# Robot Localisation Using UHF-RFID Tags: A Kalman Smoother Approach +
## 4 Results
### Experimental Results  
The effectiveness of the proposed SEKF estimator is further substantiated by the experiments conducted on a robot moving in an indoor environment, as described in Section 3.3. The RFID reader has a nominal phase noise standard deviation of 0.1 rad due to thermal noise, meaning that the measured differences of distance \(d_{i,k}\) in (8) have a nominal standard deviation of around 4 mm. In practical situations, this modelled uncertainty is increased due to the presence of multi-path interference that leads to a higher value than the nominal one, but still acceptable. Despite the phase robustness once adopted for localisation, the problem of robot displacements between consecutive measurements has a detrimental effect (as noticed and analysed in simulations in Figure 10). In practice, if the \(\dot{d}_{i,k}\) values are too close to zero, the relative impact of the measurement noises on the observation function is high. On the other hand, a relatively high sampling frequency is needed for reliable localisation using RFID signal phase to avoid problems due to the 2\(\pi\)-phase ambiguity. In other words, since the average spatial sampling (which depends on the robot speed and on the RFID system sampling rate) cannot be too large, a relative high number of tag reading for each time step must be guaranteed to compensate for the high relative impact of the measurement noise on the observation function. The variance of the uncertainties affecting vehicle velocities \(v_{k}\) and \(\omega_{k}\) has been obtained with a Type A analysis, according to the Guide to the Expression of Uncertainty in Measurement [45]. In particular, we have implemented an EKF with encoder readings and actual range measurements and compared against the ground truth range measurements, in order to derive the odometry covariance matrix. Next, we have tested the estimation procedure in simulation by running Monte Carlo experiments using a very large set of testing trajectories. At the end of the process,we found out that \(\sigma_{v}=0.1\) m/s and \(\sigma_{\omega}=0.05\) rad/s. Three different sample trajectories are considered in this section, and reported in Figure 11. The robot was driven manually through a remote control by an operator. Figure 11: Three (**a–c**) sample trajectories of the robot (dashed line), SEKF estimates (solid line) and dead reckoning only reconstructed trajectories (dotted line). In all the pictures, the starting and ending positions of the trajectories as well as the RFID tag locations are reported as well. The trajectories estimated by the SEKF (solid lines) as well as the dead reckoning results reported for comparison (dotted lines, obtained using only the encoder measurements and the model (2)) and the ground truth data (dashed lines) are plotted in Figure 11. The estimated trajectories (both using the SEKF or the encoders only) assumes perfect knowledge of the initial conditions, which is a strong assumption for any Bayesian estimator. Nonetheless, the purpose of these experiments is just to test the effect of a reduced forward velocity, set to \(v=0.23\) m/s, which has detrimental effect on the overall accuracy, as previously mentioned. From a qualitative analysis of Figure 11, it is evident how the SEKF returns a good path reconstruction, with a maximum error, happening in the turning conditions, which is approximately around 30 cm. A more detailed analysis of the localisation uncertainty for the three trajectories is reported in Figure 12, where the position RMSE shows to be almost everywhere less than 15 cm. For the orientation RMSE we have along the trajectories approximately 0.2 rad. To account for theinitial-condition uncertainty, Figure 13 reports the same trajectories shown in Figure 11 but assuming that the initial state \(\hat{s}_{0}\) is randomly generated from a Gaussian distribution with zero-mean and covariance matrix \(P_{0}^{(3,3)}\) (i.e., the marginal of the filter covariance matrix \(P_{0}\) in (15) restricted to the robot state \(\hat{s}_{k}\)).