## 4 Results
### Prediction of the Omnidirectional and Translation Capabilities
Table 5 shows the segmented errors of configurations 3B, 2A-2B, 1B-2C, 1B-2D, and 1A-2D, which have bidirectionality errors above the thresholds, thus having different matrixes \(Tv\), \(T\alpha\), and \(T\omega\) and a different combination \(Tv\alpha\omega\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Configuration** & \(Ev\) & \(E\alpha\) & \(E\omega\) \\ \hline
3B & \(Ev\) (\(\mu\)s) &  
the segmented errors shown in Table 4, which also includes the matrixes \(Tv\alpha\omega\) with the logical conjunctions that combine the segmented results for each motion command. The results in Table 4 show that the segmented errors are below the thresholds for the whole set of motion commands explored (\(Tv\alpha\omega\) is all green), so the bidirectionality compliance is satisfied and the configurations have omnidirectional capabilities. That is to say, all the set of motion commands assessed (all combinations of \(v\), \(\alpha_{i}\), and \(\omega_{j}\)) can be executed by the mobile platforms, so they can translate in any direction \(\alpha_{i}\) with or without rotation \(\omega_{j}\). In short, configurations 3A, 1A-2B, 1A-2C, 1A-1B-1C, 1A-1B-1D, 3E (Table 1) have been classified as omnidirectional according to the procedure proposed in this paper. The bidirectionality errors in the mobile platform configurations shown in Table 3 (3B, 2A-2B, 1B-2C, 1B-2D, 1A-2D) are case-dependent and their representations show a non-uniform color distribution that will be analyzed case by case. In Table 3, configuration 3B, the bidirectionality errors \(Ev\) obtained from comparing the target and calculated linear velocities of the mobile platform (\(v,v^{\prime}\)) are negligible. Similarly, the bidirectionality errors \(E\alpha\) obtained from comparing the target and calculated angular orientations of the motions (\(\alpha,\alpha^{\prime}\)) from its kinematics are also negligible. However, the bidirectionality errors \(E\omega\) obtained from comparing the target and calculated angular speeds of the motions (\(\omega,\omega^{\prime}\)) are negligible only in the set of \(\alpha_{i=1...N}\) with \(\omega_{j}=0\) rad/s, which indicates that the mechanism is uncontrollable with respect to rotation. Table 5, configuration 3B, shows these errors segmented according to the defined thresholds and the logical conjunction \"AND\" in \(Tv\alpha\omega\). In this case, some points in \(Tv\alpha\omega\) have bidirectionality errors above the thresholds (represented in white color), which means that bidirectionality compliance is not satisfied and an omnidirectional motion is not possible. Nevertheless, in the cases in the set of \(\alpha_{i=1...N}\) with \(\omega_{j}=0\) rad/s, \(Tv\alpha\omega\) show bidirectionality errors below the thresholds (represented as a green vertical line), meaning that the mobile platform is able to translate with any angular orientation without rotation. This new information about the translational capabilities of the platform is only provided by the procedure proposed in this paper. In Table 3, configurations 2A-1B, 1B-2C, 1B-2D, and 1A-2D, the bidirectionality errors \(Ev\), \(E\alpha\), and \(E\omega\) obtained from comparing the target and calculated motions of each mobile platform (comparing \(v\) and \(v^{\prime}\), \(\alpha\) and \(\alpha^{\prime}\), and \(\omega\) and \(\omega^{\prime}\)), have irregular non-zero distributions. After applying the acceptability thresholds, the representations of the combined segmented errors \(Tv\alpha\omega\) in Table 5 depict the following information. In cases 2A-1B and 1B-2D, the mobile platforms can only translate in the directions of the angular orientations \(\alpha_{i}=0^{\circ}\) and \(\alpha_{i}=180^{\circ}\) (forwards and backwards) with \(\omega_{j}=0\) rad/s, so the applicability of these configurations is very limited.