GNSS positioning calculation

The global navigation satellite system (GNSS) positioning for receiver's position is derived through the calculation steps, or algorithm, given below. In essence, a GNSS receiver measures the transmitting time of GNSS signals emitted from four or more GNSS satellites and these measurements are used to obtain its position (i.e., spatial coordinates) and reception time.

Calculation steps

  1. A global-navigation-satellite-system (GNSS) receiver measures the apparent transmitting time, \scriptstyle \tilde{t}_i, or "phase", of GNSS signals emitted from four or more GNSS satellites (\scriptstyle i \;=\; 1,\, 2,\, 3,\, 4,\, ..,\, n ), simultaneously.[1]
  2. GNSS satellites broadcast the messages of satellites' ephemeris, \scriptstyle \boldsymbol{r}_i (t), and intrinsic clock bias (i.e., clock advance), \scriptstyle\delta t_{\text{clock,sv},i} (t) as the functions of (atomic) standard time, e.g., GPST.[2]
  3. The transmitting time of GNSS satellite signals, \scriptstyle t_i, is thus derived from the non-closed-form equations \scriptstyle \tilde{t}_i \;=\; t_i \,+\, \delta t_{\text{clock},i} (t_i) and \scriptstyle \delta t_{\text{clock},i} (t_i) \;=\; \delta t_{\text{clock,sv},i} (t_i) \,+\, \delta t_{\text{orbit-relativ},\, i} (\boldsymbol{r}_i,\, \dot{\boldsymbol{r}}_i), where \scriptstyle \delta t_{\text{orbit-relativ},i} (\boldsymbol{r}_i,\, \dot{\boldsymbol{r}}_i) is the relativistic clock bias, periodically risen from the satellite's orbital eccentricity and Earth's gravity field.[2] The satellite's position and velocity are determined by \scriptstyle t_i as follows: \scriptstyle \boldsymbol{r}_i \;=\; \boldsymbol{r}_i (t_i) and \scriptstyle \dot{\boldsymbol{r}}_i \;=\; \dot{\boldsymbol{r}}_i (t_i).
  4. In the field of GNSS, "geometric range", \scriptstyle r(\boldsymbol{r}_A,\, \boldsymbol{r}_B) , is defined as straight range, or 3-dimensional distance,[3] from \scriptstyle\boldsymbol{r}_A to \scriptstyle\boldsymbol{r}_B in inertial frame (e.g., Earth-centered inertial (ECI) one), not in rotating frame.[2]
  5. The receiver's position, \scriptstyle \boldsymbol{r}_{\text{rec}}, and reception time, \scriptstyle  t_{\text{rec}}, satisfy the light-cone equation of \scriptstyle r(\boldsymbol{r}_i,\, \boldsymbol{r}_{\text{rec}}) / c \,+\, (t_i - t_{\text{rec}}) \;=\; 0 in inertial frame, where \scriptstyle c is the speed of light. The signal transit time is \scriptstyle -(t_i \,-\, t_{\text{rec}}).
  6. The above is extended to the satellite-navigation positioning equation, \scriptstyle r(\boldsymbol{r}_i,\, \boldsymbol{r}_{\text{rec}}) / c \,+\, (t_i \,-\, t_{\text{rec}}) \,+\, \delta t_{\text{atmos},i} \,-\, \delta t_{\text{meas-err},i} \;=\; 0 , where \scriptstyle \delta t_{\text{atmos},i} is atmospheric delay (= ionospheric delay + tropospheric delay) along signal path and \scriptstyle \delta t_{\text{meas-err},i} is the measurement error.
  7. The Gauss–Newton method can be used to solve the nonlinear least-squares problem for the solution: \scriptstyle (\hat{\boldsymbol{r}}_{\text{rec}},\, \hat{t}_{\text{rec}}) \;=\; \arg \min  \phi ( \boldsymbol{r}_{\text{rec}},\,  t_{\text{rec}} ), where \scriptstyle \phi ( \boldsymbol{r}_{\text{rec}},\, t_{\text{rec}} ) \;=\; \sum_{i=1}^n ( \delta t_{\text{meas-err},i} / \sigma_{\delta t_{\text{meas-err},i} } )^2 . Note that \scriptstyle \delta t_{\text{meas-err},i} should be regarded as a function of \scriptstyle \boldsymbol{r}_{\text{rec}} and \scriptstyle t_{\text{rec}}.
  8. The posterior distribution of \scriptstyle \boldsymbol{r}_{\text{rec}} and \scriptstyle t_{\text{rec}} is proportional to \scriptstyle \exp ( -\frac{1}{2} \phi ( \boldsymbol{r}_{\text{rec}},\, t_{\text{rec}} ) ), whose mode is \scriptstyle (\hat{\boldsymbol{r}}_{\text{rec}},\, \hat{t}_{\text{rec}}). Their inference is formalized as maximum a posteriori estimation.
  9. The posterior distribution of \scriptstyle \boldsymbol{r}_{\text{rec}} is proportional to \scriptstyle \int_{-\infty}^\infty \exp ( -\frac{1}{2} \phi ( \boldsymbol{r}_{\text{rec}},\, t_{\text{rec}} ) ) \, d t_{\text{rec}}.

The solution illustrated

The GPS case

 \scriptstyle \begin{cases}
  \scriptstyle \Delta t_i (t_i,\, E_i) \;\triangleq\; t_i \,+\, \delta t_{\text{clock},i} (t_i,\, E_i) \,-\, \tilde{t}_i \;=\; 0, \\
  \scriptstyle \Delta M_i (t_i,\, E_i) \;\triangleq\; M_i (t_i) \,-\, (E_i \,-\, e_i \sin E_i) \;=\; 0, 
\end{cases}

in which \scriptstyle E_i is the orbital eccentric anomaly of satellite i, \scriptstyle M_i is the mean anomaly, \scriptstyle e_i is the eccentricity, and \scriptstyle \delta t_{\text{clock},i} (t_i,\, E_i) \;=\; \delta t_{\text{clock,sv},i} (t_i) \,+\, \delta t_{\text{orbit-relativ},i} (E_i).

\scriptstyle
\begin{pmatrix}
  t_i \\
  E_i \\
\end{pmatrix}
\leftarrow 
\begin{pmatrix}
  t_i \\
  E_i \\
\end{pmatrix}
- 
\begin{pmatrix}
  1 && 0 \\
  \frac{\dot{M}_i (t_i)}{1 - e_i \cos E_i} && -\frac{1}{1 - e_i \cos E_i} \\
\end{pmatrix}
\begin{pmatrix}
  \Delta t_i \\
  \Delta M_i \\
\end{pmatrix}

The GLONASS case

Note

See also

References

  1. 1 2 Misra, P. and Enge, P., Global Positioning System: Signals, Measurements, and Performance, 2nd, Ganga-Jamuna Press, 2006.
  2. 1 2 3 4 5 6 The interface specification of NAVSTAR GLOBAL POSITIONING SYSTEM
  3. 3-dimensional distance is given by \scriptstyle r(\boldsymbol{r}_A,\, \boldsymbol{r}_B) = |\boldsymbol{r}_A - \boldsymbol{r}_B| = \sqrt{(x_A-x_B)^2+(y_A-y_B)^2+(z_A-z_B)^2} where  \scriptstyle\boldsymbol{r}_A = (x_A, y_A, z_A) and \scriptstyle\boldsymbol{r}_B = (x_B, y_B, z_B) represented in inertial frame.
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