3. Calculating the angle of the great arc of a spherical cap

3.1 Overview

For each of the reader antennas, the power received by the tag can be obtained from the Friis

equation:

If the orientation and the position of the reader antenna plus along with the position of the tag

antenna are known, it is easy to obtain the threshold value of the tag antenna gain below which

that the specific tag will not be activated. To simplify the formula, we utilize make the same

assumptions as made in Chapter 4, i.e. the reflection coefficient and are 0 and the

polarization loss factor is 0.5. Then the threshold tag antenna gain can be represented as

,

where is the minimum power required to activate the tag.

The antenna gain for a half-wave dipole antenna is only a function of only , and which

can be represented as

Therefore, a threshold value of can be obtained for each reader antenna, each of which

corresponds corresponding to the angle of the greatest arc of the spherical cap created by

that antenna. To differentiate between different reader antennas, is used to represent

the angle of spherical cap cast by reader antenna i. The bigger the value of is, the

larger the spherical cap is, and the more the number of unreadable orientations of the tag

point there are with respect to the specific reader antenna i. ) It is important to notice that

in the antenna gain function is the maximum angle between any two points on a

spherical cap and the reader axis; in other words, it is only half of the value of the great

arc angle of the spherical cap in Table 1.

3.2 Methodology

A closed form formula to calculate given the threshold value of the dipole antenna gain

has not been found in this research. The difficulty lies in the fact that is not

a typical trigonometric function. Although based on the plot of the antenna gain in Figure

6 below, it does appear to take a sine-likesinusoidal shape, it cannot be approximated

byinto a simple trigonometric function. Figure 7 plots two functions. The first one is

, which is the transformed antenna gain function obtained by dropping

the coefficient. The second one is , which is constructed in such a

way that it has the same period, and maximum and minimum values of as the first one. It

can be seen clearly that the second trigonometric function shows a different curvature

from the antenna gain function.

0 50 100 150 2000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Figure 6 Dipole antenna gain against

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60 70 80 90 100

Transformed dipoleantenna gain

sin(2θ-π/2)/2+0.5

Figure 7 Comparison between transformed antenna gain and a trigonometric function

Therefore, Iinstead of attempting a functional approximation, an off-line look-up table is

used in this research to map the value of and that of the corresponding . It is

important to realize that the threshold value of is unique only for the period of [0, π/2)

because a half-wave dipole antenna’s gain is a function of θi with a period of π and is symmetric

at θi = π/2. Therefore such a look-up table only needs to store the values for i from 0 to π/2.

Such a look-up table has severalthe following benefits. First, it is fairly fast to extract information

from because only a linear search is required once the value of antenna gain is given. Secondly,

because of the symmetry characteristics of the dipole antenna gain, the range of the search is

relatively small. All the values of the look-up table can be read directly into the memory without

compromising the system performance. Lastly, the antenna gain is monotonic monotone

increasing between 0 and π/2.

Table 1 Part of the look-up table

θi (in degree)30 0.28626531 0.30602832 0.32644833 0.34751734 0.36922935 0.39157436 0.41454137 0.43811838 0.46229139 0.48704440 0.5123641 0.5382242 0.56460443 0.59148744 0.61884545 0.646652

Table 2 shows part of the look up table for θi between 30 degrees and 34 45 degrees angle. A one-

dimension array is used to store the value of with the index from 0 to 90. In particular,

[0] is 0 because . A linear interpolation is used to obtain

the value of θi given a specific value of for . As stated before, the angle of the great arc in

the spherical cap is actually 2θi.

4. Calculating αi

αi is the angle between the plane of the great arc and the reader axis. After θi is obtained, all the

rest of the parameters in Table 1 will be calculated based on the methodology discussed in this

section. Instead of focusing on the 3-D space, a linear search along 1-D space is used [ Huang,

Tseng and Lo]. In this section, we first examine the relationship between different parameters in

the Table 1, i.e. how to calculate the values of the rest of the parametersm given the value of one

of the parameters and θi . This is important because in fact αi cannot be obtained directly in this

methodology. We tThen we introduce the algorithm in its simplified geometric form. Lastly, we

deal with issues of implemeantation such as spherical coordinate rotation, etc.

4.1 Relationship between different parameters

For conveniencet, part of the Table 1 is shown below in Table 3.

Table 2 Partial list of notation

li The arc of the intersection area that

lies in SCiri

Li The great arc that passes the end

points of li and falls into the

intersection region

Φi The arc angle of li

γi The arc angle of Li

αi The angle between the great circle

plane that contains Li and OPi

hi The distance between the center of

γi

Φi

SCiri and the center of the unit ball

bi The distance between the center of

SCiri and intersection line of Li and

SCiri

Si The lune-shape area that is formed

by Li and li

If we assume Φi is known, then

Therefore, once we can getobtain Φi , the values of the rest of the parameters are a functions of

both Φi and θi. In the next sub section, we will show thathow to obtaining the area of Si is also a

SCi

li

Li

Pi

O

αi

Si

hi

bi

function of the set of above set of parameters. However, Tthe rest of this section will be devoted

to the calculation of Φi since the calculation of θi is explained in the previous section.

4.2 Calculating Φi

4.2.1 Dot product test

Φi is the angle for the arc of the intersection area that lies in the spherical circle SCiri. To simplify

the calculation, we assume that the reader axis OPi is the z- axis and the tag point O is the origin,

so that each point q on the spherical circle SCiri can be represented as

where

Here Φi corresponds to the arc of SCiri which lies within another spherical cap SCj.

Lemma1: Let OPj be the reader axis for the jth reader with its angle of the great arc of a spherical

cap θj. Then point q is within the SCj if, and only if where and is are

the points q and OPj expressed as unit vectors format of point q and OPj.

Proof: The dot product of two units vectors and is the cosine angle of the two

vectors. If such an angle is smaller than the half of the great arc angle, then point q is

within the spherical cap SCj based on its definition.

Figure 8 An example of a point q within SCj

Therefore, for each of each point q on the spherical circle SCiri, a dot product can be compared

against the value of can be used to determine whether or not such a point is within

spherical cap SCj.

4.2.2 Algorithm

A linear search method is used to obtain the value of Φi which and is stated below.

1 ;2 _found = false;3 _found = false;4 while ( _found = true and _found = true) 5 {6 get xq, yq, zq;7

if ( ) and ( _found = false )

8 ;9 end if10

if ( ) and ( _found = true) and ( _found = false)

11 ;12 end if13 = +1;14 } loop15 ;16 if ( = 0)17 {18 ;19 _found = false;20 _found = false;21 while ( _found = true and _found = true) 22 {23 get xq, yq, zq;24

if ( ) and ( _found = false )

25 ;26 end if27

if ( ) and ( _found = true) and ( _found = false)

28 ;29 end if30 = -1;31 } loop32 ;33 }34 return ;

From lLines 4 to line 15 represent, an iteration is used to search for the range of for .

However, there is a special case that the ranger of may go across the arbitrary start/end

point which is 0 degreeHowever, it should be noted that in fact, x degrees is the same as

(360+x) degrees, and since we arbitrarily select 0/360 as the start/end point for the linear

search, we need to account for the case where 0 and 360 both lie within the range for .

To solve this special scenario as shown in Figure 9, lines 16 to line 33 repeat the linear

search from 360 downward if in the previous search the value of is 0.

Figure 9 An example of two intervals for calculating Φi

Although the search does not stop until both ends of the interval for Φi is are found, the iteration

will stop before reaches 360. This is because all the degeneracy degenerate scenarios will be

eliminated beforehand. A bisection search can could be used effectively to speed up the process

section. if If there were no special scenarios such as the above wheren the value of Φi should be is

obtained from two intervals instead of one intervals, bisection search can be convenient and fast.

In practice, the author it was found that such special scenarios are very common while the

interval length is seldom very long; therefore a simple linear search is used in the algorithm.

4.3 Coordination transformation: translation and rotation

AtIn the beginning of Sectionthe 4.2, to simply the descriptionelaboration of the algorithm, we

assumed that the reader axis OPi is is the z z-axis and the tag point O is the origin. However the

value of which is the vector form of point q will be used to obtain the dot product with OPj

(the reader axis of antenna j) in order to test whether q is within the spherical cap SCj. All vectors

should be using the same coordinate system in order to obtain the correct values. This subsection

gives the details of such the required coordinate transformation processes.

4.3.1 Coordinate translation

Overall, all calculation will be based on the translated coordinate system where the tag point is

the origin while overall there will be no rotation process involved except in the second step.

Therefore, all vectors and points should be translated into the tag-point-origin system.

Let Suppose the current tag point in the global coordinate system has the coordinates

, and let the reader antenna is be at ., then Then the unit vector should be

where r is the length of .

4.3.2 Coordinate rotation

0 36001 2

In the previous subsection, we assume that the z- axis will be aligned with OPi. Therefore, each

point q on the spherical circle SCiri can be represented as

where

However, this coordinate is based on the rotated coordinate system., in In order to obtain the dot

product , the vector should be “rotated back” to the translated system where there is no

axis rotation and the origin rests on the tag point.

In order to rotate axis z back to OPi, the rotation axis and the rotation angle should be found. The

axis of the rotation is defined as a line around which the spinning is donerotation occurs. If a

vector v1 becomes v2 after rotation, the axis of the rotation is perpendicular to the plane which

contains both vectors. Figure 10 shows an example of axis rotation.

Figure 10 Example of rotation

Let , then the axis of the rotation will be (-b, -a, 0). This can be proved

easily since both the dot product of (-b, -a, 0) and and the dot product ofat between (-

b, -a, 0) and (0, 0, 1) is are 0, which means the angle between each pair of the vectors is

90 degrees.

The rotation angle α can be represented as . Based

on geometric calculations, for a given point q(x,y,z) in the rotated local coordinate, its

original global coordinates areis

From these this set of equations, all points on the spherical circle will be converted into

the coordinate system where the tag point is as it was the originally, without any rotation.

The dot product tests with coherent coordinate system will be valid in order to obtain the

value of Φi.