# Study and Simulation of an Under-Actuated Smart Surface for Material Flow Handling
## 6 Validation
### Results of the Validation
In this case, the object velocity is inevitably reduced by the friction until the part is stopped, because the gravity is not helping the motion. The same result was highlighted before in the MATLAB environment, which confirms that the simulation environment can be used to predict these conditions. Figure 13 plots the final positions when the sorting is considered achieved for the two software packages (Figure 13a Adams, Figure 13b MATLAB). Having completed the verification of the first objective of the Adams model, it is possible to move on to the second one. Summarizing briefly, objective number two is to exploit the Adams model to show the proximity with the MATLAB results, considering that the same initial data are provided. Figure 14 shows a graphical comparison between the two models referring to the two trajectories of Figure 12a,b. In this case, the trajectories are transposed into a graph (Figure 14) with the corresponding result from MATLAB. Figure 12: Object trajectories from Adams with: (**a**) \(\gamma=10^{\circ}\), \(\dot{x}_{g0}=0\) m/s, and \(t=1\) s of simulation, (**b**) \(\gamma=0^{\circ}\), \(\dot{x}_{g0}=0.9\) m/s, and \(t=1\) s of simulation, and (**c**) \(\gamma=5^{\circ}\), \(\dot{x}_{g0}=0\) m/s, and \(t=1.8\) s of simulation. Figure 13: Final position with \(\gamma>8.5^{\circ}\) in: (**a**) Adams MSC, (**b**) MATLAB. As is visible from Figure 14a,b, the two objects follow very close trajectories and end up almost in the same position and with the same orientation. The absolute and the relative percentage errors obtained for these two simulations are: \([|e_{x}|,|e_{y}|,|e_{\theta}|]=[0.0011\text{ m},0.0001\text{ m},1.735^{\circ}]\), \([|e_{x\gamma_{0}}|,|e_{y\gamma_{0}}|,|e_{\theta\gamma_{0}}|]=[0.24,0.05,7.06]\) % when the input is \(\gamma=10^{\circ}\), \(\dot{x}_{g0}=0\text{ m}/\text{s}\) and \([|e_{x}|,|e_{y}|,|e_{\theta}|]=[0.003\text{ m},0.0037\text{ m},1.56^{\circ}]\), \([|e_{x\gamma_{0}}|,|e_{y\gamma_{0}}|,|e_{\theta\gamma_{0}}|]=[0.65,1.59,6.97]\) % when the input is \(\gamma=0^{\circ}\), \(\dot{x}_{g0}=0.9\text{ m}/\text{s}\). The errors are calculated as shown in Equation (5a,b). \[[e_{x},e_{y},e_{\theta}]=\text{FINAL DISP}_{.[PERSON]}-\text{FINAL DISP}_{. Matlab} \tag{5a}\] \[[e_{x\gamma_{0}},e_{y\gamma_{0}},e_{\theta\gamma_{0}}]=\frac{\text{FINAL DISP}_{.[PERSON]}-\text{FINAL DISP}_{.Matlab}}{\text{FINAL DISP}_{.[PERSON]}}\times 100 \tag{5b}\]  
The two examples graphically visualize the similarities and the differences between the models. However, to show and summarize all the results for the thirty-seven setups (\(\gamma=5^{\circ}\div 14^{\circ}\)), the final displacements of the Adams and MATLAB simulations together with the errors between them are reported, respectively, in Figures 15 and 16. Considering the \(x\) and \(y\) results (Figures 15 and 16), the two models (Adams, MATLAB) are not so different, and the relative percentage errors are below \(e_{x\gamma_{0}}<1\) % and \(e_{y\gamma_{0}}<8\) %. The trends of the percentage errors seem to have a slight tendency to decrease, and the higher error values occur for small \(\gamma\). In contrast, for high \(\gamma\) values, both coordinates have minimum errors, \(e_{x\gamma_{0}}<0.5\) % and \(e_{y\gamma_{0}}<3\) %. This is because small inclinations generate reduced displacements, especially along \(y\), and the sorting is not completed in the fixed simulation time (\(t=1\) s). This implies that, in the calculation of the percentage error, there are initially very small denominators, and then they become an order of magnitude larger. On the other hand, the absolute errors along \(\gamma\) values keep oscillating, with a bigger frequency for \(x\) and smaller for \(y\). For the orientation, higher \(e_{\theta\gamma_{0}}\) values are reached (\(e_{\theta\gamma_{0}}\leq 12\%\)) compared to those of \(x\) and \(y\).