## 5 Discussion and Conclusions  
This paper proposes a non-parametric calibration procedure of the effective inverse kinematic matrix of a three-wheeled omnidirectional mobile robot based on the search capabilities of genetic algorithms. The calibration procedure used a training dataset composed of five repetitions of 36 representative benchmark trajectories (180 trajectories in total). These benchmark trajectories summarize the motion performances of a three-wheeled mobile robot [17]. The calibration procedure is based on genetic algorithms because the mutation and combination performed during the search is less prone to local minima during the multivariate searches required to optimize the kinematics of a robot [14,17]. The application of this non-parametric calibration procedure to the offline computation of the odometry of the training trajectories reduced the value of the cost function from 0.1234 to 0.0215, representing an improvement of the cost function of 82.6% (see Table 2). This improvement is practically the same as that (82.6%) obtained in the parametric calibration procedure conducted previously in [17] with the same training trajectories. The analysis implemented with the validation trajectories confirmed an average improvement in the cost function of 81.8% (see Table 3) which is visually observable in the estimated trajectory of the mobile robot (see Figures 6 and 7). The calibration results obtained in this paper have a similar order of magnitude as the improvements presented by [PERSON] et al. [8] and [PERSON] et al. [23], who highlighted the importance of using curved trajectories for calibrating the kinematics of an omnidirectional mobile robot. Nevertheless, the use of different calibration trajectories precludes direct comparison of the achievements. As a summary, the non-parametric calibration of the inverse kinematic matrix of a three-wheeled omnidirectional mobile robot has two main advantages: (1) it avoids the problem of defining feasible physical-interpretable limits applied in the parametric calibration; and (2), as a consequence, this non-parametric calibration is much easier to implement and converges faster than the parametric calibration. The offline implementation of the non-parametric calibration in a high-performance workstation required an average of 637 s, while the parametric calibration conducted previously in [17] required 903 s (41% more computational time). The definition of feasible, physically interpretable limits in a parametric calibration of the kinematic matrix is a problem that cannot be neglected because of the complex numerical relationship between the parameters that define the inverse kinematic matrix of a mobile robot. The comparative analysis of the results obtained in this paper demonstrates that the improvement in odometry that can be obtained with a non-parametric calibration is practically the same as that which can be obtained with a parametric calibration. Finally, the disadvantage of performing a non-parametric calibration of the inverse kinematic matrix of a mobile robot is that the calibration result is a numerical matrix that cannot be physically interpreted. However, this disadvantage does not really exist because the parameter variation obtained in the parametric calibration performed in [17] could not be physically interpreted either. Future work will analyze the application of this non-parametric procedure in several three-wheeled omnidirectional mobile robots and the application of alternative methods based on artificial neural networks [37] to estimate the kinematics of an omnidirectional mobile robot. **Author Contributions:** Formal analysis, E.R. and E.C. ; investigation, J.P. and E.R. ; methodology, J.P.; resources, R.B. ; software, R.B. and E.C. ; writing--original draft, E.R. and E.C. ; writing--review & editing, J.P. All authors have read and agreed to the published version of the manuscript. **Funding:** This research has been partially funded by the Departament de Recerca i Universitats de la Generalitat de Catalunya: FI SDUR 2022 grant.