# Study and Simulation of an Under-Actuated Smart Surface for Material Flow Handling
## 4 Simulations
In contrast, if the velocity is less or equal to zero (\"\(\dot{x}_{g}\leq 0\) m/s\" is true), an initial positive speed is assigned to the object \(\dot{x}_{g}=0.0001\) m/s, the friction model is applied, and it is verified if the gravity effect is stronger than the friction forces in the \(x\) direction (\(\sim\sum\limits_{i=1}^{n_{i}}F_{x_{i}}+mg\sin\gamma>0\)\"). At this point, if gravity wins, the object is moving according to the process defined before (Equations (3a,b) and (4)); in contrast, if gravity is not enough, the friction forces in the \(x\) direction will be of the same magnitude of the gravity effect (\"\(\sum\limits_{i=1}^{n_{i}}F_{x_{i}}+mg\sin\gamma=0\)\"), and the displacement will be in the \(y\) direction (always according to Equations (3a,b) and (4)). To clarify the diagram of Figure 5b, the condition \"\(\sum\limits_{i=1}^{n_{i}}F_{x_{i}}+mg\sin\gamma=0\)\" permits the calculation of the \(F_{par_{i}}\) terms, while \(F_{perp}\) is the same as the previous case. For instance, this procedure permits us to obtain the motion of the object when it is placed without an initial velocity on the inclined surface, with the rotors oriented at 45\({}^{\circ}\). In fact, with a sequence of displacement in the \(y\) and \(x\) directions, the movement is achieved. The simulations conducted to test the surface are divided into three main categories, according to the application of the system:  
1. Sorting of material flows on a conveying line;
2. Slowing of material flows on a conveying line;
3. Stopping of material flows on a conveying line. The first category of simulations concerns sorting. The objective of the setup and the sorting itself is to divert an object from the transport line. The layout considered for the simulation involves an array of modules, of which the subset performing the task has the rotors with the rotation axes inclined at \(\pm\)45\({}^{\circ}\) (in Figure 6a inclined at \(+\)45\({}^{\circ}\)). The area  
Figure 5: (**a**) Iterative loop scheme for the simulation. (**b**) Block diagram of the logic process behind the model to avoid the reverse motion and to calculate friction forces (Equation (2a,b)) and equilibrium equations (Equations (3a,b) and (4)). outside the line, where the objects are directed, is simulated as an array of low-friction (\(\mu_{csurf}\)) support points, distributed as the sorting modules. In this zone, the friction is totally opposite to the velocity of the object, without considering any rotor inclination. The sorting is assumed achieved when the body moving on the line is deflected in such a way that it is only in contact with the elements outside the line. The second simulated application is the slowing activity. The rotor arrangement consists of a few modules within a line to create a controllable friction on the object to modulate its speed, without pushing it to the sides. The layout in this case can use rotors oriented with their axes in the direction of the flow (\(\varepsilon_{i}=0^{\circ}\)) (Figure 6b) or inclined at \(+45^{\circ}\) in a row and \(-45^{\circ}\) in the other (Figure 6c). The last application concerns stopping the motion of an object. This task is the extreme application with respect to the previous application of slowing down. In fact, the modules are still placed with their axes in the flow direction (Figure 6b) or oriented at \(\pm 45^{\circ}\) (Figure 6c), but aim to stop the object. These three types of simulations do not yet include either real-time control of the rotors or position or trajectory tracking for the object, because, as already indicated, the authors' objective in this paper is to verify the mechanical functioning of the surface. However, it is possible to imagine that by having sensors that recognize the object arriving at the surface, the modules can pre-arrange the rotors, as shown in Figure 6, depending on the application or sorting direction. This lays the foundation for using the system as a smart surface.