## 3 Systematic Odometry Error Evaluation and Correction
### Discussion of the Results Obtained with the Odometry Calibration Strategies  
The results of the calibration strategies evaluated in Table 4 show that the evaluation of straight trajectories usually generates worse validation results than curved trajectories. It is likely that straight target trajectories (Table A1: R1... R12) are less representative of the motion because they only use three different angular rotational velocities: \(\pm\)11.291, \(\pm\)19.557 and \(\pm\)22.583 rpm, while the curved trajectories cover six angular velocities (Table A1). In general, the best results of the iterative search are obtained with GA, probably because GA is less prone to local minimum converge. The best calibration result (\(CF_{CALIBATION}=0.0166\)) was obtained with the strategy STB-GA5, using GA and five repetitions of only six curved calibration trajectories, also with very good validation results (\(CF_{VALIDATION}=0.0232\)). A similar result was obtained with the strategy STB-GA1, using only one repetition of these six curved calibration trajectories (\(CF_{CALIBATION}=0.0170\)), confirming the representativeness of the curved calibration trajectories. The strategies STB-GA1 and STB-GA5 represent a huge improvement of 73.1% and 81.3% in the average validation of \(CF_{VALIDATION}\) relative to the uncalibrated case. The best validation result (\(CF_{VALIDATION}=0.0222\)) was obtained with the strategy STE-GA5, using GA and 5 repetitions of all 36 calibration trajectories (180 training experiments). However, a very similar result was also obtained using GA and with only 1 repetition of all training trajectories (STE-GA1, 36 experiments, \(CF_{VALIDATION}=0.0226\)). The improvements obtained in the validation of STE-GA1 and STE-GA5 were 81.8% and 82.1% respectively, so the conclusion is that both strategies are valid to calibrate the kinematic parameters of the mobile robot APR-02. Table 5 compares the differences between the nominal and calibrated kinematic parameters obtained with STE-GA5. Unexpectedly, the differences between the nominal and calibrated values of the distance from the wheels to the center of the mobile robot \((R_{a},R_{b},R_{c})\) are higher than 10%, likely caused by the bending of the structure that supports the wheels (wheelbase). The differences in the effective values of the radii of the wheels \((r_{a},r_{b},r_{c})\) are in a range from 2 to 5%, probably caused by the complex assembly of the wheels. Finally, the values of the angular orientation of the wheels \((\delta_{a},\delta_{b},\delta_{c})\) vary within a very small range (0.18% and 0.52%), confirming the good alignment of the wheels and DC motors during the assembly of the mobile robot. The exact value of the best inverse of the compact kinematic matrix \(M^{-1}\) computed from the effective value of the kinematic parameters obtained with STE-GA5 is:  
\[M_{STEGAS}{}^{-1}=\left[ \begin{array}{cc}-0.0892930568372762&-0.000560566978203& 0.0867466159313470\\ 0.0520955668635434&-0.1010213675626970&0.0533117820461363\\ 0.2364093797047320&0.2341358647406650&0.2353792947863630\\ \end{array} \right] \tag{17}\]  
Finally, Table 6 summarizes the average RMS errors and average maximum errors obtained with the uncalibrated and calibrated kinematic parameters. These validation results have been obtained with the five repetitions of the 36 calibration trajectories registered for validation (180 validation experiments). Table 6 shows that the improvement computed from the validation value of the cost function (82.1%) is representative of the trajectory improvements achieved. The application of the calibrated kinematic parameters showed an improvement of 67% in the evaluation of the RMS error distance between the odometry trajectory and the ground truth trajectory, and an improvement of 71% in the RMS error evaluation of the absolute difference between the angular orientation of the mobile robot during these trajectories.