INITIAL DATA HANDLING

Key Parameters

1. Received data is verified for validity by calculating checksums of received messages and comparing them with transmitted checksums.

In case of mismatch, the received packet is rejected. Types of checksums and methods of their calculation are presented in the previous lecture and its appendix. If parameters are valid, then their validity flags are formed \(\bigl(Пр_{i,j,k} = 0\bigr)\), where \(i\) — navigation receiver number; \(j\) — navigation satellite number; \(k\) — navigation system type: 0 — GPS; 1 — GLONASS.

2. Converting data to physical quantities format

Converting data to physical quantities format is performed by multiplying by corresponding scale factors (table 1).

Table 1. Measurement Data Fields

3. Ephemeris data from the previous lecture and its appendix is used for calculating coordinates, velocity vector components, clock offset, and drift of GPS and GLONASS satellites respectively.

Calculation is performed for time \(\,t_{\text{изл}}\), computed as:

\[ t_{\text{изл}} = rcvtime - Raw\_range \]

3.1. Calculation of GPS GNSS spacecraft motion parameters for a specified time based on ephemeris data is performed according to the following algorithm.

The input data include the parameters described in the previous lecture and its appendix.

1) Convert all values \(M_{0}\), \(\Delta n\), \(\Omega_{0}\), OMEGADOT, \(I_{0}\), IDOT to radians by multiplying by \(\pi\), where \(\pi = 3.1415926535898\).

2) Compute the moment \(\,t_{K}\) from the GPS epoch corresponding to the calculation time \(\,t\):

\[ t_{K} = t - t_{0E} \]

if \(t_{K} > 302400\) s, then \(t_{K} = t_{K} - 604800\) s (for ephemerides);

if \(t_{K} < -302400\) s, then \(t_{K} = t_{K} + 604800\) s.

3) Compute the corrected mean motion:

\[ n = \sqrt{\frac{\mu}{A^3}} \;+\; \Delta n \]

where \(\mu = 3.986005 \times 10^{14} \text{ m}^3\text{s}^{-2}\) — the gravitational constant.

4) Determine the current value of the mean anomaly \(M_{k}\) at time \(t_{k}\):

\[ M_{k} = M_{0} + n \, t_{k} \]

5) Solve Kepler’s equation for the eccentric anomaly \(E_{k}\) by Newton’s iterative method:

\( E_{k} \;-\; e_{0}\,\sin E_{k} \;=\; M_{k} \)

\( E_{k(n+1)} \;=\; E_{k(n)} \;+\; \frac{M_{k} \;-\; E_{k(n)} \;+\; e_{0}\,\sin E_{k(n)}}{1 \;-\; e_{0}\,\cos E_{k(n)}} \)

where \(n = 1, 2, 3, ...\);

\(E_{k1} = M_{k}\);

\(\bigl| E_{k(n+1)} - E_{k(n)} \bigr| < \text{eps}\);

\(\text{eps}\) — required accuracy.

6) Compute the derivative \(E_{k}'\):

\[ E_{K}' = \frac{n}{1 - e_{0} \cos E_{Kn}} \]

7) Compute the relativistic correction term:

\( \Delta t_R = C \cdot \sqrt{A} \cdot e \cdot \sin E \)

where: \(C = -4.442807633 \times 10^{-10}\).

8) Compute the true anomaly:

\[ \Theta_{K} = \arctan\!\Bigl(\tfrac{\sqrt{1 - e_{0}^{2}} \,\sin E_{K}}{\cos E_{K} \;-\; e_{0}}\Bigr)\! \]

9) Compute the argument of latitude:

\[ \Phi_{K} = \Theta_{K} \;+\; \omega \]

10) Compute the derivative of the argument of latitude:

\[ \Phi_{K}' = \tfrac{\sqrt{1 - e_{0}^{2}}\, E_{K}'}{1 \;-\; e_{0} \,\cos E_{K}} \]

11) Compute the corrected argument of latitude:

\[ U_{K} = \Phi_{K} + C_{UC}\,\cos(2\Phi_{K}) + C_{US}\,\sin(2\Phi_{K}) \]

12) Compute the derivative of the corrected argument of latitude:

\[ U_{K}' = \Phi_{K}' \Bigl( 1 \;+\; 2\bigl(C_{US}\cos 2\Phi_{K} \;-\; C_{UC}\,\sin 2\Phi_{K}\bigr)\Bigr). \]

13) Determine the current value of the corrected radius vector:

\( r_{K} = A \,(1 - e_{0}\,\cos E_{K}) + C_{RC}\,\cos(2\Phi_{K}) \;+\; C_{RS}\,\sin(2\Phi_{K}) \)

14) Compute the derivative of the radius vector:

\[ r_{K}' = A\,e_{0}\,E_{K}'\,\sin E_{K} + 2\,\Phi_{K}' \Bigl(C_{RS}\,\cos 2\Phi_{K} - C_{RC}\,\sin 2\Phi_{K}\Bigr). \]

15) Determine the corrected angle of inclination of the orbit:

\( I_{K} = I_{0} + C_{IC}\,\cos(2\Phi_{K}) \;+\; C_{IS}\,\sin(2\Phi_{K}) + IDOT \, t_{K} \)

16) Compute the derivative of the corrected orbit inclination angle:

\[ I_{K}' = IDOT \;+\; 2\,\Phi_{K}'\bigl(C_{IS}\,\cos 2\Phi_{K} - C_{IC}\,\sin 2\Phi_{K}\bigr). \]

17) Compute the satellite position vector in the orbital plane:

\(X_{\Phi_{K}} = r_{K}\,\cos U_{K}\),

\(Y_{\Phi_{K}} = r_{K}\,\sin U_{K}\).

18) Compute the derivative of this vector:

\[ X'_{\Phi_K} = r_K' \cos U_K - Y_{\Phi_K} U_K' \]

\[ Y'_{\Phi_K} = r_K' \sin U_K - X_{\Phi_K} U_K' \]

19) Compute the corrected longitude of the ascending node of the orbit:

\[ \Omega_{K} = \Omega_{0} + (OMEGADOT \;-\; \omega_{3})\,t_{K} \;-\; \omega_{3}\,t_{0E} \]

where \(\omega_{3} = 7.2921151467 \times 10^{-5}\,\text{rad}\,\cdot\,\text{s}^{-1}\) — Earth’s rotation rate.

20) Compute the derivative of the longitude of the ascending node:

\[ \Omega_{K}' = OMEGADOT \;-\; \omega_{3} \]

21) Compute the spacecraft coordinates at time \(t_{K}\):

\(X_{SVK} = X_{\Phi_{K}}\cos \Omega_{K} - Y_{\Phi_{K}}\,\cos I_{K}\,\sin \Omega_{K}\),

\(Y_{SVK} = X_{\Phi_{K}}\sin \Omega_{K} - Y_{\Phi_{K}}\,\cos I_{K}\,\cos \Omega_{K}\),

\(Z_{SVK} = Y_{\Phi_{K}}\,\sin I_{K}\).

22) Compute the velocities:

\(X'_{SVK} = -\,\Omega_{K}'\,Y_{SVK} + X'_{\Phi_{K}}\,\cos\Omega_{K} - \bigl(Y'_{\Phi_{K}}\,\cos I_{K} - Y_{\Phi_{K}}\,I'_{K}\,\sin I_{K}\bigr)\sin\Omega_{K}\),

\(Y'_{SVK} = \Omega_{K}'\,X_{SVK} + X'_{\Phi_{K}}\,\sin\Omega_{K} + \bigl(Y'_{\Phi_{K}}\,\cos I_{K} - Y_{\Phi_{K}}\,I'_{K}\,\sin I_{K}\bigr)\cos\Omega_{K}\),

\(Z'_{SVK} = Y_{\Phi_{K}}\,I'_{K}\,\sin I_{K} + Y'_{\Phi_{K}}\,\sin I_{K}\).

23) Compute the accelerations:

\[ X''_{SVK} = \Bigl(-\frac{\mu}{r_{K}^{3}} + \omega_{3}^{2}\Bigr)\,X_{SVK} \;+\; 2\,Y'_{SVK}\,\omega_{3} \]

\[ Y''_{SVK} = \Bigl(-\frac{\mu}{r_{K}^{3}} + \omega_{3}^{2}\Bigr)\,Y_{SVK} \;-\; 2\,X'_{SVK}\,\omega_{3} \]

\[ Z''_{SVK} = \frac{\mu}{r_{K}^{3}}\;Z_{SVK} \]

24) Compute the time difference:

\(t_{k} = t - t_{0c}\)

if \(t_{k} > 302400\) s, then \(t_{k} = t_{k} - 604800\) s,

if \(t_{k} < -302400\) s, then \(t_{k} = t_{k} + 604800\) s.

25) Compute the current clock offset of the spacecraft for L1 and L2 frequencies:

\[ \text{offset}_{L1} = a_{f0} + t_{K} \bigl(a_{f1} + t_{K} a_{f2}\bigr) + \Delta t_{R} - T_{GD} \]

\[ \text{offset}_{L2} = a_{f0} + t_{K} \bigl(a_{f1} + t_{K} a_{f2}\bigr) + \Delta t_{R} - 1.64694\,T_{GD} \]

26) Compute the drift:

\[ drift = a_{f1} \;+\; 2\,a_{f2}\,t_{k} \]

3.2.Calculation of GLONASS GNSS spacecraft motion parameters for a specified time based on ephemeris data.

The input data includes the parameters described in the previous lecture.

Recalculation of ephemerides from the time moment \(t_{b}\) to the time moment

\(\bigl|\tau_{i}\bigr| = \bigl|t_{i} - t_{b}\bigr| \leq 15\,\text{min}\)

is performed by numerical integration of the spacecraft differential equations of motion, whose right-hand sides account for accelerations determined by Earth’s gravitational constant, the harmonic \(C_{20}\) that characterizes Earth’s polar flattening, and the accelerations of lunar–solar gravitational perturbations. The spacecraft equations of motion are integrated in the rectangular geocentric PZ-90 coordinate system by the fourth-order Runge–Kutta method and have the form:

\(\frac{dx}{dt} = V_{x},\)

\(\frac{dy}{dt} = V_{y},\)

\(\frac{dz}{dt} = V_{z}.\)

\[ dV_{x} / dt = -\frac{\mu}{r^{3}}x + \frac{3}{2}C_{20} \frac{\mu a_{e}^{2}}{r^{5}} x \left[1 - \frac{5z^{2}}{r^{2}}\right] + \omega_{3}^{2}x + 2\omega_{3}V_{y} + \ddot{x}, \]

\[ dV_{y} / dt = -\frac{\mu}{r^{3}}y + \frac{3}{2}C_{20} \frac{\mu a_{e}^{2}}{r^{5}} y \left[1 - \frac{5z^{2}}{r^{2}}\right] + \omega_{3}^{2}y + 2\omega_{3}V_{x} + \ddot{y}, \]

\[ dV_{z} / dt = -\frac{\mu}{r^{3}}z + \frac{3}{2}C_{20} \frac{\mu a_{e}^{2}}{r^{5}} z \left[3 - \frac{5z^{2}}{r^{2}}\right] + \ddot{z}. \]

where:

• \(r = \sqrt{x^{2} + y^{2} + z^{2}}\)
• \(\mu = 398600.44 \text{ km}^{3}/\text{s}^{2}\) — Earth’s gravitational constant
• \(a_{e} = 6378.136 \text{ km}\) — Earth’s equatorial radius
• \(C_{20} = -1082.63 \times 10^{-6}\) — coefficient of the second zonal harmonic of the geopotential expansion in spherical functions
• \(\omega_{3} = 0.7292115 \times 10^{-4}\,\text{s}^{-1}\) — Earth’s angular rotation rate

The values \(\mu, a_{e}, C_{20}, \omega_{3}\) are adopted in the PZ-90 coordinate system.

Computation procedure:

1) Compute the time interval for which the prediction is made:
\(\tau_{i} = t_{i} - t_{b} \cdot 60.\)

If \(\tau_{i} > 43200\), set \(\tau_{i} = t_{i} - 86400\),
If \(\tau_{i} < -43200\), set \(\tau_{i} = t_{i} + 86400\).

2) Compute the number of integration steps:

\[ K = \frac{\tau_{i}}{h_{max}} + 1 \]

where \(h_{max}\) is the maximum allowable integration step defining the accuracy of the integral; in this task it is sufficient to take \(h_{max} = 60\) s.

3) Compute the integration step:

\[ h = \frac{\tau_{i}}{K} \]

4) Compute the coefficients \(K\) times:

\(K_{0j} = h\,F_{j}(Y_{ji});\)

\(K_{1j} = h\,F_{j}\bigl(Y_{ji} + \tfrac{1}{3}K_{0j}\bigr);\)

\(K_{2j} = h\,F_{j}\bigl(Y_{ji} + \tfrac{1}{6}K_{0j} + \tfrac{1}{6}K_{1j}\bigr);\)

\(K_{3j} = h\,F_{j}\bigl(Y_{ji} + \tfrac{1}{8}K_{0j} + \tfrac{3}{8}K_{2j}\bigr);\)

\(K_{4j} = h\,F_{j}\bigl(Y_{ji} + \tfrac{1}{2}K_{0j} - \tfrac{3}{2}K_{2j} + 2K_{3j}\bigr).\)

Where:

\(F_{1}(Y_{1i}) = \frac{dx}{dt};\)

\(F_{2}(Y_{2i}) = \frac{dy}{dt};\)

\(F_{3}(Y_{3i}) = \frac{dz}{dt};\)

\(F_{4}(Y_{4i}) = \frac{dV_{x}}{dt};\)

\(F_{5}(Y_{5i}) = \frac{dV_{y}}{dt};\)

\(F_{6}(Y_{6i}) = \frac{dV_{z}}{dt};\)

\(i = 1, \ldots, K;\)

\(j = 1, \ldots, 6.\)

The initial conditions \(Y_{j1}\) are the spacecraft coordinates and velocities from the ephemeris data at time \(t_{b}\).

After each cycle over \(i\) the new values are obtained:

\(Y_{j(i+1)} = Y_{ji} + \tfrac{1}{6}\bigl(K_{0j} + 4K_{3j} + K_{4j}\bigr).\)

The spacecraft coordinates and velocities at time \(t_{i}\) are the values \(Y_{ji}\) after completing the cycle over \(i\).

5) Compute the clock offset of the \(n\)-th spacecraft relative to the GLONASS time scale:

\(\text{offset} = \tau_{n}(t_{b}) - \gamma_{n}(t_{b}) \cdot \tau_{i}\)

6) The clock drift of the \(n\)-th spacecraft equals the parameter \(\gamma_{n}(t_{b})\):

\(\text{drift} = \gamma_{n}(t_{b}).\)

4. Almanac data from the previous lecture and its appendix are used to compute the coordinates and velocity-vector components of GPS and GLONASS spacecraft.

The calculation algorithms are given below.

4.1. Calculation of GPS GNSS spacecraft motion parameters for a specified time based on almanac data.

The input data includes the parameters described in the previous lecture and its appendix.

1) Convert all values \(M_{0}\), \(\Omega_{0}\), OMEGADOT, \(I_{0}\) to radians by multiplying by \(\pi\), where \(\pi = 3.1415926535898\).

2) Compute the moment \(t_{K}\) from the GPS epoch corresponding to the calculation time \(t\):

\(t_{K} = t - t_{0A};\)

3) Compute the corrected mean motion:

\[ n = \sqrt{\frac{\mu}{A^{3}}} \]

where \(\mu = 3.986005 \times 10^{14} \text{ m}^{3}\text{s}^{-2}\) — the gravitational constant.

4) Determine the current value of the mean anomaly \(M_{K}\) at time \(t_{K}\):

\[ M_{K} = M_{0} + n\,t_{K} \]

5) Solve Kepler’s equation for the eccentric anomaly \(E_{K}\) by Newton’s iterative method:

\( E_{K} \;-\; e_{0}\,\sin E_{K} \;=\; M_{K} \)

\( E_{K(n+1)} = E_{K(n)} + \frac{M_{K} - E_{K(n)} + e_{0}\,\sin E_{K(n)}}{1 - e_{0}\,\cos E_{K(n)}} \)

where \(n = 1, 2, 3, \ldots\)

\(E_{K1} = M_{K},\)

\(\bigl|E_{K(n+1)} - E_{K(n)}\bigr| < \text{eps},\)

\(\text{eps}\) — required accuracy.

6) Compute the derivative \(\,E_{K}'\):

\[ E_{K}' = \frac{n}{1 - e_{0}\,\cos E_{K(n)}} \]

7) Compute the true anomaly:

\[ \Theta_{K} = \arctan\!\Bigl(\frac{\sqrt{1 - e_{0}^{2}}\;\sin E_{K}}{\cos E_{K} - e_{0}}\Bigr)\! \]

8) Compute the argument of latitude:

\[ \Phi_{K} = \Theta_{K} + \omega \]

9) Compute the derivative of the argument of latitude:

\[ \Phi_{K}' = \frac{\sqrt{1 - e_{0}^{2}}\;E_{K}'}{\,1 - e_{0}\,\cos E_{K}\,} \]

10) Determine the current value of the radius vector:

\[ r_{K} = A \bigl(1 - e_{0}\,\cos E_{K}\bigr) \]

11) Compute the derivative of the radius vector:

\[ r_{K}' = A\,e_{0}\,E_{K}'\,\sin E_{K} \]

12) Compute the satellite position vector in the orbital plane:

\(X_{\Phi K} = r_{K}\,\cos \Phi_{K}, \quad Y_{\Phi K} = r_{K}\,\sin \Phi_{K}.\)

13) Compute the derivative of this vector:

\[ X'_{\Phi_K} = r'_K \cos\Phi_K - Y_{\Phi_K} \Phi'_K, \]

\[ Y'_{\Phi_K} = r'_K \sin\Phi_K - X_{\Phi_K} \Phi'_K. \]

14) Compute the corrected longitude of the ascending node of the orbit:

\[ \Omega_{K} = \Omega_{0} + \bigl(OMEGADOT - \omega_{3}\bigr)\,t_{K} \;-\; \omega_{3}\,t_{0A} \]

where \(\omega_{3} = 7.2921151467 \times 10^{-5}\,\text{rad}\cdot\text{s}^{-1}\) — Earth’s rotation rate.

15) Compute the derivative of the longitude of the ascending node:

\[ \Omega_{K}' = OMEGADOT - \omega_{3} \]

16) Compute the spacecraft coordinates at time \(t_{K}\):

\[ X_{SVK} = X_{\Phi_K} \cos\Omega_K - Y_{\Phi_K} \cos I_0 \sin\Omega_K, \]

\[ Y_{SVK} = X_{\Phi_K} \sin\Omega_K - Y_{\Phi_K} \cos I_0 \cos\Omega_K, \]

\[ Z_{SVK} = Y_{\Phi_K} \sin I_0. \]

17)Compute the velocities:

\[ X'_{SVK} = -\,\Omega'_K Y_{SVK} + X'_{\Phi_K} \cos\Omega_K - Y'_{\Phi_K} \cos I_0 \sin\Omega_K, \]

\[ Y'_{SVK} = \Omega'_K X_{SVK} + X'_{\Phi_K} \sin\Omega_K + Y'_{\Phi_K} \cos I_0 \cos\Omega_K, \]

\[ Z'_{SVK} = Y'_{\Phi_K} \sin I_0. \]

4.2. Calculation of GLONASS GNSS spacecraft motion parameters for a specified time based on almanac data.

The input data includes the parameters described in the previous lecture and its appendix.

The specified time moment is given in GLONASS time, where \(N_{T}\) is the day number within the four-year period and \(t_{ТЕК}\) is the time in seconds from the start of that day.

1)Determine the current values of the Keplerian elements and some other orbit elements:

\(i = i_{cp} + \Delta i;\quad T_{ДР} = T_{СР} + \Delta T;\)

\(n = \frac{2\pi}{T_{ДР}};\quad p = \sqrt[\,3]{\frac{\mu}{n^{2}}};\)

where

\(n\) – mean motion of the spacecraft;

\(p\) – semi-major axis of the spacecraft orbit;

\(\pi = 3.1415926536;\quad \mu = 398600.44;\)

2)Apply corrections for Earth’s nonsphericity:

\[ \lambda^{*} = \lambda + (\dot{\lambda} - \omega_{3})\,\Delta t_{n},\quad \omega^{*} = \omega + \dot{\omega}\,\Delta t_{n}; \]

\[ \Delta t_{n} = (N_{T} - N^{A}) \cdot 86400 + t_{TEK} - t_{\lambda}; \]

where

\(\dot{\lambda} = -10 \Bigl(\frac{a_{c}}{p}\Bigr)^{\tfrac{7}{2}}\,\cos(i)\,\frac{\pi}{180 \cdot 86400}\);

\(\dot{\omega} = 5 \Bigl(\frac{a_{c}}{p}\Bigr)^{\tfrac{7}{2}} \bigl(5\cos^{2}i - 1\bigr)\,\frac{\pi}{180 \cdot 86400}\);

3)Solve Kepler’s equation by Newton’s method

\[ E_{k+1} = E_{M} + \varepsilon \sin(E_{k}),\; k = 0, \ldots \text{ until } \]

\[ \bigl|E_{k+1} - E_{k}\bigr| > 3 \cdot 10^{-8}; \]

\[ E_{0} = E_{M}; \]

\[ E_{M} = n(\Delta t_{n} - \Delta T); \]

\[ \Delta T = \frac{E_{n} - \varepsilon \sin E_{n}}{n} + \begin{cases} 0, & \omega^{*} < \pi; \\ T_{др}, & \omega^{*} > \pi; \end{cases} \]

\[ E_{\pi} = 2 \arctan \left( \tan \left( \frac{\omega^{*}}{2} \right) \sqrt{\frac{1 - e}{1 + e}} \right) \]

4)Determine the spacecraft state vector in the orbital coordinate system:

\[ X^{0}_{1} = p (\cos E_{k+1} - \varepsilon), \quad \dot{X}^{0}_{1} = -\frac{np \sin E_{k+1}}{1 - \varepsilon \cos E_{k+1}}; \]

\[ X^{0}_{2} = p \sqrt{1 - \varepsilon^{2}} \sin E_{k+1}, \quad \dot{X}^{0}_{2} = \frac{np \sqrt{1 - \varepsilon^{2}} \cos E_{k+1}}{1 - \varepsilon \cos E_{k+1}}; \]

5)Calculate the unit vectors of the orbital coordinate system in the PZ-90 coordinate system:

\[ e^{0}_{11} = \cos\omega^{*} \cos\lambda^{*} - \sin\omega^{*} \sin\lambda^{*} \cos i; \]

\[ e^{0}_{12} = \cos\omega^{*} \sin\lambda^{*} + \sin\omega^{*} \cos\lambda^{*} \cos i; \]

\[ e^{0}_{13} = \sin\omega^{*} \sin i; \]

\[ e^{0}_{21} = -\sin\omega^{*} \cos\lambda^{*} - \cos\omega^{*} \sin\lambda^{*} \cos i; \]

\[ e^{0}_{22} = -\sin\omega^{*} \sin\lambda^{*} + \cos\omega^{*} \cos\lambda^{*} \cos i; \]

\[ e^{0}_{23} = \cos\omega^{*} \sin i; \]

6)Transform the spacecraft state vector from the orbital coordinate system to the PZ-90 coordinate system:

\[ \overline{X}^{S} = X^{0}_{1} \overline{e}^{0}_{1} + X^{0}_{2} \overline{e}^{0}_{2}; \]

\[ \dot{\overline{X}}^{S} = \dot{X}^{0}_{1} \overline{e}^{0}_{1} + \dot{X}^{0}_{2} \overline{e}^{0}_{2}; \]

\[ \dot{X}^{S}_{1} = \dot{X}^{S}_{1} + \omega_{1} X^{S}_{2}; \]

\[ \dot{X}^{S}_{2} = \dot{X}^{S}_{2} - \omega_{1} X^{S}_{1}; \]

\[ \dot{X}^{S}_{3} = \dot{X}^{S}_{3}. \]

5. Calculation of code pseudoranges

\[ \begin{cases} S_{C/A} = \bigl(Raw\_range_{C/A} + offset_{L1}\bigr)\,c,\\ S_{L1(L2)} = \bigl(Raw\_range_{L1(L2)} + offset_{L1(L2)}\bigr)\,c \end{cases} \quad \text{for GPS;} \]

\[ \begin{cases} S_{C/A} = \bigl(Raw\_range_{C/A} - offset\bigr)\,c,\\ S_{L1(L2)} = \bigl(Raw\_range_{L1(L2)} - offset\bigr)\,c \end{cases} \quad \text{for GLONASS;} \]

where: \( c \) is the speed of light in a vacuum; \( c = 299792458 \) m/s.

The pseudorange rate is calculated by:

\[ \dot{S}_{L1(2)} = -\,c \;\cdot\; \frac{F_{d,L1(L2)}}{f^{*}_{L1(2)}} \]

where:

\[ F_{d,L1(L2)} \text{ — Doppler frequency measured on } f_{L1} \text{ and } f_{L2} \]

(Structure of measurement data for Z18 receiver for GPS and structure of measurement data for GG24 receiver for GLONASS — see the previous lecture and its appendix for details.);

The frequency is calculated by:

\[ f^{*}_{L1(2)} = f_{L1(2)} \;\cdot\; (1 + drift); \]

\[ \begin{cases} f_{L1} = 1575.42 \cdot 10^{6}\,\text{Hz}, \\ f_{L2} = 1227.6 \cdot 10^{6}\,\text{Hz}; \end{cases} \quad \text{for GPS;} \] \[ \begin{cases} f_{L1} = 1602 \cdot 10^{6} + n \cdot 562.5 \cdot 10^{3}\,\text{Hz}, \\ f_{L2} = 1246 \cdot 10^{6} + n \cdot 437.5 \cdot 10^{3}\,\text{Hz}; \end{cases} \quad \text{for GLONASS;} \]

\(n\) is the GLONASS frequency channel number.

The phase pseudoranges are calculated using the following formulas:

\[ \begin{cases} \varphi_{L1} = \varphi_{1} \,\frac{c}{f_{L1}} + offset_{L1}\,c, \\ \varphi_{L2} = \varphi_{2} \,\frac{c}{f_{L2}} + offset_{L2}\,c \end{cases} \quad \text{for GPS;} \]

\[ \begin{cases} \varphi_{L1} = \varphi_{1} \,\frac{c}{f_{L1}} - offset\,c, \\ \varphi_{L2} = \varphi_{2} \,\frac{c}{f_{L2}} - offset\,c \end{cases} \quad \text{for GLONASS;} \]

6. Calculation of corrections accounting for Earth’s rotation

Corrections accounting for Earth’s rotation are applied only to the parameters \( S_{C/A}, S_{L1}, S_{L2}, \dot{S}_{L1}, \dot{S}_{L2}, \varphi_{L1}, \varphi_{L2} \) whose validity flags are zero \( (\text{П} p_{i,j,k} = 0) \), according to:

\[ \Delta_{\omega} \eta_{i,j,k} = \frac{\omega_3}{c} \cdot \bigl( y_{j} \cdot (x_{j} - X_{i}) - x_{j} \cdot (y_{j} - Y_{i}) \bigr); \]

\[ \Delta_{\omega} \dot{\eta}_{i,j,k} = \frac{\omega_3}{c} \cdot \bigl( -\dot{y}_{j} \cdot X_{i} + \dot{x}_{j} \cdot Y_{i} \bigr); \]

7. Sequence of measurement information processing. Software structure.

During normal operation, the Coordinate Correction Station (CCS) software executes the following sequence of measurement-information (MI) processing steps:

• MI acquisition from all Navigation Receiver Systems (NRS);
• Preliminary MI processing;
• MI filtering and statistical-characteristics calculation;
• Application of correction parameters accounting for the signal-propagation medium;
• MI analysis and navigation-field integrity monitoring;
• Generation of differential-correction and integrity (DCI) data;
• DCI quality validation and storage of results in the RPKNP Database.

The CCS software is a set of modules that populate the RPKNP DB. Core functions:

• Input, verification, and adjustment of CCS status information;
• Input, verification, and adjustment of external-system status information;
• Accounting for geophysical fields in the CCS geographic zone;
• DCI generation;
• CCS performance-and-fault monitoring;
• External-systems performance monitoring;
• Automated CCS hardware self-test via Management & Control APM commands.

// Software structure of the Coordinate-Time and Navigation Support System of Ukraine

FUNCTION main():
            // Initialize database
            rpknp_db = INIT_RPKNP_DATABASE()

            WHILE has_new_data_frame():

            // Step 1: Acquire and unpack raw data
            raw_frame = ACQUIRE_NRS_FRAME()
            unpacked_data = UNPACK_FRAME(raw_frame)

            // Step 2: Decode DI
            di_data = DECODE_DI(unpacked_data)

            // Step 3: Convert to physical parameters
            physical_params = CONVERT_DI_TO_PHYSICAL(di_data)

            // Step 4: Detect and remove outliers
            cleaned_params = DETECT_AND_REMOVE_OUTLIERS(physical_params)

            REPEAT:

            // Step 5: Filter and align
            filtered_data = APPLY_FILTERS(cleaned_params, rpknp_db)
            statistical_data = PERFORM_STATISTICAL_ANALYSIS(filtered_data)
            aligned_data = TIME_ALIGN_DI(statistical_data)

            // Step 6: Measurement corrections
            corrected_data = APPLY_MEASUREMENT_CORRECTIONS(aligned_data, rpknp_db)

            // Step 7: Navigation field integrity check
            integrity_status = NAVIGATION_FIELD_INTEGRITY_MONITOR(corrected_data, rpknp_db)

            // Step 8: Generate optimal DCI
            optimal_dci = GENERATE_OPTIMAL_DCI(corrected_data, integrity_status)

            // Step 9: Validate DCI quality
            dci_quality_ok = VALIDATE_DCI_QUALITY(optimal_dci, rpknp_db)

            UNTIL dci_quality_ok == TRUE

            // Step 10: Encode and deliver DCI
            encoded_dci = ENCODE_DCI_FOR_USERS(optimal_dci)
            DELIVER_TO_END_USERS(encoded_dci)

            // Step 11: Encode MI/DI and transfer to CCS
            encoded_midi = ENCODE_MI_DI_FOR_CCS(optimal_dci)
            TRANSFER_TO_CCS(encoded_midi)

            // Step 12: Update DB
            UPDATE_RPKNP_DATABASE(optimal_dci)

            END WHILE

            CLOSE_DATABASE(rpknp_db)

END FUNCTION

Measurement-Information Pre-processing Module shall perform:

• Data acquisition from all NRS;
• NRS mode control via commands from the Management & Control module;
• DI/MI frame unpacking and decoding received from NRS;
• MI-to-physical-parameters conversion;
• DI/MI validity check.

DCI Computation Module shall perform:

• MI filtering and statistical analysis;
• Applying signal-medium corrections to measured parameters;
• MI analysis;
• DCI formation;
• DCI quality validation.

DCI Encoding Module shall generate messages for end-user delivery (RTCM SC-104) and for upload to the RPKNP DB.

// Logic-software structure of the Coordinate-Time and Navigation Support System of Ukraine

Begin Coordinate-Time and Navigation Support System
            INITIALIZE RPKNP_DATABASE
            INITIALIZE Control_And_Management_Module

            WHILE system_is_running:

            // Управляющий модуль управляет остальными
            Control_And_Management_Module:

            ▸ Trigger MI_Preprocessing_Module
            ▸ Trigger DCI_Computation_Module
            ▸ Trigger DCI_Encoding_Module
            ▸ Monitor status of all modules

            // Модуль предварительной обработки измерительной информации
            MI_Preprocessing_Module:

            ▸ Input raw Measurement Information (MI)
            ▸ Perform preprocessing
            ▸ Write preprocessed_MI to RPKNP_Database

            // Модуль вычисления дифференциальной информации
            DCI_Computation_Module:

            ▸ Read preprocessed_MI from RPKNP_Database
            ▸ Compute DCI
            ▸ Write DCI_result to RPKNP_Database

            // Модуль кодирования дифференциальной информации
            DCI_Encoding_Module:

            ▸ Read DCI_result from RPKNP_Database
            ▸ Encode into transmission format
            ▸ Write encoded_DCI to RPKNP_Database


            END WHILE

END SYSTEM