Inter
national
J
our
nal
of
Robotics
and
A
utomation
(IJRA)
V
ol.
10,
No.
3,
September
2021,
pp.
192
∼
206
ISSN:
2089-4856,
DOI:
10.11591/ijra.v10i3.pp192-206
❒
192
New
time
delay
estimation-based
virtual
decomposition
contr
ol
f
or
n-DoF
r
obot
manipulator
Hachmia
F
aqihi
1
,
Khalid
Benjelloun
2
,
Maar
ouf
Saad
3
,
Mohammed
Benbrahim
4
,
M.Nabil
Kab
baj
5
1,2
EPTICAR,
Ecole
Mohammadia
d’Ing
´
enieurs,
Mohammed
V
Uni
v
ersity
,
Rabat,
Morocco
3
Department
of
Electrical
Engineering,
Ecole
de
T
echnologie
Sup
´
erieure,
Montreal,
Canada
4,5
LIMAS,
F
aculty
of
Sciences,
Sidi
Mohamed
Ben
Abdellah
Uni
v
ersity
,
Fez,
Morocco
Article
Inf
o
Article
history:
Recei
v
ed
Feb
16,
2021
Re
vised
Mar
6,
2021
Accepted
Apr
9,
2021
K
eyw
ords:
Free-re
gressor
Non
linear
control
Robot
manipulator
T
ime
delay
estimation
V
irtual
decomposition
control
ABSTRA
CT
One
of
the
most
ef
cient
approaches
to
control
a
multiple
de
gree-of-freedom
robot
manipulator
is
the
virtual
decomposition
control
(VDC).
Ho
we
v
er
,
the
use
of
the
re-
gressor
technique
in
the
con
v
entionnal
VDC
to
estimate
the
unkno
wn
and
uncertaities
parameters
present
some
limitations.
In
this
paper
,
a
ne
w
control
strate
gy
of
n-DoF
robot
manipulator
,
refering
t
o
reor
g
anizing
the
equation
of
the
VDC
using
the
time
delay
estimation
(TDE)
ha
v
e
been
in
v
estig
ated.
In
the
proposed
controller
,
the
VDC
equations
are
rearranged
usi
ng
the
TDE
for
unkno
wn
dynamic
estimations.
Hence,
the
decoupling
dynamic
model
for
the
manipulator
is
est
ablished.
The
stability
of
the
o
v
erall
system
is
pro
v
ed
based
on
L
yapuno
v
theory
.
The
ef
fecti
v
eness
of
the
proposed
controller
is
pro
v
ed
via
case
study
performed
on
7-DoF
robot
mani
pulator
and
com-
pared
to
the
con
v
e
ntionnal
Re
gressor
-based
VDC
according
to
some
e
v
alution
criteria.
The
results
carry
out
the
v
alidit
y
and
ef
cienc
y
of
the
proposed
time
delay
estimation-
based
virtual
decomposition
controller
(TD-VDC)
approach.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Hachmia
F
aqihi
Department
of
Electrical
and
Computer
Engineering,
EPTICAR
Ecole
Mohammadia
d’Ing
´
enieurs
Mohammed
V
Uni
v
ersity
,
Rabat,
Morocco
Email:
fhachmia@gmail.com
1.
INTR
ODUCTION
In
man
y
robotic
applications,
the
principal
technical
challenges
arise
in
control
implementations,
spe-
cialy
when
its
subject
to
hight
number
of
de
gree-of-freedom
(DoF).
Indded,
the
robotic
systems
with
high
DoF
,
can
be
modeled
by
a
set
of
coupled
highly
nonlinear
dif
ferential
equations,
with
v
ar
ious
uncertainties
and
dis-
turbances,
which
increases
the
comple
xity
of
their
control.
A
wide
range
of
approaches
ha
v
e
been
proposed
in
the
literature
to
control
robot
systems,
ranging
from
lin-
ear
to
nonlinear
techniques,
such
as
computed
torque
control
(CTC),
rob
ust
control,
passi
vity
based
control,
L
yapuno
v
stability
based
rob
ust
control,
sliding
mode
control
(SMC)
[1]-[7],
Feedback
Linearization
,
Back-
stepping
[8]-[12].
All
these
techniques
are
based
on
the
traditional
Lagrange-Euler
formulation,
which
present
inherent
incon
v
enient,
in
comple
xity
,
then
in
computational
b
urden
[13]-[18].
More
we
ha
v
e
a
high
number
of
DoF
in
the
system,
more
t
h
a
n
the
comple
xity
of
the
dynamic
model
and
the
computation
b
urden
will
increase
[19],
[20];
It
w
as
proportional
to
the
fourth
po
wer
of
the
DoF
of
the
robot
[21].
This
problem
limits
the
use
of
these
algorithms
and
reduces
the
feasibility
of
the
control
system.
Hence,
to
cope
with
the
aforementioned
problem,
a
no
v
el
theory
based
on
virtual
decomposition
con-
J
ournal
homepage:
http://ijr
a.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
193
trol
(VDC)
w
as
proposed
by
[21]
to
solv
e
the
modeling
and
control
problems
of
a
multi-DoF
robotic
system.
It
is
dened
as
adaptatif
control
approach
[22]-[25].
In
the
VDC
approach
the
entire
system
is
decomposed
virtually
into
subsystems
(
single
joint
and
single
link).
The
dynamic
interaction
between
tw
o
adjacent
subsys-
tems
is
handled
by
virtual
po
wer
o
w
(VPF),
which
leads
to
pro
v
e
the
virtual
stability
.
F
or
each
subsystem,
a
subcontroller
is
designed
independently
,
while
the
stability
of
the
global
system
is
rigorously
maintained
[22],
[24],
[26]-[30].
The
obtained
dynamic
equation
of
each
subsystem
is
relati
v
ely
simpl
e,
which
can
reduce
the
computation
b
urden
of
the
system.
Compared
to
the
Lagrangian
dynamic
model,
the
computation
of
VDC
method
is
proportional
only
to
the
number
of
subsystems
(DoFs).
In
order
to
dene
the
full
dynamics
of
the
in
v
estig
ated
robot,
the
VDC
approach
is
based
on
re
gression
technique.
Indeed,
the
dynamic
equation
of
each
subsystem
can
be
linearly
parameterized
in
terms
of
a
re
gressor
matrix,
and
a
unkno
wn
parameters
v
ectors,
using
the
required
linear/angular
v
elocity
v
ector
and
its
t
ime-deri
v
ati
v
e.
Ho
we
v
er
,
the
re
gressor
-based
VDC
control
presents
dif
culties
in
practical
implementation
due
to
the
comple
xity
of
the
re
gressor
matrix,
espe-
cially
to
estimate
the
uncertainties
parameters
[31],
[32].
Indeed,
the
re
gressor
matrix
is
kno
wn
to
be
highly
nonlinear
,
and
its
deri
v
ation
is
not
unique
and
remain
tedious,
although
the
process
is
standardized
which
increase
the
computation
comple
xity
.
In
order
to
a
v
oid
the
use
of
the
re
gressor
techniques,
some
alternati
v
e
tools
ha
v
e
been
proposed
in
the
literature
[33]-[36],
using
the
approximation-based
adapti
v
e
control
based
on
function
approximation
technique
(F
A
T).
In
order
to
approximate
the
uncertainties
parameter
v
ectors
of
the
dynamic
model,
the
F
A
T
technique
is
based
on
linear
parameterization
in
the
form
of
the
weighting
matrix
and
the
(orthogonal)
basis
function
matrix
of
the
tar
get
matrix/v
ector
.
Ho
we
v
er
,
it
presents
some
limitations.
Indeed,
the
F
A
T
is
selected
arbitrary
without
specic
criteria,
thus
an
approximation
error
can
be
produced.
Furthermore,
the
estimation
of
the
initial
v
alues
of
the
weighting
matrix
is
comple
x.
Using
the
F
A
T
for
high
DoF
can
be
generated
with
a
computational
com-
ple
xity
of
weighing
and
orthogonal
matrices
[33].
In
order
to
propose
a
suitable
solution
to
the
aforementioned
problems,
this
paper
introduces
a
ne
w
nonlinear
adapti
v
e
control
strate
gy
including
the
virtual
decomposition
control
(VDC)
[21]
and
time
delay
estimation
(TDE)
[37],
kno
wn
with
its
se
v
eral
adv
antages.
The
basic
idea
of
the
proposed
time
delay
estimation-based
virtual
decomposition
controller
(TD-VDC)
can
be
summarized
as:
a.
The
VDC
is
used
as
an
ef
cient
tool
to
handle
the
ful
l-dynamics-based
control
problem
of
n-DoF
robot
ma-
nipulator
.
This
approach
considers
the
dynamics
of
subsystems
(rigid
bodies
and
joints)
to
carry
out
a
tracking
trajectory
,
while
guaranteeing
the
stability
and
con
v
er
gence
of
the
entire
robotic
system.
b
.
The
TDE
is
used
to
estimate
simply
and
ef
fecti
v
ely
,
the
unkno
wn
parameters
v
ectors
including
uncertainties
and
e
xternal
disturbances.
It
require
a
use
of
time-delayed
information
of
the
control
torque
inputs
and
state
deri
v
ati
v
es
for
each
subsystems.
c.
Based
on
the
aforementioned
adv
antages
of
VDC
and
TDE
approachs,
the
TD
VDC
is
used
to
pro
vide
an
adapti
v
e
control
with
higher
precision,
ensuring
then
lo
w
computational
b
urden,
suitable
for
high
Dof
robotic
systems.
The
remainder
of
this
paper
is
or
g
anized
as
follo
ws:
in
Section
1,
the
dynamic
model
of
each
subsys-
tem
of
n-DOF
robot
manipulators
are
des
cribed.
In
Section
3,
the
proposed
TD
VDC
contr
o
l
ler
is
designed,
and
compared
to
the
con
v
entional
re
gressor
-based
virtual
decomposition
control
is
presented.
Ine
Section
IV
he
proposed
TD
VDC
is
designed
with
stability
analysis.
Section
V
,
a
case
study
is
performed
on
7-DoF
robot
manipulator
with
TD
VDC,
and
compared
to
the
re
gressor
-based
VDC.
The
conclusions
are
summarized
in
Section
VI.
2.
SYSTEM
MODELING
2.1.
Desription
of
system
The
equation
of
motion
of
an
n-DOF
robot
manipulators
are
described
according
to
the
Euler
-Lagrange
theory
[3],
as
(1):
τ
=
M
(
q
)
¨
q
+
C
(
q
,
˙
q
)
+
G
(
q
)
(1)
where
M
(
q
)
,
C
(
q
,
˙
q
)
,
G
(
q
)
and
G
(
q
,
˙
q
)
are
respecti
v
ely
the
manipulator’
s
mass
matrix,
t
he
Coriolis
and
cen-
trifug
al
terms
v
ector
,
the
gra
vity
terms
v
ector
,
and
the
torque
friction
v
ector
.
In
the
VDC
controller
,
the
dynamic
equations
of
the
system
can
be
e
xpressed
as
link
subsystems
and
j
oint
subsystems,
where
V
irtual
Po
wer
Flo
w
(VPF)
[21]
is
used
to
characterize
the
coupling
dynamic
interactions
among
subsystems.
The
Figure
1
represents
the
virtual
decomposition
of
serial
robot
manipulator
to
i
links
,
where
i
=
Ne
w
time
delay
estimation-based
virtual
decomposition
contr
ol
for
n-DoF
r
obot
...
(Hac
hmia
F
aqihi)
Evaluation Warning : The document was created with Spire.PDF for Python.
194
❒
ISSN:
2089-4856
1,...,n,
connected
via
mechanical
joints.
Each
link
has
one
dri
ving
cutting
point
according
to
the
frame
B
i
+1
and
one
dri
v
en
cutting
point
according
to
the
frame
B
i
.
The
i
th
joint
has
one
dri
v
en
cutting
point
according
t
o
the
frame
B
i
and
one
dri
ving
cutting
point
according
to
the
frame
T
i
.
The
dynamic
equation
of
e
v
ery
subsystem
is
deri
v
ed
with
respect
to
the
local
frame
B
i
according
to
the
Dena
vit
Hartenber
g
formalism.
Figure
1.
V
irtual
decomposition
shematic
of
serial
robot
manipulator
[21]
2.2.
Link
dynamics
The
dynamic
equation
of
the
i
th
rigid-link
subsystem
foll
o
wing
its
x
ed
frame
can
be
e
xpressed
as
(2)
[21]:
B
i
F
∗
=
M
B
i
d
dt
(
B
i
V
)
+
C
B
i
B
i
V
+
G
B
i
(2)
where
B
i
F
∗
denote
the
net
force/moment
v
ectors
applied
from
the
lo
wer
(
i
−
1)
th
link
to
the
i
th
link
e
xpressed
in
frame
B
i
.
B
i
V
is
the
generalized
linear/angular
v
elocity
,
and
M
B
i
,
C
B
i
,
G
B
i
represent
respecti
v
ely
the
inertial,
Centrifug
al/Coriolis,
and
gra
vitational
terms,
respecti
v
ely
.
Using
an
iterati
v
e
proc
ess
computation,
the
v
ector
of
the
total
generalized
force
(forces/moments)
acting
on
the
i
th
rigid
body
can
be
computed
as
(3):
B
n
F
=
B
n
F
∗
B
n
−
1
F
=
B
n
−
1
F
∗
+
B
n
−
1
U
B
n
B
n
F
∗
.
.
.
B
i
F
=
B
i
F
∗
+
B
i
U
B
i
+1
B
i
+1
F
∗
(3)
where
B
i
F
is
the
generalized
force
e
x
erted
by
body
i
+
1
on
body
i
,
B
i
+1
F
is
the
generalized
force
e
x
erted
by
body
i
+
1
on
body
i
,
B
i
V
r
is
the
v
elocity
of
body
i
.
B
i
U
T
B
i
+1
is
the
transformation
matrix,
dened
as
(4):
B
i
U
T
B
i
+1
=
B
i
R
B
i
+1
0
3
x
3
S
(
B
i
r
B
i
+1
)
B
i
R
B
i
+1
B
i
R
B
i
+1
(4)
where
B
i
R
B
i
+1
represents
the
rotation
matri
x
from
frame
B
i
to
the
frame
B
i
+1
,
0
3
x
3
is
the
null
matrix,
S
is
the
sk
e
w-symmetric
matrix
operator
performing
the
cross
product
between
tw
o
v
ectors,
and
B
i
r
B
i
+1
denotes
a
v
ector
from
the
origin
of
frame
B
i
to
the
frame
B
i
+1
,
e
xpressed
in
frame
B
i
.
Int
J
Rob
&
Autom,
V
ol.
10,
No.
3,
September
2021
:
192
–
206
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
195
2.3.
J
oint
dynamics
The
dynamic
equation
of
the
i
th
joint
subsystem
e
xpressed
in
its
x
ed
frame
is
gi
v
en
by
the
follo
wing
(5)
[21]:
τ
ij
=
I
mi
¨
q
i
+
k
ci
sig
n
(
˙
q
i
)
(5)
where
I
mi
and
k
ci
denote
the
moment
of
inertia,
and
the
Coulomb
friction
coef
cient
of
the
i
th
joint
respec-
ti
v
ely
.
Finally
the
control
torque
of
the
global
system
can
be
e
xpressed
by
(6):
τ
i
=
τ
ij
+
τ
il
(6)
τ
ij
and
τ
i
denote
the
net
torque
and
the
control
t
orque
applied
to
the
i
th
joint
respecti
v
ely
.
τ
il
denotes
the
output
torque
of
the
i
th
joint
to
w
ard
the
links,
which
can
be
computed
in
magnitude
by
the
torque
projected
from
the
links,
e
xpressed
in
(7).
τ
il
=
z
T
B
i
F
(7)
where
z
=
[0
,
0
,
0
,
0
,
0
,
1]
T
for
re
v
olute
joint.
˙
q
i
the
joint
v
elocity
v
ector
.
3.
CONTR
OLLER
DESIGN
The
main
control
objecti
v
e
in
VDC
approach
is
to
track
a
required
trajectory
such
that
the
joint
t
rack-
ing
error
between
the
actual
and
required
v
elocity
con
v
er
ges
asymptotically
to
zero
in
nite-time,
with
high
accurac
y
e
v
en
in
presence
of
uncertainties
and
e
xternal
disturbances.
In
order
to
design
the
VDC
controller
some
v
ectors
must
be
dened
related
to
this
approach.
3.1.
Requir
ed
v
ectors
The
required
v
elocity
,
is
one
of
the
concept
related
to
VDC
approach
[21],
wich
can
be
e
xpressed
as
(8):
˙
q
r
=
˙
q
d
+
λ
(
q
d
−
q
)
(8)
˙
q
d
and
q
d
denote
respecti
v
ely
the
desired
joint
v
elocity
and
the
desired
joint
angle,
λ
>
0
is
a
constant.
The
dynamic
equations
for
the
VDC
controller
design
are
based
on
the
required
joint
v
elocity
and
the
required
linear/angular
v
elocity
v
ectors.
The
required
linear/angular
v
elocity
of
the
link
can
be
computed
as
(9):
B
i
+1
V
r
=
z
˙
q
(
i
+1)
r
+
B
i
U
T
B
i
+1
B
i
V
r
(9)
Adapti
v
e
control
la
w
of
rigid-link
subsystem
3.2.
Adapti
v
e
contr
ol
law
of
link
subsystem
Referring
to
the
dynamic
link
subsystem
(2),
and
the
required
linear/angular
v
elocity
v
ector
and
its
time-deri
v
ati
v
e,
the
required
force/moment
v
ectors
are
e
xpressed
as
(10)
[21]:
B
i
F
r
∗
=
M
B
i
d
dt
(
B
i
V
r
)
+
C
B
i
B
i
V
r
+
G
B
i
(10)
Where
B
i
F
r
∗
the
required
net
force/moment
v
ectors
of
the
subsystem
links.
B
i
V
the
v
ector
of
the
generalized
v
elocities
(i.e.,
linear
and
angular
components),
wich
can
be
e
xpressed
as
(11):
B
i
V
=
z
˙
q
i
+
B
i
−
1
U
T
B
i
B
i
−
1
V
(11)
Consider
the
linear
parameterization
form,
the
link
subsystem
(2)
can
be
e
xpressed
as
(12):
B
i
F
r
∗
=
Y
l
i
θ
l
i
(12)
where
the
Y
l
i
denotes
the
i
th
re
gressor
matrix
formed
by
the
joint
v
elocity
,
the
linear/angular
v
elocity
and
its
time-deri
v
ati
v
e;
and
θ
l
i
denotes
the
i
th
parameters
v
ector
formed
by
the
uncertainties
parameter
v
ector
.
Therefore,
the
control
la
w
of
link
subsystem
is
designed
as
(13):
B
i
F
r
∗
=
Y
l
i
ˆ
θ
l
i
+
K
l
i
B
i
e
V
(13)
Ne
w
time
delay
estimation-based
virtual
decomposition
contr
ol
for
n-DoF
r
obot
...
(Hac
hmia
F
aqihi)
Evaluation Warning : The document was created with Spire.PDF for Python.
196
❒
ISSN:
2089-4856
where
K
l
i
is
a
diagonal
matrix
representing
the
g
ain
of
the
feedback
controller
,
and
B
i
e
V
is
a
measure
of
the
tracking
accurac
y
dened
by
(14):
B
i
e
V
=
B
i
V
r
−
B
i
V
(14)
ˆ
θ
l
i
is
the
estimate
of
the
uncertainties
parameter
v
ector
θ
l
i
Finally
,
the
control
la
w
of
link
subsystem,
can
be
computed
by
an
iterati
v
e
process,
as
(15):
B
n
F
r
=
Y
l
n
ˆ
θ
l
n
+
K
l
n
B
n
e
V
.
.
.
B
i
F
r
=
Y
l
i
ˆ
θ
l
i
+
K
l
i
B
i
e
V
+
B
i
U
B
i
+1
B
i
+1
F
r
∗
(15)
3.3.
Adapti
v
e
contr
ol
law
of
joint
subsystem
F
or
the
control
la
w
of
joint
subsyst
em
dened
as,
the
required
net
torque
τ
ij
r
applied
to
the
i
th
joint,
is
based
on
the
required
joint
v
elocity
v
ectors,
as
(16)
[21]:
τ
ij
r
=
I
mi
¨
q
ir
+
k
ci
sig
n
(
˙
q
ir
)
(16)
According
to
the
linear
parameterization
property
,
t
h
e
required
net
torque
τ
∗
ir
can
be
written
in
linear
form
as
(17):
τ
ij
r
=
Y
j
i
θ
j
i
(17)
where
Y
j
i
denotes
the
re
gressor
matrix
formed
by
the
joint
v
elocity
and
acceleration,
and
θ
j
i
denotes
the
parameters
v
ector
formed
by
the
ph
ysical
dynamic
parameters.
Due
to
the
dif
culty
in
kno
wing
the
e
xact
v
alue
of
the
ph
ysical
parameters
of
the
i
th
joint,
the
y
should
be
estimated.
Then
the
estimation
v
ector
denoted
by
ˆ
θ
j
i
is
used,
and
the
equation
of
control
becomes
(18):
τ
ij
r
=
Y
j
i
ˆ
θ
j
i
+
K
j
i
e
q
(18)
where
K
j
i
is
a
diagonal
matrix
representing
the
g
ain
of
the
feedback
controller
,
and
e
q
is
a
tracking
joint
error
dened
by
(19):
e
q
i
=
˙
q
ir
−
˙
q
i
(19)
Finally
,
the
total
control
torque
is
computed
using
the
required
output
torque
of
the
i
th
joint
to
w
ard
the
links,
and
the
required
control
torque
of
the
i
th
joint
as
(20):
τ
i
=
τ
ij
r
+
τ
il
r
(20)
where
τ
ij
r
denotes
the
control
torque
of
the
i
th
joint,
and
τ
il
r
the
required
output
torque
of
the
i
th
joint
to
w
ard
the
links
e
xpressed
with
the
required
force/moment
v
ectors
as
(21):
τ
il
r
=
z
T
B
i
F
r
(21)
The
control
based
VDC
approach
is
to
resolv
e
equation
(20),
where
the
v
ectors
of
parameters
estimation
ˆ
θ
l
i
and
ˆ
θ
j
i
are
used.
The
parameter
adaptation
function
should
be
chosen
to
ensure
system
stability
.
3.4.
Regr
essor
-based
VDC
contr
oller
In
the
con
v
entionnal
Re
gressor
-based
VDC
controller
,
the
uncertainties
parameter
v
ectors
ˆ
θ
j
i
and
ˆ
θ
l
i
for
joint
and
link
subs
ystems,
are
updated
using
the
projection
function
P
dened
as
a
dif
ferentiable
scalar
function
[21].
According
to
the
link
subsystem,
the
uncertainties
parameter
v
ector
is
estimated
as
(22):
ˆ
θ
j
iγ
=
P
(
s
iγ
(
t
)
,
ρ
iγ
,
a
iγ
(
t
)
,
b
iγ
(
t
)
,
t
)
(22)
where
ˆ
θ
j
iγ
denotes
the
γ
th
element
of
ˆ
θ
j
i
,
s
iγ
(
t
)
denotes
the
γ
th
element
of
s
i
(
t
)
dened
as
(23)
[21]:
s
i
(
t
)
=
Y
j
i
T
(
B
i
V
r
−
B
i
V
)
(23)
Int
J
Rob
&
Autom,
V
ol.
10,
No.
3,
September
2021
:
192
–
206
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
197
ρ
iγ
>
0
is
a
parameter
update
g
ain,
and
a
iγ
(
t
)
,
b
iγ
(
t
)
denote
the
lo
wer
and
upper
bounds
of
θ
j
iγ
.
The
projection
function
P
is
a
dif
ferentiable
sc
alar
function
dened
by
its
time
deri
v
ati
v
e
which
is
go
v
erned
by
(24)
and
(25):
˙
P
(
t
)
=
ρs
(
t
)
κ
(24)
with
κ
=
0
if
P
(
t
)
<
a
(
t
)
and
s
(
t
)
<
0
0
if
P
(
t
)
>
b
(
t
)
and
s
(
t
)
>
0
1
other
w
hise
(25)
It
is
the
same
for
ˆ
θ
l
i
of
the
required
net
torque.
The
use
of
the
projection
function
for
the
uncertainties
parameter
v
ectors
estimation,
requires
to
com-
pute
the
deri
v
ati
v
e
of
the
re
gressor
matrix
in
e
v
ery
sampling
c
ycle.
Ho
we
v
er
,
the
re
gressor
matrix
deri
v
ation
is
not
unique,
t
ho
ugh
the
process
is
standardized.
Furtheremore,
it
presents
high
comple
xity
,
then
an
additionnal
computational
b
urden
3.5.
Pr
oposed
TD
VDC
contr
oller
As
demonstrated,
re
gressor
-based
VDC
method
presents
inherent
limits
re
g
arding
the
use
of
the
pro-
jection
function
to
estimate
the
unkno
wn
parameter
v
ectors
ˆ
θ
l
i
and
ˆ
θ
j
i
for
link
and
joint
subsystem
s
respec-
ti
v
ely
.
T
o
o
v
er
come
the
issues
with
re
gressor
-based
VDC
t
echnique,
a
ne
w
control
strate
gy
combining
the
TDE
and
VDC
approachs
is
proposed.
The
idea
is
refered
to
estimate
the
dynamic
uncertainties
and
parameter
v
ectors
by
the
use
of
TDE.
Refering
to
the
dynamic
link
subsystem
gi
v
en
in
the
(2),
and
the
dynamique
joint
subsystem
gi
v
en
in
the
(5),
t
he
dynamic
uncertainties
a
n
d
unkno
wn
parameter
v
ectors
can
be
re
grouped
as
(26)
and
(27):
F
or
the
link
subsystems
B
i
F
∗
=
M
B
i
d
dt
(
B
i
V
)
+
H
l
i
(26)
F
or
the
Joint
subsystems
τ
ij
r
=
I
mi
¨
q
ir
+
H
j
i
(27)
H
l
i
and
H
j
i
represents
the
dynamic
uncertainties
and
unkno
wn
parameter
v
ectors
of
the
link
and
joint
subsys-
tems
respecti
v
ely
,
where
(28)
and
(29):
H
l
i
=
C
B
i
B
i
V
r
+
G
B
i
(28)
H
j
i
=
k
ci
sig
n
(
˙
q
ir
)
(29)
Therefore,
the
control
la
w
subsystems,
are
gi
v
en
by
(30):
(
B
i
F
r
∗
=
m
i
B
i
˙
V
+
ˆ
H
l
i
+
K
l
i
B
i
e
V
τ
ij
r
=
i
i
¨
q
ir
+
ˆ
H
j
i
+
K
j
i
e
q
i
(30)
m
i
and
i
i
are
a
constant
coef
cients
associated
to
M
B
i
,
and
I
mi
respecti
v
ely
.
The
determination
of
both
con-
stant
coef
cients
m
i
and
i
i
is
discussed
in
[37],
[38].
ˆ
H
l
i
and
ˆ
H
j
i
represents
respecti
v
ely
the
estimate
of
H
l
i
and
H
j
i
.
Ne
w
time
delay
estimation-based
virtual
decomposition
contr
ol
for
n-DoF
r
obot
...
(Hac
hmia
F
aqihi)
Evaluation Warning : The document was created with Spire.PDF for Python.
198
❒
ISSN:
2089-4856
In
order
to
design
the
TD
VDC
controller
and
carry
out
its
stability
analysis,
let
consider
the
follo
wing
assumptions:
A1:
The
joint
position
and
v
elocity
are
measured.
A2:
The
parameter
v
ectors
H
l
i
and
H
j
i
their
time
deri
v
ati
v
es
d
dt
H
l
i
and
d
dt
H
j
i
are
globally
Lipschitz
functions.
A3:
The
constant
coef
cients
m
i
and
i
i
are
chosen
assuming
that:
∥
I
n
−
M
(
q
)
m
−
1
∥
<
1
∥
I
n
−
I
(
q
)
i
−
1
∥
<
1
According
to
the
use
of
TDE
[37],
and
if
t
he
assumption
A2
is
v
eried,
we
can
estimate
H
l
i
and
H
j
i
.
Indeed
the
v
alue
of
the
function
H
l
i
and
H
j
i
are
considered
at
the
present
time
t
,
v
ery
close
to
that
at
time
(
t
−
T
)
in
the
past
for
a
small
time
delay
T
in
(31).
F
or
the
link
subsystem
ˆ
H
l
i
(
t
)
∼
=
ˆ
H
l
i
(
t
−
T
)
(31)
therefore,
using
an
i
terati
v
e
process,
the
estimate
of
the
uncertainties
parameter
v
ector
of
the
link
substem
ˆ
H
l
i
(
t
)
can
be
computed
as:
ˆ
H
l
i
(
t
)
≃
τ
il
r
(
t
−
T
)
z
−
m
i
B
i
˙
V
(
t
−
T
)
−
K
l
i
B
i
e
V
(
t
−
T
)
−
B
i
U
B
i
+1
(
t
−
T
)
ˆ
H
l
(
i
+1)
(
t
−
T
)
+
m
i
+1
B
i
+1
˙
V
(
t
−
T
)+
K
l
(
i
+1)
B
i
+1
e
V
(
t
−
T
)
ˆ
H
l
(
i
+1)
(
t
)
≃
(
τ
(
i
+1)
l
r
(
t
−
T
)
z
−
m
i
+1
B
i
+1
˙
V
(
t
−
T
)
−
K
l
i
+1
B
i
+1
e
V
(
t
−
T
)
−
B
i
+1
U
B
i
+2
(
t
−
T
)
ˆ
H
l
(
i
+2)
(
t
−
T
)+
m
i
+2
B
i
+2
˙
V
(
t
−
T
)
+
K
l
(
i
+2)
B
i
+2
e
V
(
t
−
T
)
.
.
.
ˆ
H
l
n
(
t
)
≃
τ
nl
r
(
t
−
T
)
z
−
m
n
B
n
˙
V
(
t
−
T
)
−
K
l
n
B
n
e
V
(
t
−
T
)
F
or
the
joint
substem,
the
estimates
of
the
uncertainties
parameter
v
ector
ˆ
H
j
i
(
t
)
is
gi
v
en
by
(32)
and
(33):
ˆ
H
j
i
(
t
)
∼
=
ˆ
H
j
i
(
t
−
T
)
(32)
then
ˆ
H
j
i
(
t
)
∼
=
τ
ij
r
(
t
−
T
)
−
K
j
i
e
q
i
(
t
−
T
)
(33)
where
T
is
the
estimation
time
delay
.
The
accurac
y
estimation
of
ˆ
H
l
i
(
t
)
and
ˆ
H
j
i
(
t
)
impro
v
es
for
a
small
T
.
In
practice,
the
smallest
estimation
time
delay
T
is
chosen
to
be
the
sampling
period
which
means
that
the
perfect
parameters
v
ector
are
identied
e
v
ery
sampling
period.
Finally
,
the
proposed
control
is
obtained
as
(34)-(36):
τ
i
(
t
)
=
τ
ij
r
(
t
)
+
τ
il
r
(
t
)
(34)
where
τ
ij
r
(
t
)
=
i
i
¨
q
ir
(
t
)
+
ˆ
H
j
i
(
t
)
+
K
j
i
(
˙
q
ir
(
t
)
−
˙
q
i
(
t
))
(35)
and
τ
il
r
(
t
)
=
z
T
[
m
i
B
i
˙
V
(
t
)
+
ˆ
H
l
i
(
t
)
+
K
l
i
B
i
e
V
(
t
)
−
B
i
U
B
i
+1
[
m
i
+1
B
i
+1
˙
V
(
t
)
+
ˆ
H
l
(
i
+1)
(
t
)
+
K
l
(
i
+1)
B
i
+1
e
V
(
t
)]]
(36)
The
closed-loop
control
system
based
on
the
proposed
TD
VDC
technique
is
presented
in
Figure
2.
Int
J
Rob
&
Autom,
V
ol.
10,
No.
3,
September
2021
:
192
–
206
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
199
Figure
2.
Block
diagram
of
the
proposed
controller
4.
VIR
TU
AL
ST
ABILITY
AN
AL
YSIS
According
to
the
virtual
w
ork
approach,
the
global
stability
of
the
system’
s
VDC
is
pro
v
en
through
the
virtual
stability
of
each
s
ubsystem.
Indeed,
using
the
denition
2.17
and
theorem
2.1
in
[21],
the
global
system
is
stable
in
the
sense
of
L
yapuno
v
,
if
each
subsystem
is
pro
v
ed
to
be
virtually
stable.
It
will
be
pro
v
en
that
all
the
decomposed
subsystems
of
the
studied
system
with
their
respecti
v
e
control
equations
are
virtually
stable,
leading
to
the
stability
of
the
entire
system.
Generally
,
the
stability
analysis,
in
the
sense
of
the
L
yapuno
v
approach,
ref
ers
to
dene
a
positi
v
e
candidate
function
and
then
to
sho
w
that
its
v
ariation
is
a
decreasing
function.
Considering
the
L
yapuno
v
candidate
function
for
the
entire
robot
as
summation
of
tw
o
functions
for
the
link
(
V
l
i
)
and
joint
V
j
i
)subsystems
as
(37):
i
=
{
0
,
.
.
.
,
n
}
V
=
X
i
V
l
i
+
X
i
V
j
i
(37)
4.1.
V
irtual
stability
of
the
ith
link
Let
consider
the
L
yapuno
v
candidate
function
for
the
i
th
link
as
(38):
V
l
i
=
1
2
B
i
e
V
T
M
B
i
B
i
e
V
+
1
2
(
H
l
i
−
ˆ
H
l
i
)
2
(38)
Then
from
[21]
and
the
dynamic
equation
of
the
i
th
link
gi
v
en
in
(13),
the
rst
deri
v
ati
v
e
along
time
of
V
l
i
can
be
gi
v
en
by
(39):
˙
V
l
i
=
−
B
i
e
V
T
K
l
i
B
i
e
V
+
B
i
e
V
T
(
B
i
F
r
−
B
i
F
)
+
(
H
l
i
−
ˆ
H
l
i
)(
B
i
e
V
−
˙
ˆ
H
l
i
)
(39)
where
B
i
e
V
T
C
B
i
B
i
e
V
=
0
,
since
C
B
i
dened
as
sk
e
w-symmetric.
According
to
the
TDE
use
the
˙
V
l
i
becomes
(40):
˙
V
l
i
=
−
B
i
e
V
T
K
l
i
B
i
e
V
+
B
i
e
V
T
(
B
i
F
∗
r
−
B
i
F
r
)
+
∆
H
l
i
(
Y
T
i
B
i
e
V
−
1
2
T
∆
H
l
i
)
(40)
where
∆
H
l
i
(
t
)
=
H
l
i
(
t
)
−
H
l
i
(
t
−
T
)
,
is
the
term
due
to
the
TDE
error
.
Otherwise,
as
H
l
i
(
t
)
is
a
Lipschitz
function,
then
(41):
|
∆
H
l
i
|
≤
δ
l
i
T
(41)
δ
l
i
is
the
Lipschitz
constant.
T
o
perform
the
VDC
for
each
subsystem,
the
virtual
po
wer
o
ws
are
introduced
to
characterize
the
dynamic
interaction
among
the
subsystems
at
its
cutting
points.
Indeed,
the
virtual
po
wer
o
w
is
dened
as
the
inner
Ne
w
time
delay
estimation-based
virtual
decomposition
contr
ol
for
n-DoF
r
obot
...
(Hac
hmia
F
aqihi)
Evaluation Warning : The document was created with Spire.PDF for Python.
200
❒
ISSN:
2089-4856
product
of
the
linear/angular
v
elocity
error
v
ector
and
the
force/moment
error
v
ector
,
with
respect
to
the
frame
{
A
}
,
as
(42):
p
A
=
A
e
T
V
(
A
F
∗
r
−
A
F
r
)
(42)
Therefore
from
[21],
(40)-(42),
we
obtain:
˙
V
l
i
≤
−
B
i
e
V
T
K
l
i
B
i
e
V
+
p
B
li
−
p
T
li
−
1
2
δ
l
i
(43)
where
p
B
li
and
p
T
li
represent
the
virtual
po
wer
o
ws
at
the
tw
o
cutting
points
of
each
link.
As
dened
in
[21],
according
to
an
open
chaine
structure,
for
p
B
l
1
=
0
and
p
T
ln
=
0
the
total
virtual
po
wer
o
ws
is
gi
v
en
by
(44):
X
i
n
(
p
B
li
−
p
T
li
)
=
0
(44)
Therefore
the
(43)
becomes
(45):
X
i
˙
V
l
i
≤
X
i
(
−
B
i
e
V
T
K
l
i
B
i
e
V
−
1
2
δ
l
i
)
(45)
4.2.
V
irtual
stability
of
the
ith
joint
The
positi
v
e
L
yapuno
v
candidate
function
related
to
the
joint
dynamics
can
be
chosen
according
to
the
joint
dynamic
and
its
control
la
w
,
as
(46):
V
j
i
=
1
2
I
mi
e
q
2
+
1
2
(
H
j
i
−
ˆ
H
j
i
)
2
(46)
Then,
its
time
deri
v
ati
v
e
is
(47):
˙
V
j
i
=
−
e
q
i
I
mi
˙
e
q
i
−
(
H
j
i
−
ˆ
H
j
i
)
˙
ˆ
H
j
i
(47)
with
the
TDE
use,
and
the
dynamic
equation
of
the
i
th
joint
gi
v
en
in
(18),
the
˙
V
j
i
becomes
(48):
˙
V
j
i
=
−
K
j
i
e
2
q
i
+
e
q
i
(
τ
r
ir
−
τ
ir
)
−
1
2
T
∆
H
2
j
i
(48)
According
to
[21],
(41),
(48),
and
VPF
denition,
we
obtain
(49):
˙
V
j
i
≤
−
K
j
i
e
2
q
i
−
1
2
δ
j
i
+
p
B
j
i
−
p
T
j
i
(49)
As
described
in
the
abo
v
e
section,
using
VFP
the
(49)
becomes
(50)
X
˙
V
j
i
≤
X
(
−
K
j
i
e
2
q
i
−
1
2
δ
j
i
)
(50)
4.3.
Stability
of
the
global
system
The
deri
v
ati
v
e
of
the
global
L
yapuno
v
candidate
function
(37),
is
gi
v
en
as
(51):
˙
V
=
X
˙
V
l
i
+
X
˙
V
j
i
(51)
The
˙
V
function
is
pro
v
ed
to
be
al
w
ays
decreasing
based
on
the
virtual
po
wer
as
the
inner
product
of
the
linear
angular
v
elocity
v
ector
error
and
the
force
moment
v
ector
error
presented
in
[21],
and
the
choice
of
the
parameter
function
adaptation,
where
(52):
˙
V
≤
−
X
i,j
(
B
i
e
V
T
K
l
i
B
i
e
V
+
1
2
δ
l
i
+
K
j
i
e
2
q
i
+
1
2
δ
j
)
(52)
where
δ
j
>
0
and
δ
l
>
0
are
the
Lipschitz
constants.
Since
˙
V
<
0
where
all
g
ains
are
positi
v
e,
the
system
is
asymptotically
stable
in
the
sense
of
L
yapuno
v
[21].
Int
J
Rob
&
Autom,
V
ol.
10,
No.
3,
September
2021
:
192
–
206
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
201
5.
CASE
STUD
Y
5.1.
Simulation
description
T
o
illustrat
e
the
ef
fecti
v
eness
of
the
proposed
control
strate
gy
,
in
this
section
a
case
study
is
per
formed
for
tracking
trajectory
of
7-DoF
robotic
manipulator
using
the
proposed
TD
VDC.
The
simulation
routine
is
conducted
follo
wing
the
control
architecture
gi
v
en
in
Figure
2,
wich
includes
the
desired
trajectory
gi
v
en
in
joint
space.
The
desired
joint
v
elocity
and
the
desired
joint
acceleration
are
obtained
from
the
deri
v
ation
of
the
desired
joint
position.The
equation
of
motion
for
each
link
and
joint
(i)
(
i
=
1
,
.
.
.
,
7
)
subsystems
is
deri
v
ed
with
respect
to
a
local
frame
{
B
i
}
as
sho
wn
in
Figure
1.
The
mass,
Coriolis,
and
the
gra
vity
termes
of
the
link
(i)
can
be
described
by
(53)-(55):
M
B
i
=
m
i
0
0
0
m
i
d
i
m
i
d
i
0
m
i
d
i
I
i
+
m
i
d
2
i
,
i
=
1
,
.
.
.
,
7
(53)
C
B
i
=
0
−
m
i
−
m
i
d
i
m
i
0
0
m
i
d
i
0
0
˙
q
i
,
i
=
1
,
.
.
.
,
7
(54)
G
B
i
=
m
i
sin
(
q
i
)
g
m
i
cos
(
q
i
)
g
m
i
d
i
cos
(
q
i
)
g
,
i
=
1
,
.
.
.
,
7
(55)
where
the
ph
ysical
parameters
of
the
using
robot
system
are
represented
in
T
able
1.
The
numerical
simulations
are
conducted
for
the
proposed
TD
VDC
controller
and
compared
to
the
con
v
entional
re
gressor
-based
VDC
in
order
to
pro
v
e
the
ef
fecti
v
eness
of
the
prposed
approach.
During
the
trajectory
tracking,
a
disturbances
w
as
added
to
the
torque
input
representing
5%
of
maximum
v
alue
of
the
torque
after
t
=
10s.
In
addition
an
uncertainty
function
U
(
t
)
w
as
injected
to
the
robot
dynamic
model
to
v
alidate
the
ef
fecti
v
eness
of
the
proposed
control
strate
gy
in
(56).
U
(
q
i
,
t
)
=
q
i
∗
sin
(
t
)
+
0
.
5
∗
sin
(500
∗
pi
∗
t
);
(56)
T
able
1.
Ph
ysical
parameters
l
1
=
0
.
3
m
;
l
2
=
0
.
5
m
;
l
3
=
l
4
=
0
.
21
m
;
l
5
=
0
.
25
m
;
l
6
=
0
.
5
m
m
1
=
0
.
122
K
g
;
m
2
=
0
.
66
K
g
;
m
3
=
0
.
08
K
g
;
m
4
=
0
.
175
K
g
;
m
5
=
0
.
251
K
g
;
m
6
=
0
.
023
K
g
k
c
1
=
k
c
2
=
k
c
3
=
k
c
4
=
k
c
5
=
k
c
6
=
0
.
5
N
.
m
I
1
=
I
2
=
I
3
=
I
4
=
I
5
=
I
6
=
0
.
0234
K
g
.m
2
F
or
the
proposed
TD
VDC
approach,
the
tar
get
robot
is
controlled
follo
wing
the
closed-loop
gi
v
en
in
Figure
2.
It
concerns
the
use
of
TDE
for
the
estimation
t
erms
dening
the
unkno
wn
and
uncertainties
parameter
v
ectors
of
the
robot.
The
required
linear/angular
v
elocity
and
its
time
deri
v
ati
v
e
is
computed
using
λ
constant.
The
constant
coef
cients
m
i
and
i
i
are
chosen
according
to
the
assumption
A3.
A
suitable
choose
of
these
constants
inuence
the
stabi
lity
and
at
tenuation
of
measurement
noise.
The
se
constants
are
conducted
by
the
trial
and
error
method.
The
time
delay
T
is
x
ed
as
sampling
time.
The
g
ain
parameters
of
the
feedback
controller
K
j
and
K
l
for
joint
and
link
subsystems
respecti
v
elyare
are
x
ed
ensuring
the
stability
condition.
These
parameters
v
alues
must
be
adjusted
in
order
to
obtain
the
optim
um
performance.
F
or
the
con
v
entional
re
gressor
-based
VDC
approach,
the
parameters
estimation
is
based
on
projection
function
presented
in
(22)
which
requires
the
deri
v
ation
of
the
re
gressor
matrix
in
e
v
ery
sampling
time,
as
discussed
pre
viously
.
T
o
accomplish
the
simulation
routine,
in
addi
tion
to
the
g
ains
feedback
controller
K
j
,
K
l
and
λ
,
the
parameters
ρ
,
a
,
b
are
used
for
the
projection
function.
5.2.
Simulation
r
esults
The
obtaine
d
simulation
results
of
the
tracking
trajectory
and
the
traking
errors
for
the
proposed
TD
VDC
and
the
con
v
entional
Re
gressor
-based
VDC
strate
gies
are
sho
wn
in
Figure
3,
Figure
4,
and
Figure
5
respecti
v
ely
Ne
w
time
delay
estimation-based
virtual
decomposition
contr
ol
for
n-DoF
r
obot
...
(Hac
hmia
F
aqihi)
Evaluation Warning : The document was created with Spire.PDF for Python.