# Study and Simulation of an Under-Actuated Smart Surface for Material Flow Handling
## 5 Results
### Slowing  
The second set of results, here presented, is derived from the slowing simulations. The purpose of these simulations is to simplify and speed up the determination of the number of modules required for the slowing task, which, as with sorting, is linked to the initial parameters. Similar to the sorting results, graphical examples of the functioning and an application for the design of the actual system are given in this subsection. In order to show the results of the slowing task, in this case the evolution of the object speed over time (\([\dot{x}_{g},\dot{y}_{g},\dot{\theta}_{g}]\)) is reported. Different from the sorting, here the object is not pushed away from the line, so the trajectory does not represent the course of the activity as simply as speed does. Considering the layout of the system, the only velocity component different from zero is in the \(x\) direction, which is the one plotted. Four different setups shown in Figure 9 were simulated to demonstrate the concept's operation. The graphs in Figure 10a,c show the results of the slowing simulations for the \"S-box\" and \"B-box\" cases, respectively. \begin{table}
\begin{tabular}{l l l} \hline \(\dot{x}_{g0}\) [m/s] & \(\mathbf{n^{o}}\)**Columns S-Box** & \(\mathbf{n^{o}}\)**Columns B-Box** \\ \hline  
\end{tabular}
\end{table} 
Table 3: Minimum number of rotor columns required to achieve sorting with \(\gamma=0^{o}\), but changing \(\dot{x}_{g0}\). The initial data for this analysis are the same as in Table 1, with additionally \(\gamma=0^{\circ}\) and \([\dot{x}_{g0},\dot{y}_{g0},\dot{\theta}_{g0}]=[1.5\) m/s, 0 m/s, 0 rad/s]. As visible from Figure 10a,c, both layouts are capable of reducing the speed of the object, fulfilling the slowing task. However, the speed drop is different with the orientation of the rotors. When the rotors are with \(\epsilon_{i}=0^{\circ}\), all the \(F_{par}\) forces in the \(x\) direction cause a stronger slowing, whereas with \(\epsilon_{i}=\pm 45^{\circ}\) layout, where the \(F_{par}\) are inclined like the rotor axes, the slowing is less. For these initial parameters and rotor dispositions, the ratios between the speed before and after the slowing rotors (\(r=\dot{x}_{gout}/\dot{x}_{g0}\)) are: \(r_{0}^{S-box}=0.23\), \(r_{0}^{B-box}=0.18\) and \(r_{\pm 45}^{S-box}=0.53\), \(r_{\pm 45}^{B-box}=0.52\). Additionally, Figure 10 shows the effect of \(F_{perp}\), which is slightly reducing the velocity when the rotors are with \(\epsilon_{i}=90^{\circ}\). In fact, the plots in Figure 10a,c show that the lines are a bit inclined before and after the slowing area, which is included between the \"Start slowing\" and the \"End slowing\" dotted lines. It represents the contribution of rotational friction to the deceleration. As for sorting, a parameters study was conducted within the simulation environment to derive the required number of rotor columns to achieve the slow down task of the objects. The results are displayed in Table 4. Additionally in this case, different initial speed and the two different object sizes were tested (\"S-box\", \"B-box\"). The reference layouts for the simulations are the ones in Figure 9. In Table 4, the counting of the columns number begins with the first column with inclination: \(\epsilon_{i}=0^{\circ}\) or \(\epsilon_{i}=\pm 45^{\circ}\), that the object encounters starting from the left. For example, considering Figure 9a and assuming that slowing is  
Figure 8: Sorting rates obtained with MATLAB simulations for the “S-box” and “B-box” objects, with, as input, only initial velocity (red) and only inclination of the surface (blue). verified, the result will be 2 columns (with \(\varepsilon_{i}=0^{\circ}\)). The target condition was set to at least halving the speed of the object while remaining a positive value after the slowing process (\(0.5\dot{x}_{g0}\geq\dot{x}_{gout}>0\)).