International Core Journal of Engineering 2020-26 | Page 94

We propose a line to approximate the hyperbola method
and the nodes of the three piecewise functions are the nodes of
three hyperbolas. The core idea of the line to approximate the
hyperbola method is to use three lines, three-and-four lines
and five lines respectively to approximate hyperbola.
According to the positive or negative distance difference
obtained by the TDOA measurement value, it is simple to
determine the position of mobile tag on the hyperbolic branch,
the judgment rules shown in Table .
graph with the hyperbolic focus of the anchor1 and the
anchor3.
TDOA algorithm schematic
1000
900
800
anchor3
anchor2
700
600
500
T ABLE I. J UDGMENT RULE TABLE
400
left right
d 1,3 - +
d 1,4 - +
300
d 1,2
200
100
0
anchor1
anchor4
0
200
400
600
800
1000
x
Fig. 1. TDOA algorithm schematic
In Table ,
The two-point distance equation of the TDOA algorithm
is defined as
d i , j d i  d j
( x m  x i ) 2  ( y m  y i ) 2  ( x m  x j ) 2  ( y m  y j ) 2
upper down
- +
d 1,i is the difference between the distance from
DQFKRU  to moving tag and the distance from DQFKRU L to
moving tag. The upper and down sides represent the upper
and down branches of the hyperbola with focus of the y-axis,
and the left and right sides represent the left and right
branches of the hyperbola with focus of the x-axis.
(1)
When d 1,2 <0, the mobile tag is on the upper side of the
hyperbola with the focus of the anchor1 and the anchor2.
Where ( x m , y m ) is the coordinate of the unknown mobile
tag, ( x i , y i ) and ( x j , y j ) are the coordinates of anchor i
and When d 1,2 >0, the mobile tag is on the down side of the
hyperbola with the focus of the anchor1 and the anchor2.
d i , j is the distance difference from anchor i and anchor j Similarly, when d 1,3 <0, the mobile tag is on the left side
of the hyperbola with the focus of the anchor1 and the
anchor3.
anchor j respectively, d i and d j are the distances from
anchor i and anchor j to the moving tag respectively, and
to the moving tag.
III. L INE TO APPROXIMATION HYPERBOLA ALGORITHM
When d 1,3 >0, the mobile tag is on the right side of the
hyperbola with the focus of the anchor1 and the anchor3.
In an ideal situation, if there are no obstacles and
non-line-of-sight (NLOS) in the room, the four anchors in the
four corners of the room will be considered as the focuses of
the hyperbola respectively. The hyperbolas will intersect one
point which is the coordinates of the mobile tag. However,
because of problems such as indoor blocking, NLOS
phenomenon and multipath propagation in the actual
environment, the hyperbolas do not intersect one point and
form a common overlapping area which can be considered as
the possible existing area of a mobile tag.
When d 1,4 <0, the mobile tag is on the left side of the
hyperbola with the focus of the anchor1 and anchor4.
When d 1,4 >0, the mobile tag is on the right side of the
hyperbola with the focus of the anchor1 and anchor4.
When we use three lines to approximate hyperbola, the
first line and the third line are hyperbolic asymptote lines
respectively, and the second line is the vertical line of the
hyperbolic apex. When it is judged that the mobile tag is on
one side of the hyperbola, we can use three lines to
approximate hyperbolas.
The basic principle of triangle centroid positioning
algorithm is to calculate the coordinates of the three
intersections of the common overlap of the three hyperbolas.
These three intersections form the vertices of the triangle. The
centroid of the triangle is the coordinate of the estimated
moving tag.
When we use four lines to approximate hyperbola, the first
line and the fourth line are respectively the asymptotes of
which the original hyperbola is moved some distances. The
second line is the connected line of the hyperbolic apex and
the first line. The third line is the connected line of the
hyperbolic apex and the fourth line. The formula for the
translation distance is defined as
The triangle centroid formula is defined as
x
x 1  x 2  x 3
3
y
y 1  y 2  y 3
3
(2)
Where ( x 1 , y 1 ), ( x 2 , y 2 ) and ( x 3 , y 3 ) are the
coordinates of the three vertices of the triangle, and ( x , y ) is
the coordinate of the centroid of the triangle.
L
72
( x 6  x 7 ) 2  ( y 6  y 7 ) 2
2
(3)