## 2 Materials
### Software Overview  
While the experimental setup shown in Figure 2 was used to test the so-called threshold-based control, the decentralized PID evaluation builds upon a previously established digital model developed to offer an efficient and cost-effective testing methodology. Beyond these advantages, including speed and versatility, the use of the simulation environment was also motivated by the limited sensing capabilities of the physical prototype, which would not allow for reliable PID testing at this preliminary stage. In [2], this digital model, designed to replicate a surface with specific physical characteristics, was introduced and experimentally validated. The software environment accurately simulates an object's motion based on key input parameters, such as its geometry, mass, surface layout, and friction conditions. Furthermore, the demonstrated high accuracy of the model's predictions in [2] reinforces its suitability for streamlining PID control analysis in the current study.  
To give a brief introduction of this digital model, Figure 3 presents the logic scheme that was later implemented in _Matlab_ and _Simulink_ to describe the object's motion on the under-actuated surface. Employing the flowchart in [2] and its representation style as reference, the diagram in Figure 3 incorporates additional terms related to control, including the desired configuration, errors, and the controller. In more detail, the steps followed by the controlled system are: \(S0_{1}\), \(S0_{2}\), \(S1\), \(S2\), \(S3\), \(S4\), \(S5\), \(S6\), and \(S7\). The first two steps involve the implementation of the system input data: \(S0_{1}\) pertains to the surface and the transported objects, while \(S0_{2}\) addresses the desired configuration, which is the objective of the control. Step \(S1\) focuses on determining the modules in contact, based on the object's position. This position, along with velocity and acceleration data, and the errors derived from \(S7\), are processed in step \(S2\) by the control module. This step \((S2)\) outputs the orientations of the module axes \((\epsilon_{r})\), which are then used in step \(S3\) to calculate the new friction forces and the body's equilibrium. At this point, the stop condition is verified (54). If the response is negative, the cycle continues with the integration of accelerations to velocities (55) and then to position (56). With this new information, the rotors in contact are determined again, and in step \(S7\), the differences with the desired condition are calculated, thus the errors, along with their derivatives and integrals, and in this way the loop continues.  
Figure 3: Block diagram of the system logic with the control [22].  
At this point, with the concept features and the model defined as above, the following sections illustrate the control strategies developed for the system. Specifically, the first control method, referred to as threshold-based, is explained and tested using the available physical setup of the surface. Conversely, the second method, which employs a PID controller and is designed for a system with these characteristics, is tested using the digital model of the surface, as introduced in this subsection.