# Robot Localisation Using UHF-RFID Tags: A Kalman Smoother Approach +
## 4 Results
### Experimental Results
Moreover, the roots of the elements on the diagonal of \(P_{0}^{(3,3)}\), i.e., the standard deviations, have been changed as reported in Table 1. As can be seen, the initial conditions slightly modify the reconstructed trajectory, thus showing the robustness and the consistency of the SEKF. The impact of the initial condition uncertainty is also analysed in a quantitative manner in Figure 14.  
The results have been collected by executing a Monte Carlo approach for the different initial conditions and using the same experimental data. In particular, we performed 15,000 executions for each of the initial \(P_{0}^{(3,3)}\) reported in Table 1. It can be fairly observed from the corresponding graphs that, the localisation accuracy is insensitive from the initial-condition  
Figure 12: Position RMSE time evolution during localisation for (**a**) the sample trajectory of Figure 11a, (**b**) the sample trajectory of Figure 11b, (**c**) the sample trajectory of Figure 11c. Figure 13: Trajectory estimation for (**a**) the sample trajectory of Figure 11a, (**b**) the sample trajectory of Figure 11b, (**c**) the sample trajectory of Figure 11c with different range of uncertainties about the true initial conditions. uncertainty, being the slight increase of the RMSE dictated by the first part of the trajectories (i.e., a transient effect). To assess the validity of the proposed RMSE analysis, we show also the odometry-only reconstruction that obviously report a direct proportionality with the initial conditions (i.e., the uncertainty in the initial pose is not transient). As a final empirical analysis on the experimental set-up, we analyse the problem of the number of RFID tag measurements. Intuitively, the larger is the number of sensed tags at each time step, the lower will be the RMSE. Indeed, this is empirically verified in Figure 15 considering the RMSE for the \(x\), \(y\) and \(\theta\) quantities and the associated standard deviations. These results have be collected considering the average robot speed of 0.23 m/s and considering the odometry sampling period \(T_{s}=100\) ms as the main sampling time (during the experiments, we used fixed time steps and the assumption of constant velocities at each time step). As can be seen from the figure, relying on a few number of tags, has resulted in a high localisation uncertainty, as expected. From the picture, just 10 tags provide a \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\hat{\pi_{\theta}}\) & \(\hat{\psi_{\theta}}\) & \(\hat{\theta_{\theta}}\) \\ \hline  & 0.01 & 0.01 & 0.01 \\  & 0.02 & 0.02 & 0.1 \\ \(\sigma\) & 0.04 & 0.04 & 0.2 \\  & 0.08 & 0.08 & 0.3 \\  & 0.1 & 0.1 & 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Standard deviations assumed for the initial robot state and distributed according to a Gaussian probability density function. Figure 14: Localisation estimation accuracy in presence of different uncertainties about the true initial conditions for (**a**) the position and (**b**) the orientation. Figure 15: Impact of minimum number of available phase measurements at each time step on the localisation accuracy for the current experimental setup. sufficiently accurate SEKF, at least for the purposes of trajectory analysis in the industrial domain. In an actual deployment, hence, it could be beneficial to increase the number of RFID antennas and the sampling frequency as well. Nevertheless, in such scenarios, other second-order effects will take place at the cost of a degradation in the accuracy of phase measurements, therefore such a solution is discouraged with the actual equipment. To summarise, we can conclude that it is highly recommended to have access to an average number of 7 to 10 tags readings at each time step. Of course, this is an average rate, since in the actual applications absence of measurements at some time steps cannot be ruled out, but, as reported in Figure 9, this phenomenon slightly affects the SEKF.