
s
wt
h
ip
s
wt
h
ip
P
x
x
y
x
=
b
e
th
e
d
e
s
ir
e
d
p
o
s
itio
n
o
f
th
e
s
w
i
n
g
f
o
o
t
in
th
e
lo
c
a
l
f
r
a
m
e
lo
c
a
te
d
at
t
h
e
p
o
i
n
t
f
o
o
t
.
T
h
er
e
ar
e
t
h
r
ee
cr
u
ci
al
m
o
m
en
t
s
i
n
o
n
e
s
t
a
n
ce
p
ha
s
e
a
s
s
ho
w
n
i
n
F
i
g.
4
.
Evaluation Warning : The document was created with Spire.PDF for Python.
24
I
SSN
:
20
89
-
4856
IJRA Vol. 4, No. 1, March 2015 : 19 – 30
T
he
c
o
ns
t
r
a
i
nt
e
qu
a
t
i
on
s
f
or
t
he
de
s
i
r
e
d
t
r
a
j
ect
o
r
i
es
ar
e
d
es
cr
i
b
ed
as
f
o
l
l
o
w
s
:
(
a)
.
T
h
e
b
eg
i
n
n
i
n
g
o
f
a s
t
a
n
ce
p
h
as
e:
(
)
(
)
(
)
0
d
e
s
d
e
s
s
w
t
d
e
s
d
e
s
s
w
t
d
e
s
d
e
s
s
w
t
d
e
s
LX
xL
PL
y
L
′
′
−−
−
′
−
=
=
′
−
(
21
)
(
)
(
)
(
)
0
0
d
e
s
s
w
t
d
e
s
d
e
s
s
w
t
d
e
s
d
e
s
s
w
t
d
e
s
xL
PL
y
L
′
−
′
−
=
=
′
−
(
22
)
(
b
)
.
T
he
m
o
m
e
nt
w
h
e
n
t
he
hi
p
p
a
s
s
t
ho
ug
h
t
he
s
t
a
nc
e
p
o
i
n
t
:
(
L
e
t
h
b
e
t
h
e
m
a
x
i
m
u
m
cl
ear
an
ce
o
f
t
he
s
w
i
n
g
l
e
g)
:
(
)
(
)
(
)
0
0
0
0
d
e
s
sw
t
d
e
s
sw
t
d
e
s
sw
t
x
P
h
y
=
=
(
23
)
(
)
(
)
(
)
0
0
,0
0
0
d
e
s
sw
t
d
e
s
sw
t
d
e
s
sw
t
v
x
Pv
y
=
=
≥
(
24
)
(
c
)
.
T
he
m
o
m
e
nt
w
he
n
t
he
hi
p
a
r
r
i
ve
s
a
t
X
:
(
)
(
)
(
)
0
d
e
s
d
e
s
sw
t
d
e
s
s
w
t
ma
x
d
e
s
sw
t
XL
xX
P
X
yX
+
=
=
(
25
)
(
)
(
)
(
)
0
0
d
e
s
sw
t
d
e
s
s
w
t
ma
x
d
e
s
sw
t
xX
P
X
yX
=
=
(
26
)
U
s
i
n
g
a
s
i
nu
s
oi
da
l
f
un
c
t
i
on
o
f
hi
p
x
,
t
h
e
de
s
i
r
e
d
t
r
a
j
e
c
t
o
r
y
of
s
w
i
ng
f
oot
i
s
pl
a
nn
e
d
a
s
f
ol
l
o
w
s
:
(
)
(
)
[
]
(
)
(
]
si
n
,
,
0
2
s
i
n
,
0,
2
hi
p
de
s
hi
p
de
s
d
e
s
rd
s
wt
h
ip
hi
p
de
s
hi
p
x
LX
x
L
L
x
x
x
LX
x
X
X
π
π
′′
+
∈−
′
=
+∈
(
27
)
(
)
[
]
(
]
c
o
s
,
,
0
22
c
o
s
,
0,
22
hi
p
hi
p
de
s
d
e
s
d
e
s
s
wt
h
ip
hi
p
hi
p
x
hh
xL
L
yx
x
hh
xX
X
π
π
′
+
∈−
′
=
+∈
(
28
)
W
h
er
e
d
e
s
L
a
nd
d
e
s
L
′
ar
e
t
h
e
h
o
r
i
zo
n
t
al
d
i
s
t
an
ce
b
et
w
e
e
n
th
e
d
e
s
ir
e
d
f
o
o
t
lo
c
a
tio
n
a
n
d
th
e
h
ip
in
t
h
e
cu
r
r
en
t
an
d
t
h
e
f
o
r
m
er
s
t
ep
,
r
es
p
ect
i
v
el
y
,
as
s
h
o
w
n
i
n
F
i
g
.
4
.
4.
RO
B
O
T
M
O
DE
L
AND
T
AS
K
S
P
ACE
CO
NT
RO
L
L
E
R
I
n
t
hi
s
s
e
c
t
i
o
n,
t
he
d
yna
m
i
c
s
o
f
a
p
l
a
na
r
5
-
l
i
n
k
poi
n
t
f
oot
bi
pe
d
i
s
i
n
t
r
odu
c
e
d
a
n
d
a
t
a
s
k
s
pa
c
e
c
on
t
r
ol
l
er
b
as
ed
o
n
t
h
e
d
y
n
a
m
i
cs
m
o
d
el
i
s
d
es
i
g
n
ed
t
o
r
eal
i
ze
t
h
e
t
r
aj
ect
o
r
i
es
p
r
es
en
t
ed
b
y
(
20)
,
(
27)
a
nd
(
28)
.
A
s
s
ho
w
n
i
n
F
i
g.
2
.
(
b
)
,
i
f
t
he
r
e
i
s
no
m
o
t
i
o
n
b
e
t
w
e
e
n
t
he
s
t
a
nc
e
f
o
o
t
a
nd
t
he
gr
o
und
[
2
,
6
,
10
,
21
]
,
a
nd
t
he
o
r
i
gi
n
o
f
t
h
e
l
o
cal
r
ef
er
en
ce
f
r
a
m
e
i
s
l
o
cat
ed
at
t
h
e
p
o
i
n
t
f
o
o
t
,
a s
et
o
f
g
en
er
al
i
ze
d
co
o
r
d
i
n
at
e
o
f
t
h
e
b
i
p
ed
r
o
b
o
t
can
b
e
s
el
ect
ed
as
:
[
]
;
er
qq
θ
=
(
29
)
W
he
r
e
[
]
1
2
21
;
;;
r
r
r
ll
q
θθ
θ
θ
=
i
s
t
h
e
j
oi
n
t
c
onf
i
gu
r
a
t
i
on
of
t
h
e
r
obot
.
a
n
d
θ
i
s
t
he
p
i
t
c
h
a
ng
l
e
o
f
t
he
to
r
s
o
in
th
e
lo
c
a
l
f
r
a
m
e
.
U
s
i
n
g
t
he
L
a
gr
a
nge
m
e
t
ho
d
,
w
e
c
a
n
ge
t
t
he
d
yna
m
i
c
s
o
f
t
h
e
r
o
b
o
t
[
22
]
:
(
)
(
)
(
)
,
T
e
e
e
e
e
e
M
qq
C
q
qq
G
q
S
τ
+
+
=
(
30
)
wh
e
r
e
(
)
55
e
Mq
R
×
∈
is
th
e
i
n
e
r
tia
m
a
tr
ix
o
f
th
e
r
o
b
o
t,
(
)
55
,
e
e
Cq
q
R
×
∈
is
th
e
c
e
n
tr
ip
e
ta
l
a
n
d
C
o
r
io
lis
f
o
r
ces
,
(
)
51
e
Gq
R
×
∈
i
s
t
he
gr
a
vi
t
y
f
o
r
c
e
s
,
[
]
4
4
41
0
SI
××
=
is
th
e
s
e
le
c
ti
n
g
m
a
tr
i
x
o
f
t
h
e
a
c
tu
a
te
d
j
o
in
ts
,
41
R
τ
×
∈
is
th
e
v
ect
o
r
o
f
act
u
at
ed
j
o
i
n
t
t
o
r
q
u
es
.
L
e
t
(
)
t
sk
e
P
fq
=
,
t
h
e
t
as
k
v
el
o
ci
t
i
e
s
can
b
e
cal
cu
l
at
ed
as
:
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
RA
I
S
S
N
:
2088
-
8708
Online Biped Walking
Pattern Generation with Contact Consistency (Hou Wenqi)
25
t
sk
e
t
sk
e
e
f
P
q
J
q
q
∂
=
=
∂
(
31
)
If
(
)
4
t
sk
r
ank
J
=
,
by
u
s
i
ng
t
h
e
M
oor
e
-
P
en
r
o
s
e
p
s
eu
d
o
-
i
n
v
er
s
e
o
f
t
h
e
t
a
s
k
J
a
co
b
i
an
[
23
]
:
#
e
t
sk
t
sk
q
JP
=
(
32
)
(
)
#
#
t
sk
e
t
sk
t
sk
t
sk
d
J
q
JP
P
dt
=
+
(
33
)
wh
e
r
e
(
)
1
#
TT
t
sk
t
sk
t
sk
t
sk
J
J
J
J
−
=
.
S
u
b
s
titu
ti
n
g
(
32
)
a
nd
(
33
)
in
to
(
30
)
,
t
he
d
yn
a
m
i
c
s
i
n
t
as
k
s
p
ace
can
b
e
ex
p
r
es
s
ed
as
:
(
)
(
)
(
)
(
)
#
##
,
t
sk
T
e
t
sk
t
sk
e
e
t
sk
t
sk
e
d
J
M
q
J
P
C
q
q
J
P
G
q
S
dt
τ
+
+
+
=
(
34
)
M
ul
t
i
p
l
yi
ng
e
q
ua
t
i
o
n
(
34
)
by
S
:
t
sk
t
sk
M
P
C
P
G
τ
+
+=
(
35
)
wh
e
r
e
(
)
#
e
t
sk
M
SM
q
J
=
,
(
)
(
)
(
)
#
#
,
t
sk
e
e
t
sk
e
d
J
C
S
Cq
q
J
M
q
dt
=
+
,
(
)
e
G
SG
q
=
.
E
q
ua
t
i
o
n
(
35
)
m
e
a
ns
t
he
f
ul
l
d
yna
m
c
i
s
o
f
t
he
r
o
b
o
t
c
a
n
b
e
r
e
pr
e
s
e
nt
b
y
t
h
e
t
op
por
t
i
on
s
of
t
he
e
q
ua
t
i
o
n
(
34
)
[
10
]
.
A
t
as
k
s
p
ac
e
P
D
t
y
p
e
co
n
t
r
o
l
l
er
can
b
e
d
es
i
g
n
ed
as
:
(
)
d
e
s
t
sk
t
sk
P
t
sk
D
t
sk
MP
C
P
G
M
K
e
K
e
τ
=
+
+
−
+
(
36
)
W
he
r
e
d
e
s
t
sk
t
sk
t
sk
e
PP
=
−
,
d
e
s
t
sk
t
sk
t
sk
e
PP
=
−
.
N
o
te
th
a
t
th
e
c
o
n
tr
o
lle
r
(
36
)
d
o
e
s
n
’
t
n
e
e
d
to
c
a
lc
u
la
te
t
h
e
i
n
v
e
r
s
io
n
o
f
in
e
r
tia
l
m
a
tr
i
x
,
w
h
ic
h
is
m
o
r
e
c
o
nve
ni
e
n
t
t
ha
n
t
he
m
e
t
h
od
of
[
13
,
14
]
.
A
n
d
it i
s
s
i
m
ila
r
to
th
e
c
o
n
tr
o
lle
r
s
o
f
[
10
,
21
]
,
in
w
h
ic
h
th
e
ta
s
k
s
p
aceJ
aco
b
i
an
s
h
o
u
l
d
t
ak
e
t
h
e
co
n
t
act
co
n
s
t
r
ai
n
t
s
i
n
t
o
acco
u
n
t
.
H
er
e
t
h
e
co
n
t
act
co
n
s
t
r
ai
n
t
s
ar
e
co
n
s
i
d
er
ed
i
n
e
q
u
a
tio
n
(
30
)
,
th
u
s
t
h
e
c
o
n
tr
o
ll
e
r
is
m
o
r
e
c
o
n
v
e
n
ie
n
t
to
i
m
p
le
m
e
n
t.
5.
S
I
M
UL
AT
I
O
N A
ND DI
S
C
US
S
I
O
N
S
S
i
m
u
la
tio
n
e
x
p
e
r
i
m
e
n
ts
a
r
e
i
m
p
le
m
e
n
te
d
i
n
M
A
T
L
A
B
.
B
y
u
s
i
ng
t
he
“
S
i
m
M
e
c
ha
ni
c
s
”
t
o
o
l
b
o
x,
a
v
ir
tu
a
l
5
-
l
i
nk
pl
a
n
a
r
bi
pe
d
r
ob
ot
w
i
t
h
t
w
o
s
y
m
m
e
tr
ic
le
g
s
is
b
u
ilt
a
s
s
h
o
w
n
in
F
ig
.
5
.
T
h
e
p
h
y
s
ic
a
l
p
a
r
a
m
e
te
r
s
o
f
th
e
r
o
b
o
t
a
r
e
s
h
o
w
n
i
n
T
a
b
l
e
1
,
a
n
d
w
a
lk
i
n
g
is
b
e
g
in
n
i
n
g
w
it
h
t
h
e
in
itia
l
s
ta
te
s
s
h
o
w
n
in
T
a
b
le
2
.
F
ig
u
r
e
5
.
A
v
ir
tu
a
l
5
-
l
i
n
k
pl
a
na
r
bi
pe
d
r
ob
ot
T
a
b
le
1
.
P
h
y
s
i
cal
p
ar
a
m
e
t
er
s
o
f
t
h
e
v
i
r
t
u
a
l
bi
pe
d
r
obot
M
o
d
el
p
a
r
a
m
et
er
U
n
i
ts
To
r
s
o
Th
i
g
h
S
h
a
n
k
M
a
ss
k
g
1
8
.
84
5
.
0
2
4
5
.
0
2
4
Le
n
g
t
h
m
0
.3
0
.4
0
.4
I
n
e
r
tia
2
k
g
m
⋅
0
.
2
04
1
0
.
0
6
7
6
6
0
.
0
6
7
6
6
Ma
s
s
c
e
n
te
r
to
t
h
e
jo
i
n
t
m
0
.1
0
.2
0
.2
T
a
b
le
2
.
I
n
itia
l
s
ta
te
s
o
f
t
h
e
v
i
r
tu
a
l
b
ip
e
d
r
o
b
o
t.
S
e
t
th
e
r
ig
h
t
le
g
a
s
t
h
e
s
ta
n
c
e
le
g
i
n
itia
ll
y
.
1
r
θ
2
r
θ
1
l
θ
2
l
θ
θ
h
ip
x
h
ip
y
sw
t
x
sw
t
y
(
)
,
q
r
ad
m
0
.
7
10
8
-
0
.
3
55
4
0
.
9
57
8
-
0
.
4
78
9
0
0
0
.
7
5
0
0
.
0
4
(
)
/,
/
q
r
ad
s
m
s
0
0
0
0
0
0
0
0
0
Evaluation Warning : The document was created with Spire.PDF for Python.
26
I
SSN
:
20
89
-
4856
IJRA Vol. 4, No. 1, March 2015 : 19 – 30
I
n
s
i
m
u
la
tio
n
,
w
e
s
e
t
t
h
e
d
e
s
i
r
e
d
h
e
ig
h
t
o
f
h
ip
0.75
Hm
=
.
W
ith
d
i
f
f
e
r
e
n
t
d
e
s
ir
e
d
o
r
b
ita
l
e
n
e
r
g
ie
s
(
0
d
e
s
E
=
a
nd
0.5
d
e
s
E
=
)
,
t
he
s
i
m
ul
a
t
i
o
n
r
e
s
ul
t
s
v
a
l
i
d
a
t
e
t
he
o
nl
i
ne
w
a
l
ki
ng
p
a
t
t
e
r
n
ge
ne
r
a
t
i
ng
m
e
t
ho
d
a
nd
t
he
t
a
s
k
s
pa
c
e
c
on
t
r
ol
l
e
r
.
A
h
or
i
z
on
t
a
l
di
s
t
u
r
ba
n
c
e
w
i
t
h
m
a
g
n
i
t
u
de
of
100N
i
s
i
m
pos
e
d
on
t
he
c
e
n
t
e
r
of
t
he
t
o
r
s
o
a
t
2
.
5
s
,
a
n
d
it la
s
ts
f
o
r
0
.
1
s
.
A
n
o
n
li
n
e
a
r
c
o
m
p
lia
n
t
c
o
n
ta
c
t
m
o
d
e
l
w
it
h
C
o
u
lo
m
b
f
r
ic
tio
n
is
u
s
e
d
to
m
o
d
e
l
th
e
co
n
t
act
f
o
r
ces
b
et
w
e
e
n
t
h
e
r
obot
’
s
f
oot
a
n
d
t
h
e
g
r
o
u
n
d
a
s
de
s
c
r
i
be
d
i
n
[
4
,
24
]
.
F
ig
.
6
s
h
o
w
s
th
e
o
r
b
it
e
n
e
r
g
y
o
f
th
e
h
ip
in
8
s
e
c
o
n
d
s
.
W
ith
th
e
in
itia
l
c
o
n
f
i
g
u
r
a
tio
n
in
T
a
b
le
2
,
th
e
r
o
b
o
t’
s
C
O
M
is
lo
c
a
tin
g
a
t
t
h
e
r
ig
h
t
s
id
e
of
t
h
e
s
t
a
n
c
e
poi
nt
,
du
e
t
o
t
h
e
g
r
a
vi
t
y
t
h
e
r
obot
w
i
l
l
m
o
v
e
f
or
w
a
r
d,
t
ha
t
i
s
t
he
p
r
i
m
a
r
y
a
c
t
ua
t
i
o
n
o
f
t
he
w
a
l
ki
ng.
S
o
e
ve
n
w
h
e
n
t
he
hi
p
’
s
o
r
b
i
t
e
ne
r
g
y
i
s
z
e
r
o
,
t
he
r
o
b
o
t
w
i
l
l
no
t
c
om
e
t
o
a
s
t
op
a
s
t
h
e
bl
u
e
l
i
n
e
s
h
o
w
n
i
n
F
i
g
.
6.
I
t
c
a
n
be
f
oun
d
t
h
a
t
t
h
e
or
bi
t
e
ne
r
g
y
o
f
t
he
hi
p
i
s
no
t
a
co
n
s
t
a
n
t
d
u
r
i
n
g
a
n
y
s
t
an
ce
p
h
as
e.
T
h
at
i
s
b
ecau
s
e
t
h
e
l
e
g
s
ar
e
n
o
t
m
as
s
l
es
s
,
e
s
p
eci
al
l
y
t
h
e
s
w
i
n
g
l
eg
.
W
h
i
l
e
t
h
e
i
d
eal
L
I
P
M
as
s
u
m
es
t
h
e
l
eg
s
ar
e
m
as
s
l
es
s
.
N
e
v
er
t
h
e
l
es
s
,
t
h
e
r
o
b
o
t
w
al
k
s
w
i
t
h
a
p
er
i
o
d
i
c
g
ai
t
ev
en
s
u
f
f
e
r
i
n
g
t
he
s
u
dde
n
di
s
t
u
r
ba
nc
e
.
F
i
g
ur
e
.
6
T
he
o
r
b
i
t
e
ne
r
g
y
o
f
t
he
hi
p
d
ur
i
ng
w
a
l
ki
n
g.
T
he
gr
e
e
n
l
i
n
e
s
d
en
o
t
e
t
h
e
d
es
i
r
ed
o
r
b
i
t
en
er
g
y
.
T
he
s
o
lid
lin
e
s
d
e
n
o
te
t
h
e
o
r
b
it
e
n
e
r
g
y
o
f
t
h
e
h
ip
T
he
t
a
nge
nt
i
a
l
c
o
m
p
o
ne
nt
o
f
t
he
c
o
nt
a
c
t
f
o
r
c
e
o
f
t
he
r
i
g
ht
f
o
o
t
d
ur
i
ng
w
a
l
ki
n
g
i
s
s
ho
w
n
i
n
F
i
g.
7
w
i
t
h
s
o
l
i
d
l
i
ne
s
,
a
nd
t
he
gr
e
e
n
d
a
s
he
d
l
i
ne
s
d
e
no
t
e
t
he
b
o
und
o
f
f
r
i
c
t
i
o
n
f
o
r
c
e
w
hi
c
h
i
s
c
a
l
c
ul
a
t
e
d
b
y
m
u
l
t
i
p
l
y
i
n
g
t
h
e
n
o
r
m
al
co
m
p
o
n
en
t
o
f
t
h
e
co
n
t
act
f
o
r
ce
w
i
t
h
t
h
e
co
e
f
f
i
ci
en
t
o
f
f
r
i
ct
i
o
n
u
s
ed
i
n
t
h
e
co
n
t
act
m
o
d
e
l.
W
h
e
n
t
h
e
s
o
lid
lin
e
s
(
r
ed
o
r
b
l
u
e)
l
o
cat
e
i
n
s
i
d
e
t
h
e
r
an
g
e
f
o
r
m
ed
b
y
t
h
e
zer
o
g
r
i
d
l
i
n
e
an
d
t
h
e
g
r
ee
n
d
as
h
ed
l
i
n
e,
i
t
m
ean
s
t
h
e
f
r
i
ct
i
o
n
f
o
r
ce
l
o
cat
e
i
n
s
i
d
e
t
h
e
f
r
i
ct
i
o
n
co
n
e,
t
h
u
s
t
h
e
co
n
t
act
co
n
s
i
s
t
en
c
y
i
s
p
r
es
er
v
ed
.
F
i
g
ur
e
7
.
E
f
f
ect
s
o
f
s
el
ec
t
i
n
g
d
i
f
f
er
e
n
t
s
w
i
t
ch
i
n
g
u
n
d
er
d
y
n
a
m
ic
c
o
n
d
itio
n
C
o
n
s
i
d
er
i
n
g
t
h
e
p
r
act
i
cal
r
es
t
r
i
ct
i
o
n
o
f
t
h
e
act
u
at
o
r
s
,
i
n
s
i
m
u
l
at
i
o
n
w
e
i
m
p
o
s
e
u
p
p
er
an
d
l
o
w
er
l
i
m
i
t
s
o
n
t
he
a
c
t
ua
t
i
o
n
t
o
r
q
ue
s
a
s
s
ho
w
n
i
n
F
i
g.
8
a
nd
F
i
g.
9
.
0
1
2
3
4
5
6
7
8
-
0.
06
0
0.
5
1
1.
5
E
T
i
m
e
(
s
)
E
d=
0
E0
E
d=
0.
5
E1
0
1
2
3
4
5
6
7
8
-
600
-
100
0
100
300
F
c
(
N)
(
b)
E
d=
0.
5 T
i
m
e(
s
)
0
1
2
3
4
5
6
7
8
-
350
-
100
0
100
350
F
c
(
N)
(a
) E
d
=
0
F
y
*
μ
F
x
F
y
*
μ
F
x
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
RA
I
S
S
N
:
2088
-
8708
Online Biped Walking
Pattern Generation with Contact Consistency (Hou Wenqi)
27
F
i
g
ur
e
8
.
A
c
t
u
a
t
i
on
t
or
qu
e
of
e
a
c
h
l
i
nk
du
r
i
ng
w
a
l
ki
ng
w
he
n
0
d
e
s
E
=
F
i
g
ur
e
9
.
A
c
t
u
a
t
i
on
t
or
qu
e
of
e
a
c
h
l
i
nk
du
r
i
ng
w
a
l
ki
ng
w
he
n
0.5
d
e
s
E
=
6.
CO
NCL
U
S
I
O
N
I
n
t
h
i
s
p
ap
er
,
w
e
ad
d
r
es
s
t
h
e
i
s
s
u
e
o
f
w
a
l
k
i
n
g
co
n
t
r
o
l
o
f
a p
l
an
ar
b
i
p
ed
r
o
b
o
t
w
i
t
h
co
n
t
act
c
on
s
i
s
t
e
n
c
y
.
B
y
m
ode
l
i
n
g
t
he
m
ot
i
on
of
t
h
e
h
i
p
a
s
a
2D
-
L
I
P
M
,
a
f
o
o
t
p
la
c
e
m
e
n
t
c
o
n
tr
o
lle
r
w
h
ic
h
c
a
n
p
r
es
er
v
e
t
h
e
co
n
t
act
co
n
s
i
s
t
e
n
c
y
i
s
p
r
o
p
o
s
ed
t
o
t
r
ack
t
h
e
d
es
i
r
ed
o
r
b
i
t
en
er
g
y
.
B
y
s
el
ect
i
n
g
t
h
e
h
i
p
’
s
h
o
r
i
zo
n
t
al
l
o
co
m
o
t
i
o
n
a
s
t
h
e
p
ar
am
et
er
,
t
h
e
w
al
k
i
n
g
p
at
t
er
n
i
s
g
en
er
at
ed
o
n
t
h
e
r
eal
t
i
m
e.
A
t
as
k
s
p
ace
P
D
t
y
p
e
co
n
t
r
o
l
l
er
w
i
t
h
o
u
t
cal
c
u
l
at
i
n
g
t
h
e
i
n
v
er
s
i
o
n
o
f
i
n
er
t
i
al
m
at
r
i
x
i
s
d
es
i
g
n
ed
t
o
r
eal
i
ze
t
h
e
b
i
p
ed
w
al
k
i
n
g
.
S
i
m
ul
a
t
i
o
n
r
e
s
ul
t
s
ho
w
s
t
he
p
r
o
p
o
s
e
d
m
e
t
ho
d
no
t
o
nl
y
c
a
n
c
o
nt
r
o
l
t
he
r
o
bo
t
w
a
l
ki
ng
i
nt
o
p
e
r
i
od
i
c
ga
i
t
,
b
ut
a
l
s
o
c
a
n
w
i
t
hs
t
a
nd
a
s
ud
d
e
n
d
i
s
t
ur
b
a
n
c
e
.
T
h
e
f
u
t
u
r
e
w
or
k
w
i
l
l
f
oc
u
s
o
n
e
x
t
e
n
di
ng
t
h
e
m
e
t
hod
i
n
t
o
a
3D
bi
pe
d
r
obot
.
ACK
NO
W
L
E
D
G
E
M
E
NT
S
T
h
i
s
w
or
k
h
a
s
be
e
n
f
un
de
d
by
t
he
C
r
o
s
s
i
ng
S
p
e
c
ia
ltie
s
C
o
-
ed
u
cat
i
n
g
P
ro
g
ra
m
f
o
r
P
h
D
s
t
ud
e
nt
s
in
N
a
t
i
o
na
l
U
ni
ve
r
s
i
t
y
o
f
D
e
f
e
n
s
e
T
e
c
hno
l
o
g
y
(
N
o.
k
xk
14010
1)
,
a
nd
t
he
N
at
i
o
n
al
H
i
g
h
T
ech
n
o
l
o
g
y
R
es
ear
c
h
a
n
d
D
e
v
e
l
op
m
e
n
t
P
r
og
r
a
m
of
C
h
i
n
a
(
N
o.
2011A
A
040801
)
.
0
1
2
3
4
5
6
7
8
-5
0
0
50
τ
r1
(
N*
m
)
(a
)
0
1
2
3
4
5
6
7
8
-5
0
0
50
τ
r2
(
N*
m
)
(b
)
0
1
2
3
4
5
6
7
8
-5
0
0
50
τ
l1
(
N*
m
)
(c
)
0
1
2
3
4
5
6
7
8
-5
0
0
50
τ
l2
(
N*
m
)
T
i
m
e
(
s
)
0
1
2
3
4
5
6
7
8
-5
0
0
50
τ
r1
(
N*
m
)
0
1
2
3
4
5
6
7
8
-5
0
0
50
τ
r2
(
N*
m
)
0
1
2
3
4
5
6
7
8
-5
0
0
50
τ
l1
(
N*
m
)
0
1
2
3
4
5
6
7
8
-5
0
0
50
τ
l2
(
N*
m
)
T
i
m
e
(
s
)
Evaluation Warning : The document was created with Spire.PDF for Python.
28
I
SSN
:
20
89
-
4856
IJRA Vol. 4, No. 1, March 2015 : 19 – 30
AP
P
E
NDI
X
A.
P
ro
o
f
o
f
t
h
eo
rem
1
：
Ca
s
e
1
:
W
he
n
0
0.6
aa
E
or
E
X
≤≤
,
le
t
0.6
d
e
s
LX
=
,
s
u
b
s
tit
u
te
it i
n
to
e
q
u
a
tio
n
(6
)
2
0
0.32
g
E
E
E
X
c
ons
t
ant
y
+−
∆=
−
=
=
(
37
)
T
h
at
m
ea
n
s
w
h
e
n
t
h
e
cu
r
r
en
t
o
r
b
i
t
en
er
g
y
i
s
m
uc
h
l
e
s
s
t
ha
n
t
he
d
e
s
i
r
e
d
o
ne
,
w
e
t
a
ke
a
s
t
e
p
s
uc
h
t
ha
t
t
h
e
o
r
b
i
t
en
er
g
y
i
n
cr
eas
e
w
i
t
h
a co
n
s
t
an
t
v
al
u
e.
F
o
r
t
h
e
d
es
i
r
ed
o
r
b
i
t
en
er
g
y
i
s
f
i
n
i
t
e,
s
o
af
t
er
t
ak
i
n
g
f
i
n
i
t
e
st
e
p
s,
e
.
g
.
N
,
w
e
c
a
n
ha
ve
:
2
0
0.32
0,
N
d
e
s
N
g
E
E
E
X
y
∆
=
−
∈
(
38
)
wh
e
r
e
N
E
is
th
e
o
r
b
it
e
n
e
r
g
y
w
h
e
n
(
)
12
N
，
，
s
t
ep
s
h
av
e
b
een
t
a
k
e
n
.
T
h
en
w
e
h
av
e:
(
)
(
]
2
0
2
0.6
a
N
d
e
s
y
E
E
E
X
XX
g
=
−
+∈
，
(
39
)
F
o
r
th
e
n
e
x
t
s
te
p
th
e
p
o
in
t
m
a
s
s
w
ill
c
o
m
e
to
c
a
s
e
3
.
Ca
s
e
2
:
W
he
n
1.6
a
E
X
≥
,
le
t
1.6
d
e
s
LX
=
,
s
u
b
s
tit
u
te
it i
n
to
e
q
u
a
tio
n
(6
)
:
2
0
0.78
g
E
E
E
X
c
ons
t
ant
y
+−
∆=
−
=
−
=
(
40
)
T
h
at
m
ean
s
w
h
en
t
h
e
cu
r
r
en
t
o
r
b
i
t
en
er
g
y
i
s
m
u
ch
b
i
g
g
er
t
h
an
t
h
e
d
e
s
i
r
ed
o
n
e,
w
e
t
a
k
e
a
s
t
ep
s
u
ch
t
h
at
t
h
e
o
r
b
i
t
en
er
g
y
d
ecr
eas
e
w
i
t
h
a
co
n
s
t
a
n
t
v
al
u
e.
F
o
r
t
h
e
d
es
i
r
ed
o
r
b
i
t
en
er
g
y
i
s
f
i
n
i
t
e,
s
o
af
t
er
t
ak
i
n
g
fi
n
i
t
e
s
t
ep
s
,
e.
g
.
M
,
w
e
c
a
n
ha
ve
:
2
0
0.78
,0
M
d
e
s
M
g
EE
E
X
y
∆
=
−
∈−
(
41
)
wh
e
r
e
M
E
is
th
e
o
r
b
it
e
ne
r
g
y
w
h
e
n
(
)
1,
2
,
M
s
t
ep
s
h
av
e
b
een
t
a
k
e
n
.
T
h
en
w
e
h
av
e:
(
)
[
)
2
0
2
,
1.6
a
M
d
e
s
y
E
EE
X
X
X
g
=
−
+∈
(
42
)
F
o
r
th
e
n
e
x
t
s
te
p
th
e
p
o
in
t
m
a
s
s
w
ill
c
o
m
e
to
c
a
s
e
3
,
to
o
.
Ca
s
e
3
:
W
he
n
0.6
1.6
a
X
E
X
<
<
,
le
t
d
e
s
a
LE
=
,
s
u
b
s
tit
u
te
e
q
u
a
tio
n
(
15
)
in
to
(6
)
:
0
EE
E
+−
∆=
−
=
(
43
)
T
ha
t
m
e
a
ns
w
h
e
n
t
he
p
o
in
t
m
a
s
s
ta
k
e
a
s
te
p
to
th
e
d
e
s
ir
e
d
lo
c
a
tio
n
,
th
e
o
r
b
it
e
n
e
r
g
y
w
ill
b
e
e
q
u
a
l
to
t
h
e
d
es
i
r
ed
o
n
e.
F
r
o
m
e
q
ua
t
i
o
n
(
15
)
,
w
e
can
s
ee
t
h
at
w
h
e
n
t
h
e
cu
r
r
e
n
t
o
r
b
i
t
en
er
g
y
i
s
eq
u
al
t
o
t
h
e
d
es
i
r
ed
o
n
e,
d
e
s
LX
≡
T
he
n
f
r
o
m
e
q
ua
t
i
o
n
(3
)
，
xX
x
c
ons
t
ant
=
≡
,
a
nd
t
he
n
f
r
o
m
e
q
ua
t
i
o
n
(
45)
,
t
h
e
p
er
i
o
d
o
f
s
t
an
ce
p
h
as
e
i
s
a
c
o
n
s
ta
n
t
to
o
.
N
o
t
e
t
h
at
n
o
m
at
t
er
t
h
e
ca
s
es
,
0
0.6
1.6
0.8
d
e
s
X
L
X
y
µ
≤
≤
=
(
44
)
T
h
at
m
ea
n
s
t
h
e
co
n
t
ac
t
f
o
r
ce
r
es
t
r
i
ct
i
o
n
(
10
)
w
ill
a
l
w
a
y
s
b
e
s
a
tis
f
ie
d
.
A
b
o
v
e
al
l
,
w
e
can
co
n
cl
u
d
e
t
h
at
t
h
e
d
es
i
g
n
ed
X
a
nd
d
e
s
L
c
a
n
c
o
nt
r
o
l
t
he
p
o
i
nt
m
a
s
s
m
o
ve
f
r
o
m
a
n
y
p
e
r
m
i
tte
d
s
ta
te
i
n
to
a
p
e
r
io
d
g
a
it
w
i
th
c
o
n
ta
c
t
c
o
n
s
is
te
n
c
y
.
B.
D
eri
v
at
i
on
of
t
h
e
m
axi
m
u
m
val
u
e
of
or
b
i
t
e
n
e
r
g
y
ma
x
E
G
iv
e
n
a
n
i
n
it
ia
l
s
ta
te
(
)
(
)
(
)
0,
0
xx
a
n
d
th
e
f
in
a
l
s
ta
te
(
)
(
)
(
)
,
xt
xt
,
th
e
tr
a
n
s
it
io
n
ti
m
e
c
a
n
b
e
cacu
l
at
ed
as
[
17
]
:
(
)
(
)
(
)
(
)
ln
0
0
c
c
c
xt
T
xt
tT
x
Tx
+
=
+
(
45
)
Evaluation Warning : The document was created with Spire.PDF for Python.