# Study and Simulation of an Under-Actuated Smart Surface for Material Flow Handling
## 5 Results
### Sorting  
This subsection reports the results of the first of the three categories, sorting. Initially, graphical examples of the function under varying input conditions are shown, then an application for the design of the actual system is presented, and finally a comparison with existing sorting systems is given. Figure 7a,b show the sorting towards the two directions, \(y>0\) and \(y<0\), respectively, with an initial object velocity \(\dot{x}_{g0}=1.5\) m/s and without surface inclination. Figure 7c, on the other hand, shows the sorting at different tilting angles of the surface and without an initial velocity. Each plot displays the trajectory of \(G\) (object's center of mass) during the sorting and the position and orientation of the object at the end of the simulation time. In Figure 7a, the green square and its trajectory show that, with rotors inclined at \(-45^{\circ}\), sorting towards \(y>0\) can be achieved, as the object is shifted completely off the line. The opposite sorting condition was tested as well; the only difference for the initial data was the direction of the rotors (\(+45^{\circ}\)). The result is shown in Figure 7b and the trajectory is mirrored to the first one. In fact, as expected in reality, the ending displacements of the object are equal in modulus and with opposite sign for the \(y\) and \(\theta\) displacements (Table 2). Figure 7a,b show the capability of the surface to deflect the object trajectory for sorting purposes. Furthermore, coherent results were achieved for both sorting directions, for a reasonable initial velocity, and within one second of simulation time. From now on, since the mirroring of the results has been demonstrated for a symmetric rotor arrangement, the outputs are shown only for one sorting direction. \begin{table}
\begin{tabular}{l l l} \hline
**Parameter** & **Description** & **Value** \\ \hline \(b\times h\) & object base dim. & 0.25 m\(\times\)0.25 m \\ \hline \(z_{g}\) & height of G & 0.1 m \\ \hline \(m\) & object mass & 10 kg \\ \hline \(J\) & moment of inertia & \((1/12)m(h^{2}+b^{2})\) \\ \hline \(\mu_{max}\) & max. friction coef. & 0.5 \\ \hline \(\mu_{min}\) & min. friction coef. & 0.01 \\ \hline \(\mu_{exsurf}\) & friction coef. out of the line. & 0.005 \\ \hline \(dx_{m}\) & \(x\) dist. between 2 rotors & 0.1 m \\ \hline \(dy_{m}\) & \(y\) dist. between 2 rotors & 0.1 m \\ \hline \([x_{g},y_{g},\dot{\theta}_{g}]\) & obj. initial pos. and orientation & \([b/2,1\) m, 0 rad] \\ \hline \([\dot{x}_{g},\dot{y}_{g},\dot{\theta}_{g}]\) & obj. initial acc. & \([0\) m/s\({}^{2}\), 0 m/s\({}^{2}\), 0 rad/s\({}^{2}]\) \\ \hline \end{tabular}
\end{table} 
Table 1: Initial fixed parameters for the MATLAB simulations. After this first presentation of the sorting capability for the under-actuated system, instead of using an initial object velocity on a horizontal surface, the sorting task was studied with an initially stationary object on an inclined surface with the identical rotor arrangement (Figure 7c). In Figure 7c, the first case (green) proves that, without any external actuation, the object is not moving in the simulation, as in the real word. In the second case (black square, Figure 7c), with the surface inclination of \(\gamma=5^{\circ}\), the body moves, but for the chosen simulation time (\(t=1.5\) s) it has not yet completed its sorting. This can be seen from the fact that the black square contour still contains the center of a sorting rotor (red). Finally, the third case (magenta square), where the surface inclination is \(\gamma=10^{\circ}\), reports a completed sorting activity. Summarizing the black and magenta bodies in Figure 7c demonstrate that it is possible to transport and sort parts by inclining the surface. In particular, the more it is tilted, the faster the object moves, and the quicker the sorting is achieved.