# Study and Simulation of an Under-Actuated Smart Surface for Material Flow Handling
## 5 Results
### Sorting
In Figure 7c, the values of the position and orientation reached at the end of the simulation (\(t_{black}=1.5\) s) for the \(\gamma=5^{\circ}\) case (black line) are \([x_{g},y_{g},\theta_{g}]_{black}=[0.369\) m, \(-0.171\) m, \(-36.05^{\circ}]\) (Table 2), while those of the magenta line for the same value of \(y_{g}=-0.171\) m are \([x_{g},y_{g},\theta_{g}]_{mag}=[0.402\) m, \(-0.171\) m, \(-28.62^{\circ}]\) and the simulation time for which they are obtained is \(t_{mag}=0.917\) s. Comparing these values, it can be seen that when the surface is more inclined (magenta line), it requires more space in the \(x\) direction to perform the sorting \(x_{g,mag}>x_{g,black}\), but the time required is less \(t_{black}-t_{mag}=0.583\) s. In addition, different \(\gamma\) also produce different final orientations, in this case smaller for larger \(\gamma\): \(\theta_{g,mag}<\theta_{g,black}\). Table 2 summarizes the results and the initial data about the previous sorting simulations. A practical use of the simulation is to determine the minimum number of modules necessary for sorting before building the real system. In general, as it was already presented, \begin{table}
\begin{tabular}{|l l l l|l l|} \hline \([\hat{x}_{g0},\hat{y}_{g0},\hat{\theta}_{g0}]\) & \(\gamma\) & **Sim. Time** & **Final**\([x_{g},y_{g},\theta_{g}]\) & **Figure** \\ \hline
[1.5 m/s,0 m/s,0 rad/s] & \(0^{\circ}\) & 1 s & [0.859 m,0.532 m,14.83\({}^{\circ}\)] & Figure 7a \\ \hline
[1.5 m/s,0 m/s,0 rad/s] & \(0^{\circ}\) & 1 s & [0.859 m,\(-0.532\) m,\(-14.83^{\circ}\)] & Figure 7b \\ \hline
[0 m/s,0 m/s,0 rad/s] & \(0^{\circ}\) & 1.5 s & [0 m,0 m,0\({}^{\circ}\)] & Figure 7c green \\ \hline
[0 m/s,0 m/s,0 rad/s] & \(5^{\circ}\) & 1.5 s & [0.369 m,\(-0.171\) m,\(-36.05^{\circ}\)] & Figure 7c black \\ \hline
[0 m/s,0 m/s,0 rad/s] & \(10^{\circ}\) & 1.5 s & [1.057 m,\(-0.515\) m,\(-12.16^{\circ}\)] & Figure 7c magenta \\ \hline \end{tabular}
\end{table} 
Table 2: Specific initial data and results for the sorting simulations. Figure 7: Illustration of the trajectories and the final position and orientation of the object during: (**a**) the sorting towards \(y>0\) (up) task, (**b**) the sorting towards \(y<0\) (down) task, (**c**) the sorting towards \(y<0\) (down) with different \(\gamma\). this result is affected by various initial conditions. However, it can be interesting to know how many columns of rotor units are necessary to successfully complete the different tasks for different object sizes and velocity ranges. In order to derive this relationship, a set of simulations can be performed. Table 3 reports the results of the discussed analysis for two object sizes: \(b\times\ h=0.25\ \mathrm{m}\ \times 0.25\ \mathrm{m}\), called \"S-box\", and \(b\times\ h=0.45\ \mathrm{m}\ \times\ 0.45\ \mathrm{m}\), called \"B-box\". \"B-box\" and \"S-box\" have the same mass (\(m=10\ \mathrm{kg}\)) and height (\(z_{g}=0.1\ \mathrm{m}\)). The counting of the number of columns for the sorting is done starting from the first column with the inclined rotors to the last one touched by the object before being out of the line. As is visible from Table 3, the sorting with low initial velocities is not always possible: with \(\dot{x}_{g0}\) less than \(0.8\ \mathrm{m}/\mathrm{s}\) for \"S-box\" and less than \(1.1\ \mathrm{m}/\mathrm{s}\) for \"B-box\", the boxes come to a stop within the transport line. Despite this, the speed values considered in the simulations are within the most common operating ranges of conveyors; however, any speed can be implemented in the software. In general, the missed sorting occurs because the deflecting forces also tend to slow the object, which may stop before reaching the target. In addition, the displacement to achieve to be out of the line is related to the dimension of the object, so the bigger it is, the more absolute transversal displacement is required to achieve the sorting.