## 3 Control Method
### PID Control  
For more demanding tasks such as precise position tracking, a refined control approach is required, one that continuously adjusts the control output in a progressive manner. The PID control strategy accomplishes this by constantly computing the error between the current and the desired position and generating a control signal composed of proportional, integral, and derivative components. Supported by various studies in the literature (e.g., [1,11]), this method was selected as the second control strategy for the technology under investigation to tackle more complex handling operations even under challenging conditions. Even though PID control is a simple and widely used control method, its definition and implementation are not straightforward, as the logic must be tailored to the system's specific features and operation. This section explains the formulation and underlying principles of the control approach. The analyzed system is quite complex in its control description. Firstly, the constraints linking the object to the surface vary over time (holonomic constraints) and are generally redundant. For example, depending on the pose, the object can be on 4, 5, or 6 modules (at least the boxes employed in the following tests), requiring the use of minimization algorithms to calculate the distribution of contact forces [14], thus introducing significant nonlinearities. Secondly, the surface is an under-actuated system, meaning it does not use driving forces for movement but rather friction ones that tend to stop the transported material. For example, the material flow can only be in the direction of the line, and reverse motion is not possible [14]. This feature limits the handling capability and necessitates the use of a specific friction model tailored to the concept design [12]. Thirdly, the regulation of actuation forces is not continuous but discrete, as the axes can be positioned in only a few predefined directions. This, like the redundancy of the contact points, necessitates minimization functions to calculate which discrete position produces the lowest errors. As an additional note regarding the contact force, the minimization algorithm was preferred over other studied methods, such as the spring-contact model in [15], because it prevents unrealistic negative solutions. Because of these complexities, it is not possible to define direct or inverse kinematics for the system, as is done with a serial manipulator or when the rotor velocity is controlled [1,11], nor is there a simple or explicit rule associating the desired body configuration with the orientations of the axes. The power dissipation method (PDM) must also be excluded for the dynamic description, as the system does not operate under quasi-static conditions [17]. A more reasonable approach is to use dynamic control by implementing tailored contact and friction models [14,19], and subsequently utilizing the equations of motion to determine the orientation of the module axes. In fact, it appears feasible to numerically identify the combination of axis orientations that provide the necessary forces for the desired acceleration, and consequently, for velocity and position. The procedure for achieving this is presented below, along with the formulations employed. From the Lagrangian, the vector representation of the dynamic equations of an object in equilibrium on the surface [14] can be written as:  
\[B(q)\ddot{q}+C(q,q)\dot{q}+G(q)=Q(q,\dot{q}) \tag{1}\]  
where \(q=[x_{g},y_{g},\theta_{g}]\) is the vector that contains the variables that describe the configuration of the system (in the surface reference frame, as shown in Figure 1). In particular, since \(z_{g}\) is not changing in the reference frame that is fixed on the tilted surface, \(q\) does not include it. \(Q\) consists of generalized forces and torques acting on the system, \(C\) is the Coriolis matrix, \(G\) is the vector of gravitational contributions, and \(B\) is the dynamic inertia matrix,which in this instance is independent of \(q\).