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Satellite navigation solution
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<h2 id="the-solution-illustrated">The solution illustrated</h2> <h2 id="calculation-steps">Calculation steps</h2> <ol><li>A <a href="/facts/Global_navigation_satellite_system/lVgPlF5v">global-navigation-satellite-system</a> (GNSS) receiver measures the apparent transmitting time, t ~ i {\displaystyle \displaystyle {\tilde {t}}_{i}} , or "phase", of GNSS signals emitted from four or more GNSS <a href="/facts/Satellite/PTX8c68t">satellites</a> ( i = 1 , 2 , 3 , 4 , . . , n {\displaystyle \displaystyle i\;=\;1,\,2,\,3,\,4,\,..,\,n} ), simultaneously.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li> <li>GNSS satellites broadcast the messages of satellites' <a href="/facts/Ephemeris/vjQCBG29">ephemeris</a>, r i ( t ) {\displaystyle \displaystyle {\boldsymbol {r}}_{i}(t)} , and intrinsic clock bias (i.e., clock advance), δ t clock,sv , i ( t ) {\displaystyle \displaystyle \delta t_{{\text{clock,sv}},i}(t)} as the functions of (<a href="/facts/Atomic_clock/MF8nsXDR">atomic</a>) <a href="/facts/Standard_time/3dKMYnjM">standard time</a>, e.g., <a href="/facts/GPST/DKVLnzF3">GPST</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li> <li>The transmitting time of GNSS satellite signals, t i {\displaystyle \displaystyle t_{i}} , is thus derived from the non-<a href="/facts/Closed-form_expression/y0xCXlSk">closed-form</a> <a href="/facts/Equations/pxFzLAVR">equations</a> t ~ i = t i + δ t clock , i ( t i ) {\displaystyle \displaystyle {\tilde {t}}_{i}\;=\;t_{i}\,+\,\delta t_{{\text{clock}},i}(t_{i})} and δ t clock , i ( t i ) = δ t clock,sv , i ( t i ) + δ t orbit-relativ , i ( r i , r ˙ i ) {\displaystyle \displaystyle \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 δ t orbit-relativ , i ( r i , r ˙ i ) {\displaystyle \displaystyle \delta t_{{\text{orbit-relativ}},i}({\boldsymbol {r}}_{i},\,{\dot {\boldsymbol {r}}}_{i})} is the <a href="/facts/Theory_of_relativity/qGHj3eTC">relativistic</a> clock bias, periodically risen from the satellite's <a href="/facts/Orbital_eccentricity/WGlB4NBn">orbital eccentricity</a> and Earth's <a href="/facts/Gravity_field/K1uTmJFy">gravity field</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> The satellite's position and velocity are determined by t i {\displaystyle \displaystyle t_{i}} as follows: r i = r i ( t i ) {\displaystyle \displaystyle {\boldsymbol {r}}_{i}\;=\;{\boldsymbol {r}}_{i}(t_{i})} and r ˙ i = r ˙ i ( t i ) {\displaystyle \displaystyle {\dot {\boldsymbol {r}}}_{i}\;=\;{\dot {\boldsymbol {r}}}_{i}(t_{i})} .</li> <li>In the field of GNSS, "geometric range", r ( r A , r B ) {\displaystyle \displaystyle r({\boldsymbol {r}}_{A},\,{\boldsymbol {r}}_{B})} , is defined as straight range, or 3-dimensional <a href="/facts/Distance/hbXlaEFk">distance</a>,<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> from r A {\displaystyle \displaystyle {\boldsymbol {r}}_{A}} to r B {\displaystyle \displaystyle {\boldsymbol {r}}_{B}} in <a href="/facts/Inertial_frame/1U6TrdG3">inertial frame</a> (e.g., <a href="/facts/Earth-centered_inertial/yxaIGqoT">ECI</a> one), not in <a href="/facts/Rotating_frame/X4ngwkz5">rotating frame</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a></li> <li>The receiver's position, r rec {\displaystyle \displaystyle {\boldsymbol {r}}_{\text{rec}}} , and reception time, t rec {\displaystyle \displaystyle t_{\text{rec}}} , satisfy the <a href="/facts/Light-cone/GwJ0Yhjf">light-cone</a> equation of r ( r i , r rec ) / c + ( t i − t rec ) = 0 {\displaystyle \displaystyle r({\boldsymbol {r}}_{i},\,{\boldsymbol {r}}_{\text{rec}})/c\,+\,(t_{i}-t_{\text{rec}})\;=\;0} in <a href="/facts/Inertial_frame/1U6TrdG3">inertial frame</a>, where c {\displaystyle \displaystyle c} is the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a>. The signal time of flight from satellite to receiver is − ( t i − t rec ) {\displaystyle \displaystyle -(t_{i}\,-\,t_{\text{rec}})} .</li> <li>The above is extended to the <a href="/facts/Global_navigation_satellite_system/lVgPlF5v">satellite-navigation</a> <a href="/facts/Real-time_locating_system/UAj9dcUt">positioning</a> <a href="/facts/Equation/pxFzLAVR">equation</a>, r ( r i , r rec ) / c + ( t i − t rec ) + δ t atmos , i − δ t meas-err , i = 0 {\displaystyle \displaystyle 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 δ t atmos , i {\displaystyle \displaystyle \delta t_{{\text{atmos}},i}} is <a href="/facts/Atmospheric_delay/z6putj1j">atmospheric delay</a> (= <a href="/facts/Ionospheric_delay/z6putj1j">ionospheric delay</a> + <a href="/facts/Tropospheric_delay/80y1JV4M">tropospheric delay</a>) along signal path and δ t meas-err , i {\displaystyle \displaystyle \delta t_{{\text{meas-err}},i}} is the measurement error.</li> <li>The <a href="/facts/Gauss%25E2%2580%2593Newton/7c5en5cU">Gauss–Newton</a> method can be used to solve the <a href="/facts/Nonlinear/4aYItALC">nonlinear</a> <a href="/facts/Least-squares_problem/50jIwbxC">least-squares problem</a> for the solution: ( r ^ rec , t ^ rec ) = arg min ϕ ( r rec , t rec ) {\displaystyle \displaystyle ({\hat {\boldsymbol {r}}}_{\text{rec}},\,{\hat {t}}_{\text{rec}})\;=\;\arg \min \phi ({\boldsymbol {r}}_{\text{rec}},\,t_{\text{rec}})} , where ϕ ( r rec , t rec ) = ∑ i = 1 n ( δ t meas-err , i / σ δ t meas-err , i ) 2 {\displaystyle \displaystyle \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 δ t meas-err , i {\displaystyle \displaystyle \delta t_{{\text{meas-err}},i}} should be regarded as a function of r rec {\displaystyle \displaystyle {\boldsymbol {r}}_{\text{rec}}} and t rec {\displaystyle \displaystyle t_{\text{rec}}} .</li> <li>The <a href="/facts/Posterior_distribution/fojsMD0D">posterior distribution</a> of r rec {\displaystyle \displaystyle {\boldsymbol {r}}_{\text{rec}}} and t rec {\displaystyle \displaystyle t_{\text{rec}}} is proportional to exp ( − 1 2 ϕ ( r rec , t rec ) ) {\displaystyle \displaystyle \exp(-{\frac {1}{2}}\phi ({\boldsymbol {r}}_{\text{rec}},\,t_{\text{rec}}))} , whose <a href="/facts/Mode_(statistics)/vq5tGp0G">mode</a> is ( r ^ rec , t ^ rec ) {\displaystyle \displaystyle ({\hat {\boldsymbol {r}}}_{\text{rec}},\,{\hat {t}}_{\text{rec}})} . Their inference is formalized as <a href="/facts/Maximum_a_posteriori_estimation/Y5zHyqfX">maximum a posteriori estimation</a>.</li> <li>The <a href="/facts/Posterior_distribution/fojsMD0D">posterior distribution</a> of r rec {\displaystyle \displaystyle {\boldsymbol {r}}_{\text{rec}}} is proportional to ∫ − ∞ ∞ exp ( − 1 2 ϕ ( r rec , t rec ) ) d t rec {\displaystyle \displaystyle \int _{-\infty }^{\infty }\exp(-{\frac {1}{2}}\phi ({\boldsymbol {r}}_{\text{rec}},\,t_{\text{rec}}))\,dt_{\text{rec}}} .</li></ol> <h2 id="the-gps-case">The GPS case</h2> <ul><li>For <a href="/facts/Global_Positioning_System/DKVLnzF3">Global Positioning System</a> (GPS),<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> the non-closed-form equations in step 3 result in</li></ul> { Δ t i ( t i , E i ) ≜ t i + δ t clock , i ( t i , E i ) − t ~ i = 0 , Δ M i ( t i , E i ) ≜ M i ( t i ) − ( E i − e i sin E i ) = 0 , {\displaystyle \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}}} <p>in which E i {\displaystyle \scriptstyle E_{i}} is the orbital <a href="/facts/Eccentric_anomaly/rmbJzkGl">eccentric anomaly</a> of satellite i {\displaystyle i} , M i {\displaystyle \scriptstyle M_{i}} is the <a href="/facts/Mean_anomaly/dzs98drn">mean anomaly</a>, e i {\displaystyle \scriptstyle e_{i}} is the <a href="/facts/Orbital_eccentricity/WGlB4NBn">eccentricity</a>, and δ t clock , i ( t i , E i ) = δ t clock,sv , i ( t i ) + δ t orbit-relativ , i ( E i ) {\displaystyle \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})} . </p> <ul><li>The above can be solved by using the <a href="/facts/Bivariate_analysis/skgknzMV">bivariate</a> <a href="/facts/Newton%25E2%2580%2593Raphson/VlRhI2FL">Newton–Raphson</a> method on t i {\displaystyle \scriptstyle t_{i}} and E i {\displaystyle \scriptstyle E_{i}} . Two times of iteration will be necessary and sufficient in most cases. Its iterative update will be described by using the approximated <a href="/facts/Inverse_matrix/fPqXk3V8">inverse</a> of <a href="/facts/Jacobian_matrix_and_determinant/0tNv6o8j">Jacobian</a> matrix as follows:</li></ul> <p> ( t i E i ) ← ( t i E i ) − ( 1 0 M ˙ i ( t i ) 1 − e i cos E i − 1 1 − e i cos E i ) ( Δ t i Δ M i ) {\displaystyle \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}}} </p> <ul><li><a href="/facts/Tropospheric_delay/80y1JV4M">Tropospheric delay</a> should not be ignored, while the <a href="/facts/Global_Positioning_System/DKVLnzF3">Global Positioning System</a> (GPS) specification<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> doesn't provide its detailed description.</li></ul> <h2 id="the-glonass-case">The GLONASS case</h2> <ul><li>The <a href="/facts/GLONASS/sLh1vQ6p">GLONASS</a> ephemerides don't provide clock biases δ t clock,sv , i ( t ) {\displaystyle \scriptstyle \delta t_{{\text{clock,sv}},i}(t)} , but δ t clock , i ( t ) {\displaystyle \scriptstyle \delta t_{{\text{clock}},i}(t)} .</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Dilution_of_precision_(navigation)/2B6m0r17">Dilution of precision (navigation)</a></li> <li><a href="/facts/Global_Positioning_System/DKVLnzF3">Global Positioning System § Navigation equations</a></li> <li><a href="/facts/Least_squares_adjustment/alLkR32R">Least squares adjustment</a></li> <li><a href="/facts/Precise_Point_Positioning/MpNMxYfz">Precise Point Positioning</a></li> <li><a href="/facts/Pseudo-range_multilateration/MmdIwVGx">Pseudo-range multilateration</a></li> <li><a href="/facts/Real_Time_Kinematic/MrlAvKU4">Real Time Kinematic</a></li> <li><a href="/facts/Time_to_first_fix/X1uTfwY2">Time to first fix</a></li></ul> <h2 id="notes">Notes</h2> <ul><li>In the field of GNSS, r ~ i = − c ( t ~ i − t ~ rec ) {\displaystyle \scriptstyle {\tilde {r}}_{i}\;=\;-c({\tilde {t}}_{i}\,-\,{\tilde {t}}_{\text{rec}})} is called <a href="/facts/Pseudorange/cRFhDJCQ">pseudorange</a>, where t ~ rec {\displaystyle \scriptstyle {\tilde {t}}_{\text{rec}}} is a provisional reception time of the receiver. δ t clock,rec = t ~ rec − t rec {\displaystyle \scriptstyle \delta t_{\text{clock,rec}}\;=\;{\tilde {t}}_{\text{rec}}\,-\,t_{\text{rec}}} is called receiver's clock bias (i.e., clock advance).<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a></li> <li>Standard GNSS receivers output r ~ i {\displaystyle \scriptstyle {\tilde {r}}_{i}} and t ~ rec {\displaystyle \scriptstyle {\tilde {t}}_{\text{rec}}} per an observation <a href="/facts/Epoch_(astronomy)/RDdj6SmA">epoch</a>.</li> <li>The temporal variation in the relativistic clock bias of satellite is linear if its orbit is circular (and thus its velocity is uniform in inertial frame).</li> <li>The signal time of flight from satellite to receiver is expressed as − ( t i − t rec ) = r ~ i / c + δ t clock , i − δ t clock,rec {\displaystyle \scriptstyle -(t_{i}-t_{\text{rec}})\;=\;{\tilde {r}}_{i}/c\,+\,\delta t_{{\text{clock}},i}\,-\,\delta t_{\text{clock,rec}}} , whose right side is <a href="/facts/Round-off_error/aF9N8X4I">round-off-error</a> resistive during calculation.</li> <li>The geometric range is calculated as r ( r i , r rec ) = | Ω E ( t i − t rec ) r i , ECEF − r rec,ECEF | {\displaystyle \scriptstyle r({\boldsymbol {r}}_{i},\,{\boldsymbol {r}}_{\text{rec}})\;=\;|\Omega _{\text{E}}(t_{i}\,-\,t_{\text{rec}}){\boldsymbol {r}}_{i,{\text{ECEF}}}\,-\,{\boldsymbol {r}}_{\text{rec,ECEF}}|} , where the <a href="/facts/ECEF/LuFYT2Ly">Earth-centred, Earth-fixed</a> (ECEF) rotating frame (e.g., <a href="/facts/WGS84/V9wSJnjX">WGS84</a> or <a href="/facts/International_Terrestrial_Reference_Frame/RIezq9Ms">ITRF</a>) is used in the right side and Ω E {\displaystyle \scriptstyle \Omega _{\text{E}}} is the Earth rotating matrix with the argument of the signal <a href="/facts/Time_of_flight/DSINDYyn">transit time</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> The matrix can be factorized as Ω E ( t i − t rec ) = Ω E ( δ t clock,rec ) Ω E ( − r ~ i / c − δ t clock , i ) {\displaystyle \scriptstyle \Omega _{\text{E}}(t_{i}\,-\,t_{\text{rec}})\;=\;\Omega _{\text{E}}(\delta t_{\text{clock,rec}})\Omega _{\text{E}}(-{\tilde {r}}_{i}/c\,-\,\delta t_{{\text{clock}},i})} .</li> <li>The line-of-sight unit vector of satellite observed at r rec,ECEF {\displaystyle \scriptstyle {\boldsymbol {r}}_{\text{rec,ECEF}}} is described as: e i , rec,ECEF = − ∂ r ( r i , r rec ) ∂ r rec,ECEF {\displaystyle \scriptstyle {\boldsymbol {e}}_{i,{\text{rec,ECEF}}}\;=\;-{\frac {\partial r({\boldsymbol {r}}_{i},\,{\boldsymbol {r}}_{\text{rec}})}{\partial {\boldsymbol {r}}_{\text{rec,ECEF}}}}} .</li> <li>The <a href="/facts/Global_navigation_satellite_system/lVgPlF5v">satellite-navigation</a> <a href="/facts/Real-time_locating_system/UAj9dcUt">positioning</a> <a href="/facts/Equation/pxFzLAVR">equation</a> may be expressed by using the <a href="/facts/Variable_(mathematics)/VbQlrvKz">variables</a> r rec,ECEF {\displaystyle \scriptstyle {\boldsymbol {r}}_{\text{rec,ECEF}}} and δ t clock,rec {\displaystyle \scriptstyle \delta t_{\text{clock,rec}}} .</li> <li>The <a href="/facts/Nonlinearity/4aYItALC">nonlinearity</a> of the vertical dependency of <a href="/facts/Tropospheric_delay/80y1JV4M">tropospheric delay</a> degrades the convergence efficiency in the <a href="/facts/Gauss%25E2%2580%2593Newton/7c5en5cU">Gauss–Newton</a> iterations in step 7.</li> <li>The above notation is different from that in the Wikipedia articles, 'Position calculation introduction' and 'Position calculation advanced', of <a href="/facts/Global_Positioning_System/DKVLnzF3">Global Positioning System</a> (GPS).</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://gnss-sdr.org/docs/sp-blocks/pvt/">PVT</a> (Position, Velocity, Time): Calculation procedure in the <a href="/facts/Open-source/7X9rCdz4">open-source</a> GNSS-SDR and the underlying RTKLIB</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Misra, P. and Enge, P., Global Positioning System: Signals, Measurements, and Performance, 2nd, Ganga-Jamuna Press, 2006. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>The interface specification of NAVSTAR GLOBAL POSITIONING SYSTEM <a href="http://www.navcen.uscg.gov/pdf/IS-GPS-200D.pdf" target="_blank">http://www.navcen.uscg.gov/pdf/IS-GPS-200D.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>The interface specification of NAVSTAR GLOBAL POSITIONING SYSTEM <a href="http://www.navcen.uscg.gov/pdf/IS-GPS-200D.pdf" target="_blank">http://www.navcen.uscg.gov/pdf/IS-GPS-200D.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>3-dimensional distance is given by r ( r A , r B ) = | r A − r B | = ( x A − x B ) 2 + ( y A − y B ) 2 + ( z A − z B ) 2 {\displaystyle \displaystyle 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 r A = ( x A , y A , z A ) {\displaystyle \displaystyle {\boldsymbol {r}}_{A}=(x_{A},y_{A},z_{A})} and r B = ( x B , y B , z B ) {\displaystyle \displaystyle {\boldsymbol {r}}_{B}=(x_{B},y_{B},z_{B})} represented in inertial frame. <a href="/wiki/Distance#Geometry" target="_blank">/wiki/Distance#Geometry</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>The interface specification of NAVSTAR GLOBAL POSITIONING SYSTEM <a href="http://www.navcen.uscg.gov/pdf/IS-GPS-200D.pdf" target="_blank">http://www.navcen.uscg.gov/pdf/IS-GPS-200D.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>The interface specification of NAVSTAR GLOBAL POSITIONING SYSTEM <a href="http://www.navcen.uscg.gov/pdf/IS-GPS-200D.pdf" target="_blank">http://www.navcen.uscg.gov/pdf/IS-GPS-200D.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>The interface specification of NAVSTAR GLOBAL POSITIONING SYSTEM <a href="http://www.navcen.uscg.gov/pdf/IS-GPS-200D.pdf" target="_blank">http://www.navcen.uscg.gov/pdf/IS-GPS-200D.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Misra, P. and Enge, P., Global Positioning System: Signals, Measurements, and Performance, 2nd, Ganga-Jamuna Press, 2006. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>The interface specification of NAVSTAR GLOBAL POSITIONING SYSTEM <a href="http://www.navcen.uscg.gov/pdf/IS-GPS-200D.pdf" target="_blank">http://www.navcen.uscg.gov/pdf/IS-GPS-200D.pdf</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> </ol>