# Study and Simulation of an Under-Actuated Smart Surface for Material Flow Handling
## 3 Analytical Model
### Assumptions  
Listed here are the fundamental assumptions that allowed the definition of the analytical model describing the motion of an object on the surface. 1. The first assumption states that there are only the three forces for each interaction rotor-object: two components in the plane of the device and one normal to this (Figure 2c). Considering that the contact area of the module below the object is small compared to the object itself, the previous assertion is plausible and it is translated into the possibility of neglecting the exchanged torques. 2. The second assumption is about the normal component \(N\). In fact, considering that the contact in most of the cases is on more than three rotors, the constraint is hyperstatic and the equilibrium equations of the body are not enough to determine the normal reactions. In particular, for determining the distribution of the forces \(N_{i}\), an optimization algorithm described in Equation (1) is used. \[min_{\underline{X}}\,f(\underline{X})\,\,such\,\,that \begin{cases}A_{eq}\cdot \underline{X}=\underline{b_{eq}}\\ \underline{l_{\underline{b}}}\leq\underline{X}\leq\underline{u_{\underline{b} }}\end{cases} \] (1) where the first vector equation represents the equilibrium in the vertical direction and around the two axes of rotation in the plane, and the second vector equation states the boundary conditions. The function to be minimized \(f(\underline{X})\) is the Euclidean norm of the \(N_{i}\) array. The single terms are: \[A_{eq}(3\,\,\times\,n)= \begin{bmatrix}1&1&...&1\\ y_{\bar{G}P_{1}}&y_{\bar{G}P_{2}}&...&y_{\bar{G}P_{n}}\\ -x_{\bar{G}P_{1}}&-x_{\bar{G}P_{1}}&...&-x_{\bar{G}P_{n}}\end{bmatrix} ,\quad \underline{b_{eq}}(3\,\,\times\,1)= \begin{bmatrix}mg\cos(\gamma)\\ -m\bar{y}z\\ m\bar{z}z_{g}-mgz_{g}\sin(\gamma)\end{bmatrix} ,\] \[\underline{X}(n\,\,\times\,1)= \begin{bmatrix}N_{1}\\ \,...\\ N_{i}\\ \,...\\ N_{n}\end{bmatrix} ,\quad\underline{l_{\underline{b}}}(n\,\,\times\,1)= \begin{bmatrix}0\\ 0\\...\\ 0\end{bmatrix} ,\quad\underline{u_{\underline{b}}}(n\,\,\times\,1)= \begin{bmatrix} mg\cos(\gamma)\\ mg\cos(\gamma)\\...\\ mg\cos(\gamma)\end{bmatrix} ,\quad f(\underline{X})=\|\underline{X}\|\] where \(n\) is the number of rotors under the object, \(m\) is the mass of the object, \(g\) is the gravity acceleration, and \(\gamma\) is the inclination of the surface compared to the ground (Figure 3a). In addition, \(x_{\bar{G}P_{i}}\) and \(y_{\bar{G}P_{i}}\) are the \(x\) and \(y\) components of the vector that connect \(\widetilde{G}\) to the \(i\)th contact point \(P_{i}\). 3. The third assumption asserts that the inertia of the rotors is negligible, as they are small and made of lightweight material. Instead, the rotational friction given by the spinning of the rotor around its axis is included in the calculation of contact forces. 4. The fourth assumption states that, for an object in motion on the rotors, the friction conditions at the contact points (rotors-object) are as follows: dynamic friction (sliding) in the direction of the axis of the rotors and pure rolling in the perpendicular direction. This is plausible, as the center of instantaneous rotation (CIR) of the rigid body (object) is very unlikely to be located at the center of an underlying rotor. This would guarantee a different static friction and force modeling (\(F_{par}\), \(F_{perp}\)). However, with only a few discrete points as supports, it is unlikely that the CIR is located on these. In practice, for the axis direction, this assumption justifies the choice of a smooth Coulomb friction model, which is computationally stable and reliable, as the sliding condition prevails ([40]). The same model is considered for the friction force, due to the rolling, in the perpendicular direction. This value depends on the rotational friction of the rotor axis (assumption 3), so it can also be expressed using the same Coulomb formulation.