With the previous examples and models we have laid the groundwork for the task of determining user position,Xu,Yu,Zu in ECEF coordinates and the user clock error, Tbias. Figure2.6shows the system using four SV’s and four SV clock replicas in the user receiver. We again assume the user receiver is at or near the surface of the earth (i.e., each path delay is 66–85 ms). The four SV clocks are referenced to a master clock. The receiver’s reference clock is shown with its error removed. In other words the receiver reference clock is displaying master clock time. For each SV to user distance a path delay is measured/computed. Additionally a distanceRiis assigned to the equivalent distance corresponding to path delay multiplied by the speed of light.
Figure2.7shows the set of equations. Each distance,Ri, can be computed from two different methods. The first method uses the path delay, which contains measured, computed, and SV-supplied terms. Each path delay has its own unique Tatmsv_iterm derived from estimates of user position. As user position estimates are refined, this term will also be refined in accuracy. The clock error for each SV is sent to the receiver and is identified by Terrsv_i.
Normally the Trec time would be the same for all the path delays. It is possible to have separate Trec times if the user receiver does not move appreciably from one Trec time to the next. As before we take a “snapshot” of the receiver’s reference clock and the replica clock or clocks to record Trec and Tsentsv_i. If we choose to include all the clocks in our picture, then the value of Trec could be the same for all SV path delay equations. This last point needs a bit more clarification. If the user receiver is stationary (or moving slow compared to the time to measure each path delay) then we can measure the path delay to each SV separately. That is, we could first do SV1, then SV2, etc. In other words, a sequential measurement using a single channel receiver. As long as the drift rate of the receiver’s reference clock is low enough over the time all four measurements are made, this method will work fine.
Many early GPS receivers used this method as they were single channel receivers.
30 2 Introduction to the Global Positioning System
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(Xu, Yu, Zu) Fig.2.6CalculationofuserpositionandTbiasusingfourSV’s
2.6 Solving For User Position Using Four Satellites 31
The drawback is a better reference clock is needed as the method is relying on Tbias being approximately constant over the entire measurement sequence of the four SV’s path delays. Being able to “measure” four Trec and four Tsent times simulta-neouslyrequires a four-channel receiver. This is the assumption made when writing the equations of Fig.2.7.
The distanceRican also be computed by using the distance formula as we did in the single SV example for obtaining Tbias. Each distance is computed from the user’s positionXu,Yu,Zu and the position of the SV, which is assumed to be sent from SV asXsv_i,Ysv_i,Zsv_icoordinates. By using the distance equations combined with the path delay equations we have enough information to determine the receiver’s position and the receiver clock error. Unlike our simple solution for Tbias using one SV, the equation set of Fig.2.7cannotbe solved in closed form for Tbias andXu, Yu, Zu. The reason is that the equation for distance contains the square root function. This introduces a nonlinearity which precludes a “closed form” solution. To solve the equations a iterative method is employed. An iterative solution for user position and Tbias is presented in Chap. 5.
Fig. 2.7 Equation set for solving for user position and Tbias using four SV’s
32 2 Introduction to the Global Positioning System