Inter
national
J
our
nal
of
Robotics
and
A
utomation
(IJRA)
V
ol.
10,
No.
4,
December
2021,
pp.
319
∼
325
ISSN:
2089-4856,
DOI:
10.11591/ijra.v10i4.pp319-325
❒
319
The
taxi
function
Alicia
C.
S
´
anchez
Department
of
Mathematics,
F
aculty
of
Science,
Uni
v
ersity
of
Extremadura,
Badajoz,
Spain
Article
Inf
o
Article
history:
Recei
v
ed
Feb
3,
2021
Re
vised
May
16,
2021
Accepted
Jul
23,
2021
K
eyw
ords:
Automatic
na
vig
ation
systems
Lane
k
eeping
control
Lateral
control
RAFU
functions
ABSTRA
CT
This
paper
in
v
estig
ates
the
lane
k
eeping
control
and
the
lateral
control
of
autonomous
ground
v
ehicles,
robots
or
the
lik
e
considering
the
road
agenc
y
formation
unit
(RAFU)
functions.
A
strate
gy
based
kno
wing
the
real
position
of
se
v
eral
points
of
the
trajec-
tory
is
proposed
to
achie
v
e
the
lateral
control
purpose
and
maintain
the
lane
k
eeping
errors
within
the
prescribed
performance
boundaries.
The
RAFU
functions
are
applied
to
achie
v
e
these
goals.
The
stability
of
these
functions,
their
applica
bility
to
approach
an
y
arbitrary
trajectory
and
the
easy
control
of
the
possible
error
made
on
the
approx-
imation
are
useful
adv
antages
in
practice.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Alicia
C.
S
´
anchez
Department
of
Mathematics,
F
aculty
of
Science
Uni
v
ersity
of
Extremadura
06006
Badajoz,
Spain
Email:
csanchezalicia@gmail.com
1.
INTR
ODUCTION
Intelligent
trans
p
or
t
systems
(ITS)
v
ary
in
technologies
applied:
basi
c
traf
c
si
g
na
l
control
syst
ems,
automatic
number
plate
re
cogn
i
tion,
speed
cameras
to
monitor
security
systems
and
the
li
k
e.
Mechatronics
sys-
tems
(MS)
include
a
combination
of
mechanical,
electrical,
telecommunications,
control
and
computer
science
technologies.
The
robotics
(R)
in
v
olv
es
design,
construction
and
use
of
robots
and
dra
ws
on
the
achie
v
ement
of
computer
,
mechanical
or
electronic
engineering
and
mathematics.
In
particular
ITS,
MS
and
R
include
management
metods
such
as
automatic
na
vig
ation
systems,
v
e-
hicle
control
and
automatic
dri
ving.
There
are
a
multitude
of
papers
about
this
topic.
Here
we
cite
some
of
them
as
e
xample
[1]-[5].
This
w
ork
is
conncerned
about
these
topics,
specically
about
lane
k
eeping
control
and
lateral
control.
There
is
a
wide
of
literature
on
current
de
v
elopments
in
the
eld
of
lateral
and
lane
k
eeping
control
of
autonomous
v
ehicle
motions
[6]-[8].
In
robotics,
for
e
xample,
the
concept
of
lane-k
eeping
motion
planning
algorithms
for
mobile
robots
in
order
that
the
robot
does
not
lea
v
e
the
lane
if
collision-free
motion
is
a
v
ailable
introduced
in
[9].
The
road
agenc
y
formation
unit
(RAFU)
functions
ha
v
e
bee
n
studied
i
n
Approximation
Theory
(the
interested
reader
can
see
[10]-[17]).
In
this
paper
,
the
RAFU
functions
will
be
called
the
T
axi
functions.
As
when
we
use
a
taxi,
the
taxi
dri
v
er
tak
e
us
to
the
destination,
from
kno
wledge
of
the
e
xact
posit
ions
of
se
v
eral
points
of
the
road
centerline
and
a
width
of
the
route,
our
main
goal
in
this
w
ork
is
to
obtain
RAFU
continuous
functions
that
connect
the
initial
point
with
t
he
nal
point
of
a
certain
trajectory
without
touching
the
sides
of
the
road.
T
o
achie
v
e
this
aim,
we
only
need
that
the
robot
has
the
necessary
technology
to
kno
w
the
e
xact
positions
of
some
points
of
the
route
that
it
needs
to
follo
w
at
an
y
time.
In
this
w
ork
we
are
not
concerned
about
what
the
machine
must
do
in
case
of
an
y
obstacle
appears
J
ournal
homepage:
http://ijr
a.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
320
❒
ISSN:
2089-4856
on
the
road.
On
this
subject
one
can
see
for
instance,
[18]-[22].
Until
no
w
,
RAFU
functions
ha
v
e
not
been
emplo
yed
for
this
purpose.
But
gi
v
en
the
ease
and
the
accurac
y
with
which
these
continuous
functions
approx-
imate
an
y
step
f
u
nc
tion
and,
therefore,
an
y
continuous
trajectory
,
we
think
that
the
use
of
these
functions
in
ITS,
MS
and
R
could
be
useful,
specically
in
lane
k
eeping
and
lateral
control.
Moreo
v
er
,
the
easy
control
of
the
possible
error
made
on
the
approximation
of
the
e
xact
route
is
another
adv
antage
in
practice.
In
section
2
we
solv
e
the
lane
k
eeping
and
lat
eral
control
problems
in
case
of
the
routes
are
straight
and
perpendicular
all
of
them.
Section
3
is
de
v
oted
to
solv
e
the
same
problems
in
case
that
the
trajectory
is
a
continuous
function.
The
lane
k
eeping
and
lateral
control
problems
in
case
of
an
arbitrary
trajectory
on
the
plane
is
studied
in
section
4
.
Concluding
remarks
are
in
section
5
.
This
paper
is
illustrated
with
some
e
xampl
es.
2.
TRAJECT
OR
Y
WHERE
ALL
R
OUTES
ARE
STRAIGHT
AND
PERPENDICULAR
Denition
1
Gi
v
en
an
arbitrary
function
f
dened
i
n
[
a,
b
]
and
l
et
P
s
=
{
x
0
=
a,
x
1
,
...,
x
s
=
b
}
be
a
partition,
we
dene
the
RAFU
Method
on
approximation
to
the
function
function
f
to
all
approximation
procedure
that
uses
functions
C
n
dened
in
[
a,
b
]
to
approach
the
function
f
where
the
functions
C
n
(
x
)
are
dened
by
(1).
C
n
(
x
)
=
f
(
x
1
)
+
s
X
i
=2
[
f
(
x
i
)
−
f
(
x
i
−
1
)]
·
F
n,p
(
x
i
−
1
,
x
)
(1)
being
(2).
F
n,p
(
x
i
−
1
,
x
)
=
2
n
p
+1
√
x
i
−
1
−
a
+
2
n
p
+1
√
x
−
x
i
−
1
2
n
p
+1
p
b
−
x
i
−
1
+
2
n
p
+1
√
x
i
−
1
−
a
(2)
with
p
≥
1
a
natural
number
.
The
functions
C
n
(
x
)
,
n
∈
N
are
called
RAFU
Functions
.
Suppose
we
kno
w
that
(
x
0
,
k
1
)
,
(
x
1
,
k
1
)
,
(
x
1
,
k
2
)
,
(
x
2
,
k
2
)
,
(
x
2
,
k
3
)
,
(
x
3
,
k
3
)
,...,
(
x
s
−
1
,
k
s
)
,
(
x
s
,
k
s
)
are
the
e
xact
positions
of
the
v
ertices
of
a
trajectory
E
s
(
x
)
in
which
all
streets
are
straight
and
perpendicular
and
suppose
that
it
v
eries
that
x
i
<
x
j
for
all
0
≤
i
<
j
≤
s
.
W
ith
this
notation,
the
follo
wing
Proposition
can
be
established.
Pr
oposition
1
Let
P
s
=
{
x
0
=
a,
x
1
,
...,
x
s
=
b
}
be
a
partition
of
[
a,
b
]
and
let
E
s
(
x
)
be
a
step
function
dened
in
[
a,
b
]
by
(3).
E
s
(
x
)
=
k
1
·
χ
[
x
0
,x
1
]
+
s
X
i
=2
k
i
·
χ
(
x
i
−
1
,x
i
]
(3)
with
k
i
real
numbers
and
χ
[
c,d
]
(
x
)
the
function
dened
by
χ
[
c,d
]
(
x
)
=
1
if
x
∈
[
c,
d
]
and
χ
[
c,d
]
(
x
)
=
0
if
x
/
∈
[
c,
d
]
.
F
or
all
n
≥
2
,
if
3(
b
−
a
)
n
K
≤
δ
(
s
)
,
being
δ
(
s
)
=
min
1
≤
i
≤
s
|
x
i
−
x
i
−
1
|
and
K
≥
2
a
posi
ti
v
e
inte
ger
,
it
follo
ws
that
1.
|
C
n
(
x
)
−
E
s
(
x
)
|
≤
2
K
(
M
s
−
m
s
)
n
√
n
if
x
∈
[
a,
b
]
−
S
s
−
1
i
=1
x
i
−
δ
(
s
)
3
,
x
i
+
δ
(
s
)
3
2.
|
C
n
(
x
)
−
[
k
i
(1
−
α
x
)
+
k
i
+1
α
x
]
|
≤
2
K
(
M
s
−
m
s
)
n
√
n
if
i
=
1
,
...,
s
−
1
and
x
∈
x
i
−
δ
(
s
)
3
,
x
i
+
δ
(
s
)
3
where
M
s
and
m
s
are
the
maximum
and
the
minimum
of
the
k
i
,
α
x
∈
(0
,
1)
is
a
real
number
which
depends
only
on
x
and
(
C
n
)
n
is
the
sequence
of
RAFU
functions
associated
to
E
s
according
with
(1)
and
dened
as
(4).
C
n
(
x
)
=
k
1
+
s
X
i
=2
[
k
i
−
k
i
−
1
]
·
F
n,
2
(
x
i
−
1
,
x
)
(4)
being
F
n,
2
(
x
i
−
1
,
x
)
for
each
i
=
2
,
...,
s
the
functions
dened
in
(2)
for
p
=
2
.
A
proof
of
Proposition
1
can
be
seen
in
[17].
The
e
xpression
E
r
r
or
(
n
)
=
2
K
(
M
s
−
m
s
)
n
√
n
gi
v
e
us
the
maxim
um
distance
between
the
T
axi
Function
C
n
(
x
)
and
the
real
trajectory
E
s
(
x
)
along
the
road
centerline.
In
this
sense,
x
ed
a
certain
K
,
for
an
y
n
≥
2
we
can
kno
w
E
r
r
or
(
n
)
and
reciprocally
.
Int
J
Rob
&
Autom,
V
ol.
10,
No.
4,
December
2021
:
319
–
325
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
321
In
practice
we
w
ould
w
ork
in
this
w
ay:
rst
consider
a
maximum
error
bound
E
r
r
or
(
n
)
,
then
we
calculate
δ
(
s
)
and
K
=
I
n
t
[
l
og
2
3(
b
−
a
)
δ
(
s
)
]
and
nally
we
nd
the
v
al
ue
of
n
in
order
to
determine
the
T
axi
function
C
n
(
x
)
that
the
robot
must
follo
w
.
Example
1
Suppose
that
the
trajectory
that
a
car
must
follo
w
consists
of
13
straight
and
perpendicular
streets
of
2275
m
long
in
total
and
it
is
gi
v
en
by
means
the
step
function
E
7
(
x
)
=
40
if
20
≤
x
≤
150
350
if
150
<
x
≤
600
200
if
600
<
x
≤
750
10
if
750
<
x
≤
925
80
if
925
<
x
≤
1100
250
if
1100
<
x
≤
1200
75
if
1200
<
x
≤
1400
In
this
case
δ
(
s
)
=
100
and
3
(
b
−
a
)
n
K
≤
3
(
b
−
a
)
2
K
≤
100
=
δ
(
s
)
holds
for
all
n
≥
2
when
K
=
6
According
to
Proposition
1
,
for
all
x
∈
[20
,
1400]
,
|
C
n
(
x
)
−
E
7
(
x
)
|
≤
E
r
r
or
(
n
)
=
2
K
(
M
s
−
m
s
)
n
√
n
=
2
6
(350
−
10)
n
√
n
In
Figure
1
we
represent
the
e
xact
trajectory
E
7
(
x
)
(red
color)
and
its
continuous
approximations
C
40
(
x
)
respecti
v
ely
(blue
color).
200
400
600
800
1000
1200
1400
50
100
150
200
250
300
350
Figure
1.
Approximation
to
a
step
function
In
section
2
we
ha
v
e
applied
the
result
to
the
particular
case
in
which
the
journe
y
is
dened
by
a
step
function,
b
ut
in
[10]
we
studied
ho
w
t
he
RAFU
functions
can
be
used
to
approach
an
arbitrary
discontinuous
function.
3.
TRAJECT
OR
Y
DEFINED
BY
A
CONTINUOUS
FUNCTION
Suppose
that
inside
a
machine
we
kno
w
the
e
xact
positions
of
some
points
(
x
0
,
f
(
x
0
))
,
(
x
1
,
f
(
x
1
))
,
...,
(
x
n
−
1
,
f
(
x
n
−
1
))
and
(
x
n
,
f
(
x
n
))
of
a
continuous
trajectory
f
(
x
)
.
Denition
2
Let
f
be
a
function
dened
in
[
a,
b
]
.
The
modulus
of
continuity
of
f
,
ω
(
f
,
h
)
,
is
the
maximum
of
|
f
(
x
)
−
f
(
y
)
|
for
all
a
≤
x,
y
≤
b
,
|
x
−
y
|
≤
h
.
3.1.
Case
of
a
unif
orm
net
Let
[
a,
b
]
be
an
interv
al
and
suppose
the
case
in
which
the
v
alues
x
i
v
erify
x
i
=
a
+
i
·
b
−
a
n
for
each
i
=
0
,
...,
n
.
The
taxi
function
(Alicia
C.
S
´
anc
hez)
Evaluation Warning : The document was created with Spire.PDF for Python.
322
❒
ISSN:
2089-4856
Pr
oposition
2
Let
P
n
=
{
x
0
=
a,
x
1
,
...,
x
n
=
b
}
be
a
partition
of
[
a,
b
]
with
x
j
=
a
+
j
·
b
−
a
n
,
j
=
0
,
1
,...,
n
and
let
E
n
be
the
step
function
dened
by
(5).
E
n
(
x
)
=
k
1
x
∈
[
a,
x
1
]
k
2
x
∈
(
x
1
,
x
2
]
...
k
n
x
∈
(
x
n
−
1
,
b
]
k
j
∈
R
,
j
=
1
,
...,
n
(5)
Let
C
n
be
the
RAFU
function
ass
ociated
to
E
n
dened
as
(4)
by
C
n
(
x
)
=
k
1
+
P
n
j
=2
[
k
j
−
k
j
−
1
]
·
F
n,
2
(
x
j
−
1
,
x
)
.
Then,
for
all
n
≥
2
it
follo
ws
that:
1.
|
C
n
(
x
)
−
E
n
(
x
)
|
≤
2(
M
n
−
m
n
)
n
√
n
if
x
∈
[
a,
b
]
\
∪
n
−
1
k
=1
x
k
−
b
−
a
3
n
,
x
k
+
b
−
a
3
n
2.
|
C
n
(
x
)
−
[
k
j
(1
−
α
x
)
+
k
j
+1
α
x
]
|
≤
2(
M
n
−
m
n
)
n
√
n
if
x
∈
x
j
−
b
−
a
3
n
,
x
j
+
b
−
a
3
n
and
j
=
1
,...,
n
−
1
being
M
n
and
m
n
the
maximum
and
the
minimum
of
the
k
j
respecti
v
ely
and
α
x
∈
(0
,
1)
a
number
that
depends
upon
x
.
Theor
em
1
Let
f
be
a
continuous
function
dened
in
[
a,
b
]
and
let
P
n
=
{
x
0
=
a,
x
1
,
...,
x
n
=
b
}
be
a
partition
of
[
a,
b
]
where
x
i
=
a
+
i
·
h
for
each
i
=
0
,
...,
n
and
h
=
b
−
a
n
.
Then
there
e
xists
a
sequence
of
radical
functions
(
C
n
)
n
dened
in
[
a,
b
]
as
(1)
such
that
|
C
n
(
x
)
−
f
(
x
)
|
≤
2
(
M
−
m
)
n
√
n
+
ω
(
f
,
h
)
for
all
n
≥
2
being
M
and
m
the
maximum
and
the
minimum
of
f
in
[
a,
b
]
respecti
v
ely
and
ω
(
f
,
h
)
its
modulus
of
continuity
.
The
proofs
of
Proposition
2
and
Theorem
2
can
be
seen
in
[17].
According
to
these
proofs,
we
dene
the
functions
E
n
(
x
)
as
in
Proposition
1
b
ut
taking
into
account
that
k
1
=
f
(
x
0
)
=
f
(
x
1
)
and
k
i
=
f
(
x
i
)
for
all
i
=
2
,
...,
n
and
then
its
corresponding
T
axi
functions
C
n
(
x
)
as
in
(4).
In
practice
we
w
ould
w
ork
in
the
same
w
ay
that
we
ha
v
e
mentioned
in
section
2
.
Example
2
Gi
v
en
a
route
dened
by
the
continuous
function
f
(
x
)
f
(
x
)
=
75
if
20
≤
x
≤
80
(
x
−
200)
2
288
+
25
if
80
<
x
≤
200
5
x
−
850
6
if
200
<
x
≤
350
150
if
350
<
x
≤
500
−
(
x
−
500)
2
750
+
150
if
500
<
x
≤
800
Under
the
h
ypothesis
of
Theorem
1
,
we
kno
w
that
2
(
M
−
m
)
n
√
n
+
ω
(
f
,
h
)
≤
2
(150
−
25)
n
√
n
+
5
6
·
800
−
20
n
≤
952
n
√
n
In
Figure
2
we
sho
w
the
results
for
n
=
100
.
200
400
600
800
25
50
75
100
125
150
175
Figure
2.
Approximation
to
a
continuous
function
Int
J
Rob
&
Autom,
V
ol.
10,
No.
4,
December
2021
:
319
–
325
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
❒
323
It
is
important
to
observ
e
that
it
is
not
easy
to
nd
the
error
E
r
r
or
(
n
)
in
Theorem
1
because
of
he
dif
culty
to
obtain
the
term
ω
(
f
,
h
)
.
So,
what
can
we
do?
In
practice,
we
can
calculate
∆(
f
)
=
max
1
≤
i
≤
n
|
f
(
x
i
+1
)
−
f
(
x
i
)
|
and
we
could
tak
e
this
v
alue
instead
of
ω
(
f
,
h
)
.
Some
adv
antages
of
the
use
of
the
T
axi
functions:
•
The
v
alue
of
n
for
the
functions
E
n
(
x
)
and
C
n
(
x
)
is
the
same.
•
The
stability
of
the
T
axi
functions
impro
v
es
t
he
instability
of
other
f
amilies
of
approximating
functions
and
this
is
an
important
contib
ution
of
this
w
ork.
This
can
be
inferred
from
Proposition
2
where
we
can
deduce
that
between
each
tw
o
consecuti
v
e
v
alues
x
r
and
x
r
+1
the
T
axi
functions
C
n
(
x
)
tak
e
v
alues
between
k
r
and
k
r
+1
.
•
The
functions
F
n,p
in
(2)
do
not
depend
on
the
points
x
i
b
ut
only
on
the
subindices
i
(see
p.
114
[12]).
So,
this
reduces
the
calculations
in
v
olv
ed.
•
The
T
axi
functions
C
n
(
x
)
can
be
obtained
with
the
only
condition
that
the
trajectory
to
approach
is
a
continuous
function.
•
According
to
Proposition
2
,
the
m
aximum
error
bound
made
with
the
T
axi
functions
depends
only
on
parameters
that
are
kno
wn.
3.2.
Case
of
a
non
unif
orm
net
Suppose
the
case
in
which
the
v
alues
x
i
form
a
non
uniform
net.
In
this
case
an
analogous
result
to
the
T
eorem
3
.
1
can
be
obtained.
Theor
em
2
Let
P
n
=
{
x
0
=
a,
x
1
,
...,
x
s
n
=
b
}
be
a
partition
of
[
a,
b
]
with
δ
(
s
n
)
=
min
1
≤
i
≤
s
n
|
x
i
−
x
i
−
1
|
and
∆
(
s
n
)
=
max
1
≤
i
≤
s
n
|
x
i
−
x
i
−
1
|
such
that
for
all
n
≥
2
,
3(
b
−
a
)
n
K
≤
δ
(
s
)
≤
∆
(
s
n
)
≤
h
being
h
=
b
−
a
n
and
K
≥
2
a
positi
v
e
inte
ger
.
Let
f
be
a
continuous
function
dened
in
[
a,
b
]
.
Then,
there
e
xists
a
sequence
(
C
n
)
n
dened
in
[
a,
b
]
as
(1)
for
p
=
2
such
that
|
C
n
(
x
)
−
f
(
x
)
|
≤
2
K
(
M
−
m
)
n
√
n
+
ω
(
f
,
h
)
for
all
n
≥
2
and
x
∈
[
a,
b
]
being
M
and
m
the
maximum
and
the
minimum
of
f
respecti
v
ely
and
ω
(
f
,
h
)
its
modulus
of
continuity
.
A
proof
of
Theorem
2
can
be
seen
in
[17].
T
o
obtain
the
T
axi
function
C
n
(
x
)
we
w
oul
w
ork
as
in
the
pre
vious
subsection.
The
same
adv
antages
hold
in
this
case
too.
4.
CASE
OF
AN
ARBITRAR
Y
TRAJECT
OR
Y
ON
THE
PLANE
Suppose
the
information
we
kno
w
inside
a
machine
is
the
e
xact
position
of
some
v
alues
(
x
0
,
T
(
x
0
))
,
(
x
1
,
T
(
x
1
))
,
...,
(
x
s
−
1
,
T
(
x
s
−
1
))
and
(
x
s
,
T
(
x
s
))
that
belong
to
a
continuous
trajectory
on
the
plane
(
x,
T
(
x
))
.
Here
we
study
what
we
can
do
when
this
trajectory
is
not
a
function.
As
(
x,
T
(
x
))
is
not
a
function,
without
loss
of
generality
we
can
suppose
that
x
0
<
...
<
x
i
⩾
x
i
+1
for
a
certain
i
.
There
are
tw
o
possible
cases.
1).
Case
1
.
It
v
eries
that
x
0
<
...
<
x
i
≥
x
i
+1
and
T
(
x
i
+1
)
<
T
(
x
i
)
.
Then
we
use
the
equations
of
the
rotation
around
the
center
(
x
i
,
T
(
x
i
))
and
through
the
angle
α
=
90
º
to
turn
the
points
(
x
i
+1
,
T
(
x
i
+1
))
,
...,
(
x
s
,
T
(
x
s
))
into
(
x
i
+1
,
2
,
T
(
x
i
+1
,
2
))
,
...,
(
x
s,
2
,
T
(
x
s,
2
))
respecti
v
ely
1
x
j
,
2
T
2
(
x
j
,
2
)
=
1
0
0
x
i
+
T
(
x
i
)
0
−
1
−
x
i
+
T
(
x
i
)
1
0
·
1
x
j
T
(
x
j
)
for
all
i
+
1
≤
j
≤
s
.
2).
Case
2
.
It
v
eries
that
x
0
<
...
<
x
i
≥
x
i
+1
and
T
(
x
i
+1
)
>
T
(
x
i
)
.
Then
we
use
the
equations
of
the
ro-
tation
around
the
center
(
x
i
,
T
(
x
i
))
and
through
the
angle
α
=
−
90
º
to
turn
the
points
(
x
i
+1
,
T
(
x
i
+1
))
,
...,
(
x
s
,
T
(
x
s
))
into
(
x
i
+1
,
2
,
T
(
x
i
+1
,
2
))
,
...,
(
x
s,
2
,
T
(
x
s,
2
))
respecti
v
ely
The
taxi
function
(Alicia
C.
S
´
anc
hez)
Evaluation Warning : The document was created with Spire.PDF for Python.
324
❒
ISSN:
2089-4856
1
x
j
,
2
T
2
(
x
j
,
2
)
=
1
0
0
x
i
−
T
(
x
i
)
0
1
x
i
+
T
(
x
i
)
−
1
0
·
1
x
j
T
(
x
j
)
for
all
i
+
1
≤
j
≤
s
.
In
this
w
ay
,
we
will
reproduce
the
cases
1
and
2
as
man
y
times
as
necessary
until
the
initial
points
(
x
0
,
T
(
x
0
))
,
(
x
1
,
T
(
x
1
))
,
...,
(
x
s
,
T
(
x
s
))
ha
v
e
been
turned
into
other
points
(
x
0
,
T
(
x
0
))
,
...,
(
x
s,q
,
T
q
(
x
s,q
))
in
which
x
0
<
...
<
x
s,q
.
So,
the
initial
trajectory
(
x,
T
(
x
))
becomes
a
continuous
function
F
(
x
)
to
which
we
can
apply
the
e
xplained
in
section
3
.
After
then,
we
obtain
the
T
axi
function
C
n
(
x
)
associated
to
F
(
x
)
for
a
certain
n
and
then
we
recalculate
for
each
x
∈
[
a,
b
]
its
the
corresponding
point
of
the
real
trajectory
according
to
all
the
in
v
erse
rotations
that
this
x
has
tested
in
order
to
gi
v
e
to
the
robot
the
real
information
to
follo
w
.
5.
CONCLUSION
The
RAFU
method
on
approximation
is
an
original
approximation
procedure.
In
this
w
ork
we
ha
v
e
sho
wed
ho
w
this
method
can
be
used
to
solv
e
the
lane
k
eeping
and
the
lateral
control
problems
in
intelligent
transport
systems,
specically
in
robot
na
vig
ation.
Gi
v
en
a
trajectory
on
the
plane,
the
T
axi
functions
can
be
useful
to
pro
vide
the
necessary
information
to
a
v
ehicle
in
order
to
go
from
a
place
to
another
one
autonomously
.
The
proposed
method
holds
for
an
y
trajectory
and
it
does
not
depend
on
its
re
gularity
.
Moreo
v
er
the
error
bound
that
the
T
axi
function
pro
vides
does
not
depend
on
an
y
unkno
wn
parameter
or
of
the
re
gularity
of
the
trajectory
.
Gi
v
en
the
conciseness
of
the
results
of
this
paper
,
we
belie
v
e
that
these
a
v
enues
of
research
deserv
e
some
attention.
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