# An Estimator for the Kinematic Behaviour of a Mobile Robot Subject to Large Lateral Slip
## 3 Experimental Identification of the Estimator
### Results  
Table 1 collects the test performed with the robot together with the kinematic inputs and the main parameter used as comparisons, such as the curve radiuses and the estimated track between the wheels. The theoretical radius \(R_{th}\) was computed using the kinematic relation \(R_{th}=\frac{p_{1}}{\hat{\varphi}_{1}}\), which is the expected curve radius if the mobile robot behaviour is purely kinematic. The radius \(\hat{R}\) is the radius of the trajectory curve estimated from the experimental data collected by the vision-based tracking system. As expected, the estimated radius was always larger than the theoretical one, reflecting the under-steering behaviour due to the large lateral wheel slip. The estimated track \(\hat{i}\), instead, was obtained fitting the tracked trajectory with the kinematic model described in the previous section, where the same kinematic input \(v_{1R}\) and \(v_{1L}\) that have been logged experimentally have been used as input to the model. By recalling the model described before, this trend can be interpreted as that the virtual axle used in the model is wider than the actual track in order to mathematically represent the under-steering behaviour of the robot. \(\hat{R}\) and \(\hat{i}\) are two different way to represent the robot under-steering tendency with a basic difference: \(\hat{R}\) is purely based on the fitting of the measured trajectory with a circle, while \(\hat{i}\) is obtained fitting the whole kinematic model shown before using the measured kinematic input in order to match as close as possible the robot behaviour. Figure 5 depicts a comparison between the various trajectories taken into account for each test. Figure 5a shows a comparison of the measured trajectory and the resulting curves of the two kinematic models when the measured input data are used. The difference between the two kinematic models and the actual measured data can be interpreted as the lack of dynamic phenomena modelling, such as robot inertia, wheel longitudinal, and lateral slip. By making a comparison between the two kinematic models instead, the results described before are again evident: the kinematic model with the estimated track length \(\hat{i}\) (\"Fit Kin.\") has a more under-steering behaviour compared to the kinematic model with the actual track length (\"Th. Kin.\"). Since the fitted kinematic model is a better approximation of the actual robot trajectory, it could be said that a kinematic model with virtual wheels with a slightly increased track length \(\hat{i}\) could partially compensate unmodelled dynamic behaviours better than the kinematic model with the actual track length. Figure 5b shows a comparison between two circles and the steady-state part of the measured trajectory: the actual trajectory is not an exact circle, but the two circles approximate it, one better than the other. Again, it appears from the results that the kinematic circle of radius \(R_{th}\) is smaller and a worse approximation of the actual data than the circle with the estimated radius \(\hat{R}\). This figure depicts more clearly that, if a purely steady-state comparison is made, the perfectly circular kinematic trajectories could be a good approximation of the real one. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Test**} & **Longitudinal** & **Yaw Rate** & **Theoretical** & **Estimated** & **Estimated** \\  & **Speed \(v_{1}\) [m/s]** & \(\hat{\varphi}_{1}\) [rad/s]** & **Radius \(R_{th}\) [m]** & **Radius \(R\) [m]** & **Track \(\hat{i}\) [m]** \\ \hline
1 & 0.412 & 0.961 & 0.429 & 0.452 & 0.303 \\
2 & 0.393 & 0.915 & 0.430 & 0.458 & 0.296 \\
3 & 0.310 & 0.616 & 0.504 & 0.621 & 0.319 \\
4 & 0.319 & 0.596 & 0.535 & 0.598 & 0.314 \\
5 & 0.384 & 0.940 Figure 6 highlights the latter statements even better; the figure shows the instantaneous curvature radius of the two kinematic models over time, and how the two curves reach a steady-state very close to the value of the radius of the circle described before.