Intern
ati
o
n
a
l Jo
urn
a
l
o
f
R
o
botics
a
nd Au
tom
a
tion
(I
JR
A)
V
o
l.
3, N
o
. 3
,
Sep
t
em
b
e
r
2014
, pp
. 15
1
~
16
0
I
S
SN
: 208
9-4
8
5
6
1
51
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJRA
Different Control Algorithms for a Platoon of
Autonomous Vehicles
Z
o
ran G
a
c
ovs
ki
*,
St
ojce
De
sko
v
ski
*
*
* Department of
Information
and Communication Techno
log
y
,
F
ON University
, Sk
opje, Macedonia
** Departmen
t
o
f
Engin
eering
,
University
“S
t. Kliment Ohridski”,
Bitola
,
Mac
e
do
nia
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Ja
n
6, 2014
Rev
i
sed
Jun
1
,
2
014
Accepted
Jun 20, 2014
This paper pres
ents a concep
t of
pl
atoon movement of autonomo
u
s vehicles
(s
m
a
rt cars
)
.
T
h
es
e veh
i
cl
es
h
a
ve Adap
tive
o
r
Advanced
cru
i
s
e
con
t
rol
(ACC) sy
stem
also call
e
d Inte
lligen
t cruise c
ontrol (ICC) or Adaptive
Intell
igen
t cruis
e
control (AICC)
s
y
s
t
em
. Th
e ve
hicl
es
are s
u
i
t
ab
le to fo
llow
other vehicles
on desired distance a
nd
to be organized in platoons. To
perform a research on the control a
nd stability of an AGV (
A
utomated
Guided Vehicles) string, we have de
velop
e
d a
car-following model.
To do
this, first a sing
le vehicle is modeled
and sin
ce
all cars in the p
l
atoon have
the s
a
m
e
d
y
n
a
m
i
cs
, the s
i
ngle
vehicl
e
model
is copied ten times to form
model of platoo
n (string) with ten vehi
cles
. To
control
this strin
g
, we hav
e
appli
e
d equa
l P
I
D controlle
rs
to all veh
i
cl
es
,
excep
t the l
ead
i
ng vehicl
e.
Thes
e con
t
roll
er
s
tr
y
to keep
th
e headwa
y
dis
t
a
n
ce as
cons
tant
as
pos
s
i
ble
and the v
e
locity error between
subseque
nt vehicles - small. Fo
r control of
vehicle with no
nlinear d
y
namics comb
ination
of
feedforw
ard control
and
feedba
ck con
t
ro
l approa
ch is
u
s
ed.
Feedforwar
d control
is based on the
inverse model of
nominal d
y
namics of
the
vehi
cl
e, and
fe
edback
P
I
D control
is designed based on the lin
ear
ized mode
l of the
vehicle. For simulation
an
d
analy
s
is of vehicle a
nd platoon of vehicles – we have develop
e
d
Matlab/Sim
u
link
m
odels. Sim
u
lation resu
lts, discu
ssions and conclusions are
given
at the end
of the pap
e
r.
Keyword:
Feedback Cont
rol
Fuzzy C
ontrol
Pl
at
oo
n
of
ve
hi
cl
es
Sm
art cars
String
stab
ility
Copyright ©
201
4 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Zo
r
a
n
G
a
co
vski,
Depa
rt
m
e
nt
of
In
fo
rm
at
i
on an
d C
o
m
m
uni
cati
on t
e
c
h
n
o
l
ogy
FO
N
Uni
v
ersit
y
- S
k
o
p
j
e,
Bu
l.
V
o
jv
od
ina, bb
, 100
0, Sko
p
j
e
, Macedonia
Em
a
il: g
aco
v
s
k
i
@m
t.n
e
t.
m
k
1.
INTRODUCTION
Gr
ou
pi
ng
ve
hi
cl
es i
n
t
o
pl
at
o
ons
i
s
a m
e
t
h
od
o
f
i
n
cr
easi
ng t
h
e ca
paci
t
y
of r
o
a
d
s.
A
n
aut
o
m
a
t
e
d
hi
g
h
way
sy
st
e
m
i
s
a pr
op
ose
d
t
ech
n
o
l
o
gy
f
o
r
d
o
i
n
g t
h
i
s
.
Pl
at
oo
ns
decre
a
se t
h
e
di
st
anc
e
s bet
w
e
e
n c
a
r
s
usi
n
g
electronic and
pos
sibly
m
echanical co
upling. T
h
is capabil
ity would allo
w m
a
ny cars to accelerate or bra
k
e
sim
u
ltaneously
. Instead
of
waiting a
f
ter
a traffic light
changes to
green
for dr
ive
r
s ahea
d to
react, a
sy
nch
r
o
n
i
zed
p
l
at
oon
w
o
ul
d
m
ove as o
n
e, a
l
l
o
wi
n
g
up
t
o
a
fi
ve
fol
d
i
n
c
r
ea
se i
n
t
r
a
ffi
c t
h
r
o
u
g
h
put
i
f
spa
c
i
ng
i
s
d
i
min
i
sh
ed
th
at
m
u
ch
. Th
is syste
m
a
l
so
allo
ws fo
r a
closer
headway between ve
hi
cles by eliminating reacting
di
st
ance nee
d
e
d
fo
r hum
an
re
act
i
on.
Sm
art cars with
artificial in
tellig
en
ce cou
l
d
au
toma
tically j
o
in
an
d
leav
e
p
l
ato
o
n
s
. Th
e
Au
t
o
m
a
ted
Hig
h
way
Sy
ste
m
(AHS
) is a pr
o
posal
for
one suc
h
system
,
where ca
rs orga
ni
ze t
h
em
sel
v
es i
n
t
o
pl
at
oo
ns
of
eig
h
t
to twen
ty-fiv
e
. Po
ten
tial b
e
n
e
fits fro
m
th
is AHS ar
e:
greater fuel ec
onom
y, reduce
d c
o
ngestion,
s
h
orter
co
mm
u
t
es d
u
r
i
n
g p
e
ak
p
e
riods, fewer traffic
co
llisio
n
s
, an
d
th
e
ab
ility
for v
e
h
i
cles
t
o
b
e
d
r
i
v
en
un
attend
ed.
The origi
n
of research
on
AHS was don
e b
y
a tea
m
fro
m
O
h
io
State Un
iversity led
b
y
R. E. Fen
t
on
.
Th
eir
first au
t
o
m
a
ted
v
e
h
i
cle was bu
ilt in 1
962
, an
d
is
b
e
liev
e
d
t
o
be th
e first land v
e
h
i
cle to
con
t
ain
a
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
56
IJR
A
V
o
l
.
3, N
o
. 3,
Se
pt
em
ber 20
1
4
:
15
1 – 16
0
15
2
co
m
p
u
t
er.
Steering
,
b
r
ak
ing and
sp
eed
were co
n
t
ro
lle
d
th
ro
ugh
th
e on
bo
ard electron
i
cs,
wh
ich
filled
t
h
e
t
r
u
n
k
,
ba
ck
sea
t
and
m
o
st
of t
h
e f
r
ont
of
t
h
e
passe
n
g
er si
d
e
of
t
h
e ca
r.
T
oday
– t
h
i
s
fi
el
d i
s
wi
del
y
ex
pl
o
r
ed
and i
m
pl
em
ent
e
d i
n
pract
i
ce.
SAR
T
R
E
i
s
a
Eur
o
pean C
o
m
m
i
ssi
on FP
7 c
o
-
f
u
n
d
ed
p
r
o
j
e
c
t
[1]
.
It
i
s
b
u
i
l
t
on
ex
istin
g
resu
lt
s an
d
exp
e
rien
ce and
an
al
yse th
e feasib
ility o
f
v
e
h
i
cle p
l
ato
o
n
s (con
sisting
o
f
b
o
t
h
truc
ks/busses
and pas
s
enger cars) as a
re
alistic future
t
r
an
spo
r
t an
d
m
o
b
ili
ty co
n
c
ep
t. SARTRE ai
m
s
to
exam
i
n
e t
h
e operat
i
o
n o
f
pl
a
t
oo
ns o
n
u
n
m
odi
fi
ed
pu
bl
i
c
m
o
t
o
rway
s wi
t
h
ful
l
i
n
t
e
ract
i
on
wi
t
h
ot
he
r
vehi
cl
es.
Crawford et al. [2
] ex
am
in
e th
e sen
s
o
r
y co
mb
in
ation
(G
PS, cam
eras, scann
e
rs) to fu
lfill th
e task of
fo
llowing.
Othe
r authors (Halle et al. [4]) cons
ide
r
the car platoons as
collaborativ
e m
u
lti-agent syste
m
. They propos
e a
hierarc
h
ical
a
r
chitecture base
d on
th
ree l
a
y
e
rs
(g
ui
da
nce l
a
y
e
r, m
a
nage
m
e
nt
l
a
y
e
r an
d
t
r
af
fi
c c
ont
r
o
l
l
a
y
e
r)
whi
c
h ca
n be
use
d
f
o
r si
m
u
l
a
t
i
ng a
cent
r
al
i
zed
pl
at
oo
n (
w
here a
h
ead
vehi
cl
e-a
g
ent
co
or
di
n
a
t
e
s ot
h
e
r
vehicle-a
g
e
n
ts by
applying
its
coordination rule) or
a d
ecentralized platoon (where
the
platoon is c
onsi
d
ere
d
as a team
of ve
hicle-age
n
ts tr
yin
g
to m
a
in
tai
n
th
e p
l
atoon
).
Thi
s
pa
pe
r i
s
or
ga
ni
zed as
f
o
l
l
o
w
s
. Sect
i
o
n 2
p
r
esent
s
d
e
ri
vi
n
g
of
dy
n
a
m
i
c vehi
cl
e m
odel
and i
t
s
l
i
n
eari
zat
i
on.
S
ect
i
on 3
i
s
res
e
rve
d
fo
r ve
hi
cl
e
pl
at
o
o
n
m
odel
i
n
g
an
d co
n
t
rol
.
Sect
i
o
n
4
di
scuss
e
s
si
m
u
l
a
t
i
on
resul
t
s
gi
ve
n u
s
i
ng M
a
t
l
a
b/
Si
m
u
li
nk m
odel
s
of t
h
e
ve
hi
cl
e and
pl
at
o
on
o
f
ve
hi
cl
es. Fi
n
a
l
l
y
, i
n
Sect
i
on 5
we
gi
ve
c
o
ncl
u
si
o
n
s
a
n
d di
rect
i
o
ns fo
r fut
u
re
w
o
r
k
.
2.
DYNAMIC VEHICLE MODEL
In
th
is section we p
r
esen
t
math
e
m
atica
l
m
o
d
e
l o
f
lo
ngitu
d
i
n
a
l m
o
tio
n
of th
e v
e
h
i
cle wh
ich
is
rel
e
va
nt
for
pl
at
oo
n m
odel
i
ng an
d cont
rol
.
For m
odel
i
ng i
n
t
h
i
s
case we’
v
e use
d
t
w
o co
or
di
nat
e
sy
st
em
s (see
Fi
gu
re
1):
ve
hi
cl
e-fi
x
e
d
or
bo
dy
-
f
i
x
e
d
c
o
o
r
di
nat
e
sy
st
em
,
B
(
C
;
x,z
), and Eart
h
-fi
xe
d
co
o
r
di
nat
e
s
y
st
em
,
E
(
O
;
x
o
,z
o
).
Vel
o
ci
t
y
of t
h
e
ve
hi
cl
e has c
o
m
pone
nt
s al
o
n
g
x
an
z
ax
e
s
,
i.
e.
[,
]
T
B
uv
V
. Figu
r
e
1
show
s fr
ee
bo
dy
di
ag
ram
of
a vehi
cl
e wi
t
h
m
a
ss
m
.
Vehicle is inclined
upon a
n
gle
with res
p
ect t
o
horizontal
plane
(sl
o
pe of
t
h
e ro
ad)
.
z
a
b
Gm
g
xr
F
xf
R
zr
F
z
x
o
x
o
z
xr
R
xf
F
zf
F
A
h
h
O
A
D
Fig
u
r
e
1
.
Fo
r
c
es actin
g on
a veh
i
cle
Fi
gu
re
2.
Si
m
u
l
i
nk m
odel
of t
h
e
vehi
cl
e
The
diagram
includes t
h
e si
gnifican
t forces
acting on
the
vehicle:
g
is t
h
e grav
itatio
n
a
l co
nstan
t;
D
A
is the aerody
na
m
i
c force;
G
=
mg
is the weight of the
vehicle;
F
x
is the tractive force;
R
x
is th
e r
o
llin
g-
resistance forc
e; and
ma
x
, an eq
u
i
v
a
len
t
in
ertial fo
rce, acts at
th
e cen
ter o
f
m
a
ss, C. Th
e su
b
s
crip
ts
f
and
r
refe
r to
the
fr
o
n
t (at B
)
a
n
d
re
ar (at
A
)
tire-
re
action
fo
rces,
r
e
spectively
.
Ap
plication
o
f
Newt
on
’s s
eco
nd
law
f
o
r t
h
e
x
and
z
di
rect
i
o
ns gi
ves
[
1
2]
:
sin
x
rx
f
x
r
x
f
A
mu
F
F
G
R
R
D
(1
)
0c
o
s
zf
z
r
mv
G
F
F
(2
)
The ae
ro
dy
na
m
i
c-drag
f
o
rce
depe
n
d
s
on t
h
e rel
a
t
i
v
e
vel
o
ci
t
y
bet
w
ee
n
t
h
e ve
hi
cl
e and t
h
e s
u
r
-
ro
u
ndi
ng
ai
r a
n
d i
s
gi
ve
n
by
t
h
e sem
i
-em
p
i
r
i
cal
rel
a
t
i
ons
hi
p
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA I
S
SN
:
208
9-4
8
5
6
Differen
t
Con
t
ro
l Algo
rithms
fo
r
a
Pla
t
o
o
n
of Au
tono
mou
s
Veh
i
cles (Zo
r
an
Ga
co
vski)
15
3
22
11
()
()
22
Ad
f
w
a
i
r
w
DC
A
u
u
C
u
u
(3
)
whe
r
e
ρ
i
s
t
h
e ai
r densi
t
y
(= 1.
20
2 k
g
/
m
3
at an al
t
i
t
ude
of
20
0 m
)
,
C
d
is the drag c
o
efficient,
A
f
is th
e fron
tal
area of the
vehicle,
u
is the ve
hicle-forward
velocity, and
u
w
is th
e win
d
velo
city (i.e.,positiv
e for a h
e
ad
wi
nd
and
ne
gat
i
v
e
f
o
r a
t
a
i
l
w
i
n
d
)
.
The
dra
g
c
o
e
ffi
ci
ent
f
o
r
ve
hi
cl
es ra
nges
fr
om
about
0.
2 (i
.e
., st
ream
l
i
n
ed
passe
nge
r
ve
hi
cl
es wi
t
h
u
nde
r
b
o
d
y
c
ove
r)
t
o
1.
5
(i
.e.,
t
r
uc
ks
);
0
.
4
i
s
a t
y
pi
c
a
l
val
u
e
f
o
r
pas
s
en
ger ca
rs
[
1
2
]
.
The rolling
res
i
stance arises due to
t
h
e
def
o
r
m
at
i
on on t
h
e t
i
r
e and t
h
e r
o
a
d
su
rface
, an
d i
t
i
s
rou
ghl
y
p
r
op
ortio
n
a
l t
o
th
e
n
o
rm
al fo
rce on
th
e tire:
()
c
o
s
x
x
f
x
r
r
zf
zr
r
RR
R
f
F
F
f
m
g
(4
)
whe
r
e
f
r
is th
e
ro
lling
-
resistance co
efficien
t in
th
e ran
g
e
o
f
ab
ou
t 0
.
01
to
0.4
,
with
0
.
01
5
as a typ
i
cal v
a
l
u
e fo
r
passe
nge
r ve
hi
cl
es.
For
f
u
rt
her c
o
nsi
d
e
r
at
i
on
we
use e
quat
i
o
n
(1
). E
q
uat
i
on
(1
) i
s
n
o
n
l
i
n
ea
r i
n
t
h
e
fo
r
w
a
r
d
vel
o
ci
t
y
,
()
ut
but
ot
her
w
i
s
e
i
s
a sim
p
l
e
dy
nam
i
c sy
st
em
:
i
t
onl
y
has one st
at
e vari
abl
e
. So
, w
h
at
are t
h
e
m
a
i
n
ch
allen
g
es in
cru
i
se-con
tro
l
d
e
sign
p
r
ob
lem
s
?
Th
e
d
i
ffi
cu
lties arise
main
ly fro
m
t
w
o fact
o
r
s:
(1) p
l
an
t
unce
r
t
a
i
n
t
y
d
u
e
t
o
cha
n
ge o
f
vehi
cl
e
wei
g
h
t
, and
(
2
) e
x
t
e
rnal
di
st
ur
ba
nc
es d
u
e t
o
roa
d
gra
d
e.
Th
us,
a go
o
d
cr
u
i
se-
c
on
tro
l
alg
o
r
ith
m
m
u
st
wo
rk
w
e
ll under
th
ese un
cer
t
ain
ties.
Equ
a
tio
n (1
),
usin
g (3
) and
(4) can
b
e
rewritten
:
2
1
sin
c
o
s
(
)
2
xr
a
i
r
w
mu
F
m
g
f
mg
C
u
u
(5
)
whe
r
e
ai
r
r
d
CA
C
is a cons
tant.
Usin
g (
5
) we c
r
eate n
onlinea
r
SIM
U
LI
NK
m
odel for
ve
hi
cles
in
the platoon, Figure 2. For
analysi
s
of dy
nam
i
cs a
n
d stability of
the vehi
cle and stri
ng stability of t
h
e
platoo
n we need li
nearized m
odel
of the
vehicle.
Linearization of (5) aroun
d
the specified operating
(i.e.,equilibri
um
) state is
made
using a Taylor
series e
xpa
nsio
n.
Va
riables a
n
d
fu
nctio
ns in
the e
quatio
n
(
5
)
are
pre
s
ente
d i
n
fo
rm
00
0
0
;
;
xx
uu
u
F
F
F
(6
)
whe
r
e
0
u
is
the nom
inal velocity of the vehicl
e,
0
x
F
is the nom
inal tractive force, and
0
is
the nom
inal
slope of the road. Substitu
ting (6) in
(5)
and perfor
m
i
n
g
m
a
the
m
atica
l
operatio
ns, using approximations
sin
,
c
o
s
1
, and
neglecting the sm
all quantities like
2
0
u
,
we obtain two equations:
02
1
si
n
c
os
(
)
2
oo
o
o
xra
i
r
w
mu
F
m
g
f
mg
C
u
u
(7
)
()
o
air
w
x
mu
C
u
u
u
F
d
(8
)
00
( s
i
n
-
c
o
s
)
r
dm
g
f
m
g
(9
)
whe
r
e
d
is
the
distur
ba
nce. Eq
uation (
7
) d
e
scribes n
o
m
i
n
a
l
m
o
tion o
f
t
h
e ve
hicle an
d
it has the sam
e
fo
rm
like (
5
),
an
d
(
8
)
descri
bes
pert
ur
be
d m
o
tion a
r
o
u
n
d
n
o
m
i
nal trajecto
r
y
.
If nom
i
nal
velocity
o
u
is c
onsta
nt the
n
fr
om
(7
)
we ca
n
fin
d
nom
i
nal tractive forc
e wh
ic
h is n
e
ed
ed
fo
r m
ovem
e
nt near
to
n
o
m
i
nal state:
02
1
si
n
c
o
s
(
)
2
oo
o
xr
a
i
r
w
Fm
g
f
m
g
C
u
u
(1
0)
Linearize
d
eq
uation (
9
) is of fir
s
t or
der
in whic
h
u
is state–velocity
pertur
batio
n an
d
x
F
is
pert
urbation of the tractive
force and
we ca
n use it for st
abi
lization an
d c
o
ntr
o
l o
f
the
ve
hicle by
o
b
tain
ing it
from
suitable linear controller.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
089
-48
56
IJR
A
V
o
l. 3, N
o
. 3,
Se
ptem
ber 20
1
4
:
15
1 – 16
0
15
4
Takin
g
pe
rtu
r
bation i
n
p
o
si
tion,
x
, using (8) we ca
n write next
state-s
p
ace equation for the
vehicle:
11
1
x
xu
uu
F
d
Km
m
m
(1
1)
whe
r
e
0
1/
[
(
)
]
air
w
K
Cu
u
.
In
vect
or
-m
atrix
fo
rm
(11
)
be
com
e
s:
01
0
0
11
1
0
x
xx
Fd
uu
Km
m
m
(1
2)
We ca
n
fin
d
th
e trans
f
er
f
u
nction
o
f
the
ve
hi
cle fr
om
(11
)
:
()
x
uu
K
F
d
,
(1
3)
whe
r
e
0
/[
(
)
]
ai
r
w
K
mm
C
u
u
is ti
m
e
c
onstant.
Applying La
place trans
f
orm
a
tion to
(13), a
n
d ne
glecting
disturba
nce
d, we ca
n
com
pute tra
n
sfe
r
fu
nctio
n:
()
()
()
1
x
us
K
Gs
F
ss
(1
4)
Using t
h
e
num
erical value
s
:
00
3
2
20
m
/
s,
0
,
1
000
kg
,
1
.2 kg
/
m
,
1
.
2
m
,
0.5
,
0.01
,
9
.
8
1
m
/
s,
0
fd
r
w
um
AC
f
g
u
(1
5)
we can
co
m
p
u
t
e th
e
p
a
r
a
m
e
te
r
s
in abov
e eq
uatio
n
s
(
10)
,
(
11)
,
(
12)
an
d (1
3):
24
2.
1
N
,
0.
06
94
(
m
/s
)
/
N
,
=
6
9.
4
4
s
,
o
x
FK
(1
6)
01
0
0
0
0
.0
144
0.
001
0.0
0
1
x
xx
Fd
uu
(1
7)
(
)
0.06
94
()
()
6
9
.
4
4
1
x
us
Gs
F
ss
(1
8)
3.
VEHICLE CONTROL SYSTEM
In case
s
whe
n
the real ve
hicle
is with n
o
n
linear
dy
nam
i
cs (in
ou
r case e
quatio
n
(5
) f
o
r
lon
g
itudi
nal
dy
nam
i
cs) it is very
use
f
ul t
o
im
plem
ent com
b
ination o
f
f
eed-
f
o
r
war
d
c
o
ntr
o
l an
d
feed
b
ack c
ont
rol a
p
pr
oac
h
,
prese
n
ted
o
n
Figu
re
3.
T
h
e
feed
-f
or
wa
rd
c
ont
rol is
f
o
rm
ed
on
the
inve
rse model
of t
h
e
object a
n
d
on the
gene
rato
r o
f
n
o
mi
nal traject
ories
which
generates the
desired traj
ectory
)
(
t
o
x
. This desired traj
ect
ory is
base
d o
n
the
pre
v
io
usly
p
r
e
p
are
d
data o
r
fr
om
the sy
stem
operatio
n
base
d o
n
the
m
easured
dat
a
. Fo
r
realization
of
this traj
ect
ory
it is necessary
that re
gulator in feedback i
s
pr
esent
,
whi
c
h
will generate
the
n
e
e
d
ed
con
t
ro
l
)
(
t
u
for elim
ination traj
ectory error
of the obj
ect from
the desi
re
d traj
ectory.
Th
is provides
stabilisation of the cont
rol
process
of the
obj
ect. The sum control
u
(t)
of the m
ovin
g
ob
ject f
r
om
Figu
re 3
,
when t
h
e linear regulator is
form
ed by t
h
e
m
a
trix
K
(
t
),
is given with
the following relation:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA I
S
SN:
208
9-4
8
5
6
Different Cont
rol Algorithms
for
a Plat
oon
of Autonomous
Vehicl
es (Zoran
Gacovski)
15
5
()
()
()
(
)
(
)
(
)
()
(
)
[
(
)
(
)
]
oo
oo
tt
tt
t
t
tt
t
t
uu
uu
K
x
uK
x
x
(1
9)
The sy
ntesis o
f
the c
ont
rol la
w
give
n
by
eq
uation
(
1
9)
is
perform
e
d in t
w
o steps.
In the
first step the
nom
inal co
ntr
o
l
)
(
t
o
u
is determ
ined
under assum
p
ti
on
of ideal conditions i
.
e.
when no
disturbances are
prese
n
t.
)
(
t
u
)
(
t
o
x
)
(
t
u
)
(
t
x
)
(
t
x
)
(
t
x
)
(
t
o
u
)
(
t
o
x
Figu
re
3.
C
o
nc
ept o
f
feed
-f
o
r
war
d
a
n
d fee
d
back
cont
rol system
of nonlinear
object
Figu
re 4.
Sim
u
link diag
ra
m
for the
ve
hicle cont
rol
Acco
r
d
in
g to t
h
e d
e
scri
bed c
once
p
t (
F
ig
ure
3)
, the c
o
ntr
o
l laws f
o
r
ve
hicles can
be de
velo
ped
.
I
n
this pape
r fee
d
-f
or
war
d
c
ontr
o
l is determ
ined base
d o
n
th
e (
7
),
i.
e.
(10
)
fo
r
n
o
m
in
al tractive force which is
a
nom
inal co
ntro
l.
Feedback cont
roller, whic
h provides stabilization
of
the
object around the
nom
inal trajec
tory, c
a
n
be
desig
n
e
d
usin
g
linearized
m
odel.
Un
de
r ass
u
m
p
tion
that t
h
e
dy
nam
i
c behavi
or
of
th
e ob
j
ect
with respect
to
the n
o
m
i
nal trajecto
r
y
is line
a
r, as
des
c
ribe
d wit
h
(
8
)
,
or
(1
2)
to
(1
4)
, f
o
r t
h
e c
ontr
o
l
)
(
t
u
, we ca
n a
p
ply
m
e
thods f
o
r s
y
nthesis de
velope
d f
o
r linea
r sy
stem
s: PID co
ntroller d
e
sign
, Linear
Qua
d
ratic R
e
gulato
r
(LQR
),
m
e
tho
d
s fo
r pole
pla
c
em
ent,
ada
p
tive optim
al
co
ntrol etc.[3].
In t
h
is pa
pe
r P
I
D c
o
ntr
o
l desi
gn
ap
pr
oac
h
is
us
e
d
a
nd PI
D feed
bac
k
c
ontr
o
ller
is obtaine
d based
o
n
the linear
m
odel of t
h
e
vehic
l
e deri
ved
ab
o
v
e
with
para
m
e
ters dete
rm
in
ed
usin
g
num
erical values
(
1
5)
. F
o
r
sim
u
lation an
d
testing o
f
ve
hi
cle dy
na
m
i
cs and
ve
hicle con
t
rol sy
stem
Sim
u
link m
odel is devel
ope
d w
h
ich is
sho
w
n on
Fig
u
r
e 4.
Module refe
re
nce inputs,
ge
nerate refe
rence
acceleration
a
o
, velocity
v
o
, and
po
sition
x
o
, si
m
ilar
l
i
ke
the leade
r
of the platoon. T
h
es
e signals go t
o
the PID controller where a
r
e
pr
ocesse
d acc
o
r
di
ng
to:
()
()
(
)
I
xp
o
o
D
o
K
uF
K
x
x
x
x
K
v
v
s
(2
0)
whe
r
e
,,
a
n
d
pI
D
K
KK
are
proportional, inte
gral a
n
d
de
riv
a
tive
gain
s o
f
the co
ntr
o
ller,
a
,
v
and
x
are real
acceleration, velocity and position of t
h
e
vehicle.
Mo
du
le
Nomina
l con
t
ro
l
,
Fig
u
re
4,
co
nsists
of e
q
uation
(
1
0
)
an
d m
o
d
u
le
Vehicle dy
namics,
wh
ich
is
base
d
on
f
u
ll n
onlinea
r m
odel
,
eq
uatio
n
(5
).
Sim
u
link m
o
d
e
l in Fig
u
re
4
can
be
used
f
o
r o
p
e
n
lo
o
p
, a
n
d close
d
l
o
o
p
s
i
m
u
lation o
f
th
e co
ntrolle
d
vehicle.
(i.e.
,
i
t
s ow
n m
o
tion
an
d
head
way
to the
ve
hi
cle in f
r
ont)
.
In
this pa
pe
r
we
discuss
the
ve
hicle-
following control approach,
whic
h is t
h
e focus
of m
o
st c
u
rrent
rese
a
r
ch and
devel
opm
ent work in t
h
e area
[1
2]
.
We o
b
se
rve t
h
e
m
ovem
e
nt of ve
hicles in th
e inertial (or
absolute
) c
o
o
r
dinate sy
stem
(;
,
)
oo
GO
x
y
which is fixed to the road wi
th
ori
g
in in the starting point,
O.
Positions,
i
x
, veloci
ties,
ii
vx
, and accele
r
a-
tions,
ii
av
,
,1
,
2
,
3
,
4
iL
,
m
easure
d
with res
p
ect to
(;
,
)
oo
GO
x
y
, are absolute
quantities.
Coordinate syste
m
(;
,
)
L
L
L
Lx
y
is fixed to the
vehicle-leader
with origin i
n
the cen
ter
of its
m
a
ss. Relative
position,
velocity and
acceleration of the
ve
hicles
with res
p
ect
to
(;
,
)
L
L
L
Lx
y
are
de
n
o
ted
as:
iL
i
lx
x
,
ri
L
i
vv
v
,
ri
L
i
aa
a
,
1,
2
,
3
,
4
i
respec
tively. Distances
between
vehicle
s
are
de
note
d
a
s
1
,,
1
,
2
,
3
,
4
ii
i
dx
x
x
i
L
, and
relative velocities and a
ccelerations
of
the ve
hicles wi
th respect to
ve
hicle in
front
of them
are re
spectively:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
089
-48
56
IJR
A
V
o
l. 3, N
o
. 3,
Se
ptem
ber 20
1
4
:
15
1 – 16
0
15
6
11
11
,
,,
1
,
2
,
3
,
4
.
ii
i
i
i
ii
i
i
i
dv
v
v
x
x
da
a
a
x
x
i
L
.
The m
a
in Sim
u
link
diagram
of
our m
odel is shown on Fi
gure 5. In this
m
odel each
ve
hicle gets inform
ation
about accelerat
ion, vel
o
city an
d position of t
h
e
pre
v
ious
ve
hicle, and al
so
gets the
sam
e
i
n
form
ation about the
vehicle-lea
d
er.
Figu
re
5.
M
a
tlab/Sim
u
link m
odel
o
f
the
plat
oo
n
o
f
10
ve
hi
cles.
Usin
g vehicle m
odel
(5)
,
o
r
Figu
re 2,
if
0
and
0
w
V
, we ca
n fi
nd accele
r
ation
of t
h
e ve
hicle
in this
fo
rm
:
2
0
11
()
,
2
xr
a
i
r
xx
x
ua
F
f
m
g
C
u
m
FF
F
(2
1)
Co
n
t
ro
l for
ce
x
F
is determ
ined
by a PID cont
roller, i.e.
with
equation (20).
Substituting (20) in
(21) we
can fi
nd acceleratio
n
for the
i-th ve
hicle:
11
2
10
1
[(
)
(
)
1
(
)
)],
2
Ii
ip
i
i
i
i
i
i
i
Di
i
i
x
r
a
i
r
i
K
a
K
xx
h
d
xx
h
d
ms
Kv
v
F
f
m
g
C
u
(2
2)
whe
r
e
i
hd
is consta
nt dista
n
ce bet
w
een
i
-1
-th
an
d
i
-th
ve
hicle.
Deri
vin
g
(
2
1)
we
ca
n
get je
rk
w
h
ich acts
o
n
th
e
i
-th vehicle (
0
an
d
xr
F
fm
g
are co
nstant)
,
a
n
d
usin
g
relatio
ns:
ii
x
v
(2
3)
ii
va
(2
4)
we ca
n
find:
]
)
(
)
(
)
(
[
1
1
1
1
i
o
air
i
i
Di
i
i
Pi
i
i
i
Ii
i
a
u
C
a
a
K
v
v
K
hd
x
x
K
m
a
(2
5)
Equations
(23), (24) a
n
d (25) repr
e
s
ent line
a
r state-space
m
odel of the
i
-th ve
hicle in the plato
o
n
.
Varia
b
les
11
1
,,
a
n
d
ii
i
i
x
va
a
in equatio
n (
2
5
)
ar
e input va
riabl
e
s for the
i
-th
vehicle and they are position,
velocity and ac
celerati
on
o
f
th
e p
r
evi
ous
,
or
i
-1
-th,
ve
hicle.
Equations
(23), (24) a
n
d (25) can
be use
d
for obtaining
t
h
e
state
space m
odel
of
stri
ng of seve
ral
vehicles. This m
odel is
useful for stability
analysis
of the string usi
ng t
echni
que
s of linear control theory.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA I
S
SN:
208
9-4
8
5
6
Different Cont
rol Algorithms
for
a Plat
oon
of Autonomous
Vehicl
es (Zoran
Gacovski)
15
7
Here
we f
o
rm
m
odel of strin
g
with t
h
ree
ve
hicles: ve
hicle-
leader, a
n
d tw
o ve
hicles-
f
oll
o
we
rs.
O
u
tp
uts of th
e
vehicle-lea
d
er
gene
rate inp
u
t varia
b
les,
,,
a
n
d
L
LL
x
va
, fo
r the first ve
hicle in the
string
. Othe
r two
vehi
cles ar
e
descri
bed
with
these e
quati
ons
-
fo
r t
h
e fi
rst v
e
hicle:
11
11
11
1
1
1
1
11
1
1
[(
)
(
)
()
]
,
IL
p
L
o
DL
a
i
r
xv
va
aK
x
x
h
d
K
v
v
m
Ka
a
C
u
a
(2
6)
and for the
sec
o
nd
ve
hicle:
22
22
22
1
2
2
2
1
2
21
2
2
1
[(
)
(
)
()
]
Ip
o
Da
i
r
xv
va
aK
x
x
h
d
K
v
v
m
Ka
a
C
u
a
(2
7)
Now we form
state
vector:
12
1
2
1
2
[
]
T
x
dx
v
v
a
a
x
where
state
2
dx
is
the distance
betwee
n
the
first a
n
d the se
cond
ve
hicle, a
n
d
21
2
1
2
2
dx
x
x
v
v
d
v
(2
8)
Now
we ca
n
write state-space
equation of
t
h
e strin
g
in
ve
ctor
-m
atrix (2
9):
1
1
2
2
1
1
2
2
1
11
1
1
2
2
2
22
2
2
11
00
1
0
0
0
00
1
1
0
0
00
0
0
1
0
00
0
0
0
1
00
0
0
00
0
00
0
00
0
00
0
o
Da
i
r
IP
o
Da
i
r
IP
P
D
IP
xx
dx
dx
vv
vv
KC
u
KK
aa
mm
m
aa
KC
u
KK
K
K
mm
m
m
m
KK
m
1
2
1
1
2
00
00
00
00
0
00
0
0
L
L
I
L
D
I
x
hd
v
hd
K
a
K
m
mm
K
m
(2
9)
If we
c
h
oo
se out
puts
as -
distance betwee
n vehicles,
2
dx
, and
velocities
1
v
and
2
v
, we ca
n
f
o
r
m
out
put vect
or,
21
2
[]
T
dx
v
v
y
, as:
2
1
2
0
1
00
00
000
00
1
0
00
000
00
0
1
00
000
L
L
L
dx
x
vv
va
yx
(3
0)
Stability analysis of the indi
vidual
vehicle and
platoo
n of vehicles can be made in Matla
b using their
linear m
odels and c
o
m
puting
poles
of t
h
e s
y
stem
or fi
n
d
ing
gain a
n
d p
h
ase m
a
rgin
s
with
h
e
lp
of
Nyqu
ist
plot.
Fo
r e
x
am
ple, f
o
r stri
ng
of t
w
o
ve
hicles-f
ollo
wers
, u
s
ing m
odel (
2
9)
and
p
a
r
a
m
e
ter
s
(1
5)
,
we can f
i
nd
eigen
v
alues
o
r
poles
,
p
1
,…,
p
6 :
-1.2690
,-1.2690
, -0
.5306, -0.530
6, -0
.0149, -0.01
49
whic
h a
r
e real
and ne
gative
,
a
n
d system
is stable.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
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-48
56
IJR
A
V
o
l. 3, N
o
. 3,
Se
ptem
ber 20
1
4
:
15
1 – 16
0
15
8
For
a platoon of vehicles
,
besides individual vehicl
e stabilit
y, it
is defined
stri
ng st
ability
of the
platoon [17,
18]. If t
h
e
prec
eding
ve
hicle is accelerating or
deceler
ating, t
h
e
n
the
s
p
acing error c
o
uld
be
no
nze
r
o;
we m
u
st ensure that the
spacing error attenuates as it propag
ates al
ong
the string of
vehicles
because it
propagates
upstre
a
m
towa
rd t
h
e
tail of the
string. Ta
king
(23) in
Laplace
dom
ain, a
n
d using
transfer function
1
/
ii
i
Gv
v
an
d relation fo
r ran
g
e
e
r
r
o
r
betwee
n
i-
th
an
d
i-
1th ve
hicles:
11
1
,(
)
,
ii
i
i
i
i
i
i
i
x
vv
G
s
v
x
x
D
s
(3
1)
whe
r
e
ii
i
Dh
v
den
o
tes
the desired ra
nge f
o
r the
i
th vehicle,
i
h
is a
c
onstant ti
m
e
-
h
ead
way po
licy ad
op
ted
for all vehicles, we can
f
i
nd
tran
sf
er
fu
n
c
tion
,
ik
G
from
the rang
e erro
r of
i
-t
h v
e
hicle to the range er
r
o
r o
f
i
+
k
th ve
hicle:
,1
1
1
()
1
ik
ik
ik
ik
ik
i
i
i
k
ii
i
i
Gs
h
G
Gs
G
G
G
Gs
h
G
(3
2)
For stri
ng stabi
lity
m
u
st be satisfied
[13]:
,
,(
)
1
ik
i
i
k
or
G
s
(3
3)
This
discussi
on for stri
ng
stability can be
easily a
pplied to
platoon describe
d in t
h
is
work.
In t
h
e
next
section
w
e
p
r
esent
som
e
res
u
lts f
o
r
plat
oo
n m
ovem
e
nt.
4.
R
E
SU
LTS AN
D ANA
LY
SIS
We h
a
v
e
sim
u
lated
a p
l
atoon
with
10
vehicles. A
ll vehi
cles are the sa
m
e
with para
m
e
ters (15),
desire
d
distanc
e
s am
ong
vehi
cles are d
x
i
0
=
5
0 m
.
Param
e
ters o
f
P
I
D c
ont
rollers a
r
e:
K
Pi
=700,
K
Ii
=10, and
K
Di
=1800. Ve
hicle-leade
r
ge
nerates accele
r
ation,
velo
city
and
position
whic
h a
r
e shown in t
h
e
pictures
belo
w.
Fig
u
r
e
6 s
h
o
w
s
vel
o
city
pr
ofile o
f
t
h
e ve
hi
cle leade
r
a
n
d res
p
onse
s
of
vehicles
–
followers.
Figu
re
6.
Tra
p
ezoidal c
h
a
nge
o
f
vehicle-lea
d
er
vel
o
city
and
res
p
on
ses
of
ve
hicles in
the
platoo
n.
Figu
re
7.
P
o
sitions
o
f
t
h
e
vehi
cles.
Figu
re 8 sh
o
w
s distance err
o
rs betwee
n ve
h
i
cles for
the sa
m
e
inputs as in Fig
u
re 6
.
Fig
u
re 9 s
h
o
w
s
positions of the vehicles in the platoon when each
ve
hicl
e gets inform
ation for accele
r
ation, velocit
y
and
position only from
previous
vehicle. Figure 7 shows the situation
when only last three vehicles get
inform
ation for acceleration, velocity and position from
th
e vehicle-leade
r
. In this s
ituation errors in positions
between
vehicles are sm
aller.
It is
known i
n
the literature t
h
at the info
rmation for ve
hicle-leader m
ovement
and inter-vehicle comm
unicat
ion i
n
fl
uence to
bette
r cont
rol
and stri
ng st
abilit
y of t
h
e
platoon.
0
10
20
30
40
50
60
70
80
90
10
0
18
20
22
24
26
28
t [
s
]
V
e
h
i
c
l
e
v
e
l
o
c
i
ti
e
s
, v
i
[m
/s
]
Ab
so
l
u
t
e
v
e
h
i
cl
e
v
e
l
o
ci
ti
e
s
V
e
hi
c
l
e
-
Le
ad
er
Ve
h
i
c
l
e
9
0
10
20
30
40
50
-500
0
500
1000
t [
s
]
V
ehi
c
l
e pos
i
t
i
ons
,
x
i
[
m
]
A
b
so
l
u
t
e
v
e
h
i
cl
e
p
o
si
t
i
o
n
s
V
e
hi
c
l
e-
L
eader
V
ehi
c
l
e 1
Ve
h
i
c
l
e
9
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RA I
S
SN:
208
9-4
8
5
6
Different Cont
rol Algorithms
for
a Plat
oon
of Autonomous
Vehicl
es (Zoran
Gacovski)
15
9
Figu
re
8.
Dista
n
ce e
r
r
o
rs
bet
w
een
ve
hicles
Figu
re
9.
P
o
sitions
o
f
t
h
e
vehi
cles
5.
CO
NCL
USI
O
N
In t
h
is pa
per
w
e
have
de
velo
p
e
d a n
o
n
linear
and line
a
rized
m
odel of the l
o
n
g
itu
dinal m
o
tion o
f
th
e
vehicle.
Fee
d
-
f
o
r
wa
r
d
c
ont
ro
l an
d
feed
bac
k
P
I
D c
ont
ro
l
ap
pr
oac
h
is
a
pplied
to
de
sign
v
e
hicle c
o
ntr
o
ller.
Using this
vehicle
m
odel with its de
si
g
n
ated c
ontr
o
l sy
ste
m
– we’ve
de
v
e
lope
d a m
ode
l of
plato
o
n
wi
th ten
vehicles. In t
h
is m
odel, ve
hicles can
get i
n
form
at
ion for acceleration, velocity
and position from
previous
vehicle and from
m
ove
m
e
nt
of the
vehi
cle –leader. String stability of the
platoon is discussed and
transfer
fu
nctio
n o
f
t
h
e stri
ng
use
f
ul
fo
r stabili
ty
analy
s
is
is pre
s
ente
d. B
a
sed
on
the
devel
ope
d m
odels,
M
a
tlab/Sim
u
link m
odels are
created w
h
ic
h can
be use
d
fo
r sim
u
lation an
d pe
rf
o
r
m
a
nce analy
s
is
of the
vehicle
dy
nam
i
cs an
d plat
oo
n
’
s c
ontr
o
l sy
ste
m
. This Sim
u
link
m
odels can
be
usef
ul
f
o
r
diffe
rent e
x
peri
m
e
nts
and testing of designed controllers
. Sim
u
lation res
u
lts g
i
ven in t
h
e p
r
evio
us sectio
n
sho
w
ho
w
ve
hicles
beha
ve i
n
th
e
p
l
atoon
u
n
d
er
gi
ven
co
n
d
itions
with P
I
D
c
ontr
o
llers a
pplie
d.
In fut
u
re work,
we plan
to de
ve
lop m
o
re accurate m
odels of t
h
e ve
hicles and
platoons.
We pla
n
t
o
desig
n
a
n
d te
st diff
ere
n
t-th
en P
I
D control laws
, for e
x
am
ple LQR
an
d Fuzzy logic control.
Practical
realization
using
differe
n
t se
nsors a
n
d wi
re
less com
m
unication am
ong
vehicles will be
our inte
rest in the
fut
u
re.
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NC
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Chan E
.
,
P
.
Gilh
ead,
P
.
J
e
linek
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.
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č
í; “SARTRE
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ansport S
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em
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[2]
Crawford S.A., “Performance evalu
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[9]
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n
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is
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-
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0
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40
60
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100
-1
-0
.
5
0
0.
5
1
1.
5
2
t [s
]
D
i
s
t
anc
e
e
r
r
o
r
s
bet
w
e
en v
ehi
c
l
e
s
,
[
m
]
D
i
s
t
anc
e er
r
o
r
s
b
e
t
w
e
en v
ehi
c
l
es
V
e
hi
c
l
e 9
Ve
h
i
c
l
e
1
0
10
20
30
40
50
-5
00
0
50
0
10
00
t [s
]
Ve
h
i
cl
e
p
o
s
i
ti
o
n
s
,
xi
[m
]
A
b
so
l
u
t
e
v
e
h
i
cl
e
p
o
si
t
i
o
n
s
V
e
hi
c
l
e 1
V
e
hi
c
l
e 9
V
e
hi
c
l
e - Le
ad
e
r
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
089
-48
56
IJR
A
V
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BIOGRAP
HI
ES OF
AUTH
ORS
Dr. Zoran Gaco
vski is a professor in Comput
er engineering at FON –
University
, Skopje,
Macedonia. His
teaching subjects
and his ar
eas of resear
ch
are: fu
zzy
s
y
stems, intelligent
control, mobile robots, graph
i
cal models (Pet
ri, Neur
al
and
Bay
e
sian n
e
two
r
ks), machine
learn
i
ng,
and h
u
man-computer inter
action. H
e
has earn
e
d his Ph.D. degree at Facu
lty
of
Electrical
engin
eering
,
Skopje. In his caree
r
he was awarded the Fulbright postdoctoral
fellowship (2002) for research stay
at Rutgers Un
iversity
, USA. He has also earned best-paper
award at the Baltic Ol
y
m
p
i
ad f
o
r Autom
a
tion control (2002), US NSF grant for
conducting
a
specific research
in the f
i
eld of h
u
man-computer
inter
action at Ru
tgers University
,
USA (2003),
and DAAD
grant for research stay
at University
o
f
Bremen, German
y
(2008)
. He took an active
participation in several FP7 projects. He is
an
author of 3 books, 7 journal papers, over 30
Conferenc
e
pape
rs, and was also a review
er for
IEEE journ
a
ls an
d conferen
ces.
He is the edito
r
of “Mobile Rob
o
ts: Curren
t
tr
en
ds” (Intech pub
lishing, 2010)
.
Dr. Stojce Deskovski is a profes
sor in Control S
y
st
ems, at th
e Faculty
of
Engin
e
ering, University
“
S
t. Klim
ent Ohrids
ki”, B
itol
a
.
He has
accom
p
li
s
h
ed his
m
a
s
t
ers
and P
h
D degree
s
at F
acult
y of
Ele
c
tri
cal
Eng
i
n
eering
-
Zagreb
.
His
te
aching
an
d res
ear
ch
inte
re
s
t
s
includ
e Anti-
Tank
and Air
Defens
e M
i
s
s
ile
S
y
s
t
em
s
,
Guida
n
ce and Con
t
rol
and S
y
nth
e
s
i
s
of Guidance S
y
s
t
em
s
.
He was
professor in thes
e ar
eas at Alger
i
an Milita
r
y
Academ
y
(1979-198
0), and
the Militar
y
Acad
em
y
Zagreb
(1981-19
91). From 1992
to 1995 h
e
has
estab
lished
and h
e
was elected
as
the first Dean
of Macedoni
an
Militar
y
A
cad
em
y
(1995). At
t
h
e
Mili
tar
y
Academ
y
- Skopj
e
he has dev
e
loped
the ar
eas: Ex
tern
al bal
listi
cs,
Mechanics of flight
,
Guidance and
Control of missile s
y
stems
.
For
his contributions
in the field of aeronautics (and
missile techniqu
e) in 2000 he was elected as a
m
e
m
b
er of the Am
erican Ins
tit
ute of aeron
auti
cs
and as
tronau
tics
– AIAA. He is
an activ
e
member of ETAI – Macedonian Engin
eer
ing
Society
.
He h
a
s published 2
books (Extern
a
l
Ballist
i
cs, 2000
, and Hom
i
ng-guided m
i
ssiles
for
anti-
air
def
e
nse, 2004)
an
d he has
also
published ov
er 5
0
papers
at inter
n
ation
a
l Conf
ere
n
ces
and
J
ournal
s
.
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