## 5 Conclusions  
This paper presents a procedure for estimating the motion capabilities of mobile platforms with three omni wheels, which is the minimum number of wheels required to achieve omnidirectionality. This procedure is able to differentiate between omnidirectional capabilities and translational capabilities by analyzing the bidirectionality compliance between the inverse and forward kinematics for a wide set of motion commands. The procedure has been assessed with eleven mobile platforms, each one with different omni wheel configurations. The estimation of the motion capabilities of these platforms agrees with state-of-the-art methods [17] while providing a new specific estimation of the translation capabilities and combined visual information of the motion capabilities of the platforms. These estimations have been validated by simulating the trajectories followed by the mobile platforms while executing a reduced set of motion commands. Results have shown that the procedure proposed in this paper for estimating the omnidirectionality capabilities is 869-fold faster than performing alternative trajectory simulations. This is because the computation of the bidirectionality error of a motion command requires an average time of 0.522 ms on a testbench computer, as it is based on two matrix operations (see Figure 4). Alternatively, the simulation of the execution of a trajectory by a mobile platform to verify if a motion command can be executed or not requires an average time of 0.454 s on the same testbench computer. This performance \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Platform**} & \multicolumn{6}{c}{**Experiment (Type of Motion Required)**} & \multicolumn{6}{c}{**Summary of the Simulations**} \\ \cline{2-9}  & **E1 (T)** & **E2 (T)** & **E3 (T)** & **E4 (T+R)** & **E5 (T+R)** & **E6 (R)** & **OMD** & **T** & **Trajectories** \\ \hline
1B-2D & & & & & & & & & \\  & OK & FAILED & FAILED & FAILED & FAILED & NO & NO & \\  & FAILED & OK & FAILED & FAILED & OK & NO & NO & \\  & FAILED & OK & FAILED & FAILED & OK & NO & NO & \\ \hline \hline \end{tabular}
\end{table} 
Table 9: _Cont._allows this procedure to be applied during the design stage of new mobile platforms with three omni wheels to validate their motion performances. Relative to the related scientific literature [17; 18; 51], the advantage of this proposal is that it indicates which platform configurations become uncontrollable when the angular velocity of the platform is different from zero, which indirectly conveys specific information on the translational performance of the platform under evaluation. Additionally, the graphical information on the motion capabilities of a mobile platform provides direct visual information about which motions may be successfully executed by the platform. This proposal has the limitation of not considering dynamic effects in the omni-directional motion, which will be addressed in future works. Future work will apply the procedure proposed in this paper on the optimal design of translation-only mobile platforms tailored for heavy-duty industrial applications where rotation is not needed during transportation. Conceptualization, J.P.; formal analysis, J.P.; investigation, E.R. ; methodology, E.C. ; resources, R.B. ; software, E.R., R.B. and E.C. ; supervision, J.P.; validation, E.R. ; visualization, E.R. ; writing--original draft, E.R. ; writing--review and editing, J.P. All authors have read and agreed to the published version of the manuscript. The research was supported by the predoctoral grants from the Departament de Recerca i Universitats de la Generalitat de Catalunya (AGAUR FI SDUR 2022) and the Ministerio de Ciencia, Innovacion y Universidades (FPU 2022). The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding authors. The authors declare no conflicts of interest.