Indonesian
J
our
nal
of
Electrical
Engineering
and
Computer
Science
V
ol.
37,
No.
3,
March
2025,
pp.
1515
∼
1531
ISSN:
2502-4752,
DOI:
10.11591/ijeecs.v37.i3.pp1515-1531
❒
1515
Neur
o-adapti
v
e
hierar
chical
sliding
mode
contr
ol
f
or
a
3-wheeled
mobile
r
obot
with
disturbance
r
ejection
J
ose
Carlos
V
izarr
eta
Mamani,
Bladimir
Leonel
V
era
Huamani,
Raul
Sulla
T
orr
es
Professional
School
of
Electronic
Engineering,
Uni
v
ersidad
Nacional
de
San
Agust
´
ın
de
Arequipa,
Arequipa,
Peru
Article
Inf
o
Article
history:
Recei
v
ed
May
19,
2024
Re
vised
Oct
2,
2024
Accepted
Oct
7,
2024
K
eyw
ords:
Adapti
v
e
control
Hierarchical
control
Mobile
robot
Sliding
mode
control
T
rajectory
tracking
ABSTRA
CT
High
tracking
accurac
y
and
f
ast
con
v
er
gence
are
essential
features
in
the
control
of
wheeled
mobile
robots
for
autonomous
na
vig
ation
applications.
Ho
we
v
er
,
the
design
of
a
control
system
for
these
robots
f
aces
signicant
challenges
due
to
the
inherent
comple
xity
of
their
nonlinear
dynamics
and
their
non-holonomic
underactuated
na
ture.
This
article
introduces
a
no
v
el
control
frame
w
ork
for
tra-
jectory
tracking
of
a
three-wheeled
mobil
e
robot
(3WMR),
considering
e
xternal
and
internal
disturbances.
T
o
address
the
uncertain
nonlinear
and
underactuated
non-holonomic
dynamics
of
the
3WMR,
an
adapti
v
e
hierarchical
f
ast
terminal
sliding
mode
control
(AHFTSMC)
strate
gy
is
proposed.
In
this
approach,
an
adapti
v
e
neural
netw
ork
scheme
adjusts
the
sliding
surf
ace
coef
cients
in
real
time
to
minimize
tracking
errors
and
mitig
ate
the
chattering
phenomenon.
In
addition,
a
nite
time
disturbance
observ
er
(FTDO)
is
designed
to
accurately
estimate
and
compensate
for
unkno
wn
lumped
disturbances,
which
helps
to
im-
pro
v
e
the
disturbance
rejection
capability
.
The
stability
of
the
closed-loop
sys-
tem
is
demonstrated
using
L
yapuno
v
theory
.
The
proposed
control
approach
is
v
alidated
by
numerical
simulations,
and
a
comparati
v
e
analysis
of
recent
hier
-
archical
sliding
mode
controllers
(SMC)
is
performed.
The
results
demonstrate
that
the
proposed
approach
achi
e
v
es
superior
performance
in
terms
of
f
ast
con-
v
er
gence
and
tracking
accurac
y
,
which
are
crucial
features
for
the
autonomous
na
vig
ation
of
mobile
robots.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Jose
Carlos
V
izarreta
Mamani
Professional
School
of
Electronic
Engineering,
Uni
v
ersidad
Nacional
de
San
Agust
´
ın
de
Arequipa
Arequipa
04017,
Per
´
u
Email:
jvizarretam@unsa.edu.pe
1.
INTR
ODUCTION
In
recent
years,
wheeled
mobile
robots
ha
v
e
emer
ged
as
a
signicant
eld
of
research,
dri
v
en
by
t
heir
agility
and
ener
gy
ef
cienc
y
,
which
mak
e
them
suitable
for
v
arious
applications
such
as
logistics,
security
,
and
agriculture
[1],
[2].
Despite
these
adv
antages,
controlling
wheeled
mobile
robots
presents
signicant
challenges
due
to
their
nonlinear
and
comple
x
dynamics,
as
well
as
their
underactuated
and
non-holonomic
nature,
which
means
the
y
possess
more
de
grees
of
freedom
than
a
v
ailable
control
inputs
[3].
Designing
an
autonomous
control
system
for
trajectory
tracking
in
these
robots
is
particularly
challenging
and
requires
adv
anced
control
techniques
to
ensure
ef
cient
performance.
V
arious
control
approaches
ha
v
e
been
in
v
estig
ated
in
the
literature
to
tackle
path
tracking
in
wheel
ed
mobile
robots,
including
linear
control
techniques
lik
e
proportional
inte
gral
deri
v
ati
v
e
(PID)
[4],
[5]
and
linear
quadratic
re
gulator
(LQR)
[6],
[7].
F
or
instance,
[4]
introduced
a
rob
ust
PID
control
structure
enhanced
by
J
ournal
homepage:
http://ijeecs.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
1516
❒
ISSN:
2502-4752
genetic
algorithms
for
better
performance,
while
[5]
proposed
a
double-loop
fractional-order
PID
control
to
impro
v
e
rob
ustness
and
accurac
y
.
Similarly
,
LQR
control
w
as
applied
in
[6]
using
a
cascade
control
scheme
for
a
tw
o-wheeled
robot,
and
[7]
combined
PI
and
LQR
controls
to
address
trajectory
tracking
amidst
obstacles.
Ho
we
v
er
,
these
linear
control
techniques
present
limitations
in
their
ef
fecti
v
eness
since
the
y
operate
stably
only
near
the
equilibrium
point,
where
the
dynamic
model
of
the
system
is
linearized
[8].
Moreo
v
er
,
the
y
lack
rob
ustness
in
en
vironments
with
lar
ge
e
xternal
and
internal
disturbances.
Gi
v
en
these
limitations,
nonlinear
control
techniques
ha
v
e
g
ained
rele
v
ance.
Sliding
mode
control
(SMC)
is
a
popular
nonlinear
technique
in
robotics
due
to
its
rob
ustness
ag
ainst
uncertainties
and
ability
to
manage
imprecise
models
[9],
[10].
F
or
e
xample,
second-order
SMC
w
as
applied
in
[11],
[12]
for
trajectory
tracking
of
four
-wheeled
robot
s,
incorporating
adapti
v
e
la
ws
in
[11]
and
e
xtended
state
observ
er
in
[12]
to
han-
dle
e
xternal
disturbances.
Ho
we
v
er
,
these
approaches
emplo
y
linear
sliding
surf
aces
that
can
cause
chattering
in
the
control
signal,
potentially
damaging
electric
motors
in
practical
applications
[13].
Recently
,
control
approaches
with
adv
anced
sliding
surf
aces
ha
v
e
been
proposed
to
o
v
ercome
these
limitations
and
impro
v
e
control
performance
in
mobile
robot
trajectory
tracking.
F
or
e
xample,
Labbadi
et
al.
[14],
a
fractional-order
SMC
approach
w
as
designed
for
a
dif
ferential
mobile
robotic
system,
while
[15]
proposed
an
adapti
v
e
non-singular
f
ast
terminal
SMC
scheme.
Ho
we
v
er
,
these
w
orks
[14],
[15]
did
not
tak
e
into
account
the
full
dynamic
model
of
the
mobile
robot,
limiting
themselv
es
only
to
kinematic
control,
and
although
the
y
addressed
e
xternal
disturbances,
the
y
did
not
e
v
aluate
the
rob
ustness
ag
ainst
internal
disturbances.
On
the
other
hand,
Gosw
ami
and
P
adh
y
[16],
a
dynamic
controller
based
on
a
PID-type
sliding
surf
ace
w
as
proposed
for
path
tracking
of
a
dif
ferentially
dri
v
en
mobile
robot,
taking
into
account
measure-
ment
noise,
frictional
disturbances,
and
modeling
inaccuracies.
Xie
et
al.
[17],
a
non-singular
f
ast
terminal
SMC
dynamic
controller
wit
h
adapti
v
e
la
ws
to
handle
parametric
uncertainties
and
e
xternal
disturbances
w
as
presented.
Finally
,
in
the
most
recent
w
ork
[18],
an
inno
v
ati
v
e
PI-type
adapti
v
e
SMC
grounded
in
the
dynamic
model
w
as
presented
for
the
path
tracking
of
a
mobile
robot.
Despite
these
adv
ances,
it
is
essential
to
note
that
these
approaches
[14]-[18]
f
ail
to
achie
v
e
satisf
actory
tracking
accurac
y
due
to
the
underactuation
problem
inherent
to
non-holonomic
wheeled
mobile
robots,
which
limits
their
ability
to
follo
w
trajectories
accurately
.
Gi
v
en
the
challenges
associated
with
underactuated
nonlinear
systems,
hierarchical
sliding
mode
con-
trol
(HSMC)
has
been
in
v
estig
ated
as
a
potential
solution
[19].
HSMC
preserv
es
the
benets
of
SMC
while
allo
wing
simultaneous
control
of
multiple
outputs
by
decomposing
the
system
into
subsystems
[20].
This
ap-
proach
has
been
broadly
applied
in
the
trajectory
tracking
of
v
arious
underactuated
systems,
including
in
v
erted
pendulums
[21],
aerial
robots
[22],
e
xible
roboti
c
manipulators
[23],
and
wheeled
mobile
robots
[24]-[27].
In
the
eld
of
mobile
robots,
in
[24],
an
HSMC
with
a
PID-type
sliding
surf
a
ce
w
as
proposed,
sho
wing
promis-
ing
resul
ts
in
t
racking
accurac
y
.
Ho
we
v
er
,
the
rob
ustness
of
the
control
ag
ainst
e
xternal
disturbances
w
as
not
assessed.
W
u
and
Kark
oub
[25],
a
hierarchical
fuzzy
sliding
mode
adapti
v
e
control
(HFSMA
C)
with
a
PI-
type
sliding
surf
ace
w
as
presented,
using
fourier
series-based
function
approximations
to
address
uncertainties.
More
recent
w
ork
[26],
[27]
combined
backstepping
and
sliding
mode
techniques
to
de
v
elop
a
backstepping
hierarchical
sliding
mode
control
(BHSMC)
.
Although
these
approaches
[24]-[27]
impro
v
e
trajectory
tracking
accurac
y
,
it
is
essential
to
note
that
these
w
orks
emplo
yed
con
v
entional
sliding
surf
aces,
which
lead
to
asymp-
totic
stabilit
y
and
hence
slo
w
con
v
er
gence
speed
in
tim
e
to
the
reference
trajectory
,
limiting
their
applicability
in
practical
en
vironments
where
f
ast
con
v
er
gence
in
nite
time
is
required.
The
k
e
y
ndings
from
pre
vious
studi
es
highlight
the
need
for
further
research
to
address
the
unre-
solv
ed
issues
i
n
trajectory
tracking
control
for
wheeled
mobile
robots.
Although
the
HSMC
architecture
has
pro
v
en
ef
fecti
v
e,
there
is
a
need
for
ne
w
control
strate
gies
with
nonlinear
sliding
surf
aces
that
can
achie
v
e
high
tracking
accurac
y
,
f
ast
response,
and
rob
ustness
simultaneously
.
In
this
conte
xt,
in
this
research,
we
propose
a
no
v
el
neuro-adapti
v
e
hierarchical
SMC
system
for
impro
v
ed
traject
ory
tracking
in
a
three-wheeled
mobile
robot
(3WMR).
The
main
contrib
utions
of
this
paper
are
as
follo
ws
:
rst,
unlik
e
pre
vious
w
orks
[24]-[27],
which
rel
y
on
con
v
entional
sliding
s
urf
aces,
this
research
introduces
an
adapti
v
e
hierarchical
f
ast
terminal
sliding
mode
control
(AHFTSMC)
strate
gy
.
Moreo
v
er
,
our
method
dynamically
adjusts
the
sliding
surf
ace
coef
cients
in
real
time
through
a
neural
netw
ork
to
reduce
tracking
errors
and
mitig
ate
chattering
ef
fects.
Second,
a
nite-time
dis
turbance
observ
er
(FTDO)
is
de
v
eloped
to
quickly
and
accurately
estimate
unkno
wn
lumped
disturbances.
Unlik
e
pre
vious
approaches
[24]-[27],
this
study
considers
e
xternal
and
internal
distur
-
bances
af
fecting
the
mobile
robot.
Third,
compared
to
e
xisti
n
g
control
methods
[25],
[27],
the
proposed
system
demonstrates
signicantly
superior
performance
in
trajectory
tracking,
e
videnced
by
f
aster
con
v
er
gence,
higher
accurac
y
,
and
enhanced
rob
ustness.
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
37,
No.
3,
March
2025:
1515–1531
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
❒
1517
The
remainder
of
the
document
is
or
g
anized
as
follo
ws:
section
2
details
the
research
methodology
,
which
encompasses
the
dynamic
modeling
of
the
3WMR,
the
de
v
elopment
of
the
control
system,
and
the
stability
e
v
aluation.
In
section
3
pro
vides
the
ndings
and
analysis
of
the
study
.
Lastly
,
section
4
of
fers
the
conclusions.
2.
RESEARCH
METHOD
This
section
presents
the
methodology
adopted
to
address
the
trajectory
tracking
control
problem
of
a
3WMR
in
the
pres
ence
of
e
xternal
and
internal
disturbances.
First,
the
mathematical
modeli
ng
of
the
3WMR
is
de
v
eloped,
which
establishes
the
basis
for
the
design
and
analysis
of
the
control
system.
Ne
xt,
the
design
of
the
proposed
control
system
is
introduced,
consisting
of
three
main
components:
a
FTDO,
a
nonlinear
control
strate
gy
,
and
adapti
v
e
la
ws
based
on
a
neural
netw
ork.
Finally
,
a
stability
analysis
of
the
closed-loop
system
is
conducted
to
ensure
that
the
tracking
errors
con
v
er
ge
to
zero
within
a
nite
time.
2.1.
Mathematical
model
A
3WMR
is
a
non-holonomic
robotic
system
consisting
of
tw
o
dri
ving
wheels
and
a
rotating
wheel,
allo
wing
it
to
mo
v
e
horizontally
.
This
mobile
robot
operates
with
dif
ferential
dri
v
e,
meaning
the
tw
o
rear
wheels
are
po
wered,
allo
wing
the
dri
ving
and
steering
functions
to
be
controlled
independently
.
Figure
1
sho
ws
a
top
vie
w
of
the
3WMR
in
a
horizontal
plane,
where
(
O
B
,
X
B
,
Y
B
)
represents
the
coordinate
frame
x
ed
on
the
mobile
robot,
and
(
O
G
,
X
G
,
Y
G
)
represents
the
global
coordinates.
The
position
v
ector
of
the
mobile
robot
is
denoted
as
q
=
[
x,
y
,
θ
]
T
where
x
and
y
represent
the
cartesian
position,
and
θ
represents
the
orientation.
2r
2L
Figure
1.
Non-holonomic
3WMR
Using
the
Euler
-Lagrange
formulation,
the
follo
wing
equation
describes
the
dynamics
and
kinemat
ics
of
the
non-holonomic
3WMR
[3]:
M
(
q
)
¨
q
+
C
(
q
,
˙
q
)
˙
q
=
E
(
q
)
T
−
A
T
(
q
)
λ
(1)
here
M
(
q
)
represents
a
positi
v
e
denite
inertia
matrix,
C
(
q
,
˙
q
)
denotes
the
Coriolis
ma
trix
along
with
the
centripetal
terms,
A
(
q
)
is
the
matrix
related
to
the
nonholonomic
constraints,
T
represents
the
v
ector
of
input,
and
E
(
q
)
is
the
transformation
matrix
of
the
control
inputs.
Their
e
xpressions
are
denoted
by
[27].
M
(
q
)
=
m
0
0
0
m
0
0
0
J
,
C
(
q
,
˙
q
)
=
0
0
0
,
T
=
T
1
T
2
E
(
q
)
=
1
r
cos
θ
cos
θ
sin
θ
sin
θ
L
−
L
,
A
T
(
q
)
=
1
r
−
sin
θ
cos
θ
0
(2)
The
characteristics
of
the
3WMR
are
outlined
as
follo
ws:
m
and
J
indicate
the
mass
and
moment
of
inertia
of
the
mobil
robot,
respecti
v
ely
.
T
1
and
T
2
are
the
torques
e
x
erted
by
the
right
and
left
motors
associated
with
the
dri
ving
wheels.
In
addition,
r
is
the
radius
of
the
dri
ving
wheels,
L
signies
half
the
distance
between
the
rear
wheels,
and
λ
represents
the
Lagrange
multiplier
.
T
aking
into
account
non-holonomic
constraints
and
conditions
for
anti-slip
rolling,
we
deri
v
e
the
(3).
Neur
o-adaptive
hier
ar
c
hical
sliding
mode
contr
ol
...
(J
ose
Carlos
V
izarr
eta
Mamani)
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1518
❒
ISSN:
2502-4752
˙
x
sin
θ
−
˙
y
cos
θ
=
0
(3)
The
determination
of
the
Lagrangian
bond
strength
f
actor
is
performed
as
(4)
[26].
λ
=
−
m
θ
(
˙
x
cos
θ
−
˙
y
s
in
θ
)
(4)
Therefore,
the
dynamic
model
of
the
3WMR
is
described
as
(5)
[27]:
¨
x
=
λ
m
sin
θ
+
1
r
m
cos
θ
u
1
+
d
x
(
t
)
,
¨
y
=
−
λ
m
cos
θ
+
1
r
m
sin
θ
u
1
+
d
y
(
t
)
,
¨
θ
=
L
r
J
u
2
+
d
θ
(
t
)
,
(5)
where
u
1
=
T
1
−
T
2
,
u
2
=
T
1
+
T
2
represent
the
control
inputs,
wit
h
T
1
=
0
.
5(
u
1
+
u
2
)
and
T
2
=
0
.
5(
u
2
−
u
1
)
.
The
terms
d
x
(
t
)
,
d
y
(
t
)
,
d
θ
(
t
)
denote
the
unkno
wn
lumped
disturbances
af
fecting
the
mobile
robot.
2.2.
Pr
oposed
contr
ol
system
design
This
section
outlines
the
de
v
el
opment
of
the
proposed
control
system.
Ini
tially
,
the
design
of
the
FTDO,
intended
to
estimate
and
mitig
ate
the
combined
disturbances,
is
e
xplained.
F
ollo
wing
this,
the
control
approach
is
formulated.
Based
on
the
dynamic
model
in
(5),
the
control
frame
w
ork
is
separated
into
underac-
tuated
and
fully
actuated
components.
The
underactuated
system
is
responsible
for
controlling
the
Cartesian
position
of
the
mobile
robot,
i.e.,
the
v
ariables
x
and
y
,
with
a
single
control
action.
T
o
achie
v
e
this,
an
underac-
tuated
control
architecture
called
the
HFTSMC
st
rate
gy
is
emplo
yed.
Con
v
ersely
,
the
fully
actuated
system
is
responsible
for
controlling
the
orientation
of
the
mobile
robot,
for
which
the
FTSMC
strate
gy
is
emplo
yed.
Finally
,
the
adapti
v
e
la
ws
are
designed
to
adjust
the
sliding
surf
ace
coef
cients
in
real-time
dynamically
.
Figure
2
illustrates
the
design
of
the
proposed
control
system.
POSITION
CONTROL
ORIENT
A
TION
CONTROL
Finite
T
ime
Disturbance
Observer
Neural
Network
External
Disturbances
Neural
Network
PROPOSED CONTROL
SYSTEM
Figure
2.
Proposed
control
system
structure
2.2.1.
FTDO
design
In
(5)
can
be
e
xpressed
in
the
state
space
representation
as:
˙
Π
T
=
Π
T
+1
˙
Π
T
+1
=
f
(Π
T
)
+
g
(Π
T
)
u
+
d
(
t
)
(6)
here,
Π
T
=
[
x,
˙
x,
y
,
˙
y
,
θ
,
˙
θ
]
T
denotes
the
state
v
ector
,
u
=
[
u
1
,
u
2
]
T
represents
the
control
actions,
f
(Π
T
)
and
g
(Π
T
)
are
tw
o
continuous
functions
that
depend
on
Π
T
,
and
d
(
t
)
indicates
the
aggre
g
ated
disturbances.
-
Assumption
1,
the
deri
v
ati
v
e
of
d
(
t
)
is
bounded
and
satises
lim
t
→∞
˙
d
(
t
)
=
0
.
-
Theorem
1
considering
the
dynamic
system
in
(6),
the
FTDO
design
is
dened
as
follo
ws
[8],
[28]:
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
37,
No.
3,
March
2025:
1515–1531
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
❒
1519
˙
Υ
0
=
Γ
0
+
f
(Π
T
)
+
g
(Π
T
)
u
Γ
0
=
−
ξ
1
L
1
/
3
q
|
Υ
0
−
˙
q
|
2
/
3
sign
(Υ
0
−
˙
q
)
+
Υ
1
˙
Υ
1
=
Γ
1
Γ
1
=
−
ξ
2
L
1
/
2
q
|
Υ
1
−
Γ
0
|
1
/
2
sign
(Υ
1
−
Γ
0
)
+
Υ
2
˙
Υ
2
=
−
ξ
3
L
q
sign
(Υ
2
−
Γ
1
)
(7)
where
ξ
i
,
i
=
1
,
2
,
3
,
and
L
q
are
constant
v
alues
dened
as
positi
v
e,
which
represent
the
g
ains
of
the
observ
er
.
The
v
ariables
Υ
0
,
Υ
1
,
and
Υ
2
represent
the
estimates
of
˙
q
,
d
(
t
)
and
˙
d
(
t
)
,
respecti
v
ely
,
where
˙
q
=
[
˙
x,
˙
y
,
˙
θ
]
T
denotes
the
v
elocity
of
the
mobile
robot.
This
conguration
of
FTDO
can
accurately
identify
the
lumped
disturbances
d
(
t
)
in
nite
time
[28].
-
Proof
1
we
dene
observ
er
errors
as
follo
ws:
e
obs
1
=
Υ
0
−
˙
q
,
e
obs
2
=
Υ
1
−
d
(
t
)
,
e
obs
3
=
Υ
2
−
˙
d
(
t
)
(8)
by
performing
the
time
deri
v
ati
v
e
of
(8)
and
substituting
(7),
the
follo
wing
e
xpression
is
obtained:
˙
e
obs
1
=
−
ξ
1
L
1
/
3
q
|
e
obs
1
|
2
/
3
sign
(
e
obs
1
)
+
e
obs
2
˙
e
obs
2
=
−
ξ
2
L
1
/
2
q
|
e
obs
2
−
˙
e
obs
1
|
1
/
2
sign
(
e
obs
2
−
˙
e
obs
1
)
+
e
obs
3
˙
e
obs
3
=
−
ξ
3
L
q
sign
(
e
obs
3
−
˙
e
obs
2
)
−
¨
d
(
t
)
(9)
the
rest
of
the
proof
is
similar
to
Theorem
1
of
[8],
so
it
is
omitted
for
space
reasons.
Ne
v
ertheless,
according
to
that
theorem,
the
observ
ation
errors
e
obs
1
,
e
obs
2
,
and
e
obs
3
con
v
er
ge
to
the
origin
in
nite
time
by
appropriately
selecting
the
parameters
ξ
1
,
ξ
2
,
ξ
3
,
and
L
q
,
i.e.
Υ
0
≡
b
˙
q
,
Υ
1
≡
b
d
(
t
)
,
Υ
2
≡
b
˙
d
(
t
)
.
2.2.2.
HFTSMC
strategy
design
Considering
the
dynamic
model
of
the
non-holonomic
3WMR
gi
v
en
in
(5),
the
follo
wi
ng
dynamic
model
for
the
position
control
system
is
introduced.
¨
x
=
λ
m
sin
θ
+
1
r
m
cos
θ
u
1
+
d
x
(
t
)
,
¨
y
=
−
λ
m
cos
θ
+
1
r
m
sin
θ
u
1
+
d
y
(
t
)
,
(10)
Based
on
the
HSMC
theory
[20],
the
underactuated
dynamic
system
gi
v
en
in
(10)
is
decomposed
into
tw
o
subsystems
for
each
state.
T
o
achie
v
e
this,
we
dene
the
follo
wing
state
v
ariables:
X
=
[
χ
1
,
χ
2
,
χ
3
,
χ
4
]
=
[
x,
˙
x,
y
,
˙
y
]
T
.
The
state-space
representation
of
(10)
can
then
be
e
xpressed
as
(11):
˙
χ
1
=
χ
2
,
˙
χ
2
=
f
1
(
X
)
+
g
1
(
X
)
u
1
+
d
x
(
t
)
˙
χ
3
=
χ
4
,
˙
χ
4
=
f
2
(
X
)
+
g
2
(
X
)
u
1
+
d
y
(
t
)
(11)
where,
f
1
(
X
)
=
λ
m
sin
θ
,
g
1
(
X
)
=
1
r
m
cos
θ
f
2
(
X
)
=
−
λ
m
cos
θ
,
g
2
(
X
)
=
1
r
m
sin
θ
(12)
Considering
(11),
we
dene
the
position
tracking
errors
as:
e
x
1
=
χ
1
−
x
d
,
e
x
2
=
χ
2
−
˙
x
d
e
y
1
=
χ
3
−
y
d
e
y
2
=
χ
4
−
˙
y
d
(13)
deri
ving
in
time
(13),
we
obtain
the
dynamics
of
the
errors
as
(14).
˙
e
x
1
=
e
x
2
,
˙
e
x
2
=
f
1
(
X
)
+
g
1
(
X
)
u
1
+
d
x
(
t
)
−
¨
x
d
.
;
˙
e
y
1
=
e
y
2
,
˙
e
y
2
=
f
2
(
X
)
+
g
2
(
X
)
u
1
+
d
y
(
t
)
−
¨
y
d
.
(14)
Neur
o-adaptive
hier
ar
c
hical
sliding
mode
contr
ol
...
(J
ose
Carlos
V
izarr
eta
Mamani)
Evaluation Warning : The document was created with Spire.PDF for Python.
1520
❒
ISSN:
2502-4752
T
o
ensure
accurate
performance
and
f
ast
con
v
er
gence
in
the
underactuated
control
system,
and
unlik
e
the
approaches
in
[26],
[27],
in
this
research
we
propose
a
rob
ust
nonlinear
f
ast
terminal
sliding
surf
ace
for
the
rst
layer
of
the
hierarchical
structure,
dened
as:
σ
x
=
e
x
2
+
c
1
e
x
1
+
c
2
|
e
x
1
|
α
x
sign
(
e
x
1
)
+
c
3
|
e
x
1
|
β
x
sign
(
e
x
1
)
σ
y
=
e
y
2
+
c
4
e
y
1
+
c
5
|
e
y
1
|
α
y
sign
(
e
y
1
)
+
c
6
|
e
y
1
|
β
y
sign
(
e
y
1
)
(15)
where
α
i
≥
1
and
0
<
β
i
<
1
with
i
=
x,
y
.
The
parameters
c
i
>
0
,
i
=
1
,
6
,
are
the
positi
v
e
coef
cients
of
the
sliding
surf
ace,
which
will
be
adjusted
using
a
neural
netw
ork
in
the
follo
wing
section.
Deri
ving
in
time
the
sliding
surf
aces
(15)
and
substituting
the
e
xpressions
(14),
we
obtain:
˙
σ
x
=
˙
e
x
2
+
c
1
˙
e
x
1
+
c
2
α
x
|
e
x
1
|
α
x
−
1
+
c
3
β
x
|
e
x
1
|
β
x
−
1
=
f
1
(
X
)
+
g
1
(
X
)
u
1
+
d
x
(
t
)
−
¨
x
d
+
c
1
˙
e
x
1
+
c
2
α
x
|
e
x
1
|
α
x
−
1
+
c
3
β
x
|
e
x
1
|
β
x
−
1
˙
σ
y
=
˙
e
y
2
+
c
4
˙
e
y
1
+
c
5
α
y
|
e
y
1
|
α
y
−
1
+
c
6
β
y
|
e
y
1
|
β
y
−
1
=
f
2
(
X
)
+
g
2
(
X
)
u
1
+
d
y
(
t
)
−
¨
y
d
+
c
4
˙
e
y
1
+
c
5
α
y
|
e
y
1
|
α
y
−
1
+
c
6
β
y
|
e
y
1
|
β
y
−
1
(16)
By
setting
˙
σ
i
=
0
,
i
=
x,
y
,
we
obtain
the
equi
v
alent
control
la
ws
for
each
subsystem:
u
x
eq
=
1
g
1
(
X
)
¨
x
d
−
f
1
(
X
)
−
b
d
x
−
c
1
˙
e
x
1
−
c
2
α
x
|
e
x
1
|
α
x
−
1
−
c
3
β
x
|
e
x
1
|
β
x
−
1
−
k
x
sign
(
σ
x
)
u
y
eq
=
1
g
2
(
X
)
¨
y
d
−
f
2
(
X
)
−
b
d
y
−
c
4
˙
e
y
1
−
c
5
α
y
|
e
y
1
|
α
y
−
1
−
c
6
β
y
|
e
y
1
|
β
y
−
1
−
k
y
sign
(
σ
y
)
(17)
where
k
x
,
k
y
are
the
positi
v
e
coef
cients
and
will
be
adjusted
using
a
neural
netw
ork
in
the
follo
wing
section.
The
hierarchical
total
control
la
w
for
the
position
control
system
can
be
e
xpressed
as
follo
ws
[19]:
u
1
=
u
x
eq
+
u
y
eq
+
u
x,y
sw
(18)
where
u
x
eq
,
u
y
eq
are
the
equi
v
alent
control
la
ws
of
the
rst
layer
of
the
hierarchical
structure,
and
u
x,y
sw
is
the
switching
control
la
w
.
Since
we
ha
v
e
already
established
the
tw
o
sliding
surf
aces
of
the
rst
layer
according
to
(15),
we
proceed
to
dene
the
sliding
surf
ace
of
the
second
layer
of
the
hierarchical
structure
as
follo
ws:
s
1
=
η
1
σ
x
+
η
2
σ
y
(19)
where
η
1
and
η
2
are
tw
o
positi
v
e
denite
design
parameters.
Deri
ving
in
time
(19)
and
substituting
the
e
xpres-
sions
of
(16),
we
obtain:
˙
s
1
=
η
1
˙
σ
x
+
η
2
˙
σ
y
=
η
1
f
1
(
X
)
+
g
1
(
X
)
u
1
+
b
d
x
−
¨
x
d
+
c
1
˙
e
x
1
+
c
2
α
x
|
e
x
1
|
α
x
−
1
+
c
3
β
x
|
e
x
1
|
β
x
−
1
+
η
2
f
2
(
X
)
+
g
2
(
X
)
u
1
+
b
d
y
−
¨
y
d
+
c
4
˙
e
y
1
+
c
5
α
y
|
e
y
1
|
α
y
−
1
+
c
6
β
y
|
e
y
1
|
β
y
−
1
(20)
Substituting
the
e
xpressions
o
f
(18)
into
(20)
and
performing
some
mathematical
operations,
we
obtain
the
follo
wing
e
xpression:
˙
s
1
=
η
1
f
1
(
X
)
+
g
1
(
X
)(
u
x
eq
+
u
y
eq
+
u
x,y
sw
)
+
b
d
x
−
¨
x
d
+
c
1
˙
e
x
1
+
c
2
α
x
|
e
x
1
|
α
x
−
1
+
c
3
β
x
|
e
x
1
|
β
x
−
1
+
η
2
f
2
(
X
)
+
g
2
(
X
)(
u
x
eq
+
u
y
eq
+
u
x,y
sw
)
+
b
d
y
−
¨
y
d
+
c
4
˙
e
y
1
+
c
5
α
y
|
e
y
1
|
α
y
−
1
+
c
6
β
y
|
e
y
1
|
β
y
−
1
=
η
1
g
1
(
X
)(
u
y
eq
+
u
x,y
sw
)
+
η
2
g
2
(
X
)(
u
x
eq
+
u
x,y
sw
)
(21)
T
o
ensure
a
f
ast
reach
to
the
sliding
manifold,
we
use
the
e
xponential
approach
la
w
,
dened
as:
˙
s
1
=
−
k
s
1
1
s
1
−
k
s
1
2
sign
(
s
1
)
(22)
where
k
s
1
1
and
k
s
1
2
are
tw
o
positi
v
e
denite
paramet
ers.
Therefore,
combining
e
xpressions
(21)
and
(22),
we
design
the
switching
control
la
w
as:
u
x,y
sw
=
1
η
1
g
1
(
X
)
+
η
2
g
2
(
X
)
−
η
1
g
1
(
X
)
u
y
eq
−
η
2
g
2
(
X
)
u
x
eq
−
k
s
1
1
s
1
−
k
s
1
2
sign
(
s
1
)
(23)
Finally
,
by
substituting
e
xpressions
(17)
and
(23)
into
(18),
we
obtain
the
nal
position
control
la
w
of
the
mobile
robot:
u
1
=
1
η
1
g
1
(
X
)
+
η
2
g
2
(
X
)
η
1
g
1
(
X
)
u
x
eq
+
η
2
g
2
(
X
)
u
y
eq
−
k
s
1
1
s
1
−
k
s
1
2
sign
(
s
1
)
(24)
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
37,
No.
3,
March
2025:
1515–1531
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
❒
1521
2.2.3.
FTSMC
strategy
design
Considering
the
dynamic
model
of
the
non-holonomic
3WMR
gi
v
en
in
(5),
the
follo
wing
dynamic
model
for
the
orientation
control
system
is
introduced:
¨
θ
=
L
r
J
u
2
+
d
θ
(
t
)
(25)
The
follo
wing
state
v
ariables
X
=
[
χ
5
,
χ
6
]
=
[
θ
,
˙
θ
]
T
are
dened.
Then,
the
state-space
representation
of
(25)
is
gi
v
en
as:
(
˙
x
5
=
x
6
,
˙
x
6
=
g
(
θ
)
u
2
+
d
θ
(
t
)
;
g
(
θ
)
=
L
r
J
(26)
considering
(26),
we
dene
the
tracking
errors
as:
e
θ
1
=
χ
5
−
θ
d
,
e
θ
2
=
χ
6
−
˙
θ
d
(27)
deri
ving
in
time
(35),
we
obtain
the
dynamics
of
the
errors
as
(28).
(
˙
e
θ
1
=
e
θ
2
,
˙
e
θ
2
=
g
(
θ
)
u
2
+
d
θ
(
t
)
−
¨
θ
d
(28)
T
o
ensure
accurate
performance
and
f
ast
con
v
er
gence
in
the
fully
actuated
control
system,
we
propose
a
rob
ust
nonlinear
f
ast
terminal
sliding
surf
ace,
dened
as:
s
2
=
e
θ
2
+
c
7
e
θ
1
+
c
8
|
e
θ
1
|
α
θ
sgn
(
e
θ
1
)
+
c
9
|
e
θ
1
|
β
θ
sgn
(
e
θ
1
)
(29)
where
α
θ
≥
1
and
0
<
β
θ
<
1
.
The
paramet
ers
c
i
>
0
,
i
=
7
,
9
are
the
positi
v
e
coef
cients
of
the
sliding
surf
ace,
which
will
be
adjusted
using
a
neural
netw
ork
in
the
follo
wing
section.
Deri
ving
in
time
(29)
and
substituting
the
e
xpression
of
(28),
we
obtain:
˙
s
2
=
˙
e
θ
2
+
c
7
˙
e
θ
1
+
c
8
α
θ
|
˙
e
θ
1
|
α
θ
−
1
+
c
9
β
θ
|
˙
e
θ
1
|
β
θ
−
1
=
g
(
θ
)
u
2
+
d
θ
(
t
)
−
¨
θ
d
+
c
7
˙
e
θ
1
+
c
8
α
θ
|
˙
e
θ
1
|
α
θ
−
1
+
c
9
β
θ
|
˙
e
θ
1
|
β
θ
−
1
(30)
to
ensure
f
ast
reach
of
the
sliding
manifold,
we
use
the
e
xponential
approach
la
w
,
dened
as:
˙
s
2
=
−
k
s
2
1
s
2
−
k
θ
sign
(
s
2
)
(31)
here
k
s
2
1
and
k
θ
are
tw
o
positi
v
e-denite
parameters,
with
k
θ
being
adjusted
using
a
neural
netw
ork
in
the
follo
wing
section.
Therefore,
combining
e
xpressions
(30)
and
(31),
we
design
the
ori
entation
control
la
w
of
the
mobile
robot
as:
u
2
=
1
g
(
θ
)
¨
θ
d
−
b
d
θ
−
c
7
˙
e
θ
1
−
c
8
α
θ
|
˙
e
θ
1
|
α
θ
−
1
−
c
9
β
θ
|
˙
e
θ
1
|
β
θ
−
1
−
k
s
2
1
s
2
−
k
θ
sign
(
s
2
)
(32)
2.2.4.
Design
of
adapti
v
e
laws
In
this
sect
ion,
a
neural
netw
ork
is
emplo
yed
to
dynamically
adjust
the
sliding
surf
ace
coef
cients
in
real-time.
Specically
,
the
coef
cients
ˆ
c
i
>
0
for
i
=
1
,
9
and
ˆ
k
l
>
0
for
l
=
x,
y
,
θ
are
adjusted,
where
ˆ
c
i
and
ˆ
k
l
represent
the
estimates
of
c
i
and
k
l
respecti
v
ely
.
This
approach
aims
to
minimize
tracking
errors
and
mitig
ate
the
chattering
phenomenon.
The
neural
netw
ork
is
trained
using
an
online
specialized
learning
architecture
[29],
[30],
which
uses
the
backpropag
ation
algorithm
to
dynamically
adapt
the
coef
cients
of
the
sliding
surf
aces.
T
o
this
end,
we
initially
design
the
adapti
v
e
la
ws
for
the
coef
cients
ˆ
c
i
>
0
,
i
=
1
,
3
and
ˆ
k
x
through
the
follo
wing
steps:
-
First,
we
establish
the
performance
error
E
x
=
1
2
x
−
x
d
2
,
which
is
formulated
as
a
function
of
the
dif
ference
between
the
desired
state
and
the
current
state
of
the
mobile
robot.
Neur
o-adaptive
hier
ar
c
hical
sliding
mode
contr
ol
...
(J
ose
Carlos
V
izarr
eta
Mamani)
Evaluation Warning : The document was created with Spire.PDF for Python.
1522
❒
ISSN:
2502-4752
-
Using
the
steepest
descent
approach
[29],
we
deri
v
e
the
follo
wing
adapti
v
e
equations:
ˆ
k
x
=
k
x,
0
−
ω
Z
t
0
∂
E
x
∂
ˆ
k
x
,
ˆ
c
1
=
c
1
,
0
−
ω
Z
t
0
∂
E
x
∂
ˆ
c
1
,
ˆ
c
2
=
c
2
,
0
−
ω
Z
t
0
∂
E
x
∂
ˆ
c
2
,
ˆ
c
3
=
c
3
,
0
−
ω
Z
t
0
∂
E
x
∂
ˆ
c
3
,
(33)
where
ω
is
the
learning
rate,
k
x,
0
,
c
1
,
0
,
c
2
,
0
and
c
3
,
0
are
initial
v
alues
of
ˆ
k
x
,
ˆ
c
1
,
ˆ
c
2
and
ˆ
c
3
respecti
v
ely
.
-
By
applying
the
chain
rule
for
partial
deri
v
ati
v
es,
the
follo
wing
e
xpressions
can
be
deri
v
ed:
∂
E
x
∂
ˆ
k
x
=
∂
E
x
∂
x
∂
x
∂
u
x
eq
∂
u
x
eq
∂
ˆ
k
x
=
−
(
e
x
1
)
sign
∆
x
∆
u
x
eq
1
g
1
(
X
)
sign
(
σ
x
)
,
∂
E
x
∂
ˆ
c
1
=
∂
E
x
∂
x
∂
x
∂
u
x
eq
∂
u
x
eq
∂
σ
x
∂
σ
x
∂
ˆ
c
1
=
−
sign
∆
x
∆
∂
u
x
eq
1
g
1
(
X
)
4
e
−
2
σ
x
k
x
(1
+
e
−
2
σ
x
)
2
(
e
x
1
)
2
,
∂
E
x
∂
ˆ
c
2
=
∂
E
x
∂
x
∂
x
∂
u
x
eq
∂
u
x
eq
∂
σ
x
∂
σ
x
∂
ˆ
c
2
=
−
sign
∆
x
∆
u
x
eq
1
g
1
(
X
)
4
e
−
2
σ
x
k
x
(1
+
e
−
2
σ
x
)
2
|
e
x
1
|
α
x
+1
sign
(
e
x
1
)
,
∂
E
x
∂
ˆ
c
3
=
∂
E
x
∂
x
∂
x
∂
u
x
eq
∂
u
x
eq
∂
σ
x
∂
σ
x
∂
ˆ
c
3
=
−
sign
∆
x
∆
u
x
eq
1
g
1
(
X
)
4
e
−
2
σ
x
k
x
(1
+
e
−
2
σ
x
)
2
|
e
x
1
|
β
x
+1
sign
(
e
x
1
)
.
(34)
-
Substituting
(34)
into
(33),
the
adapti
v
e
coef
cients
of
the
sliding
surf
ace
σ
x
are
designed
as
follo
ws:
ˆ
k
x
=
k
x,
0
+
ω
Z
t
0
(
e
x
1
)
sign
∆
x
∆
u
x
eq
1
g
1
(
X
)
sign
(
σ
x
)
,
ˆ
c
1
=
c
1
,
0
+
ω
Z
t
0
sign
∆
x
∆
∂
u
x
eq
1
g
1
(
X
)
4
e
−
2
σ
x
ˆ
k
x
(1
+
e
−
2
σ
x
)
2
(
e
x
1
)
2
,
ˆ
c
2
=
c
2
,
0
+
ω
Z
t
0
sign
∆
x
∆
u
x
eq
1
g
1
(
X
)
4
e
−
2
σ
x
ˆ
k
x
(1
+
e
−
2
σ
x
)
2
|
e
x
1
|
α
x
+1
sign
(
e
x
1
)
,
ˆ
c
3
=
c
3
,
0
+
ω
Z
t
0
sign
∆
x
∆
u
x
eq
1
g
1
(
X
)
4
e
−
2
σ
x
ˆ
k
x
(1
+
e
−
2
σ
x
)
2
|
e
x
1
|
β
x
+1
sign
(
e
x
1
)
.
(35)
where
∆
refers
to
the
upw
ard
dif
ference
operator
,
dened
as
∇
ζ
k
=
ζ
k
−
ζ
k
−
1
[29].
By
applying
the
same
method
as
in
the
earlier
steps,
the
coef
cients
f
o
r
the
sliding
surf
aces
σ
y
and
s
θ
are
determined
as
follo
ws:
ˆ
k
y
=
k
y
,
0
+
ω
Z
t
0
Ξ
y
(
e
y
1
)
g
2
(
X
)
sign
(
σ
y
)
,
ˆ
c
4
=
c
4
,
0
+
ω
Z
t
0
Ξ
y
(
e
y
1
)
2
g
2
(
X
)
4
e
−
2
σ
y
ˆ
k
y
(1
+
e
−
2
σ
y
)
2
,
ˆ
c
5
=
c
5
,
0
+
ω
Z
t
0
Ξ
y
sign
α
y
+1
(
e
y
1
)
g
2
(
X
)
4
e
−
2
σ
y
ˆ
k
y
(1
+
e
−
2
σ
y
)
2
,
ˆ
c
6
=
c
6
,
0
+
ω
Z
t
0
Ξ
y
sign
β
y
+1
(
e
y
1
)
g
2
(
X
)
4
e
−
2
σ
y
ˆ
k
y
(1
+
e
−
2
σ
y
)
2
.
ˆ
k
θ
=
k
θ
,
0
+
ω
Z
t
0
Ξ
θ
(
e
θ
1
)
g
2
(
θ
)
sign
(
s
θ
)
,
ˆ
c
7
=
c
7
,
0
+
ω
Z
t
0
Ξ
θ
(
e
θ
1
)
2
g
2
(
θ
)
4
e
−
2
s
θ
ˆ
k
θ
(1
+
e
−
2
s
θ
)
2
,
ˆ
c
8
=
c
8
,
0
+
ω
Z
t
0
Ξ
θ
sign
α
θ
+1
(
e
θ
1
)
g
2
(
θ
)
4
e
−
2
s
θ
ˆ
k
θ
(1
+
e
−
2
s
θ
)
2
,
ˆ
c
9
=
c
9
,
0
+
ω
Z
t
0
Ξ
sign
β
θ
+1
(
e
θ
1
)
g
2
(
θ
)
4
e
−
2
s
θ
ˆ
k
θ
(1
+
e
−
2
s
θ
)
2
.
(36)
where
Ξ
y
=
sign
∆
y
∆
u
y
eq
,
Ξ
θ
=
sign
∆
θ
∆
u
2
.
The
adapti
v
e
la
ws
de
v
eloped
in
(35)
and
(36),
together
with
the
estimation
of
the
e
xternal
di
sturbances
in
(7)
and
the
control
la
ws
dened
in
(24)
and
(32),
form
a
rob
ust
and
ef
cient
control
system
for
trajectory
tracking
in
a
3WMR.
Algorithm
1,
which
details
the
necessary
steps
to
implement
the
control
system
proposed.
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
37,
No.
3,
March
2025:
1515–1531
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
❒
1523
Algorithm
1.
Algorithm
of
the
proposed
control
system
1:
Input:
2:
Dene
the
ph
ysical
characteristics
of
the
mobile
robot:
m,
J
,
r
,
L
3:
Dene
the
parameters
of
the
FTDO
observ
er:
ξ
1
,
ξ
2
,
ξ
3
,
L
q
4:
Dene
the
parameters
of
the
control
strate
gy:
α
x
,
α
y
,
α
θ
,
β
x
,
β
y
,
β
θ
,
η
1
,
η
2
,
k
s
1
1
,
k
s
2
1
5:
Dene
the
adapti
v
e
parameters
for
the
neural
netw
ork:
k
x,
0
,
k
y
,
0
,
k
θ
,
0
,
c
1
,
0
,
c
2
,
0
,
c
3
,
0
,
c
4
,
0
,
c
5
,
0
,
c
6
,
0
,
c
7
,
0
,
c
8
,
0
,
c
9
,
0
6:
while
Repeat
do
7:
Underactuated
Contr
ol
System
8:
Acquire
the
current
states:
x,
˙
x,
y
,
˙
y
9:
Compute
the
tracking
errors:
e
x
1
,
e
x
2
,
e
y
1
,
e
y
2
10:
Dene
the
neural
netw
ork
performance
error:
E
x
,
E
y
11:
Calculate
the
adapti
v
e
parameters
for
the
sliding
surf
ace:
ˆ
k
x
,
ˆ
k
y
,
ˆ
c
1
,
ˆ
c
2
,
ˆ
c
3
,
ˆ
c
4
,
ˆ
c
5
,
ˆ
c
6
12:
Dene
the
sliding
surf
aces
of
the
rst
layer
in
the
hierarchical
structure:
σ
x
,
σ
y
13:
T
ime
deri
v
ati
v
e
of
the
sliding
surf
aces
of
the
rst
layer:
˙
σ
x
,
˙
σ
y
14:
Calculate
the
equi
v
alent
control
la
ws:
u
x
eq
,
u
y
eq
15:
Dene
the
sliding
surf
ace
of
the
second
layer
in
the
hierarchical
structure:
s
1
16:
T
ime
deri
v
ati
v
e
of
the
sliding
surf
ace
of
the
second
layer:
˙
s
1
17:
Calculate
the
switching
control
la
w:
u
x,y
sw
18:
Calculate
the
position
control
la
w:
u
1
19:
Fully
Actuated
Contr
ol
System
20:
Acquire
the
current
states:
θ
,
˙
θ
21:
Calculate
the
tracking
errors:
e
θ
1
,
e
θ
2
22:
Dene
the
neural
netw
ork
performance
error:
E
θ
23:
Calculate
the
adapti
v
e
parameters
for
the
sliding
surf
ace:
ˆ
k
θ
,
ˆ
c
7
,
ˆ
c
8
,
ˆ
c
9
24:
Dene
the
sliding
surf
ace:
s
2
25:
T
ime
deri
v
ati
v
e
of
the
sliding
surf
aces:
˙
s
2
26:
Calculate
the
orientation
control
la
w:
u
2
27:
FTDO
Obser
v
er
28:
Acquire
the
control
inputs:
u
1
,
u
2
29:
Acquire
the
current
states:
x,
˙
x,
y
,
˙
y
,
θ
,
˙
θ
30:
Calculate
the
disturbance
estimations:
b
d
x
(
t
)
,
b
d
y
(
t
)
,
b
d
θ
(
t
)
31:
end
while
2.3.
Stability
analysis
-
Theorem
2
considering
the
dynamical
system
in
(25),
together
with
the
control
la
w
designed
in
(32)
and
the
design
of
the
adapti
v
e
la
ws
dened
in
(36),
then
the
system
states
will
reach
the
sliding
surf
ace
in
a
nite
time.
As
a
result,
the
tracking
errors
in
(27)
will
approach
zero
in
a
nite
time.
-
Proof
2
considering
the
follo
wing
L
yapuno
v
candidate
function
as:
V
=
1
2
(
e
θ
1
)
2
+
1
2
(
s
2
)
2
+
1
2
(
e
k
θ
)
2
+
1
2
(
e
c
7
)
2
+
1
2
(
e
c
8
)
2
+
1
2
(
e
c
9
)
2
(37)
where
e
c
i
,
(
i
=
7
,
8
,
9)
is
the
adapti
v
e
estimation
error
,
then
e
c
i
=
ˆ
c
i
−
c
i
.
Deri
ving
in
time
(37),
we
obtain:
˙
V
=
e
θ
1
(
e
θ
2
)
+
s
2
(
˙
s
2
)
+
e
k
θ
(
˙
e
k
θ
)
+
e
c
7
(
˙
e
c
7
)
+
e
c
8
(
˙
e
c
8
)
+
e
c
9
(
˙
e
c
9
)
(38)
Substituting
the
e
xpressions
of
(28)
and
(30)
into
(38),
we
obtain
the
follo
wing:
˙
V
=
e
θ
1
(
−
ˆ
c
7
e
θ
1
−
ˆ
c
8
|
e
θ
1
|
α
θ
sgn
(
e
θ
1
)
−
ˆ
c
9
|
e
θ
1
|
β
θ
sgn
(
e
θ
1
))+
s
2
(
−
k
s
2
1
s
2
−
ˆ
k
θ
sign
(
s
2
))
+
e
k
θ
(
˙
ˆ
k
θ
)
+
e
c
7
(
˙
ˆ
c
7
)
+
e
c
8
(
˙
ˆ
c
8
)
+
e
c
9
(
˙
ˆ
c
9
)
(39)
By
performing
some
mathematical
operations
and
adding
and
subtracting
some
terms,
the
follo
wing
e
xpression
is
obtained:
˙
V
=
−
ˆ
c
7
|
e
θ
1
|
2
−
ˆ
c
8
|
e
θ
1
|
α
θ
+1
−
ˆ
c
9
|
e
θ
1
|
β
θ
+1
−
k
s
2
1
|
s
2
|
2
−
ˆ
k
θ
|
s
2
|
+
e
k
θ
(
˙
ˆ
k
θ
)
+
e
c
7
(
˙
ˆ
c
7
)
+
e
c
8
(
˙
ˆ
c
8
)+
e
c
9
(
˙
ˆ
c
9
)
+
c
7
|
e
θ
1
|
2
−
c
7
|
e
θ
1
|
2
+
c
8
|
e
θ
1
|
α
θ
+1
−
c
8
|
e
θ
1
|
α
θ
+1
+
c
9
|
e
θ
1
|
β
θ
+1
−
c
9
|
e
θ
1
|
β
θ
+1
+
k
θ
|
s
2
|
−
k
θ
|
s
2
|
=
−
e
c
7
|
e
θ
1
|
2
−
e
c
8
|
e
θ
1
|
α
θ
+1
−
e
c
9
|
e
θ
1
|
β
θ
+1
−
e
k
θ
|
s
2
|
−
k
s
2
1
|
s
2
|
2
+
e
k
θ
(
˙
ˆ
k
θ
)
+
e
c
7
(
˙
ˆ
c
7
)
+
e
c
8
(
˙
ˆ
c
8
)
+
e
c
9
(
˙
ˆ
c
9
)
−
k
θ
|
s
2
|
−
c
7
|
e
θ
1
|
2
−
c
8
|
e
θ
1
|
α
θ
+1
−
c
9
|
e
θ
1
|
β
θ
+1
(40)
Putting
together
some
common
terms,
we
obtain:
˙
V
=
−
(
c
7
+
c
8
|
e
θ
1
|
α
θ
−
1
+
c
9
|
e
θ
1
|
β
θ
−
1
)
|
e
θ
1
|
2
−
(
k
s
2
1
+
k
θ
/s
2
)
|
s
2
|
2
−
(
|
s
2
|
−
˙
ˆ
k
θ
)
|
e
k
θ
|−
(
|
e
θ
1
|
2
−
˙
ˆ
c
7
)
|
e
c
7
|
−
(
|
e
θ
1
|
α
θ
+1
−
˙
ˆ
c
8
)
|
e
c
8
|
−
(
|
e
θ
1
|
β
θ
+1
−
˙
ˆ
c
9
)
|
e
c
9
|
(41)
Neur
o-adaptive
hier
ar
c
hical
sliding
mode
contr
ol
...
(J
ose
Carlos
V
izarr
eta
Mamani)
Evaluation Warning : The document was created with Spire.PDF for Python.
1524
❒
ISSN:
2502-4752
T
o
simplify
the
e
xpression,
we
dene
the
follo
wing
v
ariables:
Φ
1
=(
c
7
+
c
8
|
e
θ
1
|
α
θ
−
1
+
c
9
|
e
θ
1
|
β
θ
−
1
)
,
Φ
2
=
(
k
s
2
1
+
k
θ
/s
2
)
,
Φ
3
=
(
|
s
2
|
−
˙
ˆ
k
θ
)
,
Φ
4
=(
|
e
θ
1
|
2
−
˙
ˆ
c
7
)
,
Φ
5
=
(
|
e
θ
1
|
α
θ
+1
−
˙
ˆ
c
8
)
,
Φ
6
=
(
|
e
θ
1
|
β
θ
+1
−
˙
ˆ
c
9
)
(42)
According
to
(42),
the
(41)
is
obtained:
˙
V
=
−
Φ
1
|
e
θ
1
|
2
−
Φ
2
|
s
2
|
2
−
Φ
3
|
e
k
θ
|
−
Φ
4
|
e
c
7
|
−
Φ
5
|
e
c
8
|
−
Φ
6
|
e
c
9
|
(43)
Performing
some
mathematical
adjustments,
we
obtain
the
follo
wing:
˙
V
=
−
Φ
1
|
e
θ
1
|
2
−
Φ
2
|
s
2
|
2
−
√
2Φ
3
|
e
k
θ
|
2
2
ε
−
√
2Φ
4
|
e
c
7
|
2
2
ε
−
√
2Φ
5
|
e
c
8
|
2
2
ε
−
√
2Φ
6
|
e
c
9
|
2
2
ε
(44)
where
ε
=
1
/
2
.
No
w
,
according
to
Lemma
3
of
[31]
and
grouping
common
terms,
we
obtain:
˙
V
=
−
Φ
1
|
e
θ
1
|
2
2
ε
−
(1
−
ε
)
ε
ε
1
−
ε
−
Φ
2
|
s
2
|
2
2
ε
−
(1
−
ε
)
ε
ε
1
−
ε
−
−
√
2Φ
3
|
e
k
θ
|
2
2
ε
−
√
2Φ
4
|
e
c
7
|
2
2
ε
−
√
2Φ
5
|
e
c
8
|
2
2
ε
−
√
2Φ
6
|
e
c
9
|
2
2
ε
=
−
Φ
1
|
e
θ
1
|
2
2
ε
−
Φ
2
|
s
2
|
2
2
ε
−
√
2Φ
3
|
e
k
θ
|
2
2
ε
−
√
2Φ
4
|
e
c
7
|
2
2
ε
−
√
2Φ
5
|
e
c
8
|
2
2
ε
−
√
2Φ
6
|
e
c
9
|
2
2
ε
+
(Φ
1
+
Φ
2
)(1
−
ε
)
ε
ε
1
−
ε
=
−
K
V
V
ε
+
P
V
(45)
where
K
V
=
Φ
1
,
Φ
2
,
√
2Φ
3
,
√
2Φ
4
,
√
2Φ
5
,
√
2Φ
6
and
P
V
=
(Φ
1
+
Φ
2
)(1
−
ε
)
ε
ε
1
−
ε
.
Therefore,
according
to
Lemma
4
of
[32]
and
for
an
y
0
<
ϱ
<
1
,
it
can
be
inferred
that
the
errors
in
(34)
can
be
stabilized
to
zero
during
the
sliding
motion
s
2
=
0
in
a
nite
time
T
f
dened
by:
T
f
=
1
(1
−
ε
)
ϱK
V
V
1
−
ε
(0)
−
P
V
(1
−
ϱ
)
K
V
1
−
ε
ε
.
(46)
Thus,
completing
the
proof.
-
Remark
1
follo
wing
the
same
procedure
as
in
the
proof
of
Theorem
2,
it
can
be
demonstrated
that
the
tracking
errors
in
(13)
stabilize
to
zero
during
the
motion
of
the
sliding
surf
ace
s
1
=
0
in
nite
time.
Figure
3
sho
ws
the
simulation
setup
of
the
control
system
proposed
in
this
paper
,
implemented
in
MA
TLAB/Simulink
softw
are
(v
ersion
2021a).
The
s
etup
includes
se
v
eral
blocks
de
v
eloped
in
this
section,
such
as
the
dynamic
modeling
of
the
3WMR,
the
FTDO
observ
er
,
the
control
strate
gy
and
the
adapti
v
e
neural
netw
ork
scheme.
Figure
3.
Simulation
setup
of
the
proposed
control
system
in
MA
TLAB/Simulink
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
37,
No.
3,
March
2025:
1515–1531
Evaluation Warning : The document was created with Spire.PDF for Python.