Performance Analysis of Navigation by the

Integration of GPS-24 with LEO & GEO Uzzal K. Acharjee, Anis Ahmed † and Shahida Rafique *

Dept. of Applied Physics, Electronics and Communication Engineering, Dhaka University, Dhaka, Bangladesh,

e-mail:uzzalus@yahoo.com † Dept. of Applied Physics, Electronics and Communication Engineering, Dhaka University, Dhaka, Bangladesh,

e-mail: anis_apedu @yahoo.com

* Dept. of Applied Physics, Electronics and Communication Engineering, Dhaka University, Dhaka, Bangladesh,

e-mail: shahidarafique @yahoo.com

Abstract — An integrated network consisting of GPS-24

(Global Positioning System), LEO (Low earth orbit) and

GEO ( Geostationary Earth Orbit) is analyzed in order to

find the position coordinates of a GPS receiver or user

position by calculating the position coordinates of GPS

satellites. The integrated network is designed to extend the

coverage area of GEO satellites up to the Polar Regions as

well as to improve the user position accuracy. Satellite

visibility through GPS receiver has been investigated using

complex simulations of failure scenarios by giving shadow

effect. An algorithm is proposed to estimate the user’s

position from the known parameters of GPS satellites.

Numerical analysis is carried out for the selection of best

four satellites to be used for the calculation of user’s

position. A new approach is also developed for the

determination of minimum Dilution of Precision (DOP)

using such integrated network.

Keywords—GPS-24, GEO, LEO, DOP, Navigation

I. INTRODUCTION

Satellite can play an important role for high-speed

communication within a large geographical area. But

either of the satellite system based on GEO, GPS or LEO

cannot provide the global coverage alone and does not

support different services simultaneously in an efficient

way. GEO satellites positioned at an altitude of about

35,786 km in equatorial plane are not visible from Polar

Regions (i.e about 800 to 900 north and south latitude) [1].

An integrated network of LEO, GPS and GEO may be

considered for worldwide coverage that supports high

quality voice, high-speed data, broadband Internet access

and multimedia services, etc. In this hybrid constellation,

the satellite of different orbits provides a backup or

complementary coverage of others if needed.

In this paper, LEO and GEO satellite constellation are

added together with the GPS-24 to produce three GPS

augmentation satellite constellation. In order to calculate

user position, four or more satellites (maximum 24) are

needed. In GPS, the position of satellite can be calculated

from the Ephemeris and Clock Data Samples in the earth-

centered, earth-fixed coordinate (ECEF) system [2]. The

GPS receiver identifies each satellite’s signal by its

distinct C/A (Coarse/Acquisition) code, then measures

the time delay for each satellite and calculates the

distance from the satellite called pseudorange [3]. There

are many factors that can degrade GPS signal and thus

affects the accuracy of user position. These are

Ionosphere and Troposphere delay, humidity, multipath

propagation, receiver clock errors, Ephemeris and clock

errors, number of satellites visible, satellite geometry and

shading, GPS jamming and relativity [3]. To analyze the

performance of navigation it is necessary to calculate the

DOP that provides positioning accuracy. Among number

visible satellites from the user’s position, a set of

satellites which are in better geometry is needed to find

minimum GDOP (Geometric Dilution of Precision).

After presenting some of the related works in sub-section

1.1, the proposed integrated network based on LEO, GPS

and GEO is described in Section 2. The user position

calculation, selection of four better geometry satellites

and DOP (Dilution of Precision) are described in Section

3. Using those four satellite positions, an algorithm to

find the user position is proposed in section 4. The

simulation model and simulation parameters are

introduced in Section 5, and finally the simulation results

and conclusions are given in Sections 7 and 8,

respectively.

A. Related Works

The Global Positioning System (GPS) is a satellite

navigation system capable of providing a highly accurate,

continuous global navigation service independent of

other positioning aids. GPS which consists of 24 active

satellites provides 24-hour, all-weather, worldwide

coverage with position, velocity and timing information.

Many of the researches [4-9] provide three dimensional

(3-D) users positioning by solving a set of trilateration

equations using pseudorange measurements and try to

find the user position accuracy based on GPS navigation.

The characteristics of GDOP of D-R (Doppler-Range)

algorithm is studied in paper [10]. Doppler and range

measurements are combined in D-R algorithm to achieve

3D positioning using three other than four satellites.

Reference [11] describes Augmentation of Global

Positioning System (GPS) using synchronized Pseudolite

technology with an illustration of continuous monitoring

1-4244-1550-0/07/$25.00 ©2007 IEEE

and position tracking. An automated method for

predicting the number of satellites visible to a GPS

receiver, at any point on the Earth's surface at any time is

presented in [12]. In this paper, a simplified calculation

method is presented considering an integrated network of

GPS, LEO and GEO satellites that can estimate user’s

position with minimum GDOP.

II. PROPOSED NETWORK ARCHITECTURE

Fig.1. Proposed integrated network

The proposed integrated network or architecture is

composed of a GEO component, a GPS-24 component

and of a LEO component [Figure 1]. In this architecture,

GEO component provides services over a typical

continental coverage area. There are some LEO satellites

which can provide services to those users located in the

areas where GEOs coverage is completely absent (poles).

The integrated network of both LEO and GEO will

support the existing GPS-24 in order to calculate accurate

user’s position globally.

III. USER POSITION CALCULATION

To measure accurate position of an user, minimum four

GPS satellite’s position are needed. In Fig. 2, there are

three known points at locations r1 (x1, y1, z1), r2 (x2, y2, z2),

and r3 (x3, y3, z3), and an unknown point at ru (xu, yu, zu)

are considered.

x1, y1, z1 z

x2, y2, z2

x3, y3, z3

x Fig.2. Use three known positions to find one unknown position.

The distances between the three known points to the

unknown point are denoted as 1 , 2 and 3 and can be

written as

2 2 2( ) ( ) ( )i i u i u i ux x y y z z

where i = 1… 3.

A. Measurement of Pseudorange

Every satellite sends a signal at a certain time tsi. The

receiver will receive the signal at a later time tu. The

distance between the user and the satellite i is

( )iT u sic t t

where c is the speed of light, iT is often referred to as

the true value of pseudorange from user to satellite i, tsi

is referred to as the true time of transmission from

satellite i, tu is the true time of reception. The measured

pseudorange i can be written as:

( ) ( )i iT i i ut i i i iD c b b c T I

where bi is the satellite position error effect on range,

Ti is the troposphere delay error, Ii is the ionosphere

delay error, i is the receiver measurement noise error,

i is the relativistic time correction. Some of these

errors can be corrected; for example, the troposphere

delay can be modeled and the ionospheric error can be

corrected in a two-frequency receiver. The errors will

cause inaccuracy of the user position. However, the user

clock error cannot be corrected through received

information. Thus, it will remain as an unknown. As a

result, equation (1) must be modified as:

2 2 2( ) ( ) ( )i i u i u i ux x y y z z + bu (2)

where i =1….4

where bu is the user clock bias error expressed in

distance, which is related to the quantity but by bu = cbut

.In equation (2), four equations are needed to solve for

four unknowns xu, yu, zu, and bu . Thus, in a GPS receiver,

y

(1)

a minimum of four satellites is required to solve for the

user position.

Linearize the equation (2) and be written as:

Equation (3) can be written in a simplified form as

x

where and x are vectors, is a matrix and these

parameters can be written as

=[ 1 2 ……. n ]T

x =[ ux uy uz ub ] T

and =

11

21

31

41

12

22

32

42

13

23

33

43

1

1

1

1

where [ ]T represents the transpose of a matrix. Since it is

not a square matrix, it cannot be inverted directly.

Equation (3) is still a linear equation. If there are more

equations than unknowns in a set of linear equations, the

least-squares approach can be used to find the solutions.

The pseudoinverse of the can be used to obtain the

solution. The solution is:

TTx1

From this equation, the values of ux , uy , uz and

ub can be found. In general, the least-squares approach

produces a better solution than the position obtained

because more data is needed.

B. Satellite Selection

A GPS receiver can simultaneously receive signals from

4 up to 24 GPS satellites, if the receiver is on the surface

of the earth. Under this condition, there are two

approaches to solve the problem. The first one is to use

all the satellites to calculate the user position. The other

approach is to choose only four satellites from the

constellation. The general purpose of the satellite

selection algorithm is to minimize the Geometric Dilution

of Precision (DOP). If there are more than four satellite

signals that can be received by a GPS receiver, a simple

way is to choose only four satellites and utilize them to

solve for the user position. The following equation is

used to find the x, y and z positions of satellite.

cos cos sin cos sin

sin cos cos cos sin

sin sin

er er

er er

r r ix

y r r i

z r i

(6)

and

where, r is the distance the from the satellite to the center

of the earth, i the inclination angle at reference time,

er is the angle between the ascending node and the

Greenwich meridian, is the true anomaly and means

argument of perigee.

The satellite position calculated in this equation is in the

ECEF frame. Therefore the satellite position is a function

of time. Using the x, y, z positions of satellites and

satellite selection algorithm that is described in reference

[13] four satellites are selected to find user’s positions.

The geometry of four satellites with minimum GDOP is

given in figure 3, where one satellite A is at zenith, and

the other three satellites B,C and D are all equally spaced

at 1200 , and placed 109.470 from A to provide a regular

tetrahedron.

Fig. 3. Geometry of GPS Satellite with minimum GDOP

C. Dilution of Precision

The dilution of precision (DOP) is often used to measure

user position accuracy. The receiver estimates the amount

of position calculation error using measurements based

on several factors that reduce accuracy. The DOP

indicators are GDOP (geometric DOP), PDOP (position

DOP), HDOP (horizontal DOP), VDOP (vertical DOP),

and TDOP (time clock offset).

(3)

Where 1i u

i

i u

x x

b2

i ui

i u

y y

b

3i u

i

i u

z z

b (4)

GDOP=2 2 2 21

x y z b

PDOP =2 2 21

x y z

HDOP=2 21

x y

VDOP =z

TDOP =b

where is the measured rms error of the pseudorange,

which has a zero mean, zyx are the measured rms

errors of the user position in the xyz directions, and b is

the measured rms user clock error expressed in distance.

Better relative geometry and higher corresponding

accuracy result in a low DOP (values between 1-3) for

calculating user position. A high DOP (4 and above)

indicates an increasing more error in the position

indicated. In selecting satellites, the DOP values should

be as small as possible in order to generate the best user

position accuracy.

IV. PROPOSED ALGORITHM

The selection of best 4 GPS satellites is described in the

following steps:

1. From the GPS satellite and user position the transit

time tt can be calculated as

2 2 2( ) ( ) ( ) /t u u ut x x y y z z c where x,

y, z, and xu, yu, zu are the coordinates of the satellite and

the user respectively, c is the speed of light.

2. Using the transit time tt , the angle er can be written

as; .

ieer er tt

where er is the angle between the ascending node and

Greenwich meridian and .

ie denotes earth rotation

speed.

3. Using the new value er in above Equation, the

position of the satellite x, y and z are calculated from Equ.

(6).

4. The above operations should be performed for every

satellite. From these values a new user xu, yu, zu position

will be calculated.

5. Now repeating steps 1, 2, 3 and 4, a new set of x,y and

z are to be calculated. When the old and new sets are

within a predetermined value, the new set can be

considered as the position of the GPS satellite in the new

coordinate system.

6. These new x, y, and z values for each 4 best geometry

GPS satellites will be used to find the user position.

V. SIMULATION SCENARIO

To investigate the efficiency of proposed integrated

constellation scenario, the simulation software MATLAB

is used [14]. GPS receiver employs a simple algorithm

[13] to select the four best geometry satellites among 24

GPS satellites. A MATLAB code is written to implement

the proposed algorithm that finds the x, y and z positions

of the satellites. The latitude, longitude & altitude of each

satellite can be calculated from the positions of the

satellite according to the World Geodetic System

(WGS84) coordinate system. Using the calculated x, y &

z positions of four satellite, user position can be solved

using the proposed algorithm.

The simulation parameters shown in Table 1 are used for

simulation. The LEO orbit consists of satellites arranged

in Teledisc configuration that is designed using 288

satellites in 12 planes each plane contains 24 satellites

[15]. GEO constellation consists of Cyberstar that is

designed using 3 GEO satellites in 3 planes each plane

contains one satellite. Finally, GPS (Global Positioning

System) constellation that is used 24 satellites in 6 planes

with 4 satellites per plane [3]. The 6 orbital planes have

approximately 550 inclination and the GPS receive can be

located anywhere of the world.

TABLE 1

SIMULATION PARAMETERS OF GEO, MEO (GPS) AND LEO

SATELLITEs

Parameters GEO MEO LEO

Satellite Name Cyberstar GPS-24

(NAVASTRAR)

Teledesic

No. of satellite 3 24 288

No. of plane 3 6 12

No. of satellite

per plane

1 4 24

Spacing

between

planes(deg)

120 30 15

Inclination of

orbital plane to

the

equator(deg)

120 55 84.7

Frequency(upli

nk/downlink)

Ka Band Ka Band Ka Band

60Ghz

Access

Technology

FDMA,

TDMA

FDMA/ TDMA MF-TDMA,

ATDMA

Modulation QPSK QPSK QPSK

VI. SIMULATION RESULT OF INTEGRATED

NETWORK

In this section, we will present the result of simulation

and discuss the accuracy of calculation for navigation

using the proposed algorithm.

A. Plotting of Satellite and User Positions in 3-D

For our calculation the known parameters were semi-

major axis, eccentricity, inclination angle, mean anomaly,

mean motion difference, rate of right ascension, argument

of perigee, reference time ephemeris of GPS satellites

and some necessary angles between earth and GPS

satellites [2].

The obtained data is shown in Table 2. From the table is

seen that the altitudes of all 4 GPS satellites are found

about 20,000 km from the earth surface. The positional

longitudes of 3 GPS satellites are 0.77, 19 and 37 degree

East, respectively and the other GPS is at -75 degree

West. On the other hand the positional latitudes of 3 GPS

satellites -10, -1.67 and - 4.31 degree South and the 4th

one is at 18 degree North. These obtained data is found

very close to our expected value.

TABLE 2

CALCULATED PARAMETERS OF BEST FOUR GPS

SATELLITES

Parameters Satellite

1

Satellite

2

Satellite

3

Satellite

4

Altitude(Km) 20,000.69 20,118.36 19,954.35 20,075.40

Longitude(Deg) -75.35 W 0.77 E 37.33 E 19.87 E

Latitude(Deg) -18.66 N -10.58 S -1.67 S -4.32 S

A 3-D graph is plotted according to the resultant values

after executing the MATLAB code in particular GPS

time. From the graph the positions of the satellite and

user can be found easily.

-2

0

2

x 109

-5

0

5

x 107

-5

0

5

10

15

x 107

z-a

xis

XYZ position of satellite & user

x-axisy-axis

s1

s2

s3

s4

u

Fig. 4. Plotting of Satellite & User Positions in GPS Time 36400s

B. Visibility of GPS satellites & Integrated Satellites

Figure-5 shows the probability of satellite visibility for a

5 deg elevation mask angle for the user latitude 90

degree. It is shown from the figure-5(a) that there are at

best 7 satellites in view. But giving shadow effect using

simulation software SaVi [16] the majority of the time

4/5 satellites in view depending on different latitude

positions. From the figure–5(b) it is revealed that satellite

visibility increases noticeably for adding several GEO-

sats and LEO-sats with the existing GPS-24 constellation.

GEO satellites cover mid latitude region, and because

they are higher latitude than the GPS satellites; fewer are

required for the same degree of coverage. And LEO

satellites are much more in number that can give a faster

and continuous coverage.

90 degree latitude

0

10

20

30

40

50

1 2 3 4 5 6 7 8 9 10 11 12 13 14

No. of satellite in view

Pro

bab

liy

(a)

0

5

10

15

20

25

30

35

40

45

1 2 3 4 5 6 7 8 9 10 11 12 13 14

No.of satellite in view

Pro

bab

lity

(b)

Fig.5. Probability of GPS satellite visibility (a) for GPS-24 satellites

(conventional) and (b) for the Integrated (GPS-24,GEO & LEO).

C. Dilution of Precision (DOP) Simulation

Figure 6 demonstrates the simulated values of indicators

of DOP (GDOP,PDOP,HDOP,VDOP & TDOP) vs. GPS

time from 36400s to 36420s with 5 seconds interval. It is

DOP vs. GPS Time

0

1

2

36400 36405 36410 36415 36420

GPS Time(s)

DO

P

GDOP PDOP HDOP

VDOP TDOP

Fig. 6. Dilution of Precision (DOP) Vs. GPS Time

used to provide an indication of quality of the

methodology that is used in this paper to calculate DOP.

It is observed from the figure 6 that the value of GDOP

lines between 1.38088 to 1.728653.It is the possible

confidence level to be used for applications demanding

highest possible precision at all times. The values of

others indicators of DOP (PDOP,HDOP,VDOP &

TDOP) are also in ideal level.

VII. CONCLUSION

It is important to note that speed variation of LEO, GPS

& GEO will not effect our obtained result. This point will

be address in our next paper. The model of integrating

GPS satellites with GEO and LEO has great potential for

improving the navigation performance. The simulation

shows an average improvement of satellite visibility and

different precision accuracy of DOP

(GDOP,PDOP,HDOP,VDOP). This additional algorithm

not only may improve the total system availability with

less computation, but also improves user’s position

accuracy. An ordinary GPS receiver needs more channels

to select a set of satellites, but a GPS receiver with

integrated algorithm may require only four-channel

calculation. GPS or GLONASS as standalone systems are

acceptable for much navigation without augmentation.

Combining the two or more systems the receiver with the

capability of selection, the best set of satellites depending

on minimum dilution of precision without the need for

differential techniques.

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