## 4 Results  
### Prediction of the Omnidirectional and Translation Capabilities  
This section presents the results obtained from the application of the procedure proposed in this paper. Tables 2 and 3 show the evaluation of the bidirectionality errors \(Ev\), \(E\alpha\), and \(E\omega\) of the mobile platforms assessed in this paper (described in Table 1). The results form Tables 2 and 3 were obtained with the following parameters: \(v=0.3\) m/s, \(\alpha_{i}\) ranging within \([0,360]^{\circ}\), and \(\omega_{j}\) ranging within \([-2,2]\)rad/s. These values are inspired by the motion commands used by a humanoid-sized assistance robot with three omni wheels using the 3A configuration [48]. Table 2 shows the graphical representation of the bidirectionality errors \(Ev\), \(E\alpha\), and \(E\omega\) of mobile platform configurations 3A, 1A-2B, 1A-2C, 1A-1B-1C, 1A-1B-1D, and 3E. Given that the order of the bidirectionality errors in Table 2 is below \(10^{-14}\), the errors can be considered numerical noise and are negligible, so they have the same matrixes \(Ev\), \(E\alpha\), and \(E\omega\) and their representation is the same for all the platform configurations assessed. The color scale used to represent the bidirectionality errors is saturated with maximum values of 0.3 m/s for \(Ev\), \(20^{\circ}\) for \(E\alpha\), and 0.3 rad/s for \(E\omega\). The term \(max\) in the color scale in Table 2 is a general representation of the saturation values for \(Ev\), \(E\alpha\), and \(E\omega\). Similarly, Table 3 shows the graphical representation of the bidirectionality errors of \(Ev\), \(E\alpha\), and \(E\omega\) of configurations 3B, 2A-2B, 1B-2C, 1B-2D, and 1A-2D, which have non-negligible bidirectionality errors. In these cases, the errors obtained depend on the configuration of the mobile platform assessed and thus the configurations have different matrixes \(Ev\), \(E\alpha\), and \(E\omega\). The color scale used in Table 3 is the same as in Table 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Configuration** & \(Ev\) & \(E\alpha\) & \(E\alpha\) & \(E\omega\) & **Scale** \\ \hline
3A & 300 & \(Ev\) [m/s] & \(E\alpha\) [T] & \(E\alpha\) [T] \\
1A-2B & 270 & 220 & 220 & 220 \\
1A-2C & 220 & 220 & 220 & 220 \\
1A-1B-1C & 900 & 900 & 900 & 900 \\
1A-1B-1D & 140 & 145 & 145 & 145 \\
3E & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mobile platform configurations with the same negligible bidirectional errors in \(Ev\), \(E\alpha\), and \(E\omega\). The maximum value in the color scale is 0.3 m/s for \(Ev\), \(20^{\circ}\) for \(E\alpha\), and 0.3 rad/s for \(E\omega\). The schematic representation of the configurations is shown in Table 1.  
The bidirectionality errors \(Ev\), \(E\alpha\), \(E\omega\) (Tables 2 and 3) are segmented using the following thresholds: \(ev_{max}=0.003\) m/s, \(e\alpha_{max}=0.2^{\circ}\), and \(e\omega_{max}=0.003\) rad/s. The images of the segmented errors \(Tv\), \(T\alpha\), \(T\omega\) are displayed in Tables 4 and 5. These tables also display the matrixes \(Tv\alpha\) storing the logical conjunction \"AND\" of the segmented errors of each motion command, which are used to evaluate the omnidirectional capabilities of the mobile platform configurations. The green color represents the combinations in which the errors are smaller than their corresponding threshold, and white color represents the combinations in which the errors are greater than their corresponding threshold. The term \(e_{max}\) in the color scale used in Tables 4 and 5 represents the thresholds \(ev_{max}\) for \(Tv\), \(e\alpha_{max}\) for \(T\alpha\), and \(e\omega_{max}\) for \(T\omega\). Table 4 shows the segmented errors of configurations 3A, 1A-2B, 1A-2C, 1A-1B-1C, 1A-1B-1D, and 3E, which have the bidirectionality errors below the thresholds, thus having the same matrixes \(Tv\), \(T\alpha\), and \(T\omega\) and the same combination \(Tv\alpha\omega\).