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Positioning System Using Low Earth Orbit Constellations
POSITIONING SYSTEM USING LOW EARTH ORBIT CONSTELLATIONS Der-Ming Ma', Shen-You Zhai^ and Huan-Ein Shen^
'Associate Professor, Department of Aerospace Engineering, Tamkang University, Tamsui, Taiwan 25137, Republic of China, Tel: -^886-2-2623-7600, Fax: +886-2-2620-9746, E-mail: [email protected]\ ' Graduate Student, Department ofAerospace Engineering, Tamkang University, Tamsui, Taiwan 25137 Republic of China. ^ Graduate Student, Department of Mechanical Engineering, Tamkang University, Tamsui, Taiwan 25137, Republic of China. Currently, in military service.
ABSTRACT A scheme of designing low Earth orbit satellite positioning system is proposed. An example low Earth orbit satellite constellation providing the positioning ftmction over certain region is presented. The proposed constellation consists of 75 satellites with four different inclinations. The satellites are in the orbit of height of about 800 Km. It can provide the positioning function over the region with radius of about 3000 Km center at the chosen site with longitude of 122° E and latitude of 25°N. The construction of the constellation is based on the requirement of good values of the Dilution of Precision over the specific region at same time. Also the study of the relativistic effects of the system shows that the requirement of keeping the orbit of the low Earth orbit satellite positioning system circle is less stringent than the GPS's. INTRODUCTION As small satellites becoming more capable through miniaturized electronic and on-board processing, the construction of low-cost satellite constellations becomes economically feasible. With the feasibility of LightSats and the possibility of smaller, relatively inexpensive launch systems, the low-altitude satellite constellations receive increased attention. The potential of low-altitude satellite constellations' applications is significant. The positioning function is one of them. The purpose of this paper is to present a low Earth orbit satellite constellation which provides the positioning function as the GPS does over certain region. In order to provide good accuracy of point positioning, the constellations have at least four satellites with good Dilution of Precision (DOP)^''^^ over the specified region at same time. The DOP factor is a measure for the geometry of the visible satellites. The good geometry should mirror a low DOP value. It is the reciprocal value of the volume of the geometric body, which is formed by the intersection points of the site-satellite vectors with the unit sphere centered at the reference site. In the application of GPS, the purposes of DOP are twofold. First, it is useful in planning a survey and, secondly, it may be helpful in interpreting processed baseline vectors. 314 —
In this paper the concept of DOP is further extended to be the design criterion in designing the low Earth orbit satellite constellation. The DOP is expressed in the topocentric coordinate system with its axes along the local north, east and vertical. The position DOP is then has tow components: HDOP, the dilution of precision in the horizontal position and the VDOP, denoting the corresponding value for the vertical component. The time dilution of precision is an another important factor to good accuracy of positioning. In the paper both position and time DOPs are considered in designing the constellation. In the next two sections the concept of geometric dilution of precision (GDOP) and then the positioning system design procedure using the idea of GDOP are present. As to the GPS, several relativistic effects those associated to the low Earth orbit satellite positioning system are non-negligible. These effects include the following: 1) gravitational field of Earth and Earth rotation; and 2) velocities of satellites. The major effects cause an increase in the satellite clock frequency as observed by a user on Earth. These relativistic effects are also discussed in the section after the designing procedure of the constellation. GEOMETRY DILUTION OF PRECISION Figure 1 illustrates an Earth-centered inertial coordinate system. At r =0, the x-axis passes through the intersection of the equator and the Greenwich meridian, the z-axis passes through the Earth rotation axis, and the >^-axis completes the right-handed coordinate system. Shown in the Figure 1 are the user position (x, y, z), and the position of satellite No. 1 (x,, >',, z,). The range distance between the user and the satellite No.l is indicated as /?,. The basic equations using four satellites are
Z .
N,^
= R,
^(x^x,y^(y-y,y^(z-z,f^T
= R,
yl{x-x,y^iy-y,)'^{z-z,y
/RANGE
x\ ^' EARTH CENTER
liSER(X,Y»Z)
\
^vAQiiATnp
GREENWICHl \ , ^ MERIDLAN
^(x-x,y^{y-y,y-^iz-z,y^T
1ST SATELLITE (X,,Y,^,)
NORTH PLOE
^y^
Y
(1) Figure 1. The relationship between the user and satellite.
^T = R,
where T is clock bias; JCJ, y, and Zj are the /th satellite position; and R^ is the pseudo-range measurements to the zth satellite. Note that the pseudo-ranges are the sum of the actual range plus the offset due to user clock error. For convenience, units have been selected such that the velocity of light is unity. In the equations above, the four pseudo-ranges are the measured quantities. The satellite positions are known, and the four unknowns are the user position (jc, y, z), and the user clock error. The equations are nonlinear. While it is possible to solve these equations directly as they are shown. The equations can be linearized. Let jCn, y^, Zn, and T^ be nominal values of JC, y, z, and T, respectively. The Ax, Ay, Az, and AT are corrections to these nominal values. 7?^, is the nominal pseudo-range measurements from the zth satellite; A/?i be the difference between the actual and nominal measurements. Therefore
— 315
jc = x„ + AJC,
(2)
z = z,^^ Az,
r = r„ + ^T, R,=R,,^AR, and (3) Substituting the Eqs. (2) and (3) into Eq. (1) yields
[ix„+Ax-xy+{y„+Ay-y,y+{z„+Az-zy]'=R„+AR,-T„-AT
(4)
By ignoring second-order error terms, these equations can be written as
[(Z
7V77^,
TTTTr
r r r . (^» -X,)AX+{y„ -y,)Ay+(z„ -z,)AZ 4ix„-x,Y =
+{y„-y,y
+{z„-z,)
(5)
R„+AR,-T„-AT.
Furthermore, substituting Eq. (3) into Eq. (5), we obtain ^" ^ A x + ^'" •^' Ay+ !" R.„ - T„ -f..III
R,.i -T,.
!: Az + AT = AR..
-r
(6)
II
The above four equations are linearized equations that relate pseudo-range measurements to the desired user navigation information as well as the user's clock bias. The known quantities of the right-hand side of the equations are actually incremental pseudo-range measurements. They are the differences between the actual measured pseudo-range and the measurements that had been predicted by the user's computer based on the knowledge of satellite position and the user's most current estimate of his position and clock bias. The quantities to be computed, Ax, A;^, Az, and AT, are corrections that the user will make to his current estimate of position and clock biases. The coefficients of these quantities on the left-hand side are the direction cosines of the line of sight from user to the satellites as projected along the x-, y-, and z-axis. These linearized equations can be conveniently expressed in matrix notation and appear as
a,, a-,
a, a, a 22 a
31
^32
V«41
^42
«-
1VAx^
r AR,^
Ay
AR2
Az
A/J3
1 ^ATj
(7)
A/f4J
Note that Ojj is the direction cosine of the angle between the range to the /th satellite and theyth coordinate. By the use of matrix notation, the above equation can be expressed as follows. Let
316 —
'a,,
^n
or,,
«21
«22
«23
1
«31
«32
«33
1
V«41
«42
«43
1.
As
0
L X = (AX ^y
Az A r y , and r = (A/?,
A/J^
Aif3
Ai?,^.
(8)
Then Ax = r
or
x = A"V.
(9)
Since this relationship is linear, it can be used to express the relationship between the errors in pseudorange measurement and the user quantities. This relationship is therefore s^^X-'s,
(10)
where e, represents the pseudo-range measurement errors and e^ the corresponding errors in user position and clock bias. Let now consider the covariance matrix of the expected errors in pseudo-range measurements and the covariance matrix of the user quantities. The first covariance measurement is a 4X4 array composed of the expected values of the squares and products of the errors in the pseudo-range measurements. The diagonal terms in the matrix, namely the squares of the expected errors, are the variances; i.e., the squares of the expected 1 G values of the pseudo-range measurement errors. The off-diagonal terms are the covariance between the pseudo-range measurement measurements and reflect the correlation to be expected in these measurements. The covariance matrix is given by cov(r) = E{£^€l} = A'' cov(x)A'^ = [A^^ cov(x)A]"'
(11)
Now making the assumption that the pseudo-range errors are uncorrelated from satellite to satellite and equal for each satellite, the pseudo-range covariance matrix reduces to cov(x) = crJl.
(12)
where I is the 4 X 4 unit vector. Thus cov(r) = o-'[A^A]-'.
(13)
We see that the position and time error variances are functions of the diagonal elements of [A^A]'. This leads to the concept of GDOP. GDOP is defined by GDO? = ^Trace[A^ A]-\
(14)
It is the amplification factor of pseudo-range error variance to the combined position and time error variance. In this paper the idea of GDOP is further extended as a criteria of designing the low Earth orbit satellite positioning system. LEO SATELLITE POSITIONING SYSTEM — 317 —
The relationship between the satellite's height, h, and the coverage area can be approximately as (15)
sinp where R^ is the radius of Earth, and fi is the angle measured at the center of the Earth of the region seen by the spacecraft (Figure 2). Note that in deriving Eq. (15) we have assumed a spherical Earth. For the satellite constellation with the height of about 800 Km, covers a region with radius of 3000 km centered at the sub-satellite point. Therefore the low Earth orbit satellite constellation provides the positioning function for the region with radius of about 3000 Km centered at the chosen site.
Horlzcn
Figure 2.
To provide a positioning capability over a certain region (for example, in the study, the region is centered at the point with longitude (A) of 122°E and latitude () of 25°N), a scheme to orbit a sufficient number of satellites to ensure that four are always electronically visible is developed. The planned constellation will provide four satellites in good geometric position 24 hours a day. Two basic requirements are needed to be ftilfilled: 1) the satellites of the constellation must be visible during one day; and 2) the four satellites visible must have good geometric relationship or with good value of GDOP. Therefore, to satisfy the requirements above the region, four sets of constellations providing the continuous coverage are required. The four satellites in good geometry position can be determined from calculation of the value of GDOP. By reviewing Eqs. (8) and (14) it is clearly that GDOP is the functions of the direction cosines of the lines of sight from the observer site to the visible satellites. The direction cosines of the line of sight from the observer site to the /th satellite in the Earth-center coordinate can be expressed as:
(16)
Note that / , y , and k are the unit vector of the Earth-center coordinate system and p, is the unit vector of the line of sight pointing from the observer to the /th satellite. The constellations with continuous coverage and having repeated ground track over the region are the candidates of the system. The procedure of designing the low Earth orbit satellite positioning system is described as following. First, using the algorithm devised in Ref. 3, the constellations with continuous coverage and having repeated ground track over the chosen site are obtained. For the chosen site, the satellite constellations with the inclinations between 24° and ST can provide the continuous coverage. Table 1 lists the orbit elements of the constellations those can provide continuous coverage and repeated ground track over the site. 318
Table 1. Constellations with Continuous Coverage and Repeated Ground Track. Inclination, /\(deg.) 24
27
Number Semi-Major of Axis, a, Sat. (Km) 19
25
7183.85
7185.48
30
23
7187.29
33
19
7189.28
36
22
7191.46
39
17
7193.81
42
18
7196.33
45
22
7199.02
48
24
7201.88
51
22
7204.90
54
29
7208.08
57
30
7211.41
Initial Node of Ascending, fi„,(deg.)
Initial Mean Anomaly,
^^«,(deg.)
1
0.00. 79.997. 159.993. 239.99, 319.987. 39.984. 119.981. 199.977. 279.974. 359.971. 79.97. 159.95. 239.96. 319.%. 39.955. 119.95. 199.95. 279.945. 359.941
|
128, 302.98. 117.97. 292.56, 107.94. 282.93. 97.91, 272.90. 87.88, 262.87. 77.85. 252.84. 67.83. 242.81. 57.80. 232.73. 47.77. 222.75. 37.74. 212.72. 27.71. 202.70.17.78. 192.67. 7.65 128. 189.84. 251.67. 313.53, 15.37, 77.22. 139.06. 200.9. 262.75. 324.59. 26.43. 88.27. 150.12. 211.96. 273.81. 335.65. 37.49. 99.33. 161.19. 223.02. 284.86. 346.70. 48.55
0.00. 70.20, 140.41. 210.61. 280.81,351.01. 61.22. 131.42. 201.62. 271.83. 342.03. 52.23, 122,43. 192.64. 262.84. 333.04. 43.25, 113.45. 183.65. 253.85. 324.06, 34.23. 104.46. 174.66. 244.87
|
128, 148, 168. 188. 208. 228. 248. 268. 288. 308. 328. 348.8.28.48.68.88. 108.128.01 128, 304.23. 120.47. 296.70. 112.94. 289.17. 105.41. 281.64. 97.88. 274.11. 90.35. 266.58. 82.82. 259.05. 75.28. 251.52. 67.75. 243.99. 60.22. 236.46. 52.69. 228.93 128, 149.18. 170.36. 191.54.212.71.233.89.255.07. 276.25. 297.43. 318.61, 339.79. 0.97. 22.15. 43.32. 64.50. 85.68. 106.86
0. 78. 159.98. 239.97, 319.96, 39.96. 19.95. 199.94. 279.93. 359.93, 79.92. 159.91. 239.90. 319.89. 39.89, 119.88, 199.87.279.86. 359.85
128. 163.41. 198.82, 234.22. 269.63. 305.04. 340.45. 15.86. 51.27. 86.68. 122.09. 157.50. 192.91, 228.32. 263.72,299.13.334.54.9.95 128. 306.70. 125.39. 304.09. 122.79. 301.49, 120.18. 298.88. 117.58. 296.28. 114.97. 293.67. 112.37. 291.06. 109.76. 288.46. 107.16. 285.85. 104.55. 283.25. 101.95. 280.64 128.00. 306.79, 125.59. 304.38. 123.17. 301.96. 120.76. 299.55, 118.34. 297.14. 115.93. 294.72. 113.51. 292.31. 111.10 . 289.89. 108.68. 287.48. 106.27. 285.06. 103.86.282.65. 101.44.280.23 128.00. 146.94, 165.89, 184.83. 203.78. 222.72. 241.66. 260.61. 279.55. 298.50. 317.44. 336.39, 355.33. 14.27. 33.22. 52.16. 71.11. 90.05. 108.99. 127.94. 146.88. 165.83 128.00. 197.73, 267.46. 337.20. 46.93. 116.66. 186.39. 256.12. 325.85. 35.59. 105.32. 175.05, 244.78. 314.51. 24.24. 93.98. 163.71. 233.44. 303.17. 12.90. 82.64. 152.37. 222.10. 291.83. 1.56. 71.29. 141.03.210.76.280.49 128. 279.66. 71.32. 222.98. 14.64. 166.30. 317.97. 109.63. 261.29. 52.95. 204.61. 356.27. 147.93, 299.59,91.25, 242.91. 34.57. 186.24. 337.90. 129.56. 281.22,72.88,224.54, 16.20. 167.86.319.52, 111.18. 262.85.54.51.206.17
0. 224.27, 88.55. 312.82. 177.10, 41.37. 265.65, 129.92, 354.19. 218.47, 82.74. 307.02. 171.29. 35.57, 259.84. 124.12.348.39.212.67
128. 148. 168. 188. 208. 228. 248. 268, 288. 308. 328. 348. 8, 28.48. 68, 88, 108. 128
0.00, 214.19. 68.38. 282.58. 136.78. 350.97, 205.16. 59.36. 273.55. 127.75. 341.94, 196.13. 50.33. 264.53. 118.72. 332.91. 187.11. 41.30. 255.50, 109.69. 323.88, 178.08. 32.27
0. 52.71. 105.43. 158.14, 210.86. 263.57. 16.29. 9. 61.72. 114.43, 167.15.219.86.272.58.325.29. 18.70.72. 123.44. 176.15. 228.87,281.58. 334.30. 27.01 0. 63.49. 126.99. 190.49. 253.98. 317.48, 20.97, 84.47. 147.%. 211.46. 274.95. 338.45. 41.95, 105.44. 168.94. 232.43. 295.93
0. 18.24. 36.47. 54.71. 72.95. 91.18. 109.42. 27.66. 145.89. 164.13. 182.37. 200.61. 218.84, 237.08. 255.32. 273.56.291.79. 310.03, 328.27, 346.50,4.74, 22.98
0.00. 16.90. 33.80. 50.70. 67.60, 84.50, 101.41, 118.31, 135.21, 152.11. 169.01. 185.91, 202.81. 219.71. 236.61, 253.51. 270.42, 287.32. 304.22. 321.12, 338.02. 354.92, 11.82,28.72 0. 94.78. 189.57. 284.35. 19.13, 113.91, 208.70. 303.48, 38.26. 133.04. 227.83. 322.61. 57.39. 152.17. 246.96, 341.74. 76.52. 171.30. 266.09. 0.87. 95.65, 190.44 0.00. 103.76. 207.51. 311.27. 55.02. 158.78. 262.53, 6.29, 110.04, 213.80. 317.55, 61.31. 165.06, 268.82, 12.57. 116.33. 220.08. 323.84, 67.59. 171.35. 275.10, 18.86, 122.62. 226.37, 330.13. 73.88. 177.64. 281.39, 25.15 0. 36.75. 73.49, 110.24. 146.99, 183.73, 220.48, 257.23. 293.98,330.72.7.47.44.22.80.96. 117.71. 154.46. 191.20, 227.95. 264.70, 301.44 . 338.19. 14.94. 51.69. 88.43. 125.18,161.93. 198.67.235.42,272.17,308.91.345.66
Second, to reduce the computation work, among the candidates the sets of four constellations with fixed inclination difference are combined and the values of GDOPs are calculated. Based on the GDOPs' values several sets with good values are selected. Several values of the fixed inclination differences are studied. Finally, several points in the region are chosen to check the values of GDOP for different sets and the set of constellation having smallest value of GDOP is the low Earth orbit satellite positioning system. Figure 4 shows the values of GDOPs during one day of the sites. Table 2 lists the satellites of the positioning constellation for the chosen site.
— 319 —
[M lyUs
TTiTTrTTiTrrmTr*
TTTi
Figure 3. The GDOP value of several points. Table 2. Low Earth Orbit Satellite Positioning System. Number Semi-Major of Axis, a, Sat. (Km)
Inclination, Mdeg.)
Initial Node of Ascending, fl,„(deg.) 128. 302.98. 117.97. 292.56. 107.94. 282.93. 97.91. 272.90. 87.88. 262.87. 77.85. 252.84. 67.83. 242.81. 57.80, 232.73. 47.77. 222.75. 37.74. 212.72. 27.71, 202.70. 17.78, 192.67. 7.65 128. 189.84. 251.67. 313.53. 15.37. 77.22. 139.06. 200.9. 262.75. 324.59. 26.43. 88.27. 150.12. 211.96. 273.81. 335.65. 37.49. 99.33. 161.19.223.02. 284.86. 346.70. 48.55
Initial Mean Anomaly, A/o^(deg.) 0.00. 70.20, 140.41. 210.61. 280.81. 351.01, 61.22, 131.42. 201.62. 271.83. 342.03, 52.23. 122.43. 192.64, 262.84. 333.04. 43.25, 113.45. 183.65. 253.85, 324.06, 34.23. 104.46, 174.66, 244.87
Set 1
27
25
7185.48
Set 2
30
23
7187.29
Set 3
33
19
7189.28
128. 148, 168. 188. 208, 228. 248. 268, 288. 308. 0. 78. 159.98. 239.97. 319.%. 39.96, 19.95. 199.94, 328, 279.93. 359.93. 79.92. 159.91. 239.90. 319.89, 348.8.28.48.68.88. 108. 128.01 39.89. 119.88. 199.87. 279.86. 359.85
Set 4
36
22
7191.46
128. 304.23. 120.47. 296.70. 112.94. 289.17. 105.41. 281.64. 97.88. 274.11. 90.35. 266.58. 82.82, 259.05. 75.28. 251.52. 67.75, 243.99. 60.22. 236.46. 52.69. 228.93
0.00. 214.19. 68.38. 282.58. 136.78, 350.97.205.16, 59.36. 273.55. 127.75. 341.94. 196.13. 50.33. 264.53. 118.72. 332.91. 187.11, 41.30. 255.50. 109.69.323.88.178.08.32.27
0. 52.71. 105.43. 158.14. 210.86.263.57. 16.29.9, 61.72. 114.43. 167.15. 219.86. 272.58, 325.29. 18. 70.72. 123.44, 176.15. 228.87, 281.58. 334.30, 27.01
THE RELATIVISTIC EFFECTS OF LEO SATELLITE POSITIONING SYSTEM Since the satellites have a large velocity, there is a non-negligible gravitational potential difference between that of satellites and that of the users on Earth's surface, and there is significant Earth rotation effect. Because clocks near Earth's surface experience a gravitational field may be in motion in addition to the motion associated with the Earth's rotation), they are subject to gravitational frequency shifts and time dilation. Let the gravitational potentials experience for the satellites and the receiver on the Earth's surface are ^r and ^« respectively, then the gravitational frequency shifts' difference is
— 320 —
/r-A=(-^^)/« c
(17)
or 1± = X^(!1ZJ^)
(18)
fr where c is the speed of light. Let the periods of the clocks of satellite and of receiver on Earth being T^ and TR respectively, then
f,JIT,_T, fr
(19)
T,
^'TT
Assumethe velocities ofsatellite and of receiver on Earth are K„ and Kj respectively, then
r«=^i-(^)'r,r, = ji-(^)^r
(20)
In Eq. (20), J is the period of zero velocity. Substituting Eq. (20) into Eq.(19), yields
Also the Doppler frequency shift is obtained as /«=
=
^fr
(22)
i_^:(^r-A) c where ^ is the unit vector along the line of sight. :^ = - — i - ^ = l + i.(K,-F,)
(23)
c Combining Eqs. (17), (21) and (23), we obtain the ratio of the receiver's and of satellite's clocks frequencies ^ = l^\(^^--^^)^2^(V^^V,')^'^^(V,-V,) fj. c' Ic c
(24)
If the satellites of the low Earth orbit satellite positioning system are on the circular orbits, then the first two terms of the right side are
— 321 —
— (^r-^/?) = —K )-( )] c c r r^ = \ .(.3.9860015£14 ^ 3.9860015£14 ^ , ,9^733,3-3 (2.997925£9)' 7.18385£6 6.378137£6
(25)
1
((4.651£2)' -(7.4489£3)') =-3.07476702"'' and TT(^'/e-f';) = 2c' '^ ^ ' 2(2.997925£9)'
(26)
respectively. The sum of the two terms is-2.29489166"''. The amount is the factory presetting to partially accommodate the relativistic effects. Besides the purposely setting lower in frequency, the time shift due to the deviation from the circular orbit is^'^ (27)
Ar' = 4.4428 X 10"'' - ^ e V a sin£ where E is the eccentric anomaly and
(28)
m^=:ju/c' = 4.435028687 X 10"'m
The amount of the time shift is proportional to the square root of the semi-major axis. Figure 4 shows the amounts of the time shift for different values of eccentricities of the LEO satellite positioning system. For the eccentricity of 0.03 the amount of shift is 20ns. It is less than the amount of 23ns for the GPS satellites with the eccentricity of 0.01. In other words, the requirement of keeping the orbits of LEO satellite positioning system being circle is less stringent. 2.5000E-10
e-0.03
2.0000E-10
\ 1.5000E-10
e=0.02
1.0000E-10 —\
e=q^l 5.0000E-11
At'
O.OOOOE+0
1
c=0
-5.0000E-11 -1.0000E-10 -1.5000E-10 —I -2.0000E-10 -2.5000E-10 0»
180»
90*
270°
360»
Figure 4. Time shift for different values of eccentricities of the LEO satellite.
322
CONCLUSION In the paper we have proposed a scheme to orbit the satellites over the region centered at point with longitude of 122° E and latitude of 25°N. The satellite constellation can be used as the positioning system as the GPS does. The proposed constellation consists of 75 satellites with four values of inclinations. The satellites are in the orbit of height of about 800 Km. It can provide the positioning function for the region with radius of about 3000 Km center at the chosen site. After the study of the relativistic effects, we find that the requirement of the keeping the orbit circle is less stringent than the GPS's and to partially accommodate the relativistic effects the satellite clock can be purposely set lower in frequency by the factor of -2.29489166'''. ACKNOWLEDGMENT The study was sponsored by the National Science Council of Republic of China under the contract number NSC85-2612.E-032-001. REFERENCES B. Hofmann-Wellenhof, H. Lichtengger and J. Collins, GPS: Theory and Practice, 2nd edition, SpringerVerlag Wien New York, 1993. P.S. Jorgensen, "Navstar/Global Positioning System 18-Satellite Constellations", Global Positioning System, Vol. II, The Institute of Navigation, 1984, pp. 1-12. D.-M. Ma and W.-C. Hsu, ''Exact Design of Partial Coverage Satellite Constellations over Oblate Earth", Journal of Spacecraft and Rockets, Vol.34, No.l, 1997, pp.29-35. Bradford W. Parkinson and James J. Spilker Jr., editors. Global Positioning System: Theory and Applications, Vol.1, AIAA, 1996, Part I, Chapter 18.
— 323
'Associate Professor, Department of Aerospace Engineering, Tamkang University, Tamsui, Taiwan 25137, Republic of China, Tel: -^886-2-2623-7600, Fax: +886-2-2620-9746, E-mail: [email protected]\ ' Graduate Student, Department ofAerospace Engineering, Tamkang University, Tamsui, Taiwan 25137 Republic of China. ^ Graduate Student, Department of Mechanical Engineering, Tamkang University, Tamsui, Taiwan 25137, Republic of China. Currently, in military service.
ABSTRACT A scheme of designing low Earth orbit satellite positioning system is proposed. An example low Earth orbit satellite constellation providing the positioning ftmction over certain region is presented. The proposed constellation consists of 75 satellites with four different inclinations. The satellites are in the orbit of height of about 800 Km. It can provide the positioning function over the region with radius of about 3000 Km center at the chosen site with longitude of 122° E and latitude of 25°N. The construction of the constellation is based on the requirement of good values of the Dilution of Precision over the specific region at same time. Also the study of the relativistic effects of the system shows that the requirement of keeping the orbit of the low Earth orbit satellite positioning system circle is less stringent than the GPS's. INTRODUCTION As small satellites becoming more capable through miniaturized electronic and on-board processing, the construction of low-cost satellite constellations becomes economically feasible. With the feasibility of LightSats and the possibility of smaller, relatively inexpensive launch systems, the low-altitude satellite constellations receive increased attention. The potential of low-altitude satellite constellations' applications is significant. The positioning function is one of them. The purpose of this paper is to present a low Earth orbit satellite constellation which provides the positioning function as the GPS does over certain region. In order to provide good accuracy of point positioning, the constellations have at least four satellites with good Dilution of Precision (DOP)^''^^ over the specified region at same time. The DOP factor is a measure for the geometry of the visible satellites. The good geometry should mirror a low DOP value. It is the reciprocal value of the volume of the geometric body, which is formed by the intersection points of the site-satellite vectors with the unit sphere centered at the reference site. In the application of GPS, the purposes of DOP are twofold. First, it is useful in planning a survey and, secondly, it may be helpful in interpreting processed baseline vectors. 314 —
In this paper the concept of DOP is further extended to be the design criterion in designing the low Earth orbit satellite constellation. The DOP is expressed in the topocentric coordinate system with its axes along the local north, east and vertical. The position DOP is then has tow components: HDOP, the dilution of precision in the horizontal position and the VDOP, denoting the corresponding value for the vertical component. The time dilution of precision is an another important factor to good accuracy of positioning. In the paper both position and time DOPs are considered in designing the constellation. In the next two sections the concept of geometric dilution of precision (GDOP) and then the positioning system design procedure using the idea of GDOP are present. As to the GPS, several relativistic effects those associated to the low Earth orbit satellite positioning system are non-negligible. These effects include the following: 1) gravitational field of Earth and Earth rotation; and 2) velocities of satellites. The major effects cause an increase in the satellite clock frequency as observed by a user on Earth. These relativistic effects are also discussed in the section after the designing procedure of the constellation. GEOMETRY DILUTION OF PRECISION Figure 1 illustrates an Earth-centered inertial coordinate system. At r =0, the x-axis passes through the intersection of the equator and the Greenwich meridian, the z-axis passes through the Earth rotation axis, and the >^-axis completes the right-handed coordinate system. Shown in the Figure 1 are the user position (x, y, z), and the position of satellite No. 1 (x,, >',, z,). The range distance between the user and the satellite No.l is indicated as /?,. The basic equations using four satellites are
Z .
N,^
= R,
^(x^x,y^(y-y,y^(z-z,f^T
= R,
yl{x-x,y^iy-y,)'^{z-z,y
/RANGE
x\ ^' EARTH CENTER
liSER(X,Y»Z)
\
^vAQiiATnp
GREENWICHl \ , ^ MERIDLAN
^(x-x,y^{y-y,y-^iz-z,y^T
1ST SATELLITE (X,,Y,^,)
NORTH PLOE
^y^
Y
(1) Figure 1. The relationship between the user and satellite.
^T = R,
where T is clock bias; JCJ, y, and Zj are the /th satellite position; and R^ is the pseudo-range measurements to the zth satellite. Note that the pseudo-ranges are the sum of the actual range plus the offset due to user clock error. For convenience, units have been selected such that the velocity of light is unity. In the equations above, the four pseudo-ranges are the measured quantities. The satellite positions are known, and the four unknowns are the user position (jc, y, z), and the user clock error. The equations are nonlinear. While it is possible to solve these equations directly as they are shown. The equations can be linearized. Let jCn, y^, Zn, and T^ be nominal values of JC, y, z, and T, respectively. The Ax, Ay, Az, and AT are corrections to these nominal values. 7?^, is the nominal pseudo-range measurements from the zth satellite; A/?i be the difference between the actual and nominal measurements. Therefore
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jc = x„ + AJC,
(2)
z = z,^^ Az,
r = r„ + ^T, R,=R,,^AR, and (3) Substituting the Eqs. (2) and (3) into Eq. (1) yields
[ix„+Ax-xy+{y„+Ay-y,y+{z„+Az-zy]'=R„+AR,-T„-AT
(4)
By ignoring second-order error terms, these equations can be written as
[(Z
7V77^,
TTTTr
r r r . (^» -X,)AX+{y„ -y,)Ay+(z„ -z,)AZ 4ix„-x,Y =
+{y„-y,y
+{z„-z,)
(5)
R„+AR,-T„-AT.
Furthermore, substituting Eq. (3) into Eq. (5), we obtain ^" ^ A x + ^'" •^' Ay+ !" R.„ - T„ -f..III
R,.i -T,.
!: Az + AT = AR..
-r
(6)
II
The above four equations are linearized equations that relate pseudo-range measurements to the desired user navigation information as well as the user's clock bias. The known quantities of the right-hand side of the equations are actually incremental pseudo-range measurements. They are the differences between the actual measured pseudo-range and the measurements that had been predicted by the user's computer based on the knowledge of satellite position and the user's most current estimate of his position and clock bias. The quantities to be computed, Ax, A;^, Az, and AT, are corrections that the user will make to his current estimate of position and clock biases. The coefficients of these quantities on the left-hand side are the direction cosines of the line of sight from user to the satellites as projected along the x-, y-, and z-axis. These linearized equations can be conveniently expressed in matrix notation and appear as
a,, a-,
a, a, a 22 a
31
^32
V«41
^42
«-
1VAx^
r AR,^
Ay
AR2
Az
A/J3
1 ^ATj
(7)
A/f4J
Note that Ojj is the direction cosine of the angle between the range to the /th satellite and theyth coordinate. By the use of matrix notation, the above equation can be expressed as follows. Let
316 —
'a,,
^n
or,,
«21
«22
«23
1
«31
«32
«33
1
V«41
«42
«43
1.
As
0
L X = (AX ^y
Az A r y , and r = (A/?,
A/J^
Aif3
Ai?,^.
(8)
Then Ax = r
or
x = A"V.
(9)
Since this relationship is linear, it can be used to express the relationship between the errors in pseudorange measurement and the user quantities. This relationship is therefore s^^X-'s,
(10)
where e, represents the pseudo-range measurement errors and e^ the corresponding errors in user position and clock bias. Let now consider the covariance matrix of the expected errors in pseudo-range measurements and the covariance matrix of the user quantities. The first covariance measurement is a 4X4 array composed of the expected values of the squares and products of the errors in the pseudo-range measurements. The diagonal terms in the matrix, namely the squares of the expected errors, are the variances; i.e., the squares of the expected 1 G values of the pseudo-range measurement errors. The off-diagonal terms are the covariance between the pseudo-range measurement measurements and reflect the correlation to be expected in these measurements. The covariance matrix is given by cov(r) = E{£^€l} = A'' cov(x)A'^ = [A^^ cov(x)A]"'
(11)
Now making the assumption that the pseudo-range errors are uncorrelated from satellite to satellite and equal for each satellite, the pseudo-range covariance matrix reduces to cov(x) = crJl.
(12)
where I is the 4 X 4 unit vector. Thus cov(r) = o-'[A^A]-'.
(13)
We see that the position and time error variances are functions of the diagonal elements of [A^A]'. This leads to the concept of GDOP. GDOP is defined by GDO? = ^Trace[A^ A]-\
(14)
It is the amplification factor of pseudo-range error variance to the combined position and time error variance. In this paper the idea of GDOP is further extended as a criteria of designing the low Earth orbit satellite positioning system. LEO SATELLITE POSITIONING SYSTEM — 317 —
The relationship between the satellite's height, h, and the coverage area can be approximately as (15)
sinp where R^ is the radius of Earth, and fi is the angle measured at the center of the Earth of the region seen by the spacecraft (Figure 2). Note that in deriving Eq. (15) we have assumed a spherical Earth. For the satellite constellation with the height of about 800 Km, covers a region with radius of 3000 km centered at the sub-satellite point. Therefore the low Earth orbit satellite constellation provides the positioning function for the region with radius of about 3000 Km centered at the chosen site.
Horlzcn
Figure 2.
To provide a positioning capability over a certain region (for example, in the study, the region is centered at the point with longitude (A) of 122°E and latitude () of 25°N), a scheme to orbit a sufficient number of satellites to ensure that four are always electronically visible is developed. The planned constellation will provide four satellites in good geometric position 24 hours a day. Two basic requirements are needed to be ftilfilled: 1) the satellites of the constellation must be visible during one day; and 2) the four satellites visible must have good geometric relationship or with good value of GDOP. Therefore, to satisfy the requirements above the region, four sets of constellations providing the continuous coverage are required. The four satellites in good geometry position can be determined from calculation of the value of GDOP. By reviewing Eqs. (8) and (14) it is clearly that GDOP is the functions of the direction cosines of the lines of sight from the observer site to the visible satellites. The direction cosines of the line of sight from the observer site to the /th satellite in the Earth-center coordinate can be expressed as:
(16)
Note that / , y , and k are the unit vector of the Earth-center coordinate system and p, is the unit vector of the line of sight pointing from the observer to the /th satellite. The constellations with continuous coverage and having repeated ground track over the region are the candidates of the system. The procedure of designing the low Earth orbit satellite positioning system is described as following. First, using the algorithm devised in Ref. 3, the constellations with continuous coverage and having repeated ground track over the chosen site are obtained. For the chosen site, the satellite constellations with the inclinations between 24° and ST can provide the continuous coverage. Table 1 lists the orbit elements of the constellations those can provide continuous coverage and repeated ground track over the site. 318
Table 1. Constellations with Continuous Coverage and Repeated Ground Track. Inclination, /\(deg.) 24
27
Number Semi-Major of Axis, a, Sat. (Km) 19
25
7183.85
7185.48
30
23
7187.29
33
19
7189.28
36
22
7191.46
39
17
7193.81
42
18
7196.33
45
22
7199.02
48
24
7201.88
51
22
7204.90
54
29
7208.08
57
30
7211.41
Initial Node of Ascending, fi„,(deg.)
Initial Mean Anomaly,
^^«,(deg.)
1
0.00. 79.997. 159.993. 239.99, 319.987. 39.984. 119.981. 199.977. 279.974. 359.971. 79.97. 159.95. 239.96. 319.%. 39.955. 119.95. 199.95. 279.945. 359.941
|
128, 302.98. 117.97. 292.56, 107.94. 282.93. 97.91, 272.90. 87.88, 262.87. 77.85. 252.84. 67.83. 242.81. 57.80. 232.73. 47.77. 222.75. 37.74. 212.72. 27.71. 202.70.17.78. 192.67. 7.65 128. 189.84. 251.67. 313.53, 15.37, 77.22. 139.06. 200.9. 262.75. 324.59. 26.43. 88.27. 150.12. 211.96. 273.81. 335.65. 37.49. 99.33. 161.19. 223.02. 284.86. 346.70. 48.55
0.00. 70.20, 140.41. 210.61. 280.81,351.01. 61.22. 131.42. 201.62. 271.83. 342.03. 52.23, 122,43. 192.64. 262.84. 333.04. 43.25, 113.45. 183.65. 253.85. 324.06, 34.23. 104.46. 174.66. 244.87
|
128, 148, 168. 188. 208. 228. 248. 268. 288. 308. 328. 348.8.28.48.68.88. 108.128.01 128, 304.23. 120.47. 296.70. 112.94. 289.17. 105.41. 281.64. 97.88. 274.11. 90.35. 266.58. 82.82. 259.05. 75.28. 251.52. 67.75. 243.99. 60.22. 236.46. 52.69. 228.93 128, 149.18. 170.36. 191.54.212.71.233.89.255.07. 276.25. 297.43. 318.61, 339.79. 0.97. 22.15. 43.32. 64.50. 85.68. 106.86
0. 78. 159.98. 239.97, 319.96, 39.96. 19.95. 199.94. 279.93. 359.93, 79.92. 159.91. 239.90. 319.89. 39.89, 119.88, 199.87.279.86. 359.85
128. 163.41. 198.82, 234.22. 269.63. 305.04. 340.45. 15.86. 51.27. 86.68. 122.09. 157.50. 192.91, 228.32. 263.72,299.13.334.54.9.95 128. 306.70. 125.39. 304.09. 122.79. 301.49, 120.18. 298.88. 117.58. 296.28. 114.97. 293.67. 112.37. 291.06. 109.76. 288.46. 107.16. 285.85. 104.55. 283.25. 101.95. 280.64 128.00. 306.79, 125.59. 304.38. 123.17. 301.96. 120.76. 299.55, 118.34. 297.14. 115.93. 294.72. 113.51. 292.31. 111.10 . 289.89. 108.68. 287.48. 106.27. 285.06. 103.86.282.65. 101.44.280.23 128.00. 146.94, 165.89, 184.83. 203.78. 222.72. 241.66. 260.61. 279.55. 298.50. 317.44. 336.39, 355.33. 14.27. 33.22. 52.16. 71.11. 90.05. 108.99. 127.94. 146.88. 165.83 128.00. 197.73, 267.46. 337.20. 46.93. 116.66. 186.39. 256.12. 325.85. 35.59. 105.32. 175.05, 244.78. 314.51. 24.24. 93.98. 163.71. 233.44. 303.17. 12.90. 82.64. 152.37. 222.10. 291.83. 1.56. 71.29. 141.03.210.76.280.49 128. 279.66. 71.32. 222.98. 14.64. 166.30. 317.97. 109.63. 261.29. 52.95. 204.61. 356.27. 147.93, 299.59,91.25, 242.91. 34.57. 186.24. 337.90. 129.56. 281.22,72.88,224.54, 16.20. 167.86.319.52, 111.18. 262.85.54.51.206.17
0. 224.27, 88.55. 312.82. 177.10, 41.37. 265.65, 129.92, 354.19. 218.47, 82.74. 307.02. 171.29. 35.57, 259.84. 124.12.348.39.212.67
128. 148. 168. 188. 208. 228. 248. 268, 288. 308. 328. 348. 8, 28.48. 68, 88, 108. 128
0.00, 214.19. 68.38. 282.58. 136.78. 350.97, 205.16. 59.36. 273.55. 127.75. 341.94, 196.13. 50.33. 264.53. 118.72. 332.91. 187.11. 41.30. 255.50, 109.69. 323.88, 178.08. 32.27
0. 52.71. 105.43. 158.14, 210.86. 263.57. 16.29. 9. 61.72. 114.43, 167.15.219.86.272.58.325.29. 18.70.72. 123.44. 176.15. 228.87,281.58. 334.30. 27.01 0. 63.49. 126.99. 190.49. 253.98. 317.48, 20.97, 84.47. 147.%. 211.46. 274.95. 338.45. 41.95, 105.44. 168.94. 232.43. 295.93
0. 18.24. 36.47. 54.71. 72.95. 91.18. 109.42. 27.66. 145.89. 164.13. 182.37. 200.61. 218.84, 237.08. 255.32. 273.56.291.79. 310.03, 328.27, 346.50,4.74, 22.98
0.00. 16.90. 33.80. 50.70. 67.60, 84.50, 101.41, 118.31, 135.21, 152.11. 169.01. 185.91, 202.81. 219.71. 236.61, 253.51. 270.42, 287.32. 304.22. 321.12, 338.02. 354.92, 11.82,28.72 0. 94.78. 189.57. 284.35. 19.13, 113.91, 208.70. 303.48, 38.26. 133.04. 227.83. 322.61. 57.39. 152.17. 246.96, 341.74. 76.52. 171.30. 266.09. 0.87. 95.65, 190.44 0.00. 103.76. 207.51. 311.27. 55.02. 158.78. 262.53, 6.29, 110.04, 213.80. 317.55, 61.31. 165.06, 268.82, 12.57. 116.33. 220.08. 323.84, 67.59. 171.35. 275.10, 18.86, 122.62. 226.37, 330.13. 73.88. 177.64. 281.39, 25.15 0. 36.75. 73.49, 110.24. 146.99, 183.73, 220.48, 257.23. 293.98,330.72.7.47.44.22.80.96. 117.71. 154.46. 191.20, 227.95. 264.70, 301.44 . 338.19. 14.94. 51.69. 88.43. 125.18,161.93. 198.67.235.42,272.17,308.91.345.66
Second, to reduce the computation work, among the candidates the sets of four constellations with fixed inclination difference are combined and the values of GDOPs are calculated. Based on the GDOPs' values several sets with good values are selected. Several values of the fixed inclination differences are studied. Finally, several points in the region are chosen to check the values of GDOP for different sets and the set of constellation having smallest value of GDOP is the low Earth orbit satellite positioning system. Figure 4 shows the values of GDOPs during one day of the sites. Table 2 lists the satellites of the positioning constellation for the chosen site.
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[M lyUs
TTiTTrTTiTrrmTr*
TTTi
Figure 3. The GDOP value of several points. Table 2. Low Earth Orbit Satellite Positioning System. Number Semi-Major of Axis, a, Sat. (Km)
Inclination, Mdeg.)
Initial Node of Ascending, fl,„(deg.) 128. 302.98. 117.97. 292.56. 107.94. 282.93. 97.91. 272.90. 87.88. 262.87. 77.85. 252.84. 67.83. 242.81. 57.80, 232.73. 47.77. 222.75. 37.74. 212.72. 27.71, 202.70. 17.78, 192.67. 7.65 128. 189.84. 251.67. 313.53. 15.37. 77.22. 139.06. 200.9. 262.75. 324.59. 26.43. 88.27. 150.12. 211.96. 273.81. 335.65. 37.49. 99.33. 161.19.223.02. 284.86. 346.70. 48.55
Initial Mean Anomaly, A/o^(deg.) 0.00. 70.20, 140.41. 210.61. 280.81. 351.01, 61.22, 131.42. 201.62. 271.83. 342.03, 52.23. 122.43. 192.64, 262.84. 333.04. 43.25, 113.45. 183.65. 253.85, 324.06, 34.23. 104.46, 174.66, 244.87
Set 1
27
25
7185.48
Set 2
30
23
7187.29
Set 3
33
19
7189.28
128. 148, 168. 188. 208, 228. 248. 268, 288. 308. 0. 78. 159.98. 239.97. 319.%. 39.96, 19.95. 199.94, 328, 279.93. 359.93. 79.92. 159.91. 239.90. 319.89, 348.8.28.48.68.88. 108. 128.01 39.89. 119.88. 199.87. 279.86. 359.85
Set 4
36
22
7191.46
128. 304.23. 120.47. 296.70. 112.94. 289.17. 105.41. 281.64. 97.88. 274.11. 90.35. 266.58. 82.82, 259.05. 75.28. 251.52. 67.75, 243.99. 60.22. 236.46. 52.69. 228.93
0.00. 214.19. 68.38. 282.58. 136.78, 350.97.205.16, 59.36. 273.55. 127.75. 341.94. 196.13. 50.33. 264.53. 118.72. 332.91. 187.11, 41.30. 255.50. 109.69.323.88.178.08.32.27
0. 52.71. 105.43. 158.14. 210.86.263.57. 16.29.9, 61.72. 114.43. 167.15. 219.86. 272.58, 325.29. 18. 70.72. 123.44, 176.15. 228.87, 281.58. 334.30, 27.01
THE RELATIVISTIC EFFECTS OF LEO SATELLITE POSITIONING SYSTEM Since the satellites have a large velocity, there is a non-negligible gravitational potential difference between that of satellites and that of the users on Earth's surface, and there is significant Earth rotation effect. Because clocks near Earth's surface experience a gravitational field may be in motion in addition to the motion associated with the Earth's rotation), they are subject to gravitational frequency shifts and time dilation. Let the gravitational potentials experience for the satellites and the receiver on the Earth's surface are ^r and ^« respectively, then the gravitational frequency shifts' difference is
— 320 —
/r-A=(-^^)/« c
(17)
or 1± = X^(!1ZJ^)
(18)
fr where c is the speed of light. Let the periods of the clocks of satellite and of receiver on Earth being T^ and TR respectively, then
f,JIT,_T, fr
(19)
T,
^'TT
Assumethe velocities ofsatellite and of receiver on Earth are K„ and Kj respectively, then
r«=^i-(^)'r,r, = ji-(^)^r
(20)
In Eq. (20), J is the period of zero velocity. Substituting Eq. (20) into Eq.(19), yields
Also the Doppler frequency shift is obtained as /«=
=
^fr
(22)
i_^:(^r-A) c where ^ is the unit vector along the line of sight. :^ = - — i - ^ = l + i.(K,-F,)
(23)
c Combining Eqs. (17), (21) and (23), we obtain the ratio of the receiver's and of satellite's clocks frequencies ^ = l^\(^^--^^)^2^(V^^V,')^'^^(V,-V,) fj. c' Ic c
(24)
If the satellites of the low Earth orbit satellite positioning system are on the circular orbits, then the first two terms of the right side are
— 321 —
— (^r-^/?) = —K )-( )] c c r r^ = \ .(.3.9860015£14 ^ 3.9860015£14 ^ , ,9^733,3-3 (2.997925£9)' 7.18385£6 6.378137£6
(25)
1
((4.651£2)' -(7.4489£3)') =-3.07476702"'' and TT(^'/e-f';) = 2c' '^ ^ ' 2(2.997925£9)'
(26)
respectively. The sum of the two terms is-2.29489166"''. The amount is the factory presetting to partially accommodate the relativistic effects. Besides the purposely setting lower in frequency, the time shift due to the deviation from the circular orbit is^'^ (27)
Ar' = 4.4428 X 10"'' - ^ e V a sin£ where E is the eccentric anomaly and
(28)
m^=:ju/c' = 4.435028687 X 10"'m
The amount of the time shift is proportional to the square root of the semi-major axis. Figure 4 shows the amounts of the time shift for different values of eccentricities of the LEO satellite positioning system. For the eccentricity of 0.03 the amount of shift is 20ns. It is less than the amount of 23ns for the GPS satellites with the eccentricity of 0.01. In other words, the requirement of keeping the orbits of LEO satellite positioning system being circle is less stringent. 2.5000E-10
e-0.03
2.0000E-10
\ 1.5000E-10
e=0.02
1.0000E-10 —\
e=q^l 5.0000E-11
At'
O.OOOOE+0
1
c=0
-5.0000E-11 -1.0000E-10 -1.5000E-10 —I -2.0000E-10 -2.5000E-10 0»
180»
90*
270°
360»
Figure 4. Time shift for different values of eccentricities of the LEO satellite.
322
CONCLUSION In the paper we have proposed a scheme to orbit the satellites over the region centered at point with longitude of 122° E and latitude of 25°N. The satellite constellation can be used as the positioning system as the GPS does. The proposed constellation consists of 75 satellites with four values of inclinations. The satellites are in the orbit of height of about 800 Km. It can provide the positioning function for the region with radius of about 3000 Km center at the chosen site. After the study of the relativistic effects, we find that the requirement of the keeping the orbit circle is less stringent than the GPS's and to partially accommodate the relativistic effects the satellite clock can be purposely set lower in frequency by the factor of -2.29489166'''. ACKNOWLEDGMENT The study was sponsored by the National Science Council of Republic of China under the contract number NSC85-2612.E-032-001. REFERENCES B. Hofmann-Wellenhof, H. Lichtengger and J. Collins, GPS: Theory and Practice, 2nd edition, SpringerVerlag Wien New York, 1993. P.S. Jorgensen, "Navstar/Global Positioning System 18-Satellite Constellations", Global Positioning System, Vol. II, The Institute of Navigation, 1984, pp. 1-12. D.-M. Ma and W.-C. Hsu, ''Exact Design of Partial Coverage Satellite Constellations over Oblate Earth", Journal of Spacecraft and Rockets, Vol.34, No.l, 1997, pp.29-35. Bradford W. Parkinson and James J. Spilker Jr., editors. Global Positioning System: Theory and Applications, Vol.1, AIAA, 1996, Part I, Chapter 18.
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