## 4 Results
### Non-Parametric Inverse Kinematic Matrix Calibrated with Genetic Algorithms
Figure 7a-d shows in detail the final position of the robot achieved in the trajectories depicted in Figure 6. Figure 7a-d shows small differences between the ground truth trajectory of the mobile robot (red line) and the odometry computed with the parametric (brown line) and non-parametric (magenta line) inverse kinematic matrices obtained for the APR-02. Finally, Tables 2 and 3 compare the performance of the non-parametric calibration performed in this paper. Table 2 presents the average values of the cost function (\(CF\)) obtained with the 180 benchmark trajectories contained in the training dataset. The reference cost function, \(CF_{\text{TIForetical IK}}\), was computed using the theoretical inverse kinematic matrix which is described in Equations (3) and (7). The values of the parametric inverse kinematic matrix (Parametric IK) were obtained in [17] and its values are presented in Equation (8). The numerical values of the non-parametric optimization of the inverse kinematic matrix performed in this paper with genetic algorithms (Non-parametric IK) are described in Equation (9). Table 2 shows that the best cost function computed with the training trajectories is practically the same for the parametric and non-parametric calibrations. Please note that the column indicating \(CF_{\text{TRAINING}}\) presents the cost function computed with the calibration trajectories included in the training dataset. Figure 6: Comparison between the ground truth trajectory followed by the mobile robot (red line) and the odometry estimated with: the theoretical IK (green line), the parametric IK (brown line) and the non-parametric IK (magenta line). Trajectories originated by the following motion commands, \((v,\alpha,\omega)\): (**a**) (0.2 m/s, 30\({}^{\circ}\), 0.35 rad/s); (**b**) (0.2 m/s, 210\({}^{\circ}\), 0.35 rad/s); (**c**) (0.2 m/s, 120\({}^{\circ}\), 0.35 rad/s); and (**d**) (0.2 m/s, 300\({}^{\circ}\), 0.35 rad/s). \begin{table}
\begin{tabular}{c c c c} \hline \hline \(M^{-1}\) & \(CF_{Theoretical\;\text{IK}}\) & \(CF_{TRAINING}\) & **Improvement** \\ \hline Theoretical IK & 0.1234 & - & - \\ Parametric IK [17] & 0.1234 & 0.0215 & 82.61\% \\ Non-parametric IK * & 0.1234 & 0.0215 * & 82.60\% * \\ \hline \hline \end{tabular}  
* Calibration result obtained in this paper.  
\end{table} 
Table 2: Values of the cost function obtained when computing the odometry of the 180 trajectories included in the training dataset with the theoretical and calibrated inverse kinematic matrices. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(M^{-1}\) & \(CF_{Theoretical\;\text{IK}}\) & \(CF_{VALID}\) & **Improvement** \\ \hline Theoretical IK & 0.1251 & - & - \\ Parametric IK [17] & 0.1251 & 0.0229 & 81.68\% \\ Non-parametric IK * & 0.1251 & 0.0227 * & 81.81\% * \\ \hline \hline \end{tabular}  
* Calibration result obtained in this paper.  
\end{table} 
Table 3: Values of the cost function obtained when computing the odometry of the 180 trajectories included in the validation dataset with the theoretical and calibrated inverse kinematic matrices. Figure 7: Detail of the final position of the robot corresponding to the full trajectories displayed in Figure 6. Ground truth trajectory of the mobile robot (red) and odometry of the mobile robot computed using the theoretical IK (green), the parametric IK [17] (brown) and the noon-parametric IK (magenta). Table 3 shows the value of the cost function evaluated with the trajectories included in the validation dataset. These validation trajectories were registered several months after the registration of the trajectories included in the training dataset. Table 3 shows that the offline computation of the odometry of the 180 trajectories included in the validation dataset offers practically the same cost function and the same improvement with both the parametric and non-parametric inverse kinematic matrices.