17 appendix

235

APPENDIX A

Coordinate Transformation

Heikkinen, M. Exact transformation method for the

conversion of ECEF coordinates to Geodetic coordinates

Heikkinen, M. had derived a set of equations for exact conversion of ECEF to

Geodetic coordinates (Zhu 1993).

Heikkinen‘s formula is

22 yxr

E2 = a2 – b2

F = 54b2z2

G = r2 + (1 - e2)z2 – e2 E2

3

24

G

rFec

3 2 21 cccs

22)11

(3 Gs

s

FP

PeQ 421

2)1(

)1()

11(

21

22222

0rP

QQ

zeP

Q

a

Q

rPer

220

2 )( zrerU

2220

2 )1()( zererV

aV

zbz

2

0

)1(2

aV

bUh

236

2

222'

b

bae

)arctan( 02'

r

zez

)arctan(z

y

where x, y, z are ECEF coordinates

ϕ, 𝝀, h are Geodetic latitude, Geodetic longitude and altitude respectively

A is the ellipsoidal equatorial radius (a=6378.138 Km for model WGS-84)

e is the eccentricity of ellipsoid (e2=0.00669437999 for model WGS-84)

b is the ellipsoidal polar radius )1( 2eab

e‘ is the second eccentricity

E is linear eccentricity

Coordinate transformation from Geodetic coordinates to

ECEF coordinates

The relations expressing ECEF coordinates (X, Y, Z) of a point ‗P‘ in terms of

Geodetic coordinates (ϕ, 𝝀, h) are

)cos()cos()( hRX

)sin()cos()( hRY

)sin())1(( 2 heZ

where R=Radius of prime vertical

)(sin1 22 e

aR

237

APPENDIX B

Satellite Position Estimation Algorithm

B.1 Satellite position estimation algorithm from the GPS satellite

broadcast ephemeris

Satellite position is computed using the six Keplerian elements ( a , e , oi , ω,

o , oM ) which describe a smooth, elliptical orbit with the satellite position

being a function of time, since ot . But, the earth is not a uniform sphere,

and there are other forces, beside gravity. Net result of these perturbing

forces such as Non-central gravitational force field, Solar radiation pressure,

Equatorial bulge changes the orbit of the GPS satellite with time, and it

should be characterized with an appropriate set of time dependent

parameters.

GPS accounts these perturbations with an expanded orbital parameter set to

retain the Keplerian look. The expanded set of quasi-Keplerian parameters

consists of 15 elements, whose values are specified relative to the reference

epoch. So, there are 16 ephemeris parameters in all. There is also an

identifier parameter, called Issue of Data- Ephemeris (IODE), associated with

each ephemeris parameter set. A change in the value of IODE between two

successive transmissions indicates that the ephemeris parameter set has

been updated. The ephemeris parameters are broadcast by each GPS

satellite and are given below Table B.1.

238

Table B.1 Ephemeris Parameters

In order to compute the satellite positions, position algorithm makes use of

the Keplerian elements along with the additional parameters ( n , , i and

six sine and cosine coefficients) which describe the derivations of an actual

Description Units

M0 Mean Anomaly at reference time semi-circles

∆n Mean motion differences from computed

value

semi-

circles/sec

e Eccentricity

a Square root of the semi major axis m½

Ω0 Longitude of ascending node of orbit plane at

weekly epoch

semi-circles

i0 Inclination angle at reference time semi-circles

ω Argument of perigee semi-circles

Rate of right ascension semi-

circles/sec

di/dt Rate of inclination angle semi-

circles/sec

Cuc Amplitude of the cosine harmonic correction

term to the argument of latitude

Radians

Cus Amplitude of the sine harmonic correction

term to the argument of latitude

Radians

Crc Amplitude of the cosine harmonic correction

term to the orbit radius

Meters

Crs amplitude of the sine harmonic correction

term to the orbit radius

Meters

Cic Amplitude of the cosine harmonic correction

term to the angle of inclination

Radians

Cis Amplitude of the sine harmonic correction

term to the angle of inclination

Radians

toe Reference time for ephemeris computation Sec

IODE Issue of data – ephemeris

239

satellite motion from this smooth ellipse. These parameters are computed

from the ephemeris data elements listed out in the Table B.1 (Kaplan, E.,

1996).

B.2 Steps of satellite position computation.

Step 1: Find the semi-major axis of elliptical orbit

2aa (meter)

Step 2: The GPS time of transmission, t corrected for transit time

sv svt t t

Step 3: Calculate the time difference ( kt ) between the time ( t ) and the epoch

time ( oet ) and must account for the beginning or end of the week

*

oek ttt

302400 604800

302400 604800

k oe k k

k oe k k

If t t t then t t

If t t t then t t

Where, t represent the coarse GPS system time, and the value can be

obtained from Time Of Week (TOW). In addition, oet can be obtained from the

ephemeris data, 302,400 is the time of half a week in seconds.

Step 4: Calculate the mean motion,

0 3n n n n

a

(rad/sec)

Step 5: From this n value the mean anomaly can be found as,

kk ntMM 0

240

Step 6: The eccentric anomaly Ek can be found from Kepler‘s equation (M = E

−e sinE) through iteration as

sink k kE M e E with, Eo = Mk

Step 7: Once Ek is obtained, the value of concise expression of the orbital

radius, rk can be found from,

(1 cos )k kr a e E

Step 8: Now let us find the true anomaly . The value can be found from,

2

1 1sin 1 sin /(1 cos )tan tan

cos (cos ) /(1 cos )

k k kk

k k k

e E e E

E e e E

Step 9: The argument of perigee (ω) can be found from the ephemeris data.

Using the definition of argument of latitude, the value of

k k

Step 10: Calculate the correction terms of argument of latitude, orbital

radius and inclination angle;

sin 2 cos 2

sin 2 cos 2

sin 2 cos 2

k us k uc k

k rs k rc k

k is k ic k

u C C

r C C

i C C

Step 11: Calculate the following correction terms as,

0

k k k

k k k

k k k

u u

r r r

i i i it

Step 12: Compute the longitude of the ascending node k by adding the

right ascension parameter, 0 and the mean earth rotation rate,

241

0 ( )k e k e oet t

Step 13: Once all the necessary parameters are obtained, the position of the

satellite can be found by applying the three rotations (through ϕ, i and Ω)

described previously. The satellite position calculated in this equation is in

the ECEF frame.

cos cos sin cos sin

cos sin sin cos cos

sin sin

k k k k k k k k

k k k r k k k k

k k k k

x r u r u i

y r u r u i

z r u i

242

APPENDIX C

Performance comparison of the three

navigation solutions after smoothening over an

hour data

The user position, receiver clock error and GDOP values after

smoothening over an hour due to the recursive least squares (RLS)

approximation and the proposed Jacobian determinant based

multipolynomial resultant technique (JMR) and Minkowski function based

absolute position (MAP) algorithms corresponding to 19th February, 2010 are

given in Table C.1, Table C.2 and Table C.3 respectively. The error in user

position estimated by all the three methods is tabulated in Table C.4. The

mean, standard deviation (σ) and variance (σ2) values for the errors estimated

by all the methods are also given in Table C.4. From Table C.4 it is clear

that, the JMR method gives better user position than the RLS method and

the MAP method provides better user position estimation than the RLS and

the JMR methods.

243

Table C.1 User position, Receiver clock error and GDOP after smoothening over an hour due to the RLS method corresponding to 19th February, 2010

S.

No.

GPS

Time

in

Hours

User position due to RLS navigation solution

(in meters) Receiver

clock

error (m)

GDOP

X position Y position Z position

1 1 706913.753 6035844.939 1929974.791 23.905 3.1275

2 2 706916.421 6035826.189 1929975.551 18.731 2.7528

3 3 706925.523 6035818.412 1929960.430 10.507 1.7314

4 4 706916.800 6035821.733 1929963.911 10.796 3.9974

5 5 706920.832 6035834.185 1929972.710 19.931 2.3259

6 6 706932.166 6035823.154 1929970.024 18.871 1.8307

7 7 706929.566 6035832.590 1929975.443 26.765 2.006

8 8 706925.258 6035841.578 1929974.582 28.264 2.4958

9 9 706927.142 6035851.078 1929971.565 36.326 1.6777

10 10 706919.133 6035848.162 1929974.525 34.739 1.8995

11 11 706923.504 6035844.498 1929979.686 36.426 1.5867

12 12 706916.469 6035840.705 1929967.466 35.68 2.4262

13 13 706921.925 6035831.350 1929969.148 23.558 2.1316

14 14 706912.560 6035818.556 1929969.984 15.619 2.6003

15 15 706917.317 6035810.326 1929979.467 12.828 2.0269

16 16 706925.892 6035810.877 1929964.705 3.6666 4.1812

17 17 706927.179 6035831.425 1929973.070 16.913 3.2442

18 18 706924.260 6035807.863 1929960.492 -1.623 6.3243

19 19 706917.307 6035828.023 1929965.507 14.016 2.1818

20 20 706920.627 6035827.443 1929969.139 12.802 2.5979

21 21 706924.526 6035820.672 1929971.435 8.2714 2.9329

22 22 706924.145 6035818.980 1929968.801 6.1742 4.5792

23 23 706920.269 6035833.131 1929974.438 17.171 1.9936

24 24 706909.534 6035823.223 1929973.547 11.442 3.6181

Mean 706921.338 6035828.712 1929970.851 18.407 2.7612

Standard

deviation (σ) 5.593 12.081 5.196 10.571 1.1181

Variance (σ2) 31.285 145.950 27.003 111.75 1.2502

244

Table C.2 User position, Receiver clock error and GDOP after smoothening

over an hour due to the JMR method corresponding to 19th February, 2010

S.

No.

GPS

Time

in

Hours

User position due to JMR navigation

solution (in meters) Receiver

clock

error (m)

GDOP


1 1 706915.409 6035836.118 1929968.837 18.303 3.1275

2 2 706917.455 6035831.793 1929973.052 19.67 2.7528

3 3 706924.784 6035824.539 1929971.653 16.199 1.7314

4 4 706918.161 6035822.429 1929965.507 11.612 3.9974

5 5 706921.787 6035833.861 1929974.483 22.711 2.3259

6 6 706925.270 6035824.049 1929972.750 20.457 1.8307

7 7 706928.227 6035836.064 1929972.962 26.277 2.006

8 8 706928.902 6035839.359 1929974.577 27.445 2.4958

9 9 706925.108 6035843.013 1929977.348 31.427 1.6777

10 10 706918.988 6035840.919 1929975.528 28.721 1.8995

11 11 706920.649 6035844.167 1929974.553 32.182 1.5867

12 12 706913.963 6035843.049 1929975.283 35.106 2.4262

13 13 706918.198 6035832.822 1929971.630 22.132 2.1316

14 14 706912.759 6035818.928 1929970.782 15.272 2.6003

15 15 706917.806 6035817.663 1929974.942 13.153 2.0269

16 16 706930.049 6035811.335 1929965.845 1.7982 4.1812

17 17 706928.270 6035831.040 1929972.129 16.821 3.2442

18 18 706925.517 6035813.683 1929963.359 2.8727 6.3243

19 19 706920.316 6035820.487 1929967.365 9.9814 2.1818

20 20 706921.517 6035825.963 1929970.849 10.872 2.5979

21 21 706924.644 6035823.892 1929972.580 9.5658 2.9329

22 22 706923.666 6035821.024 1929971.840 7.8517 4.5792

23 23 706919.852 6035833.503 1929974.730 16.718 1.9902

24 24 706916.167 6035823.084 1929972.259 8.3241 3.6181

Mean 706921.561 6035828.866 1929971.868 17.728 2.761

Standard

deviation (σ) 4.888 9.587 3.483 9.1967 1.118

Variance (σ2) 23.897 91.929 12.137 84.579 1.250

245


over an hour due to the MAP method corresponding to 19th February, 2010

S.

No.

GPS

Time

in

Hours

User position due to MAP navigation


clock

error (m)

GDOP


1 1 706947.036 6035832.468 1929969.041 18.262 3.1275

2 2 706948.586 6035828.042 1929972.572 19.608 2.7528

3 3 706956.210 6035821.883 1929971.398 16.687 1.7314

4 4 706951.547 6035822.345 1929968.762 15.731 3.5522

5 5 706953.536 6035830.467 1929974.643 22.898 2.3259

6 6 706957.208 6035820.257 1929972.657 20.405 1.8307

7 7 706959.731 6035833.943 1929973.317 27.094 2.006

8 8 706960.423 6035835.885 1929974.786 27.68 2.4958

9 9 706957.207 6035840.221 1929977.011 31.846 1.6777

10 10 706951.036 6035838.530 1929975.813 29.482 1.8995

11 11 706952.683 6035841.599 1929974.664 32.663 1.5867

12 12 706945.982 6035840.877 1929975.290 35.707 2.4262

13 13 706949.848 6035830.824 1929971.988 23.049 2.1345

14 14 706944.455 6035815.446 1929970.660 15.374 2.6003

15 15 706949.496 6035815.856 1929975.670 13.959 2.0269

16 16 706955.051 6035819.339 1929973.823 12.618 3.3867

17 17 706959.847 6035827.090 1929972.105 16.692 3.2442

18 18 706957.351 6035816.181 1929967.966 9.2612 5.8265

19 19 706952.150 6035817.281 1929967.607 10.314 2.1818

20 20 706953.271 6035823.025 1929971.023 11.357 2.5979

21 21 706956.282 6035820.304 1929972.502 9.6175 2.9329

22 22 706955.482 6035817.565 1929971.843 7.9621 4.5792

23 23 706951.850 6035831.016 1929974.606 17.355 1.9936

24 24 706948.223 6035820.619 1929972.703 9.1493 3.6181

Mean 706953.104 6035826.711 1929972.602 18.949 2.689

Standard

deviation (σ) 4.489 8.657 2.540 8.222 0.9956

Variance (σ2) 20.158 74.952 6.452 67.604 0.99123

246

Table C.4 User position errors after smoothening over an hour due to all the

three methods corresponding to 19th February, 2010

S.

No.

GPS

Time

in

Hours

RLS method JMR method MAP method

X error

(m)

Y error

(m)

Z error

(m)

X error

(m)

Y error

(m)

Z error

(m)

X error

(m)

Y error

(m)

Z error

(m)

1 1 57.156 96.083 34.791 55.5 104.9 40.745 23.873 108.55 40.541

2 2 54.488 114.83 34.03 53.453 109.23 36.529 22.323 112.98 37.01

3 3 45.386 122.61 49.152 46.125 116.48 37.928 14.699 119.14 38.184

4 4 54.108 119.29 45.671 52.748 118.59 44.075 19.362 118.68 40.82

5 5 50.077 106.84 36.872 49.122 107.16 35.099 17.373 110.55 34.939

6 6 38.743 117.87 39.558 45.638 116.97 36.832 13.701 120.77 36.925

7 7 41.343 108.43 34.138 42.682 104.96 36.62 11.178 107.08 36.264

8 8 45.651 99.444 34.999 42.007 101.66 35.005 10.485 105.14 34.795

9 9 43.767 89.944 38.017 45.801 98.009 32.234 13.702 100.8 32.57

10 10 51.776 92.86 35.057 51.921 100.1 34.054 19.873 102.49 33.768

11 11 47.405 96.524 29.896 50.26 96.855 35.029 18.226 99.423 34.917

12 12 54.44 100.32 42.115 56.946 97.973 34.298 24.926 100.15 34.291

13 13 48.984 109.67 40.433 52.711 108.2 37.952 21.061 110.2 37.593

14 14 58.349 122.47 39.598 58.15 122.09 38.799 26.454 125.58 38.922

15 15 53.592 130.7 30.115 53.103 123.36 34.639 21.413 125.17 33.912

16 16 45.017 130.15 44.876 40.86 129.69 43.737 15.858 121.68 35.758

17 17 43.73 109.6 36.512 42.639 109.98 37.453 11.062 113.93 37.477

18 18 46.649 133.16 49.089 45.392 127.34 46.223 13.558 124.84 41.616

19 19 53.602 113.0 44.074 50.593 120.54 42.216 18.759 123.74 41.975

20 20 50.282 113.58 40.443 49.392 115.06 38.733 17.637 118.0 38.558

21 21 46.383 120.35 38.147 46.265 117.13 37.002 14.627 120.72 37.079

22 22 46.763 122.04 40.78 47.243 120.0 37.741 15.427 123.46 37.739

23 23 50.64 107.89 35.144 51.056 107.52 34.851 19.058 110.01 34.975

24 24 61.375 117.8 36.035 54.742 117.94 37.323 22.686 120.4 36.879

Mean 49.571 112.30

9 38.730 49.347

112.15

6 37.713 17.804 114.31 36.979

Standard

deviation (σ) 5.593 12.081 5.196 4.888 9.587 3.483 4.489 8.657 2.540

Variance

(σ2) 31.285

145.95

0 27.003 23.897 91.929 12.137 20.158 74.952 6.452

247


smoothening over an hour due to the recursive least squares approximation

and the proposed Jacobian determinant based multipolynomial resultant

technique and Minkowski function based absolute position algorithms

corresponding to 20th February, 2010 are given in Table C.5, Table C.6 and

Table C.7 respectively. The error in user position estimated by all the three

methods is tabulated in Table C.8. From Table C.8 it is clear that, the JMR

method gives better user position than the RLS method and the MAP method

provides better user position estimation than the RLS and the JMR methods.


smoothening over an hour due to the the recursive least squares

approximation and the proposed Jacobian determinant based

multipolynomial resultant technique and Minkowski function based absolute

position algorithms corresponding to 21st February, 2010 are given in Table

C.9, Table C.10 and Table C.11 respectively. The error in user position

estimated by all the three methods is tabulated in Table C.12. The mean,

standard deviation (σ) and variance (σ2) values for the errors estimated by all

the methods are also given in Table C.12. From Table C.12 it is clear that,

the JMR method gives better user position than the RLS method and the

MAP method provides better user position estimation than the RLS and the

MAP methods.

248

Table C.5 User position, Receiver clock error and GDOP after smoothening over an hour due to the RLS method corresponding to 20th February, 2010

S.

No.

GPS

Time

in

Hours



clock

error (m)

GDOP


1 1 706911.368 6035853.214 1929975.298 27.432 3.207

2 2 706913.936 6035832.682 1929975.352 20.102 2.670

3 3 706922.988 6035826.047 1929966.197 18.509 1.716

4 4 706913.688 6035830.015 1929964.836 18.147 4.311

5 5 706921.470 6035838.911 1929971.598 25.752 2.319

6 6 706929.041 6035827.523 1929967.802 22.899 1.815

7 7 706929.654 6035833.721 1929973.289 27.267 2.042

8 8 706925.091 6035844.579 1929972.095 29.691 2.484

9 9 706926.286 6035854.867 1929972.078 36.185 1.686

10 10 706918.716 6035846.865 1929977.345 32.204 1.914

11 11 706921.212 6035844.554 1929980.523 36.72 1.589

12 12 706914.858 6035846.848 1929968.480 40.316 2.680

13 13 706918.646 6035823.040 1929970.569 11.557 2.220

14 14 706915.516 6035817.623 1929971.449 8.7076 2.558

15 15 706912.837 6035814.329 1929981.767 17.663 1.883

16 16 706921.210 6035830.186 1929975.622 23.872 4.690

17 17 706925.725 6035830.727 1929974.142 14.993 2.689

18 18 706926.735 6035826.719 1929973.248 12.739 2.294

19 19 706920.135 6035835.579 1929973.973 24.425 1.969

20 20 706915.763 6035832.802 1929969.025 19.586 2.566

21 21 706925.217 6035811.891 1929973.786 -1.4255 2.980

22 22 706920.995 6035816.181 1929972.494 2.0776 4.719

23 23 706918.587 6035827.105 1929974.173 17.475 1.683

24 24 706922.691 6035830.765 1929978.665 19.076 2.759

Mean 706920.515 6035832.366 1929973.075 21.082 2.560

Standard

deviation (σ) 5.250 11.727 4.142 10.276 0.895

Variance (σ2) 27.567 137.528 17.163 105.59 0.801

249


over an hour due to the JMR method corresponding to 20th February, 2010

S.

No.

GPS

Time

in

Hours



clock

error (m)

GDOP


1 1 706913.636 6035841.921 1929968.503 21.094 3.2078

2 2 706915.534 6035838.438 1929974.802 21.836 2.6703

3 3 706921.524 6035832.855 1929975.953 24.707 1.7161

4 4 706914.466 6035832.447 1929965.615 19.612 4.311

5 5 706920.405 6035837.200 1929973.436 27.914 2.3199

6 6 706922.521 6035828.371 1929969.725 25.474 1.815

7 7 706927.404 6035838.191 1929970.356 26.756 2.0428

8 8 706928.000 6035840.439 1929972.641 27.892 2.4847

9 9 706923.813 6035845.397 1929977.427 31.521 1.6863

10 10 706918.918 6035843.566 1929977.366 27.636 1.9147

11 11 706919.590 6035844.896 1929975.545 33.66 1.5893

12 12 706911.040 6035849.413 1929978.513 40.101 2.6809

13 13 706916.911 6035825.226 1929971.290 11.048 2.2201

14 14 706915.119 6035817.264 1929971.610 8.0545 2.5588

15 15 706916.130 6035820.198 1929976.916 17.821 1.883

16 16 706923.167 6035827.129 1929976.239 22.209 4.6907

17 17 706924.918 6035829.505 1929973.996 15.327 2.6899

18 18 706923.869 6035822.029 1929971.490 10.208 2.2947

19 19 706919.597 6035831.092 1929971.955 20.702 1.9699

20 20 706918.020 6035829.500 1929970.132 17.181 2.566

21 21 706923.270 6035814.384 1929974.810 -0.4938 2.9802

22 22 706919.955 6035820.139 1929977.188 5.1335 4.7195

23 23 706920.316 6035825.990 1929973.271 18.004 1.683

24 24 706922.566 6035823.948 1929976.844 16.415 2.7598

Mean 706920.029 6035831.647 1929973.567 20.409 2.5606

Standard

deviation (σ) 4.343 9.693 3.306 9.375 0.89518

Variance (σ2) 18.863 93.958 10.934 87.893 0.80135

250


over an hour due to the MAP method corresponding to 20th February, 2010

S.

No.

GPS

Time

in

Hours



clock

error (m)

GDOP X position Y position Z position

1 1 706945.162 6035838.126 1929968.704 20.962 3.2078

2 2 706946.669 6035834.634 1929974.257 21.733 2.6703

3 3 706952.730 6035830.432 1929975.737 25.324 1.7161

4 4 706946.024 6035829.212 1929965.700 19.933 4.2606

5 5 706952.148 6035833.431 1929973.391 27.873 2.3199

6 6 706954.474 6035824.135 1929969.589 25.21 1.815

7 7 706958.868 6035836.237 1929970.756 27.669 2.0428

8 8 706959.544 6035837.186 1929972.830 28.255 2.4847

9 9 706955.766 6035842.350 1929976.925 31.745 1.6863

10 10 706950.942 6035840.727 1929977.699 28.204 1.9147

11 11 706951.436 6035841.819 1929975.557 33.894 1.5893

12 12 706943.152 6035847.642 1929978.527 40.858 2.6809

13 13 706948.425 6035822.621 1929971.618 11.658 2.2237

14 14 706946.823 6035813.661 1929971.563 8.1015 2.5588

15 15 706947.553 6035818.282 1929977.594 18.7 1.8825

16 16 706954.686 6035824.731 1929976.858 23.142 4.7193

17 17 706956.677 6035825.466 1929973.905 15.13 2.6899

18 18 706955.815 6035818.501 1929971.683 10.362 2.2947

19 19 706951.418 6035828.015 1929972.271 21.1 1.9699

20 20 706949.617 6035826.223 1929970.196 17.455 2.566

21 21 706955.084 6035811.162 1929974.854 -0.1478 2.9577

22 22 706951.825 6035816.826 1929977.170 5.3062 4.7195

23 23 706951.703 6035822.764 1929972.952 18.205 1.683

24 24 706954.393 6035820.895 1929977.158 16.778 2.7598

Mean 706951.706 6035828.545 1929973.646 20.727 2.5589

Standard

deviation (σ) 4.362 9.811 3.305 9.444 0.89344

Variance (σ2) 19.035 96.256 10.924 89.195 0.79823

251

Table C.8 User position errors after smoothening over an hour due to all the three methods corresponding to 20th February, 2010

S.

No.

GPS

Time

in

Hours


X error

(m)

Y error

(m)

Z error

(m)

X error

(m)

Y error

(m)

Z error

(m)

X error

(m)

Y error

(m)

Z error

(m)

1 1 59.541 87.808 34.284 57.273 99.101 41.079 25.746 102.9 40.877

2 2 56.973 108.34 34.23 55.375 102.58 34.78 24.239 106.39 35.324

3 3 47.92 114.97 43.384 49.384 108.17 33.629 18.179 110.59 33.845

4 4 57.221 111.01 44.745 56.443 108.58 43.967 24.884 111.81 43.882

5 5 49.439 102.11 37.983 50.504 103.82 36.146 18.761 107.59 36.191

6 6 41.868 113.5 41.779 48.388 112.65 39.856 16.435 116.89 39.993

7 7 41.255 107.3 36.292 43.505 102.83 39.226 12.04 104.79 38.825

8 8 45.818 96.443 37.487 42.908 100.58 36.941 11.365 103.84 36.752

9 9 44.623 86.155 37.503 47.095 95.625 32.155 15.143 98.672 32.657

10 10 52.193 94.157 32.236 51.99 97.456 32.216 19.967 100.29 31.883

11 11 49.697 96.468 29.059 51.319 96.126 34.037 19.473 99.203 34.024

12 12 56.051 94.174 41.102 59.869 91.609 31.069 27.756 93.38 31.054

13 13 52.263 117.98 39.013 53.998 115.8 38.292 22.483 118.4 37.963

14 14 55.393 123.4 38.133 55.79 123.76 37.972 24.086 127.36 38.018

15 15 58.071 126.69 27.814 54.779 120.82 32.666 23.356 122.74 31.987

16 16 49.699 110.84 33.96 47.742 113.89 33.343 16.222 116.29 32.724

17 17 45.184 110.3 35.44 45.99 111.52 35.586 14.232 115.56 35.677

18 18 44.173 114.3 36.334 47.04 118.99 38.092 15.093 122.52 37.898

19 19 50.774 105.44 35.609 51.312 109.93 37.626 19.491 113.01 37.311

20 20 55.146 108.22 40.557 52.889 111.52 39.45 21.292 114.8 39.386

21 21 45.692 129.13 35.795 47.638 126.64 34.772 15.825 129.86 34.728

22 22 49.913 124.84 37.087 50.953 120.88 32.394 19.084 124.2 32.411

23 23 52.322 113.92 35.408 50.593 115.03 36.31 19.205 118.26 36.63

24 24 48.218 110.26 30.917 48.343 117.07 32.737 16.516 120.13 32.424

Mean 50.394 108.66 36.506 50.88 109.37 36.014 19.203 112.48 35.936

Standard

deviation (σ) 5.2505 11.727 4.1429 4.343 9.693 3.306 4.363 9.811 3.305

Variance (σ2) 27.568 137.53 17.163 18.864 93.959 10.935 19.036 96.256 10.925

252

Table C.9 User position, Receiver clock error and GDOP after smoothening over an hour due to the RLS method corresponding to 21st February, 2010

S.

No.

GPS

Time

in

Hours



clock

error (m)

GDOP


1 1 706916.489 6035836.222 1929970.738 14.457 3.0413

2 2 706929.258 6035814.264 1929967.444 0.53577 2.3629

3 3 706922.857 6035823.570 1929962.198 14.575 1.719

4 4 706909.096 6035823.147 1929959.616 9.7227 4.5943

5 5 706921.230 6035830.677 1929969.247 17.878 2.3004

6 6 706928.678 6035821.475 1929970.090 17.711 1.8048

7 7 706931.466 6035835.667 1929976.289 29.115 2.0716

8 8 706928.242 6035843.181 1929971.084 28.324 2.4725

9 9 706926.208 6035855.568 1929969.505 39.092 1.727

10 10 706918.307 6035850.955 1929976.061 35.604 1.9011

11 11 706922.349 6035841.359 1929978.280 33.361 1.5953

12 12 706913.702 6035845.147 1929967.027 37.521 2.915

13 13 706916.741 6035828.356 1929969.825 22.186 2.3161

14 14 706910.675 6035818.813 1929967.584 17.897 2.5267

15 15 706916.804 6035816.955 1929981.352 17.137 1.888

16 16 706921.475 6035836.190 1929976.877 25.762 5.0016

17 17 706927.893 6035830.238 1929974.628 17.625 2.6309

18 18 706927.528 6035826.435 1929974.621 14.873 2.3046

19 19 706921.855 6035831.768 1929974.216 19.19 2.0235

20 20 706920.595 6035826.448 1929970.272 12.738 2.5347

21 21 706924.206 6035823.097 1929972.540 10.425 3.0455

22 22 706925.761 6035805.324 1929964.042 -3.088 4.8285

23 23 706918.449 6035833.061 1929974.890 17.099 1.7216

24 24 706920.018 6035844.815 1929978.891 20.885 2.8551

Mean 706921.662 6035830.947 1929971.555 19.609 2.590

Standard

deviation (σ) 5.9257 12.0391 5.360 10.563 0.959

Variance (σ2) 35.1141 144.9411 28.739 111.58 0.920

253


over an hour due to the JMR method corresponding to 21st February, 2010

S.

No.

GPS

Time

in

Hours



clock

error (m)

GDOP


1 1 706919.854 6035826.239 1929967.905 9.646 3.0413

2 2 706922.708 6035824.722 1929971.358 10.096 2.3629

3 3 706922.257 6035828.803 1929972.772 18.745 1.719

4 4 706910.217 6035825.888 1929960.461 11.149 4.5943

5 5 706919.414 6035827.960 1929971.003 19.337 2.3004

6 6 706922.850 6035823.040 1929970.943 20.561 1.8048

7 7 706928.757 6035839.498 1929971.620 28.52 2.0716

8 8 706930.550 6035838.277 1929971.400 26.493 2.4725

9 9 706924.236 6035846.134 1929977.214 33.1 1.727

10 10 706919.496 6035846.643 1929977.834 30.853 1.9011

11 11 706919.377 6035844.052 1929974.639 30.296 1.5953

12 12 706910.086 6035847.411 1929977.299 37.733 2.915

13 13 706914.747 6035830.211 1929970.953 21.644 2.3161

14 14 706911.559 6035820.092 1929969.241 17.907 2.5267

15 15 706918.858 6035820.539 1929975.244 15.794 1.888

16 16 706923.795 6035835.346 1929978.032 24.873 5.0016

17 17 706926.422 6035828.665 1929974.167 17.492 2.6309

18 18 706924.913 6035821.020 1929972.476 12.155 2.3046

19 19 706920.840 6035827.596 1929971.697 14.798 2.0235

20 20 706920.502 6035826.419 1929971.330 11.799 2.5347

21 21 706923.557 6035822.454 1929972.495 9.8595 3.0455

22 22 706925.060 6035809.367 1929967.529 0.053 4.8285

23 23 706919.506 6035832.055 1929974.919 16.451 1.7216

24 24 706919.464 6035833.041 1929975.523 16.326 2.8551

Mean 706920.793 6035830.228 1929972.419 18.987 2.5909

Standard

deviation (σ) 5.217 9.604 3.888 8.903 0.95931

Variance (σ2) 27.219 92.252 15.122 79.279 0.92027

254


over an hour due to the MAP method corresponding to 21st February, 2010

S.

No.

GPS

Time

in

Hours



clock

error (m)

GDOP


1 1 706951.549 6035822.207 1929968.106 9.3755 3.0413

2 2 706954.079 6035820.940 1929971.048 10.018 2.3629

3 3 706953.750 6035826.207 1929972.506 19.269 1.719

4 4 706942.807 6035824.606 1929962.555 13.921 4.3571

5 5 706951.174 6035824.257 1929970.944 19.325 2.3004

6 6 706954.687 6035818.859 1929970.829 20.323 1.8048

7 7 706960.232 6035837.433 1929972.150 29.425 2.0716

8 8 706962.177 6035834.927 1929971.547 26.782 2.4725

9 9 706956.304 6035843.752 1929976.784 33.683 1.727

10 10 706951.502 6035844.187 1929978.124 31.555 1.9011

11 11 706951.430 6035841.597 1929974.637 30.799 1.5953

12 12 706942.206 6035845.451 1929977.258 38.37 2.915

13 13 706946.735 6035827.345 1929971.244 21.982 2.3161

14 14 706943.242 6035816.736 1929969.042 18.071 2.5267

15 15 706950.368 6035818.654 1929975.938 16.664 1.888

16 16 706955.574 6035831.953 1929978.165 25.023 5.0016

17 17 706958.199 6035824.663 1929974.114 17.324 2.6309

18 18 706956.865 6035817.457 1929972.721 12.299 2.3046

19 19 706952.782 6035824.941 1929972.195 15.436 2.0235

20 20 706952.243 6035823.214 1929971.445 12.114 2.5347

21 21 706955.278 6035819.003 1929972.583 10.036 3.0455

22 22 706956.747 6035809.784 1929969.221 3.4163 4.4425

23 23 706951.106 6035828.960 1929974.644 16.717 1.7216

24 24 706951.402 6035830.358 1929976.057 16.912 2.8551

Mean 706952.602 6035827.396 1929972.661 19.535 2.5649

Standard

deviation (σ) 5.085 9.617 3.565 8.705 0.90114

Variance (σ2) 25.858 92.500 12.714 75.79 0.81206

255

Table C.12 User position errors after smoothening over an hour due to all

the three methods corresponding to 21st February, 2010

S.

No.

GPS

Time

in

Hours


X error

(m)

Y error

(m)

Z error

(m)

X error

(m)

Y error

(m)

Z error

(m)

X error

(m)

Y error

(m)

Z error

(m)

1 1 54.419 104.8 38.844 51.055 114.78 41.677 19.36 118.81 41.476

2 2 41.651 126.76 42.138 48.2 116.3 38.223 16.83 120.08 38.533

3 3 48.052 117.45 47.384 48.652 112.22 36.81 17.159 114.82 37.076

4 4 61.813 117.88 49.965 60.691 115.13 49.12 28.102 116.42 47.026

5 5 49.678 110.35 40.334 51.495 113.06 38.578 19.735 116.76 38.637

6 6 42.231 119.55 39.491 48.059 117.98 38.638 16.222 122.16 38.753

7 7 39.442 105.36 33.293 42.152 101.52 37.961 10.677 103.59 37.432

8 8 42.667 97.841 38.497 40.359 102.75 38.182 8.7315 106.1 38.034

9 9 44.701 85.454 40.077 46.673 94.888 32.368 14.605 97.271 32.797

10 10 52.602 90.067 33.52 51.413 94.379 31.748 19.406 96.835 31.458

11 11 48.56 99.663 31.302 51.531 96.97 34.943 19.479 99.425 34.945

12 12 57.207 95.875 42.555 60.823 93.611 32.283 28.703 95.571 32.324

13 13 54.167 112.67 39.757 56.161 110.81 38.629 24.174 113.68 38.337

14 14 60.234 122.21 41.997 59.35 120.93 40.341 27.667 124.29 40.54

15 15 54.105 124.07 28.229 52.051 120.48 34.338 20.54 122.37 33.644

16 16 49.434 104.83 32.705 47.114 105.68 31.549 15.335 109.07 31.416

17 17 43.016 110.78 34.953 44.487 112.36 35.415 12.71 116.36 35.468

18 18 43.381 114.59 34.96 45.996 120.0 37.106 14.044 123.57 36.86

19 19 49.054 109.25 35.366 50.069 113.43 37.885 18.127 116.08 37.387

20 20 50.314 114.57 39.31 50.407 114.6 38.252 18.666 117.81 38.136

21 21 46.703 117.93 37.042 47.352 118.57 37.087 15.631 122.02 36.998

22 22 45.148 135.7 45.54 45.849 131.66 42.053 14.162 131.24 40.36

23 23 52.459 107.96 34.692 51.402 108.97 34.663 19.803 112.06 34.938

24 24 50.891 96.207 30.69 51.445 107.98 34.059 19.507 110.66 33.525

Mean 49.247 110.07 38.027 50.116 110.79 37.163 18.307 113.63 36.921

Standard

deviation (σ) 5.925 12.039 5.3609 5.217 9.604 3.888 5.085 9.617 3.565

Variance (σ2) 35.114 144.94 28.739 27.219 92.252 15.123 25.859 92.501 12.714

17 appendix

Documents