## 4 Results
### Non-Parametric Inverse Kinematic Matrix Calibrated with Genetic Algorithms  
As stated above, the difficult physical interpretation of the parametric optimization results obtained in [17] combined with the difficulty of establishing feasible limits to the parametric search suggested the implementation of an alternative non-parametric calibration of the inverse kinematic matrix of the mobile robot. The complete procedure used in this non-parametric calibration is described in Section 3. The optimization is based on the multivariate search capabilities of the genetic algorithms, which are less  
Figure 5: Sample trajectory followed by the APR-02 mobile robot for the motion command \((v,\alpha,\omega\
eq 0)\) = (0.2 m/s, 30\({}^{\circ}\), 0.35 rad/s). The ground truth trajectory was estimated from the information of the LIDAR (red line) while the odometry was estimated with the theoretical value of the inverse kinematic matrix of the mobile robot (blue line). prone to local minima than other gradient search algorithms. The computation of this non-parametric optimization required 637 s in the same high-performance workstation used in the reference parametric optimization [17] (29% less time). The best non-parametric inverse kinematic matrix (\(M_{\text{NON-PARAMETRIC}}\)\({}^{-1}\)) obtained with the application of the calibration procedure proposed in this paper (described in Section 3) is:  
\[M_{\text{NON-PARAMETRIC}}\)\({}^{-1}= \begin{bmatrix}-0.0883286476963123&0.0000000000000&0.0873330402667908\\ 0.0522829495809178&-0.1008246738469390&0.0536189007203205\\ 0.2357044251338890&0.2338948216993490&0.2348793373133590\end{bmatrix}  \tag{9}\]  
The following expression is used to compare the inverse kinematic matrix obtained with the parametric (Equation (8)) and non-parametric calibration (Equation (9)). This expression performs the Hadamard product (element-wise product) [36] of the difference between these two matrices. \[\left(M_{\text{NON-PARAMETRIC}}\)\({}^{-1}\right)_{ij}\).(1+M_{\text{DIFFERENCE}}\)\({}^{-1})_{ij}=\left(M_{PARAMETRIC}\)\({}^{-1}\right)_{ij} \tag{10}\]  
\[M_{\text{DIFFERENCE}}\)\({}^{-1}= \begin{bmatrix}\begin{array}{cc}1.091\%&1.505\times 10^{ 15}\%&-0.671\%\\ -0.358\%&0.195\%&-0.572\%\\ 0.299\%&0.103\%&0.212\%\end{array}\end{bmatrix}  \tag{11}\]  
The difference matrix \(M_{\text{DIFFERENCE}}\)\({}^{-1}\) shows that the results of both calibrations (parametric and non-parametric) are very similar, with differences lower than 0.7% except in the case of the coefficients m\({}_{1,1}\) (1.091%) and m\({}_{1,2}\) (\(1.505\times 10^{15}\%\)). The huge difference in the coefficient m\({}_{1,2}\) is because the non-parametric calibration maintained the original value defined in the theoretical inverse kinematic matrix (Equations (3) and (7)), with a value very close to zero: \(-1.208\)\(\times\)\(10^{-17}\). Alternatively, the result of the parametric calibration performed in [17] (displayed in Equation (8)) showed m\({}_{1,2}\) to reach the value of \(-0.00056\), which is very difficult to interpret as a parametric inaccuracy originated during the assembling the robot. The differences between the parametric and non-parametric calibration of the inverse kinematic matrices are small but cannot be neglected because the odometry is cumulatively updated ten times per second with the displacement information provided by the encoders. Figure 6 compares the offline computation of the odometry of four randomly chosen validation trajectories (from the \(36\)\(\times\)\(5\) available in the validation dataset). The ground truth trajectory computed from the LIDAR information is labeled in red and the odometry is computed using the different inverse kinematic matrices evaluated in this paper: theoretical IK (green), parametric IK (brown), and non-parametric IK (magenta).