# Study and Simulation of an Under-Actuated Smart Surface for Material Flow Handling
## 6 Validation
### Results of the Validation  
This section reports the results of the validation, which are distinguished according to the two objectives of the Adams model. Concerning the first objective, i.e., the proof of functioning of the under-actuated system, Figure 12 graphically reports the results of some Adams simulations. Considering initially only Figure 12a,b, in Figure 12a, sorting is simulated given an initial velocity of zero (\(\dot{x}_{g0}=0\) m/s) and an inclination of \(\gamma=10^{\circ}\); in Figure 12b, the initial velocity is \(\dot{x}_{g0}=0.9\) m/s and the inclination is zero (\(\gamma=0^{\circ}\)). In Figure 12a,b, it is possible to see the trajectory taken by the \(G\) of the object and its final position and orientation given the simulation time of \(t=1\) s. The object at the end of the \begin{table}
\begin{tabular}{l l l} \hline \hline  & **Contact Parameters** & \\ \hline \(St\) & stiffness cont. & \(10^{5}\) MPa \\ \hline \(Dp\) & damping cont. & \(8\) kg/s \\ \hline \(Pd\) & penetration depth cont. & \(0.1\) mm \\ \hline \(Stv\) & stiction transition vel. & \(0.1\) mm/s \\ \hline \(Ftv\) & friction transition vel. & \(1\) mm/s \\ \hline \(\mu_{max}\) & friction coef. & \(0.5\) \\ \hline \(\mu_{csurf}\) & friction coef. out the line. & \(0.005\) \\ \hline  & **Joint friction parameters** & \\ \hline \(\mu_{min}\) & min friction coef. & \(0.01\) \\ \hline \(Tv\) & transition vel. & \(0.001\) \\ \hline \(Friction\) & option chosen & _sliding only_ \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Contact and friction parameters implemented in Adams for the simulation. \begin{table}
\begin{tabular}{l l l} \hline \hline  & **Objects Parameters** & \\ \hline \(m\) & box mass & \(5.48\) kg \\ \hline \(I_{xx}\) & box m. inertia \(x\) axis & \(4.68\,\times\,10^{-2}\) kg\({}^{2}\) \\ \hline \(I_{yy}\) & box m. inertia \(y\) axis & \(4.68\,\times\,10^{-2}\) kg\({}^{2}\) \\ \hline \(I_{zz}\) & box m. inertia \(z\) axis & \(5.70\,\times\,10^{-2}\) kg\({}^{2}\) \\ \hline \(b\,\times\,h\,\times\,z_{g}\) & object dim. & \(0.25\) m\(\times 0.25\) m\(\times 0.1\) m \\ \hline \(m_{r}\) & rotor mass & \(8.61\,\times\,10^{-3}\) kg \\ \hline \(I_{rxx}\) & rot. m. inertia \(x\) axis & \(4.30\,\times\,10^{-7}\) kg\({}^{2}\) \\ \hline \(I_{yy}\) & rot. m. inertia \(y\) axis & \(2.87\,\times\,10^{-2}\) kg\({}^{2}\) \\ \hline \(I_{rzz}\) & rot. m. inertia \(z\) axis & \(2.87\,\times\,10^{-7}\) kg\({}^{2}\) \\ \hline \(r_{r}\,\times\,l\) & rot. dim. & \(0.01\) m\(\times 0.01\) m \\ \hline \(\gamma\) & inclin. of the surf. & \([5\div 15]^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Inertial and geometrical parameters implemented in Adams for the simulation. simulation is for both images (Figure 12a,b) outside the sorting zone (zone with cylinders), so the activity is considered completed. In addition to the case illustrated in Figure 12a, among all the thirty-seven simulations made by varying the inclination of the surface, the ones with \(\gamma\geq 9.5^{\circ}\) (Figure 13a,b) complete the task. For the rest, the target is not really missed, but, because of the limited simulation time (\(t=1\) s), it is not achieved yet. This time was selected because it represents a normal operating condition, for example: there is 1 m of space available in the line for sorting and the conveyor has a speed of 1 m/s. However, as a general rule, when the surface is inclined and with the rotors as in Figure 12, the motion does not self-stop, if \((\sqrt{2}/2)\mu_{min}<\sin(\gamma)\). As proof of this, Figure 12c displays the trajectory of the object with \(\gamma=5^{\circ}\) but \(t=1.8\) s of simulation time, instead of \(t=1\) s, and the sorting is clearly obtained. Therefore, the sorting is achievable with this under-actuated system, but it takes time and space. Instead, if the surface is not inclined and there is just the initial speed, the time could be not the only cause of the missed sorting.