I
AE
S In
t
er
na
t
io
na
l J
o
urna
l o
f
Ro
bo
t
ics a
nd
Aut
o
m
a
t
io
n
(
I
J
RA)
Vo
l.
1
5
,
No
.
1
,
Ma
r
ch
20
2
6
,
p
p
.
1
22
~
1
35
I
SS
N:
2722
-
2
5
8
6
,
DOI
:
1
0
.
1
1
5
9
1
/i
jr
a
.
v
1
5
i
1
.
pp
1
22
-
1
35
122
J
o
ur
na
l ho
m
ep
a
g
e
:
h
ttp
:
//ij
r
a
.
ia
esco
r
e.
co
m
M
o
deling
and
co
ntrol
o
f
a
3D
under
-
a
ctuated
bipedal
ro
bo
t
using
partia
l f
e
ed
ba
ck lineariza
tio
n
Ali G
ues
s
a
m
1,
2
,
F
o
ud
il Abde
s
s
em
ed
1
,
Abdelm
a
djid C
he
h
ha
t
2
1
LEA
L
a
b
o
r
a
t
o
r
y
,
D
e
p
a
r
t
m
e
n
t
o
f
E
l
e
c
t
r
o
n
i
c
s,
F
a
c
u
l
t
y
o
f
Te
c
h
n
o
l
o
g
y
,
U
n
i
v
e
r
si
t
y
o
f
M
o
s
t
e
f
a
B
e
n
B
o
u
l
a
i
d
,
B
a
t
n
a
,
A
l
g
e
r
i
a
2
D
e
p
a
r
t
me
n
t
o
f
M
e
c
h
a
n
i
c
a
l
E
n
g
i
n
e
e
r
i
n
g
,
F
a
c
u
l
t
y
o
f
S
c
i
e
n
c
e
s
a
n
d
Te
c
h
n
o
l
o
g
y
,
U
n
i
v
e
r
si
t
y
o
f
A
b
b
e
s
La
g
h
r
o
u
r
,
K
h
e
n
c
h
e
l
a
,
A
l
g
e
r
i
a
Art
icle
I
nfo
AB
S
T
RAC
T
A
r
ticle
his
to
r
y:
R
ec
eiv
ed
Au
g
6
,
2
0
2
5
R
ev
is
ed
J
an
2
1
,
2
0
2
6
Acc
ep
ted
Feb
2
1
,
2
0
2
6
Th
is
a
rti
c
le
p
re
se
n
ts
a
d
y
n
a
m
ic
m
o
d
e
li
n
g
a
n
d
c
o
n
tr
o
l
fra
m
e
wo
rk
fo
r
a
3
D
u
n
d
e
ra
c
tu
a
ted
fiv
e
-
li
n
k
b
i
p
e
d
a
l
r
o
b
o
t
wit
h
1
4
d
e
g
re
e
s
o
f
fre
e
d
o
m
(Do
F
)
a
n
d
e
ig
h
t
a
c
tu
a
t
o
rs.
Th
e
r
o
b
o
t
e
x
h
ib
i
ts
h
ig
h
ly
n
o
n
li
n
e
a
r,
stro
n
g
l
y
c
o
u
p
led
,
a
n
d
h
y
b
rid
d
y
n
a
m
ics
,
p
o
si
n
g
c
h
a
ll
e
n
g
e
s
fo
r
c
o
n
v
e
n
ti
o
n
a
l
c
o
n
tr
o
l
a
p
p
r
o
a
c
h
e
s.
To
a
d
d
re
ss
th
e
se
issu
e
s
a
n
d
in
tro
d
u
c
e
o
u
r
re
se
a
rc
h
c
o
n
tri
b
u
ti
o
n
,
a
p
a
rti
a
l
fe
e
d
b
a
c
k
li
n
e
a
riza
ti
o
n
(P
F
L)
-
b
a
s
e
d
trac
k
in
g
fra
m
e
wo
rk
is
p
ro
p
o
s
e
d
,
wh
ich
a
n
a
ly
ti
c
a
ll
y
d
e
c
o
u
p
les
th
e
sy
ste
m
in
to
a
c
tu
a
te
d
a
n
d
u
n
a
c
tu
a
ted
s
u
b
sy
s
tem
s,
e
n
a
b
li
n
g
e
fficie
n
t
re
a
l
-
ti
m
e
c
o
n
t
ro
l.
U
n
li
k
e
h
y
b
ri
d
z
e
ro
d
y
n
a
m
i
c
s
(HZD)
m
e
th
o
d
s
t
h
a
t
e
n
f
o
rc
e
v
irt
u
a
l
c
o
n
stra
in
ts
o
n
li
n
e
a
n
d
re
q
u
ire
o
ffli
n
e
g
a
it
o
p
ti
m
iza
ti
o
n
,
o
r
m
o
d
e
l
p
re
d
icti
v
e
c
o
n
tr
o
l
(M
P
C)
sc
h
e
m
e
s
th
a
t
a
re
o
n
li
n
e
o
p
ti
m
iza
ti
o
n
b
a
se
d
d
e
p
e
n
d
e
n
t
a
n
d
c
o
m
p
u
tatio
n
a
ll
y
d
e
m
a
n
d
in
g
,
th
e
p
ro
p
o
se
d
P
F
L
a
p
p
ro
a
c
h
a
c
h
iev
e
s
c
o
m
p
u
ta
ti
o
n
a
l
sim
p
li
c
it
y
a
n
d
fa
st
imp
le
m
e
n
tatio
n
th
ro
u
g
h
c
l
o
se
d
-
fo
rm
c
o
n
tr
o
l
law
s.
In
c
o
n
tras
t
t
o
z
e
ro
-
m
o
m
e
n
t
p
o
i
n
t
(Z
M
P
)
-
b
a
se
d
c
o
n
tr
o
ll
e
rs,
P
F
L
e
n
a
b
les
d
y
n
a
m
ic
u
n
d
e
ra
c
tu
a
ted
wa
lk
in
g
with
P
D
fe
e
d
b
a
c
k
fo
r
a
c
c
u
ra
te
traje
c
to
ry
trac
k
in
g
a
n
d
d
ist
u
rb
a
n
c
e
a
tt
e
n
u
a
ti
o
n
,
th
o
u
g
h
ro
b
u
stn
e
ss
t
o
larg
e
u
n
c
e
rtain
ti
e
s
a
n
d
d
ist
u
rb
a
n
c
e
s
m
a
y
re
q
u
ire
a
d
d
it
i
o
n
a
l
m
e
c
h
a
n
ism
s,
su
c
h
a
s
a
d
a
p
ti
v
e
c
o
n
tr
o
l,
sl
id
i
n
g
-
m
o
d
e
,
o
r
f
u
z
z
y
lo
g
ic
.
S
imu
latio
n
re
su
l
ts
o
f
th
e
a
p
p
l
ied
c
o
n
tr
o
l
m
e
th
o
d
d
e
m
o
n
stra
te
th
e
p
e
rio
d
ic
n
a
tu
re
a
n
d
sta
b
il
it
y
o
f
g
e
n
e
r
a
ted
wa
lk
in
g
g
a
it
s,
w
h
ich
p
r
o
v
e
s
th
e
e
ffe
c
ti
v
e
n
e
ss
a
n
d
re
li
a
b
il
i
ty
o
f
t
h
e
p
ro
p
o
se
d
c
o
n
tr
o
l
a
p
p
r
o
a
c
h
.
K
ey
w
o
r
d
s
:
B
ip
ed
al
r
o
b
o
ts
Hy
b
r
id
s
y
s
tem
s
No
n
lin
ea
r
co
n
tr
o
l
Par
tial f
ee
d
b
ac
k
lin
ea
r
izatio
n
Un
d
er
-
ac
tu
ated
r
o
b
o
ts
T
h
is i
s
a
n
o
p
e
n
a
c
c
e
ss
a
rticle
u
n
d
e
r th
e
CC B
Y
-
SA
li
c
e
n
se
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
Ali G
u
ess
am
Dep
ar
tm
en
t o
f
E
lectr
o
n
ics,
Fa
cu
lty
o
f
T
ec
h
n
o
l
o
g
y
,
Un
iv
er
s
i
ty
o
f
Mo
s
tef
a
B
en
B
o
u
laid
B
atn
a,
Alg
er
ia
E
m
ail: g
u
ess
am
_
ali@
u
n
iv
-
k
h
en
ch
ela.
d
z
1.
I
NT
RO
D
UCT
I
O
N
W
alk
in
g
r
o
b
o
ts
h
av
e
lo
n
g
b
ee
n
a
ce
n
t
r
al
f
o
c
u
s
o
f
r
o
b
o
tics
r
esear
ch
d
u
e
to
th
eir
ab
ilit
y
to
o
p
er
ate
in
h
u
m
an
-
ce
n
ter
ed
en
v
ir
o
n
m
en
t
s
.
T
h
eir
ca
p
ac
ity
to
r
ep
licate
h
u
m
an
-
lik
e
m
o
tio
n
en
ab
le
s
th
em
to
p
er
f
o
r
m
co
m
p
lex
task
s
in
h
az
ar
d
o
u
s
o
r
co
n
s
tr
ain
ed
s
ettin
g
s
,
in
clu
d
in
g
d
is
aster
zo
n
es,
r
ad
iatio
n
-
ex
p
o
s
ed
ar
ea
s
,
an
d
p
lan
etar
y
ex
p
lo
r
atio
n
m
is
s
io
n
s
[
1
]
–
[
3
]
.
T
h
is
ad
ap
tab
ilit
y
m
ak
es
th
em
in
d
is
p
en
s
ab
le
in
ap
p
licatio
n
s
wh
er
e
wh
ee
led
o
r
tr
ac
k
ed
s
y
s
tem
s
ar
e
h
in
d
er
e
d
b
y
ter
r
ain
ir
r
eg
u
lar
i
ties
an
d
ac
ce
s
s
ib
ilit
y
co
n
s
tr
ain
ts
.
Un
lik
e
wh
ee
led
o
r
q
u
ad
r
u
p
e
d
al
r
o
b
o
ts
,
b
ip
ed
al
s
y
s
tem
s
f
ac
e
d
is
tin
ct
ch
allen
g
es
in
m
ain
tain
in
g
b
alan
ce
,
s
tab
ilit
y
,
an
d
ef
f
icie
n
cy
,
p
a
r
ticu
lar
ly
in
3
D
en
v
i
r
o
n
m
en
ts
.
T
h
ese
ch
allen
g
es
s
tem
f
r
o
m
th
eir
h
ig
h
d
eg
r
ee
s
o
f
f
r
ee
d
o
m
(
D
o
F
)
,
s
tr
o
n
g
n
o
n
lin
ea
r
co
u
p
lin
g
,
u
n
d
er
-
ac
tu
atio
n
,
p
o
o
r
s
tab
ilit
y
,
a
n
d
h
y
b
r
id
d
y
n
am
ics
d
u
r
in
g
c
o
n
tact
tr
an
s
itio
n
s
[
4
]
.
C
o
n
tr
o
llin
g
s
u
ch
s
y
s
tem
s
,
esp
ec
ially
3
D
b
ip
ed
al
r
o
b
o
ts
,
is
an
aly
tically
an
d
co
m
p
u
tatio
n
ally
d
em
a
n
d
in
g
,
r
eq
u
ir
in
g
ac
cu
r
ate
m
o
d
elin
g
a
n
d
r
ea
l
-
tim
e
o
p
tim
izatio
n
to
e
n
s
u
r
e
ad
a
p
tiv
e
an
d
s
tab
le
lo
co
m
o
tio
n
ac
r
o
s
s
v
ar
ied
ter
r
ain
s
.
R
ec
en
t
ad
v
an
ce
s
in
co
n
tr
o
l
th
eo
r
y
,
o
p
tim
iza
tio
n
,
an
d
m
ac
h
in
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
AE
S
I
n
t
J
R
o
b
&
A
u
to
m
I
SS
N:
2722
-
2
5
8
6
Mo
d
elin
g
a
n
d
co
n
tr
o
l o
f a
3
D
u
n
d
er
-
a
ctu
a
ted
b
ip
ed
a
l ro
b
o
t
u
s
in
g
p
a
r
tia
l
feed
b
a
ck
…
(
A
li
Gu
ess
a
m
)
123
lear
n
in
g
h
av
e
d
r
iv
e
n
s
ig
n
if
ic
an
t
p
r
o
g
r
ess
,
im
p
r
o
v
in
g
m
o
tio
n
p
la
n
n
in
g
,
p
e
r
ce
p
tio
n
,
an
d
ad
ap
tiv
e
b
eh
av
io
r
s
th
r
o
u
g
h
d
ata
-
d
r
iv
en
co
n
tr
o
l
[
5
]
–
[
7
]
.
T
o
ad
d
r
ess
th
ese
ch
allen
g
es,
r
esear
ch
er
s
h
av
e
d
ev
elo
p
e
d
a
v
ar
iety
o
f
lo
co
m
o
tio
n
s
tr
ateg
ies
,
in
clu
d
in
g
o
p
tim
izatio
n
-
b
ased
co
n
tr
o
l,
b
io
-
in
s
p
ir
ed
m
eth
o
d
s
,
an
d
lear
n
in
g
-
d
r
iv
e
n
ap
p
r
o
ac
h
es.
So
m
e
r
ely
o
n
o
f
f
lin
e
tr
ajec
to
r
y
o
p
tim
izatio
n
u
s
in
g
f
u
ll
-
o
r
d
er
d
y
n
am
ic
m
o
d
els
[
8
]
,
wh
ile
o
th
er
s
u
s
e
s
im
p
lifie
d
r
ea
l
-
tim
e
m
o
d
els
th
at
ar
e
later
m
ap
p
e
d
to
f
u
ll
d
y
n
am
ics
[
9
]
.
E
ar
ly
co
n
tr
o
l
p
ar
a
d
ig
m
s
,
n
o
tab
ly
th
e
Z
er
o
Mo
m
e
n
t
Po
in
t
(
Z
MP)
cr
iter
io
n
,
estab
lis
h
ed
th
e
f
o
u
n
d
atio
n
f
o
r
m
ain
tain
i
n
g
d
y
n
am
ic
b
alan
ce
i
n
h
u
m
an
o
i
d
s
[
1
0
]
–
[
1
4
]
.
W
h
en
th
e
Z
MP
r
em
ain
s
with
in
th
e
s
u
p
p
o
r
t
p
o
ly
g
o
n
,
th
e
r
o
b
o
t
m
ain
tain
s
s
tatic
s
tab
ilit
y
—
an
ap
p
r
o
ac
h
s
u
cc
ess
f
u
lly
ap
p
lied
in
p
latf
o
r
m
s
s
u
ch
as
ASI
MO
,
H
R
P,
A
tlas
,
an
d
T
esla
Op
tim
u
s
[
1
5
]
–
[
1
9
]
.
Ho
wev
er
,
Z
MP
-
b
ased
co
n
tr
o
l
is
in
h
er
en
tly
lim
ited
t
o
q
u
asi
-
s
tatic
m
o
tio
n
an
d
ca
n
n
o
t
ac
co
m
m
o
d
ate
u
n
d
er
ac
tu
at
ed
b
ip
e
d
s
with
p
o
in
t
o
r
lin
e
f
ee
t
d
u
e
to
its
r
elian
ce
o
n
a
f
in
ite
s
u
p
p
o
r
t
r
eg
io
n
.
R
ec
en
t
s
tu
d
ies
o
n
ae
r
ial
s
y
s
tem
s
u
n
d
er
d
is
tu
r
b
an
ce
s
co
m
b
in
es
r
o
b
u
s
t
m
o
d
el
-
b
ased
co
n
tr
o
l
s
tr
ateg
ies,
s
u
ch
as
s
lid
in
g
m
o
d
e
c
o
n
tr
o
l
(
SMC
)
an
d
its
v
ar
ian
ts
,
with
in
tellig
en
t,
d
ata
-
d
r
iv
en
a
p
p
r
o
ac
h
es,
in
clu
d
i
n
g
n
eu
r
al
n
etwo
r
k
s
,
f
u
zz
y
lo
g
ic,
an
d
a
d
ap
tiv
e
n
eu
r
o
-
f
u
zz
y
in
f
e
r
en
ce
s
y
s
tem
s
[
2
0
]
–
[
2
6
]
.
T
h
is
h
y
b
r
id
f
r
am
ew
o
r
k
ef
f
ec
tiv
ely
b
alan
ce
s
r
ig
o
r
o
u
s
s
tab
ilit
y
g
u
ar
an
tees
with
th
e
ad
ap
tiv
e
lear
n
in
g
ca
p
ab
ilit
ies
r
eq
u
ir
e
d
to
h
a
n
d
le
co
m
p
lex
with
ex
ter
n
al
d
is
tu
r
b
an
ce
s
an
d
p
ar
am
ete
r
v
ar
ia
tio
n
s
,
n
o
n
lin
ea
r
,
an
d
u
n
p
r
ed
ictab
le
r
ea
l
-
wo
r
ld
d
y
n
am
ics.
I
ts
v
er
s
atility
is
d
e
m
o
n
s
tr
ated
ac
r
o
s
s
a
r
an
g
e
o
f
p
latf
o
r
m
s
,
f
r
o
m
ae
r
ial
s
y
s
tem
s
u
n
d
er
d
is
tu
r
b
a
n
ce
s
[
2
4
]
an
d
leg
g
ed
r
o
b
o
ts
[
2
1
]
to
m
o
b
ile
m
an
ip
u
lato
r
s
a
n
d
r
o
b
o
tic
ar
m
s
[
2
2
]
,
[
2
3
]
,
[
2
5
]
.
T
h
e
s
e
f
r
am
ewo
r
k
s
ar
e
p
ar
ticu
lar
ly
v
al
u
ab
le
f
o
r
en
h
an
cin
g
b
ip
ed
al
m
o
b
ilit
y
an
d
ass
is
tiv
e
tech
n
o
lo
g
ies,
in
c
lu
d
in
g
lo
wer
-
lim
b
ex
o
s
k
eleto
n
s
an
d
p
r
o
s
th
eses
[
2
0
]
,
[
2
6
]
wh
e
r
e
th
e
y
s
im
u
ltan
eo
u
s
ly
p
r
o
v
id
e
s
tab
ilit
y
,
ag
ilit
y
,
an
d
r
esil
ien
ce
i
n
au
to
n
o
m
o
u
s
s
y
s
tem
s
.
T
o
o
v
er
c
o
m
e
t
h
ese
li
m
it
ati
o
n
s
,
t
h
e
h
y
b
r
i
d
ze
r
o
d
y
n
a
m
ic
s
(
H
Z
D
)
f
r
a
m
ew
o
r
k
w
as
p
r
o
p
o
s
e
d
as
a
d
y
n
am
ics
-
c
o
n
s
is
te
n
t
alt
er
n
ati
v
e.
H
Z
D
e
n
f
o
r
c
es
v
ir
tu
al
c
o
n
s
t
r
ain
ts
t
h
at
r
e
d
u
c
e
t
h
e
s
y
s
te
m
’
s
n
o
n
li
n
ea
r
d
y
n
am
i
cs
to
a
l
o
w
-
d
i
m
e
n
s
i
o
n
al
i
n
v
a
r
i
an
t
m
a
n
i
f
o
l
d
,
e
n
ab
li
n
g
f
o
r
m
a
l
s
t
ab
i
lit
y
a
n
a
ly
s
is
a
n
d
f
e
e
d
b
ac
k
-
b
as
ed
g
a
it
d
esi
g
n
.
T
h
is
f
r
am
ew
o
r
k
h
as
d
em
o
n
s
t
r
at
e
d
p
r
o
v
a
b
l
y
s
t
a
b
le
l
o
c
o
m
o
tio
n
in
u
n
d
e
r
a
ct
u
at
e
d
r
o
b
o
ts
s
u
c
h
as
R
AB
B
I
T
,
AT
R
I
A
S,
C
ass
ie
,
an
d
th
e
k
n
e
e
-
l
ess
S
L
I
DE
R
[
2
7
]
,
[
2
8
]
,
a
n
d
h
as
e
v
o
l
v
e
d
f
r
o
m
p
l
an
ar
m
o
d
els
t
o
m
u
lti
-
d
o
m
ai
n
3
D
wal
k
i
n
g
[
2
9
]
–
[
3
1
]
.
H
o
w
e
v
e
r
,
d
es
p
it
e
its
s
u
cc
ess
,
HZ
D
r
e
m
ai
n
s
h
ig
h
l
y
m
o
d
e
l
-
d
e
p
en
d
en
t
,
o
f
t
en
r
e
q
u
ir
i
n
g
o
f
f
l
in
e
t
r
aje
ct
o
r
y
o
p
t
im
i
za
t
i
o
n
,
a
n
d
s
h
o
ws
li
m
i
te
d
r
o
b
u
s
t
n
ess
t
o
m
o
d
e
li
n
g
e
r
r
o
r
s
a
n
d
n
o
n
-
p
er
io
d
ic
d
is
t
u
r
b
a
n
ce
s
—
p
r
o
m
p
t
in
g
r
ese
ar
ch
i
n
t
o
a
d
a
p
ti
v
e
an
d
le
ar
n
i
n
g
-
b
ase
d
HZ
D
ex
te
n
s
i
o
n
s
f
o
r
e
n
h
a
n
c
e
d
f
le
x
i
b
i
lit
y
.
On
th
e
o
th
er
h
an
d
,
m
o
d
el
p
r
e
d
ictiv
e
co
n
tr
o
l
(
MPC
)
b
ased
o
n
r
e
d
u
ce
d
o
r
d
er
s
y
s
tem
,
h
as
em
er
g
ed
as
a
r
o
b
u
s
t
f
r
am
ewo
r
k
th
at
d
ir
ec
tly
h
an
d
les
co
m
p
lex
d
y
n
am
ics,
co
n
s
tr
ain
ts
,
an
d
r
ea
l
-
tim
e
o
p
tim
izatio
n
.
B
y
p
r
ed
ictin
g
f
u
t
u
r
e
s
tates
an
d
o
p
tim
izin
g
co
n
tr
o
l
in
p
u
ts
o
v
er
a
f
in
ite
h
o
r
izo
n
,
MPC
en
ab
les
ad
ap
tiv
e,
d
is
tu
r
b
an
ce
-
r
esil
ien
t
walk
in
g
ev
en
o
n
u
n
ev
en
ter
r
ain
,
an
d
s
u
p
p
o
r
ts
o
n
lin
e
g
ait
g
en
er
atio
n
f
o
r
im
p
r
o
v
e
d
r
esp
o
n
s
iv
en
ess
[
3
2
]
–
[
3
4
]
.
R
ec
en
t
p
r
o
g
r
ess
in
n
o
n
lin
ea
r
MP
C
(
NM
P
C
)
h
as
ex
ten
d
ed
th
ese
ca
p
ab
ilit
ies
to
f
u
ll
-
o
r
d
er
s
y
s
tem
s
,
ac
h
iev
in
g
wh
o
le
-
b
o
d
y
c
o
n
tr
o
l
o
n
r
o
b
o
ts
s
u
ch
as
AM
B
E
R
-
3
M,
T
AL
OS,
ANYm
al,
an
d
AT
L
AS
[
3
5
]
–
[
3
8
]
.
Fu
r
th
e
r
m
o
r
e,
h
y
b
r
id
MPC
f
r
am
ewo
r
k
s
i
n
teg
r
atin
g
r
ein
f
o
r
ce
m
en
t
lear
n
in
g
(
R
L
)
o
r
wh
o
le
-
b
o
d
y
t
o
r
q
u
e
o
p
tim
izatio
n
h
av
e
en
h
an
ce
d
ad
ap
ta
b
ilit
y
an
d
s
ta
b
ilit
y
in
u
n
p
r
ed
ictab
le
e
n
v
ir
o
n
m
en
ts
[
3
9
]
–
[
4
2
]
.
W
h
ile
HZ
D
o
f
f
er
s
f
o
r
m
al
g
u
a
r
an
tees
o
f
s
tab
ilit
y
,
it
o
f
ten
r
el
ies
o
n
in
t
r
icate
v
ir
tu
al
co
n
s
tr
ain
ts
an
d
is
m
o
r
e
s
en
s
itiv
e
to
m
o
d
el
u
n
ce
r
tain
ties
.
Mo
d
el
p
r
ed
ictiv
e
co
n
tr
o
l
(
MPC
)
,
in
co
n
tr
ast,
o
f
f
er
s
m
o
r
e
s
o
p
h
is
ticated
p
er
f
o
r
m
an
ce
an
d
p
r
o
v
i
d
es
g
r
e
ater
ad
ap
tab
ilit
y
an
d
co
n
s
tr
ain
t
m
an
ag
em
en
t,
b
u
t
in
cu
r
ex
ten
s
iv
e
co
m
p
u
tatio
n
al
r
eso
u
r
ce
s
an
d
r
ely
o
n
p
r
ec
is
e
d
y
n
am
ic
m
o
d
els,
m
ak
i
n
g
r
ea
l
-
tim
e
im
p
lem
en
tatio
n
c
h
allen
g
in
g
in
t
h
e
u
n
s
tr
u
ctu
r
ed
en
v
ir
o
n
m
e
n
ts
ty
p
ical
o
f
b
ip
ed
al
r
o
b
o
ts
.
T
o
o
v
er
c
o
m
e
th
ese
lim
itatio
n
s
,
th
is
s
tu
d
y
p
r
o
p
o
s
es
a
n
o
v
e
l,
s
im
p
le,
an
d
co
m
p
u
tatio
n
ally
ef
f
icien
t
co
n
tr
o
l
f
r
am
ewo
r
k
th
at
co
m
b
in
es
PF
L
with
PD
co
n
tr
o
l
f
o
r
u
n
d
er
ac
t
u
ated
3
D
b
ip
e
d
a
l
r
o
b
o
ts
,
e
n
ab
lin
g
d
y
n
am
ic
d
ec
o
u
p
lin
g
an
d
r
o
b
u
s
t g
ait
s
tab
ilizatio
n
with
r
ed
u
c
ed
im
p
lem
en
tatio
n
co
m
p
le
x
ity
.
T
h
e
k
ey
c
o
n
tr
ib
u
tio
n
s
o
f
th
is
r
esear
ch
p
ap
er
a
r
e
s
u
m
m
ar
ize
d
b
elo
w
˗
Der
iv
atio
n
o
f
f
lo
atin
g
-
b
ase
h
y
b
r
id
s
y
s
tem
m
o
d
el
f
o
r
u
n
d
er
ac
tu
ated
3
D
b
ip
e
d
al
r
o
b
o
ts
,
ca
p
tu
r
in
g
b
o
th
ac
tu
ated
an
d
u
n
ac
tu
ated
d
y
n
a
m
ics.
˗
Desig
n
o
f
a
n
o
v
el
n
o
n
lin
ea
r
PF
L
co
n
tr
o
ller
,
th
at
s
elec
tiv
ely
lin
ea
r
izes
th
e
ac
tu
ated
s
u
b
s
y
s
tem
wh
ile
r
ig
o
r
o
u
s
ly
ac
co
u
n
tin
g
f
o
r
th
e
d
y
n
am
ic
co
u
p
lin
g
with
it
s
u
n
ac
tu
ated
Do
F
,
p
r
o
v
ab
ly
y
ield
s
s
tab
le
d
y
n
am
ic
walk
in
g
.
˗
C
o
m
p
ar
ativ
e
e
v
alu
atio
n
o
f
th
e
p
r
o
p
o
s
ed
PF
L
+PD
f
r
am
ewo
r
k
ag
ain
s
t
HZ
D
an
d
MPC
ap
p
r
o
ac
h
es
in
ter
m
s
o
f
s
tab
ilit
y
,
r
o
b
u
s
tn
ess
,
an
d
r
ea
l
-
tim
e
f
e
asib
ilit
y
.
˗
Nu
m
er
ical
v
alid
atio
n
d
em
o
n
s
tr
ates
s
tab
le,
p
er
io
d
ic
g
ai
t
g
en
er
atio
n
,
co
n
f
ir
m
in
g
ef
f
icien
cy
,
an
d
r
o
b
u
s
tn
ess
o
f
p
r
o
p
o
s
ed
co
n
tr
o
l a
p
p
r
o
ac
h
.
T
h
e
p
r
o
p
o
s
ed
ap
p
r
o
ac
h
p
r
o
v
id
es a
d
is
tin
ct
alter
n
ativ
e
to
th
e
d
o
m
in
an
t m
o
d
el
-
b
ased
p
ar
a
d
ig
m
s
o
f
Z
MP,
HZ
D,
an
d
MPC
.
T
h
e
r
em
ain
d
er
o
f
th
e
wo
r
k
is
o
r
g
an
ized
in
th
r
ee
p
r
in
cip
al
s
ec
tio
n
s
.
Sectio
n
2
was
d
iv
id
ed
in
to
th
r
ee
m
ain
s
u
b
s
ec
tio
n
s
:
k
i
n
em
atics,
d
y
n
am
ics,
an
d
c
o
n
tr
o
l.
Sec
tio
n
3
tilt
ed
“r
esu
lts
a
n
d
d
is
cu
s
s
io
n
s
”
d
is
p
lay
s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
7
2
2
-
2
5
8
6
I
AE
S
I
n
t
J
R
o
b
&
A
u
to
m
,
Vo
l
.
1
5
,
No
.
1
,
Ma
r
ch
20
2
6
:
1
22
-
1
35
124
s
im
u
latio
n
o
u
tco
m
es.
Fin
ally
,
s
ec
tio
n
4
d
escr
ib
es,
s
u
m
m
ar
izes
th
e
s
im
u
latio
n
r
esu
lts
,
an
d
h
ig
h
lig
h
t
th
e
p
er
s
p
ec
tiv
e
r
eg
a
r
d
in
g
f
u
tu
r
e
w
o
r
k
s
.
2.
M
E
T
H
O
D
2
.
1
.
Ro
bo
t
Str
uct
ure
a
nd
co
nfig
ura
t
io
n o
f
pa
ra
m
et
er
iza
t
io
n sp
a
ce
A
3
D
b
ip
ed
r
o
b
o
t
is
ess
en
tiall
y
a
f
lo
atin
g
-
b
ased
m
u
lti
-
r
ig
id
-
b
o
d
y
s
y
s
tem
wh
o
s
e
b
o
d
ies
ar
e
co
u
p
led
in
a
k
i
n
em
atic
tr
ee
s
tr
u
ct
u
r
e.
T
o
d
escr
ib
e
th
e
r
o
b
o
t'
s
f
lo
atin
g
b
ase,
let
0
b
e
a
f
ix
e
d
wo
r
l
d
f
r
am
e,
an
d
b
e
a
co
o
r
d
in
ate
f
r
am
e
with
its
o
r
ig
in
r
ig
id
ly
f
i
x
ed
at
th
e
ce
n
ter
o
f
th
e
waist,
an
d
with
th
e
x
an
d
z
ax
is
p
o
in
tin
g
f
o
r
war
d
a
n
d
u
p
war
d
,
r
esp
ec
tiv
ely
.
T
o
th
is
e
n
d
,
th
e
g
en
er
alize
d
f
lo
atin
g
-
b
ase
c
o
o
r
d
in
ates,
=
[
,
Φ
,
]
∈
=
3
×
(
3
)
×
⊂
+
6
ar
e
u
s
ed
,
wh
er
e
=
[
,
,
]
∈
3
an
d
Φ
=
[
,
,
]
∈
(
3
)
r
ep
r
ese
n
t
th
e
g
lo
b
al
p
o
s
itio
n
a
n
d
th
e
o
r
ien
tatio
n
(
e
.
g
.
,
E
u
ler
an
g
les)
o
f
th
e
b
o
d
y
b
ase
f
r
a
m
e
,
r
elativ
e
to
th
e
wo
r
ld
f
r
a
m
e
0
.
T
h
e
r
em
ain
in
g
c
o
o
r
d
in
ates
th
at
ch
ar
ac
ter
ize
th
e
r
o
b
o
t’
s
s
h
ap
e
ar
e
g
iv
en
b
y
lo
ca
l
co
o
r
d
in
ates
an
g
les
⊂
d
ep
ictin
g
r
ev
o
lu
te
jo
in
ts
in
ter
co
n
n
ec
tin
g
r
ig
id
lin
k
s
o
f
th
e
r
o
b
o
t
as
s
h
o
wn
in
Fig
u
r
e.
1
an
d
with
th
e
r
o
b
o
t states
d
en
o
ted
b
y
=
(
,
̇
)
.
T
h
is
p
ap
er
p
r
esen
ts
a
3
D
b
ip
e
d
r
o
b
o
t
s
tr
u
ctu
r
e
em
b
o
d
ied
in
th
r
ee
k
in
em
atic
ch
ain
s
:
a
to
r
s
o
an
d
two
s
y
m
m
etr
ic
an
d
id
e
n
tical
leg
s
.
E
ac
h
leg
s
h
o
wn
in
Fig
u
r
e
1
ca
n
b
e
m
o
d
eled
as
a
k
in
em
atic
c
h
ain
with
two
lin
k
s
co
n
n
ec
ted
b
y
f
o
u
r
r
e
v
o
lu
te
jo
i
n
ts
,
n
am
ely
a
3
Do
F h
ip
,
an
d
1
Do
F k
n
ee
.
T
h
e
u
p
p
e
r
b
o
d
y
(
to
r
s
o
)
,
f
lo
atin
g
b
ase
lin
k
,
h
as si
x
Do
F
(
3
tr
an
s
latio
n
al
an
d
3
r
o
tatio
n
al)
th
a
t a
r
e
n
o
t a
ctu
a
ted
.
T
h
er
e
f
o
r
e,
th
e
p
r
e
lim
in
ar
y
k
in
em
atic
m
o
d
el
p
o
s
s
ess
e
s
1
4
Do
F
,
as sh
o
wn
in
Fig
u
r
e
1
.
Fig
u
r
e
1
.
Mo
d
el
o
f
b
ip
ed
r
o
b
o
t a
n
d
f
r
a
m
es u
s
ed
to
d
escr
ib
e
its
co
n
f
ig
u
r
atio
n
: a
f
r
am
e
R
b
is
att
ac
h
ed
to
th
e
to
r
s
o
lin
k
,
a
n
d
t
he
r
o
b
o
t’
s
p
o
s
itio
n
an
d
o
r
ien
tatio
n
a
r
e
ex
p
r
ess
ed
r
elativ
e
to
a
f
ix
e
d
wo
r
ld
f
r
am
e
0
2
.
2
.
G
ener
a
lized
co
nfig
ura
t
i
o
n
T
h
e
g
en
er
alize
d
c
o
o
r
d
i
n
ates
,
g
iv
e
n
b
y
T
a
b
le
1
,
f
o
r
1
4
-
D
o
F
b
i
p
e
d
r
o
b
o
t
ca
n
b
e
ch
o
s
en
as
(
1
)
.
=
[
,
,
,
,
,
,
ℎ
,
ℎ
,
ℎ
,
,
ℎ
,
ℎ
,
ℎ
,
]
=
(
1
,
2
,
…
,
14
)
(
1
)
T
h
e
s
t
u
d
ie
d
m
o
d
el
is
o
n
l
y
e
q
u
ip
p
e
d
w
it
h
0
8
ac
t
u
at
o
r
s
in
t
h
e
two
le
g
s
.
E
a
c
h
d
ir
ec
tl
y
c
o
n
t
r
o
l
s
its
c
o
r
r
esp
o
n
d
i
n
g
an
g
l
es
l
is
te
d
i
n
T
a
b
l
e
1
.
Ho
we
v
e
r
,
w
e
ass
u
m
e
t
h
at
th
e
tr
a
n
s
la
tio
n
a
l to
r
s
o
a
n
d
t
h
r
ee
o
t
h
e
r
D
o
F
a
r
e
p
ass
i
v
es
,
t
h
at
is
,
=
=
=
0
an
d
=
=
=
0
.
As
a
r
esu
lt,
t
h
e
1
4
-
D
o
F
r
o
b
o
t
m
o
d
el
,
t
h
a
t
we
s
t
u
d
y
,
h
as
o
n
l
y
0
8
ac
t
u
a
to
r
s
a
n
d
h
e
n
c
e,
h
as
6
d
e
g
r
e
es
o
f
u
n
d
er
a
ct
u
a
ti
o
n
.
2
.
3
.
Ro
bo
t
pa
ra
m
et
er
s
Fig
u
r
e
2
illu
s
tr
ates
d
if
f
er
e
n
t
p
ar
am
eter
s
u
s
ed
in
k
in
em
atic
s
an
d
d
y
n
am
ics,
in
clu
d
i
n
g
lin
k
len
g
th
s
,
m
ass
es,
ce
n
ter
of
m
ass
es,
an
d
in
er
tias
.
T
h
e
to
r
s
o
is
ch
a
r
ac
ter
ized
b
y
its
m
ass
,
ce
n
ter
o
f
m
ass
f
r
o
m
its
p
r
o
x
im
al
en
d
,
len
g
t
h
,
an
d
m
o
m
en
t
o
f
in
er
tia
ab
o
u
t
its
ce
n
ter
o
f
m
ass
.
E
ac
h
th
ig
h
lin
k
h
as
m
ass
ℎ
,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
AE
S
I
n
t
J
R
o
b
&
A
u
to
m
I
SS
N:
2722
-
2
5
8
6
Mo
d
elin
g
a
n
d
co
n
tr
o
l o
f a
3
D
u
n
d
er
-
a
ctu
a
ted
b
ip
ed
a
l ro
b
o
t
u
s
in
g
p
a
r
tia
l
feed
b
a
ck
…
(
A
li
Gu
ess
a
m
)
125
ce
n
ter
o
f
m
ass
at
ℎ
,
len
g
th
ℎ
,
an
d
in
er
tia
ab
o
u
t its
ce
n
ter
o
f
m
ass
is
ℎ
.
Similar
ly
,
ea
ch
s
h
an
k
lin
k
h
as m
ass
ℎ
,
ce
n
ter
o
f
m
ass
at
ℎ
,
len
g
th
ℎ
,
an
d
in
er
tia
ℎ
.
Fin
ally
,
th
e
d
is
t
an
ce
b
etwe
en
th
e
two
h
ip
jo
i
n
ts
is
d
en
o
ted
b
y
.
T
ab
le
1
.
C
o
o
r
d
in
ates d
ef
in
itio
n
f
o
r
t
h
e
p
r
o
p
o
s
ed
r
o
b
o
t
an
d
jo
in
t a
ctu
atio
n
s
C
o
o
r
d
i
n
a
t
e
D
e
scri
p
t
i
o
n
A
c
t
u
a
t
o
r
1
C
a
r
t
e
si
a
n
p
o
s
i
t
i
o
n
-
2
C
a
r
t
e
s
i
a
n
p
o
si
t
i
o
n
-
3
C
a
r
t
e
si
a
n
p
o
s
i
t
i
o
n
-
4
P
e
l
v
i
s r
o
l
l
a
n
g
l
e
-
5
P
e
l
v
i
s
p
i
t
c
h
a
n
g
l
e
-
6
P
e
l
v
i
s
y
a
w
a
n
g
l
e
ψ
-
7
Le
f
t
l
e
g
a
n
k
l
e
r
o
l
l
a
n
g
l
e
ℎ
1
8
Le
f
t
l
e
g
a
n
k
l
e
p
i
t
c
h
a
n
g
l
e
ℎ
2
9
Le
f
t
l
e
g
a
n
k
l
e
y
a
w
a
n
g
l
e
ℎ
3
10
Le
f
t
l
e
g
k
n
e
e
p
i
t
c
h
a
n
g
l
e
4
11
R
i
g
h
t
l
e
g
a
n
k
l
e
r
o
l
l
a
n
g
l
e
ℎ
5
12
R
i
g
h
t
l
e
g
a
n
k
l
e
p
i
t
c
h
a
n
g
l
e
ℎ
6
13
R
i
g
h
t
l
e
g
a
n
k
l
e
y
a
w
a
n
g
l
e
ℎ
7
14
R
i
g
h
t
l
e
g
k
n
e
e
p
i
t
c
h
a
n
g
l
e
8
Fig
u
r
e
2
.
Hu
m
a
n
o
i
d
m
o
d
el
:
m
ass
,
i
n
e
r
ti
a
ab
o
u
t
c
en
te
r
o
f
m
as
s
,
ce
n
te
r
o
f
m
ass
es
,
an
d
le
n
g
t
h
p
a
r
a
m
e
te
r
s
Gen
er
all
y
,
th
e
c
o
o
r
d
in
ates
o
f
a
r
o
b
o
t
en
d
-
ef
f
ec
to
r
tr
an
s
f
o
r
m
atio
n
m
atr
ix
r
elativ
e
to
th
e
g
l
o
b
al
f
r
am
e
ca
n
b
e
d
eter
m
i
n
ed
u
s
in
g
th
e
Den
av
it
-
Har
ten
b
er
g
(
D
-
H)
m
e
th
o
d
o
r
o
th
er
m
eth
o
d
s
.
I
n
th
is
p
ap
er
,
we
ap
p
ly
th
e
an
g
le/ax
is
m
eth
o
d
b
ased
o
n
th
e
Z
er
o
r
e
f
er
en
ce
c
o
n
f
ig
u
r
atio
n
.
2
.
4
.
Dy
na
m
ics
W
e
u
s
e
th
e
f
l
o
at
in
g
b
as
e
c
o
o
r
d
i
n
at
es
t
o
d
er
iv
e
t
h
e
d
y
n
am
ic
s
i
n
b
o
t
h
s
w
in
g
p
h
ase
an
d
i
m
p
a
ct
e
v
e
n
t
(
s
e
e
[
2
7
]
,
[
4
3
]
)
.
2
.
4
.
1
.
Co
ntinuo
us
dy
na
m
ics
T
h
e
c
o
n
ti
n
u
o
u
s
d
y
n
a
m
ics
o
f
t
h
e
s
wi
n
g
p
h
as
e
in
t
h
e
E
u
le
r
-
L
ag
r
a
n
g
e
f
o
r
m
alis
m
f
o
r
th
e
f
l
o
ati
n
g
-
b
as
e
s
y
s
te
m
[
4
4
]
,
[
4
5
]
,
ta
k
es
t
h
e
f
o
r
m
(
)
̈
+
(
,
̇
)
=
Γ
+
(
)
(
2
)
w
h
e
r
e
(
,
̇
)
=
(
,
̇
)
̇
+
(
)
(
3
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
7
2
2
-
2
5
8
6
I
AE
S
I
n
t
J
R
o
b
&
A
u
to
m
,
Vo
l
.
1
5
,
No
.
1
,
Ma
r
ch
20
2
6
:
1
22
-
1
35
126
Γ
=
[
0
6
1
1
2
.
.
8
]
=
(
4
)
w
h
er
e
n
is
th
e
n
u
m
b
e
r
o
f
r
o
b
o
t
jo
in
ts
(
=
8
in
o
u
r
m
o
d
el)
,
(
)
∈
ℝ
(
+
6
)
×
(
+
6
)
d
en
o
tes
s
y
m
m
etr
ic
in
er
tial
m
atr
ix
,
(
,
̇
)
∈
ℝ
(
+
6
)
th
e
n
o
n
-
li
n
ea
r
ter
m
s
co
n
s
is
tin
g
in
C
o
r
io
lis
an
d
C
e
n
tr
if
u
g
al
an
d
g
r
av
itatio
n
al,
∈
ℝ
s
tan
d
s
f
o
r
t
h
e
ac
tu
ate
d
jo
in
t
to
r
q
u
es,
=
[
0
6
×
,
×
]
∈
ℝ
(
+
6
)
×
is
th
e
ac
t
u
ato
r
d
is
tr
ib
u
tio
n
m
atr
ix
(
)
∈
ℝ
(
6
)
×
(
+
6
)
d
en
o
tes
a
s
u
p
p
o
r
t
J
ac
o
b
ia
n
o
f
th
e
h
o
lo
n
o
m
ic
co
n
s
tr
ain
ts
,
d
ep
en
d
in
g
o
n
th
e
n
u
m
b
e
r
o
f
s
u
p
p
o
r
ts
,
an
d
∈
ℝ
6
is
th
e
ex
ter
n
al
wr
en
ch
co
n
tain
in
g
t
h
e
g
r
o
u
n
d
r
ea
ctio
n
f
o
r
ce
s
(
GR
F)
an
d
m
o
m
en
ts
[
4
6
]
,
(
e.
g
.
,
=
2
f
o
r
r
o
b
o
ts
in
d
o
u
b
l
e
s
u
p
p
o
r
t
p
h
ase,
with
n
o
ad
d
iti
o
n
al
g
r
o
u
n
d
co
n
tact)
.
R
em
a
r
k
:
Fo
r
p
o
in
t
-
f
ee
t
b
ip
ed
al
r
o
b
o
ts
,
o
n
ly
e
x
ter
n
al
r
ea
cti
o
n
f
o
r
ce
s
a
r
e
p
r
esen
t
with
n
o
to
r
q
u
e
co
m
p
o
n
e
n
ts
in
tr
o
d
u
ce
d
.
T
h
u
s
,
we
h
av
e
∈
ℝ
3
an
d
(
)
∈
ℝ
(
3
)
×
(
+
6
)
.
C
o
n
s
eq
u
en
tly
,
th
e
s
tate
-
s
p
ac
e
r
ep
r
esen
tatio
n
o
f
th
e
d
y
n
am
ic
s
in
(
2
)
ca
n
b
e
r
ewr
itten
as
(
5
a
)
.
[
̇
]
=
[
̇
−
1
(
)
[
−
(
,
̇
)
+
(
)
]
]
⏟
(
)
+
[
0
−
1
(
)
]
⏟
(
)
(
5
a)
Fo
r
later
u
s
e
in
co
n
tr
o
l d
esig
n
an
d
s
im
u
latio
n
,
th
e
(
5
a)
is
ex
p
r
ess
ed
in
th
e
af
f
in
e
s
tate
-
s
p
ac
e
co
n
tr
o
l
f
o
r
m
as
̇
=
(
)
+
(
)
(
5
b
)
wh
er
e
≔
{
(
,
̇
)
|
∈
,
̇
∈
ℝ
14
}
is
th
e
s
ta
te
o
f
th
e
s
y
s
tem
an
d
∈
ℝ
ar
e
th
e
co
n
tr
o
l
in
p
u
ts
.
L
et
Φ
(
)
d
en
o
tes
th
e
p
o
s
itio
n
o
f
s
tan
ce
f
o
o
t.
Sin
ce
it
is
co
n
s
tr
ain
ed
to
r
em
ain
f
ix
ed
o
n
th
e
g
r
o
u
n
d
,
(
i.e
.
,
it
n
eith
er
s
lip
s
n
o
r
r
o
tates),
th
r
o
u
g
h
o
u
t th
e
walk
in
g
cy
cle,
its
v
elo
city
m
u
s
t satis
f
y
=
(
Φ
(
)
)
=
(
Φ
(
)
)
=
(
)
̇
=
0
(
6
)
Ho
lo
n
o
m
ic
c
o
n
s
tr
ain
ts
ar
e
g
u
a
r
an
teed
v
ia
e
n
f
o
r
ci
n
g
th
e
s
ec
o
n
d
o
r
d
er
d
e
r
iv
ativ
e
o
f
Φ
(
)
,
̇
,
to
b
e
z
er
o
:
̇
=
(
)
̈
+
̇
(
,
̇
)
̇
=
0
(
7
)
=
[
0
6
×
8
8
×
8
]
(
8
)
T
h
e
co
n
s
tr
ain
ed
d
y
n
am
ics
o
f
th
e
s
y
s
tem
ar
e
d
eter
m
in
e
d
b
y
s
im
u
ltan
eo
u
s
ly
c
o
m
b
in
i
n
g
b
o
th
(
2
)
a
n
d
(
7
)
in
a
co
m
p
ac
t f
o
r
m
as:
[
(
)
−
(
)
(
)
0
]
[
̈
14
×
1
3
×
1
]
=
[
−
(
,
̇
)
−
̇
(
,
̇
)
̇
]
(
9
)
w
h
er
e
is
a
v
ec
to
r
o
f
im
p
u
ls
iv
e
co
n
tact
wr
en
ch
es
f
o
r
s
tan
ce
f
o
o
t,
wh
ich
c
an
b
e
d
eter
m
in
e
d
b
y
s
o
lv
in
g
(
5
)
an
d
(
9
)
s
im
u
ltan
eo
u
s
ly
as a
f
u
n
ctio
n
o
f
th
e
s
y
s
tem
s
tate
an
d
co
n
tr
o
l in
p
u
t.
2
.
4
.
2
.
Dis
cr
et
e
ev
ent
dy
na
m
i
cs
W
h
en
th
e
s
win
g
leg
en
d
h
its
th
e
g
r
o
u
n
d
,
an
im
p
ac
t
e
v
en
t
o
c
cu
r
s
an
d
ca
n
b
e
m
o
d
eled
as
a
n
in
elastic
co
n
tact.
L
et
d
ef
in
in
g
th
e
p
r
e
-
i
m
p
ac
t
s
ta
tes
−
=
(
−
,
̇
−
)
an
d
p
o
s
t
-
i
m
p
a
ct
s
t
ates
,
+
=
(
+
,
̇
+
)
,
th
en
t
h
e
r
es
et
m
a
p
c
an
b
e
o
b
t
ai
n
e
d
as
i
n
[
4
7
]
,
+
=
(
−
,
̇
−
)
=
[
(
−
)
(
−
)
̇
−
]
(
1
0
)
T
h
e
r
elab
elin
g
p
r
o
ce
s
s
ca
n
b
e
o
b
tain
ed
as
(
1
1
)
,
(
−
)
=
−
(
1
1
)
wh
er
e
s
tan
d
s
f
o
r
th
e
r
elab
elin
g
m
atr
ix
,
a
n
d
(
−
)
r
ep
r
esen
ts
th
e
c
h
an
g
e
in
th
e
r
o
b
o
t c
o
n
f
ig
u
r
atio
n
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
AE
S
I
n
t
J
R
o
b
&
A
u
to
m
I
SS
N:
2722
-
2
5
8
6
Mo
d
elin
g
a
n
d
co
n
tr
o
l o
f a
3
D
u
n
d
er
-
a
ctu
a
ted
b
ip
ed
a
l ro
b
o
t
u
s
in
g
p
a
r
tia
l
feed
b
a
ck
…
(
A
li
Gu
ess
a
m
)
127
2
.
4
.
3
.
H
y
brid s
y
s
t
em
T
h
e
h
y
b
r
id
m
o
d
el,
illu
s
tr
ated
i
n
Fig
u
r
e
3
,
ca
n
b
e
e
x
p
r
ess
ed
i
n
an
af
f
in
e
n
o
n
lin
ea
r
co
n
tr
o
l
f
o
r
m
b
ased
o
n
its
s
tate
s
p
ac
e
d
escr
ip
tio
n
∑
:
{
̇
=
(
)
+
(
)
∉
+
=
(
−
)
−
∈
(
1
2
)
w
h
er
e
=
(
,
̇
)
∈
=
,
is
th
e
s
tate
o
f
th
e
s
y
s
te
m
,
wh
er
ea
s
:
:
→
2
(
+
6
)
an
d
:
→
2
(
+
6
)
ar
e
th
e
d
r
if
t sm
o
o
t
h
v
ec
to
r
f
ield
a
n
d
th
e
in
p
u
t m
a
p
,
r
esp
ec
tiv
ely
.
Def
in
in
g
th
e
s
witch
in
g
s
et
as
(
1
3
)
,
:
=
{
∈
2
(
+
6
)
;
=
0
,
̇
(
−
,
̇
−
)
<
0
}
(
1
3
)
wh
er
e
(
)
d
en
o
tes th
e
v
er
tical
ca
r
tesi
an
p
o
s
itio
n
o
f
th
e
s
win
g
p
o
in
t
-
f
o
o
t.
Fig
u
r
e
3
.
Hy
b
r
id
d
y
n
am
ics r
e
p
r
esen
tatio
n
o
f
b
ip
ed
r
o
b
o
t
2
.
5
.
Co
ntr
o
ller
des
ig
n
I
n
s
p
ec
tio
n
allo
ws
u
s
t
o
d
eter
m
in
e
wh
ich
jo
in
ts
ar
e
t
o
b
e
co
n
tr
o
lled
th
r
o
u
g
h
o
u
t
ea
c
h
p
h
ase
o
f
t
h
e
g
ait.
I
n
o
u
r
s
ce
n
ar
i
o
,
we
s
p
ec
i
f
y
1
1
co
n
tr
o
llab
le
r
o
tatio
n
al
jo
in
ts
as:
=
(
,
,
,
ℎ
,
ℎ
,
ℎ
,
,
ℎ
,
ℎ
,
ℎ
,
)
As
an
ex
am
p
le,
we
s
elec
t
8
Do
F
th
at
ca
n
b
e
co
n
tr
o
lled
s
im
u
ltan
eo
u
s
ly
:
to
r
s
o
o
r
ie
n
tatio
n
,
k
n
ee
an
g
les,
an
d
s
win
g
h
ip
an
g
les as f
o
llo
ws
.
,
=
(
,
,
,
ℎ
,
ℎ
,
ℎ
,
,
)
∶
f
o
r
s
tan
ce
r
ig
h
t le
g
,
,
=
(
,
,
,
ℎ
,
ℎ
,
ℎ
,
,
)
: f
o
r
s
tan
ce
lef
t le
g
.
2
.
5
.
1
.
Co
ntr
o
ller
a
na
ly
s
is
T
o
co
n
tr
o
l
t
h
e
3
D
m
o
d
el,
w
e
p
r
o
p
o
s
e
th
e
PF
L
m
eth
o
d
,
wh
o
s
e
th
e
m
ai
n
id
ea
is
to
al
g
eb
r
aica
lly
tr
an
s
f
o
r
m
th
e
n
o
n
lin
ea
r
s
y
s
tem
d
y
n
a
m
ics
in
to
a
p
a
r
tially
lin
ea
r
ized
clo
s
ed
-
l
o
o
p
s
y
s
tem
d
y
n
am
ics
,
allo
win
g
co
n
v
en
tio
n
al
lin
ea
r
m
et
h
o
d
s
t
o
b
e
ap
p
lied
.
R
ec
all
th
at
th
e
c
o
n
s
tr
ain
ed
d
y
n
am
ics eq
u
atio
n
(
9
)
,
g
i
v
en
as
[
(
)
−
(
)
(
)
0
]
⏟
[
̈
]
⏟
=
[
0
3
×
8
]
⏟
+
[
−
(
,
̇
)
−
̇
(
)
̇
]
⏟
ca
n
b
e
wr
itten
in
t
h
e
co
m
p
ac
t
af
f
in
e
f
o
r
m
as
=
+
(
1
4
)
T
h
e
ex
ten
d
e
d
in
er
tia
m
atr
ix
is
in
v
er
tib
le
s
in
ce
it
is
c
o
m
p
o
s
ed
o
f
in
er
tia
p
o
s
itiv
e
d
ef
in
ite
m
atr
ix
,
(
)
,
wh
ich
is
its
elf
in
v
er
tib
le,
an
d
th
e
J
ac
o
b
ian
,
th
at
h
as
f
u
ll
r
o
w
r
an
k
.
C
o
n
s
eq
u
en
tly
,
p
r
e
-
m
u
ltip
ly
in
g
(
1
4
)
b
y
−
1
y
ield
s
(
1
5
)
,
=
−
1
(
+
)
(
1
5
)
wh
ich
ex
p
licitly
ex
p
r
ess
es th
e
g
en
er
alize
d
ac
ce
ler
atio
n
s
an
d
co
n
s
tr
ain
t f
o
r
ce
s
in
te
r
m
s
o
f
th
e
co
n
tr
o
l in
p
u
t
an
d
th
e
s
y
s
tem
d
y
n
am
ics.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
7
2
2
-
2
5
8
6
I
AE
S
I
n
t
J
R
o
b
&
A
u
to
m
,
Vo
l
.
1
5
,
No
.
1
,
Ma
r
ch
20
2
6
:
1
22
-
1
35
128
a.
C
o
n
tr
o
lled
jo
in
ts
s
elec
tio
n
Am
o
n
g
th
e
r
o
b
o
t’
s
1
4
Do
F,
8
ar
e
s
elec
ted
as
co
n
tr
o
lle
d
v
ar
iab
les,
,
s
p
ec
if
ically
th
e
to
r
s
o
o
r
ien
tatio
n
(
,
,
)
,
th
e
s
win
g
h
ip
jo
in
ts
,
an
d
b
o
t
h
k
n
ee
an
g
les (
,
)
.
T
h
is
allo
ws u
s
to
wr
ite:
=
{
,
if
l
e
ft
s
ta
n
c
e
,
if
r
ight
s
ta
n
c
e
(
1
6
)
Fo
llo
win
g
th
e
ap
p
r
o
ac
h
d
e
v
el
o
p
ed
in
[
4
8
]
,
PFL
is
a
p
p
lied
t
o
is
o
late
th
e
ac
tu
ated
d
e
g
r
ee
s
o
f
f
r
ee
d
o
m
f
r
o
m
th
e
f
u
ll c
o
n
f
ig
u
r
atio
n
,
f
r
o
m
w
h
ich
th
e
co
n
tr
o
lled
jo
i
n
t a
cc
el
er
atio
n
s
,
̈
,
ca
n
b
e
d
eter
m
in
ed
a
s
̈
=
[
̈
]
=
(
1
7
)
Her
e,
th
e
s
elec
tio
n
m
atr
ix
∈
ℝ
8
×
17
d
ep
en
d
s
o
n
w
h
ich
leg
is
in
s
tan
ce
p
h
ase.
Acc
o
r
d
in
g
ly
,
it
ta
k
es
th
e
f
o
llo
win
g
f
o
r
m
s
f
o
r
th
e
r
i
g
h
t a
n
d
lef
t stan
ce
leg
s
,
r
esp
ec
tiv
el
y
:
=
,
=
[
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
]
(
1
8
a)
=
,
=
[
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
]
(
1
8
b
)
Su
ch
a
r
ep
r
esen
tatio
n
is
ess
en
tial
f
o
r
im
p
lem
e
n
tin
g
PF
L
in
th
e
h
y
b
r
id
b
ip
ed
al
s
y
s
tem
an
d
f
ac
ilit
ates
s
u
b
s
eq
u
en
t
co
n
tr
o
l
d
esig
n
an
d
an
aly
s
is
.
R
ec
all
th
at
th
e
o
b
jectiv
e
o
f
PF
L
is
to
lin
ea
r
iz
e
an
d
s
tab
ilize
th
e
d
y
n
am
ics
ass
o
ciate
d
with
wh
ile
leav
in
g
th
e
u
n
ac
tu
ated
s
u
b
s
y
s
tem
ev
o
lv
in
g
f
r
ee
ly
.
Su
b
s
titu
tin
g
(
1
5
)
in
to
(
1
7
)
y
ield
s
a
d
o
u
b
le
-
in
teg
r
ato
r
d
y
n
am
ics f
o
r
th
e
ac
tu
ate
d
jo
i
n
ts
:
̈
=
−
1
(
+
)
=
(
1
9
)
Her
e,
d
en
o
tes
th
e
au
x
iliar
y
c
o
n
tr
o
l
in
p
u
t
wh
ich
is
en
f
o
r
ce
d
to
f
o
llo
w
a
PD
tr
ac
k
in
g
law
with
f
ee
d
f
o
r
war
d
ac
ce
ler
atio
n
:
=
(
−
)
+
(
̇
−
̇
)
+
̈
(
2
0
)
wh
er
e
an
d
ar
e
8
×
8
d
iag
o
n
al
p
o
s
itiv
e
d
ef
in
ite
g
ain
m
atr
ices
.
T
h
e
co
n
tr
o
ller
(
2
0
)
e
n
s
u
r
es
th
at
th
e
ac
tu
ated
jo
in
ts
p
r
ec
is
ely
tr
ac
k
th
e
d
esire
d
tr
ajec
to
r
ies
,
th
er
eb
y
s
tab
ilizin
g
t
h
e
er
r
o
r
d
y
n
am
ics.
Su
b
s
eq
u
en
tly
,
th
e
r
eq
u
ir
ed
m
o
to
r
to
r
q
u
e
ca
n
b
e
co
m
p
u
ted
d
i
r
ec
tly
f
r
o
m
th
e
PF
L
f
o
r
m
u
latio
n
as f
o
llo
ws
=
−
1
[
̈
+
(
̇
−
̇
)
+
(
−
)
−
−
1
]
(
2
1
)
with
=
−
1
(
2
2
)
is
an
in
v
er
tib
le
m
atr
ix
th
at
ca
n
b
e
o
b
tain
ed
in
b
lo
c
k
f
o
r
m
,
u
s
in
g
Sch
u
r
co
m
p
lem
e
n
t,
an
d
(
,
̇
,
̈
)
s
p
ec
if
y
th
e
d
esire
d
r
ef
er
e
n
ce
p
o
s
itio
n
s
,
v
elo
cities,
an
d
ac
ce
ler
atio
n
s
f
o
r
c
o
n
tr
o
lled
jo
in
ts
.
It
i
s
i
m
p
o
r
tan
t
to
m
e
n
tio
n
th
at,
in
th
is
wo
r
k
,
th
e
r
ef
er
en
ce
tr
ajec
to
r
ies
an
d
th
eir
d
er
iv
ativ
es
ar
e
r
ep
r
esen
ted
b
y
f
if
t
h
-
o
r
d
e
r
p
o
ly
n
o
m
ial
f
u
n
ctio
n
s
.
Fo
llo
win
g
th
e
s
am
e
m
eth
o
d
o
l
o
g
y
,
th
e
u
n
ac
tu
ate
d
s
u
b
s
y
s
tem
d
y
n
am
ics,
̈
,
ca
n
b
e
o
b
tain
e
d
u
s
in
g
th
e
s
elec
tio
n
m
atr
ix
∈
ℝ
6
×
17
,
an
d
af
ter
a
p
p
r
o
p
r
iately
in
clu
d
in
g
th
e
co
n
tr
o
l in
p
u
t
f
r
o
m
(
2
1
)
,
as
(
2
3
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
AE
S
I
n
t
J
R
o
b
&
A
u
to
m
I
SS
N:
2722
-
2
5
8
6
Mo
d
elin
g
a
n
d
co
n
tr
o
l o
f a
3
D
u
n
d
er
-
a
ctu
a
ted
b
ip
ed
a
l ro
b
o
t
u
s
in
g
p
a
r
tia
l
feed
b
a
ck
…
(
A
li
Gu
ess
a
m
)
129
̈
=
=
−
1
(
+
)
(
2
3
)
R
ep
o
r
tin
g
(
2
1
)
in
t
o
(
2
3
)
,
y
iel
d
s
̈
=
−
1
[
+
(
−
1
[
̈
+
(
̇
−
̇
)
+
(
−
)
−
−
1
]
)
]
Fin
ally
,
we
g
et
̈
=
−
1
−
1
[
̈
+
(
̇
−
̇
)
+
(
−
)
]
+
−
1
[
−
−
1
−
1
]
(
2
4
)
T
h
e
in
teg
r
atio
n
o
f
(
2
4
)
o
v
e
r
th
e
walk
in
g
s
tep
allo
ws
o
b
tain
in
g
th
e
u
n
co
n
tr
o
llab
le
tr
ajec
to
r
y
.
T
h
is
f
o
r
m
u
latio
n
h
ig
h
lig
h
ts
h
o
w
th
e
u
n
ac
tu
ated
Do
F
r
esp
o
n
d
p
as
s
iv
ely
to
co
n
tr
o
l
o
f
th
e
ac
tu
at
ed
jo
in
ts
,
wh
ile
th
e
ac
tu
ated
jo
in
ts
tr
ac
k
th
e
d
esire
d
tr
ajec
to
r
ies v
ia
PF
L
.
b.
PD
g
ain
s
elec
tio
n
T
o
g
u
ar
an
tee
cr
itical
d
am
p
in
g
th
r
o
u
g
h
o
u
t
all
s
im
u
latio
n
s
,
th
e
g
ain
s
m
atr
ices
an
d
in
(
2
0
)
ar
e
s
elec
ted
as
d
iag
o
n
al,
i.e
.
,
=
.
8
×
8
an
d
=
.
8
×
8
,
with
s
ca
lar
g
ain
s
an
d
s
atis
f
y
in
g
=
2
√
I
n
s
im
u
latio
n
,
th
e
v
al
u
es
o
f
=
100
.
8
×
8
an
d
=
2
√
=
20
.
8
×
8
wer
e
ch
o
s
en
h
eu
r
is
tica
lly.
T
h
e
g
ain
s
ar
e
s
elec
ted
to
b
e
d
ia
g
o
n
al
b
ec
a
u
s
e
PF
L
y
ield
s
d
ec
o
u
p
led
d
o
u
b
le
-
in
teg
r
at
o
r
d
y
n
am
ics,
f
o
r
wh
ich
d
iag
o
n
al
PD g
ain
s
ar
e
s
tan
d
ar
d
.
2
.
5
.
2
.
P
hy
s
ica
l
pa
ra
m
e
t
er
s
o
f
t
he
ro
bo
t
T
o
v
e
r
if
y
th
e
e
f
f
ec
tiv
en
ess
o
f
th
e
p
r
o
p
o
s
ed
co
n
t
r
o
l
a
p
p
r
o
ac
h
,
a
s
im
u
latio
n
an
al
y
s
is
o
f
b
ip
e
d
walk
in
g
g
aits
f
o
r
th
e
3
D
b
ip
ed
r
o
b
o
t,
is
ca
r
r
ied
o
u
t
i
n
MA
T
L
AB
.
I
n
th
e
s
im
u
latio
n
,
th
e
r
o
b
o
t
s
t
ar
ts
f
r
o
m
th
e
f
ix
ed
p
o
in
t
o
n
th
e
g
u
ar
d
,
b
ased
o
n
o
p
tim
izatio
n
,
a
n
d
is
co
n
tr
o
lle
d
b
y
th
e
f
ee
d
b
ac
k
lin
ea
r
izatio
n
co
n
tr
o
ller
,
wh
e
r
e
all
r
o
b
o
t'
s
p
h
y
s
ical
p
ar
am
eter
s
ar
e
lis
ted
in
T
ab
le
2
.
T
ab
le
2
.
Ph
y
s
ical
p
ar
a
m
eter
s
o
f
th
e
r
o
b
o
t
U
n
i
t
To
r
so
Th
i
g
h
S
h
i
n
H
i
p
W
i
d
t
h
M
a
ss
kg
70
10
5
.
0
–
Le
n
g
t
h
m
1
.
0
0
.
5
0
.
5
0
.
1
I
n
e
r
t
i
a
[
I
ₓ
,
I
y
, I
z
]
K
g
.
m²
[
5
,
3
,
2
]
[
1
,
0
.
3
,
2
]
[
0
.
5
,
0
.
1
5
,
1
.
0
]
–
M
a
ss
C
e
n
t
e
r
m
0
.
0
6
2
0
.
0
8
6
0
.
0
5
5
–
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
T
h
e
s
tick
a
n
im
atio
n
o
v
er
f
o
u
r
walk
in
g
s
tep
s
o
f
th
e
b
ip
ed
al
r
o
b
o
t,
u
n
d
er
s
tu
d
y
with
in
s
ag
i
ttal
p
lan
e,
is
s
h
o
wn
in
Fig
u
r
e
4
.
T
h
er
ef
o
r
e,
th
e
g
e
n
er
ated
p
er
io
d
ic
p
atter
n
s
d
em
o
n
s
tr
ate
co
n
s
is
ten
t
walk
in
g
g
aits
th
a
t
m
ain
tain
d
y
n
am
ic
e
q
u
ilib
r
iu
m
th
r
o
u
g
h
th
e
l
o
co
m
o
tio
n
c
y
cle
.
As
illu
s
tr
ated
in
Fig
u
r
e
5
,
t
h
e
tim
e
ev
o
lu
tio
n
o
f
th
e
b
o
d
y
’
s
C
ar
tesi
an
p
o
s
itio
n
is
s
h
o
wn
in
Fig
u
r
e
5
(
a
)
,
wh
ile
Fig
u
r
e
5
(
b
)
d
ep
icts
th
e
co
r
r
esp
o
n
d
i
n
g
lin
ea
r
v
elo
city
o
f
t
h
e
b
o
d
y
e
x
p
r
ess
ed
in
th
e
wo
r
ld
f
r
am
e.
Fo
r
th
e
b
o
d
y
f
r
a
m
e
p
o
s
itio
n
,
it
wa
s
s
h
o
wn
th
at
at
th
e
b
eg
in
n
in
g
o
f
th
e
s
tep
,
th
e
co
m
p
o
n
en
t
is
ap
p
r
o
x
im
ately
1
,
wh
ich
m
ee
ts
to
s
tan
d
in
g
p
o
s
tu
r
e,
wh
er
e
=
ℎ
+
ℎ
=
1
,
an
d
m
ai
n
tain
s
an
av
e
r
ag
e
h
eig
h
t
o
f
ap
p
r
o
x
im
ately
0
.
9
m
with
p
er
io
d
ic
v
ar
iatio
n
s
o
f
±
0
.
0
5
m
.
As
th
e
r
o
b
o
t
m
o
v
es
f
o
r
war
d
,
th
e
co
m
p
o
n
en
t
lin
ea
r
ly
in
cr
ea
s
es
f
r
o
m
0
to
3
,
wh
ich
d
em
o
n
s
tr
ates
co
n
s
is
ten
t
f
o
r
war
d
p
r
o
g
r
ess
io
n
,
wh
ile
r
em
ain
s
n
ea
r
ly
ze
r
o
,
r
esu
ltin
g
in
s
tr
aig
h
t
-
lin
e
m
o
tio
n
.
Mo
r
eo
v
er
,
th
e
b
o
u
n
d
ed
p
e
r
io
d
ic
later
al
o
s
cillatio
n
s
m
atch
to
th
e
n
atu
r
al
s
id
e
-
to
-
s
i
d
e
weig
h
t
s
h
if
tin
g
d
u
r
in
g
b
ip
e
d
al
lo
co
m
o
tio
n
a
n
d
c
o
n
f
ir
m
s
t
h
e
ef
f
icien
c
y
o
f
th
e
3
D
co
n
tr
o
l stra
teg
y
in
m
ain
tai
n
in
g
s
tr
aig
h
t
-
lin
e
walk
i
n
g
.
B
ased
o
n
Fig
u
r
e
5
,
t
h
e
av
e
r
a
g
e
v
elo
city
o
f
th
e
b
o
d
y
f
r
a
m
e
in
th
e
x
-
d
ir
ec
tio
n
(
f
o
r
war
d
m
o
tio
n
)
is
ap
p
r
o
x
im
ately
1
,
with
a
m
ax
im
u
m
v
elo
city
n
o
t
ex
ce
e
d
in
g
1
.
5
.
T
h
e
v
er
tical
v
elo
city
(
z
-
d
ir
ec
tio
n
)
ex
h
ib
its
an
o
s
cillato
r
y
p
atter
n
with
an
am
p
litu
d
e
o
f
0
.
5
.
I
n
th
e
y
-
d
i
r
ec
tio
n
,
th
e
v
elo
city
r
e
f
lects
th
e
s
i
d
ewa
y
s
m
o
tio
n
ass
o
ciate
d
with
th
e
alter
n
atin
g
o
f
s
u
p
p
o
r
t le
g
d
u
r
in
g
walk
in
g
.
On
th
e
o
t
h
er
h
an
d
,
Fig
u
r
e
6
d
is
p
lay
s
th
e
ac
tu
al
an
g
u
la
r
p
o
s
itio
n
s
o
f
th
e
b
o
d
y
f
r
a
m
e
an
d
th
e
co
r
r
esp
o
n
d
in
g
r
e
f
er
en
ce
tr
aje
cto
r
ies,
d
em
o
n
s
tr
atin
g
ac
c
u
r
a
te
tr
ac
k
in
g
p
er
f
o
r
m
a
n
ce
.
Fig
u
r
es
6
(
a)
,
6
(
b
)
,
a
n
d
6
(
c)
,
p
r
esen
t
th
e
r
o
ll,
p
itch
,
an
d
y
aw
an
g
les,
r
esp
ec
tiv
ely
.
T
h
e
co
n
tr
o
lled
g
ait
t
r
ajec
to
r
ies
ar
e
g
en
e
r
ated
u
s
in
g
f
if
th
-
o
r
d
er
p
o
ly
n
o
m
ial
f
u
n
ct
io
n
s
.
T
h
e
b
o
d
y
(
to
r
s
o
)
an
g
le
m
ea
s
u
r
em
en
ts
r
e
v
ea
l
m
in
im
al
d
ev
iatio
n
s
-
ap
p
r
o
x
im
ately
1
0
⁻³
r
a
d
f
o
r
r
o
ll
an
g
le
an
d
1
0
⁻³
r
ad
f
o
r
p
itch
an
g
le,
in
d
icatin
g
th
at
th
e
to
r
s
o
is
ef
f
ec
tiv
ely
m
ain
tain
ed
in
a
n
u
p
r
ig
h
t th
r
o
u
g
h
o
u
t t
h
e
g
ait
cy
cle.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
7
2
2
-
2
5
8
6
I
AE
S
I
n
t
J
R
o
b
&
A
u
to
m
,
Vo
l
.
1
5
,
No
.
1
,
Ma
r
ch
20
2
6
:
1
22
-
1
35
130
(
a)
(
b
)
Fig
u
r
e
4
.
Simu
latio
n
r
esu
lts
f
o
r
(
a)
th
e
s
tick
a
n
im
atio
n
o
f
th
e
b
ip
ed
in
s
ag
ittal p
lan
e
o
v
e
r
f
o
u
r
s
tep
s
o
f
walk
in
g
an
d
(
b
)
t
h
e
f
in
al
p
o
s
tu
r
e
o
f
walk
i
n
g
g
ait.
(
a)
(
b
)
Fig
u
r
e
5
.
T
h
e
(
a)
C
ar
tesi
an
p
o
s
itio
n
an
d
(
b
)
lin
ea
r
v
elo
city
o
f
th
e
b
o
d
y
f
r
am
e
with
r
esp
ec
t t
o
th
e
wo
r
ld
f
r
am
e
(
a)
(
b
)
(
c)
Fig
u
r
e
6
.
E
u
ler
an
g
les o
f
th
e
b
o
d
y
f
r
am
e
an
d
th
eir
d
esire
d
tr
ajec
to
r
ies (
a)
ac
tu
al
r
o
ll a
n
g
le,
(
b
)
p
itc
h
an
g
le,
an
d
(
c)
y
aw
an
g
le
,
alo
n
g
with
th
eir
co
r
r
esp
o
n
d
in
g
r
ef
e
r
en
ce
s
F
i
g
u
r
e
7
p
r
e
s
e
n
ts
t
h
e
s
i
m
u
l
at
i
o
n
r
e
s
u
l
ts
f
o
r
a
n
g
u
l
a
r
v
e
l
o
c
i
ti
e
s
o
f
t
h
e
b
o
d
y
f
r
a
m
e
.
T
h
e
p
i
t
c
h
(
̇
)
a
n
d
r
o
l
l
(
̇
)
a
n
g
u
l
a
r
v
e
l
o
c
i
ti
e
s
,
s
h
o
w
n
i
n
F
i
g
u
r
e
7
(
a
)
,
r
e
m
a
i
n
cl
o
s
e
t
o
z
er
o
t
h
r
o
u
g
h
o
u
t
t
h
e
g
a
i
t
c
y
c
l
e
,
i
n
d
i
c
a
t
i
n
g
m
i
n
i
m
a
l
t
o
r
s
o
m
o
t
i
o
n
i
n
t
h
e
s
a
g
it
t
al
a
n
d
l
a
t
e
r
a
l
p
l
a
n
e
s
,
w
h
e
r
e
as
t
h
e
y
a
w
a
n
g
u
l
a
r
v
e
l
o
c
it
y
o
f
t
h
e
t
o
r
s
o
,
̇
,
r
e
a
c
h
es
p
e
a
k
v
a
l
u
e
s
o
f
a
p
p
r
o
x
i
m
a
t
e
l
y
±
1
r
a
d
/
s
,
a
s
il
l
u
s
t
r
a
t
e
d
i
n
Fi
g
u
r
e
7
(
b
)
.
T
h
i
s
c
o
n
t
r
a
s
t
h
i
g
h
li
g
h
t
s
t
h
at
t
h
e
t
o
r
s
o
m
a
i
n
t
a
i
n
s
s
t
a
b
il
i
t
y
i
n
t
h
e
s
a
g
it
t
al
a
n
d
l
a
t
e
r
a
l
d
i
r
e
c
ti
o
n
s
w
h
il
e
a
ll
o
w
i
n
g
c
o
n
t
r
o
l
l
e
d
r
o
t
a
t
i
o
n
i
n
y
aw
d
u
r
i
n
g
w
a
l
k
i
n
g
.
R
eg
ar
d
in
g
th
e
a
n
g
u
lar
p
o
s
itio
n
s
an
d
v
elo
cities
f
o
r
b
o
th
lef
t
an
d
r
ig
h
t
le
g
s
,
Fig
u
r
e
8
a
n
d
Fig
u
r
e
9
illu
s
tr
ate
th
e
co
r
r
esp
o
n
d
in
g
c
u
r
v
es,
r
esp
ec
tiv
ely
.
T
h
e
k
n
ee
an
g
les,
(
,
)
,
s
h
o
wn
in
Fig
u
r
e
8
(
a)
an
d
9
(
a
)
r
em
ain
co
n
s
is
ten
tly
n
eg
ativ
es,
r
an
g
in
g
f
r
o
m
-
1
to
0
r
ad
.
T
h
i
s
in
d
icate
s
th
at
d
u
r
i
n
g
n
atu
r
al
walk
in
g
,
th
e
k
n
ee
s
s
tay
s
lig
h
tly
b
en
t to
en
s
u
r
e
s
tab
ilit
y
an
d
en
e
r
g
y
e
f
f
icien
cy
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
AE
S
I
n
t
J
R
o
b
&
A
u
to
m
I
SS
N:
2722
-
2
5
8
6
Mo
d
elin
g
a
n
d
co
n
tr
o
l o
f a
3
D
u
n
d
er
-
a
ctu
a
ted
b
ip
ed
a
l ro
b
o
t
u
s
in
g
p
a
r
tia
l
feed
b
a
ck
…
(
A
li
Gu
ess
a
m
)
131
I
n
co
n
tr
ast,
th
e
p
itch
h
ip
an
g
l
es,
(
ℎ
,
ℎ
)
,
alter
n
ate
b
etwe
en
p
o
s
itiv
e
v
alu
es,
co
r
r
esp
o
n
d
in
g
to
h
i
p
f
lex
io
n
,
an
d
n
e
g
ativ
e
v
alu
es,
i
n
d
icatin
g
h
ip
e
x
ten
s
io
n
.
T
h
is
p
r
o
d
u
ce
s
an
o
s
cillato
r
y
m
o
tio
n
o
f
th
e
h
ip
jo
in
ts
,
as
o
b
s
er
v
ed
in
Fig
u
r
es
8
(
a)
a
n
d
9
(
a)
.
T
h
e
ev
o
lu
tio
n
o
f
th
e
a
n
g
u
lar
v
el
o
cities
is
p
r
esen
ted
in
Fig
u
r
es
8
(
b
)
an
d
9
(
b
)
.
An
aly
zin
g
th
e
r
ig
h
t
an
d
lef
t
g
aits
en
ab
les
a
k
in
em
atic
co
m
p
ar
is
o
n
o
f
lim
b
c
o
o
r
d
in
at
io
n
an
d
alter
n
atin
g
b
eh
av
io
r
th
r
o
u
g
h
o
u
t th
e
m
o
tio
n
.
(
a)
(
b
)
Fig
u
r
e
7
.
A
n
g
u
lar
v
elo
city
(
a
)
̇
,
̇
an
d
(
b
)
̇
,
o
f
th
e
b
o
d
y
f
r
am
e
Fig
u
r
e
8
.
T
h
e
(
a)
jo
in
t
an
g
les d
is
p
lace
m
en
t
an
d
(
b
)
a
n
g
u
lar
v
elo
cities o
f
lef
t le
g
Fig
u
r
e
9
.
T
h
e
(
a)
j
o
in
t a
n
g
le
p
o
s
itio
n
s
an
d
(
b
)
v
elo
cities o
f
r
i
g
h
t le
g
Gr
o
u
n
d
r
ea
ctio
n
f
o
r
ce
s
(
GR
F)
ar
e
t
h
e
f
o
r
ce
s
ex
er
ted
b
y
th
e
g
r
o
u
n
d
o
n
th
e
r
o
b
o
t’
s
p
o
in
t
f
e
et.
T
h
ese
f
o
r
ce
s
ar
e
ess
en
tial
to
u
n
d
er
s
tan
d
th
e
r
o
b
o
t
-
g
r
o
u
n
d
in
ter
ac
tio
n
an
d
cr
u
cial
f
o
r
m
ain
t
ain
in
g
b
alan
ce
an
d
s
tab
ilit
y
.
As s
h
o
wn
o
n
Fig
u
r
e
1
0
,
it c
an
b
e
o
b
s
er
v
ed
th
at
th
at
th
e
GR
Fs
ex
h
ib
it
p
atter
n
s
in
b
o
th
th
e
v
e
r
tical
(
z)
an
d
h
o
r
izo
n
tal
(
x
,
y
)
d
ir
ec
ti
o
n
s
th
at
clo
s
ely
r
esem
b
le
th
o
s
e
o
b
s
er
v
ed
in
h
u
m
an
wal
k
in
g
.
T
h
e
v
er
tical
co
m
p
o
n
en
t
in
lef
t
a
n
d
r
ig
h
t
s
tan
ce
f
o
o
t,
d
is
p
lay
s
two
p
ea
k
s
at
h
ee
l
s
tr
ik
e
(
im
p
ac
t
ab
s
o
r
p
t
io
n
)
a
n
d
at
tak
e
-
o
f
f
(
p
r
o
p
u
ls
io
n
)
,
wh
e
r
e
th
e
m
ax
im
u
m
lo
ad
is
r
ea
ch
e
d
a
r
o
u
n
d
m
id
-
s
tan
ce
,
wh
e
n
th
e
s
tan
ce
leg
f
u
lly
s
u
p
p
o
r
ts
th
e
r
o
b
o
t’
s
b
o
d
y
weig
h
t.
T
h
e
p
ea
k
f
o
r
ce
attain
s
ap
p
r
o
x
im
a
tely
1000
,
wh
ich
is
co
n
s
is
ten
t
with
b
alan
cin
g
th
e
r
o
b
o
t’
s
weig
h
t (
ab
o
u
t 1
0
0
k
g
)
d
u
r
in
g
lo
ad
tr
a
n
s
f
er
.
T
h
e
f
ac
t
t
h
at
th
e
n
o
r
m
al
f
o
r
ce
is
p
o
s
itiv
e
p
r
o
v
es
th
at
t
h
e
r
o
b
o
t
d
o
es
n
o
t
tak
e
o
f
f
f
r
o
m
t
h
e
g
r
o
u
n
d
.
Mo
r
eo
v
er
,
th
e
f
r
ictio
n
v
alu
es
ar
e
co
m
p
r
is
ed
b
etwe
en
0
.
6
an
d
0
.
1
,
as
d
is
p
lay
e
d
in
F
ig
u
r
es
1
1
.
T
h
e
n
o
-
s
lip
co
n
d
itio
n
is
g
u
ar
an
teed
wh
e
n
th
e
co
m
p
u
ted
f
r
ictio
n
r
atio
,
=
|
√
(
2
+
2
)
⁄
|
,
r
em
ain
s
s
tr
ictly
b
elo
w
th
e
m
ax
im
u
m
g
r
o
u
n
d
f
r
ictio
n
co
e
f
f
icien
t
∗
(
e.
g
.
,
∗
=
0
.
6
f
o
r
a
r
u
b
b
er
s
u
r
f
ac
e)
th
r
o
u
g
h
o
u
t
th
e
e
n
tire
walk
in
g
cy
cle.
(
a)
(
b
)
(
a)
(
b
)
Evaluation Warning : The document was created with Spire.PDF for Python.