Inter
national
J
our
nal
of
P
o
wer
Electr
onics
and
Dri
v
e
System
(IJPEDS)
V
ol.
17,
No.
2,
June
2026,
pp.
1008
∼
1024
ISSN:
2088-8694,
DOI:
10.11591/ijpeds.v17.i2.pp1008-1024
❒
1008
Hybrid
contr
ol
strategy
f
or
trajectory
tracking
and
obstacle
a
v
oidance
in
differ
ential
wheeled
r
obots:
integrating
PSO-NMPC,
GA,
and
fuzzy
logic
Abdennour
Zeghida
1
,
Lot
F
arah
2
,
Halim
Merabti
3
,
Abdelfateh
K
err
ouche
4
1
Laboratory
of
Automation
and
Signals
Annaba
(LASA),
Department
of
Electronics,
F
aculty
of
T
echnology
,
Uni
v
ersity
of
Badji
Mokhtar
,
Annaba,
Algeria
2
Electromechanical
Engineering
Laboratory
,
Department
of
Electromechanics,
F
aculty
of
T
echnology
,
Uni
v
ersity
of
Badji
Mokhtar
,
Annaba,
Algeria
3
Research
Center
in
Industrial
T
echnologies
CR
TI,
Alger
,
Algeria
4
School
of
Computing
Engineering
and
the
Built
En
vironment
Edinb
ur
gh
Napier
Uni
v
ersity
,
Edinb
ur
gh,
United
Kingdom
Article
Inf
o
Article
history:
Recei
v
ed
Aug
8,
2025
Re
vised
Feb
12,
2026
Accepted
Feb
21,
2026
K
eyw
ords:
Fuzzy
logic
Genetic
algorithm
Obstacle
a
v
oidance
PSO-NMPC
T
rajectory
tracking
Wheeled
robot
ABSTRA
CT
Mobile
robots
frequently
encounter
challenges
in
maintaining
accurate
trajectory
tracking
and
ef
fecti
v
e
obstacle
a
v
oidance
in
dynamic
and
uncertain
en
vironments.
T
raditional
control
methods,
such
as
proportional
inte
gral
deri
v
ati
v
e
(PID)
and
standard
MPC,
often
f
ail
to
pro
vide
the
necessary
adaptability
and
rob
ustness
for
comple
x
na
vig
ation
tasks.
T
o
o
v
ercome
these
limitations,
this
study
proposes
a
h
ybrid
control
frame
w
ork
for
dif
ferential-dri
v
e
wheeled
robots
that
inte
grates
particle
sw
arm
optimization–based
nonlinear
model
predicti
v
e
control
(PSO-NMPC),
adapti
v
e
neuro-fuzzy
inference
system
(ANFIS)
optimized
by
PSO,
and
genetic
algorithm
(GA)
tuning.
The
PSO-NMPC
computes
optimal
cont
rol
inputs
in
real
time
while
satisfying
system
constraints
to
ensure
precise
trajectory
tracking,
achie
ving
an
a
v
erage
RMSE
of
0.0941
m
(RMSE
x
=
0.0884
m,
RMSE
y
=
0.0812
m).
The
ANFIS-PSO
controller
manages
nonlineariti
es
and
en
vironmental
uncertainties
for
reli
able
obstacle
a
v
oidance,
with
an
o
v
erall
RMSE
of
0.1084
m
(RMSE
x
=
0.0761
m,
RMSE
y
=
0.0772
m).
The
GA
further
optimizes
k
e
y
parameters
and
trajectories,
ensuring
global
path
renement
and
rob
ust
obstacle
clearance,
achie
ving
an
o
v
erall
RMSE
of
0.1094
m
(RMSE
x
=
0.1059
m,
RMSE
y
=
0.0274
m).
Simulation
results
in
Matlab2024b
conrm
that
the
proposed
h
ybrid
frame
w
ork
pro
vides
precise
trajectory
tracking,
smooth
control,
and
rob
ust
obstacle
a
v
oidance,
making
it
a
promising
solution
for
autonomous
mobile
robots
operating
in
dynamic
and
uncertain
en
vironments.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Abdennour
Ze
ghida
Laboratory
of
Automation
and
Signals
Annaba
(LASA),
Department
of
Electronics,
F
aculty
of
T
echnology
Uni
v
ersity
of
Badji
Mokhtar
P
.O.
Box
12,
23000
Annaba,
Algeria
Email:
abdennour
.ze
ghida@uni
v-annaba.dz
1.
INTR
ODUCTION
Ov
er
the
past
fe
w
years,
ef
cient
control
of
robot
technology
has
made
signicant
progress
in
vie
w
of
the
increasing
need
for
intelligent
automation
in
medical,
agricultural,
military
,
and
industrial
applications
J
ournal
homepage:
http://ijpeds.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
❒
1009
[1].
Mobile
robots,
in
contrast
to
x
ed-base
manipulators,
can
be
mo
v
ed
and
na
vig
ate
in
both
structured
and
unstructured
en
vironments.
Dif
ferential-dri
v
e
mobile
robots
(DDMR)
are
popular
as
a
mobile
platform
due
to
their
simple
mechanical
s
tructure,
high
mobility
,
and
independent
actuation
of
the
wheels
on
both
sides
for
linear
motion
and
angular
motion
independently
[2].
Ne
v
ertheless,
precise
and
rob
ust
motion
control
of
DDMRs
is
still
an
open
challenge.
Non
l
inear
kinematics,
wheel
slippage,
actuator
saturation,
model
uncertainties,
and
dynamic
obstacles
seriously
de
grade
the
tracking
performance
and
safety
of
na
vig
ation.
DDMR
motion
relies
on
wheel
act
uators
actuated
by
electric
po
wer
and
re
gulated
using
po
wer
electronic
dri
v
e
systems,
where
sharp
or
c
ycling
control
signals
introduce
uncertainties
that
w
aste
will
increase
motor
stress,
ener
gy
w
aste,
strate
gies,
and
thermal
load.
Hence,
de
v
eloping
reacti
v
e
and
f
ast
algorithms
with
smoothness,
constraint-a
w
areness,
and
rob
ustness
is
a
necessary
step
not
only
for
reacti
v
e
collision
a
v
oidance
in
robotics
b
ut
also
to
mak
e
electric
dri
v
es
more
ener
gy-constrained
or
actuators
less
prone
to
damage.
T
rajectory
tracking
and
obstacle
a
v
oidance
problems
ha
v
e
long
been
addressed
by
traditional
cont
rol
strate
gies,
including
proportional
inte
gral
deri
v
ati
v
e
(PID)
and
linear
model
predicti
v
e
control
(MPC)
[3]–[5].
All
of
these
al
go
r
ithms
w
ork
well
when
the
systems
are
linear
and
time-in
v
ariant,
b
ut
do
not
perform
well
for
nonlinear
and
time-v
arying
scenarios.
T
o
alle
viate
such
disadv
antages,
intelligent
and
bio-inspired
control
methodologies
are
being
introduced,
such
as
fuzzy
logic
(FL),
neural
netw
orks
(NNs),
genetic
algorithms
(GAs),
and
particle
sw
arm
optimization
(PSO)
techniques
[6]–[9].
FL
controllers
are
able
to
cope
with
uncertainties
b
ut
do
not
ha
v
e
predicti
v
e
ability
.
NNs
are
procient
in
modeling
nonlinearities
b
ut
need
plenty
of
data
for
training.
Genetic
algorithm
(GA)
and
PSO
are
the
e
v
olutionary
algorithms
that
can
meet
strong
optimization,
b
ut
it
is
dif
cult
to
directly
apply
them
in
real-time
since
GA
and
PSO
ha
v
e
a
lar
ge
amount
of
computation
o
v
erhead
[10]–[13].
A
number
of
h
ybrid
control
approaches
ha
v
e
been
de
v
eloped
to
inte
grate
the
benets
of
traditi
on
a
l
and
intelligent
techniques.
Neural
netw
orks
ha
v
e
been
used
in
the
trajectory
tracking
of
adapti
v
e
systems
[14],
fuzzy
controllers
on
DDMR
control
of
motion
re
gulation
[15],
sliding
mode
control
on
ho
w
to
impro
v
e
stability
[16],
and
pre
v
enti
v
e
controllers
for
reaching
tar
gets
[17].
The
neural
netw
ork-based
predictors
and
the
impro
v
ed
MPC
schemes
were
also
reported
for
nonholonomic
mobile
robot
control
[18]–[20].
One
important
g
ap
in
the
literature
persists,
ho
we
v
er
.
Most
of
the
studies
only
consider
tr
ajectory
tracking
and
obstacle
a
v
oidance
separatel
y
,
b
ut
not
simultaneously
,
in
dynamic
or
uncertain
conditions.
In
addition,
the
inte
gration
of
predicti
v
e
control
along
with
global
optimization
and
adapti
v
e
fuzzy
intelligence
in
a
unied
manner
that
is
amenable
to
dri
v
e-constrained
robotic
systems
has
not
been
fully
in
v
estig
ated.
T
o
solv
e
these
problems,
the
present
w
ork
de
v
elops
a
h
ybrid
intelligent
control
technique
including
PSO-based
NMPC,
PSO-tuned
ANFIS
(ANFIS-PSO),
and
GA
tuning.
The
PC-based
PSO-NMPC-based
algorithm
is
utilized
to
output
optimal
control
la
ws
in
the
online
mode
for
accurate
path
tracking.
The
GA
module
optimizes
the
parameters
of
the
controller
and
path
plan
in
order
to
impro
v
e
global
performance
and
reduce
computation
cost.
Through
the
ANFIS-PSO
controller
,
which
can
compensate
for
nonlinear
and
en
vironmental
uncertainties,
adapti
v
e
obstacle
a
v
oidance
is
achie
v
ed.
The
no
v
elty
of
this
w
ork
lies
in
the
unied
multi-layer
inte
gration
of
predicti
v
e
control,
e
v
ol
utionary
optimization,
and
adapti
v
e
fuzzy
intelligence,
enabling
simultaneous
impro
v
ement
in
tracking
precision,
control
smoothness,
and
obstacle
a
v
oidance
reliability
.
This
inte
grated
strate
gy
reduces
abrupt
v
elocity
v
ariations,
which
is
benecial
for
electric
dri
v
e
ef
cienc
y
,
actuator
longe
vity
,
and
system
stability
.
The
main
contrib
utions
of
this
study
are:
–
De
v
elopment
of
a
no
v
el
h
ybrid
control
architecture
combining
PSO-NMPC,
ANFIS-PSO,
and
GA
for
DDMR
systems.
–
Real-time
optimization
of
control
parameters
under
system
constraints.
–
Inte
gration
of
adapti
v
e
fuzzy
intelligence
for
dynamic
obstacle
a
v
oidance.
–
Performance
v
alidation
demonstrating
reduced
tracking
error
,
smoother
v
elocity
proles,
and
enhanced
na
vig
ation
stability
compared
with
con
v
entional
NMPC
and
fuzzy
controllers.
The
rest
of
this
manuscript
is
structured
as
follo
ws:
i)
Section
2
describes
the
robot
modeling
and
the
proposed
control
frame
w
ork;
ii)
The
implementation
approach
is
detailed
in
section
3;
iii)
The
simulation
and
e
xperimental
results
are
presented
in
section
4;
and
i
v)
Finally
section
5
summarizes
and
concludes
the
paper
with
directions
to
future
w
ork.
Hybrid
contr
ol
str
ate
gy
for
tr
ajectory
tr
ac
king
and
obstacle
avoidance
in
...
(Abdennour
Ze
ghida)
Evaluation Warning : The document was created with Spire.PDF for Python.
1010
❒
ISSN:
2088-8694
2.
THE
PR
OPOSED
HYBRID
CONTR
OL
ALGORITHM
Examine
a
trajectory
tracking
control
problem
in
which
we
deri
v
e
control
la
ws
to
manage
the
angular
and
linear
v
elocities
(and
accelerations)
of
a
nonholonomic
wheeled
mobile
robot
to
properly
follo
w
a
specied
path.
The
control
polic
y
minimizes
the
root
mean
square
error
(RMSE)
between
the
desired
and
actual
trajectories.
T
racking
mistak
es
may
arise
from
sensor
inaccuracies,
e
xternal
disturbances,
system
noise,
and
wheel
slip.
This
nonholonomic
l
imitation
pre
v
ents
the
mobile
robot
from
instantaneously
mo
ving
in
a
direction
perpendicular
to
the
wheel
axis,
complicating
motion
control
design
compared
to
a
holonomic
system
that
allo
ws
for
unfettered
mobility
.
The
suggested
h
ybrid
control
system
combines
v
arious
intelligence
and
optimization
strate
gies
to
impro
v
e
tracking
accurac
y
,
stability
,
and
rob
ustness
in
response
to
these
obstacles.
The
h
ybrid
controller
inte
grates
the
adv
antages
of
nonlinear
model
predicti
v
e
control
(NMPC),
PSO,
GA,
and
fuzzy
logic
control
(FLC)
to
enhance
control
performance
in
di
v
erse
dynamic
and
uncertain
en
vironments.
The
suggested
h
ybrid
control
technique
amalg
amates
NMPC,
PSO,
GA,
and
an
FLC
to
ensure
resilient
trajector
y
tracking
and
obstacle
e
v
asion
in
a
dif
ferential
wheeled
robot.
Figure
1
illustrates
the
comprehensi
v
e
structure
of
the
system
[21].
The
simplest
study
of
ho
w
mechanical
systems
act
is
kine
matics.
T
o
design
ef
fecti
v
e
mobile
robots
for
tasks
and
understand
ho
w
to
de
v
elop
control
softw
are
for
a
specic
instance
of
mobile
robot
hardw
are,
we
need
to
understand
the
mechanical
beha
vior
of
the
robot
in
mobile
robotics.
The
DDMR
kinematic
modeling
notations
and
essential
parameters
are
summarized
in
T
able
1.
Throughout
the
mathematical
formulation
and
simulation
of
the
robot’
s
mot
ion,
these
abbre
viations
specify
the
fundamental
geometric
and
motion-related
v
ariables,
such
as
wheel
lengths,
radius,
and
linear
or
angular
v
elocities.
Figure
1.
Demonstration
of
a
dif
ferential
dri
v
e
mobile
robot
in
the
w
orld
coordinate
frame
T
able
1.
List
of
abbre
viations
for
DDMR
parameters
Abbre
viation
Meaning
m
Center
point
of
the
DDMR
D
Distance
between
the
tw
o
rear
wheels
of
DDMR
(m)
r
Radius
of
each
rear
wheel
of
DDMR
(m)
V
Linear
v
elocity
of
the
DDMR
(m
/s)
ω
Angular
v
el
ocity
of
the
DDMR
(rad/s)
V
lef
t
Linear
v
elocity
of
the
left
wheel
of
DDMR
(m/s)
V
r
ig
ht
Linear
v
elocity
of
the
right
wheel
of
DDMR
(m/s)
2.1.
Kinematic
model
of
the
mobile
r
obot
Kinematics
is
the
mathematical
study
of
motion
without
taking
into
account
the
forces
that
control
motion.
Robot
mo
v
ements
are
described
by
robot
kinematics.
It
discusses
ho
w
the
system’
s
geometric
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
17,
No.
2,
June
2026:
1008–1024
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
❒
1011
relationships
w
ork.
It
establishes
a
connection
between
the
cont
rol
parameters,
system
parameters,
and
space
beha
vior
.
The
kinematic
model
of
a
tw
o-wheel
mobile
robot
is
sho
wn
i
n
Figure
1.
The
DDMR
is
modeled
wi
th
tw
o
dri
ving
wheels
on
the
hub
behind
the
truck,
and
a
castor
wheel
o
v
er
the
top
of
the
truck.
DDMR
motion
and
its
direction
are
realized
with
the
right
and
left
wheel
actuators
being
2
DC
motors.
The
DDMR
motion
is
characterized
by
the
linear
v
elocities
of
its
tw
o
wheels
V
l
ef
t
and
V
r
ig
ht
.
The
DDMR
linear
and
angular
v
elocities
V
and
ω
of
the
DDMR
are
related
to
V
l
ef
t
and
V
r
ig
ht
as
gi
v
en
in
(1)
and
(2).
V
=
(
V
L
+
V
R
)
2
(1)
ω
=
(
V
L
−
V
R
)
D
(2)
A
dri
v
e
with
dif
ference
an
y
change
in
the
relati
v
e
v
elocities
of
the
tw
o
wheels
will
cause
DDMR
to
become
sensiti
v
e.
Dif
ferent
trajectories
result
from
e
v
en
a
slight
dif
ference
in
these
v
elocities.
In
the
global
coordinate
axis,
the
DDMR
kinematic
equations
are
e
xpressed
as
(3)–(5)
[22].
˙
x
(
t
)
=
(
V
R
+
V
L
)
2
cosθ
(
t
)
(3)
˙
y
(
t
)
=
(
V
R
+
V
L
)
2
sinθ
(
t
)
(4)
˙
θ
(
t
)
=
V
R
(
t
)
−
V
L
(
t
)
D
(5)
2.2.
Nonlinear
model
pr
edicti
v
e
contr
ol
The
nonlinear
model
predicti
v
e
control
(NMPC)
strate
gy
is
emplo
yed
to
ensure
precise
t
rajectory
tracking
of
the
DDMR
under
nonlinear
and
nonholonomic
constraints.
The
objecti
v
e
is
to
determine
the
control
inputs
V
(linear
v
elocity)
and
ω
(angular
v
elocity)
that
enable
the
robot
to
fol
lo
w
a
reference
trajectory
dened
by
(
x
r
ef
,
y
r
ef
)
,
as
illustrated
in
Figure
2
[23].
Figure
2.
Nonlinear
model
predicti
v
e
control
structure
for
the
mobile
robot
Consider
the
discrete
nonlinear
system
model,
as
in
(6).
x
(
k
+
1)
=
f
(
x
(
k
)
,
u
(
k
))
(6)
Hybrid
contr
ol
str
ate
gy
for
tr
ajectory
tr
ac
king
and
obstacle
avoidance
in
...
(Abdennour
Ze
ghida)
Evaluation Warning : The document was created with Spire.PDF for Python.
1012
❒
ISSN:
2088-8694
Where
x
(
k
)
i
s
the
state
v
ector
,
u
(
k
)
is
the
control
input,
and
f
(
·
)
represents
a
continuous
nonlinear
mapping.
The
control
input
is
constrained
by
(7).
u
(
k
)
∈
U
⊂
R
m
(7)
Where
U
is
a
compact
con
v
e
x
set
sati
sfying
0
∈
U
and
f
(0
,
0)
=
0
.
The
state
is
also
constrained
to
remain
within
the
admissible
re
gion,
as
(8).
x
(
k
)
∈
X
(8)
The
NMPC
optimization
problem
aims
to
re
gulate
the
robot
state
to
w
ard
the
reference
traject
ory
by
solving
a
nite-horizon
cost
minimization
problem,
as
(9).
min
u
J
N
(
x,
k
,
u
)
(9)
Subject
to
the
system
dynamics
and
constraints
gi
v
en
in
(7).
The
cost
function
is
dened
as
(10).
J
N
(
x,
k
,
u
)
=
F
(
x
(
k
+
N
))
+
k
+
N
−
1
X
i
=
k
L
(
x
(
i
)
,
u
(
i
))
(10)
Where
N
is
the
prediction
horizon.
The
terminal
state
is
required
to
lie
within
the
terminal
re
gion,
as
(11).
x
(
k
+
N
)
∈
X
f
⊂
X
(11)
The
terminal
cost
F
and
terminal
re
gion
X
f
ensure
the
closed-loop
stability
of
the
NMPC.
The
optimal
control
sequence
is
gi
v
en
by:
U
=
[
u
(
k
)
,
u
(
k
+
1)
,
.
.
.
,
u
(
k
+
N
−
1)]
∈
U
N
and
only
the
rst
control
input
u
(
k
)
is
applied
at
each
sampling
instant,
with
the
optimization
repeated
at
the
ne
xt
step.
–
Discrete-time
robot
model:
Based
on
the
kine
matic
model
of
the
DDMR
,
the
syst
em
dynamics
in
discrete
time
are
e
xpressed
as
(12).
x
k
+1
=
x
k
+
T
s
V
k
cos(
θ
k
)
y
k
+1
=
y
k
+
T
s
V
k
sin(
θ
k
)
θ
k
+1
=
θ
k
+
T
s
ω
k
(12)
Where
x
k
and
y
k
denote
the
robot
position,
θ
k
its
orientation,
V
k
the
linear
v
elocity
,
ω
k
the
angular
v
elocity
,
and
T
s
the
sampling
period.
–
System
states
and
control
inputs
as
sho
wn
in
T
able
2:
T
able
2.
System
v
ariables
and
descriptions
Symbol
Description
x
=
[
x,
y
,
θ
]
T
State
v
ector
(position
and
orientation)
u
=
[
V
,
ω
]
T
Control
input
v
ector
(linear
and
angular
v
elocities)
–
Optimization
problem
formulation:
at
each
s
ampling
instant
k
,
NMPC
solv
es
the
nite-horizon
optimization
problem,
as
(13).
min
U
J
N
(
x
k
,
U
)
=
N
−
1
X
i
=0
h
(
x
k
+
i
|
k
−
x
r
ef
)
T
Q
(
x
k
+
i
|
k
−
x
r
ef
)
+
u
T
k
+
i
|
k
R
u
k
+
i
|
k
i
+
(
x
k
+
N
|
k
−
x
r
ef
)
T
P
(
x
k
+
N
|
k
−
x
r
ef
)
(13)
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
17,
No.
2,
June
2026:
1008–1024
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
❒
1013
Subject
to
(14)–(16).
x
k
+
i
+1
|
k
=
f
(
x
k
+
i
|
k
,
u
k
+
i
|
k
)
(14)
u
min
≤
u
k
+
i
|
k
≤
u
max
(15)
x
min
≤
x
k
+
i
|
k
≤
x
max
(16)
Where
Q
,
R
,
and
P
are
positi
v
e
denite
weighting
ma
trices,
and
N
is
the
prediction
horizon.
–
Performance
metric:
the
tracking
performance
is
e
v
aluated
using
the
root
mean
square
error
(RMSE):
RMSE
=
v
u
u
t
1
N
N
X
i
=1
[(
x
r
ef
,i
−
x
i
)
2
+
(
y
r
ef
,i
−
y
i
)
2
]
(17)
–
Continuous-time
model
representation:
in
continuous
time,
the
nonlinear
kinematic
model
of
the
DDMR
is
gi
v
en
as
(18).
˙
x
(
t
)
=
V
(
t
)
cos(
θ
(
t
))
,
˙
y
(
t
)
=
V
(
t
)
sin(
θ
(
t
))
,
˙
θ
(
t
)
=
ω
(
t
)
(18)
The
linear
and
angular
v
elocities
are
related
to
the
left
and
right
wheel
v
elocities
V
L
and
V
R
by
(19).
V
(
t
)
=
V
R
(
t
)
+
V
L
(
t
)
2
,
ω
(
t
)
=
V
R
(
t
)
−
V
L
(
t
)
L
(19)
Discretizing
the
abo
v
e
model
using
a
sampling
time
∆
t
yields:
x
k
+1
=
x
k
+
∆
t
V
k
cos(
θ
k
)
,
y
k
+1
=
y
k
+
∆
t
V
k
sin(
θ
k
)
,
θ
k
+1
=
θ
k
+
∆
t
ω
k
(20)
At
each
sampling
instant,
the
NMPC
optimization
problem
can
be
formulated
as
(21).
min
U
J
=
N
−
1
X
i
=0
h
(
x
k
+
i
−
x
r
ef
k
+
i
)
T
Q
(
x
k
+
i
−
x
r
ef
k
+
i
)
+(
u
k
+
i
−
u
r
ef
k
+
i
)
T
R
(
u
k
+
i
−
u
r
ef
k
+
i
)
i
+
(
x
k
+
N
−
x
r
ef
k
+
N
)
T
Q
f
(
x
k
+
N
−
x
r
ef
k
+
N
)
(21)
Subject
to
(22)–(24).
x
k
+
i
+1
=
f
(
x
k
+
i
,
u
k
+
i
)
(22)
u
min
≤
u
k
+
i
≤
u
max
(23)
x
min
≤
x
k
+
i
≤
x
max
(24)
After
solving
the
optimization,
only
the
rst
control
input
u
∗
k
=
[
V
∗
k
,
ω
∗
k
]
T
is
applied
to
the
robot
actuators.
The
proce
ss
is
then
repeated
at
the
ne
xt
sampling
ins
tant
using
updated
state
measurements.
This
receding-horizon
strate
gy
enables
NMPC
to
ef
fecti
v
ely
handle
nonlinearities,
input
constraints,
and
disturbances,
achie
ving
smooth
and
rob
ust
trajectory
tracking
performance.
2.3.
P
article
swarm
optimization
The
particle
sw
ar
m
optimization
(PSO)
w
as
rst
presented
by
K
ennedy
and
Eberhart
in
1995
[24].
It
dra
ws
from
the
collecti
v
e
social
beha
vior
in
ocks
of
birds
or
schools
of
sh
[25].
Each
member
of
the
search
population
(particle)
is
a
possible
solution
to
the
optimization
problem,
and
it
e
v
olv
es
based
on
its
o
wn
e
xperience
and
that
of
its
neighbours.
Ev
ery
particle
has
an
indi
vidual
best
position
pbest
i
(pbest),
which
is
the
tness
v
alue
of
it
found
so
f
ar
.
In
the
global
PSO
v
ersion,
gbest
is
used
as
the
best
solution
found
by
all
particles.
F
or
each
particle.
Each
particle
mai
ntains
its
personal
best
position,
denoted
as
p
i
k
(pbest),
which
corresponds
to
the
best
position
it
has
achie
v
ed
so
f
ar
according
to
a
dened
tness
function.
Hybrid
contr
ol
str
ate
gy
for
tr
ajectory
tr
ac
king
and
obstacle
avoidance
in
...
(Abdennour
Ze
ghida)
Evaluation Warning : The document was created with Spire.PDF for Python.
1014
❒
ISSN:
2088-8694
In
the
global
v
ersion
of
PSO,
the
global
best
position
p
g
k
(gbest)
represents
the
best
solution
disco
v
ered
by
an
y
particle
in
the
sw
arm.
The
v
elocity
and
position
of
each
particle
are
updated
iterati
v
ely
using
(25)
and
(26).
V
i
k
+1
=
V
i
k
+
c
1
r
1
(
p
i
k
−
x
i
k
)
+
c
2
r
2
(
p
g
k
−
x
i
k
)
(25)
x
i
k
+1
=
x
i
k
+
V
i
k
+1
(26)
Where
the
parameters
are
dened
in
T
able
3.
T
able
3.
Notation
used
in
the
PSO
algorithm
Symbol
Description
x
i
k
Position
of
the
i
th
particle
at
iteration
k
V
i
k
V
elocity
of
the
i
th
particle
at
iteration
k
p
i
k
Local
best
position
of
the
i
th
particle
(
pbest
)
p
g
k
Global
best
position
among
all
particles
(
gbest
)
c
1
,
c
2
Acceleration
coef
cients
(cogniti
v
e
and
social
constants)
r
1
,
r
2
Random
numbers
uniformly
distrib
uted
in
[0,
1]
At
each
iteration,
t
he
v
elocity
of
each
particle
is
inuenced
by
tw
o
main
components:
the
cogniti
v
e
component,
representing
the
particle’
s
o
wn
e
xperience,
and
the
social
component,
representing
the
collecti
v
e
kno
wledge
of
the
sw
arm.
The
balance
between
these
tw
o
tendencies
enables
ef
fecti
v
e
e
xploration
and
con
v
er
gence
to
w
ard
the
global
optimum.
The
main
steps
of
the
PSO
algorithm
are
as
follo
ws:
i)
Initialize
a
sw
arm
of
N
particles
with
random
positions
x
i
and
v
elocities
V
i
;
ii)
Ev
aluate
the
tness
of
each
particle
according
to
the
objecti
v
e
function;
iii)
Update
each
particle’
s
personal
best
(
pbest
)
and
identify
the
global
best
(
g
best
);
i
v)
Update
particle
v
elociti
es
and
positions
using
(25)
and
(26);
and
v)
Repeat
until
the
stopping
condition
(e.g.,
maximum
iterations
or
minimal
error)
is
reached.
Through
this
mechanism,
PSO
ef
ciently
balances
e
xploration
and
e
xploitation
in
the
search
space,
making
it
a
rob
ust
optimization
method
for
tuning
control
parameters
in
nonlinear
systems
such
as
the
DDMR.
Inte
gration
of
PSO-NMPC:
The
combination
of
PSO
with
NMPC
aims
to
impro
v
e
the
trajectory
tracking
performance
of
the
DDMR
under
dynamic
constraints
and
model
uncertainties.
The
PSO
algorithm
is
utilized
to
optimize
the
control
inputs
and
prediction
horizon
parameters
of
the
NMPC
in
order
to
achie
v
e
a
more
accurate
and
stable
motion
response.
In
the
con
v
entional
NMPC
frame
w
ork,
the
control
problem
is
formulated
as
a
nonlinear
opt
imization
problem
that
minimizes
a
cost
function
representing
the
tracking
error
and
control
ef
fort.
Ho
we
v
er
,
due
to
t
h
e
high
nonli
nearity
of
the
robot
dynamics
and
the
presence
of
multiple
local
minima,
the
optimization
process
may
not
con
v
er
ge
to
a
global
optimum.
PSO
pro
vides
a
global
search
capability
to
enhance
the
NMPC
optimization
process
by
rening
the
control
sequence
ini
tialization
or
tuning
the
weighting
parameters
of
the
cost
function.
The
cost
function
to
be
minimized
at
each
sampling
instant
is
dened
as
(27).
J
N
(
x,
u
)
=
k
+
N
−
1
X
i
=
k
(
x
(
i
)
−
x
r
ef
(
i
))
T
Q
(
x
(
i
)
−
x
r
ef
(
i
))
+
u
(
i
)
T
R
u
(
i
)
(27)
Where:
x
(
i
)
represents
the
predicted
robot
states
(position
and
orientation);
x
r
ef
(
i
)
denotes
the
reference
trajectory
states;
u
(
i
)
is
the
control
input
v
ector
[
V
,
ω
]
T
;
Q
and
R
are
positi
v
e
denite
weighting
matrices
that
balance
tracking
accurac
y
and
control
ef
fort;
and
N
is
the
prediction
horizon
length.
In
the
proposed
PSO-NMPC
approach,
each
particle
in
the
PSO
sw
arm
represents
a
candidate
control
input
sequence
o
v
er
the
prediction
horizon:
U
=
[
u
(
k
)
,
u
(
k
+
1)
,
.
.
.
,
u
(
k
+
N
−
1)]
The
tness
of
each
particle
is
e
v
aluated
using
the
cost
function
(27),
which
reects
the
predicted
tracking
performance
o
v
er
the
horizon.
The
PSO
algorithm
searches
for
the
optimal
U
∗
that
minimizes
this
cost
while
satisfying
the
system
constraints
dened
in
the
NMPC
model.
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
17,
No.
2,
June
2026:
1008–1024
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
❒
1015
The
inte
gration
process
proceeds
as
follo
ws:
i)
Initialize
the
sw
arm
with
random
control
input
sequences
U
i
;
ii)
Ev
aluate
the
cost
function
J
N
(
x,
u
)
for
each
particle;
iii)
Update
pbest
and
g
best
based
on
indi
vidual
and
global
tness;
i
v)
Use
g
best
as
the
optimal
control
sequence
to
initialize
or
replace
NMPC
optimizer
at
the
current
time
step;
v)
Apply
the
rst
control
input
u
(
k
)
to
the
DDMR
system;
and
vi)
Repeat
the
process
at
the
ne
xt
sampling
instant
with
updated
states.
This
h
ybrid
PSO-NMPC
control
frame
w
ork
combines
the
predicti
v
e
and
constrai
nt-handling
capability
of
NMPC
with
the
global
optimization
strength
of
PSO.
The
result
is
an
adapti
v
e
and
rob
ust
controller
capable
of
maintaining
accurate
trajectory
tracking
e
v
en
under
disturbances,
slippage,
and
parameter
uncertainties
in
nonlinear
DDMR
dynamics.
3.
DESIGN
OF
THE
FUZZY
LOGIC
CONTR
OLLER
The
fuzzy
logic
controller
(FLC)
is
designed
to
manage
both
na
vig
ation
and
obstacle
a
v
oidance
tasks
simultaneously
for
the
DDMR.
Fuzzy
logic
contr
o
l
pro
vides
a
rob
ust
and
e
xible
approach
for
dealing
with
nonlinear
and
uncertain
robotic
en
vironments
where
precise
mathematical
modeling
is
dif
cult.
The
fundamental
structure
of
the
fuzzy
controller
consists
of
three
k
e
y
st
ages:
fuzzication,
inference,
and
defuzzication.
3.1.
Determination
of
input
and
output
v
ariables
In
se
v
eral
pre
vious
studies,
fuzzy
logic
has
been
successfull
y
implemented
in
mobile
robot
obsta
cle
a
v
oidance
and
path-tracking
systems.
F
or
the
proposed
DDMR
system,
the
input
v
ariables
are
dened
as
the
positional
error
and
orientation
error
relati
v
e
to
the
reference
trajectory
.
These
represent
the
de
viation
between
the
robot’
s
current
position
and
the
desired
reference
(
x
r
ef
,
y
r
ef
)
.
The
tw
o
out
p
ut
v
ariables
correspond
to
the
left
and
right
wheel
v
e
locities,
which
determine
the
robot’
s
motion
and
turning
beha
vior
.
The
fuzzy
logic
control
system
thus
generates
independent
wheel
v
elocity
commands
that
enable
the
robot
to
follo
w
the
reference
path
smoothly
while
a
v
oiding
obstacles.
i)
Right
wheel
v
elocity
(
V
right
):
The
v
elocity
command
applied
to
the
right
wheel
of
the
robot.
This
v
a
lue
adjusts
dynamically
based
on
the
error
inputs
to
guide
the
robot
to
w
ard
the
desired
trajectory
.
ii)
Left
wheel
v
elocity
(
V
left
):
The
corresponding
command
for
the
left
wheel.
By
adjusting
V
left
and
V
right
independently
,
the
DDMR
can
perform
precise
turning
and
path
correction
maneuv
ers.
3.2.
Design
of
the
initial
fuzzy
logic
contr
oller
The
initial
FLC
is
designed
with
simplicity
and
adaptability
in
mind,
emplo
ying
error
-dri
v
en
i
nputs
and
a
Sugeno-type
inference
structure
to
enable
ef
cient
l
earning
and
ne-tuning.
The
h
ybrid
ANFIS–PSO
frame
w
ork
later
renes
these
fuzzy
parameters
to
enhance
performance
in
dynamic
en
vironments.
The
proposed
fuzzy
controller
uses
tw
o
input
v
ariables
—
the
reference
X
r
ef
and
Y
r
ef
coordinates
—
representing
the
robot’
s
horizontal
and
v
ertical
positional
de
viations.
These
serv
e
as
the
main
feedback
signals
for
trajectory
tracking.
The
membership
functions
for
the
input
v
ariables
(position
and
orientation
errors)
are
dened
using
v
e
linguistic
terms:
v
ery
small
(VS),
small
(S),
medium
(M),
lar
ge
(L),
and
v
ery
lar
ge
(VL).
These
fuzzy
sets
are
modeled
using
o
v
erlapping
triangular
and
trapezoidal
membership
functions,
ensuring
smooth
transitions
and
rob
ust
control
performance.
The
v
ery
small
term
centers
around
zero,
promoting
precise
ne-tuning
near
the
reference,
while
the
wider
sets
(L
and
VL)
allo
w
ef
fecti
v
e
correction
of
lar
ge
de
viations.
trimf
(
x
;
a,
b,
c
)
=
0
,
x
≤
a
or
x
≥
c
x
−
a
b
−
a
,
a
<
x
≤
b
c
−
x
c
−
b
,
b
<
x
<
c
(28)
This
fuzzy
structure
ensures
smooth
control
transitions
and
stable
motion
e
v
en
in
the
presence
of
disturbances
or
modeling
uncertainties.
Figures
3
and
4
illustrat
e
the
input
membership
functions
used
for
the
fuzzy
logic
controllers
of
the
right
and
left
wheel
v
elocities,
respecti
v
ely
.
Hybrid
contr
ol
str
ate
gy
for
tr
ajectory
tr
ac
king
and
obstacle
avoidance
in
...
(Abdennour
Ze
ghida)
Evaluation Warning : The document was created with Spire.PDF for Python.
1016
❒
ISSN:
2088-8694
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Position Error
0
0.5
1
Membership Value
Right Wheel Controller - Input 1 (Position Error) Membership Functions
VerySmall
Small
Medium
Large
VeryLarge
-1.5
-1
-0.5
0
0.5
Orientation Error (rad)
0
0.5
1
Membership Value
Right Wheel Controller - Input 2 (Orientation Error) Membership Functions
VerySmall
Small
Medium
Large
VeryLarge
Figure
3.
Input
membership
functions
for
the
right
wheel
v
elocity
controller
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Position Error
0
0.5
1
Membership Value
Left Wheel Controller - Input 1 (Position Error) Membership Functions
VerySmall
Small
Medium
Large
VeryLarge
-1.5
-1
-0.5
0
0.5
Orientation Error (rad)
0
0.5
1
Membership Value
Left Wheel Controller - Input 2 (Orientation Error) Membership Functions
VerySmall
Small
Medium
Large
VeryLarge
Figure
4.
Input
membership
functions
for
the
left
wheel
v
elocity
controller
3.3.
Fuzzy
infer
ence
and
contr
ol
surface
generation
The
fuzzy
inference
process
applies
a
set
of
IF–THEN
rules
that
determine
the
output
wheel
v
eloci
ties
based
on
the
current
positional
and
orientation
errors.
The
defuzzication
step
then
con
v
erts
these
fuzzy
outputs
into
crisp
v
elocity
v
alues
for
V
L
and
V
R
.
Figures
5
and
6
illustrate
the
ANFIS-generated
control
surf
aces
for
the
left
and
right
wheel
v
elocities,
respecti
v
ely
.
These
surf
aces
capture
the
nonlinear
relationships
between
input
errors
(position
and
deri
v
ati
v
e
of
position)
and
the
output
v
elocity
commands.
The
surf
aces
e
xhibit
smooth
curv
ature
transitions,
reecting
consistent
control
actions
and
ensuring
stable
robot
mo
v
ement.
Steeper
gradients
corre
spond
to
stronger
correcti
v
e
actions
for
lar
ge
trajectory
de
viations,
while
atter
re
gions
indicate
ner
adjustment
s
near
the
tar
get
path.
This
fuzzy
structure,
enhanced
through
ANFIS
and
PSO
tuning,
pro
vides
reliable
and
adapti
v
e
control
performance
for
the
DDMR
under
dynamic
and
uncertain
en
vironments.
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
17,
No.
2,
June
2026:
1008–1024
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
❒
1017
Figure
5.
ANFIS
control
surf
ace
for
left
wheel
v
elocity
Figure
6.
ANFIS
control
surf
ace
for
right
wheel
v
elocity
3.4.
Genetic
algorithm
f
or
DDMR
path
planning
and
optimization
GA
is
an
e
v
olutionary
optimization
method
inspired
by
natural
selection
and
genetics.
GA
is
particularly
useful
for
global
s
earch
in
comple
x,
multimodal
spaces
and
has
been
widely
used
for
path
planning,
parameter
tuning,
and
trajectory
optimization
for
mobile
robots.
In
this
w
ork,
GA
is
applied
to
i)
generate
collision-free
paths
for
the
DDMR
and
ii)
tune
discrete
NMPC/PSO
parameters
when
required.
–
Chromosome
representation:
a
chromosome
encodes
a
path
from
start
to
goal
using
tw
o
common
methods:
i)
W
aypoint
sequence:
A
chromosome
is
a
v
ector
of
w
aypoints
{
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
,
.
.
.
,
(
x
n
,
y
n
)
}
.
Each
w
aypoint
can
be
represented
by
a
x
ed-length
binary
or
real-v
alued
gene.
ii)
Control
sequence:
A
chromosome
encodes
a
sequence
of
control
inputs
o
v
er
a
horizon
U
=
[
u
0
,
u
1
,
.
.
.
,
u
H
−
1
]
where
u
i
=
[
V
i
,
ω
i
]
.
This
is
con
v
enient
when
inte
grating
directly
with
NMPC.
–
Fitness
function:
the
tness
function
must
reect
path
quality
and
feasibility
.
A
typical
tness
F
to
maximize
(or
cost
to
minimize)
is:
F
=
−
w
1
·
Length
+
w
2
·
ClearancePenalty
+
w
3
·
Smoothness
+
w
4
·
T
ime
Where:
Length
=
P
n
i
=2
p
(
x
i
−
x
i
−
1
)
2
+
(
y
i
−
y
i
−
1
)
2
;
Cl
earancePenalty
is
a
lar
ge
penalty
if
an
y
se
gment
intersects
obstacles
(or
in
v
ersely
proportional
to
the
minimal
obstacle
distance);
Smoothness
=
P
|
∆
θ
i
|
penalizes
sharp
turns
(helps
respect
nonholonomic
constraints);
T
ime
estimates
tra
v
ersal
time
gi
v
en
wheel
limits.
W
eights
w
i
are
chosen
according
to
priority
(e.g.,
safety
w
2
lar
ge).
–
Constraints
and
repair:
ensure
chromosomes
satisfy:
i)
W
orkspace
bounds:
x
min
≤
x
≤
x
max
,
y
min
≤
y
≤
y
max
.
ii)
Collision-free:
W
aypoints
and
connecting
se
gments
must
a
v
oid
obstacles.
iii)
Kinematic
feasi
bility:
Curv
ature
and
turning
radius
constraints
consistent
with
DDMR
dynamics.
Optionally
repair
infeasible
paths
by
smoothing
or
local
replanning.
–
GA
parameters
(recommended
starting
v
alues),
as
sho
wn
in
T
able
4.
T
able
4.
Suggested
GA
parameters
for
DDMR
path
planning
P
arameter
Sugges
ted
v
alue
Population
size
30–150
Crosso
v
er
probability
0.7–0.9
–
Inte
gration
with
NMPC
and
PSO
i)
P
ath
→
NMPC:
GA
produces
a
sequence
of
w
aypoints.
NMPC
then
tracks
the
smoothed
path
by
generating
wheel
commands
while
respecting
kinematic
constraints
and
obstacles
using
predicti
v
e
constraints.
Hybrid
contr
ol
str
ate
gy
for
tr
ajectory
tr
ac
king
and
obstacle
avoidance
in
...
(Abdennour
Ze
ghida)
Evaluation Warning : The document was created with Spire.PDF for Python.