## 5 Localization Algorithm Performance
### Position Accuracy  
This subsection deals with the position accuracy achieved by the different algorithm versions over the course of the two data sets. In the beginning, Table 8 gives an overview of the most important accuracy metrics, evaluated for all three algorithm versions (single-frequency GPS, multi-frequency GPS and multi-frequency/multi-constellation GNSS) with both odometry turned off and on. As our previous research [20] has shown, the evaluation of these metrics over a whole data set combined can only provide a first impression of the achievable accuracy, but is insufficient for a detailed analysis. That is why Table 10 breaks down the RMS position error for the multi-frequency/multi-constellation algorithm with odometry aiding in six categories, depending on the reference solution quality. The subsection concludes with an evaluation of the horizontal position error and its estimated standard deviation during the passages through tunnels in Figure 6 to emphasize the importance of odometry aiding in GNSS-denied environments. All results in this subsection were obtained with the calibrated DCBs, using all types of pseudo-range observations available to the respective algorithm version simultaneously. The results in Table 8 show that the multi-frequency/multi-constellation algorithm is capable of improving all major position accuracy metrics. The arithmetic mean is given to demonstrate that position estimation with multi-frequency GNSS is less biased than with single-frequency observations. This is especially obvious when looking at the vertical coordinate's mean in data set 2: Without odometry aiding, it is \(-1.8\,\mathrm{m}\) for single-frequency GPS, \(-0.3\,\mathrm{m}\) for multi-frequency GPS and \(-0.5\,\mathrm{m}\) for multi-frequency/multi-constellation GNSS. The reduction in the mean's absolute value when including GPS pseudo-ranges on multiple frequencies confirms the ability of multi-frequency measurements to eliminate the ionospheric error, improving unbiasedness primarily in the vertical coordinate. When quantifying estimation accuracy in comparison to an external reference solution as is done here, the RMS error is more suitable than the error's standard deviation. While the standard deviation quantifies the variation around the mean, the RMS error quantifies the variation around the reference solution, delivering a metric that quantifies both precision and unbiasedness combined. In terms of the RMS error, the multi-frequency/multi-constellation algorithm delivers the best performance for all coordinates in each column. The horizontal position error quantiles in the last row of Table 8 are important because automotive positioning is mainly concerned with horizontal accuracy. Since all vehicles travel on the road, vertical accuracy is not as critical. The multi-frequency/multi-constellation algorithm without odometry aiding achieves the best performance in the 50% quantiles. When odometry is included, the 50% quantiles rise slightly (by \(0.1\,\mathrm{m}\) in each data set). The information about the vehicle's velocity provided by the odometry leads to a decreased estimated covariance of the position error (via covariance propagation through the EKF's system model). That is why the pseudo-ranges receive relatively smaller weights when odometry aiding is turned on in comparison to when odometry aiding is turned off. As a result, position errors will grow slower when GNSS is unavailable (see Figure 6), but may also decrease slower when GNSS is available. This circumstance improves positioning performance when GNSS reception is limited or GNSS measurements are faulty, but can degrade positioning performance slightly when GNSS reception changes from poor to good. This is a consequence of the relative magnitude of variances in the measurements' stochastic model. Further optimization of these variances is necessary to prevent this undesired behavior.