Inter
national
J
our
nal
of
Robotics
and
A
utomation
(IJRA)
V
ol.
10,
No.
3,
September
2021,
pp.
235
∼
260
ISSN:
2089-4856,
DOI:
10.11591/ijra.v10i3.pp235-260
❒
235
Contr
ol
of
teleoperation
systems
in
the
pr
esence
of
cyber
attacks:
A
sur
v
ey
Mutaz
M.
Hamdan,
Magdi
S.
Mahmoud
Systems
Engineering
Department,
King
F
ahd
Uni
v
ersity
of
Petroleum
&
Minerals,
Saudi
Arabia
Article
Inf
o
Article
history:
Recei
v
ed
May
28,
2021
Re
vised
Jul
9,
2021
Accepted
Jul
16,
2021
K
eyw
ords:
Bilateral
teleoperation
system
Cyber
attack
Denial
of
service
(DoS)
attack
P
assi
vity
based
control
T
elesur
gical
robot
systems
W
a
v
e
v
ariable-based
control
ABSTRA
CT
The
teleoperation
system
is
often
composed
of
a
human
operator
,
a
local
master
ma-
nipulator
,
and
a
remote
sla
v
e
manipulator
that
are
connected
by
a
communication
net-
w
ork.
This
paper
proposes
a
surv
e
y
on
feedback
control
design
for
the
bilateral
tele-
operation
systems
(BTSs)
in
nominal
situa
tions
and
in
the
presence
of
c
yber
-attacks.
The
main
i
dea
of
the
presented
me
thods
is
to
achie
v
e
the
stability
of
a
delaye
d
bilateral
teleoperation
system
in
the
presence
of
se
v
eral
kinds
of
c
yber
attacks.
In
this
paper
,
a
comprehensi
v
e
surv
e
y
on
control
systems
for
BTSs
under
c
yber
-attacks
is
discussed.
Finally
,
we
discuss
the
current
and
future
problems
in
this
eld.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
1.
INTR
ODUCTION
A
teleoperation
system
is
referred
to
a
plant
that
is
controlled
remotely
.
Man
y
teleoperation
system
s
ha
v
e
been
designed
and
used
in
the
recent
decades
to
help
people
in
performing
tasks
remotely
especially
in
hazardous
en
vironments.
These
tasks
include
w
orking
in
a
toxic
or
harmful
en
vironment,
w
orking
remotely
lik
e
telesur
gery
and
e
xplorations
of
space,
and
carrying
out
high
preci
sion
tasks
such
as
chemical
and
nuclear
reac-
tors.
In
general,
teleoperation
systems
consist
of
a
human
operator
,
a
master
and
sla
v
e
manipulators
connected
through
a
communi
cation
netw
ork
and
the
en
vironment.
The
schematic
diagram
of
a
typical
teleoperation
system
is
sho
wn
in
Figure
1.
T
eleoperation
systems
allo
w
operator
to
e
xchange
information
of
posi
tion,
v
elocity
,
and/or
force
re-
motely
to
perform
the
desired
m
otion,
sensing,
and
ph
ysical
manipulation.
The
importance
and
wide
applica-
tions
of
teleoperation
systems
attract
researchers
in
both
control
systems
and
robotics
mainly
focusing
on
the
stability
and
the
tele
presence
[1]-[3].
A
lot
of
result
are
presented
to
handle
practical
control
problems
using
se
v
eral
approaches
such
as
anti-wind
control
[4],
[5],
fuzzy
control
[6],
[7],
adapti
v
e
control
[8]-[13],
passi
vity-
based
control
[2],
[14],
[15],
and
slide-mode
control
[16],
[17].
W
e
will
discuss
these
and
other
approaches
in
details
in
section
3.
There
are
tw
o
main
types
of
teleoperation
systems:
unilateral
and
bilateral
teleoperation
system
s.
When
the
transmission
of
motion
goes
from
master
to
sla
v
e
it
is
called
a
unilateral
teleoperation
system
(UTS).
On
the
other
hand,
a
bilateral
teleoperation
system
(BTS)
includes
transmissions
in
both
the
forw
ard
and
backw
ard
directions
between
the
master
and
sla
v
e
[18].
Generally
speaking,
BTSs
ha
v
e
more
interest
among
researchers
since
the
y
are
more
used
in
places
that
are
dif
cult
for
humans
such
as
underw
ater
v
ehicles,
airspace
J
ournal
homepage:
http://ijr
a.iaescor
e
.com
Magdi
S.
Mahmoud
Systems
Engineering
Department
King
F
ahd
Uni
v
ersity
of
Petroleum
Minerals
P
.
O.
Box
5067,
Dhahran
31261,
Saudi
Arabia
Email:
msmahmoud@kfupm.edu.sa
Evaluation Warning : The document was created with Spire.PDF for Python.
236
❒
ISSN:
2089-4856
applications,
mining,
and
remote
medical
sur
geries.
Comprehensi
v
e
historical
surv
e
ys
on
BTSs
are
presented
in
[1],
[19].
In
2011,
Nu
˜
no,
Ba
sa
˜
nez
and
Orte
g
a
[2],
a
tutorial
on
passi
vity
based
control
on
BTSs
is
pro
vided.
A
surv
e
y
on
the
en
vironment
and
task-based
controller
system
is
sho
wn
in
[20].
Also,
a
re
vie
w
on
bilateral
teleoperation
with
force,
position,
po
wer
,
and
impedance
scaling
is
detailed
in
[21].
In
addition
to
the
basic
requirements
of
stability
,
a
major
aspect
to
consider
in
BTSs
is
the
trans-
parenc
y
.
Actually
,
there
is
a
tradeof
f
between
these
tw
o
concerns
mainly
as
a
result
of
the
time
delays
caused
by
the
communication
channel
[22].
The
delay-induced
ins
tability
of
BTSs
is
one
of
the
main
challenges
for
researchers
since
time
delays
e
xist
in
most
communication
channels.
Some
major
studies
based
on
time-
delayed
bilateral
teleoperation
ha
v
e
been
done
in
[12],
[23]-[25].
The
time
delays
could
be
symmetric
[26]
as
well
as
asymmetric
[27].
Some
recent
studies
on
the
stability
analysis
of
asymmetric
time-v
arying
delay
can
be
found
in
[28]-[30].
Most
of
the
recent
w
ork
on
stability
analysis
with
a
symmetric
time-v
arying
delays
emplo
yed
L
yapuno
v–Kraso
vskii
functional
to
formulate
the
relations
among
the
controller
parameters,
and
the
upper
bound
of
time-v
arying
delays
[31],
[32].
Linear
matrix
inequalities
(LMIs)
were
applied
to
present
the
stability
criteria.
So,
it
can
be
solv
ed
to
obtain
the
allo
wed
maximum
v
alues
of
delays.
Figure
1.
Block
diagram
of
the
typical
teleoperation
system
Recently
,
man
y
re
vie
ws
on
the
BTSs
were
published.
In
2018,
Gha
vifekr
,
Ghiasi
and
Badamchizadeh
[33],
discrete-time
control
methods
of
BTSs
wer
e
presented
with
concern
on
problems
of
passi
vity
,
stability
,
transparenc
y
,
and
time
delays.
The
recent
control
approaches
for
teleoperation
systems
while
considering
internet-based
communication,
unkno
wn
time-v
arying
delay
,
and
model
uncertainty
were
re
vie
wed
[34].
More
concern
w
as
gi
v
en
to
control
algorithms
that
were
applied
to
nonlinear
uncertain
systems.
The
application
of
predicti
v
e
control
schemes
in
BTSs
are
discussed
including
qualitati
v
e
and
quantitati
v
e
comparisons
among
these
methods
re
g
arding
rob
ustness,
transparenc
y
,
and
stability
[35].
In
this
paper
we
address
the
control
problem
of
bilateral
teleoperation
systems
(BTSs)
in
nominal
situations
and
in
the
presence
of
c
yber
-attacks.
The
main
contrib
utions
of
this
paper
are
summarized
as
follo
ws:
–
A
discussion
on
the
methods
of
modeling
BTSs
will
be
presented.
–
A
comprehensi
v
e
surv
e
y
on
control
schemes
for
BTSs
under
c
yber
-attacks
is
discussed.
–
The
literature
that
considers
c
yber
-attacks
in
the
design
of
the
controller
for
BTSs
is
re
vie
wed.
–
The
current
and
future
problems
in
this
eld
are
discussed.
The
remaining
of
this
paper
is
or
g
anized
as
follo
ws:
Modeling
of
BTSs
are
discussed
in
section
2.
In
section
3,
control
approaches
are
presented.
Then,
the
c
yber
-attacks
in
BTSs
are
described
in
section
4.
Finally
,
Current
and
future
research
are
listed
in
section
5.
are
deri
v
ed
by
applying
the
Lagrangian
systems
with
the
actuated
re
v
olute
joints
such
that
sho
wn
in
(1):
M
m
(
q
m
)
¨
q
m
+
C
m
(
q
m
,
˙
q
m
)
˙
q
m
+
g
m
(
q
m
)
=
J
T
m
(
q
m
)
F
m
+
τ
m
M
s
(
q
s
)
¨
q
s
+
C
s
(
q
s
,
˙
q
s
)
˙
q
s
+
g
s
(
q
s
)
=
−
J
T
s
(
q
s
)
F
s
+
τ
s
(1)
Int
J
Rob
&
Autom,
V
ol.
10,
No.
3,
September
2021
:
235
–
260
2.
MODELING
OF
BTSS
In
secti
on
2,
we
will
discuss
the
modeling
of
BTSs.
The
dynamics
of
both
the
master
and
sla
v
e
robots
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
237
with
the
subscripts
{
m
}
and
{
s
}
denote
the
master
robot
and
the
sla
v
e
robot,
respecti
v
ely
.
q
m
(
t
)
∈
n
and
q
s
(
t
)
∈
m
are
the
generalized
coordinates,
M
m
(
q
m
)
∈
n
×
n
and
M
s
(
q
s
)
∈
m
×
m
are
the
inertia
matrices.
C
m
(
q
m
,
˙
q
m
)
∈
n
×
n
and
C
s
(
q
s
,
˙
q
s
)
∈
m
×
m
are
damping
matrices.
g
m
(
q
m
)
∈
n
and
g
s
(
q
s
)
∈
m
are
the
gra
vity
forces.
τ
m
(
t
)
∈
n
and
τ
s
(
t
)
∈
m
are
the
applied
torques
which
is
deifned
sometimes
as
the
control
signals.
J
m
(
q
m
)
∈
n
×
n
and
J
s
(
q
s
)
∈
n
×
m
are
the
Jacobian
matrices,
and
F
m
(
t
)
,
F
s
(
t
)
∈
n
×
1
are
the
forces
e
x
erted
by
the
human
operator
and
the
en
vironment.
See
for
e
xample
[10],
[36],
[37].
Let
us
assume
that
the
sla
v
e
robot
is
a
redundant
manipulator
to
obtain
semi-autonomous
teleoperation
and
the
master
robot
is
non-redundant
manipulator
for
simplicity
.
Let
x
(
t
)
dened
as
the
state
v
ector
and
equal
to
[
x
m
(
t
)
˙
x
m
(
t
)
x
s
(
t
)
˙
x
s
(
t
)]
T
with
x
(
t
)
and
˙
x
are
denoting
the
position
and
speed
of
the
end
ef
fector
,
respecti
v
ely
.
And
by
considering
the
e
xternal
forces
as
disturbances,
the
general
state-space
model
for
system
(1)
is
represented
by
(2):
˙
x
(
t
)
=
Ax
(
t
)
+
B
u
(
t
)
+
B
w
w
(
t
)
(2)
with
A
=
0
1
0
0
0
−
B
m
M
m
0
0
0
0
0
1
0
0
0
−
B
s
M
s
,
B
=
0
0
1
M
m
0
0
0
0
1
M
s
=
B
1
B
2
B
1
=
0
1
M
m
0
0
,
B
2
=
0
0
0
1
M
s
,
B
w
=
0
0
1
M
m
0
0
0
0
−
1
M
s
u
(
t
)
=
F
m
(
t
)
F
s
(
t
)
T
,
w
(
t
)
=
F
h
(
t
)
F
e
(
t
)
T
(3)
In
equation
(2),
the
state
v
ariables
are
x
m
(
t
)
,
˙
x
m
(
t
)
,
x
s
(
t
)
,
and
˙
x
s
(
t
)
.
F
or
simpli
city
,
the
states
x
m
(
t
)
and
˙
x
m
(
t
)
are
called
local
state
v
ariables
and
x
s
(
t
)
and
˙
x
s
(
t
)
are
called
remote
state
v
ariables
to
the
master
,
and
con
v
ersely
with
respect
to
the
sla
v
e.
The
follo
wing
state-fee
d
ba
ck
controllers
are
designed
to
obtain
the
stability
for
the
master
and
sla
v
e
manipulators
as
(4):
F
m
(
t
)
=
K
m
x
m
(
t
)
˙
x
m
(
t
)
x
s
(
t
−
d
2
(
t
))
˙
x
s
(
t
−
d
2
(
t
))
T
F
s
(
t
)
=
K
s
x
m
(
t
−
d
1
(
t
))
˙
x
m
(
t
−
d
1
(
t
))
x
s
(
t
)
˙
x
s
(
t
)
T
(4)
where
d
1
(
t
)
and
d
2
(
t
)
are
the
forw
ard
time
delay
(the
communication
path
from
the
master
to
the
sla
v
e)
and
the
backw
ard
time
delay
(the
communication
path
from
the
sla
v
e
to
the
master),
respecti
v
ely
.
Remark
1.
It
is
mor
e
pr
actical
to
have
upper
limit
on
the
time
delays
suc
h
as
positive
scalar
s
¯
d
1
and
¯
d
2
.
So,
d
1
(
t
)
and
d
2
(
t
)
ar
e
bounded
as,
0
≤
d
1
(
t
)
≤
¯
d
1
,
0
≤
d
2
(
t
)
≤
¯
d
2
[38].
The
controllers
F
m
(
t
)
and
F
s
(
t
)
is
formulated
in
the
follo
wing
compact
system:
F
m
(
t
)
=
K
m
x
(
t
−
d
2
(
t
))
F
s
(
t
)
=
K
s
x
(
t
−
d
1
(
t
))
(5)
By
substituting
the
controllers
(5)
in
the
BTS
(3),
one
will
obtain
the
follo
wing
o
v
erall
cl
osed-loop
system
sho
wn
in
(6):
˙
x
(
t
)
=
Ax
(
t
)
+
B
1
K
m
x
(
t
−
d
2
(
t
))
+
B
2
K
s
x
(
t
−
d
1
(
t
))
+
B
w
w
(
t
)
(6)
Contr
ol
of
teleoper
ation
systems
in
the
pr
esence
of
cyber
attac
ks:
A
surve
y
(Mutaz
M.
Hamdan)
Evaluation Warning : The document was created with Spire.PDF for Python.
238
❒
ISSN:
2089-4856
3.
CONTR
OL
METHODS
FOR
BTSS
Man
y
literature
discuss
the
challenges
of
controlling
teleoperation
systems
[39].
The
focus
of
these
surv
e
ys
directed
to
adapti
v
e
control,
w
a
v
e
v
ariable,
and
predicti
v
e
control
approaches
[35],
[40],
[41].
V
arious
methodologies
were
proposed
to
handle
control
issues
in
se
v
eral
applications
of
teleoperation
systems.
Some
e
xamples
are:
nonlinear
trilateral
teleoperation
systems
[42],
PD-lik
e
controller
[43],
four
-channel
structure
[44],
operator
dynamics
considerations
[45],
output
feedback
with
force
estimation
[46],
a
s
table
and
trans-
parent
microscale
teleoperation
[47],
e
xternal
force
estimation
[48],
position
synchronization
[49],
and
ne
w
stability
criteria
using
h
ybrid
strate
gies
of
position
and
impedance
reection
[50].
Figure
2.
Control
approaches
applied
to
BTSs
3.1.
P
assi
vity
based
contr
ol
appr
oach
P
assi
v
e
systems
are
referred
to
dynamical
systems
that
ha
v
e
positi
v
e
consumption
of
total
ener
gy
.
P
assi
vity
based
control
is
a
control
approach
that
aim
s
to
maintain
the
passi
vity
beha
vior
of
the
closed-loop
system.
So,
in
this
approach,
the
controller
is
designed
and
formulated
to
k
eep
the
balance
of
the
po
wer
o
w
and
ener
gy
consumption
of
the
BTS
[51].
One
of
the
challenges
in
BTSs
is
the
occurrence
of
time
delays
in
the
communication
channel.
In
addition
to
its
ef
fect
on
the
signal,
the
time
delay
could
also
increase
the
ener
gy
in
the
communication
channel
[52].
This
could
af
fect
the
passi
vity
of
the
BTS
and
may
lead
to
instability
.
T
o
handle
this
issue,
se
v
eral
control
approaches
based
on
passi
vity
ha
v
e
been
proposed.
The
passi
vity-based
con-
trol
approach
is
suitable
and
applicable
to
nonlinear
systems,
distinctly
to
mechanical/Euler
-Lagrange
systems
[53].
Let
us
consider
the
BTS
(1),
the
input
po
wer
of
this
BTS
in
terms
of
the
passi
vity
is
gi
v
en
by
[54]:
P
in
(
t
)
=
x
T
y
(7)
where
x
is
the
input
and
y
is
the
output
of
the
system.
The
input
po
wer
o
w
in
the
BTS
(1)
is
formulated
as
(8):
P
in
(
t
)
=
˙
q
T
m
f
T
e
f
h
−
˙
q
s
.
(8)
The
passi
vity
of
system
(1)
is
assured
if
and
only
if
a
state
function
E
(
q
i
,
˙
q
i
)
≥
0
,
∀
(
q
i
,
˙
q
i
)
e
xists
and
satises
the
follo
wing
inequality
[54]
as
(9):
d
dt
E
(
q
i
,
˙
q
i
)
≤
P
in
.
(9)
Int
J
Rob
&
Autom,
V
ol.
10,
No.
3,
September
2021
:
235
–
260
As
m
entioned
in
the
Introduction,
some
recent
literature
discusses
the
control
of
BTSs
.
In
2018,
Gha
vifekr
et
al
.
[33],
discrete-time
control
methods
of
BTSs
were
presented
with
concern
on
pr
o
bl
ems
of
passi
vity
,
stabi
lity
,
transparenc
y
,
and
time
delays.
The
recent
control
approaches
for
teleoperation
systems
while
considering
internet-based
communication,
unkno
wn
time-v
arying
delay
,
and
model
uncertainty
were
re
vie
wed
[34].
More
concern
w
as
gi
v
en
to
control
algorithms
that
are
applicable
to
nonlinear
uncertain
systems.
The
application
of
predicti
v
e
control
schemes
in
BTSs
are
discussed
including
qualitati
v
e
and
quantitati
v
e
comparisons
among
these
methods
re
g
arding
rob
ustness,
transparenc
y
,
and
stability
[35].
In
section
3,
we
will
discuss
se
v
eral
kinds
of
control
approaches
applied
to
BTSs
and
the
feature
of
each
method
will
be
e
xplained.
These
approaches
are
summarized
in
Figure
2.
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
239
Man
y
literature
refers
to
the
ener
gy
storage
function
by
E
(
q
i
,
˙
q
i
)
and
the
dissipati
v
e
rate
by
d
dt
E
(
q
i
,
˙
q
i
)
[51],
[53].
The
pre
vious
inequality
(9)
is
represented
by
the
follo
wing
(10):
∂
E
(
q
i
,
˙
q
i
)
∂
q
i
˙
q
i
+
∂
E
(
q
i
,
˙
q
i
)
∂
˙
q
i
¨
q
i
≤
˙
q
T
m
f
T
e
f
h
˙
q
s
(10)
No
w
,
consider
system
(1)
and
se
v
eral
manipulations
to
describe
the
torque
control
signals
τ
i
as
(11):
∂
E
∂
˙
q
i
M
−
1
i
τ
i
≤
˙
q
T
m
f
T
e
f
h
˙
q
s
−
∂
E
∂
q
i
˙
q
i
+
∂
E
∂
˙
q
i
M
−
1
i
C
i
q
i
+
g
i
−
f
j
(11)
where
i
∈
{
m,
s
}
and
j
∈
{
h,
e
}
.
Finally
,
the
passi
vity-based
control
approach
is
summarized
as
follo
ws:
Design
a
suitable
τ
i
such
that
(11)
holds
for
all
q
i
and
˙
q
i
.
Man
y
w
orks
of
literature
in
v
estig
ate
this
control
approach.
In
E.
Kamrani
[55],
a
multi
v
ariable
control
method
with
w
a
v
e-prediction
and
passi
vity
technique
is
proposed
for
real-time
systems
in
the
presence
of
internet
delay
dynamics.
A
synchronizati
on
scheme
for
nonlinear
dynamical
netw
ork
ed
systems
including
delays
and
uncertainty
is
designed
using
passi
vity
analysis
[13].
A
tw
o-layer
control
scheme
has
been
discussed
for
maintaining
the
stability
of
BTSs
[56].
This
scheme
consists
of
one
layer
to
assure
the
transparenc
y
and
a
second
one
for
handling
the
passi
vity
of
the
system.
In
2010,
K.
Hertk
orn
et
al
.
[57],
a
general
time-domain
passi
vity
control
approach
is
applied
for
a
haptic
system
with
multi-de
gree
of
freedom.
Other
e
xamples
on
this
approach
including
stabilizing
the
ener
gy
or
position
drift
compensation
can
be
found
in
[58]-[63].
Some
adv
antages
of
this
approach
in
the
literature
include:
applying
switching
method
for
ener
gy
dissipation
[64]
or
independent
of
time-v
arying
delay
[65],
the
combination
of
passi
vity
and
transparenc
y
[56],
applying
it
to
non-ideal
system
[61],
applying
both
po
wer
-based
and
ener
gy-based
approaches
[60],
and
the
elimination
of
system
uncertainties
[66].
On
the
other
hand,
some
dra
wbacks
of
this
approach
include:
weak
transparenc
y
[64],
the
transparenc
y
is
not
guaranteed
in
the
passi
vity
layer
[56],
limited
application
and
no
consideration
of
delay
[61],
applied
to
a
system
with
constant
delay
[58],
and
limited
random
delays
[60].
More
e
xamples
on
the
passi
vity-based
control
approach
are
[2],
[14],
[15],
[67]-[71].
3.2.
W
a
v
e
v
ariable-based
contr
ol
The
rst
introduction
of
the
w
a
v
e
v
ariable-based
control
approach
w
as
in
1991
by
Nieme
yer
and
Slotine
[72].
The
follo
wing
tr
ansformation
is
applied
to
the
reference
signals
in
the
BTS,
i.e.
joints
v
elocity
and
e
xternal
forces
˙
q
i
and
f
j
,
i
∈
{
m,
s
}
and
j
∈
{
e,
h
}
to
obtain
a
w
a
v
e
formulation
sho
wn
in
(12)
and
(13).
u
m
=
1
√
2
b
f
∗
h
(
t
)
+
b
˙
q
m
(
t
)
(12)
u
s
=
1
√
2
b
f
e
(
t
)
−
b
˙
q
∗
s
(
t
)
(13)
No
w
,
the
ne
w
signals
u
m
and
u
s
are
sent
by
the
communication
channel.
Additionally
,
the
desired
signals
for
tracking
will
be
f
∗
h
and
˙
q
∗
s
,
and
the
communication
channel
has
a
characteristic
impedance
b
.
So,
the
o
w
of
po
wer
in
the
communication
channel
between
the
master
and
sla
v
e
robots
are
gi
v
en
by
(14):
P
in
(
t
)
=
f
∗
T
h
(
t
)
˙
q
m
(
t
)
−
f
T
e
(
t
)
˙
q
∗
s
(
t
)
(14)
=
1
2
u
T
m
(
t
)
u
m
(
t
)
−
v
T
m
(
t
)
v
m
(
t
)
+
1
2
u
T
s
(
t
)
u
s
(
t
)
−
v
T
s
(
t
)
v
s
(
t
)
where
v
m
(
t
)
,
v
s
(
t
)
are
the
recei
v
ed
signals
and
u
m
(
t
)
,
u
s
(
t
)
are
the
reference
signals
o
n
the
master
and
sla
v
e
sides
of
a
BTS
satisfying
by
(15)
and
(16):
f
∗
h
(
t
)
=
b
˙
q
m
(
t
)
+
√
2
bv
m
(
t
)
(15)
˙
q
∗
s
(
t
)
=
1
2
√
2
bv
s
(
t
)
−
f
e
(
t
)
.
(16)
Contr
ol
of
teleoper
ation
systems
in
the
pr
esence
of
cyber
attac
ks:
A
surve
y
(Mutaz
M.
Hamdan)
Evaluation Warning : The document was created with Spire.PDF for Python.
240
❒
ISSN:
2089-4856
The
researchers
ha
v
e
de
v
eloped
and
designed
controllers
for
BTSs
in
the
w
a
v
e
domain
due
to
the
f
act
that
intrinsic
passi
vity
is
preserv
ed
in
the
w
a
v
e
formulation
[73]-[79].
Ho
we
v
er
,
the
passi
vity
could
be
unsa
v
ed
in
the
po
wer
signals.
According
to
this
feature,
it
is
important
to
b
uild
and
place
a
passi
v
e
lter
at
the
master
robot
for
estimating
the
model
of
the
sla
v
e
robot
and
enhance
the
force
reection
in
haptic
applications.
A
similar
proposal
w
as
pro
vided
for
constant
kno
wn
delays
[78],
[79].
An
incident
named
w
a
v
e
reection
occurs
when
the
characteristic
impedance
of
the
netw
ork
channel
is
not
similar
to
that
of
the
ports
of
the
master
and
sla
v
e.
Since
it
has
a
high
impact
on
the
BTS
performance,
w
a
v
e
reection
must
be
handled
cautiously
in
the
system
design.
Normally
,
the
impedance
matching
element
b
is
used
for
tuning
this
feature
at
both
sides
of
the
communication
channel.
But,
the
performance
of
the
BTS,
mainly
the
position
tracking
could
be
af
fected
by
this
parameter
.
T
o
o
v
ercome
this
scenario
and
decrease
position
drift,
parameter
b
is
not
considered
on
the
master
side
[79].
T
o
notice
the
dif
ference
between
these
tw
o
w
ays,
the
position
tracking
drifts
for
the
master
side
when
b
is
remo
v
ed
is
gi
v
en
by
(17)
and
(18):
q
∗
m
(
t
−
d
1
(
t
))
−
q
s
(
t
)
=
1
2
b
Z
t
t
−
2
d
1
(
t
)
f
∗
e
(
η
)
dη
(17)
and
when
t
is
not
remo
v
ed
q
∗
m
(
t
−
d
1
(
t
))
−
q
s
(
t
)
=
1
b
Z
t
0
f
∗
e
(
η
)
dη
+
q
s
(
t
)
.
(18)
It
is
noted
here
that
as
b
increases,
the
position
drift
decreases.
Ne
v
ertheless,
since
more
damping
is
introduced,
the
BTS
will
ha
v
e
a
poor
performance.
Some
adv
antages
of
this
approach
in
the
literature
include:
the
impro
v
ement
of
position
tracking
[73],
optimization
of
w
a
v
e
transformation
[77],
the
impro
v
ement
of
tracking
and
po
wer
optimizati
o
n
[80],
dealing
with
passi
v
e
and
non
passi
v
e
human
model
[76],
and
guaranteed
passi
vity
[75].
The
dra
wbacks
of
this
approach
include:
the
e
xistence
of
estimation
error
[73],
no
consideration
of
time
delay
[77],
[80],
reducing
the
transparenc
y
[76],
and
considering
constant
time
delay
only
[75].
3.3.
Adapti
v
e
contr
ol
appr
oach
The
structure
of
BTSs
includes
se
v
eral
important
parameters
and
model
uncertainties
to
be
concerned
by
designers.
These
parameters
are
re
v
ealed
in
all
of
the
elements
of
the
BTS
sho
wn
in
Figure
1,
i.e.
the
master
and
sla
v
e
robots,
the
communication
netw
ork,
the
human
operator
,
and
the
en
vironment.
The
adapti
v
e
control
approach
is
widely
applied
by
researchers
to
solv
e
the
problem
of
the
e
xistence
of
the
uncertainties
in
BTSs.
Moreo
v
er
,
a
lot
of
w
orks
and
ef
forts
were
in
v
ested
in
this
approach
due
to
the
linearity
of
parameters
of
BTSs
as
noted
in
(1).
T
o
understand
this
method,
the
general
design
of
the
adapti
v
e
controllers
for
the
BTSs
is
gi
v
en
by
(19)
and
(21).
The
human
operator
and
the
en
vironment
are
pres
ented
as
a
general
nonpassi
v
e
nonhomogeneous
force
as
follo
ws
(19):
ˆ
f
j
(
t
)
=
ˆ
M
j
(
t
)
¨
x
j
(
t
)
+
ˆ
D
j
(
t
)
˙
x
j
(
t
)
+
ˆ
K
j
(
t
)
x
j
(
t
)
+
ˆ
f
j
0
(
t
)
ˆ
M
i
(
q
i
)
¨
q
i
+
ˆ
C
i
(
q
i
,
˙
q
i
)
˙
q
i
+
ˆ
g
i
(
q
i
)
+
δ
(
t
)
=
τ
i
+
f
j
(19)
where
ˆ
M
j
,
ˆ
D
j
,
ˆ
K
j
,
and
ˆ
f
j
0
are
the
estimation
parameters
of
the
mass,
damping,
stif
fness,
and
nonhomogeneous
parameters
of
the
human
operator
,
i
∈
{
m,
s
}
represents
master
and
sla
v
e
systems,
and
j
∈
{
h,
e
}
represents
the
human
and
en
vironment,
respecti
v
ely
.
Remark
2.
The
same
philosophy
could
be
applied
for
e
xternal
disturbance
r
ejection
inserting
a
bounded
term
r
epr
esenting
the
e
xternal
disturbance
δ
(
t
)
,
as
(20):
∥
δ
(
t
)
∥
≤
∆
<
∞
(20)
In
this
scenario,
an
adaptive
par
ameter
ˆ
δ
(
t
)
is
needed
to
be
added
to
the
contr
ol
signal
in
the
BTS
(1)
suc
h
that
(21)
and
(22):
τ
=
ˆ
f
j
(
t
)
+
ˆ
M
(
t
)
a
+
ˆ
C
(
q
,
˙
q
)
˙
q
+
ˆ
g
(
q
)
+
ˆ
δ
(
t
)
(21)
Int
J
Rob
&
Autom,
V
ol.
10,
No.
3,
September
2021
:
235
–
260
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
241
in
addition
to
the
term
of
the
PD
contr
oller
a
=
¨
q
des
−
K
d
˙
e
−
K
p
e
(22)
wher
e
K
d
,
K
p
>
0
ar
e
symmetric
matrices.
Then,
after
some
manipulation
and
linear
parameterizing,
the
dynamics
of
the
BTS
are
formulated
as
(23):
˜
M
(
¨
e
+
K
d
˙
e
+
K
p
e
)
=
Ψ
(
q
,
˙
q
,
¨
q
)
˜
θ
(
t
)
+
˜
δ
(
t
)
+
˜
f
j
(
t
)
(23)
with
the
estimation
error
of
the
corresponding
parameter
is
indi
cated
by
the
sign
∼
.
So,
these
estimated
parameters
ha
v
e
to
be
handled
in
the
design
of
the
controller
to
achie
v
e
or
maintain
the
stability
and
the
task
performance
of
the
BTS
in
the
presence
of
uncertainties.
The
adapti
v
e
la
w
for
˜
θ
(
t
)
and
˜
δ
(
t
)
is
obtained
by
applying
the
s
tability
analysis
such
as
the
L
yapuno
v
theorem
such
that
the
con
v
er
gence
of
the
error
signals
to
zero
is
achie
v
ed
and
the
estimation
errors
are
bounded
[34].
Additionally
,
one
of
the
general
methods
in
adapti
v
e
control
des
ign
is
to
utilize
the
linearity
in
the
parameters
of
the
BTS
(1).
Let
the
combined
error
signals
of
both
position
and
v
elocity
of
the
master
and
sla
v
e
be
dened
as
(24)
and
(25):
ϵ
m
=
˙
e
m
+
P
m
e
m
(24)
and
ϵ
s
=
˙
e
s
+
P
s
e
s
(25)
where
P
m
,
P
s
>
0
,
so,
the
control
signals
are
(26):
u
i
=
−
u
id
−
ˆ
M
i
(
q
i
)
P
i
˙
q
i
−
ˆ
C
i
(
q
i
,
˙
q
i
)
P
i
q
i
=
−
u
id
−
Ψ
i
ˆ
θ
i
,
i
∈
{
m,
s
}
(26)
and
the
closed
loop
BTS
will
be
(27):
M
i
˙
e
i
+
C
i
e
i
=
Ψ
i
˜
θ
i
+
u
id
−
f
j
,
i
∈
{
m,
s
}
,
j
∈
{
h,
e
}
(27)
will
guarantee
both
of
the
errors’
asymptotic
con
v
er
gence
and
the
parameter
estimation
errors
˜
θ
i
to
be
bounded.
Also,
the
L
yapuno
v
analysis
or
an
y
other
stability
analysis
could
be
used
to
deri
v
e
the
adaptation
la
ws
˙
ˆ
θ
i
.
Remark
3.
The
adaptive
contr
ol
appr
oac
h
could
be
combined
with
intellig
ent
and
some
nonlinear
algorithms
lik
e
fuzzy
contr
ol
and
neur
al
network
(NN).
This
featur
e
allows
r
esear
c
her
s
to
design
ne
w
algorithms
to
enhance
and
impr
o
ve
the
performance
of
the
BTSs
whic
h
mak
es
it
one
of
the
most
powerful
contr
ol
appr
oac
hes.
The
designed
control
system
using
the
combination
of
algorithms
leads
to
a
more
rob
ust
and
ef
cient
BTS.
Moreo
v
er
,
it
is
not
necessary
to
ha
v
e
the
e
xact
model
and
precise
information
about
the
BTS.
Also,
these
algorithms
allo
w
o
v
ercoming
other
restrictions
in
the
BTSs,
for
e
xample,
long
time-v
arying
delays,
input
saturation,
pack
et
loss,
force
reection
error
,
and
position
drift.
An
adapti
v
e
switched
control
considering
passi
v
e
and
non-passi
v
e
e
xternal
forces,
actuator
saturation,
and
unkno
wn
dynam
ics
is
designed
for
BTSs
including
asymmetric
time-v
arying
delay
[81].
The
obtained
scheme
has
the
capability
of
adapti
v
e
systems
whi
ch
is
indicated
by
the
achie
v
ed
bounded
position
tracking
error
of
the
BTS.
An
adapti
v
e
fuzzy
control
system
w
as
de
v
eloped
to
attain
the
state-independent
input-to-
output
stability
for
a
multilateral
asymmet
ric
teleoperation
system
[82].
Also,
similar
w
orks
are
found
such
as
[83]-[85].
Chen
et
al
.
[86],
an
adapti
v
e
nite-time
control
method
is
presented
using
subsystem
decomposition.
The
stability
is
guaranteed
by
applying
the
L
yapuno
v-Kraso
vskii
analysis
and
bounded
tracking
error
is
ob-
tained.
Rob
ust
adapti
v
e
techniques
were
proposed
for
considering
unkno
wn
parameters
the
uncertainties
of
the
BTS
[87].
More
e
xamples
on
this
approach
are
[8]-[13],
[41].
Some
adv
antages
of
the
adapti
v
e
control
approach
in
the
literature
include:
considering
non-homogenous
and
state
independent
input/output
stability
[81],
[82]
non-passi
v
e
input
forces
[24],
considering
drift
dif
fusion
impro
v
ed
haptic
[88],
and
multi
robot
sla
v
e
and
input
saturation
[89].
The
dra
wbacks
of
this
approach
include:
chattering
of
torque
[81],
[82],
[89],
does
not
consider
time
delay
[88],
constant
time
delay
[90],
and
non
assymptotic
stability
[24],
[87].
Contr
ol
of
teleoper
ation
systems
in
the
pr
esence
of
cyber
attac
ks:
A
surve
y
(Mutaz
M.
Hamdan)
Evaluation Warning : The document was created with Spire.PDF for Python.
242
❒
ISSN:
2089-4856
3.4.
Rob
ust
contr
ol
appr
oach
The
main
objecti
v
e
when
applying
the
rob
ust
control
approach
is
to
consider
the
w
orst
conditions
which
ha
v
e
ne
g
ati
v
e
ef
fects
on
the
stability
and
the
performance
of
the
BTSs.
So,
the
control
scheme
is
called
rob
ust
when
it
maintains
the
stability
and
the
performance
of
the
BTSs
af
fected
by
disturbing
f
actors.
In
addition
to
the
elements
af
fecting
normal
systems
such
as
the
uncertainties
in
the
model,
ne
glected
dynamics,
and
e
xternal
disturbances,
the
netw
ork-induced
uncertainty
is
the
most
tragic
element
tha
t
af
fects
the
stability
and
performance
in
BTSs.
In
mathmatic
w
ords,
the
rob
ust
control
problem
is
minmax
problem
such
that
it
is
a
minimization
of
the
fraction
y
δ
=
∥
y
∥
/
∥
δ
∥
while
the
term
y
x
=
∥
y
∥
/
∥
x
∥
is
maximized,
with
y
δ
refers
to
the
disturbances’
contrib
ution
δ
in
the
output
of
the
system
y
,
with
the
desired
input
x
.
∥
.
∥
is
an
y
Euclidean
norm
function.
Ideal
v
alue
for
y
δ
is
0
and
for
y
x
is
1.
The
sliding
mode
control
(SMC)
is
the
most
f
amiliar
rob
ust
control
approach.
In
this
technique,
the
controller
is
designed
to
achie
v
e
the
con
v
er
gence
of
the
error
signal
to
a
predetermined
sliding
surf
ace.
This
surf
ace
is
a
dif
ferential
equation
for
the
error
which
has
a
solution
lying
on
a
con
v
er
gent
set
or
point.
One
w
ay
for
SMC
is
to
dene
the
error
between
the
motion
of
the
master
and
sla
v
e
robots
as
e
=
x
s
−
K
x
m
.
So,
the
sliding
surf
ace
is
gi
v
en
by
(28):
s
=
˙
e
+
λe
(28)
in
which
λ
>
0
is
a
constant
used
for
determining
the
features
of
the
surf
ace
and
the
error’
s
rate
of
con
v
er
gence,
and
K
>
0
is
a
coef
cient
matrix
for
scaling.
Remark
4.
A
dissipation
condition
must
be
met
to
guar
antee
the
r
ob
ust
stability
as
the
following
foe
e
xample:
s
˙
s
≤
−
γ
∥
s
∥
<
0
wher
e
γ
>
0
is
a
constant.
Either
lar
g
e
gain
or
c
hattering
contr
ol
signals
will
be
g
ener
ated
to
obtain
this
featur
e
.
A
chattering-free
SMC
approach
is
presented
to
achie
v
e
a
rob
ust
performance
of
a
BTS
in
unce
rtain
conditions
[91].
The
number
of
the
needed
sensors
is
reduced
by
applying
a
pseudo-sensorless
approach
which
also
decreased
the
uncertainties
af
fecting
the
BTS.
A
PD
controller
is
proposed
for
specic
delayed
BTSs
using
the
linear
matrix
inequalities
(LMI)
technique
[92].
The
BTS
has
been
stabilized
re
g
ardless
of
the
occurrences
of
delays
in
the
communication
channel.
But,
the
presented
PD
controller
w
as
applied
only
on
the
sla
v
e
robot
and
designed
for
the
constant
time
delay
.
The
rob
ust
control
approach
is
more
suitable
for
linear
systems
because
of
the
norm-based
a
nalysis
and
calculations.
So,
Some
techniques
of
the
rob
ust
control
lik
e
H
2
,
H
∞
,
and
µ
-synthesis
are
applied
for
linearized
time
delay
BTSs.
These
techniques
are
normally
used
in
the
frequenc
y
domain
to
handle
the
w
orst
case
scenario,
uncertainties,
and
the
upper
limit
of
the
delay
of
the
linearized
BTSs.
Some
adv
antages
of
the
adapti
v
e
control
approach
in
the
literature
include:
reducing
time
sensi
ti
vity
and
con
v
e
x
optimization
using
H
∞
approach
[93],
achie
ving
optimal
transparenc
y
using
H
2
−
H
∞
approach
[94],
maintain
passi
vity
[86],
considering
time
v
arying
uncertainties
[95],
and
simultaneous
stabilit
y
and
trans-
parenc
y
[96].
The
dra
wbacks
of
this
approach
include:
considering
linear
model
with
constant
delay
[93],
[94],
[97],
linear
model
without
time
delay
[95],
occurrences
of
chattering
torque
[86],
[96],
constant
time
delay
[98]-[100].
3.5.
Neural
netw
ork-based
appr
oach
One
of
the
w
orst
features
af
fecting
the
stability
and
the
performance
of
BTSs
is
the
e
xistence
of
time-
v
arying
delays.
Some
literature
considering
this
situation
by
considering
the
input
saturation
with
time-v
arying
delay
using
adapti
v
e
fuzzy
control
method
[101]
or
applying
tw
o
controllers,
a
PD-lik
e
and
P-lik
e
controllers
to
handle
constant
delay
and
time-v
arying
delay
,
respecti
v
ely
[102].
The
neural
netw
ork
(NN)
based
approach
for
controlling
BTSs
has
a
great
v
alue
due
to
its
high
poten-
tial
to
learn
and
parallel
adapti
v
e
processing.
These
features
mak
e
the
NN
more
practical
for
comple
x
nonlinear
BTSs
and
suitable
to
deal
with
the
time
delays
and
uncertainties
e
xisting
in
the
BTSs.
The
basic
idea
of
NNs
comes
from
the
neuron
system
in
the
human
body
.
So,
similar
to
the
cell
model,
the
y
ha
v
e
dendrite
weights
w
j
,
a
nonlinear
function
σ
(
.
)
which
is
normally
called
the
acti
v
ation
function
of
the
cell
as
sho
wn
in
(29),
and
Int
J
Rob
&
Autom,
V
ol.
10,
No.
3,
September
2021
:
235
–
260
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
243
a
ring
threshold
w
0
.
Let
the
output
signal
of
the
n
-dimensional
input
signal
x
(
t
)
∈
R
n
to
be
y
(
t
)
.
So,
the
neuron
is
represented
by
the
follo
wing
(29):
y
(
t
)
=
σ
n
X
j
=1
w
j
x
j
(
t
)
+
w
0
.
(29)
The
insight
of
the
neuron
is
e
xci
tatory
synapses
when
w
j
>
0
and
it
is
inhibitory
synapses
when
w
j
<
0
[103].
The
acti
v
ation
function
σ
(
.
)
is
introduced
separately
and
relat
ed
to
the
requirements
of
the
current
application.
It
also
sho
ws
the
beha
vior
of
the
cell
in
the
case
that
y
(
t
)
is
in
its
range.
Also,
since
the
deri
v
ati
v
e
is
required
in
learning
algorithms
for
man
y
cases,
σ
(
.
)
has
to
be
dif
ferentiable.
Both
the
input
v
ector
and
weights
v
ector
are
assigned
to
deri
v
e
the
output
signal.
So,
we
will
get
the
follo
wing
(30):
y
(
t
)
=
σ
[
w
0
,
w
1
,
.
.
.
,
w
n
]
[1
,
x
1
,
.
.
.
,
x
n
]
T
=
σ
⃗
w
T
⃗
x
(
t
)
+
w
0
.
(30)
Remark
5.
F
or
the
case
of
the
multicell
model
of
a
neur
on
consists
of
r
cells,
with
an
input
signal
⃗
x
(
t
)
and
an
output
y
m
(
t
)
,
the
system
is
r
epr
esented
by
the
following
(31):
⃗
y
(
t
)
=
[
y
1
,
y
2
,
.
.
.
,
y
m
]
T
=
⃗
σ
W
T
⃗
x
(
t
)
+
⃗
w
0
(31)
in
whic
h
the
weight
matrix
is
given
by
(32):
W
=
[
w
ij
]
,
i
=
1
.
.
.
r
,
j
=
1
.
.
.
n.
(32)
The
features
of
the
NNs
allo
w
researchers
to
apply
them
in
a
wide
range
of
applications
to
solv
e
the
nonlinear
and
uncertainty
problem
for
both
control
and
estimation
problems
especially
in
robotics
and
BTSs
[66],
[89],
[104],
[105].
A
comprehensi
v
e
discussion
on
the
application
of
NNs
for
robotics
control
is
presented
in
[103].
NNs
were
used
in
BTSs
for
approximating
and
dealing
with
delays
and
uncertaint
ies
of
the
com-
munication
channel.
So,
NNs
are
applied
for
modeling
the
delay
in
the
system
d
f
(
t
)
and
d
b
(
t
)
,
or
the
total
transmission
between
the
master
and
the
sla
v
e.
Let
us
assume
that
the
position
signal
of
the
master
robot
x
m
(
k
)
has
to
be
transmit
ted
to
the
sla
v
e
side
at
the
sampling
time
k
.
One
can
de
v
elop
the
NNs
for
modeling
the
netw
ork
to
estimate
this
process
as
follo
ws
(33):
ˆ
x
m
(
k
)
=
T
NN
(
x
m
(
i
))
,
i
∈
[1
,
2
,
.
.
.
,
k
−
1
,
k
]
(33)
with
T
NN
(
.
)
represents
the
NN
model
obtained
by
(34):
T
NN
(
x
m
(
i
))
=
⃗
σ
W
T
x
m
(
i
)
+
⃗
w
0
.
(34)
The
local
controllers
on
the
master
and
sla
v
e
sides
are
designed
to
pro
vide
a
prediction
on
the
trans-
mitted
signals
using
the
estimator
model
of
the
communication
channel.
So,
the
control
signal
in
this
system
will
be
more
ef
cient
in
comparison
to
the
normal
case
where
the
signals
recei
v
ed
by
the
controllers
could
be
af
fecting
by
the
delay
or
the
distortion
in
the
system
[34].
A
prediction
algorithm
using
adapti
v
e
linear
NN
w
as
de
v
eloped
for
the
application
of
an
internet
time
delay
system
[106].
Auld
et
al.
[107],
a
Bayesian
NN
w
as
implemented
for
classifying
the
internet
traf
c
without
information
about
the
IP
.
The
proposed
method
pro
vided
95%
accurac
y
after
testing
it
on
a
real
training
NN
for
8
months.
A
recurrent
NN
w
as
utilized
for
modeling
and
predicting
the
Internet
end-to-end
delay
[108].
The
recurrent
NN
w
as
trained
and
v
alidated
using
the
discrete-time
data
obtained
by
measuring
the
delay
between
tw
o
dif
ferent
nodes.
NN
approach
w
as
also
implemented
for
modeling
traf
c
o
w
in
self-
similar
computer
netw
orks
which
is
statically
do
not
depend
on
time
[109]-[112].
Other
e
xamples
including
further
details
could
be
found
in
[113]-[115].
In
addition
to
the
aforementioned
applications,
NNs
were
implemented
in
the
local
cont
rollers
on
both
sides
of
the
BTSs.
A
better
performance
w
as
obtained
by
combining
the
NN
approach
with
other
control
methods,
mainly
the
adapti
v
e
control
method
[66],
[89],
[104].
Another
application
of
the
NN
in
the
BTSs
is
to
Contr
ol
of
teleoper
ation
systems
in
the
pr
esence
of
cyber
attac
ks:
A
surve
y
(Mutaz
M.
Hamdan)
Evaluation Warning : The document was created with Spire.PDF for Python.
244
❒
ISSN:
2089-4856
utilize
the
NN
in
designing
the
control
system
for
estimating
the
delayed
or
disturbed
signals
and
for
protecting
the
system
simultaneous
ly
when
it
is
subjected
to
uncertainties
and
latenc
y
[34].
Ho
we
v
er
,
it
is
w
orth
kno
wing
that
the
parameters
of
the
NN
structure
such
as
the
number
of
layers,
the
number
of
neurons,
etc.
af
fect
the
accurac
y
of
the
estimation
process.
So,
the
application
of
the
NN
control
approach
handle
the
uncertainties
and
delay
indirectly
by
gi
ving
the
best
possible
result
within
the
unkno
wn
circumstances
of
the
system.
A
radial
basis
function
(RBF)
NN-based
procedure
w
as
proposed
for
handling
v
ariable
delays
in
teleoperation
system
[36].
The
model
contains
linear
viscous
and
Coulomb
v
elocity-dependent
frictions.
But,
it
is
required
to
measure
the
acceleration
or
to
estimate
it,
and
thi
s
is
not
practical
in
man
y
cases.
The
follo
wing
RBF
Gaussian
functions
were
applied
for
approximating
the
dynamics
of
the
BTS
(1):
Φ
n
(
x
)
=
e
xp
−
1
2
H
2
n
∥
x
−
C
n
∥
2
(35)
where
H
n
is
the
width
and
C
n
is
the
center
for
the
n
th
neuron.
Dyn
i
(
x
i
)
=
W
T
i
Φ
i
(
x
i
)
+
ξ
i
(
x
i
)
(36)
where
x
i
=
[
¨
q
T
i
,
˙
q
T
i
,
q
T
i
,
¨
x
T
i
,
˙
x
T
i
,
x
T
i
]
T
.
Therefore,
the
follo
wing
approximated
dynamic
are
implemented
in
the
control
system
to
o
v
ercome
both
of
the
unkno
wn
bounded
terms
and
the
in
v
erse
dynamics
by
(37):
Dyn
i
(
x
i
)
=
f
b
i
(
q
i
,
˙
q
i
)
+
g
i
(
q
i
)
+
C
i
(
q
i
,
˙
q
i
)
˙
q
i
+
M
j
¨
x
i
+
B
j
˙
x
i
+
K
j
x
i
+
D
i
(
t
)
.
(37)
Thus,
the
control
signal
τ
i
is
calculated
for
the
follo
wing
system
sho
wn
in
(38):
M
i
(
q
i
)
¨
q
i
+
C
i
(
q
i
,
˙
q
i
)
˙
q
i
+
g
i
(
q
i
)
+
f
b
i
(
q
i
,
˙
q
i
)
+
D
i
(
t
)
=
τ
i
+
M
j
¨
x
i
+
B
j
˙
x
i
+
K
j
x
i
+
f
j
0
+
Dyn
i
(
x
i
)
=
Ψ
i
(
x
i
)Θ
i
.
(38)
The
desired
control
signal
for
system
(38)
when
the
function
Dyn
i
is
ideally
approximated,
is
obtained
using
(39):
τ
i
=
M
i
(
q
i
)
¨
q
∗
i
−
K
D
i
(
˙
q
i
−
˙
q
∗
i
)
−
K
P
i
(
q
i
−
q
∗
i
)
(39)
with
positi
v
e-denite
g
ain
matrices
K
D
i
and
K
P
i
that
by
substituting
in
(38)
leads
to
a
Hurwitz
f
u
nc
tion
sho
w
in
(40):
¨
e
i
+
K
D
i
˙
e
i
+
K
P
i
e
i
=
0
.
(40)
The
NN-based
approximations
are
inherently
intelligent
enough
to
use
the
pre
vious
e
xperiences
of
the
BTS
for
learning,
although
the
y
ha
v
e
an
error
in
real
applications.
The
accurac
y
of
the
approximation
could
be
increased
by
selecting
lar
ge
quantities
for
NN
nodes
n
in
(35)
for
function
Dyn
(
x
)
o
v
er
a
compact
set
of
the
input
x
∈
Ω
x
⊂
R
x
.
Other
methods
of
control
were
combined
with
NN
to
decrease
the
una
v
oidable
estimation
error
.
A
neural
adapti
v
e
control
w
as
proposed
for
a
single-master
multi-sla
v
e
BTS
[89].
The
use
of
the
NNvto
model
of
the
unkno
wn
nonlinear
plants
helped
to
handle
t
he
uncertainties
in
the
dynamics
of
and
the
input
of
the
BTS.
But,
se
v
eral
assumptions
were
applied
in
this
scheme
such
as
the
singularity-free
motion
of
the
sla
v
e
robot,
the
rigidity
of
the
object,
and
rigid
attachment
between
sla
v
es’
end-ef
fector
and
the
object.
Consider
the
signals
of
the
combined
error
to
be
(41):
r
m
=
˙
x
m
+
Γ
m
e
m
w
,
˙
x
mr
=
−
Γ
m
e
m
r
s
=
˙
x
s
+
Γ
s
e
s
,
˙
x
sr
=
−
Γ
s
e
s
(41)
with
error
deniti
o
ns
e
m
(
t
)
=
x
m
(
t
)
−
x
s
(
t
−
d
b
(
t
))
and
e
s
(
t
)
=
x
s
(
t
)
−
x
m
(
t
−
d
f
(
t
))
,
and
matrices
Γ
m
,
Γ
s
>
0
.
Then,
the
adapti
v
e
control
signal
is
selected
as
(42):
u
m
=
−
β
m
ˆ
Θ
T
m
Φ
m
(
Z
m
)
sgn
(
r
m
)
u
s
=
−
β
s
ˆ
Θ
T
s
Φ
s
(
Z
s
)
sgn
(
r
s
)
(42)
Int
J
Rob
&
Autom,
V
ol.
10,
No.
3,
September
2021
:
235
–
260
Evaluation Warning : The document was created with Spire.PDF for Python.