El profesorado, la mediación y los marcos interpersonales de referencia

Where: o is taken equal to 54 dB.Hz

o is the estimated C/N0 value in dB.Hz

o is the standard deviation of the observation at the zenith. Standard deviations at the zenith for code, Doppler and carrier phase measurements are expressed as follows:

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 Observations from satellites with elevations lower than 10° are discarded [Realini, 2009].  C/N0 mask values of 32 dB.Hz, 40 dB.Hz and 40dB.Hz will be used for carrier phase,

Doppler and code measurements respectively. However, other values will be experimented.

 GLONASS code observation variances are downweighted by a factor of . This factor was determined experimentally in section 3.1.1.4.

 Klobuchar model and UNB3m model are used to correct for ionospheric and tropospheric differential delay respectively.

 IGS rapid ephemeris and Russian Federal Space Agency rapid ephemeris are used for GPS and GLONASS respectively, for satellite position and satellite clock offset computation.  Variances of the accelerations in the process noise matrix were set empirically, after

testing different values: _{where ,}

, are the process noise variances of acceleration in the north, east and up direction respectively.

The performance of this baseline filter will be described in the next sections.

6.1.2

Position Error of the Baseline Solution on Data Set 1

The baseline precise positioning algorithm implemented as described in the previous section is tested using the first data set. Position error is plotted on Figure 6.1 and Figure 6.2, separating the urban data set from the beltway data set. Contrary to RTKLIB, a solution is output at every epoch in the software, even if the Kalman filter uses prediction only. Baseline solution is then less spiky than the RTKLIB

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solution presented in chapter 5, as it isn’t reinitialized any time a multipath is detected. Moreover, the null vertical velocity observation smoothed the vertical component of the solution, as shown in Appendix D.

However, the baseline solution suffers from large biases due to code pseudorange multipath. In particular, the performance on the first part of the beltway is poor, because it starts just after a very challenging environment due to tree foliage.

Figure 6.1 Difference between estimated trajectory and reference trajectory in downtown Toulouse (data set 2). Black asterisk represents epochs when ambiguity vector is validated and

fixed as integer

Figure 6.2 Difference between estimated trajectory and reference trajectory on Toulouse’s beltway (data set 2). Black asterisk represents epochs when ambiguity vector is validated and fixed as integer

Then as code measurements have a very low-weight in the processing, the bias in position is kept and propagated through multiple epochs, even if pseudoranges observations on the beltway are in general of very good quality, as seen in section 3.1.1.2. Therefore, simply down-weighting code measurements in the Kalman filter is not a very efficient solution when different environments are targeted, as the navigation will be degraded even in the period with good signal quality.

On the beltway, the speed of the vehicle is very high and the phase of the multipath changes very quickly. As a consequence Doppler-smoothed pseudoranges tend to make the position error converge to zero on the beltway, notably in the horizontal plane. However, this convergence is very slow. The covariance matrix associated to the position is too optimistic relative to the actual error as shown on Figure 6.3 and Figure 6.4. Indeed, observations multipath errors are not Gaussian and are not well modeled by the associated covariance matrix. In some cases, the estimated variance in the vertical direction is lower than in the horizontal direction. This is due to the weight of the null vertical velocity constraint which depends on the speed of the vehicle, as explained in section 4.2.1.1. At high speed such as on the beltway, this constraint has almost no effect. However when the vehicle is driven slowly, as between 9:25 and 9:30, it plays an important role in the smoothing of the solution vertical component. 09:15 09:20 09:25 09:30 09:35 09:40 -10 -8 -6 -4 -2 0 2 4 6 8 10

Error in the estimated position (Downtown Toulouse)

Time of Day (hours:minutes)

m e te rs Up East North Fix mode 09:45 09:50 09:55 -10 -8 -6 -4 -2 0 2 4 6 8 10

Error in the estimated position (Toulouse's beltway)

Time of Day (hours:minutes)

m e te rs Up East North Fix mode

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Figure 6.3 Position estimated standard deviation (3 sigma), as output by the Kalman filter in downtown Toulouse (data set 1).

Black asterisk represents epochs when ambiguity vector is validated and fixed as integer

Figure 6.4 Position estimated standard deviation (3 sigma), as output by the Kalman filter on Toulouse’s beltway (data set 1). Black asterisk represents epochs when ambiguity vector is validated and fixed as integer

Additionally, it can be seen that even with a ratio test of 3 and a minimum of 5 ambiguities available for ambiguity validation, wrong ambiguity fixing occurs frequently. During wrong fixes, positions are associated to a centimeter-level estimated standard deviation in the Kalman filter, as seen on Figure 6.3 and Figure 6.4. Then, the variance associated to the position only slowly increases, indicating the Kalman filter trusts the propagated positions deduced from the wrongly fixed ambiguities. Therefore, wrong integer ambiguity resolution has to be avoided as much as possible.