Inter
national
J
our
nal
of
Robotics
and
A
utomation
(IJRA)
V
ol.
4,
No.
2,
June
2015,
pp.
98
–
108
ISSN:
2089-4856
98
I
ns
t
it
u
t
e
o
f
A
d
v
a
nce
d
Eng
ine
e
r
i
ng
a
nd
S
cie
nce
w
w
w
.
i
a
e
s
j
o
u
r
n
a
l
.
c
o
m
A
Safe
Interaction
of
Robot
Assisted
Rehabilitation,
Based
on
Model-Fr
ee
Impedance
Contr
ol
with
Singularity
A
v
oidance
Iman
Sharifi
*
,
Heidar
Ali
T
alebi
*
,
and
Ali
Doustmohammadi
*
*
Electrical
Engineering
Department,
Amirkabir
Uni
v
ersity
of
T
echnology
Article
Inf
o
Article
history:
Recei
v
ed
Feb
9,
2015
Re
vised
May
8,
2015
Accepted
May
19,
2015
K
eyw
ord:
T
ele-rehabilitation
T
eleoperation
Haptic
feedback
Nonlinear
Control
Robotic
Control
ABSTRA
CT
In
this
paper
,
a
singularity-free
control
methodology
for
the
safe
robot-human
interaction
is
proposed
using
a
h
ybrid
control
technique
in
robotic
reha
bilitation
applications.
W
ith
the
use
of
max-plus
algebra,
a
h
ybrid
controller
is
designed
to
guarantee
fea
sible
robot
motion
in
the
vicinity
of
the
kinematic
singularities
or
going
through
and
staying
a
t
the
singular
configuration.
The
approach
tak
en
in
this
paper
is
based
on
model-free
impedance
control
and
hence
does
not
require
an
y
information
about
the
m
odel
e
xcept
the
upper
bounds
on
the
system
matrix.
The
stability
of
the
approach
is
in
v
estig
ated
using
multiple
L
yapuno
v
function
theory
.
The
proposed
control
algorithm
is
applied
to
PUMA
560
robot
arm,
a
six-
axis
industrial
robot.
The
results
demonstrate
the
v
alidity
of
the
proposed
control
scheme.
Copyright
c
2015
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Iman
Sharifi
Amirkabir
Uni
v
ersity
of
T
echnology
424
Hafez
A
v
e,
T
ehran,
Iran,
15875-4413
+98
21
64543398
imansharifi@aut.ac.ir
1.
INTR
ODUCTION
Robot
assisted
rehabilitation
is
becoming
v
ery
popular
among
people
who
ha
v
e
suf
fered
from
a
strok
e
due
to
propriocepti
v
e
neuromuscular
f
acilitation
procedure.
In
man
y
cases,
strok
e
causes
an
injury
to
the
nerv
ous
system
that
causes
disability
in
a
person.
In
cases
lik
e
this,
the
robot
is
not
only
a
good
substitute
of
a
therapist
for
performing
suitable
e
x
ercises
on
the
injured
person
b
u
t
also
it
can
perform
repetiti
v
e
and
case-oriented
e
x
erc
ises
that
are
hard
for
the
therapist
to
carryout.
Repetiti
v
e
and
task-oriented
e
x
ercises
can
impro
v
e
muscular
strength
and
mo
v
ement
coordination
in
patients
with
impairments
due
to
neurological
or
orthopedic
problems.
A
typical
repetiti
v
e
mo
v
ement
is
the
human
g
ait.
T
readmill
training
has
been
sho
wn
to
impro
v
e
g
ait
and
lo
wer
limb
motor
function
in
patients
with
locomotor
disorders
[1].
Manually
assisted
treadmill
training
w
as
first
used
as
a
re
gular
therap
y
for
patients
with
spinal
cord
injury
(SCI)
or
strok
e
around
15
years
ago.
Man
y
cl
inical
studies
pro
v
e
the
ef
fecti
v
eness
of
the
training,
particularly
in
SCI
and
strok
e
patients
[2].
Manual
e
x
ercises
ha
v
e
numerous
limitations.
The
training
is
labor
-intensi
v
e
and
therefore,
training
duration
is
usually
limited
by
f
atigue
and
therapist
shortage
[3].
Subsequently
,
the
training
sessions
are
shorter
than
what
is
required
to
achie
v
e
an
optimal
therapeutic
outcome.
Finally
,
manually-assisted
e
x
ercises
lack
repeatabil
ity
and
measurement
inde
x
e
of
patient
performance
and
progress.
In
contrast
to
manually-assisted
e
x
ercises,
using
robot
assisted
e
x
ercises,
the
duration
and
number
of
training
sessions
can
be
increased
and
the
number
of
therapists
required
per
patient
can
be
reduced.
F
urthermore,
the
robot
pro
vides
quantitati
v
e
measure
inde
x
es
and
hence
it
allo
ws
the
observ
ation
and
judgment
of
the
rehabilitation
process.
V
arious
robot
assisted
rehabilitation
systems
ha
v
e
been
de
v
eloped
to
support
therap
y
of
the
upper
e
xtremities.
Arm
trainer
from
Hesse
et
al.
[4],
the
arm
robot
from
Cozens
[5],
the
haptic
display
of
the
European
project
GENTLE/s
[6]
which
is
based
on
the
FCS
Haptic
Master
[7],
the
MIT
-Manus
[8]
and
[9],
and
the
MIME
(Mirror
Image
Mo
v
ement
Enhancer)
arm
therap
y
robot
[10]
are
e
xamples
of
such
systems.
ARMin
is
another
rehabilitation
robot
system
that
is
currently
being
de
v
eloped
for
upper
e
xtremity
treatments
[2].
Since
the
patients
limb
is
in
direct
contact
with
the
robots
J
ournal
Homepage:
http://iaesjournal.com/online/inde
x.php/IJRA
Evaluation Warning : The document was created with Spire.PDF for Python.
IJRA
ISSN:
2089-4856
99
end-ef
fector
while
training,
the
e
xcessi
v
e
interaction
force
may
beat
the
ill
limb
.
As
a
result,
the
robots
end
ef
fector
and
interaction
between
the
pat
ients
limb
must
be
deliberately
reck
oned
[11].
Recently
,
v
arious
force
and
position
control
schemes
ha
v
e
been
designed
for
robotic
interaction
tasks.
Gorce
and
Guihard
[12]
ha
v
e
proposed
a
multi-link,
position
based
impedance
controller
for
implementation
on
a
le
gged
robot.
Heinrichs
et
al.
[13]
ha
v
e
proposed
a
position
based
impedance
controller
for
an
e
xisting
h
ydraulic
industrial
robot,
confirming
their
w
ork
e
xperimentally
.
The
y
used
an
actuator
model
to
predict
the
torque
produced
from
the
pneumatic
c
ylinders
and
the
performance
of
the
controller
w
as
demonstrated
through
simulation.
Seul
June
[14]
has
used
a
double-loop
system
to
impro
v
e
the
adaptability
with
the
position
feedback
in
the
inner
loop
and
to
impro
v
e
controller
capability
by
tracking
the
desired
impedance
using
the
outer
loop.
Impedance
controllers
are
v
ery
applicable
in
the
field
of
robotics
and
human-system
interaction.
The
y
were
first
introduced
by
N.
Hog
an
in
1985
[15].
The
basic
idea
of
the
impedance
control
strate
gy
applied
to
robot-aided
treadmill
training
is
to
allo
w
a
v
ariable
de
viation
from
a
gi
v
en
le
g
trajectory
rather
than
imposing
a
rigid
g
ait
pattern.
The
de
viation
depends
on
the
patients
ef
fort
and
beha
vior
.
In
this
paper
a
ne
w
approach
for
safe
interaction
of
robot-assisted
rehabilitation
based
on
model-free
impedance
control
is
proposed.
The
impedance
controller
all
o
ws
the
robot
to
achie
v
e
a
certain
security
and
compliance.
Also,
a
singularity-free
approach
methodology
is
combined
with
such
a
controller
to
solv
e
the
problem
of
boundary
and
interior
singularity
for
safer
interac
tion.
The
rest
of
the
paper
is
or
g
anized
as
follo
ws.
In
the
ne
xt
s
ection,
Max-Plus
algebra
which
is
used
to
implement
the
h
ybrid
controller
is
briefly
discussed.
The
controller
design
is
discussed
in
Section
III
and
the
stability
of
the
controller
is
in
v
estig
ated
in
Section
IV
.
Finally
,
concluding
remarks
and
future
w
orks
are
presented
in
Section
V
.
2.
MAX-PLUS
ALGEBRA
Max-Plus
algebra
is
widely
used
to
model
beha
vior
of
systems
that
are
discrete
by
nature,
namely
Discrete
Ev
ent
Systems
(DEDS).
The
use
of
Max-Plus
algebra
structure
gi
v
es
these
problems
the
characteristic
of
a
linear
algebra
frame
w
ork
in
which
one
can
talk
about
notions
lik
e
independence,
eigen
v
alues,
eigen
v
ectors,
and
so
on
[16].
<
max
is
defined
as
<
max
=
<
[
f1g
;
denotes
the
Max
operation
and
denotes
the
Plus
operation.
Here
the
operations
and
are
defined
as
[17]:
a
b
=
max
(
a;
b
)
and
a
b
=
a
+
b:
Moreo
v
er
,
the
e
xpression
a
b
in
Max-Plus
algebra
corresponds
to
a:b
(Inner
Product)
in
classical
algebra.
In
this
article,
Max-Plus
algebra
is
used
to
switch
between
tw
o
robot
controllers
in
the
system
and
is
discuss
ed
in
more
detail
in
section
IV
.
3.
CONTR
OLLER
DESIGN
In
this
section
the
o
v
erall
architecture
of
the
controller
is
discusse
d
.
The
controller
consists
of
tw
o
major
parts.
The
first
part
is
inner
loop
controller
.
This
part
handles
the
common
problem
of
the
singularity
in
the
robotic
system
and
guarantees
the
singularity
free
motion
control.
The
second
part
is
t
he
impedance
control
which
deals
with
the
e
xact
tracking
of
both
force
and
position
tracking
together
.
3.1.
Inner
Loop
Contr
oller
Dynamical
model
of
a
robot
arm
with
n
joints
is
gi
v
en
by
M
(
q
)
•
q
+
C
(
q
;
_
q
)
_
q
+
g
(
q
)
=
u
d
(1)
where
M
(
q
)
is
the
n
n
positi
v
e
definite
manipulator
inertia
matrix,
u
is
the
n
1
v
ector
of
applied
torques,
C
(
q
;
_
q
)
_
q
is
the
n
1
centripetal
and
Coriolis
terms,
g
(
q
)
is
the
n
1
v
ector
of
gra
vity
term
and,
finally
,
q
=
f
q
1
;
q
2
;
:::;
q
n
g
T
is
the
n
1
v
ector
of
joint
displacements
.
Both
joint
space
or
task
space
can
be
used
for
controlling
the
robots.
In
joint
space,
a
robot
task
is
specified
in
an
n-dimensional
joint
space
denoted
by
q
.
Joint
le
v
el
controller
can
be
formulated
as
u
j
=
M
(
q
)(
•
q
d
+
K
V
q
e
q
2
+
K
P
q
e
q
1
)
+
(
q
;
_
q
)
_
q
+
g
(2)
A
Safe
Inter
action
of
Robot
Assisted
Rehabilitation,
Based
on
...
(Iman
Sharifi)
Evaluation Warning : The document was created with Spire.PDF for Python.
100
ISSN:
2089-4856
where
e
q
=
[
e
q
1
;
e
q
2
]
T
=
[
•
q
d
;
_
q
d
]
T
,
K
V
q
and
K
P
q
are
feedback
g
ain
matrices
.
Therefore,
equation
of
error
in
joint
space
is
as
follo
ws
_
e
q
1
=
e
q
2
_
e
q
2
=
K
P
q
e
q
1
K
V
q
eq
2
(3)
It
is
easy
to
pro
v
e
that
system
gi
v
en
by
(3)
is
locally
asymptotically
stable
for
suitable
g
ain
matrices
K
P
q
,
K
V
q
.
So,
system
(3)
can
track
an
y
trajectories
gi
v
en
as
q
d
,
_
q
d
,
•
q
d
,
re
g
ardless
of
the
current
robot
configuration.
Therefore,
the
joint
le
v
el
controller
(2)
is
v
alid
on
the
entire
w
orkspace.
Although
the
abo
v
e
f
act
is
true
on
the
entire
w
orkspace,
it
is
desired
to
design
the
robot
controllers
in
the
task
space
because
a
robot
task
is
generally
represented
by
the
desired
end-ef
fector
position
and
its
orient
ation.
So,
to
design
a
t
ask
le
v
el
controller
,
the
robot
model
needs
to
be
represented
in
terms
of
task
le
v
el
v
ariables.
Hence,
the
joint
and
task
space
acceleration
can
be
utilized
and
related
as
•
x
=
_
J
_
q
+
J
•
q
.
Thus,
the
dynamics
of
the
robot
in
terms
of
e
xtended
task
space
v
ariables
can
be
formulated
as
M
J
1
(
•
x
_
J
_
q
)
+
C
(
q
;
_
q
)
_
q
+
g
(
q
)
=
u
,
where
the
Jacobian
matrix
is
in
square
form
(
n
n
).
The
robot
controller
in
task
space
can
be
described
re
g
arding
to
the
desired
path
in
task
space,
x
d
,
as
follo
ws
u
d
=
M
J
1
(
•
x
d
_
J
_
q
+
K
V
x
e
x
2
+
K
P
x
e
x
1
)
+
C
_
q
+
g
(4)
where
e
x
=
[
e
x
1
;
e
x
2
]
T
=
[
x
d
x;
_
x
d
_
x
]
.
The
error
dynamics
in
task
space
can
be
described
as
_
e
x
1
=
e
x
2
_
e
x
2
=
K
P
x
e
x
1
K
V
x
e
x
2
(5)
It
is
also
straightforw
ard
to
pro
v
e
that
system
(5)
is
locally
asymptotically
stable
for
suitable
g
ain
matrices
K
P
x
,
K
V
x
.
3.2.
Outer
Loop
Contr
oller
(Impedance
Contr
oller)
F
or
a
safe
human-robot
interaction,
the
follo
wing
passi
v
e
impedance
model
is
imposed
to
the
robot
arm
[18]:
M
d
(
•
q
•
q
d
)
+
C
d
(
_
q
_
q
d
)
+
G
d
(
q
q
d
)
=
d
(6)
where
M
d
,
C
d
,
G
d
are
the
desired
inertia,
damping,
and
stif
fness
matrices
respecti
v
ely
and
q
d
is
the
rest
position
of
the
robot
manipulator
.
The
aim
of
the
outer
loop
controller
is
to
mak
e
the
robot
arm
(1)
dynamics
track
the
impedance
model
gi
v
en
by
(6).
The
detail
of
the
controller
design
is
gi
v
en
in
[19]
and
only
a
brief
description
of
the
design
is
presented
here.
The
follo
wing
error
signal
is
first
defined
between
the
virtual
system
and
the
real
system
with
the
specified
impedance
(6),
as
in
[20]:
w
=
M
d
•
e
+
C
d
_
e
+
G
d
e
+
d
(7)
where
e
=
q
q
d
.
The
aim
of
the
controller
is
to
mak
e
the
error
signal
go
to
zero
as
time
goes
to
infinity
,
that
is
lim
t
!1
w
(
t
)
!
0
(8)
Hence,
an
augmented
impedance
error
is
defined
as
follo
ws:
w
=
K
f
w
=
•
e
+
K
d
_
e
+
K
p
e
+
K
f
d
(9)
where
K
d
=
M
1
d
C
d
,
K
p
=
M
1
d
G
d
,
K
f
=
M
1
d
.
T
w
o
positi
v
e
definite
matrices
are
defined
as
and
E
such
that
+
E
=
K
d
_
+
E
=
K
p
:
(10)
Furthermore,
define
_
l
+
E
l
=
K
f
d
(11)
Thus,
(9)
can
be
re
written
as:
w
=
•
e
+
(
+
E)
_
e
+
(
_
+
E)
e
+
_
l
+
E
l
(12)
By
defining
z
=
_
e
+
e
+
l
(13)
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101
The
follo
wing
can
be
obtained:
w
=
_
z
+
E
z
(14)
From
(9)
and
(14),
z
=
0
will
lead
to
w
=
0
.
Finally
,
based
on
this
f
act,
the
aim
of
the
control
becomes
lim
t
!1
z
(
t
)
!
0
(15)
No
w
,
consider
the
outer
loop
control
input
as
follo
ws:
=
s
+
f
+
^
d
(16)
where
f
and
s
,
are
the
feedback
control
torque
and
switching
control
torque,
respecti
v
ely
.
The
corresponding
elaborated
e
xpressions
are
gi
v
en
belo
w
.
An
augmented
state
v
ariable
is
first
defined
as
follo
ws
_
q
r
=
_
q
d
e
^
l
(17)
where
^
l
is
the
estimator
of
l
and
satisfies
_
^
l
+
E
^
l
=
K
f
^
d
(18)
Furthermore,
z
and
~
l
are
defined
as
follo
ws
z
=
_
e
+
e
+
^
l
=
z
+
^
l
~
l
=
^
l
l
(19)
Using
(17)
and
(19)
one
can
obtain
z
=
_
q
_
q
r
(20)
The
abo
v
e
relation
will
be
used
in
control
performance
analysis
section.
The
switching
control
torque
in
(16)
is
gi
v
en
by
s
=
(
K
1
k
•
q
r
k
+
K
2
k
_
q
k
k
_
q
r
k
+
K
3
)sgn(
z
)
(21)
where
K
1
,
K
2
,
K
3
are
diagonal
and
positi
v
e
definite
matrices
of
lar
ge
enough
elements
such
that
k
1
;i
k
M
;
k
2
;i
k
C
;
k
3
;i
(
k
G
+
k
d
)
for
i
=
1
;
2
;
:::;
n
(22)
The
feedback
control
torque
(16)
is
gi
v
en
by
f
=
K
z
(23)
where
K
is
a
diagonal
positi
v
e
definite
matrix
with
elements
k
i
,
i
=
1
;
2
;
:::;
n
.
Remark
1
When
k
1
;i
,
k
2
;i
,
k
3
;i
,
k
i
ar
e
selected
identical
for
i
=
1
;
:::;
n
,
(21)
and
(23)
become
fairly
simple
con-
tr
oller
s
fr
om
implementation
point
of
vie
w
,
howe
ver
it
may
be
r
equir
ed
to
use
dif
fer
ent
k
1
;i
,
k
2
;i
,
k
3
;i
,
k
i
for
better
contr
ol
performance
in
some
cases.
4.
AN
AL
YSIS
OF
THE
SINGULARITIES
It
is
noted
that
the
e
xistence
of
task
le
vel
contr
oller
(4)
depends
on
the
e
xistence
of
J
1
0
.
Singular
configur
a-
tions
ar
e
the
configur
ations
in
whic
h
J
0
has
r
ank
deficiencies.
At
singular
configur
ations,
J
1
0
does
not
e
xist.
F
or
a
6-DOF
manipulator
,
whic
h
consists
of
a
3-DOF
spher
ical
wrist
and
a
3-DOF
for
earm,
J
,
is
a
6
6
matrix
and
a
con-
figur
ation
is
singular
if
and
only
if
det
(
J
0
)
=
0
.
Both
the
arm
and
the
wrist
singularities
can
cause
singularities
and
the
y
can
be
decoupled
into
arm
singularities
and
wrist
singularities,
r
espectively
.
The
J
acobian
could
be
decoupled
in
thr
ee
parts
instead
of
studying
the
determinant
of
J
0
[21]:
J
0
=
I
U
0
I
I
0
0
J
11
0
J
21
J
22
(24)
The
determinants
of
and
U
cannot
be
zer
o.
Hence
,
the
singularity
conditions
ar
e
det
(
J
22
)
=
0
and
det
(
J
11
)
=
0
.
det
(
J
11
)
=
0
stands
for
the
for
earm
singularity
.
T
wo
singularity
conditions
can
be
obtained
fr
om
the
for
earm
A
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action
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...
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102
ISSN:
2089-4856
Figure
1.
Frame
w
ork
for
h
ybrid
system
controllers
[22].
singularity
.
The
so
called
”boundary
singularity”
appear
s
when
the
elbow
is
fully
e
xtended
or
r
etr
acted
and
can
be
described
by
following
equation
b
=
d
4
c
3
a
3
s
3
=
0
(25)
The
other
singularity
that
is
called
”interior
singularity”,
occur
s
when
i
=
d
4
s
23
+
a
2
c
2
+
a
3
c
23
=
0
(26)
The
wrist
singularity
can
be
identified
by
c
hec
king
the
determinant
of
the
matrix
J
22
to
see
if
the
following
condition
is
satisfied
w
=
s
5
=
0
(27)
When
two
joint
axes
ar
e
collinear
,
the
wrist
singularity
occur
s.
Her
e
s
i
and
c
i
r
epr
esent
sin
q
i
and
cos
q
i
r
espectively
.
The
neighborhood
of
singularity
is
defined
by
positive
constants
"
b
;
"
i
;
and
"
w
that
could
be
e
xpr
essed
as
j
b
j
"
b
;
j
i
j
"
i
;
j
w
j
"
w
5.
R
OBO
TIC
HYBRID
CONTR
OLLER
In
hybrid
contr
ol
systems,
both
continuous
and
discr
et
e
dynamic
of
the
system
ar
e
in
volved.
The
de
velopment
of
suc
h
systems
is
given
by
equations
of
motion
that
g
ener
ally
depends
on
both
continuous
and
discr
ete
variables.
F
ig
.
1
shows
a
g
ene
r
a
l
fr
ame
work
for
the
hybrid
system
contr
oller
s.
The
hybrid
contr
oller
is
in
form
of
hier
ar
c
hical
structur
e
that
includes
a
discr
ete
and
a
continuous
layer
to
g
ether
.
A
discr
ete
switc
hing
function
is
designed
in
or
der
to
select
the
appr
opriate
continuous
function
for
use
.
Singular
configur
ation
and
the
pr
e
vious
contr
oller
status
deter
-
mines
the
situation.
When
the
r
obot
g
ets
close
to
the
singular
configur
ation,
at
the
ve
ry
fir
st
step
the
hybrid
contr
oller
uses
damped
least
squar
es
to
ac
hie
ve
an
appr
oximate
motion
of
the
end
ef
fector
and
then
it
will
switc
h
to
joint
le
vel
contr
ol.
The
g
ener
al
design
pr
ocedur
e
is
discussed
as
follows.
The
fir
st
step
in
the
hybrid
contr
oller
design
is
the
partition
of
the
workspace
and
the
singularity
analysis
of
the
manipulator
.
F
or
a
given
mec
hanical
structur
es,
the
singular
configur
ation
and
the
corr
esponding
singular
condition
vector
can
be
obtained.
The
second
step
uses
the
definition
of
the
subspaces
to
design
the
switc
hing
functions.
T
wo
switc
hing
function
vector
s
ar
e
r
equir
ed.
One
is
the
switc
hing
function
between
r
e
gion
1
and
r
e
gion
2
and
the
other
one
is
the
switc
hing
function
between
the
r
e
gular
r
e
gion
0
and
r
e
gion
1
.
The
switc
hing
functions
can
be
written
as:
m
s
1
=
sgn(
j
j
)
0
;
8
2
(28)
m
s
2
=
m
s
2
(
t
)
sgn(
+
j
j
)
0
sgn(
j
j
)
0
;
8
2
(29)
wher
e
sg
n
(
:
)
is
a
signum
function.
The
switc
hing
function
is
a
function
of
the
s
ingular
conditions
r
epr
esented
by
Max-Plus
Alg
ebr
a.
This
part
is
designed
to
rule
out
the
c
hattering
featur
e
of
a
switc
hing
contr
oller
.
The
thir
d
step
is
the
design
of
the
continuous
contr
oller
s.
In
this
step
thr
ee
important
fr
a
gments
must
be
consider
ed.
In
subspace
0
,
matrix
J
is
always
in
vertible
and
the
task
le
vel
contr
ol
(4)
is
ef
fective
in
this
subspace
.
In
subspace
2
,
the
joint
le
vel
contr
oller
is
used.
F
inally
,
in
r
e
gion
1
,
whic
h
is
close
to
a
singular
configur
ation,
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V
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2,
June
2015:
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IJRA
ISSN:
2089-4856
103
a
feasible
solution
of
in
ver
se
J
acobian
can
be
obtained
by
pseudo-in
ver
se
or
singular
r
ob
ust
in
ver
se
(SRI).
The
task
le
vel
cont
r
oller
(4)
still
can
be
used
after
r
eplacing
J
y
with
J
1
,
wher
e
J
y
is
a
kind
of
pseudo-in
ver
se
formulat
ed
by:
J
y
=
(
J
T
:J
+
m
s
1
I
)
1
J
T
;
wher
e
is
the
damping
factor
and
I
is
an
identity
matrix.
Ther
efor
e
,
t
he
contr
oller
in
r
e
gion
1
can
be
r
epr
esented
by
u
d
=
M
J
y
(
•
x
d
_
J
_
q
+
K
V
x
e
x
2
+
K
P
x
e
x
1
)
+
c
+
g
(30)
The
err
or
dynamics
of
the
system
can
be
formulated
by
_
e
x
1
=
e
x
2
_
e
x
2
=
(
I
K
)(
•
x
d
_
J
_
q
)
K
(
K
P
x
e
x
1
K
V
x
e
x
2
)
(31)
Her
e
the
value
of
K
is
defined
as
K
=
J
J
y
.
The
stability
analysis
of
contr
oller
(3.1
)
is
discussed
in
[23].
Y
et,
it
can
be
pr
o
ved
that
the
contr
oller
based
on
pseudo-in
ver
se
method
will
cause
instability
in
subspace
2
[24].
Although
subspace
2
could
be
very
small,
it
cuts
the
de
xter
ous
workspace
into
pieces.
If
the
r
obot
cannot
tr
avel
thr
ough
subspace
2
,
the
singular
configur
ation
will
gr
eatly
r
estrict
the
de
xter
ous
workspace
.
Consequently
,
joint
le
vel
contr
ol
will
be
utilized
in
subspace
2
to
stabilize
the
system.
W
ith
the
continuous
contr
oller
s
in
dif
fer
ent
subspaces
and
the
switc
h
functions,
the
ne
xt
step
is
to
formulate
the
hybrid
r
obot
motion
contr
oller
in
the
entir
e
workspace
utilizing
the
switc
hing
conditions
m
s
1
and
m
s
2
:
u
=
(1
m
s
2
)
u
d
+
m
s
2
u
j
(32)
wher
e
the
switc
h
function
m
s1
is
within
Eq.
(28).
The
values
of
m
s
1
and
m
s
2
determine
the
continuous
contr
oller
s
that
should
be
used.
m
s
2
tak
es
the
values
of
1
or
0
and
ther
efor
e
u
d
is
the
contr
oller
for
subspace
0
[
1
,
and
u
j
is
the
contr
oller
for
subspace
0
[
.
m
s
1
also
tak
es
the
values
of
1
or
0.
When
m
s
1
=
0
,
u
d
=
u
t
and
ther
efor
e
Eqs.
(4)
and
(30)
ar
e
the
same
.
When
m
s
1
equals
unity
,
the
damped
least-squar
e
method
is
instantiated,
and
the
in
ver
se
J
acobin
is
J
y
.
Thus,
the
design
of
m
s
1
and
m
s
2
is
the
discr
ete
contr
oller
as
discussed
in
the
second
step.
The
stability
of
the
hybrid
motion
contr
oller
in
the
entir
e
workspace
should
also
be
verified,
based
on
the
stability
analysis
of
the
continuous
contr
oller
s
in
their
r
espective
r
e
gion.
F
inally
,
the
last
step
of
the
hybrid
contr
oller
design
is
the
path
planning
of
the
continuous
contr
oller
s.
Given
the
desir
ed
path
in
the
task
space
(
x
d
),
contr
oller
s
(30)
and
the
task
le
vel
contr
oller
(4)
can
be
implemented.
But,
when
the
contr
oller
switc
hes
to
u
d
fr
om
u
j
,
the
task
r
epr
esentation
q
d
should
be
tr
ansformed
to
x
d
by
the
forwar
d
kinematics
equation
x
d
=
h
(
q
d
)
.
When
the
contr
oller
switc
hes
to
u
j
fr
om
u
d
,
the
task
r
epr
esentation
x
d
is
tr
ansformed
to
q
d
,
whic
h
in
volves
the
in
ver
se
kinematics
at
singular
configur
ation.
At
the
vicinity
of
singular
configur
ation,
a
suitable
incr
ement
or
decr
ement
dx
could
r
esult
in
a
lar
g
e
incr
ement
or
decr
ement
dq
.
dq
needs
to
be
r
epar
ameterized
in
joint
space
to
ac
hie
ve
the
velocity
and
acceler
ation
constr
aints.
The
continuity
of
joint
velocities
and
task
le
vel
velocities
ar
e
also
consider
ed
in
the
planning
.
After
switc
hing
fr
om
task
le
vel
contr
ol
to
joint
le
vel,
the
initial
velocity
for
e
very
joint
is
the
joint
velocity
prior
to
switc
hing
.
6.
AN
AL
YSE
OF
ST
ABILITY
This
section
discusses
the
stability
of
contr
oller
(32)
when
switc
hing
between
the
contr
oller
s
in
(30)
and
(2)
is
in
volved.
T
o
pr
oceed
further
,
the
following
assumption
and
pr
operties
ar
e
needed.
Assumption
1:
The
noise
in
tor
que
measur
ement
is
bounded
and
known
k
d
,
i.e
.,
k
~
d
k
k
d
,
wher
e
~
=
^
d
d
and
^
d
is
the
measur
ement
of
d
.
Property
1
The
matrix
_
M
(
q
)
2
C
(
q
;
_
q
)
is
a
sk
e
w
symmetric
matrix
if
C
(
q
;
_
q
)
is
in
the
Christof
fel
form,
i.e
.,
x
T
(
_
M
(
q
)
2
C
(
q
;
_
q
))
x
=
0
,
8
x
2
<
n
[25].
Property
2
The
matrix
M
(
q
)
is
symmetric
and
positive
definite
.
Property
3
k
M
(
q
)
k
6
k
M
,
k
C
(
q
;
_
q
)
_
q
k
6
k
C
k
_
q
k
and
k
G
(
q
)
k
6
k
G
,
wher
e
k
M
,
k
C
,
k
G
ar
e
positive
scalar
s
[18].
Theorem
1
Considering
the
r
obot
dynamics
described
by
(1),
under
Assumption
1,
the
contr
ol
design
(16)
with
(21)
and
(23)
guar
antees
the
following
r
esults:
(i)
lim
t
!1
w
=
0
is
bounded
by
k
d
k
M
d
k
,
i.e
.,
k
lim
t
!1
w
=
0
k
d
k
M
d
k
.
When
d
is
zer
o,
k
d
=
0
indicates
lim
t
!1
w
=
0
.
A
Safe
Inter
action
of
Robot
Assisted
Rehabilitation,
Based
on
...
(Iman
Sharifi)
Evaluation Warning : The document was created with Spire.PDF for Python.
104
ISSN:
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Figure
2.
W
orkspace
P
artition.
(ii)
all
the
signals
in
the
closed-loop
ar
e
bounded.
Proof
1
Consider
the
following
L
yapuno
v
function:
W
(
t
)
=
1
2
z
T
M
(
q
)
z
(33)
T
aking
the
derivative
of
(33)
gives
_
W
(
t
)
=
z
T
M
(
q
)
_
z
+
1
2
z
T
_
M
(
q
)
z
=
z
T
M
(
q
)
_
z
+
z
T
C
(
q
;
_
q
)
z
=
z
T
M
(
q
)(
•
q
•
q
r
)
+
z
T
C
(
q
;
_
q
)(
_
q
_
q
r
)
=
z
T
((
M
(
q
)
•
q
+
C
(
q
;
_
q
)
_
q
+
G
(
q
))
(
M
(
q
)
•
q
r
+
C
(
q
;
_
q
)
_
q
r
+
G
(
q
)))
=
z
T
(
K
z
(
K
1
k
•
q
r
k
+
K
2
k
_
q
r
k
k
_
q
k
+
K
3
)sgn(
z
)+
+
e
d
(
M
(
q
)
•
q
r
+
C
(
q
;
_
q
)
_
q
r
+
G
(
q
)))
(34)
wher
e
we
have
used
Pr
operty
1
and
Pr
operty
2
in
the
fir
s
t
equality
and
(20)
in
the
second
equality
.
Considering
Pr
operty
3,
we
have
z
T
(
M
(
q
)
•
q
r
+
C
(
q
;
_
q
)
_
q
r
+
G
(
q
)
k
z
k
(
k
M
(
q
)
•
q
r
k
+
k
C
(
q
;
_
q
)
_
q
r
k
+
k
G
(
q
)
k
k
z
k
(
k
M
(
q
)
k
k
•
q
r
k
+
k
C
(
q
;
_
q
)
k
k
_
q
r
k
+
k
G
(
q
)
k
z
T
(
k
M
k
•
q
r
k
+
k
C
k
q
r
k
k
_
q
r
k
+
k
G
)sgn(
z
)
(35)
Similarly
,
fr
om
Assumption
1,
we
obtain
z
T
d
k
d
z
T
sgn(
z
)
(36)
Substituting
(35)
and
(36)
into
(34)
r
esults
in
_
W
(
t
)
z
T
(
K
z
(
K
1
k
•
q
r
k
+
K
2
k
_
q
r
k
k
_
q
k
+
K
3
)sgn(
z
))
+
k
d
z
T
sgn(
z
)
+
z
T
(
k
M
k
•
q
r
k
+
k
C
k
_
q
r
k
k
_
q
k
+
k
G
)sgn(
z
)
=
K
z
T
z
z
T
((
K
1
k
M
I
n
)
k
•
q
r
k
+(
K
2
k
C
I
n
)
k
_
q
r
k
k
_
q
k
+(
K
3
k
G
I
n
k
d
I
n
))sgn(
z
)
=
K
z
T
z
(
K
1
k
M
I
n
)
k
•
q
r
k
z
T
sgn(
z
)
(
K
2
k
C
I
n
)
k
_
q
r
k
k
_
q
k
z
T
sgn(
z
)
(
K
3
k
G
I
n
k
d
I
n
)sgn(
z
)
0
(37)
wher
e
I
n
denotes
a
n-dimension
identity
matrix.
(37)
indicates
that
W
is
monotonically
decr
easing
.
Besides,
suppose
that
z
(0)
is
bounded,
whic
h
comes
fr
om
the
assumption
that
e
(0)
=
0
and
^
(0)
=
0
,
then
W
(0)
is
bounded
since
k
M
(
q
)
k
is
bounded.
Ther
efor
e
,
W
will
con
ver
g
e
to
a
non-ne
gative
fixed
value
,
and
thus
we
have
lim
t
!1
_
W
=
0
(38)
Immediately
,
we
have
the
following
inequality
_
W
K
z
T
z
0
IJRA
V
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2015:
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105
Figure
3.
D-H
P
arameter
of
PUMA
560.
that
leads
to
lim
t
!1
z
=
0
(39)
Furthermor
e
,
with
the
definition
of
z
in
(19),
one
can
write
lim
t
!1
z
=
lim
t
!1
~
l
(40)
F
r
om
(40),
(9),
(14)
and
Assumption
1,
we
finally
obtain
lim
t
!1
w
(
t
)
k
d
k
M
d
k
whic
h
completes
the
pr
oof
.
Theorem
2
Assume
joint
le
vel
contr
oller
(2)
is
asympt
o
t
ically
stable
in
workspace
and
task
le
vel
contr
oller
(4)
is
uniformly
ultimate
bounded
in
r
e
gion
1
.
Ther
e
e
xists
a
constant
>
0
,
as
shown
in
F
ig
.
2
and
a
,
suc
h
that
the
hybrid
r
obot
motion
contr
oller
(32)
is
ultimate-bounded
in
the
entir
e
workspace
of
the
r
obot.
7.
SIMULA
TION
Curr
ently
most
industrial
and
pr
actical
manipulator
s
ar
e
six
or
fe
wer
de
gr
ees-of-fr
eedom
(DOF).
These
manipulator
s
ar
e
usually
classified
kinematically
on
the
basis
of
the
for
e
a
r
m
or
fir
st
thr
ee
joints,
while
,
the
wrist
being
described
separ
ately
.
In
this
article
it
is
assumed
that
the
pr
ocess
of
r
ehabilitation
on
the
patient
is
accomplished
using
the
popular
PUMA
560.
The
g
ener
al
form
of
J
acobian
matrix
of
this
r
obot
is
as
follows
(41)
wher
e
J
E
is
(42)
A
constant
for
ce
is
acting
on
the
patient
arm.
The
pr
oposed
contr
oller
,
stabilizes
the
end
ef
fector
for
ce
with
a
r
eason-
able
tor
ques
acting
on
joints.
Simulation
r
esults
ar
e
sown
in
F
ig
.
(
??
).
As
can
be
seen
fr
om
the
figur
e
,.
8.
CONCLUDING
REMARKS
AND
FUTURE
W
ORKS
In
this
work,
the
human-r
obot
inter
action
has
been
in
vestigated
and
a
mixed
singular
ity-fr
ee
and
model-fr
ee
impedance
model
has
been
simulated
on
t
h
e
6-DOF
PUMA
560
r
obot
to
guar
antee
the
inter
action
stabil
ity
.
The
performance
of
the
pr
oposed
method
has
been
discussed
thr
ough
rigor
ous
analysis.
Simul
a
t
ion
r
esults
on
the
r
obot
arm
validate
the
pr
oposed
method.
A
Safe
Inter
action
of
Robot
Assisted
Rehabilitation,
Based
on
...
(Iman
Sharifi)
Evaluation Warning : The document was created with Spire.PDF for Python.
106
ISSN:
2089-4856
Figure
4.
The
6-DOF
PUMA
manipulator
[26].
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
−1
−0.5
0
0.5
1
1.5
Figure
5.
T
orques
of
the
joints
IJRA
V
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2015:
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–
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107
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A
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Based
on
...
(Iman
Sharifi)
Evaluation Warning : The document was created with Spire.PDF for Python.