## 2 Materials and Methods
### Ground Truth Trajectory
\begin{table}
\begin{tabular}{c c c} \hline \hline \(\frac{\text{AM}^{-1}}{\text{AR}_{b}}\) & \(\text{I}_{\text{g}_{\star}=0.195\text{ m}}\) & \(\frac{\text{AM}^{-1}}{\text{AR}_{b}}\) & \(\frac{\text{AM}^{-1}}{\text{AR}_{b}}\) \\ \hline \(\left[\begin{array}{cc}0.0000&0.0000&0.0000\\ 0.1687&0.1687&0.1687\\ -0.4325&-0.4325&-0.4325\end{array}\right]\) & \(\left[\begin{array}{cc}0.0000&0.0000&0.0000\\ 0.0000&-0.0667&0.0000\\ 0.0000&1.7094&0.0000\end{array}\right]\) & \(\left[\begin{array}{cc}0.0000&0.0000\\ -0.0570&0.0000&-0.0570\\ 0.1461&0.0000&-0.1461\end{array}\right]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Sensitivity of \(M^{-1}\) to the values of the nominal kinematic parameters of the wheel \(b\). \begin{table}
\begin{tabular}{c c c} \hline \hline \(\frac{\text{AM}^{-1}}{\text{AR}_{b}}\) & \(\text{I}_{\text{g}_{\star}=0.195\text{ m}}\) & \(\frac{\text{AM}^{-1}}{\text{AR}_{b}}\) & \(\frac{\text{AM}^{-1}}{\text{AR}_{b}}\) \\ \hline \(\left[\begin{array}{cc}-0.1461&-0.1461&-0.1461\\ -0.0843&-0.0843&-0.0843\\ -0.4325&-0.4325&-0.4325\end{array}\right]\) & \(\left[\begin{array}{cc}0.0000&0.0000&0.5774\\ 0.0000&0.0000&0.3333\\ 0.0000&0.0000&1.7094\end{array}\right]\) & \(\left[\begin{array}{cc}-0.0493&0.0493&0.0000\\ -0.0285&0.0285&0.000\\ -0.1461&0.1461&0.0000\end{array}\right]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Sensitivity of \(M^{-1}\) to the values of the nominal kinematic parameters of the wheel \(c\). \(t(k)\) is the time in which a new estimation of the instantaneous angular velocities of the wheels \(E(k)\) is provided by the encoder. This time can be computed as \(t(k)=k\cdot T\) or \(t(k)=t(k-1)+T\), where \(T\) is the sampling time of the PIDs; in the APR-02 this value is \(T=10\) ms. The odometry information \(O(k)\) is computed from \(E(k)\), so it is defined by the same time sequence \(t(k)\). Alternatively, \(t(p)\) is the time in which a new raw scan \(L(p)\) is provided by the LIDAR. This time can be computed as \(t(p)=t(p-1)+D_{p}\), where \(D_{p}\) ranges from 200 ms to 300 ms depending on the time required by the control system of the mobile robot to apply SLAM and estimate \(GT(p)\). This variation is because the control system waits to request a new LIDAR scan until the SLAM procedure finishes, so this time lapse will be different depending on the time needed to apply SLAM from the raw LIDAR scans. The hypothesis of this paper is then that the odometry trajectory will be correct and exact when this trajectory matches the ground truth (real) trajectory of the mobile robot. Figure 7 shows the final position results obtained when the mobile robot APR-02 repeats the same calibration trajectory five times, starting each trajectory with the same initial position and angular orientation. Figure 7 represents the zoomed information of the planned final destination of the mobile robot (black dot and line) when the mobile robot completes a simple straight trajectory, the ground truth trajectory (orange dotted line) and ground truth final position and orientation of the mobile robot (orange circle and line), and  
Figure 6: Representation of the ground truth trajectory of the mobile robot (orange line) and the trajectory estimated with the odometry (green line): (**a**) complete trajectories; (**b**) angle and distance differences between the trajectory positions estimated with the odometry and the SLAM procedure. the trajectory (green dotted line) and final mobile robot position and orientation estimated with the odometry (green square and line). The results of Figure 7 reveal the existence of systematic differences between the odometry and ground truth trajectories. This paper proposes the reduction of such differences by applying an iterative calibration procedure to the kinematic parameters used to estimate the odometry of the mobile robot APR-02.