GEOMETRIC DILUTION OF PRECISION in C#.NET

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85 GEOMETRIC DILUTION OF PRECISION
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10 GDOP 10
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1500 2000 Time, t, sec
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Figure 811: GDOP chimney for a set of four satellites
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are: SV 2 SV 24 SV 4 SV 5 SV 7 SV 9 0557466 +0469913 +0086117 +0661510 0337536 +0762094 +0829830 +0860463 +0936539 0318625 +0461389 +0267539 0024781 +0196942 0339823 0678884 0820482 0589606 +1000000 +1000000 +1000000 +1000000 +1000000 +1000000
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For this set of six satellites, there are fteen di erent combinations of four satellites, as shown in Table 86 The GDOP gures for the various sets of four satellites range from 358 to 1603 Instead of selecting the best four satellites, an all-in-view approach using all six satellites would result in a GDOP of 296 A comparison of the above discussion with the discussion of observability in Section 361 shows that the various DOP factors are measures the degree of observability of a selected set of position states are at a given time instant (and location) for a given set of satellites If observability is temporarily lost (eg, due to satellite occlusion), some of the DOP factors become in nite and a full position solution is not possible by stand-alone GPS In an aided navigation system, the integration of the high rate sensors through the vehicle kinematic equations will maintain an estimate of the vehicle state which includes the position In addition, the navigation system will maintain an estimate of the state error covariance matrix denoted as P During the time period when PDOP is in nite, some linear combination of the states will be unobservable from the available GPS measurements The
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CHAPTER 8 GLOBAL POSITIONING SYSTEM SV Combination 24, 2, 4, 5 24, 2, 4, 9 24, 2, 9, 5 2, 9, 4, 5 24, 9, 4, 5 24, 9, 4, 7 24, 9, 5, 7 24, 9, 2, 7 GDOP 372 486 509 1603 545 696 489 1310 Satellite Combination 24, 2, 4, 7 24, 2, 7, 5 2, 7, 4, 5 24, 7, 4, 5 7, 9, 4, 5 7, 9, 4, 2 7, 9, 5, 2 GDOP 625 448 358 863 859 496 392
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Table 86: Example satellite combinations and corresponding GDOP values Kalman lter will use the available measurements in an optimal fashion to maintain the estimate accuracy to the extent possible The unobservable portions of the state vector will be indicated by growth in the corresponding directions of the P matrix This could be analyzed via a SVD or eigendecomposition of P This section has worked entirely with covariance of position error Other performance metrics (eg, CEP, R95, 2drms, etc) are discussed in Section 491
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A major reason for the design of multi-frequency receivers is to allow estimation and compensation of the ionospheric delay Simultaneous L1 and L2 pseudorange observables from the same satellite and receiver can be modeled as 1 2 f2 Ia + 1 f1 f1 = + Ecm + Ia + 2 f2 = + Ecm + (883) (884)
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where Ecm is the lumped common mode errors other that the ionospheric 403 delay Ia = f1 f2 T EC, i represents the sum of the receiver noise and multipath errors, and f1 and f2 are the carrier frequencies In the analysis to follow, we will assume that 1 and 2 are independent, Gaussian random 2 variables with variance Eqns (883 884) can be manipulated to provide an ionospheric free pseudorange observable as =
2 2 f1 1 f2 2 2 f 2) (f1 2
(885)
86 TWO FREQUENCY RECEIVERS which has the measurement model = + Ecm +
2 f1 2 2 f1 f2 2 1 f 2 f 2 2 f2 1 2
(886)
Eqn (885) is not usually used directly, because as shown in eqn (886), 2 the noise variance (ie, e ect of 1 and 2 on ) is approximately 9 The following two paragraphs discuss alternative approaches to the construction of ionospheric free pseudorange observables An estimate of the ionospheric delay can be computed as Ia = f1 f2 2 2 ( 2 1 ) f1 f2 (887)
Direct substitution of eqns (883 884) into eqn (887) shows that Ia f1 f2 2 2 ( 2 1 ) f1 f2 = Ia + 1984 ( 2 1 ) = Ia + (888) (889)
which shows that Ia is unbiased The variance of Ia at each epoch is 2 approximately 8 Because of the magni cation of the receiver noise, the ionospheric estimate of eqn (887) is not used directly to compensate the pseudorange measurement Instead, because Ia changes slowly with a correlation time of a few hours, while 1 and 2 have much shorter correlation times, Ia could for example be low pass ltered by a lter with a time constant of several minutes to greatly decrease the e ect of 1 and 2 while maintaining the time variation of Ia If we denote the ltered version of a as Ia then an ionosphere free pseudorange can be computed as I = 1 which has the error model = + Ecm + 1 where the ionospheric error has been (essentially) removed and the measurement noise has not been ampli ed The remaining common-mode errors could be removed via di erential processing Another approach is discussed subsequently Simultaneous L1 and L2 phase observables from the same satellite and receiver can be modeled as 1 1 2 2 = = f2 Ia + 1 + 1 N1 f1 f1 + Ecm Ia + 2 + 2 N2 f2 + Ecm (891) (892) f2 Ia f1 (890)
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