# Study and Simulation of an Under-Actuated Smart Surface for Material Flow Handling
## 7 Conclusions  
In this paper, the authors present a new modular surface for several intra-logistical tasks. In contrast to similar existing systems, their surface is under-actuated, in particular, composed of idle instead of actuated rotors, whose axes can be fixed in defined, discrete positions within the surface plane. The surface can be used in a horizontal orientation by exploiting an object's initial velocity, in a tilted orientation by exploiting the gravitational force on the object, or with a combination of both. The authors derived an analytic model and implemented a programmable simulation environment with the software MATLAB for this modular surface. As result, the functioning of the surface concept for the sorting, stopping, and slowing activities was demonstrated, together with the capabilities of the simulation environment. In particular, the MATLAB code showed its potential for predicting, with very short calculation times (\(\approx\) or \(<\) 1 s), the number of modules required for the three handling tasks and how the transported object will behave by simply changing the initial conditions. The same code also made it possible to obtain numerical results of sorting performance and thus have a comparison with current technology, showing that the system proposed by the authors guarantees a medium sorting capacity. These results and examples highlight the usefulness of this environment for real system planning and design. In addition, a validation of the concept and of the simulation environment was conducted with the software Adams. As Adams is a highly sophisticated and well accepted commercial software for dynamic simulations frequently used by engineers to simulate and predict the physical interaction of different components adequately, it is a reasonable tool to be considered as a first reference for the comparison. In conclusion, the simplifications introduced in the surface, such as the under-actuation and the discrete number of orientations for the rotors, are not limiting the handling capabilities and the performances, but rather they are minimizing the number of constructive components requested and, thus, the costs. In fact, the same goals can be achieved with a reduced design and using in a convenient manner the external actuation, for example, gravity or previous conveyors, already in the line. Additionally, the validation with Adams also showed the accuracy of the main simulation. The differences between the two models are limited and the errors acceptable for many intra-logistics applications. This may open the way to other possible tasks for the surface integrated with the MATLAB environment, such as position and trajectory tracking. In these cases, each time the external environment requires a new position or trajectory of the object, the software calculates the orientation to be given to the rotors. The physical system must be integrated with sensors to adjust in real-time the rotors and achieve the tracking objectives. The implementation of sensors and control strategies for trajectory and position tracking greatly increases the adaptability and flexibility of the system. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline  & \(\mathbf{e_{x}\%}\) & \(\mathbf{e_{y}\%}\) & \(\mathbf{e_{\theta\%}}\) & \(\mathbf{e_{x}\) [mm]} & \(\mathbf{e_{y}\) [mm]} & \(\mathbf{e_{\theta}\) [\({}^{\circ}\)]} \\ \hline Mean & 0.02 & 1.41 & \(-\)4.47 & 0.12 & 0.25 & 0.96 \\ \hline SD & 0.44 & 3.48 & 4.36 & 1.50 & 5.20 & 1.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Mean and standard deviation of the relative percentage and absolute errors obtained from the difference between Adams and MATLAB simulations. No reference is made in the article to construction details in order to keep the validity of the work presented as general as possible. At this point, in order to provide some practical elements and above all to highlight the feasibility of the concept, some schematic solutions are proposed in Figure 17.  
Figure 17a shows an idle wheel mounted on a vertical axis of rotation.