## 4 Results
### PID--Simulated Results  
As previously stated, the analyses and tests of the decentralized PID control operation and performances were conducted employing the surface digital model summarized above and extensively explained in [2; 14]. Specifically, two main industrial cases were investigated and simulated: a sorting system with four aligned outputs (PATH 1) and one with outputs on both sides of the transport line (PATH 2). Results PATH 1  
An \"end of the line\" sorting scenario with four outputs was selected for the initial simulation of a possible use. This task emerged after a discussion with the industrial partner _Dumarey Flowmotion Technology_, during a research stay at the company. Practically, they had a line with four final codes that needed to be separated into different slots to speed up and simplify palletization with a cobot (Figure 11). Figure 11: PATH 1 task’s schematic illustration [22]. Figure 10: Trajectory (**a**) and orientation angle (**b**) of Box B with orientation control [22]. The industrial specifications required for a similar application, which in this case were provided by the company, are the number of outputs (4), their arrangement (in line as in Figure 11), the maximum footprint available (2 m \(\times\) 2 m), the parcel transported (Cardboard box: length width and height, 0.28 m \(\times\) 0.22 m \(\times\) 0.1 m, mass 7.5 kg), and its initial and final position (as in Figure 11). In line with these terms, the test was designed by defining the layout, the surface characteristics, and the package features. Figure 12 presents the graphical results of the simulations, and the chosen layout is also visible. While the green lines in the image represent the center of mass (COM) trajectories for the different simulations, each of which ends in a green rectangle of its final configuration, the green rectangle on the left depicts the package in its initial position, where it has virtually no speed (0.05 m/s in the \(x\) direction). The under-actuated modules are responsible for the movement and reorientation towards the slots (spaces between the black lines), while the inclination of the surface (8\({}^{\circ}\)) causes the motion towards \(x>0\). The red dots in the figure represent the module rotors. They are in a free-sphere state at that time if they are not accompanied by intersecting lines that denote their rotation axes. The target positions in the different simulations are indicated by the red rectangles, while the actuated axes are only visible in the final arrangement. Although only the axes of the modules in the final condition are shown for graphical clarity, what actually happens is a continuous reorientation of these axes as the box passes over. The calculation and implementation of the actuation take place at each discrete time period (0.01 s for the described simulations). The module returns to the state of an unpowered free sphere once it is no longer beneath the object. Thus, even though they have pushed the material in the direction of the targets, the spheres in the center of the image are all free. Figure 12 shows that the desired positions (called: \(P_{1}\), \(P_{1}\), \(P_{1}\), \(P_{1}\)3 and \(P_{1}\)4) are reached in all four cases, with adequate precision for the intended purpose. To quantify these values, the error trends for the various configurations are also reported (Figure 13). Figure 13 illustrates how the errors tend to zero with different oscillation levels. The required times for the selected parameters are also visible. On this matter, Table 3 introduces a standardized metric, i.e., the settling time, for further comparison among the four position tracking tasks, but also with PATH2 and other control methods once developed. These times are calculated considering the threshold of 2% around the desired equilibrium condition. The values resulting from this table indicate that the parcels settle around the equilibrium just a few instants before the simulation end, pointing out a halting condition that occurs near the ideal outcome.